Temperature elevation generated by a focused gaussian beam of ultrasound

Ultrasound in Med. & BioL Vol. 16, No. 5, pp. 489-498, 1990

0301-5629/90 $3.00 + .00 © 1990 Pergamon Press pie

Printed in the U.S.A.

OOriginal Contribution TEMPERATURE ELEVATION GAUSSIAN BEAM

GENERATED BY A FOCUSED OF ULTRASOUND

JUNRU W u a n d G O N G H U A N D u Department of Physics, University of Vermont, Burlington, VT 05405 (Received 4 August 1989; in final form 28 October 1989) Abstract--The steady state temperature elevation generated by a focused Gaussian beam, including the effect of perfusion, has been calculated along the beam axis. The medium is assumed to be a homogeneous absorbing one. The results indicate: ( 1 ) The temperature rise is an increasing function of the intensity gain of the focusing transducer, but never seems to exceed twice that at the interface of the transducer and the medium generated by its unfocused counterpart; and ( 2 ) The temperature rise at the interface of the transducer and the medium is not affected significantly by focusing. Key

Words:

Biomedical ultrasound, Safety, Focused beams, Tissue absorption, Hyperthermia.

INTRODUCTION

truncated cone in a "far field" region; the temperature distribution along the axis of the ultrasound beam was computed by using this model. In their calculations, perfusion, a cooling process due to blood flow, was not taken into account. In clinical applications of ultrasound, focused ultrasound beams are popularly utilized. It would be desirable to know the temperature fields generated by a focused ultrasound beam. In this paper we want to report the results of the calculations which we did recently for the steady state temperature distributions along the beam axis caused by continuous focused Gaussian ultrasound beams, including the effect of perfusion. As pointed out by Nyborg and Steele (1983), the capability of calculating accurately the temperature fields produced in a human patient during an application of medical ultrasound does not exist at present, because adequate knowledge of acoustical and thermal properties of tissues is still lacking. We do not anticipate that our results will give accurate predictions. However, useful insights could be obtained from the results.

When an ultrasound beam passes through an absorbing medium (e.g., tissues), the sound energy is absorbed by the medium and turned into heat; thus, temperature elevation results. Under certain circumstances, the temperature rise can be beneficial; one example is the application of ultrasound in hyperthermia (Lele 1975, 1979; H y n y n e n et al. 1987; Roemer et al. 1984). Under other conditions, for example in diagnostic applications of ultrasound, it may be harmful and should be minimized. Therefore, it is very important to study temperature distributions produced by ultrasound in the medium. Efforts have been made by many authors in this regard (Pond 1970; Robinson and Lele 1972; Chan et al. 1973; Lerner et al. 1973; Filipczynski 1978; Carstensen et al. 1981; Hynynen et al. 198'2); they were able to predict the temperature rise for some special situations. Nyborg and Steele (1983) used a simplified model in order to calculate the temperature fields produced by ultrasound in a more general case and to avoid the mathematical difficulties encountered in making calculations for the sound field generated by a piston transducer (a model commonly used to represent an unfocused ultrasound transducer). The beam was assumed to be uniform and a constant diameter in a "near field" region, but it diverged as a

CALCULATING L I M I T I N G T E M P E R A T U R E S FOR A FOCUSED GAUSSIAN BEAM It has been c o m m o n l y accepted that a linear bio-heat transfer equation initially proposed by Pennes ( 1948 ) is an adequate model for prediction of 489

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Ultrasound in Medicine and Biology

the macroscopic temperature distribution in some biological tissues. The equation can be written as ¢v T = I($72T- cv T/7" + qv,

(1)

where T is the temperature rise above the ambient level, 7~ is the rate of temperature rise, K is the thermal conductivity coefficient, c~ is the volume specific heat for the medium, 7- is the time constant for perfusion, and q¢, is the heat source function which is defined as the rate of heat production per unit volume. The time constant for perfusion, 7-, is related to the blood perfusion rate w, a quantity in units of mass divided by volume and time ( R o e m e r et al. 1984), t h r o u g h the following e q u a t i o n ( N y b o r g 1988):

7-= p~c~/(wct+),

(2)

where pb is the density of blood and C~b is the volume specific heat for blood. Another quantity, which is often used to describe perfusion, is called a perfusing length, L, for a perfusing medium, which varies with tissues; it can be as large as 2.0 cm for moderately perfused tissues (e.g., fat) and as small as 0.20 cm for vigorously perfused tissues (e.g., brain) ( R o e m e r et al. 1984). This quantity is related with r through the following relation: L = (Kr/cv) 1/2.

(3)

For a small heat source of volume dr, which has been generating heat at the rate qvdv for a sufficiently long time, the limiting temperature rise at a distance r from the source including the effect of perfusion is given by the following expression (Nyborg 1988): dT = [q~dv/(47rKr)]exp(-r/L).

(4)

Since eqn (1) is a linear equation, we can use the superposition theorem to calculate the temperature rise, 6T, at any point by adding contributions from all parts of an extended region, through which the ultrasound beam travels. This requires an integration over three dimensions, namely,

Volume16, Number 5, 1990 Fresnel field of a piston transducer; (2) sound energy in the farfield is localized in a single beam free of the diffraction lobes, characteristic of the Frauhofer field of a piston transducer; (3) its mathematical representation is simple; (4) the technique of manufacturing a Gaussian transducer is available and relatively simple (Breazeale et al. 1988; D u e t al. 1989; Wu and Du 1989). An unfocused Gaussian beam is generated by a transducer of radius, a, with a Gaussian velocity distribution. Pressure amplitude at the surface of the transducer, pa(0, ~), has axial symmetry and can be described by pa(0, ~) = p 0 e x p ( - B ~ 2 ) , where B is the Gaussian coefficient at the transducer (Du and Breazeale 1985), and ~ = p / a (Fig. la). The intensity, I, of the Gaussian beam is approximately equal to (½) p2/(oo c) (po is the density of the medium ) as long as the radius of the transducer, a, is much larger than the wavelength, ~ (this condition is satisfied for most cases in ultrasound applications) (Aanonsen et al. 1984), and can be derived from eqn (8) of the Du and Breazeale's (1985) paper as I(a, ~) = Io( 1 + B2cr2)-lexp(-2eqroa) × e x p [ - 2 B ~ 2 / ( 1 + B20-2)],

where Io is the acoustic intensity at the center of the transducer, f is the frequency, aj is the pressure attenuation coefficient, z and o are cylindrical coordinates (Fig. la). Figure 2 includes plots of the intensity vs. nondimensional radial position ( at z = 0 for different unfocused Gaussian Beams of different B but with the same power. Notice that as B increases, Io increases and the intensity becomes more radially localized. I f a planoconcave lens is cemented to the Gaussian transducer, it becomes a focusing Gaussian transducer (Fig. l a). The mathematical expression to describe the pressure amplitude o f the focused Gaussian beam should be modified accordingly (Du and Breazeale 1987); the intensity of the focused Gaussian beam can be written as the following expression: I(a, ~) = Io(AIF/B) × exp(--2alroa)exp(--2Air~2),

6T= f f f [qv/(4~Kr)]exp(-r/L)dv.

(5)

The Gaussian sound field has drawn wide attention recently because of its obvious advantages (Du and Breazeale 1985, 1987); ( 1 ) The nearfield has no intensity maxima, and minima, characteristic o f the

(6)

(7)

where AIF B[B2a 2 -t- (1 - a/aR) 2] 1 (Du and Breazeale 1987). Notice that as ~rg --~ ~ , eqn (7) reduces to eqn (6), a nonfocusing Gaussian beam case. We now return to eqn ( 5 ). To do the integration, it is c o m m o n to divide the continuous region into a =

Temperature elevation • J. Wu and G. Du

491

P

I:l

-Z

0

Fig. 1a. Illustration of the coordinates and the construction of a focusing Gaussian transducer. T: a nonfocusing Gaussian transducer; L: a planoconcave lens cemented to T. large n u m b e r o f thin discs of very large radius (Fig. lb). We consider a disc of thickness dz' whose distance from the surface of the transducer is z'. Due to the axial symmetry of a Gaussian beam, we, again, divide the thin disc into m a n y concentric rings. We n o w c h o o s e a small ring o f radial thickness dp',

whose radius is p'. Heat is generated in this ring at the rate qv(p', z')dv', which can be shown in tissues, for which absorption is mainly due to volume viscosity, equal to 2aI(p', z')27rp'dp'dz', where a is the pressure absorption coefficient of the medium and I ( p ' , _7') is the intensity at the ring. For a focused Gaussian

A -Z

Fig. lb. Illustration of the coordinates used in calculating eqn (5).

Ultrasound in Medicine and Biology

492

Volume 16, Number 5, 1990

5.00

~"

4.00

c_ ~a

2

s.oo

ca 2.00

t .00

I--I

-'-] J

- I .00

-2.00

.000

1.00

3.00

:~.00

R e l e t l v e p o s i t Ion

Fig. 2. Intensity vs. ( = p / a for nonfocusing Gaussian beams. Curve A: B 2; curve B: B = 1; curve C: B = 0.5. The dash line represents the uniform intensity distribution case, the model used by Nyborg and Steele (1983). The power emitted by all the four transducers is identical. b e a m , l ( p ' , 2 ) is given by eqn ( 7 ) in t e r m s o f nondim e n s i o n a l variables ~ a n d (. In eqn (5), r is the separation o f the source a n d the point on the axis, (0, z), at w h i c h the t e m p e r a t u r e rise is to be calculated. Therefore, for the ring chosen, r

= [pr2 q_ ( z -

2,)2

(8)

1 i/2

Substituting eqns ( 7 ) a n d ( 8 ) into eqn ( 5 ) , eqn (5) is n o w reduced to a double integral as 6T =

+ l/[8Au..(a')L']}

× erfc{ 1 / [ L ' ( 8 A , F ( a ' ) )

'/2

+ [ [ 3 [ [ 2 A i F ( o - ' ) ] l / 2 } d a ',

(10)

where error function, e r f c ( x ) is used, which is defined as erfc(x)

= 2/(7r)

1/2

f

oc

exp(-t2)dt. x

if', [3 and L ' are defined as follows:

[ 2cdo( A iF / B ) e x p ( - 2c~ro~ )

~' = Z l r o ,

exp(--2Aiv(2)]/ { 47rKIp '2 + ( z -

[32 = ( a -- a ' ) 2 r g / a 2,

2') 2 ] I/2}

e x p { - [ p '2 + (z - 2,)21 }do'de.

T h e integration for p' is integrable; this is one o f the a d v a n t a g e s o f G a u s s i a n b e a m s as p o i n t e d out earlier; m a t h e m a t i c a l l y , it is relatively easier to be handled in c o m p a r i s o n to the case o f piston transducers. After calculating this integral, eqn ( 9 ) b e c o m e s fiT(0, a) = T(0, a) - To = [(Tr/8)l/2aIoaro/(KB)]

L'= L/a.

(9)

If we let L ' a p p r o a c h c~, which is equivalent to neglecting perfusion, eqn (10) b e c o m e s 6T(0, a) = T(0, a) - To = [(Tr/8)l/2aloaro/(KB)l X

yo

[ A I F ( e ' ) ] J/2

× e x p ( - - 2 ~ l r o f f ' ) e x p { [2A~F(~')[321

fo

[AIF(ff')] l/2exp [--2cqrocr'

+ 2AIF(a')[3 2]

× erfc{ 1131[2Air(o-')] 1/2 }do'.

(11)

Temperature elevation • J. Wu and G. Du RESULTS FOR LIMITING TEMPERATURE ALONG THE BEAM AXIS T h e integration r e m a i n i n g in eqns ( 1 0 ) a n d ( 11 ) c a n n o t be readily d o n e in analitical form; they were c o m p u t e d numerically. In m a k i n g calculations f r o m eqns ( 1 0 ) a n d ( 1 1 ) , unless specified otherwise, the following values were assigned to the quantities K, Io, a, a l, a n d cv: K = 0.0060 W / ( c m ° C ) , Io = 0.1 W / c m 2, = al = 0.05 f N p / ( c m M H z ) , cv = 4.184 J / ( c m 3 ° C ) , c = 1,500 m / s .

493

F o r a focused Gaussian beam, the intensity is not u n i f o r m a n d can be described by eqn (7). Io in eqn (7) represents the intensity at the center o f the transducer. T o c o m p a r e these two cases, we assume the total powers t r a n s m i t t e d by these two transducers are equal. Considering this fact, therefore Io in eqn (7) should be replaced by 2 BIo, where B is the Gaussian coefficient at the transducer. T h e n o n d i m e n s i o n a l focal length o f a focused Gaussian transducer, aF, is d e t e r m i n e d by the following expression ( W u and D u 1990): a F = a i R / ( 1 ~- B 2 a 2 ) ~

Here for K, cv, and c the values chosen are for water. T h e expression for a a n d al is for beef liver (Frizzell et al. 1979; Cartensen et al. 1981; P o h l h a m m e r et al. 1981).

aR,

if BaR ~ 1.

(12)

T h e intensity gain o f a focused Gaussian transducer is defined as the ratio o f the intensity at the focal point to that at the center o f the transducer, a n d is given by ( W u a n d Du, 1990)

G, = ( 1 + B2a2n)l(BZa~ ) ~ 1/(B2a2), The effect o f focusing In order to see the effect o f focusing, o u r results are c o m p a r e d w i t h t h o s e o f N y b o r g a n d Steele ( 1 9 8 3 ) . In their simplified model, the b e a m is ass u m e d to be u n i f o r m in a "near-field" region. T h e intensity o f the b e a m at the t r a n s d u c e r is represented as

I=Io

for

p <_a,

=0

for

p>a.

i f B a R ~ 1.

Figure 3 includes the numerical results o f eqn ( 11 ) ( n o perfusion) for different oR; the other parameters are kept c o n s t a n t ( a = 0.6 cm, B = 1, a n d f = 3 M H z ) . T h e curve for an = ~ (GI = 1 ) represents the l i m i t i n g t e m p e r a t u r e rise vs. a for an u n f o c u s e d Gaussian beam. The position, where 6T reaches m a x i m u m , is fairly close to the transducer and agrees with N y b o r g and Steele's results. T h e m a g n i t u d e o f

o

c

2.51

A

1.5

0.5

I 0

. t00

I

(13)

L ,300

I

I

(~"

.500

Fig. 3. Temperature elevation vs. ~r = z/ro for different ~n. Curve A: an = 0.1; curve B: aR = 0.15; curve C: an = 0.2; curve D: an = oo. The other parameters used: B = 1, f = 3 MHz, a = 0.6 cm, c~ = al = 0.05 f N p / ( c m MHz). 6T at z = 0 is not affected significantly by focusing.

494

Ultrasound in Medicine and Biology

Volume 16, N u m b e r 5, 1990

°C 4.5

A O

@~

B

3.0

i"I

[,~

1.5

0 m

I'oo

. 2 0'0

. 3 0'0

. 4 0'0

. 5 0'0 ~

Fig. 4. Temperature elevation vs. a = z/ro for different a. Curve A: a - 1.0 cm; curve B: a = 0.9 cm; curve C: a = 0.8 cm; curve D: a = 0.6 cm. The other parameters used: B = 1, f = 3 MHz, c~ = c~1 = 0 . 0 5 f N p / ( c m MHz), ~rR = 0.1.

6Zmax, h o w e v e r , is a b o u t a factor o f 1.1 larger t h a n theirs. T h i s difference is e x p e c t e d , since the i n t e n s i t y d i s t r i b u t i o n o f t h e G a u s s i a n b e a m is n o t u n i f o r m (Fig. 2). F o r t wo t r a n s d u c e r s o f the s a m e power, 6 T at th e s a m e p o s i t i o n o n th e axis for th e G a u s s i a n t r a n s d u c e r s h o u l d be h i g h e r t h a n t h a t for its c o u n t e r -

p a r t by N y b o r g a n d Steele's m o d e l . T h i s will b e c o m e cl ear er w h e n we discuss t h e effect o f B later. As aR decreases o r GI increases ( t h e b e a m bec o m e s m o r e f o c u s e d ) , t h e d i s t r i b u t i o n o f Tt changes; the p o s i t i o n o f 6Tmax g r a d u a l l y m o v e s to the focal p o i n t (~F ~ aR) a n d t h e 6Tmax m e a n w h i l e increases.

o

C 3.0

A 2.5

o 2.0

i,"~

1.5

A~

1.0

0.5

1 • t00

I .200

I .300

I .400

I .500

G

Fig. 5. T e m p e r a t u r e e l e v a t i o n vs. a = z/ro for d i f f e r e n t a = a l . C u r v e A: a = 0.1 f N p / ( c m M H z ) ; c u r v e B: = 0 . 0 7 5 f N p / ( c m M H z ) ; c u r v e C: a = 0 . 0 5 f N p / ( c m M H z ) . T h e o t h e r p a r a m e t e r s u s e d : B = 1, f = 3 M H z ,

an = 0.1, a = 0.6 cm.

Temperature elevation • J. W u and G. Du

495

°c 3.0

A

2.5

m O al

2.0

N

i- 1.5 a 1.0

E0.5

I

i

I

I

.t00

.200

.300

.400

I ~

.500

Fig. 6. T e m p e r a t u r e elevation vs. ~r = z/ro for different B. C u r v e A: B = 4; curve B: B = 2; curve C: B = 1; curve D: B = 0.5. T h e o t h e r p a r a m e t e r s used: f = 3 M H z , ~R = 0.1, a = 0.6 cm, a = a~ = 0.05 f N p / ( c m M H z ) .

O u r c a l c u l a t i o n s a l s o s h o w t h a t 6Tmax g e n e r a t e d b y a focused Gaussian beam never seems to exceed twice t h a t a t a = 0, p r o d u c e d b y a n u n f o c u s e d G a u s s i a n beam of the same parameters at least in cases ofR > a. This was also pointed out earlier by Nyborg ( 1 9 8 9 ) . A n o t h e r o b s e r v a t i o n f r o m Fig. 3 is t h a t 6T a t the surface (a = 0) of the medium does not seem to

be affected significantly by focusing, which agrees with the results of Nyborg (1989).

The effects of the transducer size When we increase the radius of the transducer a n d k e e p t h e o t h e r p a r a m e t e r s u n c h a n g e d ( B = 1, a = a l = 0 . 0 5 f N p / ( c m M H z ) , aR = 0.1, f = 3 M H z ,

°c

3"0I 2.5

A

O

2.0 0

&" a

1.5

k

0.5

0

I • 100

I .200

I .300

I .400

I ~r .500

Fig. 7. T e m p e r a t u r e elevation vs. a = z/ro for different frequencies. C u r v e A: f = 3 M H z ; c u r v e B: f = 2 M H z ; c u r v e C: f = 1 M H z . T h e o t h e r p a r a m e t e r s used: B = 1, aR = 0.1, a = 0.6 cm, a = a~ = 0.05 f N p / ( c m M H z ) .

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Ultrasound in Medicine and Biology

I0 = 0.1 W / c m 2 ) , first o f all, the input power increases. T h e other two quantities, which are also related to a, are r / a n d oR (equal to R/rt). Therefore, it is evident that r0 a n d R increase accordingly (oR is kept c o n s t a n t ) . Keeping these facts in mind, it is not difficult to u n d e r s t a n d these plots shown in Fig. 4. As the input p o w e r increases by increasing a, 6Tat z = 0 increases. However, as ~ is kept constant, and the focal length (equal to rlaR) increases, the intensity at the focal point decreases; the focusing effect b e c o m e s less important. Therefore, the position o f 6Tmax gradually shifts to the position at which it w o u l d be without focusing. This kind o f shifting effect should bec o m e less i m p o r t a n t if we increase the intensity at the focal point by reducing c~; this is clearly shown in Fig. 5.

The effect of a Gaussian coefficient As we m e n t i o n e d a b o v e , t h e i n t e n s i t y o f a G a u s s i a n b e a m is n o t u n i f o r m l y d i s t r i b u t e d at a plane perpendicular to its axis, but is localized along the axis. W h e n the Gaussian coefficient at the transducer, B, increases, the intensity is m o r e radially localized. T h e radial localization o f the intensity increases the 6T at o = 0. Consequently, the effect o f focusing b e c o m e s less i m p o r t a n t ; these effects can be readily seen in Fig. 6.

The effect offrequenc.v W h e n we increase f r e q u e n c y , ~ as well as ~ (equal to 0 . 0 5 f ) increases. Therefore, the heat source

3.5

Volume 16, Number 5, 1990 function qv (equal to 2 ~ I ) increases. As a result o f that, 6T increases accordingly. In Fig. 7, three plots are shown, which correspond to three different frequencies. T h e one o f the highest frequency has the highest t e m p e r a t u r e elevation across the axis.

The effect of the boundary condition As pointed out by N y b o r g a n d Steele ( 1983 ), the b o u n d a r y condition has a large effect o f r T . E q u a t i o n (10) applies on the a s s u m p t i o n that the region o f negative z, where the transducer exists, is free o f heat sources a n d has the same t h e r m a l conductivity as the m e d i u m existing at the region o f positive o. This ass u m p t i o n certainly represents only one possibility. T h e results o f the other two possibilities are shown in Fig. 8 in c o m p a r i s o n with that o f the first (curve B). Curve A represents a case that the thermal conductivity o f the transducer assembly is so low that the temperature gradient OTt/Oa is essentially zero. Mathematically, it is equivalent to include the contribution o f a heat-source distribution existing in the space o f negative a, which is the image-source o f the real heatsource at positive z space t h r o u g h the " m i r r o r " at the plane o = 0. As expected, the t e m p e r a t u r e rise at z 0 is twice that o f curve B because o f the image contribution. Curve C represents a case when the transducer is cooled so that the t e m p e r a t u r e rise at z = 0 is kept at zero. Mathematically, it is equivalent to calculating the 6 T c o n t r i b u t e d by the heat-source at the positive z

°c A

3.0

2.5

'~

2.0

1.5

0.5

I

I

I

I

• 100

.200

.300

.400

I ~r .500

Fig. 8. Temperature elevation vs. cr - z/ro for different boundary conditions. Curve A: OTI/Oz = 0, at z = 0; curve B: this represents a case in which the region of negative z is free of heat sources and has the same thermal conductivity as the medium existing at the region of positive z; curve C: 6T = 0 at z = 0. The other parameters used: B = 1, aR = 0.1, a = 0.6 cm, c~ = al = 0.05 f N p / ( c m MHz), f = 3 MHz.

Temperature elevation • J. Wu and G. Du

497

°C 3.0

L:ce

2.5

¢I 0

)"

2.0

i~'~

.a

1.5

1.0

0.5

G • i00

.200

.300

.400

I .500

Fig. 9. Temperature elevation vs. ~r = z/ro for different L. The other parameters used: B = 1, an = 0,1, a = 0.6 cm, a = C~l = 0.05 f N p / ( c m MHz), f = 3 MHz.

space m i n u s that c o n t r i b u t e d by its i m a g e heat-source at the negative z space. T h e results show that 6 T decreases across the axis; however, 6 T at the focal p o i n t does n o t seem to be affected significantly by cooling. T h e effect o f p e r f u s i o n I n c l u d i n g the effect o f p e r f u s i o n , the i n t e g r a t i o n in e q n ( 1 0 ) s h o u l d be used to c o m p u t e 6T. T h e results o f the d i s t r i b u t i o n o f the t e m p e r a t u r e rise for the different p e r f u s i o n lengths are s h o w n in Fig. 9. T h e t o p c u r v e is for the n o - p e r f u s i o n case ( L = oo ), which is the s a m e we o b t a i n e d b y c o m p u t i n g the i n t e g r a t i o n in e q n ( 11 ). As L decreases a n d the m e d i u m is m o r e perfused, the t e m p e r a t u r e rise decreases across the axis, as expected.

Acknowledgements--We wish to thank Professor Wesley N. Nyborg for his valuable advice• This work was supported by the American Cancer Society (IN- 156), NSF, and Vermont Epscor.

a B c c~ cv R K L p Pa

P0 = pressure a m p l i t u d e at the c e n t e r of a s o u n d source qv = the heat source f u n c t i o n ro = a2o~/( 2 c ) = Rayleigh distance rl = [ a 2 o ) / ( 2 c ) ] ( 1 - c / Q ) = m o d i f i e d Rayleigh distance al = pressure a t t e n u a t i o n coefficient = pressure a b s o r p t i o n coefficient oJ = a n g u l a r f r e q u e n c y p = radial c o o r d i n a t e p0 = d e n s i t y o f a fluid m e d i u m = p / a = n o n d i m e n s i o n a l radial c o o r d i n a t e z = axial c o o r d i n a t e a = z / r o = n o n d i m e n s i o n a l axis c o o r d i n a t e an = R / r t = n o n d i m e n s i o n a l geometric focal distance r = the t i m e c o n s t a n t for p e r f u s i o n

LIST OF SYMBOLS

REFERENCES

radius of a transducer G a u s s i a n coefficient at t r a n s d u c e r speed o f s o u n d i n a fluid m e d i u m speed o f s o u n d i n solid lens the v o l u m e specific heat for the m e d i u m r a d i u s o f c u r v a t u r e o f lens, g e o m e t r i c a l focal length = t h e r m a l c o n d u c t i v i t y coefficient = perfusion length = s o u n d pressure = pressure a m p l i t u d e

Aanonsen, S. I.; Barkev, T.; Tjotta, J. N.; Tjona, S. Distortion and harmonic generation in the nearfield of a finite amplitude sound beam. J. Acoust. Soc. Am. 75:749-768; 1984. Breazeale, M. A., Wen, J. J.; Na, J. Electrode shaping to achieve Gaussian ultrasonic field distribution. J. Acoust. Soc. Am., Suppl. 1, 84:S197; 1988. Carstensen, E. L.; Becroft, S. A.; Law, W. K.; Barbee, D. B. Finite amplitude effects on the threshold for lesion production in tissue by unfocused ultrasound. J. Acoust. Soc. Am. 70:302-309; 1981. Chan, A. K.; Siegmann, R. A.; Guy, A. W.; Lehmann, J. F. Calculation by the method of finite differences of the temperature distribution in layered tissues. IEEE Trans. on Biomedical Engng. BME-20:86-90; 1973.

= = = = = =

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Ultrasound in Medicine and Biology

Du, G.; Breazeale, M. A. The ultrasonic field of a Gaussian transducer. J. Acoust. Soc. Am. 78:2083-2086; 1985. Du, G.; Breazeale, M. A. Theoretical description of a focused Gaussian ultrasonic beam in a nonlinear medium. J. Acoust. Soc. Am. 81:51-57; 1987. Du, G.; Ding, D.; Gan, C. A new type of Gaussian ultrasonic transducer. J. Acous. Soc. Am., Suppl. 1, 85:$92; 1989. Frizzell, L. A.; Carstensen, E. L.; Davis, J. D. Ultrasonic absorption and attenuation in liver tissue. J. Acoust. Soc. Am. 65:13091312; 1979. Filipczynski, L. Measurement of the temperature increases generated in soft tissues by ultrasonic diagnostic Doppler equipment. Ultrasound Med. Biol. 4:151-155; 1978. Hynynen, K.; Watmough, D. J.; Mallard, J. R. The effect of thermal conduction during local hyperthermia induced by ultrasound; a phantom study. Br. J. Cancer, Suppl. V, 45:68-70; 1982. Hynynen, K.; Watmough, D. J.; Mallard, J. R. Design of ultrasonic transducers for local hyperthermia. Ultrasound Med. Biol. 7:397-402: 1987. Lele, P. P. Hyperthermia by ultrasound. Proc. Int. Symp. on Cancer Therapy by Hyperthermia and Radiation. Washington, DC: U.S. Government Printing Office; 1975:168-176. Lele, P. P. A strategy for localized chemotherapy of tumors using ultrasonic hyperthermia. Ultrasound Meal. Biol. 5:95-96; 1979 (Letter to the Editor). Lerner, R. M.: Carstensen, E. L.; Dunn, F. Frequency dependence of thresholds for ultrasonic production of thermal lesions in tissues. J. Acoust. Soc. Am. 54:504-506: 1973.

Volume 16, Number 5, 1990 Nyborg, W. L.; Steele, R. B. Temperature elevation in a beam of ultrasound. Ultrasound Med. Biol. 9:611-620; 1983. Nyborg, W. L. Solutions of the bio-heat transfer equation. Phys. Med. Bio. 33:785-792; 1988. Nyborg, W. L. NCRP-AIUM models for temperature calculations. Ultrasound Med. Biol., Suppl. 1, 15:37-40~ 1989. Pennes, H. H. Analysis of tissue and arterial blood temperatures in the resting human forearm. J. Appl. Physiol. 1:93-122; 1948. Pohlhammer, J. D.; Edwards, C. A.; O'Brien, W. D., Jr. Phase insensitive ultrasonic attenuation coefficient determination of fresh bovine liver over an extended frequency range. Med. Phys. 8:692-694; 1981. Pond, J. B. The role of heat in the production of ultrasonic focal lesions. J. Acoust. Soc. Am. 47:1607-1611, 1970. Robinson, T. C.; Lele, P. P. An analysis of lesion development in the brain and in plastics by high-intensity focused ultrasound at low-megahertz frequencies. J. Acoust. Soc. Am. 51:1333-1357; 1972. Roemer, R. B.; Swindell, W.; Clegg, S. T.; Kress, R. L. Simulation of focused, scanned ultrasonic heating of deep-seated tumors: The effect of blood perfusion. IEEE Trans. Son. Ultrason. Su-31:457-466, 1984. Wu, J.; Du, G. A design of ultrasonic transducers with curved back-electrodes. Proceedings of IEEE Ultras. Syrup. New York: Institute of Electrical and Electronic Engineers; 1989. Wu, J.; Du, G. Acoustic radiation force on a small compressible sphere in a focused beam. Submitted to J. Acoust. Soc. Am. 87:997-1003: 1990.