Impact of velocity distribution assumption on simplified laser speckle imaging equation

: Since blood flow is tightly coupled to the health status of biological tissue, several instruments have been developed to monitor blood flow and perfusion dynamics. One such instrument is laser speckle imaging. The goal of this study was to evaluate the use of two velocity distribution assumptions (Lorentzian- and Gaussian-based) to calculate speckle flow index (SFI) values. When the normalized autocorrelation function for the Lorentzian and Gaussian velocity distributions satisfy the same definition of correlation time, then the same velocity range is predicted for low speckle contrast (0 < C < 0.6) and predict different flow velocity range for high contrast. Our derived equations form the basis for simplified calculations of SFI values.


Introduction
Noninvasive monitoring of a therapeutic intervention is desired to provide the clinician or scientist with insight into the efficacy of the treatment. Since blood flow is tightly coupled into the health status of biological tissue, several instruments have been developed to monitor blood flow and perfusion dynamics, including laser Doppler flowmetry [1,2] and Doppler optical coherence tomography [3,4].
Fercher and Briers [5] proposed a technique for flow visualization by means of singleexposure photography. It uses the spatial statistics of time-integrated speckle (essentially the speckle contrast) and was originally developed for the measurement of retinal blood flow. Recently, this laser speckle imaging (LSI) method has been developed into a digital, quasi real-time technique for the mapping of blood flow [6,7]. LSI has been used to monitor noninvasively blood flow and perfusion dynamics in the brain [1, [8][9][10][11][12], retina [5,13], and skin [14][15][16][17]. We have employed LSI to monitor blood flow dynamics during photodynamic therapy [16] and have observed marked changes in the measured speckle flow index (SFI) values which is proportional to the blood flow velocity. Cheng and Duong [18] recently proposed a simplified imaging equation that addresses a reported discrepancy [19] between the Fercher and Briers speckle imaging equation and that of Goodman [20,21]. The authors proposed that this equation is valid for T/τ c > 100, where T is image exposure time and τ c is speckle correlation time.
The goal of this study was to evaluate the use of two velocity distribution assumptions to calculate SFI [6]. To achieve this goal, we employed Mandel´s definition of correlation time [20].

Lorentzian and Gaussian velocity distributions
When coherent light is used to illuminate an object, a speckle pattern is evident. If the object contains moving optical scatterers (i.e., red blood cells), the speckles will fluctuate in time. A time-integrated image will show a reduction in the speckle contrast due to averaging of the intensity of each speckle. This reduction in contrast is related to the flow velocity. The higher the velocity, the faster the intensity fluctuations, and therefore, the larger the reduction in contrast that occurs over a given exposure time. Based on speckle statistics, Fercher and Briers [5] derived the following relationship between the speckle contrast (C) and the normalized autocorrelation function of the remitted light: where σ is the variance, I is the mean and ) (t γ is the normalized autocorrelation function of the remitted light. For a Lorentzian velocity distribution: Substitution of Eq. 2 into Eq. 1 yields [5]: Briers et al. [22] performed a similar analysis for a Gaussian velocity distribution assumption, obtaining: We postulate that Briers et al. employed the following expression for the normalized autocorrelation function representing a Gaussian velocity distribution: Substitution of Eq 5 into Eq. 1 yields Eq 4. Figure 1 demonstrates that the use of the two velocity distribution assumptions result in different SFI values (SFI = 1/τ c ∝ blood flow velocity) for given values of C. Moreover, for T/ τ c > 2, Eqs 3 and 4 can be simplified to the following algebraic expressions: where τ cl and τ cg are the correlation times for the Lorentzian and Gaussian approximations, respectively. This result is in agreement with the simplified imaging equation obtained by Cheng and Duong [18]. From Eq 6: suggesting that for T/τ c > 2, the velocity predicted by the Gaussian approximation is π times the velocity predicted by the Lorentzian approximation.
In Ref [6], the authors developed an experiment for which the actual decrease in the computed speckle contrast was from 0.15 to 0.01. Adopting the Lorentzian model (Eq. 3) the authors obtained an approximated velocity change from 55 to 120 μ m/s and adopting the Guassian model (Eq. 4), it suggested a velocity change from 100 to 200 μ m/s; this discrepancy is predicted by the Eq. 7. These results, obtained by Briers et al [6], suggest that the velocity distribution model does impact the relationship between speckle contrast and τ c . In the next paragraphs we re-derived an equation for the Gaussian model to address the

Rederived Gaussian equation
In many applications it is desirable to have a precise definition of the term "correlation time". Such a definition can be made in terms of ) (t γ , but there are different expressions of ) (t γ reported in the literature [23]. We employed Mandel´s definition of the correlation time: It is straightforward to show that Eq 2 satisfies Eq 8, but Eq 5 does not. To address this discrepancy, we propose an alternate expression for the Gaussian normalized autocorrelation function: which satisfies Eq. 8. Substituting Eq. 9 into Eq. 1, we obtain: which is similar to Eq. 4. Furthermore, the argument of the error function in Eq. 10 is similar to that described previously by Goodman (Eq. 6.1-20 in [20]) for a Gaussian spectral profile.
For T/ τ c > 2, Eq. 10 becomes: where τ cga is the correlation time for the re-derived Gaussian-based speckle imaging equation. Note that this equation is identical to that derived using the more common Lorentzian velocity distribution assumption (Eq. 6, top row).  (12) demonstrating that use of either the Lorentzian or rederived Gaussian velocity distribution assumption predict different τ c (and hence SFI) values. Although T/τ c <<1 is not encountered in typical LSI experiments, Eq. 12 demonstrates that only in this range of ratios (T/τ c <<1) will the velocity distribution assumption affect the mapping between speckle contrast and τ c . From Goodman's theory on integrated intensity [20], which takes into account the triangular averaging of the correlation function [19], it is straightforward to obtain: The relationship between τ clg and C is similar to that derived by Cheng and Duong [18]. Once again, Eq. 15 suggests that for T/ τ c > 2 (i.e., 0 < C < 0.6, see Fig. 2), Goodman's theory predicts the same SFI range for the Lorentzian and Gaussian velocity distributions. Moreover, from Eqs. 6 (top row), 11 and 15, the SFI values predicted by Goodman`s model are directly proportional to the Lorentzian and the rederived Gaussian-based speckle imaging equations.  In Ref [20], the signal-to-noise ratio associated with measurement of (1/C 2 ) is given by: The (S/N) rms associated with Eq. 15 is greater than 2 . At higher (C > 0.6) speckle contrast values, (S/N) rms is less than 2 , which is unacceptably low for practical application. Eq. 16 is valid for both the Briers and Goodman models. Briers et al. [6] first noted that experimental C values did not reach the theoretical limit of unity for completely stationary objects; they instead observed a maximum value of 0.6. Experimental data from Yuan et al. [10] also achieved a maximum C value of 0.6. Dunn et al.
[1] and Bolay et al. [12] presented experimental data taken from cortical tissue with maximum C values of ~0.15. In experimental LSI data that we acquire from rodent dorsal window chamber models [15][16][17], we typically observe C values greater than 0.6 in less than 1% of the pixels (Fig. 3). It is important to note that measured C values may differ among LSI instruments due to differences in parameters such as quality of imaging optics and camera, coherence length of incident light source, etc. Nevertheless, we believe these studies collectively justify the rationale for other researchers employing LSI to utilize the proposed simplified speckle imaging equation (Eq. 15). An advantage of Eq. 15 over either use of approximate solutions or look-up tables to extract τ c from the speckle imaging equation is that it represents an exact analytical solution for C < 0.6. Cheng and Duong [18] stated that typically-encountered ratios of T/τ c are 100 to 400. Values greater than 100 are encountered in clearly defined blood vessels, but the ratio is much lower for pixels that map to poorly-perfused regions of tissue. For example, a speckle contrast of 0.6, which is encountered experimentally, maps to a ratio of two. Our analysis demonstrates that, even for such a low ratio, the simplified imaging algorithm can be used with high accuracy.

Conclusions
When the normalized autocorrelation function for the Lorentzian and Gaussian velocity distributions satisfy the same definition of correlation time, the same SFI values are predicted for low contrast (0 < C < 0.6) conditions and different values predicted for high contrast conditions. A similar trend is predicted by Goodman`s model. Previously, the only ways to (C)