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Standard Model of Diffusion in White Matter: The SMI Toolbox

Software for robust standard-model parameter estimation from diffusion MRI data.

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We are sharing a robust implementation of parameter estimation for the standard model of diffusion in white matter along with example datasets. The toolbox can be used to analyze diffusion MRI images and extract information about various structures present in a voxel of white matter, such as axons, extra-axonal space, and cerebrospinal fluid.

The standard model is a unifying framework of white matter diffusion models, most of which are described by the same underlying physics (NMR Biomed. 2019;32[4]:e3998. doi: 10.1002/nbm.3998). In such a framework, most diffusion models can be understood as special cases of the standard model (Neuroimage. 2018;174:518-538. doi: 10.1016/j.neuroimage.2018.03.006).

White matter voxels contain signal contributions from many fiber bundles. Each bundle comprises distinct structures, which are different water compartments, such as axons, extra-axonal space, and cerebrospinal fluid. In the standard-model framework, axons are modeled as impermeable zero-radius cylinders (so-called sticks) arranged in locally coherent fascicles. The extra-axonal space of each fascicle is modeled with an axially symmetric diffusion tensor. Cerebrospinal fluid is an optional compartment modeled with an isotropic tensor. The bundle’s signal is the sum of the contributions from each compartment. Such multi-compartment bundles are distributed in a voxel according to an arbitrary fiber orientation distribution function. 

The SMI Toolbox supports:

  • Linear tensor encoding data only (i.e. single diffusion encoding) and equal echo time (TE) in all diffusion-weighted images.
  • Multiple tensor encodings and equal TE in all diffusion-weighted images.
  • Multiple tensor encodings and differing TE between all diffusion-weighted images (when multiple TE data are used, the SMI Toolbox returns axonal and extra-axonal T2 estimates).

For best results, we suggest preprocessing the diffusion-weighted data with the DESIGNER pipeline, which outputs a noise map that can be used as an input for the SMI Toolbox.

Examples of microstructure maps obtained with the standard model on a thirty-year-old normal volunteer. From left to right: axonal water fraction, intra-axonal diffusivity, parallel and perpendicular extra-axonal diffusivities, free water fraction, axonal T2 relaxation, extra-axonal T2 relaxation, and fiber dispersion.

Coelho S, Baete SH, Lemberskiy G, Ades-Aaron B, Barrol G, Veraart J, Novikov DS, Fieremans E.
Reproducibility of the Standard Model of diffusion in white matter on clinical MRI systems.
Neuroimage. 2022 May 8:119290. doi: 10.1016/j.neuroimage.2022.119290

Novikov DS, Veraart J, Jelescu IO, Fieremans E.
Rotationally-invariant mapping of scalar and orientational metrics of neuronal microstructure with diffusion MRI.
Neuroimage. 2018;174:518-538. doi: 10.1016/j.neuroimage.2018.03.006

Please cite these works if you are using the SMI Toolbox in your research.

References and Further Reading

Open this card for a handpicked list of publications grouped by standard-model features: the stick compartment model, constrained and unconstrained model versions, model degeneracies, parameter estimation with machine learning, sensitivity and specificity, model validation, reproducibility, and experimental design.

Stick Compartment Model of Axons

Kroenke CD, Ackerman JJ, Yablonskiy DA.
On the nature of the NAA diffusion attenuated MR signal in the central nervous system.
Magn Reson Med. 2004 Nov;52(5):1052-9. doi: 10.1002/mrm.20260

Jespersen SN, Kroenke CD, Østergaard L, Ackerman JJ, Yablonskiy DA.
Modeling dendrite density from magnetic resonance diffusion measurements.
Neuroimage. 2007 Feb 15;34(4):1473-86. doi: 10.1016/j.neuroimage.2006.10.037

Constrained Versions of the Standard Model

Fieremans E, Jensen JH, Helpern JA.
White matter characterization with diffusional kurtosis imaging.
Neuroimage. 2011 Sep 1;58(1):177-88. doi: 10.1016/j.neuroimage.2011.06.006

Zhang H, Schneider T, Wheeler-Kingshott CA, Alexander DC.
NODDI: practical in vivo neurite orientation dispersion and density imaging of the human brain.
Neuroimage. 2012 Jul 16;61(4):1000-16. doi: 10.1016/j.neuroimage.2012.03.072

Kaden E, Kelm ND, Carson RP, Does MD, Alexander DC.
Multi-compartment microscopic diffusion imaging.
Neuroimage. 2016 Oct 1;139:346-359. doi: 10.1016/j.neuroimage.2016.06.002

Unconstrained Standard Model Formulation

Novikov DS, Veraart J, Jelescu IO, Fieremans E.
Rotationally-invariant mapping of scalar and orientational metrics of neuronal microstructure with diffusion MRI.
Neuroimage. 2018 Jul 1;174:518-538. doi: 10.1016/j.neuroimage.2018.03.006

Novikov DS, Fieremans E, Jespersen SN, Kiselev VG.
Quantifying brain microstructure with diffusion MRI: Theory and parameter estimation.
NMR Biomed. 2019 Apr;32(4):e3998. doi: 10.1002/nbm.3998

Standard Model Degeneracies

Jelescu IO, Veraart J, Fieremans E, Novikov DS.
Degeneracy in model parameter estimation for multi-compartmental diffusion in neuronal tissue.
NMR Biomed. 2016 Jan;29(1):33-47. doi: 10.1002/nbm.3450

Coelho S, Pozo JM, Jespersen SN, Jones DK, Frangi AF.
Resolving degeneracy in diffusion MRI biophysical model parameter estimation using double diffusion encoding.
Magn Reson Med. 2019 Jul;82(1):395-410. doi: 10.1002/mrm.27714

Reisert M, Kiselev VG, Dhital B.
A unique analytical solution of the white matter standard model using linear and planar encodings.
Magn Reson Med. 019 Jun;81(6):3819-3825. doi: 10.1002/mrm.27685

Parameter Estimation with Machine Learning

Reisert M, Kellner E, Dhital B, Hennig J, Kiselev VG.
Disentangling micro from mesostructure by diffusion MRI: A Bayesian approach.
Neuroimage. 2017 Feb 15;147:964-975. doi: 10.1016/j.neuroimage.2016.09.058

Coelho S, Baete SH, Lemberskiy G, Ades-Aron B, Barrol G, Veraart J, Novikov DS, Fieremans E.
Reproducibility of the Standard Model of diffusion in white matter on clinical MRI systems.
Neuroimage. 2022 Aug 15;257:119290. doi: 10.1016/j.neuroimage.2022.119290

Coelho S, Fieremans E, Novikov DS.
How do we know we measure tissue parameters, not the prior?
Proc Intl Soc Magn Reson Med. 29 (2021). p 0397

Sensitivity Specificity Matrix for Assessing Parameter Estimation

Liao Y, Coelho S, Chen J, Ades-Aron B, Pang M, Osorio R, Shepherd T, Lui YW, Novikov DS, Fieremans E.
Mapping tissue microstructure of brain white matter in vivo in health and disease using diffusion MRI.
Imaging Neurosci (Camb). 2024; 2 1-17. doi: 10.1162/imag_a_00102

Model Validation

Coronado-Leija R, Abdollahzadeh A, Lee HH, Coelho S, Ades-Aron B, Liao Y, Salo RA, Tohka J, Sierra A, Novikov DS, Fieremans E.
Volume electron microscopy in injured rat brain validates white matter microstructure metrics from diffusion MRI.
Imaging Neurosci (Camb). 2024; 2 1-20. doi: 10.1162/imag_a_00212

Reproducibility

Coelho S, Baete SH, Lemberskiy G, Ades-Aron B, Barrol G, Veraart J, Novikov DS, Fieremans E.
Reproducibility of the Standard Model of diffusion in white matter on clinical MRI systems.
Neuroimage. 2022 Aug 15;257:119290. doi: 10.1016/j.neuroimage.2022.119290

Novikov DS, Veraart J, Jelescu IO, Fieremans E.
Rotationally-invariant mapping of scalar and orientational metrics of neuronal microstructure with diffusion MRI.
Neuroimage. 2018 Jul 1;174:518-538. doi: 10.1016/j.neuroimage.2018.03.006

Optimal Experimental Design

Coelho S, Pozo JM, Jespersen SN, Frangi AF.
Optimal experimental design for biophysical modelling in multidimensional diffusion MRI.
arXiv. Preprint posted online July 13, 2019. doi: 10.48550/arXiv.1907.06139

Lampinen B, Szczepankiewicz F, Mårtensson J, van Westen D, Hansson O, Westin CF, Nilsson M.
Towards unconstrained compartment modeling in white matter using diffusion-relaxation MRI with tensor-valued diffusion encoding.
Magn Reson Med. 2020 Sep;84(3):1605-1623. doi: 10.1002/mrm.28216

Coelho S, Baete SH, Lemberskiy G, Ades-Aaron B, Barrol G, Veraart J, Novikov DS, Fieremans E.
Reproducibility of the Standard Model of diffusion in white matter on clinical MRI systems.
Neuroimage. 2022 May 8:119290. doi: 10.1016/j.neuroimage.2022.119290

Novikov DS, Veraart J, Jelescu IO, Fieremans E.
Rotationally-invariant mapping of scalar and orientational metrics of neuronal microstructure with diffusion MRI.
Neuroimage. 2018;174:518-538. doi: 10.1016/j.neuroimage.2018.03.006

Patents

Novikov D, Jelescu I, Veraart E, Fieremans E, Kiselev V, Reisert M, inventors; Albert Ludwigs Universitaet Freiburg, New York University, assignees.
System, method and computer-accessible medium for determining brain microstructure parameters from diffusion magnetic resonance imaging signal’s rotational invariants.
US Patent 10,360,472. July 23, 2019.

Coelho S, Fieremans E, Novikov DS, inventors; New York University, assignee.
System, method and computer-accessible medium for determining rotational invariants of cumulant expansion from one or more acquisitions which can be minimal.
International patent application PCT/US2023/019,622. April 24, 2023.

Get the Code

This resource is maintained on Github as Standard Model Imaging (SMI) Toolbox.

Get the Code

Example Data

Below, we provide three example datasets for easy testing of the SMI toolbox. All data were preprocessed with DESIGNER.

  • dataset_1 contains a 2-shell diffusion MRI protocol similar to the one in the UK biobank.
  • dataset_2 contains multiple b-value b-tensor shape combinations (all with the same echo time), similar to the one we proposed in the related publication cited above.
  • dataset_3 contains multiple combinations of b-values, b-tensor shapes, and echo times.

Note that all three datasets are provided in nifti format. We used Jimmy Shen’s Tools for NIfTI and ANALYZE image toolbox to handle the data in MATLAB but users may turn to other tools of their choice and modify the lines in the example code.

Get the Data

The data available on this page are provided free of charge and comes without any warranty. CAI²R and NYU Grossman School of Medicine do not take any liability for problems or damage of any kind resulting from the use of the files provided. Operation of the software is solely at the user’s own risk. The software developments provided are not medical products and must not be used for making diagnostic decisions.

The software is provided for non-commercial, academic use only. Usage or distribution of the software for commercial purpose is prohibited. All rights belong to the author (Santiago Coelho) and NYU Grossman School of Medicine. If you use the software for academic work, please give credit to the author in publications and cite the related publications.

Please spell out your affiliation (e.g. “New York University” rather than “NYU”).

Contact

Questions about this resource may be directed to Santiago Coelho, PhD, or raised as issues on Github.

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