Proc. Natl. Acad. Sci. USAVol. 90, pp. 11944-11948, December 1993Medical Sciences

Mathematical textbook of deformable neuroanatomiesMICHAEL I. MILLER*, GARY E. CHRISTENSEN*, YALI AMITt, AND ULF GRENANDER**Department of Electrical Engineering and the Institute for Biomedical Computing, Washington University, St. Louis, MO 63130; tDepartment of Statistics,University of Chicago, Chicago, IL 60637; and *Division of Applied Mathematics, Brown University, Providence, RI 02912

Communicated by Lawrence Shepp, April 23, 1993

ABSTRACT Mathematical techniques are presented forthe transformation of digital anatomical textbooks from theideal to the individual, allowing for the representation of thevariabilities manifest in normal human anatomies. The idealtextbook is constructed on a fixed coordinate system to containall of the information currently avallable about the physicalproperties of neuroanatomies. This information is obtained viasensor probes such as magnetic resonance, as well as computedaxial and emission tomography, along with symbolc informa-tion such as white- and gray-matter tracts, nuclei, etc. Humanvariability associated with individuals is accommodated bydefining probabilistic transformations on the textbook coordi-nate system, the transformations forming mathematical trans-lation groups of high dimension. The ideal is applied to theindividual patient by finding the transformation which isconsistent with physical properties of deformable elastic solidsand which brings the coordinate system of the textbook to thatof the patient. Registration, segmentation, and fusion all resultautomatically because the textbook carries symbolic values aswell as multisensor features.

Global Shape Models and Anatomical Templates

Imaging modalities such as magnetic resonance (MR) imag-ing, x-ray computed tomography (CT), and positron emissiontomography (PET) provide exquisitely detailed in vivo infor-mation regarding the anatomy and physiological function ofspecific subjects. However, the interpretation ofthe data hasbeen hindered by the inability to expeditiously relate suchinformation to specific anatomical regions. The difficulty liesin two areas: images from atlases and other modalities mustbe registered; but more fundamentally, even when registered,normal variation in anatomy makes interpretation extremelydifficult. This paper provides algorithms for representingnormal neuroanatomies by precisely specifying the globalanatomical relationships between structures and how theycan vary from one individual to another. The goal is toprovide representations which allow for the generalization ofa single electronic anatomical textbook to the individual.Accommodating the variability manifest in human anato-

mies is an ambitious goal. There is no shortage of imageprocessing algorithms designed to improve pictures by noisesuppression or to recognize particular patterns so as tosegment pictures into subpictures. Much of the greatestsuccess has been with algorithms that model sensor variabil-ity, such as for x-ray, MR, or emission tomography.But the variability in the anatomical shapes and structures

themselves is much less well understood. The main difficultyis that human anatomies form highly complex systems.Browsing through an anatomical textbook, one is struck bythe awesome amount of information. The enormous com-plexity of biological patterns makes the design of represen-tations of even normal anatomies a difficult, not to say

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overwhelming, endeavor. Limitations of existing methodsbecome visible for such ambitious tasks as representation ofthe shape ensemble itself. Since the mid 1970s researchershave built models that attempt to incorporate structuredvariability. The early paper of Besag (1) began the line ofresearch on the use of probabilistic Markov random-fields(MRF) models for texture analysis. The MRF approach hasdemonstrated a good deal of success in image restoration andsegmentation. But this is not enough for representing theglobal relations illustrated by the typical aforementionedanatomies. Natural textures can be modeled with MRFs,since most of their variability is of a local nature: theprobabilistic dependencies extend over a quite limited range.To meet the greater challenge of representing the anatom-

ical relations between structures in human anatomies, globalrepresentations must be employed. Mathematical techniquesfor such representations began to appear in the early 1980sunder the name of "global shape" models. The global shapemodels represent image ensembles in terms of their typicalstructure via the construction of templates, and their vari-abilities by the definition of probabilistic transformations thatare applied to the templates. The transformations formgroups (translation, scale, and rotation) and are applied to thetemplate, in this case an electronic atlas, so that a rich familyof shapes may be generated with the global properties of thetemplates maintained.There has already been a vast body of work on multimo-

dality image fusion and registration (see ref. 2 for a sub-stantive introduction). The simplest methods of registrationused assume that the images or tissues being matched arehighly similar ones for which only global, course featuresmust be matched. Transformations of this type consist ofrigid global rotation and translation and/or simple scaling.We, however, are interested in accounting for very localvariability across disparate anatomies, thereby requiringhigh-dimensional transformations on the coordinate system,of dimension proportional to the size of the voxel lattice.Alternatively, many investigators have taken the approach ofdefining a small set of features-fiducial markers and/orlandmarks-which drive the registration. In the methodproposed here, the multisensor tomographic data directlyprovide the driving force for the transformation and align-ment of the coordinate systems. If fiducial markers andlandmarks are also defined, they become powerful boundaryconditions for our method, but they are not required.The work proposed herein is most akin to the physically

based modeling work ofTerzoupolos (see ref. 3, for example)in which transformations are constructed to obey certainphysical laws. Most relevant is the elegant work of Bajcsyand collaborators (4), which began in the early 1980s and inwhich deformable volume models were developed. The ap-proach presented here, while identical in spirit to both,substantially differs and extends the previous methods. First,the driving force acting throughout the continuum is the

Abbreviations: NLM, National Library of Medicine; MRF, Markovrandom field; MR, magnetic resonance; PET, positron emissiontomography.

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nonlinear variation of the distortion between the deformingtemplate and the multimodality imaging data. The full non-linear optimization is solved without linearization of thedriving force. Only under the condition that the template anddata are sufficiently similar so that the deformations are smalldo the linearized solutions give equivalent results (see ref. 5).Second, the deformation procedure is accomplished by solv-ing a sequence of optimization problems from course to finescale via parametrically defined deformation fields. This isanalogous to multigrid methods, but here the notion ofrefinement from course to fine is accomplished by increasingthe number of basis components in the series extension of thevector field. The final stage is to use the translation vectorfield over the entire continuum.Our previous work on deformable templates for biological

shape representation (6-9) involved templates of low com-plexity which could be constructed with modest effort. Forexample, organelles in electron micrographs such as mem-branes and mitochondria are generated as transformations oflinear and elliptic shapes. Likewise, amoeba are modeled astransformations of a sphere. Constructing templates for hu-man anatomies is a task orders of magnitude larger. Theproposed neurological templates consist of megabytes ofconstants associated with three-dimensional images fromsensor probes and symbolic textbook labeling. Until recentlythe construction of the template itself seemed to be the majorobstacle for the successful application of these methods. Itwas therefore a welcome surprise to learn of the The VisibleHuman (10) project undertaken by the National Library ofMedicine (NLM) in which digital anatomical templates arebeing constructed for two complete human beings. We quotefrom the NLM: "This Visible Human project would includedigital images derived from computerized tomography, mag-netic resonance imagery, and photographic images fromcryosectioning of cadavers." Initiatives such as this onemake the proposed work particularly timely.

Anatomical Textbook and Transformations

The anatomical textbook (template) is a vector functiondefined on the ideal coordinate system of the textbook. Therange of the template is real as well as symbolic in value, withthe vector of values including measures of the intrinsiccomposition of the tissue associated with the various nonin-vasive sensor modalities, as well as anatomical label andhistological information. The multivalued vector containsvalues associated with various imaging probes, MR spin-density, tl, and t2 images; CT attenuation density images;and functional PET images. Symbolic information wouldinclude the various labeled areas: white-matter tracts, gray-matter nuclei, Broca's areas, etc.The textbook is a vector mapping of the coordinate system

Q C 9P3 according to T: Ql -C 3, with the range space, C,assumed to be an M-fold product of spaces T1 X 32 X . . .fTM, where each component Tm E OTm corresponds to adifferent feature of the tissue. The triple (Ql, T, 0T) is termedthe anatomical textbook (template).There are two kinds of variations which must be accom-

modated: normal variation among humans and variation dueto diseased or abnormal states. Disease and abnormal vari-ation is not addressed in this paper. Focusing on normalhuman variation, a set of transformations h EE w on the idealcoordinate system is defined where We is the set of homeo-morphic maps h : Ql -Q fl. The homeomorphisms are gener-ated from translation groups applied to points x E Ql:

h x = (xl, X2, x3) -* [xl - u1(x), x2 - u - ,u3(x)]. [1]

The vector field u(x) = [ul(x), U2(X), U3(X)] is called thedisplacement field. The maps constructed from these trans-

formations allow for the dilation and contraction of theunderlying coordinates ofthe template into the coordinates ofthe individual anatomy at a very local level. The set ofnormalanatomies generated from the textbook (Ql, T, T) becomes {Th : h E We}, with o the composition operator.Applying the Textbook. The anatomical textbook is applied

to individual patients as follows. A patient is characterizedvia a study S, an N-valued vector function consisting ofN-characterizing data sets {Sn}N=, or substudies. Each sub-study is an examination of the patient's brain tissue via somesensor. It is assumed that all of the study types already existin the ideal textbook which implies Sn : Ql -+ 9mn,, for someMnE {1, 2, ..., M}. For the work described here, the studymodalities are assumed to be acquired in register; in general,a second processing step would be required to registermodalities from a single patient. The information in theanatomical textbook (Ql, T, T) is brought into the coordinatesofthe patient by finding the transformation h E We on fl whichregisters the studies {SnI}=l with the textbook.

Registration between the template and study is defined byusing a distance measure between the transformed textbookand the study with the distance equaling zero ifand only if thetwo are equal. For all of the MR data, the squared-errordistance 1/2o.2Iff ITn(X - U(X)) - Sn(X)I2dx is used,which is consistent with Gaussian noise models in MR.

Mechanoelastic Energy Density. To ensure that the vectorfield of Eq. 1 results in transformations which are physicallysmooth, so that structures are not broken apart, it is assumedthat they arise from a prior distribution with potential deter-mined by the kinematics of elastic solids (11). This "Bayesianview" of the estimation problem gives rise to an associatedpotential of the posterior distribution, the sum of the distanceand elasticity potentials,

1H(u)=- NN

| IT(X - U(X)) - S(X)12dxJQn=la[313auj(x)~\/auj(x)\

2 i=1 j=1 J (daxi ) a(xj )

(auj(x) +auj(x) 2]where A and u are the Lame elasticity constants and a is aLagrange multiplier. In two or three dimensions both thelikelihood term and the elasticity potential do not generateproper distributions on any function space. In practice,however, these transformations are approximated on thepixel lattice, the integrals becoming sums and the derivativesbecoming differences (see below). In that context Eq. 2 yieldsa perfectly valid potential.

Algorithm: Stochastic Gradient Search

The method of solution is to generate the transformation field u,which is the mean of the posterior distribution induced by thepotential of Eq. 2. For this stochastic gradient, algorithms areproposed which follow gradients of the potential H(u) with anadditive noise term. Under proper conditions (7, 12) averages ofthe parameter estimates generated from the gradient algorithmconverge to the mean which becomes the estimated transfor-mation field shown in all of the results below. We note thatbecause of finite averaging we are only assured that the meanofthe posterior distribution is computed locally around the largeenergy minima. For all of the results, the three-dimensionalprior is reduced to the two-dimensional prior in the foflowingstandard way. Assume that the stresses in the X3 direction andthe shear stresses in the X1,3 and X2,3 directions are zero, which

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FIG. 3. (Left) Spin-density (Upper) and t2 (Lower) images ofpatient B. (Center) The transformed textbook. (Right) Magnitudedifference images between the textbook and the patient.

with the pair of basis functions

eij1(x) = (i sin irix1 cos lrjx2j cos irix1 sin lrjx2 '

FIG. 1. Textbook components: spin-density (Upper Left), tl-weighted (Upper Right), and t2-weighted (Lower Left) MR imagesand a hand segmentation (Lower Right).

implies that 0u3/&xl = -aul/aX3, au3/ax2 = -au2/ax3, andau3/0X3 = -[A/(A + 2p)][(auU/Ox1) + (0u2/0x2)I. In two dimen-sions the constants 71 = (3AU+2,u2)/(A+ju) and 2 = A/(2A+2,u)will be extremely helpful. The estimation is accomplished in twosteps: first through a low-dimensional parametric basis (7), andsecond through a high-dimensional parameterization whichcorresponds to estimating the translation field over the pixellattice.

Low-Dimensional Coarse Refinement. The basis represen-tation of the transformation field becomes

d 21/2

.=j-o (i2 )1/2 [j,j,iej,j,i(x) + pN,j,2ej,j,2(x)] [3]

(-j sin 'rix1 cos 7jJX2ei,j2(X) = ti cos irix1 sin rjx2 / [4]

chosen to diagonalize the covariances associated with theelasticity operator. The stochastic algorithm generates theexpansion coefficient set {pL,i,, ij,2}J=O according to

1 aH(u(t))dj,j,p(t) = -- dt + dwij,p(t)2 aN,

[5]

forp = 1, 2, where wij,p(t) is a Wiener process. The gradientof the potential with respect to the coefficients becomes

a 1H(u)=2cr

f N(T.(x- U(X, t))- Sn(x))JQn=l

x VT,,(x - u(x, t))ei,,j,p(x)dx + app(i2 + j2).j,1,P(t), [6]

with a*b denoting dot product, VT = [aT/axj, aT/8x2I, A3 =71 r2/2(1- i2), and 182 = 7)1rT2/4(1 + r)).High-Dimensional Fine Search. The full optimization is

solved at the highest resolution supported by the textbook; inthis case the MR data is on a 256 x 256 pixel lattice S. Thecontinuous displacement field u is approximated by the set{u(l)h}ie at each of the 2562 lattice sites. The transformationfield is estimated by stochastic gradient search according to

1 dH(u(t))dup(l, t) = - 2- d (t) dt + dwi,p(t),2 aup(l) [7]

where the parametric solution from Eq. 5 is used as thestarting point. The partial derivatives aH(u(t))/aup(l) areobtained from the variational calculus gradient

FIG. 2. (Top) Spin-density (Left), tl-weighted (Center), andt2-weighted (Right) images of patient A. (Middle) The transformedtextbook. (Bottom) Magnitude difference images between the trans-formed textbook and the patient.

I IFIG. 4. (Left) Transformation between the textbook and patient

B applied to the rectangular grid. (Center and Right) The x and ycomponents of the transformation.

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8H(u) V2u(X, t) + _1_ V u(x, t)1

iN

-2- X (T.(x - u(x, t)) - S(x))VT.(x - u(x, t)).an=1

[8]

Here V2 is the Laplacian operator. The lattice partial deriv-atives OH(u(t))/Oup(l) (p = 1, 2) are obtained via discretiza-tion of the partial differential Eq. 8 by using standard sym-metric difference lattice approximations.For all ofthe results shown, dwas systematically increased

from 1 to 5 with 40 iterations of the stochastic gradient searchrun for each dimension. The standard deviation a and sim-ulation time step were 0.01 and 5 x 10-8, respectively. Theparametrically defined transformation field was then used asinitial conditions to the high 2 x 2562 dimensional search. Eq.7 was run to equilibrium over 200 iterations, and then 50iterations were used to generate the empirically averaged ufield. The standard deviation was kept the same, with the stepsize increased to 10-5 and the constants a71i = 0.01 and i2 =0.0. Observe that omitting the noise term in Eqs. 5 and 7transforms them into gradient descent equations leading tolocal minima corresponding to local maximum a posteriorisolutions.

Results

Constructing and Generalizing the Textbook. A two-dimensional textbook was constructed from an MR study ofa normal patient at Duke University using standard MRspin-echo sequences to generate a spin-density, tl-weighted,and t2-weighted series. A hand segmentation was also per-formed of the two-dimensional scans into the various gray-matter nuclear regions: thalamus, putamen, head of caudatenucleus, ventricle, other brain matter, and background. Thetextbook consists of a 4-tuple of three MR images and ahand-labeled segmentation. These are shown in Fig. 1. Thehand-segmented labeling ofthe textbook (Fig. 1 LowerRight)was generated by using all three MR images, along with thedetailed horizontal brain section on page 28 ofthe DeArmondet al. (13) anatomy atlas.Two different patient studies, A and B, were analyzed.

Study A consisted of N = 3 substudies (spin-density, tl-weighted, and t2-weighted MR images), and study B con-sisted ofN = 2 substudies (spin-density and t2-weighted MRimages). The studies were selected because of their similarorientations and level in the brain and not for their closenessin brain size or similarity to the textbook.The result ofestimating the transformation hA from the first

experiment is shown in Fig. 2. The top row shows thespin-density (Left), tl-weighted (Center), and t2-weightedimages (Right) from patient A. Shown in the middle row isthe MR textbook of Fig. 1, transformed to patient A: thespin-density T1 o hA (Left), the tl image T2 o hA (Center), andthe t2 image T3 o hA (Right). Notice the remarkable corre-spondence between the transformed textbook and the desti-

FIG. 5. Magnitude differenceimages between the t2 componentof patient A and the textbook(Left), the globally deformed text-book (Center), and the locally de-formed textbook (Right).

nation patient A. Shown in the bottom row is the magnitudeof the difference images between the transformed textbookand the substudies (top row minus the middle row).

Fig. 3 shows the result of estimating hB for patient B. Theleft column shows the MR spin-density (Upper) and t2(Lower) images of patient B. The middle column shows thetransformed textbook: T1 o hB and T3 o hB. The right columnshows the difference images demonstrating the near-perfectalignment.

Fig. 4 shows properties ofthe transformation hB. Fig. 4Leftshows the transformation applied to a rectangular grid, withFig. 4 Center and Right showing the x and y components ofthe transformation. Bright areas in the displacement fieldscorrespond to translation in the positive direction, and darkareas to translation in the negative direction. The mixture oflow- and high-frequency components of the transformationare a result of the low- and high-dimensional transformation.Generating the transformation required 3 min ofcomputationtime on a 64 x 64 processor DECmpp 12000Sx/model 200(MasPar).

Fig. 5 demonstrates the course to fine procedure, showingthe global as well as local flow of information as the t2component of the textbook aligns with patient A. Fig. 5 Leftshows the magnitude difference between the textbook t2component and patient A before any transformation. Noticethe large disparity in global as well as local structure. Fig. 5Center shows the correspondence after the application of theglobal transformation alone. Fig. 5 Right shows the differ-ence after both the global and local transformations areapplied. Notice that the local transformations allow for smalladjustments of the fine substructures.

FIG. 6. (Upper) Hand-labeled structures ofpatients A (Left) andB (Right). (Lower) Automatically labeled structures of patients A(Left) and B (Right).

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Once the transformations, hA and hB, from the coordinatesystem ofthe textbook to the studies ofpatients A and B havebeen found, the symbolic label information can be automat-ically mapped to the patients' coordinate system. Shown inFig. 6 is the result of applying the anatomically labeledgray-matter nuclei and ventricle information in the T4 com-ponent of the textbook to the brains of patients A and B. Thetop row shows hand-labeled structures of patients A and B,and the bottom row the automated segmentation and labelingof both patients A (T4 o hA; Left) and B (T4 o hB; Right).

Fusing Modalities. All of the information in the textbookbecomes available in the patient study, allowing for thehigh-resolution anatomical information of the textbook to befused with the physiologic studies of brain activity (14, 15).Fig. 7 demonstrates the fusion of the anatomical textbookinformation into the emission tomographic physiologic stud-ies. For this fusion study a new simulated textbook andsimulated patient study was generated. The simulated text-book consisted of a segmented image, a spin-density, and at2-weighted MR image corresponding to the hand segmenta-tion of the textbook of Fig. 1. Standard tissue parameterswere taken to represent cerebrospinal fluid, gray matter, andwhite matter, with the images generated by using the spin-echo sequence used in the Duke study.Three images were generated from the segmentation of

patient B, the first and second being spin-density and t2-weighted images. The third component of the patient studywas a PET scan simulated to correspond to the Super-PETtime-of-flight imager (16). Fig. 7 shows the result of incor-porating the anatomic information of the deformed text-book-via the MR images-into the PET reconstruction.Shown in Fig. 7 Left is the true PET tracer distribution usedfor the study. Fig. 7 Center shows use of the anatomicinformation obtained via registration oftheMRtextbook withthe patient for the calculation of the maximum a posterioriestimate of the PET tracer distribution. For this, a Good'sroughness MRF prior (17) was used to locally smooth inde-pendently over the nuclei and ventricle areas. Notice theexquisite detail and tracer accuracy in the PET reconstruc-tion. Fig. 7 Right shows the result of the same PET recon-struction algorithm with Good's smoothing globally appliedacross the boundaries of the anatomically distinct nuclei,without the use ofthe anatomic MR information. Notice howmuch anatomical detail, or resolution, is lost.

Concluions

Projects such as the NLM's Visible Human project for theconstruction of digital anatomical libraries for two completehumans has to a large part motivated the work proposedherein. Thus far, only normal neuroanatomy has been dis-cussed, although the methods must be extended to includethe abnormal variation associated with disease. Again quot-ing from the NLM: "NLM should expand upon initial image

FIG. 7. (Left) True PET tracerdistribution. (Center) Smoothed-maximum a posteriori PET recon-struction incorporating the ana-tomical MR information. (Right)Smoothed estimate generatedwithout using the anatomical in-formation.

libraries comprised of normal structure to encompass spe-cialized image collections which represent structural infor-mations, such as embryological development, normal andabnormal variations and disease-related images."The construction of digital anatomical textbooks is quickly

becoming a reality. Being able to transform the coordinatesystem of the textbook into that of any patient allows forsensor fusion as well as segmentation. All of the anatomic,histologic and pharmacologic information in the textbookthen becomes available in the study of the individual.

We are indebted to Dr. Scott Nadel (Duke University) for the MRdata, to Richard Rabbitt for help on the continuum mechanics, andto Marcus Raichle and Michael Vannier for helpful discussionsduring the development of the manuscript. M.I.M. and G.E.C. weresupported by the National Institutes of Health (DRR-1380), Office ofNaval Research (N00014-92-J-1418) and Army Research Office(DAA 03-92-6-0141). Y.A. and U.G. were supported by the ArmyResearch Office (DAAL-03-86-K-0110) and the Office of NavalResearch (N00014-88-K-289).

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Animation 1, 73-80.4. Bajcsy, R. & Kovacic, S. (1989) Comput. Vision Graphics

Image Processing 46, 1-21.5. Amit, Y. (1993) SIAM J. Sci. Comput., in press.6. Grenander, U., Chow, Y. & Keenan, D. (1990) HANDS: A

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7. Amit, Y., Grenander, U. & Piccioni, M. (1991) J. Am. Stat.Assoc. 86 (414), 376-387.

8. Miller, M. I., Maffitt, D., Shrauner, J., Roysam, B. &Grenander, U. (1991) in Proceedings of the Twenty-Fifth An-nual Conference on Information Sciences and Systems, eds.Davidson, F. & Goutsias, J. (Johns Hopkins Univ. Press,Baltimore), pp. 637-642.

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13. DeArmond, S. J., Fusco, M. M. & Dewey, M. M. (1989)Structure of the Human Brain: A Photographic Atlas (OxfordUniv. Press, New York), 3rd Ed.

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