Analyzing the requirements of high-speed camera parameters for enhanced laser speckle sensing of flow dynamics

Blood perfusion is one of the major indicators in disease and injury monitoring when tracking the evolution of pathology. Optical non-contact monitoring of tissue blood flow can provide accurate measurements with minimal invasiveness and high spatial resolution, thus making it a quick, sterile and convenient tool for medical practitioners and with respect to patient comfort [1–4].

In recent years, several methods based on the analysis of the backscattered laser have been widely adopted to estimate the erythrocyte flow velocity through the micro-capillary net in the probed volume. One approach involves the use of imaging velocimetry [5, 6]. Another common approach is based on contrast analysis of the speckle pattern images created by scattered light. These methods are known as Laser Speckle Contrast Analysis (LASCA), and Laser Speckle Contrast Imaging (LSCI) [7–9].

Another approach uses the full statistical properties of the power of the laser backscattered light as manifested by its autocorrelation function ${g}^{2}\left(\tau \right).$ Using full statistics makes it possible to differentiate between various flow regimes such as ordered flow and random Brownian motion. However, obtaining the true statistics of the flow process requires high-sampling rate single point detector hardware. Recently, new techniques for full-field coverage of speckle statistics have been developed for use with state-of-the-art high-speed video cameras [10, 11]. Nevertheless, working with a 2D pixel array obligatorily involves considering the potential effects of key imaging system parameters such as the frame rate, exposure time, speckle-to-pixel ratio, pixel binning and noise mechanism on the speckle statistic calculations.

The study reported here was designed to provide a better understanding of the effects of each of these imaging system parameters on the calculation of end speckle statistics. This clarification of the interrelationship between these parameters can help designers optimize the working point. The major benefit is a reduction in the stringent hardware requirements (e.g. high frame rate), so that more common, of-the-shelf equipment can be used. In-vitro and in-vivo experimental results confirm the feasibility of this method in real-life scenarios.

A comprehensive analysis of wide field flow statistics can be carried out when using a high-speed video camera to calculate the second order autocorrelation function, ${g}^{2}\left(\tau \right),$ at each point in the image (each pixel on the image sensor). This statistical map can later be used to evaluate the flow regime in different areas of the recording. The basic steps in the calculation process are visualized in figure 1.

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The second order autocorrelation function (ACF) is defined by equation (1) [12]:

$$\begin{eqnarray}&&{g}^{2}\left(\tau \right)\,=\,\displaystyle \frac{\left\langle I\left(t\right)I\left(t+\tau \right)\right\rangle }{{\left\langle I\left(t\right)\right\rangle }^{2}}\end{eqnarray} \tag{ 1 }$$

where ${g}^{2}\left(\tau \right)$ is the autocorrelation function of the individual pixel intensity, τ is the time lag, and I is the pixel's intensity. The decorrelation 'decay time' is defined as the time required for the distribution to decrease to e⁻¹ of its maximum value.

To reduce the effects of measurement noise, several signal pre-processing steps are frequently applied to these signals. For each calculated ACF, the first index value is eliminated since this index accounts for the autocorrelation of the noise as well as the signal. Figures 2(a)–(d) schematically depicts the rationale for discarding the first value using the example of a synthetic signal corrupted by white noise (Signal-to-Noise Ratio, SNR = 10).

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An optimal solution requires high spatial and temporal resolution. However, to obtain a clean ACF and stable τ values, noise reduction is needed. This can be achieved by spatial and/or temporal averaging but with caution since in the time domain, the averaging window must be shorter than the decorrelation dynamics process of the moving particles. Similarly, in the spatial domain, averaging of non-homogeneous regions should be avoided. Figures 3(a)–(d) illustrates the equivalence of time versus spatial averaging for a simple case of particles in Brownian motion, at a constant frame rate of 100k frames per second. To measure ACF stability quantitatively, equation (2) [13] defines the coefficient of variation (CV):

$$\begin{eqnarray}&&CV=\displaystyle \frac{\sigma }{\mu }\end{eqnarray} \tag{ 2 }$$

where σ is the standard deviation and μ is mean value. The CV improves for either spatial or temporal averaging. The averaging applied the same factor over the total voxels, using two methods: 3 × 3 pixels in the spatial domain, and 9 'time slices' (each 'time slice' equals 20 [msec]) in the temporal domain.

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This resulted in a CV reduction of the same ratio relative to the initial case (figure 3(a)). This was expected since under the assumptions mentioned above of spatial and temporal homogeneity, averaging 9 pixels spatially in a single voxel has the same effect as averaging 1 pixel temporally over 9 voxels.

Another important issue is related to the calibration of the optical system parameters to yield the proper speckle-to-pixel ratio. When an imaging condition is applied, a subjective speckle is formed. In this case the speckle size on the image plane of the camera is determined by equation (3) [14]:

$$\begin{eqnarray}&&{\rho }_{speckle}=2.44\lambda \left(1+M\right){f}_{\#}\end{eqnarray} \tag{ 3 }$$

where λ is the wavelength of light, M is the magnification of the imaging system, and ${f}_{\#}$ is the system's f number. Averaging speckles within a single pixel area will reduce the contrast and hence reduce the SNR of the ACF, as well as the SNR in the τ values. Thus, the system's optical parameters should be set to achieve a speckle-to-pixel ratio that is equal to or less than 1.

As discussed above, using a high-speed camera can directly reveal a statistical ACF model of systems containing flowing particles. The importance of an accurate ACF statistical model is well known when assessing of flow dynamics and has been extensively documented in the literature [14]. Previous works have shown that for moving particles governed mainly by Brownian motion, a single exponential model (as described in equation (4)), with n = 1, presents a good fit [15].

$$\begin{eqnarray}&&f\left(t\right)=A\cdot {e}^{B\cdot {t}^{n}}+C\end{eqnarray} \tag{ 4 }$$

where A, B and C are specific parameters, n is the model characterization feature, and t is the time. However, the above model does not work well for a more complex motion regime that includes flow combined with Brownian motion. In recent years, several models have been suggested to characterize this type of particle dynamics [16–20].

This suggests that different mathematical model may be advantageous in differentiating between different flow regimes. Here, we suggest the Dual Exponent model described in equation (5). To see intuitively how this works, we suggest thinking of each term in equation (5) as responsible for the two different forces applied on the particles; namely, flow and Brownian motion.

$$\begin{eqnarray}&&f\left(t\right)=A\cdot {e}^{B\cdot t}+C\cdot {e}^{D\cdot t}\end{eqnarray} \tag{ 5 }$$

Similar to equation (4), A, B, C, D are specific parameters and t is the time. The analysis was conducted in two ways: as a deviation from the single exponent model (n = 1) and the goodness-of-fit (GOF) of the Dual Exponent model versus other models in the literature.

In this section we describe in-vitro and in-vivo experiments to test the method presented in section 2. The novelty of optimizing system parameters is summarized and presented in the experimental results.

3.1.  In-vitro experiment

In this experiment we measured flowing particles in a tube. The aim was to demonstrate the theory described in section 2, in an in-vitro environment. The test setup is depicted in figure 4. Post-processing analysis was conducted to determine the optimal camera configuration with respect to the frame rate and exposure time.

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The optical test setup was composed of a diode-laser (780 [nm], 50 [mW], spot diameter ∼1 [mm]), a high-speed camera (Fastcam Mini AX200, by Photron) and an objective lens (F_#2, f35 [mm]). The camera frame rate was set to 100 k frames per second. The tube diameter was 4 [mm], with high transparency and was made of plastic. A compound of Intra Lipid (0.125%) and Agarose (0.4%) was used as the fluid medium. The flow rate was set to ∼90 [ml min⁻¹].

Figure 5 shows the values of τ as a function of the frame rate. The original video was recorded at 100K [fps]. All the other points on the x-axis are the averages of N frames to de-facto achieve a lower frame rate for the recorded video. Equation (6) below expresses the relationship between τ_process and the optimal frame rate, according to Nyquist theory.

$$\begin{eqnarray}&&2\cdot \displaystyle \frac{1}{{\tau }_{process}[\text{sec}]}\,=\,{f}_{Nyquist}\left[Hz\right]\end{eqnarray} \tag{ 6 }$$

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As shown, τ remained at 0.2 [msec] until the sampling rate dropped below 10K. This suggests that for this given flow, a frame rate of 10K [fps] would be sufficient, making it possible to work with a much cheaper camera.

Next, we assessed the optimal exposure time, as shown in figure 6. The decorrelation time τ was tested for several exposure times at a constant frame rate of 10K [fps]. The analysis was done numerically, by increasing the number of averaging frames, thus creating a larger de-facto camera integration period. Frames were averaged within the limit of the camera integration time, which is bounded by its 1/frame rate.

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As expected, for longer exposure times; i.e., a larger number of averaged frames, τ increased as well. This is likely due to the smearing effect on the video imposed by the increase in integration time. As a result, the contrast was reduced and the ${g}^{2}\left(\tau \right)$ function decayed slower than previously. Nevertheless, this trend was negligible in absolute values (on a scale of μsec). This suggests that it is better not to reduce the exposure time, since it has a negative effect on the SNR as well, given the smaller number of photons in each pixel.

3.2.  In-vivo experiment

Using the same optical setup as described in section 3.1, an 'occlusion test' was performed. This time instead of a tube, the laser illuminated live human tissue. The tip of the index finger was chosen as an anatomical site because it has a relatively large number of blood vessels that are close to the surface. A sphygmomanometer was used to inflate the cuff to ultra-systolic pressure to occlude blood flow, thus creating two flow regimes for each subject: normal blood flow and occluded (slow) blood flow.

Figure 7 illustrates the setup for in-vivo testing. For this test, 5 healthy females and 5 healthy males were recorded. The females ranged in age from 22 to 37, with a mean of 27. The males ranged in age from 24 to 45 with a mean of 34. Each subject started the recording with ∼4 [sec] in the occluded state and then ∼6 [sec] without occlusion.

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In this experiment all the subjects were asked to sit still to avoid any motion artifacts. To cope with a more realistic use case, one might consider increasing the frame rate, thus making it possible to work with a shorter 'time slice', while still having enough data points to generate the statistical model of the ACF curve. Another possibility would be to add a standard video capture camera which would be synchronized with the main system, and capture the flow frames only in those segments where motion is as low as possible.

A similar analysis as described in section 3.1 was applied for the in-vivo experiment. Figure 8 shows the typical change in τ as a function of fps reduction, for one subject.

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Unlike the in-vitro case, the measured flow was not necessarily laminar, but rather had a broad particle velocity distribution. This may explain the smooth rise around a frame rate of 10 K [fps], rather than the sharp peak as in the in-vitro case. Figure 9 shows deviation of τ from its original value, summarized for all 10 subjects, for normal blood flow and for occluded blood flow. Nevertheless, as shown by the blue line in figure 9(b), in the limit of 10 K frames per second, τ deviated from its original value by 100%. Obviously, if higher accuracy of τ is required, higher frame should be used, as described by figure 9.

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For the occluded blood flow, as depicted by the orange line in figure 9, the governing dynamics was Brownian motion, at a much lower magnitude, thus enabling the use of much lower frame rates, without causing large deviations in τ values.

As in the in-vitro case, we assessed the optimal exposure time for in-vivo measurements. Again we averaged the frames, at a constant frame rate of 10K [fps]. We obtained the same outcome as depicted in figure 6, for longer exposure times; i.e., a larger number of averaged frames, and τ increased as well. When repeating the analysis for all the 10 subjects, the average change in τ was only 4%, with a standard deviation of 1.3, which leads to the same conclusion that it is better not to reduce the exposure time.

In this section we describe in-vitro and in-vivo experiments that illustrate various applications of the proposed method to obtain the ACF statistical model.

4.1.  In-vitro experiment

The same setup as described in section 3.1 was used to examine the potential of the method to detect the existence of flowing particles when a static scattering layer was placed above the flow. As in diffusive correlation spectroscopy (DCS) [21], the aim of this experiment was to test in-depth vessel measurement. The added value here is the use of multi-pixel array sensor, which can exploit the statistics from several pixels in parallel. For this purpose, a tube buried in an Agarose scattering medium was built to mimic a buried blood vessel in human tissue. The sample was made of PVC consisting of a tube 4 [mm] in diameter placed at a depth of 7 [mm] within an Intra Lipid phantom (Intra Lipid 0.125%, Agarose 0.4%).

For this test, instead of illuminating with a single point at a fixed location, we scanned the phantom along its width (indicated by the black dotted line in figure 10) by moving the laser illumination spot for each recording session by 0.5 [mm].

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Figure 11 depicts how the correlation in the single exponent (n = 1) model (y-axis) changed as a function of the illumination location (x-axis) along the scanning line. There was an approximate region of 4 [mm] at the center of the phantom where the correlation values dropped relative to the outer regions. In these measurements, the illuminating photons were backscattered to the camera by the flowing particles. Thus, the single exponent (n = 1) model did not provide a good fit for these regime dynamics.

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4.2.  In-vivo experiment

We used the same procedure as described in section 3.2 to test the proposed method's ability to distinguish regions with flow versus regions with static blood. A statistical model of the flow can provide additional information that does not depend on the scattering density or temperature, as is the case when relying on the decay time τ alone.

When there was a normal blood flow, the correlation with the single exponent model (n = 1), R², was lower than when the cuff was inflated, creating an occlusion which resulted in substantial blood flow reduction. Figure 12(a) shows the typical behavior of one subject, with an occlusion release after ∼4.5 s. Figure 12(b) summarizes the measurements for 10 subjects. The mean values for the first and last 2 s were taken to represent the occluded and non-occluded states, respectively.

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There are several auto-correlation decay curve models in the literature [10, 11]. In figure 13 we compared two such models, based on data obtained on the occlusion test. Model A was chosen to be the single exponent, with t to the power of 1 (equation (4)), whereas Model B was chosen to be a similar model, but with t to the power of 0.5 (equation (7)).

$$\begin{eqnarray}&&f\left(t\right)=A\cdot {e}^{B\cdot {t}^{0.5}}+C\end{eqnarray} \tag{ 7 }$$

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Based on the data reported in Postnov et al [11], model B was expected to have a good fit with the flow state (figure 13(b)). However, if the goal was to differentiate between the two different flow regime states, Model A had a clear advantage (figure 13(a)). Figure 13(c) compares Model B to the Dual Exponent model (equation (5)) for flow state alone. As expected, both models provided very high goodness of fit, with slight superiority for model B.

This article presented a new methodology for optimizing a fast camera's working point and optimal optics when the goal is to measure flowing particles with respect to parameters such as the reduction of the fps to the necessary minimum for a normal blow flow, noise reduction, and the speckle to pixel ratio. In a series of in-vitro and in-vivo experiments, we assessed the model's ability to differentiate between Brownian motion and laminar flow motion. The results demonstrated the potential applications of identifying the diameter of buried blood vessels using the laboratory phantom.

Future work will concentrate on extending the model to other physiological parameters including the orientation and the relative speed of the flow. The main advantage of this method is its relative ease of use in terms of the required hardware complexity and post-processing analytics.

The authors declare no conflicts of interest. No funding to declare.