Fig. 1: Schematic representation of Tumor Dynamic Neural-ODE (TDNODE).

The deep learning model is designed to discover the underlying dynamical law from tumor dynamics data and use the identified kinetic parameters to predict patient overall survival (OS). Time series tumor dynamic data were split at the patient level into a training and test set. Times of last observation were obtained and the augmented patient time series data were created. The SLD data were Z-score normalized using the mean and standard deviation from the training set; the measurement times were scaled at the patient level using each patient’s last observed measurement time. Pre-treatment and truncated post-treatment tumor dynamic profiles were fed into the initial condition and parameter encoders of TDNODE, respectively. Post-treatment time series data were partitioned to improve learning of longitudinal tumor dynamics. The parameter encoder output for each patient was scaled by the corresponding time of the last measurement to produce a set of kinetic rate parameters with the interpretation of inverse time as the physical unit (please note that the neural network schematics are only representational and do not reflect the actual layers or channel dimensionalities used). The initial condition (\({\bf{z}}(0)\)) and kinetic parameters (\({\bf{p}}\)) were then used in a Neural-ODE model that represents the learned dynamical law and acts as a decoder of the system. Finally, the model solution was reduced to SLD predictions as a function of time. In parallel, the patients’ OS is predicted using the ML model XGBSE. Via SHAP-ML and PCA analysis on the kinetic rate parameter distribution, our modeling paradigm successfully links tumor dynamics and OS in a data-driven manner.