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Modeling with Differential Equations

Modeling with Differential Equations

Population dynamics can be described mathematically by considering the population size P(t) as a function of time. The rate of change of the population is then represented by the derivative of P(t). A simple assumption is that the rate of growth is proportional to the size of the population itself. This leads to an exponential growth model, where the population increases rapidly without bound. While this is a useful first approximation, it does not reflect realistic long-term...

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Related Experiment Video

Updated: Jan 17, 2026

Mapping Cortical Dynamics Using Simultaneous MEG/EEG and Anatomically-constrained Minimum-norm Estimates: an Auditory Attention Example

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Acquisition-independent deep learning for quantitative MRI parameter estimation using neural controlled differential

Daan Kuppens¹, Sebastiano Barbieri², Daisy van den Berg³

¹Department of Radiology and Nuclear Medicine, Amsterdam University Medical Center, Amsterdam, The Netherlands; Imaging and Biomarkers, Cancer Center Amsterdam, Amsterdam, The Netherlands.

Medical Image Analysis

|September 13, 2025

Summary

This summary is machine-generated.

Neural Controlled Differential Equations (NCDEs) offer a robust deep learning solution for quantitative MRI (QMRI) parameter estimation. This method improves accuracy across diverse acquisition protocols and challenging conditions, enhancing clinical applicability.

Keywords:

Deep learning Neural controlled differential equations Parameter estimation Quantitative MRI

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Area of Science:

Medical Imaging
Machine Learning
Quantitative MRI

Background:

Deep learning (DL) shows promise for quantitative MRI (QMRI) parameter estimation, outperforming traditional least squares (LSQ) fitting.
Current DL methods lack robustness to variations in MR acquisition protocols, hindering clinical adoption.
This limitation impedes the use of DL in clinical trials and practice.

Purpose of the Study:

To evaluate Neural Controlled Differential Equations (NCDEs) as a generalized deep learning tool for QMRI parameter estimation.
To assess the robustness and accuracy of NCDEs across different QMRI sequences and acquisition parameters.
To compare NCDE performance against LSQ fitting, particularly in low signal-to-noise ratio (SNR) scenarios.

Main Methods:

Implementation of NCDEs for QMRI parameter estimation.
Testing NCDEs on diverse QMRI models: variable flip angle T1-mapping, intravoxel incoherent motion MRI, and dynamic contrast-enhanced MRI.
Simulation studies and in vivo experiments in challenging anatomical regions (abdomen, leg) with varying SNR levels.

Main Results:

NCDEs demonstrated accurate QMRI parameter estimation irrespective of sequence length, independent variable configuration, or forward model.
NCDEs achieved lower mean squared error than LSQ fitting in low-SNR simulations and in vivo.
The improvement in parameter estimation accuracy with NCDEs was primarily due to reduced variance in estimation errors, especially in low-SNR conditions.

Conclusions:

NCDEs provide a robust and generalizable approach for QMRI parameter estimation, addressing limitations of current DL methods.
This technique is particularly beneficial in scenarios with high uncertainty or low image quality, such as abdominal and leg imaging.
NCDEs represent a significant advancement for the broader clinical and research application of deep learning in QMRI.