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

Aurélien Gregor; Shohei Sase; Georges Wagnieres

doi:10.3390/photonics9090597

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

Gregor, Aurélien;Sase, Shohei;Wagnieres, Georges 2022-08-23 00:00:00 hv photonics Article Optimization of the Distance between Cylindrical Light Distributors Used for Interstitial Light Delivery in Biological Tissues 1 2 1 , Aurélien Gregor , Shohei Sase and Georges Wagnieres * Laboratory for Functional and Metabolic Imaging, Swiss Federal Institute of Technology (EPFL), Station 3, 1015 Lausanne, Switzerland Medical Science and Operations Department, Rakuten Medical K.K., 2-21-1 Tamagawa, Setagaya-ku, Tokyo 158-0094, Japan * Correspondence: georges.wagnieres@epﬂ.ch 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 parame- ters 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 Citation: Gregor, A.; Sase, S.; between 11.5- and 13-mm. Variations of the absorption and reduced scattering coefﬁcients have the Wagnieres, G. Optimization of the greatest inﬂuence on CLD placements, while tissue anisotropy factor, CLD insertion geometry, CLD Distance between Cylindrical Light length, and the angular dependence of the radiance emitted by the CLDs have minimal inﬂuence. Distributors Used for Interstitial At ﬁrst approximation the inﬂuence of these optical parameters on optimal CLD-CLD distance are Light Delivery in Biological Tissues. independent. Our data also suggests it is possible to derive new treatment plans using knowledge of Photonics 2022, 9, 597. https:// previous treatment plans. doi.org/10.3390/photonics9090597 Received: 5 July 2022 Keywords: interstitial photoimmunotherapy; photodynamic therapy; cylindrical light diffusers; head Accepted: 16 August 2022 and neck cancers; Monte Carlo; light dosimetry Published: 23 August 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional afﬁl- 1. Introduction iations. Cylindrical optical ﬁber-based light diffusers (CLD) are frequently used for inter- stitial photodynamic therapy (iPDT) and photoimmunotherapy (PIT) of various solid cancers [1–4]. Although PIT and iPDT differ from many respects, these two therapies are based, in most cases, on the intravenous administration of photosensitizers (PS) followed Copyright: © 2022 by the authors. by the delivery of light to the lesions with appropriate irradiances and doses at wavelengths Licensee MDPI, Basel, Switzerland. absorbed by the PS [5]. These non-thermal illuminations, which are usually performed in This article is an open access article the red or near-infrared part of the spectrum, lead to more or less selective cancer necrosis distributed under the terms and and/or apoptosis depending on the PS and lesion types. iPDT utilizes PS with tumor conditions of the Creative Commons afﬁnity as a drug, such as Photofrin, and reactive oxygen species produced upon photoirra- Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ diation are responsible for the tumor cell death. The fundamental difference between iPDT 4.0/). and PIT is related to the fact that the latter is based on the use of speciﬁc antibodies-PS Photonics 2022, 9, 597. https://doi.org/10.3390/photonics9090597 https://www.mdpi.com/journal/photonics Photonics 2022, 9, 597 2 of 18 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 squa- mous 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 Septem- ber 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 2 2 cell lines, ranging from 4 to 30 J/cm . In the present study, the severe condition of 30 J/cm 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/cm 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 Photonics 2022, 9, 597 3 of 18 Photonics 2022, 9, x FOR PEER REVIEW 3 of 19 the treatment plan, CLDs of length 20, 60, and 80 mm were also considered. In addition, Medlight SA [17,18]. The optical properties of CLDs are such that photons backscattered the effects resulting from different angular emittance on the treatment plan were assessed by the surrounding tissue are all transmitted through the CLDs without absorption or simulating CLDs with uniform and normal to the surface emittances. 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, 2.2. Tissue Model the effects resulting from different angular emittance on the treatment plan were assessed simulating CLDs with uniform and normal to the surface emittances. The optical properties of biological tissues are usually described by four main pa- rameters: the refractive index, the absorption and scattering coefﬁcients, as well as the 2.2. Tissue Model anisotropy factor (n, m , m and g, respectively). Another useful parameter is the reduced a s The optical properties of biological tissues are usually described by four main 0 0 scattering coefﬁcient m , which combines m and g in such a way that m = m (1g). Most parameters: the refractive index, the absorption and scattering coefficients, as well as the s s s s anisotropy factor (𝑛 , µ , µ and g, respectively). Another useful parameter is the reduced biological tissues show 𝑎 𝑠 strong forward scattering in the red [19]. This is in agreement with scattering coefficient µ ′, which combines µ and g in such a way that µ ′ = µ (1 −g). 𝑠 𝑠 𝑠 𝑠 the anisotropy factor of 0.9 for HNSCC reported by Holmer et al. [20] at 690 nm. Therefore, Most biological tissues show strong forward scattering in the red [19]. This is in agreement the anisotropy factor of the “tumor” was assumed to be 0.9 in our simulations. There are with the anisotropy factor of 0.9 for HNSCC reported by Holmer et al. [20] at 690 nm. not many published values of the optical parameters for HNSCCs at 690 nm. In addition, Therefore, the anisotropy factor of the “tumor” was assumed to be 0.9 in our simulations. Th these ere are values not many present publishr ed elatively values of timportant he optical parvariations, ameters for HNS as CC can s at be 690 seen nm. In in Figure 1, which addition, these values present relatively important variations, as can be seen in Figure 1, presents 15 values (3 obtained in the oral cavity and 12 in the esophagus) of m and m which presents 15 values (3 obtained in the oral cavity and 12 in the esophagus) of µ and pairs reported by Bargo et al. [21] and Holmer et al. [20]. Since Bargo et al. measured these µ ′ pairs reported by Bargo et al. [21] and Holmer et al. [20]. Since Bargo et al. measured optical coefﬁcients at 630 nm, their values were corrected to 690 nm by extrapolations using these optical coefficients at 630 nm, their values were corrected to 690 nm by the spectral dependence of the results reported by Holmer at al. extrapolations using the spectral dependence of the results reported by Holmer at al. Reviewed values 1.2 Esophagus eff a' Selected 1.0 4 values Oral Cavity Esophagus Esophagus 0.8 Oral Cavity Esophagus Esophagus Esophagus Oral Cavity Esophagus Esophagus 0.6 Esophagus Esophagus Esophagus 0.4 Esophagus 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 -1 µ [mm ] Figure 1. Values of µ and µ ′ derive 0 d from the literature for HNSCC. The green and blue lines 𝑎 𝑠 Figure 1. Values of m and m derived from the literature for HNSCC. The green and blue lines repre- represent isocurves of µ and 𝑎 ′, respectively. The long-dashed, continuous, and dotted lines 𝑒𝑓𝑓 sent isocurves of m and a , respectively. The long-dashed, continuous, and dotted lines represent e f f represent the isocurves of the first quartile, median, and third quartile values of µ and 𝑎 ′, 𝑒𝑓𝑓 respec the isocurve tively. Five s of different the ﬁrst pairs quartile, of optical median, coefficients and were thir sel d ecquartile ted for the values simulation of m s by taking and a , respectively. Five e f f intersections of isocurves corresponding to four extreme values of µ and µ ′ pairs (points 1, 2, 4 𝑎 𝑠 different pairs of optical coefﬁcients were selected for the simulations by taking intersections of and 5) as well as the intersection of the isocurves corresponding to the medians for µ and 𝑎 ′ 𝑒𝑓𝑓 isocurves corresponding to four extreme values of m and m pairs (points 1, 2, 4 and 5) as well as the (point 3). s intersection of the isocurves corresponding to the medians for m and a (point 3). e f f Our simulations have been performed for five different pairs of µ and µ ′ defined 𝑎 𝑠 according to the following procedure: in a first step, isocurves of the first quartile, median, 0 Our simulations have been performed for ﬁve different pairs of m and m deﬁned according to the following procedure: in a ﬁrst step, isocurves of the ﬁrst quartile, median, and third quartile values of m and a , were drawn on the basis of the 15 values of the m e f f and m pairs, where m is the effective attenuation coefﬁcient (m = 3m (m + m ) and a a s e f f e f f s 0 0 s 0 a is the transport albedo a = [22] (See Figure 1). Then, pair values of m and m were 0 a m +m s a selected by taking intersections of these isocurves corresponding to four extreme values of m and m pairs (points 1, 2, 4 and 5 in Figure 1), as well as the intersection of the isocurves corresponding to the medians for m and a (point 3 in Figure 1). e f f 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 -1 µ [mm ] s' Photonics 2022, 9, 597 4 of 18 simulations are summarized in Table 1. If needed, the light propagation simulations could easily be used for other optical coefﬁcients. Table 1. Optical properties selected to simulate the propagation of light in HNSCCs at 690 nm. 1 1 1 Simulation n [mm ] ’ [mm ] [mm ] a’ a s eff 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 2.3. Simulation and Optimizing Algorithm One of the most useful approach to simulate the propagation of light in biological tis- sues 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/m , where m = m + m . The “TracePro” program from Lambda Research Corporation [25] t a s was used to simulate the propagation of light around cylindrical distributor(s) inserted in Photonics 2022, 9, x FOR PEER REVIEW 5 of 19 HNSCC treated by PIT. Each simulation was typically performed with 10 photon packets with a 0.1 0.1 0.1 mm grid. The tissue geometry we simulated consisted of a semi-inﬁnite “tumor” in which CLDs distributors; Unit: W/cm). Consequently, since the fluence rate unit is W/cm , the 2 −1 normalized fluence rate unit is (W/cm /W/cm = cm ). The threshold fluence were inserted at 90 in such a way that the proximal part of the light emitting cylindrical corresponding to the tissue destruction by PIT was on the in vitro studies [9,15,16], as section was at the level of the air-tumor interface (See Figure 2). All cylindrical light mentioned above. This fluence of 30 J/cm is assumed to be the threshold when distributors were oriented in the same direction and inserted at the same depth. The illumination conditions (Linear emittance: 0.4 W/cm at 690 nm during 250 s) are applied conﬁgurations in which the CLDs were positioned are shown in Table 2. The maximal for PIT based on the use of CLDs in previous clinical studies [12]. Therefore, the value of 2 −1 2 the normalized fluence rate threshold is 0.3 W/cm /W/cm = 0.3 cm (30 J/cm /[0.4 W/cm × number of cylindrical light distributors considered was equal to seven as a larger number 2 −1 250 s] = 0.3 W/cm /W/cm = 0.3 cm ). was considered as unsuitable for the clinical use of PIT to treat ENT cancers. These A typical normalized fluence rate distribution calculated by MC simulations, for a conﬁgurations were classiﬁed into two categories: those containing a central CLD and single CLD placed in a homogeneous medium, is shown in Figure 2 for reference. those with cyclical geometries (regular polygons). Additional distribution representations when using two, three, and four CLDs are presented in Appendix A. Figure 2. Spatial distribution of the normalized fluence rate around a single 40 mm-long CLD Figure 2. Spatial distribution of the normalized ﬂuence rate around a single 40 mm-long CLD inserted −1 −1 inserted in a homogeneous medium with µ = 0.078 mm , µ ′ = 0.70 mm and g = 0.9. The 𝑎 𝑠 1 0 1 midsection is taken along the CLD axis. The outermost contour corresponds to the threshold fluence in a homogeneous medium with m = 0.078 mm , m = 0.70 mm and g = 0.9. The midsection 2 2 rate of 0.3 W/cm /W/cm, which corresponds to a light dose of 30 J/cm when illumination conditions is taken along the CLD axis. The outermost contour corresponds to the threshold ﬂuence rate of of 0.4 W/cm for 250 s are used. The air-tissue interface is situated at 0 mm along the vertical axis. 2 2 0.3 W/cm /W/cm, which corresponds to a light dose of 30 J/cm when illumination conditions of A block diagram describing the optimizing algorithm is presented in Figure 3. In 0.4 W/cm for 250 s are used. The air-tissue interface is situated at 0 mm along the vertical axis. 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 −1 the necrosed tumors (all voxels with normalized fluence rate higher than the 0.3 cm threshold). Total necrosed volume were calculated for CLD-CLD distances between 5 and Photonics 2022, 9, 597 5 of 18 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 “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/cm , the normal- 2 1 ized ﬂuence rate unit is (W/cm /W/cm = cm ). 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/cm 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 2 1 2 2 1 0.3 W/cm /W/cm = 0.3 cm (30 J/cm /[0.4 W/cm 250 s] = 0.3 W/cm /W/cm = 0.3 cm ). 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. Addi- tional 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 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/m ) 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. Photonics 2022, 9, x FOR PEER REVIEW 6 of 19 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. Photonics 2022, 9, 597 6 of 18 Each voxel in the TracePro simulation dataset grid corresponded with a voxel in the Matlab optimizing simulation space grid. Figure 3. Block diagram of optimizing algorithm. Figure 3. Block diagram of optimizing algorithm. 3. Results A point seldom mentioned when Monte-Carlo light path tracings are employed is 3.1. Validation of the MC Simulations: Comparison between the MC and Analytical Models how fluence rate, which is the power of the photons entering a sphere divided by the An analytical model based on the diffusion approximation was used to validate the cross-sectional area of this sphere, can be obtained from a photon flux propagating MC simulations. The steady-state expression for the light propagation in a tissue is given through the cubes (voxels) of a mesh. If the angles of incidence of photons onto voxels are by [27]: random, which is the case except close (<1/µ ′) to the source, the fluence rate within each D m F(r) = q(r)/D (1) voxel can be obtained by dividing the incident flux (sum of rays entering the voxel) by the e f f cube’s average cross-sectional area. Cauchy’s surface area formula states that for every 2 2 with F(r) the ﬂuence rate (W⁄cm ), D = m /m the optical diffusivity (cm), and q the e f f convex body the average area of its parallel projections onto a plane is equal to a quarter source photon density (W⁄cm ). of its surface [26]; so for a unit cube the expected cross-sectional area is equal to 3/2. In the case of an inﬁnitely long CLD in a homogeneous volume, the solution of Therefore, the fluence rate was calculated by dividing the power of the photons entering Equation (1) for the ﬂuence rate [28] is given by: a voxel whose surface of a face is equal to unity by this factor 3/2. 2a 3. Results F(r) = K m r (2) 0 e f f D m K m a e f f 1 e f f 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 where P is the optical power per unit length (W⁄cm) of the CLD of radius a (cm). K and MC simulations. The steady-state expression for the light propagation in a tissue is given K are the modiﬁed Bessel functions of the second kind for the zeroth and ﬁrst order by [27]: 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 (Δ − µ ) Φ(r) = −q(r)/D (1) 𝑓𝑓𝑒 was compared to the simulated ﬂuence rate as a function of “r” at a depth equal to half the with Φ(r) the fluence rate (W⁄cm ), D = µ /µ the optical diffusivity (cm), and 𝑞 the length of the 40 mm-long CLD. Since the middle of the CLD is at 20 mm from either tips of 𝑓𝑓𝑒 the source CLD, phot which on dens is greater ity (W⁄ than cm ). 3 m for all optical properties simulated, edge effects e f f at this depth are negligible and the radial ﬂuence rate is expected to be similar to that of an In the case of an infinitely long CLD in a homogeneous volume, the solution of inﬁnitely long CLD [29]. Equation (1) for the fluence rate [28] is given by: 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 2πa (2) Φ(r) = K (µ r) 0 𝑓𝑓𝑒 observed. In the ﬁrst region, close to the source and known as the build-up region, the D × µ × K (µ a) 𝑓𝑓𝑒 1 𝑓𝑓𝑒 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 Photonics 2022, 9, x FOR PEER REVIEW 7 of 19 where P is the optical power per unit length (W⁄cm) of the CLD of radius a (cm). K and K 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 “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 −1 tips of the CLD, which is greater than 3 × µ for all optical properties simulated, edge 𝑓𝑓𝑒 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 Photonics 2022, 9, 597 7 of 18 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 rate at these distances are too low to affect the optimizing algorithm’s results. The agreement between this simulation and the analytical model speaks in favor of a validation excellent agreement between this simulation and the analytical model speaks in favor of of the MC algorithm used for the simulations reported in this article. a validation of the MC algorithm used for the simulations reported in this article. Figure 4. Normalized fluence rate versus radial distance from the axis of the CLD at a depth of 20 Figure 4. Normalized ﬂuence rate versus radial distance from the axis of the CLD at a depth of 20 mm −1 mm around a 40 mm long CLD. Optical properties of the medium are: µ = 0.078 mm , µ = 0.7 𝑎 𝑠 ′ 1 1 around a 40 mm long CLD. Optical properties of the medium are: m = 0.078 mm , m = 0.7 mm , −1 a s mm , g = 0.9. g = 0.9. 3.2. Influence of the Tumor Optical Parameters on the CLD-CLD Distances Maximizing the 3.2. Inﬂuence of the Tumor Optical Parameters on the CLD-CLD Distances Maximizing the Necrosed Volume Necrosed Volume Figure 5 presents the computed optimal CLD-CLD distances, and the necrosed Figure 5 presents the computed optimal CLD-CLD distances, and the necrosed volume volume per CLD at these distances, using the sets of optical properties summarized in per CLD at these distances, using the sets of optical properties summarized in Table 1 for Table 1 for 40 mm-long CLDs. The number of CLDs and their positioning geometry were 40 mm-long CLDs. The number of CLDs and their positioning geometry were setup as Photonics 2022, 9, x FOR PEER REVIEW 8 of 19 setup as described in Table 2. described in Table 2. Figure 5. (a) CLD-CLD distances maximizing the tumor volume receiving a light dose larger than Figure 5. (a) CLD-CLD distances maximizing the tumor volume receiving a light dose larger than that that corresponding to the necrosis threshold. These distances refer to the shortest distances between corresponding to the necrosis threshold. These distances refer to the shortest distances between two two CLDs axis in each geometry. For geometries with a central CLD, this refers to the distance between the central CLD and the outer CLDs. (b) Volume of necrosed tissue per CLD for these CLDs axis in each geometry. For geometries with a central CLD, this refers to the distance between distances. The numbers and color code in the legend refer to the simulations numbers given in Table the central CLD and the outer CLDs. (b) Volume of necrosed tissue per CLD for these distances. The 1 for different optical parameters of the tumor. numbers and color code in the legend refer to the simulations numbers given in Table 1 for different optical parameters of the tumor. 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 cm 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 µ (simulation 1 & 2, and 4 & 5), the 𝑓𝑓𝑒 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 µ , indicating that knowledge of µ alone is not 𝑓𝑓𝑒 𝑓𝑓𝑒 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 µ , decreases as the transport albedo increases. 𝑓𝑓𝑒 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 Photonics 2022, 9, 597 8 of 18 Looking at Figure 5 indicates that, in our conditions, the CLD-CLD distances maxi- mizing the necrosed volume range between 10 and 17 mm, leading to necrosed volumes ranging between 2.5 and about 7.5 cm 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 m (simulation 1 & 2, and 4 & 5), the one e f f 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 signiﬁ- cant as changing m , indicating that knowledge of m alone is not sufﬁcient to estimate e f f e f f 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 m , decreases as the transport albedo increases. e f f 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 0 0 40 mm-long CLD when m = m (1g) was held constant while the scattering coefﬁcient m s s and g were varied. g was set as 0.5, 0.6, 0.75, 0.9, and 0.95, while m was equal to 1.4, 1.75, 1 0 1 2.8, 7, and 14 mm , respectively, such that m = 0.7 mm . The absorption coefﬁcient was held constant at 0.078 mm 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 m constant (0.7 mm ). This is in agreement with simulations reported by Streeter et al. [29], who considered a “narrow” collimated beam illuminating an air-tissue interface. This 0 1 group observed that the ﬂuence rate “far” (i.e., more than m ) from the interface is constant when m and g are changed in such a way that m 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/(m ), 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, Photonics 2022, 9, 597 9 of 18 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 m (1g) rather than on the speciﬁc values of m and g. Therefore, in our conditions and in s s 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 m and m 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 m = 0.078 and m = 0.7 (simulation n 3). The “linear emittance” was kept constant at 0.4 W/cm and the proximal part of the CLD light Photonics 2022, 9, x FOR PEER REVIEW 10 of 19 emitting section was again at the air-tumor interface. For clarity, only geometries with a central CLD are presented. Figure 6. (a) Optimal CLD distances maximizing the volume of tumor receiving more than the Figure 6. (a) Optimal CLD distances maximizing the volume of tumor receiving more than the necrosis threshold dose. (b) Volume of necrosed tumor per CLD and per centimeter of CLD for these necrosis threshold dose. (b) Volume of necrosed tumor per CLD and per centimeter of CLD for these optimal CLD distances. Simulations were performed with CLDs of length 20, 40, 60, and 80 mm. optimal CLD distances. Simulations were performed with CLDs of length 20, 40, 60, and 80 mm. Only geometries with a central CLD are presented. µ = 0.078 and µ ′ = 0.7 (simulation n°3). 𝒂 𝑠 Only geometries with a central CLD are presented. m = 0.078 and m = 0.7 (simulation n 3). Looking at Figure 6 we see that, for simulation n°3, the CLD-CLD distances op Looking timizing at th Figur e necrosed e 6 we vo see luthat, me rfor angsimu e betlation ween 11 n 3, anthe d 13 CLD-CLD mm, leadi distances ng to necrosed optimizing volume per centimeter of CLD ranging between 0.88 and 1.24 cm , depending on the the necrosed volume range between 11 and 13 mm, leading to necrosed volume per number of CLDs and their length. For all number of CLDs, the optimal CLD distance centimeter of CLD ranging between 0.88 and 1.24 cm , depending on the number of CLDs increases with CLD length, with the biggest increase being between CLDs of 20- and 40- and their length. For all number of CLDs, the optimal CLD distance increases with CLD mm length, until reaching a plateau at 60 mm. The necrosed volume per centimeter of length, with the biggest increase being between CLDs of 20- and 40-mm length, until CLD also increases with CLD length. However, the majority of this difference in necrosed reaching a plateau at 60 mm. The necrosed volume per centimeter of CLD also increases volume can be attributed to a relatively higher percentage of emitted light escaping the with CLD length. However, the majority of this difference in necrosed volume can be medium through the air/tissue interface for short CLDs, even though the total amount of attributed to a relatively higher percentage of emitted light escaping the medium through escaping light remains constant (data not shown). The changes in optimal CLD distance the air/tissue interface for short CLDs, even though the total amount of escaping light are therefore more due to the differences in shape of the necrosed volume as edge effects remains constant (data not shown). The changes in optimal CLD distance are therefore become more significant the shorter the CLD is. As the outline of the necrosed volume more due to the differences in shape of the necrosed volume as edge effects become becomes increasingly spherical (as opposed to cylindrical) when the CLD length is more signiﬁcant the shorter the CLD is. As the outline of the necrosed volume becomes reduced, it is more optimal to place CLDs closer to each other. Comparisons of the normalized fluence rate distributions around 40 mm-long CLDs increasingly spherical (as opposed to cylindrical) when the CLD length is reduced, it is with Lambertian emission vs normal to surface emission for optical properties µ = 0.078 more optimal to place CLDs closer to each other. 𝑎 and µ ′ = 0.7 (simulation n°3) are shown in Figure 7. Although light distribution patterns Comparisons of the normalized ﬂuence rate distributions around 40 mm-long CLDs are very similar, simulations using a normal to surface emission showed a slightly greater with Lambertian emission vs normal to surface emission for optical properties m = 0.078 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). Photonics 2022, 9, 597 10 of 18 and m = 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 Photonics 2022, 9, x FOR PEER REVIEW 11 of 19 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). Photonics 2022, 9, x FOR PEER REVIEW 11 of 19 Figure 7. Midsection slice of fluence rate distributions for a single 40 mm CLD when Lambertian Figure 7. Midsection slice of ﬂuence rate distributions for a single 40 mm CLD when Lamber- emittance (a) and normal to surface emittance (b) are used. Medium optical properties are: µ = tian emittance (a) and normal to surface emittance (b) are used. Medium optical properties −1 −1 0.078 mm , µ ′ = 0.70 mm and g = 0.9 (simulation n°3). The black curve corresponds to the Figure 7. Midsecti 𝑠 on slice of fluence rate distributions for a single 40 mm CLD when Lambertian 1 0 1 are: m = 0.078 mm , m = 0.70 mm and g = 0.9 (simulation n 3). The black curve corresponds to a 2 2 threshold fluence rate of 0.3 W/cm /W/cm, which corresponds to a light dose of 30 J/cm when emittance (a) and normal to surface emittance (b) are used. Medium optical properties are: µ = 2 2 −1 −1 the illum thr inati eshold on co ﬂuence nditionsra of te 0of .4 W/cm 0.3 W/cm for 250 /W/cm, s are used which . The corr air-ti esponds ssue interf to ace a light is situated dose of at 30 0 m J/cm m when 0.078 mm , µ ′ = 0.70 mm and g = 0.9 (simulation n°3). The black curve corresponds to the 2 2 along the vertical axis. The black bar represents the CLD. threshold fluence rate of 0.3 W/cm /W/cm, which corresponds to a light dose of 30 J/cm when illumination conditions of 0.4 W/cm for 250 s are used. The air-tissue interface is situated at 0 mm illumination conditions of 0.4 W/cm for 250 s are used. The air-tissue interface is situated at 0 mm along the vertical axis. The black bar represents the CLD. along the As vertical presented axis. T in he black bar Figure 8, CLDs repres weith nts L the CL ambert D.ia n emittance consistently yielded only 2% shorter optimal CLD distances and 3% smaller necrosed volume per CLD versus using As presented in Figure 8, CLDs with Lambertian emittance consistently yielded only As presented in Figure 8, CLDs with Lambertian emittance consistently yielded only normal to surface angular emittance. Therefore, the angular distribution of light emitted 2% shorter optimal CLD distances and 3% smaller necrosed volume per CLD versus using 2% shorter optimal CLD distances and 3% smaller necrosed volume per CLD versus using from the surface of the CLD (as long as the illumination is cylindrically symmetrical) does normal to surface angular emittance. Therefore, the angular distribution of light emitted normal to surface angular emittance. Therefore, the angular distribution of light emitted not significantly influence the positioning of CLDs and leads to very similar volumes from the surface of the CLD (as long as the illumination is cylindrically symmetrical) does from the surface of the CLD (as long as the illumination is cylindrically symmetrical) does necrosed by PIT. not signiﬁcantly inﬂuence the positioning of CLDs and leads to very similar volumes not significantly influence the positioning of CLDs and leads to very similar volumes necrosed by PIT. necrosed by PIT. Figure 8. (a) Optimal CLDs distances maximizing the volume of tumor receiving more than the necrosis threshold dose. (b) Volume of necrosed tumor per CLD and per centimeter of CLD for these optimal CLD distances. Simulations were performed with CLDs having Lambertian or normal to Figure 8. (a) Optimal CLDs distances maximizing the volume of tumor receiving more than the Figure 8. (a) Optimal CLDs distances maximizing the volume of tumor receiving more than the surface emittance. necrosis threshold dose. (b) Volume of necrosed tumor per CLD and per centimeter of CLD for these necrosis threshold dose. (b) Volume of necrosed tumor per CLD and per centimeter of CLD for these optimal CLD distances. Simulations were performed with CLDs having Lambertian or normal to optimal CLD distances. Simulations were performed with CLDs having Lambertian or normal to surface emittance. surface emittance. Photonics 2022, 9, 597 11 of 18 Photonics 2022, 9, x FOR PEER REVIEW 12 of 19 4. Discussion The treatment planning scheme presented in this paper assumes that the tumor re- 4. Discussion sponse is determined entirely by the photosensitizer concentration and ﬂuence, as has The treatment planning scheme presented in this paper assumes that the tumor been done in past studies [14,30]. Since the photosensitizer concentration is assumed to response is determined entirely by the photosensitizer concentration and fluence, as has be homogeneous, the ﬂuence is the only factor that is responsible for different outcomes been done in past studies [14,30]. Since the photosensitizer concentration is assumed to be from place to place. However, in virtually all real situations, the tumor necrosis depends homogeneous, the fluence is the only factor that is responsible for different outcomes from on additional factors, including: the ﬂuence rate, the macroscopic and microscopic pho- place to place. However, in virtually all real situations, the tumor necrosis depends on tosensitizer localizations, the tumor oxygenation and the immune response [31]. In such additional factors, including: the fluence rate, the macroscopic and microscopic situations, the choice of treatment plan (laser power and treatment time) has consequences photosensitizer localizations, the tumor oxygenation and the immune response [31]. In on the outcome. Therefore, although the results presented above deserve to be validated such situations, the choice of treatment plan (laser power and treatment time) has con with sequences in vivo on experiments, the outcome. the There general fore, alconclusions though the res ar ule tscertainly presented valid abovesince deserve the toﬂuence is be validated with in vivo experiments, the general conclusions are certainly valid since the most important parameters in PIT and PDT. 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 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 studies of PIT using the conjugates with IR700. We applied the upper limit value of 30 30 J/cm as the threshold by assuming that cell death would occur more efﬁciently under J/cm 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 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 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 based on the in vitro studies can be directly applied without modification when safety is is considered. considered. When using four or more CLDs, treatment plans using geometries which don’t contain When using four or more CLDs, treatment plans using geometries which don’t a central CLD exhibit a region of sub-treated tissue in the center (see Figure 9), which contain a central CLD exhibit a region of sub-treated tissue in the center (see Figure 9), is mostly surrounded by tissue receiving sufﬁcient dose to cause necrosis. The damage 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 produced to the surrounding tissue vasculature may be sufﬁcient to occlude the blood blood vessels that feed these enveloped tumor cells, contributing to massive ischemia vessels that feed these enveloped tumor cells, contributing to massive ischemia and/or and/or vascular infarction. These effects, combined with inflammatory reactions taking vascular infarction. These effects, combined with inﬂammatory reactions taking place in place in the surrounding treated tissues, are likely to induce subsequent necrosis of this the surrounding treated tissues, are likely to induce subsequent necrosis of this region. region. Although this form of indirect tumor-cell killing might be beneficial by Although this form of indirect tumor-cell killing might be beneﬁcial by synergistically synergistically enhancing PIT and PDT, this effect is beyond the scope of this work and is enhancing PIT and PDT, this effect is beyond the scope of this work and is not considered not considered in the present model, mostly due to its complexity and its limited impact in the present model, mostly due to its complexity and its limited impact on the general on the general conclusions resulting from this study. Rather, our mathematical modelling conclusions resulting from this study. Rather, our mathematical modelling of optimal CLD of optimal CLD positioning is restricted to a simplified analysis that focuses on positioning is restricted to a simpliﬁed analysis that focuses on determining the tissue determining the tissue volume receiving above-threshold light dose in an effort to volume receiving above-threshold light dose in an effort to describe irradiation conditions describe irradiation conditions that maximize the volume of direct tumor-cell destruction that maximize the volume of direct tumor-cell destruction for a minimal number of CLDs. for a minimal number of CLDs. Figure 9. Normalized fluence rate cross-section at half the insertion depth (20 mm) at 690 nm around Figure 9. Normalized ﬂuence rate cross-section at half the insertion depth (20 mm) at 690 nm around −1 six 40 mm long CLDs positioned as a hexagon. Tumor optical properties are: µ = 0.078 mm , µ ′ 𝑎 𝑠 six 40 mm −1 long CLDs positioned as a hexagon. Tumor optical properties are: m = 0.078 mm , = 0.70 mm and g = 0.9 (simulation n°3). The black curve corresponds to the threshold fluence rate 0 1 m = 0.70 mm and g = 0.9 (simulation n 3). The black curve corresponds to the threshold ﬂu- 2 2 ence rate of 0.3 W/cm /W/cm, which corresponds to a light dose of 30 J/cm when illumination conditions of 0.4 W/cm for 250 s are used. Photonics 2022, 9, 597 12 of 18 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/cm 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 cm 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, m and m , 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 necro- sis 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. Author Contributions: Conceptualization, A.G. and G.W.; methodology, A.G. and G.W.; software, A.G.; validation, A.G. and G.W.; formal analysis, A.G., G.W. and S.S.; investigation, A.G. and G.W.; resources, G.W.; data curation, A.G.; writing—original draft preparation, A.G.; writing—review and editing, A.G., G.W. and S.S.; visualization, A.G.; supervision, G.W.; project administration, G.W.; funding acquisition, G.W. All authors have read and agreed to the published version of the manuscript. Funding: This study was supported by Rakuten Medical K.K. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are available on request from the corresponding author. The data are not publicly available due to proprietary reasons. Acknowledgments: The authors are grateful to Toshiaki Suzuki (Rakuten Medical K.K.) for his valuable comments. Conﬂicts of Interest: The following author is an employee: S.S (Rakuten Medical Inc). A.G and G.W have nothing to disclose. Photonics 2022, 9, x FOR PEER REVIEW 14 of 19 Acknowledgments: The authors are grateful to Toshiaki Suzuki (Rakuten Medical K.K.) for his Photonics 2022, 9, x FOR PEER REVIEW 14 of 19 valuable comments. Photonics 2022, 9, 597 13 of 18 Conflicts of Interest: The following author is an employee: S.S (Rakuten Medical Inc). A.G and G.W have Acknowledgments: nothing to disc The lose auth . ors are grateful to Toshiaki Suzuki (Rakuten Medical K.K.) for his valuable comments. Appendix A Appendix A Conflicts of Interest: The following author is an employee: S.S (Rakuten Medical Inc). A.G and G.W have nothing to disclose. Appendix A Figure A1. Cross-section along a horizontal plane containing two sources. Normalized fluence rate Figure Figure A1 A1. . Cr Cr oss oss-section -section alonalong g a hora izontal plane horizontal co plane ntainicontaining ng two source two s. No sour rmali ces. zed Normalized fluence rate ﬂuence around two 40 mm long cylindrical light distributor at a depth of 20 mm calculated by MC around two 40 mm long cylindrical light distributor at a depth of 20 mm calculated by MC rate around two 40 mm long cylindrical light distributor at a depth of 20 mm calculated by MC simulations. The black circle corresponds to the extent of the necrosis when the fluence simulations. The black circle corresponds to the extent of the necrosis when the fluence simulations. The black circle corresponds to the extent of the necrosis when the ﬂuence recommended recommended by Rakuten is applied (30 J/cm²; 0.4 W/cm during 250 s), i.e., 0.3 W/cm²/W/cm. In recommended by Rakuten is applied (30 J/cm²; 0.4 W/cm during 250 s), i.e., 0.3 W/cm²/W/cm. In 2 2 by Rakuten is applied (30 J/cm ; 0.4 W/cm during 250 s), i.e., 0.3 W/cm /W/cm. In other words, other words, 0.3 W/cm²/W/cm × 0.4 W/cm × 250 s = 30 J/cm² (edge effects are neglected in this other words, 0.3 W/cm²/W/cm × 0.4 W/cm × 250 s = 30 J/cm² (edge effects are neglected in this 2 2 0.3 W/cm /W/cm 0.4 W/cm 250 s = 30 J/cm (edge effects are neglected in this calculation). calculation). calculation). Figure A2. Cross-section along a vertical plane containing two sources. The air-tissue interface is at Figure A2. Cross-section along a vertical plane containing two sources. The air-tissue interface is at 0 mm on the vertical axis. The CLDs are represented by the grey cylinders (a) 2-D representation of 0 mm on the vertical axis. The CLDs are represented by the grey cylinders (a) 2-D representation of the cross-section. The black line corresponds to the necrosis threshold: 0.3 W/cm²/W/cm2 . (b) 3-D the cross-section. The black line corresponds to the necrosis threshold: 0.3 W/cm /W/cm. (b) 3-D Figure A2. Cross-section along a vertical plane containing two sources. The air-tissue interface is at 0 mm on the vertical axis. The CLDs are represented by the grey cylinders (a) 2-D representation of representation of the cross-section along a vertical plane containing the two sources. The blue surface the cross-section. The black line corresponds to the necrosis threshold: 0.3 W/cm²/W/cm. (b) 3-D corresponds to the extent of necrosis. Photonics 2022, 9, x FOR PEER REVIEW 15 of 19 Photonics 2022, 9, x FOR PEER REVIEW 15 of 19 Photonics 2022, 9, 597 14 of 18 representation of the cross-section along a vertical plane containing the two sources. The blue representation of the cross-section along a vertical plane containing the two sources. The blue surface corresponds to the extent of necrosis. surface corresponds to the extent of necrosis. Figure A3. Cross-section along a vertical plane positioned in the middle of the two fibers. (a) 2-D Figure A3. Cross-section along a vertical plane positioned in the middle of the two ﬁbers. Figure A3. Cross-section along a vertical plane positioned in the middle of the two fibers. (a) 2-D representation of the cross-section. The black line corresponds to the necrosis threshold: 0.3 (a) 2-D representation of the cross-section. The black line corresponds to the necrosis threshold: representation of the cross-section. The black line corresponds to the necrosis threshold: 0.3 W/cm²/W/cm. (b) 3-D representation of the cross-section along a vertical plane positioned in the 0.3 W/cm /W/cm. (b) 3-D representation of the cross-section along a vertical plane positioned in W/cm²/W/cm. (b) 3-D representation of the cross-section along a vertical plane positioned in the middle of the two fibers. The blue surface corresponds to the extent of necrosis. the middle of the two ﬁbers. The blue surface corresponds to the extent of necrosis. middle of the two fibers. The blue surface corresponds to the extent of necrosis. Figure A4. Same conditions as in Figure A1, but for three sources in a plane. Figure A4. Same conditions as in Figure A1, but for three sources in a plane. Figure A4. Same conditions as in Figure A1, but for three sources in a plane. Photonics 2022, 9, x FOR PEER REVIEW 16 of 19 Photonics 2022, 9, 597 15 of 18 Figure A5. Cross-section along a vertical plane containing three sources placed in a line. The air- Figure A5. Cross-section along a vertical plane containing three sources placed in a line. The air- tissue interface is at 0 mm on the vertical axis. The CLDs are represented by the grey cylinders tissue interface is at 0 mm on the vertical axis. The CLDs are represented by the grey cylinders (a) (a) 2-D representation of the cross-section. The black line corresponds to the necrosis threshold: 2-D representation of the cross-section. The black line corresponds to the necrosis threshold: 0.3 0.3 W/cm /W/cm. (b) 3-D representation of the cross-section along a vertical plane containing the W/cm²/W/cm. (b) 3-D representation of the cross-section along a vertical plane containing the three three sources. The blue surface corresponds to the extent of necrosis. sources. The blue surface corresponds to the extent of necrosis. Figure A6. Same conditions as in Figure A1, but for three equidistant sources. Figure A6. Same conditions as in Figure A1, but for three equidistant sources. Photonics 2022, 9, 597 16 of 18 Figure A7. Cross-section along a vertical plane containing one source and the middle of the other two sources. The air-tissue interface is at 0 mm on the vertical axis. The CLDs are represented by the grey cylinders (a) 2-D representation of the cross-section. The black line corresponds to the necrosis threshold: 0.3 W/cm /W/cm. (b) 3-D representation of the cross-section along a vertical plane containing the two sources. The blue surface corresponds to the extent of necrosis. Figure A8. (a) Same conditions as in Figure A1, but for four sources positioned according to a triangle plus one central source. The black line corresponds to the necrosis threshold: 0.3 W/cm /W/cm. (b) 3-D representation of the cross-section along a vertical plane containing the central source plus another one. The air-tissue interface is at 0 mm on the vertical axis. The blue surface corresponds to the extent of necrosis. The CLDs are represented by the grey cylinders. Photonics 2022, 9, 597 17 of 18 Figure A9. (a) Same conditions as in Figure A1, but for four sources positioned according to a square. The black line corresponds to the necrosis threshold: 0.3 W/cm /W/cm. (b) 3-D representation of the cross-section along a vertical plane placed in the middle of two pairs of neighboring sources. The air-tissue interface is at 0 mm on the vertical axis. The blue surface corresponds to the extent of necrosis. References 1. Kim, M.M.; Darafsheh, A. Light Sources and Dosimetry Techniques for Photodynamic Therapy. Photochem. Photobiol. 2020, 96, 280–294. [CrossRef] [PubMed] 2. Shaﬁrstein, G.; Bellnier, D.; Oakley, E.; Hamilton, S.; Potasek, M.; Beeson, K.; Parilov, E. Interstitial Photodynamic Therapy-A Focused Review. Cancers 2017, 9, 12. [CrossRef] [PubMed] 3. Okuyama, S.; Nagaya, T.; Sato, K.; Ogata, F.; Maruoka, Y.; Choyke, P.L.; Kobayashi, H. Interstitial Near-Infrared Photoimmunother- apy: Effective Treatment Areas and Light Doses Needed for Use with Fiber Optic Diffusers. Oncotarget 2018, 9, 11159–11169. [CrossRef] [PubMed] 4. Nagaya, T.; Okuyama, S.; Ogata, F.; Maruoka, Y.; Choyke, P.L.; Kobayashi, H. Near Infrared Photoimmunotherapy Using a Fiber Optic Diffuser for Treating Peritoneal Gastric Cancer Dissemination. Gastric Cancer 2019, 22, 463–472. [CrossRef] 5. Civantos, F.J.; Karakullukcu, B.; Biel, M.; Silver, C.E.; Rinaldo, A.; Saba, N.F.; Takes, R.P.; Vander Poorten, V.; Ferlito, A. A Review of Photodynamic Therapy for Neoplasms of the Head and Neck. Adv. Ther. 2018, 35, 324–340. [CrossRef] 6. van Dongen, G.A.M.S.; Visser, G.W.M.; Vrouenraets, M.B. Photosensitizer-Antibody Conjugates for Detection and Therapy of Cancer. Adv. Drug Deliv. Rev. 2004, 56, 31–52. [CrossRef] [PubMed] 7. Shirasu, N.; Nam, S.O.; Kuroki, M. Tumor-Targeted Photodynamic Therapy. Anticancer. Res. 2013, 33, 2823–2831. 8. Sato, K.; Ando, K.; Okuyama, S.; Moriguchi, S.; Ogura, T.; Totoki, S.; Hanaoka, H.; Nagaya, T.; Kokawa, R.; Takakura, H.; et al. Photoinduced Ligand Release from a Silicon Phthalocyanine Dye Conjugated with Monoclonal Antibodies: A Mechanism of Cancer Cell Cytotoxicity after Near-Infrared Photoimmunotherapy. ACS Cent. Sci. 2018, 4, 1559–1569. [CrossRef] 9. Mitsunaga, M.; Ogawa, M.; Kosaka, N.; Rosenblum, L.T.; Choyke, P.L.; Kobayashi, H. Cancer Cell–Selective in Vivo near Infrared Photoimmunotherapy Targeting Speciﬁc Membrane Molecules. Nat. Med. 2011, 17, 1685–1691. [CrossRef] 10. Rakuten Medical, Inc. A Phase 1/2a Multicenter, Open-Label, Dose-Escalation, Combination Study of RM-1929 and Photoim- munotherapy in Patients With Recurrent Head and Neck Cancer, Who in the Opinion of Their Physician, Cannot Be Satisfactorily Treated With Surgery, Radiation or Platinum Chemotherapy. Available online: Clinicaltrials.gov (accessed on 22 August 2022). 11. Rakuten Medical, Inc. A Phase 3, Randomized, Double-Arm, Open-Label, Controlled Trial of ASP-1929 Photoimmunotherapy Versus Physician’s Choice Standard of Care for the Treatment of Locoregional, Recurrent Head and Neck Squamous Cell Carcinoma in Patients Who Have Failed or Progressed On or After at Least Two Lines of Therapy, of Which at Least One Line Must Be Systemic Therapy. Available online: Clinicaltrials.gov (accessed on 22 August 2022). 12. Tahara, M.; Okano, S.; Enokida, T.; Ueda, Y.; Fujisawa, T.; Shinozaki, T.; Tomioka, T.; Okano, W.; Biel, M.A.; Ishida, K.; et al. A Phase I, Single-Center, Open-Label Study of RM-1929 Photoimmunotherapy in Japanese Patients with Recurrent Head and Neck Squamous Cell Carcinoma. Int. J. Clin. Oncol. 2021, 26, 1812–1821. [CrossRef] 13. Cognetti, D.M.; Johnson, J.M.; Curry, J.M.; Kochuparambil, S.T.; McDonald, D.; Mott, F.; Fidler, M.J.; Stenson, K.; Vasan, N.R.; Razaq, M.A.; et al. Phase 1/2a, Open-Label, Multicenter Study of RM-1929 Photoimmunotherapy in Patients with Locoregional, Recurrent Head and Neck Squamous Cell Carcinoma. Head Neck 2021, 43, 3875–3887. [CrossRef] [PubMed] Photonics 2022, 9, 597 18 of 18 14. Baran, T.M.; Foster, T.H. Comparison of Flat Cleaved and Cylindrical Diffusing Fibers as Treatment Sources for Interstitial Photodynamic Therapy. Med. Phys. 2014, 41, 022701. [CrossRef] [PubMed] 15. Nagaya, T.; Friedman, J.; Maruoka, Y.; Ogata, F.; Okuyama, S.; Clavijo, P.E.; Choyke, P.L.; Allen, C.; Kobayashi, H. Host Immunity Following Near-Infrared Photoimmunotherapy Is Enhanced with PD-1 Checkpoint Blockade to Eradicate Established Antigenic Tumors. Cancer Immunol. Res. 2019, 7, 401–413. [CrossRef] [PubMed] 16. Mitsunaga, M.; Nakajima, T.; Sano, K.; Kramer-Marek, G.; Choyke, P.L.; Kobayashi, H. Immediate in Vivo Target-Speciﬁc Cancer Cell Death after near Infrared Photoimmunotherapy. BMC Cancer 2012, 12, 345. [CrossRef] [PubMed] 17. Pitzschke, A.; Bertholet, J.; Lovisa, B.; Zellweger, M.; Wagnières, G. Determination of the Radiance of Cylindrical Light Diffusers: Design of a One-Axis Charge-Coupled Device Camera-Based Goniometer Setup. JBO 2017, 22, 035004. [CrossRef] 18. Zellweger, M.; Xiao, Y.; Jain, M.; Giraud, M.-N.; Pitzschke, A.; de Kalbermatten, M.; Berger, E.; van den Bergh, H.; Cook, S.; Wagnières, G. Optical Characterization of an Intra-Arterial Light and Drug Delivery System for Photodynamic Therapy of Atherosclerotic Plaque. Appl. Sci. 2020, 10, 4304. [CrossRef] 19. Cheong, W.F.; Prahl, S.A.; Welch, A.J. A Review of the Optical Properties of Biological Tissues. IEEE J. Quantum Electron. 1990, 26, 2166–2185. [CrossRef] 20. Holmer, C.; Lehmann, K.-S.; Wanken, J.; Reissfelder, C.; Roggan, A.; Gerhard, J.M.; Buhr, H.; Ritz, J. Optical Properties of Adenocarcinoma and Squamous Cell Carcinoma of the Gastroesophageal Junction. J. Biomed. Opt. 2007, 12, 1–8. [CrossRef] 21. Bargo, P.R.; Prahl, S.A.; Goodell, T.T.; Sleven, R.A.; Koval, G.; Blair, G.; Jacques, S.L. In Vivo Determination of Optical Properties of Normal and Tumor Tissue with White Light Reﬂectance and an Empirical Light Transport Model during Endoscopy. J. Biomed. Opt. 2005, 10, 034018. [CrossRef] 22. Niemz, M.H. Laser-Tissue Interactions; Springer: Berlin/Heidelberg, Germany, 2007; ISBN 978-3-540-72191-8. 23. Tuchin, V.V. Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis; Society of Photo-Optical Instrumentation Engineers (SPIE) Press Books: Bellingham, WA, USA, 2015; ISBN 978-1-62841-516-2. 24. Jacques, S.L.; Wang, L. Monte Carlo Modeling of Light Transport in Tissues. In Optical-Thermal Response of Laser-Irradiated Tissue; Welch, A.J., Van Gemert, M.J.C., Eds.; Lasers Photonics and Electro-Optics; Springer: Boston, MA, USA, 1995; pp. 73–100. ISBN 978-1-4757-6092-7. 25. Tracepro, Illumination and Non-Imaging Optical Design & Analysis Tool. Available online: Lambdares.com (accessed on 22 August 2022). 26. Tsukerman, E.; Veomett, E. A Simple Proof of Cauchy’s Surface Area Formula. arXiv 2016, arXiv:1604.05815. 27. Ishimaru, A. Wave Propagation and Scattering in Random Media. Volume 1—Single Scattering and Transport Theory; Academic Press: Cambridge, MA, USA, 1978; Volume 1. 28. Tromberg, B.J.; Svaasand, L.O.; Fehr, M.K.; Madsen, S.J.; Wyss, P.; Sansone, B.; Tadir, Y. A Mathematical Model for Light Dosimetry in Photodynamic Destruction of Human Endometrium. Phys. Med. Biol. 1996, 41, 223–237. [CrossRef] [PubMed] 29. Streeter, S.S.; Jacques, S.L.; Pogue, B.W. Perspective on Diffuse Light in Tissue: Subsampling Photon Populations. JBO 2021, 26, 070601. [CrossRef] [PubMed] 30. Ismael, F.S.; Amasha, H.; Bachir, W. Optimized Cylindrical Diffuser Powers for Interstitial PDT Breast Cancer Treatment Planning: A Simulation Study. BioMed Res. Int. 2020, 2020, e2061509. [CrossRef] [PubMed] 31. Robinson, D.J.; de Bruijn, H.S.; van der Veen, N.; Stringer, M.R.; Brown, S.B.; Star, W.M. Fluorescence Photobleaching of ALA- Induced Protoporphyrin IX during Photodynamic Therapy of Normal Hairless Mouse Skin: The Effect of Light Dose and Irradiance and the Resulting Biological Effect. Photochem. Photobiol. 1998, 67, 140–149. [CrossRef] [PubMed] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Photonics Multidisciplinary Digital Publishing Institute http://www.deepdyve.com/lp/multidisciplinary-digital-publishing-institute/optimization-of-the-distance-between-cylindrical-light-distributors-f2oJzXP82R

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References (22)

Kazuhide Sato, Kanta Ando, Shuhei Okuyama, S. Moriguchi, Tairo Ogura, S. Totoki, H. Hanaoka, T. Nagaya, Ryohei Kokawa, Hideo Takakura, M. Nishimura, Y. Hasegawa, P. Choyke, M. Ogawa, Hisataka Kobayashi (2018)
Photoinduced Ligand Release from a Silicon Phthalocyanine Dye Conjugated with Monoclonal Antibodies: A Mechanism of Cancer Cell Cytotoxicity after Near-Infrared Photoimmunotherapy
ACS Central Science, 4
E. Tsukerman, Ellen Veomett (2016)
A Simple Proof of Cauchy's Surface Area Formula
arXiv: Differential Geometry
Shuhei Okuyama, T. Nagaya, Kazuhide Sato, F. Ogata, Yasuhiro Maruoka, P. Choyke, Hisataka Kobayashi (2018)
Interstitial near-infrared photoimmunotherapy: effective treatment areas and light doses needed for use with fiber optic diffusers
Oncotarget, 9
P. Bargo, S. Prahl, T. Goodell, R. Sleven, G. Koval, G. Blair, Steven Jacques (2005)
In vivo determination of optical properties of normal and tumor tissue with white light reflectance and an empirical light transport model during endoscopy.
Journal of biomedical optics, 10 3
T. Baran, T. Foster (2014)
Comparison of flat cleaved and cylindrical diffusing fibers as treatment sources for interstitial photodynamic therapy.
Medical physics, 41 2
F. Civantos, B. Karakullukcu, M. Biel, C. Silver, A. Rinaldo, N. Saba, R. Takes, V. Poorten, A. Ferlito (2018)
A Review of Photodynamic Therapy for Neoplasms of the Head and Neck
Advances in Therapy, 35
G. Shafirstein, D. Bellnier, Emily Oakley, Sasheen Hamilton, M. Potasek, K. Beeson, E. Parilov (2017)
Interstitial Photodynamic Therapy—A Focused Review
Cancers, 9
N. Shirasu, S. Nam, M. Kuroki (2013)
Tumor-targeted photodynamic therapy.
Anticancer research, 33 7
F. Ismael, H. Amasha, W. Bachir (2020)
Optimized Cylindrical Diffuser Powers for Interstitial PDT Breast Cancer Treatment Planning: A Simulation Study
BioMed Research International, 2020
G. Dongen, Gerard Visser, M. Vrouenraets (2004)
Photosensitizer-antibody conjugates for detection and therapy of cancer.
Advanced drug delivery reviews, 56 1
W. Cheong, S. Prahl, A. Welch (1990)
A review of the optical properties of biological tissues
IEEE Journal of Quantum Electronics, 26
C. Holmer, K. Lehmann, Jana Wanken, C. Reissfelder, A. Roggan, G. Mueller, H. Buhr, J. Ritz (2007)
Optical properties of adenocarcinoma and squamous cell carcinoma of the gastroesophageal junction.
Journal of biomedical optics, 12 1
D. Cognetti, J. Johnson, J. Curry, S. Kochuparambil, D. McDonald, F. Mott, M. Fidler, K. Stenson, N. Vasan, M. Razaq, J. Campaña, P. Ha, G. Mann, Kosuke Ishida, M. García-Guzmán, M. Biel, A. Gillenwater (2021)
Phase 1/2a, open‐label, multicenter study of RM‐1929 photoimmunotherapy in patients with locoregional, recurrent head and neck squamous cell carcinoma
Head & Neck, 43
M. Mitsunaga, T. Nakajima, K. Sano, G. Kramer-Marek, P. Choyke, Hisataka Kobayashi (2012)
Immediate in vivo target-specific cancer cell death after near infrared photoimmunotherapy
BMC Cancer, 12
M. Tahara, S. Okano, T. Enokida, Yuri Ueda, T. Fujisawa, T. Shinozaki, Toshifumi Tomioka, W. Okano, M. Biel, Kosuke Ishida, R. Hayashi (2021)
A phase I, single-center, open-label study of RM-1929 photoimmunotherapy in Japanese patients with recurrent head and neck squamous cell carcinoma
International Journal of Clinical Oncology, 26
Michele Kim, A. Darafsheh (2020)
Light Sources and Dosimetry Techniques for Photodynamic Therapy
Photochemistry and Photobiology, 96
M. Mitsunaga, M. Ogawa, N. Kosaka, L. Rosenblum, P. Choyke, Hisataka Kobayashi (2011)
Cancer Cell-Selective In Vivo Near Infrared Photoimmunotherapy Targeting Specific Membrane Molecules
Nature medicine, 17
B. Tromberg, L. Svaasand, M. Fehr, S. Madsen, P. Wyss, B. Sansone, Y. Tadir (1996)
A mathematical model for light dosimetry in photodynamic destruction of human endometrium.
Physics in medicine and biology, 41 2
S. Jacques, Lihong Wang (1995)
Monte Carlo Modeling of Light Transport in Tissues
T. Nagaya, Shuhei Okuyama, F. Ogata, Yasuhiro Maruoka, P. Choyke, Hisataka Kobayashi (2018)
Near infrared photoimmunotherapy using a fiber optic diffuser for treating peritoneal gastric cancer dissemination
Gastric Cancer, 22
D. Robinson, H. Bruijn, N. Veen, M. Stringer, S. Brown, W. Star (1998)
Fluorescence Photobleaching of ALA‐induced Protoporphyrin IX during Photodynamic Therapy of Normal Hairless Mouse Skin: The Effect of Light Dose and Irradiance and the Resulting Biological Effect
Photochemistry and Photobiology, 67
M. Zellweger, Ying Xiao, M. Jain, M. Giraud, A. Pitzschke, M. Kalbermatten, Erwin Berger, H. Bergh, S. Cook, G. Wagnières (2020)
Optical Characterization of an Intra-Arterial Light and Drug Delivery System for Photodynamic Therapy of Atherosclerotic Plaque
Applied Sciences

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DOI: 10.3390/photonics9090597
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Abstract

hv photonics Article Optimization of the Distance between Cylindrical Light Distributors Used for Interstitial Light Delivery in Biological Tissues 1 2 1 , Aurélien Gregor , Shohei Sase and Georges Wagnieres * Laboratory for Functional and Metabolic Imaging, Swiss Federal Institute of Technology (EPFL), Station 3, 1015 Lausanne, Switzerland Medical Science and Operations Department, Rakuten Medical K.K., 2-21-1 Tamagawa, Setagaya-ku, Tokyo 158-0094, Japan * Correspondence: georges.wagnieres@epﬂ.ch 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 parame- ters 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 Citation: Gregor, A.; Sase, S.; between 11.5- and 13-mm. Variations of the absorption and reduced scattering coefﬁcients have the Wagnieres, G. Optimization of the greatest inﬂuence on CLD placements, while tissue anisotropy factor, CLD insertion geometry, CLD Distance between Cylindrical Light length, and the angular dependence of the radiance emitted by the CLDs have minimal inﬂuence. Distributors Used for Interstitial At ﬁrst approximation the inﬂuence of these optical parameters on optimal CLD-CLD distance are Light Delivery in Biological Tissues. independent. Our data also suggests it is possible to derive new treatment plans using knowledge of Photonics 2022, 9, 597. https:// previous treatment plans. doi.org/10.3390/photonics9090597 Received: 5 July 2022 Keywords: interstitial photoimmunotherapy; photodynamic therapy; cylindrical light diffusers; head Accepted: 16 August 2022 and neck cancers; Monte Carlo; light dosimetry Published: 23 August 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional afﬁl- 1. Introduction iations. Cylindrical optical ﬁber-based light diffusers (CLD) are frequently used for inter- stitial photodynamic therapy (iPDT) and photoimmunotherapy (PIT) of various solid cancers [1–4]. Although PIT and iPDT differ from many respects, these two therapies are based, in most cases, on the intravenous administration of photosensitizers (PS) followed Copyright: © 2022 by the authors. by the delivery of light to the lesions with appropriate irradiances and doses at wavelengths Licensee MDPI, Basel, Switzerland. absorbed by the PS [5]. These non-thermal illuminations, which are usually performed in This article is an open access article the red or near-infrared part of the spectrum, lead to more or less selective cancer necrosis distributed under the terms and and/or apoptosis depending on the PS and lesion types. iPDT utilizes PS with tumor conditions of the Creative Commons afﬁnity as a drug, such as Photofrin, and reactive oxygen species produced upon photoirra- Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ diation are responsible for the tumor cell death. The fundamental difference between iPDT 4.0/). and PIT is related to the fact that the latter is based on the use of speciﬁc antibodies-PS Photonics 2022, 9, 597. https://doi.org/10.3390/photonics9090597 https://www.mdpi.com/journal/photonics Photonics 2022, 9, 597 2 of 18 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 squa- mous 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 Septem- ber 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 2 2 cell lines, ranging from 4 to 30 J/cm . In the present study, the severe condition of 30 J/cm 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/cm 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 Photonics 2022, 9, 597 3 of 18 Photonics 2022, 9, x FOR PEER REVIEW 3 of 19 the treatment plan, CLDs of length 20, 60, and 80 mm were also considered. In addition, Medlight SA [17,18]. The optical properties of CLDs are such that photons backscattered the effects resulting from different angular emittance on the treatment plan were assessed by the surrounding tissue are all transmitted through the CLDs without absorption or simulating CLDs with uniform and normal to the surface emittances. 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, 2.2. Tissue Model the effects resulting from different angular emittance on the treatment plan were assessed simulating CLDs with uniform and normal to the surface emittances. The optical properties of biological tissues are usually described by four main pa- rameters: the refractive index, the absorption and scattering coefﬁcients, as well as the 2.2. Tissue Model anisotropy factor (n, m , m and g, respectively). Another useful parameter is the reduced a s The optical properties of biological tissues are usually described by four main 0 0 scattering coefﬁcient m , which combines m and g in such a way that m = m (1g). Most parameters: the refractive index, the absorption and scattering coefficients, as well as the s s s s anisotropy factor (𝑛 , µ , µ and g, respectively). Another useful parameter is the reduced biological tissues show 𝑎 𝑠 strong forward scattering in the red [19]. This is in agreement with scattering coefficient µ ′, which combines µ and g in such a way that µ ′ = µ (1 −g). 𝑠 𝑠 𝑠 𝑠 the anisotropy factor of 0.9 for HNSCC reported by Holmer et al. [20] at 690 nm. Therefore, Most biological tissues show strong forward scattering in the red [19]. This is in agreement the anisotropy factor of the “tumor” was assumed to be 0.9 in our simulations. There are with the anisotropy factor of 0.9 for HNSCC reported by Holmer et al. [20] at 690 nm. not many published values of the optical parameters for HNSCCs at 690 nm. In addition, Therefore, the anisotropy factor of the “tumor” was assumed to be 0.9 in our simulations. Th these ere are values not many present publishr ed elatively values of timportant he optical parvariations, ameters for HNS as CC can s at be 690 seen nm. In in Figure 1, which addition, these values present relatively important variations, as can be seen in Figure 1, presents 15 values (3 obtained in the oral cavity and 12 in the esophagus) of m and m which presents 15 values (3 obtained in the oral cavity and 12 in the esophagus) of µ and pairs reported by Bargo et al. [21] and Holmer et al. [20]. Since Bargo et al. measured these µ ′ pairs reported by Bargo et al. [21] and Holmer et al. [20]. Since Bargo et al. measured optical coefﬁcients at 630 nm, their values were corrected to 690 nm by extrapolations using these optical coefficients at 630 nm, their values were corrected to 690 nm by the spectral dependence of the results reported by Holmer at al. extrapolations using the spectral dependence of the results reported by Holmer at al. Reviewed values 1.2 Esophagus eff a' Selected 1.0 4 values Oral Cavity Esophagus Esophagus 0.8 Oral Cavity Esophagus Esophagus Esophagus Oral Cavity Esophagus Esophagus 0.6 Esophagus Esophagus Esophagus 0.4 Esophagus 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 -1 µ [mm ] Figure 1. Values of µ and µ ′ derive 0 d from the literature for HNSCC. The green and blue lines 𝑎 𝑠 Figure 1. Values of m and m derived from the literature for HNSCC. The green and blue lines repre- represent isocurves of µ and 𝑎 ′, respectively. The long-dashed, continuous, and dotted lines 𝑒𝑓𝑓 sent isocurves of m and a , respectively. The long-dashed, continuous, and dotted lines represent e f f represent the isocurves of the first quartile, median, and third quartile values of µ and 𝑎 ′, 𝑒𝑓𝑓 respec the isocurve tively. Five s of different the ﬁrst pairs quartile, of optical median, coefficients and were thir sel d ecquartile ted for the values simulation of m s by taking and a , respectively. Five e f f intersections of isocurves corresponding to four extreme values of µ and µ ′ pairs (points 1, 2, 4 𝑎 𝑠 different pairs of optical coefﬁcients were selected for the simulations by taking intersections of and 5) as well as the intersection of the isocurves corresponding to the medians for µ and 𝑎 ′ 𝑒𝑓𝑓 isocurves corresponding to four extreme values of m and m pairs (points 1, 2, 4 and 5) as well as the (point 3). s intersection of the isocurves corresponding to the medians for m and a (point 3). e f f Our simulations have been performed for five different pairs of µ and µ ′ defined 𝑎 𝑠 according to the following procedure: in a first step, isocurves of the first quartile, median, 0 Our simulations have been performed for ﬁve different pairs of m and m deﬁned according to the following procedure: in a ﬁrst step, isocurves of the ﬁrst quartile, median, and third quartile values of m and a , were drawn on the basis of the 15 values of the m e f f and m pairs, where m is the effective attenuation coefﬁcient (m = 3m (m + m ) and a a s e f f e f f s 0 0 s 0 a is the transport albedo a = [22] (See Figure 1). Then, pair values of m and m were 0 a m +m s a selected by taking intersections of these isocurves corresponding to four extreme values of m and m pairs (points 1, 2, 4 and 5 in Figure 1), as well as the intersection of the isocurves corresponding to the medians for m and a (point 3 in Figure 1). e f f 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 -1 µ [mm ] s' Photonics 2022, 9, 597 4 of 18 simulations are summarized in Table 1. If needed, the light propagation simulations could easily be used for other optical coefﬁcients. Table 1. Optical properties selected to simulate the propagation of light in HNSCCs at 690 nm. 1 1 1 Simulation n [mm ] ’ [mm ] [mm ] a’ a s eff 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 2.3. Simulation and Optimizing Algorithm One of the most useful approach to simulate the propagation of light in biological tis- sues 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/m , where m = m + m . The “TracePro” program from Lambda Research Corporation [25] t a s was used to simulate the propagation of light around cylindrical distributor(s) inserted in Photonics 2022, 9, x FOR PEER REVIEW 5 of 19 HNSCC treated by PIT. Each simulation was typically performed with 10 photon packets with a 0.1 0.1 0.1 mm grid. The tissue geometry we simulated consisted of a semi-inﬁnite “tumor” in which CLDs distributors; Unit: W/cm). Consequently, since the fluence rate unit is W/cm , the 2 −1 normalized fluence rate unit is (W/cm /W/cm = cm ). The threshold fluence were inserted at 90 in such a way that the proximal part of the light emitting cylindrical corresponding to the tissue destruction by PIT was on the in vitro studies [9,15,16], as section was at the level of the air-tumor interface (See Figure 2). All cylindrical light mentioned above. This fluence of 30 J/cm is assumed to be the threshold when distributors were oriented in the same direction and inserted at the same depth. The illumination conditions (Linear emittance: 0.4 W/cm at 690 nm during 250 s) are applied conﬁgurations in which the CLDs were positioned are shown in Table 2. The maximal for PIT based on the use of CLDs in previous clinical studies [12]. Therefore, the value of 2 −1 2 the normalized fluence rate threshold is 0.3 W/cm /W/cm = 0.3 cm (30 J/cm /[0.4 W/cm × number of cylindrical light distributors considered was equal to seven as a larger number 2 −1 250 s] = 0.3 W/cm /W/cm = 0.3 cm ). was considered as unsuitable for the clinical use of PIT to treat ENT cancers. These A typical normalized fluence rate distribution calculated by MC simulations, for a conﬁgurations were classiﬁed into two categories: those containing a central CLD and single CLD placed in a homogeneous medium, is shown in Figure 2 for reference. those with cyclical geometries (regular polygons). Additional distribution representations when using two, three, and four CLDs are presented in Appendix A. Figure 2. Spatial distribution of the normalized fluence rate around a single 40 mm-long CLD Figure 2. Spatial distribution of the normalized ﬂuence rate around a single 40 mm-long CLD inserted −1 −1 inserted in a homogeneous medium with µ = 0.078 mm , µ ′ = 0.70 mm and g = 0.9. The 𝑎 𝑠 1 0 1 midsection is taken along the CLD axis. The outermost contour corresponds to the threshold fluence in a homogeneous medium with m = 0.078 mm , m = 0.70 mm and g = 0.9. The midsection 2 2 rate of 0.3 W/cm /W/cm, which corresponds to a light dose of 30 J/cm when illumination conditions is taken along the CLD axis. The outermost contour corresponds to the threshold ﬂuence rate of of 0.4 W/cm for 250 s are used. The air-tissue interface is situated at 0 mm along the vertical axis. 2 2 0.3 W/cm /W/cm, which corresponds to a light dose of 30 J/cm when illumination conditions of A block diagram describing the optimizing algorithm is presented in Figure 3. In 0.4 W/cm for 250 s are used. The air-tissue interface is situated at 0 mm along the vertical axis. 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 −1 the necrosed tumors (all voxels with normalized fluence rate higher than the 0.3 cm threshold). Total necrosed volume were calculated for CLD-CLD distances between 5 and Photonics 2022, 9, 597 5 of 18 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 “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/cm , the normal- 2 1 ized ﬂuence rate unit is (W/cm /W/cm = cm ). 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/cm 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 2 1 2 2 1 0.3 W/cm /W/cm = 0.3 cm (30 J/cm /[0.4 W/cm 250 s] = 0.3 W/cm /W/cm = 0.3 cm ). 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. Addi- tional 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 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/m ) 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. Photonics 2022, 9, x FOR PEER REVIEW 6 of 19 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. Photonics 2022, 9, 597 6 of 18 Each voxel in the TracePro simulation dataset grid corresponded with a voxel in the Matlab optimizing simulation space grid. Figure 3. Block diagram of optimizing algorithm. Figure 3. Block diagram of optimizing algorithm. 3. Results A point seldom mentioned when Monte-Carlo light path tracings are employed is 3.1. Validation of the MC Simulations: Comparison between the MC and Analytical Models how fluence rate, which is the power of the photons entering a sphere divided by the An analytical model based on the diffusion approximation was used to validate the cross-sectional area of this sphere, can be obtained from a photon flux propagating MC simulations. The steady-state expression for the light propagation in a tissue is given through the cubes (voxels) of a mesh. If the angles of incidence of photons onto voxels are by [27]: random, which is the case except close (<1/µ ′) to the source, the fluence rate within each D m F(r) = q(r)/D (1) voxel can be obtained by dividing the incident flux (sum of rays entering the voxel) by the e f f cube’s average cross-sectional area. Cauchy’s surface area formula states that for every 2 2 with F(r) the ﬂuence rate (W⁄cm ), D = m /m the optical diffusivity (cm), and q the e f f convex body the average area of its parallel projections onto a plane is equal to a quarter source photon density (W⁄cm ). of its surface [26]; so for a unit cube the expected cross-sectional area is equal to 3/2. In the case of an inﬁnitely long CLD in a homogeneous volume, the solution of Therefore, the fluence rate was calculated by dividing the power of the photons entering Equation (1) for the ﬂuence rate [28] is given by: a voxel whose surface of a face is equal to unity by this factor 3/2. 2a 3. Results F(r) = K m r (2) 0 e f f D m K m a e f f 1 e f f 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 where P is the optical power per unit length (W⁄cm) of the CLD of radius a (cm). K and MC simulations. The steady-state expression for the light propagation in a tissue is given K are the modiﬁed Bessel functions of the second kind for the zeroth and ﬁrst order by [27]: 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 (Δ − µ ) Φ(r) = −q(r)/D (1) 𝑓𝑓𝑒 was compared to the simulated ﬂuence rate as a function of “r” at a depth equal to half the with Φ(r) the fluence rate (W⁄cm ), D = µ /µ the optical diffusivity (cm), and 𝑞 the length of the 40 mm-long CLD. Since the middle of the CLD is at 20 mm from either tips of 𝑓𝑓𝑒 the source CLD, phot which on dens is greater ity (W⁄ than cm ). 3 m for all optical properties simulated, edge effects e f f at this depth are negligible and the radial ﬂuence rate is expected to be similar to that of an In the case of an infinitely long CLD in a homogeneous volume, the solution of inﬁnitely long CLD [29]. Equation (1) for the fluence rate [28] is given by: 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 2πa (2) Φ(r) = K (µ r) 0 𝑓𝑓𝑒 observed. In the ﬁrst region, close to the source and known as the build-up region, the D × µ × K (µ a) 𝑓𝑓𝑒 1 𝑓𝑓𝑒 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 Photonics 2022, 9, x FOR PEER REVIEW 7 of 19 where P is the optical power per unit length (W⁄cm) of the CLD of radius a (cm). K and K 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 “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 −1 tips of the CLD, which is greater than 3 × µ for all optical properties simulated, edge 𝑓𝑓𝑒 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 Photonics 2022, 9, 597 7 of 18 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 rate at these distances are too low to affect the optimizing algorithm’s results. The agreement between this simulation and the analytical model speaks in favor of a validation excellent agreement between this simulation and the analytical model speaks in favor of of the MC algorithm used for the simulations reported in this article. a validation of the MC algorithm used for the simulations reported in this article. Figure 4. Normalized fluence rate versus radial distance from the axis of the CLD at a depth of 20 Figure 4. Normalized ﬂuence rate versus radial distance from the axis of the CLD at a depth of 20 mm −1 mm around a 40 mm long CLD. Optical properties of the medium are: µ = 0.078 mm , µ = 0.7 𝑎 𝑠 ′ 1 1 around a 40 mm long CLD. Optical properties of the medium are: m = 0.078 mm , m = 0.7 mm , −1 a s mm , g = 0.9. g = 0.9. 3.2. Influence of the Tumor Optical Parameters on the CLD-CLD Distances Maximizing the 3.2. Inﬂuence of the Tumor Optical Parameters on the CLD-CLD Distances Maximizing the Necrosed Volume Necrosed Volume Figure 5 presents the computed optimal CLD-CLD distances, and the necrosed Figure 5 presents the computed optimal CLD-CLD distances, and the necrosed volume volume per CLD at these distances, using the sets of optical properties summarized in per CLD at these distances, using the sets of optical properties summarized in Table 1 for Table 1 for 40 mm-long CLDs. The number of CLDs and their positioning geometry were 40 mm-long CLDs. The number of CLDs and their positioning geometry were setup as Photonics 2022, 9, x FOR PEER REVIEW 8 of 19 setup as described in Table 2. described in Table 2. Figure 5. (a) CLD-CLD distances maximizing the tumor volume receiving a light dose larger than Figure 5. (a) CLD-CLD distances maximizing the tumor volume receiving a light dose larger than that that corresponding to the necrosis threshold. These distances refer to the shortest distances between corresponding to the necrosis threshold. These distances refer to the shortest distances between two two CLDs axis in each geometry. For geometries with a central CLD, this refers to the distance between the central CLD and the outer CLDs. (b) Volume of necrosed tissue per CLD for these CLDs axis in each geometry. For geometries with a central CLD, this refers to the distance between distances. The numbers and color code in the legend refer to the simulations numbers given in Table the central CLD and the outer CLDs. (b) Volume of necrosed tissue per CLD for these distances. The 1 for different optical parameters of the tumor. numbers and color code in the legend refer to the simulations numbers given in Table 1 for different optical parameters of the tumor. 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 cm 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 µ (simulation 1 & 2, and 4 & 5), the 𝑓𝑓𝑒 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 µ , indicating that knowledge of µ alone is not 𝑓𝑓𝑒 𝑓𝑓𝑒 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 µ , decreases as the transport albedo increases. 𝑓𝑓𝑒 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 Photonics 2022, 9, 597 8 of 18 Looking at Figure 5 indicates that, in our conditions, the CLD-CLD distances maxi- mizing the necrosed volume range between 10 and 17 mm, leading to necrosed volumes ranging between 2.5 and about 7.5 cm 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 m (simulation 1 & 2, and 4 & 5), the one e f f 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 signiﬁ- cant as changing m , indicating that knowledge of m alone is not sufﬁcient to estimate e f f e f f 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 m , decreases as the transport albedo increases. e f f 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 0 0 40 mm-long CLD when m = m (1g) was held constant while the scattering coefﬁcient m s s and g were varied. g was set as 0.5, 0.6, 0.75, 0.9, and 0.95, while m was equal to 1.4, 1.75, 1 0 1 2.8, 7, and 14 mm , respectively, such that m = 0.7 mm . The absorption coefﬁcient was held constant at 0.078 mm 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 m constant (0.7 mm ). This is in agreement with simulations reported by Streeter et al. [29], who considered a “narrow” collimated beam illuminating an air-tissue interface. This 0 1 group observed that the ﬂuence rate “far” (i.e., more than m ) from the interface is constant when m and g are changed in such a way that m 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/(m ), 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, Photonics 2022, 9, 597 9 of 18 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 m (1g) rather than on the speciﬁc values of m and g. Therefore, in our conditions and in s s 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 m and m 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 m = 0.078 and m = 0.7 (simulation n 3). The “linear emittance” was kept constant at 0.4 W/cm and the proximal part of the CLD light Photonics 2022, 9, x FOR PEER REVIEW 10 of 19 emitting section was again at the air-tumor interface. For clarity, only geometries with a central CLD are presented. Figure 6. (a) Optimal CLD distances maximizing the volume of tumor receiving more than the Figure 6. (a) Optimal CLD distances maximizing the volume of tumor receiving more than the necrosis threshold dose. (b) Volume of necrosed tumor per CLD and per centimeter of CLD for these necrosis threshold dose. (b) Volume of necrosed tumor per CLD and per centimeter of CLD for these optimal CLD distances. Simulations were performed with CLDs of length 20, 40, 60, and 80 mm. optimal CLD distances. Simulations were performed with CLDs of length 20, 40, 60, and 80 mm. Only geometries with a central CLD are presented. µ = 0.078 and µ ′ = 0.7 (simulation n°3). 𝒂 𝑠 Only geometries with a central CLD are presented. m = 0.078 and m = 0.7 (simulation n 3). Looking at Figure 6 we see that, for simulation n°3, the CLD-CLD distances op Looking timizing at th Figur e necrosed e 6 we vo see luthat, me rfor angsimu e betlation ween 11 n 3, anthe d 13 CLD-CLD mm, leadi distances ng to necrosed optimizing volume per centimeter of CLD ranging between 0.88 and 1.24 cm , depending on the the necrosed volume range between 11 and 13 mm, leading to necrosed volume per number of CLDs and their length. For all number of CLDs, the optimal CLD distance centimeter of CLD ranging between 0.88 and 1.24 cm , depending on the number of CLDs increases with CLD length, with the biggest increase being between CLDs of 20- and 40- and their length. For all number of CLDs, the optimal CLD distance increases with CLD mm length, until reaching a plateau at 60 mm. The necrosed volume per centimeter of length, with the biggest increase being between CLDs of 20- and 40-mm length, until CLD also increases with CLD length. However, the majority of this difference in necrosed reaching a plateau at 60 mm. The necrosed volume per centimeter of CLD also increases volume can be attributed to a relatively higher percentage of emitted light escaping the with CLD length. However, the majority of this difference in necrosed volume can be medium through the air/tissue interface for short CLDs, even though the total amount of attributed to a relatively higher percentage of emitted light escaping the medium through escaping light remains constant (data not shown). The changes in optimal CLD distance the air/tissue interface for short CLDs, even though the total amount of escaping light are therefore more due to the differences in shape of the necrosed volume as edge effects remains constant (data not shown). The changes in optimal CLD distance are therefore become more significant the shorter the CLD is. As the outline of the necrosed volume more due to the differences in shape of the necrosed volume as edge effects become becomes increasingly spherical (as opposed to cylindrical) when the CLD length is more signiﬁcant the shorter the CLD is. As the outline of the necrosed volume becomes reduced, it is more optimal to place CLDs closer to each other. Comparisons of the normalized fluence rate distributions around 40 mm-long CLDs increasingly spherical (as opposed to cylindrical) when the CLD length is reduced, it is with Lambertian emission vs normal to surface emission for optical properties µ = 0.078 more optimal to place CLDs closer to each other. 𝑎 and µ ′ = 0.7 (simulation n°3) are shown in Figure 7. Although light distribution patterns Comparisons of the normalized ﬂuence rate distributions around 40 mm-long CLDs are very similar, simulations using a normal to surface emission showed a slightly greater with Lambertian emission vs normal to surface emission for optical properties m = 0.078 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). Photonics 2022, 9, 597 10 of 18 and m = 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 Photonics 2022, 9, x FOR PEER REVIEW 11 of 19 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). Photonics 2022, 9, x FOR PEER REVIEW 11 of 19 Figure 7. Midsection slice of fluence rate distributions for a single 40 mm CLD when Lambertian Figure 7. Midsection slice of ﬂuence rate distributions for a single 40 mm CLD when Lamber- emittance (a) and normal to surface emittance (b) are used. Medium optical properties are: µ = tian emittance (a) and normal to surface emittance (b) are used. Medium optical properties −1 −1 0.078 mm , µ ′ = 0.70 mm and g = 0.9 (simulation n°3). The black curve corresponds to the Figure 7. Midsecti 𝑠 on slice of fluence rate distributions for a single 40 mm CLD when Lambertian 1 0 1 are: m = 0.078 mm , m = 0.70 mm and g = 0.9 (simulation n 3). The black curve corresponds to a 2 2 threshold fluence rate of 0.3 W/cm /W/cm, which corresponds to a light dose of 30 J/cm when emittance (a) and normal to surface emittance (b) are used. Medium optical properties are: µ = 2 2 −1 −1 the illum thr inati eshold on co ﬂuence nditionsra of te 0of .4 W/cm 0.3 W/cm for 250 /W/cm, s are used which . The corr air-ti esponds ssue interf to ace a light is situated dose of at 30 0 m J/cm m when 0.078 mm , µ ′ = 0.70 mm and g = 0.9 (simulation n°3). The black curve corresponds to the 2 2 along the vertical axis. The black bar represents the CLD. threshold fluence rate of 0.3 W/cm /W/cm, which corresponds to a light dose of 30 J/cm when illumination conditions of 0.4 W/cm for 250 s are used. The air-tissue interface is situated at 0 mm illumination conditions of 0.4 W/cm for 250 s are used. The air-tissue interface is situated at 0 mm along the vertical axis. The black bar represents the CLD. along the As vertical presented axis. T in he black bar Figure 8, CLDs repres weith nts L the CL ambert D.ia n emittance consistently yielded only 2% shorter optimal CLD distances and 3% smaller necrosed volume per CLD versus using As presented in Figure 8, CLDs with Lambertian emittance consistently yielded only As presented in Figure 8, CLDs with Lambertian emittance consistently yielded only normal to surface angular emittance. Therefore, the angular distribution of light emitted 2% shorter optimal CLD distances and 3% smaller necrosed volume per CLD versus using 2% shorter optimal CLD distances and 3% smaller necrosed volume per CLD versus using from the surface of the CLD (as long as the illumination is cylindrically symmetrical) does normal to surface angular emittance. Therefore, the angular distribution of light emitted normal to surface angular emittance. Therefore, the angular distribution of light emitted not significantly influence the positioning of CLDs and leads to very similar volumes from the surface of the CLD (as long as the illumination is cylindrically symmetrical) does from the surface of the CLD (as long as the illumination is cylindrically symmetrical) does necrosed by PIT. not signiﬁcantly inﬂuence the positioning of CLDs and leads to very similar volumes not significantly influence the positioning of CLDs and leads to very similar volumes necrosed by PIT. necrosed by PIT. Figure 8. (a) Optimal CLDs distances maximizing the volume of tumor receiving more than the necrosis threshold dose. (b) Volume of necrosed tumor per CLD and per centimeter of CLD for these optimal CLD distances. Simulations were performed with CLDs having Lambertian or normal to Figure 8. (a) Optimal CLDs distances maximizing the volume of tumor receiving more than the Figure 8. (a) Optimal CLDs distances maximizing the volume of tumor receiving more than the surface emittance. necrosis threshold dose. (b) Volume of necrosed tumor per CLD and per centimeter of CLD for these necrosis threshold dose. (b) Volume of necrosed tumor per CLD and per centimeter of CLD for these optimal CLD distances. Simulations were performed with CLDs having Lambertian or normal to optimal CLD distances. Simulations were performed with CLDs having Lambertian or normal to surface emittance. surface emittance. Photonics 2022, 9, 597 11 of 18 Photonics 2022, 9, x FOR PEER REVIEW 12 of 19 4. Discussion The treatment planning scheme presented in this paper assumes that the tumor re- 4. Discussion sponse is determined entirely by the photosensitizer concentration and ﬂuence, as has The treatment planning scheme presented in this paper assumes that the tumor been done in past studies [14,30]. Since the photosensitizer concentration is assumed to response is determined entirely by the photosensitizer concentration and fluence, as has be homogeneous, the ﬂuence is the only factor that is responsible for different outcomes been done in past studies [14,30]. Since the photosensitizer concentration is assumed to be from place to place. However, in virtually all real situations, the tumor necrosis depends homogeneous, the fluence is the only factor that is responsible for different outcomes from on additional factors, including: the ﬂuence rate, the macroscopic and microscopic pho- place to place. However, in virtually all real situations, the tumor necrosis depends on tosensitizer localizations, the tumor oxygenation and the immune response [31]. In such additional factors, including: the fluence rate, the macroscopic and microscopic situations, the choice of treatment plan (laser power and treatment time) has consequences photosensitizer localizations, the tumor oxygenation and the immune response [31]. In on the outcome. Therefore, although the results presented above deserve to be validated such situations, the choice of treatment plan (laser power and treatment time) has con with sequences in vivo on experiments, the outcome. the There general fore, alconclusions though the res ar ule tscertainly presented valid abovesince deserve the toﬂuence is be validated with in vivo experiments, the general conclusions are certainly valid since the most important parameters in PIT and PDT. 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 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 studies of PIT using the conjugates with IR700. We applied the upper limit value of 30 30 J/cm as the threshold by assuming that cell death would occur more efﬁciently under J/cm 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 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 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 based on the in vitro studies can be directly applied without modification when safety is is considered. considered. When using four or more CLDs, treatment plans using geometries which don’t contain When using four or more CLDs, treatment plans using geometries which don’t a central CLD exhibit a region of sub-treated tissue in the center (see Figure 9), which contain a central CLD exhibit a region of sub-treated tissue in the center (see Figure 9), is mostly surrounded by tissue receiving sufﬁcient dose to cause necrosis. The damage 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 produced to the surrounding tissue vasculature may be sufﬁcient to occlude the blood blood vessels that feed these enveloped tumor cells, contributing to massive ischemia vessels that feed these enveloped tumor cells, contributing to massive ischemia and/or and/or vascular infarction. These effects, combined with inflammatory reactions taking vascular infarction. These effects, combined with inﬂammatory reactions taking place in place in the surrounding treated tissues, are likely to induce subsequent necrosis of this the surrounding treated tissues, are likely to induce subsequent necrosis of this region. region. Although this form of indirect tumor-cell killing might be beneficial by Although this form of indirect tumor-cell killing might be beneﬁcial by synergistically synergistically enhancing PIT and PDT, this effect is beyond the scope of this work and is enhancing PIT and PDT, this effect is beyond the scope of this work and is not considered not considered in the present model, mostly due to its complexity and its limited impact in the present model, mostly due to its complexity and its limited impact on the general on the general conclusions resulting from this study. Rather, our mathematical modelling conclusions resulting from this study. Rather, our mathematical modelling of optimal CLD of optimal CLD positioning is restricted to a simplified analysis that focuses on positioning is restricted to a simpliﬁed analysis that focuses on determining the tissue determining the tissue volume receiving above-threshold light dose in an effort to volume receiving above-threshold light dose in an effort to describe irradiation conditions describe irradiation conditions that maximize the volume of direct tumor-cell destruction that maximize the volume of direct tumor-cell destruction for a minimal number of CLDs. for a minimal number of CLDs. Figure 9. Normalized fluence rate cross-section at half the insertion depth (20 mm) at 690 nm around Figure 9. Normalized ﬂuence rate cross-section at half the insertion depth (20 mm) at 690 nm around −1 six 40 mm long CLDs positioned as a hexagon. Tumor optical properties are: µ = 0.078 mm , µ ′ 𝑎 𝑠 six 40 mm −1 long CLDs positioned as a hexagon. Tumor optical properties are: m = 0.078 mm , = 0.70 mm and g = 0.9 (simulation n°3). The black curve corresponds to the threshold fluence rate 0 1 m = 0.70 mm and g = 0.9 (simulation n 3). The black curve corresponds to the threshold ﬂu- 2 2 ence rate of 0.3 W/cm /W/cm, which corresponds to a light dose of 30 J/cm when illumination conditions of 0.4 W/cm for 250 s are used. Photonics 2022, 9, 597 12 of 18 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/cm 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 cm 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, m and m , 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 necro- sis 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. Author Contributions: Conceptualization, A.G. and G.W.; methodology, A.G. and G.W.; software, A.G.; validation, A.G. and G.W.; formal analysis, A.G., G.W. and S.S.; investigation, A.G. and G.W.; resources, G.W.; data curation, A.G.; writing—original draft preparation, A.G.; writing—review and editing, A.G., G.W. and S.S.; visualization, A.G.; supervision, G.W.; project administration, G.W.; funding acquisition, G.W. All authors have read and agreed to the published version of the manuscript. Funding: This study was supported by Rakuten Medical K.K. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are available on request from the corresponding author. The data are not publicly available due to proprietary reasons. Acknowledgments: The authors are grateful to Toshiaki Suzuki (Rakuten Medical K.K.) for his valuable comments. Conﬂicts of Interest: The following author is an employee: S.S (Rakuten Medical Inc). A.G and G.W have nothing to disclose. Photonics 2022, 9, x FOR PEER REVIEW 14 of 19 Acknowledgments: The authors are grateful to Toshiaki Suzuki (Rakuten Medical K.K.) for his Photonics 2022, 9, x FOR PEER REVIEW 14 of 19 valuable comments. Photonics 2022, 9, 597 13 of 18 Conflicts of Interest: The following author is an employee: S.S (Rakuten Medical Inc). A.G and G.W have Acknowledgments: nothing to disc The lose auth . ors are grateful to Toshiaki Suzuki (Rakuten Medical K.K.) for his valuable comments. Appendix A Appendix A Conflicts of Interest: The following author is an employee: S.S (Rakuten Medical Inc). A.G and G.W have nothing to disclose. Appendix A Figure A1. Cross-section along a horizontal plane containing two sources. Normalized fluence rate Figure Figure A1 A1. . Cr Cr oss oss-section -section alonalong g a hora izontal plane horizontal co plane ntainicontaining ng two source two s. No sour rmali ces. zed Normalized fluence rate ﬂuence around two 40 mm long cylindrical light distributor at a depth of 20 mm calculated by MC around two 40 mm long cylindrical light distributor at a depth of 20 mm calculated by MC rate around two 40 mm long cylindrical light distributor at a depth of 20 mm calculated by MC simulations. The black circle corresponds to the extent of the necrosis when the fluence simulations. The black circle corresponds to the extent of the necrosis when the fluence simulations. The black circle corresponds to the extent of the necrosis when the ﬂuence recommended recommended by Rakuten is applied (30 J/cm²; 0.4 W/cm during 250 s), i.e., 0.3 W/cm²/W/cm. In recommended by Rakuten is applied (30 J/cm²; 0.4 W/cm during 250 s), i.e., 0.3 W/cm²/W/cm. In 2 2 by Rakuten is applied (30 J/cm ; 0.4 W/cm during 250 s), i.e., 0.3 W/cm /W/cm. In other words, other words, 0.3 W/cm²/W/cm × 0.4 W/cm × 250 s = 30 J/cm² (edge effects are neglected in this other words, 0.3 W/cm²/W/cm × 0.4 W/cm × 250 s = 30 J/cm² (edge effects are neglected in this 2 2 0.3 W/cm /W/cm 0.4 W/cm 250 s = 30 J/cm (edge effects are neglected in this calculation). calculation). calculation). Figure A2. Cross-section along a vertical plane containing two sources. The air-tissue interface is at Figure A2. Cross-section along a vertical plane containing two sources. The air-tissue interface is at 0 mm on the vertical axis. The CLDs are represented by the grey cylinders (a) 2-D representation of 0 mm on the vertical axis. The CLDs are represented by the grey cylinders (a) 2-D representation of the cross-section. The black line corresponds to the necrosis threshold: 0.3 W/cm²/W/cm2 . (b) 3-D the cross-section. The black line corresponds to the necrosis threshold: 0.3 W/cm /W/cm. (b) 3-D Figure A2. Cross-section along a vertical plane containing two sources. The air-tissue interface is at 0 mm on the vertical axis. The CLDs are represented by the grey cylinders (a) 2-D representation of representation of the cross-section along a vertical plane containing the two sources. The blue surface the cross-section. The black line corresponds to the necrosis threshold: 0.3 W/cm²/W/cm. (b) 3-D corresponds to the extent of necrosis. Photonics 2022, 9, x FOR PEER REVIEW 15 of 19 Photonics 2022, 9, x FOR PEER REVIEW 15 of 19 Photonics 2022, 9, 597 14 of 18 representation of the cross-section along a vertical plane containing the two sources. The blue representation of the cross-section along a vertical plane containing the two sources. The blue surface corresponds to the extent of necrosis. surface corresponds to the extent of necrosis. Figure A3. Cross-section along a vertical plane positioned in the middle of the two fibers. (a) 2-D Figure A3. Cross-section along a vertical plane positioned in the middle of the two ﬁbers. Figure A3. Cross-section along a vertical plane positioned in the middle of the two fibers. (a) 2-D representation of the cross-section. The black line corresponds to the necrosis threshold: 0.3 (a) 2-D representation of the cross-section. The black line corresponds to the necrosis threshold: representation of the cross-section. The black line corresponds to the necrosis threshold: 0.3 W/cm²/W/cm. (b) 3-D representation of the cross-section along a vertical plane positioned in the 0.3 W/cm /W/cm. (b) 3-D representation of the cross-section along a vertical plane positioned in W/cm²/W/cm. (b) 3-D representation of the cross-section along a vertical plane positioned in the middle of the two fibers. The blue surface corresponds to the extent of necrosis. the middle of the two ﬁbers. The blue surface corresponds to the extent of necrosis. middle of the two fibers. The blue surface corresponds to the extent of necrosis. Figure A4. Same conditions as in Figure A1, but for three sources in a plane. Figure A4. Same conditions as in Figure A1, but for three sources in a plane. Figure A4. Same conditions as in Figure A1, but for three sources in a plane. Photonics 2022, 9, x FOR PEER REVIEW 16 of 19 Photonics 2022, 9, 597 15 of 18 Figure A5. Cross-section along a vertical plane containing three sources placed in a line. The air- Figure A5. Cross-section along a vertical plane containing three sources placed in a line. The air- tissue interface is at 0 mm on the vertical axis. The CLDs are represented by the grey cylinders tissue interface is at 0 mm on the vertical axis. The CLDs are represented by the grey cylinders (a) (a) 2-D representation of the cross-section. The black line corresponds to the necrosis threshold: 2-D representation of the cross-section. The black line corresponds to the necrosis threshold: 0.3 0.3 W/cm /W/cm. (b) 3-D representation of the cross-section along a vertical plane containing the W/cm²/W/cm. (b) 3-D representation of the cross-section along a vertical plane containing the three three sources. The blue surface corresponds to the extent of necrosis. sources. The blue surface corresponds to the extent of necrosis. Figure A6. Same conditions as in Figure A1, but for three equidistant sources. Figure A6. Same conditions as in Figure A1, but for three equidistant sources. Photonics 2022, 9, 597 16 of 18 Figure A7. Cross-section along a vertical plane containing one source and the middle of the other two sources. The air-tissue interface is at 0 mm on the vertical axis. The CLDs are represented by the grey cylinders (a) 2-D representation of the cross-section. The black line corresponds to the necrosis threshold: 0.3 W/cm /W/cm. (b) 3-D representation of the cross-section along a vertical plane containing the two sources. The blue surface corresponds to the extent of necrosis. Figure A8. (a) Same conditions as in Figure A1, but for four sources positioned according to a triangle plus one central source. The black line corresponds to the necrosis threshold: 0.3 W/cm /W/cm. (b) 3-D representation of the cross-section along a vertical plane containing the central source plus another one. The air-tissue interface is at 0 mm on the vertical axis. The blue surface corresponds to the extent of necrosis. The CLDs are represented by the grey cylinders. Photonics 2022, 9, 597 17 of 18 Figure A9. (a) Same conditions as in Figure A1, but for four sources positioned according to a square. The black line corresponds to the necrosis threshold: 0.3 W/cm /W/cm. (b) 3-D representation of the cross-section along a vertical plane placed in the middle of two pairs of neighboring sources. The air-tissue interface is at 0 mm on the vertical axis. The blue surface corresponds to the extent of necrosis. References 1. Kim, M.M.; Darafsheh, A. Light Sources and Dosimetry Techniques for Photodynamic Therapy. Photochem. Photobiol. 2020, 96, 280–294. [CrossRef] [PubMed] 2. Shaﬁrstein, G.; Bellnier, D.; Oakley, E.; Hamilton, S.; Potasek, M.; Beeson, K.; Parilov, E. Interstitial Photodynamic Therapy-A Focused Review. Cancers 2017, 9, 12. [CrossRef] [PubMed] 3. Okuyama, S.; Nagaya, T.; Sato, K.; Ogata, F.; Maruoka, Y.; Choyke, P.L.; Kobayashi, H. Interstitial Near-Infrared Photoimmunother- apy: Effective Treatment Areas and Light Doses Needed for Use with Fiber Optic Diffusers. Oncotarget 2018, 9, 11159–11169. [CrossRef] [PubMed] 4. Nagaya, T.; Okuyama, S.; Ogata, F.; Maruoka, Y.; Choyke, P.L.; Kobayashi, H. Near Infrared Photoimmunotherapy Using a Fiber Optic Diffuser for Treating Peritoneal Gastric Cancer Dissemination. Gastric Cancer 2019, 22, 463–472. [CrossRef] 5. Civantos, F.J.; Karakullukcu, B.; Biel, M.; Silver, C.E.; Rinaldo, A.; Saba, N.F.; Takes, R.P.; Vander Poorten, V.; Ferlito, A. A Review of Photodynamic Therapy for Neoplasms of the Head and Neck. Adv. Ther. 2018, 35, 324–340. [CrossRef] 6. van Dongen, G.A.M.S.; Visser, G.W.M.; Vrouenraets, M.B. Photosensitizer-Antibody Conjugates for Detection and Therapy of Cancer. Adv. Drug Deliv. Rev. 2004, 56, 31–52. [CrossRef] [PubMed] 7. Shirasu, N.; Nam, S.O.; Kuroki, M. Tumor-Targeted Photodynamic Therapy. Anticancer. Res. 2013, 33, 2823–2831. 8. Sato, K.; Ando, K.; Okuyama, S.; Moriguchi, S.; Ogura, T.; Totoki, S.; Hanaoka, H.; Nagaya, T.; Kokawa, R.; Takakura, H.; et al. Photoinduced Ligand Release from a Silicon Phthalocyanine Dye Conjugated with Monoclonal Antibodies: A Mechanism of Cancer Cell Cytotoxicity after Near-Infrared Photoimmunotherapy. ACS Cent. Sci. 2018, 4, 1559–1569. [CrossRef] 9. Mitsunaga, M.; Ogawa, M.; Kosaka, N.; Rosenblum, L.T.; Choyke, P.L.; Kobayashi, H. Cancer Cell–Selective in Vivo near Infrared Photoimmunotherapy Targeting Speciﬁc Membrane Molecules. Nat. Med. 2011, 17, 1685–1691. [CrossRef] 10. Rakuten Medical, Inc. A Phase 1/2a Multicenter, Open-Label, Dose-Escalation, Combination Study of RM-1929 and Photoim- munotherapy in Patients With Recurrent Head and Neck Cancer, Who in the Opinion of Their Physician, Cannot Be Satisfactorily Treated With Surgery, Radiation or Platinum Chemotherapy. Available online: Clinicaltrials.gov (accessed on 22 August 2022). 11. Rakuten Medical, Inc. A Phase 3, Randomized, Double-Arm, Open-Label, Controlled Trial of ASP-1929 Photoimmunotherapy Versus Physician’s Choice Standard of Care for the Treatment of Locoregional, Recurrent Head and Neck Squamous Cell Carcinoma in Patients Who Have Failed or Progressed On or After at Least Two Lines of Therapy, of Which at Least One Line Must Be Systemic Therapy. Available online: Clinicaltrials.gov (accessed on 22 August 2022). 12. Tahara, M.; Okano, S.; Enokida, T.; Ueda, Y.; Fujisawa, T.; Shinozaki, T.; Tomioka, T.; Okano, W.; Biel, M.A.; Ishida, K.; et al. A Phase I, Single-Center, Open-Label Study of RM-1929 Photoimmunotherapy in Japanese Patients with Recurrent Head and Neck Squamous Cell Carcinoma. Int. J. Clin. Oncol. 2021, 26, 1812–1821. [CrossRef] 13. Cognetti, D.M.; Johnson, J.M.; Curry, J.M.; Kochuparambil, S.T.; McDonald, D.; Mott, F.; Fidler, M.J.; Stenson, K.; Vasan, N.R.; Razaq, M.A.; et al. Phase 1/2a, Open-Label, Multicenter Study of RM-1929 Photoimmunotherapy in Patients with Locoregional, Recurrent Head and Neck Squamous Cell Carcinoma. Head Neck 2021, 43, 3875–3887. [CrossRef] [PubMed] Photonics 2022, 9, 597 18 of 18 14. Baran, T.M.; Foster, T.H. Comparison of Flat Cleaved and Cylindrical Diffusing Fibers as Treatment Sources for Interstitial Photodynamic Therapy. Med. Phys. 2014, 41, 022701. [CrossRef] [PubMed] 15. Nagaya, T.; Friedman, J.; Maruoka, Y.; Ogata, F.; Okuyama, S.; Clavijo, P.E.; Choyke, P.L.; Allen, C.; Kobayashi, H. Host Immunity Following Near-Infrared Photoimmunotherapy Is Enhanced with PD-1 Checkpoint Blockade to Eradicate Established Antigenic Tumors. Cancer Immunol. Res. 2019, 7, 401–413. [CrossRef] [PubMed] 16. Mitsunaga, M.; Nakajima, T.; Sano, K.; Kramer-Marek, G.; Choyke, P.L.; Kobayashi, H. Immediate in Vivo Target-Speciﬁc Cancer Cell Death after near Infrared Photoimmunotherapy. BMC Cancer 2012, 12, 345. [CrossRef] [PubMed] 17. Pitzschke, A.; Bertholet, J.; Lovisa, B.; Zellweger, M.; Wagnières, G. Determination of the Radiance of Cylindrical Light Diffusers: Design of a One-Axis Charge-Coupled Device Camera-Based Goniometer Setup. JBO 2017, 22, 035004. [CrossRef] 18. Zellweger, M.; Xiao, Y.; Jain, M.; Giraud, M.-N.; Pitzschke, A.; de Kalbermatten, M.; Berger, E.; van den Bergh, H.; Cook, S.; Wagnières, G. Optical Characterization of an Intra-Arterial Light and Drug Delivery System for Photodynamic Therapy of Atherosclerotic Plaque. Appl. Sci. 2020, 10, 4304. [CrossRef] 19. Cheong, W.F.; Prahl, S.A.; Welch, A.J. A Review of the Optical Properties of Biological Tissues. IEEE J. Quantum Electron. 1990, 26, 2166–2185. [CrossRef] 20. Holmer, C.; Lehmann, K.-S.; Wanken, J.; Reissfelder, C.; Roggan, A.; Gerhard, J.M.; Buhr, H.; Ritz, J. Optical Properties of Adenocarcinoma and Squamous Cell Carcinoma of the Gastroesophageal Junction. J. Biomed. Opt. 2007, 12, 1–8. [CrossRef] 21. Bargo, P.R.; Prahl, S.A.; Goodell, T.T.; Sleven, R.A.; Koval, G.; Blair, G.; Jacques, S.L. In Vivo Determination of Optical Properties of Normal and Tumor Tissue with White Light Reﬂectance and an Empirical Light Transport Model during Endoscopy. J. Biomed. Opt. 2005, 10, 034018. [CrossRef] 22. Niemz, M.H. Laser-Tissue Interactions; Springer: Berlin/Heidelberg, Germany, 2007; ISBN 978-3-540-72191-8. 23. Tuchin, V.V. Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis; Society of Photo-Optical Instrumentation Engineers (SPIE) Press Books: Bellingham, WA, USA, 2015; ISBN 978-1-62841-516-2. 24. Jacques, S.L.; Wang, L. Monte Carlo Modeling of Light Transport in Tissues. In Optical-Thermal Response of Laser-Irradiated Tissue; Welch, A.J., Van Gemert, M.J.C., Eds.; Lasers Photonics and Electro-Optics; Springer: Boston, MA, USA, 1995; pp. 73–100. ISBN 978-1-4757-6092-7. 25. Tracepro, Illumination and Non-Imaging Optical Design & Analysis Tool. Available online: Lambdares.com (accessed on 22 August 2022). 26. Tsukerman, E.; Veomett, E. A Simple Proof of Cauchy’s Surface Area Formula. arXiv 2016, arXiv:1604.05815. 27. Ishimaru, A. Wave Propagation and Scattering in Random Media. Volume 1—Single Scattering and Transport Theory; Academic Press: Cambridge, MA, USA, 1978; Volume 1. 28. Tromberg, B.J.; Svaasand, L.O.; Fehr, M.K.; Madsen, S.J.; Wyss, P.; Sansone, B.; Tadir, Y. A Mathematical Model for Light Dosimetry in Photodynamic Destruction of Human Endometrium. Phys. Med. Biol. 1996, 41, 223–237. [CrossRef] [PubMed] 29. Streeter, S.S.; Jacques, S.L.; Pogue, B.W. Perspective on Diffuse Light in Tissue: Subsampling Photon Populations. JBO 2021, 26, 070601. [CrossRef] [PubMed] 30. Ismael, F.S.; Amasha, H.; Bachir, W. Optimized Cylindrical Diffuser Powers for Interstitial PDT Breast Cancer Treatment Planning: A Simulation Study. BioMed Res. Int. 2020, 2020, e2061509. [CrossRef] [PubMed] 31. Robinson, D.J.; de Bruijn, H.S.; van der Veen, N.; Stringer, M.R.; Brown, S.B.; Star, W.M. Fluorescence Photobleaching of ALA- Induced Protoporphyrin IX during Photodynamic Therapy of Normal Hairless Mouse Skin: The Effect of Light Dose and Irradiance and the Resulting Biological Effect. Photochem. Photobiol. 1998, 67, 140–149. [CrossRef] [PubMed]

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Photonics – Multidisciplinary Digital Publishing Institute

Published: Aug 23, 2022

Keywords: interstitial photoimmunotherapy; photodynamic therapy; cylindrical light diffusers; head and neck cancers; Monte Carlo; light dosimetry

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

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

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

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

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