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

arXiv: https://arxiv.org/abs/2203.00628 · HTML: https://mlsanity.com/paper/2203.00628/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 78 references that are in the map.
- First posted on arXiv 1 Mar 2022.

## Builds on (most cited first, 21 of 21)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Fully Convolutional Networks for Semantic Segmentation](https://mlsanity.com/paper/1411.4038/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Deep Inside Convolutional Networks: Visualising Image Classification Models and Saliency Maps](https://mlsanity.com/paper/1312.6034/) (2013)
- [Attention U-Net: Learning Where to Look for the Pancreas](https://mlsanity.com/paper/1804.03999/) (2018)
- [Grad-CAM++: Improved Visual Explanations for Deep Convolutional Networks](https://mlsanity.com/paper/1710.11063/) (2017)
- [Score-CAM: Score-Weighted Visual Explanations for Convolutional Neural Networks](https://mlsanity.com/paper/1910.01279/) (2019)
- [Identifying the Best Machine Learning Algorithms for Brain Tumor Segmentation, Progression Assessment, and Overall Survival Prediction in the BRATS Challenge](https://mlsanity.com/paper/1811.02629/) (2018)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Explainable Deep Learning: A Field Guide for the Uninitiated](https://mlsanity.com/paper/2004.14545/) (2020)
- [Explaining Deep Neural Networks with a Polynomial Time Algorithm for Shapley Values Approximation](https://mlsanity.com/paper/1903.10992/) (2019)
- [Stiff Neural Ordinary Differential Equations](https://mlsanity.com/paper/2103.15341/) (2021)
- [Interpretable & Explorable Approximations of Black Box Models](https://mlsanity.com/paper/1707.01154/) (2017)
- [Brain tumor segmentation with self-ensembled, deeply-supervised 3D U-net neural networks: a BraTS 2020 challenge solution](https://mlsanity.com/paper/2011.01045/) (2020)
- [TorchDyn: A Neural Differential Equations Library](https://mlsanity.com/paper/2009.09346/) (2020)
- [Neural Ordinary Differential Equations for Semantic Segmentation of Individual Colon Glands](https://mlsanity.com/paper/1910.10470/) (2019)
- [Second-Order Neural ODE Optimizer](https://mlsanity.com/paper/2109.14158/) (2021)
- [Generalized Wasserstein Dice Score, Distributionally Robust Deep Learning, and Ranger for brain tumor segmentation: BraTS 2020 challenge](https://mlsanity.com/paper/2011.01614/) (2020)
- [Neural ODEs for Image Segmentation with Level Sets](https://mlsanity.com/paper/1912.11683/) (2019)
- [A Radiomics-Boosted Deep-Learning Model for COVID-19 and Non-COVID-19 Pneumonia Classification Using Chest X-ray Image](https://mlsanity.com/paper/2107.08667/) (2021)
- [Post-Radiotherapy PET Image Outcome Prediction by Deep Learning under Biological Model Guidance: A Feasibility Study of Oropharyngeal Cancer Application](https://mlsanity.com/paper/2105.10650/) (2021)

## Built on by (most cited first, 2 of 2)

- [Physics-Informed Machine Learning in Biomedical Science and Engineering](https://mlsanity.com/paper/2510.05433/) (2025)
- [Quantifying U-Net Uncertainty in Multi-Parametric MRI-based Glioma Segmentation by Spherical Image Projection](https://mlsanity.com/paper/2210.06512/) (2022)

## Abstract (arXiv)

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.