Bayesian insights into exchange and restriction in gray matter diffusion MRI.
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- [1] § Methods › Biophysical models ↔ nls_simulations_estimation.py, lines 209–226 · score 0.52 · narrow pulse approximation, microstructure, fitting, NEXI, SANDIX, models
Paper
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The authors' code
Python · 324 lines · 11 KB · no license · 1 match
- #%%
- # Fits simulated data using graymatter_swissknife (NLLS) and generates
- # Supplementary Figures 9 and 10.
- #%%
- import sys
- import argparse
- import numpy as np
- from pathlib import Path
- import matplotlib.pyplot as plt
- from graymatter_swissknife.nls.nls import nls_parallel
- from graymatter_swissknife.nls.gridsearch import find_nls_initialization
- from graymatter_swissknife.models.parameters.acq_parameters import AcquisitionParameters
- from graymatter_swissknife.models.find_model import find_model
- #%%
- # Parse arguments
- if hasattr(sys, 'ps1') == False: # Running on a terminal
- parser = argparse.ArgumentParser()
- parser.add_argument("--model",
- choices=['nexi', 'sandix'],
- help="Which model to consider.")
- parser.add_argument(
- "--protocol",
- choices=['connectom', 'extended'],
- help="Which protocol to consider: Connectom or extended \
- protocol from SMEX.")
- parser.add_argument(
- "--noise",
- choices=['rician', 'noiseless'],
- help="Whether to use the noise distribution from the data \
- or noise-free simulations.")
- args = parser.parse_args()
- if args.model == None or args.protocol == None or args.noise == None:
- print('Error: choose a biophysical model, a protocol, and a noise \
- distribution to use.')
- sys.exit(1)
- model = args.model
- protocol = args.protocol
- noise = args.noise
- nf_features = args.nf_features
- else: # Running in an interactive environment
- model = 'nexi'
- protocol = 'extended'
- noise = 'rician'
- #%%
- def postprocess_NEXI_SANDIX(samples, config):
- # Convert u0 and u1 into De_par and De_perp
- idx_u0 = np.where(np.array(list(config['prior'].keys())) == 'U0')[0]
- idx_u1 = np.where(np.array(list(config['prior'].keys())) == 'U1')[0]
- u0 = samples[:, idx_u0]
- u1 = samples[:, idx_u1]
- # Set negative values to 0, otherwise get nan values
- u0 = np.clip(u0, 0, 1)
- u1 = np.clip(u1, 0, 1)
- Di_min = 0.1
- Di_max = 3.5
- De_min = 0.1
- Di = np.sqrt((Di_max - Di_min)**2 * u0) + Di_min
- De = (Di - Di_min) * u1 + De_min
- out_samples = samples.copy()
- out_samples[:, idx_u0] = Di
- out_samples[:, idx_u1] = De
- return out_samples
- #%%
- def plot_prior_distribution(samples, prior, savename='prior_distribution.png',\
- savedir=None):
- if samples.shape[1] != len(prior):
- raise ValueError('Number of sample and prior parameters do not match.')
- # Plot the prior distribution of the parameters
- fig, axs = plt.subplots(1, len(prior), figsize=(5 * len(prior), 5))
- for i, param in enumerate(prior.keys()):
- axs[i].hist(samples[:, i], bins=50)
- axs[i].set_title(param)
- axs[i].set_xlim(prior[param][0], prior[param][1])
- plt.show()
- if savedir is not None:
- plt.savefig(savedir / savename)
- #%%
- # Path to the simulated data
- data_dir = Path.cwd() / 'simulations' / f'{protocol}' / f'{model}'
- file = 'model-{}_noise-c1_protocol-{}_limits-classic_noise-{}_set-{}.npz'
- #%%
- # Load the test data
- test_file = file.format(model, protocol, noise, 'test')
- npz_test = np.load(data_dir / test_file, allow_pickle=True)
- #%%
- # Load the test signals and parameters
- x_test = npz_test['signal']
- theta_postproc_test = npz_test['parameters']
- theta_uniform_test = theta_postproc_test.copy()
- theta_uniform_test[:, 1:3] = npz_test['U0_U1_samples']
- # Filter out samples where Di > 3 um^2/ms or De > 3 um^2/ms
- x_test = x_test[np.where((theta_postproc_test[:, 1] <= 3)
- & (theta_postproc_test[:, 2] <= 3))[0], :]
- theta_uniform_test = theta_uniform_test[np.where(
- (theta_postproc_test[:, 1] <= 3) & (theta_postproc_test[:, 2] <= 3))[0], :]
- theta_postproc_test = theta_postproc_test[np.where(
- (theta_postproc_test[:, 1] <= 3) & (theta_postproc_test[:, 2] <= 3))[0], :]
- # Limit to a certain number of voxels
- voxel_nb = 1000
- x_test = x_test[:voxel_nb, :]
- theta_test = theta_postproc_test[:voxel_nb, :]
- sigma = npz_test['sigma'][:voxel_nb]
- print(f'Theta test shape: {theta_test.shape}')
- print(f'x test shape: {x_test.shape}')
- #%%
- # Create folder to save the results
- folder_save = Path.cwd() / 'results' / model / protocol / noise / 'nls'
- folder_save.mkdir(exist_ok=True, parents=True)
- #%%
- # Load prior information
- prior_postproc_params = npz_test['parameter_names']
- prior_uniform_params = prior_postproc_params.copy()
- prior_uniform_params[1:3] = npz_test['U0_U1_names']
- prior_postproc_lim = npz_test['limits']
- prior_postproc_lim[1][1] = 3.0
- prior_postproc_lim[2][1] = 3.0
- prior_uniform_lim = prior_postproc_lim.copy()
- # prior_uniform_lim[1] = [0.0, 1.0]
- prior_uniform_lim[1] = [0.0, 9 / (3.5)**2]
- prior_uniform_lim[2] = [0.0, 1.0]
- #%%
- # Define and plot the priors
- prior_uniform = {
- p: [prior_uniform_lim[i, 0], prior_uniform_lim[i, 1]]
- for i, p in enumerate(prior_uniform_params)
- }
- print(f'Uniform prior: {prior_uniform}')
- plot_prior_distribution(theta_uniform_test,
- prior_uniform,
- savename='prior_uniform_distribution.png',
- savedir=folder_save)
- prior_postprocessing = {
- p: [prior_postproc_lim[i, 0], prior_postproc_lim[i, 1]]
- for i, p in enumerate(prior_postproc_params)
- }
- print(f'Prior postprocessing: {prior_postprocessing}')
- plot_prior_distribution(theta_postproc_test,
- prior_postprocessing,
- savename='prior_postprocessing_distribution.png',
- savedir=folder_save)
- #%%
- # Protocol parameters
- if protocol == 'connectom':
- bvals = np.array([
- 1, 2.5, 4, 6, 7.5, 1, 2.5, 4, 6, 7.5, 1, 2.5, 4, 6, 7.5, 1, 2.5, 4, 6,
- 7.5
- ])
- delta = np.array([
- 20, 20, 20, 20, 20, 29, 29, 29, 29, 29, 39, 39, 39, 39, 39, 49, 49, 49,
- 49, 49
- ])
- nb_directions = np.array([
- 13, 25, 25, 32, 65, 13, 25, 25, 32, 65, 13, 25, 25, 32, 65, 13, 25, 25,
- 32, 65
- ])
- small_delta = 9
- elif protocol == 'extended':
- bvals = np.array([
- 0.1, 0.5, 1, 2, 3, 4, 5, 6.25, 7.111, 8.163, 9.467, 11.111, 12.098,
- 13.223, 14.512, 16, 17.729, 19.753, 22.145, 25, 28.444, 32.653, 37.87,
- 44.444, 52.893, 64, 79.012, 100, 0.1, 0.5, 1, 2, 3, 4, 5, 6.25, 7.111,
- 8.163, 9.467, 11.111, 13.223, 16, 19.753, 25, 32.653, 44.444, 64, 0.1,
- 0.5, 1, 2, 3, 4, 5, 6.25, 7.111, 8.163, 9.467, 11.111, 13.223, 16,
- 19.753, 25, 32.653, 44.444
- ])
- delta = np.array([
- 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16,
- 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 11, 11, 11, 11, 11, 11, 11, 11,
- 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 7.5, 7.5, 7.5, 7.5, 7.5,
- 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5
- ])
- nb_directions = np.array([
- 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10,
- 10, 10, 30, 30, 30, 30, 30, 30, 30, 30, 10, 10, 10, 10, 10, 10, 10, 10,
- 10, 10, 10, 10, 10, 10, 10, 10, 30, 30, 30, 10, 10, 10, 10, 10, 10, 10,
- 10, 10, 10, 10, 10, 10, 10, 10, 10, 30, 30
- ])
- small_delta = 4.5
- else:
- raise ValueError(
- "The protocol type should be either 'connectom' or 'extended'")
- acq_parameters = AcquisitionParameters(bvals, delta, small_delta)
- #%%
- # NLS fitting
- if model == 'nexi' and noise == 'noiseless':
- microstruct_model = find_model('Nexi_Narrow_Pulses_Approximation')
- elif model == 'nexi' and noise == 'rician':
- microstruct_model = find_model('Nexi_Narrow_Pulses_Approximation' +
- 'RiceMean')
- elif model == 'sandix' and noise == 'noiseless':
- microstruct_model = find_model('Sandix')
- elif model == 'sandix' and noise == 'rician':
- microstruct_model = find_model('Sandix' + 'RiceMean')
- else:
- raise ValueError(f"Unknown model: {model}")
- acq_param = AcquisitionParameters(bvals, delta, small_delta=small_delta)
- if model == 'nexi':
- grid_search_nb_points = [15, 12, 8, 8]
- elif model == 'sandix':
- grid_search_nb_points = [7, 5, 5, 5, 5, 5]
- #%%
- # Define the limits for the NLS parameters including noise
- if noise == 'rician':
- new_prior = microstruct_model.param_lim.copy()
- new_prior[0:prior_postproc_lim.shape[0], :] = prior_postproc_lim
- prior_postproc_lim = new_prior
- #%%
- # Compute the initial Ground Truth to start the NLS with if requested
- initial_gt = find_nls_initialization(x_test,
- sigma,
- voxel_nb,
- acq_param,
- microstruct_model,
- prior_postproc_lim,
- grid_search_nb_points,
- debug=False)
- #%%
- # Compute the NLS estimations
- estimations, estimation_init = nls_parallel(x_test,
- voxel_nb,
- microstruct_model,
- acq_param,
- nls_param_lim=prior_postproc_lim,
- initial_gt=initial_gt,
- n_cores=-1)
- #%%
- # Save the NLS estimations
- np.savez(folder_save / f'nls_estimations_{model}_{protocol}_{noise}.npz',
- estimations=estimations,
- estimation_init=estimation_init,
- theta_test=theta_test,
- x_test=x_test,
- sigma=sigma,
- prior_postproc_params=prior_postproc_params,
- prior_postproc_lim=prior_postproc_lim)
- #%%
- if model == 'sandix':
- # Convert f to volume fraction
- idx_f = np.where(prior_postproc_params == 'f')[0]
- idx_fs = np.where(prior_postproc_params == 'fs')[0]
- f = estimations[:, idx_f]
- fs = estimations[:, idx_fs]
- f_neurites = f * (1 - fs)
- f_soma = f * fs
- estimations[:, idx_f] = f_neurites
- estimations[:, idx_fs] = f_soma
- f_gt = theta_test[:, idx_f]
- fs_gt = theta_test[:, idx_fs]
- f_neurites_gt = f_gt * (1 - fs_gt)
- f_soma_gt = f_gt * fs_gt
- theta_test[:, idx_f] = f_neurites_gt
- theta_test[:, idx_fs] = f_soma_gt
- # %%
- fontsize = 30
- ptsize = 20
- plt.figure(figsize=(5 * prior_postproc_params.shape[0], 8))
- for p, param in enumerate(prior_postproc_params):
- plt.subplot(1, prior_postproc_params.shape[0], p + 1)
- plt.plot(theta_test[:voxel_nb, p],
- theta_test[:voxel_nb, p],
- c='k',
- alpha=0.5)
- plt.scatter(theta_test[:, p], estimations[:, p], c='b', s=ptsize)
- plt.title(f'{param}', fontsize=fontsize)
- if param == 't_ex':
- plt.title(r'$t_{ex}$', fontsize=fontsize)
- elif param == 'Di':
- plt.title(r'$D_{n}^{\|}$', fontsize=fontsize)
- elif param == 'De':
- plt.title(r'$D_{e}$', fontsize=fontsize)
- elif param == 'fs':
- plt.title(r'$f_{s}$', fontsize=fontsize)
- elif param == 'f':
- plt.title(r'$f_{n}$', fontsize=fontsize)
- elif param == 'rs':
- plt.title(r'$r_{s}$', fontsize=fontsize)
- elif param == 'fs':
- plt.title(r'$f_{s}$', fontsize=fontsize)
- else:
- plt.title(fr'${param}$', fontsize=fontsize)
- plt.xlabel('Ground Truth', fontsize=fontsize)
- if model == 'sandix' and param == 'f':
- plt.xticks([0.25, 0.75]) # Set x ticks to 0.25 and 0.75
- plt.tick_params(axis='both', which='major', labelsize=fontsize - 2)
- if p == 0:
- plt.ylabel('NLLS estimation', fontsize=fontsize)
- plt.tight_layout()
- plt.savefig(folder_save / 'plot_ground_truth_vs_map_nls.png')
- # %%
nls_simulations_estimation.py at commit eec173c, no license · at the source
Overview
- Cardiff University Brain Research Imaging Centre (CUBRIC), Cardiff University, Cardiff, United Kingdom
- School of Computer Science and Informatics, Cardiff University, Cardiff, United Kingdom
- Department of Radiology, Lausanne University Hospital (CHUV), Lausanne, Switzerland
- School of Biology and Medicine, University of Lausanne, Lausanne, Switzerland
Abstract
Biophysical models in diffusion MRI (dMRI) hold promise for characterizing gray matter tissue microstructure. Yet, the reliability of their parameter estimates remains largely under studied, especially in models that incorporate water exchange. In this study, we investigate the accuracy, precision, and presence of degeneracy of two recently proposed gray matter models, NEXI and SANDIX, using established acquisition protocols, on both simulated and in vivo data. We employ µGUIDE, a Bayesian inference framework based on deep learning, to quantify parameter uncertainty and detect degeneracies, enabling a more interpretable assessment of model fits. Our results show that while some microstructural parameters, such as extra-cellular diffusivity and neurite signal fraction, are robustly estimated, others, including exchange time and soma radius, are often associated with high uncertainty and estimation bias, particularly under realistic noise conditions and reduced acquisition protocols. Comparison with non-linear least squares fitting highlights the critical advantage of uncertainty-aware methods: the ability to flag and filter out unreliable estimates. Together, these findings emphasize the need to report uncertainty and account for model degeneracies when interpreting model-based estimates. Our study advocates for the integration of probabilistic fitting approaches into imaging pipelines to improve reproducibility and biological interpretability.
Reproduced under the paper's license (CC BY), from the paper cited above.
Repository
Its files are read in the Code ↔ Paper reader above, with 1 match between paragraphs and lines of code.
mjallais/NEXI_SANDIX_uGUIDE
eec173c4730eae328c097bf64beb80c851b049c2, 17 December 2025Availability: 1 check, the latest on 27 September 2026: the link answers
- 27 September 2026: the link answers
9 files
- create_connectom_extende
d_dataset.py , Python, 342 lines - create_signals_figure2.p
y , Python, 449 lines - nls_real_data_estimation
.py , Python, 129 lines - nls_simulations_estimati
on.py , Python, 324 lines, 1 match - uGUIDE_NLLS_real_data_di
stribution.py , Python, 583 lines - uGUIDE_inference_simulat
ions.py , Python, 448 lines - uGUIDE_nb_features_tune.
py , Python, 368 lines - uGUIDE_real_data.py, Python, 332 lines
- ReadMe.md, Text, 46 lines
The paper's code and data availability statement is in the Data section.
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Data
No dataset and no data link were found in the paper.
Data and Code Availability
The code used in this study is available at https://
Reproduced under the paper's license (CC BY), from the paper cited above.
Versions
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Version 1, 27 September 2026: the first record
Recorded: type, language, journal, volume, pages, dates, 7 authors, 6 keywords, 3 funders, 72 references.
Cite
This paper
Jallais, M., Uhl, Q., Pavan, T., Molendowska, M., Jones, D. K., Jelescu, I., & Palombo, M. (2026). Bayesian insights into exchange and restriction in gray matter diffusion MRI. Imaging neuroscience (Cambridge, Mass.), 4, IMAG.a.1317. https://
BibTeX
@article{jallais2026baye
author = {Jallais, Maëliss and Uhl, Quentin and Pavan, Tommaso and Molendowska, Malwina and Jones, Derek K. and Jelescu, Ileana and Palombo, Marco},
title = {{Bayesian insights into exchange and restriction in gray matter diffusion MRI}},
journal = {Imaging neuroscience (Cambridge, Mass.)},
year = {2026},
month = jul,
volume = {4},
pages = {IMAG.a.1317},
publisher = {MIT Press},
issn = {2837-6056},
doi = {10.1162/
url = {https://
pmid = {42524206},
pmcid = {PMC13409282}
}
RIS
TY - JOUR
AU - Jallais, Maëliss
AU - Uhl, Quentin
AU - Pavan, Tommaso
AU - Molendowska, Malwina
AU - Jones, Derek K.
AU - Jelescu, Ileana
AU - Palombo, Marco
TI - Bayesian insights into exchange and restriction in gray matter diffusion MRI
T2 - Imaging neuroscience (Cambridge, Mass.)
J2 - Imaging Neurosci (Camb)
PY - 2026
DA - 2026/
VL - 4
SP - IMAG.a.1317
SN - 2837-6056
PB - MIT Press
DO - 10.1162/
UR - https://
LA - en
ER -
CSL-JSON
{
"id": "10.1162/
"type": "article-journal",
"title": "Bayesian insights into exchange and restriction in gray matter diffusion MRI",
"container-title": "Imaging neuroscience (Cambridge, Mass.)",
"author": [
{
"family": "Jallais",
"given": "Maëliss"
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"family": "Uhl",
"given": "Quentin"
},
{
"family": "Pavan",
"given": "Tommaso"
},
{
"family": "Molendowska",
"given": "Malwina"
},
{
"family": "Jones",
"given": "Derek K."
},
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"given": "Ileana"
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{
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"given": "Marco"
}
],
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"volume": "4",
"page": "IMAG.a.1317",
"DOI": "10.1162/
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"ISSN": "2837-6056",
"publisher": "MIT Press",
"URL": "https://
"language": "en",
"issued": {
"date-parts": [
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