A Coupled Fluid Naghdi Shell Windkessel Model for Pulmonary Artery Banding: Partitioned Numerical Scheme and Parameter Estimation

Abstract

We develop a 3D fluid structure interaction model for blood flow through deformable vessels with localized constrictions, motivated by pulmonary artery banding in neonates with congenital heart disease. The model couples incompressible Navier Stokes equations in an Arbitrary Lagrangian Eulerian framework with a Naghdi shell model for the arterial wall and a three element Windkessel model for downstream circulation. The shell describes a 3D cylindrical vessel wall with variable radius and is defined on a 2D reference domain. Its low regularity formulation retains membrane, bending, and transverse shear deformation while requiring only first derivatives of the reference midsurface, accommodating piecewise smooth geometries with discontinuous surface slopes. We derive the coupled variational formulation and develop a non-iterative partitioned scheme with a Robin-type interface condition incorporating arterial wall inertia. For a linearized fixed domain problem, we establish a discrete energy estimate under suitable assumptions. A unscented Kalman filter estimates proximal and distal resistances and compliance from inlet-pressure observations. Numerical experiments demonstrate consistent pressure wave propagation and wall deformation under spatial and temporal refinement. Comparisons of the standard Naghdi shell and a modified, predominantly radial shell model reveal how structural deformation mechanisms affect wave speed and displacement. An idealized pulmonary artery banding example demonstrates the coupled fluid shell Windkessel model with parameter estimation. The framework enables investigation of the hemodynamic and mechanical effects of localized constrictions and their dependence on downstream vascular properties.