arthur w. toga and paul m. thompsonusers.loni.ucla.edu/~thompson/pdf/arbe2003.pdf12 jun 2003 14:40...

12 Jun 2003 14:40 AR AR191-BE05-05.tex AR191-BE05-05.sgm LaTeX2e(2002/01/18)P1: IKH10.1146/annurev.bioeng.5.040202.121611

Annu. Rev. Biomed. Eng. 2003. 5:119–45doi: 10.1146/annurev.bioeng.5.040202.121611

Copyright c© 2003 by Annual Reviews. All rights reserved

TEMPORAL DYNAMICS OF BRAIN ANATOMY

Arthur W. Toga and Paul M. ThompsonLaboratory of Neuro Imaging, Department of Neurology, UCLA School of Medicine,Los Angeles, California 90095-1769; email: [email protected]

Key Words brain mapping, 4-D atlas, probabilistic atlas, computational anatomy

■ Abstract The brain changes profoundly in structure and function during develop-ment and as a result of diseases such as the dementias, schizophrenia, multiple sclerosis,and tumor growth. Strategies to measure, map, and visualize these brain changes are ofimmense value in basic and clinical neuroscience. Algorithms that map brain changewith sufficient spatial and temporal sensitivity can also assess drugs that aim to de-celerate or arrest these changes. In neuroscience studies, these tools can reveal subtlebrain changes in adolescence and old age and link these changes with measurable dif-ferences in brain function and cognition. Early detection of brain change in patients atrisk for dementia; tumor recurrence; or relapsing-remitting conditions, such as multi-ple sclerosis, is also vital for optimizing therapy. We review a variety of mathematicaland computational approaches to detect structural brain change with unprecedentedsensitivity, both spatially and temporally. The resulting four-dimensional (4-D) mapsof brain anatomy are warehoused in population-based brain atlases. Here, statisticaltools compare brain changes across subjects and across populations, adjusting for com-plex differences in brain structure. Brain changes in an individual can be comparedwith a normative database comprised of subjects matched for age, gender, and otherdemographic factors. These dynamic brain maps offer key biological markers for un-derstanding disease progression and testing therapeutic response. The early detectionof disease-related brain changes is also critical for possible pre-emptive interventionbefore the ravages of disease have set in.

CONTENTS

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120Applications of Mapping Brain Change. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120Novel Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120Dynamic Brain Atlases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

APPROACHES TO MAP BRAIN CHANGE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121Image Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121Volumetric Studies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122Anatomical Surface Modeling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122Longitudinal and Cross-Sectional Designs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123Serial Image Registration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123Image Subtraction Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124Brain Boundary Shift Integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

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120 TOGA ¥ THOMPSON

Optimizing Serial Image Registration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125Edge-Based Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126Serial MRI in Multiple Sclerosis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126Deformation-Based Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127Mapping Growth Patterns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127Mathematical Details. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130Fluid Modeling of Brain Change. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131Voxel Compression Mapping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132Population-Based Atlasing of Brain Change. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132Improved Dynamic Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132Random Effects Modeling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133Cortical Maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133Cortical Parameterization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134Mapping Gyral Pattern Differences in a Population. . . . . . . . . . . . . . . . . . . . . . . . . 134Mapping Gray Matter Deficits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136Dynamically Spreading Tissue Loss in Dementia. . . . . . . . . . . . . . . . . . . . . . . . . . . 137Mapping Surface Area Changes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137A Spreading Wave of Brain Change in Schizophrenia. . . . . . . . . . . . . . . . . . . . . . . 138

CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

INTRODUCTION

Applications of Mapping Brain Change

The brain changes dramatically over the human lifespan. Normal anatomicalchanges occur in brain development and aging, and these can be abnormally ac-celerated, delayed, or otherwise modified by the onset and progression of disease.

The past decade has seen tremendous advances in image analysis tools to detect,visualize, and compare these brain changes. These tools have uncovered develop-mental growth spurts in childhood and the teenage years, pointing to regions ofrapid growth and tissue loss (1–3). In studies of disease, they have revealed the dy-namic path of Alzheimer’s disease (AD) in the brain (4–6). They have also mappeda dynamically spreading wave of gray matter loss as patients develop schizophre-nia, which is correlated with increasing symptom severity (7, 8). In drug studies, theslowing of tumor growth with chemotherapeutic agents has also been mapped (9).

Novel Algorithms

The major thrust of these efforts is to design tools that map brain changes withunprecedented sensitivity (10–13). When subjects are studied longitudinally withmagnetic resonance imaging (MRI), different image analysis approaches can beapplied that are sensitive to different aspects of brain change. Simple volumetricapproaches, for example, produce global measures of change (e.g., “the hippocam-pus lost 4% of its volume in a year”). Mapping approaches, however, visualize thesechanges in detail, pinpointing where the losses occur [tensor-based morphometry(3)].


TEMPORAL DYNAMICS OF BRAIN ANATOMY 121

Dynamic Brain Atlases

Approaches to map brain change can be greatly enhanced if they are used in con-junction with a population-based brain atlas (14–17). Digital atlases can compiledata from hundreds or even thousands of subjects (15) in a standardized coordi-nate system. This information can include dynamic data on rates of brain change.Statistics can then be developed to detect how these changes are modulated indisease or by risk genes (18, 19). These statistical atlases can provide normativecriteria to help detect early brain change in patients with dementia (5, 20, 21) or inthose at genetic risk for AD (22, 23). The ability to stratify these atlases by demo-graphic or clinical criteria shows promise in uncovering how brain changes varyby age group, gender, and in disease. Subpopulations can also be selected fromthe underlying database to compare brain changes in different groups with similaror overlapping symptom profiles (24). This is advantageous for studying popula-tions of patients with complex disorders such as schizophrenia. As longitudinalstudies are performed in different neuroimaging centers worldwide, the growingdynamic atlas provides a basis to contrast chronically medicated and first-episodepatients (8), childhood-onset and adult patients (25–27), schizoaffective and bipo-lar patients, patients receiving different medications, and even family members atincreased genetic risk (28).

In the next sections, we describe the key approaches for mapping brain change.They are broadly divided into (a) image subtraction methods (which detect changesin MRI signal), (b) image deformation and tensor mapping methods (these detectchanges in brain shape), and (c) specialized approaches to map cortical change.Following the description of these major classes of algorithms, we describe addi-tional nonlinear registration and statistical mapping techniques required to registerprofiles of brain change across subjects and optimize detection of disease effects.

APPROACHES TO MAP BRAIN CHANGE

Image Acquisition

Parametric imaging (29, 30) is a type of MRI that is ideal, in some respects, forlongitudinal studies of brain change. It produces quantitative maps of tissue param-eters that can be compared over time. This differs from conventional MR imagingin which T1-weighted or T2-weighted images are optimized for tissue contrast,but are not typically designed for absolute quantitation of MR signal across timeor even across subjects. As such, the absolute signal values in conventional MRimages can change arbitrarily due to fluctuations in scanner calibration from oneacquisition to another. Parametric images, however, display scanner-independentparameters (usually the MRI signal decay constants T1 and T2). These reflectchanges in the underlying tissue lipid content, hydration, and physiology. Para-metric scans are of special interest for studying developmental brain change, aswell as steroid effects and edema in glioma patients (31). Brain development pro-duces complex changes in brain shape, as well as changes that affect the MRI



signal itself, including myelination from before birth into old age and decreases inbrain proton density, T1 and T2. These changes can be studied using relaxometry,which measures absolute changes in MR signal parameters. In a recent study ofbrain changes in glioma (32), 21 parametric T2 images were acquired from twocancer patients over 24 weeks. Consistent linear trends as small as 0.01 ms/daywere statistically detectable. The direction and consistency of these changes werevalidated using jackknifing. Tumor changes due to radiation treatment were alsomapped. Such quantitative MRI approaches show promise for detecting subtletissue changes due to brain disease progression or therapy (32, 33).

Volumetric Studies

By far the most common, and oldest, approach to study brain change has beenvolumetric image analysis. It requires only relatively simple image processing.Many groups have used manual or automated approaches, or both, to create three-dimensional (3-D) models of brain substructures, based on MRI, computed tomog-raphy (CT), or 3-D cryosection data (34–37). A variety of morphometric statisticshave been derived from these 3-D models, including shape statistics based on aver-aging parametric meshes (38), using Riemannian shape theory (39), eigenfunctiondecomposition (36, 40), or medial representations (41).

In the simplest approaches, the volume, or cross-sectional area, is computed forstructures such as the corpus callosum (19), hippocampus (8), ventricular system(16), basal ganglia (8), cerebellum (42), or individual lobes of the brain. The re-sulting volumes are subjected to volumetric analyses, such as multivariate analysisof covariance (MANCOVA) or discriminant function analysis. By fitting statisticalmodels, disease, gender, and systematic changes in these structures over time aredistinguished from fluctuations due to sampling and measurement error.

Anatomical Surface Modeling

An extension of this approach is parametric mesh modeling (18, 38, 40, 41, 43).This approach applies a regular computational grid over the 3-D boundary of astructure in the form of a triangulated mesh. By modeling the same brain structuresin multiple subjects with parametric meshes, average shape representations can bedeveloped for particular structures. Patterns of changes over time can be mappedlocally using deformation maps (described later). The surface grid format is a datatype that can be graphically visualized with surface rendering or animation tech-niques. Surface models can also be averaged across subjects. They also supportadditional computations on surface attributes, such as curvature and complexity.Parametric surface approaches are highly effective for studying changes in morecomplex surface attributes on the highly folded cerebral cortex (35, 44, 45). Strate-gies are discussed later to compute and compare dynamic changes in cortical shape(3, 43), cortical gray matter density and thickness, and surface-based functionalimaging signals (46, 47). Studies using these tools reveal increases in cortical sur-face complexity and cortical expansion during childhood (42, 48) and increases in



brain asymmetry over the human lifespan (48), as well as progressively spreadingwaves of cortical gray matter loss in dementia and psychosis (19, 27).

Longitudinal and Cross-Sectional Designs

Brain changes in development, aging, and most degenerative diseases are subtlecompared with the current resolution of MRI. Often, these changes play out overtimespans longer than a typical research study, even over entire lifetimes (48). Bycontrast, longitudinal studies, which scan the same individuals over time, typicallyuse interscan intervals of only one to two years [e.g., in development and dementia(4, 10, 49)]. Intervals as short as a week may indicate significant tumor growth inglioma studies (9, 50).

To capture brain changes occurring over longer periods, cross-sectional dataare typically used. Here, subjects of different ages are scanned once only. Due tothe large variability in brain structure across subjects, underlying brain changesover time are harder to detect. Typically, multivariate modeling and large samplesizes are used [(51) (N= 111); (48, 52)]. These multivariate models partitionthe variance in the observed data into effects of time (or age) and other influen-tial factors, such as gender or disease. Longitudinal studies typically have higherstatistical power to detect brain change, but cross-sectional studies have easierlogistics, smaller cost, and a lower subject attrition rate. To accommodate bothstyles of data acquisition, so-called mixed or random effects models have beendeveloped that can combine information from both types of data in a singleanalysis (53–55).

Serial Image Registration

Serial image registration, or digital overlay of consecutive 3-D MR images from thesame subject, has resulted in a variety of related approaches to compare anatomyover time. At the simplest level, images can be overlaid using a rigid-body trans-formation, histogram equalized, and subtracted voxel by voxel. This produces adifference image (see Figure 1,left panel). If perfect registration occurs, and if nochanges in brain shape or MRI signal occur across the interval between the twoscans, the difference image is largely noise (Figure 1,right panel) due to tempo-rally uncorrelated noise in the reconstructed MR signal from one acquisition to thenext. When subtracting images acquired during a period of significant biologicalchange, anatomical features can often appear in the difference images. Unfortu-nately, these difference images cannot usually provide reliable spatial informationon where atrophy is occurring. The features that appear may indicate localizedanatomical change, but they may also indicate registration error or an arbitrarycombination of each (56). If brain size is changing overall, for example, a rigidbody transform will best align the central regions of the brain, with more periph-eral regions successively more displaced, although the anatomical change maybe no greater there. Subsequent linear and nonlinear deformation algorithms aretherefore essential to model more localized brain change. Nonetheless, accurate



Figure 1 Image subtraction. A young normal subject was scanned at the age of 7, and againfour years later, aged 11, with the same protocol. Scan histograms were matched, rigidlyregistered, and a voxel-by-voxel map of intensity differences (left) suggests global growth.In a control experiment, identical procedures were applied to two scans from a 7-year-oldsubject acquired just two weeks apart, to detect possible artifactual change due to mechanicaleffects, and due to tissue hydration or cerebro-spinal fluid (CSF) pressure differences in theyoung subject between the scans. These artifacts were minimal, as shown by the differenceimage, which, as expected, is largely noise. Rigid registration of the scans does not localizeanatomic change, but is a precursor to more complex tensor models of structural change (seemain text), which not only map local patterns of differences or change in three dimensions,but also allow calculations of rates of dilation, contraction, shearing, and torsion (3).

rigid-body registration is usually a prerequisite before more sophisticated methodscan be applied to measure brain change.

Image Subtraction Methods

Image subtraction is used in many areas of computer vision, such as video surveil-lance and motion tracking, to measure changes in consecutive pairs of images.Various image subtraction methods have been used to measure brain changes indevelopment (57), aging, and dementia (58–63). Bromiley et al. (64) noted that in-terpretation of the resulting difference image can be problematic because the pixelvalues have no objective meaning (typically they are in gray level units unlessparametric imaging is used). To determine if changes in a subtraction image arestatistically significant, Bromiley et al. developed a nonparametric statistical testthat visualizes altered regions as a probability map. Pixel values in this map repre-sent the probability that the pairing of pixel values at that position in the originalaligned images was drawn from the bulk distribution for pixel values in the images.This distribution is estimated nonparametrically using the gray-level scattergram,



or co-occurrence matrix, of the original images. This data structure is often usedin entropy-based methods for image registration. The resulting joint distributionis invariant to differences in mean intensity between the two images and can beused to distinguish local effects from global differences or background noise.

Brain Boundary Shift Integral

Global rates of brain atrophy based on the BBSI were found to be linked withthe rate of cognitive decline in dementia patients (69). Mean atrophic rates weresignificantly faster in AD patients, at 2.8%/year (±0.9% SD), than in age-matchedcontrols [at only 0.2%/year± 0.3% SD (62)]. Power calculations (4) also sug-gested that the BBSI measure requires a smaller sample size to detect treatmenteffects than conventional measures based on manually segmenting individual brainstructures.

Fox & Freeborough (62) describe a variant of image subtraction, known asthe Brain Boundary Shift Integral (BBSI). They first isolate the brain from eachserial scan by morphological operations and manual editing, prior to rigid bodyregistration and intensity normalization. Because the scan intensities may alsobe changing over time, scan intensities are typically normalized first based onthe mean intensity of a relatively unchanging brain region, typically the whitematter. In other approaches, image intensities are first equalized across scans usinghistogram-matching and relative bias field correction (65, 66).

Image subtractions of pairs of registered scans often show intensity loss at tissueboundaries (see Figure 1). The BBSI approach (67) notes that the intensity losscorresponds to the positional shift at image edges and structure boundaries. Thevolume of lost tissue can therefore be computed by estimating the volume throughwhich the boundary has shifted. This BBSI is derived by integrating intensity losswithin a pre-defined intensity range over the external boundary between cerebro-spinal fluid (CSF) and the brain (68). Strictly speaking, the area under the intensityprofile across a boundary in image 1 is subtracted from that for image 2, and thisdifference is divided by the boundary height. This provides an accurate measureof lateral boundary shift, so long as the image contrasts are well matched in thetwo images (13).

Optimizing Serial Image Registration

Successful monitoring of brain change using image registration depends on thealgorithm used and the quality of the images. Many of the key algorithms forrigidly aligning serial images are the same ones that are routinely used for motioncorrection in functional MRI and for realigning pre- and postcontrast radiologicalexams. These registration approaches typically tune the parameters of the align-ment transformation (here, three translations and three scales for a rigid bodytransform) until a measure of the scan overlap is optimized [e.g., squared intensitydifference, cross-correlation, or mutual information; see e.g., (60, 65, 70, 71)]. Infunctional MRI, a time-series of images is typically acquired from each subject



over a period of several minutes, and retrospective image alignment is used toadjust for subject motion (70, 72). Oatridge (73) noted that the accuracy of serialimage registration depends heavily on the SNR of the images. Subvoxel accuracyis, however, often achievable if a high-order resampling method is used, such assinc or “chirp-Z” interpolation (59). Matching accuracies of better than 0.01 mmper axis and 0.01 degrees for each rotation angle can be routinely achieved inleast-squares registrations of phantoms (73).

Edge-Based Methods

Smith et al. (13) describe a related approach to estimate brain atrophy over time.Their technique is known as SIENA (Structural Image Evaluation, using Normal-ization, of Atrophy). SIENA also estimates boundary shift, but with several modifi-cations that correct for problems in prior approaches and provide greater accuracy.

To measure brain change over time, SIENA first segments brain from nonbraintissue and automatically extracts models of the exterior skull surfaces, which areused to register the two scans. Skull surfaces are registered rather than the brainitself because they are more likely to be static over time, whereas the brain maychange in overall size and shape. In aligning one scan to the other, geometricscaling is also allowed, i.e., the transform is not constrained to be rigid. Thiscompensates for any errors in the spatial calibration of the scanner (e.g., due togradient calibration drift, local field distortions, or varying head placement in thescanner). A 12-parameter (affine) registration is performed, using cross correlationto align the second scan to the first (74). Using the square root of the alignmentmatrix, both images are then resampled to a position “halfway” between them. Thisavoids differences in the level of blurring that would occur if only one scan wereresampled. Brain change is then estimated from the movement of all brain surfaceedge points (including those on the internal brain-CSF boundary). Candidate edge-points are recovered with subvoxel accuracy in both images using a gradient-basededge detector with nonmaximum suppression. The use of an edge detector avoidsthe need for intensity normalization and makes the approach more robust to overallchanges in tissue intensity and radio-frequency bias between scans. The apparentmotion of each brain edge point is computed perpendicular to the local edge.Using the recovered edge points only, the average motion perpendicular to thebrain surface is computed. Finally, this average surface shift is converted into anestimate of brain volume change, using an estimate of the brain’s volume dividedby its surface area. The resulting method measures overall brain change extremelyaccurately in longitudinal studies of dementia. Validation studies reported a 0.5%–1% error in estimating brain volume, and brain change is estimated with an erroraround 0.15% of total brain volume (13).

Serial MRI in Multiple Sclerosis

A common application of serial image subtraction is tracking disease progression inpatients with multiple sclerosis (MS) (12, 73, 75). In MS, changes of 5%–10% per



year in lesion burden can be seen in longitudinally acquired MR images. Localizedlesions differ in intensity and can be detected using tissue classification techniques(75), especially if an intravenous MRI contrast agent is used, such as GdDTPA. InMS research, the focus is on changes in image intensity observed over time, ratherthan changes in brain shape. Nonetheless, some transient brain shrinkage is seenin patients undergoing steroid treatment. Lesions are usually better detected usingserial image registration instead of manual segmenting of the lesions. Remittingand worsening lesions often can be identified in close proximity (73). In large-scalestudies, image processing pipelines (76) make it easier to evaluate how lesions varyover time in clinical trials.

Deformation-Based Methods

Maps of brain change over time also may be based on a deformation mapping con-cept. In this approach, a 3-D elastic deformation is calculated. This deformation,or warping field, drives an image of a subject’s anatomy at a baseline timepointto match its shape in a later scan (see Figure 2). Image warping techniques haveevolved over many years (77). Now, mappings can be calculated in a very exactway that matches a large number of the key functional and anatomic elements inthe scans to be matched. This results in a very complex transformation, often withup to a billion parameters, from which local volume changes in tissues can becalculated (Figure 3) (5, 24, 67, 78–92).

In capturing brain change, deformation-based methods can be complementaryto voxel-based morphometric methods (93, 94), and methods that estimate wholebrain atrophic rates (13, 95, 96). Voxel-based methods typically use a simple pixel-by-pixel subtraction of scan intensities registered rigidly across time. Deformationmethods, however, can distinguish local from global effects, and true tissue lossfrom translational shifts in anatomy, which can confound image subtraction meth-ods (56, 93).

Mapping Growth Patterns

Figure 4 shows some typical results of a deformation-based approach we developedto map brain growth in young children. An anterior-to-posterior wave of growthwas found in the brains of children scanned repeatedly between the ages of 3 and15 (3). Parametric surface meshes were built to represent anatomical structuresin a series of scans over time, and these were matched using a fully volumetricdeformation. Dilation and contraction rates, and even the principal directions ofgrowth, can be derived by examining the eigenvectors of the deformation gradienttensor or the local Jacobian matrix of the transform that maps the earlier anatomyonto the later one (Figure 3). By applying local operators to the deformationfields, tensor maps can be created to reflect the magnitude and principal directionsof tissue dilation or contraction. This mapping process is illustrated in Figure 4.The validity of the approach can also be assessed by visualizing “null maps” ofbrain change over short intervals.



In studies of brain growth using these techniques, after the age of 6, peakgrowth rates were consistently found in regions of the corpus callosum that con-nect linguistic and association cortices of the two brain hemispheres. After pu-berty, these growth rates were considerably reduced, and tissue loss was alsoidentified in subcortical regions. By characterizing the dynamics of brain devel-opment, statistical criteria can be developed to estimate the rates at which spe-cific brain regions normally develop. In developmental disorders, for example, achild may display a normal phenotype with an aberrant time-course. Similarly,if growth rates are abnormal, morphology may not be detectably different at anytime-point due to the wide variations in normal anatomy. Current developmen-tal atlasing projects make it possible to compare the dynamics of brain growthand tissue elimination in an individual or group against a database of dynamicnormative data.

←−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−Figure 2 Deforming anatomical templates with neural nets and continuum mechani-cal flows. The complex transformation required to model brain change over time can bedetermined using radial basis function neural networks [(a), (85); see (3) for details] orcontinuum-mechanical models (b). In Davis et al. (85), each of the three deformationvector components,uk(x), is the output of the neural net when the position in the imageto be deformed,x, is input to the net. Outputs of the hidden units (Gi,πm) are weightedusing synaptic weights, wik. If landmarks constrain the mapping, the weights are foundby solving a linear system. Otherwise, the weights can be tuned to optimize a mea-sure of similarity between the deforming image and the target. Continuum-mechanicalmodels (b) can also be used to compute these deformation fields (80, 82, 88, 101, 104).In (b), two line elements embedded in a linearly elastic block are slightly perturbed. TheNavier equations [shown in continuous form (b) and in discrete form (d)] are solvedto determine the values of the displacement field vectors,u(x), throughout the 2-D or3-D image. (b) Lame elasticity coefficients. Different choices of elasticity coefficients,λ andµ, in the Cauchy-Navier equations (shown in continuous form,top) result in dif-ferent deformations, even if the applied internal displacements are the same. Elasticitycoefficients can be chosen that limit the amount of curl (lower right) in the deformationfield. To emphasize differences, the displacement vector fields shown in (b) are mul-tiplied by a factor of 10. The Cauchy-Navier equations, derived using an assumptionof small displacements, are valid only when the magnitude of the deformation field issmall. Using parametric meshes to model anatomical surfaces in a registered pair ofanatomical scans, patterns of local displacement can be computed over time (caudatenucleus,bottom left panel). These displacement vectors are used to drive a complex3-D volumetric deformation of anatomy [deformed grid, (d)], which can be thought ofas a vector field (c). From this field, measures of local volumetric loss and gain can becomputed. (e) The caudate nucleus of a young child, imaged at the age of 7 and againat age 11, with a growth map overlaid in different intensities. Regions of tissue lossare found immediately adjacent to regions of dramatic tissue growth.



Figure 3 Deformation mapping. Patterns of local volumetric growth and loss canbe estimated from a deformation map that captures brain changes over time. Thedeterminant of the deformation gradient, i.e., the Jacobian or local expansion factor,is shaded. Unlike image subtraction (Figure 2), the maps distinguish local volumechanges from volume-preserving shifts in anatomy.

Applications of these deformation maps of change include measuring the statis-tics of brain growth (3), mapping tissue loss rates in dementia (97), and measuringtumor response to chemotherapy agents (9). By building probability densities onregistered tensor fields [e.g., (98)], a quantitative framework can be establishedto detect normal and aberrant brain change and its modulation by medication inclinical studies.

Mathematical Details

Deformation-based methods to track brain change have often been based on con-tinuum mechanics, which describes physical models of elastic or fluid bodies [seeFigure 4; reviewed in (77, 99); cf. (9, 31, 67, 100)]. The 3-D shape of one brain,imaged with MRI at one timepoint, is imagined to be embedded in a physicalmedium, such as an elastic block or a fluid. This deformable template is recon-figured to match its shape in a later image [intensity changes over time may alsobe modeled; q.v. (40)]. In some approaches, a complex 3-D deformation field iscomputed that matches large numbers of surface, curve, and point landmarks inthe two brains. By adding anatomical features to constrain the deformation, keyanatomic and functional interfaces can be matched up when one scan is deformedinto the shape of the other. In one approach (26, 83), parametric mesh models ofbrain structures are used to drive a 3-D deformation vector mapU:x→ u(x), whichis derived from the Navier equilibrium equations for linear elasticity:

µ∇2u+ (λ+ µ)∇(∇ · u(x))+ F(x− u(x)) = 0, ∀x ∈ R. (1)



All the terms in this equation describe forces and distortions in a 3-D materialin which the image is considered to be embedded. R is a discrete lattice repre-sentation of the scan to be transformed,∇ ·u(x)=6∂uj/∂xj is the divergence orcubical dilation of the medium,∇2 is the Laplacian operator that measures theirregularity of the deformation,F(x) is the internal force vector, and Lam´e’s coef-ficientsλ andµ refer to the elastic properties of the medium. Matching of corticalsurfaces across time and subsequently across subjects (for data averaging) can alsobe enforced. Mappings based on high-dimensional elastic and fluid models canrecover extremely complex patterns of change (Figure 4; see also (12, 67)]; theirmathematics are reviewed elsewhere (101).

Fluid Modeling of Brain Change

Building the work on elastic image registration, Christensen et al. (80) developeda compressible fluid deformation model for image registration that forces thedeformation matching the scans to be smooth and topology preserving, even underlarge deformations. Strictly speaking, the Navier equations (Equation 1) are derivedunder a small deformation assumption (which is valid for growth and atrophicprocesses), but the fluid model uses a regridding approach when necessary toensure a smooth final solution. The forces that drive one image to match theother were also designed to match regions in each dataset with high intensitysimilarity. Transformation parameters were determined by gradient descent on acost functional (Equation 2) that penalizes squared intensity mismatch betweenthe deforming template T(x− u(x, t)) and target S(x), while guaranteeing thesmoothness of the transformation:

C(T(x),S(x), u) = (1/2)∫Ä

|T(x− u(x, t))− S(x)|2dx. (2)

The driving force, which deforms the anatomic template, is defined as thevariation of the cost functional with respect to the displacement field:

F(x, u(x, t)) = −(T(x− u(x, t))− S(x))∇T|x−u(x,t), (3)

µ∇2v(x, t)+ (λ+ µ)∇(∇T · v(x, t))+ F(x, u(x, t)) = 0, (4)

∂u(x, t)/∂t = v(x, t) =∇u(x, t)v(x, t). (5)

The deformation velocity (Equation 4) is governed by the creeping flow mo-mentum equation for a Newtonian fluid, and the conventional displacement fieldin a Lagrangian reference system (Equation 5) is connected to a Eulerian veloc-ity field by the relation of material differentiation. Experimental results matchingbrain MRI volumes were excellent (80). Because fluid matching is computationallyintensive, subsequent work focused on deriving separable (and therefore computa-tionally fast) filters to approximate the continuum-mechanical filters derived above(90, 102–104). Some elastic matching algorithms in surgical applications that useintraoperative scanning (105) are now fast enough to track brain change in real time.



Voxel Compression Mapping

Freeborough et al. (68) implemented a fluid-matching algorithm to visualize howbrain structure locally contracts and expands in a longitudinal study of dementia.Calling the technique “voxel compression mapping,” they also used the Jacobianof the deformation field to compute local atrophy and expansion [see also (10, 106)].These changes were displayed as a color-coded map overlaid on the original scan.In clinical studies using this method, Fox et al. (107) found characteristic patternsof atrophy in the different dementias. AD patients showed diffuse atrophy, but moreregionally selective atrophy was found in individuals with frontotemporal demen-tia. Janssen et al. (108) suggested that voxel compression maps may even identifyregional brain atrophy prior to clinical diagnosis in both AD and frontotemporaldementia, underscoring the clinical potential of these methods.

Population-Based Atlasing of Brain Change

In different individuals, growth processes or tissue losses occur in anatomies thatare geometrically different. Additional warping techniques are needed to comparegrowth profiles across subjects. This additional warping is needed to compute av-erage profiles of growth in a group and to define statistical differences in rates ofgrowth or loss. Mathematically, ifUi(x, ti) is the 3-D displacement vector requiredto deform the anatomy at positionx in subjecti at reference time 0 to its cor-responding homologous position at timeti, then a linear approximation the localrate of volumetric growth (98) can be written in terms of the identity tensor anddisplacement gradient tensor as:

3i (x) = ∂Ji/∂t = det(I +∇Ui )/ti . (6)

If A i is the secondary deformation mapping transforming the baseline anatomyof individual i onto the atlas (Warping Fieldin Figure 4), then the set of regis-tered growth maps3i(A i(x)) (shown in the final panel of Figure 4) can be treatedas observations from a spatially parameterized random field, whose mean andvariance can be estimated. Statistical effects of age, gender, genotype, or medi-cation can then be detected using random field theory to produce statistical maps(3).

Improved Dynamic Models

In developing dynamic atlases for clinical applications, there is a particular interestin modeling developmental processes that speed up or slow down. Diseases mayaccelerate or their rate of progression may be slowed down by therapy. If individualsare scanned more than twice over large timespans, this presents the opportunityfor more accurate detection of brain change and encoding of these changes ina group atlas. To compare growth patterns in different groups of subjects, the“general linear model” (72) can be used to analyze the registered growth profiles



(or degenerative profiles). For theith individual’s jth measure we have:

Yi j = f (Ageij , β)+ εij . (7)

Here,Yi j signifies the outcome measure at a voxel or surface point, such asgrowth or tissue loss;f() denotes a constant, linear, quadratic, cubic, or otherfunction of the individual’s age for that scan; andβ denotes the regression/ANOVA(analysis of variance) coefficients to be estimated. Age (Ageij) may be replacedby time from the onset of disease, the start of medication, or the time from theonset of puberty (52). This flexibility in parameterizing the time axis allows oneto temporally register dynamic patterns using criteria that are expected to bringinto line temporal features of interest that appear systematically in a group (5). Forexample, the independent variable could be a cognitive score such as Mini-Mentalstatus (19), which declines over time in disease. In a developmental study, thisindependent variable could be a measure of physical or psychological maturitythat may better reflect the developmental stage of the subject than age alone.Parameterization of dynamic effects using measures other than time (e.g., clinicalstatus) also provides a mechanism to align new patients’ time series with a dynamicatlas (5).

Random Effects Modeling

In the above statistical model of brain change (Equation 7), the coefficient vector,β, is assumed to be constant, i.e., a fixed effect. Theεij are assumed normallydistributed and uncorrelated both between and within individuals. If multiple scansare available over time, a random effects model can also model brain changes in apopulation:

Yi j = αi + f (Ageij , β)+ εij . (8)

Here, the model is the same as the general linear model except for theαi term,which is called a random effect. It describes the correlation between an individual’smultiple scans. Random effects models may also be fitted with correlated errors(53, 54). If this is done,εij andεik (k not equal toj) are assumed correlated, wherethe correlation is a function of the time elapsed between the two measurements(52). In models whose fit is confirmed as significant, e.g., by permutation, loadingson nonlinear parameters may be visualized as attribute mapsβ(x). This revealsthe topography of accelerated or decelerated brain change. The result is a formalapproach to assess whether, and where, brain change is speeding up or slowingdown. This is a key feature in developmental or medication studies, and a keyelement of developmental atlases currently being built.

Cortical Maps

Understanding cortical anatomy and function is a major focus in brain research,and many diseases cause profound changes in the cortex. Because most imag-ing studies of brain function focus on the cortex, it is especially important to be



able to pool cortical brain mapping data from subjects whose anatomy is dif-ferent (46). Gyral pattern variation in particular (a) makes general patterns oforganization and disease effects hard to discern and (b) complicates attempts todefine statistical criteria for abnormal cortical anatomy. To simplify the com-putation of differences across individuals and changes over time, many inves-tigators have developed surface parameterization approaches, which we brieflydescribe next.

Cortical Parameterization

Several methods exist to generate surface models of the cortex from 3-D MRIscans. Some of these impose a tiled, parametric grid structure on the anatomy thatis used as a coordinate framework to support subsequent computations. In “bottom-up” approaches [e.g., (91, 109, 110)], a voxel-based segmentation of white matteris generated first using a tissue classifier or level set methods (111). Its topol-ogy is then corrected using graph theoretic methods (110). This creates a single,closed, simply connected surface homeomorphic to a sphere (109, 110, 112, 113).The surface is tiled using triangulation methods such the Marching Cubes algo-rithm (114). The gridded surface is then inflated using iterative smoothing to aspherical shape. Inverting this inflation mapping allows a spherical coordinatesystem to be projected back onto the 3-D model for subsequent computations.Alternatively, the 3-D surface may be flattened to a 2-D plane (Figure 5) (112,115–118), inducing an alternative 2-D parameterization onto the original 3-Dsurface.

A second (“top-down”) type of surface extraction method (82, 119, 120) beginswith a spherical or ellipsoidal surface that is already tiled. This parametric surfaceis successively moved under image-dependent forces, reshaping it into the complexgeometry of the cortical boundary [see (121) for work on gradient vector flow].This avoids the need for topology correction, as a single, fixed, grid structureis established at the start and mapped with a continuous deformation onto eachanatomy. Complex constraints are, however, required while deforming the surface.These ensure that the surface does not self-intersect and adapts fully to the targetgeometry. The first (bottom-up) strategy turns the cortex into a sphere, whereasthis latter approach deforms a sphere onto the cortex. Both approaches project acoordinate system onto the anatomy so that cortical locations can be referred to insurface-based coordinates.

Mapping Gyral Pattern Differences in a Population

Once cortical models are available for a large number of subjects in a common3-D coordinate space, patterns of cortical variability and cortical change over timecan be calculated. Cortical anatomy can be compared between any pair of subjectsby computing the warped mapping that elastically transforms one cortex into theshape of the other. Due to variations in gyral patterning, cortical differences amongsubjects will be severely underestimated unless elements of the gyral pattern are



matched from one subject to another. This matching is also required for corticalaveraging; otherwise, corresponding gyral features will not be averaged together.Fortunately, the major gyri and sulci of the cortical surface have a similar spatiallayout across subjects (122, 123) [see (18) for some caveats], even though theirgeometry varies substantially. Transformations can therefore be developed thatmatch large networks of gyral and sulcal features with their counterparts in thetarget brain (43, 82, 83, 109, 116).

In one approach (3), a maximal set, or template, is specified containing allprimary sulci that consistently occur in normal subjects (Figure 5b,c shows someof these). Cortical anatomy can be compared between any pair of subjects bycomputing the warped mapping that elastically transforms one cortex into theshape of the other. Due to variations in gyral patterning, cortical differencesamong subjects will be severely underestimated unless elements of the gyral pat-tern are matched from one subject to another. This matching is also required forcortical averaging; otherwise, corresponding gyral features will not be averagedtogether.

To find good matches among cortical regions, many groups perform the match-ing process in the cortical surface’s parametric space, which permits more tractablemathematics (Figure 5,right panels). This vector flow field in the parametric spaceindirectly specifies a correspondence field in 3-D, which drives one cortical surfaceinto the shape of another. This mapping not only matches overall cortical geom-etry, but matches an entire network of the 38 sulcal landmark curves with theircounterparts in the target brain, and thus is a valid encoding of cortical variation.The flow in parameter space (Figure 5) can be represented by spherical harmonics(41, 83), which are eigenfunctions of the spherical Laplacian, or by solving an elas-tic or fluid partial differential equation (PDE) that aligns sulcal/gyral landmarks(104, 124) or curvature maps (109). In one approach, based on covariant PDEs,these flows are made invariant to the way the cortical surfaces are parameterized(3). When the self-adjoint differential operator governing the PDE is discretized,fields of Christoffel symbols are derived from the metric tensor of the surfacedomain and added as correction terms. The matching fields are then independentof the surface metrics and can be used to associate signals from correspondingcortical regions across subjects.

On the sphere, the parameter shift functionu(r) :Ä→ Ä, is given by the solu-tion F:r→ r − u(r ) to a curve-driven warp in the spherical parametric spaceÄ=[0, 2π )× [0, π ). For pointsr = (r, s) in the parameter space (Figure 6), a sys-tem of simultaneous partial differential equations is written for the flow fieldu(r ):

L‡(u(r ))+ F(r − u(r )) = 0, ∀r ∈ Ä, with u(r ) = u0(r ), ∀r ∈ M0 ∪M1. (9)

Here, M0, M1 are sets of points and (sulcal or gyral) curves where displace-ment vectors,u(r ) = u0(r ), matching corresponding anatomy across subjectsare known. The flow behavior is modeled using continuum-mechanical equations.L can be any second-order self-adjoint differential operator; a common example



is the Cauchy-Navier differential operatorL = µ∇2+ (λ+µ)∇(∇T•) with bodyforceF [cf. (80, 88)]. To create mappings that are independent of the surface metrics(parameterizations), we useL‡, the covariant form of the differential operatorL.ForL‡, all of L’s partial derivatives are replaced with covariant derivatives with re-spect to the metric tensor of the surface domain where calculations are performed.The covariant derivative of a (contravariant) vector field, ui(x), is: ui

,k= ∂uj/∂xk+Γj

ik ui, where the Christoffel symbols of the second kind,Γjik , are computed from

derivatives of the metric tensor components gjk(x):

0ijk = (1/2)gil (∂glj/∂xk + ∂glk/∂xj − ∂gjk/∂xi). (10)

These correction terms are then used in the solution of PDE, producing a fam-ily of 3-D deformation maps,Ui(r ) matching each individual cortex in 3-D tothe average cortex for a group. Here,Ui is a 3-D location on theith subject’scortex, andr is the location it maps to, after warping, in the cortical parameterspace.

Mapping Gray Matter Deficits

To help understand the approach, first we describe a cross-sectional study of graymatter deficits in dementia in which each subject is imaged once; longitudinal dataare described next. Even in a study with a single scan from each subject, gyral pat-tern variation across subjects makes it difficult to infer precisely where gray matteris lost in a group. If gray matter maps are directly averaged together in stereotaxicspace, it is difficult to localize results to specific cortical regions. To address this,cortical pattern matching can help in computing group averages and statistics. Asa first step, all MRIs are corrected for radio-frequency inhomogeneity and seg-mented with a Gaussian mixture classifier, producing binary maps of gray matter.Let gi,r(x) be the “gray matter density,” i.e., the proportion of voxels classified asgray matter falling within a sphere (centerx, radiusr ) in the ith subject’s scan.Then for a point at parameter locationr on the group average cortex, gi,r(Ui(r )) isthe gray matter density at the corresponding cortical point in subjecti.

After averaging the aligned maps of gray matter density across groups of patientswith AD and healthy controls, Figure 6 reveals the spatial profile of gray matterdeficits in disease. By averaging the aligned maps and texturing them back onto agroup average model of the cortex, the average magnitude of gray matter loss wascomputed for the AD population (Figure 6,top row). Regions with up to 10%–20% reduction in the measure are demarcated from adjacent regions with littledetectable loss. The group effect size can also be measured by attaching a fieldof t statistics,t(r ), to the cortical parameter space and computing the area of thetfield on the group average cortex above a fixed threshold (p< 0.01, uncorrected).For groups that are not demographically matched, more sophisticated regressionmodels could be applied, resulting inF fields (or other nonparametric fields) thatindicate the significance of the overall fit and of how individual model parametershelp explain the loss. If whole surfaces of statistics are surveyed, there are several



approaches that are routinely used to make a multiple comparisons correction,which is required to confirm the significance of the overall effect. In permutationapproaches, the significance of the deficits can be confirmed by permuting theassignment of subjects to groups repeatly and estimating the null distribution ofstatistics on the surface. Under stronger assumptions, Gaussian field methods mayalso be used, which analyze the topology and smoothness of the statistical fieldsand their level sets.

Dynamically Spreading Tissue Loss in Dementia

Figure 6 shows these methods applied to a longitudinal study of brain change.A dynamically spreading wave of gray matter loss is visualized in the brains ofpatients with AD as it spreads over time from temporal and limbic cortices intofrontal and occipital brain regions, sparing sensorimotor cortices. The maps arebased on 52 high-resolution MRI scans of 12 AD patients (age 68.4± 1.9 years)and 14 elderly matched controls (age 71.4± 0.9 years) scanned longitudinally(two scans; interscan interval: 2.1± 0.4 years). Three key features are appar-ent: Overall, gray matter loss rates were faster in AD (5.3%± 2.3%/year) thanin healthy controls (0.9± 0.9%/year). Second, these shifting deficits are asym-metrical (left hemisphere> right hemisphere) and correlate with progressivelydeclining cognitive status. Finally, cortical tissue is lost in a well-defined sequenceas the disease progresses, mirroring the sequence of metabolic decline in positronemission tomography (PET) studies and neurofibrillary tangle accumulation seencross-sectionally at autopsy. The goal of these dynamic maps is to uncover thepath of degeneration for different brain systems and define possible MRI-basedmarkers for drug trials.

Mapping Surface Area Changes

A similar cortical matching approach has been used to map localized changesin cortical surface area over time (3, 98). Chung et al. (98) noted that ifX=X(v1, v2, t) is a parameterization of the cortical surfaceSt, its surface met-ric tensor isgi j =Xi

′Xj, where′ indicates the matrix transpose andXi= dX/dvi

denotes the partial derivative vector. The rate of local surface-area change per unitsurface area, or area-dilatation rate, is then approximated by:

d(ln L)/dt= tr[g−1(DX)′(d(DU)/dt)DX], (11)

where the local surface area element, L(t), is given by:

L(t) = det1/2(g)= (g11g22− g12g21)1/2. (12)

Here,DX = (X1, X2) is 3× 2 matrix andDU is a 3× 3 displacement gradient ma-trix. Approximating the resulting surface-based parameters ast-distributed randomfields (98) or chi-squared and Hotelling’s T2-distributed random fields (35, 117),null distributions and statistical criteria can be developed to tell where significantbrain change has occurred (3). Laplace-Beltrami smoothing (98) and statistical



flattening (125) can also help to optimize signal detection in the resulting surface-based fields.

A Spreading Wave of Brain Change in Schizophrenia

An interesting application, detecting surface-based brain changes, is compilingdynamic maps to characterize diseases with childhood or adolescent onset. In aschizophrenia study (126) (Figure 6), the gray matter mapping procedure describedabove was applied to longitudinal MRI data from 12 schizophrenic patients and 12adolescent controls scanned at both the beginning and end of a 5-year interval. Theaverage rate of gray matter loss was estimated throughout the cortex by match-ing cortical patterns and comparing changes in disease with normal changes incontrols. Cortical models and gray matter measures were elastically matched firstwithin each subject across time, to compute individual rates of loss, and then flowedinto an average configuration using flat space warping (Figure 6). The resultingmaps (Figure 6) show dynamic gray matter loss in superior parietal, sensorimotor,and some frontal brain regions (up to 5% annually) in a pattern that sweeps for-wards across the brain over time. Group differences were highly significant (p<0.01, permutation test) relative to healthy controls and nonschizophrenic controlsmatched for medication and IQ, and were linked with psychotic symptom severity(see Figure 7).

CONCLUSION

The ability to detect changes in the human brain is of great interest in basic andclinical neuroscience. Engineering challenges occur at many stages of the anal-ysis, even after serial images are acquired. Different image analysis approaches,some based on image subtraction, deformation mapping, random field theory, andanatomical surface modeling, have been developed that are sensitive to differentfeatures of brain change. Specialized approaches have also been developed to mea-sure changes in the human cortex. Statistical atlases can then store these dynamicdata and make comparisons across individuals and populations. The resulting ar-mory of tools shows enormous promise in charting the dynamics of disease and inrevealing how the brain changes over the human lifespan.

ACKNOWLEDGMENTS

Grant support was provided by a P41 Resource Grant from the National Center forResearch Resources (RR13642). Additional support for algorithm developmentwas provided by the National Library of Medicine (LM05639); the National In-stitute of Mental Health (MH65166); the National Center for Research Resources(RR00865); and by a Human Brain Project grant to the International Consortiumfor Brain Mapping, funded jointly by NIMH and NIDA (MH52176). We also thankNeal Jeffries (NINDS) for his invaluable advice on mixed effects models.



The Annual Review of Biomedical Engineeringis online athttp://bioeng.annualreviews.org

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27 Jun 2003 20:21 AR AR191-05-COLOR.tex AR191-05-COLOR.SGM LaTeX2e(2002/01/18)P1: GCE

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Figure 7 Data, statistical models, and maps. This schematic shows some of the stepsused in mapping cortical change. First, measures (Yij), such as gray matter density,asymmetry, etc. (see text), are defined that can be measured longitudinally (green dots)or once only (red dots) in a group of subjects at different ages. Fitting of statisticalmodels to these data (Statistical Model,lower right) produces estimates of parametersthat can be plotted onto the cortex using a color code. These parameters can includeage at peak (see arrow at peak of the curve); significance values; or estimated statisticalparameters, such as rates of change, and effects of demographic factors or risk genes.

arthur w. toga and paul m. thompsonusers.loni.ucla.edu/~thompson/pdf/arbe2003.pdf12 jun 2003 14:40...

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