# Computational Modeling of Ultrasound C-Scan Imaging Using Transmitted Signal Peak Density
## 2 Finite Element Analysis (FEA) Simulation
### Simulation Physics  
The soft pepper flake was considered the solid medium, and the surrounding gelatin phantom was considered the fluid medium. In the fluid medium, the acoustic wave propagation followed the Helmholtz equation shown in Equation (1) [24]. The equation provides the acoustic pressure distribution as a function of frequency inside the medium. \[\
abla^{2}p_{t}+k^{2}p_{t}=0 \tag{1}\]  
The total pressure, \(p_{t}\) is the summation of the background pressure field \((p_{b})\) and backscatter pressure field \((p_{s})\). In the model, the background pressure field was expressed as wave propagation in the \(\ell_{x}\) direction defined as \(p_{b}=p_{0}e^{-kx}\). \(k\) is the equivalent wave number defined for wave velocity \(c_{c}\) in fluid and frequency \(f\), and the attenuation coefficient \(\alpha\) is expressed in Equation (2) [24]. \[k=\frac{2\pi f}{c_{c}}-iln(10)\frac{\alpha}{20} \tag{2}\]  
The pepper flake was modeled as a linear elastic solid. The acoustic pressure that was applied to the soft flake was very small (\(\leq\)1 Pa). Therefore, the strain value of the solid flake was assumed to be too insignificant to follow a nonlinear stress-strain profile. Thus, a linear elastic approximation was considered. In the solid medium, the acoustic  
Figure 2: (**a**) Three-dimensional (3D) model with simplification details; (**b**) simplified 2D XY plane (at Z = 0). propagation was governed by [PERSON]'s equation shown in Equation (3) [24]. The equation provides solid displacement caused by the acoustic propagation at different frequencies. \[-(2\pi f)^{2}\rho_{s}\mathbf{u}=\
abla.S+\mathbf{F}e^{i\phi} \tag{3}\]  
In this equation, \(f\) is sound frequency, \(S\) is Cauchy stress tensor, \(\rho_{s}\) is solid density, \(\phi\) is the phase component of the force, \(\mathbf{F}\) is acting force on the solid, and \(\mathbf{u}\) denotes the solid displacement field. In the fluid-solid interface, in order to make sure that the sound pressure and structural acceleration got transferred between the fluid and solid, the following boundary conditions were used, which are shown in Equations (4) and (5) [24]. \[\mathbf{n}\cdot\frac{1}{\rho_{c}}\
abla p_{t}=-\mathbf{n}\cdot\mathbf{u_{tt}} \tag{4}\]  
\[\mathbf{F}=p_{t}\mathbf{n} \tag{5}\]  
\(\mathbf{n}\) is the normal unit vector to the boundary, \(\mathbf{u_{tt}}\) is structural acceleration, and \(\mathbf{F}\) is an acting load on the solid. In the fluid and solid domains, the material properties are listed in Table 1[4,25]. A free triangular mesh was created inside the fluid and solid domain. A mapped mesh was used in the perfectly matched layer. Mesh sensitivity analysis was conducted to achieve convergence in the simulation result. With decreasing mesh size, the scattered pressure was calculated for different frequency steps between 22 and 41 MHz. Convergence was achieved for the mesh element size of one-sixth of the wavelength. The simulation was conducted with a mesh of a maximum of 3,020,168 elements. A computational configuration of a Core i7-9700 processor with 16 GB RAM was used to conduct the simulation. In the experimental study, the frequency bandwidth of the high-frequency transducer was 22-41 MHz with a center frequency of 31.5 MHz [22]. Therefore, a similar frequency range was used for the computational analysis. Furthermore, the transducer property was closely approximated in the input background pressure by multiplying it with a generalized Gaussian function of pressure distribution among the frequency range (Figure 3) [26]. This enabled us to apply frequency-dependent pressure similar to a transducer with frequency bandwidth. The frequency step size was selected as 100 kHz. The model was computed for the scattered pressure at each frequency step from 22 to 41 MHz, and then all the frequency responses were compiled into a frequency spectrum at the back wall. A peak counting algorithm was used to count the number of peaks and valleys of each spectrum.