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

合作研究：检索松质骨机械性能和微观结构信息的新数学方法

• 批准号: 0920852
• 负责人: Christian Halloy
• 金额: $ 10.12万
• 依托单位: University of Tennessee Knoxville
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-15 至 2013-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0920852&HistoricalAwards=false
• 关键词: Collaborative; Research; Novel; mathematical; methods

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.

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诊断骨质疏松症和调查骨质疏松症的动力学提供一个数学背景。对于低频段(100 KHz)，所提出的方法是基于均匀化理论的。典型的骨小梁间距为0.5-2 mm，骨小梁厚度为5-150 mm。在此范围内，波长大于15 mm，因此可以应用均匀化理论。考虑了修正的Biot方程(即没有特殊的有效参数)，按Careau定律模拟的含有血液的松质骨的有效方程，以及在没有周期性微结构的限制性假设的情况下从第一原理导出的有效方程。为了超越周期性微结构假设，本文采用了两种不同的数学方法：一种是塔塔尔振荡检验函数方法的变体，另一种是随机双尺度均匀化方法。这些有效的方程将被合并到我们现有的数值求解器中，以便与实验数据进行比较。针对从超声测量中提取与骨强度相关的力学性能这一反问题，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。这种新方法是基于反齐次化的迭代格式，这是一个线性反问题。这是通过将微结构与相关算子的光谱测量的矩相关联来量化微结构来实现的，而不是直接使用微结构的特征函数。还开发了一种利用相速度的频率相关性来估计孔隙率分布的方法。该项目的动机是检测骨质疏松症的挑战，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射线成像和/或通过X射线类型的设备进行骨密度测试来完成的。然而，正如生物力学领域所熟知的，这种方法并不总是正确地预测骨折的可能性。利用显微组织的详细CT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。开发的方法将量化强度和微观结构信息(微观结构的统计信息)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，而不需要详细的微结构图像。这种微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，这可以通过超声波设备测量。收益是多方面的。首先，这种方法减少了患者对辐射的暴露。其次，超声波比X光设备便宜得多，也便于携带。第三，这种微观结构信息可以用来量化应变高度集中的区域，这是骨折等机械破坏的另一个指标。