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Hum Brain Mapp. Author manuscript; available in PMC 2010 November 8.

Published in final edited form as:

PMCID: PMC2975704

NIHMSID: NIHMS236357

Xue Hua,^{1} Alex D. Leow,^{1,}^{2} Jennifer G. Levitt,^{3} Rochelle Caplan,^{3} Paul M. Thompson,^{1} and Arthur W. Toga^{1,}^{*}

The publisher's final edited version of this article is available at Hum Brain Mapp

See other articles in PMC that cite the published article.

Previous magnetic resonance imaging (MRI)-based volumetric studies have shown age-related increases in the volume of total white matter and decreases in the volume of total gray matter of normal children. Recent adaptations of image analysis strategies enable the detection of human brain growth with improved spatial resolution. In this article, we further explore the spatio-temporal complexity of adolescent brain maturation with tensor-based morphometry. By utilizing a novel non-linear elastic intensity-based registration algorithm on the serial structural MRI scans of 13 healthy children, individual Jacobian growth maps are generated and then registered to a common anatomical space. Statistical analyses reveal significant tissue growth in cerebral white matter, contrasted with gray matter loss in parietal, temporal, and occipital lobe. In addition, a linear regression with age and gender suggests a slowing down of the growth rate in regions with the greatest white matter growth. We demonstrate that a tensor-based Jacobian map is a sensitive and reliable method to detect regional tissue changes during development.

Advancements in magnetic resonance imaging (MRI) technology and image analysis methods make the detailed characterization of human brain development during childhood and adolescence *in vivo* possible. The maturation of a normal brain represents a complex and dynamic developmental process. Detailed spatial and temporal mapping of structural changes during this process aids in understanding the basis of age-related cognitive advancement and various brain developmental disorders.

Cross sectional neuroimaging studies have consistently demonstrated reduction in overall cerebral gray matter volume and growth in white matter with relatively stable total brain volume from ages 4 to 20 years [Caviness et al., 1996; Giedd et al., 1996; Jernigan et al., 1991; Pfefferbaum et al., 1994; Reiss et al., 1996; Sowell et al., 1999a]. Fewer studies have used longitudinal data to examine the dynamic patterns of brain growth between childhood and adolescence. Giedd et al. [1999] demonstrated a linear increase in white matter and non-linear changes in gray matter volume in a large-scale longitudinal pediatric neuroimaging study. Thompson et al. [2000] applied a tensor mapping technique to a group of children repeatedly scanned between the ages of 3 and 15 years and found a rostro-caudal wave of growth at the corpus callosum in the developing human brain. Sowell et al. [2004] described correlated changes in cortical thickness and regional brain size in right frontal and bilateral parieto-occipital regions in children scanned twice between the ages of 5 and 11 years. Because the longitudinal method follows the same brain as it matures, it addresses the issue of inter-individual variations in brain structures. As such, it provides a more sensitive and finer toned measurement of individual growth patterns than cross-sectional studies [Chung et al., 2001; Giedd et al., 1999; Hand and Crowder, 1996; Sowell et al., 2004; Thompson et al., 2000].

Studies using MRI-based volumetry have provided a wealth of information for age-related changes in brain morphometry [Giedd et al., 1996; Reiss et al., 1996]. However, traditional MRI-based volumetric methods demand manual or semi-automatic image segmentations of the equivalent regions across multiple brains. Measurements obtained using these methods are usually confined to brain regions that can be clearly defined. More recently, tensor-based morphometry (TBM) has been introduced as a method to identify regional structural differences from the gradients of deformation fields (reviewed in Ashburner and Friston [2003]). A deformation field, which stores the relative positions of different brain structures, is first obtained by spatially normalizing the MRI images of one brain to another using non-linear registration. To characterize local shape differences, a tensor field, also known as a Jacobian matrix, is then calculated from the gradient of the deformation field. A determinant is taken at each point of the Jacobian matrix field to represent the volume of the unit-cube after the deformation. Jacobian determinants are usually coded by colors to indicate local volume loss or gain [Ashburner and Friston, 2003; Chung et al., 2001; Freeborough and Fox, 1998; Riddle et al., 2004; Toga, 1999]. When the TBM method is applied to longitudinal data, the deformation field is obtained by registering the MRI scans of the same subject acquired at different times. Therefore, the Jacobian map faithfully represents the developmental changes in the brain. Because the Jacobian matrix field is calculated over the entire volume without explicit delineations of regions-of-interest (ROI), the TBM method characterizes the local shape differences with a much higher spatial resolution.

In this article, TBM is used to generate detailed tissue growth maps from the longitudinal scans of 13 healthy children between the ages of 6 and 19 years. By fitting a linear model across all individual Jacobian maps voxel-by-voxel, we also evaluate age and gender-related volume changes in gray and white matter. We create three-dimensional (3D) maps that represent the dynamics of brain development, demonstrating a slowing down of the growth rate over time in regions with the greatest white matter growth.

Thirteen healthy children and adolescents (seven males and six females) aged 6–19 are included in this article (Fig. 1). The mean age for the group was 12.3 ± 2.6 years. Each subject received a series of two high-resolution 3D magnetic resonance imaging (MRI) scans with an average inter-scan interval of 2.85 ± 0.85 years (min = 1.92 years; max = 4.52 years). There was no significant age difference between males (12.3 ± 2.4 years) and females (12.3 ± 2.9 years). All subjects were recruited as part of an ongoing neuro-developmental project at the University of California, Los Angeles (UCLA). The subjects were screened for neurological, psychiatric, language, or hearing disorders by clinical interview, and for developmental history and K-SADS_PL [Kaufman et al., 1997] through interviews with the parents. This study was approved by the UCLA Human Subjects Protection committee and all parents of subjects provided written informed consent for participation.

The sensitivity and reliability of TBM depend on the contrast and stability of the scans. Thus, we followed the standard scanning procedure. All scans were conducted on a GE Signa 1.5 Tesla MRI scanner (GE Medical Systems, Milwaukee, WI). High-resolution 3D MRI scans for each subject were acquired with T1-weighted spoiled GRASS (SPGR) sequences. A sagittal plane imaging acquisition protocol was used with repetition time (TR) of 24 ms, echo time (TE) of 9 ms, and flip angle of 22°, and 2 excitations. The acquisition matrix was 256 × 256 × 124 with slice thickness of 1.2 mm and field of view of 24 cm. The image voxel size was 0.9375 × 0.9375 × 1.2 mm.

Skull and other non-brain tissues were removed using an automated program, Brain Surface Extractor (BSE), from the BrainSuite [Shattuck and Leahy, 2002]. A binary mask was generated to include brain tissues like gray matter, white matter, and cerebrospinal fluid. Non-brain tissues (such as skull, scalp, and orbits) were left out and labeled as zero. All sagittal plane images were manually edited to lower the segmentation error caused by similar image intensities between brain and non-brain tissues. A 3D radio frequency (RF) bias field correction algorithm (N3) was applied to eliminate the field inhomogeneity in MRI images caused by non-uniformities in the RF receiver coils [Sled et al., 1998]. A validation study has demonstrated the robustness of TBM with scans acquired using SPGR sequence followed by N3 correction [Leow et al., 2005b].

To account for global differences in positioning and size among individual brains, all scans were rigidly aligned to the stereotactic space defined by the International Consortium for Brain Mapping (ICBM) [Mazziotta et al., 2001]. Specifically, we used the ICBM 53 atlas which is an average of 53 T1-weighted MRI scans of young healthy adult brains. To linearly align the scans to ICBM53, a follow-up scan was linearly registered to its baseline scan with 6-parameter (6p) affine transformation, then both scans were registered to ICBM53 using a same 9-parameter (9p) transformation. Mutual information (MI), a measure of the statistical dependence between two distributions, was used as a similarity measure for the above global transformations [Maes et al., 1997]. At the step of 9p linear registration to ICBM53, all MRI images were resampled into an isotropic matrix of 199 voxels in *x*-, *y*-, and *z*-dimensions with each voxel interpolated to the size of 1 mm × 1 mm × 1 mm.

To inspect the quality of linear registrations, we used a 3D visualization tool called REGISTER, which automatically overlays the arbitrary slice geometry of each scan pairs in ICBM53 space [MacDonald, 1993]. All scans had satisfactory results for linear registration without noticeable distortion or mismatch.

Studies have suggested that registration bias can be reduced by using an unbiased group-average template. We therefore constructed a minimal deformation target (MDT) [Good et al., 2001; Kochunov et al., 2002; Kovacevic et al., 2005; Woods et al., 1998]. To compute a MDT, the first step was to create an affine average. All baseline images were registered to ICBM53 using 9-parameter affine transformations. After intensity normalization, the affine average template was created by calculating the mean intensity at each voxel of the 13 scans. The second step was to create a non-linear average. Here, each scan was non-linearly registered to the affine average template using a non-linear inverse consistent elastic intensity-based registration algorithm [Leow et al., 2005a,b; Thompson et al., 2000; Wells et al., 1996]. The deformation field was determined by maximizing the mutual information of the image intensities and minimizing the elastic energy of the deformation. A multi-resolution scheme was used. It started with a Fast Fourier Transform (FFT) resolution of 32 × 32 × 32 and was followed by 64 × 64 × 64, which corresponds to an effective voxel size of 30 mm^{3} (199 mm/64 = 3.1 mm). Convergence of MI was achieved iteratively with 300 iterations at each resolution. We then created the non-linear average brain by voxel-wise averaging the intensities of the 13 images that have been non-linearly registered to the affine average template. Finally, we created the MDT by adjusting the non-linear average with inverse geometric centering of the displacement fields [Kochunov, et al., 2002, 2005]. The MDT was subsequently used as the target for inter-subject registration. Figure 2 displays the axial slice of a randomly chosen individual brain, the group affine average, and the MDT. The MDT reveals less anatomical detail than the individual brain but it has less noise and higher contrast compared to the affine average.

The same non-linear registration algorithm as used in MDT construction was also used here to spatially normalize both intra- and inter-subject shape differences. The deformation field obtained from intra-subject scan pairs *U*_{Intra}* ^{n}* (

Individual Jacobian (growth) map. The annualized Jacobian map, representing the percentage of tissue growth per year, of an 8-year-old boy, is displayed on cortical surface model (**a**), a cortical model with the medial surface exposed (**b**), and white matter **...**

Warp diagram. Individual Jacobian maps are derived from the deformation matrix *U*_{Intra}^{n}, which is obtained by iteratively deforming the follow-up scan (Time 2) to the initial scan (Time 1) of the same subject. Variance between individual brains is explained **...**

To adjust for variable time differences between scans, we normalized the individual Jacobian maps by their inter-scan intervals. As such, each map (percent tissue change) was divided by its corresponding inter-scan interval to create the annualized Jacobian map, which represents the average change over 1 year. All results and statistical analyses are based on the annualized Jacobian maps.

All image processing steps were automated using LONI Pipeline Processing Environment which streamlines the process and allows parallelization of multiple tasks [Rex et al., 2003].

ROI were defined according to the group average in the ICBM53 space. Each brain was hand-traced by a trained anatomist to generate masks that cover frontal, parietal, temporal, and occipital regions separately. The masks were then transformed into the ICBM space and the mathematical average of the masks was used to estimate the probabilities of each region-of-interest. Brain regions were defined based on probabilities that are greater than 70%.

To assess the significance level of the Jacobian maps at each voxel, we applied the bootstrap method [Efron and Tibshirani, 1986, 1993]. Specifically, we generated 1,000 independent bootstrap samples, each of size *n* = 13 with replacements from the 13 individual Jacobian maps. The bootstrap estimate of standard error is the standard deviation of the bootstrap replications [Efron and Tibshirani, 1993]. At each voxel, a one-sample *t*-test was conducted based on the bootstrap sample mean and bootstrap estimate of standard error. Next, permutation tests were implemented to correct for multiple comparisons. Permutation tests are non-parametric in nature, and are used in this study to assess the overall significance of Jacobian maps inside each region-of-interest. In our case, the null hypothesis in consideration is that, for each ROI, there is no tissue growth or loss. Under null hypothesis, the labels of time 1 and time 2 are interchangeable within each subject. In other words, we can thus randomly multiply 1 or −1 to the Jacobian map of each subject, as each Jacobian map is simply noise under null. This procedure is, in essence, the same as what was originally proposed by Nichols [Nichols and Holmes, 2002], where group labels were permuted (in our case, the two group labels are time 1 versus time 2). At each permutation, a label of 1 or −1 was randomly assigned to each of the 13 subjects and voxels with *P* ≤ 0.05 (uncorrected) were identified. After 10,000 randomized tests, a ratio was calculated describing what fraction of time an effect of similar or greater magnitude to the real effect occurs in the random assignments. This ratio serves as an overall estimate of significance for the maps (corrected for multiple comparisons) [Nichols and Holmes, 2002]. Positive growth signals and negative tissue loss were assessed separately in each region-of-interest (frontal, parietal, temporal, and occipital lobes) (Table I). For example, the permutation test of positive Jacobian values rendered a *P*-value of 0.0054 (frontal lobe) (Table I), indicating the overall significance level of tissue expansion in the frontal lobe.

Age and gender effects were tested by modeling the data with linear regression:

$$\text{annual tissue growth rate}=\mathrm{\beta}\cdot (\text{age}-\text{mean age})+\mathrm{\alpha}$$

$$\text{annual tissue growth rate}=\mathrm{\beta}\cdot (\text{age}-\text{mean age})+(\mathrm{\chi}\cdot \text{gender}+\mathrm{\alpha})$$

The annual tissue growth rate was modeled with a constant term α (intercept) which represents the percentage of annual growth at the mean age, and a linear coefficient β (slope) that encodes the change of growth rate. To compare our result with direct modeling of gray matter volume [Giedd et al., 1999], the intercept α corresponds to the slope of the developmental trajectory of gray matter volume. Likewise, the linear coefficient β is comparable to the quadratic term of gray matter regression. Gender effect was explored by adding an additional term χ, which is a gender specific component of α. Here we assumed that males and females share the same slope term β, as developmental curves were found to have similar shapes for both genders [Giedd et al., 1999], therefore the second time derivative of the curves which is also the slope of our linear regression model would be the same. Student's *t*-tests were used to examine the voxel-level significance of the regression coefficients (uncorrected). Permutation tests were used to assess the overall significance of age correlation (ROI—the entire brain). A null distribution was constructed after 10,000 random tests. For each test, the subjects' ages were randomly permuted and voxel-wise regressions were conducted to identify voxels with age correlation coefficient more significant than *P* = 0.05. Similar to the method introduced earlier, a ratio, describing the fraction of time an random effect of similar or greater magnitude to the real effect, was calculated to quantify the overall significance of the age effect (corrected for multiple comparisons) [Chiang et al., 2007; Thompson et al., 2003; Nichols and Holmes, 2002].

To illustrate how this non-linear registration technique provides a detailed depiction of brain development, the individual tissue change map of an 8-year-old boy (initial scan conducted at age 6) is shown in Figure 3. The annualized Jacobian map is superimposed on gray or white matter surface models (generated using the tissue classification tool in BrainSuite [Shattuck and Leahy, 2002]). The annualized growth rate is displayed in color map with the hot and cold color representing local tissue expansion and atrophy respectively. Correspondingly, we visually inspected all individual Jacobian maps and they share similar features to the individual brain reported in Figure 3. However, older subjects generally display fewer changes with weaker growth signals.

Figure 5 displays the annualized mean Jacobian map of the 13 subjects after bootstrap correction. The widespread regions of tissue growth primarily correspond to the anatomical boundaries of cerebral white matter. The volume of fiber tract is also enlarged in several sub-regions of the corpus callosum including genu, isthmus (arrow in Fig. 5), and splenium.

Next, permutation tests were conducted to correct for multiple comparisons. The results provide overall significant levels of tissue expansion or atrophy in each region-of-interest. Ten thousand randomization tests were conducted and rendered significant *P*-values (*P* ≤ 0.05) of tissue expansions in the frontal, parietal, and occipital lobes. Concurrently, permutation tests validated the tissue loss in the parietal, temporal and occipital lobes. As shown in the mean map (Fig. 5), the regions of expansion correspond generally to the white matter, while the majority of atrophies are located in the gray matter (except in the primary sensorimotor areas). Thus, the permutation results of tissue expansions correspond mainly to the white matter growth and atrophies are related largely to the gray matter loss. Refer to Table I for the full results from the permutation tests.

Age effects in growth patterns are studied by fitting a linear regression model voxel-by-voxel over the Jacobian matrix maps, with Jacobian values as dependent variables and ages as explanatory variables. Figure 6 shows the regression coefficients (terms α and β) and the corresponding *P*-values at the horizontal sections. The map of α demonstrates an extensive growth of white matter tract system at a rate of 1–3% annually, at the mean age of 12.3 years. The result from the linear coefficient β suggests a slowing down of the growth rate over time in regions with the greatest white matter growth. An age effect on gray matter shrinkage was also apparent, however, the effect is not large enough to reach statistical significance. We attempted to study the gender effect by adding an additional gender specific component (χ) for the intercept. The map of χ visualizes the spatial difference in growth patterns between the two genders. It is noteworthy that the regional growth at lateral ventricles and caudate nuclei is higher for boys than girls.

Linear regression with age and gender. Panel (**a**) shows the MDT brain anatomy and the slice selections (*z* = 102, 112, or 122). Panel (**b**) displays the result of linear regression with age. Model: annual tissue growth rate = β · (age − **...**

The overall significance of the age effect in the regression was estimated by permutation tests (ROI—the entire brain). The permutation *P* value for the negative correlation between age and Jacobian map is 0.0592. This value indicates the significance level of the trend, that growth rate is slowing down as age increases.

Finally, the mean growth maps at different ages were recovered by plotting the linear model with fitted correlation coefficients (Fig. 7). The top row demonstrates the age effect in growth pattern changes, with strong white matter growth from age 7–9 years and less overall expansion at older ages. The bottom two rows exhibit the gender specific growth maps at different ages. We detected similar, age-related white matter growth, and an uneven growth pattern between males and females.

TBM is a non-linear image registration tool that measures brain changes using serial MRI scans. The Jacobian determinants of the deformation map are used to provide 3D visualization of local tissue growth or atrophy with improved spatial resolution compared to traditional volumetry-based morphometry (VBM). We demonstrate this advantage in the individual study (Fig. 3). Visual inspection suggests a localized white matter growth in the orbito-frontal area. The overall increase in frontal white matter volume is 1%, as calculated by integrating the Jacobian determinants over the region-of-interest. This result is comparable to the large scale longitudinal MRI study [Giedd et al., 1999]. Thus, TBM offers equivalent information to VBM with superior spatial details, making it a favorable method for studying age and disease-related changes in brain morphometry.

Our results from the permutation tests confirm previous developmental MRI studies that have provided evidence for age-related increases in total white matter volume and decreases in total gray matter volume [Caviness et al., 1996; Giedd et al., 1996; Jernigan et al., 1991; Pfefferbaum et al., 1994; Reiss et al., 1996; Sowell et al., 1999a]. Moreover, we provide detailed spatial mapping of the developing brain by utilizing the Jacobian matrix field to encode local volume changes. We have attained high resolution 3D mapping of white matter growth and growth dynamics. The frontiers of the tissue growth correspond well with the anatomical boundaries of brain structures such as cerebral white matter, indicating the sensitivity and specificity of our mapping approach. Global enlargement of white matter is observed across all lobes indicating an overall growth of white matter tract system in the subjects. Every brain was edited carefully by trained raters to include only brain tissues. Nevertheless, there could still be a small amount of registration errors at the edge of the brain due to imperfect segmentation and/or registration. As a result, the brain masks (ROI) used in this study do not include boundary voxels. Therefore, the results of permutation tests provide accurate estimate of the overall significance levels within each region-of-interest.

Postmortem studies have revealed an extended myelination process that continues into the third decade of life, especially for frontal and parietal regions [Yakovlev and Lecours, 1967]. Several recent findings also suggest an accelerated maturation of gray and white matter in frontal lobe during adolescence [Giedd et al., 1999; Sowell et al., 1999b]. In this study, we demonstrate a wide-spread growth in white matter, with the most prominent expansion in the frontal lobe. Inference on white matter growth using TBM, however, is based on changes in tissue boundaries or gray/white contrast, which are related to the underlying tissue microstructure. Nevertheless, it does not offer a direct measurement of the true changes to white matter integrity or myelination. MR spectroscopy [Lopez-Villegas et al., 1996] and quantitative MR relaxometry [House et al., 2006] might provide a better answer to address these questions.

The individual growth map shown in Figure 3 demonstrates cortical gray matter tissue growth over the primary sensorimotor areas. The same growth pattern is revealed again in the group average map (Fig. 5). However, other studies has shown gray matter loss in dorsal parietal and primary sensorimotor regions at this age range [Gogtay et al., 2004; Sowell et al., 2003]. This seemingly discrepant result might be explained by the partial volume effect. At the resolution of FFT 64, the size of each computational voxel is about 30 mm^{3} (199 mm/64 = 3.1 mm). This is a relatively large size that might contain both gray and white matter within the same voxel, especially for the areas where the gray matter is thin. As illustrated in Figure 8, if the voxel holds a large proportion of growing white matter with a small proportion of shrinking gray matter, the overall signal from that voxel will be dominated by the white matter growth.

Partial volume effect. A relatively large computational voxel (outlined by the dotted lines) could contain both gray matter (GM) and white matter (WM). The illustrated voxel holds a large proportion of growing white matter (red) with a small proportion **...**

Our study is based on 13 developing children with two consecutive brain scans taken as they matured in age. The data set is relatively small compared to the few medium-to-large scale neuro-imaging studies published to date [Giedd et al., 1999; Paus et al., 1999; Sowell et al., 2004]. The warp diagram (Fig. 4) is implemented to compensate for this relatively low degree of freedom and large variance among individual brains. The step of inter-subject registration is computationally intensive but it matches the local shape properties among the subjects. This enables regional comparisons and voxel-wise statistical analyses of the growth maps. A larger sample size would further improve the power, and it is necessary to track the gender specific patterns in brain development.

In this study, we map the patterns of normal brain development using TBM. TBM is a highly automated image analysis technique that delineates local tissue gain or loss at a greater spatial resolution than the traditionally used volumetry-based method. In addition to brain development, TBM has a wide spectrum of applications in longitudinal brain analysis, such as understanding how brain is affected by certain psychiatric disorders (e.g. autism, schizophrenia, depression, etc), tracking degenerative disease progression, and monitoring drug treatment effect. In future work, we look to apply TBM to a study with relatively large sample size and expand the statistical models to answer interesting questions like brain asymmetry, correlations with IQ, cognitive measures, as well as genome types.

Information on the National Centers for Biomedical Computing can be obtained from <http://nihroadmap.nih.gov/bioinformatics>. Contents are solely the responsibility of the authors and do not necessarily represent the official views of NCRR or NIH.

Contract grant sponsor: National Institutes of Health [NIH Road-map for Medical Research Grant entitled Center for Computational Biology (CCB)]; Contract grant number: U54 RR021813; Contract grant sponsor: National Center for Research Resources (NCRR), National Institutes of Health (NIH); Contract grant numbers: P41 RR013642, M01 RR000865; Contract grant sponsor: National Alliance for Autism Research; Contract grant number: 01082668; Contract grant sponsor: National Institute of Mental Health; Contract grant numbers: 5K08 MH01385, MH067187; Contract grant sponsor: National Institute of Neurological Disorders and Stroke; Contract grant numbers: 5R01 NS046018, NS32070.

Detailed spatial and temporal mapping of brain structural changes is crucial to understanding the basis of age-related cognitive advancement and various brain disorders. TBM is an image analysis method based on nonlinear registration. It identifies regional structural differences by examining the gradients of the deformation field. In general, it can be applied to detect any structural changes over time, such as during brain development and disease progression. It can also be used to map anatomical differences between different patient groups using cross-sectional data.

We explain here the application of TBM in the context of studying longitudinal brain changes. For each individual, serial MRI scans are acquired and TBM is applied to generate an individual Jacobian growth map, showing 3D profiles of tissue change over time. To formulate the process mathematically, we denote the follow-up scan as the source (*S*) and the baseline scan as the target (*T*). We compute a non-linear displacement vector field *u* = (*u _{x}, u_{y}, u_{z}*) by deforming

$$J=\left(\begin{array}{lll}\frac{\partial (x-{u}_{x})}{\partial x}\hfill & \frac{\partial (y-{u}_{y})}{\partial x}\hfill & \frac{\partial (z-{u}_{z})}{\partial x}\hfill \\ \frac{\partial (x-{u}_{x})}{\partial y}\hfill & \frac{\partial (y-{u}_{y})}{\partial y}\hfill & \frac{\partial (z-{u}_{z})}{\partial y}\hfill \\ \frac{\partial (x-{u}_{x})}{\partial z}\hfill & \frac{\partial (y-{u}_{y})}{\partial z}\hfill & \frac{\partial (z-{u}_{z})}{\partial z}\hfill \end{array}\right)$$

The determinant of the Jacobian matrix is derived from the forward deformation field. It is usually color coded to show regions of volume expansion det *J*(*r*) > 1 or tissue shrinkage det *J*(*r*) < 1 [Ashburner and Friston, 2003; Chung et al., 2001; Freeborough and Fox, 1998; Riddle et al., 2004; Thompson et al., 2000; Toga, 1999]. Once individual Jacobian maps have been created and spatially aligned to a standard coordinate system, statistical parametric maps are typically established to investigate various scientific questions such as localization of group difference, age and gender effect, correlation with disease severity, or complex interaction between effects of interest [Ashburner and Friston, 2003].

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