In this thesis, modelling and simulation of the effects of electric fields on a single spherical cell are undertaken. Of interest is the effect that different electric field waveforms have on the induced transmembrane potential of the cell and by consequence the electropermeabilization of the cell membrane in terms of pore density or the fraction of the cell area that the pores occupy. Conventional biotechnology processes of electroporation make use of unipolar electric field pulses, which are known to generate undesirable conditions such as asymmetrical electropermeabilization. These electrical protocols also contribute to lower efficiencies in electroporation based applications (in terms of uptake of molecules in the cell) by being sensitive to cell radii. Until recently, theoretical models of electroporation have lagged behind the experimental research. In order to optimize the efficiency of electroporation, it is important to consider as many biological and physical aspects as possible and it is a necessity that a variety of electric pulse parameters be tested. Thus a comprehensive model which can predict electropermeabilization as a result of any form of applied electric field and other important electroporation parameters is necessary. None of the existing theoretical modelling studies present simulations of dynamic electroporation modelling as a cell response to bipolar electric field wave-shapes. Developing such a model is the aim of this thesis. In this thesis two numerical models are developed. These models consider electroporation as a dynamic process and include the non-linear dynamic effects of membrane electropermeabilization. The first model assumes all pores are identical and small (0.76~nm radius) and is capable of simulating transmembrane potential and pore density temporally and spatially, given any form of applied electric field and other important electroporation system parameters such as external medium, membrane, and cytoplasm complex dielectric properties. The piece-wise step response model presented here is used to simulate cell response to several different applied electric field wave-shape pulses.% including a unipolar square wave, bipolar square wave, bipolar sine wave, bipolar rectangular wave (rectangular pulse train), and a bipolar triangular wave. Additional results from the first model demonstrate how the efficiency of electroporation related applications can be significantly improved by appropriately adjusting the parameters of the applied electric field and the extracellular conductivity. Emphasis is given on the normalization of the degree of electroporation (in terms of pore density) for two cell radii (7.5~$\mu$m and 15~$\mu$m). Although, these results gave a fair indication of the extent of electroporation in terms of pore density, the approximation that all pores have the same size, and do not change with time, may not be appropriate. There is a need to model electroporation so as to reflect the growth or shrinkage of pores with time, as well as efficiently handle arbitrary waveshapes of electric fields. The additional information about pore radius evolution gives a more realistic picture of the extent of electroporation, especially if one is to model for longer time (longer than 1~$\mu$s) or if an application necessarily required existence of larger pores (radius lager than 1~nm) rather than just the total pore area. Pore radius and pore numbers affect the transmembrane potential, which in turn affects pore density and pore radius. Literature includes information on spatial and temporal aspects of pore radius evolution. However, the electric fields used in these models were limited to unipolar DC pulses and details of temporal and spatial evolution of transmembrane potential and pore radius have not been reported. The second model developed in this thesis simulates spatial and temporal aspects of pore radius as an effect of any given form of applied electric field (including unipolar or bipolar), and other important electroporation system parameters. The transmembrane potential and pore radii as function of time and angular position about the cell membrane are presented. The results show that pore radii tend to be more normalized when an AC (bipolar) field is used when compared to a DC (unipolar) field (pore radii ranging from 1~nm to 8~nm for DC protocol compared with 1~nm to 3.4~nm for AC protocol when the pulse amplitude used in both cases is such as to give a similar fractional pore area at the end of 2~$\mu$s). Additional simulation results from this model are used to compare the extent of electroporation in response to sinusoidal AC (bipolar) electric field pulses of two different frequencies in a range of extracellular conductivity for two different cell radii (7.5~$\mu$m and 15~$\mu$m). It is observed that a higher frequency (1~MHz) bipolar sinusoidal applied electric field pulse reduces the relative difference in fractional pore area for the two cell sizes compared to a lower frequency (100~kHz) pulse. Nevertheless for the high frequency, a significantly higher amplitude is required to create the same level of average fractional pore area. Asymmetry of fractional pore area between the two hemispheres of the cell is observed for both field protocols.
electroporation, modelling, alternating field
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