Grow

The Grow engine applies morphological dilation to each object in a coded image:

1. Structuring element selection — A circular or square kernel of the specified radius defines the growth shape. Circular kernels produce isotropic (uniform in all directions) growth; square kernels produce slightly diamond-shaped growth.
2. Dilation — Each object's boundary is expanded outward by the kernel radius. Every background pixel within the growth distance of any object becomes part of that object's expanded region.
3. Collision resolution — Where expanded regions from different objects would overlap, the contested pixels are assigned to the nearest object (by Euclidean distance from the original boundary). This creates a Voronoi-like partition of the space between adjacent nuclei.
4. Ring creation — By running Grow at two different distances and subtracting the inner coded image from the outer, a ring-shaped region is produced. This ring represents the cytoplasmic compartment.

The output is a new coded image where each object has the same label as the original but covers a larger area. Measurements on this expanded coded image capture cytoplasmic biomarker expression rather than nuclear.

Dilation — the mathematical foundation: Gonzalez & Woods define dilation as the set of all positions where the reflected structuring element overlaps the object. In practical terms: place the structuring element centered on every boundary pixel of the object, and every pixel it touches becomes part of the expanded object. A circular structuring element of radius r pixels expands the object uniformly in all directions by r pixels.

The asymmetry of morphological operations: Solomon & Breckon note a critical asymmetry: "we cannot restore by dilation an object which has previously been completely removed by erosion." In StrataQuest's context, this means a Grow operation creates new territory (the cytoplasmic approximation) but cannot recover nuclear boundary details that were lost during detection. The grow distance should be chosen based on cell biology, not to compensate for detection errors.

Structuring element shape matters: The choice of structuring element shape is not trivial. A circular SE produces isotropic growth — the expanded boundary is equidistant from the original at all points. A square SE produces growth that extends farther along diagonals (by a factor of √2), creating slightly octagonal shapes. For cell biology, circular SEs better approximate the roughly radial distribution of cytoplasmic contents around the nucleus.

The Voronoi partition: When growing cells collide, the boundary between them follows the Voronoi diagram — the set of points equidistant from the two nearest nuclei. This is biologically reasonable: in tightly packed epithelial tissue, cell boundaries approximately bisect the space between adjacent nuclei. The Voronoi partition is the mathematically optimal division when you have no other information about where the true cell boundary lies.

Why ring measurements matter: Membrane biomarkers (PD-L1, HER2, E-cadherin) localize to the cell membrane, not the nucleus. Nuclear measurements of these markers would primarily capture background signal plus any membrane that happens to overlap the nuclear area. Cytoplasmic ring measurements, created by growing and subtracting, specifically target the peri-nuclear zone where these markers accumulate — dramatically improving signal-to-noise for membrane and cytoplasmic marker quantification.

Creating nuclear, cytoplasmic, and membrane compartments for multiplex IF analysis:

1. Nuclear: Nuclei Detection on DAPI → coded image with original nuclear boundaries
2. Perinuclear ring: Grow by 3 pixels → subtract nuclear coded image → inner cytoplasmic ring
3. Full cell: Grow by 12 pixels → approximate whole-cell region
4. Outer cytoplasm: Full cell minus perinuclear ring → outer cytoplasmic compartment

Now measure CK (cytoplasmic marker) in the ring regions, PD-L1 (membrane marker) at the full-cell boundary, and Ki-67 (nuclear marker) in the original nuclear region. Each marker is measured in the compartment where it biologically resides, maximizing signal specificity.