simple linear viscoelastic models  (Schmid GmbH)

Journal: Biomechanics and Modeling in Mechanobiology

Article Title: Modelling the rheology of living cell cytoplasm: poroviscoelasticity and fluid-to-solid transition

doi: 10.1007/s10237-024-01854-2

Figure Lengend Snippet: Rheological model of the viscoelastic filament

Article Snippet: In early theoretical models, these moduli were obtained using simple linear viscoelastic models (Schmid-Schönbein et al. ; Sato et al. ).

Techniques:

Journal: Biomechanics and Modeling in Mechanobiology

Article Title: Modelling the rheology of living cell cytoplasm: poroviscoelasticity and fluid-to-solid transition

doi: 10.1007/s10237-024-01854-2

Figure Lengend Snippet: Non-dimensional governing parameters for the mathematical model of the cytoplasm

Article Snippet: In early theoretical models, these moduli were obtained using simple linear viscoelastic models (Schmid-Schönbein et al. ; Sato et al. ).

Techniques: Expressing

Journal: Biomechanics and Modeling in Mechanobiology

Article Title: Modelling the rheology of living cell cytoplasm: poroviscoelasticity and fluid-to-solid transition

doi: 10.1007/s10237-024-01854-2

Figure Lengend Snippet: Force acting on the bead along x -direction \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F^*$$\end{document} F ∗ versus displacement of the bead \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta x^*$$\end{document} Δ x ∗ for the viscoelastic timescale \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{\text{ve}}=1.0$$\end{document} τ ve = 1.0 and the poroelastic timescale \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{{\text{pe}}}=1.0$$\end{document} τ pe = 1.0 obtained using Darcy flow model for various domain sizes \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^*=10$$\end{document} L ∗ = 10 (Red), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^*=20$$\end{document} L ∗ = 20 (Blue), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^*=40$$\end{document} L ∗ = 40 (Green), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^*=100$$\end{document} L ∗ = 100 (Cyan), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^*=200$$\end{document} L ∗ = 200 (Black)

Article Snippet: In early theoretical models, these moduli were obtained using simple linear viscoelastic models (Schmid-Schönbein et al. ; Sato et al. ).

Techniques:

Journal: Biomechanics and Modeling in Mechanobiology

Article Title: Modelling the rheology of living cell cytoplasm: poroviscoelasticity and fluid-to-solid transition

doi: 10.1007/s10237-024-01854-2

Figure Lengend Snippet: Contour plot of the apparent Young’s modulus \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_A$$\end{document} E A as a function of the viscoelastic timescale \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{\text{ve}}$$\end{document} τ ve and the poroelastic timescale \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{\text{pe}}$$\end{document} τ pe obtained from: a our numerical simulations using Darcy flow, b optical tweezers experiments of Hu et al. (Modified from the figure presented by Hu et al. ). The regimes marked above represent I Viscous, II Viscoelastic, III Elastic, IV Poroelastic, V Poroviscoelastic, VI Compressible elastic, and VII Compressible viscoelastic. The white circles in ( a ) represent the baseline cases discussed in Sect.

Article Snippet: In early theoretical models, these moduli were obtained using simple linear viscoelastic models (Schmid-Schönbein et al. ; Sato et al. ).

Techniques: Modification

Journal: Biomechanics and Modeling in Mechanobiology

Article Title: Modelling the rheology of living cell cytoplasm: poroviscoelasticity and fluid-to-solid transition

doi: 10.1007/s10237-024-01854-2

Figure Lengend Snippet: Fields of the skeleton pressure p , fluid pressure \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{\text{fl}}$$\end{document} p fl , and total pore pressure \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{\text{pore}}$$\end{document} p pore at the end of loading \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t^*=0.1$$\end{document} t ∗ = 0.1 and during relaxation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t^*=0.2$$\end{document} t ∗ = 0.2 along with the timewise variation of the skeleton force \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_{\text{sk}}$$\end{document} F sk , the pore fluid \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_{\text{pore}}$$\end{document} F pore , and the total force F in regime I (Viscous fluid) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{\text{ve}}=10^{-3}$$\end{document} τ ve = 10 - 3 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{\text{pe}}=5\times 10^3$$\end{document} τ pe = 5 × 10 3 , regime II (Viscoelastic) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{\text{ve}}=1.0$$\end{document} τ ve = 1.0 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{\text{pe}}=5\times 10^3$$\end{document} τ pe = 5 × 10 3 , regime III (Elastic) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{\text{ve}}=2\times 10^2$$\end{document} τ ve = 2 × 10 2 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{\text{pe}}=5\times 10^3$$\end{document} τ pe = 5 × 10 3 , regime IV (Poroelastic) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{\text{ve}}=2\times 10^2$$\end{document} τ ve = 2 × 10 2 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{\text{pe}}=1.0$$\end{document} τ pe = 1.0 , regime V (Poroviscoelastic) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{\text{ve}}=1.0$$\end{document} τ ve = 1.0 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{\text{pe}}=1.0$$\end{document} τ pe = 1.0 , regime VI (Compressible Elastic) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{\text{ve}}=2\times 10^2$$\end{document} τ ve = 2 × 10 2 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{\text{pe}}=10^{-4}$$\end{document} τ pe = 10 - 4 , and regime VII (Compressible Viscoelastic) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{\text{ve}}=1.0$$\end{document} τ ve = 1.0 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{\text{pe}}=10^{-4}$$\end{document} τ pe = 10 - 4

Article Snippet: In early theoretical models, these moduli were obtained using simple linear viscoelastic models (Schmid-Schönbein et al. ; Sato et al. ).

Techniques:

Journal: Biomechanics and Modeling in Mechanobiology

Article Title: Modelling the rheology of living cell cytoplasm: poroviscoelasticity and fluid-to-solid transition

doi: 10.1007/s10237-024-01854-2

Figure Lengend Snippet: a Contour plot of the logarithmic gradients of the apparent Young’s modulus (a) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$gE_{A,\text{ve}}$$\end{document} g E A , ve and b \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$gE_{A,{\text{pe}}}$$\end{document} g E A , pe as a function of the viscoelastic timescale \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{\text{ve}}$$\end{document} τ ve and poroelastic timescale \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{\text{pe}}$$\end{document} τ pe . Red lines in ( a ) and ( b ) correspond to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$gE_{A,\text{ve}}=10^{-3}$$\end{document} g E A , ve = 10 - 3 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$gE_{A,{\text{pe}}}=10^{-3}$$\end{document} g E A , pe = 10 - 3 , respectively

Article Snippet: In early theoretical models, these moduli were obtained using simple linear viscoelastic models (Schmid-Schönbein et al. ; Sato et al. ).

Techniques:

Journal: Biomechanics and Modeling in Mechanobiology

Article Title: Modelling the rheology of living cell cytoplasm: poroviscoelasticity and fluid-to-solid transition

doi: 10.1007/s10237-024-01854-2

Figure Lengend Snippet: a Contour plot of the non-dimensional apparent Young’s modulus \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_A^*$$\end{document} E A ∗ as a function of the viscoelastic timescale \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{\text{ve}}$$\end{document} τ ve and the poroelastic timescale \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{\text{pe}}$$\end{document} τ pe at viscous timescale \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{\text{vis}}=10^{-6}$$\end{document} τ vis = 10 - 6 and b variation of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_A^*$$\end{document} E A ∗ with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{\text{pe}}$$\end{document} τ pe at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{\text{ve}}=200$$\end{document} τ ve = 200 obtained using Darcy flow model and Brinkman flow model with various viscous timescales. In both figures, the red (blue) dotted straight lines show the boundaries of regimes for the Brinkman (Darcy) flow models

Article Snippet: In early theoretical models, these moduli were obtained using simple linear viscoelastic models (Schmid-Schönbein et al. ; Sato et al. ).

Techniques:

Journal: Biomechanics and Modeling in Mechanobiology

Article Title: Modelling the rheology of living cell cytoplasm: poroviscoelasticity and fluid-to-solid transition

doi: 10.1007/s10237-024-01854-2

Figure Lengend Snippet: Variation of the non-dimensional apparent Young’s modulus \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_A^*$$\end{document} E A ∗ obtained from the numerical simulations using Darcy flow model and a nonlinear curve fitting in a Viscoelastic regime at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{\text{pe}}=5\times 10^{3}$$\end{document} τ pe = 5 × 10 3 and compressible viscoelastic regimes at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{\text{pe}}=10^{-4}$$\end{document} τ pe = 10 - 4 , b Poroelastic regime at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{\text{ve}}=200$$\end{document} τ ve = 200 , and c Poroviscoelastic regime at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{\text{pe}}=10^{-2}$$\end{document} τ pe = 10 - 2 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{\text{pe}}=500$$\end{document} τ pe = 500 , and d the contour plot of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_A^*$$\end{document} E A ∗ as a function of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{\text{ve}}$$\end{document} τ ve and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{\text{pe}}$$\end{document} τ pe obtained from the nonlinear curve fitting

Article Snippet: In early theoretical models, these moduli were obtained using simple linear viscoelastic models (Schmid-Schönbein et al. ; Sato et al. ).

Techniques:

Journal: Biomechanics and Modeling in Mechanobiology

Article Title: Modelling the rheology of living cell cytoplasm: poroviscoelasticity and fluid-to-solid transition

doi: 10.1007/s10237-024-01854-2

Figure Lengend Snippet: Variation of the ratio of the apparent Young’s modulus \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_A^*$$\end{document} E A ∗ with pre-stretch to without pre-stretch \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{A,\alpha =1}^*$$\end{document} E A , α = 1 ∗ for various poroelastic timescales at a constant viscoelastic timescale \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{\text{ve}}=10^2$$\end{document} τ ve = 10 2

Article Snippet: In early theoretical models, these moduli were obtained using simple linear viscoelastic models (Schmid-Schönbein et al. ; Sato et al. ).

Techniques: