## Quantitative lateral and axial flow imaging with optical coherence microscopy and tomography |

Optics Express, Vol. 21, Issue 15, pp. 17711-17729 (2013)

http://dx.doi.org/10.1364/OE.21.017711

Acrobat PDF (2704 KB)

### 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

## 1. Introduction

1. A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography - principles and applications,” Rep. Prog. Phys. **66**, 239–303 (2003) [CrossRef] .

2. S. Makita, Y. Hong, M. Yamanari, T. Yatagai, and Y. Yasuno, “Optical coherence angiography,” Opt. Express **14**, 7821–40 (2006) [CrossRef] [PubMed] .

8. 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. Express **19**, 12141–55 (2011) [CrossRef] [PubMed] .

9. J. A. Izatt, M. D. Kulkarni, S. Yazdanfar, J. K. Barton, and A. J. Welch, “In vivo bidirectional color Doppler flow imaging of picoliter blood volumes using optical coherence tomography,” Opt. Lett. **22**, 1439–41 (1997) [CrossRef] .

18. S. Zotter, M. Pircher, T. Torzicky, M. Bonesi, E. Götzinger, R. A. Leitgeb, and C. K. Hitzenberger, “Visualization of microvasculature by dual-beam phase-resolved Doppler optical coherence tomography,” Opt. Express **19**, 1217–1227 (2011) [CrossRef] [PubMed] .

14. M. Szkulmowski, A. Szkulmowska, T. Bajraszewski, A. Kowalczyk, and M. Wojtkowski, “Flow velocity estimation using joint spectral and time-domain optical coherence tomography,” Opt. Express **16**, 6008–6025 (2008) [CrossRef] [PubMed] .

19. A. Szkulmowska, M. Szkulmowski, D. Szlag, A. Kowalczyk, and M. Wojtkowski, “Three-dimensional quantitative imaging of retinal and choroidal blood flow velocity using joint spectral and time-domain optical coherence tomography,” Opt. Express **17**, 10584–10598 (2009) [CrossRef] [PubMed] .

20. I. Grulkowski, I. Gorczynska, M. Szkulmowski, D. Szlag, A. Szkulmowska, R. A. Leitgeb, A. Kowalczyk, and M. Wojtkowski, “Scanning protocols dedicated to smart velocity ranging in spectral OCT,” Opt. Express **17**, 23736–54 (2009) [CrossRef] .

16. 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. Express **18**, 2477–94 (2010) [CrossRef] [PubMed] .

22. Y. Zhao, Z. Chen, C. Saxer, Q. Shen, S. Xiang, J. de Boer, and J. Nelson, “Doppler standard deviation imaging for clinical monitoring of in vivo human skin blood flow,” Opt. Lett. **25**, 1358–1360 (2000) [CrossRef] .

24. D. Piao, L. L. Otis, and Q. Zhu, “Doppler angle and flow velocity mapping by combined Doppler
shift and Doppler bandwidth measurements in optical Doppler tomography,”
Opt. Lett. **28**, 1120–1122 (2003) [CrossRef] [PubMed] .

24. D. Piao, L. L. Otis, and Q. Zhu, “Doppler angle and flow velocity mapping by combined Doppler
shift and Doppler bandwidth measurements in optical Doppler tomography,”
Opt. Lett. **28**, 1120–1122 (2003) [CrossRef] [PubMed] .

27. V. J. Srinivasan, H. Radhakrishnan, E. H. Lo, E. T. Mandeville, J. Y. Jiang, S. Barry, and A. E. Cable, “OCT methods for capillary velocimetry,” Biomed. Opt. Express **3**, 612–29 (2012) [CrossRef] [PubMed] .

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

25. S. G. Proskurin, Y. He, and R. K. Wang, “Determination of flow velocity vector based on Doppler
shift and spectrum broadening with optical coherence tomography,”
Opt. Lett. **28**, 1227–1229 (2003) [CrossRef] [PubMed] .

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

44. J. Lee, W. Wu, J. Y. Jiang, B. Zhu, and D. A. Boas, “Dynamic light scattering optical coherence tomography,” Opt. Express **20**, 22262–77 (2012) [CrossRef] [PubMed] .

14. M. Szkulmowski, A. Szkulmowska, T. Bajraszewski, A. Kowalczyk, and M. Wojtkowski, “Flow velocity estimation using joint spectral and time-domain optical coherence tomography,” Opt. Express **16**, 6008–6025 (2008) [CrossRef] [PubMed] .

29. 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. Express **13**(2005) [CrossRef] .

23. 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**, 409–11 (2002) [CrossRef] .

30. R. A. Leitgeb, M. Villiger, A. H. Bachmann, L. Steinmann, and T. Lasser, “Extended focus depth for Fourier domain optical coherence
microscopy,” Opt. Lett. **31**, 2450–2452 (2006) [CrossRef] [PubMed] .

## 2. Theory

### 2.1. Fourier domain coherent imaging for static scattering

*U*is the reference field amplitude and

_{r}*U*the scattered field amplitude measured by the system. Each wavenumber channel can be treated as being monochromatic. The optical system is characterized by its (achromatic) illumination mode

_{m}*m*(

_{i}*a*,

*b*) and detection mode

*m*(

_{d}*a*,

*b*), parametrized in the entrance principal plane with coordinates

**p**= (

*a*,

*b*), as in Fig. 1(a).

31. H. Gross, *Handbook of Optical Systems*, vol. 1 (Wiley-VCH, Weinheim, 2005) [CrossRef] .

**r**measured from the focal point, can be written as a superposition of plane waves with different angular directions [33]: where

*f*is the focal length of the objective,

**k**

*(*

_{i}**p**

*) describes the illumination wave vectors at the exit principal plane and*

_{i}*S*(

*k*) is the spectral envelope of the field. In the notation

*U*(

_{in}**r**;

*k*),

**r**is a variable for the function

*U*whereas

_{in}*k*is a parameter indicating the (monochromatic) wavenumber channel. The direction cosines of

**k**

*are functions of*

_{i}**p**

*= (*

_{i}*a*,

_{i}*b*): The plane waves are scattered upon interaction with the sample. Using the first order Born approximation [34], the scattered field in the principal plane is where

_{i}**r**

*(*

_{d}**p**

*) describes the exit principal plane of the objective (as shown in Fig. 1(a)) and*

_{d}*F*(

**r**;

*k*) is the scattering potential. The scattering potential can be expressed in terms of the sample susceptibility (assumed to be non-dispersive): Because |

**r**

*| =*

_{d}*f*≫ |

**r**|, we can write Finally, the coupling into the detection mode needs to be considered: For a detailed analysis of the tomogram reconstruction from Eq. (7), we refer to the works of Villiger et al. [35

35. M. Villiger and T. Lasser, “Image formation and tomogram reconstruction in optical coherence microscopy,” J. Opt. Soc. Am. A **27**, 2216–2228 (2010) [CrossRef] .

36. C. J. R. Sheppard, M. Roy, and M. D. Sharma, “Image formation in low-coherence and confocal interference
microscopes,” Appl. Opt. **43**, 1493–1502 (2004) [CrossRef] [PubMed] .

**k**

*and*

_{d}**k**

*are parallel to the optical axis, as shown in Fig. 1(b). Then,*

_{i}**k**

*= −*

_{d}**k**

*=*

_{i}*k*

**1**

*and hence the measured scattered field would be: If the scattering potential is taken to be a set of scattering layers, the classical layer model for OCT image formation is retrieved. In this work however, we will start from Eq. (7) without further approximation because the NA is crucial to measure lateral and axial flow components.*

_{z}### 2.2. Fourier domain coherent imaging for dynamic scattering

*F*on the time parameter

_{d}*t*is assumed to be on a much slower time-scale than the period of the light wave, such that the monochromatic approximation leading to Eq. (4) remains valid. A sample containing different ensembles of uniformly moving rigid scatterers can be modeled by: where the

**v**

*are the velocity vectors of the ensembles and*

_{n}*n*runs over the contributing ensembles. To evaluate the volume integral in Eq. (6) for each of the ensembles of moving scatterers, we can adopt a coordinate system

**r**

_{0}=

**r**−

**v**

*that is moving along with the ensemble. The time-dependent part*

_{n}t*U*(

_{m}*t*;

*k*) of the measured field is then: where

**K**≡

**K**(

**p**

*,*

_{d}**p**

*) =*

_{i}**k**

*(*

_{i}**p**

*)−*

_{i}**k**

*(*

_{d}**p**

*). In Eq. (11), the Fourier transformed scattering potential can be identified: From the measurement of the field*

_{d}*U*, OCT systems attempt to reconstruct the depth resolved scattering potential of the sample at each position in a lateral raster scan. Typically, an inverse Fourier transform of Eq. (1) is calculated over

_{m}*k*to this end. For this reconstruction to be accurate at some distance from the focal point, either the range of

**K**(

**p**

*,*

_{d}**p**

*) must be limited by*

_{i}*m*(

_{d}**p**

*) and*

_{d}*m*(

_{i}**p**

*) such that*

_{i}**r**

_{0}·

**K**effectively measures the optical path length along the depth direction (i.e.

**r**

_{0}·

**K**≈ 2

*zk*), or advanced reconstruction algorithms need to be applied. The former solution is valid within good approximation in the focal volume of low NA or extended focus systems [35

35. M. Villiger and T. Lasser, “Image formation and tomogram reconstruction in optical coherence microscopy,” J. Opt. Soc. Am. A **27**, 2216–2228 (2010) [CrossRef] .

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

39. C. J. R. Sheppard, S. Kou, and C. Depeursinge, “Reconstruction in interferometric synthetic aperture microscopy: comparison with optical coherence tomography and digital holographic microscopy,” J. Opt. Soc. Am. A **29**, 244–250 (2012) [CrossRef] .

*m*(

_{d}**p**

*) and*

_{d}*m*(

_{i}**p**

*) can be interpreted as distribution functions for the variables*

_{i}**p**

*and*

_{d}**p**

*. Integration over these variables creates the Doppler spectrum as follows. For each realization of*

_{i}**p**

*and*

_{d}**p**

*, the corresponding illumination and detection wave vectors can be found, as shown in Fig. 1(a). Their direction cosines are given by Eq. (3). Every such pair of illumination and detection wave vectors determines a Doppler frequency shift per wavenumber channel: The weight of each frequency in the Doppler spectrum is then given by: In the extreme case of zero NA shown in Fig. 1(b), we find that the distribution reduces to a single Doppler frequency, as expected. Moreover, lateral flow will not be detected in that case, since*

_{i}**v**

*·*

_{n}**K**= 0. Conversely, when the NA is increased,

**K**spans a range of angles due to the spread of wave vectors in the illumination and detection modes, as shown in Fig. 1(c). Therefore, a broadened Doppler frequency distribution will be seen and lateral flow can be measured. An analysis of the Doppler spectrum allows the measurement of both flow components, as will now be shown.

### 2.3. Doppler spectrum per wavenumber channel: mean and standard deviation

*μ*[

*f*;

_{D}*k*] and standard deviation

*σ*[

*f*;

_{D}*k*] of the Doppler frequency are evaluated: For the sake of simplicity, a single point scatterer has been taken into account (

*F̃*(

_{d,n}**K**) ∝

*k*

^{2}exp[−

*j*

**r**

*·*

_{n}**K**]). In other words, back-scattering is approximated to be uniform within the system’s aperture. The model in principle also allows different types of scatterers by incorporating

*F̃*(

_{d,n}**K**) in the Doppler frequency weight, but such an analysis is beyond the scope of this work. Since most optical systems exhibit rotationally symmetric illumination and detection modes, polar coordinates (

*ρ*,

*θ*) will be used to calculate the integrals in the entrance principal plane. Considering one ensemble of point scatterers with velocity vector

**v**= (

*v*,

_{x}*v*,

_{y}*v*), we find for the mean Doppler frequency for one wavenumber channel: where

_{z}*n*is the refractive index of the medium and the following notation for integrals of the illumination mode has been adopted: with

*R*the radius of the entrance pupil. Analogous notation is used for integrals of the detection mode. Eq. (18) confirms that the mean Doppler frequency is determined solely by the axial flow component. Moreover, when a moderate NA objective is used,

*m*(

*ρ*) is significantly different from zero only for

*ρ*≪

*f*and therefore

*M*≈

_{f}*fM*

_{1}. This approximation is further verified in Fig. 2(b). The mean Doppler frequency then reduces to the known formula:

*M*/

_{f}*M*

_{1}and

*M*

_{3}/

*M*

_{1}, are shown in Fig. 2 for Gaussian and Bessel modes. The dependence of the Doppler broadening on

*A*,

*f*and

*B*can be interpreted as follows. As mentioned in the introduction, it has been shown that broadening of the Doppler spectrum due to beam geometry is equivalent to broadening due to the finite transit time of scatterers moving through the focal volume [28

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

*A*, as shown in Fig. 2(c). This explains the presence of

*A*in the first term of Eq. (23). The second term measures broadening due to pure axial flow, which is caused by the angular spread of wave vectors with respect to the axial component of the velocity vector. Wave vectors with different angles measure different Doppler frequencies. Equivalently, this can be interpreted by the finite depth of focus of the system, which limits the focal volume axially. The depth of focus therefore creates Doppler broadening due to the finite axial transit time of scatterers moving through the focal volume. For high NA Gaussian systems, the depth of focus will be reduced and the factor

*C*increases, as shown in Fig. 2(d). However, this effect is only of importance when the flow is mostly axial, since

*C*≪

*A*/2, as can be seen by comparing Fig. 2(c) with (d). Note that for extended focus systems with ideal Bessel illumination and detection modes (

*m*(

_{i,d}*ρ*) =

*δ*(

*ρ*−

*ρ*

_{0})), we find

*C*= 0 for all NA.

*z*-component of

**K**for higher NA. On the other hand, a higher NA increases the Doppler broadening, as evidenced in Fig. 3(d). When a typical OCT system of NA = 0.05 is compared to a system with NA = 0.3, the Doppler broadening for lateral flow is increased about sixfold. Fig. 3(d) confirms that even for pure axial flow, some Doppler frequency broadening remains, especially for higher NA. This is caused by the second term of Eq. (23), as explained above.

### 2.4. Spatially resolved Doppler spectrum

14. M. Szkulmowski, A. Szkulmowska, T. Bajraszewski, A. Kowalczyk, and M. Wojtkowski, “Flow velocity estimation using joint spectral and time-domain optical coherence tomography,” Opt. Express **16**, 6008–6025 (2008) [CrossRef] [PubMed] .

*l*represents optical path length, and the factor 2 takes into account the double path due to the reflection configuration. As discussed above, we will assume that

*Ũ*(

_{m}*t*,

*l*) correctly represents the depth resolved sample structure. The Doppler spectrum can be estimated from the Fourier transform of Eq. (25) along

*t*. After this Fourier transform, we have a measurement of the Doppler spectrum for each depth resolution. Indeed, the integral over

*k*in Eq. (25) selects one resolution volume at depth

*l*. More details on the full algorithm are given in section 3.4. The integration over the

*k*-spectrum will cause the Doppler frequency distribution to broaden. Indeed,

*f*in Eq. (14) depends linearly on

_{D}*k*, the scattering potential in the Doppler frequency weight in Eq. (15) contains the factor

*k*

^{2}and upon inspection of Eq. (13), it is seen that the additional weighting factor

*kS*(

*k*) must be taken into account. The effect on the weighted mean and standard deviation of the Doppler frequency for a point scatterer as calculated in section 2.3, can now be evaluated.

*μ*[

*f*;

_{D}*k*], As before, for moderate NA,

*M*≈

_{f}*fM*

_{1}. When

*S*(

*k*) is a Gaussian function with mean

*k*

_{0}and standard deviation

*k*, the moments of

_{σ}*S*(

*k*) can easily be calculated. Moreover,

*S*(

*k*) to

*k*

_{0}. With these two approximations, Eq. (27) reduces again to

*nv*

_{z}k_{0}/

*π*.

*σ*

^{2}[

*f*;

_{D}*k*],

*S*(

*k*) to be Gaussian with central wavenumber

*k*

_{0}, standard deviation

*k*and

_{σ}*M*≈

_{f}*fM*

_{1}for moderate NA we find: The two terms of Eq. (30) can be interpreted as follows. The first term

*T*

_{1}is the standard deviation for a single wavenumber channel at the central wavenumber as given by Eq. (23); it accounts for broadening due to the angular spread of incident and scattered wave vectors, as discussed above. Note that

*T*

_{1}is independent of spectral width. The second term

*T*

_{2}measures an extra broadening due to the axial flow component, and depends on the spectral width

*k*. It can be interpreted using the transit time argument: a larger spectral bandwidth creates a higher axial resolution, resulting in a shorter axial transit time, and therefore a larger Doppler broadening. Indeed, the axial resolution is given by the coherence length

_{σ}*l*= 1/(

_{c}*nk*).

_{σ}### 2.5. Approximations for Gaussian modes and low NA

23. 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**, 409–11 (2002) [CrossRef] .

24. D. Piao, L. L. Otis, and Q. Zhu, “Doppler angle and flow velocity mapping by combined Doppler
shift and Doppler bandwidth measurements in optical Doppler tomography,”
Opt. Lett. **28**, 1120–1122 (2003) [CrossRef] [PubMed] .

27. V. J. Srinivasan, H. Radhakrishnan, E. H. Lo, E. T. Mandeville, J. Y. Jiang, S. Barry, and A. E. Cable, “OCT methods for capillary velocimetry,” Biomed. Opt. Express **3**, 612–29 (2012) [CrossRef] [PubMed] .

*w*

_{0}≪

*R*at the principal plane. Under this condition,

*C*≈ 0. The beam waist after focussing by the objective is then

*w′*

_{0}= 2

*f*/ (

*nk*

_{0}

*w*

_{0}), yielding for the Doppler broadening: where

23. 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**, 409–11 (2002) [CrossRef] .

**28**, 1120–1122 (2003) [CrossRef] [PubMed] .

27. V. J. Srinivasan, H. Radhakrishnan, E. H. Lo, E. T. Mandeville, J. Y. Jiang, S. Barry, and A. E. Cable, “OCT methods for capillary velocimetry,” Biomed. Opt. Express **3**, 612–29 (2012) [CrossRef] [PubMed] .

### 2.6. Doppler spectrum: numerical simulation

**p**

*for the illumination mode and*

_{i}**p**

*for the detection mode. For each combination of*

_{d}**p**

*and*

_{i}**p**

*, the Doppler frequency and its weight are calculated using Eq. (14) and Eq. (15), including the spectral weighting factor for point scatterers*

_{d}*k*

^{3}

*S*(

*k*) for a Gaussian spectrum. This calculation is repeated for each wavenumber channel. The resulting data are then binned according to their frequency and represented in a weighted histogram. Figure 5 reveals how the Doppler spectrum changes as a function of flow angle for a system with equal Gaussian modes and a system with a Bessel illumination mode and Gaussian detection mode. It is seen that the Bessel-Gauss configuration slightly increases the Doppler broadening for the same NA.

## 3. Methods

### 3.1. xfOCM

*λ*

_{0}= 780 nm, Δ

*λ*= 120 nm) is collimated and split into reference and sample beams. In the sample arm, the beam passes through an axicon lens and a telescope. The resulting Bessel-like illumination beam is guided through a relayed beam scanning system. The scanned beam enters a microscope stage, consisting of a tube lens and a 10× Zeiss Neofluar objective with a NA of 0.3. The Bessel-like illumination beam provides a uniform, high lateral resolution (1.3 μm) over an extended depth of field (400 μm). The axial resolution in tissue is 2.5 μm. The system operates with a Bessel illumination mode and a Gaussian detection mode, as shown in Fig. 2(a). A custom-built spectrometer with a high-speed line detector (Basler Sprint spL4096-140km) records the interference spectrum. For the flow measurements presented here, an A-scan rate of 20 kHz was used.

### 3.2. OCT

*λ*

_{0}= 780 nm, Δ

*λ*= 120 nm). The sample objective lens has a focal length of

*f*= 25 mm. The Gaussian mode has a full width at half maximum of 2.4 mm in the back focal plane of this lens, creating a lateral resolution of 10 μm. A custom-built spectrometer with a high-speed line detector (Basler Sprint spL4096-140km) records the interference spectrum. For the flow measurements, an A-scan rate of 100 kHz was used. More details about this set-up can be found in a previous publication [40

40. M. Szkulmowski, I. Gorczynska, D. Szlag, M. Sylwestrzak, and M. Wojtkowski, “Efficient reduction of speckle noise in Optical Coherence Tomography,” Opt. Express **20**, 15149–15169 (2012) [CrossRef] .

### 3.3. Flow system

^{4}μm

^{2}for xfOCM and 3.14 × 10

^{4}μm

^{2}for OCT experiments. A syringe pump allows to set a constant flow speed through the capillary. The system is filled with a solution of polystyrene beads (diameter 0.2 μm) in water for the xfOCM-based measurements and with an intralipid solution for the OCT-based measurements.

## 4. Experimental results

*v*is the true transverse flow velocity,

_{t}*v̂*the transverse velocity measured from the Doppler broadening and

_{t}*v*the scan speed. Another solution would be to make

_{s}*v*≪

_{s}*v*, or to stop the scan at each lateral position.

_{t}### 4.1. xfOCM

*μ*[

*f*] and standard deviation

_{D}*σ*[

*f*] of the Doppler frequency for a flow rate of

_{D}*F*= 0.45 ml/h and angle

*α*= 81°. As can be seen in 7(a), the capillary is somewhat asymmetric, with an elliptic cross-section. The axial diameter is 200 μm, but the lateral diameter is larger, accounting for a larger cross-section. The intensity gradient over the depth of the flow channel seen in 7(a) is due to three reasons. Firstly, although the extended focus maintains the lateral resolution over 400 μm, it does not have constant intensity over the whole depth, a well-known property of Bessel beams [42

42. Z. Ding, H. Ren, Y. Zhao, J. S. Nelson, and Z. Chen, “High-resolution optical coherence tomography over a large depth range with an axicon lens,” Opt. Lett. **27**, 243–245 (2002) [CrossRef] .

*μ*[

*f*] and

_{D}*σ*[

*f*] are converted to lateral and axial flow profiles as a function of depth, as shown in Fig. 8(a) and (b) for the center of the capillary. By assuming laminar flow, the volume flow rate

_{D}*F*set by the syringe pump is converted to flow velocities in the capillary. The maximum velocity

*v*of the parabolic flow profile in a cylindrical capillary is then found from

_{max}*v*= 2

_{max}*F/S*, with

*S*the measured cross-section of the capillary. The flow velocities set by the syringe pump can now be compared to the velocities measured using our model and algorithm, as shown in Fig. 8(c) and (d) for an angle

*α*= 81°. The measurements of both axial and lateral flow components are seen to be consistent with the values expected from the flow system parameters. The measured flow speeds in Fig. 8(a) and (b) show some deviation from the parabolic flow profiles most likely due to experimental noise and/or deviations from the supposed laminar flow profile. Also, note that when a particular axial velocity is higher than the expected value, the corresponding transverse velocity component is higher as well, indicating that these errors are most probably due to an inaccuracy in the actual flow rates set by the syringe pump.

### 4.2. OCT

*α*= 87°. Due to the lower lateral resolution of this system (i.e. smaller Doppler broadening), it is not as sensitive to slow lateral flow velocities as xfOCM. Hence, the flow rates used for these measurements were about ten times higher and the flow angle smaller to create larger lateral flow velocities. Nevertheless, the measurements of both axial and lateral flow components in Fig. 9(c) and (d) are again seen to match the values expected from the flow system parameters.

## 5. Discussion & conclusion

**27**, 409–11 (2002) [CrossRef] .

**28**, 1120–1122 (2003) [CrossRef] [PubMed] .

**3**, 612–29 (2012) [CrossRef] [PubMed] .

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

*F̃*(

_{d,n}**K**) as a weight in the calculation of the mean and standard deviation of the Doppler spectrum. In this way, the Doppler spectrum created by moving erythrocytes in blood could be investigated.

**16**, 6008–6025 (2008) [CrossRef] [PubMed] .

41. A. Chan, E. Lam, and V. J. Srinivasan, “Comparison of kasai autocorrelation and maximum likelihood estimators for Doppler optical coherence tomography,” IEEE T. Med. Imaging (2013) [CrossRef] .

20. I. Grulkowski, I. Gorczynska, M. Szkulmowski, D. Szlag, A. Szkulmowska, R. A. Leitgeb, A. Kowalczyk, and M. Wojtkowski, “Scanning protocols dedicated to smart velocity ranging in spectral OCT,” Opt. Express **17**, 23736–54 (2009) [CrossRef] .

**3**, 612–29 (2012) [CrossRef] [PubMed] .

43. 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] .

## Supporting information

## Acknowledgments

## References and links

1. | A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography - principles and applications,” Rep. Prog. Phys. |

2. | S. Makita, Y. Hong, M. Yamanari, T. Yatagai, and Y. Yasuno, “Optical coherence angiography,” Opt. Express |

3. | R. K. Wang, S. L. Jacques, Z. Ma, S. Hurst, S. R. Hanson, and A. Gruber, “Three dimensional optical angiography,” Opt. Express |

4. | 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. Express |

5. | A. Mariampillai, B. A. Standish, E. H. Moriyama, M. Khurana, N. R. Munce, M. 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. |

6. | L. An and R. K. Wang, “In vivo volumetric imaging of vascular perfusion within human retina and choroids with optical micro-angiography,” Opt. Express |

7. | V. J. Srinivasan, J. Jiang, M. Yaseen, H. Radhakrishnan, W. Wu, S. Barry, A. Cable, and D. Boas, “Rapid volumetric angiography of cortical microvasculature with optical coherence tomography,” Opt. Lett. |

8. | 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. Express |

9. | J. A. Izatt, M. D. Kulkarni, S. Yazdanfar, J. K. Barton, and A. J. Welch, “In vivo bidirectional color Doppler flow imaging of picoliter blood volumes using optical coherence tomography,” Opt. Lett. |

10. | 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. |

11. | 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. |

12. | B. Vakoc, S. Yun, J. de Boer, G. Tearney, and B. Bouma, “Phase-resolved optical frequency domain imaging,” Opt. Express |

13. | 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. Express |

14. | M. Szkulmowski, A. Szkulmowska, T. Bajraszewski, A. Kowalczyk, and M. Wojtkowski, “Flow velocity estimation using joint spectral and time-domain optical coherence tomography,” Opt. Express |

15. | R. K. Wang and L. An, “Doppler optical micro-angiography for volumetric imaging of vascular perfusion in vivo,” Opt. Express |

16. | 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. Express |

17. | B. Baumann, B. Potsaid, M. Kraus, J. Liu, D. Huang, J. Hornegger, A. Cable, J. Duker, and J. Fujimoto, “Total retinal blood flow measurement with ultrahigh speed swept source/Fourier domain OCT,” Biomed. Opt. Express |

18. | S. Zotter, M. Pircher, T. Torzicky, M. Bonesi, E. Götzinger, R. A. Leitgeb, and C. K. Hitzenberger, “Visualization of microvasculature by dual-beam phase-resolved Doppler optical coherence tomography,” Opt. Express |

19. | A. Szkulmowska, M. Szkulmowski, D. Szlag, A. Kowalczyk, and M. Wojtkowski, “Three-dimensional quantitative imaging of retinal and choroidal blood flow velocity using joint spectral and time-domain optical coherence tomography,” Opt. Express |

20. | I. Grulkowski, I. Gorczynska, M. Szkulmowski, D. Szlag, A. Szkulmowska, R. A. Leitgeb, A. Kowalczyk, and M. Wojtkowski, “Scanning protocols dedicated to smart velocity ranging in spectral OCT,” Opt. Express |

21. | J. Walther and E. Koch, “Enhanced joint spectral and time domain optical coherence tomography for quantitative flow velocity measurement,” in “ |

22. | Y. Zhao, Z. Chen, C. Saxer, Q. Shen, S. Xiang, J. de Boer, and J. Nelson, “Doppler standard deviation imaging for clinical monitoring of in vivo human skin blood flow,” Opt. Lett. |

23. | 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. |

24. | D. Piao, L. L. Otis, and Q. Zhu, “Doppler angle and flow velocity mapping by combined Doppler
shift and Doppler bandwidth measurements in optical Doppler tomography,”
Opt. Lett. |

25. | S. G. Proskurin, Y. He, and R. K. Wang, “Determination of flow velocity vector based on Doppler
shift and spectrum broadening with optical coherence tomography,”
Opt. Lett. |

26. | L. Yu and Z. Chen, “Doppler variance imaging for three-dimensional retina and choroid angiography,” J. Biomed. Opt. |

27. | V. J. Srinivasan, H. Radhakrishnan, E. H. Lo, E. T. Mandeville, J. Y. Jiang, S. Barry, and A. E. Cable, “OCT methods for capillary velocimetry,” Biomed. Opt. Express |

28. | R. V. Edwards, “Spectral analysis of the signal from the laser Ddoppler flowmeter: Time-independent systems,” J. Appl. Phys. |

29. | 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. Express |

30. | R. A. Leitgeb, M. Villiger, A. H. Bachmann, L. Steinmann, and T. Lasser, “Extended focus depth for Fourier domain optical coherence
microscopy,” Opt. Lett. |

31. | H. Gross, |

32. | M. Gu, |

33. | J. W. Goodman, |

34. | M. Born and E. Wolf, |

35. | M. Villiger and T. Lasser, “Image formation and tomogram reconstruction in optical coherence microscopy,” J. Opt. Soc. Am. A |

36. | C. J. R. Sheppard, M. Roy, and M. D. Sharma, “Image formation in low-coherence and confocal interference
microscopes,” Appl. Opt. |

37. | T. S. Ralston, D. L. Marks, P. Scott Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nature Phys. |

38. | S. G. Adie, B. W. Graf, A. Ahmad, P. S. Carney, and S. A. Boppart, “Computational adaptive optics for broadband optical interferometric tomography of biological tissue,” PNAS |

39. | C. J. R. Sheppard, S. Kou, and C. Depeursinge, “Reconstruction in interferometric synthetic aperture microscopy: comparison with optical coherence tomography and digital holographic microscopy,” J. Opt. Soc. Am. A |

40. | M. Szkulmowski, I. Gorczynska, D. Szlag, M. Sylwestrzak, and M. Wojtkowski, “Efficient reduction of speckle noise in Optical Coherence Tomography,” Opt. Express |

41. | A. Chan, E. Lam, and V. J. Srinivasan, “Comparison of kasai autocorrelation and maximum likelihood estimators for Doppler optical coherence tomography,” IEEE T. Med. Imaging (2013) [CrossRef] . |

42. | Z. Ding, H. Ren, Y. Zhao, J. S. Nelson, and Z. Chen, “High-resolution optical coherence tomography over a large depth range with an axicon lens,” Opt. Lett. |

43. | 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. |

44. | J. Lee, W. Wu, J. Y. Jiang, B. Zhu, and D. A. Boas, “Dynamic light scattering optical coherence tomography,” Opt. Express |

45. | M. Villiger, C. Pache, and T. Lasser, “Dark-Field optical coherence microscopy,” Opt. Lett. |

**OCIS Codes**

(170.4500) Medical optics and biotechnology : Optical coherence tomography

(170.6900) Medical optics and biotechnology : Three-dimensional microscopy

(280.2490) Remote sensing and sensors : Flow diagnostics

**ToC Category:**

Medical Optics and Biotechnology

**History**

Original Manuscript: April 17, 2013

Revised Manuscript: June 29, 2013

Manuscript Accepted: July 12, 2013

Published: July 17, 2013

**Virtual Issues**

Vol. 8, Iss. 8 *Virtual Journal for Biomedical Optics*

**Citation**

Arno Bouwens, Daniel Szlag, Maciej Szkulmowski, Tristan Bolmont, Maciej Wojtkowski, and Theo Lasser, "Quantitative lateral and axial flow imaging with optical coherence microscopy and tomography," Opt. Express **21**, 17711-17729 (2013)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-21-15-17711

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### References

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- 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]
- A. Mariampillai, B. A. Standish, E. H. Moriyama, M. Khurana, N. R. Munce, M. 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, 1530–2 (2008). [CrossRef] [PubMed]
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- M. Szkulmowski, I. Gorczynska, D. Szlag, M. Sylwestrzak, and M. Wojtkowski, “Efficient reduction of speckle noise in Optical Coherence Tomography,” Opt. Express20, 15149–15169 (2012). [CrossRef]
- A. Chan, E. Lam, and V. J. Srinivasan, “Comparison of kasai autocorrelation and maximum likelihood estimators for Doppler optical coherence tomography,” IEEE T. Med. Imaging (2013). [CrossRef]
- Z. Ding, H. Ren, Y. Zhao, J. S. Nelson, and Z. Chen, “High-resolution optical coherence tomography over a large depth range with an axicon lens,” Opt. Lett.27, 243–245 (2002). [CrossRef]
- 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. Lee, W. Wu, J. Y. Jiang, B. Zhu, and D. A. Boas, “Dynamic light scattering optical coherence tomography,” Opt. Express20, 22262–77 (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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