[Paper Review] Fractional calculus modeling of cell viscoelasticity quantifies drug response and maturation more robustly than integer order models

Anh Vo, Andrew Ekpenyong|arXiv (Cornell University)|Jan 2, 2022

Fractional Differential Equations Solutions4 citations

TL;DR

This study proposes fractional calculus-based viscoelastic models, specifically the fractional Kelvin-Voigt model, to more robustly quantify macrophage mechanical responses than traditional integer-order models. It demonstrates that fractional models better capture drug-induced changes (e.g., cytochalasin D, blebbistatin) and maturation-related viscoelastic shifts, with improved fit quality and biophysical insight into cytoskeletal dynamics.

ABSTRACT

It has recently been discovered that the viscoelastic properties of cells are inherent markers reflecting the complex biological states, functions and malfunctions of the cells. Although the extraction of model parameters from the viscoelasticity data of many cell types has been done successfully using integer order mechanical and power-law viscoelastic models, there are some cell types and conditions where the goodness of fits falls behind. Thus, fractional order viscoelastic models have been proposed as more general and better suited for such modeling. In this work, we test such proposed generality using published data already fitted by integer order models. We find that cell viscoelasticity data can be fitted using fractional order viscoelastic models in more situations than integer order. For macrophages, which are among the white blood cells that function in the immune system, the fractional order Kelvin-Voigt model best captures pharmacological interventions and maturation of the cells. The steady state viscosity of macrophages decreases following depolymerization of F-actin using the drug cytochalasin D, and also decreases following myosin II breakdown using Blebbistatin. When macrophages are treated with a bacterium-derived chemoattractant, the steady state viscosity decreases. Interestingly, both the steady state viscosity and elastic modulus are progressively altered as the cells become mature and approach senescence. Taken together, these results show that fractional viscoelastic modeling, more robustly than integer order modeling, enables the further quantification of cell function and malfunction, with potential diagnostic and therapeutic applications especially in cases of cancer and immune system dysfunctions.

Motivation & Objective

• To address limitations of integer-order viscoelastic models in capturing complex mechanical behaviors of macrophages under pharmacological and developmental conditions.
• To evaluate whether fractional calculus models provide more robust and generalizable parameterization of cell viscoelasticity than conventional models.
• To quantify how viscoelastic parameters (elastic modulus, viscosity, fractional order) change in response to cytoskeletal drugs and during macrophage maturation.
• To establish fractional models as superior tools for characterizing cell mechanical states in health and disease, particularly in immune dysfunction and cancer.

Proposed method

• Application of the fractional Kelvin-Voigt (Frac KV) model using Mittag-Leffler functions to describe viscoelastic relaxation in macrophages.
• Fitting experimental strain data from suspended macrophages to both integer-order and fractional-order viscoelastic models to compare goodness-of-fit.
• Use of time-domain stress relaxation data to extract key parameters: steady-state viscosity (η₂), elastic modulus (E₂), and fractional order (ν).
• Statistical evaluation using R-squared, adjusted R-squared, root mean square error (RMSE), and sum of squared errors (SSE) to compare model performance.
• Analysis of parameter changes across conditions: control, cytochalasin D, blebbistatin, fMLP stimulation, and time points during differentiation (24h to 96h).
• Use of fractional derivatives in the constitutive equation to model memory effects and power-law behavior in viscoelastic response.

Experimental results

Research questions

• RQ1 Can fractional calculus models provide a more robust fit to macrophage viscoelasticity data than integer-order models under diverse pharmacological and developmental conditions?
• RQ2 How do viscoelastic parameters (η₂, E₂, ν) change in response to cytoskeletal disruption via cytochalasin D and blebbistatin?
• RQ3 How do viscoelastic properties evolve during macrophage maturation and senescence, particularly around 96 hours post-differentiation?
• RQ4 To what extent does the fractional order ν reflect biologically meaningful changes in cytoskeletal organization and mechanical compliance?

Key findings

• The fractional Kelvin-Voigt model achieved superior goodness-of-fit (R² = 0.9977) for cytochalasin D-treated macrophages compared to integer-order models, with SSE = 2.83×10⁻⁵ and RMSE = 0.0007.
• Cytochalasin D treatment reduced steady-state viscosity (η₂) from 163.23 Pa·s to 29.47 Pa·s and increased elastic modulus (E₂) from 0.91 Pa to 3.88 Pa, indicating cytoskeletal softening and stiffening.
• Blebbistatin treatment decreased both η₂ (from 163.23 to 155.52 Pa·s) and E₂ (from 0.91 to 0.17 Pa), consistent with myosin II inhibition and reduced contractility.
• During maturation, η₂ decreased from 163.23 Pa·s at 24h to 96.86 Pa·s at 72h, then increased to 203.77 Pa·s at 96h, correlating with apoptosis onset.
• The fractional order ν decreased from 0.95 to 0.80 during maturation, indicating a shift toward more viscous, less elastic behavior in intermediate stages.
• The Frac KV model captured apoptosis-induced mechanical stiffening (η₂ > control) at 96h, a feature poorly resolved by integer models, highlighting its enhanced sensitivity to pathological transitions.