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Journal of Biomechanical Engineering | Volume 128 | Issue 4 | TECHNICAL PAPERS
TECHNICAL PAPERS: Cell

Frequency Response of a Viscoelastic Tensegrity Model: Structural Rearrangement Contribution to Cell Dynamics

[+] Author and Article Information
Patrick Cañadas1

CNRS UMR 5508 Laboratoire de Mécanique et Génie Civil (LMGC),  Université Montpellier II - CC 048, Place Eugène Bataillon, 34 095 Montpellier Cedex 05, Francecanadas@lmgc.univ-montp2.fr

Sylvie Wendling-Mansuy

CNRS–USR 2164 Laboratoire d’Aérodynamique et Biomécanique du Mouvement,  Université de la Méditerranée, 163 avenue de Luminy, case 918, 13288 Marseille Cedex 09, Francewendling@morille.univ-mrs.fr

Daniel Isabey

INSERM, UMR 651, Fonctions Cellulaires et Moléculaires de l’Appareil Respiratoire et des Vaisseaux, Equipe Biomécanique Cellulaire et Respiratoire, Université Paris XII, Faculté de Médecine,  ISBS Paris, 8, rue du Général Sarrail, 94010 Créteil cedex, Francedaniel.isabey@creteil.inserm.fr

1

Corresponding author.

J Biomech Eng 128(4), 487-495 (Dec 29, 2005) (9 pages) doi:10.1115/1.2205867 History: Received June 13, 2005; Revised December 29, 2005
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In an attempt to understand the role of structural rearrangement onto the cell response during imposed cyclic stresses, we simulated numerically the frequency-dependent behavior of a viscoelastic tensegrity structure (VTS model) made of 24 elastic cables and 6 rigid bars. The VTS computational model was based on the nonsmooth contact dynamics (NSCD) method in which the constitutive elements of the tensegrity structure are considered as a set of material points that mutually interact. Low amplitude oscillatory loading conditions were applied and the frequency response of the overall structure was studied in terms of frequency dependence of mechanical properties. The latter were normalized by the homogeneous properties of constitutive elements in order to capture the essential feature of spatial rearrangement. The results reveal a specific frequency-dependent contribution of elastic and viscous effects which is responsible for significant changes in the VTS model dynamical properties. The mechanism behind is related to the variable contribution of spatial rearrangement of VTS elements which is decreased from low to high frequency as dominant effects are transferred from mainly elastic to mainly viscous. More precisely, the elasticity modulus increases with frequency while the viscosity modulus decreases, each evolution corresponding to a specific power-law dependency. The satisfactorily agreement found between present numerical results and the literature data issued from in vitro cell experiments suggests that the frequency-dependent mechanism of spatial rearrangement presently described could play a significant and predictable role during oscillatory cell dynamics.
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References

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Figures

Grahic Jump Location
+Spatial view of the viscoelastic tensegrity structure (VTS) studied (6 bars and 24 viscoelastic cables). At the reference state (no external force applied to the structure), the four nodes {1, 2, 4, 8} are anchored and fixed in their spatial positions (●). The rectangular base {x,y,z} is the referential system. Oscillating forces (Fz and −Fz) are applied at nodal points {6, 11} along the z axis. The overall deformation of the VTS model is defined by the displacement along the z axis of the two nodes {6, 11} normalized by the bar length Lb which characterizes the size of the structure.
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Spatial view of the viscoelastic tensegrity structure (VTS) studied (6 bars and 24 viscoelastic cables). At the reference state (no external force applied to the structure), the four nodes {1, 2, 4, 8} are anchored and fixed in their spatial positions (●). The rectangular base {x,y,z} is the referential system. Oscillating forces (Fz and −Fz) are applied at nodal points {6, 11} along the z axis. The overall deformation of the VTS model is defined by the displacement along the z axis of the two nodes {6, 11} normalized by the bar length Lb which characterizes the size of the structure.
Figure 1

Spatial view of the viscoelastic tensegrity structure (VTS) studied (6 bars and 24 viscoelastic cables). At the reference state (no external force applied to the structure), the four nodes {1, 2, 4, 8} are anchored and fixed in their spatial positions (●). The rectangular base {x,y,z} is the referential system. Oscillating forces (Fz and −Fz) are applied at nodal points {6, 11} along the z axis. The overall deformation of the VTS model is defined by the displacement along the z axis of the two nodes {6, 11} normalized by the bar length Lb which characterizes the size of the structure.

Grahic Jump Location
+Oscillatory deformation of the overall VTS as a function of the normalized forced frequency f* in a double logarithmic scale. Three distinct zones could be distinguished: a low frequency zone (f*<0.1) of roughly constant oscillatory deformation amplitude where elastic effects are predominant; a transitional zone (0.1<f*<10) of a rapid change in the amplitude of the deformation (with a negative logarithmic slope (−0.83); and a satisfactory correlation coefficient R2=0.99) where elastic and viscous effects are balanced and a high frequency zone (f*>10) of low oscillatory deformation where viscous effects are predominant. The contribution of the spatial redistribution of the tensegrity structure appears to decrease when the frequency increases.
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Oscillatory deformation of the overall VTS as a function of the normalized forced frequency f* in a double logarithmic scale. Three distinct zones could be distinguished: a low frequency zone (f*<0.1) of roughly constant oscillatory deformation amplitude where elastic effects are predominant; a transitional zone (0.1<f*<10) of a rapid change in the amplitude of the deformation (with a negative logarithmic slope (−0.83); and a satisfactory correlation coefficient R2=0.99) where elastic and viscous effects are balanced and a high frequency zone (f*>10) of low oscillatory deformation where viscous effects are predominant. The contribution of the spatial redistribution of the tensegrity structure appears to decrease when the frequency increases.
Figure 2

Oscillatory deformation of the overall VTS as a function of the normalized forced frequency f* in a double logarithmic scale. Three distinct zones could be distinguished: a low frequency zone (f*<0.1) of roughly constant oscillatory deformation amplitude where elastic effects are predominant; a transitional zone (0.1<f*<10) of a rapid change in the amplitude of the deformation (with a negative logarithmic slope (−0.83); and a satisfactory correlation coefficient R2=0.99) where elastic and viscous effects are balanced and a high frequency zone (f*>10) of low oscillatory deformation where viscous effects are predominant. The contribution of the spatial redistribution of the tensegrity structure appears to decrease when the frequency increases.

Grahic Jump Location
+Normalized time constant τ* of the overall VTS model as a function of the normalized forced frequency f* in a double logarithmic scale with the two zones of predominant (i) elastic (f*<1) and (ii) viscous (f*>10) effects, as well as the transitional zone (0.1<f*<10). The normalized time constant of the overall tensegrity model decreases nonlinearly with the oscillatory frequency in the upper part of the transitional zone, following the power law τ*∼f*−0.42 with a correlation coefficient R2(=0.97). The decreasing contribution of spatial redistribution of VTS elements contribute to the frequency-dependent solidifyinglike process for the VTS model.
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Normalized time constant τ* of the overall VTS model as a function of the normalized forced frequency f* in a double logarithmic scale with the two zones of predominant (i) elastic (f*<1) and (ii) viscous (f*>10) effects, as well as the transitional zone (0.1<f*<10). The normalized time constant of the overall tensegrity model decreases nonlinearly with the oscillatory frequency in the upper part of the transitional zone, following the power law τ*∼f*−0.42 with a correlation coefficient R2(=0.97). The decreasing contribution of spatial redistribution of VTS elements contribute to the frequency-dependent solidifyinglike process for the VTS model.
Figure 3

Normalized time constant τ* of the overall VTS model as a function of the normalized forced frequency f* in a double logarithmic scale with the two zones of predominant (i) elastic (f*<1) and (ii) viscous (f*>10) effects, as well as the transitional zone (0.1<f*<10). The normalized time constant of the overall tensegrity model decreases nonlinearly with the oscillatory frequency in the upper part of the transitional zone, following the power law τ*∼f*−0.42 with a correlation coefficient R2(=0.97). The decreasing contribution of spatial redistribution of VTS elements contribute to the frequency-dependent solidifyinglike process for the VTS model.

Grahic Jump Location
+Normalized viscosity modulus η* and elasticity modulus E* of the overall VTS model as a function of the normalized forced frequency f* in a double logarithmic scale with the two zones of predominant elastic (f*<1) and viscous (f*>10) effects, and the transitional zone (0.1<f*<10). The normalized elasticity modulus E*(•) increases in the transitional zone with a logarithmic slope (+0.18) and the normalized viscosity modulus η*(▴) decreases with a logarithmic slope (−0.24), the two with a correlation coefficient R2(=0.99).
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Normalized viscosity modulus η* and elasticity modulus E* of the overall VTS model as a function of the normalized forced frequency f* in a double logarithmic scale with the two zones of predominant elastic (f*<1) and viscous (f*>10) effects, and the transitional zone (0.1<f*<10). The normalized elasticity modulus E*(•) increases in the transitional zone with a logarithmic slope (+0.18) and the normalized viscosity modulus η*(▴) decreases with a logarithmic slope (−0.24), the two with a correlation coefficient R2(=0.99).
Figure 4

Normalized viscosity modulus η* and elasticity modulus E* of the overall VTS model as a function of the normalized forced frequency f* in a double logarithmic scale with the two zones of predominant elastic (f*<1) and viscous (f*>10) effects, and the transitional zone (0.1<f*<10). The normalized elasticity modulus E*(•) increases in the transitional zone with a logarithmic slope (+0.18) and the normalized viscosity modulus η*(▴) decreases with a logarithmic slope (−0.24), the two with a correlation coefficient R2(=0.99).

Grahic Jump Location
+Normalized time constant τ* of the overall VTS as a function of the normalized internal tension (corresponding to the initial strain of the elastic cable) T* for three different normalized element length L* (=100, 1000, 10,000) in the transitional zone. τ* decreases following a mean power law given by (τ*∼T*−0.42; R2=0.98) whatever L*. The values of τ* decrease within a small range of [0.1-1] when T* increases by two orders of magnitude, whatever the value of L*.
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Normalized time constant τ* of the overall VTS as a function of the normalized internal tension (corresponding to the initial strain of the elastic cable) T* for three different normalized element length L* (=100, 1000, 10,000) in the transitional zone. τ* decreases following a mean power law given by (τ*∼T*−0.42; R2=0.98) whatever L*. The values of τ* decrease within a small range of [0.1-1] when T* increases by two orders of magnitude, whatever the value of L*.
Figure 5

Normalized time constant τ* of the overall VTS as a function of the normalized internal tension (corresponding to the initial strain of the elastic cable) T* for three different normalized element length L* (=100, 1000, 10,000) in the transitional zone. τ* decreases following a mean power law given by (τ*∼T*−0.42; R2=0.98) whatever L*. The values of τ* decrease within a small range of [0.1-1] when T* increases by two orders of magnitude, whatever the value of L*.

Grahic Jump Location
+Normalized elasticity modulus E* of the overall viscoelastic tensegrity structure as a function of the normalized internal tension T* for three different normalized element length L* (=100, 1000, 10,000) in the transitional zone. E* increases with T* following a power law E*∼T*0.52(R2=0.97) whatever the value of L*. Note that the values of E* decreases proportionally to the inverse of (L*2) in the overall range of T*.
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Normalized elasticity modulus E* of the overall viscoelastic tensegrity structure as a function of the normalized internal tension T* for three different normalized element length L* (=100, 1000, 10,000) in the transitional zone. E* increases with T* following a power law E*∼T*0.52(R2=0.97) whatever the value of L*. Note that the values of E* decreases proportionally to the inverse of (L*2) in the overall range of T*.
Figure 6

Normalized elasticity modulus E* of the overall viscoelastic tensegrity structure as a function of the normalized internal tension T* for three different normalized element length L* (=100, 1000, 10,000) in the transitional zone. E* increases with T* following a power law E*∼T*0.52(R2=0.97) whatever the value of L*. Note that the values of E* decreases proportionally to the inverse of (L*2) in the overall range of T*.

Grahic Jump Location
+Normalized viscosity modulus η* of the overall tensegrity model as a function of the initial internal tension T* for three different values of L* (=100, 1000, 10,000) in the transitional zone. Viscosity modulus η* is shown to increase nonsignificantly (logarithmic slope ≤+0.1;R2=0.98) with increasing T* for the three values of L*.
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Normalized viscosity modulus η* of the overall tensegrity model as a function of the initial internal tension T* for three different values of L* (=100, 1000, 10,000) in the transitional zone. Viscosity modulus η* is shown to increase nonsignificantly (logarithmic slope ≤+0.1;R2=0.98) with increasing T* for the three values of L*.
Figure 7

Normalized viscosity modulus η* of the overall tensegrity model as a function of the initial internal tension T* for three different values of L* (=100, 1000, 10,000) in the transitional zone. Viscosity modulus η* is shown to increase nonsignificantly (logarithmic slope ≤+0.1;R2=0.98) with increasing T* for the three values of L*.

Grahic Jump Location
+Normalized elasticity modulus E* as a function of the normalized length L* for five different values of the internal tension T*(=◆0.005,∎0.05,▴0.25,◆0.5,•0.75) in the transitional zone. Note that the curves show a strictly negative logarithmic slope (−2) whatever the value of T*.
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Normalized elasticity modulus E* as a function of the normalized length L* for five different values of the internal tension T*(=◆0.005,∎0.05,▴0.25,◆0.5,•0.75) in the transitional zone. Note that the curves show a strictly negative logarithmic slope (−2) whatever the value of T*.
Figure 8

Normalized elasticity modulus E* as a function of the normalized length L* for five different values of the internal tension T*(=◆0.005,∎0.05,▴0.25,◆0.5,•0.75) in the transitional zone. Note that the curves show a strictly negative logarithmic slope (−2) whatever the value of T*.

Grahic Jump Location
+Normalized viscosity modulus η* as a function of the normalized length L* for five different values of the internal tension T*(=◆0.005,∎0.05,▴0.25,◆0.5,•0.75) in the transitional zone. Note that the curves show a strictly negative logarithmic slope (−2) whatever the value of T*.
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Normalized viscosity modulus η* as a function of the normalized length L* for five different values of the internal tension T*(=◆0.005,∎0.05,▴0.25,◆0.5,•0.75) in the transitional zone. Note that the curves show a strictly negative logarithmic slope (−2) whatever the value of T*.
Figure 9

Normalized viscosity modulus η* as a function of the normalized length L* for five different values of the internal tension T*(=◆0.005,∎0.05,▴0.25,◆0.5,•0.75) in the transitional zone. Note that the curves show a strictly negative logarithmic slope (−2) whatever the value of T*.

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