2.1.

Tissue Model Geometry and Optical Properties

A two-layer cylindrical tissue model was constructed to represent human cervical epithelial tissue. In system coordinates, the longitudinal axis of the cylinder was in the $Z$ direction, and the tissue surface occupied the $x\text{-}y$ plane. The thickness of the epithelium was set at $450\mu \mathrm{m}$ to approximate the average thickness of human cervical epithelium.⁵¹ The stroma was modeled as a cylindrical slab with a $10\text{-}\mathrm{mm}$ thickness, representative of a semi-infinite tissue layer underneath the epithelium. The cylindrical model was set wide enough ( $10\text{-}\mathrm{mm}$ radius) to encompass all possible photon paths to ensure the comprehensiveness of this simulation. The profile of this layered tissue structure can be found in Fig. 1 . The optical properties of human cervical epithelial tissue (Table 1 ) were based on the literature values reported by Drezek ⁵² with additional references among other previously accepted values.^{7, 41, 49, 53, 54, 55, 56} Tissue optical properties corresponding to a wavelength of $500\mathrm{nm}$ were implemented in the simulation models.

Fig. 1

Profile view of layered tissue structure.

Table 1

Tissue optical properties implemented in this simulation model.

2.2.

Fiber-Probe Geometry

We investigated three specific fiber-probe configurations: a single fiber for illumination and collection, multiple collection fibers orthogonal to the tissue surface (parallel to the illumination), and multiple collection fibers angled with respect to the illumination axis. The illumination fibers were invariantly oriented perpendicularly to the tissue surface for all proposed probe geometries, and only the position and orientation of collection fibers were changed in this study. Schematic illustrations of the proposed fiber-probe configurations can be found in Fig. 2 . Fiber diameters were set at $100\mu \mathrm{m}$ for all fiber optics simulated in this study.

Fig. 2

(a) Schematic of the single illumination-collection fiber geometry. (b) Orthogonal fiber geometry. (c) Positively oblique fiber geometry. (d) Negatively oblique fiber geometry.

The single illumination-collection fiber was positioned at the center of the cylindrical tissue model. The source-detector separation distance (SDSD) is defined as zero for the single fiber, since SDSD measures the center-to-center distance between the illumination and collection fibers. The single-fiber probe has a complete overlap between its illumination and collection cone and is sensitive to direct backscattering from superficial tissue depths.^{38, 39, 40, 41, 56} To facilitate comparisons between various probe designs, the single-fiber geometry is designated as the reference in the evaluation of other probe geometries.

In the two other cases where separate collection fibers were used, the collection fibers were positioned at predesignated SDSD (150, 300, 500, and $800\mu \mathrm{m}$ ) from the illumination axis. 3-D solid fiber models were inserted into the simulation program to achieve realistic spatial configurations of optical fibers in fiber probes. The collection fibers were evenly spaced and positioned concentrically about the illumination axis at each SDSD, and each fiber was individually modeled as a solid 3-D object. Only photons reaching the distal facets of the collection fibers were admitted according to the specification of fiber numerical aperture: photons that came in contact with the opaque components of the fiber probes, including the exterior wall of the fiber optics, were nullified and excluded from the simulations and results. It should be noted that the concentrically arranged collection fibers at their respective SDSD effectively formed an imaginary collection circle whose circumference perpendicularly intersected the longitudinal axes of the collection fibers. Because the circumference of the imaginary collection circle was directly proportional to its radius (i.e., SDSD), the number of collection fibers on each circle also increased with respect to the values of SDSD: the numbers of collection fibers positioned at their corresponding SDSD—150, 300, 500, and $800\mu \mathrm{m}$ —were 9, 18, 31, and 50. It must be stressed that fibers on the same collection circle were geometrically equivalent due to the symmetry about the illumination axis. If the illumination angle were to deviate from its orthogonal orientation with respect to tissue surface, this symmetry and the collection-fiber equivalency would be no longer valid. The rationale of installing full circles of collection fibers was twofold. First, multiple collection fibers were implemented in the models to increase the efficiency of Monte Carlo simulation: increasing the detection area (i.e., cross-sectional area of a fiber multiplied by the number of fibers) shortened the time necessary to achieve convergence in the simulation process. Second, the signal collection efficiency of realistic fiber probes was usually maximized by adopting the same circular arrangement of multiple collection fibers around the illumination axis, thus our models would emulate circularly arranged probes of similar geometries to achieve more representative reflectance readings as with realistic probes.

The geometric configurations and arrangements of the oblique collection fibers were identical to their orthogonal counterparts as described before, with the only exception being their relative angular orientation with respect to the illumination axis. The physical axes of the collection fibers were rotated about the tissue surface to achieve an angle between the illumination and collection axes. We defined fiber facets rotating toward the illumination axis to be the positive angles [Fig. 2(c)] and negative angles as fiber facets rotating away from the illumination axis [Fig. 2(d)]. Angles of rotation implemented were 10, 20, 40, 60, 75, $-10$ , and $-25\mathrm{deg}$ .

When oriented perpendicularly to a media, the physical and optical axes of fiber optics are equivalent to each other. When implemented with an angle, the optical axes of the oblique fibers may deflect from their physical axes due to refraction. Therefore, the optical rotational angles may not be the same as the physical rotational angles specified previously. In Fig. 2, the physical rotational angle $\left({\theta }_{\mathit{pr}}\right)$ is defined as the angle between the physical axes of the illumination and collection fibers: the optical rotational angle $\left({\theta }_{\mathit{or}}\right)$ is the angle between the optical axes of the two. The location and magnitude of ${\theta }_{\mathit{or}}$ relative to ${\theta }_{\mathit{pr}}$ are dependent on the refractive index mismatch across the tissue-fiber boundary: the long-dashed line represents a hypothetical optical axis for the situation in which the external refractive index is smaller than that of the tissue, and the short-dashed line represents the opposite situation. The values of ${\theta }_{\mathit{pr}}$ and their corresponding ${\theta }_{\mathit{or}}$ used in this study are enumerated in Table 2 . Since ${\theta }_{\mathit{pr}}=75\mathrm{deg}$ exceeds the critical angle specified by Snell’s law for the glass-to-tissue interface, ${\theta }_{\mathit{or}}$ is not available for the corresponding entry in Table 2.

Table 2

Values of physical and optical rotational angles of fibers in tissue with different index-matching media calculated by Snell’s law.

Index-matching materials were used to fill the gap between the distal end of fiber probes and tissue surface. Water $(n=1.33)$ and glass $(n=1.5)$ were the two materials proposed in the simulation models. Water is one of the most widely used index-matching fluids and a good representation of realistic index-matching schemes in experimental and clinical setups. A glass optical window is a commonly used component integrated into fiber probes to provide better index matching. A glass optical window can be fabricated with specific dimensions and shapes to be fitted in probes of different geometries. For the oblique fiber probes as illustrated in Fig. 2, glass optic wedges can be affixed onto the distal ends of fiber optics in place of air or water.

Numerical aperture (NA) specifies the half angle (HA) of fiber optics via the following relationship:

Table 3

Effective NA and half angles with respect to different ambient media. Values in parentheses are degrees of angle.

2.3.

Ray Tracing Setup and TracePro® Simulation

TracePro® by Lambda Research (Littleton, Massachusetts) was the ray tracing program used for the theoretical simulations. TracePro® utilizes the well-established Monte Carlo solid modeling method to simulate photon interactions with tissue. The human interface of TracePro® is similar to that of CAD, and allows users to graphically designate the geometric details of fibers and tissue models. Details about Monte Carlo modeling can be found in many literary texts.^{57, 58, 59, 60, 61, 62} MATLAB® version 6.5 by Mathworks, Incorporated was used for postsimulation data processing.

Illumination photons were guided through a perpendicularly oriented illumination fiber at the center of the tissue surface. The nominal numerical aperture for illumination was 0.22. An evaluation of simulation convergence was conducted prior to the simulation trials. We ran ten groups of trial simulations with increasing numbers of incident rays (100 thousands to 1 million in increments of 100 thousands, 10 runs per group) and calculated the detected reflectance convergence. At the ray count of 100 thousands, the standard error of the mean sampled reflectance was less than 2.7%, and this value was further reduced to 0.78% when the ray count reached 1 million. Therefore, 1 million incident photons were sufficient enough to provide the desired data convergence, and 10 million incident photons were used for simulations to have extra data sampling.

2.4.

Analysis of the Simulation Data

The detected photons were inventoried according to their weight, path lengths, coordinates of scattering, and directional vectors between scattering events. From these quantities, we can calculate the maximal penetration depth, mean penetration depth, cumulative path length in tissue, sensitivity to each tissue layer, and exit angle of individual photons.

Unlike fluorescence emission, which has fixed fluorophores at specific tissue depths, it is more difficult to clearly characterize the depths of photon penetration in tissue for reflectance rays. We use multiple parameters to quantify the depths of photon migration in our models. The maximal penetration depth $\left(Z_{\max }\right)$ of each detected photon is defined as the greatest displacement of photon propagation in tissue in the axial direction ( $Z$ axis) measured from the point of entry $(Z_0=0)$ . Although $Z_{\max }$ does not completely represent the spatial distribution of photons in tissue because it only quantifies photon movements in the axial direction and ignores the lateral movements of photons, the concept of $Z_{\max }$ is easily visualized and unambiguous. Better quantifying photon movements is the mean penetration depth $\overline{Z}$ of each detected photon [Eq. 2], which is calculated from the mean occupancy of level $z$ by a photon. By the nomenclature of “occupancy of level $z$ ,” we mean the fraction of time spent by a photon at depth $z$ before reaching the collection fibers.^{63, 64} $\overline{Z}$ may be tangibly characterized as the weight of a photon’s probing range in tissue: a photon’s axial probing range is bounded by 0 (tissue surface) and $Z_{\max }$ (maximal penetration depth), and $\overline{Z}$ represents the central location of this probing range in the axial direction. $\overline{Z}$ may be expressed as the sum of tissue depths multiplied by the probability of a photon occupying particular depths during photon migration in tissue (from the point of entry to exit).

The sensitivities of individual photons to epithelium and stroma ( $S_{\mathrm{Ep}}$ and $S_{\mathrm{St}}$ , respectively) are defined as the following:

Thus far, we have defined the maximal penetration depth, mean penetration depth, and layer sensitivities of individual photons. To characterize the probing capability and layer sensitivities of a fiber probe, we take the weighted average of these parameters with respect to each photon’s contribution to the total detected reflectance. Let the weight of a detected photon be $f$ , the number of detected photons $NP$ , and the total detected reflectance $R$ , and then the expected values of mean probing depth and layer sensitivities of a fiber probe are:

The total reflectance was normalized against the total illumination radiation injected into the tissue. Thus, in the following sections, we use “reflectance probability” (between 0 and 1) as the measure of signal strength harvested by various probe designs. To preserve the qualitative significance of this quantity, we refer to the reflectance probability as normalized reflectance in future sections.

3.1.

Depth-Resolved Reflectance for Variable Collection Angles

Figure 3 shows the normalized reflectance profiles binned over the mean penetration depths of photons measured every $10\mu \mathrm{m}$ with respect to various collection angles. Only 0, 20, 40, 60, and $-25\mathrm{deg}$ are plotted for better readability of the graph. The fiber probes have a SDSD of $300\mu \mathrm{m}$ and a nominal NA value of 0.34. Water is the selected index-matching material between the probes and tissue. Due to the double-layer structure of the tissue model, the appearance of double-peak reflectance distribution in Fig. 3 can be attributed to the discontinuity in tissue optical properties between the epithelial and stromal layers. Evaluation of Fig. 3 indicates that the mean penetration depth of photons decreases as the collection angle increases. The $0\text{-}\mathrm{deg}$ fiber has a single reflectance peak at the mean penetration depth of approximately $320\mu \mathrm{m}$ . As the collection angle increases, the peak of reflectance shifts leftward and the peak-to-width ratio increases. This indicates the penetration depths of the detected photons become more superficial and the spatial distributions of the photons become better defined. Figure 3 also suggests that the increasing collection angles enhance the signal strength of reflectance detection in the superficial region. As the collection angle increases in the positive direction, significant reflectance increase is seen for photons with mean penetration depths less than $200\mu \mathrm{m}$ . By increasing the collection angle in the negative direction, we observe a decrease in the total reflectance and a shift into deeper depths. The spatial distribution of photons is less defined as the peak-to-width of the reflectance curve diminishes for the negative-angle fibers. Figure 4 includes the cumulative percentage distribution curves of the detected reflectance with respect to the mean penetration depths, and it is clear that collection angles have dominant effects on the spatial distributions of sampled reflectance. For fibers with positive angles, significant portions of detected photons have shallow mean penetration depths in comparison to the normal fiber, whereas the negative-angle fibers push the distribution into deeper regions of the tissue and consequently may be more sensitive to the tissue characteristics of the bottom layers.

Fig. 3

Depth-resolved normalized reflectance measured at $\mathrm{SDSD}=300\mu \mathrm{m}$ .

Fig. 4

Cumulative percentage distribution of reflectance measured at $\mathrm{SDSD}=300\mu \mathrm{m}$ .

3.2.

Depth-Resolved Reflectance for Variable Source-Detector Separation Distances

Figures 5(a), 5(b), 5(c), 5(d) display the percent reflectance distribution with respect to the photon’s mean penetration depths by fiber probes with various SDSD and collection angles. The nominal numerical aperture of the fiber probes remains at 0.34 and is index matched by water. The $x$ axis is divided into intervals of $10\mu \mathrm{m}$ , and the partial reflectance corresponding to each division is summed and weighted against the total reflectance detected by each fiber probe. The single-fiber geometry $\left(0\mu \mathrm{m}0\mathrm{deg}\right)$ was used as the reference to better facilitate contrast and comparison among the diagrams. As SDSD increases, the weight of detected reflectance persistently shifts toward the greater end of the $x$ axis, and this is observed for all collection angles. Collection angles have profound effects on the depth-resolved distribution of detected reflectance. The weight of early returning photons in the total reflectance rises with increasing collection angles, and this is manifested by the rise and leftward shift of the curves in Fig. 5. On the other hand, by implementing oblique fibers in the negative direction, deeper reflectance sampling was induced and accompanied by the diminishment of early returning photons. Diagrams in Fig. 5 have demonstrated the potential to control the probing depths and spatial sensitivities of fiber probes by manipulating geometric parameters such as SDSD and collection angles. Combinations of SDSD and collection angles can be utilized to achieve depth selectivity in reflectance sampling for various situations.

Fig. 5

Percent reflectance distribution measured at (a) 0, (b) 40, (c) 60, and (d) $-25\mathrm{deg}$ .

Figure 6(a) displays the magnitudes of reflectance detected per individual fiber placed at various SDSD. The detected reflectance is normalized to the illumination. Fiber nominal NA remains the same as in Fig. 5 (0.34) and is index matched by water. The single-fiber geometry, being positioned at the point of illumination, has the strongest reflectance per fiber, while the magnitude of reflectance quickly recedes with increasing SDSD. We also observe a consistent increase in detected reflectance per fiber when the collection angle increases from $0\text{to}40\mathrm{deg}$ . We attribute the rise in reflectance to the increasing number of detected photons and the greater individual weights of these photons: the center of reflectance detection shifts toward superficial depths as the collection angle increases, and early returning photons have higher weights due to their shorter path lengths in tissue. For collection angles of 60 and $75\mathrm{deg}$ , the total number of detected photons decreases as the collection angle continues to rise. This is primarily due to the much limited tissue sampling volume and lower probability for photons exiting the tissue surface with extremely steep trajectories. Although the detected reflectance per individual fiber diminishes as SDSD increases, multiple collection fibers can be implemented in a circular configuration as in this study to improve the detection efficiency of fiber probes. This is the advantage of implementing multiple fibers in a fiber probe with SDSD greater than 0. Figure 6(b) displays the total detected reflectance by each probe unit, and the values displayed are multiples of those shown Fig. 6(a), contingent on the number of collection fibers implemented in these probes. Note that the difference in total reflectance per probe among different SDSD configurations is greatest at collection angles of 60 and $75\mathrm{deg}$ .

3.3.

Depth-Resolved Reflectance for Variable Fiber Numerical Apertures

In Fig. 7 , the effects of numerical apertures on fiber expected probing depths are displayed and compared. The $x$ axis lists the six nominal NA of the collection fibers tested in this study, and the $y$ axis represents the expected probing depth of fiber probes. With water as the index-matching material, the effective NA of the fiber probes increases from 0.17 to 0.75. Correspondingly, the effective half angles of the fiber probes vary from $9.52\text{to}48.8\mathrm{deg}$ (Table 3). Evaluation of Fig. 7 shows that the effects of NA on the expected probing depth are subtle and contingent on the angles of collection fibers. For fibers with collection angles from $0\text{to}20\mathrm{deg}$ , the expected probing depths gradually decrease with respect to increasing NA values; a similar observation is made for the negative-angle collection fibers. The opposite is observed for fibers with collection angles from $40\text{to}75\mathrm{deg}$ , for which the expected probing depths become greater as the NA value increases.

Fig. 7

Effects of fiber NA on expected probing depths of fiber probes (index-matched by water).

3.4.

Fiber Expected Probing Depths and Layer Sensitivities

As we have shown that both collection angles and SDSD have significant influences over the spatial distributions of detected photons in tissue, collection angles and SDSD are expected to have similar effects on the expected fiber probing depths and layer sensitivities. Having the nominal fiber $\mathrm{NA}=0.34$ (index matched by water), Figs. 8(a) and 9(a) display the expected fiber probing depths and layer sensitivities under various fiber configurations, respectively. The expected probing depth of a fiber probe decreases as the collection angle increases in the positive direction, while extended probing depths can be observed for negative collection angles. A trend in the sensitivities to the epithelial layer of a fiber probe, which echoes the shifts in expected fiber probing depths in Fig. 8(a), can be observed in Fig. 9(a). Figures 8(b) and 9(b) are the respective standard deviations corresponding to the expected values shown in Figs. 8(a) and 9(a). The standard deviations of the expected probing depths and layer sensitivities generally decrease with increasing collection angles but increase with greater source-detector separation.⁶⁵ An increase in standard deviation indicates that the sampled reflectance is less spatially specific and comprises photons from a broader spread of tissue depths. The opposite effects of SDSD and collection angles create a transition in the expected probing depths of fiber probes, and when both effects are equally dominant, the expected variance would likely increase. For example, Fig. 3 shows the reflectance profiles with respect to photon’s mean penetration depths in tissue for $\mathrm{SDSD}=300\mu \mathrm{m}$ with various collection angles. At $40\mathrm{deg}$ , there are two peaks of equal height at the mean penetration depths of 110 and $260\mu \mathrm{m}$ , and the observed standard deviation for $\mathrm{SDSD}=300\mu \mathrm{m}$ is the greatest at this angle. At either 20 or $60\mathrm{deg}$ , the distributions of sampled reflectance are more definitive, and consequently we may observe a decrease in the standard deviations of the expected probing depths at these two collection angles. The opposite effects of SDSD and collection angles may explain the sudden increases in the standard deviations of the expected probing depths at $(\mathrm{SDSD},\text{angle})=(150\mu \mathrm{m},20\mathrm{deg})$ , $(300\mu \mathrm{m},40\mathrm{deg})$ , $(500\mu \mathrm{m},60\mathrm{deg})$ and $(800\mu \mathrm{m},75\mathrm{deg})$ in Fig. 8(b). To demonstrate the trend with respect to the collection angles in numerical details, the expected probing depths and sensitivities to the epithelial layer for $\mathrm{SDSD}=0$ and $300\mu \mathrm{m}$ are listed in Tables 4, 5 , in which their respective standard deviations are enclosed in parentheses.

Fig. 9

(a) Effects of collection angles and SDSD on expected sensitivities to epithelial layer of fiber probes. (b) Standard deviations of expected sensitivities to epithelial layer for probes with various collection angles and SDSD.

Fig. 8

(a) Effects of collection angles and SDSD on expected probing depths of fiber probes. (b) Standard deviations of expected probing depths for probes with various collection angles and SDSD.

Table 4

Expected probing depth (and standard deviation) of fiber probes with respect to fiber geometry and index-matching conditions.

Table 5

Expected sensitivity to epithelial layer (and standard deviation) of fiber probes with respect to fiber geometry and index-matching conditions. For entries with expected epithelial sensitivities of 1, the standard deviation shown in the parentheses denotes deviation in the minus direction.

Another informative metric used to describe the probing capabilities of fiber probes is the locations of percentiles of depth-resolved reflectance, which express the spatial distribution and center of gravity of depth-resolved reflectance. Combined with the expected probing depths, we may obtain more information in regard to the range of photon mean penetration depths in tissue. Figure 10 plots the mean penetration depths of photons, at which the 10th, 50th (median), and 80th percentiles of reflectance were recorded at a SDSD of $300\mu \mathrm{m}$ . The nominal numerical aperture of the fiber probes is 0.34 and water is the index-matching material. The downward movements of 10th, 50th, and 80th percentiles markedly show that positively oblique fibers enhance the importance of early returning photons, and the vertical spacing between the percentile markers indicates the density of photon distribution with respect to the mean penetration depths. Table 6 lists the numerical values of the mean penetration depths at which the 10th, 50th (median), and 80th percentile of depth-resolved reflectance were measured at $\mathrm{SDSD}=0$ and $300\mu \mathrm{m}$ .

Fig. 10

The 10th, 50th (median), and 80th percentile of depth-resolved reflectance measured at various collection angles $(\mathrm{SDSD}=300\mu \mathrm{m})$ .

Table 6

Mean penetration depths at which the 10th, 50th (median), and 80th percentile of depth-resolved reflectance were measured.

3.5.

Effects of Index Matching and Fiber Numerical Aperture

Transparent optical windows are used to improve the index-matching condition by eliminating the presence of air between fiber probes and tissue. Water is one of the most commonly used index-matching fluids with a refractive index of 1.33. Glass, whose refractive index is near 1.5, is a solid material commonly used in place of index-matching fluids. In this study, we investigated the effects of index matching on the depth selections of fiber probes, and the results are best observed in Table 4. We can observe a decrease in the expected probing depths of probes with positive-angle fibers when index matching is administered with glass. This effect is most prominent when fiber obliquity is high (i.e., $⩾40\mathrm{deg}$ ) and is almost nonexistent for orthogonally oriented fibers. Similar effects can be observed in Tables 5, 6, where the layer sensitivities to epithelium and centers of gravity of depth-resolved reflectance are shifted accordingly. Due to refraction, no light was detected by the $75\text{-}\mathrm{deg}$ fibers with a nominal NA of 0.22 when glass was implemented in the model.

Fiber numerical aperture imposes minor influences on the depth selection of fiber probes. Greater numerical apertures increase the half angles of fiber optics and result in more inclusive detections of light re-emitted from tissue. Figure 11 shows the normalized reflectance detected per fiber as nominal NA ranges from 0.22 to 1.0 when index matched by water. While increasing fiber NA from 0.22 to 0.34 does not significantly influence the expected probing depths of fiber probes, the magnitude of detected reflectance is doubled.

Fig. 11

Normalized reflectance per fiber detected with various fiber nominal numerical apertures (index matched by water).