## Impact of velocity distribution assumption on simplified laser speckle imaging equation

Optics Express, Vol. 16, Issue 5, pp. 3197-3203 (2008)

http://dx.doi.org/10.1364/OE.16.003197

Acrobat PDF (238 KB)

### Abstract

Since blood flow is tightly coupled to the health status of biological tissue, several instruments have been developed to monitor blood flow and perfusion dynamics. One such instrument is laser speckle imaging. The goal of this study was to evaluate the use of two velocity distribution assumptions (Lorentzian- and Gaussian-based) to calculate speckle flow index (SFI) values. When the normalized autocorrelation function for the Lorentzian and Gaussian velocity distributions satisfy the same definition of correlation time, then the same velocity range is predicted for low speckle contrast (0<*C*<0.6) and predict different flow velocity range for high contrast. Our derived equations form the basis for simplified calculations of SFI values.

© 2008 Optical Society of America

## 1. Introduction

1. A. K. Dunn, T. Bolay, M. A. Moskowitz, and D. A. Boas, “Dynamic imaging of cerebral blood flow using laser speckle,” J. Cereb. Blood Flow Metab. **21**, 195–201 (2001). [CrossRef] [PubMed]

2. S. A. Sheth, M. Nemoto, M. W. Guiou, M. A. Walker, and A. W. Toga, “Spatiotemporal evolution of functional hemodynamic changes and their relationship to neuronal activity,” J. Cereb. Blood Flow Metab **25**, 830–841 (2005). [CrossRef] [PubMed]

3. H. W. Ren, Z. H. Ding, Y. H. Zhao, J. J. Miao, J. S. Nelson, and Z. P. Chen, “Phase-resolved functional optical coherence tomography: simultaneous imaging of in situ tissue structure, blood flow velocity, standard deviation, birefringence, and Stokes vectors in human skin,” Opt. Lett. **27**, 1702–1704 (2002). [CrossRef]

4. Z. P. Chen, T. E. Milner, D. Dave, and J. S. Nelson, “Optical Doppler tomographic imaging of fluid flow velocity in highly scattering media,” Opt. Lett. **22**, 64–66 (1997). [CrossRef] [PubMed]

6. J.D. Briers and S. Webster, “Quasi real-time digital version of single-exposure speckle photography for full-field monitoring of velocity or flow fields,” Opt. Commun. **116**, 36–42 (1995). [CrossRef]

## 2. Lorentzian and Gaussian velocity distributions

5. A. F. Fercher and J. D. Briers, “Flow Visualization by Means of Single-Exposure Speckle Photography,” Opt. Commun. **37**, 326–330 (1981). [CrossRef]

*C*) and the normalized autocorrelation function of the remitted light:

*σ*is the variance, 〈

*I*〉 is the mean and

*γ*(

*t*) is the normalized autocorrelation function of the remitted light. For a Lorentzian velocity distribution:

5. A. F. Fercher and J. D. Briers, “Flow Visualization by Means of Single-Exposure Speckle Photography,” Opt. Commun. **37**, 326–330 (1981). [CrossRef]

*∝ blood flow velocity) for given values of*

_{c}*C*. Moreover, for

*T*/τ

*>2, Eqs 3 and 4 can be simplified to the following algebraic expressions:*

_{c}*and τ*

_{cl}*are the correlation times for the Lorentzian and Gaussian approximations, respectively. This result is in agreement with the simplified imaging equation obtained by Cheng and Duong [18*

_{cg}18. H. Cheng and T.Q. Duong; “Simplified laser-speckle-imaging analysis method and its application to retinal blood flow imaging,” Opt. Lett. **15**, 2188–2190 (2007). [CrossRef]

*T*/τ

*>2, the velocity predicted by the Gaussian approximation is √*

_{c}*π*times the velocity predicted by the Lorentzian approximation.

## 3. Rederived Gaussian equation

*γ*(

*t*), but there are different expressions of

*γ*(

*t*) reported in the literature [23]. We employed Mandel’s definition of the correlation time:

*is the correlation time for the re-derived Gaussian-based speckle imaging equation. Note that this equation is identical to that derived using the more common Lorentzian velocity distribution assumption (Eq. 6, top row).*

_{cga}*C*values (Fig. 2, 0<

*C*<0.6).

*(and hence SFI) values. Although T/τ*

_{c}_{c}<<1 is not encountered in typical LSI experiments, Eq. 12 demonstrates that only in this range of ratios (T/τ

_{c}<<1) will the velocity distribution assumption affect the mapping between speckle contrast and τ

_{c}.

*triangular averaging*of the correlation function [19

19. P Zakharov, A Völker, A Buck, B Weber, and F Scheffold
; “Quantitative modeling of Laser Speckle Imaging,” Opt. Lett. **31**, 3465–3467 (2006). [CrossRef] [PubMed]

*and τ*

_{clg}*are the correlation times for the Lorentzian and Gaussian approximations, respectively. For*

_{cgg}*T*/τ

*>2, Eqs. 13 and 14 can be simplified to the following expressions:*

_{c}_{clg}and C is similar to that derived by Cheng and Duong [18

18. H. Cheng and T.Q. Duong; “Simplified laser-speckle-imaging analysis method and its application to retinal blood flow imaging,” Opt. Lett. **15**, 2188–2190 (2007). [CrossRef]

*T*/τ

*>2 (i.e., 0<*

_{c}*C*<0.6, see Fig. 2), Goodman’s theory predicts the same SFI range for the Lorentzian and Gaussian velocity distributions. Moreover, from Eqs. 6 (top row), 11 and 15, the SFI values predicted by Goodman`s model are directly proportional to the Lorentzian and the rederived Gaussian-based speckle imaging equations.

*S*/

*N*)

_{rms}associated with Eq. 15 is greater than √2. At higher (

*C*>0.6) speckle contrast values, (

*S*/

*N*)

_{rms}is less than √2, which is unacceptably low for practical application. Eq. 16 is valid for both the Briers and Goodman models.

6. J.D. Briers and S. Webster, “Quasi real-time digital version of single-exposure speckle photography for full-field monitoring of velocity or flow fields,” Opt. Commun. **116**, 36–42 (1995). [CrossRef]

*C*values did not reach the theoretical limit of unity for completely stationary objects; they instead observed a maximum value of 0.6. Experimental data from Yuan et al. [10

10. S. Yuan, A. Devor, D. A. Boas, and A. K. Dunn, “Determination of optimal exposure time for imaging of blood flow changes with laser speckle contrast imaging,” Appl. Opt. **44**, 1823–1830 (2005). [CrossRef] [PubMed]

*C*value of 0.6. Dunn et al. [1

1. A. K. Dunn, T. Bolay, M. A. Moskowitz, and D. A. Boas, “Dynamic imaging of cerebral blood flow using laser speckle,” J. Cereb. Blood Flow Metab. **21**, 195–201 (2001). [CrossRef] [PubMed]

12. H. Bolay, U. Reuter, A.K. Dunn, Z.H. Huang, D.A. Boas, and M.A. Moskowitz, “Intrinsic brain activity triggers trigeminal meningeal afferents in a migraine model,” Nat. Med. **8**, 136–142 (2002). [CrossRef] [PubMed]

*C*values of ~0.15. In experimental LSI data that we acquire from rodent dorsal window chamber models [15-17

15. B. Choi, N. M. Kang, and J. S. Nelson, “Laser speckle imaging for monitoring blood flow dynamics in the in vivo rodent dorsal skinfold model,” Microvasc. Res. **68**, 143–146 (2004). [CrossRef] [PubMed]

*C*values greater than 0.6 in less than 1% of the pixels (Fig. 3). It is important to note that measured

*C*values may differ among LSI instruments due to differences in parameters such as quality of imaging optics and camera, coherence length of incident light source, etc. Nevertheless, we believe these studies collectively justify the rationale for other researchers employing LSI to utilize the proposed simplified speckle imaging equation (Eq. 15). An advantage of Eq. 15 over either use of approximate solutions or look-up tables to extract τ

*from the speckle imaging equation is that it represents an exact analytical solution for*

_{c}*C*<0.6.

18. H. Cheng and T.Q. Duong; “Simplified laser-speckle-imaging analysis method and its application to retinal blood flow imaging,” Opt. Lett. **15**, 2188–2190 (2007). [CrossRef]

_{c}are 100 to 400. Values greater than 100 are encountered in clearly defined blood vessels, but the ratio is much lower for pixels that map to poorly-perfused regions of tissue. For example, a speckle contrast of 0.6, which is encountered experimentally, maps to a ratio of two. Our analysis demonstrates that, even for such a low ratio, the simplified imaging algorithm can be used with high accuracy.

## 4. Conclusions

*from the imaging equation involved either an approximate solution to or use of look-up tables. Based on our own unpublished experimental data, we have shown that a simplified speckle imaging equation (Eq. 15) will cover the vast majority of practical experimental conditions.*

_{c}## Acknowledgements

## References

1. | A. K. Dunn, T. Bolay, M. A. Moskowitz, and D. A. Boas, “Dynamic imaging of cerebral blood flow using laser speckle,” J. Cereb. Blood Flow Metab. |

2. | S. A. Sheth, M. Nemoto, M. W. Guiou, M. A. Walker, and A. W. Toga, “Spatiotemporal evolution of functional hemodynamic changes and their relationship to neuronal activity,” J. Cereb. Blood Flow Metab |

3. | H. W. Ren, Z. H. Ding, Y. H. Zhao, J. J. Miao, J. S. Nelson, and Z. P. Chen, “Phase-resolved functional optical coherence tomography: simultaneous imaging of in situ tissue structure, blood flow velocity, standard deviation, birefringence, and Stokes vectors in human skin,” Opt. Lett. |

4. | Z. P. Chen, T. E. Milner, D. Dave, and J. S. Nelson, “Optical Doppler tomographic imaging of fluid flow velocity in highly scattering media,” Opt. Lett. |

5. | A. F. Fercher and J. D. Briers, “Flow Visualization by Means of Single-Exposure Speckle Photography,” Opt. Commun. |

6. | J.D. Briers and S. Webster, “Quasi real-time digital version of single-exposure speckle photography for full-field monitoring of velocity or flow fields,” Opt. Commun. |

7. | J. D. Briers, G. Richards, and X. W. He, “Capillary blood flow monitoring using laser speckle contrast analysis (LASCA),” J. Biomed. Opt. |

8. | H. Y. Cheng, Q. M. Luo, S. Q. Zeng, S. B. Chen, J. Cen, and H. Gong, “Modified laser speckle imaging method with improved spatial resolution,” J. Biomed. Opt. |

9. | H. Y. Cheng, Q. M. Luo, S. Q. Zeng, S. B. Chen, W. H. Luo, and H. Gong, “Hyperosmotic chemical agent’s effect on in vivo cerebral blood flow revealed by laser speckle,” Appl. Opt. |

10. | S. Yuan, A. Devor, D. A. Boas, and A. K. Dunn, “Determination of optimal exposure time for imaging of blood flow changes with laser speckle contrast imaging,” Appl. Opt. |

11. | A. K. Dunn, A. Devor, A. M. Dale, and D. A. Boas, “Spatial extent of oxygen metabolism and hemodynamic changes during functional activation of the rat somatosensory cortex,” Neuroimage |

12. | H. Bolay, U. Reuter, A.K. Dunn, Z.H. Huang, D.A. Boas, and M.A. Moskowitz, “Intrinsic brain activity triggers trigeminal meningeal afferents in a migraine model,” Nat. Med. |

13. | M. Hirao, H. Oku, W. Goto, T. Sugiyama, T. Kobayashi, and T. Ikeda, “Effects of adenosine on optic nerve head circulation in rabbits,” Exp. Eye Res. |

14. | K. R. Forrester, J. Tulip, C. Leonard, C. Stewart, and R. C. Bray, “A laser speckle imaging technique for measuring tissue perfusion,” IEEE Trans. Biomed. Eng. |

15. | B. Choi, N. M. Kang, and J. S. Nelson, “Laser speckle imaging for monitoring blood flow dynamics in the in vivo rodent dorsal skinfold model,” Microvasc. Res. |

16. | T. K. Smith, B. Choi, J. C. Ramirez-San-Juan, J. S. Nelson, K. Osann, and K. M. Kelly, “Microvascular blood flow dynamics associated with photodynamic therapy and pulsed dye laser irradiation,” Lasers Surg. Med. , |

17. | B. Choi, J.C. Ramirez-San-Juan, J. Lotfi, and J.S. Nelson, “Linear response range characterization and in vivo application of laser speckle imaging of blood flow dynamics,” J. Biomed. Opt. |

18. | H. Cheng and T.Q. Duong; “Simplified laser-speckle-imaging analysis method and its application to retinal blood flow imaging,” Opt. Lett. |

19. | P Zakharov, A Völker, A Buck, B Weber, and F Scheffold
; “Quantitative modeling of Laser Speckle Imaging,” Opt. Lett. |

20. | J. W. Goodman, |

21. | J.W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE , |

22. | J.D. Briers and A.F. Fercher
“A laser speckle technique for the visualization of retinal blood flow,” Proc. SPIE |

23. | R. Bracewell, |

**OCIS Codes**

(120.6150) Instrumentation, measurement, and metrology : Speckle imaging

(120.7250) Instrumentation, measurement, and metrology : Velocimetry

(170.3340) Medical optics and biotechnology : Laser Doppler velocimetry

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: August 20, 2007

Revised Manuscript: November 21, 2007

Manuscript Accepted: November 23, 2007

Published: February 22, 2008

**Virtual Issues**

Vol. 3, Iss. 4 *Virtual Journal for Biomedical Optics*

**Citation**

Julio C. Ramirez-San-Juan, Ruben Ramos-García, Ileana Guizar-Iturbide, Gabriel Martínez-Niconoff, and Bernard Choi, "Impact of velocity distribution assumption on simplified laser speckle imaging equation," Opt. Express **16**, 3197-3203 (2008)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-16-5-3197

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### References

- A. K. Dunn, T. Bolay, M. A. Moskowitz and D. A. Boas, "Dynamic imaging of cerebral blood flow using laser speckle," J. Cereb. Blood Flow Metab. 21, 195-201 (2001). [CrossRef] [PubMed]
- S. A. Sheth, M. Nemoto, M. W. Guiou, M. A. Walker and A. W. Toga, "Spatiotemporal evolution of functional hemodynamic changes and their relationship to neuronal activity," J. Cereb. Blood Flow Metab 25, 830-841 (2005). [CrossRef] [PubMed]
- H. W. Ren, Z. H. Ding, Y. H. Zhao, J. J. Miao, J. S. Nelson and Z. P. Chen, "Phase-resolved functional optical coherence tomography: simultaneous imaging of in situ tissue structure, blood flow velocity, standard deviation, birefringence, and Stokes vectors in human skin," Opt. Lett. 27, 1702-1704 (2002). [CrossRef]
- Z. P. Chen, T. E. Milner, D. Dave and J. S. Nelson, "Optical Doppler tomographic imaging of fluid flow velocity in highly scattering media," Opt. Lett. 22, 64-66 (1997). [CrossRef] [PubMed]
- A. F. Fercher and J. D. Briers, "Flow Visualization by Means of Single-Exposure Speckle Photography," Opt. Commun. 37, 326-330 (1981). [CrossRef]
- J. D. Briers and S. Webster, "Quasi real-time digital version of single-exposure speckle photography for full-field monitoring of velocity or flow fields," Opt. Commun. 116, 36-42 (1995). [CrossRef]
- J. D. Briers, G. Richards and X. W. He, "Capillary blood flow monitoring using laser speckle contrast analysis (LASCA)," J. Biomed. Opt. 4, 164-175 (1999). [CrossRef]
- H. Y. Cheng, Q. M. Luo, S. Q. Zeng, S. B. Chen, J. Cen and H. Gong, "Modified laser speckle imaging method with improved spatial resolution," J. Biomed. Opt. 8, 559-564 (2003). [CrossRef] [PubMed]
- H. Y. Cheng, Q. M. Luo, S. Q. Zeng, S. B. Chen, W. H. Luo and H. Gong, "Hyperosmotic chemical agent's effect on in vivo cerebral blood flow revealed by laser speckle," Appl. Opt. 43, 5772-5777 (2004). [CrossRef] [PubMed]
- S. Yuan, A. Devor, D. A. Boas and A. K. Dunn, "Determination of optimal exposure time for imaging of blood flow changes with laser speckle contrast imaging," Appl. Opt. 44, 1823-1830 (2005). [CrossRef] [PubMed]
- A. K. Dunn, A. Devor, A. M. Dale and D. A. Boas, "Spatial extent of oxygen metabolism and hemodynamic changes during functional activation of the rat somatosensory cortex," Neuroimage 27, 279-290 (2005). [CrossRef] [PubMed]
- H. Bolay, U. Reuter, A.K. Dunn, Z.H. Huang, D.A. Boas, M.A. Moskowitz, "Intrinsic brain activity triggers trigeminal meningeal afferents in a migraine model," Nat. Med. 8, 136-142 (2002). [CrossRef] [PubMed]
- M. Hirao, H. Oku, W. Goto, T. Sugiyama, T. Kobayashi and T. Ikeda, "Effects of adenosine on optic nerve head circulation in rabbits," Exp. Eye Res. 79, 729-735 (2004). [CrossRef] [PubMed]
- K. R. Forrester, J. Tulip, C. Leonard, C. Stewart and R. C. Bray, "A laser speckle imaging technique for measuring tissue perfusion," IEEE Trans. Biomed. Eng. 51, 2074-2084 (2004). [CrossRef] [PubMed]
- B. Choi, N. M. Kang and J. S. Nelson, "Laser speckle imaging for monitoring blood flow dynamics in the in vivo rodent dorsal skinfold model," Microvasc. Res. 68, 143-146 (2004). [CrossRef] [PubMed]
- T. K. Smith, B. Choi, J. C. Ramirez-San-Juan, J. S. Nelson, K. Osann and K. M. Kelly, "Microvascular blood flow dynamics associated with photodynamic therapy and pulsed dye laser irradiation," Lasers Surg. Med., 38, 532-539 (2006). [CrossRef] [PubMed]
- B. Choi, J. C. Ramirez-San-Juan, J. Lotfi, J. S. Nelson, "Linear response range characterization and in vivo application of laser speckle imaging of blood flow dynamics," J. Biomed. Opt. 11, 041129 (2006). [CrossRef] [PubMed]
- H. Cheng and T. Q. Duong, "Simplified laser-speckle-imaging analysis method and its application to retinal blood flow imaging," Opt. Lett. 15, 2188-2190 (2007). [CrossRef]
- P. Zakharov, A. Völker, A. Buck, B. Weber and F. Scheffold, "Quantitative modeling of Laser Speckle Imaging," Opt. Lett. 31, 3465-3467 (2006). [CrossRef] [PubMed]
- J. W. Goodman, Statistical Optics (John Wiley & Sons, 1985).
- J. W. Goodman, "Some effects of target-induced scintillation on optical radar performance," Proc. IEEE, 53, 1688 (1965). [CrossRef]
- J. D. Briers and A. F. Fercher, "A laser speckle technique for the visualization of retinal blood flow," Proc. SPIE 369, 22-28 (1982).
- R. Bracewell, The Fourier transform and its applications (Mc Graw-Hill, 1965).

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