Diffusion tensor imaging and tractwise fractional anisotropy statistics: quantitative analysis in white matter pathology
 HansPeter Mueller^{1}Email author,
 Alexander Unrath^{1},
 Anne D Sperfeld^{1},
 Albert C Ludolph^{1},
 Axel Riecker^{1} and
 Jan Kassubek^{1}
https://doi.org/10.1186/1475925X642
© Mueller et al; licensee BioMed Central Ltd. 2007
Received: 02 July 2007
Accepted: 09 November 2007
Published: 09 November 2007
Abstract
Background
Information on anatomical connectivity in the brain by measurements of the diffusion of water in white matter tracts lead to quantification of local tract directionality and integrity.
Methods
The combination of connectivity mapping (fibre tracking, FT) with quantitative diffusion fractional anisotropy (FA) mapping resulted in the approach of results based on groupaveraged data, named tractwise FA statistics (TFAS). The task of this study was to apply these methods to groupaveraged data from different subjects to quantify differences between normal subjects and subjects with defined alterations of the corpus callosum (CC).
Results
TFAS exhibited a significant FA reduction especially in the CC, in agreement with region of interest (ROI)based analyses.
Conclusion
In summary, the applicability of the TFAS approach to diffusion tensor imaging studies of normal and pathologically altered brains was demonstrated.
1. Background
Diffusion tensor magnetic resonance imaging (DTI) is known to be an appropriate technique to map in vivo the diffusion in human brain white matter (WM). The directional dependence of diffusion in each voxel can be characterised by a 3 × 3 matrix called the diffusion tensor $\overrightarrow{\overrightarrow{D}}$. The Eigenvectors and Eigenvalues of $\overrightarrow{\overrightarrow{D}}$ reveal the diffusivity of water in each direction and therefore can be used to quantify the diffusivity by socalled fractional anisotropy (FA) maps on a voxelwise basis. This orientational information can also be the basis for the reconstruction of the interconnectivity of brain regions by following the pathways of the fibres. This technique is known as fibre tracking (FT). The basis of FT is to connect neighboured tensors consecutively along their principal directions. Many techniques referring to this topic have been published [1–5]. Most of them focus on qualitative imaging of the FT, 3D visualisation and judgement by experienced operators. There have also been various efforts in using diffusion anisotropy as a marker for white matter tract integrity [6–8]. In these works, quantitative analysis has been performed by use of the underlying FA maps for selective statistics.
Smith et al. developed an algorithm for an alignmentinvariant tract representation to overcome normalization problems; this approach is referred to as tract based spatial statistics (TBSS) [9–15]. In the present study, an analysis technique named tractwise fractional anisotropy statistics (TFAS) is presented. Hereby, bundles of FT are used in the sense of a skeleton, which is the basis of statistical analysis of the underlying FA maps. The novel character of TFAS is that it uses averaged DTI data sets, i.e. the processing steps of normalization and averaging are not performed on FA maps but on one newly created group averaged DTI data set. Based upon this data set, one FA map is calculated. The definition of the region under observation is consequently performed by anatomical connectivity. Whereas the connectivity of white matter regions is neither under investigation in region of interest (ROI)based analysis methods nor in whole brainbased statistical analysis methods, TFAS was intended to analyse specific white matter regions as well as their connecting pathways not only in healthy brains, but also in distorted brain anatomy.
The prerequisite of statistical analysis at group level and arithmetic averaging of subject data is the normalisation to a standardised stereotactic space, e.g. the Montreal Neurological Institute (MNI) space. MNI defined a new standard brain by using a large series of MRI scans of normal controls, resulting in the MNI atlas [16]. MNI normalisation allows for arithmetic averaging of resulting FA maps.
FT with starting points in the corpus callosum (CC) was used to build skeletons for consecutive statistical analysis. The CC was chosen as the most appropriate structure in the brain since it is one of the white matter structures with an accumulation of mostly strongly directed fibres [17]. TFAS based on different skeletons was used to quantify interconnectivity and to map differences between patients with atrophy of the CC and agematched healthy controls. As a model of CC alteration, patients with complicated hereditary spastic paraparesis (cHSP) were investigated. This rare neurodegenerative disease was chosen as a prototypical alteration across the whole structure of the CC. These patients' brains frequently show a thinned CC (tCC) [18, 19]. Computer simulations that showed the validity of the applied MNI normalisation algorithms and the FT techniques complemented this work.
2. Methods
2.1. Data recording and subject population
DTI scanning protocols were performed on the same 1.5 T scanner (Symphony, Siemens Medical, Erlangen, Germany). Six healthy controls (3 men, 3 women, average age 32.7 ± 4.5), and 6 patients with (tCC) (3 men, 3 women, average age 32.5 ± 12.1) underwent the MRI protocol.
All DTI acquisitions consisted of 13 volumes (45 slices, 128 × 128 voxel, slice thickness 2.2 mm, inplane voxel size 1.5 mm × 1.5 mm) representing 12 gradient directions and one scan with gradient 0 (B _{0}). Echo time (TE) and repetition time (TR) were 93 ms and 8000 ms, respectively. b was 800 s/mm^{2}, 5 scans were kspace averaged online by the Siemens SYNGO operating software. As a morphological background, a T_{1}weighted magnetisationprepared rapidacquisition gradient echo sequence was used (MPRAGE, TR = 9.7 ms, TE = 3.93 ms, flip angle 15°, matrix size 256 × 256 mm^{2}, voxel size 1.0 × 0.96 × 0.96 mm^{3}), consisting of 160–200 sagittal partitions depending on the head size.
2.2. Data processing
2.2.1. Eddy current correction
Large discontinuities in bulk magnetic susceptibility produce local magnetic field gradients that notoriously degrade and distort DTI, particularly during echoplanar imaging [20]. These eddy current induced geometric distortions vary with the magnitude and direction of the diffusion sensitising gradients. For the correction of this distortion, the method proposed by Shen et al. [21] was applied. The technique relies on collecting pairs of images with reversed diffusion sensitising gradients – these paired images are distorted with eddy currents in opposite directions. A columnwise correction in the image domain along the phase encoding direction (anterior – posterior) was applied. This was performed by searching for the maximum value of the crosscorrelation between two corresponding columns (of two paired volumes) while one is shifted and scaled (fitting routine: Simplex method [22]). Each column was then corrected by applying opposite shifts and scales equal to half of the correction. Other techniques for the eddy current correction were described in [23, 24].
2.2.2. Transformation to isovoxels and smoothing
where I _{ t arg et }(i, J, k) was the voxel intensity at the new grid coordinates i, j, k and l, m, n were the original voxel coordinates in x, y, z direction, respectively. The factors a _{ v }were the 8 weighting factors for the interpolation.
where r _{ v }was the distance between I _{ t arg et }and I _{ v }, and g _{ t arg et }was the absolute value of the gradient at position (i,j,k). In this way, the local gradients weighted the interpolation kernel with a sharpness dependency.
2.2.3. Template generation and spatial normalisation
Spatial normalisation allowed for arithmetic averaging of the results obtained from different subjects in order to finally perform a comparison of groups of patients with certain disorders, e.g. neurodegenerative diseases, and healthy subjects. Talairach and Tournoux [28] suggested a transformation algorithm to a standard atlas involving the identification of various brain landmarks and piecemeal scaling of brain quadrants. An alternative approach was to use automated brain registration algorithms [29, 30]. In the present study, a semiautomatic spatial normalisation utilizing a study specific template was performed for the transformation into MNI space. The template used for the normalisation of the DTI data sets was created from all subjects' (b = 0) data sets of the subjects who participated in this study. The iterative algorithm has previously been described in detail [31]. In short, a first template was generated by arithmetic averaging of the data sets after an affine transformation. Fitting of all data sets to this first template using a nonaffine transformation and arithmetic averaging led to the template. Then, single subject DTI data sets could be normalised according to MNI dimensions.
Basically, a complete nonlinear MNI normalisation consisted of 3 deformation components (DC):

DC 1: A rigid brain transformation to align the basic coordinate frames. Hereby, the rotation angles had to be stored in a rotation matrix $\overrightarrow{\overrightarrow{R}}$.

DC 2: An affine deformation according to landmarks. Hereby, the 6 stretching parameters for the different brain regions had to be stored in a 6D vector $\overrightarrow{S}$.

DC 3: A nonaffine normalisation equalizing nonlinear brain shape differences. Hereby, the 3D vector shifts were different for each voxel resulting in a 6D matrix (a 3D vector for each voxel of the 3D matrix) $\overrightarrow{\overrightarrow{\overrightarrow{T}}}$.
Consequently, the resulting diffusion tensor ${\overrightarrow{\overrightarrow{D}}}_{i}$ of each voxel i had to be rotated according to all the rotations listed above.

Rotation resulting from the aligning to the basic coordinate frame (corresponding to DC 1)${\overrightarrow{\overrightarrow{D}}}_{i}\text{'}=\overrightarrow{\overrightarrow{R}}\cdot {\overrightarrow{\overrightarrow{D}}}_{i}.$(3)

Simple trigonometry gives a rotation matrix (for each voxel independently), resulting from the 3D vector shifts following the basic ideas of Alexander et al. [32]. The dilation matrices were used for the alignment of the tensor $\overrightarrow{\overrightarrow{D}}$ of each voxel to the surrounding voxels (corresponding to DC 3).${\overrightarrow{\overrightarrow{D}}}_{i}\text{'}\text{'}={\overrightarrow{\overrightarrow{t}}}_{i}\cdot {\overrightarrow{\overrightarrow{D}}}_{i}\text{'}$(4)
where ${\overrightarrow{\overrightarrow{t}}}_{i}$ are the components of $\overrightarrow{\overrightarrow{\overrightarrow{T}}}$.

The components of the Eigenvectors (${\overrightarrow{v}}_{1},{\overrightarrow{v}}_{2},{\overrightarrow{v}}_{3}$) had to be stretched according to the 6 stretching parameters of vector $\overrightarrow{S}$ (dependent on the brain region s _{ a }, a = 1...6) of the affine deformation (corresponding to DC 2). v _{ w, j }''' = s _{ a } v _{ w, j }''' (5)
with w = 1...3 and j = x, y, z. After the stretching, the Eigenvectors had to be renormalised.
These finecorrections of the tensor $\overrightarrow{\overrightarrow{D}}$ were essential for a correct FT (see section 4), and the corresponding parameters had to be stored for each subject data set independently. With the same normalisation procedure (without finecorrections), the corresponding MPRAGE were normalised.
2.2.4. Analysis methods – FA mapping and averaging of FA maps for different subjects
where $\overline{\lambda}$ was the average of all Eigenvalues.
The intensity was related to the FA and the colourcoding was as follows: red for major Eigenvector mainly in leftright direction, blue for major Eigenvector mainly in inferiorsuperior direction and green for major Eigenvector mainly in posterioranterior direction.
A ROIbased approach of defined brain regions was the method of choice for the analysis of separate areas. A ROI was defined by a sphere with an observerdefined radius r _{ s }and its centre at the userdefined focus ${\overrightarrow{f}}_{s}$. All FA values F _{ i }of the N voxels inside this sphere (position ${\overrightarrow{v}}_{i}$) were included in the following parameterisation. The ROI analysis included the following parameters:

average FA, i.e. average diffusion strength${F}_{avg}=\frac{1}{N}{\displaystyle {\sum}_{i,\left{\overrightarrow{f}}_{s}{\overrightarrow{v}}_{i}\right<{r}_{s}}^{N}{F}_{i}}$(7)

standard deviation of averaged FA${F}_{std}=\sqrt{\frac{1}{N1}{\displaystyle {\sum}_{i,\left{\overrightarrow{f}}_{s}{\overrightarrow{v}}_{i}\right<{r}_{s}}^{N}{\left({F}_{avg}{F}_{i}\right)}^{2}}}$(8)
where N was the ROI size. Extensive studies via ROI analyses about the preservation of the DTI specific parameters during the process of MNI normalisation were performed previously [31].
Group studies might be of interest if the common deficit was due to damage of one or more defined brain areas. For this task, averaging of results for different subjects was necessary. After all subjects' data had been transformed into MNI space, averaging of the results was feasible. The latter step could be performed in three ways:

The FA map was calculated separately for each subject data in MNI space, followed by arithmetic averaging of the FA maps. This led to a groupspecific FA map FAM1. Hereby, the drawback was that the directional information was lost, quantification and analysis was performed via the FA maps.

Diffusion tensor $\overrightarrow{\overrightarrow{D}}$ matrices were stored after finecorrection and an averaged diffusion tensor matrix was created. This method required 600 Mbyte for each subject and increased computation time inadequately and thus was practicable only for studies with small populations.

Each DTI data set was normalised and the whole DTI data sets were averaged before FA mapping. FA parameterisation of these groupaveraged DTI data led to a groupspecific FA map FAM2. The fine corrections were also arithmetically averaged. This resulted in an averaged fine correction to be applied to the resulting averaged DTI data set.
The third approach could be applied if the evidence that the FA values were preserved and also the orientational dependence of the Eigenvectors were preserved. ROI analysis had already shown that FA values were preserved by MNI transformation [31]. In section 4, a short excursion to computer simulation should give evidence that the orientational information could be preserved if fine corrections were applied.
2.2.5. Fibre tracking
For FT, anisotropic diffusion was characterised to determine the preferred diffusion direction. In the calculation of the diffusion spheroid, the Eigenvector corresponding to the largest Eigenvalue was the direction of fastest diffusion and indicated the fibre direction in white matter regions. Based on this directional information, different methods and algorithms had been proposed to estimate white matter connectivity. In this work, the conservative streamline tracking technique (STT) was used. STT modelled the propagation in the major Eigenvector field of the brain [20, 33].
Here, ${\overrightarrow{v}}_{new}(i,j,k)$ was the resulting vector at the new position i, j, k (float numbers), and l, m, n were the voxel coordinates (integer numbers) of the 8 neighboured voxels. The factors a _{ w }were the respective 8 weighting factors for the interpolation.
The following set of parameters was used for FT:

The threshold for the scalar product of the major Eigenvectors (angle between directions of two consecutive FT positions and the first Eigenvector directions) was set to 0.95.

The distance between two FT positions, i.e. the stepwidth, was set to 0.5 mm, corresponding to 0.5 voxels.
All the analyses were performed by the newly developed software package TIFT (Tensor Imaging and Fibre Tracking) [31]. TIFT provides various quantification and visualisation possibilities for DTI analysis. The structure of the software aims at minimisation of operatordependency providing analysis in a fast and reproducible way.
2.2.6. Tractwise fractional anisotropy statistics (TFAS)
Beside ROI analysis statistical comparison between subject groups could also be performed defining a skeleton which then was used for the selection of voxels that contribute to the statistics. FT was used to define such skeletons. When FT was performed on averaged DTI data, each voxel that is crossed by a FT was defined as 'active' for statistics. Basically, two skeletons could be built:

A skeleton based on FT in averaged normal subject data (skeleton 1). The skeleton followed well known paths and was not disturbed by neuroanatomical alterations.

A skeleton based on FT in averaged tCC subject data (skeleton 2). This skeleton was only under consideration for the sake of completeness.
where N was the respective skeleton size, which usually was different for different skeletons. Then, TFAS used F _{ avg }and F _{ std }for statistical ttest.
2.2.7 Statistical analysis
Statistical analysis (SA) by use of Student's ttest was performed in two ways:
SA1: skeleton 1 was applied to FA maps of the individual subjects. Alternatively, skeleton 2 was applied to FA maps of the individual subjects.
SA2: skeleton 1 and 2, respectively, were applied to the FA maps of the averaged DTI data sets. Hereby, the variability was calculated along the skeleton, so this approach is not an analysis of the group variability.
3. Results
3.1. Fibre track simulations
3.2. Differences between patients with tCC and controls
As an example of a disorder with a presumable affection of white matter tracts, patients with tCC were chosen. The CC was considered an important structure to investigate since a multitude of strongly directed fibres (connecting the hemispheres of the brain) is clustered in its formation. DTI should be able to detect axonal loss or at least damage of oriented white matter regions such as the CC.
3.2.1. Fractional anisotropy mapping
MNI coordinates of seeds in the corpus callosum (CC)
x/mm  y/mm  z/mm  

seed 1  0  23  8 
seed 2  0  15  23 
seed 3  0  9  36 
seed 4  0  31  29 
seed 5  0  44  16 
Region of interest (ROI) analysis of fractional anisotropy (FA) maps with radius 5 mm at seeds in the corpus callosum (CC)
FA (normals)  FA (tCC)  p  

seed 1  0.61 ± 0.13  0.19 ± 0.04  <0.001 
seed 2  0.40 ± 0.14  0.12 ± 0.01  0.02 
seed 3  0.47 ± 0.13  0.13 ± 0.03  0.004 
seed 4  0.43 ± 0.13  0.20 ± 0.04  0.06 
seed 5  0.53 ± 0.13  0.48 ± 0.05  0.67 
3.2.2. Fibre tracking on single subjects
3.2.3. Tractwise fractional anisotropy statistics

TFAS with skeleton 1: FT from healthy controls

TFAS with skeleton 2: FT from tCC patients
Tractwise fractional anisotropy statistics (TFAS) with different skeletons, based on the 5 seeds in the corpus callosum (CC) (according to Table 1).
skeleton 1  

FA (normals)  FA (tCC)  
subject 1  0.37  0.19 
subject 2  0.40  0.32 
subject 3  0.32  0.22 
subject 4  0.32  0.16 
subject 5  0.31  0.31 
subject 6  0.33  0.27 
average  0.34 ± 0.04  0.25 ± 0.06 
ttest: p < 0.01  
skeleton 2  
FA (normals)  FA (tCC)  
subject 1  0.40  0.20 
subject 2  0.43  0.36 
subject 3  0.40  0.22 
subject 4  0.35  0.17 
subject 5  0.36  0.31 
subject 6  0.38  0.28 
average  0.39 ± 0.03  0.26 ± 0.07 
ttest: p < 0.01 
Tractwise fractional anisotropy statistics (TFAS) with different skeletons, based on the 5 seeds in the corpus callosum (CC) (according to Table 1).
FA (normals)  FA (tCC)  p  

skeleton 1  0.28 ± 0.18  0.17 ± 0.09  <0.001 
skeleton 2  0.33 ± 0.19  0.17 ± 0.09  <0.001 
4. Discussion
With respect to the direction of fibres, the hypothesis could be set up that FT should give similar results for normals and patients with CC thinning. Nevertheless, it was shown that DTI and consecutive FT failed in some patients due to low FA values. Changes in orientational dependencies (FT) between the normal group and the tCC group were not related to changes in the diffusion direction; moreover, the low values in DTI (as detected by low FA values) led to errors in the diffusion tensors, and thus the orientational dependency was lost. To overcome this problem by improving the signaltonoise ratio, averaging of DTI data from patients with a similar pattern of brain alterations was performed.
ROI analysis had already shown the preservation of FA values after MNI normalisation [31], whereas preservation of the orientational dependency during MNI normalization is feasible (as shown by computer simulations). Therefore, averaging of DTI data sets was considered to be allowed. Although the drawback of the averaging method (finecorrections are averaged and thus could only partially be taken into account), the averaged DTI data set showed meaningful results for the FT. For the averaged data sets, the FT pattern looked more similar for controls and patients with tCC than the comparison of single subject data sets.
By application of tractography methods in order to obtain connectivitybased functional brain regions, as proposed by Behrens et al., regionspecific FA analysis becomes feasible [6]. In addition to ROI and whole brainbased analysis, this method takes the interconnectivity of brain regions into account. In the present study, TFAS was applied to skeletons which had been derived from different subject groups including patients with distorted brain anatomy. Here, the tractwise FA statistics again seemed to be superior to standard FA analysis methods due to its functional specifity.
It has to be kept in mind that averaging of DTI data sets was only possible if the deformations necessary for MNI normalisation were minor. Problems in cortical regions could occur as already reported by [32, 31]. Although the FA values of the averaged DTI data sets were reduced by about 20 % due to the averaging process, the differences remain statistically significant for each of the statistical analysis methods.
5. Conclusion
In order to perform the analysis methods described in this study, a software platform was needed that was able to perform all the analysis at group level in a fast, reproducible and concise way, and the TIFT software contains all prerequisites for analysis performed in this work. At group level, the resulting averaged DTI data sets showed reduced FA maps as well as a reduction of FT for patients with tCC in comparison to healthy controls. Although the skeletons were different in localization and size, the result of a significant reduction of FA values, as calculated by TFAS in controls compared with tCC patients, was independent of the skeleton. Thus, TFAS seemed to be a probate way to measure differences at group level in groups of subjects with a groupwise similar pattern of brain alterations.
Declarations
Authors’ Affiliations
References
 Conturo TE, Lori NF, Cull TS, Akbudak E, Snyder AZ, Shimony JS, McKinstry RC, Burton H, Raichle ME: Tracking neuronal fibre pathways in the living human brain. Proc Natl Acad Sci USA 1999,96(18):10422–7. 10.1073/pnas.96.18.10422View ArticleGoogle Scholar
 Lori NF, Akbudak E, Shimony JS, Cull TS, Snyder AZ, Guillory RK, Conturo TE: Diffusion tensor fibre tracking of human brain connectivity: aquisition methods, reliability analysis and biological results. NMR Biomed 2002,15(7–8):494–515. 10.1002/nbm.779View ArticleGoogle Scholar
 Mori S, van Zijl PC: Fibre tracking: principles and strategies – a technical review. NMR Biomed 2002,15(7–8):468–80. 10.1002/nbm.781View ArticleGoogle Scholar
 Lazar M, Weinstein DM, Tsuruda JS, Hasan KM, Arfanakis K, Meyerand ME, Badie B, Rowley HA, Haughton V, Field A, Alexander AL: White matter tractography using diffusion tensor deflection. Hum Brain Mapp 2003,18(4):306–21. 10.1002/hbm.10102View ArticleGoogle Scholar
 Behrens TE, JohansenBerg H, Woolrich MW, Smith SM, WheelerKingshott CA, Boulby PA, Barker GJ, Sillery EL, Sheehan K, Ciccarelli O, Thompson AJ, Brady JM, Matthews PM: Noninvasive mapping of connections between human thalamus and cortex using diffusion imaging. Nat Neurosci 2003,6(7):750–7. 10.1038/nn1075View ArticleGoogle Scholar
 Pagani E, Filippi M, Rocca MA, Horsfield MA: A method for obtaining tractspecific diffusion tensor MRI measurements in the presence of disease: Application to patients with clinically isolated syndromes suggestive of multiple sclerosis. Neuroimage 2005,26(1):258–265. 10.1016/j.neuroimage.2005.01.008View ArticleGoogle Scholar
 Wilson M, Tench CR, Morgan PS, Blumhardt LD: Pyramidal tract mapping by diffusion tensor magnetic resonance imaging in multiple sclerosis: improving correlations with disability. J Neurol Neurosurg Psychiatry 2003, 74: 203–207. 10.1136/jnnp.74.2.203View ArticleGoogle Scholar
 Kanaan RA, Shergill SS, Barker GJ, Catani M, NG VW, Howard R, McGuire PK, Jones DK: Psychiatry Res. 2006,146(1):73–82. 10.1016/j.pscychresns.2005.11.002View ArticleGoogle Scholar
 Behrens TE, Berg HJ, Jbabdi S, Rushworth MF, Woolrich MW: Probabilistic diffusion tractography with multiple fibre orientations: What can we gain? Neuroimage 2007,34(1):144–55. 10.1016/j.neuroimage.2006.09.018View ArticleGoogle Scholar
 Smith SM, Jenkinson M, JohansenBerg H, Rueckert D, Nichols TE, Mackay CE, Watkins KE, Ciccarelli O, Cader MZ, Matthews PM, Behrens TE: Tractbased spatial statistics: voxelwise analysis of multisubject diffusion data. Neuroimage 2006,31(4):1487–505. 10.1016/j.neuroimage.2006.02.024View ArticleGoogle Scholar
 Smith SM, JohansenBerg H, Jenkinson M, Rueckert D, Nichols TE, Miller KL, Robson MD, Jones DK, Klein JC, Bartsch AJ, Behrens TE: Acquisition and voxelwise analysis of multisubject diffusion data with tractbased spatial statistics. Nat Protoc 2007,2(3):499–503. 10.1038/nprot.2007.45View ArticleGoogle Scholar
 JohansenBerg H, Behrens TE: Just pretty pictures? What diffusion tractography can add in clinical neuroscience. Curr Opin Neurol 2006,19(4):379–85. 10.1097/01.wco.0000236618.82086.01View ArticleGoogle Scholar
 Corouge I, Fletcher PT, Joshi S, Gilmore JH, Gerig G: Fibre tractoriented statistics for quantitative diffusion tensor MRI analysis. Med Image Comput Comput Assist Interv Int Conf Med Image Comput Comput Assist Interv 2005,8(Pt 1):131–9.Google Scholar
 Corouge I, Fletcher PT, Joshi S, Gouttard S, Gerig G: Fibre tractoriented statistics for quantitative diffusion tensor MRI analysis. Med Image Anal 2006,10(5):786–98. 10.1016/j.media.2006.07.003View ArticleGoogle Scholar
 Kochunov P, Thompson PM, Lancaster JL, Bartzokis G, Smith S, Coyle T, Royall DR, Laird A, Fox PT: Relationship between white matter fractional anisotropy and other indices of cerebral health in normal aging: tractbased spatial statistics study of aging. Neuroimage 2007,35(2):478–87. 10.1016/j.neuroimage.2006.12.021View ArticleGoogle Scholar
 Brett M, Johnsrude IS, Owen AM: The problem of functional localization in the human brain. Nat Rev Neurosci 2002,3(3):243–9. 10.1038/nrn756View ArticleGoogle Scholar
 Sullivan EV, Adalsteinsson E, Pfefferbaum A: Selective agerelated degradation of anterior callosal fibre bundles quantified in vivo with fibre tracking. Cerebral Cortex 2006, 16: 1030–1039. 10.1093/cercor/bhj045View ArticleGoogle Scholar
 Kassubek J, Juengling FD, Baumgartner A, Ludolph AC, Sperfeld AD: Different regional brain volume loss in pure and complicated hereditary spastic paraparesis: a voxelbased morphometry study, Amyotrophic Lateral Sclerosis. 2007, in press.Google Scholar
 Fink JK: Hereditary spastic paraplegia. Curr Neurol Neurosci Rep 2006,6(1):65–76. 10.1007/s1191099600111View ArticleGoogle Scholar
 Basser PJ, Jones DK: Diffusiontensor MRI: theory, experimental design and data analysis – a technical review. NMR Biomed 2002, 15: 456–67. 10.1002/nbm.783View ArticleGoogle Scholar
 Shen Y, Larkman DJ, Counsell S, Pu IM, Edwards D, Hajnal JV: Correction of HighOrder Eddy Current Induced Geometric Distortion in DiffusionWeighted EchoPlanar images. Magn Reson Med 2004, 52: 1184–1189. 10.1002/mrm.20267View ArticleGoogle Scholar
 Press WH, Flannery BP, Teukolsky SA, Vetterling WT: Numerical Recipes in C : The Art of Scientific Computing. Cambridge University Press, New York; 1992.Google Scholar
 Morgan PS, Bowtell RW, McIntyre DJ, Worthington BS: Correction of spatial distortion in EPI due to inhomogeneous static magnetic fields using the reversed gradient method. J Magn Reson Imaging 2004, 19: 499–507. 10.1002/jmri.20032View ArticleGoogle Scholar
 Andersson JLR, Skare S, Ashburner J: How to correct susceptibility distortions in spinecho echoplanar images: application to diffusion tensor imaging. Neuroimage 2003, 20: 870–888. 10.1016/S10538119(03)003367View ArticleGoogle Scholar
 Lehmann TM, Gonner C, Spitzer K: Survey: interpolation methods in medical image processing. IEEE Trans Med Imaging 1999,18(11):1049–75. 10.1109/42.816070View ArticleGoogle Scholar
 Mishra A, Lu Y, Meng J, Anderson AW, Ding Z: Unified framework for anisotropic interpolation and smoothing of diffusion tensor images. Neuroimage 2006,31(4):1525–35. 10.1016/j.neuroimage.2006.02.031View ArticleGoogle Scholar
 Mishra A, Lu Y, Choe AS, Aldroubi A, Gore JC, Anderson AW, Ding Z: An imageprocessing toolset for diffusion tensor tractography. Magn Reson Imaging 2007,25(3):365–76. 10.1016/j.mri.2006.10.006View ArticleGoogle Scholar
 Talairach J, Tournoux P: Coplanar stereotaxic atlas of the human brain. New York: Thieme Medical; 1988.Google Scholar
 Collins DL, Neelin P, Peters TM, Evans AC: Automatic 3D intersubject registration of MR volumetric data in standardized Talairach space. J Comput Assist Tomogr 1994, 18: 192–205. 10.1097/0000472819940300000005View ArticleGoogle Scholar
 Friston KJ, Ashburner J, Frith CD, Poline JB, Heather JD, Frackowiak RSJ: Spatial registration and normalization of images. Human Brain Mapp 1995, 2: 165–189. 10.1002/hbm.460030303View ArticleGoogle Scholar
 Müller HP, Unrath A, Ludolph AC, Kassubek J: Preservation of Diffusion Tensor Properties during Spatial Normalization by use of Tensor Imaging and Fibre Tracking on a Normal Brain Database. Phys Med Biol 2007, 52: N99–109. 10.1088/00319155/52/6/N01View ArticleGoogle Scholar
 Alexander DC, Pierpaoli C, Basser PJ, Gee JC: Spatial transformations of diffusion tensor magnetic resonance images. IEEE Trans Med Imaging 2001,20(11):1131–1139. 10.1109/42.963816View ArticleGoogle Scholar
 Mori S, Cain B, Chacko VP, van Zijl PCM: Three dimensional tracking of axonal projections in the brain by magnetic resonance imaging. Ann Neurol 1999, 45: 265–269. Publisher Full Text 10.1002/15318249(199902)45:2<265::AIDANA21>3.0.CO;23View ArticleGoogle Scholar
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