The classic liver lobule – most clearly visualized in the normal porcine liver, in which dense and well-delineated interlobular fibrous septa linking portal tracts establish an easily identifiable boundary to each classic lobule – has traditionally been characterized and illustrated as uniform hexagonal structures, with portal tracts positioned in each of the six vertices and a central vein located in its center (Fig. 1). The human liver is generally thought to be organized in a similar fashion, although the delineation of the classic lobule cannot be properly visualized on routines stains. Functionally, in both human and non-human mammalian species, and based on the microanatomy of the classic liver lobule, hepatocytes are subdivided into three distinct zones (Rappaport lobule, Fig. 1). In spite of the typical lobular representation as juxtaposed regular hexagons, histologic examination of porcine liver sections (or human liver sections stained with zone-specific immunohistochemical markers) reveals a decidedly more complex picture: liver lobules of variable sizes and shapes - commonly but not always roughly hexagonal – with a non-uniform number of portal tracts surrounding each central vein. This relatively high degree of variability also extends to the central veins themselves, which are often somewhat eccentrically located within the liver lobules, show a seemingly haphazard orientation, and frequently bifurcate or trifurcate.
Although the tri-dimensional structure of the liver lobule is fairly complex and proper orientation of portal tracts and central veins is not practically feasible histologically, evaluation of sections with the aid of zone-specific immunohistochemical markers and utilization of digital tools enabled us to hypothesize a novel concept – whereby the two dimensional classic lobular architecture of the liver is neither random nor uniform; rather, it is generally organized following the mathematical/geometric principles of Voronoid diagram and Delaunay triangulation.
Some of the basic concepts related to Voronoi diagrams were investigated as early as 1644 by the French philosopher and mathematician René Descartes, as a method of describing the distribution of matter throughout the solar system and the universe (Supplemental Fig. 1). This method was later applied by the German mathematician Johan Dirichlet in 1850 to the study of quadratic forms. Voronoi diagrams (also known as Voronoi tessellation, Voronoid partition, or Dirichlet tessellation) were named after the Russian mathematician Georgy Voronoy, who expanded Dirichlet’s formalized concepts of diagrams in the two- and three-dimensional cases to the n-dimensional case. A Voronoi diagram is defined as the partition of a plane with n generating seeds (also referred to as “sites”, “points”, or “generators”) into convex polygons (known as “regions”, or “cells”), in which each polygon has exactly one seed and every specific location within a given polygon is nearest to its generating seed than to any other seed.20–22
Voronoi patterns are quite ubiquitous in nature – as exemplified by the crystalloid structure of some minerals, the puzzle-like pattern of a giraffe’s fur, the delicate ridges on a dragonfly’s wings, and tortoise shell plates (Supplemental Fig. 5). This pattern arises in many cases due to expansion (or growth in case of biological tissue) from the “seed”/originating point of the Voronoi region outwards (Supplemental Fig. 6). Gomez-Galvez et al.13 have proposed that a three-dimensional geometrical shape named “scutoid” (resembling the scutellum of a beetle) represents the optimal configuration for energy efficiency and three-dimensional packing of epithelial cells. This unique geometrical shape was predicted by Voronoid tessellation models and subsequently verified in various types of epithelium by the same group. Voronoi diagrams have also been utilized to study the density and spatial distribution of neurons,14 to design 3D scaffolds for bone tissue bioengineering,15 and to explain tissue self-organization and cytomorphology,16 and to model human tumor tissue growth.17
Our data demonstrated that the overall two-dimensional lobular architecture of the liver in both pigs and humans is characterized by a pattern that closely approximates that of a Voronoi diagram, with central veins representing originating sites. This knowledge seems particularly helpful since, in humans, neither the classic lobule nor the hepatic acinus is specifically demarcated histologically, and even histochemical/ immunohistochemical zonal markers are not practically useful with respect to precise lobular delineation. Although Voronoi (and Voronoi-like) diagrams can be constructed using slightly different methods in this context, all approaches utilized by our group were able to describe the known classic lobular architecture of porcine livers with an accuracy greater than 85%, as assessed by digital image analysis and, in humans, voronoi diagrams were able to place GS-positive areas in zones 3 and portal tracts in zones 1 with an overall accuracy ranging from 85% to nearly 90%. Therefore, our data strongly indicates that the typical characterization of the two-dimensional lobular architecture of the liver as juxtaposed regular hexagons is inaccurate or, at best, oversimplified. Rather, the hepatic lobular organization is best described as a “Voronoi pattern” in which polygons with 5–7 sides predominate (Fig. 8).
In pigs, our model showed pentagons to actually represent the most common shape of Voronoi polygons modeling classic lobules (40%), followed by hexagons (32.5%), with number of sides varying from 3 to 8. These observations are essentially in agreement with early meticulous descriptions by E. G. White in 193923, who also noted pentagons (47%) and hexagons (37%) to be the most common shapes among 650 adult porcine lobules, with number of sides varying from 3 to 8. Our model also showed a mean lobular area of 0.69 mm2 in pigs. While the area of lobules was not calculated in classic studies, the works of White,23 Johnson,24 and Mall5 mention average diameters of 1.5 mm, 1.2 mm, and 1.2 mm, respectively (compared to 1.2 mm in our model).
In humans, information regarding the shape and size of human classic lobules is fairly scarce in literature. The radius of the human lobules has been recently measured at 491 µm (diameter of approximately 1 mm) based on portal tract-central vein distances by Hall et al.,25 which is in keeping with the previously stated lobular diameters ranging from 1.0 to 1.3 mm.26–28 Based on these numbers, using an circumcircle radius of 0.491 to 0.65 mm for a regular heptagon (considering this as a prototypical human lobule), the resulting polygon area would be 0.65 to 1.15 mm2 (or 0.89 mm2 using the average [0.57 mm] of these previously reported values), which is the exact average cross-sectional area obtained by our model. Using alkaline phosphatase histochemical stain, Teutsch29 reported human liver lobules to be polyhedral, with seven to nine facets. In our model, the most common shapes were heptagons (34%) and hexagons (32%), with number of polygon sides varying from 3 to 8. In addition, as predicted by Voronoi tessellation, the non-equidistant central veins in sections of human livers are often eccentrically placed in their respective polygons (lobules) rather than always being at or around their center, as would be the case for a honeycomb pattern formed exclusively by regular hexagons.
Aside from representing a more accurate descriptive model of the classic lobular architecture of the liver, the geometric properties proposed in this study also have implications to other liver unit models. Regarding liver zonation, for instance, the precise boundaries between the different zones of the Rappaport lobule can be established mathematically (or computationally) based on the location of central veins – especially if aided by central zone-specific immunostain or immunofluorescence markers. Currently, zonation of liver tissue cannot reliably be established by any method, especially in areas away from the immediate vicinity of central veins and portal tracts. As part of this study, and based on Voronoi diagrams, we have designed a digital image analysis algorithm that was able to delineate the borders of classic lobules in both pigs and humans as well as subdivide each lobule into different zones, in a fashion analogous to the Rappaport acinar model. We were also able to test the performance of our algorithm in human livers given the known location of zones 3 (GS-positive centrilobular areas) and positions of portal tracts (within zone 1). Using these structures as zonal landmarks, our best model had an accuracy of nearly 90%. Hence, this method – or future refinements thereof – could be used to more accurately and objectively study not only the size and shape of liver lobules in normal conditions (and how these may change in different diseases) but also the zonality of normal phenomena in hepatic physiology and zonal involvement by common pathologic processes such as steatosis, inflammation, necrosis, and liver fibrosis. In addition, using a computational geometry algorithmic approach, the delineation and definition of lobular zones (i.e., size and configuration of each zone) can be customized to specific needs, according to which physiologic or pathologic process is being studied.
Finally, an important implication derived from the Voronoid organization of the classic liver lobules relates to its embryology. Although the specific dynamics of hepatocyte tissue growth during embryological and post-natal development is highly complex – and beyond the scope of this study - the Voronoid organization of the classic liver lobules would indicate that cellular growth of primordial hepatocytes starts at the vicinity of the central vein and proceeds outwards radially, with portal tracts (and fibrous septa in some species) forming along the expanding edge of the nascent lobules – and eventually settling along the border of two or three classic lobular units.
In summary, our work describes the relationship between Voronoi diagrams and the liver microarchitecture in both pigs and humans. We have presented histologic and mathematical evidence that Voronoi diagrams accurately describe the basic two-dimensional organization of the normal liver. This method seems especially relevant to the study human livers, since reliable histologic landmarks of lobular boundaries are absent. We have also designed an algorithm based on Voronoi diagrams that enabled us to delineate boundaries of zones 1–3 within the classic lobules in humans, hence representing a method that would allow for more precise quantitative analysis of both physiologic and pathologic zonal processes, regardless of how, exactly, the hepatic zones are defined. Therefore, in addition to a better understanding of the liver microstructure itself, the utilization of this mathematical/computational tool opens numerous possibilities of relevant applications in the study of the normal liver and liver diseases, especially in view of the increasing utilization of digital pathology and artificial intelligence-assisted histologic evaluation.