Fractional Derivatives Embody Essential Features of Cell Rheological Behavior

Abstract

Mechanical moduli of cultured airway smooth muscle cells (Fabry, B., et al. Phys. Rev. Lett. 87:148102, 2001) reveal that the frequency dependence of cell rheological behavior conforms to a weak power-law relationship over a wide range of frequency (10(-2)-10(3) Hz). Such a behavior cannot be accounted for by standard viscoelastic models characterized by a discrete number of time constants that have been commonly used in previous studies of cell viscoelasticity. Fractional calculus, by contrast, provides a natural framework for describing weak power-law relationships and requires no assumptions about the type of material, the time constant distribution or the time/frequency interval in which rheological observations are made. In this study, we developed a rheological model of the cell using methods of fractional calculus. We used a least-squares technique to fit the model to data previously obtained from measurements on airway smooth muscle cells. The fit provided an excellent correspondence to the data, and the estimated values of model parameters were physically plausible. The model leads to a novel and unexpected empirical link between dynamic viscoelastic behavior of the cytoskeleton and the static contractile stress that it bears.

A fractional K-BKZ constitutive formulation for describing the nonlinear rheology of multiscale complex fluids

“…Both the FMM as well as the FKVM are characterized by only four parameters -two powerlaw exponents, which control the scaling for the temporal and frequency response, and two quasi-properties, which set the scales for magnitude of the stresses in these multiscale materials. Examples of the successful application of these two canonical fractional models to describe the linear rheology of complex multiscale materials include red blood cell membranes (Craiem and Magin (2010)), smooth muscle cells (Djordjević et al (2003)), food gums (Ma and Barbosa-Canovas (1996)) and comb-shaped polymers (Friedrich (1992)). However we note that while the linear viscoelastic predictions of fractional models have now been extensively studied, there is an absence of fractional constitutive equations that are able to predict the nonlinear rheological response of these complex materials observed at large strain.…”

Hopf bifurcation analysis in a fractional-order survival red blood cells model and PDα$\mathit{PD}^{\alpha} $ control

“…3 School of Mathematics and Statistics, Shandong Normal University, Ji'nan, 250014, China. 4 College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, 400067, China.…”

“…Compared with integer-order derivatives, it has been found that fractional derivatives have the superiority of accuracy and flexibility when used to describe some non-classical phenomena in natural science and engineering applications such as neurons [1], finance systems [2], biological systems [3], and so on. Especially in biological systems, fractional calculus has more advantages than traditional integer-order calculus in describing molecular dynamics with memory characteristics and historical dependence [4,5].…”

A Neural Network Approach for Solving Fractional-Order Model of HIV Infection of CD4+T-Cells

“…In biology, it has been deduced that the membranes of cells of biological organism have fractionalorder electrical conductance [7] and then are classified in groups of non-integer order models. Fractional derivatives embody essential features of cell rheological behavior and have enjoyed greatest success in the field of rheology [12]. The reason of using fractional order differential equationsisthattheyarenaturally related to systems with memory which exists in most biological systems and they are closely related to fractals which are abundant in biological systems.…”