What Visualization Researchers Should Know About HARDI Models

Transcription

1 What Visualization Researchers Should Know About HARDI Models Thomas Schultz October 26, 2010

2 The Diffusion MRI (dmri) Signal ADC Modeling Diffusion Propagator Fiber Models Diffusion Tensor (DT-MRI) Higher-Order Models of Diffusivity Diffusion Spectrum Imaging (DSI) Diffusion Orientation Transform (DOT) Q-Ball Spherical Deconvolution Multi-Tensor Models

3 The Diffusion MRI (dmri) Signal ADC Modeling Diffusion Propagator Fiber Models Diffusion Tensor (DT-MRI) Higher-Order Models of Diffusivity Diffusion Spectrum Imaging (DSI) Diffusion Orientation Transform (DOT) Q-Ball Spherical Deconvolution Multi-Tensor Models

4 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 Free Isotropic Diffusion Hindered Anisotropic Diffusion

5 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

6 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

7 The Diffusion MRI (dmri) Signal ADC Modeling Diffusion Propagator Fiber Models Diffusion Tensor (DT-MRI) Higher-Order Models of Diffusivity Diffusion Spectrum Imaging (DSI) Diffusion Orientation Transform (DOT) Q-Ball Spherical Deconvolution Multi-Tensor Models

8 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:

9 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 Order 2 Order 4 Alternative basis of functions on the sphere Equivalent to Spherical Harmonics Order 6

10 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!

11 The Diffusion MRI (dmri) Signal ADC Modeling Diffusion Propagator Fiber Models Diffusion Tensor (DT-MRI) Higher-Order Models of Diffusivity Diffusion Spectrum Imaging (DSI) Diffusion Orientation Transform (DOT) Q-Ball Spherical Deconvolution Multi-Tensor Models

12 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) ψ DSI by integrating out r: Both P and ψdsi integrate to unity:

13 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)

14 Advantages of DSI: Pros and Cons 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

15 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 higherorder models of apparent diffusivity

16 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

17 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]

18 Q-Ball vs. DSI Tuch [2004] claims that Q-Ball ODF approximates Compared to DSI ODF the factor r 2 from Cartesian Spherical is neglected Barnett [2009] shows that ψqball approximates neither ψ Tuch nor ψ DSI Still provides meaningful information about anisotropy Broad peaks make Q-Balls less suitable for tractography

19 The Diffusion MRI (dmri) Signal ADC Modeling Diffusion Propagator Fiber Models Diffusion Tensor (DT-MRI) Higher-Order Models of Diffusivity Diffusion Spectrum Imaging (DSI) Diffusion Orientation Transform (DOT) Q-Ball Spherical Deconvolution Multi-Tensor Models

20 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]: Signal Kernel fodf When modeling the dmri signal in Spherical Harmonics, deconvolution amounts to division

21 Interference of fodf Maxima When adding approximated delta peaks, maxima get... shifted + = + = masked

22 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

23 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

24 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

25 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%)

26 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

27 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

28 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.

29 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]

30 URLs Camino: MRtrix: FSL: Teem: OpenWalnut: DTK / TrackVis: MedINRIA: 3D Slicer: ExploreDTI: LONI MiND:

31 References: Surveys Alexander: Multiple-Fiber Reconstruction Algorithms for Diffusion MRI. Ann. N. Y. Acad. Sci. 1064: , 2005 Alexander: An Introduction to Computational Diffusion MRI: the Diffusion Tensor and Beyond. In Weickert/Hagen (Eds): Visualization and Processing of Tensor Fields, pp , 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

32 References: DT-MRI Basser, Jones: Diffusion-Tensor MRI: Theory, Experimental Design and Data Analysis A Technical Review NMR in Biomedicine 15(7-8): , 2002 Fillard, Pennec, Arsigny, Ayache: Clinical DT-MRI Estimation, Smoothing, and Fiber Tracking With Log-Euclidean Metrics IEEE Trans. on Medical Imaging 26(11): , 2007 Pasternak, Sochen, Basser: The Effect of Metric Selection on the Analysis of Diffusion Tensor MRI Data NeuroImage 49: , 2010 Welk, Weickert, Becker, Schnörr, Feddern, Burgeth: Median and Related Local Filters for Tensor-Valued Images Signal Processing 87(2): , 2007

33 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: , 2002 Frank: Characterization of Anisotropy in High Angular Resolution Diffusion-Weighted MRI Magnetic Resonance in Medicine 47: , 2002 Liu, Bammer, Acar, Moseley: Characterizing Non-Gaussian Diffusion by Using Generalized Diffusion Tensors Magnetic Resonance in Medicine 51: , 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: , 2003

34 References: DSI and DOT Canales-Rodríguez, Lin, Iturria-Medina, Yeh, Cho, Melie-García: Diffusion Orientation Transform Revisited NeuroImage 49: , 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: , 2006 Wedeen, Hagmann, Tseng, Reese, Weisskoff: Mapping Complex Tissue Architecture with Diffusion Spectrum Magnetic Resonance Imaging Magnetic Resonance in Medicine 54: , 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): , 2010

35 References: Q-Ball Anderson: Measurement of Fiber Orientation Distributions Using High Angular Resolution Diffusion Imaging Magnetic Resonance in Medicine 54: , 2005 Barnett: Theory of Q-Ball Imaging Redux: Implications for Fiber Tracking Magnetic Resonance in Medicine 62: , 2009 Descoteaux, Angelino, Fitzgibbons, Deriche: Regularized, Fast, and Robust Analytical Q-Ball Imaging Magnetic Resonance in Medicine 58: , 2007 Hess, Mukherjee, Han, Xu, Vigneron: Q-Ball Reconstruction of Multimodal Fiber Orientations Using the Spherical Harmonics Basis Magnetic Resonance in Medicine 56: , 2006 Tuch: Q-Ball Imaging Magnetic Resonance in Medicine 52: , 2004

36 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: , 2003 Schultz, Seidel: Estimating Crossing Fibers: A Tensor Decomposition Approach IEEE Trans. Vis. Comp. Graphics 14(6): , 2008 Tournier, Calamante, Gadian, Connelly: Direct Estimation of the Fiber Orientation Density Function from Diffusion-Weighted MRI Data Using Spherical Deconvolution NeuroImage 23: , 2004 Tournier, Calamante, Connelly: Robust Determination of the Fibre Orientation Distribution in Diffusion MRI: Non-negativity Constrained Super-Resolved Spherical Deconvolution NeuroImage 35: , 2007

37 References: Multi-Compartment Models Alexander, Hasan, Lazar, Tsuruda, Parker: Analysis of Partial Volume Effects in Diffusion-Tensor MRI Magnetic Resonance in Medicine 45: , 2001 Behrens et al (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 , 2010 Data Format for dmri data: Patel, Dinov, Van Horn, Thompson, Toga: LONI MiND: metadata in NifTI for DWI. NeuroImage 51(2): , 2010

38 Questions? Find the slides at

39 Additional Slides

40 Preserving Positive Definiteness Negative ADCs are physically impossible, so noisefree 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]

41 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]

42 Antipodal Symmetry: P(x)=P(-x) Theoretically justified only for free diffusion In theory, lifting this constraint could detect nonsymmetric 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

43 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