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1 Laboratory of Neuro Imaging, Department of Neurology, Division of Brain Mapping and
2 Alzheimer's Disease Center, UCLA School of Medicine, Los Angeles, CA, USA
Abstract |
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Top Abstract Introduction Materials and Methods Results Discussion Notes Appendix References |
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Introduction |
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Top Abstract Introduction Materials and Methods Results Discussion Notes Appendix References |
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We use a new computational strategy to compute gyral pattern variations across subjects. This approach is central to the study. It was used, rather than a more conventional stereotaxic approach, so that profiles of gray matter loss could be related to the gyral anatomy of each individual and not be confounded by the spatial variability within a stereotaxic template. Measures of gray matter made in each subject could then be averaged across regions of cortex that corresponded anatomically, rather than just stereotaxically. The resulting maps allow measures of gray matter loss to be plotted relative to the gyral anatomy of the cortex. Profiles of gray matter loss (and the spatial variability of the cortical pattern) can then be plotted on average models of the cortex for each group. In these average maps gyral features are well resolved and appear in their mean spatial locations. In this way, profiles of early gray matter loss are mathematically separated from underlying differences in cortical patterns and registration mismatch. Group differences are then displayed visually, using color coded 3D maps.
Hypotheses
We hypothesized that population-based averaging of anatomy would reveal a region of earliest tissue loss in the left temporal and parietal cortices, with a comparative sparing of the sensorimotor and occipital cortices. We also predicted that the temporoparietal cortices, specifically the left perisylvian language regions, would exhibit the greatest spatial variability, making it difficult to resolve these early structural changes without specialized approaches to control for such high anatomical variance (Thompson et al., 1997). These new approaches were also designed to allow mapping of cortical asymmetries important in evaluating evidence for the asymmetrical progression of the disease. In this way, true differences in gray matter distribution and cortical metabolism can be distinguished from individual or hemispheric differences in cortical organization.
Materials and Methods |
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Top Abstract Introduction Materials and Methods Results Discussion Notes Appendix References |
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Diagnosis
The 26 patients met the National Institute of Neurological and Communicative Disorders and Stroke/Alzheimer's Disease and Related Disorders Association (NINCDS-ARDRA) criteria for AD (McKhann et al., 1984). In addition, the patients had an acquired persistent decline involving at least three of the following domains: language, memory, visuospatial skills, cognition, emotion or personality (Cummings et al., 1980). Exclusion criteria for all subjects were the presence of a focal lesion on brain MRI, history of head trauma, past psychiatric history or an active medical problem.
Demographics
Patients were matched for age (75.8 ± 1.7 years, 14 females/12 males), educational level (15.2 ± 0.4 years), disease severity and handedness (all right-handed). Their mean Mini-Mental State Exam score of 20.0 ± 0.9 (maximum score 30) (Folstein et al., 1975) was carefully matched across the patient cohort to reflect the mild AD population typically presenting initially to a clinic (Murphy et al., 1993). The 20 elderly control subjects were matched with the patients for age, sex, educational level and handedness (mean age 72.4 ± 1.3 years, 8 females/12 males, mean educational level 15.4 ± 0.5 years, all right-handed).
Image Alignment and Pre-processing
3D Image Alignment
Imaging data were first aligned to a standard anatomical image template, specially constructed to reflect the average morphology of an elderly population. The construction of this template has been described in detail (Thompson et al., 2000a). It forms the core of a growing disease-specific atlas of the brain in AD (Thompson et al., 2000a,b,c; Mega et al., 1997, 1998, 2000a,b). Briefly, the average MRI brain template (Fig. 1c) was constructed to have the average shape and size for a group of elderly subjects. Specialized approaches for anatomical averaging were used to generate an average MRI scan with well-resolved cortical features in their mean spatial locations. The resulting template reflects the average morphology of an elderly group of subjects and better reflects the anatomy of the subjects in this study than imaging templates based on young normals (Evans et al., 1994) (Fig. 1a) or post-mortem data (Talairach and Tournoux, 1988).
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RF Inhomogeneity Correction
MRI volumes were corrected for potential non-uniformities in MR signal intensity due to field inhomogeneities in the scanner. An automated algorithm (Zijdenbos and Dawant, 1994; Sled et al., 1998) derived a 3D map of these low frequency signal fluctuations and divided by this scalar field to correct for any errors at the voxel level.
Tissue Classification and Mapping
For all 46 subjects maps of gray matter, white matter and cerebrospinal fluid (CSF) were generated, so that regional differences in gray matter could be identified. Briefly, samples of each tissue class were interactively tagged to compute the parameters of a Gaussian mixture distribution that reflects statistical variability in the intensity of each tissue type (Sled et al., 1998). A nearest neighbor tissue classifier then assigned each image voxel to a particular tissue class (gray, white or CSF) or to a background class (representing extracerebral voxels in the image). The inter/intra-rater reliability of this protocol and its robustness to changes in image acquisition parameters have been described previously (Sowell et al., 1999a,b). Gray matter maps were retained for subsequent analysis.
Cortical Surface Extraction
Next, a high resolution shape representation of the cortex was automatically extracted for each subject, as described previously (Thompson et al., 1997). This algorithm successively deforms a spherical surface into the configuration of a given subject's cortex, resolving the gyral pattern. The ability of the 3D active surface extraction algorithm (MacDonald et al., 1993, 1994, 1998) to extract high fidelity surface representations of the CSF/gray matter and gray/white matter interfaces in high resolution MR data has been extensively tested and validated in prior studies (Holmes et al., 1996; Thompson et al., 1997, 1998; MacDonald, 1998). The resulting cortical model consists of a mesh of discrete triangular elements that tile the surface. The intensity value at which the gray matter/CSF interface occurred was determined in 2030 cortical regions and the threshold set to the mean. This threshold was independently validated in each case by comparing the automatically extracted surface boundaries with the manually determined surface of the gray matterCSF boundary, identified at high magnification in the corresponding 3D MR volumes.
Gyral Pattern Modeling
To determine the patterns of variability for individual regions of cortex, 36 additional cortical structures per brain were traced in all 46 subjects. These 36 major external fissures and sulci in the brain (Table 1) were manually outlined on a highly magnified surface-rendered image of each cortex. Priority was given to biological features whose topological consistency has been demonstrated across normal populations (Ono et al., 1990; Le Goualher et al., 1996; MacDonald et al., 1997). Detailed anatomical criteria were applied as previously set out (Steinmetz et al., 1989, 1990; Missir et al., 1989; Leonard, 1996; Thompson and Toga, 1997; Thompson et al., 1997; Kennedy et al., 1998) and as in a sulcal atlas (Ono et al., 1990). In both hemispheres 3D curves were drawn to represent the superior and inferior frontal, central, post-central, intraparietal, superior and inferior temporal, collateral, olfactory and occipito-temporal sulci, as well as the Sylvian fissures. Additional 3D curves were drawn to represent gyral limits at the interhemispheric margin1 (Thompson et al., 1997). Stereotaxic locations of contour points derived from the data volume were re-digitized to produce 36 uniformly parameterized cortical contours per brain, representing the primary gyral pattern of each subject (Thompson et al., 1997; Thompson and Toga, 1998).
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Computer Platform
All algorithms were written in C and executed on Silicon Graphics O2 R10000 workstations running IRIX 6.5, except for the algorithms for cortical extraction and matching, which were parallelized and executed on a networked cluster of 14 workstations and a Silicon Graphics RealityMonster with 32 internal processors.
Results |
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Top Abstract Introduction Materials and Methods Results Discussion Notes Appendix References |
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A pervasive left greater than right hemisphere reduction in gray matter was found (with up to 2030% loss locally; see Fig. 7), consistent with the suggestion from metabolic studies (Loewenstein et al., 1989) that the left hemisphere is, on average, more severely affected at this stage of the disease. The occipital cortices were comparatively spared bilaterally, as were the sensorimotor cortices (05% loss, P > 0.05). There was also severe gray matter loss (2030%, P < 0.0010.0001) in the middle frontal gyrus, in the vicinity of areas 9 and 46 (Rajkowska and Goldman-Rakic, 1995). We further investigated whether the regions of more significant gray matter loss reflected a correspondingly greater average reduction in the local gray matter index (Fig. 7). This was important, as a greater significance value can result either from (i) a genuinely greater percent reduction in the mean gray matter in AD or (ii) a local reduction in the variance of the gray matter index across the group, which translates into a greater detection sensitivity. Interestingly, a map of the percentage reduction in average gray matter (Fig. 7) followed approximately the same anatomical pattern, suggesting that there is indeed a hierarchy in the severity of gray matter loss at this stage of the disease, rather than a fluctuation in the local power of the statistical model to detect it. Again, the temporal and temporo-parietal cortex exhibited severe (1030%) reductions in gray matter. This contrasted with a comparative sparing of the superior margins of the central and post-central gyri and occipital poles (05% loss). Although diffuse gray matter loss is likely to occur across the majority of the cortex, it is interesting that the superior central and post-central gyri and occipital poles show very little reduction in gray matter when adjacent posterior temporal cortex and the parietal operculum are severely affected, in both the percentage loss and statistical anatomical maps.
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Tensor Maps Reveal Directional Biases in Cortical Variability
For each region of cortex clear directional biases emerged in the principal directions of gyral pattern variability (Fig. 9a,b). Gyral patterns did not vary equally in all directions and the statistical distribution that describes the location of a cortical region in space was elongated in a particular direction, which also varied locally across the cortex. To visualize this, cortical variations were modeled as vector field displacements of an average cortical model and ellipsoids of constant probability density were computed for positions of cortical regions (relative to the average cortex). Figure 9c shows the shape of a 3D Gaussian distribution fitted at each point on the average normal cortex, reflecting the cross-subject variation of points from equivalent gyral regions. The shape of this distribution at each cortical point is described by the covariance tensor of the 3D distribution. Its value determines a set of nested ellipsoids that represent confidence limits for the locations of corresponding anatomical points in stereotaxic space (Thompson et al., 1997; Cao and Worsley, 2000). These ellipsoids (Fig. 9c) are colored by the determinant of the covariance tensor, for which larger values (pink) represent greater 3D variability and small values (blue) represent regions whose morphology is highly conserved across subjects.
Anatomical variations in the temporo-parietal regions displayed the greatest anisotropy, with a strong tendency to vary in a plane oriented upwards at a 45° angle to the horizontal plane (see Fig. 9). In several cortical regions the principal directions of variability (along which the glyphs are elongated in Fig. 9) were approximately orthogonal to the primary gyral pattern. This directional trend was similar in some respects to the torquing, or petalia, which causes cortical regions in the right hemisphere to be situated slightly anterior to their counterparts on the left (Galaburda and Geschwind, 1981; Bilder et al., 1994). The region of highly anisotropic variability was strongly localized to the temporo-parietal cortex and did not extend anteriorly into the post-central and central gyri. A marked anatomical division occurred at the post-central gyrus, where variability was reduced and was spatially more isotropic. The component of variability normal to the average cortex was greatest at the temporal poles, where gyral patterns are relatively stable and variations in temporal lobe size may dominate. Importantly, this directional cortical variability is controlled by surface matching within the continuum-mechanical atlas, thereby allowing accurate maps of disease-related gray matter loss to be constructed (Figs 5,7).
Cortical Pattern Asymmetry
Figure 10 illustrates the group average patterns of cortical asymmetry, highlighting regional trends. In a previous study we found Sylvian fissure asymmetry to be significantly greater in AD (P < 0.05) than in controls matched for age, gender and handedness (Thompson et al., 1998). Although these asymmetries are not apparent in every individual, a localized region can be clearly defined in which major asymmetrical trends are present (Fig. 10). Severe asymmetry exhibited by the posterior Sylvian fissure (up to 10 mm) contrasted with negligible asymmetry in the frontal, parietal and occipital cortex (12 mm). The group average anatomy (Fig. 10) shows the average Sylvian fissure terminating more posteriorly (P < 0.0002) and oriented more horizontally on the left than the right, corroborating postmortem measurements of the planum temporale (Geschwind and Levitsky, 1968, Witelson and Kigar, 1992; Galaburda, 1995). The average right Sylvian fissure also shows an upward turn at its posterior limit (Fig. 10) and is anterior to the posterior limit on the left (Thompson et al., 1998).
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Discussion |
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Top Abstract Introduction Materials and Methods Results Discussion Notes Appendix References |
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Hemispheric Differences
Interestingly, patterns of greater gray matter loss in the left hemisphere corroborate earlier reports (Loewenstein et al., 1989) of predominant left hemisphere metabolic dysfunction in mild to moderate AD, when cerebral glucose utilization is measured by positron emission tomography (PET). Structural, perfusion and metabolic studies suggest that the left hemisphere may be more susceptible to neuronal loss, instead of the alternative explanation that equivalent neuronal loss may result in greater functional deficits on one side, due to asymmetrical cortical organization. Greatest gray matter loss in the temporo-parietal cortex may underlie the prominent temporal-parietal hypometabolism that is consistently found at this stage of AD, often asymmetrically (Friedland and Luxenberg, 1988; Johnson et al., 1998). Although the focus of this study was to determine patterns of gray matter loss in vivo, immunocytochemical studies have reported between 11 and 50% synaptic loss in the superior temporal and inferior parietal cortices, with a comparative sparing of occipital cortices (cf. Figs 5,7). Relatively greater atrophy is often reported in the temporal lobe relative to overall cerebral volume (Murphy et al., 1993). The early progression of AD pathology into the parietal and frontal association cortices suggests a degeneration of synaptically linked cortical pathways, and this pattern correlates with symptoms of memory impairment, aphasias, apraxias, personality changes and spatial deficits (Roberts et al., 1993). Interestingly, gray matter loss at autopsy is predominantly cortical in Alzheimer's patients under 80 years of age (Hubbard and Anderson, 1981), when volumes of subcortical nuclei are not significantly different between patients and controls (De La Monte, 1989). Nonetheless, atrophy of the amygdala and basal nuclei (Cuénod et al., 1993) may ultimately be followed by alterations in thalamic nuclei (Jernigan et al., 1991), induced perhaps by degeneration of their cortical projection areas.
Profiles of Tissue Loss
While the lateral temporal and parietal cortices exhibit diffuse gray matter loss, some regions of the central and paracentral cortex appear to have several foci of average gray matter loss in territory that is otherwise comparatively spared (Fig. 7). Gray matter loss within a gyrus may be a multifocal process (as, for example, the discrete lesions in vascular dementia) or may occur rather uniformly within individual gyri. Clearly, some features occur at small spatial scales in both the statistical (P value) maps and the average loss maps. This multifocal effect does not appear to be attributable to sampling error in estimating the variance for the gray matter measure, as these variance values are spatially quite homogeneous. Structural and functional features with a spatial scale smaller than a gyrus may begin to be resolved if data from corresponding gyri are better aligned across subjects when averaging features from a population (Thompson et al., 2000a; Zeineh et al., 2000). Conversely, gyral features may be blurred out (cf. Fig. 3) when these correspondences are not taken into account (Evans et al., 1994). We did not hypothesize this multifocal effect in advance, so we did not test for its significance specifically. Longitudinal studies may allow us to better understand the scale and consistency of these localized changes over time and may reveal whether gray matter loss is an inherently diffuse or multifocal process within individual cortical gyri.
Advantages of Gray Matter Maps
Cortical gray matter is lost in AD in a pattern that is temporally stereotyped and, initially, regionally specific. By resolving this pattern across the cortex, a detailed evaluation of degenerative change can be made in living populations. Conventional volumetric analysis of MRI data shows substantial overlap in both lobar volumes and gray matter measures between patients and controls, often because of difficulties in identifying equivalent areas of cortex. Overall structure volumes also display considerable variability. High dimensional registration (i.e. elastic matching) of cortical maps offers a solution to this difficulty, in that local measurements of gray matter can be calibrated against a local measure of tissue variance. Large differences in cortical organization are also readily accommodated.
Cortical Pattern Matching
The goal of the cortical matching procedure is to bring cortical regions into correspondence, so that data from corresponding regions can be averaged together across subjects. Without a procedure to align cortical structures, such as the one described in this paper, an averaging procedure applied voxel-by-voxel in stereotaxic space does not always average data from the same region of cortex and, in principle, data from the temporal cortex of some subjects could be averaged with data from the frontal cortex of other subjects. The gyral matching procedure alleviates this problem to a degree, although it does not solve it completely. Gyral matching does not guarantee that data from corresponding cytoarchitectonic regions will be averaged together. However, many functional regions of the cortex defined by PET and functional MRI (Watson et al., 1993), as well as many cyto-architectonic regions (Rademacher et al., 1993), bear a consistent relationship to macroanatomical landmarks of the gyral pattern. The degree to which cortical pattern matching reduces architectonic and functional variation can be evaluated by quantifying residual variability of functional or cellular landmarks after normalizing gross anatomical features (Rajkowska and Goldman-Rakic, 1995; Van Essen and Drury, 1997; Fox et al., 1999; Geyer et al., 2000). Differences in the topological layout of architectonic regions within the cortical sheet ultimately preclude the mapping of discrete cortical regions from one subject to another, so an important intermediate goal has been to identify and match a comprehensive network of sulcal and gyral elements which are consistent in their incidence and topology across subjects (Ono et al., 1990; Rademacher et al., 1993; Thompson et al., 1996a, 1997). While gyral matching substantially reduces the variability in cortical organization across subjects, in the future functional and architectonic landmarks may be definable in vivo that better guarantee matching of the cortical mantle from one subject to another in population studies (Dumoulin et al., 2000).
At this stage, the pathological burden of AD may be greater in terms of functional deficits, and synaptic loss, in the heteromodal cortex than in the idiotypic cortex. In our prior studies AD patients exhibited significantly greater asymmetry and structural variability in the deep perisylvian cortex, relative to controls matched for age, gender, educational level and handedness (P < 0.05) (Thompson et al., 1998). Clear differences in both AD cortical variation and gray matter distribution suggest the need for disease-specific brain atlases that better reflect the disease-related anatomy of patients and calibrate individual loss against statistical data from normative populations.
Emerging Patterns
In both groups anatomical features emerged that are not observed in individual representations due to their considerable variability. As shown in Figure 10, the marked anatomical asymmetry in the posterior perisylvian cortex (Geschwind and Levitsky, 1968) extends rostrally into the post-central cortex. The posterior bank of the post-central gyrus is thrust forward by 89 mm on the right compared with the left (Fig. 10). This asymmetry extends caudally across the lateral convexity into the superior and inferior temporal cortex. As shown by averaging models of ventricular anatomy (Thompson et al., 2000d), this asymmetrical trend penetrates subcortically into the occipital horns of the lateral ventricles, but not into adjacent parieto-occipital and calcarine cortex (Thompson et al., 1998). In contrast with existing brain atlases based on a single brain hemisphere (Talairach and Tournoux, 1988), population-based atlases encode information on asymmetry and its group variation, so that departures from normal patterns in individuals or groups can be identified (Thompson et al., 1997; Thirion et al., 1998; Thompson and Toga, 1998; Cao and Worsley, 2000). There is a substantial literature on Sylvian fissure cortical surface asymmetries (Eberstaller, 1884; Cunningham, 1892; Geschwind and Levitsky, 1968; Davidson and Hugdahl, 1994) and their relation to functional lateralization (Strauss et al., 1983), handedness (Witelson and Kigar, 1992), language function (Davidson and Hugdahl, 1994), asymmetries of associated cytoarchitectonic fields (Galaburda and Geschwind, 1981) and their thalamic projection areas (Eidelberg and Galaburda, 1982), However, no prior reports have mapped the asymmetry profile across the cortex in three dimensions. These localized patterns of asymmetry in cortical morphology clearly have multiple determinants. We previously found Sylvian fissure asymmetry to be significantly greater in AD patients than in controls matched for age, gender, educational level and handedness (P < 0.05) (Thompson et al., 1998), suggesting that AD pathology asymmetrically disrupts the anatomy of the temporo-parietal cortex. The improved ability to localize asymmetries of cortical organization or tissue loss in a group atlas presents opportunities to analyze diseases with asymmetrical progression, including different stages of AD, and to map hypothesized alterations in cortical and hippocampal asymmetry in disease states such as schizophrenia (Falkai et al., 1992; Kikinis et al., 1994; Kulynych et al., 1996; Csernansky et al., 1998).
Population-based Brain Templates
From a practical standpoint, approaches for anatomical averaging also provide an average anatomical image template to represent a particular clinical group. In contrast to earlier studies, we matched cortical patterns across subjects to resolve fundamental anatomical features across a group. Similar approaches are under active development to create average brain representations for the macaque (Grenander and Miller, 1998) and for individual structures such as the corpus callosum (Gee et al., 1995; Davatzikos, 1996), central sulcus (Manceaux-Demiau et al., 1998), cingulate and paracingulate sulci (Paus et al., 1996), hippocampus (Haller et al., 1997; Csernansky et al., 1998; Joshi et al., 1998) and for transformed representations of the human and macaque cortex (Drury and Van Essen, 1997; Grenander and Miller, 1998; Fischl et al., 1999). The resulting averages provide templates in which multimodality brain maps can be integrated (Mazziotta et al., 1995; Toga and Thompson, 1998). The probabil-istic information they contain can also guide Bayesian approaches for automatically identifying anatomical structures (Gee et al., 1995; Mangin et al., 1995; Royackkers et al., 1996; Pitiot et al., 2000). Finally, these probabilistic atlases can constrain the search space for activations in functional imaging experiments (Dinov et al., 2000).
A group-specific atlas of the brain in early AD enables functional, metabolic and tissue distribution data to be analyzed in an anatomical framework that reflects AD morphology. The effects of morphological variation can also be controlled. However, the strategy described here is applicable, in principle, to any population. Since AD is a progressive disease, a homogeneous patient group was selected for this study, matched for age and educational level, at a stage in the disease when patients often present for initial evaluation and where MR, PET and SPECT scans may have maximal diagnostic value. By expanding the underlying patient database and stratifying the population according to different criteria, atlases to represent the more advanced stages of AD, or other clinically defined groups, could also be developed.
Longitudinal Studies
Longitudinal studies, in which a cohort of subjects is scanned repeatedly over time, show considerable promise in tracking the dynamics of normal aging and dementia. The mean rate of brain atrophy in AD, based on MRI measures of total cerebral volumes, was recently reported to be 2.4 ± 1.1% per year in AD, compared with 0.4 ± 0.5% per year in matched elderly controls (MMSE 19.6 ± 4.1 and 29.2 ± 1.0 at baseline, for patients and controls, respectively) (Fox et al., 2000). Higher rates of atrophy and tissue loss have been estimated for specific structures, including the hippocampus (Kaye et al., 1997; Jack et al., 1998; Laakso et al., 2000). Four-dimensional maps of degenerative rates may also be derived by computing a deformation field that elastically transforms a subject's anatomy from its earlier configuration to its shape in a later scan (Fox et al., 1996, 2000; Thompson et al., 2000b,d). We are currently extending the mapping approach described here to store detailed population-based maps of degenerative rates across time and explore linkages between these maps and cognitive variables (Thompson et al., 2000d), as well as therapeutic and genetic factors [e.g. ApoE genotype (Small et al., 2000)].
Accurate mapping of gray matter changes in a living population with AD holds significant promise for genetic, longitudinal and interventional studies of dementia. In any study where staging of the disease is required, the ability to calibrate gray matter integrity against a reference population is paramount. The patient cohort on which our atlas is based is being expanded to accommodate groups at different stages of dementia. By following the same patients longitudinally (Thompson et al., 2000b), statistical maps of gray matter loss at multiple time points will ultimately provide a dynamic frame-work to help understand the progression of the disease and to gauge therapeutic, disease-modifying response in an individual or clinically defined group.
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Address correspondence to Paul Thompson, Room 4238, Reed Neurological Research Center, Laboratory of Neuro Imaging, Department of Neurology, UCLA School of Medicine, 710 Westwood Plaza, Los Angeles, CA 900951769, USA. Email: thompson@loni.ucla.edu
Appendix |
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Top Abstract Introduction Materials and Methods Results Discussion Notes Appendix References |
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(1) |
defined at each mesh node (u,v), where rµ(u,v) is the average surface. The square root of this function gives the standard deviation in stereotaxic position as a 3D r.m.s. distance for each internal surface point. The appropriate numerical value, at each grid point, is given by the root mean square magnitude of the 3D displacement vectors assigned to that point, in the n surface maps from the individual to average. The variability measure is visualized using a color code to illustrate the profile of variability across the anatomy.
Elastic Matching of Gyral Patterns using Flow Fields
Differences in cortical patterns between any pair of subjects were determined by deforming one cortical model to match the other. This procedure has been covered in detail by Thompson and co-workers (Thompson et al., 2000a) and is summarized here for completeness. Since each cortical model is obtained by deforming a spherical surface into the shape of the cortex, gyral features can be mapped back onto a sphere and, subsequently, to a plane (Fig. 2). This simplifies computation of anatomical correspondences. Anatomical correspondences can therefore be computed by defining a flow field in the flat, 2D parameter space that matches gyral features from one subject to another (Figs 2, 3) (Davatzikos et al., 1996; Thompson et al., 1996a, 1997, 2000d; Fischl et al., 1999). The flow is given by the solution to a curve-driven warp in the flat parametric space of the cortex (Thompson et al., 1996, 1998, 2000). The flow behavior is modeled using equations derived from continuum mechanics and these equations are governed by the CauchyNavier differential operator L = µ2 + ( + µ)(T) (Davatzikos et al., 1996; Thompson et al., 1996, 1998, 2000d; Grenander and Miller, 1998).
Technical Details
Specifically, for points r = (r,s) in the cortical parameter space = [0,2) x [0,), a system of simultaneous partial differential equations can be written for the flow field u (r):
(2) |
Here M0, M1 are sets of points and (sulcal or gyral) curves where displacement vectors u(r) = u0(r) matching the corresponding anatomy across subjects are known. The flow behavior is governed by the CauchyNavier differential operator L = µ2 + ( + µ)(T) with body force F (Thompson et al., 1996, 1998, 2000; Grenander and Miller, 1998). In solving this governing equation matching sulcal networks across subjects, dependencies between the metric tensors of the surface parameterizations and the matching field are eliminated with an approach known as covariant regularization, which uses generalized coordinates and correction terms known as Christoffel symbols (Thompson and Toga, 2000a,d). Because of the intrinsic curvature of the cortex, this means that the covariant form L of the differential operator L is used when solving these equations (Thompson and Toga, 1998; Thompson et al., 2000d). This adjustment also makes sure that cortical surfaces are matched in a way that is actually independent of the way the surfaces are flattened; in other words, the matching procedure is parameterization-invariant. In the partial differential equations (2) we replace L by the covariant differential operator L. In L all L value partial derivatives are replaced by covariant derivatives. These covariant derivatives are defined with respect to the metric tensor of the surface domain where calculations are performed. The covariant derivative of a (contravariant) vector field, ui(x), is defined as ui,k = uj/xk + jik ui, where the Christoffel symbols of the second kind, jik, are computed from derivatives of the metric tensor components gjk(x):
(3) |
These correction terms are then used in the elastic transformation used to match one cortex with another.
Finally, because 3D cortical positions are encoded in color on the flat maps, the surface matching transformation is recovered in 3D as a mapping that drives one cortex onto another. A color code (Fig. 2e) representing 3D cortical point locations in an individual subject is convected along with the flow that drives the sulcal pattern into the average configuration for the group (Fig. 2f). Once this is done in all subjects at a particular location in the flat map (Fig. 2f), points on each individual's cortex are recovered that have the same relative location to the primary folding pattern in all subjects. Averaging of these corresponding points results in a crisp average cortex (Fig. 3, bottom row). The corresponding 3D displacement is recovered between the cortical models. This displacement matches a large network of sulcal features and thus is a valid encoding of gyral pattern differences.
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Footnotes |
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2 Significance levels. If there had been no pre-existing hypothesis on the localization of significant gray matter loss, which was expected in the temporal and temporo-parietal cortex, a correction for multiple comparisons can be made. The significance threshold can be set at a level derived from the effective number of resolution elements in the statistical field (RESELs) (Worsley, 1994). This corrected P value depends on the smoothness tensor of the residuals of the statistical model, which can also be estimated from the surface data, using an approach known as statistical flattening (Worsley et al., 1999; Thompson et al., 2000d).
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