Time domain computational modelling of 1D arterial networks in monochorionic placentas

Franke, V.; Parker, Kim H.; Wee, L.; Fisk, Nicholas M.; Sherwin, Spencer J.

doi:10.1051/m2an:2003047

Cited by 20 publications

(24 citation statements)

References 17 publications

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“…The advent of new imaging modalities, such as Magnetic Resonance Imaging, and the availability of computational methods developed for compressible flows now offer the potential to solve efficiently the 1-D models in anatomically correct,patient specific arterial systems [30,31,27]. Furthermore the relatively inexpensive cost of these numerical methods for large networks (i.e.…”

Section: Discussion and Concluding Remarksmentioning

confidence: 99%

“…In the above definition we assume that W 1 and W 2 have been defined with respect to the equilibrium state as is the case in equations (27), (29) and (31). A value of R t = 1 represents a full reflection of the outgoing wave whereas R t = 0 corresponds to no reflection or an absorbing outflow.…”

Section: Terminal Resistance R Tmentioning

confidence: 99%

“…To relate the above definition to their work, we recall that the form of the linearised characteristic variables is given in terms of (u , p) by equation (31) and note that the terminal resistance for this system can be defined as…”

Section: Terminal Resistance R Tmentioning

confidence: 99%

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

Sherwin

Franke

Peiró

et al. 2003

Journal of Engineering Mathematics

397

497

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Abstract. 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.

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Section: Discussion and Concluding Remarksmentioning

confidence: 99%

Section: Terminal Resistance R Tmentioning

confidence: 99%

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

Sherwin

Franke

Peiró

et al. 2003

Journal of Engineering Mathematics

397

497

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“…(ii) Neumann problem: Under the Hypothesis 3.1, the coupled fluid-structure problem (1), (2), (18), (7), (24) and (25), satisfies the following a priori energy estimate, for all t ∈ I…”

Section: D Non-linear Elastic Modelmentioning

confidence: 99%

“…These models are described by hyperbolic systems of partial differential equations and, despite having a lower level of accuracy compared to the fully 3D model, they capture effectively the pulse waves at a much lower computational cost. This allows one to properly simulate pressure waves propagation in large regions of the arterial tree by a network of these models coupled together [16][17][18]36].…”

mentioning

confidence: 99%

On the stability of the coupling of 3D and 1D fluid-structure interaction models for blood flow simulations

Formaggia¹,

Moura²,

Nobile³

2007

ESAIM: M2AN

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Abstract.We consider the coupling between three-dimensional (3D) and one-dimensional (1D) fluidstructure interaction (FSI) models describing blood flow inside compliant vessels. The 1D model is a hyperbolic system of partial differential equations. The 3D model consists of the Navier-Stokes equations for incompressible Newtonian fluids coupled with a model for the vessel wall dynamics. A non standard formulation for the Navier-Stokes equations is adopted to have suitable boundary conditions for the coupling of the models. With this we derive an energy estimate for the fully 3D-1D FSI coupling. We consider several possible models for the mechanics of the vessel wall in the 3D problem and show how the 3D-1D coupling depends on them. Several comparative numerical tests illustrating the coupling are presented.Mathematics Subject Classification. 65M12, 65M60, 92C50, 74F10, 76Z05.

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Anatomically based simulation of hepatic perfusion in the human liver

Hunter

Cousins³

et al. 2019

Numer Methods Biomed Eng

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Liver structures of a healthy subject are digitised and segmented from computed tomography (CT) images, and hepatic perfusion is modelled in the hepatic artery and portal vein of the healthy subject with structured tree‐based outflow boundary conditions. This self‐similar structured tree is widely used in the literature, eg, blood flow simulation in larger systemic arteries and cerebral circulation, and is used in this study to model the effect of the smaller hepatic arteries and arterioles, as well as the smaller hepatic portal veins and portal venules. Physiologically reasonable results are obtained. Since the structured tree terminates at the size of the microvasculature system in liver lobules, the structured tree boundary condition will enable the proposed organ‐level model of hepatic arterial flow to be easily connected to tissue‐level models of liver lobules. Blood flow in the hepatic vein is also modelled in this subject with three‐element Windkessel model as outflow boundary conditions. The benefit of integrating the perfusion in all hepatic vascular vessels is that it helps us analyse some complicated clinical phenomenon more efficiently, eg, one possible application is to obtain the portal pressure gradient (PPG) to help examine the reliability of hepatic venous pressure gradient (HVPG) as an indirect measure of portal pressure. Moreover, since four to six generations of hepatic vessels, which are sufficient for liver classification analysis, were employed in the model, this study is setting the computational foundation of a potentially handy surgical tool.

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Time domain computational modelling of 1D arterial networks in monochorionic placentas

Cited by 20 publications

References 17 publications

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

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

On the stability of the coupling of 3D and 1D fluid-structure interaction models for blood flow simulations

Anatomically based simulation of hepatic perfusion in the human liver

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