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: 0920850
• 负责人: Robert Gilbert
• 金额: $ 28.71万
• 依托单位: University of Delaware
• 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=0920850&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年的《美国复苏与再投资法》（公法111-5）资助的。该研究的目的是提供数学背景，以理解使用超声波疏松症诊断的超声波化方法以及对骨质疏松症动态的调查。对于低频范围（100 kHz），提出的方法基于均化理论。小梁骨间距的典型尺度在0.5至2mm之间，小梁厚度为5至150 mm。在此范围内，波长超过15毫米；因此，可以应用均质理论。修改后的生物剂方程（即没有临时有效参数），具有毛孔空间的取消骨骼的有效方程，该方程式填充了由Careau Law建模的血液，而这些方程式是从第一原则中得出的，而没有定期微观结构的限制性假设。采用两种不同的数学方法，以超越周期性的微观结构假设：一种是通过牙垢振荡测试函数的一种变体，另一个通过随机-two级均质化。这些有效的方程式应将其纳入我们现有的数值求解器中，以与实验数据进行比较。 提出了一种新型的数学方法，用于检索与超声测量的骨骼强度有关的机械性能的反问题。 这是一个不良的，非线性的反问题。 这种新颖的方法是基于利用非原子化的迭代方案，这是一个线性的反问题。 通过将微观结构与关联运算符的光谱度量的矩相关联，而不是直接使用微结构的特征函数来实现这一可能。 还开发了一种通过利用其频率依赖性来估计孔隙率分布的方法。该项目是由检测骨质疏松症的挑战所激发的，骨质疏松症是一种主要的公共卫生威胁，影响了4400万美国人。治疗和预防疾病都取决于患者骨骼状况的最佳评估。目前，使用X射线矿物质密度测试通过X射线类型的设备进行诊断。但是，正如在生物力学领域所熟知的那样，这种方法并不总是正确地预测骨断裂的可能性。使用微观结构的详细CT图像，可以使用数值方法计算各种有效参数，但这非常昂贵。开发的方法将量化强度和微观结构信息（微结构的统计数据）之间的数学联系，而不是详细的图像。反过来，这将导致发现一种计算骨骼强度的方法，而无需详细的微观结构图像。 该微观结构信息将通过利用骨有效机械性能的频率依赖性获得，可以通过超声设备进行测量。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射线型设备便宜得多，便携。第三，该微观结构信息可用于量化高浓度应变面积，这是机械故障的另一个指标，例如骨断裂。