Indentation Measurements to Validate Dynamic Elasticity Imaging Methods

Abstract

We describe macro-indentation techniques for estimating the elastic modulus of soft hydrogels. Our study describes (a) conditions under which quasi-static indentation can validate dynamic shear-wave imaging estimates and (b) how each of these techniques uniquely biases modulus estimates as they couple to the sample geometry. Harmonic shear waves between 25 and 400 Hz were imaged using ultrasonic Doppler and optical coherence tomography methods to estimate shear dispersion. From the shear-wave speed of sound, average elastic moduli of homogeneous samples were estimated. These results are compared directly with macroscopic indentation measurements measured two ways. One set of measurements applied Hertzian theory to the loading phase of the force-displacement curves using samples treated to minimize surface adhesion forces. A second set of measurements applied Johnson-Kendall-Roberts theory to the unloading phase of the force-displacement curve when surface adhesions were significant. All measurements were made using gelatin hydrogel samples of different sizes and concentrations. Agreement within 5% among elastic modulus estimates was achieved for a range of experimental conditions. Consequently, a simple quasi-static indentation measurement using a common gel can provide elastic modulus measurements that help validate dynamic shear-wave imaging estimates.

Ramp-hold relaxation solutions for the KVFD model applied to soft viscoelastic media

“…The general rule of thumb for spherical indentation is for the diameter of the probe to be less than one tenth of the sample diameter. We confirmed the validity of this rule for representative gelatin samples recently [32]. Indented samples were 50 mm in diameter for a 5 mm probe diameter.…”

“…There are no established standards for calibrating viscoelastic measurements in soft materials, although comparisons among independent measurement techniques can test for precision and consistency [32]. A model-based solution can be valuable for predicting viscoelastic properties from load-relaxation time series if it satisfies the following coupled criteria: (1) The model reliably predicts sample responses with few parameters; (2) Parameter estimates are robust to changes in experimental variables and conditions; (3) The same model applies to different materials having a variety of mechanical properties in a range of interest.…”

“…Specifically, force–displacement measurements made with water on the top sample surface and processed according to Hertzian contact theory to find an elastic modulus were found to be equivalent to force–displacement measurements made without water on the sample and processed according to JKR theory. The agreement was within measurement error provided the measurements were completed within an hour [32]. …”

Assessing composition and structure of soft biphasic media from Kelvin–Voigt fractional derivative model parameters

“…The elastic modulus measured by applications of the Kevin-Voigt (integer-order derivative) model approximates the shear modulus [24, 25]. To show this is also true for the KVFD model, we compared E 0 in Eq (3) with the elastic modulus determined from Hertzian contact theory and wave-propagation measurements.…”

“… * Measurements from [24] in pure gelatin at concentrations of 2%, 4%, 6%, and 8% were fit to the polynomial y = 178.75 x 2 − 426.5 x + 485 to interpolate and estimate a value at 5%. Thus no error bar is reported.…”