On the exact solution of the Riemann problem for blood flow in human veins, including collapse

Spiller, C.; Toro, Eleuterio F.; Vzquez-Cendn, M.E.; Contarino, Christian

doi:10.1016/j.amc.2017.01.024

Cited by 15 publications

(8 citation statements)

References 26 publications

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“…These three structural layers, from innermost to outermost, are called tunica interna (intima), tunica media and tunica externa [48]. At a macroscopic level, the arterial wall can be seen as a complex multi-layer viscoelastic structure which deforms under the action of blood pressure [34,51], even collapsing, in the case of veins, under certain circumstances [41,47,43]. The modeling of the interaction between blood flow and vessel wall mechanics requires the definition of a constitutive law which has to correctly describe the energy transfer between the two means, to accurately represent wave propagation phenomena [18,22].…”

Section: Introductionmentioning

confidence: 99%

Modeling blood flow in viscoelastic vessels: the 1D augmented fluid–structure interaction system

Bertaglia

Caleffi

Valiani

2020

Computer Methods in Applied Mechanics and Engineering

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Nowadays mathematical models and numerical simulations are widely used in the field of hemodynamics, representing a valuable resource to better understand physiological and pathological processes in different medical sectors. The theory behind blood flow modeling is closely related to the study of incompressible flow through compliant thin-walled tubes, starting from the incompressible Navier-Stokes equations. Furthermore, the mechanical interaction between blood flow and vessels wall must be properly described by the model. Recent works showed the benefits of characterizing the rheology of the vessel wall through a viscoelastic law. Taking into account the viscous contribution of the wall material and not simply the elastic one leads to a more realistic representation of the vessel behavior, which manifests not only an instantaneous elastic strain but also a viscous damping effect on pulse pressure waves, coupled to energy losses. In this context, the aim of this work is to propose an easily extensible one-dimensional mathematical model able to accurately capture fluid-structure interactions. The originality of the model lies in the introduction of a viscoelastic tube law in PDE form, valid for both arterial and venous networks, leading to an augmented fluid-structure interaction system. In contrast to well established mathematical models, the proposed one is natively hyperbolic. The model is solved with an efficient and robust second-order numerical scheme; the time integration is based on an Implicit-Explicit Runge-Kutta scheme conceived for applications to hyperbolic systems with stiff relaxation terms. The validation of the proposed model is performed on several different test cases. Results obtained in Riemann problems, adopting a simple elastic tube law for the characterization of the vessel wall, are compared with available exact solutions. To validate the contribution given by the viscoelastic term, the Method of Manufactured Solutions has been applied. Specific tests have been designed to verify the well-balancing with respect to fluid-at-rest condition and the accuracy-preserving property of the scheme. Finally, a specific test case with an inlet pulse pressure wave has been designed to assess the effects of viscoelasticity with respect to a simple elastic behavior of the vessel wall. The complete code, written in MATLAB (MathWorks Inc.) language, with the implemented test cases, is made available in Mendeley Data repository.

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Section: Introductionmentioning

confidence: 99%

Modeling blood flow in viscoelastic vessels: the 1D augmented fluid–structure interaction system

Bertaglia

Caleffi

Valiani

2020

Computer Methods in Applied Mechanics and Engineering

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“…Regarding the SLSM parameters, the estimation procedure presented in [14] is followed. Notice that the relaxation time τ r used for the artery is greater than the one used for the vein because the latter is characterized by a more restrained viscoelastic behavior, which leads to less dissipation of energy, since the venous walls are very little rigid, indeed tending to collapse when relatively high external pressures act [46,59]. It is worth underlining that few works have been proposed concerning viscoelasticity applied to veins.…”

Section: -Vessels Junctionmentioning

confidence: 99%

“…Thus, to properly and accurately investigate the propagation of blood in circulatory systems, mathematical models need to consider that blood mechanically interacts with vessel walls and tissues. Vessel walls are indeed deeply affected by internal pressure, undergoing strong deformations, even collapsing in the case of veins under specific circumstances [46,66,59]. The fluid-structure interaction (FSI) in blood propagation phenomena requires the introduction of a constitutive law which defines the transfer of energy between the two means [12,29,34,56].…”

Section: Introductionmentioning

confidence: 99%

Modeling blood flow in networks of viscoelastic vessels with the 1-D augmented fluid-structure interaction system

Piccioli,

Bertaglia,

Valiani

et al. 2021

Preprint

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A noteworthy aspect in blood flow modeling is the definition of the mechanical interaction between the fluid flow and the biological structure that contains it, namely the vessel wall. Particularly, it has been demonstrated that the addition of a viscous contribution to the mechanical characterization of vessels brings positive results when compared to in-vivo measurements. In this context, the implementation of boundary conditions able to keep memory of the viscoelastic contribution of vessel walls assumes an important role, especially when dealing with large circulatory systems. In this work, viscoelasticity is taken into account in entire networks via the Standard Linear Solid Model. The implementation of the viscoelastic contribution at boundaries, namely inlet, outlet and junction, is carried out considering the natively hyperbolic nature of the mathematical model. Specifically, a non-linear system is established based on the definition of the Riemann Problem at junctions, characterized by rarefaction waves separated by contact discontinuity waves, among which the mass and the total energy are conserved. Basic junction test cases are analyzed, such as a trivial 2vessels junction, for both a generic artery and a generic vein, and a simple 3-vessels junction, considering an aortic bifurcation scenario. The chosen IMEX-SSP2 Finite Volume scheme is demonstrated to be second-order accurate in the whole domain and well-balanced, even when including junctions. Two benchmark arterial networks are then implemented, differing in number of vessels and in viscoelastic parameters. Comparison of the results obtained in the two networks underlines the high sensitivity of the model to the chosen viscoelastic parameters. In these numerical tests, the network-wise conservation of the contribution provided by the viscoelastic characterization of vessel walls is assessed. Finally, numerical results of pressure waveforms in the common carotid artery and in the femoral artery of the second network simulated are compared with in-vivo data recorded from different healthy subjects.

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“…If using this type of law for modelling flow in veins, when veins collapse the blood flow is greatly reduced and a negative value of transmural pressure is observed. Therefore, classical waves in the conventional Riemann problem cannot be connected to the zero state with a zero value of vessel area [31] . When approaching a collapsed state it is of utmost importance to control the approximations made in the numerical scheme.…”

Section: Article In Pressmentioning

confidence: 99%

“…To the best of our knowledge, there are not previous works focused on the application and design of numerical solvers devoted to the solution of Riemann problems in collapsed vessels. Recently, the presence of collapsed states in the exact solution has been analyzed in [31] , remarking the differences between veins and arteries. Another novelty of this work is the use of the HLLS solver.…”

Section: Article In Pressmentioning

confidence: 99%

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

2019

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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.

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On the exact solution of the Riemann problem for blood flow in human veins, including collapse

Cited by 15 publications

References 26 publications

Modeling blood flow in viscoelastic vessels: the 1D augmented fluid–structure interaction system

Modeling blood flow in viscoelastic vessels: the 1D augmented fluid–structure interaction system

Modeling blood flow in networks of viscoelastic vessels with the 1-D augmented fluid-structure interaction system

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

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