H4 - Bone-driven Simulation for Orthognathic Surgery Using Physics-Informed Neural Networks

Introduction: : Cranio-maxillofacial (CMF) deformities are congenital or acquired abnormalities of the facial skeleton and its surrounding soft-tissue. To restore occlusion, CMF patients usually undergo orthognathic surgery, in which bony segments are repositioned to restore a normal occlusion [1,2,3]. Soft tissue then passively adapts to the new bony framework; however, the relationship between bony displacement and soft-tissue response is highly nonlinear and patient-specific, so normalizing bone shape does not automatically result in a normal facial contour. Therefore, accurate bone-driven simulations are essential for achieving desired surgical outcomes.
Finite Element Method (FEM) has long been pursued for this task because it can explicitly encode material properties and boundary conditions. In practice, FEM simulation demands substantial computational resources, commonly requiring 10 to 20 minutes [4,5], which limits their clinical utility. Data-driven based deep learning approaches [6,7,8,9] have been explored to accelerate prediction; although they reduce computation to seconds, their performance collapses when training data are limited or distributed differently from clinical reality.
We address this limitation with a physics-informed neural network (PINN) that combines the speed of deep learning with the accuracy of physics-guided models. During training, the network minimizes not only data-driven loss but also residuals of the governing partial differential equations for soft-tissue mechanics and defined boundary-condition losses [10,11]. Embedding these physical priors guides the model to consider fundamental biomechanical laws, delivering more accurate soft-tissue predictions while preserving real-time performance even with limited data.

Materials and

Methods: : The facial region from nose to mandible was extracted to focus on the surgical field and discretized it into a hexahedral mesh with 3960 nodes. For each patient, pre-operative soft-tissue node coordinates together with the corresponding adjacency matrix and edge weights were fed to a Soft-tissue GNN subnetwork. The same topological template was reused to encode skeletal motion: inner-surface nodes were replaced by the bone displacement vectors obtained from the planned osteotomies, providing the input features for a Bone GNN subnetwork.
Both subnetworks contain six graph neural network (GNN) layers with tanh activations. To mitigate over-smoothing, jumping-knowledge was inserted after the third layer of each branch. The resulting soft-tissue and bone features were concatenated and passed through two fully connected layers with tanh activation to predict the soft-tissue displacement vector per node.
Training minimized a loss composed of three terms. (1) data-driven loss: the mean-squared error (MSE) between the network output and the gold-standard FEM simulation displacement vectors (2) physics loss: the residual of the Navier-Cauchy equation for homogeneous linear elasticity, computed from gradients of the predicted displacement (3) boundary-condition loss: nodes belonging to the midface were constrained to zero displacement, whereas nodes rigidly attached to the bones were constrained to match given displacement vectors. The overall architecture and loss formulation are illustrated in Figure 1.
Experiments were conducted and evaluated on 42 orthognathic-surgery cases using five-fold cross-validation. Accuracy was quantified by the volumetric distance and outer-surface distance between network predictions and FEM simulation results.

Results, Conclusions, and Discussions:: Results
Table 1 reports the volumetric and surface-distance errors for four anatomical subregions with the whole face. Using only data-driven supervision, Lampen et al. [7] obtains whole-face errors of 0.644 mm (volume) and 0.846 mm (outer surface distance). Our PINN-based simulation reduces these to 0.617 mm and 0.809 mm, a relative drop of about 4 percent in both metrics. Two-tailed paired t-tests across the 42 cases confirm that every reduction, including those in the individual nose, lips, cheek and chin regions, is significant at p < 0.05, highlighting the benefit of embedding physical constraints. Moreover, our model achieves a mean inference time of 0.05s per case on an NVIDIA A100 GPU, thereby enabling real-time interactive surgical planning.
Figure 2 presents qualitative results for a randomly selected patient. Front and profile views of the predicted soft tissue closely follow the FEM simulation, and the accompanying error maps show small local deviations, indicating that global shape and regional details are preserved even in areas of large chin displacement.

Discussion
The key observation is that performance gains extend beyond the highly deformable chin to the entire face. Regularizing every node with the same physics law enables the network to generalize well despite the limited training set. This stands in contrast to conventional data-driven approaches, which often overfit sparsely sampled motions. A current limitation is the relatively coarse mesh; the lips and surrounding mouth region are represented by few nodes, which restricts fine-grained evaluation. We anticipate that a denser mesh resolution, combined with patient-specific soft-tissue material properties, will further enhance predictive accuracy, especially for subtle surface features that are clinically relevant for aesthetic outcomes.

Conclusion
Embedding biomechanical equations into deep learning network yields a simulator that is both fast and accurate for orthognathic surgery planning. The proposed PINN-based simulation outperforms a data-driven model across all facial regions while retaining real-time inference. Future work will focus on higher-resolution meshes and the incorporation of patient-specific tissue properties to achieve even finer predictive detail.

Acknowledgements and/or References (Optional):: References
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