Ultrasound Elastography for Biomedical Applications and Medicine

5 Continuum Mechanics Tensor Calculus and Solutions to Wave Equations

Luiz Vasconcelos¹, Jean‐Luc Gennisson² and Ivan Nenadic¹

¹ Department of Physiology and Biomedical Engineering, Mayo Clinic, Rochester, MN, USA

² Imagerie par Résonance Magnétique Médicale et Multi‐Modalités, Université Paris‐Saclay, Orsay, France

5.1 Introduction

The phenomenon of wave propagation is well known and extremely common to our lives. One's speech is propagated through sound using the air as medium, for example. Words originate from a forced motion on a portion of a deformable medium; from this point forward the movement is propagated from particle to particle, creating a wave. This process may also be understood as an energy transmission that can be done within small bodies, like medical ultrasound, or large structures, like earthquakes. By analyzing how those waves are transmitted through the medium and the interfaces, a series of properties can be determined and so too can its viscoelastic properties [1].

Many ultrasound elastography methods rely on these transmission characteristics to rheologically define biological tissues, and therefore analyze how they are behaving in comparison to healthy tissue parameters. This process is supported by a very strong mathematical basis – a toolkit of equations that describe the laws of mechanical wave propagation. Once those equations are established all conditions and space configurations can be evaluated and new, and more specific, equations ...