Fractal dimension and architecture of trabecular bone

Abstract

The fractal dimension of trabecular bone was determined for biopsies from the proximal femur of 25 subjects undergoing hip arthroplasty. The average age was 67·7 years. A binary profile of the trabecular bone in the biopsy was obtained from a digitized image. A program written for the Quantimet 520 performed the fractal analysis. The fractal dimension was calculated for each specimen, using boxes whose sides ranged from 65 to 1000 μm in length. The mean fractal dimension for the 25 subjects was 1·195+0·064 and shows that in Euclidean terms the surface extent of trabecular bone is indeterminate. The Quantimet 520 was also used to perform bone histomorphometric measurements. These were bone volume/total volume (BV/TV) (per cent)=11·05+4·38, bone surface/total volume (BS/TV) (mm2/mm3)=1·90+0·51, trabecular thickness (Tb.Th) (mm)=0·12+0·03, trabecular spacing (Tb.Sp) (mm)=1·03+0·36, and trabecular number (Tb.N) (number/mm)=0·95+0·25. Pearsons' correlation coefficients showed a statistically significant relationship between the fractal dimension and all the histomorphometric parameters, with BV/TV (r=0·85,P0·0001), BS/TV (r=0·74,P0·0001), Tb.Th (r=0·50,P0·02), Tb.Sp (r=−0·81,P0·0001), and Tb.N (r=0·76,P0·0001). This method for calculating fractal dimension shows that trabecular bone exhibits fractal properties over a defined box size, which is within the dimensions of a structural unit for trabecular bone. Therefore, the fractal dimension of trabecular bone provides a measure which does not rely on Euclidean descriptors in order to describe a complex geometry.