Corpus Callosum Shape Analysis with Application to Dyslexia

Abstract

Morphometric studies of the corpus callosum suggest its involvement in a number of psychiatric conditions. In the present study we introduce a novel pattern recognition technique that offers a point-by-point shape descriptor of the corpus callosum. The method uses arc lengths of electric field lines in order to avoid discontinuities caused by folding anatomical contours. We tested this technique by comparing the shape of the corpus callosum in a series of dyslexic men (n = 16) and age-matched controls (n = 14). The results indicate a generalized increase in size of the corpus callosum in dyslexia with a concomitant diminution at its rostral and caudal poles. The reported shape analysis and 2D-reconstruction provide information of anatomical importance that would otherwise passed unnoticed when analyzing size information alone.

Introduction

The corpus callosum is a commissural tract connecting homologous brain areas of the two cerebral hemispheres. During corticalization, the relative reduction in the size of this structure has been associated to cerebral dominance (laterality) and the emergence of associated higher cognitive faculties, e.g., language [1, 2]. Changes in corpus callosum size during corticalization provide for a reduction in the total number of longer commisural fibers. This bias in long vs. short connections propitiates novel parcellation schemes based on a scale-free connectivity map [3]. It is therefore unsurprising that abnormalities in both corpus callosum size and interhemispheric anatomical asymmetries have been implicated in psychiatric conditions characterized, in part, by language disorder [4, 5].

Reading is an acquired skill that depends on the establishment of a facilitating circuit. In multifactorial conditions such as dyslexia, both genetic and epigenetic influences disturb the establishment of this circuit [6], thus resulting in faulty phonological awareness and a reading disability [7]. Recent studies suggest that in dyslexia this altered pattern of connectivity is accompanied by equally prominent changes in gross morphological features of the brain [8–10].

The majority of imaging studies detailing corpus callosum morphometry have been based upon measurements of a single mid-sagittal image [11]. In these studies, parcellation attempts that used anatomical divisions (e.g. genu, body, splenium) have gradually been replaced by operational criteria that rely on geometrical considerations. More recently, shape analysis has been used to complement the information derived from structural studies focusing exclusively on size. The first shape descriptors (or shape representations) equated anatomical structures to a particular geometrical figure. Thus, when measuring the roundness of an object, also known as the sphericity index, objects were compared to a sphere [12]. These algorithms had limited usefulness and were primarily used in both cytology and histology for the purpose of assessing nuclear pleomorphism. In effect, for several decades computerized image analysis primarily served as a way of quantitating the grade of dysplasia in a variety of different tumors [13].

Shape descriptors were applied to neuroimaging in an effort to mathematically define the contour of curves and surfaces. These indices, meant to reflect the amount of elongation and/or convexity of a structure [14], were first used to quantitate the curvature of the corpus callosum [14] and to provide an idea of overall brain gyrification [15]. More recently, exact Fourier descriptors were introduced to neuroimaging by Casanova et al. [11, 16, 17]. Ever since the introduction of exact shape descriptors, there has been disagreement as to the best signature (i.e., complex coordinates, centroid distance, tangent angle) for deriving representations and normalization [18]. It has been noted that the use of a Fourier expansion series is limited by the fact that the variables analyzed (i.e., phase angle, harmonic amplitude) are intuitively abstruse and do a poor job at translating results into anatomical representations. Furthermore, in any given study it is arguable how many harmonic moments are necessary for analysis, as it is readily acknowledged that higher moments within a Fourier series relate poorly to shape [18].

Purely empirical shape models take a different approach than series expansions. Here, the surface of interest is deformed so as to be congruent with a reference object or with another surface for comparison. Quantities of interest are then derived from the function that maps points on the original surface to the reference, x⃗_ref = f (x⃗). For example, the Jacobian |∂f/∂x⃗| is a scalar function that measures the local volumetric deformation (expansion or contraction) at any give point on the original surface as it is transformed to the reference [19]. Alternatively, let f be considered as the functional composition of a deformation (non-rigid transformation) with an alignment (rigid transformation and uniform scaling). Then the norm of the deformation component, integrated over the aligned surface, provides a metric of the difference in shape between the two surfaces [20].

The existing centerlines from 3D objects approaches, can be classified as, distance transform methods [21–23], topological thinning methods [24–27], and hybrid methods [28–30] for volumetric data and Voronoi-based methods [31, 32] for polygonal data. Below, we review only some representative methods of each category.

Zhou and Toga [23] proposed a voxel coding technique in which a discrete wave front propagates through the entire object starting from a manually selected reference point. The wave divides the object into set of clusters that are approximately normal to the centerlines. Bitter et al. [21] proposed a penalized-distance algorithm to extract centerlines. Bouix et al. [30] extracted centerlines by thinning the object’s medial surface, which is computed by thresholding the negative average outward flux of the gradient field of the distance map. Attali et al. [31] compute the medial surface of an object from a finite set of points sampling its closed boundary and then prune it based on geometric criteria to yield its centerlines.

Each existing technique for extracting centerlines suffers from at least one of the following shortcomings: (1) dependence on the accuracy of determining the medial surface, (2) computational complexity, (3) lack of robustness, or (4) sensitivity to boundary noise. In this paper, we present a new level set based centerline extraction framework that addresses these shortcomings. The key idea is to propagate from a centerline wave fronts with a fast speed at central points such that centerlines intersect the propagating fronts at those points of maximum positive curvature and located at maximum distance from the object boundary.

In the present article, the authors expand on a new method (see above) to examine the shape of the corpus callosum. The method can be used to analyze the boundaries of anatomical regions of interest while being invariant to translation, rotation, and scaling. The study compares the shape of the corpus callosum in a series of dyslexic patients and controls. Previous studies have shown significant, localized increase or decrease (dependent on region) in corpus callosum cross section in dyslexic patients [33, 34]. The focus of this study was on further establishing the morphometric nature of this size variation, i.e., whether the reported size variability was the result of focal or generalized changes.

Experimental Procedures

Corpus callosum segmentation and mapping

Probabilistic model of the CC shape

Most of the recent works on image segmentation use level set-based representations of shape, where an individual region is bounded by a set of pixels (voxels in 3D) {x | F (x) = 0} for some suitable distance function F. Then the region’s shape is approximated by the closest linear combination of training shapes. The main drawback of this representation is that the space of signed distance functions is not closed with respect to linear operations. Linear combinations of the distance functions may relate to invalid or even physically impossible boundaries.

To circumvent this limitation, we represent the shape of the CC having been learned from a training set of co-registered MRI with the probabilistic 3D model s: R → [0,1] where s(x, y, z) is the empirical probability that the voxel (x, y, z) ∈ R belongs to the CC. The soft template is constructed as follows:

1. Co-align the training set of MRI using a rigid 3D registration with mutual information as a similarity measure [35].

2. Manually segment each CC from the aligned set.

3. Estimate the voxel-wise probabilities s(x, y, z) as the fraction of aligned images where voxel (x, y, z) was labeled as belonging to the CC.

Segmentation algorithm

Let g : R→{0,1,...,255} denote a grayscale MRI, and let m denote a binary region map, i.e., m is the indicator function of CC ⊆ R. In total, the proposed CC segmentation is obtained by the following algorithm:

1. Perform an affine alignment of a given g to an arbitrary prototype CC from the training set using mutual information as a similarity measure.

2. Estimate the conditional intensity model P(g|m) by fitting a bimodal linear combination of discrete Gaussians.

3. From P(g|m) and the learned probabilistic shape model s, perform an initial segmentation of the CC, i.e. construct an initial region map m.

4. Use the initial region map to identify the Markov-Gibbs random field (MGRF) model P(m) of region maps, and update the conditional intensity P(g|m).

5. Perform the final Bayesian segmentation of the CC in accord with the updated joint MGRF model P(g, m).

Centerline extraction from the CC

The problem of extracting the centerline, or curvilinear axis joining the splenium A to the rostrum B, can be formulated as a variational problem: find the path C that minimizes the total cost T of traveling from the starting point to the destination.

$$T=\underset{C}{min}\underset{0}{\overset{L}{\int }}W(C(s))ds,$$

where C(0) = A, C(L) = B, and W(x, y, z) is a cost function. It can be shown that the minimizer of T satisfies the Eikonal equation

$$\left|\right|\nabla t(C)\left|\right|·\left|\right|\mathbf{v}(C)\left|\right|=1,$$

where t(x, y, z) is the travel time for a wavefront propagating from point A with velocity v(x, y, z). These considerations led to the following algorithm to extract the CC centerline (Figure 1):

Figure 1.

Cross-section of a propagating wavefront at several time points. The points of maximum curvature at each time, which determine the centerline of the CC, are circled.

1. Compute the Euclidean distance map of the interior of the CC, i.e. the minimum distance D(x, y, z) from each point in the CC to the boundary ∂CC.

2. Choose a starting point A in the splenium.

3. Propagate an orthogonal wavefront from point A with speed ||v(x, y, z)|| = exp(−D(x, y, z)) by solving the Eikonal equation.

4. Track the point of maximal curvature of the wavefront as it propagates [36]. In the event of multiple relative maximal curvature, select the one farthest from ∂CC.

5. Terminate when the point of maximum curvature intersects the rostrum. This will be the point B, and the trajectory of maximum curvature is the centerline C.

6. Repeat the process in the reverse direction, starting from B and terminating in the splenium, so as to eliminate any dependence on the somewhat arbitrary choice of A.

Mapping

Comparison of the CC between different subjects and groups is facilitated by mapping their respective boundaries into a standardized coordinate system. We employed a pseudocylindrical system (ρ, ϕ, z) based upon the centerline of the CC (Figure 3). Any point on ∂CC lies on a plane intersecting C(s) and orthogonal to the tangent vector at s for some value of path length s. Let this path length be the z coordinate. Then ϕ is the polar angle in the orthogonal plane measured clockwise from anatomical right, and ρ is distance from C(s). The boundary of each CC was thus transformed into a two dimensional map ρ(ϕ, z).

Figure 3.

Mapping the surface of the corpus callosum. In the pseudocylindrical coordinate system, the z coordinate is the arc length along the centerline from the splenium (left). Radius and polar angle are then measured within the plane orthogonal to the centerline at z.

MRI data set

MRI data were obtained from 16 right-handed dyslexic men aged 18 to 40 years and 14 controls matched for sex, age, educational level, handedness, socioeconomic background, and general intelligence (Table 1). All gave informed consent before participation. All were physically healthy and free of any history of neurologic disease, head injury, significant uncorrected sensory deficit, severe psychiatric disorder, and chronic substance abuse. All scans were acquired with the same 1.5 T Signa MRI scanner, configured to produce T1-weighted volumes with a pixel dimensions of 0.9375 mm and slice spacing 1.5 mm. Full details of the study participants and the MRI acquisition protocol are provided in our prior article on gyrification in dyslexia [10].

Table 1.

Summary of study participants. All participants were right handed, male, and Caucasian. Values are given as mean ± std. deviation. Acronyms: GFW, Goldman-Fristoe-Woodcock Sound Symbol Tests; GORT-3, Gray Oral Reading Test, 3^rd edition; WAIS-R, Wechsler Adult Intelligence Scale, Revised; WRAT-3, Wide Range Achievement Test, 3^rd ed.

	Dyslexic	Control
N	16	14
Age (years), mean	28.2	25.1
Range	18.5–40.4	17.8–40.6
Education (years)	14 ± 3	14 ± 2
Social class	All middle to upper middle class
WAIS-R IQ	113 ± 7	111 ± 12
GORT-3 Passage	4 ± 2	13 ± 2
Comprehension	13 ± 2	11 ± 2
GFW Reading	41 ± 4	51 ± 4
Spelling	43 ± 8	51 ± 8
WRAT-3 Reading	91 ± 11	107 ± 7
Spelling	74 ± 14	106 ± 7
Arithmetic	97 ± 13	112 ± 11
LAC total	80 ± 12	96 ± 5

Validation of the shape model

Segmentation via the present algorithm (above) was validated against manual delineation of the CC by an expert. Fifteen MRI scans, which were not part of the training set for our algorithm, were selected for this purpose. For comparison, the same CCs were also segmented using the level-set approach of Tsai et al. [37], and an active shape model (ASM) [38]. Algorithm performance was measured using relative error, i.e. volume of misclassified voxels (type I or type II error) divided by the manually segmented CC volume.

Statistics

The transformed surfaces were aggregated pointwise to provide mean corpus callosum shape maps ρ_dyslexia and ρ_control. The difference Δρ = ρ_dyslexia − ρ_control, divided pointwise by their pooled standard error, yielded a statistical parametric map of t statistics with 28 degrees of freedom. Regions of statistical significance were derived from the associated P-values using the method of Benjamini and Hochberg [39] with a false discovery rate q^* = 0.05.

Results

Our segmentation routine (Figure 2) outperformed both the level-set and ASM methods (Table 2). The mean error was gauged to be significantly less according to the t-tests (both P < 0.0001).

Figure 2.

Cross sections (outlined) of four CCs segmented using our algorithm. The two scans on the right are from dyslexic individuals, while the two on the left come from normative controls. Axial and coronal sections are shown in “neurological” orientation with the left hemisphere to the left.

Table 2.

Segmentation accuracy of the proposed algorithm and two other methods from the literature, taking the manual segmentation by an expert as “ground truth”. Minima, maxima, etc. are over a test set of 15 MRI scans. To get fair comparison we used one third of the scans as training data for the all three approaches. The training subset comprised those scans whose CC shapes had the greatest eigenvalues in a principal component analysis.

Relative error (%)	Algorithm
	This paper	Level-set [37]	ASM [38]
Minimum	0.11	4.5	8.5
Maximum	1.87	11.8	19.1
Mean	0.97	5.1	10.1
Std. Deviation	1.07	3.1	9.1

White matter in dyslexia was greater than controls, bilaterally along the body of the corpus callosum (Figure 4). Where differences attained statistical significance, the Δρ achieved values up to 6.4 mm on the right and 7.0 mm on the left. There was, however, a reduction at the anterior and posterior extrema of the structure in dyslexia. In particular, significant values of Δρ were observed as low as −6.5 mm in the genu and −4.7 mm in the splenium. There was otherwise no significant difference around the medial sagittal plane (Figure 5).

Figure 4.

The map of Δρ, i.e. the mean excess in ρ(ϕ, z) in the dyslexic cases compared with controls. Statistically significant differences (q^* = 0.05) are outlined.

Figure 5.

The map of regions where Δρ reached statistical significance was back-projected into anatomical space by inverting the map for one of the control cases. Areas where ρ was significantly larger in dyslexia are cross-hatched, while regions where ρ was significantly smaller appear in black. Top: superior view; bottom: inferior view. The genu is toward the right.

Discussion

Relevance to dyslexia

Previous work by our research team has related dyslexia to a minicolumnopathy [40] and a bias in corticocortical connectivity that emphasizes long connections at the expense of shorter ones [5, 10, 40]. This is manifested as a decrease in the outer radiate white matter compartment (short arcuate connections) and an increase in size of the corpus callosum (long commissural fibers) [41, 42]. Corresponding changes have been found in gyral white matter depth when used as a proxy measurement for the gyral window [10].

The present study expands on previous findings by illustrating in graphical manner the nature of the corpus callosum disturbance in dyslexia. To the authors’ knowledge this method has not been previously described in the medical literature. Preliminary attempts at “conformal mapping” have measured straight-line distances from the axis to the boundary of the region of interest as a function of polar angle. This is not a conformal (angle-preserving) map, and more importantly, may not be invertible. The cross sections of many anatomical structures are not perfectly convex, or even star-shaped, so a ray from the axis may intersect the boundary at more than one point. These discontinuities may provide for spurious shape distortions within given segments of the analyzed outline.

Previous studies suggest size differences of the corpus callosum in patients with dyslexia [33]. The nature of this abnormality has been variously ascribed to all major segments of this commissural tract [33, 34, 43]. While comments on corpus callosum size difference abound in the dyslexia literature, references as to the shape of this structure are, by comparison, scant. A morphometric study by Robichon and Habib [44] indicated that the corpus callosum of dyslexic patients was more rounded and evenly thicker. The latter authors suggested that their finding, increased size of the corpus callosum and a corresponding increase in commissural connections, could explain reports of reduced cortical asymmetries in this condition [45]. Another study by Robichon et al. [46] showed significant differences in the angulation of the posterior segment of the corpus callosum. The finding was discussed in relation to parietal asymmetries and possible hormonal effects occurring in utero or during the early postnatal period in this patient population.

Our findings are in agreement with those of Robichon and Habib [44]. Shape reconstruction within our series indicates a generalized increase in size of the corpus callosum in dyslexia (Figure 5). Size reductions at both poles may help average out any size increase and provide an explanation to negative studies of corpus callosum morphology in dyslexia [47–49]. In this regard, shape analysis offers complementary information to areal measurement in neuroimaging studies.

Future directions

The method we implemented measures the distance from a point on the boundary to the axis by the arc length of an electric field line inside a corpus callosum-shaped conducting surface with a point charge on the axis. Besides avoiding the problem of rays intersecting the boundary at more than one point (see above) this approach can be used directly on three-dimensional volumes by solving the Laplace equation with a line charge on the axis of the two- dimensional boundary surface at zero potential. The point of intersection with the axis provides that point’s z-coordinate in the projection space. Applying the method to the present series revealed abnormalities in corpus callosum size with an overall preservation of its general outline.