Some records ofblood pressure-time history are given in Figure 2. We have two questions: how to understand these curves, and how to use them. These are discussed in the current and following sections.

According to physics, we can write down a partial differential equation to describe the motion of a fluid. This equation is known as the Navier-Stokes equation (one vector equation, 3 equations in components). Originally intended for applications to the flow of fluids like air and water (Newtonian fluids), the Navier-Stokes equation has been extended to the treatment of blood flow, with blood rheology taken into consideration (3, 7, 9, 23). The boundary conditions of blood flow are those imposed by blood vessels and the heart. There are solid bodies. The vectorial partial differential equation that describes the motion of elastic solids is called Navier's equation. Originally used to treat solids like steel, the Navier's equation has been extended to treat blood vessels, with full vascular rheology taken into consideration (1, 7, 9).

Time (hours) Time (hours)

Time (hours) Time (hours)

Figure 2. A: A record of the pulmonary arterial blood pressure of a rat subjected to step lowering of oxygen concentration in breathing gas. B: Typical mean trend of the blood pressure and its best-fit formulaf(t) (reprinted from Ref. 20). As it is explained in the ref., the IMF method, means of various orders can be defined rigorously, the one's with the higher order has fewer zero crossings. In this figure, the order of the mean is 12.

When nonlinear rheology and convective acceleration (acceleration originated by moving a particle in a nonuniform velocity field) are considered, both the Navier-Stokes and the Navier's equations are nonlinear. These nonlinear equations have few exact solutions. Ifwe wish to read the blood pressure history illustrated in Figure 2 according to physics, we do not have the exact solutions to help us.

Without exact solutions of the full equations, are there exact solutions for simplified cases? The answer is, fortunately, yes. One important simplified case is the Poiseuille flow which assumes that the blood vessel is a circular cylindrical tube of uniform cross section with fixed, nonvarying inner wall (endothelium) of radius a, that the blood pressure, p, is fixed at the ends of a vessel of finite length, L, and that the velocity is zero on the wall. Then the pressure gradient is constant, the flow is steady, the velocity profile in the tube is parabolic, the shear strain gradient and the shear stress of the blood on the wall are constant, and the flow rate is governed by a formula frequently quoted in this book: pressure gradient = flow x resistance. Flow is the movement of volume per unit time across any cross section. Pressure gradient is the difference ofthe pressure at the ends divided by L. Under the simplifying assumptions named above, the resistance, R, is equal to:

R = (8/n) x (coefficient of viscosity) x (vessel length) x (radius of tube)'4

The minus 4th power ofthe radius in this formula speaks for the significance of the change of the radius of the blood vessel on the arterial blood pressure. For example, a 10% reduction in radius would cause a 34.4% increase in resistance. It is this formula that laid the responsibility of hypoxic hypertension in pulmonary arteries on the contraction of the smooth muscle.

The second important simplified case is that of wave propagation of a nonviscous fluid in an elastic tube. The simplifying assumptions are that the tube is linearly elastic and cylindrical, the convective acceleration can be neglected, and that the viscosity effect is negligible. Under these assumptions the Navier- Stokes equations for the blood, and the Navier equations of the wall are linearized, and the equation of continuity (conservation of mass) becomes a simple equation relating the temporal rate of change of vessel cross sectional area to the spatial rate of change of blood flow, whereas the equation of motion is reduced to a linear equation between the local acceleration duldtlo the pressure gradient dpldx. Combining these two equations yields a wave equation whose solution is an arbitrary wave moving forward or backward:

Here p is blood pressure, p0, pg, w0, u0' are arbitrary constants, f and g are arbitrary functions of the variables x-ct and x+ct, and c is the velocity of the waves in which E is the Young's modulus of elasticity of the vessel wall, h is the thickness of the vessel wall, ai is its inner radius, p is the density of the blood (see Ref. 9, p. 144). Notice the difference in the sign ofthe last terms in Eq. [1]. This leads to important relations:

These formulas say that, for a wave propagating in the direction of flow, the pressure rise is equal to the velocity rise multiplied by the product ofthe density of the blood and the speed of sound in the blood. For a wave propagating in the opposite direction of the flow, the pressure rise has the same magnitude but opposite sign.

The Poiseuille solution takes care of the effect of blood viscosity in a steady flow. The wave solution takes care of the transient oscillation in a nonviscous fluid. The sum of these two solutions is not an exact solution of the total Navier-Stokes and Navier equations, because the two problems are linearized differently. Womersley and many other biomechanics researchers have worked very hard to find improved solutions (23, 24, 27, 28). Complexity can be expected, but the major features are given by these two simplified solutions. The principles are quite clear, immediate progress can be counted on with the use of supercomputers.

Was this article helpful?

Your heart pumps blood throughout your body using a network of tubing called arteries and capillaries which return the blood back to your heart via your veins. Blood pressure is the force of the blood pushing against the walls of your arteries as your heart beats.Learn more...

## Post a comment