Collaborative Research: Novel mathematical methods for retrieving mechanical properties and microstructural information of cancellous bones

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The goal of this research is to provide a mathematical background for understanding the use of ultrasound methodology for osteoporosis diagnosis and for the investigations of the dynamics of osteoporosis. For low frequency range ( 100 kHz), the proposed methods are based on homogenization theory. A typical scale for trabecular bone spacing is between 0.5 to 2mm with trabecular thicknesses of 5 to 150 mm. In this range, the wavelength is longer than 15 mm; hence, homogenization theory can be applied. Modified Biot equations (i.e. no ad hoc effective parameters), effective equations for cancellous bone with pore space filled with blood-marrow modeled by the Careau law, and those derived from first principles without the restrictive assumption of periodic microstructure are considered. Two different mathematical approaches are applied in order to go beyond the periodic microstructure assumption: one by a variant of Tartar's method of oscillating test functions, the other by stochastic-two-scale homogenization. These effective equations are to be incorporated into our existing numerical solver for comparison with experimental data. A novel mathematical method is proposed for the inverse problem of retrieving mechanical properties relevant to the strength of bone from ultrasound measurement. This is an ill-posed, nonlinear inverse problem. This novel method is based on an iteration scheme utilizing dehomogenization, which is a linear inverse problem. This is made possible by quantifying the microstructure through relating it to moments of the spectral measure of the associated operator, rather than using directly the characteristic function of the microstructure. A method for estimating porosity distribution from phase velocity by utilizing its frequency dependence is also developed.This project is motivated by the challenges in detecting osteoporosis, a major public health threat affecting more than 44 million Americans. Both treatment and prevention of the disease rely on the best assessment possible of the condition of the patient's bones. Currently, the diagnosis is done using X-ray images and/or bone mineral density test by X-ray type of device. However, as is well known in the field of biomechanics, this approach does not always correctly predict the likelihood of bone fracture. Using a detailed CT image of microstructure, various effective parameters can be computed using numerical method but this is very expensive. The method developed will quantify the mathematical link between strength and microstructural information (statistics of the microstructure), rather than the detailed image. This in turn will lead to the discovery of a way of calculating the strength of bones without detailed image of microstructure. This microstructural information will be obtained by utilizing the frequency dependence of the effective mechanical properties of bone, which can be measured by an ultrasound device. The gains are manifold. First, this approach reduces the patients' exposure to radiation. Secondly, ultrasound is much cheaper and portable than an X-ray type device. Thirdly, this microstructural information can be used for quantifying the area of high concentration of strain, which is another indicator for mechanical failure such as bone fracture.