## Dynamic ray tracing for modeling optical cell manipulation |

Optics Express, Vol. 18, Issue 16, pp. 16702-16714 (2010)

http://dx.doi.org/10.1364/OE.18.016702

Acrobat PDF (1173 KB)

### Abstract

Current methods for predicting stress distribution on a cell surface due to optical trapping forces are based on a traditional ray optics scheme for fixed geometries. Cells are typically modeled as solid spheres as this facilitates optical force calculation. Under such applied forces however, real and non-rigid cells can deform, so assumptions inherent in traditional ray optics methods begin to break down. In this work, we implement a dynamic ray tracing technique to calculate the stress distribution on a deformable cell induced by optical trapping. Here, cells are modeled as three-dimensional elastic capsules with a discretized surface with associated hydrodynamic forces calculated using the Immersed Boundary Method. We use this approach to simulate the transient deformation of spherical, ellipsoidal and biconcave capsules due to external optical forces induced by a single diode bar optical trap for a range of optical powers.

© 2010 OSA

## 1. Introduction

1. S. Suresh, J. Spatz, J. P. Mills, A. Micoulet, M. Dao, C. T. Lim, M. Beil, and T. Seufferlein, “Connections between single-cell biomechanics
and human disease states: gastrointestinal cancer and
malaria,” Acta Biomater. **1**(1), 15–30
(2005). [CrossRef]

2. G. Y. H. Lee and C. T. Lim, “Biomechanics approaches to studying human
diseases,” Trends Biotechnol. **25**(3), 111–118
(2007). [CrossRef] [PubMed]

3. J. P. Mills, M. Diez-Silva, D. J. Quinn, M. Dao, M. J. Lang, K. S. Tan, C. T. Lim, G. Milon, P. H. David, O. Mercereau-Puijalon, S. Bonnefoy, and S. Suresh, “Effect of plasmodial RESA protein on
deformability of human red blood cells harboring *Plasmodium
falciparum*,” Proc. Natl. Acad. Sci. U.S.A. **104**(22),
9213–9217 (2007). [CrossRef] [PubMed]

4. R. M. Hochmuth, “Micropipette aspiration of living
cells,” J. Biomech. **33**(1), 15–22
(2000). [CrossRef]

5. A. L. Weisenhorn, M. Khorsandi, S. Kasas, V. Gotzos, and H.-J. Butt, “Deformation and height anomaly of soft
surfaces studied with an AFM,”
Nanotechnology **4**(2), 106–113
(1993). [CrossRef]

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]

7. R. R. Huruta, M. L. Barjas-Castro, S. T. O. Saad, F. F. Costa, A. Fontes, L. C. Barbosa, and C. L. Cesar, “Mechanical properties of stored red blood
cells using optical tweezers,” Blood **92**(8),
2975–2977 (1998). [PubMed]

8. 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]

9. A. Ashkin, “Acceleration and trapping of particles by
radiation pressure,” Phys. Rev. Lett. **24**(4), 156–159
(1970). [CrossRef]

*et al.*deformed RBCs into a parachute shape using three traps, two at the ends and one in the middle [6

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]

*et al.*deformed RBCs using two silica beads that were adhered to opposite sides of the cell surface [8

8. 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]

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]

11. J. Guck, S. Schinkinger, B. Lincoln, F. Wottawah, S. Ebert, M. Romeyke, D. Lenz, H. M. Erickson, R. Ananthakrishnan, D. Mitchell, J. Käs, S. Ulvick, and C. Bilby, “Optical deformability as an inherent cell
marker for testing malignant transformation and metastatic
competence,” Biophys. J. **88**(5),
3689–3698 (2005). [CrossRef] [PubMed]

*et al.*[12

12. R. W. Applegate, J. Squier, T. Vestad, J. Oakey, and D. W. M. Marr, “Fiber-focused diode bar optical trapping for
microfluidic flow manipulation,” Appl. Phys.
Lett. **92**(1), 013904
(2008). [CrossRef]

13. I. Sraj, J. Chichester, E. Hoover, R. Jimenez, J. Squier, C. D. Eggleton, and D. W. M. Marr, “Cell deformation cytometry using diode-bar optical stretchers,” J. Biomed. Opt. in press. [PubMed]

14. A. Ashkin, “Forces of a single-beam gradient laser trap
on a dielectric sphere in the ray optics regime,”
Biophys. J. **61**(2), 569–582
(1992). [CrossRef] [PubMed]

*et al.*used the RO method for a dual-beam fiber-optic stretcher and used it to determine the stiffness of RBCs [10

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]

15. 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),
12503–12509 (2006). [CrossRef]

16. 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]

## 2. Numerical method

17. C. Peskin and D. McQueen, “A three dimensional computational method for
blood flow in the heart I. Immersed elastic fibers in a viscous incompressible
fluid,” J. Comput. Phys. **81**(2), 372–405
(1989). [CrossRef]

19. P. Parag, S. Jadhav, C. D. Eggleton, and K. Konstantopoulos, “Roles of cell and microvillus deformation
and receptor-ligand binding kinetics in cell rolling,”
Am. J. Physiol. Heart Circ. Physiol. **295**(4),
H1439–H1450 (2008). [CrossRef]

18. C. D. Eggleton and A. S. Popel, “Large deformation of red blood cell ghosts
in a simple shear flow,” Phys. Fluids **10**(8),
1834–1845 (1998). [CrossRef]

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]

## 3. Traditional ray optics

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

12. R. W. Applegate, J. Squier, T. Vestad, J. Oakey, and D. W. M. Marr, “Fiber-focused diode bar optical trapping for
microfluidic flow manipulation,” Appl. Phys.
Lett. **92**(1), 013904
(2008). [CrossRef]

15. 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),
12503–12509 (2006). [CrossRef]

*a*is the radius of the sphere and λ the laser wavelength. For cells of typical radius greater than 3 µm and laser wavelengths less than 1 µm this condition is satisfied [10

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

*n*is the index of refraction of the buffer medium,

_{m}*c*the speed of light in vacuum,

*P*the laser power, and

*Q*is the trapping efficiency, a dimensionless factor representing the amount of momentum transferred.

*Q*is independent of the power applied and depends only on object geometry and the reflectance of the medium. Calculation of

*Q*to date has been for spherical rigid cells [14

14. A. Ashkin, “Forces of a single-beam gradient laser trap
on a dielectric sphere in the ray optics regime,”
Biophys. J. **61**(2), 569–582
(1992). [CrossRef] [PubMed]

23. J. Y. Walz and D. C. Prieve, “Prediction and measurement of the optical
trapping forces on a microscopic dielectric sphere,”
Langmuir **8**(12),
3073–3082 (1992). [CrossRef]

24. J. Guck, R. Ananthakrishnan, T. J. Moon, C. C. Cunningham, and J. Käs, “Optical deformability of soft biological
dielectrics,” Phys. Rev. Lett. **84**(23),
5451–5454 (2000). [CrossRef] [PubMed]

25. F. Wottawah, S. Schinkinger, B. Lincoln, R. Ananthakrishnan, M. Romeyke, J. Guck, and J. Käs, “Optical rheology of biological
cells,” Phys. Rev. Lett. **94**(9), 098103
(2005). [CrossRef] [PubMed]

*Q*is calculated across the front and back sphere surfaces separately as a function of refractive index and incident laser beam profile, an approach valid because refracted angles can be determined at both front and back interfaces from Snell’s law:where

*α*and

*β*and are the incident and refracted ray angles (Fig. 1 ) and

*Q*at any point on the surface can then be expressed as a function of the reflectance

*R*, the original incident angle

*α*, and the refracted angle

*β*. For simplicity

*Q*is typically decomposed into two components (parallel to and perpendicular to the laser axis) as:

*n =*

*R<0.005*for all rays. The magnitude at any point is found via

*Q*are valid only on spherical cells and thus, in using TRO, the corresponding optical forces are calculated once initially and kept constant throughout the simulations.

## 4. Dynamic ray tracing

*a priori*. Thus, the traditional ray tracing approach is not suitable and another technique is required to calculate the optical forces on evolving complex geometries. We show here how DRT can be used to calculate the optical force distribution on the cell in both its initial stress-free geometry as well as the transient morphologies as the cell deforms in response to combined hydrodynamic and optical forces. In DRT a finite number of rays are issued from a light source with a given intensity and known direction and travel linearly from the source until they intersect with a surface (defined as an interface between two media with different refractive indices). To implement this, we divide the surface into triangular elements with the same surface elements used for both elastic and optical stress calculations. From Eqs. (3) and (4),

*Q*can be determined from incident, refracted and reflected ray angles with the laser axis angle at any point of ray-surface intersection. Thus the elemental optical forces at the first refraction can be expressed as:where

*i*,

*r*and

*t*are the angles formed between the incident, reflected, and transmitted ray with the laser axis (Fig. 2 ). The angle between the incident ray and the surface normal

*α*remains the same in the calculation of reflectance

*R*. Note here that at any surface

*Q*is multiplied by a factor of

*1-R*to account for energy loss from previous refractions and thus the general form of

*Q*from a medium

*x*to another medium

*y*can be written as:where α’ is the incident angle of the ray in the previous refraction. This general form is useful in DRT where a ray is traced throughout complex shaped cells.

26. M. Lofsted and T. Moller, “An evaluation framework for ray-triangle
intersection algorithms,” J. Graphics, GPU, Games
Tools **10**(2) 13–26 (2005).
[CrossRef]

**I**is modeled as a vector with a given origin

**e**and direction

**d, I = e +**

*s*

**d**where

*s*is the distance along the ray measured from the origin in the positive direction (

*s > 0*, semi-line). We assume rays are initially parallel to the laser axis, an assumption that works well for non- or weakly-diverging laser diodes (Fig. 2); however, any initial ray direction can be simulated by choosing appropriate ray parameters. We then discretize the surface of the cell into a finite number of non-overlapping 2D triangular elements. In this, each single ray emanating from the source intersects the object at a maximum of one triangle and may not intersect the surface at all. For a triangle with three vertices

*A, B*and

*C*(Fig. 2), each point on a triangle can be defined using its barycentric coordinates:

*P*to be inside/on the triangle the sum of the barycentric coordinates must equal one; therefore,

*P*can be represented as

*P = (1-u-v)A + uB + vC = A + u(B-A) + v(C-A)*with

**I**then intersects the triangle at

*P*when

*P =*

*e**+ s*

**for a given**

*d**s*. Combining both conditions we have

*s, u*and

*v*to find the intersection point and the distance of intersection [26

26. M. Lofsted and T. Moller, “An evaluation framework for ray-triangle
intersection algorithms,” J. Graphics, GPU, Games
Tools **10**(2) 13–26 (2005).
[CrossRef]

*s*for each element. Smaller values of

*s*indicate closer objects to the light source; if two objects are both intersected, the one with the smaller value of

*s*is recorded as the nearest point of intersection

*P*. This procedure can be computationally expensive depending on the number of triangular elements and number of rays. Advanced techniques for saving computational time that avoid calculating ray-triangle intersections over elements in the shadow of elements on the “front-side” of the surface are available in the literature [29].

**N**, the incident angle can be calculated as

*β*and the direction vector of both reflected and refracted rays via:

*P*) and new intensity. Therefore

**I**

*=*

**T**and the previous procedure repeated to find a new contact location. A spherical cell typically has only two intersection locations, one at the front and one at the back surface; however, this approach works for any cell geometry where a ray might encounter multiple external or internal surfaces. For example, in a biconcave capsule, some rays encounter up to four hits at different surfaces. It is important to note that when a ray changes direction it is moving from one medium to another and therefore the indices of refraction are swapped after each hit as the direction of the force changes.

## 5. Dynamic ray tracer validation

### 5.1 Trapping efficiency calculation

*Q*on a sphere using the two methods. The trapping efficiency

*Q*is independent of the light source and is only function of the cell shape and the index of refraction of the cell and the medium

*n*1.37 and

_{p}=*n*1.335 [15

_{m}=15. 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),
12503–12509 (2006). [CrossRef]

*Q*can be found as a function of the polar angle on both the front and back surface as shown in Fig. 3 . The 3D DRT is based on shape discretization and we employ a mesh of 20480 triangles. For comparison, we plot only the calculated

*Q*at the first and second quadrant at a plane parallel to the laser axis passing through the center of the cell representing the front and back surfaces. In these calculations we see an average error of 1.43% across the surface while the maximum error calculated across this plane is 10% at the back surface at the peak value, attributed to the convergence of rays on the back surface. The error on the front surface is relatively small because the gradients in the incoming beam are small and each element is intersected by one ray. Transmitted rays however converge on the backside, causing multiple intersections on some elements and no intersections on other elements, increasing optical force gradients and leading to larger localized error.

### 5.2 Net optical forces

*Q*distribution, one can now calculate induced optical forces by defining the cell size and light source. A spherical cell representing a swollen erythrocyte has a radius

*a*of 3.3 μm. We simulate a linear diode bar light source that is 200 μm long and 1 μm wide [13

13. I. Sraj, J. Chichester, E. Hoover, R. Jimenez, J. Squier, C. D. Eggleton, and D. W. M. Marr, “Cell deformation cytometry using diode-bar optical stretchers,” J. Biomed. Opt. in press. [PubMed]

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

## 6. Transient cell deformation

*L*and

*B*are the lengths of the major and minor axes of elongated cells in a specified plane. Here we define

*DF*in the

*x-z*plane as the rays are assumed coming parallel to the z axis and the perpendicular component of the optical forces is in the

*x*direction.

*DF*describes the geometrical deformation from perfect spheres (

*DF = 0*) to highly elongated morphologies.

*Eh*= 0.1 dyn/cm, an unstressed cell is initially placed in a fluid of density

*ρ*= 1 g/cm

^{3}and dynamic viscosity

*µ*= 0.8 cP. The fluid domain is a cube with a side 8x the cell radius

*a*with periodic boundary conditions. The grid used in the simulations has 64

^{3}nodes with a uniform grid spacing

*h = a/8*. The hydrodynamic intrinsic time scale for cell deformation is

*t*=

_{c}*μa/Eh*[30

30. A. Diaz, N. Pelekasis, and D. Barthes-Biesel, “Transient response of a capsule subjected to
varying flow conditions: Effect of internal fluid viscosity and membrane
elasticity,” Phys. Fluids **12**(5), 948–957
(2000). [CrossRef]

*t*~10

_{c}^{−3}s for our simulation parameters. With this, we use a time step of 10

^{−5}s to ensure resolution and numerical stability. In Fig. 4 we show the force at both the front and back surface along with the total net force on the cell function of time. From this we see that both the front and back forces decrease as the cell deforms; however, the net force increases by 6%, using the 12 mW/μm laser power to a constant value when the cell reaches a steady-state shape. In Fig. 5 we show that the net translation force is in the z-direction, pushing the cell away from the light source and deforming the cell as expected for non-converging traps. We note that the initial forces at time

*t* = 0*are the forces that remain constant using TRO on a sphere.

*Q*changes with cell deformation. Here, calculated values of F

_{net}compared to those when neglecting shape change differ by 42% even though the corresponding

*DF*changes only 3.65% at the higher laser power. Note that, at lower laser intensities, the two techniques converge as expected.

_{xz}## 7. Non-Spherical Cell Shapes

31. E. Evans and Y. C. Fung, “Improved measurements of the erythrocyte
geometry,” Microvasc. Res. **4**(4), 335–347
(1972). [CrossRef] [PubMed]

*C*= 0.207161,

_{0}*C*= 2.002558 and

_{1}*C*= −1.122762, the radius

_{2}*a*is 4 μm, and the surface area is 140 μm

^{2}. In many experimental studies, RBCs are osmotically swollen and assume a spherical shape. To demonstrate the flexibility of DRT, we model both unswollen (biconcave) and swollen (oblate spheroid) RBCs. For the swollen case, we assume an average radius of 2.8 μm, corresponding to a perfect sphere of the same surface area and radius 3.3 μm, and average deformation parameter

*DF*

_{xz}= 0.126 as seen in experimental work for swollen bovine RBCs [13

13. I. Sraj, J. Chichester, E. Hoover, R. Jimenez, J. Squier, C. D. Eggleton, and D. W. M. Marr, “Cell deformation cytometry using diode-bar optical stretchers,” J. Biomed. Opt. in press. [PubMed]

16. 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]

*DF*

_{xz}is recorded. Here, the optical forces applied by the diode laser bar are the only external forces on the cell and any flow is induced only by the cell deformability and its translation. Beginning initially with spherical cells for comparison and using a range of laser powers, 12 mW/μm to 72 mW/μm, we see the net optical force induced by a 12 mW/μm laser power is 0.57 pN with a final induced deformation of

*DF*

_{xz}= 0.0158. Oblate spheroids on the other hand are subjected to more force at identical powers. For example a net force of 0.79 pN is calculated when using the same 12 mW/μm power with a net deformation of

*DF*

_{xz}= 0.0177, higher than the spherical case (Fig. 8 ).

*DF*was zero). For the normal RBC case, we observe that the net optical force is significantly higher for the same 12 mW/μm power with a 1.29 pN net optical force induced by rays intersecting the surface at four different positions with stretching forces on both the back and front surface. The net deformation however is smaller

*DF*

_{xz}= 0.0062. Figures 9(a) and 9(b) (Media 2 and Media 3) shows the steady-state shape of swollen and unswollen RBCs. Table 1 summarizes the net optical forces on different geometries for the same power and corresponding net deformation. From these measurements it is clear that the shape of the cell strongly influences net cell stretching force and deformation.

## 8. Conclusions

## Acknowledgments

## References and links

1. | S. Suresh, J. Spatz, J. P. Mills, A. Micoulet, M. Dao, C. T. Lim, M. Beil, and T. Seufferlein, “Connections between single-cell biomechanics
and human disease states: gastrointestinal cancer and
malaria,” Acta Biomater. |

2. | G. Y. H. Lee and C. T. Lim, “Biomechanics approaches to studying human
diseases,” Trends Biotechnol. |

3. | J. P. Mills, M. Diez-Silva, D. J. Quinn, M. Dao, M. J. Lang, K. S. Tan, C. T. Lim, G. Milon, P. H. David, O. Mercereau-Puijalon, S. Bonnefoy, and S. Suresh, “Effect of plasmodial RESA protein on
deformability of human red blood cells harboring |

4. | R. M. Hochmuth, “Micropipette aspiration of living
cells,” J. Biomech. |

5. | A. L. Weisenhorn, M. Khorsandi, S. Kasas, V. Gotzos, and H.-J. Butt, “Deformation and height anomaly of soft
surfaces studied with an AFM,”
Nanotechnology |

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. | R. R. Huruta, M. L. Barjas-Castro, S. T. O. Saad, F. F. Costa, A. Fontes, L. C. Barbosa, and C. L. Cesar, “Mechanical properties of stored red blood
cells using optical tweezers,” Blood |

8. | 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. |

9. | A. Ashkin, “Acceleration and trapping of particles by
radiation pressure,” Phys. Rev. Lett. |

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. | J. Guck, S. Schinkinger, B. Lincoln, F. Wottawah, S. Ebert, M. Romeyke, D. Lenz, H. M. Erickson, R. Ananthakrishnan, D. Mitchell, J. Käs, S. Ulvick, and C. Bilby, “Optical deformability as an inherent cell
marker for testing malignant transformation and metastatic
competence,” Biophys. J. |

12. | R. W. Applegate, J. Squier, T. Vestad, J. Oakey, and D. W. M. Marr, “Fiber-focused diode bar optical trapping for
microfluidic flow manipulation,” Appl. Phys.
Lett. |

13. | I. Sraj, J. Chichester, E. Hoover, R. Jimenez, J. Squier, C. D. Eggleton, and D. W. M. Marr, “Cell deformation cytometry using diode-bar optical stretchers,” J. Biomed. Opt. in press. [PubMed] |

14. | A. Ashkin, “Forces of a single-beam gradient laser trap
on a dielectric sphere in the ray optics regime,”
Biophys. J. |

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

16. | 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 |

17. | C. Peskin and D. McQueen, “A three dimensional computational method for
blood flow in the heart I. Immersed elastic fibers in a viscous incompressible
fluid,” J. Comput. Phys. |

18. | C. D. Eggleton and A. S. Popel, “Large deformation of red blood cell ghosts
in a simple shear flow,” Phys. Fluids |

19. | P. Parag, S. Jadhav, C. D. Eggleton, and K. Konstantopoulos, “Roles of cell and microvillus deformation
and receptor-ligand binding kinetics in cell rolling,”
Am. J. Physiol. Heart Circ. Physiol. |

20. | E. Lac, D. Barthes-Biesel, N. A. Pelekasis, and J. Tsamopoulos, “Spherical capsules in three-dimensional
unbounded Stokes flows: Effect of the membrane constitutive law and onset of
buckling,” J. Fluid Mech. |

21. | P. Bagchi, P. C. Johnson, and A. S. Popel, “Computational fluid dynamic simulation of
aggregation of deformable cells in a shear flow,”
J. Biomech. Eng. |

22. | H. C. Van de Hulst, “Light scattering by small particles,” John Wiley and Sons, New York. 172–176 (1957). |

23. | J. Y. Walz and D. C. Prieve, “Prediction and measurement of the optical
trapping forces on a microscopic dielectric sphere,”
Langmuir |

24. | J. Guck, R. Ananthakrishnan, T. J. Moon, C. C. Cunningham, and J. Käs, “Optical deformability of soft biological
dielectrics,” Phys. Rev. Lett. |

25. | F. Wottawah, S. Schinkinger, B. Lincoln, R. Ananthakrishnan, M. Romeyke, J. Guck, and J. Käs, “Optical rheology of biological
cells,” Phys. Rev. Lett. |

26. | M. Lofsted and T. Moller, “An evaluation framework for ray-triangle
intersection algorithms,” J. Graphics, GPU, Games
Tools |

27. | D. Badouel, “An efficient ray-polygon
intersection,” in |

28. | T. Moller and B. Trumbore, “Fast, minimum storage ray-triangle
intersection,” J. Graphics, GPU, Games
Tools |

29. | E. Reinhard, B. E. Smits, and C. Hansen, “Dynamic acceleration structures for interactive ray tracing,” Proceedings of the Eurographics workshop on rendering techniques, Brno, Czech Republic, 299–306 (2000). |

30. | A. Diaz, N. Pelekasis, and D. Barthes-Biesel, “Transient response of a capsule subjected to
varying flow conditions: Effect of internal fluid viscosity and membrane
elasticity,” Phys. Fluids |

31. | E. Evans and Y. C. Fung, “Improved measurements of the erythrocyte
geometry,” Microvasc. Res. |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(020.7010) Atomic and molecular physics : Laser trapping

(140.2020) Lasers and laser optics : Diode lasers

(170.1530) Medical optics and biotechnology : Cell analysis

**ToC Category:**

Optical Trapping and Manipulation

**History**

Original Manuscript: May 17, 2010

Revised Manuscript: June 24, 2010

Manuscript Accepted: July 11, 2010

Published: July 23, 2010

**Virtual Issues**

Vol. 5, Iss. 12 *Virtual Journal for Biomedical Optics*

**Citation**

Ihab Sraj, Alex C. Szatmary, David W. M. Marr, and Charles D. Eggleton, "Dynamic ray tracing for modeling optical cell manipulation," Opt. Express **18**, 16702-16714 (2010)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-18-16-16702

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