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Medical Image Analysis 10 (2006) 752–763

Deformable registration of brain tumor images via a statistical modelof tumor-induced deformation

Ashraf Mohamed a,b,*, Evangelia I. Zacharaki b, Dinggang Shen a,b, Christos Davatzikos a,b

a Section of Biomedical Image Analysis, University of Pennsylvania School of Medicine, Philadelphia, PA, USAb CISST NSF Engineering Research Center, Department of Computer Science, Johns Hopkins University, Baltimore, MD, USA

Received 30 January 2006; received in revised form 4 May 2006; accepted 8 June 2006Available online 24 July 2006

Abstract

An approach to the deformable registration of three-dimensional brain tumor images to a normal brain atlas is presented. Theapproach involves the integration of three components: a biomechanical model of tumor mass-effect, a statistical approach to estimatethe model’s parameters, and a deformable image registration method. Statistical properties of the sought deformation map from the atlasto the image of a tumor patient are first obtained through tumor mass-effect simulations on normal brain images. This map is decom-posed into the sum of two components in orthogonal subspaces, one representing inter-individual differences in brain shape, and theother representing tumor-induced deformation. For a new tumor case, a partial observation of the sought deformation map is obtainedvia deformable image registration and is decomposed into the aforementioned spaces in order to estimate the mass-effect model param-eters. Using this estimate, a simulation of tumor mass-effect is performed on the atlas image in order to generate an image that is similarto tumor patient’s image, thereby facilitating the atlas registration process. Results for a real tumor case and a number of simulatedtumor cases indicate significant reduction in the registration error due to the presented approach as compared to the direct use of deform-able image registration.� 2006 Elsevier B.V. All rights reserved.

Keywords: Brain image registration; Brain tumor; Finite element model; Atlas registration; Statistical deformation model; Neurosurgical planning

1. Introduction

Deformable registration of normal brain images into acommon stereotactic space makes possible the constructionof statistical atlases that are based on collective morpholog-ical, functional, and pathological information (Davatzikos,1997). Similar atlases constructed from tumor patients’images can act as tools for optimal planning of therapeuticand neuro-surgical approaches that deal with tumors bystatistically linking functional and structural neuroanatomyto variables such as the tumor size, location, and gradeto the surgical or treatment approach and outcomes

1361-8415/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.media.2006.06.005

* Corresponding author. Present address: Siemens Corporate ResearchInc., Imaging and Visualization, 755 College Road East, Princeton, NJ08540, USA. Tel.: +1 609 233 8826; fax: +1 832 355 7923.

E-mail address: ashraf.mohamed@siemens.com (A. Mohamed).

(Kyriacou et al., 1999; Mohamed et al., 2001; Dawantet al., 2002; Cuadra et al., 2004).

A major hurdle preventing the construction of suchbrain tumor atlases is the unsuitability of currently avail-able deformable registration methods for adapting atumor-bearing image to the stereotactic space of a normalneuroanatomy atlas image. This is due to the substantialdissimilarity between the two images resulting from topo-logical differences, tissue death and resorption, the con-founding effects of edema and tissue infiltration, andsevere deformation in the vicinity of the tumor beyond nat-ural anatomical variability. These image dissimilarities typ-ically result in the failure of currently available imageregistration methods to produce an accurate registrationin the vicinity of the tumor.

To account for topological differences between the atlasand the patient’s images Dawant et al. (2002) proposed the

Fig. 1. Illustration of the deformation maps involved in the proposedapproach. uf is the map from the atlas to a subject’s tumor-bearing image.Regions TSD and DSD denote the bulk tumor and edema regions in thesubject’s images, and TA, DA are the corresponding regions in the atlas. uc

is the mapping from the atlas to the subject’s image before tumor mass-effect simulation (BS is not known for non-simulated cases), and ud is thatobtained through the simulation of tumor mass-effect. Simulating thetumor mass-effect on the atlas results in ua and a deformed atlas imagewhich can then be registered to the deformed subject’s image through ub.

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introduction of a tumor ‘‘seed’’ in the atlas image andrelied on image features to drive the registration. Cuadraet al. (2004) extended this idea by adding a radially sym-metric model of tumor growth. The lack of a physicallyrealistic model of tumor-induced deformation, as well asthe approximate determination of the seed location resultsin limited accuracy of these approaches for large tumorcases. In an earlier attempt at solving this deformationregistration problem, Kyriacou et al. (1999) used a bio-mechanical model of the deformation caused by tumorsto register images of non-infiltrative tumor patients to ana-tomical atlases. However, this approach was only imple-mented in 2D and relied on a computationally expensiveregression procedure to solve the inverse problem of esti-mating the tumor location in the atlas.

In order to register brain tumor images to a normal ana-tomical brain atlas, here we present an approach thatinvolves the integration of three components. The first isa biomechanical three-dimensional 3D model for the soft-tissue deformation caused by the bulk tumor and peri-tumor edema. This model is implemented using the finiteelement (FE) method and it is used to generate a numberof examples of deformed brain anatomies due to tumorsstarting from normal brain images. The second componentis a statistical model of the desired deformation map. Thestatistical model approximates this map via the sum oftwo components which lie in orthogonal subspaces, andwhich have different statistical properties. The first of thesetwo components represents normal inter-individual differ-ences in brain shape and the second represents the defor-mation induced by a tumor (tumor mass-effect). For anyparticular tumor case that should be registered to the atlas,a partial observation of the sought deformation map isobtained via a deformable image registration method,which is the third component of the presented approach.Based on the constructed statistical model of the deforma-tion, this partial observation is used to estimate the corre-sponding mass-effect model parameters that correspond tothe observed deformation. Finally, the sought deformationmap is obtained by applying the mass-effect model to theatlas image and the use of deformable image registrationto match it to the subject’s image.

The rest of this paper is organized as follows. The detailsof the proposed approach are presented in Section 2. Thisincludes a description of the mass-effect model, statisticaltraining, and statistical estimation approaches. In Section3, we demonstrate the overall approach on a real and anumber of simulated tumor cases, and we show that theregistration error decreases significantly with our approachas compared to the direct use of a readily available imageregistration method. The paper is concluded with a discus-sion of the results and future work in Section 4.

2. Methods

The proposed approach is explained with the aid ofFig. 1. The subject’s brain BSD includes regions TSD (bulk

tumor), and possibly DSD (peri-tumor edema). The maingoal of the deformable registration problem is to find theinvertible transformation

uf : BA n T A ! BSD n T SD

which maps points with coordinates XA in the atlas imageto points with coordinates XSD in the subject image. An-other goal is to identify TA, which corresponds to brain tis-sue that is no longer present in the subject’s image (died orinvaded by tumor).

As will be demonstrated later in this paper, the use of areadily available brain image registration approach toobtain uf produces an inaccurate warping in and aroundtumor area (the area denoted by MA on the atlas imagein Fig. 1). This is mainly due to the inability of such imagematching approaches to account for topological differencesbetween the two images, the effects of edema, tumor infil-tration, signal difference between the two images in anaround the tumor region, and tumor mass-effect whichcauses severe deformation atypical of natural inter-subjectvariability to which these approaches are tuned.

If an accurate model of the deformation induced by thetumor is available, it can be used to simulate this deforma-tion in the atlas and to obtain ua, followed by the applica-tion of deformable image registration to get ub, andtherefore uf = ub � ua. A model of the mass-effect causedby tumor growth is described in Section 2.1. Estimates ofregion TA as well as the other parameters affecting themodel’s behavior, such as the extent of peri-tumor edemaand the mass-effect of the bulk tumor, are still needed inorder to apply this approach. Here, we solve this inverseestimation problem by exploiting the statistical dependencybetween uf and the mass-effect model parameters.

754 A. Mohamed et al. / Medical Image Analysis 10 (2006) 752–763

Although an approximation of uf obtained through thedirect application of deformable image registration isincorrect in and around the tumor, the pattern of thisdeformation outside that region can guide the estimationof the tumor model parameters.

The overall approach for deformable image registrationbetween an atlas and a brain tumor image is summarized inFig. 2. In Section 2.2, we explain the collection of the sta-tistics on uf = ud � uc through tumor mass-effect simula-tions on images of normal subjects. The use of thesestatistics to estimate the tumor mass-effect model parame-ters is explained in Section 2.3.

2.1. Tumor mass-effect model

In Mohamed and Davatzikos (2005), we described a 3DFE model of brain tumor mass-effect. This model is initial-ized with a 3D normal brain image (free of tumor) and itproduces an estimate of the deformation due to the mass-effect of a simulated tumor. Here, we briefly review thedescription of this model by assuming that it is applied tothe atlas image. As explained later in this paper, thismass-effect model is also applied to other normal imagesfor purposes of statistical model training.

Since tumor growth is not a purely mechanical process,but it involves a host of interacting biological, biochemicaland mechanical mechanisms at different scales, it is essen-tial to initialize the mass-effect model simulations with aconfiguration for the brain from which the target configu-ration (that deformed by the tumor at the desired stage oftumor growth) is reachable by solving a mechanical bound-ary value problem (BVP).

Fig. 2. A flowchart summarizing the overall steps involved in the described imthe atlas to the patient’s brain obtained through direct image registration. eH isimulation on the atlas followed by deformable image registration produces thproduce, uf , the estimated atlas to tumor image deformation map.

With the assumption that the tumor mass-effect is dueto the bulk tumor and the peri-tumor edema only, regionsTA and DA are defined in the undeformed (normal) atlasimage. These correspond to the bulk tumor and peri-tumor edema regions, respectively, in the deformed atlasat the end of the simulation. The bulk tumor is assumedto be composed of proliferative, quiescent, and necrotictumor cells (Kansal et al., 2000; Wasserman and Acharya,1996; Greenspan, 1972). The extent of peri-tumor edemafor brain tumors is highly variable and it depends onthe type of the involved tumor and its grade (Lamszus,2004).

Given the initial shape of the brain, regions TA and DA,the stresses caused by the tumor, and the amount of swell-ing due to edema, the deformation map ua can be obtainedby solving the mechanical BVP. For real tumor cases, val-ues of the parameters which are responsible for producingthe deformation in the patient’s brain are not known, butcan be estimated via the statistical approach as explainedin Section 2.3. To simulate the mass-effect of tumors start-ing with normal brain images, approximations of theseparameters must be used as explained in the followingsection.

2.1.1. Specifying TA and DA

TA corresponds to brain tissue that is no longer presentat the target stage of tumor growth (died or infiltrated bytumor cells), while DA corresponds to brain tissue that isswollen by edema. Although regions TA and DA are highlyvariable for different tumor cases and are not known ingeneral, here for tractability, we assume that these regionsare both spherical and concentric with center ct and radii rt

age registration approach. ~uf is an initial rough estimate of the map froms an estimate of the parameters of tumor mass-effect model. A mass-effecte two deformation maps ua and ub, respectively, which are composed to

A. Mohamed et al. / Medical Image Analysis 10 (2006) 752–763 755

and rd, respectively. ct, rt, and rd will now be treated asparameters of the mass-effect model.

It is important to note that the assumptions on the shapeof regions TA and DA do not restrict our approach to deal-ing with spherical tumors only since the final simulatedtumor shapes depend on the elastic constraints aroundthe tumor (Kyriacou et al., 1999). As illustrated later in thisarticle in Fig. 3, tumors generated by the described mass-effect model need not be spherical. Also, it is worth men-tioning that for the deformable registration of real braintumor cases to brain atlases, estimates of regions TA andDA are later deformed through the deformable image reg-istration component of our approach.

2.1.2. Bulk tumor mass-effect and edema

To account for the mass-effect of the bulk tumor, we fol-low the work of Wasserman and Acharya (1996) andassume that the expansive force of the neoplasm is gener-ated by the proliferative tumor cells only. Accordingly,for model simulations, brain tissue in the region TA isremoved, and a constant outward pressure P is appliednormal to the boundary of Tr. P is a model parameter that

Fig. 3. Illustration of a tumor mass-effect simulation and the associated displawith an introduced small tumor, and resulting image after simulation of tumormap ud � XS, and displacement map uf. Bottom figure: displacement map ud.

determines the mass-effect exerted by the bulk tumor, andtherefore, to a large extent, the final tumor size.

Depending on the type of tumor and its aggressiveness,TA may be surrounded with a peri-tumor edema region DA.Peri-tumor edema is usually of the vasogenic type, and itcauses swelling of white matter (WM) only, mostly perpen-dicular to the direction in which the fibers run (Kuroiwaet al., 1994; Nagashima et al., 1994). Here, no knowledgeof WM fibers’ orientation is assumed and therefore, an iso-tropic expansive strain e is applied to WM in DA by usinganalogy to thermal expansion. A thermal conductivityvalue of zero for brain tissues prevents this expansion fromspreading outside DA. Studies that measured a volumeexpansion of 200–300% in WM due to edema (Kuroiwaet al., 1994; Nagashima et al., 1994) imply thate 2 [0.26,0.44]. For simulations starting with normal brainscans, a value of e = 0.35 is adopted.

2.1.3. Boundary value problem statement

Given the time scale of the tumor growth process, whichis at least a few days, the deformation of brain tissues maybe modeled as a quasi-static process. Additionally, if body

cement maps. Upper row (left to right): atlas image, normal subject’s MRImass-effect. Middle row (left to right): displacement map uc, displacement

756 A. Mohamed et al. / Medical Image Analysis 10 (2006) 752–763

forces, such as gravity, are ignored, the required deforma-tion mapping ua can be found by solving the static equilib-rium equation (Marsden and Hughes, 1983):

DivðSÞ ¼ 0 ð1Þ

where S is the first Piola–Kirchhoff tensor, which is relatedto strain via the material constitutive law (ABAQUSVersion 6.4, 2003).

FE simulations were run to compare the behaviour ofthe linear elastic material model (Wasserman and Acharya,1996; Skrinjar et al., 2002; Clatz et al., 2005) and fourhyperelastic material models Kyriacou et al. (1999); Men-dis et al. (1995); Miller and Chinzei (2002); Prange et al.(2002) used for brain tissues in the literature. In all simula-tions, nonlinearities caused by large deformations weretaken into account. The stability of the tested materialmodels at strain levels encountered during tumor mass-effect simulations, the ability of these simulations to con-verge, and the error in predicting deformations observedin real tumor cases were the factors used to select a materialmodel to use in our mass-effect model. Based on theseexperiments, which are described in Mohamed (2005), weadopted the hyperelastic model proposed by Miller andChinzei (2002) while relaxing the perfect incompressibilityassumption. Since the cell doubling time of typical braintumors such as gliomas has been estimated to be between1 week and 12 months (Swanson et al., 2000), and sincethe viscous time constants of normal soft tissues are atmost of the order of 1 min (Miller and Chinzei, 2002;Prange et al., 2002), viscous effects of the brain tissuesmay be ignored. Under these assumptions, the strainenergy density function of the brain material becomes(ABAQUS Version 6.4, 2003):

W ¼ 2la2ðka

1 þ ka2 þ ka

3 � 3Þ þ 1

D1

ðJ=J th � 1Þ2; ð2Þ

where ki ¼ J�1=3ki, ki, i = 1, 2, 3 are the principal materialstretches, J ¼ detð~F Þ is the volume ratio, ~F is the deforma-tion gradient, Jth = (1 + eth)3 is the thermal volume ratio,and eth is the thermal strain. The constants l, D1 are relatedto the Young’s modulus at zero strain Eo, and Poisson’sratio m by

l ¼ Eo

2ð1þ mÞ and D1 ¼6ð1� 2mÞ

Eo

: ð3Þ

The value a = �4.7 determined in Miller and Chinzei(2002) was adopted here. Based on the arguments andthe experiments explained in Mohamed and Davatzikos(2005) and Mohamed (2005), we adopted the value ofl = 842 Pa suggested by Miller and Chinzei (2002) and aPoisson’s ratio of m = 0.485.

A sliding boundary condition is used over the wholebrain surface except for locations, where the falx meetsthe inner surface of the skull which are assumed pinned(Miga et al., 1999a). The sliding boundary conditionapproximates the interaction between the brain and theinner surface of the skull by preventing motion in the direc-

tion normal to the brain surface and permitting motion inthe tangential direction (Miga et al., 1999b). For simula-tions in this paper, the brain ventricles are assumed to bevoid with a zero intra-ventricular pressure (Kyriacouet al., 1999). These boundary conditions, the static equilib-rium Eq. (1), and the material constitutive relationship (2)formulate a mechanical BVP which is solved via the FEmethod using ABAQUS (ABAQUS Version 6.4, 2003).FE meshes are generated using the approach described inMohamed and Davatzikos (2004).

2.2. Statistical model training

Let the parameters of the mass-effect model described inSection 2.1 be collectively referred to by H ” (ct, rt, rd, P).The values of these parameters are not known for a realtumor case. The goal of the work described in the sectionis to create a statistical model for the deformation uf thatwill be useful in the estimation of H based on the braintumor image for a patient. Since uc is not related to thetumor model parameters, we collect statistics for ud anduc separately as follows.

First, the deformation maps uci; i ¼ 1; . . . ; ns between

the atlas and MRI images of ns normal subjects areobtained using a deformable image registration approach(Shen and Davatzikos, 2002). Simulations of the mass-effect of tumor growth are then conducted for each subjecti for values Hj, j = 1,. . ., nm covering a range of themodel parameters to produce the deformations udi;j

;i ¼ 1; . . . ; ns; j ¼ 1; . . . ; nm.

A problem preventing the collection of statistics on udi;j

directly is that the domains of these maps are different fordifferent values of i and j. This precludes the point-to-pointcomparison of these deformation maps. To overcome thisproblem, for all tumor model simulations, regions T Aj

and DAj are defined in the atlas space based on Hj andmapped to each subject’s space via uci

; i ¼ 1; . . . ; ns.Now, for XA 2 BA n T Aj ; i ¼ 1; . . . ; ns; j ¼ 1; . . . ; nm, wedefine

ufi;jðXAÞ � udi;j

� uciðXAÞ ¼ udi;j

ðuciðXAÞÞ ð4Þ

ufi;jðXAÞ � ufi;jðXAÞ � XA ð5Þ

uciðXAÞ � uciðXAÞ � XA ð6Þ

udi;jðXAÞ � ufi;jðXAÞ � uci

ðXAÞ ð7Þ

Eqs. (5)–(7) imply that

ufi;jðXAÞ ¼ uciðXAÞ þ udi;jðXAÞ: ð8Þ

Therefore, we have decomposed ufi;j , the displacement mapfrom the atlas to a tumor image, into the sum of two dis-placement maps, uci and udi;j , which are both defined inthe atlas space. The displacement uci is due to differencesbetween the shape of the brain of subject i and the atlas.The displacement udi;j is due to the tumor mass-effect sim-ulation with tumor model parameters Hj on the brain ofsubject i. For different i = 1, . . ., ns, but the same

A. Mohamed et al. / Medical Image Analysis 10 (2006) 752–763 757

j = 1, . . ., nm, the domains of udi;j are exactly the same. InFig. 3, we show an example of a tumor mass-effect simula-tion on a normal brain image and we also show the in-volved displacement maps.

We construct discrete versions of the displacement mapsuci and udi;j by sampling their Cartesian components for allvoxels in the atlas in BAnMA to yield vectors uci and udi;j ,respectively, for i = 1, . . ., ns, j = 1, . . ., nm. Assuming thatuci ; i ¼ 1; . . . ; ns are independent realizations of a Gaussianrandom vector, principal component analysis (PCA) isapplied to these vectors to yield the mean lc and the matrixVc whose columns are eigenvectors corresponding to thefirst mc principal components (mc 6 ns � 1). Through thisassumption, we have approximated the displacement mapfrom the atlas to a normal brain image via a linear sub-space in the space of all possible displacements from theatlas.

Next, we construct a statistical model of the displace-ment due to the tumor mass-effect. We compute the com-ponent of udi;j in the subspace orthogonal to the columnsof Vc as

u0di;j¼ Udi;j � VcV

Tc Udi;j : ð9Þ

We further assume that, for each j, u0di;j; i ¼ 1; . . . ; ns are

independent realizations of a Gaussian random vectorand we perform PCA on these vectors to yield the meanldj

and the matrices Vdj whose columns are eigenvectorscorresponding to the first mdj principal components associ-ated with eigenvalues kdj;l; l ¼ 1; . . . ;mdjðmdj 6 ns � 1Þ.Now, we can approximate the discrete displacement mapUf between the atlas and a subject with a simulated tumorwith parameters Hj, j = 1,. . .,nm as follows:

Uf � lc þ Vcaþ ldjþ Vdj bj: ð10Þ

Therefore, we can now represent the displacement from theatlas to a image of a tumor patient as the sum of two com-ponents with different statistical properties that can belearnt from readily available training data. The first com-ponent (lc + Vca) represents normal deviation of the shapeof the brain of the patient from that of the atlas. The sec-ond component ðldj

þ Vdj bjÞ represents tumor mass-effect,and is chosen, by construction, to be orthogonal to thespace of the first component. This orthogonality, althoughnot strictly essential, makes the estimation of the tumormass-effect model parameters for a tumor patient easieras will be explained below.

The use of PCA implies that the vectors a, andbj ¼ ½bj;1; . . . ; bj;mdj

�T each follows a Gaussian distributionwith decorrelated components. The probability densityfunction of bj explicitly stated here as:

fjðbjÞ ¼1Qmdj

l¼1

ffiffiffiffiffiffiffiffiffiffiffiffiffi2pkdj;l

p exp �0:5Xmdj

l¼1

b2j;l

kdj;l

!

for j ¼ 1; . . . ; nm: ð11Þ

2.3. Statistical estimation

Given an approximate deformation map ~uf (between areal tumor patient’s images and the atlas) obtained throughthe direct use of deformable image registration, the goalof the methods presented here is to obtain an estimate bHof the tumor model parameters. The displacement map~uf defined in a similar manner to Eq. (5) is also discretizedover all the atlas voxels in BAnMA and represented by avector ~Uf . Owing to the orthogonality of Vdj to Vc for allj, we can compute the component of this displacement thatis caused by the tumor by projection as:

~Ud ¼ ~Uf � lc � Vc~a; ð12Þwhere ~a ¼ VT

c ð~Uf � lcÞ. The likelihood that ~Ud could begenerated with tumor model parameters Hj is defined as:

Lj � fjð~bjÞ; ð13Þwhere

~bj ¼ VTd;jð~Ud � ld ;jÞ for j ¼ 1; . . . ; nm: ð14Þ

High Lj implies that the corresponding Hj is likely accord-ing to the observed tumor-induced deformation. One pos-sible approach to estimating H would involve computingthe likelihoods for all Hj that were used for training andreturning the one with the highest Lj. This estimator wouldhowever provide only discrete values of H – one of thoseused during training. To produce an estimator with contin-uous values of H within the range used in training, we pro-pose to estimate the tumor mass-effect model parametersas:

bH ¼ Xnm

j¼1

LjHj

!, Xnm

j¼1

Lj

!: ð15Þ

This estimator favors values of Hj used in training and hav-ing high likelihood, while allowing the estimated H to takecontinuous values.

3. Experiments and results

Results of applying the approach described above arereported here for a real tumor image and for eight simu-lated tumor images. The real tumor image is an MRI ofpatient with a glioma and a large region of peri-tumoredema. The simulated tumor images are obtained by apply-ing the mass-effect model described in Section 2.1 to MRimages of normal subjects. All images used are T1-weighted MRI. The atlas image dimensions are256 · 256 · 198 and a voxel size of 1 · 1 · 1 mm. The usedand the simulated tumor images are of dimensions256 · 256 · 124 and voxel size 0.9375 · 0.9375 · 1.5 mm.

The FE tumor mass-effect model simulations are themost computationally intensive step of the presentedapproach. The simulations were carried out using the FEsoftware ABAQUS (ABAQUS Version 6.4, 2003). Theaverage time needed to perform one simulation was

758 A. Mohamed et al. / Medical Image Analysis 10 (2006) 752–763

approximately 35 min on a 900 MHz processor SGImachine. In order to make the statistical training step trac-table, we performed tumor simulations on ns = 20 MRIbrain images of normal subjects. For each subject,nm = 64 simulations were performed with two values ofeach of the six model parameters covering the rangeexpected for the real tumor case. The parameter valueswere rt 2 {3, 5} mm, rd 2 {20, 27} mm, P 2 {2, 5} kPaand corners of a cube in the atlas for the simulated tumorcenter locations. Therefore, in total, 1280 simulations wereperformed for statistical training. For the results reportedhere, all principal components of the displacement Uc wereretained and we used mdj ¼ 1, for j = 1, . . ., nm.

3.1. Tumor patient image

In Figs. 4 and 5, the results of applying the proposedapproach to register the image of the real tumor subjectto the atlas are illustrated. With the use of deformable reg-istration to directly register the (normal) atlas image to thepatient’s MR image, the warping result is inaccurate in thetumor area. Gray matter from the right cingulate regionand adjacent cortical CSF in the atlas were stretched tomatch the tumor and the surrounding edema in thepatient’s image. To apply the statistical estimate approachdescribed in Section 2.3, the region MA, where the registra-

Fig. 4. Three orthogonal 2D images of the atlas and the real tumor patient beimage, atlas image warped to the patient’s space via direct deformable registraspace via the use of the proposed approach.

tion is inaccurate was manually outlined. The estimatedtumor model parameters were ct ¼ ð109; 86; 126Þ, rt ¼3:9 mm, rd ¼ 24 mm and bP ¼ 3:55 kPa.

In order to quantitatively assess the improvement in theregistration accuracy due to the proposed approach, 21landmark points were selected around the tumor area inthe patient’s images and corresponding points were identi-fied by an expert in the atlas. The point coordinates weremapped through the resulting deformation map with directdeformable registration, and with the approach describedabove. The location of one landmark point in the atlasimage and the patient’s image before and after registrationis shown in Fig. 6. The results for all 21 points are pre-sented in Table 1. The maximum error was reduced by71% by the use of our approach while the mean errorwas reduced by 57.6%.

3.2. Simulated tumor images

Deformable registration experiments were performedfor simulated tumor images generated from MRI scans ofnormal brain images. The brain tumor mass-effect modeldescribed in Section 2.1 was used for these simulations.Values of the mass-effect model parameters used werechosen to be in the range of the parameter values usedfor training the statistical model. The use of simulated

fore and after deformable registration. Left to right: atlas image, subject’stion between the two images, and the warped atlas image in the patient’s

Fig. 5. Three orthogonal 2D images of the atlas and the real tumor patient before and after deformable registration. Some labels associated with the atlasimage are warped and superimposed on the patient’s images. Left to right: atlas image with all labels, atlas image with five selected labels near the tumorarea, patient’s image, patient’s image with superimposed labels warped from the atlas via direct deformable registration between the two images, and thepatient’s image with superimposed labels warped from the atlas via the proposed approach. The five selected labels are: the right middle frontal gyrus(green), the right medial frontal gyrus (dark green), the right superior frontal gyrus (cyan), the right cingulate region (magenta), and the left cingulateregion (brown).

Fig. 6. Locations of one manually selected landmark point in three orthogonal 2D images of the atlas and the patient before and after deformableregistration. The landmark point is marked with the green cross. Left to right: atlas image, patient’s image with the manually selected landmark point,patient’s image with the warped landmark point from the atlas with the direct use of deformable registration, and the patient’s image with the warpedlandmark point from the atlas with the proposed approach.

A. Mohamed et al. / Medical Image Analysis 10 (2006) 752–763 759

Table 1Deformable registration error statistics for landmark points in the real tumor (RT) and simulated tumor (ST) cases

Minimum Mean Maximum Standard deviation

RT no model, mm 1.06 8.70 24.87 6.19RT with model, mm 0.47 3.69 7.19 1.83ST no model, mm 2.54 6.39 10.91 2.62ST with model, mm 0.61 3.90 7.79 2.01

For each case, the errors are provided for the direct deformable image registration to the atlas (no model), and the registration using the approachdescribed in this paper (with model). 21 landmark points were used for RT and 25 were used for ST.

760 A. Mohamed et al. / Medical Image Analysis 10 (2006) 752–763

mass-effect images provides two benefits. First, since forsimulated cases, both uc and ud are available, the mappingud � uc is also available. Therefore, ud � uc is treated asground truth for uf and is used for evaluating the accuracyof the deformable registration, i.e., ub � ua. Second, sincevalues of the mass-effect model parameters are known forthe simulated cases, this offers a method for evaluatingthe accuracy of the statistical estimator of the modelparameters presented in Section 2.3.

First, we report the results for one simulated tumor casewith the parameters ct = (106, 86, 128), rt = 4.5 mm,rd = 21 mm and P = 4.5 kPa. Using the statisticalapproach described above, the estimated values of theseparameters were ct ¼ ð109; 85; 128Þ, rt ¼ 4:1 mm, rd ¼23 mm and bP ¼ 3:6 kPa. The subject’s image and that ofthe atlas before and after registration are shown inFig. 7. The warping of a few selected labels associated withthe atlas to the subject’s image near the tumor area areshown in Fig. 8. Visually, the results indicate an improve-

Fig. 7. Three orthogonal 2D images of the atlas and the simulated tumor susubject’s image before simulation of tumor mass-effect, subject’s image after simdirect deformable registration between the two images, and the warped atlas i

ment in the image registration accuracy due to theapproach described in this paper.

To quantitatively evaluate the registration accuracy in away similar to that used in the real tumor case, 25 pointswere selected arbitrarily in the area around the simulatedtumor, and their corresponding coordinates (foundthrough ud � uc) were computed in the atlas image andtreated as ground truth. The errors for the direct deform-able registration and that obtained by the proposedapproach are presented in Table 1 next to the results ofthe real tumor case. On average, for the simulated case,the maximum error was reduced by 29% using the pro-posed approach and the corresponding average error wasreduced by 39%.

In another experiment, seven simulated tumor cases weregenerated with mass-effect model parameters ct = (109, 84,129), rt = 4.3 mm, rd = 25 mm and P = 4 kPa starting withbrain images of seven normal humans. The estimatedmodel parameters in each case is provided in Table 2. The

bject before and after deformable registration. Left to right: atlas image,ulation of tumor mass-effect, atlas image warped to the subject’s space via

mage in the subject’s space via the use of the proposed approach.

Fig. 8. Three orthogonal 2D images of the atlas and the simulated tumor subject before and after deformable registration. Some labels associated with theatlas image are warped and superimposed on the subject’s images. Left to right: atlas image with all labels, atlas image with five selected labels, subject’simage, subject’s image with superimposed labels warped from the atlas via direct deformable registration between the two images, and the subject’s imagewith superimposed labels warped from the atlas via the proposed approach. The five selected labels are the same as those shown in Fig. 5 for the real tumorcase.

Table 2Estimated values of the tumor mass-effect model parameters for sevensimulated tumor images

ct rt, mm rd, mm P, Pa

Subj1 (109, 85, 127) 4.0 23 3456Subj2 (109, 86, 127) 4.0 23 3432Subj3 (109, 85, 127) 4.0 23 3463Subj4 (109, 85, 127) 4.0 23 3481Subj5 (109, 85, 127) 4.0 23 3467Subj6 (109, 86, 125) 4.0 24 3544Subj7 (109, 83, 126) 4.0 24 3476

The actual values of mass-effect model parameters used to generate theseimages were ct = (109, 84, 129), rt = 4.3 mm, rd = 25 mm and P = 4 kPa.

Table 3Minimum, maximum, and root-mean-square errors (mm) in the deformable r

No model

Minimum Maximum r.m.s

Subj1 0.028 19.70 7.54Subj2 0.117 18.07 8.26Subj3 0.138 22.08 6.19Subj4 0.074 18.39 5.55Subj5 0.063 22.58 6.87Subj6 0.090 22.52 6.50Subj7 0.071 16.37 4.97Mean 0.083 19.96 6.55

Results are reported for direct deformable registration between the atlas and thModel). Voxels in the atlas over which the error statistics are computed lie wi

A. Mohamed et al. / Medical Image Analysis 10 (2006) 752–763 761

deformable registration accuracy was evaluated as follows.A spherical region of radius 2.5 cm was selected in the atlasaround the center of the simulated tumor but outside thetumor itself. This region, which is the same region whereedema is assumed to be present, includes voxels of theatlas, where the direct deformable registration betweenthe atlas the tumor image is likely to be inaccurate. Themaximum, minimum, and root-mean-squares (r.m.s.) errorin the registration for all voxels in this region are reportedin Table 3. Despite differences between the estimated mass-effect model parameters and the true values used for simu-lations, the deformable registration results indicate that on

egistration for seven simulated tumor images

With model

. Minimum Maximum r.m.s.

0.031 16.33 3.620.049 16.47 4.430.043 14.18 3.440.042 14.71 4.340.098 18.76 4.080.080 9.91 3.620.078 18.22 4.470.060 15.51 4.00

e tumor image (No Model) and the approach described in this paper (Withthin 25 mm from the center of the simulated tumors.

762 A. Mohamed et al. / Medical Image Analysis 10 (2006) 752–763

average the r.m.s. error in the registration is reduced by38.9% and the maximum is reduced by 22.3%. In one case(Subj6), the maximum error was reduced by 56% with acorresponding 44% reduction in the r.m.s. error.

4. Discussion and future work

In this paper, we described an approach for the deform-able registration of a brain atlas to brain tumor images.The approach uses a 3D biomechanical FE model oftumor-induced deformation to introduce and simulate thetumor in the atlas followed by the use of a readily availabledeformable image registration method. In addition to theproposed overall approach for deformable registrationfor tumor images, the contributions of this work includeintroducing a statistical approach for solving the inverseproblem of determining the mass-effect model parameters.This statistical approach relies on the decomposition of thedesired deformation map (between the atlas and the tumor-bearing patient’s image) into the sum of two maps inorthogonal subspaces, defined on the same domain, buthaving different statistical properties. The first deformationmap is from the atlas to another normal brain image. Thesecond, is the deformation map from the normal brainimage to one that is deformed by the biomechanical modelof tumor mass-effect. The statistical properties of both ofthese deformation maps are learned via PCA from a num-ber of training samples. Owing to the orthogonality of thetwo components of the modeled deformation map, an ini-tial rough estimate of this map is projected onto the sub-space representing tumor-induced deformation and isused to estimate the tumor mass-effect model parameters.

The results of applying the proposed approach to a realtumor image and simulated tumor images indicate a signif-icant reduction in the registration error. These experimentsshould be regarded as a proof-of-concept study. More val-idation experiments are needed to asses the viability of theproposed approach for a variety of tumor cases of differentgrades, types and sizes. In particular, more experimentswith real tumor patient images are needed, with an assess-ment of inter-rater and intra-rater variability in the identi-fication of homologous landmark points between the atlasand the patient images. In addition, the sensitivity of thestatistical estimator of the mass-effect model parametersto the number of used principal components and the num-ber of training samples also present important directionsfor future investigations.

Understanding the effects of the introduction of thetumor and edema into the atlas image, the biomechanicalmass-effect model simulation, and the statistical estimatoron the stability of the overall image registration approachneeds further analysis. Visual examination of the imageof the atlas after warping to the real and simulatedpatient’s images reveals no significant differences in the reg-istration with and without the proposed approach awayfrom the tumor, in the ipsilateral and contralateral hemi-

spheres. Quantitative analysis of the stability of the overallapproach is subject to future work.

One of the difficulties associated with the deformableregistration of a normal brain atlas image to a brain tumorpatient’s image is the topological differences between thetwo brains. These differences arise from loss of normal tis-sue from the patient’s brain (due to its death or invasion bytumor) and its replacement with tumor tissue. To compen-sate for these differences, the approach described in thispaper introduces a region of simple spherical geometry intothe atlas image. This region approximates tissue that waslost from the patients brain due to tumor-related effectsand that was replaced by tumor tissue. The model approx-imates peri-tumor edema with another spherical region.The two deformations applied to register the atlas to abrain tumor image do not change the topology of thesesimple spherical regions introduced into the atlas image.While mechanical factors included in the mass-effect simu-lations can make the final simulated tumor shape non-spherical, in real tumors, other factors that are notmechanical in nature (e.g., tumor infiltration and edemaspread), may also play a role in determining the tumorshape and may cause it to deviate from radial symmetry.More accurate mass-effect simulations may require the def-inition of bulk tumor and edema regions that are notspherical and that may vary from one simulation toanother. To overcome this limitation of the current model,two approaches may be investigated in future work. First,models of tumor infiltration and edema spread (e.g., seeClatz et al., 2005; Nagashima et al., 1994), combined withadditional image information (e.g., diffusion tensorimages), may be used to obtain patient-specific initializa-tions of the mass-effect model. In the second approach, seg-mentations of the tumor and edema regions in the patientimage, if available, may be mapped back to the atlas viathe most current estimate of the deformation map uf andused for a new initialization of the mass-effect model. Thus,the bulk tumor and edema regions will not be necessarilyspherical, but will be derived from the shapes of the bulktumor and edema regions in the patient. This suggests aniterative approach, where the initialization of the mass-effect model is hopefully improved as better estimates ofthe sought deformation map are obtained.

Another complication associated with the deformableregistration of brain tumor images is the significant signalchanges associated with edema in MR images. Edema typ-ically causes hypointensity changes in T1-weighted images,which makes it difficult to discern cortical sulci in theaffected brain regions. It is therefore not possible to obtainan accurate deformable registration in these regions basedon image matching alone. The coordinates of brain struc-tures masked by edema may however, be estimated fromthe known structures (outside the edema region) throughthe statistical estimation approaches (Liu et al., 2004).

Although the implementation presented above relied onmanual outlining of the region MA (where direct deform-able registration is inaccurate), automatic identification of

A. Mohamed et al. / Medical Image Analysis 10 (2006) 752–763 763

this region can be achieved based on the elastic stretchingenergy or the matching criterion of many deformableimage registration algorithms.

The developed biomechanical mass-effect model playsan important role in the overall presented approach forimage registration. Improvements of this mass-effect modelare expected to improve the registration results. For exam-ple, improving the accuracy of the boundary conditions byusing a stiffer material for the falx compared to the rest ofthe brain and allowing the brain tissue to slide against itcan be expected to produce more accurate simulations ofsubcortical tissue deformation. Other improvements tothe mass-effect model may involve the use of biphasic con-solidation theory material model which has been suggestedin the literature for modeling edema expansion (Nagashimaet al., 1994) and brain shift (Miga et al., 1999a; Miga et al.,1999b). While this constitutive model offers a direct way tosimulate dilatation due to edema, the implementation ofthis material model in ABAQUS requires the use of sec-ond-order tetrahedral FE meshes, which are one orderhigher than the meshes used in the current mass-effect sim-ulations. To avoid the significantly higher computationalcost associated with second-order tetrahedral meshes, wechose to use a single phase material model in this work.The comparison of biphasic models to the adopted singlephase model with quadratic tetrahedral meshes is an inter-esting point for future work.

Acknowledgements

The authors thank Dr. Nick Fox at the University Col-lege London, UK, for providing the tumor patient’simages. We also thank Xiaoying Wu at the Section of Bio-medical Image Analysis at the University of Pennsylvaniafor her help in processing the used data. This work wassupported in part by the National Science Foundation un-der Engineering Research Center Grant EEC9731478, andby the National Institutes of Health Grant R01NS42645.

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