Exponential taper in arteries: an exact solution of its effect on blood flow velocity waveforms and impedance

The dimensions and wall elasticity commonly taper along the length of mammalian arteries. The effects of taper on flow velocity waveforms can be included by either of two methods; to theoretically divide the artery into short sections wherein the properties are assumed constant (the approximate solution); or to find an exact solution incorporating the effects of taper. In this paper, an exact solution to the resulting, and previously unsolved nonlinear Ricatti equation for the impedance, is obtained by a process of substitutions. This solution is utilised to develop an exact expression for the flow velocity in the artery. The transmission line equations are then combined into a single integral expression for the entire artery and an exact solution to this is evaluated. This is the first solution to simultaneously account for both geometric and elastic taper, and it has been validated by comparing simulations of flow in the aorta of a dog to those using an infinitesimal approximate solution. The Pulsatility Index of the approximate solution requires at least 10 segments to converge to within 5% of that using the exact solution. The exact solution thus accurately accounts for the effects of exponential taper, and may be used to improve existing arterial models, which use the less accurate and more computationally cumbersome approximate solution.