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Fisher Statistics for Analysis of Diffusion Tensor Directional

Information

Elizabeth B. Hutchinsona,*, Paul A. Ruteckia,b, Andrew L. Alexanderc, and Thomas P.

Sutulaa

a Department of Neurology, University of Wisconsin UW Medical Foundation Centennial Building

1685 Highland Ave Madison, WI 53705 USA

b Department of Neurology, William S. Middleton Memorial VA Hospital 2500 Overlook Terrace

Madison, WI 53705 USA

c Department of Medical Physics, University of Wisconsin Wisconsin Institutes Medical Research

1111 Highland Ave Madison, WI 53705 USA

1 Introduction

Diffusion tensor imaging (DTI)1 has emerged as a unique MRI-based probe of brain

microstructure able to sensitively distinguish heterogeneous tissue types and detect

structural abnormalities in human populations and experimental disease models. DTI maps

are constructed from estimation of the diffusion tensor (DT), which describes the 3-

dimensional self-diffusion of water, at every MRI voxel (Basser et al., 1994). The

eigenvalues (λ1, λ2 and λ3) and eigenvectors (ε1, ε2 and ε3) of the DT can be used to

characterize differences in magnitude and shape of diffusion in DTI index maps, such as

fractional anisotropy (FA) and mean diffusion (MD). Although FA and MD are the primary

DTI indices employed in brain research, the diffusion tensor is a rich source of structural

information with the potential to provide additional metrics relevant to a range of

neurobiological questions including alterations in tissue orientation properties consistent

with developmental or pathological reorganizatiht on of circuitry.

In conventional analysis, the orientation of ε1 has been depicted by directionally encoded

color (DEC) maps (Pajevic & Pierpaoli, 1999) enabling qualitative visual comparisons.

While this type of subjective observation can provide qualitative insight into the nature of

tissue abnormality, there is a general need for quantitative statistcal methods to assess the

likelihood that any observed alterations differ from random variation. A range of approaches

have been suggested for this purpose (Jones et al., 2002), however the existing frameworks

(Wu et al., 2004; Wang et al., 2008) are based on the coherence and dispersion of ε1, and

© 2012 Elsevier B.V. All rights reserved

*corresponding author 1760 Medical Sciences Center 1300 University Ave Madison, WI 53706 phone: 00 1 608 265 9568 fax: 00 1

608 265 5321.

Author email addresses ehutchinson@wisc.edu rutecki@neurology.wisc.edu alalexander2@wisc.edu sutula@neurology.wisc.edu

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1Abbreviations: Diffusion tensor imaging, DTI; diffusion tensor, DT; fractional anisotropy, FA; mean diffusion, MD; directionally

encoded color maps, DEC maps; probability density function, pdf; region of interest, ROI; status epilepticus, SE; traumatic brain

injury, TBI; controlled cortical impact, CCI; corpus callosum, CC.

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Author Manuscript

J Neurosci Methods. Author manuscript; available in PMC 2013 April 30.

Published in final edited form as:

J Neurosci Methods. 2012 April 30; 206(1): 40–45. doi:10.1016/j.jneumeth.2012.02.004.

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hemispheric symmetry of white matter tracts and do not make possible groupwise statistical

comparison of regional orientation information.

The objective of this work was to develop a straightforward statistical approach for the

quantitative analysis of directional DTI information and to interrogate robust subjective

visual observations of abnormal tissue orientation on DEC maps in the hilus region of the

hippocampal dentate gyrus following injury. The Fisher probability density function (Fisher,

1953), which is the spherical coordinate system analogue of the familiar Gaussian

probability density function, was chosen as the basis for this framework. Historically, Fisher

statistics have been widely implemented in the field of paleomagnetism (Butler & Butler,

1992; Tauxe, 2010) to characterize magnetization vectors measured from rock samples and

used to determine the history of the Earth's magnetic field. Here we report for the first time

the theoretical and practical application of Fisher statistics for analysis of DTI directional

information and demonstrate the utility of this approach to detect directional abnormalities

in the dentate gyrus following injury.

2 Theory and Calculations

2.1 Terminology and assumptions

The Fisher statistical method can be used to estimate descriptive and inferential statistics

from a set of directional observations. Just as one-dimensional statistics are based on the

Gaussian probability density function with true mean (μ) and variance (σ2), these directional

statistics are based on the Fisher probability density function, with true direction (υ) and

precision parameter (κ).

The type of directional data that can be interrogated using Fisher statistics are normally

distributed sets of unit vectors where each vector has a length of 1 and originates from

x=y=z=0 so that each set may be represented by a group of points on the unit sphere. For a

set of N vectors, V = (v1, …, vN), a mean direction vector, vm, can be determined and each

vector in the set, vi = (xi, yi, zi), can be described by it's angle, αi, relative to vm using the dot

product αi,=cos−1(vi˙ vm).

Ultimately, an estimate can be made for the confidence angle, α(1-p), within which (1-p)% of

all directional observations can be expected to be found and F-statistic hypothesis testing

may be performed to make inferences about group differences in orientation.

2.2 The Fisher distribution

The Fisher probability density function (pdf) (Fisher, 1953) describes the probability of a

point on a sphere of unit radius within an angular area dA (in steradians) centered at an

angle α from υ:

(1)

where κ is a precision parameter that is inversely proportional to the dispersion.

Since it is more useful for this type of statistics to consider the pdf in terms of the spatial

variables α and φ, the azimuthal angle, and because dA=sin(α)dαdϕ, the pdf is more often

written as:

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(2)

which gives the probability of finding a direction vector in the circular strip on the unit

sphere between the angles α and α+dα from the true direction.

2.3 Descriptive statistics

The Fisher probability distribution is the basis for the estimation of descriptive and

inferential statistics presented in this section.

2.3.1 Mean direction and resultant vector—The mean direction vector, vm, describing

the average of a set of N unit vectors can be calculated as follows:

where

(3)

and R is the length of the resultant vector sum of all observations, which is given by:

(4)

2.3.2 The precision parameter—The precision parameter, κ, is a population parameter

that describes the concentration of the pdf around the true direction. As κ approaches 0, the

distribution becomes uniform over the unit sphere and as κ approaches∞, the distribution

becomes singular at the true direction. The sample estimate of κ is k (Fisher, 1953;

McFadden, 1980) given by:

(5)

where N is the number of unit vectors in the sample and R is the length of the resultant

vector calculated in (4).

2.3.3 Confidence angles—For a confidence level of (1-p)% (typically 95% or p=0.05), a

circle can be defined by all points at an angle of α(1-p) from the calculated mean direction,

vm. α(1-p) is given by:

(6)

where N is the number of unit vectors in the sample, R is the resultant vector calculated in

(4) and p is defined according to confidence level.

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2.5 Hypothesis testing

In order to test the null hypothesis - that sample observations from 2 or more groups are

taken from the same population - the mean direction vectors and confidence angles may be

compared across groups or inferential statistics may be calculated.

An intuitive comparison of group directional information is to use the circle of confidence

defined by α95. The clearest inferences can be made when: 1) α95 circles for the groups do

not overlap, then the null hypothesis is unlikely or 2) the mean direction vector from one

group lies inside α95 of the other, then the null hypothesis is likely. A less clear case occurs

when α95 circles for the groups do overlap, but the mean from each group lies outside the

α95-circle of the other.

A more quantitative way to perform hypothesis testing is by calculation of the F statistic.

The following equation was derived by Watson (Watson, 1956), to compare two groups with

N1 and N2 observed unit vectors respectively and resultant vectors of length R1 and R2

respectively:

(7)

where N=N1+N2 and R is the length of the resultant vector for the pooled direction vector

observations from both groups. The larger the value of F, the more different the two group

mean directions and, a p-value may be obtained using the appropriate degrees of freedom (2

and 2(N-2) respectively). Equation 7 also extends to hypothesis testing of more than 2

groups.

2.5 Special considerations for DTI data

Although Fisher statistics are generalizable to the analysis of many types of directional data

sets, it is important to consider certain inherent attributes of DTI data for accurate directional

analysis. Here, it is suggested that the potential pitfalls of DTI data be avoided or accounted

for prior to statistical analysis so that the input for Fisher statistics are groups of direction

vectors where each vector accurately represents the tissue orientation within a given region

of interest (ROI) for one brain sample and is collected and calculated consistently across all

samples and groups.

2.5.1 Sources of experimental error—Sources of variance that may contribute to

erroneous directional measurement include: positioning of the sample within the scanner

reference frame, gross brain structure abnormalities (e.g. by uneven fixation) and ROI mask

placement variations. To the extent possible, these should be reduced by experimental

practices or corrected for by post-processing techniques, for example DTI-appropriate

registration to standard space (Jones et al., 2002).

2.5.2 DEC ambiguity and lateralization—Most DEC maps report directional coloration

based on an absolute value algorithm, which creates maps that are qualitatively intuitive and

do not suffer from discontinuity artifacts (Pajevic & Pierpaoli, 1999), however this approach

suffers from ambiguity in that each color represents 4 unique directions. For example, if

<R,G,B> = < |x|, |y|, |z|>, then the directions: x,y,z = (0.6667, 0.3333, 0.6667), (−0.6667,

0.3333, 0.6667), (0.6667, −0.3333, 0.6667) and (−0.6667, −0.3333, 0.6667) are all different

from one another, but coded for by the same shade of “pink”. This is particularly evident for

bilateral structures, such as the fimbria (see figures 1a and e), which are encoded for by the

same color in both brain hemispheres when in fact the ε1 vectors for symmetric anatomical

locations are oriented differently, albeit symmetrically. Consequently, care should be taken

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to consider ROIs from different hemispheres separately or carefully transform directional

data from anatomically symmetric regions to a single hemisphere.

2.5.3 Antipodal symmetry—Perhaps the most substantial issue in the analysis of

directional DTI data arises from the arbitrary sign of the DTI eigenvectors (i.e. ε1=−ε1)

which implies that the collection of eigenvectors in an ROI may be clustered about two

points, known as antipodes, that are exactly opposite one another on the unit sphere

(illustrated in figure 1b). Uncorrected, this could lead to severe misestimation of the

direction vector. To avert this, it is possible to map all vectors to a single antipode before

determining the direction vector. Similarly, direction vectors for all observations within and

across groups should be taken from antipodes in the same hemisphere, although choice of

the hemisphere will depend on the orientation of the structure in question.

3. Methods

3.1 DTI

3.1.1 Brain samples—Ex-vivo brain samples from two rat models of brain injury, status

epilepticus (SE) and traumatic brain injury (TBI), were compared to control samples. DTI

images from these models were selected specifically for the development of the Fisher

statistics method based on qualitative identification of abnormalities on a DEC map. All rats

were from the Sprague Dawley strain and all animals were housed and treated according to

national guidelines and institutional oversight and approval.

SE was induced by kainic acid administered intraperitoneally (8–10 mg/kg) to juvenile

(P=35, n=4) and adult (P70, n=2) male rats according to conventional procedures (Ben-Ari,

1985). For inclusion, rats must have demonstrated sustained class V seizure behavior and the

groups were combined based on similarity of the brain images and identified abnormalities

on the DEC map. TBI was induced in adult rats (n=6 male and 4 female) by controlled

cortical impact (CCI) according to conventional procedures (Dixon et al., 1991) using a

stereotaxic impactor with 3mm cylindrical bit placed over the temporo-parietal cortex. CCI

was performed with a velocity of 4 m/s and penetration depth between 1–3 mm. Controls

(n=3) were normal adult male rats.

At greater than 6 months following SE or CCI each rat was transcardially perfused with 4%

paraformaldehyde (PFA) and the brain was removed from the skull and stored in PFA for a

minimum of 2 weeks. Prior to imaging, brains were rinsed in 0.9% saline for 48 hours.

During imaging, brain samples were placed in a custom built sample holder and immersed in

Fluorinert (FC-3283; 3M, St. Paul, MN).

3.1.2 Acquisition—For each 7 hour and 2 minute ex-vivo DTI acquisition, 3 brains were

simultaneously imaged using a 4.7 T Agilent MRI system and 3.5cm diameter quadrature

volume RF coil. A series of multi-slice, diffusion-weighted, spin echo images were acquired

with three non-weighted (b~0) and 30 diffusion weighted (b~1200 s/mm2, δ=4.5 ms, Δ=11.2

ms and G=23.84 G/cm), using non-colinear weighting directions (selected from the GE

medical systems tensor.dat direction set by Bryan Mock). Other imaging parameters were

TE/TR=24.17/2000ms, FOV = 30×30 mm2, matrix = 192×192 reconstructed to 256×256,

slice thickness = 0.5mm, number of slices = 35 and 2 signal averages.

3.1.3 Post-processing—DTI maps were created offline using a combination of FSL

software and custom Matlab code (version 7.8, R2009a). The diffusion tensor was

determined at each voxel using a non-linear least squares fitting algorithm (Koay et al.,

2006). FA and DEC maps were generated as well as images containing the Cartesian vector

coordinates for ε1.

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3.1.4 Experimental error considerations—To address the sources of experimental

error described in section 2.5.1, no brains were included with compromised perfusion

surgeries and design of the sample holder and positioning of the samples in the scanner

reference frame were intended to minimize variations in sample orientation. No registration

of the images was performed to make post-processing corrections for sample misalignment.

ROI generation was carefully planned and performed to ensure consistent placement.

3.2 Direction analysis

3.2.1 ROI eigenvector analysis—Multi-slice ROI masks for the left and right hilus of

the dorsal hippocampal dentate gyrus, midline corpus callosum (CC) and left and right

fimbria were manually created for each sample in the native space of the acquired DTI

images based on anatomical landmarks as defined by standard coordinates of a rat brain

atlas (Watson & Paxinos, 1986). The ROI masks were then used to extract ε1 for every voxel

in the ROI.

Based on the suggestions made in 2.5.3, all ε1 measurements were mapped to a single

antipode and a single unit direction vector for each ROI for each sample was determined as

follows. First, a reference “pole” was defined automatically by pooling all ε1 measurements

and their −ε1 pairs across all voxels and all subjects for a given ROI. Then, cluster analysis

was performed to determine a pair of necessarily opposite vectors and the vector in the

positive z hemisphere was selected for use as the “pole”. Next, each (ε1, −ε1) pair in the ROI

was evaluated and only the vector nearest to the “pole” was included. Finally, the unit

direction vector describing average direction in the ROI for each sample was defined as the

vector average of the mapped eigenvectors within the ROI and the collection of sample unit

direction vectors across experimental groups were subjected to Fisher statistical analysis.

3.2.2 Fisher statistics implementation—All statistical analysis and generation of

spherical plots was performed using custom Matlab code. For each group, the resultant

vector length and mean direction vector were calculated from the collection of observed

direction vectors according to equations (3) and (4). For visualization and hypothesis testing,

the mean direction vector and circle defined by α95, calculated using equation (6), were

plotted for each group and the direction vectors for each sample in all three groups were

plotted. The F-statistic was calculated for group comparisons according to equation (7) and a

p-value was determined from the F distribution table.

4. Results

Sets of directional DTI data were measured and analyzed for the CC, fimbria and hilus of

rats following SE, TBI and normal controls. The technique and results are illustrated in

figures 1 and 2.

There was no significant difference in CC mean direction between controls and SE model

animals (F=0.198, p=0.823) or TBI model animals (F=0.228, p=0.798). There were also no

directional differences found in the left or right fimbria between controls and SE (left:

F=3.325, p=0.066; right: F=0.782, p=0.476) or TBI (left: F=0.041, p=0.960; right: F=0.196,

p=0.823) groups.

Significant directional differences were found in the hilus between the control group and the

SE model group (left: F=16.520, p<0.001; right: F=15.189, p<0.001) as well as the TBI

model group on the side of injury (right: F=24.8560, p<0.001), but not the contralateral side

(left: F=2.230, p=0.131).

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5. Discussion

The Fisher statistical method was applied to obtain descriptive statistics of DTI directional

information in three groups of ex-vivo rat brain samples and enabled quantitative inferences

about directional differences in 3 different brain structures. No directional abnormalities

were found following injury in the CC or fimbria, but a significant difference was found for

the hilus that confirmed subjective visual observations of ε1 directional shift from anterior-

posterior to dorsal-ventral. Although the structural substrates of this directional alteration

remain to be shown, the hilus is known to be highly susceptible to cell loss and aberrant

reorganization following injury and during epileptogenesis (Dudek et al., 2002). Directional

analysis using Fisher statistics appears to be sensitive to tissue change following both TBI

and SE and may complement and extend the findings of other post-SE DTI investigations,

which have shown increased FA in the dentate gyrus of the hippocampus (Laitinen et al.,

2010) and correlations of DTI measures with histological measures of mossy fiber sprouting

(Kuo et al., 2008; Laitinen et al., 2010). This approach also quantitatively distinguished the

expected spatial patterns of directional abnormalities between the two models by showing

significant bilateral differences from control values in the SE model and differences in the

TBI model that were restricted to the side of injury.

Fisher statistics are potentially useful for the comparison of direction vectors between

experimental groups, however they may not be suitable for all types of directional DTI data.

The requirement that directional data are normally distributed about the mean direction

vector and independently observed precludes analysis within a single sample (e.g.

eigenvectors in an ROI). For data that do not conform to the requirements of Fisher

statistics, other approaches exist, such as the Kent distribution for elliptically distributed

directions, the Bingham distribution for bimodal data or non-parametric methods (a useful

review of these methods exists in Tauxe, 2010).

6. Conclusion

A framework for the statistical description and inference of directional DTI data has been

presented based on the Fisher statistical method and was demonstrated to be effective for

distinguishing ROIs with altered directional properties following injury. Quantitative

investigation of directional properties of brain tissue are potentially useful in both animal

and human studies to identify regions of tumor infiltration, white matter tract disruption,

developmental structural organization or any other process that affects the directional

microstructure of brain tissue.

Acknowledgments

The authors thank Sue Osting and Craig Levenick for their technical assistance with animal surgeries and sample

preparation. This work was funded by the following sources: NINDS R01 25020 and Department of Defense

DR080424. EH is funded in part by an award from the Lily's Fund for Epilepsy Research.

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Figure 1.

Determination of DTI directional information. (a) Representative ROIs for the CC and

fimbria are shown on a DEC map. The approach for determination of single sample (b) and

group (c) directional information for the CC is shown, where red points are included

eigenvector measurements, black points are excluded (antipodal) measurements, red lines

are single sample direction vectors and the black line is the group mean direction vector.

Group analysis is shown for the CC (d) and the fimbria (e) where each colored point

represents one sample direction vector and the colored circles correspond to α95 for each

group. All plots are on the unit sphere and color coding is consistent with DEC maps (red-

left/right; green-dorsal/ventral; blue-anterior/posterior)

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Figure 2.

Orientation differences in the hilus of the hippocampal dentate gyrus. ROI analysis of the

hilus (red mask, a) was used to confirm subjective observations (white arrows, b) of altered

tissue orientation following SE or TBI. Spherical plots of the mean direction vectors (lines)

and α95 (circles) for each experimental group and colored points representing the observed

direction vectors for all samples in each of the groups are shown on the unit sphere for the

right (c) and left (d) hilus.

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