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Diffusion tensor imaging (DTI): A basic introduction to data acquisition and analysis

Matthew Cykowski, MDPostdoctoral fellow

Research Imaging CenterUTHSCSA

Room 2.320cykowski@uthscsa.edu

PART I:Acquiring DTI images

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Diffusion as a probe into tissue architecture

• The diffusion of water molecules can be used as a probe intotissue architecture– Not just for brain tissue, but also in skeletal muscle or

cardiac tissue– In tissues where diffusion is restricted to due some ordered

structure (e.g., myelinated nerve fibers) diffusion is said tobe anisotropic

• In MRI, diffusion is estimated by comparing the signal ofimages with and without (Bo or So image) dephasing gradients

Restricted diffusion

• Due to (see figure):

– Closed spaces (A)• e.g., within cells or axons

– Tortuous extracellular pathways (B)

– Exchange between intra- andextracellular compartments (C)

• At the voxel level, what is measuredis “apparent diffusion” Le Bihan (2003) Nature Reviews Neuroscience

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• Diffusion-weighted images have signal intensities that reflect the random motion of watermolecules in tissues

– This sensitivity is achieved by applying a pair of dephasing & rephasing gradientpulses

Diffusion-sensitizing gradients

Mori and Zhang (2006) Neuron

Calculating an “apparent” diffusion coefficient

• In the most basic case, an “apparent diffusioncoefficient” per voxel can be calculated if:

– a set of DWIs has been acquired– One or more images without diffusion

weighting have been acquired– Features of the gradient pulse application

(summarized in the “b value”) are known

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Fun stuff

http://www.hbm2008.com/podcast2.php

For a nice lecture:

Diffusion-sensitizing gradients

• Diffusion gradients– Above is a plot of a DTI experiment at the RIC:

• The first three rows are acquisitions without DW• Rows 3-89 represent the application of DW along various non-collinear

directions (this should appear relatively random!)

X Y Z (in magnet coordinate system)

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X Y Z

So

Dxx Dyy Dzz

Dxy Dyz Dxz

Sample gradients

… and the images formed…

Row 1 of “sample gradients”

Row 2, etc…

Back to the ADCs

So

Dxx

Slope = D or ADC

*

* Here are the definitions for the above:

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ADC mapping: limitations

• If you took three ADC measurements like this…

– You could provide a fair estimate of the structure's anisotropy.

• However, if the three measurements were taken like this…

– You’d have no measurements of the ADC that are eitherperpendicular or parallel to the long axis of the structure

zz

xx

yy

yy

xx

zz

Gradient applications:

Gradient applications:

Tissue A scannerorientation

Tissue B scannerorientation

Is there a more accurate way?

• “Effective” diffusion tensor follows from “apparent” diffusion coefficient– ADC measurements alone can underestimate diffusion anisotropy

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In a word: DT-MRI v. ADC mapping

• The diffusion tensor, D, removes the “orientationbias” in measures of tissue anisotropy

• Also, traditional ADC measurements imply abelief in the cylindrical symmetry of tissues– According to this “belief”, only parallel and

perpendicular measurements are needed

Diffusion tensor MRI (DT-MRI)

Peter Basser, NIH

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DT-MRI

• The diffusion tensor, D, is simply a tool to account for diffusivity along the principal axes (x, y, z) of themagnet, as well as diffusivity along non-principal axes.

– It can be used to represent the size (e.g., mean diffusivity), the shape (e.g., fractionalanisotropy), and the directional orientation (e.g., eigenvectors) of a diffusion ellipsoid.

• Whereas ADC was calculated with univariate linear regression, D, is estimated by multivariateregression** with the following known values:

– Signal intensity in DW images– A “b-matrix” from:

• Vectors representing gradient application for each DWI• b values used during each DW image acquisition

• Solve for:– 6 non-redundant elements of D (in bold -->)– S0 (intercept)

**there is a whole literature just on the various methods to do this..ln(S) = ln(S0) - D*b-matrix

DT-MRI: the crux of the biscuit

• IMPORTANT: above are images of the tensor components AFTER these (previously) unknowns have been calculated.#1: These are NOT DW images! Therefore the intensity image corresponds to high diffusion coefficients - make sure this

representation makes sense to you…#2: Notice that the “off-diagonal” elements (e.g., Dxy or Dyz) have some seriously high intensities - this is expected and

means that Peter Basser’s original justification for DT-MRI (rather than ADC mapping) is pretty sound!

Dxx DzzDyy

Dxz/zxDxy/yx Dyz/zy

So (no DW)

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The tensor of diffusion coefficients..

Dxx DzzDyy

Dxz/zxDxy/yx Dyz/zy

The tensor ismanuallyreconstructedto test thecalculation ofscalarparameters..

An orientation-independent representation ofanisotropy..

• *This procedure generates a representation of D in terms of its eigenvectors & eigenvalues - thisrepresentation of the data now has an “intrinsic” coordinate system.

!e eigenvalues “furnish the effective diffusion distances in each of the three principal directions within the anisotropicmedium” - P Basser

D (diff tensor)

Diagonalization*

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Anisotropy

FSL’s FA image value = 0.9068

• With the tensor components(previous 2 slides), we canmanually produce (inmatlab) a rough estimate ofFA

A “rough estimate ” of diffusion ellipsoid

PART II:DTI Parameters Commonly Used in Image Analysis

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Quantities derived from D:What are they? Why use them?

• Utility:– #1 : These are “rotationally invariant” representations:

• by calculating D you have removed the importance of theirorientation in the patient (or monkey’s) head!

– #2 : They are easily calculated by “diagonalizing” the diffusiontensor (in the simplest case a 3x3 matrix) and obtaining theeigenvalues (AKA, the principal diffusivities or L1, L2, and L3)

• These are the major diffusion coefficients of the intrinsiccoordinate system of the voxel/ tissue

– #3 : Scalar parameters are more amenable to univariatestatistical tests (a Student’s t-test) than are matrices/ tensors tomultivariate statistical tests.

– #4 : The principal diffusion directions (eigenvectors of thetensor) can be utilized to reconstruct white matter fiber tracts.

Tissue A scannerorientation

Tissue B scannerorientation

DTI parameters or “scalar invariants” (FA, TrD, MD)

• Fractional anisotropy– Relative anisotropy

• The trace of the tensor– Mean diffusivity

• Their principal eigenvalues– “Axial diffusivity”

• Some combination of non-principal eigenvalues– “Radial diffusivity”

• Their principal eigenvectors

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Synonyms

• Principal diffusion direction = 1st eigenvector = v1

• Highest diffusivity = l1 = L1 = ADCII = “axialdiffusivity”

• Intermediate and lowest diffusivity = l2 & l3 = L2 andL3 = ADC = “transverse diffusivities”

– An image formed from the L2 & L3 images istermed a “radial diffusivity” image

L1

(L2+L3)*0.5V1 V1

Fractional anisotropy

• FA:– A ratio of the squared

difference betweeneigenvalues and their mean,to the sum of squaredeigenvalues

Fractional anisotropy

Relative anisotropy

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Anisotropy

• Some WM tissues exhibit veryhigh L1 measurementsrelevant to L2/L3– Corpus callosum &

pyramidal tracts

• Other WM regions exhibitlower anisotropy– Centrum semiovale

Trace of the tensor & mean diffusivity

• The trace is the sum of tensor eigenvalues and Tr(D) / 3represents average or mean diffusivity (across all orientations)

• In healthy subjects, no statistically significant differences existin the Trace measurement for gray/ white matter structures,except for the cerebral cortex

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Principal eigenvectors (v1-v3)

• Though not discussed today, this data can be utilized in reconstructing & visualizingmajor fiber tracts

– Many of the algorithms that do this utilize some knowledge of human fibercharacteristics, or will allow a probabilistic representation, or employ other restrictivefeatures (e.g., not allowing a fiber tract to be created across the lateral ventricles)

Axial & radial diffusivity: are they markers?

• "Although measures of anisotropy derived from DTI are commonly used as biomarkers of white matter pathology, an increase in radial diffusivity or a decrease inaxial diffusivity will both cause a decrease in anisotropy.

– Therefore, although anisotropy is an exquisitely sensitive marker of pathology, it is not specific to either axonal damage or demyelination. Theavailable evidence suggests that axial and radial diffusivities hold promise as specific biomarkers of white matter pathology.

– Specifically, demyelination is associated with an increase in radial diffusivity, presumably because the loss of myelin membrane integrity permitsincreased water movement perpendicular to the myelin layers.

– In contrast, axonal damage is associated with a decrease in axial diffusivity. Although the underlying mechanism is still unclear, the decreasedaxial diffusivity may result from the loss of coherent organization of the axon and many other structural and physiological mechanisms associated withaxon damage and degeneration.” Budde et al, (NMR Biomed., 2008)

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PART III:Aligning DTI parameter images for analysis

Issues

• The major concerns prior to carrying out voxelwiseanalysis are intersubject image registration &image smoothing– Whole brain, VBM-style analyses

• Whole brain, VBM-style analyses using information aboutthe location of WM

– ROI analyses (with or without alignment)– Tract-Based Spatial Statistics (TBSS)

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Tract-Based Spatial Statistics

Linear

Nonlinear

Nonlinear

Target image MNI-152

Mean FA image (of study) Skeletonized FA images

FA > 0.3 FA > 0.2

Tract-Based Spatial Statistics

Subject (4d file

)

Smith et al. (2006) Neuroimage

Smith et al. (2006) Neuroimage

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PART IV:Statistical inference on DTI parameters

General linear model (GLM)

• X = dependent (DV) or response variable or regressand– where i = scan, j = voxel

• g = explanatory (‘EV’) or independent or predictor variable or regressor– Can be a coding or dummy variable (young = 1 or not young = 0)– Can be a continuous covariate (age)– Could be a ‘nuisance variable’ (regressor of no interest)

• β = parameter estimate or “effect size coefficient”– This is the value that g must be multiplied by to fit the observed data (X)

• Some options for significance testing:– Question 1: Is B1 significantly non-zero: t = B1 / error of B1’s estimation– Question 2: Is B1 significantly > B2 : t = B1 - B2 / combined standard error

– If we had fMRI data, Question 1 would ask where does something activate on average (where doesthe model fit the data really well)

– If we had DTI data, Question 2 would ask, where does the model fit the observed FA values for group1 better than group 2

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Sample design matrix: 2 groups, unpaired t-test

(A) FSL Model (B) “Manual” model with real data

(C) Raw fit…

Raw results…

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Assessing the significance of the raw result & then correcting for multiplecomparisons

• We can generate a permutationdistribution to assess the significanceof the prior result– P value = 1/N where N is the total

number of “labelings” performed

• The data is still not corrected for thenumber of comparisons made acrossthe TBSS skeleton…so there arevarious approaches:– Cluster size approach– Cluster “mass” approach– Maximum statistic approach– False discovery rate (FDR)

Theory underlying permutation testing

“ If there is no experimental effect, then the labeling of observations by thecorresponding experimental condition is arbitrary, because the same datawould have arisen whatever the condition. "ese labels can be anyrelevant attribute: condition “tags,” such as “rest” or “active”; a covariate,such as task difficulty or response time; or a label, indicating groupmembership. ”

-Nichols & Holmes (2002)

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Maximum statistic

• Basic principles:– With a single corrected threshold, t, you can control the FWER for the entire image at 5 %– An example of how to calculate:

• With the “true” condition, and for each permutation, record the maximum statistic to create the nulldistribution of TMAX

• Determine the threshold (t) as: alpha (e.g., 0.05) * permutations (e.g., 5000) and add 1 to this count (250+1 =251). The 251st member of the TMAX distribution is the threshold to apply to the “true condition” SPI

• Report the corrected p-value as the proportion of the TMAX distribution more extreme than the observed t

Cluster size corrections

• Basic principles:– Define an arbitrary threshold (t = 2.3) and use this to threshold every SPI (resulting from the “true” and permutation

labels).– After applying the threshold, determine the volume of the largest continuous cluster of supra-threshold voxels (supra-

threshold cluster size or ‘STCS’). Using this procedure, form the null distribution of STCSMAX

– As with the maximal statistic approach (corrected threshold is the cluster size at = alpha * N labelings + 1). Thecorrected p-value is similarly reported as the area of cluster size null distribution more extreme than this threshold.

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Cluster mass corrections

• Basic principles:– Again, generate SPIs with the original and permutation labelings.– As with cluster size correction, define an arbitrary threshold and find the suprathreshold cluster as before. This time

though, subtract the arbitrary threshold from the observed voxel statistics (per voxel) in the STC - take the sum of theresult as the cluster or exceedance mass (see excerpt from Bullmore et al., 1999, above left).

– With the permutation distribution formed, determine the threshold and correct p as with STCS and maximum statisticapproaches.

False discovery rate (FDR)

• Basic principles:– A Bonferroni correction across an entire SPI might look like:

• Probability of rejecting null hypothesis when true (false positives) < α (e.g., 0.05)– In contrast, an FDR correction would imply that:

• Proportion of false positives in all rejected null hypotheses < the rate specified as q (e.g., 0.05)

..& in practice.. In theory…

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What does reduced FA represent?

• Caution must be taken when interpreting DTI studies in disorderssuch as PDS. This is especially important as the ‘interpretation ofanisotropy change may not be as straightforward as themeasurements’ and the exact mechanisms governing anisotropy arenot known (S. Mori in Brain Mapping: the Methods, p. 392).

• The contributions to diffusion from “intra-axonal and extracellularspaces, and the myelin sheath” and the exact mechanisms ofanisotropy are unknown (D. Le Bihan in NRN).– I encourage you to see out Le Bihan’s publications for experimental

studies on contribution of various compartments to the obtained signal!

References & further reading 1. Le Bihan D. Looking into the functional architecture of the brain with diffusion

MRI. Nature reviews 2003;4(6):469-480. 2. Genovese CR, Lazar NA, Nichols T. Thresholding of statistical maps in

functional neuroimaging using the false discovery rate. NeuroImage 2002;15(4):870-878.

3. Basser PJ, Jones DK. Diffusion-tensor MRI: theory, experimental design and data analysis - a technical review. NMR in biomedicine 2002;15(7-8):456-467.

4. Basser PJ, Mattiello J, LeBihan D. Estimation of the effective self-diffusion tensor from the NMR spin echo. Journal of magnetic resonance 1994;103(3):247-254.

5. Basser PJ, Pierpaoli C. Microstructural and physiological features of tissues elucidated by quantitative-diffusion-tensor MRI. Journal of magnetic resonance 1996;111(3):209-219.

6. Pierpaoli C, Jezzard P, Basser PJ, Barnett A, Di Chiro G. Diffusion tensor MR imaging of the human brain. Radiology 1996;201(3):637-648.

7. Le Bihan D, Poupon C, Amadon A, Lethimonnier F. Artifacts and pitfalls in diffusion MRI. Journal of Magnetic Resonance Imaging 2006;24(3):478-488.

8. Le Bihan D, Mangin JF, Poupon C, Clark CA, Pappata S, Molko N, Chabriat H. Diffusion tensor imaging: Concepts and applications. Journal of Magnetic Resonance Imaging 2001;13(4):534-546.

9. Nichols TE, Holmes AP. Nonparametric permutation tests for functional neuroimaging: A primer with examples. Human brain mapping 2002;15(1):1-25.

10. Smith SM, Jenkinson M, Johansen-Berg H, Rueckert D, Nichols TE, Mackay CE, Watkins KE, Ciccarelli O, Cader MZ, Matthews PM, Behrens TE. Tract-based spatial statistics: voxelwise analysis of multi-subject diffusion data. NeuroImage 2006;31(4):1487-1505.

11. Smith SM, Johansen-Berg H, Jenkinson M, Rueckert D, Nichols TE, Miller KL, Robson MD, Jones DK, Klein JC, Bartsch AJ, Behrens TE. Acquisition and voxelwise analysis of multi-subject diffusion data with Tract-Based Spatial Statistics. Nature protocols 2007;2(3):499-503.

12. Bullmore ET, Suckling J, Overmeyer S, Rabe-Hesketh S, Taylor E, Brammer MJ. Global, voxel, and cluster tests, by theory and permutation, for a difference between two groups of structural MR images of the brain. Ieee Transactions on Medical Imaging 1999;18(1):32-42.