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Adaptive tumor growth forecasting via neural & universal ODEs

Kavya Subramanian, Prathamesh Dinesh Joshi, Raj Abhijit Dandekar, Rajat Dandekar, Sreedath Panat

TL;DR

This work addresses the challenge of forecasting tumor growth under limited data by integrating mechanistic Gompertz dynamics with data-driven Neural ODEs and Universal Differential Equations. By preprocessing data with sigmoid interpolation and employing symbolic recovery, the authors produce adaptive, interpretable models that outperform fixed-parameter approaches. Key contributions include robust forecasting under data scarcity, and the extraction of explicit, biologically plausible ODE expressions that combine logistic and Gompertz terms. The results, validated across multiple subjects, demonstrate the potential of SciML to inform personalized treatment planning in oncology.

Abstract

Forecasting tumor growth is critical for optimizing treatment. Classical growth models such as the Gompertz and Bertalanffy equations capture general tumor dynamics but may fail to adapt to patient-specific variability, particularly with limited data available. In this study, we leverage Neural Ordinary Differential Equations (Neural ODEs) and Universal Differential Equations (UDEs), two pillars of Scientific Machine Learning (SciML), to construct adaptive tumor growth models capable of learning from experimental data. Using the Gompertz model as a baseline, we replace rigid terms with adaptive neural networks to capture hidden dynamics through robust modeling in the Julia programming language. We use our models to perform forecasting under data constraints and symbolic recovery to transform the learned dynamics into explicit mathematical expressions. Our approach has the potential to improve predictive accuracy, guiding dynamic and effective treatment strategies for improved clinical outcomes.

Adaptive tumor growth forecasting via neural & universal ODEs

TL;DR

This work addresses the challenge of forecasting tumor growth under limited data by integrating mechanistic Gompertz dynamics with data-driven Neural ODEs and Universal Differential Equations. By preprocessing data with sigmoid interpolation and employing symbolic recovery, the authors produce adaptive, interpretable models that outperform fixed-parameter approaches. Key contributions include robust forecasting under data scarcity, and the extraction of explicit, biologically plausible ODE expressions that combine logistic and Gompertz terms. The results, validated across multiple subjects, demonstrate the potential of SciML to inform personalized treatment planning in oncology.

Abstract

Forecasting tumor growth is critical for optimizing treatment. Classical growth models such as the Gompertz and Bertalanffy equations capture general tumor dynamics but may fail to adapt to patient-specific variability, particularly with limited data available. In this study, we leverage Neural Ordinary Differential Equations (Neural ODEs) and Universal Differential Equations (UDEs), two pillars of Scientific Machine Learning (SciML), to construct adaptive tumor growth models capable of learning from experimental data. Using the Gompertz model as a baseline, we replace rigid terms with adaptive neural networks to capture hidden dynamics through robust modeling in the Julia programming language. We use our models to perform forecasting under data constraints and symbolic recovery to transform the learned dynamics into explicit mathematical expressions. Our approach has the potential to improve predictive accuracy, guiding dynamic and effective treatment strategies for improved clinical outcomes.

Paper Structure

This paper contains 21 sections, 6 equations, 25 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Data Preprocessing and Interpolation
  4. Gompertz ODE
  5. Neural ODE
  6. UDE
  7. Symbolic Recovery
  8. Results
  9. Gompertz ODE
  10. Neural ODE
  11. UDE
  12. Forecasting future growth
  13. Training with $90\%$ of the data
  14. Training with $80\%$ of the data
  15. Training with $70\%$ of the data
  16. ...and 6 more sections

Figures (25)

  • Figure 1: Experimental tumor volume measurements against the sigmoid interpolation curve.
  • Figure 2: Solution to the Gompertz growth model against the interpolated experimental data.
  • Figure 3: Solution to the Neural ODE model against the interpolated experimental data.
  • Figure 4: Solution to the UDE model against the interpolated experimental data.
  • Figure 5: Forecasting using 90% of the data. (a) Neural ODE model's forecast. (b) UDE model's forecast.
  • ...and 20 more figures