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Resistor–capacitor modeling of the cell membrane: A multiphysics analysis

In this Tutorial, we provide a discussion of “What are cell membrane resistance (MR) and capacitance (MC)?” and then give a number of examples to illustrate how cell membranes constitute nature's ultimate stretchable resistor–capacitor network. There are many approaches to the analysis of the electric field effects in cell membranes, but a particularly intuitive and conceptually straightforward method is to use the biophysically inspired lumped parameter resistor (R)–capacitor (C) network in order to simulate the charging and discharging processes. By developing advanced multiphysics and multiscale numerical analysis, we expect to learn many cross-properties of biological materials which involve multiple spatial or temporal scales. These include electrodeformation (ED) and electroporation (EP) biophysical processes occurring in the cell membrane. In a first stage, we present electric and mechanical circuit analog models of cell membranes and examine their predictions and limitations. An important parameter that researchers can tune with these deterministic approaches is the strength of the transmembrane voltage Vm: at low values of Vm, MC varies quadratically as a function of Vm and MR is infinite, but as Vm is increased at a value below the EP threshold, the membrane should be considered as a nonlinear capacitor. Over the EP threshold, there is a decrease in Vm and MR due to the charge transport across the membrane. Mechanical and electrical stresses, singly or in combination, can result in damage and eventually breakdown of the membrane. In a second stage, the parameters in the finite element (FE) modeling that we present are linked to scales we know should be associated with EP and ED processes. We present simulation data and attempt to determine whether the MC and MR behaviors compare well with experimental observations and/or trends from analytical approaches. MC and MR are correlated with the dielectric, mechanical, and morphological information of cells. For an initially spherical cell exposed to an electric field, monitoring MC and MR reflects a quadratic and then higher order nonlinear behavior as a function of Vm. The quadratic regime scales with spheroidal morphologies of the stressed cell up to a critical value of Vm beyond which higher order nonlinearities arise, and the cell shape is no longer described by a spheroid. Furthermore, we consider the present challenges of connecting electrostatic stress, strain energy in multi-cellular environments to sub-cellular scale material properties, and show that they have the potential to explain the ED and EP of cell membranes via multi-physics and multi-scale numerical analysis. The emergence of Vm as a reporter of neighboring cell interactions is also discussed in a theory-based method for constructing realistic models of tissues based on densely packed environments made by irregularly shaped cells. Of particular interest is the proximity-induced ED and capacitive coupling between neighboring cells, and the subsequent correlation that this has upon anisotropic local ED distribution over a wide range of conditions. For future studies, we identify significant challenges, opportunities, and a sampling of a few used case studies for the development of tissue ED and EP modeling in the coming years.