Table Of ContentUniversidade de Aveiro Departamento de Electr(cid:19)onica e Telecomunica(cid:24)c~oes
1999
Universidade de Aveiro Departamento de Electr(cid:19)onica e Telecomunica(cid:24)c~oes
1999
Carlos Alberto da A model for the simulation of Doppler ultrasound
Costa Bastos signals from pulsatile blood (cid:13)ow
Um modelo para a simula(cid:24)c~ao de sinais Doppler
ultra-s(cid:19)onicos provenientes de (cid:13)uxo sangu(cid:19)(cid:16)neo
puls(cid:19)atil
Thesis submitted to the Universidade de Aveiro for the degree of Doctor of
PhilosophyinElectricalEngineeringunderthesupervisionofMr. PeterFish,
Reader in the School of Electronic Engineering and Computer Systems of
the University of Wales{Bangor, United Kingdom, and Dr. Francisco Vaz,
Professor of the Departamento de Electro(cid:19)nica e Telecomunicac(cid:24)~oes of the
Universidade de Aveiro
Disserta(cid:24)c~ao apresentada (cid:18)a Universidade de Aveiro para cumprimento dos
requesitos necess(cid:19)arios (cid:18)a obtenc(cid:24)~ao do grau de Doutor em Engenharia Elec-
trot(cid:19)ecnica, realizadasobaorienta(cid:24)c~ao cient(cid:19)(cid:16)(cid:12)ca deMr. PeterFish,Professor
daSchool ofElectronic Engineering andComputerSystemsdaUniversityof
Wales{Bangor, Reino Unido, e do Dr. Francisco Vaz, Professor Catedr(cid:19)atico
do Departamento de Electr(cid:19)onica e Telecomunicac(cid:24)~oes da Universidade de
Aveiro
o jur(cid:19)(cid:16) / examiners committee Prof. Doutor Casimiro Adri~ao Pio
presidente / president professor catedr(cid:19)atico de Universidade de Aveiro por delega(cid:24)c~ao do Reitor da
Universidade de Aveiro
Prof. Doutor Francisco Ant(cid:19)onio Cardoso Vaz
professor catedr(cid:19)atico da Universidade de Aveiro (orientador)
Prof. Doutor Jos(cid:19)e Alberto dos Santos Rafael
professor associado da Universidade de Aveiro
Prof. Doutor Ant(cid:19)onio Miguel Pontes Pimenta Monteiro
professor auxiliar da Faculdade de Engenharia da Universidade do Porto
Prof. Doutor Augusto Marques Ferreira da Silva
professor auxiliar da Universidade de Aveiro
Prof. Doutor Jos(cid:19)e Carlos da Silva Cardoso
professor auxiliar da Universidade de Tr(cid:19)as-os-Montes e Alto Douro
Mr. Peter John Fish
Reader na School of Electronic Engineering and Computer Systems da Uni-
versity of Wales-Bangor, Reino Unido (co-orientador)
agradecimentos / I would like to express my most sincere thanks to Mr. Peter Fish and
acknowledgements Prof. Dr. Francisco Vaz for their supervision, critical suggestions, support,
assistance,patienceandadvicethroughoutthecourseofthiswork. Without
their help and support this work would probably never exist.
I thank all my colleagues but specially Tom(cid:19)as Oliveira e Silva, Osvaldo
Pacheco and Lu(cid:19)(cid:16)s Almeida at Aveiro, and Robin Steel and Jos(cid:19)e Carlos
Cardoso at Bangor, for their friendship, support and encouragement. The
help of Tom(cid:19)as with the Latex word processor and daily incentive during the
last stages of writing up are gratefully acknowledged. Also Robin’s help in
(cid:12)nding a solution for the integral in Appendix A has to be mentioned.
IthanktheUniversidadedeAveiro,theDepartamentodeElectro(cid:19)nicaeTele-
comunica(cid:24)c~oes, the School of Electronic Engineering andComputer Systems
of the University of Wales{Bangor, and INESC Aveiro, for providing the
means and the environment that made this work possible. I extend my
thanks to the sta(cid:11) members of these institutions that contributed in any
way to my work.
The (cid:12)nancial support of Funda(cid:24)c~ao para a Ci^encia e Tecnologia (formerly
JNICT) through a CIENCIA grant extended to aPRAXISgrantisgratefully
acknowledged. ThePRAXISgrantbene(cid:12)tedfromthesupportoftheScience
nd
and Technology Subprogram of the 2 Community Support Framework
o
(Sub-Programa Ci^encia e Tecnologia do 2 Quadro Comunit(cid:19)ario de Apoio).
Thanks are also due to Fundo Social Europeu for supporting Universidade
de Aveiro with a PRODEP grant that provided the (cid:12)nancial means for part
of my work.
I would like to thank my parents, my sister and my grandparents for their
love, support and encouragement during the course of this work.
Veryspecialthanksandloveto mywifeOlgaandmydaughterIn^es for their
unconditional love and patience throughout the course of this work and the
long absences at Bangor. Olga’s sacri(cid:12)ce of her own professional career to
join me in Bangor for a complete year is also gratefully acknowledged.
Resumo O detector ultra-s(cid:19)onico de (cid:13)uxo sangu(cid:19)(cid:16)neo usa o efeito Doppler para estimar de
forman~aoinvasivaavelocidadedosanguenacircula(cid:24)c~ao. Temsidobastanteusado
nas (cid:19)ultimas quatro d(cid:19)ecadas para detectar a presen(cid:24)ca de estenoses.
Odesenvolvimentodenovast(cid:19)ecnicasdeprocessamentodosinal Dopplernecessita
de sinais de teste cujas caracter(cid:19)(cid:16)sticas sejam conhecidas ou possam ser medidas
com precis~ao. Isto (cid:19)e dif(cid:19)(cid:16)cil de obter com sinais Doppler medidos in vivo devido
(cid:18)a elevada varia(cid:24)c~ao do (cid:13)uxo sangu(cid:19)(cid:16)neo de pessoa para pessoa e tamb(cid:19)em com o
estado (cid:12)siol(cid:19)ogico da pessoa no momento da medida, por exemplo a tens~ao arte-
rial in(cid:13)uencia signi(cid:12)cativamente o (cid:13)uxo sangu(cid:19)(cid:16)neo. Um modelo para gerar sinais
Dopplersimuladoscujascaracter(cid:19)(cid:16)sticassejamcontrol(cid:19)aveise/oumensur(cid:19)aveis(cid:19)euma
ferramenta bastante (cid:19)util,pois permite que as novas t(cid:19)ecnicasde processamento do
sinalDopplersejamtestadasemcondic(cid:24)~oescontroladas. Permite,tamb(cid:19)em,estudar
oefeitodev(cid:19)ariosfactoresqueafectamoespectrodosinalDoppler. Habitualmente
oefeito individualdosv(cid:19)ariosfactoresn~aopodeseridenti(cid:12)cadoquandos~ao usados
sinais medidos in vivo.
Nestetrabalho foi desenvolvidoum modeloparagerar sinais Doppler ultra-so(cid:19)nicos
simulados. O modelo cont(cid:19)em dois sub-modelos, um para o (cid:13)uxo sangu(cid:19)(cid:16)neo nos
membros inferiores de um ser humano e outro para gerar os sinais simulados a
partir do campo de velocidades do sangue e das caracter(cid:19)(cid:16)sticas do instrumento.
O (cid:13)uxo sangu(cid:19)(cid:16)neo nos membros inferiores foi simulado com um an(cid:19)alogo el(cid:19)ectrico
para a rede vascular dos membros inferiores. Cada art(cid:19)eria foi simulada por uma
linha de transmiss~ao com perdas e as redes vasculares perif(cid:19)ericas por um circuito
Windkessel com tr^es elementos. O circuito el(cid:19)ectrico foi implementado com o
simulador de circuitos SPICE.
Para simular a interac(cid:24)c~ao entre os gl(cid:19)obulos vermelhos e o campo de ultra-sons o
vaso sangu(cid:19)(cid:16)neo foi dividido em pequenos volumes elementares. As contribuic(cid:24)~oes
dosvolumeselementaresforamtodassomadasparagerarosinalDopplersimulado.
O modelo fez algumas aproxima(cid:24)c~oescomo sejam, por exemplo, considerar o (cid:13)uxo
sangu(cid:19)(cid:16)neo laminar e sem rota(cid:24)c~ao.
As caracter(cid:19)(cid:16)sticas dos sinais gerados pelo modelo s~ao bastante parecidas com as
esperadasparaosinalDopplerreal. Omodelodesenvolvidofoiusadoparaestudar
a in(cid:13)u^encia que a acelera(cid:24)c~ao sangu(cid:19)(cid:16)nea, o tamanho do volume de amostragem e
a dura(cid:24)c~ao da janela de amostragem t^em na largura de banda e(cid:12)caz do espectro
do sinal Doppler. Foi deduzida uma f(cid:19)ormula que estima a largura de banda e(cid:12)caz
a partir das contribui(cid:24)c~oes individuais do alargamento espectral devido (cid:18)a n~ao esta-
cionaridade, do alargamentoespectral intr(cid:19)(cid:16)nseco, do alargamentoespectral devido
(cid:18)a dura(cid:24)c~ao da janela de amostragem e ainda da gama das velocidades que passam
pelo volume de amostragem.
Foram, ainda,deduzidasexpress~oesemformafechadaparaoespectrodepot^encia
do sinal Doppler devido unicamente (cid:18)a gama de velocidades que atravessam um
volume de amostragem com forma Gaussiana colocado num per(cid:12)l de velocidades
com forma expon^encial. Foram, tamb(cid:19)em, obtidas express~oes para a largura de
banda e(cid:12)caz no caso especial do volume de amostragem Gaussiano ter simetria
esf(cid:19)erica e estar colocado no centro do vaso sangu(cid:19)(cid:16)neo.
Abstract The Doppler ultrasonic blood (cid:13)ow detector estimates non-invasively the
velocity of blood in the circulatory system. It has been extensively used in
the last four decades for the detection of stenoses in the circulation.
Thedevelopment ofnewsignal processing techniques for theDopplersignal
requires test signals with known or measurable characteristics. This is very
di(cid:14)cult to achieve with Doppler signals obtained in vivo because of the
variability of blood (cid:13)ow between persons and with physiological state, for
example blood pressure. A model for generating simulated Doppler signals
whose characteristics are controllable and/or measurable is a useful tool
because it permits the test of new processing techniques under controlled
conditions. It permits also the study of the e(cid:11)ect of various factors on the
Doppler spectrum. Usually these e(cid:11)ects cannot be isolated with in vivo
measurements.
DuringthisworkamodelforthegenerationofsimulatedDopplerultrasound
signals was developed. It comprised two sub-models one for blood (cid:13)ow in
the human lower limb and the other for generating simulated signals from
the blood velocity (cid:12)eld and the instrument’s characteristics.
Blood (cid:13)ow in the lower limb was modelled by an electric analogue for the
lower limb vascular tree. Each artery was modelled by a lossy transmission
line and the peripheral vascular beds by three{element Windkessel mod-
els. The electric analogue circuit was implemented with the SPICE circuit
simulator.
To simulate the inter-action of the blood cells with the ultrasonic (cid:12)eld the
vessel was divided into small elemental volumes whose contributions were
added together to generate the simulated Doppler signal. The model as-
sumed irrotational laminar (cid:13)ow and some other simplifying approximations.
The characteristics of the signals generated by the model were similar to
those expected for the Doppler signal. The model was used to study the in-
(cid:13)uenceofbloodacceleration,samplevolumesizeanddatasegmentduration
on the root mean square (rms) width of the Doppler spectrum. A simple
formula was derived for estimating the Doppler rms spectral width from the
individual contribution of non-stationarity broadening, intrinsic broadening,
window broadening and the range of blood velocities passing through the
sample volume.
InadditionclosedformexpressionswerederivedfortheDopplerpowerspec-
trum due solely to the range of blood velocities passing through a Gaussian
sample volumes placed in irrotational laminar (cid:13)ow with a velocity pro(cid:12)le
obeying a simple power law. Closed form expressions were also obtained
for the root mean square spectral width in the special case of a spherically
symmetric Gaussian sample volume placed in the centre of the vessel.
(cid:18)
A Olga,
(cid:18)a In^es, (cid:18)a Ana Raquel
e aos meus pais
Contents
List of Figures v
List of Tables ix
List of Symbols xi
List of Acronyms xix
List of Publications xxi
1 Introduction 1
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Thesis organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Main contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Background 7
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Blood (cid:13)ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.1 The circulatory system . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.2 Types of blood (cid:13)ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.3 The e(cid:11)ects of geometric changes . . . . . . . . . . . . . . . . . . . . . 15
2.2.4 Models of arterial blood (cid:13)ow . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Doppler ultrasound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.1 Ultrasound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.2 The Doppler e(cid:11)ect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.3 Doppler ultrasound instruments. . . . . . . . . . . . . . . . . . . . . . 25
2.3.4 The Doppler spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3.5 Models for the Doppler signal backscattered from moving blood. . . . 38
2.4 Doppler signal spectral estimation . . . . . . . . . . . . . . . . . . . . . . . . 42
2.4.1 Spectral estimation basics . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.4.2 The periodogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.4.3 Parametric methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
i.
ii Contents
2.4.4 Time-frequency transforms . . . . . . . . . . . . . . . . . . . . . . . . 50
2.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3 Model of blood (cid:13)ow in the human lower limb 55
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2 Lower limb arterial bed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2.1 Some characteristics of the pressure and (cid:13)ow pulses in the lower limb 57
3.3 Introduction to the SPICE circuit simulator . . . . . . . . . . . . . . . . . . . 59
3.4 SPICE model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.4.1 The input waveform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.4.2 Arteries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.4.3 Peripheral arterial beds . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.4.4 Adjustment of model parameters . . . . . . . . . . . . . . . . . . . . . 67
3.5 Assessment of model results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.5.1 The complete model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.5.2 Input impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.5.3 Pressure and (cid:13)ow waveforms . . . . . . . . . . . . . . . . . . . . . . . 70
3.5.4 Pulsatility Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.6 Stenoses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4 Doppler ultrasound signal model 77
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2 Model description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.2.1 Signal from a single scatterer . . . . . . . . . . . . . . . . . . . . . . . 79
4.2.2 Signal from an elemental volume . . . . . . . . . . . . . . . . . . . . . 80
4.3 Ensemble averaged Doppler spectrum . . . . . . . . . . . . . . . . . . . . . . 82
4.4 Time-varying blood velocity pro(cid:12)les . . . . . . . . . . . . . . . . . . . . . . . 84
4.5 Implementation issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.6 Simulation experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.6.1 Assessment of model results . . . . . . . . . . . . . . . . . . . . . . . . 89
4.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5 Doppler power spectrum from a Gaussian sample volume 97
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.2 Derivation of the Doppler spectrum. . . . . . . . . . . . . . . . . . . . . . . . 98
5.2.1 Wide uniform beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.2.2 Gaussian sample volume . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.2.3 Symmetric sample volume . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.2.4 Sample volume centred in the vessel . . . . . . . . . . . . . . . . . . . 103
5.2.5 Symmetric sample volume centred in the vessel . . . . . . . . . . . . . 103
Contents iii
5.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.4 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.4.1 Non-symmetric sample volumes . . . . . . . . . . . . . . . . . . . . . . 107
5.4.2 Sample volumes with some symmetry . . . . . . . . . . . . . . . . . . 110
5.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6 Spectral broadening in the Doppler signal|a model based study 113
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.2 Separation of factors a(cid:11)ecting the Doppler spectrum . . . . . . . . . . . . . . 114
6.2.1 E(cid:11)ect of window and acceleration. . . . . . . . . . . . . . . . . . . . . 115
6.2.2 E(cid:11)ect of velocity pro(cid:12)le and sample volume size. . . . . . . . . . . . . 119
6.2.3 Intrinsic spectral broadening. . . . . . . . . . . . . . . . . . . . . . . . 120
6.2.4 Variation of acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.2.5 Approximate spectral width. . . . . . . . . . . . . . . . . . . . . . . . 121
6.3 Simulation experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.4 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.4.1 Single streamline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.4.2 Velocity pro(cid:12)le . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7 Conclusion 133
7.1 General conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.2 Recommendations for future work . . . . . . . . . . . . . . . . . . . . . . . . 135
A Evaluation of function M(a;b;(cid:12)) from chapter 5 137
References 139
