Non‐parametric graphnet‐regularized representation of dMRI in space and time

Highlightsq&tgr;‐dMRI signal representation without making biophysical assumptions.Non‐parametric, time‐dependent reconstruction of the diffusion signal and EAP.Effective GraphNet regularization to reduce q&tgr;‐DWI sampling requirements.Analytic estimation of time‐dependent q‐space index functions (q&tgr;‐indices).Good test‐retest reproducibility of q&tgr;‐index trends in two wild‐type mice. Graphical abstract Figure. No Caption available. Abstract Effective representation of the four‐dimensional diffusion MRI signal – varying over three‐dimensional q‐space and diffusion time &tgr; – is a sought‐after and still unsolved challenge in diffusion MRI (dMRI). We propose a functional basis approach that is specifically designed to represent the dMRI signal in this q&tgr;‐space. Following recent terminology, we refer to our q&tgr;‐functional basis as “q&tgr;‐dMRI”. q&tgr;‐dMRI can be seen as a time‐dependent realization of q‐space imaging by Paul Callaghan and colleagues. We use GraphNet regularization – imposing both signal smoothness and sparsity – to drastically reduce the number of diffusion‐weighted images (DWIs) that is needed to represent the dMRI signal in the q&tgr;‐space. As the main contribution, q&tgr;‐dMRI provides the framework to – without making biophysical assumptions – represent the q&tgr;‐space signal and estimate time‐dependent q‐space indices (q&tgr;‐indices), providing a new means for studying diffusion in nervous tissue. We validate our method on both in‐silico generated data using Monte–Carlo simulations and an in‐vivo test‐retest study of two C57Bl6 wild‐type mice, where we found good reproducibility of estimated q&tgr;‐index values and trends. In the hopes of opening up new &tgr;‐dependent venues of studying nervous tissues, q&tgr;‐dMRI is the first of its kind in being specifically designed to provide open interpretation of the q&tgr;‐diffusion signal.

[1]  Rachid Deriche,et al.  AxTract: Microstructure-Driven Tractography Based on the Ensemble Average Propagator , 2015, IPMI.

[2]  P. Hagmann,et al.  Mapping complex tissue architecture with diffusion spectrum magnetic resonance imaging , 2005, Magnetic resonance in medicine.

[3]  Jörg Kärger,et al.  The propagator representation of molecular transport in microporous crystallites , 1983 .

[4]  Rachid Deriche,et al.  Diffusion MRI signal reconstruction with continuity constraint and optimal regularization , 2012, Medical Image Anal..

[5]  Rachid Deriche,et al.  A Unifying Framework for Spatial and Temporal Diffusion in Diffusion MRI , 2015, IPMI.

[6]  Y. Cohen,et al.  High b‐value q‐space analyzed diffusion‐weighted MRS and MRI in neuronal tissues – a technical review , 2002, NMR in biomedicine.

[7]  Alessandro Daducci,et al.  Microstructure Informed Tractography: Pitfalls and Open Challenges , 2016, Front. Neurosci..

[8]  Carl-Fredrik Westin,et al.  Estimation of fiber Orientation Probability Density Functions in High Angular Resolution Diffusion Imaging , 2009, NeuroImage.

[9]  Jian Cheng,et al.  Estimation and Processing of Ensemble Average Propagator and Its Features in Diffusion MRI , 2012 .

[10]  Andrew L. Alexander,et al.  Computation of Diffusion Function Measures in $q$ -Space Using Magnetic Resonance Hybrid Diffusion Imaging , 2008, IEEE Transactions on Medical Imaging.

[11]  P. Basser,et al.  Axcaliber: A method for measuring axon diameter distribution from diffusion MRI , 2008, Magnetic resonance in medicine.

[12]  Yogesh Rathi,et al.  On Approximation of Orientation Distributions by Means of Spherical Ridgelets , 2008, IEEE Transactions on Image Processing.

[13]  Charles S Springer,et al.  Equilibrium water exchange between the intra‐ and extracellular spaces of mammalian brain , 2003, Magnetic resonance in medicine.

[14]  Cheng Guan Koay,et al.  A signal transformational framework for breaking the noise floor and its applications in MRI. , 2009, Journal of magnetic resonance.

[15]  Rachid Deriche,et al.  Multi-Spherical Diffusion MRI: Exploring Diffusion Time Using Signal Sparsity , 2016, MICCAI 2016.

[16]  D. Le Bihan.  Molecular diffusion, tissue microdynamics and microstructure. , 1995, NMR in biomedicine.

[17]  Rachid Deriche,et al.  A sensitivity analysis of q-space indices with respect to changes in axonal diameter, dispersion and tissue composition , 2016, 2016 IEEE 13th International Symposium on Biomedical Imaging (ISBI).

[18]  P. Rakić,et al.  Cytological and quantitative characteristics of four cerebral commissures in the rhesus monkey , 1990, The Journal of comparative neurology.

[19]  Cheng Guan Koay,et al.  Nuclear magnetic resonance characterization of general compartment size distributions , 2011 .

[20]  P. Grenier,et al.  MR imaging of intravoxel incoherent motions: application to diffusion and perfusion in neurologic disorders. , 1986, Radiology.

[21]  Carlo Pierpaoli,et al.  Mean apparent propagator (MAP) MRI: A novel diffusion imaging method for mapping tissue microstructure , 2013, NeuroImage.

[22]  D. Louis Collins,et al.  Diffusion Weighted Image Denoising Using Overcomplete Local PCA , 2013, PloS one.

[23]  R. Deriche,et al.  Regularized, fast, and robust analytical Q‐ball imaging , 2007, Magnetic resonance in medicine.

[24]  A. Dale,et al.  Quantitative Histological Validation of Diffusion MRI Fiber Orientation Distributions in the Rat Brain , 2010, PloS one.

[25]  Peter Craven,et al.  Smoothing noisy data with spline functions , 1978 .

[26]  Stephen P. Boyd,et al.  CVXPY: A Python-Embedded Modeling Language for Convex Optimization , 2016, J. Mach. Learn. Res..

[27]  Rachid Deriche,et al.  MAPL: Tissue microstructure estimation using Laplacian-regularized MAP-MRI and its application to HCP data , 2016, NeuroImage.

[28]  C. Beaulieu,et al.  The basis of anisotropic water diffusion in the nervous system – a technical review , 2002, NMR in biomedicine.

[29]  Maxime Descoteaux,et al.  Non Local Spatial and Angular Matching : Enabling higher spatial resolution diffusion MRI datasets through adaptive denoising , 2016, Medical Image Anal..

[30]  J. E. Tanner,et al.  Restricted Self‐Diffusion of Protons in Colloidal Systems by the Pulsed‐Gradient, Spin‐Echo Method , 1968 .

[31]  Derek K. Jones,et al.  Including diffusion time dependence in the extra-axonal space improves in vivo estimates of axonal diameter and density in human white matter , 2016, NeuroImage.

[32]  D. Tuch.  Q‐ball imaging , 2004, Magnetic resonance in medicine.

[33]  Baba C. Vemuri,et al.  Leveraging EAP-Sparsity for Compressed Sensing of MS-HARDI in (k, q)-Space , 2015, IPMI.

[34]  H. Zou,et al.  Regularization and variable selection via the elastic net , 2005 .

[35]  Stamatios N. Sotiropoulos,et al.  An integrated approach to correction for off-resonance effects and subject movement in diffusion MR imaging , 2016, NeuroImage.

[36]  Cheng Guan Koay,et al.  Temporal scaling characteristics of diffusion as a new MRI contrast: Findings in rat hippocampus , 2012, NeuroImage.

[37]  F. Ståhlberg,et al.  Diffusion‐weighted MRI measurements on stroke patients reveal water‐exchange mechanisms in sub‐acute ischaemic lesions , 2009, NMR in biomedicine.

[38]  J. Veraart,et al.  Universal power-law scaling of water diffusion in human brain defines what we see with MRI , 2016, 1609.09145.

[39]  E. Stejskal.  Use of Spin Echoes in a Pulsed Magnetic‐Field Gradient to Study Anisotropic, Restricted Diffusion and Flow , 1965 .

[40]  Philippe Hantraye,et al.  New paradigm to assess brain cell morphology by diffusion-weighted MR spectroscopy in vivo , 2016, Proceedings of the National Academy of Sciences.

[41]  D Le Bihan,et al.  Is water diffusion restricted in human brain white matter? An echo-planar NMR imaging study. , 1993, Neuroreport.

[42]  Alan Connelly,et al.  Robust determination of the fibre orientation distribution in diffusion MRI: Non-negativity constrained super-resolved spherical deconvolution , 2007, NeuroImage.

[43]  P. Basser,et al.  New modeling and experimental framework to characterize hindered and restricted water diffusion in brain white matter , 2004, Magnetic resonance in medicine.

[44]  Moo K. Chung,et al.  A 4D hyperspherical interpretation of q-space , 2015, Medical Image Anal..

[45]  P van Gelderen,et al.  Restricted and anisotropic displacement of water in healthy cat brain and in stroke studied by NMR diffusion imaging , 1991, Magnetic resonance in medicine.

[46]  William S. Price,et al.  Pulsed-field gradient nuclear magnetic resonance as a tool for studying translational diffusion, part 1: basic theory , 1997 .

[47]  Rachid Deriche,et al.  Sparse Reconstruction Challenge for diffusion MRI: Validation on a physical phantom to determine which acquisition scheme and analysis method to use? , 2015, Medical Image Anal..

[48]  A. Scheibel,et al.  Fiber composition of the human corpus callosum , 1992, Brain Research.

[49]  D. Bihan,et al.  Relationship between the diffusion time and the diffusion MRI signal observed at 17.2 tesla in the healthy rat brain cortex , 2014, Magnetic resonance in medicine.

[50]  Carl-Fredrik Westin,et al.  New insights about time‐varying diffusivity and its estimation from diffusion MRI , 2017, Magnetic resonance in medicine.

[51]  Dinggang Shen,et al.  Tensorial Spherical Polar Fourier Diffusion MRI with Optimal Dictionary Learning , 2015, MICCAI.

[52]  J. E. Tanner.  Transient diffusion in a system partitioned by permeable barriers. Application to NMR measurements with a pulsed field gradient , 1978 .

[53]  Cheng Guan Koay,et al.  Simple Harmonic Oscillator Based Reconstruction and Estimation for One-Dimensional q-Space Magnetic Resonance (1D-SHORE) , 2013 .

[54]  Paul T. Callaghan,et al.  Pulsed-Gradient Spin-Echo NMR for Planar, Cylindrical, and Spherical Pores under Conditions of Wall Relaxation , 1995 .

[55]  H. L. Dryden,et al.  Investigations on the Theory of the Brownian Movement , 1957 .

[56]  Jelle Veraart,et al.  In vivo observation and biophysical interpretation of time-dependent diffusion in human white matter , 2016, NeuroImage.

[57]  Els Fieremans,et al.  Revealing mesoscopic structural universality with diffusion , 2014, Proceedings of the National Academy of Sciences.

[58]  Bohdan Paszkowski,et al.  A+A+L , 1964 .

[59]  Dmitry S. Novikov,et al.  Mesoscopic structure of neuronal tracts from time-dependent diffusion , 2015, NeuroImage.

[60]  P. Ellen Grant,et al.  Multi-shell diffusion signal recovery from sparse measurements , 2014, Medical Image Anal..

[61]  Tim B. Dyrby,et al.  Orientationally invariant indices of axon diameter and density from diffusion MRI , 2010, NeuroImage.

[62]  Rachid Deriche,et al.  Noise Floor Removal via Phase Correction of Complex Diffusion-Weighted Images: Influence on DTI and q-Space Metrics , 2016, MICCAI 2016.

[63]  Rachid Deriche,et al.  Diffusion MRI microstructure models with in vivo human brain Connectom data: results from a multi-group comparison , 2016, 1604.07287.

[64]  Simon Ameer-Beg,et al.  Biomedical Imaging: From Nano to Macro , 2008 .

[65]  R. Deriche,et al.  Design of multishell sampling schemes with uniform coverage in diffusion MRI , 2013, Magnetic resonance in medicine.

[66]  Emmanuel J. Candès,et al.  Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information , 2004, IEEE Transactions on Information Theory.

[67]  P. Callaghan.  Principles of Nuclear Magnetic Resonance Microscopy , 1991 .

[68]  Partha P. Mitra,et al.  Pulsed-Field-Gradient NMR Measurements of Restricted Diffusion and the Return-to-the-Origin Probability , 1995 .

[69]  Daniel C. Alexander,et al.  Camino: Open-Source Diffusion-MRI Reconstruction and Processing , 2006 .

[70]  Y. Cohen,et al.  In vivo and in vitro bi‐exponential diffusion of N ‐acetyl aspartate (NAA) in rat brain: a potential structural probe? , 1998, NMR in biomedicine.

[71]  R. Deriche,et al.  Comparison of sampling strategies and sparsifying transforms to improve compressed sensing diffusion spectrum imaging , 2015, Magnetic resonance in medicine.

[72]  Carlo Pierpaoli,et al.  Clinical feasibility of using mean apparent propagator (MAP) MRI to characterize brain tissue microstructure , 2016, NeuroImage.

[73]  Jonathan E. Taylor,et al.  Interpretable whole-brain prediction analysis with GraphNet , 2013, NeuroImage.

[74]  G. Wahba.  Smoothing noisy data with spline functions , 1975 .

[75]  Mathews Jacob,et al.  Acceleration of high angular and spatial resolution diffusion imaging using compressed sensing with multichannel spiral data , 2015, Magnetic resonance in medicine.

[76]  D. Donoho,et al.  Sparse MRI: The application of compressed sensing for rapid MR imaging , 2007, Magnetic resonance in medicine.

[77]  Daniel C. Alexander,et al.  An Introduction to Computational Diffusion MRI: the Diffusion Tensor and Beyond , 2006, Visualization and Processing of Tensor Fields.

[78]  Rachid Deriche,et al.  Continuous diffusion signal, EAP and ODF estimation via Compressive Sensing in diffusion MRI , 2013, Medical Image Anal..

[79]  Cristina Granziera,et al.  What lies beneath? Diffusion EAP-based study of brain tissue microstructure , 2016, Medical Image Anal..

[80]  M. Budde,et al.  Microstructural organization of axons in the human corpus callosum quantified by diffusion-weighted magnetic resonance spectroscopy of N-acetylaspartate and post-mortem histology , 2014, Brain Structure and Function.

[81]  Luc Brun,et al.  Efficient and robust computation of PDF features from diffusion MR signal , 2009, Medical Image Anal..

[82]  P. Basser,et al.  MR diffusion tensor spectroscopy and imaging. , 1994, Biophysical journal.

[83]  Peter J Basser,et al.  The effect of the diffusion time and pulse gradient duration ratio on the diffraction pattern and the structural information estimated from q-space diffusion MR: experiments and simulations. , 2008, Journal of magnetic resonance.

[84]  Schwartz,et al.  Probability of return to the origin at short times: A probe of microstructure in porous media. , 1995, Physical review. B, Condensed matter.

[85]  J. Helpern,et al.  Monte Carlo study of a two‐compartment exchange model of diffusion , 2010, NMR in biomedicine.

[86]  P. Basser.  Relationships between diffusion tensor and q‐space MRI † , 2002, Magnetic resonance in medicine.

[87]  Dinggang Shen,et al.  Joint 6D k-q Space Compressed Sensing for Accelerated High Angular Resolution Diffusion MRI , 2015, IPMI.

[88]  Peter J Basser,et al.  Observation of anomalous diffusion in excised tissue by characterizing the diffusion-time dependence of the MR signal. , 2006, Journal of magnetic resonance.

[89]  V. Kiselev,et al.  Effective medium theory of a diffusion‐weighted signal , 2010, NMR in biomedicine.

[90]  Peter F. Neher,et al.  Tractography-based connectomes are dominated by false-positive connections , 2016, bioRxiv.

[91]  Rachid Deriche,et al.  Multiple q-shell diffusion propagator imaging , 2011, Medical Image Anal..

[92]  Rachid Deriche,et al.  Laplacian-regularized MAP-MRI: Improving axonal caliber estimation , 2015, 2015 IEEE 12th International Symposium on Biomedical Imaging (ISBI).

[93]  Martin Bendszus,et al.  Mono-Exponential Fitting in T2-Relaxometry: Relevance of Offset and First Echo , 2015, PloS one.

[94]  G. Sapiro,et al.  Reconstruction of the orientation distribution function in single‐ and multiple‐shell q‐ball imaging within constant solid angle , 2010, Magnetic resonance in medicine.

[95]  V. Kiselev,et al.  Quantifying brain microstructure with diffusion MRI: Theory and parameter estimation , 2016, NMR in biomedicine.

[96]  Maxime Descoteaux,et al.  Dipy, a library for the analysis of diffusion MRI data , 2014, Front. Neuroinform..