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

This paper presents a coupled fluid-structure interaction model combining Navier-Stokes equations, a low-regularity Naghdi shell, and a Windkessel boundary condition to simulate pulmonary artery banding, featuring a non-iterative partitioned numerical scheme and parameter estimation via an unscented Kalman filter to analyze hemodynamic effects of localized constrictions.

Original authors: Yifan Wang, Josip Tambaca, Suncica Canic

Published 2026-10-16

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CC BY 4.0

Original authors: Yifan Wang, Josip Tambaca, Suncica Canic

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the delicate world of newborn medicine, some infants are born with a hole between the two main pumping chambers of their heart. This opening allows blood to flow freely between the left and right sides, often sending too much blood rushing into the lungs. When the lungs receive more blood than they can handle, they become congested, and the heart struggles to pump effectively. To buy time for these babies until a full repair is possible, doctors sometimes perform a palliative procedure called pulmonary artery banding. This involves placing a tight band around the main artery leading to the lungs to restrict the flow, much like pinching a garden hose to reduce the water pressure. However, finding the perfect tightness is a difficult balancing act. If the band is too loose, it fails to protect the lungs; if it is too tight, it starves the lungs of blood and overworks the heart. Currently, surgeons rely on experience and limited measurements taken during surgery to adjust the band, but these snapshots cannot fully predict how the complex network of blood vessels downstream will react to the change.

To address this uncertainty, a team of researchers has developed a sophisticated computer simulation that acts as a virtual testing ground for these procedures. Their work creates a detailed digital twin of a baby's pulmonary artery, complete with a band and a representation of the entire downstream circulatory system. Unlike previous models that treated blood vessel walls as rigid pipes, this new approach recognizes that arteries are flexible and move with the pulse of the blood. The researchers built a system that couples the physics of flowing blood with the mechanics of a stretching, bending vessel wall. They also included a mathematical model of the tiny blood vessels further down the line, which act like a sponge and a resistance valve, absorbing and slowing the flow. By connecting these three elements—the moving fluid, the flexible wall, and the downstream resistance—the team created a unified framework that can simulate how pressure waves travel through the system and how the vessel deforms under the stress of the band.

The core of this achievement lies in how the researchers described the vessel wall. Instead of using a simplified model that ignores certain types of bending or twisting, they employed a more comprehensive mathematical description known as a Naghdi shell model. This approach allows the simulation to handle the sharp, sudden changes in shape that occur where the band is placed, including the transition zones where the vessel narrows and then widens again. These areas often have corners or kinks that simpler models struggle to represent accurately. By using this more robust description, the simulation can capture not just the stretching of the vessel but also its bending and shearing, providing a more realistic picture of the mechanical forces at play. The team tested this new method by simulating a pressure pulse traveling through a straight, flexible tube. They found that their model produced consistent and stable results, accurately tracking how the pressure wave moved and how the tube expanded and contracted, even as they refined the grid of the computer mesh to higher levels of detail.

A critical part of their work was comparing this advanced model against a simpler, more traditional version that assumes the vessel wall moves only in a radial direction, like a balloon inflating and deflating. While both models produced similar patterns of expansion, the researchers discovered a significant difference in the speed at which the pressure wave traveled. The more complex model, which accounts for the full range of wall movements, showed the pressure wave moving more slowly than the simplified version. This finding highlights that the way a vessel deforms—whether it stretches, bends, or shears—directly influences how fast pressure signals travel through the circulatory system. This distinction is vital for understanding the true mechanical environment inside a banding procedure, where the speed of these waves affects how the heart and lungs interact.

To make this tool useful for clinical decision-making, the researchers added a method to estimate the hidden properties of the downstream circulation. In a real patient, doctors cannot easily measure the resistance and elasticity of the tiny vessels deep within the lungs. The team's simulation uses a statistical technique to work backward from a single, easily measurable value: the pressure at the entrance of the artery. By feeding the observed inlet pressure into their model, the system iteratively adjusts the estimated resistance and compliance of the downstream network until the simulation matches the real-world measurement. In a test case involving an idealized banding scenario, the system successfully updated its estimates of the downstream resistance and compliance, converging on a stable set of values that explained the observed pressure. The simulation showed that the proximal resistance, the resistance closest to the band, changed significantly during the estimation process, while the distal resistance and compliance remained relatively steady.

The ultimate goal of this framework is to separate the local effects of the band from the uncertainty of the distant circulation. By accurately estimating the downstream properties, surgeons could theoretically test different band sizes and positions in a virtual environment before touching the patient. They could predict how a specific band configuration would alter the pressure drop across the vessel, the speed of the blood flow, and the stress on the vessel wall. While this study serves as a proof of concept using idealized geometries rather than a specific patient's anatomy, it demonstrates that such a coupled approach is mathematically sound and computationally feasible. The researchers have laid the groundwork for a future where virtual simulations can guide the precise tuning of life-saving interventions, moving beyond guesswork toward a more predictable and personalized approach to treating congenital heart defects.

Technical Summary: A Coupled Fluid–Naghdi Shell–Windkessel Model for Pulmonary Artery Banding

Problem Statement and Motivation
This work addresses the hemodynamic and mechanical challenges associated with pulmonary artery banding (PAB), a palliative surgical intervention used to limit excessive pulmonary blood flow in neonates with congenital heart disease, specifically those with large ventricular septal defects. Current clinical practice relies on empirical sizing and intraoperative measurements that cannot uniquely determine the optimal band configuration or the properties of the distal pulmonary circulation. Existing computational models often utilize rigid-wall assumptions or prescribed outlet conditions, failing to capture the feedback between arterial wall deformation and blood flow, or to distinguish local band effects from downstream vascular impedance.

The authors propose a mathematical framework to separate the local mechanical effects of the band from the influence of the unresolved downstream vasculature. The specific challenges addressed include:

Modeling the arterial wall as a thin, deformable structure capable of handling piecewise-smooth geometries with localized constrictions (discontinuous surface slopes).
Coupling a 3D fluid domain with a 2D shell model and a 0D downstream circulation model (Windkessel) without suffering from added-mass instability.
Estimating unknown downstream vascular parameters (proximal resistance, distal resistance, and compliance) from limited inlet pressure observations.

Methodology
The paper develops a coupled three-dimensional (3D) fluid-structure interaction (FSI) model integrated with a zero-dimensional (0D) Windkessel model.

Fluid Domain: Blood flow is modeled using the incompressible Navier-Stokes equations within an Arbitrary Lagrangian-Eulerian (ALE) framework to handle the moving fluid domain caused by wall deformation.
Structural Domain (Shell): The arterial wall is represented by a 3D Naghdi shell model defined on a 2D reference domain. Unlike classical Koiter shells, the Naghdi formulation treats the midsurface displacement and shell-director rotation as independent unknowns. This allows the model to retain membrane, bending, and transverse-shear deformation mechanisms. Crucially, the formulation requires only first derivatives of the reference midsurface ( $W^{1,\infty}$ regularity), enabling the modeling of piecewise-smooth geometries with geometric corners (such as the transition zones of a band) without requiring higher-order continuity.
Downstream Circulation: The distal pulmonary circulation is modeled using a three-element Windkessel model, relating outlet flow and pressure through proximal resistance ( $R_p$ ), distal resistance ( $R_d$ ), and compliance ( $C$ ).
Coupling Scheme: The system is solved using a non-iterative, kinematically coupled partitioned scheme. To mitigate added-mass instability, the scheme incorporates the translational inertia of the shell into the fluid subproblem via a Robin-type interface condition. The algorithm proceeds in three steps per time step: (1) solving the structure subproblem using a fraction of the previous fluid traction; (2) updating the ALE mesh; and (3) solving the fluid and Windkessel subproblems together on the updated domain.
Parameter Estimation: To address uncertainty in downstream parameters, a reduced-state unscented Kalman filter (UKF) is employed. The filter state consists only of the logarithms of the three Windkessel parameters ( $R_p, R_d, C$ ) to ensure positivity. The filter updates these estimates based on the discrepancy between simulated and observed inlet pressures, avoiding the need for tangent or adjoint equations.

Key Contributions

Low-Regularity Naghdi Shell Formulation: The authors derive a weak formulation for a Naghdi shell that accommodates piecewise-smooth midsurfaces with discontinuous slopes. This is essential for accurately modeling the geometric transitions inherent in pulmonary artery banding.
Stable Partitioned Scheme: A non-iterative splitting scheme is developed that incorporates shell inertia via a Robin interface condition. A discrete energy estimate is derived for a linearized fixed-domain problem, demonstrating unconditional stability with respect to added-mass coupling for the splitting parameter $\beta \in [0, 1]$ .
Reduced-State Data Assimilation: The integration of a reduced-state UKF allows for the estimation of downstream vascular properties from inlet pressure data alone, decoupling the local band effects from distal circulation uncertainty.
Comparative Analysis: The study compares the standard Naghdi model against a modified, predominantly radial (Koiter-like) shell model to quantify the influence of transverse shear and axial deformation on pressure-wave propagation speeds.

Numerical Results
The framework was validated through three numerical experiments:

Straight-Tube Benchmark: A 3D extension of a pressure-pulse benchmark verified the implementation. Both spatial and temporal convergence studies demonstrated consistent pressure-wave propagation and wall deformation.
Model Comparison: Comparing the standard Naghdi shell with a penalized, radial-dominant model revealed that while both reproduced expected pressure-wave and wall-expansion behaviors, the modified (penalized) model produced a faster-moving pressure front than the standard Naghdi model. Theoretical analysis indicated that 3D models exhibit wave speeds approximately $1/\sqrt{2}$ times slower than their 2D counterparts under equivalent radial stiffness assumptions.
Pulmonary Artery Banding with Estimation: An idealized banded pulmonary artery was coupled to a Windkessel outlet. The reduced-state UKF successfully estimated the proximal resistance, distal resistance, and compliance from a constant inlet pressure observation. The results showed a significant adjustment in the estimated proximal resistance (increasing by a factor of ~22) to match the observed pressure drop, while distal resistance and compliance remained relatively stable. The simulated inlet pressure converged to within 2.69% of the target observation.

Significance and Claims
The authors position this work as a foundational step toward patient-specific optimization of pulmonary artery banding. They claim that the proposed framework provides a basis for investigating the hemodynamic and mechanical effects of localized vascular constrictions while explicitly accounting for the influence of downstream vascular properties.

The paper modestly frames the numerical experiments as a "proof of concept" for the coupled solver and parameter-estimation framework rather than a completed patient-specific optimization study. The significance lies in the ability to distinguish local band mechanics from distal impedance, a capability the authors argue is missing in current rigid-wall or prescribed-outlet models. Future applications envisioned by the authors include the use of pulsatile patient-specific data and image-based geometries to systematically investigate band configurations and arterial wall properties.

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