Optimization of the Distance between Cylindrical Light Distributors Used for Interstitial Light Delivery in Biological Tissues

Abstract

Cylindrical light diffusers (CLDs) are often employed for the treatment of large tumors by interstitial photodynamic therapy (iPDT) and photoimmunotherapy (PIT), which involves careful treatment planning to maximize therapeutic dose coverage while minimizing the number of CLDs used. There is, however, a lack of general guidelines regarding optimal positioning of CLDs, in particular when they are inserted in parallel to treat head and neck squamous cell cancer (HNSCC). Therefore, the purpose of this study is to determine the CLD-CLD distances maximizing the necrosis for different geometries of CLD positions and shed light on the influence of different optical parameters on this distance, in particular when HNSCCs are treated by interstitial PIT with cetuximab–IR700 using up to seven CLDs. To that end, Monte-Carlo simulations of the light propagation around CLDs inserted perpendicularly in a semi-infinite tumor were performed to determine the volume receiving a fluence larger than a therapeutic threshold. An optimization algorithm was then used to calculate and maximize the necrosed tumor volumes. Tumor optical properties were derived from published data. Our findings suggest that optimal CLD positioning maximizing the volume of necrosed tumor during interstitial PIT for typical HNSCC optical properties corresponds to a CLD-CLD distance between 11.5- and 13-mm. Variations of the absorption and reduced scattering coefficients have the greatest influence on CLD placements, while tissue anisotropy factor, CLD insertion geometry, CLD length, and the angular dependence of the radiance emitted by the CLDs have minimal influence. At first approximation the influence of these optical parameters on optimal CLD-CLD distance are independent. Our data also suggests it is possible to derive new treatment plans using knowledge of previous treatment plans.

1. Introduction

Cylindrical optical fiber-based light diffusers (CLD) are frequently used for interstitial photodynamic therapy (iPDT) and photoimmunotherapy (PIT) of various solid cancers [1,2,3,4]. Although PIT and iPDT differ from many respects, these two therapies are based, in most cases, on the intravenous administration of photosensitizers (PS) followed by the delivery of light to the lesions with appropriate irradiances and doses at wavelengths absorbed by the PS [5]. These non-thermal illuminations, which are usually performed in the red or near-infrared part of the spectrum, lead to more or less selective cancer necrosis and/or apoptosis depending on the PS and lesion types. iPDT utilizes PS with tumor affinity as a drug, such as Photofrin, and reactive oxygen species produced upon photoirradiation are responsible for the tumor cell death. The fundamental difference between iPDT and PIT is related to the fact that the latter is based on the use of specific antibodies-PS conjugates to improve the cancer targeting [6], whereas more general targeting approaches, if any, are used in iPDT [7]. More precisely, PIT takes advantage of the high recognition properties of antibodies toward antigens expressed in tumor cells to increase treatment selectivity. One illustrative example is a conjugate involving an antibody targeting the tumor cell membrane and a phthalocyanine-based PS, IR700, which induces a necrotic cell death caused by a physicochemical process upon illumination [8]. Since the first pioneering report of this compound by Kobayashi et al. in 2011 [9], preclinical studies of PIT have been intensively investigated utilizing conjugates with various antibodies. In addition, a clinical application has already been started in 2015 with the compound cetuximab–IR700 [10]. A global phase 3 clinical trial for unresectable locoregional or recurrent head and neck squamous cell cancer (HNSCC) patients after at least two lines of therapy is ongoing [11], where cetuximab–IR700 is used to target epidermal growth factor receptors (EGFR). In September 2020, the first marketing approval was obtained in Japan to treat unresectable locally advanced or recurrent head and neck cancers based on results of two clinical trials [12,13].

For treatment sites that are bulky or not accessible to surface illumination, one or several CLDs are inserted directly into the tumor to improve the light delivery in its whole volume. This insertion-based modality is primarily used because it maximizes the illuminated volume and thus minimizes both the number of CLDs that need to be inserted, and the treatment time [14]. However, the success of this modality requires improved understanding of the light propagation within the tumor, together with a planned arrangement of these CLDs inside tumors. More precisely, key questions addressed for such planning consist to determine the optimal number, pattern and distances separating CLDs, for a given tumor geometry. Surprisingly, there is still a lack of general strategies in the literature regarding optimal interstitial positioning of CLDs, in particular when they are inserted in parallel. It should be noted that this optimal positioning not only depends on the tissue optical properties and tumor geometry, but also on the photosensitizer potency. Herein, we have focused the study presented here on conditions corresponding to those used to treat advanced HNSCC by PIT with cetuximab–IR700 [13]. Concerning the threshold of fluence, we set a value based on the in vitro studies of PIT utilizing a conjugate with IR700 [9,15,16]. It was shown that the fluence that induces cell death varies among cell lines, ranging from 4 to 30 J/cm². In the present study, the severe condition of 30 J/cm² was applied because it is considered that cell death is more likely to occur efficiently under the upper limit conditions.

Therefore, the purpose of this article is to highlight the combinatory effects of using multiple CLDs positioned in parallel, and the consequences this combination has on CLD-CLD distances when trying to maximize the volume of treated tumors with a minimal number of CLDs. This computational study is, consequently, of interest to minimize the number of CLDs, inserted in parallel in the tumor, while maximizing the volume destroyed by interstitial PIT. The propagation of the light delivered by CLDs presenting a Lambertian emission was simulated by Monte-Carlo for optical coefficients, derived from the literature, corresponding to those of HNSCCs at 690 nm. It should be noted that the results of our study can be extrapolated to other solid cancers treated by PIT or PDT, providing that their optical properties are the same and that the necrosis threshold dose of 30 J/cm² also applies.

2. Materials and Methods

2.1. Cylindrical Light Diffusers

The dimensions of the CLDs simulated in this study corresponds to those used for the treatment of certain HNSCCs by interstitial PIT with cetuximab–IR700. The length of the emitting window is 40 mm, and the outer diameter is 1.47 mm. The surface of these CLDs has a uniform and Lambertian emittance, as it is the case for diffusers produced by Medlight SA [17,18]. The optical properties of CLDs are such that photons backscattered by the surrounding tissue are all transmitted through the CLDs without absorption or scattering (100% transmission). To assess the effects of the light emitting window length on the treatment plan, CLDs of length 20, 60, and 80 mm were also considered. In addition, the effects resulting from different angular emittance on the treatment plan were assessed simulating CLDs with uniform and normal to the surface emittances.

2.2. Tissue Model

The optical properties of biological tissues are usually described by four main parameters: the refractive index, the absorption and scattering coefficients, as well as the anisotropy factor ($n$, ${\mu }_a$, ${\mu }_s$ and g, respectively). Another useful parameter is the reduced scattering coefficient ${\mu }_s’$, which combines ${\mu }_s$ and g in such a way that ${\mu }_s’={\mu }_s(1-$g$)$. Most biological tissues show strong forward scattering in the red [19]. This is in agreement with the anisotropy factor of 0.9 for HNSCC reported by Holmer et al. [20] at 690 nm. Therefore, the anisotropy factor of the “tumor” was assumed to be 0.9 in our simulations. There are not many published values of the optical parameters for HNSCCs at 690 nm. In addition, these values present relatively important variations, as can be seen in Figure 1, which presents 15 values (3 obtained in the oral cavity and 12 in the esophagus) of ${\mu }_a$ and ${\mu }_s’$ pairs reported by Bargo et al. [21] and Holmer et al. [20]. Since Bargo et al. measured these optical coefficients at 630 nm, their values were corrected to 690 nm by extrapolations using the spectral dependence of the results reported by Holmer at al.

Our simulations have been performed for five different pairs of ${\mu }_a$ and ${\mu }_s’$ defined according to the following procedure: in a first step, isocurves of the first quartile, median, and third quartile values of ${\mu }_{eff}$ and $a’$, were drawn on the basis of the 15 values of the ${\mu }_a$ and ${\mu }_s’$ pairs, where ${\mu }_{eff}$ is the effective attenuation coefficient (${\mu }_{eff}$ $=\sqrt{3{\mu }_a \left({\mu }_a+{\mu }_s’\right)}$ and $a’$ is the transport albedo $a^{\prime }=\frac{{\mu }_s’}{{\mu }_s’+{\mu }_a}$ [22] (See Figure 1). Then, pair values of ${\mu }_a$ and ${\mu }_s’$ were selected by taking intersections of these isocurves corresponding to four extreme values of ${\mu }_a$ and ${\mu }_s’$ pairs (points 1, 2, 4 and 5 in Figure 1), as well as the intersection of the isocurves corresponding to the medians for ${\mu }_{eff}$and $a’$ (point 3 in Figure 1).

In these simulations, the refractive index $n$ of HNSCC was fixed at 1.34 since this is a typical value for many tissues in the red [23]. The optical coefficients used for our simulations are summarized in

Table 1. Optical properties selected to simulate the propagation of light in HNSCCs at 690 nm.

Simulation n°	${\mu }_{\mathit{a}} \left[{\mathbf{mm}}^{-1}\right]$	${\mu }_{\mathit{s}}’ \left[{\mathbf{mm}}^{-1}\right]$	${\mu }_{\mathit{e}\mathit{f}\mathit{f}} \left[{\mathbf{mm}}^{-1}\right]$	a’
1	0.052	0.64	0.33	0.925
2	0.071	0.44	0.33	0.861
3	0.078	0.70	0.42	0.899
4	0.082	1.00	0.52	0.925
5	0.111	0.69	0.52	0.861

. If needed, the light propagation simulations could easily be used for other optical coefficients.

2.3. Simulation and Optimizing Algorithm

One of the most useful approach to simulate the propagation of light in biological tissues is based on the use of Monte-Carlo (MC) methods that enable, among other, to trace the trajectory of individual photons [24]. They allow accurate descriptions of light propagation in complex geometries providing that the tissues dimensions are much larger than $1/{\mu }_t$, where ${\mu }_t={\mu }_a+{\mu }_s$. The “TracePro” program from Lambda Research Corporation [25] was used to simulate the propagation of light around cylindrical distributor(s) inserted in HNSCC treated by PIT. Each simulation was typically performed with 10⁶ photon packets with a 0.1 × 0.1 × 0.1 mm³ grid.

The tissue geometry we simulated consisted of a semi-infinite “tumor” in which CLDs were inserted at 90° in such a way that the proximal part of the light emitting cylindrical section was at the level of the air-tumor interface (See Figure 2). All cylindrical light distributors were oriented in the same direction and inserted at the same depth. The configurations in which the CLDs were positioned are shown in

Table 2. CLDs insertion geometries considered, viewed in a direction perpendicular to the air-tumor interface, for different number of CLDs. The black dots represent the CLDs.

Number of CLDs
One	Two		Three	Four	Five	Six	Seven
		Central CLD:
		Cyclical:

. The maximal number of cylindrical light distributors considered was equal to seven as a larger number was considered as unsuitable for the clinical use of PIT to treat ENT cancers. These configurations were classified into two categories: those containing a central CLD and those with cyclical geometries (regular polygons).

The “normalized fluence rates” presented in some of the following figures are given so that they can be easily extrapolated for various “linear emittances” (For cylindrical light distributors; Unit: W/cm). Consequently, since the fluence rate unit is W/cm², the normalized fluence rate unit is (W/cm²/W/cm = cm⁻¹). The threshold fluence corresponding to the tissue destruction by PIT was on the in vitro studies [9,15,16], as mentioned above. This fluence of 30 J/cm² is assumed to be the threshold when illumination conditions (Linear emittance: 0.4 W/cm at 690 nm during 250 s) are applied for PIT based on the use of CLDs in previous clinical studies [12]. Therefore, the value of the normalized fluence rate threshold is 0.3 W/cm²/W/cm = 0.3 cm⁻¹ (30 J/cm²/[0.4 W/cm × 250 s] = 0.3 W/cm²/W/cm = 0.3 cm⁻¹).

A typical normalized fluence rate distribution calculated by MC simulations, for a single CLD placed in a homogeneous medium, is shown in Figure 2 for reference. Additional distribution representations when using two, three, and four CLDs are presented in Appendix A.

A block diagram describing the optimizing algorithm is presented in Figure 3. In more details, the simulations were performed using TracePro for a single CLD for each set of optical coefficients, which were then duplicated and translated to form the various CLD configurations presented in Table 2 using MATLAB. Producing the insertion geometries of multiple CLDs this way greatly reduces the number of simulations needed, at the expense of neglecting the interaction between the light emitted from one CLD with the other CLDs. However, for the inter-CLD distances considered here, and due to the rapid decay of light intensity with distance from the source, this contribution is several orders of magnitude lower than the necrosis threshold value and negligibly affects the optimal inter-CLD distances calculated. The optimal distances between light distributors positioned according to different pattern was determined by maximizing the volume of the necrosed tumors (all voxels with normalized fluence rate higher than the 0.3 cm⁻¹ threshold). Total necrosed volume were calculated for CLD-CLD distances between 5 and 30 mm, increasing by increments of 0.1 mm, and the optimal distance was the one maximizing this volume. Each CLD delivered a total of 400 J to the sample (0.4 W/cm × 250 s × 40 mm). No abscopal (distant) phototoxic effects were modeled in this analysis. Each voxel in the TracePro simulation dataset grid corresponded with a voxel in the Matlab optimizing simulation space grid.

A point seldom mentioned when Monte-Carlo light path tracings are employed is how fluence rate, which is the power of the photons entering a sphere divided by the cross-sectional area of this sphere, can be obtained from a photon flux propagating through the cubes (voxels) of a mesh. If the angles of incidence of photons onto voxels are random, which is the case except close (<1/${\mu }_s’$) to the source, the fluence rate within each voxel can be obtained by dividing the incident flux (sum of rays entering the voxel) by the cube’s average cross-sectional area. Cauchy’s surface area formula states that for every convex body the average area of its parallel projections onto a plane is equal to a quarter of its surface [26]; so for a unit cube the expected cross-sectional area is equal to 3/2. Therefore, the fluence rate was calculated by dividing the power of the photons entering a voxel whose surface of a face is equal to unity by this factor 3/2.

3. Results

3.1. Validation of the MC Simulations: Comparison between the MC and Analytical Models

An analytical model based on the diffusion approximation was used to validate the MC simulations. The steady-state expression for the light propagation in a tissue is given by [27]:

$$\left(\mathsf{\Delta }-{\mu }^2{}_{eff}\right)\mathsf{\Phi }\left(\mathrm{r}\right)=-\mathrm{q}\left(\mathrm{r}\right)/\mathrm{D}$$

(1)

with $\mathsf{\Phi }\left(\mathrm{r}\right)$ the fluence rate (W⁄cm²), $\mathrm{D}={\mu }_a/{\mu }^2{}_{eff}$ the optical diffusivity (cm), and $q$ the source photon density (W⁄cm³).

In the case of an infinitely long CLD in a homogeneous volume, the solution of Equation (1) for the fluence rate [28] is given by:

$$\mathsf{\Phi }\left(\mathrm{r}\right)=\frac{\frac{\mathrm{P}}{2\mathsf{\pi }\mathrm{a}}}{\mathrm{D}\times {\mu }_{eff}\times {\mathrm{K}}_1\left({\mu }_{eff}\mathrm{a}\right)}{\mathrm{K}}_0\left({\mu }_{eff}\mathrm{r}\right)$$

(2)

where $\mathrm{P}$ is the optical power per unit length (W⁄cm) of the CLD of radius $\mathrm{a}$ (cm). ${\mathrm{K}}_0$ and ${\mathrm{K}}_1$ are the modified Bessel functions of the second kind for the zeroth and first order respectively. The radial distance from the CLD axis is expressed as r (cm).

The fluence rate derived from this analytical expression for an infinitely long CLD was compared to the simulated fluence rate as a function of “$\mathrm{r}$” at a depth equal to half the length of the 40 mm-long CLD. Since the middle of the CLD is at 20 mm from either tips of the CLD, which is greater than $3\times {\mu }_{eff}$⁻¹ for all optical properties simulated, edge effects at this depth are negligible and the radial fluence rate is expected to be similar to that of an infinitely long CLD [29].

Figure 4 shows the simulated (red curve) and analytically calculated (blue curve) fluence rates in the semi-infinite tumor. In the MC simulation, two different regions were observed. In the first region, close to the source and known as the build-up region, the fluence rate is large due to the scattering properties of the medium. A small difference can be observed between the simulated and analytically calculated normalized fluence rate close to the surface in this first region because the diffusion approximation is not valid close to boundaries. In the second region, the fluence rate can be approximated by an exponential function decreasing with depth. Deviations only started to appear at large radial distances (>25 mm) from the CLDs due to random noise which results from the limited number of photons used for the MC simulation. However, the normalized fluence rate at these distances are too low to affect the optimizing algorithm’s results. The excellent agreement between this simulation and the analytical model speaks in favor of a validation of the MC algorithm used for the simulations reported in this article.

3.2. Influence of the Tumor Optical Parameters on the CLD-CLD Distances Maximizing the Necrosed Volume

Figure 5 presents the computed optimal CLD-CLD distances, and the necrosed volume per CLD at these distances, using the sets of optical properties summarized in Table 1 for 40 mm-long CLDs. The number of CLDs and their positioning geometry were setup as described in Table 2.

Looking at Figure 5 indicates that, in our conditions, the CLD-CLD distances maximizing the necrosed volume range between 10 and 17 mm, leading to necrosed volumes ranging between 2.5 and about 7.5 cm³ per CLD, depending on the CLD number and tumor optical properties. As presented in Figure 5b, for all optical properties simulated, the optimized configuration which increases the necrosed volume per CLD the most is the seven-CLD hexagon with a central CLD, with an average increase of 31% +/− 4% in necrosed volume per CLD, compared to using a single CLD. It is of interest to note that the amount of necrosed volume per CLD increases monotonically with the number of CLD for geometries with a central CLD and for all sets of optical properties simulated (see Figure 5b). The situation is different for the cyclical (no central CLD) geometry where the necrosed volume per CLD reaches a plateau with more than four CLDs, which corresponds to an average increase of about 22% in necrosed volume per CLD, compared to using a single CLD. This volume even slowly decreases for higher numbers of CLDs as an untreated volume appears in the center of the geometry, since the CLDs opposite from each other are too distant. Finally, it is interesting to note that geometries with a central CLD consistently have a lower necrosed volume per CLD than cyclical configurations when using three, four and five CLDs, but are higher for six and seven CLDs.

For simulations using tissues with the same ${\mu }_{eff}$ (simulation 1 & 2, and 4 & 5), the one with a larger transport albedo yielded a larger necrosed volume per CLD, and consequently required CLDs to be positioned further apart. For the optical properties simulated, the differences in fluence rate intensity due to changes of the transport albedo were as significant as changing ${\mu }_{eff}$, indicating that knowledge of ${\mu }_{eff}$ alone is not sufficient to estimate optimal CLD placement. This is because in the region of “diffuse” photons, illumination intensity (Equation (2)) becomes inversely proportional to the optical diffusivity D, which for a given ${\mu }_{eff}$, decreases as the transport albedo increases.

The CLD geometries that lead to the largest distance between CLDs are the 3-CLD cyclical (equilateral triangle) and 7-CLD (hexagon with central CLD) geometries. These geometries lead to increases of the CLD-CLD distances maximizing the necrosed volume compared to the distance resulting from the use of two CLDs by 9% and 11%, respectively. If we discount these two geometries, these distances for the remaining geometries are all within 4% of the value for two CLDs. For the simulations performed with the set of optical parameters n°3, the average optimal distance calculated for these remaining geometries is 11.8 +/− 0.2 mm.

Since the distances maximizing the necrosed volume is the same between two CLDs and three CLDs placed in the same plan, it is safe to assume that this distance will hold for larger numbers of CLDs positioned in a same plan.

One question of interest consists to determine if the results presented in Figure 5 depend on the value of the anisotropy factor g, which was fixed at 0.9 in all simulations. This question was addressed by performing simulations of the light propagation around a 40 mm-long CLD when ${\mu }_s’={\mu }_s(1-$g$)$ was held constant while the scattering coefficient ${\mu }_s’$ and g were varied. g was set as 0.5, 0.6, 0.75, 0.9, and 0.95, while ${\mu }_s$ was equal to 1.4, 1.75, 2.8, 7, and 14 mm⁻¹, respectively, such that ${\mu }_s’$ = 0.7 mm⁻¹. The absorption coefficient was held constant at 0.078 mm⁻¹ and the illumination setup remained the same as described above. When a 40 mm long CLD was simulated, there were no measurable difference in the CLDs optimal distances and corresponding necrosed volumes when using a strong forward scattering anisotropy g value of 0.95 and a low value of 0.5, while maintaining ${\mu }_s’$ constant (0.7 mm⁻¹). This is in agreement with simulations reported by Streeter et al. [29], who considered a “narrow” collimated beam illuminating an air-tissue interface. This group observed that the fluence rate “far” (i.e., more than ${\mu }_s’$⁻¹) from the interface is constant when ${\mu }_s$ and g are changed in such a way that ${\mu }_s’$ was held constant. This is also in agreement with the studies performed by Tromberg at al. [28] who reported that, for volume elements that are at depth and/or radial distance greater than 1/(${\mu }_s’$), the photon trajectories are randomized, and photons are effectively diffusing outward from the beam along a concentration gradient, so the photons can be called “diffuse”. In other words, in this region, if there are sufficient scattering events before an absorption event occurs, then the average photon propagation after many scattering events becomes dependent on ${\mu }_s(1-$g$)$ rather than on the specific values of ${\mu }_s$ and g. Therefore, in our conditions and in most applications of interstitial PIT, the distance at which the light flux becomes diffuse is at least an order of magnitude smaller than the distance between CLDs. Consequently, the tissue optical coefficients that affect the extent of necrosis and subsequently the optimal distance between CLDs are only ${\mu }_a$ and ${\mu }_s’$ in our conditions.

3.3. Influence of CLDs Properties on the CLD-CLD Distances Maximizing the Necrosed Volume

Another question of interest consists to determine if the results presented in Figure 5 depend on the CLDs length and the angular distribution of their emittances. Figure 6 illustrates how different CLD lengths affect their optimal distances as well as the resulting necrosed volume per centimeter of CLD when ${\mu }_a$ = 0.078 and ${\mu }_s’$ = 0.7 (simulation n°3). The “linear emittance” was kept constant at 0.4 W/cm and the proximal part of the CLD light emitting section was again at the air-tumor interface. For clarity, only geometries with a central CLD are presented.

Looking at Figure 6 we see that, for simulation n°3, the CLD-CLD distances optimizing the necrosed volume range between 11 and 13 mm, leading to necrosed volume per centimeter of CLD ranging between 0.88 and 1.24 cm³, depending on the number of CLDs and their length. For all number of CLDs, the optimal CLD distance increases with CLD length, with the biggest increase being between CLDs of 20- and 40-mm length, until reaching a plateau at 60 mm. The necrosed volume per centimeter of CLD also increases with CLD length. However, the majority of this difference in necrosed volume can be attributed to a relatively higher percentage of emitted light escaping the medium through the air/tissue interface for short CLDs, even though the total amount of escaping light remains constant (data not shown). The changes in optimal CLD distance are therefore more due to the differences in shape of the necrosed volume as edge effects become more significant the shorter the CLD is. As the outline of the necrosed volume becomes increasingly spherical (as opposed to cylindrical) when the CLD length is reduced, it is more optimal to place CLDs closer to each other.

Comparisons of the normalized fluence rate distributions around 40 mm-long CLDs with Lambertian emission vs normal to surface emission for optical properties ${\mu }_a$ = 0.078 and ${\mu }_s’$ = 0.7 (simulation n°3) are shown in Figure 7. Although light distribution patterns are very similar, simulations using a normal to surface emission showed a slightly greater light penetration in the tumor as more photons are emitted in the direction perpendicular to the CLD surface. This effect is more marked where edge effects are present (at the tip of the CLD and the air/tissue interface).

As presented in Figure 8, CLDs with Lambertian emittance consistently yielded only 2% shorter optimal CLD distances and 3% smaller necrosed volume per CLD versus using normal to surface angular emittance. Therefore, the angular distribution of light emitted from the surface of the CLD (as long as the illumination is cylindrically symmetrical) does not significantly influence the positioning of CLDs and leads to very similar volumes necrosed by PIT.

4. Discussion

The treatment planning scheme presented in this paper assumes that the tumor response is determined entirely by the photosensitizer concentration and fluence, as has been done in past studies [14,30]. Since the photosensitizer concentration is assumed to be homogeneous, the fluence is the only factor that is responsible for different outcomes from place to place. However, in virtually all real situations, the tumor necrosis depends on additional factors, including: the fluence rate, the macroscopic and microscopic photosensitizer localizations, the tumor oxygenation and the immune response [31]. In such situations, the choice of treatment plan (laser power and treatment time) has consequences on the outcome. Therefore, although the results presented above deserve to be validated with in vivo experiments, the general conclusions are certainly valid since the fluence is the most important parameters in PIT and PDT.

To evaluate the necrosis volume, we set the fluence threshold based on the in vitro studies of PIT using the conjugates with IR700. We applied the upper limit value of 30 J/cm² as the threshold by assuming that cell death would occur more efficiently under such severe conditions. Nevertheless, in the clinical setting, the background of the tissue and the biology of the tumor are heterogeneous, and it is not clear whether the threshold based on the in vitro studies can be directly applied without modification when safety is considered.

When using four or more CLDs, treatment plans using geometries which don’t contain a central CLD exhibit a region of sub-treated tissue in the center (see Figure 9), which is mostly surrounded by tissue receiving sufficient dose to cause necrosis. The damage produced to the surrounding tissue vasculature may be sufficient to occlude the blood vessels that feed these enveloped tumor cells, contributing to massive ischemia and/or vascular infarction. These effects, combined with inflammatory reactions taking place in the surrounding treated tissues, are likely to induce subsequent necrosis of this region. Although this form of indirect tumor-cell killing might be beneficial by synergistically enhancing PIT and PDT, this effect is beyond the scope of this work and is not considered in the present model, mostly due to its complexity and its limited impact on the general conclusions resulting from this study. Rather, our mathematical modelling of optimal CLD positioning is restricted to a simplified analysis that focuses on determining the tissue volume receiving above-threshold light dose in an effort to describe irradiation conditions that maximize the volume of direct tumor-cell destruction for a minimal number of CLDs.

At first approximation, the variations in optimal CLD distance due to changes in CLD length, optical properties of the tissue, and choice of geometry, are independent, e.g., the relative difference in optimal CLD distance between using a 2 cm and 4 cm CLD is mostly independent of the other variables studied in this study, namely the optical properties of the tissue and the CLD number or geometry.

5. Conclusions

This study, based on the use of Monte-Carlo simulations to model the propagation of light at 690 nm in HNSCC, enabled us to determine the distances separating CLDs, inserted in parallel, to maximize the volume of tumor necrosed by PIT. Therefore, this simulation study represents an essential part in the development of PIT or PDT treatment plans. An illustrative result of our study is that, for optical properties most typical of HNSCC (corresponding to the intersection of median effective attenuation coefficient and transport albedo; simulation parameters n°3) and in the conditions used to treat HNSCC by PIT with RM technology within a fluence threshold of 30 J/cm² that is set based on in vitro study, this separation distance must be ~13 mm when using three CLD (positioned as equilateral triangle) or seven CLD (positioned as hexagon with central CLD), and 11.8 +/− 0.2 mm for the other CLD numbers and positioning considered in this study, inducing a necrosed volume of about 1 cm³ per CLD and per cm of CLD length.

The cumulative doses received in a tumor volume element from different CLDs explain the global slight increase of their optimal distances with the number of CLDs. This increase is much more pronounced for the volume of the necrosis per CLD.

The conclusions drawn in this study regarding the optimal distances separating CLDs are also valid for other treatments where the tissue necrosis only depends on the fluence. In particular, the general principle, procedures and steps enabling to determine these optimal distances can easily be adapted to other therapeutics, including PDT.

Among the parameters varied in this paper, the absorption and reduced scattering coefficients of the tumor, ${\mu }_a$ and ${\mu }_s’$, had the greatest effect on necrosed volume and optimal CLD distances, while tissue anisotropy factor g, angular dependence of the radiance emitted by the CLD, and CLD length, minimally affected their placements.

Using this data, by knowing the optical properties of the tumor, or the volume of necrosis obtained from treatment using a single CLD, we may estimate the optimal distance and number of CLDs to use for a tumor of known volume during interstitial PIT. Knowledge of the optimal distance between two CLDs may also be sufficient to determine the positioning of a greater number of CLDs as the relative difference in optimal distance between the various CLD insertion geometries is consistent for a wide range of CLD characteristics and tissue optical parameters.