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Purpose: To develop a neural ordinary differential equation (ODE) model forvisualizing deep neural network (DNN) behavior during multi-parametric MRI(mp-MRI) based glioma segmentation as a method to enhance deep learningexplainability. Methods: By hypothesizing that deep feature extraction can bemodeled as a spatiotemporally continuous process, we designed a novel deeplearning model, neural ODE, in which deep feature extraction was governed by anODE without explicit expression. The dynamics of 1) MR images afterinteractions with DNN and 2) segmentation formation can be visualized aftersolving ODE. An accumulative contribution curve (ACC) was designed toquantitatively evaluate the utilization of each MRI by DNN towards the finalsegmentation results. The proposed neural ODE model was demonstrated using 369glioma patients with a 4-modality mp-MRI protocol: T1, contrast-enhanced T1(T1-Ce), T2, and FLAIR. Three neural ODE models were trained to segmentenhancing tumor (ET), tumor core (TC), and whole tumor (WT). The key MRmodalities with significant utilization by DNN were identified based on ACCanalysis. Segmentation results by DNN using only the key MR modalities werecompared to the ones using all 4 MR modalities. Results: All neural ODE modelssuccessfully illustrated image dynamics as expected. ACC analysis identifiedT1-Ce as the only key modality in ET and TC segmentations, while both FLAIR andT2 were key modalities in WT segmentation. Compared to the U-Net results usingall 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 differenceswithout significance. Conclusion: The neural ODE model offers a new tool foroptimizing the deep learning model inputs with enhanced explainability. Thepresented methodology can be generalized to other medical image-related deeplearning applications.

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