#### 4.1. SOED and SOLD Intercomparison

The SOED calculated [

^{1}O_{2}] value in solution in

Figure 4a corresponded to the volumetric averaged instantaneous singlet oxygen concentration over a volume of 1 cm depth and 1 cm

^{2} area. SOED-calculated [

^{1}O_{2}]

_{rx} in

Figure 4b corresponded to the volumetric average reacted singlet oxygen concentration of the same 1 cm

^{3} volume. In these solutions, the light fluence was calculated by introducing attenuation that was only due to the photosensitizer absorption, since no scatterer was added and solutions were of pure Photofrin:

φ = φ_{0} exp(

−μ_{a} × d), where

φ_{0} is the light fluence rate (mW/cm

^{2}) measured directly in the back of the front wall of the solution facing the laser. Absorption coefficients (

μ_{a}) were 0.15, 0.45, and 0.74 cm

^{−1} for Photofrin concentrations of 17, 50, and 83 mM, respectively, at 523 nm.

In order to experimentally determine the photophysical parameters of the spontaneous phosphorescence rate of

^{1}O

_{2} to

^{3}O

_{2} (

k_{6}) and the bimolecular reaction rate of

^{1}O

_{2} with the substrate (

k_{7}) in Photofrin phantoms, photosensitizer triplet-state and singlet oxygen lifetime measurements were obtained using the SOLD system. Varying amounts of the singlet oxygen quencher, sodium azide (NaN

_{3}), were added to the Photofrin–MeOH solutions. The resulting fits to obtain

k_{6} and

k_{7} are shown in

Figure 2. For Photofrin with NaN

_{3},

k_{6} was found to be 1.14 × 10

^{5} s

^{−1} (the intercept of the line of best fit in

Figure 2) and

k_{7} was found to be 235 μM

^{−1}·s

^{−1} (the slope of the line of best fit in

Figure 2).

k_{7} is pH-dependent, but is in the range of the reported value of 300–400 μM

^{−1}·s

^{−1} for the quenching rate constant in water [

38]. These values were used to calculate

τ_{∆} for the in vitro condition (without NaN

_{3}) and the in vivo condition (taken from the literature for biological tissue [

34]). Assuming that

k_{7} for NaN

_{3} is greater than or equal to that of in vivo conditions (assuming biological tissue is less efficient than NaN

_{3} in quenching

^{1}O

_{2}), it can be estimated that in vivo acceptor concentration [

A] ≥ 10

^{7} (s

^{−1})/235 mM

^{−1}·s

^{−1} = 42 mM. This value is much higher than [

A] = 0.83 mM in the literature [

10], but we feel that it is more reasonable since the singlet oxygen lifetime in vivo,

τ_{∆}, does not change for reacted singlet oxygen concentrations [

^{1}O_{2}]

_{rx} as high as 12 mM [

16], indicating there are still plenty of acceptors in vivo at this level.

The light fluence rate distribution in a semi-infinite medium as a function of distance (

d) was calculated by a Monte Carlo (MC) simulation [

39] of a circular parallel beam (diameter 0.8 cm,

Figure 5a) and broad beam (diameter 16 cm,

Figure 5b) for absorption coefficient (

μ_{a}) of 0.09, 0.18, and 0.58 cm

^{−1}, and reduced scattering coefficient (

μ_{s}’) of 0.2 cm

^{−1}. The resulting

φ/

φ_{0} is shown in

Figure 5 along with an exponential fit based on

μ_{a}. For the tissue-simulating phantoms with Photofrin shown in

Figure 3,

φ_{0} is the measured local fluence rate at the front inner surface of the phantom facing the laser and

d is the depth from surface. At 630 nm,

μ_{a} = 0.09, 0.18, and 0.58 cm

^{−1} for Photofrin concentrations of 27, 50, and 167 mM, respectively. It is clear that the function

e^{−μad}, while working well for the broad beam, does not work very well for the 0.8 cm diameter beam at the deepest depths investigated. As a result, MC-generated light fluence rate

φ/

φ_{0} was used directly for the SOED calculations in phantom.

SOED calculations of singlet oxygen concentration are highly dependent on the photophysical parameters used as input (

Table 1). In turn, these parameters depend on the photosensitizer used, as well as the treatment environment. The necessary parameters for Photofrin-mediated PDT for in vitro studies were validated with explicit measurements of the [

^{3}O_{2}] and [

S_{0}]. In particular, the consumption rate of [

S_{0}] per PDT dose was used to determine a more accurate value of

σ (slope/

ξ) and

δ (intercept/slope) for the experimental setup used (

Figure 3c and Equation (7)). This was used to determine

δ and

σ using a method from Reference [

40] that is also described in

Section 2.1. Photosensitizer concentration was measured over time to determine the photobleaching rate (

−d[

S_{0}]/

dt) and [

S_{0}]. Along with the measured [

^{3}O_{2}], the PS photobleaching rate per PDT dose can be calculated and plotted as in

Figure 3c. The slope and intercept of the fit to the data are used to calculate

δ and

σ. The value for

ξ in vitro was calculated by the definition of

ξ provided in

Table 1. The resulting values were

δ = 25 ± 4.3 μM and

σ = (6.6 ± 7) × 10

^{−5} μM

^{−1}. The value of

β was set to be 11.9 μM for this set of experiments [

34].

Figure 3a,b show the SOED calculations using Equations (2) and (3), which agree with [

^{3}O_{2}](

t) and [

S_{0}](

t) measurements at surface (

d = 0 cm) of the Intralipid phantom.

Figure 3d shows the magnitude of SOED-calculated [

^{1}O_{2}]

_{rx} using Equation (4) for Photofrin to be in the mM range.

It can be concluded from the intercomparison of SOED and SOLD in Photofrin solutions (

Figure 5b) that the cumulative SOLD [

^{1}O_{2}] counts, [

SOLD], and SOED-calculated [

^{1}O_{2}]

_{rx} values track each other very well (

R^{2} = 0.98) for Photofrin, with a conversion factor of the following form:

The ratio of slopes between the two panels ((a) and (b)) in

Figure 4 is 9.6 × 10

^{−6} s, which is consistent with the value of

τ_{Δ} obtained (9.4 × 10

^{−6} s). The reason for the intercept is not known, and a linear fit without intercept reduces the correlation (

R^{2} = 0.86). The good correlation of SOED-calculated [

^{1}O_{2}] and [

SOLD] demonstrates that SOED can be utilized in scenarios where direct phosphorescence measurement of

^{1}O

_{2} is difficult.

#### 4.2. Feasibility of Using SOLD at 523 nm for Predicting [^{1}O_{2}]_{rx} at 630 nm

Currently, the only available pulsed laser suitable for the SOLD application (CrystaLaser, QL-523-200-S, CrystaLaser, Reno, NV, USA) is at 523 nm. As a result, the effective tissue-sampling depth for [

^{1}O_{2}] is not the same as that of the 630 nm treatment light used clinically with Photofrin.

Figure 6 shows the measured values

μ_{a} and

μ_{s}’ in various sites measured in vivo in patients, including the anterior chest wall, apex of the heart (apex), posterior chest wall, diaphragm (diaph), serratus (ser), anterior sulcus, posterior sulcus, pericardium (peri), and normal (norm) tissue. Patients were undergoing an institutional review board (IRB)-approved pleural mesothelioma Photofrin-PDT clinical protocol at the University of Pennsylvania. These optical properties were measured using a custom-built multifiber contact probe for absorption spectroscopy [

27]. The measured optical properties include

μ_{a} and

μ_{s}’ for tissue as well as Photofrin.

Using an analytical fit [

28] to MC simulations [

29,

30,

31,

32], the longitudinal distribution of

φ in tissue with different optical properties was calculated.

Figure 7 shows the ratio of

φ and in-air fluence rate (

φ_{air}) versus tumor depth for (a) 523 nm and (b) 630 nm. The gray area shows the region of

φ/

φ_{air} with the upper and lower bounds of the tissue optical properties obtained in vivo as dark blue and light blue, respectively. The dashed black lines show the calculated light fluence distribution using the mean optical properties of

μ_{a} = 5.52 cm

^{−1} and

μ_{s}’ = 17.61 cm

^{−1} for 523 nm and

μ_{a} = 0.58 cm

^{−1} and

μ_{s}’ = 15.61 cm

^{−1} for 630 nm. As expected, the optical penetration is much deeper at 630 nm than at 523 nm in in vivo malignant pleural mesothelioma (MPM) patients.

The

φ distributions were then used to calculate the reacted singlet oxygen concentration for the two wavelengths, in order to study whether SOLD signals measured at 523 nm can be used to monitor [

^{1}O_{2}]

_{rx} at 630 nm. MPM PDT is currently performed at 630 nm. Correlation between the calculated [

^{1}O_{2}]

_{rx} for 630 and 523 nm is shown in

Figure 8.

μ_{a} ranges from 0.66 to 23.1 cm

^{−1} at 523 nm and 0.17 to 1.35 cm

^{−1} at 630 nm, while

μ_{s}’ ranges from 2.80 to 73.7 cm

^{−1} at 523 nm and 2.55 to 30.5 cm

^{−1} at 630 nm (

Figure 6). In order to investigate the effects of different

φ on the [

^{1}O_{2}]

_{rx}, the SOED calculations were repeated for

φ = 5, 25, 50, 75, and 150 mW/cm

^{2}. Different colors of symbols represent different

φ. The black solid lines are the best fits in

Figure 8b. At 523 nm, the range of [

^{1}O_{2}]

_{rx} changed from 0–0.1, 0–0.63, and 0–5.6 mM for PS concentrations of 0.21, 2.1, and 21 µM, respectively, while the range of [

^{1}O_{2}]

_{rx} at 630 nm changed from 0–0.25, 0–2.5, and 0–20 mM, respectively, for the same PS concentrations.

The resulting correspondence for a range of PS concentrations (

c) of 0.21, 2.1, and 20.1 mM (based on the average Photofrin concentration obtained in patients) and light fluence of 10–120 J/cm

^{2} [

33] can be expressed as:

where

and

where

c is the PS concentration (in μM) and

φ is the light fluence (in J/cm

^{2}). We thus conclude that SOLD measurement at 532 nm can be used to monitor [

^{1}O_{2}]

_{rx} at 630 nm if a conversion formula (Equations (11)–(13)) is used to convert the measured SOLD signal.

When SOLD signal from patients are used to determine the generation of singlet oxygen, it is important to develop a tissue optical properties correction factor to account for the absorption and scattering of luminescence by tissue, similar to the optical properties correction factor needed for using fluorescence to determine the PS concentration [

14]. This is beyond the scope of this paper.