We start with images already registered in MNI space.
- Nonlinear registration produces a deformation (warp) field
- For each voxel, this encodes:
- local expansion (inflation)
- local contraction (shrinkage)
👉 This deformation serves as a proxy for brain morphology
- Neurodegenerative diseases (e.g. Parkinson’s, Alzheimer’s)
- Expect atrophy patterns (systematic shrinkage)
- Deformation fields reveal where structural changes occur
- Voxel-level data is too large (~2M voxels)
- Morphological changes are not independent, but structured across regions
- Apply brain parcellation
- Reduce dimensionality to region-level representation
Typical scale:
- ~1k – 10k parcels (explored experimentally)
👉 Each parcel aggregates multiple voxels → first compression step
-
Split into macro compartments:
- Left cortex
- Right cortex
- Subcortical regions
-
Perform micro-parcellation within each compartment
-
Recombine into full-brain parcel set
- Test multiple resolutions (low / medium / high)
- Avoid jumping directly to maximum resolution
Instead of reducing each parcel to a single summary statistic such as the mean, we represent each parcel by the distribution of log-Jacobian determinant values measured across all voxels in that parcel.
For each parcel:
- Extract all voxel-wise log-Jacobian determinant values from the nonlinear warp
- Sort the values and interpolate them onto a common 15-point quantile grid
- Use that quantile representation as the parcel distribution P(x)
👉 Result:
- Each parcel is described by its full morphometric profile, not just by one number
🎯 Purpose:
- Preserve the shape of the deformation pattern
- Distinguish uniform abnormalities from heterogeneous or patchy abnormalities
- Retain richer information for downstream network construction
Build subject-level parcel graphs using multiple definitions of pairwise morphometric similarity so that downstream validation does not depend on a single edge metric.
For each pair of parcels A and B, compute edges using:
- Wasserstein distance
- KL divergence (Gaussian version)
- Median + IQR
For each pair of parcels A and B, compute the symmetric Wasserstein-1 distance
between their common quantile representations. The resulting graph is
undirected:
- For Wasserstein distance, convert distance into a similarity score before
adding the edge to the graph. The current transforms are
$\exp(-W)$ and$1/(1+W)$ . - For KL divergence (Gaussian version), use the divergence-derived pairwise comparison defined on Gaussian parcel summaries.
- For Median + IQR, use robust summary-statistic comparisons rather than full-density matching.
👉 Result:
- Nodes = parcels
- Edge weights = pairwise parcel relationships under three alternative morphometric definitions
For each subject and for each edge definition:
- build a graph using the corresponding parcel-to-parcel edge weights
- start with a fully dense graph
- later, evaluate 2-3 different thresholding strategies
Recommended workflow:
- construct the full dense weighted graph
- compute graph statistics on the dense version first
- then test a small set of thresholded variants for sensitivity analysis
This makes it possible to compare how stable downstream results are with respect to sparsification.
For each graph, compute the weighted degree of every parcel:
where:
-
$w_{ij}$ is the edge weight between parcel$i$ and parcel$j$ -
$N$ is the number of nodes in the graph - the diagonal self-similarity is excluded
👉 Primary graph-level readout:
- weighted degree = mean similarity to all other nodes
- High weighted degree means a parcel is morphometrically similar to many other parcels.
- Low weighted degree means a parcel is unusual relative to the subject's overall parcel population. It is an anomaly score, not a direct or specific measurement of atrophy.
- Diseased parcels are expected to be more atrophied and to show deformation patterns that deviate from the rest of the brain.
- Therefore, diseased parcels are hypothesized to show lower weighted degree than healthy parcels.
- Weighted degree depends on graph composition. Large compartments such as cortex and CSF contribute many comparison nodes, so anatomical compartment and parcel counts must be considered during group-level interpretation.
Validation will be carried out on the following datasets:
- OASIS
- Epilepsy dataset
- PPMI
- Psychiatric datasets
These datasets provide the test bed for comparing graph-derived parcel metrics between healthy and diseased groups.
For each subject-level graph:
- compute the weighted degree for every parcel
- split subjects into:
- Healthy
- Diseased (137 subjects)
- for each parcel:
- compare weighted degree between healthy and diseased groups
- run a t-test
- compute effect size
Repeat this analysis for:
- all parcels (approximately 90k)
- all three edge metrics
👉 Core validation question:
- which parcels show the strongest group differences in weighted degree, and how consistent are those differences across edge definitions?
- Build one graph per subject for each of the three edge definitions
- Compute parcel-wise weighted degree
- Perform parcel-wise healthy vs diseased comparisons
- Record both significance and effect size
- Start from fully dense graphs
- Later test 2-3 threshold choices
- Re-run the weighted-degree validation after thresholding
- Perform the analysis across the full parcellation
- Target scale: approximately 90k parcels
The following methods remain part of the project as legacy approaches from earlier exploration, but they are no longer the primary validation pathway:
- Sparse weighted graph construction
- Leiden community detection
Historical purpose:
- identify communities of parcels with coordinated deformation patterns
Current status:
- retained for comparison and historical reference
- not the main validation endpoint
- Mini-batch K-means on parcel-level features
- optional dimensionality reduction before clustering
Historical purpose:
- fast scalable clustering baseline
Current status:
- retained as a legacy baseline
- not the main analysis for the current project description
- Treating one edge definition as definitive before validation
- Discarding the dense-graph analysis too early
- Relying only on clustering outputs when the current endpoint is parcel-wise statistical validation
- Gradient-based / deep learning approaches (for now)
👉 The current emphasis is on validating parcel-wise graph statistics across multiple edge definitions and datasets.
- t-statistics
- effect sizes
- parcel-wise healthy vs diseased differences
- weighted degree per parcel
- sensitivity to threshold choice
- consistency across the three edge definitions
- runtime
- peak RAM
- scalability to approximately 90k parcels
- Build parcel graphs under three alternative edge definitions
- Compute parcel-wise weighted degree for each subject
- Identify parcels whose weighted degree differs between healthy and diseased groups
- Quantify how robust those findings are across datasets and threshold choices
Represent each parcel through morphometric summaries derived from the warp, construct dense subject-level graphs using three alternative edge definitions, compute parcel-wise weighted degree, and validate group differences between healthy and diseased subjects.
- Wasserstein-based graph
- Gaussian KL-based graph
- Median + IQR-based graph
- Weighted-degree validation across parcels and datasets
- Leiden community detection → legacy graph method
- Mini-batch K-means → legacy clustering baseline