3.1.

Contact Spatially Resolved Diffuse Reflectance Spectroscopy

Figure 2 schematically illustrates the well-established contact DRSsr setup and the distal end of the probe used.¹¹ A tungsten halogen (T–H) Lamp (HL2000 Ocean Optics) is used as the source, and a QE65000 (Ocean Optics) is used as the spectrometer detector (cooled down to $-15°\mathrm{C}$ to reduce dark noise). The spectral working range of the instrument lies between 470 and 880 nm which is appropriate to address superficial signals originating at less than a few millimeters deep. The probe features a central illumination fiber and concentric detection fibers at six different distances $D6-D1$ ranging from 300 to $2488\mu \mathrm{m}$ (center-to-center). These distances are adapted to separate and thereby quantify absorption and scattering effects in the visible spectrum. Indeed, it is known that diffuse reflectance signals close and far from the light source have different sensitivities to the medium’s optical properties. For instance, at distances smaller than one transport length, $l_{\mathrm{tr}}$, reflectance strongly depends on the scattering properties, phase function, and anisotropy factor and is less affected by absorption. On the contrary, larger distances ($>5l_{\mathrm{tr}}$) measure a reflectance that has a high dependence on absorption and minimal dependence on scattering effects.^16–18 The detection fiber rings are referred to as $F6-F1$, with $F6$ and $F1$ being the closest ($D6$) and furthest ($D1$) ring to the source, respectively (see Fig. 2). Correspondingly, the diffused reflectance signals $S$ measured by the fibers are referred to as $S6-S1$. The core diameter of the central excitation fiber ($E$) and that of each detection fiber is 500 and $100\mu \mathrm{m}$, respectively.

Fig. 2

Contact probe DRSsr setup with detailed illustration of the distal end of the measuring probe.

Measurements are taken by placing the probe in contact with the sample under constant ambient darkness conditions. The setup allows the measurement of the source signal $S_{\text{source}}$ just after $S6-S1$ measurements to control the slight signal intensity and spectral variability. Integration time $t$ of each signal ($S_{\text{source}}$ and $S6-S1$) is adapted to the dynamic range of the spectrometer.

3.2.

Parallel Noncontact Probe-Based Spatially Resolved Diffuse Reflectance Spectroscopy

Compared to the contact DRSsr modality, the noncontact DRSsr modality considers different clinical interests. For instance, noncontact measurements of sensitive samples such as injured or infected tissues are completely noninvasive and, therefore, sterile. Moreover, the noncontact measurement modality is not affected by the coupling variations which arise from the fluctuation of pressure exerted on the sample by the contact probe and directly influence the estimation of optical properties.¹⁹ In addition, the use of optics in a noncontact setup provides higher flexibility in the choice of geometrical dimensions of the projected illumination beam and detection areas of the resulting reflectance.

We built a noncontact DRSsr setup by placing an achromatic doublet pair between the measuring probe (see Fig. 2) and the sample as shown in Fig. 3.

Fig. 3

Parallel noncontact probe DRSsr setup.

The focal distance of the doublets is chosen to be the same to guarantee magnification equal to 1 at a convenient distance of 100 mm. The doublet pair has an antireflection coating that ensures optimal transmission between 400 and 700 nm. Parasite reflections consequently arise after 700 nm and should be measured as explained in Sec. 4.2.1.

Focusing on the phantom is achieved in two steps. In the first step, the height of the adjustable support is accommodated to a visually acceptable first focusing range. In the second step, $S6$ is used since it is a direct measurement of the signal close to the borders of the projected illumination source point that decrease at the focal plane. Consequently, the height obtaining a minimal $S6$ within the first focusing range is determined to be the methodical focal plane. All phantoms are placed at this focal plane using the same container and keeping the phantom’s volume constant.

Diffused reflectance, parasite reflections, and offset are measured under the same ambient darkness conditions.

3.3.

Folded Noncontact CCD-Based Spatially Resolved Diffuse Reflectance Spectroscopy

To further test the calibration algorithm, we built a CCD-based DRSsr setup. The use of a CCD as detector (replacing the probe and spectrometer) requires the separation of the illumination and detection paths. To do so, we use a $50:50$ beamsplitter that ensures an orthogonal projection of the illumination orthogonal on the sample’s surface, as shown in Fig. 4. Two achromatic doublets establish a $500\text{-}\mu \mathrm{m}$ diameter of the projected illumination point.

Fig. 4

Folded noncontact CCD-based DRSsr setup with separated detection and illumination paths with an additional filter support for spectral scanning.

Once this architecture was validated with a probe detector, we replaced it with a 12-bit monochrome PixelFly VGA CCD and fixed a filter between a collimator just after the source, as shown in Fig. 4. Since this work aims for a proof of concept, we consider only four different Thorlabs band-pass filters ($\mathrm{FWHM}=10\mathrm{nm}$): 450, 550, 650, and 750 nm.

Focusing is achieved through video-mode for each filtered light. To allow measurement of the diffused reflectance decay up to 2.49 mm ($D1$) with the constrained dynamic range of the CCD, several acquisition times, $t$, adapted to a specific signal intensity, are used (see Fig. 5). We consider the effect of pixel blooming around the saturated zones of the image to be negligible after having checked signal linearity at all detectable distances. Each integration time determines a detectable area in the image with a nonsaturated gray level value higher than the limit of detection, being three times the offset’s standard deviation.

Fig. 5

Adaptive integration time needed to recover the complete reflectance decay with the limited CCD dynamic range.

All measurements are taken under constant ambient darkness. Background images are acquired with no light source for some acquisition times, $t$. Average values of each background image are used to obtain the linear offset-$t$ curve which is interpolated for all other $t$s. Detectable areas of all phantom images are corrected from the background at their corresponding acquisition time $t$ and then combined to construct the reflectance decay (see Fig. 5).

4.1.

Monte Carlo Simulation

A home-made Monte Carlo simulation in MATLAB^® has been used to precompute an LUT, used as a numerical solution of the RTE because, contrary to the diffusion approximation, it is valid for the highly absorbing phantoms we use and at the close detection distance $D6$ from the source. The simulation describes photons reaching the medium’s surface through the excitation fiber $E$ (with a numerical aperture of 0.22), traveling through the semi-infinite medium, and measured back at the surface at different SD distances from the source (F6–F1), under contact conditions (see Fig. 6). Therefore, the instrumental effect of the achromatic doublet (present in the noncontact modality) is not considered. Photons start with an initial weight and are injected perpendicularly to the sample’s surface. Absorption events of photons are considered through a cumulative function that decreases the photon’s weight. Scattering events deviate photons by an elementary step $1/{\mu }_{\mathrm{s}}^{\prime }$ and by an angle calculated with the Henyey–Greenstein function.

Fig. 6

Illustration of the illumination profile and reflectance detection according to the Monte Carlo simulation conditions.

The geometrical features of the illumination and detection correspond to the DRSsr probe shown in Fig. 2. The range of optical properties considered in the simulated medium is ${\mu }_{\mathrm{a}}\epsilon [0.05,25]{\mathrm{cm}}^{-1}$ and ${\mu }_{\mathrm{s}}\epsilon [\mathrm{10,400}]{\mathrm{cm}}^{-1}$ covering that of all evaluated phantoms. A constant anisotropy factor $g=0.8$ throughout the spectrum is used and its wavelength dependence is considered in a following scaling procedure. The refraction index of the refractive medium $n_{\mathrm{r}}$ is set to 1.37, which is characteristic of skin.²⁰ The refraction index of the incident medium $n_{\mathrm{i}}$ is set to that of silica (main material of the fibers’ core) and equal to 1.45.

The resulting simulated reflectance $R^{\mathrm{LUT}}$ is saved under an LUT which is used for comparison and data fitting with the measured and normalized reflectance $S_{\mathrm{N}}$ [see Eq. (2)].

4.2.

Signal Treatment

The existing method that treats $S_{\mathrm{N}}$ to derive the estimation of optical properties is based on the instrumental calibration with a normalized reflectance $S_{\mathrm{N},\mathrm{calib}}$ of a single reference phantom for which the optical properties are known. $S_{\mathrm{N}}$ and $S_{\mathrm{N},\mathrm{calib}}$ are obtained through Eq. 2 by scaling reflectance measurements $S$ and $S_{\mathrm{calib}}$ of unknown and reference phantoms, respectively, from the source signal $S_{\mathrm{N},\text{source}}$ [calculated with Eq. (1)] and acquisition time $t$ (proper to each signal) and by subtracting the additive signal $S_{\text{additive}}$ (measured at the same $t$) which includes instrumental offset and parasite reflections (see Sec. 4.2.1).

Thus, a calibration factor (CF) is calculated as the ratio of the simulated reflectance of the reference phantom $R_{\mathrm{calib}}^{\mathrm{LUT}}$ and $S_{\mathrm{N},\mathrm{calib}}$ [see Eq. (3)], for each wavelength and distance.

Normalized reflectance (at each distance) of unknown phantoms $S_{\mathrm{N}}$ is corrected with CF according to

The resulting corrected reflectance $R$ is then fitted to $R^{\mathrm{LUT}}$ with a least-squared minimization that estimates the unknown optical properties. This method is valid only if the measured reflectance signals $S$ and $S_{\mathrm{calib}}$ are taken under the same conditions of ambient light and instrumental configuration. Only then it is possible to compensate for additive effects through Eq. (2) (Sec. 4.2.1) and constant multiplicative effects through a single CF calculated through Eq. (3) (Sec. 4.2.2). However, in a noncontact setup, the method is unable to make up for the modification of the illumination beam profile (Sec. 4.2.3) which depends on the noncontact architecture and influences the measured reflectance depending on the optical properties of the sample.

Since further distances from the source are most sensitive to the absorption of photons in the medium, ${\mu }_{\mathrm{a}}$ is estimated with further fibers $F5-F1$. On the contrary, scattering sensitivity is highest at short distances from the source justifying the selection of $F5-F3$ for ${\mu }_{\mathrm{s}}^{\prime }$ estimation. In the noncontact DRSsr setups, $F6$ is strongly illuminated by the profile extension and cannot be used for optical properties estimation (see Sec. 4.2.3).

4.2.3.

Effect of the projected illumination beam profile

The plate doublets and beamsplitter (depending on the noncontact DRSsr setup) induce optical aberrations that degrade the projection of the illumination point on the sample, shown in Fig. 7. Indeed, it is seen that $F6$ is greatly affected by the intense specular reflection of the inaccurate point borders that are not considered in the contact Monte Carlo simulation (Sec. 4.1). After having confirmed the capacity of the contact probe DRSsr system to resolve optical properties without the $S6$ signal (measured by $F6$), it was decided not to use it for the noncontact DRSsr systems. The rest of the fibers measure a combined diffused reflectance signal resulting from the modeled central illumination and the nonmodeled optical aberrations of the projected point which include the extended point borders and, in the case of the folded illumination setup (Fig. 4), the parasite reflections provoked by the beamsplitter.

Fig. 7

Illumination profiles of all DRSsr setups: contact DRSsr (Fig. 2), parallel noncontact (NC) DRSsr (Fig. 3), folded noncontact (NC) DRSsr without and with a 650-nm filter (Fig. 4).

It is seen that the influence of the nonmodeled signal is strongest for the closest distances and negligible for further ones. Considering that the closest fibers are the most sensitive to ${\mu }_{\mathrm{s}}^{\prime }$ and the furthest to ${\mu }_{\mathrm{a}}$,^16–18 it is possible to differentiate ${\mu }_{\mathrm{a}}$ with no illumination profile correction only if both the reference and unknown phantoms have the same ${\mu }_{\mathrm{s}}^{\prime }$, meaning that the overall instrumental effect can be calibrated through the multiplicative correction of Eq. (3). Adversely, when the phantoms have different ${\mu }_{\mathrm{s}}^{\prime }$, the nonmodeled signal on the closest fibers will vary and with it the required multiplicative correction. In this case, the correction of the illumination profile is crucial for the appropriate optical properties estimation. In other words, for the noncontact DRSsr setups, correction of the nonmodeled signal is mainly related to ${\mu }_{\mathrm{s}}^{\prime }$.

A possible correction strategy consists of considering the noncontact projected illumination beam profile in the Monte Carlo simulations as it is done in Ref. 21. An alternative solution considers a calibration approach. We opted for a dichotomic calibration algorithm described in Sec. 5.

5.1.

Interpolation

If $\widetilde{\mathrm{CF}}$ estimates $\widetilde{{\mu }_{\mathrm{s}}^{\prime }}*$ which lies between two neighboring ${\mu }_{\mathrm{s},\mathrm{ref}}^{\prime }$, an interpolation step follows. Neighboring ${\mu }_{\mathrm{s},\mathrm{ref}}^{\prime }$ are used to define an interpolated correction factor $\widehat{\mathrm{CF}}$ for each $S$ and $\lambda$. This final correction factor $\widehat{\mathrm{CF}}$ obtains the optimal estimation of both optical properties $\widehat{{\mu }_{\mathrm{s}}^{\prime }}$ and $\widehat{{\mu }_{\mathrm{a}}}$.

5.2.

Correction of Instrumental Variations

For the optimal performance of ACA-Pro the measurement of a CF reference base set should be taken under the same conditions of the unknown phantom measurements. Thus, a CF reference base appropriate to each experiment is needed. Since the optical properties of intralipid liquid phantoms depend on the temperature,²² the water quality, and the Intralipid emulsifier, and further mutate after some time²³ through processes such as cream formation,²⁴ experiments outside this time period ($t_0$) require the manufacturing of new fresh phantoms.

To allow a faster and less burdensome instrumental calibration between different experiments, a new calibration step has been determined. The latter makes use of a single measurement of a common material $S_k$ that is taken for each experiment and characterizes individual experiment conditions. For this, it is important to guarantee spatially homogeneous and constant optical properties of the material over indefinite time periods separating experiments. Homogeneous liquids should, therefore, be fresh and manufactured in the same way for each experiment. An attractive alternative is the use of homogeneous solid materials with negligible surface variations and temporal optical stability. The advantages of solid materials is that they do not pose the problems related to liquid phantoms including tedious and constant fabrication, short-term conservation at low temperatures, and demanding manipulation. However, because the optical properties of solid materials are not easily characterized, they cannot be used to directly build the CF reference base. Therefore, liquid phantoms are manufactured and measured only once to build a single CF reference base and a measurement of a solid material taken at each experiment is used to compensate for experimental variations.

The calibration is performed by calculating the ratio $K_{t0-t1}$ between the normalized signal of the solid material $S_{Nk}$ [calculated with Eq. (5)] obtained at a specific experiment or moment $t_0$, and that obtained for a second experiment or moment $t_1$ [see Eq. (6)].

Hence, the unknown phantom reflectance $S_N$ at $t_1$ ($S_{N,t_1}$) is adapted to the original instrumental conditions of the single CF reference base built with measurements at $t_0$. The corresponding signal at $t_0$ ($S_{N,t_0}$) is obtained through the correction with the ratio $K_{t_0-t_1}$ [see Eq. (7)].

The flowchart of Fig. 8 and the results shown in Sec. 6 facilitate the understanding of ACA-Pro’s performance in the method for optical properties estimation.

Fig. 8

Summary of ACA-Pro execution with the main and interpolation approaches in purple and the correction of instrumental variations with the use of a solid material (mat.) in green.

6.1.

Calibration Factor Reference Base

We built a comprehensive CF six-reference base with six-reference phantoms having common ${\mu }_{\mathrm{a},\mathrm{ref}}=0.4{\mathrm{cm}}^{-1}$ at 600 nm and different ${\mu }_{\mathrm{s},\mathrm{ref}}^{\prime }$ corresponding to ${\mathrm{IL}}_{\mathrm{ref}}=0.5\%$, 1%, 1.5%, 2%, 2.5%, and 3%. The CF six-reference base shows the variability of the relationship between Monte Carlo simulated and measured reflectance of phantoms having different scattering properties. Figure 9 shows examples of the CF six-reference base obtained with the contact (Fig. 2) and parallel noncontact (Fig. 3) probe DRSs, respectively, for $F5-F1$, some wavelengths $\lambda$, and with two different measurement sets $M1$ and $M2$. All CFs are normalized with respect to a common CF at ${\mathrm{IL}}_{\mathrm{ref}}=1\%$. Ideally, if no calibration were necessary, all normalized CF should be equal to 1. It is, therefore, not surprising that because of the smaller difference between measured and simulated reflectance, normalized contact CFs are closer to 1 than noncontact CFs. The high difference of contact M1 and M2 at ${\mathrm{IL}}_{\mathrm{ref}}=0.5\%$ (mostly seen for $F3-F4$) is probably related to the limited accuracy of the Monte Carlo simulation at such a low scattering value. This will be analyzed in future work.

Fig. 9

CF six-reference base for contact and parallel noncontact probe-based DRSsr setups measured twice (M1 and M2).

Moreover, notice that noncontact CFs tend to be slightly more repeatable than those measured with contact DRSsr. This can be explained by the higher difference between simulated $R_{\mathrm{calib}}^{\mathrm{LUT}}$ and the noncontact measurements that render experimental noise negligible. Contrastingly, the smaller difference between contact reflectance measurements and $R_{\mathrm{calib}}^{\mathrm{LUT}}$ accords a more significant role to the experimental noise. Moreover, contact measurements are affected by the higher dynamic changes of the intralipid phantom, induced by the positioning of the probe. This is not the case for the noncontact measurement modality.

6.2.

Estimation of Optical Properties with the Calibration Factor Reference Base

The interest of ACA-Pro with the CF six-reference base set is shown for an unknown phantom having theoretical properties ${\mathrm{IL}}_{\mathrm{theo}}=1\%$ and ${\mu }_{\mathrm{a},\mathrm{theo}}=1{\mathrm{cm}}^{-1}$ at 600 nm. Optical properties $\widetilde{{\mu }_{\mathrm{s}}^{\prime }}$ and $\widetilde{{\mu }_{\mathrm{a}}}$ are estimated with all CF from the different ${\mathrm{IL}}_{\mathrm{ref}}=X\%$ ($\mathrm{CF}X\%$) of the CF six-reference base.

The estimated scattering and absorption properties obtained with the contact and the parallel noncontact probe-based DRSsr setups are shown in Fig. 10.

Fig. 10

Estimation of optical properties of an unknown phantom (${\mathrm{IL}}_{\mathrm{theo}}=1\%$ and ${\mu }_{\mathrm{a},\mathrm{theo}}=1{\mathrm{cm}}^{-1}$ at 600 nm) with the CF six-reference base for each DRSsr setup. (a) Contact probe DRSsr ${\mu }_{\mathrm{s}}^{\prime }$ estimation. (b) Contact probe DRSsr ${\mu }_{\mathrm{a}}$ estimation. (c) Parallel noncontact probe DRSsr ${\mu }_{\mathrm{s}}^{\prime }$ estimation. (d) Parallel noncontact probe DRSsr ${\mu }_{\mathrm{a}}$ estimation.

When looking at the $\widetilde{{\mu }_{\mathrm{s}}^{\prime }}$ estimated with all $\mathrm{CF}X\%$ [see Figs. 10(a) and 10(c)], it is clear that most of them are closest to the ${\mu }_{\mathrm{s},\mathrm{ref}}^{\prime }$ curve proper of ${\mathrm{IL}}_{\mathrm{ref}}=1\%$ (dotted violet curve) because ${\mathrm{IL}}_{\mathrm{theo}}=1\%$. Not surprisingly, when comparing all $\widetilde{{\mu }_{\mathrm{s}}^{\prime }}$ separately, the $\widetilde{{\mu }_{\mathrm{s}}^{\prime }}$ estimated with $\mathrm{CF}1\%$ (circle-violet curve) is the nearest to ${\mathrm{IL}}_{\mathrm{theo}}$. The algorithm, therefore, chooses $\mathrm{CF}1\%$ as $\widehat{\mathrm{CF}}$.

In terms of absorption [see Figs. 10(b) and 10(d)], $\widetilde{{\mu }_{\mathrm{a}}}$ estimated with $\mathrm{CF}1\%$ (circle-violet curve) obtains the minimal error with respect to ${\mu }_{\mathrm{a},\mathrm{theo}}$ (dotted-black curve). This is consistent with both DRSsr setups, which validates the new ACA-Pro algorithm.

Figure 11 summarizes the average percentage error of optical properties estimations of Fig. 10. The average relative error throughout the spectrum is calculated for each $\widetilde{{\mu }_{\mathrm{s}}^{\prime }}$ and $\widetilde{{\mu }_{\mathrm{a}}}$ with corresponding ${\mu }_{\mathrm{s},\mathrm{ref}}^{\prime }$ (determining $\mathrm{CF}X\%$) and ${\mu }_{\mathrm{a},\mathrm{theo}}$, respectively. The errors give a general idea of the ACA-Pro performance and confirm that minimal errors are achieved with $\widehat{\mathrm{CF}}=\mathrm{CF}1\%$ for all setups. Table 1 displays these minimal errors.

Table 1

Average relative percentage errors achieved with CF^.

Fig. 11

Average relative percentage errors over the working spectrum of optical properties estimation with each $\mathrm{CF}X\%$.

Estimation of unknown optical properties is best achieved for noncontact DRSsr because of its robust CF-reference base. Correspondingly, contact DRSsr displays a slightly higher error because of the vulnerability to experimental noise of its CF-reference base (see Fig. 9).

When comparing optical properties estimation of phantoms having ${\mu }_{\mathrm{a}}>1{\mathrm{cm}}^{-1}$ (even further from ${\mu }_{\mathrm{a},\mathrm{ref}}=0.4{\mathrm{cm}}^{-1}$) and the same ${\mathrm{IL}}_{\mathrm{theo}}={\mathrm{IL}}_{\mathrm{ref}}=1\%$, the average relative errors shown in Fig. 12 are obtained.

Fig. 12

Average relative percentage errors over the working spectrum of optical properties estimation for different unknown phantoms with ${\mathrm{IL}}_{\mathrm{theo}}=1\%$ and ${\mu }_{\mathrm{a},\mathrm{theo}}=1$, 2, and $3{\mathrm{cm}}^{-1}$.

All ${\mu }_{\mathrm{s}}^{\prime }$ errors are considered below 4% for both probe-based DRSsr setups as a result of the ${\mu }_{\mathrm{s}}^{\prime }$-based ACA-Pro algorithm. For those properties that concern the ${\mu }_{\mathrm{a}}$ error, it is seen that for contact DRSsr it varies little, while for the noncontact setup it increases proportionally to the deviation from ${\mu }_{\mathrm{a},\mathrm{ref}}=0.4{\mathrm{cm}}^{-1}$. However, all ${\mu }_{\mathrm{a}}$ errors are below 8% for both probe-based DRSsr setups.

6.3.

Interpolation

To validate the interpolation method of ACA-Pro, we built a CF two-reference base set with two reference phantoms (having common ${\mu }_{\mathrm{a},\mathrm{ref}}=0.4{\mathrm{cm}}^{-1}$ and ${\mathrm{IL}}_{\mathrm{ref}}=0.5\%$ and 1.5%) and used it to estimate the optical properties of an unknown phantom with ${\mathrm{IL}}_{\mathrm{theo}}=1\%$ in an aqueous blue pigment solution.

Recall from Sec. 5.1 that interpolated estimations are calculated when the noninterpolated $\widetilde{{\mu }_{\mathrm{s}}^{\prime }}*$ estimation lies between two neighboring $\widetilde{{\mu }_{\mathrm{ref}}^{\prime }}$. Figure 13 shows absorption estimations with interpolated $\widehat{\mathrm{CF}}$ between CF0.5% and CF1.5% and noninterpolated $\widetilde{\mathrm{CF}}=\mathrm{CF}1.5\%$ compared to the ideal absorption estimation obtained with the noninterpolated $\widehat{\mathrm{CF}}=\mathrm{CF}1\%$. Results are shown not only for the contact and noncontact probe-based DRSsr setups, but also for the CCD-based DRSsr setup (Fig. 4) for an extended validation of ACA-Pro.

Fig. 13

Absorption ${\mu }_{\mathrm{a}}$ estimation with interpolated ($\widehat{\mathrm{CF}}=\mathrm{CF}[0.5,1.5]\%$), noninterpolated ($\widetilde{\mathrm{CF}}=\mathrm{CF}1.5\%$), and ideal ($\widehat{\mathrm{CF}}=\mathrm{CF}1\%$) CFs for a phantom having ${\mathrm{IL}}_{\mathrm{theo}}=1\%$ and ${\mu }_{\mathrm{a},\mathrm{theo}}$ characteristic of blue pigment depicted in dash-dot lines. (a) Contact DRSsr. (b) Parallel noncontact DRSsr. (c) Folded CCD-based DRSsr.

Figure 14 summarizes the average relative percentage error for both optical properties and the three setups: contact, (parallel) noncontact, and (folded) CCD-based DRSsr. Keep in mind that these errors are not impartial because only $4\lambda \mathrm{s}$ (filters) are considered for the CCD-based setup while $11\lambda \mathrm{s}$ are considered for the probe-based setups, reducing the weight of a single $\lambda$ error.

Fig. 14

Average relative percentage error for noninterpolated ($\widetilde{\mathrm{CF}}=\mathrm{CF}1.5\%$), interpolated ($\widehat{\mathrm{CF}}=\mathrm{CF}[0.5,1.5]\%$) and noninterpolated ideal ($\widehat{\mathrm{CF}}=\mathrm{CF}1\%$) estimations of optical properties of a phantom with ${\mathrm{IL}}_{\mathrm{theo}}=1\%$ and aqueous blue pigment.

The results validate the interpolation strategy of the ACA-Pro algorithm in all considered setups since it improves the accuracy of optical properties when compared to noninterpolated estimations.

A more detailed analysis of ${\mu }_{\mathrm{s}}^{\prime }$ estimations (left Fig. 14) obtained with the interpolated $\widehat{\mathrm{CF}}=\mathrm{CF}[0.5,1.5]\%$ of the contact DRSsr setup shows an exception for which the noninterpolated ${\mu }_{\mathrm{s}}^{\prime }$ estimation is slightly better. This is clearly related to the discontinuity of the contact reference base between $\mathrm{IL}=0.5\%$ and $\mathrm{IL}=1\%$) which is mostly visible with signals measured by $F3-F5$ (see Fig. 9). To improve this effect, as mentioned earlier, the Monte Carlo simulation should be refined at these low scattering values.

6.4.

Correction of Instrumental Variations in Contact Spatially Resolved Diffuse Reflectance Spectroscopy Measurements

The extension of the ACA-Pro calibration approach between various experiments considers the use of a single measurement of a common homogeneous material that characterizes instrumental conditions to calibrate from variations that occur between different experiments (see Sec. 5.2). To illustrate this calibration algorithm, we consider the contact DRSsr setup only.

In this study, a severe experimental variation is simulated by deliberately provoking a change in the signal intensities of $S3$ and $S4$ to affect the sensitivities of both ${\mu }_{\mathrm{s}}^{\prime }$ and ${\mu }_{\mathrm{a}}$ estimation. This is done by decentering the spectrometer fiber from the detection fiber bundles $F3$ and $F4$. A solid piece of white resin²⁵ having stable and homogeneous unknown optical properties is measured under contact conditions and used to characterize experimental conditions before and after the decentralization of fibers.

A reference phantom having ${\mu }_{\mathrm{a},\mathrm{ref}}=0.4{\mathrm{cm}}^{-1}$ at 600 nm and ${\mathrm{IL}}_{\mathrm{ref}}=1\%$ is used. To show the validation of the correction strategy in the ${\mu }_{\mathrm{s}}^{\prime }$ estimation, three different phantoms with common ${\mu }_{\mathrm{a},\mathrm{theo}}=1{\mathrm{cm}}^{-1}$ at 600 nm and ${\mathrm{IL}}_{\mathrm{theo}}=1\%$, 2%, and 3% are used. To show the validation of the ${\mu }_{\mathrm{a}}$ estimation, two different phantoms with ${\mu }_{\mathrm{a},\mathrm{theo}}=1$ and $3{\mathrm{cm}}^{-1}$ at 600 nm and a common ${\mathrm{IL}}_{\mathrm{theo}}=1\%$ are used.

Figure 15 shows $\widehat{{\mu }_{\mathrm{s}}^{\prime }}$ and $\widehat{{\mu }_{\mathrm{a}}}$ of the different phantoms (different colors and shapes) with the noncorrected signals (empty shapes) and corrected signals (black-filled shapes). Estimations under the same experimental conditions of the CF six-reference base are illustrated with slashed black lines while theoretical optical properties are represented with black dotted lines.

Fig. 15

Estimations of (a) ${\mu }_{\mathrm{s}}^{\prime }$ of phantoms with different ${\mathrm{IL}}_{\mathrm{theo}}=1\%$, 2%, and 3% and common ${\mu }_{\mathrm{a},\mathrm{theo}}=1{\mathrm{cm}}^{-1}$. (b) ${\mu }_{\mathrm{a}}$ of phantoms with ${\mu }_{\mathrm{a},\mathrm{theo}}=1$ and $3{\mathrm{cm}}^{-1}$ and common ${\mathrm{IL}}_{\mathrm{theo}}=1\%$. Different phantoms are represented with different colors. Measurements are taken under the same experimental conditions/fiber positions (slashed black lines) or different to the ones set to build the CF six-reference base. The latter are corrected (filled shapes) or noncorrected (black-filled shapes). The legend is given for one phantom, but the same line type code applies to all other phantoms. (a) Reduced scattering coefficient. (b) Absorption coefficient.

Figure 16 sums up the average (over the whole spectrum) relative errors of optical properties estimations shown in Fig. 15.

Fig. 16

Average relative percentage errors with noncorrected and corrected signals from measurements taken under different and same experimental conditions to those of the CF six-reference base.

Figures 15 and 16 show the accuracy improvement in both optical properties estimation of corrected measurements with respect to noncorrected measurements. Estimations with corrected measurements lie much closer than the noncorrected ones to those taken under the same experimental conditions. These results validate the correction strategy of ACA-Pro for experimental conditions variations.

For ${\mu }_{\mathrm{s}}^{\prime }$ estimation, we note that relative errors with the interpolation strategy increase with $\mathrm{IL}(\%)$ [see Fig. 16(a)]. To better understand this effect, $K_{t_0-t_1}$ [calculated according to Eq. (6)] for all $F6-F1$ is shown in Fig. 17. Notice that because of the decentralization of $F3$ and $F4$, $S3$ and $S4$ decrease by $\sim 50\%$ while other nondecentralized fibers’ signals remain close to 1. Nevertheless, an unexpected alteration of $S5$ is noticed (see Fig. 17), most probably related to the surface heterogeneities of the solid material.

Fig. 17

$K_{t_0-t_1}$ ratio of signals taken with centralized and decentralized $F3$ and $F4$ fibers to model different experimental conditions.

Therefore, the ${\mu }_{\mathrm{s}}^{\prime }$ estimation degradation noticed with increasing IL(%) [see Fig. 16(a)] is related to the high sensitivity of ${\mu }_{\mathrm{s}}^{\prime }$ to $S5$ being at a close distance from the source (see Sec. 4.2). This points out the sensitivity of the calibration step to $K_{t_0-t_1}$ and thereby the importance of its robustness to avoid the erroneous alteration of signals. To ensure a robust $K_{t_0-t_1}$ unaffected by the surface heterogeneities of the solid material, measurements could be repeated at different sites and averaged. Alternatively, a better protocol, ensuring measurement repeatability, could be established. We developed a new measurement protocol through which the probe is not placed in direct contact with the material but $\sim 100\mu \mathrm{m}$ away from it. Following the same decentralization procedure for $F2$ and $F4$, we calculated the $K_{t_0-t_1}$ ratio of signals between centralized and decentralized measurements. Figure 18 shows the improved results. It is clear that repeatability of measurements is achieved and that changes are only due to expected experimental variations ($F2$ and $F4$ decentralization).

Fig. 18

$K_{t_0-t_1}$ ratio of signals taken with centralized and decentralized $F2$ and $F4$ fibers and an optimized measurement protocol.

To summarize, we have simulated an extreme change (unlikely to occur naturally) in experimental conditions by decentralizing certain fibers and reducing their signal to $\sim 50\%$ to test the capability of our strategy to correct from experimental variations. The obtained estimation results validate the correction approach. This work has demonstrated that the adoption of a solid material, measured only once to characterize each experimental condition, has the potential to allow the use of a single reference base for optimal optical properties estimation.