Mathematical analysis of the quasilinear effects in a hyperbolic model blood flow through compliant axi‐symmetric vessels

Abstract

SUMMARYIn this paper, we present a mathematical analysis of the quasilinear e ects arising in a hyperbolic system of partial di erential equations modelling blood ow through large compliant vessels. The equations are derived using asymptotic reduction of the incompressible Navier-Stokes equations in narrow, long channels.To guarantee strict hyperbolicity we ÿrst derive the estimates on the initial and boundary data which imply strict hyperbolicity in the region of smooth ow. We then prove a general theorem which provides conditions under which an initial-boundary value problem for a quasilinear hyperbolic system admits a smooth solution. Using this result we show that pulsatile ow boundary data always give rise to shock formation (high gradients in the velocity and inner vessel radius). We estimate the time and the location of the ÿrst shock formation and show that in a healthy individual, shocks form well outside the physiologically interesting region (2:8 m downstream from the inlet boundary). In the end we present a study of the in uence of vessel tapering on shock formation. We obtain a surprising result: vessel tapering postpones shock formation. We provide an explanation for why this is the case.

Blood flow mechanics and oxygen transport and delivery in the retinal microcirculation: multiscale mathematical modeling and numerical simulation

“…Some of the authors of this paper have already theoretically investigated the Starling resistor effect of the retinal venules using a lumped model to describe the retinal circulation (Guidoboni et al 2014b), and it would be very interesting to extend the present model by incorporating such an effect. Even though in the present contribution the simulations have been carried out in the case of rigid vessel walls, reduced models for large compliant and collapsible vessels are already available in the literature Formaggia et al (2009),Čanić andKim (2003) and George and Liu (1981). The main challenge from the mathematical viewpoint is to deal with a nonlinear hyperbolic system, particularly at the junctions between branches, where there might be the occurrence of spurious reflected waves to be properly handled by the numerical scheme.…”

On the Mechanics Underlying the Reservoir-Excess Separation in Systemic Arteries and their Implications for Pulse Wave Analysis

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Review of Zero-D and 1-D Models of Blood Flow in the Cardiovascular System

“…For this situation 1D modelling can offer greater advantages in revealing the pressure and flow changes along the full length of the vessel studied. Canic and Kim [115] have studied in detail the characteristics of the axisymmetric form of the Navier-Stokes equations. They demonstrate that, providing the radius of the vessel is small relative to a characteristic wavelength, the radial momentum equation dictates that the pressure is constant over any cross-section and that, on integrating the axial momentum equation over the cross-sectional area of the vessel, the radial velocity terms are subsumed into an area term.…”

“…For the 1D, 2D or 3D distributed parameter models, the boundary conditions can be either prescribed values for variables in the governing equations, or prescribed values for the derivatives of variables, or prescribed values for the linear combination of variables and their derivatives. The 0D and 1D models use pressure and velocity (or flow rate) as basic variables, spatially averaged (or integrated) over the transverse plane (as discussed earlier, uniform pressure on a cross-section is a consequence of the radial momentum equation, whilst the velocity distribution in the nonlinear convective term can be represented by a correction coefficient [115]). The 2D and 3D models often utilise pressure and velocity as primitive variables.…”