Blood flow.

Blood flow and gas exchange is examined in Chapter 38 of your text and this chapter will provide good background for the following discussion. Blood flows through our body because of pressure differences at different parts of the system. The rate of flow between any two points in the system is proportional to the pressure difference between those points, and the direction of flow is always from the point of highest pressure to that of lowest pressure. This can be represented mathematically by:

$$\mbox{Flow rate} \propto (P_1-P_2), \mbox{ or,}$$ (1)

F = K (P₁-P₂) (2)

The coefficient, K, is a proportionality constant that quantifies the efficiency with which the pressure difference is translated into fluid flow. K must always be less than, and is typically considerably less than 1. It is a measure of the degree of resistance (R) in the system: i.e. K=1/R. This resistance is a function of the cross- sectional area of the vessel, the length of the vessel, the nature of the vessel wall (e.g., is it smooth?) and properties of the fluid moving through the vessel (especially viscosity).

How does cross-sectional area of the vessel affect fluid flow? Suppose we consider flow at 2 points in a non-uniform pipe (Fig. 1). At point 1, fluid is flowing at a rate v₁ and in a time period t advances in the pipe a distance of $\Delta x_1=v_1 t$. At point 2, the fluid flows at rate v₂and advances $\Delta x_2= v_2 t$ in time period t. Let the cross- sectional area at the two points be A₁ and A₂ respectively. Now the mass of fluid flowing through the pipe is conserved. That is for every gram of fluid that flows by point 1, a gram must flow by point 2. This point actually implies that the fluid is incompressible. Although this assumption is probably not hard to accept with respect to water (or blood), you may question its validity for air. Nevertheless, air flow in lungs can still be adequately modelled in this way.

The mass, M₁, of fluid flowing by point 1 in time t is equal to the fluid density ($\rho$) times the volume of the fluid in question, i.e. $M_1=\rho A_1 \Delta x_1 = \rho A_1 v_1 t$. In similar fashion we can calculate the mass of fluid flowing by point 2 during time period t, $M_2=\rho A_2 v_2 t$. According to the equation of continuity,

M₁ = M₂

$$\rho A_1 v_1 t \rho A_2 v_2 t$$ =

A₁ v₁ = A₂ v₂. (3)

The above implies that the velocity of fluid in a pipe will increase if the cross-sectional area is constricted. The nozzle on a garden hose makes use of this principal: constriction of the diameter at the end of the hose increases the speed in the water spout.

The blood vascular system of vertebrates is designed to get O₂ (among other things) to all cells of the body and to remove waste products from these tissues. It includes large vessels which move blood to the tissues and capillaries where diffusion of O₂ into the tissues occurs. Ideally we would like blood to be transported rapidly in large vessels and much more slowly in capillaries but this runs counter to the flow principles we have just discussed: ceteris paribus blood should flow more rapidly through the narrow vessels. Why doesn't this happen?

The vertebrate blood system contains a preponderance of capillaries. That is, the total cross-sectional area in the capillary beds greatly exceeds the cross-sectional area at any other point in the system. It is for this reason that blood flows slowly through capillary beds. Only a portion of the capillary beds are in use at any one time. Muscles in the arterial walls determine how much blood will flow through different parts of the capillary system.

Blood pressure is determined by cardiac output (C_O) and total peripheral resistance (P_R) according to the simple relationship, BP = C_O P_R. Cardiac output can be changed by increasing heart rate or by increasing the force of contraction. However, pressure variations within the circulatory system are largely determined by peripheral resistance and the biggest drop occurs in the arterial system (Fig. 2).

According to the Poiseuille-Hagen equation, the volumetric flow rate through a cylindrical blood vessel is approximated by:

$$V = \frac{\Delta P \pi r^{4}}{8 l \mu},$$

where $\Delta P/l$ is the pressure gradient, $\mu$ the viscosity of blood, and r the vessel radius. The resistance, R, in the system (assuming ( $V = \Delta P/R$) is

$$R =\frac{8 l \mu}{\pi r^{4}}.$$

Thus, regulation of vessel radius is a potent regulator of blood flow. If the radius is reduced by half, blood flow is reduced 16-fold!

The applicability of this equation is subject to the following assumptions:

1.

the tube is a rigid cylinder where $l \gg r$.

Blood vessels are not rigid but elastic. Nevertheless, they respond in such a way as to approximate well to the model.

2.

the fluid is ideal in that $\mu$ is not dependant upon shear.

Again this is not true, since blood viscosity is dependant upon shear. However, the general qualitative principle still holds.

3.

flow is laminar and steady.

Blood flow is pulsatile especially in larger vessels but probably approximates well to laminar steady flow in most smaller vessels.

4.

Velocity is zero at the vessel wall.