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**|**Biomed Opt Express**|**v.2(6); 2011 June 1**|**PMC3114234

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Biomed Opt Express. 2011 June 1; 2(6): 1687–1702.

Published online 2011 May 25. doi: 10.1364/BOE.2.001687

PMCID: PMC3114234

S. C. Kanick,^{1,}^{2,}^{*} U. A. Gamm,^{1,}^{2} M. Schouten,^{1} H. J. C. M. Sterenborg,^{1} D. J. Robinson,^{1} and A. Amelink^{1}

Received 2011 March 24; Revised 2011 May 19; Accepted 2011 May 19.

Copyright ©2011 Optical Society
of America

This is an open-access article distributed under the terms of the Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 Unported License, which permits download and redistribution, provided that the original work is properly cited. This license restricts the article from being modified or used commercially.

This article has been cited by other articles in PMC.

This paper presents a relationship between the intensity collected by a single fiber reflectance device (*R _{SF}*) and the fiber diameter (

Reflectance spectroscopy is widely used for noninvasive measurement of the tissue absorption and scattering coefficients, which can provide diagnostic information about tissue function and structure, respectively [1–9]. Scattering in tissue originates from spatial heterogeneities of the optical refractive index that occur on size scales ranging from a few nanometers to a few millimetres [9]. Scattering properties of tissue include the scattering coefficient (*μ _{s}*), which describes the mean free path between scattering events, and the scattering phase function (

The influence of phase function on reflectance intensity measured close to the source has been described in the literature. For example, Mourant et al. [14] reported significant differences in the Monte Carlo simulated reflectance intensities for a small source-detector probe for different phase functions (Henyey-Greenstein vs. Mie) but with the same anisotropy. Kienle et al. [15] also reported large deviations in the calculation of *μ _{a}* and
${\mu}_{s}^{\prime}$ after simulating the reflectances for a source-detector separation of < 1 mm for multiple phase functions by applying a standard solution for the diffusion equation [10] to extract the optical properties. Bevilacqua and Depeursinge [16, 17] systematically investigated the influence of higher order moments of the phase functions for several source-detector separations on the reflectance intensities. They introduced a phase function dependent coefficient

Recently, our group reported on the use of experimentally validated Monte Carlo (MC) simulations to investigate the effect that optical properties have on the propagation of photons that are collected during a single fiber reflectance (SFR) measurement [23]. SFR spectroscopy uses a single fiber for the delivery and detection of light; a schematic of the device setup is shown in Fig. 1. The advantages of an SFR device are the simplicity of the setup machinery and the small dimensions of the fiber-optic probes, which allows *e.g.* measurement through the lumen of FNA-needles [24, 25]. In the past, single fiber geometries have been used for the investigation of fluorescence in tissue and tissue simulation phantoms [26–29] and for particle size analysis [30, 31]. The relationship between single fiber reflectance intensity and optical properties was studied by Moffit and Prahl [19]; however, their analysis did not yield quantitative estimates of
${\mu}_{s}^{\prime}$ or characterize phase-function dependence of the signal. Further studies have focused on technical aspects of single fiber reflectance, such as the need to remove internal specular reflections from the collected signal [32] and the dependence of the collection efficiency of a single fiber on the optical properties of the sampled medium [33]. Our previous work has focused on utilizing SFR spectroscopy to quantitate the absorption coefficient (*μ _{a}*), and the corresponding chromophore concentrations, in turbid media. These studies introduced an empirical function that describes the dependence of the effective photon path length [23,34] on the optical properties of the sampled medium. A limitation of our method is that the path length relation requires knowledge of
${\mu}_{s}^{\prime}$, a parameter that is currently not quantitatively extracted from the measured spectra. The current spectral analysis algorithm uses estimated values for the wavelength-dependent
${\mu}_{s}^{\prime}$ and the uncertainty introduced has been documented [24]. A quantitative measurement of
${\mu}_{s}^{\prime}$ would improve the accuracy of our chromophore concentration estimates. Moreover, quantitative measurement of
${\mu}_{s}^{\prime}$ using an SFR device is inherently useful, because it may provide information about the tissue microarchitecture [9]; information that is complementary to the physiological information extracted from

There are two challenges associated with quantitative extraction of
${\mu}_{s}^{\prime}$ from an SFR measurement. First, a calibration procedure must be designed that allows absolute measurement of the reflectance (*R _{SF}*) of a turbid medium from an SFR measurement. Second,

Our methodology to experimentally extract
${\mu}_{s}^{\prime}$ of a turbid medium from SFR spectra, using a calibration phantom and a mathematical model of
${R}_{\mathit{\text{SF}}}\hspace{0.17em}=\hspace{0.17em}f\left({\mu}_{s}^{\prime}{d}_{\mathit{\text{fib}}}\right)$, is validated using SFR measurements on Intralipid phantoms. We show that with knowledge of the wavelength-dependent phase function, the model function can be used to accurately extract
${\mu}_{s}^{\prime}$ from SFR measurements for a wide range of reduced scattering coefficients (
${\mu}_{s}^{\prime}\hspace{0.17em}=\hspace{0.17em}\left[0.05\hspace{0.17em}-\hspace{0.17em}8\right]\hspace{0.17em}{\text{mm}}^{-1}$) and fiber diameters (*d _{fib}* = [0.2, 0.4, 0.6, 0.8, 1] mm).

The Monte Carlo (MC) code used to simulate single fiber reflectance spectroscopy in this study has been described in detail previously [23]. In short, our MC code is based on the MCML code [35] that stochastically simulates photon propagation within a turbid medium. During photon propagation, each photon step size was selected from an exponential distribution weighted by the scattering coefficient, and each scattering angle was selected from a user-specified phase function. Reflection and refraction due to the index of refraction mismatch at the medium/fiber and medium/air interface were calculated using the Fresnel equations and Snells law. The index of refraction (*n*) of the medium and fiber were specified at 1.37 and 1.5, respectively. The numerical aperture (NA) of the fiber was set as 0.22. Photons were initialized by selecting a location from a uniform distribution on the single fiber face in contact with the turbid medium (*z* = 0), and the launch direction was selected from a uniform distribution of angles within the fiber cone of acceptance, where the acceptance angle was given as
${\Theta}_{a}\hspace{0.17em}=\hspace{0.17em}\mathit{\text{asin}}\hspace{0.17em}\left(\frac{\mathit{\text{NA}}}{{n}_{\mathit{\text{medium}}}}\right)$. For photons propagating within the turbid medium that cross the medium interfacial boundary at *z* = 0, it was checked if they hit the fiber face; those in contact with the fiber face and traveling at an angle within the fiber cone of acceptance were collected, the rest were terminated. Photons propagating within the medium far from the fiber face do not contribute to the collected reflectance intensity and were terminated at a hemispherical limit from the fiber face of
$10\frac{{d}_{\mathit{\text{fib}}}}{{\mu}_{s}^{\prime}}$; a limit that was confirmed to not influence model outputs for the range of optical properties investigated in this study. For each simulation, the MC code outputted the reflectance intensity as the ratio of the total number of photons collected (TPC) to the total number of photons launched (TPL) during the MC simulation, calculated as:

$${R}_{\mathit{\text{SF}}}^{\mathit{\text{MC}}}\hspace{0.17em}=\hspace{0.17em}\frac{\mathit{\text{TPC}}}{\mathit{\text{TPL}}}$$

(1)

MC simulations were run over a broad range of reduced scattering coefficient values (
${\mu}_{s}^{\prime}\hspace{0.17em}=\hspace{0.17em}\left[0.1,\hspace{0.17em}0.3,\hspace{0.17em}1.3,\hspace{0.17em}10,\hspace{0.17em}30,\hspace{0.17em}100,\hspace{0.17em}300\right]\hspace{0.17em}{\text{mm}}^{-1}$ and 5 different fiber diameters [0.2, 0.4, 0.6, 0.8, 1.0] mm. To investigate the dependence of the collected signal on phase function, sets of simulations were run that investigated variations in both phase function and anisotropy values. Figure 2 shows the the angular scattering probability of selected phase functions. The Henyey-Greenstein-phase function (HG), commonly used to describe photon scattering in tissue, was simulated for *g*-values of [0.5, 0.7, 0.8, 0.9]. The Modified Henyey-Greenstein phase function (MHG), which contains a pronounced backscattering feature, was simulated with an effective anisotropy value of *g* = 0.9 as in [21, 23]. In order to compare simulations with experimental measurements in Intralipid 20%, simulations were performed with the wavelength-dependent phase functions of Intralipid as given by Michaels et al. [36]. Simulations were run for wavelengths from 400 to 900 nm in steps of 50 nm; corresponding anisotropy values for these phase functions were in the range of [0.47 – 0.82]. At least 2 million photons were launched for each MC simulation.

Inspection of simulated reflectance data over the range of simulated scattering properties yielded an observable relationship between ${R}_{\mathit{\text{SF}}}^{\mathit{\text{MC}}}$ and the product ${\mu}_{s}^{\prime}{d}_{\mathit{\text{fib}}}$; this product is referred to throughout this paper as dimensionless scattering. A semi-empirical model is utilize to describe this relationship, given by the following set of Equations,

$${R}_{\mathit{\text{SF}}}^{\mathit{\text{Model}}}\hspace{0.17em}=\hspace{0.17em}\frac{\mathit{\text{TPC}}}{\mathit{\text{TPL}}}\hspace{0.17em}=\hspace{0.17em}{\eta}_{c}\Phi $$

(2)

$$\Phi \hspace{0.17em}=\hspace{0.17em}\frac{\mathit{\text{TPH}}}{\mathit{\text{TPL}}}\hspace{0.17em}=\hspace{0.17em}\left[\frac{{\left({\mu}_{s}^{\prime}{d}_{\mathit{\text{fib}}}\right)}^{{\rho}_{2}}}{{\rho}_{1}\hspace{0.17em}+\hspace{0.17em}{\left({\mu}_{s}^{\prime}{d}_{\mathit{\text{fib}}}\right)}^{{\rho}_{2}}}\right]$$

(3)

$${\eta}_{c}\hspace{0.17em}=\hspace{0.17em}\frac{\mathit{\text{TPC}}}{\mathit{\text{TPH}}}\hspace{0.17em}=\hspace{0.17em}{\eta}_{\mathit{\text{limit}}}\hspace{0.17em}\left(1\hspace{0.17em}+\hspace{0.17em}{\rho}_{3}{e}^{-{\rho}_{1}\hspace{0.17em}\left({\mu}_{s}^{\prime}{d}_{\mathit{\text{fib}}}\right)}\right)$$

(4)

The collected reflectance intensity (
${R}_{\mathit{\text{SF}}}^{\mathit{\text{Model}}}$) is calculated as the ratio of TPC to TPL during the MC simulation, as given in Eq. (2). This ratio can be described by the the product of the fraction of photons remitted from the medium that contact the fiber face (Φ), and the collection efficiency (*η _{c}*) of the fiber. Equation (3) presents Φ as the ratio of photons that are remitted from the medium and contact the fiber face (

The parameters [*ρ*_{1}, *ρ*_{2}, *ρ*_{3}] were fitted by minimizing the weighted residual error between the simulated (
${R}_{\mathit{\text{SF}}}^{\mathit{\text{MC}}}$) and model-estimated (
${R}_{\mathit{\text{SF}}}^{\mathit{\text{Model}}}$) reflectance intensities, with each point weighted by the inverse of the simulated data point. It is important to note that neither Φ nor *η _{c}* are observable from our experiments, and while they were estimated by the MC model, the fitted parameter values [

Figure 1 shows a schematic of the single fiber reflectance spectroscopy device setup, which consists of a halogen light source (HL-2000-FHSA, Ocean Optics, Duiven, Netherlands), a spectrometer (SD2000, Ocean Optics, Duiven, Netherlands) and a solid core single fiber probe. The single fiber is connected to the light source and the spectrometer via a bifurcated fiber. During measurements, light travels from the light source through the fiber and exit into the probe tip that is in contact with the sampled medium. Photons that backscatter to the fiber face at an incident angle that is within the cone of acceptance enter the fiber core and travel to the spectrometer. To remove specular reflections within the collected reflectance intensity due to index of refraction mismatch at the probe/medium interface tip, the probe tip is polished at an angle of 15 degrees [34]. Variations in the output of the lamp, transmission characteristics of the fiber, and sensitivity of the spectrometer, as well as remaining specular reflectance and other internal reflections are being accounted for by performing a calibration procedure that we have described previously [34]. The calibration includes measurements of white and black spectralon standards (Labsphere SRS-99 and SRS-02, spectrally flat, with 99% and 2% reflectance, respectively) and a measurement of water in a black container. The single fiber reflectance signal can be calculated as

$${R}_{\mathit{\text{SF}}}^{\mathit{\text{meas}}\u2013\mathit{\text{rel}}}\hspace{0.17em}=\hspace{0.17em}\left[\frac{I\hspace{0.17em}-\hspace{0.17em}{I}_{\mathit{\text{water}}}}{{I}_{\mathit{\text{white}}}\hspace{0.17em}-\hspace{0.17em}{I}_{\mathit{\text{black}}}}\right]$$

(5)

Here *I _{water}*,

Solid phantoms were prepared according to the recipe reported by de Bruin et al. [40]. Silicone [Sylgard 184 Silicone Elastomer Kit (Dow Corning Europe SA, Seneffe, Belgium)] served as a stable matrix and Titanium dioxide (TiO_{2}) (Sigma Aldrich, Zwijndrecht, Netherlands) was used as a scatterer. All solid phantoms were prepared with a weight of 20 g, including different amounts of TiO_{2} to create a series of different highly scattering phantoms (TiO_{2} weight percentage= [0, 2.5, 5, 10, 20, 30] %). The silicone kit consisted of the silicone compound and a curing agent which was always mixed in a ratio of 9:1. To prepare the phantoms the silicone compound and TiO_{2} were weighed, mixed with a Dremel 300 (Dremel, Leinfelden-Echterdingen, Germany) and placed into an ultrasound bath for 10 min, to assure that the TiO_{2} particles were homogeneously distributed within the silicone. Then the curing agent was added and the composition was mixed again until the agent was completely mixed into the compound. The mass was poured into the mold and placed in a vacuum desiccator (Duran Group GmbH, Mainz, Germany) and held under vacuum until remaining air bubbles were removed from the phantom (1 – 3 hours). The phantoms were cured after approximately 12 hours at room temperature.

During the single fiber reflectance measurements the single fiber probe tip was brought into contact with the phantom surface, with no gap between fiber face and phantom surface. Three spectra were recorded at 3 different locations on the phantom surface and averaged. Each phantom was measured with the 5 different fiber probes with diameters of [0.2, 0.4, 0.6, 0.8, 1.0] mm. To test the homogeneity of the phantoms and the reproducibility of the preparation, 6 solid phantoms with a weight percentage of 30 % TiO_{2} were prepared and single fiber spectra were measured at 10 different locations on each phantom surface. The maximum relative deviation between measurements on two phantoms was 0.66 %, while the intra-phantom variability was even lower.

The liquid phantoms were prepared by mixing 0.9 % NaCl (Baxter, Utrecht, Netherlands) with different amounts of Intralipid 20% (Fresenius Kabi, s-Hertogenbosch, Netherlands). The reduced scattering coefficient of the Intralipid stock solution was determined to be ${\mu}_{s}^{\prime}\left(800\text{nm}\right)\hspace{0.17em}=\hspace{0.17em}18.1\hspace{0.17em}{\text{mm}}^{-1}$ by using the formula given by Michels et al. [36]. The dependence of ${\mu}_{s}^{\prime}$ on the intralipid concentration was assumed to be linear and phantoms with ${\mu}_{s}^{\prime}\left(800\text{nm}\right)\hspace{0.17em}=\hspace{0.17em}\left[0.072,\hspace{0.17em}0.144,\hspace{0.17em}0.24,\hspace{0.17em}0.36,\hspace{0.17em}0.48,\hspace{0.17em}0.72,\hspace{0.17em}0.96,\hspace{0.17em}1.2,\hspace{0.17em}2.4,\hspace{0.17em}3.6\right]\hspace{0.17em}{\text{mm}}^{-1}$ were prepared to a volume of 20 ml. Reflectance measurements were performed by submerging the probe tip a few millimeters into the phantom and suspending it multiple centimeters from the container bottom or sides, such that the probe would not collect reflections from container surfaces. The measurement was repeated 3 times for each phantom, with these spectra averaged for each phantom. All measurements were performed with 5 fiber probes varying in diameter: [0.2, 0.4, 0.6, 0.8, 1] mm. The phantom was gently agitated before collecting each spectrum to ensure homogeneity.

Measurement of single fiber reflectance intensity from the set of solid TiO_{2} phantoms (outlined in Section 2.4) yielded
${R}_{\mathit{\text{SF}}}^{\mathit{\text{meas}}\u2013\mathit{\text{rel}}}$ vs. TiO_{2} weight percentage (wt %) for a range of TiO_{2} wt % and fiber diameters. TiO_{2} wt % was converted to
${\mu}_{s}^{\prime}$ by estimating a linear conversion factor. In order to do this, measurements on the set of solid phantoms from all fibers were interpolated onto a curve that was normalized to the highest measured intensity in the set; this conversion corrected for different probe-spectralon distances, as noted in Section 2.3. Then a linear conversion factor (
${\mu}_{s}^{\prime}\hspace{0.17em}/\hspace{0.17em}{\text{TiO}}_{2}\hspace{0.17em}\text{wt}\hspace{0.17em}\%$) was estimated by scaling the normalized reflectance vs. TiO_{2} wt % data to fit the reflectance predicted by the mathematical model, presented in Eq. (2). This calculation yields a
${\mu}_{s}^{\prime}$ for each TiO_{2} phantom.

Measurements of liquid Intralipid phantoms were converted from a value relative to spec-tralon distance, ${R}_{\mathit{\text{SF}}}^{\mathit{\text{meas}}\u2013\mathit{\text{rel}}}\hspace{0.17em}\left(\mathit{\text{IL}}\right)$, to an absolute percentage scale, ${R}_{\mathit{\text{SF}}}^{\mathit{\text{meas}}\u2013\mathit{\text{abs}}}\hspace{0.17em}\left(\mathit{\text{IL}}\right)$, by,

$${R}_{\mathit{\text{SF}}}^{\mathit{\text{meas}}\u2013\mathit{\text{abs}}}\hspace{0.17em}\left(\mathit{\text{IL}}\right)\hspace{0.17em}=\hspace{0.17em}{R}_{\mathit{\text{SF}}}^{\mathit{\text{meas}}\u2013\mathit{\text{rel}}}\hspace{0.17em}\left(\mathit{\text{IL}}\right)\hspace{0.17em}\left[\frac{{R}_{\mathit{\text{SF}}}^{\mathit{\text{meas}}\u2013\mathit{\text{abs}}}\hspace{0.17em}\left(\mathit{\text{phantom}}\right)}{{R}_{\mathit{\text{SF}}}^{\mathit{\text{meas}}\u2013\mathit{\text{rel}}}\hspace{0.17em}\left(\mathit{\text{phantom}}\right)}\right]$$

(6)

where
${R}_{\mathit{\text{SF}}}^{\mathit{\text{meas}}-\mathit{\text{rel}}}\hspace{0.17em}\left(\mathit{\text{phantom}}\right)$ is the relative reflectance intensity measured from a solid TiO_{2} phantom after the calibration procedure given in Eq. (5) and
${R}_{\mathit{\text{SF}}}^{\mathit{\text{meas}}\u2013\mathit{\text{absl}}}\hspace{0.17em}\left(\mathit{\text{phantom}}\right)$ is the absolute reflectance on that phantom expressed as a percentage of incident photons, which is calculated from the known
${\mu}_{s}^{\prime}$ of the phantom and the mathematical model of reflectance given in Eq. (2) (using values for [*ρ*_{1}, *ρ*_{2}, *ρ*_{3}] specified in Section 3.2.1).

Figure 3 shows the MC simulated reflectance intensity (
${R}_{\mathit{\text{SF}}}^{\mathit{\text{MC}}}$) vs. dimensionless scattering (
${\mu}_{s}^{\prime}{d}_{\mathit{\text{fib}}}$) with combinations of a wide range of
${\mu}_{s}^{\prime}\hspace{0.17em}\left[0.1\hspace{0.17em}-\hspace{0.17em}300\right]\hspace{0.17em}{\text{mm}}^{-1}$ and *d _{fib}* [0.2 – 1.0] mm. Here,
${R}_{\mathit{\text{SF}}}^{\mathit{\text{MC}}}$ is expressed as a percentage of the incident photons. In Fig. 3, symbols differentiate the specified phase function, including
${R}_{\mathit{\text{SF}}}^{\mathit{\text{MC}}}$ data from the HG (

The observed relationship between
${R}_{\mathit{\text{SF}}}^{\mathit{\text{MC}}}$ and
${\mu}_{s}^{\prime}{d}_{\mathit{\text{fib}}}$ was described using the mathematical model introduced in Section 2.2. Figure 4(A) shows
${R}_{\mathit{\text{SF}}}^{\mathit{\text{MC}}}$ for the HG phase function with *g* = 0.8 for
${\mu}_{s}^{\prime}\hspace{0.17em}=\hspace{0.17em}\left[0.1\hspace{0.17em}-\hspace{0.17em}300\right]\hspace{0.17em}{\text{mm}}^{-1}$ and *d _{fib}* = [0.2 – 1.0] mm (blue cross marks), and the corresponding model predicted reflectance (
${R}_{\mathit{\text{SF}}}^{\mathit{\text{Model}}}$) estimated from Eq. (2) (black line). The estimated parameter values, given in Table 1, result in predictions of
${R}_{\mathit{\text{SF}}}^{\mathit{\text{Model}}}$ that are highly correlated with simulated reflectance intensities over the full dimensionless scattering range, as evidenced by a Pearson Correlation Coefficient of

(A) Single fiber reflectance and mathematical model fit for HG phase function (g=0.8). Model estimates of incident photons contacting fiber face (B) and collection efficiency of fiber (C).

Equation (2) expresses
${R}_{\mathit{\text{SF}}}^{\mathit{\text{Model}}}$ as the product of Φ, the fraction of incident photons that are remitted from the medium in contact with the probe face, and *η _{c}*, the fraction of photons that contact the fiber face within the cone of acceptance and are collected. Figures 4(B) and 4(C) show the MC model estimates (green squares and red diamonds) and mathematical model predictions (black lines) of Φ and

Figure 5 shows
${R}_{\mathit{\text{SF}}}^{\mathit{\text{Model}}}$ vs.
${R}_{\mathit{\text{SF}}}^{\mathit{\text{MC}}}$ for
${\mu}_{s}^{\prime}\hspace{0.17em}=\hspace{0.17em}\left[0.1\hspace{0.17em}-\hspace{0.17em}300\right]\hspace{0.17em}{\text{mm}}^{-1}$ and *d _{fib}* = [0.2 – 1.0] mm, with symbols distinguishing each phase function, including HG (

Reflectance measurements on TiO_{2} phantoms yielded
${R}_{\mathit{\text{SF}}}^{\mathit{\text{meas}}\u2013\mathit{\text{rel}}}$ vs. TiO_{2} wt % data in the range [2.5 – 30] wt %. As outlined in Section 2.6, these data were scaled to
${R}_{\mathit{\text{SF}}}^{\mathit{\text{meas}}\u2013\mathit{\text{abs}}}$ using the semi-empirical reflectance model. Because the exact phase function of the TiO_{2} phantoms is unknown, we used the averages of the fitted parameter values: *ρ*_{1} = 7.9, *ρ*_{2} = 1.08, *ρ*_{3} = 2.4. Using these parameter values, a linear conversion factor of (
${\mu}_{s}^{\prime}\hspace{0.17em}/\hspace{0.17em}{\text{TiO}}_{2}\hspace{0.17em}\text{wt}\hspace{0.17em}\%$) = 7.42 ± 0.35 was found. Figure 6 shows the
${R}_{\mathit{\text{SF}}}^{\mathit{\text{meas}}\u2013\mathit{\text{abs}}}$ (800nm) from all measurements on TiO_{2} phantoms vs.
${\mu}_{s}^{\prime}{d}_{\mathit{\text{fib}}}$ as measured with 5 probes with diameters in the range [0.2 – 1.0] mm, using the 7.42 (
${\mu}_{s}^{\prime}\hspace{0.17em}/\hspace{0.17em}{\text{TiO}}_{2}\hspace{0.17em}\text{wt}\hspace{0.17em}\%$) ratio. Also displayed on the plot are
${R}_{\mathit{\text{SF}}}^{\mathit{\text{MC}}}$ vs
${\mu}_{s}^{\prime}{d}_{\mathit{\text{fib}}}$ from simulations of an HG phase function with *g* = 0.8, which shows good agreement between experimental and simulated data. Note that due to the high reduced scattering coefficients of the phantoms, these measurements are in the high dimensionless scattering region (
${\mu}_{s}^{\prime}{d}_{\mathit{\text{fib}}}\hspace{0.17em}>\hspace{0.17em}10$), which showed an insensitivity to phase function. Varying the parameters *ρ*_{1} – *ρ*_{3} in Eq. (2) to be representative of HG (*g* = [0.5 – 0.9]) and MHG (*g* = [0.9]) phase functions (see Table 1) induced only a 2.2% mean residual error between
${R}_{\mathit{\text{SF}}}^{\mathit{\text{meas}}\u2013\mathit{\text{abs}}}$ (based on the average *ρ*_{1} – *ρ*_{3}) and
${R}_{\mathit{\text{SF}}}^{\mathit{\text{Model}}}$ (based on the different *ρ*_{1} – *ρ*_{3} combinations listed in Table 1). These results led to the selection of the 30% TiO_{2} phantom (which equated to
${\mu}_{s}^{\prime}\hspace{0.17em}=\hspace{0.17em}222.6\hspace{0.17em}\pm \hspace{0.17em}10.5\hspace{0.17em}{\text{mm}}^{-1}$) as the calibration phantom for subsequent measurements on Intralipid.

Figure 7(A) shows
${R}_{\mathit{\text{SF}}}^{\mathit{\text{meas}}\u2013\mathit{\text{abs}}}$ from measurements of Intralipid at multiple wavelengths (*λ* = [400 – 900] nm in steps of 100 nm) vs. dimensionless scattering. Here the
${R}_{\mathit{\text{SF}}}^{\mathit{\text{meas}}\u2013\mathit{\text{abs}}}$ values represent the raw intensities scaled by measurements on the solid calibration phantom, allowing the conversion to an absolute scale. This calibration technique allows reflectance intensities from measurements with 5 fibers with different diameters to fall onto the same curve. Figure 7(B) shows
${R}_{\mathit{\text{SF}}}^{\mathit{\text{MC}}}$ vs.
${\mu}_{s}^{\prime}{d}_{\mathit{\text{fib}}}$ from MC simulations of the reflectance measurements in Intralipid over the same range of wavelengths. Here, the phase function for each wavelength was specified using an empirical expression (as in [36]), and the corresponding anisotropy values were in the range *g* = [0.477 – 0.818]. For both measurements and simulations, the *R _{SF}* values measured at different wavelengths (and therefore different phase functions) show stratification at low dimensionless scattering values, and gradually collapse towards a limiting value; these features are similar to the data shown in Fig. 3 for HG and MHG phase functions. Figure 8(A) displays the corresponding simulated
${R}_{\mathit{\text{SF}}}^{\mathit{\text{MC}}}$ vs. measured
${R}_{\mathit{\text{SF}}}^{\mathit{\text{meas}}\u2013\mathit{\text{abs}}}$ values. These data are highly correlated (

Single fiber reflectance measured (A) and simulated (B) from Intralipid optical phantoms at multiple wavelengths *λ* = [400, 500, 600, 800, 900] nm; other wavelengths measured in this range not shown to improve clarity.

The semi-empirical reflectance model given in Eq. (2) was fitted to
${R}_{\mathit{\text{SF}}}^{\mathit{\text{MC}}}$ simulated in Intralipid for each wavelength-dependent phase function. Table 1 (bottom) lists detailed data for each simulation, including the anisotropy, estimated parameters values, and Pearson correlation coefficients for each fit. These model fits were then used to analyze
${R}_{\mathit{\text{SF}}}^{\mathit{\text{meas}}\u2013\mathit{\text{abs}}}$ measured experimentally from Intralipid phantoms and estimate
${\mu}_{s}^{\prime}$. Prior to this calculation, the
${R}_{\mathit{\text{SF}}}^{\mathit{\text{meas}}\u2013\mathit{\text{abs}}}$ data are corrected for slight offset with MC simulations, as noted in Section 3.2.2. Figure 8(B) shows
${\mu}_{s}^{\prime}$ estimated using
${R}_{\mathit{\text{SF}}}^{\mathit{\text{meas}}\u2013\mathit{\text{abs}}}$ and the mathematical model vs. the known experimental
${\mu}_{s}^{\prime}$ value; these data are highly correlated (*r* = 0.997) and show good agreement, with a mean residual of 9%. As observed in the reflectance data, the relative error in
${\mu}_{s}^{\prime}$ estimates increases as dimensionless scattering decreases; this is discussed in Section 4.

This paper presents a methodology to quantitatively extract
${\mu}_{s}^{\prime}$ from the reflectance intensity collected by a single fiber reflectance device (*R _{SF}*). Monte Carlo simulations were used to investigate the dependence of

Monte Carlo simulations were used to investigate the effect of
${\mu}_{s}^{\prime}$, *d _{fib}*, and phase function on

The experimental application of this theory motivated the construction of a solid optical phantom with a large concentration of scattering material (TiO_{2}) that corresponded to
${\mu}_{s}^{\prime}$ values that were within the high dimensionless scattering (and phase function insensitive) regime. These solid phantoms allow single fiber reflectance intensities measured from different fibers to be calibrated onto the same absolute scale; previously, this was difficult due to the fiber-diameter dependent effect of and variations in probe-spectralon distance utilized in our previous calibration technique (as noted in Section 2.3). In this study reflectance measurements on the set of phantoms investigated were scaled to a reflectance model to determine the relationship between TiO_{2} wt % and
${\mu}_{s}^{\prime}$. We have assumed that this conversion factor is linear. While it is possible that the relation may not be linear, the data shown on Fig. 6 present a convincing argument that this assumption is appropriate. Uncertainty associated with the
${\mu}_{s}^{\prime}\hspace{0.17em}/\hspace{0.17em}{\text{TiO}}_{2}\hspace{0.17em}\text{wt}\hspace{0.17em}\%$ ratio is not expected to introduce error. This is because the highest scattering phantom (30 wt % TiO_{2}) was selected for scaling subsequent measurements (on liquid phantoms in this study, or on tissue in future work), and in this dimensionless scattering region, a mis-estimation of
${\mu}_{s}^{\prime}$ for this phantom are not expected be significant due to the relatively small changes in *R _{SF}* corresponding to changes in
${\mu}_{s}^{\prime}$ in this regime.

The semi-empirical model function used to describe the relation between *R _{SF}* and dimensionless scattering for each investigated phase function contains 3 fitted parameters, given by Eqs. (2)–(4). The development of this mathematical model provided insight into the factors contributing to the collected signal. The model represents the light collected during measurement as a product of the photons contacting the fiber surface (Φ) and the fraction of those photons within the fiber cone of acceptance (

This study includes an experimental proof of concept for the extraction of
${\mu}_{s}^{\prime}$ from experimental measurements of optical phantoms containing Intralipid. As part of that investigation, *R _{SF}* was simulated by a MC model that emulated the experimental measurements in optical phantoms. These techniques yielded

The goal for performing the absolute *R _{SF}* calibration was to extract an absolute value for
${\mu}_{s}^{\prime}$ within the sampled turbid medium from the measured single fiber reflectance intensity. This was achieved in this study by performing measurements on Intralipid phantoms, calibrating the measured reflectance with measurements on the highly scattering solid calibration phantom, and analyzing the data using a phase-function specific mathematical model describing the dependence of reflectance on dimensionless scattering. The work in this study calculated
${\mu}_{s}^{\prime}$ for 5 fiber diameters, on Intralipid phantoms of 10 different concentrations at 11 different wavelengths (range [400–900] nm), representing a wide range of phase functions and
${\mu}_{s}^{\prime}\hspace{0.17em}=\hspace{0.17em}\left[0.05\hspace{0.17em}-\hspace{0.17em}8\right]\hspace{0.17em}{\text{mm}}^{-1}$. Because this calculation utilized the model fit to MC simulations to analyze measured reflectance data, the measurements were corrected for the observed 4% offset prior to
${\mu}_{s}^{\prime}$ estimation, as described in detail earlier in this Section. After this correction we were able to accurately extract the reduced scattering coefficient from Intralipid measurements, with a mean residual of 9% over the entire range of
${\mu}_{s}^{\prime}$; this
${\mu}_{s}^{\prime}$ range is wider than experienced in tissue, and therefore error associated with biologically relevant
${\mu}_{s}^{\prime}$ values is expected to be < 9%. In this study the semi-empirical model was used in conjunction with measured reflectance on phantoms to provide absolute estimates of
${\mu}_{s}^{\prime}$, estimates that have been demonstrated to be accurate for a given phase function. In tissue, the phase function is generally unknown and assumptions about the values for

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