Multiscale modelling of metabolism and transport phenomena in living tissues

The range of applications of mathematical modelling in biosciences has recently expanded to encompass problems posed by biomedicine and sport sciences. Topics of interest are for instance the prediction of the response of an athlete to exercise, the pharmacokinetics of a chemical compound, or the detection of illicit drugs. In this work, we consider some of these problems, related to metabolism, circulation and mass transport in tissues. First, we address the quantitative analysis of the biochemical reactions that are responsible of energy production in muscle cells. These reactions are strictly dependent on chemical exchanges between blood and tissues, by several physiological auto-regulation mechanisms. For this reason, we consider coupled problems in which the reaction phenomena are influenced by transport in blood. In particular, the problem of local blood perfusion and supply of substrates to tissues is studied in detail. The processes underlying the interaction between metabolism and circulation feature a multiscale nature: for instance, although metabolism takes place in cells, it modifies the hemodynamics of peripheral (capillaries) and central (heart) circulation. Therefore, we will set up a hierarchy of models, corresponding to these different scales. At first, we adopt an integrative approach, based on a compartmental model of the whole-body response to exercise, or more generally to variations in skeletal muscle metabolism. This model is the higher level of the hierarchy, describing the interactions between organs. Then, we increase the level of detail and focus on isolated tissues and vessels, considering more accurate one-dimensional models for blood flow and mass transport, as well as coupled 1D-3D models of tissue perfusion. In the latter models, the microvascular matrix is represented as a three-dimensional homogeneous medium, where larger vessels are described as 1D networks: circulation, transport and reaction of biochemical species are modelled at both the scales. The models considered in this work may provide a multi-scale analysis of metabolic processes, such as those induced by exercise, that often begin at cellular level, progressively propagate up through the hierarchy of scales, until adaptation of the whole body is reached. Examples of simulations, dealing with exercise protocols or clinical study cases, are provided to support the range of applications.