Layered patterns in nature, medicine, and materials: quantifying anisotropic structures and cyclicity.

Various natural patterns-such as terrestrial sand dune ripples, lamellae in vertebrate bones, growth increments in fish scales and corals, aortas and lamellar corpuscles in humans and animals-comprise layers of different thicknesses and lengths. Microstructures in manmade materials-such as alloys, perlite steels, polymers, ceramics, and ripples induced by laser on the surface of graphen-also exhibit layered structures. These layered patterns form a record of internal and external factors regulating pattern formation in their various systems, making it potentially possible to recognize and identify in their incremental sequences trends, periodicities, and events in the formation history of these systems. The morphology of layered systems plays a vital role in developing new materials and in biomimetic research. The structures and sizes of these two-dimensional (2D) patterns are characteristically anisotropic: That is, the number of layers and their absolute thicknesses vary significantly in different directions. The present work develops a method to quantify the morphological characteristics of 2D layered patterns that accounts for anisotropy in the object of study. To reach this goal, we use Boolean functions and an N-partite graph to formalize layer structure and thickness across a 2D plane and to construct charts of (1) "layer thickness vs. layer number" and (2) "layer area vs. layer number." We present a parameter disorder of layer structure (DStr) to describe the deviation of a study object's anisotropic structure from an isotropic analog and illustrate that charts and DStr could be used as local and global morphological characteristics describing various layered systems such as images of, for example, geological, atmospheric, medical, materials, forensic, plants, and animals. Suggested future experiments could lead to new insights into layered pattern formation.

Enamel-calibrated lamellar bone reveals long period growth rate variability in humans.

Mammalian teeth exhibit incremental structures representing successive forming fronts of enamel at varying time scales, including a short daily increment called a cross striation and a long period called a stria of Retzius, the latter of which, in humans, occurs on average every 8-9 days. The number of daily increments between striae is called the repeat interval, which is the same period as that required to form one increment of bone, i.e. the lamella, the fundamental - if not archetypal - unit of bone. Lamellae of known formation time nevertheless vary in width, and thus their measures provide time-calibrated growth rate variability. We measured growth rate variability for as many as 6 years of continuously forming primary incremental lamellar bone from midshaft femur histological sections of sub-Saharan Africans of Bantu origin and known life history. We observed periodic growth rate variability in approximately 6- to 8-week intervals, and in some cases annual rhythms were visible. Endogenous biological periodicities, cycles manifest in the external environment, and/or perturbations of development are all potentially contained within growth rate variability studies of lamellar incremental patterns. Because lamellae are formed within defined periods of time, quantitative measures of widths of individual lamellae provide time-resolved growth rate variability that may reveal rhythms in human bone growth heretofore unknown.

Lamellar bone is an incremental tissue reconciling enamel rhythms, body size, and organismal life history.

Mammalian enamel formation is periodic, including fluctuations attributable to the daily biological clock as well as longer-period oscillations that enigmatically correlate with body mass. Because the scaling of bone mass to body mass is an axiom of vertebrate hard tissue biology, we consider that long-period enamel formation rhythms may reflect corresponding and heretofore unrecognized rhythms in bone growth. The principal aim of this study is to seek a rhythm in bone growth demonstrably related to enamel oscillatory development. Our analytical approach is based in morphology, using a variety of hard tissue microscopy techniques. We first ascertain the relationship among long-period enamel rhythms, the striae of Retzius, and body mass using a large sample of mammalian taxa. In addition, we test whether osteocyte lacuna density (a surrogate for rates of cell proliferation) in bone is correlated with mammalian body mass. Finally, using fluorescently labeled developing bone tissues, we investigate whether the bone lamella, a fundamental microanatomical unit of bone, relates to rhythmic enamel growth increments. Our results confirm a positive correlation between long-period enamel rhythms and body mass and a negative correlation between osteocyte density and body mass. We also confirm that lamellar bone is an incremental tissue, one lamella formed in the species-specific time dependency of striae of Retzius formation. We conclude by contextualizing our morphological research with a current understanding of autonomic regulatory control of the skeleton and body mass, suggesting a central contribution to the coordination of organismal life history and body mass.