^{1,a),b)}, Maxim Deminsky

^{2}, V. Bogdan Neculaes

^{1}, V. Chashihin

^{2}, Andrey Knizhnik

^{2}and Boris Potapkin

^{2}

### Abstract

While electromagnetic fields induce structural changes in cell membranes, particularly electroporation, much remains to be understood about membrane level temperature gradients. For instance, microwaves induce cell membrane temperature gradients (∇T) and bioeffects with little bulk temperature change. Recent calculations suggest that nanosecond pulsed electric fields (nsPEFs) may also induce such gradients that may additionally impact the electroporation threshold. Here, we analytically and numerically calculate the induced ∇T as a function of pulse duration and pulse repetition rate. We relate ∇T to the thermally induced cell membrane electric field (Em ) by assuming the membrane behaves as a thermoelectric such that Em ∼ ∇T. Focusing initially on applying nsPEFs to a uniform membrane, we show that reducing pulse duration and increasing pulse repetition rate (or using higher frequency for alternating current (AC) fields) maximizes the magnitude and duration of ∇T and, concomitantly, Em . The maximum ∇T initially occurs at the interface between the cell membrane and extracellular fluid before becoming uniform across the membrane, potentially enabling initial molecular penetration and subsequent transport across the membrane. These results, which are equally applicable to AC fields, motivate further studies to elucidate thermoelectric behavior in a model membrane system and the coupling of the Em induced by ∇T with that created directly by the applied field.

I. INTRODUCTION

II. CELL MEMBRANE TEMPERATURE GRADIENTS—ANALYTICAL MODEL

A. Analytic approximation of cell membrane temperature gradient

B. Spatial and temporal progression of cell membrane temperature gradient

III. 3D AND 1D NUMERICAL CALCULATIONS OF CELL HEATING DYNAMICS

IV. MULTIPLE PULSE REGIME

V. CONCLUDING REMARKS

### Key Topics

- Cell membranes
- 122.0
- Electric fields
- 42.0
- Bioelectrochemistry
- 32.0
- Electroporation
- 31.0
- Cell membrane transport
- 18.0

## Figures

Simplified schematic of a spherical biological cell with a concentric spherical nucleus (e.g., Refs. 1 and 44 ). The cell consists of five regions characterized by thermal conductivity (λ), electric conductivity (σ), and permittivity (ε): the nucleoplasm (np), nuclear envelope (ne), cytoplasm (cp), and cell membrane (m). The nuclear and cell radii are given by R1 and R2 , respectively, and thickness of the nuclear envelope and cell membrane are denoted by d1 and d2 , respectively.

Simplified schematic of a spherical biological cell with a concentric spherical nucleus (e.g., Refs. 1 and 44 ). The cell consists of five regions characterized by thermal conductivity (λ), electric conductivity (σ), and permittivity (ε): the nucleoplasm (np), nuclear envelope (ne), cytoplasm (cp), and cell membrane (m). The nuclear and cell radii are given by R1 and R2 , respectively, and thickness of the nuclear envelope and cell membrane are denoted by d1 and d2 , respectively.

Lumped equivalent circuit for electric field application to biological cells, where Rm , Rc , Rfluid , and RL are the resistances of the plasma membrane, cytoplasm, extracellular fluid, and load, respectively, C is the plasma membrane capacitance, and U is the applied voltage. Because RL is much less than the other resistances, it is neglected in analyses.

Lumped equivalent circuit for electric field application to biological cells, where Rm , Rc , Rfluid , and RL are the resistances of the plasma membrane, cytoplasm, extracellular fluid, and load, respectively, C is the plasma membrane capacitance, and U is the applied voltage. Because RL is much less than the other resistances, it is neglected in analyses.

Analytic calculation of the spatial evolution of the temperature across the cell for 10 ns (solid) and 10 μs (dashed) pulses after (a) 18 ns, (b) 18 μs, and (c) 85 μs. The characteristic heat conduction frequency for each pulse condition is τc −1 = 3 × 104 s−1.

Analytic calculation of the spatial evolution of the temperature across the cell for 10 ns (solid) and 10 μs (dashed) pulses after (a) 18 ns, (b) 18 μs, and (c) 85 μs. The characteristic heat conduction frequency for each pulse condition is τc −1 = 3 × 104 s−1.

Analytic calculation of the spatial evolution of the temperature gradient (∇T) across the cell membrane (x = 1 represents the intersection of the membrane with the extracellular fluid) calculated for 10 ns (solid) and 10 μs (dashed) pulses after (a) 18 ns, (b) 18 μs, and (c) 85 μs. The characteristic heat conduction frequency for each pulse condition is τc −1 = 3 × 104 s−1.

Analytic calculation of the spatial evolution of the temperature gradient (∇T) across the cell membrane (x = 1 represents the intersection of the membrane with the extracellular fluid) calculated for 10 ns (solid) and 10 μs (dashed) pulses after (a) 18 ns, (b) 18 μs, and (c) 85 μs. The characteristic heat conduction frequency for each pulse condition is τc −1 = 3 × 104 s−1.

(a) Time and (b) spatial dependence of a Gaussian heat source (with r = 0 the center of the cell) used for determining cell and membrane temperature gradients.

(a) Time and (b) spatial dependence of a Gaussian heat source (with r = 0 the center of the cell) used for determining cell and membrane temperature gradients.

Temperature across the cell and extracellular fluid normalized to the maximum temperature increase due to a single pulse of duration τ pulse /τdiff = 0.01 at different (dimensionless) times using a 3D model of the cell. Arc length represents location with the center of the cell at zero and the inner edge of the membrane at one.

Temperature across the cell and extracellular fluid normalized to the maximum temperature increase due to a single pulse of duration τ pulse /τdiff = 0.01 at different (dimensionless) times using a 3D model of the cell. Arc length represents location with the center of the cell at zero and the inner edge of the membrane at one.

Temperature gradient in the system normalized to the maximum ratio of the maximum temperature increase to cell radius for exposure to pulses of duration τ pulse/τ dif = 0.01 at different (dimensionless) times using a 3D model of the cell. Arc length represents location with the center of the cell at zero and the inner edge of the membrane at one.

Temperature gradient in the system normalized to the maximum ratio of the maximum temperature increase to cell radius for exposure to pulses of duration τ pulse/τ dif = 0.01 at different (dimensionless) times using a 3D model of the cell. Arc length represents location with the center of the cell at zero and the inner edge of the membrane at one.

Temperature gradient normalized to the ratio of membrane temperature change due to a single pulse and the cell radius (ΔT pulse/R2 ) as a function of pulse duration normalized to thermal diffusion time. The analytic gradient law agrees well with 1D and 3D numerical calculations at longer pulses and permit estimation when numerical approaches diverge at shorter pulse duration due to meshing constraints.

Temperature gradient normalized to the ratio of membrane temperature change due to a single pulse and the cell radius (ΔT pulse/R2 ) as a function of pulse duration normalized to thermal diffusion time. The analytic gradient law agrees well with 1D and 3D numerical calculations at longer pulses and permit estimation when numerical approaches diverge at shorter pulse duration due to meshing constraints.

1D simulation (symbols) and analytic calculations (lines) of the maximum and minimum temperature gradient normalized to the ratio of membrane temperature change due to a single pulse and the cell radius (ΔT pulse/R2 ) as a function of time normalized to thermal diffusion time, τdiff , for pulse trains with duration τpulse = 10−6 τdiff and pulse repetition times, τrep , of 2.5 × 10−5 τdiff and 2.5 × 10−2 τdiff , where τdiff = 7 × 10−4 s is the thermal diffusion time.

1D simulation (symbols) and analytic calculations (lines) of the maximum and minimum temperature gradient normalized to the ratio of membrane temperature change due to a single pulse and the cell radius (ΔT pulse/R2 ) as a function of time normalized to thermal diffusion time, τdiff , for pulse trains with duration τpulse = 10−6 τdiff and pulse repetition times, τrep , of 2.5 × 10−5 τdiff and 2.5 × 10−2 τdiff , where τdiff = 7 × 10−4 s is the thermal diffusion time.

## Tables

The parameters of the model cell under investigation. The physical quantities relate to the temperature of 320 K. 44

The parameters of the model cell under investigation. The physical quantities relate to the temperature of 320 K. 44

Maximum temperature difference, temperature gradient, and induced electric field across the cell membrane due to pulsed electric field application onto biological cells. Note that we apply Eq. (3) in the text to the multiple pulse conditions because the time between pulses far exceeds the thermal diffusion time.

Maximum temperature difference, temperature gradient, and induced electric field across the cell membrane due to pulsed electric field application onto biological cells. Note that we apply Eq. (3) in the text to the multiple pulse conditions because the time between pulses far exceeds the thermal diffusion time.

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