2.1.1.

Spectroscopy system

The DOS system consisted of: a xenon flash lamp (Model: E6611, Hamamatsu) as the light source; a spectrograph (Model: SP2150i, Princeton Instruments) and camera (CoolSNAP™, Photometrics) as the spectrometer; and a fiber optic probe ($\text{dia}\mathrm{.}=20\mu \mathrm{m}$, $\mathrm{NA}=0.22$; FiberTechOptica, Ontario, Canada) to illuminate and collect light. For the initial LUT optimization and validation studies, we used a six around one configuration, using the central fiber for illumination and the six peripheral fibers for collection. The S-D separation for this configuration was 0.25 mm (center to center distance), as shown in Fig. 1(a). For the studies involving large S-D separations we used a probe with fibers arranged in a linear array [Fig. 1(b)]. These fibers were spaced 0.37 mm apart, providing S-D separations in multiples of 0.37 mm, depending on the choice of the source and collection fiber.

Fig. 1

Schematic of probes: (a) Six around 1 configuration with a S-D separation of 0.25 mm. (b) Linear configuration with potential S-D separations ranging from 0.37 to 3.7 mm (only four of 11 fibers are shown to emphasize probe geometry).

2.1.2.

LUT construction

The basics of the LUT have been described in detail previously.¹³ Briefly, we generated the LUT by measuring the functional form of the reflectance using tissue phantoms with known optical properties (calibration set). The initial LUT was fabricated using phantoms containing polystyrene microspheres ($\text{diameter}=1\mu \mathrm{m}$; Polysciences, Warrington, PA) and India ink (Salis International) dissolved in water to simulate scattering and absorption, respectively. This LUT contained 24 phantoms, which was deemed to be sufficient for overlap in the wavelength-dependent scattering and absorption properties for different phantoms to cover the entire range of the LUT matrix. For example, for consecutive phantoms of increasing ${\mu }_s^{\prime }$, the lowest ${\mu }_s^{\prime }$ of each subsequent phantom overlapped with the highest ${\mu }_s^{\prime }$ of the previous phantom. To fabricate the tissue phantoms for the calibration set, we began with purely scattering phantoms that corresponded to different reduced scattering coefficients at 630 nm, [${\mu }_s^{\prime }({\lambda }_0=630\mathrm{nm})=0.5$, 1, 2, and $4{\mathrm{mm}}^{-1}$]. Subsequently, we added aliquots of absorbing dye (India ink) to each phantom to increase the absorption coefficient [${\mu }_a({\lambda }_{\max })=0$ to $5.33{\mathrm{mm}}^{-1}$]. ${\lambda }_{\max }$ refers to the wavelength corresponding to the highest absorption coefficient of India ink.

To determine the smallest number of calibration standards required to encompass the same range of optical properties as in the initial study, we used an absorber with absorbance values spanning the entire range of absorption coefficients with a single concentration. Using red dye as the absorber, we fabricated a new LUT that spanned the same range of optical properties as the previous one. As we did previously, we used Mie theory to calculate ${\mu }_s^{\prime }$ of the tissue phantoms. We measured the optical density of the red dye solution using a spectrophotometer (DU 720, Beckman Coulter) and calculated ${\mu }_a$ using Beer’s law. We initially fabricated three tissue phantoms with different scattering parameters [${\mu }_s^{\prime }(\lambda )=0.3–5{\mathrm{mm}}^{-1}$; $g({\lambda }_0=630\mathrm{nm})=0.85–0.96$]. After measuring the reflectance, we added a single aliquot of red dye to each phantom to obtain varying absorption values [${\mu }_a(\lambda )=0–5.3{\mathrm{mm}}^{-1}$], that were consistent with previously used values. The volume and concentration of red dye added was calculated based on the desired final ${\mu }_a$ of the solution ($5.3{\mathrm{mm}}^{-1}$). Because the addition of red dye dilutes the solution, a small change in ${\mu }_s^{\prime }$ was expected and accounted for when calculating the known ${\mu }_s^{\prime }$ of the tissue phantoms. The probe was placed in contact with the surface of the tissue phantoms, and white light spectra from the phantoms were recorded in the wavelength range of 400 to 700 nm. Reflectance was calculated by dividing white light intensity measured from the phantom by white light intensity from a reflectance standard (Spectralon, Labsphere). Both measurements were background corrected to account for CCD dark current and ambient light. For each phantom, we measured the background intensity with the light source shuttered and the probe in contact with the surface of the tissue phantom. The measurement-measurement variation in the diffuse reflectance spectra was less than 2 percent.

2.2.1.

Scattering and absorption

To determine the accuracy of the optimized LUT model in measuring scattering and absorption, we fabricated a set of tissue simulating phantoms with known optical properties. As described for the construction of the LUT, we used Mie theory to generate the theoretical values for scattering and optical density for absorption. Accuracy was computed as the root mean square (RMS) error of the measured values for scattering and absorption as compared to the expected values.

We used ferrous hemoglobin powder (H0267, Sigma, MO) as the absorber in our validation phantoms. We fabricated a matrix of $18(3\times 6)$ tissue phantoms with hemoglobin (Hb) as the sole absorber and polystyrene beads ($\text{diameter}=1\mu \mathrm{m}$) as the scatterer. We varied the reduced scattering coefficient and Hb concentration over a physiologically relevant range of values. Specifically, we used 6 different Hb concentrations (0, 0.5, 1, 1.5, 2, 2.5, and $3\mathrm{mg}/\mathrm{ml}$) and three different values of reduced scattering coefficient [${\mu }_s^{\prime }({\lambda }_0=630\mathrm{nm})=1$, 2, and $3{\mathrm{mm}}^{-1}$] to generate the validation matrix. We created a stock solution of $20\mathrm{mg}/\mathrm{ml}$ of Hb and added aliquots to purely scattering phantoms to obtain the different Hb concentrations. These values correspond to ${\mu }_s^{\prime }(\lambda )=0.72–4.91{\mathrm{mm}}^{-1}$ and ${\mu }_a(\lambda )=0–2.29{\mathrm{mm}}^{-1}$ over the entire wavelength range. As mentioned earlier, we accounted for the stepwise decrease in scattering levels due to multiple additions of Hb.

2.2.2.

Hemoglobin saturation

To determine the performance of the LUT model in measuring Hb saturation from a given spectrum, we measured reflectance spectra and partial pressure of oxygen at regular intervals of time as the partial pressure of oxygen decreased in a tissue phantom. Baker’s yeast (2.5 g) was added to water (10 mL, $T=80°\mathrm{C}$) to create an active yeast solution and a trace amount (5 μl) was added to the phantom. A dissolved oxygen sensor (inO2, Innovative Instruments) was positioned near the center of the phantom, adjacent to the fiber optic probe. The current across the ${\mathrm{pO}}_2$ sensor tip was recorded with an NI 6802 DAQ board (National Instruments). A spectrum and its corresponding current measurement were simultaneously recorded every 2 sec for a total of 25 min. After data acquisition, the current reading was converted to ${\mathrm{pO}}_2$ using a simple linear relationship. We used both oxy- and deoxy-hemoglobin as part of the absorption equation. The absorption coefficient was calculated using the absorption cross-sections (${\sigma }_{\mathrm{HbO}2}$ and ${\sigma }_{\mathrm{Hbde}}$) of these chromophores as ${\mu }_a(\lambda )=[\mathrm{Hb}][\alpha {\sigma }_{\mathrm{HbO}2}+(1-\alpha ){\sigma }_{\mathrm{Hbde}}]$, where $\alpha$ is the hemoglobin oxygen saturation and $[\mathrm{Hb}]$ is the total hemoglobin concentration.

To validate the measurement of Hb saturation, we followed the method described by Hull and Foster,¹⁹ wherein the measured saturation is fit to the Hill equation $[\alpha ={{\mathrm{pO}}_2}^n/({{\mathrm{pO}}_2}^n+p_{50})]$, where ${\mathrm{pO}}_2$ is the partial pressure due to oxygen, $p_{50}$ is the partial pressure of oxygen when $\alpha$ is half maximum, and $n$ is the Hill coefficient. We calculated the expected saturation curve as a function of oxygen partial pressure using the subroutine described by Kelman.²⁰ Based on this subroutine and the Hill equation, we calculated the expected values for the exponent ($n$) and the partial pressure of oxygen at which the Hb saturation is 50 percent ($p_{50}$). The pH of the tissue phantom was 6.9 and the ambient temperature was 24°C. The partial pressure of ${\mathrm{CO}}_2$ was determined based on the level of ${\mathrm{CO}}_2$ present in air.

2.2.3.

Presence of multiple absorbers

Most tissue sites consist of multiple absorbers such as melanin, beta-carotene and hemoglobin. Therefore, we determined the accuracy of the LUT to measure scattering and absorber concentrations in the presence of more than 1 absorber. We fabricated two validation sets containing two absorbers—hemoglobin and blue food coloring dye. The absorber kept constant in each set was maximally concentrated to determine the measurement accuracy of the target absorber in the presence of a strong, secondary absorber. In one set, we maintained a constant hemoglobin concentration of $3.5\mathrm{mg}/\mathrm{ml}$ and varied the concentration [0, 0.2, 0.4, 0.6, and $0.8\%(\mathrm{v}/\mathrm{v})$] of the coloring dye. For the second set, we maintained a constant dye concentration of 0.8 percent and varied the hemoglobin concentration (0, 0.5, 1.5, 2.5, and $3.5\mathrm{mg}/\mathrm{ml}$). For both sets, we used 1 μm (diameter) polystyrene spheres as the scatterer. The values for ${\mu }_s^{\prime }({\lambda }_0)$ and ${\mu }_a(\lambda )$ were $1{\mathrm{mm}}^{-1}$ and $0–3.4{\mathrm{mm}}^{-1}$, respectively.

2.2.4.

Effect of anisotropy

We used polystyrene microspheres of different sizes to simulate changes in anisotropy in our third validation experiment. We fabricated validation sets with the same optical properties as that described in Sec. 2.2. However, each validation set contained microspheres of a specific size ($\text{radius}=0.045$, 0.09, 0.175, 0.23, and 0.5 μm) as the scatterer, corresponding to a wide range of anisotropy values [$g({\lambda }_0=630\mathrm{nm})=0.2$, 0.4, 0.75, 0.8, 0.9, respectively]. Measurement error was again determined by calculating the RMS error of the deviation from the theoretical value.

2.2.5.

Depth sensitivity

The validation experiments described in the previous section involve the shortest S-D distance used in these studies (0.25 mm). The LUT is system specific: Altering the S-D separation or geometry warrants creating a new LUT. To determine the accuracy of the LUT at different S-D separations, we generated LUTs for four additional separation distances (0.37, 0.74, 1.11, and 1.48 mm). We did not investigate larger distances due to declining signal to noise of the diffusely reflected light. As in the previous cases, we tested these on the same set of validation phantoms.

In addition to testing the accuracy of these LUTs, we also determined the sampling depth for each set of S-D separations. These experiments were performed to present the advantage of using several S-D separations to sample multiple depths in tissue. To determine the sampling depth, we measured the depth-sensitive response of a fiber optic probe with detector fibers spaced at multiples of 0.37 mm from the source fiber ($\mathrm{S}\text{-}\mathrm{D}\text{separation}=0.37$, 0.74, 1.11, 1.48, 1.85 mm). To measure the depth response, we fabricated tissue-simulating phantoms using 10 percent intralipid (a fat emulsion liquid) to simulate scattering and hemoglobin to simulate absorption by blood. The scattering and absorption properties of the phantom were ${\mu }_s^{\prime }=1.87{\mathrm{mm}}^{-1}$ and ${\mu }_a=0.4{\mathrm{mm}}^{-1}$ ($\lambda =430\mathrm{nm}$). The bottom of the vessel containing the phantom was painted flat black to absorb any incident light on the bottom surface. The fiber probe was placed within the tissue phantom normal to and in contact with the bottom surface ($\text{elevation}=0$). The probe was then translated vertically with a motorized translational stage, and reflectance measurements were collected at intervals of 3.5 μm. By processing the measured reflectance as a function of the depth at which it was measured, we can determine the mean sampling depth of the probe.

3.2.1.

Scattering and absorption measurements

To compare the accuracy of the optimized LUT to the original, we measured the optical properties for a set of tissue phantoms with known optical properties, as described previously. The optimized LUT inverse model measured the reduced scattering and absorption coefficients over a wide range [${\mu }_s^{\prime }(\lambda )=0.72–4.91{\mathrm{mm}}^{-1}$ and ${\mu }_a(\lambda )=0–2.29{\mathrm{mm}}^{-1}$] with excellent mean RMS errors (calculated across all wavelengths and phantoms) of 2.5 and 4 percent, respectively. This was achieved using only six calibration standards; one-quarter of what was initially required. Figure 4 shows the measured ${\mu }_s^{\prime }(\lambda )$ and ${\mu }_a(\lambda )$ for the entire validation set.

Fig. 4

Scatter plots of the known versus the measured values of ${\mu }_s^{\prime }$ (a) and ${\mu }_a$ (b) for the entire set of validation phantoms. The diagonal line indicates perfect agreement.

3.2.2.

Hemoglobin saturation measurements

The measured hemoglobin oxygen saturation was in good agreement with that predicted by theory. Figure 5(a) presents reflectance spectra measured from the phantom in oxygenated and deoxygenated states. Two oxygenated spectra are presented to indicate the subtle decrease in reflectance that occurred immediately after the addition of yeast. The oxygen saturation ($\alpha$) was allowed to vary between $0.001$ and 1 in the inverse model and was accurately measured over the entire saturation range with errors less than 5 percent [Fig. 5(b)]. The expected values of $n$ and $p_{50}$ for the phantom were 2.61 and 26.11, respectively. Based on the measured values of Hb saturation, the measured values for $n$ and $p_{50}$ were within 5 percent of the expected value (2.57 and 25.76, respectively). Figure 5(c) demonstrates the changes in measured oxy and deoxy-hemoglobin concentrations as a function of phantom ${\mathrm{pO}}_2$.

Fig. 5

(a) Representative reflectance spectra in the oxygenated and de-oxygenated states. Notice two reflectance spectra for the oxygenated state, one before and one after the addition of yeast. (b) Comparison of expected and measured $\alpha$ as a function of phantom ${\mathrm{pO}}_2$. (c) Measured ${\mathrm{Hb}}_{\mathrm{de}}$ and ${\mathrm{HbO}}_2$ concentrations.

3.2.3.

Effect of multiple absorbers

The presence of multiple absorbers has negligible effect on accuracy of measuring specific absorber concentrations. Figure 6(a) shows reflectance spectra and corresponding fits from phantoms containing only one of each absorber. The reflectance spectrum shown in Fig. 6(b) illustrates the effect of two absorbers on the reflectance spectrum, with the $Q$-bands (542 and 577 nm) of oxygenated hemoglobin superimposed on the larger absorption trough of blue dye. The LUT fit shows excellent agreement with the data. Figure 5(c) and 5(d) presents the measured ${\mu }_a$ over all phantoms and all wavelengths in both sets of validation phantoms. The data indicate good agreement between the expected and measured values of the added absorber. The LUT model measured the concentration of both absorbers with a mean error of less than 5 percent. The error in scattering in both validation sets was less than 5 percent (data not shown).

Fig. 6

(a) Representative fits of LUT model to reflectance spectra from phantoms containing hemoglobin or blue dye exclusively and (b) a phantom containing both absorbers. (c) Expected versus measured ${\mu }_a$ for the blue dye while holding the ${\mathrm{HbO}}_2$ concentration constant. (d) Expected versus measured ${\mu }_a$ for hemoglobin while holding the blue dye concentration constant. The diagonal line in (c) and (d) indicates perfect agreement.

3.2.4.

Effect of anisotropy

We found the measured $B$ (scattering slope) to be in good agreement with theoretical values generated using Mie theory. Additionally, we determined that particle sizes corresponding to a physiologically relevant range of anisotropy values do not introduce error in measuring ${\mu }_s^{\prime }$ and ${\mu }_a$. Each reflectance spectrum shown in Fig. 7(a) has the same optical properties, with ${\mu }_s^{\prime }({\lambda }_0)=1.75{\mathrm{mm}}^{-1}$ and ${\mu }_a({\lambda }_0)=1{\mathrm{mm}}^{-1}$. As the sphere size decreases, the anisotropy decreases as well, indicating an increased contribution of backscattered light in the reflectance. This additional reflectance contribution manifests as an increase in the power law slope, $B$. However, we noticed that the smallest sphere radius of 0.045 μm did not conform to this intuitive trend. On the contrary, the magnitude of reflectance was far lower than that for other sphere sizes.

Fig. 7

(a) Example fits showing changes in reflectance with variations in bead size ($\text{radius}=0.045$, 0.09, 0.175, 0.23, and 0.5 μm). (b) Theoretical (solid line) and measured (shapes) dependence of ${\mu }_s^{\prime }$ on $\lambda$ as a function of bead size. (c) Error in recovering scattering and absorption properties as a function of anisotropy. Error bars represent one standard deviation.

Based on the theoretical values of ${\mu }_s^{\prime }(\lambda )$ calculated using Mie theory, we measured the scattering slope ($B$) for several sphere diameters by fitting ${\mu }_s^{\prime }(\lambda )$ generated from Mie theory to the equation ${\mu }_s^{\prime }(\lambda )={\mu }_s^{\prime }{(\lambda /{\lambda }_0)}^{-B}$ and generated a theoretical curve for $B$ as a function of sphere radius [Fig. 7(b)]. The measured values of $B$ from the different phantom sets are also plotted and indicate excellent agreement with the expected value of $B$, with the exception of the smallest sphere size. We suspect this anomaly is inherent to the short source-detector separation used; the small sample volume will be considerably sensitive to changes in anisotropy. Beginning with the next set of validation phantoms ($r=0.175\mu \mathrm{m}$), the LUT provided accurate measures of the optical properties. For $r=0.09\mu \mathrm{m}$, the LUT measured ${\mu }_a$ and ${\mu }_s^{\prime }$ with errors of 15 and 4%, respectively. For $r=0.175$, 0.23 and 0.5 μm, all of which correspond to $g>0.7$, the LUT measured both ${\mu }_a$ and ${\mu }_s^{\prime }$ with errors less than 5 percent.

3.2.5.

Effect of source-detector separation distance

The measured mean sampling depth was linearly related to the source-detector separation. Figure 8(a) shows the normalized reflectance at one wavelength (430 nm) for each S-D separation, corresponding to the cumulative distribution function (CDF) at that wavelength. The curve is constructed from reflectance values measured as the probe is drawn up from the bottom of the container. As the probe is translated vertically away from the bottom, the probe begins to sample light scattered in the limited volume beneath the probe, and the reflectance increases until it asymptotically approaches its maximum value, at which point the probe is no longer sampling the bottom surface of the phantom. The points at which the probe first registers reflected light in the collection fiber and when it no longer detects any change in reflectance with axial translation represent the approximate lower and upper limits of light travel (‘banana curve’), respectively. To determine the mean sampling depth, we take the derivative of the CDF with respect to elevation to obtain the probability distribution function [PDF; Fig. 8(b)]. The maximum of the PDF represents the elevation where the majority of the photons are located. The squares in Fig. 8(c) represent the elevation for which the PDF was at its maximum, and thus indicate the mean sampling depth; the upper and lower bars represent the sampling depths at which 90 and 10 percent of the reflected light was detected, respectively. In this way, we can measure the mean sampling depth and the upper and lower bounds for each S-D separation. The results clearly indicate that the sampling depth increases with increasing S-D separation, as is expected. The shortest S-D separation is sensitive to changes at the tissue surface with a mean sampling depth of approximately 0.40 mm with sensitivity up to approximately 0.70 mm. On the other hand, the longest S-D separation had a mean sampling depth of 1 mm with sensitivity up to 1.5 mm.

Fig. 8

Measures of the sampling depth for the probe design shown in Fig. 1 with S-D separations of 0.37, 0.74, 1.11, 1.48, and 1.85 mm. (a) Plot of the normalized reflectance at one wavelength (${\lambda }_0=430\mathrm{nm}$) as the probe is translated vertically starting at the bottom of the tissue phantom. (b) Derivative of (a) with respect to depth indicating sampling volume for each S-D separation measured. (c) Mean sampling depth versus S-D separation (error bars represent elevations corresponding to 10 and 90 percent of the maximum reflectance detected). ${\mu }_a$ and ${\mu }_s^{\prime }$ for the tissue phantom at 430 nm were 0.4 and $1.5{\mathrm{mm}}^{-1}$, respectively.

As the S-D separation is varied, the overall shape of the LUT is changed. Figure 9 presents LUTs developed for different S-D separations. The LUT is no longer monotonically decreasing for the same range of optical properties at source detector separations beyond 0.40 mm. As the S-D separation increases, the reflectance increases with scattering but appears to plateau at much lower values of scattering. For example, at 0.37 mm [Fig. 9(a)], the reflectance is still monotonically increasing as a function of scattering at the highest value of scattering reported; however, larger S-D separation distances [Fig. 9(b) and 9(c)] result in reflectance that is no longer monotonic as a function of scattering.

Fig. 9

Representative look-up tables for multiple source detector separations: (a) 0.37 mm, (b) 0.74 mm, and (c) 1.11 mm. (d) Average error in measuring scattering and absorption properties of tissue phantoms as a function of S-D separation. Errors bars represent one standard deviation. The validation phantoms were constructed over the same range of ${\mu }_a$ and ${\mu }_s^{\prime }$ as described in Sec. 2.1.

We fit the reflectance spectra measured from the validation phantoms to the respective LUTs and found that the errors over all S-D separations were less than 15 percent [Fig. 9(d)]. Although the average scattering error increases as the S-D separation increases, the upper limit of the average error across the entire data set is less than 10 percent. The error in estimating absorption properties remained nearly constant with change in S-D separation.