Stochastically perturbed model of cell electropermeabilization

1. Problem Statement

The paper addresses the mathematical modeling of electroporation (reversible electropermeabilization), a phenomenon where short, high-voltage electric pulses transiently increase the permeability of cell membranes. While deterministic models exist (specifically the phenomenological model by Kavian et al. [11]), they fail to account for inherent random effects such as temperature fluctuations, microscopic thermal noise, or uncertainty in the applied electric field.

The authors aim to:

1. Formulate a stochastically perturbed version of the deterministic electroporation model.
2. Prove the well-posedness (existence and uniqueness) of the resulting coupled system, which consists of a Stochastic Partial Differential Equation (SPDE) for the transmembrane potential and an Ordinary Differential Equation (ODE) for the membrane porosity.
3. Investigate the long-term behavior of the system through numerical simulations to provide evidence for the existence of an invariant measure (ergodicity).

2. Mathematical Model

The model describes a single cell (or a tissue of cells) with an intracellular domain ($G_i$), an extracellular domain ($G_e$), and a membrane interface ($\Gamma$).

• Deterministic Core: The system couples the electrostatic equations for the electric potential $u$ in the bulk domains with a nonlinear evolution law for the transmembrane potential jump $v = [u] = u_i - u_e$ and the porosity degree $w$.
  • The membrane conductivity is modeled as $S_m(v, w) = (S_0 + S_1 w)v$.
  • The porosity $w$ evolves according to a nonlinear ODE driven by a function $f(v, w)$ involving characteristic electroporation ($\tau_{ep}$) and resealing ($\tau_{res}$) times.
• Stochastic Perturbation: The authors introduce noise specifically on the membrane potential equation to model fluctuations. The resulting system (Equation 9) is:
  $$\begin{aligned} c_m dv &= -(Lv + S_m(v, w) + \sigma_i \nabla p \cdot n) dt + b(v) dW_t, \\ dw &= f(v, w) dt, \end{aligned}$$
  where:
  • $L$ is a Dirichlet-to-Neumann type operator derived from the electrostatics.
  • $W_t$ is a $Q$-Wiener process.
  • $b(v)$ is a noise coefficient that can be additive or multiplicative.
  • The noise is degenerate, acting only on the SPDE component ($v$), while the ODE component ($w$) is driven indirectly through the nonlinear coupling.

3. Methodology

The primary mathematical challenge is that the nonlinear coupling term ($v \cdot w$) is neither Lipschitz continuous nor monotone in the classical sense, which prevents the direct application of standard monotone operator theory.

To overcome this, the authors employ a variational approach based on the generalized monotonicity and coercivity conditions developed by Liu, Röckner, and Prévôt [19].

Key Steps in the Proof:

1. Galerkin Approximation: The infinite-dimensional SPDE-ODE system is approximated by a finite-dimensional system of SDEs using a projection onto a finite basis.
2. Uniform Estimates: The authors derive uniform a priori estimates for the Galerkin approximations. Crucially, they utilize a generalized local monotonicity condition:
   $$\langle A(U_1) - A(U_2), U_1 - U_2 \rangle \leq K(1 + \|v_2\|^2_{H^{1/2}}) (\|U_1 - U_2\|^2)$$
   This condition holds along the solutions rather than for arbitrary functions in the space.
3. Compactness and Limit Passage: Using these estimates, they establish weak convergence of the approximations and pass to the limit to obtain a variational solution.
4. Uniqueness: Uniqueness is proven using a weighted Itô formula with a product rule to handle the non-Lipschitz nature of the drift, effectively neutralizing the random factor in the monotonicity condition.

4. Key Contributions

• Well-posedness of Degenerate SPDE-ODE Systems: The paper establishes the existence and uniqueness of a variational solution for a coupled system where the stochastic forcing is multiplicative and degenerate (acting on only one variable). This is a significant theoretical advancement for systems where noise enters only part of the coupled dynamics.
• Handling Non-Lipschitz Nonlinearities: The work demonstrates how to apply generalized monotonicity conditions to electroporation models where the nonlinearity ($v \cdot w$) violates classical assumptions.
• Mild Solution Regularity: Under stronger regularity assumptions on the domain and data, the variational solution is shown to be a mild solution.
• Numerical Evidence of Ergodicity: The paper provides the first numerical investigation of the stochastic electroporation model, computing time averages and standard deviations to suggest the existence of an invariant measure.

5. Results

Theoretical Results:

• Theorem 3.2: Proves that for any initial data in $L^2$, there exists a unique variational solution $U = (v, w)$ such that the trajectories belong to $L^2(0, T; V) \cap C([0, T]; H)$ almost surely, with finite second moments.
• The solution is pathwise unique.

Numerical Results:

• Simulation Setup: Simulations were performed on a 2D periodic cell using the FEniCSx package with a semi-implicit Euler method.
• Noise Types: Both additive noise (global temporal fluctuation) and linear multiplicative noise (dependent on the potential) were tested.
• Dynamics:
  • The membrane potential $v$ rises rapidly, peaks, and then decays as the porosity $w$ increases (allowing current leakage).
  • Ergodicity Indication: The authors computed time averages of $\|v\|^2$ and $\|w\|^2$. The sample standard deviation of these time averages across multiple Monte Carlo runs was observed to decay as $T^{-1/2}$.
  • This decay rate is a strong numerical indicator that the system is ergodic and possesses a unique invariant measure, although the analytical proof of the invariant measure remains a future task.

6. Significance

This work bridges the gap between deterministic biophysical models and stochastic reality. By rigorously proving the well-posedness of the stochastic model, it provides a solid mathematical foundation for future studies on:

• Optimization of Electroporation Protocols: Understanding how noise affects the efficiency of drug delivery or gene transfection.
• Cancer Treatment: Improving non-thermal ablation techniques by accounting for biological variability.
• Stochastic Analysis in Biology: Offering a framework for analyzing coupled SPDE-ODE systems with degenerate noise and non-monotone nonlinearities, which are common in biological modeling but mathematically challenging.

The paper successfully moves the field from purely deterministic descriptions to a stochastic framework, acknowledging that biological systems are inherently noisy, and provides the necessary analytical tools to study them.