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

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, 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. Thus, there are twofold challenges in modeling tumor dynamics: (i) accurate estimation of model parameters and (ii) discovery of the mathematical equations governing the physical and biological systems. These types of problems are referred to as gray-box identification areas, where both experimental data and partial system knowledge are used to recover unknown parameters and missing components. 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 mechanisms from sparse and noisy measurements. We test the performance of the CBINN model on three distinct nonlinear compartmental tumour-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, 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. Structural and practical identifiablility of the model parameters are also discussed using computational and Fisher information matrix based analysis. This work provides valuable guidance for researchers addressing inverse problems and gray-box identification challenges in complex dynamical systems.