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

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。本研究的目的是为理解使用超声方法进行骨质疏松症诊断和骨质疏松症动力学研究提供数学背景。对于低频率范围（100khz），提出的方法是基于均匀化理论。骨小梁间距的典型尺度为0.5至2mm，骨小梁厚度为5至150 mm。在此范围内，波长长于15mm；因此，均质化理论可以应用。本文考虑了修正的Biot方程（即没有特别有效参数）、基于Careau定律的孔内填充骨髓的松质骨有效方程以及不考虑周期微观结构约束假设的第一原理方程。为了超越周期性微观结构假设，应用了两种不同的数学方法：一种是通过Tartar的振荡测试函数方法的变体，另一种是通过随机双尺度均匀化。这些有效方程将并入我们现有的数值解算器中，以便与实验数据进行比较。针对超声测量中与骨强度相关的力学性能反演问题，提出了一种新的数学方法。这是一个不适定的非线性逆问题。该方法基于一种利用去均匀化的迭代方案，是一个线性逆问题。这可以通过将微观结构与相关算子的光谱测量的矩相关联来量化微观结构，而不是直接使用微观结构的特征函数。本文还提出了一种利用相速度的频率依赖性来估计孔隙度分布的方法。骨质疏松症是影响4400多万美国人的主要公共健康威胁。这种疾病的治疗和预防都依赖于对患者骨骼状况的最佳评估。目前，诊断是通过x射线图像和/或x射线设备进行骨矿物质密度测试来完成的。然而，众所周知，在生物力学领域，这种方法并不总是正确地预测骨折的可能性。利用详细的CT显微结构图像，可以用数值方法计算各种有效参数，但这是非常昂贵的。所开发的方法将量化强度和微观结构信息（微观结构统计）之间的数学联系，而不是详细的图像。这反过来将导致发现一种计算骨骼强度的方法，而不需要详细的微观结构图像。这种微观结构信息将通过利用骨的有效机械性能的频率依赖性来获得，这可以通过超声设备来测量。收益是多方面的。首先，这种方法减少了患者的辐射暴露。其次，超声波比x光设备更便宜，更便携。第三，该显微组织信息可用于量化应变高浓度区域，这是骨折等机械失效的另一个指标。

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.