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Bayesian insights into exchange and restriction in gray matter diffusion MRI.

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  1. [1] § Methods › Biophysical models ↔ nls_simulations_estimation.py, lines 209–226 · score 0.52 · narrow pulse approximation, microstructure, fitting, NEXI, SANDIX, models

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The authors' code

Python · 324 lines · 11 KB · no license · 1 match

  1. #%%
  2. # Fits simulated data using graymatter_swissknife (NLLS) and generates
  3. # Supplementary Figures 9 and 10.
  4. #%%
  5. import sys
  6. import argparse
  7. import numpy as np
  8. from pathlib import Path
  9. import matplotlib.pyplot as plt
  10. from graymatter_swissknife.nls.nls import nls_parallel
  11. from graymatter_swissknife.nls.gridsearch import find_nls_initialization
  12. from graymatter_swissknife.models.parameters.acq_parameters import AcquisitionParameters
  13. from graymatter_swissknife.models.find_model import find_model
  14. #%%
  15. # Parse arguments
  16. if hasattr(sys, 'ps1') == False: # Running on a terminal
  17. parser = argparse.ArgumentParser()
  18. parser.add_argument("--model",
  19. choices=['nexi', 'sandix'],
  20. help="Which model to consider.")
  21. parser.add_argument(
  22. "--protocol",
  23. choices=['connectom', 'extended'],
  24. help="Which protocol to consider: Connectom or extended \
  25. protocol from SMEX.")
  26. parser.add_argument(
  27. "--noise",
  28. choices=['rician', 'noiseless'],
  29. help="Whether to use the noise distribution from the data \
  30. or noise-free simulations.")
  31. args = parser.parse_args()
  32. if args.model == None or args.protocol == None or args.noise == None:
  33. print('Error: choose a biophysical model, a protocol, and a noise \
  34. distribution to use.')
  35. sys.exit(1)
  36. model = args.model
  37. protocol = args.protocol
  38. noise = args.noise
  39. nf_features = args.nf_features
  40. else: # Running in an interactive environment
  41. model = 'nexi'
  42. protocol = 'extended'
  43. noise = 'rician'
  44. #%%
  45. def postprocess_NEXI_SANDIX(samples, config):
  46. # Convert u0 and u1 into De_par and De_perp
  47. idx_u0 = np.where(np.array(list(config['prior'].keys())) == 'U0')[0]
  48. idx_u1 = np.where(np.array(list(config['prior'].keys())) == 'U1')[0]
  49. u0 = samples[:, idx_u0]
  50. u1 = samples[:, idx_u1]
  51. # Set negative values to 0, otherwise get nan values
  52. u0 = np.clip(u0, 0, 1)
  53. u1 = np.clip(u1, 0, 1)
  54. Di_min = 0.1
  55. Di_max = 3.5
  56. De_min = 0.1
  57. Di = np.sqrt((Di_max - Di_min)**2 * u0) + Di_min
  58. De = (Di - Di_min) * u1 + De_min
  59. out_samples = samples.copy()
  60. out_samples[:, idx_u0] = Di
  61. out_samples[:, idx_u1] = De
  62. return out_samples
  63. #%%
  64. def plot_prior_distribution(samples, prior, savename='prior_distribution.png',\
  65. savedir=None):
  66. if samples.shape[1] != len(prior):
  67. raise ValueError('Number of sample and prior parameters do not match.')
  68. # Plot the prior distribution of the parameters
  69. fig, axs = plt.subplots(1, len(prior), figsize=(5 * len(prior), 5))
  70. for i, param in enumerate(prior.keys()):
  71. axs[i].hist(samples[:, i], bins=50)
  72. axs[i].set_title(param)
  73. axs[i].set_xlim(prior[param][0], prior[param][1])
  74. plt.show()
  75. if savedir is not None:
  76. plt.savefig(savedir / savename)
  77. #%%
  78. # Path to the simulated data
  79. data_dir = Path.cwd() / 'simulations' / f'{protocol}' / f'{model}'
  80. file = 'model-{}_noise-c1_protocol-{}_limits-classic_noise-{}_set-{}.npz'
  81. #%%
  82. # Load the test data
  83. test_file = file.format(model, protocol, noise, 'test')
  84. npz_test = np.load(data_dir / test_file, allow_pickle=True)
  85. #%%
  86. # Load the test signals and parameters
  87. x_test = npz_test['signal']
  88. theta_postproc_test = npz_test['parameters']
  89. theta_uniform_test = theta_postproc_test.copy()
  90. theta_uniform_test[:, 1:3] = npz_test['U0_U1_samples']
  91. # Filter out samples where Di > 3 um^2/ms or De > 3 um^2/ms
  92. x_test = x_test[np.where((theta_postproc_test[:, 1] <= 3)
  93. & (theta_postproc_test[:, 2] <= 3))[0], :]
  94. theta_uniform_test = theta_uniform_test[np.where(
  95. (theta_postproc_test[:, 1] <= 3) & (theta_postproc_test[:, 2] <= 3))[0], :]
  96. theta_postproc_test = theta_postproc_test[np.where(
  97. (theta_postproc_test[:, 1] <= 3) & (theta_postproc_test[:, 2] <= 3))[0], :]
  98. # Limit to a certain number of voxels
  99. voxel_nb = 1000
  100. x_test = x_test[:voxel_nb, :]
  101. theta_test = theta_postproc_test[:voxel_nb, :]
  102. sigma = npz_test['sigma'][:voxel_nb]
  103. print(f'Theta test shape: {theta_test.shape}')
  104. print(f'x test shape: {x_test.shape}')
  105. #%%
  106. # Create folder to save the results
  107. folder_save = Path.cwd() / 'results' / model / protocol / noise / 'nls'
  108. folder_save.mkdir(exist_ok=True, parents=True)
  109. #%%
  110. # Load prior information
  111. prior_postproc_params = npz_test['parameter_names']
  112. prior_uniform_params = prior_postproc_params.copy()
  113. prior_uniform_params[1:3] = npz_test['U0_U1_names']
  114. prior_postproc_lim = npz_test['limits']
  115. prior_postproc_lim[1][1] = 3.0
  116. prior_postproc_lim[2][1] = 3.0
  117. prior_uniform_lim = prior_postproc_lim.copy()
  118. # prior_uniform_lim[1] = [0.0, 1.0]
  119. prior_uniform_lim[1] = [0.0, 9 / (3.5)**2]
  120. prior_uniform_lim[2] = [0.0, 1.0]
  121. #%%
  122. # Define and plot the priors
  123. prior_uniform = {
  124. p: [prior_uniform_lim[i, 0], prior_uniform_lim[i, 1]]
  125. for i, p in enumerate(prior_uniform_params)
  126. }
  127. print(f'Uniform prior: {prior_uniform}')
  128. plot_prior_distribution(theta_uniform_test,
  129. prior_uniform,
  130. savename='prior_uniform_distribution.png',
  131. savedir=folder_save)
  132. prior_postprocessing = {
  133. p: [prior_postproc_lim[i, 0], prior_postproc_lim[i, 1]]
  134. for i, p in enumerate(prior_postproc_params)
  135. }
  136. print(f'Prior postprocessing: {prior_postprocessing}')
  137. plot_prior_distribution(theta_postproc_test,
  138. prior_postprocessing,
  139. savename='prior_postprocessing_distribution.png',
  140. savedir=folder_save)
  141. #%%
  142. # Protocol parameters
  143. if protocol == 'connectom':
  144. bvals = np.array([
  145. 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,
  146. 7.5
  147. ])
  148. delta = np.array([
  149. 20, 20, 20, 20, 20, 29, 29, 29, 29, 29, 39, 39, 39, 39, 39, 49, 49, 49,
  150. 49, 49
  151. ])
  152. nb_directions = np.array([
  153. 13, 25, 25, 32, 65, 13, 25, 25, 32, 65, 13, 25, 25, 32, 65, 13, 25, 25,
  154. 32, 65
  155. ])
  156. small_delta = 9
  157. elif protocol == 'extended':
  158. bvals = np.array([
  159. 0.1, 0.5, 1, 2, 3, 4, 5, 6.25, 7.111, 8.163, 9.467, 11.111, 12.098,
  160. 13.223, 14.512, 16, 17.729, 19.753, 22.145, 25, 28.444, 32.653, 37.87,
  161. 44.444, 52.893, 64, 79.012, 100, 0.1, 0.5, 1, 2, 3, 4, 5, 6.25, 7.111,
  162. 8.163, 9.467, 11.111, 13.223, 16, 19.753, 25, 32.653, 44.444, 64, 0.1,
  163. 0.5, 1, 2, 3, 4, 5, 6.25, 7.111, 8.163, 9.467, 11.111, 13.223, 16,
  164. 19.753, 25, 32.653, 44.444
  165. ])
  166. delta = np.array([
  167. 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16,
  168. 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 11, 11, 11, 11, 11, 11, 11, 11,
  169. 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 7.5, 7.5, 7.5, 7.5, 7.5,
  170. 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
  171. ])
  172. nb_directions = np.array([
  173. 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10,
  174. 10, 10, 30, 30, 30, 30, 30, 30, 30, 30, 10, 10, 10, 10, 10, 10, 10, 10,
  175. 10, 10, 10, 10, 10, 10, 10, 10, 30, 30, 30, 10, 10, 10, 10, 10, 10, 10,
  176. 10, 10, 10, 10, 10, 10, 10, 10, 10, 30, 30
  177. ])
  178. small_delta = 4.5
  179. else:
  180. raise ValueError(
  181. "The protocol type should be either 'connectom' or 'extended'")
  182. acq_parameters = AcquisitionParameters(bvals, delta, small_delta)
  183. #%%
  184. # NLS fitting
  185. if model == 'nexi' and noise == 'noiseless':
  186. microstruct_model = find_model('Nexi_Narrow_Pulses_Approximation')
  187. elif model == 'nexi' and noise == 'rician':
  188. microstruct_model = find_model('Nexi_Narrow_Pulses_Approximation' +
  189. 'RiceMean')
  190. elif model == 'sandix' and noise == 'noiseless':
  191. microstruct_model = find_model('Sandix')
  192. elif model == 'sandix' and noise == 'rician':
  193. microstruct_model = find_model('Sandix' + 'RiceMean')
  194. else:
  195. raise ValueError(f"Unknown model: {model}")
  196. acq_param = AcquisitionParameters(bvals, delta, small_delta=small_delta)
  197. if model == 'nexi':
  198. grid_search_nb_points = [15, 12, 8, 8]
  199. elif model == 'sandix':
  200. grid_search_nb_points = [7, 5, 5, 5, 5, 5]
  201. #%%
  202. # Define the limits for the NLS parameters including noise
  203. if noise == 'rician':
  204. new_prior = microstruct_model.param_lim.copy()
  205. new_prior[0:prior_postproc_lim.shape[0], :] = prior_postproc_lim
  206. prior_postproc_lim = new_prior
  207. #%%
  208. # Compute the initial Ground Truth to start the NLS with if requested
  209. initial_gt = find_nls_initialization(x_test,
  210. sigma,
  211. voxel_nb,
  212. acq_param,
  213. microstruct_model,
  214. prior_postproc_lim,
  215. grid_search_nb_points,
  216. debug=False)
  217. #%%
  218. # Compute the NLS estimations
  219. estimations, estimation_init = nls_parallel(x_test,
  220. voxel_nb,
  221. microstruct_model,
  222. acq_param,
  223. nls_param_lim=prior_postproc_lim,
  224. initial_gt=initial_gt,
  225. n_cores=-1)
  226. #%%
  227. # Save the NLS estimations
  228. np.savez(folder_save / f'nls_estimations_{model}_{protocol}_{noise}.npz',
  229. estimations=estimations,
  230. estimation_init=estimation_init,
  231. theta_test=theta_test,
  232. x_test=x_test,
  233. sigma=sigma,
  234. prior_postproc_params=prior_postproc_params,
  235. prior_postproc_lim=prior_postproc_lim)
  236. #%%
  237. if model == 'sandix':
  238. # Convert f to volume fraction
  239. idx_f = np.where(prior_postproc_params == 'f')[0]
  240. idx_fs = np.where(prior_postproc_params == 'fs')[0]
  241. f = estimations[:, idx_f]
  242. fs = estimations[:, idx_fs]
  243. f_neurites = f * (1 - fs)
  244. f_soma = f * fs
  245. estimations[:, idx_f] = f_neurites
  246. estimations[:, idx_fs] = f_soma
  247. f_gt = theta_test[:, idx_f]
  248. fs_gt = theta_test[:, idx_fs]
  249. f_neurites_gt = f_gt * (1 - fs_gt)
  250. f_soma_gt = f_gt * fs_gt
  251. theta_test[:, idx_f] = f_neurites_gt
  252. theta_test[:, idx_fs] = f_soma_gt
  253. # %%
  254. fontsize = 30
  255. ptsize = 20
  256. plt.figure(figsize=(5 * prior_postproc_params.shape[0], 8))
  257. for p, param in enumerate(prior_postproc_params):
  258. plt.subplot(1, prior_postproc_params.shape[0], p + 1)
  259. plt.plot(theta_test[:voxel_nb, p],
  260. theta_test[:voxel_nb, p],
  261. c='k',
  262. alpha=0.5)
  263. plt.scatter(theta_test[:, p], estimations[:, p], c='b', s=ptsize)
  264. plt.title(f'{param}', fontsize=fontsize)
  265. if param == 't_ex':
  266. plt.title(r'$t_{ex}$', fontsize=fontsize)
  267. elif param == 'Di':
  268. plt.title(r'$D_{n}^{\|}$', fontsize=fontsize)
  269. elif param == 'De':
  270. plt.title(r'$D_{e}$', fontsize=fontsize)
  271. elif param == 'fs':
  272. plt.title(r'$f_{s}$', fontsize=fontsize)
  273. elif param == 'f':
  274. plt.title(r'$f_{n}$', fontsize=fontsize)
  275. elif param == 'rs':
  276. plt.title(r'$r_{s}$', fontsize=fontsize)
  277. elif param == 'fs':
  278. plt.title(r'$f_{s}$', fontsize=fontsize)
  279. else:
  280. plt.title(fr'${param}$', fontsize=fontsize)
  281. plt.xlabel('Ground Truth', fontsize=fontsize)
  282. if model == 'sandix' and param == 'f':
  283. plt.xticks([0.25, 0.75]) # Set x ticks to 0.25 and 0.75
  284. plt.tick_params(axis='both', which='major', labelsize=fontsize - 2)
  285. if p == 0:
  286. plt.ylabel('NLLS estimation', fontsize=fontsize)
  287. plt.tight_layout()
  288. plt.savefig(folder_save / 'plot_ground_truth_vs_map_nls.png')
  289. # %%

nls_simulations_estimation.py at commit eec173c, no license · at the source

Overview

  1. Cardiff University Brain Research Imaging Centre (CUBRIC), Cardiff University, Cardiff, United Kingdom
  2. School of Computer Science and Informatics, Cardiff University, Cardiff, United Kingdom
  3. Department of Radiology, Lausanne University Hospital (CHUV), Lausanne, Switzerland
  4. School of Biology and Medicine, University of Lausanne, Lausanne, Switzerland
Institutions: Cardiff University (United Kingdom); Centre Hospitalier Universitaire Vaudois (Switzerland); University of Lausanne (Switzerland)
Journal: Imaging neuroscience (Cambridge, Mass.), volume 4, article IMAG.a.1317
Dates: received 15 December 2025; accepted 29 June 2026; published online 27 July 2026
Type: Research article · Language: English
License: CC BY
Identifiers: DOI 10.1162/imag.a.1317 · PMID 42524206 · PMCID PMC13409282 · OpenAlex W4417145599
Open access: diamond, a free copy (OpenAlex)
Status: code verified
Categories: structural MRI / diffusion (modality), computational (subfield)
Methods: Smoothing, state filtering, decompositions, Spectral & time-frequency, Machine learning
Keywords: diffusion MRI, gray matter, water exchange, Bayesian inference, uncertainty, degeneracy
Topic: Advanced Neuroimaging Techniques and Applications (Radiology, Nuclear Medicine and Imaging, Medicine), according to OpenAlex
Funding: UK Research and Innovation Future (MR/T020296/2, UKRI1073); Swiss National Science Foundation (194260); Wellcome Trust (104943/Z/14/Z, 227882/Z/23/Z, 317797/Z/24/Z)
Citations: cited by 1 paper (Europe PMC); 72 references in the paper

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

License: none: the authors keep all their rights
State: the link answers, verified on 27 September 2026
Evidence: files inventoried
Commit: eec173c4730eae328c097bf64beb80c851b049c2, 17 December 2025
Languages: Python (8)
Size: 10 files, 8 scripts
Software Heritage: not archived
Found in: “Data and Code Availability”
Holds: README
Not found: license file, CITATION.cff, environment file, tests, continuous integration, documentation
Tools: NumPy (8 files), Matplotlib (6 files), NiBabel (4 files), Plotly (1 file)
Availability: 1 check, the latest on 27 September 2026: the link answers
  • 27 September 2026: the link answers
9 files

The paper's code and data availability statement is in the Data section.

Tracing map

Proposed by the machine: these links were found in the paper and verified at the source, without human review. The map will receive a Zenodo DOI once one of the paper's authors has validated it with their ORCID.

What the map holds:

  • 1 repository of the authors' code, each at its verified commit, with its license and how the link was found in the paper;
  • 8 scripts, each with its path and the digest of its content;
  • 1 match between paragraphs of the paper and lines of the code (method lexical-v1);
  • neither the text of the paper nor the code itself.

Its JSON (tracing-map.json) is deposited on Zenodo with its DOI once the map is validated.

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://github.com/mjallais/NEXI_SANDIX_uGUIDE. No new data was created during this study. The data analyzed are from a previously published study and are available from the original data custodians upon reasonable request, subject to a data sharing agreement and ethics approval, as described in https://doi.org/10.1162/imag_a_00104.

Reproduced under the paper's license (CC BY), from the paper cited above.

Versions

The history of this record: each version stored by the harvester or made by a correction of its authors or of the maintainers of its code, and what changed in its facts. The texts of the paper (its abstract, its availability statements) are not part of it; versions that changed only those are not listed.

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://doi.org/10.1162/imag.a.1317

BibTeX

@article{jallais2026bayesian,
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/imag.a.1317},
url = {https://doi.org/10.1162/imag.a.1317},
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 - 2026/07/27
VL - 4
SP - IMAG.a.1317
SN - 2837-6056
PB - MIT Press
DO - 10.1162/imag.a.1317
UR - https://doi.org/10.1162/imag.a.1317
LA - en
ER -

CSL-JSON

{
"id": "10.1162/imag.a.1317",
"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"
},
{
"family": "Uhl",
"given": "Quentin"
},
{
"family": "Pavan",
"given": "Tommaso"
},
{
"family": "Molendowska",
"given": "Malwina"
},
{
"family": "Jones",
"given": "Derek K."
},
{
"family": "Jelescu",
"given": "Ileana"
},
{
"family": "Palombo",
"given": "Marco"
}
],
"container-title-short": "Imaging Neurosci (Camb)",
"volume": "4",
"page": "IMAG.a.1317",
"DOI": "10.1162/imag.a.1317",
"PMID": "42524206",
"PMCID": "PMC13409282",
"ISSN": "2837-6056",
"publisher": "MIT Press",
"URL": "https://doi.org/10.1162/imag.a.1317",
"language": "en",
"issued": {
"date-parts": [
[
2026,
7,
27
]
]
}
}

The tracing map gets a citation of its own once an author has validated it and it has a DOI.

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Myo-inositol concentration in the medial prefrontal cortex is associated with changes in brain white matter microstructure in early psychosis.
Journal: Translational psychiatry
In common: NiBabel, NumPy, structural MRI / diffusion, 2 references, 2 authors
[8] doi:10.1002/hbm.70553 [code]
Axon Diameter Mapping in the Living Human Brain with Ultra-High-Gradient Diffusion MRI at 500 mT/m Gradient Strength.
Journal: Human brain mapping
In common: NiBabel, NumPy, structural MRI / diffusion, 8 references
[9] doi:10.1371/journal.pone.0346132 [code]
Analysis of cortical dysplasias using b-tensor encoding diffusion MRI in an animal model.
Journal: PloS one
In common: NiBabel, Matplotlib, NumPy, structural MRI / diffusion, 8 references
[10] doi:10.1038/s41598-026-39162-7 [code]
White matter microstructure differences in obstructive sleep apnea severity groups assessed by diffusion tensor metrics and biophysical modeling.
Journal: Scientific reports
In common: NiBabel, NumPy, structural MRI / diffusion, 7 references

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