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NeuroImage 14, 000–000 (2001)doi:10.1006/nimg.2001.0862, available online at http://www.idealibrary.com on

A Unified Statistical Approach to Deformation-Based MorphometryM. K. Chung,* K. J. Worsley,*,† T. Paus,† C. Cherif,‡ D. L. Collins,† J. N. Giedd,§

J. L. Rapoport,§ and A. C. Evans†*Department of Mathematics and Statistics and †Montreal Neurological Institute, McGill University, Montreal, Quebec, Canada;

‡Departement de Mathematiques, Ecole Polytechnique Federale de Lausanne, Switzerland; and §Child Psychiatry Branch,National Institute of Mental Health, NIH, Bethesda, Maryland 20892

Received August 14, 2000

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We present a unified statistical framework for ana-lyzing temporally varying brain morphology using the3D displacement vector field from a nonlinear defor-mation required to register a subject’s brain to anatlas brain. The unification comes from a single modelfor structural change, rather than two separate mod-els, one for displacement and one for volume changes.The displacement velocity field rather than the dis-placement itself is used to set up a linear model toaccount for temporal variations. By introducing therate of the Jacobian change of the deformation, thelocal volume change at each voxel can be computedand used to measure possible brain tissue growth orloss. We have applied this method to detecting regionsof a morphological change in a group of children andadolescents. Using structural magnetic resonance im-ages for 28 children and adolescents taken at differenttime intervals, we demonstrate how this methodworks. © 2001 Academic Press

Key Words: volume change; volumetry; brain growth;morphometry; atrophy; deformation; brain develop-ment.

INTRODUCTION

Temporally varying morphological differences in thebrain have been examined primarily by MRI-basedvolumetry. Classical MRI-based volumetry requiressegmentation of the identical region of interest, eithermanually or by spatial normalization, in two MR im-ages taken at different times t1 and t2. Then the totalvolumes V1 and V2 of the homologous regions are cal-culated by counting the total number of voxels. After-ward, the volume variation DV 5 V2 2 V1 is used as anindex of morphological changes (Giedd et al., 1996a;Rajapakse et al., 1996; Reiss et al., 1996; Thirion andCalmon, 1999).

As a part of deformation-based morphometry, a newtechnique called deformation-based volumetry ismerging; this method does not require segmentationf a priori regions of interest (Davatzikos, 1999; Ash-

1

urner and Friston, 2000). In deformation-based volu-etry, the Jacobian of the deformation field that is

equired to register one brain to another is used toetect volumetric changes. By definition, the Jacobianf the deformation is the volume of the unit-cube afterhe deformation. Assuming that one can find the defor-ation field at any voxel, volume change can be de-

ected at a voxel level. So the advantage of this tech-ique over the classical MRI-based volumetry is that itoes not require a priori knowledge of the region ofnterest to perform the morphological analysis. More-ver, the deformation-based volumetry improves theower of detecting the regions of volume change withinhe limits of the accuracy of the registration algorithm.hese two advantages of the deformation-based volu-etry over the standard MRI-based volumetry have

lso been noted by Davatzikos (1999) and Ashburnernd Friston (2000).Because the deformation-based morphometry (DBM)

s a relatively new method, very few morphologicaltudies have used the Jacobian for local volumehange. Davatzikos et al. (1996) used the Jacobian ofhe 2D deformation field as a measure of local areahange in 2D cross-sections of the corpus callosum toest gender-specific shape differences. Thompson andoga (1999) applied the Jacobian of 3D deformations asmeasure of the regional growth of the corpus callo-

um. Also volume dilatation, which is the first-orderpproximation of the Jacobian, has been used insteadf the Jacobian itself to measure local volume change.hirion and Calmon (1999) used the divergence of theisplacement vector field, which is equivalent to theilatation, for detecting growth of brain tumors.hompson et al. (2000) used local rates of dilatation,ontraction, and shearing from the deformation field toetect morphological changes in brain development.There has also been a parallel development in de-

ecting morphological changes without volumetry us-ng Hotelling’s T2 statistic for the displacement field

(Thompson et al., 1997; Joshi, 1998; Collins et al., 1998;Gaser et al., 1999; Cao and Worsley, 1999). Although it

1053-8119/01 $35.00Copyright © 2001 by Academic Press

All rights of reproduction in any form reserved.

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2 CHUNG ET AL.

seems that there are many different ways of detectingmorphological changes in deformation-based mor-phometry, a translation, a rotation, and a strain aresufficient for detecting a relatively small displacementand, in turn, for characterization of morphologicalchanges over time.

In this paper, we present a unified statistical frame-work for detecting brain tissue growth and loss is tem-porally varying brain morphology. As an illustration,we will demonstrate how the method can be applied todetecting regions of tissue growth and loss in brainimages longitudinally collected in a group of childrenand adolescents.

METHODS

Statistical Model

Unlike other brain morphological studies that try tocharacterize the structural variabilities among differ-ent individuals of similar age groups, morphologicalstudies of temporally varying brain structure have anextra temporal dimension. Therefore, a different ap-

FIG. 1. The statistical analysis of local volume change data onives an incorrect impression that the local volume change only occurb) t map of local volume change. Local maxima appear around the c

t map of local volume change after 10-mm Gaussian kernel smoothinsignal-to-noise ratio improves. (d) Thresholded t map superimposed ovolume increase. When the corrected threshold of t . 6.5 is applied,isthmus and splenium of the corpus callosum.

proach to morphometry is required to fully understandthe spatiotemporal complexity of brain development.

Let U(x, t) 5 (U1, U2, U3) be the 3D displacementvector field required to move the structure at positionx 5 (x1, x2, x3) [ R3 and at the reference time 0 of asubject brain to the corresponding position after time t.Thus the structure at x deforms to x 1 U(x, t) withrespect to a fixed reference coordinate. The displace-ment field U(x, t) at fixed time t is usually estimatedvia volume-based nonlinear registration techniques ontwo images taken at time 0 and at time t. Then wepropose to test the following stochastic model of braindevelopment,

­U

­t~x, t! 5 L~U! 1 S 1/2~x!e~x!, (1)

where L is a partial differential operator involvingpatial components and S(x) is the 3 3 3 symmetric

positive-definite covariance matrix, which allows cor-relations between components of the deformations anddepends on the spatial coordinates x only. Since S is

midsagittal section. (a) The sample mean dilatation rate Mvolume. Itear the outer cortical boundaries due, perhaps, to registration error.us callosum. A lot of noise on the cortical boundaries disappears. (c)The smoothing is applied directly to the displacement fields and thehe midsagittal section of the atlas brain. The corpus callosum showsst of the red regions disappears except for the local maximum in the

thes norpg.n tmo

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3DEFORMATION-BASED MORPHOMETRY

symmetric positive-definite, the square root of S al-ays exists. The components of the error vector e are

ndependent and identically distributed as smooth sta-ionary Gaussian random fields with zero mean andnit standard deviation. The error structure S1/2e was

first introduced in Worsley (1996) and Cao and Worsley(1999). An equation of type (1) is called a stochasticevolution equation and it models how the structureevolves over time. Any smooth morphological changecan be completely described by (1) within the bound setby the error structure S1/2e. Modeling the rate ofchange as a differential equation originates from New-ton. If the deformation is assumed to follow a diffusingbehavior, then L can be chosen as the Laplacian oper-ator

L 5 s 2S ­ 2

­x 12

1­ 2

­x 22

1­ 2

­x 32D .

If the morphological changes are assumed to follow afluid dynamics model, L becomes a Navier–Stokes op-erator given in Landau and Lifshitz (1989).

Longitudinal analysis based on (1) is essentially theinverse problem of brain registration. This analysistries to determine the partial differential operator L

FIG. 2. Square grid under translation, rotation, and volume chanranslation. (a) Square grid under no deformation. (b) Horizontal tranotation. The rotation induces the outer region of the center of the roo radially translate outward.

ith given displacement fields. On the other hand, inrain registration, the objective is to find the displace-ent field U that matches homologous points between

wo images based on minimizing a cost function orctually solving partial differential equations. Theost widely used physical models that have been used

n brain registration are elastic deformations and fluidynamics models (Christensen et al., 1993; Thompsont al., 1999; Davatzikos, 1999; Gee and Bajcsy, 1999).uppose that the displacement field U is obtained as aolution of the elastic deformation equation given by

­U

­t5 Lelastic~U! 1 S1/2e,

where the elastic operator

Lelastic~U! 5 l1¹2U 1 l2¹~¹ z U! 1 F

is defined in Warfield et al. (1999). Then using thisdisplacement field U as given data, we try to estimate(1) which minimizes a certain error criterion based onS1/2e. Then the best estimator of L is heavily biasedtoward the prior operator Lelastic. It indicates that theestimation of (1) should be based on an image registra-

Red, volume increase; blue, volume decrease; gray, rotation; yellow,tion caused by local volume increase on the left side. (c) 45° clockwiseion to translate. (d) Volume expansion in the middle causes the grid

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4 CHUNG ET AL.

tion method that does not assume an a priori physicalmodel or on an empirical Bayesian framework. We willuse intensity-based registration algorithms that do nothave explicit physical model assumptions to warp onebrain to another (Collins et al., 1995; Ashburner et al.,1997), but there should be further comparative studiesof the different image registration methods to drawany general conclusions.

It can be assumed that, in the case of morphologicalchanges occurring in a healthy brain over a relativelyshort period of time, deformation occurs continuouslyand smoothly, so the higher order temporal derivativesof the displacement U are relatively small compared tothe displacement itself. In such a case, the first-orderapproximation to L(U) is sufficient to capture most ofthe morphological variabilities over time. Therefore,we approximate ­U/­t with only a first-order term m0(x)which is constant over time; i.e.,

­U

­t~x, t! 5 m0~x! 1 S 1/2~x!e~x!. (2)

By taking the expectation E on the both sides of (2),e see that m0 5 E ­U/­t, the mean displacement rate.

Under the linear model (2), the problem of detectinglocal displacement can be solved with a simple hypoth-esis test:

H0 : m0~x! 5 0 vs H1 : m0~x! Þ 0.

If one wishes to see the convexity of the growthcurve, an additional second-order term is needed in (2).Unlike estimating the first-order linear term m0, theproblem of estimating the second-order nonlinear termrequires a large amount of data to have a statisticallystable result due to intrasubject variabilities acrossspatial and temporal dimensions. In this paper, wehave limited our discussion to the detection of thefirst-order morphological changes and we will not at-tempt to analyze the full model (1).

Detecting Local Displacement

We are interested in detecting regions with statisti-cally significant changes in displacement using thelinear model (2) under a Gaussian error structure. Thisis a standard multivariate statistical inference prob-lem and solved using Hotelling’s T2 statistic (Thompsont al., 1997; Joshi, 1998; Gaser et al., 1999; Cao andorsley, 1999).Let Uj(x, tj) be the 3D displacement vector field re-

quired to deform the structure at the reference time 0of the brain of subject j to the corresponding homolo-gous position after time tj. Let

V j~x! 5U j~x, tj!

t

j

be the displacement velocity of subject j. Then thesample mean displacement velocity V# is given by

V# ~x! 51

n Oj51

n

V j~x!,

while the sample covariance matrix C of the displace-ment velocity is given by

C~x! 51

n 2 1 Oj51

n

~V j~x! 2 V# ~x!!~V j~x! 2 V# ~x!! t,

where the superscript t denotes the matrix transpose.Then Hotelling’s T2 field H(x) is defined as

H~x! 5 nV# t~x!C 21~x!V# ~x!. (3)

At each voxel x, under the hypothesis of no mean dis-placement velocity, i.e., m0(x) 5 0, H(x) is distributedas a multiple of an F distribution with (3, n 2 3)degrees of freedom; i.e.,

H~x! , 3n 2 1

n 2 3F3,n23.

Then the P value of the maxima of H(x), which correctsfor searching across a whole brain volume, is used tolocalize the region of statistically significant structuraldisplacement (Cao and Worsley, 1999). As pointed outin Ashburner and Friston (2000), Hotelling’s T2 statis-ic based on the displacement field does not directlyocalize regions within different structures, but ratherdentifies brain structures that have translated to dif-erent positions. It measures relative position of twoarticular voxels before and after the deformation.herefore, in the context of temporally varying brainorphology where the brain volume change is an im-

ortant concern, the statistic based on the displace-ent field should be taken as an indirect measure of

rain growth. The more direct morphological criterionhat corresponds to the actual brain tissue growth oross is the Jacobian of the deformation field, which weill look at the next section.

etecting Local Volume Change

The deformation in the Lagrangian coordinate sys-em, i.e., fixed coordinate system at time t, is

x 3 x 1 U~x, t!.

he local volume change of the deformation in theeighborhood of a point x and at time t is determinedy the Jacobian J, which is defined as J(x, t) 5 det(I 1

w¹

dg

5DEFORMATION-BASED MORPHOMETRY

­U/­x), where I denotes an identity matrix and ­U/­x isthe 3 3 3 displacement gradient matrix of U given by

­U

­x~x, t! 5 1

­U1

­x1

­U1

­x2

­U1

­x3

­U2

­x1

­U2

­x2

­U2

­x3

­U3

­x1

­U3

­x2

­U3

­x3

2 .

The component ­Uj/­xi is called the displacement ten-sor and, in tensor-based morphometry (Ashburner andFriston, 2000), these nine components form scalarfields used to measure the second-order morphologicalvariabilities. Note that local translation captures thefirst-order morphological variability. A statisticalmodel for the displacement gradient ­U/­x can be di-rectly derived from (1) by taking the partial derivativewith respect to the spatial coordinates x. Hence, bymodeling the morphological changes in the randomfields (Adler, 1981; Worsley et al., 1996), the situationof having two possibly incompatible statistical modelson the displacement U and the displacement gradient­U/­x can be avoided. In our unified statistical model-ing approach using (1), all possible statistical distribu-tions of morphological test criteria can be directly de-rived and easily manipulated from (1).

Since the Jacobian J measures the volume of thedeformed unit-cube after time t, the rate of the changeof the Jacobian J, i.e., ­J/­t, is the rate of the localvolume change. In brain imaging, a voxel can be con-sidered the unit-cube; therefore, ­J/­t(x) essentiallymeasures the change in the volume of voxel x after thedeformation.

Expanding the Jacobian J, we get

J 5 det~I 1 ¹U!

5 1 1 tr~¹U! 1 detr2~¹U! 1 det~¹U!,

here detr2(¹U) is the sum of 2 3 2 principal minors ofU. For relatively small displacements, which is the

case in brain development, we may neglect the higherorder terms and get J ' 1 1 tr(¹U). Taking the partial

erivative with respect to the temporal coordinate t, weet

­J

­t<

­ 2U1

­t­x11

­ 2U2

­t­x21

­ 2U3

­t­x3

5­

­t~¹ z U!

5 ¹ z S­UD ,

­t

where ¹z is the divergence operator. In elastic theory,the volume dilatation is defined as Qvolume(x) 5 ¹ z U(Marsden and Hughes, 1983). Therefore, the rate of theJacobian change is approximately the rate of the vol-ume dilatation change for relatively small displace-ments, i.e.,

­J

­t<

­

­tQvolume~x! 5 Lvolume~x!,

where we term Lvolume to be the volume dilatation rate.Since derivatives of a Gaussian field and the sum ofcomponents of a multivariate Gaussian field are againGaussian field, from (2), we have a linear model on thevolume dilatation rate Lvolume given by

Lvolume~x! 5 lvolume~x! 1 evolume~x!, (4)

where lvolume is the mean volume dilatation rate andevolume is a Gaussian random field with zero mean. Whenlvolume(x) 5 0 in the neighborhood of x, the deformationis incompressible so there is no volume change. How-ever, if lvolume(x) . 0, the volume increases whilelvolume(x) , 0, the volume decreases after the deforma-tion. In certain registration algorithms, the Jacobian Jis forced to be larger than a certain threshold to ensurethe homologous correspondence between two brains(Christensen et al., 1997). When such a registrationalgorithm is used, the power of detecting the region ofstatistically significant volume change may be reduced.Statistical inference on the linear model (4) is easierthan that of (2) since it is a univariate Gaussian. Todetect statistically significant local volume change, theT random field with its P value of the maximum fieldcan be used (Worsley et al., 1994).

Let Qvolumej denote the volume dilatation of the dis-

placement Uj 5 (U1j , U2

j , U3j ) for subject j after time tj.

Then the volume dilatation rate or growth rate Lvolumej of

subject j is

L volumej 5

1

tjQ volume

j 51

tjS­U 1

j

­x11

­U 2j

­x21

­U 3j

­x3D .

In the actual numerical implementation, the displace-ment tensor ­Ui

j/­xi can be computed by the finitedifference on rectangular grid. For example, at voxelposition x 5 (x1, x2, x3),

­U 1j

­x1<

U 1j ~x1 1 dx1, x2, x3! 2 U 1

j ~x1, x2, x3!

dx1,

where dx1 is the length of the edge of a voxel along thex axis. Then the T random field is defined as

1

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6 CHUNG ET AL.

T~x! 5 ÎnMvolume~x!

Svolume~x!, (5)

where Mvolume and Svolume are the sample mean andstandard deviation of Lvolume

j . Under the assumption ofno local volume change at x, i.e., lvolume(x) 5 0, T(x) ;tn21, a Student t distribution with n 2 1 degrees offreedom. As we shall see under Results, the samplemean dilatation rate Mvolume does not provide accuratenformation about where the brain growth is dominantut the T field does (Fig. 1). Then the P value of theaxima of T(x), which corrects for searching across ahole brain volume, is used to localize the region of

tatistically significant structural displacement (Wors-ey, 1994).

mportant Measures in Brain Development

We have presented two different statistics, (3) and5), based on local translation and local volumehanges to measures morphological changes over time.ne might ask if these two statistics are sufficient to

apture temporally varying morphological changes inrain and how one statistic is related to the other. Dohey measure common morphological properties or dif-erent aspects of morphological changes? In this sec-ion, we will give some answers to these questions.

FIG. 3. The procedure for computing the displacement vector fiecan registered onto the atlas brain Vatlas. (2) Compute the displacem

(3) Compute the difference Uj 5 US1j 2 US2

j , which is then automatic

For relatively small displacement, neglecting higherrder terms in the Taylor expansion, the displacement

at x 1 dx can be written as

U~x 1 dx, t! < U~x, t! 1­U

­x~x, t!dx.

As we have pointed out, in tensor-based morphometrysome of all elements of the 3 3 3 displacement gradientmatrix ­U/­x are used to measure morphologicalchanges (Thirion and Calmon, 1997; Ashburner et al.,2000; Thompson et al., 2000). The displacement tensorcan be further decomposed into two parts depending onwhether it is symmetric or antisymmetric:

­Uj

­xi5

1

2 S­Uj

­xi2

­Ui

­xjD 1

1

2 S­Uj

­xi1

­Ui

­xjD .

The antisymmetric first part corresponds to a rotationor vorticity of the deformation and the symmetric sec-ond part corresponds to a strain. Then the displace-ment at x 1 dx can be decomposed into three parts,

U~x 1 dx, t! < U~x, t! 2 w~x, t! 3 dx 1 e~x, t!dx,

j for subject j. (1) Compute the displacement field US1j from the first

field US2j from the second scan registered onto the atlas brain Vatlas.

y defined at each voxel x [ Vatlas.

ld Uentall

tm

­

tc(

7DEFORMATION-BASED MORPHOMETRY

where w 5 12 (¹ 3 U) is the vorticity vector and e 5

(eij) 5 12 [­U/­x 1 (­U/­x)t] is the strain matrix. By

aking the temporal derivative, we have the displace-ent velocity decomposed into three parts:

U

­t~x 1 dx, t! <

­U

­t~x, t! 2

­w

­t~x, t! 3 dx

1­e

­t~x, t!dx. (6)

Equation (6) captures most of the spatiotemporal vari-abilities of the displacement velocity into three compo-

FIG. 4. (Left) 3D statistical parametric maps of local volume incrhresholded at the probability 0.025, 0.025, 0.05 (corrected). (Right) Soronal sections of the atlas brain MRI. The cross-sections are tasomatosensory and motor cortex). The white lines indicate where t

nents: the rate of changes in a translation, a rotation,and a strain for relatively small displacements.

The strain-rate tensor ­eij/­t can be further sepa-rated into two parts: the diagonal elements ­eij/­t de-scribing the length change of the volume element ineach x1, x2, and x3 coordinate, and the off-diagonalelements ­eij/­t (i Þ j) describing the shearing rate ofthe volume element. The volume element is a mathe-matical abstraction defined as an infinitesimally smallcube, but because the smallest unit in brain imaging isa voxel, we may take the voxel as the volume element.Shearing is the deformation that preserves the volumeof a voxel but distorts its shape. Note that the sum of

e (red), volume decrease (blue), and structural displacement (yellow)istical parametric maps are superimposed on the axial, sagittal, andn at the interior of the largest red cluster inside the purple boxross-sections are taken.

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8 CHUNG ET AL.

the diagonal elements of the strain rate is the first-order approximation to the rate of the Jacobianchange; i.e.,

­J

­t<

­K

­t5

­e11

­t1

­e22

­t1

­e33

­t.

It seems that we may have to consider transla-tional, rotational, and strain changes for a completemorphological description. However, the most mean-ingful measurement of brain tissue growth or loss isthe rate of the Jacobian change because it directlymeasures the volumetric changes in the brain. Thelocal translation, the local rotation, and the localshearing change can all be considered as readjust-ments and reorientations of the local brain structuredue to the volumetric changes in the neighboringregions (Fig. 2). In between-subject morphologicalstudies of different clinical populations, such mea-surements might be useful criteria of shape differ-ences. However, in temporally varying within-sub-ject brain morphological studies, we are moreinterested in regions of brain tissue growth or lossthat cause the volumetric changes. Hence, the rate ofthe Jacobian change is the most meaningful morpho-logical measure of brain tissue growth or loss indeformation-based morphometry.

Finally, the dilatation statistic that consists ofspatial derivatives of the displacement field is statis-tically independent from the local translation statis-tic. To see this, note that any partial derivative of astationary Gaussian random field is statistically in-dependent from the field itself (Adler, 1981). Sincethe dilatation consists of spatial derivatives of thedisplacement, it must be statistically independent ofthe displacement. So Hotelling’s T 2 field of the dis-placement and the T field of the dilatation measuremorphologically different properties at the samevoxel.

Detecting Global Volume Change

Standard MRI-based volumetry, where we are in-terested in detecting volume changes of the regionsof interest (ROI), can be considered a special case ofdeformation-based volumetry. Let V t

ROI be the 3Dregion of interest with smooth 2D boundary ­V t

ROI attime t. The region V0

ROI deforms to V tROI under the

eformation x 3 x 1 U( x, t). Note that the volume oftROI is given by

iV tROIi 5 E

V tROI

dx 5 EV 0

ROI

J~x, t!dx.

Then the ROI volume-dilatation rate LROI is given by

LROI 51

iV0ROIi

­

­tiV t

ROIi

51

iV 0ROIi E

V 0ROI

­J

­tdx

<1

iV 0ROIi E

V 0ROI

Lvolumedx.

Therefore, the global ROI volume dilatation rate LROI isequivalent to the average of the local volume dilatationrate Lvolume taken over all V0

ROI. Since Lvolume is distrib-uted as a Gaussian random field, LROI becomes aGaussian random variable. So testing the hypothesiswhether there is any volume change between V0

ROI andVt

ROI can be performed through a simple t test.It is also possible to test the global volume change

via surface-based deformation analysis. Gauss’s Diver-gence Theorem states that

EV 0

ROI

¹ z Udx 5 E­V 0

ROI

U z ndA, (7)

where n is a unit normal vector on the surface ]V0ROI

and dA is the surface area element (Marsden andHughes, 1983). It follows that

LROI <1

iV0ROIi E

­V 0ROI

V z ndA,

where V 5 ­U/­t is the surface displacement velocitydefined on the boundary ]V0

ROI. Since V is distributed asa Gaussian random field, LROI is distributed as aGaussian random variable and statistical inferencewill be again based on a simple t test.

RESULTS

Twenty-eight normal subjects were selected based onhe same physical, neurological, and psychological cri-eria described in Giedd et al. (1996a). Two T1-

weighted MR scans were acquired for each subject atdifferent times on the same GE Sigma 1.5-T supercon-ducting magnet system. The first scan was obtained atthe age of 11.5 6 3.1 years (min 7.0 years, max 17.8ears) and the second scan was obtained at the age of6.1 6 3.2 years (min 10.6 years, max 21.8 years). Theime difference between the first and the second scanas 4.6 6 0.9 years (min time difference 2.2 years, max

mcmtl

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9DEFORMATION-BASED MORPHOMETRY

time difference 6.4 years). Using the automatic image-processing pipeline (Zijdenbos et al., 1998), a total of 56MR images were transformed into standardized stereo-tactic space via a global affine transformation (Ta-lairach and Tournoux, 1988) followed by a nonlineardeformation to match the atlas brain Vatlas. The globalaffine transformation removes most of the intra- andintersubject global differences in brain sizes; adultbrains are approximately 5% larger than those of5-year-old children (Dekaban, 1977; Dekaban andShadowsky, 1978). Because we are only interested infinding local morphological changes, these global mor-phological variabilities should be removed via globalaffine transform in order to improve the power of de-tection. These registration procedures are based on anautomatic multiresolution intensity matching algo-rithm (Collins et al., 1995; Collins and Evans, 1999).Unlike other registration algorithms that assume acertain fluid dynamics or an elastic deformation model,the intensity-based registration does not assume anyexplicit physical model in which the deformation fromthe subject brain to the atlas brain should follow (Geeand Bajcsy, 1999; Thompson et al., 2000). So the defor-

ation fields obtained from these registration pro-esses can be considered free of any explicit physicalodel assumption although there might be some in-

ensity-based model assumption, which somehow re-ates to a physical model.

If US1j and US2

j are the displacement obtained fromthe nonlinear registration of the first and the secondscan of subject j to the atlas brain Vatlas at time tS1

j andtS2

j , the actual displacement Uj between the first scanand the second scan is Uj 5 US1

j 2 US2j and the time

difference is tj 5 tS2j 2 tS1

j (Fig. 3). It is true that if thefirst scan were directly registered to the second scanwithout going through the atlas brain, the registrationerror would be smaller. However, the displacementfields obtained by the direct registration method stillmust be registered onto the atlas brain in order to formstatistical parametric maps. The reason for such sta-tistical treatment to analyze the structural data isobvious considering that the displacement field ob-tained from image registration algorithms for braindevelopment contains a fairly large component of error.The length of the displacement velocity we have ob-served for the spatially normalized MR scans of 28normal subjects is usually less than 1 mm/year, i.e.,m0 5 E(­U/­t) # 1 mm/year in average. Optimisticallyassuming that the image registration algorithm is ac-curate to within one voxel distance (usually 1 or 2 mm),the registration error seems to be relatively large inbrain development. So one may be skeptical aboutwhether the deformation-based morphometry can pos-sibly detect such small changes. Nevertheless it is stillpossible to pick out the signal when there are enoughdata; Figure 1 illustrates how image smoothing andthe statistical treatments improve the power of detec-

tion. Statistical treatments compensate for some ofsuch registration errors. Finally the displacement ve-locity field is smoothed with a 10 mm full width athalf-maximum (FWHM) Gaussian kernel to increasethe signal to noise ratio. Gaussian kernel smoothingwith FWHM 5 4(ln 2)1/2=t of the signal f(x), x [ R3 is

efined as the convolution of the signal f with theGaussian kernel:

F~x, t! 51

~4pt! 3/2 ER 3

e 2~x2y! 2/4tf~y!dy.

Without the smoothing, it may have been more difficultto detect morphological patterns illustrated in Fig. 1.However, the Gaussian kernel smoothing sometimestends to blur the fine details of deformation pattern.

The regions of statistically significant displacementhave been detected (Fig. 4, yellow) by Hotelling’s T2 fieldwith the corrected threshold (Cao and Worsley, 1999):

P~ maxx[Vatlas

H~x! . 60.0! < 0.05.

Most of the structural movements were observed in thefrontal lobe without any accompanying significantchange in local volume. This may indicate that thereare continued readjustments of the exact position ofbrain structures in the frontal lobe without any braintissue growth or loss in adolescence. Also note that thestatistically significant displacement occurs evenly andshows some degree of symmetry between the left andthe right hemispheres. Because the local translationstatistic measures the relative displacement of brainstructure, it does not truly reflect the brain tissuegrowth process. However, it does indicate the principaldirection of the brain growth as shown in the purplebox in Fig. 4 and enlarged in Fig. 5. Hence, the localtranslation statistic should be used in conjunction withthe local volume change statistic to fully understandthe complex dynamics of temporally changing morpho-logical pattern.

Previous developmental MRI studies have providedevidence for age-related increase in total white mattervolume and decrease in total gray matter volume(Jernigan et al., 1991; Pfefferbaum et al., 1994; Ra-japakse et al., 1996; Riess et al., 1996; Courchesne etal., 2000), but the analytic procedures used in thesestudies did not allow the investigators to detect localvolume change. The local volume change statistic T(x)is computed using the formula (5) with tj 5 tS1

j 2 tS2j .

he t statistic map is thresholded at

P~ maxx[Vatlas

T~x! . 6.5! < 0.025,

P~ minx[Vatlas

T~x! , 26.5! < 0.025.

gTOtar(sbstdtystsltd

ls

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10 CHUNG ET AL.

At this threshold, most of the local volume increaseobserved around the corpus callosum in Fig. 1 disap-pears except for very few localized statistically signif-icantly “peaks” in the isthmus and splenium. Therewas no volume change detected in the rostrum andgenu. Figure 4 also shows the localized growth in thecorpus callosum on the coronal section (the single reddot). Therefore, we observe a highly focused region ofbrain tissue growth in the isthmus and splenium of thecorpus callosum. Pujol et al. (1993), Giedd et al. (1996b,1999), and Thompson et al. (2000) reported similarresults of growth pattern at the corpus callosum.

The growth at the corpus callosum seems relativelysmall when compared to the global peaks observedpredominantly in somatosensory and motor cortex (thelargest red cluster in Fig. 4). Localized brain tissue losswas also detected at the same time as tissue growth.This tissue loss was highly localized in the subcorticalregion of the left hemisphere (Fig. 4, blue). Similarresults were also reported in Thompson et al. (2000),where the extent of the peak growth was wider and lesslocalized than our study has found. It seems our sta-tistical treatments based on the large sample size (n 528) tend to remove a lot of intrasubject variabilitiesand pick out the common morphological pattern among

FIG. 5. A close-up of part of the outer left hemisphere insidedisplacement velocity subsampled every 10 mm and scaled by 50 mmlocal volume expansion (red) causes the translational movement of tarrows are manually enhanced to clearly indicate the direction of th

subjects compared to the smaller sample size (n 5 6)studied in Thompson et al. (2000). Slightly differentrowth patterns observed between our study andhompson et al. (2000) may be due to many factors.ur approach is based on the systematic statistical

reatments of large sample size (n 5 28) with a lessccurate intensity-based automatic registration algo-ithm. While the approach taken in Thompson et al.2000) is based on a sample size of six without anytatistical approach, a more accurate elastic modelased registration algorithm with manually matchedulcal landmarks was used. However, the most impor-ant difference between the two studies is the ageistribution of the subjects. In Thompson et al. (2000),he age distribution of the six subjects is in most partounger than our mean age of 11.5 years for the firstcan and 16.1 years for the second scan. So althoughhere are similar growth patterns common to bothtudies such as predominant growth at parietal cortex,ocalized peak growth at the corpus callosum, etc., thewo studies are detecting morphological changes inifferent but nonexclusive age groups.We have observed very interesting relations between

ocal displacement change and local volume changetatistics as illustrated in Figs. 4 and 5. Figure 5 is the

e purple box in Fig. 4. Black arrows represent the sample meanr. The direction of the mean displacement velocity suggests how the

structure (yellow) toward the region of atrophy (blue). The heads ofisplacement.

th/yeahe

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A

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11DEFORMATION-BASED MORPHOMETRY

close-up of the parietal region of the left hemisphere(the purple box in Fig. 4), showing a large local dis-placement from the region of local volume increase(gray matter) to a region of local volume decrease(white matter), indicating how the structure boundary(inner cortical surface) has moved from the increasingvolume to the decreasing volume. This phenomenon isalso schematically illustrated in Fig. 1b, where thesquare grid is undergoing a horizontal translation fromthe region of volume increase of the left to the region ofvolume decrease on the right, and Fig. 2d, where thevolume expansion in the middle causes the neighbor-ing structures to radially translate outward. It seemsthat by studying these two statistical parametric mapssimultaneously, the complex dynamic patterns in tem-porally varying brain morphology can be captured.

CONCLUSIONS

The deformation-based volumetry presented herecan localize the regions where local volume growth orloss occurs over temporally varying brain morphologyby measuring the rate of local volume changes. Byusing the displacement velocity instead of the displace-ment itself in detecting the anatomical changes, tem-poral variabilities in MR images for different agegroups and different time intervals can be accountedfor. As an illustration, we have applied the method toMR scans of 28 normal children and adolescents anddetected regions of tissue growth in the corpus callo-sum and somatosensory and motor cortex.

Our unified statistical framework based on the de-formation-based volumetry can be also used as a toolfor future investigations of neurodevelopmental disor-ders where volumetric analysis would be relevant. Itcan also be applied to a general morphological studies,such as testing for structural shape differences be-tween two different groups of subjects.

ACKNOWLEDGMENTS

Steve Robbins at Montreal Neurological Institute provided techni-cal assistance with the atlas brain. The valuable comments of JohnAshburner at the Wellcome Department of Cognitive Neurology;Luke Oh at the Department of Neuroscience, University of Connect-icut Health Center; and Jim Ramsay at the Department of Psychol-ogy, McGill University, are also acknowledged. The Matlab programused in computing the corrected thresholds of the T random field andHotelling’s T2 random field can be found at http://www.math.mcgill.ca/keith/BICstat.

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