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**|**Biomed Opt Express**|**v.2(7); 2011 July 1**|**PMC3130588

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Biomed Opt Express. 2011 July 1; 2(7): 2047–2054.

Published online 2011 June 24. doi: 10.1364/BOE.2.002047

PMCID: PMC3130588

Martinos Center for Biomedical Imaging, Massachusetts General Hospital, Harvard Medical School, Building 149, 13th Street, Charlestown, Massachusetts 02129, USA

Received 2011 April 4; Revised 2011 June 13; Accepted 2011 June 17.

Copyright ©2011 Optical Society
of America

This is an open-access article distributed under the terms of the Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 Unported License, which permits download and redistribution, provided that the original work is properly cited. This license restricts the article from being modified or used commercially.

This article has been cited by other articles in PMC.

We suggest that Diffuse Correlation Spectroscopy (DCS) measurements of tissue blood flow primarily probe relative red blood cell (RBC) motion, due to the occurrence of multiple sequential scattering events within blood vessels. The magnitude of RBC shear-induced diffusion is known to correlate with flow velocity, explaining previous reports of linear scaling of the DCS “blood flow index” with tissue perfusion despite the observed diffusion-like auto-correlation decay. Further, by modeling RBC mean square displacement using a formulation that captures the transition from ballistic to diffusive motion, we improve the fit to experimental data and recover effective diffusion coefficients and velocity de-correlation time scales in the range expected from previous blood rheology studies.

Light scattering methods have been used to probe the motion of suspended particles for the last several decades, in either single [1] or multiple scattering regimes [2]. The latter technique, known as diffusing wave spectroscopy (DWS) has been extended to heterogeneous multiple-scattering media by Boas *et al.* [3, 4] and has gained acceptance as a method to measure perfusion in bulk tissue under the name of Diffuse Correlation Spectroscopy (DCS) [5]. By measuring the intensity fluctuations of light diffusely reflected from tissue, DCS can offer a measure of microvascular blood flow and has been successfully validated against other blood flow measurement techniques, such as arterial spin labeling (ASL) magnetic resonance imaging (MRI) [6–8], Doppler Ultrasound [9, 10], Xenon-CT [11] and fluorescent microspheres [12]. Given the three dimensional micro-topography of vasculature, red blood cell motion has been expected to have the characteristics of ballistic random flow with a uniform spatial velocity distribution [13]. Surprisingly though, the good agreement seen in the validation studies cited above requires modeling particle movement as a diffusion random-walk. Two recent studies have probed the characteristics of DCS signals more deeply, looking at whether the pulsatile nature of blood flow and/or the presence of extravascular tissue shearing contribute to the observed diffusive scatterer dynamics. Using parallel detection of independent speckles Dietsche *et al.* [14] have measured the auto-correlation of light intensity fluctuations with 26 ms temporal resolution, and noted up to 240% variation in the auto-correlation decay time during one heart pulsation, as well as a somewhat stronger curvature of the decay curve at the systolic maximum flow compared to the diastolic minimum flow. While, this “super-diffusive” decay appears to have a slight ballistic flow quality, the data presented by the authors in Fig. 7 of Ref. [14] indicates the scatterer motion remains predominantly diffusive throughout the pulsation cycle. Further, Ninck *et al.* [15], using an *ex-vivo* artificially perfused porcine kidney model has shown that, in the absence of blood, the DCS signal carries the signature of extravascular tissue shearing. Nevertheless, during pulsatile blood perfusion, the correlation decay curves are approximately described by diffusion even as they vary during the pulsation cycle because the contribution of extravascular tissue shearing is small. Taking into account the results of these studies, Ninck *et al.* [15] conclude in their discussion that the discrepancy between expected red blood cell ballistic flow and diffusion dynamics measured by DCS remains unexplained, but perhaps DCS signals might reflect erythrocyte diffusion in the direction perpendicular to flow, as has been observed through particle tracking experiments [16], with a magnitude proportional to the flow shear rate.

In this paper we revisit the assumptions made in obtaining blood flow estimates from DCS data. In particular we argue that the occurrence of multiple sequential scattering within blood vessels would render the DCS measurements sensitive to relative red blood cell motions. As noted above, these motions are diffusive in nature, and their magnitude scales nearly linearly with blood flow velocity [16], in good agreement with published DCS studies. We also show that an effective hydrodynamic diffusion model capturing the transition between early ballistic and subsequent diffusive motion results in a modest, but significant improvement in the fit to experimental data. In addition to an effective diffusion coefficient proportional to blood flow velocity, this model also provides a measure of the particle velocity randomization time scale, a potentially useful tool for blood rheology studies addressing dynamics faster than the millisecond range currently accessible using video microscopy methods.

In the context of DWS/DCS, the normalized temporal electric field **E**(**t**) auto-correlation function *g*_{1}(*τ*) = **E**(**0**)**E**^{*}(*τ*)/|**E**(**0**)|** ^{2}** is given by [4]:

$${g}_{1}\mathrm{(\tau )\mathrm{=\mathrm{{\int}_{0}^{\infty}P(s)\mathrm{\text{exp}\mathrm{[-1/3\mathrm{{k}_{0}^{2}\mathrm{\Delta {r}^{2}\mathrm{(\tau )}\mathrm{\frac{s}{l}}]\mathrm{\mathit{\text{ds}}}}}}}}}}$$

(1)

where *τ* is the correlation delay time, *P*(*s*) is the normalized probability distribution of detected photon path lengths *s*, *k*_{0} = 2*π*/*λ*, with *λ* the wavelength of the illumination laser, Δ*r*^{2}(*τ*) is the mean-squared displacement of scattering particles in time *τ*, and *l* = 1/*μ _{s}* is the photon random-walk step length which is equal to the inverse of the scattering coefficient

Red blood cell dynamics have been the subject of numerous blood rheology studies. As also noted by Ninck *et al.* [15], video microscopy has been used *ex vivo* to track the motion of hemoglobin-depleted ghost RBCs [16] and whole blood [21] in microchannels, as well as *in vivo* in rat venules [22]. It was found that RBCs undergo shear-induced displacements in the bulk flow frame of reference that can be characterized by an effective diffusion coefficient *D*_{eff} on the order of 10^{−5} mm^{2}/s, much higher than the Brownian diffusion coefficient expected for the RBCs in plasma ~ 5 × 10^{−8} mm^{2}/s [22]. Most importantly, *D*_{eff} appears to scale linearly with the shear rate. We postulate that this mechanism underlies the measurement of tissue blood flow using Diffuse Correlation Spectroscopy.

While the simplified Brownian displacement formulation used in DCS literature (Δ*r*^{2}(*τ*) = 6*D _{b}τ*) appears to work well, it assumes that the ballistic to random-walk hydrodynamic transition in the RBC diffusion occurs at time scales shorter than those probed by DCS measurements. Since there is no data to support this assumption, we remove it by using the Langevin formulation for RBC mean squared displacement [1]:

$$\Delta {r}^{2}\mathrm{(\tau )}\mathrm{=\mathrm{6{D}_{\text{eff}}\mathrm{(\tau \mathrm{-\mathrm{{\tau}_{c}(1\mathrm{-\mathrm{\text{exp}(-\tau /{\tau}_{c})))}}}}}}}$$

(2)

where *D*_{eff} is the effective diffusion coefficient and *τ _{c}* is the time scale for the randomization of velocity vectors associated with RBC scattering events. By Taylor expanding Eq. (2), it can be shown that this formulation of the displacement term describes ballistic motion at short delay times, and diffusive motion at long delay times. Note that we are referring here to the short time scale ballistic motion contained within any diffusive process (including that of erythrocytes in the bulk flow frame of reference), and not to the bulk ballistic motion of erythrocytes in vasculature as seen in the laboratory frame of reference.

To evaluate the performance of the proposed model, we have applied it to the infant data previously reported by our group in Ref. [10]. Briefly, 11 premature infants, between 28 and 34.5 weeks gestational age were measured several times at weekly intervals, for a total of 66 study visits. At each visit, measurements were obtained in up to seven areas of the head using a handheld multi-distance probe with source detector distances of 1–2.5 cm. Both frequency domain diffuse reflectance as well as diffuse correlation spectroscopy data was acquired. A frequency domain instrument emitting 110 MHz modulated light at 8 wavelengths between 659 and 825 nm was used to estimate absolute optical properties, while the DCS system employed a long-coherence length laser operating at 785 nm (CrystaLaser, Reno, NV, USA) and four photon-counting avalanche photodiodes (Perkin-Elmer, Quebec, Canada) connected to a digital auto-correlator operating at delay times from 200 ns to 1 s (Correlator.com, Bridgewater, NJ, USA). Additional experimental details can be found in Ref. [10].

To analyze the DCS data we used the semi-infinite medium correlation diffusion formulation [5]:

$${G}_{1}\mathrm{(\rho ,\mathrm{\tau )\mathrm{=\mathrm{\frac{3{\mu}_{s}^{\prime}}{4\pi}\mathrm{(\frac{\text{exp}(-K{r}_{1})}{{r}_{1}}\mathrm{-\mathrm{\frac{\text{exp}(-K{r}_{2})}{{r}_{2}}})}}}}}}$$

(3)

where
${K}^{2}\mathrm{=\mathrm{3{\mu}_{a}{\mu}_{s}^{\prime}\mathrm{+\mathrm{\alpha {\mu}_{s}^{\prime 2}{k}_{0}^{2}\Delta {r}^{2}\mathrm{(\tau )}}}}}$, *α* is the probability of scattering from a moving particle (RBC), *ρ* is the source-detector separation,
${r}_{1}\mathrm{=\mathrm{{({\rho}^{2}\mathrm{+\mathrm{{z}_{0}^{2})}}1/2}^{}}}$, *r*_{2} = (*ρ*^{2} + (*z*_{0} + 2*z _{b}*)

$${g}_{2}\mathrm{(\rho ,\mathrm{\tau )\mathrm{=\mathrm{1\mathrm{+\mathrm{\beta \mathrm{{({g}_{1}\mathrm{(\rho ,\mathrm{\tau ))}2};\mathrm{{g}_{1}\mathrm{(\rho ,\mathrm{\tau )\mathrm{=\mathrm{|\frac{{G}_{1}\mathrm{(\rho ,\mathrm{\tau )}{G}_{1}\mathrm{(\rho ,\mathrm{0)}}|}}{}}}}}}}^{}}}}}}}}$$

(4)

Figure 1 shows a sample intensity auto-correlation curve *g*_{2}(*τ*) that exemplifies the different fits obtained using displacement formulations corresponding to simplified Brownian diffusion, hydrodynamic diffusion (Eq. (2)) and random flow, respectively. For this data, it is clear that only hydrodynamic diffusion provides a good match to the shape of the auto-correlation decay. The decay predicted by the simplified Brownian diffusion appears too “slow”, while the decay predicted by random flow is too “fast”, leading to incorrect estimation of the *β* factor as well as of the decay rate. To quantify the quality of the fit, we use the statistical metric “fraction of variance unexplained” (FVU), defined as the ratio of the model mean squared error to the variance of the experimental data. For the data in Fig. 1, the hydrodynamic diffusion model has the lowest residuals (FVU=0.04%), followed by Brownian diffusion (FVU=0.48%), and random flow (FVU=0.76%). The average values of FVU over the entire set of experimental measurements are 0.36% for hydrodynamic diffusion, 0.46% for Brownian diffusion, and 2.32% for random flow, respectively. As seen in previous studies, the simplified Brownian diffusion model is found to have significantly lower fit errors compared to random flow. The same direct comparison cannot be made between the simplified Brownian and full hydrodynamic diffusion models because of their different number of parameters. Instead, we perform a statistical F-test to determine if the improvement in the hydrodynamic model exceeds the reduction in unexplained variance expected from adding an additional fitting parameter (*τ _{c}*) :

$$F\mathrm{=\mathrm{\frac{({\mathit{\text{SSE}}}_{\mathit{\text{Db}}}\mathrm{-\mathrm{{\mathit{\text{SSE}}}_{D\text{eff}})}{\mathit{\text{SSE}}}_{D\text{eff}}}\frac{{\mathit{\text{DoF}}}_{D\text{eff}}}{({\mathit{\text{DoF}}}_{D\text{eff}}\mathrm{-\mathrm{{\mathit{\text{DoF}}}_{\mathit{\text{Db}}})}}}}{}}}$$

(5)

where *SSE* is the sum of the squares of the residuals and *DoF* is the number of degrees of freedom (number of correlation time bins (137 in our case) minus the number of model parameters (2 for Brownian diffusion (*D _{b}* and

Comparison of data fit errors using the Brownian diffusion (dotted, FVU=0.48%), hydrodynamic diffusion (solid, FVU=0.04%) and random flow (dashed, FVU=0.76%) mean square displacement models.

When Eq. (2) is substituted into Eqs. (3) and (4), a product forms between the probability of scattering from a red blood cell *α* and the diffusion coefficient. This quantity has been used as a “blood flow index” in DCS studies, because the value of *α* generally cannot be estimated independently. Figure 2 shows a scatter plot of the *αD*_{eff} vs. the *αD _{b}* values obtained from each of our measurements. We observe an approximate relationship of

Scatter plot of *αD*_{eff} vs. *αD*_{b} across the entire data set. The dotted line indicates the diagonal of the plot (ratio=1).

With respect to *τ _{c}*, we observed a range between 0.06 and 7.7

In conclusion we have provided a mechanistic explanation for the diffusion-like scatterer dynamics observed in diffuse correlation spectroscopy measurements of biological tissue based on multiple-scattering within individual blood vessels. We have also shown that experimental data is best fit with a hydrodynamic diffusion formulation that includes both ballistic flow at short times and diffusive motion at later times. Both the effective diffusion coefficients and the transition time scale are within an order of magnitude of expected values. Finally, our results imply that, in addition to tissue perfusion measurements, DCS may be a useful tool for blood rheology studies.

The authors acknowledge funding from NIH grants K99EB011889, R01HD042908, P41RR14075 and K99NS067050.

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