[Paper Review] Cortical Geometry Network and Topology Markers for Parkinson's Disease

Amanmeet Garg, Donghuan Lu+2 more|arXiv (Cornell University)|Nov 14, 2016

Topological and Geometric Data Analysis7 references3 citations

TL;DR

This paper introduces a novel Geometry Networks (GN) framework that models cortical surface geometry in Parkinson’s disease (PD) using 3D coordinates of anatomically defined cortical parcels. By constructing inter-regional distance matrices and analyzing their topology via persistent homology, the method identifies significant differences in Betti numbers (β₁ and β₃) between PD and healthy controls (p < 0.05), demonstrating its potential as a non-invasive diagnostic marker for neurodegenerative changes in cortical geometry.

ABSTRACT

Neurodegeneration affects cortical gray matter leading to loss of cortical mantle volume. As a result of such volume loss, the geometrical arrangement of the regions on the cortical surface is expected to be altered in comparison to healthy brains. Here we present a novel method to study the alterations in brain cortical surface geometry in Parkinson's disease (PD) subjects with a \emph{Geometry Networks (GN)} framework. The local geometrical arrangement of the cortical surface is captured as the 3D coordinates of the centroids of anatomically defined parcels on the surface. The inter-regional distance between cortical patches is the signal of interest and is captured as a geometry network. We study its topology by computing the dimensionality of simplicial complexes induced on a filtration of binary undirected networks for each geometry network. In a permutation statistics test, a statistically significant ($p&lt;0.05$) difference was observed in the homology features between PD and healthy control groups highlighting its potential to differentiate between the groups and their potential utility in disease diagnosis.

Motivation & Objective

• To investigate how Parkinson’s disease alters the 3D geometric arrangement of cortical regions due to neurodegeneration and gray matter atrophy.
• To develop a network-based framework that captures polyadic (many-to-many) interactions between cortical regions, moving beyond traditional dyadic network models.
• To apply persistent homology to inter-regional distance matrices to extract topological features sensitive to subtle structural changes in PD.
• To evaluate the discriminative power of these topological features in differentiating PD patients from healthy controls using statistical testing.
• To establish a generalizable, non-invasive imaging marker for neurodegenerative diseases based on cortical surface geometry and topology.

Proposed method

• Cortical surfaces are parcellated into anatomically defined regions, with centroids used as 3D spatial coordinates to represent each region’s location.
• Inter-regional Euclidean distances between parcel centroids form a weighted undirected graph, serving as the input for network filtration.
• A network filtration is applied using increasing threshold values (εk), generating a hierarchy of binary undirected graphs from fully disconnected to fully connected states.
• Vietoris-Rips simplicial complexes are induced on the filtered networks to compute persistent homology features, including Betti numbers β₀, β₁, β₂, and β₃.
• Topological features (Betti numbers) are extracted across the filtration to capture multiway interactions and persistent structural patterns in the cortical geometry.
• Statistical significance of group differences (PD vs. healthy controls) is assessed using permutation-based Fisher’s exact test and Hotelling’s T² test on the homology features.

Experimental results

Research questions

• RQ1 How does Parkinson’s disease alter the 3D geometric arrangement of cortical regions as reflected in inter-regional distances?
• RQ2 Can topological features derived from persistent homology of cortical geometry networks effectively differentiate PD patients from healthy controls?
• RQ3 Do polyadic interactions captured by persistent homology provide more discriminative power than traditional dyadic network metrics in PD?
• RQ4 Which specific Betti numbers (β₀, β₁, β₂, β₃) show statistically significant differences between PD and control groups?
• RQ5 Can the proposed Geometry Networks framework serve as a robust, non-invasive imaging biomarker for neurodegenerative disease progression?

Key findings

• The persistent homology feature β₁ (number of 1-dimensional holes) showed a statistically significant difference between PD and healthy control groups (p = 0.017).
• The β₃ (number of 3-dimensional holes) feature demonstrated an even stronger statistical difference (p = 0.005), indicating high sensitivity to disease-related geometric changes.
• Combining β₁ and β₃ features improved discrimination performance, with a p-value of 0.005 in the Fisher’s exact test, suggesting synergistic diagnostic potential.
• Classical network metrics such as nodal degree, clustering coefficient, and local efficiency failed to show statistical significance (p > 0.05), highlighting limitations of dyadic network models.
• The Betti number features β₁ and β₃ were the only ones to achieve p < 0.05 in both permutation testing and Hotelling’s T² test, confirming their robustness and discriminative power.
• The results support that PD-related neurodegeneration induces detectable, non-random changes in the topological structure of cortical geometry, particularly in higher-order simplicial features.