Mathematical modelling of the cardiovascular system and skeletal muscle interaction during exercise

Abstract

Abstract. In this work we present a mathematical model for the coupling of cardiovascular system and metabolism under sport efforts. Effects such as oxygenation, heart rates, flow rates, blood pressure and mass transfer to tissues will be considered. A special attention will be payed to the modelling of the regulatory mechanisms (baroreflex and chemoreflex effectors) affecting the heart and the peripheral circulation. We aim at extending the existing models to describe metabolism, hemodynamics and mass transport during exercise, providing a framework to understand their mutual interactions. Numerical issues are presented for different workloads, and a quantitative analysis of dynamic aerobic exercise is discussed.Résumé. Dans ce travail nous présentons un modèle mathématique de couplage du système cardiovasculaire au métabolisme lors d'efforts intenses de type sportif. Les conséquences au plan cardiaque, au plan des pressions et débits sanguins, de l'oxygénation, et des transferts de matière au sein des tissus sont prises en compte. Une attention spéciale est apportéeà la modélisation des mécanismes cardiovasculaires de régulation mis en jeu (baroréflexe et chémoréflexe), d'un point de vue cardiaque ou vasculaire périphérique. Nous nous proposons d'élargir les modèles existants de métabolisme, de l'hémodynamique et des transferts de matière lors de l'exercice, apportant ainsi une architecture utileà la compréhension de leurs interactions mutuelles. Les résultats de simulations numériques sont présentés pour différents niveaux de charge d'exercice, et l'analyse quantitative de l'exercice dynamique aérobie est discutée.

Cardiovascular mathematics

“…where the Kedem-Katchalsky equation (19) has been rewritten in terms of the volume averaged concentration. Finally, we observe that the multilayer model is described by a set of equations similar to (20) and (21). Precisely, its fluid dynamics part is obtained by adding to problem (20) a further domain, describing the intima, that will be coupled to the lumen and the media prescribing that the normal velocity across the interface between these domains is continuous and equal to the flux J v , defined in (18).…”

“…Chemoreflex and baroreflex effects can be included in the differential models presented so far through another coupled ODE system that models these feedback mechanisms (see, e.g., [21], [20] and the references therein).…”

“…Analogously, the extension of equation (21) to the multilayer case features a third advection-diffusion equation defined in the intima and coupled to the rest of the system by imposing that the total flux of chemical (namely (−D∇c + γūc/ε) · n) is continuous across the interfaces and equal to J s , defined in (19). Besides their interest for bio-medical applications, (20) and (21) represent a difficult system of nonlinear partial differential equations whose analysis has been specifically addressed in [105], [79], [80]. Irregularities in the flow field across the arterial wall influence the concentration distribution within the wall.…”