Geometric and topological characterization of the cytoarchitecture of islets of Langerhans

Abstract

The islets of Langerhans are critical endocrine micro-organs that secrete hormones regulating energy metabolism in animals. Insulin and glucagon, secreted by beta and alpha cells, respectively, are responsible for metabolic switching between fat and glucose utilization. Dysfunction in their secretion and/or counter-regulatory influence leads to diabetes. Debate in the field centers on the cytoarchitecture of islets, as the signaling that governs hormonal secretion depends on structural and functional factors, including electrical connectivity, innervation, vascularization, and physical proximity. Much effort has therefore been devoted to elucidating which architectural features are significant for function and how derangements in these features are correlated or causative for dysfunction, especially using quantitative network science or graph theory characterizations. Here, we ask if there are non-local features in islet cytoarchitecture, going beyond standard network statistics, that are relevant to islet function. An example is ring structures, or cycles, of α and δ cells surrounding β cell clusters or the opposite, β cells surrounding α and δ cells. These could appear in two-dimensional islet section images if a sphere consisting of one cell type surrounds a cluster of another cell type. To address these issues, we developed two independent computational approaches, geometric and topological, for such characterizations. For the latter, we introduce an application of topological data analysis to determine locations of topological features that are biologically significant. We show that both approaches, applied to a large collection of islet sections, are in complete agreement in the context both of developmental and diabetes-related changes in islet characteristics. The topological approach can be applied to three-dimensional imaging data for islets as well.

Fig 1. Examples of geometric and PH-cycles.

(A) αδ-cycle around a NS β-component. (B) β-cycle around a NS αδ-component.

Fig 2. Comparing cell composition across developmental stages.

(A) KDE plots of the distribution of islets characterized by the total number of cells and β-cell fraction from stage 0 to stage 3. The number between every pair of plots shows the KL-divergence between respective KDEs. The KL-divergence between stages 0 and 1 and between 2 and 3 are at most 0.07 as compared to at least 0.27 for every other pairwise comparison. The mode of the density estimate is marked by a red star in each plot. Peaks are at (0.38, 3.27), (0.49, 2.95), (0.66, 3.27), and (0.75, 3.6) for stages 0 to 3. A higher proportion of islets in the last two stages have a higher β-cell fraction. (B) KDE plots of distribution of islets characterized by number of α and δ-cells. In the later developmental stages (2 and 3) there is a large proportion of islets with more α-cells than δ-cells (bright regions in the KDE under the y = x white dashed line).

Fig 3. Changes in cycle around core formation during development correlate with changes in the topology of islets.

(A) Percentages of islets with at least one NS β-component in a αδ-cycle (blue) and those with at least one NS αδ-component in a β-cycle (red). The trends of the two are distinct. (B) Percentiles of distributions of maximum persistences of αδ-cells (blue) and β-cells (red) in islets. Trends in (B) correlate with those in (A) from stage 1 onwards.

Fig 4. Comparing KDEs of distributions of islets from developmental data set with (A) at least one NS β-comp inside an αδ-cycle and (B) at least one NS αδ-comp inside an β-cycle.

Stages 0 to 3 in clockwise from the left-most plot for stage 0. Peaks in (A) are at (0.31, 4.95), (0.26, 4.31), (0.27, 4.63), and (0.29, 5.14) and in (B) are at (0.47, 5.53), (0.54, 4.95), (0.72, 4.63), and (0.66, 4.69), for stages 0 to 3.

Fig 5. Comparing features of islets between control and diabetic subjects.

(A) KDEs for control (left column) and diabetic (right column) subjects for all islets (top row), islets with at least one NS β-component in an αδ-cycle (middle row), and islets with at least one NS αδ-component in a β-cycle (bottom row). Numbers show the KL-divergence. (B) The percentage of islets that have at least one cycle around a NS component is lower in diabetic subjects as compared to non-diabetic subjects. Percentiles of maximum dimension-1 persistence of islets also are lower for diabetic subjects.

Fig 6. Analysis of cycles with respect to islet periphery and center.

(A) KDEs for a minimal distance of cycles from islet periphery vs. islet center. The majority of the cycles are below the y = x line (white dashed) showing that their minimal distance from the periphery is less than that from the islet center. (B) Minimal distances of cycles from islet-periphery vs. islet’s estimated area. αδ-cycles in small islets touch the periphery and very few cycles contain the islet center inside them. There are cycles in larger islets that are far from the periphery.

Fig 7. Examples of closed polyhedral structures found by PH in 3D islets.

(A) β-cycles that contain αδ-cells inside them in a human islet. β-cells are in green and αδ-cells are in red. (B) αδ-cycles that contain β-cells inside them in a human islet. (C) β-cycles thet contain αδ-cells inside them in a mouse islet.