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Homogenization Theories and Inverse Problems

  • Robert P. GilbertEmail author
  • Ana Vasilic
  • Sandra Ilic
Chapter

Abstract

Various approaches are presented for modelling the acoustic response of cancellous bone to ultrasound interrogation. As the characteristic pore size in cancellous bone is much smaller than a typical bone sample, there is a clear scale separation (micro versus macro). Thus, our modelling methods are mainly based on homogenization techniques and numerical upscaling. First, we consider the so-called direct problems and present models for both periodically perforated domain and a domain with random distribution of pores, as well as nonlinear model with a shear-thinning viscoelastic material emulating the blood-marrow mixture. A numerical procedure is given for the upscaling of a diphasic mixture using different trabeculae thicknesses and various frequencies for the ultrasound excitation. Finally, the results of a quite accurate two-dimensional inversion for the Biot parameters are presented. Further details for these different problems are amply described in the literature cited in the bibliography.

Keywords

Homogenization Cancellous bone Inverse problems 

Notes

Acknowledgements

This work was partially supported by the NSF under Grant DMS-0920850.

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Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of DelawareNewarkUSA
  2. 2.Department of Mathematical SciencesUAE UniversityAbu DhabiUAE
  3. 3.Lehrstuhl für Allgemeine MechanikRuhr Universität Bochum IA 3/129BochumDeutschland

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