An efficient and accurate method for modeling nonlinear fractional viscoelastic biomaterials

Abstract

Computational biomechanics plays an important role in biomedical engineering: using modeling to understand pathophysiology, treatment and device design. While experimental evidence indicates that the mechanical response of most tissues is viscoelasticity, current biomechanical models in the computation community often assume only hyperelasticity. Fractional viscoelastic constitutive models have been successfully used in literature to capture the material response. However, the translation of these models into computational platforms remains limited. Many experimentally derived viscoelastic constitutive models are not suitable for three-dimensional simulations. Furthermore, the use of fractional derivatives can be computationally prohibitive, with a number of current numerical approximations having a computational cost that is O(N 2 T ) and a storage cost that is O(N T ) (N T denotes the number of time steps). In this paper, we present a novel numerical approximation to the Caputo derivative which exploits a recurrence relation similar to those used to discretize classic temporal derivatives, giving a computational cost that is O(N ) and a storage cost that is fixed over time. The approximation is optimized for numerical applications, and the error estimate is presented to demonstrate efficacy of the method. The method is shown to be unconditionally stable in the linear viscoelastic case. It was then integrated into a computational biomechanical framework, with several numerical examples verifying accuracy and computational efficiency of the method, including in an analytic test, in an analytic fractional differential equation, as well as in a computational biomechanical model problem. Fractional Derivatives and their Approximations Caputo fractional derivativeThe concept of fractional calculus started with questions about about the generalization of integral and differential operators by L'Hospital and Leibniz [68] from the set of integers to the set of real numbers.Subsequently, many prominent mathematicians focused on fractional calculus (for reviews, see, Ross [68] and Machado [51]). Within the field, many different definitions for fractional differential and integral operators of arbitrary order have been introduced [64]. In this work, we focus on the Caputo definition where the fractional derivative, D α t (with α > 0), of a n-times differentiable function f can be written as,

Fast Solution Methods for Fractional Differential Equations in the Modeling of Viscoelastic Materials

“…with certain functions h 1 , h 2 : (0, ∞) → R. Many different specific choices for these functions are admissible and have been suggested in the literature; cf., e.g., [1,3,6,8,9,19,20,24,[29][30][31][32][33].…”

Trends, directions for further research, and some open problems of fractional calculus

“…In view of the observations recalled in the introduction to Sect. 3, a number of novel numerical methods for solving fractional differential equations have been developed; see, e.g., [65][66][67][68][69][70][71][72][73][74][75][76][77]. Although there are various differences in the details, all these methods share the common feature that they are based on some nonclassical representation of the Caputo-type fractional differential operator C D α a+ of order α > 0 that appears in the differential equations under consideration, i.e., instead of one of the traditional forms…”

A Viscoelastic Model for Human Myocardium

“…To examine the behavior and uniqueness of the parameter space of the proposed viscoelastic model, a parameter sweep was performed. To generate model predictions, each experiment had to be simulated based on the given kinematics to solve the fractional differential equation forward in time (see [35,93] for details). In this case, the model predicted response depended linearly on parameters θ l given a set of parameters, θ * .…”