Imaging and quantifying transverse flow velocity with the Doppler bandwidth in a phase-resolved functional optical coherence tomography.

The Doppler bandwidth extracted from the standard deviation of the frequency shift in phase-resolved func- tional optical coherence tomography (F-OCT) was used to image the velocity component that is transverse to the optical probing beam. It was found that above a certain threshold level the Doppler bandwidth is a linear function of flow velocity and that the effective numerical aperture of the optical objective in the sample arm determines the slope of this dependence. The Doppler bandwidth permits accurate measurement of flow velocity without the need for precise determination of flow direction when the Doppler flow angle is within 6 15 ± perpendicular to the probing beam. Such an approach extends the dynamic range of flow velocity measurements obtained with the phase-resolved F-OCT. © 2002 Optical Society America

Optical Doppler tomography (ODT) combines Doppler velocimetry with optical coherence tomography 1 (OCT) for noninvasive localization and measurement of particle-f low velocity in highly scattering media with micrometer-scale spatial resolution. 2 -6 ODT uses a low-coherence source and an optical interferometer to obtain high-spatial-resolution gating of photons at a user-specified depth in biological tissues or other turbid media. In combination with a high-speed scanning device such as a rapid-scanning optical delay line, 7,8 ODT permits ranging of microstructure and particle motion. Several ODT algorithms and hardware schemes have been developed to detect the Doppler frequency shift produced by moving particles. The most straightforward method for determining frequency shift is to use a short-time fast Fourier transformation. 2 -4 However, the sensitivity of this method depends mainly on the fast Fourier transform time window, which limits axial scanning speed and spatial resolution when one is measuring slow-moving blood f low in small vessels. A phase-resolved technique 5,6 can decouple the Doppler sensitivity and spatial resolution while maintaining high axial scanning speed. A previous Letter 6 on phase-resolved functional OCT (F-OCT) demonstrated how one can use the standard deviation of the Doppler spectrum to locate the microvasculature. In this Letter we demonstrate that one can use the spectral bandwidth of the Doppler frequency shift (Doppler bandwidth) to provide quantitative information on f low velocity.
There are a number of sources, including velocity gradient, turbulence, Brownian motion, speckle, and probe-beam geometry, that contribute to broadening of the Doppler spectrum. When f low velocity is low, Brownian motion dominates the broadening of the Doppler spectrum. When velocity is high, probe-beam geometry dominates. The contribution of focusing-beam geometry can be derived from geometrical optics, as shown in Fig. 1. When the Doppler angle is larger than tan 21 ͑2w͞l c ͒, where w is the waist radius of the Gaussian optical beam at the focal point and l c is the source coherence length, the Doppler bandwidth is determined by the difference between the two extreme average Doppler frequency shifts f a and f b , caused by the two optical rays at the two outer boundaries of the probe beam: where B d is the Doppler bandwidth def ined by geometrical optics, V is the f low velocity, f is the optical aperture angle, u is the Doppler angle, l is the center wavelength of the light source, and NA eff is the effective numerical aperture. For a Gaussian optical beam, the Doppler bandwidth B 1/e (full width at 1͞e of maximum spectrum amplitude) is the inverse of the transit time spent by particles passing through the focus zone: Considering the relationship between standard deviation s and the Doppler bandwidth for a Gaussian optical beam, If contributions from Brownian motion and other sources that are independent of the macroscopic f low velocity are included, Eq. (4) can be modif ied as where b accounts for spectrum broadening owing to Brownian motion, velocity gradient, turbulence, and the speckle nature of the F-OCT signal. Equation (5) indicates that, above a threshold value, the Doppler bandwidth is a linear function of f low velocity. The effective numerical aperture NA eff of the optical objective in the sample arm determines the slope of this linear dependence. When the f low velocity is much higher than this threshold value and NA eff is known, the linear dependence of the Doppler bandwidth permits accurate measurement of the f low velocity that is transverse to the optical axis of the probe beam.
The experimental system used for Doppler bandwidth measurements was described previously. 5,6 The interferometer uses a low-coherence amplified spontaneous emission broadband source whose output power, center wavelength, and bandwidth are 5 mW, 1300 nm, and 65 nm, respectively. The source light is coupled into the interferometer and split into reference and sample arms by a 3-dB 2 3 2 coupler. A rapid-scanning optical delay line in a group-delay scanning mode is used for A-line scanning at 500 Hz. An electro-optical phase modulator is used in the reference arm to generate a 500-kHz carrier frequency. In the sample arm shown in Fig. 1, interchangeable optical collimator and optical objective (203; numerical aperture, 0.35) are used to focus the beam at the center of a capillary tube with an inner diameter of 900 mm for M-mode imaging. A 0.1% Intralipid solution composed of particles of 0.356-mm diameter is used as the turbid liquid medium. A syringe pump driven by a high-resolution translation stage controls the f low of the Intralipid solution through the capillary tube. In the detection arm of the interferometer, the signal from the photodetector is amplified with a bandpass preamplifier and then sent to a 12-bit analog-to-digital converter and a data-acquisition board sampling at 5 MHz. The number of data points for each A-line data acquisition is 4096.
The dependence of the Doppler bandwidth of the Intralipid f low at the center of the capillary tube on numerical aperture NA eff , Doppler f low angle u, and f low velocity V was measured. The standard deviation prof iles for Intralipid f low in the capillary tube for three different f low velocities are shown in Fig. 2 (NA eff 0.09 and u 77 ± ). The background b value for these curves is approximately 72 Hz. The measured and predicted standard deviations of the Doppler spectrum as a function of f low velocity are shown in Fig. 3 (NA eff 0.09 and u 77 ± ). We extracted the data from the center of the tube that corresponds to the top of the parabolic f low prof ile to study the relationship between Doppler bandwidth and velocity. The standard deviation of the Doppler spectrum from experimental measurements (open circles for NA eff 0.09 and open triangles for NA eff 0.05) and the theoretical predications from Eq. (5) (solid curves) are in good agreement for f low velocities larger than 300 mm͞s. At low f low velocities, the standard deviation is constant because the Doppler bandwidth is dominated by broadening caused by Brownian motion, which is independent of f low velocity. For velocities higher than 300 mm͞s, the Doppler bandwidth is dominated by broadening that is due to the probe beam's geometry. The Doppler bandwidth is a linear function of velocity, with the slope determined by NA eff . The measured and predicated standard deviation values are shown as a function of Doppler angles when the f low velocity is 698 mm͞s in Fig. 4. The results indicate that the Doppler bandwidth is insensitive to variation in the Doppler angles when the angle is within 615 ± perpendicular to the probe beam.
Phase-resolved F-OCT increases the sensitivity of f low velocity detection by more than 2 orders of   magnitude while at the same time increasing the imaging frame rate. However, F-OCT has a limited dynamic range because of an aliasing phenomenon caused by 2p ambiguity in the arctangent function. The maximum unambiguous velocity V max fd detected by F-OCT for a given A-line scan frequency f s is given by The f low velocity measured by the Doppler bandwidth has a larger dynamic range than that measured by direct Doppler shift. The standard deviation is similarly limited by the aliasing phenomenon described by s # f s . However, because s is only a small fraction of the average Doppler frequency shift encountered in most clinical situations, the unambiguous detection range of velocities for the Doppler bandwidth method can be as much as 10-20 times larger than that of the corresponding phase-resolved method with the Doppler angle in the range of 70 ± -120 ± . Such an increase in dynamic range was confirmed by our experiments. As shown in Fig. 3, there is no aliasing phenomenon even though the velocity has reached 960 mm͞s, which is much higher than the measurable velocity range.
In summary, Doppler bandwidth can be used to measure f low velocity that is transverse to the probing beam. Above a threshold value, the Doppler bandwidth is linearly dependent on f low velocity, and the effective numerical aperture of the optical objective determines the slope of this dependence. When the Doppler f low angle is within 615 ± perpendicular to the probe beam, the average Doppler frequency shift is highly sensitive to angle position, but the Doppler bandwidth is insensitive to f low direction. Therefore the linear dependence of f low velocity on Doppler bandwidth permits accurate measurement of the f low velocity without precise determination of f low direction.