A study of the electric field distribution in erythrocyte and rod shape cells from direct RF exposure

Mathematical models of naturally "morphed" human erythrocytes: stomatocytes and echinocytes

We present two mathematical models that describe human red blood cells (RBCs) with morphologies that are attained naturally under certain patho-physiological conditions, namely stomatocytes and echinocytes. Muñoz San Martín et al. (Bioelectromagnetics 27:521-527, 2006) recently presented models of these shapes based on our previous set of parametric equations (Kuchel and Fackerell, Bull. Math. Biol. 61:209-220, 1999) that involve Jacobi elliptic functions and integrals. Thus, both discocytes and stomatocytes are described. Here, we derived the Cartesian forms of these new equations; and, in addition, present a realistic model of a Type III echinocyte, using prolate spheroids 'decorating' a central sphere at the vertices of an internal dodecahedron. The RBC models based on Cartesian equations have been used for representing the shape changes (morphological transformations or "morphing") that occur in RBCs under various experimental conditions; specifically, when the shape changes have been monitored by nuclear magnetic resonance (NMR) micro-imaging.

Polarizability of red blood cells with an anisotropic membrane

We predict the complex polarizability of a realistic model of a red blood cell (RBC), with an inhomogeneous dispersive and anisotropic membrane. In this model, the frequency-dependent complex electrical parameters of the individual cell layers are described by the Debye equation while the dielectric anisotropy of the cell membrane is taken into account by the different permittivities along directions normal and tangential to the membrane surface. The realistic shape of the RBC is described in terms of the Jacobi elliptic functions. To calculate the polarizability, we evoke the effective dipole moment method to determine the cell internal electric field distribution, employing an adaptive finite-element numerical approach. We have furthermore investigated the influence of the anisotropic membrane and dispersive electrical parameters of each individual cell layer on the total complex polarizability. Our findings suggest that the individual layer contribution depends on two factors: the volume of the layer and the associated induced electric field, which in turn is influenced by other layers of the cell. These results further show that the average polarizability spectra of the cell are significantly impacted by the anisotropy and associated dispersion of the cellular compartments.

Polarizability of shelled particles of arbitrary shape in lossy media with an application to hematic cells

We show that within the dipole approximation the complex polarizability of shelled particles of arbitrary shape can be written as the volume of the particle times a weighted average of the electric field in the particle, with weights determined by the differences in permittivities between the shells and the external, possibly lossy media. To calculate the electric field we use an adaptive-mesh finite-element method which is very effective in handling the irregular domains, material inhomogeneities, and complex boundary conditions usually found in biophysical applications. After extensive tests with exactly solvable models, we apply the method to four types of hematic cells: platelets, T-lymphocytes, erythrocytes, and stomatocytes. Realistic shapes of erythrocytes and stomatocytes are generated by a parametrization in terms of Jacobi elliptic functions. Our results show, for example, that if the average polarizability is the main concern, a confocal ellipsoid may be used as a model for a normal erythrocyte, but not for a stomatocyte. A comparison with experimental electrorotation data shows quantitatively the effect of an accurate geometry in the derivation of electrical cell parameters from fittings of theoretical models to the experimental data.

Analysis of the electric field induced forces in erythrocyte membrane pores using a realistic cell model

We calculate the induced electric stress forces on transient hydrophobic pores in the membrane of an erythrocyte exposed to an electric field. For this purpose, we use a finite element numerical technique and a realistic shape for the biconcave erythrocyte represented by a set of parametric equations in terms of Jacobi elliptic functions. The results clearly show that the electrical forces on the base and sidewalls of the pore favour the opening of the pore. A comparison of the force densities obtained for an unstretched flat membrane and for the realistic erythrocyte model shows that the thinning and curvature of the membrane cannot be neglected. We also show that the pore deformation depends strongly on the orientation of the pore with respect to the external field, and in particular is very small when the field is tangent to the membrane surface.

Model of a confined spherical cell in uniform and heterogeneous applied electric fields

Cells exposed to electric fields are often confined to a small volume within a solid tissue or within or near a device. Here we report on an approach to describing the frequency and time domain electrical responses of a spatially confined spherical cell by using a transport lattice system model. Two cases are considered: (1) a uniform applied field created by parallel plane electrodes, and (2) a heterogeneous applied field created by a planar electrode and a sharp microelectrode. Here fixed conductivities and dielectric permittivities of the extra- and intracellular media and of the membrane are used to create local transport models that are interconnected to create the system model. Consistent with traditional analytical solutions for spherical cells in an electrolyte of infinite extent, in the frequency domain the field amplification, G(m) (f) is large at low frequencies, f<1 MHz. G(m) (f) gradually decreases above 1 MHz and reaches a lower plateau at about 300 MHz, with the cell becoming almost "electrically invisible". In the time domain the application of a field pulse can result in altered localized transmembrane voltage changes due to a single microelectrode. The transport lattice approach provides modular, multiscale modeling capability that here ranges from cell membranes (5 nm scale) to the cell confinement volume ( approximately 40 microm scale).

Modeling normal and altered human erythrocyte shapes by a new parametric equation: Application to the calculation of induced transmembrane potentials

We present simple parametric equations in terms of Jacobi elliptic functions that provide a realistic model of abnormal variations in size which maintain the biconcave shape of a normal erythrocyte (anisocytosis) and abnormal variations in shape which maintain the original volume of the erythrocyte (poikilocytosis), as well as continuous deformations from the normal to the altered shapes. We illustrate our results with parameterizations of microcytes, macrocytes, and stomatocytes, and we apply these parameterizations to the numerical calculation of the induced transmembrane voltage in microcytes, macrocytes, and stomatocytes exposed to an external electromagnetic field of 1800 MHz.

Erythrocyte rouleau formation under polarized electromagnetic fields

We study the influence of an external electromagnetic field of 1.8 GHz in the formation or disaggregation of long rouleau of identical erythrocyte cells. In particular we calculate the variation of the transmembrane potential of an individual erythrocyte illuminated by the external field due to the presence of the neighboring erythrocytes in the rouleau, and compare the total electric energy of isolated cells with the total electric energy of the rouleau. We show that the polarization of the external electromagnetic field plays a fundamental role in the total energy variation of the cell system, and consequently in the formation or disaggregation of rouleau.

Modelling the internal field distribution in human erythrocytes exposed to MW radiation

This paper studies the internal electric field distribution in human erythrocytes exposed to MW radiation. For this purpose, an erythrocyte cell model is exposed to linearly polarized electromagnetic (EM) plane waves of frequency 900 MHz and the electric field within the cell is calculated by using a finite element (FE) technique with adaptive meshing. The results obtained show the dependence of the induced electric field distribution on the main modelling parameters, i.e., the electrical properties (permittivity and conductivity) of the membrane and cytoplasm and the orientation of the cell with respect to the applied field. It is found that for certain orientations, the field amplification within the membrane of the erythrocyte shape cell can be higher than the one observed in an equivalent simple spheroidal geometry cell, commonly used in bioelectromagnetism. The present work shows that a better insight of the interaction of electromagnetic fields with basic biological structures is obtained when the most possible realistic cell shape is used.