A brief survey on homogenization with a physical application

It is well known that electric potentials can be used in diagnostic devices to investigate the properties of biological tissues. Such techniques are essentially based on the possibility of determining the physiological properties of a living body by means of the knowledge of its electrical resistence. However, it has been observed that, applying high frequency alternating potentials to the body, a capacitive behaviour takes place, due to the electric polarization at the interface of the cell membranes, which act as capacitors. In this regards, the biological tissue is modelled as a composite media with a periodic microscopic structure with two finely mixed phases (intra and extra cellular) separated by an imperfect interface (cellular membrane). Hence, the homogenization theory seems to be the appropriate tool in order to discover the macroscopic behaviour of the tissue. In these notes we present the basic ideas of homogenization theory with the purpose of enabling the reader to apply this theory in physical problems. An example of such applications is given in the framework of electrical conduction in living tissues.