## 4 Finite Element Analysis
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Nodule Position (Center)** & **LB [m]** & **MB [m]** \\ \hline x & \(-0.045\) & \(-0.03\) \\ y & \(0.055\) & \(0.04\) \\
**Variable** & **[m]** & **[m]** \\ \hline a & \(0.07\) & \(0.05\) \\ b & \(0.14\) & \(0.1\) \\ c & \(0.02\) & \(0.015\) \\ d & \(0.01\) & \(0.01\) \\
**Mesh Elements Degree of** & \(914\) & \(790\) \\
**Freedom (DoF)** & \(3758\) & \(3270\) \\ \hline \hline \end{tabular}
\begin{tabular}{c c} \hline \hline
**Boundary Number** & **Boundary Condition** \\ \hline \hline \end{tabular}
\begin{tabular}{c c} \hline \hline
1 & Load \\ \(\mathrm{Fx}=-10000^{*}\mathrm{nx\_pn}\) [Pa] \\ \(\mathrm{Fy}=-10000^{*}\mathrm{ny\_pn}\) [Pa] \\ \(\mathrm{Constraint\ Rx}=0\), \(\mathrm{Ry}=0\) \\ \(2\),\(3\),\(4\),\(6\),\(7\),\(8\) & \(\mathrm{No\ Constraint}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Model parameters for the 2D finite element (FE) analysis. The expressions “nx_pn” and “ny_pn” are internal variables of COMSOL to describe the normal direction with respect to the indicated boundary. Figure 4: Two-dimensional (2D) model with 10-mm diameter inclusion embedded in breast tissue (large breast, LB). Blue numbers indicate the boundary of the breast, while red letters define the breast’s dimensions. A description of the parameters is given in Table 1.  
Figure 6 represents the total displacement in response to an external pressure on the 2D breast model--according to the model, the maximum difference in displacement between healthy and abnormal breast tissue ranges from 2.5 mm up to 3 mm depending on the Young's modulus of the inclusion. Figure 7 highlights the total displacement of boundary #1 of the 2D breast model for different positions of the inclusion--the position of the maximum difference in displacement between a healthy and abnormal breast changes with the position of the inclusion, and reaches a higher value when the inclusion is closer to the surface of the breast. \begin{table}
\begin{tabular}{c c c} \hline
**Nodule Position (Center)** & **LB [m]** & **MB [m]** \\ \hline x & \(-0.02\) & \(0.01\) \\ y & \(0.02\) & \(0.02\) \\ z & \(-0.05\) & \(-0.03\) \\ \hline
**Mesh Elements Degree of** & 9320 & 8635 \\
**Freedom (DoF)** & 41,328 & 38,355 \\ \hline \end{tabular}
\end{table} 
Table 2: Model parameters for the 3D FE analysis. Figure 5: Three-dimensional (3D) model for FE analysis with 10-mm diameter inclusion embedded in breast tissue (LB) (Axes are in (m)). Figure 6: Displacement field of the LB 2D model, with a 10 mm diameter inclusion and an applied pressure of 10 kPa. (**a**) Surface plot of the displacement with an inclusion with E\({}_{\text{nod}}\) = 125 kPa; and (**b**) total displacement plot of boundary #1 of the breast for different E\({}_{\text{nod}}\) values. The blue line shows the response of a healthy breast. The 3D FE model was used to evaluate the spatial resolution necessary to clearly identify the presence of the inclusion (Figure 8). Figure 8: (**a**) Total displacement in the 3D FE model of the breast with a 10-mm inclusion with Young’s modulus of 125 kPa. The 2D map of the Von Mises stress with a superimposed 5 mm spaced grid for LB (**b**) and medium breast (MB) model (**c**). The bar scale is in (Pa). Figure 7: Differential displacement between abnormal and healthy breast as a function of nodule position evaluated along the boundary #1, indicated in red in the inset. (**a**) 125 kPa nodule placed superficially and closer to the boundary (see inset); and (**b**) 125 kPa nodule placed deeper in the breast and near the center (see inset). According to the 3D FE model, differences in the Von Mises stress between the healthy and abnormal tissue (Figure 8b,c) indicated that a sensor matrix able to detect a change in pressure of 600 Pa over a distance of 5 mm was sufficient to detect the presence of a stiffer inclusion of 10-mm diameter located inside a breast.