Overview

This chapter is dedicated to studying and simulating blood pressures and flows in the circulatory system.We have already seen how transport phenomena are central to the operation of biological systems. In the previous chapter we saw how the pumping of the heart is responsible for driving blood flow to transport solutes throughout the body. Here we focus on the mechanics of the heart and circulatory system themselves.

Pumping of the heart and flow of blood throughout the circulatory system represent a critical life-support system in man. Malfunction of the heart and/or the circulatory system is associated with a great number of diseases and pathophysiological conditions. For example, hypertension – chronic systemic high blood pressure – puts stress on the heart that can ultimately lead to its failure. Here we will see that the functioning and malfunctioning of the circulatory system are best understood in terms of mathematical models that capture the key mechanistic underpinnings of its anatomy and physiology.

Our modeling and analysis in this chapter will rely on lumped parameter circuit models, analogous to electrical circuits made up of resistors, capacitors, and inductors. Readers not familiar with simple circuit analysis may choose to review Section 9.7 of the Appendices, which provides a short background on the subject, before undertaking this chapter.

We will begin our study of the circulatory system with an analysis of the main pump responsible for moving blood through the circuit described in Section 3.2 of the previous chapter.

Overview

Transport of mass, into, out of, and within biological systems (including single cells, multicellular organisms, and even ecological systems) is fundamental to their operation. The subject of transport phenomena is treated in great depth in classic texts [10], as well as in books focused on biological systems [62]. Here we explore a number of examples that allow us to see how fundamental transport phenomena are accounted for in a wide range of biological systems. Specifically, we develop and apply basic frameworks for simulating transport in the following sorts of systems:

• Well-mixed systems. The defining characteristic of these systems is that they are fluid systems (often aqueous solutions in biological application) with the solutes of interest distributed homogeneously (i.e., well mixed) over the timescales of interest. An example of a well-mixed system is the aquarium studied in the previous chapter. Other examples are chemical reaction systems inside cells or compartments within cells when spatial gradients of the intracellular reactants do not significantly influence the behaviors that are simulated. Models of well-mixed systems (or models that adopt the well-mixed assumption) do not explicitly account for the spatial distribution of the variables simulated. For biochemical systems this means that, at any given time, concentrations are constant throughout a compartment. The kinetics of such systems are typically described by ordinary differential equations, as in the examples of Section 2.1 of this chapter and in Chapter 3.