1. Introduction

Photodynamic therapy (PDT) is dynamic and multifaceted because it is based on the interactions between a treatment light at a particular wavelength, a photosensitizer, and tissue oxygenation ([³O₂]) [1,2]. PDT is uniquely advantageous because its recovery time is comparatively short relative to other treatment modalities and not associated with the development of delayed toxicities. Moreover, PDT provides for good cosmesis. Unlike radiotherapy, however, dosimetric platforms have not been commercialized to guide the delivery of PDT. In fact, there remains much research on a well-defined dose metric that accurately predicts biological response in PDT. ROS are accepted as the cytotoxic agents responsible for therapeutic effect in PDT, and the direct detection of ROS can provide an accurate quantity to guide treatments and predict treatment outcomes. However, in vivo detection of ROS during clinical PDT is very challenging due to its weak signal and short lifetime. To overcome this, a macroscopic reactive oxygen species explicit dosimetry (ROSED) model was recently developed to calculate for the accumulated reacted ROS concentration ([ROS]_rx) that is predictive of PDT treatment outcome [3,4].

Photofrin (Porfimer sodium), is a first-generation photosensitizer purified from hematoporphyrin derivative (HpD) and consists of a mixture of oligomers of hematoporphyrin joined to each other by ether and ester bonds [5]. Photofrin was approved by the US Food and Drug Administration (FDA) in 1995 for the treatment of advanced obstructive esophageal cancer, and in 1998 for the treatment of early-stage endobronchial non-small-cell lung cancer, and in 2003 for the ablation of high-grade dysplasia in Barrett’s esophagus. Photofrin is a type II photosensitizer in which the triplet state transfers energy to ground state tissue oxygen (³O₂) to produce highly reactive singlet oxygen (¹O₂) [6]. Generated ¹O₂ causes cytotoxicity and eventually cell death and/or therapeutic effects [1]. Photofrin-mediated PDT is currently being studied in several clinical trials [2,7,8].

Using macroscopic ROSED model of light fluence (rate), Photofrin drug concentration, and tissue oxygen concentration ([³O₂]), [ROS]_rx can be determined to evaluate its effectiveness as a dosimetric predictor for Photofrin-mediated PDT outcome. To our knowledge, this study is the first to demonstrate that [ROS]_rx,meas, which is based on real-time measurements of light fluence (rate), Photofrin concentration, and measured [³O₂], correlates with the cure index of Photofrin-PDT at 14 days after the treatment of mice. In addition, we investigate the relationship between various dose metrics (fluence, PDT dose, [ROS]_{rx, calc}, and [ROS]_{rx, meas}) and cure index at 14 days in the same RIF mouse model. The major photochemical parameters in the macroscopic ROSED model have been determined for the photosensitizer Photofrin and a drug-light interval of 24 hours (see Table 1) [3,9]

Table 1. Photochemical parameters obtained from literature [9].

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2. Materials and methods

2.1 Tumor model

Radiation-induced fibrosarcomas (RIF) cells were cultured and injected intradermally over the right shoulders of 6-8 weeks old female C₃H mice (NCI-Frederick, Frederick, MD). A cell solution of 30 μl was injected at a concentration of 1×10⁷ cells/ml, as described previously [9]. Animals were under the care of the University of Pennsylvania Laboratory Animal Resources. All studies were approved by the University of Pennsylvania Institutional Animal Care and Use Committee. The fur of the treatment region was clipped prior to cell inoculation, and the treatment area was depilated with Nair (Church & Dwight Co., Inc., Ewing, NJ) at least 24 hours prior to measurements and treatment. Tumors were treated when they were 3∼5 mm in diameter. Mice were given a chlorophyll-free (alfalfa-free) rodent diet (Harlan laboratories Inc., Indianapolis, IN) at least 10 days prior to treatment to eliminate the fluorescence signal from chlorophyll-breakdown products, which have a similar range to the photosensitizer spectra obtained in this study. The photosensitizer fluorescence was used to determine the in vivo concentrations in this study, using methods described previously [9,10]. Throughout treatment, mice were maintained under anesthesia on a heating pad at 38°C (see Fig. 1(a)).

 

Fig. 1. Experiment setup with the (a) collimated beam treatment of RIF tumor on mouse shoulder, (b) A handheld broadband reflectance spectroscopy contact probe was used to measured the optical properties and drug concentration before and after PDT. (c) Oxylite Pro oxygen monitor with a fluorescence-based bare-fiber oxygen probe (Oxford Optronix, Oxford, UK)

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2.2 Photodynamic therapy treatment protocol

Photofrin (Pinnacle Biologics, Chicago, Illinois) at a dose of 5 mg∕kg was injected through the mouse tail vein as described previously [9,11]. At 18-24 hours drug–light interval, superficial irradiation of the tumor was performed with a 630-nm laser (Biolitec Inc., A-1030, East Longmeadow, MA, USA). A microlens fiber was coupled to the laser to irradiate the tumor uniformly (see Fig. 1(a)). Mice were treated within in-air fluence rates (ϕ_air) of 75 or 150 mW/cm² and total in-air fluences of 50–150 J/cm² to induce different PDT outcomes. The “in-air fluence rate” is defined as the calculated irradiance determined by laser power divided by the treatment area (1 cm diameter spot size). The “in-air fluence” was calculated by multiplying the “in-air fluence rate” by the treatment time. The light dose group was divided into 8 treatment groups, each comprised of 1 to 4 mice, for different in-air fluence, ϕ_air, and mean [³O₂], see Table 2. For the same light fluence and ϕ_air, mean [³O₂] assigning to different groups are different by 1.8 times. RIF-bearing mice that received neither light irradiation nor Photofrin were used as controls (n = 5). Treatment conditions are summarized in Table 2.

Table 2. In-air light fluence, in-air light fluence rate, ϕ_air, Mean oxygen concentration over PDT treatment, [³O₂]_mean, photosensitizer concentrations pre and post PDT, [PS]_pre and [PS]_post, fluence at 3 mm, PDT dose at 3 mm depth, calculated and measured reacted reactive oxygen species concentrations, [ROS]_rx,calc, and [ROS]_rx,meas at 3 mm depth, tumor re-growth rate, k, and cure index, CI, for each PDT treatment group. Number of mice per group is shown in the second column.

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2.3 Oxygen measurements

The in vivo tissue oxygen partial pressure pO₂ was measured during PDT using a phosphorescence-based ³O₂ probe (OxyLite Pro, Oxford Optronix, Oxford, United Kingdom). A bare-fiber-type probe (NX-BF/O/E, Oxford Optronix, Oxford, United Kingdom) was placed inside the tumor at a 3 mm depth from the treatment surface (Fig. 1(c)). The ³O₂ concentration ([³O₂]) was calculated by multiplying the measured pO₂ with the ³O₂ solubility in tissue, which is 1.295 μM/mmHg [12,13] . Measured [³O₂]₀ and [³O₂](t) was used to calculate for [ROS]_rx by the macroscopic ROSED model [9].

2.4 Measurement of Photofrin concentration

At 18–24 hours after Photofrin administration, measurements of light fluence rate, photosensitizer concentration and [³O₂] were performed. In-vivo Photofrin fluorescence spectra was obtained using a custom-made multi-fiber contact probe ( Fig. 1(b)) before and after PDT at surface [9,14]. The probe was connected to a 405 nm laser (Power Technology Inc., Little Rock, AR, USA) for the fluorescence excitation of Photofrin and a multichannel CCD spectrograph (InSpectrum, Princeton Instruments, Trenton, NJ, USA) for the collection of the fluorescence spectra. The distance between the source and detector fibers for fluorescence is 2.01 mm. Due to the properties of light propagation in tissue, the predominant region that is probed by these measurements will be ∼one-quarter to one-half (mean ∼ 1/3) of the source-detector separation distance and therefore a depth ∼ 0.5–1 mm (mean ∼0.67 mm) below the tissue surface [15]. In this way, signal is dominated by Photofrin levels in the superficial part of the tumor, with less contribution from the overlaying skin or deeper tumor tissue. Therefore, fluorescence contribution is reduced, if not completely eliminated, from the skin. Due to the shorter optical penetration depth of our excitation wavelength (405nm) compared to the wavelength used in the quoted study (532 and 633 nm) [15], one would expect more fluorescence originates in the immediate region of source on tissue, resulting in an asymmetrical “banana-shaped” fluorescence source. Since the entire tumor is not probed, but rather signal is generated from predominantly the superficial part of the tumor [15], an assumption of homogenous Photofrin concentration distribution is needed. Thus, we additionally performed an ex vivo validation relative to Photofrin concentration in the whole tumor, as is described in section 2.5 below. In order to provide absolute quantities for probe-measured in vivo photosensitizer concentration, the in vivo measured Photofrin spectra was compared with those of tissue-simulating (intralipid) phantoms with known photosensitizer concentrations using an SVD method [16], as described in detail elsewhere [9]. The attenuation of the fluorescence signal due to the light absorption and scattering by tissues was corrected by applying an empirical correction factor described elsewhere [9]. Average tissue optical properties of mice, which were within 10% compared to the individual tissue optical properties in terms of predicted light fluence rate vs. depth for depth up to 3 mm at emission wavelength (632 nm) [9], were used for the correction factor. The accuracy of the contact probe has been previously validated in both preclinical [9,17] and clinical [18] studies for Photofrin, as well as other photosensitizers [10,19].

2.5 Ex vivo validation of photofrin concentration

In vivo fluorescence measurements of the photosensitizer concentration as described above were performed for all tumors before PDT. To evaluate the accuracy of the in vivo fluorescence measurements, ex vivo measurements of the Photofrin concentration were performed in a separate set of mice and compared with the Photofrin concentration determined from in vivo measurements. Five mice were administered Photofrin at different doses between (2.5 to 13 mg/kg). In vivo fluorescence measurements were taken of tumor in each mouse at 18 - 24 hours after Photofrin administration. After fluorescence measurements, mice were sacrificed and the tumors (∼ 1 mm³) were excised, protected from light, and stored at −80°C. For ex vivo analyses, homogenized solutions of the tumors were prepared using Solvable (PerkinElmer, Waltham, Massachusetts). The fluorescence of the homogenized sample was measured by a spectrofluorometer (FluoroMax-3; Jobin Yvon, Inc.) with an excitation wavelength of 405 nm and an emission range from 630 to 750 nm with two emission maximums at 635 nm and 705 nm. The photosensitizer concentration in the tissue was calculated based on the change in fluorescence resulting from the addition of a known amount of Photofrin to each sample after its initial reading. The in vivo measurements were correlated to ex vivo data using a linear fit to examine their agreement based on the goodness of the fit (R²) [See Fig. 2].

 

Fig. 2. Comparison between in vivo and ex vivo measured Photofrin concentrations. Solid line is a linear fit for ex vivo vs. in vivo measured Photofrin concentration (symbols) at 24 hours. Dished line is for y = x.

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The ex vivo measurements of Photofrin concentration were compared to those obtained in vivo using the contact probe method to evaluate the accuracy of the in vivo acquired Photofrin concentrations. The linear fit of ex vivo to in vivo measured Photofrin levels (solid line) shows close agreement between the in vivo and ex vivo Photofrin concentrations, y = 0.985x, with a fitting goodness of R²= 0.99. The dashed line represents the line for y = x, if the two measurements were completely in agreement. Five measurements were taken from each tumor. Each data point in Fig. 2 represents the mean value of these measurements. The horizontal and vertical error bars represent the standard deviation of 5 ex vivo and in vivo measurements.

2.6 Tumor regrowth rate analysis

Tumor volumes were tracked daily, for 14 days, after PDT. Width (a) and length (b) were measured with slide calibers, and tumor volumes (V) were calculated using V =π/6 × a² × b [20]. Tumor regrowth factor (k) was calculated by the best exponential fit [with a form f(d) = Ae^kd] to the measured volumes over the days (d). CI was calculated for each treatment group as

2.7 Reactive oxygen species explicit dosimetry

The PDT process can be described by a set of kinetic equations which can be simplified to describe the creation of [ROS]_rx [3,4,21]. These equations are dependent on the temporal and spatial distribution of ϕ, photosensitizer concentration ([S₀]), ground state oxygen concentration ([³O₂]), oxygen supply rate (g), and the photosensitizer-specific reaction-rate parameters (δ, β, σ, and ξ). The relevant equations are (see Appendix for detailed derivation):

The details of the five parameters involved in the kinetic equations can be found elsewhere (see Table 1) [9]. ξ is the photochemical oxygen consumption rate per light fluence rate and photosensitizer concentration under ample ³O₂ supply. σ is the probability ratio of a ROS molecule to react with ground state photosensitizer compared to the ROS molecule reacting with a cellular target. β represents the ratio of the monomolecular decay rate of the triplet state photosensitizer to the bimolecular rate of the triplet photosensitizer quenching by ³O₂. δ is the low concentration correction factor, and g is the maximum macroscopic oxygen perfusion rate. [ROS]_rx can be calculated in two ways: [ROS]_rx,calc is calculated by solving Eqs. (2)–(4) using measured light fluence (rate) and [S₀], i.e., [³O₂](t) is determined from Eq. (3) using an initial ground state oxygen concentration of ([³O₂]₀) based on measurement; [ROS]_rx,meas is determined by integrating the term of the right-hand side of Eq. (4) over the time course of PDT treatment using the measured ϕ, [S₀] and [³O₂]:

Notice that the effect of photobleaching and oxygen depletion during PDT are fully accounted for since the temporal dependence of both [S₀] and [³O₂] are included in Eq. (5). Both [ROS]_rx,calc and [ROS]_rx,meas are determined at 3 mm depth using the (calculated) ϕ and calculated and measured [³O₂] concentration at 3 mm depth, respectively, to ensure that their minimum values cover the maximum extent of RIF tumors used in the study. [S₀] is assumed to be uniform throughout the tumor. If one uses 1 mm or 2 mm instead of 3 mm, the value of [ROS]_{rx, calc} (or [ROS]_{rx, meas}) will increase, thus the resulting threshold [ROS]_{rx, calc} (or [ROS]_{rx, meas}) value but the general curve shape (Fig. 6(c) and 6(d)) will not change.

Monte-Carlo (MC) statistical analysis was used to produce the grey area due to uncertainty in the parameters a, b for the sigmoid curve y = 1/(1+a*exp(-bx)). The means and standard deviations of the simulation parameters a and b were obtained using the global optimization toolbox of Matlab(cftool.m). In the MC simulation, we selected 1000 parameter pairs (a, b) within the standard deviation (δa, δb) using a random number generator with normal distributions, the resulting calculated y = 1/(1+a*exp(-bx)) for all (a, b) for a particular dosimetry metric (fluence, PDT dose, [ROS]_rx,calc or [ROS]_rx,meas) was used to generate an cumulative probability distribution for each of the y values between [0, 1]. We then found corresponding x values for the 5% tiles and 95% tiles of the cumulative probability distribution; these formed the two bounds of the grey zone, and the left and right bounds were joined to form the uncertainty (grey) areas.

3. Results

Photofrin-mediated PDT with various in-air fluences and ϕ_air were performed in mice bearing RIF tumors. Light fluence rate, photosensitizer concentration, and tissue oxygenation were measured to calculate fluence, PDT dose, and [ROS]_rx,calc using the macroscopic model (Eqs. (2)–(4) based on the measured light fluence (rate) and PS concentrations, and [ROS]_rx,meas. Table 2 summarizes all treatment conditions as well as the measured and calculated quantities using the photochemical parameters summarized in Table 1.

To compare the regrowth rate between different tumors, volumes were normalized to average tumor volume on day 0 (∼12 mm³). Figure 3 shows the normalized tumor volume versus time (in days) for the 8 treatment groups and the control group along with the fits to the data with an exponential growth equation. These exponential fits to the data determine the value of k for each treatment group of mice. The statistical analyses showed all treated groups to experience a reduction in tumor regrowth rate compared to the control (all with p < 0.05).

 

Fig. 3. Tumor volumes (V) over days after V = V₀e^kd PDT treatment, V=π/6a²b, where a and b are the width and length of tumor. Solid lines are the exponential fit to the data with a functional form of e^kd, where d is days ater PDT treatment. The resulting tumor regrowth rates, k, and its uncentrainty, δk, are listed in Table 2. The legend for each group lists:in-air fluence rate, in-air fluence, treatment time (in seconds), Photofrin concentration (in μM), tumor re-growth rate, and R² of the fitting the the exponential equation.

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Figure 4 shows the temporal dependence of the measured photosensitizer concentrations (symbols) and the macroscopic model calculated photosensitizer concentrations (lines).

 

Fig. 4. The temporal changes of Photofrin concentration vs. light fluence at 3 mm depth for the treatment conditions. The lines represent calculations of Photofrin concentration based on Eqs. (2)–(3) during PDT treatment. The symbols represent measured PS concentration ([S₀]). The average of [S₀] is shown in the figure legend for each condition. The uncertainty of [S₀] is listed in Table 2.

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Measured [³O₂] was used to refine the photochemical parameters previously determined for the singlet oxygen explicit dosimetry model used to determine [ROS]_rx.meas. Measured data are shown with solid lines in Fig. 5. For comparison, temporal variation of oxygen concentration [³O₂] during treatment, based on the macroscopic model (Eqs. (2)–(4), are also plotted as dashed lines. Fluence, PDT dose, calculated [ROS]_rx,calc, and measured [ROS]_rx,meas at 3 mm were compared as dosimetric quantities for their correlation with treatment outcome for Photofrin-mediated PDT of RIF tumors in Fig. 6.

 

Fig. 5. The temporal dependence of [³O₂] concentration for different treatment conditions. The dashed lines represent the calculated temporal dependence of [³O₂] based on macroscopic model (Eqs. (2)–(4) using the photochemical parameters in Table 1. The solid lines are the represnetative temporal dependence of measured [³O₂] for the treatment conditions.

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Fig. 6. CI plotted against (a) fluence at a 3 mm tumor depth, (b) PDT dose at 3 mm depth, (c) calculated reacted reactive oxygen species concentration ([ROS]_{rx, calc}) at 3 mm depth calculated using Eqs. (2)–(4) and the parameters summarized in Table 1, and (d) measured reacted oxygen species ([ROS]_{rx, meas}) at 3 mm depth. The solid lines show the best-fit to the data with functional forms CI = 1/(1 + 223.5e^−0.05269x), 1/(1 + 9961e^−0.02097x), 1/(1 + 396.6e^−5.854x), and 1/(1 + 6517e^−9.157x) with R² = 0.777, 0.874, 0.921, and 0.972 for (a), (b), (c) and (d), respectively. The gray region indicates the upper and lower bounds of the fit with 90% confidence level. The delimiters of grey regions are obtained from a Monte -Carlo statistical simulation of the sigmoid model (see text in section 2.7 for details).

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4. Discussion

As shown in Fig. 2, the accuracy of the in vivo Photofrin concentration determined by our fluorescence spectroscopy is validated by comparing it with ex vivo measurements in mice without PDT. In vivo fluorescence measurements were taken at 18-24 hours after Photofrin administration in mice without PDT, followed by tumor excision and ex vivo evaluation in the same mice tumor. The agreement between in vivo and ex vivo Photofrin concentration at 18 - 24 hours is within 1.5% because they are linear to each other and the slope of the fit to the in vivo and ex vivo Photofrin concentration is 0.985.

Compared to control mice, all mice treated with total fluences greater than 50 J/cm² had significant control of the tumor re-growth after PDT (see Fig. 3). We found that regrowth rate decreased when in-air light fluence increased (e.g., comparing the group of mice treated to 250 J/cm² and that of 50 J/cm²). However, in-air light fluence alone is not a good predictor of tumor regrowth because of significant variations in the in vivo concentrations of PS and/or oxygen ([³O₂]). As a result, we have split each (in-air fluence rate, in-air fluence) group further into subgroups as a function of PS concentration and [³O₂], whenever there was a significant difference.

Based on a previous study [22], an empirical six-parameter fitting equation was used to fit the depth (d) dependence of the Monte-Carlo simulated data for a 1 cm diameter field, with μ_a = 0.9 cm⁻¹ and μ_s’ = 8.4 cm⁻¹[9]. The equation is of the following form [19]:

Figure 4 shows temporal variations in PS concentration during Photofrin-mediated PDT. There is good agreement between the measurement and the macroscopic model predictions.

Figure 5 shows marked difference between measured (solid lines) and calculated (dashed lines) temporal dependence of [³O₂]. This is probably caused by biological and physiological effects of PDT that are not included in the model, e.g., it is known that the blood flow will change after PDT [23] but our model currently assumes a constant average blood perfusion rate, g. So, this assumption of our model does not represent reality, thus the need for direct oxygen measurement in-vivo.

Fluence, PDT dose at 3 mm, and [ROS]_rx at 3 mm were compared as dosimetric quantities to estimate the outcome of Photofrin-mediated PDT of murine RIF tumors. Outcome was evaluated by the calculation of CI. No tumor re-growth up to 14 days after treatment resulted in a CI of 1. The goodness of the fit and the corresponding upper and lower bounds of the fits (gray area) to the fluence, PDT dose, [ROS]_rx,calc, and [ROS]_rx,meas are presented in Fig. 6. PDT dose is calculated as a time integral of the product of PS uptake and measured light fluence rate at 3 mm. It is proportional to light fluence rate. We used Eqs. (2)–(4) and the photophysiological parameters shown in Table 1 to calculate [ROS]_rx,calc. We used Eq. (5) and measured light fluence, [PS] and [³O₂] to determine [ROS]_{rx, meas}. The goodness of the fit and the corresponding upper and lower bounds of the fit with 90% confidence interval (gray area) to the fluence, PDT dose, the calculated [ROS]_{rx, calc}, and the measured [ROS]_rx,meas are presented in Fig. 6. Figure 6(a) plots the sigmoid curve by which fluence correlates with PDT outcome; it exhibits large uncertainties as defined by the large bounds of the gray area as well as by the low value of R² = 0.777. As shown in Fig. 6(b), PDT dose allows for reduced subject variation and improved predictive efficacy as compared to fluence. PDT dose showed a better correlation with CI with a higher value of R² = 0.874 and a narrower band of gray area as it accounts for both light dose and tissue [PS] levels. However, PDT dose overestimates [ROS]_rx in the presence of hypoxia as it does not account for the oxygen dependence of ROS (mostly ¹O₂) quantum yield. The goodness of fit R² = 0.91 and 0.972 and the narrowest gray area in Fig. 6(c) and 6(d) shows that the [ROS]_rx,calc and [ROS]_rx,meas correlates better with CI than either fluence and PDT dose. Unlike BPD-PDT where [ROS]_rx,calc agrees with [ROS]_rx,meas [19], there were difference between model calculated [ROS]_rx,calc and measured [ROS]_rx,meas for Photofrin due to difference in oxygen temporal variation from the model prediction (Fig. 5), as discovered in previous studies [17,19]. However, the difference between [ROS]_rx,calc (Fig. 6(c)) and [ROS]_rx,meas (Fig. 6(d)) is not statistically significant at the current study, probably due to limited animals and treatment conditions used.

Based on the findings of this study, PDT dose, and [ROS]_rx exhibit threshold dose behavior as they can be fit by a sigmoid function (S(x) = 1/(1+e(-(x-x₀)/w₀)), where x₀ = 439 μM J/cm² with uncertainty w₀ = 48 μM J/cm² for PDT dose, x₀ = 1.02 mM with uncertainty w₀ = 0.17 for [ROS]_rx,calc, and x₀ = 0.96 mM with uncertainty w₀ = 0.11 for [ROS]_rx,meas, respectively. For PDT dose, x₀ can be converted to the absorbed dose by Photofrin by multiplying by the extinction coefficient (ε = 0.0035 µM⁻¹ cm⁻¹), resulting in 1.53 ± 0.17 J/cm³, which corresponds to (4.9 ± 0.5)×10¹⁸ photons/cm³ (by dividing the energy per photon hcλ = 3.15 × 10⁻¹⁹ J for λ = 630 nm). This PDT dose threshold value is comparable to the 3.4×10¹⁸ photons/cm³ reported by others for Photofrin in-vitro [24,25]. However, it is smaller than the 6.7×10¹⁸ photons/cm³ reported in our previous study for the same CI at 14 days at CI = 50% [11]. This is probably caused by the smaller size of tumor used in the current study—tumors of < 100 mm³ were evaluated in the present study, compared to tumors of 100–400 mm³ in the previous work. The threshold [ROS]_rx of 0.96 mM is in good agreement with our previous studies for Photofrin (0.92 mM) [11], BPD (0.98 mM) [19], and HPPH (0.98 mM) [10] for CI at 14 days.

5. Conclusion

The response of mouse RIF tumors to PDT depends on the tissue oxygenation, photosensitizer uptake, total energy delivered, and the ϕ at which the treatment is delivered. An accurate dosimetry quantity for the evaluation of the treatment outcome should account for all of these parameters. This study evaluated the efficacy and outcomes of different PDT treatments and how fluence, PDT dose, [ROS]_rx,calc, and [ROS]_rx,meas compare as dosimetric quantities. The correlation between CI and [ROS]_rx suggests that [ROS]_{rx, meas} at 3 mm is the best quantity to predict the treatment outcome for a clinically relevant tumor regrowth endpoint. PDT dose is a better dosimetric quantity than fluence, but is worse than [ROS]_rx,calc and [ROS]_rx,meas as it does not account for the consumption of [³O₂] for different ϕ. For Photofrin in RIF tumors, the temporal dependence of in vivo oxygen concentration during PDT can’t be well modeled by our macroscopic model, thus measuring [³O₂] during PDT to obtain [ROS]_rx,meas can improve the estimation of [ROS]_rx (vs. [ROS]_rx,calc) and provide better prediction of PDT outcome (CI at 14 days). We have shown that it is possible to measure oxygen using bloodflow to predict [ROS]_rx in human clinical trial [18].

Appendix

Mathematical modeling of PDT photochemical reactions

The photosensitizer molecule in its ground state [S₀] is a singlet such that all electron spins are paired. When ground state photosensitizer molecule absorbs a photon with appropriate quantum energy or wavelength, an electron is promoted to a higher energy level in the same spin orientation as it was in the ground state as shown by the Jablonski diagram in Fig. 7. The excited singlet state photosensitizer is very unstable and short lived. It loses its excess energy by production of heat through internal conversion and returns to its ground state by emitting a fluorescent photon. The excited singlet state photosensitizer molecule may also undergo intersystem crossing, in which the excited electron spin is reversed to form a more stable triplet state photosensitizer [T₁]. The triplet state photosensitizer molecule can decay back to ground state by emitting a phosphorescent photon. This decay is slow since it involves a spin forbidden process, with a lifetime of microseconds compared to nanoseconds for the decay of an excited singlet state photosensitizer. The excited triplet state photosensitizer can undergo two types of photochemical reactions, characterized as type I and type II reactions that create cytotoxic reactive oxygen species that ultimately result in photodynamic damage. Both type I and type II reactions can occur simultaneously and the ratio between these processes depends on the type of photosensitizer, the concentration of substrate and the availability of oxygen.

 

Fig. 7. Diagram of the photoactivation of photosensitizer in the presence of oxygen and biomolecules. S₀, S₁, and T₁ are the ground state, excited singlet state and excited triple state photosensitizer, respectively. ³O₂ and ¹O₂ are the ground state triplet and excited singlet oxygen. O₂^−• is the superoxide anions and A is the biological acceptor of ROS. k₀ to k₈ are photochemical reaction rate constants.

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For both type I and II primary photochemical reactions can be described using a set of coupled differential equations (Eqs. (7–13)) as shown below [4]:

(7)$$\frac{{d\left[ {{S_0}} \right]}}{{dt}} = - {k_0}\left[ {{S_0}} \right] - {k_{12}}\left[ {{} ^1{O_2}} \right]\left( {\left[ {{S_0}} \right] + \delta } \right) - {k_{11}}\left[ {O_2^{ - \cdot }} \right]\left( {\left[ {{S_0}} \right] + \delta } \right) + {k_2}\left[ {{T_1}} \right]\left[ {{} ^3{O_2}} \right] + {k_3}\left[ {{S_1}} \right] + {k_4}\left[ {{T_1}} \right],$$

(8)$$\frac{{d[{{S_1}} ]}}{{dt}} ={-} ({{k_3} + {k_5}} )[{{S_1}} ]+ {k_0}[{{S_0}} ],$$

(9)$$\frac{{d[{{T_1}} ]}}{{dt}} ={-} {k_2}[{{T_1}} ][{{}^3{O_2}} ]- {k_4}[{{T_1}} ]+ {k_5}[{{S_1}} ]- {k_8}[{{T_1}} ][A ],$$

(10)$$\frac{{d[{{}^3{O_2}} ]}}{{dt}} ={-} {S_\Delta }{k_2}[{{T_1}} ][{{}^3{O_2}} ]- {S_I}{k_2}[{{T_1}} ][{{}^3{O_2}} ]+ {k_6}[{{}^1{O_2}} ]+ \Gamma ,$$

(11)$$\frac{{d[{{}^1{O_2}} ]}}{{dt}} ={-} {k_{12}}[{{}^1{O_2}} ]({[{{S_0}} ]+ \delta } )+ {S_\Delta }{k_2}[{{T_1}} ][{{}^3{O_2}} ]- {k_6}[{{}^1{O_2}} ]- {k_{72}}[A ][{{}^1{O_2}} ],$$

(12)$$\frac{{d[{O_2^{ -{\cdot} }} ]}}{{dt}} ={-} {k_{11}}[{O_2^{ -{\cdot} }} ]({[{{S_0}} ]+ \delta } )+ {S_I}{k_2}[{{T_1}} ][{{}^3{O_2}} ]- {k_{71}}[A ][{O_2^{ -{\cdot} }} ],$$

(13)$$\frac{{d[A ]}}{{dt}} ={-} {k_{72}}[A ][{{}^1{O_2}} ]- {k_{71}}[A ][{O_2^{ -{\cdot} }} ]- {k_8}[{{T_1}} ][A ].$$

k₀ to k₈ are rate constants of photochemical processes as illustrated in Fig. 7 and their definitions are listed in Table 3. [S₀], [S₁], and [T₁] are the concentrations of ground state, excited singlet state and excited triple state photosensitizers, respectively. [³O₂] is the concentration of ground state triple oxygen. [$O_2^{ - \cdot }$] and [¹O₂] are the concentration of superoxide anions and singlet oxygen molecules produced in type I and II reactions, respectively. [A] is the concentration of biological substrate that reacts with $O_2^{ - \cdot }$ and ¹O₂ and Γ is the oxygen supply rate.

Table 3. Definition of photochemical reaction rate constants.

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If one only cares about the dynamic processes of PDT in the time scale of a few seconds to hours, then the time derivative on the right hand sides of equations (8), (9), (11), and (12) can be set to zero because these processes are known to be very fast (µs or less) and converge to equilibrium states. Solving for this equilibrium state, the equations become

(14)$$[{{S_1}} ]= \frac{1}{{{k_3} + {k_5}}}\frac{\varepsilon }{{hv}}\phi [{{S_0}} ],$$

(15)$$[{{T_1}} ]= \frac{{{k_5}}}{{{k_3} + {k_5}}}\frac{1}{{{k_2}}}\frac{1}{{[{{}^3{O_2}} ]+ \beta }}\frac{\varepsilon }{{hv}}\phi [{{S_0}} ],$$

(16)$$[{{}^1{O_2}} ]= \frac{1}{{{k_{12}}({[{{S_0}} ]+ \delta } )+ {k_6} + {k_{72}}[A ]+ {k_9}[Q ]}}{\xi _{II}}\frac{{[{{}^3{O_2}} ]}}{{[{{}^3{O_2}} ]+ \beta }}\phi [{{S_0}} ],$$

(17)$$[{O_2^{ -{\cdot} }} ]= \frac{1}{{{k_{11}}({[{{S_0}} ]+ \delta } )+ {k_{71}}[A ]}}{\xi _I}\frac{{[{{}^3{O_2}} ]}}{{[{{}^3{O_2}} ]+ \beta }}\phi [{{S_0}} ],$$

(18)$$\frac{{d[{{S_0}} ]}}{{dt}} ={-} ({{\xi_{II}}{\sigma_{II}} + {\xi_I}{\sigma_I}} )\frac{{[{{}^3{O_2}} ]}}{{[{{}^3{O_2}} ]+ \beta }}({[{{S_0}} ]+ \delta } )\phi [{{S_0}} ]- \eta \frac{1}{{[{{}^3{O_2}} ]+ \beta }}\phi [{{S_0}} ],$$

(19)$$\frac{{d[{{}^3{O_2}} ]}}{{dt}} = [{ - ({{\xi_{II}} + {\xi_I}} )+ {\xi_{II}}{\tau_\triangle }({{k_6}} )} ]\frac{{[{{}^3{O_2}} ]}}{{[{{}^3{O_2}} ]+ \beta }}\phi [{{S_0}} ] + {\Gamma},$$

(20)$$\frac{{d[A ]}}{{dt}} ={-} ({k_{72}}{\xi _{II}}{\tau _\triangle } + {k_{71}}{\xi _I}{\tau _S})[A ]\frac{{[{{}^3{O_2}} ]}}{{[{{}^3{O_2}} ]+ \beta }}\phi [{{S_0}} ]- \eta \frac{1}{{[{{}^3{O_2}} ]+ \beta }}\phi [{{S_0}} ].$$

In the macroscopic reactive oxygen species explicit dosimetry (ROSED), the oxygen supply term Γ in Eqs. (10) and (19) is expressed as [13]

(21)$$\varGamma = g\left( {1 - \frac{{[{{}^3{O_2}} ]}}{{[{{}^3{O_2}} ]({t = 0} )}}} \right),$$

where [³O₂] (t=0) is the initial oxygen concentration before PDT treatment and g is the maximum oxygen supply rate when there is no oxygen gradient. g can be a function of time to account for blood flow change during PDT. All of the parameters ($\xi $, ${\xi _I}$, ${\xi _{II}}$, $\sigma $, ${\sigma _I}$, ${\sigma _{II}}$, ${\tau _f}$, ${\tau _\triangle }$, ${\tau _S}$) are given in Table 4. $\sigma = ({{\xi_{II}}{\sigma_{II}} + {\xi_I}{\sigma_I}} )/\xi $ where $\xi = {\xi _{II}} + {\xi _I}$. For the in vivo scenario, it is assumed that the concentration of biological acceptors is large, so ${k_{72}}[A ]{\tau _\triangle } \approx 1$ and ${k_{71}}[A ]{\tau _S} \approx 1$. Furthermore, ${\sigma _{II}}({[{{S_0}} ]+ \delta } )\ll 1$.

Table 4. Definition of some key parameters used in PDT modeling.

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Utilizing Eq. (20), the amount of biological acceptor that has reacted with a reactive oxygen species ([ROS]_rx) can be defined by the following

(22)$$\frac{{d{{[{ROS} ]}_{rx}}}}{{dt}} = f\left\{ { - \xi \frac{{[{{}^3{O_2}} ]}}{{[{{}^3{O_2}} ]+ \beta }}\phi [{{S_0}} ]- \eta \frac{1}{{[{{}^3{O_2}} ]+ \beta }}\phi [{{S_0}} ]} \right\}$$

where f is the fraction of ROS interacting with [A]. Here, the first term relates to the fraction of acceptors that reacted due to ROS-mediated reactions, and the second term relates to the fraction that reacts under hypoxic conditions or any other non-oxygen-mediated reactions, such as triplet interactions.

In cases where triplet interaction is negligible ($\eta = 0$), the reactive oxygen species ([ROS]_rx) can be described by (assuming f = 1) as Eq. (4). Similarly, since $\eta = 0,\; $ k₆τ_Δ<<1, $\xi = {\xi _{II}} + {\xi _I}$, and $\sigma \xi = ({{\xi_{II}}{\sigma_{II}} + {\xi_I}{\sigma_I}} ),$ one can rewrite Eqs. (18) and (19) as Eqs. (2) and (3).

The required photochemical parameters can be reduced from 11 (δ, g, k0, …, k8) to 5 (δ, β, ξ, σ, g), with some of the latter expressed as ratios of the former. The definitions for the photochemical parameters, ξ, β, η, δ, and σ, are shown in Table 4, along with their relationships to the reaction rate constants.

Funding

National Cancer Institute (P01 087971), (R01 CA236362), (R01 CA154562), (R44 CA183236).

Acknowledgments

The authors would like to thank Min Yuan, Joann Miller, and Shirron Carter for their advice concerning the mouse studies and protocols. We thank Prof. Wensheng Guo for help with statistics analysis.

Disclosures

T.C. Zhu and T.M. Busch declares a role on the Advisory Board for Simphotek, Inc. There is no conflict of interest to declare for all other authors.

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