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Modeling and Analysis of Ultrasound Backscattering by Red Blood Cell Aggregates with

a System-Based Approach

Beng-Ghee Teh Department of Electrical Engineering

McGill University, Montreal

March, 1998

A thesis submitted to the Faculty of Graduate Studies and Research in partial fuüiliment of the requirements for the degree of Master of Engineering.

O Beng-Ghee Teh, 1998

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McGill University

Faculty of Graduate Studies and Research

Modeiing and Analysis of Ultrasound Backscattering by Red Blood Ceii

Aggngates witb a System-Based Approaeh

Thesis presented by:

Beng-Ghee Teh

and evaluated by:

Dr. Guy Cloutier supervisor

Dr. Howard C. Lee CO-supervisor

Dr. Richard S. C . Cobbold extemal examiner

Thesis accepted on April20,1998

The fear ofGod is the beginnzng of knowledge ...

Proverbs Ir 7

Abstract

The present study concemi the modeling and analysis of ultrasound bacb t te r ing

by red blood ce11 aggregates, which under pathological conditions play a signincant role in

the rheology of blood within human vessels. A theoretical model based on the convolution

between a tissue matrix and a point spread funciion, representing respectively the RBC

aggregates and the characteristics of the ultrasound system, was used to examine the

influence of the scatterer shape and size toward the backscattered power. Both scatterers

in the fom of clumps of red biood ceil aggregates and rouleaux were modeled. The

sirnulated results were used to postulate potential scatterer shape and sue teading to the

"black hole" phenomenon, a hypoechoic zone observed in ultrasonic B-mode images. It is

concluded that the ultrasound backscattered power does not always increase with the size

of the aggregates, especially when they are no longer Rayleigh scatterers. New potential

causes of the "black hole" phenomenon werr also proposed based on the model, in

addition to the proposais suggested by earlier researchers.

Résumé

Ce mémoire porte sur la modélisation et l'analyse du signal ultrasonore rétrodiffisé

par des agrégats de globules rouges qui, dans des conditions pathologiques, jouent un rôle

important dans la rhéologie du sang dans les vaisseaux sanguins. Un modèle théorique,

basé sur la convolution d'une matrice tissulaire servant à décrire les propriétés des

agrégats et d'une matrice représentant les propriétés du système d'ultrason, a été utilisé

pour examiner l'effet de la forme et de la taille des d i b u i s sur la puissance ultrasonore

rétrodi ffisée. Des difhseurs ayant la fonne d'agrégats sphériques de globules rouges et la

forme de rouleaux ont été modelis&. Les simulatioas avaient principaiement pour but

l'étude de l'effet de la fonne et de la taille des agrégats sur les caractéristiques d'un

phénomène acoustique nommé "trou noir". Ce phénomène observé sur des images

échographiques en mode B de vaisseaux sanguins simulés est caractérisé par des variations

de l'intensité de l'écho à l'intérieur de l'image et la présence d'une zone de faible intensité

au centre du tube. Les résultats montrent, lorsque la relation entre la taille des agrégats et

la longueur d'onde du faisceau d'ultrason ne suit plus la loi de Rayleigh, que la puissance

rétrodiffusée n'augmente pas nécessairement avec la taille de ces derniers. De nouvelles

hypothèses portant sur la genèse du ''trou noir" sont proposées aimi que la suggestion

faite par d'autres chercheurs à ce sujet au cours des dernières années.

Acknowledgment

1 would like to express my utmost gratitude to God for His grace and mercy, and

also for pmviding such a manrelous opp tun i ty to pursue post-graduate studies in this

laboratory. 1 would also like to give th& to Him for the constant source of strength,

encouragement and peace 1 fimi in His Word during this period of tirne- Without Him, this

degree would not have been possible.

1 would Iike to thank my thesis supervisor Prof. Guy Cloutier for his technical

leadership, patience, and encouragement throughout this entire period. His guidance

played a pivotal role in the sumssful completion of this thesis. 1 thadc Guy for providing

me with the opportunity to explore different technical areas, and for always k i n g there to

help and support. 1 would also like to thank my other thesis supewisor, Prof. Howard C.

Lee for his guidance, assistance and encouragement, especiall y du ring the beginning of my

M.Eng. program. 1 wish to thank both of my thesis supervisors for their careful revision of

this thesis. Without them, this thesis would not be in its current form today. I would also

li ke to thank Prof. Ooi for k i n g my CO-supervisor with Prof. Cloutier during the last stage

of my M.Eng. program.

1 am indebted to Prof. Michel Bertrand at the Institut de Génie Biomédical (IGB)

of the École Polytechnique de Montréal for always k i n g there to prwide technical

assistance and to share his insight. 1 also wish to thank Michel for providing the access to

the workstations as well as the cornputer programs developed at IGB.

1 would like to express my th& to Jocelyn Durand, Louis Allard, Luc

Dandeneau, and Richard Cimon at the Laboratory of Biomedical Engineering of the

Institut de recherches cliniques de Montréal (IRCM) for their patience and help in the

building of the flow loop mode1 as well as the interface circuit for data acquisition. Their

willingnes to go the extra mile to help was invaluable to the successful completion of the

experimental setup that was complementary to this thesis.

1 wish to take this opporhmity to expnss my gratitude to Prof. Lxniis-Gilles

Durand, Francine Durand, Dr. Zhenyu Guo, Marjan Yazdanpanah, Dr. Raja Bedi,

Danmin Chen, Xuan Zhang, Dany Leong Kon, Zhao Qin, Dr. Xiaoduan Weng, Isabelle

Fontaine, Dr. Herkole Sava, Damien Garcia, Frederic Sakr and Dr. Philippe Pibarot for

their many insightful discussioos, their friendship, and for making this laboratory at IRCM

a stimulating environment for reseaxch and advance 1eamiDg.

1 am also indebted to the Medical Research Council of Canada, the Whitaker

Foundation of USA, and the Heart and Stroke Foundation of Quebec for providing the

research funds throughout the entire course of my studies.

Finally, many t h h are due to my pastors, ail my Enends at the Christian

fellowships at McGill as well as the Taiwanese Presbyterian Church of Montreal, and my

family for their constant support, prayers and encouragement.

Contents

Abbreviated Lbt of Symbob œoœœœooooœœœœoœœwoœœoœœœoaœœoooooœoœoœœoœœœooooooaœœooœœoooooœooœoœoœœœœwœooœo-aœoœooo~ii

List of F i g ~ m ~ooo~œœaoœœœœœœ~œœoœooooœœœœœwœooœœœoooœoooœooooœoœœoœoœoœœoœœooooœoooœoœooœooœœoœœœœœœoœœooowœœooœoœœœœm~iii

Lbt OC Tables oœooœœœoœoœooœooœœœœœœ.œœœœœoœœoœoœeoeooœoœœooooooaœœœoaoooaooaoœoaoœœooooœooœooœœoœœooœœ.œoooœmœœœoœœmœœoœœœoo~fi

Chapter 1: Bickgt~uid and Obje~tive~ œ~aœo~œœ~oœœœoooœoœœoooœœœoœoooœoœoo~œœœoooœooooœoœœooooaœoooooo 1

1.1 The Rationale Behind the Study of RBC Aggregation ............................................ 1

1.2 Methods Roposed for the Study of RBC Aggregation .......................................... 5

1.2.1 Microscopie Obsenration ................................................................................ -5

......................................................... 1.2.2 Erythrocyte Sedimentation Rate (ESR) 6

.............................................................. 1.2.3 Direct Observation Under Shear Flow 6

................................................ 1.2.4 Laser Light Reflection Method ............... .... 7

................................................................. 1.2.5 Ultrasound Backscatteriog Methods 8

................................................................................................... 1.3 Proposed Study 10

1.3.1 A Brief Review on the Hypoechogenic Zone Discovery ................................. 10 1.3.2 Summary .................................................................................................. 14

1.4 Objectives ............................................................................................................ 15 Chapter 2: Tbeory, Implementation and Methodology ................................O............ 16

2.1 Scattering of Ultrasound by Blood ...................... .. ...................................... 1 6

..................................................... 2.1.1 RBCS as Scattering Targets ........... . 1 6

........................................................................................ 2.1.2 Rayleigh Scattering 17

2.1.3 The Modeling of Ultrasound Bacbcattering by RBCs .................................... 19 .................................... 2.1.3.1 The Particle Approach ...................... .. .... ......... 1 9

........................................ 2.1.3.2 The Continuum Approach

2.1.3.3 The Hybnd Appmach .................. .. ..................................................... -22

2.2 The Modeling of Ultrasound Backscattering by RBCs with the Bystem-

........................................................................ Based Approach ................ ....... 22

2.3 Implementation ..................................................................................................... 29

2.4 Methodology ........................................................................................................ 36

....................................................... 2.4.1 Verification of the Validity of the Mode1 3 8

2.4.1.1 Isotropy / Anisotropy Due to the Structure of the Scatterers .................... 38

....................... ............ 2.4.1 2 Power vs . Scatterer Volume Relationship ...... -42

2.4.2 The Effects of the Scatterer Structure and Size on the Signal Power .............. 43

2.4.3 The Potential Causes of the "Black Hole" Phenomenon ................................. 44

2.5 Summary ........................ ... ............................................................................... 44

Chapter 3: Resulîs oooœoooœooooœœoooœœooooœoooœooooooooœooœoooooooœoooooœooœoooooooooooœoooooœoooooooœooootooooooo.oooœo-œ45

3.1 Verification of the Validity of the System-Based Model ..................................... -45

.......................... 3.1.1 Isotropy / Anisotropy Due to the Structure of the Scatterers 45

3.1.2 Power vs . Scatterer Volume Relationship ....................................................... 47

3.1.3 Summary ....................................................................................... ............. 48

3.2 The Effects of the Scatterer Size and Structure on the Power ............................... 49

................................................. 3.3 The "Black Hole" Phenomenon: Potential Causes 52

3.4 Summary ........................................................................................................... 6 6

Chapter 4: D ~ ~ ~ ~ ~ b ~ ~ o ~ a o o o . o o ~ o o o o ~ o o œ o o o o o o o o o o o o œ o o œ o o o o o œ o o o o o o o o o o œ o o o o œ o o o o o o œ o œ o o o o œ œ o o o o o o o o o o 67

......................................................................................... 4.1 Analysis of the Results -67

................. 4.1.1 Power vs . Volume Relationship €rom a System-Based Perspective 67

4.1.1.1 Anisotropic cylindrical scatterers ......................... .... ........................... 67

..................... 4.1.1.2 Effects of the insonififation angle for anisotropic scatterers 70

.................................................................... 4.1.1.3 Isotropie spherical scatterers 71

4.1.1.4 Other considerations ................................................................................ 72

4.1.1.5 Simulation of a more realistic tissue image .......................... .... ..... 7 3

4.1.2 Most Probable Causes of the "Black Hole" Phenornenon ............................ ...75 4.2 The Strengths and Limitations of the Mode1 ......................................................... 79

4.3 Summary .......................... .. ........................................................................... 8 0

Chnpter 5: Conelusbn ................................................................................................. 82

References .................................................................................................................... 84

Abbreviated List of Symbols

BSC

ESR

FFT

IFFr

MAI

PSF

i? RBC

RF

ROI

s 10

s 6 0

tA

tF

VOI

backscattering coefncient

erythrocyte sedimentation rate

fast Fourier transform

inverse fast Fourier transfonn

micrmcopic aggregation index

point spread function

correlation coefficient

red blood ceIl

radio frequency

region of interest

the mean kinetic index at 10 seconds

the mean kinetic index at 60 seconds

the primary aggregation time

the final aggregation time

volume of interest

the scattering cross-section

List of Figures

Fig. 1.1 Diluted blood sampIe (20 % hematoctit) nom a patient with coronary

artery disease show ing heavil y networked RBC aggregates (magnified

330 times). Couaesy of Dr. Xiaoduan Weng .......................,.... ..........O............ 2

Fig. 1.2 Diluted blood sample (20 % hematocrit) fiom a normal subject showing

some rouleaux of a few to several RBCs (magnified 375 times) 1531 .................. 3

Fig. 1.3 The vicious cyde triggered by reduced blood flow due to increased RBC

aggregation level. Adapted h m (511. ................... ....... .................... 4

Fig. 1.4 B-mode images of 28 % hematocrit porcine whole blood shwing the

hypoechogenic zone ("black hole") at the center of the vessei. Both

images were obtained at an entrance distance 60 times the diameter of

the tube with a mean velocity at 1.4 cm/s ..................... .. ....................... 1 2

Fig. 2.1 The rotation of the X-Y plane about the z-axis to simulate different

Fig. 2.2

Fig. 2.3

Fig. 2.4

Fig. 2.5

Fig. 2.6

insonification angles. Y represents the direction of propagation of the

ultrasonic waves and X is the axis corresponding to the beamwidth. ................ 32

An illustration of the angle of insonification fiom a physical perspective. ......... 32

A drawing of a small RBC clump and a few rouleaux. ................................... 37

The PSF of the transducer used in the present study (Eq. 2.19). ...................... 40

Sampies of tissue images at an angle of zero degree (Eqs. 2.13 and 2.21). ....... 40

RF images of RBC clump and rouleau mirnics at an angle of zero degree

(Eq. 2.6). ....................... .. .......................................................................... 41

Fig. 2.7 B-mode images of RBC clump and rouleau des (Hilbert

...................................... transformation of Eq. 2.6) at an angle of zero degree 41

Fig. 3.1 Cornparison of the mean backscattered power between simulations of

RBC clumps (60 pm in diameter) and rouleaux (60 pm in leagth). The

error bars for each angle represent one standard deviation obtained from

30 tissue matrices. ......................................................................................... 46

Fig. 3.2

Fig. 3.3

Linear power-volume relationship for isotropie spherical scatteres. The

error bars for each scatterer size represent one standard deviation

obtained nom 30 tissue matrices. The correlation coefficient (I) was

....................... fitted onto the mean backscattered power values. ... ..... ..... 47

Linear power-volume relaîiomhip for anisotropic cy lindrical scatterers.

The error bars for each scatterer size represent one standard deviation

obtained Oom 30 tissue matrices. The correlation coefficient (4 was

fitted oato the mean backscattered power values. .................................... 4 8

Fig. 3.4a Power vs. the diameter of RBC clumps. The data are for 5 different

insonification angles (O0, 22.S0, 4S0, 67.5" and 904. Since no angular

dependence is observed for these angles, al1 data were pooled together

and plotted as one cucve. The error bars for each scatterer size represent

one standard deviation obtained from 30 tissue matrices. ................................. 49

Fig. 3.4b Power vs. RBC rouleau length at 5 different insonification angles. The

Fig. 3.5

Fig. 3.6

Fig. 3.7

Fig. 3.7

Fig. 3.8

Fig. 3.9

error bars for each scatterer size represent one standard deviation

obtained kom 30 tissue matrices .............................. .,l ............................... 50

Power vs. the number of RBCs per aggregate across 5 different

insonification angles. For the RBC clump mimic, data were pooled for

angles of O", 22.S0, 49, 67.5" and 90'. The error bars represent one

standard deviation obtained fkom 30 tissue matrices ......................................... 51

.......... An illustration of the size distribution of RBC rouleaux across the tube. 53

.............................................................. Simulated Wack hole" for Case la. 5 3

........................................................................................ (coat.) ............. .. 54

An illustration of the size distribution of RBC rouleaux across the tube

for Case lb ......... .................................... .................................................. 55

Simulated %lack hole" with the hypo-echogenic ring, appeariag at low

.............................. ............................ angies (O0, 22S0, 45") for Case lb. ... 56

Fig. 3.10 An illustration of the ske distribution of RBC rouleaux across the tube

...... for Case 2. ....................... ... ............,..... 57

........................ ............................... Fig. 3.1 1 Simulateci "black hole" for Case 2. ..... .58

Fig. 3.12 An illustration of the size distribution of RBC rouleaux acr(36s the tube

................................................................................................... for Case 3a 5 9

.............................................................. Fig. 3.13 Simulated "black hole" for Case 3a. 6 0

Fig. 3.14 An illustration of the size distribution of RBC rouleaux across the tube

for Case 36 ............................................................................................. 6 1

Fig. 3.15 Simulated "black hole" with the hypolechogenic ring for Case 3b ..... ............... 61 Fig. 3.16 An illustration of the size distribution of RBC aggregates across the tube

...................................................................................................... for Case 4. 62

Fig. 3.17 Sirnulated %la& hole" for Case 4. ............................................................. -.-63

Fig. 3.18A graphitai illustration of the scatterer structure and mean size

........................... ............................................ arrangement for Case 5. ... 64

...................... Fig. 3.19 Simuiated "black hole" for Case 5. ....................................... 6 5

Fig. 4.1 The transducer PSF (2nd order derivative of H(4y)) in the spatial domain

....................... with the comsponding magnitude spectmm (zoomed version). 6û

Fig. 4.2 Tissue matrix mimicking 20 pm RBC rouleaux at 10 % hematocrit with

the corresponding magnitude spectnim. Note that the DC wmponent has

been removed from the spectnim for better visualization. The RBC

........................................ ........... rouleaux are aligned parallel to the y-axis. .. 6 û

Fig. 4.3 Tissue matrix mimicking RBC clumps of 20 Fm diameter and 10 %

hematocrit with the corresponding magnitude spectmm. Note that the

DC cornponent has been removed from the spectrum for better . . visualization. .................. ..... ........... .. ......... 71

Fig. 4.4 A sample tissue matrix rnimicking RBC rouleaux of 250 pn long with

the corresponding magnitude spectrum. Note that the DC cornponent has

been removed ftom the spectrum for better visuaiization. The RBC

rouleaux are digneci at an average angle of 4S0, with a random

.......................... camponent of I 5". ......................... Fig. 4.5 Aggregation index of normal human RBCs at 45 % hematocrit as a

fundion of the shear rate (adapted fkom [IO]). RBCs were separated

h m the plasma and were suspended in a dextran saline solution. .................... 76

Fig. 4.6 "Black hole" magnitude vs, imnification angle wmputed from the

simulation of Case la The results were expresseci in terrns of mean t one

standard deviation and were averaged over 30 tissue matrices. ... ..................... 78

List of Tables

....... Table 2.1 The sias and concentration of major blood particles (adapted h m [18D 17

.......... Table 2.2 Acoustic properties of blood constituents at 20 O C (adapted nom [49D 27

............................ Table 2.3 The parameters used in the simulation for the present study 36

Chapter 1

Background and Objectives

This thesis is a study of red blood ceIl (RBC) aggregation in human blood by using

ultrasound. In this chapter, the essential background and the objectives will be pcesented.

1.1 The Rationde Behind the Study of RBC Aggregation

The human blood is composeci of plasma and cells. The plasma, in which the cells

are suspended, contains proteins such as dbumin, globulin and fibrinogen; nutrients,

hormones, mineral electrolytes and metabolic end products. There are mainly three types

of cells in blood: leukocytes (the white blood cells), platelets, and erythrocytes (the red

blood cells) which make up over 99 % of the cells in blood [52].

Under nomal physiologicai conditions, the red blood cells (RBCs) may aggregate

into stach called rouleurci; which is a result of the interaction between plasma proteins

and the RBC membrane. The rouleaux may further interact with other rouleaux to form

rouleau networks. Fibrinogen plays a very important part in rouleau cohesion and size

1511, and the RBC aggregation rate is observed to be accelerated with increased

fibrinogen concentration 1321. Aggregates formed in stationary Bow or low flow

conditions will disaggregate at higher flow rates due to the increase in shear forces.

Because RBC aggregation is a reversible process, reducing the shear forces wiii result in

the reaggregation of RBCs.

The aggregability of RBCs, i.e. the tendency of RBCs to form aggregates, has

been shown to play a major role in b l d flow, especially in the microcirculation [3],

where 80 % of the £iow resistance is. It is also known that RBC aggregability is a major

determinant of the viscosity of blood [9]. Under pathological conditions where RBC

aggregability is increased, the blood is more viscous and the adhesive strength between

RBCs forming an aggregate is increased. This c m result in reduced, or even absence of

flow in localized microvasçular regions. Such condition leads to increaseà flow resistance,

the reduction of tissue perfusion, and ischemia, a condition where the tissues are deprived

of the blood supply. In larger vessels, RBC aggregation is believed to be implicated in

thrombus formation 131, where a blood dot formed within a b l d vesse1 perturbs the

flow, may detach to block totally a smaller vessel, and consequently, causes tissue

ischemia Figs. 1.1 and 13 compare a pathological state blood sample to a normal one.

Although the two photographs were taken under static condition, the ciifference between

the two are expected to remain under flowing condition. Reduced blood flow due to the

increased RBC aggregation level may also trigger a vicious cycle [SI] which further

enhances RBC aggregation, as shown in Fig. 13.

Fig. 1.1 Diluted blood sample (20 % hematmritl) m m a patient with coronary artery

disease shawing huvily networked RBC aggregates (magnified 330 times).

Courtesy of Dr. Xiaoduan Weng.

Fig. 1.2 Diluted blood sample (20 % hematocrit) from a normal subject showing some

rouleaux of a few to several RBCs (magnified 375 times) [53].

I Local Acidosis

Fig. 1.3 The vicious cycle triggered by reduced blood flow due to increased RBC

aggregation level. Adapted h m [SI].

RBC aggregation levels, among othec hemorheologic parameters such as the levels

of whole blood and plasma viscosity, are found to be higher in patients with coronary

heart disease than in heaithy subjects [16,23,42]. It is also observed that the RBC

aggregation level provides a good indication of cardiwascular risk [47, as well as

microvascular hemorheological disorders [34]. An enhanœd level of RBC aggregation has

also k e n observed in patients suffering from diseases such as hypertension [46,57],

diabetes mellitus [30], gynacological malignancies [371, and canceis [28]. Thus, the study

of RBC aggregation is very important in clinical hemorheology .

1.2 Methods Proposed for the Study of RBC Aggregatïon

With the exception of the ultrasound baclrscattering method descnbed later, al1 the

methods described below require the withdrawal of blood fiom the patient for the

determination of the RBC aggregability. This may affect the tendency of RBCs to fom

aggregates and consequently the level of aggregation may ciifter fkom that present in vivo

within vessels, Only the erythrocyte sedimentation rate method is used clinically (described

in Section 1.2.2).

1.2.1 Microscopie Observation @]

This method is a static method, and the quantification of RBC aggregation is

achieved by dividing the RBCs into two separate groups, the first k i n g suspendeci in

plasma o r other maaomolecular solutions, and the other in a Knge? solution containing

0.5 % of serum albumin3. The albumin preserves the shape of the RBCs without inducing

aay aggregation.

M e r the RBCs have sedimented to the bottom of a container, photomicrographie

images are taken so that the number of cell units per volume of suspension can be

counted. Note that each ce11 can be a single red blood ce11 or an aggregate of RBCs. Using

the same RBC concentration in both solutions, the average number of red cells in each unit

can be expresseci in ternis of an index called the microswpic aggregation Utdk (MAI):

Number of cells in the Ringer solution MAI =

Num ber of wits in plasma or other macromolecular solutions

A balanctd saline solution uscd in physiologial experimtnfs to provide an isotonic medium for living tissucs. A simple proiein that is bat-coagulabk and watcr-soluble, and prcsent in blood plasma or senim.

Thus, when there is no RBC aggregation, MAI equals to 1. The drawbacb aPsociated

with this method are that the ce11 concentration in the solution must be low (e.g. 1 %) and

only the static condition is reflected, which is not the case in vivo.

1.22 Erythroeyte Sedimentation Rate (ESR) W

As the name suggests, this method relies on the rate of sedimentation of RBCs. A

plot of plasma-cell interface versus time is created, and the maximum sedimentation rate is

obtained f b m the steepest dope of the curve. The ESR should be determined at a Gxed

hematocrit, which provides a convenience in that the ESR can be determined UI witro

simply by using a hematocrit level similar to that in the normal circulation.

ESR reflects the degree of RBC aggregation as larger aggregates sediment €aster.

Again, the weakness of this method is that only the static condition is coosidered, which is

not really relevant physiologically. Moreover, the visc~sity of the plasma, which may differ

between samples of different patients, has an influence on the rate of sedimentation.

1.2*3 Direct Observation Under Shear Flow [8]

Chen [a] pmposed a method to obtain quantitative measures of RBC aggregability.

A computerized image analyzer was developed to acquire images of a single layer of

aggregates flowing in a chamber with controllable flow conditions. The flow chamber was

constructed by sandwichhg a thin layer of metal sheet that was approximately 40 Pm thick

between two transparent plates. A rectangular window of 1 x 20 cm was cut in the metal

sheet to form the flow chamber. Two holes were drilled in one of the transparent plates to

form the inlet and outlet of the fluid. The images of the RBCs in flow were magnified by a

microscope, monitored by a CCD TV carnera and were digitized. The images of the RBC

aggregates were di&rentiated by segmentatiori of the background, and that would give

the projected area of each aggregate. The volume of the aggregate was obtained by

integrating the projedeci are% and the size of the aggregate was obtained by dividiag the

volume by the average volume of a single r d blood c d .

With this system, aggregate size distribution curves at various shear stresses could

be obtained, and this prwided aggregability parametes such as the average or median size

of the aggregate population, the peak aggregate size, as well as the aggregation kinetics.

The shear stress required for complete disaggregation could be known by varying the

shear stress applied.

The problem with this method is that the chamber imposes a constraint on the

aggregate volume and size as only a single layer of aggregate is allowed to pass through.

This effectively forces a three dimensional network of aggiegates into a two dimensional

version, which is not entirely valid in physiologicaf conditions, especidly for large vessels.

The thickaess of 40 pn of the flow chamber may mimic the conditions in the

microcirculation, but the width of 1 cm does not; therefore the aggregate size estimated

and al1 the statistics derived rnay not be relevant to what adually occurs Ui vivo. This is

especially true for RBC aggregates in pathological states where the cross-section of an

aggregate may mach beyond the dimension of the chamber.

1.2.4 Laser Llght Reflection Method [17]

This method relies on the study of the variation in laser intensity refleded by the

scatterers (RBCs). Two series of measurements c m be performed. The first is to apply a

variable shear rate to the blood sample and measure the refleded light intensity at each

shear rate. As the shear rate increases, the aggregate ske decreases, leading to the

decrease in scatterer spacing and the increase of the refleded intensity. A curve showing

the refleded light inteusity versus the shear rate can be obtained for each blood sarnple.

From the curve, parameters on the adhesive strength holding the RBCs together can be

measured.

The second set of measurement gives the aggregation kinetics of the b l d simple,

Le. the tendency of RBCs to form rouleaux at zero shear rate. The blood is s h e d for 10

seconds at 550 S-' to disrupt rouleaux and orient RBCs with the flow. The refleded light

intensity at that stage is recorded. The shear stress is then terminated abmptiy, which

results in the reagpegation of RBQ. The variation of the refleded light intensity is

recorded during the rouleau formation process and a curve showing the reflected Iight

intensity versus time is obtained. From this curve, the primary aggregation time (tA), the

finai aggregation time (tF), and mean kiaetic indices at 10 s (Sio) and 60 s (&O) are

evaiuated. The aggregatioa proass is assumed to be completed 2 min. after the Bow

stoppage. T o date, the laser light refledion method is the most reliable approach for the

measurement of the dynamics of RBC aggregation in vitro. However, as other methods,

the disadvantage is that blood withdrawal is still required.

1.15 Ultrasouad Backscattering Methds

Each of the methods described above has its own advantages, but none of them

permits the study of RBC aggregability in vivo. Oniy noninvasive methods such as

ultrasound has the potential to do so, and in red time.

Several research groups have shown that ultraswnd is sensitive to the presence of

RBC aggregation. Boynaid et al. [SI used ultrasound in an in vitro set up to measure the

backscattering intensity of an RBC saline suspension, which do not produce RBC

aggregation. They made an attempt to relate the mean size of RBC aggregates to the

backscattered intensity [4]. They have also compared the RBC aggregation in eldedy and

young subjects using ul trasonic i nterferometry and backscatteri ng met hods [24], and were

able to observe the difference in RBC aggregation levei between these two groups of

subjects. However, their expenments were not conducted under flowing conditions and

relied on the sedimentation of RBCs.

Kim et aL 1291 demonstrated in an experiment that the ultrasonic baclcpcattered

intensity increased afler the blood sample in an oscillatory tlow was abruptly stopped.

They also observed a rapid decrease in echo intensity when the fluw was resumed. Sirniiar

results were obseived by Yuan and Shung [55] where the ultrasoaic backatter h m

flowing whole blood was fouad to be different fkom that of RBC saline suspensions in that

the former is shear rate dependent. The experiments were conducted with animal blood

ftom different specis, and the rcsults pointed to a species dependent backatter which

could be explained based on the degree of aggregation of the species' blood sample. In

addition, the same group also sbowed that ultrasonic backscatter Eiom flowing whole

blood was dependent on the concentration of fibriwgen when RBC aggregation exists

[54]. Shehada et al. [48] also reported an inverse relatiooship between the ultrasonic

echogenicity and the applied shear rates in an experiment conducted with 28 % hematocrit

porcine whole blood.

AU the studies conducted above were in vitro studies. Recently however, Cloutier

et al. [14] demonstrated that a difference in RBC aggregation levels between veiis and

artenes of normal subjects and patients with hyperlipidemia4 could be observed in vivo

with the use of power Doppler ultrasound. Thus, this study contïrms the possibility of

assessing RBC agpgability in vivo and noninvasively.

Al1 the results prwided by the different research groups above can be summarized

as follows: A correlation exists between the size of RBC aggregates and the ultrasonic

backscattered power. When the blood sample is in motion, the shearing effects of the flow

cause the RBC aggregates to break apart, which lead to smaller aggregates and weaker

backscattered power. Cowersely, when the flow is ceased or reduced, the shearing effeds

decay accordingly and consequently, the RBCs reaggregate, which is refiected by a

stronger backscattered power. Blood samples with M e or no aggregates such as bovine

whole blood or RBCs suspended in saline solutions are independent of the shearing effeds

- -

A condition where an excessive amount of fat or lipids is pxcscnt in the blood.

of the flow. Since fibrinogen auxlerates the RBC aggregation rate [32], it is no surprise

that the fibrinogen concentration affects the backscattered power.

13 Proposed Study

Fmm the above information, it is clear that the ultrasound backscattered power

can be used as a signature to study RBC aggregation. However, although it is h o w n h m

the literature t hat RBC aggregation increases ultrasound backîcattenng, the exact

mechankm by which the power is iaawed is unlmown. The volume of the aggregates

certainly has an effect, but other factors such as the hematocrit, the packing structure of

the aggregates, the variance in the aggregate volume, and the fluctuation of ail these

parametes in time and space can al1 contribute. In the present study, the iduence of the

volume of the scatterers and their structure will be specifically addressed. The contribution

of both factors in explaining what is Lnown as the "bblack hole" phenomenon will also be

i nvestigated.

1.3.1 A Brief Review on the Hypoechogenic Zone Discovery

The observation of blood echogedcity clifferences in ultrasonic B-mode images

was first reported by Sigel et aL [SOI. An ultrasonic scanner was used to scan the

surgicdly exposed inferior vena cava and portal veim of eight anesthetized dogs, and

hypoechogenic zones were observed immediately downstream fkom the entry of the two

tributary veins. Such hypoechogenic zones could be traced back to the tributaries since the

echo in the large vein was strooger. The cause of such hypoechogenic zones was

attnbuted to the absence of RBC aggregation in blood originating from those tributary

veins in which the shear rates were higher.

More recently, Yuan and Shung [56] observed the presence of a hypoechogenic

zone at the center of the flow conduit while imaging porcine whole blood under laminar

fiow. This observation, hiown today as the "black hole", was obtained by collecting data

at a position well beyond the entrance length of the vessel. They reported that such a zone

appeared to be more liiœly to occur at higher hematoait levels. Under similar Qow

conditions, no signïficant echogenicity variation was observed for bovine whole blood,

which does not have a stioag tendency to form aggregata. The authors suggested that the

cause was more likiely to be due to a low level of ultrasonic backscatter rather than

attenuation, and that a very low local hematocrit couid be the reason. They îuxther

proposed that local motion of RBCs in the b l d flow could result in such variations in

hematocrit- Radial migratory behavior of particles in a fluid is a phenornenon Lnowa as the

Segre-Silberberg effect. It was pstulated that as the erythrocytes migrated radiaily h m

the center axis to the wail and vice versa, an equilibrium was reached, resulting in a

maximum ce11 concentration region somewhere between the tube axis and the wall, thus

leading to the hypoechogenic zone in the center of the vessel.

la another study conducted by Mo et al. 1401, porcine whole blood at 28 %

hematocrit was circulated in a flow loop model mimicking a large vessel, and cross-

sectional and horizontal-plane B-mode images were obtained at different entrance lengths

with an ultrasound Iinear array transducer. The flow loop model had shear rates ranging

from O at the tube center to 5.3 s-' at the wall. Again, a hypoechogenic zone was present

at the center of the vessel, but the zone was observed to develop slowly as the entrance

length increased. Near the entrance of the tube, the echogenicity was very low; further

downstream, the echogenicity increased around the tube axis and the hypoechogenic zone

was developed. Experiments were also conducted with porcine RBCs suspended in saiine-

plasma mixtures with proportions ranging from pure plasma to pure saline solution. The

hematocrit was kept constant at 28 % in al1 mixtures. It was noticed that the

hyperechogenic ring around the hypoechogenic central zone decreased in echogenicity as

the proportion of saline increased, leading to a l e s distinct "black hole". From these

observations, the aut hors proposed the following:

Echogenic variations across the tube are related to the degree of RBC aggregation.

The hypoechogenic zone arises h m disaggregated RBCs having imufficient time to

significanly reaggcegate by the time they reached the other end of the vessel.

Aithough high shear rates (> 10 s-') generijliy nsult in disaggregation, the aggregation

level for low shear rates (e.g. 02 S-3 could be greater than that at zero shear rate

[10,15] due to the imreased i n t e d o n among the RBCs. Thus having the shear rate

ranging fkom O to 53 s-' might produce a condition where the degxee of RBC

aggregation is maximum at an intermediate radial location leading to a hyperechogenic

ring.

Fig. 1.4 shows sample B-mode images of the hypoechogenic zone adapted from

[40]. The image on the ieft was obtained by positioning the transduar dong the tube axis

whereas the one on the right was acquired cross-sectionally:

"Black hole"

Fig. 1.4 B-mode images of 28 % hematocrit porcine whole blood showing the

hypoechogenic zone ("black hole") at the center of the vessel. Both images were

obtained at an entrance distance 60 times the diameter of the tube with a mean

velocity at 1.4 cm/s.

Shehada et aL [48] further investigated the "black hoie" phemenoa, and they

proposed that the shear rate was high enough to dismpt the RBC aggregates at the tube

entrance, leaciing to a unifom echogenicity. Ar the tlow was M y developed huiher

downstream, the appearance of a hypoechogenic zone at the center of the tube was

proposed to be caused by shear rates lower than 0.05 s-'. They further suggested that the

hyperechogenic ring surrounding the "black hole" was the result of shear rates favorable

for the formation of RBC aggregates (0.05 to 2 9'. approximately).

Using a Doppler ultrasound system, Qin et aL [43,44] obsenred a hypoechogenic

zone at the center of the tube in experiments conducted with home blood. The entrance

length used in the experiments was sufficiently long to allow rouleau build-up, and the

flow rates were vacied €tom 102 ml/min. to 1250 &min. to obtain a wide range of shear

rate across the tube. The data was wllected at 5" increments kom 40" to 70" where the

angle is between the tube axis and the transducer. That was to confinu their hypothesis

that the orientation of large rouleaux in addition to their size contributes to the "black

hole" phenomenon. Since the refkaction of sound waves at the vessel wall could affect the

Doppler backscattend power as the angle was changed, only relative power was

measured. The ratio between the maximum and minimum power across the vesse1 was

used as a measure of the "black hole" magnitude. It was found that the power drop at the

center of the tube was more pronounced for hyperaggregating blood samples, and that it

was not related to the flow rates. In addition, the magnitude of the "black hole" increased

as the angle was raised, suggesting the psibi i i ty that rouleau orientation contributes to

such phenomenon.

Finally, in a set of experiments conducted by Cloutier and Qin [12], it was

observed that the hypoechogenic zone was oniy observed occasionally in the middle of the

vessel in experiments conducted with porcine blood. Those blood samples exhibiting such

a zone did not show any trend of hyperaggregation. However, the 'Wack hole" was almost

always observed in experiments conducted with home blood, as noted by Qin et al.

[43,44]. H o m blood is known to have a strong tendency to fonn long chairs of rouleaux

[53], however this may not be the case for porcine whole blood.

1.3.2 Summary

To the best of our howledge, Sigel et aL [SOI were the first t o identify

hypoechogenic zones downstream from the entry tributary veins in dogs. Since the

scanneci position was relatively close to the eatrance, they proposed that their observation

was most pmbably due to the lack of RBC aggregation as most of the aggregates got

disrupted at the entrance of the receiving veim where the shear rates were high. The cause

of the phenornenon did not appear to be that simple when the hypoechogenic m n e was

obsewed at positiom far beyond the entrance Iength and at the center of the tube where

the shear rate is minimum [56]. It was then hypothesized that the hypoechogenic zone o r

the "black hole" was due to the Segre-Silberberg effect, a phenornenon where particles in

a fluid cross-migrate radially such that an equilibrium was achieved and resulted in the

hypoechogenic zone.

Using porcine RBCs suspended in saline-plasma mixtures at a fhed hematocrit

with varying amount of saline content, Mo et aL [40] later confirmeci that the echogenic

variations obsewed in B-mode images were related to the degree of RBC aggregation,

thus casting some doubts on the Segre-Silberberg effect proposed by Yuan and Shung

1561. The dependence on the degree of RBC aggregation was also observed by Qin et al.

143,441. Mo et ai. [40] suggested that the hypoechogenic zone arose from the

insufficiency in time for the RBCs to reaggregate as they reach the end of the vessel, and

together with Shehada et al. [48], they proposed that the hypoechogenic zone was caused

by shear rates lower than 0.05 s-l at the center of the tube. In the last two studies, the

hyperechogenic ring around the Wack hole" was postulated to be due to shear rates

favorable for the formation of RBC aggregation (0.05 i1 to 2 s", depending on the flow

rate). Closer to the wall of the tube, the higher shear rate reduced the aggregate sizes and

the echogenicity.

Qin et al. [43,44] postulated the prrsence of organized structure in the

hypoechogenic zone, and proposeci that the shape and organization of the scatterers may

contribute to the %la& hole". Thus, the proposais by Mo and Shehada rnay not be the

only reasons. Aithough the experiments by Qin were conducted at much higher shear rates

( b m 1 s" to 55 d), the blood samples used were home whole blood, charaderiad by

strong intercellular lioks. This may explain why the "black hole" wuld be observed at a

high shear rate.

1.4 Objectives

Qin et al. [43,44] postulated that the shape and structural organization of RBC

rouleaux may play a role in the formation of the "black hole". In the present study, we

propose to study such phenomenon with a theoretical model based on a system approach.

The model provides the conditions of an ideal environment, where parameters ordinarily

beyond human control in an actuai laboratory setting can be managed. Examples of such

parameters are the size, orientation and number of scatterers within a given sample volume

of the blood vessel. Using a model also allows the possibility of performing "experiments"

beyond the capabilities of devices currently available in the laboratory, thus providing the

freedom from the constraints of existing apparatus and hardware. The following are the

objectives of the present thesis:

1. To study the effkds of the scatterer shape and size toward the backscattered power

with a theoretical simulation at various imnification angles.

2. To propose possible scatterer organization, shape and size that may contribute to the

"black hole" phenomenon.

Chapter 2

Theory, Implementation and Methodology

In this chapter, a brief introduction on the scattering of ultrasound by blood is first

presented. This is then followed by a description of the system-based mode1 used in the

present study. The parameters involved in the implementation and the methodology wiil be

presented Iast.

2.1 Scattering of Ultrasound by Blood

2.1.1 RBCs as Scattering Targets

Fluids have the two essential characteristics to support the acoustic wave

propagation: elasticity and inettia, which can also be expressed in terms of compressibility

K and mass density p, respediveiy. Elasticity implies that any deviation of the fiuid

molecules fkom the state of equilibrium wili tend to be corrected in the opposite direction.

Inertia implies that such correction will have the tendency to overshoot the state of

equilibrium and thus, requiring the need for a correction in the opposiie direction.

When an acoustic wave propagates through a homogeneous lossless medium, it

can travel indefinitely, However, in reai biological tissues, neither the compressibility nor

the density is constant throughout the entire media. Rather, they fluctuate €tom one spatial

location to another. An example is blood, where RBCs and other types of cells and

macroproteim are suspended in plasma. It is the differences in the compressibility and

density between these particles and the surrounding plasma that result in the scattered

acoustic waves. In general, other processes do occur when an acoustic wave propagates

through an inhomogeneous medium, and these indude the reflection and reîkaction of the

wave by tissue intertaces, as well as the absorption of the acoustic energy and its

conversion into heat by the medium. The backscattering of ultrasound by blood is largely

due to the RBCs since they are significantly more numerous than the slightly larger

leukocytes, and much larger and more wmerous than the platelets [18] (Table 2.1).

Erythrocytes (RBCs)

Concentrat ion

@articles/mm~

5 x 106

Approximate

dimensions (pm)

% of the total blood

voiume

Table 2.1 The sizes and concentration of major blood particles (adapted h m [18D.

The scattering of ultrasound by RBCs depends on the size of the individual

scattering target, the RBC concentration, and the acoustic properties of the scatterers. In

most commercially available clinical ultrasound instruments, the frequency range of the

acoustic wave is generally between 2 and 30 MHz, which corresponds to wavelengths of

approximately 785 pm and 52 Pm, respectively. Note that such wavelengths are much

Iarger than the greatest dimension of an RBC, which imply Rayleigh scattering.

2.1.2 Rayleigh Scattering

The backscattering coefficient (BSC) is a common parameter used to characterize

ultrasound scattering by blood and tissues. BSC is defined as the powr bacb t t e r ed by a

unit volume of scatterers per unit incident intemity, per unit solid angle in the direction

opposite of the incident wave [49]. For the case when Rayleigh scattering occurs, i-e.

when the incident acoustic wave encounters a particle much smdler than its wavelength, a

portion of the acoustic energy is scattered uniformly in ail directions. Such scattering was

first studied by Lord Rayleigh in 1871, and thus the oame Rayleigh scatterbtg is

commonly used today to refer to such phenomenon 1451.

When an incident wave encounters a group of scatterers in the medium, the

acoustic waves scattered uniformly 6rom each particle will result in mulîiple scattering, a

phenomenon where a scattered wave €tom one particle encounters other particles in the

area and is rescattered again. In theory, multiple scattering may go on indefinitely. In

practice however, multiple scatteri ng can be safely ignored especiail y for weak scatterers

such as RBCs since acoustic waves rescattered by neighboring RBCs are usually so weak

that their effects are negligible. Multiple scattering will not be considered in the

simulations of the present study.

Another commonly used parameter in the characterization of the ultrasound

scatteriag properties of blood is the scattering cross-section a@), which is the ratio of the

total scattered power, S, to the intemity 1 of the incident wave on one single target [33];

Le. a = SM. Under Rayleigh scattering condition, a@) is defined as:

where u(B) is the scattering cross-sectional area of a sikgle scatterer, B is the angular

direction of the scattered wave measured with respect to the direction of the incident

wave, V, is the scatterer volume, h is the wavelength of the incident wave, and K, p, and

K, p, are the compressibility and density of the scatterer and of the surrounding medium,

respeaively. Note that the scatterer is assumed to be spherical in shape in this model.

From Eq. 2.1, one can see that the baclcrcattering cross-section, ah for 0 = 180" is

predicted to be proportional to the square of the particle volume, and to the fourth power

of the signal âequency Cf = c h where c, the speed of sound in blood is appmximately

1570 m/s). In pradice, the b a c b t t e r e d power due to a single scatterer cannot be

measureci. In the following sections, it will be shown that the presence of several scatterers

and their spatial-temporal arrangements affect the backscattered power of the signal.

2.13 The Modeüng of Ultrasound Bockscattering by RBCs

Under normal physiological conditions, RBCs in human blood are densely packed,

and on average, the separation between adjacent cells is l e s than 10 % of the diameter of

a single RBC [49]. Such condition implies that the positions of any pair of RBCs are

neither totally uncomlateâ, nor perfectly correlateà, Le. they are partially conelated. This

essentially means that the acoustic wave scattered from each individual RBC can either

interact comtructively or destructively, making the modeling of ultrasound signals very

difficult. Such unpredictable interaction of the backscattered waves has a direct influence

on the total backattered signal power, and this can be d ina ly litked to the spatial-

temporal arrangements of the RB& at the time of insonification.

Several approaches have been pmposed in the past to model ultrasound

backscattering from RBCs, and they can be roughly categorized into the particle, the

continuum, and the hybrid methods. The following sub-sections were adapted fcom

Chapter 5 of [49].

2.13.1 The Particle Approach

T o the b a t of our howledge, Brody and Meindl [6] and Albright and Hanis [l]

were the first to model ultrasound scattering by the particle approach. The scattering

medium for [6] was assumed to be made up of a collection of identical, independent point

scatterers which scatter isotropically. It was also assumed that the average number of

scatterers per cubic miliiliter at any given time was Poisson distributed. Both research

groups treated the backscattered acoustic waves as the sumat ion of the contribution

€rom each individual scatterer.

Mo and Cobbold [38] developed a more general mode1 by tceating blood as a

suspension of RBC aggregates whose e&divc volume was a random variable, rather than

independent, point-sized paxticles. This approach was used to simulate the random

distribution of RBC aggregate sizes. The size of the agpgates in this model was

considered much smaller than the wavelength. Based on this model, the backscattering

coefficient, BSC, was given by:

where ob is the backscattering cross-section, H is the hematocrit, V, is the mean volume

of the scatterers, and W is the packing factor, which can be perceived as a measure of the

spatial orderliness of RBCs. The more "orderly" the scatterers are, the lower is the value

of W. Thus W is unity when the packhg of the scatterers is wmpletely randorn, and

gradually approacha zero when the correlation between RBCs increares. In terms of the

hematocrit, increasing the number of scatterers in a 6xed volume wiil invariably result in a

greater orderliness in terms of the spatial arrangement of the scatterers. At high hematocrit

levels, one can always find a scatterer that interfere destructively with the backscattered

acoustic waves fkom another scatterer, leading to a reduced BSC. Therefore, at a given

hematocrit, BSC is proportional to the scatterer volume weighted by the packing factor,

and to the fouah power of fkequency (see Eqs. 2.1 and 2.2). At very low hematocrit

levels, W = 1 and BSC is proportional to H. In Eq. 2.1, the values of the density and

compressibility of the medium are fixed, which can only be either the value iaside o r

outside the particle. Most of the particle-based models can be considered as special cases

of Eq. 2.2, w here W is expressed as an explicit hinction of

2J3.2 The Continuum Approach

This approach treats blood as a continuous medium where the backscattered

waves originate fiom local fluctuations in density and compressibility rather than nom

individual particles. This is based on the argument that the spatial cesolution of

conventional ulttasonic transducers is limited, and that individual RBCs and small RBC

aggregates are much smaller than the wavelength such that they ~ n n o t be tesolved

individually by these transducers.

Compared to the particte approach, the continuum approach models the

backscattering of ultrasouad as the summation of al1 contributions fkom the entire acoustic

field rather than tracking the position of each RBC in the sample volume. Thus, the

variations in the density and compfessibility are d o m for the continuum approach,

reflecting the random spatial distribution of the scatterers in the medium. Mo and Cobbold

(Chapter 5 of [49D proposed an expression for the BSC using the continuum approach:

where the subscript m in the equation indicates the average of the conesponding

parameter in the random medium; IL,,, = cJf is the average ultrasonic wavelength in the

random medium where c is the average speed of sound and f is the ûequency; V. =

( 4 d ) / 3 is the effedive ce11 volume such that r is the effective radius; K- p. and ic, p, are

the compressibiIity and density of the scatterer and of the surrounding medium,

respectively; ne is the volume of a voxel, which is a fixed elemental blood volume; and

var(n) is the variance in the number of scatterers in Re obtained by averaging over space

and time.

The average wavelength is used because the speed of sound depends on the

medium it propagates in. Thus when the ultrasonic wave propagates through a random

medium, the wavelength changes accordingly. This explaim why A,,, depends on the

hematocrit level in this model.

2.133 The Hybrid Appmach

Mo and Cobbold 1391 proposed this approach which combines the stmngths of the

two approaches above. In this model, the RBCs are no longer treated as individual

scatterers as in the particle approach, rather the sample volume is divided into elemeotal

acoustic volumes d l e d wxels, such that each voxel contains a variable number of RBCs

treated as a single scattering unit. Each voxel mwes with a single velocity. The BSC for

this approach is obtained by determining the contribution Grom a single scattering unit, and

theo summing the cont~butions nom ali the voxels. The influence of the mean number of

scatterers per voxel as well as its variation in tirne is included as part of the ~onsiderations~

Mo et al. propased the BSC for this approach to be:

where ab is the backscattering cross-section as defined in Eq. 2.1 for 0 = 180°, var@) is

the variance in the number of scatterers per voxel, and 8, is the volume of a voxel. In this

model, a voxel is defined in terms of a &action of the wavelength A.

2.2

the

The Modeüng of Ultrasound Backscattering by RBCs with

System-Bmed Approach (adapted fiom [35,36])

This approach to the modeling of ultrasound backscattering was selected for the

present study mainly because it provides the afcess to the specification of the shape, size,

orientation and number of scatterers, as well as the possibüity of insoniQing a sample

tissue image at any desired angle. Such features, which may not be readily available with

the models described above, make this approach very suitable for the objectives of the

present study. Other strengths and limitations of this approach are presented in Section

4.2. The following is a brie€ description of the formulation and the theory behiad the

system-based approach, which uses the principle of superposition applicable to linear

systems.

Assume that an ultrasound system is used to transmit a pulse of acoustic wave into

a fluid medium containhg only a small particle suspended in it. Then the backscattered

acoustic wave is chacaderized by the differences in the compressibility and density

between the particle and its sumunding fluid (Section 2.1.1), and the ultrasound system is

characterized by the radio frrquency (RF) signal at the output of the system, which is

essentially the point spread funcîion (PSF) of the system. If one is able to model the

acoustic irnpedance of a particle (which depends on the differences in the compressibility

and density between the particle and its sumiunding Buid), and the PSF of the ultrasound

system, then a simulated version of the backscattered RF signals can be obtained: When

there are more than one particle suspended in the fluid (under certain assumptions to be

given later), the backscattered R F signal at the output of the ultrasound system is

essentially the weighted PSFs that are summed over the space where the particles are

Iocated. Each weight essentially represents the acoustic irnpedance of each individual

scatterer at its respective location.

Since each scatterer is wnsidered in this model, it can be perceived as a particle

approach. However, the modeling is totally different €rom that presented in Eq. 2.2. In the

present study, if the region interrogated is sufficiently small, the PSF is essentially space-

invariant, and thus the summation over d l scatterers can be replaced by a convolution

operation. Therefore, the backscattered RF signal from a volume of interest (VOI),

W3D(&y,z), c m be expressed as:

where H&(%yYz) is the 3-D PSF of the ultrasound system, O denotes the convolution

operation, and Zm(%y,z) is the acoustic impedance hinction. The second order derivative

with respect to y refers to the direction of propagation of the acoustic waves. Note that

RFzo(~y,z) is a "volume" of RF signais- For the ease of amputation, W'(x;y,z) can be

well reprcsented by a 2-diwmional version of the RF image, RFm(5y). This is shown to

be possible in [35,36] if H3D(qy,z) is separable, that is, Hui(qy,z)=H(%y)H(t)- H&y,z)

wiii be defined later and wiii be shown to be separable. Thus the RF image cm be

described as:

Since the region of interest is composed of a coiledion of scatterers, the t

impedance, Z&(&yy~) can be beumed to be an ensemble of small scatterer

inhomogeneities:

where n indicates the location of the scatterer in the VOI. Ideaily, if the scatterers al1 have

the same size and shape, then only a universal scatterer prototype is required to generate

Z(i,y) [35]. In that case, a scatterer prototype function can be defined and repeated at

each scatterer location, i.e.:

where a. represents the echogenicity of the scatterer at position n and S3&) is the

scatterer prototype huiction. Substituthg (2.9) into (2.8) and then into (2.7) gives:

For the case where the scatterer size is small compared to the thickness of the beam in the

z direction, then the value of H(-z) will remain constant within each scatterer volume.

With respect to each scatterer, this implies that H(-z) varies only from scatterer to

scatterer and not in the z direction &&in any particular scatterer location. Therefore H(-z)

c m be taken out of the integral, which results in:

n

w here

This means that the 2-dimemional tissue impedance function Z(x,y) c m be generated by

projecting the scatterer prototype fundion dong the >axis and then weighting the result

with H(-zJ in each scatterer location. Z&y) c m also be expressed as the Z-dirnensional

convolution between S(x;y) and a position matrix P(k,y):

where

n

P(i,y) is essentially a matrix with randomly positioned Dirac delta fiindions weighted by

H(-zJ and a.. Eq. 2.13 meam that one wiii get an "ideal" tissue matrix, where ail

scatterers simulated have the same s h , shape and orientation. Having obtained Z(l;y), the

2-dimensional RF image M'&%y) can be computed according to Eq. 2.6.

In the event that a more realistic tissue image is required, i.e. a unique shape, size

and orientation for each scatterer, then each of the scatterers has to be integrated

independently and summed together to create the tissue matrix Z(l;y). A 2-D tissue

irnpedanœ matrix for each scatterer would then be obtained by:

and

where S.(gy,z) detemines the shape, size and orientation of the n * scatterer. This

essentially considers the entire V01 to contain only that particular scatterer, and have it

weighted with a, and H(-zJ, and have the result integrated along the z-axis. That wiil

generate a 2-0 tissue impedance matrix for that specific scatterer (Eq. 2.15). This

procedure is then repeated for each of the scatterers, and al1 the 2-D impedance matrices

are summed to produce the tissue impedance matrix, Z(qy) (Eq. 2.16). Note that the

creation of Zky) involves a position matrix as well. The 2dimeosional RF image

RFu>(Sy) can then be obtained h m Eq. 2.6.

The following are the main assumptions used in the system-based approach in the

modeling of ultrasound b a c b t t e r i n g by RBCs.

1. The medium is weMy scattering, thus multiple scattering is negligible.

There is no aîtenuation by the medium.

The noise level is zero.

The scatterers are in the fhr field of the transducer.

The PSF of the ultrasound system can be cepresented by a cosine modulated by a 3-

dimensional Gaussian envelope.

The ultrasound beam is sufficiently large such that the scatterers are small wmpared to

the beam thickness, i.e. H(-z) in Eq. 2.10 cemains constant within each scatteter

volume.

AH scatterers have equal echogenicity a,.

The region of interest (ROI) is sufficiently small such that the PSF is space invariant.

The assumption of weak scatterers essentiall y means that the mismatches in deaîity

and compressibility between an RBC and the plasma is Edidy smdl. As recognized in the

literature 1491 and shown in Table 4.1, this condition is satisfied for blood:

Medium

RBCS

Plasma

0.9 % saline

Distilled water

Table 2.2 Acoustic properties of blood constituents at 20 O C (adapted from [49]).

Mass density, p (g/cm3)

1.092

1.021

1.005

0.998

A weak scattering medium also implies that multiple scattering does aot have a major

contribution on the backscattered power. This is because the scattering from a particle is

so weak that when the acoustic wave bounces on neighboring particles, the rescattered

waves are negligible. The attenuation of ultrasound in blood and tissues is attributed to

scattering and absorption. The contribution to attenuation due to scattering is srnail

whereas absorption in blood is a p p d m a t e l y linearly proportional to the hemoglobin

Adiabatic compressibility, K (IO-= cm/dyne)

34.1

40.9

44.3

46.1

concentration b e l w 15 % [22]. In the cumnt study, the depth position of the ROI was

not considered thus the attenuation within the blood sample could not be wmpensated.

The assumption of zero noise level is just for the sake of simplicity, it can be easily

incorporated into the model by introducing some noise into the RF image. Under normal

clinical conditions, ultrasound measunments are perfomied in the far field region, thus

assumption 4 is consistent 4 t h what is done in ceality. Since the PSF can be seen as the

echo obtained when the scatterer is a small point object, a m i n e function modulated by a

Gaussian envelope can be a good appmrrimaîion. Note that a Gaussian fwction is a h

separable (see Eqs. 2.17 and 2.18 on page 30), which was another motivation for choosing

this model.

For the simulations of RBC rouleaux, the change in the scatterer volume was

refleded in the length of the rouleau while its diameter remained constant. For the RBC

clump mimic, the diameter of the spherical clump increased with the scatterer volume. In

the present study, as described later, the largest clump mimic used in the simulations had a

diameter of 120 p, which was still considerably smaller than the beam thicimess of a

typicat ultrasound transducer which is on the order of a millimeter. Thus, the assumption

that the barn thiclmess was large enough such that If(-z) remained constant within each

scatterer was valid for our study. As long as this assumption holds, H(-z) can be omitted

fiorn the mode1 because it has no effect on the variation of the backscattered power as a

fundion of the scatterer size and structure. Assumption 7 is reasooable because in

practice, although the echogenicity related to the density and compressibility of RBCs may

change, this has never been demonstrated.

In practice, the point spread function (PSF) of an ultrasound system in the far field

camot be considered space invariant and it is hiown to decrease in intensity with depth

[Il]. Moreover, a point scatterer at different depths wiU appear to have different sizes on

a B-mode image because of the divergence of the beam. However, if the interrogated area

is sufficiently srnall (1.28 mm by 1.28 mm in this study), the hypothesis of the space-

invariance of the PSF may be valid Using this hypothesis allows the possibility of

producing the RF image by convoluting the same PSF with a tissue image (Eq. 2.6), for

each position x. and y. of the scatterers within the ROI.

To implement the system-based mode1 of Eq. 2.6 on computer, the task can be

summarized into the following steps:

I. Create the PSF matrix % ~ ( x , y) (Eq. 2.6), and rotate the PSF about the center ay

of the matrix to simulate different angles of insonification. In the present study,

angles varying between O and 90" were tested.

II. Create the tissue matrix Z(qy) (Eq. 2.13).

A. Create a scatterer prototype S(i,y) (Eq. 2.12).

B. Create a position matrix P(i,y) (Eq. 2.14).

C. Compute the discrete convolution of S(5y) and P k y ) in the fkquency

domain.

III. Create the 2-dimemional RF image RFm(&y) by computing the discrete

a 2 convolution between - H(x, y) and Z(x,y) in the fiequency domain (Eq. 2.6).

ay

IV. Compute the average power of M'&,y) as a measure of the backscattering

coefficient.

The functions used to generate the matrices H(gy), Z(qy), S(i,y) and P(k,y) will be

defined below. AU convolutions were perfomied in the fkequency domain to enhance the

computational speed. AU matrices had the same size which reflects the size of the sample

volume.

For the present study, al1 scatterers in each tissue image were assumed to be

onented in the same direction and they dl had the same shape and size. Such an ideai

condition was used in order to reduce tissue image generation time, as convoluting each of

the scatterers one at a time, as described in Eq. 2.16, is computationally very intensive.

The justification of using such ideal conditions is presented in Section 4.1.15.

Step 1:

a 2 The focus of this step is to create the PSF fiinction -H(x,y) in Eq. 2.6. In this

?Y2

study, an approximation of the k-field PSF was used. In Eq. 2.5, the 3-dùnensional PSF,

H'.&,y,z), could be represented by a cosine function modulated by a 3-D Gaussian

envelope fwction, as justified before:

Expressing H3&,y,z) in tenns of H(x,y)H(z), we obtain:

and H(z) - e

where ly, , I#,, , and 2LZ were the standard deviations of the 3-dimeasional separable

Gaussian function which were the parameters controlling the beamwidth, the bandwidth

(length of the PSF) and the ultrasound beam thichess, respective1 y.

From Eq. 2.17, the first order denvative of H(i,y) is:

where E(x, y) = e

and the second order derivative of N(;re;y) is:

From the equation above,

Having obtained the second order derivative of H(x,y), its rotation about the origin

could be accomplished by first rotating the a i s by a desired angle, and then implemeating

the function on the rotated axes, as illustrated in Fig. 2.1.

Fig. 2.1 The rotation of the X-Y plane about the z-axis to simulate different insonification

angles. Y represents the direction of propagation of the ultrasonic waves and X is

the axis corresponding to the beamwidth.

Any arbitrary point on the X-Y plane was mapped onto the P-Q plane through the

celationship shown in Fig. 2.1. Since the 2-dimeasional Gaussian hindion was applied

onto the rotated axes, no inteplation was perfomed on the PSF matrix and therefore,

there were no interpolation errors introduced. Fig. 2.2 shows the angle of iosonification

with respect to the vesse1 fiom a physical perspective, where 8 is the angle of

insoni fi cation.

Tube wall

Flow

direction

1 Tube wall

Direction of / Y'' ul trasound

d propagation

Fig. 2.2 An illustration of the angle of iasonification h m a physical perspective.

Step 2:

aL Refemng to Eq. 2.6, having obtained ZH(x9y), the focus of this step is to

dy

create the tissue impedance function Z(qy), which is the comrolution between a scatterer

prototype function S(x,y) and a position matrix P&y) (Eq. 2.13).

Step 2a:

The focus of this section is on the creation of the scatterer prototype function

S(4y). A 3-dimensional separable Gaussian furiction was used to mode1 S 4 5 y , r ) (to be

justified in Section 4.1.1.4):

Frorn Eq. 2.12,

which is a 2-dimemional Gaussian function weighted by the thickness of the scatterer in

the z-direction. Note that a, O, and a= are the standard deviations of the 3dimensional

Gaussian fundion representing the width, length and depth of the scatterer prototype. The

2-dimeasional scatterer prototype matrix S(+) can be used to npresent eit her individual

RBCs, RBC rouleaux, or clumps of RBCs in pathological cases. This result in a tissue

matrix composed of scatterers al1 identical to one another in temis of size, shape and

orientation.

Step 26:

From Eq. 2.13, the task now is to obtain the scatterer position matrk P(5y).

Refemng to Eq. 2.14, P(qy) is a 2dimensional matrix with randomly positioned Dirac

delta fundions scaled by a,, and H(-r$. If the scatterers have equal echogenicity a,, P(5y)

can be modeled as a Poisson distributed 2dimensional matrix modeling the random

number of scatterers per pixel in the sample volume. The distribution used is dependent on

the number of scatterers per pixel. For a k e d hematocrit kvel of 10 96, the scatterer

number per pixel is generaily l e s than 12 (for scatterers greater than approximately 4 Pm

in length o r diameter), which can be adequately modeled by the Poisson distribution. The

number of scatterers per pixel is determined by the hematocrit level simulated. This matrix

implies that two or more scatterers may overlap on top of one another when the 3-

dimensional matrix is projected dong the z axis and is collapsed into a 2-dimensional

version in Eq. 2.6 above. Note that even though the scatterers may overlap, the

contribution of each of them is considered in the model. Moreover, note that H(-z,,) is a

funaion representing a transducer parameter that is considered constant within the

thickness of the beam. Thus this parameter is ignored in the computation of the signal

power since it is the relative change in signal power due to the variations in the scatterer

size and structure that is of interest and not the absolute power.

Step 2c:

Having created the scatterer position matrix P(x;y), the tissue impedanc~ lunaion

Z(i,y) in Eq. 2.13 can be obtained by computing the product of the Fast Fourier

Transfomi (FFI') of both S(*;y) and P(l;y) matrices, and then computing the inverse FET

(IFFI") of the product. In order to minimize computation, the IFFT was not computed,

and the tissue impedance, Z(X,n was ieft in the frequency domain.

Step 3:

a Having created - f f (x ,y) in step 1 aird Z(X,,Y) in step 2, the 2dimensioaal RF

av

image RF'(i,y) in Eg. 2.6 can be obtained by taking the FFI' of %H(x, y) , by au

multiplying the result with Z(X,,E3. and by computing the WFi' of the produa. The B-

mode image is esentially the envelope of the RFtD(i,y) image, which can be derived by

taking the absolute value of the Hilben transfonn of RFm(*;y). In order to reduce

computatioa, the fast IFFI' was dso not computed when evduating the b a c h t t e c e d

power, as specified below.

Step 4:

The BSC defined in Eq. 2.4 is essentially the backscattered power, which can be

wmputed in the Etequency domain from the RF image obtained in step 3. The power

spectrum of the RF image, POW(X,v, is computed by:

where X, Y are the 2-dimensional ftequency variables, and N is the number of sample dong

each side of the image. The average power of the RF image in dB was computed as

follows:

where k is the frequency sample.

Aithough the scatterers are aiways moving in vivo, such motion was not directly

sirnulated as the motion of the Eluid was reflected in the shape, size and orientation of the

scatterers. Thus simulating a snap shot of a "fiozen image" of the scatterers is consistent

with reality . Simulating several snaps ho& for averaging purposes may reflect position

changes of the scatterers as a fuaction of time, althwgh no specific time-variation pattern

was sirnulated in the present study. Table 2 2 lis& the mode1 specifications used for the

simulations:

1 Transducer frtquency (9: 1 O M H z 1 Ultrax>und velocity (c): 1570 m/s I

Table 2.3 The parameters used in the simulation for the present study.

Transducer beam width (21pJ: 0.5 mm

Length of the PSF (29,): 652 pm

Transducer beam thickness (21pJ: 0.5 mm

2.4 Methodology

Size of the ROI: 1.28 mm x 1.28 mm

Number of pixels in the ROI: 512 x 512

image resolution: 2 5 p m

This section describes the steps taken to accomplish the objectives stated in

Chapter 1. Since RBC aggregates usually exist in small rouleaux a s in normal human

blood, or in the form of rouleau networks as in hyper-aggcegating blood samples, it would

be adequate to simulate the former in the shape of a cylinder, and the latter in the form of

a spherical clump. This approach is especially valid s i n e the spatial resolution of a 10

MHz system is not high enough to differentiate one rouleau from another in a clump of

aggregate. An approximate diameter of 7 Pm and a thickness of 2 pm was used to

represent the physical dimension of an RBC in the simulation. The pixel size into which

the scatterers were positioned according to the Poisson distribution ninction had a volume

of 3125 (25 pm x 2 5 p m x 0.5 mm).

For al1 simulations, the center Eequency of the PSF remained fixed at 10 Mm and

the hematocrit level stayed constant at 10 % so that the influence of the packing factor in

Eq. 2.2 could be negleded. This is mainly because scatterers with di&rent shape and size

have different packing factors, which may not necessatily be known. The use of a low

hematocrit such as 10 % in al1 simulations is to minimize the effects of the paclring factor

on the backscattered power. Although the shape and size of scatterers simulated in this

study do not reflea the actual physiological condition at 10 % hematocrit, the present

study represents a first step in approximating the bacb t te red power from scatterers of

different shape and size, and also in examining the potential causes of the "black holen

phenornenon. Note that a constant hematocrit also implies a variable number of scatterers

per pixel as the scatterer volumes changed. Amrding to Eq. 2.14, it is the position of the

scatterers not their entire volume that is considered in each pixel. By using a Poisson

distribution (step 2b), the variance in the number of scatterers per pixel is equal to its

mean number, which reflects the low hematocrit level simulated. Fig. 2.3 shows a drawing

of an RBC ctump and rouleaux:

Fig. 2 3 A drawing of a small RBC clump and a few rouleaux.

2.4.1 Verifidon of the Vaiidity of the Mode1

Before any theoretid model can be used for meaninml investigations, its vdidity

must first be veri fied. The following two subsections present the methods to examine two

different aspects of the system-based model. The results will be shown in Chapter 3.

2.4.1.1 Isotropy / hksotropy Due to the Structure of the Scatterers

Lntrasonic baclcrcattered power fkom scatterers with an asymmetnc structure has

been shown in the past to be angular dependent. where the angle is between the ultrasonic

beam and the longitudinal axis of the scatterers. Some examples of such observation can

be found in experiments conducted with myocardial tissue [31,41], bovine liver [q, human

Achilles tendon [25] and rend parenchyma [26]. AU of the results showed a maximum

ultrasonic bacbatter when the direction of the ultrasonic beam was perpendicular to the

longitudinal axis of the scatterers, and a minimum when the beam direction was parallel to

the axis of the scatterers.

Such anisotropic behavior was also observed in flow experiments conducted with

porcine whole blood, as well as with carbon fibers suspended in a saline-glycerol solution

[2]. Each carbon fiber was approximately 7 pm in diameter and 250 Pm in length,

mimicking a long rouleau of RBCs. However, aaisotropy was not observed for porcine

RBCs suspended in a saline solution 121, where the scatterers existed in the form of

individual RBCs not forming aggregates. Therefore, a proper theoretical model should

exhibit the following characteristics:

1. The power should be angular dependent when the structure of the scatterers is

anisotropic and large ewugh in cornparison with the wavelength. Such an anisotropy

may exist only if the longitudiaai axis of each scatterer is aligned approximately parallel

to one another.

2. The power should be angular independent when the structure of the scatterers is

isotropic, independent of their size.

In order to verify these characteristics, the signai power fkom two different sets of

tissue matrices repiesenting two different types of scatterer structures was compared: one

was a group of isotropic scatterers; the other was anisotropic. Thirty tissue matrices for

each set were generated for the purpose of establishing the error margin. AU of them had

the same scatterer size and hematodt Ievel; the only difference king the location of the

scatterers. The power variation was also examined at diffèrent insonification angles

ranging fiom 0" to 90" at 10' increments. This was accomplished by convoluting each of

the scatterer matrices with the PSF rotated across these angles. Note that at O", the

direction of propagation is parailel to the long axis of the rouleaux.

The scatterers in each set of the tissue matrices were generated with two-

dimensional Gaussian functions (Eq. 2.21), one with identical standard deviations to mimic

the shape of a spherical clump, and the other with the standard deviation in one direction

greater than the other to mimic a rouleau, More specifically, the width and depth of each

rouleau which is determined by a, and uz in Eq. 2.21 were set to 7 Pm (i.e. a, and O= were

both set to 3.5 pm), while the length of each rouleau, which is determined by a, was set to

60 pm (Le. a- was set to 30 pm). This was to simulate a group of long rouleaux each

consisting of 30 RBCs stacked together in the form of a rod.

For the case of RBC clump mimic, since a sphere is used to mimic such type of

aggregates, a, a, and 4 were al1 set to 30 pm, thus rnirnicking a group of RBC clumps

each with a 60 pm diameter. Both scatterer types were simulatecl at such dimension so

that anisotropy can be better observed at 10 MHz. Figs. 2.4 to 2.7 show the PSF, one of

the 30 gray scale images for each type of scatterer, as well as the correspondhg RF and

B-mode images. The results of the comparison between these two types of scatterers are

presented in Section 3.1.1.

Trarieducer PSF

Fig. 2.4 The PSF of the tmnsducer used in the pnsent study (Eq. 2.19).

Fig. 2.5 Samples of tissue images at an angle of zero degree (Eqs. 2.13 and 2.21).

RF iman. of dmp rninûc RF imga of mc rodeau mimic

Fig. 2.6 RF images of RBC clump and rouleau mimics at an angle of zero degree (Eq.

2.6).

Fig. 2.7 B-mode images of RBC clump and rouleau mimics (Hilbert transformation of Eq.

2.6) at an angle of zero degree.

2.4.1.2 Power W. Scattlrer Volume Ueïatlonship

It was mentioned earlier that for a fixed low hematocrit level at 10 %, the influence

of the packing factor Win Eq. 2.2 could be neglected. Substituting Eq. 2.1 into Eq. 2.2,

one gets BSC to be l i d y proportional to the scatterer volume V,. Thus a linear

relationship is expected between the signal power and V, for Rayleigh scatterers if the

system-based model is valid.

In order to verify this characteristic, the signal power fmm 10 difirent sets of

tissue matrices was computed with each set representing a different volume. Each set

consisted of 30 tissue matrices with identical scatterer volume but different spatial

distribution so that an error margin could be established. As in Section 2.4.1.1, this was

done for both scatteirer types, Le. each of the rouleau and clump mimic had those 10 sets

of volumes. Both scatterer types were generated with 24mensional Gaussian fundions as

in Eq. 2.21.

With the carrier frequency at 10 MHz and a sound velocity in blood at 1570 m/s,

the corresponding wavelength is 157 Pm. Since Rayleigh scattering occurs at sizes

approximately l e s than one tenth of the wavelength [49], the dimension of the scatterers

should be less than 15.7 pm- For the rouleau mimic, both a, and a, in Eq. 2.21 remained

at 3.5 Pm, and the change in volume was reflected in the length of the rouleau, determined

by a,. The length of the rouleau mimic was preset to range £kom 4 pm to 13 Pm, at 1 (rm

increment. This corresponded to O- varying fkom 2 Pm to 6 5 Fm, and an approximate

range of scatterer volume Ekom 154 to 500 respectively. The diameter of the

clump mimic was also preset to the same range with the corresponding scatterer volume

ranging from 34 to 1150 approximately. The increase in scanerer volume for

the clump mimic was reflected in the increase of am a, and o, which were identical to one

another at al1 times. The results of the power-volume relationship is presented in Section

3.1.2.

2.4.2 The Effects d the Scatterer Structure and Size on the Signal Power

In this section, the method is described to examine the influence of the scatterer

structure and size on the signal power. Several factors affect the backscattered power,

among them are the scatterer size, number, shape and their orientation. Physiologically, ail

these parameters cannot be easily controlled either in the human circulation or in an

experimental setting. With the present model, the influence of these parameters could be

determined.

The backscattered power for 30 different scatterer volumes was wmpared for both

RBC rouleau and clump mimics. Referring to Eq 2.21, both a, and a, remained at 3.5 Pm

for the rouleau mimic, and the change in volume was reflected in a- which raaged nom 2

um to 60 Pm, mrresponding to 2 RBCs t o 60 RBCs stacked together in the shape of a

rod. For the RBC clump mimic, a, a, and a= ail had identical values, and they ranged

from 2 Pm to 60 Pm as well. This corresponded approximately to a range between l e s

than 1 to 11,755 RBCs in a clump, since the number of RBCs was approximated by

dividing the volume of the clump by that of a human RBC which is around 77 pn3.

The simulated range was sdely for the purpose of examining the power at that

size. The range does not imply their physical existence physiologically. Such a range was

also selected to observe the power in the non-Rayleigh scattering situation. The denvation

of this model made an implicit use of the Born approximation [35], which is valid when

the medium is weakly scattering and the scatterer size is rnuch smaller than the

wavelength. However, agreements have been found to exist between experiments and

theory using the Born approximation even if the scatterers were of the size on the order of

the wavelength 1271. Thus the model should still be valid up to the dimension mentioned

above, which is close to the size of the wavelength but no longer in the realm of Rayleigh

scattering.

As before, 30 tissue matrices were generated for each volume of each scattenr

type in order to establish the error margin. Al1 tissue matrices were iasoni6ed h m O" to

90" at 22.5" increment so that the effect of the scatterer structure on the signal power

could be observecl. The results are presemted in Section 3.2.

2.4.3 The Potential Causes O€ the "Black Ede" Phenornenon

information on the influence of the scatterer size and structure on the

corresponding backscattered power enables us to iwestigate the possible composition of

scatterer structures and s i s that produce the "black hole" phenomeoon. Based on the

results in Section 3.2, five separate possible cases are pmposed. The scatterers are either

made up of strictly RBC rouleau mimic or clump mimic, or a combination between the

two. These p r o p a l s and their graphical representatioos are presented in Section 3.3.

Several models have been proposed in the past to predict the ultrasound

backscattering €rom blood, but most of them dedt with the Rayleigh scattering condition,

and did not prwide the freedom to specify the shape, the size, the orientation a s well as

the number of scatterers in the insonified region. A system-based model, which is fkee

fiom the abwe mentioned limitatioas was presented in this chapter. The implementation of

this model on computer as well as the essential parametes used in the simulation were

also described. The model was used to examine the behavior of the backscattered power

as the scatterers Vary in their size and structure, and aiso to iwestigate the potential

causes of the '%lack hole" phenornenon observed on B-mode images.

Chapter 3

Results

In this chapter, the results on the verification of the validity of the model is

presented fist. This is followed by the resule on the effects of the scattenr shape and size

toward the backscattered power. Proposais on possible scatterer structure and size that

may cause the Wack hole" phenornenon are presented in the third section.

3.1 Verifkation of the VaISdity of the System-Based Mode1

The following sections present simulation results for the venfication of the model,

as demibed in Section 2.4.1. Two aspects of the model were examined. Aithough there

may be other aspects of the mode1 that can be tested, these verificatioos are adequate for

the objectives of the present study. Further justification of the model can be found in [58],

where the model was validated agaimt experimental results and results based on the hybnd

t heoretical approach for bot h Rayleigh as well as non-Ray lei& scatterers.

3.1.1 Isotropy 1 Anisotropy Due to the Structure O€ the Scatterers

Fig. 3.1 compares the mean power computed from 30 tissue matrices mimicking

RBC clumps and rouleaux.

RBC clump mimic A

RBC rouleau mimic

O 20 40 60 80 1 O0

Angle (degree)

Fig. 3.1 Cornparison of the mean backscattered power between simulations of RBC

clumps (60 Fm in diameter) and rouleaux (60 p in length). The error bars for each

angle represent one standard deviation obtained from 30 tissue matrices.

From this graph, it is apparent that the difference in the scatterer shape is refleded in the

power with this model. A change of approximately 25 dB in the mean power for RBC

rouleau rnimic was observed as a hinction of the angle. Virtually no change in the mean

power was found for the RBC clumps. This coofimis the consistency of the model with

experimentai obsewations reported in the literature as describcd in Section 2.4.1.1.

3.1.2 Power vs. Scatterer Vdume Relatioaship

Under the conditions described in Section 2.4.1.2, a linear relationship is expected

between the signal power and the volume of the scatterers, V, if the system-based mode1

is valid. The figure below shows the power versus volume nlatiomhip for the RBC clump

rnimic. As shown, a satisfactory linear relationship (8 = 0.99) is obtained for the scatterer

volumes considered.

1 Computed power l# - Fitted line

O 2000 4000 6000 8000 10000

Scatterer volume

Fig. 3.2 Linear power-volume relationship for isotropie spherical scatterers. The error bars

for each scatterer size represent one standard deviation obtained from 30 tissue

matrices. The comlation coefficient (2) was fitted onto the mean baclrscattered

power values.

Fig. 3.3 shows the power-volume relatiooship for anisotropic cylindrical scatterers:

100 150 200 250 300 350 400 450 500 550

Scatterer volume @m3) Fig. 3.3 Linear power-volume relationship for anisotropic cylindrical scatterers. The error

bars for each scatterer size represent one standard deviation obtained from 30 tissue

matrices. The currelation coefficient (8) was fitted onto the mean backscattered

power values.

The linear relatiomhip (8 = 0.97) between the power and the scatterer volume is obvious,

thus showing the wnsistency of the model.

3.1 J Summary

Two aspects of the mode1 have been examined. The first is the change in signal

power when the target scatterers have different structures, and the second is the

relationship between the signal power and the scatterer volume. This section has

demonstrated t hat the mode1 is consistent with experimental observations repoaed in the

literature and the results expected fiom the Rayleigh scattering theory.

3.2 The Effects of the Scatterer Size and Structure on the

Power

Figs. 3.4a and 3.4b compare the power obtained for both RBC clump and rouleau

mimics for a range of sizes.

240

O 20 40 60 80 100 120 140

Diameter of RBC clumps (pm) Fig. 3.4a Power vs. the diameter of RBC clumps. The data are for 5 different

insonification angles (O0, 22.S0, 4S0, 67.5" and 903. Since no angular dependence is

observed for these angles, al1 data were pooled together and plotted as one curve.

The error bars for each scatterer size represent one standard deviation obtaiued h m

30 tissue matrices.

O 20 40 60 80 100 120 140

Length of RBC rouleaux (pm) Fig. 3.4b Power vs. RBC rouleau Iength at 5 different iosonification angles. The enor bars

for each scatterer size represent one standard deviation obtained from 30 tissue

matrices.

From Figs. 3.4a and 3.4b. one c m see that the signal power increases up to a peak

as the scatterer volume increases, but any funher increase in the diameter or length of the

scatterers results in a decrease in the signal power, except when the scatterer structure is

in the fom of a rouleau at the iosonification angle of 90". In such a case, the s igd power

continues to iacrease with the scatterer kngth. The position of the peak is also observed

to change depending on the iasoaification angle, and the differenas across the

insonification angles appear to be enhanœd for longer rouleaux. Note that the number of

RBCs in a clump is greater than that in a rouleau. Considering the small standard

deviations obtained in Ag. 3.4% it can be concluded that sphenfal clumps produa

isotropic scattering as expected.

Fig. 3.5 compares the power €rom both structures in tenns of the number of RBCs

per aggregate:

240

- RBC clump mimic 22.5'

- RBC rouleau mirnic o0

O 10 20 30 40 50 60 70 80

Number of RBCs per aggregate Fig. 3.5 Power vs. the number of RBCs pex aggregate across 5 different iosonification

angles. For the RBC clump mimic, data were pooled for angles of O", 22.9, 4S0,

675" and 90". The enor bars represent one standard deviation obtained nom 30

tissue matrices.

From Fig. 3.5, it is apparent that when the number of RBCs per aggregate is the

same, the structure of the aggregate has a significant impact on the bacbt te red power.

For every number of RBCs, spherical ciumps produce larger bacbt tered power than

rouleaux. As the number of RBCs increasa, the signai power may change depending on

the structure of the aggregates formed and the insonification angle.

3 3 The Wack Hole" Phenornenon: Potential Causes

Based on the results obtained in Section 3.2, this section postulates the possible

composition of scatterer structures and sizes that may produce the Wack hole"

phenornenon. More specifically, the scatterer structure and size that lead to a Wack hole"

were investigated, based on the results obtained in Figs. 3.4a and 3.4b. Five different cases

are presented. Case la is based on the proposal by Mo et a l and Shehada et al. [40,48],

the experimental observatiois by Qin et aL 143,441 are also incorporated in this case. Case

3a is based solely on the proposal by Mo et aL and Shehada et al. 140,481, while the

remaining cases are new proposais. Note that for ail figures in this section, R and -R

represent the vessel walls, and the "0" on the abscissa is the center of the vessel. The error

bars at each radial position represent one standard deviation obtained fiom 30 tissue

matrices.

Case la:

The "black hole" is the result of the presence of small aggregates of RBCs at the

center of the vessel, as proposed by Mo et aL and Shehada et al. [4û,48]. The condition

proposed is that the shear rate at the center of the vessel is unfavorable for large aggregate

formation. As the shear rate increases radially towards the vesse1 wdl, a region between

the tube center and the wall exists such that the shear rate promotes the aggregation of

RBCs. This results in a hyperechogenic ring, making the region at the tube center a "black

hole" in contrast. Closer to the wall, it was suggested that the higher shear rates produced

disaggregation of RBQ. The size distribution of rouleaux shown in Fig. 3.6 was used in

the simulation study to represent the above proposed condition. Fig. 3.7 shows the

correspondhg simulated 'Wack hole" region for different insonification angles:

M e Caltu of ïùbe d the tube d

Fig. 3.6 An illustration of the size distribution of RBC rouleaux across the tube.

Insonificaüon angle: 00 Insonificrition angle: 22.50

-R O R -R O R

Spatial location across the tube Spatial location across the tube

Fig. 3.7 Simulated "black hole" for Case la.

InsonHfcadlon angle: & hroniiication angle: 6ï.s"

Spatial location across the tube Spatial location across the tube

Spatial location across the tube

Fig. 3.7 (cont.)

Each rouleau in Fig. 3.6 represents the mean rodeau size in a regwn within the tube. As

shown in Fig. 3.7, a Wack hole" is observed at the center of the vesse1 for every angle

tested. The magnitude of the "black hole" is affected by the iiwnification angle and

increases as the ultnisound beam is oriented more perpendicuiar with the axis of the

vessel.

Case lb:

It is interesting to show that if the region promoting the aggregation of RBCs

results in longer rouleaux (e-g. greater than 10 RBCs per aggregate), a slightly dithxent

backscattered power distribution across the tube may be obtained. It is shown below that a

hypo-echogenic ring sandwiched between two hyper-echogenic rings around the "black

hole" cm occur at low insoaification angies. Figs. 3.8 and 39 show the rouleau

arrangement dong with the backscattered power distribution that corresponds to this

simulation:

Tube wall

C e of the tube

Tube Wall

Fig. 3.8 An illustration of the size distribution of RBC rouleaux across the tube for Case

lb.

Insoniffcation angle: O0

-R O R Spatial location across the tube

Insonification angle: 450

-R O R Spatial location across the tube

InsonMcation angle: 22.50

Spatial location across the tube

-R O R Spatial location across the tube

Insonification angle: 90°

-R O R Spatial location across the tube

Fig. 3.9 Simulated "black hole" with the hypo-echogenic ring, appearing at low angles (OoY

22Soy 45") for Case lb.

Refemng to Fig. 3.9, the magnitude of the %la& hole" was obsenred to increase with the

iasonification angle, whereas the magnitude of the hypwchogenic ring decreased as the

angle increased. The two peaks amund the 'Wack hole" were greater in power than that of

Case 1% reûecting the incceased rouleau sizes in that region.

Case 2:

The '%la& hole" may a h be due to the presence of a group of nlatively long

RBC rouleaux at the center of the vessel. Refemng to Fig. 3.4b, the power is observed to

increase up to a peak and then decreases as the length of RBC rouleaux continues to

increase for angles below 90". Thus the "black hole" phenomewn could be the result of

having a group of relatîvely long rouleaux at the tube center and smaller rouleaux towards

the wall. The figure below illustrates the aggregate structure and size arrangement that

corresponâs to t his case:

Tube waii

Ce* of tbe tube

Tube wan

#

Fig. 3.10 An illustration of the size distribution of RBC rouleaux across the tube for Case

2.

Fig. 3.11 shows the simulated "black hole" region with such a rouleau arrangement:

Insonification angle: 0'

-R O R Spatial location across the tube

Insonificiition angk: 4 5 O

InsonHication angk: 22.50

-R O R Spatial location across the tube

Insonifïcation angk: 67.s0

-R O R Spatial location across the tube

-R O R Spatial location across the tube

Insonification angle: 90'

-R O R Spatial location across the tube

Fig. 3.11 Simulated "black hole" for Case 2.

The Wack hole7' does appear but seems to be more prominent at lower angles (iess than

454. Its magnitude is observed to de- with the iasonification angle and disappears

completely at higher iasonification angles, which is watrary to what was observed h

Cases la and lb.

Case 3a:

Another passible situation muld aise following the same logic as in Case 1, but

having rouleau networks as aggregates rather than just plain rouleaux. Such a condition

could be possible especially for the case of hyper-aggregating RBCs at extremely low flow

rates. The figure below illustrates the aggregate stnidure and size arrangement used in the

simulation:

Ce* of the tube

I

Fig. 3.12 An illustration of the size distribution of RBC rouleaux across the tube for Case

3a.

Due to the fact that the scatterers are isotropic, the simulaîed 'Wack hole" is angular

independent. Fig. 3.13 shows the simulateci "Mack hole" region foc such isotropic

spherical scatterers. The simulated "black hole" has a magnitude of approximately 3 dB

for d l insonification angles.

-R O R

Spatial location across the tube

Fig. 3.13 Simulated Wack hole" for Case 3a.

Case 36:

As in Case 1, such scatterer arrangement may also lead to the presence of a hypo-

echogenic ring around the '"black hole" region. However, the likelihood of such

occurrence is unlaiown as the scatterers in the region where RBC aggregation exists must

be relatively huge (e.g. an RBC clump with a 40 Pm diameter) in order for such

phenornenon to appear. The figure below shows the aggregate structure and size

arrangement that was used to generate the "black hole" and the hypo-echogenic ring

illustrated in Fig. 3.15. For dl insonification angles, the magnitude of the "black hole" as

well as the hypo-echogenic ring were approximately 2 dB. Due to the size of the RBC

clumps, only those scatterers in the lefi half of the tube are shown.

Cmtm of me the tube W1P

Fig. 3.14 An illustration of the size distribution of RBC rouleaux across the tube for Case

3b.

-R O R Spatial location across the tube

Fig. 3.15 Simulated "black hole" with the hypoechogenic ring for Case 3b.

Case 4:

In Fig. 3.44 we see that when large clumps decrease in volume, the backscattered

power reaches a peak and then declines. Thus moving ftom the center of the tube, such

break-up would result in an increase in signal power uatil a peak is reached, and as the

shear rate continues to increlse at positions closer to the wall, the clurnps get broken into

smaller pieces and the signal power eventually declines. At a position close to the wall, the

aggregates could either exist in the f o m of small clumps or small rouleaux. This situation

results in a rapid àecline in signal power at mgions closer to the vesse1 wall. Fig. 3.16

shows the simulated scatterer arrangement for Case 4:

i Nmbu of6LBCs (rppra-1

r L+=&=-of~wfPfi-(Crm)

Fig. 3.16 An illustration of the size distribution of RBC aggregates across the tube for

Case 4.

Due to the €ad that RBC aggregates in the form of clumps are isotropic, the magnitude of

the "black hole" in Fig. 3.17 is iodependent of the insonification angle. The simulated

Wack hole" has a magnitude of appmximately 4 dB.

Case 5:

Spatial location across the tube

Fin. 3.17 Simulated Wack hole" for Case 4.

This case was derived strictly bom the mode[, the chances of its existence under

physiological conditions are unhiown. This final alternative has large clumps of RBC

aggregates at the center of the tube, and as the shear rate increases radially towards the

wall, the clumps get broken into small RBC rouleaux forming the hyperechogenic ring. As

the shear rate continues to increase towards the wall, the rouleaux get disrupted by the

high shear rate and the signal power eventually declines. Fig. 3.18 is an illustration of the

aggregate structure and size arrangement that produces the "black hoIe7' in Fig. 3.19:

Tube tbe tube

Fig. 3.18 A graphical illustration of the scatterer structure and mean size arrangement for

Case 5.

As shown in Fig. 3.19, the "black hole" is observed at the center of the vesse1 for

every angle tested. The magnitude of the "black hole" is also observed to be influenced by

the insonifkation angle. It increases in magnitude fkom approximately 2 dB to over 5 dB

as the angle approaches 90". Note that such observation is very similar to that of Case la,

except that the scatterer arrangement is totally different.

Relative power (dB)

Relative power (dB)

2 2 2 2 I U P Q ) O P O "D-

Relative power (dB)

Relative power (dB)

3 :: a

! O a PD 3 m 0 m 0 .

- - - - I

.I

C

Relative power (dB)

4 o

L I 1 1 1

In this chapter, the system-bascd mode1 was first verified, and then used to

simulate scatterers with predetemined volumes and structures. The signal power

computed fkom these scatterer-mimic was then used to estimate the passible orgaaization

of scatterers that wwld result in the "black hole" phenomenon observed experimentally by

other researchers. Five different cases that could pdentially cause such a phenomenon

were proped. These results shall be discussed in the following chapter.

Chapter 4

Discussion

This chapter consists of two main sections, the analysis of the resula in Chapter 3

will first be presented while the strengths as well as the limitations of the mode1 will be

discussed in the second section.

4.1 Analysis of the Results

In the following, the results obtained in Section 3.2 will be analyzed, and this will

be followed by a discussion on the proposed causes of the Wack hole" phenomenon.

4.1.1 Power vs. Volume Relatioaship €rom a System-Based Perspdve

Figs. 3.4a and 3.4b show that the backscattered power from a 6xed hematocrit

tissue sample increases up to a peak and then decreases as the scatterer volume continues

to increase. From a system-based perspective, such behavior can be better understood in

the frequency domain.

4.1.1.1 Anisotropic cylindrical scatterers

The convolution operation described in Eq. 2.6 is the multiplication in the

frequency domain of the Fourier transform of the PSF and the Fourier transfom of the

tissue image. Figs. 4.1 and 4.2 show these images with their comsponding magnitude

spedrum for RBC rouleau mimic at an arbitrary irisonification angle of O":

Fig. 4.1 The traosducer PSF (2nd order derivative of w)) in the spatial domain with the

corresponding magnitude spedrum (zoomed version).

Fig. 4.2 Tissue matrix mimicking 20 pm RBC rouleaux at 10 % hematocrit with the

coriesponding magnitude spectmm. Note that the DC component has been removed

nom the spectnim for better visualization. The RBC rouleaux are aligned paralle1 to

the y-axis.

Ail figures are s h o w 4 t h the origin at the center of the image. In Fig. 4.1, the

displacement of the two cinilar spots in the spedrum nom the origin reflects the carrier

frequency of the PSF. The spectrum in Fig. 4 2 is esccntiaily a matrix with 2-dimensional

delta functions weighted by the scattenr prototype fundion (Eq. 2.12).

The product of the two spectra in Fi*. 4.1 and 4.2 can be better perceiveci by

supenmposing the two spectra on top of one another. The overlapping regions between

these two spectra wiU produce a non-zero produd; anywhere else in the spectrum would

yield a nul1 product. Thus, the baclscattered power is diredy affecteci by the overlappiag

area between the PSF spectnim and the tissue image spectrum.

From the Fourier transfocm relationship between space and hequency, it is knawn

that scaling in the spatial domain leads to an inverse relationship in the kquency domain:

where f(m) is the fiinction in the spatial domain, a is the scaling factor, x is the spatial

domain variable, FT is the Fourier transform, F is the Cunction in the fiequency domain,

and o is the Erequency domain variable. Eq. 4.1 shows t hat as the rouleau size is increased

along the y-direction in the spatial domain, the tissue image spectrum will shcink along the

same direction in the fiequency domain and its amplitude wiU increase. If the rouleau size

is reduced, the ellipsoid in the spectrum of the tissue image wiU tend to be more circular,

thus increasing the overlap with the two spots in the spectrum of the PSF. As the length of

the scatterers is increased, the inccease in the amplitude of the spectrum offsets such

scaling effect in the fkequency domain and as a nsult, the backscattered power is increased

as obseived for the case of Rayleigh scattering (Fis. 3.3 and 3.4b for rouleau lengths l e s

than 13 pm).

In Fig. 3.4b, as the scatterer size continues to increase, such behavior of the

magnitude spectrum wiii result in a decrease in the overlapping region between the PSF

spectrum and the ellipsoidal tissue image spectrurn, leading to the decrease in

backscattered power. In general, one can wnclude that, as the length of rouleaux is

increaseà, the baclrscattered power depends on the i m a s e in amplitude of the tissue

spedrum, the decrease in the ovedapping region between both spectra, and the aumber of

scatterers in the region of interest. At the limit, when the scatterers are relatively large

(e.g. 120 pm) and the angle is close to zero degree, the overlapped area is so smdl that

the product between the two spectra is a d y nulk, leading to a reduction of the

bacbcattered powrer by several decibels. The number of scatterers in the region of interest

for large scatterem is also not as numems as when the scatterers are smaller, since the

hematocrit is kept constant. As a consequence, fewer scatterers wiil produce less

backscattered signai, thus reducing the b a c b t t e r e d power.

4.1.1.2 Eff'ects of the insonifkation angle for anisotropic scatterets

In the system-based model, the change in the insoaification angle was represented

by the rotation of the PSF about its origin, which is the center of the image. According to

Fig. 3.4b, increasing the angle leads to an increase in the b a c b t t e r e d power. This can

be explained by the fact that the two spots on the PSF spectmm wiU increasingly overlap

with the region oaxipied by the ellipsoidal tissue image spectrum as the angle is changed

fkom O" to 90". Note that at 90°, the power drop is not observed because the volume

change of RBC rouleaux is refleded in the length, which is dong the y-sis . In the

bequency domain, the change in the scatterer volume does not affect the overlapping

region between both spectra because the two spots of the PSF are aligned dong the x-axis

and superimposed over the tissue spedrum. Thus, the iacrease in the amplitude of the

tissue spectnim as the rouleau length is increased raises the backscattered power.

Fig. 4.3 shaws the tissue image and the corresponding magnitude spedrum for

spherical scatterers mimicking RBC clumps:

Fig. 4.3 Tissue matrix mimifang RBC clump of 20 Pm diameter and 10 % hematocrit

with the comsponding magnitude spectmm. Note that the DC component has been

removed h m the spectnim for better visualization.

Using a 3-dimemional Gaussian hindion with identical standard deviations in the

simulation of RBC clumps results in a spectmm that is circular in shape. As the clump in

the spatial domain increascs in size, the circular spectmm of the tissue image will decrease

in its diameter, following the same relatiomhip as in Eq. 4.1. The same explmation for the

behavior of the backscattered power used for the rouleau mimic applies to the clump

mimic, except that the latter is angular independent and appears to have a stronger

backscattered power compared to the former when the number of RBCs per aggregate is

the same (Fig. 3.9.

From the spectrum in Fig. 4.3, it is not difficult to see that the backscattered power

fkom such a tissue mimic is angular independent, since the isotropie stmcture of the

scatterers is refleded in the frequency domain. As the PSF is rotated across the different

iosonification angles, the symmetry of the tissue image spectrum causes the ovedapped

region to remain unchangea leading to the aagular independent backscattered power. For

a given number of RBCs per aggregate, the stronger backxatteced power for clumps c m

be explainecl by the increase in the scatterer size in dl three dimeasions rather than just

dong the y-direction as in the case of the RBC rouleau mimic.

4.1.1.4 Otber considerations

It is important to note that the position of the peaks observed in Figs. 3.4a and

3.4b is dependent on the carrier frequency of the PSF. In this study, a PSF carrier

frequency of 10 MHz was used for al1 simulations. However, the peak would be expeded

to occur at a larger scatterer size if the carrier frequency had k e n reduced and viœ versa.

Refemng to the magnitude spectrum of the PSF in Fig. 4.1, the carrier frequeacy is

reflected in the position of the two "spots" with respect to the origin. Reducing the carrier

fiequency would mean that it would take a larger scatterer size to decrease the

overlapping region mentioned above, thus resulting in the shift of the peak position. From

a physical point of view, lower carrier frequency implies greater wavelength, which means

that larger scatterers can be considered as Rayleigh scatteres, where the power is related

to the volume at a fixed hematocrit.

With regards to the effects of the system parameters toward the bacb t t e r ed

power, the bandwidth and beamwidth of the ultrasound system, which corresponds

respectively to the standard deviations and I#, in Eq. 2.17, are also of importance

because both variables affect the spectrum of the PSF. However, any power variations due

to these system parameters wiil not affect the trend of the bacbcattered power due to the

change in the scatterer parameters as observed in Figs. 3.4 and 3.5.

As shown in Eq. 2.20, 3J-dimeosional Gaussian shaped scatterers were used in this

study. This assumption was used mainly for the sake of simplicity, s i n e the Fourier

transfonn of a Gaussian fuaction is still a Gaussian function, and it can be truncated with

negligible error at the third o r fourth standard deviations. In addition, the transducer may

not be able to r e d v e the edges of the scatterers in practice, thus, a hinction that decays

smoothly should be sufficient as an approximation. One could have used other

mathematid functions such as the ~c tangular window to represent the scatterers. But the

edges of such a fundion would introduœ side lobes in the fkquency domain. This is fine if

the scatterers are large, since the PSF would overlap only with smail side lobes toward the

"tail" of the spedrum. However, if the scatterers are small, the side lobes may complicate

the intecpretation of the tesults since the PSF would overlap with large si& Lobes close to

the baseband of the spectrum. Thus, the use of mathematical fundons that may introduce

side lobes in the hequency domain needs to be considered very carefully, especially when

the scatterers are very small. The use of the Gaussian fundion can be coasidered

appropriate sinœ the simulated results agree well w ith experimental observations reported

in the literature (described in Section 24.1.1) and the theoretical prediction as presented in

Section 3.1.

4.1.1.5 Simulation of a more redistic tksue image

Eqs. 2.15 and 2.16 can be used to simulate a more realistic tissue image. For

illustration, such a sample tissue image can be created by introducing a random component

in the length of each scatterer and in its angle with respect to the tube ais (O0 represents

the long axis of the tube). Having the scatterers aligned at an average angle other than O"

will result in a change in the length and angle with respect to the tube axis when projecting

the scatterer onto the 2-dimemional plane. The simulated scatterers had a diameter of 7

pm and a length of 250 Pm. AU of them were aligned at an average angle of 4S0, with a

random angular component of r 5". A random component cm be incorporated in the

length of the scatterers a s well, but this was not done for the puipose of the present

illustration. Fig. 4.4 shows the sample tissue image with the componding magnitude

spectmm:

Fig. 4.4 A sample tissue mat& mimicting RBC rouleaux of 250 Pm long with the

corresponding magnitude spectmm. Note that the DC component has been removed

fiom the spectnim for better visualization. The RBC rouleaux are aligned at an

average angle of 4S0, with a random compownt of * 5".

The system-based mode1 is capable of producing realistic tissue matrices, but at the

expense of long coxnputational time because each scatterer must be added iodiMdually

onto the tissue matrix with its individual projected scatterer length, angle of alignment,

and position. From the magnitude spedrum in Fig. 4.4, as long as the variance in the

angular alignment of each scatterer with respect to the tube axis is not too large, the

magnitude spectnim will not be very different fkom an ideal situation (where al1 scatterers

are assumed to have the same length and orientation), in which case the spectrum would

appear as an ellipsoid (e.g. Fig. 4.2) aligned at 45". Thus the trend of the backscattered

power due to the changes in scatterer parameters for both ideal and non-ided cases should

be fairly similar, i.e. we do not expect any drastic changes in the trend of the results

presented in Figs. 3.4 and 3.5 when more realistic tissue matrices are used in the

simulation. Thus the use of ideal tissue matrices should be sufficient for the objectives of

the present study.

Fiuctuations in the angular alignment of the scatterers may occur when the Bow is

turbulent, or when their concentration is low, as observed by Goldsmith et oL 119-211. It

was observed that a low concentration of elongated particles had the tendency to rotate

and deform under shear flow, with the angular velocity k i n g dependent o n the particle

length The histogram of the number of elongated particles as a function of their

corresponding angles showed that most of them were aiigned with the BOW at a given

time. As mentioncd in Section 2.3, d l scatterers were alignai with the flow at O" without

any random angular fluctuatioas for the simulations of the present study. However, the

random scatterer le* component is not expeded to wntribute signincantly because it is

averaged out when many (e.g. 30) tissue images are used to compute the average power.

4.1.2 Most Probable Causes of the "Black Hole" Phenornenon

In order to examine the validity of each of the cases presented in Section 3.3, one

way is to compare the results to experimental observations. Using porcine whole bled,

Mo et al. [40] and Shehada et aL [48] pmposed that there should exist a location

somewhere between the tube axis and the waif that promotes RBC aggregation. As a

result, a hyperechogenic ring is formed around the tube am's, leading to the "black hole"

phenornenon. The range of shear rates used in their studies was €tom 0.001 to 5.3 s".

From the results obtained by Chien 1101 on the aggregation of normal human RBCs at 45

% hematocrit, it was observed that a range of shear rates promoting the aggregation of

RBCs does exist. Fig. 4.5 shows a plot of the aggregation index computed by Chien as a

function of the shear rate:

AGGREGAnOhJ OF NORMAL RBC (n - 46 %) IN DE%TFUN W (4g 1100 ml)

œ

-1.

Q2-

SHEAR RATE (sr&)

Fig. 4.5 Aggregation index of normal human RBCs at 45 % hematocrit as a function of

the shear rate (adapted fiom [IO]). RBCs were separated nom the plasma and were

suspended in a dextran saline solution.

One can see from Fig. 4 5 that zero shear rate does not necessarily produce the largest

aggregates. The same celationship as in Fig. 4.5 was found for porcine whole blood 1481.

Thus, as the shear rate increases from the tube a i s to the wall, the "black hole" region as

well as the hyperechogenic ring around the tube axis would correspond to the region to

the lef3 of the peak in Fig. 4.5. Closer to the wall, the shear rate is increased beyond the

peak of Fig. 4 5 and the echogenicity would be reduced, thus emphasizing the

hyperechogenic ring. In the studies by Mo et aL and Shehada et aL, spherical aggregates

were considered. The explanation given above shows that their hypothesis is lihly to be

valid. This hypothesis was simulateù in Case 3a (Fig. 3.12).

Perfonning expnments in a smaller diameter tube under sirnilar flow conditions,

Cloutier et aL [12,13] observed that the "black hole" phenomenon was not always present

with porcine whole blood. This awld be exp la id by the fact that the minimum shear rate

value present was approximatefy 2 i1 at the center of the vessel. Depending on the

variance of the shear rate at the center of the vessel, this region could be favorable or

unfavorable for the aggregation of porcine RBG. In the oivorable case, t his would lead to

the situation where the region arwnd the tube axis had the largest aggregates. As the

shear rate was i n c d toward the wall, the aggregates got broken apart and the

backscatteced power declined. Note that the size of the aggregates at the center is not

expeded to be as large as in Cases 4 or 5 (Fip. 3.16 and 3-18), as that would lead to the

presence of a %la& hole". If the shear rate was unfitvotable for the occurrence of the

largest aggregates, then the situation would be as described by Mo et aL [40] and Shehada

et al. [48].

Usiog horse blood, Qin et aL [43,44] suggested that the orientation of large,

organized rouleau structure contributes to the formation of the '8lack hole" (Section

1.3.1). The magnitude of the "black hole" was observed to inccease with the insonification

angle. With that, it was proposed that the RBC aggregates would appear as shown in Case

l a (Fig. 3.6). Fig. 4.6 shows the plot of the magnitude of the %Iack hole" as a huiction of

the insonification angle for Case la:

lnsonification angle (degree) Fig. 4.6 "Black hole" magnitude vs. insoaification angle computed eorn the simulation of

Case la. The results were expressed in terms of mean t one standard deviation and

were averaged over 30 tissue matrices.

The simulated results in Fig. 4.6 follav a similar trend to those obtained by Qin et al.

[43,44]. ln that study, the magniîude of the Wack hole" varied nom 1 to 2 dB for angles

varying fiom 4û0 to 70°. The angle was defineà between the ultrasound beam and the tube

axis. Because the RBC rouleaux were probably not aligned completely with the flow in the

experiments conducted by Qin et al , the quantitative cornparison with Fig. 4.6 is diffiailt

since the angle with respect to the long axis of rouleaux was unknown in t hat study. In the

present study, the magnitude of the "black hole", M, was computed as follows:

where Pr is the power at the pak representing the hyperechogenic ring and Pz is the

power in the valley representing the "black hole". Note that Qin et aL took the average

power of the hvo peaks in the computation of the magnitude as the values w e n d i e n n t

experimentall y because of the ultrasound attenuation by blood.

Since the magnitude of the "bIack hole" for Case 2 decreases with the angle of

insonificatioa, this case seems unlikely to be vdid because the angular dependence is

contrary to the results obtained experimenially with home blood 143,441. Due to the fact

that no experimental data is avdable with regards to the angular dependence of the "black

hole" for porcine whole blood, the scatterer structure and organization could appear as in

Case la (Fig. 3.6) or Case 3a (Fig. 3.12) or a combination of both, but the conclusion is

yet to be determiued. As mentioned eariier, the largest aggregates are expected to be at

the center of the tube in cases where the Wack hole" is not observed for porcine whole

blood. However, if hyper-aggregating RBCs is present at the center of the tube as in Cases

2,4 and 5 (Fie. 3.10,3.15 and 3.18), a "black hole" may also occur. Only Cases 1 , 2 and

5 present angular dependence in the mapitude of the "black hole", with the magnitude

change of Case 2 in the opposite direction of Cases 1 and 5 as the insonificatioa angle

i ncreases.

4.2 The Strengths and Limitations of the Model

The system-based approach is a good model in the sense that it not only provides

the fieedom to specib the shape, structure, orientation and number of scatterers three

dimensionally, but it also allows one to visualize the actual tissue to be imaged. The model

provides the freedom of insoniQing a tissue image at any angles which is not always

possible in an actual experimental or clinical setting. Although b l d motion was not

simulated in the present study, this is not beyond the capabilities of the model. Mwing

scatterers can be simulateû by creating successive "fkozen images" by shifting and

interpolating the position maaix. A motion pictuce can be generated to create a realistic

and intuitive "mwie" of the scatterers in motion. Other advantages of the model include

the ability to specify the system parametes such as the bandwidth and beamwidth, as weli

as the ability to retcieve either the RF or B-mode images for other processictg purposes.

One limitation is the assumption of wealt scatterers, making the modeling of other

particles such as ultrasonic wntrast agents rather diffiatlt. The packing factor is aaother

parameter that ne& to be incorporated into the model if the modeling of higher levels of

hematocrit is of interest. niis may be very chdlenging, especially if RBC aggregates are

considered. The modeling of a focused tramducer which operates at the near field region

may be beyond the reach of this approach as the scatterers are assumed to be in the far

field of the transducer,

Another weabiess of the model may be its enormous appetite for computational

power. However, the model can be implemented one-dimensionally with good aariracy.

That is, imtead of having a PSF and a tissue matrix, one has an impulse response and a

tissue vector on which the 3D scatterer prototype is projected. But one will have to forfeit

the freedom ofchanging the iosonification angle. Thus, unless the iosonification angle is of

no interest for al1 simulation studies, the mode1 has to be implemented in 2D. Any attempt

to include in the model scatterers as small as an RBC requires a very high spatial

resolution. Since the physical size of the sample volume camot be infïnitely small, that

translates to the use of images large enough to accommodate the sample volume size, with

sufficient number of pixels to avoid spatial aliasing of the RBCs. The computing of the

two-dimensional convolution of images with large number of pixels is the major

determinant of the computation time. This disachantage however, is system-dependent,

and will fade away when more powerful processors are available in the future.

In this chapter, the results shown in Section 3.2 were analyzed in the ûequency

domain to explain the behavior of the backscattered power with respect to the change in

scatterer size and structure. The effacts of the system parameters on the b a c b t t e r e d

power were also disaissed. An example of a more realistic tissue image was provideâ

together with the justification of using a more idealized tissue image in al1 simulation.

Conccming the "black hole" phenomeaon, Cases l a and 3a were deemed to be more Iikely

valid. Both Cases l a and 5 agreed with the angular dependence of the "black hole"

presented by Qin et oL [43,44], with Cases 4 and 5 king new proposais solely based on

the system-based model presented in this study. Chse 2 was wncluded to be uniikely to

occur based on the experimental results by Qin et aL. The strengths and limitations of the

model were discussed in the final section.

Chapter 5

Conclusion

Motivated by the potential clinical use of the ultrasound bacb t t e r ed power in

the assessment of patients' RBC aggregation level, and the dismvery of the 'Wack hole"

phenomenon by other researchers, the objectives of the present study have been the

assessment of the influence of the scatterer characteristics on the backscattered power, as

well as the potentiai causes of the Wack hole" phenomenon.

It can be concluded that, besicles the previously known dependencies of the

ultrasound backscattered power on the scatterer volume, orientation, and shape, the

measured power could be a deceptive parameter especially when the scatterers are

relatively large and are no longer Rayleigh scatteres. Using a system-based model, the

backscattered power was observed to drop beyond certain scatterer size at a carrier

frequency of 10 MHz. The power difference at different insonification angles was also

observed to increase when the scatter size is larger. Further research is needed before

ultrasound backscattered power can becorne a usehl clinical tool in the assessment of

RBC aggregation level in patients.

The proposai of Mo et al. [40] and Shehada et al. [48] as described in Case l a and

3a was tested with the system-based model, and the results of simulation were compared

to the experimeotal results obtained by Qin et al. [43,44]. It can be mncluded that the

hypotheses proposed by Qin et al. in these studies are very likely valid, as the simulation

and experimental results matched very well. The mode1 was also used to create several

other scenarios t hat could potentially lead to a "black hole". The conditions leading to the

possible occurrence of a hypo-echogenic ring around the 'Wack hole" were also

presented. Based on the results of this thesis, it can be concluded that the Wack hole"

phenomenon could have more than one potential cause. Further work will have to be

conducted both experimentdly and with the system-bd mode1 to ver@ the causes of

this phenornenon.

References

[l] R. J. Albright and J. H. Harris, "Diagrmis of urethral Bow parametes by ultnsonic

backatter," IEEE Tram Bwmed Eng, vol. BME-22, pp. 1-11, 1975.

[2] L. Allard, G. Cloutier. and L G. Durand, ''EEcd of the iasonification angle on the

Doppler backscattend power under red blood ceIl aggregation conditions9" IEEE Tram

U h o n Ferroelec Freq Cmt, vol. 43, pp. 211-219,1996.

[3] H. 1. Bicher, BIood c d aggregarrion Ui thronrbotic processes. 1st ed. Illinois: Chades

C. Thomas, 1972.

[4] M. Boynard and J. C. Lelievre, "Size determination of red blood cell aggregates

induœd by dextran using ultrasound bacbttering phenornenon," Bwrheology, vol. 27,

pp. 39046,1990.

[SI M. Boynard, J. C. Lelime, and R. Guillet, "Aggregation of red blood cells shidied by

ultrasound backscattering," Bwrhedogy, vol. 24, pp. 45 1-461, 1987.

[6] W. R. Brody and J. D. Meindl, Theoretical analysis of the CW Doppler ultrasonic

flowmeter," IEEE Tram Biomed Eng, vol. BME-21, pp. 183492,1974.

[7] T. M. Burke, E. L. Madsen, and J. A. Zagzebski, "A prelirninary study on the angular

distri but ion of scattered ultrasound fkom bovine liver and m yocardium," Ulîrasonic

Imaging, vol. 9, pp. 132-145, 1987.

[8] S. Chen, "halysis of organization dynamics of red blood cells under flow, in health

and disease, by computerized imaging", Ph.D. thesis, Hebrew University, Jerusalem, pp.

1-82, 1994.

[9] S. Chien. "Biophysical behavior of red cells in suspensions," in me red blood ceil.

D.M. Surgenor Ed. New York, San Francisco, London: Academic Press, 1975, vol. II, pp.

1032-1133.

[ I O ] S. Chien, ''Electrochemical interactions between erythrocyte surfaces," Thrombosis

Research, vol. 8, pp. 189-202, 1976.

[Il] D. A. Christensen, Ultrasonic BwUistrumentotwn. 1st ed. New York, Chichester,

Brisbane, Toronto, Siagapore: John Wiley & Sons, 1988.

[12] G. Cloutier and Z. Qin. "Shear rate dependence of nonnal, hypo-, and hyper-

aggregating erythrocytes studied with power Doppler ultrasound," in Acousticul Imuging.

S. Lees and L. A. Fenari Ed. New York: Plenum Publishing Corporation, 1997, vol. 23,

pp. 291-296.

[13] G. Cloutier, 2. Qin, L. G. Durand, and B. G. Teh, "Power Doppler ultrasound

evaluation of the shear rate and shear stress dependences of red blood cell aggregation,"

IEEE Trans Bwmed Eng, vol. 43, pp. 441450,1996.

1141 G. Cloutier, X. Weng, G. 0. Roederer, L. Allard, F. Tardif, and R. Beaulieu,

''Differences in the erythrocyte aggregation level between veins and arteries of

normolipidemic and hyperlipidemic individuals," Ultrasound Med Bwf, vol. 23, pp. 1383-

1393,1997.

[IS] A. L. Copley, R. G. King, and C. R. Huang. '?3ythmcyte sedimentation of human

blood at varying shear rates," in Microcircularion. J. Grayson and W. Zingg Eds. New

York: Plenum Press, 1976, vol. 1, pp- 133-134.

[16] H. Demiroglu, I. Barista, and S. Diindar, "Erythrocyte aggregability in patients with

coronary heart disease," Cfinicul Hemorheology, vol. 16, pp. 313-317, 1996.

[17] M. Domer, M. Siadat, and J. F. Stoltz, ''Erythrocyte aggregation: Approach by light

scattering determination," Bwrheology, vol. 25, pp. 367-375, 1988.

[18] D. H. Evans, W. N. McDicken, R. Skidmore, and J. P. Woodcock, Doppkr

Ultrasowtd. Physics, Instrumentation, and Cfinicol Applicatwns. 1 st ed. Chic hester,

New York, Brisbane, Toronto, Singapore: John Wiley & Sons, 1989.

[19] H. L Goldsmith, "The microcirculatory society Eugene M. Landis award lecture. The

microrheology of human blood," Microvascuhr Res, vol. 31, pp. 121-142, 1986.

[20] H. L. Goldsmith and J. Marlow, "Flow behavior of erythrocytes. I. Rotation and

deformation in dilute suspensions," Proc Royal Soc (London), vol. 182, pp. 351-384,

1972.

[21] H. L. Goldsmith and S. G. Mason. "Some model experiments in hemodynamia. 111,"

in Hemorheology. Proceedings of the first hintnational conference. A.L. Copley Ed.

New York: Pergamon Press, 1966, pp. 237-254.

[22] J. F. Greenleaf, Tbsue characteriurtion wdh ultrasound. 1st ed. Boca Raton,

Fiorida: CRC Press, 1986.

1231 R. Hahn, P. M. Müller-Seydlitz, K. H. Jtkkel, H. Hubert, and P. Heimburg,

"Viscoelasticity and n d blood ce11 aggregation in patients with foronary heart disease,"

Angiofogy, vol. 40, pp. 914-920,1989.

[24] H. Hammi, P. Penotin, R. Guillet, and M. Boynard, ''Determination of red blood a U

aggregation in young and eldedy subjects evaiuated by ultrasound: Influence of

dih ydroergocryptine mesylate," C h i c a l ffemorheology, vol. 14, pp. 1 17.126, 1994.

[25] B. K Hofnneister, A. K. Wwg, E. D. Verdonk, S. k Wickiine, and J. G. Miller,

"Anisotropy of ultrasonic backscatter from human tendon compared to that fiom normal

human myocardium," Ultrasmicr Symposimz, vol- 2, pp. 1127-1 131, 199 1.

[26] M. F. Insana, T. J. Hall, and J. L Fishback, 'Identifying acoustic scattering sources in

normal noal parenchyma €tom the anisotmpy in acoustic properties," Ultrasuund Med

Bwl, vol. 17, pp. 613-626, 1991.

[27l M. F. Insana, R. F. Wagner, D. O. Brown, and T. J. Hall, 'Pesnibing small-scale

structure in random media using pulse-echo ultrasound," J Acowt Soc Am, vol. 87, pp.

179-192,1990.

[28] M. M. Khan, R. R. Puniyani, N. G. Huilgol, M. A. Hussain, and G. G. Ranade,

''Hemorheo1ogical profila in cancer patients," Clhical Hemorheology, vol. 15, pp. 37-44,

1995.

[29] S. Y. Kim, 1. F. Miller, B. Sigel, P. M. Coosigny, and J. Justin, "Ultrasonic evaluation

of eryt hrocyte aggregation d ynamics," Biorheology , vol. 26, pp. 723-736, 1989.

[30] C. Le Devehat, M. Vimeux, G. Bondoux, and T. Khodabandehlou, "Red blood cell

aggregation in diabetes mellitus," Int Angiol, vol. 9, pp. 11-15,1990.

[31] E. 1. Madaras, J. Perez, B. E. Sobel, J. G. Mottley, and J. G. Miller, "Anisotropy of

the ultrasonic backatter of myocardial tissue: II. Measurement in vivo." J Acoust Soc

Am, vol. 83, pp. 7620769,1988.

[32] N. Maeda, K. Imaizumi, M. Sekiya, and T. Shiga, "Rhedogical characteristics of

desialylated erythrocytes in relation to fibrinogen-induced aggregation," Biochimica &

Bwphysica Acta, vol. 776, pp. 151-158, 1984.

[33] W. N. McDicken, Diagnostic uftrasonLcs: Prihciples and use of Urstruments. 3rd ed.

Edinburgh, Lnndoa, Melbourne & New York: Churchill Livingstone, 1991.

(343 G. Mchedlishvili, R. Shalrarishvili, M. Aioeva, and N. Momtselidze, "Elaborated - "Georgian index" of erythrocyte aggregability characterizing the microrhedogical

disorders associated wit h brain infimt,'' C h i a L Hemrheology, vol. 15, pp. 783-793,

1995.

[35] J. Meunier and M. Bertrand, "Echographic image mean gray level changes with tissue

dynamia: A system-based mode1 study," IEEE Tram Biomed Eng, vol. 42, pp. 403-410,

1995.

[36] J. Meunier and M. Bertrand, "Ultrasonic texture motion analysis: Theory and

simulation," i E E Trans Med ImagUrg, vol. 14, pp. 293-300, 1995.

[37J B. Miller and L. Heilmam, "Hemorheological parameters in patients with

gynecologic malignancies," Gynecologic O n c o l . , vol. 33, pp. 177-181, 1989.

1381 L. Y. L. Mo and R. S. C. Cobbold, "A stochastic mode1 of the backscattered Doppler

ultrasound from blood," IEEE Tram Bwmed Eng, vol. BME-33, pp. 20-27, 1986.

[39] L. Y. L. Mo and R. S. C. Cobbold, "A unified approach to modeling the

backscattered Doppler ultrasound îkom blood," IEEE Trans Biomed Eng, vol. 39, pp.

450461,1992.

[40] L. Y. L. Mo, G. Yip, R S. C. Cobbold, C. Gutt, M. Joy, G. Santyr, and K. K. Shung,

cWon-newtooian behavior of whole blood in a large diameter tube." Bwrheology, vol. 28,

pp. 421-427,1991.

1411 J. G. Mottley and J. G. Miller, "Anisotropy of the ultrasonic backatter of

m yocardial tissue: 1. Theory and measurements in vitro." J Acoccst Soc Am, vol. 83, pp.

755-761,1988-

[42] F. J. Neumann, H. A. Katus, E. Hoberg, P. Roebmck, M. Braun, H. M. Haupt, H.

Tillmanns, and W. Kübler, "Increased plasma viscosity and erythrocyte aggregation:

lndicators of an unfavourable clinid outcome in patients with unstable angina pectons,"

Br Heurt J, vol. 66, pp. 425430,199 1.

[43] Z. Qin, "Characterization of red blood ce11 aggregation dynamics with Doppler

ultrasound", M.Sc. thesis, Université de Montréal, Institut de recherches cliniques de

Montréal, Montreal, pp. 1-130, 1997.

[44] 2. Qin, L. G. Durand, aad G. Cloutier, "Kinetics of the "black hole" phenornenon in

ultrasound backscattenng measuremenis with red blood ce11 aggregation," Ultrasuund

Med Biol, vol. 24, pp. 245-256, 1998.

[45] J. W. S. Rayleigh 'Vibrations of solid bodies," in 'Ineory of s d . New York:

Dover Publications, 1945, vol. 2, pp. 415431.

[46] S. M. Razavian, M. Del Pino, A. Simon, and J. Levcrison, Yncrease in erythrocyte

disaggregation shear stress in hypertension," Hypertension, vol. 20, pp. 247-252, 1992.

[47l G. Ruhemtroth-Bauer, O. MBssmer, J. 0111, S. Koenig-Erich, and 0. Heinemann,

"Highly significant negative correlations between erythrocyte aggregation value and senim

concentration of high density lipopmteia cholesteml in a sample fiom a nomal population

and in patients with coronary heart disease," Eur J Ciin Invest, vol. 17, pp. 275-279,

1987.

[48] R. E. N. Shehaàa, R. S. C. Cobbolà, and L. Y. L. Mo, 'Aggregation efftxts in whole

blood: Influence of time and shear rate measured using ultrasound," Biorieology, vol. 31,

pp. 115-135,1994.

[49] K. K. Shung and G. A. Thieme, Uftrasonic scattering i~ biologicul trssues. 1st ed.

Boca Raton, A m Arbor, London, Tokyo: CRC Press, 1993.

[SOI B. Sigel, J. Machi, J. C. Beitler, J. R. Justin, and J. C. U. Coelho, 'Variable

ultrasound echogenicity in flowing bfood," Science, vol. 218, pp. 1321-1323, 1982.

[SI] J. F. Stoltz and M. Donner, Wemorheology: Importance of erythrocyte

aggregation," C h i c a l Hemorheology, vol. 7, pp. 15-23, 1987.

[52] A. J. Vander, J. H. Sherman, and D. S. Luciano, Humun physwfogy. The

mechaniFms of body funcrion. 6th ed. New York, St. h i s , San Francisco, London.

Toronto, Tokyo: McGraw-Hill, Inc. 1997.

[53] X. Weng, G. Cloutier, P. Pibarot, and L. G. Durand, "Cornparison and simulation of

different levels of erythcocyte aggregation with pig, home, sheep, cal€, and nonnal human

blood," Biorheobgy, vol. 33, pp. 365-377, 1996.

[54] Y. W. Yuan and K K. Shung, Wltrasonic backscatter nom Bowing whole b l d : II.

Dependence on kequency and fibriaogen macentration." J Acourt Soc Am, vol. 84, pp.

1195-1200,1988.

[SS] Y. W. Yuan and K. K. Shung, 'Uîrasonic backscatter nom flowing whole blood: 1.

Dependence on shear rate and hematocrit," JAwwt SocAm, vol. 84, pp. 52-58,1988.

[56] Y. W. Yuan and K. K S hung, "Echoici ty of whole blood," J Ultrasound Med, vol. 8,

pp. 425-434,1989.

[57] F. Zamad, P. Voisin, F. Brunette, J. F. Bnintz, J. F. Stoltz, and J. M. Gilgenkrantz,

c~aemorheoiogical aboomialities in artend hypertension and their relation to d i a c

hypertrophy," J H y e w n vol. 6, pp. 2939297,1988.

1581 1. Fontaine, M. Bertrand, G. Cloutier, 'Validation of a system-based approach to

mode1 the ultrasound signal b a c b t t e r e d by red blood cells," to be submitted to IEEE

T m Biomed Eng.