The paper simulates arterial blood pressure in a cylindrical artery of constant radius and constant thickness. The artery is assumed to be sufficiently remote from the heart for a damped co-sinusoidal model to be appropriate. The damping effects are seen to make the velocity of propagation of a pressure wave depend locally on the pressure itself, thus making the second order hyperbolic partial differential equation governing the pressure non-linear.
The relative merits of employing a finite difference or a characteristic mesh for computing the arterial pressure are discussed, and the Method of Characteristics is described for computing the pressure along the length of the artery for steady contractions of the heart.
Numerical results for variable wave velocity are reported and the model is seen to cope adequately with the damping effects. The results are compared with a mathematical model which assumes constant wave velocity. Arterial blood pressure Characteristics Co-sinusoidal model Hyperbolic equation PHYSIOLOGICAL AND MATHEXATICAL DISCUSSION OF THE MODEL It is often difficult to estimate values of arterial blood pressure in healthy arteries, other than at post mortem after untimely death, and the building of a mathematical model to compute this pressure is therefore justified. A mathematical model can estimate the change in pressure due to a number of possible physiological changes in the artery.
The model to be considered in this paper will assume a cylindrical artery of length L, inner radius R, and thickness H. Young's modulus for the arterial wall will be denoted by E, Poisson's ratio for the wall will be denoted by C, and the density of blood will be denoted by p. It will also be assumed that the artery is sufficiently remote from the heart for a co-sinusoidal model to be appropriate, and that the heart beats steadily at z contractions per second. The formula developed will give the pressure at points along the length of the artery, which will be assumed to be straight, between successive contractions of the heart.
The model will be capable of dealing with changes in the velocity of propagation of a pressure wave, and in the arterial pressure itself, due to a number of physiological changes. These include the changes in the elastic properties and in the inner radius and thickness of the wall due to arteriosclerosis or hardening of the arteries, the changes in density and viscosity of the blood due to coagulation, and the gradual blockage of the artery due to thrombosis.
The limitations of the model are that it does not take into account junctions or curves in the artery or reflected pressure waves, nor does it deal successfully with very localized changes in the inner radius or thickness of the arterial wall. Such changes must be uniformly distributed along the length of the artery, though piecewise application of the model along the length of the artery would produce a successful mathematical treatment of such local aberrations. Also the model is appropriate only when the arterial pressure may be approximated by a cosine curve.
The arterial blood pressure u at a point x units along the length of the artery at time t is governed by the differential equation c 2 s2u d2U 8X2 at2 ' (0)