A Coupled Fluid–Naghdi Shell–Windkessel Model for Pulmonary Artery Banding: Partitioned Numerical Scheme and Parameter Estimation
Yifan Wang, Josip Tambaca, Suncica Canic
Abstract
We develop a coupled three-dimensional (3D) fluid-structure interaction model for blood flow through deformable vessels with localized constriction, motivated by pulmonary artery banding in neonates with congenital heart disease. The model couples the incompressible Navier-Stokes
equations in an Arbitrary Lagrangian Eulerian (ALE) framework with a 3D Naghdi Shell for the arterial wall, defined on a two-dimensional (2D) reference domain, and a zero-dimensional threeelement Windkessel model for downstream circulation. A central feature of the formulation is
the use of a low-regularity Naghdi shell model that retains membrane, bending, and transverseshear deformation while requiring only first derivatives of the reference midsurface. This allows the model to accommodate piecewise-smooth vascular geometries, including localized constrictions
with discontinuous surface slopes.
We derive the coupled variational formulation and develop a non-iterative pratitioned scheme
to solve the coupled system. The coupling incorporates the inertia of the arterial wall through
a Robin-type interface condition. For a linearized fixed-domain problem, we establish a discrete
energy estimate under suitable assumptions. To account for uncertainty in the downstream circulation, we further couple the model with a reduced-state unscented Kalman filter to estimate the
proximal resistance, distal resistance, and compliance from inlet-pressure observations. Our numerical experiments demonstrate consistent pressure-wave propagation and wall deformation under
spatial and temporal refinement. Comparisons between the standard Naghdi shell and a modified,
predominantly radial shell model illustrate the influence of structural deformation mechanisms on
pressure-wave speed and displacement. Moreover, an idealized pulmonary artery banding example further demonstrates the usage and application of the coupled Fluid-Naghdi shell-Windkessel
model with parameter estimation. The resulting framework provides a basis for investigating the
hemodynamic and mechanical effects of localized vascular constrictions while accounting for the
influence of downstream vascular properties.