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 , 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}) :

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

*β*), 3 for hydrodynamic diffusion (

*D*_{eff},

*τ*_{c} and

*β*)). For hydrodynamic diffusion to be a better model than Brownian diffusion, the corresponding F-number must exceed a critical F-value, which for

*p* < 0.05 is 3.91 (as calculated using the finv function in Matlab (Mathworks, Natick,MA)). This is true for 80.1% of the individual measurements from our data set, and the whole set F-number is 33.8, corresponding to a highly significant p-value of 4 × 10

^{−8} (calculated using fcdf in Matlab).

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. 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

*αD*_{eff} = 1.07

*αD*_{b} + 2.1 × 10

^{−7}(mm

^{2}/s).

*α**D*_{eff} has a substantially linear relationship with

*αD*_{b}, with a nearly-zero intercept, indicating both parameters can serve as relative blood flow indices, but

*αD*_{eff} is expected to provide a more accurate absolute measure. Encouragingly, by assuming

*α* = 0.1 [

23] for brain tissue, our estimated effective diffusion coefficients fall within the same range as those determined from video microscopy blood flow studies (10

^{−5} – 10

^{−4} mm

^{2}/s) [

16,

21,

22].

With respect to

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

*μ*s, with an average of 1.3

*μ*s (however values lower than ~ 0.4

*μ**s* are not reliable because of the limited time resolution of our correlator). A rough estimation of expected

*τ*_{c} values may be obtained from hydrodynamic diffusion theory. For a rigid spherical particle the velocity decorrelation characteristic time is on the order of

*τ*_{v} =

*ρa*^{2}*/**η*, where

*a* is the particle size,

*ρ* is the fluid density and

*η* is the fluid viscosity. Assuming a red blood cell diameter of

*a* = 4

*μm*,

*η* = 1.2 cP [

22] and the density of water,

*τ*_{v} = 13

*μ*s, within an order of magnitude of our measurements. While the deformable nature of RBCs and the complexity of blood flow make this comparison less meaningful, our results do indicate the ballistic motion time scale (

*τ* <

*τ*_{c}) of the RBC hydrodynamic diffusion process is observable in most DCS measurements. Thus the full

Eq. (2) should be used to reduce variance in the obtained blood flow velocity estimates and to characterize the diffusive transition time scale. To further characterize

*τ*_{c} we plot in )

*τ*_{c} vs. flow velocity, represented by

*αD*_{eff} and b)

*τ*_{c} vs. inter-RBC distance (expected to be proportional to the inverse cube root of the blood hemoglobin concentration

*HGB*^{−1/3}), for all the measurements where

*R*^{2} of the hydrodynamic fit was greater than 0.999 (giving us confidence in the estimation of

*τ*_{c}). We observe a weak but statistically significant decrease in

*τ*_{c} with increased blood flow, as well a weak decrease with increased inter-particle distance that does not reach a p<0.01 significance level. The inverse proportionality between

*τ*_{c} and

*D*_{eff} (and hence blood flow) could be explained as an acceleration of the interaction time scale. It is also expected from the short

*τ* Taylor expansion of the mean square displacement expression (

Eq. (2)):

. Assuming the short

*τ* ballistic displacement takes the form

Δ

*r*^{2}(

*τ*)

=

*v*^{2}*τ*^{2}, the early ballistic velocity

*v* is proportional to

. As a first approximation, one could expect

*v* to be fairly constant (i.e. dependent on blood viscosity and temperature, but not on the speed of the bulk flow), suggesting an inverse-proportional relationship between

*τ*_{c} and

*D*_{eff}. A similar intuitive explanation is not apparent for the observed decrease in

*τ*_{c} with increased inter-particle distance. This trend is likely due to complex blood flow mechanisms. Note though that

*τ*_{c} is known to be affected by particle deformability in colloids [

24], thus it may become a useful tool to monitor RBC mechanical properties. Such changes can occur due to physiological and pathological mechanism, such as the fetal-to-adult hemoglobin replacement, and disease states such as sickle-cell anemia.