Multi-Structure Shape Analysis
MR-based morphometric assessment of the human brain has been widely employed in neuroimaging studies. Compared to volumetric assessments of the brain, the patterns of anatomical shape abnormalities provide richer information to distinguish diseases. It has thus become an attractive research area in the field of medical image analysis. A shape analysis pipeline under the large deformation diffeomorphic metric mapping (LDDMM) framework aims at assessing the anatomical connectivity in the multiple subcortical structures on the basis of shape expansion or compression.
Figure 1. Schematic of the multi-structure shape analysis pipeline.
The shape
analysis pipeline (in Figure 1) depicts the momentum extraction in the
population of shapes. The template segmentation is mapped onto the target
segmentation via the diffeomorphic mapping 
.
The target shape of each structure is generated according to
,
.
The initial momentum maps indexed over the template coordinates
are
extracted by the LDDMM surface mapping from 
to
,
which are modeled as random fields.
Shape Generation via LDDMM Image Mapping:
The anatomical parcellation are generated by automatic or manual segmentation algorithms (e.g, Figure 2(a,e) show FreeSurfer segmented hippocampi in the volume representation, Figure 2(b,f) are the surface representations). The topological and global shape properties of the CFA subcortical templates (see template webpage) are injected into the parcellation by solving the LDDMM image mapping. The mapped template segmentations are “denoised” approximations of the target parcellation as shown in Figure 2(c,g) and (d.h).
Figure 2. Examples of shape denoising.
Parametric Shape Encoding via Normal Surface Momentum Maps:
For homogeneous subcortical subvolumes the representation of shapes can be reduced to a representation of scalar fields that are concentrated on the boundary of the different homogeneous subvolumes. These scalar fields indexed over the bounding surface determine the “momentum” of the LDDMM mapped template shapes into the population and become the variables for encoding the shape variation of the homogeneous subvolumes. This parametric reduction follows from two key properties of the geodesic connection of the template to the target under LDDMM: 1) the conservation of the momentum property, and 2) the normality property. We can thus write the initial momentum as
Random Field Modeling of the Momentum Maps:
A hierarchical statistical analysis is examined by first building random fields within each subcortical structure based on its geometry and then reducing the data dimensionality via principal component analysis (PCA) for expressing the correlation pattern of shape variations in the multiple subcortical structures in order to identify anatomical connectivity between regions on the basis of similar shape change pattern (e.g. compression or expansion). In the first level, the scalar momentum is modeled as a linear combination of the Laplace-Beltrami (LB) bases on the template surface in the form of
The examples of the LB bases on the amygdala and hippocampus are shown in Figure 3.
In the second
level, a shape vector, 
is
constructed. Principal component analysis (PCA) is used to model the shape
correlation of the multiple subcortical structures through the shape vector:
i.e. 
.
Figure 3. Examples of the Laplace-Beltrami basis functions on the amygdala (top row) and the hippocampus (bottom row). From the left to right, panels respectively show the second, fifth, and seventh basis functions.
REFERENCES:
Methodological Papers:
1. Anqi Qiu, Timothy Brown, Bruce Fischl, Anthony Kolasny, Jun Ma, Randy L. Buckner, Michael I. Miller, “Subcortical Structure Template Generation with its Applications in Shape Analysis”, NeuroImage, in revision.
2.Anqi Qiu, Michael I. Miller, “Multi-Structure Network Shape Analysis via Normal Surface Momentum Maps”, NeuroImage, accepted.
Applications in Neuropsychiatric and Neurodegenerative Diseases:
1. Anqi Qiu, Deana Crocetti, Marcy Adler, Mark Mahone, Martha Deckla, Michael I. Miller, and Stewart H. Mostofsy, “Basal ganglia volume and shape in children with ADHD”, Am J Psychiatry, submitted.
