An inverse technique to deduce the elasticity of a large artery

Our purpose is to build an inverse method which best fits a model of artery flow and experimental measurements (we assume that we are able to measure the displacement of the artery as a function of time at three stations). Ha ving no clinical data, we simulate these measurements with the numerical co mputations from a "boundary layer" code. First, we revisit the system of Li ng and Atabek of boundary layer type for the transmission of a pressure pul se in the arterial system for the case of an elastic wail (but we solve it without arty simplification in the u partial derivative u/partial derivativ e x term). Then, using a method analogous to the well known Von Karman-Pohl hausen method from aeronautics but transposed here for a pulsatile flow, we build a system of three coupled non-linear partial differential equations depending only on time and axial co-ordinate. This system governs the dynam ics of internal artery radius, centre velocity and a quantity related to th e presence of viscous effects. These two methods give nearly the same numer ical results. Second, we construct an inverse method: the aim is to find fo r the simple integral model, the physical parameters to put in the "boundar y layer" code (simulating clinical data). This is done by varying in the in tegral model the viscosity and elasticity in order to fit best with the dat a. To achieve this in a rational way, we have to minimise a cost function, which involves the computation of the adjoint system of the integral method . The good set of parameters (i.e. viscosity, and two coefficients of a wal l law) is effectively found again. It opens the perspective for application in real clinical cases of this new non-invasive method for evaluating the viscosity of the flow and elasticity of the wall.