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

Abstract

Kelvin-Voigt fractional derivative (KVFD) model parameters have been used to describe viscoelastic properties of soft tissues. However, translating model parameters into a concise set of intrinsic mechanical properties related to tissue composition and structure remains challenging. This paper begins by exploring these relationships using a biphasic emulsion materials with known composition. Mechanical properties are measured by analyzing data from two indentation techniques – ramp-stress relaxation and load-unload hysteresis tests. Material composition is predictably correlated with viscoelastic model parameters. Model parameters estimated from the tests reveal that elastic modulus E0 closely approximates the shear modulus for pure gelatin. Fractional-order parameter α and time constant τ vary monotonically with the volume fraction of the material’s fluid component. α characterizes medium fluidity and the rate of energy dissipation, and τ is a viscous time constant. Numerical simulations suggest that the viscous coefficient η is proportional to the energy lost during quasi-static force-displacement cycles, EA. The slope of EA versus η is determined by α and the applied indentation ramp time Tr. Experimental measurements from phantom and ex vivo liver data show close agreement with theoretical predictions of the η − EA relation. The relative error is less than 20% for emulsions 22% for liver. We find that KVFD model parameters form a concise features space for biphasic medium characterization that described time-varying mechanical properties.

Modeling ramp-hold indentation measurements based on Kelvin–Voigt fractional derivative model

“…The KVFD model provides an accurate and flexible method for describing material properties with only three parameters [14,15,31]. The constitutive equation relating stress σ ( t ) to strain ε ( t ) for the fractional-derivative Kelvin-Voigt model is expressed as [30,35]…”

“…The rigid molds used to form samples for spherical indentation testing are 50 mm in diameter and 20 mm in height. Sample sizes were shown to be large enough to avoid significant boundary influences [14].…”

“…However, it is difficult to apply a near-step strain without creating oscillating transients that are more representative of the measurement device than the material. In contrast, ramp-hold loading strategy can dramatically improve experimental data stability and improve the reliability of model fitting but only if the ramp-hold stage is included in the model [14,34, 43][45, 46].…”

“…Among fractional derivative models, KVFD has shown an ability to represent time-dependent mechanical behavior for several biological materials and tissues. It offers a good balance between model simplicity and experimental fitting accuracy [14,15,31].…”

“…In our previous work, we derived the KVFD model in closed form for the ramp-hold relaxation experiment, and showed it can accurately represent the entire experimental data record. We also found that KVFD model parameters can be interpreted in terms of tissue composition and structure [14].…”