Framework for the Statistical Shape Analysis of Brain Structures using SPHARM-PDM

Abstract

Shape analysis has become of increasing interest to the neuroimaging community due to its potential to precisely locate morphological changes between healthy and pathological structures. This manuscript presents a comprehensive set of tools for the computation of 3D structural statistical shape analysis. It has been applied in several studies on brain morphometry, but can potentially be employed in other 3D shape problems. Its main limitations is the necessity of spherical topology.The input of the proposed shape analysis is a set of binary segmentation of a single brain structure, such as the hippocampus or caudate. These segmentations are converted into a corresponding spherical harmonic description (SPHARM), which is then sampled into a triangulated surfaces (SPHARM-PDM). After alignment, differences between groups of surfaces are computed using the Hotelling T(2) two sample metric. Statistical p-values, both raw and corrected for multiple comparisons, result in significance maps. Additional visualization of the group tests are provided via mean difference magnitude and vector maps, as well as maps of the group covariance information.The correction for multiple comparisons is performed via two separate methods that each have a distinct view of the problem. The first one aims to control the family-wise error rate (FWER) or false-positives via the extrema histogram of non-parametric permutations. The second method controls the false discovery rate and results in a less conservative estimate of the false-negatives.

Figure 1

Schematic view of our SPHARM-PDM based shape analysis pipeline.

Figure 2

The SPHARM shape description of a human lateral ventricle shown at 4 different degrees ( 1, 3, 6, 10 harmonics).

Figure 3

Left: Illustration of SPHARM correspondence via alignment of the spherical parametrization using the first order ellipsoid meridian and equator. Right: Set of 6 caudate structures with correspondence shown at selected locations via colored spheres.

Figure 4

Icosahedron subdivision for different levels of sampling. From left to right: Base icosahedron, subdivision factors 2, 4 and 6.

Figure 5

Left: Schematic view of testing procedure: A set of features per surface point (shown as colormaps) are analyzed using uni- or multi-variate statistics, resulting in a raw significance map (bottom) .Right: The raw significance maps are overly optimistic and need to be corrected to control for the multiple comparison problem. The corrected significance maps on the other hand are commonly pessimistic estimates.

Figure 6

Left: Scheme of P-value computation via permutation tests. The real group difference S₀ (e.g. Hotelling T²) is compared to that the group differences S_j of random permutations of the group labels. The quantile in the S_j histogram associated with S₀ is the p-value. Right: Example run on test data with 1000 random permutations. On the left the individual S_j are plotted and on the left the corresponding S_j histogram is shown. The black line indicates the value of S₀.

Figure 7

Workflow of the shape analysis pipeline starting at the segmentation of the brain structures to the local final statistical analysis.

Figure 8

Quality Control visualization of the SPHARM correspondence with KWMeshVisu using the Phi-attribute file shown on 3 randomly selected cases. Same color represent the same ϕ parameter value of the spherical parametrization.

Figure 9

Descriptive statistics of right caudate analysis. The left visualization shows the mean difference as a colormap from green (0 mm difference) to red (1.6 mm difference) as well an vectorfield on the overall mean surface mesh. For two regions with large differences a zoomed view is provided. The right visualization shows the covariance ellipsoid colored with the magnitude of the largest ellipsoid axis (axis of largest covariance). The same two zoomed views are provided. The variance in the first zoom section is markedly reduced compared to the second zoom section.

Figure 10

Significance testing on right caudate study