Ординатура / Офтальмология / Английские материалы / Handbook of Optical Coherence Tomography_Bouma, Tearney_2002

yellow or cyan, respectively [19]). In work completed, Doppler OCT variance images are presented separately.

8.3DOPPLER OCT OF IN VITRO SAMPLES

Early demonstration and development of Doppler OCT required testing and experimentation using samples in vitro. These studies and experiments improved our understanding of the effect of instrumentation parameters and optical properties of the media on the measured Doppler OCT signal. In this section we first describe results of a study using a Monte Carlo simulation of the light detection process in an OCT system to identify factors that affect the Doppler signal current, id ðtÞ. Then we indicate some of the early imaging experiments completed using low speed (100–200 s/image) scanning systems that demonstrated the feasibility of using Doppler OCT to image flow in turbid media with optical properties similar to those of tissue.

8.3.1Parameters Influencing Doppler Frequency Spectra

Better understanding of how instrumentation parameters and medium properties affect the Doppler signal current ½id ðtÞ& may be gained by Monte Carlo simulation of light propagation in the turbid medium under consideration. The results presented in this section are based on Monte Carlo simulations of an experiment involving an in vitro phantom, where blood flow is confined to a vessel submerged at a fixed depth and oriented parallel to the air/tissue interface in a scattering medium [20]. Simulated photons are launched from the probe in such a way that the numerical aperture is filled with a Gaussian intensity profile. The probe acts as a confocal detector, because the antenna theorem for a heterodyne receiver limits the effective size of an interferometric receiver to twice the diameter of a confocal detector [21,22]. This feature is implemented through the confocality angle . Intralipid was chosen as a model medium that could easily be used experimentally to verify predictions resulting from the Monte Carlo study. Both Intralipid and blood were modeled as homogeneous media, each characterized by three optical parameters: the absorption coefficient a, the scattering coefficient s, and the anisotropy parameter g. Based

a ¼ 0:75 mm , s ¼ 150 mm , g ¼ 0:99.

The geometric model representing the combined emission and detection probe of a fiber-optic Doppler OCT instrument is shown in Fig. 1. The probe is pointed at an incidence angle of (typically 15 ), and we simulate Doppler OCT measurements of blood flow in a vessel immersed at an axial depth of zo ¼ 250 m in the scattering medium. The probe is aimed in the direction of point P on the z axis, but owing to refraction at the surface of the medium the focus is shifted to point P0 at depth z0, where z0 ¼ cz. The probe has a numerical aperture NA ¼ sin , defined by the focusing lens L and the divergence of the light from the fiber core F. The surfaces S1 and S2 define the spherical wave fronts of the beam diverging from the fiber tip and being focused by the lens L to converge toward point P (in air). Photons that are backscattered from the turbid medium meet three criteria for detection: (1) They intersect the surface S2 defined by the numerical aperture; (2) they deviate from normal

Figure 1 Geometry of Doppler OCT flow phantom for Monte Carlo simulations. A 100m diameter wall-less vessel is positioned horizontally in a 2% Intralipid solution at an axial depth of 250 m. The y axis is taken to be parallel to the vessel axis. The probe is oriented at an incidence angle of at various focus positions z0 on the z axis through the center of the vessel lumen. Blood flow is assumed to have a parabolic velocity profile with on-axis velocity Vo.

incidence on S2 by less than the confocality angle ; and (3) they have a total optical path length that deviates by less than the source coherence length Lc from the nominal pathlength for a photon launched from S2 and scattered from P0 back to S2 in a single backscattering event.

The Monte Carlo simulation consisted of propagating photons into the turbid medium while keeping an account of the accumulated pathlength in medium-equiva- lent physical distance units. Details of the sampling and phase function used in the simulation have been described [23]. At each scattering interaction with the moving medium in the vessel the resulting Doppler shift ð kÞ was determined according to Eq. (9), k ¼ DðkÞ. The accumulated Doppler shift from all nj scattering interactions in the medium was defined as the Doppler frequency, D, of the detected photon:

Xnj

The average Doppler frequency for photons detected at each position in a longitudinal scan was computed as

where ni is the number of photons detected for each scan position.

The simulated geometry corresponds to an optical depth from the surface to the top of the vessel of 0.8 mean-free-path (mfp) unit in Intralipid (g ¼ 0:7) and an additional optical depth in blood (g ¼ 0:99), increasing to 15 mfp units through the full diameter of the vessel. To account for the nonvertical path, these values are multiplied by the factors 1.02 and 1.05 for incidence angles of 10:9 and 18 in the medium, corresponding to incidence angles of 15 and 25 , respectively in air. The final optical pathlengths of detected photons are actually twice the above values due to the round-trip path to the focus position and back to the detector. In the simulations, the number of photons detected was found to be nearly proportional to the source coherence length and the squares of numerical aperture and confocality angle.

Figure 2 shows average Doppler frequencies along a vertical scan through the vessel at its axis. The average Doppler frequency for photons detected from a particular focus position was determined with an accuracy of about 5% deviation from the mathematically expected Doppler frequency as shown by the middle panels in Fig. 2. This result is remarkable considering that the distributions of observed Doppler frequencies were quite broad, with standard deviations typically exceeding 250 Hz (top panels, filled symbols, Fig. 2), compared to the nominal Doppler frequencies of D ¼ 1212 and D ¼ 1979 Hz on the flow axis for ¼ 15 and 25 incidence angle. Evidently, the averaging of Doppler frequencies over many photons for each focus position represents such a robust estimation procedure that the resulting mean value quite accurately approximates the true frequency. The frequency profiles show the expected increase in maximum value with increasing incidence angle of the probe.

Standard deviations of Doppler frequencies measured within the vessel increase with increasing numerical aperture of the detector (Fig. 2, top panels, filled symbols). This can be explained by simple consideration of the variability in Doppler frequency shifts represented by backscattering events involving marginal rays at minimum and maximum angles with the flow axis, i.e., (90 Þ and ð90 þ ), respectively, giving rise to maximal and minimal Doppler frequencies with differences increasing with NA.

Standard deviations show a maximum around the center of the vessel and diminish away from the flow axis, suggesting a value proportional to the parabolic velocity profile within the vessel. This is evidenced by the precision profiles in the lower panels, which indicate that the coefficients of variance (CVs) are nearly independent of position along the cross section of the vessel. The lower CV for lower NA is apparent, and since the standard deviations show insignificant variation with increasing incidence angle, the CV decreases with increasing incidence angle due to increased mean Doppler frequency (lower right versus lower left panel). The precision of the estimated average Doppler frequency will therefore be considerably improved by using a low NA detection geometry. Results of the Monte Carlo simulations confirm that the precision (i.e., relative standard deviation) of the Doppler frequency on the flow axis will be best for large incidence angles and low numerical aperture.

Closer examination of the individual histories of photons backscattered from the central region of flow reveal that each photon experiences a series of stochastic Doppler shifts on the downward path through the vessel, a large positive Doppler shift upon backscattering, and a series of stochastic Doppler shifts on the upward path through the vessel. Figure 3 shows spectra of Doppler shifts from individual

Figure 2 Upper panels, open symbols show Doppler frequency profiles along the z axis obtained from Monte Carlo simulations with the probe at incidence angles of 15 (left) and 25 (right). Data are shown for Lc ¼ 14 m and ¼ 0:5 . Different symbols correspond to different numerical apertures: (*,*) NA ¼ 0:4; (!,!) NA ¼ 0:2; (&,&) NA ¼ 0:1. The broken line shows the frequency profile determined by the parabolic velocity profile specified as part of the input data for the simulation. Filled symbols show standard deviations of simulated Doppler frequency spectra obtained for incidence angles of 15 (left) and 25 (right). Data are shown for Lc ¼ 28 m and ¼ 0:5. Use of Lc ¼ 14 m gave similar results but greater variability due to fewer detected photons. Middle panels show the accuracy of estimated mean Doppler frequencies expressed as relative differences between estimated and theoretical values from upper panels. Symbols indicate detector NA values as noted for upper panels. Lower panels show relative standard deviations (CVs) of estimated Doppler frequencies, i.e., standard deviations divided by mean values from upper panels. Symbols as before.

interactions (open circles) as well as the accumulated Doppler frequencies for detected photons (filled symbols). The bimodal distributions demonstrate the clear distinction between the pronounced Doppler shifts due to backscattering and many small, random Doppler shifts from individual forward-scattering events. The distinc-

Figure 3 Distributions of Doppler shifts from individual interactions (*) and accumulated Doppler frequencies for photons (*) detected with the probe focus on the flow axis at 250 m depth. Results are shown for probe incidence angles of 15 (left) and 25 (right). Other detection parameters were NA ¼ 0:2, Lc ¼ 14 m, and ¼ 0:5 . Detected photons had experienced an average of 18 and 20 scattering interactions for incidence angles of 15 and 25 , respectively.

tion between Doppler noise due to forward scattering in the flow region and the Doppler shift due to backscattering becomes even more pronounced as the probe incidence angle is increased from 15 (left panel) to 25 (right panel). The distributions of accumulated Doppler frequencies (filled symbols) approximate the distributions of backscattering Doppler shifts but show less variability.

8.3.2Shadowing by Doppler Noise Below Flow Regions

Data from below the vessel (Fig. 4) show similar distributions for the two angles of incidence considered. Both Doppler shifts for individual interactions (open circles) and accumulated Doppler frequencies for detected photons (filled symbols) are distributed around zero, with similar widths for both incidence angles of the probe. The width corresponds to the Doppler noise contribution seen in spectra at positions on the flow axis (Fig. 3). If the individual Doppler shifts are assumed to be stochastically independent, their standard deviations, , can be used to express the standard deviation of accumulated Doppler frequencies, D, as follows:

where mj is the mean number of scattering interactions within the flowing medium for each photon. Estimates of mj obtained by inserting pairs of values and D for photons backscattered from below the vessel were found to be in good agreement with the expected number of interactions:

Here the integral is taken along the photon path, S, at an incidence angle of 0 back and forth through the flowing medium of vertical thickness T.

Figure 4 Distributions of Doppler shifts from individual interactions (*) and accumulated Doppler frequencies for photons (*) detected with the probe focus aimed below the vessel at 340 m depth. Results are shown for probe incidence angles of 15 (left) and 25 (right). Other detection parameters were NA ¼ 0:2, Lc ¼ 14 m, and ¼ 0:568. Detected photons had experienced an average of 34 and 36 scattering interactions for incidence angles of 15 and 25 , respectively.

The influence of some measurement parameters on the Doppler noise generated by multiple forward scattering of photons traversing a flow region can be deduced by a simple analysis based on the geometry in Fig. 5.

The Doppler shift resulting from a single interaction k for photon j when the

k ¼ 1; 2; 3; . . . ; mj

where V is the velocity of the scattering center at the position under consideration and mj is the number of scattering interactions in the flowing medium for photon j (as opposed to the total number of scattering events for photon j; nj). The expected value, i.e., mean value, of k is h ki ¼ 0, because the azimuthal angle varies uniformly between 0 and 2 . The variance of k is

By combining Eqs. (20), (21), and (23) we obtain

Figure 5 Vector diagram illustrating forward scattering according to Eq. (22).

where Vave denotes the flow velocity averaged over the scattering locations along the photon path.

Theoretical values according to Eq. (24) were calculated for various flow conditions and plotted to show the correlation with corresponding Monte Carlo results. The fitted regression line in Fig. 6 indicates a proportionality relationship. Although the data show reasonably good correlation, the proportionality constant (0.43) is quite different from the value of 1.0 that would be expected if Eq. (24) were a correct expression for the standard deviations of Doppler noise spectra for detected photons.

An explanation for systematically smaller observed standard deviations than predicted by Eq. (24) may be that Doppler OCT selectively detects photons that are extremely forward-scattered (apart from the backscattering event). Such selective detection of minimally scattered photons was observed in OCT Monte Carlo simulations [23]. If we define an effective anisotropy parameter geff ¼ cos ’ for detected

photons, a value of geff ¼ 0:998 (as opposed to g ¼ 0:99) inserted into Eq. (24) would yield theoretical standard deviations similar to those observed in the Monte Carlo

experiments. The fact that values for the 2 diluted blood (filled triangles, Fig. 6) fall below the regression line might indicate that for less optical thickness ( sT), for- ward-scattered photons are even more selectively detected and characterized by an even higher value for geff .

This figure is equivalent to Fig. 11 in Lindmo et al. [20] except that k0 ¼ nk was used instead of k in the present analysis, where k0 ¼ 1:37k

Figure 6 Correlation between theoretical values for standard deviations of Doppler noise from below the vessel according to Eq. (24) and corresponding results from Monte Carlo simulations. Open symbols represent NA ¼ 0:2, filled symbols NA ¼ 0:4. (*,*) Standard flow conditions (parabolic velocity profile with Vo ¼ 2 mm/s) at ¼ 10–30 angles of incidence; (~,~) standard flow of 2 diluted blood at 15 incidence angle; (^,^) 50% increased velocity under otherwise standard conditions (Vo ¼ 3 mm/s); (&,&) flat velocity profile (Vo ¼ 2 mm/s) using 15 incidence angle. The broken line through the origin is fitted to illustrate a proportionality relationship (y ¼ 0:43xÞ.

It is interesting to observe that the correlation between simulated and theoretical values in Fig. 6 is better for NA ¼ 0:2 (open circles) than for NA ¼ 0:4 (filled circles). In fact, in the analysis leading to Eq. (24), the numerical aperture of the probe was assumed to be vanishingly small, corresponding to photon paths close to the optical axis of the probe at incidence angle . The results in Fig. 6 (filled versus open circles) suggest that there may be an effect on numerical aperture for large incidence angles that is not contained in Eq. (24).

The Monte Carlo simulation results are supported by experimental investigations. Figure 7 shows experimental Doppler signal power spectra representing regions above and below the vessel as well as at the center of the lumen in a geometry corresponding to Fig. 1. Power spectra representing backscattering from static Intralipid are distributed around the carrier frequency (1600 Hz) with mean values in the range 1560–1650 Hz, whereas spectra representing the center of the lumen are shifted to higher frequencies, more so for the larger angle of incidence (Fig. 7, right versus left). Standard deviations of the spectra above the vessel were 310 and 350 Hz for the incidence angles of ¼ 15 and 25 , respectively (dashed lines), whereas corresponding values for spectra below the vessel were 610 and 650 Hz (thick solid lines).

Standard deviations of experimental Doppler noise spectra recorded at positions below the vessel were thus greater than those of corresponding spectra above the vessel. Although several sources of noise contribute to the width of experimental Doppler signal power spectra, the increased standard deviations of spectra at positions below the vessel are taken to indicate Doppler broadening caused by the blood flow. In agreement with Monte Carlo results, standard deviations of experimental