Boundary Control for an Arterial System

Abstract

-We consider a boundary control problem arising in the study of the dynamics of an arterial system which consists of one arterial segment (modeling the aorta in the cardiovascular system) coupled at the inflow with a pressurized chamber (modeling the left ventricle) via a valve. The opening and closing of the valve is dynamically determined by the pressure difference between the left ventricular and aortic pressures. Mathematically, this is described by a 1D system of coupled PDEs for the pressure and flow in the arterial segment, with a Dirichlet boundary condition imposed on the flow (when valve is closed) or on the pressure (when valve is open). At the outflow we impose a peripheral resistance model, which leads to a non-homogeneous Dirichlet condition. A numerical scheme based on the discontinuous Galerkin method is used to approximate the solution of the resulting system. We then use this methodology to simulate the heart rate variability observed in real physiological systems, by optimizing the timing of the heartbeat and the peripheral resistance, modeled using a terminal reflection coefficient, with the goal of achieving a prescribed mean pressure in the system.

Flow optimization in vascular networks

“…When considering additional effects, such as visco-elasticity or wall inertia, the pressure term may include also wall velocity η t and wall acceleration η tt , respectively, which leads to different fluidstructure interaction models. One such model is reported elsewhere [7].…”

“…with U k e (t) the time-varying coefficients of the expansion. Replacing (8) in (7) and letting Φ δ e = U δ e , we obtain 2K differential equations to be solved for each Ω e , e = 1, ..., N el : F (U δ e ) n−1 , k = 0, ..., K, i = 1, 2, e = 1, ..., N el , in which ∆t is the time step and n the number of every time step. To calculate the integrals we use a Gauss quadrature formula of order q ≥ K + 1.…”

On boundary optimal control problem for an arterial system: First-order optimality conditions

“…where ρ is the blood density. Considering the relation η t = − 1 2 r 0 u x and rearranging terms in u we get the system in the form (2)- (3). It remains to furnish the system by corresponding initial and boundary conditions which we propose to take in the form (3)- (4).…”

“…Since η t ≈ − 1 2 r 0 u x (see [3]) and we suppose that νη xx should be small enough, it easily follows from (11) that the given cost functional (10) can be reduced to the tracking type (1).…”

“…However, there seems to be no studies that focus on both aspects at the same time: a detailed description of the four chambers of the heart and on the spatial dynamics in the aorta. Some numerical aspects of optimizing the dynamics of the pressure and flow in the aorta as well as the heart rate variability, taking into account the elasticity of the aorta together with an aortic valve model at the inflow and a peripheral resistance model at the outflow, based on the discontinuous Galerkin method and a two-step time integration scheme of Adam-Bashfort, were recently treated in [3] for the Boussinesq system like (2). More broadly, theory and applications of optimization and control in spatial networks, basing on the different types of conservation laws have been extensively developed in literature, have been successfully applied to telecommunications, transportation or supply networks ( [6,7]).…”