Macroscopic Modeling of the singlet oxygen production during PDT Timothy C Zhu*, Jarod C. Finlay, Xiaodong Zhou, Jun Li Department of Radiation Oncology, University of Pennsylvania, Philadelphia, PA 19104 ABSTRACT Photodynamic therapy (PDT) dose, D, is defined as the absorbed dose by the photosensitizer during photodynamic therapy. It is proportional to the product of photosensitizer concentration and the light fluence. This quantity can be directly characterized during PDT and is considered to be predictive of photodynamic efficacy under ample oxygen supply. For type-II photodynamic interaction, the cell killing is caused by the reaction of cellular acceptors with singlet oxygen. The production of singlet oxygen can be expressed as ηD, where η is the singlet oxygen quantum yield and is a constant under ample oxygen supply. For most PDT, it is desirable to also take into account the effect of tissue oxygenation. We have modeled the coupled kinetics equation of the concentrations of the singlet oxygen, the photosensitizers in ground and triplet states, the oxygen, and tissue acceptors along with the diffusion equation governing the light transport in turbid medium. We have shown that it is possible to express η as a function of local oxygen concentration during PDT and this expression is a good approximation to predict the production of singlet oxygen during PDT. Theoretical estimation of the correlation between the tissue oxygen concentration and hemoglobin concentration, oxygen saturation, and blood flow is presented. Keywords: photodynamic therapy, singlet oxygen production, oxygen dependence of singlet oxygen quantum yield.
1. INTRODUCTION Photodynamic therapy (PDT) is a cancer treatment modality based on the interaction of light, a photosensitizing drug, and oxygen.1 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.2 To quantify the photodynamic efficacy, the current start of art in clinical PDT dosimetry uses a quantity called PDT dose,3 defined as the number of photons absorbed by photosensitizing drug per gram of tissue [ph/g], i.e., D = ∫0t εc ⋅
φ (t ' ) 1 ⋅ dt ' , hν ρ
(1)
where ρ is the density of tissue [g/cm3], φ is the light fluence rate [W/cm2], hν is the energy of a photon [J/ph], c is the drug concentration in tissue [µM], ε is the extinction coefficient of the photosensitizer [1/cm/µM]. The photodynamic dose (D) does not consider the quantum yield (η) of oxidative radicals, the effect of tissue oxygenation on η, or the fraction (f) of radicals that oxidize cellular targets. Thus, it is only applicable in cases where ample oxygen supplies exist. For type-II photodynamic interaction, production of singlet oxygen is responsible for the PDT damage. The production of singlet oxygen that has damaged the cellular targets can be expressed as: 4 [1 O2 ] rx = f ⋅η ⋅ ρD ,
(2)
where f depends on the localization of the photosensitizer at the cell level and thus depends on the photosensitizer and tissue types, the quantum yield η gives the number of singlet oxygen molecules produced per an absorbed photon, which is a constant under ample oxygen supply. However, when insufficient oxygen supply exists, η is also a function of the
*
[email protected]; phone 1 215 662-4043; fax 1 215 349-5978. Optical Methods for Tumor Treatment and Detection: Mechanisms and Techniques in Photodynamic Therapy XVI, edited by David Kessel, Proc. of SPIE Vol. 6427, 642708, (2007) · 1605-7422/07/$18 · doi: 10.1117/12.701387
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oxygen concentration, or pO2, in tissue. (We explain later why the PDT dose multiplied the tissue mass density, or ρD [ph/cm3] is used in Eq. (2)). The purpose of this study is to establish the relationship between the singlet oxygen concentration and the PDT dose under arbitrary oxygen supply conditions. This is achieved by solving the coupled kinetics equations describing the macroscopic process of the generation of singlet oxygen and the diffusion equations describing the light transport assuming the 3-dimensional distribution of photosensitizers can be independently characterized during PDT. Since the oxygen concentration in tissue is a local quantity that cannot be easily measured. We have also presented a model that can be used to determine the local oxygen concentration by measuring the macroscopic quantities related to oxygen supplies in the blood vessel, i.e., hemoglobin concentration, oxygen saturation, and blood flow.
2. METHOD The majority of photosensitizers available for PDT utilize Type II photodynamic processes, i.e., the photodynamic effect is achieved through the production of singlet oxygen.5, 6 The Jablonski diagram shown in Fig. 1 summarizes the underlying physical processes involved in type-II PDT. The process begins with the absorption of a photon by photosensitizer in its ground state, promoting it to an excited state. The photosensitizer molecule can return to its ground state by emission of a fluorescence photon, which can be used for fluorescence detection. Alternatively, the molecule may convert to a triplet state, a process known as intersystem crossing (ISC). A high intersystem-crossing yield is an essential feature of a good Type II photosensitizer. Once in its triplet state, the molecule may undergo a collisional energy transfer with ground state molecular oxygen (type II) or with the substrate (type I). In type II interaction, the photosensitzer returns to its ground state, and oxygen is promoted from its ground state (a triplet state) to its excited (singlet) state. Since the photosensitizer is not consumed in this process, the same photosensitizer molecule may create many singlet oxygen molecules.
S1
Intersystem crossing k5 T
k0, absorption
k7, Reaction with targets [A] Energy transfer to O2 k2
1
O2
k3,fluorescence k4, phosphorescence
k6, phosphorescence λ = 1260 nm
k1, photobleaching
S0
Sensitizer
3
O2
Oxygen
Figure 1: Jablonski diagram of photosensitized singlet oxygen formation by Type II photosensitizer. The rate constants for monomolecular transition (solid lines) and bimolecular energy transfer (dashed lines) are indicated. Once the singlet oxygen is created, it reacts almost immediately with cellular targets in its immediate vicinity. The majorities 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, singlet oxygen may react with the photosensitizer itself. This leads to its irreversible destruction (photobleaching). Photobleaching can decrease the effectiveness of PDT by reducing the photosensitizer concentration, however it can also be useful for dosimetry.7 Because of its high reactivity, singlet
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oxygen has a very short lifetime in tissue. However, a small fraction of the singlet oxygen produced may return to its ground state via emission of a phosphorescence photon, which can be detected optically.8, 9 2.1 Macroscopic kinetics rate equations We adopted the rate equation approach first proposed by Foster et al 10 and later refined by Hu et al 11 to describe the PDT kinetics process. We use ki (i = 0, 1, …, 7) to designate the reaction rate. The details of the reactions associated with the reaction rates and the species involved are summarized below: The PDT process started by the absorption of light by the photosensitizer in the ground state, S0, and it excites to the singlet state S1. The S1 state will spontaneously decay to the ground state, S0, by emission of a fluorescent photon. k0
[ S0 ] ⇔[ S1 ] k3
This is a reversible process. The monomolecular reaction rate k0 (1/s) is proportional to the light fluence φ (in unit of W/cm2): k0 = ε ⋅ ( φ ) , where ε is the extinction coefficient (in unit of cm-1/µM), hν is the photon energy (in unit of Joule). hν
The monomolecular reaction rate k3= 1/τ3 (in units of 1/s) is the inverse of the relaxation time τ1 from S1 to S0 due to fluorescence. The photosensitizer in the ground state can interact with the singlet oxygen and produce photoproduct [SO2]. The bimolecular reaction rate, k1 (in unit of 1/s/µM), describes the irreversible photobleaching reaction of the photosensitizer: k1 [ S 0 ] +[1O2 ] ⇒[ SO2 ]
Similarly, the bimolecular reaction rate, S∆k2 (in unit of 1/s/µM) describe the production of singlet oxygen caused by the collisions between the photosensitizer triplet state [T] and the ground state oxygen [3O2] to create the singlet oxygen [1O2]. The second equation illustrates that a fraction (S∆) of interaction between [T] and [3O2] does not yield singlet oxygen. S∆ k2 [T ] +[ 3O2 ] ⇒ [ S 0 ] +[1O2 ]
[T ] +[3O2 ]
(1− S∆ )k2 ⇒ [S0 ] +[3O2 ]
The phosphorescence and inter-system crossing of the photosensitizer are described by the monomolecular reaction rate k4 and k5 (in units of 1/s), respectively: k4 [T ] ⇒[ S 0 ]
k5 [ S1] ⇒[T ] The phosphorescence of singlet oxygen and the oxidation of acceptors [A] are described by the reaction rates k6 and k7, respectively: k6 [1O2 ] ⇒[3O2 ] k7
[ A] +[1O2 ] ⇒[ AO2 ]
Here [S0], [S1], and [T] also describes the concentration of the ground, excited, and triplet state of the photosensitizer molecule, respectively. [1O2] and [3O2] is the concentration of singlet and triplet (ground) state of the oxygen,
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respectively, and [A] is the concentration of all singlet oxygen acceptors except for the photosensitizer itself. [SO2], [AO2] are the concentrations of oxidation products. The kinetics of the above processes can be described by the coupled differential equations: d[S0 ] dt
= −k0[S0 ] − k1[1O2 ][S0 ] + k2[T ][3O2 ] + k3[S1] + k4[T ]
d [ S1 ] = −(k3 + k5 )[S1] + k0[ S0 ] dt d [T ] = −k2 [T ][3O2 ] − k4 [T ] + k5[ S1 ] dt d [ 3O 2 ]
= − S ∆ k 2 [T ][ 3O2 ] + k 6 [1O2 ] + P dt 1 d [ O2 ] = −k1[ S 0 ][1O2 ] + S ∆ k 2 [T ][3O2 ] − k6 [1O2 ] − k7 [ A][1O2 ] dt d [ A] = −k7 [ A][1O2 ] . dt
(3) (4) (5) (6) (7) (8)
Where P is the oxygen diffusion and perfusion rate and can be treated as a known constant. S∆ is the fraction of interaction between [3O2] and [T] that resulted in production of singlet oxygen. The values of the various parameters for Photofrin are listed in Table 1. Table 1: Parameters used in the macroscopic kinetics equations for Photofrin. Definition Values References 12 1.9 s-1 PS absorb. rate at φ =100 mW/cm2 11 5 Photobleaching rate 1.2 × 10 1/µM·s 3 13 Reaction rate of O2 with T 100 1/µM·s for k4/k2=12.1 µM 13 Rate of S1 to S0 for k5/(k5+k3)= 0.80 2.0 × 107 1/s 10 Rate of T to S0 1250 1/s 11 7 Rate of S1 to T 8.0 × 10 1/s 11 1 3 6 Rate of O2 to O2 1 × 10 1/s 1 13 6 Reaction rate of O2 with [A] for k1/k7[A]=56.5M-1 2.6 × 10 1/µM·s 12 -1 Extinction coefficient of [S0] 0.0036 cm /µM ε 13 1 3 0.5 Fraction O2 from reaction [T] and [ O2] S∆ 11 P Oxygen Diffusion and Perfusion rate 1.66 × 10-2 µM/s 14 [S0]i PS concentration 8.5 µM* (= 5 mg/kg) 11 Init. Con. [S1]i 0 µM 11 Init. Con. [T]i 0 µM 13 [3O2]i Init. Con. 83 µM 11 1 Init. Con. [ O2]i 0 µM 11 Init. Con. [A]i 830 µM 14 -3 * 1 µM = 6.022 × 10 cm . Sym. k0 k1 k2 k3 k4 k5 k6 k7
It should be emphasized that the first term in Eq. (3), k0[S0] is the rate of PDT dose, as that defined in Eq. (1), multiplied by the tissue mass density ρ. The light fluence rate, φ, describes the light propagation in tissue and follows a diffusion equation: ∇(1 / 3µ s ' ) D∇φ − µ a φ = S , (9) where µa = µ0 + ε [S0 ] (in cm-1) and µs’ (in cm-1) are the absorption and reduced scattering coefficients in tissue, and S (in W) is the source strength. For a point source in an infinite turbid medium, one can provide an analytical solution to Eq. (9) as:
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φ=
3µ s ' S − µeff r e , 4πr
(10)
where S is the light source strength (in W), r is the radial distance from the point source, and µeff = 3µa µ s ' is the effective attenuation coefficient. For a given photosensitizer, the reaction rates (k0 – k7) are considered constants. The rate equation is implemented in the COMSOL finite-element modeling (FEM) package (COMSOL AB, Stockholm, Sweden). The calculation time is in seconds for the rate equation alone and tens of minutes for the time and spatially coupled differential equations. 2.2 Oxygen dependence of the singlet oxygen quantum yield The life time of the singlet and triplet states of photosensitizer ([S1] and [T]) and the singlet oxygen (1O2) 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 for Eqs. (4), (5), (7). Thus, the coupled kinetics equations (3) – (8) can be rewritten as: k5 1 [ 3O2 ] [1O2 ] = S ∆ ( )⋅( 3 )⋅( ) ⋅ k0 [ S0 ] (11) k5 + k3 [ O2 ] + k 4 / k 2 k6 + k7 [ A] 1 ) ⋅ k 0 [S 0 ] k5 + k3 k5 1 [T ] = ( )⋅( 3 ) ⋅ k0 [ S 0 ] k5 + k3 k 2 [ O2 ] + k 4 [ S1 ] = (
(12)
(13)
d [S0 ] k5 k [S ] [3O2 ] = −( )( 3 )( 1 0 ) ⋅ k 0 [ S 0 ] dt k5 + k3 [ O2 ] + k 4 / k 2 k6 + k7 [ A]
(14)
k5 k7 [ A] d [3O2 ] [3O2 ] = −S∆ ( )( 3 )( ) ⋅ k0 [ S 0 ] + P dt k5 + k3 [ O2 ] + k 4 / k 2 k6 + k7 [ A]
(15)
and
We have replaced k1[S0]+k7[A] by k7[A] in Eq. 15 since k1[S0]
1. INTRODUCTION Photodynamic therapy (PDT) is a cancer treatment modality based on the interaction of light, a photosensitizing drug, and oxygen.1 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.2 To quantify the photodynamic efficacy, the current start of art in clinical PDT dosimetry uses a quantity called PDT dose,3 defined as the number of photons absorbed by photosensitizing drug per gram of tissue [ph/g], i.e., D = ∫0t εc ⋅
φ (t ' ) 1 ⋅ dt ' , hν ρ
(1)
where ρ is the density of tissue [g/cm3], φ is the light fluence rate [W/cm2], hν is the energy of a photon [J/ph], c is the drug concentration in tissue [µM], ε is the extinction coefficient of the photosensitizer [1/cm/µM]. The photodynamic dose (D) does not consider the quantum yield (η) of oxidative radicals, the effect of tissue oxygenation on η, or the fraction (f) of radicals that oxidize cellular targets. Thus, it is only applicable in cases where ample oxygen supplies exist. For type-II photodynamic interaction, production of singlet oxygen is responsible for the PDT damage. The production of singlet oxygen that has damaged the cellular targets can be expressed as: 4 [1 O2 ] rx = f ⋅η ⋅ ρD ,
(2)
where f depends on the localization of the photosensitizer at the cell level and thus depends on the photosensitizer and tissue types, the quantum yield η gives the number of singlet oxygen molecules produced per an absorbed photon, which is a constant under ample oxygen supply. However, when insufficient oxygen supply exists, η is also a function of the
*
[email protected]; phone 1 215 662-4043; fax 1 215 349-5978. Optical Methods for Tumor Treatment and Detection: Mechanisms and Techniques in Photodynamic Therapy XVI, edited by David Kessel, Proc. of SPIE Vol. 6427, 642708, (2007) · 1605-7422/07/$18 · doi: 10.1117/12.701387
Proc. of SPIE Vol. 6427 642708-1 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 08/10/2015 Terms of Use: http://spiedigitallibrary.org/ss/TermsOfUse.aspx
oxygen concentration, or pO2, in tissue. (We explain later why the PDT dose multiplied the tissue mass density, or ρD [ph/cm3] is used in Eq. (2)). The purpose of this study is to establish the relationship between the singlet oxygen concentration and the PDT dose under arbitrary oxygen supply conditions. This is achieved by solving the coupled kinetics equations describing the macroscopic process of the generation of singlet oxygen and the diffusion equations describing the light transport assuming the 3-dimensional distribution of photosensitizers can be independently characterized during PDT. Since the oxygen concentration in tissue is a local quantity that cannot be easily measured. We have also presented a model that can be used to determine the local oxygen concentration by measuring the macroscopic quantities related to oxygen supplies in the blood vessel, i.e., hemoglobin concentration, oxygen saturation, and blood flow.
2. METHOD The majority of photosensitizers available for PDT utilize Type II photodynamic processes, i.e., the photodynamic effect is achieved through the production of singlet oxygen.5, 6 The Jablonski diagram shown in Fig. 1 summarizes the underlying physical processes involved in type-II PDT. The process begins with the absorption of a photon by photosensitizer in its ground state, promoting it to an excited state. The photosensitizer molecule can return to its ground state by emission of a fluorescence photon, which can be used for fluorescence detection. Alternatively, the molecule may convert to a triplet state, a process known as intersystem crossing (ISC). A high intersystem-crossing yield is an essential feature of a good Type II photosensitizer. Once in its triplet state, the molecule may undergo a collisional energy transfer with ground state molecular oxygen (type II) or with the substrate (type I). In type II interaction, the photosensitzer returns to its ground state, and oxygen is promoted from its ground state (a triplet state) to its excited (singlet) state. Since the photosensitizer is not consumed in this process, the same photosensitizer molecule may create many singlet oxygen molecules.
S1
Intersystem crossing k5 T
k0, absorption
k7, Reaction with targets [A] Energy transfer to O2 k2
1
O2
k3,fluorescence k4, phosphorescence
k6, phosphorescence λ = 1260 nm
k1, photobleaching
S0
Sensitizer
3
O2
Oxygen
Figure 1: Jablonski diagram of photosensitized singlet oxygen formation by Type II photosensitizer. The rate constants for monomolecular transition (solid lines) and bimolecular energy transfer (dashed lines) are indicated. Once the singlet oxygen is created, it reacts almost immediately with cellular targets in its immediate vicinity. The majorities 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, singlet oxygen may react with the photosensitizer itself. This leads to its irreversible destruction (photobleaching). Photobleaching can decrease the effectiveness of PDT by reducing the photosensitizer concentration, however it can also be useful for dosimetry.7 Because of its high reactivity, singlet
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oxygen has a very short lifetime in tissue. However, a small fraction of the singlet oxygen produced may return to its ground state via emission of a phosphorescence photon, which can be detected optically.8, 9 2.1 Macroscopic kinetics rate equations We adopted the rate equation approach first proposed by Foster et al 10 and later refined by Hu et al 11 to describe the PDT kinetics process. We use ki (i = 0, 1, …, 7) to designate the reaction rate. The details of the reactions associated with the reaction rates and the species involved are summarized below: The PDT process started by the absorption of light by the photosensitizer in the ground state, S0, and it excites to the singlet state S1. The S1 state will spontaneously decay to the ground state, S0, by emission of a fluorescent photon. k0
[ S0 ] ⇔[ S1 ] k3
This is a reversible process. The monomolecular reaction rate k0 (1/s) is proportional to the light fluence φ (in unit of W/cm2): k0 = ε ⋅ ( φ ) , where ε is the extinction coefficient (in unit of cm-1/µM), hν is the photon energy (in unit of Joule). hν
The monomolecular reaction rate k3= 1/τ3 (in units of 1/s) is the inverse of the relaxation time τ1 from S1 to S0 due to fluorescence. The photosensitizer in the ground state can interact with the singlet oxygen and produce photoproduct [SO2]. The bimolecular reaction rate, k1 (in unit of 1/s/µM), describes the irreversible photobleaching reaction of the photosensitizer: k1 [ S 0 ] +[1O2 ] ⇒[ SO2 ]
Similarly, the bimolecular reaction rate, S∆k2 (in unit of 1/s/µM) describe the production of singlet oxygen caused by the collisions between the photosensitizer triplet state [T] and the ground state oxygen [3O2] to create the singlet oxygen [1O2]. The second equation illustrates that a fraction (S∆) of interaction between [T] and [3O2] does not yield singlet oxygen. S∆ k2 [T ] +[ 3O2 ] ⇒ [ S 0 ] +[1O2 ]
[T ] +[3O2 ]
(1− S∆ )k2 ⇒ [S0 ] +[3O2 ]
The phosphorescence and inter-system crossing of the photosensitizer are described by the monomolecular reaction rate k4 and k5 (in units of 1/s), respectively: k4 [T ] ⇒[ S 0 ]
k5 [ S1] ⇒[T ] The phosphorescence of singlet oxygen and the oxidation of acceptors [A] are described by the reaction rates k6 and k7, respectively: k6 [1O2 ] ⇒[3O2 ] k7
[ A] +[1O2 ] ⇒[ AO2 ]
Here [S0], [S1], and [T] also describes the concentration of the ground, excited, and triplet state of the photosensitizer molecule, respectively. [1O2] and [3O2] is the concentration of singlet and triplet (ground) state of the oxygen,
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respectively, and [A] is the concentration of all singlet oxygen acceptors except for the photosensitizer itself. [SO2], [AO2] are the concentrations of oxidation products. The kinetics of the above processes can be described by the coupled differential equations: d[S0 ] dt
= −k0[S0 ] − k1[1O2 ][S0 ] + k2[T ][3O2 ] + k3[S1] + k4[T ]
d [ S1 ] = −(k3 + k5 )[S1] + k0[ S0 ] dt d [T ] = −k2 [T ][3O2 ] − k4 [T ] + k5[ S1 ] dt d [ 3O 2 ]
= − S ∆ k 2 [T ][ 3O2 ] + k 6 [1O2 ] + P dt 1 d [ O2 ] = −k1[ S 0 ][1O2 ] + S ∆ k 2 [T ][3O2 ] − k6 [1O2 ] − k7 [ A][1O2 ] dt d [ A] = −k7 [ A][1O2 ] . dt
(3) (4) (5) (6) (7) (8)
Where P is the oxygen diffusion and perfusion rate and can be treated as a known constant. S∆ is the fraction of interaction between [3O2] and [T] that resulted in production of singlet oxygen. The values of the various parameters for Photofrin are listed in Table 1. Table 1: Parameters used in the macroscopic kinetics equations for Photofrin. Definition Values References 12 1.9 s-1 PS absorb. rate at φ =100 mW/cm2 11 5 Photobleaching rate 1.2 × 10 1/µM·s 3 13 Reaction rate of O2 with T 100 1/µM·s for k4/k2=12.1 µM 13 Rate of S1 to S0 for k5/(k5+k3)= 0.80 2.0 × 107 1/s 10 Rate of T to S0 1250 1/s 11 7 Rate of S1 to T 8.0 × 10 1/s 11 1 3 6 Rate of O2 to O2 1 × 10 1/s 1 13 6 Reaction rate of O2 with [A] for k1/k7[A]=56.5M-1 2.6 × 10 1/µM·s 12 -1 Extinction coefficient of [S0] 0.0036 cm /µM ε 13 1 3 0.5 Fraction O2 from reaction [T] and [ O2] S∆ 11 P Oxygen Diffusion and Perfusion rate 1.66 × 10-2 µM/s 14 [S0]i PS concentration 8.5 µM* (= 5 mg/kg) 11 Init. Con. [S1]i 0 µM 11 Init. Con. [T]i 0 µM 13 [3O2]i Init. Con. 83 µM 11 1 Init. Con. [ O2]i 0 µM 11 Init. Con. [A]i 830 µM 14 -3 * 1 µM = 6.022 × 10 cm . Sym. k0 k1 k2 k3 k4 k5 k6 k7
It should be emphasized that the first term in Eq. (3), k0[S0] is the rate of PDT dose, as that defined in Eq. (1), multiplied by the tissue mass density ρ. The light fluence rate, φ, describes the light propagation in tissue and follows a diffusion equation: ∇(1 / 3µ s ' ) D∇φ − µ a φ = S , (9) where µa = µ0 + ε [S0 ] (in cm-1) and µs’ (in cm-1) are the absorption and reduced scattering coefficients in tissue, and S (in W) is the source strength. For a point source in an infinite turbid medium, one can provide an analytical solution to Eq. (9) as:
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φ=
3µ s ' S − µeff r e , 4πr
(10)
where S is the light source strength (in W), r is the radial distance from the point source, and µeff = 3µa µ s ' is the effective attenuation coefficient. For a given photosensitizer, the reaction rates (k0 – k7) are considered constants. The rate equation is implemented in the COMSOL finite-element modeling (FEM) package (COMSOL AB, Stockholm, Sweden). The calculation time is in seconds for the rate equation alone and tens of minutes for the time and spatially coupled differential equations. 2.2 Oxygen dependence of the singlet oxygen quantum yield The life time of the singlet and triplet states of photosensitizer ([S1] and [T]) and the singlet oxygen (1O2) 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 for Eqs. (4), (5), (7). Thus, the coupled kinetics equations (3) – (8) can be rewritten as: k5 1 [ 3O2 ] [1O2 ] = S ∆ ( )⋅( 3 )⋅( ) ⋅ k0 [ S0 ] (11) k5 + k3 [ O2 ] + k 4 / k 2 k6 + k7 [ A] 1 ) ⋅ k 0 [S 0 ] k5 + k3 k5 1 [T ] = ( )⋅( 3 ) ⋅ k0 [ S 0 ] k5 + k3 k 2 [ O2 ] + k 4 [ S1 ] = (
(12)
(13)
d [S0 ] k5 k [S ] [3O2 ] = −( )( 3 )( 1 0 ) ⋅ k 0 [ S 0 ] dt k5 + k3 [ O2 ] + k 4 / k 2 k6 + k7 [ A]
(14)
k5 k7 [ A] d [3O2 ] [3O2 ] = −S∆ ( )( 3 )( ) ⋅ k0 [ S 0 ] + P dt k5 + k3 [ O2 ] + k 4 / k 2 k6 + k7 [ A]
(15)
and
We have replaced k1[S0]+k7[A] by k7[A] in Eq. 15 since k1[S0]
