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OPTICS LETTERS / Vol. 39, No. 3 / February 1, 2014
Simple correction factor for laser speckle imaging of flow dynamics J. C. Ramirez-San-Juan,1,* R. Ramos-Garcia,1 G. Martinez-Niconoff,1 and B. Choi2,3,4 1
2
3
Optics Department, INAOE, Luis Enrique Erro No. 1, Tonantzintla, Puebla 72840, Mexico Beckman Laser Institute and Medical Clinic, Department of Surgery, University of California, Irvine, 1002 Health Sciences Road East, Irvine, California 92612, USA
Department of Biomedical Engineering, University of California, Irvine, 3120 Natural Sciences II, Irvine, California 92697, USA 4 Edwards Life Sciences Center for Advanced Cardiovascular Technology, University of California, Irvine, 2400 Engineering Hall, Irvine, California 92697, USA *Corresponding author:
[email protected] Received November 6, 2013; revised December 20, 2013; accepted December 23, 2013; posted December 24, 2013 (Doc. ID 200844); published January 30, 2014 One of the major constraints facing laser speckle imaging for blood-flow measurement is reliable measurement of the correlation time (τC ) of the back-scattered light and, hence, the blood’s speed in blood vessels. In this Letter, we present a new model expression for integrated speckle contrast, which accounts not only for temporal integration but spatial integration, too, due to the finite size of the pixel of the CCD camera; as a result, we find that a correction factor should be introduced to the measured speckle contrast to properly determine τC ; otherwise, the measured blood’s speed is overestimated. Experimental results support our theoretical model. © 2014 Optical Society of America OCIS codes: (120.6150) Speckle imaging; (170.3880) Medical and biological imaging. http://dx.doi.org/10.1364/OL.39.000678
Light propagation in scattering media produces speckle patterns. If the media contain moving scatterers, the scattered intensity field will fluctuate in proportion to the speed of scattering centers. Typically, CCD cameras are used to image the ensuing pattern. If the camera exposure time is large compared to the speckle correlation time, then the speckle visibility is reduced. Using this fact, Fercher and Briers [1] employed speckle imaging of tissue to determine blood flow. Due in part to the simplicity and low cost of this approach, researchers have rapidly integrated laser speckle imaging (LSI) in their studies, which cover a wide range of applications, including ophthalmology [2], dermatology [3,4], dentistry [5,6], and neurobiology [7,8], among others. The methods to extract blood-flow information from the imaged speckle patterns have been refined over the years [9,10]. For example, Parthasarathy et al. [11] recently derived a new relationship between speckle contrast and speckle correlation time that takes into account contributions from stationary and dynamic scattering centers. In this Letter, we demonstrate that the size of the camera pixel plays an important role in blood flow measurements. We use principles first introduced by Goodman [12] to derive a new model expression that accounts for the finite size of camera pixels. We demonstrate that, if the finite size is not taken into account, the correlation time of the backscattered light is underestimated, and hence blood flow is overestimated. We present experimental data that support our new model. The electric field re-emitted from a scattering object depends on the superposition of fields associated with the spatiotemporal distribution of optical scattering centers within the object. The Siegert relation [10] describes the relationship between the electric field and intensity autocorrelation functions (g1 and g2 , respectively): 0146-9592/14/030678-04$15.00/0
g2 τ 1 jg1 τj2 :
(1)
Starting with this equation, Bandyopadhyay et al. [10] derived a second-generation relationship between speckle contrast K and correlation time τc , which takes into account the finite size of the optical detector: K2 β
e−2x − 1 2x ; 2x2
(2)
where β is a correction factor [13], x T∕τc , and T is the camera’s exposure time. Recently, research groups [10,11,13,14] demonstrated the need to further modify the model to account for the presence of light scattered from stationary optical scatterers. The scattered electric field is the superposition of a fluctuating (E f ) plus a static (E s ) component [11]: Et E f te−iwt Es e−iwt ;
(3)
where ω is the optical frequency of the excitation source. Using the Siegert relation, Parthasarathy et al. [11] derived the following third-generation relationship between K and τc : K β
1∕2
1∕2 e−2x − 1 2x e−x − 1 x 4ρ1 − ρ 1 − ρ2 ρ2 2 2 2x x
Cn;
(4)
where ρ I f ∕I f I s is the fraction of total light that interacts with moving scatterers, I f hE f E f i is the intensity of light interacting with moving scatterers, I s hE s E s i is the intensity of light interacting with stationary scatterers, and C n is a term that accounts for noise contributions to the measurement. We propose a © 2014 Optical Society of America
February 1, 2014 / Vol. 39, No. 3 / OPTICS LETTERS
related approach to study the effects of stationary and moving scatterers on the remitted speckle pattern. We model the re-emitted electric field as a superposition of a stationary but spatially dependent component E s x0 ; y0 and a fluctuating component E f x0 ; y0 ; t, which has scattered at least once from a moving scatterer. Similar to Eq. (3), the resultant electric field is Ex0 ; y0 ; t E f x0 ; y0 ; te−iwt E s x0 ; y0 e−iwt :
(5)
Following the approach of Boas [14], we substituted Eq. (5) into Eq. (1) to arrive at the following equation: g2 Δx0 ; Δy0 ; τ 1 jg1 Δx0 ; Δy0 ; τj2 hEx01 ; y01 ; tE x01 Δx0 ; y01 Δy0 ; t τi2 1 α hEx0 ; y0 ; tE x0 ; y0 ; ti 1
1
1
α1 − ρ2 jg1;s Δx0 ; Δy0 j2 C 2n ;
hE f x01 ; y01 ;tE f x01 Δx0 ; y01 Δy0 ;t τi g1;f Δx0 ;Δy0 ; τ hE f x01 ;y01 ; tE f x01 ;y01 ; ti hE x0 ; y0 E x0 Δx0 ;y0 Δy0 i . (7) g1;s Δx0 ;Δy0 s 1 1 0s 0 1 0 10 hE f x1 ;y1 E s x1 ; y1 i The intensity correlation function [Eq. (6)] is a general form of the function derived by Parthasarathy et al. [11]. It represents a new expression for the Siegert relation, taking into account not only temporal variations but also spatial variations in the speckle pattern resulting from a mixture of moving and stationary scatterers. Note that for constant values of Δx0 and Δy0 , Eq. (6) reduces to the Siegert relation derived by Boas and Dunn [15]: g2 τ 1 ρ2 jg1;f τj2 β 2ρ1 − ρjg1;f τjβ
1 2 2 AD T
Z Z
T
Z Z
0
∞
Z Z
−∞
∞ −∞
Dx01 ; y01 Dx02 ; y02
× hIx01 ; y01 ; t1 Ix02 ; y02 ; t2 idx01 dy01 dx02 dy02 dt1 dt2 Z ZTZ Z∞Z Z∞ 1 2 2 Dx01 ; y01 Dx02 ; y02 AD T 0 −∞ −∞ × hIi2 g2 Δx0 ; Δy0 ; τdx01 dy01 dx02 dy02 dt1 dt2 ;
(9)
where Dx0 ; y0 is a real and positive weighting function that represents the spatial distribution of detector photosensitivity. For a uniformly sensitive photodetector: 0
0
Dx ; y
1
in the sensitive area
0
outside of the sensitive area
Z Z
(6)
where α is a normalization parameter that accounts for effects (e.g., polarization) that reduce speckle contrast and that differ from spatial sampling of the speckle pattern; g1;f and g1;s are the normalized correlation functions of the fluctuating and static electric fields, respectively; and Δx0 and Δy0 are the distances between two arbitrary points (x01 , y01 ) and (x02 , y02 ) on the detector surface. The constant term (C 2n ) was introduced to account for contributions of noise [15]. g1;f and g1;s are given as
1 −
hI 2 i
:
(10)
The sensitive area of the photodetector AD is
2αρ1 − ρjg1;f Δx0 ; Δy0 ; τjjg1;s Δx0 ; Δy0 j
C 2n :
correlation function [Eq. (6)] and a map of the spatial detector photosensitivity:
1
1 αρ2 jg1;f Δx0 ; Δy0 ; τj2
ρ2 β
679
(8)
We now use Eq. (6) to derive a new expression relating K and τc . Based on Goodman [12], the second moment of measured intensity depends on the spatial intensity
AD
∞ −∞
Dx0 ; y0 dx0 dy0 :
(11)
Substituting Eq. (6) into (9), we obtain hI 2 i − hIi2 hIi2 Z Z TZ Z ∞ ρ2 α 2 2 K D Δx0 ;Δy0 jg1;f Δx0 ;Δy0 ; t2 − t1 j2 AD T 0 −∞
K2 ≡
× dΔx0 dΔy0 dt1 dt2 Z Z TZ Z ∞ 2αρ1 − ρ K D Δx0 ;Δy0 jg1;s Δx0 ; Δy0 j A2D T 2 0 −∞ × jg1;f Δx0 ;Δy0 ;t2 − t1 jdΔx0 dΔy0 dt1 dt2 Z Z TZ Z ∞ α1 − ρ2 2 2 K D Δx0 ;Δy0 jg1;s Δx0 ; Δy0 j2 AD T 0 −∞ × dΔx0 dΔy0 dt1 dt2 ;
(12)
where K D Δx0 ;Δy0
Z Z
∞ −∞
Dx01 ;y01 Dx01 −Δx0 ;y01 −Δy0 dx01 dy01 : (13)
For a spatially incoherent source model that represents many scattering systems, the electric-field correlation function can be separated into terms describing the stationary and moving scatterers [16]: jg1;f Δx0 ; Δy0 ; t2 − t1 j jg1;s Δx0 ; Δy0 jjgf t2 − t1 j: (14)
OPTICS LETTERS / Vol. 39, No. 3 / February 1, 2014
We use Eq. (14) to rewrite Eq. (12) as
1 K α 2 AD 2
Z Z
∞ −∞
K D Δx0 ; Δy0 jg1;s Δx0 ; Δy0 j2
× dΔx0 dΔy0 Z Z T 1 ρ2 jgf t2 − t1 j ρ1 − ρjgf t2 − t1 j × 2 T 0 (15) 1 − ρ2 dt1 dt2 :
For a square detector with uniform photosensitivity and a Gaussian-shaped intensity pattern, the first term of Eq. (15) is simplified to [12] 1 A2D
Z Z
∞ −∞
K D Δx0 ; Δy0 jgs Δx0 ; Δy0 j2 dΔx0 dΔy0
r 2 p 1 1 erf πM − 1 − e−πM ≡ βM; (16) M πM
where M AD ∕AC , and AC is the correlation area of the intensity (effectively, the speckle size) on the detector: Z Z AC
∞ −∞
jgs Δx0 ; Δy0 j2 dΔx0 dΔy0 :
(17)
The second factor of Eq. (15) is rewritten as [11,15] Z ZT 1 ρ2 jgf t2 − t1 j ρ1 − ρjgf t2 − t1 j 1 − ρ2 dt1 dt2 T2 0 ρ2
e−2x − 1 2x e−x − 1 x 4ρ1 − ρ 1 − ρ2 ≡ K t x: 2 2x x2 (18)
Substituting Eqs. (16) and (18) into Eq. (15) and adding a noise term K n , we obtain K α1∕2
r p 1 1 erf πM − 1−e−πM M πM
1∕2 −2x e −12x e−x −1x 2 4ρ1−ρ 1−ρ K n × ρ2 2x2 x2 α1∕2 β1∕2 MK t1∕2 xK n :
(19)
consisted of a microchannel (inner diameter of 300 μm) placed at the surface of a rigid polymer resin that contained TiO2 particles (particle size
