Abstract

Vocal fold vibration is vital in voice production and the correct pitch of speech. We have developed a high speed functional optical coherence tomography (OCT) system with a center wavelength of 1050 nm and an imaging speed of 100,000 A-lines per second. We imaged the vibration of an ex-vivo swine vocal fold. At an imaging speed of 100 frames per second, we demonstrated high quality vocal fold images during vibration. Functional information, such as vibration frequency and vibration amplitude, was obtained by analyzing the tissue surface during vibration. The axial direction velocity distribution in the cross-sectional images of the vibrating vocal folds was obtained with the Doppler OCT. The quantitative transverse direction velocity distribution in the cross-sectional images was obtained with the Doppler variance images.

© 2011 OSA

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2010 (4)

M. Dedecjus, Z. Adamczewski, J. Brzeziński, and A. Lewiński, “Real-time, high-resolution ultrasonography of the vocal folds--a prospective pilot study in patients before and after thyroidectomy,” Langenbecks Arch. Surg. 395(7), 859–864 (2010).
[CrossRef] [PubMed]

J. B. Kobler, E. W. Chang, S. M. Zeitels, and S. H. Yun, “Dynamic imaging of vocal fold oscillation with four-dimensional optical coherence tomography,” Laryngoscope 120(7), 1354–1362 (2010).
[CrossRef] [PubMed]

B. Potsaid, B. Baumann, D. Huang, S. Barry, A. E. Cable, J. S. Schuman, J. S. Duker, and J. G. Fujimoto, “Ultrahigh speed 1050nm swept source/Fourier domain OCT retinal and anterior segment imaging at 100,000 to 400,000 axial scans per second,” Opt. Express 18(19), 20029–20048 (2010), http://www.opticsinfobase.org/abstract.cfm?URI=oe-18-19-20029 .
[CrossRef] [PubMed]

L. An, H. M. Subhush, D. J. Wilson, and R. K. Wang, “High-resolution wide-field imaging of retinal and choroidal blood perfusion with optical microangiography,” J. Biomed. Opt. 15(2), 026011 (2010).
[CrossRef] [PubMed]

2009 (2)

C. G. Tsai, J. H. Chen, Y. W. Shau, and T. Y. Hsiao, “Dynamic B-mode ultrasound imaging of vocal fold vibration during phonation,” Ultrasound Med. Biol. 35(11), 1812–1818 (2009).
[CrossRef] [PubMed]

L. Yu, G. Liu, M. Rubinstein, A. Saidi, B. J. Wong, and Z. Chen, “Office-based dynamic imaging of vocal cords in awake patients with swept-source optical coherence tomography,” J. Biomed. Opt. 14(6), 064020 (2009).
[CrossRef]

2008 (3)

R. Patel, S. Dailey, and D. Bless, “Comparison of high-speed digital imaging with stroboscopy for laryngeal imaging of glottal disorders,” Ann. Otol. Rhinol. Laryngol. 117(6), 413–424 (2008).
[PubMed]

J. Lohscheller, U. Eysholdt, H. Toy, and M. Dollinger, “Phonovibrography: mapping high-speed movies of vocal fold vibrations into 2-D diagrams for visualizing and analyzing the underlying laryngeal dynamics,” IEEE Trans. Med. Imaging 27(3), 300–309 (2008).
[CrossRef] [PubMed]

J. Lohscheller and U. Eysholdt, “Phonovibrogram visualization of entire vocal fold dynamics,” Laryngoscope 118(4), 753–758 (2008).
[CrossRef] [PubMed]

2007 (1)

2006 (2)

S. Makita, Y. Hong, M. Yamanari, T. Yatagai, and Y. Yasuno, “Optical coherence angiography,” Opt. Express 14(17), 7821–7840 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-17-7821 .
[CrossRef] [PubMed]

K. Lüerßen, H. Lubatschowski, N. Radicke, and M. Ptok, “Optical characterization of vocal folds using optical coherence tomography,” Med. Laser Appl. 21(3), 185–190 (2006).
[CrossRef]

2005 (2)

2004 (1)

2003 (1)

2002 (2)

2000 (2)

1997 (3)

1991 (1)

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. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Adamczewski, Z.

M. Dedecjus, Z. Adamczewski, J. Brzeziński, and A. Lewiński, “Real-time, high-resolution ultrasonography of the vocal folds--a prospective pilot study in patients before and after thyroidectomy,” Langenbecks Arch. Surg. 395(7), 859–864 (2010).
[CrossRef] [PubMed]

Adler, D. C.

An, L.

L. An, H. M. Subhush, D. J. Wilson, and R. K. Wang, “High-resolution wide-field imaging of retinal and choroidal blood perfusion with optical microangiography,” J. Biomed. Opt. 15(2), 026011 (2010).
[CrossRef] [PubMed]

Barry, S.

Barton, J. K.

Baumann, B.

Bless, D.

R. Patel, S. Dailey, and D. Bless, “Comparison of high-speed digital imaging with stroboscopy for laryngeal imaging of glottal disorders,” Ann. Otol. Rhinol. Laryngol. 117(6), 413–424 (2008).
[PubMed]

Bouma, B.

Brecke, K. M.

Brzezinski, J.

M. Dedecjus, Z. Adamczewski, J. Brzeziński, and A. Lewiński, “Real-time, high-resolution ultrasonography of the vocal folds--a prospective pilot study in patients before and after thyroidectomy,” Langenbecks Arch. Surg. 395(7), 859–864 (2010).
[CrossRef] [PubMed]

Cable, A. E.

Cense, B.

Chang, E. W.

J. B. Kobler, E. W. Chang, S. M. Zeitels, and S. H. Yun, “Dynamic imaging of vocal fold oscillation with four-dimensional optical coherence tomography,” Laryngoscope 120(7), 1354–1362 (2010).
[CrossRef] [PubMed]

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. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Chen, C. N.

T. Y. Hsiao, C. L. Wang, C. N. Chen, F. J. Hsieh, and Y. W. Shau, “Elasticity of human vocal folds measured in vivo using color Doppler imaging,” Ultrasound Med. Biol. 28(9), 1145–1152 (2002).
[CrossRef] [PubMed]

Chen, J. H.

C. G. Tsai, J. H. Chen, Y. W. Shau, and T. Y. Hsiao, “Dynamic B-mode ultrasound imaging of vocal fold vibration during phonation,” Ultrasound Med. Biol. 35(11), 1812–1818 (2009).
[CrossRef] [PubMed]

Chen, T.

Chen, Z.

L. Yu, G. Liu, M. Rubinstein, A. Saidi, B. J. Wong, and Z. Chen, “Office-based dynamic imaging of vocal cords in awake patients with swept-source optical coherence tomography,” J. Biomed. Opt. 14(6), 064020 (2009).
[CrossRef]

J. Zhang and Z. Chen, “In vivo blood flow imaging by a swept laser source based Fourier domain optical Doppler tomography,” Opt. Express 13(19), 7449–7457 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-19-7449 .
[CrossRef] [PubMed]

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), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-27-6-409 .
[CrossRef]

Y. Zhao, Z. Chen, C. Saxer, Q. Shen, S. Xiang, J. F. de Boer, and J. S. Nelson, “Doppler standard deviation imaging for clinical monitoring of in vivo human skin blood flow,” Opt. Lett. 25(18), 1358–1360 (2000), http://www.opticsinfobase.org/abstract.cfm?URI=ol-25-18-1358 .
[CrossRef]

Y. Zhao, Z. Chen, C. Saxer, S. Xiang, J. F. de Boer, and J. S. Nelson, “Phase-resolved optical coherence tomography and optical Doppler tomography for imaging blood flow in human skin with fast scanning speed and high velocity sensitivity,” Opt. Lett. 25(2), 114–116, (2000), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-25-2-114 .
[CrossRef]

Z. 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(1), 64–66 (1997), http://www.opticsinfobase.org/abstract.cfm?URI=ol-22-1-64 .
[CrossRef] [PubMed]

Z. Chen, T. E. Milner, S. Srinivas, X. J. Wang, A. Malekafzali, M. J. C. van Gemert, and J. S. Nelson, “Noninvasive imaging of in vivo blood flow velocity using optical Doppler tomography,” Opt. Lett. 22(14), 1119–1121 (1997), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-22-14-1119 .
[CrossRef] [PubMed]

Dailey, S.

R. Patel, S. Dailey, and D. Bless, “Comparison of high-speed digital imaging with stroboscopy for laryngeal imaging of glottal disorders,” Ann. Otol. Rhinol. Laryngol. 117(6), 413–424 (2008).
[PubMed]

Dave, D.

de Boer, J.

de Boer, J. F.

Dedecjus, M.

M. Dedecjus, Z. Adamczewski, J. Brzeziński, and A. Lewiński, “Real-time, high-resolution ultrasonography of the vocal folds--a prospective pilot study in patients before and after thyroidectomy,” Langenbecks Arch. Surg. 395(7), 859–864 (2010).
[CrossRef] [PubMed]

Ding, Z.

Dollinger, M.

J. Lohscheller, U. Eysholdt, H. Toy, and M. Dollinger, “Phonovibrography: mapping high-speed movies of vocal fold vibrations into 2-D diagrams for visualizing and analyzing the underlying laryngeal dynamics,” IEEE Trans. Med. Imaging 27(3), 300–309 (2008).
[CrossRef] [PubMed]

Duker, J. S.

Eysholdt, U.

J. Lohscheller, U. Eysholdt, H. Toy, and M. Dollinger, “Phonovibrography: mapping high-speed movies of vocal fold vibrations into 2-D diagrams for visualizing and analyzing the underlying laryngeal dynamics,” IEEE Trans. Med. Imaging 27(3), 300–309 (2008).
[CrossRef] [PubMed]

J. Lohscheller and U. Eysholdt, “Phonovibrogram visualization of entire vocal fold dynamics,” Laryngoscope 118(4), 753–758 (2008).
[CrossRef] [PubMed]

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. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Fujimoto, J. G.

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. G. 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. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Hong, Y.

Hsiao, T. Y.

C. G. Tsai, J. H. Chen, Y. W. Shau, and T. Y. Hsiao, “Dynamic B-mode ultrasound imaging of vocal fold vibration during phonation,” Ultrasound Med. Biol. 35(11), 1812–1818 (2009).
[CrossRef] [PubMed]

T. Y. Hsiao, C. L. Wang, C. N. Chen, F. J. Hsieh, and Y. W. Shau, “Elasticity of human vocal folds measured in vivo using color Doppler imaging,” Ultrasound Med. Biol. 28(9), 1145–1152 (2002).
[CrossRef] [PubMed]

Hsieh, F. J.

T. Y. Hsiao, C. L. Wang, C. N. Chen, F. J. Hsieh, and Y. W. Shau, “Elasticity of human vocal folds measured in vivo using color Doppler imaging,” Ultrasound Med. Biol. 28(9), 1145–1152 (2002).
[CrossRef] [PubMed]

Huang, D.

Huber, R.

Izatt, J. A.

Kobler, J. B.

J. B. Kobler, E. W. Chang, S. M. Zeitels, and S. H. Yun, “Dynamic imaging of vocal fold oscillation with four-dimensional optical coherence tomography,” Laryngoscope 120(7), 1354–1362 (2010).
[CrossRef] [PubMed]

Kulkarni, M. D.

Lewinski, A.

M. Dedecjus, Z. Adamczewski, J. Brzeziński, and A. Lewiński, “Real-time, high-resolution ultrasonography of the vocal folds--a prospective pilot study in patients before and after thyroidectomy,” Langenbecks Arch. Surg. 395(7), 859–864 (2010).
[CrossRef] [PubMed]

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. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Liu, G.

L. Yu, G. Liu, M. Rubinstein, A. Saidi, B. J. Wong, and Z. Chen, “Office-based dynamic imaging of vocal cords in awake patients with swept-source optical coherence tomography,” J. Biomed. Opt. 14(6), 064020 (2009).
[CrossRef]

Lohscheller, J.

J. Lohscheller and U. Eysholdt, “Phonovibrogram visualization of entire vocal fold dynamics,” Laryngoscope 118(4), 753–758 (2008).
[CrossRef] [PubMed]

J. Lohscheller, U. Eysholdt, H. Toy, and M. Dollinger, “Phonovibrography: mapping high-speed movies of vocal fold vibrations into 2-D diagrams for visualizing and analyzing the underlying laryngeal dynamics,” IEEE Trans. Med. Imaging 27(3), 300–309 (2008).
[CrossRef] [PubMed]

Lubatschowski, H.

K. Lüerßen, H. Lubatschowski, N. Radicke, and M. Ptok, “Optical characterization of vocal folds using optical coherence tomography,” Med. Laser Appl. 21(3), 185–190 (2006).
[CrossRef]

Lüerßen, K.

K. Lüerßen, H. Lubatschowski, N. Radicke, and M. Ptok, “Optical characterization of vocal folds using optical coherence tomography,” Med. Laser Appl. 21(3), 185–190 (2006).
[CrossRef]

Makita, S.

Malekafzali, A.

Milner, T. E.

Nassif, N.

Nelson, J. S.

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), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-27-6-409 .
[CrossRef]

Y. Zhao, Z. Chen, C. Saxer, S. Xiang, J. F. de Boer, and J. S. Nelson, “Phase-resolved optical coherence tomography and optical Doppler tomography for imaging blood flow in human skin with fast scanning speed and high velocity sensitivity,” Opt. Lett. 25(2), 114–116, (2000), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-25-2-114 .
[CrossRef]

Y. Zhao, Z. Chen, C. Saxer, Q. Shen, S. Xiang, J. F. de Boer, and J. S. Nelson, “Doppler standard deviation imaging for clinical monitoring of in vivo human skin blood flow,” Opt. Lett. 25(18), 1358–1360 (2000), http://www.opticsinfobase.org/abstract.cfm?URI=ol-25-18-1358 .
[CrossRef]

Z. Chen, T. E. Milner, S. Srinivas, X. J. Wang, A. Malekafzali, M. J. C. van Gemert, and J. S. Nelson, “Noninvasive imaging of in vivo blood flow velocity using optical Doppler tomography,” Opt. Lett. 22(14), 1119–1121 (1997), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-22-14-1119 .
[CrossRef] [PubMed]

Z. 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(1), 64–66 (1997), http://www.opticsinfobase.org/abstract.cfm?URI=ol-22-1-64 .
[CrossRef] [PubMed]

Park, B.

Patel, R.

R. Patel, S. Dailey, and D. Bless, “Comparison of high-speed digital imaging with stroboscopy for laryngeal imaging of glottal disorders,” Ann. Otol. Rhinol. Laryngol. 117(6), 413–424 (2008).
[PubMed]

Pierce, M.

Potsaid, B.

Ptok, M.

K. Lüerßen, H. Lubatschowski, N. Radicke, and M. Ptok, “Optical characterization of vocal folds using optical coherence tomography,” Med. Laser Appl. 21(3), 185–190 (2006).
[CrossRef]

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. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Radicke, N.

K. Lüerßen, H. Lubatschowski, N. Radicke, and M. Ptok, “Optical characterization of vocal folds using optical coherence tomography,” Med. Laser Appl. 21(3), 185–190 (2006).
[CrossRef]

Ren, H.

Rubinstein, M.

L. Yu, G. Liu, M. Rubinstein, A. Saidi, B. J. Wong, and Z. Chen, “Office-based dynamic imaging of vocal cords in awake patients with swept-source optical coherence tomography,” J. Biomed. Opt. 14(6), 064020 (2009).
[CrossRef]

Saidi, A.

L. Yu, G. Liu, M. Rubinstein, A. Saidi, B. J. Wong, and Z. Chen, “Office-based dynamic imaging of vocal cords in awake patients with swept-source optical coherence tomography,” J. Biomed. Opt. 14(6), 064020 (2009).
[CrossRef]

Saxer, C.

Schuman, J. S.

Shau, Y. W.

C. G. Tsai, J. H. Chen, Y. W. Shau, and T. Y. Hsiao, “Dynamic B-mode ultrasound imaging of vocal fold vibration during phonation,” Ultrasound Med. Biol. 35(11), 1812–1818 (2009).
[CrossRef] [PubMed]

T. Y. Hsiao, C. L. Wang, C. N. Chen, F. J. Hsieh, and Y. W. Shau, “Elasticity of human vocal folds measured in vivo using color Doppler imaging,” Ultrasound Med. Biol. 28(9), 1145–1152 (2002).
[CrossRef] [PubMed]

Shen, Q.

Srinivas, S.

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. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Subhush, H. M.

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C. G. Tsai, J. H. Chen, Y. W. Shau, and T. Y. Hsiao, “Dynamic B-mode ultrasound imaging of vocal fold vibration during phonation,” Ultrasound Med. Biol. 35(11), 1812–1818 (2009).
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Wang, C. L.

T. Y. Hsiao, C. L. Wang, C. N. Chen, F. J. Hsieh, and Y. W. Shau, “Elasticity of human vocal folds measured in vivo using color Doppler imaging,” Ultrasound Med. Biol. 28(9), 1145–1152 (2002).
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L. An, H. M. Subhush, D. J. Wilson, and R. K. Wang, “High-resolution wide-field imaging of retinal and choroidal blood perfusion with optical microangiography,” J. Biomed. Opt. 15(2), 026011 (2010).
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L. An, H. M. Subhush, D. J. Wilson, and R. K. Wang, “High-resolution wide-field imaging of retinal and choroidal blood perfusion with optical microangiography,” J. Biomed. Opt. 15(2), 026011 (2010).
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L. Yu, G. Liu, M. Rubinstein, A. Saidi, B. J. Wong, and Z. Chen, “Office-based dynamic imaging of vocal cords in awake patients with swept-source optical coherence tomography,” J. Biomed. Opt. 14(6), 064020 (2009).
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L. An, H. M. Subhush, D. J. Wilson, and R. K. Wang, “High-resolution wide-field imaging of retinal and choroidal blood perfusion with optical microangiography,” J. Biomed. Opt. 15(2), 026011 (2010).
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Science (1)

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. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Ultrasound Med. Biol. (2)

T. Y. Hsiao, C. L. Wang, C. N. Chen, F. J. Hsieh, and Y. W. Shau, “Elasticity of human vocal folds measured in vivo using color Doppler imaging,” Ultrasound Med. Biol. 28(9), 1145–1152 (2002).
[CrossRef] [PubMed]

C. G. Tsai, J. H. Chen, Y. W. Shau, and T. Y. Hsiao, “Dynamic B-mode ultrasound imaging of vocal fold vibration during phonation,” Ultrasound Med. Biol. 35(11), 1812–1818 (2009).
[CrossRef] [PubMed]

Supplementary Material (3)

» Media 1: AVI (5893 KB)     
» Media 2: AVI (3119 KB)     
» Media 3: AVI (4225 KB)     

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Figures (10)

Fig. 1
Fig. 1

Schematic of the high speed OCT and ODT system. GM: galvanometer mirror; CG: cover glass; BD: balanced photo detector.

Fig. 2
Fig. 2

(a) OCT images of a mirror; (b) Adjacent A-line phase difference before correction; (c) Adjacent A-line phase difference after correction.

Fig. 3
Fig. 3

(a) Photograph of side view of the porcine larynx. (b) Photograph of top view of the porcine larynx.

Fig. 4
Fig. 4

M-mode OCT image of the vibrating vocal fold.

Fig. 5
Fig. 5

Fitting of the surface of the vibrating vocal fold with Eq. (1).

Fig. 6
Fig. 6

Color Doppler image of the vibrating vocal fold.

Fig. 7
Fig. 7

OCT images of a vibrating vocal fold with frequency of around 94.3Hz. (a) B-mode OCT structure image. (b) B-mode color Doppler OCT image. (c) B-mode Doppler variance OCT image. Scale bar: 500  μm .

Fig. 8
Fig. 8

OCT images of a vibrating vocal fold with frequency of around 1.1Hz. (a) B-mode OCT structure image. (b) B-mode color Doppler OCT image. (c) B-mode Doppler variance OCT image. Scale bar: 500  μm .

Fig. 9
Fig. 9

OCT Movies of a vibrating vocal fold with frequency of around 1.1Hz. (a) B-mode OCT structure image (Media 1). (b) B-mode color Doppler OCT image (Media 2). (c) B-mode Doppler variance OCT image (Media 3).

Fig. 10
Fig. 10

B-mode color Doppler images of a slow vibrating vocal fold. Scale bar: 500 um.

Equations (5)

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

y ( t ) = y 0 + A sin [ π ( t t c ) w ] ,
v ( t ) = d [ y ( t ) ] d t = A π w cos [ π ( t t c ) w ] ,
v max = A π w .
f d = 2 v z λ c = Δ θ 2 π T ,
v z = λ c Δ θ 4 π T .

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