1. Introduction

Several optical approaches have been used for non-invasive blood flow measurements for the last several decades, in either single or multiple scattering regimes. The former technique, known as laser speckle contrast imaging (LSCI) [1–4], uses the spatio-temporal blurring of the speckle imaging defined as speckle contrast K to measure blood flow information in superficial tissues. LSCI is performed by the illumination of the biological tissue with a coherent light source and imaging of the reflected speckle by a camera. This technique allows the use of a simple experimental setup with the advantages of a relatively high spatio-temporal resolution, but under the condition of single or few scattering events it limits the photon penetration depth in the tissue [5].

On the other hand, diffuse correlation spectroscopy (DCS) [6, 7] has been developed for probing deep tissue blood flow by monitoring the light intensity fluctuations of the reflected diffuse speckle from the tissue. DCS offers continuous blood flow monitoring and has been extensively used for various clinical applications such as brain [8–10], cancer [11, 12] and muscle [13, 14]. Usually, DCS needs the single mode fiber with a smaller diameter, single photon counting avalanche photodiode and photon correlator to detect an independent speckle. To obtain a higher signal-to-noise (SNR), it is necessary to use multiple detectors to simultaneously acquire multiple independent speckles, which results in the increasing of the cost and complexity of DCS.

Recently, an alternative method called diffuse speckle contrast analysis (DSCA) [15–18] was developed for measuring blood flow in the deep tissue by the combination of LSCI and DCS. DSCA holds deep tissue probing capabilities of DCS and simple hardware instrument of LSCI. Generally, DSCA utilizes the dependence of measured speckle contrasts on source-detector distances or exposure time [19] to decouple the effects of absorption and scattering [20, 21] from the dynamics, and, therefore, extracts a quantitative estimate of blood flow in vivo [22]. Besides, the fast cameras with a high frame rate [23, 24] can rapidly improve acquisition speed of speckle images.

In fact, DSCA and DCS are examining different aspects of the same entity [25], i.e., the temporal electric autocorrelation function of the diffusely reflected speckles caused by the movement of red blood cells (RBCs). Currently, the most common model for describing the RBCs motion is a Brownian diffusion-like process, which provides a better characterization of the autocorrelation function signals in a range of subjects and tissue types than the expected random flow model which describes the RBCs motion is ballistic. Based on Brownian motion model, we have developed and tested a thorough speckle contrast model [26–28] in DSCA for quantitative measurement of the RBCs motion. Although the blood flow index (BFI) in Brownian motion model has been shown to be approximately proportional to absolute blood flow BF_abs, the interpretation of the BFI is complicated due to its units of cm²/s and the value of this proportionality between BFI and BF_abs is not clear in DSCA. Therefore, establishing the quantitative relationship between BFI in DSCA and BF_abs is urgently necessary and significant.

For this quantification, firstly it is necessary to clearly understand intravascular RBCs dynamics in the tissues. It has been found [29, 30] that RBCs undergo shear-induced displacement in blood flow, and the calculated electric autocorrelation function largely depends on this shear-induced diffusive RBCs motions, while the effect of correlated advective RBCs motion is relatively smaller [31–33]. The challenge of understanding the effect of diffusive and advective RBCs motions on DSCA signals stems from the fact that the speckle contrast signal is obtained from dynamic scattering inside the vessels and the vessels comprise only a small fraction of the tissue volume. Therefore, it is necessary to separate the intravascular scattering events from the extravascular scattering, and explicitly record the location of each dynamic scattering event in the vessels. Monte Carlo simulation [5, 31] can accurately model dynamic scattering event in the realistic and complex tissue.

In this paper, the reduction in speckle contrast due to the relative contribution of diffusive and correlated advective RBCs motions is studied by theory and Monte Carlo simulations. We derive the linear approximation expression for speckle contrast in the heterogeneous tissue with varying vessel radii and blood flow profiles, and establish the quantitative relationship between the slope k_slope of this linear function with the absolute blood flow BF_abs. Meanwhile, we employ three-dimensional Monte Carlo simulations of photons propagation through tissues to support these results. This linear approximation expression can be also used to model the speckle contrast signals in the realistic tissue with the heterogeneous blood flow profiles and a complex vascular network.

2. Theory and methods

2.1 Intravascular red blood cell motion

The question of the proper statistical model for describing intravascular RBCs motion and ultimately relating this model to the measured electric autocorrelation function or speckle contrast have been the subject of numerous studies [30, 34–36]. Usually, RBCs displacement has been modeled as random flow with <Δr²(τ)> = V²τ², where <Δr²(τ)> is the mean-squared displacement (MSD) of RBCs in time τ and V² is the second moment of the velocity of RBCs, or as Brownian motion with <Δr²(τ)> = 6D_Bτ, where D_B is the effective Brownian diffusion coefficient. Unexpectedly, Brownian motion model leads to better description of the experimental results in validation studies compared with random flow model.

It should be noted that both models mentioned above have generally assumed that the dynamic scattering events with RBCs in vessels are uncorrelated. This assumption is valid if the photons scatter only once inside a vessel and then encounter another vessel before their direction becomes randomized by the scattering from extravascular tissue. In fact, the scattering length of the photons inside the vessel at the common laser wavelengths (790-810nm) used in DCS or DSCA is ~12μm [37], which is smaller than some vessel diameters in the tissue. This leads to the breakdown of this uncorrelated scattering assumption because the photons in a large vessel will most likely undergo multiple scattering events which are in fact highly correlated. When multiple scattering in a single vessel is present, the contribution of each scattering event to dynamic information depends on the velocity difference between subsequent correlated scattering events [38].

To account for the effect of correlated RBCs motions on the speckle contrast signal, it is necessary to understand RBCs dynamic in the vessels. Usually the spatial flow speed profile of the vessels has the form as [39]

Equation (1) describes a general form of advective flow in vessels and is experimentally verified by the method of dynamic light scattering – optical coherence tomography (DLS-OCT) [33]. Usually, the flow profile of bigger vessels (R>20μm) [33] can be considered to be parabolic (i.e., m = 2) and there is more bluntness in smaller vessels. Meanwhile, recent studies [31, 32] found that in addition to advective motions RBCs also undergo shear-induced diffusive motions in the vessels. The shear-induced diffusion coefficient D(r) is proportional to the shear rate α_ss [40], i.e.,

Based on Eq. (1) and (2), we can determine the velocity of the location of each scattering event in the vessel taking into account both diffusive and advective RBCs motions. In the following section, we will show the effect of correlated RBC motion on electric autocorrelation function and speckle contrast.

2.2 Diffuse speckle contrast analysis (DSCA)

A speckle pattern is a random interference pattern generated when photons from coherent light are scattered in a random medium. If the scatterers in the medium are mobile, as in the case of RBCs, the time integrated speckle pattern recorded by the camera becomes blurred. DSCA qualifies this degree of blurring by the ratio of the measured standard variance (${\sigma }_I$) to the mean intensity (${\mu }_I$) in spatial or temporal domain, as

Formally, the transport of G₁(ρ,τ) in multiple scattering media is well modeled by the correlation diffusion equation (CDE) [7, 42]. Sakadžić et al. [32] recently considered the effect of the correlated RBCs motion with diffusive and advective dynamics on the CDE, and derived the expression of G₁(ρ,τ) in a realistic vascular network with a heterogeneous distribution of the vessels with different radii and flow velocities. For simplicity, we consider only one vessel type with the same radius R and blood flow in the tissue and the analytic solution of G₁(ρ,τ) for the semi-infinite media is given by [32]

Based on Eq. (4)-(9), we can determine the theoretical dependence of speckle contrast on the SD separation or exposure time under the condition of the combination of diffusive and convective RBCs motion. Meanwhile, we note that it is difficult to obtain the analytical expression for the speckle contrast function due to the complex computation process of integral equation, and the necessary steps to be taken to overcome it.

2.3 Linear approximation for diffuse speckle contrast analysis

Our previous works [27] have demonstrated that the relation between 1/K² and exposure time can be described by a linear approximation equation and the slope k_slope is equal to the inverse of the correlation time (τ_c) of autocorrelation function g₁(ρ,τ). In this section, we derive the expressions for the correlation time τ_c and this linear approximation model considering diffusive and advective motion.

To obtain correlation time in Eq. (5), we firstly note that the function G₁(ρ,τ) is derived from the radiative transfer equation using diffusion approximation. Since diffusion approximation is usually valid when the SD separation ρ is much larger than the transport mean-free path z₀, r₂ in Eq. (5) can be approximately given by

Meanwhile, we can introduce another approximation$\exp \left[-\kappa \left({\tau }_c\right)z_1^2/r_1\right]\approx 1-\kappa \left({\tau }_c\right)z_1^2/r_1$ when $\kappa \left({\tau }_c\right)z_1^2/r_1\ll 1$, which is usually satisfied at the larger SD separation. This procedure yields the expression for G₁(ρ,τ_c)

Equation (17) is a general result that determines the correlation time in DCS considering the combination of diffusive and advective RBCs motion. Meanwhile, we have demonstrated that the slope k_slope of the linear approximation model for DSCA is equal to the inverse of the correlation time [27], i.e., k_slope = 1/τ_c. Thus, the corresponding speckle contrast 1/K² can be approximatively expressed as

In order to clearly understand the relationship between the blood flow index k_slope in Eq. (19) and absolute blood flow BF_abs, it is important to have a clear definition of BF_abs used in DCS and DSCA. Usually BF_abs is defined as the volume of blood transiting through a given cross-sectional area per second, which is given by cross-sectional area times average RBCs speed, i.e., ${\text{BF}}_{\text{abs}}=\pi R^2{v}_{\text{av}}$. Therefore, a clear linear relationship between k_slope and BF_abs can be established by

From Eq. (20), it is apparent that the relative change in BF_abs can be rapidly obtained by calculating the corresponding change in k_slope. Meanwhile, knowledge of this proportionality in Eq. (20) is required to determine BF_abs form the measured k_slope. However, this proportionality is complicated and related to multiple parameters such as the vessel radius, RBCs rheology (a, m, and α_ss), optical properties of the tissue, δ_ves and SD separation, etc. Some parameters in tissue are now not accessible by experimental method and further studies are necessary to discuss this problem. Besides, an important assumption for deriving Eq. (20) is that the first order expansion $\exp \left[-\kappa \left({\tau }_c\right)z_1^2/r_1\right]\approx 1-\kappa \left({\tau }_c\right)z_1^2/r_1$ is a good approximation. In fact, this assumption is valid for the large SD separation, i.e., $\rho \gg 1/{{\mu }^{\prime }}_s$. Analytical and numerical computations of the influence of this error due to this approximation on k_slope are given in the following section.

2.4 Monte Carlo simulation

For studying photon traversal inside the heterogeneous tissue and considering the effect of correlated RBCs motion on the speckle contrast, Monte Carlo simulation can be used to track the consecutive scattering events in the tissue. Our Monte Carlo code is based on a three dimensional voxelized model developed in Ref [47]. Each voxel of the Monte Carlo geometry is assigned the optical properties that determine the distribution of photon scattering lengths and the degree of the absorption. The Henyey-Greenstein phase function with corresponding anisotropy g [48, 49] within the voxel containing the scattering event is used to calculate the scattering angle.

The point source of photons with a defined direction is incident on the surface of the medium. The photons exiting the medium at different separations from the point source are detected. For each detected photon, the vector of momentum transfer $\overrightarrow{q}$ and the radial location of each scattering in different vessels, along with the absorption weight in the medium are saved. The momentum transfer $\overrightarrow{q}$ is defined as $\overrightarrow{q}={\overrightarrow{k}}_s-{\overrightarrow{k}}_i$, where ${\overrightarrow{k}}_s$ and ${\overrightarrow{k}}_i$ are the scattered and incident field wavevector, respectively. $\overrightarrow{q}$ has a magnitude of $q=2k_0\sin \left(\theta /2\right)$, where $\theta$ is the scattering angle of the photon. The saved radial positon of the scattering event is used to calculate advective and diffusive dynamics of RBCs by Eq. (1) and (2). Based on these recorded values, the correlation function G₁(ρ,τ) can be given by summing over all detected photons [31]

3. MC simulation geometry and RBCs motion

To demonstrate the flexibility and validity of linear approximation model for DSCA in the heterogeneous media, a voxelized tissue-mimicking geometry in Fig. 1 is used in conjunction with Monte Carlo simulation to generate the speckle contrast measurements. Figure 1 shows the (x, z) cross section of the geometry with a size of 6 × 3cm. For simplicity, the blood vessels have the same radius R and are all oriented along the y-axis with an equal spacing h = 200μm [31] in x-axis and z-axis. The values of the radius R and the vessel spacing h determine the volume fraction of the vessels. If needed, other values for h can be further used to consider different volume fraction of the vessels. Therefore, the geometry has an infinite extent in y-axis due to the translation symmetry.

 

Fig. 1 (x, z) cross section of Monte Carlo simulation geometry.

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In our simulations, the blood flow profiles in Eq. (1) are assumed to be constant in time and approximately parabolic flow for the vessels with a larger radius (R>20μm), i.e., a = 1 and m = 2. In general, the vessels with a smaller radius tend to have a uniform flow profile and in this case RBCs motion is dominated by diffusive motion. Increasing the vessel radius and considering parabolic flow profiles can find the influence of correlated RBCs motion on speckle contrast signals. If needed, other values for a and m can be further used to consider different blood flow profiles in the future. Besides, the value of shear rate α_ss is set to 0.24 × 10⁻⁶mm² from Ref [33]. measured in vivo.

For each simulation, 10⁸ photons at a point are used to illuminate the surface of the geometry along z-axis and a 0.5mm × 0.5mm square region of interest (ROI) at different distances ρ from the source point is used as the detector. All of the information for calculating Eq. (21) and (4) are recorded by our MC simulation. Besides, the intravascular and extravascular optical properties at 800nm for a blood hematocrit 40% [37] are shown in Table 1.

Table 1. Optical properties for MC geometry

View Table

4. Results

4.1 Corrected k_slopeexpression

Firstly, we test the accuracy of the expression for correlation time τ_c using theoretical data as shown in Fig. 2. An important assumption in this expression is that there are smaller errors arising from truncating SD separation terms in the Taylor Series expansion $\exp \left[-\kappa \left({\tau }_c\right)z_1^2/r_1\right]$ to first order. This assumption is valid when using large SD separations ρ, i.e., $\rho \gg 1/{{\mu }^{\prime }}_s$. Therefore, it is necessary to quantify the range of SD separation for which this expression can be accurately employed. The relative error in τ_c due to this assumption is defined as

 

Fig. 2 Dependence of relative error in τ_c on (a)${\mu }_a$, (b)${{\mu }^{\prime }}_s$ and$\rho {{\mu }^{\prime }}_s$, and corresponding corrected electric field autocorrelation function results $\ln \left[g_1\left(\rho ,{\tau }_{cr}\right)\right]$ at the modified correlation time τ_cr. Note that other parameters do not induce the change of the relative error. The change in relative error only depends on ${\mu }_a$,${{\mu }^{\prime }}_s$ and SD separations.

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We control the fractional changes in all parameters used for the calculation of correlation time τ_c in Eq. (17) and isolate the influence of all parameters on the relative error in τ_c. The theoretical results show that the error in τ_c only depends on tissue optical properties (${\mu }_a$and${{\mu }^{\prime }}_s$) and SD separation, and the error signal is not sensitive to other parameters changes (such as v_max, a, m, etc.). As shown in Fig. 2, the change in relative error is greatest due to the combination term $\rho {{\mu }^{\prime }}_s$ changes, and the changes in absorption ${\mu }_a$ and scattering ${{\mu }^{\prime }}_s$ for the constant $\rho {{\mu }^{\prime }}_s$ have the negligible influence on the relative error. Meanwhile, the error is relatively small (<5%) for the range $\rho {{\mu }^{\prime }}_s\ge 10$.

In order to correct this error due to this assumption, the modified expression for correlation time τ_cr is introduced by

Figure 2 also shows the corresponding corrected electric field autocorrelation function results $\ln \left[g_1\left(\rho ,{\tau }_{cr}\right)\right]$ at the modified correlation time τ_cr. We note that this error is relatively small and can be negligible, even for dimensionless term approaching $\rho {{\mu }^{\prime }}_s=5$, which in turn implies higher accuracy and wider range for the modified τ_cr expression. Therefore, the corresponding expression for k_slope and BF_abs can be further corrected by Eq. (23)

4.2 Validation with MC simulation data

In this section, we validate the linear approximation model by MC simulation results. We vary RBCs speed (v_max from 1mm/s to 6mm/s with a step of 1mm/s) and vessel radius (R = 20, 25, 30, 35, 40μm) with a fixed vessel spacing (h = 200μm) in MC simulations to test the dependence of k_slope on BF_abs. For all simulations, the intravascular and extravascular optical properties at 800nm for a blood hematocrit 40% are shown in Table 1. Then the average absorption ${\mu }_a$and reduced scattering coefficients ${{\mu }^{\prime }}_s$ of the mimicking tissue are given by the volume fraction weighted average of the intravascular (${\mu }_{a,\text{in}}$and ${{\mu }^{\prime }}_{s,\text{ex}}$) and extravascular (${\mu }_{a,\text{ex}}$and ${{\mu }^{\prime }}_{s,\text{ex}}$) optical coefficients, i.e., ${\mu }_a={\delta }_{ves}{\mu }_{a,\text{in}}+\left(1-{\delta }_{ves}\right){\mu }_{a,\text{ex}}$ and ${{\mu }^{\prime }}_s={\delta }_{ves}{{\mu }^{\prime }}_{s,\text{in}}+\left(1-{\delta }_{ves}\right){{\mu }^{\prime }}_{s,\text{ex}}$, where ${\delta }_{ves}=\pi R^2/h^2$is the volume fraction of blood vessel. Besides, the RBCs dynamics in vessels and other parameters are detailed in Sec. 3.

Figure 3(a) and 3(b) show K² and 1/K² results from MC simulations and theoretical calculations, respectively. In this example, the vessel radius is set to 30μm and SD separation is 2cm. Note that the theoretical results are numerically obtained by Eq. (4)-(9). These results show that the theoretical speckle contrast model fits MC simulation results well and the linear relation between 1/K² and exposure time is obvious. Figure 3(c) shows the fitted values of k_slope versus BF_abs, compared with those calculated by the corrected relation using Eq. (24). These results in Fig. 3 demonstrate the validity of linear approximation model for DSCA and confirm that k_slope increases linearly with BF_abs.

 

Fig. 3 MC simulation and theoretical results for (a) K² and (b) 1/K² at different flow speeds v_max. The symbols and solid lines in (a) and (b) represent MC simulation and theoretical results, respectively. The symbols in (c) show the linear fitting results of k_slope versus BF_abs. The solid line in (c) is calculated by the corrected relation Eq. (24). Here, we use R = 30μm vessels and SD separation is 2cm.

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Figure 4 shows the calculated absolute blood flow BF_abs from the linear fitting k_slope results at different SD separations. Note that the changes in the absolute blood flow BF_abs are accomplished by varying the vessel radius (from 20 to 40μm) with a fixed flow speed v_max = 3mm/s and vessel spacing h = 200μm. The solid lines in Fig. 4 represent the expected BF_abs results for SD separations of 1.0, 1.5 and 2.0cm. It is clearly found that this slope k_slope of the linear approximation model predict well the absolute blood flow BF_abs. The change in BF_abs can be simply obtained by calculating the change in k_slope.

 

Fig. 4 Calculated BF_abs results for MC simulation and theoretical calculation versus linear fitting k_slope results. The solid lines at different SD separations are calculated by Eq. (24). The absolute blood flow BF_abs is accomplished by increasing the vessel radius from 20 to 40μm with a fixed flow speed v_max = 3mm/s and a fixed vessel spacing h = 200μm.

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5. Discussion

Diffuse speckle contrast analysis (DSCA) has been employed extensively in blood flow measurement, in large part because of its simplicity. Usually the calculated speckle contrast fluctuations are driven by the RBCs motion and thus by blood flow. The question as to which model best characterizes RBCs motion within vessels has been widely studied. In this paper, we derive linear approximation expression for 1/K² taking account both shear-induced diffusive and advective RBCs motions. To validate this theoretical model, we have performed Monte Carlo simulations in a simple tissue-mimicking geometry with parabolic flow profile and varying vessel radius. These theoretical and simulation results allow us to write a linear relation between the measured k_slope and absolute blood flow BF_abs. Meanwhile, in order to connect k_slope calculation to BF_abs some effects on this model need to be further discussed in actual applications.

It should be noted that our previous works [27] have deduced an analytical expression k_slope-DB for diffusive RBCs motions, i.e.,

It is worthwhile to determine the relative contribution of diffusive motion on k_slope in Eq. (24). We simply quantify this contribution by calculating the ratio between k_slope-DB and k_slope as shown in Fig. 5. Here we have used a fixed vessel volume fraction δ_ves = 0.02, parabolic flow profile, a shear rate α_ss = 0.24 × 10⁻⁶mm² and the same optical properties in Table 1. Figure 5 shows that increasing the vessel radius and decreasing SD separation both result in reduced importance of diffusive RBCs motion. Especially for the large vessel radius at the short SD separation [50], the advective RBCs motion may dominate k_slope. For the small vessel radius, k_slope mainly depends on diffusive RBCs motion. Furthermore, this contribution of diffusive motion also depends on other parameters of the vessels, such as shear rate α_ss and blood flow profile (a and m). So far the measured values of α_ss have large difference from in vivo and ex vivo [33, 40]. One can note that the higher the shear rate α_ss, the larger contribution of RBCs motion. The influence of blood flow profile on this contribution can be calculated by Eq. (24) and k_slope-DB. Equation (24) suggests an approach to determine the relative contributions of diffusive and advective RBCs motions on the reduction in speckle contrast.

 

Fig. 5 Relative contributions of diffusive RBCs motions on the k_slope for different vessel radiuses and SD separations.

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The source-detector (SD) separations used in MC simulation results are such that SD direction is perpendicular to the blood flow direction as shown in Fig. 1. In fact, the actual vessel directions in the tissue are random and complex. Figure 6(a) shows the effect of specific SD direction, which is perpendicular or parallel to vessel direction, and all used SD directions on the obtained speckle contrast 1/K² results from the same MC simulations. It is obvious that the results 1/K² tend to be the same and the corresponding absolute percentage changes in 1/K² compared with the perpendicular case are shown in Fig. 6(b). The relative error in 1/K² is no more than 2%. Meanwhile, Ref [31] have verified that the preferred direction of the vessels only introduce a small bias for the correlation function g₁(ρ,τ) compared with a truly random direction of the vessels. It is apparent that the calculated 1/K² at a certain SD separation doesn’t depend on the relative direction of SD separation with respect to vessel direction.

 

Fig. 6 Effect of SD direction with respect to vessel direction on the obtained speckle contrast 1/K² results from MC simulations. The red and black lines in (a) indicate 1/K² from SD separations that is perpendicular and parallel to vessel direction, respectively. The blue lines in (a) indicate 1/K² from SD separations in all directions. The corresponding percentage change in 1/K² at different SD separations compared with perpendicular case is shown in (b). Here, we use R = 30μm vessels with a fixed flow speed v_max = 3mm/s and a fixed vessel spacing h = 200μm.

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For simplicity, we neglect the variation in radius and in flow speed among different vessels, and consider only one type of the vessel in this paper. In fact, when performing DSCA or DCS measurements in the tissue, the vessels encountered are different and complicated. For instance, capillaries with a small radius (<5μm) tend to be have similar velocities, but veins, venules, arterioles, and arteries all have different radii and flow speeds. Usually the photons come across multiple vessels along their path, each with different speeds and radiuses, and often interact with extravascular compartments. These contributions taken together cause the decay of speckle contrast function. Note that the analytic solutions for G₁(ρ,τ) [32] in Eq. (5)-(7) also can be used to predict DSCA signals in a realistic vascular network with a heterogeneous distribution of vessels with different radii and blood flow velocities. In this case, F(τ) in Eq. (7) can be expressed as [32]

Our results show that we can linearly relate the measured k_slope to absolute blood flow BF_abs. Note that the proportionality is related to the vascular morphology, RBCs rheology, optical properties of the tissue, and SD separation. In order to accomplish the quantitative measurement of BF_abs from the measured k_slope, we need to have a detailed knowledge of these parameters. However, some parameters now are not readily accessible. But the progress in the experimental methods may make these parameters available in the future. Further research is needed to study the quantitative measurement of BF_abs.

6. Conclusion

In summary, we have presented the theoretical derivations for linear approximation model in DSCA that consider both diffusive and correlated advective RBCs motion, and demonstrated that the slope k_slope of this linear model has a linear relation with absolute blood flow BF_abs, which are in agreement with our Monte Carlo simulation results. Through this expression for k_slope, we can determine the relative contribution of diffusive motion on the reduction of speckle contrast. We expect further studies to use this relation to obtain a direct measurement of BF_abs in the more complex and realistic configurations of the tissue.