## 3 Results and Discussions
### Dispersive Phantom  
To incorporate the phantom model into the study, the acoustic parameters of the phantom was firstly characterized according to procedures described in earlier studies [48; 49]. In brief, the acoustic phase velocity \(c(f)\) inside the phantom material was determined by [48]  
\[c(f)=c_{\mathrm{w}}\left[1+2\frac{\theta_{\mathrm{w}}(f)-\theta_{\mathrm{s}}(f )}{\theta_{2}(f)-\theta_{1}(f)}\right], \tag{17}\]  
where the sound velocity in water \(c_{\mathrm{w}}\) was considered as 1500 m/s. When the sound velocity inside the phantom was measured, the acoustic signals \(p_{\mathrm{s}}\) and \(p_{\mathrm{f}}\) were the acoustic pressure acquired by a transducer before and after the phantom was inserted into the acoustic path. While \(p_{1}\) and \(p_{2}\) were the pressure of the reflected signals from the first and second water/phantom interfaces, respectively, \(\theta_{\mathrm{w}}(f)\), \(\theta_{\mathrm{s}}(f)\), \(\theta_{1}(f)\) and \(\theta_{2}(f)\) were the corresponding phase spectra. The frequency dependence of the attenuation coefficient \(\alpha(f)\) was calculated according to [48; 49]  
\[\alpha(f)=\frac{1}{L_{\mathrm{s}}}\left[\ln\left(\frac{A_{1}}{A_{2}}\right)- \ln\left(\frac{A_{\mathrm{w}}}{A_{\mathrm{s}}}\right)\right], \tag{18}\]  
where \(A_{\mathrm{w}}\), \(A_{\mathrm{s}}\), \(A_{1}\) and \(A_{2}\) were the amplitude spectra corresponding to the above-mentioned phase responses. Figure 3 plots the measured attenuation coefficient and acoustic velocity as a function of frequency. In the frequency range 1.5-3.1 MHz, the sound velocity increased by about 26 m/s. The acoustic attenuation coefficient increased from 0.59 dB/cm at 1.5 MHz to 1.79 dB/cm at 3.1 MHz, indicating the attenuation factor being \(y=1.83\) for the phantom. The attenuation factor was obtained from the exponential fitting using a curve fitting toolbox in Matlab. It should be noted that sound velocity could fluctuate due to the thermal effect of focused ultrasound. However, in this work the duty cycle was as low as 0.18%, and a medium-level surface pressure of up to 0.4 MPa was chosen for the transducer. Thus, no significant temperature elevation was observed during the experiments, and the thermal-induced change in sound velocity could be neglected [50]. It should also be mentioned, that to account for the thermal-effect, an appropriate bio-heat transferring equation should be incorporated with the current model. The main concern then lies in the dispersion due to microsphere scattering. In clinical applications, the major dispersion originates from tissues like fat, whose attenuation coefficient is generally larger than that of body tissue. Therefore, in comparison with other research in which the parameters of the phantom were nearly the same as body tissue (e.g., liver and spleen), the choice of phantoms with larger attenuation coefficients in the present experiments might provide results in better agreement with the actual situation. Meanwhile, since the measured physical parameters are close to those of dense fat, this setup could be regarded as a simplified mimic of the abdomen in HIFU therapies. Figure 3: Measured frequency-dependent acoustic velocity and attenuation coefficient of the phantom sample. As is illustrated in Figure 4, the acoustic field distribution along the axial direction is examined by sitting the phantom on the acoustic path of the transducer. The results show the comparison between measured data and simulated results obtained from both the modified and traditional KZK models. Figure 4a describes the total pressure and Figure 4b-d shows the fundamental, second harmonic and third harmonic components, respectively. For the total pressure distribution, it is clearly observed that the peak-pressure location predicted through the mKZK model agrees well with that acquired in experiments, while the data calculated from the traditional KZK model show a deviation from the previous two groups.