Implicit neural representations for accurate estimation of the standard model of white matter

TL;DR

This work addresses the challenge of accurately estimating the Standard Model parameters of white matter from diffusion MRI, which is hampered by high dimensionality and noise. It proposes an implicit neural representation framework that maps 3D coordinates to SM kernel and FOD parameters using Fourier-feature spatial encoding, a small MLP, and per-parameter heads, enabling self-supervised joint estimation and continuous spatial upsampling. Across synthetic and in vivo data, the INR method demonstrates superior accuracy and noise robustness compared with supervised and NLLS baselines, supports SH orders up to at least eight, and can incorporate gradient non-uniformity corrections during fitting. The approach offers a flexible, data-efficient tool for diffusion MRI analysis that yields spatially coherent parameter maps and better supports downstream tasks like tractography, while avoiding training-data biases inherent to supervised methods.

Abstract

Diffusion magnetic resonance imaging (dMRI) enables non-invasive investigation of tissue microstructure. The Standard Model (SM) of white matter aims to disentangle dMRI signal contributions from intra- and extra-axonal water compartments. However, due to the model its high-dimensional nature, accurately estimating its parameters poses a complex problem and remains an active field of research, in which different (machine learning) strategies have been proposed. This work introduces an estimation framework based on implicit neural representations (INRs), which incorporate spatial regularization through the sinusoidal encoding of the input coordinates. The INR method is evaluated on both synthetic and in vivo datasets and compared to existing methods. Results demonstrate superior accuracy of the INR method in estimating SM parameters, particularly in low signal-to-noise conditions. Additionally, spatial upsampling of the INR can represent the underlying dataset anatomically plausibly in a continuous way. The INR is self-supervised, eliminating the need for labeled training data. It achieves fast inference, is robust to noise, supports joint estimation of SM kernel parameters and the fiber orientation distribution function with spherical harmonics orders up to at least 8, and accommodates gradient non-uniformity corrections. The combination of these properties positions INRs as a potentially important tool for analyzing and interpreting diffusion MRI data.

Figures (9)

• Figure 1: INR network architecture. a) shows how the input coordinates are mapped to a higher dimensional frequency space. b) These values are then forwarded to the MLP. c) Output layer of the MLP is converted to SM parameters.
• Figure 2: Visualization of the INR fitting process. The input coordinates $\bm{x}$ are input in the INR (architecture shown in Figure \ref{['fig:architecture']}) and are mapped to a parameter estimate $\bm{\hat{k}}$. Using $\bm{\hat{k}}$ the signal estimate $\hat{S}$ is reconstructed following (3). The loss between $\hat{S}$ and the measured signal $S$ is used to update the INR weights.
• Figure 3: Experiment 1 (SNR 20): a) Scatter density plots of ground truth versus parameter estimations of all methods. The titles of the subplots indicate $\rho$ and RMSE. b) SM parameter maps corresponding to the results in a. Bottom row shows the ground truth (GT).
• Figure 4: Effect of Rician Loss Likelihood function (SNR = 20). a) Scatter plots for estimation with MSE and Rician loss. b) Brain parameter maps of the predictions from a. Difference maps are with the ground truth parameters.
• Figure 5: SM parameter maps fitted on in vivo data where each row corresponds to a different method.