Statistical geometry of pancreatic islets

Fractal nature of human gastrointestinal system: Exploring a new era

The morphological complexity of cells and tissues, whether normal or pathological, is characterized by two primary attributes: Irregularity and self-similarity across different scales. When an object exhibits self-similarity, its shape remains unchanged as the scales of measurement vary because any part of it resembles the whole. On the other hand, the size and geometric characteristics of an irregular object vary as the resolution increases, revealing more intricate details. Despite numerous attempts, a reliable and accurate method for quantifying the morphological features of gastrointestinal organs, tissues, cells, their dynamic changes, and pathological disorders has not yet been established. However, fractal geometry, which studies shapes and patterns that exhibit self-similarity, holds promise in providing a quantitative measure of the irregularly shaped morphologies and their underlying self-similar temporal behaviors. In this context, we explore the fractal nature of the gastrointestinal system and the potential of fractal geometry as a robust descriptor of its complex forms and functions. Additionally, we examine the practical applications of fractal geometry in clinical gastroenterology and hepatology practice.

Fully Automated Islet Cell Counter (ICC) for the Assessment of Islet Mass, Purity, and Size Distribution by Digital Image Analysis

For isolated pancreatic islet cell preparations, it is important to be able to reliably assess their mass and quality, and for clinical applications, it is part of the regulatory requirement. Accurate assessment, however, is difficult because islets are spheroid-like cell aggregates of different sizes (<50 to 500 μm) resulting in possible thousandfold differences between the mass contribution of individual particles. The current standard manual counting method that uses size-based group classification is known to be error prone and operator dependent. Digital image analysis (DIA)-based methods can provide less subjective, more reproducible, and better-documented islet cell mass (IEQ) estimates; however, so far, none has become widely accepted or used. Here we present results obtained using a compact, self-contained islet cell counter (ICC3) that includes both the hardware and software needed for automated islet counting and requires minimal operator training and input; hence, it can be easily adapted at any center and could provide a convenient standardized cGMP-compliant IEQ assessment. Using cross-validated sample counting, we found that for most human islet cell preparations, ICC3 provides islet mass (IEQ) estimates that correlate well with those obtained by trained operators using the current manual SOP method ( r² = 0.78, slope = 1.02). Variability and reproducibility are also improved compared to the manual method, and most of the remaining variability (CV = 8.9%) results from the rearrangement of the islet particles due to movement of the sample between counts. Characterization of the size distribution is also important, and the present digitally collected data allow more detailed analysis and coverage of a wider size range. We found again that for human islet cell preparations, a Weibull distribution function provides good description of the particle size.

The fractal spatial distribution of pancreatic islets in three dimensions: a self-avoiding growth model

The islets of Langerhans, responsible for controlling blood glucose levels, are dispersed within the pancreas. A universal power law governing the fractal spatial distribution of islets in two-dimensional pancreatic sections has been reported. However, the fractal geometry in the actual three-dimensional pancreas volume, and the developmental process that gives rise to such a self-similar structure, has not been investigated. Here, we examined the three-dimensional spatial distribution of islets in intact mouse pancreata using optical projection tomography and found a power law with a fractal dimension of 2.1. Furthermore, based on two-dimensional pancreatic sections of human autopsies, we found that the distribution of human islets also follows a universal power law with a fractal dimension of 1.5 in adult pancreata, which agrees with the value previously reported in smaller mammalian pancreas sections. Finally, we developed a self-avoiding growth model for the development of the islet distribution and found that the fractal nature of the spatial islet distribution may be associated with the self-avoidance in the branching process of vascularization in the pancreas.

Quantitative assessment of islet cell products: estimating the accuracy of the existing protocol and accounting for islet size distribution

The ability to consistently and reliably assess the total number and the size distribution of isolated pancreatic islet cells from a small sample is of crucial relevance for the adequate characterization of islet cell preparations used for research or transplantation purposes. Here, data from a large number of isolations were used to establish a continuous probability density function describing the size distribution of human pancreatic islets. This function was then used to generate a polymeric microsphere mixture with a composition resembling those of isolated islets, which, in turn, was used to quantitatively assess the accuracy, reliability, and operator-dependent variability of the currently utilized manual standard procedure of quantification of islet cell preparation. Furthermore, on the basis of the best fit probability density function, which corresponds to a Weibull distribution, a slightly modified scale of islet equivalent number (IEQ) conversion factors is proposed that incorporates the size distribution of islets and accounts for the decreasing probability of finding larger islets within each size group. Compared to the current calculation method, these factors introduce a 4-8% downward correction of the total IEQ estimate, but they reflect a statistically more accurate contribution of differently sized islets.

Distribution of gland-like structures in human gallbladder adenocarcinomas possesses fractal dimension

BACKGROUND AND OBJECTIVES

Epithelial cells form tissue patterns of higher order such as gland-like structures. A question arises whether distribution of those patterns in adenocarcinomas is subject to certain regularity.

METHODS

Due to the pilot nature of this study, gallbladder adenocarcinomas were preselected by histopathological, immunohistochemical, and morphometric analysis to ensure relative homogeneity of the patterns analyzed. A box-counting method was applied to investigate a relationship between a number of gland-like structures and a radius of the expanding family of the concentric circles.

RESULTS

The coefficient of linear regression characterizing that relationship possesses noninteger value. It is 1. 585 (well-differentiated adenocarcinomas, standard deviation (SD) = 0.038, n = 100 sections), and 1.340 (moderately differentiated adenocarcinomas, SD = 0.044, n = 100 sections). While both nuclear area and nucleo-cytoplasmic ratio in those tissues remain within a similar range (Analysis of Variance (ANOVA), F(0) = 0.791 < F(alpha) = 3.84, P = 3 x 10(-3) and F(0) = 0.077 < F(alpha) = 3.84, P = 10(-6), respectively, for k = 20,000 cells, in which F(0) is a value of the test function, F(alpha) is a critical, limit value of the F-test at the constant confidence value alpha = 0.05), a difference of fractal dimension is significant (F(0) = 3.94 > F(alpha) = 0.693, n = 100 sections, P = 2 x 10(-3)). Also, variablity of fractal dimension between tumor sections is significant (moderately differentiated adenocarcinomas, F(0) = 1.9856 > F(alpha) = 1.4262, n = 100 sections, P = 0.189).

CONCLUSIONS

There is fractal regularity in distribution of gland-like structures in human gallbladder adenocarcinomas. Fractal dimension is a holistic parameter which can be applied to evaluate tumor grading in a quantitative manner.

On the holistic approach in cellular and cancer biology: nonlinearity, complexity, and quasi-determinism of the dynamic cellular network

A keystone of the molecular reductionist approach to cellular biology is a specific deductive strategy relating genotype to phenotype-two distinct categories. This relationship is based on the assumption that the intermediary cellular network of actively transcribed genes and their regulatory elements is deterministic (i.e., a link between expression of a gene and a phenotypic trait can always be identified, and evolution of the network in time is predetermined). However, experimental data suggest that the relationship between genotype and phenotype is nonbijective (i.e., a gene can contribute to the emergence of more than just one phenotypic trait or a phenotypic trait can be determined by expression of several genes). This implies nonlinearity (i.e., lack of the proportional relationship between input and the outcome), complexity (i.e. emergence of the hierarchical network of multiple cross-interacting elements that is sensitive to initial conditions, possesses multiple equilibria, organizes spontaneously into different morphological patterns, and is controlled in dispersed rather than centralized manner), and quasi-determinism (i.e., coexistence of deterministic and nondeterministic events) of the network. Nonlinearity within the space of the cellular molecular events underlies the existence of a fractal structure within a number of metabolic processes, and patterns of tissue growth, which is measured experimentally as a fractal dimension. Because of its complexity, the same phenotype can be associated with a number of alternative sequences of cellular events. Moreover, the primary cause initiating phenotypic evolution of cells such as malignant transformation can be favored probabilistically, but not identified unequivocally. Thermodynamic fluctuations of energy rather than gene mutations, the material traits of the fluctuations alter both the molecular and informational structure of the network. Then, the interplay between deterministic chaos, complexity, self-organization, and natural selection drives formation of malignant phenotype. This concept offers a novel perspective for investigation of tumorigenesis without invalidating current molecular findings. The essay integrates the ideas of the sciences of complexity in a biological context.

The spatial distribution of pancreatic islets follows a universal power law

In a previous report we quantified the apparently complex and irregular distribution of pancreatic islets in the guinea pig by showing that they formed a set of cluster or correlation dimension approximately 2.5. Here we show that this distribution holds in a wide range of mammalian species and through ontogenetic development in the guinea pig. These results strongly suggest that islet formation follows an iterative or fractal rule which is universal among mammals.