Single fiber reflectance (SFR) spectroscopy is a technique that is sensitive to small-scale changes in tissue. An additional benefit is that SFR measurements can be performed through endoscopes or biopsy needles. In SFR spectroscopy, a single fiber emits and collects light. Tissue optical properties can be extracted from SFR spectra and related to the disease state of tissue. However, the model currently used to extract optical properties was derived for tissues with modified Henyey-Greenstein phase functions only and is inadequate for other tissue phase functions. Here, we will present a model for SFR spectroscopy that provides accurate results for a large range of tissue phase functions, reduced scattering coefficients, and absorption coefficients. Our model predicts the reflectance with a median error of 5.6% compared to 19.3% for the currently used model. For two simulated tissue spectra, our model fit provides accurate results.
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Reflectance spectroscopy techniques can provide rapid information related to the structural and biochemical composition of tissue. Single Fiber Reflectance (SFR) spectroscopy is a technique where light is emitted and collected through a single fiber, which is connected to a broadband light source and a spectrograph using a bifurcated fiber or a beam splitter. SFR measures the steady-state reflectance versus wavelength. Due to its small footprint, SFR measurements can be performed through endoscopes or biopsy needles [1–3]. SFR spectroscopy has been studied for medical diagnostics mainly in the field of oncology [2–8], but also in e.g. saturation monitoring  and orthopedics [1,10].
SFR spectroscopy is especially suitable to detect small-scale changes in tissue, due to its relatively small sampling volume, ∼(100 µm)3, and sensitivity to the tissue phase function. Since light is emitted and collected through the same fiber, photon path lengths are biased towards short pathlengths and diffusion theory alone cannot describe the measured reflectance. SFR spectroscopy is a subdiffuse technique and, therefore, measurements are sensitive to the tissue phase function [11–13], which is related to the nanoscale architecture of tissue. When light propagation is described as a random walk, the phase function (p[θ]) describes the probability of a photon scattering at an angle θ relative to the photon’s previous trajectory for each scattering event.
With an appropriate model, optical properties can be extracted from SFR spectra, which can be related to the disease state of tissue. Until now, only a single model was available for SFR spectroscopy to relate the measured reflectance to scattering and absorption properties of tissue, which was derived by Kanick et al. [14,15]. The model of Kanick et al. was derived using Monte Carlo (MC) simulations with modified Henyey Greenstein (MHG) phase functions. Unfortunately, even though the MHG phase function describes the phase function of skin tissue , for many other tissues other phase functions have been measured, e.g. the two-term Henyey-Greenstein (TTHG) for liver [17,18], uterus , brain , breast  and muscle  and the Reynolds McCormick phase function (RMC, also known as Gegenbauer) for blood [22,23]. In practice, the phase function of a specific tissue under investigation is generally not known. Therefore, a model that is valid for the wide range of tissue phase functions is essential.
Recently, we took the first steps towards the development of a new model for the SFR reflectance that provides accurate results for the wide range of tissue phase functions that can be encountered. We developed a new parameter (psb) to capture the phase function influence on the measured reflectance and we developed a model for the reflectance as a function of tissue scattering properties in the absence of absorption . However, due to the presence of absorbers in tissue such as blood, fat, and water, it is essential that absorption is included in the model to enable accurate extraction of optical properties from tissue measurements.
In this paper, we will develop a comprehensive model for the SFR reflectance as a function of both scattering and absorption properties of tissue. Furthermore, we will validate our model based on MC simulations for the wide range of reduced scattering coefficients, absorption coefficients, and phase functions that can be encountered in tissue. We will demonstrate that our comprehensive model for SFR spectroscopy predicts the measured reflectance substantially better than the model from Kanick et al. [14,15]. Finally, we will demonstrate the use of our model to determine optical properties from SFR measurements by simulating spectra of two tissue types.
2.1 SFR model
Diffusion theory can accurately describe the reflectance when photon path lengths are larger than several transport mean free paths, where the transport mean free path is the inverse of the reduced scattering coefficient: 1/µs’ and the reduced scattering coefficient equals µs’=µs(1-g1), where µs is the scattering coefficient and g1 is the scattering anisotropy (the first Legendre moment of the phase function). In SFR spectroscopy, photon path lengths are generally less than one transport mean free path since light is emitted and collected through the same fiber. In this so-called subdiffuse regime, the measured reflectance is the sum of photons undergoing a large number of scattering events (diffuse photons) and photons that undergo a few scattering events (semiballistic photons). Therefore, we model the reflectance as the sum of a semiballistic reflectance (RSFR,sb) and a diffuse reflectance (RSFR,dif):
Semiballistic photons are defined here as detected photons that underwent a single backscattering event in combination with an arbitrary number of forward scattering events .
Diffuse photons undergo many scattering events before they are detected and, therefore, their direction is randomized. Thus diffuse reflectance does not depend on the details of the tissue phase function, but only on the scattering anisotropy g1. Semiballistic photons undergo only a few scattering events and, therefore, their direction is not fully randomized. The semiballistic contribution to the reflectance is thus more sensitive to the details of the phase function. To capture the influence of the phase function on the measured reflectance, we incorporate the previously derived phase function parameter psb  into our model.
2.1.1. Diffuse reflectance RSFR,dif
For the contribution of diffuse photons to the reflectance, we make use of the diffusion approximation to the radiative transport equation, which describes the reflectance versus radial distance, R(ρ), for a pencil beam illumination. The implementation of the radiative transport equation for an overlapping source and detection fiber was derived by Faber et al., which models the diffuse reflectance as a single integral over the fiber diameter , where the integral of the diffuse reflectance versus radial distance is performed over the probability density function of distances over the fiber face p(ρ):
Equation (4) describes the distribution of distances between two randomly placed points on a disk with diameter d, which is a classic problem in the field of geometric probability [26,27]. For R(ρ) we will use the diffuse reflectance versus radial distance as a function of the reduced scattering coefficient and the absorption coefficient for a pencil beam illumination using the extended boundary condition as proposed by Farrell et al. :
Not all diffuse photons that reach the fiber face will be transported through the fiber to the detector. Only photons that arrive at an angle smaller than or equal to the acceptance angle of the fiber (θacc) will be detected, where θacc = asin(NA/n), NA is the fiber numerical aperture and n is the tissue refractive index. The fraction of photons arriving at the fiber face that is detected is described by the collection efficiency ηc . Thus, the diffuse reflectance (RSFR,dif) equals the collection efficiency of the fiber (ηc) times the fraction of diffuse photons that reach the fiber face (Rdif):24].
2.1.2. Ratio semiballistic to diffuse photons X
In Eq. (1), X is the ratio between the semiballistic and diffuse reflectance. Since semiballistic photons undergo only a few scattering events before detection, the semiballistic reflectance is sensitive to the tissue phase function p(θ). Therefore, the influence of the phase function on the semiballistic reflectance needs to be captured in the formula for X. Previously, we introduced the parameter psb to model the semiballistic contribution to the reflectance :
In short, the parameter psb describes the detection probability of photons that undergo a single backscatter event and multiple forward scattering events (photons that we defined as semiballistic). The integration angles were based on an analysis of MC simulations employing a large range of phase functions. To ensure the simulated reflectance was in the semiballistic regime, we had used a low reduced scattering coefficient (µs’d = 0.1) and no absorption in the MC simulations. The full derivation of psb and the choice of integration angles can be found in .
To derive a model for the ratio X in the presence of absorption, we start from the observation that the reflectance can be written as the product of the reflectance in the absence of absorption (R0) and the integral of the photon path length distribution (p(l)) weighted by the Beer-Lambert law:
2.2 Monte Carlo simulations
We will develop a model for f(µa/µs’) based on MC simulations. Photons were launched at locations based on a uniform distribution across the fiber with an angle from a uniform angular distribution within the acceptance angle of the fiber θacc, where θacc = arcsin(NA/n). Photons were detected if they reached the fiber face at an angle within θacc. For all MC simulations, the NA was 0.22 and the refractive index was 1.35 for the tissue, 1.45 for the fiber face, and 1.00 for the medium above the tissue. We ran each simulation three times and had chosen the number of launched photons such that the standard deviation over the mean of the reflectance for each set of three simulations was less than approximately 2%.
To derive the model for f(µa/µs’) in Eq. (11) of our model, we performed simulations with 10 different phase functions (Table 1), chosen such that they cover a wide range of psb values, g1 values, and phase function types. With these phase functions, we performed simulations for a fiber diameter of 0.01 cm, with 20 values of µs’ from 10 to 10000 cm−1 (equally spaced in 20 steps on a logarithmic scale) and 68 values of µa between 0.1 and 500 cm−1 (0.1 to 1 in steps of 0.1, 1 to 10 in steps of 1, 10 to 500 in steps of 10).
To determine the accuracy of our model for a wide range of phase functions, we performed additional simulations using the following 207 phase functions: 15 modified Henyey Greenstein (MHG), 146 two-term Henyey-Greenstein (TTHG), and 46 Reynolds McCormick phase functions (RMC), employing the parameters specified in Table 2 and applying the restrictions g1≥0.5 and g2<0.9 to exclude biologically unreasonable phase functions. These restrictions resulted in phase functions with fairly equally distributed g1 values between 0.5 and 0.94. We performed simulations for a fiber diameter of 0.01 cm and all the combinations of µa = [1,5,10,30, 100] cm−1, µs’ = [1,10, 50, 100] cm−1, as well as for a fiber diameter of 0.1 cm and all the combinations of µa = [1,5,10,30] cm−1, µs’ = [1,10, 50, 100] cm−1, resulting in 36 sets of simulations, with 207 phase functions each. We tested the accuracy of our model for the combination of the set of 10 phase functions and the set with 207 phase functions (21052 simulations in total). We compared this to the accuracy obtained with the currently used model of Kanick et al. [14,15].
To demonstrate the use of our model to determine optical properties from SFR spectra, we performed MC simulations with optical properties that can be encountered in tissue and performed a fit on these spectra using our model. We modeled spectra of two tissues, from 400 to 900 nm in steps of 5 nm. We based the reduced scattering coefficients and absorption coefficients on the review by Jacques , which are summarized in Table 4. The first tissue was simulated to resemble skin, with µs’ = 46(λ/500)−1.421 cm−1, a blood volume fraction of 0.01, an oxygen saturation level of 98%, and an MHG phase function where g1 increases linearly over the wavelengths from 0.7 to 0.77. The second tissue that we simulated resembles soft tissue, with µs’ = 18.9(λ/500)−1.3 cm−1, a blood volume fraction of 0.05, an oxygen saturation level of 98%, and a TTHG with α = 0.95, gb = -0.05, and gf linearly increasing over the wavelengths from 0.85 to 0.9.
For robust fit results, the number of fit parameters should be substantially smaller than the number of data points. Therefore, we performed MC simulations of spectra for two different fiber diameters. This implementation of SFR is referred to as multi-diameter SFR (MD-SFR) . We performed simulations with fiber diameters of 300 and 600 microns. For a single SFR measurement, the number of data points (reflectance values) equals the number of wavelengths, i. Clearly, separate values for µs’, µa, and psb per wavelength cannot be determined directly using a fit on a single SFR spectrum since the number of fit parameters would equal 3i. Therefore, we reduce the number of fit parameters by modeling the reduced scattering coefficient as µs’ = a· (λ/500)−b, where a is the scattering amplitude and b the scattering slope. The absorption coefficient is modeled as a sum of absorption spectra of different absorbers present in the tissue times the concentration of these absorbers: µa,tissue = Σcj·µa,j(λ), e.g. for tissue with blood and water: µa,tissue = cblood·µa,blood(λ)+ cwater·µa,water (λ). Since the wavelength-dependence of psb is unknown, we fit a value of psb for each wavelength within the spectrum. For a single measured spectrum, the number of data points equals i and the number of fit parameters equals 2+i + j (2 for µs’, i for psb, and j for the number of absorbers, respectively). Such an under-determined system will not provide robust fit results. The use of two separate measurements with two fibers of different diameters overcomes this issue. In that case, the number of data points will equal 2i, while the number of fit parameters remains 2+i + j.
Figure 2 shows the simulated reflectance as a function of µs’d for three different values of psb (colored lines) and two different values of µad. The dashed black line indicates the diffuse reflectance (RSFR,dif) for the corresponding µs’d and µad values. For high values of µs’d the reflectance equals the diffuse reflectance (dashed line). For lower values of µs’d there is an additional semiballistic contribution to the reflectance. With increasing psb and µad the fraction of semiballistic photons increases.
The diffuse contribution to the reflectance depends on µs’d and µad only. To investigate whether the semiballistic contribution to the reflectance can be modeled as a function of µs’d, µad and psb, we compared the semiballistic reflectance for simulations with different values of µs’ and µa, but the same values of µs’d, µad for all 207 phase functions (Table 2). Here, the semiballistic reflectance is the simulated reflectance minus RSFR,dif. Figure 3 shows the results for µs’d = 1.0 and µad = 1.0, as well as for µs’d = 0.1 and µad = 0.1 – using two different fiber diameters of 0.01 and 0.1 cm. The semiballistic reflectance values for the same µs’d, µad, and phase functions nearly overlap. Therefore, we will model the semiballistic contribution to the reflectance using µs’d, µad, and psb. The small differences in the reflectance can be explained by the fact that there is an uncertainty in our measured reflectance because we performed MC simulations such that the standard deviation over the mean of 3 simulations was less than 2%. It can be seen that for higher values of psb, there is some variation in the reflectance for the same psb values. This variation will likely result in a less accurate model for higher values of psb.
Based on our simulations we searched for a model for f(µa/µs’) based on visual inspection of the data. Figure 4 shows an example of the visualization of two sets of simulations, with µs’d = 0.1 and 0.144, with µad values of 0.001 to 1 and an MHG phase function. Based on the visual inspection we arrived at the following model:
Figure 5 shows the simulated reflectance versus the modeled reflectance for the combination of the set of 10 phase functions and the set with 207 phase functions (using b1,2 from Table 3). For our model [Fig. 5(a)] the median error is 5.6% with a standard deviation of 8.8%. Figure 5(b) shows the results for the model of Kanick et al., where the median error is 19.3%, with a standard deviation of 43.2%.
To obtain a better understanding of the relationship between the error in the reflectance and the optical properties, Fig. 6 depicts color maps of the median error in the reflectance versus µs’d, µad, and psb for the simulations with 10 phase functions. For a large range of optical properties, the median error is below 10%. Median errors increase up to 25% for µad values above 4, in combination with lower values of µs’d below 5. Also, median errors increase above 20% for high values of psb in combination with µs’d values above 3. We determined the median error for each type of phase function for the 36 sets of simulations with the 207 phase functions (Table 2). The median errors were 6.4% for the TTHG, 8.6% for the MHG, and 5.1% for the RMC.
Figure 7 shows two sets of simulated tissue spectra and the fit results for µs’, µa, and psb. Table 4 lists the resulting fit parameters related to blood (blood volume fraction and oxygen saturation) and scattering (scattering amplitude and slope). For both simulated tissues the fit results for µs’ are very close to the simulated µs’ values [Figs. 7(c), 7(h)]. The average differences in µs’ over all wavelengths were 7% and 4% for tissue 1 and 2, respectively. The difference in the blood volume fraction was 20% for tissue 1 and 2% for tissue 2. The blood oxygenation was underestimated by 1% for tissue 1 and overestimated by 2% for tissue 2. The average differences in psb over all wavelengths were 9% and 4% for tissue 1 and 2, respectively [Figs. 7(e), 7(j)]. Comparing the fit result for µa [Figs. 7(d), 7(i)] and psb, it seems that these fit parameters are competing. The absorption dips of hemoglobin are visible in the fit results for psb.
SFR spectroscopy is a spectroscopic technique especially suitable to detect small-scale changes in tissue. From the measured reflectance, tissue optical properties can be extracted and related to the disease state of tissue. However, the currently used model of Kanick et al.  to obtain optical properties from SFR measurements is limited to tissues with MHG phase functions. However, many tissues have different types of phase functions and often the phase function of a specific tissue under investigation is not known. Therefore, a model that is valid for a wide range of tissue phase functions is vital. Here, we developed a comprehensive model for the SFR reflectance as a function of both scattering and absorption properties of tissue that provides accurate results for a wide range of tissue phase functions. This new model predicts the measured reflectance substantially better compared to the model of Kanick et al.  (5.6 vs. 19.3% median error) and is valid for a wider range of phase functions.
We modeled the diffuse contribution to the reflectance by solving the model for spatially resolved reflectance of Farrell et al.  for an overlapping source and detection fiber  by integrating over the probability density function of distances over the fiber face. For the diffuse reflectance in the absence of absorption, a closed form of Rdif has been derived , but this has not yet been done in the presence of absorption. If a closed form of Rdif is derived this could increase the speed of a fitting procedure for MDSFR spectra. Here, we showed that the diffuse reflectance depends on µs’d and µa/µs’, and using MC simulations, we determined that the total reflectance can be described using µs’d, µad, and psb.
We investigated the median error in the reflectance versus µs’d, µad, and psb. For a large range of optical properties, the median error was below 10%. Median errors increase above 20% from a µad value of 4, in combination with lower values of µs’d below 5. The main absorbers in tissue are blood, fat, and water. A µad value above 4 will not be reached as a result of the fat or water content of tissue. For pure fat the highest absorption coefficient from 400-1100 nm is 0.13 cm−1 , for a large fiber diameter of 0.1 cm, this would result in a µad value of 0.013. For pure water, assuming a large fiber diameter of 0.1 cm, µad only becomes larger than 4 for wavelengths above 1880 nm . The increased errors for µad > 4 are relevant for the absorption by blood. The absorption spectrum of blood has one high peak from 400-450 nm and two lower peaks from 500-600 nm. To increase the range of blood volume fractions for which the model accurately predicts the absorption, a smaller fiber can be used. For a fiber diameter of 0.05 cm, a blood volume fraction of 28% will still result in µad < 4 for the spectrum above 450 nm . For most tissue types, the blood volume fraction is in the order of 1-5% , therefore, the model will provide accurate results for the spectrum above 450. The blood absorption spectrum also has a high peak from 400-450 nm. Assuming a fiber diameter of 0.05 cm, µad will be below 4 from 400-450 nm for blood volume fractions up to 3%. Nevertheless, the entire absorption spectrum of blood is fitted to a measured SFR spectrum. Therefore, we expect that even for higher blood volume fractions the fit will provide accurate estimates of the blood volume fraction since the majority of the spectrum will be accurately modeled. This is demonstrated by our results for the simulated tissue with a blood volume fraction of 5%. Even though the residual of the fit was high for 400-450 nm (Fig. 7(g)), the fit result for the blood volume fraction was accurate (Table 4).
We also found that median errors increase above 20% for high values of psb in combination with µs’d value s above 3. These higher errors can be explained by the fact that for higher values of psb there is more variation in the reflectance values obtained for different phase functions with similar psb values (Fig. 3). Higher errors for higher values of psb are thus an inevitable result when psb is used to model SFR measurements. Nevertheless, we showed previously that the variation in the reflectance was lowest for psb, compared to other parameters (σ , γ , δ  and RpNA ) that have been used to incorporate the phase function influence into models for subdiffuse reflectance . Therefore, modeling the reflectance using psb will provide more accurate results than using σ, γ, δ and RpNA.
To extract optical properties from SFR measurements, we used two different fiber diameters in an approach known as MDSFR. In our analysis in this paper, we assumed that both fibers sample a volume with the same optical properties. In clinical applications, when the two fibers are placed next to each other and tissue is inhomogeneous, this is not necessarily the case. Compared to DRS – where it is also assumed the tissue within the sampling volume is homogenous – MDSFR has the advantage that the sampling volume is much smaller and, therefore, the assumption of a homogenous sampling volume is more likely to hold. If two fibers are placed next to each other, the sampling volume will be shifted sideways with respect to the tissue surface by only a few hundred micrometers. Nevertheless, in MDSFR measurements the larger fiber will sample a deeper tissue volume, which is especially relevant for the absorption of light by the microvasculature. This can be accounted for by fitting separate parameters related to absorption for each fiber diameter .
For the MDSFR approach in this paper, we modeled the reduced scattering coefficient as µs’=a· (λ/500)−b and the absorption coefficient as a sum of absorption spectra of different absorbers present in the tissue times the concentration of these absorbers. Currently, the wavelength dependence of the phase function in general, and psb specifically, is not well-characterized. Therefore, a value of psb was fitted for each wavelength in the spectrum. It seems that µa and psb currently compete in the fit, leading to less accurate results. The robustness of the fit is expected to increase if a model for psb is used that decreases the number of fit parameters.
We developed a model for SFR spectroscopy to describe the reflectance as a function of tissue scattering and absorption properties, which provides accurate results over a wide range of phase functions. The new model predicts the measured reflectance substantially better compared to the currently used model of Kanick et al.  which was developed for tissues with MHG phase functions only. The phase function of a specific tissue under investigation is generally not known. Therefore, a model that is valid for the wide range of phase functions that can be encountered in tissue is essential.
KWF Kankerbestrijding (2014-7009); Nederlandse Organisatie voor Wetenschappelijk Onderzoek (iMIT-PROSPECT grant number 12707).
The authors declare no conflicts of interest.
See Supplement 1 for supporting content.
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