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Hyperelastic Cardiovascular NN–FE: A Framework for Integrating Nonlinear Finite Element Solvers with Neural Networks for the Hyperelastic Modeling of Cardiac Valve Tissue

This paper presents a hybrid NN–FE framework that integrates a feedforward neural network with a nonlinear finite element solver to create a computationally efficient, physics-consistent surrogate model for accurately predicting the hyperelastic deformation of cardiac valve tissue under various biaxial loading conditions.

Original authors: Maedeh Makki, Gerhard Holzapfel, Maziar Raissi, Chung-Hao Lee

Published 2026-09-14

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CC BY 4.0

Original authors: Maedeh Makki, Gerhard Holzapfel, Maziar Raissi, Chung-Hao Lee

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

The human heart is a tireless pump, but its valves are the quiet gatekeepers that ensure blood flows in only one direction. These delicate flaps of tissue, known as leaflets, must withstand constant pressure, stretching and snapping back with every beat. To understand how they work, or to design artificial replacements that won't fail, scientists need to predict exactly how this tissue deforms under stress. Traditionally, this has been done using complex computer simulations that treat the tissue like a mathematical puzzle. These simulations are incredibly accurate but also incredibly slow, often taking hours or even days to run a single scenario. This speed limit makes them impractical for real-time uses, such as guiding a surgeon during an operation or testing thousands of device designs in a single afternoon. On the other hand, newer methods that rely purely on data and artificial intelligence can be fast, but they often lack a deep understanding of the physical laws governing the tissue, sometimes producing results that look plausible but are physically impossible.

A team of researchers has now bridged this gap by creating a new hybrid system that combines the speed of artificial intelligence with the rigorous physical accuracy of traditional engineering software. Their work focuses on the tricuspid valve, specifically the posterior leaflet, which is a thin, flexible sheet of tissue found in the right side of the heart. The researchers wanted to see if they could train a computer program to predict how this tissue would stretch and bend under various forces, without having to run the slow, heavy simulations every time. They did not simply teach the computer to guess based on past pictures; instead, they built a system where the computer's guesses are constantly checked and corrected by a physics engine. This approach allows the system to learn the true mechanical behavior of the tissue while maintaining the speed necessary for real-world applications.

To build this system, the team started with real biological data. They took a sample of tissue from a pig heart, a common model for human heart mechanics, and cut a small square piece from the center of the valve. They then placed this sample in a machine that pulled on it from four sides, stretching it in different combinations of force. They tested seven distinct patterns of pulling, ranging from pulling equally in all directions to pulling much harder in one direction than the other. By measuring how the tissue moved and deformed under these specific loads, they created a detailed map of its mechanical behavior. They confirmed that the tissue behaves like a hyperelastic material, meaning it returns to its original shape after being stretched, but it gets stiffer the more it is pulled, a trait essential for heart valves to function properly.

The core of their new framework is a partnership between two different types of computer programs. The first is a neural network, a type of artificial intelligence designed to recognize patterns. This network acts as a fast predictor, taking the location on the tissue and the amount of force applied, and instantly guessing how much that specific spot will move. However, because the tissue is complex and the forces are nonlinear, a simple guess is rarely perfect. This is where the second part of the system comes in: a traditional finite element solver. This is the heavy-duty engineering software that calculates the laws of physics to ensure the tissue is in a state of balance. In this new framework, the neural network makes a quick guess, and the engineering software immediately steps in to correct it, ensuring that the laws of physics are satisfied. The neural network then learns from this correction, adjusting its internal logic to make better guesses next time.

This "solver-in-the-loop" approach proved to be highly effective. The researchers tested their system by hiding one of the seven stretching patterns from the training data and asking the system to predict the outcome for that hidden pattern. In most cases, the system successfully predicted the movement of the tissue with a high degree of accuracy. For the overall shape and movement of the tissue, the predictions were off by less than 15 percent on average, and the timing of the movement matched the real physics almost perfectly. The system was even more precise when predicting how much the tissue stretched, with errors of less than 1.4 percent. When it came to predicting the internal stress, or the force felt within the tissue fibers, the error remained below 7.8 percent. These results suggest that the system has truly learned the underlying rules of how the tissue behaves, rather than just memorizing the data it was shown.

The study also revealed where the system struggles. The predictions were most accurate in the middle of the tissue sample, where the movement is smooth and predictable. The errors tended to cluster near the edges, where the tissue was gripped by the testing machine. This is a known challenge in mechanics; the sharp constraints at the edges create complex stress concentrations that are difficult for any model to resolve perfectly. Furthermore, the system found it slightly harder to predict the tissue's behavior when the forces were extremely unbalanced, such as pulling much harder in one direction than the other, especially if that specific type of imbalance was rare in the training data. Despite these localized challenges, the system consistently reproduced the characteristic "J-shaped" curve of the tissue's response, capturing both the easy initial stretch and the rapid stiffening that occurs at high loads.

The significance of this work lies in its potential to change how biomedical engineers approach complex problems. By combining the flexibility of neural networks with the reliability of physics-based solvers, the researchers have created a tool that is both fast and trustworthy. Unlike purely data-driven models that might fail when faced with a new situation, this hybrid system is grounded in the physical laws of the material. This makes it a promising candidate for real-time applications, such as helping surgeons plan procedures or allowing engineers to rapidly test new heart valve designs. While the current study was limited to a single tissue sample and static loading conditions, the framework demonstrates a clear path forward. It shows that it is possible to build intelligent systems that respect the laws of physics, offering a powerful new way to simulate the intricate mechanics of the human body without sacrificing speed or accuracy.

Technical Summary: Hyperelastic Cardiovascular NN–FE Framework

Problem Statement
Accurate prediction of the mechanical behavior of cardiac valve tissues is essential for computational biomechanics and surgical planning. However, conventional finite element (FE) methods, while physically rigorous, incur high computational costs that hinder real-time applications. Conversely, purely data-driven neural network approaches often lack physical consistency and interpretability. While Physics-Informed Neural Networks (PINNs) attempt to bridge this gap by embedding partial differential equations (PDEs) into the loss function, they frequently encounter convergence difficulties and spectral bias when applied to strongly nonlinear, hyperelastic materials under large strains. This study addresses the need for a computationally efficient surrogate model that maintains physical fidelity without the convergence instabilities associated with standard PINNs in hyperelastic regimes.

Methodology
The authors propose a hybrid framework, termed NN–FE, which integrates a feedforward neural network with a nonlinear FE solver to model the anisotropic hyperelastic response of the tricuspid valve posterior leaflet (TVPL) under quasi-static biaxial loading.

Experimental Data: TVPL tissues from porcine hearts were subjected to displacement-controlled biaxial tests at seven distinct loading ratios ( $F_X:F_Y$ ranging from 1:1 to 0.25:1). Tissue response was characterized using a Fung-type hyperelastic constitutive model, with material parameters identified via differential evolution optimization.
Neural Network Architecture: A six-layer feedforward neural network (five hidden layers with 128 neurons each, using hyperbolic tangent activation) approximates the pointwise displacement field as a function of spatial coordinates, time, and the loading ratio.
Solver-in-the-Loop Integration: Unlike standard PINNs that enforce equilibrium via PDE residuals, the NN–FE framework employs a "solver-in-the-loop" approach. The neural network generates initial displacement predictions, which are then refined by a nonlinear FE solver (Abaqus) to satisfy the equilibrium equations via Newton-Raphson iterations.
Training Strategy: The FE-corrected displacement fields serve as physics-consistent training targets. The loss function minimizes the mean squared error between the neural network's raw predictions and the FE-corrected fields. This approach eliminates the need for explicit PDE residual terms or empirical weighting of multiple physics-based losses. The framework was evaluated using a leave-one-ratio-out cross-validation strategy across six case studies, including both in-distribution (asymmetric ratios) and out-of-distribution (equibiaxial 1:1) scenarios.

Key Contributions

Hybrid Architecture: The paper introduces a novel coupling of neural networks and commercial FE solvers where the solver acts as a non-differentiable black box to enforce equilibrium, thereby avoiding the convergence bottlenecks of strong-form PINNs in large-strain hyperelasticity.
Simplified Loss Function: By relying on the FE solver to enforce governing equations, the training objective requires only a single residual term (displacement discrepancy), removing the complexity of balancing multiple physics-based loss weights.
Generalization Capability: The framework demonstrates the ability to learn constitutive behaviors and generalize to unseen loading ratios, including equibiaxial conditions trained on highly asymmetric data.

Results
The framework was validated across six case studies with the following performance metrics:

Displacement Prediction: The model achieved mean absolute percentage errors (MAPE) of less than 15% for displacement predictions. Temporal coefficients of determination ( $R^2$ ) exceeded 0.85 for out-of-distribution scenarios, reaching up to 0.997 for in-distribution cases.
Stress and Stretch Accuracy: Tissue stretch predictions demonstrated superior accuracy with a MAPE < 1.4%. Stress predictions exhibited a MAPE of less than 7.8%, though errors were slightly higher due to the exponential amplification inherent in the Fung model at peak loads.
Error Localization: Analysis revealed that prediction errors were spatially localized near the constrained boundaries (BioRakes interfaces) and temporally peaked at maximum loading. This is attributed to the exponential stiffening of the tissue and the difficulty in resolving high displacement gradients at boundaries.
Constitutive Learning: The framework successfully reproduced the characteristic J-shaped stress-stretch curves, capturing both the compliant toe region and exponential stiffening, confirming that the network learned the underlying constitutive behavior.

Significance and Claims
The authors claim that the NN–FE framework represents a computationally efficient surrogate model suitable for real-time applications in cardiovascular biomechanics, such as surgical planning and device design optimization. The study highlights that this approach successfully overcomes the "stiff" equation challenges and spectral bias often reported in PINNs for hyperelastic problems.

The paper modestly acknowledges limitations, noting that the current workflow is restricted to quasi-static conditions and relies on a single tissue specimen, which limits immediate generalizability to patient-specific heterogeneity. Furthermore, the authors note that while the framework performs robustly in the interior domain, boundary regions remain challenging due to the highly nonlinear nature of the tissue response and sparse sampling of the anisotropic stress-strain space in the experimental protocol. Despite these limitations, the work establishes a foundational step toward integrating FE methods and neural networks for efficient, physics-consistent modeling of complex cardiovascular tissues.

Technical Summary: Hyperelastic Cardiovascular NN–FE Framework

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