Abstract

Optical coherence tomography (OCT) and optical coherence microscopy (OCM) allow the acquisition of quantitative three-dimensional axial flow by estimating the Doppler shift caused by moving scatterers. Measuring the velocity of red blood cells is currently the principal application of these methods. In many biological tissues, blood flow is often perpendicular to the optical axis, creating the need for a quantitative measurement of lateral flow. Previous work has shown that lateral flow can be measured from the Doppler bandwidth, albeit only for simplified optical systems. In this work, we present a generalized model to analyze the influence of relevant OCT/OCM system parameters such as light source spectrum, numerical aperture and beam geometry on the Doppler spectrum. Our analysis results in a general framework relating the mean and variance of the Doppler frequency to the axial and lateral flow velocity components. Based on this model, we present an optimized acquisition protocol and algorithm to reconstruct quantitative measurements of lateral and axial flow from the Doppler spectrum for any given OCT/OCM system. To validate this approach, Doppler spectrum analysis is employed to quantitatively measure flow in a capillary with both extended focus OCM and OCT.

© 2013 OSA

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de Boer, J.

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Moriyama, E. H.

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T. Bolmont, A. Bouwens, C. Pache, M. Dimitrov, C. Berclaz, M. Villiger, B. M. Wegenast-Braun, T. Lasser, and P. C. Fraering, “Label-free imaging of cerebral beta-amyloidosis with extended-focus optical coherence microscopy,” J. Neurosci.32, 14548–14556 (2012).
[CrossRef] [PubMed]

M. Villiger, C. Pache, and T. Lasser, “Dark-Field optical coherence microscopy,” Opt. Lett.35, 3489–3491 (2010).
[CrossRef] [PubMed]

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B. Park, M. Pierce, B. Cense, and S. Yun, “Real-time fiber-based multi-functional spectral-domain optical coherence tomography at 1.3 m,” Opt. Express13(2005).
[CrossRef]

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Pierce, M.

B. Park, M. Pierce, B. Cense, and S. Yun, “Real-time fiber-based multi-functional spectral-domain optical coherence tomography at 1.3 m,” Opt. Express13(2005).
[CrossRef]

Pircher, M.

Potsaid, B.

Proskurin, S. G.

Radhakrishnan, H.

Ralston, T. S.

T. S. Ralston, D. L. Marks, P. Scott Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nature Phys.3, 129–134 (2007).
[CrossRef]

Ren, H.

Roy, M.

Ruvinskaya, S.

Sakadzic, S.

Saxer, C.

Schmetterer, L.

Schwartz, D.

Scott Carney, P.

T. S. Ralston, D. L. Marks, P. Scott Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nature Phys.3, 129–134 (2007).
[CrossRef]

Sharma, M. D.

Shen, Q.

Sheppard, C. J. R.

Srinivasan, V. J.

Standish, B. A.

Steinmann, L.

Sylwestrzak, M.

Szkulmowska, A.

Szkulmowski, M.

Szlag, D.

Tearney, G.

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Vakoc, B.

Villiger, M.

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Vitkin, I. A.

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J. Walther and E. Koch, “Enhanced joint spectral and time domain optical coherence tomography for quantitative flow velocity measurement,” in “Optical Coherence Tomography and Coherence Techniques V,”, R. A. Leitgeb and B. E. Bouma, eds. (Proc. of SPIE, 2011), p. 80910L.

Wang, R. K.

Wegenast-Braun, B. M.

T. Bolmont, A. Bouwens, C. Pache, M. Dimitrov, C. Berclaz, M. Villiger, B. M. Wegenast-Braun, T. Lasser, and P. C. Fraering, “Label-free imaging of cerebral beta-amyloidosis with extended-focus optical coherence microscopy,” J. Neurosci.32, 14548–14556 (2012).
[CrossRef] [PubMed]

Welch, A. J.

Wieser, W.

Wilson, B. C.

Wojtkowski, M.

Wolf, E.

M. Born and E. Wolf, Principles of Optics : Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, Cambridge, 2009), 7th ed.

Wu, W.

Xiang, S.

Yamanari, M.

Yang, C.

Yang, V. X. D.

Yaseen, M.

Yasuno, Y.

Yatagai, T.

Yazdanfar, S.

Yu, L.

L. Yu and Z. Chen, “Doppler variance imaging for three-dimensional retina and choroid angiography,” J. Biomed. Opt.15, 016029 (2011).
[CrossRef]

Yun, S.

B. Park, M. Pierce, B. Cense, and S. Yun, “Real-time fiber-based multi-functional spectral-domain optical coherence tomography at 1.3 m,” Opt. Express13(2005).
[CrossRef]

B. Vakoc, S. Yun, J. de Boer, G. Tearney, and B. Bouma, “Phase-resolved optical frequency domain imaging,” Opt. Express13, 5483–93 (2005).
[CrossRef] [PubMed]

Zhao, Y.

Zhu, B.

Zhu, Q.

Zotter, S.

Appl. Opt.

Biomed. Opt. Express

J. Appl. Phys.

R. V. Edwards, “Spectral analysis of the signal from the laser Ddoppler flowmeter: Time-independent systems,” J. Appl. Phys.42, 837 (1971).
[CrossRef]

J. Biomed. Opt.

L. Yu and Z. Chen, “Doppler variance imaging for three-dimensional retina and choroid angiography,” J. Biomed. Opt.15, 016029 (2011).
[CrossRef]

J. Neurosci.

T. Bolmont, A. Bouwens, C. Pache, M. Dimitrov, C. Berclaz, M. Villiger, B. M. Wegenast-Braun, T. Lasser, and P. C. Fraering, “Label-free imaging of cerebral beta-amyloidosis with extended-focus optical coherence microscopy,” J. Neurosci.32, 14548–14556 (2012).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

Nature Phys.

T. S. Ralston, D. L. Marks, P. Scott Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nature Phys.3, 129–134 (2007).
[CrossRef]

Opt. Express

B. Park, M. Pierce, B. Cense, and S. Yun, “Real-time fiber-based multi-functional spectral-domain optical coherence tomography at 1.3 m,” Opt. Express13(2005).
[CrossRef]

C. Blatter, B. Grajciar, C. M. Eigenwillig, W. Wieser, B. R. Biedermann, R. Huber, and R. A. Leitgeb, “Extended focus high-speed swept source OCT with self-reconstructive illumination,” Opt. Express19, 12141–55 (2011).
[CrossRef] [PubMed]

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, 2477–94 (2010).
[CrossRef] [PubMed]

B. Vakoc, S. Yun, J. de Boer, G. Tearney, and B. Bouma, “Phase-resolved optical frequency domain imaging,” Opt. Express13, 5483–93 (2005).
[CrossRef] [PubMed]

S. Makita, Y. Hong, M. Yamanari, T. Yatagai, and Y. Yasuno, “Optical coherence angiography,” Opt. Express14, 7821–40 (2006).
[CrossRef] [PubMed]

A. H. Bachmann, M. L. Villiger, C. Blatter, T. Lasser, and R. A. Leitgeb, “Resonant Doppler flow imaging and optical vivisection of retinal blood vessels,” Opt. Express15, 408–22 (2007).
[CrossRef] [PubMed]

R. K. Wang, S. L. Jacques, Z. Ma, S. Hurst, S. R. Hanson, and A. Gruber, “Three dimensional optical angiography,” Opt. Express15, 4083–97 (2007).
[CrossRef] [PubMed]

J. Fingler, D. Schwartz, C. Yang, and S. E. Fraser, “Mobility and transverse flow visualization using phase variance contrast with spectral domain optical coherence tomography,” Opt. Express15, 12636–53 (2007).
[CrossRef] [PubMed]

M. Szkulmowski, A. Szkulmowska, T. Bajraszewski, A. Kowalczyk, and M. Wojtkowski, “Flow velocity estimation using joint spectral and time-domain optical coherence tomography,” Opt. Express16, 6008–6025 (2008).
[CrossRef] [PubMed]

L. An and R. K. Wang, “In vivo volumetric imaging of vascular perfusion within human retina and choroids with optical micro-angiography,” Opt. Express16, 11438–11452 (2008).
[CrossRef] [PubMed]

R. K. Wang and L. An, “Doppler optical micro-angiography for volumetric imaging of vascular perfusion in vivo,” Opt. Express17, 8926–40 (2009).
[CrossRef] [PubMed]

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Figures (9)

Fig. 1
Fig. 1

Schematic representations of wave vectors and the principal planes of an objective. (a) The axes and definitions used in the model. P and Q are the entrance and exit principal planes of the objective, respectively. (b) A low NA system has a small angular spread of wave vectors; lateral flow is therefore not detected because the flow velocity vector vn is then perpendicular to K = ki(pi) − kd(pd). Equal Gaussian illumination and detection modes are shown in the schematic. (c) Higher NA systems have a larger angular spread of wave vectors and the scalar product is no longer always zero, allowing the detection of lateral flow.

Fig. 2
Fig. 2

(a) Illumination or detection mode in the principal plane for Gaussian mode (blue) and Bessel mode (green) in a system with NA = 0.3. (b) The ratio Mf / M1 normalized to the focal length f for Gaussian and Bessel modes as a function of NA. For lower NA, Mf / M1 approach f. (c) M3 / M1 normalized to f2 as a function of NA. (d) C = 2f2AB normalized to f2 as a function of NA for Gaussian illumination and detection modes. Note that CM3/M1.

Fig. 3
Fig. 3

(a) The velocity vector v of a point scatterer makes an angle α with the optical axis. (b) Equal Gaussian illumination and detection modes for an optical system with f = 16.4 mm. NA = 0.05, 0.3 and 0.5 for the blue, green and red curves respectively. (c) Mean Doppler frequency as a function of α. Parameters: λ = 780 nm, |v| = 1 mm/s, n = 1.33. (d) Standard deviation of the Doppler frequency.

Fig. 4
Fig. 4

(a) Mean Doppler frequency for a system with equal Gaussian illumination and detection modes (Gauss-Gauss). Parameters: NA = 0.05, |v| = 1 mm/s, λ0 = 780 nm, n = 1.33. Color code for the graphs is given in panel b. (b) Standard deviation of the Doppler frequency for different spectral bandwidths Δλ (full-width at half-maximum of Gaussian S(λ)). NA = 0.05. (c) Mean Doppler frequency for a Gauss-Gauss system with NA = 0.3. (d) Standard deviation of the Doppler frequency. NA = 0.3.

Fig. 5
Fig. 5

(a) Mean Doppler frequency as a function of flow velocity angle α. Blue and green curves represent a system with equal Gaussian modes (GG) and a system with Bessel illumination mode and Gaussian detection mode (BG) respectively. Asterisks correspond to values obtained by the numerical simulations shown in panels c and d, and solid lines are calculated using Eq. (27) and Eq. (29). Parameters: NA = 0.3, |v| = 1 mm/s, λ0 = 780 nm, Δλ = 120 nm, n = 1.33. (b) Standard deviation of the Doppler frequency. (c) Doppler spectra for different flow velocity angles α, calculated by numerical simulation for a system with equal Gaussian modes. (d) Doppler spectra for a system with a Bessel illumination mode and Gaussian detection mode, NA = 0.3.

Fig. 6
Fig. 6

(a) Overview of the algorithm used to obtain functional tomograms containing spatially resolved Doppler frequency spectra and flow velocity components. (b) The interference spectra I(λ, x) are acquired in an over-sampled lateral (x) scan. δx represents the lateral resolution. (c) A window is applied around each x-position in the lateral scan such that the windowed data represent time samples Iw(λ, t) at that position. (d) Spatially resolved Doppler spectra Ĩ(z, fD) are obtained after spectral background subtraction, λ to k mapping and 2D FFT. (e) The center of mass μ of the measured Doppler spectrum (blue graph) is calculated and the re-centered data are fitted to Eq. (32) (green graph). The standard deviation σ of the Doppler spectrum is calculated using the fitted curve as weighting function. (f) μ and σ are converted to lateral and axial flow components vt and vz using Eq. (27) and Eq. (29), or their appropriate approximations.

Fig. 7
Fig. 7

(a) xfOCM tomogram of the capillary used for flow measurements, dynamic range is 20dB. The channel has an angle α = 81° and the flow rate is F = 0.45 ml/h. The lateral sampling frequency is 16 times the Nyquist frequency. (b) xfOCM tomogram of the mean Doppler frequencies. (c) xfOCM tomogram of the standard deviation of the Doppler frequencies. 10 B-scans were averaged for the images in (b) and (c).

Fig. 8
Fig. 8

Depth profiles at the center of the capillary (α = 81°) measured with xfOCM of (a) the axial flow component and (b) the lateral flow component, for different flow rates. Parabolae are fitted to the measurements, assuming v = 0 at the capillary wall. The maxima of the parabolae are extracted and compared with the expected velocity set by the syringe pump under laminar flow conditions for (c) the axial flow velocity component and (d) the lateral component. Error bars represent ± standard deviation over 10 measurements.

Fig. 9
Fig. 9

Depth profiles at the center of the capillary (α = 87°) measured with OCT of (a) the axial flow component and (b) the lateral flow component, for different flow rates. The maxima of the parabolae are compared with the expected velocity set by the syringe pump for (c) the axial flow velocity component and (d) the lateral component. Error bars represent ± standard deviation over 10 measurements.

Tables (1)

Tables Icon

Table 1 Approximations and corresponding equations for μ[fD] and σ2[fD].

Equations (32)

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I ( k ) [ U r * ( k ) U m ( k ) ] ,
U in ( r ; k ) = j k 2 π f S ( k ) P m i ( p i ) exp [ j r k i ( p i ) ] d 2 p i ,
α i = a i f , β i = b i f , γ i = 1 α i 2 β i 2 .
U s ( p d ; k ) = V U in ( r ; k ) F ( r ; k ) exp ( j k | r d ( p d ) r | ) | r d ( p d ) r | d 3 r ,
F ( r ; k ) = k 2 4 π χ ( r ) = k 2 4 π ( n 2 ( r ) 1 )
U s ( p d ; k ) e j k f f V U in ( r ; k ) F ( r ; k ) exp [ j r k d ( p d ) ] d 3 r
U m ( k ) = P m d ( p d ) U s ( p d ; k ) d 2 p d .
U m ( k ) k S ( k ) V F ( r ; k ) exp ( 2 j k z ) d 3 r .
F ( r ; t , k ) = F s ( r ; k ) + F d ( r ; t , k ) .
F d ( r ; t , k ) = n = 0 N F d , n ( r v n t ; k ) ,
U m ( t , k ) n = 0 N k S ( k ) V d 3 r 0 [ F d , n ( r 0 ; k ) P m d ( p d ) P m i ( p i ) exp [ j ( r 0 + v n t ) K ] d 2 p i d 2 p d ] ,
F ˜ d , n ( K ) = F d , n ( r 0 ; k ) e j r 0 K d 3 r 0 .
U m ( t , k ) n = 0 N k S ( k ) P m d ( p d ) P m i ( p i ) F ˜ d , n ( K ) exp ( j v n t K ) d 2 p i d 2 p d .
f D ( v n , p d , p i ) = 1 2 π v n K ( p d , p i ) .
m d ( p d ) m i ( p i ) F ˜ d , n ( K ) .
μ [ f D ; k ] = P P f D ( v n , p d , p i ) m d ( p d ) m i ( p i ) d 2 p i d 2 p d P P m d ( p d ) m i ( p i ) d 2 p i d 2 p d ,
σ 2 [ f D ; k ] = μ [ f D 2 ] μ 2 [ f D ] .
μ [ f D ; k ] = n k 2 π f v z ( M f , i M 1 , i + M f , d M 1 , d ) ,
M 1 , i = 0 R ρ m i ( ρ ) d ρ ,
M 3 , i = 0 R ρ 3 m i ( ρ ) d ρ ,
M f , i = 0 R f 2 ρ 2 ρ m i ( ρ ) d ρ ,
μ [ f D ; k ] n v z k / π .
σ 2 [ f D ; k ] = n 2 k 2 4 π 2 f 2 A 2 ( v x 2 + v y 2 ) + n 2 k 2 4 π 2 f 2 C v z 2 ,
A = M 3 , i M 1 , i + M 3 , d M 1 , d , B = M f , i 2 M 1 , i 2 + M f , d 2 M 1 , d 2 , C = 2 f 2 A B .
U ˜ m ( t , l ) = 1 2 π U m ( t , k ) e j 2 k l d k ,
μ [ f D ] = k 3 S ( k ) μ [ f D ; k ] d k k 3 S ( k ) d k .
μ [ f D ] = n 2 π f v z ( M f , i M 1 , i + M f , d M 1 , d ) k 4 S ( k ) d k k 3 S ( k ) d k .
σ 2 [ f D ] = k 3 S ( k ) σ 2 [ f D ; k ] d k k 3 S ( k ) d k + σ k 2 [ μ [ f D ; k ] ] .
σ 2 [ f D ] = n 2 4 π 2 f 2 ( A 2 ( v x 2 + v y 2 ) + C v z 2 ) k 5 S ( k ) d k k 3 S ( k ) d k + ( k 5 S ( k ) d k k 3 S ( k ) d k ( k 4 S ( k ) d k k 3 S ( k ) d k ) 2 ) ( n 2 π f v z ( M f , i M 1 , i + M f , d M 1 , d ) ) 2 .
σ 2 [ f D ] n 2 k 0 2 4 π 2 f 2 ( A 2 ( v x 2 + v y 2 ) + C v z 2 ) T 1 + k σ 2 n 2 v z 2 π 2 T 2 .
σ 2 [ f D ] v t 2 π 2 w 0 2 + v z 2 π 2 l c 2 ,
p ( f D ) = a exp ( f D 2 / b 2 f D 4 / c 4 ) + d .

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