2.1.

Point Radiance Spectroscopy Set-Up

The details of the experimental set-up for radiance measurements in Intralipid-1% have been extensively described elsewhere.^19,20 A schematic of the experimental setup is shown in Fig. 1. Briefly, the phantom Lucite box (with blackened 18 cm walls) was filled with Intralipid-1% solution and accommodated a fiber with a 800-micron spherical diffuser (connected to a white light source) and a 600-micron side firing fiber (the radiance detector). The side firing fiber had a well-defined angular acceptance window of $\sim 10\mathrm{deg}$ in water. Both fibers were threaded through $\sim 1.1\mathrm{mm}$ stainless steel tubes for mechanical stability. Source-detector separation was varied from 6.5 to 30.5 mm in various measurements. The radiance profiles were acquired by rotating the detector over a 360 deg range with a 2 deg step. The side firing fiber was connected to a USB 4000 spectrometer (Ocean Optics) that collected spectra at every angular step. Intralipid-1% solution was prepared by volume dilution of the stock solution of Intralipid-20% from Sigma Aldrich Canada.

Fig. 1

The schematic of the experimental setup. The inset shows a tip of the side firing fiber.

2.2.

Set-Ups For Basic Characterization of Intralipid-1%

Basic characterization included measurements of the absorption coefficient (${\mu }_a(\lambda )$), the reduced scattering coefficient (${\mu }_s^{\prime }(\lambda )$), and the scattering coefficient (${\mu }_s(\lambda )$) for the Intralipid sample using the collimated transmission and the spatially resolved reflectance set-ups. The experimental set-ups and the measurement procedure have been described previously.^22,23 Anisotropy factor, $g(\lambda )$, was calculated as $g(\lambda )=1-{\mu }_s^{\prime }(\lambda )/{\mu }_s(\lambda )$.

Figure 2 displays optical parameters of Intralipid-1% that were obtained from the basic characterization measurements: ${\mu }_s^{\prime }(\lambda )$ (a), ${\mu }_a(\lambda )$(b), ${\mu }_s(\lambda )$(c), and $g(\lambda )$(d). Error bars for the reduced scattering coefficient combine the standard deviation of three measurements and 5% systematic error expected by using the diffusion theory for retrieving the optical properties from the measurements. The absorption coefficient demonstrates a good agreement with the absorption of water for wavelengths $>700\mathrm{nm}$.²⁴ For wavelengths $<600\mathrm{nm}$, the absorption of soy bean oil is the main contribution to the measured absorption spectrum.²² Both the scattering and the reduced scattering coefficients are well described by the power law (fitted red solid lines). The error for the scattering coefficient (not shown) equals the standard deviation of five measurements.

Fig. 2

Optical parameters of Intralipid-1% from basic characterization measurements: (a) the reduced scattering coefficient,${\mu }_s^{\prime }(\lambda )$, (b) the absorption coefficient,${\mu }_a(\lambda )$, c) the scattering coefficient,${\mu }_s(\lambda )$, d) the anisotropy factor, $g(\lambda )$.

2.3.

Modeling Approach

A recently derived¹⁸ analytical solution of the RTE for the radiance caused by an isotropic light source placed inside an infinitely extended anisotropic scattering medium was used for comparison with experimental data. Because of the excellent agreement between the solution of the RTE and the Monte-Carlo method using the Henyey-Greenstein phase function, the proposed $P_N(N=\infty )$ method was considered as a reference for the present study. We chose a sufficiently high number of summation terms $N=17$ to closely match the $P_{\infty }$ approximation.

For the purpose of this article, the radiance ($I$) has been expressed as a function of optical properties of the Intralipid sample [${\mu }_a(\lambda )$, ${\mu }_s^{\prime }(\lambda )$ and $g(\lambda )$] that can be considered as fixed parameters. These parameters were obtained directly from the basic characterization measurements. The radiance is also a function of the cosine of the angle between the direction of propagation and the direction of scattering ($\theta$), the distance between the source and the detector ($r$), and the wavelength ($\lambda$) (implicitly) that we consider as variables. This distinction between the parameters and the variables is of artificial origin that merely facilitates a direct comparison with experimental data. For example, by fixing all parameters and variables except for the wavelength, we can obtain the spectral radiance as a function of wavelength. Similarly, by fixing everything but the angle we can obtain the angular radiance. By analogy, the distance dependent radiance can be obtained by fixing everything but the source-detector separation.

The radiance obtained from the diffusion equation ($P_1$ approximation) is known to be valid under certain limited conditions, and it has been discussed extensively in the literature.²⁵ The reduced scattering coefficient must be much larger than the absorption coefficient, and the product of the reduced scattering coefficient and the measurement distance should be large ($\gg 1$). Also, the discrepancies for the angular radiance $I(\theta )$ obtained from $P_1$, $P_3$, $P_{\infty }$, and Monte-Carlo simulations are well known and documented.¹⁸ Less studied is the distance dependent radiance, $I(r)$, its combined analysis in spectral and angular domains, and applicability of $P_1$ approximation for its description. For example, absolute distance dependent measurements of the fluence are known to provide the absorption and the reduced scattering coefficients thanks to the simplistic expression for the fluence.^6,8 An attractive feature of the $P_1$ approximation is that it offers a clear physical framework for data interpretation and straightforward parameter extraction during a fitting procedure. We tested the applicability of the diffusion approximation in the current study by fitting both simulated and experimental distance dependent data to the expressions derived from the diffusion approximation for radiance:

We assigned the shortest source-detector separation from our measurements to be $r_0$. Equation (2) was also used in the fitting procedure.

3.1.

Radiance at a Fixed (12 mm) Source-Detector Separation

As a starting point of most tissue or phantom characterizations, measurements are usually performed at a fixed source-detector separation. Detailed radiance measurements were done at a 12 mm source-detector separation. Filling the phantom box with the Intralipid medium changes the spectrum collected by the detector ($I_{\text{Intralipid}}$) as compared with the one measured in air ($I_{\text{air}}$). To exclude spectral features of the white light source and the detector, the ratio of two spectra, $I_{\text{Intralipid}}/I_{\text{air}}$ was constructed. It is plotted in Fig. 3(a) as an experimental spectral radiance. Two spectral profiles of the radiance are shown: one taken with the detector facing the light source (corresponding to 0 deg polar angle) and the other with the detector facing opposite the light source (corresponding to 178 deg polar angle). (Experimental data were corrected by a difference in light refraction on the curved fiber surface in air and water). The plot also shows two theoretical curves that were computed using the analytical radiance model ($P_{\infty }$) with input parameters from the basic characterization measurements as shown in Fig. 2. Once a single amplitude scaling factor is introduced, the agreement between the experimental and theoretical data is quite good. A scaling factor is required because modeling was done using a unit power, and no absolute calibration of radiance in experiment was performed. All spectral profiles show characteristic features of water absorption: a dip at 750 nm and a steep decrease toward 1000 nm. The peak around 550 nm is formed by overlapping of decreasing with wavelength absorption of Intralipid and increasing with wavelength water absorption (see Intralipid absorption spectrum in Fig. 2). Since this shape is a convolution of responses of two chromophores (soybean and water) with different absorption coefficients, the overall absorption spectrum of the turbid medium will be a summation of products of the corresponding absorption coefficient and the effective optical path taken by photons before reaching the detector for every chromophore. However, because of the multiple scattering the spectral shape will be a distance-variant feature.

Fig. 3

(a) Spectral radiance (experimental and theoretical) for Intralipid-1% at 0 deg and 178 deg. (b) Angular radiance (normalized, experimental and theoretical) for Intralipid-1% for 550 nm and 850 nm. Inset: light beams coming from the side firing fiber.

Radiance pattern as in Fig. 3(a) can be considered a function of three parameters: the absorption coefficient, the reduced scattering coefficient, and the anisotropy factor. The excellent agreement between the experiment and modeling indicates a possibility of parameter extraction from experiments by developing a suitable fitting algorithm. In order to recover three independent parameters, one would need technically only three relations. Because these parameters are distance- and angle-invariant, the robustness of the recovery will be greatly improved by fitting at the same time not three but all experimental angular-dependent radiance data sets (180 in this study). Multiple curve fitting can be a challenge, but a multi-spectral approach that has similar issues is commonly used in DOT.¹⁷ Hence, once achieved, it can bring radiance-based cw techniques closer to those of time- and frequency-domain in terms of being able to separate absorption from scattering in a single measurement for a single source-detector pair.

The angular radiance (measured and computed) for two selected wavelengths (550 nm and 850 nm) is shown in Fig. 3(b). In order to facilitate the comparison, the data were normalized to the maximum value at 0 deg. Experimental curves demonstrate slightly lower values than theoretical ones. The maximum difference occurs at 178 deg and is greater for shorter wavelengths ($\sim 8\%$ for 550 nm and $\sim 4\%$ for 850 nm). The difference is due to a number of factors. A finite width of a $\sim 1.1\mathrm{mm}$ diameter stainless steel tube that accommodates the side firing fiber obscures a path of photons in Intralipid. In particular, photons that would have reached the fiber after being backscattered undergo additional scattering and absorption when hitting the tube. Because of a three-dimensional nature of scattering, the entire part of the tube immersed into Intralipid acts as a perturbation. As a result, the experimental radiance detected at 178 deg is lower than the theoretical one. To confirm this effect we replaced the 600-μm side firing fiber in the 1.1 mm tube with a bare (no tube) 400 μm diameter side firing fiber and repeated the angular measurements. The resulting values for the radiance measured at 178 deg were then greater than the theoretical values: by $\sim 7\%$ and $\sim 15\%$ for 550 and 850 nm, respectively. The increase is accounted by a contribution of light that does not undergo a total internal reflectance in the fiber tip. The inset in Fig. 3(b) demonstrates the light emission from the side firing fiber when coupled to a white light source in a weakly scattering medium. Aside from the main beam coming at $\sim 90\text{\hspace{0.17em}deg}$ angle to the left, there are three more weak beams. The one that exits at $\sim 90\text{\hspace{0.17em}deg}$ angle to the right is most likely a Fresnel reflection of the main beam at the glass/air interface inside the cap. Two other beams are formed by light that does not undergo total internal reflection (as confirmed by a manufacturer). It means that light would be able to couple back to a fiber following these pathways. All three beams contribute to the measured radiance by providing an increased background and a weakened angular sensitivity. By placing a Si detector in air close to the fiber top and the fiber side, we estimated the amount of light leaking through the top to be within $\sim 12\%$ range of the total signal which agrees well with the measured differences. All the fibers tested have light patterns similar to the one shown on the inset of Fig. 3(b). It appears that the contribution from all secondary beams was almost overcompensated by the blocking effect of the tube showing radiance values less than expected for the 600 μm fiber in the tube. Once the tube was removed and the thinner 400 μm fiber was tested, the contribution from the secondary beams became more evident. To clarify the matter it is worth noting that the total amplitude of a signal collected by the 400 μm fiber is always less than that collected by the 600 μm fiber. All further experiments reported in the article were performed with the 600 μm side firing fiber in the tube, therefore, accepting up to $\sim 8\%$ difference with modeling results.

Since both experimental spectral and angular radiance data sets have been validated with modeling, they can be considered as elemental building blocks for more complex representations. For 360 deg spectral collection, Intralipid data were converted into an “Intralipid” matrix (angle versus wavelength) with individual spectra serving as columns. Spectral radiance was measured every 2 deg in (0 to 360) deg interval in the same spectral range for all angles. An “air” matrix was constructed by filling all (angular) columns with the spectrum measured in air at 0 deg. The result of the division of two matrices (Intralipid/air) is presented as a contour plot (normalized) in Fig. 4(a) and demonstrates light distribution inside Intralipid-1%. This type of referencing to air does not introduce any angular skew but consistently removes instrumental spectral features. Contour lines and same color represent equal intensity of light. The light wavelengths that undergo the least attenuation in the Intralipid correspond to red color. Contour lines clearly delineate the transmission peaks (i.e., around 550 nm) and the valleys (i.e., around 740 nm). Figure 4(a) not only shows all spectral features observed in Fig. 3(a) but presents the corresponding angular distribution. An angular spread due to multiple scattering of light is clearly observed. The effect of light spreading in Intralipid can be monitored by fixing the wavelength on the $Y$-axis and following the intensity (color) variation with angle. Such angular information is not achievable with fluence measurements.

Fig. 4

(a) Experimental contour plot (normalized) of the spectro-angular distribution of radiance in Intralipid-1%. (b) Experimental surface plot (normalized) of the spectro-angular distribution of radiance in Intralipid-1%. All for 12 mm source-detector separation (the same color coding for both plots).

We found that different Intralipid batches from the same manufacturer, Sigma-Aldrich Canada exhibit some variations in the spectral response in the 450 to 550 nm range. This range is very sensitive to the Intralipid composition which yields a different absorption. All wavelengths exhibit the maximum of light distribution along a 0 deg direction that is consistent with a preferential forward scattering in a homogenous medium. Cross-sections of the surface plot along $Y$-axis at constant $X$ (angle) produces spectral profiles similar to those shown in Fig. 3(a). Sectioning the plot at a constant $Y$-axis (wavelength) along $X$-axis generates angular radiance distributions similar to those shown in Fig. 3(a). A 2-D analysis of Fig. 4(a) provides for the identification of the wavelength and the direction of the least attenuation, i.e., $\sim 550\mathrm{nm}$ and 0 deg, correspondingly.

Details of the spectral variation with angle become more evident when data are presented in the form of a surface plot in Fig. 4(b) that allows seeing more clearly a spectral content. A symmetrical around 0 deg distribution of light in the Intralipid phantom is not surprising giving a homogenous composition of the solution. As one can see from Fig. 4(b), other than a reduction in intensity, the backscattered spectrum undergoes relatively minor changes in overall shape compared to the forward scattered spectrum. Therefore, the nature of the signals is the same and reflects a distribution of the photon density due to an increased optical pathlength of backscattered photons.

A comparison of Fig. 4(a) and 4(b) with a similar representation of radiance obtained from modeling under $P_{\infty }$ approximation [Fig. 5(a) and 5(b)] demonstrates a remarkable degree of correspondence between these complex plots. Since experimentally obtained data are not bound to any approximations (diffusion in particular), a good correspondence to modeling under $P_{\infty }$ approximation indicates that the spectro-angular approach is a valid way to present and analyze radiance data. It is important to note that simulated radiance does not involve any referencing and provides absolute values. As was described earlier, experimentally obtained radiance is a ratio of two spectra and therefore, has a relative nature.

Fig. 5

(a) Simulated contour plot (normalized) of the spectro-angular distribution of radiance in Intralipid-1%. (b) Simulated surface plot (normalized) of the spectro-angular distribution of radiance in Intralipid-1%. All for 12 mm source-detector separation (the same color coding for both plots).

While for homogenous phantoms and tissues quantitative information about optical properties can be extracted from spectro-angular snapshots via numerical analysis by solving the inverse problem (which is beyond the scope of the current article), some qualitative trends can be predicted when comparing several snapshots. Spectral variations due to changes in wavelength-dependent absorption coefficients are easily seen in the forward scattering direction where the signal is maximal. It can be applied to track changes in tissue properties as a result of coagulation or drug action (i.e., before and after PDT). On the other hand, an increase in the backscattered signal would be indicative of a decreased anisotropy factor, decreased absorption coefficient, or increased reduced scattering coefficient.¹

It is evident that a spectro-angular snapshot of a homogenous turbid medium has a symmetric pattern exhibiting a maximum for forward scattered photons (i.e., 0 deg). As was shown earlier,²¹ introducing a localized optical inhomogeneity via Au nanoparticles disrupts the symmetry and produces a new spectroscopic signature in the direction of the inhomogeneity. Therefore, in a single 360 deg scan a single radiance detector positioned at a fixed distance from a single light source can map a localized inhomogeneity in the spectro-angular domain. Such a target inhomogeneity cannot be localized or identified from a single fluence measurement. That is why various combinations of multiple sources and/or detectors are employed in DOT imaging to visualize the inhomogeneity in $X$,$Y$-domain using selected wavelengths of light. Hence, trade-offs between multiple point superficial measurements required by DOT to obtain an accurate image of the inclusion at a selected wavelength and a single point interstitial radiance scan to achieve angular localization of the target should be clearly understood. Moreover, spectro-angular mapping using radiance is not a typical imaging approach, but rather provides additional spatial information absent in other spectroscopic techniques.

Many in vivo PDT treatments in prostate, including treatment planning and dosimetry, utilize interstitial fluence measurements,^11,14,26 with source and probe fibers in various combinations inserted into multiple tissue segments, thus increasing invasiveness. Alternatively, radiance applications for prostate characterization hold the potential for requiring only a single interstitial placement of a rotating radiance probe in the urethra and an illumination source in the rectum, to acquire spatial information about spectral properties of the surrounding tissue. This illumination-detection geometry can be considered more favorable than the fluence-based approach because it offers a minimally invasive application for prostate diagnostics or PDT monitoring. Hence, radiance may be a viable alternative, minimally invasive method for tissue diagnostics especially when offered as an endoscopic technique.

It is known that prostate tissue heterogeneity represents an important problem for PDT dosimetry because it contributes to the effective attenuation coefficient averaged over a large portion of the prostate gland that can complicate the delivery of the desired PDT dose to tissues. Translating detecting fibers along the thickness of the prostate gland or using multiple sites for combined illumination-detection fluence measurements can provide indications of a presence of such heterogeneity rather than incorporating it in a tissue model. However, spectro-angular mapping with radiance may be better suited for characterization of localized heterogeneities or inclusions, providing the needed spatial information in a less invasive way. Radiance cannot replace fluence in PDT applications at the current stage, but with developments in radiance-based dosimetry and treatment planning algorithms it may serve as a new basis for PDT.

3.2.

Distance-Dependent Measurements in Intralipid-1%

Tissue diagnostics and imaging require a clear understanding of how measured optical radiance is affected by variations in the source-detector separation. Distance-dependent radiance measurements were completed by changing the source-detector separation from 6.5 to 30.5 mm with a 2 mm step and performing either a full 360 deg angular scan at every step or keeping the angle fixed in certain experiments. Initially, we considered the case when the detector is facing the light source (i.e., 0 deg angle, no rotation).

We started the study with simulating distance-dependent radiance for the same source-detector separation range as in the experiment. The spatial dependencies for selected wavelengths are shown in Fig. 6(a) with colored symbols. We tested the applicability of the expression for the distance dependent radiance under diffusion approximation and fitted simulated data with Eq. (1) using $D(\lambda )$ and ${\mu }_{\mathrm{eff}}(\lambda )$ as parameters. Solid lines in Fig. 6(a) represent results of fittings to Eq. (1) with the unit power ($P_0=1$). Since all the parameters of Intralipid-1% have been available from the basic characterization experiments, we also calculated $D(\lambda )$ and ${\mu }_{\mathrm{eff}}(\lambda )$ from the experimental values shown in Fig. 2. Fittings produced excellent agreement with experimental values. For example, for 450 nm wavelength the values were: reduced ${\chi }^2=3.1\times {10}^{-6}$, ${\mu }_{\text{eff model}}(450)=0.07502\pm 6.8\times {10}^{-5}{\mathrm{mm}}^{-1}$, $D_{\text{model}}(450)=0.17737\pm 2.5\times {10}^{-4}\mathrm{mm}$ and ${\mu }_{\mathrm{eff}\exp }(450)=0.07468{\mathrm{mm}}^{-1}$, and $D_{\exp }(450)=0.17926\mathrm{mm}$. A full comparison for ${\mu }_{\mathrm{eff}}(\lambda )$ is shown in Figure 6(b) where the solid line corresponds to values directly calculated from basic characterization measurements and red open circles—to values from fittings of the simulated radiance. Therefore, even though simulations were done under the $P_{\infty }$ approximation, distance dependent radiance can be described well by the radiance under diffusion approximation allowing for important parameter extraction. (It is important to emphasize that the diffusion approximation still fails in providing an adequate description of angular dependent radiance).

Fig. 6

(a) Simulated distance dependent radiance for selected wavelengths in Intralipid-1%. (b) The effective attenuation coefficient of Intralipid-1%.

In experiments, the input source power is not unity. Moreover, providing calibrated absolute radiance is a difficult task. To circumvent the problem, we explored the concept of relative measurements that would cancel the dependence on $P_0$ but preserve two optical parameters of interest. For this purpose we divided all simulated radiance values by the value at the shortest distance, i.e., 6.5 mm and fitted the normalized distance-dependent radiance to Eq. (2). In spite of an apparent simplicity of the approach and encouragingly small reduced ${\chi }^2$, the results of fitting were not as expected: ${\chi }^2=5.7\times {10}^{-8}$, ${\mu }_{\mathrm{eff}}(450)=0.07451\pm 1.9\times {10}^{-5}{\mathrm{mm}}^{-1}$, and $D(450)=0.227\pm 0.001\mathrm{mm}$. The value of the slope (i.e., which represents the effective attenuation coefficient) was successfully recovered, however; the diffusion coefficient error was $\sim 26\%$. The error increased up to $\sim 80\%$ for longer wavelengths. Apparently, normalization leads to an error accumulation at every fitting step which is manifested in the parameter responsible for the amplitude [i.e., $D(\lambda )$]. Finally, an attempt to introduce a third unconstrained parameter to the model (that would recover $P_0$) of non-normalized radiance data failed to recover $P_0$ and $D(\lambda )$ but again yielded accurate values for ${\mu }_{\mathrm{eff}}(\lambda )$. Extending the source-detector range up to 100 mm distance in simulations increased the dynamic range of radiance up to $\sim 10$ orders of magnitude but did not improve the recovery of $P_0$ and $D(\lambda )$. Hence, in the analysis of the distant-dependent experimental radiance, our focus was on the recovery of ${\mu }_{\mathrm{eff}}(\lambda )$ using the normalization approach and a two-parameter model.

Fitting experimental distance dependent normalized radiance produced results almost indistinguishable from modeling. Values of the effective attenuation coefficient extracted from the experimental radiance are presented in Fig. 6(b) as triangles. Again there is excellent agreement between the actual values (from basic characterization measurements) and those from the fits of both simulated and experimental radiance.

We applied the same analysis to the radiance obtained at 178 deg. In principle, due to the isotropy of the medium, the effective attenuation coefficient does not depend on the angle of detection. Indeed, the values of ${\mu }_{\mathrm{eff}}(\lambda )$ obtained from experimental backscattering data were similar to those shown in Fig. 6(b). Therefore, a rate of radiance change with distance is identical for all observation angles and equal to the effective attenuation coefficient for a homogenous medium. This approach offers the possibility of obtaining angular variations in the effective attenuation coefficient or its analogy for turbid media with a localized optical inhomogeneity.

The spectral shape of the effective attenuation coefficient, as observed in Fig. 6(b) carries important information about the wavelength corresponding to the least attenuation inside turbid media. While its shape resembles the absorption coefficient of Intralipid, it is distorted due to the contribution of the reduced scattering coefficient. It is the effective attenuation coefficient and its spectral shape that are ultimately responsible for the variation in radiance with distance. Such a variation is presented in Fig. 7. Continuing the analysis in multiple domains, we combined experimental distance-dependent data measured at 0 deg into a spectro-spatial contour plot as in Fig. 7 (note a log scale for radiance intensity variations). Modeling produces almost analogous plot (data not shown due to space limitations). In Fig. 7, the light wavelengths that are less attenuated are represented by a red color map. According to Fig. 7, the optimal transparency window for Intralipid-1% is in the 500 to 700 nm range. Working in the 700 to 900 nm spectral range should be avoided because of the detrimental impact of water absorption. In biological tissues, one of the important chromophores that defines the limits of the biomedical transparency window is hemoglobin, and it is absent in the present study. Clearly, the actual range of the biomedical transparency window depends on the nature of chromophores involved and should not be taken for granted as “one-fits-for-all” model. We’d like to point to a very important distinction between the information content of Fig. 6(b) and that of Fig. 7. The effective attenuation coefficient is a computed quantity that is valid for conditions satisfying the diffusion approximation. On the other hand, spectro-spatial mapping of the distance-dependent radiance is obtained directly from the experiment and is not bound by any approximations. Therefore, it will be able to provide an important insight into details of light extinction in different directions inside turbid media even when a concept of the effective attenuation coefficient is not applicable. The selected distance range (6.5 to 30.5 mm) targets prostate probing applications. For an average prostate diameter of 5 cm, illuminating through rectal wall and detecting with urethra-placed fiber will ensure probing prostate in a minimally invasive way. When a standard brachytherapy grid is used for interstitial placement of the source fiber in various locations of the prostate, spectro-spatial representation would allow determining the biomedical transparency window for the prostate, estimating the effective attenuation coefficient and relating spectral changes to distance. This information is likely useful for diagnostics and treatment planning. Fluence measurements done in the same geometry can also benefit from spectro-spatial analysis, especially when the diffusion approximation is not valid. It should be noted that both fluence and radiance measurements performed in this way share an increased degree of invasiveness because of a need of multiple distance measurements.

Fig. 7

Spectro-spatial contour plot of experimental radiance (log scale) for distance-dependent measurements at 0 deg.

When spectro-angular representation is applied to the distance-dependent radiance, it enables a unique ability to visualize angular transformations in the radiance distribution with depth inside turbid media. Presenting intensity variations of radiance simultaneously in spectral, spatial, and angular domains in a single quazi-3D plot is extremely difficult. To demonstrate dynamics of changes in light distribution, we combined a series of experimental spectro-angular snapshots [similar to one shown in Fig. 4(a)] measured every 2 mm in a 6.5 to 30.5 mm source-detector separation range in a short animation with 12 frames (Video 1). To facilitate the comparison and smooth transformations, every individual spectro-angular snapshot (i.e., a frame) was normalized to a maximum signal. Therefore, the identical color code covers the same 0 to 1 range for all distances. The short animation demonstrates spectral and angular changes in a photon distribution as the detector moves away from the source. In addition to expected angular broadening due to multiple scattering effects, spectral changes also accompany variations in distance. Similar to the analysis done for a 12 mm source-detector separation, we confirmed all experimental plots via simulation (data not shown due to space limitations). The origin of spectral changes is analyzed below.

Video 1

Transformation of spectro-angular light distribution with distance in Intralipid-1% (from 6.5 to 30.5 mm, 2 mm step) (MPG, 2.54 MB) [ http://dx.doi.org/10.1117/1.JBO.17.6.067007.1].

The first snapshot taken at shortest 6.5 mm source-detector separation somewhere resembles the one shown at Fig. 4(a). Since this approach removes spectral features inherent to the source and the detector, all signatures that are visible are due to Intralipid only. The spectrum measured at 6.5 mm shows a broadly defined maximum in the 480 to 550 nm range with a gradual slope towards the water absorption dip at 750 nm and a steeper slope starting around $\sim 900\mathrm{nm}$ At short distances from the source, the effects of the two chromophores (soybean oil and water) on the transmission spectrum are not very distinct because photons reaching the detector interact with a relatively small volume of Intralipid around the source. Even though concentrations of both chromophores do not change, a relatively short effective optical pathlength is responsible for spectral changes. Therefore, the interaction between the photons and the medium can be relatively insensitive to weak absorption by chromophores at short distances. In the angular domain, the transmission peak spans a 50 deg. range (according to the first contour line) and rests on $\sim 0.7$ in magnitude background provided by backscattered photons.

The distribution obtained at 30.5 mm away from the source (Fig. 8) undergoes substantial changes. As was mentioned earlier, spectral features due to various absorption components become more prominent with distance. Because of the increased effective optical pathlength, photons interact with a larger volume of Intralipid. The transmission peaks become sharper and more well-defined because they are now bounded by regions of stronger absorption. At the same time, the relative amount of light contained in the spectral range $>700\mathrm{nm}$ falls from $\sim 50\%$ to $\sim 10\%$. Distance-dependent plots indicate that the transmission spectrum of Intralipid changes with distance, although the rate of change is expected to be less at very large distances. It implies that when translated to biological tissues one should not expect to deal with a universal spectrum that is distance-invariant and can be thought of as a characteristic fingerprint of a tissue. Rather, contributions from present chromophores will continuously reshape the spectrum with distance making it recognizable but not identical.

Fig. 8

Experimental contour plot of the spectro-angular distribution of radiance (a) and experimental surface plot of the spectro-angular distribution of radiance (b) in Intralipid-1% for 30.5 mm source-detector separation (the same color coding for both plots).

This effect, however, can be turned into a benefit by linking the distinct spectral and spatial evolutions to the source-detector separation and extracting a polar distance coordinate in addition to the angular one. This is important for solving the inverse problem when the accuracy of distance knowledge impacts values of recovered optical parameters, for example. Also, linking characteristic spectro-spatial changes to distance may help in determining a distance between a localized inclusion and a detector that is usually unknown but of high interest. Information reach spectro-angular data manifest two clear trends: angular broadening and spectral narrowing with distance. Rather than analyzing the distance behavior at any particular wavelength, we attempted to look at the behavior of the entire spectro-angular shape which would demonstrate some degree of universality while losing any explicit wavelength dependence. We’ve selected two contour lines (corresponding to 0.95 and 0.9 levels) from experimental spectro-angular contour plots and followed a distance variation of spectral ($\mathrm{\Delta }\lambda$) and angular ($\mathrm{\Delta }\theta$) ranges bounded by the outermost values in the contour lines. The results are presented in Fig. 9(a) and 9(b). The values of the spectral range ($\mathrm{\Delta }\lambda$) versus distance can be independently predicted from a spectro-spatial plot of experimental transmission at 0 deg as in Fig. 7 after some modifications. A common feature of all spectro-angular snapshots is that the widest spectral range occurs at 0 deg for all contour lines. A modification of the plot from Fig. 7 included normalization to a maximum value at every distance, and the new contour plot is shown in Fig. 9(c). This plot also features the contour lines of interest (0.9 and 0.95, the latter in white). Intersection of the contour lines with the corresponding distance values on $Y$-axis not only produces the correct values of $\mathrm{\Delta }\lambda$ from $X$-axis but the outermost values of the wavelengths that were used to compute $\mathrm{\Delta }\lambda$ The same prediction can be made from the similar plot obtained from radiance modeling. Presenting a plot for analogous angular predictions is more challenging because it requires combining data from angular, wavelength, and distance domains. The curves like those in Fig. 9(a) and 9(b) can be used alone or in some combination to relate observed changes to distance and generate an empirical curve for distance calibration. For example, Fig. 9(d) demonstrates a constructed ratio of $\mathrm{\Delta }\lambda /\mathrm{\Delta }\theta$ that clearly shows $(\text{distance}{)}^{-1}$ behavior (note a double-log scale). It is understood that in order to implement distance extraction algorithms, a detailed study of the effect of optical properties of the medium on observed trends via either analytical solution of RTE or Monte-Carlo modeling is required. Nevertheless, we believe that the demonstrated experimental results indicate an interesting possibility of using the shape analysis as a tool for distance extraction.

Fig. 9

Distance dependence for the spectral, $\mathrm{\Delta }\lambda$ (a) and angular, $\mathrm{\Delta }\theta$ (b) ranges produced by contour lines at 0.9 and 0.95 levels of the experimental spectro-angular contour plots. Part (c) shows a spectro-spatial contour plot that correctly predicts $\mathrm{\Delta }\lambda$ (contour line for 0.95 level is in white) Part (d) shows a ratio of $\mathrm{\Delta }\lambda /\mathrm{\Delta }\theta .$