Multiple minima in inverse hyperthermia temperature estimation problems

Using one-, two-, and three-dimensional numerical simulation models it is shown that multiple minima solutions exist for some inverse hyperthermia temperature estimation problems. This is a new observation that has important implications for all potential applications of these inverse techniques. The general conditions under which these multiple minima occur are shown to be solely due to the existence of symmetries in the bio-heat transfer model used to solve the inverse problem. General rules for determining the number of these global minimum points in the unknown parameter (perfusion) space are obtained for several geometrically symmetric (with respect to the sensor placement and the inverse case blood perfusion model) one-, two-, and three-dimensional problem formulations with multiple perfusion regions when no model mismatch is present. As the amount of this symmetry is successively reduced, all but one of these global minima caused by symmetry become local minima. A general approach for (a) detecting when the inverse algorithm has converged to a local minimum, and (b) for using that knowledge to direct the search algorithm toward the global minimum is presented. A three-dimensional, random perfusion distribution example is given which illustrates the effects of the multiple minima on the performance of a state and parameter estimation algorithm. This algorithm attempts to reconstruct the entire temperature field during simulated hyperthermia treatments based on knowledge of measured temperatures from a limited number of locations.