8 Rheological Model‐based Methods for Estimating Tissue Viscoelasticity

Jean‐Luc Gennisson

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

8.1 Introduction

As was shown in Chapter 5 in this book, the wave equation can be expressed in terms of the different pair parameters of elastic properties of tissue. It was shown that two types of waves propagate in the solid: compressional waves and shear waves. Each of these waves will be intuitively linked to the speed propagation (c P, c S), but they can also be connected to the pair of bulk and shear moduli (κ, μ). Thus the equations for the compression and shear waves can be reformulated as

(8.1) equation
(8.2) equation

where ρ, u p, u s are respectively the density, the longitudinal displacement, and the shear displacement.

It is then possible to make an analogy with the oscillations of a spring or even the behavior of any unamortized oscillator. The left‐hand side is the product of the mass of the object and its acceleration, whereas the right‐hand side corresponds to the restoring force which allows the oscillations of the object considered. As stated, the formalism of this equation is the equation of motion for an undamped oscillator (i.e. oscillations exist indefinitely). In ...

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