Abstract
We introduce a data-driven fractional modeling framework for complex materials, and particularly bio-tissues. From multi-step relaxation experiments of distinct anatomical locations of porcine urinary bladder, we identify an anomalous relaxation character, with two power-law-like behaviors for short/long long times, and nonlinearity for strains greater than 25%. The first component of our framework is an existence study, to determine admissible fractional viscoelastic models that qualitatively describe linear relaxation. After the linear viscoelastic model is selected, the second stage adds large-strain effects to the framework through a fractional quasi-linear viscoelastic approach for the nonlinear elastic response of the bio-tissue of interest. From single-step relaxation data of the urinary bladder, a fractional Maxwell model captures both short/long-term behaviors with two fractional orders, being the most suitable model for small strains at the first stage. For the second stage, multi-step relaxation data under large strains were employed to calibrate a four-parameter fractional quasi-linear viscoelastic model, that combines a Scott-Blair relaxation function and an exponential instantaneous stress response, to describe the elastin/collagen phases of bladder rheology. Our obtained results demonstrate that the employed fractional quasi-linear model, with a single fractional order in the range α = 0.25–0.30, is suitable for the porcine urinary bladder, producing errors below 2% without need for recalibration over subsequent applied strains. We conclude that fractional models are attractive tools to capture the bladder tissue behavior under small-to-large strains and multiple time scales, therefore being potential alternatives to describe multiple stages of bladder functionality.
Highlights
Bio-tissues are complex and multi-functional materials, optimized for their specific host organisms, and constrained by limited set of building blocks and available resources [1]
We develop an existence study that considers a set of linear fractional building block models (Scott-Blair, fractional Kelvin-Voigt, fractional Maxwell), which are selected according to the multi-power-law nature of the relaxation data and calibration errors at the linear viscoelastic regime
We note that the fractional Kelvin-Voigt (FKV) model pragmatically recovered the SB model in all instances, where the particle-swarm optimization (PSO) algorithm obtained optimal values for the fractional orders that are close to the SB model
Summary
Bio-tissues are complex and multi-functional materials, optimized for their specific host organisms, and constrained by limited set of building blocks and available resources [1]. Power-law viscoelastic rheology is a complex response observed in many bio-tissues such as arteries [2], cartilage [3], lungs [4], smooth muscle [5], liver and kidneys [6], among other classes of materials. These power-law materials, termed anomalous, exhibit one or more powerlaw scalings for creep/relaxation in the form J (t) ∝ t β and G (t) ∝ t− β across multiple time-scales. The origin of this power-law behavior at the continuum
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