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The usual vector derivative constructs (∇, ∇, ∇×) in terms of tensor differentiation, to put dyads (e.g., ∇v) into proper context, to understand how to derive certain identities involving tensors, and finally, the true test, how to program a realistic viscous tensor to endow a fluid. Scalars, Vectors and Tensors. A scalar is a physical quantity that it represented by a dimensional num- ber at a particular point in space and time. Examples are hydrostatic pres- sure and temperature. A vector is a bookkeeping tool to keep track of two pieces of information (typically magnitude and direction) for a physical quantity.
Choose Language.AbstractDiffusion tensor imaging (DTI) techniques provide information on the microstructural processes of the cerebral white matter (WM) in vivo. The present applications are designed to investigate differences of WM involvement patterns in different brain diseases, especially neurodegenerative disorders, by use of different DTI analyses in comparison with matched controls.DTI data analysis is performed in a variate fashion, i.e. Voxelwise comparison of regional diffusion direction-based metrics such as fractional anisotropy (FA), together with fiber tracking (FT) accompanied by tractwise fractional anisotropy statistics (TFAS) at the group level in order to identify differences in FA along WM structures, aiming at the definition of regional patterns of WM alterations at the group level. Transformation into a stereotaxic standard space is a prerequisite for group studies and requires thorough data processing to preserve directional inter-dependencies.
The present applications show optimized technical approaches for this preservation of quantitative and directional information during spatial normalization in data analyses at the group level. On this basis, FT techniques can be applied to group averaged data in order to quantify metrics information as defined by FT. Additionally, application of DTI methods, i.e. Differences in FA-maps after stereotaxic alignment, in a longitudinal analysis at an individual subject basis reveal information about the progression of neurological disorders. Further quality improvement of DTI based results can be obtained during preprocessing by application of a controlled elimination of gradient directions with high noise levels.In summary, DTI is used to define a distinct WM pathoanatomy of different brain diseases by the combination of whole brain-based and tract-based DTI analysis. Diffusion tensor imaging in the human brainThe white matter (WM) tracts in the central nervous system consist of densely packed axons in addition to various types of neuroglia and other small populations of cells.
The axonal membrane as well as the well-aligned protein fibers within an axon restricts water diffusion perpendicular to the fiber orientation, leading to anisotropic water diffusion in brain WM 1. Myelin sheaths around the axons may also contribute to the anisotropy for both intra- and extracellular water 2.The quantitative description of this anisotropy could be detected by diffusion tensor imaging (DTI). DTI produces images of tissues weighted with the local microstructural characteristics of water diffusion. The image-intensities at each position are attenuated, depending on the strength and direction of the so-called magnetic diffusion gradient (represented in the b-value), as well as on the local microstructure in which the water molecules diffuse 3, the diffusion coefficient D, a scalar value:However, in the presence of anisotropy in WM, diffusion can no longer be characterized by a single scalar coefficient, but requires a tensor which in first approximation describes molecular mobility along each direction and correlation between these directions 4. Diffusion anisotropy is mainly caused by the orientation of fiber tracts in WM and is influenced by its micro- and macrostructural features. Of the microstructural features, intraaxonal organization appears to be of greatest influence on diffusion anisotropy, besides the density of fiber and cell packing, degree of myelination, and individual fiber diameter. On a macroscopic scale, the variability in the orientation of all WM tracts in an imaging voxel influences its degree of anisotropy 5.In typical DTI measurements, the voxel dimensions are in the order of millimeters.
Thus, a voxel always contains the averaged information of the water molecules inside the detected volume that usually covers several axons as well as the surrounding water molecules. Despite this multidirectional environment, DTI is sensitive to the orientation of the largest principal axis which aligns to the predominant axonal direction, i.e. The axonal contribution dominates the measured signal 2.DTI provides two types of information about the property of water diffusion: first, the orientation-independent extent of diffusion anisotropy 5 and second, the predominant direction of water diffusion in image voxels, i.e. Analysis Methods: Pre- and PostprocessingThe task of the following protocol is to analyze diffusion properties voxelwise within white matter tracts which could be - due to the voxelwise detection - either isotropic or anisotropic, resulting in prolate or oblate diffusion tensors for the respective voxels. The parameterization of the voxel tensors is used for either the calculation of FA-maps or the identification of fibertracts ( Figure 1).In order to obtain analysis results as shown in the following, use the software package Tensor Imaging and Fiber Tracking (TIFT) 17.
TIFT provides analysis tools for the following requirements:. analysis in terms of DTI metrics, e.g. FA-maps,. stereotaxic normalization. group comparison in terms of FA or other DTI metrics. various analysis approaches of FT. FT on group averaged DTI data and the corresponding statistical analysis.These features allow a variety of analyses in one software environment 17,29,30,31.
The TIFT software is constantly under development for new options in DTI data analysis.Figure 2 gives a schematic overview how to analyze DTI data at the group level after spatial normalization by two complementary approaches, i.e. Both by WBSS and by TFAS to finally obtain differences between subject samples at the group level, e.g. Diseased brains versus healthy controls. Here, WBSS aims at a voxelwise unbiased detection of areas with differences at the group level, whereas TFAS is based upon pre-defined fibertracts; the TFAS starting areas can either be freely chosen or can be derived from the WBSS results (`hotspots` of significantly altered FA).Individual longitudinal comparison of FA-maps is performed by detecting differences in FA-maps of measurements at different timepoints after affine stereotaxic alignment ( Figure 2). Quality check (QC) including correction for corrupted gradient directionsIn case of motion disturbances during the acquisition, i.e. In case of corrupted volumes, an SNR increase is obtained by omitting single gradient directions (GD) for tensor calculation.
For that purpose, a quality check (QC) algorithm 32 was developed. In brief, for scans that contained corrupted volumes, an SNR increase is achieved by omitting single gradient directions one at a time before tensor estimation: for each GD, the weighted variance is computed from all remaining directions in the sequence by weighting with the angle in which they differed from the index GD. Perform an artifact correction by detecting GD with at least one slice showing decreased intensity, i.e. Motion artifacts caused by spontaneous subject movement ( Figure 3, upper panel). For any diffusion weighted volume, compute the mean intensity for each slice and compare its intensity with the same slice in all other volumes by using a weighted average approach - the weighting factor is the dot product of vectors of two GD:denotes the arithmetic average intensity of the slice under observation and a slice for comparison.
The relative average intensity deviation is weighted by the dot product of the GD. Thus, in order to define a global parameter:reflects the minimum of slicewise comparisons of all slices. If Q is under a certain threshold (in the example, a threshold of 0.8 is used for this purpose), eliminate that whole volume, or GD.
A threshold of 0.8 is considered a stable solution 32. Figure 3 illustrates motion artifacts visible in sagittal reconstructions and detected by the QC algorithm. In this example, out of the total number of GD (blue dots in Figure 3c), 17 were below the red line which corresponds to Q = 0.8 and should be eliminated. An example of a volume elimination statistics for a whole study is presented in Figure 3d. In this exemplary study, DTI data of 29 presymptomatic HD subjects were compared to DTI data of 30 controls. Further details of this algorithm are presented in 32, 33. Preprocessing and spatial normalization.
Perform the correction of eddy current-induced geometric distortions of the echo-planar imaging data sets by the method proposed by 34. For the stereotaxic normalization, create a study-specific (b = 0) - template and an FA-template as previously described 17,28,31. Basically, a complete non-linear stereotaxic normalization consists of three deformation components. Consequently, the resulting diffusion tensor of each voxel i has to be rotated according to all the rotations listed above ( Figure 4):. Figure 4a shows a rigid brain transformation to align the basic coordinate frames.
The rotation resulting from the aligning to the basic coordinate frame has to be applied. Figure 4b shows a linear deformation according to landmarks. The components of the Eigenvectors have to be adapted according to the six normalization parameters of S (dependent on the brain region s a, a=1.6) of the linear deformation.v w,j'=s av w,j'w=1,2,3 and j=x,y,z.
Figure 4c shows a non-linear normalization equalizing non-linear brain shape differences. The 3-D vector shifts are different for each voxel leading to a separate transformation for each voxel of the 3-D voxel array ). Standard trigonometry gives a rotation matrix independently for each voxel, resulting from the 3-D vector shifts following the concepts of 16 in order to preserve the directional relations between Eigenvectors of neighbored voxels. Thus, different shifts of two neighbored voxels result in rotations of the corresponding Eigenvectors. Use the dilation matrices for the alignment of the tensor of each voxel to the surrounding voxels.are the components ofThe whole normalization process is iterative, i.e.
Create a scanner- and sequence-specific (b = 0) - template for this study in the first step by arithmetically averaging the (b = 0) - volumes of all subjects after linear transformation according to manually set landmarks. After this first normalization, create improved templates in order to optimize the normalization matrices. The following steps 1.2.3 up to 1.2.5 are schematically visualized in Figure 5a.
After this individual normalization procedure (step (i) - DTI-data I 0), use all individual DTI data sets for creating a study-specific (b = 0) - template and an FA-template (step (ii) - templates T 1). QC and correction for corrupted gradient directions in application to data of patients with hyperkinetic disordersAs an example for the effect of the application of QC and subsequent volume exclusion (as a consequence from the correction for corrupted GD), Figure 8 shows differences in whole brain based spatial statistics with and without volume exclusion for group comparison of 29 premanifest Huntington's disease subjects vs.
30 age and gender matched controls. The scanning protocol was performed on a 1.5 Tesla Magnetom Symphony (Siemens Medical, Erlangen, Germany). The DTI study protocol was identical for patients and controls and consisted of 72 volumes (40 slices, 96 x 96 pixels, slice thickness 2.3 mm, pixel size 2.3 x 2.3 mm), representing 64 gradient directions (b = 1,000 sec/mm 2) and 8 scans with minimal diffusion weighting (b = 100 sec/mm 2).
The echo time (TE) and repetition time (TR) were 90 msec and 8,000 msec, respectively.2. DTI in xeroderma pigmentosumXeroderma pigmentosum (XP) is a rare autosomal recessive progeroid syndrome where the underlying DNA repair defect plays a central role in the aging process 39,40. A multiparametric MRI approach to characterize the cerebromorphological phenotype was used in seven XP patients of different subtypes in order to assess the macrostructural and microstructural cerebral morphology in comparison to controls 41, including DTI, volumetric measurements, and MR spectroscopy ( 1H MRS).The MRI protocol was acquired on a 1.5 Tesla MR system (Magnetom Symphony, Siemens, Erlangen, Germany), equipped with a standard headcoil. T1 weighted (T1w) scans consisted of 196 slices with a slice thickness of 1.0 mm (256 x 256 pixels, pixel size 1.0 x 1.0 mm). TE and TR were 12 msec and 456 msec, respectively. The DTI study protocol consisted of 13 volumes (45 slices, 128 x 128 pixels, slice thickness 2.2 mm, pixel size 1.5 x 1.5 mm), representing 12 gradient directions and one scan with gradient 0 (b = 0). TE and TR were 93 msec and 8,000 msec, respectively; b was 800 sec/mm 2 and five scans were averaged online by the scanner software in image space.Due to the clinical and demographic heterogeneity of the XP-subjects, the comparison was not performed at the group level but rather in a pairwise manner, i.e.
Each XP-subject was analyzed in comparison to an age- and gender matched control. FA-map comparison was performed for ROIs located in the thalamus, in the upper corticospinal tract, in the internal capsule, and in the corpus callosum. Furthermore, directionality changes were compared pairwise for FT and consecutive TFAS, with starting points in the thalamus. DTI demonstrated significantly reduced WM directionality in all regions investigated, i.e.
The thalamus, the corticospinal tracts and the dorsal corpus callosum, with volume and directionality reductions of the fiber projections involving both the craniocaudal fibers and the interhemispheric connections ( Figure 9). These findings, although heterogeneous among the study sample, could be correlated with the clinico-neurological symptoms. The imaging findings support the position that myelin structures degrade prematurely in the brain of XP patients, as discussed in 41.DTI in neurodegeneration (motor neuron diseases)Morphological changes in amyotrophic lateral sclerosis (ALS) patients by structural MRI analysis 42,43 as well as sensorimotor functional connectivity changes in ALS patients 21 have been reported recently. In this work, as an example for application of the analysis methods WBSS and TFAS, twenty ALS patients were investigated by multiparametric MRI.
Severity of physical symptoms as measured with the revised ALS functional rating scale (ALS-FRS-R) was in the range of mild to moderate (35.9 ± 8.0), and none of the patients showed neuropsychological signs of frontotemporal dementia. As control group twenty age- and gender matched healthy controls were scanned.The scanning protocol was performed on a 1.5 Tesla Magnetom Symphony (Siemens Medical, Erlangen, Germany). The DTI study protocol was identical for patients and controls and consisted of 13 volumes (45 slices, 128 x 128 pixels, slice thickness 2.2 mm, pixel size 1.5 x 1.5 mm), representing 12 gradient directions and one scan with gradient 0 (b = 0).
This book combines relativity, astrophysics, and cosmology in a single volume, providing an introduction to each subject that enables students to understand more detailed treatises as well as the current literature. The section on general relativity gives the case for a curved space-time, presents the mathematical background (tensor calculus, Riemannian geometry), discusses the Einstein equation and its solutions (including black holes, Penrose processes, and similar topics), and considers the energy-momentum tensor for various solutions. The next section on relativistic astrophysics discusses stellar contraction and collapse, neutron stars and their equations of state, black holes, and accretion onto collapsed objects. Lastly, the section on cosmology discusses various cosmological models, observational tests, and scenarios for the early universe. Clearly combines relativity, astrophysics, and cosmology in a single volume so students can understand more detailed treatises and current literature.
Extensive introductions to each section are followed by relevant examples and numerous exercises. Provides an easy-to-understand approach to this advanced field of mathematics and modern physics by providing highly detailed derivations of all equations and results.