JuliaCon 2025

Tumor-immune dynamics modeling requires integrating biological principles with data-driven learning. This work presents a physics-informed neural ordinary differential equation (PNODE) approach implemented in Julia for accurate tumor-immune interaction modeling.

Methodology
Our approach combines established Gompertz growth dynamics with neural-learned immune effects using Kevin Atsou's tumor growth datasets. The dual neural network architecture separates immune-mediated suppression from general model corrections. The immune response network processes normalized tumor volume and immune cell fractions to predict immune suppression effects, while the correction network uses volume and time inputs to capture unmodeled residuals.

The physics-informed loss function incorporates five components: volume prediction accuracy, derivative matching for physics consistency, growth smoothness penalties, biological constraints ensuring volume positivity, and L2 regularization. This multi-objective approach ensures learned dynamics respect fundamental biological laws while maintaining predictive accuracy.

Julia Implementation
The implementation leverages six core Julia packages demonstrating the ecosystem's scientific computing capabilities:

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DifferentialEquations.jl: Robust ODE solving with automatic differentiation support

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Flux.jl: Flexible neural network framework with seamless ecosystem integration

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SciMLSensitivity.jl: Efficient gradient computation through differential equation solutions

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Optimization.jl: Unified optimization interface for training coordination

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OptimizationOptimJL.jl: Advanced LBFGS optimization algorithms

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Zygote.jl: Automatic differentiation enabling end-to-end gradient computation

Training employs a two-phase strategy combining AdamW (1000 iterations) for global exploration and LBFGS (1000 iterations) for local refinement, achieving convergence in under one hour.

Results
The model demonstrates strong performance across multiple evaluation metrics:

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Prediction accuracy exceeding R² = 0.85 across tumor scenarios C2-C4 and cases T1-T5

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Captured 42.3% tumor volume reduction through immune effect modeling

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Strong negative correlation (r = -0.82, p < 0.001) between immune response strength and tumor growth rates

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Real-time prediction capabilities suitable for interactive analysis

The 3D immune response surface learned by the neural networks reveals complex non-linear relationships between tumor burden and immune activation, providing insights into tumor-immune interaction mechanisms.

Technical Contributions
The work demonstrates several technical innovations:

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Integration of physics-informed constraints with neural ODE modeling

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Dual network architecture separating biological mechanisms

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Multi-objective loss function balancing accuracy and biological realism

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Efficient Julia implementation showcasing ecosystem interoperability

The approach successfully models tumor volume trajectories under varying immune conditions, capturing both the underlying Gompertz growth dynamics and immune-mediated modifications. The physics-informed framework ensures biological plausibility while the neural components provide flexibility to capture complex, non-linear relationships.

Dataset and Validation
Model training and validation used Kevin Atsou's comprehensive tumor-immune interaction datasets, including time-series tumor volume measurements, corresponding immune cell levels, and multiple immune parameter scenarios. The dataset encompasses tumor scenarios T1-T5 under immune conditions C2-C4, enabling robust evaluation across diverse biological conditions.

Julia Ecosystem Impact
This work contributes to the Julia scientific computing ecosystem by demonstrating effective integration of differential equations, neural networks, and optimization for biological modeling. The approach showcases Julia's unique advantages for scientific machine learning applications, achieving both high performance and code clarity essential for complex biological system modeling.

The methodology provides a foundation for physics-informed modeling of other biological systems where mechanistic understanding must be combined with data-driven learning. The modular design enables adaptation to different biological processes and disease systems.