| We propose several mathematical models describing effective acoustic properties of cancellous bone. Cancellous bone is a highly porous, spongy inner layer of bone consisting of two components: a calcified bone (forming a matrix) and a fatty bone marrow (fluid filling the pores). To understand the changes in bone structure due to osteoporosis and to make use of ultrasound methodology in early detection of osteoporosis, there is a need for better mathematical models describing acoustic properties of cancellous bone.;While most of the existing models of cancellous bone are based on the Biot theory, we propose homogenization approach. This method is more rigorous and provides better mathematical basis for understanding acoustic behavior of cancellous bone.;In this work, we first propose models where the microstructure contains a periodic arrangement of fluid-saturated pores inside the solid matrix. Due to the small size of the pores, there are two significantly different characteristic length scales: the macroscopic scale determined by the overall sample size and the microscopic scale defined by the size of the periodicity cell. We extract the macroscopic scale behavior of the system in the limit as the ratio of the length scales, epsilon, converges to zero. To pass to the limit in the equations governing wave propagation, we make use of the two-scale convergence (and other weak convergence methods).;We first develop a linear theory for the time harmonic excitation of cancellous bone, considering two physical cases: the monophasic case (when fluid and solid move in phase), and the diphasic case (when fluid and solid move out of phase). Next, we extend our study of the acoustic behavior of cancellous bone by considering non-linear constitutive equations for the fluid phase. Modeling bone marrow as non-Newtonian shear thinning fluid better reflects the real situation. Moreover, this problem is mathematically more complex and more interesting than the linear models. Finally, in the last chapter, we consider a random non-periodic arrangement of pores and employ the stochastic two-scale convergence in the mean to obtain effective equations. |
