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

## TL;DR

This paper demonstrates that complex fractional, poroelastic, and poroviscoelastic models in elastography can be represented using simple spring-damper systems, simplifying their interpretation and parameter estimation.

## Contribution

It shows that various advanced tissue models can be expressed as elementary viscoelastic models, reducing parameter complexity and enabling more physical interpretation.

## Key findings

- Poroelastic shear wave modeled by a three-component Zener model.
- Extended poroviscoelastic model equivalent to a four-parameter model.
- Complex models can be interpreted through elementary viscoelastic components.

## Abstract

In MR elastography it is common to use an elastic model for the tissue's response in order to properly interpret the results. 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 all may 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, hinting at a fractional distribution of time constants or relaxation frequencies. This may open up for a more physical interpretation of the 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. As long as 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.