Directly measuring absolute flow speed by frequency-domain laser speckle imaging

Hao Li; Qi Liu; Hongyang Lu; Yao Li; Hao F. Zhang; Shanbao Tong

doi:10.1364/OE.22.021079

1. Introduction

Laser speckle contrast imaging (LSCI) is an attractive functional biomedical imaging method that maps blood flow by using simple setups [1–3]. In LSCI, random speckle patterns from bio-tissues are generated under coherent illumination and imaged by a digital camera. Blood flow speed is estimated by motion induced image blur through the speckle contrast K, which is the ratio of the speckle standard deviation to its mean intensity (K = σ/‹I›). The simplicity and practicality of LSCI has led to a wide range of applications in biomedical research, clinical diagnosis, and surgical guidance in dermatology, dentistry, ophthalmology, and neurophysiology [4, 5].

Despite its success in biomedical applications, traditional LSCI has limited quantitative analytical capabilities because speed estimation from speckle blur is strongly affected by experimental conditions (e.g., ambient light, illumination, and camera performance) and light scattering in biological samples (e.g., type of flow distribution, static speckle from surrounding tissues or other scatterers, and multiple scattering). As a result, directly measuring absolute flow speed is fundamentally difficult using LSCI [5, 6].

The complications of speed estimation from speckle pattern blurring can be described by dynamic light scattering theory. The statistical properties of moving particles can be described by the autocovariance of scattered light [7, 8]:

C_{t} (τ) = 〈 {[I (t) - 〈 I 〉]}^{*} [I (t + τ) - 〈 I 〉] 〉 = {〈 I {(t)}^{*} I (t + τ) 〉}_{t} - {〈 I 〉}^{2}

where I(t) is the intensity of scattered lights as a function of time t; τ is the time lag and ‹···› denotes time averaging operation. C_t(τ) statistically describes the correlation between light intensities with a time lag τ. The motion of scatterers increases the speckle variation and decreases the intensity similarity and thus, causes the decay of C_t(τ) over τ. When the speckle pattern is “fully developed” in ideal conditions, the K value ranges from 0 to 1 and is related to the exposure time T and C_t(τ) by [9]

K^{2} = \frac{2}{T {〈 I 〉}^{2}} \int_{0}^{T} (1 - \frac{τ}{T}) C_{t} (τ) d τ .

Assuming that the type of speed distribution is known (e.g. Brownian motion or Gaussian distribution [10]), the equation for C_t(τ) will be known and the speed value can be uniquely determined by K. However, in practice, static speckle, environmental background contributions, and noise from the camera always present in LSCI images. These non-ergodic signal sources reduce speckle contrast to be much less than 1. In addition, the C_t(τ) equation is highly dependent on the flow distribution model. One K value may lead to multiple results when using different flow models. Altogether, these factors result in a complicated correlation between speckle contrast and the absolute flow speed. Even under careful pre-calibrations using controlled phantoms, it is still challenging to directly estimate the absolute flow speed in LSCI [5, 10, 11].

In the past years, researchers have been investigating measurement methodologies as well as blood flow models to improve the estimation of flow speed in LSCI [10–12]. Recently, multiple-exposure speckle imaging (MESI) has shown capability in reducing the influence of static scattering and improving the accuracy of decorrelation time estimation [13, 14]. Furthermore, as pointed out by Thompson and Andrews [15], the C_t(τ) function obtained from MESI can be transformed to a “Doppler-like” spectrum. As a result, absolute flow estimation could be achieved via the same algorithm used in laser Doppler flowmetry (LDF). Although these efforts improved the evaluation of the speckle contrast, most LSCI applications only quantify the relative blood flow change in arbitrary unit [11, 16]. To the best of our knowledge, no direct measurement of absolute speed from speckle images has been reported yet [11].

In this work, we present a novel method to analyze speckle variation in frequency domain, referred to as frequency-domain laser speckle imaging (FDLSI), that can obtain the autocovariance function curve without the influence of static scattering or illumination intensity. We also built an analytical flow model, upon which absolute flow speed can be fitted from the autocovariance function. The phantom experiment showed that the flow speed obtained by FDLSI was reliably within a 10% deviation from the preset actual values. Furthermore, in vivo blood vessel imaging showed that FDLSI was immune to illumination changes. FDLSI enables quantitative comparison of blood flow imaging from different subjects at multiple time points, which could further extend the limits of traditional LSCI for chronic and longitudinal applications.

2. Theory

2.1 Autocovariance of speckle intensity in frequency domain

I(t) can be written in frequency domain [8]:

{\begin{cases} \tilde{I} (ω) = \frac{1}{2 π} \int I (t) e^{- i ω t} d t \\ I (t) = \int \tilde{I} (ω) e^{i ω t} d ω \end{cases} .

Substituting Eq. (3) into Eq. (1), autocovariance can be written in the frequency domain based on the Wiener-Khinchin theorem [17]:

\begin{array}{l} C_{t} (τ) = \lim_{T \to \infty} \frac{1}{2 T} \int_{- T}^{T} (\iint (\tilde{I} {(ω_{1})}^{*} e^{- i ω_{1} t}) (\tilde{I} (ω_{2}) e^{i ω_{2} (t + τ)}) d ω_{1} d ω_{2}) d t - {〈 I 〉}^{2} \\ = \iint δ_{ω_{1}, ω_{2}} \tilde{I} {(ω_{1})}^{*} \tilde{I} (ω_{2}) e^{i ω_{2} τ} d ω_{1} d ω_{2} - {〈 I 〉}^{2}, \\ = \int {| \tilde{I} (ω) |}^{2} e^{i ω τ} d ω - {〈 I 〉}^{2} \end{array}

where δ_ω1_,_ω2 is the Kronecker delta function.

Static speckle only contributes to the DC term in Eq. (4) and can be directly subtracted after Fourier transform. Furthermore, the AC term which contains information of the speckle variation can be normalized to eliminate influences from different illumination intensities [14]. Thus, from frequency-domain analysis, we can obtain a “pure” autocovariance function. The estimation of flow speed will be “immune” to either static speckle or illumination intensity.

C_t(τ) in Eq. (4) is different form autocovariance in LDF model, which is obtained from the Fourier transform of Ĩ(ω) [18]. We believe this mathematical difference is from their distinct assumptions of the origin of intensity variations. In LDF, it is assumed that photons scattered from moving scatterers change their frequencies as a result of Doppler effect. The interference between the frequency-shifted and unshifted photons creates an intensity variation [18, 19]. On the other hand, either traditional LSCI or FDLSI assumes no frequency shift in scattered photons. The speckle variation is considered mainly from the scattering pattern changes caused by particle motions [11, 19]. Therefore, in LDF, autocovariance reflects the electric field frequency shift, and in LSCI, it reflects scattering strength and pattern variation.

2.2 Autocovariance of moving particles with single speed

We first investigate a single particle moving in x-y plane at position (x₀, y₀) at t = 0, with a velocity of v along the x axis. Under coherent illumination and ignoring multi-scattering, the scattered electric field amplitude distribution can be written as δ (x-x₀-vt, y-y₀)*U_sc(x,y) where U_sc(x,y) is the scattering potential and * represents convolution integral [20]. Without loss of generality, we assume that the resolution of imaging system is comparable or lower than scatterer’s radius, as applied in most LSCI systems for perfusion imaging [15, 21, 22] The shape of the scatterers has less impact on scattering and we can simplify the scattering amplitude as δ (x-x₀-vt, y-y₀). In an imaging system with finite numerical aperture (NA), the electric filed amplitude distributions on image plane is [23]

\begin{array}{l} U (x, y) = δ (x - M x_{0} - M v t, y - M y_{0}) * h (x, y), \\ = h (x - M x_{0} - M v t, y - M y_{0}) \end{array}

where M is the magnification factor, h(x,y) is the impulse response of the image system. If the imaging aberration is negligible, h(x,y) can be expressed by the point spread function of the system [23]:

h (x, y) = A_{0} \frac{J_{1} (2 π \frac{NA}{λ M} \sqrt{x^{2} + y^{2}})}{2 π \frac{NA}{λ M} \sqrt{x^{2} + y^{2}}},

where J₁ is the 1st order Bessel function of the first kind, A₀ is a constant and λ is the wavelength of the illuminating light. h(x,y) is also known as Airy diffraction pattern. The radius of Airy disk r₀ on image plane is 0.61 × λM/NA [23]. r₀ also defines the resolution of the image system by r₀/M.

Considering multiple particles under the same speed, normalized C_t(τ) in an arbitrary point (x₁, y₁) can be obtained by substituting Eqs. (5) and (6) into Eq. (1):

C_{t} (v, τ) = 〈 \frac{| h {(x_{1} - M x_{0}, y_{1} - M y_{0})}^{*} h (x_{1} - M x_{0} - M v τ, y_{1} - M y_{0}) |}{{| h (x_{1} - M x_{0}, y_{1} - M y_{0}) |}^{2}} 〉 .

Considering a sufficient number of particles with the same velocity v but with random initial positions, the time averaging in Eq. (7) can be transferred to the space averaging over the entire object plane:

C_{t} (v, τ) = \frac{\iint {| h {(x, y)}^{*} h (x + M v τ, y) |}^{2} d x d y}{\iint {| h (x, y) |}^{2} d x d y},

where x₁-Mx₀ and y₁-My₀ are replaced by x and y respectively.

According to Eq. (8), C_t (v,τ) can be approximately considered as the overlap area of two Airy discs with a distance of Mvτ as shown in Fig. 1(a).

Fig. 1 Simulation results. (a) Optical configuration of FDLSI imaging system. The Airy disc is on the imaging plan. The image pattern displacement reflects the particle movement. Autocovariance function decay curve under single speed can be fitted well by Gaussian function (Adj. R-square: 0.99946). (b) Decorrelation distance l₀ as a function of NA⁻¹ numerically calculated from Eq. (8). r₀: Airy disk radius of the image system. (c) l₀ as a function of CCD single pixel size a. l₀ keeps constant when a is smaller than r₀ and gradually approaches to a-2r₀ when a is larger than r₀. (d) Comparison of autocovariance function decay curves under different flow models. Rapid flow model: ordered flow with Gaussian distribution. Slow flow model: hybrid flow described by Eq. (13). In single speed, rapid flow and slow flow models, mean speed is 0.5 mm/s. In rapid Gaussian flow, slow Gaussian flow and Brownian motion models, root-mean-square speed is 0.4 mm/s. (1 × , NA = 0.2).

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In order to derive the analytical expression of autocovariance, we define decorrelation displacement l₀ = Mvτ₀, where τ₀ is the decorrelation time when C_t (v,τ) = e^-¹C_t (v,τ₀). l₀ denotes the distance that the overlap area of two Airy discs is e^-¹ of the original area. As shown in Fig. 1(a), the profile of C_t (v,τ) can be approximately expressed in Gaussian function:

C_{t} (v, τ) = e^{\frac{- {(M v τ)}^{2}}{l_{0}^{2}}} .

Figure 1(b) compares l₀ and r₀ as a function of NA. Similar with r₀, l₀ is inversely proportional to NA. By comparing two slopes, we can derive the relation l₀~0.67r₀ = 0.41Mλ/NA.

In the above discussion, we assumed the pixel size of the camera is much smaller than the speckle size, which is not always true. Taking the pixel size into consideration, Eq. (8) becomes

{\begin{cases} C_{t} (v, τ) = \frac{\iint {| h^{'} {(x, y)}^{*} h^{'} (x + M v τ, y) |}^{2} d x d y}{\iint {| h^{'} (x, y) |}^{2} d x d y} \\ h^{'} (x, y) = \iint h (x - x^{'}, y - y^{'}) d x^{'} d y^{'} \end{cases},

where the integration of x’, y’ take place on one pixel area.

We set the pixel as a square with the side length of a and calculated the corresponding l₀. As shown in Fig. 1(c), when a is smaller than or comparable with r₀, the pixel-size effect on l₀ is not obvious. When a is much larger than r₀, l₀ will follow the relation l₀ = a-2r₀. Figure 1(c) suggests that a should be at least comparable with r₀. This conclusion is consistent with the previous reports that sampling of CCD should satisfy Nyquist-criterion [6]. When a< r₀, we can simply use Eq. (8) to replace Eq. (10).

It is worthwhile to note that in Eqs. (8) and (9), v only represents the speed in the x-y plane. C_t (v,τ) is insensitive to motions along the z direction. Also, Eq. (9) is derived from point scatterers. It is only valid when the optical resolution is comparable or worse than the scatterer size. When image is taken under a high optical resolution or with larger scatterers, Eq. (9) needs to be modified by considering the scattering potential [20].

2.3 Analytical blood flow model

Given a speed distribution P(v), the autocovariance on image plane can be written as [24, 25]

C_{t} (τ) = \int_{0}^{+ \infty} P (v) C_{t} (v, τ) d v .

The blood flow distribution varies drastically in different type of vessels. In microvascular circulation, it is believed that the distribution is a combination of Brownian motion and highly ordered motion with Gaussian distribution [10, 11]. Thus, we propose a P(v) by considering both distributions as

P (v) = \frac{2}{\sqrt{π} {\tilde{v}}^{3}} {(v - v_{0})}^{2} e^{- \frac{{(v - v_{0})}^{2}}{{\tilde{v}}^{2}}},

where v₀ is the mean speed and ṽ is the root mean square speed. With Eqs. (9) and (12), we can obtain the analytical expression of Eq. (11) by calculating the Cauchy principle value (the integration in (-∞, 0) is considered to be 0):

C_{t} (τ) = e^{- \frac{M^{2} v_{0}^{2} τ^{2}}{l_{0}^{2} + M^{2} {\tilde{v}}^{2} τ^{2}}} [\frac{l_{0}^{3}}{{(l_{0}^{2} + M^{2} {\tilde{v}}^{2} τ^{2})}^{\frac{3}{2}}} + \frac{2 M^{4} v_{0}^{2} {\tilde{v}}^{2} l_{0} τ^{4}}{{(l_{0}^{2} + M^{2} {\tilde{v}}^{2} τ^{2})}^{\frac{5}{2}}}] .

Its typical decay curve is compared with that of Brownian motion and Gaussian distribution in Fig. 1(d). We can see that the curve of our model lies between those for Gaussian distribution and Brownian motion distribution, which is consistent with the “rigid-body mode” derived by Duncan et al. [10]. Equation (13) has only two unknown variables (v₀ and ṽ), so we can fit the experimental C_t (τ) curve to obtain absolute flow speed.

3. Phantom experiment on absolute flow speed measurement

The schematic of the experimental setup is shown in Fig. 2. Deionized water containing polystyrene particles (0.1% w/v, average diameter: 3.2 μm) were pumped into a polyethylenetube (PE-50, outer diameter: 0.97 mm; inner diameter: 0.58 mm). The tube was illuminated by a 780 nm CW laser and imaged by a high speed CCD (microscope: Leica, S8-APO, 4 × , NA = 0.2; CCD: Basler, acA2040-180km, pixel size 5.5 × 5.5 μm2, 2 × 2 binning). This imaging system was also used in later experiments.

Fig. 2 Experimental setup

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As shown in Fig. 3(a), the selected stripe (10 × 480 pixels) for analysis covers the cross section of the tube. Speckle intensity I(t) was recorded for 1 sec (fps: 1k Hz; exposure time: 200 μs). The C_t(τ) curve on each pixel was calculated from I(t) by Eq. (4) and then fitted to Eq. (13) to compute absolute flow speed v₀.

Fig. 3 Phantom test of FDLSI to quantify absolute flow speed. (a) Image of a micro-tube under coherent illumination. The white box highlights the capture area by FDLSI. (b) Flow profiles at different speeds. Solid lines are parabolic fitting results. (c) Comparison between FDLSI measured center flow speed and preset actual values obtained from syringe pump.

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Figure 3(b) shows the spatial distribution of absolute mean speed profiles. The real speed values were obtained from syringe pump. Inside the tube, the speed profiles fit closely to theoretical parabolic curves. Figure 3(c) is the fitted center speed as a function of preset real speed. The deviation is less than 10%, indicating a good accuracy in measuring absolute flow speed using FDLSI.

The temporal resolution of our experiments was 1 second, which is acceptable in most biomedical applications [26]. This resolution can be improved by using fewer frames at the cost of lower accuracy. In cases that higher temporal resolution is needed, we suggest using FDLSI as a pre-calibration for conventional LSCI by sharing the same optical setup. The lowest detectable flow speed is 0.008 mm/s, which is determined by the background noise in Fig. 3(b). The upper limit is determined when the decorrelation time is comparable with thetime sampling interval of the camera. In our setup, the upper limit was estimated to be ~2mm/s. Higher upper limits could be achieved by using high-frame-rate camera, e.g., 10k fps provides measureable speeds of up to 20 mm/s. That dynamic range could cover flow measurement of arterioles with diameters up to 50 μm and most venules [27].

5. Robustness testing in vivo

Figure 4 demonstrates the robustness of FDLSI in varying illumination conditions in in vivo experiments. Figure 4(a) is the schematic of our experimental setup. A Sprague Dawley rat was anesthetized (chloral hydrate intraperitoneal injection, 70 mg/mL, initial dose: 5 mL/kg, following dose: 1.5 mL/kg per hour), depilated on the right ear, and immobilized under the stereomicroscope. The corresponding animal experiment protocols were approved by the Animal care and use committee of Med-X Institute of Shanghai Jiao Tong University. The laser diode was placed in one of four different angles (θ₁ = 52.43°, θ₂ = 58.47°, θ₃ = 63.09°, θ₄ = 66.50°) to illuminate the right ear (see Fig. 2). Those angles were arbitrarily chosen to mimic illumination angles and intensity variations in practice. The blood vessels of the right ear were imaged by both traditional LSCI (temporal domain, exposure time: 5 ms, total frames: 100) and FDLSI. During the whole imaging period (<5 min), the animal remained in a stable anesthesia state without any external interventions. Therefore, we hypothesized that blood flow speed was stable.

Fig. 4 In vivo FDLSI measurements of blood flow speed under different illumination conditions. (a) Schematic of experimental setup. The illumination condition was altered by changing the illumination angle. (b) Traditional LSCI image of rat ear overlaid with a pseudo-colored FDLSI image. (c) Center flow speed measured by FDLSI. (d) 1/K² measured by traditional LSCI. (e) and (f) Blood flow distributions under angle θ₂ and θ₃ by FDLSI and traditional LSCI respectively. Solid lines are parabolic fitting results.

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Figure 4(b) is the image of the blood vessels under the illumination angle of θ₁. Both LSCI and FDLSI showed similar flow distribution profiles. Figures 4(c) and 4(d) show one vessel’s center blood flow under four illumination angles by FDLSI and traditional LSCI, respectively. The flow profiles corresponding to θ₂ and θ₃ are shown in Figs. 4(e) and 4(f), respectively. We can observe that 1/K², representing the flow speed in LSCI ([10, 28]), increases almost two- fold under different illuminations, indicating the severe influence from illumination condition changes. The results agree with previous reports [10] and partiallyexplain why conventional LSCI was mainly used for quantifying blood flow speed with arbitrary units. It is difficult to directly compare the LSCI values from different subjects, illumination conditions, or imaging operators [5, 10, 11].

In contrast, the flow speeds measured by FDLSI exhibit very small deviations, with almost identical centerline speeds at 1.3 mm/s and flow profiles regardless of illumination changes (see Figs. 4(c) and 4(e)). Therefore, it can be used to quantitatively compare blood flow data from different experiments and imaging systems. It is noted that in both Figs. 4(e) and 4(f), the speed values in the area off the visible vessels are not zero, which might be caused by blood flow in deep capillaries and other biological motions [21, 29].

Compared with tradition LSCI, FDLSI dynamically measures the variation of laser speckle. FDLSI requires a high speed camera instead of low-cost video-rate camera. Fortunately, consumer-grade high speed cameras became recently available (e.g., Casio ZR800). By equipping such cameras, FDLSI would keep LSCI’s advantage as a simple and low-cost modality. Compared with MESI, FDLSI should obtain equivalent C_t(τ) curves. However, unlike MESI, FDLSI does not require adjusting illumination intensity to fit exposure time and thus, should demonstrate higher robustness to imaging illumination [5, 13]. Similar to tradition LSCI, FDLSI is a two dimensional imaging method and can’t resolve flow speed perpendicular to the object plane. Future work of integrating scanning speckle imaging into FDLSI may provide an opportunity of quantitative measurement of three-dimensional flow velocity [11, 30].

7. Conclusion

In conclusion, we presented an FDLSI method that could directly measure the absolute flow speed in laser speckle imaging. Phantom experiments showed that the measured speed was reliably within a 10% deviation from the real values. In vivo blood vessel imaging showed that FDLSI was immune to illumination condition changes. FDLSI, therefore, could substantially expand the biomedical applications of current LSCI methods, particularly, into chronic or longitudinal studies and enable the quantitative comparison of blood flow from different experiments or subjects.

Acknowledgment

The authors sincerely acknowledge the generous financial supports from the Chinese “111 Project” (B08020), the National Natural Foundation of China NSFC (61371018).and the US National Science Foundation CBET (CBET-1055379, CBET-1066776).

References and links

1. A. F. Fercher and J. D. Briers, “Flow visualization by means of single exposure speckle photography,” Opt. Commun. 37(5), 326–330 (1981). [CrossRef]

2. L. M. Richards, S. M. S. Kazmi, J. L. Davis, K. E. Olin, and A. K. Dunn, “Low-cost laser speckle contrast imaging of blood flow using a webcam,” Biomed. Opt. Express 4(10), 2269–2283 (2013). [CrossRef] [PubMed]

3. O. Yang and B. Choi, “Laser speckle imaging using a consumer-grade color camera,” Opt. Lett. 37(19), 3957–3959 (2012). [CrossRef] [PubMed]

4. A. Ponticorvo, D. Cardenas, A. K. Dunn, D. Ts’o, and T. Q. Duong, “Laser speckle contrast imaging of blood flow in rat retinas using an endoscope,” J. Biomed. Opt. 18(9), 090501 (2013). [CrossRef] [PubMed]

5. A. K. Dunn, “Laser speckle contrast imaging of cerebral blood flow,” Ann. Biomed. Eng. 40(2), 367–377 (2012). [CrossRef] [PubMed]

6. J. C. Ramirez-San-Juan, E. Mendez-Aguilar, N. Salazar-Hermenegildo, A. Fuentes-Garcia, R. Ramos-Garcia, and B. Choi, “Effects of speckle/pixel size ratio on temporal and spatial speckle-contrast analysis of dynamic scattering systems: Implications for measurements of blood-flow dynamics,” Biomed. Opt. Express 4(10), 1883–1889 (2013). [CrossRef] [PubMed]

7. D. A. Boas and A. K. Dunn, “Laser speckle contrast imaging in biomedical optics,” J. Biomed. Opt. 15(1), 011109 (2010). [CrossRef] [PubMed]

8. B. J. Berne, Dynamic light scattering: With applications to chemistry, biology and physics (Courier Dover Publications, 1976).

9. R. Bandyopadhyay, A. S. Gittings, S. S. Suh, P. K. Dixon, and D. J. Durian, “Speckle-visibility spectroscopy: A tool to study time-varying dynamics,” Rev. Sci. Instrum. 76(9), 093110 (2005). [CrossRef]

10. D. D. Duncan and S. J. Kirkpatrick, “Can laser speckle flowmetry be made a quantitative tool?” J. Opt. Soc. Am. A 25(8), 2088–2094 (2008). [CrossRef] [PubMed]

11. D. Briers, D. D. Duncan, E. Hirst, S. J. Kirkpatrick, M. Larsson, W. Steenbergen, T. Stromberg, and O. B. Thompson, “Laser speckle contrast imaging: Theoretical and practical limitations,” J. Biomed. Opt. 18(6), 066018 (2013). [CrossRef] [PubMed]

12. D. D. Duncan, S. J. Kirkpatrick, and R. K. K. Wang, “Statistics of local speckle contrast,” J. Opt. Soc. Am. A 25(1), 9–15 (2008). [CrossRef] [PubMed]

13. A. B. Parthasarathy, W. J. Tom, A. Gopal, X. J. Zhang, and A. K. Dunn, “Robust flow measurement with multi-exposure speckle imaging,” Opt. Express 16(3), 1975–1989 (2008). [CrossRef] [PubMed]

14. S. M. S. Kazmi, A. B. Parthasarthy, N. E. Song, T. A. Jones, and A. K. Dunn, “Chronic imaging of cortical blood flow using multi-exposure speckle imaging,” J. Cereb. Blood Flow Metab. 33(6), 798–808 (2013). [CrossRef] [PubMed]

15. O. B. Thompson and M. K. Andrews, “Tissue perfusion measurements: Multiple-exposure laser speckle analysis generates laser doppler-like spectra,” J. Biomed. Opt. 15(2), 027015 (2010). [CrossRef] [PubMed]

16. J. Senarathna, A. Rege, N. Li, and N. V. Thakor, “Laser speckle contrast imaging: Theory, instrumentation and applications,” IEEE Rev. Biomed. Eng. 6, 99–110 (2013). [CrossRef] [PubMed]

17. J. W. Goodman, Speckle phenomena in optics: Theory and applications (Roberts and Company Publishers, 2007).

18. R. Bonner and R. Nossal, “Model for laser doppler measurements of blood flow in tissue,” Appl. Opt. 20(12), 2097–2107 (1981). [CrossRef] [PubMed]

19. J. D. Briers, “Laser doppler, speckle and related techniques for blood perfusion mapping and imaging,” Physiol. Meas. 22(4), R35–R66 (2001). [CrossRef] [PubMed]

20. J. Lim, H. F. Ding, M. Mir, R. Y. Zhu, K. Tangella, and G. Popescu, “Born approximation model for light scattering by red blood cells,” Biomed. Opt. Express 2(10), 2784–2791 (2011). [CrossRef] [PubMed]

21. J. D. Briers and S. Webster, “Laser speckle contrast analysis (lasca): A nonscanning, full-field technique for monitoring capillary blood flow,” J. Biomed. Opt. 1(2), 174–179 (1996). [CrossRef] [PubMed]

22. P. Miao, H. Y. Lu, Q. Liu, Y. Li, and S. B. Tong, “Laser speckle contrast imaging of cerebral blood flow in freely moving animals,” J. Biomed. Opt. 16(9), 090502 (2011). [CrossRef] [PubMed]

23. J. W. Goodman, Introduction to fourier optics (Roberts and Company Publishers, 2005).

24. A. Oulamara, G. Tribillon, and J. Duvernoy, “Biological-activity measurement on botanical specimen surfaces using a temporal decorrelation effect of laser speckle,” J. Mod. Opt. 36(2), 165–179 (1989). [CrossRef]

25. J. Briers and A. Fercher, “Laser speckle technique for the visualization of retinal blood flow,” in The Max Born Centenary Conference(International Society for Optics and Photonics, 1983), pp. 22–28. [CrossRef]

26. S. M. Daly and M. J. Leahy, “‘Go with the flow ’: A review of methods and advancements in blood flow imaging,” J Biophoton. 6(3), 217–255 (2013). [CrossRef] [PubMed]

27. Y. P. Ma, A. Koo, H. C. Kwan, and K. K. Cheng, “On-line measurement of the dynamic velocity of erythrocytes in the cerebral microvessels in the rat,” Microvasc. Res. 8(1), 1–13 (1974). [CrossRef] [PubMed]

28. H. Y. Cheng and T. Q. Duong, “Simplified laser-speckle-imaging analysis method and its application to retinal blood flow imaging,” Opt. Lett. 32(15), 2188–2190 (2007). [CrossRef] [PubMed]

29. W. Maï, C. T. Badea, C. T. Wheeler, L. W. Hedlund, and G. A. Johnson, “Effects of breathing and cardiac motion on spatial resolution in the microscopic imaging of rodents,” Magn. Reson. Med. 53(4), 858–865 (2005). [CrossRef] [PubMed]

30. R. Z. Bi, J. Dong, and K. Lee, “Deep tissue flowmetry based on diffuse speckle contrast analysis,” Opt. Lett. 38(9), 1401–1403 (2013). [CrossRef] [PubMed]

Directly measuring absolute flow speed by frequency-domain laser speckle imaging

Abstract

1. Introduction

2. Theory

2.1 Autocovariance of speckle intensity in frequency domain

2.2 Autocovariance of moving particles with single speed

2.3 Analytical blood flow model

3. Phantom experiment on absolute flow speed measurement

5. Robustness testing in vivo

7. Conclusion

Acknowledgment

References and links

Cited By

Figures (4)

Equations (13)

Optics Express