homology-morphometry-brain-atrophy

name

homology-morphometry-brain-atrophy

description

Homology-based Morphometry (HBM) methodology for analyzing brain atrophy using persistent homology. Two complementary pipelines for quantifying multiscale geometric features of structural T1-weighted MRI scans: Pipeline 1 for regional thinning via Euclidean distance transform, Pipeline 2 for structural similarity via α-filtrations. Use for Alzheimer's disease detection, longitudinal brain change tracking, and topological biomarker extraction. Keywords: brain atrophy, persistent homology, TDA, MRI morphometry, Alzheimer's disease, topological biomarker.

Homology-based Morphometry of Brain Atrophy

Homology-based Morphometry (HBM) is a topological data analysis framework for quantifying multiscale geometric features of structural T1-weighted MRI scans without requiring nonlinear registration to a standard template.

Overview

Contemporary structural brain analysis relies on voxel-based morphometry (VBM) which requires normalization to a standard template, potentially obscuring subject-specific geometric features. HBM addresses this limitation using persistent homology (PH), a tool from topological data analysis.

Core Methodology

Pipeline 1: Regional Thinning Analysis

Purpose: Quantify regional thinning via Euclidean distance transform

Process:

1. Apply Euclidean distance transform to tissue masks in a slice-wise manner
2. Capture regional thinning patterns
3. Best suited for between-subject analyses

Key Applications:

• Cross-sectional group comparisons
• Alzheimer's disease vs cognitively normal classification
• ROC-AUC = 0.895 on ADNI dataset

Pipeline 2: Structural Similarity Analysis

Purpose: Measure structural similarity between pairs of scans

Process:

1. Use α-filtrations to capture multiscale geometric features
2. Detect sulcal widening and ventricular enlargement
3. Best suited for within-subject designs

Key Applications:

• Longitudinal change tracking
• Disease progression monitoring
• Follow-up scan comparison

Methodological Advantages

Feature	Traditional VBM	HBM (This Method)
Registration	Required	Not required
Subject-specific features	May be obscured	Preserved
Pathological cases	Problematic	Handled robustly
Interpretability	Statistical	Topological
Multiscale analysis	Limited	Native support

Implementation Workflow

Prerequisites

import numpy as np
import gudhi  # For persistent homology
from scipy.ndimage import distance_transform_edt
import nibabel as nib

Pipeline 1: Regional Thinning

def pipeline1_regional_thinning(tissue_mask, slice_axis=2):
    """
    Quantify regional thinning using Euclidean distance transform
    
    Args:
        tissue_mask: Binary tissue mask (WM/GM segmentation)
        slice_axis: Axis for slice-wise processing (default: 2)
    
    Returns:
        Persistence diagrams for each slice
    """
    n_slices = tissue_mask.shape[slice_axis]
    persistence_diagrams = []
    
    for i in range(n_slices):
        # Extract slice
        if slice_axis == 0:
            slice_mask = tissue_mask[i, :, :]
        elif slice_axis == 1:
            slice_mask = tissue_mask[:, i, :]
        else:
            slice_mask = tissue_mask[:, :, i]
        
        # Apply Euclidean distance transform
        distance_map = distance_transform_edt(slice_mask)
        
        # Build cubical complex and compute persistence
        cc = gudhi.CubicalComplex(
            dimensions=slice_mask.shape,
            top_dimensional_cells=distance_map.flatten()
        )
        persistence = cc.persistence()
        persistence_diagrams.append(persistence)
    
    return persistence_diagrams

Pipeline 2: Structural Similarity

def pipeline2_structural_similarity(scan1, scan2, max_alpha=100):
    """
    Measure structural similarity between two scans using α-filtrations
    
    Args:
        scan1, scan2: Point cloud representations of brain structure
        max_alpha: Maximum α value for filtration
    
    Returns:
        Similarity score and persistence comparison
    """
    # Build α-complex for both scans
    alpha_complex1 = gudhi.AlphaComplex(points=scan1)
    simplex_tree1 = alpha_complex1.create_simplex_tree(max_alpha_square=max_alpha**2)
    persistence1 = simplex_tree1.persistence()
    
    alpha_complex2 = gudhi.AlphaComplex(points=scan2)
    simplex_tree2 = alpha_complex2.create_simplex_tree(max_alpha_square=max_alpha**2)
    persistence2 = simplex_tree2.persistence()
    
    # Compute persistence-based similarity
    similarity = compute_persistence_similarity(persistence1, persistence2)
    
    return similarity, persistence1, persistence2

Performance Metrics

Pipeline 1 Results (ADNI Dataset)

• ROC-AUC: 0.895 for AD vs CN classification
• Peak effects: Localized to medial temporal regions
• Input: Single-modality T1-weighted MRI
• No nonlinear registration required

Pipeline 2 Results (Longitudinal)

• Follow-up scans remain closest to their own baselines
• AD subjects show greater short-interval change than CN subjects
• Captures disease-related longitudinal change patterns

Interpretable Topological Biomarkers

Key Topological Features

1. Birth-death pairs: Indicate scale of geometric features
2. Persistence: Measures feature significance
3. Betti numbers: Count of topological features (connected components, holes, voids)

Clinical Interpretation

Biomarker	Clinical Significance
High persistence in temporal regions	Hippocampal atrophy
Increased void persistence	Ventricular enlargement
Reduced surface persistence	Cortical thinning
Altered connectivity patterns	White matter degradation

Use Cases

Case 1: Alzheimer's Detection

# Load MRI scan
mri_scan = nib.load('subject_t1w.nii.gz')
brain_mask = extract_brain_mask(mri_scan)

# Apply Pipeline 1
pdgm = pipeline1_regional_thinning(brain_mask)

# Extract features for classification
features = extract_persistence_features(pdgm)
prediction = classifier.predict(features)  # AD vs CN

Case 2: Longitudinal Monitoring

# Compare baseline and follow-up scans
baseline_scan = load_point_cloud('baseline.ply')
followup_scan = load_point_cloud('followup_6months.ply')

# Apply Pipeline 2
similarity, _, _ = pipeline2_structural_similarity(
    baseline_scan, followup_scan
)

# Track change over time
if similarity < threshold:
    print("Significant structural change detected")

Integration with Existing Pipelines

BIDS Compatibility

# Load BIDS-formatted data
from bids import BIDSLayout

layout = BIDSLayout('/data/adni_bids')
t1w_files = layout.get(suffix='T1w', extension='nii.gz')

for t1w in t1w_files:
    # Process with HBM
    results = process_hbm(t1w.path)

Comparison with VBM

HBM complements traditional VBM by:

• Providing registration-free analysis
• Preserving subject-specific morphological features
• Enabling robust analysis of pathological brains
• Offering interpretable topological biomarkers

References

• Paper: arXiv:2604.24714v1 [math.AT]
• Title: "Homology-based Morphometry of Brain Atrophy: Methods and Applications"
• Authors: Donato Quiccione, Mariam Pirashvili, Nathan Broomhead, et al.
• Dataset: Alzheimer's Disease Neuroimaging Initiative (ADNI)

Related Skills

• brain-network-controllability: Network control theory for brain networks
• topological-ml-eeg-classification: Topological methods for EEG analysis
• brain-graph-neural: GNN methods for brain connectivity

Activation Keywords

• brain atrophy analysis
• persistent homology MRI
• topological morphometry
• homology-based brain measurement
• TDA neuroimaging
• 脑萎缩同调分析
• 拓扑脑形态学