## Comparison of the discrete dipole approximation and the discrete source method for simulation of light scattering by red blood cells

Optics Express, Vol. 18, Issue 6, pp. 5681-5690 (2010)

http://dx.doi.org/10.1364/OE.18.005681

Acrobat PDF (883 KB)

### Abstract

The discrete sources method (DSM) and the discrete dipole approximation (DDA) were compared for simulation of light scattering by a red blood cell (RBC) model. We considered RBCs with diameters up to 8 μm (size parameter up to 38), relative refractive indices 1.03 and 1.06, and two different orientations. The agreement in the angle-resolved *S*_{11} element of the Mueller matrix obtained by these methods is generally good, but it deteriorates with increasing scattering angle, diameter and refractive index of a RBC. Based on the DDA simulations with very fine discretization (up to 93 dipoles per wavelength) for a single RBC, we attributed most of the disagreement to the DSM, which results contain high-frequency ripples. For a single orientation of a RBC the DDA is comparable to or faster than the DSM. However, the relation is reversed when a set of particle orientations need to be simulated at once. Moreover, the DSM requires about an order of magnitude less computer memory. At present, application of the DSM for massive calculation of light scattering patterns of RBCs is hampered by its limitations in size parameter of a RBC due to the high number of harmonics used for calculations.

© 2010 OSA

## 1. Introduction

3. S. V. Tsinopoulos and D. Polyzos, “Scattering of He-Ne laser light by an average-sized red blood cell,” Appl. Opt. **38**(25), 5499–5510 (1999). [CrossRef]

7. T. Wriedt, J. Hellmers, E. Eremina, and R. Schuh, “Light scattering by single erythrocyte: comparison of different methods,” J. Quant. Spectrosc. Radiat. Transf. **100**(1-3), 444–456 (2006). [CrossRef]

3. S. V. Tsinopoulos and D. Polyzos, “Scattering of He-Ne laser light by an average-sized red blood cell,” Appl. Opt. **38**(25), 5499–5510 (1999). [CrossRef]

4. J. Q. Lu, P. Yang, and X. H. Hu, “Simulations of light scattering from a biconcave red blood cell using the finite-difference time-domain method,” J. Biomed. Opt. **10**(2), 024022 (2005). [CrossRef] [PubMed]

6. A. Karlsson, J. P. He, J. Swartling, and S. Andersson-Engels, “Numerical simulations of light scattering by red blood cells,” IEEE Trans. Biomed. Eng. **52**(1), 13–18 (2005). [CrossRef] [PubMed]

5. M. A. Yurkin, K. A. Semyanov, P. A. Tarasov, A. V. Chernyshev, A. G. Hoekstra, and V. P. Maltsev, “Experimental and theoretical study of light scattering by individual mature red blood cells by use of scanning flow cytometry and a discrete dipole approximation,” Appl. Opt. **44**(25), 5249–5256 (2005). [CrossRef] [PubMed]

6. A. Karlsson, J. P. He, J. Swartling, and S. Andersson-Engels, “Numerical simulations of light scattering by red blood cells,” IEEE Trans. Biomed. Eng. **52**(1), 13–18 (2005). [CrossRef] [PubMed]

8. E. Eremina, Y. Eremin, and T. Wriedt, “Analysis of light scattering by erythrocyte based on discrete sources method,” Opt. Commun. **244**(1-6), 15–23 (2005). [CrossRef]

10. E. Eremina, “Light scattering by an erythrocyte based on Discrete Sources Method: shape and refractive index influence,” J. Quant. Spectrosc. Radiat. Transf. **110**(14-16), 1526–1534 (2009). [CrossRef]

11. A. M. K. Nilsson, P. Alsholm, A. Karlsson, and S. Andersson-Engels, “T-Matrix Computations of Light Scattering by Red Blood Cells,” Appl. Opt. **37**(13), 2735–2748 (1998). [CrossRef]

7. T. Wriedt, J. Hellmers, E. Eremina, and R. Schuh, “Light scattering by single erythrocyte: comparison of different methods,” J. Quant. Spectrosc. Radiat. Transf. **100**(1-3), 444–456 (2006). [CrossRef]

6. A. Karlsson, J. P. He, J. Swartling, and S. Andersson-Engels, “Numerical simulations of light scattering by red blood cells,” IEEE Trans. Biomed. Eng. **52**(1), 13–18 (2005). [CrossRef] [PubMed]

12. A. N. Shvalov, J. T. Soini, A. V. Chernyshev, P. A. Tarasov, E. Soini, and V. P. Maltsev, “Light-scattering properties of individual erythrocytes,” Appl. Opt. **38**(1), 230–235 (1999). [CrossRef]

*S*

_{11}of the Mueller matrix calculated with both methods varying cellular characteristics within typical biological ranges. A special attention is paid to limitations of both methods in terms of size parameter of a RBC.

## 2. Simulation methods

### 2.1 The discrete sources method

9. E. Eremina, J. Hellmers, Y. Eremin, and T. Wriedt, “Different shape models for erythrocyte: Light scattering analysis based on the discrete sources method,” J. Quant. Spectrosc. Radiat. Transf. **102**(1), 3–10 (2006). [CrossRef]

### 2.2 The discrete dipole approximation

15. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A **11**(4), 1491–1499 (1994). [CrossRef]

17. “ADDA - light scattering simulator using the discrete dipole approximation”, http://code.google.com/p/a-dda/ (2009).

18. M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “The discrete dipole approximation for simulation of light scattering by particles much larger than the wavelength,” J. Quant. Spectrosc. Radiat. Transf. **106**(1-3), 546–557 (2007). [CrossRef]

18. M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “The discrete dipole approximation for simulation of light scattering by particles much larger than the wavelength,” J. Quant. Spectrosc. Radiat. Transf. **106**(1-3), 546–557 (2007). [CrossRef]

## 3. Optical model of a RBC

19. P. Mazeron, S. Muller, and H. El Azouzi, “Deformation of erythrocytes under shear: a small-angle light scattering study,” Biorheology **34**(2), 99–110 (1997). [CrossRef] [PubMed]

*et al.*[20

20. R. Skalak, A. Tozeren, R. P. Zarda, and S. Chien, “Strain energy function of red blood cell membranes,” Biophys. J. **13**(3), 245–264 (1973). [CrossRef] [PubMed]

*et al.*[21

21. Y. C. Fung, W. C. Tsang, and P. Patitucci, “High-resolution data on the geometry of red blood cells,” Biorheology **18**(3-6), 369–385 (1981). [PubMed]

*et al.*[21

21. Y. C. Fung, W. C. Tsang, and P. Patitucci, “High-resolution data on the geometry of red blood cells,” Biorheology **18**(3-6), 369–385 (1981). [PubMed]

*z*and

*ρ*are cylindrical coordinates and

*D*= 2

*R*is the diameter of a RBC (the only free parameter in this model). A particular case of

*D*= 6 μm is shown in Fig. 1 . The morphology data for RBCs from different sources has been reviewed by Yurkin [22, Chapter 4.1]. The diameter of a RBC typically varies from 6 to 9 μm, and the real part of the refractive index at

*λ*= 0.66 μm falls between 1.39 and 1.41. Imaginary part of the refractive index is about 10

^{−4}[23

23. K. A. Semyanov, P. A. Tarasov, J. T. Soini, A. K. Petrov, and V. P. Maltsev, “Calibration-free method to determine the size and hemoglobin concentration of individual red blood cells from light scattering,” Appl. Opt. **39**(31), 5884–5889 (2000). [CrossRef]

24. D. H. Tycko, M. H. Metz, E. A. Epstein, and A. Grinbaum, “Flow-cytometric light scattering measurement of red blood cell volume and hemoglobin concentration,” Appl. Opt. **24**(9), 1355–1365 (1985). [CrossRef] [PubMed]

*m*are 1.03 and 1.06, and wavelength in the medium is fixed at

*λ*= 0.4936 μm (0.66 mm in vacuum). Size parameter of the simulated RBCs is from 28 to 38. Incident radiation propagates along the

*z*-axis, and we calculate dependence of

*S*

_{11}on the scattering angle

*θ*in the

*xz*-plane. Orientation of a RBC is with symmetry axis along either

*z*-axis or

*x*-axis (the Euler angle

*β*equals 0° or 90° respectively).

25. M. A. Yurkin, and A. G. Hoekstra, “User manual for the discrete dipole approximation code ADDA v.0.79,” http://a-dda.googlecode.com/svn/tags/rel_0_79/doc/manual.pdf (2009).

## 4. Results and discussion

*S*

_{11}(

*θ*) calculated using the DSM and the DDA varying diameter, refractive index and orientation of a RBC. The agreement between the two methods is generally good however it strongly depends on problem parameters. General tendency is that differences increases with

*θ*,

*D*,

*β*, and

*m*. Refractive index has the least effect on the difference, when varied inside biological range for a RBC. For

*β*=0° the agreement is good along the whole

*θ*range when

*D*≤ 7 μm, but there is significant (order of magnitude) disagreement for 90° ≤

*θ*≤ 120° when

*D*= 7.5 μm. For

*β*=90° and

*D*≤ 7.5 μm the agreement is good only up to

*θ*=60°-70°. The largest tested diameter (8 μm) shows principal disagreement for all test cases, except the near-forward scattering (up to 15).

*λ*= 0.4936 μm 45 harmonics are needed. This causes numerical instability for large orientation angles

*β*. For example, the surface residual for all RBC diameters presented in paper does not exceed 0,2% for

*β*=0°. At the same time the surface residual for

*β*= 90° varies from 3% for

*D*= 6 µm to 8% for

*D*= 8 µm. Although the surface residual can be used as an internal quality test, it is only an approximate measure of the accuracy of the final scattering quantities.

26. M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “Convergence of the discrete dipole approximation. I. Theoretical analysis,” J. Opt. Soc. Am. A **23**(10), 2578–2591 (2006). [CrossRef]

*D*= 7.5 μm,

*m*= 1.03) and two orientations (

*β*= 0° and 90°) varying dipole size

*d*from

*λ*/8 to

*λ*/93. Number of dipoles per grid varied from 128 to 1408 respectively, and total number of dipoles was up to 6×10

^{8}. These huge simulations were carried out on the Dutch compute cluster LISA [27

27. “Description of the national compute cluster Lisa,” https://subtrac.sara.nl/userdoc/wiki/lisa/description (2009).

*θ*= 120°) are shown in Fig. 6 . Although this is one of the worst convergence among all

*θ*(data not shown), one can clearly see that the DDA results indeed converge with decreasing dipole size. However, this convergence is oscillating, complicating the choice of a particular reference value or interval. For instance, using extrapolation technique as described by Yurkin

*et al.*[28

28. M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “Convergence of the discrete dipole approximation. II. An extrapolation technique to increase the accuracy,” J. Opt. Soc. Am. A **23**(10), 2592–2601 (2006). [CrossRef]

^{−2}for

*S*

_{11}(120°) for the case of

*β*= 0°, which is too wide for practical purposes. The extrapolation technique was originally tested on scatterers with wavelength-sized scatterers [28

28. M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “Convergence of the discrete dipole approximation. II. An extrapolation technique to increase the accuracy,” J. Opt. Soc. Am. A **23**(10), 2592–2601 (2006). [CrossRef]

*λ*/16. Such confidence intervals for

*S*

_{11}(120°) are [1.59,2.43]×10

^{−2}and [2.06,2.32]×10

^{−1}for

*β*= 0° and 90°respectively (also shown in Fig. 6). One can see that they seem reliable, i.e. further decrease of dipole size should not move the DDA results out of these intervals. Similar conclusions can be reached for other

*θ*(data not shown), however, this reliability is empirical rather than rigorously proven. The confidence bounds obtained by this method for all scattering angles are shown in Fig. 7 together with DSM and DDA (with default discretization) results for the same RBCs. Confidence bounds are very narrow and coincident the DDA results for default discretization over the major part of the angular range. Assuming the reliability of the confidence bounds, we conclude that the DSM is significantly less accurate than the DDA for this particular RBC, at least in the range of

*θ*with the largest difference between the two methods. Accuracy of the DDA is expected to have small dependence on particle size for fixed dipole size [18

18. M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “The discrete dipole approximation for simulation of light scattering by particles much larger than the wavelength,” J. Quant. Spectrosc. Radiat. Transf. **106**(1-3), 546–557 (2007). [CrossRef]

*β*(from 0° to 90° with step 10°), which is a typical task when constructing a database of light scattering patterns [5

5. M. A. Yurkin, K. A. Semyanov, P. A. Tarasov, A. V. Chernyshev, A. G. Hoekstra, and V. P. Maltsev, “Experimental and theoretical study of light scattering by individual mature red blood cells by use of scanning flow cytometry and a discrete dipole approximation,” Appl. Opt. **44**(25), 5249–5256 (2005). [CrossRef] [PubMed]

*t*(0°) and

*t*(90°) for two orientations as 9

*t*(0°) + 5

*t*(90°), assuming linear dependence of the number of iterations in the DDA on

*β*. The latter implies that simulation time for

*β*≠ 0° linearly changes from 2

*t*(0°) to

*t*(90°) when

*β*increases from 0° to 90°. Memory usage for the DSM method is approximately 60 MB for all studied cases (not shown in Table 1). Memory requirements of the DDA increase with

*D*and are about an order of magnitude larger than that of the DSM. However, it is still small enough to fit into a standard desktop computer.

*m*is markedly different for the compared methods. The DSM speed is comparable or even slightly faster for

*m*= 1.06 than for 1.03, while the DDA is about 50% slower for the larger

*m*. This agrees with previous observations that the DDA performance is especially good for index-matching particles [18

**106**(1-3), 546–557 (2007). [CrossRef]

29. M. A. Yurkin, A. G. Hoekstra, R. S. Brock, and J. Q. Lu, “Systematic comparison of the discrete dipole approximation and the finite difference time domain method for large dielectric scatterers,” Opt. Express **15**(26), 17902–17911 (2007). [CrossRef] [PubMed]

*β*= 0°, which is a special symmetric case for both methods (see Section 2). For a single non-symmetric orientation (

*β*= 90°) the DDA is 1.4-3.4 times faster. However, the DSM is 1.5-3 times faster for the set of 10 orientations.

25. M. A. Yurkin, and A. G. Hoekstra, “User manual for the discrete dipole approximation code ADDA v.0.79,” http://a-dda.googlecode.com/svn/tags/rel_0_79/doc/manual.pdf (2009).

30. Y. Okada, I. Mann, I. Sano, and S. Mukai, “Acceleration of the iterative solver in the discrete dipole approximation: Application to the orientation variation of irregularly shaped particles,” J. Quant. Spectrosc. Radiat. Transf. **109**(8), 1461–1473 (2008). [CrossRef]

## 5. Conclusion

*S*

_{11}element of the Mueller matrix between the two methods is generally good, however it deteriorates with increasing scattering angle, diameter and refractive index of a RBC, and when switching from symmetric to non-symmetric orientation of a RBC with respect to the incident radiation. Separate convergence study of the DDA for a single RBC involving up to 6 × 10

^{8}dipoles showed that most of the disagreement between the methods can be attributed to the DSM. The relatively worse accuracy of the DSM is also evidenced by its internal error estimates and high-frequency ripples present in its results, especially for larger RBC diameter.

## Acknowledgements

## References and links

1. | J. P. Greer, J. Foerster, and J. N. Lukens, eds., |

2. | V. P. Maltsev, and K. A. Semyanov, |

3. | S. V. Tsinopoulos and D. Polyzos, “Scattering of He-Ne laser light by an average-sized red blood cell,” Appl. Opt. |

4. | J. Q. Lu, P. Yang, and X. H. Hu, “Simulations of light scattering from a biconcave red blood cell using the finite-difference time-domain method,” J. Biomed. Opt. |

5. | M. A. Yurkin, K. A. Semyanov, P. A. Tarasov, A. V. Chernyshev, A. G. Hoekstra, and V. P. Maltsev, “Experimental and theoretical study of light scattering by individual mature red blood cells by use of scanning flow cytometry and a discrete dipole approximation,” Appl. Opt. |

6. | A. Karlsson, J. P. He, J. Swartling, and S. Andersson-Engels, “Numerical simulations of light scattering by red blood cells,” IEEE Trans. Biomed. Eng. |

7. | T. Wriedt, J. Hellmers, E. Eremina, and R. Schuh, “Light scattering by single erythrocyte: comparison of different methods,” J. Quant. Spectrosc. Radiat. Transf. |

8. | E. Eremina, Y. Eremin, and T. Wriedt, “Analysis of light scattering by erythrocyte based on discrete sources method,” Opt. Commun. |

9. | E. Eremina, J. Hellmers, Y. Eremin, and T. Wriedt, “Different shape models for erythrocyte: Light scattering analysis based on the discrete sources method,” J. Quant. Spectrosc. Radiat. Transf. |

10. | E. Eremina, “Light scattering by an erythrocyte based on Discrete Sources Method: shape and refractive index influence,” J. Quant. Spectrosc. Radiat. Transf. |

11. | A. M. K. Nilsson, P. Alsholm, A. Karlsson, and S. Andersson-Engels, “T-Matrix Computations of Light Scattering by Red Blood Cells,” Appl. Opt. |

12. | A. N. Shvalov, J. T. Soini, A. V. Chernyshev, P. A. Tarasov, E. Soini, and V. P. Maltsev, “Light-scattering properties of individual erythrocytes,” Appl. Opt. |

13. | Yu. Eremin, “The method of discrete sources in electromagnetic scattering by axially symmetric structures,” J. Commun. Technol. Electron. |

14. | Y. Eremin, N. Orlov, and A. Sveshnikov, “Models of electromagnetic scattering problems based on discrete sources method” in: |

15. | B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A |

16. | M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: An overview and recent developments,” JQSRT |

17. | “ADDA - light scattering simulator using the discrete dipole approximation”, http://code.google.com/p/a-dda/ (2009). |

18. | M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “The discrete dipole approximation for simulation of light scattering by particles much larger than the wavelength,” J. Quant. Spectrosc. Radiat. Transf. |

19. | P. Mazeron, S. Muller, and H. El Azouzi, “Deformation of erythrocytes under shear: a small-angle light scattering study,” Biorheology |

20. | R. Skalak, A. Tozeren, R. P. Zarda, and S. Chien, “Strain energy function of red blood cell membranes,” Biophys. J. |

21. | Y. C. Fung, W. C. Tsang, and P. Patitucci, “High-resolution data on the geometry of red blood cells,” Biorheology |

22. | M. A. Yurkin, “ |

23. | K. A. Semyanov, P. A. Tarasov, J. T. Soini, A. K. Petrov, and V. P. Maltsev, “Calibration-free method to determine the size and hemoglobin concentration of individual red blood cells from light scattering,” Appl. Opt. |

24. | D. H. Tycko, M. H. Metz, E. A. Epstein, and A. Grinbaum, “Flow-cytometric light scattering measurement of red blood cell volume and hemoglobin concentration,” Appl. Opt. |

25. | M. A. Yurkin, and A. G. Hoekstra, “User manual for the discrete dipole approximation code ADDA v.0.79,” http://a-dda.googlecode.com/svn/tags/rel_0_79/doc/manual.pdf (2009). |

26. | M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “Convergence of the discrete dipole approximation. I. Theoretical analysis,” J. Opt. Soc. Am. A |

27. | “Description of the national compute cluster Lisa,” https://subtrac.sara.nl/userdoc/wiki/lisa/description (2009). |

28. | M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “Convergence of the discrete dipole approximation. II. An extrapolation technique to increase the accuracy,” J. Opt. Soc. Am. A |

29. | M. A. Yurkin, A. G. Hoekstra, R. S. Brock, and J. Q. Lu, “Systematic comparison of the discrete dipole approximation and the finite difference time domain method for large dielectric scatterers,” Opt. Express |

30. | Y. Okada, I. Mann, I. Sano, and S. Mukai, “Acceleration of the iterative solver in the discrete dipole approximation: Application to the orientation variation of irregularly shaped particles,” J. Quant. Spectrosc. Radiat. Transf. |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(170.1530) Medical optics and biotechnology : Cell analysis

(290.5850) Scattering : Scattering, particles

**ToC Category:**

Scattering

**History**

Original Manuscript: November 19, 2009

Revised Manuscript: January 3, 2010

Manuscript Accepted: January 6, 2010

Published: March 5, 2010

**Virtual Issues**

Vol. 5, Iss. 7 *Virtual Journal for Biomedical Optics*

**Citation**

Konstantin V. Gilev, Elena Eremina, Maxim A. Yurkin, and Valeri P. Maltsev, "Comparison of the discrete dipole approximation and the discrete source method for simulation of light scattering by red blood cells," Opt. Express **18**, 5681-5690 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-6-5681

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### References

- J. P. Greer, J. Foerster, and J. N. Lukens, eds., Wintrobe's Clinical Hematology, (Lippincott Williams & Wilkins Publishers, Baltimore, USA, 2003).
- V. P. Maltsev, and K. A. Semyanov, Characterisation of Bio-Particles from Light Scattering, Inverse and Ill-Posed Problems Series (VSP, Utrecht, 2004).
- S. V. Tsinopoulos and D. Polyzos, “Scattering of He-Ne laser light by an average-sized red blood cell,” Appl. Opt. 38(25), 5499–5510 (1999). [CrossRef]
- J. Q. Lu, P. Yang, and X. H. Hu, “Simulations of light scattering from a biconcave red blood cell using the finite-difference time-domain method,” J. Biomed. Opt. 10(2), 024022 (2005). [CrossRef] [PubMed]
- M. A. Yurkin, K. A. Semyanov, P. A. Tarasov, A. V. Chernyshev, A. G. Hoekstra, and V. P. Maltsev, “Experimental and theoretical study of light scattering by individual mature red blood cells by use of scanning flow cytometry and a discrete dipole approximation,” Appl. Opt. 44(25), 5249–5256 (2005). [CrossRef] [PubMed]
- A. Karlsson, J. P. He, J. Swartling, and S. Andersson-Engels, “Numerical simulations of light scattering by red blood cells,” IEEE Trans. Biomed. Eng. 52(1), 13–18 (2005). [CrossRef] [PubMed]
- T. Wriedt, J. Hellmers, E. Eremina, and R. Schuh, “Light scattering by single erythrocyte: comparison of different methods,” J. Quant. Spectrosc. Radiat. Transf. 100(1-3), 444–456 (2006). [CrossRef]
- E. Eremina, Y. Eremin, and T. Wriedt, “Analysis of light scattering by erythrocyte based on discrete sources method,” Opt. Commun. 244(1-6), 15–23 (2005). [CrossRef]
- E. Eremina, J. Hellmers, Y. Eremin, and T. Wriedt, “Different shape models for erythrocyte: Light scattering analysis based on the discrete sources method,” J. Quant. Spectrosc. Radiat. Transf. 102(1), 3–10 (2006). [CrossRef]
- E. Eremina, “Light scattering by an erythrocyte based on Discrete Sources Method: shape and refractive index influence,” J. Quant. Spectrosc. Radiat. Transf. 110(14-16), 1526–1534 (2009). [CrossRef]
- A. M. K. Nilsson, P. Alsholm, A. Karlsson, and S. Andersson-Engels, “T-Matrix Computations of Light Scattering by Red Blood Cells,” Appl. Opt. 37(13), 2735–2748 (1998). [CrossRef]
- A. N. Shvalov, J. T. Soini, A. V. Chernyshev, P. A. Tarasov, E. Soini, and V. P. Maltsev, “Light-scattering properties of individual erythrocytes,” Appl. Opt. 38(1), 230–235 (1999). [CrossRef]
- Yu. Eremin, “The method of discrete sources in electromagnetic scattering by axially symmetric structures,” J. Commun. Technol. Electron. 45(Suppl.2), 269–280 (2000).
- Y. Eremin, N. Orlov, and A. Sveshnikov, “Models of electromagnetic scattering problems based on discrete sources method” in: Generalizes Multipole Techniques for Electromagnetic and Light Scattering, T. Wriedt, ed. (Elsevier, Amsterdam, 1999), Chapter 4.
- B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11(4), 1491–1499 (1994). [CrossRef]
- M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: An overview and recent developments,” JQSRT 106, 558–589 (2007).
- “ADDA - light scattering simulator using the discrete dipole approximation”, http://code.google.com/p/a-dda/ (2009).
- M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “The discrete dipole approximation for simulation of light scattering by particles much larger than the wavelength,” J. Quant. Spectrosc. Radiat. Transf. 106(1-3), 546–557 (2007). [CrossRef]
- P. Mazeron, S. Muller, and H. El Azouzi, “Deformation of erythrocytes under shear: a small-angle light scattering study,” Biorheology 34(2), 99–110 (1997). [CrossRef] [PubMed]
- R. Skalak, A. Tozeren, R. P. Zarda, and S. Chien, “Strain energy function of red blood cell membranes,” Biophys. J. 13(3), 245–264 (1973). [CrossRef] [PubMed]
- Y. C. Fung, W. C. Tsang, and P. Patitucci, “High-resolution data on the geometry of red blood cells,” Biorheology 18(3-6), 369–385 (1981). [PubMed]
- M. A. Yurkin, “Discrete dipole simulations of light scattering by blood cells” PhD thesis, (University of Amsterdam, 2007).
- K. A. Semyanov, P. A. Tarasov, J. T. Soini, A. K. Petrov, and V. P. Maltsev, “Calibration-free method to determine the size and hemoglobin concentration of individual red blood cells from light scattering,” Appl. Opt. 39(31), 5884–5889 (2000). [CrossRef]
- D. H. Tycko, M. H. Metz, E. A. Epstein, and A. Grinbaum, “Flow-cytometric light scattering measurement of red blood cell volume and hemoglobin concentration,” Appl. Opt. 24(9), 1355–1365 (1985). [CrossRef] [PubMed]
- M. A. Yurkin, and A. G. Hoekstra, “User manual for the discrete dipole approximation code ADDA v.0.79,” http://a-dda.googlecode.com/svn/tags/rel_0_79/doc/manual.pdf (2009).
- M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “Convergence of the discrete dipole approximation. I. Theoretical analysis,” J. Opt. Soc. Am. A 23(10), 2578–2591 (2006). [CrossRef]
- “Description of the national compute cluster Lisa,” https://subtrac.sara.nl/userdoc/wiki/lisa/description (2009).
- M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “Convergence of the discrete dipole approximation. II. An extrapolation technique to increase the accuracy,” J. Opt. Soc. Am. A 23(10), 2592–2601 (2006). [CrossRef]
- M. A. Yurkin, A. G. Hoekstra, R. S. Brock, and J. Q. Lu, “Systematic comparison of the discrete dipole approximation and the finite difference time domain method for large dielectric scatterers,” Opt. Express 15(26), 17902–17911 (2007). [CrossRef] [PubMed]
- Y. Okada, I. Mann, I. Sano, and S. Mukai, “Acceleration of the iterative solver in the discrete dipole approximation: Application to the orientation variation of irregularly shaped particles,” J. Quant. Spectrosc. Radiat. Transf. 109(8), 1461–1473 (2008). [CrossRef]

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