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3 Ansichten152 SeitenParragh Stephanie - 2016 - Mathematical Models for Pulse Wave Analysis..

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3 Ansichten152 SeitenParragh Stephanie - 2016 - Mathematical Models for Pulse Wave Analysis..

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Technischen Universität Wien aufgestellt und

zugänglich.

http://www.ub.tuwien.ac.at

available at the main library of the Vienna

University of Technology.

http://www.ub.tuwien.ac.at/eng

Dissertation

considering ventriculo-arterial coupling in

systolic heart failure

Doktorin der technischen Wissenschaften

unter der Leitung von

E101

Institut für Analysis und Scientific Computing

Fakultät für Mathematik und Geoinformation

von

0526658

Währinger Gürtel 166/2/4

1090 Wien

Zusammenfassung

bestimmen, die wichtige Informationen über den Status des kardiovaskulären Systems einer Per-

son liefern. Für die Risikostratifizierung werden insbesondere immer öfter nichtinvasiv gemes-

sene aortale Druckkurven und modellierte Flusskurven eingesetzt. In Patienten mit systolischer

Herzinsuffizienz (SHI) ist die Interpretation der PWA Parameter allerdings schwierig und be-

kannte Risikofaktoren sind nicht anwendbar. Außerdem ist nur wenig über die Verwendbarkeit

von bereits bestehenden Flussmodellen in diesen Patienten bekannt.

SHI wird durch eine Veränderung der ventrikulo-arteriellen Kopplung, d.h. dem Zusammenspiel

von Herz und Arteriensystem, charakterisiert, die oft mit einem veränderten Auswurfmuster

des linken Ventrikels einhergeht. Um diese Charakteristik richtig darstellen zu können wird in

dieser Arbeit ein neues Flussmodell präsentiert. Dieses basiert auf einem Kompartmentmodell

des arteriellen Systems, dem sogenannten 4-elementigen Windkessel, das verwendet wird um

eine parametrische Darstellung der Transferfunktion zwischen Blutdruck und Blutfluss im Fre-

quenzbereich zu erhalten. Ausgehend von einer aortalen Druckkurve, kann der Blutfluss mittels

einer Parameteridentifikation somit direkt berechnet werden. Simulationsergebnisse zeigen, dass

es mit diesem Modell tatsächlich möglich ist, sowohl physiologische als auch pathologische Fluss-

verläufe darzustellen. Die Resultate einer Sensitivitätsanalyse weisen weiters darauf hin, dass

die beschriebene Methode robust auf Änderungen der Eingangsgröße sowie der Modellparameter

reagiert. Das verwendete Windkesselmodell berücksichtigt jedoch keine Wellenreflexionen, wes-

halb das Flussmodell für einen bestimmten Typ von Druckkurven, der vor allem in Patienten mit

normaler systolischer Pumpleistung vorkommt, nicht geeignet ist. Deshalb wird eine Möglichkeit

vorgestellt, den Windkesselfluss mit einem bereits bestehenden Flussmodell zu kombinieren. Als

Entscheidungskriterium dient dabei ein Formfaktor der Druckkurve, der im Frequenzbereich be-

stimmt wird.

Um die Rolle der ventrikulo-arteriellen Kopplung für die arterielle Hämodynamik weiter zu unter-

suchen, wurde der Einfluss von SHI auf die PWA Parameter analysiert. Dafür wurden gemessene

Fluss- und Druckverläufe von 61 Patienten mit SHI und einer Kontrollgruppe bestehend aus 122

Patienten mit normaler systolischer Pumpleistung verwendet. Das Ausmaß der Wellenreflexionen

war in der SHI Gruppe scheinbar niedriger als in der Kontrollgruppe, was durch die Reduktion

der Auswurfdauer, als Ausdruck der gestörten Kopplung zwischen Ventrikel und Arteriensystem,

i

ii

erklärt werden konnte. PWA wird allgemein zur Quantifizierung der Gefäßeigenschaften verwen-

det. Die Ergebnisse dieser Arbeit zeigen jedoch, dass die PWA Parameter in SHI stark von der

Herzfunktion beeinflusst werden, was für die Risikostratifizierung in diesen Patienten unbedingt

berücksichtigt werden sollte.

Schließlich wurden verschiedene bereits bestehende Flussmodelle aus der Literatur, sowie der neu

vorgestellte Ansatz in derselben Studienpopulation auf ihre Anwendbarkeit für PWA untersucht.

Die Ergebnisse zeigen, dass ein Flussmodell eine gewisse Flexibilität in der Flussform sowie einen

physiologischen Verlauf aufweisen sollte um akkurate Schätzer der PWA Parameter zu liefern, die

dasselbe qualitative Verhalten aufweisen wie Parameter, die mittels gemessenem Fluss bestimmt

wurden. Der kombinierte Windkesselfluss erfüllt diese Kriterien und war das einzige Modell, das

sowohl in den beiden Patientengruppen separat als auch im Vergleich beider Gruppen eine akzep-

table Übereinstimmung mit gemessenem Fluss erreichte. Diese Resultate stellen allerdings nur

einen Machbarkeitsnachweis dar, da dieselben Daten zur Modellentwicklung und -evaluierung

verwendet wurden.

Zusammenfassend unterstreichen die Resultate die Wichtigkeit der Herzfunktion für die Inter-

pretation druckbasierter Parameter und zeigen, dass nur mit einem akkuraten Flussmodell auch

akkurate Schätzer der PWA Parameter erreicht werden können. Mit dem vorgestellten Ansatz

konnten erste, sehr vielversprechende Ergebnisse erzielt werden, die eine vergleichbare Präzision

für Patienten mit normaler und eingeschränkter systolischer Funktion aufzeigen.

Abstract

Parameters gained from pulse wave analysis (PWA) of the pressure and flow waveforms in the

human aorta yield important information about the status of the cardiovascular system. In par-

ticular, non-invasively measured aortic blood pressure together with modelled flow are more and

more used for the stratification of cardiovascular risk. However, in patients with systolic heart

failure (SHF), the interpretation of PWA parameters is puzzling and general risk indicators are

not applicable. Moroever, little is known about the feasibility of using existing flow models for

PWA in these patients.

SHF is characterized by an alteration in the ventriculo-arterial coupling (i.e. the interplay of the

heart and the arterial system), which often results in a modified ejection pattern. To properly

describe these characteristics, a new flow model is presented in this thesis. In brief, the proposed

approach is based on a one compartment model of the arterial system, the so-called 4-element

Windkessel, from which a parametric equation for the transfer function between pressure and

flow in the frequency domain is derived. Using non-invasively assessed aortic pressure as input,

a parameter identification then allows for a direct computation of blood flow. Simulation runs

and the results of a sensitivity analysis indicate that the presented model is able to reproduce

physiological and pathological ejection patterns and is robust against changes in the input values

and model parameters. However, the Windkessel model does not account for the effects of wave

reflections, making the flow model unsuitable for a specific type of pressure waves found mainly

in subjects with normal systolic function. Therefore, a combination of the Windkessel flow with

an already established flow model is proposed based on a form factor of the pressure wave derived

in the frequency domain.

To further investigate the role of ventriculo-arterial coupling for arterial heamodynamics, the

impact of an impaired systolic function on the PWA parameters derived from measured pressure

and flow was investigated in 61 patients with SHF and 122 controls. Most parameters quanti-

fying wave reflections were reduced in SHF, which could be attributed to a shortening of the

ejection duration as a manifestation of an impaired coupling between the ventricle and the ar-

terial system. PWA is commonly used to quantify arterial function only. However, the results

demonstrate that the derived parameters are susceptible to cardiac function in SHF, which has

to be kept in mind for risk stratification in these patients.

iii

iv

Finally, the performance of different existing flow models as well as of the novel approach for

PWA was examined in the same study population. The results indicate that for a flow model,

both a physiological waveform as well as the capability to adapt in shape are necessary to yield

accurate PWA parameters that show the same qualitative behaviour as parameters obtained with

measured flow. The combined Windkessel flow fulfils these criteria and was the only flow model

that achieved an acceptable agreement to measured flow in and across both groups of patients.

However, it should be emphasised that these results represent a proof of concept only because

the same study population was used for model development and evaluation.

In conclusion, the results underline the importance of cardiac function for the interpretation

of pressure-derived parameters and stress that an accurate flow estimate is needed to derive

accurate PWA parameters. First promising results could be achieved with the novel flow model,

yielding parameter estimates with a comparable accuracy and precision in both, patients with

normal and impaired systolic function.

Acknowledgements

First of all, I want to thank Prof. Felix Breitenecker for offering me the opportunity and privilege

to do a doctorate and for involving me in his research group, which meant a lot to me.

I would also like to express my gratitude to Sigi Wassertheurer, who enabled me to carry out my

thesis at the Austrian Institute of Technology and guided me throughout this project.

Furthermore, my special thanks belong to Bernhard Hametner from the AIT for the many in-

sightful discussions, for critically questioning (almost) everything, for the helpful advice and

suggestions when I was stuck, and, last but not least, for proofreading this thesis.

The four years that I have spent at the AIT would not have been the same without my great

colleagues Martin, Christopher, Brigitte, Stefan (x3), Andi, Evi and (again) Bernhard, who made

the days in the office always pleasant and most often delightful. Thank you!

Moreover, I would like to extend my thanks to my collaborator Thomas Weber from the Klinikum

Wels-Grieskirchen, who not only provided the data used in this thesis but also his scientific ad-

vice and medical expertise in several common projects.

I would also like to say thank you to my family and friends for supporting me all the way - I am

more than lucky to have you! Especially, I wish to thank my parents, Rita and Laci for the love

and care they have given me and for always believing in me, as well as my siblings Sophie and

Nico plus families for being there for me no matter what.

Finally, finishing this thesis would have been all the more difficult were it not for my boyfriend

Dominik, who endured all my moods and my ups and downs with an incredible patience and

encouraged me over and over again. I don’t know what I would have done without you, and I

want to thank you from the bottom of my heart.

Stephanie Parragh

v

Contents

List of Figures ix

1 Introduction 1

1.1 Background and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Aim of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Heart failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4.1 The heart failure epidemic . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4.2 Systolic heart failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 Pressure pulse wave analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Timing information and pressure levels . . . . . . . . . . . . . . . . . . . . 9

2.1.2 Parameters quantifying the propagation of the pressure pulse throughout

the body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.3 Pressure time integrals and wasted energy . . . . . . . . . . . . . . . . . . 13

2.2 Impedance analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.1 Fourier analysis and input impedance . . . . . . . . . . . . . . . . . . . . 15

2.2.2 Wave separation analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.3 Estimating aortic characteristic impedance . . . . . . . . . . . . . . . . . 20

2.3 Windkessel models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.1 Two-element Windkessel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.2 Three- and four-element Windkessel . . . . . . . . . . . . . . . . . . . . . 24

2.3.3 Diastolic decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4 Wave intensity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.4.1 Deriving a 1D model of blood flow and pressure in an elastic vessel . . . . 29

2.4.2 Solution using the method of characteristics . . . . . . . . . . . . . . . . . 33

2.4.3 Wave intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.4.4 Waterhammer equations and wave separation . . . . . . . . . . . . . . . . 37

vi

Contents vii

3 Models of aortic blood flow based on pressure alone: existing methods and a

novel approach 41

3.1 Triangular flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2 Averaged flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3 ARCSolver flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4 Windkessel flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.4.1 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.4.2 Methods and implementation . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.4.3 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.4.4 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.4.5 Combination with ARCSolver flow . . . . . . . . . . . . . . . . . . . . . . 64

of the art and data analysis 66

4.1 State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.2 Aim of the study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5 Using flow models based on pressure alone in patients with systolic heart

failure: state of the art and data analysis 85

5.1 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.2 Aim of the study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

models 107

A.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

A.1.1 Simulation runs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

A.1.2 Fitting performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

A.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

A.2.1 Simulation runs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

A.2.2 Fitting performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

A.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

B.1 Moens-Korteweg and Bramwell-Hill . . . . . . . . . . . . . . . . . . . . . . . . . . 115

B.2 Investigating the assumption of constant pulse wave velocity . . . . . . . . . . . . 116

Bibliography 119

Contents viii

Abbreviations 133

Nomenclature 134

List of Figures

1.1 Hospitalisations due to heart failure in different European countries and the United

States. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Schematic representation of a normal heart compared to systolic and diastolic

dysfunction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Determination of timing information and pressure levels . . . . . . . . . . . . . . 9

2.3 Computation of AIx. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Assessment of carotid-femoral PWV using sequential pressure measurements and

a simultaneous electrocardiogram. . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.5 Computation of the time indices representing the areas under different portions

of the pressure wave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.6 Fourier decomposition and approximation of aortic pressure and flow for the first

10 harmonics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.7 Modulus and phase of arterial input impedance. . . . . . . . . . . . . . . . . . . . 17

2.8 Aortic pressure and flow separated in their forward and backward components

using wave separation analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.9 Estimation of aortic characteristic impedance. . . . . . . . . . . . . . . . . . . . . 21

2.10 Representation of the two-element Windkessel model as an electrical circuit. . . . 23

2.11 Representation of the three- and four-element Windkessel model as electrical circuits. 25

2.12 Comparison of the aortic pressure waves modelled by the three different Wind-

kessel models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.13 PRC and PLZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.14 Comparison of the modulus and phase of the arterial input impedances described

by the three different Windkessel models. . . . . . . . . . . . . . . . . . . . . . . 28

2.15 A straight arterial segment with length l, oriented along the x-axis. . . . . . . . . 30

2.16 Schematic representation of the characteristic directions and the corresponding

Riemann invariants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.17 Wave intensity obtained from measured pressure and flow velocity. . . . . . . . . 36

2.18 A single forward wave travelling in a uniform, lossless vessel along the character-

istic direction x̂+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.19 Aortic pressure and flow separated in their forward and backward components

using wave intensity analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

ix

List of Figures x

3.1 Triangular approximation and averaged waveform as estimates of aortic blood flow. 42

3.2 Estimation of initial values used for the computation of blood flow. . . . . . . . . 50

3.3 Exemplary data of a patient with normal EF and one with reduced EF used for

sensitivity analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.4 Sensitivity analysis showing the effect of changes in the model parameters Ca , Zc

and L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.5 Sensitivity analysis showing the effect of changes in the model parameters P∞ , Rp

and ñ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.6 Boxplots showing the changes in the objective function relative to its baseline for

varying Ca (a), Zc (b) and L (c) from 85% to 115% of their respective baseline

values while keeping all other parameters constant. . . . . . . . . . . . . . . . . . 55

3.7 Analysis of the effect of the grid-search approach for ñ. . . . . . . . . . . . . . . . 56

3.8 Analysis of the effect of the upper boundary of P∞ used in the optimisation. . . 57

3.9 Influence of the upper boundary (ub) of P∞ on an exemplary pressure wave from

a patient with normal (a) and reduced EF (b). . . . . . . . . . . . . . . . . . . . 58

3.10 Analysis of the influence of changes in DBP, SBP and MBP. . . . . . . . . . . . . 59

3.11 Sensitivity analysis showing the effect of changes in the magnitude of ±5 mmHg

in the input pressure levels DBP, SBP and MBP. . . . . . . . . . . . . . . . . . . 60

3.12 Exemplary simulation results for patients with reduced EF. . . . . . . . . . . . . 62

3.13 Exemplary simulation results for patients with normal EF. . . . . . . . . . . . . . 63

3.14 Phase angle of central pressure for patients with normal and reduced EF. . . . . 64

ventricular volume. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.2 Assessment of aortic blood pressure and flow. . . . . . . . . . . . . . . . . . . . . 72

4.3 Averaged flow and pressure waveform for patients with reduced and normal EF. 74

4.4 Difference of the mean values of wave reflection parameters between patients with

reduced and normal EF adjusted for temporal characteristics. . . . . . . . . . . . 79

4.5 Difference of the mean values of the S to D ratio between patients with reduced

and normal EF adjusted for temporal characteristics. . . . . . . . . . . . . . . . . 79

5.1 Examples of Doppler flow waves and the corresponding model estimates for pa-

tients with reduced EF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.2 Examples of Doppler flow waves and the corresponding model estimates for controls. 90

5.3 Bland-Altman plots comparing |Pf |. . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.4 Bland-Altman plots comparing |Pb |. . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.5 Bland-Altman plots comparing RM. . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.6 Bland-Altman plots comparing the S wave energy. . . . . . . . . . . . . . . . . . 97

5.7 Bland-Altman plots comparing the R wave energy. . . . . . . . . . . . . . . . . . 98

5.8 Bland-Altman plots comparing the ratio of the R to S wave energy. . . . . . . . . 98

5.9 Bland-Altman plots comparing the D wave energy. . . . . . . . . . . . . . . . . . 99

5.10 Bland-Altman plots comparing the ratio of the S to D wave energy. . . . . . . . . 99

List of Figures xi

A.1 Exemplary aortic pressure signal including a missing heart beat. . . . . . . . . . 109

A.2 A: typical flow curve used as input to the models. B-D: simulated pressure waves

for P∞ ranging from 0 to 75 mmHg obtained with the WK2 (B), WK3 (C) and

the WK4p (D). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

A.3 Difference between modulus and phase of modelled input impedance for P∞ = 0

and P∞ = 75 mmHg for the three different Windkessel models. . . . . . . . . . . 111

A.4 Exemplary result of fitting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

A.5 Mean RMSE between analytical function and regular and prolonged diastole when

fitted to the regular and prolonged part respectively. . . . . . . . . . . . . . . . . 113

B.1 Distension of the cross-sectional area for a constant pulse wave velocity as a func-

tion of pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

List of Tables

2.1 Nomenclature used for the dominating waves according to the sign of dI and dP . 37

3.2 Results of the parameter identification. . . . . . . . . . . . . . . . . . . . . . . . . 61

4.2 Clinical measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.3 PWA parameters derived by the SphygmoCor system and pressure time indices. 76

4.4 WSA and WIA parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.5 Correlations between PWA, WSA and WIA parameters. . . . . . . . . . . . . . . 78

4.6 Correlations between measures of cardiac function and structure and haemody-

namic parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.1 Comparison of the flow shape and the WSA parameters obtained by the different

blood flow models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.2 Comparison of the WIA energies obtained by the different blood flow models. . . 94

5.3 Comparison of the WIA peaks obtained by the different blood flow models. . . . 95

A.1 The input impedances Zin described by the three different Windkessel models. . 108

A.2 Parameter values for the standard parametrisation. . . . . . . . . . . . . . . . . . 109

xii

Chapter 1

Introduction

Arterial blood pressure has been used for the stratification of cardiovascular risk for literally

thousands of years. For a long time, its assessment was restricted to palpation only [79]. The

non-invasive measurement of both the systolic and diastolic blood pressure became feasible in

1905 and is still in use today in conventional blood pressure readings. Systolic blood pressure

(SBP) describes the maximum and diastolic blood pressure (DBP) the minimum of the arterial

pressure waveform. It is well accepted that this ”usual blood pressure is strongly and directly

related to vascular (and overall) mortality” [102, p. 1903], as demonstrated in a multitude of

studies and meta-analyses [102]. However, besides the absolute values of brachial SBP and DBP,

also an excessive pulsatility of arterial pressure, given by the pulse pressure PP=SBP-DBP, was

found to be a strong indicator of cardiovascular risk [157] and has been included as an additional

risk factor in the elderly by the current European guidelines for the treatment of hypertension

[61]. Even though these relations were first reported for peripheral pressure levels only, first

evidence exists that central or more precisely aortic pressure (measured or synthesised) might be

even superior [1, 65, 110]. This is in line with the idea that central and not peripheral pressure

is the one directly affecting the organs and determining the load the heart has to eject against.

Analysis of the central pressure and flow waveforms moreover helped to gain a deeper under-

standing of the mechanisms underlying physiological ageing or the pathogenesis of cardiovascular

disease, showing the importance and contribution of arterial stiffness and wave reflections to

pulsatile haemodynamics [54, 77]. For this purpose, various mathematical models have been de-

veloped, adapted or simplified to describe the relation between properties of the arterial system

and the corresponding pressure and flow dynamics [28, 91, 148, 152]. Based on these models,

different methodologies were subsequently introduced to derive information about the status of

the cardiovascular system of a specific person from (non-invasively) measured pressure and/or

flow, often summarised under the general term pulse wave analysis (PWA) [84]. Nowadays, PWA

parameters are more and more finding their way into clinical research and increased arterial stiff-

1

Chapter 1. Introduction 2

ness and excessive wave reflections have been shown to be related to a worse prognosis in the

general population [17, 133] as well as in different groups of high risk patients [111].

However, the situation changes when the contractility of the left ventricle is impaired and the

ventriculo-arterial coupling, in other words the interplay of the heart and the arterial system,

is therefore altered. The corresponding clinical syndrome, characterised by a left ventricular

systolic dysfunction accompanied by typical symptoms and signs, is called systolic heart failure

(SHF) or heart failure with reduced ejection fraction (EF). The latter refers to the imperfect

emptying of the left ventricle due to the impaired contractility [5], whereby the EF is defined as

the volume of blood that is ejected during one cardiac cycle relative to the left ventricular filling

volume prior to ejection.

Even though high blood pressure [127], excessive wave reflections [17] and increased arterial

stiffness [128] have been found to be important predictors for the new onset of heart failure,

the behaviour of conventional pressure-based risk indicators changes as soon as systolic heart

failure develops. The relation of peripheral pulse pressure to outcome was found to be reversed

[4, 26, 42, 67, 107] or U-shaped [53] and a low systolic blood pressure was associated to a

worse prognosis in a recent meta-analysis including more than 8000 heart failure patients [106].

Moreover, indices of wave reflections assessed by PWA were repeatedly reported to be lower in

patients with systolic heart failure compared to controls [21, 137], complicating their interpreta-

tion and usability as risk indicators. Overall, as stated in the 2016 guidelines for the diagnosis

and treatment of acute and chronic heart failure of the European Society of Cardiology (ESC),

the ”precise risk stratification in HF remains challenging” [101, p.10]. These observations un-

derline the importance of ventriculo-arterial coupling for all haemodynamic parameters and the

necessity to understand and ideally to quantify the impact of cardiac function on the derived

PWA parameters in order to use them for risk assessment in these patients.

The umbrella term PWA comprises a large variety of models with varying complexities, ranging

from the simple detection of an inflection point over linear transmission line theory to one-

dimensional models describing the conservation of mass and balance of momentum in an arterial

segment. The most simple approaches are thereby based on a single pressure measurement, which

will be referred to as pressure pulse wave analysis, while the more advanced methods use both

pressure and flow. Two prominent representatives of the latter class are wave separation analysis

(WSA), introduced by Westerhof and coworkers in 1972 [148], and wave intensity analysis (WIA),

proposed by Parker et al. in 1990 [91]. However, independent of their level of sophistication, all

of these methods are intended to help in the stratification of cardiovascular risk and the early de-

tection of cardiovascular disease. Therefore, they should ideally be easily applicable at a primary

care level and should present no risk for the patients, i.e. non-invasive measurements are prefer-

able. For blood pressure, validated transfer functions exist to synthesise central pressure from

peripheral readings [47], which can be performed using a simple pressure cuff [134, 135]. The

non-invasive acquisition of aortic blood flow, on the other hand, is more complicated. Doppler

ultrasound can be used to non-invasively measure the blood flow velocity in the left ventricular

Chapter 1. Introduction 3

outflow tract [105], but this requires dedicated equipment and specially trained operators which

limits its widespread use. To facilitate the acquisition of blood flow, different flow models based

on pressure alone have therefore been introduced in literature [35, 49, 142].

Data regarding the applicability of these flow models for WSA is limited to patients with normal

systolic function so far [35, 49, 142]. Even though one approach has already been used to compute

WSA in SHF as well [122], these patients often show a modified ejection pattern compared

to controls and the feasibility is therefore more than questionable. With regards to WIA, no

systematic evaluation of any of the flow models exists to date, independent of the cardiac function.

The use of a blood flow model facilitating the computation of WSA or WIA parameters from

pressure alone is a key element in order to enable a widespread use of these techniques, including

in patients with SHF. Therefore, further research is needed to evaluate the feasibility of the

existing approaches. Moreover, because of the unique features found in SHF, a new flow model

might be needed that is able to accurately reproduce also pathological flow patterns.

The aim of this thesis can be separated into three individual, yet interrelated goals: 1) to

analyse the differences in the PWA, WSA and WIA parameters in SHF compared to controls

and to investigate how they can be attributed to alterations in ventriculo-arterial coupling, 2)

to develop a new model of blood flow based on pressure alone that is capable to capture the

pathological flow patterns found in patients with SHF and 3) to investigate and to compare the

performance of different existing flow models as well as of the novel approach for WSA and WIA

in SHF patients and controls.

In chapter 1, the topic of the thesis is motivated and the specific objectives are formulated.

Furthermore, a short introduction to heart failure is given including its medical definition, epi-

demiology and significance as socio-economic health burden. Chapter 2 provides an overview of

the different methodologies to derive parameters for the stratification of cardiovascular risk from

measured pressure and flow waves, with a main focus on their mathematical background and

physiological interpretation. In chapter 3, three existing models of blood flow based on pressure

alone are presented and a new approach for the computation of blood flow is introduced. Chap-

ter 4 is dedicated to the analysis of the parameters derived by the different methods given in

chapter 2 in patients with systolic heart failure compared to controls. For this purpose, clinical

data of 183 patients is analysed. In chapter 5, the same study population is used to test and to

evaluate the performance of the different blood flow models introduced in chapter 3. Finally, a

short summary of the results and derived conclusions is given in chapter 6.

Chapter 1. Introduction 4

Figure 1.1: Hospitalisations due to heart failure in different European countries and the United

States. (1°), (2°), (any) indicates HF being the primary, secondary or any diagnosis; total

describes the total number of (all-cause) hospital addmissions per year; LoS: average length of

hospital stay in days. Reproduced from [18], licensed under CC BY-NC-ND 4.0. Please refer

to the original work for the corresponding data sources which are indicated by numbers in the

graphic.

Heart failure (HF) describes any abnormality of cardiac function or structure that results in a

reduction of cardiac output or an elevation of the cardiac pressure levels and causes typical symp-

toms like breathlessness or fatigue [101]. In this section, a short overview over its epidemiology

as well as the definition of systolic heart failure will be given.

cardiovascular epidemic of the 21st century”. -Thomas F. Lüscher, 2015 [57]

Worldwide, more than 23 million people are suffering from HF [109]. The total prevalence in

the adult population equals 1-2% in western countries and increases with age, with a prevalence

of more than 10% in persons aged 70 or older [101]. As a result of the ageing society, HF is

expected to become even more frequent in the future, whereby projections in the U.S. show an

increase in prevalence of up to 46% from 2012 to 2030 [71]. In the Rotterdam study, the lifetime

risk for developing HF was found to be 30% for a person aged 55, whereby men showed a higher

risk than women (33% vs. 28%) [8] and from data in the U.S., a lifetime risk of 20% at the age

of 40 was determined [71]. The health related quality of life is significantly reduced for patients

Chapter 1. Introduction 5

living with HF, particularly with regards to physical activity and vitality [155], and mortality is

high. Even though modern treatment could improve the mortality rates of HF within 5 years of

the diagnosis, which were as high as 60-70% before 1990 [5], they still reach approximately 50%

[71].

hospitalisation. Therefore HF had become ”the single most frequent cause of hospitalization

in persons 65 years of age or older”[10, p. 1365] in 1997. In Austria, more than 25000 hospi-

tal admissions due to HF as primary cause were reported in 2010 based on data collected by

the Statistik Österreich [18], representing 1% of all hospital admissions in this year, see figure

1.1. For patients with stable HF, i.e. patients without any changes for at least one month, the

12-month hospitalisation rate equals 32%, meaning that almost one third will be admitted to

a hospital within the next year. For patients with acute HF, who are currently hospitalised,

this rate increases to 44% [101] and the 1-month (all-cause) readmission rate reaches 25% in

these patients [155]. Figure 1.1 gives an overview over total annual hospitalisations due to HF

in Europe and the U.S.. The high amount of hospital admissions associated to HF of course

represents a considerable economic burden. For example for the U.S., hospitalisation costs are

in the range of 15$ billion per year, which equals approximately half of the total costs caused by

HF comprising healthcare, medication and lost productivity [155].

Heart failure is very heterogenous in itself and various subtypes exist. The most prominent

classification considers two different types, namely HF with reduced EF, normally defined as

EF< 40%, and HF with preserved EF, i.e. EF> 50% [101] 1 . The first type is also referred to as

systolic and the latter as diastolic heart failure, relating to the impairment in systolic ejection

(systolic dysfunction) and diastolic filling (diastolic dysfunction) respectively. In systolic heart

failure, the ventricle typically dilates and the contractility is decreased, while in diastolic heart

failure, the thickness of the ventricular wall and therefore its stiffness increases and its filling

capacity is decreased, compare figure 1.2. However, it should be noted that diastolic dysfunction

may be present in systolic heart failure and subtle systolic dysfunction in diastolic heart failure,

which is why the ESC recommends the use of heart failure with reduced and preserved EF instead

[101]. Nevertheless, in this thesis, the terms heart failure with reduced EF and systolic heart

failure will be used interchangeably. HF with reduced EF constitutes approximately one half of

all HF cases and generally shows a worse prognosis than HF with preserved EF [101].

1 Patients with HF and EF between 40% and 50% represent a borderline group, which should be treated as a

Chapter 1. Introduction 6

Figure 1.2: Schematic representation of a normal heart (A) compared to systolic (B) and diastolic

dysfunction (C). Reproduced from [18], licensed under CC BY-NC-ND 4.0.

Chapter 2

cardiovascular risk

The haemodynamics in the arterial system result from a complex, dynamic interaction of the

heart and the vasculature. Analysis of the corresponding pressure and flow waveforms can

therefore provide important information about the cardiovascular health status of a specific

person. Various methodological approaches exist to gain parameters from measured pressure

and/or flow waves for the assessment of cardiovascular risk and some of the most widely used

techniques and their mathematical background will be presented in this chapter.

High blood pressure has been identified as a risk factor for stroke as early as 4000 years ago.

Back then, the diagnosis was based on the ’hardness’ of the pulse felt by palpation of superficial

arteries. This diagnostic procedure did not change until the first sphygmographs were developed

in the 19th century enabling the non-invasive measurement and recording of the arterial pressure

pulse. [79]

Based on these sphygmographical measurements, Frederick Akbar Mahomed1 was already able

to describe the effects of ageing and hypertension on the shape of the radial pressure wave in the

middle of the nineteenth century, thereby laying the foundation of modern pulse wave analysis.

Moreover, he found that the pulse form changes during its propagation through the body, a

phenomenon that was almost one hundred years later explained by McDonald2 and Womersley3

with the occurence of wave reflections. [86, 87]

1 Frederick

Henry Horatio Akbar Mahomed (1849-1889), British physician

2 Donald Arthur McDonald (1917-1973), British physician

3 John Ronald Womersley (1907-1958), British mathematician

7

Chapter 2. Parameters for the assessment of cardiovascular risk 8

Figure 2.1: Examples of different measurement techniques to acquire arterial pressure. (a)

invasive measurement of intra-aortic pressure during cardiac catheterisation, (b) non-invasive

recording of radial pressure by applanation tonometry, (c) non-invasive brachial pressure reading

using an oscillometric pressure cuff. Partly adapted from [59]

However, at the turn of the 20th century, the sphygmographs were replaced by cuff sphygmo-

manometers, which were easier to handle but provided the extreme values of pressure only. The

method introduced by Riva-Rocci4 in 1896 was based on the occlusion of the brachial artery

by an inflatable cuff around the upper arm, a manometer to measure the pressure applied and

palpation of the radial pulse to determine the inflation pressure needed for a total occlusion,

which equals the maximum, systolic blood pressure. By replacing palpation by auscultation

as proposed by Korotkov5 , the measurement of both maximum and minimum (diastolic) blood

pressure became feasible. This principle is still in use today, but also automatic devices based

on oscillometric pressure cuffs evolved which determine systolic and diastolic blood pressure by

measuring the oscillations in the cuff introduced by the semi-occluded artery. [79, 80]

During most parts of the 20th century, entire pressure waveforms were therefore primarily as-

sessed invasively during cardiac catheterisation until, starting in the 1980ies, new sphygmograph-

ical techniques were developed. These include applanation tonometry, photo-plethysmography

and methods based on oscillometric pressure cuffs to non-invasively record peripheral pressure

waveforms [23, 79, 134], compare figure 2.1. Furthermore, the investigation of the relation of

peripheral to central pressure in the frequency domain led to generalised transfer functions that

enable the estimation of the central pressure wave from a peripheral reading [47, 135]. Thus,

nowadays, peripheral and central pressure waves can be obtained non-invasively and be used for

cardiovascular risk stratification, in line with the idea of Mahomed dating back to 1872 [86]:

”...the information which the pulse affords is of so great importance, and so often

consulted, surely it must be to our advantage to appreciate fully all it tells us, and

to draw from it every detail that it is capable of imparting.” [60, p. 62]

This section focuses on the computation and interpretation of parameters that can be directly

derived from measurements of the arterial pressure waveform.

4 Scipione Riva-Rocci (1863-1937), Italian internist

5 Nikolai Sergeyewich Korotkov (1874-1920), Russian surgeon

Chapter 2. Parameters for the assessment of cardiovascular risk 9

(a) (b)

115

Radial Pressure

110 SBP

105

T 100

0 1 2 3 4 5 6 7 8 9 PP

Time, s

95

Aortic Pressure

90

85

DBP

T 80 ED

0 1 2 3 4 5 6 7 8 9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Time, s Time, s

Figure 2.2: Determination of timing information and pressure levels. (a): Exemplary radial

pressure signal (upper panel) assessed by applanation tonometry and the corresponding aortic

pressure (lower panel) obtained by a generalised transfer function (SphygmoCor device, AtCor

Medical, Sydney, Australia). The red dots indicate the feet of the pressure waves at the beginning

of systolic upstroke which are used to identify single heartbeats and to compute the heartbeat

duration T . (b): aortic pressure wave obtained from the measurement shown on the left by taking

the average over several heartbeats (SphygmoCor). Systolic (SBP) and diastolic blood pressure

(DBP) as well as the amplitude of the pressure wave, the pulse pressure (PP), are shown. By

identifying the dicrotic notch at the end of systole, the ejection duration (ED) can be derived.

A typical pressure signal measured at the radial artery and the corresponding aortic pressure

derived with a generalised transfer function are shown in figure 2.2 (a). Left ventricular ejection

results in a fast increase in blood pressure and by identifying the beginning of upstroke, it is

possible to discriminate single heartbeats in the signal. The heartbeat duration T is then given

by the time distance between the feet of two successive pressure waves and the heart rate (HR)

subsequently by HR = 60/T . The closure of the aortic valve at the end of systole is normally

marked by an incisura in the pressure wave called the dicrotic notch that can be used to deter-

mine the ejection duration (ED) as the time between the foot and the incisura.

The pressure wave P (t), t ∈ [0, T ], corresponding to one single heartbeat or, in order to reduce

the influence of noise, the average over several heartbeats can be further analysed as shown in

figure 2.2 (b). As mentioned before, the most famous measures are the maximum and minimum

value. The maximum is always reached during ventricular ejection in systole and is therefore

called systolic blood pressure (SBP). When ejection stops, pressure decreases until it reaches

its minimum at the end of diastole, called the diastolic blood pressure (DBP). The difference

between SBP and DBP equals the amplitude of the pressure wave and is termed pulse pressure

(PP).

PP = SBP − DBP = max P (t) − min P (t) (2.1)

t∈[0,T ] t∈[0,T ]

PP represents the pulsatile component of pressure whereas the mean blood pressure (MBP)

Chapter 2. Parameters for the assessment of cardiovascular risk 10

Z T

1

MBP = P (t) dt. (2.2)

T 0

throughout the body

During systole, the ejecting heart generates pressure waves that propagate throughout the body

and result in the characteristic pulse that is palpable at any superficial artery. The initial waves

originating at the ventricle are travelling away from the heart in forward direction. At any

bifurcation, any irregularity in the vessel wall or more generally discontinuity of the arterial

tree, parts of these waves are reflected resulting in backward travelling waves. On the way back

towards the heart, re-reflections occur, giving again rise to forward waves and so forth. The

observable, measured waveforms at any arterial site are therefore composed of multiple forward

and backward running waves. For this and all of the other concepts presented in this chapter,

only one forward and one backward wave will be considered which represent the cumulation of

all forward and backward waves respectively.

The morphology of the pressure wave thus depends on the magnitude and relative timing of both

the forward and the reflected wave. To quantify the contribution of each, the so-called augmen-

tation index (AIx) was introduced. From data obtained in invasive studies using catheters to

measure pressure and flow in the ascending aorta, it was observed that peak flow corresponds

to the first shoulder ”defined as the first concavity on the upstroke” of the pressure wave [48,

p.1654]. The first shoulder can either lay before or be equal to the maximum which implies that

the corresponding pressure level P1 ≤ SBP.

Up to the first shoulder, pressure is mainly caused by blood flow and therefore directly by

ventricular ejection. Due to the return of the reflected wave, this initial pressure is further

increased, resulting in a second peak or shoulder P2 , where either P1 or P2 equals SBP. For

SBP = P2 > P1 , wave reflections cause an augmentation of total maximum pressure by P2 − P1 ,

which is termed augmentation or augmented pressure (AP). Naturally, the magnitude of AP

depends on the absolute pressure level and amplitude of a specific person and is therefore indexed

to PP in order to obtain a relative and comparable measure, the augmentation index (AIx).

P2 − P1 AP

AIx = = (2.3)

PP PP

For SBP = P1 > P2 , AIx is defined analogously, resulting in negative values of both AP and AIx

without direct physiological interpretation. Depending on the value of AIx, an aortic pressure

waveform can be classified as type A for AIx > 0.12, type B for 0 < AIx ≤ 0.12 or type C for

AIx ≤ 0 [72].

Chapter 2. Parameters for the assessment of cardiovascular risk 11

115

P2=SBP

110 AP

P1

105

100

AIx=0.14

95

90

85

80

Tr DBP

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Time, s

Figure 2.3: Computation of AIx. By identifying the two shoulders in the pressure signal, the

corresponding pressure levels P1 and P2 can be obtained. In this case, P1 < P2 and the returning

reflected wave therefore causes an augmentation of total pressure by AP, resulting in an AIx of

0.14. The pressure wave can thus be classified as a type A waveform. The inflection point marks

the return of the reflected wave, which is used to derive the round-trip travel time Tr .

The arrival of the backward wave corresponds to an inflection point in the pressure signal [83],

which can be used to determine the time it takes for a wave to reach a reflection site and

subsequently return to the heart, called round trip travel time Tr . Figure 2.3 shows an exemplary

type A pressure wave and the corresponding pressure levels P1 and P2 as well as Tr .

For the computation of AP and AIx, commonly higher order derivatives of the pressure signal

are employed. In particular, the zero-crossings of the fourth derivative can be used to detect

the first or second shoulder depending on whether P1 or P2 equals SBP [20, 48, 124]. Another

approach is to use the zero crossings of the second derivative to identify the inflection point

and to compute AIx from the inflection pressure [114]6 . Even though quantitative differences

exist between these two approaches, the corresponding values of AIx were shown to be highly

correlated [114]. For Tr , in contrast, differences resulting from applying the second or the fourth

derivative to identify the inflection point are more pronounced [114, 143].

The round trip travel time Tr is affected by the distance to the reflection site as well as the

propagation speed, the pulse wave velocity (PWV). PWV in turn depends on the properties

of the vessel and the blood within as well as on the pressure level. Under some simplifying

assumptions7 , local PWV can be directly related to the stiffness of the arterial wall by the

Moens-Korteweg equation [11]. s

Eh

PWV = (2.4)

2ρr0

E thereby denotes the elastic modulus of the vessel wall, ρ the blood density, h the stationary wall

6 It should be noted that Segers et al. [114] did not use the mathematical definition of inflection point but

defined it as the mid-point between two consecutive zero-crossings of the second derivative.

7 The main assumptions include blood to be inviscous and the tube to be thin-walled. For a more detailed

description of the derivation as well as the simplifications applied, see e.g. [11].

Chapter 2. Parameters for the assessment of cardiovascular risk 12

Pressure

Δt1

Carotis

ECG

Δx

Pressure

Δt2

Femoralis

ECG

Time, s

Figure 2.4: Assessment of carotid-femoral PWV using sequential pressure measurements and

a simultaneous electrocardiogram (SphygmoCor). For both measurement sites, the time delay

between the R-peak in the ECG (indicated by a black circle in the pressure signal) and the arrival

of the foot of the pressure wave (red diamonds) are computed resulting in the total travel time

∆t = ∆t2 − ∆t1 . The corresponding travel distance ∆x is obtained directly from the patient,

either by approximating it by the total distance between the two sites as depicted in this figure,

or by various other approaches accounting for the fact that the waves are actually travelling in

different directions [136]. In either case, PWV is subsequently determined by PWV = ∆x/∆t.

Partly adapted from [59]

thickness and r0 the stationary lumen radius. According to the above relation, the propagation

speed increases with arterial stiffening (increasing E), leading to an earlier return of the reflected

wave and a shift of its arrival from diastole to late systole. Consequentially, blood pressure

augmentation and therefore also SBP and PP are increased. Arterial stiffening is a normal

consequence of ageing, mainly affecting the large conduit arteries and in particular the aorta.

Aortic PWV might be more than twice as high in old age compared to youth. However, also

cardiovascular diseases can cause premature or exaggerated arterial stiffness making PWV an

important indicator. [85, 86]

PWV is commonly assessed by the time delay or foot-to-foot method. For this approach, pressure

is measured (simultaneously or sequentially with a simultaneous electrocardiographic recording)

at two different sites and the distance between the sites is divided by the time delay between

the arrival of the foot of the pressure wave at the respective sites [54], compare figure 2.4. In

contrast to the Moens-Korteweg PWV, the so-obtained PWV represents the average velocity

over the whole path from site A to site B.

The current gold standard in the assessment of aortic pulse wave velocity is its measurement

during cardiac catheterisation. In this case, the time delay method is employed during catheter

pullback with both measurement sites located in the aorta. A non-invasive alternative is given by

Chapter 2. Parameters for the assessment of cardiovascular risk 13

the carotid-femoral PWV, which uses tonometric or oscillatory pressure measurements at both

the carotis and the femoralis to cover the aortic path except for the ascending aorta [138]. Even

though this method is ”generally accepted as the most simple, non-invasive, robust and repro-

ducible method to determine arterial stiffness” [54, p.2591], special attention has to be laid on

the estimation and measurement of the travel distance, which strongly influences the computed

velocity [136, 138].

Recently, also a one-point estimate of aortic PWV based on a single oscillometric pressure read-

ing, i.e. a reading performed at one site only, has been proposed and implemented as part

of the ARCSolver method (AIT, Austrian Institute of Technology GmbH) [34]. The estima-

tion procedure combines parameters of pulse wave analysis, age and model-based characteristic

impedance, a parameter of impedance analysis which will be introduced in section 2.2. Estimated

values showed acceptable agreement with invasively assessed PWV [34, 138].

The elasticity of the arterial wall characterised by the elastic modulus E in the Moens-Korteweg

equation is a local property that varies throughout the body. The large conduit arteries close

to the heart show the lowest values, whereas E increases in distal direction away from the heart

[77]. Although radius and wall thickness are decreasing too, the term Eh/r0 was found to grow

almost exponentially with decreasing r0 in the main arteries [81] resulting in a high increment

of PWV according to the Moens-Korteweg equation (2.4). Exemplary values taken from a study

by Wang et al. [132] predict a 1.6-fold increase in PWV from the ascending aorta to the femoral

artery, if the blood density is taken to be constant ρ = 1060 kg/m3 [143].

Invasive measurements show a similar boost in PWV towards the periphery [77]. Moreover, the

length of the arteries decreases and the branching increases away from the heart leading to shorter

distances to the reflection sites. As a result, reflected waves return much earlier during the cycle

in the peripheral than in the central arteries, amplifying both systolic and pulse pressure [54].

This phenomenon can be quantified by the ratio of peripheral to central PP, the pulse pressure

amplification (PPAmp).

peripheral PP

PPAmp = (2.5)

central PP

PPAmp is highest in young ages, when aortic PWV is low and the pressure augmentation mainly

occurs during diastole. With age and stiffening of the central arteries, central PP increases and

the amplification diminishes. [77, 85]

Pressure time integrals or indices represent different areas under the pressure curve obtained by

integration over a specific time interval. The systolic pressure time index (SPTI), as the name

Chapter 2. Parameters for the assessment of cardiovascular risk 14

115

Ew

110

SPTI=Ew+ΔPSA+ΔSPTI

105

100

95

ΔPSA

90

DPTI

85

ΔSPTI

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Time, s

Figure 2.5: Computation of the time indices representing the areas under different portions of

the pressure wave. The systolic index (SPTI) can furthermore be separated in the area below

DBP (∆SPTI) and, if the inflection point occurs before the maximum as depicted in this figure,

the area below (∆PSA) and above the inflection point (Ew ).

Z ts

SPTI = P (t) dt, (2.6)

0

where ts denotes the time of the end of systole. SPTI is also called tension time index and is

supposed to be directly related to the cardiac oxygen demand [29], since the pressure developed

during cardiac ejection greatly affects the cardiac workload. Due to an early return of reflected

waves during systole, as observed in type-A and type-B waveforms, the left ventricular afterload

is augmented and additional energy is required for blood ejection. In line with the concepts

introduced in the last section, this wasted effort Ew can be estimated by the portion of systolic

pressure above the inflection point [39].

Hence, SPTI can be separated into the rectangular area below DBP denoted as ∆SPTI =

DBP · ED and the area above DBP called pressure systolic area (PSA). If the inflection point

occurs before the maximum, PSA can be further subdivided into the area below (∆PSA) and

above the inflection point (Ew ). [21]

The diastolic pressure time index (DPTI) denotes the corresponding area under the diastolic

part of the pressure wave

Z T

DPTI = P (t) dt, (2.7)

ts

and is related to coronary blood flow. Coronary perfusion, particularly with regards to the left

heart, mainly occurs during diastole when the heart is relaxed. Driven by a pressure gradient, the

blood is pushed into the myocardium and DPTI provides an estimate of this perfusion gradient.

The ratio of DPTI to SPTI finally represents an index of coronary perfusion termed myocardial

Chapter 2. Parameters for the assessment of cardiovascular risk 15

viability ratio, that indicates if the cardiac oxygen demand is met by the supply. [12]

Modern arterial haemodynamics started with the work of McDonald and Womersley describing

the propagation of waves in the arterial system in the frequency domain. Based on the (non-

linear) Navier-Stokes equations, Womersley formulated the mathematical conditions needed for

linearisation8 [156]. Thereby he showed that nonlinearities are negligible in larger arteries in a

first approximation under most pathophysiological conditions. These theoretical considerations

were later corroborated by experimental results [76], justifying the assumption of linearity. If the

arterial system is furthermore assumed to be in steady state, i.e. a regularly beating heart with

both pressure and flow being periodic functions, it can be completely characterised by the ratio

of pressure to flow at the inlet of the system in the frequency domain using Fourier analysis [6].

Referring to electrical engineering, this transfer function is called arterial or input impedance Zin .

Based on this concept, a method to separate measured aortic waveforms in their forward and

backward travelling components, called wave separation analysis (WSA), was introduced by

Westerhof and coworkers in 1972 [148] and will be presented in the following sections.

The real-valued, periodic, time-varying functions of both pressure P (t) and flow Q(t) can be

represented by their Fourier series9

∞

X

P (t) = P + (an cos(ωn t) + bn sin(ωn t)) ,

n=1

X∞

Q(t) = Q + (cn cos(ωn t) + dn sin(ωn t)) ,

n=1

Z T Z T

2 2

an := P (t) cos(ωn t) dt, bn := P (t) sin(ωn t) dt, (2.8)

T 0 T 0

Z T Z T

2 2

cn := Q(t) cos(ωn t) dt, dn := Q(t) sin(ωn t) dt. (2.9)

T 0 T 0

T and T denotes the period, i.e. the heartbeat

duration. P and Q are the zero-frequency coefficients and represent the mean, steady components

8 The main assumptions include (1) blood to be a homogenous, incompressible, Newtonian fluid with constant

viscosity (2) the arteries to be circular and axisymmetric and (3) the wavelength of the considered pressure and

flow waves to be much larger than the radius of the tube, as well as the mean flow velocity to be much smaller

than the pulse wave velocity, see [156] for details.

9 Natural signals are generally smooth functions, in other words it can be assumed that P and Q are at least

Chapter 2. Parameters for the assessment of cardiovascular risk 16

98 120 76

400

0 96 100 74 200

0

94 80 72

20 120 200

400

1 0 100 0 200

0

−20 80 −200

10 120 200

400

2 0 100 0 200

0

−10 80 −200

5 120 100

400

3 0 100 0 200

0

−5 80 −100

1 120 50

400

4 0 100 0 200

Harmonics

0

−1 80 −50

1 120 50

400

5 0 100 0 200

0

−1 80 −50

1 120 50

400

6 0 100 0 200

0

−1 80 −50

1 120 20

400

7 0 100 0 200

0

−1 80 −20

0.5 120 10

400

8 0 100 0 200

0

−0.5 80 −10

0.5 120 10

400

9 0 100 0 200

0

−0.5 80 −10

0.2 120 5

400

10 0 100 0 200

0

−0.2 80 −5

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

Time, s Time, s Time, s Time, s

Figure 2.6: Fourier decomposition and approximation of aortic pressure and flow for the first 10

harmonics. The single sinusoids for harmonics 0-10 are shown in the respective left panel and

the waveforms corresponding to the sum of all preceding harmonics (blue solid line) as well as

the original measured ones (black, dashed line) in the respective right panel.

Z T

1

P := P (t) dt, (2.10)

T 0

Z T

1

Q := Q(t) dt. (2.11)

T 0

Thus in particular it holds that P = MBP. For each harmonic describing the oscillations with a

frequency equal to an integer multiple of the fundamental frequency 1/T , the corresponding real

coefficients can be summarised in the following complex coefficients

Pbn := an − ibn , Q

b n := cn − idn , (2.12)

Chapter 2. Parameters for the assessment of cardiovascular risk 17

|Zin|, mmHg⋅s/ml

1

0.5

0

0 1 2 3 4 5 6 7 8 9 10

Harmonic

π/4

arg(Zin), rad

−π/4

0 1 2 3 4 5 6 7 8 9 10

Harmonic

Figure 2.7: Modulus (upper panel) and phase (lower panel) of the arterial input impedance

computed from the flow and pressure waves shown in figure 2.6 for the first 10 harmonics.

∞ ∞

!

X X

iωn t

P (t) = P + Re Pbn e =P + Pn cos ωn t + arg Pbn , (2.13)

b

n=1 n=1

∞ ∞

!

X X

Q(t) = Q + Re b n eiωn t

Q =Q+ Qn cos ωn t + arg Q

b bn , (2.14)

n=1 n=1

where i denotes the imaginary unit, Re(.) the real part and arg(.) the argument of a complex

number. For measured pressure and flow signals, where the continuous functions P (t) and Q(t)

are represented by a discrete sequence of N equidistant data points per heartbeat depending on

b n for the harmonics n = 1, . . . , b(N − 1)/2c can easily be obtained

the sampling rate dt, Pbn and Q

by applying the fast Fourier transform. To reconstruct the original signal, usually the first 10-15

harmonics are sufficient [90, 123], compare figure 2.6.

b 0 = Q, the input impedance Zin (ωn ) is now defined by

Pbn

Zin (ωn ) = , n ≥ 0. (2.15)

Qbn

Figure 2.7 shows an example of Zin obtained from measured pressure and flow. The zeroth

frequency component of Zin equals the ratio of mean pressure to mean flow and represents

the systemic vascular resistance (SVR) or, in other words, the total resistance the vasculature

opposes to flow.

Chapter 2. Parameters for the assessment of cardiovascular risk 18

For the approach of wave separation analysis, the arterial system is considered a network of

elastic tubes transmitting the waves generated by the left ventricle to the periphery. The system

is assumed to be linear, implying that ”a single sinusoid of flow as input produces a sinusoid of

pressure as output with identical frequency and with a phase shift” [6, p.150]. Therefore each

harmonic of pressure and flow can be treated separately and concepts of linear transmission line

theory can be applied. In this framework, pressure corresponds to voltage and flow to current.

[148]

For the following analysis, only the oscillatory components and therefore the coefficients for

n ≥ 1 will be considered while the mean parts are omitted. In the reflectionless case, e.g. for

an infinitely long, uniform tube, the ratio of pressure to flow is the same at any instance and at

any location along the tube and is given by the frequency-dependent characteristic impedance

P

Z0 (ωn ) = bn , n ≥ 1. Z0 (ωn ) is determined by the (cross-sectional) geometry and material

bn

Q

properties of the tube and the blood within and is independent of its actual length [113].

Generally, Z0 is a complex quantity, meaning that pressure and flow waves differ not only in

their amplitude but are also shifted in phase, i.e. a time delay occurs. However, for lossless

tubes (tubes without friction), Z0 (ωn ) is a real, frequency-independent constant Zc . In this case,

pressure and flow are in phase, have the same overall shape and, because no reflections occur,

equal the incident (superscript i), forward (superscript f ) waves generated at the inlet.

bn = Q b in

When the uniform tube is finite in length and loaded with an impedance ZL (ωn ) at the end,

parts of the forward running waves are reflected according to the pressure reflection coefficient

ZL −Z0

Γ= ZL +Z0 . Γ defines the ratio of backward to forward pressure with an absolute value between

0, no reflections, and 1, total reflection. The reflection coefficient of flow has the same magnitude

but opposite sign, i.e. equals −Γ. In either case, because of the presumed linearity, the total

waveforms at the inlet of the tube are given by the superposition of the composite forward and

backward (superscript b) travelling waves. [148]

bn = Q

n

bb

n (2.16)

Due to the reflections, Zin no longer equals Z0 . However, looking at the forward and backward

waves separately, they can be interpreted as single waves, travelling undisturbedly through the

tube and their relation is therefore again given by

Pbnf Pbnb

= Z 0 (ωn ) = − . (2.17)

Qb fn Qb bn

Chapter 2. Parameters for the assessment of cardiovascular risk 19

30 600

P measured Q measured

Pf Qf

Pb Qb

400

Pressure above DBP, mmHg

20

Flow, ml/s

|Pf| 200

10

0

|Pb|

0

−200

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time, s Time, s

Figure 2.8: Aortic pressure and flow separated in their forward (black) and backward (red)

components using wave separation analysis. For illustration, Pf,b and Qf,b were shifted to start

at 0. Zc was estimated in the frequency domain using harmonics 4-10.

The negative sign on the right side results from the fact that flow waves are reflected 180 degrees

out of phase compared to pressure waves, as already mentioned before.

In the context of the arterial system, the tube represents a short, presumably uniform segment

of the ascending aorta and ZL the impedance of the branching arterial tree further downstream

[148]. Jager and colleagues [43] showed that losses due to friction are very small for large, conduit

arteries. Aortic Z0 (ωn ) can therefore be regarded a real constant Zc [148]. This simplification

implies that both the viscosity of the blood as well as the viscoelasticity of the arterial wall are

neglected [149].

From equations (2.16) and (2.17), the following expression can be derived for the forward trav-

elling aortic pressure wave Pbf n

b bn =

b f ) = Pbn + Zc Q

bn − Q

= Pbn + Zc (Q b n − Pbf ,

n n

and hence

Pbn + Zc Q

bn

Pbnf = . (2.18)

2

Pbn − Zc Q

bn

b fn = Pn + Zc Qn ,

b b

b bn = − Pn − Zc Qn .

b b

Pbnb = , Q Q (2.19)

2 2Zc 2Zc

The corresponding temporal waveforms Pf,b (t), Qf,b (t) can be recovered from the Fourier series,

see (2.13,2.14). Moreover, since Zc is real, the separation can also be performed directly in the

Chapter 2. Parameters for the assessment of cardiovascular risk 20

∞

! !

X Pbn + Zc Q

bn

iωn t

Pf (t) = Re e

n=1

2

∞ ∞

1 X Z X

c

= Re Pbn eiωn t + Re Qb n eiωn t =

2 n=1 2 n=1

P (t) − P + Zc Q(t) − Q

=

2

Figure 2.8 shows an exemplary aortic pressure and flow wave and the corresponding forward and

backward waves obtained by WSA. To actually quantify the reflections present in the system,

the amplitudes of Pf and Pb can be computed

t∈[0,T ] t∈[0,T ]

Relative measures can be obtained by determining the reflection magnitude (RM) and the re-

flection index (RI), which are defined by

|Pb |

RM = , (2.21)

|Pf |

|Pb |

RI = . (2.22)

|Pf | + |Pb |

In summary, the total (measured) waveforms of pressure and flow at the aortic root can be

separated in their forward and backward components if only aortic characteristic impedance is

known. In order to obtain an estimate of Zc , different approaches have been suggested, two of

which will be introduced in the following section.

The first approach is based in the frequency domain and its explanation will follow the one given

by Westerhof et al. [144]. Measurements of the reflection coefficients encountered by forward

travelling waves at the branching points in larger arteries were found to be ”mostly in the order

of 0.1 or smaller, approximately real, and only slightly frequency dependent” [144, p.139]. These

values increase up to 0.5 in the periphery, meaning that reflections mainly occur in the more

peripheral arteries. For backward travelling waves, the reflection coefficients attain much higher

values, implying that reflected waves loose much of their amplitude on their way back to the heart.

Furthermore, the way how reflected waves interact with the incident waves depends on the dis-

tance to the reflection site relative to the wavelength λ, which is given by the phase velocity

divided by the frequency. For low frequencies, i.e. below 3 Hz, λ is long and the waves reach

the periphery and return to the heart during one cycle. Therefore, they add in phase amplifying

incident pressure. Higher frequency components, in contrast, are reflected at different instances

of different cycles. Thus, on the one hand, the large backward reflection coefficients greatly

Chapter 2. Parameters for the assessment of cardiovascular risk 21

(a) (b)

0.09 115

110

0.08

105

0.07

|Z |, mmHg⋅s/ml

Pressure, mmHg

100

0.06

95

in

0.05

90

Zc≈ΔP/ΔQ

0.04

85

Zc

0.03 80

1 2 3 4 5 6 7 8 9 10 −50 0 50 100 150 200 250 300 350 400 450

Harmonic Flow, ml/s

Figure 2.9: Estimation of aortic characteristic impedance. (a) average of the modulus of input

impedance for harmonics 4-10 (indicated by stars), (b) slope of the P Q-loop in early systole

assessed from the first 8 data-points ≈ 55ms (indicated by stars).

diminish their magnitude and on the other hand, they tend to cancel out when superimposed

on their way back towards the heart. In consequence, almost no reflections of higher frequency

waves are present in the ascending aorta and arterial input impedance should therefore approx-

imate aortic characteristic impedance Zc .

The characteristic pattern of the modulus of arterial input impedance with frequency observed

in humans indeed corresponds well with this concept, see figure 2.9 (a). Starting at its maximum

value, |Zin | rapidly decreases and then oscillates around an approximately constant value, while

its phase approaches 0 for increasing frequency [6]. Thus, to obtain an estimate of Zc , |Zin | is

averaged over an appropriate frequency band to eliminate the influence of oscillations [24].

For the second approach, this concept is translated to the time domain. At the beginning of

ventricular ejection, both pressure and flow show a rapid upstroke which can be predominantly

attributed to the high-frequency components. Their relation should therefore be given by the

characteristic impedance [24]. Plotting pressure over flow indeed shows an almost linear rela-

tionship during early systole, the slope of which can be used to estimate Zc , i.e. Zc ≈ ∆P/∆Q

compare figure 2.9 (b). Values obtained with this estimation procedure were shown to correspond

well with their counterparts determined in the frequency domain [24].

Another explanation to use the relation between pressure and flow in early systole is that during

this phase reflections have not yet returned and both pressure and flow are therefore composed

of forward travelling waves only. The ratio of their slopes should thus approximate Zc , i.e.

Zc ≈ dP

dt / dQ

dt [149]. An overview and comparison of different time-domain estimates can be

found in a study by Lucas et al. [58].

Chapter 2. Parameters for the assessment of cardiovascular risk 22

Windkessel models belong to the most simple class of cardiovascular models, namely the 0-D

models. Thus, in contrast to the wave transmission approach introduced in the previous sec-

tion, space and therefore also wave travel phenomena are not considered. The properties of the

arterial tree are lumped into single parameters and the whole arterial system is modelled as

one compartment. Despite their simplicity, they are useful tools to study the dynamic relation

between pressure and flow requiring only a few and, what is even more important, physiological

meaningful parameters. [145]

Windkessel models owe their name to the so-called Windkessel-effect, which describes the damp-

ing and smoothing of pulsatile flow caused by a reservoir coupled in between the pump and

the outflow. In the cardiovascular system, the heart represents the pulsatile pump whereas the

large elastic arteries and in particular the aorta constitute the reservoir. During left ventricular

ejection, their walls expand, thereby storing blood which is released again during diastole when

pressure decreases and the walls constrict.

The first mathematical model based on this concept was develeped by Otto Frank in 1899 [28]

and consisted of only two parameters. Nowadays, various extensions of this initial two-element

Windkessel exist and two of them will be introduced in the next sections.

In the original two-element formulation, the arterial system is characterized by the contained

blood volume V (t) and the two parameters peripheral resistance Rp and arterial compliance Ca .

Rp represents the resistances to blood flow of all vessels in the system added up to a single value,

with the biggest contribution coming from the small peripheral arteries and arterioles. Ca is a

measure of arterial distensibility and therefore mainly characterises the large elastic arteries.

To obtain the model equations, the arterial system is considered a conservative compartment with

only one inflow and one outflow. Changes in V (t) must therefore equal the difference between

the blood flow entering the system from the heart Q(t) and the blood flow out of the system to

the periphery Qout (t).

dV

(t) = Q(t) − Qout (t) (2.23)

dt

Because the system is distensible, changes in volume are furthermore related to changes in

pressure PRC (t). Under the assumption of linear-elasticity, this relation can be characterized by

the constant factor Ca ,

dV

Ca = . (2.24)

dPRC

Chapter 2. Parameters for the assessment of cardiovascular risk 23

Q Qout

Ca Rp

P PRC

P∞

Figure 2.10: Representation of the two-element Windkessel model as an electrical circuit. Pres-

sure P thereby corresponds to voltage and blood flow Q to current flow. The two parameters

Rp and Ca are represented by a resistor and capacitor respectively, P∞ by a voltage source.

dV dV dPRC dPRC

Q − Qout = = = Ca . (2.25)

dt dPRC dt dt

According to the hydraulic equivalent of Ohm’s law, outflow Qout can be related to pressure PRC

via the resistance Rp ,

PRC = Rp Qout . (2.26)

Combining equations (2.25) and (2.26) yields the following linear ordinary differential equation

(ODE) for the pressure PRC .

dPRC 1 1

(t) + PRC (t) = Q(t) (2.27)

dt Rp Ca Ca

Total aortic pressure P (t) is then given by P (t) = PRC (t) + P∞ , where P∞ represents an asymp-

totic pressure level that is maintained by the vascular system even without excitation from the

heart. However, disunity exists about the interpretation of P∞ and whether it should be included

or not. Its impact on the Windkessel models is discussed in more detail in Appendix A.

Figure 2.10 shows an electrical analogue representation of the two-element Windkessel model

and the mathematical definition is given below.

For Rp , Ca > 0 and P∞ ≥ 0, the dynamic relation between aortic pressure P (t) and flow Q(t)

is modelled as

dPRC 1 1

(t) = Q(t) − PRC (t). (2.29)

dt Ca Rp Ca

If the system is furthermore assumed to be in steady state, the described input impedance as

defined in (2.15) can easily be derived from the model equations when bearing in mind that the

Fourier transform is linear and that the Fourier coefficients (2.12) of the first derivative equal

Chapter 2. Parameters for the assessment of cardiovascular risk 24

iωn times the original ones. Thus, from equation (2.29) it follows that

1 b 1 b Rp

iωn PbRC,n = Qn − PRC,n ⇒ PbRC,n = bn , n ≥ 0

Q (2.30)

Ca Rp C a 1 + iωn Rp Ca

whereby the zero-frequency component (n = 0) denotes the respective mean value, e.g. Q

b 0 = Q.

PRC fully describes the dynamic, oscillatory behaviour of P whereas P∞ affects the steady part

only10 , i.e. in particular Pbn = PbRC,n for n ≥ 1 and P = P RC + P∞ for n = 0, see equation

(2.28). Hence, the input impedance can be computed as:

P P RC + P∞ PbRC,0 P∞ P∞

= = + = Rp + , n=0

Q Q Q0

b Q Q

Zin,WK2 (ωn ) = (2.31)

Pb Pb Rp

n = RC,n =

, n≥1

Qn

b Qn

b 1 + iωn Rp C a

Comparison of the input impedance modelled by the WK2 (2.31) with measured data showed

a good agreement in the low frequency range. However, as discussed in the last section, the

arterial system behaves almost like a uniform, reflectionless tube for higher frequencies with

|Zin (ωn )| ≈ Zc , whereas limn→∞ |Zin,WK2 (ωn )| = 0. The two-element Windkessel therefore fails

to describe the arterial input impedance over the whole frequency range.

The Windkessel theory, which assumes that all changes in pressure happen instantaneously

throughout the arterial system (infinite pulse wave velocity) and the wave transmission models

or more precisely the reflectionless tube model (finite pulse wave velocity) have long been consid-

ered contradictory. However, Westerhof et al. [144] and later Quick et al. [104] showed that they

cover different frequency ranges. While Zin (ωn ) = Zc for n ≥ 1 (uniform, infinitely long tube)

provides a poor description for lower frequencies, the WK2 reveals its weaknesses in the medium

to high frequency range, which become evident as a slow upstroke in the beginning of systole

in the time domain, see figure 2.12. Therefore, Westerhof proposed adding aortic characteristic

impedance Zc as a third element to the WK2, in an attempt to unify these two competing con-

cepts [104].

Zc enters the model as an additional resistance coupled in series with the RC-component, as

shown in fig. 2.11, whereby the ODE for PRC remains unaltered. The resulting model equations

are given in the following definition.

For Rp , Ca > 0 and Zc , P∞ ≥ 0 the dynamic relation between aortic pressure P (t) and flow Q(t)

10 From dP dPRC

dt

= dt

it follows that iωn Pbn = iωn PbRC,n and therefore Pbn = PbRC,n for iωn 6= 0 ⇔ n 6= 0.

Chapter 2. Parameters for the assessment of cardiovascular risk 25

(a) (b)

Zc

Q Zc Qout Q Qout

QL L

Ca Rp Ca Rp

P PRC P PRC

P∞ P∞

Figure 2.11: Representation of the three- (a) and four-element Windkessel model (b) as elec-

trical circuits. The characteristic impedance Zc is modelled as a resistor in series with the

RC-component, arterial inertia L by an inductor in parallel with Zc .

is modelled as

dPRC 1 1

(t) = Q(t) − PRC (t). (2.33)

dt Ca Rp Ca

P∞ , while the dynamic behaviour is determined by Zc Q + PRC . The input impedance modelled

by the WK3 is therefore given by:

P = Zc Q + P RC + P∞ = Zc + Rp + P∞ ,

n=0

Q Q Q

Zin,WK3 (ωn ) = (2.34)

Pn

b Zc Qn + PRC,n

b b Rp

= = Zc + , n≥1

1 + iω n Rp Ca

Qn

b Qn

b

The inclusion of Zc indeed diminishes the high frequency error as limn→∞ |Zin,WK3 (ωn )| = Zc

and the temporal waveforms of modelled pressure show a faster systolic upstroke and a better

correspondence to measured data [145], compare figure 2.12. However, Zc is technically a wave

resistance that describes the relation of the oscillatory components of pressure and flow only.

Representing it as a resistor also alters the low and in particular the zeroth frequency component

of Zin , see equation (2.34). As a consequence, the WK3 was found to overestimate Ca and to

underestimate Zc in parameter identification studies [115, 145].

Burattini and Gnudi therefore proposed the inclusion of an inductance L in parallel with Zc ,

as shown in figure 2.11, resulting in the (parallel) four-element Windkessel model WK4p [13].

Initially, L was considered solely an additional degree of freedom, which improved the accuracy

of the fitted parameters but was lacking a direct physical interpretation. However, Stergiopulos

and coworkers were able to demonstrate that L indeed represents the local arterial inertances

added up to a single value and therefore termed it total arterial inertance [119].

By including L, inertial effects are taken into account, relating pressure to an acceleration or de-

Chapter 2. Parameters for the assessment of cardiovascular risk 26

120

WK2

115 WK3

WK4p

110

105

100

95

90

85

80

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time, s

Figure 2.12: Comparison of the aortic pressure waves modelled by the three different Windkessel

models. A typical flow curve shown in figure 2.8 was taken as input and the same parameters

were used for all three models.

celeration of blood. To derive the model equations of this four-element Windkessel, the auxiliary

quantity PLZ describing the pressure at the LZ-component is introduced. If the flow through L

is denoted by QL the following relations hold according to Kirchoff’s circuit laws.

dQL

PLZ = L

dt

PLZ = Zc (Q − QL )

dQL

From the first equation, an expression for dt can be obtained, which, when plugged in the time

derivative of the second equation, yields the following ODE for PLZ .

dPLZ dQ Zc

= Zc − PLZ (2.35)

dt dt L

The model equations of the WK4p therefore consist of two ODEs, namely one for PRC (2.27)

and one for PLZ (2.35). Exemplary solutions for both are shown in figure 2.13.

For Rp , Ca , L, Zc > 0 and P∞ ≥ 0, the relation between aortic pressure P (t) and flow Q(t) is

modelled as

dPLZ dQ Zc

(t) = Zc (t) − PLZ (t), (2.37)

dt dt L

dPRC 1 1

(t) = Q(t) − PRC (t). (2.38)

dt Ca Rp Ca

From equation (2.37), the following relation between the Fourier coefficients of PLZ and Q can

Chapter 2. Parameters for the assessment of cardiovascular risk 27

60 15

55

10

50

PRC, mmHg

PLZ, mmHg

5

45

40

0

35

−5

30

25 −10

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

Time, s Time, s

Figure 2.13: PRC (left) and PLZ (right) as components of the pressure modelled by the parallel

4-element Windkessel (WK4p) shown in figure 2.12.

be derived:

iωn ZcL b

PbLZ,n = Qn (2.39)

iωn L + Zc

The modelled input impedance is thus given by:

P PbRC,0 + PbLZ,0 + P∞ P∞

Q = = Rp + , n=0

Q Q

Zin,WK4p (ωn ) = (2.40)

Pbn PbRC,n + PbLZ,n Rp iωn Zc L

= = + , n≥1

Qn

b Qn

b 1 + iω R C

n p a iω n L + Zc

From the above equation it can be seen that Zc no longer affects Zin,WK4p (0) while it is still

fulfilled that limn→∞ |Zin,WK4p | = Zc . Thus, the WK4p ”switches” between the WK2 in the

low-frequency range and the WK3 in the high-frequency range thereby unifying the advantages

of both. Moreover, it holds that Zin,WK4p → Zin,WK3 for L → ∞ and Zin,WK4p = Zin,WK2 for

L = 0. A comparison of the input impedances described by the different Windkessel models is

given in figure 2.14.

During diastole, the aortic valve is closed and therefore Q(t) ≡ 0. In this phase, the ODEs for

both PRC (2.27) and PLZ (2.35) become homogeneous, describing an exponential decay with

time constants Rp Ca and L/Zc respectively. The resulting diastolic pressure drop modelled by

the WK2 and WK3 is given in equation (2.41) and that by WK4p in equation (2.42).

ts − t

Pdias (t) = PRC (ts ) exp + P∞ (2.41)

Rp Ca

WK4p (ts − t)Zc

Pdias (t) = Pdias (t) + PLZ (ts ) exp (2.42)

L

Chapter 2. Parameters for the assessment of cardiovascular risk 28

0.15

WK2

|Z |, mmHg⋅s/ml

WK3

0.1

WK4p

0.05

in

0

0 1 2 3 4 5 6 7 8 9 10

Harmonic

WK2

WK3

arg(Z ), rad

π/4

WK4p

0

in

−π/4

−π/2

0 1 2 3 4 5 6 7 8 9 10

Harmonic

Figure 2.14: Comparison of the modulus (upper panel) and phase (lower panel) of the arterial

input impedances described by the three different Windkessel models. The same parameters

were used as for the pressure waves obtained in the time-domain shown in figure 2.12.

ts again denotes the duration of mechanical systole and T the length of the heartbeat. In both

cases, the exponential terms converge to 0 for t → ∞, thus, without further excitation from the

heart, the pressure decreases towards P∞ , justifying its name asymptotic pressure level.

Wave intensity analysis (WIA) was proposed by Parker and Jones in 1990 [91] as an alternative

to impedance analysis to investigate the wave-phenomena in the arterial system. Impedance

analysis, as presented in section 2.2, is based in the frequency-domain and waves are considered

sinusoidal wave trains that add up to the measured waveforms when superimposed, assuming a

linear system. These periodic waves transmit the energy of cardiac ejection, thereby causing a

displacement of blood yet without net mass transport. Wave intensity analysis, in contrast, is a

time-domain method where successive wavefronts describing the incremental changes in pressure

dP and velocity dU are analysed.

Wave intensity analysis is based on the solution of a one-dimensional model of blood flow using

the method of characteristics, as will be presented in the following sections.

Chapter 2. Parameters for the assessment of cardiovascular risk 29

vessel

ment will be derived based on the approach presented by Sherwin et al. [117]. Generally, blood

flow velocity ~u and pressure p in an artery are functions of space ~x = (x, y, z) and time t, i.e.

~u = ~u(t, ~x), p = (t, ~x) with the components of ~u = (u, v, w) denoting the velocity in x−, y− and

z−direction. In order to derive a one-dimensional formulation, several simplifications have to

be applied to reduce the spatial dimension. First of all, it is assumed that the vessel is straight

and that the x-direction corresponds to the axial, forward flow direction as shown in figure 2.15.

Furthermore, for the purpose of this model, only the velocity in the axial direction u(t, ~x) will

be considered.

To eliminate the dependency of velocity and pressure on y and z, both state variables are averaged

over the cross-sectional area A. Therefore, it is assumed that the vessel is axisymmetric and

deformations apply in radial direction only, meaning that the cross section is always circular and

the length of the vessel constant. Hence, A depends on time and x only: A = A(t, x). The

average velocity U and pressure P are now defined as

Z

1

U (t, x) := u(t, ~x) dA, (2.43)

A(t, x) A(t,x)

Z

1

P (t, x) := p(t, ~x) dA. (2.44)

A(t, x) A(t,x)

In contrast to the previous sections, mean blood flow velocity U and not volumetric blood flow

R

Q is used. However, these two state variables are intrinsically linked by Q = A u dA = AU .

A last set of assumptions relates to the properties of blood. In reality, blood is a suspension of

particles (e.g. white and red blood cells) in an aqueous solution, the blood plasma. Yet, this

property is most relevant at a capillary level, whereas in large arteries, blood can be regarded

a homogeneous fluid. Also the non-Newtonian behaviour of blood is negligible in large arteries,

where large relates to vessel diameters above 100µm [15]. Therefore, blood will be considered a

homogenous, Newtonian fluid. Morevoer, it is assumed to be incompressible with constant den-

sity, which represents a good approximation for physiological flow rates [15], and viscous effects

will not be taken into account.

Overall, the model describes the relation between flow, pressure and cross-sectional area in a

straight, axisymmetric, elastic vessel with blood being a homogeneous, inviscid and incompress-

ible fluid, i.e. viscosity is assumed to be zero µ = 0 and density to be constant ρ(t, ~x) = ρ.

The three state variables (A, P, U ) are thereby assumed to be sufficiently smooth, in particular

continuously differentiable in both t and x. The governing equations will be derived by applying

conservation of mass and balance of momentum to a control volume CV (t) defined as the arterial

segment shown in figure 2.15 with constant, yet arbitrary length l.

Chapter 2. Parameters for the assessment of cardiovascular risk 30

U (t, x) A(t, x)

Figure 2.15: A straight arterial segment with length l, oriented along the x-axis.

Conservation of mass

For mass to be conserved, the change in mass within the control volume MCV with time has to

dMCV

equal the mass flow across the surface. For constant density ρ, dt (t) is given by

Z Z l Z l

dMCV d d ∂A

(t) = ρ dV = ρ A dx = ρ dx.

dt dt CV (t) dt 0 0 ∂t

The last equality follows from the assumption that A(t, x) is continuously differentiable in t for

any x and thus integration and differentiation can be interchanged.

If it is further assumed that the walls of the vessel are impermeable, mass flow in and out of the

control volume is given by

Z Z

ρU dA − ρU dA = ρ(A(t, 0)U (t, 0) − A(t, l)U (t, l))

A(t,0) A(t,l)

Z l

∂AU

=− ρ dx.

0 ∂x

Z l

∂A ∂AU

ρ +ρ dx = 0.

0 ∂t ∂x

This equality has to hold for any length l and hence the integrand has to be zero, yielding the

equation for conservation of mass in differential form

∂A ∂AU

+ = 0. (2.45)

∂t ∂x

Balance of momentum

According to Newton’s second law, the change of momentum with respect to time equals the sum

of forces acting on the control volume. As before, the total change in momentum in x-direction

is given by the change within the control volume minus the momentum flux across the surface,

taking into account that particles entering or leaving the system take momentum with them. The

Chapter 2. Parameters for the assessment of cardiovascular risk 31

Z Z lZ Z l

d d ∂(AU )

ρU dV = ρU dA dx = ρ dx.

dt CV (t) dt 0 A(t,x) 0 ∂t

Z Z

(ρU )U dA − (ρU )U dA = ρ(A(t, 0)U 2 (t, 0) − A(t, l)U 2 (t, l)) (2.46)

A(t,0) A(t,l)

l

∂AU 2

Z

=− ρ dx. (2.47)

0 ∂x

The total force F~ = (Fx , Fy , Fz ) acting on the control volume is comprised of external forces (e.g.

gravity) pressure forces and frictional forces. If external forces are neglected, only pressure forces

remain since viscosity is not considered, i.e. no friction occurs. Pressure acts on the surface of

the volume and the total force is therefore given by

Z

F~ = − P ~n dS,

∂CV (t)

with ~n = (nx , ny , nz ) denoting the unit normal vector on the surface pointing in outward direc-

tion. For the purpose of this model, again only the x-component Fx will be considered. nx at

the left and right boundary equals −1 and 1 respectively. To determine its value along the vessel

walls, denote the radius of the vessel as r(t, x). In cylindrical coordinates, the vessel wall is given

φ, r(t, x)
sin φ) for 0 ≤ x ≤ l and 0 ≤ φ ≤ 2π. The unit normal can

by ~x(t, x, φ) = (x, r(t, x) cos

be computed as ∂x × ∂φ /
∂xx × ∂φ

∂~

x ∂~

x
∂~ ∂~x
∂r

and the x-component thus equals nx = − ∂x ∂~

x

r/k ∂x ∂~

x

× ∂φ k.

∂A ∂r

Keeping in mind that A(t, x) = πr2 (t, x) and therefore ∂x = 2πr ∂x yields

Z

Fx = − P nx dS

∂CV (t)

Z lZ !

Z Z 2π

∂~x ∂~x

=− − P dA + P dA + P nx

×
dφ dx

A(t,0) A(t,l) 0 0 ∂x ∂φ

Z l

∂r

= −(A(t, l)P (t, l) − A(t, 0)P (t, 0)) + P 2πr dx

0 ∂x

Z l Z l

∂AP ∂A ∂P

= − +P dx = −A dx.

0 ∂x ∂x 0 ∂x

∂AU ∂AU 2 ∂P

ρ +ρ +A = 0. (2.48)

∂t ∂x ∂x

Chapter 2. Parameters for the assessment of cardiovascular risk 32

Conservation of mass (2.45) and balance of momentum (2.48) lead to the following set of equations

to describe the haemodynamics in terms of the three state variables (A, P, U ).

∂A ∂AU

+ =0 (2.49)

∂t ∂x

∂AU ∂AU 2 ∂P

ρ +ρ +A =0 (2.50)

∂t ∂x ∂x

The equation for balance of momentum (2.50) can be further simplified by explicitly computing

∂AU ∂(AU )U

the derivatives of ∂t and ∂x and using conservation of mass (2.49).

∂A ∂U ∂AU ∂U ∂P

0 = ρU + ρA + ρU + ρAU +A

∂t

∂t ∂x

∂x ∂x

∂A ∂AU ∂U ∂U 1 ∂P

= ρU + +ρA +U +

∂t ∂x ∂t ∂x ρ ∂x

| {z }

=0

∂U ∂U 1 ∂P

⇒0= +U +

∂t ∂x ρ ∂x

If it is now assumed that the cross-sectional area is a function of pressure and location and not

explicitly time and location11 , A(t, x) = A(P (t, x), x), conservation of mass can be reformulated

to become

∂A ∂P ∂A ∂P ∂A ∂U

0= +U + +A

∂P ∂t ∂P ∂x ∂x ∂x

∂A ∂P ∂P ∂A ∂U

= +U +U +A .

∂P ∂t ∂x ∂x ∂x

!

A(P (t, x), x)

c(P (t, x), x)2 := ∂A

, (2.51)

ρ ∂P (P (t, x), x)

ρc2 U Ax

Pt + U Px + ρc2 Ux = − (2.52)

A

1

Ut + U Ux + Px = 0, (2.53)

ρ

where sub-indices indicate partial derivatives. c has the unit of velocity and represents the speed

at which the waves travel through the artery, i.e. the pulse wave velocity (PWV). [90] [91]

11 Strictlyspeaking, the resulting function should be denoted differently, i.e. A(t, x) = Ã(P (t, x), x) and therefore

∂A ∂ Ã ∂P

∂t

= ∂P ∂t

and ∂A

∂x

∂ Ã ∂P

= ∂P ∂x

+ ∂∂x

Ã

. However, for the sake of ease of notation and readability, they are called

the same.

Chapter 2. Parameters for the assessment of cardiovascular risk 33

The 1D-equations for blood flow and pressure in an artery were originally derived by Euler12 in

1755. However, only after Riemann13 introduced the method of characteristics to find a general

solution for hyperbolic partial differential equation (PDE) over 100 years later, the theoretical

framework was developed to actually solve these equations [89]. In this section the solution using

the method of characteristics will be presented following [91] and [90].

The quasilinear, hyperbolic PDEs defined in equations (2.52) and (2.53) can be written in matrix

form. ! ! ! !

Pt U ρc2 Px 0

+ = 2

Ut 1

ρ U Ux − ρc AU Ax

The Eigenvalues of the matrix of coefficients for the spatial derivatives are given by

(U − λ)2 − c2 = 0 ⇒ λ1,2 = U ± c

dx±

= U ± c. (2.54)

dt

Hence, the axial location along the tube is parametrised by t. These paths correspond to a ve-

locity of U + c (forward, x+ ) or U − c (backward, x− ) and for every initial value x± (0) = x0 one

unique forward and backward solution exists. As stated before, c represents the velocity of the

pulse waves travelling through the vessel. The total velocity of a forward travelling pulse wave is

then given by c plus the velocity of the fluid itself, i.e. U + c, which equals the velocity along the

forward direction. Analogously, for a backward wave travelling with speed −c, the total velocity

is given by U − c, which equals the velocity along the backward characteristic direction.

Along these directions, the total derivatives of P (x± (t), t) and U (x± (t), t) with respect to time

are given by

dP dx±

= Px + Pt = Px · (U ± c) + Pt ,

dt dt

dU dx

= Ux + Ut = Ux · (U ± c) + Ut .

dt dt

By rearranging the terms, an expression for the partial derivatives Pt and Ut can be obtained.

dP

Pt = − Px · (U ± c)

dt

dU

Ut = − Ux · (U ± c)

dt

12 Leonhard Euler (1707-1783), Swiss mathematician and physicist

13 Georg Friedrich Bernhard Riemann (1826-1866), German mathematician

Chapter 2. Parameters for the assessment of cardiovascular risk 34

These relations can now be plugged in the original system of PDEs (2.52) and (2.53), yielding

dP ρc2 U Ax

∓ cPx + ρc2 Ux = − ,

dt A

dU 1

∓ cUx + Px = 0.

dt ρ

For the forward direction (U + c), the first equation is divided by ρc and added to the second

one.

)

I 1 dP 1

ρc dt − ρ Px + cUx = − cUAAx dU 1 dP cU Ax

dU 1

I+II: + =− (2.55)

II dt + ρ Px − cUx = 0 dt ρc dt A

Similarly, for the backward direction (U − c), the first equation is divided by (−ρc) and again

added to the second one.

dU 1 dP cU Ax

− = (2.56)

dt ρc dt A

Along the characteristic directions, the relation between U and P is therefore given by ordinary

differential equations. These can be expressed in terms of the so called Riemann variables R± .

dR± dU 1 dP

(t) := (t, x± (t)) ± (t, x± (t)) (2.57)

dt dt ρc(P (t, x± (t)), x± (t)) dt

For small arterial segments without bifurcations or discontinuities, it can be assumed that the

vessel is uniform in space, i.e. Ax = 0 [91]. Then, the right hand sides of equations (2.55) and

(2.56) become zero, implying that R± is constant along the characteristic directions. In this case,

R± are called Riemann invariants. The pulse wave velocity c is a function of pressure P and

space x: c = c(P, x). Yet for a uniform vessel with A(P, x) = A(P ), c is dependent on pressure

only c = c(P ), see equation (2.51).

Z t Z P

1 dP 1

R± = U ± dt = U ± dP.

t0 ρc dt P0 ρc

If it is furthermore assumed that c is constant, which will be discussed in more detail in appendix

B, the expression for R± reduces to

1

R± = U (t, x± (t)) ± P (t, x± (t)). (2.58)

ρc

In the following, the Riemann invariants belonging to the characteristic directions x̂± satisfying

the initial value condition x̂± (t0 ) = x0 , will be denoted by R± (t0 , x0 ) or R± (x̂± ).

From equation (2.58), it follows that for any point (t, x), characteristic directions x̂+ and x̂−

with x̂+ (t) = x = x̂− (t) and corresponding Riemann-invariants R+ (x̂+ ) and R− (x̂− ), the flow

Chapter 2. Parameters for the assessment of cardiovascular risk 35

R

R −(

)

−( t0 0

t0

,x + ,x ) ŷ+

0)

∆

t, (t 0 ,x

0

x0 R+ t

) ∆

+

(t 0

R+

(t1 , x1 )

x

(t0 , x0 ) (t0 + ∆t, x0 )

t

Figure 2.16: Schematic representation of the characteristic directions and the corresponding

Riemann invariants. Two points lying on different characteristics (x̂± , ŷ± ) at different times (t0 ,

t0 + ∆t) at the same location x0 are shown. The corresponding differences can be obtained from

the differences between the Riemann invariants R± (t0 , x0 ) and R± (t0 + ∆t, x0 ).

1

U (t, x) = (R+ (x̂+ ) + R− (x̂− )), (2.59)

2

ρc

P (t, x) = (R+ (x̂+ ) − R− (x̂− )). (2.60)

2

Hence, to actually solve the system of PDEs defined in equations (2.52) and (2.53) with a zero

right-hand side, boundary conditions for U and P at both ends of the vessel under consideration

have to be prescribed, compare [3] and figure 2.16. From these, the values of the Riemann invari-

ants for any characteristic direction can be obtained according to equation (2.58). Subsequently

P and U at any point (t, x) within the vessel can be determined from the Riemann invariants

belonging to the characteristic directions intersecting at (t, x), see equations (2.59) and (2.60).

However, for wave intensity analysis, the focus does not lie on the propagation of flow and pres-

sure, but on the interpretation and analysis of the temporal changes at a specific site x0 , as will

be discussed in the next section.

Chapter 2. Parameters for the assessment of cardiovascular risk 36

35

30

25

20

15

10

−5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time, s

Figure 2.17: Wave intensity obtained from measured pressure and flow velocity.

Equations (2.59) and (2.60) can be used to express the temporal differences of pressure and flow

at a specific location x0 in terms of the Riemann-invariants by

1

= (R+ (t + ∆t, x0 ) + R− (t + ∆t, x0 ) − R+ (t, x0 ) − R− (t, x0 )) (2.62)

2

1

= (dR+ (t, x0 , ∆t) + dR− (t, x0 , ∆t)) (2.63)

2

and

ρc

= (R+ (t + ∆t, x0 ) − R− (t + ∆t, x0 ) − R+ (t, x0 ) + R− (t, x0 )) (2.65)

2

ρc

= (dR+ (t, x0 , ∆t) − dR− (t, x0 , ∆t)). (2.66)

2

1 ρc

dI := dU dP = (dR+ + dR− ) (dR+ − dR− ) = (2.67)

2 2

ρc 2 2

= (dR+ − dR− ). (2.68)

4

dI has the SI-unit kg/s3 = W/m2 and corresponds to the rate of energy transfer per unit area

associated with the wavefronts dU and dP [91]. The property that makes dI appealing for

analysis is that the contribution of forward waves is always positive, whereas backward waves

enter with a negative sign, see equation (2.68). The sign of dI therefore holds information on

which waves dominate at an instance t. Depending on whether pressure is increased (dP > 0)

or decreased (dP < 0), the dominating waves can be further classified as summarised in table 2.1.

Chapter 2. Parameters for the assessment of cardiovascular risk 37

dI > 0 dP > 0 dU > 0 forward compression wave

dI > 0 dP < 0 dU < 0 forward expansion wave

dI < 0 dP > 0 dU < 0 backward compression wave

dI < 0 dP < 0 dU > 0 backward expansion wave

Table 2.1: Nomenclature used for the dominating waves according to the sign of dI and dP .

To compute the wave intensity dI from a measured pair of pressure and flow velocity, the dif-

ferences for each time step determined by the sampling rate are used. Figure 2.17 shows an

example of the so-obtained wave intensity. Typically, two positive peaks can be distinguished

corresponding to an early-systolic compression wave and a late-systolic expansion wave, which

are attributed to the ejection and relaxation dynamics of the left ventricle respectively. The first

represents the initial impulse coming from the heart, increasing pressure and accelerating blood

flow, whereas the second causes a decrease in pressure and a flow reversal leading to the closure

of the aortic valve at the beginning of ventricular relaxation. The negative peak in mid-systole

implies that this phase is dominated by reflections of the initial forward wave. [44, 90, 92, 120]

For a single wave travelling forward along the characteristic direction x̂+ in a uniform and lossless

vessel, it has to hold that Uf (t0 , x̂+ (t0 )) = Uf (t1 , x̂+ (t1 )), as no energy is lost and no reflections

are present. Expressed in terms of the Riemann invariants, this means that (x̂+ (t0 ) = x0 ,

x̂+ (t1 ) = x1 ):

z }| { z }| {

1 1

(R+ (t0 , x0 ) + R− (t0 , x0 )) = (R+ (t1 , x1 ) + R− (t1 , x1 )) (2.69)

2 2

R− (t0 , x0 ) = R− (t1 , x1 ) (2.70)

Denote the characteristic directions intersecting at (t0 , x0 ) by x̂± and those intersecting at (t0 +

∆t, x0 ) by ŷ± . x̂+ and ŷ− intersect at a point (t1 , x1 ) with t1 < t0 + ∆t and x1 > x0 , see figure

2.16. Because of equation (2.70), the Riemann invariants along x̂− : R− (t0 , x0 ), and along ŷ− :

R− (t1 , x1 ) = R− (t0 + ∆t, x0 ), have to be equal and therefore it holds that R− (t0 + ∆t, x0 ) =

R− (t0 , x0 ). Thus, for a single, forward travelling wave, (2.62) simplifies to

1

dUf (t0 , x0 , ∆t) = (R+ (t0 + ∆t, x0 ) + R− (t0 + ∆t, x0 ) − R+ (t0 , x0 ) − R− (t0 , x0 )) (2.71)

2

1

= (R+ (t0 + ∆t, x0 ) − R+ (t0 , x0 )). (2.72)

2

Chapter 2. Parameters for the assessment of cardiovascular risk 38

Uf (t, x)

U

dUf

x

x̂+

Figure 2.18: A single forward wave travelling in a uniform, lossless vessel along the characteristic

direction x̂+ .

ρc

dPf (t0 , x0 , ∆t) = (R+ (t0 + ∆t, x0 ) − R− (t0 + ∆t, x0 ) − R+ (t0 , x0 ) + R− (t0 , x0 )) (2.73)

2

ρc

= (R+ (t + ∆t, x0 ) − R+ (t, x0 )) (2.74)

2

and therefore

The relations

dPf,b = ±ρcdUf,b (2.77)

are called the Waterhammer equations. Assuming that the interaction between forward and

backward travelling waves is linear

the Waterhammer equations (2.77) can be used to separate the wavefronts in their forward and

backward travelling components, yielding

dP ± ρcdU

dPf,b = , (2.79)

2

ρcdU ± dP

dUf,b = . (2.80)

2ρc

The separated total waveforms Pf,b and Uf,b can be obtained by integration or summation of

the successive differences

X X

Pf,b = dPf,b , Uf,b = dUf,b . (2.81)

Chapter 2. Parameters for the assessment of cardiovascular risk 39

30 140

P measured U measured

Pf 120 Uf

25

Pb 100 Ub

Pressure above DBP, mmHg

20 80

60

15

40

10

20

5 0

−20

0

−40

−5 −60

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time, s Time, s

Figure 2.19: Aortic pressure and flow separated in their forward (black) and backward (red)

components using wave intensity analysis. ρc was estimated in the time domain from the PU-

loop using the first 8 data points ≈ 55 ms.

These expressions are very similar to the ones obtained in the frequency domain in section 2.2.

Indeed, if the characteristic impedance Zc is identified with ρc/A and the relation Q = U A is

used, both approaches yield formally identical results with differences arising primarily from the

chosen approach to estimate ρc or Zc respectively [41, 147], compare figures 2.19 and 2.8.

With the same argumentation as in section 2.2.3, ρc is commonly assessed from the early systolic

part of the PU-loop, when reflections are assumed to have not yet returned [90].

From the separated wavefronts, also the forward and backward wave intensity can be computed

±1 2

dIf,b = dPf,b dUf,b = (dP ± ρcdU ) , (2.82)

4ρc

the sum of which again yields the net wave intensity dI.

= dPf dUf + dPb dUb + dPf dUb + dPb dUf

= dIf + dIb + ρcdUf dUb − ρcdUb dUf

= dIf + dIb

For dIf and dIb the same characteristics as for dI can be observed, see figure 2.20. The forward

intensity shows two peaks denoted as S wave (early systolic) and D wave (late systolic/early

diastolic). The backward intensity is most prominent during mid-systole, resulting in a negative

R wave representing the reflections in the system. For quantification, usually the amplitudes or

the areas under the respective portions are used. The latter has the unit of Joule per square

meter (mmHg·cm≈ 1.3332 J/m2 ) and represents the ”absolute energy carried per unit cross-

sectional area by a wave” [19, p. H557]. As a relative index of wave reflection, the ratio of the

Chapter 2. Parameters for the assessment of cardiovascular risk 40

35

dI

f

30

S dI

b

25

20

15

10

5 D

R

−5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time, s

Figure 2.20: Forward (black) and backward (red) wave intensity. Forward intensity shows two

peaks S and D related to ventricular contraction and relaxation respectively. Reflections rep-

resented by backward wave intensity are most prominent in mid-systole resulting in a negative

peak R.

However, it has to be kept in mind that the magnitude of the computed wave intensity strongly

depends on the chosen time step dt, restricting the comparability of the derived parameters

between different investigators. Therefore, the use of the time derivative or difference quotient

instead of the absolute temporal change was proposed [120].

Chapter 3

on pressure alone: existing

methods and a novel approach

In the last chapter, different concepts to describe and to quantify the mechanisms in the arterial

system based on the waveforms of aortic pressure and/or flow were introduced. The derived

parameters are intended to help in the assessment of cardiovascular risk and the early detection

of cardiovascular disease. To enable a widespread use, the corresponding measurements should

therefore be easy to realise and should present no risk for the patient, thus non-invasive methods

are desirable. With regards to pressure, validated transfer functions exist to generate aortic

pressure waveforms from non-invasive peripheral readings [47] and different commercial devices

are available incorporating these algorithms.

Unfortunately, the non-invasive assessment of aortic root flow is more complicated. One of

the most widely used techniques is echocardiography, particularly the use of pulsed-wave or

continuous wave Doppler ultrasound [105]. However, echocardiographic examination requires

dedicated devices and specially trained operators, both limiting its widespread use at a primary

care setting. Therefore, different blood flow models based on pressure alone were introduced in

literature and three of them will be presented in the following sections. Subsequently, a novel

model of blood flow will be proposed which is based on the 4-element parallel Windkessel to

compute flow from pressure in the frequency domain.

Typically, aortic blood flow shows a fast systolic upstroke followed by a, more or less linear,

monotone decrease. The overall waveshape might thus be considered triangular. Westerhof and

41

Chapter 3. Blood flow models 42

Flow, AU

Flow, AU

0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8

Time, s Time, s

Figure 3.1: Triangular approximation (a) and averaged waveform (b) as estimates of aortic

blood flow. In both cases, the ejection duration derived from the pressure wave (indicated in

gray) determines the width of the respective waveform. For the triangulation, the timing of peak

flow is furthermore set to the timing of the inflection point in the pressure signal.

coworkers [142] therefore proposed the use of a triangular function based on timing information

obtained from the pressure wave to estimate the shape of aortic blood flow for the use in wave

separation analysis (WSA).

The idea is based on the mechanisms described in section 2.1.2 for the computation of the aug-

mentation index (AIx). The width of the triangle is defined by the ejection duration (ED) and

the maximum is aligned with the first shoulder in the pressure signal derived by the fourth

derivative [142], see figure 3.1(a).

As a proof of concept, Westerhof et al. applied their flow model on a dataset of 29 simultaneously

measured flow and pressure waves of 19 different patients assessed during cardiac catheterisation.

They found a good agreement between the WSA-parameters derived from the measured flow wave

and the triangular approximation and concluded that it might be suitable also for non-invasive

data [142]. In the following, their method which is often referred to as flow triangulation has

been used for WSA in a variety of non-invasive studies, e.g. [56, 75, 133].

Shortly after the introduction of the triangulation method, Kips and coworkers [49] proposed the

use of an averaged waveform as an estimate of aortic blood flow to ensure a physiologic shape.

Therefore, Doppler flow measurements of 74 healthy subjects (32 female) were collected from

the Asklepios population [108], normalised in time and amplitude and finally averaged. The

resulting flow waveform is shown in figure 3.1(b). In order to derive a patient-specific estimate,

Chapter 3. Blood flow models 43

Kips et al. tested their approximation in the Asklepios population on a dataset comprising non-

invasive (carotid) pressure measurements of more than 2300 patients and compared it to Doppler

flow as well as the triangular estimate. They found it to be superior to the triangle with regards

to the waveshape and the correlation of Doppler and approximated reflection magnitude (RM),

yet still showing considerable deviations. [49]

Nevertheless, RM derived from the averaged waveform was shown to be a strong predictor for

the onset of congestive heart failure in a subsequent noninvasive study performed by Chirinos et

al. [17].

Another class of flow models is based on the arterial Windkessel coupled with an optimality

criterion. Generally, the Windkessel models require either one of the two state variables pressure

or flow as well as given parameter values to derive the other one. However, if an additional

linkage between pressure and flow is included, both state variables can be retrieved at the same

time for a given parameter set. This allows for identification of the Windkessel parameters and

computation of aortic flow based on a single pressure measurement.

The general procedure for this type of flow model can be summarised in four steps:

criterion and suitable boundary conditions as well as constraints.

an optimal pressure and flow contour.

3. Identification of the Windkessel parameters by minimising the error between the optimal

pressure contour from step 2 and the measured one.

4. Computation of the optimal flow time-course using the parameters from step 3.

Since the 1970ies, a variety of models based on this concept was developed with more or less

promising results, differing in the Windkessel model used, the optimality criterion demanded

as well as the boundary conditions and constraints applied, see e.g. [25, 31, 78, 96, 100, 154].

Starting point for all of them was the assumption of minimal ventricular stroke work or en-

ergy dissipation, but also extensions by penalty terms were considered. Moreover, the different

presumptions led to different mathematical tools needed for solving, in particular calculus of

variation and optimal control theory. A more detailed overview in German can be found in [93].

Also the ARCSolver method for the computation of aortic flow that was developed at the AIT

Chapter 3. Blood flow models 44

Austrian Institute of Technology GmbH and is part of the ARCSolver algorithms for pulse wave

analysis is based on this principle using a 3-element Windkessel with P∞ = 0 and a minimal

cardiac work criterion [35, 135]. The cardiac work or stroke work SW is given by the integral of

pressure times flow over one heartbeat

Z T Z ts

SW (P, Q) = P (t)Q(t) dt = P (t)Q(t) dt. (3.1)

0 0

The second equality follows from the presumption of zero flow during diastole. Using the 3-

element Windkessel equations

dPRC 1 1

(t) = Q(t) − PRC (t), (3.3)

dt Ca Rp Ca

PRC (t)

both pressure and flow can be expressed in terms of the peripheral outflow Qout (t) = Rp by

dQout

Q(t) = Ca Rp (t) + Qout (t). (3.5)

dt

Hence, stroke work SW can be formulated as a functional of Qout , i.e. SW (P, Q) = SW (Qout ).

Under the additional constraint that a given stroke volume

Z T Z ts

SV = Q(t) dt = Q(t) dt > 0 (3.6)

0 0

L(x,ẋ)

Rt z }| {

minimise SW (x) = 0 s Zc Ca2 Rp2 ẋ(t)2 + Ca Rp (2Zc + 1)ẋ(t)x(t) + (Zc + Rp )x(t)2 dt

x

R ts

subject to G(x) = 0

Ca Rp ẋ(t) + x(t) dt = SV

| {z }

K(x,ẋ)

The constraint has the same form as the functional that is to be minimised and the optimisa-

tion problem therefore represents an isoperimetric problem that can be solved using calculus of

variation. More specifically, since both Lagrangians L and K are polynomials in x and ẋ and

therefore smooth functions, the method of Lagrangian multipliers can be applied. If it is further-

more assumed that x = Qout is at least two times continuously differentiable, the Euler-Lagrange

equations yield the following, necessary condition for a function x minimising SW .

∂ d ∂

(L + λK) − (L + λK) = 0, (3.7)

∂x dt ∂ ẋ

where λ ∈ R denotes the Lagrange multiplier. This leads to a linear, inhomogeneous, second

Chapter 3. Blood flow models 45

Zc + Rp λ

ẍ(t) − x(t) = . (3.8)

Zc Ca2 Rp2 2Zc Ca2 Rp2

q

Zc +Rp

with µ denoting the Eigenvalue of the ODE, i.e. µ = 2.

Zc Ca2 Rp The three degrees of freedom

A, B, C have to be determined from the boundary conditions and constraints.

For blood flow Q, a possible choice of boundary values is Q(0) = Q(ts ) = 0, ensuring flow to be

zero during diastole. If the system is in steady state, boundary values for the blood pressure P

can be obtained from the assumption of periodicity, in particular P0 = P (0) = P (T ). From the

exponential decay predicted by the 3-element Windkessel (WK3) during diastole, this can also

ts −T

be translated to a condition for P (ts ) by P (ts ) = P0 e Rp Ca . Hence, together with the constraint

of a fixed stroke volume (SV), 5 possible conditions can be formulated to determine the three

constants A, B, C, leading to an over-determined system of linear equations, which is in gen-

eral not solvable. Different options exist which conditions to include, resulting in substantially

differing behaviour of the obtained optimal pressure and flow contours, which can be qualita-

tively categorised into three classes depending on the chosen conditions for Q [33]. Neglecting

Q(0) = 0 thereby yields the best results in terms of the pressure waveshape and this also repre-

sents the approach that was chosen for all flow models presented in literature so far [25, 100, 154].

ts −T

P̄ ·T

In the ARCSolver routine, the conditions Q(ts ) = 0, P (ts ) = P0 e Rp Ca and SV = Rp are used.

The so obtained optimal pressure contours do indeed resemble measured ones, which allows

for identification of the Windkessel parameters, whereby the area under the systolic pressure

curve after accounting for wave reflections serves as error measure [64]. However, by neglecting

Q(0) = 0, no physiological flow waveform can be achieved. To overcome this drawback, the

simulated flow wave is further modified using a discrete second order delay element to attain

Q(0) = 0 [35].

The ARCSolver flow was found to provide accurate estimates compared with Doppler flow mea-

surements in patients with preserved systolic function [35]. Also the derived parameters of WSA

as well as wave intensity analysis (WIA) were shown to be in good agreement with the reference

values obtained from Doppler flow [35–37].

Chapter 3. Blood flow models 46

3.4.1 Concept

The arterial input impedance Zin offers a complete description of the arterial system, linking

flow to pressure independently of the heart. Knowledge of Zin therefore provides the means to

compute pressure from flow or vice versa according to the relation given in section 2.2.1.

Pbn Pbn

Zin (ωn ) = ⇒Q

bn = , n≥0 (3.10)

Qbn Zin (ωn )

However, since Zin is normally not available it has to be estimated in order to obtain a flow curve

based on pressure alone. Therefore, a Windkessel model is used as a parametric approximation of

Zin . The parallel 4-element Windkessel (WK4p) seems to be the best choice for this purpose as

it combines the strengths of both the 2-element Windkessel (WK2) regarding the lower frequency

range and the WK3 regarding the higher frequencies.

Nevertheless, the WK4p naturally represents a vast simplification of the real impedance and,

as stated by Burkhoff and coworkers in 1988 referring to the WK3, ”...while the Windkessel

model captures many of the gross features of the real impedance, it fails to reproduce many of

its details” [14, p. H742]. One of these details is the behaviour of the phase angle of Zin . From

experimental data, arg(Zin ) was found to be negative for low frequencies, cross zero, become

positive and either stay positive, oscillate around zero or become zero for higher frequencies

[72, 76]. The phase angle of the modelled input impedance, in contrast, remains always negative

for all frequencies, approaching zero from below.

To reduce the error introduced by the wrong phase shift, an approach is chosen that is inspired

by a work by Quick et al. [104] on the possibilities to determine arterial system geometry and

characteristics from measured pressure and flow profiles referred to as the ”hemodynamic inverse

problem”. In their work, Quick and co-workers argue that while the Windkessel behaviour given

by the WK2 provides an accurate description for the very low frequencies and the characteristic

impedance Zc for the high frequencies, the intermediate frequency range depends on arterial

topology and wave reflections. To analyse the contribution of each to experimentally determined

impedance spectra, they propose to use a hybrid approximation of Zin , with Zhybrid = Zin,WK2

for frequencies lower than a threshold fb and Zhybrid = Zc for frequencies above fb , where fb is

individually determined to best fit the data.

Even though their goal is different from the aim of this work, their idea of a hybrid approximation

of Zin with variable threshold is adopted to improve the approximation of arg(Zin ). More

precisely, the model impedance Zmodel is defined such that

(

Zin,WK4p (ωn ) for n < ñ

Zmodel (ωn ) = (3.11)

|Zin,WK4p (ωn )| for n ≥ ñ

Chapter 3. Blood flow models 47

thereby artificially forcing arg(Zmodel (ωn )) = 0 for n greater than an individual threshold ñ.

By the use of the WK4p as an estimate of Zin , the number of unknowns is greatly reduced.

However, the question remains how to identify the 5 parameters needed, namely Rc , Ca , Zc , L

and P∞ and if they can be uniquely determined. For this purpose, different features of the aortic

flow wave will be considered. First of all, there should be no ventricular outflow during diastole

and thus in particular also at the start of systole which implies

A second condition can be derived from the argumentation used for the estimation of character-

istic impedance presented in section 2.2.3. Namely, at the beginning of ejection, pressure above

diastolic level and flow are supposed to be proportional with proportionality factor Zc , yielding

Qearly systole = . (3.13)

Zc

These conditions are now combined in a weighted sum to form the cost function used for param-

eter identification

X

error(~x) =w1 · Q̃(0)2 + w2 · Q̃(ts )2 + w3 · Q̃(t)2 +

diastole

2

X P (t) − P (0)

+ w4 · − Q̃(t) , (3.14)

Zc

early systole

where Q̃ denotes the estimated flow curve, wi ∈ R+ , i = 1, . . . , 4, the weighting factors and ~x the

parameters considered. Condition (3.12) is split up into three parts in order to allow for differ-

ent weights being laid on Q(0) and Q(ts ), which are crucial for the overall shape of the flow wave.

In the above conditions, no constraint regarding the flow level, i.e. maximum or mean flow,

is considered. However, looking again at equation (3.10), it can be seen that multiplying the

impedance with an arbitrary constant c > 0 results in a flow wave with the same shape yet scaled

by the factor 1/c for a given pressure contour. From

P∞

Rp + , n=0

Q

Zin,WK4p (ωn ) = (3.15)

Rp iωn Zc L

n≥1

+ ,

1 + iωn Rp Ca iωn L + Zc

it furthermore follows that scaling the parameters Rp , Zc and L by c and Ca as well as mean

flow Q by 1/c yields c · Zin,WK4p . This means that (1) flow can only be computed qualitatively,

in other words only the shape can be obtained as is the case for the triangular as well as the

average approximation and (2) the value of at least one parameter has to be fixed in order to

allow for identification of the others.

Chapter 3. Blood flow models 48

In conclusion, by minimising the cost function (3.14) with respect to the Windkessel parameters

assuming a fixed flow level, it should be possible to identify the parameters and at the same time

compute a flow profile from a given pressure signal.

To develop and to test the proposed flow model, a dataset comprising pressure and flow waves

of 183 persons was used. Of these 183, 61 were diagnosed as having severely reduced ejection

fraction (EF), while the remaining 122 had normal EF. Pressure waveforms were acquired by

radial tonometry and central pressure was obtained by a generalised transfer function (Sphygmo-

Cor, Atcor Medical, Sydney, Australia). The SphygmoCor system provides one peripheral and

one central pressure time-contour, which represent the ensemble average over several heartbeats.

These are stored in a proprietary database with a sampling rate of 128 Hz and can be saved as a

.txt file with the inbuilt export function for further processing. Aortic blood flow was acquired

by Doppler ultrasound and subsequently manually digitised. A more detailed description of the

population and the measurements performed can be found in section 4.3 in the next chapter.

All computations were realised in Matlab R2011b (The MathWorks Inc, Natick, Massachusetts,

United States).

In a first step, ED was estimated from the peripheral pressure wave using an algorithm that

was developed by the cardiovascular diagnostics reasearch group at the AIT. From the estimate

of the time of valve closure ts , the diastolic pressure portion was determined. Then, the ex-

ponential diastolic decay modelled by the WK4p, compare equation (2.42), was fitted to the

experimental data in order to obtain first estimates of the two time constants τ = Rp Ca and

σ = L/Zc as well as P∞ . To reduce the influence of wave activities, the fitting was not applied to

the whole diastolic pressure wave but started with a prespecified offset as proposed in literature

[118, 131, 145]. Reported values for this delay range from 10% [145] to 33% [118, 131] of the total

diastolic duration when using the monoexponential decay of the 2- or 3-element Windkessel. For

this work, the offset was set to 20%. Moreover, the last 5 data points corresponding to the last

39 ms were neglected because the SphygmoCor signals often show a slight upstroke in the end,

see figure 3.2.

Fitting of the diastolic decay was performed using the objective function

N −5 2

X T − tj ts − tj

Pm (tj ) − P∞ − (DBP − P∞ ) exp − PLZ (ts ) exp , (3.16)

j=nn

τ σ

where Pm denotes the measured pressure signal, N the number of datapoints and nn the index

corresponding to the time tnn = ts + 0.2(T − ts ), i.e. the offset. This function was minimised

Chapter 3. Blood flow models 49

with respect to τ, σ, P∞ and PLZ (ts ) using Matlab’s lsqnonlin algorithm. The initial values for

optimisation were set to [0.5, 0.02, 0.5 · DBP, −1] and lower and upper bounds were defined by

0.1 ≤τ ≤ 2, (3.17)

0.01 ≤σ ≤ 10, (3.18)

0 ≤P∞ ≤ 0.9 · DBP, (3.19)

−100 ≤PLZ (ts ) ≤ 0. (3.20)

In the objective function (3.16), the periodicity of the pressure wave was used for the first expo-

nential term in order to reduce the number of parameters to be identified, neglecting the end-

diastolic value of PLZ . Strictly speaking, the multiplicative term before the exponential should

read (DBP − PLZ (T ) − P∞ ) instead of (DBP − P∞ ), since PRC (T ) = P (T ) − PLZ (T ) − P∞ .

However, PLZ (T ) is presumed to be very small, compare figure 2.13.

From the diastolic decay, initial estimates for τ , σ and P∞ were obtained. As discussed above,

the flow level has to be fixed to ensure identifiability of the Windkessel parameters. Therefore,

the stroke volume SV was set to 70 ml, which is within the normal range for healthy adults at

rest [73, p. 211]. Even though this choice is arbitrary, using a physiological flow level ensures

that also the other parameters are within the range observed in other studies.

Together with the heartbeat duration T and the mean pressure P derived from the measured

signal, the resistance Rp was obtained by Rp = (P − P∞ )/Q and subsequently Ca by τ /Rp . The

only missing parameter values are therefore Zc and L. Without further information, one of these

has to be fixed in order to derive the other one from the estimate of their ratio. Zc represent the

more widely-used parameter and its value has been investigated more extensively. Therefore, Zc

was set to 0.07 mmHg·s/ml, based on previously reported values for the WK4p [115, 119]. L was

subsequently computed as Zc · σ. Therewith, initial values for all parameters of the WK4p were

available. The estimation procedure is summarised in figure 3.2.

Computation of flow

The computation of aortic flow was carried out in the frequency domain. Therefore, the Fourier

coefficients of the pressure signal were computed by the fast Fourier transform using Matlab’s

inbuilt fft function. For the further computations, the first 15 harmonics were used, compare

section 2.2.

well as the individual threshold ñ, the model impedance was computed according to equations

(3.15) and (3.11) and the corresponding flow harmonics were obtained by Q

b n = Pbn /Zmodel (ωn ).

Chapter 3. Blood flow models 50

135

ts

Aortic pressure, mmHg T

P = MPB Q = SV/T

125 P∞ Rp = (P-P∞)/Q

τ = RpCa Ca = τ/Rp

MBP

115 σ = L/Zc Zc = 0.07

L = σZc

105 T SV= 70

Time, s

Figure 3.2: Estimation of initial values. From the measured pressure contour (blue), the heart-

beat duration T , time of end of systole ts and mean blood pressure MBP = P are computed.

Then the exponential decay of the WK4p (red) is fitted to the measured data during diastole

(gray area) yielding estimates of the two time constants τ and σ as well as of the asymptotic

pressure level P∞ . By setting SV to 70 ml and Zc to 0.07 mmHg·s/ml (indicated by bold letters),

initial estimates for all Windkessel parameters can be derived.

14

!

X

Q̃(tj ) = Q + Re b n eiωn tj

Q . (3.21)

n=1

In order to identify the Windkessel parameters, the cost function defined in equation (3.14) was

used. More precisely, the function

N

2 1 X 2

error(τ, Zc , L, P∞ ) =4 · Q̃(0) + 4 · Q̃(ts ) + · Q̃(tj )2 +

4 j=s+1

9 2

X P (tj ) − P (0)

+ − Q̃(tj ) , (3.22)

j=2

Zc

which equals ||~e(τ, Zc , L, P∞ )||22 for ~e defined accordingly, was minimised using Matlab’s lsqnonlin

algorithm and the initial values obtained before. s thereby denotes the index corresponding to

the end of systole and Rp and Ca were calculated in each step as described above, compare figure

3.2. Lower and upper bounds were moreover specified as

0.1 ≤τ ≤ 10,

0.005 ≤Zc ≤ 1,

0.000001 ≤L ≤ 0.3,

0 ≤P∞ ≤ 0.9 · DBP.

τ instead of Ca was included because its value is independent of the absolute scaling of blood

flow and boundary conditions are therefore easier to define. The individual threshold ñ used for

Chapter 3. Blood flow models 51

the definition of Zmodel , compare equation (3.11), was not included in the optimisation procedure

because of its discrete values. Instead, parameter identification was performed for each ñ varying

from 2 to 10 and the result showing the lowest absolute error, i.e. ||~e||21 , was selected.

Model parameters

To analyse the behaviour of the proposed model, as a first step, its sensitivity on changes in

the Windkessel parameters as well as in the individual threshold ñ was investigated. For this

purpose, an exemplary pressure waveform corresponding to a patient with normal EF, as well

as one pertaining to a patient with reduced EF were used. From these, the parameters were

identified as described in the last section and a flow wave was computed, as shown in figure 3.3.

Then, the identified values given in table 3.1 were chosen as baseline and the effect of variations

in single parameters was examined as shown in figures 3.4 and 3.5.

Rp peripheral resistance mmHg·s/ml 0.373 0.235

Ca total arterial compliance ml/mmHg 0.995 2.052

Zc characteristic impedance mmHg·s/ml 0.0593 0.0246

L total arterial inductance mmHg·s2 /ml 0.0049 0.0018

P∞ asymptotic pressure level mmHg 57.6 72.7

ñ individual threshold, see eqn. (3.11) - 4 5

SVR total systemic vascular resistance mmHg·s/ml 1.25 1.15

HR heart rate bpm 56 68

P = MBP mean blood pressure mmHg 82.0 91.3

ED ejection duration ms 336 288

SV stroke volume ml 70 70

The baseline parameters differed substantially between the two datasets, resulting in very distinct

flow waveforms that bore a strong resemblance to the Doppler flow measurements in both cases,

as shown in figure 3.3. It should be kept in mind though that the stroke volume was chosen

equally for both patients and the parameters are therefore wrongly scaled. Despite their different

appearances and baseline values, the qualitative influence of variations in single parameters on the

resulting waveform was very similar for both datasets, and will be discussed for each parameter

separately in the following.

The arterial compliance Ca . Ca was varied over a wide range of values that covers the

physiological range reported in literature, see e.g. [115]. In theory, the more compliant the

arteries are, the more blood has to be ejected during systole in order to reach a given pressure

level. This explains the large increase or decrease observed for peak flow when changing Ca

while keeping all other parameters at their baseline values. Moreover, since SV was fixed, this

Chapter 3. Blood flow models 52

Flow, AU

110 0.2

100

measured

|Zin|, AU

100 modelled 0.1

80

90 60 0

2 4 6 8 10 12 14

40 2

80

arg(Zin), rad

20

70 0

0

60 −2

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 2 4 6 8 10 12 14

Time, s Time, s Harmonics, n

110 0.1

100

measured

|Zin|, AU

105 modelled

80 0.05

100

60 0

2 4 6 8 10 12 14

95

40 2

arg(Zin), rad

90

20

0

85

0

80 −2

0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 2 4 6 8 10 12 14

Time, s Time, s Harmonics, n

Figure 3.3: Exemplary data of a patient with normal EF (a) and one with reduced EF (b)

used for sensitivity analysis. The synthesised aortic pressure (left panels) is used to compute

the Windkessel flow (middle panels, blue) with the parameter values given in table 3.1. For

comparison, also the Doppler flow waveforms (middle panels, black) as well as the corresponding

input impedances (right panels) are shown.

resulted in either a continuation of blood flow or a backflow during diastole if the values were

chosen too low or too high respectively. For the patient with reduced EF and high baseline Ca ,

it can furthermore be seen that the resulting flow wave resembles the pressure wave for very low

values of Ca , approaching the situation in a rigid tube. The induced changes in the shape of the

flow waveform were ”monotonic” for both datasets, indicating that the optimal value represents

a global minimum of the objective function with respect to Ca . This is corroborated by the

results obtained in the whole study population depicted in figure 3.6(a). For this analysis, the

error, more precisely ||~e||2 , obtained with the baseline value of Ca was taken as reference and its

relative change caused by a change in Ca of up to 15% relative to its baseline value was computed

per patient. For the medians, a parabolic shape can be observed with a symmetric increase in

both directions, whereby baseline Ca represents the vertex.

The characteristic impedance Zc . Zc had a strong impact on the position of peak flow,

with a shift to the right for higher values in both datasets, as depicted in figure 3.4. Similar to

the behaviour observed for Ca , a marked increase in the square root of the objective function

Chapter 3. Blood flow models 53

Normal EF Reduced EF

500

600 Ca=0.50 Ca=0.50

400

Ca=0.99 Ca=1.00

400

Ca=1.50 300 Ca=1.50

Flow, ml/s

Flow, ml/s

Ca=2.00 Ca=2.05

200 200

Ca=2.50 Ca=2.50

0 100

0

−200

−100

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8

Time, s Time, s

400 600

Zc=0.02 500 Zc=0.02

300 Zc=0.04 Zc=0.04

400

Zc=0.06 Zc=0.06

Flow, ml/s

Flow, ml/s

200 300

Zc=0.08 Zc=0.08

200

100 Zc=0.10 Zc=0.10

100

0

0

−100

−100 −200

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8

Time, s Time, s

500 500

L=0.0001 L=0.0001

400 L=0.0010 400 L=0.0010

L=0.0049 L=0.0018

300 L=0.0100 300 L=0.0100

Flow, ml/s

Flow, ml/s

L=0.1000 L=0.1000

200 200

100 100

0 0

−100 −100

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8

Time, s Time, s

Figure 3.4: Sensitivity analysis showing the effect of changes in the model parameters Ca , Zc

and L on an exemplary flow waveform of a patient with normal (left) and reduced (right) EF.

The respective baseline parameters are indicated in gray.

was found for both an increase as well as a decrease in Zc relative to its baseline value, see figure

3.6(b). However, this increase seemed to be less symmetric than for Ca . Also in the simulation

results, a distinct behaviour for values below and above the baseline could be observed, with a

more pronounced change in shape in the first case. In contrast to Ca , which mainly affects the low

Chapter 3. Blood flow models 54

Normal EF Reduced EF

400 500

P∞=0⋅DBP P∞=0⋅DBP

400

300 P∞=0.2⋅DBP P∞=0.2⋅DBP

P∞=0.5⋅DBP 300 P∞=0.5⋅DBP

Flow, ml/s

Flow, ml/s

200

P∞=0.7⋅DBP P∞=0.7⋅DBP

200

100 P∞=0.9⋅DBP P∞=0.9⋅DBP

100

0

0

−100 −100

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8

Time, s Time, s

400 500

Rp=0.37 Rp=0.23

400

300 Rp=0.50 Rp=0.50

Rp=0.75 300 Rp=0.75

Flow, ml/s

Flow, ml/s

200

Rp=1.00 Rp=1.00

200

100 Rp=1.25 Rp=1.15

100

0

0

−100 −100

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8

Time, s Time, s

400 500

ñ=2 ñ=2

ñ=3 400 ñ=3

300

ñ=4 ñ=5

ñ=7 300 ñ=7

Flow, ml/s

Flow, ml/s

200

ñ=10 ñ=10

200

100

100

0

0

−100 −100

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8

Time, s Time, s

Figure 3.5: Sensitivity analysis showing the effect of changes in the model parameters P∞ , Rp

and ñ on an exemplary flow waveform of a patient with normal (left) and reduced (right) EF.

The respective baseline parameters are indicated in gray.

frequency range, Zc describes the relation of the high frequency components of pressure and flow.

Thus, for higher values, the fast pressure oscillations, that are responsible for the fast changes in

the waveform like the upstroke, are damped, explaining the slower upstroke and shifted peak in

the flow wave. For low values, on the contrary, they are amplified, resulting in a faster upstroke

Chapter 3. Blood flow models 55

(a) Error relative to baseline (b) Error relative to baseline (c) Error relative to baseline

2.2 2.2 2.2

2 2 2

1.8 1.8 1.8

1.6 1.6 1.6

1.4 1.4 1.4

1.2 1.2 1.2

1 1 1

0.8 0.8 0.8

0.85 0.9 0.95 1 1.05 1.1 1.15 0.85 0.9 0.95 1 1.05 1.1 1.15 0.85 0.9 0.95 1 1.05 1.1 1.15

Fraction of baseline Ca Fraction of baseline Zc Fraction of baseline L

Figure 3.6: Boxplots showing the changes in the objective function relative to its baseline for

varying Ca (a), Zc (b) and L (c) from 85% to 115% of their respective baseline values while

keeping all other parameters constant.

and an earlier peak. However, these effects of course depend on the specific pressure wave and the

corresponding higher frequency amplitudes. As before for Ca , the considered range for variation

was wide and the induced changes in the waveform moderate in comparison.

The arterial inertance L. The influence of L was similar to that of Zc , showing a more

pronounced qualitative change for values below the baseline. For L = 0, the WK4p equals the

WK2, thus, the results obtained for very low values represent the transition from one model to

the other. Except for the baseline, L was increased by a factor 10 per variation step, indicating

that the influence of L on the flow waveform is overall rather small. Nevertheless, figure 3.6(c)

shows, that also for L, the baseline value seems to represent a uniquely defined minimum of the

objective function.

The asymptotic pressure P∞ and the peripheral resistance Rp . P∞ and Rp show the

same effect, because systemic vascular resistance (SVR) as well as its components Q and P were

fixed and their values are therefore coupled, see equation (3.15). The two plots are included

separately for the sake of completeness only. Even though the values of P∞ were varied over the

whole range considered in the optimisation, i.e. from 0 to 0.9·DBP, the induced changes in the

flow waveforms were rather subtle, with a slight shift in maximum and a more pronounced shift

in minimum flow. In contrast to the other Windkessel parameters, which all represented inner

minima of the marginal error, P∞ was optimal at its boundary, i.e. P∞ = 0.9 · DBP, for most

of the datasets, including those shown in figure 3.5. This of course challenges the choice of the

upper boundary, as will be discussed in the next subsection.

The threshold ñ. ñ appears in the formula for the computation of the model impedance

Zmodel , see equation (3.11). It represents the harmonic above which the phase of modelled input

impedance is set to zero. Choosing it too low resulted in strong oscillations during diastole for

both patients, whereas the influence of choosing it higher than the baseline was rather small,

Chapter 3. Blood flow models 56

(a) (b)

70

60 5

50

4

Frequency

40

30 3

20

2

10

1

0

2 3 4 5 6 7 8 9 10 2 3 4 5 6 7 8 9 10

Threshold ñ Fixed value of threshold ñ

Figure 3.7: Analysis of the effect of the grid-search approach for ñ. (a): histogram showing the

distribution of the individual threshold ñ in patients with reduced EF (red) and controls (blue).

(b): boxplots showing the relative error when fixing ñ at one value compared to the grid-search

approach.

compare figure 3.5. Therefore, the question arises if the computationally expensive grid search

approach for ñ is expedient at all. Figure 3.7(a) shows a histogram of the distribution of the

optimal ñ values determined in the whole population of 183 patients. Even though it was equal

to its prespecified upper boundary for many, a second peak could be observed around the fourth

harmonic, resulting in a higher number of patients with an optimal threshold ñ < 10 than ñ = 10.

This finding was furthermore independent of the EF-status. Thus, despite the small changes in

the flow waveform observed for the two datasets considered for higher ñ, the histogram provides a

first indication that allowing for different values of the threshold might be beneficial. To quantify

this effect, the absolute error, i.e. ||~e||21 , obtained with the grid search was taken as baseline and

its relative changes were analysed when fixing ñ between 2 and 10 for all patients. The results

are shown in figure 3.7(b). Since the threshold with the lowest error is chosen in the original

approach, in other words a second optimisation is performed with respect to ñ, naturally no

improvement can be achieved when fixing its value implying that all relative errors are ≥ 1, as

can be observed in the boxplots. In line with the behaviour found for the two datasets shown in

figure 3.5, low values of ñ were associated with a considerable increase in error, while for higher

values of ñ the relative error decreased. However, also for ñ = 10, the median error was slightly

higher than baseline, and significantly higher for several datasets. Hence, a constant threshold

might work well for most patients, but it could still corrupt the results for some, overall justifying

the grid search approach.

Boundary conditions

For the Windkessel parameters Ca , Zc and L, the optimal values correspond to inner minima of

the objective function for all datasets considered. Therefore, the boundary conditions have no

influence on the optimal solution. For P∞ on the contrary, its upper boundary value was optimal

Chapter 3. Blood flow models 57

7

15

RMSE, AU

10

6.5

5

6

Relative error ||e||2

1.5

1 5.5

0.5

5

0.85 0.9 0.95 1 0.85 0.9 0.95 1

Upper boundary of P as fraction of DBP Upper boundary of P as fraction of DBP

∞ ∞

Figure 3.8: Analysis of the effect of the upper boundary of P∞ used in the optimisation. The

median RMSE between measured and modelled flow waves scaled to 100 AU for the upper

boundary varying from 0.1 to 1 times DBP in steps of 0.1 (left, upper panel) as well as from 0.85

to 1 in steps of 0.005 (right panel) is shown for the whole study population (black) as well as for

patients with reduced (red) and normal (blue) EF separately. Left, lower panel: error relative

to the baseline value of 0.9·DBP. Error bars represent the respective 2.5% and 97.5% quantile.

in the great majority of the datasets. In other words, the imposed restrictions have an impact

on the modelled flow waveforms in this case.

The choice of 0.9 · DBP was motivated by several considerations discussed in appendix A. First

of all, from a model perspective, P∞ is supposed to represent the pressure that is maintained by

the vasculature when the heart stops beating and P∞ should therefore be ≤DBP. From the pres-

sure waves for missing heartbeats presented in appendix A, it can furthermore be seen that even

though the pressure drop slows down, pressure continues to decrease without excitation, indicat-

ing that P∞ should be less than DBP. Moreover, values reported in literature were substantially

smaller than DBP. However, higher values seemed to be beneficial for the fitting quality, compare

figure A.5. Therefore, the upper boundary of 0.9·DBP was chosen as a compromise between the

physiological interpretability and a good fitting result.

To investigate how this choice influences the modelled flow waveforms, the upper boundary was

varied from 0.1 to 1 and the resulting value of ||~e||2 relative to its baseline value, i.e. 0.9 · DBP,

was computed. Additionally, the root mean square error (RMSE) to the measured flow wave

was calculated for each dataset. Therefore, both modelled and measured flow were scaled to

the same peak value of 100 arbitrary unit (AU) first. As expected, the objective function was

on average decreasing with respect to the upper boundary for P∞ , as shown in figure 3.8 for

values between 0.85 and 1 times DBP. The number of optimal P∞ that were equal to the upper

boundary started to decrease rapidly for higher values, dropping from 174 at 0.9 to 17 out of

183 for an upper boundary equal to DBP. The average deviation from measured flow measured

Chapter 3. Blood flow models 58

Normal EF Reduced EF

350 500

400

ub=0.3⋅DBP ub=0.3⋅DBP

250

ub=0.5⋅DBP 300 ub=0.5⋅DBP

Flow, ml/s

Flow, ml/s

200

ub=0.7⋅DBP ub=0.7⋅DBP

150 ub=0.9⋅DBP 200 ub=0.9⋅DBP

100 ub=1⋅DBP ub=1⋅DBP

100

50

0

0

−50 −100

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8

Time, s Time, s

Figure 3.9: Influence of the upper boundary (ub) of P∞ on an exemplary pressure wave from a

patient with normal (a) and reduced EF (b).

by the RMSE, in contrast, was decreasing at first, but increased again for very high values. This

behaviour is depicted in figure 3.8 for the upper boundary ranging from 0.1 to 1 relative to DBP

in incremental steps of 0.1, as well as from 0.85 to 1 in steps of 0.005 to allow for a more detailed

resolution of the relevant region. Moreover, in addition to the results obtained in the whole

population, the RMSE is indicated for the two groups of patients separately. Despite a marked

offset, with a substantially smaller error in patients with reduced EF that will be addressed in

the next section, the behaviour with respect to the upper boundary of P∞ was similar between

the groups. Splitting the RMSE into its systolic and diastolic part furthermore showed that

the increase for higher values was driven by the systolic part only, whereas the diastolic error

resembles the behaviour of ||~e||2 , i.e. it declines until reaching a plateau for very high values.

Thus, while both the systolic and the diastolic RMSE decrease at first for increasing values of the

upper boundary, this behaviour changes for values above 0.9. However, this qualitative change in

the systolic error is not captured by the chosen objective function, which mainly represents the

diastolic part, and restricting P∞ therefore seems legitimate not only from a model perspective

but also from the achieved output.

Overall, in both groups as well for the total median, the choice of 0.9·DBP seems reasonable

with regards to the quality of the flow estimate. Furthermore, these results demonstrate that

the model is robust against changes in P∞ . Figure 3.9 finally shows the influence of variations

in the upper boundary of P∞ on the two exemplary waveforms used before, again underlining

the rather small effect on the derived waveforms.

Input

Last but not least the sensitivity of the model on changes in the input values, that is to say the

pressure waves, will be analysed. For this purpose, the systolic, diastolic and mean pressure were

Chapter 3. Blood flow models 59

1.5

DBP SBP MBP

0.5

−5 −4 −3 −2 −1 0 +1 +2 +3 +4 +5

Increment from baseline, mmHg

Figure 3.10: Analysis of the influence of changes in DBP, SBP and MBP on the modelled flow

waveforms, assessed by the RMSE to the flow contour obtained at baseline. Median values are

shown and error bars represent the respective 2.5% and 97.5% quantile.

x

PDBP = (P − DBP) · 1 − + DBP + x,

SBP − DBP

x

PSBP = (P − DBP) · 1 + + DBP,

SBP − DBP

x

PMBP =P · 1+ ,

MBP

with x = −5, −4, . . . , 4, 5 mmHg. ±5 mmHg represents the average accuracy accepted for non-

invasive blood pressure measurement devices by the ISO 81060-2. From the equations given

above it follows that DBP remains unaltered when changing SBP and vice-versa while MBP of

course changes accordingly. Likewise, SBP as well as DBP are affected by changes in MBP. To

quantify the effect of the different pressure levels, the flow waveform obtained with the original

pressure wave was taken as baseline and the deviation in the modelled waveforms induced by

changes in SBP, DBP or MBP was assessed by the RMSE. All flow contours were again nor-

malised to 100 AU to make the values comparable between the patients. The results are shown

in figure 3.10 for the whole study population as well as exemplarily for the two datasets in figure

3.11.

The median RMSE as well as its deviation were very small for both DBP and SBP and changes

in MBP had no impact at all on the corresponding flow waveforms, indicating that the model

is robust against errors in the absolute values of blood pressure. The last result may not be

surprising since the same argumentation used for the scaling of blood flow, compare equation

3.15, can of course be used for pressure as well. More precisely, for a fixed level of mean flow

Q, the same flow waveform is obtained for MBP and for c·MPB with an arbitrary positive

constant c > 0, if P∞ , Rp , Zc and L are multiplied by c and Ca by 1/c. In other words, if a

flow contour is optimal for MBP, the same flow contour should be optimal for c·MBP. The fact

Chapter 3. Blood flow models 60

Normal EF Reduced EF

350 500

DBP−3 400 DBP−3

250 DBP DBP

DBP+3 300 DBP+3

Flow, ml/s

Flow, ml/s

200

DBP+5 DBP+5

150 200

100

100

50

0

0

−50 −100

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8

Time, s Time, s

350 500

SBP−3 400 SBP−3

250 SBP SBP

SBP+3 300 SBP+3

Flow, ml/s

Flow, ml/s

200

SBP+5 SBP+5

150 200

100

100

50

0

0

−50 −100

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8

Time, s Time, s

350 500

MBP−3 400 MBP−3

250 MBP MBP

MBP+3 300 MBP+3

Flow, ml/s

Flow, ml/s

200

MBP+5 MBP+5

150 200

100

100

50

0

0

−50 −100

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8

Time, s Time, s

Figure 3.11: Sensitivity analysis showing the effect of changes in the magnitude of ±5 mmHg in

the input pressure levels DBP, SBP and MBP on an exemplary flow waveform of a patient with

normal (left) and reduced (right) EF.

that the parameter identification procedure indeed yielded the same waveforms in both cases

demonstrates that not only the model itself but also the numerical realisation is robust.

Chapter 3. Blood flow models 61

In this section, the capability of the model to reproduce different flow wave shapes and to provide

accurate estimates of aortic flow will be investigated based on first simulation results. There-

fore, the model was applied to all 183 datasets1 and the resulting flow waveforms were visually

assessed. The identified parameters are summarised in table 3.2 and exemplary waveforms are

shown in figures 3.12 and 3.13 for patients with normal and reduced EF respectively. In addition

to the modelled flow waves, the corresponding pressure waves computed from measured flow

using Zmodel determined from the identified parameters are shown. These can be seen as an

indicator for the quality of identified parameters.

Ca , ml/mmHg 1.221 0.903 to 1.694 0.356 7.937

Rp , mmHg·s/ml 0.3 0.255 to 0.356 0.111 0.994

Zc , mmHg·s/ml 0.0527 0.0411 to 0.0670 0.0205 0.117

L, mmHg·s2 /ml 0.00468 0.00358 to 0.00632 0.00107 0.3

P∞ , mmHg 71.01 64.43 to 77.28 0 102.38

ñ 5 4 to 10 2 10

SVR, mmHg·s/ml 1.209 1.082 to 1.396 0.597 1.848

Table 3.2: Results of the parameter identification. The median, inter-quartile range as well as

total range are given for each parameter.

The results depicted in figures 3.12 and 3.13 were selected to provide a comprehensive impression

of the variety of possible (measured) pressure and flow waveforms and of the resulting simulated

flow waves. In most cases and in particular in patients with reduced EF, the proposed approach

indeed yielded qualitatively correct estimates and the modelled input impedances mostly were

in good qualitative agreement with the measured ones. However, the proposed approach and the

underlying Windkessel model of course have their limitations, as evidenced by the flow waves

shown in figure 3.13(c)-(d), presenting an exaggerated nose or even second peak during the flow

decline. This behaviour is related to the upstroke of the pressure wave, more specifically, it is

mainly found for waveforms with a pronounced shoulder in early systole. As presented in the

last chapter, this shoulder is considered to be caused by the return of the reflected waves and

the resulting pressure augmentation. However, the Windkessel models do not incorporate wave

phenomena and the only explanation for a bump in the pressure wave is therefore a correspond-

ing one in the flow wave. How pronounced the bump will appear in the flow wave then again

depends on the specific parameter values. The fact that the flow model yielded better results

for patients with reduced EF is well in line with this observation, since previous results showed

a lower AIx for these patients compared to controls [21, 137], as will be discussed in more detail

in the next chapters.

1 The computations took 102s on a personal computer with an Intel©Corei7-5600 CPU with 2.60GHz and

8GB RAM.

Chapter 3. Blood flow models 62

130 0.2

100

measured measured

|Zin|, AU

125

computed modelled 0.1

80

120

60 0

115

2 4 6 8 10 12 14

110 40 4

arg(Zin), rad

105 20 2

100 0

0

95 −2

0 0.2 0.4 0.6 0 0.2 0.4 0.6 2 4 6 8 10 12 14

Time, s Time, s Harmonics, n

(b) Pressure, mmHg Flow, AU Input Impedance

120 0.1

100

measured measured

|Zin|, AU

115

computed modelled 0.05

80

110

60 0

105

2 4 6 8 10 12 14

100 40 2

arg(Zin), rad

95 20 0

90 −2

0

85 −4

0 0.2 0.4 0.6 0 0.2 0.4 0.6 2 4 6 8 10 12 14

Time, s Time, s Harmonics, n

(c) Pressure, mmHg Flow, AU Input Impedance

115 0.1

100

measured measured

|Zin|, AU

110

computed modelled 0.05

80

105

60 0

100

2 4 6 8 10 12 14

95 40 5

arg(Zin), rad

90 20

0

85

0

80 −5

0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 2 4 6 8 10 12 14

Time, s Time, s Harmonics, n

(d)

Pressure, mmHg Flow, AU Input Impedance

130 0.4

100

measured measured

|Zin|, AU

120

computed modelled 0.2

80

110

60 0

100

2 4 6 8 10 12 14

90 40 2

arg(Zin), rad

80 20

0

70

0

60 −2

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 2 4 6 8 10 12 14

Time, s Time, s Harmonics, n

Figure 3.12: Exemplary simulation results for patients with reduced EF. Left panels: measured

aortic pressure (black) and pressure computed from the identified parameters and the measured

flow wave (blue). Middle panels: measured flow (black) and modelled approximation (blue).

Right panels: measured (black) and modelled (blue) input impedance.

Chapter 3. Blood flow models 63

0.2

100

110 measured measured

|Zin|, AU

computed 80

modelled 0.1

100

60 0

2 4 6 8 10 12 14

90

40 0

arg(Zin), rad

80 −0.5

20

−1

70 0

−1.5

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 2 4 6 8 10 12 14

Time, s Time, s Harmonics, n

(b) Pressure, mmHg Flow, AU Input Impedance

110 0.2

100

measured measured

|Zin|, AU

105

computed modelled 0.1

80

100

95 60 0

2 4 6 8 10 12 14

90

40 2

85

arg(Zin), rad

20 0

80

75 −2

0

70 −4

0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 2 4 6 8 10 12 14

Time, s Time, s Harmonics, n

(c) Pressure, mmHg Flow, AU Input Impedance

0.2

100

120 measured measured

|Zin|, AU

80

110

60 0

2 4 6 8 10 12 14

100

40 1

arg(Zin), rad

90 0

20

−1

80 0

−2

0 0.2 0.4 0.6 0 0.2 0.4 0.6 2 4 6 8 10 12 14

Time, s Time, s Harmonics, n

(d)

Pressure, mmHg Flow, AU Input Impedance

1

150 100

measured measured

|Zin|, AU

80

130

60 0

120

2 4 6 8 10 12 14

110 40 5

arg(Zin), rad

100

20

90 0

80 0

−5

0 0.2 0.4 0.6 0 0.2 0.4 0.6 2 4 6 8 10 12 14

Time, s Time, s Harmonics, n

Figure 3.13: Exemplary simulation results for patients with normal EF. Left panels: measured

aortic pressure (black) and pressure computed from the identified parameters and the measured

flow wave (blue). Middle panels: measured flow (black) and modelled approximation (blue).

Right panels: measured (black) and modelled (blue) input impedance.

Chapter 3. Blood flow models 64

0

−1

−3

−4

−5

−6

0 1 2 3 4 5 6 7 8

Frequency, Hz

Figure 3.14: Phase angle of central pressure for patients with normal (blue) and reduced (red)

EF as well as their corresponding averaged curve (bold line). The dashed lines indicate a phase

angle of −π at 5Hz.

The simulation results as well as the sensitivity analysis show that the proposed model is able to

reproduce physiological and pathological ejection patterns based on pressure alone and is robust

against changes in the model parameters, the input values as well as the boundary conditions

applied. The fact that different types of flow contours can be obtained depending on the shape

of the input pressure represents one of the main strengths of this approach, since this feature

potentially enables its use also for systolic heart failure patients. However, on the downside,

because of this flexibility, not all modelled waveforms match the measured ones or at least show

the typical characteristics that one would expect from aortic blood flow. In particular, an exag-

gerated nose or even a second peak in the flow wave were observed in many patients with normal

EF, making the waveforms unsuitable for further computations.

In an attempt to identify those pressure waveforms that result in the unwanted behaviour, a

Fourier-analysis was performed and differences in the phase angle of pressure at around 5Hz

were observed for patients with reduced and normal EF respectively, compare figure 3.14. In

particular, for patients with normal EF, the corresponding phase angle tended to be > −π,

while for patients with reduced EF it was on average < −π, implying that the first peak of the

corresponding harmonic occurs earlier in the control group2 , which might result in the shoulder

in the pressure upstroke. A further analysis indeed indicated an association between high values

of the phase angle at 5Hz and the occurrence of a strong bump in the pressure upstroke.

The ARCSolver flow was shown to provide reliable estimates for patients with normal EF [35].

Therefore, a combination of these two approaches might be expedient to obtain suitable estimates

2 A wave with a frequency of 5Hz and a phase angle equal to −π, i.e. a sine wave, would reach its first peak

at 0.1s. An additional phase shift by ±0.5 rad would result in a time delay of ∓0.016s.

Chapter 3. Blood flow models 65

of aortic flow, independent of cardiac function. From the considerations described above, −π

was chosen as threshold for the phase angle of pressure at 5Hz to select either the ARCSolver

flow (values above the threshold) or the Windkessel flow (values below the threshold). According

to this criterion, the ARCSolver flow was chosen in 20 out of 61 patients with reduced EF and

in 107 out of 122 patients with normal EF, eliminating all problematic flow waves. As a last

finalising step, the modelled flow waveforms obtained with the Windkessel flow were set to 0 at

the beginning and end of systole as well as during diastole.

Chapter 4

Non-invasive quantification of

wave reflections in systolic heart

failure: state of the art and data

analysis

This chapter is dedicated to the examination of parameters derived by pressure pulse wave analy-

sis (PWA), wave separation analysis (WSA) and wave intensity analysis (WIA) in patients with

systolic heart failure. First, a brief summary of the current state of the art is given followed

by a data analysis investigating the differences between patients with systolic heart failure and

controls.

Most results presented in this chapter were previously published in the article ”Determinants and

covariates of central pressures and wave reflections in systolic heart failure” in the International

Journal of Cardiology in 2015 [94].

The mechanical factors leading to and resulting from heart failure (HF) as well as the resulting

compensation mechanisms have been a main interest of cardiovascular research for many decades.

With the development of high fidelity micromanometer catheters and electromagnetic velocity

probes mounted at the catheter-tip in the second half of the 20th century, the instantaneous

measurement of both aortic blood pressure and flow velocity at sufficient quality became feasi-

ble. Their analysis yielded valuable insights in the contractile state of the myocardium and the

induced alterations in ventricular haemodynamics in HF. [63]

66

Chapter 4. Wave reflections in systolic heart failure 67

Various invasive studies were subsequently conducted during the 1970ies and 1980ies, comparing

the aortic input impedance of patients with HF with reduced ejection fraction (EF) and controls

[51, 52, 66, 99]. In all of these studies, the systemic vascular resistance (SVR), as a measure of the

steady part of vascular load, was found to be significantly greater in patients with HF. However,

with regards to the aortic characteristic impedance Zc , i.e. the pulsatile component of vascular

load related to aortic stiffness, findings were less conclusive. While Pepine et al. [99] reported a

significant elevation of Zc in HF patients, the other investigators found no statistical evidence for

a difference between patients with and without HF, although numerical values were consistently

higher [51, 52, 66]. Also wave reflections, assessed by the first modulus of the complex aortic

reflection coefficient [52] or the reflection magnitude obtained by WSA [51], were (statistically)

comparable between the groups. However, the considered study populations were small (10-17

subjects per group) and might have been statistically underpowered.

In contrast to these first, pioneering studies, most modern cross-sectional studies comparing the

pulsatile hemodynamics in patients with HF to controls were based on non-invasively acquired

data and thus included larger samples of patients [19, 21, 68, 88, 137]. Moreover, while patients

were treated primarily with diuretics and digitalis in the earlier studies, recent heart failure

patients were also receiving angiotensin-converting enzyme inhibitors or angiotensin receptor

blockers and betablockers, in line with the current guidelines [101]. Furthermore, patients in-

cluded in the more recent studies were generally older (mean age was approximately 60 years

versus 40 years in the older studies) and had more comorbidities in comparison to those that

were investigated 40 years ago, which might indicate a different aetiology and phenotype of HF

in the patients considered [153].

In these contemporary, well treated patients, ejection duration (ED) was markedly shorter (with

a mean difference in the range of 30 to 40 ms) in HF patients versus controls in most investi-

gations [21, 68, 137], an observation that was already reported by Weissler and colleagues [141]

in 1968. This reduction seems to be related to the severity of left ventricular (LV) systolic dys-

function, since Paglia et al. [88], who studied patients with moderately to severely reduced EF,

could not confirm this difference in their data at first. However, when only patients with severe

left ventricular systolic dysfunction (LVSD) defined as EF≤ 30% were considered, a significant

shortening in ED became again apparent. Even Mitchell et al. [68] reported a significant re-

duction in ED despite pooling patients with heart failure with reduced and preserved EF for

analysis. This might have actually led to an underestimation of the real effect since ED was

found to be shorter in patients with HF with reduced than with preserved EF [125], a tendency

that was also confirmed by Mitchell et al. in subgroup analysis. Indeed, ED was reported to be

even prolonged in patients with heart failure with preserved EF compared to subjects without

HF [139]. A rather small, yet still significant, reduction in ED was observed by Denardo and

colleagues [21] (≈ 20ms), which is possibly due to the intrinsic link between heart rate (HR)

and ED and the fact that HR served as a matching criterion for patients with HF and controls

in their study. This is again in line with the results presented by Weissler et al. showing an

alteration in the association between HR and ED for patients with systolic HF with lower values

Chapter 4. Wave reflections in systolic heart failure 68

Another difference that was repeatedly reported was a significantly lower augmentation index

(AIx) in HF patients [21, 68, 137]. In the study by Curtis et al. [19], in contrast, HF was

associated with an increased AIx, albeit not significantly. However, compared to the aforemen-

tioned works, patients with HF were older than controls in their study and the LV impairment

assessed by EF was on average less severe with more female patients included in the HF group

which might explain the discrepancy. This is corroborated by the results reported by Paglia and

colleagues [88], who found higher values of AIx in patients with moderate than with severe LVSD

or controls, which resulted in no difference to controls when looking at the HF patients as a whole.

With regards to the pressure levels, rather contradictory results were obtained. While both pe-

ripheral and estimated central systolic blood pressure were either lower [21, 88] or comparable

[68, 88, 137] for HF patients versus controls, all three possible cases were reported for the pulse

pressure.

Besides the classic pulse wave analysis parameters, Denardo and coworkers [21] as well as Paglia

and colleagues [88] additionally reported pressure time integrals and Mitchell and coworkers [68]

as well as Curtis et al. [19] included Doppler flow measurements to perform WSA and WIA

respectively. It should be noted though that the two last mentioned works both used tonometri-

cally measured carotid pressure waveforms as estimates of central pressure and combined it with

either aortic [68] or carotid [19] Doppler flow waveforms.

In keeping with the lower values of AIx, Denardo and coworkers [21] reported a reduction in

wasted energy in their HF group, which primarily consisted of patients with severe LVSD. Paglia

et al. [88] observed the same behaviour for patients with severe dysfunction but not for their

entire group of HF patients. Denardo et al. furthermore found a decrease in systolic pressure

time index (SPTI) multiplied by HR, which represents a measure of the LV oxygen requirement

per minute (i.e. normalised to HR), and a higher myocardial viability ratio (DPTI/SPTI) in

HF, which they speculated to ”serve as a compensatory mechanism in severe LVSD to optimize

efficiency of the failing LV”[21, p.153].

Mitchell et al. [68] followed the example set by the older invasive studies and examined the char-

acteristic impedance Zc , although they used a time domain approach for estimation while prior

studies were set in the frequency domain. They observed an elevation of Zc in HF and argued

that the stiffness of the central conduit arteries might be increased, even though no difference in

carotid-femoral pulse wave velocity (PWV) was found. In contrast to the prior studies, SVR was

comparable between the groups, as was total arterial compliance. Furthermore, the amplitude

of the forward travelling pressure wave |Pf | was increased, while |Pb | was similar, resulting in a

decreased reflection magnitude (RM) in patients with HF compared to controls.

Chapter 4. Wave reflections in systolic heart failure 69

With respect to the WIA parameters, a totally different behaviour was found by Curtis et al.

[19]: the energy of the forward S wave resulting from ventricular ejection was reduced, while

the R and D wave energy, related to wave reflections and ventricular relaxation dynamics, were

unaltered in patients with systolic HF compared to controls, which led to a significant elevation

of relative wave reflections, i.e. the ratio of the R to S wave energy. Also Sugawara et al. [120]

reported a significant reduction of the first but not the second peak of wave intensity in patients

with dilated cardiomyopathy compared to controls. Unfortunately though, no further informa-

tion about the study is available, since data was not published.

In summary, studies performed in patients with systolic heart failure yielded rather inconclusive

results regarding alterations in arterial stiffness or wave reflections. The common findings of the

older studies, i.e. a higher peripheral resistance and a tendency towards higher aortic stiffness

in systolic HF patients, could not be confirmed in the more recent ones, which might be due

to the differences in medication or even differences in the phenotype and aetiology of HF in

the patients considered. The results obtained in the newer studies seem to imply that wave

reflections are reduced in patients with HF. However, the composition of the HF groups included

was diverging between the non-invasive studies, ranging from a combination of HF with preserved

and reduced EF [68], to purely systolic HF patients yet with differing severity levels of LV

dysfunction [21, 88, 137]. Furthermore, different matching criteria were applied to select the

control group, including age [21, 88, 137], gender [21, 88, 137], anthropometric measures (height,

weight, body surface area or body mass index) [21, 137], heart rate [21], brachial pressure levels

[88, 137] or none [19, 68], making the results less comparable.

haemodynamics in patients with systolic heart failure is very limited, especially with regards to

the methods based on pressure and flow, namely WSA and WIA. Moreover, the reported results

were inconsistent. Therefore, the aim of this study is threefold: (1) to investigate the differences

in PWA, WSA and WIA parameters between patients with systolic HF and controls, (2) to

compare the different methods against each other and finally (3) to analyse the relation between

measures of cardiac and arterial function.

4.3 Methods

Study population

Patients were collected between 2009 and 2011 from a cohort undergoing diagnostic coronary

angiography for suspected coronary artery disease at the university teaching hospital Wels-

Grieskirchen in Wels, Austria. All measurements were performed in the hospital Wels-Grieskirchen

within the framework of ongoing studies on the role of pulsatile haemodynamics in cardiology,

Chapter 4. Wave reflections in systolic heart failure 70

volume. Left: left ventriculogram during diastole taken from [16], licensed under CC BY-NC

4.0, right: echocardiogram obtained in the apical 4 chamber view during diastole taken from

[62], licensed under CC BY 2.0.

which were approved by the local ethics committee. Participants provided written informed

consent prior to enrolment. Exclusion criteria included arrhythmias (mainly atrial fibrillation),

valvular heart disease exceeding a mild severity level and unstable clinical conditions.

The study population considered throughout this thesis consisted of 61 patients diagnosed with

severely reduced EF (rEF) and a control group comprising 122 patients with normal EF (nEF),

matched for gender, age, body mass index and brachial blood pressure levels.

Data acquisition

Pressure waveforms were recorded non-invasively at the radial artery using applanation tonom-

etry (Millar SPT 301, Millar, Inc., Houston, Texas, US) and calibrated with absolute brachial

pressure values obtained with an automated oscillometric sphygmomanometer (Omron M5-I,

Omron Healthcare, Kyoto, Japan). Aortic pressure was subsequently derived by the Sphygmo-

Cor system (AtCor Medical Pty. Ltd, West Ryde, Australia) and its inbuilt generalised transfer

function from ensemble averaged peripheral pressure waves. The waveforms of peripheral and

central pressure processed by the SphygmoCor system were finally exported and converted to a

sampling rate of 83.3̄ Hz, which represents the sampling rate used in the ARCSolver algorithms,

compare figure 4.2. All further computations were carried out in Matlab R2011b.

Besides the pressure reading, coronary angiography, echocardiography as well as a blood exam-

Chapter 4. Wave reflections in systolic heart failure 71

ination were performed in all patients within a maximum of 4 days of each other. 6 French

fluid filled pigtail catheters were used for cardiac catheterisation, enabling the invasive mea-

surement of LV and aortic pressure levels before contrast cineangiography. Additional pressure

related indices like maximum LV pressure rise, a measure of cardiac contractility, were auto-

matically provided by the coronary angiography system (Siemens Artis Zee with AXIOM Sensis

hemodynamic recording system, Siemens healthcare, Erlangen, Germany). From the monoplane

cineangiograms in RAO view, cardiac volumes and EF were determined, compare figure 4.1.

During catheter pullback, aortic PWV was quantified by the foot-to-foot technique using pres-

sure measured at the ascending aorta and the aortic bifurcation. The corresponding positions

were marked with a tape on the catheter to derive the exact distance, compare [138].

gram was carried out immediately before or after pressure measurement using a Philips iE33

Ultrasound machine (Philips Medical Systems, Andover, Massachusetts). LV systolic function

and geometry were quantified according to the recommendations of the American Society of

Echocardiography [50], compare figure 4.1. Blood flow velocity at the LV outflow tract was

measured by pulsed wave Doppler ultrasound and the spectral density recordings were manually

digitised to obtain aortic blood flow waveforms, whereby the width of the flow was scaled to

match the estimated ejection duration provided by the SphygmoCor system. Thereafter, flow

and pressure were carefully aligned in time using the respective upstrokes as visual indicators,

see figure 4.2. The so-obtained flow velocity U was normalised to an amplitude of 100 arbitrary

units (AU) to be used as a qualitative waveform of volumetric blood flow Q.

From the blood samples, plasma levels of N-terminal pro-B-type natriuretic peptides (NT-

proBNP) were determined with the electrochemiluminescence immunoassay ”ECLIA” on the

Elecsys 1020 analyser (Roche Diagnostics, Mannheim, Germany). NT-proBNP is produced by

the heart ”and released into the circulation in response to increased wall tension” [7, p. 150].

An elevation of the NT-proBNP level represents an important indicator of cardiac dysfunction

and diagnostic marker for heart failure [101].

Pressure levels and timing information, i.e. HR and ED, were automatically derived by the

SphygmoCor system. From these estimates, the gender-specific left ventricular ejection time

index (LVETI) was computed according to the formulas LVETI = 1.7 · HR + ED for male and

LVETI = 1.6 · HR + ED for female patients [30], which are based on the work by Weissler et al.

[141]. SphygmoCor’s inbuilt PWA algorithms furthermore provided the AIx and all associated

parameters, compare section 2.1.2.

In addition, the pressure time indices and the wasted energy introduced in section 2.1.3 were

computed from the central pressure waveforms using the SphygmoCor parameters to determine

the respective portions and Matlab’s trapz function for integration. These values obviously

Chapter 4. Wave reflections in systolic heart failure 72

Transfer function

Radial Pressure, mmHg (SphygmoCor) Aortic Pressure, mmHg

110 110

P2=SBP

100 100 AP

P1

90 90 PP

80 80

Time, s ED Time, s

Manual digitization

Doppler Ultrasound & alignment Aortic Flow, cm/s

150

100

50

0 ED

0 0.2 0.4 0.6 0.8

Time, s

Figure 4.2: Assessment of aortic blood pressure and flow. The tonometrically measured radial

pulse waveforms are processed with the SphygmoCor software to obtain aortic pressure as well

as the PWA parameters (upper panel). Blood flow velocity is measured by Doppler ultrasound

in the LV outflow tract, then manually digitized and aligned with the synthesised aortic pressure

waveform (lower panel). Adapted from [95], © Institute of Physics and Engineering in Medicine.

Reproduced with permission. All rights reserved.

depend on the integration intervals, i.e. the heart beat duration. To diminish this influence, the

pressure time indices per minute were calculated by multiplication with HR, compare [21].

For wave separation analysis (WSA), central pressure and flow were Fourier transformed using

Matlab’s fft function and Zin was computed. Zc was then estimated as the average of the

harmonics > 3Hz up to approximately 10Hz, excluding values bigger than 3 times the median to

control for outliers [116]. More precisely, the averaging started at the first harmonic correspond-

ing to a frequency higher than 3Hz and ended at the harmonic being closest to 10Hz.

Reported procedures for the estimation of Zc most often involve a fixed harmonic range for all

patients, e.g. harmonics 4 to 10 [37, 123]. However, the idea behind this estimation procedure is

based on the wavelength and therefore the absolute and not the relative frequency with respect to

the heart rate, that is to say the harmonic, compare section 2.2.3. Moreover, while the difference

in the frequencies considered might be small for similar heart rates, failing of the left ventricle is

often reflected in an increase in HR [45]. This implies that, for a fixed harmonic range, different

Chapter 4. Wave reflections in systolic heart failure 73

frequencies might be used for the computation of Zc in the two groups of patients included in

this work. To ensure comparability, the frequency and not the harmonic range was therefore

prescribed in the present analysis, as in the older invasive studies [51, 66, 99], even though this

means that the number of harmonics used for estimation could vary between the patients.

From the estimate of Zc , forward and backward pressure waves were computed in the time

domain, as described in section 2.2, and RM as well as the reflection index (RI) were computed.

It should be noted that the normalisation of aortic flow has no impact on the form or magnitude

of either Pf or Pb , as Zc changes inversely proportional with the normalisation factor and the

product Zc Q therefore remains unchanged.

The wavefronts dP and dU were computed as the incremental changes in P and U per time step

dt = 0.012 s resulting from the chosen temporal resolution. For the separation, the wave speed c

times the blood density ρ was estimated from the slope of the PU-loop during early systole using

the first 6 datapoints (≈60 ms). From the forward and backward wave intensities, the peaks of

and areas under the S, D and R wave were calculated. For the R wave, absolute values were

used, i.e. both the peak and energy are reported as positive values.

Statistics

For the statistical analyses, the statistical software MedCalc version 16.2.1 (MedCalc Software

bvba, Ostend, Belgium) as well as Matlab 2011b were used. Continuous data is summarised as

mean (standard deviation) if normally distributed and median [inter quartile range] else-wise,

depending on the results of a Shapiro-Wilk (platykurtic data) or Shapiro-Francia (leptokurtic

data) test for normality1 . Differences between the groups were assessed using an unpaired t-

test (normally or log-normally distributed data with equal variances), Welch-test (normally or

log-normally distributed data with unequal variances) or Mann-Whitney-U test (non-normally

distributed data), as appropriate. Categorical data is presented as number (percentage) and the

chi-squared test was used for group-wise comparison. Multiple regression analysis was performed

to correct for confounders (ANCOVA) and associations between different parameters were eval-

uated with Pearson’s correlation coefficient. For all tests, significance was assumed at a 5%

level.

4.4 Results

The study population consisted of predominantly male patients (92%) and mean age was 60.3

(range 27 to 87) years for patients with reduced EF and 59.5 (range 33 to 80) for controls. The

1 This test was performed using the swtest function by Ahmed Ben Saı̈da, retrieved from https://de.

mathworks.com/matlabcentral/fileexchange/13964-shapiro-wilk-and-shapiro-francia-normality-tests,

December 2015.

Chapter 4. Wave reflections in systolic heart failure 74

normal EF

80 110

Flow, AU

60 100

40

90

20

80

0

70

0 0.1 0.2 0.3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time, s Time, s

Figure 4.3: Averaged flow and pressure waveform for patients with reduced (blue) and normal

(red) EF. Pressure levels, heart rate and ejection duration were set to the respective group

means.

Patients, n 61 122

Female gender, n (%) 61/122 5 (8 %) 10 (8 %) 1.00

Age, years 61/122 60.3 (12 SD) 59.5 (10.6 SD) 0.65

Height, cm 61/122 174 [170,178] 174 [170,178] 0.77

Weight, kg 61/122 86 [76.8,97] 85 [76,96] 0.83

Body mass index, kg/m2 61/122 28.4 [25.8,31.6] 28.3 [25.1,31.1] 0.68

SBP brachial, mmHg 61/122 125 (20.3 SD) 127 (13.7 SD) 0.53

DBP brachial, mmHg 61/122 78.6 (13.4 SD) 79.1 (9.42 SD) 0.78

Hypertension, n (%) 61/122 40 (66 %) 80 (66 %) 1.00

Diabetes, n (%) 61/122 21 (34 %) 20 (16 %) 0.006

Coronary artery disease, n (%) 61/122 33 (54 %) 50 (41 %) 0.09

Medication

ACE/ARB, n (%) 61/122 52 (85 %) 47 (39 %) < 0.0001

Betablocker, n (%) 61/122 51 (84 %) 58 (48 %) < 0.0001

Calcium channel blocker, n (%) 61/122 6 (10 %) 11 (9 %) 0.86

Diuretics, n (%) 61/122 35 (57 %) 31 (25 %) < 0.0001

NO donor, n (%) 61/122 9 (15 %) 11 (9 %) 0.24

Acetylsalicylic acid, n (%) 61/122 44 (72 %) 97 (80 %) 0.26

Statin, n (%) 61/122 22 (36 %) 52 (43 %) 0.39

Table 4.1: Baseline characteristics of the study population. ACE, angiotensin converting en-

zyme inhibitors; ARB, angiotensin receptor blocker. P-values indicate results of a group-wise

comparison.

Chapter 4. Wave reflections in systolic heart failure 75

Cardiac catheterisation, angiography

LVEDV, ml 58/113 250 (79.5 SD) 130 (30.9 SD) < 0.0001

LVESV, ml 57/113 171 [124,242] 33 [25,45] < 0.0001

SV, ml 57/113 58 [48,79] 93 [75,113] < 0.0001

EF, % 57/113 27.5 (10.5 SD) 72.6 (9.46 SD) < 0.0001

LV systolic BP, mmHg 60/119 123 (26.5 SD) 137 (22.3 SD) 0.0005

LV end-diastolic BP, mmHg 60/119 22.9 (8.39 SD) 15.2 (5.22 SD) < 0.0001

LV dP/dt max, mmHg/s 57/113 1130 [898,1440] 1873 [1627,2148] < 0.0001

SBP aortic, mmHg 61/120 123 [111,136] 134 [122,147] 0.0006

DBP aortic, mmHg 61/120 70 [59,80.5] 68.5 [61,75.5] 0.71

PP aortic, mmHg 61/120 48 [37,66.8] 65 [53,75] < 0.0001

PWV, m/s 60/117 8.5 [6.6,9.9] 7.7 [6.6,9.2] 0.24

2-dimensional echocardiography

LVEDV, ml 61/120 170 [133,246] 81.5 [66.5,95] < 0.0001

LVESV, ml 61/120 118 [90,175] 25 [17,31] < 0.0001

SV, ml 61/121 51.9 (21.4 SD) 55.7 (15.6 SD) 0.23

EF, % 61/121 27.9 (8.65 SD) 69.4 (7.55 SD) < 0.0001

LV mass, g 54/115 257 [206,317] 169 [145,190] < 0.0001

Laboratory analysis

NT-proBNP, pg/ml 61/119 1766 [810,5211] 90 [44.8,174] < 0.0001

volume. Echocardiographic LV volumes were assessed by the modified Simpson’s rule, compare

[50]. P-values indicate results of a group-wise comparison.

two groups were well matched with regards to gender, age, body mass index and brachial systolic

and diastolic pressure levels. Coronary artery disease as well as hypertension was as common

in rEF patients as in controls, but diabetes was more frequent in rEF. Moreover, the usage of

diuretics, betablockers and inhibitors of the renin–angiotensin system was proportionally higher

in rEF. Baseline characteristics of the study population are summarised in table 4.1.

Results of the clinical measurements are presented in table 4.2. As expected, all indices of sys-

tolic ventricular function were altered in patients with reduced compared to normal EF. More

precisely, levels of NT-proBNP were significantly elevated and maximum LV dP/dt was reduced

in the low EF group versus controls. Besides the maximum pressure rise, also the maximum

(systolic) absolute pressure in the left ventricle was lower while LV enddiastolic pressure was

higher in rEF.

fore ejection) and endsystolic (after ejection) volume as well as LV mass were larger in rEF than

in nEF, whereas the resulting stroke volume (i.e. the difference between enddiastolic and endsys-

tolic volume) was lower. Even though the reduction in SV only reached statistical significance

for the angiographic and not the echocardiographic value, the corresponding difference in EF

Chapter 4. Wave reflections in systolic heart failure 76

HR, bpm 61/122 71 [63,85] 63 [57,69] < 0.0001

ED, ms 61/122 273 [245,284] 312 [293,321] < 0.0001

LVETI, ms 61/122 393 (21 SD) 416 (18.6 SD) < 0.0001

SBP central, mmHg 61/122 112 (18.3 SD) 116 (12.7 SD) 0.13

DBP central, mmHg 61/122 79.2 (13.6 SD) 79.9 (9.45 SD) 0.73

PP brachial, mmHg 61/122 45 [34.8,56] 46 [39,53] 0.30

PP central, mmHg 61/122 32 [23,41] 35.5 [30,42] 0.02

PPAmp, - 61/122 1.41 [1.32,1.57] 1.29 [1.21,1.41] < 0.0001

P1 height, mmHg 61/122 26 [19,32] 26.5 [22,32] 0.45

AP, mmHg 57/122 5 [3,9.25] 9 [6,12] < 0.0001

AIx, - 57/122 0.18 [0.12,0.24] 0.26 [0.18,0.31] < 0.0001

Tr, ms 56/122 139 [132,145] 141 [134,146] 0.23

SPTI, mmHg·s 61/122 27.5 (5.19 SD) 32.5 (4.07 SD) < 0.0001

DPTI, mmHg·s 61/122 46.3 [38.4,54.5] 54.8 [47.7,63.3] < 0.0001

∆PSA, mmHg·s 61/122 5.77 [3.96,7.08] 6.44 [5.42,7.81] 0.0004

Ew , mmHg·s 61/122 0.401 [0.157,0.882] 1.05 [0.671,1.7] < 0.0001

SPTI·HR, mmHg/60 61/122 1915 [1647,2275] 2039 [1849,2295] 0.06

DPTI·HR, mmHg/60 61/122 3406 (516 SD) 3481 (418 SD) 0.29

∆PSA·HR, mmHg/60 61/122 395 [286,476] 401 [351,496] 0.12

Ew ·HR, mmHg/60 61/122 25.4 [13.4,59.6] 66.2 [42.8,103] < 0.0001

DPTI/SPTI, - 61/122 1.72 [1.51,1.98] 1.69 [1.51,1.89] 0.53

Ew /∆PSA, - 61/122 0.0721 [0.035,0.141] 0.166 [0.089,0.238] < 0.0001

Table 4.3: PWA parameters derived by the SphygmoCor system and pressure time indices.

P-values indicate results of a group-wise comparison.

was very distinct for both measurement techniques. Moreover, the LV ejection pattern showed,

on average, a later maximum for patients with impaired ventricular function, compare figure 4.3.

Invasively measured aortic systolic blood pressure (SBP) was higher in controls than in the rEF

group, while diastolic blood pressure (DBP) was comparable and pulse pressure (PP) therefore

increased. For aortic PWV, no statistical significant difference was observed.

PWA parameters obtained with the SphygmoCor system are presented in table 4.3. Temporal

characteristics showed significant differences between the groups with a higher HR and a shorter

ED as well as LVETI, i.e. ED indexed to HR, in rEF patients vs. controls. Estimated central

blood pressures were similar in both groups, with a slightly lower systolic and pulse pressure in

patients with rEF, compare also figure 4.3. Both augmented pressure (AP) and AIx were signif-

icantly lower in the rEF group, while the height of incident pressure P1 as well as the round-trip

travel time Tr were very similar.

SPTI, diastolic pressure time index (DPTI), ∆PSA and Ew were all less in the rEF group than

in controls, whereas when multiplied by HR to account for the shorter heart beat duration in

rEF, only for Ew a difference remained. Also the ratio of Ew to ∆PSA was significantly lower

in rEF, whereas the myocardial viability ratio was comparable, see table 4.3.

Chapter 4. Wave reflections in systolic heart failure 77

Wave separation analysis

|Pf |, mmHg 61/122 23.3 [18.4,28.4] 23.8 [19.1,28.6] 0.63

|Pb |, mmHg 61/122 13.3 [9.1,17.2] 15 [12.4,17.9] 0.02

RM, - 61/122 0.564 (0.115 SD) 0.625 (0.0984 SD) 0.0002

RI, - 61/122 0.357 [0.323,0.388] 0.386 [0.358,0.413] 0.0002

Wave intensity analysis

S energy, mmHg·cm 61/122 3.24 [2.25,4.39] 4.58 [3.71,5.76] < 0.0001

D energy, mmHg·cm 61/122 1 [0.723,1.3] 1.02 [0.783,1.35] 0.28

R energy, mmHg·cm 61/122 0.568 [0.339,0.862] 0.978 [0.699,1.24] < 0.0001

R/S energy, - 61/122 0.183 [0.134,0.236] 0.201 [0.15,0.262] 0.17

S/D energy, - 61/122 3.2 [2.56,4.31] 4.64 [3.89,5.24] < 0.0001

S peak, mmHg·cm/s 61/122 49.3 [31.9,71.8] 81 [59.6,113] < 0.0001

D peak, mmHg·cm/s 61/122 19.3 [13.8,24.7] 16.2 [11.8,21.4] 0.07

R peak, mmHg·cm/s 61/122 6.23 [4.44,8.98] 9.91 [7.22,13.7] 0.0001

R/S peaks, - 61/122 0.137 [0.0938,0.191] 0.118 [0.0864,0.155] 0.03

S/D peaks, - 61/122 2.63 [1.73,3.59] 5.08 [4.11,6.65] < 0.0001

Table 4.4: WSA and WIA parameters. P-values indicate results of a group-wise comparison.

Table 4.4 finally depicts the results obtained by WSA and WIA. Similar to the behaviour observed

for PWA, the forward pressure amplitude |Pf | was comparable whereas backward amplitude and

therefore also RM and RI were lower for patients with reduced EF. The WIA parameters, in

contrast, showed a significant reduction not only in the intensity of the backward R but also of

the initial forward travelling S wave, both for the wave energy and the peak value. No signifi-

cant reduction of relative wave reflections assessed by WIA was found for rEF patients. On the

contrary, for amplitudes, R/S was even increased. Moreover, the S to D ratio was significantly

less in the low EF group than in controls.

To compare the different approaches for the quantification of wave reflections, Pearson’s cor-

relation coefficient r between the corresponding parameters was computed per group, as given

in table 4.5. The peaks of forward and backward wave intensities showed the weakest (linear)

association to the other methods in both groups, whereas all other parameters were highly corre-

lated with r > 0.6 each. The association of measures describing the backward wave was highest

between AP and |Pb | in both groups (0.88 and 0.82 in rEF and nEF respectively), whereas the

correlation of the WIA parameters to the other methodologies was markedly weaker, especially

in the control group, with values around 0.8 in rEF and 0.7 in nEF patients for the R wave

energy. Regarding the forward wave parameters, the same behaviour could be observed, with

r > 0.9 between P1 and |Pf | in both groups. For the relative measures of wave reflection, in

contrast, the largest r was found between R/S energy and RM in both groups.

In order to account for the possible confounding effect of the temporal characteristics on the

haemodynamic parameters, the different measures of wave reflection were finally adjusted for

HR, ED as well as HR in combination with ED, as presented in figure 4.4. HR alone did not

Chapter 4. Wave reflections in systolic heart failure 78

P1 height - 0.93 [0.90,0.95] 0.71 [0.61,0.79] 0.60 [0.48,0.71]

|Pf | 0.92 [0.87,0.95] - 0.64 [0.52,0.73] 0.53 [0.39,0.64]

S energy 0.85 [0.77,0.91] 0.82 [0.71,0.89] - 0.96 [0.94,0.97]

S peak 0.75 [0.61,0.84] 0.72 [0.58,0.83] 0.96 [0.93,0.98] -

AP |Pb | R energy R peak

AP - 0.82 [0.75,0.87] 0.68 [0.57,0.76] 0.45 [0.30,0.58]

|Pb | 0.88 [0.80,0.92] - 0.72 [0.62,0.80] 0.56 [0.42,0.67]

R energy 0.81 [0.70,0.88] 0.83 [0.73,0.89] - 0.87 [0.82,0.91]

R peak 0.64 [0.45,0.77] 0.68 [0.52,0.80] 0.92 [0.87,0.95] -

AIx RM R/S energy R/S peaks

AIx - 0.73 [0.63,0.80] 0.73 [0.64,0.80] 0.52 [0.37,0.64]

RM 0.69 [0.52,0.80] - 0.87 [0.82,0.91] 0.65 [0.54,0.75]

R/S energy 0.69 [0.53,0.81] 0.82 [0.71,0.89] - 0.87 [0.82,0.91]

R/S peaks 0.38 [0.13,0.58] 0.46 [0.24,0.64] 0.73 [0.59,0.83] -

Table 4.5: Correlations between PWA, WSA and WIA parameters. The first block represents

the respective measures of the initial forward wave, the second those of the reflected backward

wave and the third finally the relative indices of wave reflection. Values in the lower triangular

matrix pertain to patients with reduced EF whereas those in the upper part (highlighted in gray)

pertain to patients with normal EF.

suffice to explain the differences in pulse pressure amplification (PPAmp) and AIx, while after

adjustment for ED or HR in combination with ED, PPAmp, AIx and RM became comparable

between the groups. In contrast, the R/S energy, which did initially not differ between the

groups, became significantly higher in patients with reduced EF than in controls when adjusted

for ED.

The same adjustment was performed for the S to D ratio of both the peak values and the wave

energies, see figure 4.5. In contrast to the wave reflection parameters, adjustment for ED or ED

in combination with HR had no notable effect on the difference between patients with normal

and reduced EF. Accounting for HR alone slightly attenuated the gap between the patients,

however, the difference remained highly significant.

The correlations between measures of cardiac function and structure and parameters describing

the pulsatile haemodynamics in the arterial system are presented in table 4.6. In the rEF group,

ventricular contractility assessed by maximum LV dP/dt was positively associated with all in-

dices describing the forward and backward waves, a relation that could not be found in controls.

The association between EF and the temporal characteristics HR and ED was similar in both

groups. However, in the low EF group, higher values of EF were moreover related to a lower

PPAmp and a higher AIx, RM and R wave energy. In both groups, ED was associated to all

parameters of wave reflection (with the only exception of peak intensities in controls) but not

to aortic PWV. Nevertheless, for the low EF group, correlations were markedly higher than in

controls and ED was additionally related to all indices describing the initial forward wave. LV

Chapter 4. Wave reflections in systolic heart failure 79

raw data

HR

ED

HR, ED

ΔPPAmp ΔAIx ΔRM ΔR/S energy

Figure 4.4: Difference of the mean values of wave reflection parameters between patients with

reduced and normal EF for the unadjusted data and when adjusted for HR, ED as well as HR

in combination with ED. Values represent the regression coefficient of the EF-status in multiple

regression analysis and error bars indicate the corresponding 95% confidence interval.

raw data

HR

ED

HR, ED

−3 −2 −1 0 −3 −2 −1 0

∆S/D energy ∆S/D peaks

Figure 4.5: Difference of the mean values of the S to D ratio between patients with reduced and

normal EF for the unadjusted data and when adjusted for HR, ED as well as HR in combination

with ED. Values represent the regression coefficient of the EF-status in multiple regression

analysis and error bars indicate the corresponding 95% confidence interval.

Chapter 4. Wave reflections in systolic heart failure 80

volumes were highly negatively correlated with PWV in patients with reduced but not in patients

with normal EF. Moreover, although a negative association of LV volumes to most reflection

indices could be observed in both groups, only for patients with normal systolic function, this

association was also found for stroke volume (SV). In the control group, levels of NT-proBNP

were associated with most haemodynamic parameters as well as PWV, whereas in the rEF group,

only for the S/D ratio, S peak and PWV a significant association could be found.

4.5 Discussion

In this work, differences in clinical and pulse wave characteristics as well as their interrelations

were examined in patients with systolic heart failure versus controls. The enlargement of the

left ventricle reflected in the increased values of LV mass and filling volumes, and the impaired

contractility assessed by LV dP/dt observed in the rEF group are characteristic for patients with

a systolic dysfunction [5]. The elevated enddiastolic LV filling pressure may additionally indicate

a decrease in LV compliance and thus an impairement of LV relaxation (diastolic dysfunction).

Alternatively, it may be ascribed to the elevated filling volume caused by the impaired emptying

[63].

The previously observed reduction of ED in patients with systolic HF [21, 68, 88, 125, 137, 141]

could again be confirmed in this work. Furthermore, despite the higher prevalence of betablocker

use, patients with rEF had a significantly elevated HR. This elevation is in keeping with the

older invasive studies [52, 66], whereas in the more recent works, HR was consistently higher, too,

yet mostly without reaching statistical significance [19, 125, 137]. The reduction in LVETI, i.e.

ED indexed to HR, moreover implies that ED was reduced beyond the effect of HR alone, in line

with earlier results [21, 68]. The alterations in the temporal characteristics might be explained

by the inability of the failing ventricle to overcome late systolic load, resulting in a premature

stop of ejection (shorter ED) and an increase in HR to compensate the consequently lower stroke

volume [146].

To enable a direct comparison of all pressure-dependent parameters, the rEF and nEF patients

were matched for brachial blood pressure levels in this study. Nevertheless, estimated central

SBP tended to be lower in the rEF group and pulse pressure was significantly reduced, leading

to a higher PPAmp. For the invasively measured aortic blood pressure levels, this reduction was

even more pronounced. Weber et al. [137] previously reported an almost identical behaviour in a

very similar study population. Unfortunately, though, no brachial pressure levels were assessed

at the time of angiography for a direct comparison. A possible explanation for the differences in

estimated central PP can be sought in the elevated HR in rEF patients, since PPAmp assessed

with the SphygmoCor system has been found to increase linearly with HR [151]. However,

Denardo and coworkers [21] observed a higher PPAmp in rEF patients despite matching for

HR and HR alone could not completely explain the difference in PPAmp in multiple regression

Chapter 4. Wave reflections in systolic heart failure 81

Reduced EF

HR −0.71∗ −0.29∗ −0.09 0.00 −0.30∗ 0.01 0.28∗

ED 1.00∗ 0.27∗ 0.05 −0.04 0.25 0.15 −0.24

LVETI 0.65∗ 0.10 −0.04 −0.07 0.05 0.25 −0.08

PWV −0.09 0.08 −0.42∗ −0.36∗ −0.25 0.29∗ 0.36∗

PP central 0.55∗ 0.22 −0.25 −0.27∗ −0.00 0.46∗ −0.08

PPAmp −0.70∗ −0.40∗ 0.19 0.28∗ −0.22 −0.24 0.18

P1 height 0.35∗ 0.16 −0.23 −0.22 −0.06 0.41∗ −0.10

AP 0.63∗ 0.23 −0.22 −0.25 0.05 0.36∗ 0.03

Aix 0.62∗ 0.28∗ −0.17 −0.24 0.14 0.25 0.01

|Pf | 0.39∗ 0.03 −0.13 −0.10 −0.12 0.38∗ −0.08

|Pb | 0.49∗ 0.22 −0.34∗ −0.34∗ −0.07 0.49∗ −0.02

RM 0.30∗ 0.32∗ −0.38∗ −0.42∗ 0.04 0.30∗ 0.09

S energy 0.29∗ 0.18 −0.14 −0.18 0.08 0.42∗ −0.06

D energy 0.33∗ 0.03 0.09 0.08 0.05 0.25 −0.19

R energy 0.50∗ 0.31∗ −0.26∗ −0.32∗ 0.11 0.47∗ −0.03

R/S energy 0.48∗ 0.24 −0.22 −0.25 0.05 0.26∗ 0.03

S/D energy 0.01 0.19 −0.24 −0.28∗ 0.07 0.15 0.27∗

S peak 0.20 0.15 −0.18 −0.21 0.05 0.40∗ 0.01

D peak 0.23 0.07 −0.05 −0.04 −0.02 0.26 −0.27∗

R peak 0.43∗ 0.23 −0.19 −0.24 0.11 0.35∗ −0.03

R/S peaks 0.35∗ 0.02 0.01 0.01 0.00 −0.06 0.03

S/D peaks 0.08 0.13 −0.18 −0.21 0.04 0.15 0.28∗

Normal EF

HR −0.67∗ −0.22∗ −0.21∗ 0.07 −0.24∗ −0.14 −0.06

ED 1.00∗ 0.21∗ 0.08 −0.15 0.12 0.10 0.02

LVETI 0.74∗ 0.07 −0.07 −0.13 −0.05 0.03 −0.03

PWV −0.02 0.03 −0.19 −0.16 −0.12 0.16 0.45∗

PP central 0.28∗ 0.03 0.02 −0.03 −0.04 0.13 0.32∗

PPAmp −0.51∗ −0.07 0.17 0.16 0.14 −0.12 −0.24∗

P1 height 0.14 0.01 0.12 0.04 0.04 0.14 0.26∗

AP 0.36∗ 0.03 −0.14 −0.11 −0.17 0.06 0.29∗

Aix 0.34∗ 0.03 −0.21∗ −0.13 −0.19∗ 0.05 0.21∗

|Pf | 0.15 0.02 0.11 0.04 0.02 0.08 0.31∗

|Pb | 0.28∗ 0.05 −0.04 −0.09 −0.07 0.13 0.30∗

RM 0.27∗ 0.06 −0.25∗ −0.21∗ −0.16 0.13 0.05

S energy 0.01 0.01 0.11 0.05 0.02 0.12 0.19∗

D energy 0.09 0.02 0.08 0.00 −0.01 0.01 0.31∗

R energy 0.28∗ 0.09 −0.18 −0.21∗ −0.14 0.17 0.25∗

R/S energy 0.27∗ 0.04 −0.27∗ −0.20∗ −0.19∗ 0.07 0.08

S/D energy −0.10 0.01 0.07 0.04 0.08 0.17 −0.25∗

S peak −0.11 −0.08 0.04 0.09 −0.04 0.13 0.19∗

D peak −0.10 0.01 0.03 0.02 −0.05 −0.03 0.33∗

R peak 0.09 0.09 −0.16 −0.20∗ −0.12 0.22∗ 0.18

R/S peaks 0.18 0.09 −0.23∗ −0.22∗ −0.14 0.11 0.00

S/D peaks 0.00 −0.06 0.05 0.07 0.05 0.18 −0.18∗

Table 4.6: Correlations between measures of cardiac function and structure and haemodynamic

parameters. Values obtained by angiography were used for EF, LVEDV, LVESV, SV and LV

dP/dt. Moderate to high correlations with |r| ≥ 0.3 are highlighted in gray, * indicates Pearson’s

r being significantly different from 0 at a 5% level.

Chapter 4. Wave reflections in systolic heart failure 82

analysis in the present work, see figure 4.4. Hence, the difference in central PP is only partially

attributable to HR. Adjustment for ED or both ED and HR on the other hand, fully balanced

PPAmp between the groups.

In line with the lower central pressure amplitude, the absolute parameters quantifying the wave

reflections present in the system assessed by PWA and WSA, namely AP and |Pb |, were reduced

in patients with rEF versus controls. In contrast, measures of the incident forward wave did not

differ between rEF and nEF patients. The resulting lower values of AIx and RM are consistent

with previous studies [21, 51, 68, 137]. Of note, numerical values of RM reported by Laskey

and coworkers [51], who worked with a very similar study population regarding age and pressure

levels but used invasive measurements of pressure and flow, were almost identical to the values

found in this work, namely 0.61 (0.14 SD) and 0.53 (0.14 SD) for normal and reduced EF re-

spectively.

The behaviour of the WIA parameters differed from PWA and WSA in that, besides the reflected

wave energy, also the energy of the initial forward compression wave, i.e. the S wave, was lower

in rEF. The S wave originates from ventricular ejection and its energy has been found to be

positively associated to the contractile state of the left ventricle [27, 44, 120]. The reduction of

the S wave energy for patients with impaired systolic ventricular function found in this as well as

previous works [19, 120] agrees well with this concept. Also in keeping with literature [19, 120],

no difference in the forward decompression wave, the D wave, was observed between the groups.

For WIA, the incremental changes in pressure and flow are used, making the computations sus-

ceptible to noise, in particular when using the peak values for quantification. This might explain

the rather low correlations of the peak intensities with the other methods, whereas the wave

energies, which are more robust because the whole wave and not only a single point is used,

showed a stronger association to the PWA and the WSA parameters. PWA and WSA are both

based on the absolute values of pressure, which might explain why the respective forward and

backward indices were more closely related to each other than to WIA. Nevertheless, for the

relative measures of wave reflections, correlation between the R/S energy and RM was higher

than between AIx and RM in both groups.

The present results show that the relation between measures of cardiac and arterial function

differs between patients with normal and impaired LV systolic function. In the first case, aortic

root flow at rest looks very similar between individuals, compare e.g. figure 5.2 in the next

chapter, which means that the LV ejection pattern is relatively independent of the opposed af-

terload. Therefore, differences in the shape of the flow waveform are more or less negligible for

the interpretation of the haemodynamic parameters, and characteristics of the pressure wave-

form can be used to derive information about arterial function only. Changes in the contractile

behaviour of the ventricle leading to alterations in the flow waveform, however, directly affect the

pressure wave and therefore the derived parameters, compare [46]. This might explain why EF,

Chapter 4. Wave reflections in systolic heart failure 83

showed no association to pulsatile parameters in subjects with normal LV function, whereas a

strong correlation was found in patients with LV systolic dysfunction. Thus, while for example

a low pulse pressure might be an indicator of healthy, elastic arteries for normal hearts, it rather

reflects the inability of the ventricle to pump against the afterload for the failing heart, without

revealing much about the status of the arteries. The observed relation between a shortening of

the ejection duration, as a manifestation of ventricular failure, and an attenuation of the forward

pressure indices in the rEF patients is well in line with this concept too. For nEF, in contrast,

no relation of ED to the forward indices was found, in keeping with findings by Namasivayam

and coworkers [74]. Westerhof and O’Rourke described this phenomenon in terms of the me-

chanical pump function of the heart [146]. They argued that the normal heart acts almost like

a flow source, being able to eject against the load presented by the arterial system, whereas the

failing heart behaves more like a pressure source. Therefore, while wave reflections returning

during systole augment pressure but have only little effect on the flow wave in the first case,

they diminish flow and thus affect pressure to a lesser extent in the second case. The idea that

wave reflections, although equally strong, manifest themselves less in the pressure wave when

the ventricle is failing, might explain the lower reflection indices in patients with rEF observed

in this study, despite the similar aortic stiffness assessed by PWV between the groups.

Aortic PWV was statistically comparable between the groups in this as well as previous works

[68, 137]. Nevertheless, absolute values tended to be higher for patients with low EF. Because

of the interplay between elastin and collagen fibers in the arterial media, arteries become stiffer

when they dilate [82], meaning in particular that PWV increases with increasing blood pressure

[126]. Taking into account that invasively measured aortic blood pressure at the time of PWV

measurement was significantly lower in rEF patients, this could indicate that aortic stiffness

might even be elevated in rEF patients. However, this notion has to be investigated in future

studies.

AIx depends on both the timing and the magnitude of the reflected waves and is known to be

strongly influenced by temporal characteristics, which was suggested to be mainly due to alter-

ations in ED [117, 150]. This could explain the differences observed between the rEF and nEF

group. Similar to the behaviour of PPAmp, adjustment for ED or ED in combination with HR

indeed resulted in comparable AIx between the groups. The same held for RM, whereas the

R/S energy became even higher in rEF, indicating that wave reflections may at least be equally

strong in both groups.

Results regarding the pressure time indices found in previous studies [21, 88] could not all be

confirmed in the present work: Ew and SPTI were indeed lower in rEF versus controls, but the

myocardial viability ratio, which was reported to be increased to optimise LV efficiency [21], was

very similar between the groups.

Chapter 4. Wave reflections in systolic heart failure 84

Another interesting parameter, not directly related to wave reflections, is the S to D ratio of

forward wave intensity, which was first analysed by Hametner et al. [38] as a marker of systolic

heart failure in the same population. The S and D wave are supposed to describe the ventricular

ejection and relaxation dynamics respectively. Their ratio could thus be seen an index char-

acterising the ventricular dynamics over the whole mechanical systole. For both the energies

and the peak values, a significant and very pronounced reduction in this index was found for

patients with impaired ventricular function, which might qualify the parameter as a non-invasive

indicator of systolic function as proposed in [38]. Even though the reduction was very distinct

in patients with low compared to normal EF, no association to EF was found when looking at

the groups separately, compare table 4.6. In other words, the S to D ratio seems to depend on

whether EF is normal or reduced as a categorical variable, but not on how low or how high the

EF really is. Moreover, in contrast to the wave reflection parameters, it seems to be independent

of ED, both within the groups as shown in table 4.6 as well as between the groups, since the

difference could not be explained by ED, compare figure 4.5. The same holds for the heart rate.

Furthermore, the S to D ratio was positively related to NT-proBNP for reduced EF, while the

association was negative for controls. However, the mechanisms behind this behaviour still have

to clarified.

The main strength of this study is that patients were matched for brachial blood pressure,

thereby enabling the direct comparison of all pressure-dependent parameters. Moreover, a vari-

ety of measurements was available including clinical and haemodynamic parameters, which made

it possible to analyse the relationships between cardiac and arterial function and the respective

differences arising from ventricular impairment. However, also several limitations have to con-

sidered. The number of women included in the study was very low and gender differences could

therefore neither be analysed nor accounted for. Previous studies in systolic heart failure showed

a similar, unbalanced gender distribution [21, 125] (between 80% and 97% male), which might

be due to the unequal prevalence of LVSD in men and women, which was found to be 2.5 times

higher in men [70]. Another limitation is that measurements were partly performed on different

days, possibly limiting the comparability between the parameters. Moreover, aortic pressure

waveforms were synthesised by the means of a generalised transfer function and flow waveforms

were manually digitised, an assessment method that is prone to a certain degree of subjectivity.

Conclusion

In conclusion, the current study showed that haemodynamic parameters are susceptible to cardiac

function in patients with ventricular impairment. This has to be kept in mind for interpreta-

tion and risk stratification. In particular, parameters of wave reflections might not be suitable

indicators of arterial stiffness in patients with severely reduced ejection fraction. However, these

intrinsic differences in non-invasively assessed pulsatile haemodynamics might potentially be used

as indicator of a systolic dysfunction in the future.

Chapter 5

pressure alone in patients with

systolic heart failure: state of the

art and data analysis

In this chapter, the use of blood flow models in patients with systolic heart failure is investigated

with respect to their applicability in wave separation analysis (WSA) and wave intensity analysis

(WIA). While the last chapter dealt with differences in the parameters derived from measured

pressure and flow between patients with systolic heart failure and controls, this chapter focuses

on the ability of different flow models to provide accurate estimates and, probably even more

important, to reproduce these differences. For this purpose, first, again a summary of the rel-

evant literature is given. Then, the same study population as in the previous chapter is used

to evaluate and to compare the performance of the four blood flow models presented in chapter 3.

Parts of this chapter are based on the work ”Non-invasive wave reflection quantification in pa-

tients with reduced ejection fraction” published in ”Physiological Measurements” in 2015 [95].

The previously reported results were extended by a comparison with the Windkessel flow intro-

duced in section 3.4.

In 2006, Westerhof and coworkers [142] introduced their triangulation method for flow approxi-

mation and compared their estimates with invasive flow measurements collected from previously

published works. They found a relatively good agreement of reflection magnitude (RM) and

reflection index (RI) derived by their flow model with the ones obtained with measured flow and

85

Chapter 5. Flow models in systolic heart failure 86

concluded that, with their approach, it might be feasible to perform WSA based on non-invasively

assessed pressure alone. Thereafter, several investigators applied their method for wave sepa-

ration in both cross-sectional [56, 74, 75] as well as longitudinal studies [40, 121, 133]: Lieber

and colleagues [56], for example, used it to investigate gender-related differences in aortic wave

reflections, while Namasivayam and coworkers [75] examined the contribution of the forward and

backward wave to the increase in aortic pulse pressure with age. Wang et al. [133] studied the

WSA parameters derived with the triangulation method with regards to their prognostic value

in a large, community-based population comprising more than 1000 participants. They found

the backward wave amplitude |Pb | to be a strong predictor of 15-year cardiovascular mortality,

independent of brachial pressure levels, conventional risk factors and arterial stiffness. Qasem

and Avolio [103] moreover proposed a new approach for the one-point estimation of pulse transit

time and pulse wave velocity from pressure alone based on wave separation using the triangu-

lation method. Last but not least, AtCor Medical incorporated a WSA algorithm based on the

triangulation method in their SphygmoCor software, which has already been used in a variety of

studies, e.g. [22, 55].

However, in their original work, Westerhof and coworkers used a very small set of data to proof

their concept, consisting of measurements of only 19 different individuals. In 2009, Kips et

al. [49] applied the triangulation method to carotid pressure waveforms of 2325 subjects of the

Asklepios cohort and compared it to Doppler flow measurements of the aortic root flow. For

the construction of the triangle, they used timing information that was directly derived from the

measured flow wave to investigate the best possible approximation. Nevertheless, they found

only a moderate agreement for Pb , Pf and RM. With their averaged waveform, in contrast,

agreement could be markedly improved. Furthermore, RM derived with the averaged waveform

was shown to be independently associated to cardiovascular events as well as new onset heart

failure in 5960 participants of the MESA study with an average follow up duration of 7.6 years

[17].

In 2013, Hametner and coworkers published another comparison of the triangular approxima-

tion and the averaged waveform in 148 patients with preserved systolic function [35, 37]. Using

Doppler measurements of aortic blood flow as reference, they again found the averaged waveform

to be superior to the triangle for wave separation. Additionally, they investigated the perfor-

mance of the ARCSolver flow for WSA and found a similar agreement to Doppler flow as for

the averaged waveform. In a longitudinal study performed by Weber et al. [140], |Pb | derived

using the ARCSolver flow was furthermore shown to be predictive of cardiovascular events, in-

dependently of brachial pressure levels, and to be associated to end-organ damage in a high-risk

population of 725 patients.

To summarise, for subjects with normal left ventricular (LV) systolic function, the different blood

flow models have already been (successfully) used for WSA in a hand-full of studies. However,

the question remains if these results can be transfered to patients with systolic heart failure,

who often show modified ejection patterns [77]. The only relevant study in this regard might

Chapter 5. Flow models in systolic heart failure 87

be the one performed by Sung and colleagues, who used the triangular flow approximation to

determine forward and backward amplitude as well as RM in a cohort hospitalised for acute

heart failure [122]. They found the value of |Pb | on admission to be a strong predictor of adverse

outcome in both patients with heart failure with preserved (n=48) and reduced ejection fraction

(n=72). However, a thorough search of the relevant literature yielded no results with respect to

a direct comparison of modelled to measured flow waves or the derived parameters in patients

with systolic heart failure.

Regarding the use of blood flow models for WIA, data from literature was restricted to the inves-

tigations performed at the Austrian Institute of Technology using the ARCSolver flow [32, 36, 38].

These preliminary results indicated a good agreement between the forward wave energy obtained

by Doppler and modelled flow in patients with normal systolic function [36], as well as a similar

reduction in the S to D ratio of the respective peak values for patients with systolic heart failure

compared to controls [38].

Overall, while the available data on the applicability of the three different blood flow models for

wave separation is already limited for patients with normal ventricular function, it is extremely

scarce for patients with systolic heart failure. Even though Sung and co-workers were indeed able

to demonstrate the prognostic value of a WSA parameter obtained with the triangular flow in

heart failure patients, the accuracy of the flow estimate itself as well as of the derived parameters

has not yet been investigated in patients with reduced ejection fraction (EF). In their initial

study population, Westerhof and colleagues included two patients with heart failure, but this

sample is too small to draw any conclusions and results were not reported separately. Also for the

other two approaches, an evaluation of their performance in heart failure patients is still missing.

For WIA, apparently no data exists about the feasibility of using the triangular or averaged

blood flow for computation, regardless of ventricular function. However, first results obtained

with the ARCSolver flow imply that it might be feasible to replace the flow measurements also

for WIA.

The aim of this study is to investigate and to compare the performance of the different blood

flow models introduced in chapter 3 in patients with systolic heart failure and controls. More

specifically, the agreement of the waveforms themselves as well as of the derived WSA and

WIA parameters to the corresponding values obtained from Doppler flow measurements will be

examined.

Chapter 5. Flow models in systolic heart failure 88

5.3 Methods

The same study population as in the previous chapter was used and the reader is therefore re-

ferred to the corresponding methods section 4.3 for a thorough description of the population as

well as of the measurements performed.

The 4 different blood flow models included in this analysis were already presented in detail in

chapter 3. Therefore, only a few comments on the numerical realisation will be given in the

following. The duration of cardiac ejection ts was estimated from the peripheral pressure wave

using an algorithm developed by the cardiovascular diagnostics research group at the AIT and

was used for all flow models to ensure comparability. The width of the averaged waveform de-

rived by Kips et al. [49] was scaled to ts by linear interpolation. For the triangular estimate,

the base of the triangle was set to ts and its apex to the point in time corresponding to the

inflection point in the pressure signal1 determined by the SphygmoCor system. Moreover, all

flow estimates were normalised to 100 arbitrary units (AU). The term ”combWK flow” will be

used throughout this section to refer to the combination of the Windkessel flow model and the

ARCSolver flow, as presented in section 3.4.5.

To evaluate the performance of the different blood flow models, the digitised Doppler flow mea-

surements scaled to 100 AU (compare again section 4.3) served as reference. All parameters were

summarised as median [95% central range], independently of their actual distribution, to facili-

tate the comparison between the models. Moreover, the Mann-Whitney-U test (non-parametric)

was used to statistically assess between-group differences for all parameters. The root mean

square error (RMSE) between measured and estimated flow during systole was computed to

compare the shape of the waveforms, whereas the derived parameters were assessed quantita-

tively by calculating the median (accuracy) and 95% central range (precision) of the difference

to the reference values, as well as Pearson’s correlation coefficient. Results obtained with the

ARCSolver and combWK flow were furthermore compared graphically to the reference method

by Bland-Altman plots using mean and standard deviation of the differences to determine the

limits of agreement.

In contrast to the previous chapter, also for WIA the flow waveform was normalised to 100

AU. Therefore, all absolute parameters, i.e. the peaks and time-integrals of wave intensity, are

wrongly scaled. This has to be kept in mind for quantitative comparisons. Relative measures

like ratios are of course unaffected by the scaling of blood flow.

1 This point was identified from the augmentation index (AIx).

Chapter 5. Flow models in systolic heart failure 89

Figure 5.1: Examples of Doppler flow waves (black solid line) and the corresponding triangular

(gray) and averaged (teal) estimates (upper panel) as well as the ARCSolver (red) and combWK

(blue) flow (lower panel) for patients with reduced EF. The flow waves are arranged according

to the timing of maximum flow relative to the respective ejection duration. All flow waves were

scaled to the same peak value and the x-axis depicts 0.4 seconds.

5.4 Results

Flow waveform

Examples of the flow wave estimates obtained by the different models and the corresponding

Doppler-based flow waves are shown in figure 5.1 for patients with reduced EF (rEF) and in

figure 5.2 for patients with normal EF (nEF). Also cases where combWK and ARCSolver flow

coincide are shown, which occurred in 20 patients (33%) with reduced and 107 (88%) patients

with normal EF. In the rEF group, a much greater variation in shape can be observed for the

measured flow waves as compared to controls, in particular with respect to the timing of peak

flow. On average, maximum flow is reached significantly later relative to the ejection duration

for patients with ventricular impairment, as presented in table 5.1. The same behaviour could be

observed for the triangular, ARCSolver and combWK flow, but not for the averaged waveform,

which, by definition, does not vary in shape. In both groups, the highest correlation between the

position of modelled and measured maximum flow was found for the ARCSolver, closely followed

by the combWK flow. The latter method furthermore reached the lowest median RMSE between

Doppler and modelled waveform for both normal and reduced EF, which was almost identical in

both groups.

Table 5.1 summarises the results obtained by WSA with the different flow estimates in compar-

ison to the reference method. For the amplitude of the forward travelling pressure wave |Pf |,

correlation was high with r ≥ 0.9 for all flow models and all but the triangular estimate even

Chapter 5. Flow models in systolic heart failure 90

Figure 5.2: Examples of Doppler flow waves (black solid line) and the corresponding triangular

(gray) and averaged (teal) estimates (upper panel) as well as the ARCSolver (red) and combWK

(blue) flow (lower panel) for controls. For most of the controls, the combWK flow coincides with

the ARCSolver flow. All flow waves were scaled to the same peak value and the x-axis depicts

0.4 seconds.

attained r ≥ 0.95. However, both the triangle and the averaged waveform yielded significantly

lower values of |Pf | for patients with reduced EF versus controls, while the reference values were

comparable. For the backward amplitude, correlations were even higher than for the forward

amplitude for the averaged, ARCSolver and combWK flow in both groups. Moreover, all models

were able to reproduce the reduction in |Pb | for rEF compared to controls, although it did not

reach statistical significance for the averaged flow. With regards to RM, correlation was high-

est for the combWK flow independent of the EF-status. Even though RM obtained with the

ARCSolver estimate and averaged waveform yielded very similar correlations with the reference

method, using the averaged flow for computation resulted in no difference in RM between pa-

tients with rEF and nEF, in contrast to the behaviour of the Doppler measurements and the

other flow models.

For both the forward and backward amplitudes, the widest scatter in the deviation from the

reference values was found for the triangular approximation in patients with reduced as well

as normal ejection fraction, whereby estimates were even less precise in the latter group. The

averaged waveform provided the most precise and most accurate estimate of |Pf | for patients

with normal EF. In the rEF group, accuracy was highest for the combWK flow, while precision

was similar for the combWK and ARCSolver method, with slightly better results obtained by

the latter as also shown in figure 5.3. Looking at the whole study population, precision was very

similar for both methods, but accuracy was again better for the combWK flow. With regards to

|Pb |, precision was highest for the combWK flow in both groups, followed by the ARCSolver flow.

A direct comparison of these two methods for patients with rEF as well as for the whole study

population is shown in figure 5.4. In line with the ordering implied by the correlation coefficients,

Chapter 5. Flow models in systolic heart failure 91

the combWK method yielded the most accurate and most precise values of RM, closely followed

by the ARCSolver flow, see figure 5.5. Also for the whole study population, a closer agreement

could be observed for the combWK than for the ARCSolver flow, as shown again in figure 5.5.

Furthermore, the Bland-Altman plots revealed no systematic trends for neither the ARCSolver

nor the combWK method for any WSA parameter.

The WIA parameters derived with the different flow estimates are depicted in tables 5.2 and

5.3 for the wave energies and peak intensities respectively. Again it should be noted that all

absolute values are wrongly scaled because of the normalisation of blood flow velocity. Similar to

the findings for |Pf |, correlation of the S wave energy obtained with the different flow models to

the reference value was very high in both groups, i.e. r ≥ 0.94. Peak values showed a lower, but

still high correlation with r ≥ 0.83 for all flow models and r ≥ 0.90 for ARCSolver and combWK

flow. However, the reduction in S wave energy observed for patients with impaired ventricular

function when using the Doppler flow measurements was not captured well by the models. For

the peak values on the other hand, the combWK flow was able to imitate this behaviour. Nev-

ertheless, the deviation from the reference value differed significantly between the groups, with

an almost three times higher median deviation for rEF compared to nEF patients for all but

the triangular flow. A comparison of the S wave energy obtained by ARCSolver and combWK

method to Doppler flow is shown in figure 5.6.

The R wave energy was consistently lower in the rEF than in the nEF group when using the

Doppler, triangular, ARCSolver or combWK flow. This reduction was significant for all but the

ARCSolver method. Using the averaged waveform, however, resulted in the opposite behaviour.

Correlations between modelled and measured R wave energy were generally higher for patients

with reduced than with normal EF, with the strongest association found for the combWK flow

in the rEF group, while the correlation was comparable for averaged, ARCSolver and combWK

flow in controls. Also the 95% central interval of the difference to the reference was by far nar-

rowest for the combWK flow in the rEF group, and almost identical to the corresponding 95%

central interval found in controls. In comparison, the range was almost twice as large in rEF for

the other methods, yet similar to the combWK for the averaged and ARCSolver flow in the nEF

group, compare figure 5.7.

Regarding the peak values of the R wave, no statistical evidence for a difference between patients

with normal and reduced EF could be found for Doppler, triangular or combWK flow, whereas

both the ARCSolver and averaged waveform resulted in significantly greater values for controls.

Accuracy was worst for the averaged waveform, while precision was best for the combWK flow

in both groups. Correlations were throughout lower than for the wave energies.

The relative magnitude of wave reflections assessed by the R/S energy was similar between the

groups for the ARCSolver and combWK method, in line with the results obtained by Doppler

Chapter 5. Flow models in systolic heart failure 92

flow. The median error found for the combWK method was furthermore identical between the

groups, as was the observed scatter. Compared to the ARCSolver method, accuracy and preci-

sion could therefore be slightly improved for both the rEF group alone as well as for the whole

study population, see figure 5.8. Also the correlation was highest for the combWK method in

both groups. The lowest correlation and largest deviation from the reference in terms of both

accuracy and precision was found for the triangle, with r = 0.66 in the reduced EF and r = 0.20

in the normal EF group. For the peak values, correlations were again lower.

The increase of the energy of the second forward wave, i.e. the D wave, in the rEF group

found using Doppler measurements2 could also be observed for the triangular, ARCSolver and

combWK flow but not for the averaged waveform, despite showing the highest correlation and

highest precision to the reference method in both groups. The same held true for the peak in-

tensities. Bland-Altman plots for the ARCSolver and combWK flow are depicted in figure 5.9.

For the ratio of the S to D energy, finally, correlations were markedly lower than for the other

WIA parameters, with values in the range of 0.2 for the triangle, 0.5-0.7 for the ARCSolver and

combWK flow and 0.7 for the averaged waveform. The averaged waveform tended to overesti-

mate the ratio for rEF and underestimate it for nEF, resulting in virtually equal values in both

groups, contrary to the reduction in rEF observed for the Doppler, triangular, ARCSolver and

combWK flow. However, even though the triangulation method could reproduce this difference

with respect to the median, the obtained values scattered excessively around the reference. In the

low EF group, using the combWK flow could improve the precision compared to the ARCSolver

flow, which resulted in a slight improvement over the whole study population, see figure 5.10.

Looking at the peak values, correlations were doubled for the triangle but further attenuated for

all other methods with r ≈ 0.4 for the averaged waveform and r ≈ 0.3 for the ARCSolver flow

in both groups. The combWK flow resulted in the highest correlation (r = 0.57) and highest

precision for patients with reduced EF. However, by overestimating the S to D ratio for rEF and

underestimating it for nEF, the very distinct difference between the groups found for the refer-

ence values was strongly diminished. This effect could be observed for all flow models, although

in varying degrees.

Overall, for most of the WIA parameters, estimates of the peak values obtained with the different

flow models were markedly less precise than estimates of the wave energies.

5.5 Discussion

With the increasing popularity of wave separation and wave intensity analysis, blood flow mod-

els are more and more finding their way into clinical research, facilitating the quantification of

wave reflections from pressure alone [17, 133, 140]. Different commercially available pulse wave

2 The discrepancy to the previous chapter, where no difference in D wave energy or peak intensity was observed

between patients with rEF and nEF, is due to the fact that flow was scaled to 100 AU in the present analysis.

Doppler Triangular Averaged ARCSolver combWK

Reduced EF

RMSE flow, AU - 11.9 [7.51,24.1]* 16.7 [9.67,26.2]* 9.69 [4.32,20.5] 7.06 [3.74,15.3]

tmax/ED, - 0.387 [0.259,0.533]* 0.42 [0.306,0.632]* 0.274 [0.247,0.301] 0.343 [0.274,0.434]* 0.368 [0.274,0.474]*

∆ - 0.0465 [-0.0985,0.215] -0.112 [-0.253,0]* -0.046 [-0.168,0.0634]* 0 [-0.147,0.0928]

r - 0.48 [0.27,0.66] -0.07 [-0.32,0.19] 0.63 [0.45,0.76] 0.60 [0.40,0.74]

|Pf |, mmHg 23.3 [12.3,45.5] 26.1 [13.7,45.8]* 20.6 [10.9,40.6]* 22.1 [11.5,41.9] 22.8 [12.1,41.3]

∆ - 1.15 [-4.33,8.64]* -2.24 [-7.43,1.59]* -1.32 [-5.68,2.32]* -0.377 [-5.67,2.69]*

r - 0.93 [0.89,0.96] 0.96 [0.94,0.98] 0.96 [0.94,0.98] 0.96 [0.94,0.98]

|Pb |, mmHg 13.3 [5.89,25.7]* 12.6 [5.33,31.1]* 15 [7.54,29.7] 11.7 [5.27,24.5]* 12.2 [5.59,25.1]*

∆ - -0.506 [-2.87,5.55]* 2.3 [-0.0165,5.08] -1.02 [-3.42,1.5] -0.568 [-3,1.28]

r - 0.95 [0.92,0.97] 0.98 [0.96,0.99] 0.98 [0.97,0.99] 0.99 [0.98,0.99]

RM, - 0.555 [0.356,0.815]* 0.499 [0.337,0.711]* 0.731 [0.597,0.855] 0.535 [0.324,0.711]* 0.545 [0.301,0.788]*

∆ - -0.0403 [-0.171,0.0616]* 0.172 [-0.0323,0.284]* -0.0216 [-0.111,0.0725] -0.0176 [-0.0984,0.0856]

r - 0.76 [0.63,0.85] 0.82 [0.71,0.89] 0.83 [0.74,0.90] 0.91 [0.85,0.94]

Normal EF

Chapter 5. Flow models in systolic heart failure

tmax/ED 0.272 [0.216,0.368]* 0.344 [0.298,0.589]* 0.273 [0.258,0.296] 0.289 [0.247,0.347]* 0.29 [0.247,0.37]*

∆ - 0.0752 [0,0.316] 0 [-0.0926,0.0698]* 0 [-0.0549,0.0722]* 0 [-0.0463,0.0858]

r - 0.25 [0.08,0.41] 0.00 [-0.17,0.18] 0.48 [0.33,0.61] 0.45 [0.30,0.58]

|Pf |, mmHg 23.8 [13.8,39.2] 28.4 [16.4,52.1]* 23.5 [13.6,39]* 23.2 [14.1,38.5] 23.5 [14.1,38.5]

∆ - 4.85 [-1.13,14.8]* -0.0922 [-4.7,2.12]* 0.215 [-5.02,3.02]* 0.392 [-4.59,3.7]*

r - 0.90 [0.86,0.93] 0.97 [0.96,0.98] 0.96 [0.94,0.97] 0.96 [0.94,0.97]

|Pb |, mmHg 15 [7.69,25.2]* 15.9 [7.57,27.5]* 17.2 [9.41,27.8] 14.1 [7.2,25.6]* 14.1 [7.2,25.6]*

∆ - 0.164 [-2.21,7.69]* 1.85 [-0.0485,5.99] -0.708 [-2.94,2.02] -0.661 [-2.8,1.54]

r - 0.85 [0.79,0.89] 0.95 [0.93,0.97] 0.97 [0.95,0.98] 0.97 [0.96,0.98]

RM, - 0.628 [0.448,0.799]* 0.548 [0.38,0.736]* 0.728 [0.58,0.833] 0.596 [0.401,0.767]* 0.593 [0.401,0.767]*

∆ - -0.0813 [-0.238,0.0589]* 0.102 [-0.0226,0.221]* -0.0404 [-0.148,0.124] -0.0401 [-0.148,0.111]

r - 0.66 [0.55,0.75] 0.73 [0.64,0.81] 0.75 [0.65,0.82] 0.78 [0.69,0.84]

Table 5.1: Comparison of the flow shape and the WSA parameters obtained by the different blood flow models. Values are given as median [95%

central range] and correlations as Pearson’s correlation coefficient and 95% confidence interval (CI). ∆ indicates the difference and r the correlation

to the Doppler derived values. * indicate a significant difference of a group-wise comparison rEF vs nEF.

93

Doppler Triangular Averaged ARCSolver combWK

Reduced EF

S energy, AU 3.76 [2.03,6.87]* 3.49 [1.77,6.09] 4.43 [2.42,7.54] 4.21 [2.29,7.42] 4.04 [2.18,7.3]

∆ - -0.425 [-1.33,0.127]* 0.364 [0.0214,1.07]* 0.225 [-0.0732,0.864]* 0.131 [-0.205,0.914]*

r - 0.97 [0.95,0.98] 0.98 [0.97,0.99] 0.99 [0.98,0.99] 0.98 [0.97,0.99]

D energy, AU 1.16 [0.416,2.19]* 0.988 [0.209,2.67]* 1.09 [0.429,2.17] 1.14 [0.545,2.1]* 1.09 [0.482,2.27]*

∆ - -0.224 [-1.02,1.3]* -0.0892 [-0.48,0.3]* -0.0494 [-0.64,0.778] -0.0404 [-0.638,0.301]*

r - 0.52 [0.31,0.68] 0.90 [0.84,0.94] 0.73 [0.59,0.83] 0.89 [0.82,0.93]

R energy, AU 0.619 [0.25,1.98]* 0.665 [0.0585,1.66]* 1.46 [0.68,3.17]* 0.651 [0.327,1.73] 0.558 [0.177,1.67]*

∆ - 0.0041 [-1.18,0.41] 0.675 [0.155,1.44]* 0.00529 [-1.2,0.338]* -0.0974 [-0.557,0.223]

r - 0.83 [0.72,0.89] 0.89 [0.82,0.93] 0.92 [0.87,0.95] 0.96 [0.94,0.98]

R/S energy, - 0.183 [0.0756,0.44] 0.196 [0.0209,0.427]* 0.346 [0.174,0.614]* 0.17 [0.0917,0.335] 0.156 [0.0635,0.337]

∆ - 0.0184 [-0.238,0.289] 0.137 [0.0249,0.346]* -0.0113 [-0.236,0.0542]* -0.0319 [-0.12,0.0325]

r - 0.66 [0.49,0.78] 0.82 [0.71,0.89] 0.90 [0.83,0.94] 0.95 [0.92,0.97]

S/D energy, - 3.2 [1.87,6.04]* 3.65 [1.01,17.3]* 3.9 [2.59,6.15] 3.44 [2.06,5.4]* 3.91 [1.87,5.67]*

∆ - 0.503 [-2.13,14.9]* 0.647 [-1.19,2.47]* 0.382 [-2.56,2.07]* 0.264 [-1.18,2.53]*

r - 0.26 [0.01,0.48] 0.68 [0.52,0.80] 0.52 [0.31,0.68] 0.69 [0.53,0.80]

Normal EF

Chapter 5. Flow models in systolic heart failure

S energy, AU 4.17 [2.57,7.8]* 3.55 [2.09,6.39] 4.4 [2.78,7.7] 4.28 [2.7,7.33] 4.25 [2.7,7.33]

∆ - -0.585 [-1.92,-0.13]* 0.133 [-0.34,0.687]* 0.0415 [-0.536,0.528]* 0.0315 [-0.536,0.528]*

r - 0.94 [0.92,0.96] 0.98 [0.98,0.99] 0.98 [0.98,0.99] 0.98 [0.98,0.99]

D energy, AU 0.946 [0.499,1.89]* 0.805 [0.452,2.38]* 1.14 [0.658,2.15] 0.953 [0.547,1.71]* 0.953 [0.547,1.72]*

∆ - -0.109 [-0.75,1.23]* 0.231 [-0.111,0.493]* 0.00559 [-0.594,0.283] 0.0119 [-0.594,0.345]*

r - 0.56 [0.42,0.67] 0.91 [0.88,0.94] 0.85 [0.79,0.89] 0.84 [0.77,0.88]

R energy, AU 0.924 [0.294,1.88]* 0.849 [0.367,1.52]* 1.29 [0.641,2.17]* 0.79 [0.298,1.72] 0.766 [0.272,1.72]*

∆ - -0.0219 [-0.657,0.884] 0.374 [-0.0772,0.882]* -0.0969 [-0.54,0.283]* -0.124 [-0.54,0.269]

r - 0.54 [0.40,0.65] 0.81 [0.74,0.86] 0.81 [0.74,0.87] 0.81 [0.74,0.86]

R/S energy, - 0.201 [0.0844,0.433] 0.235 [0.133,0.507]* 0.276 [0.16,0.486]* 0.174 [0.0756,0.366] 0.172 [0.0756,0.366]

∆ - 0.0233 [-0.109,0.414] 0.078 [-0.014,0.188]* -0.025 [-0.107,0.0358]* -0.03 [-0.118,0.0334]

r - 0.20 [0.02,0.37] 0.84 [0.78,0.89] 0.85 [0.79,0.89] 0.85 [0.79,0.89]

S/D energy, - 4.64 [2.97,7.09]* 4.53 [0.987,6.84]* 3.86 [2.92,5.25] 4.53 [3.3,6.57]* 4.5 [3.01,6.57]*

∆ - -0.0862 [-2.5,2.1]* -0.738 [-2.32,0.743]* 0.00895 [-1.75,1.59]* -0.00719 [-1.97,1.59]*

r - 0.23 [0.06,0.39] 0.73 [0.64,0.81] 0.58 [0.45,0.69] 0.58 [0.44,0.68]

Table 5.2: Comparison of the WIA energies obtained by the different blood flow models. Values are given as median [95% central range] and

correlations as Pearson’s correlation coefficient and 95% CI. ∆ indicates the difference and r the correlation to the Doppler derived values. *

94

Doppler Triangular Averaged ARCSolver combWK

Reduced EF

S peak, AU 67.6 [23.5,115]* 44.6 [19.4,102]* 101 [40.8,182] 74.7 [34.3,151] 69.9 [31.6,155]*

∆ - -15.8 [-43.5,-0.18]* 38.9 [-6.05,79.9]* 12.2 [-12.1,40.6]* 9.48 [-9.58,41.5]*

r - 0.89 [0.82,0.93] 0.83 [0.73,0.89] 0.92 [0.86,0.95] 0.91 [0.85,0.95]

D peak, AU 23.1 [8.91,48.4]* 13 [4.81,29.1]* 30.8 [12.8,54.6] 28.8 [13.4,55.2]* 21.8 [8.64,41.1]*

∆ - -10.5 [-36.6,6.02]* 7.81 [-17,20.9]* 4.47 [-21.1,34] -2.34 [-25.9,13.8]*

r - 0.52 [0.31,0.68] 0.70 [0.54,0.81] 0.28 [0.03,0.50] 0.50 [0.28,0.67]

R peak, AU 7.79 [2.8,25.4] 9.57 [1.34,21] 19.2 [10.4,38.8]* 8.96 [3.74,29.7]* 6.59 [1.73,18.8]

∆ - 1.56 [-13.4,8.72]* 10.3 [-3.59,24.5]* 0.841 [-17,11.4]* -2.04 [-11.5,8.36]

r - 0.64 [0.47,0.77] 0.49 [0.27,0.66] 0.51 [0.29,0.67] 0.75 [0.62,0.85]

R/S peaks, - 0.137 [0.0557,0.562] 0.218 [0.0441,0.566]* 0.197 [0.108,0.4]* 0.113 [0.0564,0.406]* 0.0897 [0.0319,0.403]

∆ - 0.0785 [-0.186,0.262]* 0.0638 [-0.177,0.176] -0.00797 [-0.194,0.202]* -0.0504 [-0.194,0.102]*

r - 0.46 [0.24,0.64] 0.54 [0.34,0.70] 0.45 [0.22,0.63] 0.56 [0.36,0.71]

S/D peaks, - 2.63 [1.25,8.04]* 3.61 [1.26,7.01]* 3.29 [1.88,4.48]* 2.62 [1.22,6.18]* 3.6 [1.23,7.01]*

∆ - 0.957 [-2.97,3.41]* 0.609 [-4.11,2.7]* 0.211 [-5.14,3.11]* 0.664 [-1.69,3.54]*

r - 0.53 [0.33,0.69] 0.42 [0.18,0.60] 0.30 [0.05,0.51] 0.57 [0.37,0.72]

Normal EF

Chapter 5. Flow models in systolic heart failure

S peak, AU 76.2 [41.4,178]* 47.1 [25.2,104]* 90.8 [52,185] 80.7 [46.5,157] 80.5 [46.5,157]*

∆ - -24.9 [-94,-5.85]* 12.6 [-24.6,51.9]* 4.25 [-35.9,29.1]* 3.87 [-35.9,29.1]*

r - 0.85 [0.79,0.89] 0.87 [0.82,0.91] 0.90 [0.86,0.93] 0.90 [0.86,0.93]

D peak, AU 15.1 [7.27,45.1]* 10.6 [6.3,23.2]* 30 [18.4,55.7] 18.4 [8.37,38.7]* 18 [8.37,37]*

∆ - -3.47 [-29.3,2.02]* 14.9 [1.09,27.5]* 2.81 [-21.2,16.5] 2.4 [-21.2,15.5]*

r - 0.59 [0.47,0.70] 0.75 [0.66,0.82] 0.48 [0.33,0.61] 0.51 [0.36,0.63]

R peak, AU 9.43 [3.44,20.8] 8.75 [3.75,16.6] 15.9 [10,26.1]* 7.69 [3.33,15.5]* 7.04 [2.74,15.5]

∆ - -0.137 [-11.3,8.04]* 6.4 [-1.51,14.6]* -1.28 [-10.1,3.97]* -1.42 [-10.2,2.85]

r - 0.53 [0.38,0.64] 0.43 [0.28,0.57] 0.62 [0.50,0.72] 0.59 [0.46,0.69]

R/S peaks, - 0.118 [0.0472,0.3] 0.169 [0.0942,0.392]* 0.174 [0.107,0.295]* 0.0912 [0.0413,0.212]* 0.0856 [0.0413,0.212]

∆ - 0.0524 [-0.0639,0.254]* 0.0563 [-0.0702,0.156] -0.0232 [-0.117,0.0345]* -0.0271 [-0.137,0.0222]*

r - 0.34 [0.18,0.49] 0.53 [0.38,0.64] 0.77 [0.68,0.83] 0.75 [0.66,0.82]

S/D peaks, - 5.08 [2.47,8.93]* 4.71 [1.29,8.33]* 2.99 [2.05,4.5]* 4.41 [2.25,9.29]* 4.32 [2.56,9.29]*

∆ - -0.477 [-4.19,3.35]* -2.29 [-5.69,0.808]* -0.677 [-3.99,4.37]* -0.572 [-3.99,4.37]*

r - 0.44 [0.29,0.58] 0.40 [0.24,0.54] 0.30 [0.13,0.45] 0.30 [0.13,0.45]

Table 5.3: Comparison of the WIA peaks obtained by the different blood flow models. Values are given as median [95% central range] and correlations

as Pearson’s correlation coefficient and 95% CI. ∆ indicates the difference and r the correlation to the Doppler derived values. * indicates a significant

95

Chapter 5. Flow models in systolic heart failure 96

(a) (b)

+1.96 SD +1.96 SD

5 5.3 5

Difference of |Pf | by

Difference of |Pf | by

4.8

mean mean

0 1.1 0 0.61

−1.96 SD −1.96 SD

−3.1 −3.6

−5 −5

20 30 40 50 60 20 30 40 50 60

Mean of |Pf | by Mean of |Pf | by

ARCSolver and Doppler flow combWK and Doppler flow

(c) (d)

ARCSolver and Doppler flow

5 +1.96 SD 5 +1.96 SD

Difference of |Pf | by

Difference of |Pf | by

4.4 4.1

mean mean

0 0.42 0 0.14

−1.96 SD −1.96 SD

−3.6 −3.8

−5 −5

20 30 40 50 60 20 30 40 50 60

Mean of |Pf | by Mean of |Pf | by

ARCSolver and Doppler flow combWK and Doppler flow

Figure 5.3: Bland-Altman plots comparing |Pf | derived by Doppler and ARCSolver flow (a,c)

and by Doppler and combWK flow (b,d) for patients with reduced EF (a,b) and the whole study

population (rEF blue, nEF red) (c,d).

(a) (b)

ARCSolver and Doppler flow

4 4

combWK and Doppler flow

+1.96 SD

Difference of |Pb | by

Difference of |Pb | by

3.1 +1.96 SD

2 2 2.5

mean mean

1 0.72

0 0

−1.96 SD −1.96 SD

−1.1 −1.1

−2 −2

10 20 30 40 10 20 30 40

Mean of |Pb | by Mean of |Pb | by

ARCSolver and Doppler flow combWK and Doppler flow

(c) (d)

ARCSolver and Doppler flow

4 4

combWK and Doppler flow

+1.96 SD

Difference of |Pb | by

Difference of |Pb | by

+1.96 SD

2.9 2.6

2 2

mean mean

0.81 0.69

0 0

−1.96 SD −1.96 SD

−1.3 −1.2

−2 −2

10 20 30 40 10 20 30 40

Mean of |Pb | by Mean of |Pb | by

ARCSolver and Doppler flow combWK and Doppler flow

Figure 5.4: Bland-Altman plots comparing |Pb | derived by Doppler and ARCSolver flow (a,c)

and by Doppler and combWK flow (b,d) for patients with reduced EF (a,b) and the whole study

population (rEF blue, nEF red) (c,d).

Chapter 5. Flow models in systolic heart failure 97

0.2 0.2

Difference of RM by

Difference of RM by

+1.96 SD

0.15 +1.96 SD

0.1 0.1 0.11

mean mean

0 0.022 0 0.019

−1.96 SD

−0.1 −1.96 SD −0.1 −0.075

−0.1

−0.2 −0.2

0.4 0.6 0.8 1 0.4 0.6 0.8 1

Mean of RM by Mean of RM by

ARCSolver and Doppler flow combWK and Doppler flow

(c) (d)

ARCSolver and Doppler flow

0.2 0.2

Difference of RM by

Difference of RM by

+1.96 SD +1.96 SD

0.16 0.15

0.1 0.1

mean mean

0 0.029 0 0.029

−1.96 SD −1.96 SD

−0.1 −0.1 −0.088

−0.1

−0.2 −0.2

0.4 0.6 0.8 1 0.4 0.6 0.8 1

Mean of RM by Mean of RM by

ARCSolver and Doppler flow combWK and Doppler flow

Figure 5.5: Bland-Altman plots comparing RM derived by Doppler and ARCSolver flow (a,c)

and by Doppler and combWK flow (b,d) for patients with reduced EF (a,b) and the whole study

population (rEF blue, nEF red) (c,d).

(a) (b)

ARCSolver and Doppler flow

0.5 0.5

Difference of S energy by

Difference of S energy by

+1.96 SD

+1.96 SD 0.33

0.19

0 0

mean

mean −0.17

−0.26

−0.5 −0.5

−1.96 SD −1.96 SD

−0.71 −0.66

−1 −1

4 6 8 10 4 6 8 10

Mean of S energy by Mean of S energy by

ARCSolver and Doppler flow combWK and Doppler flow

(c) (d)

ARCSolver and Doppler flow

0.5 0.5

Difference of S energy by

Difference of S energy by

+1.96 SD +1.96 SD

0.37 0.4

0 mean 0 mean

−0.12 −0.082

−0.61 −0.56

−1 −1

4 6 8 10 4 6 8 10

Mean of S energy by Mean of S energy by

ARCSolver and Doppler flow combWK and Doppler flow

Figure 5.6: Bland-Altman plots comparing the S wave energy derived by Doppler and ARCSolver

flow (a,c) and by Doppler and combWK flow (b,d) for patients with reduced EF (a,b) and the

whole study population (rEF blue, nEF red) (c,d).

Chapter 5. Flow models in systolic heart failure 98

Difference of R energy by

Difference of R energy by

1 1

+1.96 SD +1.96 SD

0.5 0.62 0.5 0.59

mean mean

0 0.052 0 0.13

−1.96 SD

−0.5 −1.96 SD −0.5 −0.32

−0.52

1 2 3 4 5 1 2 3 4 5

Mean of R energy by Mean of R energy by

ARCSolver and Doppler flow combWK and Doppler flow

(c) (d)

ARCSolver and Doppler flow

Difference of R energy by

Difference of R energy by

1 1

+1.96 SD +1.96 SD

0.5 0.58 0.5 0.58

mean mean

0 0.09 0 0.13

−1.96 SD −1.96 SD

−0.5 −0.4 −0.5 −0.32

1 2 3 4 5 1 2 3 4 5

Mean of R energy by Mean of R energy by

ARCSolver and Doppler flow combWK and Doppler flow

Figure 5.7: Bland-Altman plots comparing the R wave energy derived by Doppler and ARCSolver

flow (a,c) and by Doppler and combWK flow (b,d) for patients with reduced EF (a,b) and the

whole study population (rEF blue, nEF red) (c,d).

0.3 0.3

(a) (b)

ARCSolver and Doppler flow

Difference of R/S energy by

combWK and Doppler flow

0.2 0.2

+1.96 SD +1.96 SD

0.1 0.14 0.1 0.13

mean mean

0 0.021 0 0.038

−1.96 SD

−1.96 SD −0.052

−0.1 −0.098 −0.1

−0.2 −0.2

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

Mean of R/S energy by Mean of R/S energy by

ARCSolver and Doppler flow combWK and Doppler flow

0.3 0.3

(c) (d)

ARCSolver and Doppler flow

Difference of R/S energy by

combWK and Doppler flow

0.2 0.2

+1.96 SD +1.96 SD

0.1 0.13 0.1 0.13

mean mean

0 0.027 0 0.035

−1.96 SD −1.96 SD

−0.075 −0.057

−0.1 −0.1

−0.2 −0.2

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

Mean of R/S energy by Mean of R/S energy by

ARCSolver and Doppler flow combWK and Doppler flow

Figure 5.8: Bland-Altman plots comparing the ratio of the R to S wave energy derived by Doppler

and ARCSolver flow (a,c) and by Doppler and combWK flow (b,d) for patients with reduced EF

(a,b) and the whole study population (rEF blue, nEF red) (c,d).

Chapter 5. Flow models in systolic heart failure 99

1 (a) 1 (b)

Difference of D energy by

Difference of D energy by

+1.96 SD

0.5 0.63 0.5 +1.96 SD

0.48

mean mean

0 0.025 0 0.064

−1.96 SD

−0.5 −1.96 SD −0.5 −0.35

−0.58

Mean of D energy by Mean of D energy by

ARCSolver and Doppler flow combWK and Doppler flow

(c) (d)

ARCSolver and Doppler flow

1 1

Difference of D energy by

Difference of D energy by

0.5 +1.96 SD 0.5 +1.96 SD

0.48 0.43

mean mean

0 0.014 0 0.02

−1.96 SD −1.96 SD

−0.5 −0.46 −0.5 −0.39

Mean of D energy by Mean of D energy by

ARCSolver and Doppler flow combWK and Doppler flow

Figure 5.9: Bland-Altman plots comparing the D wave energy derived by Doppler and ARCSolver

flow (a,c) and by Doppler and combWK flow (b,d) for patients with reduced EF (a,b) and the

whole study population (rEF blue, nEF red) (c,d).

(a) (b)

ARCSolver and Doppler flow

Difference of S/D energy by

combWK and Doppler flow

3 3

2 +1.96 SD 2

1.8 +1.96 SD

1 1 1.4

0 mean 0 mean

−0.18 −0.32

−1 −1

−1.96 SD −1.96 SD

−2 −2

−2.2 −2

−3 −3

2 4 6 8 2 4 6 8

Mean of S/D energy by Mean of S/D energy by

ARCSolver and Doppler flow combWK and Doppler flow

(c) (d)

ARCSolver and Doppler flow

Difference of S/D energy by

combWK and Doppler flow

3 3

2 +1.96 SD 2 +1.96 SD

1.8 1.7

1 1

mean mean

0 0

−0.034 −0.054

−1 −1

−1.96 SD −1.96 SD

−2 −1.9 −2 −1.8

−3 −3

2 4 6 8 2 4 6 8

Mean of S/D energy by Mean of S/D energy by

ARCSolver and Doppler flow combWK and Doppler flow

Figure 5.10: Bland-Altman plots comparing the ratio of the S to D wave energy derived by

Doppler and ARCSolver flow (a,c) and by Doppler and combWK flow (b,d) for patients with

reduced EF (a,b) and the whole study population (rEF blue, nEF red) (c,d).

Chapter 5. Flow models in systolic heart failure 100

analysis (PWA) devices already incorporate flow estimation procedures and also for heart failure

patients, first studies have been performed based on modelled blood flow [122]. Nonetheless, in

fact little is known about the applicability of these methods for WSA and WIA in patients with

systolic heart failure so far. This was the starting point of the present work with the aim of

providing first data regarding the performance of four different approaches for the estimation of

aortic blood flow from pressure alone in patients with impaired systolic LV function. For this

purpose, derived parameters were compared against the reference values obtained with Doppler

flow measurements on the one hand, and between patients with reduced and normal EF on the

other hand.

In their initial study, Westerhof and coworkers [142] observed that the triangular approxima-

tion tended to overestimate |Pf | and slightly underestimate |Pb | compared to the amplitudes

obtained with measured flow. This is in accordance with the present findings in the rEF group.

Subsequent studies [35, 49] then again reported an overestimation of both the backward and,

even more pronounced, the forward amplitude when using the triangle, which is in line with the

results obtained in patients with normal EF in this work. A possible reason for this discrepancy

between the groups might be sought in the fact that Westerhof and coworkers [142] used a sample

comprising different pathologies including heart failure, while both Kips and coworkers [49] and

Hametner et al. [35] investigated patients with normal systolic function only. Correlations be-

tween the WSA parameters obtained with the triangular flow and the measured waveform were

almost identical to those found by Hametner and coworkers in the control group [35], overall less

than those reported by Westerhof [142], but greater than those found by Kips when using timing

information derived from the pressure wave as in the present work [49].

By using the inflection point of the pressure wave to determine the position of the apex, the

triangular flow was able to reproduce also belated flow peaks in the rEF group, compare figure

5.1. The estimated timing of maximum flow even showed a closer agreement to the measured

one in patients with reduced EF than in controls. This might explain why the triangulation

method provided better estimates of the WSA parameters for patients with reduced than for

patients with normal EF, despite a higher median RMSE from the measured flow waveform for

low EF. However, in both groups, modelled maximum flow tended to occur later than measured

one. This might be due to the temporal difference between the first shoulder in the pressure

wave, which was found to coincide well with peak flow [48], and the inflection point, which always

occurs afterwards. Also Westerhof and coworkers [142] reported a slight overestimation as well

as a wider scatter when using the inflection point to estimate the time of peak flow compared to

measured values.

For the averaged waveform, correlations of |Pf | and |Pb | for estimated and measured flow in the

control group were very similar to the earlier reported values [35, 49]. Furthermore, this approach

performed better in patients with normal than with reduced EF, which might not be surprising

considering that the waveform was derived from healthy subjects with normal systolic function.

In these patients, the relative position of peak flow was well described by the averaged flow. Also

Chapter 5. Flow models in systolic heart failure 101

in line with previous findings [35, 49], the averaged waveform was superior to the triangle for

wave separation in terms of both precision and correlation. Interestingly though, this held true

for both patients with normal as well as reduced EF, despite the considerable variations in flow

wave morphology introduced by the impairment of LV systolic function, as depicted in figure 5.1.

Actually, one might expect that a flow model that is capable to adapt in shape would provide

a more accurate approximation of aortic blood flow in patients with systolic heart failure. The

averaged waveform, which does not allow for any changes in shape beyond a scaling in width,

indeed resulted in the highest RMSE of all flow models in the rEF group. Nonetheless, the de-

rived WSA parameters showed a smaller deviation from the reference values than those obtained

with the triangulation method when looking at patients with reduced and normal EF separately.

However, when considering both groups, the deficiency of the averaged waveform became ap-

parent, namely its inability to reproduce the reduction in reflection magnitude associated with

ventricular impairment found with the Doppler flow. This indicates that a constant waveform is

in fact too simplistic to capture all group characteristics. Moreover, it underlines the importance

of the flow wave morphology for the computation of the WSA parameters, and in particular the

dependency on the position of maximum flow.

The ARCSolver as well as the Windkessel flow are based on Windkessel models to describe the

dynamic relation between pressure and flow. Therefore, in contrast to both the triangular and the

averaged waveform, the whole pressure pulse is taken into account to construct the corresponding

flow contour. As a consequence, both methods were indeed able to mimic the ejection patterns

found for normal and failing hearts, compare figures 5.1 and 5.2. Moreover, both approaches

yield physiological flow waveforms, thereby eluding the drawbacks of the triangular approxi-

mation (no physiological shape) as well as of the averaged waveform (no adoption in shape).

For the ARCSolver routine, some additional postprocessing steps are applied to obtain the final

waveform [35], which is why its shape is more restricted than the Windkessel flow. Additionally,

the ARCSolver flow is computed using a 3-element Windkessel model while the Windkessel flow

is based on a 4-element model extended by P∞ , resulting in two additional degrees of freedom,

which again allow for a greater variation in shape. However, this higher degree of flexibility

comes with the price of a loss in robustness compared to the ARCSolver flow, see section 3.4.

In an attempt to unify the strengths of both, these two approaches were therefore combined in

the combWK flow used in this chapter. In fact, the combWK flow provided the most accurate

estimate of aortic flow assessed by the RMSE in both groups, and even more importantly, was

the only method that resulted in the same RMSE for patients with reduced and normal EF. In

other words, the quality of the estimate did not depend on the EF status, while the error became

larger in patients with reduced EF for all other models. This is also reflected in the position

of peak flow, which was most accurately estimated by the combWK model in both groups of

patients.

The correlations found for |Pf | and |Pb | in controls when using the ARCSolver flow are well in

line with the values reported by Hametner et al. [35] and are very similar to those obtained

with the combWK flow as well as with the averaged waveform. Also in terms of accuracy and

Chapter 5. Flow models in systolic heart failure 102

precision, the three approaches showed a comparable behaviour for patients with normal systolic

function, whereby the averaged waveform performed slightly better than the other two models

with regards to |Pf |, while the ARCSolver and combWK flow yielded better results for |Pb |.

In patients with reduced EF, on the other hand, both the ARCSolver and combWK flow were

favourable to the averaged waveform and, with the combWK flow, estimates of |Pb | and RM

could be further improved compared to the ARCSolver flow, as shown in figures 5.4 and 5.5.

In the control group, the combWK flow equals the ARCSolver flow in 88% of the cases. There-

fore, the parameters derived with the two flow models are very similar for patients with normal

EF, which is why the nEF group is not depicted separately in Bland-Altman plots. Instead, a

comparison for the whole study population was included. With the aim of developing a flow

model that is applicable in the general population, the quality and behaviour of the estimates

should ideally be independent of the EF-status. Even though some differences in performance of

course persisted between the groups, the limits of agreement could be noticeable tightened for

|Pb | and RM when using the combWK flow compared to the ARCSolver flow, not only for the

rEF group but also for the whole study population.

As already discussed in the previous chapter, WIA can be considered less robust than WSA

because the computations are based on the product of the incremental changes in pressure and

flow and not the absolute values. While robustness referred to the effect of measurement noise

or errors in the last chapter, with respect to the flow models it means that also small deviations

from the measured flow wave might have a large effect on the derived parameters. These ef-

fects can be expected to be particularly pronounced when using peak intensities, while the wave

energies should be less affected. Moreover, different portions of the flow wave affect different

parameters: those associated to the S wave depend mostly on the early systolic upstroke up to

the flow peak, those describing the D wave on the late systolic decline and for the R wave, the

mid-systolic part is most relevant. However, of course the shape of the pressure waveform, i.e.

its raising and falling, also has an impact on which specific aspects or portions of the flow wave

are most emphasised.

For the triangular flow, both upstroke and decline are always linear resulting in a poor corre-

spondence to the measured waveforms in this regard. This is also reflected in the derived wave

energies, since accuracy and precision were, of all methods included, by far worst for the trian-

gular approximation independent of ventricular function. The median difference to the reference

value was up to a factor 10 greater than for the best estimate and the limits of agreement were

partly more than twice as wide. Because of its non-physiological shape, the triangular flow wave

therefore represents no suitable approximation of blood flow for WIA, neither for patients with

normal nor with reduced ejection fraction.

The averaged waveform, in contrast, was computed from measured flow waves and therefore

displays all the typical features of a physiological ejection pattern. However, from figure 5.2 it

Chapter 5. Flow models in systolic heart failure 103

can be seen that, also for controls, the upstroke tends to be more concave than in the Doppler

flow waves used in this work, resulting in a larger maximum slope. This might explain the

overestimation of peak forward wave intensity and wave energy during early systole. For pa-

tients with reduced EF and belated maximum, peak flow is moreover reached too early, the

upstroke is therefore too steep and the duration of the forward compression wave, i.e. the time

to peak flow, is consequently shorter than for the measured waves. The first two arguments

justify the excessive overestimation of the S peak found for the averaged waveform. This effect

is less pronounced for the wave energy, probably because of the shorter duration of the forward

compression wave compared to the reference method. For the decline, similar observations can

be made. In particular, in patients with impaired ventricular function, the premature flow peak

causes a prolongation of the backward compression wave, possibly explaining the enormous over-

estimation of the R wave energy. The beginning of the forward decompression wave, in contrast,

depends on the pressure contour only, which might explain why estimates of the D wave energy

were more precise in both groups of patients. However, by assuming flow to be qualitatively

constant in all patients, differences between the groups observed for the Doppler flow could not

be reproduced. Thus, in line with the results obtained for WSA, not all characteristics were

captured. The present data therefore suggests that, even though the averaged waveform might

represent an acceptable approximation for patients with normal systolic function only, it might

not the best choice for the use in the general population, where subjects with different degrees

of ventricular impairment might be included.

WIA estimates obtained with the ARCSolver and combWK flow were more accurate than those

derived using the averaged or triangular waveform, indicating that the respective modelled flow

waves showed less systematic deviation from the measured one. Furthermore, both methods were

overall superior to the triangle in the present dataset and, in contrast to the averaged waveform,

differences in wave energies and their ratios between the groups observed for the Doppler flow

were well captured, with the only exception of the reduction in S wave energy. However, the

combWK showed a tendency towards lower values in the rEF group, yet without reaching sta-

tistical significance. Comparison of the ARCSolver and combWK flow showed an improvement

for all but the S wave energy, both at a group and a population level when using the combWK

flow. Nevertheless, it should be stressed again that the limits of agreement were relatively large

for all WIA parameters and in particular for the peak values.

To summarise, for WSA, it seems feasible to replace aortic blood flow by modelled waveforms

also for patients with systolic heart failure. For this purpose, the model should be able to adopt

in shape and at the same time provide a physiological waveform in order to attain high precision

as well as the correct qualitative behaviour, as observed for the combWK flow in this work. The

triangulation method has already been used for WSA in a longitudinal study in patients with

systolic heart failure and a prognostic value of |Pb | for the occurence of cardiovascular events has

been found [122]. The results obtained in this work show that estimates of |Pb | can be markedly

improved when applying a more sophisticated flow model. Therefore, also its association to ad-

verse events and its value for risk prediction might potentially be enhanced by the use of a more

Chapter 5. Flow models in systolic heart failure 104

As expected, WIA was more sensitive than WSA and the considered flow estimates resulted in

rather large deviations from the reference values. In order to improve the performance, mea-

surements with higher sampling rates might be beneficial. Alternatively, a postprocessing, e.g.

smoothing or averaging, of the data could help to make the parameters more reliable. However,

peak values might generally be too sensitive on the particular waveform to be used with flow

models and wave energies therefore seem to be preferable. Nonetheless, despite these drawbacks,

some of the main features observed for the WIA parameters derived with Doppler measurements

could again be reproduced using the ARCSolver or combWK flow.

With regards to the combWK flow, it has to be emphasised that this work represents a proof of

concept only. The same study population was used to develop and to test the method, which

might of course introduce considerable bias. Also the criterion when to use the ARCSolver and

when to use the Windkessel flow was derived from this population, even if it was not optimised

to avoid a possible overfitting. Moreover, it should be kept in mind that the criterion might not

work in different data sets, especially when using central pressure waves assessed by a different

device. The age of the study population represents another limitation. Ideally, the combWK

flow should be tested in a larger sample, including also younger subjects. In particular because

it has been suggested that with age and the associated increase in wavelength, the arterial

system ”degenerates” into a Windkessel [69]. Thus it is possible that the model fits middle-

aged to elderly subjects with increased stiffness better than the young and healthy ones. For

patients with systolic heart failure, on the other hand, the considered population seems to be

representative for this cohort, at least when compared to other data found in literature, e.g.

[21, 88, 137]. However, also in this regard, the behaviour of the model for different severity levels

of ventricular impairment should be examined.

Conclusion

In conclusion, with the proposed novel approach first promising results could be obtained. As

discussed in the last chapter, the change in flow wave morphology associated with ventricular

impairment had a considerable impact on the derived WSA and WIA parameters, which could

not be reproduced by all flow models considered. This should be kept in mind when applying

a flow model, not only in patients with systolic heart failure but also in the general population,

where subjects with differing LV systolic performance might be included. Nevertheless, for

WSA it seems feasible to replace flow measurements with modelled flow waves derived from

non-invasively recorded pressure alone also in patients with systolic heart failure. This would

greatly facilitate the assessment of WSA parameters and would therefore enable its use in large

population studies or, in a next step, its inclusion in regular patient check-ups or screening

strategies based on a single non-invasive pressure measurement. WIA parameters were found to

be extremely sensitive on the flow waveforms, unfortunately resulting in a rather poor quality of

the estimated parameters.

Chapter 6

The haemodynamics in the arterial system result from a complex interaction of the ejecting

heart and the vasculature, i.e. the ventriculo-arterial coupling. For healthy hearts, the ventricu-

lar component is more or less negligible because the way how the heart ejects is (qualitatively)

very similar across individuals as well as for different loading conditions. In systolic heart failure

(SHF), however, the interaction becomes much more relevant, making these patients especially

interesting but at the same time challenging from a haemodynamic point of view.

One of the aims of this thesis was to analyse the role of cardiac function for the parameters of

pulsatile haemodynamics derived by PWA, WSA and WIA. The obtained results indicate that

the derived parameters can indeed be used to characterise the properties of the arteries only in

subjects with normal systolic function. For patients with SHF, on the contrary, cardiac perfor-

mance was associated to most haemodynamic measures, including indices of wave reflections,

which complicates their interpretability. The apparent reduction in wave reflections in SHF pa-

tients compared to controls could be explained by a shortening of the ejection duration as a

manifestation of the impaired coupling between the ventricle and the arterial system. Hence,

the magnitude of wave reflections might not be a suitable indicator of arterial stiffness when the

heart is failing. These observations underscore the importance of considering cardiac function

for risk stratification. This applies in particular for patients with SHF and more widely for the

general population, where asymptomatic subjects with varying degrees of systolic impairment

might be included.

SHF is very heterogeneous in its appearance, including in the resulting ejection patterns. There-

fore, a new model of blood flow (combWK) was developed in the course of this work, with a special

focus on its ability to reproduce also pathological flow waveforms. This method was compared to

already existing models of blood flow, namely the triangular, averaged and ARCSolver flow and

their performance in patients with SHF and controls was investigated using measured flow as

reference. The triangular flow represented a rather poor description in both groups with respect

to the precision of the derived estimates of WSA and WIA compared to the reference. This was

105

Chapter 6. Summary and conclusions 106

most probably caused by the unphysiological shape of the flow wave, which became especially

noticeable for the WIA parameters. The averaged waveform achieved a better precision, but it

was too simplistic to capture the qualitative differences between the groups. The ARCSolver

and the combWK flow are both based on mathematical models to describe the dynamic relation

between pressure and flow and provide a physiological waveform as well as a certain flexibility

in shape. Nonetheless, the good performance of the ARCSolver flow observed for patients with

normal systolic function could not be fully transferred to patients with SHF. This drawback could

be eliminated with the proposed combined model, for which the accuracy of the estimated flow

waves and of the derived parameters was almost independent of cardiac function. However, these

results present a proof of concept only, because the same study population was used to develop

and to test the model. Further research is needed to evaluate the performance of the model on

different datasets. Moreover, there is still room for improvement regarding the parameter identi-

fication procedure and in particular the definition of the objective function used for optimisation.

Overall, replacing the flow measurement by an approximation derived from non-invasively as-

sessed central pressure seems feasible for WSA in both patients with SHF and controls. WIA

is very sensitive on the flow waveforms and the parameter estimates were less accurate than for

WSA, especially when using peak intensities for quantification. The use of wave energies might

therefore be preferable. Nonetheless, also for WIA, distinct differences between SHF and controls

found with measured flow could be reproduced with modelled flow waveforms. These differences

might potentially be used as indicators for a systolic dysfunction in the future, enabling the

detection of SHF from non-invasively measured pressure alone as well as an improvement of risk

stratification in patients with impaired systolic function.

Appendix A

Asymptotic pressure as an

additional parameter in the

Windkessel models

The Windkessel models surely belong to the most extensively investigated models of the car-

diovascular system. Nevertheless, disunity exists whether aortic pressure should be modelled as

resulting from ventricular ejection only or if an asymptotic pressure level P∞ should be included.

P∞ thereby represents a pressure that is sustained by the vasculature even without further ex-

citation from the heart.

The aim of this chapter is therefore twofold. First, the influence of an additional model pa-

rameter P∞ on the overall behaviour of the three Windkessel models presented in section 2.3 is

examined. For this purpose, simulated pressure contours with and without P∞ are compared in

the time as well as in the frequency domain based on a talk given at the MATHMOD 2015 and

published in [97].

Second, the question whether pressure should be modelled as dropping to zero or as approaching

an asymptotic level P∞ when the heart stops beating is investigated. Therefore, invasively

acquired pressure contours of patients presenting with missing heart beats and thus prolonged

diastolic pressure drops are examined. More precisely, the fitting performance of the (mono-)

exponential decay described by the Windkessel models is analysed with respect to P∞ . This part

is based on a poster presented at the ARTERY conference 2014 [98].

107

Appendix A. Asymptotic pressure in the Windkessel models 108

P∞ Rp

WK2, 2-element Windkessel Rp +

Q 1 + iωn Rp Ca

P∞ Rp

WK3, 3-element Windkessel Rp + Zc + Zc +

Q 1 + iωn Rp Ca

P∞ iωn Zc L Rp

WK4p, parallel 4-element Windkessel Rp + +

Q iωn L + Zc 1 + iωn Rp Ca

Table A.1: The input impedances Zin described by the three different Windkessel models.

A.1 Methods

In order to investigate the influence of P∞ , first of all it is important to understand its effect on

the other model parameters. P∞ directly affects the steady part of pressure only and therefore

the zero frequency component of the arterial input impedance Zin , which is usually identified

with the systemic vascular resistance (SVR). Strictly speaking, the pressure drop between mean

aortic P and right atrial pressure P ra should be used for the computation of SVR, but, since

P ra P , right atrial pressure is usually neglected [77].

P − P ra P

SVR = ≈ = Zin (0) (A.1)

Q Q

Originally, the model parameter Rp was supposed to represent SVR. However, even without

including P∞ , the identity Rp = SVR only holds for the 2-element (WK2) and parallel 4-element

Windkessel (WK4p), but not for the 3-element Windkessel (WK3), where SVR equals Rp + Zc ,

see table A.1. Thus, for the further investigation, SVR and Rp were considered two distinct

parameters, which are connected via the following relations in dependency of P∞ .

P − P∞ P∞

WK2, WK4p: Rp = = SV R − (A.2)

Q Q

P − P∞ P∞

WK3: Rp = − Zc = SV R − − Zc (A.3)

Q Q

If both pressure and flow of a specific person are known, SVR as well as Zc can be estimated,

compare equation (A.1) and section 2.2.3. Considering these parameters to be given, the above

equations thus imply that only Rp is directly affected by P∞ , whereby higher values of P∞ result

in lower values of Rp . This means that even though P∞ explicitly appears in the formula for

Zin (0) only, its effect on Zin will be observable over the whole frequency range due to its quan-

R

titative influence on Rp . However, the term including Rp equals 1+iωnpRp Ca in all three models,

compare table A.1, which converges to 0 for n → ∞. It is therefore expected that differences in

Zin will be most noticeable in the lower frequency range.

To enable a direct comparison of the simulated pressure waves with and without P∞ , the pa-

Appendix A. Asymptotic pressure in the Windkessel models 109

SVR systemic vascular resistance WK2,WK3,WK4p 1.4 mmHg · s/ml

Ca total arterial compliance WK2,WK3,WK4p 1.6 ml/mmHg

Zc characteristic impedance WK3,WK4p 0.04 mmHg · s/ml

L total arterial inductance WK4p 0.014 mmHg · s2 /ml

Q̄ mean flow WK2,WK3,WK4p 73.6 ml/s

DBP diastolic blood pressure WK2,WK3,WK4p 90.1 mmHg

rameters Ca , Zc , L and SVR were fixed and Rp was varied according to the above relations for

P∞ ∈ {0, 25, 50, 75} mmHg. A typical flow curve depicted in figure A.2 was taken as input to

the models and the parameters were chosen based on values reported in literature for a healthy,

adult population [115] as given in table A.2.

Aortic blood pressure waveforms were assessed invasively using high-fidelity pressure-tip catheters

(5F Millar SPC-454D) at the university teaching hospital Wels-Grieskirchen in Wels, Austria.

The recordings of 5 different patients presenting with missing heartbeats were used for analysis,

resulting in a total of 35 data sets of irregular heartbeats. The regular heartbeat duration T was

determined from the preceding beat and diastolic blood pressure (DBP) was set to P (0). An

examplary pressure reading is depicted in figure A.1.

dicrotic notch,

140

beginning of

diastole

120

Pressure (mmHg)

100

80 regular

diastole

60 prolonged

diastole

T ts

40

Time (sec)

Figure A.1: Exemplary aortic pressure signal including a missing heart beat. The duration of a

regular cardiac cycle T is determined from the preceding beat.

The decrease of aortic pressure has been found to be approximately exponential during diastole,

a feature that is well caught by the Windkessel models, compare section 2.3.3. Therefore, the

Appendix A. Asymptotic pressure in the Windkessel models 110

ts − t

Pdias (t) = P∞ + PRC (ts ) exp

Rp C a

with time constant τ = Rp Ca was used to describe the diastolic pressure. In order to investigate

how P∞ influences the fitting performance, the exponential decline in its equivalent formulation

(assuming periodicity)

T −t

P∞ + (P∞ − DBP) · exp

Rp Ca

was fitted to both the regular and the prolonged diastole of the experimental data for P∞

increasing from 0 to 100% of DBP in steps of 5 percentage points. The fitting was performed

in Matlab R2011b (The MathWorks Inc, Natick, Massachusetts, United States) by minimising

the squared error with respect to τ using the inbuilt fminsearch function with initial estimate

τ = 1.5 s. To compare the results for different values of P∞ , the root mean square error (RMSE)

between fitted and measured pressure over both the regular and the prolonged diastole was

computed. Results were finally averaged first per patient and subsequently over the group to

account for the different number of individual measurements.

A.2 Results

The results of the simulation runs for the three different Windkessel models and P∞ varying

from 0 to 75 mmHg are depicted in figure A.2. In all three cases, the effect is most prominent

during diastole with a steeper decline for higher values of P∞ . The systolic upstroke, in contrast,

remains almost unchanged with the largest changes observed for the WK4p.

The differences between the modelled input impedances for P∞ = 0 and P∞ = 75 mmHg are

presented in figure A.3. As expected, the differences are largest for low frequencies and decline

for increasing n.

The prolonged heart beats were on average 1.81 (range 1.49 to 2.02) times longer than the pre-

ceding regular beats and the mean duration of diastole was more than doubled. More precisely,

regular diastole endured on average 0.52 s (range 0.39 to 0.69 s), while the mean duration of a

prolonged diastole was 1.22 s (range 0.86 to 1.66 s), resulting in a 2.3-fold increase (range 1.81

to 2.62) in diastolic duration. Mean diastolic blood pressure was 77 mmHg (7.2 SD) and mean

systolic blood pressure 137 mmHg (11.6 SD).

Figure A.4 shows a representative example of the results obtained with the two different fitting

Appendix A. Asymptotic pressure in the Windkessel models 111

A B

130

400

P∞=0

Aortic flow, ml/s P∞=25

300

P∞=50

200 110 P∞=75

100

0

90

0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8

Time, s Time, s

C D

130 130

Aortic pressure, mmHg

P∞=0 P∞=0

P∞=25 P∞=25

P∞=50 P∞=50

110 P∞=75 110 P∞=75

90 90

0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8

Time, s Time, s

Figure A.2: A: typical flow curve used as input to the models. B-D: simulated pressure waves

for P∞ ranging from 0 to 75 mmHg obtained with the WK2 (B), WK3 (C) and the WK4p (D).

−3

x 10

4

∆ |Zin|, mmHg⋅s/ml

−2

−4

0 1 2 3 4 5 6 7 8 9 10

Harmonic

0

∆ arg(Zin), rad

−0.1

WK2

WK3

WK4p

−0.2

0 1 2 3 4 5 6 7 8 9 10

Harmonic

Figure A.3: Difference between modulus (top) and phase (bottom) of modelled input impedance

for P∞ = 0 and P∞ = 75 mmHg for the three different Windkessel models. Relative to the

respective Zin for P∞ = 0 as reference, the maximum deviation equals 5% for the modulus and

15% for the phase angle, compare also figure 2.14.

Appendix A. Asymptotic pressure in the Windkessel models 112

140 P∞ = 0

fitting P∞ = 0.5·DBP

P∞ = 0.7·DBP

Aortic pressure, mmHg

120 P∞ = 0.9·DBP

100

80 DBP

60

T

40

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Time, s

140 P∞ = 0

fitting P∞ = 0.5·DBP

P∞ = 0.7·DBP

Aortic pressure, mmHg

120 P∞ = 0.9·DBP

100

80 DBP

60

T

40

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Time, s

Figure A.4: Exemplary result of fitting. Top: fitted over the duration of the regular diastole,

bottom: fitted over prolonged diastole.

Appendix A. Asymptotic pressure in the Windkessel models 113

7 7

RMSE, regular diastole

6 6

5 5

4 4

3 3

2 2

1 1

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 0% 10% 20% 30% 40% 50% 60% 70% 80% 90%

P∞, % of DBP P∞, % of DBP

Figure A.5: Mean RMSE between analytical function and regular (left) and prolonged (right)

diastole when fitted to the regular (red) and prolonged (blue) part respectively.

procedures. Independently of the part chosen for fitting, mean RMSE was lowest for P∞ =

0.7 · DBP, compare figure A.5. When P∞ was not included, the RMSE between modelled

pressure drop and prolonged diastole was more than twice as high as for P∞ = 0.7 · DBP ≈ 54

mmHg (5.1 SD), regardless of the part chosen for fitting.

A.3 Discussion

The simulation runs show that the temporal waveforms of modelled pressure are only moderately

affected by P∞ if SVR is fixed and Rp varied accordingly. Because of the stronger impact of

P∞ on the lower frequency range of the modelled input impedance, the biggest changes could

thereby be observed during diastole, with a steeper decline for higher values of P∞ . The fast

systolic upstroke, in contrast, remained nearly unaltered, reflecting the negligible influence of

P∞ on Zin for higher frequencies.

Because SVR and Ca were fixed, higher values of P∞ resulted in lower values of Rp and therefore

smaller time constants τ . The corresponding pressure waveforms thus showed a faster diastolic

drop towards a higher asymptotic value, which explains the greater curvature observed in figure

A.2. The results obtained with the fitting experiments suggest that this effect might indeed

improve the accuracy of an exponential decline as a model of the diastolic pressure decay. The

deviation from measured data could be markedly reduced when pressure was not assumed to

drop to zero, compare figure A.4. This reduction was particularly notable with regards to the

prolonged diastole, where the RMSE could be more than halved by including P∞ . However, even

when only the regular diastole was considered, an improvement in the RMSE could be achieved.

The physiological meaning and magnitude of P∞ has been investigated before [2, 129, 131]. From

the definition of SVR in equation (A.1), it follows that choosing P∞ equal to mean right atrial

pressure P ra ≈ 5 mmHg would agree with the classical physiological interpretation of Rp as total

Appendix A. Asymptotic pressure in the Windkessel models 114

vascular resistance SVR. This approach was adopted by Aguado-Sierra et al. [2] in a simulation

study investigating wave reflections in the arterial system. However, Wang et al. [130] observed

that P∞ is influenced by the vasoactive state of the arteries and suggested that P∞ is thus ”a

function of the local biochemical, humoral, and mechanical milieu” [130, p.H162], i.e. a varying

parameter that does not coincide with venous or right atrial pressure. Moroever, when P∞ was

estimated together with τ from the diastolic decay in experimental data, indeed values greater

than P ra were obtained. In a study in mongrel dogs, values between 30-40 mmHg were found for

a mean aortic pressure of 83 mmHg [131]. Schipke et al. [112] analysed human aortic pressure

waves during periods of up to 20s of controlled cardiac arrest and reported values in the range of

25 mmHg for the constant term in the exponential decay when fitted to the measured diastolic

pressure drop. Also the fitting results obtained in this work show that a value of P∞ above right

atrial pressure might be preferable. However P∞ = 0.7 · DBP, which corresponds to a minimum

RMSE in this work, is markedly higher than the values previously proposed.

As mentioned before, the equality Rp = SVR is lost for P∞ 6= P ra . However, it should be noted

that Rp and SVR were considered two distinct parameters in this work and the value of SVR, as

an important characteristic of the arterial system, stays completely unaffected by P∞ . For a clear

distinction between the two parameters, Rp has previously been termed ”effective resistance of

the peripheral systemic circulation” [2, 131]. Although the physiological meaning and anatomic

counterpart of Rp are not yet entirely clear, it was suggested to represent the resistance of the

small arteries with a diameter > 60µm [129].

Overall, the results suggest that the fitting performance might be improved by including P∞

and that this inclusion does not affect the modelled characteristics of the arterial system, like

resistance SVR and compliance Ca . The so-called decay time method is a common technique

for the estimation of arterial compliance [145]. Since the time constant τ of the exponential

pressure drop described by the Windkessel models equals Rp times Ca , τ is supposed to hold

information on both peripheral resistance and arterial compliance. Thus, to obtain an estimate

of Ca , the decay time τ is identified by fitting the analytic function to the measured diastolic

pressure. If Rp is known, Ca can subsequently be derived. Even though P∞ is not considered in

the classic decay time method [118], the results of this work demonstrate that, by determining

both τ and P∞ from a measured diastolic pressure drop, the estimate of τ might become more

accurate. From this estimate, as with the classic decay time method without P∞ , Ca can then

be computed from the knowledge of SVR using the relation given in equation (A.2) to derive Rp .

In conclusion, P∞ influences the model parameter Rp only and mainly affects the diastolic part

of the pressure described by the three different Windkessel models. Thereby, P∞ might actually

improve the model accuracy during diastole, while leaving the general qualitative behaviour

unchanged, which could be beneficial for any kind of parameter identification. However, the

physiological meaning as well as the appropriate size of P∞ still need to be determined in future

studies.

Appendix B

velocity

In order to solve the 1D model of blood flow introduced in section 2.4.1, it was assumed that the

pulse wave velocity c defined in equation 2.51 is constant. This assumption will be investigated

in more in detail in this chapter. Therefore, first, an equation for pulse wave velocity introduced

by Bramwell and Hill in 1922 [9] will be derived from the Moens-Korteweg pulse wave velocity

(PWV) given in equation (2.4). Then it will be shown that, for small increases in area, c is

approximately equal to the Bramwell-Hill PWV if it is assumed to be constant.

For an inviscid fluid, the Moens-Korteweg equation introduced in equation (2.4) states that the

pulse wave velocity can be computed as

s

Eh

PWVM K = ,

2ρr0

with E denoting the incremental (angular) elastic modulus, ρ the blood density, h the stationary

wall thickness and r0 the stationary inner radius of the artery. The incremental elastic modulus

E is defined as the slope of the (presumably linear) stress-strain relation

σθ

E= ,

θ

where σθ denotes the stress and θ the strain, i.e. the deformation, in angular direction. For a

thin-walled cylinder (thickness of the wall is at least one tenth of the radius) the circumferential

or hoop stress created by an inner change in uniform pressure ∆P is given by

∆P r0

σθ = , (B.1)

h

115

Appendix B. Remarks on the pulse wave velocity 116

compare [11]. The deformation caused by the stress equals the change in circumference divided

by the stationary circumference.

θ = =

2πr0 r0

E is therefore given by

∆P r0

σθ h ∆P r02

E= = ∆r

= .

θ r0

h∆r

∆P r02

∆r = (B.2)

Eh

Hence, because of the assumed linear elasticity, r is increasing linearly with P . For the cross-

sectional area of the vessel it holds that

2 !

2∆r ∆r

A = r2 π = (r0 + ∆r)2 π = r02 π 1 + + .

r0 r0

∆r

For rather small relative changes in radius r0 , the last quadratic term can be neglected, which

equals a linearisation of A with respect to r, i.e. A is increasing linearly instead of quadratically

with r and therefore P . Using equation (B.2) leads to

2∆r 2r0

A ≈ A0 1 + = A0 1 + ∆P

r0 Eh

and hence

∂A 2r0 A0 Eh

≈ A0 ⇒ ∂A

≈ ,

∂P Eh ∂P

2r0

which finally yields the Bramwell-Hill equation for pulse wave velocity

s s

Eh A0

PWVM K = ≈ ∂A

= PWVBH . (B.3)

2ρr0 ρ ∂P

velocity

A

c2 = ∂A

,

ρ ∂P

∂A 1

a constant c implies that ∂P A is constant. This term describes the relative change in area

caused by a change in pressure, in other words the distensibility of the artery [90]. Constant c

thus implies constant distensibility.

Appendix B. Remarks on the pulse wave velocity 117

∂A 1

The rearranged equation ∂P = ρc2 A describes an exponential increase of A with pressure

P −P0

A(P ) = A0 e ρc2 ,

for A(P0 ) = A0 . Examples for the resulting distension of the cross sectional area for different

values of c are shown in figure B.1.

Linearisation of the above equation at the stationary pressure P0 , i.e. approximating the expo-

−P0 P −P0

nential term by exp( Pρc 2 ) ∼ 1 + ρc2 , yields the Bramwell-Hill PWV

P −P0 ∆P ∂A 1 A0

A(P ) = A0 e ρc2 ∼ A0 1 + 2 ⇒ ∼ A0 2 ⇒ c2 ∼ ∂A = PWV2BH . (B.4)

ρc ∂P ρc ρ ∂P

The error introduced by linearisation depends on the magnitude of the factor ∆P/ρc2 and con-

sequently on the resulting relative dilation of the artery A/A0 , compare figure B.1. Hence, for

small (relative) changes in cross-sectional area A, it holds that

c ≈ PWVBH .

In the previous section, a very similar condition was needed to derive the Bramwell-Hill PWV

from the Moens-Korteweg PWV, namely a small relative change in radius r. Hence, taken

together, it follows that

c ≈ PWVBH ≈ PWVM K ,

for small relative changes in the cross-sectional area. The pulse wave velocity obtained by

applying the method of characteristics to the 1D-model of blood flow derived in section 2.4.1 is

thus, in a first approximation, equal to the pulse wave velocities derived by Bramwell and Hill

or Moens and Korteweg.

Appendix B. Remarks on the pulse wave velocity 118

c=5 m/s

c=7.5 m/s

1.6

c=10 m/s

1.5

1.4

A/A0

1.3

1.2

1.1

1

0 10 20 30 40 50 60 70 80 90 100

P−P0, mmHg

Figure B.1: Distension of the cross-sectional area for a constant pulse wave velocity as a function

of pressure. ρ was set to 1060kg/m3 . For a stiffer artery (c = 10m/s, red line), an increase

in pressure by 100 mmHg leads to a distension of 1.13 times the original area, whereas in the

more elastic case (c = 5m/s, blue line), the area increases by a factor of 1.67. The dashed lines

represent the linearised equations.

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Abbreviations

AP Augmented Pressure PPAmp Pulse Pressure Amplification

AU Arbitrary Unit PSA Pressure Systolic Area

CI Confidence Interval PWA Pulse Wave Analysis

DBP Diastolic Blood Pressure PWV Pulse Wave Velocity

DPTI Diastolic Pressure Time Index rEF Reduced Ejection Fraction

ED Ejection Duration RI Reflection Index

EF Ejection Fraction

RM Reflection Magnitude

ESC European Society of Cardiology

RMSE Root Mean Square Error

HF Heart Failure

SBP Systolic Blood Pressure

HR Heart Rate

SHF Systolic Heart Failure

LV Left Ventricle; Left Ventricular

SPTI Systolic Pressure Time Index

LVETI Left Ventricular Ejection Time Index

LVSD Left Ventricular Systolic Dysfunction SV Stroke Volume

MBP Mean Blood Pressure SVR Systemic Vascular Resistance

nEF Normal Ejection Fraction WIA Wave Intensity Analysis

NT-proBNP N-terminal pro-B-type natri- WK2 2-element Windkessel

uretic peptides WK3 3-element Windkessel

ODE Ordinary Differential Equation WK4p parallel 4-element Windkessel

PDE Partial Differential Equation WSA Wave Separation Analysis

133

Nomenclature

AIx augmentation index - 10

AP augmented pressure mmHg 10

c wave speed cm/s 32

Ca total arterial compliance ml/mmHg 22

D wave late systolic/early diastolic peak in forward intensity mmHg·cm/s 39

DBP diastolic blood pressure mmHg 1

dI wave intensity mmHg·cm/s 36

dIf,b forward and backward wave intensity mmHg·cm/s 39

DPTI diastolic pressure time index mmHg·s 14

E Young’s modulus, elastic modulus mmHg 11

~e error vector used in the objective function of the parame- 50

ter identification procedure

Ew wasted effort s·mmHg 14

ED ejection duration ms 9

EF ejection fraction % 2

h wall thickness cm 11

HR heart rate bpm 9

L total arterial inductance mmHg·s2 /ml 109

LVETI left ventricular ejection time index ms 71

MBP mean blood pressure mmHg 9

ñ individual threshold used for the definition of Zmodel 47

ωn n-th harmonic angular frequency rad/s 15

P arterial blood pressure mmHg 9

P∞ asymptotic pressure level mmHg 23

PLZ auxiliary variable used in the 4-element Windkessel model mmHg 26

to describe the pressure across the LZ-component

PRC auxiliary variable used in the Windkessel models to de- mmHg 22

scribe the pressure across the RC-component

Pbn n-th complex Fourier coefficient (phasor) of pressure P 16

P = Pb0 , mean blood pressure mmHg 15

134

Nomenclature 135

P2 second shoulder in the pressure wave mmHg 10

Pf,b forward, backward travelling pressure wave mmHg 19

|Pf,b | amplitude of Pf,b mmHg 20

PP pulse pressure mmHg 9

PPAmp pulse pressure amplification - 13

∆PSA pressure systolic area related to ventricular ejection mmHg·s 14

PSA pressure systolic area mmHg·s 14

PWV pulse wave velocity m/s 11

Q volumetric blood flow ml/s 15

Q

bn n-th complex Fourier coefficient (phasor) of blood flow Q 16

Q =Q

b 0 , mean blood flow ml/s 15

Q̃ modelled blood flow ml/s 47

R wave mid-systolic peak in backward intensity mmHg·cm/s 39

r radius cm 11

Rp peripheral resistance s·mmHg/ml 22

ρ blood density g/ml 11

RI reflection index - 20

RM reflection magnitude - 20

S wave early systolic peak in forward intensity mmHg·cm/s 39

SBP systolic blood pressure mmHg 1

σ = L/Zc , time constant of the exponential decay of PLZ s 48

during diastole

∆SPTI pressure systolic area below DBP mmHg·s 14

SPTI systolic pressure time index mmHg·s 13

SV stroke volume ml 45

SVR systemic vascular resistance mmHg·s/ml 17

T heartbeat duration s 9

Tr round trip travel time ms 11

ts time of end of systole s 14

τ = Rp Ca , time constant of the exponential decay of PRC s 48

during diastole

U arterial blood flow velocity cm/s 29

Z0 complex characteristic impedance, frequency-dependent 18

Zc real characteristic impedance, frequency-independent mmHg·s/ml 18

Zin complex input impedance, frequency-dependent 17

Zmodel complex model impedance, frequency-dependent 46

List of Publications

Journal articles

T. Weber. Determinants and covariates of central pressures and wave reflections in systolic

heart failure. International Journal of Cardiology, 190(0):308–314, 2015.

invasive wave reflection quantification in patients with reduced ejection fraction. Physio-

logical Measurement, 36(2):179–190, 2015.

on the Windkessel Models of the Arterial System. IFAC-PapersOnLine, 48(1):17–22, 2015.

[4] S. Parragh, B. Hametner, and S. Wassertheurer. Simulating aortic blood flow and pressure

by an optimal control model. SNE Simulation Notes Europe, 23(2):101–106, 2013.

Y. von Kodolitsch, and S. Wassertheurer. Ambulatory (24 h) blood pressure and arterial

stiffness measurement in marfan syndrome patients: a case control feasibility and pilot

study. BMC Cardiovascular Disorders, 16:81, 2016.

E. Rietzschel, and S. Wassertheurer. Assessment of model based (input) impedance, pulse

wave velocity, and wave reflection in the asklepios cohort. PLoS ONE, 10(10):e0141656,

10 2015.

to assess pulse wave velocity: comparison with the invasive gold standard and relationship

with organ damage. Journal of Hypertension, 33(5):1023–1031, 2015.

stiffening possible? American Journal of Hypertension, Invited commentary, submitted for

publication.

136

List of Publications 137

Zusammenhang zwischen aortalem Rerservoirdruck und kardiovaskulären Ereignissen in

Patienten mit systolischer Herzinsuffizienz. Poster presentation at the ÖGH Jahrestagung

2016, Vienna, Austria, 2016-11-18 – 2016-11-19.

tionship of reservoir pressure to cardiovascular events in patients with heart failure with

reduced ejection fraction. In Artery Research, 16:55, 2016. Talk at the Artery 16, Copen-

hagen, Denmark, 2016-10-13 – 2016-10-15.

invasive wave reflection quantification in patients with reduced ejection fraction. Invited

talk at the MPEC, Medical Physics and Engineering Conference 16, Manchester, UK,

2016-09-12 – 2016-09-14.

on the Windkessel Models of the Arterial System. In IFAC-PapersOnLine, 48(1):17–22,

2015. Talk at the MATHMOD 2015 - 8th Vienna International Conference on Mathemat-

ical Modelling, Vienna, Austria, 2015-02-08 – 2015-02-20.

[5] S. Parragh, B. Hametner, T. Weber, B. Eber, and S. Wassertheurer. The decay of aortic

blood pressure during diastole: Influence of an asymptotic pressure level on the exponential

fit. In Artery Research, 8(4):162, 2014. Poster presentation at the Artery 14, Maastricht,

Netherlands, 2014-10-09 – 2014-10-11.

Curriculum Vitæ

Personal Information

Citizenship Austria

Email stephanie.parragh@gmx.at

Education

Doctoral programme in Engineering Sciences - Technical Mathematics

Diploma programme Technical Mathematics

Diploma thesis ”Modelle zur Bestimmung des aortalen Blutflusses basierend auf

Optimalitätsbedingungen”

Graduated as Diplom-Ingenieurin

Applied Modelling and Simulation and Decision Making (AMSDM) Masterschool

Erasmus Programme

EF Academic Year Abroad

Humanistic secondary school

School pilot project with a main focus on languages

Graduated with honours

138

Curriculum Vitæ 139

Work Experience

Lecturer at the Institute for Analysis and Scientific Computing

Courses: “Modelling and Simulation of the Heart Circulation” and

“Preparatory Course Mathematics”

Health & Environment Department, Biomedical Systems

Research Fellow

Health & Environment Department, Biomedical Systems

Internship (FEMtech scolarship)

Teaching Assistant at the Institute for Analysis and Scientific Computing

Courses: “Mathematics for Electrical Engineers” (1 and 2) and

“Preparatory Course Mathematics”

Student member of the Association for Research into Arterial Structure and Phys-

iology.

for the best paper published in Physiological Measurements in 2015, awarded by

IOP Publishing in association with IPEM (Institute of Physics and Engineering

in Medicine).

Research Skills

Medical statistics

ematica, AnyLogic, R

Language Skills