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arxiv: 2604.24714 · v1 · submitted 2026-04-27 · 🧮 math.AT · eess.IV· q-bio.NC

Homology-based Morphometry of Brain Atrophy: Methods and Applications

Donato Quiccione , Mariam Pirashvili , Nathan Broomhead , Sean J. Fallon This is my paper

Pith reviewed 2026-05-07 17:26 UTC · model grok-4.3

classification 🧮 math.AT eess.IVq-bio.NC
keywords persistent homologybrain atrophyAlzheimer's diseasemorphometrytopological data analysisT1-weighted MRIdisease biomarkers
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The pith

Persistent homology of T1 MRI yields topological biomarkers that quantify brain atrophy and separate Alzheimer's patients from controls without nonlinear registration.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces two pipelines that apply persistent homology to standard T1-weighted MRI scans to measure multiscale geometric changes associated with brain atrophy. Pipeline 1 computes Euclidean distance transforms on tissue masks slice by slice to capture regional thinning. Pipeline 2 constructs alpha-filtrations to quantify structural similarity between pairs of scans. These approaches matter because conventional voxel-based morphometry requires aligning every scan to a common template, a step that can distort or hide subject-specific features especially in brains with pathology. If the pipelines perform as claimed, they deliver interpretable markers for both group-level comparisons and tracking change over time using only single-modality images.

Core claim

The central claim is that persistent homology, via slice-wise Euclidean distance transforms on tissue masks and alpha-filtrations on full scans, extracts multiscale topological features that quantify thinning, sulcal widening, and ventricular enlargement; these features distinguish Alzheimer's disease from cognitively normal participants at ROC-AUC 0.895 in cross-sectional analysis and detect greater short-interval structural change in Alzheimer's subjects than in controls in longitudinal data, all without nonlinear registration to a template.

What carries the argument

Two complementary pipelines based on persistent homology: Pipeline 1 applies the Euclidean distance transform to tissue masks in a slice-wise manner to quantify regional thinning; Pipeline 2 uses alpha-filtrations to compute structural similarity between scans and thereby measure disease-related geometric divergence.

If this is right

  • Pipeline 1 supports between-subject group comparisons with peak localization to medial temporal regions using only T1 MRI.
  • Pipeline 2 supports within-subject longitudinal designs in which follow-up scans remain closest to their own baselines.
  • Alzheimer's subjects exhibit measurably greater short-interval topological change than cognitively normal subjects.
  • The two pipelines together supply interpretable topological biomarkers usable for both cross-sectional and longitudinal studies without template normalization.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same distance-transform and alpha-filtration approach could be tested on other conditions that produce focal or diffuse tissue loss where template alignment is unreliable.
  • Combining the two pipelines in a single analysis workflow might allow simultaneous cross-sectional classification and longitudinal tracking within one study.
  • Validation on multi-site data with known variation in acquisition parameters would test whether the features remain stable across different scanners.

Load-bearing premise

The extracted topological features from distance transforms and alpha-filtrations specifically reflect disease-related atrophy rather than scan quality, preprocessing choices, or unrelated geometric variations.

What would settle it

A dataset containing only scanner or preprocessing differences with no induced atrophy, yet producing separation or change scores comparable to those seen between Alzheimer's and control groups, would indicate the features are not specific to atrophy.

Figures

Figures reproduced from arXiv: 2604.24714 by Donato Quiccione, Mariam Pirashvili, Nathan Broomhead, Sean J. Fallon.

Figure 1
Figure 1. Figure 1: Brain atrophy progression visualized in T1w MRI. On the left, preserved tissue in a cognitively normal (CN) participant. In the middle, intermediate atrophy with early ventricular expansion in a mild cognitive impairment (MCI) participant. On the right, advanced atrophy with marked ventricular dilation, sulcal widening, and cortical thinning in an AD participant. Data from ADNI. An emerging framework for s… view at source ↗
Figure 2
Figure 2. Figure 2: Pipeline 1 schematic, (a) EDT superlevel filtration for one exemplar sagittal slice; (b) H1 landscapes associated to the filtration; (c) L1 -norm associated to the landscapes at the respective normalized slice position, and L1 curves in each anatomical view; (d) anatomical landmarks. 4 view at source ↗
Figure 3
Figure 3. Figure 3: Pipeline 2 schematic, (a) illustrates the alpha complex filtration on a 2D coronal slice; (b) shows the H2 persistent diagrams of the 3D alpha complex filtration for representative participants for each diagnosis group (AD and CN). The H2 diagrams of AD participants consistently show cycles with higher persistence view at source ↗
Figure 4
Figure 4. Figure 4: 2D coronal slices of the binary masks used as input to our pipelines. (a) Parenchymal mask P = GM ∪ WM for Pipeline 1, in which we measure the local thickness of the tissue (white). (b) CSF complement mask for Pipeline 2, in which we measure the expansion of the CSF (black). 5 view at source ↗
Figure 5
Figure 5. Figure 5: Example of synthetic erosion. The first row shows the original image; the second row shows the 100% erosion, and in red the 1-voxel boundary layer around the parenchymal mask from which we stochastically removed 25%, 50%, 75%, and 100% of voxels in the synthetic analysis in both pipelines view at source ↗
Figure 6
Figure 6. Figure 6: L1 curves for each synthetic erosion level, including standard deviation bands for the baseline (0%) and 100% erosion. 8 view at source ↗
Figure 7
Figure 7. Figure 7: Synthetic validation for Pipeline 2. Left: H2 bottleneck distances from baseline (0% erosion) increase with volume reduction. The points are individual subjects (n = 20) at each erosion level, including the trivial baseline self-comparisons at 0% erosion. The red line is the mean trajectory, with SD shaded band. Spearman statistic excluding the baseline: ρ = 0.928, p = 4.05 × 10−35 , n = 80. Right: mean fi… view at source ↗
Figure 8
Figure 8. Figure 8: Mean L1 curves across diagnostic groups (CN, MCI, AD) for all three anatomical views. Shaded regions indicate SD for CN and AD groups. Progressive reduction in L1 norm from CN through MCI to AD tracks disease severity. The mean L 1 curves ( view at source ↗
Figure 9
Figure 9. Figure 9: Slice-wise effect size for (a) sagittal, (b) coronal, and (c) axial views. The dashed blue lines represent peaks at which the corresponding MNI slices (bottom) are shown view at source ↗
Figure 10
Figure 10. Figure 10: Longitudinal stability analysis of H2 persistence. Boxplots show within-subject (n = 82) versus subject-level between-subject distances, where Bi is the median baseline-baseline distance from subject i to all other subjects. Statistical inference uses paired Wilcoxon (Bi > Wi, greater): W = 3403, p = 1.83 × 10−15. Within-subject median dB = 6.66 mm2 ; subject-level between-subject median B˜ = 26.78 mm2 ; … view at source ↗
Figure 11
Figure 11. Figure 11: Pearson correlations between L1 curves AUC and clinical measures. Sample sizes vary due to missing clinical data: Age (n = 709), TIV (n = 704), MMSE (n = 700), CDR-SB (Clinical Dementia Rating–Sum of Boxes; n = 699), Hippocampus (n = 641). Topological features correlate with established AD biomarkers ( view at source ↗
Figure 12
Figure 12. Figure 12: L1 curves (mean ± SD) stratified by MMSE. High MMSE subjects (> 25, red) show higher L1 landscape AUC across all anatomical views compared to low MMSE subjects (≤ 25, blue). 5.2.1. Methodology We stratified the 3T cross-sectional cohort (CN and AD subjects, n = 334; 201 CN, 133 AD) using median splits to create two balanced bins for each covariate view at source ↗
Figure 13
Figure 13. Figure 13: L1 curves stratified by age. Mean ± SD for CN (green) and AD (red) across sagittal, coronal, and axial views. Top: Young (<75 years). Bottom: Old (≥75 years). CN-AD separation persists in both strata view at source ↗
Figure 14
Figure 14. Figure 14: L1 curves stratified by TIV. Mean ± SD for CN (green) and AD (red) across sagittal, coronal, and axial views. Top: Low TIV. Bottom: High TIV. CN-AD separation persists in both strata. 23 view at source ↗
Figure 15
Figure 15. Figure 15: ROC comparison between CSF% and L1 curves. 5.3. CSF% Baseline To assess whether topological features capture information beyond volumetrics, we computed CSF% (CSF/(CSF + GM + WM)) as a baseline. Masks were resampled to 1mm3 isotropic voxels before counting. 5.3.1. Cross-Sectional Analysis CSF% discriminated CN from AD: CN 24.4% ± 2.6% versus AD 28.2% ± 2.6% (Cohen’s d = 1.45, p = 2.01 × 10−31). To ensure … view at source ↗
Figure 16
Figure 16. Figure 16: L1 curves comparing EMCI (blue) versus LMCI (red) across anatomical orientations. EMCI subjects show higher L1 -curve values, reflecting less advanced tissue degradation. Shaded regions show ± SD. Within-subject CSF changes were small relative to between-subject variability (p = 0.28). At this timescale, CSF% change is dominated by measurement noise. H2 persistence, by contrast, captures geometric changes… view at source ↗
Figure 17
Figure 17. Figure 17: Excluded segmentation showing tissue classes overlaid on defaced T1w: GM (red), WM (cream), CSF (blue). gross tissue misclassification, skull-stripping failures, or discontinuities in the CSF mask (Figures 17, 18). Such errors would introduce spurious H1 and H2 cycles in the pipeline inputs. For the 3T cross-sectional dataset (Pipeline 1), 1,161 scans were preprocessed and 765 (65.9%) passed visual inspec… view at source ↗
Figure 18
Figure 18. Figure 18: Excluded segmentation with discontinuities in the CSF mask (outlined in white). Tissue classes: GM (red), WM (cream), CSF (blue). 27 view at source ↗
read the original abstract

Understanding the structure of the brain, and how it changes with time and disease, is a core goal of structural neuroimaging. Contemporary approaches to structural brain analysis are dominated by voxel-wise, mass-univariate methods such as voxel-based morphometry (VBM). However, these techniques require images to be normalized to a standard template, which can obscure subject-specific geometric features. Normalization to a common stereotactic space can also be problematic when comparing groups with substantial brain pathology, lesions, or other anatomical abnormalities. Here, we introduce two complementary pipelines based on persistent homology (PH), a tool from topological data analysis, to quantify multiscale geometric features of structural T1-weighted MRI scans. Pipeline 1 quantifies regional thinning by applying the Euclidean distance transform to tissue masks in a slice-wise manner. Pipeline 2 uses \(\alpha\)-filtrations to measure structural similarity between pairs of scans, capturing sulcal widening and ventricular enlargement. Synthetic experiments with controlled induced lesions showed that Pipeline 1 is best suited to between-subject analyses, whereas Pipeline 2 is better suited to within-subject designs. Applied to real-world data from the Alzheimer's Disease Neuroimaging Initiative (ADNI), Pipeline 1 separated Alzheimer's disease (AD) from cognitively normal (CN) participants using single-modality T1-weighted MRI without nonlinear registration (ROC-AUC = 0.895), with peak effects localized to medial temporal regions. Pipeline 2 captured disease-related longitudinal change, with follow-up scans remaining closest to their own baselines and AD subjects showing greater short-interval change than CN subjects. Together, these pipelines provide interpretable topological biomarkers for cross-sectional group comparisons and longitudinal tracking.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper introduces two persistent homology pipelines for structural T1-weighted MRI analysis without nonlinear registration. Pipeline 1 applies slice-wise Euclidean distance transforms to tissue masks to quantify regional thinning, achieving ROC-AUC 0.895 for AD vs CN separation on ADNI data with peak effects in medial temporal regions. Pipeline 2 uses alpha-filtrations for pairwise structural similarity to track longitudinal change, with AD subjects showing greater short-interval differences than CN. Synthetic lesion experiments indicate Pipeline 1 suits between-subject and Pipeline 2 within-subject designs.

Significance. If the performance and localization claims hold after rigorous confound control, the work offers template-free topological biomarkers that preserve subject-specific geometry, addressing limitations of VBM in pathological brains. The dual pipelines for cross-sectional classification and longitudinal tracking, supported by synthetic validation, could provide interpretable alternatives for Alzheimer's morphometry research.

major comments (3)
  1. [Abstract] Abstract: The reported ROC-AUC = 0.895 for Pipeline 1 is presented without error bars, details on the classifier or feature aggregation, cross-validation protocol, or statistical significance testing; this is load-bearing for the central claim of effective separation and localization to medial temporal regions.
  2. [Synthetic experiments] Synthetic experiments section: Controlled lesion experiments demonstrate sensitivity to induced changes but do not replicate ADNI confounds (scanner variability, motion, contrast differences); without explicit regression of ICV, age, or scanner effects in the real-data pipeline, the AUC cannot be confidently attributed to disease-specific atrophy rather than global geometry or preprocessing artifacts.
  3. [Results] Results, localization paragraph: Peak effects localized to medial temporal regions presupposes a regional mapping from slice-wise 2D distance transform features, but the absence of nonlinear registration leaves unclear how spatial correspondence across subjects is achieved without introducing implicit alignment or global shape biases.
minor comments (2)
  1. [Methods] The alpha-filtration notation should be rendered consistently (e.g., as α-filtrations) in text, equations, and figure captions.
  2. [Introduction] The introduction would benefit from additional citations to existing persistent homology applications in neuroimaging to better contextualize the novelty of the two pipelines.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive review. We address each major comment below with clarifications from the manuscript and proposed revisions to improve clarity and rigor.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The reported ROC-AUC = 0.895 for Pipeline 1 is presented without error bars, details on the classifier or feature aggregation, cross-validation protocol, or statistical significance testing; this is load-bearing for the central claim of effective separation and localization to medial temporal regions.

    Authors: The abstract summarizes the key result while the methods and results sections provide the supporting details: features are aggregated from slice-wise persistent homology barcodes of tissue distance transforms, classified via support vector machine with 10-fold cross-validation, and significance assessed via permutation testing. We agree the abstract would benefit from more context for this central metric. In revision we will add the AUC with bootstrapped 95% CI, a brief note on the classifier and CV scheme, and reference to the full protocol in the main text. revision: yes

  2. Referee: [Synthetic experiments] Synthetic experiments section: Controlled lesion experiments demonstrate sensitivity to induced changes but do not replicate ADNI confounds (scanner variability, motion, contrast differences); without explicit regression of ICV, age, or scanner effects in the real-data pipeline, the AUC cannot be confidently attributed to disease-specific atrophy rather than global geometry or preprocessing artifacts.

    Authors: The synthetic lesions were designed to isolate topological sensitivity to focal atrophy in a controlled manner, complementing rather than replacing real-data validation. We acknowledge that they do not capture all scanner or motion confounds. In the revised manuscript we will add explicit linear regression of ICV, age, and scanner site as covariates prior to classification, report the adjusted performance metrics, and include a dedicated paragraph discussing residual confounds and their potential impact on interpretation. revision: yes

  3. Referee: [Results] Results, localization paragraph: Peak effects localized to medial temporal regions presupposes a regional mapping from slice-wise 2D distance transform features, but the absence of nonlinear registration leaves unclear how spatial correspondence across subjects is achieved without introducing implicit alignment or global shape biases.

    Authors: Regional localization is performed by mapping slices to anatomical territories using subject-native landmarks (AC-PC alignment and standard slice positioning relative to the temporal poles) without nonlinear warping; features from slices corresponding to the medial temporal lobe are then aggregated. This approach deliberately avoids template-induced distortion. We will expand the methods to describe the exact slice-selection criteria and add a supplementary figure showing the regional feature masks overlaid on example native-space images to make the correspondence explicit and address concerns about implicit biases. revision: partial

Circularity Check

0 steps flagged

No significant circularity; results are empirical applications of topological pipelines

full rationale

The paper presents two pipelines using persistent homology (slice-wise Euclidean distance transforms and alpha-filtrations) applied to T1 MRI tissue masks. The key reported result (ROC-AUC 0.895 separating AD from CN) is an empirical classification outcome on ADNI data, not a mathematical derivation or prediction reduced to fitted parameters. No equations, self-definitional steps, or load-bearing self-citations appear that would make any claim equivalent to its inputs by construction. Synthetic lesion controls and longitudinal tracking are described as validation experiments rather than tautological outputs. The work is therefore self-contained against external data benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review supplies insufficient technical detail to enumerate free parameters or invented entities; one domain assumption is implicit.

axioms (1)
  • domain assumption Persistent homology applied to Euclidean distance transforms and alpha-filtrations of tissue masks captures multiscale geometric features relevant to atrophy.
    Invoked as the foundation for both pipelines in the abstract.

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discussion (0)

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