Statistical geometry of pancreatic islets

Abstract

Quantitative histomorphometric studies of the dynamics of growth and development of pancreatic islets in normal and pathological states pose substantial methodological and conceptual problems. We address these problems with the geometry of random fractals, and apply our methods to the analysis of islet regeneration in the alloxan-treated guinea-pig. In both experimental islet-regenerated and control animals, islet centres are found to cluster in similar fractal subsets of dimension strictly less than 3, in agreement with the postulated origin of islets along a system of ductules, and suggesting that regeneration follows the same mathematical dynamics as original islet formation.

Book reviews

“…Fractal analysis can be easily automated, and quantitates patterns in terms of fractal exponents and their standard deviations, in contrast with more intensive and qualitative "conventional methods." The reviewer and others (Hastings et al, 1992) had previously applied fractal analysis to quantify the apparently random distribution of pancreatic islets.…”

“…(Mandelbrot, 1982) and in retinal neurons (Caserta et al, 1990), as well as the distribution of pancreatic islets (Hastings et al, 1992) and the structure of aortic valve leaflets (Peskin, 1994). More recently, Weibel described the role of fractal geometry as a "design principle for living organisms" (Weibel, 1991).…”

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

“…The numerical experimental data were fitted to the equation of linear regression log y(x) ‫ס‬ log a + b log x, in which y is a number of gland-like structures within a circle of the grid with a given radius x, a is a scaling coefficient, and b is fractal dimension using software Sigma Plot version 4.0 (SPSS Inc. USA). Arbitrarily, a square root of 2 was applied as the base of all logarithms [15]. Then, the mean value of the parameter b for the 100 samples in each statistical group was calculated.…”

Abrupt fragmentation thresholds of eight zonal forest types in mainland Spain

“…The relationships between event size and frequency can be described by a power function with an exponent that varies in a small interval for the phenomenon in question. This method was tested for the first time by Korcak to describe the Aegean Islands (Imre and Novotný, 2016;Korcak, 1938), and has since then been applied to other problems, including such diverse fields as the magnitude of earthquakes (Lai, 2000), porosity of soil in forests (Menéndez et al, 2005) or pancreatic islets (Hastings et al, 1992). Hastings et al (1982) made a pioneering application for finding changes in vitality of forest masses in the Okefenokee Swamp (Georgia-Florida, USA), and since then, the use of hyperbolic exponents has been generalized to measure fragmentation (Peralta and Mather, 2000;Scanlon et al, 2007) or degradation (Kéfi et al, 2007) in vegetation cover.…”