Deep Learning for PK/PD and Disease Progression Time-Course Predictions using Neural-ODE
James Lu (1), Brendan Bender (1), Yuanfang Guan (2), Pascal Chanu (1), Dan Lu (1), Joy C. Hsu (1), Jin Y. Jin (1)
(1) Modeling & Simulation/Clinical Pharmacology, Genentech Inc, USA, (2) Department of Computational Medicine & Bioinformatics, University of Michigan, USA.
Introduction: The modeling of pharmacokinetic/pharmacodynamic (PK/PD) and disease progression are currently performed using population approaches. However, the recent development of neural ordinary differential equations (neural-ODE)  offers a promising way to advance modeling by bringing the benefits of ODE formulation together with the power of deep learning. Rather than pre-specifying the structure of the ODEs as typically done in the current modeling paradigm, this formulation enables the generation of differential equations directly from data via neural networks . This methodology may enable the modeling of complex and variable data, shorten the turnaround time and improve the accuracy of individualized predictions.
Methods: Neural-ODE methodologies were developed and applied to two different contexts: PK/PD and disease progression modeling. Firstly, we developed a neural-PK/PD formulation that combines neural-ODE  with pharmacological principles . PK and PD data are encoded via networks consisting of gated recurrent units into low dimensional vectors. The vector field of the neural-PK/PD model is represented by a multilayer perceptron (MLP) . We applied the model to the PK and platelet count data for patients treated with trastuzumab emtansine (T-DM1) collected across clinical trials . The complete data is split by individual patients into a training (80%) and a test (20%) set. The network weights are optimized using the training set and the predictive performance is evaluated on the unseen test set. For the second application of disease progression modeling, we used the neural-ODE formulation to build a single dynamical system model describing all patients. Each patient is mapped to an initial disease state, via neural network encoders that transform the early observed data as well as the baseline covariates into a low-dimensional vector. Subsequently, the patients’ disease dynamics evolves autonomously, based on a vector field represented by an MLP. We applied the network to model geographic atrophy areas (GA)  and performed a cross-trial evaluation. For all cases, the predictive performance of models was evaluated using the R2 and RMSE metrics.
Results: For the first task of platelet count prediction, the neural-PK/PD methodology generated an ODE model whose qualitative dynamics was similar to the existing population-PK/PD model . In predicting future platelet count from observing only cycle 1 data, the population-PK/PD model showed predictive performance of R2=0.40±0.03, RMSE=62±1. In comparison, neural-PK/PD demonstrated improved results, with R2=0.51±0.02, RMSE=56±1. We also obtained similar findings using observation data up to cycles 2 and 3 to predict later cycles: the R2 for population-PK/PD are 0.39±0.02 and 0.46±0.02 respectively, while for neural-PK/PD they are 0.52±0.01 and 0.55±0.02 respectively. We further demonstrate the ability of the neural-PK/PD model, which was trained on patients most of whom were treated with the Q3W regimen, to simulate alternative dosing regimens of Q1W and Q3D. For the second task of predicting the disease progression of GA area, the neural-ODE model was able to make prediction reliably across trials: using the baseline covariates and one additional GA area measurement post baseline (at a median time of 5.5 months), the subsequent change in GA area at ~18 months (since the post baseline measurement) can be predicted with R2=0.35±0.05 and RMSE=1.47±0.2. We demonstrated that the proposed methodology can generalize well, by showing that when the training data is truncated to a shorter observation duration the model exhibits little loss of predictivity.
Conclusions: We demonstrated the favorable performance of neural-ODE models for making individualized patient time-course predictions. By building upon the strengths of both ODE formulation and deep learning, the neural-ODE formulation has the benefit of being interpretable (e.g., via the visualizations of the identified vector fields), generalizable (e.g., across dosing regimens and the duration of observation data), as well as being able to explore in an automated manner a large space of equations in describing complex data. The promising results and the positive attributes of the neural-ODE warrant further development and assessment, with potential future applications in advancing personalized medicine.
 Chen R.T.Q., et al, Advances in Neural Information Processing Systems (2018)
 Lu J., et al, Nature Machine Intelligence (accepted, 2021)
 Bender, B.C., et al, Cancer chemotherapy and pharmacology (2012)
 Chanu, P., et al, PAGE 2019