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Influences of tissue absorption and scattering on diffuse correlation spectroscopy blood flow measurements

In this study we evaluate the influences of optical property assumptions on near-infrared diffuse correlation spectroscopy (DCS) flow index measurements. The optical properties, absorption coefficient (µa) and reduced scattering coefficient (µs′), are independently varied using liquid phantoms and measured concurrently with the flow index using a hybrid optical system combining a dual-wavelength DCS flow device with a commercial frequency-domain tissue-oximeter. DCS flow indices are calculated at two wavelengths (785 and 830 nm) using measured µa and µs′ or assumed constant µa and µs′. Inaccurate µs′ assumptions resulted in much greater flow index errors than inaccurate µa. Underestimated/overestimated µs′ from −35%/+175% lead to flow index errors of +110%/−80%, whereas underestimated/overestimated µa from −40%/+150% lead to −20%/+40%, regardless of the wavelengths used. Examination of a clinical study involving human head and neck tumors indicates up to +280% flow index errors resulted from inter-patient optical property variations. These findings suggest that studies involving significant µa and µs′ changes should concurrently measure flow index and optical properties for accurate extraction of blood flow information.


Introduction
Near-infrared (NIR) light has been recently employed in the noninvasive acquisition of blood flow information from deep tissues (up to several centimeters), which is referred to as NIR Diffuse Correlation Spectroscopy (DCS) [1][2][3][4] or Diffuse Wave Spectroscopy (DWS) [5,6]. DCS measures relative change of tissue blood flow (rBF) which has been extensively validated in various tissues through comparisons with laser Doppler flowmetry (LDF) [7], Doppler ultrasound [8,9] The use of NIR light for deep tissue measurements stems from the exploitation of a spectral region (650-950 nm) wherein light absorption of the biological tissue is relatively low. When using NIR spectroscopy (NIRS) to detect optical properties of deep tissues, a pair of source and detector fibers is usually placed along the tissue surface with a distance of a few centimeters. NIR light generated by a laser emits into tissues through the source fiber and is detected by a photodetector through the detector fiber. Photon migration in tissue is now known to be a diffusive process [2,7]. During this migration, photons encounter absorption and, more commonly, scattering events. The probabilities of these events are described by an absorption coefficient, μ a , and a reduced scattering coefficient, μ s , also referred to as the optical properties, intrinsic to the probed tissue volume. The penetration depth of NIR light in biological tissues is approximately half of the source-detector separation. NIR DCS flow measurements are accomplished by monitoring speckle fluctuations of photons emitted at the tissue surface. In non-muscular tissues moving red blood cells (RBCs) inside vessels are primarily responsible for these fluctuations [4,5,[7][8][9][10][11][12][13][14]16,17,19,20,27,29,[34][35][36][37][38][39][40][41][42], but complications such as tissue shearing and motion artifacts can arise for muscular tissues [28,32]. Blood flow indices and rBF can be calculated from the changes in the speckle patterns. Ensuing calculations of blood flow using DCS measurements include a dependence on the optical properties (μ a and μ s ) and are thus potentially influenced by variations thereof (see the details in Section 2). DCS is not inherently capable of measuring absolute values of absorption and scattering coefficients. Solutions to this issue have typically been approached using two general methods: optical property assumptions or optical property measurements with separate instrumentation. Some studies have chosen to use the values of μ a and μ s  from the literature [8,32], respective to the tissue type (e.g., brain or muscle), or assumed a constant μ s  while examining changes in μ a [13,30,40]. These assumptions may be susceptible to deviations in optical properties that occur transiently, longitudinally, due to subject differences or from differences in literatures. A few of the recent studies have employed hybrid instrumentation allowing for measurement of both sets of information to extract accurate blood flow [9,19,29,43]. However, a generalization of potential flow index errors due to the inaccurate estimation of optical properties has not been investigated for the DCS flow measurements. In addition to optical properties, another potential influence on DCS flow indices is determined by selection of the laser wavelength.
Our lab has recently built a hybrid instrument capable of the simultaneous measurement of absolute μ a , μ s  and flow indices at multiple wavelengths, through combining a commercial frequency-domain NIR tissue-oximeter, the Imagent (ISS, Inc., IL, USA) [44,45], and a custom-made NIR DCS flow-oximeter [31,33,41]. This newly developed hybrid instrument allows us to quantify the influences of optical properties on DCS flow indices measured at different wavelengths. In this study, homogeneous liquid phantoms with controlled variations of optical properties were created, attempting to isolate the influence of each optical property parameter (i.e., μ a or μ s ) on DCS flow indices. The usage of tissue-like phantoms for instrument calibration and experimental validation of NIRS and DCS techniques is common [2,4,[44][45][46][47][48][49][50][51]. In DCS measurements, the dynamic scatterer motions (typically microvasculature RBCs) are best modeled by Brownian diffusion as opposed to random ballistic flow, which has been determined empirically, but for reasons currently unknown [4,7,8,10,11,20,27,40]. An effective Brownian diffusion coefficient is calculated as the blood flow index when measuring in biological tissues and is usually distinct from the conventional Brownian diffusion coefficient predicted by Einstein [52]. However, when utilizing liquid phantoms with Intralipid particles to provide Brownian motion, the two diffusion coefficients are expected to be equivalent. Through this special case using liquid phantoms, DCS flow indices calculated using measured or assumed optical properties can be compared to the Einstein prediction (as a true flow index) for Brownian particles suspended in liquid. Measurement errors are then determined through these comparisons for DCS flow indices at different wavelengths.
Simultaneous measurements of optical properties and blood flow indices are essential for extracting accurate hemodynamic information in tissues with transient, longitudinal and intersubject differences in optical properties. To this end, we show a clinical study using the hybrid instrument to accurately quantify tissue optical properties and blood flow indices in head and neck tumors. The measurement errors in tumor blood flow indices induced by potential inaccurate estimations of tissue optical properties are ultimately discussed and compared to the phantom study results to determine the in vivo applicability thereof.

Diffuse correlation spectroscopy (DCS) for blood flow measurement
DCS flow indices are quantified by a dual-wavelength DCS system [31] with two long coherence length continuous-wave (CW) NIR laser sources at 785 and 830 nm (100 mW, Crystalaser, Inc., NV, USA). The DCS sources emit light alternately into the tissue via two multi-mode optical fibers bundled at the same location on the tissue surface (see Fig. 1a). Four detector fibers are tightly bundled and placed on the tissue surface at a distance of 1.5 cm away from the source fibers, and each is connected to a single photon-counting avalanche photodiode (APD) (PerkinElmer Inc., Canada). The outputs of 4 APDs are sent to a 4-channel autocorrelator board (Correlator.com, NJ, USA) producing normalized light intensity temporal autocorrelation functions (g 2 ) which are averaged to improve the signal-noise-ratio. The averaged g 2 from DCS is related to the normalized electric field temporal autocorrelation function (g 1 ) through the following Siegert relation [53]: where τ is the delay time, r is the position vector, and β depends on laser stability and coherence length and the number of speckles detected. Scatterer motion is directly associated with the unnormalized electric field temporal autocorrelation function (G 1 ) which obeys a correlation diffusion equation, derived rigorously elsewhere [2,46] and defined as follows for homogeneous media using a CW source (steady state): (0-1) term is added to account for not all scatterers being dynamic and is defined as the ratio of moving scatterers to total scatterers. The combined term, αD B , is referred to as the blood flow index in biological tissues and is commonly used to calculate the relative blood flow (compared to the baseline flow index before physiological changes). In contrast to tissue samples where scatterers may be static (e.g., organelle, mitochondria) or dynamic (moving RBC), all scatterers in liquid phantom solutions (see Section 2.4) are considered dynamic with α 1 and the flow index is thus reported as simply D B . The homogeneous CW solution to Eq.
(2) for semi-infinite geometry is  [7,46,54,55]. The R eff term accounts for the mismatch between the medium and the air indices of refraction with n being the ratio between them.
For semi-infinite geometry, the collimating laser source at (0, 0, 0) and detector at (ρ, 0, 0) are placed on the tissue surface with z = 0 (see Fig. 1b). This solution (Eq. (3)) involves an extrapolated zero boundary condition including an isotropic source at z = z 0 and negative isotropic imaging source at z = (z 0 + 2z b ). The position vector, r , from Eq. (2) considers the point source at (0, 0, z 0 ) and the negative imaging source at (0, 0, (z 0 + 2z b )). The superposition of solutions to these two sources using infinite geometry provides the resulting Eq. (3) where now the semi-infinite boundary is modeled by the scalar parameter, ρ. Further details can be found elsewhere [46,55].
Flow index calculations begin with using Eq. (1) to first determine the β. Using the g 2 data at earliest τ and letting g 1 1 (i.e., g  , calculated β and Eq. (1), 1 ( , ) g  is calculated for all τ. Equation (3) is then used with the unknown parameter αD B (flow index) to fit the g 1 derived from DCS measurements (see Fig. 1c). For a complete frame of DCS data acquisitions at two wavelengths, two flow indices are obtained sequentially.

Frequency-domain spatially resolved NIRS for tissue optical property measurement
Quantification of absolute μ a and μ s  is performed by a frequency-domain multi-distance spatially resolved spectroscopy, i.e., the Imagent. Two wavelengths (780 and 830 nm) of a four-wavelength (690, 750, 780 and 830 nm) Imagent system are chosen to match the DCS lasers available (785 and 830 nm). The Imagent emits sinusoidally modulated light into tissue through 8 optical fibers (4 per wavelength) placed at four pre-determined distances (2.0, 2.5, 3.0, and 3.5 cm) from a detector fiber bundle connected to a photomultiplier tube (PMT) (see Fig. 1a). Source light is modulated at 110 MHz resulting in detected AC, DC and Phase (φ) information from multiple distances. A simplified solution based on semi-infinite geometry for the photon diffusion equation exposes linear relationships between φ, logarithmic AC or logarithmic DC and spatial distances [44,45]. From the fitting slopes (S AC , S DC , S φ ) of the linear relationships μ a and μ s  can be extracted at each wavelength. Different source-detector separations generally provide measurements at different depths/regions based on diffusion theory [3]. However, the depth/regional difference has minimal effect on measurement of optical properties of homogeneous phantoms.

Brownian motion of spherical particles in liquid phantoms
where k B is the Boltzmann constant, T is the phantom temperature, R is the radius of the spherical particles and η is the viscosity [52]. In order to obtain the temperature and viscosity parameters, a temperature sensor (Physitemp, NJ, USA) is attached near the optical probe and viscosity is measured using a viscometer (Brookfield, MA, USA). Viscosity is reported in units of cP (centipoise), where 1 cP = 1 mPa•s (millipascal•second) = 0.001 kg•m 1 s 1 [56]. The radius of Intralipid particles is estimated as 196 nm (see Section 2.5).

Liquid phantoms with varied optical properties
Liquid phantoms are comprised of distilled water, India ink (Black India 44201, Higgins, MA, USA) and Intralipid (30%, Fresenius Kabi, Uppsala, Sweden). India ink is used to manipulate the absorption coefficient of the phantom, μ a (λ), where λ is the laser source wavelength. India ink is first diluted to a 10% solution with distilled water. The 10% ink solution (instead of pure ink) is used to create phantoms. Intralipid provides particle Brownian motion and control of the reduced scattering coefficient of the phantom, μ s  (λ). Setup of the liquid phantom is shown in Fig. 1b. A hybrid fiber-optic probe is placed on the surface of the liquid phantom solution contained inside a glass aquarium. A custom-made probe holder attached to a lab stand holds the probe at the center of the solution to simulate a semi-infinite geometry. For creating phantoms with μ a variation, a list of μ a (λ) and a constant μ s  (λ) are chosen (see details in Section 2.5). The μ a (λ) and μ s  (λ) of distilled water, 10% ink solution, and 30% Intralipid are first determined. These values in combination with titration equations provide the necessary volumes of water, ink and Intralipid to achieve desired phantom optical properties [57]. The subscripts "ink", "Intralipid", and "water" are used in this paper to denote 10% ink solution, 30% Intralipid, and distilled water, respectively. The μ a ink (λ) is derived from spectrometer (Beckman Coulter, CA, USA) measurements. Since the 10% ink solution is out of the measurable range of the spectrometer, further dilution is performed to get a 0.025% ink solution for spectrometer absorbance measurements. Absorbance measured from multiple 0.025% ink solution samples are averaged and converted to the μ a ink (λ) [57]. The μ a water (λ) is taken from the literature [58]. The μ a Intralipid (λ) is assumed to be equivalent to that of water. Distilled water and 10% ink solution are both assumed to have no contributions to the phantom μ s  (λ), i.e., μ s  water (λ) = μ s  ink (λ) = 0 cm 1 . The μ s  (λ) of 10% Intralipid is calculated using a Mie theory approximation [59]. The theory and details including the Intralipid particle radius and refractive index were described in the original derivation [59] which has been extensively used for quantification of Intralipid-based liquid phantoms [2,7,57,60,61]. For 30% Intralipid used in this study, μ s  Intralipid (λ) is the μ s  (λ) of 10% Intralipid multiplied by a factor of three [57].

Phantom experimental protocols and data analysis μ a variation.
In this protocol, variation of µ a was induced while maintaining a constant µ s . Thirteen steps were performed to cover µ a (830 nm) from 0.05 to 0.20 cm 1 with a step size of 0.0125 cm 1 (i.e., µ a (830 nm) = 0.05, 0.0625, 0.075, …, 0.20 cm 1 ) and µ s  (830 nm) = 10 cm 1 . Prior to beginning the Imagent requires calibration to a phantom of known optical properties. During this process, corrections are made to account for the efficiency of optical coupling among the lasers/detector, optical fibers, and phantom [45,62]. The Imagent was calibrated to a liquid phantom of equivalent composition and optical properties at the midpoint (step 7) of the experimental range, i.e., µ a (830 nm) = 0.125 cm 1 and µ s  (830 nm) = 10 cm 1 . The combined probe was then placed upon a liquid phantom at the lowest optical property step, i.e., µ a (830 nm) = 0.05 cm 1 . For each of 13 steps the following actions were taken. Desired volume of ink solution was added to the liquid phantom, mixed, and allowed to stabilize for 10 minutes. Viscosity data was attained during this time by extracting three 500 µl samples. The three samples were carefully taken from the left, right and middle of the solution (at the surface) to minimize spatial variations without submerging the pipette. Room light was then turned off and the experimental setup was covered with black plastic to reduce ambient light. Measurements were taken by the hybrid optical instrument and temperature sensor for a 5 minute interval. μ s  variation. Variation of μ s  immediately followed the performing of μ a variation experiments. Between experiments the phantom from μ a variation was disposed of and replaced with a new initial phantom for μ s  variation. The hybrid probe was cleaned with alcohol pads and repositioned on the surface of the second phantom for μ s  variation study. Neither Imagent nor DCS were shut down between protocols. Variation of µ s  was induced while maintaining a constant µ a . A scattering range of µ s  (830 nm) from 4 to 16 cm 1 with a step size of 1 cm 1 (i.e., µ s  (830 nm) = 4, 5, 6, …, 16 cm 1 ) and µ a (830 nm) = 0.125 cm 1 was performed over thirteen steps. The volume of Intralipid required to increase μ s  as desired could potentially reduce μ a as well as influence the level of probe submersion. To remedy the first difficulty, ink was added with the Intralipid to maintain the µ a of the phantom. For the second, an equivalent amount of phantom solution was removed after being mixed at each step. Viscosity, temperature and hybrid optical measurements were taken similarly to those during μ a variation.
Data analysis and presentation. Each 5 minute interval measurement involves post calculations of interval averages of μ a and μ s  at each wavelength along with the temperature, sample average of viscosity, and three diffusion coefficients (D B s). Data between intervals (i.e., adding/taking solution, stirring) are excluded from data analysis. The µ a and µ s  are measured by Imagent using the spatially resolved (slope) method (see Section 2.2) and averaged over the 5 minute interval. The averaged µ a and µ s  are then used as known parameters in fitting DCS measured g 2 s using Eq. (1) and (3)   s at 830 and 785 nm, respectively. The influence of wavelength mismatch (5 nm) between 780 and 785 nm is considered to be minor. The third D B (D B-Einstein ) is calculated using Eq. (4) with the estimated particle radius, interval averaged temperature and three sample averaged viscosity. The estimated particle radius of 196 nm is determined to exhibit the least errors between the measured DCS flow indices (D B-dynamic ) and calculated D B-Einstein at the calibration point (step 7). This estimation falls within the range of Intralipid particle size reported in the literatures [2,59].
Measurement errors are characterized by calculating percentage errors between the measured values and predictions. P-values from Student t-tests are included for comparisons of measurement errors and the criterion for significance is p < 0.05. Results are presented in figures and tables to visualize measurement variations, differences between expected and obtained values, and the optical property influences on DCS flow indices. Data are depicted as mean ± standard deviation (SD) in tables and SD is illustrated by error bars in figures.

In vivo quantification of head and neck tumor hemodynamics
The possible errors of assuming constant optical properties in calculation of flow indices may be more readily visualized through analysis of real tissue measurements. In order to evaluate such influences in in vivo measurements, tissue hemodynamic properties of head and neck tumors in 10 patients were measured using a hybrid optical instrument similar to that in the phantom study. Institutional review board (IRB) approval was given by the University of Kentucky and consent forms were obtained before subject participation. Only subjects with Stage III-IVb Squamous Cell Carcinoma of the Head and Neck (SCCHN) were included. Neck lymph nodes measuring more than 1 cm and clinically thought to be involved by tumor were selected to study.
The Imagent used 690 and 830 nm source wavelengths whereas DCS used 785 and 854 nm lasers. Other wavelengths used for phantom studies were not available for the tumor study. Thus, data from 830 nm for Imagent and 854 nm for DCS are analyzed for this tumor study as these wavelengths provide the best match. By contrast, the second pair of wavelengths (785 nm versus 690 nm) is excluded for data analysis due to the significance of wavelength mismatch (95 nm). The Imagent source-detector separations used (2.0, 2.5, 3.0, and 3.5 cm) are identical to the phantom study. DCS utilized 3 detector fibers at 1.5, 2.4, and 2.8 cm separations from the two source fibers. The probe was held by hand and secured on the subjects in the center of the area identified as tumor node while tumor optical properties and DCS flow data were obtained for ~2 minutes. DCS data from the 2.8 cm separation are examined, comparable to the tissue region/depth probed by the Imagent. Using different sets of optical properties measured by the Imagent, four DCS flow indices for each of 10 subjects are calculated and then averaged over the 2-minute measurement interval. The μ a and μ s  are averaged over the measurement duration (2 minutes) for each subject and used in calculating the first DCS blood flow index (αD B-dynamic ), which is considered as a true flow index. The minimum, mean and maximum μ a and μ s  over 10 subjects are determined and used for calculating the respective remaining three DCS blood flow indices (αD B-min , αD B-mean , and αD B-max respectively) for comparisons with the true flow index (αD B-dynamic ). Data in figures are presented by interval mean ± SD, where SD is depicted by error bars.

μ a variation
In order to evaluate the influence of µ a variation on flow indices, thirteen steps of liquid phantoms were performed to cover µ a (830 nm) from 0.05 to 0.20 cm 1 with a step size of 0.0125 cm 1 while maintaining a constant µ s  (830 nm) = 10 cm 1 . For each step/interval of measurements over 5 minutes, the means and SDs of viscosity (from three samples), temperature and calculated D B-Einstein are displayed as data sets (means) and error bars (SDs) in Fig. 2. D B-Einstein (see Fig. 2c) is calculated using the measured temperature (see Fig. 2b) per interval along with the associated viscosity (see Fig. 2a) and estimated particle radius (196 nm).
The interval means and SDs (error bars) of μ a , μ s , D B-Einstein , D B-mid , and D B-dynamic throughout the 13 steps of μ a variation are displayed in Fig. 3. The measured values of μ a (see Figs. 3a and 3d) and μ s  (see Figs. 3b and 3e) at two wavelengths by the Imagent are compared to predictions calculated using spectrometer and Mie theory for the evaluation of measurement errors, respectively. Two DCS flow indices (D B-mid and D B-dynamic ) are compared to the D B-Einstein (as a true flow index) for both wavelengths (see Figs. 3c and 3f). The D B-mid or D B-dynamic at each wavelength is calculated using the DCS measurement with averaged µ a and µ s  from the middle interval [µ a (830 nm) = 0.125 cm 1 and µ s  (830 nm) = 10 cm 1 ] or from the corresponding interval.

μ s  variation
Similar to µ a variation, a scattering range of µ s  (830 nm) from 4 to 16 cm 1 with a step size of 1 cm 1 was performed over thirteen steps while maintaining a constant µ a (830 nm) = 0.125 cm 1 . Results for μ s  variation are plotted in a similar fashion as μ a variation (see Section 3.1). Fig. 4. Stability of Brownian motion (DB-Einstein) during μs variation, depicted as means ± SDs (as error bars). Viscosity (a) is averaged from three samples corresponding to each step; temperature (b) and DB-Einstein (c) are averaged respectively over 5-minute intervals.

Quantification of μ a and μ s  influences on flow indices
Influence of μ a and μ s  variations on D B-Einstein . Table 1 lists the means ± SDs and coefficients of variation (CVs) for viscosity, temperature, and D B-Einstein over the entire range of μ a and μ a  variations, calculated based on the data shown in Figs. 2 and 4. The CV is defined as a percentage of SD/mean, indicating the variation of the mean values over steps. The CVs of temperature, viscosity, and D B-Einstein are less than 2.2%, indicating the minor influences of μ a and μ a  variations on these variables. The D B-Einstein is thus used as a true flow index to evaluate the DCS flow measurement errors. Mean measurement errors in μ a , μ s  and DCS flow indices. Table 2 lists the means ± SDs of absolute percentage errors in measurements of μ a , μ s , and DCS flow indices (D B-dynamic and D B-mid ) over the entire range of μ a and μ a  variations, calculated based on the data shown in Figs. 3 and 5. Absolute percentage error is defined as (|Estimate-True|/True) X 100%. For μ a and μ s , the Imagent measured values are considered estimates while the spectrometer and Mie theory, respectively, are used as true values. For flow indices, D B-dynamic and D B-mid are considered estimates and D B-Einstein as true. The measurement errors for μ a , μ s , and D B-dynamic are small, averaging less than ~7%, whereas those of D B-mid are found to be larger, averaging up to 12.89% and 49.63% for μ a and μ s  variations, respectively. The influences of μ s  variation can be seen to produce greater percentage errors on flow indices than those of μ a variation. In Table 2, mean measurement errors between wavelengths are also compared using 2-sample unequal variance, two-tailed t-tests with significant differences considered for pvalue < 0.05 and denoted with * prefix. Significant differences in mean measurement errors between wavelengths are found in μ a (p = 0.01) during μ a variation and in μ s  (p = 0.04) during μ s  variation. These differences between wavelengths are most likely associated with the intrinsic feature of the instrument (Imagent) in detection accuracy at different wavelengths. No significant differences in mean measurement errors between wavelengths are found for both D B-dynamic and D B-mid.  Table 3 provides p-value results for comparisons of the mean measurement errors of D Bdynamic and D B-mid at two wavelengths during μ a and μ s  variations. It is apparent that there are significant (though it is borderline at 785 nm during μ a variation) differences between the measuring (D B-dynamic ) and assuming (D B-mid ) variables; D B-dynamic is more accurate (with less measurement errors, see Table 2) than D B-mid . The much lower p-values for μ s  variation as compared with those for μ a variation again suggest μ s  as a greater influence factor on DCS flow indices. Fig. 6, overlaying the results from both μ a and μ s  variations over 13 steps (see Figs. 3 and 5). The percentage errors for μ a during μ a variation and for μ s  during μ s  variation are defined as [(μ a -mid  μ a -dynamic )/μ a -dynamic ] X 100% and [(μ s  -mid  μ s  -dynamic )/μ s  -dynamic ] X 100%, respectively. The subscripts "mid" and "dynamic" correspond to assumed constant (middle-interval) and dynamic optical properties.

Influence of tissue optical properties on head and neck tumor blood flow index
The means ± SDs of measured tumor optical properties (μ a and μ s ) and blood flow indices (αD B-dynamic , αD B-min , αD B-mean , and αD B-max ) along with corresponding percentage errors for 10 patients with head and neck tumors are shown in Fig. 7. Patients are shown in order of increasing αD B-dynamic (as true flow index), designated with a black line, for comparison of trend differences when using optical property assumptions (i.e., αD B-min , αD B-mean , and αD Bmax ). Note that the patient numbers represent indices to illustrate the trend rather than actual patient numbers corresponding to the measurement sequence. The mean optical properties over subjects are: μ a (830 nm) = 0.12 ± 0.03 cm 1 and μ s  (830 nm) = 7.80 ± 2.64 cm 1 . Maximum and minimum optical properties out of all subjects at 830 nm are indicated using the red and blue dots, respectively, in Figs. 7a and 7b. DCS blood flow indices calculated using DCS data at 854 nm with different optical properties are represented in Fig. 7c. Without considering the tissue optical property influence, the trends of flow indices (αD B-min , αD B-mean , and αD B-max ) are not the same as the true flow index (αD B-dynamic ). Percentage αD B errors are calculated between the αD B-dynamic (true) and the estimated αD B-min , αD B-mean , and αD B-max .
Large ranges of percentage errors are found for αD B estimates: αD B-min from 8.07 to 278.15%, αD B-mean from 39.48 to 149.01%, and αD B-max from 70.26 to 22.59%. The tendency to overestimate or underestimate the blood flow indices follow the same trends as shown for μ s  variation in Fig. 6, supporting that μ s  has a greater influence on DCS flow indices than μ a .

μ a and μ s  variation influences on D B-Einstein
The Einstein-Stokes formula calculation, Eq. (4), provides the Einstein diffuse coefficient (D B-Einstein ) for spherical particles moving in liquid phantoms. The D B-Einstein is determined by the temperature and viscosity of the liquid as well as the particle radius of Intralipid in the liquid phantoms. Only slight variations are exhibited in overall average temperature (CV < 1.1%), viscosity (CV < 2.2%), and D B-Einstein (CV < 2.2%) during both μ a and μ s  variations, as seen in Table 1 and Figs. 2 and 4. The particle radius of Intralipid should not change during both μ a and μ s  variations. The changes in temperature are likely due to room temperature increase over the ~4.5 hour experimental durations. These include contributions by heat from equipment in the confined room. The cause of small variations in viscosity is likely due to the measurement variations by the viscometer. With these small variations in temperature and viscosity, it is thus expected that the derived D B-Einstein from Eq. (4) is stable over the large variations of optical properties.
On the other hand, increases of μ a (ink concentration) during μ a variation are expected to have no contribution to D B-Einstein , as ink provides no particle motion. Similarly, increases of μ s  (Intralipid concentration) during μ s  variation do not show significant influence on D B-Einstein , which is expected as all scatterers (Intralipid particles) provide equivalent motion in liquid phantoms and the ratio of moving scatterers to all scatterers (α) remains unchanged (α = 1). Due to the independence of optical properties and high stability throughout, D B-Einstein is considered reasonable as the true flow index for spherical particles moving in liquid phantoms.

Measurement errors of μ a , μ s , and D B-dynamic
In agreement with expectations, ink contributes only to increasing the absorption of the phantom during μ a variation. Additions of equivalent amounts of ink per interval resulted in linear increases in μ a for both Imagent wavelengths (see Figs. 3a and 3d). Only minimal variations occurred in μ s  at both wavelengths during μ a variation (see Figs. 3b and 3e). Calibration at the midpoint [μ a (830 nm) = 0.125 cm 1 , μ s  (830 nm) = 10 cm 1 ] may influence the variation patterns seen in Imagent measurements. Intralipid contributes linearly to μ s  increases, as expected, during μ s  variation for both Imagent wavelengths (see Figs. 5b and 5e). The μ a stayed relatively constant with minimal variations during μ s  variation (see Figs. 5a and 5d). Measured μ a and μ s  during both experiments are consistent with predictions from spectrometer measurements and Mie theory, respectively. More specifically, the μ s  measurement errors were less than 6% (see Table 2) for both experiments and wavelengths, which are comparable to those obtained from the literature using the Mie theory estimation (see Section 2.4) [59]. Overall, average measurement errors of μ a and μ s  were small during μ a (< 4%) and μ s  (< 6%) variations (see Table 2), which are in agreement with those found in previous studies using frequency-domain spatially resolved NIRS [44,45]. For μ a variation, significant difference was found between measurement errors of μ a at 780 and 830 nm. For μ s  variation, there was significant difference between measurement errors of μ s  at 780 and 830 nm. These differences are likely attributable to the intrinsic instrument (Imagent) feature in detection accuracy at separate wavelengths.
Average measurement errors for D B-dynamic (< 7%) compared to D B-Einstein at both wavelengths are similar to those obtained for optical properties (< 6%) during both experiments (see Table 2), suggesting the influence of optical properties on D B .

Resulting D B errors from optical property assumptions
When using assumed constant optical properties (i.e., middle-interval μ a and μ s ) to calculate D B-mid , mean D B-mid measurement errors during μ a (~13%) and μ s  (~50%) variations (see Tables 2 and 3) were significantly higher than those of D B-dynamic (~7%). It is evident that the influence of μ s  on the DCS flow index is much greater than that of μ a . This result is further supported by the great difference in p-values, where p-values during μ s  variation are much less than p-values during μ a variation (see Table 3). Also, looking at Fig. 6, the range of D B percentage errors for inaccurate estimations of μ s  is much wider than that for μ a . This result is expected due to DCS flow indices being derived from light speckle fluctuations, originated from photon phase shifts by dynamic scatterers. Upon examination of the K 2 definition (see Eq. (3)), μ s  should have a more significant influence than μ a given the μ s  2 term along with the much larger scattering over absorption (i.e., μ s  >> μ a ) in biological tissues and the liquid phantoms. No significant difference was found between wavelengths in D B-dynamic and D B-mid measurement errors during both experiments (see Table 2). This indicates that wavelength may not be a critical factor in determining the importance of optical property influence on D B measurement, although further investigations using a large range of wavelengths are needed for making a solid conclusion. The trends of D B estimation errors when using D B-mid were found to be different between the μ a and μ s  variations (see Figs. 3c, 3f, 5c, 5f and 6). For μ a variation, overestimated or underestimated μ a results in overestimated or underestimated D B . By contrast, for μ s  variation overestimated or underestimated μ s  results in underestimated or overestimated D B . Extreme examples of incorrect estimations of D B can be seen in Fig. 6. Overestimation errors of μ a up to ~+150% during μ a variation resulted in percentage errors up to ~+40% and underestimation errors up of ~40% resulted in percentage errors up to ~20%. When overestimation errors of μ s  reach up to ~+175% during μ s  variation, D B percentage errors were up to ~80%. For underestimation errors of μ s  up to ~35%, D B percentage errors reach up to ~+110%. Note that these estimation errors in optical properties and resulting D B may be affected by the selection of phantom properties for calibration.

In vivo tumor study data in comparison to phantom study results
In the tumor study, measured μ a and μ s  show large variations between subjects (see Figs. 7a and 7b). The range of variations, μ a (830 nm) from 0.07 to 0.16 cm 1 and μ s  (830 nm) from 5.35 to 13.1 cm 1 , is within the range studied using liquid phantoms. The influence of the μ s  variations on flow indices was found to be greater than that of μ a , supporting the phantom study results. This is exemplified by the trends shown in Fig. 7c. The overestimation of optical properties (using maximum μ a and μ s ) leads to underestimation of DCS flow index (αD B-max ) and underestimation (using minimum μ a and μ s ) leads to overestimation of DCS flow index (αD B-min ). These are in agreement with the trends of D B estimation errors using inaccurate μ s  in liquid phantoms (see Figs. 5c, 5f and 6). Percentage αD B errors range greatly, from ~70% up to ~ +280%, depending on optical properties assumed. Errors in flow indices (see Fig. 7d) produce an incorrect observation of trends in the αD B magnitudes among patients (see Fig. 7c). It is evident that lack of consideration for optical property influences can lead to invalid results in similar studies.

Conclusions
The advent of DCS technology as a safe and quick alternative for measurement of blood flow in deep tissues has brought the need to further investigate potential errors, notably by the assumption of constant optical properties, μ a and μ s . The flow index produced by DCS measurement is based on a solution to the correlation diffusion equation which includes parameters of μ a and μ s . Utilizing a novel hybrid optical equipment setup, capable of measuring all three parameters of interest (i.e., flow index, μ a , and μ s ), with liquid phantom experimental protocols has made it possible to perform this investigation. The present study evaluates the influences of tissue optical properties on DCS flow indices through isolated variations of μ a and μ s  in liquid phantoms. It is found that the particle motions in liquid phantoms are not influenced by the variations in optical properties, and the usage of Einstein particle Brownian motion coefficient (D B-Einstein ) as true flow index is reasonable for comparison with DCS flow indices. During μ a and μ s  variations, μ s  has a much greater influence on DCS flow indices than μ a , regardless of the wavelengths used. Studies involving significant μ a and μ s  changes should concurrently measure flow index and optical properties for accurate extraction of blood flow information in tissue. The flow index errors resulted from the optical property assumptions in the tumor study elicit such need for concurrent monitoring of optical properties. Incorporation of laser sources at wavelengths beyond those tested in this study may be the subject of future investigation. The range of optical properties tested in the phantoms may also be extended to encompass a wider variety of tissues.