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

Yifan Wang, Josip Tambaca, Suncica Canic

arXiv:2610.20966v1math.NAcs.NAmath-phmath.MP

TL;DR

Pulmonary artery banding outcomes depend on both constriction characteristics and downstream pulmonary circulation, while existing models leave wall motion and distal impedance insufficiently resolved. The paper couples moving-domain fluid flow, a Naghdi shell, and a Windkessel outlet, then tests the framework and parameter estimation in idealized numerical studies. The reported comparisons show similar dominant pressure and radial-displacement responses across shell models, but differences in wave speed and local displacement.

  • Problem

    Banding effects depend on local vessel and band properties as well as unresolved downstream circulation, which routine measurements cannot uniquely characterize for patient-specific planning.

    Source: §1 · p. 2

  • Method

    The framework couples moving-domain Navier–Stokes flow, a Naghdi shell, and a three-element Windkessel outlet, with non-iterative partitioning and a reduced-state unscented Kalman filter for parameter estimation.

    Source: §6 · p. 24, §8 · p. 42

  • Results

    The dominant pressure and radial-displacement responses were similar across shell models, while structural model choice and scaling affected pressure-wave speed and local displacement.

    Source: §8 · p. 42

  • Takeaways & Limitations

    The numerical studies support the Naghdi model for representing transverse shear, axial deformation, and localized constrictions in the examined framework.

    Source: §8 · p. 42

    The banding example is an idealized single compliant artery segment with one downstream Windkessel outlet, so it is a proof of concept rather than a direct reproduction of the cited multiscale study.

    Source: §7.3 · p. 36

Abstract

from arXiv · show

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.

1 Introduction

Pulmonary artery banding can limit excessive pulmonary flow, but its effects depend on both the local constriction and downstream vascular properties. The paper motivates a coupled fluid–structure model with a Windkessel outlet to represent these interacting influences.

1 Introduction

  • Pulmonary and systemic vascular resistance balance influences shunt direction and magnitude, while falling pulmonary resistance after birth generally increases left-to-right shunting and pulmonary overcirculation.Overcirculation can cause pulmonary congestion and left-heart volume overload.Source: §1 · p. 2
  • A band that is too loose may leave pulmonary overcirculation insufficiently reduced, while one that is too tight may raise right-ventricular afterload and reduce pulmonary flow and systemic oxygen delivery.Band effects also depend on its geometry and position, vessel properties, incoming flow, and downstream resistance and compliance; routine measurements cannot uniquely determine patient-specific band configuration or distal properties.Source: §1 · p. 2
  • Prior banding simulations used rigid walls or multiscale models, leaving wall-deformation feedback and the distinction between local band effects and distal-circulation impedance insufficiently resolved.The proposed framework pairs a moving-domain 3D fluid–structure interaction model with a three-element Windkessel outlet representing proximal resistance, distal resistance, and compliance.Source: §1 · p. 2
  • The Naghdi shell retains membrane, bending, and transverse-shear deformation and remains well defined for piecewise-smooth vessel surfaces with limited regularity.Its formulation uses only first derivatives of the reference midsurface and permits standard continuous Lagrange finite elements.Source: §1 · p. 3
  • The proposed workflow combines a non-iterative partitioned solver with a reduced-state unscented Kalman filter for estimating three Windkessel parameters from pressure observations.The filter state contains proximal resistance, distal resistance, and compliance; future virtual comparisons assume downstream vascular properties remain unchanged over the intervention time scale.Source: §1 · pp. 3–4

2 Mathematical Model

This section formulates the coupled vessel fluid–shell model, defining its variable-radius geometry, ALE fluid flow with a Windkessel outlet, and low-regularity Naghdi shell for piecewise-smooth midsurfaces. It specifies the geometrically admissible fluid–structure interface, matching fluid and shell velocities and applying fluid traction to the shell midsurface without a separate couple.

2.1 Fluid and shell domains

  • A localized band is represented by a continuous radius profile that returns to the inlet/outlet radius, while piecewise-C1 segments may meet at slope jumps and form a connected Lipschitz midsurface.This regularity is compatible with the Naghdi-type formulation for W 1,∞ parameterizations.Source: §2.1 · p. 5

2.2 Fluid equations in the ALE formulation

  • In the ALE formulation, incompressible Newtonian fluid flow has prescribed inlet velocity and a three-element Windkessel outlet whose pressure depends on outlet flow.The outlet model includes compliance, proximal and distal resistance, and downstream venous pressure; the parameters are estimated from inlet-pressure measurements using an unscented Kalman filter.Source: §2.2 · pp. 8–10

2.3 Naghdi shell model

  • The low-regularity Naghdi shell formulation uses only first weak derivatives of the reference midsurface, so it remains defined for piecewise-C1 geometries with slope discontinuities.The shell unknowns are midsurface displacement and cross-section rotation; its geometric coefficients are interpreted almost everywhere.Source: §2.3 · p. 10
  • The low-regularity Naghdi model uses the full matrix A(η, ω), retaining drilling shear in addition to the displacement-dependent covariant membrane strain.Shell coefficients depend only on first derivatives of the reference map; global H1 regularity enforces trace continuity across band transitions, representing a connected, welded shell without separate transition-curve boundary conditions.Source: §2.3 · pp. 11, 13

2.4 Fluid-structure interface coupling conditions

  • The current shell map is restricted to be bi-Lipschitz, orientation-preserving, and to have surface Jacobian bounded away from zero, ensuring a well-defined interface geometry.These conditions prevent self-intersection and local collapse and ensure the normal, inverse surface map, and surface change-of-variables formula are well defined.Source: §2.4 · p. 13
  • No-slip coupling sets fluid velocity equal to shell midsurface velocity, with no independent fluid kinematic condition for shell rotation.The geometric condition also makes the fluid-mesh boundary and shell midsurface occupy the same current location.Source: §2.4 · p. 13
  • Fluid traction acts directly on the shell midsurface as a surface-force resultant, with no independent interface couple or direct virtual-work term for rotation.The force can still generate rotation indirectly because displacement and rotation are coupled in the shell bilinear forms.Source: §2.4 · p. 15
  • Band-transition curves require no extra condition or surface-integral term for regular distributed traction, and the model omits concentrated line forces and couples.A transition curve has zero surface measure; line loads or couples, if present physically, would require explicit line terms.Source: §2.4 · p. 15

3 Weak Formulation

The coupled weak formulation places the fluid equations on the moving domain and the shell equations on a fixed parameter domain. Compatible interface traces make the interface work cancel when the fluid and shell formulations are combined.

Fluid weak form

  • Fluid trial and test spaces are defined on the current domain by pulling fields back to the reference fluid domain, with test functions vanishing on the Dirichlet boundary.The prescribed inlet velocity is imposed strongly, leaving no inlet-traction term in the fluid weak form.Source: §3 · pp. 15–16

Coupled weak formulation

  • Admissible fluid and shell virtual velocities share the same interface trace, matching the strong kinematic coupling condition.This compatibility ensures that the explicit interface-work terms cancel when the fluid and shell equations are added.Source: §3 · pp. 15–16

Shell weak form

  • The shell weak form uses displacement and rotation test functions in the Naghdi shell space, with clamped and circumferential-periodicity conditions.The shell equations are posed on the fixed parameter domain, unlike the fluid equations on the current time-dependent domain.Source: §3 · pp. 15–16

Coupled weak formulation

  • The combined variational formulation couples fluid velocity, pressure, shell displacement and rotation, and outlet pressure through the Windkessel term.The unknown outlet pressure is denoted P_B; the outlet traction term also includes the resistance-weighted flow rate.Source: §3 · p. 16

4 The loosely coupled splitting scheme

The time-splitting scheme advances the coupled system through structure, ALE mesh-update, and fluid subproblems. Its fluid stage incorporates remaining shell translational inertia through a Robin interface condition.

4 The loosely coupled splitting scheme

  • The scheme advances each time level through three substeps: structure, ALE mesh update, then fluid and Windkessel solution.The scheme is described as a loosely coupled time-splitting method.Source: §4 · p. 17

Step 1: Solve the structure subproblem

  • The structure stage applies a fraction β of the previously computed shell load and updates displacement, rotation, and their velocities.The completed interface velocity is supplied to the next structure subproblem, while angular velocity is completed in this stage.Source: Step 1: Solve the structure subproblem · p. 17

Step 2: Calculate the ALE update

  • The ALE update solves a harmonic mesh-extension problem, and the updated mesh follows the structure-stage velocity.The passage clarifies that this identity concerns mesh motion only; the completed translational shell velocity comes from the fluid trace in Step 3.Source: Step 2: Calculate the ALE update · p. 18

Step 3: Solve the Fluid subproblem

  • The fluid subproblem incorporates remaining translational shell inertia through a Robin interface condition, replacing the interface-traction work in the fluid variational formulation.The Windkessel state is updated by backward Euler, while the updated mesh is held fixed during the fluid solve.Source: Step 3: Solve the Fluid subproblem · pp. 18–19

5 Stability analysis of the splitting scheme

The analysis derives a discrete energy estimate for the kinematically coupled scheme on a linearized fixed domain. Under its stated assumptions, the estimate gives stability with respect to added-mass coupling for every β in [0, 1], but does not establish stability for the fully nonlinear moving-domain problem.

5 Stability analysis of the splitting scheme

  • The discrete estimate controls fluid, shell, interface, and Windkessel energy contributions, with the Robin split generating interface-traction energy and nonnegative β-splitting dissipation.The derivation combines the fluid, shell, interface, and Windkessel identities.Source: §5 · pp. 19, 23
  • On an open domain, the standard semi-implicit convective form is not automatically energy neutral; the skew-symmetric form is energy neutral, or the boundary flux must meet the stated condition.The standard form may be retained when the boundary flux is nonnegative or treated as prescribed kinetic-energy flux.Source: §5 · p. 21
  • For every β ∈[0, 1], the fixed-domain scheme is unconditionally stable with respect to added-mass coupling, subject to the theorem’s convection and boundary-work assumptions.The estimate is for a linearized problem on a fixed domain; it does not establish stability for the fully nonlinear moving-domain problem.Source: §5 · pp. 19, 23
  • For prescribed nonzero inlet velocity, an H1 lifting introduces additional work terms involving the lifting and its discrete time difference, which must remain in the estimate.The prescribed inlet does not invalidate the energy method, but its work cannot be omitted from the right-hand side.Source: §5 · p. 23

6 Estimation of the Windkessel parameters using data assimilation

The parameter-estimation method uses inlet-pressure observations to update a reduced filter state containing only the three Windkessel parameters. It uses log coordinates and sigma-point simulations, while treating those parameters as constant between measurements.

6 Estimation of the Windkessel parameters using data assimilation

  • The reduced-state unscented Kalman filter estimates only proximal resistance, distal resistance, and compliance; fluid and shell variables are computed by the deterministic solver.The filter uses the solver to provide the nonlinear parameter-to-observation map.Source: §6 · p. 24
  • Logarithmic parameter coordinates ensure that estimated proximal resistance, distal resistance, and compliance remain strictly positive.Their logarithms are applied to numerical values expressed in fixed units.Source: §6 · p. 24
  • Between pressure measurements, the prior parameter mean and covariance carry forward unchanged, treating all three Windkessel parameters as unknown but constant in time.No process-noise covariance is included in the parameter-evolution model.Source: §6 · p. 24
  • The filter compares averaged inlet pressure predicted by each candidate parameter vector with measured inlet pressure to form its scalar observation.The inlet pressure is selected as output because inlet velocity is prescribed strongly.Source: §6 · p. 24
  • For Nθ = 3, the unscented transform uses 2Nθ + 1 = 7 sigma points; one candidate fluid-Windkessel problem is computed per point, producing predicted observations for the Kalman update of the posterior logarithmic parameter mean.Source: §6 · pp. 25–26
  • After the measurement update, candidate solutions are discarded and one final solve with the posterior estimate supplies the accepted fluid and Windkessel state.That accepted state is carried into the next fluid–structure interaction time step.Source: §6 · p. 26

7 Numerical results

The numerical results examine spatial and temporal convergence, compare standard and penalized Naghdi-shell wave propagation, and present an idealized pulmonary artery banding setup for parameter estimation. They also describe the reduced reference cross-sectional area, the need for conforming fluid and shell meshes in long-time simulations, and the distinction between resistance in the 3D banded segment and downstream Windkessel resistance.

7.1 Straight-tube pressure-pulse benchmark

  • Spatial and temporal refinement show convergence in averaged pressure and mean-diameter profiles; comparisons with the published benchmark focus on the tube interior because clamped ends may create boundary layers and the shell omits damping.The original benchmark uses absorbing radial-displacement conditions and includes structural damping, unlike this vector-valued Naghdi-shell model.Source: §7.1 · pp. 27, 28, 30

7.2 Comparison with Koiter shell with radial displacement only

  • The modified shell uses penalties to suppress transverse shear, axial motion, and antisymmetric tangential deformation; its profiles converge and are consistent with except for wave speed.The reported penalty parameters are γsh = 10, γx = 10, and γsym = 10.Source: §7.2 · p. 33
  • The penalized Naghdi pressure front propagates faster than the standard model, but its 3D wave speed remains about a factor of 2 below the corresponding 2D Koiter-shell speed.Figure 14 compares pressure-wave snapshots from the standard and penalized models.Source: §7.2 · p. 36, Figure 14 · p. 37

7.3 Pulmonary artery banding with Windkessel parameter estimation

  • The banding example is a proof-of-concept, not a reproduction of Baker et al., using one idealized pulmonary-artery segment with a 5-mm native radius, 2.5-mm band radius, and 3-mm constriction.The segment is coupled to a single downstream Windkessel outlet.Source: §7.3 · p. 36
  • Non-conforming interface transfers can accumulate error, excite spurious rotation, distort the fluid mesh, and ultimately stall or terminate long-time simulations.Building the shell mesh from the fluid-interface facets preserves geometric and topological correspondence and suppresses artificial interface torque.Source: §7.3 · p. 39
  • The band produces a reduction in the reference cross-sectional area.Source: §7.3 · p. 39
  • The effective inlet-to-venous resistance includes pressure loss in the resolved 3D banded segment and downstream Windkessel resistance, so it need not equal the reference pulmonary vascular resistance.Under the initial scaling, their difference represents the effective resistance contribution required from the resolved 3D domain.Source: §7.3 · p. 40
  • The simulation runs for 1 s, and Figure 16 presents shell displacement magnitude and fluid velocity magnitude at the final time.Source: Numerical results · p. 40

Numerical results

The simulation estimates the three Windkessel parameters and produces a stable inlet pressure that remains slightly different from the prescribed observation. The results also show downstream-directed axial shell displacement and a substantial downstream contribution to the inlet-to-venous pressure difference.

Numerical results

  • At t = 1 s, the shell displacement is predominantly axial, with the banded and adjacent ramp regions displaced downstream.Source: Figure 16 · p. 40
  • Rp increases by a factor of approximately 22.1, while Rd and C change by approximately 0.51% and 0.72%, respectively.The proximal-resistance fraction rises from 0.099 to 0.708, while the distal time constant remains nearly unchanged.Source: Numerical results · p. 41
  • At t = 1 s, the estimated parameters are Rp = 2.214 × 10^7 Pa s m−3, Rd = 9.146 × 10^6 Pa s m−3, and C = 2.085 × 10^−8 m3 Pa−1.Source: Numerical results · p. 40, Figure 17 · p. 41
  • 13.35 mmHg is the stable final simulated inlet pressure, differing from the prescribed observation by approximately 0.35 mmHg, or 2.69%.The remaining innovation may reflect measurement- and process-noise covariances, initial parameter covariance, and the available observation window.Source: Numerical results · p. 41, Figure 17 · p. 41
  • 7.82 mmHg of the total 8.35 mmHg inlet-to-venous pressure difference comes from the downstream Windkessel.This conditional estimate assumes outlet flow has approached Q0 = 2.0 L/min and capacitor pressure is approximately steady.Source: Numerical results · p. 42
  • A single pressure observable estimates three parameters, so converged parameter traces may not establish their independent identifiability.The reported changes in Rd and C are small, and their product RdC remains nearly unchanged; parameter correlation and identifiability are left for future work.Source: Numerical results · p. 42

8 Conclusion

The framework couples deformable-artery blood flow, Naghdi shell mechanics, and downstream Windkessel circulation, with a non-iterative partitioned solver. Numerical studies demonstrate pressure-wave propagation, shell-model effects, and an illustrative banding application with parameter estimation.

8 Conclusion

  • The framework couples moving-domain incompressible Navier-Stokes flow, a Naghdi arterial-wall shell, and three-element Windkessel downstream circulation through interface conditions and outlet flow and pressure.The shell retains membrane, bending, and transverse-shear effects for spatially varying vessel radii and localized constrictions.Source: §8 · p. 42
  • A non-iterative partitioned algorithm advances shell, ALE mapping, fluid, and downstream circulation without subiterations; a fixed-domain linearized problem has a discrete energy estimate under stated assumptions.The scheme preserves modularity while avoiding the computational cost of a monolithic formulation.Source: §8 · p. 42
  • The banding example coupled the fluid-Naghdi shell model to a three-element Windkessel outlet and updated Rp, Rd, and C from inlet-pressure observations using a reduced-order unscented Kalman filter.In this illustrative calculation, estimated parameters and simulated inlet pressure approached stable final-time values.Source: §8 · p. 42
  • Tube tests showed stable pressure-wave propagation under spatial and temporal refinement, while shell-model choice and scaling affected wave speed and local displacement; patient-specific data and image-based geometries remain future extensions.The banding calculation is presented as a small proof of concept, and future work includes uncertainty and parameter-sensitivity analyses.Source: §8 · p. 42
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