## 3 Results and Discussions
### Discussion  
In previous studies, shifts between the predicted and observed focal regions have been frequently reported in HIFU-related studies. Usually, the shift was measured to be around 1-3 mm for 1-MHz HIFU transducers [24], 4-5 mm for \(\sim\)2.2-MHz excitation [51], and an empirical formula was also used to calculate the focus shift [22]. Although different possible mechanisms have been proposed and a series of correlation studies have been carried out [21; 23; 24; 25; 26; 27], detailed theoretical proof that could quantitatively explain the inherent mechanisms of the observed focal shifts is still lacking. The proposed model modification here clarifies how the acoustic dispersion played a role in the complicated physics of this problem.  
The experiments carried out here have demonstrated that the observed focal shift should come from the dispersive behavior of sound velocity and attenuation in media. For different harmonic components in the HIFU beam, their sound velocities could be different at the gel/water interface. Speculating from [PERSON]'s law, these different components could actually propagate along slightly different paths in the phantom, causing the focal point to shift its location. That is also why the main difference between the two models is observed for the harmonics in Figures 4 and 5--because the sound velocity of the harmonics deviated further away as their frequencies were higher. During this process, the attenuation should also have contributed in a dispersive way. However, the influence of dispersive attenuation could not be separated from that of sound velocity.  
Figure 6: Illustration of the focus shift induced by the presence of phantom sample: (**a**) the theoretical \(-3\) dB area; (**b**) the measured \(-3\) dB area.  
Despite the general recognition that the Westervelt equation and SBE provide higher precision in non-dispersive sound field prediction than KZK, difficulties in predicting accurate sound field distribution in strong dispersion media still remains a great challenge [22]. Moreover, due to the existing complexity of the Westervelt equation and SBE, adding more modifications to these two equations could be over-whelming and/or time-consuming for clinical applications that require real-time monitoring. Considering that KZK shows a balance between accuracy and computation burden, this paper chose to further modify the KZK equation as a straightforward way to achieve improved simulation of HIFU propagation, especially to work out how the focal-shift could be predicted accurately. In this work, for consistency of the experiment, the mKZK and KZK in ordinary media was first confirmed. Then, when measuring the parameters in the strong dispersion medium, mKZK gave a more accurate prediction of the focal shift. Thus, by using the mKZK equation we can quickly and accurately predict the HIFU field in complex media.  
However, further theoretical studies are still needed since the current model is unable to eradicate all the possible unfavorable factors in predicting the characteristics of HIFU. For example, defocusing effects or shifts in focal position might be caused by the layered tissue effect, where sound speed/attenuation may vary in different tissues layers. The thermal lesion effect, in which the tissue properties change due to the heating of HIFU, could be more difficult to include in the modeling. A full solution could be even more challenging if other possible mechanisms, including acoustic radiation force, acoustic cavitation and inconsistent thermal deposition [27] are also considered.  
The focal shifts observed in phased-array-based HIFU devices are also notable. For instance, a focal shift of about 2-mm was observed along the transducer axis in multiple-layered soft tissues sonicated with a 65-element phased array transducer [22]. To overcome this problem in phased-array HIFU, possible solutions could be obtained by drawing lessons from the underlying mechanisms discussed above.