## 5 Conclusions  
The effects of a circular inclusion on temperature distribution in a tissue-bone composite structure during ultrasound diathermy were investigated in this study. This composite system was simplified as a bilayer system consisting of a soft tissue attached with a bone having a pre-specified curvature. A hybrid computational framework was proposed herein to investigate the effect of double circular inclusions on the temperature distribution during ultrasound diathermy. Numerical results demonstrate that the presence of the inclusion changes both the peak temperature and the temperature distribution when the inclusion is sufficiently large relative to the ultrasonic wavelength. The peak temperature in the composite system decreases when the inclusion is closer to the bone layer. The peak temperature of the composite system with a circular HDPE inclusion is lower than with a circular stainless-steel inclusion. Finally, the temperature of the composite system with a circular stainless-steel inclusion with a 3 MHz ultrasound is lower than under a 1 MHz ultrasound.  
Our simulation results indicate that determining the temperature distribution in the human body is a complex challenge. The scattering or focusing of ultrasonic energy is dominated by the geometries of the inclusion and bone layer. The proposed computational framework could also provide fundamental understanding of the thermal therapy and lay the foundation for implant design and clinical strategies in related treatments. Clearly, the proposed hybrid scheme is applicable to inclusions with simple geometric shapes. This is because the near-field acoustic pressure computation relies on analytical solutions derived or those obtained in the literature. The 2-D extended topics cover semicircular inclusions, elliptical inclusions, semi-elliptical inclusions, truncated circular inclusions, truncated elliptical inclusions, multiple inclusions (with above shapes), multilayer composites, coated cavities, and any combination of the above.  
The following supporting information can be downloaded at: [[https://www.mdpi.com/article/10.3390/app12105166/s1](https://www.mdpi.com/article/10.3390/app12105166/s1)]([https://www.mdpi.com/article/10.3390/app12105166/s1](https://www.mdpi.com/article/10.3390/app12105166/s1)), Section S1: Formulation for Pressure Fields; Section S2: Formulation for Temperature Fields; Section S3: Verification; Section S4: Limitations of the Proposed Computational Framework; Figure S1: Temperature history curves and temperature distribution curves; Figure S2: Pressure distribution; Figure S3: Temperature distribution. References [25-30] are cited in the Supplementary Materials.  
[PERSON] organized this work and performed numerical simulations. [PERSON] derived analytical solutions for the pressure fields. [PERSON] and [PERSON] wrote the manuscript. All authors have read and agreed to the published version of the manuscript.  
The authors acknowledge the financial support of the Ministry of Science and Technology (MOST 106-2221-E-033-034-MY3, MOST 109-2224-E-080-001, and MOST 105-2221-E-033-008), Taiwan. This research is also partially supported by the John and Wen Su Term Research Acceleration Fund, and by Su Development.  
Not applicable.  
The data presented in this study are available on request from the corresponding author.  
The authors are grateful to Simutech Solution Corporation and the National Center for High-performance Computing for providing computational resources. Special thanks goes to the support from the Office of Research and Development at Chung Yuan Christian University (Project Nos. 107011041, 107011042, 107011051, 107011052, and 110-CYCU-RG2-14).  
The authors declare no conflict of interest.