Shannon's entropy and fractal dimension provide an objective account of bone tissue organization during calvarial bone regeneration

Abstract

The regeneration of compact bone involves the deposition of a poorly organized connective tissue template that remodels into compact lamellar bone. An objective description of this process is difficult because classical histomorphometry is unable to correctly characterize qualitative changes in tissue complexity. In this study, we demonstrated the use of two distinct methods of image texture analysis, the Shannon's entropy [standard error (SE)], and the fractal dimension (FD) to characterize the formation and remodeling of newly formed compact bone within two different polyanionic collagen-elastin matrices. The matrices were implanted in defects created into parietal bones of rats. The SE and FD were calculated for histological images of the experimental groups collected 3, 7, 15, 30, 60, and 365 days postsurgery and for the original bone only at day 365. Results showed that the SE and the FD initially increased and then diminished for all groups from day 3 to day 365 approaching the values of the original bone. These results are consistent with poor tissue organization during early osteogenesis that remodels into an organized lamellar structure, showing that these methods can be valuable tools to describe bone tissue remodeling during the regeneration process of compact bones.

Assessment of the protective effects of oral tocotrienols in arginine chronic-like pancreatitis

“…Patterns of nonlinear scaling rules define fractals, which can be measured in terms of complexity and homogeneity. Fractal geometry has been used to quantitate biological processes, such as tumor boundaries, vascular architecture, angiogenesis (8,10,23,24), bone tissue organization during regeneration (39), and liver fibrosis in rats and humans (5,12,25). Fractal analysis of collagen images provided quantitative measurements for the morphological observations.…”

Fractal dimension and Shannon's entropy analyses of the architectural complexity caused by the inflammatory reactions induced by highly crystalline poly(vinyl alcohol) microspheres implanted in subcutaneous tissues of the Wistar rats

“…Fractal Dimension (FD), as defined by the fractal geometry theory, is an alternative and attractive tool to solving impossibilities presented by classical Euclidian geometry in describing irregular and discontinuous shapes observed in the nature, specially problems involving structural complexities and irregularities in biological systems 7, 8. Representative examples of FD analysis application can be found in different areas such as ophthalmology,9 molecular biology, neuropathology, physiological modeling,10, 11 dentistry,12 neoplastic tissues analyses,13–16 bone regeneration evaluation,17, 18 cellular rejection in postcardiac transplantations,19 and structural complexity analyses of blasts and promyelocytes present in myelodysplastic syndrome 20…”

Symmetry in the Language of Gene Expression: A Survey of Gene Promoter Networks in Multiple Bacterial Species and Non-σ Regulons

“…Many cellular interaction networks also exhibit scale-free architectures including protein-protein interaction and metabolic networks [18,[20][21][22][23]. There is fractal structure found at the level of tissue organization [24], as well as whole-organism physiology [25] and development [26], and there are patterns of allometric scaling that explain biological pattern ranging from biochemistry to ecosystem organization [27][28][29].…”