On the Comparative Suitability of Strain Relaxation and Stress Relaxation Compression for Ultrasound Poroelastic Tissue Characterization

Poroelastic tissue strain imaging measures the time-varying and spatially varying deformation of a soft-tissue matrix during compression as the tissue fluid flows out of the compartmental boundaries. With the help of ultrasound, it has been carried out by observing the evolution of the images of the ultrasound echo strain over time, which shows that, in a stress-relaxation experiment (constantly applied global axial strain), a front of negative dilatation (volumetric strain) propagates slowly from the boundaries of a sample toward the center of the compressed region. The fitting of equations that predict this behavior to experimental data has earlier allowed quantitative imaging of the product of aggregate modulus and permeability of a tissue phantom, HAk, and its Poisson's ratio, ν. An ability to image and measure such novel tissue characteristics is likely to benefit biomedical research and have a wide range of clinical applications, including the assessment of lymphoedema, the diagnosis of cancer, the prediction of anticancer drug effectiveness, and monitoring of the tissue response to various treatments. This method is problematic, however, for application in vivo because the calculation of the volumetric strain requires the lateral and elevational strains, which are not easily measured accurately with conventional ultrasound strain imaging. This paper investigates for the first time whether the ultrasound observation in a strain-relaxation experiment (constantly applied uniaxial stress) could be used to observe the same mechanical behavior and provide the same information about the properties of a poroelastic sample as in a stress-relaxation experiment. The analytical theory was used to demonstrate that the propagation of dilatation shown in stress relaxation should also be observable in strain relaxation and that it should be detectable using axial strain, which is relatively easily measured in vivo. Finite element modeling (FEM) was employed to simulate all strain components within a homogeneous poroelastic material first during strain relaxation and then during stress relaxation, again demonstrating their equivalence for the observation of the propagation of a dilatation. The validity of using the strain relaxation conditions as an alternative to stress relaxation for measuring a poroelastic material's response was further confirmed by a fitting of the analytical models to the results of FEM. This allowed for an inversion of the time-varying volumetric strain, to recover the images of HAk and ν, for either loading configuration. The strain-relaxation configuration offers not only an opportunity to derive the same important quantitative poroelastic properties of the tissue as stress relaxation but also the potential to avoid the difficulties and errors associated with the estimation of strain along the axes perpendicular to the imaging axis, thus offering opportunities for easier clinical translation.

A Study of Stress Relaxation with Finite Strain

Two simple types of constitutive equations appropriate to materials exhibiting viscoelasticity are presented, one of a basic solid nature and one of a basic fluid nature. The predictions of the equations for a stress relaxation experiment are worked out and compared to the data from some experiments on various elastomers. The fluid theory is shown to be most appropriate in a certain sense.