Three dimensional transport lattice model for describing action potentials in axons stimulated by external electrodes

Electrodiffusion of molecules in aqueous media: a robust, discretized description for electroporation and other transport phenomena

Electrically driven transport of molecules and ions within aqueous electrolytes is of long-standing interest, with direct relevance to applications that include the delivery/release of biologically active solutes to/from cells and tissues. Examples include iontophoretic and electroporation-mediated drug delivery. Here, we describe a robust method for characterizing electrodiffusive transport in physiologic aqueous media. Specifically, we treat the case of solute present in sufficiently low concentration as to negligibly contribute to the total ionic current within the system. In this limiting case, which applies to many systems of interest, the predominant electrical behavior due to small ions is decoupled from solute transport. Thus, electrical behavior may be characterized using existing methods and treated as known in characterizing electrodiffusive molecular transport. First, we present traditional continuum equations governing electrodiffusion of charged solutes within aqueous electrolytes and then adapt them to discretized systems. Second, we examine the time-dependent and steady-state interfacial concentration gradients that result from the combination of diffusion and electrical drift. Third, we show how interfacial concentration gradients are related to electric field strength and duration. Finally, we examine how discretization size affects the accuracy of these methods. Overall these methods are motivated by and well suited to addressing an outstanding goal: estimation of the net ionic and molecular transport facilitated by electroporation in biological systems.

Frequency-dependent interaction of ultrashort E-fields with nociceptor membranes and proteins

We examined the influence of ultrashort pulses (USP) on sensory neurons. Single and high frequency bursts of 12 ns E-fields were presented to rat skin nociceptors that expressed distinct combinations of voltage-sensitive proteins. A single E-field pulse produced action potentials in all nociceptor subtypes at a critical threshold (E(c) ) of 403 V/cm. When configured into high frequency bursts, USP charge integrated to reduce the action potential threshold in a frequency and burst duration-dependent manner with E(c) as low as 16 V/cm (4000 Hz, 25 ms burst). There was no evidence of electroporation at field intensities near the E(c) for nociceptor activation. USP bursts activated a late, persistent Ca(++) flux that was identified as a dantrolene-sensitive Ca(++) -induced Ca(++) release (CICR). Influx of Ca(++) into the cell was required for the CICR and resulted in a reduction of the single pulse E(c) by about 50%.

Hybrid finite element method for describing the electrical response of biological cells to applied fields

A novel hybrid finite element method (FEM) for modeling the response of passive and active biological membranes to external stimuli is presented. The method is based on the differential equations that describe the conservation of electric flux and membrane currents. By introducing the electric flux through the cell membrane as an additional variable, the algorithm decouples the linear partial differential equation part from the nonlinear ordinary differential equation part that defines the membrane dynamics of interest. This conveniently results in two subproblems: a linear interface problem and a nonlinear initial value problem. The linear interface problem is solved with a hybrid FEM. The initial value problem is integrated by a standard ordinary differential equation solver such as the Euler and Runge-Kutta methods. During time integration, these two subproblems are solved alternatively. The algorithm can be used to model the interaction of stimuli with multiple cells of almost arbitrary geometries and complex ion-channel gating at the plasma membrane. Numerical experiments are presented demonstrating the uses of the method for modeling field stimulation and action potential propagation.

Electrical behavior and pore accumulation in a multicellular model for conventional and supra-electroporation

Extremely large but very short (20 kV/cm, 300 ns) electric field pulses were reported recently to non-thermally destroy melanoma tumors. The stated mechanism for field penetration into cells is pulse characteristic times faster than charge redistribution (displacement currents). Here we use a multicellular model with irregularly shaped, closely spaced cells to show that instead overwhelming pore creation (supra-electroporation) is dominant, with field penetration due to pores (ionic conduction currents) during most of the pulse. Moreover, the model's maximum membrane potential (about 1.2 V) is consistent with recent experimental observations on isolated cells. We also use the model to show that conventional electroporation resulting from 100 microsecond, 1 kV/cm pulses yields a spatially heterogeneous electroporation distribution. In contrast, the melanoma-destroying pulses cause nearly homogeneous electroporation of cells and their nuclear membranes. Electropores can persist for times much longer than the pulses, and are likely to be an important mechanism contributing to cell death.

Microdosimetry for conventional and supra-electroporation in cells with organelles

Conventional electroporation (EP) by 0.1 to 1 kV/cm pulses longer than 100 micros, and supra-electroporation by 10 to 300 kV/cm pulses shorter than 1 micros cause different cellular effects. Conventional EP delivers DNA, proteins, small drugs, and fluorescent indicators across the plasma membrane (PM) and causes moderate levels of phosphatidylserine (PS) translocation at the PM. We hypothesize that supra-EP is central to intracellular effects such as apoptosis induction and higher levels of PS translocation. Our cell system model has 20,000 interconnected local models for small areas of the PM and organelle membranes, small regions of aqueous media, appropriate resting potentials, and the asymptotic EP model. Conventional EP primarily affects the PM, but with a hint of endoplasmic reticulum involvement. Supra-EP can involve all of a cell's membrane at the largest fields. Conventional EP fields tend to go around cells, but supra-EP fields go through cells, extensively penetrating organelles.