Mathematical description of the heat transfer in living tissue (Part I)

Abstract

In the present monograph we formulate a simple model for heat transfer in living tissue with self - regulation. The initial point of the model is the governing equations describing heat transfer in living tissue at the mesoscopic level, i.e. considering different vessels individually. Then, basing on the well known equivalence of the diffusion type process and random walks, we develop a certain regular procedure that enables us to average these mesoscopic equations practically over all scales of the hierarchical vascular network. The microscopic governing equations obtained in this way describe living tissue in terms of an active medium with continuously distributed self - regulation. One of the interesting results obtained in the present monograph is that there can be the phenomena of ideal self - regulation in large active hierarchical systems. Large hierarchical systems are characterized by such a great information flow that none of its elements can possess whole information required of governing the system behavior. Nevertheless, there exists a cooperative mechanism of regulation which involves individual response of each element to the corresponding hierarchical piece of information and leads to ideal system response due to self - processing of information. The particular results are obtained for bioheat transfer. However, self - regulation in other natural hierarchical systems seems to be organized in a similar way. The characteristics of large hierarchical systems occurring in nature are discussed from the stand point of regulation problems. By way of example, some ecological and economic systems are considered. An cooperative mechanism of self-regulation which enables the system to function ideally is proposed.