Effect of the object 3D shape on the viscoelastic testing in optical tweezers

Lingyao Yu; Yunlong Sheng

doi:10.1364/OE.23.006020

1. Introduction

The ability of biological cells to react to mechanical stimuli is critical for basic cellular activities like migration, growth, differentiation, and cell lineage development. To investigate the biomechanics of the cells, experimental tests have been conducted using micropipette aspiration, flow chamber, microplate traction, micropillar plat, atomic force microscopy, cytoindentation, magnetic tweezers, optical tweezers and optical stretcher etc. A great body of modelling for a variety of cell mechanical testing has been performed. In addition to classical mechanical testing and modeling, difficulties in the cell mechanics lie in the fact that as biophysical entities the living cells can continuously remodel their internal structure and the mechanical properties of their material types to adapt to the physiological environment and the external stimuli. Moreover, by the mechano-transduction effect, the mechanical stress and strain may be converted to biochemical signals transmitted within the cell to change its mechanical properties. Thus, progress in the cell mechanics research requires more detailed and accurate modeling for each new testing technique [1–3].

In this paper we focus on the application of the optical tweezers and optical stretcher for investigating cell mechanical properties, and on the effect of the object 3D morphological shape on the viscoelastic testing in the optical tweezers. The optical tweezers and stretcher were proposed for high throughput label-free cell flow cytometry and cell sorting using the cell deformation and its material viscoelasticity as biomarkers. Historically, the viscoelastic behavior of the biological cells has been modeled by the spring-dashpot network model or power-law model [3–5]. Moreover, in most optical tweezers and optical stretcher experiments the radiation stress applied to the cell was mostly 1D, and only the 1D elongation of the cell was measured. Therefore, in several recent publications the cell viscoelastic properties were still modeled by the spring-dashpot network model or power-law model without considering the morphological shapes of the cells [6, 7]. However, we must be aware that the deformed cells are 3D. The spring-dashpot model or power-law model can characterize only the homogeneous, isotropic viscoelastic material, but not the cell itself. In this paper we report the mechanical analysis and full 3D analysis with the finite element method (FEM) for both creep testing and dynamic testing. We show that the object deformations are different significantly for different 3D object morphological shapes, in the creep testing. We find that in the dynamic testing the loss tangent measurable directly in the experiment is not sensitive to the 3D shape of the object. The relation between the loss tangent and the oscillation frequency can characterize the trapped cells. But the stress-strain hysteresis loop still depends on the 3D object shape, because of the involvement of the value of the strain oscillation. Our analysis is closely related to the experimental works in the recent publications and can fit to experimental data reported in Ref [8].

2. Creep response to step load

A suspended cell or other biological object can be trapped and stretched by the dual trap optical tweezers, optical jumping tweezers or the optical stretcher [9–13]. The latter consists of two non-focused counter-propagating laser beams. The experiments for both large and small deformation of the erythrocyte have been successfully modeled for computing the radiation stress and the cell deformation, mostly with the elastic membrane and liquid core model of the cell of 3D spherical [12] or biconcave shapes [14]. The finite element method was used in the numerical analysis [13, 15, 16]. Recently, the viscoelastic property of the cell has been considered in investigation [17–19] taking into account the temporal behavior of the cell. The viscoelastic testing includes measuring the strain creep with the given stress and the stress relaxation with given strain, and the dynamic testing. In recent publications reporting the experimental testing, the cell viscoelastic behavior was studied with the spring-dashpot network model or the power-law model, which are in fact the 1D viscoelastic material models [6–8]. The spring-dashpot model was even used in the stiffness testing of the adherent cells, which were seeded on the fibronectin (FN) gel coated substrate and stretched by a trapping beam via a FN-coated microbead bounded to the cell surface [20]. However, although the homogeneous and isotropic viscoelastic material is modeled by the 1D material models, the trapped and deformed object is 3D. Indeed, in the optical tweezers and stretcher experiments the radiation forces to the cell are 1D approximately, and only 1D elongations of the cell are measured. This does not mean that the 3D object can be modeled as a 1D object. Even if the cell viscoelastic behavior was investigated using the 1D material model in early days, in the more detailed and accurate investigation the 3D morphological shape of the object must be taken into account.

To show the effect of the cell 3D shape to the experimental result, we performed the FEM simulation for a creep testing of three objects filled with the same viscoelastic materials but had different morphological shapes. The three objects were cylinder, biconcave and sphere, as shown in Fig. 1. The creep testing uses the transient load. When a step stress is applied and is maintained constant afterward, the object will deform continuously following an exponential creep function. The FEM was fully 3D and based on the continuum mechanics with the material filled in the objects described by the 1D spring-dashpot network model. To analyze the effect of the 3D shape of the object on the experimentally measured cell’s elongation, we assume that a step stress, σ₀ = 1 Pa in (+/−) x-direction was applied to the two extremities of the objects. The stress magnitude was the same for the three objects and with the same loading area to stretch the objects. We computed the time-dependent creep behaviors of three objects using the full 3D FEM, and recorded only the elongations in the x-direction of the three objects, as that done in most experiments. The cylinder had a diameter of 2.3 μm and a length of 7.8 μm along the x-axis. The biconcave shaped object was built according to the analytical expression given in [14], with the diameter of the platelet disc of 7.8 μm. The sphere had a diameter of 7.8 μm. The loading areas were kept the same as 4.2 μm² for the three objects, so that the total forces were the same for the three objects. For the cylinder, the loading region was the entire end face. For the biconcave object, it was the region of |x| ≥ 3.6 μm. For the sphere, it was the region |x| ≥ 3.73 μm. The material was linear, viscoelastic, homogeneous and isotropic and was modeled with the generalized Maxwell model, which consists of one main elastic branch of the spring elastic modulus G and N parallel spring-dashpot branches of the spring modulus G_m and the relaxation time $τ_{m}$ . The generalized Maxwell model contains the N + 1 branches to fit various stress and strain relations in the experimental results. We chose the standard linear solid (SLS) model, which is the generalized Maxwell model with N = 1. The strain was solved by the constitutive equation of the SLS model [4] and written by [13]

ε (t) = \frac{σ_{0}}{G} - \frac{σ_{0} G}{(G + G_{1}) G_{1}} e^{- t G_{1} G / [η_{1} (G_{1} + G)]}

The viscoelastic parameters were set as G₁ = 156 Pa, G = 15.6 Pa, η₁ = 15.6 Pa·s, and the relaxation time

τ_{1} = G_{1} / η_{1} = 0.1 s

. Figure 2 shows the simulation results. As the applied stress was uniform over the cross section of the cylinder the problem is 1D. For the cylinder of L₀ = 7.8 μm length, the result was in perfect agreement with the creep function of the SLS model. At t = 0 the strain jumped to ε(0) = σ₀/(G₁ + G) = 0.58%. At the equilibrium state the strain was ε(t_∞) = σ₀/G = 6.4%, with the final elongation of L(t_∞) = L₀ε(t_∞) = 0.5 μm.

Fig. 1 Three morphologically different objects: cylinder (a); biconcave (b); and sphere (c).

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Fig. 2 Creep behaviors of three 3D objects and 1D SLS model.

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The final elongations of the three objects differed significantly, as seen easily in Fig. 2. The cylinder experienced the largest elongation, while the sphere had the smallest deformation in x-direction. Table 1 shows a comparison of the elongations of the three objects at the equilibrium state. This result implies that the 1D spring-dashpot material model cannot predict the elongation of a 3D object correctly. The deformation of a 3D object does not depend only on the viscoelastic parameters and the loadings, but also on its 3D shape. In fact, according to the continuum mechanics when an object is acted upon by external forces at its surfaces, the internal contact forces, which act on the virtual internal interfaces of the body elements, are generated as a result of the mechanical interactions between the body elements, and are transmitted from point to point inside the body to balance the external loads. The continuum mechanics assumes that the distribution of the internal contact forces throughout the volume of the object is continuous. The internal contact stress is associated to the local strain according to the viscoelasticity of the material, and the 3D deformation of the object is computed from the local strains in the object through the constitution equations.

Table 1. Characteristic creep testing for three solid body objects

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To explain the results shown in Fig. 2 one may simply imagine that the sphere physically contains the volume of the cylinder. Therefore, while the cylinder was deformed in free space, the same virtual cylinder contained in the sphere was attached with the surrounding material and had a reduced elongation in x-direction because of the resistance to elongation of the surrounding material, which bored also the external loads. The biconcave object had an elongation less than that of the cylinder, but larger than that of the sphere, as the biconcave object did not physically contain the entire cylinder and there was less material attached to that virtual cylinder.

Figure 3 shows the component σ_xx of the 3D stress tensor in the cylinder, biconcave object and sphere under the step stress load σ₀ at time t = 8 s. For the cylinder, σ_xx was uniformly distributed through the whole volume, forming a perfect 1D model. For the sphere, however, the local stress σ_xx on the x-axis decreased quickly from σ₀ = 1 Pa in the loading surfaces to 0.2 Pa at the center of the sphere. The σ_xx was dispersed into the whole volume of the sphere depending on the distance from the x-axis. In the cross section passing through the center of the sphere and parallel to the yz-plane the local stress contained only the σ_xx components for the raison of geometrical symmetry. In other cross sections parallel to the yz-plane the local stress tensor contained other directional components. The integration of σ_xx over each cross section parallel to the yz-plane would be equal to the external load A₀σ₀ with the loading area A₀, so that the σ_xx decreased along the x-axis from the loading surfaces to the center of the sphere. At the sphere center σ_xx was minimal as the area of the cross section passing through the center of the sphere was a maximum. For a biconcave object the local stress σ_xx in the x-axis also decreased from the loading surface to the center of the object, and was also dispersed into the object volume depending on the distance from the x-axis. There was a concentration of σ_xx in the concaved region, as shown in Fig. 3, so that the elongation of the biconcave object was less than that of the cylinder, but larger than that of the sphere.

Fig. 3 3D distribution of local stress σ_xx component in three solid body objects.

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Then, the same experiments were performed for the same three objects, but with the viscoelastic membrane model of the cell. The cell membrane with the underneath spectrin network play an important role in the cell deformation, but the liquid cytosol dose not present any resistance to the deformation. Figure 4 shows the local stress σ_xx distribution on the membrane of three objects, whose membranes were modelled with the viscoelastic material with the parameters as G = 500 Pa, G₁ = 1500 Pa, and τ₁ = 0.1 s as well as the thickness of membrane h = 0.2 μm. The loading stress of 1 Pa and the loading areas were in the same regions of the three objects as before. The local stresses in the membranes were much larger than that with the solid model because of the small thickness of the membranes. As the membrane had a finite thickness h = 0.2 μm the stress σ_xx had opposite signs on the two sides of the shell. In addition, results in Table 2 show that sphere has the largest elongation while the cylinder experienced the smallest elongation. The elongations of three objects were different significantly.

Fig. 4 3D Volumetric distribution of local stress σ_xx component in the membrane of three objects.

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Table 2. Characteristic creep testing for three objects of viscoelastic membrane

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3. Dynamic response to sinusoidal load

Some recent experiments with optical tweezers performed the dynamic testing for viscoelastic object [8, 21]. In the dynamic testing the external stimulus is vibrating as a sinusoidal function of time. For the linear viscoelastic material, the strain is also a sinusoidal function of time, and there is a phase shift between the stress and strain.

In the experiments of Ref [8], two trapping beams were focused at the edges of a single erythrocyte. One trap was static. Another trap was oscillated in the position of its focal point in a frequency range from 50 Hz to 1 kHz. Two extra laser beams were used to detect the vibrations of the two edges of the cell using two quadrant photodiodes. The phase difference in the displacements of the two opposite edges of the cell was recorded as a function of the frequency of the oscillation. In a linear optical tweezers Ref [21]. a laser beam was focused one-dimensionally to create a 1D stretching stress distribution on the cell. In another dimension was the microfluidic channel allowing high-throughput cell sorting. The stress was vibrating with the laser power oscillating in a frequency range of nearly three orders of magnitude. The cell’s deformation was recorded by a high speed CCD camera. The phase shift between the stimulus and cell response as a function of the oscillation frequency was found to be able to distinguish the infected and uninfected erythrocytes. Slightly different dynamic test was performed in Ref [22]. with a triangle wave trap. Two beads were attached to two extremities of an erythrocyte. The cell was stretched by moving one bead at given speeds to a given displacement, while the trapping force was monitored by the displacement of another stably trapped bead with respect to the known laser position in order to determine the stiffness of the cell. This was a stress relaxation test.

An advantage of the dynamic testing is that as all the stress and strain components are harmonic functions of time, the differential equations governing the viscoelastic behavior of the object become simple algebra equations. Moreover, the phase shift between the external load and the measured deformation, as the phase of the complex dynamic modulus, is less sensitive to the 3D shape of the object. In fact, in the 3D analysis, the stress tensor $σ$ consists of three diagonal elements representing the pressure components normal to the surfaces of an infinitesimal element, and other six diagonally symmetric elements representing the shear stress components. The deviatory stress tensor is left over after the pressure components are subtracted from the stress tensor, $s_{d} = σ + p I$ , where $I$ is the identity tensor and $p = (1 / 3) t r a c e (σ)$ . Correspondingly, the deviatory strain tensor related to the object deformation is presented as $ε_{d} = ε - (1 / 3) ε_{v} Ι$ , with the volumetric strain $ε_{v} = (1 / 3) t r a c e (ε)$ . On the other hand, in the generalized Maxwell model, the springs are perfectly elastic with $s_{m} = G_{m} ε_{m}$ where G_m is the elastic modulus of the m-th spring, and the dashpots are perfectly viscous with the velocity of the strain proportional to the stress $s_{m} = η_{m} {\dot{ε}}_{m}$ where $η_{m}$ is the viscosity of the m-th dashpot and is related to the relaxation time by $η_{m} = G_{m} τ_{m}$ . The N + 1 parallel branches share the total stress, so that the 3D deviatory stress tensor can be expressed as

s_{d} = G ε_{d} + \sum_{m = 1}^{N} G_{m} q_{m}

where q_m is the extension of the m-th spring. Moreover, the spring and dashpot in the m-th branch receive the same stress s_m = G_mq_m, and their respective extensions are summed up. As all the branches experience the same strain, we have

{\dot{ε}}_{d} = \frac{{\dot{s}}_{m}}{G_{m}} + \frac{s_{m}}{η_{m}}, a n d G_{m} {\dot{ε}}_{d} = {\dot{q}}_{m} + \frac{q_{m}}{τ_{m}}

In the dynamic testing the relations between the stress and strain are much simplified in the frequency domain as

{\tilde{s}}_{d} = G_{0} {\tilde{ε}}_{d}

where

{\tilde{s}}_{d}

and

{\tilde{ε}}_{d}

are the complex amplitudes of the vibrating deviatory stress tensor

s_{d} = i m a g {{\tilde{s}}_{d} \exp (j ω t)}

and the deviatory strain tensor

ε_{d} = i m a g {{\tilde{ε}}_{d} \exp (j ω t)}

, respectively. Note that the tensor Eq. (4) holds for each component in the deviatory stress tensor and strain tensor at all locations in the 3D object. With the time derivatives of

s_{d}

and

ε_{d}

easily obtained, the constitutive Eqs. (2) and (3) are solved for the complex modulus G₀, which characterizes the material viscoelasticity and is expressed with its real part as the storage modulus

G_{0}^{'}

, and imaginary part as the loss modulus

G_{0}^{"}

:

G_{0}^{'} = G + \sum_{m = 1}^{N} G_{m} \frac{{(ω τ_{m})}^{2}}{1 + {(ω τ_{m})}^{2}} a n d G_{0}^{''} = \sum_{m = 1}^{N} G_{m} \frac{ω τ_{m}}{1 + {(ω τ_{m})}^{2}}

In the SLS with N = 1, Eq. (5) becomes

G_{0}^{'} = G + G_{1} \frac{{(ω τ)}^{2}}{1 + {(ω τ)}^{2}} a n d G_{0}^{''} = G_{1} \frac{ω τ}{1 + {(ω τ)}^{2}}

and the loss tangent is given by

\tan δ \equiv \frac{G_{0}^{"}}{G_{0}^{'}} = \frac{G_{1} ω τ_{}}{G [1 + {(ω τ)}^{2}] + G_{1} {(ω τ)}^{2}} = \frac{1}{[1 + (1 / K)] ω τ + [1 / (K ω τ)]}

From Eqs. (5)–(7) the loss tangent is a function of the angular frequency ω, the relaxation time τ, and the ratio K = G₁/G. For a given viscoelastic material the loss tangent depends only on the oscillation frequency ω, and is independent of the values of the stress or the strain, and therefore independent of the 3D shapes of the objects. In the experiments the phase shift δ between the stress and strain as a function of the frequency is directly measurable, provided that the local displacement measured in the experiment represents the local strain in the region where the stress is applied. This is true for 1D objects, but is approximate for the objects of other shapes, where the local strains are not equal to the measured object elongation divided by their lengths.

We simulated the dynamic testing for the three 3D objects: cylinder, biconcave object and sphere with the FEM. The vibrating stress was applied at the same ending regions of the three objects as that in Section 2, but the opposite ending regions were fixed. The strain is calculated as the ratio of the total elongation of the objects in x-axis over the original length. The stress oscillation frequency varied from 200 Hz to 1000 Hz at an interval of 200 Hz. The three objects were the solid bodies and filled with the same viscoelastic material as that in the Section 2 with the SLS model. Considering the loss tangent ranging from 0 to 1 [8, 21] and the ratio of the spring elastic moduli K = G₁/G typically ranging from 0.1 to 7 [5, 23–27] by fitting to the creep or frequency-dependent dynamic testing results for biological cells to the SLS or to the generalized Maxwell models, we set K = G₁/G = 10. Ref [8]. reported the respond time τ’ = 640 μs for the normal erythrocytes measured in their dynamic testing. In Ref [8]. the two laser traps and the cell were all modeled by the Kelvin solid model and the resulting relation for the loss tangent was tanδ = ωτ’/2π. In the SLS model for the oscillation frequency of 1 KHz as $ω τ$ = 0.07 and K = 10, Eq. (7) gives $\tan δ \approx K ω τ$ , so that we set the relaxation time $τ = τ' / (2 π K)$ = 11 μs. By this setting our model could fit the experimental results in Ref [8]. The object models were meshed and solved with the total number of 23,739, 80,397, and 93,717 degrees of freedom for the cylinder, the sphere and the biconcave objects respectively, with the meshes including the tetrahedral, the triangular, the edge and the vertex elements, which was performed by using Comsol multiphysics^TM.

In the experiments [8, 21] the loss tangent was obtained by measuring the phase delay between the strain and stress in the time domain, as shown in Fig. 5(a). The loss tangent can be also computed in the time domain by plotting the stress-strain curve, as a hysteresis loop. Figure 5(b) was obtained in the FEM simulation for the cylinder in the time range of 0 to 3 ms at a time resolution of 0.01 ms with the loading oscillation period of 1 ms. From the hysteresis loop the loss tangent is computed by $\tan δ$ = OA/OB in the figure, as explained in Appendix. Note that Fig. 5(b) was plot from real data with all positively valued stress, so that the origin O is shifted to (ε₀/2, σ₀/2). Figure 6(a) shows the hysteresis loops for the three objects of the same linear viscoelastic material computed by the FEM. A careful examination shows that the three ellipses in Fig. 6(a) have the same value of the loss tangent. In Fig. 6(b), the purple line is the loss tangent as a function of oscillatory frequency calculated with Eq. (7) from the 1D SLS material model. The points represent the loss tangents of the 3D cylinder, biconcave and sphere objects resulted from the values of tanδ = OA/OB in FEM simulation. It is noted that the loss tangents, for the three objects were overlapped for the frequency range up to 1KHz, and were in perfect agreement with Eq. (7). Apparently, compared to the elongation in creep, the loss tangent is less sensitive to the 3D morphological shapes of the object.

Fig. 5 (a): Stress and strain as functions of time with arbitrary units; (b): Stress versus strain curve of the simulated cylinder loaded by a sinusoidal load at a frequency of 1 KHz with the illustrations of mean lines and intercepts.

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Fig. 6 (a): Stress versus strain curves of three 3D objects in the dynamic testing at 1 KHz by FEM. (b): Loss tangent as a function of frequency computed by the SLS model (purple line); and by FEM for three 3D objects.

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The three loops in Fig. 6(a) contain different areas, representing different dissipated energies in the three objects. Moreover, the real parts $G_{0}^{'}$ of the complex modulus, represented by the slopes from origin to the maximum strain point on the loops, as explained in Appendix, had different values for the three objects, although they were of the same viscoelastic material. This error was caused by the fact that in the dynamic testing, the value of $G_{0}^{'}$ obtained from the stress-strain curve depends on the values of the maximal strain ε₀. As the three objects were of different 3D morphological shapes, their deformations and elongations were different significantly. Note that the measurement of the phase delay alone in the dynamic testing may be not sufficient to determine the complex modulus of the viscoelastic materials. A second separated measurement for the cell elongation under the static forces and the cell elasticity corresponding to G, was performed in Ref [8]. This measurement however is still sensitive to the 3D morphological shape of the object.

4. Conclusion

We have conducted the full 3D finite element analysis for testing the viscoelastic properties of biological cells in both creep and dynamic regimes in the optical tweezers and stretcher with the material in the cells modeled by the general Maxwell models. We have shown that the deformations of the trapped 3D objects of different morphological shapes are different significantly, when a static load is applied to the object.

Although historically, the biological cells have been modeled by the spring-dashpot network model in the investigation of their viscoelastic behavior, and the radiation stress applied to the cell is mostly 1D and only the 1D elongation of the cell was measured in most experiments with the optical tweezers and stretcher, we must be aware that the deformed cells are 3D. The spring-dashpot model or the power-law model, which are 1D models, can only characterize the viscoelastic property of the homogeneous, isotropic viscoelastic material, but not that of the cell itself. The results from experimental testing and theoretical modeling with the 1D spring-dashpot model or the power-law model of the cell would correspond only to those of a 1D objects as the cylinder, but not of the real 3D cell.

In the case of dynamic testing with the sinusoidal loadings on the object, we found that the loss tangent is not sensitive to the 3D morphological shape of the object. The loss tangent is related to the phase of the dynamic complex modulus of the viscoelastic material and can be measured directly in the experiments in the time domain. Its relation to the oscillation frequency can be used to characterize the trapped cells. However, the stress-strain curve, as the hysteresis loop, still depends on the 3D shape of the objects, because the hysteresis loop depends on the value of the strain oscillation amplitude.

Appendix

With the stress expressed as $σ = σ_{0} \sin ω t$ and the strain as $ε = ε_{0} \sin (ω t - δ)$ the work done by the external load in one oscillation cycle is expressed as

W = \int_{0}^{ε_{0}} σ d ε = \int_{0}^{ω / 2 π} σ \frac{d ε}{d t} d t = ε_{0} σ_{0} (\frac{\cos δ}{2} + \frac{π \sin δ}{4})

The first term represents the stored energy and the second term represents dissipated energy, which equals to zero if the phase delay δ = 0. Thus, the ratio of the dissipated energy over the stored energy is equal to (π/2) tanδ, representing the loss tangent. In Fig. 5(b), at the intersect point A, σ_A = σ₀sinωt_A = 0, ωt_A = π and ε_A = ε₀sin δ, at point D the stress is the maximum σ_D = σ₀, so that ωt_D = π/2, ε_B = ε₀cos δ. Thus,

\tan δ

= OA/OB. In Fig. 5(b) the maximum strain occurs at point P where ε_P = ε₀sin(ωt_P-δ) = ε₀, ωt_p = π/2 + δ and σ_P =

G_{0}^{}

ε_P = ε_{0 $G_{0}^{'}$} = σ₀cosδ, so that the stored energy corresponds to the area in the triangle OPQ. Moreover, from the relation

σ_{0}^{*}

= σ’ + iσ” = σ₀cosδ + iσ₀sinδ, we have σ’ = σ₀cosδ = ε_{0 $G_{0}^{'}$}, so that

G_{0}^{'}

is represented as the slope from origin to the maximum strain point P.

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	Diameter or length in x-axis (μm)	Loading area (μm²)	Loading stress (Pa)	Volume (μm³)	Final elongation (μm)
Cylinder	7.8	4.2	1	33	0.50
Biconcave	7.8	4.2	1	93	0.35
Sphere	7.8	4.2	1	248	0.21

	Membrane thickness (μm)	Loading area (μm²)	Loading stress (Pa)	Volume (μm³)	Final elongation (μm)
Cylinder	0.2	4.2	1	11.5	0.16
Biconcave	0.2	4.2	1	25.1	0.19
Sphere	0.2	4.2	1	36.3	0.29

	Diameter or length in x-axis (μm)	Loading area (μm²)	Loading stress (Pa)	Volume (μm³)	Final elongation (μm)
Cylinder	7.8	4.2	1	33	0.50
Biconcave	7.8	4.2	1	93	0.35
Sphere	7.8	4.2	1	248	0.21

	Membrane thickness (μm)	Loading area (μm²)	Loading stress (Pa)	Volume (μm³)	Final elongation (μm)
Cylinder	0.2	4.2	1	11.5	0.16
Biconcave	0.2	4.2	1	25.1	0.19
Sphere	0.2	4.2	1	36.3	0.29

Effect of the object 3D shape on the viscoelastic testing in optical tweezers

Abstract

1. Introduction

2. Creep response to step load

3. Dynamic response to sinusoidal load

4. Conclusion

Appendix

References and links

Cited By

Figures (6)

Tables (2)

Equations (8)

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