On the propagation of pressure and flow waves through the patient specific arterial system

Abstract

For pre-operative decision making in cardiovascular surgery, patient-specific physiological data are needed. These data (e.g. pressure, flow and wall shear stress) can be obtained using a computational model of the arterial system. Because of the high computational costs involved with fully three-dimensional models of the total arterial tree, one-dimensional wave propagation models are more suited to provide clinically relevant information. Current models of the arterial system are based on assumptions concerning the frictional and convection forces in the one-dimensional momentum balance that yield an inaccurate representation of the physiological situation. Moreover, the constitutive law, relating the local pressure to the local cross-sectional area, is usually based on purely elastic material properties of the arterial wall, whereas arteries are known to possess viscoelastic properties as well. Furthermore, standard one-dimensional wave propagation methods are based on the assumption of fluid flow through straight or slightly tapered vessels where the velocity component in the radial direction is negligibly small with respect to its axial counterpart. In pathological regions such as stenoses and aneurysms these assumption do not hold. In the current study, a one-dimensional wave propagation model is developed, using an approximate velocity profile function to provide an estimate for the frictional forces and the non-linear term. The resulting wall shear stress and convection forces are compared to the analytical solution for pulsatile flow in a rigid tube showing good agreement. With respect to the arterial wall, a constitutive law, based on the viscoelastic behaviour of the standard linear solid model is introduced, that relates the local cross-sectional area of the vessel lumen to the local blood pressure. The resulting one-dimensional wave propagation model is validated by a comparison to data obtained from an experimental setup, modelling fluid flow through straight and tapered polyurethane vessels. In order to apply the one-dimensional wave propagation model to patient-specific arterial systems, a bifurcation model is implemented to relate the pressure and flow of the parent artery to the pressure and flow of the child arteries. Also, terminal impedances based on a three-element Windkessel model are introduced to obtain appropriate boundary conditions at the truncated ends of the arterial network. Furthermore, to accurately model the fluid dynamics near pathological regions, such as stenoses and aneurysms, relations between the pressure drop and flow characteristics as a function of the local geometry are developed. These relations are based on the results of a computational study of blood flow through two-dimensional axisymmetric stenoses and aneurysm models. The final model is applied to an idealised arterial network known from literature to investigate the influence of the different model assumptions made on the pressure, the flow and on the wall shear stress. The pressure and flow waves computed using the approximate velocity profile function, show only moderate changes with respect to those obtained using Poiseuille profiles. The resulting wall shear stress, however, does differ significantly. The introduced viscoelastic properties of the arterial wall are shown to significantly contribute to the pressure and flow wave attenuation and the influence of a femoral stenoses and an abdominal aortic aneurysms has been demonstrated. In conclusion, the resulting one-dimensional wave propagation model can be used to obtain clinically relevant information that may be crucial in surgical planning.