Deciphering Elastic Waves in Deformed Materials

In review in Biomechanics and Modeling in Mechanobiology (contact D. Nordsletten for Pre-Print).

Wave behavior is a fascinating subject and increasingly important in biorheology.  Increasingly, waves are being used to probe the depths of our tissues and sample the material properties within — a technique called Elastography.  Producing meaningful numbers is an art, whereby momentum and stress are brought to equal order and used to locally invert linear elasticity equations to construct tissue stiffness.

While applied to liver, brain and breast tissues to much success, translation of Elastography to muscles is tricky.  Muscles tend to be stiff, anisotropic, capable of actively altering their stiffness properties and are often under load — factors which make reconstruction of tissue properties extremely challenging.

In work recently submitted to BMMB, we examined one of these challenges: the effects of load.  Changes in tissue load can dramatically influence measured apparent stiffness.  Deformation of a simple isotropic homogenous material can yield apparent spatial variability and anisotropy.  To explain this phenomena, we applied Perturbation Theory (PT) to the large deformation mechanics equations, aiming to link intrinsic biomechanical parameters to their apparent parameters.  Or, in other words, explain how loading a material changes the behavior of wave motion.  For this we:

  • Derived a First order approximation for small amplitude waves using PT
  • Derived its weak form and examined the influence of large-scale deformations on wellposedness
  • Derived a finite element method for the solution of the system
  • Derived two analytic solutions to test the implementation

The final step was to conduct an experiment where we compare wave behavior in loaded and unloaded samples of silicon gel phantom.  Measurements were done in a 3T Philips MR scanner, providing images of the waves in three dimensions.  Wave motion data in the undeformed configuration was used to calibrate the model at rest.  Large deformation mechanics were then applied to deform the sample, and wave behavior was predicted using the system derived.  Results on the right show the predicted wave behavior by the mathematical model (top) and the experimentally measured results (bottom).  The mathematical model was able to predict both the change in wavelength and decreased attenuation observed in the experiment.

These results were recently presented at:

VPH 2014, Trondheim (Honorable Mention for Poster award)

Bioeng 2014, London