A memory-efficient computer procedure to estimate the fractal dimension of trabecular bone

The fractal dimension is a measure of the space filling characteristics of a particular object. In this paper we describe a method to quantify the fractal dimension (D) of three-dimensionally reconstructed samples from serial sections based on the two-dimensional mass-radius relation of the object embedded in the sections. The advantage of performing the analysis in this way is that it requires little computer memory to hold the data in comparison to the procedure that calculates the mass-radius dimension from the three-dimensional data array. In this case, the test datum is a human first lumbar vertebra embedded in black resin, then horizontally sectioned 256 times and each section photographed using a digital imaging system. The captured images were converted to binary images and then analysed using the 2-D and 3-D mass-radius relation to determine the fractal dimension (D) of the trabecular bone. D was 3.01 using the cubic structuring element and 2.92 using the spherical structuring element. The method may be used to quantify in an objective manner the complex structure of trabecular bone, to model and design new materials (bone substitutes), to understand the physical properties of bone and model the patterns of radiographic images.