## 3 Results
### Simulation Results of BFRs  
The BFR is the irreversible coagulation necrosis induced by HIFU exposure. In order to include this concept to numerical simulation, the method of equivalent thermal dose was introduced. Numerous studies had confirmed the application of this method in different tissues and its reliability in clinical tumor hyperthermia [26; 27].  
Simulated BFRs of phantoms after ultrasonication with different combinations of acoustic power and time are shown in Figure 4. In order to explore the relationship between the AFR and BFRs with different combinations of acoustic power and time in the phantom, the shapes of BFRs and exposure energy were calculated when the lengths of BFRs were approximately equal to that of the AFR, as shown in Figure 5.  
For the phantom, the shape of the biological focal region produced by 'high power \(\times\) short time' is closer to that of the physical focal region of the transducer compared with 'low power \(\times\) long time'. For example, in the phantom (Figure 5), when the lengths of BFRs were equal to that of the AFR, the difference between the width of the AFR and that of the BFR was 0.9 mm with an acoustic power of 150 W, and the difference was 2.05 mm with an acoustic power of 50 W. The axis length ratio of the BFR was 7.9 with an acoustic power of 150 W and was 6.3 with an acoustic power of 50 W. It can be seen from Figure 4 that when the lengths of the BFRs were approximately equal to that of the AFR, the exposure time decreased with the increase in acoustic power, and the energy of the BFR was less at a higher acoustic power than that at a lower acoustic power.  
Figure 4: Simulated BFRs of phantom after ultrasonication with different combinations of acoustic power and time: (**a**) 50 W \(\times\) 25.3 s, (**b**) 100 W \(\times\)10.6 s, (**c**) 150 W \(\times\) 6.3 s.  
Figure 3: The simulation distribution of (**a**) peak positive pressure and (**b**) acoustic focal region in the r-z plane generated by the transducer.  
Simulated BFRs of bovine liver after ultrasonication with different combinations of acoustic power and time are shown in Figure 6. The shapes of biological focal regions and exposure energy in bovine liver are described in Figure 7. The shape of the BFR induced by 'high power \(\times\) short time' exposure was closer to that of AFR than the shape of the BFR induced by 'low power \(\times\) long time' exposure when the lengths of these BFRs were approximately equal. As the power increased from 50 W to 150 W, the exposure time required to produce BFRs of the same length decreased from 60.2 s to 12.8 s, and the exposure energy decreased from 3008 J to 1918 J.  
Figure 5: (**a**) Lengths, (**b**) widths, (**c**) axis length ratios of the BFRs, and (**d**) energy after ultrasonication with different combinations of acoustic power and time in a phantom simulation.  
Figure 6: Simulated BFRs of bovine liver after ultrasonication with different combinations of acoustic power and time: (**a**) 50 W \(\times\) 60.2 s, (**b**) 100 W \(\times\)21.1 s, (**c**) 150 W \(\times\) 12.8 s.  
Figure 7: (**a**) Lengths (**b**) widths (**c**) axis length ratios of the BFRs, and (**d**) energy after ultrasonication with different combinations of acoustic power and time in bovine liver simulation.