Mathematical techniques for the assessment of damage and nonlinear behaviour in bone

骨质疏松症是一种严重的健康问题，影响着大约40%的女性和20%的50岁以上的男性。它会导致骨质量的退化，导致骨折风险的严重增加。因此，预测骨质疏松症的发生、发展和随后的骨折风险在医学上具有重要意义。这种预测的标准方法是使用电离x射线密度测量来评估骨矿物质密度（BMD）。线性超声也被用作预测骨密度的替代方法，它具有非电离的优点。骨密度评估被认为是预测骨折风险的最佳临床技术，但骨的其他结构和材料特性在评估整体骨强度bbb时极为重要。特别是由于日常日常活动如步行、抬举等引起的微损伤的积累是非常重要的，特别是考虑到裂纹密度随着年龄的增长呈指数增长。重要的是，x射线密度测量和线性超声评估都对增加的微损伤不敏感，直到材料失效点。最近，实验非线性声学技术已经开始利用现在普遍接受的骨在小应变下的非线性本构行为，以评估骨[2]，[3]的微损伤程度。这项资助的研究将集中在使用数学技术来模拟骨骼的本构行为，并模拟骨骼中的非线性声波传播。特别是非线性均质化和微观力学技术将被开发和应用，以推导骨骼的宏观本构定律，包括弹性非线性和滞后（非弹性或不可逆）效应。在声学背景下，我们将重点关注初始大预应力如何影响随后的波传播。在线性声学系统中，这种预应力影响波[3]的后续飞行时间（波速）。在共振情况下，我们将研究它如何改变骨的共振特性。我们还将通过非线性模态混合[4]来评估骨中的非线性波相互作用。这可能成为评估骨微损伤的有用工具。这项工作将与法国巴黎的两个世界领先的研究机构（巴黎第六大学的影像参数实验室和巴黎第十二大学的力学物理实验室）合作完成。[1]Sambrook， P.和Cooper， C. 2006骨质疏松症，柳叶刀367,2010-2018刘建军，刘建军，刘建军，等。2008非线性超声检测骨损伤的研究进展[j] .中国生物医学工程学报。41 (1):1062-1068雷诺，G.，卡尔，S.，雷梅涅拉斯，j . p .。and Defontaine， M. 2008基于飞行时间调制的骨小梁非线性弹性的探索，IEEE Ultra。铁。频率控制55,1497-1507.[4]Hillis, a.j., Neild, s.a., Drinkwater, b.w.和Wilcox， P.D. 2006，使用双谱分析的全球裂纹检测，Proc. Roy。Soc。462, 1515-1530。

Osteoporosis is a serious health problem which affects approximately 40% of women and 20% of men over the age of 50 [1]. It causes a degradation of bone quality which leads to a severe increase in fracture risk. The prediction of the onset of osteoporosis, its development and subsequent fracture risk is therefore of great importance in medical science. The standard approach for this prediction is to use ionizing x-ray densitometry in order to assess bone mineral density (BMD). Linear ultrasound has also been used as an alternative method by which to predict BMD and this has the advantage of being non-ionizing. BMD assessment is considered to be the best clinical technique for fracture risk prediction but other structural and material properties of bone are extremely important in assessing overall bone strength [2]. In particular the accumulation of microdamage which is induced by constant standard daily activities such as walking, lifting, etc. is of great importance, especially given that crack density increases exponentially with age. Importantly, it has been shown that x-ray densitometry and linear ultrasound assessment are both insensitive to increased microdamage until the point of material failure.Recently experimental nonlinear acoustic techniques have been initiated which exploit the now commonly accepted nonlinear constitutive behaviour of bone at small strain, in order to assess the extent of microdamage in bone [2], [3]. Research in this grant will focus on the use of mathematical techniques in order to model the constitutive behaviour of bone and also to model nonlinear acoustic wave propagation in bone. In particular nonlinear homogenization and micromechanics techniques will be developed and applied in order to derive macroscopic constitutive laws for bone, including elastic nonlinearity and hysteretic (inelastic or irreversible) effects. In the acoustic context we will focus on how an initial large pre-stress affects subsequent wave propagation. In the linear acoustic regime this pre-stress affects the subsequent time of flight (wavespeed) of the wave [3]. In the resonance situation we will investigate how it modifies the resonance properties of bone. We will also assess nonlinear wave interaction in bone via nonlinear mode mixing [4]. This could become a useful tool by which to assess microdamage in bone.This work will be done in collaboration with two world-leading research institutes in Paris, France (Laboratoire d'Imagerie Parametrique at Universite Paris VI and Laboratoire de Mechanique Physique at Universite Paris XII). [1] Sambrook, P. and Cooper, C. 2006 Osteoporosis, Lancet 367, 2010-2018.[2] Muller, M., Mitton, D., Talmant, M., Johnson, P.A. and Laugier, P. 2008 Nonlinear ultrasound can detect accumulated damage in human bone, J. Biomech. 41, 1062-1068.[3] Renaud, G., Calle, S., Remenieras, J-P. and Defontaine, M. 2008 Exploration of trabecular bone nonlinear elasticity using time-of-flight modulation, IEEE Ultra. Ferro. Freq. Control 55, 1497-1507.[4] Hillis, A.J., Neild, S.A., Drinkwater, B.W., and Wilcox, P.D. 2006, Global crack detection using bispectral analysis, Proc. Roy. Soc. A. 462, 1515-1530.