There has been a general research trend towards exploring possible health hazards arising from wireless communication systems, and with it an increasing interest in electrical modeling of tissues from knowledge of sub-tissue and cellular details and exploring possible effects [1]. Schwan et al. [2,3], were among the first to provide a framework for understanding the electrical behavior of tissues as an assembly of cells, and to employ a spherical cell model and the solution of Laplace's equation to arrive at an analytical model for the transmembrane potential (TMP) induced through the external field across the membrane model. Theoretical estimation of bio-impedance and induced membrane voltages are useful in studying electroporation in membranes, which might be applicable for treatment of certain cancers [4].
While analytical formulas might provide good TMP models for simple cell shapes, they fail to provide the accurate TMP distribution for more complex cell shapes. The need to provide estimates for more realistic cell shapes have been shown [5, 6, 7], and numerical electromagnetic modeling has been employed together with several cell models [8, 9, 10, 11]. In these works, however, in order to evaluate the induced TMP, the membranes are often included in the computational mesh directly. Obviously, this increases the computational complexity because the numerical grid (mesh) has objects with extremely large contrasts in size: the membrane model on the order of several nanometers, and the inner volume mesh-cells on the order of micrometers. With most electromagnetic simulation engines, such an arrangement can lead to increased computational load, and poor quality results [12]. To remedy this problem at very low frequencies, it has been shown that the membranes can be substituted by a zero-thickness boundary condition [12]. Upon numerical simulation, the new model will produce the same field results in all regions, and a voltage drop at the interface equal to that of the original model. This method is applicable to low frequencies, however.
This paper shows why the presence of membranes loses significance in the overall impedance, and can be neglected in theoretical estimations of tissue impedance after a certain frequency, and further shows how to calculate the induced TMP without directly modeling the membranes at such frequencies. This obviously reduces the number of mesh elements, allows uniform element sizes for better matrix-conditioning, and makes computer simulation of multiple-cell tissue models possible.
In order to investigate the effect of thin cell membranes on the impedance at sufficiently high frequencies, we first assume fairly general conditions:
The membranes can be considered as insulating, and
They occupy a sufficiently small volume fraction.
We will then find conditions under which the absence of membranes does not affect the impedance and fields obtained at non-membrane regions in the model.
Intuitively, at low frequencies, charging and discharging can occur at the site of poorly conducting membranes in response to the externally induced electric fields, which act more or less as barriers for the free electrolyte charges in the cytoplasm and the extra-cellular aqueous medium. Since the membranes are extremely thin (about 5.5 nm), this capacitive charging results in extremely high concentrations of electric field within the membrane, known as the membrane amplification effect [5]. At higher frequencies, where displacement currents dominate over conduction currents, this effect gradually vanishes. Above the transition frequency, the presence of cell membranes in the electrical model has almost no effect on the overall electric field distribution and impedance.
To investigate this assumption more rigorously, let us consider the local model around the membranes [5,14] as presented in Figure 1. The local field distribution in this region is quasi-electrostatic and can be solved by defining a voltage excitation
We assume all materials to be non-magnetic, i.e. μ =1is assumed for all regions. We assume
We now compare the impedance of the models with and without membranes in Figure 1 for both polarizations. For fields parallel to the membrane, corresponding to the local models in Figure 1a, the fields in non-membrane regions are obviously independent of the presence of membrane. This is justified by analogy to circuit concepts; parallel capacitors have the same potential, independent of each other and equal to that of the common terminal. For field polarizations perpendicular to the membrane, the series model in Figure 1b can be considered. For the fields in non-membrane regions to be independent of membrane presence, the total current density in models with and without membranes should be the same under equal voltage excitations, which means that the two models should have the same impedance, or that the admittance ratio
where
For sufficiently low frequencies (below the first relaxation of pure water around 20 GHz), we assume the aqueous phase to be described by ε
Intuitively, the displacement current dominates in the total current density for ω ≫ σ/εrε0, and the conduction current (and the blocking effect of the membrane for free charges) loses significance in determining the overall field. At the same time, even though the conduction current in the aqueous phases gradually increases (due to the increasing field penetration into the cytoplasm, upper-bounded by an average field value in the tissue), the accumulated charge on either side of the membrane unboundedly decreases with frequency, since ρ
For
The constant real-part model used for ε
As a side note, we see that heating and thermal absorption of energy from the applied field (proportional to
Having obtained the correct field distributions at all regions (except the membrane) using analytical or theoretical methods, one can obtain the externally-induced transmembrane voltages as a post-processing step by considering (Eq.3) from Maxwell's equations:
The expression inside the brackets represents the addition of the conduction and displacement currents, which we denote by
from which the electric field across the membrane
The induced transmembrane voltage due to the external field;
which can be used to recover the membrane potentials (or electric fields) after obtaining the numerical field results.
To verify the proposed procedure, we can compare a direct computation of TMP with the one obtained by removing the membranes from the numerical mesh and using (Eq.6). We can do this numerical experiment for a non-trivial example cell shape.
To elaborate, we use a biconcave cell model with a diameter of 16 μm, whose least thickness at the center is 4 μm. The interior and exterior are filled with the same aqueous medium ε
At 1 MHz, we are far below the characteristic frequency of
From a computational point of view, this will obviously reduce the number of elements and, more remarkably, help maintain a uniform mesh size across the model. For tightly packed tissue models composed of membranes separating cytoplasmic regions, such simplification may drastically reduce the computational effort by allowing us to solve an almost homogeneous tissue model. Consequently, larger multiple-cell scenarios can be solved with implications for tissue-level electroporation studies [5].
We shall underline our basic assumption of dm ≪ daq , or equivalently fm ≪ 1, which means that the local volume fraction of lipid membranes should be rather small. This condition is essential for constructing a local model such as that in Figure 1, and thus for the whole conclusion to hold. This condition usually holds for cell membranes, since cells have dimensions typically extending to and beyond several micrometers. It does not hold for dense intracellular arrangements of membranes or the outer segment of retinal rod cells, which have a stacked arrangement of membranous disks. In such cases, the effective medium approach might be used to simplify the computational model.