Cell Electropermeabilization Modeling via Multiple Traces Formulation and Time Semi-Implicit Coupling

Abstract

We simulate the electrical response of multiple disjoint biological 3D cells undergoing an electropermeabilization process. Instead of solving the boundary value problem in the unbounded volume, we reduce it to a system of boundary integrals equations--the local Multiple Traces Formulation--coupled with nonlinear dynamics on the cell membranes. Though in time the model is highly non-linear and poorly regular, the smooth geometry allows for boundary unknowns to be spatially approximated by spherical harmonics. This leads to spectral convergence rates in space. In time, we use a multistep semi-implicit scheme. To ensure stability, the time step needs to be bounded by the smallest characteristic time of the system. Numerical results are provided to validate our claims and future enhancements are pointed out.

Figures (15)

• Figure 1: A system of three cells $\mathcal{N}=3$.
• Figure 2: Error convergence for traces in Example 1 (Section \ref{['sec:mtf-veri']}). The relative error $L^2(\Gamma_1)$\ref{['re_2']} is computed against the analytic solution with parameter values in Table \ref{['space-convergence-parameters']}.
• Figure 3: Field $u_0^{50}$ of Example 2, Section \ref{['sec:mtf-veri']} with parameters from Table \ref{['space-convergence-2spheres-parameters']}.
• Figure 4: Absolute error in $L^2(\Gamma_1)$ between $\overline{v}_1^{25}$ (discrete approximation) and $v_1$ (analytic solution), as well as $\frac{\tau^2}{4} \left\| \partial^2_{t}v_1(t) \right\| _{L^2(\Gamma_j)}$, plotted to verify the bound given by Theorem \ref{['teo-time-converge']} for the time scheme from Section \ref{['sec:semiimpveri']} where linear dynamics are assumed. The time step $\tau$ is $2.5\cdot10^{-2}$$\mu$s and the rest of the parameters used are in Table \ref{['time-valitadion-parameters']}.
• Figure 5: Absolute values of the analytically obtained second derivatives for the three most significant coefficients for the linear dynamics example from Section \ref{['sec:semiimpveri']} with $\phi_{time-ext}(t)=e^{-t}$. It can be seen that the dip for $l=1$ matches the dip from Figure \ref{['time-verification-results-01']}(a).

Theorems & Definitions (15)

• Remark 2.1
• Theorem 2.3
• Theorem 2.4: Existence, uniqueness and stability
• Theorem 2.5: Lemma 4.3 in HJH18
• Remark 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4
• proof
• Corollary 3.5