Magnetic resonance elastography

MRE quantitatively determines the stiffness of biological tissues by measuring its mechanical response to an external stress.[4] Specifically, MRE calculates the shear modulus of a tissue from its shear-wave displacement measurements.[3] The elastic modulus quantifies the stiffness of a material, or how well it resists elastic deformation as a force is applied. For elastic materials, strain is directly proportional to stress within an elastic region. The elastic modulus is seen as the proportionality constant between stress and strain within this region. Unlike purely elastic materials, biological tissues are viscoelastic, meaning that it has characteristics of both elastic solids and viscous liquids. Their mechanical responses depend on the magnitude of the applied stress as well as the strain rate. The stress-strain curve for a viscoelastic material exhibits hysteresis. The area of the hysteresis loop represents the amount of energy lost as heat when a viscoelastic material undergoes an applied stress and is distorted. For these materials, the elastic modulus is complex and can be separated into two components: a storage modulus and a loss modulus. The storage modulus expresses the contribution from elastic solid behavior while the loss modulus expresses the contribution from viscous liquid behavior. Conversely, elastic materials exhibit a pure solid response. When a force is applied, these materials elastically store and release energy, which does not result in energy loss in the form of heat.[6]

Yet, MRE and other elastography imaging techniques typically utilize a mechanical parameter estimation that assumes biological tissues to be linearly elastic and isotropic for simplicity purposes.[1] The effective shear modulus ${\displaystyle \mu }$ can be expressed with the following equation:

${\displaystyle \mu =E/[2(1+\nu )]}$

where ${\displaystyle E}$ is the elastic modulus of the material and ${\displaystyle \nu }$ is the Poisson’s ratio.

The Poisson’s ratio for soft tissues is approximated to equal 0.5, resulting in the ratio between the elastic modulus and shear modulus to equal 3.[7] This relationship can be used to estimate the stiffness of biological tissues based on the calculated shear modulus from shear-wave propagation measurements. A driver system produces and transmits acoustic waves set at a specific frequency (50–500 Hz) to the tissue sample. At these frequencies, the velocity of shear waves can be about 1–10 m/s.[8][9] The effective shear modulus can be calculated from the shear wave velocity with the following:[10]

${\displaystyle \mu =\rho {v_{s}}^{2}}$

where ${\displaystyle \rho }$ is the tissue density and ${\displaystyle v_{s}}$ is the shear wave velocity.

Recent studies have been focused on incorporating mechanical parameter estimations into post-processing inverse algorithms that account for the complex viscoelastic behavior of soft tissues. Creating new parameters could potentially increase the specificity of MRE measurements and diagnostic testing.[11][12]

• Strain–encoded magnetic resonance imaging