Abstract

We introduce an integration of dynamic light scattering (DLS) and optical coherence tomography (OCT) for high-resolution 3D imaging of heterogeneous diffusion and flow. DLS analyzes fluctuations in light scattered by particles to measure diffusion or flow of the particles, and OCT uses coherence gating to collect light only scattered from a small volume for high-resolution structural imaging. Therefore, the integration of DLS and OCT enables high-resolution 3D imaging of diffusion and flow. We derived a theory under the assumption that static and moving particles are mixed within the OCT resolution volume and the moving particles can exhibit either diffusive or translational motion. Based on this theory, we developed a fitting algorithm to estimate dynamic parameters including the axial and transverse velocities and the diffusion coefficient. We validated DLS-OCT measurements of diffusion and flow through numerical simulations and phantom experiments. As an example application, we performed DLS-OCT imaging of the living animal brain, resulting in 3D maps of the absolute and axial velocities, the diffusion coefficient, and the coefficient of determination.

© 2012 OSA

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  20. U. Dirnagl, B. Kaplan, M. Jacewicz, and W. Pulsinelli, “Continuous measurement of cerebral cortical blood flow by laser-Doppler flowmetry in a rat stroke model,” J. Cereb. Blood Flow Metab. 9(5), 589–596 (1989).
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2011 (1)

2009 (1)

2006 (1)

2005 (1)

2003 (2)

1999 (1)

B. M. Ances, J. H. Greenberg, and J. A. Detre, “Laser Doppler imaging of activation-flow coupling in the rat somatosensory cortex,” Neuroimage 10(6), 716–723 (1999).
[Crossref] [PubMed]

1998 (1)

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” Siam J Optimiz 9(1), 112–147 (1998).
[Crossref]

1997 (3)

1996 (1)

P. Abry and F. Sellan, “The wavelet-based synthesis for fractional Brownian motion proposed by F. Sellan and Y. Meyer: remarks and fast implementation,” Appl. Comput. Harmon. Anal. 3(4), 377–383 (1996).
[Crossref]

1991 (2)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

D. J. Durian, D. A. Weitz, and D. J. Pine, “Multiple light-scattering probes of foam structure and dynamics,” Science 252(5006), 686–688 (1991).
[Crossref] [PubMed]

1990 (1)

J. G. H. Joosten, E. T. F. Geladé, and P. N. Pusey, “Dynamic light scattering by nonergodic media: Brownian particles trapped in polyacrylamide gels,” Phys. Rev. A 42(4), 2161–2175 (1990).
[Crossref] [PubMed]

1989 (2)

P. N. Pusey and W. Van Megen, “Dynamic light scattering by non-ergodic media,” Physica A 157(2), 705–741 (1989).
[Crossref]

U. Dirnagl, B. Kaplan, M. Jacewicz, and W. Pulsinelli, “Continuous measurement of cerebral cortical blood flow by laser-Doppler flowmetry in a rat stroke model,” J. Cereb. Blood Flow Metab. 9(5), 589–596 (1989).
[Crossref] [PubMed]

1986 (1)

1984 (1)

1975 (1)

M. D. Stern, “In vivo evaluation of microcirculation by coherent light scattering,” Nature 254(5495), 56–58 (1975).
[Crossref] [PubMed]

1971 (1)

R. V. Edwards, J. C. Angus, and M. J. French, “Spectral analysis of the signal from the laser Doppler flowmeter: time-independent systems,” J. Appl. Phys. 42(2), 837–850 (1971).
[Crossref]

1970 (1)

N. A. Clark, J. H. Lunacek, and G. B. Benedek, “A study of Brownian motion using light scattering,” Am. J. Phys. 38(5), 575–585 (1970).
[Crossref]

1954 (1)

L. Van Hove, “Correlations in space and time and Born approximation scattering in systems of interacting particles,” Phys. Rev. 95(1), 249–262 (1954).
[Crossref]

Abry, P.

P. Abry and F. Sellan, “The wavelet-based synthesis for fractional Brownian motion proposed by F. Sellan and Y. Meyer: remarks and fast implementation,” Appl. Comput. Harmon. Anal. 3(4), 377–383 (1996).
[Crossref]

Ances, B. M.

B. M. Ances, J. H. Greenberg, and J. A. Detre, “Laser Doppler imaging of activation-flow coupling in the rat somatosensory cortex,” Neuroimage 10(6), 716–723 (1999).
[Crossref] [PubMed]

Angus, J. C.

R. V. Edwards, J. C. Angus, and M. J. French, “Spectral analysis of the signal from the laser Doppler flowmeter: time-independent systems,” J. Appl. Phys. 42(2), 837–850 (1971).
[Crossref]

Ansari, R. R.

Barton, J. K.

Benedek, G. B.

N. A. Clark, J. H. Lunacek, and G. B. Benedek, “A study of Brownian motion using light scattering,” Am. J. Phys. 38(5), 575–585 (1970).
[Crossref]

Boas, D. A.

Boppart, S. A.

G. J. Tearney, M. E. Brezinski, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, and J. G. Fujimoto, “In vivo endoscopic optical biopsy with optical coherence tomography,” Science 276(5321), 2037–2039 (1997).
[Crossref] [PubMed]

Bouma, B. E.

J. F. de Boer, B. Cense, B. H. Park, M. C. Pierce, G. J. Tearney, and B. E. Bouma, “Improved signal-to-noise ratio in spectral-domain compared with time-domain optical coherence tomography,” Opt. Lett. 28(21), 2067–2069 (2003).
[Crossref] [PubMed]

G. J. Tearney, M. E. Brezinski, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, and J. G. Fujimoto, “In vivo endoscopic optical biopsy with optical coherence tomography,” Science 276(5321), 2037–2039 (1997).
[Crossref] [PubMed]

Bourquin, S.

Brezinski, M. E.

G. J. Tearney, M. E. Brezinski, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, and J. G. Fujimoto, “In vivo endoscopic optical biopsy with optical coherence tomography,” Science 276(5321), 2037–2039 (1997).
[Crossref] [PubMed]

Cense, B.

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Chen, Y.

Chowdhury, D. P.

Clark, N. A.

N. A. Clark, J. H. Lunacek, and G. B. Benedek, “A study of Brownian motion using light scattering,” Am. J. Phys. 38(5), 575–585 (1970).
[Crossref]

de Boer, J. F.

Detre, J. A.

B. M. Ances, J. H. Greenberg, and J. A. Detre, “Laser Doppler imaging of activation-flow coupling in the rat somatosensory cortex,” Neuroimage 10(6), 716–723 (1999).
[Crossref] [PubMed]

Dirnagl, U.

U. Dirnagl, B. Kaplan, M. Jacewicz, and W. Pulsinelli, “Continuous measurement of cerebral cortical blood flow by laser-Doppler flowmetry in a rat stroke model,” J. Cereb. Blood Flow Metab. 9(5), 589–596 (1989).
[Crossref] [PubMed]

Duker, J. S.

Durian, D. J.

D. J. Durian, D. A. Weitz, and D. J. Pine, “Multiple light-scattering probes of foam structure and dynamics,” Science 252(5006), 686–688 (1991).
[Crossref] [PubMed]

Edwards, R. V.

R. V. Edwards, J. C. Angus, and M. J. French, “Spectral analysis of the signal from the laser Doppler flowmeter: time-independent systems,” J. Appl. Phys. 42(2), 837–850 (1971).
[Crossref]

Fercher, A.

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

French, M. J.

R. V. Edwards, J. C. Angus, and M. J. French, “Spectral analysis of the signal from the laser Doppler flowmeter: time-independent systems,” J. Appl. Phys. 42(2), 837–850 (1971).
[Crossref]

Fujimoto, J.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Fujimoto, J. G.

Y. Chen, L. N. Vuong, J. Liu, J. Ho, V. J. Srinivasan, I. Gorczynska, A. J. Witkin, J. S. Duker, J. Schuman, and J. G. Fujimoto, “Three-dimensional ultrahigh resolution optical coherence tomography imaging of age-related macular degeneration,” Opt. Express 17(5), 4046–4060 (2009).
[Crossref] [PubMed]

G. J. Tearney, M. E. Brezinski, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, and J. G. Fujimoto, “In vivo endoscopic optical biopsy with optical coherence tomography,” Science 276(5321), 2037–2039 (1997).
[Crossref] [PubMed]

Geladé, E. T. F.

J. G. H. Joosten, E. T. F. Geladé, and P. N. Pusey, “Dynamic light scattering by nonergodic media: Brownian particles trapped in polyacrylamide gels,” Phys. Rev. A 42(4), 2161–2175 (1990).
[Crossref] [PubMed]

Gorczynska, I.

Greenberg, J. H.

B. M. Ances, J. H. Greenberg, and J. A. Detre, “Laser Doppler imaging of activation-flow coupling in the rat somatosensory cortex,” Neuroimage 10(6), 716–723 (1999).
[Crossref] [PubMed]

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Hee, M. R.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Hitzenberger, C.

Ho, J.

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Izatt, J. A.

Jacewicz, M.

U. Dirnagl, B. Kaplan, M. Jacewicz, and W. Pulsinelli, “Continuous measurement of cerebral cortical blood flow by laser-Doppler flowmetry in a rat stroke model,” J. Cereb. Blood Flow Metab. 9(5), 589–596 (1989).
[Crossref] [PubMed]

Joosten, J. G. H.

J. G. H. Joosten, E. T. F. Geladé, and P. N. Pusey, “Dynamic light scattering by nonergodic media: Brownian particles trapped in polyacrylamide gels,” Phys. Rev. A 42(4), 2161–2175 (1990).
[Crossref] [PubMed]

Kaplan, B.

U. Dirnagl, B. Kaplan, M. Jacewicz, and W. Pulsinelli, “Continuous measurement of cerebral cortical blood flow by laser-Doppler flowmetry in a rat stroke model,” J. Cereb. Blood Flow Metab. 9(5), 589–596 (1989).
[Crossref] [PubMed]

Karamata, B.

Kulkarni, M. D.

Lagarias, J. C.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” Siam J Optimiz 9(1), 112–147 (1998).
[Crossref]

Lambelet, P.

Lasser, T.

Laubscher, M.

Lee, J.

Leitgeb, R.

Lester, T. W.

Leung, A. B.

Leutenegger, M.

Lin, C. P.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Liu, J.

Lunacek, J. H.

N. A. Clark, J. H. Lunacek, and G. B. Benedek, “A study of Brownian motion using light scattering,” Am. J. Phys. 38(5), 575–585 (1970).
[Crossref]

Merklin, J. F.

Park, B. H.

Pierce, M. C.

Pine, D. J.

D. J. Durian, D. A. Weitz, and D. J. Pine, “Multiple light-scattering probes of foam structure and dynamics,” Science 252(5006), 686–688 (1991).
[Crossref] [PubMed]

Pitris, C.

G. J. Tearney, M. E. Brezinski, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, and J. G. Fujimoto, “In vivo endoscopic optical biopsy with optical coherence tomography,” Science 276(5321), 2037–2039 (1997).
[Crossref] [PubMed]

Puliafito, C. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Pulsinelli, W.

U. Dirnagl, B. Kaplan, M. Jacewicz, and W. Pulsinelli, “Continuous measurement of cerebral cortical blood flow by laser-Doppler flowmetry in a rat stroke model,” J. Cereb. Blood Flow Metab. 9(5), 589–596 (1989).
[Crossref] [PubMed]

Pusey, P. N.

J. G. H. Joosten, E. T. F. Geladé, and P. N. Pusey, “Dynamic light scattering by nonergodic media: Brownian particles trapped in polyacrylamide gels,” Phys. Rev. A 42(4), 2161–2175 (1990).
[Crossref] [PubMed]

P. N. Pusey and W. Van Megen, “Dynamic light scattering by non-ergodic media,” Physica A 157(2), 705–741 (1989).
[Crossref]

Radhakrishnan, H.

Reeds, J. A.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” Siam J Optimiz 9(1), 112–147 (1998).
[Crossref]

Schuman, J.

Schuman, J. S.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Sellan, F.

P. Abry and F. Sellan, “The wavelet-based synthesis for fractional Brownian motion proposed by F. Sellan and Y. Meyer: remarks and fast implementation,” Appl. Comput. Harmon. Anal. 3(4), 377–383 (1996).
[Crossref]

Sorensen, C. M.

Southern, J. F.

G. J. Tearney, M. E. Brezinski, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, and J. G. Fujimoto, “In vivo endoscopic optical biopsy with optical coherence tomography,” Science 276(5321), 2037–2039 (1997).
[Crossref] [PubMed]

Srinivasan, V.

Srinivasan, V. J.

Stern, M. D.

M. D. Stern, “In vivo evaluation of microcirculation by coherent light scattering,” Nature 254(5495), 56–58 (1975).
[Crossref] [PubMed]

Stinson, W. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Suh, K. I.

Swanson, E. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Taylor, T. W.

Tearney, G. J.

J. F. de Boer, B. Cense, B. H. Park, M. C. Pierce, G. J. Tearney, and B. E. Bouma, “Improved signal-to-noise ratio in spectral-domain compared with time-domain optical coherence tomography,” Opt. Lett. 28(21), 2067–2069 (2003).
[Crossref] [PubMed]

G. J. Tearney, M. E. Brezinski, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, and J. G. Fujimoto, “In vivo endoscopic optical biopsy with optical coherence tomography,” Science 276(5321), 2037–2039 (1997).
[Crossref] [PubMed]

Van Hove, L.

L. Van Hove, “Correlations in space and time and Born approximation scattering in systems of interacting particles,” Phys. Rev. 95(1), 249–262 (1954).
[Crossref]

Van Megen, W.

P. N. Pusey and W. Van Megen, “Dynamic light scattering by non-ergodic media,” Physica A 157(2), 705–741 (1989).
[Crossref]

Vuong, L. N.

Weitz, D. A.

D. J. Durian, D. A. Weitz, and D. J. Pine, “Multiple light-scattering probes of foam structure and dynamics,” Science 252(5006), 686–688 (1991).
[Crossref] [PubMed]

Welch, A. J.

Witkin, A. J.

Wright, M. H.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” Siam J Optimiz 9(1), 112–147 (1998).
[Crossref]

Wright, P. E.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” Siam J Optimiz 9(1), 112–147 (1998).
[Crossref]

Yazdanfar, S.

Yodh, A. G.

Am. J. Phys. (1)

N. A. Clark, J. H. Lunacek, and G. B. Benedek, “A study of Brownian motion using light scattering,” Am. J. Phys. 38(5), 575–585 (1970).
[Crossref]

Appl. Comput. Harmon. Anal. (1)

P. Abry and F. Sellan, “The wavelet-based synthesis for fractional Brownian motion proposed by F. Sellan and Y. Meyer: remarks and fast implementation,” Appl. Comput. Harmon. Anal. 3(4), 377–383 (1996).
[Crossref]

Appl. Opt. (3)

J. Appl. Phys. (1)

R. V. Edwards, J. C. Angus, and M. J. French, “Spectral analysis of the signal from the laser Doppler flowmeter: time-independent systems,” J. Appl. Phys. 42(2), 837–850 (1971).
[Crossref]

J. Cereb. Blood Flow Metab. (1)

U. Dirnagl, B. Kaplan, M. Jacewicz, and W. Pulsinelli, “Continuous measurement of cerebral cortical blood flow by laser-Doppler flowmetry in a rat stroke model,” J. Cereb. Blood Flow Metab. 9(5), 589–596 (1989).
[Crossref] [PubMed]

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

Nature (1)

M. D. Stern, “In vivo evaluation of microcirculation by coherent light scattering,” Nature 254(5495), 56–58 (1975).
[Crossref] [PubMed]

Neuroimage (1)

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Figures (9)

Fig. 1
Fig. 1

Conceptual illustration of the DLS-OCT theory. (A) Particles within the OCT resolution volume can be categorized into three groups: static, flowing or diffusing, and entering or exiting particles. For clarification, entering/exiting particles enter into or exit out of the voxel during a single measurement time step, resulting in stochastic fluctuations of the OCT signal. (B) The general behavior of the field autocorrelation function in the complex plane predicted by our model. MS and MF are approximately proportional to the fractions of static and flowing/diffusing particles, respectively, weighted by their scattering cross-sections.

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Fig. 2
Fig. 2

The performance of the simple fitting algorithm for an isotropic voxel (A) and an anisotropic voxel (B). ht = 4h was used for the anisotropic voxel. The performance was tested for a total of 25,200 combinations (36 number densities, 10 diffusion coefficients, 10 velocities and 7 flow angles). ME = 1 – MS – MF. Data are presented in mean ± SD. Each point represents the mean and error for the combination of other parameters (e.g., the point of MS = 0.5 in the MS plot shows the mean and error of the 700 results from the combination of 10 diffusion coefficients, 10 velocities and 7 flow angles where MS = 0.5).

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Fig. 3
Fig. 3

The algorithm of estimating the dynamic parameters from the field autocorrelation function.

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Fig. 4
Fig. 4

The performance of the advanced fitting algorithm for an isotropic voxel (A) and an anisotropic voxel (B). ht = 4h was used for the anisotropic voxel. The performance was tested for a total of 25,200 combinations as described in the caption of Fig. 2. Data are presented in mean ± SD. Each point represents the mean and error for the combination of other parameters as in Fig. 2, but the error was too small to be seen.

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Fig. 5
Fig. 5

Process of numerical simulation for validating the DLS-OCT theory. True values of the diffusion coefficient and velocity were determined by fitting the mean square displacement (MSD) of the numerical position data.

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Fig. 6
Fig. 6

Numerical validation of the velocity estimation. (A) Examples of the autocorrelation functions obtained from the numerical position data and their fitting results. D = 0 μm2/s. (B) Results of the estimation of the absolute velocity and the axial velocity. The estimation was tested for a total of 525 combinations (15 number densities, 5 velocities and 7 flow angles). Data are presented in mean ± SD in the top, where each point represents the mean and error across flow angles. Meanwhile, in the bottom, all estimations for 525 cases are presented and the data was fit to a line (red), resulting in 1.03 × (true) + 0.001 (r2 = 0.99) where r2 is the coefficient of determination. The autocorrelation function was averaged across 100 random initial distributions of particles.

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Fig. 7
Fig. 7

Numerical validation of the diffusion estimation. (A) Examples of decays of the autocorrelation functions obtained from the numerical position data and their fitting results. v = 0 mm/s and vz = 0 mm/s. (B) Results of the estimation of the diffusion coefficient. The estimation was tested for a total of 75 combinations (15 number densities and 5 diffusion coefficients). The estimation data was fit to a line (red), resulting in 1.04 × (true) + 0.30 (r2 = 0.89). The autocorrelation function was averaged across 1,000 random initial distributions of particles. (C) Decays of the autocorrelation functions between flowing and diffusing particles. The solid line shows decay of the MF-term of the autocorrelation function of diffusion particles (NS/N = 0.1, NF/N = 0.8, D = 10 μm2/s, v = 0, and vz = 0), while the dotted line shows that of flowing particles (NS/N = 0.1, NF/N = 0.8, D = 0, v = 5 mm/s, and vz = 0).

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Fig. 8
Fig. 8

Experimental validation of DLS-OCT measurements of the flow velocity (A) and diffusion (B). Various combinations of the lateral scanning speeds and the axial piezo speeds implemented several equivalent flow angles, θ = arctan(vz/vt). The gray line in (B) shows the Einstein-Stokes equation. Data are presented in mean ± SD. The horizontal error bar in (A) resulted from the variation in the piezoelectric actuation.

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Fig. 9
Fig. 9

DLS-OCT imaging of the living rodent cortex. (A) The first image presents the maximum projection (MP) of the 3D map of the absolute velocity along the depth (i.e., en face), and the second image presents the en face signed maximum projection (SMP) of the 3D map of the axial velocity. The depth of focus was ~300 μm. The images with the green boundary show the MP of absolute velocity and the SMP of the axial velocity along the transverse direction (i.e., cross-sectional) over the volume indicated as the green box in the en face images. The SMPs of the axial velocity are presented over the range from −5/√2 to 5/√2 mm/s, where a negative velocity (blue color) means that blood flows toward the surface of the cortex. (B) The first image presents the en face MP of the 3D map of the diffusion coefficient. In the second image, the diffusion image (yellow) is overlaid with the absolute velocity image (red). The 3X magnified images of the cyan boxes are presented to clearly show the characteristic dynamics of the vessel boundaries. This merged image is presented with smaller ranges of the velocity and diffusion coefficient for higher contrast. (C) The single planes of the merged map and the coefficient of determination map at the depth of 120 μm are presented. (D) Examples of the autocorrelation function are presented for three voxels (cyan crosses) that are located in the plane indicated as the cyan box in (C). The middle row shows the autocorrelation function data (black dots) and their fits (red lines) in the complex plane, where the estimated MS and MS + MF are presented as the blue and green circles, respectively. The bottom row shows decay of the MF-terms. The coefficients of determination of these three voxels are R2 = 0.999, 0.988, and 0.647. All scale bars, 100 μm.

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Tables (2)

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Table 1 Physical quantities used in this study.

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Table 2 Parameters used in the numerical simulation.

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Equations (18)

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R 1 (t)= R 01 e − 1 2 (Δq) 2 z 1 2 (t) e iq z 1 (t)
R 1 (t)= R 01 e −2 h t 2 [ x 1 2 (t)+ y 1 2 (t)]−2 h 2 z 1 2 (t) e iq z 1 (t)  .
P 1 ( r f , t f | r i , t i )= 1 2 πD( t f − t i ) e − | r f − r i −v( t f − t i ) | 2 4D( t f − t i )  .
g(τ)=E[ 〈 R 1 * (t) R 1 (t+τ) 〉 t 〈 R 1 * (t) R 1 (t) 〉 t ] = 1 1+4 h t 2 Dτ 1 1+4 h 2 Dτ e − h t 2 ( v t τ) 2 1+4 h t 2 Dτ − h 2 ( v z τ) 2 1+4 h 2 Dτ e − q 2 Dτ 1+4 h 2 Dτ e iq v z τ 1+4 h 2 Dτ
g(τ)= e − h t 2 v t 2 τ 2 − h 2 v z 2 τ 2 e − q 2 Dτ e iq v z τ  .
R(t)= ∑ j=1 N R j (t) = ∑ j=1 N R 0j e −2 h t 2 [ x j 2 (t)+ y j 2 (t)]−2 h 2 z j 2 (t) e iq z j (t)
g(τ)= e − h t 2 v t 2 τ 2 − h 2 v z 2 τ 2 e − q 2 Dτ e iq v z τ
R(t)= ∑ j=1 N S R S0j e −2 h 2 r Sj 2 e iq z Sj + ∑ j=1 N F R F0j e −2 h 2 r Fj 2 (t) e iq z Fj (t) + ∑ j=1 N E R E0j e −2 h 2 r Ej 2 (t) e iq z Ej (t)
R(t)= ∑ j=1 N S e −2 h 2 r Sj 2 e iq z Sj + ∑ j=1 N F e −2 h 2 r Fj 2 (t) e iq z Fj (t) + ∑ j=1 N E e −2 h 2 r Ej 2 (t) e iq z Ej (t) ≡ R S + R F (t)+ R E w(t)
〈 R * (t)R(t+τ) 〉 =〈 [ R S * + R F * (t)+ R E * w * (t)] [ R S + R F (t+τ)+ R E w(t+τ)] 〉 = | R S | 2 +〈 R F * (t) R F (t+τ) 〉+ | R E | 2 〈 w * (t)w(t+τ) 〉             (self coupled terms)     + R S * 〈 R F (t+τ) 〉+ R S 〈 R F * (t) 〉                                                    (S−F coupled terms)     + R S * R E 〈 w(t+τ) 〉+ R S R E * 〈 w * (t) 〉                                          (S−E coupled terms)     + R E 〈 R F * (t)w(t+τ) 〉+ R E * 〈 R F (t+τ) w * (t) 〉                   (F−E coupled terms)
E[ | R S | 2 ]= N S π π 8 h 3 E[〈 R F * (t) R F (t+τ) 〉]= N F π π 8 h 3 e − h 2 v 2 τ 2 e − q 2 Dτ e iq v z τ E[ | R E | 2 〈 w * (t)w(t+τ) 〉]= N E δ(τ) E[ R S * 〈 R F (t+τ) 〉]= N S π π 2 2 h 3 e − 3 8 ( q h ) 2 { ∑ k=1 N F 1 T ∫ τ τ+T dt' ∏ j=x,y,z e − [qDt'− i 2 ( r jFk0 + v j t')] 2 8 h 2 D 2 t ' 2 +Dt' 1+8 h 2 Dt' } → ( q h ) 2 >>1 0 E[ R S 〈 R F * (t) 〉] → ( q h ) 2 >>1 0 E[ R S * R E 〈 w(t+τ) 〉]=E[ R S R E * 〈 w * (t) 〉]=0 E[ R E 〈 R F * (t)w(t+τ) 〉]= N S N E π 3 8 h 6 e − 3 4 ( q h ) 2 → ( q h ) 2 >>1 0 E[ R E * 〈 R F (t+τ) w * (t) 〉]= → ( q h ) 2 >>1 0
P( r Fk ,t+τ| r Sm ,t)= P[( r Fk ,t+τ)∩( r Sm ,t)] P( r Sm ,t) =P( r Fk ,t+τ) .
g(τ)= M S + M F   e − h t 2 v t 2 τ 2 − h 2 v z 2 τ 2 e − q 2 Dτ e iq v z τ +(1− M S − M F )δ(τ) .
g(r,τ)= 〈 R * (r,t) R(r,t+τ) 〉 t 〈 R * (r,t) R(r,t) 〉 t = M S (r)+ M F (r)  e − h t 2 v t 2 (r)  τ 2 − h 2 v z 2 (r)  τ 2 e − q 2 D(r) τ e iq v z (r) τ   +[1− M S (r)− M F (r)]δ(τ) .
R 2 =1− 〈 | g(τ)− M S − M F   e − h t 2 v t 2 τ 2 − h 2 v z 2 τ 2 e − q 2 Dτ e iq v z τ −(1− M S − M F )δ(τ) | 2 〉 〈 | g(τ)−〈 g(τ) 〉 | 2 〉  .
v z = 1 q 〈 arg[g(τ)− M S ] τ | g(τ)− M S | 〉 τ>0 〈 τ 2 | g(τ)− M S | 〉 τ>0 g 1 (τ)= 1 M F  | g(τ)− M S |  e h 2 v z 2 τ 2 [ 〈 τ 4 g 1 (τ) 〉 τ>0 〈 τ 3 g 1 (τ) 〉 τ>0 〈 τ 3 g 1 (τ) 〉 τ>0 〈 τ 2 g 1 (τ) 〉 τ>0 ][ − h t 2 v t 2 − q 2 D ]=[ 〈 τ 2 g 1 (τ)ln g 1 (τ) 〉 τ>0 〈 τ  g 1 (τ)ln g 1 (τ) 〉 τ>0 ] .
[ τ 1 2 τ 1 1 τ 2 2 τ 2 1 τ 3 2 τ 3 1 ][ c 1 c 2 c 3 ]=[ | g( τ 1 ) | | g( τ 2 ) | | g( τ 3 ) | ] M E =1− c 3
[ 〈 a k 2 〉− a ¯ 2 〈 ( a k − a ¯ ) b k 〉 〈 ( b k − b ¯ ) a k 〉 〈 b k 2 〉− b ¯ 2 ][ A B ]=[ 〈 a k 3 〉− a ¯ 〈 a k 2 〉+〈 ( a k − a ¯ ) b k 2 〉 〈 b k 3 〉− b ¯ 〈 b k 2 〉+〈 ( b k − b ¯ ) a k 2 〉 ] M S =A

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