A reflectance spectroscopic device that utilizes a single fiber for both light delivery and collection has advantages over classical multi-fiber probes. This study presents a novel empirical relationship between the single fiber path length and the combined effect of both the absorption coefficient, μa (range: 0.1–6 mm-1), and the reduced scattering coefficient, μ′s (range: 0.3 – 10 mm-1), for different anisotropy values (0.75 and 0.92), and is applicable to probes containing a wide range of fiber diameters (range: 200 – 2000 μm). The results indicate that the model is capable of accurately predicting the single fiber path length over a wide range (r = 0.995; range: 180 – 3940 μm) and predictions do not show bias as a function of either μa or μ′s.
©2009 Optical Society of America
Reflectance spectroscopy is a method that is capable of noninvasively determining tissue scattering and absorption properties [1, 2]; information that can be used to estimate tissue chromophore concentrations and for diagnosis of abnormal or malignant tissue . Classical reflectance spectroscopy devices often utilized multiple optical fibers to deliver and collect light during measurement [4, 5], however, the potential advantages of reflectance probes with a single optical fiber to deliver/collect light have been suggested previously [6, 7]. Advantages of the single fiber design include small probe size and simple device design, making it more-suitable than multi-fiber probes for some specific clinical applications, such as optical biopsy of potential malignancies via endoscopy. Such a procedure has the potential to assist clinicians in the selection of tissue extraction during standard biopsy procedures , and is applicable in tissues such as the breast , brain , and lymph nodes . While the biopsy needle at the end of the endoscope catheter is often too small for multi-fiber probes to fit (with an inner diameter of only ≈ 400 μm), this is not a concern for small single fiber probes with fiber diameters in the range of 200 – 400 μm. The small diameter (≤ 400 μm) single fiber probes are well-suited for endoscopic measurements, however, specific applications may require longer photon path lengths (that are more sensitive to changes in tissue absorption coefficient) or optical sampling of deep tissue volumes (potentially interrogating millimeters below the tissue surface); these factors can be addressed by designing larger diameter single fiber probes in the range of 400 – 2000 μm. These larger fibers are suitable for applications requiring measurement of optical properties from deeper tissue layers, which is desired when noninvasively assessing the morphology and physiology of e.g. oral mucosa .
A fundamental limitation of reflectance spectroscopy has been the following: knowledge of the photon path length is required to accurately estimate the tissue optical properties, however, the path length is dependent on the optical properties of the sampled tissue. Multiple groups have addressed this problem for classical multi-fiber diffuse reflectance spectroscopy devices [4, 12, 13], by describing the effect of absorption and scattering on the photon path length. Previously, our group addressed this problem by developing differential path length spectroscopy (DPS), which utilizes a specialized two fiber geometry that collects light with both the delivery fiber and an adjacent collection fiber. The resulting differential signal (calculated as the difference between the signals collected by the two fibers) selectively sampled shallow tissue depths, and resulted in a photon path length that remains constant over a wide range of optical properties [14, 15]. The motivation for these studies is clear; quantitative analysis of reflectance spectra required the knowledge of photon path length during measurement of tissue in vivo. Previous studies that have suggested the use of single fiber spectroscopy have lacked a formal description of the photon path length as a function of optical properties [6, 7], and this must be addressed in order for the measurement to return quantitative information.
This study develops an empirical relationship between the single fiber reflectance photon path length and both the absorption coefficient (μa) and reduced scattering coefficient (μ′s) within an optically sampled turbid medium. The empirical formula presented here is capable of predicting the single fiber path length over a wide range of optical properties, and the relationship holds for a wide range of fiber diameters.
2.1. Single fiber reflectance spectroscopy
Figure 1 shows the schematics for the two single fiber device probe setups used to measure optical phantoms in this study, which includes (A) the single fiber probe, for fiber diameters of 200 – 1000 μm, and (B) the single fiber bundle probe, for a fiber diameter of 2000 μm. Both probe setups utilized a spectrophotometer (SD 2000; Ocean Optics; Duiven, the Netherlands) and halogen light source (HL-2000-FHSA; Ocean Optics; Duiven, the Netherlands). The single fiber probe, as shown in Fig. 1(A), contains a single optical fiber that both delivers light to the tissue and collects light remitted from the tissue. During measurement, photons travel from the light source through one arm of a bifurcated fiber and through the single fiber, after which it exits into the sample. Reflected photons that are collected by the single fiber travel through the second arm of the bifurcated fiber and into the spectrophotometer. The fiber diameters in each arm of the bifurcated fiber are one-half of the diameter of the single fiber diameter, allowing the maximum overlap between the fibers at the interface. Spectral reflections at the probe tip due to refractive index mismatch between fiber and sample are minimized by polishing the single fiber probe tip at an angle of 15 degrees . A calibration procedure was utilized to account for other internal reflections, variability in lamp-specific output, and in fiber-specific transmission properties. The calibration involves measurement of both white and black Spectralon standards (Labsphere SRS-99 and SRS-02, with spectrally flat 99% and 2% reflectance, respectively) in air and measurement of water within a dark container. These measurements are used to calculate the single fiber reflectance signal, RSF, as
Here, the measured intensity (I) is reduced by the intensity attributable to internal reflections (I water) and normalized by the difference between intensities measured from white and black spectralons. All symbols marked in boldface type are considered wavelength-dependent. The calibration constant, c cal accounts for the specific distance between the fiber probe tip and spectralon surface during calibration measurements.
The single fiber bundle probe, as shown in Fig. 1(B), contains a bundle of small optical fibers (with fiber diameters of 200 μm), of which half of the fibers are connected to the light source, and the other half to the spectrophotometer. The assignment of individual fibers to source/spectrograph is randomized, as visualized in the schematic by the shaded or unshaded fibers, such that during measurement, the photons effectively enter and exit the tissue across the entire 2000 μm surface of the probe. Spectral reflection is not a concern for this probe, because the individual source fibers are not collecting light, and therefore, the probe surface is polished at 0 degrees. The calibration procedure is slightly different for the single fiber bundle probe, and does not require measurement of either the black spectralon or water. In this case, the single fiber bundle reflectance signal, RSF, is given as
Here, the single fiber reflectance intensity as measured by the single fiber bundle, RSF, is the same quantity as returned in Eq. (1).
For reflectance intensities measured by all probes, attenuation in response to the addition of a chromophore can be described using the Beer-Lambert law as follows,
where RSF0 is the single fiber reflectance prior to addition of the chromophore, μia and Ci are the specific absorption coefficient and concentration of the chromophore i, respectively, and τ SF is the mean path length of photons contributing to the single fiber signal, RSF. Here, it is important to note that τ SF is dependent on μ a and μs′.
2.2. Optical phantom preparation
Optical phantoms were prepared by mixing Intralipid 20% (Fresenius Kabi AG, Bad Homburg, Germany), Evans Blue powder (Sigma-Aldrich Inc., Vienna, Austria), and saline solution (0.9%). The μa of each phantom was selected by varying the concentration of Evans Blue, which has a specific absorption (μ a EB) maximum of 18 L/(g mm) at 611 nm. The μ′s of each phantom was selected by varying the amount of Intralipid 20%, which has a μs of 80 mm-1 and an anisotropy of g = 0.75 (resulting in a μ′s of 20 mm-1) undiluted at 611 nm. Because Intralipid is not an optical standard, the optical properties of the batch utilized in this study were verified using a spatially resolved diffuse reflectance measurement, as has been done previously .
Optical phantoms were constructed with reduced scattering coefficients of: μ′s=[0.38,0.75,1.5,2.25,3,5,10.5] mm-1 and absorption coefficients of: μa=[0.1,0.4,1,3] mm-1. The 200 μm single fiber probe was used to measured phantoms with μa=[0.4, 1,3] mm-1, and the 2000 μm probe measured phantoms with μa=[0.1,0.4,1] mm-1. Therefore 21 combinations of optical properties were measured by each probe. Additional phantoms were prepared at each selected μ s′ with no Evans Blue added, which were utilized to obtain baseline measurements of RSFO that represented μa=0 mm-1 at 611 nm.
This study also incorporated single fiber path length measurements that had been originally performed for analysis of differential path length spectroscopy . Phantoms in that previous study were constructed in the same manner as described here. Those data included variations in absorption coefficient μa=[0.1,0.2,0.4,0.8,1.6,3.2,6.4] mm-1 for a constant reduced scattering coefficient (μ s′=3.75 mm-1), and also variations in reduced scattering coefficient μ s′=[0.38,0.75,1.5,2.25,3,5,10.5] mm-1 for a constant absorption coefficient (μa=0.4 mm-1). These phantoms were measured with single fiber probes containing fiber diameters of: d fiber=[200,400,600,800,1000] μm.
During measurement by the single fiber device, the probe was lowered into the phantom so that the probe tip was below the meniscus of the phantom surface. Each phantom consisted of a 10 ml or 20 ml sample (for 200 and 2000 μm probes, respectively) contained within a 24 mm diameter cylindrical container. Boundary effects were assumed to be negligible after measurements were not shown to be different than 40 ml phantoms. Each phantom was prepared 3 independent times, and each preparation was measured 10 sequential times with the integration time adjusted to obtain adequate collected light intensity from measurement of each phantom.
2.3. Analysis of single fiber path length
The Beer-Lambert law was used to describe attenuation of the single fiber reflectance signal, RSF, due to the addition of an absorber, as in Eq. (3). Changes between RSF (with Evans Blue) and RSFo (without Evans Blue) were assumed to be attributable to the difference in absorption coefficient between the two samples. Therefore, the difference between RSF and RSFo was measured at a wavelength where the optical properties were known (at 611 nm). This was calculated by normalizing RSFo and RSF over the wavelength range [750 – 800] nm. This normalization procedure accounts for small differences in fiber transmission due to differences in fiber bending for the paired phantom measurements. Typically, these transmission differences are less than 1%, but for small absorption coefficients, such small differences would have a large effect on the calculated path lengths. The single fiber path length was calculated at 611 nm as:
2.4. Empirical model of single fiber path length
Observations of the effect of μs′ and μa on the single fiber path length led to the selection of the following model:
Here, the parameter set [p 1,p 2,p 3,p 4] is fitted by minimizing the residual error between measured and predicted dimensionless path length (τSF/d fiber), with each point weighted by the inverse of the standard error of the mean. Confidence intervals on parameter estimates were calculated from the square of the diagonal of the covariance matrix, which is described in detail elsewhere . Parameter estimation was achieved using a Levenberg-Marquardt algorithm  that was scripted into LabView code (Version 7.1.1; National Instruments).
3.1. Single fiber reflectance data
Single fiber reflectance spectra were measured at a resolution of 2048 pixels over the wavelength range [340 – 1027] nm, and the data were smoothed by averaging data points into bins of 10 pixels, which allowed the calculation of a standard deviation that represents noise within the signal . Figure 2 shows the normalized single fiber reflectance intensity data from measurements with d fiber=2000 μm on optical phantoms without Evans Blue, RSFo (μ′s=5 mm-1 and μa=0 mm-1), and with Evans Blue, RSF (μ′s=5 mm-1 and μa=0.4 mm-1).
3.2. Single fiber path length dependence on μ′s
Figures 3(A) and (B) show the single fiber path length, extracted from the empirical data using Eq. (4), vs. μ′s for various selected values of μa as measured by probes with fiber diameters of 200 and 2000 μm, respectively. Here, data points represent the mean from single fiber measurements of 3 independent optical phantoms, and the error bars indicate one standard deviation about the mean. The data show the general trend that τSF increases as μ′s decreases, which is an expected result, due to photons travelling deeper into the phantom before being backscattered to the fiber. In order to compare the effect of μ′s on τSF over a range of fiber diameters, the data are transformed into dimensionless path length (τSF/d fiber) and dimensionless reduced scattering (μ′s d fiber), as shown for 200 and 2000 μm fiber diameters on log-log scales in Fig. 3 (C) and (D), respectively. Additionally, Fig. 3(E) shows (τSF/d fiber) vs. (μ′s fiber) for measurements of optical phantoms with constant μa (0.4 mm-1), as measured by single fiber probes with d fiber over the range [200 – 1000] μm. These data show that the dimensionless single fiber path length increases as dimensionless reduced scattering decreases, with vertical stratification of the dimensionless path lengths due to the effect of different μa values.
3.3. Single fiber path length dependence on μa
Figures 4(A) and (B) show τSF vs. μa for various selected values of μ′s as measured by probes with fiber diameters of 200 and 2000 μm, respectively. Again, data points represent the mean from single fiber measurements of 3 independent optical phantoms, and the error bars indicate one standard deviation about the mean. The data show a general trend of decreasing τSF in response to increasing μa, an expected result as photons with longer path lengths are more likely to be absorbed, and therefore, less likely to be detected, resulting in the reduction of the path length. Figures 4(C) and (D) show (τSF /d fiber) as a function of dimensionless absorption (μad fiber) on log-log scales, as measured by the 200 and 2000 μm probes, respectively. Figure 4(E) shows (τSF/d fiber)vs. (μad fiber) for measurements of optical phantoms with constant μ′s (3.75 mm-1), as measured by single fiber probes with d fiber over the range [200 – 1000] μm. These data show that the dimensionless single fiber path length decreases as dimensionless absorption decreases, with vertical stratification of the dimensionless path lengths due the effect of different μ′s values.
3.4. Empirical model of single fiber photon path length as function of μ′s and μa
Figure 5(A) shows τSF model, the single fiber path length calculated by Eq. (5), vs. τSF, the single fiber path length measured empirically. The estimated parameter values of p 1 = 1.34 ± 0.16, p 2 = 0.17 ±0.08, p 3 = 0.23 ±0.05, and p 4 = 0.52 ±0.16 resulted in the smallest weighted residual error between measured data and model predictions. Model predictions were significantly correlated with measured values, as evidenced by a Pearson product correlation coefficient of r = 0.995; this effect is observable in the plot as the data are scattered about the line of unity. Figure 5(B) plots the data on a log-log scale, which shows that the relationship holds over the wide range of single fiber path lengths (range: 180 – 3940 μm). Figure 5(C) shows the residual error as a fraction of the measured single fiber path length value (residual percentage= 100 × (τSF - τSF model)/τSF), vs. the measured τSF values, with error bars calculated as the ratio of the standard deviation to the mean, representing the uncertainty in each measured data point. The mean absolute residual error percentage is 4.1 ±3.5%, and all 110 data points have an absolute residual error that is ≤ 20% of the measured path length value. The plot shows the residual error scattered about zero across the range of measured path lengths, with no observable trends. The residuals showed no correlation with either (μ′s d fiber) or (μad fiber), yielding Pearson product correlation coefficients of r = -0.072 and r = 0.061, respectively. This result indicates that the model structure is appropriate, and that incorporation of additional fitted parameters would only describe noise within the data.
3.5. Effect of the anisotropy on the single fiber path length
The empirical relationship for single fiber path length, presented in Eq. (5), is a function of reduced scattering coefficient, μ′s, and not total scattering coefficient, μs. This distinction was made by investigating the effect that variation in the phase function, and anisotropy (g) value, within an optical phantom would have on the path length for constant μs values. This was investigated by preparing a series of optical phantoms that used polystyrene beads (size 1 μm) as the scattering material, instead of Intralipid 20%. The optical phantoms were prepared with μa = 1 mm-1, μs=[1.5,3,6,9,12,20,42] mm-1 (to match the total scattering coefficient as measured in the Intralipid phantoms), but with a g = 0.92 (compared with the g = 0.75 for Intralipid 20%), the reduced scattering coefficient became: μ′s =0.12,0.24,0.48,0.72,0.96,1.6,3.36] mm-1. Figure 6 shows the τSF data as measured by a single fiber probe with d fiber = 200 μm as a function of μ′s as measured in optical phantoms with both Intralipid and polystyrene beads. The data show a dependence on the reduced scattering coefficient, with measurements following the model predicted values. There is a strong agreement between the data measured in the polystyrene bead phantoms and τSFmodle predictions at the corresponding μ′s values, with a the Pearson product correlation coefficient of r = 0.995. This result is novel, yet not unexpected, because the path traveled by photons propagating through the medium is dependent on both μs and g, so the combined effect of each is represented in μ′s. It should be noted that the data measured in polystyrene bead phantoms was not used to fit Eq. (5), and these results indicate the quality of the predictions of this model.
4. Discussion and conclusions
Quantitative analysis of single fiber reflectance measurements of tissue requires knowledge of the photon path length. However, this is complicated by the dependence of the photon path length on the optical properties of the optically sampled medium. This study presents a novel empirical formula that describes the dependence of the single fiber reflectance photon path length on both reduced scattering and absorption coefficients. This relationship is valid for a wide range of μ′s (range: 0.3 – 10 mm-1) and μa (range: 0.1 – 6.4 mm-1), values that are representative of the optical properties of tissue in the ultra-violet / visible / near-infrared (UV/VIS/NIR) wavelength regions [20, 21]. The equation is applicable to probes containing a wide range of fiber diameters (range: 200 – 2000 μm), meaning that this description of path length can be used for probes that sample either shallow or deeply into tissue (with the depth of optical sampling dependent on the fiber diameter and tissue optical properties). Moreover, the error associated with model predictions is very low, with a mean absolute error of only 4.1 ±3.5% from measurements that spanned a 64-fold change in μa, a 28-fold change in μ′s, and a 10-fold change in d fiber.
Previous publications have described the concept of single fiber reflectance spectroscopy and the advantage compared with multi-fiber devices [6, 7]. Our group has utilized single fiber reflectance probes to investigate measurement of single scattering events , and utilized the measurement to describe changes in tissue associated with photodynamic therapy . However, these studies were not able to present quantitative analysis of single fiber reflectance spectra measured in tissue in vivo because the path length was not accurately known. To the best of the authors’ knowledge, this is the first study to present a mathematical description of the photon path length for a single fiber reflectance probe; a relationship that allows quantitative analysis of the optical properties from measured reflectance spectra.
The relationship describing single fiber path length as a function of μ′s and μa is useful in analyzing spectra measured from tissue in vivo. It is expected that the single fiber reflectance spectra can be described analytically, by specifying a background scattering model, such as Mie and Rayleigh scattering [18, 23], and utilizing the Beer-Lambert law to account for absorption from tissue chromophores, as in Eq. (3). This mathematical method allows reasonable estimation of μa across the fitted wavelength range (assuming that the chromophores contributing to the detected absorbance have been specified correctly) . This calculation will also require estimation of the absolute values of μ′s across the fitted wavelength range, which can be difficult to estimate accurately, but one that can be approached sensibly. One approach would be to specify the underlying scattering model and assume the absolute value of μ′s at one wavelength (preferably in a region where scattering does not vary much), allowing the model to describe μ′s as a function of wavelength. While this is a potential source of error, the magnitude of this error could be easily estimated by varying the assumed μ′s value and determining the effect on estimated parameters. Another approach is to utilize measurements of multiple fiber diameters (on the same tissue) to mathematically estimate the value of μ′s, removing the need to estimate the value of μ′s at a single wavelength; this is an issue that will be addressed in future studies. It is also worth noting that many tissues contain μa < 0.1 mm-1 within the UV/VIS wavelength region, values which are below the lower absorption coefficient investigated experimentally within this study. This is not expected to be a source of error within an estimate of reflectance, because it is the product of the absorption coefficient and path length (μaτSF) that appears in Eq. (3). Therefore, in the logical limit of an μa approaching zero, the product μaτSF also approaches zero, and the effect of the path length estimate in this case will have near zero effect on the attenuation term.
The current study defines the single fiber reflectance path length dependence on optical properties, allowing quantitative analysis of spectra. This development has the potential to not only allow probes with individual single fibers to be utilized quantitatively, but it would be possible to incorporate the single fiber measurement into classical multi-fiber devices, if the source fiber were also used as a detector. Such resulting devices would allow optical properties to be accurately estimated from multiple tissue volumes with a single measurement (assuming that the path length is well-defined for the other collection fiber geometries as well).
The results presented in this study indicate that the single fiber photon path length will vary across wavelengths from measurements of tissue in the UV/VIS/NIR wavelength ranges. This may complicate analysis of spectra, since the single fiber reflectance measurement may optically sample different tissue volumes at different wavelengths (based on the magnitude of the tissue optical properties). But, the results also suggest that measurements of multiple fiber diameters on the same tissue volume may provide information about the depth-dependence of the optical properties. Future studies will investigate the (potentially complex) relationship between single fiber photon path length and optically sampled tissue volume.
The authors would like to thank B. Kruijt and F. van Zaane for assistance in constructing the single fiber probes, and to thank D.J. Robinson for helpful discussions. This research is supported by the Dutch Technology Foundation STW, applied science division of NWO and the Technology Program of the Ministry of Economic Affairs.
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