Formulation of exactly balanced solvers for blood flow in elastic vessels and their application to collapsed states

In this work, numerical solvers based on extensions of the Roe and HLL schemes are adapted to deal with test cases involving extreme collapsing conditions in elastic vessels. To achieve this goal, the system is transformed to provide a conservation–law form, allowing to define Rankine–Hugoniot conditions. The approximate solvers allow to describe the inner states of the solution. Therefore, source term fixes can be used to prevent unphysical values of vessel area and, at the same time, the eigenvalues of the system control stability. Numerical solvers of different order are tested using a wide variety of Riemann problems, including extreme vessel collapse and blockage. In all cases, the robustness of the approximate solvers presented here is checked using first and third order methods in time and space, using the WENO reconstruction scheme in combination with the TVDRK3 method

Elastic jump propagation across a blood vessel junction

The theory of small-amplitude waves propagating across a blood vessel junction has been well established with linear analysis. In this study we consider the propagation of large-amplitude, nonlinear waves (i.e. shocks and rarefactions) through a junction from a parent vessel into two (identical) daughter vessels using a combination of three approaches: numerical computations using a Godunov method with patching across the junction, analysis of a nonlinear Riemann problem in the neighbourhood of the junction and an analytical theory which extends the linear analysis to the following order in amplitude. A unified picture emerges: an abrupt (prescribed) increase in pressure at the inlet to the parent vessel generates a propagating shock wave along the parent vessel which interacts with the junction. For modest driving, this shock wave divides into propagating shock waves along the two daughter vessels and reflects a rarefaction wave back towards the inlet. However, for larger driving the reflected rarefaction wave becomes transcritical, generating an additional shock wave. Just beyond criticality this new shock wave has zero speed, pinned to the junction, but for further increases in driving this additional shock divides into two new propagating shock waves in the daughter vessels

Multiscale constitutive framework of 1D blood flow modeling: Asymptotic limits and numerical methods

In this paper, a multiscale constitutive framework for one-dimensional blood flow modeling is presented and discussed. By analyzing the asymptotic limits of the proposed model, it is shown that different types of blood propagation phenomena in arteries and veins can be described through an appropriate choice of scaling parameters, which are related to distinct characterizations of the fluid-structure interaction mechanism (whether elastic or viscoelastic) that exist between vessel walls and blood flow. In these asymptotic limits, well-known blood flow models from the literature are recovered. Additionally, by analyzing the perturbation of the local elastic equilibrium of the system, a new viscoelastic blood flow model is derived. The proposed approach is highly flexible and suitable for studying the human cardiovascular system, which is composed of vessels with high morphological and mechanical variability. The resulting multiscale hyperbolic model of blood flow is solved using an asymptotic-preserving Implicit-Explicit Runge-Kutta Finite Volume method, which ensures the consistency of the numerical scheme with the different asymptotic limits of the mathematical model without affecting the choice of the time step by restrictions related to the smallness of the scaling parameters. Several numerical tests confirm the validity of the proposed methodology, including a case study investigating the hemodynamics of a thoracic aorta in the presence of a stent

Numerical coupling of 0D and 1D models in networks of vessels including transonic flow conditions. Application to short-term transient and stationary hemodynamic simulation of postural changes

When modeling complex fluid networks using one‐dimensional (1D) approaches, boundary conditions can be imposed using zero‐dimensional (0D) models. An application case is the modeling of the entire human circulation using closed‐loop models. These models can be considered as a tool to investigate short‐term transient and stationary hemodynamic responses to postural changes. The first shortcoming of existing 1D modeling methods in simulating these sudden maneuvers is their inability to deal with rapid variations in flow conditions, as they are limited to the subsonic case. On the other hand, numerical modeling of 0D models representing microvascular beds, venous valves or heart chambers is also currently modeled assuming subsonic flow conditions in 1D connecting vessels, failing when transonic and supersonic flow conditions appear. Therefore, if numerical simulation of sudden maneuvers is a goal in closed‐loop models, it is necessary to reformulate the current methodologies used when coupling 0D and 1D models, allowing the correct handling of flow evolution for both subsonic and transonic conditions. This work focuses on the extension of the general methodology for the Junction Riemann Problem (JRP) when coupling 0D and 1D models. As an example of application, the short‐term transient response to head‐up tilt (HUT) from supine to upright position of a closed‐loop model is shown, demonstrating the potential, capability and necessity of the presented numerical models when dealing with sudden maneuvers

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

<p>Abstract</p> <p>Background</p> <p>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.</p> <p>Method and Results</p> <p>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 <url>http://models.cellml.org/</url>. They are freely available to the research and education communities.</p> <p>Conclusion</p> <p>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.</p

High-order fully well-balanced numerical methods for one-dimensional blood flow with discontinuous properties

In this paper, we are interested in the numerical study of the one-dimensional blood flow model with discontinuous mechanical and geometrical properties. We present the mathematical model together with its nondimensional form. We do an exhaustive investigation of all its stationary solutions and we propose high-order fully well-balanced numerical methods that are able to preserve all of them. They are based on the combination of the Generalized Hydrostatic Reconstruction and well-balanced reconstruction operators. These methods are able to deal with more than one discontinuous parameter. Several numerical tests are shown to prove its well-balanced and high-order properties, and its convergence to the exact solutions.The research of EPG and CP was partially supported by the Spanish Government (SG), the European Regional Development Fund (ERDF), the Regional Government of Andalusia (RGA), and the University of Málaga (UMA) through the projects of reference RTI2018-096064-B-C21 (SG-ERDF), UMA18-Federja-161 (RGA-ERDF-UMA), and P18-RT-3163 (RGA-ERDF). EPG was also financed by the European Union – NextGenerationEU. // Funding for open access charge: Universidad de Málaga / CBU

Exact solutions and conservation lawsof a one-dimensional PDE model for a blood vessel

Two aspects of a widely used 1D model of blood flow in a single blood vessel are studied by symmetry analysis, where the variables in the model are the blood pressure and the cross-section area of the blood vessel. As one main result, all travelling wave solutions are found by explicit quadrature of the model. The features, behaviour, and boundary conditions for these solutions are discussed. Solutions of interest include shock waves and sharp wave-front pulses for the pressure and the blood flow. Another main result is that three new conservation laws are derived for inviscid flows. Compared to the well-known conservation laws in 1D compressible fluid flow, they describe generalized momentum and generalized axial and volumetric energies. For viscous flows, these conservation laws get replaced by conservation balance equations which contain a dissipative term proportional to the friction coefficient in the model.Comment: 23 pages; 9 figure

Riemann problems in the retinal circulation

Retinal haemorrhage (abnormal bleeding of the blood vessels in the retina) is often observed following traumatic brain injury. The retinal blood vessels, known as the retinal circulation, are supplied by the central retinal artery (CRA) and the central retinal vein (CRV), which pass along the optic nerve after passing through a region of ceresbrospinal fluid (CSF) in the nerve sheath. In this thesis we develop a theoretical model for blood flow in the retinal blood vessels which forms the foundation of a predictive model of retinal haemorrhage. We first consider a case where sudden change in CSF pressure (i.e. following brain injury) drives a large amplitude pressure perturbation (shock waves), treating the blood vessel as a single compliant tube with uniform material properties. We develop a Riemann problem to consider the response to an instantaneous discontinuity in flow and/or tube cross-sectional area across an interface, and examine the flow propagation. We identify the four classical flow structures that emerge from the discontinuity. We extend our Riemann problem to consider an abrupt change in material properties coincident with the initial discontinuity, as a model for the central retinal vessels crossing between regions (e.g. between the optic nerve and the retina). Our Riemann problem exhibits the four classical solutions each with an additional stationary wave across the point of discontinuity. However, these classical solutions are not sufficient to cover the entire parameter space, and we demonstrate the existence of transcritical states resulting from a rarefaction wave resonating with the stationary wave, producing additional shock waves. These new resonant solutions were enough to close the gaps in our parameter space left by the classical solutions. Finally we investigate the propagation of these large-amplitude pressure waves through a symmetric junction (mimicking blood flow through the retinal vessel networks). Our Riemann problem exhibits analogs of the four classical wave structures (again with a discontinuity across the stationary wave at the junction) as well as analogs of the transcritical (resonant) states identified in flow through a single vessel with a discontinuity in material properties. Furthermore, we find that the onset of resonance is delayed when the daughter vessels are chosen to be less stiff than the parent vessel and can be eliminated entirely if the reduction in stiffness is sufficiently large

Cardiovascular System: Modelling and Optimization

2014 - 2015A conservation law is a partial di_erential equation, in which the variable is a quantity which remains constant in time, that is it cannot be created and destroyed. Thanks to the conservation laws it is possible to de_ne models able to describe real systems in which something is stored. Fluid dynamic models, which are based on them, have a wide range of applications, because they can be used to describe blood ows, tra_c evolution on street networks of big cities or on motorways of big states, data ows on telecommunication networks, ows of goods on supply chains, electric networks, etc. ... [edited by author]XIV n.s