How can persistent homology model longitudinal brain MRI topology?

Betti curves become the outcome, not a feature

Most brain-imaging uses of persistent homology reduce a scan to a summary and then ask whether that summary separates groups or predicts an outcome. This study reverses the direction. For each homology dimension it keeps the whole Betti curve, which counts how many topological features are present at each value of the filtration scale, and treats that curve as the thing to be explained.

The distinction matters because a Betti curve is a function over a continuum of scales, not a single number. Modeling it as a response preserves where along the scale axis a covariate acts, instead of collapsing that information into one coefficient. The paper's stated motivation is exactly this: topological summaries are usually predictors or classification features, and their functional nature across filtration values is not fully exploited.

Two properties of the response shape the model. Betti counts are discrete and potentially overdispersed, so a negative-binomial distribution replaces a Gaussian one. And because each person contributes several scans, the model needs a multilevel structure rather than a single error term.

What the filtration and homology dimensions measure

Persistent homology tracks when topological features appear and disappear as a threshold, or scale parameter, sweeps through the image. A filtration is a nested sequence of complexes indexed by that scale; for images on regular voxel grids the paper uses a cubical complex, which lets persistent homology be computed directly from image data.

Homology dimension p names the kind of feature being counted. H0 counts connected components, H1 counts loops or one-dimensional holes, and H2 counts voids or two-dimensional holes. A Betti curve for dimension p therefore reports the number of p-dimensional features alive at each filtration value.

The paper fits H0, H1 and H2 separately, each with its own coefficient functions. This separation is deliberate but also a stated limitation: the authors note that joint modeling of multiple homology dimensions could account for their dependence.

Separating between-person from between-visit variation

The regression links covariates to the log expected Betti curve through coefficient functions that themselves vary over the filtration scale s. Baseline age is centered at 75 years, sex enters with female as the reference category, follow-up time is measured in years from each participant's first MRI visit with the first visit set to zero, and dementia severity enters as CDR indicators with CDR = 0 as reference and separate terms for CDR = 0.5 (very mild) and CDR = 1 (mild).

Beyond fixed effects, the model adds two sets of latent functional terms: subject-level components shared across a person's visits and visit-level components specific to each scan occasion. The paper retains three components at each level, chosen from a sensitivity analysis. Functional principal component analysis then describes the dominant modes of variation at each level.

Estimation is Bayesian and joint, so the functional regression effects and the multilevel principal components are estimated together, with uncertainty quantified for both. The authors report convergence diagnostics for all three fitted models, including trace plots, posterior densities, R-hat values close to 1 and adequate effective sample sizes.

Covariate effects shift location and shape across scales

The fixed-effect results are not uniform shifts. For H1, the baseline-age coefficient stays close to zero across most of s with credible intervals generally including zero, which the authors read as limited evidence of a baseline age effect. The sex coefficient is more pronounced at lower values of s, suggesting higher expected H1 Betti counts for males than females in that region. Follow-up time and the CDR coefficients vary around zero, with wider credible intervals at higher filtration values.

For H2 the pattern differs. The age coefficient again stays near zero, while the sex coefficient varies more over s, positive in the earlier and later portions and negative in the middle range. The follow-up-time coefficient is positive at lower s and approaches zero thereafter. CDR coefficients also vary over s, again with relatively wide credible intervals.

On the original count scale, with other covariates at reference values and latent effects set to zero, the paper describes increasing baseline age as raising the first H1 peak and lowering the second, so the age effect changes sign across the scale rather than producing a uniform increase or decrease. Males show substantially higher expected H1 counts around both major peaks, and longer follow-up is associated with progressively higher counts around the peaks while peak locations stay similar. For H2, CDR = 0.5 is associated with a lower dominant peak than CDR = 0, whereas CDR = 1 is associated with a higher peak at a slightly larger value of s.

Visit-level variation dominates, but less so for H2

After population-level covariates, the multilevel decomposition attributes most remaining functional variation to the visit level: about 97.03% for H0, 91.22% for H1 and 81.78% for H2, with 95% credible intervals reported for each. Subject-level shares rise across dimensions, from 2.97% for H0 to 8.78% for H1 and 18.26% for H2.

The authors interpret this as within-subject variation across visits remaining dominant while between-subject differences become increasingly prominent in higher-dimensional topological features. That is a statement about unexplained functional variability in this sample, not about which source is clinically more important.

The principal component results add a second distinction. For H2, the first subject-level component explains 53.34% of variation and mainly reflects the prominence of the dominant peak, while the second explains 29.67% and captures shape and relative prominence of the early and dominant peaks. At the visit level the first two components explain 57.28% and 33.08%. Compared with H0 and H1, H2 variation is less concentrated in the first component, so several distinct modes contribute at both levels.

Twenty-two participants bound what can be claimed

The application uses OASIS-2, a publicly available longitudinal neuroimaging dataset. The release contains T1-weighted structural MRI from 150 older adults aged 60 to 96 across 373 imaging sessions. The analysis restricts to participants with three MRI visits and focuses on those with dementia, including converters from nondemented to demented during follow-up, yielding 22 participants and 66 examinations.

That selection is the main boundary on the findings. The covariate associations describe this small dementia-focused subset with exactly three visits, and the wide credible intervals for several CDR and follow-up coefficients already signal limited precision. The paper reports associations, not causal effects of age, sex, follow-up time or dementia severity on topology.

A second boundary is scope: the analysis uses structural MRI alone and models each homology dimension separately. The authors list joint modeling across dimensions and extension to multimodal imaging, including diffusion and functional MRI, as future work rather than completed results.

The scale-resolved comparison is the real contribution

The most useful result is not that covariates are associated with topology but where along the filtration they act and how that location differs by homology dimension. Age raising one H1 peak while lowering another, sex acting at low s for H1 but across early, middle and late ranges for H2, and CDR = 0.5 versus CDR = 1 producing opposite changes in the H2 dominant peak are all scale-specific patterns that a single summary statistic would average away.

This also sets the priority for anyone reusing the approach: keep the curve as the response, choose a count distribution that matches Betti counts, and separate subject-level from visit-level variation before interpreting covariate effects. The visit-level share of roughly 82% to 97% indicates that most unexplained variation in these curves sits within participants across visits, so a model that ignored repeated measurement would misattribute that variation.

The paper's own next step follows from the same logic. Because H2 shows both the largest subject-level share and the least concentrated component structure, joint modeling of homology dimensions is the extension most directly motivated by these results.

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