† Corresponding author. E-mail:

Project supported by the National Basic Research Program of China (Grant Nos. 2012CB921504 and 2011CB707902), the National Natural Science Foundation of China (Grant No. 11274166), the Fundamental Research Funds for the Central Universities, China (Grant No. 020414380001), the Fund from State Key Laboratory of Acoustics, Chinese Academy of Sciences (Grant No. SKLA201401), China Postdoctoral Science Foundation (Grant No. 2013M531313), and the Priority Academic Program Development of Jiangsu Higher Education Institutions and SRF for ROCS, SEM.

We used the spheroidal beam equation to calculate the sound field created by focusing a transducer with a wide aperture angle to obtain the heat deposition, and then we used the Pennes bioheat equation to calculate the temperature field in biological tissue with ribs and to ascertain the effects of rib parameters on the temperature field. The results show that the location and the gap width between the ribs have a great influence on the axial and radial temperature rise of multilayer biological tissue. With a decreasing gap width, the location of the maximum temperature rise moves forward; as the ribs are closer to the transducer surface, the sound energy that passes through the gap between the ribs at the focus decreases, the maximum temperature rise decreases, and the location of the maximum temperature rise moves forward with the ribs.

Biomedical ultrasound is one of the most active fields in modern ultrasonic technology, with ultrasonic diagnosis and treatment now widespread in biomedicine. The main application of biomedical ultrasound in clinical diagnosis is ultrasonic imaging such as B-ultrasound and D-ultrasound. Clinical treatment applications are mainly limited to ultrasonic lithotripsy and high intensity focused ultrasound based on cavitation and heat effects.^{[1]} High intensity focused ultrasound can make a focal area of tissue produce relatively large vibrations which are able to break up stones,^{[2,3]} and it is also an important medical technology in modern cancer treatment. The treatment principle of this technique is to achieve high sound intensity (10^{3} W/cm^{2}–10^{4} W/cm^{2}) by focusing sound energy generated by a large ultrasonic focusing transducer to achieve a high temperature (above 65 °C) in a focal area in a short period of time, causing tumor tissue to undergo thermal coagulation necrosis thus killing cancerous cells.^{[4,5]} High frequency ultrasound has better directivity and most of the sound energy can be focused on a target area without affecting surrounding tissues by accurate control of the ultrasonic focus, therefore it can realize noninvasive treatment.^{[6,7]} In practical applications, a sound wave generated by the transducer will pass through multiple biological media before reaching tumor tissue, and it is difficult to avoid reflection and scattering of sound energy because of the mismatch of acoustic impedance between different media, thus affecting the sound field of the target area, and distorting the sound field accordingly. Particularly in the treatment of liver cancer, the presence of the ribs can greatly influence the sound and temperature field, and ultimately affect the efficacy of the treatment. In early clinical treatments, surgical removal of ribs was often used to prevent reflections of ultrasound energy from affecting the stability of the sound field. Consequently, effective prediction of the distribution of the sound and temperature fields behind the ribs becomes very necessary in order to realize noninvasive treatment. Kamakura *et al.* used an ellipsoidal coordinate system to develop the spheroidal beam equation (SBE) based on the Westervelt equation.^{[8]} The SBE includes nonlinearity, diffraction, and absorption effects, and can be used to predict the sound field of a transducer with a wide aperture angle.^{[8,9]} Li *et al.* used the implicit-prediction correction method to calculate the influence of the ribs on a sound field under a cylindrical coordinate system using the Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation^{[10]} while Liu *et al.* studied its temperature field.^{[11]} Lin *et al*. simulated and obtained the sound field of tissues behind ribs by focusing a transducer with a wide aperture angle under a three-dimensional ellipsoidal coordinate system using the SBE model.^{[12]} Fan *et al.* calculated the nonlinear sound field of a strong focusing transducer with a wide aperture angle using the SBE model,^{[13,14]} while Qian *et al*. investigated the influence of sound source parameters and material characteristics on the nonlinear effect of a concave spherical self-focusing sound field using the SBE model.^{[15]}

Previous articles generally analyze the sound field of a focusing transducer with a small aperture angle.^{[8–10]} Liu *et al.* studied the temperature field caused by weakly focused ultrasound^{[11]} and Lin *et al*. simulated and obtained the sound field of tissues behind ribs by focusing a transducer with a wide aperture angle.^{[12]} However, the innovative point of this paper is that we analyze the temperature field of a focusing transducer with a wide aperture angle on biological tissue containing ribs using the SBE model. In the paper, we first used the SBE to calculate the sound field produced by a focusing transducer with a wide aperture angle to obtain heat deposition, and then solved the Pennes bioheat equation to obtain the resulting temperature field in a biological tissue with ribs and study the effects of rib parameters on the temperature field.

The KZK equation is only suitable for a transducer with a half-aperture angle less than 16.6° due to the use of the parabolic or paraxial approximation, and the SBE equation can be used for a transducer with a wide half-aperture angle. A focusing transducer with a wide aperture angle is a spherical-shell focusing transducer with a spherical half-angle greater than 16.6°.^{[16]} Because such a wide angle focusing transducer has a larger radiating area, stronger focusing effect, smaller focal area radius and better penetrability than that of a small angle focusing transducer, a greater temperature rise can be generated within a tumor while reducing damage to other normal tissues.^{[17]}

Figure *a* is the geometric radius of the radiating surface of the focusing transducer, *α* is the half-angle of the transducer, and 2*b* is the distance between two focal points of the ellipsoidal coordinate system. The horizontal axis represents the *z* axis as the *σ* direction, and the vertical axis represents the *x*–*y* plane. *σ*, *η*, and *φ* are the three coordinates of the ellipsoidal coordinate system, and *σ* = *z* = 0 is the geometric focal point.

As in Fig. *et al.* divided the paraxial sound field of a focusing transducer into a spherical wave area and a planar wave area.^{[8]} In this model, the cut-off point between the two areas is *σ*_{0}.^{[18,19]} The area less than *σ*_{0} is the spherical wave area, which is close to the surface of the focusing transducer and forms a mainly spherical wave. The area larger than *σ*_{0} is the planar wave area and this area is close to the focal point of the focusing transducer such that the planar wave is dominant.

We calculated the sound field under the ellipsoidal coordinate system, and established the conversion relationship between the three coordinates (*σ*, *η*, *ϕ*) of the ellipsoidal coordinate system and the three coordinates (*x*, *y*, *z*) of the rectangular coordinate system:^{[12]}

*σ*< +∞,0 ≤

*η*≤ 1,0 ≤

*ϕ*≤ 2

*π*. To facilitate the numerical calculation, we introduced

*η*= cos

*θ*, where

*θ*is the angle between the

*x*axis and the planar projection in the

*z*–

*z*plane of the connecting line between an observation point in the coordinate system and the focal point.

The Westervelt equation can be written as follows:^{[16]}

*p*is the amplitude of the sound pressure,

*β*is the nonlinear coefficient of the medium,

*ρ*

_{0}is the density of the medium,

*c*

_{0}is the sound velocity of the medium,

*δ*is the diffusion rate of the sound wave, and

*t*is the radiation time.

In the spherical wave area, the retarded time pertinent to the spherical wave is introduced:

*R*is the distance between an observation point and the geometric focus

*O*.

In the planar wave area, the retarded time pertinent to the planar wave is introduced:

*σ*=

*σ*

_{0},

*z*is the axis component of the distance between an observation point and the geometric focus

*O*,

*k*= 2

*πf*/

*c*

_{0}is the wave number, and

*f*is the frequency of the transducer. Substituting Eq. (

*p̅*=

*p*/

*p*

_{0}(

*p*

_{0}is the sound pressure on the source surface) as follows:

^{[12]}

*ɛ*= 1/2

*kb*,

*ω*= 2

*πf*.

Consolidating Eq. (

*E*= (

*σ*

^{2}+ cos

^{2}

*θ*)/1 +

*σ*

^{2}.

The normalized sound pressure is represented as the following Fourier series expansion:

*C*

_{n}is the nth complex amplitude of a harmonic wave, and

*C*

_{n}.

*α*

_{*}is the sound absorption coefficient of the medium when the sound frequency is

*f*

_{*},

*f*

_{0}is the frequency of the source,

*μ*is 2 in water and 1.1–1.3 in biological tissue. Then use the Alternating Direction Implicit Procedure method to solve Eq. (

The heat deposition can be expressed as:

*I*

_{0}=

*p*

^{2}/

*ρ*

_{0}

*c*

_{0}is the sound intensity of the source.

The Pennes bioheat equation^{[20]} can be written as:

*c*

_{t}is the specific heat of a tissue,

*τ*

_{h}is the heating time,

*K*

_{t}is tissue thermal conductivity,

*T*is the temperature of the tissue,

*w*

_{b}is the blood perfusion rate,

*c*

_{b}is the specific heat of blood,

*T*

_{b}is the temperature of blood, and

*Q*

_{m}is the rate of heat generation per tissue unit volume.

Because we study biological tissues *in vitro*, no heat flow from blood circulation exists, so that *w*_{b} can be neglected, *Q*_{m} can also be neglected in dead tissues. So the Pennes equation can be written as:

*T̅*=

*T*−

*T*

_{0},

*T*

_{0}is the temperature before heating.

In the oblate spheroidal coordinate system, the Laplace operator

By substituting Eq. (

*H*,

*K*are the step sizes of the Crank–Nicolson implicit difference method. That is:

*Σ*

_{1}=

*K*

_{t}

*τ*

_{h}/

*b*

^{2}

*ρ*

_{0}

*c*

_{t}

*H*

^{2},

*Σ*

_{2}=

*K*

_{t}

*τ*

_{h}/

*b*

^{2}

*ρ*

_{0}

*c*

_{t}

*K*

^{2}.

The matrix chase-after method (Thomas algorithm) is used to solve Eq. (^{[22]}

(i) In this paper, Tables

(ii) Temperature rise in biological tissue without ribs

In order to guarantee the correctness of the simulation model, we should first compare the simulation results with actual values reported in the literature.^{[22]}

Figure

Figure

Figure *z* = −17 mm, liver-1 at an axial location of *z* = −6.7 mm, and liver-2 at an axial location of *z* = 0.3 mm. From Fig.

(iii) Temperature rise of a biological tissue with two ribs located symmetrically on either side of the axis

Next, we investigated the influences of the initial sound pressure, frequency, half-angle of the transducer, gap width and the location of ribs on the temperature rise of biological tissues under the condition that the half-angle is 17.5°, 23.6°, and 30° respectively. The locations of fat, liver and two ribs are shown in Fig.

Figure

Figure *z* = −23mm, an axial location of liver-1 *z* = −8.8mm and an axial location of liver-2 *z* = 0.3 mm. From Fig.

Figure *μ* = 1.1–1.3). Taking into account the above two factors, no change in the temperature rise of the liver with frequency is evident. The maximum temperature rise of fat increases because the reflective energy induced by the ribs increases when the diffraction effect of the sound field decreases, and the focusing effect is enhanced as the frequency increases, and moves forward slightly.

Figure

Figure

Figure

From the results of our simulation, it can be seen that the influence of ribs on the temperature field in biological tissue is very obvious. The location and the gap width of the ribs both have a great influence on the temperature rise of biological tissue. As two ribs are distributed on either side of the axis, a sound wave can pass through the rib gap, so there is a relatively high temperature rise in the liver. As the gap width between the ribs increases, the sound energy that passes through increases, so the temperature rise of the liver also increases. As the ribs are closer to the focusing transducer, the focusing effect decreases and less sound energy passes through the rib gap, so the temperature rise of the liver decreases. Consequently if we want to achieve a higher temperature rise in the liver, in addition to increasing the initial sound pressure or the half-angle of the focusing transducer, we can also increase the distance between the transducer and the ribs. This work can provide guidance for the practical treatment of liver cancer.

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