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Deep operator network models for predicting post-burn contraction

Selma Husanovic, Ginger Egberts, Alexander Heinlein, Fred Vermolen

TL;DR

It is suggested that DeepONets can effectively serve as a surrogate for traditional finite element methods in simulating post-burn wound evolution, with potential applications in medical treatment planning.

Abstract

Burn injuries present a significant global health challenge. Among the most severe long-term consequences are contractures, which can lead to functional impairments and disfigurement. Understanding and predicting the evolution of post-burn wounds is essential for developing effective treatment strategies. Traditional mathematical models, while accurate, are often computationally expensive and time-consuming, limiting their practical application. Recent advancements in machine learning, particularly in deep learning, offer promising alternatives for accelerating these predictions. This study explores the use of a deep operator network (DeepONet), a type of neural operator, as a surrogate model for finite element simulations, aimed at predicting post-burn contraction across multiple wound shapes. A DeepONet was trained on three distinct initial wound shapes, with enhancement made to the architecture by incorporating initial wound shape information and applying sine augmentation to enforce boundary conditions. The performance of the trained DeepONet was evaluated on a test set including finite element simulations based on convex combinations of the three basic wound shapes. The model achieved an $R^2$ score of $0.99$, indicating strong predictive accuracy and generalization. Moreover, the model provided reliable predictions over an extended period of up to one year, with speedups of up to 128-fold on CPU and 235-fold on GPU, compared to the numerical model. These findings suggest that DeepONets can effectively serve as a surrogate for traditional finite element methods in simulating post-burn wound evolution, with potential applications in medical treatment planning.

Deep operator network models for predicting post-burn contraction

TL;DR

It is suggested that DeepONets can effectively serve as a surrogate for traditional finite element methods in simulating post-burn wound evolution, with potential applications in medical treatment planning.

Abstract

Burn injuries present a significant global health challenge. Among the most severe long-term consequences are contractures, which can lead to functional impairments and disfigurement. Understanding and predicting the evolution of post-burn wounds is essential for developing effective treatment strategies. Traditional mathematical models, while accurate, are often computationally expensive and time-consuming, limiting their practical application. Recent advancements in machine learning, particularly in deep learning, offer promising alternatives for accelerating these predictions. This study explores the use of a deep operator network (DeepONet), a type of neural operator, as a surrogate model for finite element simulations, aimed at predicting post-burn contraction across multiple wound shapes. A DeepONet was trained on three distinct initial wound shapes, with enhancement made to the architecture by incorporating initial wound shape information and applying sine augmentation to enforce boundary conditions. The performance of the trained DeepONet was evaluated on a test set including finite element simulations based on convex combinations of the three basic wound shapes. The model achieved an score of , indicating strong predictive accuracy and generalization. Moreover, the model provided reliable predictions over an extended period of up to one year, with speedups of up to 128-fold on CPU and 235-fold on GPU, compared to the numerical model. These findings suggest that DeepONets can effectively serve as a surrogate for traditional finite element methods in simulating post-burn wound evolution, with potential applications in medical treatment planning.

Paper Structure

This paper contains 29 sections, 1 theorem, 35 equations, 10 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Numerical simulations
  4. Mathematical morphoelastic model
  5. The fibroblasts
  6. The myofibroblasts
  7. The signaling molecules
  8. The collagen molecules
  9. The mechanical components
  10. Boundary conditions
  11. Initial conditions
  12. Numerical discretizations and solvers
  13. Relative surface area
  14. Machine learning models
  15. Neural networks
  16. ...and 14 more sections

Key Result

Theorem 1

Let $\sigma$ be a continuous nonpolynomial function, $X$ be a Banach space, $K_1 \subset X$ and $K_2 \subset \mathbb{R}^d$ two compact subsets, respectively, $V$ be a compact subset of $\mathcal{C}(K_1)$, and $G$ be a nonlinear continuous operator $V \to \mathcal{C}(K_2)$. Then, for any $\varepsilon for all $v \in V$ and $y \in K_2$.

Figures (10)

  • Figure 1: Nonlinear activation functions (a) and a feedforward neural network with two hidden layers (b)
  • Figure 2: (Unstacked) DeepONet architecture; based on lu_deeponet_2021
  • Figure 3: The DeepONet architecture used for predicting dermal displacement. The legend at the top describes four other cases against which we have tested our proposed model; see \ref{['tab:performance']}
  • Figure 4: The three initial wound shapes used for training: rectangle (a), rhombus (b), and ellipse (c). Due to symmetry, a quarter of the complete domain is considered. Additionally, a visualization of the quadruple $(y_{cut}, x_m, y_m, x_{cut})$, which serves as extra input to the trunk. The coordinate $(x_l, y_l)$ is required for the sine augmentation step
  • Figure 5: Enforcement of boundary conditions as a result of sine augmentation (a) and a convex combination of the three basic wound shapes (b)
  • ...and 5 more figures

Theorems & Definitions (1)

  • Theorem 1: Universal approximation for operators chen_universal_1995lu_deeponet_2021