## 2 Theory and Experiments
### Experimental Methods
#### 2.4.2 Experimental Setup  
The thickness of each phantom sample was \(L_{\mathrm{s}}\) = 42.3 mm, with a density of \(\rho_{1}\) = 1000 kg/m\({}^{3}\) so that it could stably suspend in the water. The distance between transducer and the phantom was 37.7 mm. Following the same protocol used in our previous work [45], the nonlinearity parameter was measured as \(\beta\) = 4.2. Since the only difference in gel recipe between the two works is the introduction of amino polystyrene microspheres in this paper, the same \(\beta\) value indicates that microspheres did not influence the nonlinear propagation of waves. To confirm this, we measured the ratio between the second harmonic and fundamental components, and found it was identical to that in [45] under the same sonication conditions. The attenuation coefficient and sound velocity of phantom, as functions of frequency, were measured through a broadband spectrum method [48; 49]. In the measurement, two planar piston transducers (Immersion, Unfocused, Panametrics, Waltham, MA, USA) calibrated with a needle hydrophone (HNC-1000, ONDA Corp., Sunnyvale, CA, USA), were placed on the opposite sides of the phantom, with one of them driven by a broadband pulse generator (5900 PR, Panametrics). The reflected and transmitted acoustic signals were then acquired by the transducers and digitalized with a digital oscilloscope (54830B, Agilent, Santa Clara, CA, USA). (Device connection was similar to that in [49]). In the measurements, 8 continuous pulse sequences were acquired and averaged to reduce the signal-to-noise ratio (SNR).  
Prior to the HIFU experiments, a low-level driving voltage was used to drive a customized HIFU transducer (Chongqing Haifu Med. Tech. Co., Ltd., Chongqing, China) to emit a linear sound field. Simultaneously, effective parameters of the transducer, such as effective radius, radiation profile, and angle of divergence were obtained by adjusting the transducer parameters in the KZK calculations, such that the linear field predicted via KZK was consistent with that measured [45]. The effective parameters were then used in the simulations of both the KZK and mKZK models. As a result, the HIFU transducer (working frequency 1.12 MHz) had an effective aperture radius of 48.6 mm and a geometrical focal length of 101.5 mm. As illustrated in Figure 1, the transducer was immersed in water and driven with signals from a signal generator (33250A, Agilent, Santa Clara, CA, USA) amplified by a broadband power amplifier (2200L, E&I, Rochester, NY, USA). The input voltage was set as 465 mV (20 cycles; burst period, 10 ms; duty cycle, 0.18%). Another needle hydrophone (HNA-0400, ONDA Corp., Sunnyvale, CA, USA) was mounted on a customized three-dimensional (3D) scanning system (Controller Model: XPS-C8, Newport, CA, USA) to scan the HIFU field. To suppress possible acoustic cavitation in surrounding liquid, the water was processed with a self-developed water degassing and deionizing system. The temporal and spatial scanning procedure was controlled via the GPIB interface (National Instruments, Austin, TX, USA).  
Figure 1: The experimental setup.