Structural shape characterization via exploratory factor analysis
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.
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 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.
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.
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 Jöreskog [34] presented a thorough discussion on the 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.
Section snippets
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,
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 is rotated in order to find the orthogonal axes along which the data is maximally spread out. The new p-dimensional basis is achieved by multiplying the original variables by an orthogonal matrix : Each new variable z, or component, is a linear combination of
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.
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 isotropic volumes (flip angle = 35°, TR=35 ms, TE=6 ms, field of view = 24 cm, mm in-plane resolution, 1.0 mm slice thickness, no gap). The
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
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, ρ, is in the confidence interval of 0.257<ρ<0.683. The value of 0.5 is also sufficient to reject the hypothesis that ρ
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
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
Acknowledgements
This work was partially supported by CNPq-Brazil, grants 471077/01-1 and 350750/1994-7, by the Pontifical Catholic University of Minas Gerais, grant FIP 2002/20TLE and by the USPHS under grants NS33662, AG15116, AG17586, and LM03504. The authors are grateful to the Mental Health Clinical Research Center of the University of Pennsylvania for sharing the corpus callosum data.
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