A Neural Ordinary Differential Equation Model for Visualizing Deep Neural Network Behaviors in Multi-Parametric MRI based Glioma Segmentation

Purpose: To develop a neural ordinary differential equation (ODE) model for visualizing deep neural network (DNN) behavior during multi-parametric MRI (mp-MRI) based glioma segmentation as a method to enhance deep learning explainability. Methods: By hypothesizing that deep feature extraction can be modeled as a spatiotemporally continuous process, we designed a novel deep learning model, neural ODE, in which deep feature extraction was governed by an ODE without explicit expression. The dynamics of 1) MR images after interactions with DNN and 2) segmentation formation can be visualized after solving ODE. An accumulative contribution curve (ACC) was designed to quantitatively evaluate the utilization of each MRI by DNN towards the final segmentation results. The proposed neural ODE model was demonstrated using 369 glioma patients with a 4-modality mp-MRI protocol: T1, contrast-enhanced T1 (T1-Ce), T2, and FLAIR. Three neural ODE models were trained to segment enhancing tumor (ET), tumor core (TC), and whole tumor (WT). The key MR modalities with significant utilization by DNN were identified based on ACC analysis. Segmentation results by DNN using only the key MR modalities were compared to the ones using all 4 MR modalities. Results: All neural ODE models successfully illustrated image dynamics as expected. ACC analysis identified T1-Ce as the only key modality in ET and TC segmentations, while both FLAIR and T2 were key modalities in WT segmentation. Compared to the U-Net results using all 4 MR modalities, Dice coefficient of ET (0.784->0.775), TC (0.760->0.758), and WT (0.841->0.837) using the key modalities only had minimal differences without significance. Conclusion: The neural ODE model offers a new tool for optimizing the deep learning model inputs with enhanced explainability. The presented methodology can be generalized to other medical image-related deep learning applications.

Heavy Ball Neural Ordinary Differential Equations

We propose heavy ball neural ordinary differential equations (HBNODEs), leveraging the continuous limit of the classical momentum accelerated gradient descent, to improve neural ODEs (NODEs) training and inference. HBNODEs have two properties that imply practical advantages over NODEs: (i) The adjoint state of an HBNODE also satisfies an HBNODE, accelerating both forward and backward ODE solvers, thus significantly reducing the number of function evaluations (NFEs) and improving the utility of the trained models. (ii) The spectrum of HBNODEs is well structured, enabling effective learning of long-term dependencies from complex sequential data. We verify the advantages of HBNODEs over NODEs on benchmark tasks, including image classification, learning complex dynamics, and sequential modeling. Our method requires remarkably fewer forward and backward NFEs, is more accurate, and learns long-term dependencies more effectively than the other ODE-based neural network models. Code is available at \url{https://github.com/hedixia/HeavyBallNODE}.

Post-Radiotherapy PET Image Outcome Prediction by Deep Learning under Biological Model Guidance: A Feasibility Study of Oropharyngeal Cancer Application

This paper develops a method of biologically guided deep learning for post-radiation FDG-PET image outcome prediction based on pre-radiation images and radiotherapy dose information. Based on the classic reaction-diffusion mechanism, a novel biological model was proposed using a partial differential equation that incorporates spatial radiation dose distribution as a patient-specific treatment information variable. A 7-layer encoder-decoder-based convolutional neural network (CNN) was designed and trained to learn the proposed biological model. As such, the model could generate post-radiation FDG-PET image outcome predictions with possible time-series transition from pre-radiotherapy image states to post-radiotherapy states. The proposed method was developed using 64 oropharyngeal patients with paired FDG-PET studies before and after 20Gy delivery (2Gy/daily fraction) by IMRT. In a two-branch deep learning execution, the proposed CNN learns specific terms in the biological model from paired FDG-PET images and spatial dose distribution as in one branch, and the biological model generates post-20Gy FDG-PET image prediction in the other branch. The proposed method successfully generated post-20Gy FDG-PET image outcome prediction with breakdown illustrations of biological model components. Time-series FDG-PET image predictions were generated to demonstrate the feasibility of disease response rendering. The developed biologically guided deep learning method achieved post-20Gy FDG-PET image outcome predictions in good agreement with ground-truth results. With break-down biological modeling components, the outcome image predictions could be used in adaptive radiotherapy decision-making to optimize personalized plans for the best outcome in the future.