Source-linked AI summary
Persistent homology analysis of brain artery trees
Paul Bendich, J. S. Marron, Ezra Miller, Alex Pieloch, Sean Skwerer
TL;DR
Statistical analysis of three-dimensional brain artery trees is difficult because the observations are structured objects rather than vectors. The paper uses persistent-homology representations and summaries to study age and sex associations, finding stronger anatomical correlations than earlier representations and retaining age correlation after adjustment for total artery length.
Problem
Statistical analysis is challenging when observations are structured three-dimensional artery trees rather than vectors.
Method
Persistent homology records multiscale vessel connectedness and loops in persistence diagrams, which are converted into feature vectors for statistical analysis.
Results
TDA finds age correlations with artery bending independent of total artery length and stronger sex effects than prior analyses.
Takeaways & Limitations
Persistent-homology representations provide anatomical information unavailable from earlier representations of the brain artery trees.
Takeaways & Limitations
The study does not carefully account for multiple-comparison issues, so its lessons are intended to demonstrate methodological power rather than establish anatomical conclusions.
Abstract
from arXiv · showhide
New representations of tree-structured data objects, using ideas from topological data analysis, enable improved statistical analyses of a population of brain artery trees. A number of representations of each data tree arise from persistence diagrams that quantify branching and looping of vessels at multiple scales. Novel approaches to the statistical analysis, through various summaries of the persistence diagrams, lead to heightened correlations with covariates such as age and sex, relative to earlier analyses of this data set. The correlation with age continues to be significant even after controlling for correlations from earlier significant summaries
1 Introduction
The paper applies persistent homology to represent three-dimensional brain artery trees and analyze their structure statistically. These representations reveal age- and sex-related anatomical patterns beyond earlier analyses.
- Motivation: Brain artery trees are structured three-dimensional data objects with attributes such as vessel thickness, making conventional vector-based statistical analysis challenging.The dataset contains artery trees reconstructed from human-brain MRI data.
- Motivation: Earlier analyses correlated some artery-tree features with age and provided hints of sex effects, motivating richer representations.
- Findings: Topological data analysis finds age correlations with artery bending that are independent of earlier correlations with total artery length, while also finding stronger sex effects than prior work.
- Approach: Persistent homology records changes in vessel connectedness beneath rising horizontal planes and the evolution of loops as neighborhood thickness increases.Each tree produces a persistence diagram, which is then converted into feature vectors for statistical analysis.
- Approach: The workflow transforms three-dimensional tubular images into persistence diagrams and then feature vectors suitable for standard linear statistics.This represents the data object while retaining topological information before statistical modeling.
2 Brain artery trees
The study analyzes 98 reconstructed brain artery trees while addressing the difficulty of comparing richly structured, three-dimensional trees. Earlier methods used simpler summaries or two-dimensional embeddings, whereas alternative tree representations faced their own structural constraints.
- Dataset: The dataset contains 98 artery trees reconstructed from three-dimensional MRA images using automatic and manual assembly.The subjects range from 18 to 72 years old, with pathological cases excluded.
- Earlier analyses: Earlier analyses used summaries such as total branch length and average branch thickness, which were significantly correlated with age.
- Earlier analyses: Dyck-path representations enabled functional-data analysis and produced stronger age correlations plus the first indication of a significant sex effect.
- Representation challenges: Prior approaches required embedding three-dimensional trees into two dimensions, forcing arbitrary child orderings and potentially losing anatomically consistent information across subjects.
- Representation challenges: Phylogenetic-tree methods face a fixed-leaf-set requirement that conflicts with brain artery trees, whose leaves vary with imaging resolution.
- Related approaches: Other tree-data methods were developed to avoid planar embedding and fixed-leaf-set problems, or to extend principal component analysis and central-limit theory to tree spaces.
3 Persistent homology analysis of brain arteries
The study applies persistent homology and statistical summaries of persistence-derived features to brain artery trees, revealing age and sex effects in their three-dimensional geometry. Age associations remain after controlling for total artery length, while loop-based features show stronger age and sex signals.
- Persistent homology representations: TDA represents each brain artery tree through persistence-derived numerical features that quantify bends, loops, and how arterial structure occupies three-dimensional space.Zero-dimensional persistence measures bend sizes, while one-dimensional persistence measures loop lifetimes as vessel thickness increases.
- 3.2 Age effects: PCA of zero-dimensional persistence features shows a clear age correlation in PC1, while PC2 and PC3 show little apparent age connection.The first component captures the dominant variation, with PC1 scores visibly correlated with age.
- 3.2 Age effects: ρ = 0.53 with p-value < 10^-7 for the correlation between zero-dimensional-feature PC1 and age.The correlation is reported using Pearson correlation and a Gaussian-based test against no correlation.
- 3.2 Age effects: ρ = 0.61 with p-value < 10^-10 for the correlation between one-dimensional loop-persistence PC1 and age, stronger than for the zero-dimensional features.The loop-based feature set produces the stronger reported age association.
- 3.3 Total artery length / 3.4 Sex effects: After controlling for total artery length, the one-dimensional persistence residuals retain an age correlation of ρ = 0.35, while loop features yield a sex-difference p-value of 0.032.The persistent-homology age signal therefore is not limited to coarse total-length variation; feature selection produces still lower sex-effect p-values.
4 Topological data analysis methods
Persistence diagrams provide compact records of geometric and topological changes as objects are built in stages. This paper uses diagrams tracking connected components in filtered graphs and loops in thickenings.
- Persistence diagrams compactly record geometric and topological changes occurring as an object is built in stages.
- The paper focuses on connected components in filtered graphs and loops forming and disappearing in thickened objects.
- For a graph with height function h, Dgm0(h) summarizes component evolution across threshold sets.
4.1 Height functions and connected components
Height-based persistent homology tracks connected components as threshold sets of a graph expand. The resulting persistence diagram records component births, deaths, and lifetimes, while remaining robust to small input perturbations.
- Height functions and threshold sets: A height function h on graph vertices is extended to edges using the maximum endpoint height, defining threshold subgraphs.
- Connected-component evolution: Persistent homology records how the number β0(i) of connected components changes as vertices are added in ascending height order.
- Critical values: When a lower link is empty, a new component is born; when it has k + 1 vertices, k components die and one remains.
- Persistence diagrams: A dot (a, b) in Dgm0(h) represents components born at a and dying at b, with persistence b −a measuring independent lifetime.
- Interpretation and stability: Small persistence often suggests noise, but persistence indicates reliability rather than importance, and moderately persistent dots can distinguish cases best.
- Stability: The p-th Wasserstein distance compares diagrams through minimum-cost bijections and is bounded by K||f −g||∞ under mild conditions.
4.2 Thickening and loops
One-dimensional persistent homology studies loops in progressively thickened point sets. Loop birth and death times form persistence diagrams that remain stable under perturbations and can approximate diagrams from dense subsamples.
- Thickening: For a compact set Y, thickened neighborhoods Yα expand with α, causing loops to appear and later fill in.
- Loop persistence diagrams: The persistence diagram Dgm1(Y) records the birth and death times of loops as dots in the plane.
- Diagram interpretation: In a point-cloud example, the highest-persistence dots correspond to the larger loops, while near-diagonal dots represent short-lived small loops.
- Stability: Stability bounds Wasserstein distance between loop diagrams by K · dH(Y, Y′), supporting approximation from small dense subsamples.
4.3 From trees to diagrams
The study converts brain artery trees into zero- and one-dimensional persistence diagrams. Height-based diagrams are computed directly, while loop diagrams use branch subsampling supported by stability guarantees.
- Zero-dimensional persistence: For each tree, height-based persistence uses the vertex z-coordinate and computes Dgm0(h) with a union-find algorithm in O(N log N).
- One-dimensional persistence: Full-resolution Dgm1(T) is not computed because one-dimensional persistence is much slower than the zero-dimensional calculation.
- Subsampling: Each tree is subsampled to 3000 total vertices, compared with roughly 10^5 original vertices distributed across about 200–300 branches.
- Subsampling justification: The stability theorem provides theoretical guarantees that subsampling does not greatly change the resulting persistence diagrams.
- Illustrative data objects: Figures 13 and 14 display brain trees alongside zero- and one-dimensional diagrams for 24-year-old and 68-year-old subjects.
5 Detailed analysis of brain artery data
The analysis converts persistence diagrams into vector summaries and tests how feature scale affects age correlations and sex differences. Medium-scale features produce strong age associations, while sufficiently broad loop-feature vectors consistently separate male and female subjects.
- Feature representations: Persistence diagrams are summarized either as feature vectors for standard statistics or analyzed directly with diagram-based metrics.The paper treats vector summaries as a middle ground between overly simple numerical summaries and direct population-level diagram analysis.
- Feature representations: Each zero-dimensional diagram yields a vector of the 100 largest dot persistences, with coordinate p_i representing the size of the i-th largest bend.For feature-scale analysis, the study constructs contiguous subvectors p_n,N and q_n,N for positive integers n < N ≤ 200; the original vectors are p_1,100 and q_1,100.
- Age correlations: ρ = 0.26 for the two highest persistences alone, whereas the 28th-longest bar alone yields near-optimal age correlation.Across the lower-triangular feature ranges, age correlation is fairly uniform and high; the heat map encodes ρ(n, N) over n and N.
- Age correlations: Medium-scale persistence features, rather than only the most persistent features, provide strong age effects in the zero-dimensional analysis.The one-dimensional feature analysis shows a similar pattern, although its corresponding heat map is not shown.
- Sex differences: p(1, 2) = 0.21, but every one-dimensional loop-vector value with N > 30 is below 0.05, reaching a minimum of p(189, 192) = 0.013.The permutation-test heat map displays these sex-difference significances, with darker colors indicating lower and therefore more significant values.
6 Discusssion
Persistent homology provides a new representation of brain artery trees that yields stronger age and sex findings than earlier approaches. The study also reports age associations independent of total artery length, while cautioning that its findings are not anatomically conclusive.
- Discussion: Persistent homology takes analysis of brain artery tree data in a new direction and produces stronger results than earlier representations.The approach is presented as a topological method for representing structured artery-tree data.
- Discussion: The study reports significant results after controlling for total artery length, which it identifies as a first for this data set.This conclusion extends the earlier age-related findings based on simpler artery summaries.
- Limitations and future work: Multiple-comparison issues were not carefully accounted for, so the lessons are intended to demonstrate tool potential rather than provide anatomically conclusive findings.The paper points to future applications on other tree-structured data sets, including airway data.
- Limitations and future work: The analyzed trees came from tube-tracking algorithm outputs rather than directly observed artery sets, creating concern that artifacts may contribute to the results.The concern applies especially to zero-dimensional analysis; persistent-homology stability offers protection for the loop analysis under slight tree modifications.