what visualization researchers should know about hardi...

What Visualization ResearchersShould Know About HARDI Models

Thomas Schultz <[email protected]>

October 26, 2010

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ADC Modeling

Diffusion Tensor(DT-MRI)

Higher-Order Modelsof Diffusivity

Diffusion Propagator

Diffusion SpectrumImaging (DSI)

Diffusion OrientationTransform (DOT)

Q-Ball

Fiber ModelsSpherical

Deconvolution

Multi-Tensor Models

Diffusion as a Probe for Tissue Structure

● Water produces an MR signal● Molecules perform a random heat motion (“diffusion”)● Spin displacement along a diffusion sensitizing

gradient leads to MR signal attenuation (“dMRI”)● In fibrous tissue (e.g., muscles or nerves),

displacements are anisotropic

Thomas Schultz <[email protected]> October 26, 2010

Free Isotropic Diffusion Hindered Anisotropic Diffusion

Main Parameters of a dMRI Measurement

● Spatial resolution● Example: 1mm x 1mm x 2mm

● Voxel anisotropy● Angular resolution

● Number and distribution of gradient directions● Example: 60 directions, uniformly distributed on hemisphere

● Strength of diffusion weighting● q value: Gradient strength x gradient duration● b value: q value x effective diffusion time


Partial Voluming

● Typical voxel edge length: ≈1mm● Finer resolution impossible at state of the art

● Typical axon diameter: ≈1μm● Inevitably leads to averaging over complex fiber

configurations, including:


Crossing Fibers Passing Fibers Diverging Fibers

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ADC Modeling






Q-Ball


Deconvolution

Multi-Tensor Models

The Diffusion Tensor Model (DT-MRI)

● The Apparent Diffusion Coefficient (ADC) quantifies molecular mobility.

● The second-order diffusion tensor D is the most common model of anisotropic apparent diffusivity● Quadratic form yields apparent diffusion coefficients● Real, symmetric 3x3 matrix● Related to dMRI attenuation A via Stejskal-Tanner Equation


● When a single direction is dominant, the principle eigenvector indicates it:

Higher-Order ADC Models

● Frank [2002] and Alexander et al. [2002] use spherical harmonics (SH) to model multimodal ADC profiles● Orthonormal basis of functions on the sphere● Alexander et al. [2002] use statistical F-Test to

decide on SH order (complexity of the model)

● Özarslan et al. [2003]: Higher-Order Diffusion Tensors● Generalization of the Diffusion Tensor● Alternative basis of functions on the sphere● Equivalent to Spherical Harmonics


Order 2

Order 4

Order 6

ADC Maxima and Fiber Directions

Caveat: When a voxel contains multiple compartments, their apparent diffusivities do not add linearly.


Important consequence:

When there is more than one fiber compartment,ADC maxima do not approximate fiber directions!

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ADC Modeling






Q-Ball


Deconvolution

Multi-Tensor Models

The Diffusion Propagator

● Spin displacements x are described by a probability distribution function P(x), the diffusion propagator

● For tractography, P(x) is reduced to an orientation distribution function (ODF) ψDSI by integrating out r:

● Both P and ψDSI integrate to unity:


Diffusion Spectrum Imaging (DSI)

● The diffusion propagator P(x) is related to the measured attenuation A(q) via a 3D Fourier integral:

● Wavevector q is the Fourier dual of the displacement vector x; it lives in “q space”


● Diffusion Spectrum Imaging [Wedeen et al. 2005] samples A(q) on a Cartesian grid and performs a FFT to obtain P(x)

Pros and Cons of DSI

Advantages of DSI:● Acquires detailed information about the diffusion process● Makes very few prior assumptions● Conceptually simple and well-founded

Disadvantages of DSI:● High measurement effort (many directions, strong gradients)● Requires even larger voxels to make measurement feasible● Numerical ODF integration requires interpolation● Much information is “thrown away” by taking the ODF


DT-MRI vs. DSI

● DT-MRI assumes that the diffusion propagator P is a trivariate Gaussian:

● The propagator is completely determined by diffusion time t (fixed in measurement) and diffusion tensor D

● Similarly, a propagator can be derived from higher-order models of apparent diffusivity


Diffusion Orientation Transform (DOT)

Özarslan et al. [2006]:● Assume monoexponential attenuation


● Predict A(q) from D(x), Fourier Transform to get P(x)● Analytical solution when D(x) given in

Spherical Harmonics

Q-Ball

● Tuch [2004] proposes Q-Ball ODF● Measure A(θ,φ) on spherical shell in q space (single b value)● Compute an ODF via the Funk-Radon transform

● Treats each point as a pole, assigns integral over equator


● Efficient and regularized analytic implementations use spherical harmonics● Anderson [2005], Hess et al. [2006],

Descoteaux et al. [2007]

Q-Ball vs. DSI

● Tuch [2004] claims that Q-Ball ODF approximates

● Compared to DSI ODF

the factor r2 from Cartesian→Spherical is neglected

● Barnett [2009] shows that ψQball approximates neither ψTuch nor ψDSI

● Still provides meaningful information about anisotropy● Broad peaks make Q-Balls less suitable for tractography


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Q-Ball


Deconvolution

Multi-Tensor Models

Spherical Deconvolution

Observation by Behrens et al. [2003]:● Diffusion MR signal can be modeled as the

convolution of a fiber distribution function with a kernel that reflects the effect of● Fibers on the Diffusion● Diffusion on the Signal

Practical result by Tournier et al. [2004]:● When modeling the dMRI signal in Spherical

Harmonics, deconvolution amounts to division


Signal Kernel fODF

Interference of fODF Maxima


When adding approximated delta peaks, maxima get...

+

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…shifted

…masked

Inferring Fibers with Higher-Order Tensors


● Schultz et al. [2008]:● Higher-Order Tensor Formalism● Approximate ODF with rank-k tensor (k=fiber number)● Iterative nonlinear estimation of principal directions

Multi-Compartment Models


Alternative for HARDI interpretation:● Multi-Tensor-Models (Alexander et al. [2001])

Frequent Variant:● Ball-and-Stick (Behrens et al. [2003])● 1 perfectly isotropic “ball” compartment● n perfectly linear “stick” compartments● Same diffusivity in all compartments

Pros and Cons of Multi-Tensor Models

Advantages of Multi-Compartment Models:● Parameters indicate directions for fiber tracking● Peak interference is taken into account automatically

Disadvantages of Multi-Compartment Models:● Nonlinear fitting can be unreliable and inefficient● Need to select the appropriate number of fibers


Advanced Multi-Tensor-Fitting


[Schultz et al. 2010] Deconvolve and perform rank-k approximation to “kick-start” multi-tensor fitting

Advantages:● Deconvolution, tensor approximation and subsequent

fitting is twice as fast as fitting from random seed● Convergence to global optimum in >98% (two-fiber)

or >93% (three-fiber) of all cases (before: 90% / 83%)

Summary: Tensors in HARDI

● Many models involve functions on the sphere● DT-MRI, ADC, ODFs from DSI, DOT, Q-Ball, SD

● Spherical Harmonics and Symmetric (Higher-Order) Tensors are equivalent bases for such functions● Advantage of Spherical Harmonics:

– Simplifies deconvolution [Tournier et al. 2004]

● Advantages of Higher-Order Tensors:– Representation of single fibers as rank-1 terms [Schultz et al. 2008]

– Generalization of tensor ellipsoid as ODF glyph [Schultz et al. 2010]

● Conversion between SH and Tensors is as simple as a matrix-vector product● Always use the most convenient form for the task at hand


Open Source Software for HARDI

● Camino (Alexander, University College London)● Model fitting (Higher-Order ADC, Q-Ball, Deconvolution,

Multi-Tensor Models)● Tractography based on above models● Data synthesis (Model-based, Monte Carlo)

● MRtrix (Tournier, Brain Research Institute)● Linear and constrained deconvolution● Tractography based on ODFs● Basic visualization tools for ODF and tract visualization

● FSL (FMRIB Centre, University of Oxford)● Preprocessing and ball-and-multistick tractography


More Open Source Software for HARDI

● Teem (Kindlmann/Schultz, University of Chicago)● Two-tensor fitting and tractography● Higher-order tensor approximations● Efficient ODF glyphs (polar plot and HOME glyph)

● OpenWalnut (University of Leipzig and MPI CBS)● Fitting Spherical Harmonics● ODF glyphs (based on Teem)

Both projects currently extend their HARDI support.


Other Software for HARDI

● DTK / TrackVis (Wedeen, MA General Hospital)● DSI and Q-Ball reconstruction and tractography● Interactive fiber visualization, filtering, and statistics● Binary only, but free of charge for research use

● Others include MedINRIA, 3D slicer, ExploreDTI● Not sure about extent of HARDI support

● Problem: Each package uses its own data format● “LONI MiND” - tries to standardize HARDI metadata in NIfTI

[Patel 2010]


URLs

● Camino: http://www.cs.ucl.ac.uk/research/medic/camino/● MRtrix: http://www.nitrc.org/projects/mrtrix/● FSL: http://www.fmrib.ox.ac.uk/fsl/● Teem: http://teem.sourceforge.net/● OpenWalnut: http://www.openwalnut.org/● DTK / TrackVis: http://www.trackvis.org/● MedINRIA:

http://www-sop.inria.fr/asclepios/software/MedINRIA/● 3D Slicer: http://www.slicer.org/● ExploreDTI: http://www.exploredti.com/● LONI MiND: http://mind.loni.ucla.edu/


http://www.cs.ucl.ac.uk/research/medic/camino/

http://www.nitrc.org/projects/mrtrix/

http://www.fmrib.ox.ac.uk/fsl/

http://teem.sourceforge.net/

http://www.openwalnut.org/

http://www.trackvis.org/

http://www-sop.inria.fr/asclepios/software/MedINRIA/

http://www.slicer.org/

http://www.exploredti.com/

http://mind.loni.ucla.edu/

References: Surveys

● Alexander: “Multiple-Fiber Reconstruction Algorithms for Diffusion MRI”. Ann. N. Y. Acad. Sci. 1064:113-133, 2005

● Alexander: “An Introduction to Computational Diffusion MRI: the Diffusion Tensor and Beyond”. In Weickert/Hagen (Eds): Visualization and Processing of Tensor Fields, pp. 83-106, Springer, 2006

● Hagmann, Jonasson, Maeder, Thiran, Wedeen, Meuli: “Understanding Diffusion MR Imaging Techniques: From Scalar Diffusion-weighted Imaging to Diffusion Tensor Imaging and Beyond” RadioGraphics 26:S205-S223, 2006


References: DT-MRI

● Basser, Jones: Diffusion-Tensor MRI: “Theory, Experimental Design and Data Analysis – A Technical Review” NMR in Biomedicine 15(7-8):456-467, 2002

● Fillard, Pennec, Arsigny, Ayache: “Clinical DT-MRI Estimation, Smoothing, and Fiber Tracking With Log-Euclidean Metrics” IEEE Trans. on Medical Imaging 26(11):1472-1482, 2007

● Pasternak, Sochen, Basser: “The Effect of Metric Selection on the Analysis of Diffusion Tensor MRI Data” NeuroImage 49:2190-2204, 2010

● Welk, Weickert, Becker, Schnörr, Feddern, Burgeth: “Median and Related Local Filters for Tensor-Valued Images” Signal Processing 87(2):291-308, 2007


References: Higher-Order ADC Models

● Alexander, Barker, Arridge: “Detection and Modeling of Non-Guaussian Apparent Diffusion Coefficient Profiles in Human Brain Data” Magnetic Resonance in Medicine 48:331-340, 2002

● Frank: “Characterization of Anisotropy in High Angular Resolution Diffusion-Weighted MRI” Magnetic Resonance in Medicine 47:1083-1099, 2002

● Liu, Bammer, Acar, Moseley: “Characterizing Non-Gaussian Diffusion by Using Generalized Diffusion Tensors” Magnetic Resonance in Medicine 51:924-937, 2004

● Özarslan, Mareci: “Generalized Diffusion Tensor Imaging and Analytical relationships between Diffusion Tensor Imaging and High Angular Resolution Diffusion Imaging” Magnetic Resonance in Medicine 50:955-965, 2003


References: DSI and DOT

● Canales-Rodríguez, Lin, Iturria-Medina, Yeh, Cho, Melie-García: “Diffusion Orientation Transform Revisited” NeuroImage 49:1326-1339, 2010

● Mitra, Halperin: “Effects of Finite Gradient-Pulse Widths in Pulsed-Field-Gradient Diffusion Measurements”, J. Magnetic Resonance A 113:94-101, 1995

● Özarslan, Shepherd, Vemuri, Blackband, Mareci: “Resolution of Complex Tissue Microarchitecture Using the Diffusion Orientation Transform (DOT)” NeuroImage 31:1086-1103, 2006

● Wedeen, Hagmann, Tseng, Reese, Weisskoff: “Mapping Complex Tissue Architecture with Diffusion Spectrum Magnetic Resonance Imaging” Magnetic Resonance in Medicine 54:1377-1386, 2005

● Yeh, Tournier, Cho, Lin, Calamante, Connelly: “The effect of Finite Diffusion Gradient Pulse Duration on Fibre Orientation Estimation in Diffusion MRI” NeuroImage 51(2):743-751, 2010


References: Q-Ball

● Anderson: “Measurement of Fiber Orientation Distributions Using High Angular Resolution Diffusion Imaging” Magnetic Resonance in Medicine 54:1194-1206, 2005

● Barnett: “Theory of Q-Ball Imaging Redux: Implications for Fiber Tracking” Magnetic Resonance in Medicine 62:910-923, 2009

● Descoteaux, Angelino, Fitzgibbons, Deriche: “Regularized, Fast, and Robust Analytical Q-Ball Imaging” Magnetic Resonance in Medicine 58:497-510, 2007

● Hess, Mukherjee, Han, Xu, Vigneron: “Q-Ball Reconstruction of Multimodal Fiber Orientations Using the Spherical Harmonics Basis” Magnetic Resonance in Medicine 56:104-117, 2006

● Tuch: “Q-Ball Imaging” Magnetic Resonance in Medicine 52:1358-1372, 2004


References: Spherical Deconvolution

● Behrens, Woolrich, Jenkinson, Johansen-Berg, Nunes, Clare, Matthews, Brady, Smith: “Characterization and Propagation of Uncertainty in Diffusion-Weighted MR Imaging” Magnetic Resonance in Medicine 50:1077-1088, 2003

● Schultz, Seidel: “Estimating Crossing Fibers: A Tensor Decomposition Approach” IEEE Trans. Vis. Comp. Graphics 14(6):1635-1642, 2008

● Tournier, Calamante, Gadian, Connelly: “Direct Estimation of the Fiber Orientation Density Function from Diffusion-Weighted MRI Data Using Spherical Deconvolution” NeuroImage 23:1176-1185, 2004

● Tournier, Calamante, Connelly: “Robust Determination of the Fibre Orientation Distribution in Diffusion MRI: Non-negativity Constrained Super-Resolved Spherical Deconvolution” NeuroImage 35:1459-1472, 2007


References: Multi-Compartment Models

● Alexander, Hasan, Lazar, Tsuruda, Parker: “Analysis of Partial Volume Effects in Diffusion-Tensor MRI” Magnetic Resonance in Medicine 45:770-780, 2001

● Behrens et al. 2003 (see previous slide)

● Nedjati-Gilani, Parker, Alexander: “Mapping the Number of Fibre Orientations per Voxel in Diffusion MRI” Proc. ISMRM, p. 3169, 2006

● Schultz, Westin, Kindlmann: “Multi-Diffusion-Tensor Fitting via Spherical Deconvolution: A Unifying Framework” Proc. MICCAI, LNCS 6361, pp. 673-680, 2010

Data Format for dMRI data:

● Patel, Dinov, Van Horn, Thompson, Toga: “LONI MiND: metadata in NifTI for DWI.” NeuroImage 51(2):665-676, 2010


Questions?


Find the slides athttp://www.ci.uchicago.edu/~schultz/visweek10/

Additional Slides


Preserving Positive Definiteness

● Negative ADCs are physically impossible, so noise-free diffusion tensors should be positive semidefinite

● Should we enforce this constraint? If so, how?● Fillard et al. [2007] propose a “Log-Euclidean” framework for

tensor estimation, smoothing, and tractography● Pasternak et al. [2010] argue against generally enforcing

positive ADCs in diffusion tensor estimation

● Should we preserve this constraint when met in input?● Interpolation or post-processing that only take convex

combinations do this automatically [Welk et al. 2007]


Narrow Pulse Condition

● Gradient length δ much shorter than diffusion time Δ● Violated in clinical practice● Consequence:

● P(x) describes center of mass rather than individual spins [Mitra et al. 1995]

● Leads to a stronger apparent anisotropy [Yeh et al. 2010]


Antipodal Symmetry: P(x)=P(-x)

● Theoretically justified only for free diffusion● In theory, lifting this constraint could detect non-

symmetric tissue geometry [Liu et al. 2004]● E.g., Y-shaped junctions

● Deviation from symmetry encoded in dMRI phase● Corrupted by measurement artifacts● Discarded by taking the modulus of the complex signal


ODF-based Fiber Selection

Statistical Tests to detect overfitting(e.g., [Nedjati-Gilani et al. 2006])● Bayesian / Akaike Information Criterion● Cascade of F-Tests

Alternative: Infer Fiber Number from Deconvolution ODF● Based on weights from Rank-k Approximations● Roughly comparable to peak counting

Results:● More reliable on synthetic data (at high b-value)● Less likely to overfit bending or fanning bundles


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