John Rick Manzanares Banyuhay
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Doctoral research · Interactive explanation

Can shape help explain bone strength?

Trabecular bone is not simply more bone or less bone. It is a network in three dimensions. Connectivity, tunnels, cavities, orientation, local geometry, and load paths all contribute to its architecture.

Explore the topology See the research pipeline

What this page does

It explains the research idea in layers.

The page begins with intuition, then connects a synthetic filtration to persistence, introduces the mathematical structure, and finally shows how the ideas enter the research workflow.

What this page does not do

It does not present unpublished outcome results.

No research specimen images, identifying information, or detailed numerical results from the manuscript are shown in the teaching interaction.

Browser inspection

Open a prepared bone surface without ImageJ.

A separate prototype viewer opens a small overview first, then refines the specimen while keeping the camera in place. Local mode reads prepared surfaces directly from your computer, so research data do not need to be published to the website.

Open the trabecular bone viewer

Start with the idea

Topology asks a different structural question.

Conventional morphometry measures quantities such as bone volume fraction, trabecular thickness, separation, connectivity density, and anisotropy. Persistent homology studies how topological features appear and disappear across a changing threshold or scalar field.

H0

Components

Tracks separate pieces and the thresholds at which they merge.

H1

Tunnels

Tracks loop structure surrounding a passage that has not been filled.

H2

Cavities

Tracks voids enclosed by surrounding structure in three dimensions.

Interactive concept

Connect a filtration to persistence

The example is deliberately small. Its purpose is to make birth, death, merging, and persistence visible before the same ideas are applied to images in three dimensions.

  1. 1Move the threshold and watch cells enter the filtration.
  2. 2Compare the Betti numbers with the persistence diagram.
  3. 3Select a persistence point to inspect its interval and an explanatory witness.

Filtration

Synthetic cubical filtration in two dimensions A teaching grid containing an early component, a ring, a later bridge, and an interior that eventually fills the ring. Cells enter as the threshold increases.

Persistence diagram

Persistence diagram for the finite classes in the teaching filtration The diagram contains a dimension 0 point born at 28 and dying at 45, and a dimension 1 point born at 28 and dying at 68. The earlier dimension 0 component is essential and is listed separately because its death time is infinite. birth death H0 H1

The shaded region contains classes whose birth is at most the current threshold and whose death is greater than it. These are the finite classes that are alive at that threshold.

H0 H1 alive at the current threshold
H0
Essential component birth 18 · death ∞
  1. 18 first component
  2. 28 ring and void
  3. 45 components merge
  4. 68 void fills
  5. 90 grid fills
β0 0 components under edge adjacency
β1 0 enclosed voids

Adjust the threshold to see components appear, merge, and enclose a void.

Selected persistence class

Select a persistence point

Dimension
Not selected
Birth
Not selected
Death
Not selected
Persistence
Not selected
At current threshold
Not selected

Choose either point in the diagram or one of the equivalent buttons below it.

This is a teaching construction, not research specimen data.

The scalar values were chosen so that the finite classes are known and easy to inspect. The H0 interval is [28, 45), the H1 interval is [28, 68), and the first H0 class is essential.

When a persistence point is selected, highlighted cells provide one explanatory witness for this construction. In general, a persistence class does not come with a unique canonical geometric representative.

From the teaching example to the research data

The object changes, but the logic is the same.

01

Start with an image volume

A calibrated micro CT volume provides intensity and spatial information inside a defined analysis region.

02

Build a scalar filtration

Image intensity or signed distance values determine when cells enter a cubical complex.

03

Track topology across thresholds

Persistent homology records which structural features survive as the filtration changes.

Read at the level you need

One project, three ways in.

The scientific question is the same in every view. Only the level of detail changes.

Intuition

A single threshold can hide how structure changes.

A binary segmentation gives one answer at one chosen threshold. Persistent homology instead studies a sequence of related shapes. A feature that disappears almost immediately behaves differently from one that remains present across a broad part of the filtration.

Persistence is therefore about stability across change. The aim in this project is not to replace morphometry, but to ask whether topology provides complementary information about architecture.

Mathematics

A filtration is a nested family of complexes.

Let K be a cubical complex and let f assign a scalar filtration value to its cells. For a threshold t, define the sublevel complex

Kt = { Q : f(Q) ≤ t } .

If s is at most t, then Ks is contained in Kt. Inclusion induces a map on homology:

Hk (Ks) → Hk (Kt) .

Persistent homology records when homology classes appear and when they disappear. A finite class can be represented by a point (birth, death) in a persistence diagram. The difference between death and birth is its persistence.

Implementation

Images become scalar fields, then cubical filtrations.

Image volumes are restricted to an analysis region and converted into scalar filtrations based on quantities such as image intensity or signed distance. Cubical persistent homology then produces interval or diagram data that can be transformed into representations suitable for modelling.

The conventional morphometry code is deliberately kept separate from the persistent homology workflow. The public bone_morphometry package handles morphometry and other scalar descriptors. Topological analysis remains independently inspectable in the associated research repository.

bone_morphometry → Topological analysis code →

Research provenance

Make the path from image to model visible.

Each stage changes the object being analysed. Showing those changes helps separate image processing, mathematical description, and statistical modelling.

  1. 01

    Image volume and analysis region

    Begin with a calibrated volume and an explicitly defined region of interest.

  2. 02

    Preprocessing and segmentation

    Construct a bone representation while preserving the intended analysis domain.

  3. 03

    Scalar filtration

    Use image intensity or distance information to determine when cells enter the complex.

  4. 04

    Persistent homology

    Track topological classes in dimensions 0, 1, and 2 across the filtration.

  5. 05

    Numerical representation

    Transform persistence information into features that can be compared or used in statistical models.

  6. 06

    Validation and strength modelling

    Compare topological and morphometric feature families with procedures appropriate for a small cohort.

Public now

Methods, software, and the conceptual interface

The mathematical idea, the teaching interaction, and public software can be inspected without releasing detailed outcome comparisons before the corresponding manuscript is public.

Inspect the Rust package →

Still in preparation

Detailed numerical comparisons

Strength prediction tables, manuscript figures, and other detailed outcome results are intentionally absent from this public teaching page for now.

Questions that follow

The first descriptor is not the end of the problem.

Bone structure also raises questions about direction, invariance, mechanics, and field learning.

Direction

How should topology respond when trabecular architecture has a preferred orientation?

Invariance

Which descriptors retain their meaning when the coordinate frame changes?

Mechanics

How do structural changes relate to load paths, stiffness, and local weakness?

Operator learning

Can topology help learned field models preserve information that matters to the physics?

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