## A Pulsed Finite-Difference Time-Domain (FDTD) Method for Calculating Light Scattering from Biological Cells Over Broad Wavelength Ranges

Optics Express, Vol. 6, Issue 7, pp. 147-157 (2000)

http://dx.doi.org/10.1364/OE.6.000147

Acrobat PDF (546 KB)

### Abstract

We combine the finite-difference time-domain method with pulse response techniques in order to calculate the light scattering properties of biological cells over a range of wavelengths simultaneously. The method we describe can be used to compute the scattering patterns of cells containing multiple heterogeneous organelles, providing greater geometric flexibility than Mie theory solutions. Using a desktop computer, we calculate the scattering patterns for common homogeneous models of biological cells and also for more complex representations of cellular morphology. We find that the geometry chosen significantly impacts scattering properties, emphasizing the need for careful consideration of appropriate theoretical models of cellular scattering and for accurate microscopic determination of optical properties.

© Optical Society of America

## 1. Introduction

*et al*. found that the scatterer sizes in biological cells ranged from 0.4 to 2.0 µm, a size consistent with organelles smaller than a cell’s nucleus [10

10. J. Mourant, J. Freyer, A. Hielscher, A. Eick, D. Shen, and T. Johnson, “Mechanisms of light scattering from biological cells relevant to noninvasive optical-tissue diagnostics,” Appl. Opt. **37**, 3586–3593 (1998). [CrossRef]

*et al*. clearly demonstrated that forward scattering of lymphocytes varied inversely with cell volume. This would not be expected from a Mie theory model of scattering and emphasizes that cellular light scattering cannot always be adequately described using the simplest conceivable geometric model [11

11. L. McGann, M. Walterson, and L. Hogg, “Light scattering and cell volumes in osmotically stressed and frozen thawed cells,” Cytometry. **9**, 33–38 (1988). [CrossRef] [PubMed]

12. A. Brunsting and P. Mullaney, “Light scattering from coated spheres: model for biological cells,” Appl. Opt. **3**, 675–680 (1972). [CrossRef]

13. P. Sloot and C. Figdor, “Elastic light scattering from nucleated blood cells: rapid numerical analysis,” Appl. Opt. **25**, 3559–3565 (1986). [CrossRef] [PubMed]

## 2. Methods

16. J Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. **114**, 185–200 (1994). [CrossRef]

*λ*/10 or smaller. At each grid point, the permittivity and conductivity of the medium is specified. The cell is constructed by assigning permittivity values to each cell component. A range of values may be assigned to a particular component if that component is inhomogeneous. Details regarding the application of the FDTD method specifically to biological cells, including refractive index values for specific cellular constituents, may be found in [17

17. A. Dunn, C. Smithpeter, A. Welch, and R. Richards-Kortum, “Finite-Difference Time-Domain Simulation of Light Scattering from Single Cells,” J. Biomed. Opt. **2**, 262–266 (1997). [CrossRef] [PubMed]

19. R. Drezek, A. Dunn, and R. Richards-Kortum,” Light scattering from cells: finite-difference time-domain simulations and goniometric measurements,” Appl. Opt. **38**, 3651–3661 (1999). [CrossRef]

20. C. Furse, S. Mathur, and O. Gandi, “Improvements to the finite-difference time-domain method for calculating the radar cross section of a perfectly conducting target,” IEEE Trans. Microwave Theory Tech. **38**, 919–927 (1990). [CrossRef]

21. C. Britt, “Solution of electromagnetic scattering problems using time domain techniques,” IEEE Trans. Antennas Propagat. **37**, 1181–1192 (1989). [CrossRef]

20. C. Furse, S. Mathur, and O. Gandi, “Improvements to the finite-difference time-domain method for calculating the radar cross section of a perfectly conducting target,” IEEE Trans. Microwave Theory Tech. **38**, 919–927 (1990). [CrossRef]

21. C. Britt, “Solution of electromagnetic scattering problems using time domain techniques,” IEEE Trans. Antennas Propagat. **37**, 1181–1192 (1989). [CrossRef]

20. C. Furse, S. Mathur, and O. Gandi, “Improvements to the finite-difference time-domain method for calculating the radar cross section of a perfectly conducting target,” IEEE Trans. Microwave Theory Tech. **38**, 919–927 (1990). [CrossRef]

*o*nal requirements of a single frequency run. The grid dimensions and incident pulse are determined based on the lowest wavelength of interest. Either a Gaussian or raised cosine pulse can be used as the excitation source for pulsed FDTD; in this work, a Gaussian pulse was employed. To obtain the frequency response, the magnitude and phase of the time domain waveforms are determined using a discrete Fourier transform (DFT). The discrete Fourier transform is expressed as shown in Eq. (1):

**38**, 919–927 (1990). [CrossRef]

## 3. Results

### 3.1 Code Verification

*m*=1.02). The scattering diagrams for wavelengths ranging from 1 µm to 2 µm with a 5 nm increment were computed. The use of the terms scattering diagram,

*a*,

*n*,

*x*, and

*m*follow the definitions of van de Hulst [22]. The term “scattering diagram” refers to a plot of the scattered intensity as a function of angle. When the scattering diagram is normalized so that a value of one is obtained when integrated over 4π steradians, the curve is referred to as a phase function. Throughout this paper, the quantity

*m*refers to relative refractive index,

*a*refers to the scatterer radius,

*λ*=

*λ*

_{0}/

*n*where

*n*is the index of refraction of the medium surrounding the object and λ

_{0}is the wavelength in vacuum, and x=2π

*a*/λ. FDTD data were compared to an analytical solution computed using Mie theory. Results are shown in Figure 1. The FDTD curves agree closely with theoretical predictions. The discrepancy between Mie theory and FDTD data at the highest angles is due to imperfect boundary conditions. The two curves shown in Figure 1 demonstrate that in some cases there is a precise match with theoretical predictions at all angles while in other instances imperfections in the boundary conditions can influence the high angle data (>165°). The top curve in Figure 1 shows worst case data; the bottom curve shows best case data. It should be pointed out that high angle discrepancies between FDTD data and theory are most prominent when the dynamic range of the problem is large.

^{2}[22]. There were not high angle discrepancies in this simulation.

## 3.2 Comparison of Cell Geometries

12. A. Brunsting and P. Mullaney, “Light scattering from coated spheres: model for biological cells,” Appl. Opt. **3**, 675–680 (1972). [CrossRef]

23. R. Meyer, “Light scattering from biological cells: dependence of backscatter radiation on membrane thickness and refractive index,” Appl. Opt. **18**, 585–590 (1979). [CrossRef] [PubMed]

## 3.3 Heterogeneous Cell Geometries: Normal versus Pre-cancerous Cells

26. B. Palcic, D. Garner, and C. MacAulay, “Image cytometry and chemoprevention in cervical cancer,” J Cell Biochem (Suppl). **23**, 43–54 (1995). [CrossRef]

^{-1}providing a fine, heterogeneous chromatin structure. In the dysplastic cell, nuclear index variations were distributed between Δn=±0.04 about the mean nuclear index, n=1.42, at spatial frequencies ranging from 3–30 µm

^{-1}providing a coarser, more heterogeneous chromatin structure. Both normal and dysplastic cells contained several hundred organelles (radii ranging from 50 nm to 0.5 µm; n=1.38 to 1.40) randomly distributed throughout the cytoplasm. Wavelengths spanned from 600 nm to 1000 nm with a 5 nm increment.

## 4. Discussion and Conclusions

16. J Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. **114**, 185–200 (1994). [CrossRef]

19. R. Drezek, A. Dunn, and R. Richards-Kortum,” Light scattering from cells: finite-difference time-domain simulations and goniometric measurements,” Appl. Opt. **38**, 3651–3661 (1999). [CrossRef]

5. G. Zonios, L. Perelman, V. Backman, R. Manoharan, M. Fitzmaurice, J. Van Dam, and M. Feld, “Diffuse reflectance spectroscopy of human adenomatous colon polyps *in vivo*,” Appl. Opt. **38**, 6628–6637 (1999). [CrossRef]

7. L. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. Crawford, and M. Feld, “Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution,” Opt. Lett. **80**, 627–630.

## Acknowledgements

## References and links

1. | R. Marchesini, M. Brambilla, C. Clemente, M. Maniezzo, A. Sichirollo, A. Testori, D. Venturoli, and N. Cascinelli, “In vivo spectrophotometric evaluation of neoplastic and non-neoplastic skin pigmented lesions. I, Reflectance measurements,” Photochem. Photobiol. |

2. | J. Mourant, I. Bigio, J. Boyer, T. Johnson, R. Conn, T. Johnson, and T. Shimada, “Spectroscopic diagnosis of bladder cancer with elastic light scattering,” Lasers Surg. Med. |

3. | J. Mourant, I. Bigio, J. Boyer, T. Johnson, and J. Lacey, “Elastic scattering spectroscopy as a diagnostic for differentiating pathologies in the gastrointestinal tract: preliminary testing,” J. Biomed. Opt. |

4. | Z. Ge, K. Schomacker, and N. Nishioka, “Identification of colonic dysplasia and neoplasia by diffuse reflectance spectroscopy and pattern recognition techniques,” Appl. Spectrosc. |

5. | G. Zonios, L. Perelman, V. Backman, R. Manoharan, M. Fitzmaurice, J. Van Dam, and M. Feld, “Diffuse reflectance spectroscopy of human adenomatous colon polyps |

6. | J. Mourant, T. Fuselier, J. Boyer, T. Johnson, and I. Bigio, “Predictions and measurements of scattering and absorption over broad wavelength ranges in tissue phantoms,” Appl. Opt. |

7. | L. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. Crawford, and M. Feld, “Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution,” Opt. Lett. |

8. | V. Backman, R. Gurjar, K. Badizadegan, I. Itzkan, R. Dasari, L. Perelman, and M. Feld, “Polarized light scattering spectroscopy for quantitative measurement of epithelial structures in situ,” IEEE J. Sel. Topics Quantum Electron. |

9. | K. Sokolov, R. Drezek, K. Gossage, and R. Richards-Kortum, “Reflectance spectroscopy with polarized light: is it sensitive to cellular and nuclear morphology,” Opt. Lett. |

10. | J. Mourant, J. Freyer, A. Hielscher, A. Eick, D. Shen, and T. Johnson, “Mechanisms of light scattering from biological cells relevant to noninvasive optical-tissue diagnostics,” Appl. Opt. |

11. | L. McGann, M. Walterson, and L. Hogg, “Light scattering and cell volumes in osmotically stressed and frozen thawed cells,” Cytometry. |

12. | A. Brunsting and P. Mullaney, “Light scattering from coated spheres: model for biological cells,” Appl. Opt. |

13. | P. Sloot and C. Figdor, “Elastic light scattering from nucleated blood cells: rapid numerical analysis,” Appl. Opt. |

14. | A. Taflove, Computational Electrodynamics: The Finite Difference Time Domain Method (Artech, Boston, 1995). |

15. | Z. Liao, H. Wong, B. Yang, and Y. Yuan, “A transmitting boundary for transient wave analysis,” Sci. Sin. Ser. A. |

16. | J Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. |

17. | A. Dunn, C. Smithpeter, A. Welch, and R. Richards-Kortum, “Finite-Difference Time-Domain Simulation of Light Scattering from Single Cells,” J. Biomed. Opt. |

18. | A. Dunn and R. Richards-Kortum, “Three-dimensional computation of light scattering from cells,” IEEE J. Sel. Topics Quantum Electron. |

19. | R. Drezek, A. Dunn, and R. Richards-Kortum,” Light scattering from cells: finite-difference time-domain simulations and goniometric measurements,” Appl. Opt. |

20. | C. Furse, S. Mathur, and O. Gandi, “Improvements to the finite-difference time-domain method for calculating the radar cross section of a perfectly conducting target,” IEEE Trans. Microwave Theory Tech. |

21. | C. Britt, “Solution of electromagnetic scattering problems using time domain techniques,” IEEE Trans. Antennas Propagat. |

22. | H. C. van de Hulst, |

23. | R. Meyer, “Light scattering from biological cells: dependence of backscatter radiation on membrane thickness and refractive index,” Appl. Opt. |

24. | M. Anderson, J. Jordon, A. Morse, and F. Sharp, |

25. | C. MacAulay and B. Palcic, “Fractal texture features based on optical density surface area: use in image analysis of cervical cells,” Analyt. Quant. Cytol. Histo. |

26. | B. Palcic, D. Garner, and C. MacAulay, “Image cytometry and chemoprevention in cervical cancer,” J Cell Biochem (Suppl). |

27. | A. Taflove, Advances in Computational Electrodynamics: The Finite Difference Time Domain Method (Artech, Boston, 1998). |

28. | A. Dunn, |

**OCIS Codes**

(170.6510) Medical optics and biotechnology : Spectroscopy, tissue diagnostics

(290.0290) Scattering : Scattering

**ToC Category:**

Research Papers

**History**

Original Manuscript: February 11, 2000

Published: March 27, 2000

**Citation**

Rebekah Drezek, Andrew Dunn, and Rebecca Richards-Kortum, "A pulsed finite-difference time-domain (FDTD) method for calculating light scattering from biological cells over broad wavelength ranges," Opt. Express **6**, 147-157 (2000)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-6-7-147

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### References

- R. Marchesini, M. Brambilla, C. Clemente, M. Maniezzo, A. Sichirollo, A. Testori, D. Venturoli, and N. Cascinelli, "In vivo spectrophotometric evaluation of neoplastic and non-neoplastic skin pigmented lesions. I, Reflectance measurements," Photochem. Photobiol. 53, 77-84 (1991). [CrossRef] [PubMed]
- J. Mourant, I. Bigio, J. Boyer, T. Johnson, R. Conn, T. Johnson, and T. Shimada, "Spectroscopic diagnosis of bladder cancer with elastic light scattering," Lasers Surg. Med. 17, 350-357 (1995). [CrossRef] [PubMed]
- J. Mourant, I. Bigio, J. Boyer, T. Johnson, and J. Lacey, "Elastic scattering spectroscopy as a diagnostic for differentiating pathologies in the gastrointestinal tract: preliminary testing," J. Biomed. Opt. 1, 192-199 (1996). [CrossRef] [PubMed]
- Z. Ge, K. Schomacker, and N. Nishioka, "Identification of colonic dysplasia and neoplasia by diffuse reflectance spectroscopy and pattern recognition techniques," Appl. Spectrosc. 52, 833-839 (1998). [CrossRef]
- G. Zonios, L. Perelman, V. Backman, R. Manoharan, M. Fitzmaurice, J. Van Dam, and M. Feld, "Diffuse reflectance spectroscopy of human adenomatous colon polyps in vivo," Appl. Opt. 38, 6628-6637 (1999). [CrossRef]
- J. Mourant, T. Fuselier, J. Boyer, T. Johnson, and I. Bigio, "Predictions and measurements of scattering and absorption over broad wavelength ranges in tissue phantoms," Appl. Opt. 36, 949-957 (1997). [CrossRef] [PubMed]
- L. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. Crawford, and M. Feld, "Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution," Opt. Lett. 80, 627-630.
- V. Backman, R. Gurjar, K. Badizadegan, I. Itzkan, R. Dasari, L. Perelman, and M. Feld, "Polarized light scattering spectroscopy for quantitative measurement of epithelial structures in situ," IEEE J. Sel. Topics Quantum Electron. 5, (1999). [CrossRef]
- K. Sokolov, R. Drezek, K. Gossage, and R. Richards-Kortum, "Reflectance spectroscopy with polarized light: is it sensitive to cellular and nuclear morphology," Opt. Lett. 5, 302-317 (1999).
- J. Mourant, J. Freyer, A. Hielscher, A. Eick, D. Shen, and T. Johnson, "Mechanisms of light scattering from biological cells relevant to noninvasive optical-tissue diagnostics," Appl. Opt. 37, 3586-3593 (1998). [CrossRef]
- L. McGann, M. Walterson, L. Hogg, "Light scattering and cell volumes in osmotically stressed and frozen thawed cells," Cytometry. 9, 33-38 (1988). [CrossRef] [PubMed]
- A. Brunsting and P. Mullaney, "Light scattering from coated spheres: model for biological cells," Appl. Opt. 3, 675-680 (1972). [CrossRef]
- P. Sloot, and C. Figdor, "Elastic light scattering from nucleated blood cells: rapid numerical analysis," Appl. Opt. 25, 3559-3565 (1986). [CrossRef] [PubMed]
- A. Taflove, Computational Electrodynamics: The Finite Difference Time Domain Method (Artech, Boston, 1995).
- Z. Liao, H. Wong, B. Yang, and Y. Yuan, "A transmitting boundary for transient wave analysis," Sci. Sin. Ser. A. 27, 1063-1076 (1984).
- J Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys. 114, 185-200 (1994). [CrossRef]
- A. Dunn, C. Smithpeter, A. Welch, and R. Richards-Kortum, "Finite-Difference Time-Domain Simulation of Light Scattering from Single Cells," J. Biomed. Opt. 2, 262-266 (1997). [CrossRef] [PubMed]
- A. Dunn, and R. Richards-Kortum, "Three-dimensional computation of light scattering from cells," IEEE J. Sel. Topics Quantum Electron. 2, 898-894 (1996). [CrossRef]
- R. Drezek, A. Dunn, and R. Richards-Kortum," Light scattering from cells: finite-difference time-domain simulations and goniometric measurements," Appl. Opt. 38, 3651-3661 (1999). [CrossRef]
- C. Furse, S. Mathur, and O. Gandi, "Improvements to the finite-difference time-domain method for calculating the radar cross section of a perfectly conducting target," IEEE Trans. Microwave Theory Tech. 38, 919-927 (1990). [CrossRef]
- C. Britt, "Solution of electromagnetic scattering problems using time domain techniques," IEEE Trans. Antennas Propagat. 37, 1181-1192 (1989). [CrossRef]
- H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957).
- R. Meyer, "Light scattering from biological cells: dependence of backscatter radiation on membrane thickness and refractive index," Appl. Opt. 18, 585-590 (1979). [CrossRef] [PubMed]
- M. Anderson, J. Jordon, A. Morse, and F. Sharp, A Text and Atlas of Integrated Colposcopy. (Mosby, St. Louis, 1993).
- C. MacAulay, and B. Palcic, "Fractal texture features based on optical density surface area: use in image analysis of cervical cells," Analyt. Quant. Cytol. Histo. 12, 394-398 (1990).
- B. Palcic, D. Garner, and C. MacAulay, "Image cytometry and chemoprevention in cervical cancer," J Cell Biochem (Suppl). 23, 43-54 (1995). [CrossRef]
- A. Taflove, Advances in Computational Electrodynamics: The Finite Difference Time Domain Method (Artech, Boston, 1998).
- A. Dunn, Light Scattering Properties of Cells. PhD Dissertation, (University of Texas at Austin, Austin, TX, 1997).

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