## Dynamic deformation of red blood cell in Dual-trap Optical Tweezers

Optics Express, Vol. 18, Issue 10, pp. 10462-10472 (2010)

http://dx.doi.org/10.1364/OE.18.010462

Acrobat PDF (1388 KB)

### Abstract

Three-dimensional dynamic deformation of a red blood cell in a dual-trap optical tweezers is computed with the elastic membrane theory and is compared with the experimental results. When a soft particle is trapped by a laser beam, the particle is deformed depending on the radiation stress distribution whereas the stress distribution on the particle in turn depends on the deformation of its morphological shape. We compute the stress re-distribution on the deformed cell and its subsequent deformations recursively until a final equilibrium state solution is achieved. The experiment is done with the red blood cells in suspension swollen to spherical shape. The cell membrane elasticity coefficient is obtained by fitting the theoretical prediction with the experimental data. This approach allows us to evaluate up to 20% deformation of cell’s shape

© 2010 OSA

## 1. Introduction

1. A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature **330**(6150), 769–771 (1987). [CrossRef] [PubMed]

3. J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. **81**(2), 767–784 (2001). [CrossRef] [PubMed]

4. M. Dao, C. T. Lim, and S. Suresh, “Mechanics of the human red blood cell deformed by optical tweezers,” J. Mech. Phys. Solids **51**(11-12), 2259–2280 (2003). [CrossRef]

5. S. Hénon, G. Lenormand, A. Richert, and F. Gallet, “A new determination of the shear modulus of the human erythrocyte membrane using optical tweezers,” Biophys. J. **76**(2), 1145–1151 (1999). [CrossRef] [PubMed]

6. P. J. H. Bronkhorst, G. J. Streekstra, J. Grimbergen, E. J. Nijhof, J. J. Sixma, and G. J. Brakenhoff, “A new method to study shape recovery of red blood cells using multiple optical trapping,” Biophys. J. **69**(5), 1666–1673 (1995). [CrossRef] [PubMed]

8. G. B. Liao, P. B. Bareil, Y. Sheng, and A. Chiou, “One-dimensional jumping optical tweezers for optical stretching of bi-concave human red blood cells,” Opt. Express **16**(3), 1996–2004 (2008). [CrossRef] [PubMed]

6. P. J. H. Bronkhorst, G. J. Streekstra, J. Grimbergen, E. J. Nijhof, J. J. Sixma, and G. J. Brakenhoff, “A new method to study shape recovery of red blood cells using multiple optical trapping,” Biophys. J. **69**(5), 1666–1673 (1995). [CrossRef] [PubMed]

8. G. B. Liao, P. B. Bareil, Y. Sheng, and A. Chiou, “One-dimensional jumping optical tweezers for optical stretching of bi-concave human red blood cells,” Opt. Express **16**(3), 1996–2004 (2008). [CrossRef] [PubMed]

9. P. B. Bareil, Y. Sheng, and A. Chiou, “Local scattering stress distribution on surface of a spherical cell in optical stretcher,” Opt. Express **14**(25), 12503 (2006). [CrossRef] [PubMed]

10. J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. **81**(2), 767–784 (2001). [CrossRef] [PubMed]

8. G. B. Liao, P. B. Bareil, Y. Sheng, and A. Chiou, “One-dimensional jumping optical tweezers for optical stretching of bi-concave human red blood cells,” Opt. Express **16**(3), 1996–2004 (2008). [CrossRef] [PubMed]

9. P. B. Bareil, Y. Sheng, and A. Chiou, “Local scattering stress distribution on surface of a spherical cell in optical stretcher,” Opt. Express **14**(25), 12503 (2006). [CrossRef] [PubMed]

10. J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. **81**(2), 767–784 (2001). [CrossRef] [PubMed]

## 2. Stress distribution

*x*= ± D/2,

*y*=

*z*= 0), respectively. As the RBC diameter 2ρ ~7

*µm*is larger than the wavelength λ = 1.06

*µm*, the ray optics approach was used to compute the radiation pressure approximately [1

1. A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature **330**(6150), 769–771 (1987). [CrossRef] [PubMed]

3. J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. **81**(2), 767–784 (2001). [CrossRef] [PubMed]

9. P. B. Bareil, Y. Sheng, and A. Chiou, “Local scattering stress distribution on surface of a spherical cell in optical stretcher,” Opt. Express **14**(25), 12503 (2006). [CrossRef] [PubMed]

*A*is the area covered by the beam,

*E*is the incident beam energy, P is the beam power,

_{i}*c*is the speed of light,

*n = n*where

_{2}/n_{1}*n*is the refractive index of the medium surrounding the cell, and

_{1}= 1.335*n*=

_{2}*1.378*is that inside the cell,

*T*and

*R*are the Fresnel transmission and reflection coefficient respectively, and

*n*the stress

_{2}/n_{1}>1,*z*= 0. However, we ignored this slight offset in the calculation of the stress and the deformation, for the sake of simplicity. The mathematical formulas for calculating the stress on the sphere by a shifted focusing beam using the ray-tracing approach are given in Appendix A.

3. J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. **81**(2), 767–784 (2001). [CrossRef] [PubMed]

**14**(25), 12503 (2006). [CrossRef] [PubMed]

*Q*,

_{x}*Q*and

_{y}*Q*in Eq. (A1) were not required. The Fresnel reflection coefficients are still computed with Eq. (A2). The two approaches gave identical result for the spherical cell. However, the latter approach is simpler. Moreover, when the cell is deformed, the vector

_{z}*T*and neglecting the absorption by the cytosol. As

*n*~0.043 is very small, the reflectance

_{2}-n_{1}*R*at the cell/buffer interface is small, we neglected the radiation stress associated with the third and subsequent reflections inside the cell.

*x*= ± D/2,

*y*=

*z*= 0), respectively and then added the two stress distributions. The results are shown in Fig. 1 When the beam separation

*D*= 0, the stress distribution is revolutionary symmetric with respect to the z-axis, and there is a crown-shaped region around the equator of the sphere in the

*x-y*plane, on which only few or no optical rays incident, so that the stress is low or equal to zero, as shown in Fig. 1(a). The width of this region depends on the numerical aperture of the trapping beam and the size of the cell. We then shift the beam focus along the x-axis. We observed that the stress distribution remains practically unchanged in the range of

*D*< 2 μm. In the view of projection along the

*z*axis the elongation of the cell along ± x axis is negligible for

*D*< 2 μm. This is in agreement with the experimental observation [8

**16**(3), 1996–2004 (2008). [CrossRef] [PubMed]

*D*= 1,27 – 2,54 μm. When

*D*continues to increase the stress is concentrated gradually towards the

*x-y*plane and towards the ±

*x*directions with the magnitude of the stress peaks increasing significantly, as shown in Fig. 1(b)–1(d).

11. T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A, Pure Appl. Opt. **9**(8), S196–S203 (2007). [CrossRef]

^{TM}). Then, the radiation stress was computed using the Maxwell stress tensor. As shown in Fig. 1(d)–1(f), the ray-tracing, the T-matrix and the FDTD method produced similar results with the computed peak stress at the first surface of the cell as 0.979, 0.923 and 0.980 N/m

^{2}, respectively, for the beam separation

*D*= 5.07 μm. The differences in the stress values are mostly due to the beam intensity distributions and the beam models. In fact, in the ray tracing, the trapping beam was modeled as a bundle of the focused optical rays with the Gaussian intensity distribution. In the FDTD a paraxial Gaussian beam model was used. On the other hand, in the GLMT the trapping beam was modeled as the TEM

_{00}(or LG

_{00}) beam satisfying the vector Helmholtz equation for the strongly focus beam and was then expanded by the multipole expansion into the orthogonal set of vector spherical wave-functions (VCWF) taking into account the high NA. Moreover, the expansion coefficients were determined by matching the far field distribution of the VSWF expansion to the far field Gaussian beam [12

12. T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transf. **79–80**, 1005–1017 (2003). [CrossRef]

*n*[13

13. M. Mansuripur, “Electromagnetic stress tensor in ponderable media,” Opt. Express **16**(8), 5193–5198 (2008). [CrossRef] [PubMed]

*n*= 1.335 in the buffer and

_{1}*n*= 1.378 inside the cell and the reflectance and transmittance,

_{2}*R*and

*T,*at the normal incidence as:

*n*and

_{2}> n_{1}*R*> 0, we have σ

_{rear}> σ

_{front}. However, the difference in σ

_{rear}and σ

_{front}has no direct consequence on our experiment, as only the cell’s elongation along the ±

*x*axis will be taken as a measure for cell’s deformation. In Section 4 we will choose the ray tracing approach and embed its Matlab

^{TM}codes into the Comsol multiphysics

^{TM}modules for computing the dynamic deformation with the finite element method, for the sake of simplicity.

## 3. Static deformations

14. F. Xu, J. A. Lock, G. Gouesbet, and C. Tropea, “Optical stress on the surface of a particle: homogeneous sphere,” Phys. Rev. A **79**(5), 053808 (2009). [CrossRef]

15. Y. C. Fung, “Theoretical considerations of the elasticity of red cells and small blood vessels,” Fed. Proc. **25**(6), 1761–1772 (1966). [PubMed]

*h*of the membrane and the cell radius

*ρ, h*/

*ρ*~1% [3

**81**(2), 767–784 (2001). [CrossRef] [PubMed]

*ρ*with only the radial stress σ

_{r}(φ,θ), the equilibrium equations are given by [16

16. P. B. Bareil, Y. Sheng, Y. Q. Chen, and A. Chiou, “Calculation of spherical red blood cell deformation in a dual-beam optical stretcher,” Opt. Express **15**(24), 16029–16034 (2007). [CrossRef] [PubMed]

*N*and

_{ϕ}*N*are the membrane forces per unit length, applied normally to the boundaries of a differential membrane element in the zenith and the azimuth directions, respectively, and

_{θ}*S*is the in-plane shear force per unit length, tangential to the boundaries of the element. One can solve

*N*,

_{ϕ}*N*and

_{θ}*S*by Eqs. (2a)–(2c), and then associate the stresses to the local strains prescribed by Hooke’s law in the limit of the linear elasticity. The deformation of the cell is described as displacement vectors of the material points on the membrane and can be solved from the local strain by the constitutive equations, which are for the spherical membrane where (

*u,υ,w*) are the displacement in the directions of the curvilinear coordinate system (

*U,V,W*) centered at the differential membrane element, with the

*W*-axis directed inward to the origin of the sphere and (

*U, V*) tangential to the zenith and the azimuth directions, φ and θ, and

*E*is the modulus of elasticity and

*v*is the Poisson coefficient representing the volume change of the membrane due to the deformation. For small deformation the membrane is isotropic and

*v*= 0.5 approximately.

_{r}= σ

_{r}(φ,θ) is not symmetric to the z-axis due to the lateral shift of the two beams. In this case we can still solve the equilibrium and the constitutive equations analytically by using the Fourier expansions with respect to θ. The mathematical detail is given in Appendix B.

## 4. Dynamic deformation

*D*is large and the cell’s deformation is significant, the re-distribution of the radiation stress on the deformed surface needs to be considered. Furthermore, the stress re-distribution will lead to cell’s re-deformation, which in its turn will lead to new re-distribution of the stress. The process continues until a final equilibrium state is reached for a given incoming optical field. In this case the equilibrium and the constitutive equations for the deformed cell can no longer take the simple forms as Eqs. (2) and (3) and analytical solutions would be more difficult. We used the finite element solution in Comsol

^{TM}structured mechanics module to compute the linear elastic deformations of the deformed cell. To implement the computation we embedded our Matlab

^{TM}codes for computing the radiation stress with the ray optics approach to the Comsol

^{TM}module.

*h*, a test value of the elasticity coefficient

*E*and the beam separation value

*D*, and we selected a uniform deformable moving mesh on the cell surface. As in our model the net radiation force applied to the cell is not zero, the cell’s position would be shifted in the computing. However, this shift is not of interest when computing cell’s deformation. We set the constraints that the 6 poles of intersection of the cell surface with the

*x*,

*y*and

*z*axes can move only along the

*x*,

*y*and

*z*axis respectively for preventing the cell from shift in the space. We launched the stationary solution solver in the Comsol

^{TM}. The stress distribution was first computed by our Matlab

^{TM}codes, and then the cell’s deformation under the computed stress distribution was computed by a linear solver SPOOLES. Once SPOOLES completed its execution, the cell was deformed along with the moving mesh, so that the stresses on the deformed mesh nodes were re-computed using the embedded Matlab

^{TM}codes. Then, the SPOOLES was launched again to compute the deformation under the new stress distribution of the deformed mesh nodes. The iterative process continued until a convergence of the solution was finally reached after 4-5 iterations, and the deformed cell reached a final equilibrium state. We found that the cell’s deformation computed in the first iteration of the linear solver SPOOLES with the stress distribution on the initially spherical cell is identical to the static deformation computed by the analytical method described in the Appendix B. This agreement validates our Comsol

^{TM}solution.

*D*, which are now ready to fit the experimental results.

## 5. Experiments

_{4}cw laser) for optical trapping passed through a beam expander (BE) which expanded and collimated the beam such that the beam diameter overfilled the back aperture of an oil immersion microscope objective (

*NA*= 1.25, 100X). The laser beam was split into two via a polarizing beam splitter (PBS) and recombined by a second polarizing beam splitter (PBS) to form two parallel trapping beams, focused in the focal plane of the objective lens; the optical powers of the two beams can be adjusted byrotating the half-wave (HW) plate. One of the laser beams passed through a telescope consisting of two lenses with 200

*mm*focal length. The first lens was mounted on a motorized translational stage (MS) for displacement in the direction transversal to the optical axis. The telescopic imaging arrangement transformed the transverse displacement of the first lens into a lateral shift of the focal spot of the trapping beam without any beam walk-off at the entrance pupil of the objective lens. The RBC was trapped and stretched in the focal plane. Wide-field images of the RBC were captured by a CCD camera for observation and analysis.

*D*between the focal spots of the two trapping beams was increased from 0 μm to 6.34 μm in 6 discrete steps. As the cell’s reaction to the external load is typically slow, in the experiments we increased the beam separation D stepwise and always allowed enough time for the cell to reach equilibrium before the next increase of D. In each step we measured the length of the major axis of the RBC along the direction of the beam separation with a pre-calibrated length-scale on the CCD image. The elongation was measured for the cell without drug treatment and with 1mM N-ethylmaleimide (NEM) treatment for 30 minutes as a function of the dual beam separation distance

*D*; Experimental micrographic image examples are shown in Fig. 5(a) and 5(b) for the RBCs without drug treatment and with 1mM NEM treatment for 30 minutes, and for the different dual beam separation distances respectively.

*x-y*plane are the ellipses elongated in the ±

*x*directions as observed in the experiments, and shown in Fig. 5(a) and 5(b). The cell’s deformation along the

*z*-axis was not readily observable in the experiments. The fitting parameters were the product of the membrane elasticity and the membrane thickness

*Eh*and the cell’s initial radius

*ρ*. As

*ρ*can vary in the experiments among the group of 30 samples, we took

*ρ*as a fitting parameter. We found

*Eh*= 15.99 (μN/m) and

*ρ =*3.554µm for the normal RBC’s and

*Eh =*24.39 (μN/m) and

*ρ =*3.697µm for the RBCs with the drug treatment. The corresponding shear modulus are

*Gh*=

*Eh*/2(1 +

*v*) = 5.33 and 8.13 (μN/m), respectively.

15. Y. C. Fung, “Theoretical considerations of the elasticity of red cells and small blood vessels,” Fed. Proc. **25**(6), 1761–1772 (1966). [PubMed]

## 6. Conclusion

^{TM}modules along with embedded codes to compute the stress distribution on the deformed cell. Our tool thus combines the geometrical optics and the structure mechanics into the single calculation. This allows computing the stress re-distribution and cell’s re-deformation in the iterations. Our theoretical predictions fit to the experimental data permitting differentiating the red blood cells with and without the drug treatment and obtaining the cell membrane’s elasticity coefficients. This approach allows us to evaluate large deformation of the cells in the dual-trap optical tweezers.

## Appendix A

*B*(

*r*,

*φ*,

*θ*) and refracted by the cell/buffer interface toward the focal point

*F*(

*x*= D/2,

*y*=

*z*= 0), as shown in Fig. 6. Take the refracted ray

*B*. In this plane the refracted angle may be computed, according to the Snell’s law, as

^{−1}(

*n*/

_{2}sinβ*n*). Then the vector

_{1}*ϕ*between

*x*-axis is given by

*ϕ*= tan

^{−1}((

*z*)

^{2}+y^{2}^{1/2}/

*x*). The Fresnel reflection coefficients are computed for the polarization normal to and parallel to in the incident plane, respectively, as

*T*=

*1*-

*R*were used in Eq. (1). The nominal power

*P*in Eq. (1) of the incident ray in the Gaussian beam is determined in the parallel plane to the

*x-y*plane that sections the trapping beam at the incident point

*B,*

*w*depends on the distance

*z*from the plane of section of the beam to the focal spot. Thus the stress σ(φ,θ) can be computed from Eqs. (1) and (A1).

## Appendix B

*N = N*+

_{φ}*S*and apply the Fourier series expansions with respect to θ as

*N*as

_{m}*L*=

*N*-

_{φ}*S*we obtain and solve the second differential equation for

*L*, which is the Fourier expansion coefficient of

_{m}*L*, in the similar ways as

*A*and

_{m}*B*are integration constants. Sum and subtract

_{m}*N*and

_{m}*L*allow us to obtain

_{m}*N*and

_{φ}*S*can be obtained by the inverse Fourier transform of Eq. (2) and

*N*is obtained from

_{θ}*N*and σ

_{φ}_{r}by Eq. (2c).

*u, υ, w*) in the similar ways as solving Eqs. (2a)–(2c) results in the Fourier series coefficients

*C*and

_{m}*D*are integration constants

_{m}*u, υ, w*) are reconstructed by the Fourier series with respect to

*θ*using (

*u*). The integration constants A

_{m}, υ_{m}, w_{m}_{m}, B

_{m}, C

_{m}, and D

_{m}are useful to reconstruct the function A, B, C and D, which were trimmed in order to keep the cell’s volume constant during the deformation as the liquid inside is incompressible.

## References and links

1. | A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature |

2. | S. M. Block, “Making light work with optical tweezers,” Nature |

3. | J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. |

4. | M. Dao, C. T. Lim, and S. Suresh, “Mechanics of the human red blood cell deformed by optical tweezers,” J. Mech. Phys. Solids |

5. | S. Hénon, G. Lenormand, A. Richert, and F. Gallet, “A new determination of the shear modulus of the human erythrocyte membrane using optical tweezers,” Biophys. J. |

6. | P. J. H. Bronkhorst, G. J. Streekstra, J. Grimbergen, E. J. Nijhof, J. J. Sixma, and G. J. Brakenhoff, “A new method to study shape recovery of red blood cells using multiple optical trapping,” Biophys. J. |

7. | W. G. Lee, H. Bang, J. Park, S. Chung, K. Cho, C. Chung, D.-C. Han, and J. K. Chang, “Combined microchannel-type erythrocyte deformability test with optical tweezers,” |

8. | G. B. Liao, P. B. Bareil, Y. Sheng, and A. Chiou, “One-dimensional jumping optical tweezers for optical stretching of bi-concave human red blood cells,” Opt. Express |

9. | P. B. Bareil, Y. Sheng, and A. Chiou, “Local scattering stress distribution on surface of a spherical cell in optical stretcher,” Opt. Express |

10. | J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. |

11. | T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A, Pure Appl. Opt. |

12. | T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transf. |

13. | M. Mansuripur, “Electromagnetic stress tensor in ponderable media,” Opt. Express |

14. | F. Xu, J. A. Lock, G. Gouesbet, and C. Tropea, “Optical stress on the surface of a particle: homogeneous sphere,” Phys. Rev. A |

15. | Y. C. Fung, “Theoretical considerations of the elasticity of red cells and small blood vessels,” Fed. Proc. |

16. | P. B. Bareil, Y. Sheng, Y. Q. Chen, and A. Chiou, “Calculation of spherical red blood cell deformation in a dual-beam optical stretcher,” Opt. Express |

17. | E. Ventsel, and T. Krauthammer, |

**OCIS Codes**

(140.7010) Lasers and laser optics : Laser trapping

(170.1530) Medical optics and biotechnology : Cell analysis

(170.4520) Medical optics and biotechnology : Optical confinement and manipulation

(350.4855) Other areas of optics : Optical tweezers or optical manipulation

**ToC Category:**

Optical Trapping and Manipulation

**History**

Original Manuscript: February 11, 2010

Revised Manuscript: March 18, 2010

Manuscript Accepted: April 22, 2010

Published: May 5, 2010

**Virtual Issues**

Vol. 5, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

Sebastien Rancourt-Grenier, Ming-Tzo Wei, Jar-Jin Bai, Arthur Chiou, Paul P. Bareil, Pierre-Luc Duval, and Yunlong Sheng, "Dynamic deformation of red blood cell
in Dual-trap Optical Tweezers," Opt. Express **18**, 10462-10472 (2010)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-18-10-10462

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### References

- A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330(6150), 769–771 (1987). [CrossRef]
- S. M. Block, “Making light work with optical tweezers,” Nature 360(6403), 493–495 (1992). [CrossRef]
- J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81(2), 767–784 (2001). [CrossRef]
- M. Dao, C. T. Lim, and S. Suresh, “Mechanics of the human red blood cell deformed by optical tweezers,” J. Mech. Phys. Solids 51(11-12), 2259–2280 (2003). [CrossRef]
- S. Hénon, G. Lenormand, A. Richert, and F. Gallet, “A new determination of the shear modulus of the human erythrocyte membrane using optical tweezers,” Biophys. J. 76(2), 1145–1151 (1999). [CrossRef]
- P. J. H. Bronkhorst, G. J. Streekstra, J. Grimbergen, E. J. Nijhof, J. J. Sixma, and G. J. Brakenhoff, “A new method to study shape recovery of red blood cells using multiple optical trapping,” Biophys. J. 69(5), 1666–1673 (1995). [CrossRef]
- W. G. Lee, H. Bang, J. Park, S. Chung, K. Cho, C. Chung, D.-C. Han, and J. K. Chang, “Combined microchannel-type erythrocyte deformability test with optical tweezers,” Proc. of SPIE.6088, 608813–1-12, (2006).
- G. B. Liao, P. B. Bareil, Y. Sheng, and A. Chiou, “One-dimensional jumping optical tweezers for optical stretching of bi-concave human red blood cells,” Opt. Express 16(3), 1996–2004 (2008). [CrossRef]
- P. B. Bareil, Y. Sheng, and A. Chiou, “Local scattering stress distribution on surface of a spherical cell in optical stretcher,” Opt. Express 14(25), 12503 (2006). [CrossRef]
- J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81(2), 767–784 (2001). [CrossRef]
- T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A, Pure Appl. Opt. 9(8), S196–S203 (2007). [CrossRef]
- T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transf. 79–80, 1005–1017 (2003). [CrossRef]
- M. Mansuripur, “Electromagnetic stress tensor in ponderable media,” Opt. Express 16(8), 5193–5198 (2008). [CrossRef]
- F. Xu, J. A. Lock, G. Gouesbet, and C. Tropea, “Optical stress on the surface of a particle: homogeneous sphere,” Phys. Rev. A 79(5), 053808 (2009). [CrossRef]
- Y. C. Fung, “Theoretical considerations of the elasticity of red cells and small blood vessels,” Fed. Proc. 25(6), 1761–1772 (1966).
- P. B. Bareil, Y. Sheng, Y. Q. Chen, and A. Chiou, “Calculation of spherical red blood cell deformation in a dual-beam optical stretcher,” Opt. Express 15(24), 16029–16034 (2007). [CrossRef]
- E. Ventsel, and T. Krauthammer, Thin plate and shells (Marcel Dekket, New York, 2001).

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