# Numerical Simulation of Electric Field Distribution around an Instrumented Total Hip Stem
## 2 Materials and Methods
### Numerical Simulations  
The simulations were performed using commercial software SIMULIA CST Studio Suite (Dassault Systemes, www.3 ds.com/simulia, accessed on 10 May 2021), and the electric field within the femur model was computed. Here, the specific solver used was based on the Finite Integration Technique (FIT). The assembly of the computational domain was based on the reconstruction of the femur and the CAD model of the hip stem. Both were fitted into the numerical model, with the coordinates being taken by the coordinate measuring arm of the experimental setup for the sake of better comparison of computed and measured results. The dielectric properties of the different tissues (bone marrow, cancellous bone, and cortical bone) were defined using the parameters from [PERSON] et al. [22] and [PERSON] et al. [23] as provided for human bone as no data were available for the porcine bone. It had to be assumed, though, that the dielectric properties of the adolescent porcine bone tissues differ from adult human bone tissues. This simulation is referred to as case study 1. Because of the almost insulating properties of the bone marrow when using parameters from [PERSON] et al. [22], a second simulation was performed using the conductivity of the cancellous bone within the domain of the bone marrow. This simulation is referred to as case study 2. Furthermore, case study 3 was performed using the parameters from [PERSON] et al. [23] but with red bone marrow. As the electric conductivity of titanium is considerably higher than that of the biological tissue, a Perfect Electric Conductor (PEC) could be used rather than including titanium into the computational domain, which reduced the computational effort. Conversely, the insulator has a considerably lower conductivity compared to the biological tissue. Due to this reason, it was defined as a perfect electric insulator (a vacuum). Figure 1: CAD model of the porcine femur with the modified total hip stem. The inset displays the measurement points at the cross section of the porcine bone along two lines. The points are allocated on two parallel lines at 2, 3, 5, and 7 mm from the first electrode (line A) and from the interface between insulator and implant (line B). The electric double layer, which exists at the interface between electrodes and organic tissue, as well as between the implant surface and organic tissue, was characterized by its impedance \(Z\) based on the approximate formula below using measured data of its surface \(A\) along with the impedance \(Z_{Sample}\) measured for a sample electrode with the surface \(A_{Sample}\). \[Z\approx\frac{A_{Sample}}{A}\;Z_{Sample}. \tag{1}\]  
Using this approximation, an impedance of 3 k\(\Omega\) at the electrode and 120 \(\Omega\) at the hip stem was calculated. After the first simulation, the determination of the impedance of the bone sample got possible. For the dielectric tissue parameters as proposed by [PERSON] et al. [22], the resultant impedance was 13.25 k\(\Omega\). To consider the effects of the double layer, its impedances were considered in a serial connection with the impedance of the organic tissue. Using the equation  
\[\frac{U}{U_{total}}=\frac{Z}{Z_{total}}, \tag{2}\]  
the boundary conditions for the simulation were reset to 0.044 V instead of 0.000 V at the hip stem and 4.900 V instead of 6.000 V at the electrode. The problem was discretized on a hexahedral mesh in conjunction with the Enhanced Fast Perfect Boundary Approximation (EFPBA), which provided an accurate geometrical approximation of the continuous structure. In particular, the mesh around the electrode and the insulator was enhanced to ensure an optimal resolution in the area of interest. The mesh around the readout locations was also densified to avoid interpolation errors. During the first simulations, the adaptive mesh refinement was used to improve the hexahedral mesh automatically. As a result of the convergence study, the final mesh consisted of approximately 9.55 million mesh cells.