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

Murillo, J.; Navas-Montilla, Adrián; García‐Navarro, P.

doi:10.1016/j.compfluid.2019.04.008

Cited by 16 publications

(19 citation statements)

References 59 publications

(121 reference statements)

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“…When dealing with veins, their possible collapse in case of large negative transmural pressures has to be considered [11,47,32]. The collapsed state for veins is identified by a cross-sectional area assuming a buckled, dumbbell shape configuration, in which opposite sides of the interior wall touch each other, still leaving some fluid in the two extremes [11,43].…”

Section: Elastic Constitutive Tube Lawmentioning

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: Elastic Constitutive Tube Lawmentioning

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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“…with A a suitable value of vessel area defined depending on the flow conditions [28] andf the integral of the friction force in the control volume. Using a discrete approach of the speed at the cell edge, c i+1/2 [28], the discrete source term G i+1/2 can be expressed as:…”

Section: Numerical Computation Of Fluxes At the Inner Interfacesmentioning

confidence: 99%

“…Under this condition, the following approximate semi-discrete RP in conservation-law form is presented [28]:…”

Section: Numerical Computation Of Fluxes At the Inner Interfacesmentioning

confidence: 99%

“…The HLLS method can be formulated by appropriately imposing the Rankine-Hugoniot (RH) conditions over the slowest (λ L ) and fastest (λ R ) wave celerities of the RP in (32), and the RH relation for the steady contact wave, of speed λ 0 = 0, at x = 0, leading to the evolved fluxesF − e,i andF + e,i+1 at the left and the right sides of the RP [27,28]:…”

Section: The Hlls Solvermentioning

confidence: 99%

“…The explicit scheme selected in this work to compute the updating fluxes in the cells within the 1D domain is the first order HLLS solver [27,28], based on the HLL numerical scheme developed by Harten et al [29] and extended to deal with discontinuous source terms [27,30]. The first and third order in time and space HLLS was successfully applied in [28] to model flow in collapsible tubes with discontinuous mechanical properties including collapsed states including subsonic, transonic and supersonic flow under conditions of extreme vessel collapse and sonic blockage.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Solution of the Junction Riemann Problem for 1D Hyperbolic Balance Laws in Networks including Supersonic Flow Conditions on Elastic Collapsible Tubes

Murillo

García‐Navarro

2021

Symmetry

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The numerical modeling of one-dimensional (1D) domains joined by symmetric or asymmetric bifurcations or arbitrary junctions is still a challenge in the context of hyperbolic balance laws with application to flow in pipes, open channels or blood vessels, among others. The formulation of the Junction Riemann Problem (JRP) under subsonic conditions in 1D flow is clearly defined and solved by current methods, but they fail when sonic or supersonic conditions appear. Formulations coupling the 1D model for the vessels or pipes with other container-like formulations for junctions have been presented, requiring extra information such as assumed bulk mechanical properties and geometrical properties or the extension to more dimensions. To the best of our knowledge, in this work, the JRP is solved for the first time allowing solutions for all types of transitions and for any number of vessels, without requiring the definition of any extra information. The resulting JRP solver is theoretically well-founded, robust and simple, and returns the evolving state for the conserved variables in all vessels, allowing the use of any numerical method in the resolution of the inner cells used for the space-discretization of the vessels. The methodology of the proposed solver is presented in detail. The JRP solver is directly applicable if energy losses at the junctions are defined. Straightforward extension to other 1D hyperbolic flows can be performed.

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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

Murillo

García‐Navarro

2023

Numer Methods Biomed Eng

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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.

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

Cited by 16 publications

References 59 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

A Solution of the Junction Riemann Problem for 1D Hyperbolic Balance Laws in Networks including Supersonic Flow Conditions on Elastic Collapsible Tubes

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

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