Numerical method of characteristics for one-dimensional blood flow

Mathematical modeling at the level of the full cardiovascular system requires the numerical approximation of solutions to a one-dimensional nonlinear hyperbolic system describing flow in a single vessel. This model is often simulated by computationally intensive methods like finite elements and discontinuous Galerkin, while some recent applications require more efficient approaches (e.g. for real-time clinical decision support, phenomena occurring over multiple cardiac cycles, iterative solutions to optimization/inverse problems, and uncertainty quantification). Further, the high speed of pressure waves in blood vessels greatly restricts the time step needed for stability in explicit schemes. We address both cost and stability by presenting an efficient and unconditionally stable method for approximating solutions to diagonal nonlinear hyperbolic systems. Theoretical analysis of the algorithm is given along with a comparison of our method to a discontinuous Galerkin implementation. Lastly, we demonstrate the utility of the proposed method by implementing it on small and large arterial networks of vessels whose elastic and geometrical parameters are physiologically relevant

Numerical Study of a Left Ventricular Assist Device (LVAD) With Different Blade Heights and Tip Clearances

One treatment modality for heart failure is to employ a mechanical heart assist device to increase blood flow to peripheral organs. There are various kinds of axial and centrifugal type mechanical pumps available for implantation depending on patient condition. Axial pumps are smaller in size comparatively, although centrifugal pumps have the advantages of lower rotational speed as well as better maintaining any native blood flow pulsatility. This work presents the results of the numerical study of the centrifugal blood pump configured as a Left Ventricular Assist Device (LVAD). The pump design utilized standard industrial centrifugal pump design principles but applied to smaller sized blood pumps. Flow characteristics are modelled using 3-dimensional steady state models operating at design speed of 2000 rpm using Newtonian blood properties for the fluid. Two design parameters of the pump are studied, the impeller blade height and tip clearance resulting in nine model variants. Analysis includes the hydrodynamic performance of the pump and the flow characteristics in the pump. A haemolysis prediction model quantifying red blood cell stress from exposure time and shear stress was used for quantitative predictions of haemolysis within the blood pump. Blood damage estimation was calculated along each path-line and averaged to a single value. By using a ranked selection method, the model with the 15 mm blade height and 800 µm tip clearance was selected as the preferred configuration with Haemolysis Index of 0.01 mmHg, efficiency of 58% at 104 mmHg outlet pressure

Oscillatory flow in a tube with time-dependent wall deformation and its application to Myocardial Bridges

In this paper we numerically investigate a one-dimensional model of blood flow in the human coronary arteries. The nonlinear hyperbolic system is expressed in terms of the cross-sectional area, flow velocity and pressure (A, u, p). The more widely studied linearised system is also discussed where conservation of static pressure, instead of total pressure, is enforced. The method of outgoing characteristics is used to satisfy the interface conditions, while a three-element windkessel model is adopted as outflow condition at the terminals of the network. Inside the segmental domain the leap-frog method is used for numerical integration. Within the context of this model we pay particular attention to the case when abrupt or smooth, space and time dependent variation of cross-sectional area of an artery is caused by externally prescribed motion of the vessel walls (e.g. myocardial bridge, flow watch). The derivation of the model and the numerical implementation are detailed. They are applied to model numerical experiments of the arterial system. Additionally to a system studied in [10, 15, 22, 28] the coronary arteries are parameterised. The main features of the flow through myocardial bridges are discussed

Modeling blood flow in the arterial system

The investigation of arterial blood flow has always been an important topic in physiology. Large proportion of the cardiovascular diseases (arteriosclerosis, vasoconstriction) affects the arterial part of the circulatory system. Physiologists place huge efforts in the improvement of the diagnosis and the treatment of these diseases. During the last decades fluid mechanics has become a powerful tool in the analysis of arterial blood flow. In the current paper a numerical approach for the calculation of one dimensional unsteady blood flow in the arterial network is presented. The continuity and momentum equations are solved using the method of characteristics (MOC). Furthermore a viscoelastic material model is applied to describe the behavior of the arterial walls. The network model of the arterial system is set up using 45 viscoelastic branches. The heart is modeled using an unsteady volume flow rate boundary condition. Arterioles and capillaries are treated as linear resistance boundary conditions. The parame ters of the viscoelastic material model were fine-tuned by comparing the results of the calculation with previously measured blood pressure curves. The resulting model is capable for simulating arteriosclerosis in an arbitrary part of the arterial network. The viscoelastic material model and the calculation method are presented in detail. Results of the calculation are presented and discussed

Numerical methods and applications for reduced models of blood flow

The human cardiovascular system is a vastly complex collection of interacting components, including vessels, organ systems, valves, regulatory mechanisms, microcirculations, remodeling tissue, and electrophysiological signals. Experimental, mathematical, and computational research efforts have explored various hemodynamic questions; the scope of this literature is a testament to the intricate nature of cardiovascular physiology. In this work, we focus on computational modeling of blood flow in the major vessels of the human body. We consider theoretical questions related to the numerical approximation of reduced models for blood flow, posed as nonlinear hyperbolic systems in one space dimension. Further, we apply this modeling framework to abnormal physiologies resulting from surgical intervention in patients with congenital heart defects. This thesis contains three main parts: (i) a discussion of the implementation and analysis for numerical discretizations of reduced models for blood flow, (ii) an investigation of solutions to different classes of models in the realm of smooth and discontinuous solutions, and (iii) an application of these models within a multiscale framework for simulating flow in patients with hypoplastic left heart syndrome. The two numerical discretizations studied in this thesis are a characteristics-based method for approximating the Riemann-invariants of reduced blood flow models, and a discontinuous Galerkin scheme for approximating solutions to the reduced models directly. A priori error estimates are derived in particular cases for both methods. Further, two classes of hyperbolic systems for blood flow, namely the mass-momentum and the mass-velocity formulations, are systematically compared with each numerical method and physiologically relevant networks of vessels and boundary conditions. Lastly, closed loop vessel network models of various Fontan physiologies are constructed. Arterial and venous trees are built from networks of one-dimensional vessels while the heart, valves, vessel junctions, and organ beds are modeled by systems of algebraic and ordinary differential equations

Direct numerical simulation of a pulsatile flow in a stenotic channel using immersed boundary method

A three-dimensional direct numerical simulation model coupled with the immersed boundary method has been developed to simulate a pulsatile flow in a planar channel with single and double one-sided semicircular constrictions. For relevance to blood flow in large arteries, simulations have been performed at Reynolds numbers of 750 and 1000. Flow physics and resultant wall shear stress (WSS)-based hemodynamic parameters are presented. The instantaneous vortex dynamics, mean flow characteristics, and turbulent energy spectra are evaluated for flow physics. Subsequently, three WSS-based parameters, namely the time-averaged WSS, oscillatory shear index, and relative residence time, are calculated over the stenotic wall and correlated with flow physics to identify the regions prone to atherosclerotic plaque progression. Results show that the double stenotic channel leads to high-intensity and broadband turbulent characteristics downstream, promoting critical values of the WSS-based parameters in the post-stenotic areas. In addition, the inter-space area between two stenoses displays multiple strong recirculations, making this area highly prone to atherosclerosis progression. The effect of stenosis degree on the WSS-based parameters is studied up to 60% degree. As the degree of occlusion is increased, larger regions are involved with the nonphysiological ranges of the WSS-based parameters

Non-Newtonian Rheology in Blood Circulation

Blood is a complex suspension that demonstrates several non-Newtonian rheological characteristics such as deformation-rate dependency, viscoelasticity and yield stress. In this paper we outline some issues related to the non-Newtonian effects in blood circulation system and present modeling approaches based mostly on the past work in this field.Comment: 26 pages, 5 figures, 2 table

Mathematical analysis of blood flow model through channels with flexible walls

A simplified mathematical model of blood flow through flexible arteries is developed and analyzed. The resulting system of non-linear, non-homogeneous PDE\u27s is analyzed numerically using the Richtmyer Lax-Wendroff method. Numerical and theoretical results show excellent agreement suggesting that in physiologically relevant situations shocks only develop outside the domain of interest. These results suggest that when the model assumptions are satisfied the model provides sufficient regularity to yield a physically reasonable representation of flow through a flexible artery. We conclude with a discussion of future directions for this model

The Yield Condition in the Mobilization of Yield-Stress Materials in Distensible Tubes

In this paper we investigate the yield condition in the mobilization of yield-stress materials in distensible tubes. We discuss the two possibilities for modeling the yield-stress materials prior to yield: solid-like materials and highly-viscous fluids and identify the logical consequences of these two approaches on the yield condition. As part of this investigation we derive an analytical expression for the pressure field inside a distensible tube with a Newtonian flow using a one-dimensional Navier-Stokes flow model in conjunction with a pressure-area constitutive relation based on elastic tube wall characteristics.Comment: 28 pages, 1 figur