Equation-based models of dynamic biological systems

The endothelium serves barrier, synthetic and catalytic functions and is a site of complex interacting processes involving a large number of biological components. Mathematical modeling might provide valuable insight into the global integration of those interactions into tissue function. The purpose of this chapter is to provide a nontechnical review of a well-established modeling platform, namely differential equations, that harnesses the powerful tools of calculus to analyze the time-dependent behavior of dynamical systems. Differential equations have been abundantly used by modelers. Yet, this framework is largely unknown to basic and clinical scientists. We will briefly describe this framework, provide examples that relate to endothelium modeling, and discuss its strengths and weaknesses (Figure 191.1). DYNAMICAL SYSTEMS: A dynamical system is an amalgam of interacting components together with a set of rules for how the states of the components evolve in time, and so the notion of time evolution is key when thinking about such a system. Many primary or calculated useful physiological quantities, such as cardiac output and vascular resistance, are related in a static fashion. In other words, one can relate these quantities by means of algebraic equations of varying complexity. The equations resulting from drawing an analogy between electrical circuits and the circulation have led to additional appealing concepts, such as peripheral vascular resistance and vascular capacitance. However, the clinician is clearly aware that these quantities change over time as the “system” adapts to changing external and internal conditions such as fluid administration, local concentration of effectors, or drug dose.