One-dimensional modelling of a vascular network in space-time variables

Abstract

Abstract. In this paper a one-dimensional model of a vascular network based on space-time variables is investigated. Although the one-dimensional system has been more widely studied using a space-frequency decomposition, the space-time formulation offers a more direct physical interpretation of the dynamics of the system. The objective of the paper is to highlight how the space-time representation of the linear and nonlinear one-dimensional system can be theoretically and numerically modelled.In deriving the governing equations from first principles, the assumptions involved in constructing the system in terms of area-mass flux (A, Q), area-velocity (A, u), pressure-velocity (p, u) and pressure-mass flux(p, Q) variables are discussed. For the nonlinear hyperbolic system expressed in terms of the (A, u) variables the extension of the single vessel model to a network of vessels is achieved using a characteristic decomposition combined with conservation of mass and total pressure. The more widely studied linearised system is also discussed where conservation of static pressure, instead of total pressure, is enforced in the extension to a network. Consideration of the linearised system also allows for the derivation of a reflection coefficient analogous to the approach adopted in acoustics and surface waves.The derivation of the fundamental equations in conservative and characteristic variables provides the basic information for many numerical approaches. In the current work the linear and nonlinear systems have been solved using a spectral/hp element spatial discretisation with a discontinuous Galerkin formulation and a second order Adams-Bashforth time integration scheme. The numerical scheme is then applied to a model arterial network of the human vascular system previously studied by Wang and Parker (To appear in J. Biomech. (2004)).Using this numerical model the role of nonlinearity is also considered by comparison of the linearised and nonlinearised results. Similar to previous work only secondary contributions are observed from the nonlinear effects under physiological conditions in the systemic system. Finally, the effect of the reflection coefficient on reversal of the flow waveform in the parent vessel of a bifurcation is considered for a system with a low terminal resistance as observed in vessels such as the umbilical arteries.

Population persistence in river networks

“…Potential applications in biology have been noted (Carlson 2006;Maury et al 2009;Nicaise 1985;Sherwin et al 2003), but the literature here is more sporadic and undeveloped. Population dynamics in rivers is typically modeled using extensions of reaction-diffusion-advection (RDA) partial differential equations, which are parabolic initial value problems.…”

Geometric multiscale modeling of the cardiovascular system, between theory and practice

“…These conditions descend from the continuity of mass and momentum, respectively. With these interface conditions at the bifurcations, the 1D network undergoes a stability estimate that ensures energy conservation (up to the dissipative terms), see [2,60,178].…”

“…A high-order discontinuous Galerkin approximation is considered in [177,178], allowing to propagate waves of different frequencies without suffering from excessive dispersion and diffusion errors, so to reliably capture the reflection at the junctions induced by tapering. Alternatively, a high-order finite volume scheme is presented in [129] and a space-time finite element method is proposed in [204].…”

Coupling 3D and 1D fluid–structure‐interaction models for wave propagation in flexible vessels using a finite volume pressure‐correction scheme

“…The difference lies in the choice of the dependent variables that can be: area (A), velocity (u), pressure ( p) or flow rate (Q). The derivation of (A, Q), (A, u), ( p, u) and ( p, Q) models from first principles (mass conservation and momentum balance) can be found in [12]. In the present paper, we select the variables ( p, u) because, as will be seen later, they are very convenient for the coupling.…”

“…Using this pressure-area relation and substituting into (4), we find that the distensibility D is given by System (3) with pressure-area relation (5) is a non-linear hyperbolic system. It can be transformed into a system of (2) decoupled equations (the methodology and the intermediate steps are well described in the literature, for example [12,17]) as…”