MR Imaging and Osteoporosis: Fractal Lacunarity Analysis of Trabecular Bone

Abstract

We develop a method of magnetic resonance (MR) image analysis able to provide parameter(s) sensitive to bone microarchitecture changes in aging, and to osteoporosis onset and progression. The method has been built taking into account fractal properties of many anatomic and physiologic structures. Fractal lacunarity analysis has been used to determine relevant parameter(s) to differentiate among three types of trabecular bone structure (healthy young, healthy perimenopausal, and osteoporotic patients) from lumbar vertebra MR images. In particular, we propose to approximate the lacunarity function by a hyperbola model function that depends on three coefficients, alpha, beta, and gamma, and to compute these coefficients as the solution of a least squares problem. This triplet of coefficients provides a model function that better represents the variation of mass density of pixels in the image considered. Clinical application of this preliminary version of our method suggests that one of the three coefficients, beta, may represent a standard for the evaluation of trabecular bone architecture and a potentially useful parametric index for the early diagnosis of osteoporosis.

Fractal lacunarity of trabecular bone and magnetic resonance imaging: New perspectives for osteoporotic fracture risk assessment

“…A correlation between parameter β and age can also be observed with a decreasing trend of β values from young to old subjects [58] . It has to point out that the healthy old subject (in this context without clinical signs of osteoporosis) shows a β value higher than the younger osteoporotic patients.…”

“…In fact, hyperbola formula contains three coefficients (α , β, γ) that represent our suitable numerical indices, where α correlate with the fractal dimension, and β, related to the concavity, characterizes the lacunarity. The result is a triplet of parameters (α* β * γ *) that univocally characterizes any single TBA analyzed [57,58] . Application of the method to several TBA from 25 female subjects with different age and physiopathologic status (4 young, 5 premenopause, 6 postmenopause, 10 osteoporotic with an agerange of 3137, 4252, 5181, and 5975 respectively) has highlighted that, among the three coefficients, parameter β is particularly sensitive to both age and physiopathologic changes.…”

“…The introduction in biomedical field of paradigms, such as theory of complexity, chaos and fractals, suggests new approaches and provides innovative NewmanKeuls multiple range test) in the three classes of TBA considered (healthy young, perimenopausal, and osteoporotic) [57,58] . A correlation between parameter β and age can also be observed with a decreasing trend of β values from young to old subjects [58] .…”

Histomorphometric, fractal and lacunarity comparative analysis of sheep (Ovis aries), goat (Capra hircus) and roe deer (Capreolus capreolus) compact bone samples

“…Studies regarding the fractal characteristics of the bone tissue were conducted on histological slides [1,14] or on radiographic, computed tomography and magnetic resonance images [32], but apparently not on the manually ground bone tissue samples. This bone processing method was able to maintain the structural architecture for further analysis.…”

On the Relations Between 2D and 3D Fractal Dimensions: Theoretical Approach and Clinical Application in Bone Imaging

“…In clinical practice, the diagnosis of osteoporosis is mainly based on the measurement of bone mineral density. However, the risk of bone fragility results also from the changes in the bone architecture that is independent of bone density [20].…”