Abstract

Laser Speckle Contrast Imaging (LSCI) is a minimally invasive full field optical technique used to generate blood flow maps with high spatial and temporal resolution. The lack of quantitative accuracy and the inability to predict flows in the presence of static scatterers such as an intact or thinned skull have been the primary limitation of LSCI. We present a new Multi-Exposure Speckle Imaging (MESI) instrument that has potential to obtain quantitative baseline flow measures. We show that the MESI instrument extends the range over which relative flow measurements are linear. We also present a new speckle model which can discriminate flows in the presence of static scatters. We show that in the presence of static scatterers the new model used along with the new MESI instrument can predict correlation times of flow consistently to within 10% of the value without static scatterers compared to an average deviation of more than 100% from the value without static scatterers using traditional LSCI. We also show that the new speckle model used with the MESI instrument can maintain the linearity of relative flow measurements in the presence of static scatterers.

© 2008 Optical Society of America

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References

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  1. A. Fercher and J. Briers, "Flow visualization by means of single-exposure speckle photography," Opt. Commun. 37, 326-330 (1981).
    [CrossRef]
  2. A. Dunn, H. Bolay, M. Moskowitz, and D. Boas, "Dynamic Imaging of Cerebral Blood Flow Using Laser Speckle," J. Cereb. Blood Flow Metab. 21, 195-201 (2001).
    [CrossRef] [PubMed]
  3. B. Weber, C. Burger, M. Wyss, G. von Schulthess, F. Scheffold, and A. Buck, "Optical imaging of the spatiotemporal dynamics of cerebral blood flow and oxidative metabolism in the rat barrel cortex," Eur. J. Neurosci. 20, 2664-2670 (2004).
    [CrossRef] [PubMed]
  4. T. Durduran, M. Burnett, G. Yu, C. Zhou, D. Furuya, A. Yodh, J. Detre, and J. Greenberg, "Spatiotemporal Quantification of Cerebral Blood Flow During Functional Activation in Rat Somatosensory Cortex Using Laser-Speckle Flowmetry," J. Cereb. Blood Flow Metab. 24, 518-525 (2004).
    [CrossRef] [PubMed]
  5. D. Atochin, J. Murciano, Y. Gursoy-Ozdemir, T. Krasik, F. Noda, C. Ayata, A. Dunn, M. Moskowitz, P. Huang, and V. Muzykantov, "Mouse Model of Microembolic Stroke and Reperfusion," Stroke 35, 2177-2182 (2004).
    [CrossRef] [PubMed]
  6. A. Kharlamov, B. Brown, K. Easley, and S. Jones, "Heterogeneous response of cerebral blood flow to hypotension demonstrated by laser speckle imaging flowmetry in rats," Neurosci. Lett 368, 151-156 (2004).
    [CrossRef] [PubMed]
  7. K. Murari, N. Li, A. Rege, X. Jia, A. All, and N. Thakor, "Contrast-enhanced imaging of cerebral vasculature with laser speckle," Appl. Opt. 46, 5340-5346 (2007).
    [CrossRef] [PubMed]
  8. B. Ruth, "Measuring the steady-state value and the dynamics of the skin blood flow using the non-contact laser speckle method." Med. Eng. Phys. 16, 105-11 (1994).
    [CrossRef] [PubMed]
  9. K. Forrester, C. Stewart, J. Tulip, C. Leonard, and R. Bray, "Comparison of laser speckle and laser Doppler perfusion imaging: Measurement in human skin and rabbit articular tissue," Med. Biol. Engg. Comp. 40, 687-697 (2002).
    [CrossRef]
  10. B. Choi, N. Kang, and J. Nelson, "Laser speckle imaging for monitoring blood flow dynamics in the in vivo rodent dorsal skin fold model," Microvasc. Res 68, 143-146 (2004).
    [CrossRef] [PubMed]
  11. K. Yaoeda, M. Shirakashi, S. Funaki, H. Funaki, T. Nakatsue, A. Fukushima, and H. Abe, "Measurement of microcirculation in optic nerve head by laser speckle flowgraphy in normal volunteers," Am. J. Opthalmol. 130, 606-610 (2000).
    [CrossRef]
  12. C. Ayata, A. Dunn, Y. Gursoy-Ozdemir, Z. Huang, D. Boas, and M. Moskowitz, "Laser speckle flowmetry for the study of cerebrovascular physiology in normal and ischemic mouse cortex," J. Cereb. Blood Flow Metab. 24, 744-755 (2004).
    [CrossRef] [PubMed]
  13. A. Strong, E. Bezzina, P. Anderson, M. Boutelle, S. Hopwood, and A. Dunn, "Evaluation of laser speckle flowmetry for imaging cortical perfusion in experimental stroke studies: quantitation of perfusion and detection of periinfarct depolarisations," J. Cereb. Blood Flow Metab. 26, 645-53 (2006).
    [CrossRef]
  14. A. Dunn, A. Devor, A. Dale, and D. Boas, "Spatial extent of oxygen metabolism and hemodynamic changes during functional activation of the rat somatosensory cortex," NeuroImage 27, 279-290 (2005).
    [CrossRef] [PubMed]
  15. S. Yuan, A. Devor, D. Boas, and A. 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]
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    [CrossRef] [PubMed]
  18. R. Bandyopadhyay, A. Gittings, S. Suh, P. Dixon, and D. Durian, "Speckle-visibility spectroscopy: A tool to study time-varying dynamics," Rev. Sci. Instrum. 76, 093110 (2005).
    [CrossRef]
  19. B. Berne and R. Pecora, Dynamic Light Scattering : with Applications to Chemistry, Biology and Physics (Dover Publications, 2000).
  20. R. Bonner and R. Nossal, "Model for laser Doppler measurements of blood flow in tissue," Appl. Opt. 20, 2097-2107 (1981).
    [CrossRef] [PubMed]
  21. K. Forrester, J. Tulip, C. Leonard, C. Stewart, and R. Bray, "A laser speckle imaging technique for measuring tissue perfusion," IEEE Trans. Biomed. Engg. 51, 2074-2084 (2004).
    [CrossRef]
  22. P. Dixon and D. Durian, "Speckle Visibility Spectroscopy and Variable Granular Fluidization," Phys. Rev. Lett. 90, 184302 (2003).
    [CrossRef] [PubMed]
  23. P. Lemieux and D. Durian, "Investigating non-Gaussian scattering processes by using n th-order intensity correlation functions," J. Opt. Soc. Am. A 16, 1651-1664 (1999).
    [CrossRef]
  24. H. Cheng, Q. Luo, S. Zeng, S. Chen, J. Cen, and H. Gong, "Modified laser speckle imaging method with improved spatial resolution," J. Biomed. Opt. 8, 559-564 (2003).
    [CrossRef] [PubMed]
  25. P. Li, S. Ni, L. Zhang, S. Zeng, and Q. Luo, "Imaging cerebral blood flow through the intact rat skull with temporal laser speckle imaging," Opt. Lett. 31, 1824-1826 (2006).
    [CrossRef] [PubMed]
  26. 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]
  27. D. Boas and A. Yodh, "Spatially varying dynamical properties of turbid media probed with diffusing temporal light correlation," J. Opt. Soc. Am. A 14, 192-215 (1997).
    [CrossRef]
  28. J. Anderson, D. Chiu, R. Jackman, O. Cherniavskaya, J. McDonald, H. Wu, S. Whitesides, and G. Whitesides, "Fabrication of Topologically Complex Three-Dimensional Microfluidic Systems in PDMS by Rapid Prototyping," Science 261, 895 (1993).
  29. A. Oldenburg, F. Toublan, K. Suslick, A. Wei, and S. Boppart, "Magnetomotive contrast for in vivo optical coherence tomography," Opt. Express 13, 6597-6614 (2005).
    [CrossRef] [PubMed]

2007 (2)

K. Murari, N. Li, A. Rege, X. Jia, A. All, and N. Thakor, "Contrast-enhanced imaging of cerebral vasculature with laser speckle," Appl. Opt. 46, 5340-5346 (2007).
[CrossRef] [PubMed]

H. Shin, P. Jones, M. Garcia-Alloza, L. Borrelli, S. Greenberg, B. Bacskai, M. Frosch, B. Hyman, M. Moskowitz, and C. Ayata, "Age-dependent cerebrovascular dysfunction in a transgenic mouse model of cerebral amyloid angiopathy," Brain 130, 2310-2319 (2007).
[CrossRef] [PubMed]

2006 (3)

P. Li, S. Ni, L. Zhang, S. Zeng, and Q. Luo, "Imaging cerebral blood flow through the intact rat skull with temporal laser speckle imaging," Opt. Lett. 31, 1824-1826 (2006).
[CrossRef] [PubMed]

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]

A. Strong, E. Bezzina, P. Anderson, M. Boutelle, S. Hopwood, and A. Dunn, "Evaluation of laser speckle flowmetry for imaging cortical perfusion in experimental stroke studies: quantitation of perfusion and detection of periinfarct depolarisations," J. Cereb. Blood Flow Metab. 26, 645-53 (2006).
[CrossRef]

2005 (4)

A. Dunn, A. Devor, A. Dale, and D. Boas, "Spatial extent of oxygen metabolism and hemodynamic changes during functional activation of the rat somatosensory cortex," NeuroImage 27, 279-290 (2005).
[CrossRef] [PubMed]

S. Yuan, A. Devor, D. Boas, and A. 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]

R. Bandyopadhyay, A. Gittings, S. Suh, P. Dixon, and D. Durian, "Speckle-visibility spectroscopy: A tool to study time-varying dynamics," Rev. Sci. Instrum. 76, 093110 (2005).
[CrossRef]

A. Oldenburg, F. Toublan, K. Suslick, A. Wei, and S. Boppart, "Magnetomotive contrast for in vivo optical coherence tomography," Opt. Express 13, 6597-6614 (2005).
[CrossRef] [PubMed]

2004 (7)

K. Forrester, J. Tulip, C. Leonard, C. Stewart, and R. Bray, "A laser speckle imaging technique for measuring tissue perfusion," IEEE Trans. Biomed. Engg. 51, 2074-2084 (2004).
[CrossRef]

B. Choi, N. Kang, and J. Nelson, "Laser speckle imaging for monitoring blood flow dynamics in the in vivo rodent dorsal skin fold model," Microvasc. Res 68, 143-146 (2004).
[CrossRef] [PubMed]

C. Ayata, A. Dunn, Y. Gursoy-Ozdemir, Z. Huang, D. Boas, and M. Moskowitz, "Laser speckle flowmetry for the study of cerebrovascular physiology in normal and ischemic mouse cortex," J. Cereb. Blood Flow Metab. 24, 744-755 (2004).
[CrossRef] [PubMed]

B. Weber, C. Burger, M. Wyss, G. von Schulthess, F. Scheffold, and A. Buck, "Optical imaging of the spatiotemporal dynamics of cerebral blood flow and oxidative metabolism in the rat barrel cortex," Eur. J. Neurosci. 20, 2664-2670 (2004).
[CrossRef] [PubMed]

T. Durduran, M. Burnett, G. Yu, C. Zhou, D. Furuya, A. Yodh, J. Detre, and J. Greenberg, "Spatiotemporal Quantification of Cerebral Blood Flow During Functional Activation in Rat Somatosensory Cortex Using Laser-Speckle Flowmetry," J. Cereb. Blood Flow Metab. 24, 518-525 (2004).
[CrossRef] [PubMed]

D. Atochin, J. Murciano, Y. Gursoy-Ozdemir, T. Krasik, F. Noda, C. Ayata, A. Dunn, M. Moskowitz, P. Huang, and V. Muzykantov, "Mouse Model of Microembolic Stroke and Reperfusion," Stroke 35, 2177-2182 (2004).
[CrossRef] [PubMed]

A. Kharlamov, B. Brown, K. Easley, and S. Jones, "Heterogeneous response of cerebral blood flow to hypotension demonstrated by laser speckle imaging flowmetry in rats," Neurosci. Lett 368, 151-156 (2004).
[CrossRef] [PubMed]

2003 (2)

P. Dixon and D. Durian, "Speckle Visibility Spectroscopy and Variable Granular Fluidization," Phys. Rev. Lett. 90, 184302 (2003).
[CrossRef] [PubMed]

H. Cheng, Q. Luo, S. Zeng, S. Chen, J. Cen, and H. Gong, "Modified laser speckle imaging method with improved spatial resolution," J. Biomed. Opt. 8, 559-564 (2003).
[CrossRef] [PubMed]

2002 (1)

K. Forrester, C. Stewart, J. Tulip, C. Leonard, and R. Bray, "Comparison of laser speckle and laser Doppler perfusion imaging: Measurement in human skin and rabbit articular tissue," Med. Biol. Engg. Comp. 40, 687-697 (2002).
[CrossRef]

2001 (1)

A. Dunn, H. Bolay, M. Moskowitz, and D. Boas, "Dynamic Imaging of Cerebral Blood Flow Using Laser Speckle," J. Cereb. Blood Flow Metab. 21, 195-201 (2001).
[CrossRef] [PubMed]

2000 (1)

K. Yaoeda, M. Shirakashi, S. Funaki, H. Funaki, T. Nakatsue, A. Fukushima, and H. Abe, "Measurement of microcirculation in optic nerve head by laser speckle flowgraphy in normal volunteers," Am. J. Opthalmol. 130, 606-610 (2000).
[CrossRef]

1999 (1)

1997 (1)

1994 (1)

B. Ruth, "Measuring the steady-state value and the dynamics of the skin blood flow using the non-contact laser speckle method." Med. Eng. Phys. 16, 105-11 (1994).
[CrossRef] [PubMed]

1993 (1)

J. Anderson, D. Chiu, R. Jackman, O. Cherniavskaya, J. McDonald, H. Wu, S. Whitesides, and G. Whitesides, "Fabrication of Topologically Complex Three-Dimensional Microfluidic Systems in PDMS by Rapid Prototyping," Science 261, 895 (1993).

1981 (2)

R. Bonner and R. Nossal, "Model for laser Doppler measurements of blood flow in tissue," Appl. Opt. 20, 2097-2107 (1981).
[CrossRef] [PubMed]

A. Fercher and J. Briers, "Flow visualization by means of single-exposure speckle photography," Opt. Commun. 37, 326-330 (1981).
[CrossRef]

1975 (1)

M. Stern, "In vivo evaluation of microcirculation by coherent light scattering," Nature (London) 254, 56-8 (1975).
[CrossRef] [PubMed]

Am. J. Opthalmol. (1)

K. Yaoeda, M. Shirakashi, S. Funaki, H. Funaki, T. Nakatsue, A. Fukushima, and H. Abe, "Measurement of microcirculation in optic nerve head by laser speckle flowgraphy in normal volunteers," Am. J. Opthalmol. 130, 606-610 (2000).
[CrossRef]

Appl. Opt. (3)

Brain (1)

H. Shin, P. Jones, M. Garcia-Alloza, L. Borrelli, S. Greenberg, B. Bacskai, M. Frosch, B. Hyman, M. Moskowitz, and C. Ayata, "Age-dependent cerebrovascular dysfunction in a transgenic mouse model of cerebral amyloid angiopathy," Brain 130, 2310-2319 (2007).
[CrossRef] [PubMed]

Eur. J. Neurosci. (1)

B. Weber, C. Burger, M. Wyss, G. von Schulthess, F. Scheffold, and A. Buck, "Optical imaging of the spatiotemporal dynamics of cerebral blood flow and oxidative metabolism in the rat barrel cortex," Eur. J. Neurosci. 20, 2664-2670 (2004).
[CrossRef] [PubMed]

IEEE Trans. Biomed. Engg. (1)

K. Forrester, J. Tulip, C. Leonard, C. Stewart, and R. Bray, "A laser speckle imaging technique for measuring tissue perfusion," IEEE Trans. Biomed. Engg. 51, 2074-2084 (2004).
[CrossRef]

J. Biomed. Opt. (1)

H. Cheng, Q. Luo, S. Zeng, S. Chen, J. Cen, and H. Gong, "Modified laser speckle imaging method with improved spatial resolution," J. Biomed. Opt. 8, 559-564 (2003).
[CrossRef] [PubMed]

J. Cereb. Blood Flow Metab. (4)

T. Durduran, M. Burnett, G. Yu, C. Zhou, D. Furuya, A. Yodh, J. Detre, and J. Greenberg, "Spatiotemporal Quantification of Cerebral Blood Flow During Functional Activation in Rat Somatosensory Cortex Using Laser-Speckle Flowmetry," J. Cereb. Blood Flow Metab. 24, 518-525 (2004).
[CrossRef] [PubMed]

A. Dunn, H. Bolay, M. Moskowitz, and D. Boas, "Dynamic Imaging of Cerebral Blood Flow Using Laser Speckle," J. Cereb. Blood Flow Metab. 21, 195-201 (2001).
[CrossRef] [PubMed]

C. Ayata, A. Dunn, Y. Gursoy-Ozdemir, Z. Huang, D. Boas, and M. Moskowitz, "Laser speckle flowmetry for the study of cerebrovascular physiology in normal and ischemic mouse cortex," J. Cereb. Blood Flow Metab. 24, 744-755 (2004).
[CrossRef] [PubMed]

A. Strong, E. Bezzina, P. Anderson, M. Boutelle, S. Hopwood, and A. Dunn, "Evaluation of laser speckle flowmetry for imaging cortical perfusion in experimental stroke studies: quantitation of perfusion and detection of periinfarct depolarisations," J. Cereb. Blood Flow Metab. 26, 645-53 (2006).
[CrossRef]

J. Opt. Soc. Am. A (2)

Med. Biol. Engg. Comp. (1)

K. Forrester, C. Stewart, J. Tulip, C. Leonard, and R. Bray, "Comparison of laser speckle and laser Doppler perfusion imaging: Measurement in human skin and rabbit articular tissue," Med. Biol. Engg. Comp. 40, 687-697 (2002).
[CrossRef]

Med. Eng. Phys. (1)

B. Ruth, "Measuring the steady-state value and the dynamics of the skin blood flow using the non-contact laser speckle method." Med. Eng. Phys. 16, 105-11 (1994).
[CrossRef] [PubMed]

Microvasc. Res (1)

B. Choi, N. Kang, and J. Nelson, "Laser speckle imaging for monitoring blood flow dynamics in the in vivo rodent dorsal skin fold model," Microvasc. Res 68, 143-146 (2004).
[CrossRef] [PubMed]

Nature (London) (1)

M. Stern, "In vivo evaluation of microcirculation by coherent light scattering," Nature (London) 254, 56-8 (1975).
[CrossRef] [PubMed]

NeuroImage (1)

A. Dunn, A. Devor, A. Dale, and D. Boas, "Spatial extent of oxygen metabolism and hemodynamic changes during functional activation of the rat somatosensory cortex," NeuroImage 27, 279-290 (2005).
[CrossRef] [PubMed]

Neurosci. Lett (1)

A. Kharlamov, B. Brown, K. Easley, and S. Jones, "Heterogeneous response of cerebral blood flow to hypotension demonstrated by laser speckle imaging flowmetry in rats," Neurosci. Lett 368, 151-156 (2004).
[CrossRef] [PubMed]

Opt. Commun. (1)

A. Fercher and J. Briers, "Flow visualization by means of single-exposure speckle photography," Opt. Commun. 37, 326-330 (1981).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. Lett. (1)

P. Dixon and D. Durian, "Speckle Visibility Spectroscopy and Variable Granular Fluidization," Phys. Rev. Lett. 90, 184302 (2003).
[CrossRef] [PubMed]

Rev. Sci. Instrum. (1)

R. Bandyopadhyay, A. Gittings, S. Suh, P. Dixon, and D. Durian, "Speckle-visibility spectroscopy: A tool to study time-varying dynamics," Rev. Sci. Instrum. 76, 093110 (2005).
[CrossRef]

Science (1)

J. Anderson, D. Chiu, R. Jackman, O. Cherniavskaya, J. McDonald, H. Wu, S. Whitesides, and G. Whitesides, "Fabrication of Topologically Complex Three-Dimensional Microfluidic Systems in PDMS by Rapid Prototyping," Science 261, 895 (1993).

Stroke (1)

D. Atochin, J. Murciano, Y. Gursoy-Ozdemir, T. Krasik, F. Noda, C. Ayata, A. Dunn, M. Moskowitz, P. Huang, and V. Muzykantov, "Mouse Model of Microembolic Stroke and Reperfusion," Stroke 35, 2177-2182 (2004).
[CrossRef] [PubMed]

Other (1)

B. Berne and R. Pecora, Dynamic Light Scattering : with Applications to Chemistry, Biology and Physics (Dover Publications, 2000).

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Figures (8)

Fig. 1.
Fig. 1.

Multi-Exposure Speckle Imaging instrument (a) Schematic (b) Speckle Contrast image at 0.1 ms exposure duration (c) Speckle Contrast image at 5 ms exposure duration (d) Speckle Contrast image at 40 ms exposure duration (scale bar = 50 µm)

Fig. 2.
Fig. 2.

Cross-section of microfluidic flow phantom (not to scale) (a) Without Static scattering layer, (b) With Static scattering layer. Samples were imaged from the top

Fig. 3.
Fig. 3.

Multi-Exposure Speckle Contrast data fit to new speckle model. Speckle variance as a function of exposure duration for different speeds. Measurements were made on samples with no static scattering layer (Fig. 2(a))

Fig. 4.
Fig. 4.

Multi-Exposure Speckle Contrast data analyzed by spatial (ensemble) sampling (Solid lines) and temporal (time) sampling (dotted lines). Measurements were made at 2 mm/sec. The three curves for each analysis technique represent different amounts of static scattering. μ s values refer to the reduced scattering coefficient in the 200 µm static scattering layer. μ s =0 cm 1 : No static scattering layer (Fig. 2(a)), μ s =4 cm 1 0.9 mg/g of TiO2 in static scattering layer (Fig. 2(b)), μ s =8 cm 1 : 1.8 mg/g of TiO2 in static scattering layer (Fig. 2(b)). Speckle variance curves show that the nonergodic variance νne is absent in all three temporally sampled curves and in the completely dynamic spatially (ensemble) sampled curve. νne is significant in the cases with a static scattered layer, when the data is analyzed by spatial (ensemble) sampling.

Fig. 5.
Fig. 5.

Multi-Exposure Speckle Contrast data from two samples fit to the new speckle model. Speckle variance as a function of exposure duration for two different speeds and two levels of static scattering. Solid lines represent measurements made on sample without static scattering layer. Dotted lines represent measurements made on sample with static scattering layer. μ s values refer to the reduced scattering coefficient in the 200 µm static scattering layer. μ s =0 cm 1 : No static scattering layer (Fig. 2(a)), μ s =8 cm 1 : 1.8 mg/g of TiO2 in static scattering layer (Fig. 2(b)).

Fig. 6.
Fig. 6.

Percentage deviation in τc over changes in amount of static scattering for different speeds (estimated using Eq. 12). Data from all three static scattering cases μ s =0 cm 1 : No static scattering layer (Fig. 2(a)), μ s =4 cm 1 : 0.9 mg/g of TiO2 in static scattering layer (Fig. 2(b)), μ s =8 cm 1 : 1.8 mg/g of TiO2 in static scattering layer (Fig. 2(b)) was used in this analysis. τc estimates with the new speckle model have extremely low deviation

Fig. 7.
Fig. 7.

Performance of different models to relative flow. Baseline speed : 2 mm/sec. Plot of relative τc to relative speed. Plot should ideally be a straight line (dashed line). Multi-Exposure estimates extend linear range of relative τc estimates. Error bars indicate standard error in relative correlation time estimates. Measurements made using microfluidic phantom with no static scattering layer (Fig. 2(a)).

Fig. 8.
Fig. 8.

Quantifying the effect of static scattering on relative τc measurements. Plot of relative correlation time (Eq. 13) to relative speed. Baseline Speed - 2 mm/sec. The three curves on each graph represent different amounts of static scattering. Error bars on (b) indicate standard error in estimates of relative correlation times. μ s values refer to the reduced scattering coefficient in the 200 µm static scattering layer. μ s =0 cm 1 : No static scattering layer (Fig. 2(a)), μ s =4 cm 1 : 0.9 mg/g of TiO2 in static scattering layer (Fig. 2(b)), μ s =8 cm 1 : 1.8 mg/g of TiO2 in static scattering layer (Fig. 2(b)). New speckle model retains the linearity of relative τc estimates.

Equations (13)

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K = σ s I ,
g 2 ( τ ) = 1 + β g 1 ( τ ) 2 ,
K ( T , τ c ) = ( 1 e 2 x 2 x ) 1 2 ,
K ( T , τ c ) = ( β e 2 x 1 + 2 x 2 x 2 ) 1 2 .
E h ( t ) = E ( t ) + E s e i ω 0 t ,
g 2 h ( τ ) = 1 + β ( I f + I s ) 2 [ I f 2 g 1 ( τ ) 2 + 2 I f I s g 1 ( τ ) ] = 1 + A β g 1 ( τ ) 2 + B β g 1 ( τ ) ,
I 2 T 0 T 0 T I i ( t ) I i ( t ) dt dt T 2 i = I 2 0 T 0 T [ 1 + A β ( g 1 ( t t ) ) 2 + B β g 1 ( t t ) ] dt dt T 2 .
ν 2 ( T ) 0 T 0 T [ A β ( g 1 ( t t ) ) 2 + B β g 1 ( t t ) ] dt dt T 2 .
ν 2 ( T ) = A β 0 T 2 ( 1 t T ) [ g 1 ( t ) ] 2 dt T + B β 0 T 2 ( 1 t T ) [ g 1 ( t ) ] dt T .
K ( T , τ c ) = { β ρ 2 e 2 x 1 + 2 x 2 x 2 + 4 β ρ ( 1 ρ ) e x 1 + x x 2 } 1 2 ,
K ( T , τ c ) = { β ρ 2 e 2 x 1 + 2 x 2 x 2 + 4 β ρ ( 1 ρ ) e x 1 + x x 2 + ν ne + ν noise } 1 2 ,
% Deviation in τ c = Standard deviation in τ c τ c in the absence of static scatterers × 100
relative τ c = τ co τ c ,

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