2.1.

Doppler Phase Shift and Mean Signal

A detailed description of the theoretical models for the Doppler phase shift and the signal power decrease as functions of the absolute velocity of an obliquely moving sample can be found in Refs. 18, 19, 27. Therefore, the improved theory is mentioned briefly in this section.

The oblique sample motion is described by an axial $\Delta z$ and transverse $\Delta x$ component during the integration time $T_{\mathrm{A}\text{-scan}}$ of the line detector of the SD OCT system. These parameters as well as the time are transformed into dimensionless coordinates as shown in Eqs. 2, 3, 4, where $w_0$ is the beam width (FWHM) of the Gaussian sample beam, ${\lambda }_0$ is the center wavelength, and $n$ is the refractive index of the investigated sample.

Fig. 2

(a) Contour plot of the average phase shift between sequential A-scans as a function of the normalized transverse $\delta x$ (horizontal) and axial $\delta z$ (vertical) displacement of the oblique sample motion and (b) the damping of the mean signal $10\log \left(I_{\text{mean}}\right)$ due to oblique sample motion is also shown by a universally valid contour plot. The presented lines with ${\vartheta }^{\prime }$ values of 39 and $58\mathrm{deg}$ correspond to theoretical values compared with the experimental data presented in Sec. 4 [cf. Figs. 5, 6, 10, 10].

Fig. 5

(a) Points: measured phase shifts in the interval of $[0,2\pi ]$ as a function of the calculated velocity $v$ assuming a parabolic flow profile. The Doppler angle $\vartheta$ is $2.2\mathrm{deg}$ , resulting in ${\vartheta }^{\prime }$ of $39\mathrm{deg}$ in the contour plots in Fig. 2. The velocity range of the 12 single measurements corresponds to $v_{\text{mean}}$ from $6.9\text{to}84.4\mathrm{mm}∕\mathrm{s}$ . Dashed line, calculated phase shift assuming a linear dependence of the phase shift from the axial velocity (classic Doppler model); solid line, $\Delta \phi$ as a function of $v$ in accordance to the new Doppler model by considering the measured beam width $w_0$ of $6.7\mu \mathrm{m}$ . (b) Flow velocities (index $M$ , points with interpolated lines) determined by considering the new Doppler model as a function of the radial position $r$ compared to the theoretical prediction (index $T$ , solid line). Since the velocity can not be quantified unambiguously for $\vartheta =2.2\mathrm{deg}$ and $v$ larger than $80\mathrm{mm}∕\mathrm{s}$ , velocity profiles are presented for $v_{\text{mean}}$ up to $31.1\mathrm{mm}∕\mathrm{s}$ .

Fig. 6

(a) Points: signal power decrease of the 1% Intralipid flow at $v_{\text{mean}}$ from $6.9\text{to}84.4\mathrm{mm}∕\mathrm{s}$ as a function of the absolute flow velocity $v$ at $\vartheta$ of $2.2\mathrm{deg}$ . Solid line, theoretical prediction of the signal damping of the new model. As supposed by the simulation [Fig. 2], the signal power decreases monotonously with increasing flow velocity. (b) Flow profiles (index $M$ ) quantified by the signal damping and its theoretical prediction (index $T$ ) for $v_{\text{mean}}$ ranging from $26.7\text{to}84.4\mathrm{mm}∕\mathrm{s}$ . For the measurements with $v_{\text{mean}}$ starting from $31.1\mathrm{mm}∕\mathrm{s}$ , the Doppler SD OCT is no longer applicable [cf. Fig 5].

Fig. 10

(a) Relationship of the phase shift $\Delta \phi$ in the interval of $[0,2\pi ]$ and the absolute sample velocity $v$ . The Doppler angle $\vartheta$ of $4.1\mathrm{deg}$ corresponds to ${\vartheta }^{\prime }$ of $58\mathrm{deg}$ in the contour plots in Fig. 2: solid line, result of the new Doppler model considering the spot size $w_0$ of $6.7\mu \mathrm{m}$ ; dashed line, result of the classic Doppler model. (b) Corresponding damping of the mean signal $-10\log \left(I_{\text{mean}}\right)$ as a function of the velocity $v$ for $\vartheta$ of $4.1\mathrm{deg}$ .

Astonishingly, for the case of oblique sample motion, $\Delta \phi$ and $I_{\text{mean}}$ do not follow the widely used classic Doppler model, which considers just the sum of the axial and transverse effect. As a result, for ${\vartheta }^{\prime }⩽60\mathrm{deg}$ , the phase shift $\Delta \phi$ never reaches $2\pi$ and oscillates at most around $\pi$ . With decreasing angles ${\vartheta }^{\prime }$ , $\Delta \phi$ approaches a constant value at higher velocities, making it difficult to quantify flow velocities on the basis of $\Delta \phi$ . The result of $I_{\text{mean}}$ in Fig. 2 shows that points of total fringe washout at $\delta z=j$ do not occur for any Doppler angle set. Instead of this, only oscillations in the signal power decrease appear if the axial displacement $\Delta z$ during $T_{\mathrm{A}\text{-scan}}$ is at least ${\lambda }_0∕2n$ within the sample beam with the spot size $w_0$ . The explanation for this unexpected characteristic of the phase shift $\Delta \phi$ and the mean signal $I_{\text{mean}}$ is based on the fact that the scattering particles are not present in the sample beam during the entire integration time $T_{\mathrm{A}\text{-scan}}$ leading to a reduced effective axial displacement $\Delta z$ .

2.2.

Flow Measurement by Signal Power Decrease

Considering the results of the numerical simulation, it becomes apparent that the Doppler flow measurement is limited in use for ${\vartheta }^{\prime }<45\mathrm{deg}$ because $\Delta \phi$ reaches a value less than or equal to $\pi$ for higher velocities. Since this angle range can not be prevented, particularly in the quantitative in vivo blood flow measurement, we propose to extend the limited velocity detection range of the Doppler analysis in SD OCT by taking the monotone signal damping into account. For this kind of flow measurement, a few conditions must be fulfilled. First, the backscattering signal of the flowing medium with the homogenously distributed scattering particles must be higher than the noise level of the OCT system. Second, the signal-damping method requires a reference that contains the signal power at a defined velocity to calculate the signal decrease at a velocity indeterminable for Doppler SD OCT. For this, the often-measured arterial pulsatile blood flow offers a large velocity range between systole and diastole, where the latter is usable as a reference. The diastolic flow velocity is very slow, so that $\delta x$ and $\delta z$ are generally much smaller than 1 and with this are quantifiable using the Doppler analysis. The higher systolic blood flow velocity is then quantified by calculating the signal decrease relative to the diastolic point of time and assuming that the scattering properties of blood do not change between systole and diastole. The experimental flow measurement using the signal decay is presented first by a flow phantom model and second in the in vivo mouse model in Sec. 4.

3.1.

SD OCT System

The measurement system used in this study is based on fiber-coupled SD OCT with a free-space Michelson interferometer (see Fig. 3 ) and was previously described.²⁷ The light source is a superluminescent diode (SLD 371 MP, Superlumdiodes Ltd., Russia) with a spectral bandwidth of $50\mathrm{nm}$ (FWHM) and a center wavelength ${\lambda }_0$ of $845\mathrm{nm}$ . The light of the SLD is guided to the 3-D scanner head by an optical circulator (OC, Thorlabs, USA). The scanner head contains a collimator (C1) to generate a free-space beam, which is divided into a reference arm and a sample arm by a 20:80 beamsplitter (BS). The sample beam is deflected by two galvanometer scanners ( $xy$ GS, Cambridge Technology Inc., USA) and focused on the sample with telecentric optics and a measured FWHM of the intensity profile $w_0$ of $6.7\mu \mathrm{m}$ . The backscattered light is superimposed with the reference light and again coupled into the optical circulator. The self-designed spectrometer in the detection arm contains a collimator (C2), a reflective gold grating (G), and the lens system (L3) for focusing the interference fringes at a CCD line scan detector (DALSA IL C6, DALSA, USA). The integration time of the line detector amounts to $T_{\mathrm{A}\text{-scan}}=84\mu \mathrm{s},$ which corresponds to an A-scan rate of $11.88\mathrm{kHz}$ at a duty cycle of 100%. The galvanometer scanners and the detector are triggered by an analog input/output card (National Instruments, USA). The acquisition and the processing of the experimental data are realized by means of a personal computer and custom software developed with LabVIEW (National Instruments, USA).

Fig. 3

Schematic of the SD OCT system used in this study: SLD, superluminescent diode; OC, optical circulator; C1 and C2, collimators; L1 to L3, lenses; BS, beamsplitter; RM, reference mirror; $xy$ GS, galvanometer scanners; G, reflective gold grating; CCD, DALSA line scan detector.

3.2.

In Vitro Capillary Model

To present the potential of the signal power decrease in the SD OCT for extending the limited Doppler velocity detection range at angles ${\vartheta }^{\prime }<45\mathrm{deg}$ , a flow phantom model was used. In this experiment, a 1% Intralipid emulsion flowing through a glass capillary with an inner diameter of $320\mu \mathrm{m}$ (Paul Marienfeld GmbH & Co. KG, Germany) was imaged two-dimensionally. The laminar flow of the turbid emulsion was ensured by an infusion pump (Fresenius Kabi AG, Germany) and a Reynolds number $\mathrm{Re}<10$ for all experiments. The flow rates were set from $2.0\text{to}24\mathrm{ml}∕\mathrm{h}$ , corresponding to mean velocities of $6.9\text{to}84.4\mathrm{mm}∕\mathrm{s}$ . The capillary was submerged in water to reduce optical distortion effects. The center of the capillary was positioned at the sample arm focus. Before imaging the flowing Intralipid, a volume scan was detected to measure the Doppler angle $\vartheta$ resulting in $2.2\mathrm{deg},$ which corresponds to ${\vartheta }^{\prime }$ of $39\mathrm{deg}$ in the contour plots in Fig. 2. For the flow quantification by Doppler SD OCT and SD OCT signal damping, time-resolved B-scans were acquired with a transverse displacement of the sample beam of $0.5\mu \mathrm{m}$ . The resulting oversampling effect was revoked in the signal processing.

3.3.

In Vivo Mouse Model

The experiment in the in vivo mouse model represents a pilot study as part of a main research project considering the vasodynamics and its influence on the hemodynamics at the early stage of atherosclerosis.²⁸ In this feasibility study, the combination of the phase-resolved Doppler SD OCT and the SD OCT signal power damping method were tested by experimental data of the saphenous artery of the right leg of a male C57BL/6 mouse under resting conditions without the use of vasoactive stimuli. Before examination, the mouse has been narcotized by intraperitoneal application of 95% ketamine $(10\mathrm{mg}∕\mathrm{ml})$ combined with 5% xylazine $(20\mathrm{mg}∕\mathrm{ml})$ using a dose of $10\mu \mathrm{l}$ per $1\mathrm{g}$ of body weight. Because of the highly scattering properties of the fur covering the saphenous artery, the skin of the right hind leg was incised to gain access to the vessel. Therefore, only a connective tissue layer of about $50\mu \mathrm{m}$ sits above the artery. To quantify the blood flow velocities, temporally resolved B-scans with a transverse displacement of the sample beam of $0.5\mu \mathrm{m}$ were acquired. The Doppler angle $\vartheta$ of $4.1\mathrm{deg}$ was measured from a 3-D data set and corresponds to ${\vartheta }^{\prime }$ of $58\mathrm{deg}$ in the contour plots in Fig. 2. The procedure was approved by the Institutional Ethics Commission for Animal Experiments of the medical faculty of the University of Technology Dresden and the government of Saxony.

4.1.

In Vitro Capillary Model

In Fig. 4 , cross-sectional Doppler flow and structural SD OCT images of the flowing Intralipid through a $320\mu \mathrm{m}$ glass capillary are presented for six different mean velocities $v_{\text{mean}}$ ranging from $6.9\text{to}84.4\mathrm{mm}∕\mathrm{s}$ and a set Doppler angle $\vartheta$ of $2.2\mathrm{deg}$ . In the Doppler SD OCT images, the determined phase shifts $\Delta \phi$ are shown by a grayscale for the range of 0 to $2\pi$ . Here, $\Delta \phi \left(z\right)$ is calculated by multiplying the complex Fourier coefficient of one A-scan $\Gamma _J{}_{+1}\left(z\right)$ with the complex conjugate coefficient of the subsequent A-scan ${\Gamma }_J^{*}\left(z\right)$ in each depth $z$ , where $J$ is the A-scan number.^{10, 12, 18, 29} The result, in turn, is a complex value ${\Gamma }_{\mathrm{res}}\left(z\right)$ with $\Delta \phi \left(z\right)$ as the argument. For the averaging of $\Delta \phi$ of adjacent A-scans, the complex data ${\Gamma }_{\mathrm{res}}$ is averaged and the mean value of $\Delta \phi$ is computed from the resulting argument. For one image, nine adjacent complex A-scans and 15 complex B-scans were averaged to eliminate the oversampling effect and to reduce the strong speckle noise. As seen in the image series at the top (Fig. 4), emanating from the capillary center the phase shift $\Delta \phi$ becomes larger with increasing $v_{\text{mean}}$ from $6.9\text{to}16.5\mathrm{mm}∕\mathrm{s}$ . For a $v_{\text{mean}}$ of $31.1\mathrm{mm}∕\mathrm{s}$ , $\Delta \phi$ exceeds $\pi$ . As predicted by the new Doppler model, for higher flow velocities $\Delta \phi$ does not reach and exceed $2\pi$ and consequently does not wrap to the primary interval of $[0,2\pi ]$ as expected by considering the classic Doppler model. Instead, $\Delta \phi$ amounts to a value of about $\pi$ independent of the flow velocity. Another interesting effect is the phase shift at the backside of the capillary, which was also seen in a previous study.³⁰ In contrast, the front side of the capillary does not show this shift. Currently, this feature can only be explained by the assumption that light scattered forward from fast moving scatterers is reflected at the capillary backside, resulting in a phase shift different from zero and must be investigated in a future research.

Fig. 4

Cross-sectional Doppler flow (top) and structural SD OCT images (bottom) of the Intralipid flow through a $320\text{-}\mu \mathrm{m}$ glass capillary immersed in water at a Doppler angle $\vartheta$ of $2.2\mathrm{deg}$ and six different mean velocities $v_{\text{mean}}$ from $6.9\text{to}84.4\mathrm{mm}∕\mathrm{s}$ . The phase shift $\Delta \phi$ is presented in the range from 0 to $2\pi$ . The gray logarithmic amplitude scale represents a range of $0\text{to}50\mathrm{dB}$ . The scale bar at the top of the images on the left shows a distance of $50\mu \mathrm{m}$ .

For the structural SD OCT images presented as series at the bottom of Fig. 4, the real parts of the complex Fourier coefficients ${\Gamma }_J\left(z\right)$ of nine adjacent A-scans are averaged. Subsequently, 15 of the resulting real-valued B-scans were also averaged. The signal power presented in the logarithmic scale is ranging up to $50\mathrm{dB}$ . It becomes apparent that a monotonously decreasing signal power occurs with increasing mean flow velocity $v_{\text{mean}}$ up to the highest flow velocity measured.

For the analysis of the cross section through the capillary center, 64 instead of only 15 B-scans were averaged in the manner already described to average a total of 576 single values for the mean values of $\Delta \phi$ and $I_{\text{mean}}$ . Figure 5 shows the phase shifts $\Delta \phi$ of 12 measurements with mean velocities ranging from $6.9\text{to}84.4\mathrm{mm}∕\mathrm{s}$ as a function of the calculated sample velocity $v$ assuming a parabolic flow profile. As seen, the data spans a velocity range up to approximately $170\mathrm{mm}∕\mathrm{s},$ which corresponds to maximum displacements $\delta x$ of 2.1 and $\delta z$ of 1.7. The prediction of the classic Doppler model calculated by Eq. 1 is presented by the dashed line, showing large deviations at higher flow velocities.

In addition, $\Delta \phi$ calculated from the new model is plotted as a black solid line and corresponds to the values of the linear slope at ${\vartheta }^{\prime }=39\mathrm{deg}$ drawn in Fig. 2. We can see that the measured data correspond very well to the phase shift of the new Doppler model. Despite of an identical averaging used for the entire velocity range, the data at higher velocities show more noise, most likely caused by the random phases of particles present in the sample beam at only one of the considered A-scans for the Doppler calculation. Figure 5 presents the flow velocity profiles (index $M$ ) calculated by using the $\Delta \phi$ to $v$ relationship of the new Doppler model for $\vartheta =2.2\mathrm{deg}$ [solid line in Fig. 5] as a function of the radial position $r$ of the capillary center in comparison to the theoretical prediction (index $T$ ) of the flow velocity according to the Hagen-Poiseuilles law. Here, the positive values of $r$ represent the backside of the capillary. As shown in Fig. 5, the flow velocity according to the new Doppler model can only be calculated unambiguously up to $80\mathrm{mm}∕\mathrm{s}$ corresponding to a phase shift $\Delta \phi$ of $3.03\mathrm{rad}$ . Unfortunately, higher flow velocities can not be determined by using $\Delta \phi$ for the presented experiment.

As shown in the preceding section, a quantitative flow measurement by using the Doppler analysis is limited in use for ${\vartheta }^{\prime }<45\mathrm{deg}$ at high flow velocities. Therefore, the signal damping as a function of the absolute sample velocity for the 12 measurements with $v_{\text{mean}}$ ranging from $6.9\text{to}84.4\mathrm{mm}∕\mathrm{s}$ is determined in the next step. As a reference, the signal power measured at zero flow is chosen. Figure 6 plots the logarithmized mean signal $-10\log \left(I_{\text{mean}}\right)$ of the averaged A-scan at the capillary center against the calculated velocity $v$ , assuming a parabolic flow as well. As for the Doppler analysis, each of the shown data points relates to a mean value of 576 single measurements. The solid curve corresponds to the simulated values of the line with ${\vartheta }^{\prime }=39\mathrm{deg}$ in Fig. 2. Despite the speckle noise, the measured data agree very well with the theory of the new model. Because the signal power decreases monotonously, the flow velocity can be calculated unambiguously. Figure 6 presents the resulting flow profiles (index $M$ ) in comparison to the theoretical profiles (index $T$ ). As seen, the measured values are in good agreement with the theory. At the border area of the capillary lumen, small deviations can be noticed, which are caused by the strong speckle noise relative to the minor signal damping at small flow velocities.

By considering the experimental results, the question at which flow velocity the signal damping is preferred to the Doppler analysis is addressed in the following section. A weighted mean value of the velocities calculated by using the Doppler and the signal damping method can be determined by taking the error of both methods into account. For this, the weights $w_{v_D}$ of the velocity $v_D$ calculated by the Doppler phase shift $\Delta \phi$ and $w_{v_S}$ of the velocity $v_S$ determined by the signal damping are ascertained as presented in Eqs. 11, 12

Fig. 7

(a) Normalized weight of the velocity $v_D$ determined by the Doppler analysis ( $w_{v_D}$ , dashed line) and of the velocity $v_S$ computed by using the signal damping method ( $w_{v_S}$ , solid line) as a function of the absolute flow velocity $v$ ranging from zero to the Doppler uniqueness limit of $80\mathrm{mm}∕\mathrm{s}$ for $\vartheta =2.2\mathrm{deg}$ . The intersection point at $v=46\mathrm{mm}∕\mathrm{s}$ represents the point from which the flow velocity is more precisely calculated by the signal damping method. The range, in which the combined method can be used, extends the shown velocity range. (b) Weighted mean value $v_{\text{weight}}$ of the flow velocities of both methods, the Doppler analysis and the signal damping, are calculated by taking the errors of $v_D$ and $v_S$ into account.

4.2.

In Vivo Mouse Model

In this experiment, data were acquired from the murine saphenous artery in vivo. The resulting grayscale cross-sectional Doppler flow and structural SD OCT images of three consecutive B-scans describing one cardiac cycle are presented in Fig. 8 . The measurement in Fig. 8 shows the systole, that in Fig. 8 relates to the point of time between systole and diastole, and that in Fig. 8 corresponds to the diastole. For one image, 10 adjacent A-scans were averaged in the way described in Sec. 4.1. In contrast to the experiments with the capillary model, a significant signal power attenuation of the flowing blood with increasing depth $z$ can be observed and is caused by the highly scattering properties of the blood for the wavelength range used.³³ On closer examination of the Doppler flow images at the top of Fig. 8, we can notice that the phase shift $\Delta \phi$ shows a strong noise even at a sufficiently high signal power and despite of averaging 10 single values. The reason for this effect may be multiple scattering events causing a significant signal in larger depth and probably a noisy phase shift.³⁴ Accordingly, the phase shift $\Delta \phi$ can be determined reliably only near the upper side of the vessel lumen. The damping of the mean signal $I_{\text{mean}}$ is caused by the continuous phase change during the integration time $T_{\mathrm{A}\text{-scan}}$ and is consequentially related to the phase shift $\Delta \phi$ . From this it follows that the analyzable depth range for the signal damping is limited to the upper vessel part as well. To determine the feasibility of the blood flow measurement by the signal damping in combination with the Doppler method for SD OCT, the following analysis refers to the limited depth range of the vessel lumen.

Fig. 8

Gray-scale Doppler flow (top) and structural SD OCT images (bottom) of three consecutive B-scans of the saphenous artery in vivo under resting conditions. The scale bar represents a distance of $50\mu \mathrm{m}$ . (a) Relates to the systole, (b) shows the point of time between systole and diastole and corresponds to the intermediate, and (c) represents the diastole.

Figure 9 shows the phase shift $\Delta \phi$ of the A-scan at the vessel center in the interval $[0,2\pi ]$ as a function of the radial position $r$ inside the artery for the intermediate (white diamonds) and the diastolic point of time (black diamonds). Here, the negative values of $r$ relate to the upper side of the vessel. Because at the systole only $3\text{pixels}$ at the upper vessel part show reliable values of the phase shift, this measurement can be used neither for the Doppler analysis nor for the signal damping method. For the intermediate and the diastole, only the phase shifts for $r$ ranging from $-137.5\text{to}-100\mu \mathrm{m}$ corresponding to 11 depth pixels are reliable and are used for fitting a parabolic profile. By means of this range, we can notice that the flow velocity is decreased from the intermediate to the diastole as expected. Figure 9 shows the corresponding signal power and the fit of a polynomial of second degree to the reliable measurement points against the parameter $r$ . Even for this small analyzable depth range, a signal power damping can be observed for the intermediate in comparison to the smaller diastolic blood flow.

Fig. 9

(a) Phase shift $\Delta \phi$ of the A-scan at the vessel center for the intermediate (white diamonds) and the diastole (black diamonds) as a function of the radial position $r$ . The diameter of the artery was determined by assuming a refractive index of blood ${\mathrm{n}}_{\text{blood}}$ of 1.4 and amounts to $275\mu \mathrm{m}$ . The reliable values of $\Delta \phi$ can be found in the range $\mathrm{r}=-137.5\text{to}-100\mu \mathrm{m}$ . (b) Related signal power for the intermediate and the diastole and the fitted polynomials for the calculation of the absolute signal power damping of the intermediate.

For the measured reliable phase shifts of the intermediate and the diastolic point of time [cf. Fig. 9], the flow velocity values were determined by taking the result of the new Doppler model into account. The used relationship of the phase shift $\Delta \phi$ and the sample velocity $v$ for $\vartheta =4.1\mathrm{deg}$ is presented by the solid line in Fig. 10 . In comparison, the result of the classic model is shown by the dashed line. The parabolic fit to the flow velocity of the diastole results in a maximum velocity of $v\left(0\right)=41.5\mathrm{mm}∕\mathrm{s}$ at the vessel center $(r=0)$ and an exponent of $K=2.1$ :

Figure 11 presents the resulting blood flow velocities as a function of the radial position $r=-137.5\text{to}0\mu \mathrm{m}$ . There, the flow velocities of the intermediate and the diastole calculated by the new Doppler model (index $D$ ) are shown by the white and black diamonds, respectively. The fitted parabolic profile of the diastole with $v\left(0\right)=41.5\mathrm{mm}∕\mathrm{s}$ and $K=2.1$ corresponds to the thin solid line. The blood flow velocities of the intermediate point of time determined by the signal damping (index sd) are shown by the gray points, which are in excellent agreement with the reliable values determined with the Doppler analysis. The fit of the intermediate flow velocities results in $v\left(0\right)=59.7\mathrm{mm}∕\mathrm{s}$ and $K=2.1$ for both the Doppler and signal damping analyzed values.

Fig. 11

Detailed view of the flow velocity against the radial position $r$ in the range of $-137.5\text{to}0\mu \mathrm{m}$ : index $D,$ flow velocities of the intermediate (white diamonds) and the diastole (black diamonds) calculated by considering the new Doppler model; index sd, the result of the velocity quantification of the intermediate flow by using the signal damping with the Doppler analyzed diastolic flow as a reference.