## 3 Results and Discussions
### Ultrasound Operation Frequency  
The hydrogel-implant-bone composite specimen in the literature [12] was selected as the benchmark example. The transparent hydrogel layer has a thickness of 15 mm. The metal implant, made of 316 stainless steel, has a thickness of 2 mm. The average thickness of the flat bovine bone is about 10 mm. The material properties of the hydrogel phantom, metal implant, and bone used in the analytical solutions and numerical simulations are listed in Table 1.  
In addition to the proposed analytical solution, two-dimensional finite element models were also created to simulate the in vitro experiments using the commercial finite element package Abaqus [29]. A two-dimensional finite-element model was constructed using quadratic thermal elements (D2 TH8 in Abaqus) with an element size of \(1\times 10^{-4}\) m. The dimensions of the hydrogel-implant-bone system are shown in Figure 2. The front surface of the composite system was set to a constant initial temperature \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Hydrogel Phantom** & **316 Stainless Steel** & **Bone** \\ \hline Density (kg/m\({}^{3}\)) & 1190 & 8000 & 1975 \\ Specific heat (J/kg\({}^{\circ}\)C) & 3431 & 502 & 1313 \\ Sound velocity (m/s) & 1512 & 5600 & 3476 \\ Thermal conductivity (W/m\({}^{\circ}\)C) & 0.6 & 16.27 & 0.32 \\ Attenuation coefficient (dB/m) & 54 & 110 & 690 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties of the N-isopropyl acrylamide (NIPAM)-based hydrogel phantom, implants, and bone [12]. while the other three outer surfaces were set as adiabatic. The area beneath the ultrasound probe was the heating region with the heat source given by Equation (6). The time increment was 0.5 s in the thermal transient analyses. Figures 3 and 4 show the temperature distribution along the acoustic axis at 30 s during the ultrasound diathermy corresponding to 1 MHz and 3 MHz, respectively. For an operational ultrasound frequency of 1 MHz, the peak temperature occurs in the hydrogel phantom of the bio-composite system. Alternatively, the peak temperature occurs in the bone of the bio-composite system for an operational frequency of 3 MHz. In addition, the peak temperature in the hydrogel under 1 MHz is slightly higher than that in the hydrogel under 3 MHz. The peak temperature in the bone under 3 MHz is higher than that in the bone under 1 MHz. As shown in Figures 3 and 4, the analytical results coincide well with the numerical  
Since material and geometric properties of these two composite systems are the same, the differences in temperature distribution in Figures 3 and 4 mainly resulted from the changes in the pressure field under different ultrasound operation frequencies. Figure 5 shows the pressure amplitude distributions along the acoustic axis corresponding to 1 MHz and 3 MHz. One can observe that the pressure amplitude in hydrogel for 1 MHz is higher than that for 3 MHz, which implies that more energy is input into the hydrogel in the case of 1 MHz and the ultrasound of 1 MHz can propagate deeper in tissues. As a result, the temperature in hydrogel in the case of 1 MHz is higher than that for 3 MHz. However, the temperature in the stainless implant for 1 MHz is higher than that for 3 MHz, although the pressure amplitude in the implant for 1 MHz is lower than that for 3 MHz. This result is likely because the temperature in the medium depends not only on the input energy from the heat source but also on the heat conduction from the neighboring medium (hydrogel). Finally, since the area enclosed by the pressure amplitude curve in the bone layer for 1 MHz is smaller than that for 3 MHz, the temperature in bone layer for 1 MHz is lower than that for 3 MHz. Figures 6 and 7 represent the temperature distribution, obtained from the analytical solution, in the three-layer composite system in the first five seconds corresponding to the low (1 MHz) and high (3 MHz) ultrasound frequencies, respectively. As observed in Figures 6 and 7, the standing wave pattern occurred in the hydrogel layer.