## 2 Materials and Methods
### Evaluation of Pressure Fields  
As shown in Figure 1, with the model consists of a composite solid with three parallel layers, labeled by regions 1, 2, and 3, respectively. Each layer has a thickness \(L_{j}\), for which the subscript \(j\) denotes the region number. The one-dimensional coordinate system origin is located at the surface of the composite system. As a result, the interfacial layers are positioned at \(x=L_{1}\) and \(L_{1}+L_{2}\). Each layer material is assumed to be isotropic, homogeneous, and linearly elastic. Although the heterogeneities of the complex tissue-implant-bone system are not taken into consideration, these assumptions for biological tissues can provide acceptable results for ultrasonic thermal therapies [21]. For these three layers, the mass density, the longitudinal wave velocity, and the acoustic characteristic impedance are given, respectively, by \(\rho_{j}\), \(v_{j}\), and \(R_{j}\). The incident acoustic wave was considered to be harmonic with an angular frequency \(\omega\) so that the velocity potential \(\Phi_{j}(x,t)\) describing the motion in each region is given by  
\[\Phi_{j}(x,\ t)=\text{Re}\big{[}\phi_{j}(x)\exp(-i\omega t)\big{]},\ j=1,\ 2,\ 3, \tag{1}\]  
where Re[ ] signifies the real part of the complex number, \(\phi_{j}(x)\) is the spatial potential function, and \(i\) is the unit imaginary number [22]. Based on linear acoustics, the solutions of the spatial potential function can be solved by the 1-D reduced wave equation, namely  
\[\frac{\partial^{2}\phi_{j}}{\partial x^{2}}+k_{j}^{2}\phi_{j}=0,\ j=1,\ 2,\ 3, \tag{2}\]  
where \(k_{j}=\omega/v_{j}\) is the wave number.  
The acoustic waves, generated by the ultrasound probe, are considered as an infinite train of plane waves horizontally incident from \(x=0\)[23; 24]. For each layer, the pressure fields of the left- or right-going waves can be expressed as follows [25]:  
\[P_{1}=A_{1i}\exp(ik_{1}x)+A_{1r}\exp(-ik_{1}x), \tag{3a}\]  
Figure 1: Schematic layout of a three-layer composite medium subjected to ultrasound waves.  
\[P_{2}=A_{2i}\exp[ik_{2}(x-L_{1})]+A_{2r}\exp[-ik_{2}(x-L_{1})], \tag{3b}\] \[P_{3}=A_{3i}\exp[ik_{3}[x-(L_{1}+L_{2})]], \tag{3c}\]  
where \(A_{1i}\), \(A_{1r}\), \(A_{2i}\), \(A_{2r}\), and \(A_{3i}\) are the unknown amplitude coefficients. The subscripts \(i\) and \(r\) represent the amplitudes of the incident wave and the reflection wave, respectively. It should be noted that multiple reflections appearing in each medium are neglected for simplicity.  
The pressure and velocity continuity conditions must be satisfied on each bilayer interface. This gives rise to a linear system of four equations with four unknowns. Using [PERSON]'s rule and normalizing yields the following explicit forms:  
\[\frac{A_{1r}}{A_{1i}}=\frac{-iR_{2}(R_{1}-R_{3})\cos(k_{2}L_{2})+(R_{2}^{2}-R_{ 1}R_{3})\sin(k_{2}L_{2})}{iR_{2}(R_{1}+R_{3})\cos(k_{2}L_{2})+(R_{2}^{2}+R_{1} R_{3})\sin(k_{2}L_{2})}\exp(2 ik_{1}L_{1}), \tag{4a}\] \[\frac{A_{2i}}{A_{1i}}=\frac{iR_{2}(R_{2}+R_{3})}{iR_{2}(R_{1}+R_{3})\cos(k_{2}L _{2})+(R_{2}^{2}+R_{1}R_{3})\sin(k_{2}L_{2})}\exp[i(k_{1}L_{1}-k_{2}L_{2})],\] (4b) \[\frac{A_{2r}}{A_{1i}}=\frac{R_{2}(R_{2}-R_{3})}{-R_{2}(R_{1}+R_{3})\cos(k_{2}L _{2})+i(R_{2}^{2}+R_{1}R_{3})\sin(k_{2}L_{2})}\exp[i(k_{1}L_{1}+k_{2}L_{2})],\] (4c) \[\frac{A_{3i}}{A_{1i}}=\frac{2 iR_{2}R_{3}}{iR_{2}(R_{1}+R_{3})\cos(k_{2}L_{2})+(R_{2}^{2 }+R_{1}R_{3})\sin(k_{2}L_{2})}\exp(ik_{1}L_{1}), \tag{4d}\]