Measures of complexity for cancellous bone

Abstract

The problem of quantifying the structure of cancellous bone has been addressed in the past by histomorphometry and more recently by imaging techniques using X-ray attenuation. The current approaches compute and describe parts of the construction of the trabecular net. We developed a new technique which quantifies cancellous bone of human lumbar vertebrae as a whole. The interactions, transactions, and interrelationships of all parts of the structural composition of the trabeculae are accounted for and quantified. The method is based on the concept of structural complexity within the framework of nonlinear dynamics. The methodology was developed by using axial high resolution computed tomography images. The technique was transferred to quantitative computed tomography images and is based on the non-invasive assessment of 50 human L3 specimens. The value of Houndsfield units per pixel representing trabecular bone of the vertebrae was transformed into color-encoded and alphabet-encoded symbols. The procedure of transformation of the X-ray attenuation pixels into symbols was necessary as a basis on which measures of complexity were introduced to assess the composition of symbols within the images. The development of a generalization of symbolic dynamics, a mathematical method, to work with two-dimensional images was a prerequisite. The results of this study demonstrate that the structural composition of cancellous bone declines more rapidly than bone mineral density during the loss of bone. This outcome strongly suggests an exponential relationship between bone mineral density and the architectural composition of cancellous bone. Normal trabecular bone has a complex ordered structure. The structural composition during the osteopenic phase of bone loss is characterized by lower structural complexity and a significantly higher level of architectural disorder. A high grade of osteoporosis leads again to an ordered structure, although its structural complexity is minimal.