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.

/pdf/one-dimensional-modelling-of-a-vascular-network-in-space-50c22v2hd3.pdf

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

Zero-dimensional (lumped parameter) and one dimensional models, based on simplified representations of the components of the cardiovascular system, can contribute strongly to our understanding of circulatory physiology. Zero-D models provide a concise way to evaluate the haemodynamic interactions among the cardiovascular organs, whilst one-D (distributed parameter) models add the facility to represent efficiently the effects of pulse wave transmission in the arterial network at greatly reduced computational expense compared to higher dimensional computational fluid dynamics studies. There is extensive literature on both types of models. The purpose of this review article is to summarise published 0D and 1D models of the cardiovascular system, to explore their limitations and range of application, and to provide an indication of the physiological phenomena that can be included in these representations. The review on 0D models collects together in one place a description of the range of models that have been used to describe the various characteristics of cardiovascular response, together with the factors that influence it. Such models generally feature the major components of the system, such as the heart, the heart valves and the vasculature. The models are categorised in terms of the features of the system that they are able to represent, their complexity and range of application: representations of effects including pressure-dependent vessel properties, interaction between the heart chambers, neuro-regulation and auto-regulation are explored. The examination on 1D models covers various methods for the assembly, discretisation and solution of the governing equations, in conjunction with a report of the definition and treatment of boundary conditions. Increasingly, 0D and 1D models are used in multi-scale models, in which their primary role is to provide boundary conditions for sophisticate, and often patient-specific, 2D and 3D models, and this application is also addressed. As an example of 0D cardiovascular modelling, a small selection of simple models have been represented in the CellML mark-up language and uploaded to the CellML model repository http://models.cellml.org/
 . They are freely available to the research and education communities. Each published cardiovascular model has merit for particular applications. This review categorises 0D and 1D models, highlights their advantages and disadvantages, and thus provides guidance on the selection of models to assist various cardiovascular modelling studies. It also identifies directions for further development, as well as current challenges in the wider use of these models including service to represent boundary conditions for local 3D models and translation to clinical application.

/pdf/review-of-zero-d-and-1-d-models-of-blood-flow-in-the-4ollfgzfpg.pdf

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

We numerically investigate a one-dimensional model of blood flow in human arteries using both a discontinuous Galerkin and a Taylor-Galerkin formulation. The derivation of the model and the numerical schemes are detailed and applied to two model numerical experiments. We first study the effect of an intervention, such the implantation of a vascular prosthesis (e.g. a stent), which leads to an abrupt variation of the mechanical characteristics of an artery. We then discuss the simulation of the propagation of pressure and velocity waveforms in the human arterial tree using a simplified model consisting of the 55 main arteries

/pdf/computational-modelling-of-1d-blood-flow-with-variable-1e4b07e7et.pdf

Computational modelling of 1D blood flow with variable mechanical properties and its application to the simulation of wave propagation in the human arterial system

Publisher Summary The development of mathematical models, algorithms and numerical simulation tools for the investigation of the human cardiovascular system has received a great impulse in the past years This chapter addresses the problem of developing models for the numerical simulation of the human circulatory system It particularly focuses on the problem of hemodynamics in large human arteries There are several important aspects, which require the use of sophisticated mathematical and numerical tools, such as the reconstruction of geometries from medical data; the transport of biochemicals in blood and vessel wall tissue; the heart dynamics; and blood rheology Besides, the need of validating the models calls for development of accurate in-vivo measurement techniques The number and complexity of the mathematical, numerical and technological problems involved makes the development of tools for accurate, reliable and efficient simulations of the human cardiovascular system one of the challenges of the next decades

Mathematical Modelling and Numerical Simulation of the Cardiovascular System

A (2+1)-dimensional quasilinear system is said to be ‘integrable’ if it can be decoupled in infinitely many ways into a pair of compatible n-component one-dimensional systems in Riemann invariants. Exact solutions described by these reductions, known as nonlinear interactions of planar simple waves, can be viewed as natural dispersionless analogues of n-gap solutions. It is demonstrated that the requirement of the existence of ‘sufficiently many’ n-component reductions provides the effective classification criterion. As an example of this approach we classify integrable (2+1)-dimensional systems of conservation laws possessing a convex quadratic entropy.

On the Integrability of (2+1)-Dimensional Quasilinear Systems

In this paper, we present a mathematical analysis of the quasilinear effects arising in a hyperbolic system of partial differential equations modelling blood flow 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 first derive the estimates on the initial and boundary data which imply strict hyperbolicity in the region of smooth flow. 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 flow 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 first shock formation and show that in a healthy individual, shocks form well outside the physiologically interesting region (2.8m downstream from the inlet boundary). In the end we present a study of the influence 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. Copyright © 2003 John Wiley & Sons, Ltd.

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

We prove the existence of a solution to the weak regular reflection problem for the unsteady transonic small disturbance (UTSD) model for shock reflection by a wedge. In weak regular reflection, the state immediately behind the reflected shock is supersonic and constant. The flow becomes subsonic further downstream; the equation in self-similar coordinates is degenerate at the sonic line. The reflected shock becomes transonic and begins to curve there; its position is the solution to a free boundary problem for the degenerate equation. Using the Rankine-Hugoniot conditions along the reflected shock, we derive an evolution equation for the transonic shock, and an oblique derivative boundary condition at the unknown shock position. By regularizing the degenerate problem, we construct uniform bounds; we apply local compactness arguments to extract a limit that solves the problem. The solution is smooth in the interior and continuous up to the degenerate boundary.



This work completes a stage in our program to construct self-similar solutions of two-dimensional Riemann problems. In a series of papers, we developed techniques for solving the degenerate elliptic equations that arise in self-similar reductions of hyperbolic conservation laws. In other papers, especially in joint work with Gary Lieberman, we developed techniques for solving free boundary problems of the type that arise from Rankine-Hugoniot relations. For the first time, in this paper, we combine these approaches and show that they are compatible. Although our construction is limited to a finite part of the unbounded subsonic region, it suggests that this approach has the potential to solve a variety of problems in weak shock reflection, including Mach and von Neumann reflection in the UTSD equation, and the analogous problems for the unsteady full potential equation. © 2002 John Wiley & Sons, Inc.

A free boundary problem for a quasi-linear degenerate elliptic equation: Regular reflection of weak shocks

We formulate and solve a transonic regular reflection problem for the unsteady transonic small disturbance equation, using a free boundary problem approach. Our method applies to self-similar shock reflection when the incident shock angle is large enough to permit a regular reflection configuration with a subsonic state behind the reflected shock. For the small-disturbance approximation in weak shock reflection, this corresponds to relatively large wedge angles. One contribution of this paper is the development of an asymptotic formula for the reflected shock, far from the reflection point, and for the subsonic state far downstream. These asymptotic series are valid for the small-disturbance approximation, for any incident shock angles. The main result in the paper is an existence theorem for the nonuniform subsonic flow behind the reflected shock. The flow velocity satisfies a quasilinear elliptic equation which is coupled to the Rankine-Hugoniot equations for the reflected shock, forming a free boundary problem on part of the boundary. Because the equation is not uniformly elliptic in the entire domain, we introduce a cut off to give a bounded domain, and also cut offs to the coefficients. Our result is incomplete in the following sense: we have been unable to remove the cut offs entirely. However, we prove that the flow we have constructed solves the original problem in a domain of finite size around the reflection point.

/pdf/free-boundary-problems-for-the-unsteady-transonic-small-3ec69cnroc.pdf

Free Boundary Problems for the Unsteady Transonic Small Disturbance Equation: Transonic Regular Reflection

We study a family of two-dimensional Riemann problems for compressible flow modeled by the nonlinear wave system. The initial constant states are separated by two jump discontinuities, $x = \pm \ka...

Free boundary problems for nonlinear wave systems : Mach stems for interacting shocks

From the observation that self-similar solutions of conservation laws in two space dimensions change type, it follows that for systems of more than two equations, such as the equations of gas dynamics, the reduced systems will be of mixed hyperbolic-elliptic type, in some regions of space In this paper, we derive mixed systems for the isentropic and adiabatic equations of compressible gas dynamics We show that the mixed systems which arise exhibit complicated nonlinear dependence In a prototype system, the nonlinear wave system, this behavior is much simplified, and we outline the solution to some typical Riemann problems

Eun Heui Kim

Papers

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

A free boundary problem for a quasi-linear degenerate elliptic equation: Regular reflection of weak shocks

Free Boundary Problems for the Unsteady Transonic Small Disturbance Equation: Transonic Regular Reflection

Free boundary problems for nonlinear wave systems : Mach stems for interacting shocks

Mixed hyperbolic-elliptic systems in self-similar flows