Abstract

Quantitative velocity estimations in optical coherence tomography requires the estimation of the axial and lateral flow components. Optical coherence tomography measures the depth resolved complex field reflected from a sample. While the axial velocity component can be determined from the Doppler shift or phase shift between a pair of consecutive measurements at the same location, the estimation of the lateral component for in vivo applications is still challenging. One approach to determine lateral velocity is multiple simultaneous measurements at different angles. In another approach the lateral component can be retrieved through repeated measurements at (nearly) the same location by an analysis of the decorrelation over time. In this paper we follow the latter approach. We describe a model for the complex field changes between consecutive measurements and use it to predict the uncertainties for amplitude-based, phase-based and complex algorithms. The uncertainty of the flow estimations follows from a statistical analysis and is determined by the number of available measurements and the applied analysis method. The model is verified in phantom measurements and the dynamic range of velocity estimations is investigated. We demonstrate that phase-based and complex (phasor) based lateral flow estimation methods are superior to amplitude-based algorithms.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (3)

2018 (9)

J. Tang, S. E. Erdener, B. Li, B. Fu, S. Sakadzic, S. A. Carp, J. Lee, and D. A. Boas, “Shear-induced diffusion of red blood cells measured with dynamic light scattering-optical coherence tomography,” J. Biophotonics 11(2), e201700070 (2018).
[Crossref]

M. Salas, M. Augustin, F. Felberer, A. Wartak, M. Laslandes, L. Ginner, M. Niederleithner, J. Ensher, M. P. Minneman, R. A. Leitgeb, W. Drexler, X. Levecq, U. Schmidt-Erfurth, and M. Pircher, “Compact akinetic swept source optical coherence tomography angiography at 1060 nm supporting a wide field of view and adaptive optics imaging modes of the posterior eye,” Biomed. Opt. Express 9(4), 1871–1892 (2018).
[Crossref]

M. Sugita, R. A. Brown, I. Popov, and A. Vitkin, “K-distribution three-dimensional mapping of biological tissues in optical coherence tomography,” J. Biophotonics 11(3), e201700055 (2018).
[Crossref]

M. T. Bernucci, C. W. Merkle, and V. J. Srinivasan, “Investigation of artifacts in retinal and choroidal OCT angiography with a contrast agent,” Biomed. Opt. Express 9(3), 1020–1040 (2018).
[Crossref]

T. Park, S. J. Jang, M. Han, S. Ryu, and W. Y. Oh, “Wide dynamic range high-speed three-dimensional quantitative OCT angiography with a hybrid-beam scan,” Opt. Lett. 43(10), 2237–2240 (2018).
[Crossref]

A. Akif, K. Walek, C. Polucha, and J. Lee, “Doppler OCT clutter rejection using variance minimization and offset extrapolation,” Biomed. Opt. Express 9(11), 5340–5352 (2018).
[Crossref]

H. Spahr, C. Pfaffle, P. Koch, H. Sudkamp, G. Huttmann, and D. Hillmann, “Interferometric detection of 3D motion using computational subapertures in optical coherence tomography,” Opt. Express 26(15), 18803–18816 (2018).
[Crossref]

X. X. Li, W. Wu, H. Zhou, J. J. Deng, M. Y. Zhao, T. W. Qian, C. Yan, X. Xu, and S. Q. Yu, “A quantitative comparison of five optical coherence tomography angiography systems in clinical performance,” Int. J. Ophthalmol. (Engl. Ed.) 11(11), 1784–1795 (2018).
[Crossref]

A. Rabiolo, F. Gelormini, R. Sacconi, M. V. Cicinelli, G. Triolo, P. Bettin, K. Nouri-Mahdavi, F. Bandello, and G. Querques, “Comparison of methods to quantify macular and peripapillary vessel density in optical coherence tomography angiography,” PLoS One 13(10), e0205773 (2018).
[Crossref]

2017 (3)

2016 (5)

R. Haindl, W. Trasischker, A. Wartak, B. Baumann, M. Pircher, and C. K. Hitzenberger, “Total retinal blood flow measurement by three beam Doppler optical coherence tomography,” Biomed. Opt. Express 7(2), 287–301 (2016).
[Crossref]

A. Wartak, R. Haindl, W. Trasischker, B. Baumann, M. Pircher, and C. K. Hitzenberger, “Active-passive path-length encoded (APPLE) Doppler OCT,” Biomed. Opt. Express 7(12), 5233–5251 (2016).
[Crossref]

N. Uribe-Patarroyo and B. E. Bouma, “Velocity gradients in spatially resolved laser Doppler flowmetry and dynamic light scattering with confocal and coherence gating,” Phys. Rev. E 94(2), 022604 (2016).
[Crossref]

S. B. Ploner, E. M. Moult, W. Choi, N. K. Waheed, B. Lee, E. A. Novais, E. D. Cole, B. Potsaid, L. Husvogt, J. Schottenhamml, A. Maier, P. J. Rosenfeld, J. S. Duker, J. Hornegger, and J. G. Fujimoto, “Toward quantitative optical coherence tomography angiography: Visualizing Blood Flow Speeds in Ocular Pathology Using Variable Interscan Time Analysis,” Retina 36(Suppl 1), S118–S126 (2016).
[Crossref]

M. Rucci, P. V. McGraw, and R. J. Krauzlis, “Fixational eye movements and perception,” Vision Res. 118, 1–4 (2016).
[Crossref]

2015 (1)

R. F. Spaide, J. G. Fujimoto, and N. K. Waheed, “Image Artifacts in Optical Coherence Tomography Angiography,” Retina 35(11), 2163–2180 (2015).
[Crossref]

2014 (5)

2013 (3)

2012 (3)

2011 (1)

P. Cimalla, J. Walther, M. Mittasch, and E. Koch, “Shear flow-induced optical inhomogeneity of blood assessed in vivo and in vitro by spectral domain optical coherence tomography in the 1.3 mu m wavelength range,” J. Biomed. Opt. 16(11), 116020 (2011).
[Crossref]

2009 (1)

B. J. Vakoc, G. J. Tearney, and B. E. Bouma, “Statistical Properties of Phase-Decorrelation in Phase-Resolved Doppler Optical Coherence Tomography,” IEEE Trans. Med. Imaging 28(6), 814–821 (2009).
[Crossref]

2008 (3)

2005 (2)

2004 (2)

M. Emre, S. Orgul, K. Gugleta, and J. Flammer, “Ocular blood flow alteration in glaucoma is related to systemic vascular dysregulation,” Br. J. Ophthalmol. 88(5), 662–666 (2004).
[Crossref]

S. Martinez-Conde, S. L. Macknik, and D. H. Hubel, “The role of fixational eye movements in visual perception,” Nat. Rev. Neurosci. 5(3), 229–240 (2004).
[Crossref]

2002 (2)

2000 (1)

1997 (2)

1995 (1)

1992 (1)

V. Patel, S. Rassam, R. Newsom, J. Wiek, and E. Kohner, “Retinal Blood-Flow in Diabetic-Retinopathy,” Br. Med. J. 305(6855), 678–683 (1992).
[Crossref]

1925 (1)

R. A. Fisher, “Theory of statistical estimation,” Math. Proc. Cambridge Philos. Soc. 22(5), 700–725 (1925).
[Crossref]

Akcay, C.

Akif, A.

Augustin, M.

Bajraszewski, T.

Bandello, F.

A. Rabiolo, F. Gelormini, R. Sacconi, M. V. Cicinelli, G. Triolo, P. Bettin, K. Nouri-Mahdavi, F. Bandello, and G. Querques, “Comparison of methods to quantify macular and peripapillary vessel density in optical coherence tomography angiography,” PLoS One 13(10), e0205773 (2018).
[Crossref]

Barry, S.

Barton, J. K.

Baumann, B.

Berclaz, C.

Bernucci, M. T.

Bettin, P.

A. Rabiolo, F. Gelormini, R. Sacconi, M. V. Cicinelli, G. Triolo, P. Bettin, K. Nouri-Mahdavi, F. Bandello, and G. Querques, “Comparison of methods to quantify macular and peripapillary vessel density in optical coherence tomography angiography,” PLoS One 13(10), e0205773 (2018).
[Crossref]

Boas, D. A.

J. Tang, S. E. Erdener, B. Li, B. Fu, S. Sakadzic, S. A. Carp, J. Lee, and D. A. Boas, “Shear-induced diffusion of red blood cells measured with dynamic light scattering-optical coherence tomography,” J. Biophotonics 11(2), e201700070 (2018).
[Crossref]

J. Lee, W. Wu, J. Y. Jiang, B. Zhu, and D. A. Boas, “Dynamic light scattering optical coherence tomography,” Opt. Express 20(20), 22262–22277 (2012).
[Crossref]

Bolmont, T.

Bosscha, M.

Bouma, B. E.

N. Uribe-Patarroyo and B. E. Bouma, “Velocity gradients in spatially resolved laser Doppler flowmetry and dynamic light scattering with confocal and coherence gating,” Phys. Rev. E 94(2), 022604 (2016).
[Crossref]

N. Uribe-Patarroyo, M. Villiger, and B. E. Bouma, “Quantitative technique for robust and noise-tolerant speed measurements based on speckle decorrelation in optical coherence tomography,” Opt. Express 22(20), 24411–24429 (2014).
[Crossref]

B. J. Vakoc, G. J. Tearney, and B. E. Bouma, “Statistical Properties of Phase-Decorrelation in Phase-Resolved Doppler Optical Coherence Tomography,” IEEE Trans. Med. Imaging 28(6), 814–821 (2009).
[Crossref]

B. H. Park, M. C. Pierce, B. Cense, S. H. Yun, M. Mujat, G. J. Tearney, B. E. Bouma, and J. F. de Boer, “Real-time fiber-based multi-functional spectral-domain optical coherence tomography at 1.3 mu m,” Opt. Express 13(11), 3931–3944 (2005).
[Crossref]

Bourquin, S.

Bouwens, A.

Braaf, B.

Brecke, K. M.

Brown, R. A.

M. Sugita, R. A. Brown, I. Popov, and A. Vitkin, “K-distribution three-dimensional mapping of biological tissues in optical coherence tomography,” J. Biophotonics 11(3), e201700055 (2018).
[Crossref]

Cable, A. E.

Carp, S. A.

J. Tang, S. E. Erdener, B. Li, B. Fu, S. Sakadzic, S. A. Carp, J. Lee, and D. A. Boas, “Shear-induced diffusion of red blood cells measured with dynamic light scattering-optical coherence tomography,” J. Biophotonics 11(2), e201700070 (2018).
[Crossref]

Cense, B.

Chan, A. C.

A. C. Chan, V. J. Srinivasan, and E. Y. Lam, “Maximum Likelihood Doppler Frequency Estimation Under Decorrelation Noise for Quantifying Flow in Optical Coherence Tomography,” IEEE Trans. Med. Imaging 33(6), 1313–1323 (2014).
[Crossref]

Chen, C. L.

Chen, Z. P.

Choi, W.

S. B. Ploner, E. M. Moult, W. Choi, N. K. Waheed, B. Lee, E. A. Novais, E. D. Cole, B. Potsaid, L. Husvogt, J. Schottenhamml, A. Maier, P. J. Rosenfeld, J. S. Duker, J. Hornegger, and J. G. Fujimoto, “Toward quantitative optical coherence tomography angiography: Visualizing Blood Flow Speeds in Ocular Pathology Using Variable Interscan Time Analysis,” Retina 36(Suppl 1), S118–S126 (2016).
[Crossref]

Cicinelli, M. V.

A. Rabiolo, F. Gelormini, R. Sacconi, M. V. Cicinelli, G. Triolo, P. Bettin, K. Nouri-Mahdavi, F. Bandello, and G. Querques, “Comparison of methods to quantify macular and peripapillary vessel density in optical coherence tomography angiography,” PLoS One 13(10), e0205773 (2018).
[Crossref]

Cimalla, P.

P. Cimalla, J. Walther, M. Mittasch, and E. Koch, “Shear flow-induced optical inhomogeneity of blood assessed in vivo and in vitro by spectral domain optical coherence tomography in the 1.3 mu m wavelength range,” J. Biomed. Opt. 16(11), 116020 (2011).
[Crossref]

Ciulla, T. A.

R. Ehrlich, A. Harris, N. S. Kheradiya, D. M. Winston, T. A. Ciulla, and B. Wirostko, “Age-related macular degeneration and the aging eye,” Clin. Interventions Aging 3, 473–482 (2008).
[Crossref]

Cole, E. D.

S. B. Ploner, E. M. Moult, W. Choi, N. K. Waheed, B. Lee, E. A. Novais, E. D. Cole, B. Potsaid, L. Husvogt, J. Schottenhamml, A. Maier, P. J. Rosenfeld, J. S. Duker, J. Hornegger, and J. G. Fujimoto, “Toward quantitative optical coherence tomography angiography: Visualizing Blood Flow Speeds in Ocular Pathology Using Variable Interscan Time Analysis,” Retina 36(Suppl 1), S118–S126 (2016).
[Crossref]

Dave, D.

Davidoiu, V.

Davis, A. M.

de Boer, J. F.

M. G. O. Gräfe, M. Gondre, and J. F. de Boer, “Precision analysis and optimization in phase decorrelation OCT velocimetry,” Biomed. Opt. Express 10(3), 1297–1314 (2019).
[Crossref]

O. Nadiarnykh, V. Davidoiu, M. G. O. Gräfe, M. Bosscha, A. C. Moll, and J. F. de Boer, “Phase-based OCT angiography in diagnostic imaging of pediatric retinoblastoma patients: abnormal blood vessels in post-treatment regression patterns,” Biomed. Opt. Express 10(5), 2213–2226 (2019).
[Crossref]

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Supplementary Material (1)

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» Visualization 1       Simulation of the complex E-field when a collection of point scatterers move (laterally) through the detection volume of an OCT device

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Figures (8)

Fig. 1.
Fig. 1. Simulation of the complex E-field when a collection of point scatterers move (laterally) through the detection volume. Left: visualization of point scatterers. The sphere indicates where the collection efficiency of the OCT drops to 1/e2. Units are displayed in w0 (laterally) and $\sqrt 2$lc (axially). Right: the E-field which is a coherent summation of the backscattering and collection of all point scatterers. Each step represents a motion of δx/w0=0.5 (see Visualization 1)
Fig. 2.
Fig. 2. Plots of PDFs for different normalized motion β. (a) marginal PDF for phase differences; (b) marginal PDF for amplitude ratios; (c), (d) and (e) show the joint PDF for phase differences and amplitude ratios for the same values of β as in (a) and (b). The maximum of the color scale (“parula”, Mathworks Inc.) was set to the maximum of each PDF. When the motion becomes stronger, β becomes larger and therefore the phasors decorrelate more. This behavior is expressed as a broadening of the PDFs.
Fig. 3.
Fig. 3. (a) CRLBs for phasor-based (black), purely phase-based (red) and purely amplitude-based (blue) decorrelation estimation methods for N = 1 measurements. The phasor-based methods have the lowest CRLB. The CRLBs decrease proportional to $\sqrt N$. b) The square of the ratio of the CRLB of the phase-based and amplitude-based methods with respect to the phasor-based method, which expresses how many more measurements are needed by these methods as a function of β to reach the same uncertainty as the phasor-based approach. The purely phase-based analysis performs nearly as well as the phasor-based but the purely amplitude-based needs an order of magnitude more measurements especially in the high velocities rang ($\beta \approx 1$) to reach the same uncertainty. The black horizontal line indicates a ratio of one.
Fig. 4.
Fig. 4. a) schematic drawing of the setup. The scan module of an OCT system emitting a collimated beam was mounted above a lens and retina phantom. The phantom consist of a glass vessel embedded in a scattering medium. The capillary was connected to a syringe pump which pumped an aqueous intralipid with a constant flow rate. b) Structural B-scan of the phantom with indicated location of capillary and area of static scattering medium.
Fig. 5.
Fig. 5. Illustration of the derivative of the joint likelihood which is used to find combinations of β and θ fulfilling Eq. (20) for one measured ensemble at β=1.2
Fig. 6.
Fig. 6. Verification of linear estimation behavior and dynamic range of motion estimation with phasor-based MLE for (a) axial motion and (b) lateral motion. The vertical axis shows the estimated parameter β. Lines were fitted to the linear parts of the graphs. The upper limit of the dynamic range (linear regime) is dependent on the number of phasor pairs (N) per ensemble. The lower limit is independent on N and determined by noise sources.
Fig. 7.
Fig. 7. Uncertainty estimations as standard deviations for measurements (dots) of decorrelation for flow of intralipid through a glass vessel. The corresponding CRLBs are displayed as solid lines of the same color of the respective measurements. (a) all results for the uncertainty estimation with N = 200, (b) results from (a) binned by a binning factor of 9 for a more concise display, (c) effect on standard deviation when randomness of phasor pairs is increased (N = 50), (d) display of outliers in amplitude-based estimation when PDF is sampled insufficiently (N = 50).
Fig. 8.
Fig. 8. Illustration of the combination of multiple dynamic ranges and their relative error (standard deviation divided by expectation value) for phasor-based (solid black), phase-based (red) and amplitude-based (blue) analysis for a number of N = 25 phasor pairs. The vertical dashed lines indicate point in the velocity regime where the individual dynamic ranges are merged. The phasor-based analysis delivers the most constant relative error making the standard deviation a fixed percentage of the expectation value

Tables (1)

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Table 1. Conversion of β under the assumption of a beam radius of 10µm

Equations (20)

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Γ B = | α | Γ A + 1 α α Γ C
α = e i 2 k 0 δ z e δ x 2 w 0 2 e δ z 2 2 l c 2 ,
P | α | Γ A ( x 1 , y 1 ) = 1 π α α exp ( x 1 2 + y 1 2 α α )
P 1 α α Γ C ( Δ x , Δ y ) = 1 π ( 1 α α ) exp ( Δ x 2 + Δ y 2 ( 1 α α ) )
P ( x 1 , y 1 , Δ x , Δ y ) = 1 π α α exp ( x 1 2 + y 1 2 α α ) 1 π ( 1 α α ) exp ( Δ x 2 + Δ y 2 ( 1 α α ) ) .
P ( x A , y A , x C , y C ) = 1 π 2 e x A 2 y A 2 e x C 2 y C 2 .
P ( x A , y A , x B , y B ) = e x A 2 y A 2 π 2 ( 1 α α ) exp [ ( x B x A | α | ( 1 α α ) ) 2 ( y B y A | α | ( 1 α α ) ) 2 ] .
( x B y B ) = R ( x B y B ) = [ cos θ sin θ sin θ cos θ ] ( x B y B ) = ( x B cos θ y B sin θ x B sin θ + y B cos θ ) .
P ( A A , φ A , A B , φ B ) = A A A B e A A 2 π 2 ( 1 α α ) × exp [ ( A B cos ( φ B + θ ) | α | A A cos φ A ( 1 α α ) ) 2 ] × exp [ ( A B sin ( φ B + θ ) | α | A A sin φ A ( 1 α α ) ) 2 ]
P ( A A , A B , φ ) = 2 A A A B e A A 2 π ( 1 α α ) exp [ A B 2 + α 2 A A 2 2 | α | A A A B cos ( φ + θ ) ( 1 α α ) ] .
P ( p , q , φ ) = e p / q π ( 1 α α ) p q exp ( p q + | α | p q 2 | α | p cos ( φ + θ ) ( 1 α α ) ) .
P ( p , φ ) = 2 p π ( 1 α α ) exp ( 2 p | α | cos ( φ + θ ) ( 1 α α ) ) K 0 ( 2 p ( 1 α α ) )
P ( q , φ ) = q ( 1 α α ) π ( q 2 + 1 2 q | α | cos ( φ + θ ) ) 2
E l + 1 E l = p e i φ P ( p , φ ) d p d φ = α .
P ( ϕ ) = 1 α α 2 π ( 1 α α cos 2 ϕ ) [ 1 + | α | cos ϕ 1 α α cos 2 ϕ ( π cos 1 [ | α | cos ϕ ] ) ] ,
P ( q ) = 2 q ( 1 + q 2 ) ( 1 α α ) ( 1 + q 2 + 2 q | α | ) 3 ( 1 + q 2 2 q | α | ) 3 ,
α = exp ( i θ β 2 ) .
I = E [ log P ( x | α ) α ] ,
β M L E = arg max β j log P ( q j , φ j | β ) ,
2 N e β M L E 2 1 e 2 β M L E 2 + j 4 q j cos ( φ j + θ ) 1 + q j 2 2 q j e β M L E 2 cos ( φ j + θ ) = 0.