On a fluid-structure interaction problem for plaque growth

Abstract

We study a free-boundary fluid-structure interaction problem with growth, which arises from the plaque formation in blood vessels. The fluid is described by the incompressible Navier-Stokes equation, while the structure is considered as a viscoelastic incompressible neo-Hookean material. Moreover, the growth due to the biochemical process is taken into account. Applying the maximal regularity theory to a linearization of the equations, along with a deformation mapping, we prove the well-posedness of the full nonlinear problem via the contraction mapping principle.

On a fluid-structure interaction problem for plaque growth: cylindrical domain

“…In this paper, we focus on a 3d free-boundary fluid-structure interaction problem for plaque growth, which was also addressed in the authors' previous work [1]. To be more precise, the blood is assumed to be the incompressible Navier-Stokes equation and the artery is modeled by an elastic equation with viscoelasticity, while inside the blood flow and vessel wall react the cells, leading to the plaque formation, see e.g.…”

“…Moreover, Σ t is assumed to be perpendicular to G t at ∂Σ t as well. In such setting, the domain is endowed with the fixed contact line ∂S and moving contact line ∂Σ t with ninety-degree contact angles for a short time, while in [1], we considered a smooth domain without contact.…”

“…σ f (v f , π f ) = −π f I + ν f (∇v f + ∇ ⊤ v f ) denotes the Cauchy stress tensor, π f is the unknown fluid pressure and ν f > 0 represents the fluid viscosity. Same as in [1], the motion of vessel wall is captured by the incompressible neo-Hookean material with viscoelasticity…”

“…[45,50]. Here F s is the inverse deformation gradient and F s,e denotes an inverse elastic tensor under the assumption of growth as in authors' previous work [1]. Similarly, the Kelvin-Voigt stress tensor σ e s is introduced, see e.g.…”

“…Mielke-Roubíček [31]. For more discussions about it, readers are referred to [1,Remark 1.1]. Moreover, the second equation of (1.2) results from the mass balance, where f g s = γβc s with β, γ > 0, is called growth function and represents the rate of mass growth per unit volume due to the formation of the plaque, which comes from the reaction of cells in the whole domain.…”