Reinhard Mahnke

Power Laws and Skew Distributions - Volume 12 Issue 3 - Reinhard Mahnke, Jevgenijs Kaupužs, Mārtiņš Brics

The present paper is devoted to mathematical description of the vascular network response to local perturbations in the cellular tissue state, being one of the basic mechanisms controlling the inner environment of human body. Keeping in mind individual organs we propose a model for distributed self-regulation of living tissue, which is regarded as an active hierarchical system without any controlling center. This model is based on the self-processing of information about the cellular tissue state and cooperative interaction of blood vessels governing redistribution of blood flow over the vascular network. The information self-processing is implemented via mass conservation, i.e., conservation of blood flow as well as special biochemical compounds called activators transported by blood. The cooperative interaction of blood vessels stems from the response of individual vessels to activators in blood flowing through them. The general regularities are used to specify the vessel behavior. The arterial and venous beds are considered to be individually of the tree form. The constructed governing equations are analyzed numerically. In particular, first, we show that the blood perfusion rate approximately (in the analyzed case within 10% accuracy) depends only on the local concentration of activators in the cellular tissue. It is due to the hierarchical structure of the vascular network rather than the ideal behavior of individual vessels accepted previously. Second, we demonstrate the distinction between the reaction thresholds of individual vessels and that of the vascular network as a whole. The latter effect is the cause for introducing the notion of activators instead of using such quantities as temperature in describing the living tissue self-regulation.

Self-regulation of living tissue as an example of self-organization phenomena in hierarchical systems of biological, ecological, and social nature is under consideration. The characteristic feature of these systems is the absence of any governing center and, thereby, their self-regulation is based on a cooperative interaction of all the elements. The work develops a mathematical theory of a vascular network response to local effects on scales of individual units of peripheral circulation.

Self-regulation of living tissue as an example of self-organization phenomena in active fractal systems of biological, ecological, and social nature is under consideration. The characteristic feature of these systems is the absence of any governing center and, thereby, their self-regulation is based on a cooperative interaction of all the elements. The paper develops a mathematical theory of a vascular network response to local effects on scales of individual units of peripheral circulation. First, it formulates a model for the self-processing of information about the cellular tissue state and cooperative interaction of blood vessels governing redistribution of blood flow over the vascular network. Mass conservation (conservation of blood flow as well as transported biochemical compounds) plays the key role in implementing these processes. The vascular network is considered to be of the tree form and the blood vessels are assumed to respond individually to an activator in blood flowing though them. Second, the constructed governing equations are analyzed numerically. It is shown that at the first approximation the blood perfusion rate depends locally on the activator concentration in the cellular tissue, which is due to the hierarchical structure of the vascular network. Then the distinction between the reaction threshold of individual vessels and that of the vascular network as a whole is demonstrated. In addition, the nonlocal component of the dependence of the blood perfusion rate on the activator concentration is found to change its form as the activator concentration increases.