Fractal geometry and vertebral compression fractures


  • Robert S. Weinstein M.D.,

    Corresponding author
    1. Metabolic Bone Disease Laboratory, Department of Medicine, University of Arkansas for Medical Sciences, Little Rock, Arkansas
    • Medical College of Georgia HB-5027 Augusta, GA 30912
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  • Sharmila Majumdar

    1. Department of Radiology, University of California, San Francisco, California
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Cancellous bone in postmenopausal osteoporosis is characterized by widely spaced and disconnected trabeculae. These architectural changes may disproportionately increase bone fragility compared with the decrease in bone mass alone. To determine whether there is an independent architectural contribution to fracture risk, we applied fractal geometry to the cancellous bone in osteoporosis. Fractal objects have recurrent, branching patterns that are quantified by a fractal dimension D, that describes how the object fills space. Photomicrographs of transiliac bone biopsy specimens were digitized, and D was calculated from the negative of the straight portion of the slope of the log of the number of pixels containing boundary points versus the log of the pixel size over a range of pixel sizes. The results were compared with the cancellous bone histomorphometry in 31 individuals, aged 19–80 years, who were healthy before sudden death. D was inversely related to age (r = −0.72, p < 0.0001) and trabecular spacing (r = −0.87; p < 0.0001) and directly related to bone area (r = 0.86, p < 0.0001) and trabecular number (r = 0.83, p < 0.0001). We then examined 12 healthy volunteers and 12 patients with vertebral compression fractures as a result of osteoporosis who were matched for age and cancellous bone area. The patients with fractures had a 7.9% lower mean value for D (1.224 ± 0.085 SD versus 1.329 ± 0.125, p < 0.026). Other markers of the cancellous architecture, such as trabecular width, separation, and number, were not different between the matched groups. D was more than 1 SD below the normal mean value in 50% of the fracture cases and in only 1 of the nonfracture cases (p < 0.025). Applied to cancellous bone, fractal geometry may discriminate more effectively between fracture and nonfracture cases than measurement of bone area and help accent the impact of architecture on bone fragility. The single expression, D, greatly simplifies description of the spatial distribution of cancellous bone and thus has a potential role in the computer-assisted histologic investigation of osteoporosis.