Abstract

Optical coherence Doppler tomography (ODT) is a promising neurotechnique that permits 3D imaging of the cerebral blood flow (CBF) network; however, quantitative CBF velocity (CBFv) imaging remains challenging. Here we present a simple phase summation method to enhance slow capillary flow detection sensitivity without sacrificing dynamic range for fast flow and vessel tracking to improve angle correction for absolute CBFv quantification. Flow phantom validation indicated that the CBFv quantification accuracy increased from 15% to 91% and the coefficient of variation (CV) decreased 9.3-fold; in vivo mouse brain validation showed that CV decreased 4.4-/10.8- fold for venular/arteriolar flows. ODT was able to identify cocaine-elicited microischemia and quantify CBFv disruption in branch vessels and capillaries that otherwise would have not been possible.

© 2014 Optical Society of America

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2014 (1)

2013 (4)

W. Trasischker, R. M. Werkmeister, S. Zotter, B. Baumann, T. Torzicky, M. Pircher, and C. K. Hitzenberger, “In vitro and in vivo three-dimensional velocity vector measurement by three-beam spectral-domain Doppler optical coherence tomography,” J. Biomed. Opt.18(11), 116010 (2013).
[CrossRef] [PubMed]

A. Bouwens, D. Szlag, M. Szkulmowski, T. Bolmont, M. Wojtkowski, and T. Lasser, “Quantitative lateral and axial flow imaging with optical coherence microscopy and tomography,” Opt. Express21(15), 17711–17729 (2013).
[CrossRef] [PubMed]

J. Lee, W. Wu, F. Lesage, and D. A. Boas, “Multiple-capillary measurement of RBC speed, flux, and density with optical coherence tomography,” J. Cereb. Blood Flow Metab.33(11), 1707–1710 (2013).

H. Radhakrishnan and V. J. Srinivasan, “Compartment-resolved imaging of cortical functional hyperemia with OCT angiography,” Biomed. Opt. Express4(8), 1255–1268 (2013).
[CrossRef] [PubMed]

2012 (5)

H. Ren, C. Du, and Y. Pan, “Cerebral blood flow imaged with ultrahigh-resolution optical coherence angiography and Doppler tomography,” Opt. Lett.37(8), 1388–1390 (2012).
[CrossRef] [PubMed]

A. Devor, S. Sakadzic, V. J. Srinivasan, M. A. Yaseen, K. Nizar, P. A. Saisan, P. Tian, A. M. Dale, S. A. Vinogradov, M. A. Franceschini, and D. A. Boas, “Frontiers in optical imaging of cerebral blood flow and metabolism,” J. Cereb. Blood Flow Metab.32(7), 1259–1276 (2012).

A. Y. Shih, J. D. Driscoll, P. J. Drew, N. Nishimura, C. B. Schaffer, and D. Kleinfeld, “Two-photon microscopy as a tool to study blood flow and neurovascular coupling in the rodent brain,” J. Cereb. Blood Flow Metab.32(7), 1277–1309 (2012).

H.-G. Ren, C.-W. Du, Z.-J. Yuan, K. Park, N. D. Volkow, and Y.-T. Pan, “Cocaine-induced cortical microischemia in the rodent brain: clinical implications,” Mol. Psychiatry17(10), 1017–1025 (2012).
[CrossRef] [PubMed]

T. N. Kim, P. W. Goodwill, Y. Chen, S. M. Conolly, C. B. Schaffer, D. Liepmann, and R. A. Wang, “Line-Scanning Particle Image Velocimetry: An Optical Approach for Quantifying a Wide Range of Blood Flow Speeds in Live Animals,” PLoS ONE7(6), e38590 (2012).
[CrossRef] [PubMed]

2011 (3)

2010 (3)

2009 (2)

2008 (1)

2007 (2)

R. K. Wang, S. L. Jacques, Z. Ma, S. Hurst, S. R. Hanson, and A. Gruber, “Three dimensional optical angiography,” Opt. Express15(7), 4083–4097 (2007).
[CrossRef] [PubMed]

Y. T. Pan, Z. L. Wu, Z. J. Yuan, Z. G. Wang, and C. W. Du, “Subcellular imaging of epithelium with time-lapse optical coherence tomography,” J. Biomed. Opt.12(5), 050504 (2007).
[CrossRef] [PubMed]

2006 (1)

C. B. Schaffer, B. Friedman, N. Nishimura, L. F. Schroeder, P. S. Tsai, F. F. Ebner, P. D. Lyden, and D. Kleinfeld, “Two-Photon Imaging of Cortical Surface Microvessels Reveals a Robust Redistribution in Blood Flow after Vascular Occlusion,” PLoS Biol.4(2), e22 (2006).
[CrossRef] [PubMed]

2005 (1)

2004 (4)

W. Drexler, “Ultrahigh-resolution optical coherence tomography,” J. Biomed. Opt.9(1), 47–74 (2004).
[CrossRef] [PubMed]

R. A. Leitgeb, L. Schmetterer, C. K. Hitzenberger, A. F. Fercher, F. Berisha, M. Wojtkowski, and T. Bajraszewski, “Real-time measurement of in vitro flow by Fourier-domain color Doppler optical coherence tomography,” Opt. Lett.29(2), 171–173 (2004).
[CrossRef] [PubMed]

C. Iadecola, “Neurovascular regulation in the normal brain and in Alzheimer’s disease,” Nat. Rev. Neurosci.5(5), 347–360 (2004).
[CrossRef] [PubMed]

R. G. Shulman, D. L. Rothman, K. L. Behar, and F. Hyder, “Energetic basis of brain activity: implications for neuroimaging,” Trends Neurosci.27(8), 489–495 (2004).
[CrossRef] [PubMed]

2003 (3)

2002 (2)

H. Ren, K. M. Brecke, Z. Ding, Y. Zhao, J. S. Nelson, and Z. Chen, “Imaging and quantifying transverse flow velocity with the Doppler bandwidth in a phase-resolved functional optical coherence tomography,” Opt. Lett.27(6), 409–411 (2002).
[CrossRef] [PubMed]

V. X. D. Yang, M. L. Gordon, A. Mok, Y. Zhao, Z. Chen, R. S. C. Cobbold, B. C. Wilson, and I. Alex Vitkin, “Improved phase-resolved optical Doppler tomography using the Kasai velocity estimator and histogram segmentation,” Opt. Commun.208(4–6), 209–214 (2002).
[CrossRef]

2000 (1)

1999 (1)

1994 (1)

A. Villringer, A. Them, U. Lindauer, K. Einhäupl, and U. Dirnagl, “Capillary perfusion of the rat brain cortex. An in vivo confocal microscopy study,” Circ. Res.75(1), 55–62 (1994).
[CrossRef] [PubMed]

1990 (1)

S. Ogawa, T. M. Lee, A. R. Kay, and D. W. Tank, “Brain magnetic resonance imaging with contrast dependent on blood oxygenation,” Proc. Natl. Acad. Sci. U.S.A.87(24), 9868–9872 (1990).
[CrossRef] [PubMed]

1982 (1)

Alex Vitkin, I.

V. X. D. Yang, M. L. Gordon, A. Mok, Y. Zhao, Z. Chen, R. S. C. Cobbold, B. C. Wilson, and I. Alex Vitkin, “Improved phase-resolved optical Doppler tomography using the Kasai velocity estimator and histogram segmentation,” Opt. Commun.208(4–6), 209–214 (2002).
[CrossRef]

Atochin, D. N.

V. J. Srinivasan, D. N. Atochin, H. Radhakrishnan, J. Y. Jiang, S. Ruvinskaya, W. Wu, S. Barry, A. E. Cable, C. Ayata, P. L. Huang, and D. A. Boas, “Optical coherence tomography for the quantitative study of cerebrovascular physiology,” J. Cereb. Blood Flow Metab.31(6), 1339–1345 (2011).

Ayata, C.

V. J. Srinivasan, D. N. Atochin, H. Radhakrishnan, J. Y. Jiang, S. Ruvinskaya, W. Wu, S. Barry, A. E. Cable, C. Ayata, P. L. Huang, and D. A. Boas, “Optical coherence tomography for the quantitative study of cerebrovascular physiology,” J. Cereb. Blood Flow Metab.31(6), 1339–1345 (2011).

Bajraszewski, T.

Barry, S.

V. J. Srinivasan, D. N. Atochin, H. Radhakrishnan, J. Y. Jiang, S. Ruvinskaya, W. Wu, S. Barry, A. E. Cable, C. Ayata, P. L. Huang, and D. A. Boas, “Optical coherence tomography for the quantitative study of cerebrovascular physiology,” J. Cereb. Blood Flow Metab.31(6), 1339–1345 (2011).

Baumann, B.

W. Trasischker, R. M. Werkmeister, S. Zotter, B. Baumann, T. Torzicky, M. Pircher, and C. K. Hitzenberger, “In vitro and in vivo three-dimensional velocity vector measurement by three-beam spectral-domain Doppler optical coherence tomography,” J. Biomed. Opt.18(11), 116010 (2013).
[CrossRef] [PubMed]

Behar, K. L.

R. G. Shulman, D. L. Rothman, K. L. Behar, and F. Hyder, “Energetic basis of brain activity: implications for neuroimaging,” Trends Neurosci.27(8), 489–495 (2004).
[CrossRef] [PubMed]

Berisha, F.

Boas, D. A.

J. Lee, W. Wu, F. Lesage, and D. A. Boas, “Multiple-capillary measurement of RBC speed, flux, and density with optical coherence tomography,” J. Cereb. Blood Flow Metab.33(11), 1707–1710 (2013).

A. Devor, S. Sakadzic, V. J. Srinivasan, M. A. Yaseen, K. Nizar, P. A. Saisan, P. Tian, A. M. Dale, S. A. Vinogradov, M. A. Franceschini, and D. A. Boas, “Frontiers in optical imaging of cerebral blood flow and metabolism,” J. Cereb. Blood Flow Metab.32(7), 1259–1276 (2012).

V. J. Srinivasan, D. N. Atochin, H. Radhakrishnan, J. Y. Jiang, S. Ruvinskaya, W. Wu, S. Barry, A. E. Cable, C. Ayata, P. L. Huang, and D. A. Boas, “Optical coherence tomography for the quantitative study of cerebrovascular physiology,” J. Cereb. Blood Flow Metab.31(6), 1339–1345 (2011).

V. J. Srinivasan, S. Sakadzić, I. Gorczynska, S. Ruvinskaya, W. Wu, J. G. Fujimoto, and D. A. Boas, “Quantitative cerebral blood flow with Optical Coherence Tomography,” Opt. Express18(3), 2477–2494 (2010).
[CrossRef] [PubMed]

Bolmont, T.

Boppart, S. A.

Bouma, B.

Bouwens, A.

Brecke, K. M.

Cable, A.

Cable, A. E.

V. J. Srinivasan, D. N. Atochin, H. Radhakrishnan, J. Y. Jiang, S. Ruvinskaya, W. Wu, S. Barry, A. E. Cable, C. Ayata, P. L. Huang, and D. A. Boas, “Optical coherence tomography for the quantitative study of cerebrovascular physiology,” J. Cereb. Blood Flow Metab.31(6), 1339–1345 (2011).

Chen, M.

Chen, Y.

T. N. Kim, P. W. Goodwill, Y. Chen, S. M. Conolly, C. B. Schaffer, D. Liepmann, and R. A. Wang, “Line-Scanning Particle Image Velocimetry: An Optical Approach for Quantifying a Wide Range of Blood Flow Speeds in Live Animals,” PLoS ONE7(6), e38590 (2012).
[CrossRef] [PubMed]

Chen, Z.

Cobbold, R. S. C.

V. X. D. Yang, M. L. Gordon, A. Mok, Y. Zhao, Z. Chen, R. S. C. Cobbold, B. C. Wilson, and I. Alex Vitkin, “Improved phase-resolved optical Doppler tomography using the Kasai velocity estimator and histogram segmentation,” Opt. Commun.208(4–6), 209–214 (2002).
[CrossRef]

Conolly, S. M.

T. N. Kim, P. W. Goodwill, Y. Chen, S. M. Conolly, C. B. Schaffer, D. Liepmann, and R. A. Wang, “Line-Scanning Particle Image Velocimetry: An Optical Approach for Quantifying a Wide Range of Blood Flow Speeds in Live Animals,” PLoS ONE7(6), e38590 (2012).
[CrossRef] [PubMed]

Cuevas, M.

E. Koch, J. Walther, and M. Cuevas, “Limits of Fourier domain Doppler-OCT at high velocities,” Sens. Actuators A Phys.156(1), 8–13 (2009).
[CrossRef]

Dale, A. M.

A. Devor, S. Sakadzic, V. J. Srinivasan, M. A. Yaseen, K. Nizar, P. A. Saisan, P. Tian, A. M. Dale, S. A. Vinogradov, M. A. Franceschini, and D. A. Boas, “Frontiers in optical imaging of cerebral blood flow and metabolism,” J. Cereb. Blood Flow Metab.32(7), 1259–1276 (2012).

de Boer, J.

de Boer, J. F.

Devor, A.

A. Devor, S. Sakadzic, V. J. Srinivasan, M. A. Yaseen, K. Nizar, P. A. Saisan, P. Tian, A. M. Dale, S. A. Vinogradov, M. A. Franceschini, and D. A. Boas, “Frontiers in optical imaging of cerebral blood flow and metabolism,” J. Cereb. Blood Flow Metab.32(7), 1259–1276 (2012).

Dhalla, A.-H.

Ding, Z.

Dirnagl, U.

A. Villringer, A. Them, U. Lindauer, K. Einhäupl, and U. Dirnagl, “Capillary perfusion of the rat brain cortex. An in vivo confocal microscopy study,” Circ. Res.75(1), 55–62 (1994).
[CrossRef] [PubMed]

Drew, P. J.

A. Y. Shih, J. D. Driscoll, P. J. Drew, N. Nishimura, C. B. Schaffer, and D. Kleinfeld, “Two-photon microscopy as a tool to study blood flow and neurovascular coupling in the rodent brain,” J. Cereb. Blood Flow Metab.32(7), 1277–1309 (2012).

Drexler, W.

Driscoll, J. D.

A. Y. Shih, J. D. Driscoll, P. J. Drew, N. Nishimura, C. B. Schaffer, and D. Kleinfeld, “Two-photon microscopy as a tool to study blood flow and neurovascular coupling in the rodent brain,” J. Cereb. Blood Flow Metab.32(7), 1277–1309 (2012).

Du, C.

Du, C. W.

Z. Yuan, Z. C. Luo, H. G. Ren, C. W. Du, and Y. Pan, “A digital frequency ramping method for enhancing Doppler flow imaging in Fourier-domain optical coherence tomography,” Opt. Express17(5), 3951–3963 (2009).
[CrossRef] [PubMed]

Y. T. Pan, Z. L. Wu, Z. J. Yuan, Z. G. Wang, and C. W. Du, “Subcellular imaging of epithelium with time-lapse optical coherence tomography,” J. Biomed. Opt.12(5), 050504 (2007).
[CrossRef] [PubMed]

Du, C.-W.

H.-G. Ren, C.-W. Du, Z.-J. Yuan, K. Park, N. D. Volkow, and Y.-T. Pan, “Cocaine-induced cortical microischemia in the rodent brain: clinical implications,” Mol. Psychiatry17(10), 1017–1025 (2012).
[CrossRef] [PubMed]

Ebner, F. F.

C. B. Schaffer, B. Friedman, N. Nishimura, L. F. Schroeder, P. S. Tsai, F. F. Ebner, P. D. Lyden, and D. Kleinfeld, “Two-Photon Imaging of Cortical Surface Microvessels Reveals a Robust Redistribution in Blood Flow after Vascular Occlusion,” PLoS Biol.4(2), e22 (2006).
[CrossRef] [PubMed]

Einhäupl, K.

A. Villringer, A. Them, U. Lindauer, K. Einhäupl, and U. Dirnagl, “Capillary perfusion of the rat brain cortex. An in vivo confocal microscopy study,” Circ. Res.75(1), 55–62 (1994).
[CrossRef] [PubMed]

Fercher, A. F.

Franceschini, M. A.

A. Devor, S. Sakadzic, V. J. Srinivasan, M. A. Yaseen, K. Nizar, P. A. Saisan, P. Tian, A. M. Dale, S. A. Vinogradov, M. A. Franceschini, and D. A. Boas, “Frontiers in optical imaging of cerebral blood flow and metabolism,” J. Cereb. Blood Flow Metab.32(7), 1259–1276 (2012).

Friedman, B.

C. B. Schaffer, B. Friedman, N. Nishimura, L. F. Schroeder, P. S. Tsai, F. F. Ebner, P. D. Lyden, and D. Kleinfeld, “Two-Photon Imaging of Cortical Surface Microvessels Reveals a Robust Redistribution in Blood Flow after Vascular Occlusion,” PLoS Biol.4(2), e22 (2006).
[CrossRef] [PubMed]

Fujimoto, J. G.

Goodwill, P. W.

T. N. Kim, P. W. Goodwill, Y. Chen, S. M. Conolly, C. B. Schaffer, D. Liepmann, and R. A. Wang, “Line-Scanning Particle Image Velocimetry: An Optical Approach for Quantifying a Wide Range of Blood Flow Speeds in Live Animals,” PLoS ONE7(6), e38590 (2012).
[CrossRef] [PubMed]

Gorczynska, I.

Gordon, M.

Gordon, M. L.

V. X. D. Yang, M. L. Gordon, A. Mok, Y. Zhao, Z. Chen, R. S. C. Cobbold, B. C. Wilson, and I. Alex Vitkin, “Improved phase-resolved optical Doppler tomography using the Kasai velocity estimator and histogram segmentation,” Opt. Commun.208(4–6), 209–214 (2002).
[CrossRef]

Gruber, A.

Gu, S.

Hanson, S. R.

Hasselbalch, S. G.

O. B. Paulson, S. G. Hasselbalch, E. Rostrup, G. M. Knudsen, and D. Pelligrino, “Cerebral blood flow response to functional activation,” J. Cereb. Blood Flow Metab.30(1), 2–14 (2010).

He, Y.

Hendargo, H. C.

Hitzenberger, C. K.

W. Trasischker, R. M. Werkmeister, S. Zotter, B. Baumann, T. Torzicky, M. Pircher, and C. K. Hitzenberger, “In vitro and in vivo three-dimensional velocity vector measurement by three-beam spectral-domain Doppler optical coherence tomography,” J. Biomed. Opt.18(11), 116010 (2013).
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R. A. Leitgeb, L. Schmetterer, C. K. Hitzenberger, A. F. Fercher, F. Berisha, M. Wojtkowski, and T. Bajraszewski, “Real-time measurement of in vitro flow by Fourier-domain color Doppler optical coherence tomography,” Opt. Lett.29(2), 171–173 (2004).
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Huang, P. L.

V. J. Srinivasan, D. N. Atochin, H. Radhakrishnan, J. Y. Jiang, S. Ruvinskaya, W. Wu, S. Barry, A. E. Cable, C. Ayata, P. L. Huang, and D. A. Boas, “Optical coherence tomography for the quantitative study of cerebrovascular physiology,” J. Cereb. Blood Flow Metab.31(6), 1339–1345 (2011).

Hurst, S.

Hyder, F.

R. G. Shulman, D. L. Rothman, K. L. Behar, and F. Hyder, “Energetic basis of brain activity: implications for neuroimaging,” Trends Neurosci.27(8), 489–495 (2004).
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C. Iadecola, “Neurovascular regulation in the normal brain and in Alzheimer’s disease,” Nat. Rev. Neurosci.5(5), 347–360 (2004).
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Itoh, K.

Izatt, J. A.

Jacques, S. L.

Jarvi, M.

Jenkins, M. W.

Jiang, J.

Jiang, J. Y.

V. J. Srinivasan, D. N. Atochin, H. Radhakrishnan, J. Y. Jiang, S. Ruvinskaya, W. Wu, S. Barry, A. E. Cable, C. Ayata, P. L. Huang, and D. A. Boas, “Optical coherence tomography for the quantitative study of cerebrovascular physiology,” J. Cereb. Blood Flow Metab.31(6), 1339–1345 (2011).

Kärtner, F. X.

Kay, A. R.

S. Ogawa, T. M. Lee, A. R. Kay, and D. W. Tank, “Brain magnetic resonance imaging with contrast dependent on blood oxygenation,” Proc. Natl. Acad. Sci. U.S.A.87(24), 9868–9872 (1990).
[CrossRef] [PubMed]

Khurana, M.

Kim, T. N.

T. N. Kim, P. W. Goodwill, Y. Chen, S. M. Conolly, C. B. Schaffer, D. Liepmann, and R. A. Wang, “Line-Scanning Particle Image Velocimetry: An Optical Approach for Quantifying a Wide Range of Blood Flow Speeds in Live Animals,” PLoS ONE7(6), e38590 (2012).
[CrossRef] [PubMed]

Kleinfeld, D.

A. Y. Shih, J. D. Driscoll, P. J. Drew, N. Nishimura, C. B. Schaffer, and D. Kleinfeld, “Two-photon microscopy as a tool to study blood flow and neurovascular coupling in the rodent brain,” J. Cereb. Blood Flow Metab.32(7), 1277–1309 (2012).

C. B. Schaffer, B. Friedman, N. Nishimura, L. F. Schroeder, P. S. Tsai, F. F. Ebner, P. D. Lyden, and D. Kleinfeld, “Two-Photon Imaging of Cortical Surface Microvessels Reveals a Robust Redistribution in Blood Flow after Vascular Occlusion,” PLoS Biol.4(2), e22 (2006).
[CrossRef] [PubMed]

Knudsen, G. M.

O. B. Paulson, S. G. Hasselbalch, E. Rostrup, G. M. Knudsen, and D. Pelligrino, “Cerebral blood flow response to functional activation,” J. Cereb. Blood Flow Metab.30(1), 2–14 (2010).

Koch, E.

E. Koch, J. Walther, and M. Cuevas, “Limits of Fourier domain Doppler-OCT at high velocities,” Sens. Actuators A Phys.156(1), 8–13 (2009).
[CrossRef]

Lasser, T.

Lee, J.

J. Lee, W. Wu, F. Lesage, and D. A. Boas, “Multiple-capillary measurement of RBC speed, flux, and density with optical coherence tomography,” J. Cereb. Blood Flow Metab.33(11), 1707–1710 (2013).

Lee, K.

Lee, T. M.

S. Ogawa, T. M. Lee, A. R. Kay, and D. W. Tank, “Brain magnetic resonance imaging with contrast dependent on blood oxygenation,” Proc. Natl. Acad. Sci. U.S.A.87(24), 9868–9872 (1990).
[CrossRef] [PubMed]

Leitgeb, R. A.

Lesage, F.

J. Lee, W. Wu, F. Lesage, and D. A. Boas, “Multiple-capillary measurement of RBC speed, flux, and density with optical coherence tomography,” J. Cereb. Blood Flow Metab.33(11), 1707–1710 (2013).

Leung, M. K.

Leung, M. K. K.

Li, X. D.

Liepmann, D.

T. N. Kim, P. W. Goodwill, Y. Chen, S. M. Conolly, C. B. Schaffer, D. Liepmann, and R. A. Wang, “Line-Scanning Particle Image Velocimetry: An Optical Approach for Quantifying a Wide Range of Blood Flow Speeds in Live Animals,” PLoS ONE7(6), e38590 (2012).
[CrossRef] [PubMed]

Lindauer, U.

A. Villringer, A. Them, U. Lindauer, K. Einhäupl, and U. Dirnagl, “Capillary perfusion of the rat brain cortex. An in vivo confocal microscopy study,” Circ. Res.75(1), 55–62 (1994).
[CrossRef] [PubMed]

Lo, S.

Luo, Z. C.

Lyden, P. D.

C. B. Schaffer, B. Friedman, N. Nishimura, L. F. Schroeder, P. S. Tsai, F. F. Ebner, P. D. Lyden, and D. Kleinfeld, “Two-Photon Imaging of Cortical Surface Microvessels Reveals a Robust Redistribution in Blood Flow after Vascular Occlusion,” PLoS Biol.4(2), e22 (2006).
[CrossRef] [PubMed]

Ma, Z.

Mariampillai, A.

McNabb, R. P.

Mok, A.

V. Yang, M. Gordon, B. Qi, J. Pekar, S. Lo, E. Seng-Yue, A. Mok, B. Wilson, and I. Vitkin, “High speed, wide velocity dynamic range Doppler optical coherence tomography (Part I): System design, signal processing, and performance,” Opt. Express11(7), 794–809 (2003).
[CrossRef] [PubMed]

V. X. D. Yang, M. L. Gordon, A. Mok, Y. Zhao, Z. Chen, R. S. C. Cobbold, B. C. Wilson, and I. Alex Vitkin, “Improved phase-resolved optical Doppler tomography using the Kasai velocity estimator and histogram segmentation,” Opt. Commun.208(4–6), 209–214 (2002).
[CrossRef]

Morgner, U.

Moriyama, E. H.

Munce, N. R.

Nelson, J. S.

Nishimura, N.

A. Y. Shih, J. D. Driscoll, P. J. Drew, N. Nishimura, C. B. Schaffer, and D. Kleinfeld, “Two-photon microscopy as a tool to study blood flow and neurovascular coupling in the rodent brain,” J. Cereb. Blood Flow Metab.32(7), 1277–1309 (2012).

C. B. Schaffer, B. Friedman, N. Nishimura, L. F. Schroeder, P. S. Tsai, F. F. Ebner, P. D. Lyden, and D. Kleinfeld, “Two-Photon Imaging of Cortical Surface Microvessels Reveals a Robust Redistribution in Blood Flow after Vascular Occlusion,” PLoS Biol.4(2), e22 (2006).
[CrossRef] [PubMed]

Nizar, K.

A. Devor, S. Sakadzic, V. J. Srinivasan, M. A. Yaseen, K. Nizar, P. A. Saisan, P. Tian, A. M. Dale, S. A. Vinogradov, M. A. Franceschini, and D. A. Boas, “Frontiers in optical imaging of cerebral blood flow and metabolism,” J. Cereb. Blood Flow Metab.32(7), 1259–1276 (2012).

Ogawa, S.

S. Ogawa, T. M. Lee, A. R. Kay, and D. W. Tank, “Brain magnetic resonance imaging with contrast dependent on blood oxygenation,” Proc. Natl. Acad. Sci. U.S.A.87(24), 9868–9872 (1990).
[CrossRef] [PubMed]

Otis, L. L.

Pan, Y.

Pan, Y. T.

Y. T. Pan, Z. L. Wu, Z. J. Yuan, Z. G. Wang, and C. W. Du, “Subcellular imaging of epithelium with time-lapse optical coherence tomography,” J. Biomed. Opt.12(5), 050504 (2007).
[CrossRef] [PubMed]

Pan, Y.-T.

H.-G. Ren, C.-W. Du, Z.-J. Yuan, K. Park, N. D. Volkow, and Y.-T. Pan, “Cocaine-induced cortical microischemia in the rodent brain: clinical implications,” Mol. Psychiatry17(10), 1017–1025 (2012).
[CrossRef] [PubMed]

Park, K.

H.-G. Ren, C.-W. Du, Z.-J. Yuan, K. Park, N. D. Volkow, and Y.-T. Pan, “Cocaine-induced cortical microischemia in the rodent brain: clinical implications,” Mol. Psychiatry17(10), 1017–1025 (2012).
[CrossRef] [PubMed]

Paulson, O. B.

O. B. Paulson, S. G. Hasselbalch, E. Rostrup, G. M. Knudsen, and D. Pelligrino, “Cerebral blood flow response to functional activation,” J. Cereb. Blood Flow Metab.30(1), 2–14 (2010).

Pekar, J.

Pelligrino, D.

O. B. Paulson, S. G. Hasselbalch, E. Rostrup, G. M. Knudsen, and D. Pelligrino, “Cerebral blood flow response to functional activation,” J. Cereb. Blood Flow Metab.30(1), 2–14 (2010).

Peterson, L. M.

Piao, D.

Pircher, M.

W. Trasischker, R. M. Werkmeister, S. Zotter, B. Baumann, T. Torzicky, M. Pircher, and C. K. Hitzenberger, “In vitro and in vivo three-dimensional velocity vector measurement by three-beam spectral-domain Doppler optical coherence tomography,” J. Biomed. Opt.18(11), 116010 (2013).
[CrossRef] [PubMed]

Pitris, C.

Proskurin, S. G.

Qi, B.

Radhakrishnan, H.

H. Radhakrishnan and V. J. Srinivasan, “Compartment-resolved imaging of cortical functional hyperemia with OCT angiography,” Biomed. Opt. Express4(8), 1255–1268 (2013).
[CrossRef] [PubMed]

V. J. Srinivasan, D. N. Atochin, H. Radhakrishnan, J. Y. Jiang, S. Ruvinskaya, W. Wu, S. Barry, A. E. Cable, C. Ayata, P. L. Huang, and D. A. Boas, “Optical coherence tomography for the quantitative study of cerebrovascular physiology,” J. Cereb. Blood Flow Metab.31(6), 1339–1345 (2011).

Ren, H.

Ren, H. G.

Ren, H.-G.

H.-G. Ren, C.-W. Du, Z.-J. Yuan, K. Park, N. D. Volkow, and Y.-T. Pan, “Cocaine-induced cortical microischemia in the rodent brain: clinical implications,” Mol. Psychiatry17(10), 1017–1025 (2012).
[CrossRef] [PubMed]

Rollins, A. M.

Rostrup, E.

O. B. Paulson, S. G. Hasselbalch, E. Rostrup, G. M. Knudsen, and D. Pelligrino, “Cerebral blood flow response to functional activation,” J. Cereb. Blood Flow Metab.30(1), 2–14 (2010).

Rothman, D. L.

R. G. Shulman, D. L. Rothman, K. L. Behar, and F. Hyder, “Energetic basis of brain activity: implications for neuroimaging,” Trends Neurosci.27(8), 489–495 (2004).
[CrossRef] [PubMed]

Ruvinskaya, S.

V. J. Srinivasan, D. N. Atochin, H. Radhakrishnan, J. Y. Jiang, S. Ruvinskaya, W. Wu, S. Barry, A. E. Cable, C. Ayata, P. L. Huang, and D. A. Boas, “Optical coherence tomography for the quantitative study of cerebrovascular physiology,” J. Cereb. Blood Flow Metab.31(6), 1339–1345 (2011).

V. J. Srinivasan, S. Sakadzić, I. Gorczynska, S. Ruvinskaya, W. Wu, J. G. Fujimoto, and D. A. Boas, “Quantitative cerebral blood flow with Optical Coherence Tomography,” Opt. Express18(3), 2477–2494 (2010).
[CrossRef] [PubMed]

Saisan, P. A.

A. Devor, S. Sakadzic, V. J. Srinivasan, M. A. Yaseen, K. Nizar, P. A. Saisan, P. Tian, A. M. Dale, S. A. Vinogradov, M. A. Franceschini, and D. A. Boas, “Frontiers in optical imaging of cerebral blood flow and metabolism,” J. Cereb. Blood Flow Metab.32(7), 1259–1276 (2012).

Sakadzic, S.

A. Devor, S. Sakadzic, V. J. Srinivasan, M. A. Yaseen, K. Nizar, P. A. Saisan, P. Tian, A. M. Dale, S. A. Vinogradov, M. A. Franceschini, and D. A. Boas, “Frontiers in optical imaging of cerebral blood flow and metabolism,” J. Cereb. Blood Flow Metab.32(7), 1259–1276 (2012).

V. J. Srinivasan, S. Sakadzić, I. Gorczynska, S. Ruvinskaya, W. Wu, J. G. Fujimoto, and D. A. Boas, “Quantitative cerebral blood flow with Optical Coherence Tomography,” Opt. Express18(3), 2477–2494 (2010).
[CrossRef] [PubMed]

Saxer, C.

Schaffer, C. B.

A. Y. Shih, J. D. Driscoll, P. J. Drew, N. Nishimura, C. B. Schaffer, and D. Kleinfeld, “Two-photon microscopy as a tool to study blood flow and neurovascular coupling in the rodent brain,” J. Cereb. Blood Flow Metab.32(7), 1277–1309 (2012).

T. N. Kim, P. W. Goodwill, Y. Chen, S. M. Conolly, C. B. Schaffer, D. Liepmann, and R. A. Wang, “Line-Scanning Particle Image Velocimetry: An Optical Approach for Quantifying a Wide Range of Blood Flow Speeds in Live Animals,” PLoS ONE7(6), e38590 (2012).
[CrossRef] [PubMed]

C. B. Schaffer, B. Friedman, N. Nishimura, L. F. Schroeder, P. S. Tsai, F. F. Ebner, P. D. Lyden, and D. Kleinfeld, “Two-Photon Imaging of Cortical Surface Microvessels Reveals a Robust Redistribution in Blood Flow after Vascular Occlusion,” PLoS Biol.4(2), e22 (2006).
[CrossRef] [PubMed]

Schmetterer, L.

Schroeder, L. F.

C. B. Schaffer, B. Friedman, N. Nishimura, L. F. Schroeder, P. S. Tsai, F. F. Ebner, P. D. Lyden, and D. Kleinfeld, “Two-Photon Imaging of Cortical Surface Microvessels Reveals a Robust Redistribution in Blood Flow after Vascular Occlusion,” PLoS Biol.4(2), e22 (2006).
[CrossRef] [PubMed]

Seng-Yue, E.

Shepherd, N.

Shih, A. Y.

A. Y. Shih, J. D. Driscoll, P. J. Drew, N. Nishimura, C. B. Schaffer, and D. Kleinfeld, “Two-photon microscopy as a tool to study blood flow and neurovascular coupling in the rodent brain,” J. Cereb. Blood Flow Metab.32(7), 1277–1309 (2012).

Shulman, R. G.

R. G. Shulman, D. L. Rothman, K. L. Behar, and F. Hyder, “Energetic basis of brain activity: implications for neuroimaging,” Trends Neurosci.27(8), 489–495 (2004).
[CrossRef] [PubMed]

Srinivasan, V. J.

H. Radhakrishnan and V. J. Srinivasan, “Compartment-resolved imaging of cortical functional hyperemia with OCT angiography,” Biomed. Opt. Express4(8), 1255–1268 (2013).
[CrossRef] [PubMed]

A. Devor, S. Sakadzic, V. J. Srinivasan, M. A. Yaseen, K. Nizar, P. A. Saisan, P. Tian, A. M. Dale, S. A. Vinogradov, M. A. Franceschini, and D. A. Boas, “Frontiers in optical imaging of cerebral blood flow and metabolism,” J. Cereb. Blood Flow Metab.32(7), 1259–1276 (2012).

V. J. Srinivasan, D. N. Atochin, H. Radhakrishnan, J. Y. Jiang, S. Ruvinskaya, W. Wu, S. Barry, A. E. Cable, C. Ayata, P. L. Huang, and D. A. Boas, “Optical coherence tomography for the quantitative study of cerebrovascular physiology,” J. Cereb. Blood Flow Metab.31(6), 1339–1345 (2011).

V. J. Srinivasan, S. Sakadzić, I. Gorczynska, S. Ruvinskaya, W. Wu, J. G. Fujimoto, and D. A. Boas, “Quantitative cerebral blood flow with Optical Coherence Tomography,” Opt. Express18(3), 2477–2494 (2010).
[CrossRef] [PubMed]

Standish, B. A.

Szkulmowski, M.

Szlag, D.

Tank, D. W.

S. Ogawa, T. M. Lee, A. R. Kay, and D. W. Tank, “Brain magnetic resonance imaging with contrast dependent on blood oxygenation,” Proc. Natl. Acad. Sci. U.S.A.87(24), 9868–9872 (1990).
[CrossRef] [PubMed]

Tearney, G.

Them, A.

A. Villringer, A. Them, U. Lindauer, K. Einhäupl, and U. Dirnagl, “Capillary perfusion of the rat brain cortex. An in vivo confocal microscopy study,” Circ. Res.75(1), 55–62 (1994).
[CrossRef] [PubMed]

Tian, P.

A. Devor, S. Sakadzic, V. J. Srinivasan, M. A. Yaseen, K. Nizar, P. A. Saisan, P. Tian, A. M. Dale, S. A. Vinogradov, M. A. Franceschini, and D. A. Boas, “Frontiers in optical imaging of cerebral blood flow and metabolism,” J. Cereb. Blood Flow Metab.32(7), 1259–1276 (2012).

Torzicky, T.

W. Trasischker, R. M. Werkmeister, S. Zotter, B. Baumann, T. Torzicky, M. Pircher, and C. K. Hitzenberger, “In vitro and in vivo three-dimensional velocity vector measurement by three-beam spectral-domain Doppler optical coherence tomography,” J. Biomed. Opt.18(11), 116010 (2013).
[CrossRef] [PubMed]

Trasischker, W.

W. Trasischker, R. M. Werkmeister, S. Zotter, B. Baumann, T. Torzicky, M. Pircher, and C. K. Hitzenberger, “In vitro and in vivo three-dimensional velocity vector measurement by three-beam spectral-domain Doppler optical coherence tomography,” J. Biomed. Opt.18(11), 116010 (2013).
[CrossRef] [PubMed]

Tsai, P. S.

C. B. Schaffer, B. Friedman, N. Nishimura, L. F. Schroeder, P. S. Tsai, F. F. Ebner, P. D. Lyden, and D. Kleinfeld, “Two-Photon Imaging of Cortical Surface Microvessels Reveals a Robust Redistribution in Blood Flow after Vascular Occlusion,” PLoS Biol.4(2), e22 (2006).
[CrossRef] [PubMed]

Vakoc, B.

Villringer, A.

A. Villringer, A. Them, U. Lindauer, K. Einhäupl, and U. Dirnagl, “Capillary perfusion of the rat brain cortex. An in vivo confocal microscopy study,” Circ. Res.75(1), 55–62 (1994).
[CrossRef] [PubMed]

Vinogradov, S. A.

A. Devor, S. Sakadzic, V. J. Srinivasan, M. A. Yaseen, K. Nizar, P. A. Saisan, P. Tian, A. M. Dale, S. A. Vinogradov, M. A. Franceschini, and D. A. Boas, “Frontiers in optical imaging of cerebral blood flow and metabolism,” J. Cereb. Blood Flow Metab.32(7), 1259–1276 (2012).

Vitkin, A.

Vitkin, I.

Vitkin, I. A.

Volkow, N. D.

H.-G. Ren, C.-W. Du, Z.-J. Yuan, K. Park, N. D. Volkow, and Y.-T. Pan, “Cocaine-induced cortical microischemia in the rodent brain: clinical implications,” Mol. Psychiatry17(10), 1017–1025 (2012).
[CrossRef] [PubMed]

Walther, J.

E. Koch, J. Walther, and M. Cuevas, “Limits of Fourier domain Doppler-OCT at high velocities,” Sens. Actuators A Phys.156(1), 8–13 (2009).
[CrossRef]

Wang, L.

Wang, R. A.

T. N. Kim, P. W. Goodwill, Y. Chen, S. M. Conolly, C. B. Schaffer, D. Liepmann, and R. A. Wang, “Line-Scanning Particle Image Velocimetry: An Optical Approach for Quantifying a Wide Range of Blood Flow Speeds in Live Animals,” PLoS ONE7(6), e38590 (2012).
[CrossRef] [PubMed]

Wang, R. K.

Wang, Z. G.

Y. T. Pan, Z. L. Wu, Z. J. Yuan, Z. G. Wang, and C. W. Du, “Subcellular imaging of epithelium with time-lapse optical coherence tomography,” J. Biomed. Opt.12(5), 050504 (2007).
[CrossRef] [PubMed]

Werkmeister, R. M.

W. Trasischker, R. M. Werkmeister, S. Zotter, B. Baumann, T. Torzicky, M. Pircher, and C. K. Hitzenberger, “In vitro and in vivo three-dimensional velocity vector measurement by three-beam spectral-domain Doppler optical coherence tomography,” J. Biomed. Opt.18(11), 116010 (2013).
[CrossRef] [PubMed]

Wilson, B.

Wilson, B. C.

Wojtkowski, M.

Wu, T.

Wu, W.

J. Lee, W. Wu, F. Lesage, and D. A. Boas, “Multiple-capillary measurement of RBC speed, flux, and density with optical coherence tomography,” J. Cereb. Blood Flow Metab.33(11), 1707–1710 (2013).

V. J. Srinivasan, D. N. Atochin, H. Radhakrishnan, J. Y. Jiang, S. Ruvinskaya, W. Wu, S. Barry, A. E. Cable, C. Ayata, P. L. Huang, and D. A. Boas, “Optical coherence tomography for the quantitative study of cerebrovascular physiology,” J. Cereb. Blood Flow Metab.31(6), 1339–1345 (2011).

V. J. Srinivasan, S. Sakadzić, I. Gorczynska, S. Ruvinskaya, W. Wu, J. G. Fujimoto, and D. A. Boas, “Quantitative cerebral blood flow with Optical Coherence Tomography,” Opt. Express18(3), 2477–2494 (2010).
[CrossRef] [PubMed]

Wu, Z. L.

Y. T. Pan, Z. L. Wu, Z. J. Yuan, Z. G. Wang, and C. W. Du, “Subcellular imaging of epithelium with time-lapse optical coherence tomography,” J. Biomed. Opt.12(5), 050504 (2007).
[CrossRef] [PubMed]

Xiang, S.

Yang, V.

Yang, V. X.

Yang, V. X. D.

A. Mariampillai, B. A. Standish, E. H. Moriyama, M. Khurana, N. R. Munce, M. K. K. Leung, J. Jiang, A. Cable, B. C. Wilson, I. A. Vitkin, and V. X. D. Yang, “Speckle variance detection of microvasculature using swept-source optical coherence tomography,” Opt. Lett.33(13), 1530–1532 (2008).
[CrossRef] [PubMed]

V. X. D. Yang, M. L. Gordon, A. Mok, Y. Zhao, Z. Chen, R. S. C. Cobbold, B. C. Wilson, and I. Alex Vitkin, “Improved phase-resolved optical Doppler tomography using the Kasai velocity estimator and histogram segmentation,” Opt. Commun.208(4–6), 209–214 (2002).
[CrossRef]

Yaseen, M. A.

A. Devor, S. Sakadzic, V. J. Srinivasan, M. A. Yaseen, K. Nizar, P. A. Saisan, P. Tian, A. M. Dale, S. A. Vinogradov, M. A. Franceschini, and D. A. Boas, “Frontiers in optical imaging of cerebral blood flow and metabolism,” J. Cereb. Blood Flow Metab.32(7), 1259–1276 (2012).

Yuan, Z.

Yuan, Z. J.

Y. T. Pan, Z. L. Wu, Z. J. Yuan, Z. G. Wang, and C. W. Du, “Subcellular imaging of epithelium with time-lapse optical coherence tomography,” J. Biomed. Opt.12(5), 050504 (2007).
[CrossRef] [PubMed]

Yuan, Z.-J.

H.-G. Ren, C.-W. Du, Z.-J. Yuan, K. Park, N. D. Volkow, and Y.-T. Pan, “Cocaine-induced cortical microischemia in the rodent brain: clinical implications,” Mol. Psychiatry17(10), 1017–1025 (2012).
[CrossRef] [PubMed]

Yun, S.

Zhao, Y.

Zhu, Q.

Zotter, S.

W. Trasischker, R. M. Werkmeister, S. Zotter, B. Baumann, T. Torzicky, M. Pircher, and C. K. Hitzenberger, “In vitro and in vivo three-dimensional velocity vector measurement by three-beam spectral-domain Doppler optical coherence tomography,” J. Biomed. Opt.18(11), 116010 (2013).
[CrossRef] [PubMed]

Appl. Opt. (1)

Biomed. Opt. Express (3)

Circ. Res. (1)

A. Villringer, A. Them, U. Lindauer, K. Einhäupl, and U. Dirnagl, “Capillary perfusion of the rat brain cortex. An in vivo confocal microscopy study,” Circ. Res.75(1), 55–62 (1994).
[CrossRef] [PubMed]

J. Biomed. Opt. (3)

W. Trasischker, R. M. Werkmeister, S. Zotter, B. Baumann, T. Torzicky, M. Pircher, and C. K. Hitzenberger, “In vitro and in vivo three-dimensional velocity vector measurement by three-beam spectral-domain Doppler optical coherence tomography,” J. Biomed. Opt.18(11), 116010 (2013).
[CrossRef] [PubMed]

W. Drexler, “Ultrahigh-resolution optical coherence tomography,” J. Biomed. Opt.9(1), 47–74 (2004).
[CrossRef] [PubMed]

Y. T. Pan, Z. L. Wu, Z. J. Yuan, Z. G. Wang, and C. W. Du, “Subcellular imaging of epithelium with time-lapse optical coherence tomography,” J. Biomed. Opt.12(5), 050504 (2007).
[CrossRef] [PubMed]

J. Cereb. Blood Flow Metab. (5)

O. B. Paulson, S. G. Hasselbalch, E. Rostrup, G. M. Knudsen, and D. Pelligrino, “Cerebral blood flow response to functional activation,” J. Cereb. Blood Flow Metab.30(1), 2–14 (2010).

A. Y. Shih, J. D. Driscoll, P. J. Drew, N. Nishimura, C. B. Schaffer, and D. Kleinfeld, “Two-photon microscopy as a tool to study blood flow and neurovascular coupling in the rodent brain,” J. Cereb. Blood Flow Metab.32(7), 1277–1309 (2012).

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H.-G. Ren, C.-W. Du, Z.-J. Yuan, K. Park, N. D. Volkow, and Y.-T. Pan, “Cocaine-induced cortical microischemia in the rodent brain: clinical implications,” Mol. Psychiatry17(10), 1017–1025 (2012).
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C. Iadecola, “Neurovascular regulation in the normal brain and in Alzheimer’s disease,” Nat. Rev. Neurosci.5(5), 347–360 (2004).
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Opt. Commun. (1)

V. X. D. Yang, M. L. Gordon, A. Mok, Y. Zhao, Z. Chen, R. S. C. Cobbold, B. C. Wilson, and I. Alex Vitkin, “Improved phase-resolved optical Doppler tomography using the Kasai velocity estimator and histogram segmentation,” Opt. Commun.208(4–6), 209–214 (2002).
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Opt. Express (7)

V. Yang, M. Gordon, B. Qi, J. Pekar, S. Lo, E. Seng-Yue, A. Mok, B. Wilson, and I. Vitkin, “High speed, wide velocity dynamic range Doppler optical coherence tomography (Part I): System design, signal processing, and performance,” Opt. Express11(7), 794–809 (2003).
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Figures (9)

Fig. 1
Fig. 1

A schematic of spectral-domain μOCT system for simultaneous 3D μODT and μOCA imaging. CM: collimator, D: dispersion compensator; FPC: fiber polarization controller, L1, L2, L3: achromatic lenses, G: servo mirrors. BS: beam splitter.

Fig. 2
Fig. 2

A sketch of new image acquisition and processing GUI to enable instantaneous display of 3D ODT of the quantitative CBFv network. a) Diagram of multi-threading structure with 6 sub-threads running separately with inter-thread data synchronization. b) Screen capture of GUI control panel during scanning CBFv network in vivo.

Fig. 3
Fig. 3

Numerical method for vessel tracking and least-squares fitted angle correction. Vessel skeleton (b) was generated by centroid tracking of the raw data (a). 2D (c) and 3D (d) curve fitting was applied to smoothen raw vessel skeleton extracted by centroid tracking technique, i.e., blue dots in (d). As result of two-step fitting, a vessel skeleton f(x,y,z) was obtained in (e). (f): Doppler angle |cosθz(x,y,z)| was calculated by Eq. (9) and further smoothened by Fourier curve (red line) to avoid extreme case when cosθz = 0.

Fig. 4
Fig. 4

Comparison of time partition between data acquisition thread and phase reconstruction threads.

Fig. 5
Fig. 5

Phantom flow study (1% intralipid, ϕ280µm tubing) to demonstrate that phase summation method enhances the sensitivity for slow flow detection and increases the dynamic range for fast flow detection. All images are projected onto same phase scale [0, π] for comparison.

Fig. 6
Fig. 6

Results of flow phantom study (1% intralipid, ϕ280µm tubing) to show that angle correction using gradient vessel tracking dramatically ~9.3-fold reduces the error in flow rate quantification.

Fig. 7
Fig. 7

3D ODT image of quantitative CBFv network on a mouse somatosensory motor cortex (1.9 × 1.5 × 1mm3). Left panel: 3D CBFv image without angle correction; Right panel: comparison of flow rate correction for a vein and an artery.

Fig. 8
Fig. 8

3D OCA images (upper panels) and 3D ODT images (lower panels) of the somatosensory motor cortex (2 × 1.5 × 1mm3) from a control mouse (left) and a chronic cocaine treated mouse (right), showing significant decreases in the entire CBFv network. Dashed circles show examples of vasoconstriction.

Fig. 9
Fig. 9

Ratio image to quickly track the ROIs, e.g., vessels with drastic flow decrease after repeated cocaine administration (2.5mg/kg/ea, iv). a) baseline CBFv image, b) currently updating CBFv image, c) ratio image of (b)/(a) to identify flow disruptions, d) statistical analysis of CBFv decrease. White dashed lines: ROIs with diminishing flows.

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

v= λ 0 Δϕ 4πnTcosθ
Δ ϕ T,z (i)=arctan{ Im( A i+1,z )Re( A i,z )Im( A i,z )Re( A i,+1z ) Im( A i+1,z )Re( A i,z )+Im( A i,z )Re( A i+1,z ) },πΔ ϕ T,z (i)π
Δ ϕ 2T,z (i)=Δ ϕ T,z (i)+Δ ϕ T,z (i+1),2πΔ ϕ 2T,z (i)2π
v wrap = λ 0 Δ ϕ max 4πnTcosθ
v min = λ 0 σ Δϕ 4πnT = λ 0 (SNR) 1/2 4πnT
VDR[dB]=20log( V wrap v min )
x=ay+b
z(x,y)= p 00 + p 10 y+ p 01 x+ p 20 y 2 + p 11 xy+ p 02 x 2
cos θ z (x,y,z)= f z ( f x ) 2 + ( f y ) 2 + ( f z ) 2
v z = λ 0 Δϕ 4πnT
v T = v z cos θ z
v(r)= v max (1 r 2 R 2 )= 2F π R 2 (1 r 2 R 2 )

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