2.1.

Whittle-Matérn Model of Tissue Refractive-Index Distribution

The Whittle-Matérn correlation family is a three-parameter model that has been used to describe spatially continuous refractive-index fluctuations and is represented by the equation:

2.2.

Polarization-Sensitive MC Simulations Using the Whittle-Matérn Model

Publicly available polarized-light MC algorithms developed by Ramella-Roman et al.¹⁹ were used to simulate the propagation and collection of polarized light. The implementation of this algorithm to track photons as functions of polarization, radius, ${\theta }_c$, and maximum depth travelled ($z$) has been described in detail by Turzhitsky et al.¹⁷ Reflection, refraction, and alteration of the Stokes vector at the sample/environment interface was computed using Snell’s Law and the Stokes formalism of the Fresnel equations.²⁰ To employ the Whittle-Matérn model described in the previous section, we used a phase function derived from Eq. (1). First, the spectral density $\mathrm{\Phi }$ at spatial frequency $\kappa$ can be calculated from the Fourier transform of $B_n(r)$ yielding

We designed the MC simulations to mimic the geometry of the polarization-gated probe we have used in previous studies. This probe has overlapping illumination-collection areas of radius 400 µm and a collection angle of $\sim 14\mathrm{deg}$.¹⁷ The Stokes vectors of each photon packet were tracked and incoherently summed into corresponding bins of radius ($r$) giving the response to an input pencil beam. Mueller matrix multiplication was used to obtain the output intensity after a polarizer was oriented at 0 or 90 deg ($I_{\Vert }$ or $I_{\perp }$, respectively) with respect to the incident polarization. Convolution was employed to extend the beam from an infinitely narrow source to a circular illumination spot with finite area and radius R. The main objective of this MC study was to determine how the $I_{\Vert }$ or $I_{\perp }$ signals behaved as a function of ${\mu }_s^{\prime }$ and $m$. Thus, $m$ is varied between 1.1 and 2. ${\mu }_s^{\prime }$ is varied between 1.75 and $118{\mathrm{cm}}^{-1}$ for different anisotropy factors ($g=0.75$, 0.8, 0.85, 0.9, 0.93). By recording the signal intensity of the polarization-gated signals as a function of ${\mu }_s^{\prime }$, we developed empirical mathematical relations between the signal intensity, wavelength, and $m$ according to Eq. (2).

To confirm that we could depth-selectively recover $m$, we also created a two-layer MC model where the first layer had an $m=1.1$ and the second layer had an $m=1.7$. We then calculated the value $m$ from the $I_{\Vert }$ and $I_{\perp }$ signal intensities. As the $I_{\perp }$ signal probes a deeper penetration depth than the $I_{\Vert }$ signal, we expected the $I_{\Vert }$ signal to be more sensitive to the $m$ of the top layer and the $I_{\perp }$ signal to be more sensitive to the $m$ of the bottom layer. We also varied the thickness of the top layer to ascertain the precise thickness at which the individual $m$’s could be distinguished by independent analysis of the $I_{\Vert }$ and $I_{\perp }$ signal intensities.

2.3.

Determination of Length-Scale Sensitivity of the $m$ Measurement

We followed a procedure outlined by Yi et al. to determine the range of length scales that the measurement of $m$ is sensitive to.²² The wavelength dependence of the polarization-gated signals is primarily determined by ${\mu }_s^{\prime }(\lambda )$ which in turn can be calculated from the index correlation function. Thus to determine the length-scale sensitivity of the $m$ measurement we perturbed the index correlation function at different length scales and calculated the corresponding change in ${\mu }_s^{\prime }(\lambda )$. To determine the inner length scale $r_{\min }$, the index correlation function was truncated at $r_{\min }$ such that $B_n(r<r_{\min })=B_n(r_{\min })$. This perturbed index correlation function was then used to calculate $B_n^{\prime }$. To do this, the spectral density $\mathrm{\Phi }$ was taken as the Fourier transform of $B_n^{\prime }$ and the differential scattering cross section $\sigma$ was calculated as:

Next, the scattering coefficient ${\mu }_s$ was computed by integrating the differential scattering section over all angles:

2.4.

Animal Model

This study utilized the AOM-induced rat carcinogenesis model that we have previously employed to detect morphological alterations associated with field carcinogenesis.¹⁵ Animal experimental protocols were approved by the Institutional Animal Care and Use Committee (IACUC) of NorthShore University HealthSystem and Northwestern University. Twenty-four male Fisher 344-strain rats were housed in NorthShore’s animal research facility and provided with a diet of AIN-76A (Harlan Teklad, Madison, WI) rodent chow. Animals were housed in polycarbonate, solid-bottomed cages with Alpha-dry bedding which were changed weekly. Each solid-bottom cage was supplemented with an automatic water system and ventilation ports. Each cage housed two rats in a 12-h light/dark cycle animal room at a temperature between 18 and 26ºC with minimal humidity. Animals were monitored for one week prior to injections as an acclimation period for the rodents. Rats were randomized to either a saline or AOM treatment group. At eight weeks of age, 18 rats received two weekly intraperitoneal injections of 15-mg-per-kg-of-bodyweight 13.4-moles/liter AOM (Sigma, St. Louis, MO) while the remaining six rats instead received intraperitoneal saline injections to serve as a control group.

2.5.

Animal Colonoscopy and Polarization-Gated Spectroscopy Measurements

At 18 and 40 weeks post initial AOM/saline injection, rats underwent colonoscopy and polarization-gated spectroscopy measurement. Animals were fasted overnight prior to colonoscopic procedures. Rats were anaesthetized using the Veta Mac Lab Animal Anesthesia System (Veta Mac, Rossville, IN). Each animal was anaesthetized individually in an anesthesia induction chamber supplemented with (vaporized) 1–3% isoflorane (Piramal Healthcare Limited, Andhra Pradesh, India) and oxygen mixture.

After complete sedation was achieved, the rat was removed from the chamber and placed on its ventral. A nose cone was placed on the animal to continuously deliver isofluorane and maintain sedation. The colon was then lavaged to remove fecal debris by rectal injection of saline (with a ball-point needle and 50-mL syringe). Karl Storz veterinary (straight) endoscope (Storz, Goleta, CA) was inserted into the rectum and the rat colon was visualized on the video screen. The presence of polyps, their size, and their relative location were noted. Representative endoscopic images of a rat with a normal colon and a rat harboring a polyp are shown in Fig. 1(a) and 1(b), respectively. The scope was then removed and a 2.45-mm diameter fiber-optic polarization-gated spectroscopy probe was inserted into the rectum approximately 5 cm into the animal colon. Five measurements were then taken with the probe from the distal colonic mucosa. Probe measurements were taken in areas devoid of polyps as determined by the colonoscopy visualization. The probe and nose cone were then removed and the rat was monitored until effects of anesthetic were no longer noticeable.

Fig. 1

Endoscopic images of a normal saline control (a) and an AOM-treated rat with a tumor (b).

2.6.

Polarization-Gated System and Data Analysis

The details of the polarization-gated probe and signal-collection system have been described previously.¹⁷ In brief, the probe consists of 3 fibers arranged in an equilateral triangle. One of these fibers delivers a linearly polarized and collimated beam of light onto the tissue surface. The two remaining fibers receive light backscattered from the tissue with one fiber collecting light polarized parallel $(I_{\Vert })$ to the incident beam and the fiber collecting light polarized perpendicular $(I_{\perp })$ to the incident beam. Each collection fiber is connected to a fiber-optic spectrometer so that the signal intensity can be analyzed as a function of wavelength. The spectrum $I(\lambda )$ from each tissue site and both polarization channels was analyzed according to the following relationship:

3.1.

MC-Derived Relationship between Polarization-Gated Signal Intensities and ${\mu }_s^{\prime }$

In Fig. 2(a), we plot the $I_{\Vert }$ signal intensity versus ${\mu }_s^{\prime }$ for several different anisotropy factors and $m=1.5$. For $g>0.85$, the individual curves for different $g$ are within 5%, indicating that the $I_{\Vert }$ signal can be written as a function of ${\mu }_s^{\prime }$ rather than a combination of $g$ and ${\mu }_s^{\prime }$. In Fig. 2(b), we plot the same $I_{\Vert }$ signal intensity versus ${\mu }_s^{\prime }$ curves but for different $m$ and a single anisotropy factor ($g=0.9$). These curves clearly show that the value of $m$ influences the $I_{\Vert }$-signal-intensity-versus-${\mu }_s^{\prime }$-relationship. However, we found that each curve could be fit with a power law such that $I_{\Vert }\propto {({\mu }_s^{\prime })}^{C(m)}$ where the exponent $C(m)$ is a function of $m$. The power law fit each of the curves well with an $R^2>0.99$ in all cases. The exponent $C(m)$ was found to be a linear function of $m$ with an $R^2>0.99$ as shown in Fig. 2(c). Using the proportionality between ${\mu }_s^{\prime }$ and ${\lambda }^{2m-4}$ derived from the Whittle-Matern model and the linear fit to $C(m)$, we can express the wavelength dependence of the copolarized signal $I_{\Vert }(\lambda )$ as:

Fig. 2

Extraction of the index correlation-function shape ($m$) from the copolarized intensity (top row) and crosspolarized intensity (bottom row). (a) For $g>0.85$, the intensity is a function of and independent of $g$. A power law can be fit to this dependence for different $m$ as shown in (b). The exponent of the power law has a linear relationship with $m$ as shown in (c). (d–f) are analogous to (a–c) but for the crosspolarized intensity rather than copolarized intensity.

3.2.

Measurement of $m$ from a Dual-Layer MC Model

The analysis so far assumes a homogeneous medium defined by a single $m$ value. In reality, biological tissue is a layered medium most likely defined by more than a single m. Since the copolarized and crosspolarized signals sample different penetration depths (350 µm and 575 µm on average, respectively), it is likely that the $m$ values measured by these individual signals will be different, with the copolarized signal measuring the $m$ value of superficial layers of and the crosspolarized signal measuring the $m$ value of deeper layers. To test this hypothesis, we created a two-layer MC model in which the top layer had an $m$ of 1.1 and the bottom layer an $m$ of 1.7. The optical thickness $\tau$ of the first layer, defined as the product of the physical thickness and the scattering coefficient ${\mu }_s$, was varied between 0 and 100 while the overall $\tau$ of the medium was fixed at 1000. Figure 3 shows the $m$ values measured from the co- and crosspolarized intensities as a function of the optical thickness of the top layer. When $\tau$ of the top layer is less than $\sim 5$, both co- and cross-signals predominantly interrogate the bottom layer ($m=1.7$). For $\tau$ greater than $\sim 15$, both signals probe the top layer ($m=1.1$). For $5<\tau <15$, the copolarized signal probes mainly the top layer and thus $m$ is close to 1.1. In contrast, the crosspolarized signal measures the bottom layer and has an $m$ value closer to 1.7.

Fig. 3

Depth-selective extraction of index correlation shape ($m$) from a dual-layer Monte Carlo model. The top layer has $m=1.1$ and bottom layer an $m$ of 1.7. The $m$ values from the copolarized and crosspolarized signals are recorded as the optical thickness ($\tau$) of the top layer is varied. For $5<\tau <15$, the copolarized $m$ parameter is sensitive to the top layer and the crosspolarized $m$ parameter is sensitive to the bottom layer.

3.3.

Length-Scale Sensitivity of $m$ Measurements

We determined the minimum and maximum length scale over which our spectral determination of $m$ is sensitive to by perturbing the index correlation function with different $r_{\min }$ and $r_{\max }$ values as outlined in Sec. 2.3. Figure 4(a) and 4(b) display the percent error between the input $m$ value and the measured $m$ value from the perturbed index correlation function for different $r_{\min }$ and $r_{\max }$, respectively. We can define a percent error threshold whereby if the percent error is greater than this threshold for a particular $r_{\min }$ and $r_{\max }$, then the length scales to which the measurement is sensitive are between $r{}_{\min }$ and $r_{\max }$. Defining a threshold of 5% error suggests that the measurement of the $m$ value is sensitive to length scales between 45 nm and 1.5 µm according to Fig. 4.

Fig. 4

Length-scale sensitivity of index correlation function shape ($m$) measurement. (a) Percent between measured and input $m$ values after correlation function was truncated at a particular $r_{\min }$ value. (b) Percent error between measured and input $m$ values after index correlation function was perturbed at various $r_{\max }$ values according to Eq. (9).

3.4.

Tumor Development in Saline/AOM-treated Rats

Rats were assessed at 18 and 40 weeks for tumor development by colonoscopy. The number of rats who had developed tumors by each time point is summarized in Table 1. As expected, none of the saline-treated rats developed tumors over the study time period. 77% of the AOM-treated rats possessed tumors by the end of 40 weeks, with more than half developing tumors by the 18-week time point. This is consistent with earlier studies on tumor development in the AOM-treated rat.²⁷ These results confirm that the saline-treated rats could be reliably regarded as controls and that the AOM treatment was a potent initiator of carcinogenesis.

Table 1

Number of rats that developed tumors at the 18- and 40- week timepoints.

3.5.

Measurement of the Refractive-Index Correlation-Function Shape from Salineand AOM Treated Rats

After colonoscopy at the 18-week time point, five polarization-gated probe measurements were taken throughout the visually normal distal mucosa of each saline and AOM-treated rat. For each tissue site, an $m$ value was extracted from the copolarized and crosspolarized signals according to Eqs. (7) and (8). The median $m$ value from these five sites was taken and used to characterize the mucosa of each individual rat. A Welch’s t-test was used to determine statistical significance between groups. The first comparison we made was between saline-treated rats and all rats treated with AOM. Figure 5(a) shows that in rats treated with AOM, there is a statistically significant increase in the $m$ parameter from the visually normal mucosa for both the co- and crosspolarized signals ($P$ $\text{value}=0.007$ and 0.003, respectively). The $\text{average}\pm \text{standard deviation}$ $m$ values for saline controls were $1.77\pm 0.04$ and $1.64\pm 0.06$ from the co- and crosspolarized signals, respectively, while for AOM they were $1.9\pm 0.17$ and $1.79\pm 0.16$ from the co- and crosspolarized signals, respectively. Figure 5(b) illustrated what the shape of the refractive-index correlation would look like for saline and AOM-treated rats given the $m$ values from the crosspolarized signal. Both curves can be represented by stretched exponentials with the AOM curve decaying more slowly, suggesting that the AOM-rat tissue structure has correlations at longer length scales than the saline-rat tissue structure does. We next investigated whether the $m$ parameter could be a predictor of both concurrent and future risk of neoplasia. In Fig. 5(c), we compare the m parameter values measured at 18 weeks among saline controls, AOM rats that developed tumors by 18 weeks (tumor presence), and AOM rats that had not developed tumors by 18 weeks (no tumor presence). This figure demonstrates that the $m$ parameter mirrors neoplasia risk. For both the copolarized and crosspolarized $m$ parameter, there is an increase in $m$ for rats that had not developed tumors by the 18-week time point but were still at risk of developing tumors because of the AOM treatment ($P=0.1$ and 0.007, respectively). There is a further increase in $m$ in rats that harbored neoplasia at the time of measurement as shown in Fig. 5(b). These results demonstrate that the $m$ parameter is most sensitive to the presence of neoplasia, but that it is also predictive of the risk of future neoplastic development.

Fig. 5

The shape of the index correlation function is sensitive to concurrent and future risk of neoplasia in the AOM-treated rat. (a) Comparison of the $m$ values from the copolarized and crosspolarized signals for saline versus all AOM-treated rats shows a statistical significant increase in $m$ ($P$ $\text{value}=0.007$ and 0.003). (b). Comparison of the shape of index correlation function shape between saline and AOM-treated rats. (c). Comparison of $m$ values between saline controls and AOM-treated rats who had and had not developed tumors at the 18-week measurement point. The $m$ value is highest for rats harboring a tumor but is also significant for AOM rats that had not yet developed a tumor. Solid circle points are group means and error bars are 95% CI.