Structural shape characterization via exploratory factor analysis

Artificial Intelligence in Medicine 30 (2004) 97–118

Structural shape characterization via exploratory factor analysis Alexei M.C. Machadoa,*, James C. Geeb, Mario F.M. Camposc a

Graduate Program on Electrical Engineering, Pontifical Catholic University of Minas Gerais, Av. Dom Jose Gaspar 500, 30535-610 Belo Horizonte, MG, Brazil b Department of Radiology, University of Pennsylvania, 3400 Spruce Street, Philadelphia, PA 19104, USA c Department of Computer Science, Federal University of Minas Gerais, Caixa Postal 702, 30161-610 Belo Horizonte, MG, Brazil

Received 12 September 2002; received in revised form 27 December 2002; accepted 17 March 2003

Abstract This article presents an exploratory factor analytic approach to morphometry in which a highdimensional set of shape-related variables is examined with the purpose of finding clusters with strong correlation. This clustering can potentially identify regions that have anatomic significance and thus lend insight to knowledge discovery and morphometric investigations. Methods: The information about regional shape is extracted by registering a reference image to a set of test images. Based on the displacement fields obtained form image registration, the amount of pointwise volume enlargement or reduction is computed and statistically analyzed with the purpose of extracting a reduced set of common factors. Experiments: The effectiveness and robustness of the method is demonstrated in a study of gender-related differences of the human corpus callosum anatomy, based on a sample of 84 right-handed normal controls. Results: The method is able to automatically partition the structure into regions of interest, in which the most relevant shape differences can be observed. The confidence of results is evaluated by analyzing the statistical fit of the model and compared to previous experimental works. # 2003 Elsevier Science B.V. All rights reserved. Keywords: Morphometry; Factor analysis; Corpus callosum; Knowledge discovery; Image registration; Magnetic resonance imaging

* Corresponding author. Tel.: þ55-31-33194002; fax: þ55-31-91526355 E-mail addresses: [email protected] (A.M.C. Machado), [email protected] (J.C. Gee), [email protected] (M.F.M. Campos).

0933-3657/$ – see front matter # 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0933-3657(03)00039-3

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1. Introduction An important problem related to medical image analysis is the representation of the large amount of data provided by imaging modalities and the extraction of relevant shape-related knowledge from the dataset. The data should not only be represented in a manageable way, but also facilitate hypothesis-driven explorations of regional shape differences and lend deeper insight to morphometric investigation. This work presents a novel method for knowledge discovery, based on unsupervised learning, that explores the correlation among morphometric variables and the possible anatomic significance of these relationships. The analysis of large datasets is a difficult task even for experts and can be done manually by the registration of a limited set of landmarks. Imaging modalities may provide too much data for manual pointwise registration, what motivates the development of automatic methods that implement computer vision algorithms. The result of registration may, nevertheless, increase the amount of data to be analyzed. The method proposed in this article explores the data with the purpose of discovering underlined connections among the variables that can be used to support hypothesis on the morphology and functionality of structures. Our approach is based on the analysis of high-dimensional sets of vector variables obtained from non-rigidly registering or deforming an image, taken as a reference, so as to align its anatomy with the subject anatomy of a group, depicted in MRI studies. The result of registration is a set of displacement fields from which the amount of volume enlargement or reduction at each point of the image can be computed and statistically analyzed with the purpose of extracting a reduced set of common factors. In addition to the morphometric variables, clinical and demographic information can be considered in the analytic model and contribute to explore the relationship between regions in the image and pathologies or features of special interest. We demonstrate the exploratory potential of the method in a study of morphologic differences among the corpora callosa of normal males and females. 1.1. The main hypothesis The association of structural morphometry and functionality has become a research issue of great interest in the past decades. The development of non-invasive imaging modalities opens a new perspective for in vivo studies, in which anatomical and functional aspects can be jointly observed. There have been evidences that the anatomy of structures such as the hyppocampus [39], midbrain [30], cerebellar vermis [18] and thalamus [19] play an important role on the characterization of neurological activities and pathologies. Traditionally, the analysis of structural shape variation has been performed after the regions of interest are delimited. However, this is not an easy task when the structures are not clearly bounded, as is the case of the hyppocampus. The corpus callosum is another example in which ad hoc segmentation may be required. Although the segmentation of the outline section in the midsagittal plane is usually simple, the partition of the fiber tracts that connect different regions of the hemispheres is difficult, since no borders or texture exist to facilitate segmentation. Studies based on postmortem analysis have been presented, in which the corpus callosum is divided into segments roughly associated with cortical regions. The subregions are measured and statistical distributions are built to support

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Fig. 1. Topology of the corpus callosum, adapted from Witelson [42]. The regions of the callosal structures are the rostrum (1), genu (2), rostral body (3), anterior midbody (4), posterior midbody (5), isthmus (6) and splenium (7).

specific hypotheses relating functional specialization of cerebral hemispheres with demographic and clinical variables. An important study of the corpus callosum morphometry was presented by Witelson [42], based on postmortem analysis. The corpus callosum was first divided into seven regions, determined as posterior and anterior halves, thirds and fifths of the callosal length. After segmentation, the regions of each subject were measured with the purpose of building statistical distributions. Witelson hypothesized that the anatomical variation of the corpus callosum would support functional asymmetry and specialization of cerebral hemispheres related to gender and handiness. Fig. 1 shows a schematic of the callosal midsagittal plane subdivisions, in which the seven regions of interest are defined. The subdivision was proposed based on studies of autoradiographic tracing methods, with monkeys and clinical studies with humans [3,7,20,29,35,38]. The regions of the corpus callosum were related to specific regions of the cortex, although presenting considerable overlap: (1) rostrum— related to the caudal and orbital prefrontal regions and the inferior premotor cortical region; (2) genu—prefrontal region; (3) rostral body—related to the premotor and supplementary motor regions; (4) anterior midbody—related to the motor cortical region; (5) posterior midbody—somaesthetic and posterior parietal regions of the cortex; (6) isthmus—superior temporal and posterior parietal lobes; (7) splenium—related to the occipital and inferior temporal lobes. In this work, we hypothesize that it is possible to automatically determine subregions in a structure, even if there are no borders or texture, by analyzing its overall shape variation. If the hypothesis holds, the direction of the process is inverted: the information obtained from shape analysis is used to segment the structure into regions of interest. We show that unsupervised learning can be used to explore the anatomy of substructures and facilitate segmentation. The method, applied to a study of the corpus callosum, presented results that are in accordance to the topology inferred by means of postmortem analysis. 1.2. Background One of the most relevant mathematical frameworks used to describe general shape variation has been the principal component analysis (PCA). Marcus [27] used PCA to study the variation in the skull measurements of rodent and bird species. The resulting principal modes of variation were subjectively interpreted as size and gross shape components. Marcus concluded that no specific interpretation should be expected from the method, since it did not embed a biological model.

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Cootes et al. [8] applied the theory of PCA to build a statistical shape model of organs based on manually chosen landmarks. The organs were represented by a set of labeled points located at particular regions in order to outline their characteristic shape. The model provided the average positions of the points and the principal modes of variation computed from the dataset. The ability of the method to locate structures in medical images was demonstrated in a set of experiments with echocardiograms, brain ventricle tracking and prostate segmentation. Generalizing the use of PCA to high-dimensional sets of variables, Le Briquer and Gee [22] applied the method to analyze the displacement fields obtained from registering a reference image volume of the brain to a set of subjects, based on the elastic matching framework [2]. The analysis provided the inference of morphological variability within a population and was the basis for the construction of a statistical model for the brain shape, which could be used as prior information to guide the registration process. Davatzikos et al. [11] showed how the results obtained from matching boundaries of structures could be interpolated to determine an estimate for the complete displacement field. The method was useful in the registration of structures such as the corpus callosum, whose contour was of more interest than its inner texture. Further analysis on the gradient determinant of the resulting displacement fields showed the amount of area enlargement/ reduction while deforming the reference image to match the images in the study. The method was applied to a small set of images of the human corpus callosum, revealing gender-related morphological differences. Using the same dataset, Machado and Gee [24] performed elastic matching to both the boundary and the interior of the structure. Based on the displacements fields obtained from image registration, the method was able to reproduce Davatzikos’ results and additionally determine the principal modes of callosal shape variation between sexes. Martin et al. [28] also applied PCA to the shape characterization of the brain, in a study of schizophrenia and Alzheimer’s disease. The putamen and ventricles were modeled as a linear elastic material and warped to match the same structures of a normal brain volume. The principal modes of variation computed from the results of image registration were fed to a Gaussian quadratic classifier. The experiments showed that using principal components instead of gross volume as the features for the classifier increased the rate of correct classification from 60 to 72% while discriminating the putamen of normal and schizophrenic patients. The major objective of PCA is to represent data in a new basis whose axes correspond to the principal modes of the sample variance. However, when the purpose is to explore the covariance among the variables, factor analysis (FA) may be considered an appropriate alternative [36]. On exploring the morphology of a specific structure, one may be concerned with the relationship between regions of interest. FA may reveal aspects about the correlation between those regions and facilitates interpretation. Nonetheless, the use of FA in morphometry has been restricted to the representation of gross measurements and landmarks, regardless of exploring the relationship between pointwise shape-related variables, as the ones obtained from image registration. Marcus [27] compared the application of PCA and FA on a set of length measurements for several hundreds skeletons of birds. The extracted factors were interpreted as general features related to the overall size of the subjects. Reyment and Jo¨ reskog [34] presented a thorough discussion on the

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factor analysis of shape-related landmarks. Scalar features such as the distances between landmarks in the carapace of ostracod species were considered in the analysis. Some of the resulting factors were interpreted as shape-changes in specific regions of the shell, location of eye tubercles and valves. Other factors, however, were related to global features such as the dimensions and curvature of the shell. Stievenart et al. [41] applied FA to study the correlation among parts of the corpus callosum, whose boundary curvature was measured at 11 different positions. The results revealed three factors that explained 69% of the variation of the original curvature values. The first and second values were clearly related to the curvature of the isthmus and posterior region of the splenium, respectively. Another relevant work on the factor analysis of the corpus callosum was presented by Deneberg et al. [12], in which the structure was divided into 100 segments taken along equally spaced intervals of the longitudinal axis. Although the structure partitioning criteria was deliberately chosen to result on transversal segments, the study was able to identify regions in the corpus callosum, particularly the isthmus, which presented morphological differences related to gender and handiness. This article is structured as follows. In the next section, the factor analytic model and its relationship to PCA are presented. The method is demonstrated in a study of differences in the callosal morphometry between sexes and the results are compared to other published findings, followed by discussion and conclusions.

2. The factor analytic model The purpose of FA is to explore the correlation among the variables of a problem. Similarly to PCA, FA is a powerful method of data reduction, which makes it possible to manage the large amount of information obtained from image registration. A fundamental feature of FA is that, in addition to data reduction, it may favor data interpretation. In this work, we show how the factors obtained in the analysis of shape-related variables can be associated to specific regions of interest in the images, evidencing morphological differences between populations. In FA, a p-dimensional set of original variables, y ¼ ðy1 ; . . . ; yp ÞT , is represented as linear combinations of m hypothetical constructs called factors: y ¼ Af þ ;

(1) T

T

where f ¼ ð f1 ; . . . ; fm Þ is a vector of common factors, ¼ ðE1 ; . . . ; Ep Þ are the unique factors or residual terms which account for the portion of y that is not common to other variables, and A ¼ ðða11 ; . . . ; a1m ÞT ; . . . ; ðap1 ; . . . ; apm ÞT ÞT is the loading matrix. The coefficients aij , called loadings, express the covariance between variable yi and factor fj . Variables y are standardized, so that their expected values are 0 with variances equal to 1. The covariance matrix for the population in the original basis, R, can be defined as R ¼ covðyÞ ¼ covðAf þ Þ:

(2)

The factor analytic model imposes certain assumptions on f and . Since the expected value Eð yÞ is the null vector, EðAf þ Þ must also be 0. It is assumed that Eð Þ is 0. In order

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for the factors to account for all the correlation among the variables y, the covariances among unique factor terms and common factors are 0. The covariances among unique factor terms are represented by the diagonal matrix W ¼ diagðc1 ; . . . ; cp Þ. Factors are not correlated, so that the corresponding covariance matrix is the identity. It should also be noticed that FA, as well as PCA, can be completely modeled from the information represented in the covariance matrix. In other words, FA is implemented in the context of the classical assumption of Gaussianity. Determining if the data fit a multivariate Gaussian distribution is an additional aspect of the problem, since the population parameters should be estimated from the sample. In order to justify the choice of a specific model, a test of Gaussianity should be made (e.g. forth-order cumulants [9,23]), so that the results can be considered reliable. Considering the assumptions in the factor analytic model, the variance s2i of a given variable yi can be decomposed into components due to the m common factors, a2i1 þ    þ a2im , called the communality, and a specific variance ci : m X a2ij þ ci : (3) s2i ¼ j¼1

The population covariance matrix R defined in Eq. (2) can thus be represented in a simpler way. Since Af and are not correlated, the covariance matrix of their sum is the sum of the covariance matrix of each term. Also, since covð f Þ ¼ I, the relationship between R, A and W can be written as R ¼ covðA f þ Þ ¼ covðA f Þ þ covð Þ ¼ A covð f ÞAT þ W ¼ AAT þ W:

(4)

3. Principal components and factor analysis Although the main objective of PCA and FA is data reduction, they differ fundamentally on two aspects: the algebraic model of the transformation and how data reduction is achieved. In PCA, the set of original variables y is rotated in order to find the orthogonal axes along which the data is maximally spread out. The new p-dimensional basis z ¼ ðz1 ; . . . ; zp ÞT is achieved by multiplying the original variables by an orthogonal matrix B: z ¼ By:

(5)

Each new variable z, or component, is a linear combination of the original variables y. The algebraic models for PCA and FA evidence a major difference. In FA, the original variables are represented as a linear combination of new variables (factors), while in PCA, the new variables (principal components) are linear combinations of the original variables. In PCA, data reduction is achieved by changing the basis of the variable space, so that the new orthogonal axes represent most of the variance embedded in the dataset. The objective of PCA can be defined as maximizing the variance of a linear combination of the original variables. Data reduction is obtained (with possible loss of information) by ignoring the axes in which the data present small variance. In contrast, FA aims to find a new low-dimensional set of non-observed variables that maximally represents the covariance (or correlation) among the original variables.

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Another difference to be highlighted is that PCA is closely related to the dataset behavior, while FA aims to understand the relationship among the many variables of the problem. In this sense, FA can be considered a more useful exploratory tool, rather than just a data reduction technique. The parameters of FA can be estimated from the sample, by replacing the population covariance matrix R in Eq. (4) by the covariance matrix S obtained from the dataset: ^ ^A ^ T þ W; SA ^ are estimations for the loading matrix and unique factor covariance matrix, ^ and W where A respectively, since they are computed from the sample. Many techniques have been ^ and factor S using spectral ^ The simplest one is to neglect W proposed to determine A. decomposition: ^A ^ T  QKQT  ðQK1=2 ÞðQK1=2 ÞT ; SA 1=2

where K1=2 ¼ diagðl1 ; . . . ; l1=2 p Þ is the diagonal matrix with the square root of the eigenvalues of R and Q is the matrix of the corresponding eigenvectors. The loading matrix can then be estimated based on the sample covariance matrix as ^ ¼ QK1=2 : A

(6)

The method described above for estimating the loading matrix A is very similar to PCA and is known as principal factor method. The models would be identical if the covariance matrix for the factors could not be assumed to be the identity and the loading matrix further rotated. In PCA, the model must represent both the diagonal and off-diagonal elements of the covariance matrix, so the diagonal elements of S must be 1, otherwise the analysis of variance will not be properly performed. In contrast, the aim of FA is to represent only the off-diagonal elements that account for the correlation among variables. By neglecting the ^ the factor analysis of the covariance matrix is performed by specific variance matrix W, placing communalities in the diagonal elements. In this case, the recovered covariance ^A ^ T will have its off-diagonal elements affected. Hence, it is important to measure matrix A this error in order to have a robust estimation of whether the data fit the model. Other techniques such as the maximum likelihood methods can also be used for determining the loading matrix [17]. These methods have, nevertheless, the drawback of requiring inverting the covariance matrix. When the number of variables is greater than the number of subjects in the sample, as it is the case of morphometry studies, the covariance matrix is always singular. In PCA, data reduction is obtained after the computation of A by eliminating the components that do not contribute significantly to the representation of variance. In principle, the dimension of the new basis z is the same of y, so data reduction is achieved by discarding components, with possible loss of information. In fact, since PCA is directed to the representation of variance, the covariance information is guaranteed to be preserved only if all components are kept. Furthermore, data reduction is possible only when the original variables y are correlated. In the case of independent variables, all the eigenvalues will have similar values. This contrasts with FA, in which dimension reduction is accomplished during the computation of A. The number of factors to be

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considered must be chosen in order to compute the loading matrix. As a consequence, the elements of A changes as the number of factors m considered in the analysis varies. The choice of the number of factors to be employed is another fundamental aspect of the problem and will be discussed in Section 4. As in the case of PCA, in which the number of components to be kept should also be defined, the number of factors must also be chosen. The results of FA should therefore be evaluated by analyzing the statistical fit of the data. An important property of the loading matrix A is that it can be rotated and still be able to represent the covariance among factors and original variables [17]. The rotation of loadings plays an important role in factor interpretation, as it is possible to obtain a matrix that assigns few high loading for each variable, keeping the other loadings small. If such matrix is obtained, each variable will be related to a single factor or at least to few ones. Since the variables are related to pixels in the image, the resulting factors can be visually identified as regions in the structure. On the other hand, in the PCA framework, further rotations of the principal components go against the main purpose of the analysis, which is to find a configuration in which the data variance is optimally represented. PCA is also not invariant to variable scaling, as this may change the shape of the distribution and the values of variance. As a consequence, the PCA of the covariance matrix and its corresponding correlation matrix may yield different results. The differences in the way data is reduced impact particularly in the interpretation of the results. Since PCA aims to maximally represent the variance of the data, the resulting components may be linear combinations of original variables that do not share strong relationship. On the other hand, the goal of factors analysis is to group correlated variables. The variables are partitioned into groups that are associated to different factors, and can be visually observed as regions in the image. For exploratory studies, this may be essentially helpful, as it reveals regions of interest in which shape variability behaves in a correlated way.

4. Methods The rationale for structural shape characterization is to provide a quantitative description of the morphometric differences between structures that present a gross common anatomy. Shape description can be achieved by taking a reference image and warping it as to align its anatomy with the anatomy of each individual in the study. The spatial transformation obtained in the warping process can be analyzed and yields immediate knowledge about the anatomic variation among the subjects of the sample. 4.1. Image registration Image registration aims to determine a correspondence between each pixel q in the reference image IR to a pixel p in the test image IT . The problem of matching can be stated as finding two functions h and g such that IR ðqÞ ¼ gðIT ðhðqÞÞÞ:

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The intensity transformation function g establishes a correspondence between the two image spectra. The spatial transformation function h maps corresponding pixels between the two images: hðqÞ ¼ q þ uðqÞ ¼ p; where u is a displacement field [14]. Image registration may be performed by first applying rigid transformations (translation and rotation), in order to approximately register corresponding features, and then warping the template to match the subject. Image volumes may be described as continuous media to which a constitutive model will be prescribed. The linear elasticity model [2], in which the image is deformed as an elastic body, guarantees smoothness to the deformation, so that neighboring structures in the reference image will be matched to neighboring structures in the subject’s image, preserving the gross anatomy common to the majority of individuals in the population. Other models of spatial transformation include modal matching [37], active shape models [8,13], fluid mechanics [6] and active contour models [10]. From the set of displacement fields obtained by registering the reference image to each subject image, the average displacement field can be computed as ðqÞ ¼ u

N 1X ui ðqÞ; N i¼1

where ui ðqÞ is the displacement field associated to the ith subject and N is the sample size. 4.2. Jacobian analysis When the reference image is warped to match a subject image, some regions may get enlarged and some may be reduced. It is possible to determine the amount of scaling applied to an infinitesimal area around each point of the reference image, by computing the Jacobian determinant of the spatial transformation. In the case of two-dimensional images, the displacement vector field from the reference to subject i, ui , can be decomposed into its components ui and vi . Similarly, q can be expressed in terms of its coordinates ðx; yÞ. The Jacobian determinant Ji ðqÞ is defined as the determinant of the gradient of the mapping function q þ ui ðqÞ:    @ui ðqÞ @ui ðqÞ    þ 1   @x @y : Ji ðqÞ ¼ jrðq þ ui ðqÞÞj ¼   @vi ðqÞ  @vi ðqÞ þ 1   @x @y The value of J is strictly positive, as the registration is constrained to produce a continuous one-to-one mapping. The set of pointwise Jacobian determinants is the input to FA. Since the result of image registration is a smooth displacement field, it is expected that the factors be correlated to the Jacobian determinants of neighboring points. The regions of interest found in FA can be compared to the effect size eðqÞ, computed for the size differences

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between two samples as eðqÞ ¼

J1 ðqÞ J2 ðqÞ ðs2 ðJðqÞÞÞ1=2

;

where Ji ðqÞ is the mean determinant value, averaged over the ith group, and s2 ðJðqÞÞ is the variance computed over all of the subjects. The effect size is a simple but useful measure used to evidence regional differences between the means of two populations, normalized by their joint standard deviation. Regions in which the effect size present large values may be considered a discriminant feature for shape variation analysis. If it is possible to find a correspondence between these regions and the results of FA, the confidence on the factoranalytic approach to morphometry is strengthened. 4.3. Statistical fit of the factor analytic model Another important issue that affects FA is the number of factors to be considered. Harman [17] presented a comprehensive discussion on how to determine the number of factors that best represent a dataset. Machado et al. [25] proposed an iterative algorithm for choosing which factors to retain by evaluating the number of observed variables associated to each factor. In the experiments, only those factors with correlation greater than 0.5 with at least two variables were considered informative. The initial number of factors (number of columns in A) was determined as the number of eigenvalues greater than 1, since they accounted for the variation of at least one variable. The number of factors was reduced, at each iteration, by discarding factors which did not present high correlation to at least two variables. Convergence was achieved when the same number of factors was determined at two consecutive iterations. In this work, we also discuss the influence of the choice of the number of factors in the results, based on the quantitative analysis of the statistical fit of the model. The discussion is based on three parameters. (1) The completeness of factor analysis If the original variables are naturally correlated so that clusters of variables related to independent factors can be found, the specific factor terms should be minimum. Since the variables are standardized, the variance for each variable yi is 1. From Eq. (3), we have that m X

^ ^ a2ij ¼ 1 c i

j¼1

for each variable yi , meaning that the communality tends to unity, when the variance can be completely explained by the factors and consequently there is no significant ^ . The completeness of the FA can thus be defined as the average of specific variance c i all communalities: m 1XX ^ a2 : p i¼1 j¼1 ij p

completeness ¼

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(2) Analysis of the communality for specific features of interest For exploratory purpose, FA can be applied to a heterogeneous variable set composed of the Jacobian determinants and other measurements or features of interest. In this case, the p-dimensional variable set, y, will be composed of the Jacobian determinants JðqÞ, for all pixels q in the structure being investigated, appended by the k feature variables f1 ; . . . ; fk : y ¼ ðJðq1 Þ; . . . ; Jðqp k Þ; f1 ; . . . ; fk Þ: The loadings related to these specific features of interest can be separately investigated. If the number of factors is not properly chosen, the specific variance may be too large, meaning that the model is not able to correlate the feature of interest to the morphometric variables. Large values for the specific variance may alternatively reveal that the feature is actually not correlated to the shape of the structure. The communality for feature i is defined as h2i ¼

m X

^ a2ij ;

j¼1

where i is the index of the feature variable in the variable vector. (3) Analysis of the residual correlations ^ is a good estimate A straightforward way to analyze whether the loading matrix A of the correlation between factors and variables is to compute the distribution of the ^A ^ T should be a good correlation residuals. The recovered correlation matrix A approximation to S. Let rjk be the observed correlation between variables yj and yk , j 6 k, and ^r jk be the corresponding recovered correlation determined as ^r jk ¼

m X

aji aki :

i¼1

The residual correlations, rjk ^r jk , should have a distribution similar to that of a zero correlation, whose standard deviation is computed as sr¼0 ¼ N 1=2 : ^ can be considered a good approximation if the standard The loading matrix A deviation for the residuals is smaller or equal to sr¼0 . This threshold may, however, be too high when the size of the sample N is small compared to the number of variables. In fact, a thorough investigation of the statistical fitness of the model should jointly take into account all the parameters discussed in this section.

5. Materials The MRI images used in the experiments, gently shared by the Mental Health Clinical Research Center of the University of Pennsylvania, are normal controls recruited for a larger study on schizophrenia. The images were acquired in the axial plane on a GE 1.5 T instrument, using a spoiled GRASS pulse sequence optimized for high resolution, near

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isotropic volumes (flip angle ¼ 35 , TR ¼ 35 ms, TE ¼ 6 ms, field of view ¼ 24 cm, 0:9375 mm 0:9375 mm in-plane resolution, 1.0 mm slice thickness, no gap). The middlemost sagittal slices were extracted and reformatted into 256 256 8-bit images after the volumes have been realigned to a standard position by determining the anterior commissure to posterior commissure line and interhemispheric fissure. The sample used in the study is composed of 42 male and 42 female right-handed controls. The age of the subjects is in the range of 19–68 years (mean  S:D:, 30:4  11:8) for males and 18–68 years (26:5  9:0) for the females. The sample was chosen in order to provide an approximated distribution of age and race for both groups and to guarantee a minimum influence of these features in the analysis. No expressive correlation was detected, in the sample, between gender and age (0.18) and between gender and race (0.06).

6. Experimental procedure The images were first segmented by supervised thresholding. The process was performed twice by a single rater who was blind to subject gender. The reliability of the segmentation was measured by computing the intraclass correlation coefficient (ICC) based on the area of the corpus callosum. The ICC value for the dataset was 0.888. Additionally, the same dataset was segmented by a second rater who was also blind to demographic information. Using the adaptive K-means clustering algorithm of Pappas [32], the images were partitioned into white matter, gray matter and cerebro-spinal fluid components, from which the corpus callosum structure was extracted by manual delineation. The interrater variability, measured based on the ICC was 0.884. Global registration was performed by translating and rotating each callosal structure to align with the template, without scaling, through landmark matching. A set of three landmarks were manually chosen in each subject and in the template: the intersection of the subject’s callosum boundary to its minimum enclosing rectangle at the most inferior point of the splenium, the most anterior point of the genu and the most superior point of the callosum body. The rigid transformation for each subject was achieved by jointly minimizing the distance between the subject’s landmarks and the corresponding points at the template. The process was repeated three times by a single rater, over all the dataset, yielding ICC values greater than 0.980 for all coordinates. Local registration was performed by warping the template to match each globally aligned subject callosum. The warping algorithm is a Bayesian generalization of the Bajcsy and Kovacˇ icˇ multi-resolution elastic matching technique [2,16]. In this approach, the reference brain is modeled as an elastic body, which is deformed to match the brain of a subject in the study. The method implements an optimization procedure which finds an equilibrium among the external forces represented by the likelihood of matching a point in the reference to a point in the subject and the internal forces which rule the smoothness and topological constraints of warping. Elastic matching is usually applied as a fully volumetric procedure for the whole brain volume, explicitly taking into account the morphology and topology of all its individual structures. In this study, however, since the analysis was focused on the corpus callosum midsagittal profile, the registration was specifically centered in the callosum region to improve the effectiveness of the elastic matching.

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The displacement fields obtained from image registration were the basis for the computation of the Jacobian determinants. Vector y was formed by the determinant of the Jacobian matrix at each of the 851 pixels in the callosal template, together with the feature gender, and used as input to FA. The normal male and female controls were given a value of 0 and 1, respectively, for the gender variable. Since FA assumes that the variables fit Gaussian distributions, a test of normality was performed by estimating the skewness and kurtosis of the distribution for the populations. For the male sample, with a level of significance of 0.01, there were no evidences to reject the hypothesis of normality for 84.6% of the variables, based on the skewness, and for 89.0% of the variables, based on the kurtosis of the distribution. The results for the females were similar: the hypothesis could not be rejected for 83.0% and 90.0% of variables, respectively for the skewness and ^ (Eq. (6)), it was rotated in order kurtosis analyses. After determining the loading matrix A to maximize the variance of the squared loadings in each column, so that each variable presented high loading for fewer factors (varimax algorithm) [34]. A brief discussion should be made about the impact of the steps performed prior to factor analysis. The reliability of the results of multivariate analysis depends on the quality of the data provided by the registration algorithm. The elastic matching method, as well as other registration methods, requires the choice of a set of parameters, such as the stiffness of the medium. The models that are based on continuum mechanics, such as linear elasticity and fluid mechanics make strong assumptions on the medium behavior. It is common sense that registration is a visual task, depending on human perception, and that brain tissues cannot be expected to behave as elastic or fluid medium. Nevertheless, these models have a desired property to perform registration preserving gross anatomy. When implemented in a multiresolution way, the number of parameters to be specified increases. Parameter estimation is a whole field for research and is not addressed in this study. We have experimentally determined a set of parameters that yield good results for the registration of the corpus callosum. In the case of elastic matching, the global registration is used as a first approximation of corresponding anatomical regions, since it is known that linear elasticity presents better results for smaller deformations.

7. Experimental results The algorithm used to determine the number of factors, described in Section 4.3, took nine iterations to converge from 78 to 11 factors with correlation magnitude greater than 0.5 among at least two variables. With a level of significance of 0.01, a correlation coefficient magnitude of 0.5 computed for the sample gives an estimation that the population correlation coefficient, r, is in the confidence interval of 0:257 < r < 0:683. The value of 0.5 is also sufficient to reject the hypothesis that r ¼ 0 with level of significance a < 0:001. Fig. 2 shows, in the first column, the 11 factors that are correlated to the largest number of pixels in the callosal structure, from 78 factors determined at the first iteration. The 11 factors obtained after the algorithm’s convergence is shown in the second column. For each factor, the regions in the callosal structure that have loading values greater than 0.5 are shown in white and the regions that have loadings smaller than 0:5 are shown in light gray. The topology proposed by Witelson is superimposed to the

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Fig. 2. Results of factor analysis using different numbers of factors. Each row shows the regions of the callosal structure related to a given factor, for different values of m. In the first column, in which m ¼ 78, only the 11 most significant factors are shown. The remaining columns show the regions related to each factor, as m is reduced to 11, 9, 7 and 5, respectively. The parts of the structure, which have loading values greater than 0.5, are shown in white and the regions that have loadings smaller than 0:5 are shown in light gray. The topology proposed by Witelson is also superimposed to the images.

images and will be discussed later. It can be seen, from the images, that the reduction of m, from 78 to 11 factors, does not impact significantly on the region of the structure that they represent, except for factor 2. The number of variables related to each factor is displayed in Table 1. Fig. 2 also shows what happens to the factors when m is further reduced to 9, 7 and 5. The results of PCA are shown in Fig. 3. The principal modes of variance are ordered from left to right, top to down, by the amount of variance they represent. Each variable is assigned to the mode for which it presents the greatest absolute coefficient. Variables with positive coefficient are shown in white and the ones with negative values are shown in light gray. The 11 principal components accounted for 64.3% of the total variance. The contribution of each component is given in Table 2.

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Table 1 Number of variables related to each factor when m is equal to 78 (A78 ) and 11 (A11 ) Factor

A78 A11 A11 /A78

f0

f1

f2

f3

f4

f5

f6

f7

f8

f9

f10

90 93 1.03

76 85 1.12

86 154 1.79

50 61 1.22

46 52 1.13

48 55 1.15

53 63 1.19

26 31 1.19

54 58 1.07

16 12 0.75

17 17 1.00

z8 2.39

z9 2.15

z10 1.91

The third row shows the ratio between the values of the second and first rows.

Table 2 Variance accounted for each mode of variation Mode Variance

z0 15.30

z1 12.10

z2 9.74

z3 6.07

z4 5.07

z5 3.67

z6 3.31

z7 2.58

8. Discussion The differences between the average callosal shape of male and female populations have been frequently addressed in the literature, although the reported results are still inconclusive. Some studies reveal shape differences, mainly in the splenium region of the corpus callosum [1,21,40], while others report no relevant differences [4,5,31]. In previous works, we have detected morphological differences related to sex in studies with smaller number of subjects. The first study, based on a sample of 16 elder subjects, revealed substantial differences in the shape of the splenium [24]. The results were in accordance to previous findings of Davatzikos et al., which examined the same images using a different computational approach [11]. A second study based on a sample of 28 subjects with average age of 27 years revealed a smaller difference in the splenium region [15,26]. For the present work, in which a larger dataset in examined, the differences in the average shape of the corpus callosum for males and females, as depicted in Fig. 4, are inconspicuous. The null hypothesis of equal means for the male and female populations was tested for each variable, using two-tailed t-tests with different levels of significance. In only 2.7% of the tests there

Fig. 3. Results of principal component analysis. Modes of variance are ordered left to right, top to bottom, according to the amount of variance they represent. Each variable is assigned to the mode for which it presents the greatest coefficient. Variables with positive coefficients are shown in white and the ones with negative values are shown in light gray. The topology proposed by Witelson is superimposed to the images.

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Fig. 4. Average callosal shape for the males (left) and females (right). The topology proposed by Witelson is superimposed to the images.

Fig. 5. Results of t-mean tests. The images show, in white, the variables for which the hypothesis of equal means between the male and female populations is rejected, with significance levels of 0.01 (a), 0.05 (b) and 0.10 (c). The topology proposed by Witelson is superimposed to the images.

was enough evidence to reject the null hypothesis, with a ¼ 0:01. With levels of significance of 0.05 and 0.10, the proportion of variables for which the hypothesis is rejected is 11.0 and 34.9%, respectively. Fig. 5 shows the variables for which the hypothesis is rejected, at different levels of significance. The hypothesis of equal population variances were also tested with F-variance tests. With levels of significance of 0.01, 0.05 and 0.10, the null hypothesis was rejected for 4.3, 16.6 and 35.7% of the variables, respectively (Fig. 6). Rather than taking specific conclusions about the influence of gender or other feature on the callosal morphology, the aim of this work is to discuss the appropriateness of FA for exploratory purposes and to propose a robust methodology to evaluate experimental results. The analysis of studies on gender differences in the callosal morphometry are nevertheless useful for comparison. The discussion of the results obtained in the experiments will be based on the exploratory aspects of FA and compared to the effect size analysis, statistical fit of the model and other published studies. 8.1. Interpretation of factors The use of FA as a statistical model may be questionable when the interpretation of the factors is not straightforward [33]. Since factors are non-observed variables, they should have a natural meaning in order to provide new information about the data, otherwise the method would serve only as a data reduction tool. In the case of morphological studies, the factors are visually interpreted as regions in the structure. This property was explored by

Fig. 6. Results of F-variance tests. The images show, in white, the variables for which the hypothesis of equal variances between the male and female populations is rejected, with significance levels of 0.01 (a), 0.05 (b) and 0.10 (c). The topology proposed by Witelson is superimposed to the images.

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Denenberg et al. [12], who applied FA to the study of the corpus callosum by partitioning the structure into 100 segments, following the callosal axis at equally spaced intervals. The width of each interval, together with other scalar variables such as the callosal area, perimeter and axis length were examined. Factor analysis were based on the principal factor method with oblique rotation and the criterion for retaining factors was similar to the one used in our study: eigenvalues should be greater than unity and the loading value greater than 0.6 for a variable to be assigned to a given factor. The main problem on Denenberg’s experimental procedure is that no registration was performed and, consequently, a precise correspondence between segments and substructures of the corpus callosum could not be established. The splenium, for instance, is a region that varies substantially with respect to length. If we take Denenberg’s method to compare one corpus callosum with long splenium to another with short splenium (proportional to the whole structure), we will have segments of the splenium of the first one matched to the isthmus of the second. With the elastic matching algorithm, on the other hand, local shape differences are taken into account and are reflected in the resulting Jacobian determinants. In this case, if a short splenium is registered to a long one, there will be an dilatation of the pixels on the first one in order to be matched to the former. This may explain why three out of the seven factors revealed by the study of Denenberg et al. were located between segments number 77 and 99. Compared to the results of our study, they correspond to factor0, which encompasses the splenium region. The remaining factors determined in Denenberg’s study correspond to factors number 4, 8, 5 and 1, respectively related to the isthmus, posterior midbody, rostral body and genu regions of the corpus callosum (see Fig. 2). The results of the exploratory FA presented in our study are also in accordance with the topology proposed by Witelson [42], regarding the subdivision of the callosal structure. Compared to the factors depicted in Fig. 2, it is possible to relate the rostrum with factor 9, the genu with factors 1 and 3, the rostral body with factor 5, the anterior midbody with factor 6, the posterior midbody with factor 8, the isthmus with factors 4 and 7, and the splenium with factors 0 and 10. Factor 2, which represents the contour of the callosal structure, is probably related to error in the segmentation and registration steps—in the elastic matching model, the choice of the stiffness parameter may hinder perfect matching of high-frequency details in the boundary. In the case of factor 9, related to the rostrum, it may reflect error in the segmentation step, as this part of the corpus callosum is frequently linked to the fornix. The results show that the subdivision of the corpus callosum proposed by Witelson, based on experimental procedures, are very similar to the ones obtained by our method, which were achieved by unsupervised learning. A comparison between the results of FA and PCA can also be made based on Figs. 2 and 3. It is clear that FA provides results that can be interpreted as regions in the structure that present correlated shape variation. In the case of PCA, however, the morphological meaning of the results is not straightforward, as PCA focus on the representation of the sample variance. 8.2. Analysis of communality for the gender variable The values of the loadings for the sex-related variable were naturally driven by the varimax algorithm to reveal its relationship with the morphometric variables. For this

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Table 3 Loadings for the gender variable, showing how it is correlated to each factor, when m is equal to 11 Factor Loading

f0 0:23

f1 0:21

f2 0:12

f3 0.00

f4 0.07

f5 0.09

f6 0.21

f7 0:13

f8 0.17

f9 0:12

f10 0:06

dataset, the factors that presented greater loading values to gender were number 0, 6, 1 and 8, in that order, although the magnitude of the correlation was not greater than 0.23. The loadings for the sex-related variable, based on the 11 factors shown in the second column of Fig. 2, are displayed in Table 3. These results are coherent with average callosal shapes shown in Fig. 4, in which no relevant morphological differences can be visually observed. Although the regions that present stronger correlation to gender are located at the splenium, genu and midbody, the magnitude of correlation is small. 8.3. Effect size and factor analysis The effect size results obtained for the study is depicted in Fig. 7, where the white regions correspond to the areas in which one group presents larger dilation than the other group. In the first row, the effect size between males and females is shown for values of 0, 0.05, 0.10, 0.15 and 0.20. In the second row, the effect size between females and males is shown within the same range of values. An advantageous feature of FA over effect size and PCA analyses is that FA allows for the observation of substructural regions that present a coherent behavior regarding dilation during the elastic matching. In PCA, all regions are jointly represented by the principal modes of variance. In effect size analysis, the correlation among the pointwise deformation is also neglected and only the difference between the amount of dilatation is examined. Nevertheless, the result of comparing Fig. 2 with Fig. 7 can be used to highlight the relationship between factors and the areas in which female and male morphology differ. Factor 0, shown in Fig. 2, is related to the major gender-related difference on the splenium anatomy (slightly larger in the female sample, on average). Small regions at the top and bottom portion of the splenium also appear in factors 3 and 10, which are not strongly correlated to gender (loading values of 0.00 and 0.05, respectively). These regions were also observed in a previous study [25], but since the area they represent is small, it may not be asserted whether they are related to errors in the experimental procedure or actually

Fig. 7. Results of effect size analysis. The images in the first row show the regions within which the effect size between the male and female groups are greater than 0 (a), 0.05 (b), 0.1 (c), 0.15 (d) and 0.2 (e). The second row shows the analogous analysis of the effect size between females and males. The topology proposed by Witelson is superimposed to the images.

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have a morphological meaning. Other subregions related to factors number 1, 6 and 8, which present greater loading magnitude for the gender variable, can also be found a correspondence in the effect size analysis, as Fig. 7 is compared to Fig. 2. The results of t-tests are in agreement to the ones obtained by the effect size analysis, which is expected, since female and male samples have equal size and the hypothesis of equal means can not be rejected for most variables. It should also be noticed that the isthmus, related to factor 4, seems to present greater variability with respect to gender, as shown in Fig. 6. 8.4. Impact of the number of factors The impact of parameter m (number of factors) was investigated by choosing different values, computing the corresponding loading matrix and analyzing the statistical fit of the model. Table 4 shows the completeness of factor analysis, the communality of the gender variable, the average magnitude, mean and standard deviation for the analysis of the correlation residual at different values of m. Since the expected standard deviation for a zero correlation distribution for the same sample size is given as sr¼0 ¼ N 1=2 ¼ 84 1=2 ¼ 0:109, all analyses with the number of factors greater than or equal to 4 could be accepted. The analysis based on the 11 factors determined by the iterative algorithm presented about half the expected standard deviation, showing the satisfactory statistical fitness of this representation. The behavior of the communality of gender variable is also informative and is depicted in Fig. 8. It shows that the curve has roughly a constant slope as m decreases from 78 to 16, drops quickly when m equals 13, and then stays stable until m reaches value 5, when another significant change in slope occurs. If these results are compared with the factors obtained by varying m from 78 to 5 (Fig. 2), one may notice that reducing the number of factors in the analysis, from 11 to 9, causes f9 and f10 to be eliminated, with no significant changes to the other factors. The further reduction from 9 to 7 causes f7 and f8 to be attached to factors 4 and 0, respectively. The reduction from 7 to 5 is much more relevant, as the regions represented by factors f1 and f3 change significantly. Table 4 Statistical fit of the model m

Completeness

Communality

Average magnitude

Mean

S.D.

3 4 5 6 7 8 9 10 11 13 78

0.371 0.432 0.483 0.519 0.552 0.578 0.602 0.624 0.643 0.675 0.995

0.098 0.103 0.109 0.211 0.211 0.214 0.221 0.233 0.234 0.247 0.982

0.0814 0.0706 0.0619 0.0552 0.0500 0.0463 0.0425 0.0398 0.0376 0.0303 0.0000

0.0193 0.0107 0.0101 0.0082 0.0078 0.0055 0.0018 0.0005 0.0004 0.0003 0.0000

0.1130 0.0970 0.0831 0.0751 0.0677 0.0630 0.0588 0.0549 0.0517 0.0468 0.0022

For each number of factors m, the table shows the completeness of factor analysis, communality of the diagnosis variable, average magnitude, mean and standard deviation of the correlation residual distribution.

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Fig. 8. Communality of the gender variable as a function of m.

9. Conclusion A novel approach to morphometry was presented, in which the relationship among parts of anatomies were explored. The method is based on the factorial analytic model, in which the covariances between variables are represented by a new variable set of lower cardinality. Applied to high-dimensional vector representations of the anatomy, the method is able to provide concise description and allow exploratory analysis of the correlation between regions of interest. The application of this approach to the study of the human corpus callosum revealed strong agreement with previous published results. The ability of FA to provide prior modeling and the effectiveness of factors as discriminant variable sets are a vast field for future work. The current results are preliminary in nature and much work remains, including studies relating the effect of smoothness in the spatial and intensity transformation and their impact on the correlational structure modeled by the factors, a careful examination of the anatomy underlying the factors for a given study, the implications for the particular condition being investigated and the effectiveness of the method for exploring pathologies. The accordance of the results with respect to the topology of the corpus callosum are nevertheless encouraging. It showed the ability of the method to discover knowledge related to anatomical characterization, based on unsupervised learning. A remark should be made about complexity: Factor analysis can be straightforwardly applied to 3D images and larger structures as far as the number of variables considered in the analysis is controlled, since the computation of the correlation matrix presents quadratic complexity with respect to computational time and storage space. The application of FA to high-dimensional representations of the anatomy are particularly advantageous, since the method facilitates the interpretation of the results. The factors can be visually identified as regions that embed strong correlation. The final number of factors to be used in the model can be related to the desired degree of details in the analysis and, consequently, to its completeness: reducing the number of factors results on larger regions with coarser correlation, whereas a larger number of factors may represent smaller regions with stronger correlation. Even when applied to small datasets, FA was able

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to provide results that are in accordance to the ones obtained by computing the effect size, and additionally provided information about the correlation among morphological variables in a region of interest.

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