In this section, we present results from the statistical analysis performed to assess the shape variation of the corpus callosum in association with sex, age, schizophrenia and genetic predisposition for schizophrenia.

Data

The population data included structural MRI scans from 218 subjects enrolled in the UCLA Family Study (

Nuechterlein et al., 2002;

Yang et al., 2010) (age: 39.16±16 years, sex: 116 males, 102 females) – obtained after approval by the UCLA Institutional Review Board (IRB). Specifically, the subjects included 48 patients with a DSM-IV diagnosis of schizophrenia (36 M/13 F; mean age: 31.8±8.9 SD), 66 unaffected first-degree relatives of patients (29 M/41 F; 46.84±16.13), and 104 community comparison (C) subjects and their first-degree relatives (50 M/55 F; 37.52±16.53) sampled from 96 separate families. shows the demographic and clinical details of subjects. Groups defined by biological risk for schizophrenia (schizophrenia patients, patient relatives and controls) differed in age (

*F*(5, 215)=15.04,

*p*<.001) and gender (χ

^{2}(5, 215)=11.56,

*p*<.001), but not handedness or years of education completed. These subjects underwent high-resolution T1-weighted structural MRI scanning on a Siemens 1.5 T Sonata system using a 3D MPRAGE sequence (FOV= 256; TR/TE=1900 ms/28 ms; voxel size=1 mm×1 mm×1 mm; TI=1100; matrix size=256×256×160; flip angle: 15°, averages: 4). The corpus callosum was manually contoured on the sagittal slices for each subject. Intra and inter-rater reliability for manual tracing, determined by repeat contouring of the callosum in six randomly chosen brain volumes, was

*r*_{1}>0.98. Additionally, using the parcellation scheme provided by Freesurfer, the callosum was also separated into five subregions to determine the effects of sex, age and schizophrenia for midsagittal callosal area.

| **Table 2**Demographic and MRI measures for schizophrenia patients, non-psychotic relatives of patients, C (community comparison) probands, and non-psychotic relatives of C probands |

Mean shapes for population

The procedures described in the Karcher mean section were applied to the corpus callosum traces for healthy controls, schizophrenia patients, and all of the subjects together. The resulting mean shapes are displayed in , where the top panels show the mean shapes for each group separately and the bottom panel shows the three mean shapes overlaid for better comparison. The callosal shapes for healthy controls appear flatter, whereas those for the schizophrenia patients are more arched.

TPCA variation

Visualization of the group mean shapes of the different subsets is qualitative and not instructive about the directions in the underlying shape variation in the population. To analyze the covariance structure of shapes, we applied the tangent principal component analysis described in the Tangent space PCA for shapes section above. The first ten (

*k*=10) eigen-projections (accounting for around 70% of the variance) were included as shape covariates for each subject for subsequent statistical analysis. The cut-off choice

*k*=10 was chosen to reduce the possibility of type II errors for the statistical analysis. The distribution of the eigen values was heavy-tailed, suggesting that the shape analysis method was highly sensitive to local changes in shape. Algorithm 3 above was used to visualize the eigen shape variation from the TPCA model. shows the eigen shape variation for ±3 standard deviations about the mean shape, as well as the closest corresponding original shapes from the population for the first five eigen directions. As illustrated in this figure, each direction captures specific morphological characteristics of the underlying population shape variation. For example, going from top to bottom, the first eigen mode reflects arching of the corpus callosum and also the extent to which the splenium is bulbous in shape. The second eigen mode reflects the shape of the genu becoming more bulbous, when going from bottom to top. The closest original shapes corresponding to the eigen shapes exhibit similar morphological variation thus confirming that the TPCA model captures salient shape characteristics in the population. To visualize the effect of the extreme eigen shape variation, overlays the eigen shapes at the two extremities

.

Mixed effects modeling

Linear regression and mixed model analyses were used to examine genetic-liability effects by comparing schizophrenia patients, relatives of patients, and control subjects and their relatives. To examine effects of schizophrenia, patients with schizophrenia were compared to community comparison subjects and their relatives. Sex and age were included as covariates in all statistical analyses. Since some subjects were biologically related, family membership was included as a random factor for all analyses including related individuals. Significant schizophrenia effects were followed-up by examining effects of disease-related genetic liability (comparing patient relatives to controls) and disease-specific processes (comparing patients and patient relatives) in pairwise contrasts. Examining the first five PCA sources describing callosal shape variation, significant sex effects were observed for the first (*p*=0.04), second (*p*=0.014), and the third (*p*=0.025) eigen projections. Significant age effects were observed for the first and the fourth eigen projections (*p*<0.0001). Significant schizophrenia effects (schizophrenia patients versus C subjects) were observed for the first eigen projection (*p*=0.023). Additionally, the first eigen projection showed a significant disease-specific effect (*p*=0.004) when schizophrenia probands were compared with non-psychotic first-degree relatives of patients. The third eigen projection showed both significant genetic liability effects (*p*=0.021) when non-psychotic patient siblings were compared to healthy control probands and siblings, as well as disease-specific effects (*p*=0.019). The statistical results highlighting main effects for age, sex, and schizophrenia are summarized in .

| **Table 3**Summary of significant main effects (highlighted in red) for age, sex, schizophrenia (patients compared to controls and control relatives), genetic liability (patient relatives compared to controls and control relatives) and disease-related effects (patients (more ...) |

To visualize the corpus callosum shape variation related to sex, age, and biological risk for schizophrenia, we color-coded the original shapes closest to the first eigen projection according to their gender, age, and diagnosis (). Subject age was divided into four quartiles over the total age range of 12–80 years, while diagnosis was coded according to whether a subject was a healthy control, a control relative, a schizophrenia patient, or a patient relative. From , we observe that the tail end of the first eigen projection indicated that older male subjects and predominantly patients were more likely to show an arching of the callosum.

To determine the effect of sex, age and diagnosis on local shape changes in the corpus callosum, we chose the first eigen projection and reconstructed the callosal shapes using Algorithm 3 described above. Since the shape is reconstructed from the average, we computed the magnitude of the deformation field from the mean shape to the generated shape. This deformation field is computed at each poin talong the shape and thus represents a local measure of distortion of the shape about the mean. We then used the general linear model at each point to test the main effects of sex, age, and biological risk for schizophrenia on the local changes in shape, including each of the other remaining covariates in the model. We found significant effects for both sex and age for the local shape changes reflected in the deformation field due to the first eigen direction. Additionally, we analyzed the residual contributions for eigen projections 2–10 using the same statistical model including false discovery rate (FDR) correction. Significant age (*p*_{FDR}=0.00018) and sex (*p*_{FDR}=0.0043) effects were found locally along the shape for the first eigen projection. We also found significant age (*p*_{FDR}=0.0052) and sex (*p*_{FDR}=0.00625) effects for local shape when analyzing the residual variation for the remaining (2–10) eigen projections. Though some locations showed significant effects of diagnosis, the results did not survive the threshold for multiple comparisons. shows the significant p-values overlaid on the mean shape for sex and age for the first eigen projection as well as the residual (2–10) projections. While we observe significant effects due to sex almost all over the shape for the dominant eigen projection, the effects are localized to the genu and partially to the splenium for age.

Comparison with callosal thickness

We computed the local thickness (see

Luders et al., 2010;

Narr et al., 2000;

Nasrallah et al., 1986 for the description of methods) of the callosal shape and projected it at each point along the contour. We used the same models previously used for the analysis of local deformation fields for analyzing thickness, but we failed to find any significant effects for sex, age and biological risk for schizophrenia.

Comparison with regional callosal areas

We compared our results with previously established regional measures of callosal anatomy by partitioning the corpus callosum into the posterior, mid-posterior, central, mid-anterior, and the anterior regions using Freesurfer (

Dale et al., 1999). The regional areas as well as the total callosal area that have been used in prior studies in schizophrenia (

Downhill et al., 2000) were measured for each subject. We used the same statistical models to examine sex, age, and schizophrenia effects and biological risk for the total and partitioned callosal areas. Whole brain and tissue volume measurements were also compared for descriptive purposes. Only significant age effects for the mid-anterior (

*p*<0.0001), central (

*p*<0.0001), the mid-posterior (

*p*=0.001), and the total area (

*p*=0.012) were observed. Without correction for brain volume, males had larger callosal areas than females (

*p*<0.0001).

Shape correlation with sub-regional callosal areas

We also assessed the relationship of the intrinsic statistical shape covariates with the callosal area measures across the population. Significant associations were observed between the posterior region and the scores from the first (*r*=0.191, *p*=0.005), sixth (*r*=0.196, *p*=0.004), seventh (*r*=0.319, *p*<1e–5), and the eighth (*r*=−0.159, *p*=0.0019) eigen projections. In particular, eigen projection four was significantly correlated with the mid-posterior (*r*=−0.420, p<1e–5), central (*r*=−0.289, *p*<1e–5), mid-anterior (*r*=−0.3, p<1e–5), and anterior (*r*=−0.338, *p*<1e–5) regions as well as the total area (*r*=−0.354, *p*<1e–5). Similarly we observed significant correlations for the remaining regions with scores from multiple eigen shape projections as shown in . Importantly we also observed significant correlations of total white matter volume with the first (*r*=0.168, *p*=0.013), fourth (*r*=−0.201, *p*=0.003), and the tenth (*r*=0.138, *p*=0.042) eigen projections, as well as significant correlations of the cerebrospinal fluid (CSF) volume with the second (*r*=−0.249, *p*<1e–5), third (*r*=−0.221, *p*=0.001), fourth (*r*=0.179, *p*=0.008), and the ninth (*r*=−0.196, *p*=0.004) eigen shape projections.

| **Table 4**Summary of significant (highlighted in red) correlations of eigen shape projections with corpus callosum regional areas |