Mathematical Modeling of Vascular Stents

Tambača, Josip; Kosor, Mate; Čanić, Sunčica; Paniagua, David

doi:10.1137/080722618

Cited by 43 publications

(47 citation statements)

References 13 publications

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“…The ubiquity of this interaction type has led to a rapidly growing research interest in this model. Its mathematical solvability [5,9,15,17,22,21,26,32,33,34,37,38,49,55,52], numerical approximations [23,29,39,54,53,44,46], and stability [14,27,28,31] have been intensively studied.Mathematically, this interaction is described by a partial differential equation (PDE) system that couples the parabolic (fluid) and hyperbolic (elasticity) phases, where the key issue is that the traces of the elastic component at the energy level are not defined via the standard trace theory. The loss of regularity induced by the hyperbolic component left the basic question of existence open until recently.…”

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confidence: 99%

Well-Posedness Analysis for a Linearization of a Fluid-Elasticity Interaction

Bociu¹,

Toundykov²,

Zolésio³

2015

SIAM J. Math. Anal.

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We study the well-posedness of a total linearization, with respect to a perturbation of the external forcing, of a free-boundary nonlinear elasticity-incompressible fluid interaction. The total linearization for the coupling modeled by the Navier-Stokes equations and the nonlinear equations of elastodynamics was obtained recently in [L. Bociu and J.-P. Zolésio, Evol. Equ. Control Theory, 2 (2013), pp. 55-79]. The equations and the free boundary were linearized together, and the result turned out to be quite different from the usual coupling of classical linear models. New additional terms are present on the common interface, some of them involving boundary curvatures and boundary acceleration. These terms play an important role in the final linearized system and cannot be neglected; their presence also introduces new challenges in the well-posedness analysis, which proceeds to establish that the evolution operator associated to the linearized system can be represented as a bounded perturbation of a maximal dissipative semigroup generator. Introduction.A typical fluid-structure interaction system is composed of a solid either surrounding a fluid flow (arteries, pipelines, reservoir tanks, etc.) or surrounded by a fluid or gas (blood cells, marine animals, bridge supports, aircraft, automobiles, etc.). This model falls in the category of free boundary problems, since the motion of the common boundary between the solid and the fluid is one of the unknowns in the coupled system. The ubiquity of this interaction type has led to a rapidly growing research interest in this model. Its mathematical solvability [5,9,15,17,22,21,26,32,33,34,37,38,49,55,52], numerical approximations [23,29,39,54,53,44,46], and stability [14,27,28,31] have been intensively studied.Mathematically, this interaction is described by a partial differential equation (PDE) system that couples the parabolic (fluid) and hyperbolic (elasticity) phases, where the key issue is that the traces of the elastic component at the energy level are not defined via the standard trace theory. The loss of regularity induced by the hyperbolic component left the basic question of existence open until recently.The first major development appeared in [16], where the authors established local existence and regularity of solutions for the coupling of Navier-Stokes equations and a linear elasticity model, describing a body submerged in a fluid, under the assumptions of smooth initial data; specifically, the initial fluid velocity w 0 has Sobolev regularity H 5 , and the initial data for the elastic component (ϕ 0 , ϕ 1 ) belongs to H 3 × H 2 . Moreover, due to the incompressibility condition on the fluid, the uniqueness result

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confidence: 99%

Well-Posedness Analysis for a Linearization of a Fluid-Elasticity Interaction

Bociu¹,

Toundykov²,

Zolésio³

2015

SIAM J. Math. Anal.

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“…Extensions of the proposed algorithm to systems of hyperbolic conservation laws on networks, such as those, e.g., in [19,18], are subject to future investigations.…”

Section: Resultsmentioning

confidence: 99%

Runge–Kutta Discontinuous Galerkin Method for Traffic Flow Model on Networks

et al. 2014

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Abstract. We propose a bound-preserving Runge-Kutta (RK) discontinuous Galerkin (DG) method as an efficient, effective and compact numerical approach for numerical simulation of traffic flow problems on networks, with arbitrary high order accuracy. Road networks are modeled by graphs, composed of a finite number of roads that meet at junctions. On each road, a scalar conservation law describes the dynamics, while coupling conditions are specified at junctions to define flow separation or convergence at the points where roads meet. We incorporate such coupling conditions in the RK DG framework, and apply an arbitrary high order bound preserving limiter to the RK DG method to preserve the physical bounds on the network solutions (car density). We showcase the proposed algorithm on several benchmark test cases from the literature, as well as several new challenging examples with rich solution structures. Modeling and simulation of Cauchy problems for traffic flows on networks is notorious for lack of uniqueness or (Lipschitz) continuous dependence. The discontinuous Galerkin method proposed here deals elegantly with these problems, and is perhaps the only realistic and efficient high-order method for network problems.

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“…The effect of the number of struts and the ratio between the coated area was researched by Delfour et al, and they attempted to optimize the effect of the dose (Delfour et al, 2005). A mathematical model for the study of the mechanical properties of endovascular stents in their expanded state has also been proposed (Tambaca et al, 2010). Three-dimensional models of stent expansion have been presented by, among others, (Zunino et al, 2009) and (Horner et al, 2010).…”

Section: Introductionmentioning

confidence: 99%

Simulating Drug-eluting Stents - Progress Made and the Way Forward

McGinty¹,

McCormick²,

McKee³

et al. 2013

Proceedings of the 3rd International Conference on Simulation and Modeling Methodologies, Technologies and Applications

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Abstract:Drug-eluting stents have significantly improved the treatment of coronary artery disease. Compared with their bare metal predecessors, they offer reduced rates of restenosis and thus represent the current gold standard in percutaneous coronary interventions. Drug-eluting stents have been around for over a decade, and while progress is continually being made, they are not suitable in all patients and lesion types. Furthermore there are still real concerns over incomplete healing and late stent thrombosis. In this paper, some modelling approaches are reviewed and the future of modelling and simulation in this field is discussed.

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Mathematical Modeling of Vascular Stents

Cited by 43 publications

References 13 publications

Well-Posedness Analysis for a Linearization of a Fluid-Elasticity Interaction

Well-Posedness Analysis for a Linearization of a Fluid-Elasticity Interaction

Runge–Kutta Discontinuous Galerkin Method for Traffic Flow Model on Networks

Simulating Drug-eluting Stents - Progress Made and the Way Forward

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