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

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

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