A time-dependent numerical model of transmembrane voltage inducement and electroporation of irregularly shaped cells

Multiple nanosecond electric pulses increase the number but not the size of long-lived nanopores in the cell membrane

Exposure to intense, nanosecond-duration electric pulses (nsEP) opens small but long-lived pores in the plasma membrane. We quantified the cell uptake of two membrane integrity marker dyes, YO-PRO-1 (YP) and propidium (Pr) in order to test whether the pore size is affected by the number of nsEP. The fluorescence of the dyes was calibrated against their concentrations by confocal imaging of stained homogenates of the cells. The calibrations revealed a two-phase dependence of Pr emission on the concentration (with a slower rise at<4μM) and a linear dependence for YP. CHO cells were exposed to nsEP trains (1 to 100 pulses, 60ns, 13.2kV/cm, 10Hz) with Pr and YP in the medium, and the uptake of the dyes was monitored by time-lapse imaging for 3min. Even a single nsEP triggered a modest but detectable entry of both dyes, which increased linearly when more pulses were applied. The influx of Pr per pulse was constant and independent of the pulse number. The influx of YP per pulse was highest with 1- and 2-pulse exposures, decreasing to about twice the Pr level for trains from 5 to 100 pulses. The constant YP/Pr influx ratio for trains of 5 to 100 pulses suggests that increasing the number of pulses permeabilizes cells to a greater extent by increasing the pore number and not the pore diameter.

A microdevice to locally electroporate embryos with high efficiency and reduced cell damage

The ability to follow and modify cell behaviour with accurate spatiotemporal resolution is a prerequisite to study morphogenesis in developing organisms. Electroporation, the delivery of exogenous molecules into targeted cell populations through electric permeation of the plasma membrane, has been used with this aim in different model systems. However, current localised electroporation strategies suffer from insufficient reproducibility and mediocre survival when applied to small and delicate organisms such as early post-implantation mouse embryos. We introduce here a microdevice to achieve localised electroporation with high efficiency and reduced cell damage. In silico simulations using a simple electrical model of mouse embryos indicated that a dielectric guide-based design would improve on existing alternatives. Such a device was microfabricated and its capacities tested by targeting the distal visceral endoderm (DVE), a migrating cell population essential for anterior-posterior axis establishment. Transfection was efficiently and reproducibly restricted to fewer than four visceral endoderm cells without compromising cell behaviour and embryo survival. Combining targeted mosaic expression of fluorescent markers with live imaging in transgenic embryos revealed that, like leading DVE cells, non-leading ones send long basal projections and intercalate during their migration. Finally, we show that the use of our microsystem can be extended to a variety of embryological contexts, from preimplantation stages to organ explants. Hence, we have experimentally validated an approach delivering a tailor-made tool for the study of morphogenesis in the mouse embryo. Furthermore, we have delineated a comprehensive strategy for the development of ad hoc electroporation devices.

Dual-porosity model of solute diffusion in biological tissue modified by electroporation

In many electroporation applications mass transport in biological tissue is of primary concern. This paper presents a theoretical advancement in the field and gives some examples of model use in electroporation applications. The study focuses on post-treatment solute diffusion. We use a dual-porosity approach to describe solute diffusion in electroporated biological tissue. The cellular membrane presents a hindrance to solute transport into the extracellular space and is modeled as electroporation-dependent porosity, assigned to the intracellular space (the finite rate of mass transfer within an individual cell is not accounted for, for reasons that we elaborate on). The second porosity is that of the extracellular space, through which solute vacates a block of tissue. The model can be used to study extraction out of or introduction of solutes into tissue, and we give three examples of application, a full account of model construction, validation with experiments, and a parametrical analysis. To facilitate easy implementation and experimentation by the reader, the complete derivation of the analytical solution for a simplified example is presented. Validation is done by comparing model results to experimentally-obtained data; we modeled kinetics of sucrose extraction by diffusion from sugar beet tissue in laboratory-scale experiments. The parametrical analysis demonstrates the importance of selected physicochemical and geometrical properties of the system, illustrating possible outcomes of applying the model to different electroporation applications. The proposed model is a new platform that supports rapid extension by state-of-the-art models of electroporation phenomena, developed as latest achievements in the field of electroporation.

Shape-dependent optoelectronic cell lysis

We show an electrical method to break open living cells amongst a population of different cell types, where cell selection is based upon their shape. We implement the technique on an optoelectronic platform, where light, focused onto a semiconductor surface from a video projector creates a reconfigurable pattern of electrodes. One can choose the area of cells to be lysed in real-time, from single cells to large areas, simply by redrawing the projected pattern. We show that the method, based on the "electrical shadow" that the cell casts, allows the detection of rare cell types in blood (including sleeping sickness parasites), and has the potential to enable single cell studies for advanced molecular diagnostics, as well as wider applications in analytical chemistry.

Electropermeabilization of the cell membrane

Membrane electropermeabilization is the observation that the permeability of a cell membrane can be transiently increased when a micro-millisecond external electric field pulse is applied on a cell suspension or on a tissue. Applicative aspects for the transfer of foreign molecules (macromolecules) into the cytoplasm are routinely used. But only a limited knowledge about what is really occurring in the cell and its membranes at the molecular levels is available. This chapter is a critical attempt to report the present state of the art and to point out some of the still open problems. The experimental facts associated to membrane electropermeabilization are firstly reported. They are valid on biological and model systems. Secondly, soft matter approaches give access to the bioelectrochemical description of the thermodynamical constraints supporting the destabilization of simplified models of the biological membrane. It is indeed described as a thin dielectric leaflet, where a molecular transport takes place by electrophoresis and then diffusion. This naïve approach is due to the lack of details on the structural aspects affecting the living systems as shown in a third part. Membranes are part of the cell machinery. The critical property of cells as being an open system from the thermodynamical point of view is almost never present. Computer simulations are now contributing to our knowledge on electropermeabilization. The last part of this chapter is a (very) critical report of all the efforts that have been performed. The final conclusion remains that we still do not know all the details on the reversible structural and dynamical alterations of the cell membrane (and cytoplasm) supporting its electropermeabilization. We have a long way in basic and translational researches to reach a pertinent description.

Cell electrofusion using nanosecond electric pulses

Electrofusion is an efficient method for fusing cells using short-duration high-voltage electric pulses. However, electrofusion yields are very low when fusion partner cells differ considerably in their size, since the extent of electroporation (consequently membrane fusogenic state) with conventionally used microsecond pulses depends proportionally on the cell radius. We here propose a new and innovative approach to fuse cells with shorter, nanosecond (ns) pulses. Using numerical calculations we demonstrate that ns pulses can induce selective electroporation of the contact areas between cells (i.e. the target areas), regardless of the cell size. We then confirm experimentally on B16-F1 and CHO cell lines that electrofusion of cells with either equal or different size by using ns pulses is indeed feasible. Based on our results we expect that ns pulses can improve fusion yields in electrofusion of cells with different size, such as myeloma cells and B lymphocytes in hybridoma technology.

DERIVING AN ELECTRIC CIRCUIT EQUIVALENT MODEL OF CELL MEMBRANE PORES IN ELECTROPORATION

Electroporation is the formation of reversible pores in cell membranes under a brief pulse of high electric field. Dynamics of pore formation during electroporation suggests that the transmembrane potential would settle approximately at the threshold transmembrane potential and the transmembrane resistance would decrease significantly from the state of relaxation. The current electric circuit equivalent models for electroporation containing time-invariant, static and passive components are unable to capture the pore dynamics. A biophysically-inspired electric circuit equivalent model containing dynamic components for membrane pores has been derived using biological parameters. The model contains a voltage-controlled resistor driven by a two-stage cascaded integrator that is activated through a voltage-gated switch. Simulation results with the derived model showed higher accuracy compared to a commonly used model, where the transmembrane resistance decreased million-fold at the onset of electroporation and the transmembrane potential settled at 99.5% of the critical transmembrane potential, thus enabling improved dynamic behavior modeling ability of the pores in electroporation. The derived model allows fast and reliable analysis of this biophysical phenomenon and potentially aids in optimization of various parameters involved in electroporation.

Electroporation of intracellular liposomes using nanosecond electric pulses--a theoretical study

Nanosecond (ns) electric pulses of sufficient amplitude can provoke electroporation of intracellular organelles. This paper investigates whether such pulses could provide a method for controlled intracellular release of a content of small internalized artificial lipid vesicles (liposomes). To estimate the pulse parameters needed to selectively electroporate liposomes while keeping the plasma and nuclear membranes intact, we constructed a numerical model of a biological cell containing a nucleus and liposomes of different sizes (with radii from 50 to 500 nm), which were placed in various sites in the cytoplasm. Our results show that under physiological conditions selective electroporation is only possible for the largest liposomes and when using very short pulses (few ns). By increasing the liposome interior conductivity and/or decreasing the cytoplasmic conductivity, selective electroporation of even smaller liposomes could be achieved. The location of the liposomes inside the cell does not play a significant role, meaning that liposomes of similar size could all be electroporated simultaneously. Our results indicate the possibility of using ns pulse treatment for liposomal drug release.

Microdosimetric study for nanosecond pulsed electric fields on a cell circuit model with nucleus

Recently, scientific interest in electric pulses, always more intense and shorter and able to induce biological effects on both plasma and nuclear membranes, has greatly increased. Hence, microdosimetric models that include internal organelles like the nucleus have assumed increasing importance. In this work, a circuit model of the cell including the nucleus is proposed, which accounts for the dielectric dispersion of all cell compartments. The setup of the dielectric model of the nucleus is of fundamental importance in determining the transmembrane potential (TMP) induced on the nuclear membrane; here, this is demonstrated by comparing results for three different sets of nuclear dielectric properties present in the literature. The results have been compared, even including or disregarding the dielectric dispersion of the nucleus. The main differences have been found when using pulses shorter than 10 ns. This is due to the fact that the high spectral components of the shortest pulses are differently taken into account by the nuclear membrane transfer functions computed with and without nuclear dielectric dispersion. The shortest pulses are also the most effective in porating the intracellular structures, as confirmed by the time courses of the TMP calculated across the plasma and nuclear membranes. We show how dispersive nucleus models are unavoidable when dealing with pulses shorter than 10 ns because of the large spectral contents arriving above 100 MHz, i.e., over the typical relaxation frequencies of the dipolar mechanism of the molecules constituting the nuclear membrane and the subcellular cell compartments.

Computationally efficient simulation of electrical activity at cell membranes interacting with self-generated and externally imposed electric fields

OBJECTIVE

We present a computational method that implements a reduced set of Maxwell's equations to allow simulation of cells under realistic conditions: sub-micron cell morphology, a conductive non-homogeneous space and various ion channel properties and distributions.

APPROACH

While a reduced set of Maxwell's equations can be used to couple membrane currents to extra- and intracellular potentials, this approach is rarely taken, most likely because adequate computational tools are missing. By using these equations, and introducing an implicit solver, numerical stability is attained even with large time steps. The time steps are limited only by the time development of the membrane potentials.

MAIN RESULTS

This method allows simulation times of tens of minutes instead of weeks, even for complex problems. The extracellular fields are accurately represented, including secondary fields, which originate at inhomogeneities of the extracellular space and can reach several millivolts. We present a set of instructive examples that show how this method can be used to obtain reference solutions for problems, which might not be accurately captured by the traditional approaches. This includes the simulation of realistic magnitudes of extracellular action potential signals in restricted extracellular space.

SIGNIFICANCE

The electric activity of neurons creates extracellular potentials. Recent findings show that these endogenous fields act back onto the neurons, contributing to the synchronization of population activity. The influence of endogenous fields is also relevant for understanding therapeutic approaches such as transcranial direct current, transcranial magnetic and deep brain stimulation. The mutual interaction between fields and membrane currents is not captured by today's concepts of cellular electrophysiology, including the commonly used activation function, as those concepts are based on isolated membranes in an infinite, isopotential extracellular space. The presented tool makes simulations with detailed morphology and implicit interactions of currents and fields available to the electrophysiology community.

Nonthermal irreversible electroporation: fundamentals, applications, and challenges

Tissue ablation is an essential procedure for the treatment of many diseases. In the last decade, a nonthermal tissue ablation using intensive pulsed electric fields, called nonthermal irreversible electroporation (NTIRE), has rapidly emerged. The exact mechanisms responsible for cell death by NTIRE, however, are currently unknown. Nevertheless, the technique's remarkable ability to ablate tissue in the proximity of larger blood vessels, to preserve tissue architecture, short procedure duration, and shortened postoperative recovery period rapidly moved NTIRE from bench to bed side. This work provides an overview on the development of NTIRE, its current state-of-the-art, challenges, and future needs.

Modeling extracellular electrical stimulation: II. Computational validation and numerical results

The validity of approximate equations describing the membrane potential under extracellular electrical stimulation (Meffin et al 2012 J. Neural Eng. 9 065005) is investigated through finite element analysis in this paper. To this end, the finite element method is used to simulate a cylindrical neurite under extracellular stimulation. Laplace's equations with appropriate boundary conditions are solved numerically in three dimensions and the results are compared to the approximate analytic solutions. Simulation results are in agreement with the approximate analytic expressions for longitudinal and transverse modes of stimulation. The range of validity of the equations describing the membrane potential for different values of stimulation and neurite parameters are presented as well. The results indicate that the analytic approach can be used to model extracellular electrical stimulation for realistic physiological parameters with a high level of accuracy.

Modeling extracellular electrical stimulation: I. Derivation and interpretation of neurite equations

Neuroprosthetic devices, such as cochlear and retinal implants, work by directly stimulating neurons with extracellular electrodes. This is commonly modeled using the cable equation with an applied extracellular voltage. In this paper a framework for modeling extracellular electrical stimulation is presented. To this end, a cylindrical neurite with confined extracellular space in the subthreshold regime is modeled in three-dimensional space. Through cylindrical harmonic expansion of Laplace's equation, we derive the spatio-temporal equations governing different modes of stimulation, referred to as longitudinal and transverse modes, under types of boundary conditions. The longitudinal mode is described by the well-known cable equation, however, the transverse modes are described by a novel ordinary differential equation. For the longitudinal mode, we find that different electrotonic length constants apply under the two different boundary conditions. Equations connecting current density to voltage boundary conditions are derived that are used to calculate the trans-impedance of the neurite-plus-thin-extracellular-sheath. A detailed explanation on depolarization mechanisms and the dominant current pathway under different modes of stimulation is provided. The analytic results derived here enable the estimation of a neurite's membrane potential under extracellular stimulation, hence bypassing the heavy computational cost of using numerical methods.

Novel passive element circuits for microdosimetry of nanosecond pulsed electric fields

Microdosimetric models for biological cells have assumed increasing significance in the development of nanosecond pulsed electric field technology for medical applications. In this paper, novel passive element circuits, able to take into account the dielectric dispersion of the cell, are provided. The circuital analyses are performed on a set of input pulses classified in accordance with the current literature. Accurate data in terms of transmembrane potential are obtained in both time and frequency domains for different cell models. In addition, a sensitivity study of the transfer function for the cell geometrical and dielectric parameters has been carried out. This analysis offers a new, simple, and efficient tool to characterize the nsPEFs' action at the cellular level.

Asymptotic model of electrical stimulation of nerve fibers

We present a novel theory and computational algorithm for modeling electrical stimulation of nerve fibers in three dimensions. Our approach uses singular perturbation to separate the full 3D boundary value problem into a set of 2D "transverse" problems coupled with a 1D "longitudinal" problem. The resulting asymptotic model contains not one but two activating functions (AF): the longitudinal AF that drives the slow development of the mean transmembrane potential and the transverse AF that drives the rapid polarization of the fiber in the transverse direction. The asymptotic model is implemented for a prototype 3D cylindrical fiber with a passive membrane in an isotropic extracellular region. The validity of this approach is tested by comparing the numerical solution of the asymptotic model to the analytical solutions. The results show that the asymptotic model predicts steady-state transmembrane potential directly under the electrodes with the root mean square error of 0.539 mV, i.e., 1.04% of the maximum transmembrane potential. Thus, this work has created a computationally efficient algorithm that facilitates studies of the complete spatiotemporal dynamics of nerve fibers in three dimensions.

Effects of nanosecond pulsed electric fields on the activity of a Hodgkin and Huxley neuron model

The cell membrane poration is one of the main assessed biological effects of nanosecond pulsed electric fields (nsPEF). This structural change of the cell membrane appears soon after the pulse delivery and lasts for a time period long enough to modify the electrical activity of excitable membranes in neurons. Inserting such a phenomenon in a Hodgkin and Huxley neuron model by means of an enhanced time varying conductance resulted in the temporary inhibition of the action potential generation. The inhibition time is a function of the level of poration, the pore resealing time and the background stimulation level of the neuron. Such results suggest that the neuronal activity may be efficiently modulated by the delivery of repeated pulses. This opens the way to the use of nsPEFs as a stimulation technique alternative to the conventional direct electric stimulation for medical applications such as chronic pain treatment.

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.

Equivalent pulse parameters for electroporation

Electroporation-based applications require the use of specific pulse parameters for a successful outcome. When recommended values of pulse parameters cannot be set, similar outcomes can be obtained by using equivalent pulse parameters. We determined the relations between the amplitude and duration/number of pulses resulting in the same fraction of electroporated cells. Pulse duration was varied from 150 ns to 100 ms, and the number of pulses from 1 to 128. Fura 2-AM was used to determine electroporation of cells to Ca(2+). With longer pulses or higher number of pulses, lower amplitudes are needed for the same fraction of electroporated cells. The expression derived from the model of electroporation could describe the measured data on the whole interval of pulse durations. In a narrower range (0.1-100 ms), less complex, logarithmic or power functions could be used instead. The relation between amplitude and number of pulses could best be described with a power function or an exponential function. We show that relatively simple two-parameter power or logarithmic functions are useful when equivalent pulse parameters for electroporation are sought. Such mathematical relations between pulse parameters can be important in planning of electroporation-based treatments, such as electrochemotherapy and nonthermal irreversible electroporation.

Microdosimetry for nanosecond pulsed electric field applications: a parametric study for a single cell

A microdosimetric study of nanosecond pulsed electric fields, including dielectric dispersivity of cell compartments, is proposed in our paper. A quasi-static solution based on the Laplace equation was adapted to wideband signals and used to address the problem of electric field estimation at cellular level. The electric solution was coupled with an asymptotic electroporation model able to predict membrane pore density. An initial result of our paper is the relevance of the dielectric dispersivity, providing evidence that both the transmembrane potential and the pore density are strongly influenced by the choice of modeling used. We note the crucial role played by the dielectric properties of the membrane that can greatly impact on the poration of the cell. This can partly explain the selective action reported on cancerous cells in mixed populations, if one considers that tumor cells may present different dielectric responses. Moreover, these kinds of studies can be useful to determine the appropriate setting of nsPEF generators as well as for the design and optimization of new-generation devices.

Induced transmembrane voltage and its correlation with electroporation-mediated molecular transport

Exposure of a cell to an electric field results in inducement of a voltage across its membrane (induced transmembrane voltage, DeltaPsi (m)) and, for sufficiently strong fields, in a transient increase of membrane permeability (electroporation). We review the analytical, numerical and experimental methods for determination of DeltaPsi (m) and a method for monitoring of transmembrane transport. We then combine these methods to investigate the correlation between DeltaPsi (m) and molecular transport through an electroporated membrane for isolated cells of regular and irregular shapes, for cells in dense suspensions as well as for cells in monolayer clusters. Our experiments on isolated cells of both regular and irregular shapes confirm the theoretical prediction that the highest absolute values of DeltaPsi (m) are found in the membrane regions facing the electrodes and that electroporation-mediated transport is confined to these same regions. For cells in clusters, the location of transport regions implies that, at the field strengths sufficient for electroporation, the cells behave as electrically insulated (i.e., as individual) cells. In contrast, with substantially weaker, nonelectroporating fields, potentiometric measurements show that the cells in these same clusters behave as electrically interconnected cells (i.e., as one large cell). These results suggest that sufficiently high electric fields affect the intercellular pathways and thus alter the electric behavior of the cells with respect to their normal physiological state.

Gene electrotransfer: from biophysical mechanisms to in vivo applications : Part 2 - In vivo developments and present clinical applications

Gene electrotransfer can be obtained not just on single cells in diluted suspension. For more than 10 years, this is a quasi routine strategy in tissue on the living animal and a few clinical trials have now been approved. New problems have been brought by the close contacts of cells in tissue both on the local field distribution and on the access of DNA to target cells. They need to be solved to provide a further improvement in the efficacy and safety of protein expression. There is a competition between gene transfer and cell destruction. Nevertheless, present results are indicative that electrotransfer is a promising approach for gene therapy. High level and long-lived expression of proteins can be obtained in muscles. This is used for a successful method of electrovaccination.

Measuring the induced membrane voltage with Di-8-ANEPPS

Placement of a cell into an external electric field causes a local charge redistribution inside and outside of the cell in the vicinity of the cell membrane, resulting in a voltage across the membrane. This voltage, termed the induced membrane voltage (also induced transmembrane voltage, or induced transmembrane potential difference) and denoted by DeltaPhi, exists only as long as the external field is present. If the resting voltage is present on the membrane, the induced voltage superimposes (adds) onto it. By using one of the potentiometric fluorescent dyes, such as di-8-ANEPPS, it is possible to observe the variations of DeltaPhi on the cell membrane and to measure its value noninvasively. di-8-ANEPPS becomes strongly fluorescent when bound to the lipid bilayer of the cell membrane, with the change of the fluorescence intensity proportional to the change of DeltaPhi. This video shows the protocol for measuring DeltaPhi using di-8-ANEPPS and also demonstrates the influence of cell shape on the amplitude and spatial distribution of DeltaPhi.