CBINNS: Cancer Biology-Informed Neural Network for Unknown Parameter Estimation and Missing Physics Identification

TL;DR

CBINN introduces a cancer biology-informed neural network framework to simultaneously estimate unknown parameters and uncover missing physics in tumor–immune ODE models from sparse, noisy data. By testing on three representative compartmental models with distinct dynamical regimes, CBINN demonstrates robust parameter recovery and accurate discovery of hidden dynamics, even when only partial state measurements are available. Structural identifiability analyses guide which parameters can be uniquely recovered, while Fisher information matrix assessments probe practical identifiability under noise. The method shows promise for enhancing predictive oncology models and guiding experimental design, with potential extensions to spatially resolved PDEs and incorporation of symbolic regression for interpretability.

Abstract

The dynamics of tumor-immune interactions within a complex tumor microenvironment are typically modeled using a system of ordinary differential equations or partial differential equations. These models introduce some unknown parameters that need to be estimated accurately and efficiently from the limited and noisy experimental data. Moreover, due to the intricate biological complexity and limitations in experimental measurements, tumor-immune dynamics are not fully understood, and therefore, only partial knowledge of the underlying physics may be available, resulting in unknown or missing terms within the system of equations. In this study, we develop a cancer biology-informed neural network model(CBINN) to infer the unknown parameters in the system of equations as well as to discover the missing physics from sparse and noisy measurements. We test the performance of the CBINN model on three distinct nonlinear compartmental tumor-immune models and evaluate its robustness across multiple synthetic noise levels. By harnessing these highly nonlinear dynamics, our CBINN framework effectively estimates the unknown model parameters and uncovers the underlying physical laws or mathematical structures that govern these biological systems, even from scattered and noisy measurements. The models chosen here represent the dynamic patterns commonly observed in compartmental models of tumor-immune interactions, thereby validating the generalizability and efficacy of our methodology.

Figures (24)

• Figure 1: The workflow for the identification of model parameters. If any parameters are not structurally identifiable, repeat data collection step to obtain additional data, or fix some parameters using prior knowledge. If the Julia StructuralIdentifiability test indicates that some parameters are only locally identifiable, then in this case we restrict the search ranges of those parameters. The parameters are estimated using the proposed CBINN and then the practical identifiability analysis is done using Fischer Information Matrix. In case of singular FIM certain parameter needs to be fixed or more data is to be collected.
• Figure 2: The output of local structural identifiability test for model (\ref{['eq1']}) - (\ref{['eq2']}) using Julia library StructuralIdentifiability dong2023differential. We find that if $x(t)$ is observable, then fixing $\eta$, all the other parameters of the model are locally identifiable (market as true in the output of the local structural identifiability test).
• Figure 3: Schematic of the CBINN algorithm for discovering unknown term $f(t; \theta_2)$. The scaled input $\hat{t}$ is fed to two fully-connected sub-networks, a solution network $\hat{u}(t;\theta_1)$ (top) that predicts the states and a dynamics network $f(t;\theta_2)$ (bottom) that predicts the missing terms each composed of several hidden layers with nonlinear activation function $\sigma$. The time derivative $d\hat{u}/dt$ is computed by automatic differentiation and combined with $f$ to form an ODE residual, and training minimizes the weighted sum of the data, initial-condition and ODE residual losses to estimate $\theta_1,\theta_2$. The loss terms are defined as mean squared error (MSE) losses.
• Figure 4: Schematic of the CBINN algorithm for predicting the unknown term $h(t; \theta_2)$ and $i(t; \theta_2)$. We have two networks, one to approximate the ODE solution and the other to approximate the unknown functions.
• Figure 5: CBINN inferred dynamics of tumor-immune model $(\ref{['eq11']})$--$(\ref{['eq21']})$ from noisy observations compared with the true solution. True solutions are generated by solving the system (\ref{['eq11']})-(\ref{['eq21']}) using the LSODA algorithm implemented in Python’s scipy.integrate.odeint function with the target parameter values. Predictions are performed on equally-spaced time instants in the interval of $0 - 100$ days. 200 measurements are corrupted by a zero-mean Gaussian noise and standard deviation of $\sigma^* = 0.05c$ where $c$ is the standard deviation of the observable variable ($x$) over the observation time window.