_{a}) and total scattering (μ

_{s}) coefficients, which are common within the ultraviolet/visible(UV/VIS) wavelength region. This study extends the description of τ to larger μ

_{a}and smaller μ

_{s}values, optical properties that are representative of the near-infrared region (NIR), a region where the DPS path length may be dependent on both coefficients. This study presents a novel empirical relationship between τ and the combined effect of both μ

_{a}(range: 0.1–12 mm

^{-1}) and μ

_{s}(range: 1.5–42 mm

^{-1}), anisotropy of 0.8, and is applicable to DPS probes containing a wide range of fiber diameters (range: 100–1000 μm). The results indicate that the simple empirical formula, including only one fitted parameter, is capable of accurately predicting over a wide range (r=0.985; range: 80–940 μm) and predictions are not biased versus μ

_{a}or μ

_{s}. This novel relationship is applicable to analysis of DPS measurements of tissue in both the UV/VIS and NIR wavelength regions and may provide information about the wavelength-specific tissue volume optically sampled during measurement.

## 1.

## Introduction

Diffuse reflectance measurements of tissue are complicated by the effect that tissue optical properties have on the photon path length.^{1, 2, 3} As photons propagate through tissue, they undergo scattering and absorption events, with the likelihood of these interactions given by the total scattering and absorption coefficients (
${\mu}_{\mathrm{s}}$
and
${\mu}_{\mathrm{a}}$
, respectively). ^{1, 2} Classical diffuse reflectance devices collect photons with path lengths that are heavily dependent on the
${\mu}_{\mathrm{s}}$
and
${\mu}_{\mathrm{a}}$
values within the tissue being optically sampled.^{2, 3} This complicates spectral analysis, because the Beer–Lambert law requires accurate knowledge of the photon path length in order to estimate chromophore concentrations within the optically sampled tissue volume.^{3} Moreover, the dependence of path length on optical properties has an effect on the tissue volume sampled during measurement, with measurements of different tissues potentially interrogating different volumes, or measurements of multiple wavelengths on the same tissue sample interrogating different volumes at different wavelengths, due to the wavelength-specific optical properties.

Our group has previously developed differential path length spectroscopy (DPS), which utilizes a unique fiber geometry to selectively sample photons that have propagated shallow depths into the tissue, making the path length approximately equal to the fiber diameter.^{4} This fiber geometry allows accurate control of the path length and overcomes the classical limitation of unknown photon path length during diffuse reflectance measurements. The DPS path length,
$\tau $
, is insensitive to variations in both
${\mu}_{\mathrm{s}}$
and
${\mu}_{\mathrm{a}}$
over a wide range of values commonly experienced in the ultraviolet/visible (UV/VIS) wavelength region, a phenomenon that has been investigated by both Monte Carlo modeling^{4} and experimental measurement.^{5} In the latter study,^{5} it was noted that relatively large
${\mu}_{\mathrm{a}}$
values for the UV/VIS region
$({\mu}_{\mathrm{a}}>1\phantom{\rule{0.3em}{0ex}}{\mathrm{mm}}^{-1})$
caused a reduction in the DPS path length. These absorption values could be observed in highly vascularized tissue due to strong absorption bands of hemoglobin in the
$300\u2013600\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
wavelength region. Therefore, the relationship was described by an empirical formula relating the dimensionless DPS path length
$(\tau \u2215{d}_{\text{fiber}})$
to the dimensionless absorption coefficient, given as the product
${\mu}_{\mathrm{a}}{d}_{\text{fiber}}$
. This formula held for a wide range of
${\mu}_{\mathrm{a}}$
values (range:
$0.1\u20136.4\phantom{\rule{0.3em}{0ex}}{\mathrm{mm}}^{-1}$
) and fiber diameters (range:
$200\u20131000\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$
). The same study^{5} also reported that very small
${\mu}_{\mathrm{s}}$
values for the UV/VIS region
$({\mu}_{\mathrm{s}}<5\phantom{\rule{0.3em}{0ex}}{\mathrm{mm}}^{-1})$
increased the DPS path length, an effect that was not modeled because such small scattering values are not commonly experienced in the UV/VIS range. These results indicated that the DPS path length could be accurately estimated during analysis of tissue spectra in the UV/VIS region.

However, the findings indicated that the current description of DPS path length was not directly extensible to the near-infrared (NIR) wavelength region. This is because the NIR region contains larger
${\mu}_{\mathrm{a}}$
$(\approx 10\phantom{\rule{0.3em}{0ex}}{\mathrm{mm}}^{-1})$
and smaller
${\mu}_{\mathrm{s}}$
$(\approx 2\phantom{\rule{0.3em}{0ex}}{\mathrm{mm}}^{-1})$
values than the UV/VIS region,^{6, 7, 8} potentially making the NIR-DPS path length more dependent on the absolute value of these coefficients. This study reports DPS path lengths measured in optical phantoms that contain variations in the combinations of
${\mu}_{\mathrm{s}}$
and
${\mu}_{\mathrm{a}}$
values that are representative of the NIR wavelength region. The measurements are made with DPS probe fiber diameters of 100 and
$200\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$
, small fibers specifically selected to collect appreciable amounts of light from measurements of a medium containing large absorption coefficients. Additionally, this study utilizes previously reported data^{5} that showed the effect of independent variations in
${\mu}_{\mathrm{s}}$
and
${\mu}_{\mathrm{a}}$
on the DPS path length for fiber diameters of
$200\u20131000\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$
. The data are utilized to develop a novel empirical formula that describes the combined effect of both absorption and scattering on the DPS path length for a wide range of optical properties representative of the NIR wavelength region in tissue.

## 2.

## Methods

## 2.1.

### DPS

DPS utilizes a specialized fiber geometry to selectively sample superficial layers of tissue with a known photon path length. The DPS probe contains two fibers: one used for delivery and collection $\left(dc\right)$ and an adjacent fiber used for collection $\left(c\right)$ . During measurement, photons exit the $dc$ fiber, scatter throughout the tissue, and are collected by both the $dc$ and $c$ fibers. The $dc$ fiber collects photons that have scattered a range of different distances throughout the tissue, a subset of which has traveled short distances before being backscattered by the superficial layers of tissue. Comparatively, the $c$ fiber signal collects approximately the same number of deeply scattered photons, but this probe does not contain a subset of superficially backscattered photons. The difference between the signals collected by the $dc$ and $c$ fiber contains the contribution from photons that interrogated shallow tissue depths. This relationship can be expressed as follows:

where $\mathbf{I}$ is the light intensity collected by the $dc$ fiber, $\mathbf{J}$ is the light intensity collected by the $c$ fiber, and $\mathbf{R}$ is the differential reflectance. All symbols marked in boldface type are considered wavelength dependent. Attenuation of the $\mathbf{R}$ signal by addition of a chromophore can be described using the Beer–Lambert law as follows:## Eq. 2

$$\mathbf{R}={\mathbf{R}}_{\mathbf{o}}\phantom{\rule{0.3em}{0ex}}\mathrm{exp}(-{\mathbf{\mu}}_{\mathbf{a}}^{\mathbf{i}}{C}_{i}\mathbf{\tau}),$$## 3.

## DPS Instrumentation

Figure 1
shows a schematic of the DPS setup used to measure optical phantoms in this study. The device contains a spectrophotometer (SD 2000; Ocean Optics; Duiven, The Netherlands) and halogen light source (HL-2000-FHSA; Ocean Optics; Duiven, The Netherlands). During measurement, photons travel from the light source through one arm of a bifurcated fiber and through the
$dc$
fiber, after which it exits into the sample. Reflected photons that are collected by the
$dc$
fiber travel through the second arm of the bifurcated fiber and into the first channel of the spectrophotometer. Reflected photons that are collected by the
$c$
fiber travel directly into the second channel of the spectrophotometer. Spectral reflections at the probe tip due to refractive index mismatch between the fiber and sample are minimized by polishing the DPS probe tip at an angle of
$15\phantom{\rule{0.3em}{0ex}}\mathrm{deg}$
.^{9} A calibration procedure, described in detail elsewhere,^{9} 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) in air and measurement of water within a dark container. These measurements are used to calculate the differential reflectance signal,
$\mathbf{R}$
, as

## Eq. 3

$$\mathbf{R}={c}_{\mathrm{cal}}[\frac{(\mathbf{I}-{\mathbf{I}}_{\text{water}})}{({\mathbf{I}}_{\text{white}}-{\mathbf{I}}_{\text{black}})}-\frac{\mathbf{J}}{({\mathbf{J}}_{\text{white}}-{\mathbf{J}}_{\text{black}})}].$$## 3.1.

### 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
${\mu}_{\mathrm{a}}$
of each phantom was selected by varying the concentration of Evans Blue, which has an absorption
$\left({\mu}_{\mathrm{a}}^{\mathrm{EB}}\right)$
maximum of
$18\phantom{\rule{0.3em}{0ex}}\mathrm{L}\u2215\left(\mathrm{g}\phantom{\rule{0.3em}{0ex}}\mathrm{mm}\right)$
at
$611\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
. The
${\mu}_{\mathrm{s}}$
of each phantom was selected by varying the amount of Intralipid 20%, which has a
${\mu}_{\mathrm{s}}$
of
$80\phantom{\rule{0.3em}{0ex}}{\mathrm{mm}}^{-1}$
and an anisotropy of
$g\approx 0.8$
when undiluted at
$611\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
. 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.^{10}

Phantoms were constructed with scattering coefficients of ${\mu}_{s}=1.5$ , 3, 6, 9, 12, 20, and $42\phantom{\rule{0.3em}{0ex}}{\mathrm{mm}}^{-1}$ , for each absorption coefficient of: ${\mu}_{\mathrm{a}}=0.4$ , 1, 3, and $12\phantom{\rule{0.3em}{0ex}}{\mathrm{mm}}^{-1}$ , representing 28 combinations of optical properties. Also, additional phantoms were prepared at each selected ${\mu}_{\mathrm{s}}$ with no Evans Blue added, which were utilized to obtain baseline measurements of ${\mathbf{R}}_{\mathbf{o}}$ that represented ${\mu}_{\mathrm{a}}=0\phantom{\rule{0.3em}{0ex}}{\mathrm{mm}}^{\text{-}1}$ at $611\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ . Each phantom consisted of a $10\text{-}\mathrm{mL}$ sample contained within a $24\text{-}\mathrm{mm}$ -diameter cylindrical container. Phantoms for each paired value of optical properties were independently prepared three times and measured by DPS probes with ${d}_{\text{fiber}}$ of 100 and $200\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$ .

This study also incorporated DPS path length data reported previously.^{5} Phantoms in that previous study were constructed in the same manner as described here. Those data included variations in the absorption coefficient
${\mu}_{\mathrm{a}}=0.1$
, 0.2, 0.4, 0.8, 1.6, 3.2, and
$6.4\phantom{\rule{0.3em}{0ex}}{\mathrm{mm}}^{-1}$
for a constant scattering coefficient
${\mu}_{\mathrm{s}}=15\phantom{\rule{0.3em}{0ex}}{\mathrm{mm}}^{-1}$
, and also variations in the scattering coefficient
${\mu}_{\mathrm{s}}=1.5$
, 3, 6, 9, 12, 15, 23, and
$42\phantom{\rule{0.3em}{0ex}}{\mathrm{mm}}^{-1}$
for a constant absorption coefficient of
${\mu}_{\mathrm{a}}=0.4\phantom{\rule{0.3em}{0ex}}{\mathrm{mm}}^{-1}$
. These phantoms were measured with DPS probes containing fiber diameters of
${d}_{\text{fiber}}=200$
, 400, 600, 800, and
$1000\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$
.

During measurement by DPS, the probe was lowered into the phantom so that the probe tip was below the meniscus of the phantom surface. Boundary effects were assumed to be negligible after measurement of $10\phantom{\rule{0.3em}{0ex}}\mathrm{mL}$ phantoms was not shown to be different than $40\text{-}\mathrm{mL}$ phantoms. During the measurement of each phantom, the DPS recorded 10 sequential measurements on each phantom with the integration time adjusted to obtain adequate collected light intensity from measurement of each phantom.

## 3.2.

### Data Analysis

The Beer–Lambert law was used to describe attenuation of the DPS reflectance signal, $\mathbf{R}$ , due to the addition of an absorber, as in Eq. 2. Basis reflectance measurements of ${\mathbf{R}}_{\mathbf{o}}$ were made on phantoms prepared for each scattering coefficient with no Evans Blue added, and measurements of $\mathbf{R}$ were made on phantoms prepared with the same scattering coefficient plus the addition of Evans Blue. Changes between $\mathbf{R}$ and ${\mathbf{R}}_{\mathbf{o}}$ were assumed to be attributable to the difference in absorption coefficient between the two samples. Therefore, the difference between $\mathbf{R}$ and ${\mathbf{R}}_{\mathbf{o}}$ was measured at a wavelength where the optical properties were known (at $611\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ ). This was calculated by normalizing ${\mathbf{R}}_{\mathbf{o}}$ and $\mathbf{R}$ over the wavelength range $750\u2013800\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ (to account for small vertical shifts in the collected light intensities). 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 DPS path length was calculated at $611\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ as

## 3.3.

### Empirical Model of DPS Path Length

Observations of the effect of variations in ${\mu}_{\mathrm{s}}$ and ${\mu}_{\mathrm{a}}$ on the DPS path length led to the selection of the following model:

## Eq. 5

$$\frac{{\tau}_{\text{model}}}{{d}_{\text{fiber}}}=\frac{(1+{\left({\mu}_{\mathrm{s}}{d}_{\text{fiber}}\right)}^{-\mathrm{a}})}{(1+{\left({\mu}_{\mathrm{a}}{d}_{\text{fiber}}\right)}^{\mathrm{a}})}.$$^{11}Parameter estimation was achieved using a Levenberg–Marquardt algorithm

^{12}that was scripted into LabView code (vers. 7.1.1, National Instruments).

## 4.

## Results

## 4.1.

### DPS Reflectance Data

DPS reflectance spectra were measured at a resolution of
$2048\phantom{\rule{0.3em}{0ex}}\text{pixels}$
over the wavelength range
$340-1027\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
. The data were smoothed by averaging data into bins of
$10\phantom{\rule{0.3em}{0ex}}\text{pixels}$
, which allowed the calculation of a standard deviation that represents noise within the signal.^{5} Figure 2
shows the normalized DPS reflectance intensity data over the range
$400\u2013900\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
from measurements with
${d}_{\text{fiber}}=200\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$
on optical phantoms without Evans Blue,
${\mathbf{R}}_{\mathbf{o}}$
(
${\mu}_{\mathrm{s}}=42\phantom{\rule{0.3em}{0ex}}{\mathrm{mm}}^{-1}$
and
${\mu}_{\mathrm{a}}=0\phantom{\rule{0.3em}{0ex}}{\mathrm{mm}}^{-1}$
), and with Evans Blue,
$\mathbf{R}$
(
${\mu}_{\mathrm{s}}=42\phantom{\rule{0.3em}{0ex}}{\mathrm{mm}}^{-1}$
and
${\mu}_{\mathrm{a}}=1\phantom{\rule{0.3em}{0ex}}{\mathrm{mm}}^{-1}$
).

## 4.2.

### DPS Path Length Dependence on ${\mu}_{\mathrm{s}}$

Figures 3a and 3b
show the DPS path length, extracted from the measurement using Eq. 4, versus
${\mu}_{\mathrm{s}}$
for various selected values of
${\mu}_{\mathrm{a}}$
as measured by DPS probes with fiber diameters of 100 and
$200\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$
, respectively. Here, data points represent the mean from DPS measurements of three independent optical phantoms, and the error bars indicate one standard deviation about the mean. Inspection of the data show two regimes: (*i*)
$\tau $
is sensitive to scattering changes for relatively small
${\mu}_{\mathrm{s}}$
, and (*ii*)
$\tau $
is insensitive to changes over a range of larger
${\mu}_{\mathrm{s}}$
. In order to properly compare the effect of
${\mu}_{\mathrm{s}}$
on
$\tau $
over a range of fiber diameters, it is important to consider the relationship between dimensionless DPS path length
$(\tau \u2215{d}_{\text{fiber}})$
and dimensionless scattering
$\left({\mu}_{\mathrm{s}}{d}_{\text{fiber}}\right)$
. These relationships are shown on log–log scale for the 100 and
$200\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$
data in Figs. 3c and 3d, respectively. Additionally, Fig. 3e shows
$(\tau \u2215{d}_{\text{fiber}})$
versus
$\left({\mu}_{\mathrm{s}}{d}_{\text{fiber}}\right)$
for measurements of optical phantoms with constant
${\mu}_{\mathrm{a}}=0.4\phantom{\rule{0.3em}{0ex}}{\mathrm{mm}}^{\text{-}1}$
, as measured by DPS probes with
${d}_{\text{fiber}}$
over the range
$200\u20131000\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$
. The dimensionless data show that the transition of the DPS path length from sensitive to insensitive occurs at
$({\mu}_{\mathrm{s}}{d}_{\text{fiber}}<\approx 2)$
. This transition region is also observable for the
${d}_{\text{fiber}}$
of 100 and
$200\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$
probes on Figs. 3c and 3d, with the trend affected by the magnitude of
${\mu}_{\mathrm{a}}$
. It is worth noting that the effect of scattering on DPS path length was investigated by varying
${\mu}_{\mathrm{s}}$
while holding
${\mu}_{\mathrm{a}}$
constant, however, the path length is dependent on dimensionless absorption. Therefore, dimensionless path lengths for a constant
${\mu}_{\mathrm{a}}$
that are measured with different
${d}_{\text{fiber}}$
values do not uniformly collapse onto one another; instead, the larger fiber diameters will have a more pronounced effect from absorption. This phenomenon accounts for differences in the dimensionless path lengths as measured by 100 and
$200\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$
[visible in Figs. 3c and 3d]. This effect is also evident in Fig. 3e, where the dimensionless path for larger DPS fiber diameters of 800 and
$1000\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$
deviate from the rest of the group.

## 4.3.

### DPS Path Length Dependence on ${\mu}_{\mathrm{a}}$

Figures 4a and 4b
show the DPS path length versus
${\mu}_{\mathrm{a}}$
for various selected values of
${\mu}_{\mathrm{s}}$
, as measured by DPS probes with
${d}_{\text{fiber}}$
of 100 and
$200\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$
, respectively. Again, data points represent the mean from DPS measurements of three independent optical phantoms, and the error bars indicate one standard deviation about the mean. The data show an expected trend, with an increase in
${\mu}_{\mathrm{a}}$
causing a reduction in
$\tau $
. Here, comparison of data over multiple fiber diameters is aided by visualizing
$\tau \u2215{d}_{\text{fiber}}$
versus the dimensionless absorption
${\mu}_{\mathrm{a}}{d}_{\text{fiber}}$
. These data are shown on log–log scale for the 100 and
$200\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$
data in Figs. 4c and 4d, respectively. These data show a general reduction in dimensionless DPS path length as dimensionless absorption increases, with vertical stratification of the dimensionless path lengths at constant values of dimensionless scattering. Figure 4e shows
$(\tau \u2215{d}_{\text{fiber}})$
versus
$\left({\mu}_{\mathrm{a}}{d}_{\text{fiber}}\right)$
for measurements of optical phantoms with constant
${\mu}_{\mathrm{s}}=15\phantom{\rule{0.3em}{0ex}}{\mathrm{mm}}^{-1}$
, as measured by DPS probes with
${d}_{\text{fiber}}$
over the range
$200\u20131000\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$
. The data presented in plot Fig. 4e were reported in our previous study,^{5} and these data were used to model the previously published relationship between
$\tau $
and
${\mu}_{\mathrm{a}}$
. The data show a small effect of
${\mu}_{\mathrm{a}}$
variation on
$\tau $
for small dimensionless absorption coefficients, with more pronounced effects when for
${\mu}_{\mathrm{a}}{d}_{\text{fiber}}>0.6$
.

## 4.4.

### Empirical Model of DPS Photon Path Length as a Function of ${\mu}_{\mathrm{s}}$ and ${\mu}_{\mathrm{a}}$

Figure 5a shows ${\tau}_{\text{model}}$ , the DPS path length predicted by Eq. 5, versus $\tau $ , the DPS path length measured empirically. The estimated parameter value of $a=0.53\pm 0.09$ resulted in the smallest weighted residual error between data and model predictions. Model predictions were significantly correlated with measured values, as evidenced by a Pearson product correlation coefficient of $r=0.985$ ; this effect is observable in the plot as the data are scattered about the line of unity. Figure 5b plots the data on a log–log scale, which shows that the relationship holds for small path lengths. In order to observe the quality of model predictions over a wide range of DPS path lengths (range: $80\u2013940\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$ ), Fig. 5c shows the residual error as a fraction of the measured DPS path length value [residual $\text{percentage}=100\times (\tau -{\tau}_{\text{model}})\u2215\tau $ ], versus the measured $\tau $ values, with error bars calculated as the ratio of the standard deviation to the mean, representing the uncertainty in each measured data point. This plot shows the residual error scattered about zero across the range of DPS path lengths. The mean absolute residual error percentage is $8.5\pm 6.9\%$ . Here, 97% of the data points have an absolute residual error that is $\u2a7d20\%$ within the measurement error. The remaining 3% of the data points are characterized by extreme optical properties: large ${\mu}_{\mathrm{a}}$ , small ${\mu}_{\mathrm{s}}$ , small ${d}_{\text{fiber}}$ .

Additionally, the residuals displayed on Fig. 5c were not correlated with either ${\mu}_{\mathrm{s}}{d}_{\text{fiber}}$ or ${\mu}_{\mathrm{a}}{d}_{\text{fiber}}$ , with Pearson product correlation coefficients of $r=-0.170$ and $-0.028$ , respectively. This indicates that the model predictions are not biased as a function of either dimensionless scattering or dimensionless absorption, which confirms that the selected model structure is appropriate and that incorporation of additional fitted parameters would only describe noise within the data.

## 5.

## Discussion and Conclusions

The DPS device utilizes a specific fiber geometry to make the photon path length insensitive to changes in the tissue optical properties and instead dependent on the fiber diameter. Previously reported data support this claim for ranges of optical properties common within the UV/VIS wavelength region.^{4, 5} However, this assumption is not suitable for the large absorption and small scattering coefficients that are common within the NIR wavelength region.^{7, 8} This study presents a novel empirical formula that describes the dependence of the DPS photon path length on both total scattering and absorption coefficients over the range of values experienced within the NIR wavelength region. This relationship is valid for a wide range of
${\mu}_{\mathrm{s}}$
(range:
$1.5\u201342\phantom{\rule{0.3em}{0ex}}{\mathrm{mm}}^{-1}$
) and
${\mu}_{\mathrm{a}}$
(range:
$0.1\u201312\phantom{\rule{0.3em}{0ex}}{\mathrm{mm}}^{-1}$
) values and for a wide range of DPS fiber diameters
$(100\u20131000\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m})$
. This formula allows analysis of DPS spectra from tissue containing a wider range of biological measurements than previously possible, including those properties common to the NIR wavelength range.

The empirical model presented in Eq. 5 describes an intuitive relationship between the DPS path length and both ${\mu}_{\mathrm{a}}$ and ${\mu}_{\mathrm{s}}$ . As noted previously, the DPS path length is approximately equal to the fiber diameter for tissue measurements over a range of optical properties commonly experienced in the UV-VIS wavelength region. However, it has been shown that small ${\mu}_{\mathrm{s}}$ values result in a DPS path length that is greater than the fiber diameter. This phenomenon is captured by the numerator of Eq. 5, where for large ${\mu}_{\mathrm{s}}$ values the numerator $[1+{\left({\mu}_{\mathrm{s}}{d}_{\text{fiber}}\right)}^{-a}]$ approaches 1, resulting in a DPS path length that approximates the fiber diameter; while for small ${\mu}_{\mathrm{s}}$ values, the numerator becomes $>1$ , resulting in an increased DPS path length. Conversely, large ${\mu}_{\mathrm{a}}$ values result in a decreased DPS path length. This phenomenon is captured by the denominator of Eq. 5, where for small ${\mu}_{\mathrm{a}}$ values, the denominator $[1+{\left({\mu}_{\mathrm{a}}{d}_{\text{fiber}}\right)}^{a}]$ approaches 1, resulting in a DPS path length that approximates the fiber diameter, while for large ${\mu}_{\mathrm{a}}$ values, the numerator becomes $<1$ , resulting in a reduced DPS path length. The resulting model structure is capable of capturing the effects of both ${\mu}_{\mathrm{s}}$ and ${\mu}_{\mathrm{a}}$ with only one fitted parameter.

The results presented in this paper suggest that the empirical model presented in Eq. 5 is able to accurately describe the effect of variations in both ${\mu}_{\mathrm{s}}$ and ${\mu}_{\mathrm{a}}$ . This conclusion is based on the high correlation between measured and predicted DPS path lengths $(r=0.985)$ , the observation that the residuals are scattered about zero [with no observable trends, as shown in Fig. 5c], and the lack of correlation of residuals with either ${\mu}_{\mathrm{s}}$ or ${\mu}_{\mathrm{a}}$ . Moreover, the mean absolute residual error of 8.4% appears reasonable when considering that the model contains only one fitted parameter and that the model was fitted to data values that spanned very large ranges: a 120-fold change in ${\mu}_{\mathrm{a}}$ , a 28-fold change in ${\mu}_{\mathrm{s}}$ , and a 10-fold change in ${d}_{\text{fiber}}$ . Therefore, the authors expect that the error associated with the experimental construction and measurement of the optical phantoms may be equivalent to any inadequacy within the model.

The work presented here is a logical extension of our previous study, which quantified the effect that independent variation of either
${\mu}_{\mathrm{s}}$
or
${\mu}_{\mathrm{a}}$
had on the DPS path length.^{5} In that previous study^{5} it was reported that the DPS path length is insensitive to changes in scattering coefficient over a broad range, with the path length varying only 16% over the range
$5\phantom{\rule{0.3em}{0ex}}{\mathrm{mm}}^{-1}<{\mu}_{\mathrm{s}}<50\phantom{\rule{0.3em}{0ex}}{\mathrm{mm}}^{-1}$
for fiber diameters in the range
$400\u20131000\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$
. The change was more pronounced for the fiber diameter of
$200\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$
, which had 25% variation, and it was hypothesized that this effect would be magnified for a DPS fiber diameter of
$100\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$
. This study investigated and confirmed that hypothesis, finding 38% variation for the
$100\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$
over the same
${\mu}_{\mathrm{s}}$
range. Moreover, this study incorporated the effect that changes in
${\mu}_{\mathrm{s}}$
have on the DPS path length, which were shown to be appreciable for
${\mu}_{\mathrm{s}}{d}_{\text{fiber}}<2$
. This study develops the correlation between DPS path length and
${\mu}_{\mathrm{s}}$
for a constant anisotropy value of
$g\approx 0.8$
, and the effect of anisotropy variation over the range of values expected within tissue in both the UV-VIS and NIR wavelength regions (range:
$g\approx 0.7\u20130.95$
) on the DPS path length is not well-characterized. The authors expect anisotropy to have a more pronounced effect on the DPS path length for samples with small
${\mu}_{\mathrm{s}}$
and for measurements with small fiber diameters. In these situations, collected photons have undergone relatively few scattering events and therefore, the direction of individual scattering events is expected to have a more pronounced effect on the mean DPS photon path length.

Also in the previous study,^{5} changes in absorption coefficient over the range
$0\phantom{\rule{0.3em}{0ex}}{\mathrm{mm}}^{-1}<{\mu}_{\mathrm{a}}<1\phantom{\rule{0.3em}{0ex}}{\mathrm{mm}}^{-1}$
caused only a 15% variation in the DPS path length. This effect increased for increasing
${\mu}_{\mathrm{a}}$
, and while the DPS path length was insensitive to
${\mu}_{\mathrm{a}}$
for
${\mu}_{\mathrm{a}}{d}_{\text{fiber}}<0.6$
, the effect was appreciable for
${\mu}_{\mathrm{a}}{d}_{\text{fiber}}>0.6$
. An empirical formula was utilized to describe the relationship over a wide range of absorption values (range
${\mu}_{\mathrm{a}}=0.1\u20136.4\phantom{\rule{0.3em}{0ex}}{\mathrm{mm}}^{-1}$
), specifically, the model was utilized to describe the data presented in Fig. 4e. Comparing the predictions from that previous model with the model presented in Eq. 5, the predictions are nearly identical for estimation of the original data set (with a Pearson correlation coefficient of
$r=0.990$
). This indicates that the novel formula presented in this study is capable of capturing the previously modeled effect with the same accuracy, with the enhanced aspect of incorporating scattering effects on the path length. The tissue
${\mu}_{\mathrm{s}}$
in the UV-VIS region is typically in the range
$5\u201350\phantom{\rule{0.3em}{0ex}}{\mathrm{mm}}^{-1}$
, and recalculation of the data presented in Fig. 4e (originally calculated with
${\mu}_{\mathrm{s}}=15\phantom{\rule{0.3em}{0ex}}{\mathrm{mm}}^{-1}$
) for a
${\mu}_{\mathrm{s}}$
of either 5 or
$50\phantom{\rule{0.3em}{0ex}}{\mathrm{mm}}^{-1}$
results in a change to the model predicted path length of
$\u2a7d20\%$
, for either case. This source of error is similar to that introduced by the assumption of insensitivity of the DPS path length to physiologically relevant changes in scattering, as required by the model presented by Kaspers
^{5} These results indicate that the novel formula has potential application for analysis of DPS tissue spectra measured in both the UV/VIS and NIR wavelength regions.

It is important to consider the assumptions required to incorporate the novel pathlength formula into a spectral analysis algorithm. Previously, the algorithm used to analyze DPS spectra was able to capture the effects of scattering on the reflectance by specifying a background scattering model (such as Mie scattering,^{13, 14} or a combination of Mie and Rayleigh scattering.)^{11, 15} The analysis included the assumption that the DPS path length was independent of the total scattering coefficient. Therefore, no estimation of an absolute value for
${\mu}_{\mathrm{s}}$
was required (which was valid for measurements in the UV/VIS range). However, the formula for DPS path length presented in this study [in Eq. 5] is dependent on the absolute
${\mu}_{\mathrm{s}}$
value, necessitating the estimation (or calculation) of
${\mu}_{\mathrm{s}}$
, an assumption that may have a pronounced effect on DPS path length in the NIR wavelength region. This may be addressed by specifying the scattering model (as before) and simply assuming an absolute value of
${\mu}_{\mathrm{s}}$
at one wavelength, allowing the algorithm to describe wavelength-dependent changes within the spectra; however, this may be a source of error in the analysis, because the absolute value of
${\mu}_{\mathrm{s}}$
may not be known with high certainty. Hypothetically, this limitation could be addressed through a calibration procedure that would allow accurate estimation of the total scattering coefficient; this is an area of future work.

This novel description of DPS path length as a function of both absorption and total scattering coefficients may enhance analysis of clinical data. This relationship is required for analysis of DPS measurements in the NIR wavelength region, where the relative extreme optical properties (compared with the UV/VIS wavelength region) would affect the DPS path length and would otherwise introduce error into estimates of absorber concentrations within the tissue. This formula also allows estimation of a wavelength-specific DPS photon path length, which may provide information about the volume of tissue optically sampled during measurement, a factor that may vary between UV/VIS and NIR measurements on the same tissue. In a broader sense, the formula presented here may allow estimation of changes in the total scattering coefficient within tissue in response to a clinical procedure such as photodynamic therapy, or allow accurate estimation of differences in the total scattering coefficient between normal and diseased tissue, useful in tissue diagnostics.

## Acknowledgments

The authors thank B. Kruijt and F. van Zaane for assistance in constructing the DPS fiber probes, and 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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