ISSN 10637710, Acoustical Physics, 2010, Vol. 56, No. 1, pp. 105–114. © Pleiades Publishing, Ltd., 2010. Original Russian Text © A.N. Reznik, P.V. Subochev, 2010, published in Akusticheskiі Zhurnal, 2010, Vol. 56, No. 1, pp. 113–125.
ACOUSTICS OF ANIMATE SYSTEMS. BIOACOUSTICS
On the Theory of Acoustothermometry of Waterlike Media: Effects of the QuasiStatic Field, Strong Absorption, and Radiation Pattern A. N. Reznika and P. V. Subochevb a
Institute for Physics of Microstructures, Russian Academy of Sciences, Nizhni Novgorod, 603950 Russia email:
[email protected]nnov.ru b Institute of Applied Physics, Russian Academy of Sciences, ul. Ul’yanova 46, Nizhni Novgorod, 603950 Russia email:
[email protected] Received December 26, 2008
Abstract—An integral equation relating the thermal noise’s pressure measured by an acoustothermometer to the onedimensional temperature profile of a waterlike medium is derived by applying the fluctuationdissi pative theorem to hydrodynamic equations. On the basis of this equation, effects of the quasistationary field of thermal radiation, a strong absorption of acoustic waves in the medium, and a finite beam width of the receiving antenna are investigated. Conditions under which the solution to the equation coincides with the result of the transfer theory, which ignores the aforementioned effects, are determined. The role of the effects under study in acoustothermometry of biological media is investigated. A method is proposed for controlling the depth from which the radiation is received. This method allows the retrieval of the temperature profile of the medium from the data of acoustic sounding. DOI: 10.1134/S106377101001015X
1. INTRODUCTION The internal thermodynamic temperature of the human organism is an important parameter whose monitoring and control are necessary in medicine. Based on the information on the internal tempera ture’s distribution, it is possible to judge the state and functioning of organs and systems and to trace the response of the organism to various external actions. The internal temperature can be monitored in differ ent ways, but passive and noninvasive methods are of most interest to medicine, i.e., methods that allow diagnosing without surgical invasion or adverse effects on the organism. One of such methods is the acoustic brightness thermometry [1–7], which allows deep internal temperature measurements to within 0.1 K by receiving radiation of acoustic noise. In this case, the internal temperature of the medium is determined from the measured pressure of thermal radiation by using the analytic relationship between these charac teristics. In most of the publications on acoustother mometry [1, 3–7], this relationship is represented by the integral equation derived in [8] from the transfer equation for the ray’s acoustic intensity with sources corresponding to thermal radiation:
(
Ta ( f ) = 1 − Rray
∞
2
) ∫T ( z ') γ ( f ) exp (−γ ( f ) z ') dz ', 0
(1)
where Ta is the acoustic brightness temperature of the halfspace (the thermodynamic temperature of a blackbody generating the same radiation flux as the object under study), γ ( f ) is the frequencydependent absorption coefficient for sound intensity, T ( z ') is the desired distribution of the thermodynamic tempera ture, Rray = (ρ1c1 − ρ2c2 ) (ρ1c1 + ρ2c2 ) is the Fresnel reflection’s coefficient for the plane wave’s reflection from the boundary between the transparent medium and the radiating one, and ρ1,2 and c1,2 are the densities and the velocities of sound in these media. The case of ρ1c1 ≠ ρ2c2 was not considered in [8], but the corre sponding generalization can be easily performed with the use of the transfer theory (see, e.g., [9]). However, there are a number of effects that are not included in Eq. (1) but may considerably affect the results of acoustothermometry. First, near the surface of an absorbing medium, in addition to the wave field, a quasistatic thermal field (near field) is present [10], which cannot be described by the ray theory used to derive Eq. (1). Interacting with the receiving system, this field may affect the results of acoustic measure ments. Second, the transfer equation is inapplicable in the case of a strong absorption [11], and, hence, the contribution of strongly attenuating waves may also be inadequately described by the ray theory.
105
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REZNIK, SUBOCHEV
Measurement of |P|2
ρ1, c1 ≈ ρ, c z' = z
z' = 0
ξ, η ρ, c
Heated absorbing half space
ξ1, η1 = 0
T(z')
r
However, the effects related to strong absorption and presence of the QTF were not considered in the cited papers.
z' Fig. 1. Geometry of the problem.
The existence of the quasistatic thermal field (QTF) was first predicted by S.M. Rytov for electro magnetic radiation as the consequence of his theory of thermal fluctuations in electrodynamics [12]. This field is characterized by the absence of energy transfer in free space and by a sharp decrease in the mean energy’s density with distance from the surface of the radiating body. In [13], it was shown that, if an antenna with a small wave size (i.e., with the size of its aperture being R λ, where λ is the wavelength) is introduced in the region of the QTF localization, a QTFinduced energy flux exceeding the wave field’s contribution appears in the reception channel. An experimental observation of the electromagnetic QTF was reported in [14]. In addition, in [13, 14], it was demonstrated that QTF measurements can make radiometric data more informative. Studies of the aforementioned effects have become possible owing to the inclusion of the QTF in consideration in [13, 14], which resulted in a qualitative transformation of Eq. (1). Radiometry of strongly absorbing media was also first implemented in electrodynamics [15]. In this study, Eq. (1) was modified: it acquired the effective absorption coefficient, γ → γeff, which changes to the ordinary coefficient γ from the transfer equation when the absorption in the radiating medium is weak. Since the problems of describing thermal acoustic fields are close to similar problems of electrodynam ics, it is of interest to consider the corresponding effects in acoustics. To study the aforementioned effects, it is necessary to use an approach that is more general than the transfer theory and is based on the application of the fluctuationdissipative theorem to hydrodynamic equations. This approach was earlier applied to hydrodynamic equations in [16–19] to con struct the theory of correlation acoustothermometry.
2. THE THEORY OF HYDRODYNAMIC FLUCTUATIONS Let us consider the problem of the thermal acoustic radiation of a homogeneous halfspace z' > 0 filled with a waterlike absorbing medium. The geometry of the problem is shown in Fig. 1, where T ( z ') is the tem perature distribution in the radiating medium and ξ ( f ) , η( f ) are the bulk and shear viscosities. A piezo electric transducer is positioned above the surface of the medium under testing and performs measure ments of noise pressure in the plane z ' = z . The region z' < 0 is filled with an immersion liquid, which is acoustically transparent ((ξ1, η1 = 0) ) and has a density ρ1 and sound velocity с1 coinciding with the corre sponding parameters ρ and с of the radiating medium (in this case, in Eq. (1), we have Rray = 0 ). The theory of acoustic fluctuations in an absorb ing medium is based on linearized hydrodynamic equations in terms of complex amplitudes with noise
forces F . From these equations, after elimination of velocity and density, an equation for the fluctuation pressure Р was derived [18, 19]: ΔP +
2
|| k0 P = div F , a a
(2)
where a = 1 + iε , ε = k0b ρ c , k0 = ω c , ω is the cyclic frequency, and b = ξ + 4 3 η is the effective viscosity. According to Eq. (2), the source of fluctuation of the thermal pressure is only the longitudinal (potential) component of the total noise’s force F = F ⊥ + F ||, || where div F ⊥ = 0 and curl F = 0 . Equation (2) yields the following expression for the pressure [19]: ∞
∫
∞
∫ (
)
|| P( r , z ) = − d r ' dz ' F ( r ', z ) ∇ ' G ( r − r ', z , z ') . (3) −∞
2
0
Here, the Green function G represents the pressure field produced in the absorbing medium by a point force acting in the transparent medium. This function can be expanded in plane waves [20]. With allowance for the boundary between the media (see Fig. 1), this expansion has the form ACOUSTICAL PHYSICS
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G(r − r ', z, z ') i exp ⎛⎜ iκ ' ( r − r ') − iz k02 − κ '2 ⎞⎟ ⎝ ⎠ = W ( κ') 2 2 2π k0 − κ ' −∞ ∞
∫
(4)
⎛ ⎞ k2 × exp ⎜⎜ iz ' 0 − κ '2 ⎟⎟ d 2κ' , a ⎝ ⎠ where the coefficient of transmission from the trans parent medium to the absorbing one differs from unity because of the difference in viscosities:
where the subscripts m, n = {x, y , z} indicate different projections on the coordinate axes. Using Eq. (6), we proceed from Eq. (8) to the expression for the correla tion functions of the twodimensional spectral com ponents of random forces:
Fm||(κ, z ')F n||*(κ ', z '')
= 4πbT(z ')Amnδ ( κ − κ ') δ ( z ' − z '') ,
(5) . k02 2 2 ' ' −κ + −κ a Formulas (3)–(5) do not take into account the Kon stantinov absorption’s effects, which, for the case under study (ρ1 = ρ) are insignificant [21]. We introduce the spatial spectrum of random forces by using the Fourier transform in the transverse coor dinate r:
k02
Amn
∫ F (r, z)exp (−iκr) d r.
F (κ, z) =
||
2
∞
∞
P( r , z )
0
= − ⎡⎢ ∂ G(κ, z, z ')⎤⎥ δ ( z ' − z '') dz '. ⎣∂z ' ⎦
∫
∫ d κd κ ' G*(κ ', z, z '') 2
P (z)
−∞
⎛ × G(κ, z, z ')exp [ir ( κ − κ ')] ⎜ Fr(κ, z ')κ − Fz(κ, z ') (7) ⎝ *⎞ ⎞⎛ k2 k2 × 0 − κ 2 ⎟⎟ ⎜ Fr*(κ ', z '')κ ' − Fz*(κ ', z '') 0 − κ '2 ⎟ , ⎟ a a ⎠ ⎜⎝ ⎠ where
(
exp −iz k0 − κ 2
k0 − κ 2
2
)
2
2 ⎛ k 2⎞ × exp ⎜⎜ iz ' 0 − κ ⎟⎟ . a ⎝ ⎠ Applying the fluctuationdissipative theorem (see, e.g., [18]), we obtain the expression for the correlation functions of random forces:
Fm||(r, z ')F n||*(r ', z '')
(
(8)
)
= T(z ')bπ −1 ∇'m∇''n δ ( r − r ') δ ( z ' − z '') , ACOUSTICAL PHYSICS
0
Then, after simplification, we obtain 2
G(κ, z, z ') = W ( κ)
⎤
∞
∞
∫
⎡∂
0
Using Eqs. (3)–(6), we write the expression for the mean square of pressure (the asterisk denotes complex conjugation):
= 1 2 dz ' dz '' (2π)
iκ x ∂ ⎞⎟ ∂z '' ⎟ iκ y ∂ ⎟. ∂z '' ⎟ ∂ ∂ ⎟ ⎟ ∂z ' ∂z '' ⎠
∫ ⎢⎣∂z ' δ ( z '− z '')⎥⎦ G(κ, z, z ') dz '
(6)
−∞
2
⎛ κ κ' κ xκ 'y ⎜ x x ⎜ κ yκ 'y = ⎜ κ yκ 'x ⎜ ⎜ ∂ ∂ ⎜ − iκ 'x − iκ 'y ∂z ' ⎝ ∂z '
Substituting Eq. (9) in Eq. (7), we take into account that
∞ ||
(9)
where the matrix elements Аmn have the form
2 k02 − κ '2
W ( κ') =
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2
=
ρω2γ 2 πc a
∞
∞
∫
(10a)
∫
× G(κ, z ', z) PS ( κR) κd κ T ( z ') dz ', 2
2
0
0
2 ⎛ k 2⎞ exp ⎜⎜ z '2Im 0 − κ ⎟⎟ a 2 2 ⎝ ⎠ G(κ, z ', z) = W ( κ) 2 2 2 k0 − κ
(10b)
1, κ ≤ k0 ⎫ ⎧⎪ ⎪ ×⎨ ⎬. 2 2 − > exp 2 z k , k κ κ 0 0⎪ ⎪⎩ ⎭
(
)
In Eq. (10a), γ = ω2b/ (ρc 3) is the power absorption’s coefficient for a plane homogeneous wave in the case of a small loss ε 1 [22]. Precisely this coefficient γ appears in the transfer equation whose solution is 2 described by formula (1). The factor PS ( κR) is intro duced in Eq. (10a) to allow for the finite size of the piezoelectric transducer, which is assumed to be a disk of radius R positioned in the plane z ' = z < 0. The
108
REZNIK, SUBOCHEV
transfer function of such a piezoelectric transducer, when normalized by its area, has the form 2π
Ps ( κR) =
R
∫ dϕ∫ exp (iκr cos ϕ) rdr 0
0
2π
R
∫ dϕ∫ rdr 0
=
2J 1 ( κR) . κR
(11)
0
2
The function PS( κ R ) noticeably differs from zero only in the region of spatial frequencies κR < π. We also take into account that, for transfer function (11), the integrand involved in Eq. (10a) depends on κ = κ alone. Then, the integral over the space κ in Eq. (10a) is reduced to a single integral over dκ. The more general expression (10) allows us to ana lyze several additional effects that are not taken into account by Eq. (1). These are the effects of a strong absorption, the radiation pattern, and the near field. Below, we consider them in more detail. 3. THERMAL ACOUSTIC EMISSION OF A TEMPERATURESTRATIFIED MEDIUM 3.1. Effects of the QuasiStatic Field, Strong Absorption, and Radiation Pattern The inner integral in Eq. (10a) consists of two parts. The region κ ≤ k0 corresponds to wave fields, whereas the region κ > k0 corresponds to near fields. The inte grand in Eq. (10a) tends to zero in the region κ > k0 if one of the two conditions is satisfied: either R λ or z λ, where λ = 2π/k0. The first of these conditions means filtering of spatial frequencies κ > k0 by the piezoelectric transducer characterized by transfer function (11). The second condition restricts the con tribution of the quasistatic field by the decrease in its power with distance from the surface of the radiating medium, which is described by the factor enclosed in brackets in Eq. (10b). If the measurements are per formed by a small piezoelectric transducer with the size R λ, and if this transducer is positioned fairly close to the radiating surface, i.e., at z λ, the inte grand in Eq. (10a) is nonzero in the region κ > k0 . In this case, one should expect a noticeable effect of the near field on the results of measurements. Relations (10) are valid for arbitrary acoustic absorption, because they are derived from the hydro dynamic theory where no limitations are imposed on the magnitude of absorption. Unlike Eq. (10), Eq. (1) is derived from the transfer equation, which is only valid for a weak absorption ε 1 [11]. To go beyond the limits of the transfer theory and to solve the direct acoustothermometry problem for the case of an arbi trary loss in the medium is possible using the method of an auxiliary plane wave. This method was applied by V.I. Passechnik in [23, 24]. Below, we study in detail
the conditions of the transition from Eq. (10) to Eq. (1) and give quantitative estimates of the effect of a strong absorption in comparison with the results obtained from the transfer theory (this was not done in the cited publications). In addition, Eq. (10) takes into account the effects related to the nonzero beam’s width of the receiving antenna, which in our case is represented by the sur face of the piezoelectric transducer. These effects manifest themselves for R < λ and are not described by Eq. (1). However, it should be noted that the effects of the radiation pattern can also be taken into account within the ray theory. From the transfer equations and the boundary conditions at the surface z ' = 0, it is pos sible to derive the thermal radiation’s intensity of the halfspace as a function of the observation angle. Inte grating this expression in combination with the radia tion pattern over the angular variables, we arrive at the desired generalization of Eq. (1). Such procedures were performed in radiometric studies (see, e.g., [25]). For acoustothermometry, the corresponding expres sion will be derived below from Eq. (10). The above speculations suggest that general for mula (10) should yield particular results of the ray the ory for ε 1 and R λ or for ε 1 and z λ. Let us consider more closely this issue. We transform the Eq. (10a) via introducing the acoustic brightness tem perature, which is determined as the thermodynamic temperature of a uniformly heated blackbody provid ing the reception of the same power as that received from the object under study: 0
∫ Ψ ( z ', z, R)T ( z ') dz ',
Ta ( z ) =
(12)
n
−∞
where ∞
Ψ n ( z ', z, R) = ∞
∫ G(κ, z ', z) 0
2
PS ( κR) κd κ 2
,
∞
∫ dz '∫ G(κ, z ', z) 0
2
(13)
PS ( κR) κd κ 2
0
2
and G(κ, z ', z) is given by Eq. (10b). In the case of R λ or z λ, we can set the limit of integration over κ in Eqs. (10a) and (13) to be k0, because, under these conditions, the QTF contribution that is determined by the region of integration κ > k0 is vanishingly small. Then, we change the integration variable in Eq. (13): κ = k0 sin θ , d κ = k0 cos θ dθ , where 0 ≤ θ ≤ π/2. We also take into account that
(
W ( κ) = 1 − R0 2
2
)
a
2
k0 − κ k2 2 2Im 0 − κ . (14) k 0γ a 2
ACOUSTICAL PHYSICS
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Here, R0( κ) =
k02 − κ 2 − k02/ a − κ 2
k02 − κ 2 + k02/ a − κ 2 transformation, we obtain
. After some
∞
109
ε 1.5
1.0
∫
Ta = Ψ p(z ', R)T ( z ') dz ',
(15)
3
2
0
0.5
where
1
π /2
∫ (1 − R (θ) ) 2
Ψ p(z ', R) =
0
(16)
0
× sin θ Φ (θ , R)γ eff(θ )exp(γ eff(θ )z ')dθ is the kernel of integral equation (15); −1
(17) R0 (θ) = cos θ − a −1 − sin 2 θ cos θ + a − sin θ is the Fresnel coefficient of reflection of a plane wave from the boundary between the absorbing and nonab sorbing media; 2
PS ( k0R sin θ)
Φ (θ) = π /2
∫ P (k R sin θ) S
0
2
2
sin θdθ
0
is the radiation pattern of the receiving antenna (the piezoelectric transducer) with normalization chosen π /2
so as to satisfy the condition
∫ Φ (θ)sin θdθ = 1; and 0
(1 + ε ) cos θ − iε − ε 2
γ eff (θ) = 2k0 Im
2
2
(18) 1 + ε2 is the effective absorption coefficient of the absorbing medium, which is obtained as a result of the transfor mation of the factor 2Im k0 2a −1 − κ 2 involved in Eqs. (10b) and (14) after the change of variable κ = k0 sin θ with allowance for the relation a = 1 + iε . We note that Eq. (16) fully coincides with its elec trodynamic analog [15] with the only difference that γeff(θ) has the form of Eq. (18) specific to the hydrody namic problem under consideration. The passage to the results of the ray theory occurs when ε 1 . In this case, instead of Eq. (18), we have (19) γ eff (θ) ≈ −γ cos −1 θ, i.e., we obtain the absorption coefficient γ appearing in the transfer equation and characterizing the absorp tion of the power of a plane homogeneous wave prop agating in a homogeneous medium at an angle θ to the z axis. Thus, if ε 1 , Eqs. (15) and (16) represent the generalization of Eq. (1) to the case where the antenna beam’s width is nonzero. If R λ, it is possible to apply the approximation of a narrowbeam antenna. ACOUSTICAL PHYSICS
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0
20
40
60
80 100 b, kg/(m s)
Fig. 2. Dissipation parameter ε = k 0b (ρ c ) versus the effec tive viscosity b . The operating frequencies of the acousto thermograph are (1) 1.5, (2) 3, and (3) 5 MHz.
In this case, Eqs. (15) and (16) are transformed to 2 2 Eq. (1) correct to the change Rray → R0 , where R0 = R0(0) is the complex reflection’s coefficient (17) of a plane wave at normal incidence θ = 0 . We note that, when ε 1 , from Eq. (17) we obtain R0 ~ ε2, which allows us to set R0 = Rray = 0 in Eq. (16) (as in Eq. (1)) for the particular case under consideration: ρ1 = ρ and c1 = c . Hence, in the limit of a small dissipative loss and a narrowbeam receiving antenna, Eq. (15) fully coincides with Eq. (1). It should be noted that, unlike the more general formula (12), in Eq. (15) Та is not a function of the antenna height z. The corresponding dependence is a consequence of the effect of the QTF on Та, which is taken into account in Eq. (12) but is absent in Eq. (15). From Eq. (12) with allowance for normalization (13), we readily see that Ta = Ta(z) ≠ const only for an inho mogeneous temperature profile of the radiating medium: T(z) ≠ const. Interpretation of this effect is given below. 3.2. Results of Numerical Analysis We now analyze the validity of the assumptions made in passing from Eq. (12) to Eq. (1) for the actual parameters of media and antennas used in medical biological studies. 3.2.1. The effects of strong absorption. We begin with the condition of weak absorption ε 1 , which allows us in Eq. (15) to use approximate representa tion (19) of the effective absorption coefficient instead of exact formula (18) and also to neglect reflection coefficient (17). Figure 2 shows the results of calculating the dimen sionless dissipation parameter versus the effective vis cosity b. As seen from Fig. 2, the condition ε 1 ceases being satisfied for liquids with the effective viscosity values b > 10 kg/(m s) (this corresponds to ε > 0.2).
110
REZNIK, SUBOCHEV |γ/k0cosθ|, |γeff(θ/k0)| 3.5 3.0 2.5 2.0 3
1.5 1.0 3
2
0.5
1
2 1
0
0.1
0.2
0.3
0.4
0.5 ε
Fig. 3. The ray absorption coefficient γ k 0 cos θ (the dot ted lines) and effective absorption coefficient γ eff (θ ) k 0 (the solid lines) versus the dissipation parameter ε at ρ = 1000 kg/m3 and c = 1500 m/s. The angle of thermal radia tion receiving is (1) θ = 0°, (2) 60°, and (3) 80°.
∫
(20)
0
3 (a)
1.06 2 1.04 1 1.02
10−1
1 (b)
3
1.06
2
1.04
1 1.02
1.00
∞
⎧d ε ⎫ ⎧Ψ ε( z ', R )⎫ ⎨ ⎬ = dz ' z ' ⎨ ⎬, ⎩d p ⎭ ⎩Ψ p( z ', R )⎭
|dp/dε|
1.00 1.08
In this case, the error of the ray theory becomes noticeable for all the thermal radiation’s reception angles θ ≥ 0 (the angle θ is measured with respect to the z axis). However, for flat angles θ, deviations of Eq. (19) from Eq. (18) become noticeable at smaller values of b. The corresponding result is illustrated in Fig. 3, which shows the angular dependences of the absorption coefficients calculated by exact formula (18) and approximate formula (19). In particular, for θ = 80° and b = 10 kg/(m s), the function γeff(θ) differs from –γ/cosθ by a factor of two. Such values of effec tive viscosity are typical of glycerin and gelatin solu tions [26, 27], which are often used to imitate biologi cal tissues in laboratory experiments. Thus, a weak absorption approximation (19) may lead to inaccurate retrieval of the temperature profile by acoustother mometry. Let us estimate the error in temperature determi nation for viscous media. For this purpose, we intro duce the parameter characterizing the probing depth of an acoustothermograph:
10−1
1 R/λ
Fig. 4. Probing depth ratio (the strong absorption effect) versus (a) the radius of a diskshaped transducer and (b) the width of a ring transducer. The operating frequency is 1 MHz. The effective viscosity is b = (1) 2, (2) 5, and (3) 10 kg/(m s).
where the kernel Ψр is given by Eq. (16) and Ψε is determined by the same expression taken at R0 = 0 with the replacement of γeff by –γ/cosθ according to Eq. (19). As it was noted above, the kernel Ψε repre sents the result of solving the transfer equation for matched halfspaces at a nonzero beam width of the receiving antenna. As one can see from the behavior of the quantity d p d ε (Fig. 4a), the effect of strong absorption leads to the situation where the approximate expression for the kernel of Eq. (15), Ψр instead of Ψε, which ignores these effects, underestimates the probing depth. The necessary correction exceeds 2% if the measurements are performed by a transducer with the size R < 0.3λ in a medium with the viscosity b > 2 kg/(m s). In this case, application of Eq. (19) yields a displaced esti mate of the depthaverage temperature of the medium under test. Let us determine the displacement δТ for the linear temperature’s profile T(z) = T0 + αz. For such a profile, one easily obtains δТ = α(dp – dε). For our estimates, we use α = 1°C/cm. Such temperature gradients may occur within a depth of several centime ters in the human body, because the skin temperature on the surface of the body is about 34°С while, in the case of internal inflammation, the temperature of the affected tissue may reach 40–42°С. For the parameters of the medium that were used in calcula tions for Fig. 4a, we have dp/dε ≈ 1.025 at the maxi mum value dp ≈ 4 cm (this corresponds to the case of b = 2 kg/(m s)). Then, we obtain the displacement of the temperature estimate δТ ≈ 0.1°С. Such errors may be significant in acoustothermometry of biolog ical tissues. ACOUSTICAL PHYSICS
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The character of the dependence of dp/dε on R in Fig. 4a is a consequence of the results of calcula tions given in Fig. 3 for ε < 10–1, which corresponds to b < 10 kg/(m s). Indeed, for a narrowbeam antenna (R λ) of the given orientation (see Fig. 1), prob ing is performed in the direction of the z axis, i.e., at θ = 0°. In this case, γeff ≈ γ (see curve 1 in Fig. 3); i.e., the effect of strong absorption does not manifest itself for such values of R. For a broadbeam antenna (R λ), according to definition (20), the quantity d is a certain weighted averaged value over all the observa tion angles 0° < θ < 90°. For angles θ > 60°, a consid erable deviation of γeff(θ) from γ/cosθ takes place (see Fig. 3), which leads to the growth of the ratio dp/dε in the region R < 0.5λ. Evidently, more significant displacements occur for the estimates of the depthaverage temperature of viscous liquids when probing is performed by antennas whose radiation patterns have higher sidelobe levels, as compared to diskshaped piezoelectric transducers. These are, e.g., ring antennas, which are used when it is necessary to focus the sound field in the direction of the z axis. Figure 4b shows the results of calculating dp/dε as a function of R for a ring with the inner diam eter R1 = 3 R and outer diameter R2 = 4 R , so that R characterizes the transverse size of the working area of the antenna at the fixed ratio R1/ R2 = 0.75. The trans fer function of such an antenna has the form PSring ( κR) = [8J 1 (4κR) − 6J 1 (3κR)] (7κR) . From Fig. 4b it follows that, for a liquid with the viscosity b = 2 kg/(m s), the error δТ ≈ 0.1°С (corresponds to dp/dε ≈ 1.025) occurs already for R < 2λ. The effects of strong absorption considered here may also be significant for acoustothermometry of biological media on the basis of the receiving of trans verse thermal waves, which are caused by shear forces arising in such media [21]. To study these waves, it is necessary to consider a more general system of hydro dynamic equations taking into account the forces and the displacements in the direction that is transverse with respect to the wave vector. 3.2.2. Effects of the near field and the radiation pat tern. Now, let us proceed to considering the effects caused by the QTF and the radiation pattern. These effects are most significant in the case of using small size piezoelectric transducers with R < λ. The use of smallsize antennas is especially expedient in acoustic thermotomography [6], because it allows one to increase the amount of data by placing a greater num ber of piezoelectric elements within the same area. Modern technical means allow the production of piezoelectric receivers with the operating frequency band within 1–5 MHz and a diameter of up to 0.25 mm. Such receivers are small for frequencies below 1 MHz (R/λ < 0.2). In general, the antenna temperature Та is deter mined by integral equation (12), whose kernel Ψn(z', z, R) ACOUSTICAL PHYSICS
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is given by Eq. (13) with allowance for Eq. (10b). As we have shown in Section 3.1, the QTF should be neglected in the case of passing to the limit: Ψ p(z ', R) = Ψ n(z ', z → ∞, R) . In this case, Eq. (12) is transformed to Eq. (15) with kernel (16), which takes into account the effect of the radiation pattern. We note that, here, we are not interested in the effects of strong absorption, i.e., we consider the situation where, ε 1, we have R0(θ) 1 and γeff(θ) ≈ –γ/cosθ. Then, the kernel Ψp(z', R) takes the form
Ψ p(z ', R, ε 1) π /2
=
∫ dθ sin θΦ(θ, R)(cos θ) exp (− cos θ). γ
γz '
(21)
0
Finally, if we neglect the effects of both the QTF and radiation pattern in Eq. (12), after the passage to the limit Ψ 0(z ') = Ψ n(z ', z → ∞, R → ∞), we obtain (22) Ψ 0(z ') = γ exp(−γ z '), where the function Ψ0(z') represents the kernel of Eq. (1) at Rray = 0. By analogy with Eq. (20), we introduce the effec tive probing depths for Та calculated from Eqs. (1), (12), and (15):
⎧d 0 ⎫ ∞ ⎧ Ψ 0( z ') ⎫ ⎪ ⎪ ⎪ ⎪ (23) ⎨d p ⎬ = dz ' z ' ⎨ Ψ p( z ', R ) ⎬. ⎪⎩d n ⎪⎭ 0 ⎪⎩Ψ n(z ', z , R)⎪⎭ Since the function Ψn given by Eq. (13) is obtained without imposing any limitations on z and R, the parameter dn determines the probing depth for all the possible values of the height and size of the antenna. Under the limiting conditions determined above for the functions Ψn, p,0 obtain the transitions: dn → dp and dn → d0. The character of these transitions can be determined by using the results of calculations from Fig. 5, which shows the dependences dn(R)/dp(R), dn(R)/d0, and dp(R)/d0 for several fixed values of z/λ. The calculations were performed for ω/2π = 1 MHz, ρ = 1000 kg/m3, c = 1500 m/s, and b = 2 kg/(m s). From Fig. 5, one can see that the effects of the radi ation pattern are significant for antennas with the size R/λ < 1 irrespective of their distance to the radiating surface z/λ. At the same time, for R/λ < 0.1, we have an isotropic radiation pattern, for which dp/d0 ≈ 0.5 (Fig. 5c). If the conditions R/λ < 1 and z/λ < 1 are sat isfied simultaneously, it is important to take into account the effect of the QTF, which is greater the smaller the parameters R/λ and z/λ are. In this case, we have dn/d0 < dp/d0; in addition, for z = 0, we have the asymptotics dn → 0 at R → 0 (curve 1 in Figs. 5a and 5b), which is caused by the effect of the QTF alone; i.e., it cannot be obtained from Eqs. (1) or (15) ignoring this effect. Thus, the effect of the QTF and radiation pattern is reduced to the fact that Eq. (1) ignoring these effects
∫
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overestimates the probing depth of the acoustother mograph. As is seen from Fig. 5b, the difference in the depths is greater than threefold if a transducer with the radius R ≤ 0.1λ is positioned at the distance z ≤ 0.1λ from the radiating surface. The result of applying Eq. (1) in this case is a displacement of the depth average temperature of the medium under study. Let us determine the displacement δТ for the linear tem perature’s profile T(z) = T0 + αz when δТ = α(d0 – dn). For the parameters of the medium that were used in Fig. 5, we obtain d0 ≈ 4 cm. Taking α = 1°C/cm, for dn/d0 ≤ 0.975 we obtain δТ > 0.1°С, which exceeds the errors allowed in medical diagnostics for determina tion of the internal temperature of the body. Sum marizing the aforesaid, we represent in Fig. 6 the z/λ – R/λ plane separated into regions of values so that, with the accuracy δТ < 0.1°С at the chosen value of the temperature gradient α, in some of these regions it is possible to use the approximate descriptions of the kernel Ψ given by Eqs. (21) and (22), whereas in other regions it is necessary to use more general for mulas (13) and (10b). These regions are indicated in Fig. 6 by white, light grey, and dark grey colors, respectively. The complex shape of the boundaries separating these regions is caused by the oscillatory nature of the functions shown in Fig. 5, which, in its turn, is caused by the behavior of instrument func tion (11) in the region κR > π. From Fig. 6, one can see that, at z = 0 (this geometry is common to acous tothermometry), the effects of the QTF and radiation pattern should be taken into account for transducers with R λ < 4 .
|dn/dp| 1.0 3 0.8 0.6 2 0.4 1
(a)
0.2 0 −2 |dn/d0| 10 1.0
10−1
1
10
0.8 0.6 3 0.4 2
(b)
0.2 1 0 −2 |dp/d0| 10 1.0
10−1
1
10
4. NEARFIELD ACOUSTIC THERMOMETRY 0.8
The effects considered in this paper suggest new possibilities for acoustic tomography of radiating media. In application to the problem under consider ation, the tomographic methods consist in retrieval of the profile T(z) from acoustothermometric data. Such methods are based on the inversion of Eq. (1) [3–8]. The fundamental possibility of reconstructing T(z) is determined by the fact that Eq. (1) relates the mea sured acoustic brightness temperature Та to the tem perature of the medium that is averaged with a certain weight over the depth within 0 < z < d0. Controlling the maximal depth of averaging, it is possible to obtain information on the distribution T(z) at different depths. The retrieval of the T(z) profile from Eq. (1) is performed using the data of measuring Та for several different values of the parameter d0. Under the condi tions where the QTF effects are insignificant, i.e., Eq. (15) with kernel (16) or (21) is applicable; the main method of controlling the parameter d0 is the use of focusing acoustic systems [28, 29]. If we measure the thermal field by a piezoelectric transducer that is small compared to λ and is positioned near the radiat ing surface or in contact with it, the acoustic bright
0.6 1–3 0.4 (c) 0.2 0
10−2
10−1 R/λ
1
10
Fig. 5. Probing depth ratio versus the radius of a disk shaped antenna with allowance for the effects of the (a) QTF, (b) QTF and radiation pattern, and (c) radiation pattern alone. The height of the antenna above the surface of the medium under test is z/λ = (1) 0, (2) 0.1, and (3) 1.
ness temperature Та will be related to T(z) by Eq. (12). In this case, additional possibilities arise for control ling the probing depth, which now is represented by ACOUSTICAL PHYSICS
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ON THE THEORY OF ACOUSTOTHERMOMETRY OF WATERLIKE MEDIA z/λ 1.5
1.0 1
2
3 0.5
0 0
1
2
3
4
5
6
7 R/λ
Fig. 6. Regions of values of the parameters z/λ and R/λ within which (1) the effects of both radiation pattern and QTF are insignificant, (2) only the finite beam width of the antenna should be taken into account, and (3) both finite beam width and QTF should be taken into account.
the parameter dn depending on z and R according to Eqs. (23) and (13). Varying the parameters z and R, it is possible to vary the probing depth within 0 < dn < d0 (see Fig. 5) and, in this way, to reconstruct the T(z) profile. We note that such methods were used in [14], where the quasistationary component of the thermal electromagnetic field of the radiating medium was measured. Since Та in Eq. (12) represents the average temper ature of the medium in the region 0 < z' < dn(z, R), we obtain the dependence Та = Та(z, R) for T(z') ≠ const that is characterized by Eq. (12). If the QTF effect is ignored, we have dn(z, R) → dp(R); i.e., Та is no more a function of the receiver height z. This also follows from the fact that the thermal wave field’s intensity does not depend on the distance z to the surface of the radiating halfspace [10]. With allowance made for the QTF, the situation changes and we obtain Та = Та(z) for the radiation of a nonuniformly heated medium. 5. CONCLUSIONS Thus, in this paper, on the basis of the theory of hydrodynamic thermal fluctuations, we derived an integral equation relating the acoustic brightness tem perature measured by the acoustothermograph to the onedimensional temperature profile of the radiating medium. This equation is valid for an antenna of arbi trary size and describes the new effects of acoustother mometry: the effects of the near acoustic fields, radia tion pattern of the antenna, and strong absorption in ACOUSTICAL PHYSICS
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the medium. We determined the conditions for the size of the antenna and its distance to the radiating surface that allow the passage to the result of the trans fer theory ignoring the aforementioned effects. We showed that the neglect of these effects may cause errors greater than 0.1°С in the results of measure ments of the internal temperature for the medium under testing, such errors being considerable for med ical diagnostics. We proposed a method for recon structing the subsurface temperature profile of the medium on the basis of measuring the near field of its own acoustic radiation. The problem of detecting the QTF, which was for mulated by Rytov in [12], is an interesting and impor tant problem. In [14], experimental data confirming the existence of an electromagnetic QTF were reported. The results of our study show that similar measurements can be carried out in acoustics. For such measurements, a promising object is a tempera turestratified liquid medium. The indication of the presence of the acoustic QTF should be the detection of the effect predicted in this paper: dn < dp ≈ 0.5d0. Evidently, for such measurements, it is expedient to generate the maximal possible temperature gradient α in the medium. In [14], the experiments were per formed with α ≈ 5–8°C/cm, which considerably exceeds the values of α observed in biological media and used in our estimates above. For the aforemen tioned gradient values, the effects of interest can be detected by the commercial ultrasonic transducers with R ≈ 2–3 mm at the sensitivity of the acoustother mograph being δТ ≥ 0.5°С. Electrodynamic studies [14] showed that, in the case of detecting the QTF effects with implementation of the corresponding nearfield methods, an important factor is the efficiency of the electrically small receiving antennas. The problem of the efficiency of acoustic antennas is beyond the scope of this paper, but it requires special investigation because of the evident analogy between the corre sponding acoustic and electrodynamic problems. We only described the general idea of the nearfield acoustothermometry. The prospects of this method should be investigated on the basis of solving the inverse problem of temperature profile retrieval, which is for mulated with the use of Eq. (12). This problem is ill posed and requires special methods for its solution. By computer simulation of the retrieval process, it is pos sible to determine the accuracy of solutions and the requirements on the measuring system. Above, we noted that the shear forces, which are usually neglected in liquids, also serve as sources of thermal acoustic noise. However, the problem is that many biological media cannot be considered as liq uids. Their description should be based on the equa tions of the acoustics of solids, in which the shear forces play an important role being the source of radi ation of a transverse thermal wave. Attenuation of these waves in the medium presumably should lead to the situation where the effects of strong attenuation in
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a liquid that were considered above will be still more significant for solids. REFERENCES 1. V. I. Pasechnik, A. A. Anosov, and M. G. Isrefilov, Int. J. Hyperthermia 15 (2), 123 (1999). 2. V. A. Burov, E. E. Kasatkina, O. D. Rumyantseva, and S. A. Filimonov, Akust. Zh. 49, 167 (2003) [Acoust. Phys. 49, 134 (2003)]. 3. E. V. Krotov, S. Yu. Ksenofontov, A. D. Mansfel’d, A. M. Reiman, A. G. Sanin, and M. B. Prudnikov, Izv. Vyssh. Uchebn. Zaved., Ser. Radiofiz. 42, 479 (1999). 4. A. A. Anosov and V. I. Pasechnik, Akust. Zh. 40, 885 (1994) [Acoust. Phys. 40, 781 (1994)]. 5. A. A. Anosov, K. M. Bograchev, and V. I. Pasechnik, Akust. Zh. 44, 299 (1998) [Acoust. Phys. 44, 248 (1998)]. 6. Yu. V. Gulyaev, K. M. Bograchev, I. P. Borovikov, Yu. B. Obukhov, and V. I. Pasechnik, Radiotekh. Elek tron. (Moscow) 43 (9), 140 (1998). 7. E. V. Krotov, M. V. Zhadobov, and A. M. Reyman, Appl. Phys. Lett. 81, 3918 (2002). 8. V. I. Babii, Mor. Gidrofiz. Issled. 65, 189 (1974). 9. A. Isimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978; Mir, Moscow, 1981), Vol. 1. 10. S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Intro duction to Statistical Radio Physics (Nauka, Moscow, 1978), Vol. 2 [in Russian]. 11. L. A. Apresyan and Yu. A. Kravtsov, Theory of Emission Transfer (Nauka, Moscow, 1983) [in Russian]. 12. S. M. Rytov, Theory of Electrical Fluctuations and Ther mal Emission (Akad. Nauk SSSR, Moscow, 1953) [in Russian]. 13. A. N. Reznik and N. V. Yurasova, Izv. Vyssh. Uchebn. Zaved., Ser. Radiofiz. 19, 1039 (2001).
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Translated by E. Golyamina
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