A probabilistic approach to bone fracture analysis a)
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I. INTRODUCTION
Bones are made of biological structural material designed to bear certain load depending on the function and ontogeny. Unlike man-made structures, the bones are made of dynamically adaptable tissues that continuously undergo subtle remodeling to meet the changing external or internal constraints both in terms of load and function. Load carrying capacity and strength of bones, therefore, vary considerably from animal to animal and from bone to bone in the same animal.1 Additionally, the bone strength depends on the stage of evolution of the animal and the utility of the bone in the animal's normal course of daily life. Study of the fracture strength of bones has received considerable attention from biologists, mechanical engineers, and orthopedic surgeons, because of the frequent occurrence of bone fracture.2"5 Most bone fractures occur during automobile accidents and sports, particularly in children. Also, because of old age and disease, they lose their strength and become fragile. Orthopedic surgeons have a special interest in the study of the fracture strength of bones for establishing specifications for selection and use of compatible surgical implants. The understanding of bone structure and its strength variation requires, therefore, an interdisciplinary approach.6
HAVERSIAN CANAL
OSTEOCYTES
OSTEONS
II. BONE MICROSTRUCTURE
Determination of a critical fracture stress for bones is complicated because of nonuniformity in size, shape, and microstructure. Figure 1 shows schematically the microstructure of a typical bone. Bones can be considered as a complex natural composite material with a
FIG. 1. Schematic illustration of microstructure of a typical bone.
"'Presented at the MRS symposium, November 1988.
nonuniform distribution of non-load bearing, but functionally needed, constituents such as Haversian and
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© 1991 Materials Research Society
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J. Mater. Res., Vol. 6, No. 1, Jan 1991 Downloaded: 17 Mar 2015
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R. Sadananda: A probabilistic approach to bone fracture analysis
Volkmann canals. In addition there is also a distribution of load bearing constituents7 such as osteons and interstitial lamellae that are bonded with somewhat weaker material. Even the osteons and the interstitial lamella are microstructurally inhomogeneous. They are made up of fibers of organic constituent, collagen, and inorganic constituent, small crystallites of hydroxyapatite epitaxially formed on collagen fibers. These complex microstructural variations result in statistically large variations in strength. Estimation of fracture strength in terms of statistical averages has very little meaning when the standard deviation in the measurement is nearly of the same magnitude as the average value. In this report a different approach involving probabilistic fracture analysis is presented to quantify the fracture strength using Weibull statistics.8'9 III. WEIBULL STATISTICS
Use of Weibull statistics for the analysis of fracture is not new, since materials sc
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