Effect of Mode Conversion on Ultrasonic Heating at Tissue Interfaces B. A. Haken, BS, L.A. Frizzell, PhD, E. L. Carstensen, PhD
A number of investigators have observed localized heating by ultrasound beams near impedance discon· tinuities within tissues. It has been suggested that mode conversion to shear waves at impedance discontinuities and subsequent absorption of these waves in a very small distance was the explanation for this heating. A mathematical model for mode conversion at a plane interface between two viscoelastic media is presented.
M
any investigators have reported evidence for localized heating at tissue interfaces which have been irradiated by ultrasound beams. 1- 5 • The possible role of mode conversion as a source of this interfacial heating has been discussed.6" 8 Pain associated with excess bone heating is a limiting factor in the hyperthermic treatment of some tumors. 9 Some pulsed Doppler instruments emit enough power to produce significant heating of bone in situations of minimum attenuation by overlying tissues, which is of particular concern for fetal exposures. Finally, an understanding of absorbed power distribution within bone is fundamental to the determination of the mechanisms involved in accelerated healing of bone by ultrasound beams. Chan and associates 10 indicated that shear waves
Received from the Bioacoustics Research Laboratory (B.A.H., L.A.F.), Department of Electrical and Computer Engineering, University of Illinois, Urbana, lllinois, and the Department of Electrical Engineering (E.L.C.), University of Rochester, Rochester, New York. Address correspondence and reprint requests to Leon A. Frizzell, PhD, Bioacoustics Research Laboratory, Department of Electrical and Computer Engineering, University of Illinois, 1406 West Green Street, Urbana, IL 61801. This work was supported in part by U.S. Public Health Service Grant No. GM 09933, a grant from the UIUC Hines VA Satellite Facility, and a National Science Foundation Graduate Fellowship.
Longitudinal and shear properties are used to calculate the amount of mode conversion that occurs at muscleair and muscle-bone interfaces. Shear waves in bone are found to be an important source of heating, but shear waves in the muscle provide a negligible effect on heating at the interface. KEY woRos: Ultrasound; Hyperthermia; Bone; Shear waves; Mode conversion.
generated in the bone when a longitudinal wave impinges on a muscle-bone interface must be considered in an analysis of heating by ultrasound waves. In their analysis, Chan and colleagues considered shear waves in the soft tissues to be negligible. Filipczynski 11 has examined the role of shear waves at an air boundary and concluded that their contribution to heating is negligible in this special case. Frizzell and Carstensen 12 examined the more general case by allowing the possibility of shear waves in media on both sides of an interface. They studied mode conversion in the cases of longitudinal waves in muscle incident on four different materials: air, water, cholesteryl linoleate, and bone. An extension of this work is reported here. Calculations in this study are based on measurements of the shear properties of soft body tissues at low megahertz frequencies. 13 The results of these measurements have since been confirmed and extended by Madsen and coworkers. 14 Utilizing these measurements, mode conversion and heating at muscle interfaces with bone and with air have been computed for oblique incidence on a plane interface. The power deposition is calculated as a function of distance from the interface in each of these cases. Theoretical results confirm that shear waves contribute significantly to heating in hard tissues such as bone, but shear waves in muscle provide a negligible contribution to heating at the interfaces examined.
© 1992 by the American Institute of Ultrasound in Medicine• J Ultrasound Med 11:393-405, 1992 • 0278-4297 /92/$3.50
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ULTRASONIC HEATING EFFECT
THEORY A longitudinal plane wave of infinite lateral exten t is assumed to be obliquely incident on an infinite plane interface. By choosing a coordinate system (Fig. 1) so that the propagation vectors for the waves lie in the xi. x2 plane, the analysis is greatly simplified with no loss in generality. The two media are characterized by their complex longitudinal and shear propagation constants Ki., Kl and Ks, Ks and their densities p, p', respectively, where K
=-"'c - Ja.
rcncctcd 1hc111
incident
tcflcctcd lorifitudinal
(1)
and w is the angular frequency, c is the real ph ase speed of the wave, and a is its absorption coefficient. It is equivalent to describe the properties of the media in terms of complex propaga tion speeds
V = -"'
(2)
K
This leads to the following expressions for the com· plex longitudinal and complex shear propagation constants in terms of the complex bulk, 8, and complex shear,µ, moduli of elasticity. 15 (3a)
Ks =~
(3b)
B
= 81 + jwB2
(4a)
µ
=
+ jwµ2
(4b)
where 8 1 and µ 1 are the bulk and shear stiffness, respectively, and 82 and µ 2 are the bulk and shear viscosity, respectively. In general, the particle displacement vector~ may be defined as the gradient of a scalar plus the curl of a vector: ~ =
V¢
+Vx
A
the incident longitudinal, ¢1
~~ exp[-jKL(x1sin 0 -
=
~L 4'1expUwt),
x2cos O)expUwt] (6a)
~~R exp[-jKt(X1sin OR + x2cosOR))expUwt]
= ~L ~RexpUwt],
(6b)
the transmitted longitudinal,
' = ~~T exp[-jKL(x 1sin O' - x2cos O'))expUwt)
= ~L ~·exp[jwt],
(5)
where the scalar potential ¢, which is associated with a longitudinal wave, and the vector potential A, which is associated with a shear wave, are solutions of the wave equation for the medium. In the chosen coordinate system, only the component of A along the x3 axis will be nonzero, It will be represented by the scalar a. Using the terminology of Cooper,16 the potentials are:
=
the reflected longitudinal,
tPR=
The complex moduli may be expressed as
µI
Figure 1 Reflected and transmitted longitudinal and shear waves f or obliqu.e incidence of a longitudinal wave on an interface between two semi-infinite media.
(6c)
the reflected shear, a= =
~R exp[-jKs(X1Sin 1/1 + X2COS 1/l)JexpUwt] ~s A expUwt),
(6d)
J Ultrasound Med 11:393-405, 1992
HAKEN ET AL
and the transmitted shear, ,
L1ST
a=-
Ks
· exp[-jK5(x1sin "1'
- x2cos ,P')]expLJwt]
(6e)
Ks
Sij -
(12)
(7a)
"1 = Ktsin
(J
(7b)
(13)
= Ktsin (J K5sin "1 ' = KLsin 0 Ki.sin 8'
(7c)
(7d)
(note that OR = 0). Using Equation 7, the boundary conditions simplify to the following relations, which determine the coefficients of the potentials LR, LT, SR> and ST: cosl{I sinO -sinl{I cosO -pVLcos2,P pVsSin 21/; sin20
~) + dX1
(J
Kssin
2
axl
KLsin OR = KLsin
where the angles are defined in Figure 1 and the potential coefficients LR, LT, SR, and ~ are in general complex. Application of the boundary conditions of continuity of particle displacement and continuity of force yields the Snell's law relations that can be used to determine the complex angles of reflection and refraction,
v:
2
and the asterisk (*) and an underline denoting the complex conjugate and a tensor, respectively. The relatively lengthy expressions for the intensities in terms of the potential coefficients are provided in Appendix A. Because of the choice of coordinates, the stress tensor can be characterized by only two numbers, one corresponding to normal stress and the other to tangential stress at the boundary between the media. Finally, the power loss, or power deposition, per unit volume is
· expUwt],
pV
fourth-order stiffness tensor for the medium, which can be defined in terms of the complex velocities of the longitudinal and shear waves and the density for an isotropic medium,~ being the second-order strain tensor with components
_.!.(()~I
=_!_A'
395
-sinO' cosO' p'Vi.cos 2y/
cosy/ sin '11' p'V5sin 21{1'
p'V'2s . , pVscos21{1 - - s i n 28 p'V5cos2,P' VL -sin 0 cos 0 pV5cos21{1 (8) pVs2
"""V;:"" sin 20 Here, the coefficient L1 has been chosen such that the magnitude of the incident intensity is unity. As noted by Chan and associates, 10 the total ultrasonic intensity is (9)
where
aE at
u=-
(10)
is the particle velocity and
-T= -G · -S
(11)
is the second-order stress tensor, with G being the
Thus, Equations 9 and 13 contain terms that can be associated with purely longitudinal waves (terms involving products of scalar potentials) and purely shear waves (terms involving products of vector potentials), but in addition, several ·cross terms· involve products of the vector and scalar potentials, or products of the incident and reflected scalar potentials (see Appendix A). In lossless media, the cross terms cancel each other, and the two types of waves are resolved into shear and longitudinal contributions to the intensity. This investigation, however, considered the general case where both media are ·1ossy• (have nonzero absorption coefficients). Under these conditions the cross terms do not cancel and must be included to maintain energy conservation at the interface and to obtain the correct form for the power deposition. Equation 13 gives the power deposition per unit volume in the general case. It would be interesting to know what fractions of this power deposition and resultant heating should be attributed to longitudinal and to shear waves. In the past, two similar approaches have been taken to answer this question. In each, the intensities were divided into longitudinal and shear components and separately inserted into Equation 13. Since the intensities contain cross terms, the assignment of these terms presented a problem. Frizzell and Carstensen 12 assigned the cross terms to the longitudinal and shear intensities in proportion to the purely longitudinal and the purely shear terms of the intensity. Chan and coworkers 10 assigned these cross terms in proportion to the amplitudes of the vector and scalar potentials and used the results to calculate power deposition. As will be seen, this arbitrary simplification of the problem results in loss of information about the power deposition and demonstrates, in part, that the
396
JUltrasound Med 11:393-405, 1992
ULTRASONIC HEATING EFFECT
J.2
heating rate involves interactions of shear and longitudinal particle velocities and stresses.
1.0
TESTS OF THEORY Several tests of the theory were performed. First, it was determined that the sum of the normal components of the transmitted and reflected intensities equaled the normal component of the incident intensity, satisfying the requirement for conservation of energy at the boundary. Second, for the cases without loss, numerical results for the normal components of intensities (i.e., the transmitted and reflected power coefficients) obtained using this theory agreed with those of Ewing and colleagues. 11 The muscle· bone interface problem was examined by using the same properties and assumptions (no shear waves in muscle) as Chan.16 The resulting particle displacement potential coefficients agreed with those reported by Chan. Nyborg 19 developed relations for the power deposition in media with no shear viscosity and with no bulk viscosity where two longitudinal plane waves propagate at right angles to each other. This is equivalent to the situation in the unprimed medium when the angle of incidence is 45 degrees, L.11 = l, Sa • LT= ST= 0, and µ 2 • 0 or B2 = 0, respectively. Unlike our case, in Nyborg's analysis the wave amplitude does not decay with distance. However, a meaningful comparison was possible by considering small values of the absorption coefficient and small distances from the interface. The plots in Figure 2 show that numerical results from each theory are virtually identical. The numerical values of the media parameters used in these calculations may be found in Table 1. These tests show that the theory presented in this paper provides solutions that agree with results previously reported for less general theories.
f"I
e
..!:! ~
i:
~
0.6
Cl
0.4
"iiD.
.. .. ; Cl
c.
The power disposition, Pa., was computed near muscleair and muscle-bone interfaces directly without separating the intensity into shear and longitudinal com· ponents. Results obtained by this direct calculation are compared in Figure 3 to those obtained by using the method of Chan and coHeagues 10 for separation of energy into shear and longitudinal components at the interface (i.e., attributing cross term intensities to shear and Jongitudinal based on the ratios of the magnitudes of the respective potential coefficients). The power depositions as a function of distance from the interface for the method of Chan and associates were then
0.2
0.0 0.02
0.04
0.06
0.08
0. 10
0.08
O. IO
x 1 (cm)
A
0.04
:i" ..i .... Cl
~
0.02
B
RESULTS AND DISCUSSION
0.8
0.04
0.06 x 1 (cm)
Figure 2 Power deposition versus distance from the interface in the incident medium for zero shear viscosity (A) and zero bulk viscosity (B) according to Equation 13 (-~-) and according to Nyborg19 ( - ) . The angle of incidence is 45 degrees, and all energy is reflected in the longitudinal wave. Frequency is 2 MHz; incident intensity is 1 W/cm2 • The two curves in each figure (A and B) are indistinguishable.
calculated by applying the appropriate absorption coefficient to both shear and longitudinal waves and using ray tracing to determine the contribution from each wave. Then the contribution from each wave compo· nent was summed to obtain the total power deposition as a function of distance from the interface.
JUltrasound Med 11:393-405, 1992
HAKEN ET AL
397
Table 1: Numerical Values Used in the Comparison to Nyborg 19 in Figure 2 A and B (frequency, 2 MHz; incident intensity, 1 W/cm2) Zero Shear Viscosity (Fig. 2A) B (g/s2) ll (g/52) KL (cm- 1)
Ks (cm-
1 )
2.608681 x 3.989381 x 81.59986 2.086669 x
The results are plotted in Figure 3 as a function of x2 for incident angles of 0, 30, and 60 degrees and an incident intensity of 1 W/cm 2 (see Table 2 for properties used). Results obtained using Chan's method of separating intensities do not show the spatially oscillating nature of the power deposition that is exhibited by Pi., the directly calculated power deposition. It is seen in Figure 3A, C, and E that within the muscle, the results using Chan's method obtained approximate the spatial average of PL. However, Figure 30 and F show that within the bone the results from Chan's method for oblique incidence are not a good approximation to P1.t especially at large angles of incidence. Because power deposition calculated using Chan's method was not in particularly good agreement with Pi., it seems that this method of classifying the energies as shear or longitudinal is not very useful. Another way of studying the effect of shear waves on power deposition is to compare the intensity and the power deposition in media that are the same except for their ability to support shear waves. Instead of solving a new set of equations in which shear waves are not allowed in one or both media, shear waves were vir· tually eliminated by changing the shear absorption coefficient to an arbitrarily large value (as or a 5 [or both]= 108 cm- 1). As shown by Equation 3b, a medium with zero shear stiffness and zero shear viscosity has an infinite value for Ks (i.e., a 5 - oo and Cs = 0). However, shear waves are eliminated just by letting a s approach an infinite value. As is shown in Appendix B, all the terms in the expressions for power deposition (found in Appendix A) that involve shear waves (i.e., those which involve vector potentials), go to zero in the limit that the shear absorption coefficient goes to infinity. A material in which shear waves have been eliminated in this manner will be referred to as "shearless.' Power deposition was computed as a function of angle of incidence immediately adjacent to the interface for muscle-bone and muscle-air interfaces (incident wave in muscle), as shown in Figures 4 and 5, respectively. In each figure, four cases are plotted: a calculation using measured values for the shear absorption coefficients and three calculations using the specified large value for as, or both, to make the
as,
Zero Bulk Viscosity (Fig. 28)
10 + j8.993428 x 10 105 + jO j0.1406545 104 + jO 10
7
2.608661 x 3.989381 X 81.60022 5.711987 x
1010 + jO 105 + j2.9877995 X 10~ j6.230474 x 10 103 - jS.000000 x 103 4
'
media each individually or both shearless. It is clear from Figure 4A that substituting shearless muscle or shearless bone or making both media shearless immensely reduces the power deposition in the muscle. Thus, shear waves do contribute significantly to power deposition in soft tissue immediately adjacent to an interface with another medium that supports shear waves. In the bone, power deposition is significantly smaller for a muscle-shearless bone interface than for a muscle-bone interface, as expected, but the shearless muscle-bone case also is indistinguishable from the muscle-bone case (Fig. 4B). Thus, existence of shear waves in the muscle has no effect on power deposition in the bone. For comparison it is interesting to examine an inter· face between muscle and air (Fig. 5). When the air is shearless, the power deposition is not appreciably smaller in the muscle than it is for a muscle-air interface (Fig. SA), but, as might be expected, is also noticeably smaller in the air (Fig. SB). For interfaces between shearless muscle and air or shearless muscle and shearless air, the power deposition is significantly smaller in both media. This again shows that the contribution of shear waves to power deposition for oblique incidence in soft tissue, immediately adjacent to the interface, is significant. Note that the power deposition in the mus· de immediately adjacent to the interface is very low at all angles of incidence and near zero at normal incidence because the air interface approximates a free surface, where the acoustic pressure and the power deposition would be zero. In Figure 6, the power depositions are plotted versus distance from the interface at angles of incidence of 30 and 60 degrees for the four different types of muscle· bone interfaces. Figure 6A and B are both plots of power deposition within the muscle for an incident angle of 30 degrees. Figure 6A shows that the contribution to PL from the shear wave has disappeared at approximately 5 µm from the interface. Figure 68 shows the behavior of PL at distances greater than 5 µm from the interface. It can be seen clearly in Figure 6B that the results for the muscle-bone and the shearless muscle-bone interfaces are indistinguishable at distances greater than a few micrometers from the interface. A similar result was obtained for shear waves
398
J Ultrasound Med 11:393-405, 1992
ULTRASONIC HEATING EFFECT
0.8
...
E
u
i
ICI
!·;;
.. ..
I
0.4
Cl
0.
Q
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il c
a..
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~.
.
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0.0
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..,
~
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~
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8
eu
i
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ij I
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.. ...
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,
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Cl
.
'
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'
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0.5
0 -2
x2 (cm)
D
A
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~
c:
.5!
~0.
. .
4
...
E u
3
i
2
.5!
ICI
.=
.."' ...
0.6 0.4
Cl
0.
Q
Q
il
0.2
~
Cl
A.
0.8
l.
0 -2
-1
0.0
0
0.0
x2(cm)
....E
u
~
B
E
0.8
24
..,
0.6 .
-
0.4
Q
0.2 .
Ill
.5!
I
.
~
-
Cl
0.
"'
...
\
il Cl
A.
0.0 0.0
r
J
Al
\v
\
.
0.1
~ I /\f
I
i
~ J~ ' \ I
0.2
.
I
I
0.3
0.4
I
I
\I
\
v o.s
c
e
u
~
II
Cl.
8
..
Q
cu ~
c
a..
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x 2 ( cm )
0.4
0.5
16 12
.
0.2
20
.52
="'c
0.1
4 0 -2
-1 x 2 (cm )
0
F
Figure 3 Results from calculations of power deposition Pi. in muscle at 0 (A), 30 (C), and 60 degrees (E) angle of incidence and in bone at 0 (B), 30 (D), and 60 degrees (F) angle of incidence using Equation 13 (-·)and dividing cross term intensities as done by Chan and colleagues 1G (- )for a 2 MHz longitudinal wave incident from the muscle on a muscle.. bone interface with intensity of 1 W/cm2 •
in muscle at a muscle- air interface. Also, the results for the muscle-shearless bone and the shearless musde-shearless bone interfaces are indistinguishable, as noted previously.
Figure 6C shows the power deposition within bone for an angle of incidence of 30 degrees. Here it can also be seen that the results for muscle-bone and shearless muscle-bone are indistinguishable as are the
JUltrasound Med 11:393-405, 1992
HAKEN ET AL
399
Table 2: Properties of Tissues at 2 MHz Used for Mode Conversion Computations Air
Muscle CL (cm/s) aL (cm- 1) Cs (cm/s) as (cm- 1) KL(cm- 1) Ks (cm- 1) p (g/cm 3)
154,000 17' 18* 0.1517,18 2,200 10 5,000 10 81.6 - j0.15 5,710 - j5,000 1.1
34,400 20 o.820 2,000t 6,500t 365 - j0.8 6,280 - j6,500 0.0012
Bone 336,000 17•18 2.51 7,16 179,000 17'18 4.017.18 37.4 - j2.5 70.2-j4.0 1.7
"The superscript numbers refer to the reference used. t The shear properties of air were calculated based on its shear viscosity.
Figure 4 Power deposition versus angle of incidence. A, At x~ = 10-5 cm (in the ·muscle•) for a muscle-bone interface (X), a muscle-shearless bone interface (0), a shearless muscle-bone interface (0), and a shearless muscle-shearless bone interface(+). B, At x2 = -10- 5 cm (in the •bone") for a muscle-bone interface (X), a muscle-shearless bone interface llO
60
30
30 Angle
or Incidence
A
90
60 ( degrees )
IS
10 I:
·~
l..
....
Q
s
~
...
oL-_..~_._~....._~....__..~_._~..._~-=-
B
0
30 60 Angle or Incidence ( degrees )
90
(0), a shearless muscle-bone interface (0), and a shearless muscle-shearless bone interface (+). Frequency is 2 MHz, incident intensity is 1 W/cm2, and the incident wave is in the •muscle:
results for muscle-shearless bone and shearless muscle-shearless bone. Thus, the existence of shear waves in the muscle has no effect on power deposition within the bone. Figure 60, E, and F show that results at an incident angle of 60 degrees are qualitatively similar to those for a 30 degree angle of incidence. The results discussed to this point show that shear waves in muscle contribute significantly to power deposition only within a few micrometers of the interface. It is desirable to determine the contribution of these shear waves to temperature rise near the interface. To estimate their contribution, Pt was integrated from x2 = -1 cm to x2 = 1 cm to determine the total energy deposited within 1 cm of the interface per unit area of the interface. A distance of 1 cm was chosen to be large enough to include all the contribution from shear waves in muscle and bone. Yet, 1 cm is small enough that the energy deposited within that distance will, by conduction, contribute to the temperature rise near the interface after several minutes. The results for the integrated power deposition for 1 cm of bone and 1 cm of muscle are plotted separately as functions of angle of incidence in Figure 7A and B to provide more detail. It is clear from an examination of these results that the existence of shear waves in the muscle makes a negligible contribution to the total energy deposited within 1 cm of the interface and therefore to the temperature rise near the interface. However, the existence of shear waves in the bone decreases the energy deposited in the muscle because more of the incident energy is transmitted into bone than into shearless bone. It is also apparent from Figure 78 that the integrated power deposition in bone does not significantly increase with angle of incidence, as might be erroneously concluded intuitively, on the basis of conversion of energy to more highly absorbed shear waves. In fact, the integrated power deposition begins to decrease with angles of incidence greater than approxi-
400
J Ultr~sound Med 11:393-405, 1992
ULTRASONIC HEATING EFFECT
45
0.02
40
...-
...
35
E
~"'
30
c
·~ ·:;;
c
25
....c
20
E
~
·! ·:;;
...c..
0.01
..
....
Q
Q
II
iS
=
iS
D..
l.
JO
60
Angle of Incidence ( degrees )
0
o.uooo
90
0.0025
-
0.01120
c
0.8
0 . 0006
x 2 (cm)
O.OOOK
0.011111
c
·~
l..
~§
..
0.0010
; g.
c
11.0005
O.OOCHI
0.6
Q
iS
=-
"'
~
~
...
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1.0
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~
~
0.0004
1.2
E
...
0.0002
A
A
...
10
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1111•--1111111--------•
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15
0.4
0.2
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0
_..._.....__ _ _ __..J_
JO
60
__.__;;;:mq111
An:,le of Incidence ( degrees )
0.0 tl.00
'JO
0.05
x 2 {cm )
0.1 0
0.15
B
B
Figure 5 Power deposition versus angle of incidence. A, At 5 X ? = ']0- cm (in the •muscle•) for a muscle-air inter(ace (X), a muscle-shearless air interface (0), a shearless muscle-air interface (0), and a shearless muscle-shearless air interface (+). B, At x2 = -10- 5 cm (in the air) for a muscle-air interface (X), a muscle-shearless air interface (0 ), a shearless muscleair anterface (0), and a shearless muscle- shearless air inter· face(+). Frequency is 2 MHz, incident intensity is 1 W/cmi, and the incident wave is in the ·muscle:
Figure 6 Power deposition versus distance from the interface for a muscle-bone interface (X), a muscle-shearless bone interface (0), a shearless muscle- bone interface (0), and a shearless muscle- shearless bone interface(+). A, B, At 0 = 30 degrees in the ·muscle." C, 8 = 30 degrees in the ·bone: D, E, At 0 a 60 degrees in the •muscle: F, At 0 = 60 degrees •n the ·bone.· Frequency is 2 MHz, incident intensity is 1 W/ cm 2, and the incident wave is in the ·muscle:
mately 35 degrees, apparently due to greater reflection of energy at the bone interface. Yet, shear waves in the bone do contribute significantly to heating because the integrated power deposition for shearless bone begins to decrease rapidly with angles of incidence greater than approximately 15 degrees. Results at an interface between muscle and air also
indicated that shear waves in the muscle made a negligible contribution to the integrated power deposition near the interface. This confirms the result of Filipczynski, 11 who used a less general theory to show that shear waves in the soft tissue at a gas interface provide a negligible contribution to temperature rise compared to longitudinal waves.
HAKEN ET AL
JUltrasound Med 11:393-405, 1992
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x2 (cm)
F
D
Figure 6 Cand D
Determination of the temperature rise close to the interface for short times would require modeling using the bioheat transfer equation. Such an analysis is beyond the scope of this paper.
CONCLUSIONS The results of this study show that the assignment of energy to shear and longitudinal waves as done by Chan and associates 10 yields an inaccurate representation of power deposition. The details of the spatial resolution are lost, and the magnitude is incorrect in many instances compared to the true power deposition as determined by direct calculation without consider-
Figure6 Eand F
ation of whether the energy is in a shear or longitudinal wave. Shear waves in bone contribute significantly to heat• ing, but shear waves in soft body tissues such as muscle make a negligible contribution to heating at tissue interfaces. This is a quantitative validation of the assumption by Chan and colleagues, ic and confirms the results of Filipczynski 11 for interfaces with gas, that shear waves in soft tissues such as muscle can be ignored in studies of interfacial heating. Presumably the properties of other tissues such as tendon are intermediate between muscle and bone. However, the data required are not available to place tendon quantitatively in perspective with the results of the present study.
402
-
J Ultrasound Med 11:393-405, 1992
ULTRASONIC HEATING EFFECT
0.8
APPENDIX A
....
E
~
0.6
• l!I
0.4
•
The following are the intensity equations developed in L.A. Frizzell's thesis, 'Ultrasonic Heating of Tissues'.11 A prime (') symbol denotes the primed or transmitting medium. A subscript 1 refers to the direction parallel to the interface; a subscript 2 refers to the direction perpendicular to the interface. The complex Lame constant Xequals B - 2h µ. An asterisk (*) denotes complex conjugate. Re means the real part of the complex number that follows in parentheses. The potentials cI>1, clJR, A, cl>', and A' and other symbols are defined in the body of the paper .
•
•
•
II
~
l.
....6
c
0.2
Cl.
0.0 ______.__........._......__ _ _....___._
0
_.._
30 60 Angle of Incidence ( degrees )
_.
911
11
~ Rel(>. + 2µ)KLsin O(cl>1cI>T + cI>RcI>U + (2µK 5 sin l/t cos l/t cos*i/t +
A
-
µKsSin•iJ!(sin 21/t - cos2iJ!))AA•
0.8
....
0.6
+
(XKtsin 0 + 2µKLsin 8(sin 20 - cos28))
• •
+ cl>TcI>R) + ((X + 2µ)K Lsin 28 cos*i/t + XKLcos28 cos*t/; + cI>1cI>~
2µKLsin 0 cos 0 sin*l/t)cI>1A• + ((X + 2µ)KLsin 20 cos*i/t + XKL cos20 cos•tJ! -
0.4
2µKLsin 0 cos 8 sin•l/t)cI>RA• + (2µKssin l/t cos l/t sin 0 + 0.2
•
-0.0 .____._ _.__ _,__ 0
• • _.__ _ _• .____...................._
30 60 Angle of Incidence ( degrees )
µKscos O(sin 21/t - cos 21/t))cl>TA (2µKssin l/t cos
+
"1 sin 0 -
__.
µKscos O(sin 2t/; - cos2iJ!))4>lAI
90
B
Figure 1 Power deposition for a muscle-bone interface (X), a muscle-shearless bone interface (0), a shearless musclebone interface (0), and a shearless muscle- shearless bone interface (+) at x1 = 0 numerically integrated over x2 from 10- 10 cm to 1 cm in the •muscle" (A) and from -10-• cm to - 1 cm in the ·bone· (B). Step size is 10- 1 cm, except in the ·muscle· for distances from the interface between 10- 111 and 10- l cm, where the step size is 10-! cm. Frequency is 2 MHz, incident intensity is 1 W/cm 2, and the incident wave is in the 'muscle: The integrated power deposition, which is related to temperature rise, in the bone does not significantly increase with angle of incidence, as might erroneously be concluded from the conversion of energy to more highly absorbed shear waves. The integrated power deposition in the bone begins to decrease with angles of incidence greater than approximately 35 degrees, due to greater reflection of energy at the bone interface.
(2µKssin t/; sin•t/;cos*i/t µKscos*i/t(sin 2,P - cos 21/t ))AA•
+
(XKLcos 8 + 2µKLcos O(cos28 - sin 20)) ( '1>1cl>l - lf>Tcl>R)
(-(X
2
2µKLsin 0 cos 0 (-(X
+
+ 2µ)KLcos 0 sin*t/; - XKtsin 8 sin•i/t 2
+ 2µ)KLcos 20 sin•l/t -
cos*i/t)cI>~·
AKtsin 28 sin•iJ!
+
+
2µKLsin 0 cos 8 cos•tJ! )4>RA•
+
(2µKssin l/t cos if cos 0 µKssin O(sin21/t - cos 21/t))cI>rA
+
(-2µKsSin l/t cos 1/1 cos O µKssin O(sin 2"1
-
cos 21/t))4>lAI
J Ultrasound Med 11:393-405, 1992
HAKEN ET AL
APPENDIXB
11· =~Rel(Kl(>•' + 2µ')sin*O' sin 20' + 2
X'KL.sin*O'cos 28' + 2µ'KL.sin O'cos O'cos•IJ')4>'4>'* + (2µ'K5sin 1/t' cos 1/t' cos*•// µ'K5sin*1/t'cos 21/t' (-(X'
+ µ'Ks·sin•1/t'sin 21/t')A'A'• +
+ 2µ')Kl.sin 20'cos•1/t'
X'Klcos28'cos•1//
+ 2µ'Klsin
-
O'cos O'sin*1//)4> 1 A'•
(-2µ'K5sin if;'cos 1/t'sin*O'
+
+
µ' K5cos•o '(sin21/I'
For a shearless material, an arbitrarily large value for shear absorption is chosen because, as will be shown in the following material, all terms in the expression for power deposition (see Appendix A) that involve reflected (transmitted) shear waves go to zero in the limit in which as (as) goes to infinity. These limits are taken at x 1 = 0. ·shear heating• in the unprimed medium (x2 > 0) = (w/2>1Ksl 2ImlµllAl 2cosh(4 Iml1/tl> 2w Imlµl(RelKLKs(cos 8 cos•.p - sin0sin*1/t)
- cos21/t' ))4>*A')
(cos8sin•.p + sin 8 cos*1/t)cl>1A •j
h· = ~ Rel(-2µ'Klsin O'cos O'sin•IJ' -
(cos 0 sin•lJ- - sin 8 cos*ljl)RA•J] ·shear heating• in the primed medium (x2 < 0) = (w/2)1Ksl 2Imlµ')IA'l 2cosh(4Iml1/t'I> -
+
2wlmlµ'JRel(IKd 2sin28 + K{.K5*cos 8' cos•i//)
2µ'K5sin 1/t'sin•l/t'cos 1/t')A' A'* + (2µ'Klsin O'cos O'cos*l/I' (X'
X'Klsin 28'sin*1/t')4>'A'• + (-µ'K5sin*O'(sin 21/t' - cos2i/I') 2µ'K5sin 1/t' cos if;' cos•o 1 )«1>'• A' I The following are the equations for the power deposition derived from the foregoing intensity equations using Equation 13. w 2 [IKd 2ImlX + 2µj(l4>i] 2 +
(sin O' cos*1// - cos
(J'
sin*i/J')' A'*I
Each term in the foregoing expressions is examined individually below to determine the limit.
+ 2µ')Klcos 28'sin•1/t' -
Pt=
+
RelKtK*5(cos 0 cos•1f + sin 0 sin*ljl)
(X' + 2µ')Klcos•IJ'cos 20' - X'Klcos•IJ'sin 20') 4>'4>'*
403
• . . .r. I1:nut.,.,-s1n .,,
limit.,.,--cos 1/t = .Jl - sin 2.p =
J1 -
(0)2 = 1
limit~.--sin 1/t' = (~Dsin 8 = (w/cl, - .jal) . 8 =0 , Sin w Cs - Jas
l4>Rl 2) -
I
IKsl 2ImlµJIAl 2cosh(4Imli/ll)] -
limit~.-cos i/I' =
Jt -
sin21/t' = Jl - (0) 2 = 1
= limit~.-.P' - 0 . . 2pw 3cs3as hrmt.,.,-ImlµJ = 2 _,_ 2 2 = 0 limit~.--1/t
2wlmfµl[RelKLKs(cos O cos*i/I sin 8 sin*1/t)(cos 0 sin*1/t + sin 0 cos*l/1)4>1A•I +
(w
Re(KLKs(cos 8 cos*1/t + sin 8 sin•!Ji) (cos 0 sin*l/t - sin 0 cos•1f)cJJRA*JI w!Kd 2 ImlX + 2µ(1 - 2 sin 28)2 1Re(cJJ1cJJ=I
. O = (w/Ct . 8= = (KL) -K sin I - :jat)sm s w Cs - Jas
+
Pl=~ [IKl.l 21mlX' + 2µ'cosh 2(21m(1/t'l)llcI>'l 2 -
limit..,--!Ksl 21m(µJIAl 2 = (
w2
Cs2
+ as2)
IKsl 2Imlµ'llA'l 2cosh(4Iml1/t'l)] 2wlm(µ'jRel(IKd2sin28 + KlKs·•cos 8' cos*1/t') (sin O' cos*•// - cos O'sin*1/t')cI>' A'*I
+ q;as)
-o
2pw3CzJas IS,.l2IL1!2 (w2 + c..2as2)2
0
404
ULTRASONIC HEATING EFFECT
J Ultrasound Med 11:393- 405, 1992
3. Taylor KJW, Connolly CC: Differing hepatic lesions caused by the same dose of ultrasound. J Pathol 98:291, 1969
2P I w3Cs13 asI 2 2 [( ' . w ' ) ISTI IL1I exp as + J( -;) ) x2cos if; + (w2 +cs12as12 2 cs
4. Curtis JC: Action of intense ultrasound on the intact mouse liver. In Kelley E (ed): Ultrasonic Energy. Urbana, IL, University of Illinois Press, 1965, p 85 5. Linke CA, Carstensen EL, Frizzell LA, et al: Localized tissue destruction by high intensity focused ultrasound. Arch Surg 107:887, 1973 6. Carstensen EL! The mechanism of the absorption of ultrasound in biological materials. IRE Trans Med Elect ME-7:158, 1960 7. O'Brien WO, Jr, Shore ML, Fred RK, et al: On the assessment of risk to ultrasound. In de Klerk J (ed): 1972 Ultrasonic Symposium Proceedings (IEEE Cat# 72 CHO 708-8SU), 1972, pp 486- 490
The coefficien.ts of the potentials (4>1, R, A, cl>', A') also depend on the shear absorption coefficient. If the limit as as (a~) goes to infinity of each component of the matrix in Equation 8 and of each component of the vector on the right-hand side of Equation 8 is taken before solving the system of equations, the values of SR and L• (ST and Lr) will be finite, not infinite, so the '"shear heating.. terms are zero, as indicated above.
REFERENCES
8. Sarvazyan AP: Acoustic properties of tissues relevant to therapeutic applications. Br J Cancer 45(Suppl V):52, 1982 9. Hynynen K: Hot spots created at skin-air interfaces during ultrasound hyperthennia. Int JHyperthermia 6:1005, 1990 10. Chan AK, Sigelmann RA, Guy AW: Calculations of therapeutic heat generated by ultrasound in fat-musclebone layers. IEEE Trans Biomed Eng BME· 21:280, 1974 11. Filipczynski L! Absorption of longitudinal and shear waves and generation of heat in soft tissues. Ultrasound Med Biol 12:223, 1986 12. Frizzell LA, Carstensen EL: Ultrasonic heating of tissues. Electrical Engineering Technical Report No. GM 0993320, University of Rochester, New York, 1975 (same as Frizzell LA: PhD Thesis, University of Rochester, Roch• ester, NY 1975) 13. Frizzell LA, Carstensen EL, Dyro JF: Shear properties of mammalian tissues at low megahertz frequencies. J Acoust Soc Am 60:1409, 1976 14. Madsen EL, Sathoff HJ, Zagzebski JA: Ultrasonic shear wave properties of soft tissues and tissuelike materials. J Acoust Soc Am 74:1346, 1983 15. Thurston RN: Wave propagation in fluids and normal solids. In Mason WP (ed): Physical Acoustics. Vol 1, Part A. New York, Academic Press, 1964, p 74 16. Cooper HF Jr: Reflection and transmission of oblique plane waves at a plane interface between viscoelastic media. J Acoust Soc Am 60:1064, 1967 17. Ewing WM, Jardetsky WS, Press F: Elastic waves in layered media. New York, McGraw-Hill, 1957, p 83 18. Chan AK: Thermal effects due to the propagation of acoustic waves in biological tissues. PhD Thesis, Univer·· sity of Washington, Seattle, 1971 19. Nyborg WL: Sonically produced heat in a fluid with bulk viscosity and shear viscosity. J Acoust Soc Am 80:1133, 1986
1. Patzold J, Born H: Behandlung biologische Gewebe mit gebundetten Ultraschall. Strahlentherape 76:486, 1945
20. Dyro JF: Ultrasonic study of material related to athero· sclerotic plaque--dynamic viscoelastic properties of cho· lesteric esters. PhD Thesis. University of Pennsylvania, Philadelphia, 1972
2. Bell E: The action of u•trasound on the mouse liver. J Cell Comp Physiol 50:83, 1957
21. Carslaw HS, Jaeger JC: Conduction of heat in solids. 2nd Ed. Oxford, England, Clarendon, 1959, p 75
J Ultrasound Med 11:393-405, 1992
22. Goss SA, Johnston RL, Dunn F: Comprehensive compi· lation of empirical ultrasonic properties of mammalian tissues. J Acoust Soc Am 64:423, 1978 23. Goss SA, Johnston RA, Dunn F: Comprehensive compi• lation of empirical ultrasonic properties of mammalian tissues: II. J Acoust Soc Am 68:93, 1980 24. Del Grosso VA, Mader CW: Speed of sound in pure water. J Acoust Soc Am 52:1442, 1972
HAKEN ET AL 405
25. Beranek L: The Medium. In Beranek L (ed): Acoustic Measurements. New York, John Wiley & Sons, 1949, p 68 26. Dyro JF, Edmonds PD: Ultrasonic absorption and dispersion of cholesteryl esters. Mol Cryst Liq Cryst 25:175, 1974 27. Dyro JF, Edmonds PD: Dynamic viscoelastic properties of cholesteric liquid crystals. Mol Cryst Liq Cryst 29:263, 1975
A number of investigators have observed localized heating by ultrasound beams near impedance discon· tinuities within tissues. It has been suggested that mode conversion to shear waves at impedance discontinuities and subsequent absorption of these waves in a very small distance was the explanation for this heating. A mathematical model for mode conversion at a plane interface between two viscoelastic media is presented.
M
any investigators have reported evidence for localized heating at tissue interfaces which have been irradiated by ultrasound beams. 1- 5 • The possible role of mode conversion as a source of this interfacial heating has been discussed.6" 8 Pain associated with excess bone heating is a limiting factor in the hyperthermic treatment of some tumors. 9 Some pulsed Doppler instruments emit enough power to produce significant heating of bone in situations of minimum attenuation by overlying tissues, which is of particular concern for fetal exposures. Finally, an understanding of absorbed power distribution within bone is fundamental to the determination of the mechanisms involved in accelerated healing of bone by ultrasound beams. Chan and associates 10 indicated that shear waves
Received from the Bioacoustics Research Laboratory (B.A.H., L.A.F.), Department of Electrical and Computer Engineering, University of Illinois, Urbana, lllinois, and the Department of Electrical Engineering (E.L.C.), University of Rochester, Rochester, New York. Address correspondence and reprint requests to Leon A. Frizzell, PhD, Bioacoustics Research Laboratory, Department of Electrical and Computer Engineering, University of Illinois, 1406 West Green Street, Urbana, IL 61801. This work was supported in part by U.S. Public Health Service Grant No. GM 09933, a grant from the UIUC Hines VA Satellite Facility, and a National Science Foundation Graduate Fellowship.
Longitudinal and shear properties are used to calculate the amount of mode conversion that occurs at muscleair and muscle-bone interfaces. Shear waves in bone are found to be an important source of heating, but shear waves in the muscle provide a negligible effect on heating at the interface. KEY woRos: Ultrasound; Hyperthermia; Bone; Shear waves; Mode conversion.
generated in the bone when a longitudinal wave impinges on a muscle-bone interface must be considered in an analysis of heating by ultrasound waves. In their analysis, Chan and colleagues considered shear waves in the soft tissues to be negligible. Filipczynski 11 has examined the role of shear waves at an air boundary and concluded that their contribution to heating is negligible in this special case. Frizzell and Carstensen 12 examined the more general case by allowing the possibility of shear waves in media on both sides of an interface. They studied mode conversion in the cases of longitudinal waves in muscle incident on four different materials: air, water, cholesteryl linoleate, and bone. An extension of this work is reported here. Calculations in this study are based on measurements of the shear properties of soft body tissues at low megahertz frequencies. 13 The results of these measurements have since been confirmed and extended by Madsen and coworkers. 14 Utilizing these measurements, mode conversion and heating at muscle interfaces with bone and with air have been computed for oblique incidence on a plane interface. The power deposition is calculated as a function of distance from the interface in each of these cases. Theoretical results confirm that shear waves contribute significantly to heating in hard tissues such as bone, but shear waves in muscle provide a negligible contribution to heating at the interfaces examined.
© 1992 by the American Institute of Ultrasound in Medicine• J Ultrasound Med 11:393-405, 1992 • 0278-4297 /92/$3.50
394
J Ultrasound Med 11:393-405, 1992
ULTRASONIC HEATING EFFECT
THEORY A longitudinal plane wave of infinite lateral exten t is assumed to be obliquely incident on an infinite plane interface. By choosing a coordinate system (Fig. 1) so that the propagation vectors for the waves lie in the xi. x2 plane, the analysis is greatly simplified with no loss in generality. The two media are characterized by their complex longitudinal and shear propagation constants Ki., Kl and Ks, Ks and their densities p, p', respectively, where K
=-"'c - Ja.
rcncctcd 1hc111
incident
tcflcctcd lorifitudinal
(1)
and w is the angular frequency, c is the real ph ase speed of the wave, and a is its absorption coefficient. It is equivalent to describe the properties of the media in terms of complex propaga tion speeds
V = -"'
(2)
K
This leads to the following expressions for the com· plex longitudinal and complex shear propagation constants in terms of the complex bulk, 8, and complex shear,µ, moduli of elasticity. 15 (3a)
Ks =~
(3b)
B
= 81 + jwB2
(4a)
µ
=
+ jwµ2
(4b)
where 8 1 and µ 1 are the bulk and shear stiffness, respectively, and 82 and µ 2 are the bulk and shear viscosity, respectively. In general, the particle displacement vector~ may be defined as the gradient of a scalar plus the curl of a vector: ~ =
V¢
+Vx
A
the incident longitudinal, ¢1
~~ exp[-jKL(x1sin 0 -
=
~L 4'1expUwt),
x2cos O)expUwt] (6a)
~~R exp[-jKt(X1sin OR + x2cosOR))expUwt]
= ~L ~RexpUwt],
(6b)
the transmitted longitudinal,
' = ~~T exp[-jKL(x 1sin O' - x2cos O'))expUwt)
= ~L ~·exp[jwt],
(5)
where the scalar potential ¢, which is associated with a longitudinal wave, and the vector potential A, which is associated with a shear wave, are solutions of the wave equation for the medium. In the chosen coordinate system, only the component of A along the x3 axis will be nonzero, It will be represented by the scalar a. Using the terminology of Cooper,16 the potentials are:
=
the reflected longitudinal,
tPR=
The complex moduli may be expressed as
µI
Figure 1 Reflected and transmitted longitudinal and shear waves f or obliqu.e incidence of a longitudinal wave on an interface between two semi-infinite media.
(6c)
the reflected shear, a= =
~R exp[-jKs(X1Sin 1/1 + X2COS 1/l)JexpUwt] ~s A expUwt),
(6d)
J Ultrasound Med 11:393-405, 1992
HAKEN ET AL
and the transmitted shear, ,
L1ST
a=-
Ks
· exp[-jK5(x1sin "1'
- x2cos ,P')]expLJwt]
(6e)
Ks
Sij -
(12)
(7a)
"1 = Ktsin
(J
(7b)
(13)
= Ktsin (J K5sin "1 ' = KLsin 0 Ki.sin 8'
(7c)
(7d)
(note that OR = 0). Using Equation 7, the boundary conditions simplify to the following relations, which determine the coefficients of the potentials LR, LT, SR> and ST: cosl{I sinO -sinl{I cosO -pVLcos2,P pVsSin 21/; sin20
~) + dX1
(J
Kssin
2
axl
KLsin OR = KLsin
where the angles are defined in Figure 1 and the potential coefficients LR, LT, SR, and ~ are in general complex. Application of the boundary conditions of continuity of particle displacement and continuity of force yields the Snell's law relations that can be used to determine the complex angles of reflection and refraction,
v:
2
and the asterisk (*) and an underline denoting the complex conjugate and a tensor, respectively. The relatively lengthy expressions for the intensities in terms of the potential coefficients are provided in Appendix A. Because of the choice of coordinates, the stress tensor can be characterized by only two numbers, one corresponding to normal stress and the other to tangential stress at the boundary between the media. Finally, the power loss, or power deposition, per unit volume is
· expUwt],
pV
fourth-order stiffness tensor for the medium, which can be defined in terms of the complex velocities of the longitudinal and shear waves and the density for an isotropic medium,~ being the second-order strain tensor with components
_.!.(()~I
=_!_A'
395
-sinO' cosO' p'Vi.cos 2y/
cosy/ sin '11' p'V5sin 21{1'
p'V'2s . , pVscos21{1 - - s i n 28 p'V5cos2,P' VL -sin 0 cos 0 pV5cos21{1 (8) pVs2
"""V;:"" sin 20 Here, the coefficient L1 has been chosen such that the magnitude of the incident intensity is unity. As noted by Chan and associates, 10 the total ultrasonic intensity is (9)
where
aE at
u=-
(10)
is the particle velocity and
-T= -G · -S
(11)
is the second-order stress tensor, with G being the
Thus, Equations 9 and 13 contain terms that can be associated with purely longitudinal waves (terms involving products of scalar potentials) and purely shear waves (terms involving products of vector potentials), but in addition, several ·cross terms· involve products of the vector and scalar potentials, or products of the incident and reflected scalar potentials (see Appendix A). In lossless media, the cross terms cancel each other, and the two types of waves are resolved into shear and longitudinal contributions to the intensity. This investigation, however, considered the general case where both media are ·1ossy• (have nonzero absorption coefficients). Under these conditions the cross terms do not cancel and must be included to maintain energy conservation at the interface and to obtain the correct form for the power deposition. Equation 13 gives the power deposition per unit volume in the general case. It would be interesting to know what fractions of this power deposition and resultant heating should be attributed to longitudinal and to shear waves. In the past, two similar approaches have been taken to answer this question. In each, the intensities were divided into longitudinal and shear components and separately inserted into Equation 13. Since the intensities contain cross terms, the assignment of these terms presented a problem. Frizzell and Carstensen 12 assigned the cross terms to the longitudinal and shear intensities in proportion to the purely longitudinal and the purely shear terms of the intensity. Chan and coworkers 10 assigned these cross terms in proportion to the amplitudes of the vector and scalar potentials and used the results to calculate power deposition. As will be seen, this arbitrary simplification of the problem results in loss of information about the power deposition and demonstrates, in part, that the
396
JUltrasound Med 11:393-405, 1992
ULTRASONIC HEATING EFFECT
J.2
heating rate involves interactions of shear and longitudinal particle velocities and stresses.
1.0
TESTS OF THEORY Several tests of the theory were performed. First, it was determined that the sum of the normal components of the transmitted and reflected intensities equaled the normal component of the incident intensity, satisfying the requirement for conservation of energy at the boundary. Second, for the cases without loss, numerical results for the normal components of intensities (i.e., the transmitted and reflected power coefficients) obtained using this theory agreed with those of Ewing and colleagues. 11 The muscle· bone interface problem was examined by using the same properties and assumptions (no shear waves in muscle) as Chan.16 The resulting particle displacement potential coefficients agreed with those reported by Chan. Nyborg 19 developed relations for the power deposition in media with no shear viscosity and with no bulk viscosity where two longitudinal plane waves propagate at right angles to each other. This is equivalent to the situation in the unprimed medium when the angle of incidence is 45 degrees, L.11 = l, Sa • LT= ST= 0, and µ 2 • 0 or B2 = 0, respectively. Unlike our case, in Nyborg's analysis the wave amplitude does not decay with distance. However, a meaningful comparison was possible by considering small values of the absorption coefficient and small distances from the interface. The plots in Figure 2 show that numerical results from each theory are virtually identical. The numerical values of the media parameters used in these calculations may be found in Table 1. These tests show that the theory presented in this paper provides solutions that agree with results previously reported for less general theories.
f"I
e
..!:! ~
i:
~
0.6
Cl
0.4
"iiD.
.. .. ; Cl
c.
The power disposition, Pa., was computed near muscleair and muscle-bone interfaces directly without separating the intensity into shear and longitudinal com· ponents. Results obtained by this direct calculation are compared in Figure 3 to those obtained by using the method of Chan and coHeagues 10 for separation of energy into shear and longitudinal components at the interface (i.e., attributing cross term intensities to shear and Jongitudinal based on the ratios of the magnitudes of the respective potential coefficients). The power depositions as a function of distance from the interface for the method of Chan and associates were then
0.2
0.0 0.02
0.04
0.06
0.08
0. 10
0.08
O. IO
x 1 (cm)
A
0.04
:i" ..i .... Cl
~
0.02
B
RESULTS AND DISCUSSION
0.8
0.04
0.06 x 1 (cm)
Figure 2 Power deposition versus distance from the interface in the incident medium for zero shear viscosity (A) and zero bulk viscosity (B) according to Equation 13 (-~-) and according to Nyborg19 ( - ) . The angle of incidence is 45 degrees, and all energy is reflected in the longitudinal wave. Frequency is 2 MHz; incident intensity is 1 W/cm2 • The two curves in each figure (A and B) are indistinguishable.
calculated by applying the appropriate absorption coefficient to both shear and longitudinal waves and using ray tracing to determine the contribution from each wave. Then the contribution from each wave compo· nent was summed to obtain the total power deposition as a function of distance from the interface.
JUltrasound Med 11:393-405, 1992
HAKEN ET AL
397
Table 1: Numerical Values Used in the Comparison to Nyborg 19 in Figure 2 A and B (frequency, 2 MHz; incident intensity, 1 W/cm2) Zero Shear Viscosity (Fig. 2A) B (g/s2) ll (g/52) KL (cm- 1)
Ks (cm-
1 )
2.608681 x 3.989381 x 81.59986 2.086669 x
The results are plotted in Figure 3 as a function of x2 for incident angles of 0, 30, and 60 degrees and an incident intensity of 1 W/cm 2 (see Table 2 for properties used). Results obtained using Chan's method of separating intensities do not show the spatially oscillating nature of the power deposition that is exhibited by Pi., the directly calculated power deposition. It is seen in Figure 3A, C, and E that within the muscle, the results using Chan's method obtained approximate the spatial average of PL. However, Figure 30 and F show that within the bone the results from Chan's method for oblique incidence are not a good approximation to P1.t especially at large angles of incidence. Because power deposition calculated using Chan's method was not in particularly good agreement with Pi., it seems that this method of classifying the energies as shear or longitudinal is not very useful. Another way of studying the effect of shear waves on power deposition is to compare the intensity and the power deposition in media that are the same except for their ability to support shear waves. Instead of solving a new set of equations in which shear waves are not allowed in one or both media, shear waves were vir· tually eliminated by changing the shear absorption coefficient to an arbitrarily large value (as or a 5 [or both]= 108 cm- 1). As shown by Equation 3b, a medium with zero shear stiffness and zero shear viscosity has an infinite value for Ks (i.e., a 5 - oo and Cs = 0). However, shear waves are eliminated just by letting a s approach an infinite value. As is shown in Appendix B, all the terms in the expressions for power deposition (found in Appendix A) that involve shear waves (i.e., those which involve vector potentials), go to zero in the limit that the shear absorption coefficient goes to infinity. A material in which shear waves have been eliminated in this manner will be referred to as "shearless.' Power deposition was computed as a function of angle of incidence immediately adjacent to the interface for muscle-bone and muscle-air interfaces (incident wave in muscle), as shown in Figures 4 and 5, respectively. In each figure, four cases are plotted: a calculation using measured values for the shear absorption coefficients and three calculations using the specified large value for as, or both, to make the
as,
Zero Bulk Viscosity (Fig. 28)
10 + j8.993428 x 10 105 + jO j0.1406545 104 + jO 10
7
2.608661 x 3.989381 X 81.60022 5.711987 x
1010 + jO 105 + j2.9877995 X 10~ j6.230474 x 10 103 - jS.000000 x 103 4
'
media each individually or both shearless. It is clear from Figure 4A that substituting shearless muscle or shearless bone or making both media shearless immensely reduces the power deposition in the muscle. Thus, shear waves do contribute significantly to power deposition in soft tissue immediately adjacent to an interface with another medium that supports shear waves. In the bone, power deposition is significantly smaller for a muscle-shearless bone interface than for a muscle-bone interface, as expected, but the shearless muscle-bone case also is indistinguishable from the muscle-bone case (Fig. 4B). Thus, existence of shear waves in the muscle has no effect on power deposition in the bone. For comparison it is interesting to examine an inter· face between muscle and air (Fig. 5). When the air is shearless, the power deposition is not appreciably smaller in the muscle than it is for a muscle-air interface (Fig. SA), but, as might be expected, is also noticeably smaller in the air (Fig. SB). For interfaces between shearless muscle and air or shearless muscle and shearless air, the power deposition is significantly smaller in both media. This again shows that the contribution of shear waves to power deposition for oblique incidence in soft tissue, immediately adjacent to the interface, is significant. Note that the power deposition in the mus· de immediately adjacent to the interface is very low at all angles of incidence and near zero at normal incidence because the air interface approximates a free surface, where the acoustic pressure and the power deposition would be zero. In Figure 6, the power depositions are plotted versus distance from the interface at angles of incidence of 30 and 60 degrees for the four different types of muscle· bone interfaces. Figure 6A and B are both plots of power deposition within the muscle for an incident angle of 30 degrees. Figure 6A shows that the contribution to PL from the shear wave has disappeared at approximately 5 µm from the interface. Figure 68 shows the behavior of PL at distances greater than 5 µm from the interface. It can be seen clearly in Figure 6B that the results for the muscle-bone and the shearless muscle-bone interfaces are indistinguishable at distances greater than a few micrometers from the interface. A similar result was obtained for shear waves
398
J Ultrasound Med 11:393-405, 1992
ULTRASONIC HEATING EFFECT
0.8
...
E
u
i
ICI
!·;;
.. ..
I
0.4
Cl
0.
Q
0.2
il c
a..
0.0
~.
.
0.1
0.0
" '
..,
~
~
~
0.6 .
8
eu
i
ICI
~
ij I
\
.. ...
4
-~
I
,
6
0. Q
2
~
Cl
.
'
0.2
0.3
a..
'
0.4
0.5
0 -2
x2 (cm)
D
A
...
E
~
c:
.5!
~0.
. .
4
...
E u
3
i
2
.5!
ICI
.=
.."' ...
0.6 0.4
Cl
0.
Q
Q
il
0.2
~
Cl
A.
0.8
l.
0 -2
-1
0.0
0
0.0
x2(cm)
....E
u
~
B
E
0.8
24
..,
0.6 .
-
0.4
Q
0.2 .
Ill
.5!
I
.
~
-
Cl
0.
"'
...
\
il Cl
A.
0.0 0.0
r
J
Al
\v
\
.
0.1
~ I /\f
I
i
~ J~ ' \ I
0.2
.
I
I
0.3
0.4
I
I
\I
\
v o.s
c
e
u
~
II
Cl.
8
..
Q
cu ~
c
a..
0.3
x 2 ( cm )
0.4
0.5
16 12
.
0.2
20
.52
="'c
0.1
4 0 -2
-1 x 2 (cm )
0
F
Figure 3 Results from calculations of power deposition Pi. in muscle at 0 (A), 30 (C), and 60 degrees (E) angle of incidence and in bone at 0 (B), 30 (D), and 60 degrees (F) angle of incidence using Equation 13 (-·)and dividing cross term intensities as done by Chan and colleagues 1G (- )for a 2 MHz longitudinal wave incident from the muscle on a muscle.. bone interface with intensity of 1 W/cm2 •
in muscle at a muscle- air interface. Also, the results for the muscle-shearless bone and the shearless musde-shearless bone interfaces are indistinguishable, as noted previously.
Figure 6C shows the power deposition within bone for an angle of incidence of 30 degrees. Here it can also be seen that the results for muscle-bone and shearless muscle-bone are indistinguishable as are the
JUltrasound Med 11:393-405, 1992
HAKEN ET AL
399
Table 2: Properties of Tissues at 2 MHz Used for Mode Conversion Computations Air
Muscle CL (cm/s) aL (cm- 1) Cs (cm/s) as (cm- 1) KL(cm- 1) Ks (cm- 1) p (g/cm 3)
154,000 17' 18* 0.1517,18 2,200 10 5,000 10 81.6 - j0.15 5,710 - j5,000 1.1
34,400 20 o.820 2,000t 6,500t 365 - j0.8 6,280 - j6,500 0.0012
Bone 336,000 17•18 2.51 7,16 179,000 17'18 4.017.18 37.4 - j2.5 70.2-j4.0 1.7
"The superscript numbers refer to the reference used. t The shear properties of air were calculated based on its shear viscosity.
Figure 4 Power deposition versus angle of incidence. A, At x~ = 10-5 cm (in the ·muscle•) for a muscle-bone interface (X), a muscle-shearless bone interface (0), a shearless muscle-bone interface (0), and a shearless muscle-shearless bone interface(+). B, At x2 = -10- 5 cm (in the •bone") for a muscle-bone interface (X), a muscle-shearless bone interface llO
60
30
30 Angle
or Incidence
A
90
60 ( degrees )
IS
10 I:
·~
l..
....
Q
s
~
...
oL-_..~_._~....._~....__..~_._~..._~-=-
B
0
30 60 Angle or Incidence ( degrees )
90
(0), a shearless muscle-bone interface (0), and a shearless muscle-shearless bone interface (+). Frequency is 2 MHz, incident intensity is 1 W/cm2, and the incident wave is in the •muscle:
results for muscle-shearless bone and shearless muscle-shearless bone. Thus, the existence of shear waves in the muscle has no effect on power deposition within the bone. Figure 60, E, and F show that results at an incident angle of 60 degrees are qualitatively similar to those for a 30 degree angle of incidence. The results discussed to this point show that shear waves in muscle contribute significantly to power deposition only within a few micrometers of the interface. It is desirable to determine the contribution of these shear waves to temperature rise near the interface. To estimate their contribution, Pt was integrated from x2 = -1 cm to x2 = 1 cm to determine the total energy deposited within 1 cm of the interface per unit area of the interface. A distance of 1 cm was chosen to be large enough to include all the contribution from shear waves in muscle and bone. Yet, 1 cm is small enough that the energy deposited within that distance will, by conduction, contribute to the temperature rise near the interface after several minutes. The results for the integrated power deposition for 1 cm of bone and 1 cm of muscle are plotted separately as functions of angle of incidence in Figure 7A and B to provide more detail. It is clear from an examination of these results that the existence of shear waves in the muscle makes a negligible contribution to the total energy deposited within 1 cm of the interface and therefore to the temperature rise near the interface. However, the existence of shear waves in the bone decreases the energy deposited in the muscle because more of the incident energy is transmitted into bone than into shearless bone. It is also apparent from Figure 78 that the integrated power deposition in bone does not significantly increase with angle of incidence, as might be erroneously concluded intuitively, on the basis of conversion of energy to more highly absorbed shear waves. In fact, the integrated power deposition begins to decrease with angles of incidence greater than approxi-
400
J Ultr~sound Med 11:393-405, 1992
ULTRASONIC HEATING EFFECT
45
0.02
40
...-
...
35
E
~"'
30
c
·~ ·:;;
c
25
....c
20
E
~
·! ·:;;
...c..
0.01
..
....
Q
Q
II
iS
=
iS
D..
l.
JO
60
Angle of Incidence ( degrees )
0
o.uooo
90
0.0025
-
0.01120
c
0.8
0 . 0006
x 2 (cm)
O.OOOK
0.011111
c
·~
l..
~§
..
0.0010
; g.
c
11.0005
O.OOCHI
0.6
Q
iS
=-
"'
~
~
...
...-
1.0
E
~
~
0.0004
1.2
E
...
0.0002
A
A
...
10
5
1111•--1111111--------•
o.oo 0
15
0.4
0.2
L..___._
0
_..._.....__ _ _ __..J_
JO
60
__.__;;;:mq111
An:,le of Incidence ( degrees )
0.0 tl.00
'JO
0.05
x 2 {cm )
0.1 0
0.15
B
B
Figure 5 Power deposition versus angle of incidence. A, At 5 X ? = ']0- cm (in the •muscle•) for a muscle-air inter(ace (X), a muscle-shearless air interface (0), a shearless muscle-air interface (0), and a shearless muscle-shearless air interface (+). B, At x2 = -10- 5 cm (in the air) for a muscle-air interface (X), a muscle-shearless air interface (0 ), a shearless muscleair anterface (0), and a shearless muscle- shearless air inter· face(+). Frequency is 2 MHz, incident intensity is 1 W/cmi, and the incident wave is in the ·muscle:
Figure 6 Power deposition versus distance from the interface for a muscle-bone interface (X), a muscle-shearless bone interface (0), a shearless muscle- bone interface (0), and a shearless muscle- shearless bone interface(+). A, B, At 0 = 30 degrees in the ·muscle." C, 8 = 30 degrees in the ·bone: D, E, At 0 a 60 degrees in the •muscle: F, At 0 = 60 degrees •n the ·bone.· Frequency is 2 MHz, incident intensity is 1 W/ cm 2, and the incident wave is in the ·muscle:
mately 35 degrees, apparently due to greater reflection of energy at the bone interface. Yet, shear waves in the bone do contribute significantly to heating because the integrated power deposition for shearless bone begins to decrease rapidly with angles of incidence greater than approximately 15 degrees. Results at an interface between muscle and air also
indicated that shear waves in the muscle made a negligible contribution to the integrated power deposition near the interface. This confirms the result of Filipczynski, 11 who used a less general theory to show that shear waves in the soft tissue at a gas interface provide a negligible contribution to temperature rise compared to longitudinal waves.
HAKEN ET AL
JUltrasound Med 11:393-405, 1992
...
....
6
E
401
E
~
~
~
= ·! ·;;
= ..=
·~
.. ..
·;;;
= Q.
Q.
...
0.8
0.6
Q
,Q
0.4
~
~
~
2
0.2
0
-u.s
-0 .4
-0.3
-0.2
x l ( cm )
.0.1
o.o u.oo
0.0
c
=
..,
E
40
a: ·!= ~
30
10
·;;
..l ....
0.
...
0.15
-
...
~ ..=
0.10
IS
50
~
x2 (cm)
E 60
~
0.05
Q
Q
20
~
=
5
~
c.
c.=
IO
0
o.uooo
0.0002
0.0004 ~!
0.0006
(cm )
O.OOOK
0
0.0010
.u.s
-0.4
-0.3
-0.2
.0.1
0.0
x2 (cm)
F
D
Figure 6 Cand D
Determination of the temperature rise close to the interface for short times would require modeling using the bioheat transfer equation. Such an analysis is beyond the scope of this paper.
CONCLUSIONS The results of this study show that the assignment of energy to shear and longitudinal waves as done by Chan and associates 10 yields an inaccurate representation of power deposition. The details of the spatial resolution are lost, and the magnitude is incorrect in many instances compared to the true power deposition as determined by direct calculation without consider-
Figure6 Eand F
ation of whether the energy is in a shear or longitudinal wave. Shear waves in bone contribute significantly to heat• ing, but shear waves in soft body tissues such as muscle make a negligible contribution to heating at tissue interfaces. This is a quantitative validation of the assumption by Chan and colleagues, ic and confirms the results of Filipczynski 11 for interfaces with gas, that shear waves in soft tissues such as muscle can be ignored in studies of interfacial heating. Presumably the properties of other tissues such as tendon are intermediate between muscle and bone. However, the data required are not available to place tendon quantitatively in perspective with the results of the present study.
402
-
J Ultrasound Med 11:393-405, 1992
ULTRASONIC HEATING EFFECT
0.8
APPENDIX A
....
E
~
0.6
• l!I
0.4
•
The following are the intensity equations developed in L.A. Frizzell's thesis, 'Ultrasonic Heating of Tissues'.11 A prime (') symbol denotes the primed or transmitting medium. A subscript 1 refers to the direction parallel to the interface; a subscript 2 refers to the direction perpendicular to the interface. The complex Lame constant Xequals B - 2h µ. An asterisk (*) denotes complex conjugate. Re means the real part of the complex number that follows in parentheses. The potentials cI>1, clJR, A, cl>', and A' and other symbols are defined in the body of the paper .
•
•
•
II
~
l.
....6
c
0.2
Cl.
0.0 ______.__........._......__ _ _....___._
0
_.._
30 60 Angle of Incidence ( degrees )
_.
911
11
~ Rel(>. + 2µ)KLsin O(cl>1cI>T + cI>RcI>U + (2µK 5 sin l/t cos l/t cos*i/t +
A
-
µKsSin•iJ!(sin 21/t - cos2iJ!))AA•
0.8
....
0.6
+
(XKtsin 0 + 2µKLsin 8(sin 20 - cos28))
• •
+ cl>TcI>R) + ((X + 2µ)K Lsin 28 cos*i/t + XKLcos28 cos*t/; + cI>1cI>~
2µKLsin 0 cos 0 sin*l/t)cI>1A• + ((X + 2µ)KLsin 20 cos*i/t + XKL cos20 cos•tJ! -
0.4
2µKLsin 0 cos 8 sin•l/t)cI>RA• + (2µKssin l/t cos l/t sin 0 + 0.2
•
-0.0 .____._ _.__ _,__ 0
• • _.__ _ _• .____...................._
30 60 Angle of Incidence ( degrees )
µKscos O(sin 21/t - cos 21/t))cl>TA (2µKssin l/t cos
+
"1 sin 0 -
__.
µKscos O(sin 2t/; - cos2iJ!))4>lAI
90
B
Figure 1 Power deposition for a muscle-bone interface (X), a muscle-shearless bone interface (0), a shearless musclebone interface (0), and a shearless muscle- shearless bone interface (+) at x1 = 0 numerically integrated over x2 from 10- 10 cm to 1 cm in the •muscle" (A) and from -10-• cm to - 1 cm in the ·bone· (B). Step size is 10- 1 cm, except in the ·muscle· for distances from the interface between 10- 111 and 10- l cm, where the step size is 10-! cm. Frequency is 2 MHz, incident intensity is 1 W/cm 2, and the incident wave is in the 'muscle: The integrated power deposition, which is related to temperature rise, in the bone does not significantly increase with angle of incidence, as might erroneously be concluded from the conversion of energy to more highly absorbed shear waves. The integrated power deposition in the bone begins to decrease with angles of incidence greater than approximately 35 degrees, due to greater reflection of energy at the bone interface.
(2µKssin t/; sin•t/;cos*i/t µKscos*i/t(sin 2,P - cos 21/t ))AA•
+
(XKLcos 8 + 2µKLcos O(cos28 - sin 20)) ( '1>1cl>l - lf>Tcl>R)
(-(X
2
2µKLsin 0 cos 0 (-(X
+
+ 2µ)KLcos 0 sin*t/; - XKtsin 8 sin•i/t 2
+ 2µ)KLcos 20 sin•l/t -
cos*i/t)cI>~·
AKtsin 28 sin•iJ!
+
+
2µKLsin 0 cos 8 cos•tJ! )4>RA•
+
(2µKssin l/t cos if cos 0 µKssin O(sin21/t - cos 21/t))cI>rA
+
(-2µKsSin l/t cos 1/1 cos O µKssin O(sin 2"1
-
cos 21/t))4>lAI
J Ultrasound Med 11:393-405, 1992
HAKEN ET AL
APPENDIXB
11· =~Rel(Kl(>•' + 2µ')sin*O' sin 20' + 2
X'KL.sin*O'cos 28' + 2µ'KL.sin O'cos O'cos•IJ')4>'4>'* + (2µ'K5sin 1/t' cos 1/t' cos*•// µ'K5sin*1/t'cos 21/t' (-(X'
+ µ'Ks·sin•1/t'sin 21/t')A'A'• +
+ 2µ')Kl.sin 20'cos•1/t'
X'Klcos28'cos•1//
+ 2µ'Klsin
-
O'cos O'sin*1//)4> 1 A'•
(-2µ'K5sin if;'cos 1/t'sin*O'
+
+
µ' K5cos•o '(sin21/I'
For a shearless material, an arbitrarily large value for shear absorption is chosen because, as will be shown in the following material, all terms in the expression for power deposition (see Appendix A) that involve reflected (transmitted) shear waves go to zero in the limit in which as (as) goes to infinity. These limits are taken at x 1 = 0. ·shear heating• in the unprimed medium (x2 > 0) = (w/2>1Ksl 2ImlµllAl 2cosh(4 Iml1/tl> 2w Imlµl(RelKLKs(cos 8 cos•.p - sin0sin*1/t)
- cos21/t' ))4>*A')
(cos8sin•.p + sin 8 cos*1/t)cl>1A •j
h· = ~ Rel(-2µ'Klsin O'cos O'sin•IJ' -
(cos 0 sin•lJ- - sin 8 cos*ljl)RA•J] ·shear heating• in the primed medium (x2 < 0) = (w/2)1Ksl 2Imlµ')IA'l 2cosh(4Iml1/t'I> -
+
2wlmlµ'JRel(IKd 2sin28 + K{.K5*cos 8' cos•i//)
2µ'K5sin 1/t'sin•l/t'cos 1/t')A' A'* + (2µ'Klsin O'cos O'cos*l/I' (X'
X'Klsin 28'sin*1/t')4>'A'• + (-µ'K5sin*O'(sin 21/t' - cos2i/I') 2µ'K5sin 1/t' cos if;' cos•o 1 )«1>'• A' I The following are the equations for the power deposition derived from the foregoing intensity equations using Equation 13. w 2 [IKd 2ImlX + 2µj(l4>i] 2 +
(sin O' cos*1// - cos
(J'
sin*i/J')' A'*I
Each term in the foregoing expressions is examined individually below to determine the limit.
+ 2µ')Klcos 28'sin•1/t' -
Pt=
+
RelKtK*5(cos 0 cos•1f + sin 0 sin*ljl)
(X' + 2µ')Klcos•IJ'cos 20' - X'Klcos•IJ'sin 20') 4>'4>'*
403
• . . .r. I1:nut.,.,-s1n .,,
limit.,.,--cos 1/t = .Jl - sin 2.p =
J1 -
(0)2 = 1
limit~.--sin 1/t' = (~Dsin 8 = (w/cl, - .jal) . 8 =0 , Sin w Cs - Jas
l4>Rl 2) -
I
IKsl 2ImlµJIAl 2cosh(4Imli/ll)] -
limit~.-cos i/I' =
Jt -
sin21/t' = Jl - (0) 2 = 1
= limit~.-.P' - 0 . . 2pw 3cs3as hrmt.,.,-ImlµJ = 2 _,_ 2 2 = 0 limit~.--1/t
2wlmfµl[RelKLKs(cos O cos*i/I sin 8 sin*1/t)(cos 0 sin*1/t + sin 0 cos*l/1)4>1A•I +
(w
Re(KLKs(cos 8 cos*1/t + sin 8 sin•!Ji) (cos 0 sin*l/t - sin 0 cos•1f)cJJRA*JI w!Kd 2 ImlX + 2µ(1 - 2 sin 28)2 1Re(cJJ1cJJ=I
. O = (w/Ct . 8= = (KL) -K sin I - :jat)sm s w Cs - Jas
+
Pl=~ [IKl.l 21mlX' + 2µ'cosh 2(21m(1/t'l)llcI>'l 2 -
limit..,--!Ksl 21m(µJIAl 2 = (
w2
Cs2
+ as2)
IKsl 2Imlµ'llA'l 2cosh(4Iml1/t'l)] 2wlm(µ'jRel(IKd2sin28 + KlKs·•cos 8' cos*1/t') (sin O' cos*•// - cos O'sin*1/t')cI>' A'*I
+ q;as)
-o
2pw3CzJas IS,.l2IL1!2 (w2 + c..2as2)2
0
404
ULTRASONIC HEATING EFFECT
J Ultrasound Med 11:393- 405, 1992
3. Taylor KJW, Connolly CC: Differing hepatic lesions caused by the same dose of ultrasound. J Pathol 98:291, 1969
2P I w3Cs13 asI 2 2 [( ' . w ' ) ISTI IL1I exp as + J( -;) ) x2cos if; + (w2 +cs12as12 2 cs
4. Curtis JC: Action of intense ultrasound on the intact mouse liver. In Kelley E (ed): Ultrasonic Energy. Urbana, IL, University of Illinois Press, 1965, p 85 5. Linke CA, Carstensen EL, Frizzell LA, et al: Localized tissue destruction by high intensity focused ultrasound. Arch Surg 107:887, 1973 6. Carstensen EL! The mechanism of the absorption of ultrasound in biological materials. IRE Trans Med Elect ME-7:158, 1960 7. O'Brien WO, Jr, Shore ML, Fred RK, et al: On the assessment of risk to ultrasound. In de Klerk J (ed): 1972 Ultrasonic Symposium Proceedings (IEEE Cat# 72 CHO 708-8SU), 1972, pp 486- 490
The coefficien.ts of the potentials (4>1, R, A, cl>', A') also depend on the shear absorption coefficient. If the limit as as (a~) goes to infinity of each component of the matrix in Equation 8 and of each component of the vector on the right-hand side of Equation 8 is taken before solving the system of equations, the values of SR and L• (ST and Lr) will be finite, not infinite, so the '"shear heating.. terms are zero, as indicated above.
REFERENCES
8. Sarvazyan AP: Acoustic properties of tissues relevant to therapeutic applications. Br J Cancer 45(Suppl V):52, 1982 9. Hynynen K: Hot spots created at skin-air interfaces during ultrasound hyperthennia. Int JHyperthermia 6:1005, 1990 10. Chan AK, Sigelmann RA, Guy AW: Calculations of therapeutic heat generated by ultrasound in fat-musclebone layers. IEEE Trans Biomed Eng BME· 21:280, 1974 11. Filipczynski L! Absorption of longitudinal and shear waves and generation of heat in soft tissues. Ultrasound Med Biol 12:223, 1986 12. Frizzell LA, Carstensen EL: Ultrasonic heating of tissues. Electrical Engineering Technical Report No. GM 0993320, University of Rochester, New York, 1975 (same as Frizzell LA: PhD Thesis, University of Rochester, Roch• ester, NY 1975) 13. Frizzell LA, Carstensen EL, Dyro JF: Shear properties of mammalian tissues at low megahertz frequencies. J Acoust Soc Am 60:1409, 1976 14. Madsen EL, Sathoff HJ, Zagzebski JA: Ultrasonic shear wave properties of soft tissues and tissuelike materials. J Acoust Soc Am 74:1346, 1983 15. Thurston RN: Wave propagation in fluids and normal solids. In Mason WP (ed): Physical Acoustics. Vol 1, Part A. New York, Academic Press, 1964, p 74 16. Cooper HF Jr: Reflection and transmission of oblique plane waves at a plane interface between viscoelastic media. J Acoust Soc Am 60:1064, 1967 17. Ewing WM, Jardetsky WS, Press F: Elastic waves in layered media. New York, McGraw-Hill, 1957, p 83 18. Chan AK: Thermal effects due to the propagation of acoustic waves in biological tissues. PhD Thesis, Univer·· sity of Washington, Seattle, 1971 19. Nyborg WL: Sonically produced heat in a fluid with bulk viscosity and shear viscosity. J Acoust Soc Am 80:1133, 1986
1. Patzold J, Born H: Behandlung biologische Gewebe mit gebundetten Ultraschall. Strahlentherape 76:486, 1945
20. Dyro JF: Ultrasonic study of material related to athero· sclerotic plaque--dynamic viscoelastic properties of cho· lesteric esters. PhD Thesis. University of Pennsylvania, Philadelphia, 1972
2. Bell E: The action of u•trasound on the mouse liver. J Cell Comp Physiol 50:83, 1957
21. Carslaw HS, Jaeger JC: Conduction of heat in solids. 2nd Ed. Oxford, England, Clarendon, 1959, p 75
J Ultrasound Med 11:393-405, 1992
22. Goss SA, Johnston RL, Dunn F: Comprehensive compi· lation of empirical ultrasonic properties of mammalian tissues. J Acoust Soc Am 64:423, 1978 23. Goss SA, Johnston RA, Dunn F: Comprehensive compi• lation of empirical ultrasonic properties of mammalian tissues: II. J Acoust Soc Am 68:93, 1980 24. Del Grosso VA, Mader CW: Speed of sound in pure water. J Acoust Soc Am 52:1442, 1972
HAKEN ET AL 405
25. Beranek L: The Medium. In Beranek L (ed): Acoustic Measurements. New York, John Wiley & Sons, 1949, p 68 26. Dyro JF, Edmonds PD: Ultrasonic absorption and dispersion of cholesteryl esters. Mol Cryst Liq Cryst 25:175, 1974 27. Dyro JF, Edmonds PD: Dynamic viscoelastic properties of cholesteric liquid crystals. Mol Cryst Liq Cryst 29:263, 1975
