Temperature distribution in tissues from a regular array of hot source implants: an analytical approximation

Abstract
An approximate analytical model based upon the bioheat transfer equation is derived and used to calculate temperatures within a perfused region implanted regularly with dielectrically coated hot source implants; for example, hot water tubes, electrically heated rods, or inductively heated ferromagnetic implants. The effect of a regular array of mutually parallel heat sources of cylindrical shape is approximated by idealizing one of the boundary conditions. The solution, as could be expected, is in terms of modified Bessel functions. In calculating the temperature of each thermoregulating source in the array, the steady state power balance is enforced. The important feature of the model is that the finite size of implant diameter and its dielectric coating can be incorporated. The effect of thickness and thermal conductivity of the coating on the source and tissue temperatures along with various other interesting features are deduced from this model. The analytically calculated implant and tissue temperatures are compared with those of a numerical 3-D finite difference model. The analytical model also is used to define a range of parameters such that minimal therapeutic temperatures will be achieved in the implanted volume without exceeding prescribed maximum temperatures. This approach leads to a simple means of selecting implant spacing and regulation temperatures of hot source methods prospectively.