# Pol Grasland-Mongrain$${}^{a}$$, Cyril Lafon$${}^{b}$$, (a) Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon, France (b) 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 18th 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. While the first impedance measurements were focused on cardiovascular parameters, techniques are now being applied to the whole body, as electrical impedance present important variations through the human body.

A few techniques have been proposed since to image this parameter. Combining electrical current, magnetic fields, and acoustics in various ways have led to interesting techniques, which are presented here. It includes Electrical Impedance Tomography, Magnetic Resonance Electrical Impedance Tomography, Acousto-Electric Imaging, Lorentz Force Electrical Impedance Tomography and Magneto-Acoustic Tomography.

## The electrical properties of a medium

The electrical properties of a medium can be described by its global impedance $$Z$$, linking voltage $$V$$ and intensity $$I$$ with Ohm’s law $$U=ZI$$, or locally by its impeditivity: $$\mathbf{E}(\mathbf{r},t)=\rho(\mathbf{r})\mathbf{j}(\mathbf{r},t)$$, where $$\mathbf{E}$$ is the electric field in a given point $$\mathbf{r}$$ at a time $$t$$, $$\mathbf{j}$$ the current density at this point and time and $$\rho$$ the local impeditivity11if $$\rho$$ is complex, it can be separated as the sum of a real part, the resistivity, and an imaginary part, the reactivity. But for electrical impedance imaging techniques, authors prefer often to use the admittivity $$\sigma$$, defined as $$\mathbf{j}(\mathbf{r},t)=\sigma(\mathbf{r})\mathbf{E}(\mathbf{r},t)$$, a quantity defined in Siemens per meter (S.m $${}^{-1}$$)

The admittivity is written as the sum of a real and imaginary part according to the relation $$\sigma=\sigma^{\prime}+i\sigma^{\prime\prime}$$, where $$\sigma^{\prime}$$ is the electrical conductivity and $$\sigma^{\prime\prime}$$ the susceptivity. Admittivity and electrical conductivity are often mixed up when imaginary part plays a minor role.

The conductivity $$\sigma^{\prime}$$ 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, $$\sigma^{\prime}_{s}$$ which represents the resistive losses in the medium (resistance to free particles motion) and $$\sigma^{\prime}_{d}$$ which represents the dielectric losses (resistance to the movement of bound particles).

The susceptivity $$\sigma^{\prime\prime}$$ 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(Jackson 1975). 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 $$\epsilon_{r}$$ can be defined as the susceptivity divided by the pulsation: $$\epsilon_{r}=\frac{\sigma^{\prime\prime}}{\epsilon_{0}\omega}$$. This magnitude is more often used because it has generally a smaller amplitude variation than $$\sigma^{\prime\prime}$$.

More details can be found in (Jackson 1975).

## Electrical properties of biological tissues

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

Biological tissues electrical impedance is due to the movement of ions. Generally speaking, tissues which have a low concentration of ions, such as adipose tissues, will be less conductive than ones with a high concentration, such as muscles.

As an example, the conductivity and the relative permittivity of four biological media (muscle, blood, fat, bone) are shown in Figure \ref{figConductivityPermittivity}-(A) and -(B): at 1 kHz, fat, muscle, blood and bone have respectively a conductivity of .04 S.m$${}^{-1}$$, .08 S.m$${}^{-1}$$, 0.3 S.m$${}^{-1}$$ and 0.7 S.m$${}^{-1}$$, and a relative permittivity of 7.10$${}^{3}$$, 20.10$${}^{3}$$, 30.10$${}^{3}$$ and 700.10$${}^{3}$$.

Detailed measurements in many organs can be found in (Gabriel 1996), (Gabriel 1996a), considered as a reference.