Neural Network Surrogate and Projected Gradient Descent for Fast and Reliable Finite Element Model Calibration: a Case Study on an Intervertebral Disc

TL;DR

This work tackles the heavy computational burden of calibrating finite element models in biomechanics by introducing a neural-network surrogate to predict RoM for a $L4-L5$ IVD FE model and a Projected Gradient Descent (PGD) calibration that enforces parameter feasibility. The NN surrogate enables near-instant RoM predictions, and PGD, with a projection step, efficiently identifies 13 material parameters by matching RoM measurements while keeping parameters within bounds. On synthetic data, PGD w/ NN achieves MAE ≈ $0.06$ and $ar{R}^2 \,=\, 0.99$, outperforming a GA-w/NN baseline and an inverse model; on experimental specimens, it outperforms GA w/ NN in five of six cases and delivers end-to-end calibration times of a few seconds after initial dataset generation. The approach demonstrates substantial speedups (end-to-end) and improved reliability, paving the way for extending to more complex FE models and patient-specific simulations, with future work addressing active learning, uncertainty, and geometry-informed surrogates.

Abstract

Accurate calibration of finite element (FE) models is essential across various biomechanical applications, including human intervertebral discs (IVDs), to ensure their reliability and use in diagnosing and planning treatments. However, traditional calibration methods are computationally intensive, requiring iterative, derivative-free optimization algorithms that often take days to converge. This study addresses these challenges by introducing a novel, efficient, and effective calibration method demonstrated on a human L4-L5 IVD FE model as a case study using a neural network (NN) surrogate. The NN surrogate predicts simulation outcomes with high accuracy, outperforming other machine learning models, and significantly reduces the computational cost associated with traditional FE simulations. Next, a Projected Gradient Descent (PGD) approach guided by gradients of the NN surrogate is proposed to efficiently calibrate FE models. Our method explicitly enforces feasibility with a projection step, thus maintaining material bounds throughout the optimization process. The proposed method is evaluated against SOTA Genetic Algorithm and inverse model baselines on synthetic and in vitro experimental datasets. Our approach demonstrates superior performance on synthetic data, achieving an MAE of 0.06 compared to the baselines' MAE of 0.18 and 0.54, respectively. On experimental specimens, our method outperforms the baseline in 5 out of 6 cases. While our approach requires initial dataset generation and surrogate training, these steps are performed only once, and the actual calibration takes under three seconds. In contrast, traditional calibration time scales linearly with the number of specimens, taking up to 8 days in the worst-case. Such efficiency paves the way for applying more complex FE models, potentially extending beyond IVDs, and enabling accurate patient-specific simulations.

Figures (13)

• Figure 1: Calibrating an L4-L5 intervertebral disc (IVD) finite element (FE) model to match in vitro measurements. (1) Create a dataset by sampling material parameters within feasible bounds using Latin hypercube sampling (LHS) mckay2000comparison and obtaining corresponding range of motion (RoM) values with FE simulations. (2) Train a neural network (NN) surrogate to minimize the Mean Absolute Error (MAE) between the predicted and simulated RoM. (3) Freeze the network weights and optimize the NN input parameters to match the predicted RoM to the measurements. Projected Gradient Descent (PGD) ensures the optimized parameters remain within feasible bounds. Finally, (4) validate the calibrated material parameters by comparing simulated intradiscal pressure (IDP) to experimental measurements.
• Figure 2: Meshed geometry of the L4-L5 IVD and the strain energy density functions in the FE model nicolini2022experimentalgruber12comparative. The figure shows the separation into nucleus pulposus (gray) and annulus fibrosus (blue), both with hyperelastic material definitions. The strain energy density functions for the nucleus ($\mathit{W_{n}}$, Mooney-Rivlin) and annulus ($\mathit{W_{a}}$, Holzapfel-Gasser-Ogden) are provided, with the latter specifically accounting for the anisotropic properties resulting from the collagen fibers in the annulus. The calibration process involved 13 parameters, with five in the strain energy density functions (bold). The remaining parameters, used in functions not provided here, describe fiber dispersion, angle variations, and stiffness changes across the annulus. For a more detailed explanation, refer to our previous publications nicolini2022experimentalgruber12comparative.
• Figure 3: Illustration of Projected Gradient Descent (PGD) in the loss landscape of two calibrated parameters. The contour plot shows the RoM MAE between the surrogate NN's predictions and the third specimen by nicolini2022effects. Only two input parameters are varied for illustration, while the others are fixed. The dashed box represents the feasible input parameter area. The algorithm follows these steps: (1) It starts from the black point, indicating the current step's configuration. (2) Using gradients of the loss, the red points representing local minima for specific load case and moment combinations are obtained, and their mean is computed, shown by the blue point. (3) This mean is then projected into the feasible area, shown by the green point. (4) Finally, the algorithm updates the configuration toward the projection, leading to the orange point, with the step size depending on the learning rate.
• Figure 4: Interpolation and extrapolation abilities of surrogate models across moments. The background colors correspond to the ranges evaluated: white for trained moment, peach for extrapolation, and blue for interpolation. Both abilities are essential for the calibration of specimens. The line shadings represent the standard deviation across the cross-validated folds. The NN surrogate outperforms the other models for all but moments of 0.5 Nm.
• Figure 5: Example calibration results for samples 1 (top) and 3 (bottom) from experiments by nicolini2022effects. The plots show the RoM values for the in vitro measurements (blue), the calibrated configuration using the proposed method (orange), and the GA w/ NN baseline (green). The proposed method performs better on sample 1 than on sample 3. The poorer performance on sample 3 is attributed to the large variability in RoM across load cases in the target configuration, which differs significantly from the surrogate's training data. See \ref{['appendix:detailed_results']} for the other specimens.