# Pol Grasland-Mongrain, Cyril Lafon$${}^{a}$$, (a) Inserm, U1032, LabTau, Lyon, F-69003, France ; Université de Lyon, Lyon, F-69003, France

Effects of electrical currents on biological tissues have been observed since the end of the XVIIIe century, with the observation of frog muscular movements due to electricity by Luigi Galvani. To predict the behavior of electrical currents in a medium, physicists are using the electrical impedance. Electrical impedance refers indeed to the ”difficulty” for an electrical current to spread through the medium. The impedance (and its inverse the conductance) refers to a global parameter, i.e., for a given volume or medium. The resistivity (and its inverse the conductivity) refers to a local parameter, at a given position. These different names are used depending on the context, and techniques presented in this review sometime use terms of impedance or conductivity without much difference.

Interestingly, the electrical impedance present important variations through the human body. This parameter have drawn interest for diagnosis, and many techniques have been proposed to map it. New technological concepts have been introduced recently, in the last 20 years, which have been made possible notably by improvements in other fields of physics, namely acoustics and magnetic resonance.

This review aims to list this bloom of new techniques. They have been grouped here into a few categories based on the main underlying physical phenomenon: use of electrical currents only, of magnetic fields, of a combination of acoustics and electrical currents, etc.

## The electrical properties of the biological tissues

From the electromagnetic point of view, biological tissues are complex media, quite different from other materials

Most biological tissues are diamagnetic - except some proteins such as methaemoglobin which are paramagnetic. The human body is therefore generally considered as insensitive to the magnetic field, and a magnetic permeability equal to that of vacuum is almost systematically assumed (Bernard 2007).

The electrical properties of a medium are described by its complex electrical conductivity $$\underline{\sigma}(\omega)$$, defined as:

$$\mathbf{j}=\underline{\sigma}(\omega)\mathbf{E}\\$$

where $$\mathbf{j}$$ is the current density at a point, $$\mathbf{E}$$ the electric field at this point and $$\omega$$ the pulsation of the electric field. This complex electrical conductivity can be written as the sum of a real and imaginary part according to the relation $$\underline{\sigma}(\omega)=\sigma^{\prime}(\omega)+i\sigma^{\prime}$$, where $$\sigma^{\prime\prime}(\omega)$$ is the real electrical conductivity, $$i$$ the complex number whose square is -1 and $$\sigma^{\prime\prime}(\omega)$$ the imaginary electrical conductivity. This quantity represents the conductance of a volume, and has the unit Siemens per meter (S.m $${}^{-1}$$). The resistivity $$\rho$$ is defined as the inverse of $$\underline{\sigma}(\omega)$$, with $$\Omega$$.m as the unit.

The real electrical conductivity $$\sigma^{\prime}(\omega)$$ represents the loss of current due to the displacement of the free and bound charged particles. This quantity can be separated into the sum of two components, $$\sigma_{S}$$ which represents the resistive losses in the medium (resistance to free particles motion) and $$\sigma_{d}$$ which represents the dielectric losses (resistance to the movement of bound particles). Note that contrary to $$\sigma_{S}$$, the dielectric losses highly depend on frequency.

The imaginary electric conductivity $$\sigma^{\prime\prime}(\omega)$$ represents the polarizability of the material, and therefore its ability to store energy in the form of an electric field. Several polarization mechanisms exist, such as orientational polarization of rigid dipole molecules, ionic polarization, deformation of the electron cloud of atoms… Each type of polarization has a maximum response to a defined electric field frequency: at this frequency, a resonance mechanism leads to an absorption of energy in the medium. The relative permittivity $$\epsilon_{r}^{\prime}$$ can be defined as the imaginary electrical conductivity divided by the pulsation: $$\epsilon^{\prime}=\frac{\sigma^{\prime\prime}}{\epsilon_{0}\omega}$$. This magnitude is more often used for biological tissues because it has a smaller amplitude of variation than $$\sigma^{\prime\prime}(\omega)$$.

As illustrated in figure \ref{figLFEITIntroConducElecTissus1}-(A) from a model given by Schwan et al. (Schwan 1957), (Schwan 1957a), four major relaxation phenomena are observed experimentally in biological tissues at frequencies below the gighahertz: they are respectively called $$\alpha$$, $$\beta$$, $$\delta$$ and $$\gamma$$ (Grimnes 2008).

\label{figLFEITIntroConducElecTissus1}(A) Variation of the electrical conductivity and the relative permittivity typical of a tissue as a function of frequency. We observe several jumps, corresponding to relaxation phenomena, called $$\alpha$$, $$\beta$$, $$\delta$$ and $$\gamma$$ (adapted from (Schwan 1957)). Variation of the electrical conductivity (B) and relative permittivity (C) of fat, blood, muscle and bone as a function of frequency (adapted from (citation not found: Bernard_2007))

The relaxation $$\alpha$$ appears between a few hertz and a few kilohertz. It corresponds to a diffusion of the ions at the cell membrane. Due to this phenomenon, a double layer of ions appears at the surface of the membranes, which create local electrical dipoles. The real part of the electrical conductivity varies little while the relative permittivity $$\epsilon_{r}$$ decreases sharply.

The relaxation $$\beta$$ is between 500 kHz and 20 MHz. At these frequencies, the membrane cannot be considered as an insulator, and the real part of electrical conductivity becomes representative of the intracellular and extracellular conductivities. On the other hand, the membrane is less polarized, so the relative permittivity decreases.

The relaxation $$\delta$$, around 50 MHz, is rather weak and is due to the relaxation of water molecules in the vicinity of macromolecules.

Finally, the relaxation $$\gamma$$ intervenes only at very high frequency, around 17 GHz, and is due to the dipole orientation of the water molecules.

## The different electrical conductivity models of biological tissues

Biological fluids are often well modeled by electrolytes. Different empirical models have been proposed to model the frequency dependence o