The Circulatory System

Part of the Interdisciplinary Applied Mathematics book series (IAM, volume 8/2)


The circulatory system forms a closed loop for the flow of blood that carries oxygen from the lungs to the tissues of the body and carries carbon dioxide from the tissues back to the lungs (Figs. 11.1 and 11.2). There are two pumps to overcome the resistance and maintain a constant flow. The left heart receives oxygen-rich blood from the lungs and pumps this blood into the systemic arteries. The systemic arteries form a tree of progressively smaller vessels, beginning with the aorta, branching to the small arteries, then to the arterioles, and finally to the capillaries. The exchange of gases takes place in the capillaries. Leaving the systemic capillaries, the blood enters the systemic veins, through which it flows in vessels of progressively increasing size toward the right heart. The systemic veins consist of venules, small veins, and the venae cavae. The right heart pumps blood into the pulmonary arteries, which form a tree that distributes the blood to the lungs. The smallest branches of this tree are the pulmonary capillaries, where carbon dioxide leaves and oxygen enters the blood. Leaving the pulmonary capillaries, the oxygenated blood is collected by the pulmonary veins, through which it flows back to the left heart. It takes about a minute for a red blood cell to complete this circuit.

While there is an apparent structural symmetry between the pulmonary and sys temic circulations, there are significant quantitative differences in pressure and blood volume. Nevertheless, the output of the right and left sides of the heart must always balance, even though the cardiac output, or total amount of blood pumped by the heart, varies widely in response to the metabolic needs of the body. One of the goals of this chapter is to understand how the cardiac output is determined and regulated in re sponse to the metabolic needs of the body. Questions of this nature have been studied for many years, and many books have been written on the subject (see, for example, Guyton, 1963, or Reeve and Guyton, 1967. A more recent book that discusses several mathematical models of the blood and circulatory system is Ottesen et al., 2004). Here, we consider only the simplest models of the control of cardiac output. Figure 11.1 Schematic diagram of the circulatory system, showing the systemic and pulmonary circulations, the chambers of the heart, and the distribution of blood volume throughout the system. (Guyton and Hall, 1996, Fig. 14-1, p. 162.)


Cardiac Output Pressure Drop Arterial Pressure Left Heart Arterial Pulse 
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© Springer-Verlag Berlin Heidelberg 2009

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