M.I.T. Media Laboratory Perceptual ComputingTechnical Report No. 331 May 1995Submitted to IEEE PAMI, February 1995Characterization of Neuropathological Shape DeformationsJohn Martin, Alex Pentland, Stan Sclaro�, and Ron KikinisSurgical Planning Laboratory, Department of RadiologyBrigham and Women's Hospital75 Francis Street, Boston, MA 02115Perceptual Computing Section, The Media LaboratoryMassachusetts Institute of Technology20 Ames Street, Cambridge, MA 02139Email: martin@bwh.harvard.edu, sandy@media.mit.edu,sclaro�@cs.bu.edu, kikinis@bwh.harvard.eduAbstractWe present a framework for analyzing the shape de-formation of structures within the human brain. Amathematical model is developed describing the defor-mation of any brain structure whose shape is a�ectedby both gross and detailed physical processes. Usingour technique, the total shape deformation is decom-posed into analytic modes of variation obtained from�nite element modeling, and statistical modes of vari-ation obtained from sample data.Our method is general, and can be applied to manyproblems where the goal is to separate out importantfrom unimportant shape variation across a class of ob-jects. In this paper, we focus on the analysis of dis-eases that a�ect the shape of brain structures. Becausethe shape of these structures is a�ected not only bypathology but also by overall brain shape, disease dis-crimination is di�cult. By modeling the brain's elasticproperties, we are able to compensate for some of thenonpathological modes of shape variation. This allowsus to experimentally characterize modes of variationthat are indicative of disease processes.We apply our technique to magnetic resonanceimages of the brains of individuals with schizophre-nia, Alzheimer's disease, and normal-pressure hydro-cephalus, as well as to healthy volunteers. Classi�ca-tion results are presented.

1 IntroductionVarious neurological disorders a�ect the grossanatomical shape of di�erent brain structures. Thesechanges have been studied for several decades, usingboth postmortem and invasive in vivo methods. Re-cent advances in the contrast and resolution of mag-netic resonance (MR) scanners now make it possibleto study these shape e�ects in vivo and noninvasively,with the potential for better diagnosis and treatment.Our aim is to quantitatively describe these pathologi-cal shape deformations.Previous studies of neuropathological morphologysu�er from two drawbacks. First, these studies haveused just linear [1-5], planar [6], and/or volumetric[5,7-17] measurements in order to characterize neu-ropathological shape changes. Research that has usedmore general shape measures has been qualitative, e.g.having a user manually grade the uniformity of corpuscallosum thinning [3][18] and/or smoothness [3]. Noneof these previous shape descriptions is both generaland quantitative.The second drawback of previous work involvesthe method of normalizing for nonpathological inter-patient di�erences. These di�erences are a result ofboth genetic and environmental factors, which causebiological structures to have a large range of normalvariation. To properly study pathological deforma-1

(a)(b)(c)Figure 1: Schematic representation of head and ven-tricles. (a) Di�erent shaped heads, no ventricular dis-ease present. The only ventricular shape di�erence isdue to the di�erence in head shape. (b) Same shapedheads, with ventricular disease. The lower tips of theventricles are expanded due to the disease's physicalprocesses. (c) Di�erent shaped heads, with ventricu-lar disease. The pathological di�erence in ventricularshape is partially masked out by the nonpathologicaldi�erence due to head shape.

tions, these nonpathological di�erences must �rst betaken into account. Previous studies addressed this bynormalizing brain structure measurements for overallbrain size. While this is the correct �rst-order approx-imation, a more thorough normalization would takeinto account the complete brain shape.We overcome these limitations by creating a math-ematical framework that (1) separates out disease de-formation from deformation due to head shape, (2)uses the complete head shape to normalize cranial con-tents, and (3) represents pathological deformation in ageneral and natural manner. Our shape description isin terms of physical and statistical deformation modes.These modes can be displayed to show how structuresdeform due to both head shape and pathology, and canbe used in pattern recognition algorithms to classifydiseases based on shape changes.2 The Basic IdeaThis paper addresses the general problem of sepa-rating out interesting from uninteresting shape defor-mations in a class of objects. The method we develophas applications in such areas as understanding bio-logical shapes, and in distinguishing important fromunimportant shape di�erences in the encoding of func-tional classes. In this paper we focus on the speci�cexample of separating out nonpathological shape vari-ation from the pathological deformations caused byvarious neurological disorders.Figure 1 demonstrates our framework. Two peoplewith di�erent head shapes will tend to have di�erentventricular shapes, even in the absence of pathology.This is illustrated in Figure 1a. With disease, how-ever, two people with the same head shape and origi-nally the same ventricular shape will end up with dif-ferent ventricular shape, as illustrated in Figure 1b.In the most general case, both the e�ects of headshape and ventricular pathology will be present simul-taneously, complicating diagnosis based on ventricularshape. Figure 1c shows this case.To make this precise, we represent the shape of anaverage, healthy brain structure as a set of 3-D pointpositions XA. This list will contain one entry for ev-ery spatial location included in the model. This couldbe every voxel in the volume, just the surface voxels,or even just a small set of landmarks. The particularchoice is an implementation issue. Then, for any par-ticular patient p, the observed deformation up;i awayfrom any point xi of XA can be separated into twodistinct components:up;i(xi) = uHp;i(xi) + uDp;i(xi) (1)2

(a) (b)Figure 2: Reconstructions of the left putamen created from MR images. (a) Normal, healthy adult. (b) Patientwith schizophrenia.(a) (b) (c)Figure 3: Reconstructions of the cerebral ventricles created from MR images. (a) Normal, healthy adult. (b)Patient with Alzheimer's disease. (c) Patient with normal-pressure hydrocephalus.where uHp;i(xi) is the deformation due to global ef-fects that are correlated with overall head shape, anduDp;i(xi) is the deformation caused by disease and in-dividual local variation. For the entire point set wehave Up =UHp +UDp (2)where Up is the 3V � 1 vectorUp = 26666664 up;1(x1)up;2(x2)���up;V (xV ) 37777775 (3)and UHp and UDp are de�ned similarly. V is the num-ber of points in the model.What is needed then is a method that separatesout these two types of deformations, allowing just thepathological deformationsUDp to be analyzed. We ac-complish this by using the �nite element method tocreate a physical model that describes the macroscopice�ects caused by di�erent head shapes. After elasti-cally warping the cranial contents according to this

physical model, we are left with residual shape di�er-ences across patients that are largely independent ofhead shape. Once an entire database of patients hasbeen normalized for head shape in this manner, statis-tical techniques are then used in order to characterizepathological shape variation.We apply modal analysis to the physical modeling,and principal component analysis to the experimen-tal observations. Both are eigenanalysis techniques,and represent shape in terms of deformation modes[19][20][21]. These modes represent unique, naturalcoordinates in which to express the shape and defor-mation of brain structures.2.1 An exampleTo demonstrate our method, we examine defor-mations in the shape of the left putamen caused byschizophrenia, and deformations in the shape of thecerebral ventricles caused by Alzheimer's disease (AD)and by normal-pressure hydrocephalus (NPH). Figure2 shows the left putamen of a healthy volunteer andof a patient with schizophrenia, while Figure 3 showsthe ventricles of a healthy volunteer, an AD patient,and an NPH patient. Recent studies have shown thatschizophrenia can cause the putamen to enlarge [17],3

and that both AD and NPH cause the ventricles toenlarge [22]. While these studies treated just volu-metric changes, we seek to explore other pathologicaldeformations in addition to just volume.As a reference point for the methodology to be de-veloped in the following sections, we consider here pos-sible ways of classifying patients into the two classesshown in Figure 2. Given a data set consisting ofsamples from these two categories, the most straight-forward classi�cation procedure would be to use justone feature, putamen volume. However, since a per-son with a larger head will tend to have larger puta-men, even if healthy he or she may be misclassi�ed asschizophrenic. Therefore the second procedure to trywould be to normalize each person's putamen volumeby his or her overall intra-cranial cavity (ICC) volume.Using each of the above two features, we ran aGaussian linear classi�er on a data set consisting of 13schizophrenics and 12 normal control subjects. As Ta-ble I shows, normalizing for overall ICC volume actu-ally causes the classi�cation rate to slightly decrease.While this decrease in performance is probably mostlydue to our small sample size, it also points to possi-ble problems in our normalization. While head size iscertainly important, the complete head shape is reallywhat we ought to use in the normalization.TABLE I

Putamen Volume Classification Rates

Feature(s) %Correct

Putamen volume 60Normalized putamen volume 56With this in mind, our technique can be viewedas a more sophisticated version of the two features ofTable I. Instead of using just brain structure volume,a principal component analysis of the brain structuredeformation is calculated, providing other importantdiscriminating features in addition to volume. Also,instead of normalizing for just head size, we normalizefor the complete head shape.2.2 Organization of the paperThe remainder of this paper is organized as follows.Section 3 describes our procedure for head shape nor-malization. Once a database of patients has been nor-malized, we then statistically characterize the patho-logical deformation, as described in Section 4. Section

5 presents the results of applying our technique to twodi�erent medical data sets. In Section 6 we discuss ourmethod and experimental results. Section 7 comparesour technique to other research, and Section 8 sum-marizes the work done.3 Head Shape NormalizationTo characterize the global deformations due to headshape, we model the ICC as a linear elastic material,and then set up equations describing its behavior. Onereason for using a physically-based model is that wecan formulate an approximate physical model for theICC as a whole. A second and more practical rea-son is that the �nite element implementation of physi-cal modeling provides analytic interpolation functionsthat allow us to relate deformations at one point toforces and deformations throughout an object. Thesefunctions make the task of accurately warping and re-sampling the data straightforward, allowing us to re-late each data set to a standard or normative headshape.3.1 The �nite element methodThe most common numerical approach for solv-ing elastic deformation problems is the �nite elementmethod (FEM) [23]. The major advantage of the FEMis that it uses the Galerkin method of surface inter-polation. This provides an analytic characterizationof shape and elastic properties over the whole sur-face, rather than just at the nodes. The ability tointegrate material properties over the whole surfacealleviates problems caused by irregular sampling offeature points. It also allows variation of the elasticbody's properties in order to weigh reliable featuresmore than noisy ones, or to express a priori constraintson size, orientation, smoothness, etc. In Galerkin'smethod, we set up a system of polynomial shape func-tions that relate the displacement of a single point tothe relative displacements of all the other nodes of anobject. By using these functions, we can calculate thedeformations which spread over the body as a functionof its constitutive parameters.In the isoparametric FEM formulation, polynomialshape functions H are de�ned in a parametric spacer = (r; s; t)T , with both positions and displacementsmapped from parametric to element coordinates usingthe same shape functions:x(r) = H(r)X (4)u(r) = H(r)U: (5)4

Here X and U denote the nodal position and displace-ment vectors, respectively, and are de�ned in the ele-ment (object) coordinate system, x = (x; y; z)T is anypoint in the element (object), and u is the displace-ment at x. Note that although u is the displacement inthe element coordinate system, because x is a functionof r, u can be written as a function of either x or r.(Throughout this paper, a vector with a \hat" (^) de-notes a set of FEM nodal positions or displacements,while a vector without a \hat" denotes positions ordisplacements at a set of non-nodal points.)For most applications it is necessary to calculate thestrain due to deformation. Strain � is de�ned as theratio of displacement to the actual length. The poly-nomial shape functions can be used to calculate thestrains (�) over the body provided the displacementsat the node points are known:�(x) = B(x)U (6)where the strain displacement matrix B is computedby taking the appropriate derivatives of the interpola-tion matrix H. Because B is a function of x, and His a function of r, the chain rule must be invoked inorder to perform the di�erentiation. This requires theuse of the Jacobian matrix J:J = 26666664 @x@r @y@r @z@r@x@s @y@s @z@s@x@t @y@t @z@t 37777775: (7)As mentioned earlier, we need to solve the prob-lem of deforming an elastic body subjected to externalforces. This requires solving the equilibrium equationKU = R (8)for the set of nodal displacements U. Here R is theload vector whose entries are external forces acting onthe nodes, and K is the sti�ness matrix. K is com-puted directly from the strain displacement matrix byintegrating over the object's volume:K =ZV BTCB dV; (9)where C is the material matrix, which expresses thematerial's particular stress-strain law. See Bathe [23]for more details on setting up FEM integrals and equa-tions.

3.2 Modal analysisIt is often more convenient to represent the nodaldisplacement vector U in the modal coordinate sys-tem, in which displacements are represented as lin-ear combinations of an object's free vibration modes.These modes provide a unique, natural, and compactcoordinate system in which to represent shape andshape change, are computationally e�cient to calcu-late, and have convenient robustness properties withrespect to sampling irregularities and measurementnoise [19].To compute the free vibration modes, Equation 8is diagonalized via an orthogonal transform �:U = � ~U (10)where ~U is a vector of generalized displacements inthe new coordinate system. The columns of � are thebasis vectors of this new coordinate system.Substituting Equation 10 into Equation 8 and pre-multiplying by �T yields~K ~U = ~R (11)where ~K = �TK� and ~R = �TR.The optimal transformation matrix � is derivedfrom the eigenvalue problemK�i = �i�i (12)which, for a discretization with N nodes, has 3N so-lutions (�1;�1); (�2;�2); :::; (�3N;�3N ). For dynamicsystems, these eigenvectors are called the free vibra-tion modes of the system, with the correspondingeigenvalues giving the square of the vibrational fre-quency.We see then that the transformation matrix � hasfor its columns the eigenvectors of K,� = [�1;�2;�3; � � � ;�3N ] ; (13)and that ~K is a diagonal matrix with the eigenvalueson its diagonal:~K = �TK� = � = 26666664 �1 �2 : : : �3N 37777775 :(14)Because the sti�ness matrix has been diagonalized,the resulting system of equations is decoupled andtherefore computationally much simpler. Also, the5

high frequency modes often can and should be dis-carded for two reasons. First, the low frequency (loweigenvalue) modes contain more information than thehigh frequency (high eigenvalue) modes in the sensethat their amplitudes are larger and therefore for ob-ject discrimination they are typically more powerful.Second, because of noise considerations the low fre-quency modes are more reliably estimated than thehigh frequency modes.3.3 FEM model of average ICCIn order to create an FEM model of the averageICC, we �rst construct a voxel-based average frompatient data sets. This is done by automatically seg-menting the ICC from each of the data sets [24], per-forming a rigid body alignment between all the ex-tracted ICCs [25], and then averaging spatial occu-pancy over the aligned ICCs.Once the average ICC is constructed, a deformableFEM model is then warped so that its shape approxi-mates that of the average ICC. This procedure startsby extracting theM surface voxels of the average ICC:XICCA = [x1; y1; z1; � � � ; xM ; yM ; zM ]T : (15)Next we attach virtual springs between each of thesevoxel coordinates and the closest point on the surfaceof our �nite element model. These virtual springs de-�ne forces acting on the deformable model:fm = d(xm; FEM surface) (16)where d(�) is the displacement vector between thegiven point and the nearest point on the FEM surface.In general, a force fm will not act at an FEM node;however we can use the FEM interpolation functionsH to distribute the force to the surrounding nodes ina physically meaningful way. The load vector of Equa-tion 8 is therefore constructed asRA = HTFA: (17)where FA is a vector consisting of all the individualforces fm.Equation 8 can then be solved for the nodal dis-placements that give the FEM model a shape approx-imating that of the average ICC:UICCA =K�1RA: (18)These nodal displacements can be added to the orig-inal nodal positions to obtain the nodal positions ofthe average ICC:XICCA = X + UICCA ; (19)

where X is the original, undeformed nodal positionvector of Equation 4.We can then transform into the modal coordinatesystem via Equation 10:~UICCA = �T UICCA (20)where the modal coe�cient vector ~UICCA speci�es howmuch of each deformation mode is contained in theshape of the average ICC. If only the modal coe�-cients ~UICCA and not the nodal displacements UICCAare required, then we can skip solving for UICCA , andinstead solve directly for ~UICCA , using Equation 11.As mentioned earlier, because the system of equationsis decoupled in the modal coordinate system, usingEquation 11 directly is much faster.Figure 4 shows several of the resulting physicaldeformation modes of the average ICC. In order toobtain additional computational advantages, thesemodes were computed using the \idealized modes"technique described in [19]. It should be noted thatwhile these idealized modes provide a physical frame-work and a very useful �rst approximation to the ac-tual physical modes of the ICC, we cannot claim thatthese are necessarily the same as the actual modes.The use of more recent implementations of modal �t-ting [26] will recover modes that are closer to the ac-tual ICC modes.See [19] for a more detailed description of modalanalysis. Software that uses the FEM and modalanalysis to recover and describe shapes is avail-able from whitechapel.media.mit.edu in the �le/pub/modal.tar.Z.3.4 FEM �tting to patient ICCsJust as the deformable FEM model was warpedto the shape of the average ICC, the model can alsobe warped to �t the ICC of any particular patient p.First, the ICC surface points XICCp of the patient areextracted. Virtual springs are then attached betweenXICCp and the deformable model, generating a forcevector Fp. Next these forces are distributed to theFEM nodes, creating the nodal load vector Rp. Asbefore, the nodal displacements can now be found:UICCp = K�1Rp; (21)and then used to �nd the nodal positions for patientp's ICC: XICCp = X+ UICCp : (22)Alternatively, we can recover the modal amplitudes~UICCp via Equation 11.6

(a) (b) (c)(d) (e) (f)Figure 4: The average ICC (a) and several of the ICC's physical deformation modes (b-f).Note that although each ~UICCp (or, equivalently,each UICCp ) represents a di�erent patient p, each pa-tient is warping the same original set of FEM nodes.The distinction between patients comes from the par-ticular amount and type of deformation that the set ofnodes undergoes; the nodes themselves all start out inthe same position. The net e�ect is that each patient'sICC surface points have now been referred back to thesame set of nodes. Thus the original MR sampling dif-ferences between patients' ICCs have been removed.3.5 WarpingThe recovered displacement vectors can now beused to normalize each patient's cranial contents inorder to account for his or her particular head shape.We have available the set of nodal positions XICCAand displacements UICCA of the average ICC, as wellas XICCp and UICCp for each patient p. To avoid creat-ing gaps (unde�ned voxels) when warping, each voxelcoordinate in the average ICC coordinate system ismapped into the patient's coordinate system [27]. Be-cause this will produce non-integer coordinates in thepatient space, interpolation is necessary in order tocalculate an intensity value for the voxel. Repeatingthis procedure for every voxel position in the averageICC space completely �lls up that coordinate systemwith values from the ICC of patient p.Recall that the �tting was not done directly from

patient ICC to average ICC, but rather from unde-formed model to patient ICC, and from undeformedmodel to average ICC. Thus the required mappingfrom average ICC space to patient p space must bedone in two steps, as described in the next two sec-tions. The �nal section shows how the other availableinformation - the modal amplitudes ~UICCA and ~UICCp- can be used to perform the warping directly in themodal coordinate system.3.5.1 From average ICC space to parameterspaceGiven a voxel position x in the average ICC coordi-nate system, the �rst step is to transform it into theparametric coordinate system. This can be done withEquation 4, using the nodal positions of the averageICC: x(r) =H(r)XICCA : (23)Note, however, that Equation 23 must be inverted -given x we need to �nd r.This is accomplished as follows. According toEquation 23, once the nodal positions XICCA areknown, then x is just a function of r:x = g(r) (24)where g is the system of three polynomials given inEquation 23. To solve for r, g must be inverted:r = g�1(x): (25)7

The solution can be found iteratively using Newton'sMethod: Jk �xk+1 � xk�+ gk = 0 (26)where J is the Jacobian matrix de�ned in Equation 7,and k represents iteration [28][29].3.5.2 From parameter space to patient spaceOnce the parametric coordinates r are known for x,they must be converted into the coordinate systemof patient p. Once again this is accomplished usingEquation 4, but this time we use it directly, with theknown r and the ICC nodal positions of patient p:x0(r) =H(r)XICCp : (27)Once x0 is known, we can simply look into patientp's data set in order to assign a value to x. If theoriginal gray scale data is being warped, then trilin-ear interpolation can be used to calculate the value.However, because our data sets are segmented, thevalue is just set to that of the integer coordinates thatare nearest to x0. To avoid the aliasing that this in-troduces, the segmented data can be smoothed beforewarping.3.5.3 Modal warpingIn our implementation, we recover the modal am-plitudes directly, without ever calculating the nodalpositions and displacements. As already mentioned,modes o�er two important advantages: they decouplethe FEM equations to yield improved computationalperformance, and they provide a unique, canonical co-ordinate system in which to represent shape.In the modal coordinate system, instead of Equa-tion 4, the interpolation and warping is done by com-bining Equations 5 and 10:u(r) = H(r)� ~U: (28)To further increase computational e�ciency, the poly-nomial deformations of Equation 28 are approximatedby a 3 � 3 modal deformation matrix D(r; ~U) [19],which is used to map from parametric to element co-ordinates: x(r) = D(r; ~U)r: (29)Thus for a given voxel position x in the average ICCcoordinate system,x(r) = D(r; ~UICCA )r (30)is inverted and solved via Newton's Method to �nd r,and then x0(r) = D(r; ~UICCp )r (31)

is solved directly to �nd x0.Equation 30, Equation 31, and the appropriate in-terpolation scheme can be applied to every voxel posi-tion in the average ICC coordinate system. The �nalresult is that locations inside a patient's cranium aredisplaced to the position that they would occupy ifthe patient had an average-shaped ICC. By mappingbetween patient ICC and average ICC in this man-ner, we account for the geometric di�erences due toindividual head shape, as well as the MR samplingdi�erences between patients.3.6 Example: Healthy subject with largecraniumWarping of the cranial contents can result in ven-tricles that are closer to the average. This is demon-strated in Figure 5. Figures 5a and 5b show the av-erage ICC and ventricles, while Figures 5c and 5dshow the ICC and ventricles of one of the healthy sub-jects. This particular subject's ICC is larger than av-erage, particularly in the front-to-back direction (left-to-right in the �gure). This ICC shape di�erenceis propagated down to the ventricles, where we seesimilar shape di�erences between the two ventricularsystems. Calculating the ICC physical deformationmodes that warp this subject's ICC to the shape of theaverage ICC, and then applying that warping to thesubject's ventricles, produces the warped ventricles inFigure 5e. As can be seen, these warped ventricles aremore similar to the average ventricles in Figure 5b.4 Characterization of Disease States4.1 Shape representationAs described in Equations 1-3, our shape represen-tation for a brain structure of a particular patient pis a list of displacements Up away from a set of aver-age point positions XA. These average positions arefound by �rst constructing a volumetric average of thestructure from a group of patient data sets, and thenextracting the surface voxels.The displacement vector Up is then constructed asfollows. Using the original, unwarped data, the surfacevoxels Xp of the structure under study are extracted.Then the displacement at a point xi of XA is com-puted by �nding the nearest point in Xp. By doingthis for each of the voxels on the averaged structure'ssurface, we can compute the displacement vector Upof a patient p in the data set.The above procedure can also be applied to thewarped patient data. Because nonpathological defor-mation is removed by the head shape normalization,8

(a) (b)(c) (d) (e)Figure 5: Normalizing ventricular shape by cranium shape. (a) Average ICC (computed from all the data sets).(b) Average normal ventricles (computed from just the healthy subjects). (c),(d) ICC and ventricles of one ofthe healthy subjects. Compared to (a) this ICC is larger than average, especially in the front-to-back direction(left-to-right from this viewing direction). The ventricles exhibit similar di�erences as compared to the averageventricles in (b). (e) Subject's ventricles after warping her ICC in (c) to the shape of the average ICC in (a).The ventricles have decreased in size, most notably in the front-to-back direction. Because we have normalizedfor head shape, the ventricles are now more similar to the average ventricles in (b).the deformation remaining after warping is due to dis-ease processes (along with any nonpathological di�er-ences unaccounted for by head shape). The displace-ment vector constructed is thus UDp of Equation 2.We have therefore met our original goal of sepa-rating the total deformation into its pathological andnonpathological components. Furthermore, just aswith the FEM ICC �tting, because we have referredeach patient's particular coordinate system back tothe same standard coordinate system, sampling dif-ferences between patients' brain structures have beenremoved. In the remainder of this section, we focuson using the set of pathological displacement vectorsUDp to characterize disease.4.2 Eigenanalysis of shape variationWe can transform the pathological displacementvectors into a coordinate system in which deforma-tions are more naturally represented. This is accom-plished through the use of principal component anal-ysis.Principal component analysis is a statistical tech-nique that �nds the directions of maximumvariability

inherent in a data set. When applied to 2-D outline or3-D surface data, the principal components are calledthe eigenshapes of the structure under study. Unlikethe physical modes we have been using throughoutthis paper, eigenshapes are derived solely from a dataset, without the aid of an underlying physical model.The eigenshapes of our data sets are found as fol-lows. From our P patients, each with a 3V � 1 patho-logical displacement vectorUDp , the sample covariancematrix is formed asS = 1P � 1 PXp=1(UDp �UD)(UDp �UD)T ; (32)where UD is the mean of the P displacement vec-tors. Solving for the eigenvectors of S yields theprincipal components, or eigenshapes, of the data set.These eigenvectors are ordered according to their cor-responding eigenvalues.The principal components can be assembled as thecolumns in a matrix . Any patient's displacementvector can be written as a linear combination of the9

principal components:UDp = bp; (33)where bp is the vector of projections onto the principalcomponents for patient p.4.3 Classi�er DesignThere are 25 patients in each of our two data sets,and hence 24 principal components. Employing anengineering rule of thumb which states that the top14 eigenvectors of a sample covariance matrix are re-liably estimated, there are six principal componentswith which to perform classi�cation.Designing a Gaussian quadratic classi�er requiresestimating the 6 � 6 covariance matrix �i of each ofthe C classes present. However, with only between 7and 13 members per class, the �i cannot be reliablyestimated. The quadratic classi�er will consequentlybe overparametrized, with the result that the trainingset will not be a good predictor for new cases. Wetherefore use a Gaussian linear classi�er, which as-sumes that �1 = �2 = � � � = �C = �. Thus onlythe overall � has to be estimated, which can be doneusing all the data sets.In Gaussian linear classi�cation, a discriminantfunction is computed for each of the C classes1:gi(bp) = 2mTi ��1bp �mTi ��1mi (34)where mi is the mean of class i and bp represents thedata set to be classi�ed. In our case bp is a vectorof the projections of patient p onto the �rst six prin-cipal components (see Equation 33), and mi is the6 � 1 mean vector computed by averaging the pro-jections over disease class i. The classi�cation rule isthen to choose the class i which has the largest gi, or,equivalently, the maximum probability density whenevaluated at bp [30].In order to separate the training stage from the clas-si�cation stage, we use the leave-one-out [31] method.In this procedure, the sample to be classi�ed is with-held from the other samples, which are then used todesign the classi�er. The held-out sample is then clas-si�ed. These two steps are repeated for each of thesamples, and the results tallied to arrive at the classi-�cation rate. Use of this procedure prevents an arti�-cial in ation of the classi�cation rate.1Equation 34 assumes that the prior probabilities of allclasses are equal.

In summary, the steps of our classi�cation are:� For each of the 25 patients:1. Use the remaining 24 patients to calculate �and the C class means mi.2. Compute the C discriminant functions gi(Equation 34).3. Choose as the winner the class i with thelargest gi.5 Experimental Results5.1 SchizophreniaThirteen schizophrenic patients and 12 healthy con-trol subjects, matched for gender, age, and handed-ness, underwent an MR brain scan. As part of an on-going volumetric study [7][17], the basal ganglia weremanually segmented from these scans. Because re-sults of the volumetric study indicated that the basalganglia of schizophrenics may increase in volume, wedecided to examine the basal ganglia for other typesof shape changes. We chose the putamen because itsrelatively large size and simple shape are attractivefeatures when attempting to extract a shape descrip-tion.First, using the techniques of Section 3, the cranialcontents of each patient were warped in order to nor-malize the database for head shape. Next, the patho-logical displacement vectorUDp was calculated for eachpatient's left putamen, as described in Section 4.1. Aprincipal component analysis of the 25 displacementvectors was then performed.Figure 6 shows the �rst two putamen principal com-ponents. The �rst mode (Figure 6a-e) is a contrastbetween the size of the top of the putamen and thesize of the bottom, while the second (Figure 6f-j) con-trasts the size of the front and back of the putamen(left and right in the �gure).Next we input the top six principal components intoour Gaussian linear classi�er. Table II shows the re-sults, along with the classi�cation rate using just puta-men volume. A 12% improvement in the classi�cationrate occurs when using the putamen principal compo-nents instead of just volume.10

(a) (b) (c) (d) (e)(f) (g) (h) (i) (j)Figure 6: (a)-(e) First principal component of the putamen data set. The amplitude of the mode is increasingfrom (a) to (e). (f)-(j) Second principal component of the putamen data set. The amplitude of the mode isincreasing from (f) to (j).TABLE II

Putamen Classification Rates

Feature(s) %Correct

Putamen volume 60Putamen principal components 72

11

(a) (b) (c) (d) (e)(f) (g) (h) (i) (j)Figure 7: (a)-(e) First principal component of the ventricular data set. The amplitude of the mode is increasingfrom (a) to (e). (f)-(j) Second principal component of the ventricular data set. The amplitude of the mode isincreasing from (f) to (j).5.2 Ventricular disordersNine patients with Alzheimer's disease, 7 patientswith normal-pressure hydrocephalus, and 9 healthycontrol subjects, all matched for age, underwent anMR brain scan as part of a previous volumetric study[22] in which the ventricles were segmented using asemi-automatic procedure. Using precisely the samesteps that were applied to the putamen, we estimatedthe principal components of the ventricular data set.Figure 7 shows two of the principal components.The �rst mode (Figure 7a-e) is just a measure of theoverall size of the ventricles, while the second (Figure7f-j) is the degree of extension of the posterior hornsof the lateral ventricles.Table III shows the results of running our Gaussianlinear classi�er on the top six principal components,along with the classi�cation rate obtained using justventricular volume. As before, principal componentanalysis has provided a feature space with improvedclass separability.6 Discussion6.1 E�ect of head shape normalizationTo test the e�ect of head shape normalization, wecomputed the putamen principal components without�rst warping all patients' cranial contents to the samemodel. For the putamen data set, this dropped the

TABLE III

Ventricle Classification Rates

Feature(s) %Correct

Ventricle volume 80Ventricle principal components 88classi�cation rate from 72 to 64%, indicating that ournormalization did remove some nonpathological puta-men deformation.Since the classi�cation is performed in a six-dimensional space, it is di�cult to visualize. We there-fore plotted just the top two principal components forall 25 patients in the putamen data set, both with-out and with the head shape normalization. Figure8a shows the projections onto the top two modes ofthe unwarped data, while Figure 8b shows the projec-tions onto the top two modes computed after warp-ing. Note that b1 and b2 of Figure 8b are the �rsttwo components of the projection vector b of Equa-tion 33, and that the eigenvectors onto which they areprojected were shown earlier in Figure 6. Similarly,a1 and a2 of Figure 8a are projections onto the twohighest-variance eigenvectors of the sample covariancematrix of the original, unwarped putamen.Consistent with the decrease in classi�cation rate,12

-2 2

-2

2

a1

a2

-2 2

-2

2

b1

b2

-2 2

-2

2

ICC

ICC

U~

p1

U~

p2(a) (b) (c)Figure 8: Projections onto modes computed from the schizophrenia database. Each schizophrenic patient isdenoted by a +, and each healthy volunteer is denoted by a �. (a) Putamen principal components, computedfrom the original data. (b) Putamen principal components, computed after �rst normalizing for head shape. (c)ICC physical deformation modes.Figure 8a shows less class separability than Figure 8b.This is because the head shape normalization has re-moved some of the nonpathological shape variation be-tween patients. This nonpathological deformation isrepresented by the projections onto the �rst two ICCdeformation modes, shown in Figure 8c. As can beseen, the projections onto the ICC modes show littleclass distinction. This is to be expected, since craniumshape is uncorrelated with disease state.We repeated this procedure for the ventriculardatabase. This time the classi�cation rate did notdrop, staying at 88%. Figure 9 shows the projectionsonto the top two modes for the unwarped ventricles,the warped ventricles, and the ICC. No improvementis seen in class separability between Figures 9a and9b.One possible explanation for the lack of improve-ment when normalizing for head shape is the alreadyhigh classi�cation rate (88%) obtainable without re-moving any nonpathological e�ects. Coupled with thesmall sample size, there is little room for improve-ment. Another factor is our simple nearest-point cor-respondence scheme. For calculating the correspon-dence between two ICCs or two putamen, this proce-dure is adequate. For structures as complicated as theventricular system, however, nearest-point techniqueswill provide only a very coarse approximation to thetrue correspondence. A third possible cause is thatthe ventricular data set was not controlled for gender.On average the male cranium is larger than the fe-male, but interior structures do not necessarily scaleby precisely the same amount [22]. Since normalizingan entire database to one standard head shape does

not take into account gender-based shape variation,this gender-based variation may be interfering withthe analysis of pathological shape di�erences.To further investigate this problem, we constructeda mean square measure of ventricular similarity, bycomputing the distance between each voxel on the av-erage ventricular surface and the nearest point on apatient's ventricular surface. The sum of these for aparticular patient is a measure of the similarity of thepatient to the average. If the ICC warping is truly re-moving some of the variation between ventricles, thenthis measure should decrease. Figure 10 shows the re-sults for our 25 ventricular patients. Only a moderatedecrease is seen with warping.6.2 An alternative to �nite elementmodelingIn our current implementation, the ICC is modeledas a homogeneous, linear elastic object. This enabledus to use general �nite element methods to analyticallycharacterize the entire ICC. Of course the ICC is nothomogeneous, and so our simple physical assumptionswill lead to inaccuracies. However, it is possible to by-pass this dependence on physical assumptions by us-ing the following connection between the eigenshapescalculated using principal component analysis and thephysical deformationmodes computed via modal anal-ysis.To derive this relationship, we begin by interpretingEquation 8 and its solutionU = K�1R (35)13

-2 2

-2

2

a1

a2

-2 2

-2

2

b1

b2

-2 2

-2

2

ICC

ICC

U~

p1

U~

p2(a) (b) (c)Figure 9: Projections onto modes computed from the ventricular database. Each Alzheimer's patient is denotedby a +, each normal-pressure hydrocephalus patient by an �, and each healthy volunteer by a �. (a) Ventricularprincipal components, computed from the original data. (b) Ventricular principal components, computed after�rst normalizing for head shape. (c) ICC physical deformation modes.24

0

Unwarped WarpedFigure 10: Mean square distance (in mm2) betweenventricles before and after head shape normalization.from a probabilistic viewpoint. Treating U and R asrandom vectors related by the linear transform K�1,we have that �U = K�1�R �K�1�T (36)where �U and �R are the covariance matrices of Uand R. Because K is positive semide�nite we canwrite �U = K�1�RK�1: (37)Under the assumption that the elements of R are

independent and have variance �2, then�U = �2K�2: (38)We can form the sample covariance matrix S � �Ufrom a set of observations of U, then use Equation 38to obtain the estimateS � �2K�2: (39)Thus by collecting samples of U we can approximatethe sti�ness matrix K.This connection leads to several useful observations.First, using a physical model is equivalent to makingassumptions about the distribution of samples we ex-pect to see. Not using any model and just collectingdata, on the other hand, requires no a priori knowl-edge of this distribution and instead represents an at-tempt to statistically approximate it through experi-mental observation.Second, we have the following result. By applyingto S the orthogonal transform from Section 3.2 thatdiagonalized K, we have�TS� � �2�TK�1K�1�= �2 ��TK�1�� ��TK�1��= �2��1��1= �2 266666664 1�21 1�22 : : : 1�23N 377777775 (40)14

where the second to last step holds because the eigen-values of the inverse of any nonsingular matrix arejust the reciprocals of the eigenvalues of the originalmatrix. This shows that the orthogonal transform �also diagonalizes S, which implies that � consists ofthe eigenvectors of S. Therefore the eigenvectors ofS converge to those of K, which says that the an-alytic and estimated modes are the same under theassumption of an independent distribution of loadsR. Because of the reciprocal relationship between theeigenvalues of S and K, the high-variance directions(large eigenvalues) estimated using sample data arethe low-frequency directions (small eigenvalues) in amodal decomposition.Thus we can forgo the reliance on particular physi-cal assumptions and instead compute the sti�ness ma-trix directly from medical imaging data, in the follow-ing way. The ICCs from a large database of normalsubjects can be used to compute displacement datathat relates each patient's ICC surface to a model ofan average ICC. Equations 35-39 can then be used toestimate the sti�ness matrix. The major drawbackto this approach is that without the physical modeland the FEM, there will be no physical interpolationfunctions with which to warp interior structures.7 Related WorkIn the introduction we discussed previous work in-volving shape measurements of neuropathologies. Wethen proceeded to present our method of characteriz-ing neuropathological deformation by using both themodes of physical models and of statistical observa-tions.While our method of using both types of modesis novel, as is its application to neuropathology, bothphysical modeling and statistical techniques have beenused previously in image analysis. In the medical do-main, they have been used primarily for registration,segmentation, and/or shape description. Although thegoals of these three applications di�er, the mathemati-cal techniques employed are often very similar. In thissection we review relevant literature from all three ap-plication areas, and draw comparisons to our work onshape description.7.1 RegistrationBajcsy [32] used an elastic model, combining it withcross correlation measures in order to align raw grayscale brain scans with a simpli�ed brain atlas. Collinset al. [33][34] also applied a cross correlation measureto align raw patient data with a brain atlas, using both

gradient and intensity measures. The allowable defor-mations were not enforced through physically-basedelasticity constraints, but rather by limiting deforma-tions to be on the order of the current scale in theirmultiresolution approach. Christensen et al. [35] im-plemented both elastic and viscous uid models of de-formation in order to warp patient data to a brainatlas. In their technique, the elastic constraints areused as the prior distribution in a Bayesian formula-tion. The likelihood function incorporates the agree-ment between patient data and atlas through a simi-larity function resembling cross correlation. Minimummean square estimation (MMSE) is then used to �ndthe posterior distribution, giving the parameters of theelastic transformation from patient to atlas.Although the above physically-based and proba-bilistic approaches are formulated quite di�erently, ithas been shown that, under certain conditions, theyare in fact equivalent [36][37]. Thus the importantdi�erences between the above approaches lie more inthe methods of implementation, and thus in speed andconvergence properties, than in the particular formu-lation employed.While the above approaches have shown promisingresults, they all su�er from one major drawback. Thepriors of these models, whether in the form of elasticconstants or prior probabilities, are typically chosenin an ad hoc fashion, often for numerical convenience.The result is either that the physical models deform ina physically unrealistic way, or that the prior probabil-ities in the Bayesian models are poor approximationsof the true priors.7.2 SegmentationBoth physical and probabilistic methods have alsobeen used for medical image segmentation. The pri-mary way in which physical models have been used hasbeen through the use of snakes [38] and their variants.Cohen [39] augmented the original snake formulationwith a balloon force to help it avoid local minima.Staib and Duncan [40] used \Fourier snakes", basedon a Fourier decomposition of an object's shape. In-stead of relying on the elastic constraints of a phys-ical model, they approximated the probability distri-butions of the Fourier parameters using a training setof manually traced contours, and then applied Bayesrule in order to �nd the best set of parameter val-ues. Sz�ekely et al. [41] extended this approach to3-D using the spherical harmonic technique developedby Brechb�uhler [42], with statistical deviations in theshape parameters derived using a training set of hand-segmented surfaces.15

As in image registration, one of the main problemsin segmentation is the construction of realistic priormodels. Both Staib and Duncan [40] and Sz�ekely etal. [41] addressed this by using training examples toform the priors. This method of constructing priorsis also an active research area in shape description, aswill be described in the next section.In addition to segmentation, snake models can alsobe used for shape description. In this domain, how-ever, the mesh-like snake approaches su�er from twoproblems [19]. First, because the parameters of mostsnakes can be arbitrarily de�ned, the recovered shapedescriptions are not unique. Second, because the pa-rameters are coupled, the descriptions are also notcompact. Both of these drawbacks limit the useful-ness of snakes for object recognition.7.3 Shape Description7.3.1 Previous WorkBecause of the above limitations of mesh-like ap-proaches, researchers have developed other physically-based shape representations. Pentland and Sclaro�[19], on which our work is partially based, representedshape in terms of an object's physical deformationmodes. Instead of using the modes of a particular ob-ject, Bookstein described shape deformation in termsof the physical deformation modes of an in�nite thinplate. Although his original work [43] required cor-responding landmarks, more recent e�orts [44] havefocused on automatically obtaining required points,edges, and surfaces from the image data.Instead of physically modeling the structure understudy, researchers have also sought to obtain shape de-scriptions directly from sample data. Turk and Pent-land [45] have used principle components to describefacial variation and have been able to use this ap-proach to reliably recognize people's faces. Cootes etal. [46] used principal components to experimentallydescribe the modes of variation inherent in a trainingset of 2-D heart images. Hill et al. [47] extended thistechnique to 3-D and analyzed the cerebral ventriclesfor purposes of segmentation and representation.Along with ourselves [48], other researchers havealso begun investigating the relationship betweenphysical and experimental modeling. Cootes [21] hasexamined both physical and statistical shape models,with the goal of smoothly transitioning from a physi-cal to a statistical shape description as more and moredata become available. Zhu and Yuille [49] have alsoconsidered both physical and statistical shape models,in the context of representing and recognizing objectsfrom their 2-D silhouettes.

7.3.2 Comparison to Our MethodLike modal analysis, all of the above shape descrip-tors satisfy the requirements of being unique and com-pact. As argued by Sclaro� and Pentland [20], how-ever, physical deformation modes o�er the additionaladvantage that they are an orthogonal basis for a �-nite element model. Thus there is a connection withthe underlying physics, which is useful for simula-tion, regularization, and for including a priori infor-mation about the material properties of the object un-der study. While Bookstein's thin-plate modes are alsophysical, they come from a 2-D model and are derivedfrom a �nite di�erence formulation. A �nite elementmodel is more general, and has better convergenceproperties. Also, as mentioned earlier, using the FEMprovides interpolation functions that can be used to re-fer all patients' data back to the same model, therebyremoving sampling di�erences between patients.A more novel aspect of this paper is the connec-tion between FEM modes and principal componentanalysis, presented in Section 6.2. Using that devel-opment, prior models can be constructed either bymaking physical assumptions, or through the use oftraining data. These ideas are similar to the recentwork of Cootes [21].The �nal original feature of our work is the decom-position of shape deformation into two distinct com-ponents. This allows us to remove just the nonpatho-logical deformation, and therefore to focus on patho-logical morphology. This contrasts with the abovework, where only healthy brains are used, and thegoal is usually to remove all morphological di�erencesbetween the individual brains. As discussed in theintroduction, previous work that has examined patho-logical brain morphology has used just linear, planar,and/or volumetric shape measurements.8 SummaryWe have presented a new method that addressesthe general problem of separating out normal shapevariations across a class of objects from those varia-tions that carry special importance. Using both phys-ical modeling and statistical techniques, our methoddescribes shapes in terms of modes of deformation.When applied to the human brain, our technique isable to separate out pathological from normal shapedeformation, allowing better representation and anal-ysis of the deformations due to disease. The represen-tation is in the form of a disease's deformation modes,which provide a very natural basis set in which to ex-amine pathological shape deformation. The analysissuggests that by �rst discounting the experimentally-16

derived modes of a brain structure by the physicalmodes of the intra-cranial cavity, it may be possibleto improve disease classi�cation.Our method was applied to schizophrenia, Alz-heimer's disease, and normal-pressure hydrocephalus.The putamen of schizophrenics, although initially verysimilar to those of normal controls, were easier to dif-ferentiate from the control putamen once head shapewas taken into account. Conversely, the ventricles ofAlzheimer's patients, normal-pressure hydrocephaluspatients, and normal controls were somewhat di�er-entiable to begin with, but this separability did notmarkedly improve when cranial contents were normal-ized for head shape.The limitations of our method involve the accuracyof physical models of brain sti�ness, the ability to de-termine the correct correspondence between points onstructures, and the degree to which nonpathologicaland possibly pathological morphology are correlatedwith overall head shape. Overcoming these limitationswill require better implementation, further investiga-tion of the brain's material properties, and shape cor-relation studies involving large numbers of patients.In summary, there are two main contributions ofthis work. First, we have developed a method ofshape analysis that is useful for separating out in-teresting from uninteresting shape variation. By ap-plying modal analysis to the physical modeling, andprinciple component analysis to the experimental ob-servations, all shape variations were consistently de-scribed in terms of deformation modes. Second, whenour method is applied to neuropathology, it may bepossible to improve disease classi�cation by �rst nor-malizing for the physical modes associated with headshape. In addition to serving as features for classi�-cation, putamen and ventricle eigenshapes were alsodisplayed in order to illustrate the pathological defor-mation modes caused by schizophrenia, Alzheimer'sdisease, and normal-pressure hydrocephalus.AcknowledgmentsThe authors would like to thank Irfan Essa for hismany helpful comments, Mike Matsumae for providingthe ventricular MR brain data, and Martha Shentonfor providing the schizophrenic MR brain data.References[1] D. Bartelt, C.E. Jordan, E. Strecker, andA.E. James. Comparison of ventricular en-largement and radiopharmaceutical retention:

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