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IHESIS

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3 1293 01554 651

 

LIBRARY

Michigan State
University

 

 

 

This is to certify that the
dissertation entitled
ENERGY AND FIELD DISTRIBUTIONS IN MATERIAL
UNDER ACOUSTIC WAVE IRRADIATION

presented by
YAN CHANG CHANG

has been accepted towards fulfillment
of the requirements for

Ph.D

4/45“

Date 1/ 1 1/ 95

MS U is an Affirmative Action/Equal Opportunity Institution 0- 12771

degreein Electrical Engineering-

 

PLACE IN RETURN 30X to remove thie checkout from your record.
TO AVOID FINES return on or betore date due.

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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___ll__-
fiflF—j

MSU 18 An Affirmative Action/Emil Opportunity Inefltmion

-
.l—T
l__J

 

 

 

 

 

 

 

 

 

 

 

\7

ENERGY AND FIELD DISTRIBUTIONS IN MATERIAL
UNDER ACOUSTIC WAVE IRRADIATION

By

Yaw-Chang Chang

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Electrical Engineering
1995

ABSTRACT

ENERGY AND FIELD DISTRIBUTIONS IN MATERIAL
UNDER ACOUSTIC WAVE IRRADIATION

By

Yaw-Chang Chang

Ultrasonics has found increasing attention in clinical diagnostic and therapeutic
applications in recent years. Heat generation, increasing temperature rate and temperature
profile during ultrasonic therapeutics play important roles in determining the processing
time as well as the optimum and safety acoustic intensity levels. In this dissertation, an
analytic model using the Green’s function of a multi-layered structure has been
developed. This particular model takes into account all the possible multiple reflections
in the biological interfaces which are commonly ignored by other investigators for
simplicities. This study uses intensity point of view to derive the Green’s function of a
structure. The analysis of a fat-muscle—bone model is also included in the study to

validatetheapproach.

In order to account for the total energy carried by an acoustic wave traveling
through a medium, an energy balance relation was formulated. An acoustic Poynting
theorem for an inhomogeneous and moving medium which includes the equivalent forcing
terms of spatial and temporal variations of density and stiffness has been developed.

Thermal diffusion, particle collision, drift, and heat flow effects have also been taken
into account in this investigation. A general acoustic wave solution using the Green’s
function technique is also established. A transmission-line model for acoustic wave
propagation in an inhomogeneous moving medium has been developed. This model takes
into account thermal diffusion, collision, drifi, and heat flow effects as the equivalent
current sources for the transmission-line equations. This model can readily be reduced
to the results reported elsewhere for appropriate parameters. A computational method for
evaluating the ultrasonic displacement field within an inhomogeneous, arbitrarily shaped

scatter is also developed in this dissertation.

ACKNOWLEDGMENTS

The author wouldlike to express his sincere thanks to Professor B. Ho, the
author’s adviser, for bringing the topic to the author’s attention through his teaching,
constant guidance, stimulating discussions, encouragement, valuable suggestions _
throughout the course of this study. Thanks are also given to the guidance committe:
members, Professors W. E. Kuan, D. Liu and H. R. Zapp for their guidance,
suggestions and taking time to serve on the committee.

Finally, the support from my parents and my wife, Hsing-Ming, during this study
are greatly appreciated.

iv

TABLE OF CONTENTS

LIST OF TABLE
LIST OF FIGURES

CHAPTER 1: INTRODUCTION
1.1 Introduction
1.2 Outline of the dissertation

CHAPTER 2: BACKGROUND
2.1 Acoustic wave behavior
2.2 Reflection and transmission of acoustic waves normal to a boundary
2.3 Reflection and transmission of acoustic waves at oblique incidence
2.4 Wave attenuation for propagating through a medium
2.4.1 Absorption reduction
2.4.2 Scattering effects
2.4.3 Frequency dependent
2.5 Acoustic analogue to the electromagnetic system
2.6 Electromagnetic Poynting’s theorem
2.7 Acoustic Poynting’s theorem
2.8 Temperature elevation by acoustic absorption

CHAPTER 3: INTENSITY DISTRIBUTION AND TEMPERATURE

RISE OF MULTILAYERED STRUCTURE

3,1 Introduction

3.2 Green’s function formulation
3.2.1 Green’s function of multi-layered structure
3.2.2 Heat generated and initial increasing temperature rate

3.3 Biologic target modeling and results

3.4 Discussions and summary

CHAPTER 4: THE ACOUSTIC POYNTING THEOREM IN AN
INHOMOGENEOUS MOVING MEDIUM

4.1 Poynting theorem description
4.2 Basic acoustic wave equations
4.3 Acoustic Poynting’s theorem
4.4 Time harmonic excitation
4.4.1 Green’s function solution to the time harmonic problem
4.5 Summary

CHAPTER 5: TRANSMISSION-LINE MODEL FOR ACOUSTIC WAVES
5.1 Distributed circuit transmission line model

5.1.lInss1ess transmission line ease

5.1.3 Finite transmission line with a resistance termination case

5.2 Trammission-line model for inhomogeneous moving acoustic medium

38&38%

idfl$$

CHAPTERfiACOUS'HCFIELDWITHINANARBITRARILYSHAPED
SCATTERER
6.1 Introduction
6.2 Green’s function description of the scattered field
6.3 Matrix formulation procedures
6.4 Determination of transformation matrix elements
6.4.1 Off-diagonal elements
6.4.2 Diagonal elements
6.5 Computer simulation for the scattered field
6.5.1 Pixel dimension description
6.5.2 Computational evaluation of inhomogeneous field distribution 93

6.6 Summary 102

CHAPTER ‘7: EXPERIMENTAL MEASUREMENTS AND SIMULATION
RESULTS

BSSSSKSBU

7.1 Experimental results 104
7.1.1 Acoustic property measurements 104
7.1.1.a'I‘hepropagation speed 109
7.1.l.b The attenuation coefficient 109
7.1.1.c The reflection coefficient and acoustic impedance 110
7.1.2 Experimental effects of temperature variation on
acoustic parameters 115
7.1.3 Experimental results on delamination 123
7.2 Simulation results 126
CHAPTER 8: CONCLUSIONS 131
BIBLIOGRAPHY 136

2.1:
3.1:
6.1:

6.2:

7.1:

LIST OF TABLES

An analogue between acoustics, circuits and electromagnetics
Parameters for fat, bone and muscle

The central displacement field of a homogeneous cube with various n(r),
insonified by a plane ultrasonic wave of different frequencies.

'I'hespecifiedvaluesoftissueproperties underaharmonicfrequencyof
frequency of 1 MHz were used in this computational stimulations.

Temperature effect on ultrasonic propagation speed in water.

28

52

95

121

2.1:

2.2:

2.3:
2.4:
3.1:
3.2:
3.3:
3.4:
3.5:
3.6:
3.7:
3.8:
3.9:
3.10:
5.1:

5.2:

LIST OF FIGURES

DeformationofaslabofareaA and thicknessALsubjectedtoa
time-varying stress propagating in the z-direction.

Normal incident wave at an interface between two different media with
different acoustic parameters.

Oblique incident wave at an interface between two different media.
A slab of material with A2 subjected to an acoustic pressure.

A multilayered biological system.

Field position, within layer n, in a multilayered structure.
Multiple reflections for determining R,A in layer n.

Multiple reflections for determining R,” in layer n.

Multiple reflections for determining Tu“ from layers n to n+1.
Multiple reflections for determining T“, from layers 11 to n-l .
Ultrasonic field intensity distribution for 1 MHz and for 0.8 MHz.
Heat generated by ultrasonic irradiation at 0.8 MHz.

Heat generated by ultrasonic irradiation at 1 MHz.

Initial increasing temperature rate at 1.0 MHz and at 0.8 MHz.

A T-type transmission-line model for an electric circuit.

A T-type transmission-line model for an inhomogeneous acoustic

viii

16
21
24
37
41

43

t

47

55
56
57
7O

78

6.1: An inhomogeneous scatterer insonified by a plane ultrasonic wave. 82
6.2: Different pixel size configurations of a homogeneous bar with n(r) = 91

10, f = 1 MHz.
6.3: The ultrasonic field u,(r) along the z axis. 92
6.4: A fat-muscle plate model configuration. 96
6.5: The field distribution within the fat plate of Figure 6.4. 97
6.6: The field distribution within the muscle plate of Figure 6.4. 98
6.7: The fat-muscle-fat model configuration. 100
6.8: The x-component of the acoustic field along the x—direction of a

fat—musclefat model. 101
7 . 1: Experimental configuration of the ultrasonic imaging system. 105
7 .2: Acoustic characteristics measurement setup. 107
7.3: The returned echo from the top interface showing round trip delay. 111

7.4: The returned echo from the bottom interface showing round trip delay. 112
7.5: The returned echo from the top interface showing the peak magnitude. 113

7.6: The returned echo from the bottom interface showing the peak magnitude. 114

7.7a: Ultrasonic propagation speed at 24°C. 116
7.7b: Ultrasonic propagation speed at 26°C. 117
7.7c: Ultrasonic propagation speed at 30°C. 118
7.7d: Ultrasonic propagation speed at 33°C. 119
7.7c: Ultrasonic propagation speed at 38°C. 120
7.8: Temperature dependent ultrasonic propagation speed in water. 122

7.9:

7.10:
7.11:
7.12:
7.13:

7.14:

Nondestructive evaluation setup for honeycomb structure. 124
Nondestructive evaluation of honeycomb structure. 125
Ultrasonic intensity distribution of a five-layer model at normal incidence. 127
Ultrasonic intensity distribution of a five-layer model at 30° incidence. 128
Ultrasonic intensity distribution of a three-layer model at normal incidence. 129

Ultrasonic intensity distribution of a three—layer model at 30" incidence. 130

CHAPTER 1

INTRODUCTION

1.me

Ultrasonic techniques has found increasing attention in clinieal diagnostics (both
as a diagnostic tool and for therapeutic applications), nondestructive evaluation and many
other applications in recent years [1]. Ultrasound is a preferred diagnostic tool, because
it is noninvasive, nonhazardous, non-traumatic and low cost. Compared to other existing
diagnostic instruments, it is safer. The noninvasive character of ultrasound and its ability
to distinguish interfaces between tissues of different acoustical impedanees has been its
main attraction [2]. In contrast, X-ray only responds to atomic weight differences and
may require the injection of a more dense contrast medium for visualization of specific

tissues.

Several techniques have come to the forefront for medical and nondestructive
applications as stated previously, namely X-ray tomography, nuclear magnetic resonance
imaging (MRI), and ultrasound. Each has certain advantages and drawbacks of given

applications. For example, both X-ray tomography and MRI imaging systems are capable

2
ofproducingeacellartpicturesofawidelydiverseclassoftestobjects,andinsome

cases can even be used to give a three-dimensional reconstruction of these tested subjects.
Also, ancxcellentclarityoftheboundaries scparatingtheinternalpartsofthenorm,
which is invaluable in medical diagnostic tests, particularly in delicate areas where
exploratory surgery is not feasible or possible without patient risk. Commercial systems
are rather well developed, with many features available, and are quite widespread.
However, there are certain disadvantages with the above systems. For example, the
exceedinglyhighcouandadminedlyunhrownheumfisksposedbyexposunmhigh
intardtyenergyunissionsusedmmtermgateflwobjectbeingprobed.1hewmd
”interrogation” generally refers to the type of verbal questioning. From physics point of
view, ultrasonic interrogation refers to a procedure to collect data about the internal
structure of an object and then generate an image of some otherwise hidden properties

of the object.

In contrast to the above technologies, ultrasound is an alternative in certain
medical and/or industrial applications. Ultrasonic technique is a nondestructive nrethod
in which ultrasonic beam of high-frequency pressure wave, introduced into the material
beinginspccted, forthepurposeofdetecting the surfaceandsub-surfaceflaws. The
ultrasonic waves travel through the medium with certain amount of attenuation and are
reflectedatthebormdarybctweendifferentmedia. Thereflected sigrralthenbedetected
and analyzed to define the presence and the location of defects. The defects of materials

that ultrasonic technique is able to detect could be cracks, delaminations, shrinkage

3
cavity, pores and other discontinuities that act as metal-gas interfaces.

Many applications of ultrasound involve human exposure to ultrasound
envirmmartbydthuincidarmuymasapanofatherapcuficprocedureofcfinical
applications. Knowledge concerning the ultrasonic energy distribution and the ultrasonic
energyabsorbedwillbeverycrucial indetermining theheatand temperatureprofiles
inside the medium. Especially in therapeutic applications, how the ultrasonic energy
distributes is of major interest. Therefore, a detailed and complete model of biological

structure exposed to the ultrasonic irradiation deserves further exploration.

The knowledge of precise ultrasonic intensity distribution in clinical applications
isessentialfromthefollowingviewpoints. Firstofall, itdetermines theamountofthe
ultrasonic arergy reaching at various depths of biological structures. For example,
knowing the ultrasonic intensity distribution in a mother’s abdomen and the uterine cavity
determines the amount of ultrasonic energy reaching the fetus [3]. This is extremely
important for safety reason during ultrasonic diagnostic procedure, because the use of
diagnosficflfiawundismpervasivethatevenasnmnpotenfialhazardconcause
unnecessary damage. Secondly, it is important in therapeutic process that part of the
ultrasonic energy will be permanently converted (an irreversible heat process) to heat and
causing temperature arise due to the absorption process. In particular, knowing the
amountofwmpuaMmfiseinfissuewiHbeveryhelpfiflindetenmningmethaapwfic
dosage in hyperthermia treatment of tumors, for example.

4
Before developing the multi-layered model, the Green’s function of a given multi-

layered biological structure by considering all the possible wave reflections needs to be
determined. Since heat generation is the prime concern, the Green’s function derived is
direcflyfiomcmfideringulfiawnicmtensitymmermmfiommfiasonicpressumpoint
of view [4,5 , 6,7]. Consequently, the calculation of intensity field distribution and initial
temperamreincreasingmtecanbeobtainedmamthersnaightforwardmanner.
Ultrasonic field distribution in biological tissues has been investigated in recent years.
However, the models used in the reported work were either deriving from pressure
viewpoint or ignoring the multiple reflections in various tissues boundaries [8,9,10]. In
reality, however, multiple reflections do occur in tissues’ interfaces. In order to obtain
a closer cstinration of the ultrasonic energy distribution in biological tissues under
ultrasonic irradiation, an analytical approach to consider all the possible multiple

reflections in a layered biological structure will be considered in this study.

The acoustic Poynting’s theorem interrelates the power supplied by acoustic
sources, the time rate of change of energy stored in the medium, the power dissipated
in the medium through propagation, and the power flow of the system. Understanding
where ultrasonic energy comes from, goes to is the fundamental requirement for
ultrasonic study. The basic acoustic Poynting’s theorem for stationary medium has been
developed over the years [11,12,13]. However, for moving medium and the treatment
of spatial and time dependencies of density and stiffness of medium are commonly

ignored.Inordertogainanoverallpictureofenergybalancewherranacousticwave

5
fiavdsthmughamovingnwdiumboflrtemporalandspafialvariafionsofmaterial

properties have to be considered. In addition, effects such as thermal diffusion, collision,
andmediumdriftshouldalsobetakenintoaccount.

W

The dissertation is organized as follows:

Chapter 2 is the analytical foundation of the dissertation in which the acoustic
wave equation is developed. Some acoustic parameters, such as acoustical impedance,
particle velocity, density, acoustic intensity, refection coefficient, transmission coefficient
and attenuation coefficient are defined and developed. Frequency dependent attenuation
coefficient is also stated and utilized. Some fundamental and important techniques such
as electrical and acoustic Poynting’s theorems, acoustic energy stored, energy dissipated
and heat generation are also included. An electromagnetic—acoustic analogue is also
presented for better understanding the physical properties between electromagnetic waves

and acoustic waves.

Chapter 3 is about the intensity distribution and temperature rise for multilayered
structure. A generalized Green’s function, including all possible transmissions and
reflections occurred at the interface between different media, is formulated. A computer
simulation of fat-muscle-bone system is developed here to verify the theoretic work.

6
Chapter 4 is a more generalized Poynting’s theorem for inhomogeneous, moving

acoustic system. A detailed breakdown equation of the acoustic energy conservation
theorem is developed in this chapter. The thermal diffusion, particle collision, drift speed
and heat flow in acoustic system are commonly ignored for simplicity reason [11,12,13].

They are all been tam into account in this chapter.

Chapter 5 involves the development of a transmission-line model for the acoustic
wave propagation in an electric system and an acoustic system. The transnrission-linc
model offers an excellent overall picture for understanding the acoustic wave travels

throughamedium.

Chapter 6 deals with formulation of the ultrasonic displacement field within an
arbitrarily shaped scatter. A dyadic Green’s function technique is used for the

development.

Chapter 7 presents the experimental and simulation results and chapter 8 is the

conclusions and some suggestions for future study.

CHAPTER 2

BACKGROUND

Todiscussdrereseamhundertakeninflfisdissertafionfitwiflbenecessaryto
reviewwmeprdiminaryconccpuofacousficwave.Wefirstdewfibedrebasicwave
behaviors of acoustic wave travelling in a medium. Secondly, the acoustic wave
propagafimpropafieswinbediscussed.Fmanywedewfibemeanaruafingphmmrena
when acoustic wave propagating through a lossy material.

WM

Unlike the electromagnetic waves, the acoustic waves require a medium through
which the acoustic energy can be transferred. Therefore, the medium where the acoustic
wave travelling through plays an important role in acoustic wave applications, such as
ultrasonics and underwater acoustics. The most important acoustic parameters are
pressure, particle displacement (or particle velocity) and medium density.

If the driving source produces the particle movement in the wave propagation
direction, the wave is called a compression wave or longitudinal wave. On the other

8
hand,ifdresourcennvesflreparficleindremediumpcrpardiculartodrewave
propagationdirecfion,itiscalledashearorfiansversewave.Formostofthe acoustic
applicafions,mchuulnasonicwavesuscdinmedicalandmaterialevaluadm
applications, weconsidcr the ultrasonic energy is travelling in one direction only, usually
in the particle longitudinal vibration direction. As a result, only longitudinal wave is
considered in this dissertation for simplicity. Furthermore, there is an experimental
evidence[14,15]tosuggestthatshearwavesarestronglydampedthatonlylongitudinal
wavesneedtobeconsideredindiagnosticmedicine(0.5-20MI-Iz)andmaterial

evaluation.

In order to simply the analysis, a one-dimensional wave propagation is considered
asshowninFigure2.l, whereaslabofmaterialofthicknessALissubjectedtoan
externalforce-Tperunitarea. Thestressand strainofinelasticmaterialaredefinedas

follows

‘i
I

(2.1)

to
I
<4: sale:

where Fisthefcrceappliedtothe medium,A is the unitareaand Visthevolume.

Theuactionforceistransnfitteddrmughflremediumandeausethemediumto

deform.BytheNewton’ssecondlaw,theforce(F)actingonthisslabcanbeshownas

z+u z+AL+u+Au
l
l
l
I

-—.IU.- -c- U+ «.—

Au

 

 

 

 

—T"_*—"‘+T _""T+ AT

2 'z+AL

 

 

Figure 2.1 Deformation of a slab of area A and thickness AL subjected to a time-
varying stress propagating in z-direction.

F=Ma ‘3'”
or
. 61'
F A 3; AL
M- pAAL ta.”
. .52
at?

where Tisthestressappliedtothemedium,Misthemass,aistheacceleration, uisthe
particle displacement and p is the medium density respectively. Applying Eq. (2.3) to
Eq. (2.2), one obtains

61‘ 62a av
35 3 p a; 3 9 E (2.4)

where v is the particle velocity of medium.

By applying the definition of strain (3) of the medium and the incremental change
of particle velocity over the thickness AL of the slab, one yields the following equation

65' 3 av . (2-5)

at E

By applying the linear Hook’s law

T333 (2c‘)

11
where B is the elastic stiffness constant, Eq. (2.5) can be modified as follows

161' . 6v
3.3.5 67.7 . (3.7)

Both Eqs. (2.4) and (2.7) are coupled equations between T and v. To develop a
general acoustic wave equation, we can decouple the stress 1‘ and the particle velocity

vbynkingderivationwithrespecttotimeandspaceonemoretime.

fl ., av
623 5231:
(2.8)
fl :3 62V
at:2 52%
ThewaveequationforTbecomes
EFT .. a it:
E; B t3 . (2.!)
Thewaveequationcanthenbeexpressedas
as _ , 62
$51.12, C) - C2 Egg-712, t)
(2.10)

CHI-é
p

where c is the acoustic wave velocity. Both acoustic wave field quantities stress (7) and

particle velocity (v) can be expressed as Eq. (2.10), one-dimensional acoustic wave

12

equation.

Let <I>(z,t) denote the acoustic field quantity which satisfies Eq. (2.10). <I>(z,t) has

a general solution form of

o (z, t) = D ei‘mm (2.11)

where D is amplitude, a) is radian frequency, 1: = We is the wave number, ’ +’ sign
represents a backward (- z direction) propagation wave and ’-’ sign represents a forward
(+ z direction) propagation wave. The movement of the medium particle moving back
and forth forms the particle velocity. It should be noted that this velocity is different
from the rate of energy propagating through the medium, which is defined as the group
velocity or the sound propagation speed. We shall characterize the acoustic wave by a
pressure field P(z,t). In the case of a harmonic wave in a homogeneous medium, the

particle displacement velocity in the acoustic field is given as:

(VP) = i .ejhatartz) . (2.12)

V = ,
Jmp pc

Analogous to the electromagnetic waves, it is possible to define an acoustic

impedance of an acoustic wave travelling in the +2 direction as

13

 

- T p Bailout")
2 a — n — = = c
V D . ails-stats) P (2'13,
DC

TheboundarycondifionscanbeobtainedbyintegratingEq.(2.5)acrossthe
boundary

fl ‘2‘
3T _ av .

0‘0 I-‘O

For as - 0 (Le. boundary conditions), the boundary conditions for T or P can be shown

r, = T, ,- p, = P, . (2.15)

Assuming the interface is an ideal contact, that is, the particle displacement at the
boundary has to be continuous or
P, = I"2 pressure is continuous, and

v, = v; particle velocity is continuous.

The intensity of a wave is defined as the average power carried by the wave per
unit area normal to the direction of propagation. Applying the analogies of pressure with
voltage and particle velocity with current of electric circuit theory, we can define the rate

ofenergy flow in theacoustic system. At aparticular location, say 20, the kinetic energy

l4
ispropordmaltothesquamofflrcparficlevelocityandthepownfialarergyis
proportional to the square of pressure. The average rate of energy flow past a given
locafionisproporfimaltothemeanvalueofthepmductofpressureandparficle
velocity. Foracoustic wave propagation, the acoustic wave time-varying intensity, i(r),

is related to the medium particle velocity and the pressure by the following equation

.i(t) =P(t) °v(t) . (3-1‘)

For sinusoidal excitation, the time average acoustic intensity, I , can be carried out by

averaging i(t) over a period of time as

I a g p, v(, (2.17)

where Po and vo represent the peak values. However, most of ultrasonic imaging
applications use pulsed mode operations and unfortunately the intensity of the ultrasonic
beam is inherently non-uniform. As a result, there are two ultrasonic intensity definitions
commonly used in the pulsed ultrasonic systems: spatial average-temporal average
intensity (SATA) where the temporal average intensity is averaged over the beam cross
section in a specified plane, it may be approximated as the ratio of ultrasonic power to
the beam cross section area; and spatial peak-temporal average intensity (SPTA) where
thevalueoftheternporalaverage intensity is taken atapointin theacousticfieldwhere
the spatial peak intensity is maximum.

 

When a plane wave impinges normally on an interface between two different
mediawithdifferentacousticcharacteristics, itwillbepartiallyreflectedandpartially
transmitted as shown in Figure 2.2. Let P,, P,, and P, represent the incident, reflected

and transmitted acoustic waves, respectively, or

p, s an, t) = 9,91 “”1"
p, - p,(z, e) - p,e“°"*:" “out

p, I p,(z. t) . 0,91 “"9"

The symbols 0,, D, and D, represent acoustic wave pressure amplitudes for
incident, reflected and transmitted wave. at, and x, are the wave numbers for the two
media. From Eq. (2.12), one can obtain the particle velocities v,, v, and v, for the

incident, reflected and transmitted acoustic waves, respectively, or

 

 

 

V (VPI) = _ D1 . eflrsc-qs)
1 Iii-’91 Prcr

v a (VP!) .. _D—‘. ~e""‘*“1 " (2.19)
‘ jaws Plcl

v (vpt) , _ Dc . adult-gs)
c jwpa 92c:

 

At the interface, as stated previously, the following boundary conditions must be
satisfied at all times:

16

 

P. >

 

 

PP:

 

 

20

Figure 2.2 Normal incident wave at an interface between two different media
with different acoustic parameters.

17
(1) mordchprua'veconfinuitthepressureattheinterfacemustbedresameon

bothsides,or

P,(o,t) . P,(o,t) + P,(o,t) , (2.20)

(2) Parficlevelocifiumrmalwdreinterfacemustbeequalonbothndesmdrerwise

thetwomcdiawouldnotremainincontact

vc(0,t:) - v1(0,t:) + v,(0,t) . (2.21)

From the above boundary conditions, one ean define the pressure reflection, R,

and the pressure transmission coefficient, T, as follows

P,(0.t)
P1(0.t)

P.(0.t)
PAC. 1:)

R s
(2.22)
TI

Substituting Eqs. (2.18) and (2.19) into Eqs. (2.20) and (2.21), one obtains the reflection

and transmission coefficierrts for normally incident acoustic waves as

, Paca " Pics . za " Zr
9292 T 01C; Za + Zr

 

(2.23)

and

2(92031 _ 222
9202 "' Prcr 2: T Z:

 

 

= 1 + R . (2.24)

18
In order to derive the dual interrogation relationship, the reflection coefficient (R ')

and the transmission coefficient (1") from the opposite direction, i.e. from medium 2
incidenttomedium l,csnalsobeobtained.1nasimilarmanner,onecaneasilyarrive
atthefollowingrelstionship

R’=R
Wal-T.

(2.25)

As stated previously, the acoustic intensity I is proportional to the square of the

pressure, i.e.

I a |P(z, t) |2 . (2.“)

We can also define the acoustic intensity reflection coefficient (7) and the acoustic
transmission coefficient (t) by applying the pressure reflection and transmission
coefficients into Eq. (2.26). At the interface, the ratio of the acoustic intensity of the
reflected wave to that of the incident wave defines the intensity reflection coefficient (7),
andtheratiooftheinwnsityofthettansmittedwavetothatofincidcntwaveisthe

intensity transmission coefficient (t).

 

2 _ 2
y I 5.1;: - IP'(O' ”I a [u] (3.37)

It Vume ‘%+%

From the conservation of energy at the interface the acoustic intensity transmission

coefficimt becomes

t :2 - 4%

I1 ZZ + Z1

I
H

I
.<
II

(2.28)

where 1,, I, and I, are the acoustic intensity of the incident, reflected and transmitted
waves, respectively. It is interesting to note that the reciprocity applies also to the
acoustic intensity reflection and transmission coefficients. In other words, 7 and t are the
same regardless the acoustic wave propagates from medium 1 to medium 2 or from

medium 2 to medium 1. That is

= Y I (202’)

and

tl=1—Y/=1-Y=t' (2.30)

where 7 ’ and t ’ represent the acoustic intensity reflection and transmission coefficients
from medium 2 to medium 1, respectively. The behavior of both the pressure and the

intensity at the boundary plays an important role in ultrasonic interrogation.

 

In general, acoustic wave may be considered to be propagating at an incident

angle to the interface between different media as shown in Figure 2.3. The acoustic wave

20

wifltharbepufiaflyreflectedtosamemediumandparfiauymsmiuedmthenext

medium as in the normal incident situation discussed in Sec. 2.2. The boundary condition

attheinterfaceisobtainedfromthefactthattheparticlevelocityalongtheinterfacemust

be continuous. Three conditions have to be satisfied under this boundary condition:

(a)

(b)

(c)

The tangential components of incident and reflected wave velocities must be

equal, or

v, sine, = v, sine, (2.31)

where 0, and 0, are the incident angle and reflected angle of acoustic wave,

respectively.
The tangential components of incident and transmitted wave velocities must be

equal. or

v, sine, 8 VC sinfic (2.32)

where 0, and 0, are the incident angle and transmitted angle of acoustic wave,

respectively.
The normal component of the particle velocity must be continuous, otherwise, the

medium will not remain in contact.

v, (20301 - v, c080, - vc c080,; . (2.33)

By Eqs. (2.31), (2.32) and (2.33) and the definition of particle velocity, one

21

Medium 1 Medium 2
Pl 2C1 Farce

3'0?) -
W u‘CF")

 

 

 

 

Figure 2.3 Oblique Incident Wave at Interface Between two Difiuurt Media.

22

 

arrivesatthefollowingresults
e, . o, , (2.3.)
3(a) . .5; . w (3.3;)
1 p2 c:2 C0891 + 91 (21 C080: '
and
Pt 292 c2 C0801
T(0) . P1 92 c2 C0891 + 91 c1 cosOc (2.36)

where R(0) and T0) are the pressure reflection and transmission coefficients of the
oblique incidart wave, respectively. By defining a general acoustic medium characteristic
impedance as shown in Eq. (2.37)

I (2.37)

one can conclude that the reflection and transmission coefficients at a normal

incident interface appeared on Eqs. (2.23) and (2.24) are the special cases of Eqs. (2.34)

and (2.35) with the incident angle 0 equals to o.

 

When an acoustic wave propagating through a medium, its pressure magnitude

23
decreases due to the acoustic wave interaction with the material. Two important types of

attenuationareahsorptionandscattering.

2|] 5] . I .
WhenanacousficpressureisappfiedbaslabofmatefialasshowninFigure2A.
Theamountofprunuedecreasedalongadistancedzispmporfionalmmeinifial

pressureflz)andthethicknessoftheslabAz,or

AP . a P(z) Az “-3”

where a is the attenuation coefficient of the medium. The decrease of pressure can also
beexpressed as

_ _62
AP 5; AZ . (2.29)

Equating Eqs. (2.38) and (2.39), the location varying pressure P(z) can be obtained as

PM!) = P(0) e’" . (2-30)

In other words, the magnitude of acoustic pressure decreases with distance
exponentially. Since the acoustic intensity I is proportional to the square of the pressure,
it decreases in the manner of 6". The absorbed acoustic energy inside the medium is
transformed into another form of energy. The absorption loss in liquids and solids, so

24

 

 

 

 

 

1” I J
I
V’ I ’ ‘
’ I
I u
d
y I’I ’I ”
1’ 4
I I
1’ I ”
I I
V’ I ’ I‘
I ’/ ’
I
V’ ’l’ ,’ ’I’
I I I I
I
p(Z)——.y’ I I’ ’I -—. P(Z+AZ)
/’ ’ I
V I’ [I I,
I I ’I I
V, II I ”
I ’ I ‘
V’ I
_ AZ ‘
I I
V I
1” 1” 1’ 4
r 2’ 2’ I A /z\

 

Figure2.4 Aslabofrrmerialwithdzsubjectstoanacousticpressure.

25

called relaxation absorption, occurs when the acoustic wave propagates through material.

2 4 rin ff

When an acoustic wave arrives at a boundary, the wave is partially reflected and
partially transmitted. The acoustic energy which is scattered no longer travels in the
original propagation direction. Scattering of an ultrasonic wave occurs because most of
materials are not truly homogeneous. Crystal discontinuities at medium boundary tend
to deflect small amounts of ultrasonic energy. As a result, part of the energy will not be
able to be recovered. Such losses can also be considered as wave attenuation. In some
instances, determination of the degree of scattering can be used as a basis for acceptance

or rejection of materials.

W

The attenuation coefficient a is a strong function of the operating frequency. In

general, it can be written as follows [52]

a(f) = (10 f” (2.41)

where
(10 = attenuation coefficient, and
n = exponent dependency; l < n < 2 typically.
As can be seen from equation (2.41), low frequency acoustic signals with long

wavelength should be employed for deep penetration application. Unfortunately, the

26

resolution of detecting objects is in the order of the operating wavelength A. Shorter
wavelength will be desirable. Clearly, there is a trade off between the depth of
penetration and the resolution. If the incident wave is a narrow-band signal which is
commonly used in laboratory for good resolution, the attenuation will not change
significantly over a range of frequencies around the center frequency. A narrow-band
signal is very difficult to realize in conventional systems, therefore, for more accurate

analysis, a frequency dependent attenuation coefficient needs to be considered.

 

There is a greatly similarity between acoustic wave and electromagnetic wave.
This can easily be observed by comparing the acoustic wave equations which are mainly
shown in Eqs. (2.4) - (2.6) and the well-known Maxwell’s equations for
electromagnetics. The field problems of acoustic waves are of the same nature as those
occurred in electromagnetic waves, such as uniform plane wave propagation, guided
waves, periodic waves and coupled modes. Presentation of the acoustic field equations
in a form analogous to Maxwell’s equations of electromagnetism simplifies the analytic
procedures in solving acoustic problems. With this in mind, the analogy between acoustic
waves, electric circuits and electromagnetic waves is presented in Table 2.1 in hope that
one can obtain more insight of the acoustic wave properties. The following glossary

shows the definition and the unit used in Table 2.1:

volts/m

amp/ m

coulomb/m2

farad /m
henry/m
N / m2
N/m2

m/ sec
unitless
N. sec/m3
kg/m3
N/m2
watt

volt

amp

ohm

watt

27

electric field
magnetic field
displacement field
permittivity
permeability
stress

pressure

particle velocity
strain

impedance
density

stiffness constant
intensity

voltage

current
resistance

power

28

 

Electro-

 

 

 

 

 

 

 

 

Acsoustics Magnetics Circuits
-T=P E V
v H i.
-_P_ _ E _ v
Z- v Z-T-T R-‘i—
P ,u _.
1
B E ’
-l __
S-BT D-EE -
I - :3

 

 

 

Table 2.1 An analogue between acoustics, circuits and electromagnetics

 

29
W

The electromagnetic field equations, i.e. the well-known Maxwell’s equations, are
shown in Eqs. (2.42) through (2.45), [12]

-vxs=£ (2.42)

at
VXH=a—:-+J,+J, (2.43)
V°D=p (2.44)
V-B-O (2.45)

where J, is the conduction current density, J, is the source current density and B is the

magnetic flux density.

To make the set of field equations complete, the following constitutive relations

areneeded

D=€'E an“)

B=u°H. (2.47)

30

Substituting the constitutive relations into Eqs. (2.42) and (2.43) and using the

conduction current relation

J = o °E (2.48)

the homogeneous Maxwell’s equations become

-vxs=n-%¥ (2.49)

and

VXH=e-%‘:3+o-E+J,. (2.50)

The homogeneous electromagnetic Poynting theorem can be derived by following
the steps below: First, taking the scalar product of equation (2.42) with -H, then adding
to it the scalar product of equation (2.43) with —E, finally using the vector identity and
integrating over a volume V. The electromagnetic Poynting theorem is obtained as

GB
— +

£(EXH)°fidS=*fV(H' at

E - a—D) dV
‘9‘ (2.51)

-fy(5 '1‘) dV+fy(—E J) (W,

where S is the surface enclosing V and n is a unit vector in the outward normal direction

to the surface. With some identifications, Eq. (2.51) can be further expressed in an

31

energy conservation form

fs(EXH)-fidS+%-:+P,=P,, (2.52)

where
U is the energy stored in the volume being considered,
P, is the dissipated power, and

P, is the power supplied by the source.

The first term on the left-hand side of Eq. (2.52) can be interpreted as the total power

flow outward through the closed surface S.

7Ai "hm

There are two acoustic field equations, namely ”strain-displacement relation"

shown in Eq. (2.53) and "equation of motion" shown in Eq. (2.54) [12]

S = Vu . (2.53)

V°T=p§X-F , (2.54)
at

where F, is the external force. The elastic constitutive equation for acoustic medium

which relates T and S is shown in equation (2.55)

32
T=BS+11§-g-. (2.55)
at

Applying a similar procedures as given in Sec. 2.6 and using the electromagnetic-
acoustics analogy, one can easily arrives at the following the acoustic Poynting’s theorem

for homogeneous medium
fS(-v-1)-ad5+%’+p,=p,, (2.56)

where
U is the stored energy which includes kinetic energy and potential energy,
P, is the total viscous power loss in the system,
P, is the power supplied to the system, and
-v . T is called acoustic Poynting vector similar to the concept of its counterpart
in electromagnetic field.

The U, P, and P, are defined in Eq. (2.57).

= 6S2
P. - fyn (3;) «W

P zfyv-FdV

S

U a [y (uv + up dV (2.57)
_ .1. 2
av 2pv
u, - 11352
2

 

When the acoustic intensity or energy is absorbed in a medium, the temperature
of the medium will tend to rise. The rate of heat generated per unit volume is equal to
the loss of acoustic power

= _ @
Q dz (2.58)

2 01 1(2)

The rate of temperature rise can be shown in the following expression

__ _ _ , (2.59)

where H is the heat capacity per unit mass. For water and most of soft tissues, pH is

approximately equal to 4.2 joule/cm3.°C.

CHAPTER 3

INTENSITY DISTRIBUTION AND
TEMPERATURE RISE OF
MULTILAYERED STRUCTURE

Heatgeneration,temperamreincreasingrateandtemperaturepmfileduring
ultrasonic therapeutics play important roles in determining the processing duration time
aswellastheoptimumandsafetyacousticintensitylevels. Inthischapter, ananalytical
model using the Green’s function of a multi-layered structure has been developed. This
model tales into account all the possible multiple reflections in the biological boundaries
which are commonly ignored by other investigators [6,8]. This study uses intensity point
of view to derive the Green’s function of a structure. Traditionally, such a Green’s
function is derived from ultrasonic pressure point of view. The analysis of a simple fat-

muscle—bone lay-up is also included in this chapter to validate the approach.

W

Knowledge concerning the ultrasonic energy distribution and the ultrasonic energy
absorbed willbeverycrucialindetermining the heat and temperatureprofilesinsidethe

34

35
medium. Especially in therapeutic applications, how the ultrasonic energy distributes and

being absorbed are of major interest. Therefore, a detail and complete model of
biologicalstructureeaposedtotheultrasonic irradiation deservesfurtherexploration.

The knowledge ofprecise ultrasonic intensity distribution in clinical applications
isessential fromthefollowingviewpoints. Firstofall, itdeterminestheamountofthe
ultrasonic energy reaching at various depths of biological structures. For example,
knowing the ultrasonic intensity distribution in a mother’s abdomen and the uterine cavity
determines the amount of ultrasonic energy reaching the fetus [3]. This is extremely
important for safety reason during ultrasonic diagnostic procedure, because the use of
diagnosficulfiamundiswpavasiveflratevenasmaflpotenfialhawdwuldcause
unnecessary damage. Secondly, it is important in therapeutic process that part of the
ultrasonicena'gywillbepermanently converted (an irreversibleheatprocess) toheatand
causingtemperauu'eriseduetotheabsorptionprocess. Inparticular, knowingtheamount
oftemperatureriseintissuewillbevery helpfulindeterminingthetherapeutic dosage

in hyperthermia treatment of tumors, for example.

Beforedevelopingthismodel, weneedtofirstdeterminetheGreen’sfunctionof
a multilayered biological structure by considering all the possible energy reflections.
Since heat generation is the prime concern, the Green’s function derived should be
directly from considering ultrasonic intensity rather than from ultrasonic pressure point
of view [4,5,6,7]. Using the Green’s function, the calculation of intensity field

36
disuibufimandinififltunpaatunincreasingmtecanbeobminedmammersuaight
forward manner. Ultrasonic field distribution in biological tissues has been investigated
in recent years. However, the models used in the reported work were either deriving
from pressure viewpoint or ignoring the multiple reflections in various tissues boundaries
[8,9,16]. In reality, however, multiple reflections do occur in tissues’ interfaces. In order
to obtain a more accurate estimation of the ultrasonic energy distribution in biological
tissuesunderultrasonicirradiation, wepresentananalytiealapproachbytakingall
possible multiple reflections into account in a layered biological structure in this chapter.

WW
321E ,fl .fl H l

A Multilayered biological structure is intended for this investigation. Figure 3.1
shows the structure, where Z is the characteristic impedance, C is the propagation speed,
a is the attenuation coefficient, H is the heat capacity, and d is the thickness. In our
analysis, the structure is infinitely extended in both 1: and y directions. In other words,
a one-dimensional problem is being considered. This structure is irradiated by an oblique
ultrasonic continuous wave (CW) with incident angle 0, . The ultrasonic field intensity

can be determined by the following Green’s function formulation

10'.) - f fwflr’y G(r.,r’) dzr’ (3.1)

37

 

 

 

 

 

 

21. Ct. at. H1. at Layer 1
22. Ca. Ola. H2. d2 Layer 2
2.-1,Cn-t,an-t_ H.-t,d.-t Layer n-1
2... C... at... H... d,, Layer n
Z..t,C..t,d...t, H..t,d..t Layer n+1

0 ’2‘ Layer n+2 A

Figure 3.1

 

A Multilayered biological System.

x)

38
where

f(r') is ultrasonic source distribution;
G(r,,r') is intensity Green’s function due to multilayered structure;

10‘.) is ultrasonic field intensity at 1;.

Using an appropriate Green’s function, the ultrasonic intensity field can then be
obtained by Eq. (3.1). Suppose the ultrasonic source is situated in the first layer and is
locatedattheoriginofthecartesiancoordinates,ourgoalistofindtheintensity
distribution in the individual layers. A study of ultrasonic scattering from multilayered
elastic structures was investigated by Aymé-Bellegarda er 01. [4,5]. Both longitudinal and
shearwaveswereoonsideredintheiranalysis sincethematerialsusedintheirmodelare
of elastic nature. For clinical applications, however, only longitudinal wave is considered
since biological tissues support mainly longitudinal wave. For simplicity, we assume the
medium is homogenous in all layers. Our model can be extended to include
inhomogeneous medium by dividing inhomogeneous layer into sublayers with
infinitesimal thickness. An ultrasonic plane wave is incident upon the interface of layers
1 and 2. Part of the ultrasonic energy is transmitted into the second layer while part of
the energy reflected back. The transmitted energy will be attenuated when it travels
through the second medium. This forward traveling wave will then be partially reflected
when it hits the next interface. This process repeats at each interface. As a result,
multiple reflections will exist in each layer. The sum of all the forward and backward

traveling pressure waves will contribute to the total heat generation in that layer.

39
When an ultrasonic wave propagates through a medium, its intensity is dictated
by the reflection and transmission coefficients at the boundaries and the absorption
characteristics of the medium. The transmission and reflection coefficients can be
obtained by applying the boundary conditions at the interfaces which states that the
normal component of ultrasonic pressure and the particle velocity must be continuous in
the interface. The intensity refection coefficient (r) and the intensity transmission

coefficient(t)fromlayer1tolayechanbeexpressedas:

It. .ELLEIZ; 5H=4_z,_z,_ (3.2)
If (23 + Z1): ’1 (z; + z‘)’
where
Z, I p,C,/cos (0,), 0, is incident angle.
22 I p,C,/cos(0,), 0, is transmission angle.

The intensity of a plane ultrasound wave attenuates through medium in the following

manner,

1(2) = 10 .4“ (3.3)

where
[(2) (W/cm’) is the ultrasonic field intensity amplitude at a position 2 (cm),

la is the constant ultrasonic field intensity amplitude at z=0, and

40
a (NP/cm) is the attenuation coefficient of the medium.

Figure 3.2 shows the upward and downward traveling waves within an arbitrary
layer It in a multilayered system. R,‘ is the global reflection coefficient which includes
all the possible multiple reflections and transmissions contributed from all the layers
above the layer )1. Similarly R,’ is the global reflection coefficient which includes all the
possible multiple reflections and transmissions contributed from all the layers below the
layer n. W,* is the propagation wave from the upper interface between layer (ll-1) and
layer It to the field point. Similarly W; is the propagation wave from the lower interface

between layer It and layer (n+1) to the field point. “1," and W; can be expressed as

W: - Im-jknm’
W; = chp<-ik.~r.)l’

(3.4)

where k, =B,-ja, is the wave number of layer )1, r,+ is field position for the downward

propagating wave and r,’ is field position for the upward propagating wave.

R.‘ can be determined by considering all multiple reflections from each layer as
indieatedinFigure3.3andexpressedasaninfiniteseries:
R_‘= (I)+(II)+(III)+(IV)+ ...... , where

(1)= r,___, .

(n) = n-IJLo-IRp-IALrlth-I 2

41

Layer n-1

 

 

I
I
I
l
+ I
W" ;
I
ul (.12 US U4 5
I
I
Layer!“ :1 l2 :3
I
I -
E w,
I
'e 'a ‘8 1 We '3 'a
Rn Rn Rn : Rf) Rn Rn
Layer n+1

Figure 3.2 Field position, within Layer )1, in a multilayered structure.

42
(III) " tad.-(LberIJLI-lrn-IJJLa-IRn-IALa-Iturl t
(IV) = t.....(Ar-”ML...r...)’L.tR.z‘L.zt..-z ,

(V) = ten.(L.1R.1‘L.tr..tJ’I-.1R..1‘L..tt...t. - . -

Inasimilarmanner,R_'canbeexpressedasaninfiniteseriesas showninFigure3.4,
R,‘ = (i) +(ii)+(iii)+(iv)+(v) + ...... , where

(i = ’...+1 .

(ii) 3 t,+,,L.+,R,+,'L.+,tu+, .

(iii): ‘.+1..(la+rkn+r.L.-1’.+1JL.“Rama-4’.»1 .

(iv)= n+1,-(LI+IRQ+I.Ln-Ira+lJ2Ln+an+I.Ln-Itrs.u+1 .

(V)= tn+1mma+IRI+I.Lrlrn+lJ’La+IRn+InLn-1tn,n+ls °°°°°°

The reflection coefficient r’s and the transmission coefficients t’s are defined as
r", is the intensity reflection coefficient from layer It to layer 21-] ,
h...) is the intensity reflection coefficient from layer It to layer 1) +1 ,
I”, is the intensity transmission coefficient from layer It to layer n—I ,
I...” is the intensity transmission coefficient from layer It to layer n+1,
L.= |exp(-jk,. r,‘) | 2 is the attenuating factor when passing through layer It, with

n = 2,3,4,... and r.‘=£x_+2d. .

43

 

 

Rf.) Rf.) R11, R1, R1, Layer n-Z
L..-)
tn.-. r..-) r..... r..... rm. LOW" ”-1
/\r.,,-, \t..... \i..-. \t...-. 1...... '\ '
(v) <11) (m) (N) (V)... ,
. Layer n

 

Figure 3.3 Multiple reflections for determining R.A in layer It .

 

 

 

Layer n .
(i) (ii) (iii) (iv) (v)
ram) /t mm) / t rum /tn.n+t /t (1.1») /
tam,” rmmt F...“ roan) rmrwt
Layer: n+1
L...
B .
Road») REM RnB-t-I Ros-+1 RM») Layer n+2

Figure 3.4 Multiple Reflections for Determining R" in Layer )1 .

45
In order to obtain the field intensity distribution in layer It, a calculation of the

intensity transmitted from layer 1 to layer (21-1) and layer )1 interface after considering
multiple reflections is shown in Figure 3.5. The global transmission function from layer
ntolayer n+1 canbeexpressedasaninfinite series:
Tu“ =- (1) + (2) + (3) + (4) + , where

(1) = n... . .

(2)= ....,(L...R...'L...t.....).

(3)= .,...(L...R...'L...t.....)’.

(4)= .,...(L...R...'L...t....)’ ......

Similarly, 1;“:- (a)+(b)+(c)+(d)+ ...... is global transmission function from layer
It to layer (ll-I) shown in Figure 3.6, where

(a) = a... .

(b) = I...) (lintRu'cht...1).

(e) = .azLLI'Lstt...)’.

(d) = ,,,(L,,R,,’L,,t,, ......

By summing previous infinite series and provided that It....|. um. IR.'I. W and
I’...+1I < l,one arrives at

Rs. 3 a..+1 + (n+1.- (La-+12 R"). (n+1) (1 ' L"): Ron. run-)4 .

R.‘ = r .,.-1 + tat-r,- (Ln-ran-rdtnpl) (1 ' L...’R..1‘rHJ’.

7:1,"! = tan-+1 (1 ' Ln+lst+r’rn+1Jl s and

46

Layer n

 

 

I'M)... I'M”) rue-1.0
tn...
(1) (2) (3) (4) ' ~
Layer n+1 L...M
B
R... RE. RE. RE.
Layer n+2

Figure 3.5 Multiple reflections for determining Tu“ from layers n to 11 +1.

47

 

 

Layer n-2
RnA-l R:.1 R:.1
Layer n-1 Ln-l
(9) (b) (C) (d)
tan-t
rn-t.n I'll-1.0 rfl-IJI
Layer n /

Figure 3.6 Multiple reflections for determining T“, from layer It to layer n—I .

48
Tam-I g 3.3-1. (I'Lb-Ile-IA"I-I,I).I '

To calculate the ultrasonic field intensity I,ll (r) at a field point r within layer I), one
needs to determine the Green’s function of the multilayered structure depicted in Figure
3.1. This multilayer structure Green’s function will be composed by two sources, U =
ul+u2+u3+u4+u5...andL=ll+12+13+l4+...,asshowninFigure
3.2. The part U composes of all the components travelling downward from (rt-1) and )1
interface toward the field position, and the part L is the sum of all the components
traveling upward from bottom interface to the field position. After a lengthy
manipulation, the ul, u2, u3, can be expressed as follows

“1 = W3 .taLndfl‘ualLaz)...fl'uLz)(T 1.215).

“2 = WI.“ (Ia’RfRf) (T .1111) m-ZJ-ILn-J- .. (T 2.3L?) (73,215).

"3 = WT Q’E‘K'fflazkdflamlJJ-u fl'zsLdfliaPz).

”4 = “11+aIIRuARn.)’(TrIJl%l)(Tu-Zm-1Luoz)°°°”ZHUIJPI)’ 311d

Pl 3 |exp(-jk,.(r,‘-r’)|’ -

By summing upul, 142, 113, , thepart U becomes

U = Wn+ (I'ankndRa.)-I (1.51..er aans—ILu-T)‘ ° ° (1.2.110) (TLZPI) . W3+A.B.s (305)
where
A‘ I (1-L,’R.‘R,’)" is the multiple refection factor of layer It and

B' I (T‘,,_,L,_,)... (Tun) is the transmission factor from layer 1 to layer It.

49
Similarly, the part L can be expressed as L = W,’A"B"L,R,'. Finally, the Green’s

function of the multilayered biological structure can be expressed in the form of

G(r..r’) - W.’A “B ' + mu '1) “an, . (3.6)

'I’heultrasonicfielddistributionatlayerncanthenbeevaluatedbycarryingoutthe
integral given by Eq. (3.1).

 

Equation (3.3) shows the ultrasonic intensity attenuation along the path of
propagation. The attenuated ultrasonic energy will be absorbed by the medium. The
absorption of ultrasonic energy is accompanied by an irreversible process of heat transfer
in the medium. In the therapeutic application, this heat will be transferred to raise tissue
temperature by which the therapeutic procedure fulfills. The rate of heat generation per
unit volume (Q) is defined as the loss of the intensity per unit length [7,17,18,52] in

equation (3.7)

Q = if}? = 2a 1(2) (chm’) . (3.1)

The relationship between temperature, heat generated and intensity can be obtained from

the bio-heat equation [7.18]

50

32

a l
_ = — _ - 3.8
RU) K 21k,” + HIQ(Z,I) C (2.1)] I )

where
x is the theer conductivity,
H is the heat capacity, and
C(z,t) is a cooling function which is the heat carried away by blood circulation

or other cooling agents.

Thesituationthatisofinterestinthischapteristheinitialprocess stageinwhich
nosignificantheatconductionandblood coolingaretakingplace, sotheinitialrateof

temperature rise for a given Q can be reduced to

91' = .0. = __2°"(Z) . 3.
a pH pH (C/sec) . ( 9)

Eq. (3.9) is employed to determine the increasing temperature rate at the very first stage

process which the diffusion effect won’t happen yet.

 

A triple—layered structure of biological structure, fat(lcm)-muscle(5cm)-
bone(2cm), is used as a simple model for this analysis. The parameters used for this
model are shown in Table 3.1. A continuous ultrasound wave source with constant

intensity (Io-10 W/cm’) is entering the triple-layered biological structure in a normal

51
incidence manner. For normal incidence, only the longitudinal ultrasonic wave needed

to be considered [14,15]. Since the attenuation coefficients are strongly frequency
dependent, two frequencies (0.8 MHz and 1 MHz) are used to observe the frequency
effects. The resultant ultrasonic intensity distributions, heat generated by ultrasonic
irradiationandinitialtemperatureincreasing ratesinvarious mediaofthestructureare
shown in Figures 3.7 through 3.10. Ultrasonic field intensity distributions in the structure
with different frequencies are shown in Figure 3.7. Figures 3.8 and 3.9 show the
heatingpatternsfortheaetwofrequencies. Theresultsofasimilarstructurefromother
investigators are also shown for the sake of comparison [8,9,16]. Finally, Figure 3.10

shows the initial increasing temperature rates for this structure.

WW

The ultrasonic field intensity is significantly attenuated while passing through
biological media, only 32% of the incident intensity is transmitted to the bone region at
lMHz. Theresults shownverysmallincreaseintransmitted intensity(39% intothe
bone regions, an increase of 7% , see Figure 3.7) by a slightly lower frequency (0.8
MHz). This result indeed confirms the fact that lower frequency permits higher energy
transmission through material, since the attenuation coefficients are strongly frequency
dependent. The ultrasonic characteristics of fat and muscle are quite similar,
consequently, most of the ultrasonic energy (99 96) will pass through the fat-muscle
interface while only 1% being reflected. This can readily be seen from Figure 3.7 that

52

 

 

 

C Z a p H

(mlsec) (102 g/cm’.s) (NP/cm) (g/cm3} (J/g.“C)

propagation character. attenuation density heat

speed impedance coefficient capacity

Fat 1,485 1,366 0.071“ 0.97 0.62
Muscle 1,590 1,685 0.11f‘m 1.07 0.75
Bone 3,360 5,800 2.5 (1 MHz) 1.70 0.504

1.51 (0.8 MHz)
Table 3.1 Parameters for fat, bone, muscle [3,19,20]

53
there is almost no abrupt change in intensity at the fat-muscle boundary. However, only

42% of the ultrasonic energy at 0.8 MHz is successfully transferred through the muscle-
bone region due to the drastic change in acoustic impedance at the interface. As
expected, there is a distinct drop in intensity at the muscle-bone interface.

The heat generated by ultrasonic irradiation shows the ability of heat absorption
by the biological media. The heating patterns generated at different frequencies (shown
in Figures 3.8 and 3.9) from our results are compared closely with those published
elsewhere [8,9,16]. In our analysis, multiple reflections at interfaces have been
considered. It accounts for about 10% of the total regional intensity as our results
indicated. We believe that the published results slightly deviated from ours is due to the
fact that in their analyses the multiple reflections have been ignored. It is important to
have an accurate knowledge of the intensity distribution, since it directly relates the
ultrasonic dosage for a given therapeutic setting. It can be seen from Figures 3.8 and 3.9
that heat generated in the bone region is much higher than those in the muscle or fat
region, although the intensity level in bone is much lower than that in muscle or fat.
The reason is that the bone has a much higher ability to absorb ultrasonic energy.
Figures 3.8 and 3.9 demonstrate the fact that bone is more effective in converting
acoustic energy into heat than muscle or fat especially at higher frequencies. These

results are in agreement with Schwan er al. ’s prediction [3] and Chan et al. ’s result [8].

The initial rates of temperature rise are shown in Figure 3.10. The initial

54

12 v

 

U I V

_:lMHz; --:0.8MI-Ia

Initial Intensity: 10 WW2

Intensity (WW2)

 

 

 

 

Depth (cm)

Figure 3.7 Ultrasonic field intensity distribution for 1 MHz and for 0.8 MHz

55

6 .

 

Frequency: .8 MI-lz '
5 _ I- I Published Data at .9 MB: (Games) I
' O 0 Published Data at .8 MHz (Chan at 01.)

@

Heat Generated (Wicm‘3)
w

M

 

 

 

 

Depth (cm)

Figure 3.8 Heat generated by ultrasonic irradiation at 0.8 MHz.

56

 

Frequency: 1 MHz I
+ -I Published Data at 1 MHz (Chan er cl.)
0 0 Published Data at .9 MHz (Gunner)

7 - 9H: Published Data at l MI-Iz (Schwans)

O)
r

 
 
   

Us
I

_ 1:3, 0 Muscle

Heat Generated .(W/cm‘3)
3a.

 

 

 

 

Figure 3.9 Heat Generated by Ultrasonic Irradiation at 1.0 MHz.

57

 

 

2.5 . .
Initial Intensity: 10 W/cm’Q
_: 1 MHz: - : 0.8 MB:
A 2 h «1
i3.
0
ii
ID
é 1.5 " . .l
8 l
' l
a I
3 IT
2 I
g l r- 53! Muscle |t Bone .
a '1
C
is
5
I-

 

 

 

 

 

Figurc 3.10 Initial increasing temperature rate at 1.0 MHz and at 0.8 MHz

58
increasing temperature rate in bone tissue near bone-muscle interface is 2.4 “C/sec (1.48

’C/sec) for 1 MHz (0.8 MHz), then drops dramatically to 0.017 “C/sec (0.072 °C/sec)
atlcmawayfromtheinterface. Thisresultrevealsthatat 1 MHzthissystemhashigher
temperature increasing rate at the muscle-bone interface than at 0.8 MHz. This again
confirms with the results from Schwan and Chan [8,9]. At both frequencies, temperature
increasing rate in the bone region is much higher than that in the fat or muscle regions.
It is about 4.5 times (2.7 times) higher temperature increasing rates in bone compared
to fat-muscle boundary at 1 MHz (0.8 MHz). One should also notice that although higher
frequency would give rise to higher temperature increase but in the meantime it loses the
ability of pemtrating into material. Depending upon application, one should optimized

the operating frequency for the best therapeutic effects.

This model offers a method to determine the Green’s function of a multilayered
biological structure. Using this Green’s function, several important physical quantities
such as the ultrasonic field intensity distribution, the heat generated by ultrasonic
irradiation and the initial temperature increasing rate can be evaluated. The proposed
approach takes all multiple reflections into account, which are commonly ignored to
avoid the complexity it may introduce. The results of the fat-muscle-bone structure used
for this method are in good agreement with the published results. For a more accurate
evaluation of ultrasonic energy distribution and absorption in human tissues, multiple
reflections should no longer be ignored. This is especially important in determining the

ultrasonic dosage in the hyperthermia treatment of tumor.

CHAPTER 4

THE ACOUSTIC POYNTIN G THEOREM IN AN
INHOMOGENEOUS MOVING MEDIUM

An acoustic Poynting theorem for an inhomogeneous and moving medium which
includes the equivalent forcing terms of spatial and temporal variations of density and
stiffness has been developed. Thermal diffusion, particle collision, drift, and heat flow
effects have also been taken into account in this development. A general acoustic wave

solution using Green’s function technique is also formulated.

W

The acoustic Poynting theorem interrelates the power supplied by acoustic
sources, the time rate of change of energy stored in the medium, the power dissipated
in the medium through propagation, and the power flow of the system. The basic
acoustic Poynting theorem for stationary medium has been developed over the years.
However, for a moving medium and the treatment of spatial and temporal dependencies
of density and stiffness of a medium are commonly ignored [11,12,13]. To have an

overall picture of energy balance when an acoustic wave travels through a moving

59

60
medium,bothwmporalandspafialvafiafionsofthemediumpmperfiesmustbe
considered. In addition, thermal diffusion, particle collision, and medium drift should
also be talmn into account.

42W

Two fundamental acoustic field equations are used to derive the acoustic
Poynting’s theorem: the Euler’s equation of motion and the continuity equation. The
Euler’sequationofnlotionisagenerallaw statingthatthemasstimestheacceleration
ofcenterofmassequalsthenetapparentforceexerted on the surfacesandbythebody

force. It can be expressed as follows [11]

glllmdwlim 40.1"” (in

where
f’istheapparentsurfaceforceperunitarea,
F‘isthebodyforce,
pisthedensityofthemedium,and
vistheparticlevclocity.

By applying Reynold ’3 transport theorem [11] and including the drift velocity of
the medium, the effect of particle collision, thermal diffusion and gravity force [11],
equation (4.1) becomes

61
V ' T = 9%"- + p[(uo+V)-Vlv - 98 My - 9ch - F‘ (43)

where

T is the acoustic stress,

3 is the gravitational acceleration,

CD I (M0) is the thermal diffusion constant,

at, is the Boltzmann’s constant,

K is the temperature,

tn is the particle mass,

no is the ambient density,

u,, is the drift velocity of the medium, and

F’is the foreeapplied to the system.
Thedriftvelocityuoisassumed tobeaconstantwhich isindependentoftimeand spatial

location.

The continuity equation is deduced from the conservation ofmass stating that the
net mass per unit time entering a volume V through an enclosed surface A is equal to the

timeratcofchangeofthemassfll]

gown 41. «mm- M

For an inhomogeneous medium, the continuity equation can be expressed as

62

2g— = W + lev (4.4)
a: p

whereSistheacousticstrainfield.
Therelationbetweenthestrainandthestressinalossymedium(orviscousmedium)can

be coupled by the constitutive equation [12]

r=53+n§§ (4.5)
at

whereBisthestiffnessofthemediumandnistheviscositycoefficientofthematerial.
Heatflowisalsoincludedinthcoverall energy balance relationinthisstudy. Theheat

equationcanbeexpressedasIll]

CC

= x, K V'K (4-5)

ails

where
c, I (poK)/c, is the entropy constant,
c, is the specific heat coefficient,
k, I MT, is the thermal conducting constant,

k, & T0 are the thermal conductivity and the ambient temperature respectively, and

c is the specific entropy.

W

The acoustic Poynting’s theorem can be obtained by multiplying Eq. (4.2) with

63
theconjugateofparticlevelocity, v‘, then,addingtoittheacousticstress Ttimesthe
conjugate of Eq. (4.4). Onearrives at

a 1 . 1 .6 .
V(-—-——)--—[5(pv‘v‘ +BSS +c,ce ‘)]+5(vv £68

%)-%ln (§%')+x.(vx)’l+§pv's+§cppv‘°Vp-l .pvv‘ “-71

1 . BS . n 66' . .
"-1" ‘K “WWW—V 'Vp- ——Vp 'v +— —v ‘°F.‘
2 In 29‘ 29' a 2

After integrating over the entire volume of interest and applying the Green’s theorem,
dleacousficPoynfing’stheoremcanfllalbeexpressedinanenergyconservafim form

ff Pads+_+p,=p+ +1», (4.1))

where 11 is the unit vector normal to the surface which encloses the medium.

3 as in Eq. (4.8) is the equivalent acoustic Poynting vector which includes two terms:
3.I-(v‘T)I2 is the acoustic Poynting vector, and

F. I -(x,KVK)/2 is the equivalent Poynting vector due to the heat flow.

U is energy stored in the medium which has three components:
UkIfffv (pv-v‘)l2 dV is the kinetic energy stored in the medium,

U,I.U[v BSS'IZ dVisthepotential energy (strain energy) stored in the medium, and

64
KIN-[V c,ee'/2 W is the equivalent energy stored in the medium due to entropy.

P,ispowerlossthroughpropagation, inwhich
P‘s-IN; rescuer/at) dV is power loss due to the viscous damping, and
P.-fffy(K,NK)1)/2 dV is the equivalent power loss due to the heat flow.

The right side of equation (4.8) are the power sources:

P,I P..+ P. + P“ +P. + P. +P,,andP,I P,,,+ P,.,+ P,» + P,,, where
P,Ifffv (v‘-F ')/2 dV is the power supplied by the acoustic source,

P.I.va (pv'og)l2 W is the power source due to gravity,

P,Ifffv (-f,pv' ~v)l2 dV is the equivalent power source due to particle collision,
PoIfffchpv'-Vp)l2 dV is the equivalent power source due to thermal diffusion,
P,I[ffv [~pv'-(u,,-V)v]/2 dV is the equivalent power source due to drift of medium,
P,Ifffv [-pv° - (v ~V)v]/2 dV is the equivalent power source due to the particle
velocity,

PM Ifffv (v‘ - vap/at)/2 dV is the equivalent power due to the temporal varying density,
P... IN]. (s’saB/aryz dV is the equivalent power due to the temporal varying stiffness,
P,.,I.va-(BSv'-Vp‘)l(2p’)dv is the equivalent power due to spatial variation of
density, and

P", IL”. -[1)(6S/at)v' . Vp‘]/(2p’) dV is the equivalent power due to gradient of density

65
Eqs. (4.7) and (4.8) offer a detail consideration of the energy conservation

theorem including inhomogeneity of medium density and stiffness, particle collision,
diffusion, drifi, heat flow, and gravity. With this energy balance theorem, we are able
toidentifythepowersupplied, thcenergy stored, andenergy dissipatedinthemedium

of an acoustic propagation system.
W
For time-harmonic excitation, where em” dependence is assumed, the acoustic

field solution can be obtained by taking gradient of Eq. (4.2), multiplying (12)) of Eq.

(4.4), and utilizing the relation given by Eq. (4.5). After steps of manipulation, one

arrives at
V'Wk’v- --@-(V'V)+&[F'+Vn(VV)-p(V°V)v-p(uo'V)v+nV'v
B B (4.9)
+ps-f.pv+cppvpl
where
k’ I (030MB.

1: is the acoustic wave number, and

bistheradianfrequency.

Eq. (4.9) can be solved by using Green’s function technique. The result can be

expressed as Eq. (4.10) which is decomposed into components due to various effects,

66

v=v+v +v +v +v +v+v+v+v
0 VB ad w a c D
"‘ ‘ (4.1a)
-vo+v,

where
Va is the ambient particle velocity, and

vj (j = vq, vB, vd, w, s, g, c, and D) are the component velocities.

The ambient particle velocity v0 satisfies the homogeneous Helmholtz equation,

Viv, + 1:sz . o. (4.11)

The v,’s ( j = V», vB, vd, w, s, g, c, and D) satisfy the inhomogeneous Helmholtz

equation and can be formulated by using the Green’s function technique,

(4.12)

 

The Green’s function G(r,r') of the system depends strongly on the structure’s

geometry. It satisfies the following inhomogeneous Helmholtz equation,

V'G(r.r’) + k’G(r,r’) =- -6(r-r’) (4.13)

67
where 6(r-r') is the impulse function. The 1; ’s in Eq. (4.12) are the forcing functions
whichincludethebodyforcesandaflthespafialandtemporalvaryingmedium
parameters. These equivalent forces are defined as:
a. f..I(ib/B)VqV-v due to the spatial variations of particle velocity and viscosity,
b. f..I-(VB/B)V-v due to the spatial variations of particle velocity and stiffness,
c. f,.I(-;i&)/B)p(uo-V)v due to the spatial variation of particle velocity for a moving
medium,
(1. L. I (-j&)/B) p (v-V) v due to the spatial variation of particle velocity,
e. f,, I (ib/BhV’v due to the particle velocity distribution,
f. f, - (fro/B)!” applied force.
8- f. - (iii/B) as due to the gravity,
h. f, I (-ja)/B) f,pv due to the particle collision, and

i. fl, I (ib/B)pcDVp due to the nonuniform medium density.

The inhomogeneous Helmholtz Eq. (4.13) can be solved by the moment method or the
point matching techniques.

M

An energy conservation theorem for acoustic wave propagation in an
inhomogeneous and moving medium has been developed. The spatial and temporal
variations of density and stiffness, effect of thermal diffusion, particle collision, gravity,

68
heat flow and moving medium have also been included in this development. This result

gives an overall picture of the energy balance of an acoustic wave. The gravity effect can
be neglected in most cases, except for very low frequency situations [11]. The moving
mediumeffectcouldbcasignificantfactorincertaincasessuchasacousticwave
propagation in underwater with drift current and Doppler detection of blood flow in
human body. When the medium is homogeneous and stationary, and with the effects of
thermal diffusion, heat flow, collision and gravity ignored, the result can be reduced to
those given by Auld [12] and Ristic [13]. If consider the heat flow effect alone, the result

can be reduced to those given by Pierce [11].

CHAPTER 5

TRANSMISSION-LINE MODEL FOR
ACOUSTIC WAVES

An equivalent electric circuit of a transmission-line model provides a good way
to realize the energy balance, wave propagation and analytic solution for transmission
line. There are two major parameters in the transmission-line equations, voltage and
current. The voltage and current on the transmission line are functions of both distance

along the transmission line and time.

Let L, C, R, and G be the inductance, capacitance, resistance and conductance
per unit length of transmission line. An equivalent circuit for the transmission line with
differential distance Az is shown in Figure 5.1 [22]. Let V(z,t) and V(z+Az,t) be the
voltages at z and z+Az respectively. Similarly, let I(z,t) and I(z+Az,t) be the currents
at z and z +Az respectively. The Kirchhoff’ 5 loop equation for the circuit shown in

Figure 5.1 is

69

70

 

 

 

Figure 5.1 A T-type transmission-line model for an electric circuit.

71

V(z,t) - LAzgflzJ) - RAzI(z,t) - V(z+Az,t) + vsAz = O . (5-1)

0n dividing Eq. (5.1) by A2, taking the limit as Az goes to zero, and using the definition

of derivative, we find that [23]

a a
.. _ = _ + - s .2
V(z,t) L I(z,t) RI(z,t) V; , ( )

where v, is the external voltage source applied to the transmission line system.

The Kirchhoff’s node equation for the upper node in Figure 5.1 is

[(2.0 - CA2§V(2+Az,t) - GAzV(z+Az,t) - I(z+Az,t) = o . (5.3)

A procedure similar to that used in Eq. (5.1), when applied to Eq. (5.3), yields

gnu) = Cit/(2.0 + 01(le . (5-4)

Eqs. (5.2) and (5.4) are the first order transmission-line equations. We may obtain
second-order partial differential equations containing either the voltage or the current in
the following manner. We partially differentiate both side of Eq. (5.2) with respect to

z and replace the term (a/az)I(z,t), which is given in Eq. (5.4), then obtain

72

3:214”) = w%WU) + (LG+CR)2V(ZJ) + RGV(z,t) + —a—v’ . (5.5)
62 az 3‘ a:

In a similar manner, the second-order current transmission-line equation is expressed as

a2 a2

—I(z.t) = LC—I(z,t) + (LG +RC) 531w) + RGI(z,t) - (c3 +G)v, . (5.6)
az2 3:2 at a:

The second-order transmission-line equations can be utilized to solve the transmission-

line problem. Two special cases, lossless and with termination, are described as follows:
1 1 1 i i lin
In the special case of lossless transmission lines corresponding to R = G = 0,

and with no external voltage source, v, = O, the transmission line equations in Eqs. (5.2)

and (5.4) become

6 a
__ = __ 5.
Wu) L I(z,t) , ( '7)
and
a a
_ _ = ‘ _ V . 5 .8

The second-order partial differential equations (5.5) and (5.6) become

73

considerably simpler for the lossless case and can be expressed as a general second-order

equation shown in Eq. (5.9)

12412.!) = liiz-Mzi) , (53)
az c at

where
¢(z,t) is the transmission lien parameter. It could be either I (z, t) or V(z, t).

c =1/(LC)"2 is the velocity of propagation of the voltage and current.

 

Consider a finite transmission line with intrinsic resistance R0 terminated with a
resistor in one end. When the voltage and current waves travel in the positive 2 direction
reaching the termination of transmission line, the voltage and current waves will reflect
back to travel in the negative 2 direction. By Kirchhoff’s circuit equations, the ratio
between the magnitude of the backward propagating wave and that of the forward

propagating wave is defined as the reflection coefficient, K

 

R = Rb ' R0 . (5.10)
RL+R0

The ratio between the magnitude of forward wave and that of the termination resistor is

defined as the transmission coefficient, T

74

T _ 2 R1.
R1. + R0 .
Eqs. (5.10) and (5.11) define the extent of reflected and transmitted propagating wave

(5.11)

 

along the transmission-line.

5,; l,ll=.l_‘lll_.H-l‘ lIH‘ fl'l‘ (A 1'2V‘ Irina); !' 4, LI .'!'”')"|"x'.

mm [51]

Equation (4.7) in chapter 4 offers a breakdown picture of the energy conservation
theorem. It includes the inhomogeneity of medium density and stiffness, particle
collision, diffusion, drift, heat flow, and gravity. With this energy balance theorem, one

is able to identify the power supplied, the energy stored, and energy dissipated in the

system separately.

For acoustic wave excited by the acoustic power sources and/ or other equivalent
sources, this problem can be treated efficiently by using analogue of an electrical
transmission-line model. In an isotropic and inhomogeneous medium the propagation
direction can be assumed in the z direction without loss of generality and the problem
is then reduced to one dimension for simplicity. An equivalent stress T“, can be defined

as

75

l
T” a T + TD — 1;, = T + Eopp2 - puov (5.12)

where
T, = puov is the moving medium equivalent drift stress, and

TD = 1/2 ch’ is the equivalent thermal diffusion stress.

The transmission-line model of this system including previous considerations can

be deduced from equations (4.2), (4.4) and (4.5),

3(‘T 6v 6p
_22=___- -_ + +1." (5.13)
oz 9 at W» no az)v 98
and

6v 6(-T 3 av 1

'6—2' = "C—jg ‘G(-T.?) +C(ro‘CDP)Ep +CUOPE -§Gch2 +Gpu0V (5°14)
where

C as (B+n6/3t)", and

G a (B+nd/3t)'2[6B/6t+(8n/8t)(3/6t)+n(3’/6t2)].

Eq. (5.13) and Eq. (5.14) are the acoustic transmission-line equations with
equivalent electrical transmission-line variables Tea and v, which are analogously
identified as V (voltage) and I (current) in the electrical circuit system, respectively. This
system will be analogue to an electrical transmission-line model [22,23] by introducing

an equivalent resistance RAz , an equivalent conductance GAz , an equivalent voltage

76

source V, , and four equivalent current sources as shown in Eq. (5.15),

fl=-L§-z,i+v,
dz 3!
(5015)
§=-ca—V-GV+II+Iz-13—I4
62 at

where L, R, C ,G , V,, 11, 12, I3, and 14 are the equivalent acoustic transmission-line
parameters with the following definitions:
(1) L E p,

(2) R

fvp - Voap/az,

(3) C E (B+n6/6t)",

(4) G a (B+n6/at)'2[6B/dt+(an/at)(6/6t)+n(62/0t2)],
(5) V. 5 pg + F',

(6) Il Cuopav/at,

(7) 12 E Gpro.
(8)13 a (ch-uov)C8p/8t and

(9) I4 "=- Gch2/ 2.
This analogue transmission-line model of acoustic inhomogeneous moving medium
with the temporal and spatial variations, the particle collision, the thermal diffusion, and

the gravity effects is depicted in Figure 5.2.

A detailed T-type electrical transmission-line model developed here provides a

77
deeper insight of the system behavior due to the spatial and temporal variations of density

and stiffness of the medium. It is interesting to note that for a homogeneous medium, if
the density of medium and the stiffness are independent of time and space, and with the
diffusion, collision, gravity effects are ignored, then the proposed model is reduced to

the isotropic, homogeneous transmission-line model given by Auld [12].

78

 

 

 

 

 

 

 

§az #42 2%: Quiz
.0r3——-’V\r-—’0‘ ——’\/V-’0‘—®-9
-7" SW 05162 reassign
l T
.c A ’e

 

 

 

Figure 5 .2 A T-type transmission-line model for an inhomogeneous acoustic moving
medium.

CHAPTER 6

ACOUSTIC FIELD WITHIN AN ARBITRARILY
SHAPED SCATTERER

W

Knowledge concerning the ultrasonic energy distribution will be very crucial in
detaminingflreheatandtemperamrepmfilesinsidethemedium.Especiaflyin
therapeutic applications, for example, in hyperthermia treatment of tumor [24,25 .26], the
induced energy distribution is of major interest. The temperature profile inside the tissue
under hyperthermia treatment is directly coupled to the ultrasonic displacement field
distribution. Therefore, ultrasonic displacement field distribution plays an extremely

important role in therapeutic process.

Theevaluafionofscaueredacousficdisplacementfieldhasbeendevdopedin
related studies [12,27-29]. Also, the dyadic Green’s function technique has also been
employed to calculate the acoustic field distribution externally to the scatterer [30-33] for

years. The acoustic field distribution within the scatterer is commonly ignored because
of singularity problem and the ratter complex boundary conditions of irregularly shaped

79

80
scatterer. However, without having the information of the internal acoustic field

distribution, the heat generated by acoustic wave irradiation and consequently the
temperature profile within a medium under investigation can hardly be fulfilled.
Evaluating the internal field distribution of a planar layered homogeneous medium has
been reported in elsewhere [34-36] because of simple boundary conditions. For
underwater acoustics or seismology studies [37-39], the boundary conditions can be
ignored because of a large-scale consideration. In recent investigations, the scattering by
a scatter embedded in an inhomogeneous layered structure can be solved by employing
the dyadic Green’s function with Born approximation [5,40,41]. However, little attention
has been paid to find an analytic solution of the displacement field distribution within the

scatter because of the complexity of singularity problem.

We would like to present an analytic proposal to solve the scattering field
distribution within an arbitrarily shaped scatterer by avoiding the singularity of the
integral equation. A general three dimensional forward scattering formulation for the
purposeofwalmfingthescafieredfieldofanarbifiafilyshapedscafiererisfirst
developed. Then we decompose the displacement field into an incident field and a
scattered field of interest [42-44]. The incident field results from the original excitation
in the background medium on the absence of the scatterer whereas the scattered field
results from the scatterer. Then we consider an arbitrarily shaped medium embedded in
a homogeneous background, and develop an expression for the three-dimensional

displacement field induced by a uniform plane acoustic wave. We use the displacement

81
field to numerically implement the corresponding element’s coefficients of the

msformafimmauix.1hesimuhfionresultsshowthatthefieldsobminedfiommis

model are reasonable quantitatively over a wide range of pixel dimensions.

 

Let’s consider a finite scatterer of arbitrary shape with density p,(7) and
compressibility c.( 7 ), illuminated in a background medium by a longitudinal plane
ultrasonic wave as shown in Figure 6.1. The induced ultrasonic force in the scatterer
gives rise to a scattered ultrasonic displacement field TI '( 'r‘ ), which might be accounted

for by replacing an equivalent background force density function f,( 'r‘) [1, 12,42,431,

 

 

f.,(i') - - [09.0) - Mn] 1'4?)
- [c.(mim - mi] an
- 292:9 _ 1 [0‘4] pm (6.1)
onto
- no) [c.xi] are)
= - me) am

where
ltd?) = 3.6-r.) ~ja.(7 ) is the wave number.
v,( 7) = [c,( 7 )/p,,,( 7 )]"2 is the wave propagation speed,

a.( 7) is the attenuation coefficient,

82

 

 

 

 

 

 
    
       
   

€<7>=nce>a<el

Vs

[P,(Tc'), 47)]
Inhomogeneous
Scatter

CC?) = (I'm + C???)

lfimmeGJ. .AnummmqymemnammnmfinxmfihdbyaphmeMnmnmbsame

83
p... c, are the density and compressibility of the background medium, respectively,
n(? ) is the complex deviation ratio of the scatterer and background as
defined in liq. (6.1), and
'J (7) is the total displacement field inside the scatterer.

In general, the scattered displacement field, TI '(7) inside the scatterer can be

expressedinter'moff,,(7)bytheuseofthedyadicGreerr’sfimctim6(7/7 ’)asfollows

arm - [Mrs not") at”
l e vac-r1 (6.2)
V V

007'") =
(029, 4r: If-f’l

 

 

where the dyadic Green’s function 0(7 /7 ’) represents the transfer function of the
background medium, and | 7 0| = t.u[)tl,,c,,]"2 denotes the wave number of the background

medium.

Inthecasewhenthefieldpointisinsidethescatterer, I; s(7)slrouldbeevaluated
with special care because of the existence of singularity and the requirement of
unrquenesspropertles'lhescattered field ‘17 ’(7)atanarbitrarypointinsidethescatterer

can be written as [42-45]

84

mm Pvaf~(i”)-G(i',i")dV’-J— “49— . (6.3)

3019

where the symbol 'PV' denotes the principal value of the integral [42-46]. We may now
write the total ultrasonic displacement field If (7) inside the scatterer as the sum ofthe

incidentfield 'J‘andthcscatteredlield 'u‘ ‘as
so) . s‘m + it"(r) . (6.4)
Substituting Eq. (6.3) into Eq. (6.4) and rearranging terms gives the integral equation

for 7(7):

[1 - —"'—‘—” am Wf m(?’)fi’(i”)G(i77’)dV’= rm. (6.5)

303.9

In Eq. (6.5), if ‘( 7) is the incident ultrasonic field and is considered to be known
quantity, 7: (‘r’ ) is the unknown total ultrasonic field inside the scatterer, which can be

discretized to yield a linear system of equations with complex coefficients.

6W

Assumingthat it. (7)andm(7)aresmoothvaryingfunctionsofspacewithinthe

volume of the scatter, we can partition the medium into N pixels, and treat the 7 ( 7 ) and

85

m('r‘)ssconstantsinencbpirte1. Therefore, theintegralEq. (6.5)canbetreatedasa

linear system. With denoting the m“ sub-volume by V. and the position of V. by 7 .,

wecanrewriteEq.(6.5),usingx1=x,x2=y,andx3 =z,as

3 N
—-,— «47.) + 2: 2: map
0 p,

q-l a-l

l-

 

evfnov'tryrw’] r447.) - ago.)

wherep,q=l,2,3and

 

 

 

82 e'fl‘q'fl
G r r = j
’1'! J I) 02p. ax'ax

, “tn-F1
For simplicity, one can define

A; = UK?) -Pny.GV'(7,/7’)dl" ,

then, Eq. (6.6) becomes

3 N

 

”+55 l-"m 7=‘7
«23;; ' " "[ 3mzPsHut'h) “4.)

wherem=1,2,...,Nandp,q=l,2,3.

Denote [1 ml“. as a matrix with its elements assigned by

(6.0

(6.7)

(6.8)

(6.9)

86

 

 

 

. r. "(7.)
A“ g 4,}. + ems” [I - 3‘02“ (6.10)
wherem,n= 1,2, ...,Nandp,q= 1,2,3.
Let [u,] and [u,’] be N-dimensional vectors given by
(1:47,) 14'0"!)
[U5] = 2 . [up] = . “-1”
. “4?”. L “$79)

 

 

 

where p = 1, 2, 3. For all possible values of m and p in equation (6.10), the matrix

representation of equation (6.9) can be expressed in a matrix form as

 

“.1 an] [1.1 [us] [all
VLJ w (1,1 W] = lu,‘l (6.12:)
“n1 “o1 {In} ["2] [14:]
01'
[1] [u] = [u'] (6.121))

where [ A ] is a 3Nx3N matrix, while [u] and [u‘] have dimension 3N expressing the total

displacement field and the incident displacement field at the centers of the N pixels.

Note that in this way, a three-dimensional image matrix is represented by a one-

87
dimensional vector staclmd by rows. The inverse problem becomes that ofdetermining

thetntalultrasormddisplacemmtfield T(?)rromthemeasurements 72 ‘(7).'I‘herefore,
one may evaluate the field distribution inside the medium by solving Eq. (6.12) with

available methods [43,44,47-50].

 

lnordertoevaluatethecoefficientsofthe matrix [A w,], one mustbecarefulin
calculating the principle value of Eq. (6.8). Examining the off-diagonal elements of
[A In], weclearly see thatr_doesnotbelong to V; and Gm(7./7 ’) iscontinuous
throughout V}. Therefore, we may omit the principal value operation from our
evaluations. Howeva', when we deal with the diagonal elements of [ A flu], r,, belongs to

V," and we need to make some approximations.

W 142-44]

First, let’s consider the off-diagonal elements of [ A m] , where p, q = l, 2, 3.
From Eq. (6.9), it is clear that

1;. = A; = ”(f').fv.G’r'c(7J7’) at" man. (6.13)

With the same approximation described in Sec. 6.3, we have

1;, = .3, = M7,) thwi') AV. mm, (6.14)

where AV, = v. dV’. By the definition of Eq. (6.7), we can rewrite Eq. (6.14) as

 

1;: (I, “If. 1:) . [3_(‘oR-.)2+j3xoR~]_(l+M-)a"
‘ (6.15)

l MFJAV'e'M" ;m¢n ,

039.

 

whereR_ - |7_- 7,| while 7.and 7.aredenotedas
r.-2x.'+rx:+es'. r.-txt'+rxa'+ba'. (‘40
When N is a large number, the approximation given by Eqs. (6.14) and (6.15)

should yield reasonable results. For better accuracy, however, Eq. (6.13) should be
integrated numerically by refined methods.

W 142-451

For the diagonal elements of [J Am] in Eq. (6.9), we have

89

 

 

 

r. (7.)
.41 =rrrtr opv G,(rr')dv’+6 1-'" .
v. .> f ,, J ..[ ,pr (m
no.)
- r-lim G rrav’w l-
“) my!“ "5', I) I" [ 303p,

Thevolume Khasbeenapproximatedbyaspherehaving thesamevolumeasthecubic
pixelwhichiscenteredatnforbetteraccuracy. Now,1eta,,betheradiusofthesphere,
after steps of manipulations [42,44], Eq. (6.17) becomes

 

.. i' ._ 7
45-5,, —"—‘f’;[tl.noapc--t]+[r - ”‘9 I] (6.18)

36.»2 3039,

where a. = [3AV,,/4‘r]"3 . If the actual shape of V. differs significantly from that of a
sphere, the approximation may lead to errors. In such case, the integration throughout
a small sphere surrounding r, should be performed by procedures of principle value

evaluation; the integration throughout the remainder of V; has to be carried numerically.

 

Numerical simulations of some simple composite models are performed to test the
validitythetheorydeveloped. Othermodelswillalsobeusedtoexplorethescattering
effects within an arbitrarily shaped inhomogeneous medium.

“12.“. . I ..

When using a discretized pulse expansion in a linear system of equations, it is
crucial to establish a suitable limitation on the dimensions of the sub-volumes. In order
to arrive at an optimal choice, we consider a homogeneous bar with deviation ration of
at?) a 10, as showninFigure 6.2, illunrinatedbya 1 MI-Izplane wave with unit
strength. Expressed in wavelengths, the dimension of this homogeneous bar is 6 x 1/2
x 1/2 X3 .

The homogeneous medium is partitioned into cubical pixels, the induced field is
then evaluated in each pixel. Models with different pixel numbers (N = 12, 96, and 324)
have dimensions of 1/2 )t, 1/4 A, 1/6 h respectively are used in this study.
Computational results confirm our assumption that the induced ultrasound field has
indeed had only x-component (comparing to x-component, the magnitudes of y and 2

components are insignificant).

Figure 6.3 shows the axial ultrasound fields along the z—axis for each model. For
the 1/2 A, case, the field intensity distribution is deviated from the other two cases. This
is because too large a pixel dimension was chosen. For the 1/4 h and 1/6 h cases, the
fieldsaresymmetrictotheaxialaxis. Astanding wavepatternisestablishedalongthe
plate due to the finite boundary condition. Since none of the pixels, as shown in the 1/4

A case, lie along the z-axis, we plotted the average field intensity in its surroundings to

91

”T”
1 W3)

 

 

 

 

 

 

_/
/
Uni‘t Cell Unit Cell Unit Cell
3 s _1_ s
'8' k ‘61? 1 are."

Figure 6.2 Different pixel size configurations of a homogeneous bar with 11(7)-10,
f=l NIHZ.

 

oer

 

 

 

 

0.45?
- ' " ' 2N6 C359 ‘
0.4)-

. \ 2N4 case ‘.
_o.35..- j '--' 9% case ' _l'
:3: . . ,
g 0.3} _ .- J

I ' , '
g 0.25? \l ‘ I" +‘- \b ‘k I ‘3 I,
g " I \ .l \ .’ ‘hJ’ \. .I \ I t i ’ T
3 .i -\ I t I \ i \. _I \ I l i ‘ i ..
g 0.2)- \¥ «l», + V ‘4! 4 ‘4 y.
D
0.15l' .
o~‘ "
OO1L e‘~e—"‘o-‘~O—--O"'O--.Opv’°'~-”,.
0.05'
93 -2 -1 O ‘l 2 3
z-axis- (mm)

Figure 6.3 The ultrasonic field u,(7) along the z axis.

93
facilitateacomparisnnwiththecasesofll2h and l/6>..'lheresultsindicatethat

pixels with dimension of 1/4 h produce quite accurate field distribution. It is therefore
not necessary to consida' further smaller pixel size. Using larger pixel size reduces

computational load drastically.

Wehavealsoinvesfigatedtheeffectsoffiequencyanddeviafionrafioupondre
field distributions. A small cube (1.5 x 1.5 x 1.5 mm’) with combinations of different
harmonic frequencies and deviation ratios was used. Since the cube presents a
symmetrical cross section to the incident field, we only need to evaluate the induced field
in one quadrant. The results are summarized in Table 6.1. We observed that the field

strength at the center of cube is nearly independent of the harmonic frequency but is

highlyrelatedtothedeviationratio.

 

Two models composed of biological materials with their properties given in Table
6.2 [7,20], are used in our simulation.

(1) Model A: fat-muscle plate.

(2) Model B: fat-muscle-fat bar.

For the planar model, as shown in Figure 6.4 the incident ultrasonic field is
normal to the surface of the model (i.e., x-axis). From the results of previous section,

there is no transverse components needed to be considered since the incident wave is a

94

 

7L... Displacement

n(r) Field 0’:

MHz mm mm cubicle centefl
- heron-Jame _
1' 1.5 0.5 2.25 u,=.0179-J.0021

' ux=.lO7-J.085
0.1 15 0.5 82.5 Uy=.021'J.005

ux=-.O47+J.010
0.1 15 5.0 . 90.0 u,=-.ooa+J.ool

 

 

 

 

 

 

 

 

 

 

Table 6.1: The central displacement field of a homogeneous cube with various n(r),
insonified by a plane ultrasonic wave of different harmonic frequencies.

Table 6.2:

95

 

 

 

 

M t i l Velocity Attenuation Density
° 9” m/sec Np/n ' Kg/n'
Fat 4.485 .7 970

Muscle #1590 ~11 1070

 

 

 

 

 

The specified values of tissue properties, under a harmonic frequency of
1 MHz, were used in this computational stimulations.

96

 

 

 

 

 

 

 

 

 

 

Figure 6.4 A fat-muscle plate model configuration.

97

 
    

   
  
  
   
    

, ./
sly/”I
I" ' /
e ‘3 “||'I‘0.: [WI/I I

\‘ \

 
   
  

      
    
    

   
   
    
  

 

   

      

 

     
      

           

     
  
    
 
  

      
 

      
 

 
 
    

 
  
 
  
  

. ’I O
‘ - «It I
o s ’49.
' r "u. ~‘\“ ‘
//;’;‘:““\ ‘3- ‘ ‘ “‘\‘“ ‘.‘.‘
”II”...‘\‘\‘\\\- [I ‘ ‘\ \\ a"
0'4 ”’c’é‘u‘rrr‘\‘\\“(\\“:::r"’o ,
. ’o o 6‘ u“ ‘t\ l‘m‘. ~ w '
t \ \‘rt av ,,,,.
0 'e‘ f “ \\\\N\\\\t‘u ‘ " > « 4’s? ,, , .
'2 ““\\\\\\\\‘,‘.l; ‘ ’0’0‘.‘ a‘ .3. 7”: a :““‘ .
o ¢ , ’4 , \\.
\180‘ ”IX! s‘ , . I:';--3‘ss‘.6‘o‘o"/ 7;”,"t““w.~;,
’l/ll’ ". ‘ ,O'O“‘..\“ “\\| 9 O ’l//"/ " ’. ‘\‘\\In.
0 mm, I o ‘r \

“an“:-
I “u“-..
o

        

  

\‘a‘ ' ‘ 7
‘ \Y‘q’ 7”" .
an: “ “’7’ :17” '9.
‘t‘t‘t‘r‘l‘S‘oj”i{:’.’:‘:‘.‘“.‘_‘.“‘
‘ u “c, "-’—‘ "‘
a “ NV

  

Figure 6.5 The field distribution within the fat plate of Figure 6.4.

98

 
  
      
 

  

“‘ /////’
a — A109.
“M?” 15.01““
‘ “luff"~ 7,» I),

 

““‘\“9““;; {‘3 “u M W 9“
’:

ohm” “I, \“ 1 sh
'9.’ 0‘ 3.. “‘90 \‘\‘\‘ m,” lt‘. ,le'ftttuto,
\..“‘ /; ““ 9: \\\\‘\\;[s" ‘“

.: ‘\ \ -” III
“:rI/VI o :9“ \\ 39““ ll’ ’ “$1 .‘\\ \/ ~\9\ \\ “:\ \‘"I O“ \ \\\‘\\‘I‘.’{///// u

 
        
     
  
 

  
 
 

-://’ .
257/ ’I II.‘ . x
5:10: ’6’," a“ 3‘? ’

  

Figure 6.6 The field distribution within the muscle plate of Figure 6.4.

longitudinal ultrasonic waves.

The fat-muscle square plate has a dimension of 4.5 x 4.5 x 0.75 mm’. The
distributions of the ultrasonic field for both layers, versus the 9 - and i - axes, are shown
in Figures 6.5 and 6.6. As expected, the field distributions at the fat-muscle interface are
discontinuous. The field intensity is higher in the fat region than that in the muscle

region.

Figure 6.7 shown a 13.5 mm long bar of fat with a 1.125 mm layer of muscle
sandwiches at its center. The axial field distribution is shown in Figure 6.8. It can readily
be seen that the field contains a forward propagation and a reflected components. These
waves attenuate exponentially as they travel along. As a result, a standing wave pattern
is established between the boundaries. For homogeneous medium, the field intensity is
the lowest at the middle ofthe structure. It is as expected, since there is no boundary
exist inside the structure which could cause reflection and multiple reflections to trap
acoustic energy. 0n the other hand, a structure consists of different materials produces

local maximum of the field intensity.

For non-dispersive medium, the acoustic pressure field P( -r. ) can be expressed as

a linear function ofthe phase constant B as

100

muscle

K) |
X)

‘F 't ".33.. F of 1435""

WV 3:... . - 2;; . i Ar
:3: ' 1.12500

as; ‘ _L

r— 5.685nn——-' r-—5.625nn-—-l
-—1 |—-2.25nn

 

 

 

 

 

 

Figure 6.7 The fat-muscle-fat model configuration.

101

 

1 T f f V V '

0.9} m ", '—' : Homogeneous father -r
1 , '- -' : Fat-muscta-tat bar
|
1

 

r:
d’

 

 

 

 

 

 

 

 

 

 

 

 

0.21- '
0.1 » '
00 i 5:- 8 a 10 12 14

x-axis (mm)

Figure 6. 8 The x-component of the acoustic field along the x—direction of a fat-
muscle-fat model.

102

M = 9409?- = 109.09 «(a = mm 2.09 am (6.19)

mm?) =-p.17)v.(7)isdieacousticirnpcdanceandv,(7)istheacousticvelocity
in the medium. The power absorbed by the medium becomes

3
powerm = [1“ [ 75%;}; (6.20)

wherehistheabsorptioncoefficientm.

am

We have developed a method to determine the acoustic field distribution around
and inside a scatterer. Dyadic Green’s function has been used in the evaluation process.
A special arrangement to handle the singularity in the Green’s function integration has
also been demonstrated. Two models of simple structures made of biological materials
were used in the field distribution analysis. In choosing the pixel size for computer
simulation, it is obvious that smaller pixel size will give more accurate result. However,
it will consume enormous computing time. We found that pixel size of 1/4 x cube gives

acceptable result with reasonable computing effort.

In most of the published work, the factor e‘“ is assumed in the solution to account

for the attenuation effect. Strictly speaking, as indicated in our analysis, the attenuation

103
andphasecharactaisficsshouflbehmdledasammplexpmpagafionfacbraummugh

the analysis.

Itisinterestingtonotethatthepowerabsorbedinthemuscleregionofan
inhomogeneous bar is several times larger than those absorbed by the surrounding fat
layers. Specially, the material having high field attenuation and high mass density will
absorb more power and consequently reaches higher temperature. A good example is in
the case of hyperthermia treatment of tumor. Tumor has high mass density and higher
attenuation than the surrounding soft tissues. Together with the fact that tumor site has
poor blood circulation, its temperature could easily exceed the cell survival level under
ultrasonic irradiation.

CHAPTER 7

EXPERIMENTAL MEASUREMENTS AND RESULTS

Characterintion of material properties can be achieved by using acoustic waves.
Typically,reflectiontechniqueisused. Oncetheacousticpropertiesofthematerialare
determined, one can then use either experimental method or simulation method to
determine the acoustic intensity distributions and/or temperature profiles within the
material. 'l‘hereflectiontechniquerequiresonlyoneacoustic transducerwhichactsas
both the transmitter and the receiver of the acoustic signal. The experimental setup
configuration is shown in Figure 7.1. The incident pulse is produced by a 2.25 MHz
transducer (Parametric) which is driven by a pulser/receiver. The reflected signal is then
sampled by an AID converter which operates at a sampling rate of 40 MHz with 8-bit

resolution. The returned echoes were averaged 50 times for the purpose of improving the

104

105

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

Panametrics A/ nv

505° ' D co arter . Data processing

Pulsar/Receiver 40 MHz and Display
F

Transducer Stepping Motor X -—'* M°‘°' °°""°'

;-.° . ""9”“.
I Stepping Motor Y ’ IBM PC
compatible
microcomputer

 

 

 

 

 

 

 

 

 

 

 

Water Tank

Figure 7.1 Experimental configuration of the ultrasonic imaging system.

106
signal-to-noise(S/N)ratio.Ingeneral,thenoisehaszeromeansothattheavenging

prooesslowerstherelativenoise.

To measure the acoustic characteristics such as the attenuation coefficient and
acousticimpedanoe, wesetuptheexperimental configurationas depictedinFigure7.2.
Two materials with different thicknesses were aligned in the water tank. The magnitude
ofthereflectedsignalrelatestothatoftheincidentsignalandeanbeexpressedasEqs.

(7.1) to (7.4)

A, - R 4,, (W'- , (7.1)

where
A0 is the incident pulse amplitude,
0:0 is the attenuation coefficient of coupling medium which is water in this setup,
doisthedistancebetweenthen'ansducerand the material undertested, and
R is the reflection coefficient at the boundary between the coupling medium and

A, :- R (R1 - 1) A0 6"“- (W! , (7.2)

107

 

 

 

 

 

moving .

--------

 

-------—

 

6
\ m--. A0
ll"
transducer’81>

82/

 

water
tank

 

 

 

water
tank

 

Figure 7.2 Acoustic characteristic measurement setup.

 

108

Bi'RAo‘.M , (7-3)

a, - R (R1 - 1) A, :4". (“MW , (7.4)
where
a, is the attenuation coefficient of the testing material, and

duarethethiclmessoftesting material.

Dividing Eq. (7.4) by Eq. (7.2), the attenuation coefficient of the testing material
can be determined as

a = i In {$1 . (7.5)

Moreover, the reflection coefficient, R, at the boundary between the coupling medium
andthetestingmaterialcanbeshowntobe

R=J1+$.[fi]‘fl’, (7.6)
A1 82

 

Theacousticimpedanceofthematerialundertestedcanthenbedetermincdaslongas

the acoustic impedance of the coupling medium is known.

2 = 20 1*: , (7.7)

 

109
where

Z,istheacousticimpedanceofthematerialundertestcd, and

la is the acoustic impedance of the coupling medium, which is a known quantity.

Tintesfingmaterialsusedinthisexpedmwmlsempareplexiglassphteswidt
thicknessesof27mmand39mm.TheyaresituatedinthewatertankasshowninFigure
7.2. The echoes returned from bottom plates are shown in Figures 7.3, 7.4, 7.5 and 7.6.
The acoustic characteristics of plexiglass plate can be determined from the experimental
data described below.

WM

Theacousticpropagationspeedvinplexiglassplatecanbedetenninedby
observing the time between echoes, Figures 7.3 and 7.4, and the known thickness of the

plate.

2‘11 _ 2x2.7cm

c:—

- = 2.755 bit/sec . (7-8)
At 1.96 x 10" sec

 

It is in good agreement with those given by Ristic [l3] and Selfridge [58], 2.68 kin/sec

and 2. 75 bit/SOC respectively.

2111]] . fifi‘

l 10
The attenuation coefficient of the plexiglass plate can be obtained from Figures

7.5 and 7.6 and Eq. (7.5) as

l -O.131 Np
= ____ ln = _ , (7.9)
2 x 1.2 cm -0.076 0227 cm

 

The attenuation coefiicient was also in close agreement with 0. 216 Np/cm reported by

Kuc [56].

 

The reflection coefficient of the water-plexiglass plate can be calculated from

Figures (7.5) and (7.6) and Eq. (7.6) as

 

 

 

R =J1 + -o.131 . —o.131 ’11 = 0.364 . (7.10)
0.514 -o.076

Once reflection coefficient is determined, one can obtain the acoustic impedance of

plexiglass plate by Eqs. (7.7) and (7.10)

z=15xto‘x :———'—*3::—3;—=322x10‘—3—. (7.11)
- m2

This acoustic impedance value is within 2 96 of those reported by Ristic [13] and

Selfridge [58], 3.17 kg/m’.sec and 3.26 kg/m’. sec respectively.

111

 

0.5 -(

0.4 —:

0.3 a

0.2 _

0.1...

"O.’ -4

Relative Amplitude
O

-O.2 q

-O..3 -

-O.4 —(

-O.5 -1

 

 

ar=1 .96 ——..

 

 

 

 

 

i l l i l l I

 

l0.0

11.0 12.0
Time (tops )

13.0

Figure 7 .3 The returned echo from the top interface showing round trip delay.

 

112

 

 

 

 

 

 

 

 

 

 

 

0.5 A
; AT=2.83
0.4 -l
0.3 -I
0.2 -l
0
'0
,3 0.1 -
'3.
S o r
0
.2
:‘9 -0.1 -
0
O!
-0.2 -
-0.3 -1
-0.4 —
-0.5 d
l l l l l l l l L
10.0 i 1.0 12.0 13.0

Time ( lOHS )

Figure 7.4 The returned echo from the bottom interface showing round trip delay time.

113

 

A, 0.514
0.5- f

0.‘ ..
0.3 -

0.2 —,

' 0.108
0.1- \

 

-O.l _.

Relative Amplitude
O
1
1i
l
i
l
l

 

 

-0.J -

-0.4 _

-0.5 -1

 

I I I I I I1 I I I

 

 

10.0 1 1.0 12.0 13.0
Time ( 101.13 )

Figure 7.5 The returned echo from the top interface showing the peak amplitude.

114

 

a. 0.514
0.5- f

0.4 -
0.3 -1
0.2 _.

0.1.,

 

Relative Amplitude
O
I

-0.1 -
a, .-0.076 -/

-0.2 —l

 

 

-0.3 -l

-0.4 _.

-0-5. .—

 

 

1 L I I J I I I I
10.0 11.0 12.0 13.0

 

Time ( 10M: )

Figure 7.6 The returned echo from the bottom interface showing the peak magnitude.

115
Itisintaesdngmnotethatthereisaphaserevemalbenveendremmmedechoes

form the top plate (water-plexiglass interface) and the bottom plate (pleXiSWwater
intmface). Observing A1 (top plate echo) and A2 (bottom plate echo) of Figure 7.5, one
caneasilynoticethatthereisa lSO’phasedifferencebetweentwoechoes. Thatis, when
tlrepeakofechoA,isapositivesignal, thepeekofAzwillbeanegativeone. This
phenomena is the result of acoustic wave propagation from dense to less dense media.

 

In ultrasonic applications, it is desirable to know the temperature profiles and
ultrasonic heating pattern. In other words, the temperature effect plays an important role
in ultrasonic applications inside a structure, especially in clinical application such as
hyperthermia treatment of tumors. The acoustic properties changes in accordance to the
change of temperature in a material under tested or treated. In this study, an experiment
regarding the temperature effect on the acoustic propagation speed was designed. The
experiment setup is shown in Figure 7.2 except that only one plexiglass plate is situated
in the water tank. The distance between the transducer and the water-plexiglass plate
interface, do, is 50 mm and the thickness (d,) of the plexiglass is 12 mm. The water tank
washeated suchthatthetemperaturewasraised from 24°Cto38°C. Thedataweretaken
when the water temperature was 24°C, 26°C, 30°C, 33°C and 38°C. The results are
shown in Figures (7.7a-e). As can be seen from these figures, the traveling time from

the transducer to the water-plexiglass plate interface is shorter as the temperature is

Relative Amplitude

0.8 —

0.6 -1

0.4 ...

0.2 .

116

(a)

 

Temperature - 24 Degrees

 

 

-0.2 -~

-0.4 .-

_008 ‘

 

 

 

 

 

__,)1 1 I l 1 I 1 1
’20.0 20.2 20.4 20.6 20.3 21.0 21.2

Time ( 1on3 )

Figure 7.7 Ultrasonic propagation speed of water at 24°C.

Relative Amplitude

1.1

117

— Figure 7.7 cont’d -

(b)

 

0.8 -1

 

0.6 -l

0.4 ...1

0.2 .1

' Temperoture =- 26 Degrees

 

' v—v‘ ——-— ‘___~_ v

 

-0.2 —
-0.2 —
-o.4 _.

-0.8 -

 

 

 

 

J 1 l I I
20.4 20.6 20.8 21.0 21.2

Time ( 1011s )

 

Relative Amplitude

0.8 d

0.6 _l

0.4 _.

0.2 _.

118

- Figure 7.7 cont’d -

(C)

 

Temperature 8 30 Degrees

 

 

 

 

 

L I I I 1 1
20.2 20.4 20.6 20.8 21.0 21.2

Time ( mus )

 

Relative Amplitude

119

- Figure 7.7 cont’d -

(d)

 

0.8 q

0.6 —

0.4 _.

0.2 .1

 

-O.2 -1

-0.2 —

-o.4 —

-0.8 —«

 

 

 

 

I I

J

Temperature = 33 Degrees

I

 

Time ( 1011s )

20.6 20.8

21.0

21.2

 

Relative Amplitude

120

- Figure 7.7 cont’d -

(e)

 

0.8

 

0.6 —

0.4 ._.

0.2 ...i

 

Temperature - 58 Degrees

 

 

 

 

 

))l 1 I 'i I I L

 

fi’zoo 20.2 20.4 20.6 20.8 21.0 21.2
Time ( 1011s )

 

121

 

 

 

 

 

 

 

 

 

 

TEMPERAruPE TRAVELING TIME PROPAGATION SPEED
(°C) (10» SEC) (M/SEC)
24 6.720 1,488
26 6.707 1,491
30 6.671 1.499
33 6.645 1.505
38 6.566 1.523

 

Table 7 .1 Temperature effect an ultrasonic propagation spwd in water.

 

122

i
1540 4
i

 

 

 

‘1' 13001131100 0010 [57]
1530 "l 0 Experimental data
A
o x
8
\ 1520 -
E
v
§‘ 1510 -+
U
2
G)
> 1500 -4
1490 -
x
1 l . I
23 29 - 35 41

Temperature ( °C)

Figure 7. 8 Temperature dependent ultrasonic propagation speed in water.

123
raised. Inotherwords, thepropagationspeedinwaterincreesesasthetemperamreis
raised, which is in agreement with the observation elsewhere [57]. Use the propagation
speed formula given by Eq. (7.8), the ultrasonic propagation speed as a function of water
temperature is tabulated in Table 7.1. The experimental data are then compared with
previously published data, as shown in Figure 7.8. The results are in good agreement

with those given by others [57].

 

A honeycomb structure of solar cell is employed as the specimen of ultrasonic
nondestructive evaluation in this study. The experiment setup is shown in Figure 7.9.
There are three samples under tested: (a) without surface panel, (b) with surface panel
and (c) with surface panel having a slot-cut open as shown in Figures (7 .9a-c). Figure
(7 .9b) shows the honeycomb structure with a 1 mm delamination. Figure (7 .9c) shows
that the honeycomb structure has a slot-cut damage. The frequency of the transducer was
5 MHz. The specimen were scanned along the top surface for both x— and y- directions
alternately. The results are shown in Figures (7.10a—c). The results were very good.

Delamination as small as 150 um has been detected.

 

124

transducer

/

moving water
(00909.” °" ' ‘ °‘°' tank

1 speciment
under testing

 

 

 

 

water
tank

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(a) without surface panel

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

‘Elmm
AI’I” ---------- T
(b) with surface panel
£1mm

 

---------
’I’ ’ I,
I [III/I
" ‘

(c) with surface panel and slot-cut open

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 7.9 Nondestructive evaluation setup for honeycomb structure.

 

125

 

 

 

 

 

 

(a) without surface plate (b) with surface plate

 

 

 

(c) with surface plate and a slot-cut open

Figure 7.10 Nondestructive evaluation of honeycomb structures.

126
W

Upon the availability of acoustic characteristics such as attenuation coefficient,
reflection coefficient and acoustic impedance, one can utilize computer technique to
simulate the intensity distribution inside a structure under ultrasonic irradiation. Two
models are used for the simulation, 11 three-layer (water-plexiglass-water) and a five-layer
(water’PchiBIISI-water-plexiglass-water) structures, each with water of 1.5 cm thickness
and plexiglass plate of 1.0 cm thickness. These models are subjected to a continuous
ultrasonic wave (CW) with initial intensity of 4 mW/cm’ and with different incident
angles (0 degree and 30degrees). The results are shown in Figures 7.11, 7.12, 7.13 and
7.14. The results show that the intensity distributions in the plexiglass plates are far less
than the initial intensity for both cases due to the energy reflection occurred at the water-

plexiglass plate interface.

 

127

 

5 . r
4.5- - ..
4" .
3.5- '-

U
l
L

 

Intensity (rnW/emcm)
1'2

 

 

 

 

21- .1
156 .
1. 1
05- ~
G. . 1 g a I. g t 7
Propegationdireetion (z-axis): cm

Figure 7.11 Ultrasonic intensity distribution of a five-layer model at normal incidence.

128

 

 

 

 

6 —T t
5: 4... ..
.m.‘
i
A0 l 1
1
g 1
3 i
53- i -
3
- i
i
_ 1
2 1 ...... ‘
"1
E
1
1" 3 er
3. ..............
i
0 J l I. ......... : ‘‘‘‘‘‘‘‘‘‘ - - ‘
0 1 2 3 4 5 6 7
Propagation direction (z-axis): cm

Figure 7.12 Ultrasonic intensity distribution of a five-layer model at 30° incidence.

129

 

 

 

 

 

 

s T 1 1 #—
4.5 '1
4 I- It
’6
g 3.5 7 a
3
e 3-
'2‘
a 2.5 - .
a at
2 1-
105 - q
l " ct
0.5 L r L r t s 's a
0 0.5 l 1.5 2 2.5 3 3.3 4 4.5

Propagation direction (z-axis): cm

Figure 7 . 13 Ultrasonic intensity distribution of a three-layer model at normal incidence.

130

 

 

 

 

6 v ‘
5 In. ........ ‘
A i
3 4 " ' J
l
i
3 !
g 3 1- i '1
i
. i
g s
.s 2 - 5 d
...... .3
i D
1- i .
1
i
i ...................
0 l g t L ‘ ‘ l L
0 05 1 15 2 25 3 35 4 *5
Propagation direction (z-axis): cm

Figure 7.14 Ultrasonic intensity field distribution of a three-layer model at 30° incidence.

CHAPTER 8

CONCLUSIONS

A model to determine the Green’s function of a multilayered biological structure
has been developed in chapter 3. Employing this model, one is able to evaluate the
ultrasonic field intensity distributions, the heat generated by ultrasonic irradiation and the
initial increasing temperature rate of the multilayered biological structure. This model
takes into account all the multiple reflections within a layer which are commonly ignored
by other investigators. A triple-layered biologieel structure, fat-muscle-bone, is used as
an example to verify the theory developed. The results are compared closely with those
previously published elsewhere. This method offers a more accurate evaluation of
ultrasonic energy distribution and energy absorption in human tissues, since the multiple
reflections within the material have also been taken into account. An accurate evaluation
of temperature profile inside biological structure is very crucial to certain applications

such as hyperthermia treatment 0f tumors.

Understanding the detail energy balance of an acoustic wave propagating through
a medium plays an important role in many applications. Knowing how the energy flows

and where the ultrasonic energy stored, dissipated and supplied, one ean minimize the

131

132
energy loss and maximize the ultrasonic efficiency. An energy conservation theorem or

Poynting’s theorem for acoustic wave propagation in an inhomogeneous and moving
medium has been proposed in chapter 4 [54]. The conventional acoustic Poynting’s
theorem considers only the homogeneous medium and ignores the effect of moving
medium which in some eases may play an important role, such as in underwater
acoustics. The acoustic Poynting’s theorem developed here provides a detail acoustic
energy model for both inhomogeneous and homogeneous, both moving and stationary
media. The spatial and temporal variations of density and stiffness, effects of thermal
diffusion, particle collisions, gravity and heat flow have also been included in this
development. The result in this particular model gives an overall breakdown term-by-
term of energy balance of an propagating acoustic wave. The gravity effect can be
neglected in most applications, except for very low frequency situations [11]. The
movingrnediumeffectcouldbeasiflficantfacmrincerminappficafionsnwhas
acoustic wave propagation in underwater with drift current and Doppler detection of
blood flow in human body. This model can be readily reduced to those given by other
investigators. For example, when the acoustic medium is homogeneous and stationary,
and with thermal diffusion, heat flow, particle collision and gravity effects ignored, the
result presented in this study could be simplified to those given by Auld’s [12] and
Ristic’s [13]. If the medium is assumed homogeneous and stationary, but considering the
heat flow effect, the result could be reduced to those given by Pierce’s [11].

A T-type transmission-line model, as shown in Figure 5.2 for an acoustic wave

133
propagation in inhomogeneous and moving medium was developed. This model offers
adeeperinsightoftheaoousticsystembehaviordueto thespatialandternporal variations
ofthe medium density and stiffness of the medium by introducing the equivalent source
terms of the transmission-line equations. It is interesting to note that for a homogeneous
medium, ifthe density and stiffness of the medium are independent oftime and space and
with the thermal diffusion, particle collision and gravity effects are ignored, this model
is then reduced to that given by Auld’s [12]. By this transmission-line model and along
with the electromagnetic-acoustic analogy and/or acoustic-electric circuit analogy, one

can analytically solve the acoustic wave propagating problems with much less effort.

An computation model of evaluating the acoustic displacement field within an
inhomogeneously, arbitrarily shaped scatterer has been formulated in this study. The
dyadic Green’s function technique has been employed in the formulation process. This
model will be useful in clinical applications, because most of the human tissues are
irregularly shaped structures. Two biological models were used in the ultrasonic field
distribution analysis, model A : fat-muscle plate as shown in Figure 6.4 and model B:
fat-muscle-fat bar, shown in Figure 6.7. The distributions of the ultrasonic field of model
A: fat-muscle plate for both layers versus the y-axis and z-axis are displayed in Figures
6.6 and 6.7. The axial field distribution of model B: fat-muscle-fat bar is shown in
Figure 6.8. It can be readily seen that the field contains a forward and backward
propagating. In choosing the pixel size for computer simulation, it is obvious that smaller

pixel size will have more accurate results. However, it consumed enormous computing

134
time. It is interesting to note that the power absorbed in the muscle region of an

inhomogeneousbarisseveraltimeslargerthanthosepowers absorbedinthesurrounding
fat layers. Especially, the materials having high attenuation coefficient and high density
will absorb more energy and consequently convert into heat with higher temperature.
This effect will be very suitable for hyperthermia treatment of tumor. It is well
established fact that tumors and cancerous cells will not be able to survive at a
temperature higher than 45°C. Under ultrasonic irradiation, the temperature at the tumor
site can easily reach that critical level, due to the poor blood circulation at the tumor site
as a result of its higher tumor attenuation and mass density that the surrounding soft

tissues.

For future study along the general direction of this dissertation, the following

suggestions are made.

Experiments of measuring the ultrasonic intensity and/or energy distribution of
biological multilayered structures should be designed and performed. Before
carrying out the actual experiment, it is advised that a computer simulation should

be carried out first to obtain a general idea of the distributions.

Experimentally, an ultrasonic exposure system with well-calibrated ultrasonic
irradiation has to be designed and constructed. Precise ultrasonic dosage has to

be monitored. In addition, sensitive temperature measurements, such as using

135
fluoro-optic thermometer have to be employed for minimum disturbance of the

system. Before clinical evaluation, animal samples such as mice with implanted
tumor should be tested first.

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