A 1D PDE boundary control model successfully simulates heart rate variability and peripheral resistance adjustments to maintain constant mean arterial pressure.
Theoretical arterial-valve PDE model extends 1D control frameworks; leaves open clinical hemodynamic validation.
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.
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Cascaval et al. (2016) studied this question.
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