Reactive oxygen species explicit dosimetry to predict local tumor growth for Photofrin-mediated photodynamic therapy

: Although photodynamic therapy (PDT) is an established modality for cancer treatment, current dosimetric quantities, such as light ﬂuence and PDT dose, do not account for the diﬀerences in PDT oxygen consumption for diﬀerent ﬂuence rates ( φ ). A macroscopic model was adopted to calculate reactive oxygen species concentration ([ ROS ] rx ) to predict Photofrin-PDT outcome in mice bearing radiation-induced ﬁbrosarcoma (RIF) tumors. Singlet oxygen is the primary cytotoxic species for ROS, which is responsible for cell death in type II PDT, although other type I ROS is included in the parameters used in our model. Using a combination of ﬂuences (50-250 J/cm 2 ) and φ (75 or 150 mW/cm 2 ), tumor regrowth rate, k , was determined for each condition by ﬁtting the tumor volume versus time to V 0 * exp(k * t) . Treatment was delivered with a collimated laser beam of 1 cm diameter at 630 nm. Explicit dosimetry of light ﬂuence rate on tissue surface, tissue oxygen concentration, tissue optical properties, and Photofrin concentration were performed. Light ﬂuence rate at 3 mm depth ( φ 3mm ) was determined for the treatment volume based on Monte-Carlo simulations and measured tissue optical properties. Initial tissue oxygenation [ 3 O 2 ] 0 was measured by an Oxylite oxygen probe before PDT and used to calculate [ ROS ] rx,calc. This value was compared to [ ROS ] rx,meas as calculated with the entire tissue oxygen spectrum [ 3 O 2 ](t), measured over the duration of light delivery for PDT. Cure index, CI = 1- k/k ctr , for tumor growth up to 14 days after PDT was predicted by four dose metrics: light ﬂuence, PDT dose, and [ ROS ] rx,calc , and [ ROS ] rx,meas . PDT dose was deﬁned as the product of the time-integral of photosensitizer concentration and φ at a 3 mm tumor depth. These studies show that [ ROS ] rx,meas best correlates with CI and is an eﬀective dosimetric quantity that can predict treatment outcome.


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 ([ 3 O 2 ]) [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 ( 3 O 2 ) to produce highly reactive singlet oxygen ( 1 O 2 ) [6]. Generated 1 O 2 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 ([ 3 O 2 ]), [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 [ 3 O 2 ], 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]

Tumor model
Radiation-induced fibrosarcomas (RIF) cells were cultured and injected intradermally over the right shoulders of 6-8 weeks old female C 3 H mice (NCI-Frederick, Frederick, MD). A cell solution of 30 µl was injected at a concentration of 1×10 7 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)).

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/cm 2 and total in-air fluences of 50-150 J/cm 2 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 [ 3 O 2 ], see Table 2.
For the same light fluence and φ air , mean [ 3 O 2 ] 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.

Oxygen measurements
The in vivo tissue oxygen partial pressure pO 2 was measured during PDT using a phosphorescencebased 3

Measurement of Photofrin concentration
At 18-24 hours after Photofrin administration, measurements of light fluence rate, photosensitizer concentration and [ 3 O 2 ] 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   Table 1.
rx,calc is calculated using Eqs. (2)-(4) using the parameters listed in Table 1. 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].

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 mm 3 ) 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 (R 2 ) [See Fig. 2].
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 R 2 = 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.

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 × a 2 × 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 where k is the tumor regrowth factor for each group and k ctr is the regrowth factor for the control group, which received no injection of Photofrin and light illumination. Previous studies by our group have shown that this works well for Photofrin [9], HPPH [10] and BPD [19]. Those with growth delay but no regrowth within 14 days are counted as a cure.

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 ([S 0 ]), ground state oxygen concentration ([ 3 O 2 ]), 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 3 O 2 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 3 O 2 . δ is the low concentration correction factor, and g is the maximum macroscopic oxygen perfusion rate.
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.

Results
Photofrin-mediated PDT with various in-air fluences and φ air were performed in mice bearing RIF  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 mm 3 ). 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). Figure 4 shows the temporal dependence of the measured photosensitizer concentrations (symbols) and the macroscopic model calculated photosensitizer concentrations (lines).  Table 2.   Fig. 6.

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/cm 2 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/cm 2 and that of 50 J/cm 2 ). 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 ([ 3 O 2 ]). 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 [ 3 O 2 ], 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]: where the parameters λ 1 (12.041/cm), λ 2 (5.661/cm), λ 3 (13.131/cm), b (0.45), C 2 (4.6), and C 3 (−0.01) are functions of µ a (0.91/cm) and µ s '(8.41/cm) and details of each can be found elsewhere [22]. INV = (SSD/(SSD + d)) 2 , where the source-to-surface distance, SSD = 9.34 cm based on the measurement of light fluence rate in water for the same collimated beam as a function of depth. The INV at 3 mm tumor depth is 0.94 and the effective φ/φ air at 3 mm tumor depth determined using Eq. (6) is approximately 0.78. 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 [ 3 O 2 ]. 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 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 R 2 = 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 R 2 = 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 1 O 2 ) quantum yield. The goodness of fit R 2 = 0.91 and 0.972 and the narrowest gray area in Fig. 6 [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-x 0 )/w 0 )), where x 0 = 439 µM J/cm 2 with uncertainty w 0 = 48 µM J/cm 2 for PDT dose, x 0 = 1.02 mM with uncertainty w 0 = 0.17 for [ROS] rx,calc , and x 0 = 0.96 mM with uncertainty w 0 = 0.11 for [ROS] rx,meas , respectively. For PDT dose, x 0 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/cm 3 , which corresponds to (4.9 ± 0.5)×10 18 photons/cm 3 (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 3.4×10 18 photons/cm 3 reported by others for Photofrin in-vitro [24,25]. However, it is smaller than the 6.7×10 18 photons/cm 3 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 mm 3 were evaluated in the present study, compared to tumors of 100-400 mm 3 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.

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]  [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].

Mathematical modeling of PDT photochemical reactions
The photosensitizer molecule in its ground state [S 0 ] 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 [T 1 ]. 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. For both type I and II primary photochemical reactions can be described using a set of coupled differential equations (Eqs. (7)(8)(9)(10)(11)(12)(13)) as shown below [4]: k 0 to k 8 are rate constants of photochemical processes as illustrated in Fig. 7 and their definitions are listed in Table 3  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 [ In the macroscopic reactive oxygen species explicit dosimetry (ROSED), the oxygen supply term Γ in Eqs. (10) and (19) is expressed as [13] where [ 3 O 2 ] (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 (ξ, ξ I , ξ II , σ, σ I , σ II , τ f , τ , τ S ) are given in Table 4. σ = (ξ II σ II + ξ I σ I )/ξ where ξ = ξ II + ξ I . For the in vivo scenario, it is assumed that the concentration of biological acceptors is large, so k 72 [A]τ ≈ 1 and k 71 [A]τ S ≈ 1. Furthermore, σ II ([S 0 ] + δ) 1. , where k 3R is fluorescence radiative decay rate between S I and S 0 Φ t Triplet quantum yield k 5 k 3 +k 5 Utilizing Eq. (20), the amount of biological acceptor that has reacted with a reactive oxygen species ([ROS] rx ) can be defined by the following 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 (η = 0), the reactive oxygen species ([ROS] rx ) can be described by (assuming f = 1) as Eq. (4). Similarly, since η = 0, k 6 τ ∆ <<1, ξ = ξ II + ξ I , and σξ = (ξ II σ II + ξ I σ 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.