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

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.