Mathematical modeling of human secondary osteons.

This investigation explores the structural dimensions and patterns within single secondary osteons, with consideration of their biological variation. New data from images obtained previously of osteons observed through linearly polarized light, electron microscopy, and micro-x-ray, combined with recent findings on lamellae by circularly polarized light, confocal microscopy, synchrotron x-ray diffraction, and micro-x-ray, provide the basis for novel computerized models of single osteons and single lamellae. The novelty of such models is the concurrent representation of (1) collagen-hydroxyapatite orientation, (2) relative hydroxyapatite percentage, (3) distributions of osteocytes' lacunae and canaliculae, and (4) biological variations in dimensions of the relevant structures. The mathematical software Maple realizes the computerized models. While the parts of the models are constructed on a personal computer, the voluminous data associated with the representation of lacunar and canalicular distributions require a supercomputer for assembly of the models and final analysis. The programming used to define the models affords the option to randomize the dimensional specifications of osteons, lamellae, lacunae, and canaliculae within the experimentally observed numeric ranges and distributions. Through this option, the program can operate so that each run of the file produces a unique random model within the observed biological variations. The program can also be run to implement specific dimensional requirements. The modeling has applications in the microstructural study of fracture propagation and remodeling, as well as in the simulation of mechanical testing. The approach taken here is of wide application and could be of value in other areas of microscopy such as scanning electron microscopy, microcomputerized tomography scan, and magnetic resonance imaging on cancellous bone structures.