Geometric Classes

Beyond statistical mechanics, the GLAM framework integrates descriptors from geometry and topology to characterize the spatial arrangement and complexity of biological textures. These metrics are organized into three primary domains:

Structural and Directional Organization

These metrics characterize how tissue components are packed and oriented within the tumor microenvironment.

Coordination Number (CN)

The Coordination Number (CN) measures the local packing or clustering of gray-levels. Adapted from atomic coordination in materials science, it represents the average number of \(\beta\)-voxels surrounding a reference \(\alpha\)-voxel within the first coordination shell:

\[CN_{\alpha,\beta} = 4\pi\rho_\beta \int_{0}^{r_{min}} g_{\alpha\beta}(r) r^2 dr\]

where \(\rho_\beta\) is the mean voxel density and \(r_{min}\) is the first RDF minimum beyond the primary peak. Diagonal terms describe local self-clustering (e.g., tumor cell density), whereas off-diagonal elements quantify the degree of direct interfacing between cancerous and stromal tissue.

Second Virial Coefficient — **Figure:** Coordination Number matrices derived from four co-registered MRI sequences: pre-contrast T1-weighted (T1), post-contrast T1-weighted (T1c), T2-weighted (T2), and Fluid-Attenuated Inversion Recovery (FLAIR).

Interpretation: This metric asks, “On average, how many immediate neighbors of type B does a central voxel of type A physically touch?”
Advantage: It provides a direct, intuitive measure of local packing density and the physical extent of the immediate contact boundary between different tissue types.

Correlation Length (\(\xi\))

While \(B_2\) quantifies the magnitude of spatial order, the correlation length \(\xi(\alpha, \beta)\) characterizes its spatial extent. It describes the range over which voxel interactions persist before structural memory is lost:

\[h_{\alpha\beta}(r) \propto A \exp \left( -\frac{r}{\xi_{\alpha,\beta}} \right), \quad r > r_{peak}\]

Longer correlation lengths indicate coherent, organized structures, while shorter ones imply more localized or disordered texture.

Interpretation: This metric asks, “How far away does a voxel still ‘feel’ the structural influence of a reference voxel before the tissue arrangement becomes completely random?”
Advantage: It quantifies the absolute physical size of coherent biological structures (like tumor nests or stromal bands) independently of their intensity values.

Anisotropy Indices

GLAM captures directional organization (e.g., in aligned stromal bands) using gyration and nematic ordering tensors.

Positional Anisotropy: Uses the Relative Shape Anisotropy index, \(A_{\alpha,\beta}\), to quantify geometric elongation derived from the eigenvalues of the local gyration tensor.
Orientational Anisotropy: Calculates the Nematic Order Parameter (:math:`S`), analogous to liquid crystal physics, to quantify the degree of alignment of local intensity gradients. \(S=0\) represents random orientation, while \(S=1\) indicates perfect alignment.

Interpretation: This metric asks, “Are the tissue structures stretched out and aligned in a specific direction, or are they perfectly round and directionless?”
Advantage: It captures directional gradients and structural elongation, which are critical for identifying invasive fronts, aligned collagen tracks, or collective cell migration patterns.

Complexity and Topology

These measures describe the roughness, multiscale nature, and connectivity of image features.

Fractal and Multifractal Dimensions

GLAM utilizes a 3D box-counting algorithm to quantify multiscale self-similarity and complexity.

Volume Fractal Dimension (:math:`D_V`): Calculated for voxels of a single gray level, indicating its space-filling capacity.
Interface Fractal Dimension (:math:`D_I`): Measures the roughness and invasiveness of boundaries between two tissue types.
Multifractal Spectrum: Employs Generalized Dimensions (\(D_q\)) to characterize tissues where scaling properties vary across the region. The spectrum width (\(\Delta\alpha\)) quantifies the diversity of scaling behaviors, representing the “heterogeneous chaos” of the tissue.

Interpretation: This metric asks, “How complex, branching, and space-filling is this tissue structure across different zoom levels?”
Advantage: It captures the self-similar “roughness” of biological tissues, allowing for robust differentiation between smooth, encapsulated tumors and highly invasive, branching morphologies.

Lacunarity (\(\Lambda\))

While fractal dimension quantifies space-filling, Lacunarity measures the “gappiness” or heterogeneity of void spaces within the tissue architecture. High Lacunarity indicates large, irregular gaps, while low values suggest a uniform, homogeneous distribution.

Topological Invariants (Betti Numbers)

GLAM uses algebraic topology to count discrete features that are invariant under continuous deformation:

Betti-0 (:math:`B_0`): Counts fragmented “islands” of a specific gray level.
Betti-1 (:math:`B_1`): Counts tunnels or loops (e.g., vascular networks).
Betti-2 (:math:`B_2`): Counts enclosed internal cavities or voids.
Euler Characteristic (:math:`chi`): A classic measure of topological complexity, where \(\chi = B_0 - B_1 + B_2\).

Interpretation: This metric asks, “How many distinct islands, connective tunnels, and hollow voids exist within the tissue, regardless of their exact physical shape?”
Advantage: Topology is completely invariant to stretching, bending, or scaling. This makes Betti numbers incredibly robust against patient positioning differences, organ deformation, and imaging variations.

Discrete Morphology

This category extracts explicit geometric descriptors for distinct tissue clusters identified by gray-level thresholds.

Sphericity and Solidity: Measure the compactness and “ruggedness” of individual gray-level isosurfaces.
Interface Area: Quantifies the total surface of direct contact between two distinct tissue types, representing the extent of physical infiltration.
Centroid Distance: Measures the Euclidean distance between the centers of mass of different tissue components.

Interpretation: This metric asks, “What are the tangible physical dimensions, roundness, and contact areas of these specific tissue clumps?”
Advantage: It provides highly tangible, classic geometric descriptors that correlate directly with standard visual pathological assessments.