Photodynamic therapy (PDT) is an anti-cancer treatment modality based on the interaction of light, a photosensitizing drug, and oxygen . PDT has been approved by the US Food and Drug Administration for the treatment of microinvasive lung cancer, obstructing lung cancer, and obstructing esophageal cancer and Barrett’s esophagus with high-grade dysplasia, as well as for age-related macular degeneration and actinic keratosis .
PDT is inherently a complex process in which the photosensitizer (PS), light and oxygen vary dynamically and interdependently on timescales specific to an individual treatment condition. Thus, the distribution of light is determined by the light source characteristics and the effective tissue optical absorption and scattering at the treatment wavelength. The effective absorption is affected by the local concentration of PS and the concentration and oxygenation status of the blood. The oxygen distribution is altered by photodynamic consumption and any PDT-induced changes in blood flow . The effective PS concentration and distribution may change due to photobleaching. On the one hand, these effects make accurate PDT dosimetry extremely challenging, on the other hand, this is critical to achieve optimal efficacy and safety, particularly when there is curative intent, as is the case of PDT of multiple types of malignancies [4, 5].
Most PSs available for PDT utilize type-II photodynamic processes, i.e., the photodynamic effect is achieved through the production of singlet oxygen (1O2) [6, 7]. The energy level diagram shown in Figure 1 summarizes the underlying physical processes involved in type-II PDT with reaction rates (ki) defined in Table 1. The process begins with the absorption of a photon (k0) by PS in its ground state (S0), promoting it to an excited state (S1). The PS molecule can return to its ground state by emission of a fluorescence photon (k3), which can be used for fluorescence detection. Alternatively, the molecule may convert to a triplet state, a process known as intersystem crossing (k5). A high intersystem crossing yield is an essential feature of a good type-II PS. Once in its triplet state, the molecule may undergo a collisional energy transfer with ground state molecular oxygen (type II, k2) or with the substrate (type-I) . In type-II interaction, the PS returns to its ground state, and oxygen is promoted from its ground state (a triplet state) to its excited (singlet) state. Since the PS is not consumed in this process, the same PS molecule may create many 1O2 molecules. The PS triplet state can also return to the ground state spontaneously without interacting with oxygen by emitting a phosphorescence photon instead (k4). Once the singlet oxygen is created, it reacts almost immediately with cellular targets in its immediate vicinity. The majority of these reactions are irreversible, and lead to consumption of oxygen. This consumption of oxygen is efficient enough to cause measurable decreases in tissue oxygenation when the incident light intensity is high enough. In addition to its reactions with cellular targets (k7), singlet oxygen may react with the PS itself (k1). This leads to its irreversible destruction (photobleaching). Photobleaching can decrease the effectiveness of PDT by reducing the PS concentration; however it can also be useful for dosimetry, so-called “singlet oxygen implicit dosimetry” (SOID) . Because of its high reactivity, 1O2 has a very short lifetime (τΔ) in tissue. However, a small fraction of the 1O2 produced may return to its ground state via emission of a phosphorescence photon (k6, λ=1270 nm), which can be detected optically, the so-called “singlet oxygen luminescence dosimetry” (SOLD) [10, 11].
For type-II PSs, excited-state singlet oxygen, the major cytotoxic species causing biological effects, is generated upon the absorption of light by the PS in the presence of ground state molecular oxygen (3O2). Thus, direct measurement of singlet oxygen constitutes the ultimate PDT dosimetry method to correlate with PDT outcome. One method of determining the singlet oxygen is through in-vivo measurements of light and PS concentration, followed by calculation of the generation of singlet oxygen via a dynamic model, so-called “singlet oxygen explicit dosimetry” (SOED). Compared to SOID and SOLD, SOED has the advantage that it is a more quantitative indication of the 1O2 concentration that is directly interacting with tissue and is a natural extension of the existing in-vivo PDT dosimetry of light and PS concentration. However, SOED requires knowledge of photochemical parameters and the singlet oxygen threshold dose [1O2]rx,sh. The purpose of this study is to use SOED to estimate the magnitude of the reacted singlet oxygen threshold concentration where tissue necrosis occurs in an in-vivo model.
2 Materials and methods
In this study, ki (i=0, 1, …, 7) are used to designate the reaction rate. The definitions associated with the reaction rates are summarized in Table 1.
2.1 Macroscopic kinetics rate equations
A rate equation approach first proposed by Foster et al.  and later refined by Hu et al.  was adopted to describe the PDT kinetics processes. The complete set of rate equations is published elsewhere [8, 14]. The lifetime of the singlet and triplet states of PS ([S1] and [T]) and the singlet oxygen are very short (ns – μs time scale) since they either decay or react with cellular targets immediately after they are created. Thus, it is reasonable to set the time dependences, d[S1]/dt, d[T]/dt, and d[1O2]/dt to be zero. The simplified rate equations can be expressed as follows [8, 14]:
Note that a macroscopic oxygen perfusion rate, is used to replace the microscopic diffusion and metabolic consumption of oxygen in tissue in a uniform vascular structure in the macroscopic model [8, 14], where g is the oxygen maximum perfusion rate. The specific oxygen consumption rate, is the PDT oxygen consumption rate per fluence rate per PS concentration. The specific photobleaching ratio, σ=k1/k7[A], is the ratio of photobleaching rate by singlet oxygen between PS and all other acceptors [A] per PS concentration. The oxygen quenching threshold concentration, β=k4/k2, is the value of oxygen concentration where the quantum efficiency of singlet oxygen generation is reduced by half, following the oxygen dependence as Here the reacted singlet oxygen concentration is defined as where f is the fraction of singlet oxygen interacting with target [A], and k7[A]=1/τΔ is the inverse of the singlet oxygen lifetime in tissue. In this paper singlet oxygen f=1 is used.
The corresponding expressions for the instantaneous concentrations of singlet oxygen, as well as singlet, and triplet states of PS are:
The rate equations are implemented in the MATLAB code. The calculation time is in seconds for the rate equation alone [Eqn. (2)–(7)] and minutes for the time and spatially coupled differential equation [Eq. (1)]. All these quantities are a function of space (x, y, z) and time (t).
2.2 Necrosis experiment and SOED
Through in-vivo studies of a murine radiation-induced fibrosarcoma (RIF) tumor model, the extent of necrosis that is generated by PDT with benzoporphyrin derivative monoacid ring A (BPD) was measured to determine [1O2]rx,sh. Two drug-light intervals (DLI), 15 min and 3 h, were used. The treatment laser wavelength was 690 nm. Table 2 summarizes the conditions that were evaluated for in-vivo PDT.
The RIF tumors were grown by the intradermal injection of 3×105 cells on the right shoulders of the mice and studies were initiated approximately 10 days later when tumors were 8–12 mm in diameter. Animal husbandry was provided by the University of Pennsylvania Laboratory Animal Resources in Association for Assessment and Accreditation of Laboratory Animal Care (AALAC)-accredited facilities according to protocols approved by the University of Pennsylvania Institutional Animal Care and Use Committee. PDT was performed using a 1-cm long cylindrical diffusing fiber (CDF) to deliver light at various fluence rates and total treatment times . The mice were sacrificed 24 h after PDT and the tumors resected perpendicular to the CDF. The experimental set-up is shown in Figure 2A and B.
Tumors were then sectioned perpendicularly to the direction of light delivery at 200-μm separations and 4–10 sections were placed on slides and subsequently stained with hematoxylin and eosin (H&E) to show the necrotic area. Slides were then digitally scanned so that the area, Area, and radius, r, of necrosis could be outlined and determined. The radius of necrosis was found by the following calculation:
All tumors were measured immediately post-excision and just prior to sectioning for each mouse to determine the shrinkage. From this data an average shrinkage factor of 30% was incorporated into the determination of the necrotic area.
The standard deviation of the radius of necrosis was found by:
N is the total number of necrosis radii examined.
For areas of necrosis that were ellipsoidal in shape, multiple radii were measured from the location of the light source (indicated by the hole left behind by the CDF catheter in the tumor section) to the edge of necrosis. Standard deviations were added to the standard deviations for the radius of necrosis for each group of mice that were treated under the same conditions. For the set of experiments performed with individual mice, a set standard deviation of 0.5 mm was applied. The necrosis radius due to insertion of the 1-cm CDF alone was measured as 0.5 or 1.0 mm in nine unilluminated control mice. The radius in these controls was observed to be a function of whether a plastic catheter was used to house the 1-cm CDF during its insertion. A catheter was not used for CDF placement in most PDT-treated animals except for the ones also used for optical properties measurement. These uncertainties were included in the data analysis of the measured necrosis radius.
The optical properties (absorption, μa, and reduced scattering, coefficients) of the tumor environment were determined using a two catheter method . A 2-mm point source and an isotropic detector were inserted into the tumor using two parallel catheters (Figure 2A). Light fluence data was scanned along the point source by moving the isotropic detector at various source positions. The data was then fitted with the diffusion approximation:
where S is the source strength of the point source (in mW), ϕ(r) is the fluence rate at position r, and [15, 16]. The light fluence rate (data not presented) at the second catheter per mouse was also measured and compared with the calculated light fluence rate using the measured optical properties as a second consistent check of the optical properties values. The light fluence rate at the radius of necrosis was calculated for each mouse along the radial axis with respect to the center of the CDF:
where r is the distance to the point of interest along the radial axis given the center of the CDF as origin, l is the length of the CDF, and s is the source strength of the CDF (in mW/cm).
The PS concentration in the tumor was determined using fluorescence spectra. For this, a polished side-firing fiber was inserted into the source position catheter to excite the PS with 405-nm light and collect the fluorescence signal through a dichroic beam splitter to be collected by a multi-channel spectrometer. Spectra were analyzed by using singular value decomposition methods with known phantom fluorescence spectra . Optical properties correction of fluorescence signal reduction due to tissue absorption and scattering was performed using an empirical correction method .
2.3 Optimization algorithm to determine in-vivo singlet oxygen threshold dose
The optimization routine was performed by calculating the time series solution for reacted singlet oxygen, [1O2]rx. The PDT equations were solved by inputting initial estimates for the modeling parameters g, ξ, and σ, along with the experimental light fluence rate, the initial oxygen concentration, and the initial PS concentration. Two parameters, β and δ, were held at fixed values based on the literature [19, 20], while g, ξ, σ, and the singlet oxygen threshold dose [1O2]rx,sh were free parameters. The initial oxygen concentration was assumed to be 83 μm, with 100% of the oxygen concentration present in the treatment environment reacting with the PS. The effect of initial oxygen concentration is beyond the scope of the current study. After calculating the time series solution for [1O2]rx, this value at the radius of necrosis with the given input parameters is compared to the calculated [1O2]rx,sh value, by minimizing a standard deviation according to the following equation:
where N is the total number of groups (or individual mouse for an earlier experiment) and ri is the measured radius of necrosis for group (mouse) i. In our most recent experiments, each group consisted of 3 mice with the same treatment conditions for which the radii of necrosis and other measured parameters were averaged (Table 3A). In an earlier experiment (see Table 3B) individual mice were used. Multi-variable optimization uses the functional minimization function “fminsearch” from MATLAB and is implemented in the same way as described elsewhere . The quality of the fitted results was evaluated by examining the deviation of [1O2]rx for the total number of mice around the singlet oxygen threshold [1O2]rx,sh dose fitted with our algorithm. Multiple fitting runs were performed with a range of initial parameter estimates in a physiologically reasonable range based on literature values. The specific oxygen consumption rate, ξ, was set to initially range between 20×10-3 and 100×10-3 cm2mW-1s-1. The oxygen maximum perfusion rate, g, was initially varied from 0.5 to 2.5 μm/s. The specific photobleaching ratio, σ, was estimated to initially be between 0.5×10-5 and 2.5×10-5 μm-1. Finally, we also varied the initial threshold singlet oxygen dose, [1O2]rx,sh, from 0.5 to 1.5 mm. For the multiple runs, the maximum relative deviation of reacted singlet oxygen were compared, defined as:
By evaluating Eqn. (13) for varying the initial estimate of parameters to be fit, the standard deviations of the fitted parameters were found. The sensitivity of the parameter determination depends also on the variation of necrosis for the light fluence rate and total fluence used. For a DLI of 15 min, the necrosis radii are all near 4.5 mm without substantial variability. The parameter set determined for BPD with a DLI of 15 min was considered to be “preliminary” due to the narrow range of experimental conditions that were performed. Additional experiments should be added to the study with a DLI of 3 h to improve the accuracy of the resulting parameters (ξ, σ, g, β) but it is not expected to change the conclusion that [1O2]rx,sh for the BPD, DLI=15 min study is much lower than that of the BPD, DLI=3 h study.
3 Results and discussion
3.1 Macroscopic modeling of singlet oxygen threshold dose
Figures 3 and 4 show the fitting results of (A) the reacted singlet oxygen concentration [1O2]rx versus radius of necrosis and (B) the comparison of predicted and measured radius of necrosis for BPD for a DLI of 3 h and 15 min, respectively. The recovered photochemical parameters for the rate equations are listed in Table 5 for BPD and several other photosensitizers from the literature.
Each data point was evaluated for their quality after the fitting runs was performed. Mice, which had a lower than expected physiological concentrations of the PS were eliminated from the data analysis. In addition, light fluence rates were calculated for the distance of the detector away from the treatment linear source, using the source strength and the measured optical properties. If the calculated fluence rates deviated more than 50% from the measured fluence rate during treatment, then the data point was not considered for analysis. Table 6 summarizes the singlet oxygen threshold dose, [1O2]rx,sh, calculated from the fits for BPD and other photosensitizers based either on similar data analysis of the existing in-vivo data or on the data analysis of in-vitro spheroid data in the literature. For comparison, [1O2]rx,sh for PDT from other studies are included. Clearly, there is a difference in the [1O2]rx,sh values between in-vivo results measured in mice (0.4–0.7 mm) and from spheroid measurements (8–12 mm). The value for BPD15 min is ignored since it is completely vascular in nature. The current study is consistent with other in-vivo mouse studies in mTHPC where a threshold dose of 0.4 mm in vivo was obtained . However, the in-vivo results are at least 20 times smaller than those observed in spheroids, indicating that factors other than singlet oxygen-mediated tumor cell kill may contribute to PDT damage for these treatment conditions . The singlet oxygen threshold dose obtained for BPD with a DLI of 15 min (BPD 1 mg/kg, 690 nm) was also included, where a vascular effect is predominant . The lower singlet oxygen threshold dose of 0.12 mm that is associated with these conditions is to be expected, given the BPD concentration (∼0.15 μm) in tissue after a DLI of 15 min is much lower than that (∼0.6 μm) after administration of BPD with a DLI of 3 h (which is associated with less of a vascular effect). The fact of the matter is that for BPD with a DLI of 15 min, most of the BPD remains in the vasculature and has not yet diffused into the surrounding tissue. As a result, damage of the vasculature is the dominant mechanism of PDT effect after a DLI of 15 min and the measured BPD concentration in tissue is only used as a surrogate to quantify the corresponding BPD concentration in the vasculature. For this reason, the singlet oxygen threshold dose obtained for BPD with a DLI of 15 min will be excluded in our final analysis of in vivo singlet oxygen threshold dose for direct tissue damage due to singlet oxygen.
3.2 Uncertainty in singlet oxygen threshold dose determination
The mean values and standard deviations of singlet oxygen threshold dose were determined to be 0.72±0.21 mm and 0.12±0.08 mm for BPD with DLI 3 h and 15 min, respectively. They were shown as dashed lines in Figures 3 and 4, respectively. Similarly the value was found to be 0.56±0.26 mm for Photofrin® with a DLI of 24 h . The standard deviation for the fitted [1O2]rx,sh was found by evaluating the quality of each data point obtained. In Figure 3, the mean [1O2]rx,sh is indicated with the black dashed line, and the standard deviation is indicated by the shaded grey region. The points numbered 12 and 13 were disregarded for the analysis due to their lower than physiologically sound PS concentrations compared to other data for similar fluence rate and fluence conditions (represented in data points 2 and 5). In Figure 4, the data set appears to show two distinct model trends, further illustrating the need for a more detailed study. Most of the data points had extremely large areas of necrosis (with irregular shapes exceeding the boundary of the tumor). The standard deviation of singlet oxygen threshold dose (gray area) is very large and requires further study to improve it. However, it is clear that the [1O2]rx,sh values for a DLI of 15 min is substantially lower than that for a DLI of 3 h.
To examine both the error in the experiments and determine the singlet oxygen threshold dose, all the individual measurements were plotted between the reacted singlet oxygen and necrosis radius. Figure 5 shows all the necrosis results for BPD for a DLI of 3 h (Figure 5A) and a DLI of 15 min (Figure 5B). From this, it can be concluded that the maximum spread in singlet oxygen threshold dose is 0.07–1.60 mm for BPD with a 3-h DLI and 0.01–0.2 mm for BPD with a 15-min DLI. The upper limit of [1O2]rx,sh in vivo is 1.6 mm and 0.2 mm for BPD with a DLI of 3 h and 15 min, respectively, from our studies (Figure 5A and B).
3.3 Theoretical consideration of singlet oxygen threshold dose between in vivo and in vitro
Microscopically, one can write down the relationship between the reacted singlet oxygen concentration [1O2]rx and individual cell survival probability in vitro as :
where [1O2]rx0 is the reacted singlet oxygen concentration when the cell survival is dropped to 1/e, SF is the cell survival fraction of each individual cell (SF=1 means all cell survive, and SF=0.1 means 1-in-10-cells survival). Given that one needs to kill k cells to produce an observable necrosis in vivo, then the chance of all k cells not surviving is expressed as:
Inserting Eqn. (14) into Eqn. (15), one can determine the relationship between P and [1O2]rx. Figure 6A shows that P exhibits a threshold dose behavior, i.e., there is a very rapid increase in P when [1O2]rx is above a threshold dose. This value of threshold dose [1O2]rx,sh, defined as P=0.5, is proportional to [1O2]rx0 and the proportional constant, m, is a function of ln(k) (see Figure 6B, linear fit of data):
For k=108 cells, m=18.78; m varies between 13.8–32.2 for k varying between 106 and 1014, with the former corresponding to the number of cells in 2 mg of tissue and the later corresponding to the number of cells in 200 kg of tissue (a heavy human body weight), respectively. Using the direct measurement of the number of RIF tumor cells in vivo, 5×108 cm-3, the singlet oxygen threshold dose [1O2]rx0 for 1/e cell kill can be converted to between mm and the number of singlet oxygen molecules per cell:
Eqn. (16) and (17) can be used to obtain singlet oxygen threshold dose from in vivo tissue necrosis to 1/e cell killing for BPD from 0.72 mm for tissue necrosis to 4.6×107 (0.72×1.20×109/m, where m=18.78 is used assuming k=108) or equivalently 0.064 mm (0.72/m) for 1/e cell killing. In comparison, in-vitro measurement in MAT-LyLu (MLL) cells using m-tetrahydroxyphenylchlorin (mTHPC) gives this value as 9×108 molecules of singlet oxygen per cell , while in-vitro measurement for MLL cells using BPD gave 1.3 mm  to reduce the surviving fraction by 1/e. The in vivo singlet oxygen threshold dose for necrosis (4.6×107/cell) is about 20 times smaller than that determined in vitro for 1/e cell killing (9×108/cell).
In conclusion, the singlet oxygen threshold dose measured in vivo is approximately 0.56 mm [(0.56+0.72+0.4)/3] to produce necrosis or 3.6×107 molecules of singlet oxygen per cell to reduce the surviving fraction by 1/e in studies of the RIF tumor model. This value is 20 times smaller than that determined from the spheroid model (∼7.9 mm)  to produce necrosis or an in-vitro model of MLL cells  in which 9×108 molecules of singlet oxygen per cell were needed to reduce the surviving fraction by 1/e using mTHPC.
It was shown that using a set of rate equations and fitting the data to the necrosis radius in a RIF mouse model, the singlet oxygen threshold dose in vivo can be determined. Preliminary studies have identified that the singlet oxygen threshold dose is in the range of 0.56–0.72 mm for two photosensitizers studied: Photofrin® and BPD (Table 6). This value is about 20 times smaller than that determined in vitro. It is concluded that PDT is more potent in vivo than that in vitro, as was already pointed out by Wang et al. .
We thank the technical supports from Dr. Baochang Liu for helps in BPD mice measurements and data analysis. We also would like to thank Ken K.-H. Wang, for his help in developing the optimization algorithm for determination of the photochemical parameters.
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About the article
Published Online: 2015-02-27
Published in Print: 2015-02-01
Funding: National cancer Institute of Health (NIH) (Grant/Award Number: “R01 CA154562-03”, “R01 CA085831”, and “P01 CA87971”).