Spring-damper equivalents of the fractional, poroelastic, and poroviscoelastic models for elastography

Abstract

In MR elastography, it is common to use an elastic model for the tissue's response in order to interpret the results properly. More complex models, such as viscoelastic, fractional viscoelastic, poroelastic, or poroviscoelastic ones, are also used. These models appear at first sight to be very different, but here it is shown that they may all be expressed in terms of elementary viscoelastic models. For a medium expressed with fractional models, many elementary spring-damper combinations are added, each of them weighted according to a long-tailed distribution of time constants or relaxation frequencies. This may open up a more physical interpretation of fractional models. The shear-wave component of the poroelastic model is shown to be modeled exactly by a three-component Zener model. The extended poroviscoelastic model is found to be equivalent to what is called a non-standard four-parameter model. Accordingly, the large number of parameters in the porous models can be reduced to the same number as in their viscoelastic equivalents. While the individual displacements from the solid and fluid parts cannot be measured individually, the main use of the poro(visco)elastic models is therefore as a physics-based method for determining parameters in a viscoelastic model.

Keywords: MR elastography (MRE); fractional viscoelasticity; poroelasticity.