## 5 Conclusions  
Some of the inferences that can be drawn from the current study are as follows:  
1. A new HBEM model was used to describe the nonlinear fractional biomechanical interactions in FGA biological tissues.
2. The bioheat governing equation was solved by implementing the LRBFCM and GBEM for obtaining the temperature, and then the poroelastic governing equation was solved using the BEM to calculate the displacement at each time step.
3. An efficient partitioned semi-implicit coupling algorithm was implemented with the GMSS to solve equations arising from the boundary element discretization.
4. The numerical findings were depicted graphically to display the influences of the graded parameter, fractional parameter, and anisotropic property on the bio-thermal stress.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**CAM-Arnoldi**[31; 32; 33]} & \multicolumn{2}{c}{**Regularized**[34; 35]} & \multicolumn{2}{c}{**GMSS**[36; 37]} \\ \cline{3-7}
**Discretization** & **Preconditioning** & **Process Time** & **Number of** **Iterations** & **Process Time** & **Number of** **Iterations** & **Process Time** & **Number of** **Iterations** \\ \hline
1 (32) & 0 & 0.07 & 6 & 0.07 & 6 & 0.07 & 6 \\ \hline
2 (56) & 0 & 0.2 & 10 & 0.2 & 10 & 0.2 & 10 \\
1 & 0.16 & 8 & 0.16 & 8 & 0.16 & 8 \\ \hline
3 (104) & 0 & 0.5 & 13 & 0.64 & 14 & 0.4 & 12 \\
3 (104) & 1 & 0.46 & 10 & 0.6 & 11 & 0.32 & 7 \\
2 & 0.4 & 7 & 0.56 & 9 & 0.28 & 5 \\ \hline
4 (200) & 0 & 2.48 & 14 & 2.84 & 18 & 1.64 & 14 \\
4 (200) & 1 & 2.04 & 12 & 2.62 & 16 & 1.48 & 9 \\
2 & 1.68 & 8 & 1.96 & 11 & 1.24 & 7 \\
3 & 1.46 & 7 & 51.82 & 4 & 0.98 & 3 \\ \hline
5 (392) & 0 & 12.2 & 22 & 14.26 & 28 & 6.85 & 16 \\
1 & 10.06 & 20 & 12.36 & 26 & 5.79 & 14 \\
5 (392) & 2 & 9.38 & 18 & 10.4 & 22 & 4.98 & 12 \\
3 & 8.28 & 14 & 9.48 & 16 & 4.04 & 10 \\
4 & 7.62 & 10 & 8.1 & 14 & 3.64 & 4 \\ \hline
0 & 40.8 & 20 & 46.4 & 26 & 36.5 & 15 \\
1 & 38.5 & 18 & 42.2 & 24 & 32.4 & 13 \\
2 & 36.6 & 16 & 40.3 & 22 & 30.8 & 11 \\
3 & 32.5 & 14 & 36.8 & 16 & 26.2 & 9 \\
4 & 30.4 & 12 & 34.3 & 14 & 20.3 & 5 \\
5 & 28.2 & 8 & 32.9 & 12 & 18.2 & 3 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Processing times and numbers of iterations for the CA-Arnoldi, regularized, and GMSS methods.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **FDM**[40] & **FEM**[41] & **Present** \\ \hline Number of nodes & 62,000 & 60,000 & 64 \\ Number of elements & 14,800 & 12,400 & 32 \\ CPU time (min) & 250 & 240 & 4 \\ Memory (MByte) & 220 & 230 & 1 \\ Disc space (MByte) & 310 & 330 & 0 \\ Accuracy of results (\%) & 2.4 & 2.2 & 1.2 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison of computer resource requirements for the considered computations.  
5. The numerical findings also show the differences between the Fourier, single-phase-lag, and dual-phase-lag bioheat models, and verified the validity, accuracy, and effectiveness of the developed HBEM.
6. The main advantages of the current HBEM model are its generality and simplicity.
7. The numerical findings supported the claim that the proposed method offers more advantages than other domain discretization techniques.  
The findings of this paper contribute to the development of mathematical models that can be applied in bio-thermal engineering applications, such as blood perfusion, temperature measurement during cryosurgery, analysis of microvascular heat transfer, thermal injury, cryotherapy, hyperthermia, and the influence of large blood vessels.  
This research was funded by the Deanship of Scientific Research at Umm Al-Qura University, grant number [22 UQU4340548 DSR07], and the APC was funded by the Deanship of Scientific Research at Umm Al-Qura University.  
Not applicable.  
All data generated or analyzed during this study are included in this published article.  
The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work via grant code: (22 UQU4340548 DSR07).  
The authors declare no conflict of interest.