Asymptotic model of electroporation

Electroporation is described mathematically by a partial differential equation (PDE) that governs the distribution of pores as a function of their radius and time. This PDE does not have an analytical solution and, because of the presence of disparate spatial and temporal scales, numerical solutions are hard to obtain. These difficulties limit the application of the PDE only to experimental setups with a uniformly polarized membrane. This study performs a rigorous, asymptotic reduction of the PDE to an ordinary differential equation (ODE) that describes the dynamics of the pore density $N\left(t\right).$ Given $N\left(t\right),$ the precise distribution of the pores in the space of their radii can be determined by an asymptotic approximation. Thus, the asymptotic ODE represents most of the phenomenology contained in the PDE. It is easy to solve numerically, which makes it a powerful tool to study electroporation in experimental setups with significant spatial dependence, such vesicles or cells in an external field.