Inverse problems for poroelastic composites with applications in bone health monitoring

University Of Delaware, Newark DE

Investigators

Abstract

This research project is directed towards developing mathematical and computational methods arising from quantitative ultrasound techniques (QUT) for detecting osteoporosis, a major public health threat affecting more than 44 million Americans. Understanding the bio-mechanical advance of the disease and developing quantitative mathematical techniques for its monitoring is the first step towards efficient treatment and prevention. Diagnostics based on QUT is relatively inexpensive and, unlike X-ray densitometry, it does not ionize the tissue. The strength of cancellous bone plays an essential role in the diagnosis of osteoporosis. The proposed work aims at developing mathematical tools for determining various parameters that describe the strength of porous-elastic materials, by probing the materials with mechanical waves such as ultrasound. The educational component emphasizes the interdisciplinary nature of the proposed research in several STEM fields. Through joint work with the PI's collaborators in mechanical and biomedical engineering, facilitated by the Center for Biomedical Engineering Research at the University of Delaware, a new interdisciplinary educational initiative linking biomedical engineering and mathematics for undergraduate and graduate student training is planned. The results from this project will be of interest to researchers who conduct laboratory experiments to study the relation between bone strength and bone micro-architecture and will open an avenue for a more accurate and reliable ultrasound devices for measuring properties of porous-elastic materials. The goal of the research is to utilize the dehomogenization techniques to study the mathematical characterization of effective parameters relevant to mechanical properties of poroelastic materials and the governing equations of wave propagation in these materials. Poroelastic composites are two-phase composite materials consisting of elastic solid frames with fluid-saturated pore space, e.g. fluid saturated rocks, sea ice, and cancellous bone. The study of poroelasticity plays an important role in biomechanics, seismology, and geophysics. The physical properties of these composites depend not only on the constituent materials but also on the microstructure of pore space and how the viscous pore fluid interacts with the solid frame. The bulk properties are described by various 'effective parameters' that appear as coefficients of the integro-differential equations governing wave propagation through poroelastic materials when the wavelength is much larger than the scale of the microstructure. Retrieval of effective parameters through a non-destructive test using waves is essential in monitoring the health of these composites. In the course of this project, a numerical paradigm for simulating wave propagation in poroelastic composites and the inverse problem of the parameter retrieval will be developed, analyzed, and implemented. By utilizing the fact that all the effective properties in the poroelastic wave equations for a given bone sample are underpinned by the same microstructure, the computational model will be constructed from micro-CT scans in a mathematically consistent way.