1.

Introduction

Diffuse correlation spectroscopy (DCS) is a relatively new technique based on temporal speckle fluctuations of the diffuse light for blood flow measurements in the microvasculature.^1,2 DCS has been widely used in different medical applications such as neurology and oncology.^3–5 In traditional DCS analysis, the optical properties of the medium, absorption (${\mu }_{\mathrm{a}}$), and reduced scattering (${\mu }_{\mathrm{s}}^{\prime }$) coefficients should be known and the errors in the estimation of these parameters influence the calculated blood flow index (BFI). Most significant errors come from errors in estimating the scattering coefficient, whereas the results are less sensitive to the absorption coefficient.⁶ DCS is often applied in tandem with a near-infrared diffuse optical spectroscopy (NIRS/DOS) system in order to get more physiological information and also because ${\mu }_{\mathrm{a}}$ and ${\mu }_{\mathrm{s}}^{\prime }$ change in time.²

Previously, it has been shown that simultaneously fitting ${\mu }_{\mathrm{s}}^{\prime }$ and/or ${\mu }_{\mathrm{a}}$ with BFI was impractical from a single intensity autocorrelation measurement, i.e., from a single source-detector pair,⁷ due to strong coupling between the parameters. We have hypothesized and observed that the effect of this coupling is dependent on the source-detector separation which has led us to explore if there is a regime where multiple source-detector separations in an analogous fashion to continuous-wave or frequency-domain NIRS/DOS can be used to completely or partially decouple these parameters.

In this work, we describe a multidistance DCS (MD-DCS) algorithm for simultaneous measurement of ${\mu }_{\mathrm{a}}$, ${\mu }_{\mathrm{s}}^{\prime }$, and BFI in small source-detector separations using noise-added simulated data, phantom measurements, and on a longitudinal in vivo study of a mouse tumor.

2.

Theory, Algorithms, and Experimental Methods

Information about the dynamics of the scatterers, in most cases about the motion of red blood cells, i.e., blood flow, can be derived from the measurement of the intensity autocorrelation of a single speckle generated by diffuse light in tissues.² For DCS analysis, we converted measured intensity autocorrelation $g_2$ to field autocorrelation function using the Siegert relation $g_1(\tau )=\{\sqrt{[g_2(\tau )-1]/\beta }\}$, where $\beta$ is a constant determined primarily by the collection optics of the experiment. We note that in our experiments, we use nonpolarized light sources and single-mode fibers without polarizers to collect the diffuse light. Therefore, in our experiments, $\beta \approx 0.5$.

The electric field autocorrelation function ($G_1$), Eq. (1), and the normalized field autocorrelation function ($g_1$) can then be derived and a correlation diffusion model can be used to fit the data and extract the blood flow information.² The Green’s function solution of the correlation diffusion equation for semi-infinite boundary conditions is given by

Here, $\tau$ is the delay time, $r_1=\sqrt{{(1/{\mu }_{\mathrm{s}}^{\prime })}^2+{\rho }^2}$, $r_b=\sqrt{{(2z_b+1/{\mu }_{\mathrm{s}}^{\prime })}^2+{(1/{\mu }_{\mathrm{s}}^{\prime })}^2+{\rho }^2}$, and $K(\tau )=\sqrt{3{\mu }_{\mathrm{a}}{\mu }_{\mathrm{s}}^{\prime }+6{\mu }_{\mathrm{s}}^{\prime 2}{\kappa }^2\alpha \tau \mathrm{BFI}}$. $\alpha$ is the fraction of photon scattering events that occur from moving particles in the medium and $\kappa$ is the wave-number of light in the medium.

Here, we posit that the source-detector separation dependence of the $g_1$ allows us to decouple the contribution of static (${\mu }_{\mathrm{a}}$ and ${\mu }_{\mathrm{s}}^{\prime }$) and dynamic, BFI, properties of the medium under certain conditions. Therefore, we fit for all desired variables (${\mu }_{\mathrm{a}}$, ${\mu }_{\mathrm{s}}^{\prime }$, and BFI) using a single penalty function, ${\chi }^2$, defined as

There is more information in the MD-DCS measurements since we are recording the average detected intensity in an analogous fashion to continuous-wave NIRS/DOS. This measured intensity at each detector position from the continuous-wave DCS laser can also be utilized to obtain the effective attenuation coefficient (${\mu }_{\mathrm{eff}}=\sqrt{3{\mu }_{\mathrm{a}}{\mu }_{\mathrm{s}}^{\prime }}$) to improve the robustness of the algorithm by constraining the relationship between absorption and scattering. We will demonstrate the utility of these in experimental results.

There is one important caveat that we should mention here: DCS uses the intensity autocorrelation functions and, as such, it does not suffer from the so-called source-detector coupling problems. NIRS/DOS, on the other hand, uses the source-detector separation dependence of the intensity alone to measure ${\mu }_{\mathrm{eff}}$ which requires that these coefficients are calibrated. For the purposes of this work, we did not construct a new probe and our existing probe configuration does not allow the measurement of ${\mu }_{\mathrm{eff}}$ through the DCS fibers due to this calibration issue.^8,9 However, it contains a set of self-calibrating NIRS/DOS fibers that can be used for the measurement of ${\mu }_{\mathrm{eff}}$ and we utilized them in some of the experimental data. It is beyond the scope of this paper to overcome this limitation.

3.

Results

3.1.

Penalty Function in Single-Distance and Multidistance Diffuse Correlation Spectroscopy Fitting

As described above, a simulated DCS measurement with ${\mu }_{\mathrm{a}}=0.1{\mathrm{cm}}^{-1}$, ${\mu }_{\mathrm{s}}^{\prime }=10{\mathrm{cm}}^{-1}$, and $\mathrm{BFI}={10}^{-8}{\mathrm{cm}}^2/\mathrm{s}$ was used to calculate the penalty function ${\chi }^2$ defined in Eq. (2) for a range of values. The ${\chi }^2$ for the expected ${\mu }_{\mathrm{a}}$ plane, i.e., the plane where ${\mu }_{\mathrm{a}}=0.1{\mathrm{cm}}^{-1}$, is shown in Fig. 1 for SD-DCS (a) and MD-DCS (b).

Fig. 1

The penalty function of (a) SD-DCS and (b) MD-DCS. ${\chi }^2$ is plotted for the expected ${\mu }_{\mathrm{a}}$ plane, i.e., the plane where ${\mu }_{\mathrm{a}}=0.1{\mathrm{cm}}^{-1}$.

In the single distance fitting, the minimum of the penalty function occurs over a wide range of values of optical parameters and BFI as previously reported.⁷ Adding more measurements from multiple source-detector separations, i.e., the MD-DCS approach, confines ${\chi }^2$ to a smaller area which leads to better convergence of the algorithm.

3.2.

Multidistance and Single-Distance Diffuse Correlation Spectroscopy Fitting on the Simulated Data

3.2.1.

Noise-free simulations

We have applied SD-DCS and MD-DCS to noise-free data for eight source-detector separations from 2.5 to 5.0 mm (2.5, 2.9, 3.2, 3.6, 3.9, 4.3, 4.6, and 5.0 mm). Figure 2 shows that both SD-DCS and MD-DCS can accurately retrieve all parameters except for one point with lowest ${\mu }_{\mathrm{s}}^{\prime }/\mathrm{BFI}$ and highest ${\mu }_{\mathrm{a}}$ that SD-DCS fails to fit. We have performed a similar fitting for intermediate (10 to 20 mm) and large (20 to 34 mm) separations, and the results are shown in Appendix A. While MD-DCS fits all the parameters (${\mu }_{\mathrm{a}}$, ${\mu }_{\mathrm{s}}^{\prime }$, and BFI) without any error, SD-DCS has considerable error even in noise-free data and the error rises by increasing the source-detector separation (Figs. 7 and 8).

Fig. 2

The values of fitted ${\mu }_{\mathrm{s}}^{\prime }$, ${\mu }_{\mathrm{a}}$, and BFI versus the expected values in noise-free, small source-detector separations (2.5 to 5.0 mm). The top three plots are the results of SD-DCS and the bottom ones are from MD-DCS fitting. The slope (expected slope is 1) and intercept (expected intercept is 0) of the fitted values versus expected ones for each parameter are displayed in the inset. Each color represents one ${\mu }_{\mathrm{s}}^{\prime }$ (red: ${\mu }_{\mathrm{s}}^{\prime }=4$, green: ${\mu }_{\mathrm{s}}^{\prime }=7$, blue: ${\mu }_{\mathrm{s}}^{\prime }=10$, cyan: ${\mu }_{\mathrm{s}}^{\prime }=13$, and black: ${\mu }_{\mathrm{s}}^{\prime }=16$) and each ${\mu }_{\mathrm{a}}$ is shown by a different marker (star: ${\mu }_{\mathrm{a}}=0.04$, circle: ${\mu }_{\mathrm{a}}=0.07$, square: ${\mu }_{\mathrm{a}}=0.10$, cross: ${\mu }_{\mathrm{a}}=0.13$, and triangle: ${\mu }_{\mathrm{a}}=0.16$). Dodging is used (moving the points slightly across the $x$-axis) for better visualization of BFI values where for each ${\mu }_{\mathrm{s}}^{\prime }$ or ${\mu }_{\mathrm{a}}$, BFI increases from left to right. (a) SD-DCS and (b) MD-DCS retrieve all three parameters ${\mu }_{\mathrm{s}}^{\prime }$ (left), ${\mu }_{\mathrm{a}}$ (middle), and BFI (right) accurately except for one point with the lowest ${\mu }_{\mathrm{s}}^{\prime }/\mathrm{BFI}$ and highest ${\mu }_{\mathrm{a}}$ that SD-DCS fails to fit.

3.2.2.

Noise-added simulations

We have applied SD-DCS and MD-DCS to noise-added simulated data in small optode distances (2.5 to 5 mm), and the results are presented in Fig. 3. It shows that SD-DCS has a considerable error when dealing with realistic experimental noise, whereas MD-DCS fits all parameters with a high accuracy (less than 5% error).

Fig. 3

The values of fitted ${\mu }_{\mathrm{s}}^{\prime }$, ${\mu }_{\mathrm{a}}$, and BFI by (a) SD-DCS and (b) MD-DCS versus the expected values in small source-detector separations (2.5 to 5.0 mm). The slope (expected slope is 1) and intercept (expected intercept is 0) of the fitted values versus the expected ones for each parameter are displayed in the inset. (a) SD-DCS has a fairly large error in fitting ${\mu }_{\mathrm{a}}$, ${\mu }_{\mathrm{s}}^{\prime }$, and BFI. (b) In MD-DCS, the fitted line for all parameters has a slope of 1 and 0 intercept, which indicate the complete agreement of the fitted and expected values. Each color represents one ${\mu }_{\mathrm{s}}^{\prime }$ (red: ${\mu }_{\mathrm{s}}^{\prime }=4$, green: ${\mu }_{\mathrm{s}}^{\prime }=7$, blue: ${\mu }_{\mathrm{s}}^{\prime }=10$, cyan: ${\mu }_{\mathrm{s}}^{\prime }=13$, and black: ${\mu }_{\mathrm{s}}^{\prime }=16$) and each ${\mu }_{\mathrm{a}}$ is shown by a different marker (star: ${\mu }_{\mathrm{a}}=0.04$, circle: ${\mu }_{\mathrm{a}}=0.07$, square: ${\mu }_{\mathrm{a}}=0.10$, cross: ${\mu }_{\mathrm{a}}=0.13$, and triangle: ${\mu }_{\mathrm{a}}=0.16$). Dodging is used (moving the points slightly across the $x$-axis) for better visualization of BFI values where for each ${\mu }_{\mathrm{s}}^{\prime }$ or ${\mu }_{\mathrm{a}}$, BFI increases from left to right.

The results of fitting for intermediate and large separations are shown in Appendix B. In these simulations, it was observed that the accuracy of the fitted parameters by MD-DCS decreases when the source-detector separation is larger than $\sim 2\mathrm{cm}$. Therefore, from this point on, we focus on small source-detector separations for both phantom and in vivo validations.

3.3.

Multidistance Diffuse Correlation Spectroscopy Fitting on the Liquid Phantom Data

As explained above, to validate the algorithm in experimental conditions, we have measured homogenous liquid phantoms with both MD-DCS and NIRS. Figure 4(a) shows the measured (fitted by MD-DCS) ${\mu }_{\mathrm{s}}^{\prime }$ and the agreement with the expected values ($\text{slope}=0.9$, $\text{intercept}=0.4$). The absorption coefficient of all phantoms is expected to be the same. As shown in Fig. 4(b), the fitted absorption coefficient of all phantoms ($0.0282\pm 0.0007{\mathrm{cm}}^{-1}$) is the same and is in the range of the expected ${\mu }_{\mathrm{a}}$ of water (${\mu }_{\mathrm{a}}^{785\mathrm{nm}}=0.02$ to $0.03{\mathrm{cm}}^{-1}$). Since the changes in viscosity due to Lipofundin titration are negligible, it is expected that BFI also does not change by Lipofundin titration. Figure 4(c) shows that the measured BFI ($1.83\times {10}^{-8}\pm 0.06\times {10}^{-8}{\mathrm{cm}}^2/\mathrm{s}$) is constant over all phantoms. We note that the relatively large error bars are due to the protocol that was chosen which, as explained above, involves the lifting and replacement of the probe as it is done for the in vivo measurements.

Fig. 4

The measured ${\mu }_{\mathrm{a}}$, ${\mu }_{\mathrm{s}}^{\prime }$, and BFI by MD-DCS versus expected values in the liquid phantom measurement. On each box, the central mark is the median and the edges of the box are the 25’th and 75’th percentiles over all 10 relocating measurements. (a) The measured (fitted by MD-DCS) ${\mu }_{\mathrm{s}}^{\prime }$ versus expected values and their increase in each step of titration. The linear fitting equation of fitted ${\mu }_{\mathrm{s}}^{\prime }$ versus expected is presented ($\text{slope}=0.9$, $\text{intercept}=0.4$). (b) The measured (fitted by MD-DCS) absorption coefficient of all phantoms is the same and in the range of the expected ${\mu }_{\mathrm{a}}$ for water (${\mu }_{\mathrm{a}}^{785\mathrm{nm}}=0.02$ to $0.03{\mathrm{cm}}^{-1}$). (c) The measured (fitted by MD-DCS fitting) BFI is constant over all phantoms.

3.4.

Multidistance Diffuse Correlation Spectroscopy Fitting on the Longitudinal In Vivo Data from a Mouse Tumor

To validate the MD-DCS technique in a more realistic scenario where the heterogeneity of the medium optical properties may pose difficulties to a multidistance measurement, we have applied the method on a mouse tumor. The results of fitting values with both methods (MD-DCS and NIRS/DOS) for all three parameters over 24 days are presented in Fig. 5. Although the fitted ${\mu }_{\mathrm{a}}$ and ${\mu }_{\mathrm{s}}^{\prime }$ from NIRS/DOS and MD-DCS deviate slightly and MD-DCS has a significantly higher standard deviation, the fitted BFI from both fittings shows a fair agreement.

Fig. 5

The fitted values from both methods (MD-DCS and NIRS/DOS) for ${\mu }_{\mathrm{s}}^{\prime }$, ${\mu }_{\mathrm{a}}$, and BFI over 24 days on a mouse tumor. The central markers represent the mean values and the error bars show the standard deviation at each day of the measurement. (a) The fitted ${\mu }_{\mathrm{s}}^{\prime }$ from NIRS/DOS and MD-DCS. (b) The fitted ${\mu }_{\mathrm{a}}$ from NIRS/DOS and MD-DCS. (c) The fitted BFI from hybrid device and standalone MD-DCS. The blue stars show where the null hypothesis of equal means is rejected.

We have examined the null hypothesis that the BFI values measured by standalone MD-DCS and hybrid device have equal means. The null hypothesis was rejected for only 3 days which are shown by blue stars in Fig. 5.

4.

Discussion

In this study, we have introduced an algorithm that utilizes MD-DCS to simultaneously estimate the absolute values of static (reduced scattering and absorption coefficients) and dynamic (blood flow) properties of the tissue. The main goal of this algorithm is to be able to obtain a robust estimate of the BFI even without an independent measurement of the optical properties.

As it was previously shown⁷ in traditional DCS fitting (SD-DCS), the cross-talk between optical properties (${\mu }_{\mathrm{s}}^{\prime }$ and ${\mu }_{\mathrm{a}}$) and BFI does not allow the simultaneous fitting of all three parameters. Here, we show that the source-detector separation dependence of $g_1$ allows us to decouple completely (in case of noise-free data) or partially (under the influence of noise) the contribution of ${\mu }_{\mathrm{a}}$, ${\mu }_{\mathrm{s}}^{\prime }$, and BFI by fitting for multiple separations aiming to minimize a single cost function. Figure 1 illustrates this point in an objective manner where it is observed that in the case of the single distance fitting, the minimum of penalty function (${\chi }^2$) occurs over a wide range of values which can be translated as the high cross-talk between ${\mu }_{\mathrm{s}}^{\prime }$ and BFI in SD-DCS. By considering multiple separations, MD-DCS, ${\chi }^2$ is confined to a smaller area which leads to a better convergence of the algorithm.

To demonstrate that this can be utilized in a fitting algorithm, we have applied SD-DCS and MD-DCS to noise-free data for small, intermediate, and large source-detector separations (Figs. 2, 7, and 8) showing that MD-DCS can accurately retrieve all parameters while SD-DCS has considerable error even in noise-free data and the error increases by increasing the source-detector separation. Since the data are noise-free and the signal-to-noise ratio is not affected by the optode distance, these results demonstrate that, as expected from the cost function, SD-DCS is not a reliable method to simultaneously fit multiple parameters. These results are in agreement with previous work.⁷

Furthermore, SD-DCS has a considerable error when dealing with realistic noise in the simulated data as shown in Figs. 3, 9, and 10. On the other hand, in the noise-added simulated data, we have observed that in the small source-detector regime, MD-DCS fits all parameters with high accuracy (less than 5% error), but by increasing the source-detector separations, the accuracy of the fitted parameters decreases.

We attribute this to the interplay of several factors: for the larger separations the signal-to-noise ratio is worse, the autocorrelation curve decays faster, and the bin size for each delay time is finite. As shown in Eq. (1), the decay of the autocorrelation function depends on the optical properties and BFI via $K(\tau )$ and the source-detector separation $\rho$. The couplings of the optical properties, in particular ${\mu }_{\mathrm{s}}^{\prime }$, and BFI are evident in this equation but the strength of the coupling depends on both $\rho$ and the relative values of the $\tau$ dependent and independent parts of $K$. As the signal-to-noise ratio gets worse and the curve decays faster, the sensitivity to different parameters decreases, which becomes even worse by the finite bin size. In the future, a more advanced theoretical analysis may allow for better algorithms to decouple the different parameters.

Our next step was to accept these findings, limit ourselves to smaller source-detector separations, and validate MD-DCS on liquid phantoms with ${\mu }_{\mathrm{s}}^{\prime }$ titration while ${\mu }_{\mathrm{a}}$ and BFI were kept constant. As was shown in Fig. 4, the measured (fitted by MD-DCS) ${\mu }_{\mathrm{a}}$, ${\mu }_{\mathrm{s}}^{\prime }$, and BFI agreed with the expected values. We once more stress that this measurement has validated the algorithm in challenging conditions which resemble the in vivo measurements. The liquid phantoms were measured by a handheld probe and for each measurement the probe was removed and then replaced 10 times.

We have then tested the algorithm on our previous measurements⁹ from a mouse tumor undergone a longitudinal study after receiving a cancer therapy. In Fig. 5, we have compared the fitted ${\mu }_{\mathrm{a}}$ and ${\mu }_{\mathrm{s}}^{\prime }$ by NIRS/DOS and MD-DCS, which has demonstrated that the results of MD-DCS alone are noisier than a dedicated NIRS/DOS measurement. This was expected since the NIRS/DOS system used a multidistance, broadband dataset with optimized detector sizes for an improved signal-to-noise ratio and a great deal of redundancy. MD-DCS, on the other hand, used small ($\approx 6\mu \mathrm{m}$) single-mode fibers which are not optimized to collect the maximum intensity. However, the BFI results that were shown in Fig. 5 were in good agreement over the majority of the days with a comparable variance. We argue that these results are satisfactory since the goal of this algorithm is to obtain reliable BFI measurements rather than to replace NIRS/DOS.

Finally, we note that for the in vivo results we have used estimates of ${\mu }_{\mathrm{eff}}$ to constrain the optical properties. We have argued that the continuous-wave DCS laser and the mean intensity could also be used if the source-detector coupling coefficients are calibrated. This illustrates both a possible shortcoming, i.e., the need to constrain the values, and also a potential solution, i.e., the use of the MD-DCS data to estimate that value. To explore this further, in noise-added simulated data, we have tested all the source-detector separation ranges (small, intermediate, and large) and have evaluated the error in fitted BFI (${\epsilon }_{\mathrm{BFI}}$) deploying Eq. (4) with and without the knowledge of ${\mu }_{\mathrm{eff}}$ as a prior. The estimated error for noise-added data is presented in Fig. 6. It shows that in simulated homogenous media, providing ${\mu }_{\mathrm{eff}}$ does not affect the accuracy of MD-DCS considerably when the conditions are near-ideal. However, MD-DCS is highly sensitive to the source-detector separation regime (small, intermediate, and large) and its accuracy dramatically decreases in large optode distances. This means there is more room for further improvement in the experimental implementation of this algorithm.

Fig. 6

The error in fitted BFI by MD-DCS on noise-added simulated data for different source-detector separation regimes (small, intermediate, and large). For each source-detector separation range, the estimated error with and without providing ${\mu }_{\mathrm{eff}}$ is presented.

5.

Conclusion

We have investigated the possibility of simultaneous extraction of absorption coefficient (${\mu }_{\mathrm{a}}$), reduced scattering coefficient (${\mu }_{\mathrm{s}}^{\prime }$), and BFI of the tissue using MD-DCS. Numerical simulations demonstrate that the fitting is more robust in relatively smaller source-detector separations (from 2 to 20 mm). MD-DCS is then validated on tissue simulating phantoms. Finally, it is applied for in vivo measurements on a mouse tumor. The results suggest that MD-DCS can be used to decouple the absorption and scattering coefficients from BFI in small source-detector separations to allow simpler, DCS-only approaches in cases where BFI is the primary parameter of interest.