Nonlinear photodynamic therapy. Photochemical dose levelling within a tumor by saturating a photosensitizer’s triplet states Boris Ya. Kogan State Research Center of the Russian Federation, Research Institute of Organic Intermediates and Dyes, 1 ul. B. Sadovaya, 123995 Moscow, Russian Federation. E-mail:
[email protected] Received 21st November 2003, Accepted 15th January 2004 First published as an Advance Article on the web 10th February 2004
The photodynamic therapy involving saturation of triplet state of a photosensitizer by short high-energy light pulses was studied theoretically. The possibility of creating a uniform therapeutic dose distribution throughout a large tumor using both surface and interstitial irradiation is shown. Possible thickness of the treated tissue layer and the required duration of the PDT session are estimated.
DOI: 10.1039/ b315112c
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Introduction
A major drawback of conventional photodynamic therapy (PDT) is essentially nonuniform spatial distribution of the photochemical dose (PD). This is due to nonuniformity of the light dose distribution owing to absorption of light by tissues. The layers located closer to a light source receive a too high PD, which entails some negative effects (e.g., predominance of necrosis over the more preferable apoptosis mechanism of cell death 1 and enhancement of light absorption by the tissue 2). Such an overdosage is particularly high when attempting to attain a therapeutic effect at a great depth under surface irradiation or at a long distance from the light source under interstitial irradiation. The PD received near the light source can be hundreds of times higher than the required dose. Creating a uniform PD distribution throughout a thick tissue layer requires the use of nonlinear PDT (here, no linear correlation between the PD and the light dose occurs). Earlier,2 we have reported a PDT technique, which involves a total photobleaching of photosensitizer (PS) and helps both to avoid overdosage and to obtain a uniform PD distribution within the tissue. The procedure allows operation with conventional light sources but demands special PS with appropriate photobleaching parameters. In the present paper we consider a modification of the PDT method, which leads to the same results using any PS by saturating its triplet states. Practical implementation of the technique requires the use of pulsed light sources capable of producing high-energy pulses. It is necessary to remark that neither the method of total PS photobleaching nor the triplet states saturation (TSS) mode require an increase of the needed complete light dose during PDT session as it can seem at first thought. Generally, pulsed light sources have been widely used in PDT but the pulse energies are low. A number of studies on comparison of continuous-wave and pulsed lasers for conventional PDT are available.3–18 The results obtained were found to be ambiguous. Sterenborg and van Gemert have theoretically analyzed the results of this comparison.19 The basic conclusion was that the effectiveness of pulsed excitation of haematoporphyrin in PDT is identical to that of CW excitation for fluence rates below 40 kW cm⫺2. Above this threshold the effectiveness drops significantly. In practice this effect can only occur with pulsed lasers with high pulse energy and low repetition frequency. Authors 19 have noted that “the advantage of saturation is that the photodynamic effect is only dependent on the local concentration of the photosensitizer”. This feature can be practically implemented to level the PD throughout the tumor volume and also to facilitate the solution of the PDT dosimetry P h o t o c h e m . P h o t o b i o l . S c i . , 2 0 0 4 , 3, 3 6 0 – 3 6 5
problem, which has been the subject of considerable research in the last two decades.20–30 The concern that this benefit can be reduced to zero by photobleaching of PS is not substantiated as all used PS are too stable. The increase of a quantum yield of photobleaching only promotes a PD levelling. We analyzed this problem earlier in details.2 The practical application of TSS method needs knowledge of spatial PD distribution within a tumor depending on an irradiation mode. In this paper we report on mathematical treatment of the PD distribution in a tissue during a PDT session using pulsed lasers with high pulse energies much exceeding the TSS threshold and with pulse durations that are shorter than the triplet lifetime of PS. This last condition is valid for the majority of pulsed lasers. Such short pulses are most suitable for the TSS mode. Both surface and interstitial irradiation are considered.
2 2.1
Theory Photochemical dose accumulation
Let us define the PD at any point of a tissue as the concentration of defects due to photochemical oxidation.2 This concentration is equal to the concentration of photochemically bound oxygen. In the text below we will use the simplest diffusion theory,31 which uses a concept of “photon concentration” for the description of propagation of a light throughout a tissue. We will denote the concentrations of the PS molecules, photons, and oxygen at a certain point of a tissue as n, U, and [O2], respectively. The change in the concentration of the PS triplet states, n2, can be described by the following equation: dn2/dt = k1(n ⫺ n2)U ⫺ n2(k2[O2] ⫹ k3)
(1)
Here, k1 and k2 are respectively the second-order rate constants for formation (by light) and quenching (by oxygen) of triplet states, and k3 is the first-order rate constant for triplet relaxation. We will denote the light pulse duration tp. It will be recalled that this paper deals with the short-pulse case: tp 1
(5)
At k1Φp >> 1, almost all PS molecules go to the triplet state by the end of the pulse. The photochemical processes proceed after pulse termination. Quenching of triplet states can be described by the equation: dn2/dt = ⫺ n2(k2[O2] ⫹ k3)
(6)
with the initial conditions n2(0) = n(1 ⫺ exp(⫺k1Φp)) and [O2](0) = [O2]0. The oxygen depletion rate due to biosubstrate oxidation equals the PD accumulation rate: -d[O2]/dt = θn2k2[O2] = dPD/dt
(7)
Here θ is the probability of oxidation after quenching of the triplet state by oxygen. From eqn. (6) and (7) one gets: (1 ⫹ k3(k2[O2])⫺1) d[O2] = θ dn2
(8)
Integration of eqn. (8) from n2(0) up to 0 gives the PD per pulse (PDp): PDp ⫺ k3k2⫺1ln(1 ⫺ PDp/[O2]0) = θn(1 ⫺ exp(⫺k1Φp))
Propagation of light throughout a tissue can be described by the stationary diffusion equation: 31 ∆U ⫺ β 2U = 0
(13)
where β 2 = væ/D, D is the photon diffusion coefficient, v is the velocity of light in the medium, and æ is the absorption coefficient. We assume that PS contributes insignificantly to light absorption by the tissue and that β is independent of n. This is a quite reasonable practical approximation, which is undoubtedly valid under TSS conditions. Let us consider the distributions of photons in tissue for both superficial and interstitial irradiations. We will restrict ourselves to the simplest cases, viz., a plane light wave for superficial and a spherical source for interstitial irradiation. Practically, a restricted light beam falling on a tissue surface and a cylindrical light source placed in depth of a tissue are more often used. However, the solutions of the corresponding equations are cumbersome while qualitative results are similar. 2.2.1 Plane light wave falling on a tissue surface. In this case the solution of eqn. (13) has the form: U(x) = U(0)exp(⫺βx)
(9)
Fig. 1 illustrates the dependence of PDp on Φp (eqn. (9)) at different θn([O2]0)⫺1 values and k3(k2θn)⫺1 = 10 (see Section 3.1).
Distribution of photons in tissue
(14)
where x is the distance from the surface. 2.2.2 Spherical light source placed in depth of a tissue. In this case the solution of eqn. (13) is given by the following expression: U(r) = U(r0)r0r⫺1exp(⫺β(r ⫺ r0))
(15)
where r0 is the source radius (below we will consider only the case βr0 r0).
Fig. 1 PDp saturation at k3(k2θn)⫺1 = 10 and θn([O2]0)⫺1 = 3 × 10⫺1 (1), 10⫺1 (2), 3 × 10⫺2 (3), 10⫺2 (4), and 3 × 10⫺3 (5).
As can be seen in Fig. 1, the PDp reaches a saturation value (PDp) when Φp meets condition (5). The higher the [O2]0 concentration, the closer the PDp value to θn. This can be derived immediately from eqn. (9). At θn r0). The distribution of photons at a distance r from the center of the source (r > L) can be obtained with sufficient accuracy using eqn. (15). Generally, the distribution of photons around a cylindrical source can be derived as superposition of solutions of eqn. (15) for a number of spherical sources located on the cylinder axis. 2.3
Photochemical dose distribution within tissue
2.3.1 Plane light wave falling on a tissue surface. In this case the dependence of PDp on the depth x can be obtained from eqn. (9) and (14). A simpler but correct solution can be derived by substituting eqn. (14) into eqn. (10): PDp = θn(1 ⫺ exp(⫺k1Φp(0)exp(⫺βx)))(1 ⫹ A)⫺1
(16)
The Φp(0) value correlates with the fluence per pulse, Ep(0), by ratio Φp(0) ≅ (1 ⫹ R∞)Ep(0)2(vhν)⫺1 (R∞ is the reflection coefficient of a thick tissue layer, and hν is the photon’s energy).2 The PDp dependence on x according to eqn. (16) at various fluence values is illustrated by some examples shown in Fig. 2. P h o t o c h e m . P h o t o b i o l . S c i . , 2 0 0 4 , 3, 3 6 0 – 3 6 5
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from 0 up to tp taking into account the relationships R∞ = (1 ⫺ βd)(1 ⫹ βd)⫺1 and d = 2D/v: c∆T = β(1 ⫺ R∞)Ep(0)
(19)
where c is the heat capacity per unit volume of the tissue and ∆T is the increase in the temperature due to the action of the light pulse. Tissue heating near a spherical source can be determined similarly from eqn. (18): c∆T = β 2εp(4πr0)⫺1
(20)
⫺2
Fig. 2 PDp distribution in depth of a tissue at Ep(0) = 10 (1), 10⫺1 (2), 1 (3), and 10 J cm⫺2 (4); k1 = 2.3 × 10⫺6 cm3 s⫺1, R∞ = 0.43, v = 2.2 × 1010 cm s⫺1, hν = 3 × 10⫺19 J, β = 4 cm⫺1, and A = 0.1.
The numerical values of the equation parameters are as follows: k1 = 2.3 × 10⫺6 cm3 s⫺1, R∞ = 0.43, v = 2.2 × 1010 cm s⫺1, hν = 3 × 10⫺19 J, β = 4 cm⫺1, and A = 0.1. 2.3.2 Spherical light source in depth of a tissue. The PDp distribution around a spherical source can be obtained by substituting eqn. (15) into eqn. (9) or eqn. (10). In the latter case one gets a simpler solution: PDp ≅ θn(1 ⫺ exp(⫺k1Φp(r0)r0r⫺1exp(⫺βr)))(1 ⫹ A)⫺1
(17)
The Φp(r0) value correlates with the pulse energy, εp, by ratio Φp(r0) = εp(4πr0Dhν)⫺1.2 Fig. 3 shows the dependence of PDp on r according to eqn. (17) at various energies εp. The r0 value has little effect on the PDp distribution at βr0 NthEp(0)I(0)a⫺1 = PDthEp(0)(PDpI(0)a)⫺1
where I(0)a is the maximum permissible average fluence rate and PDth is the required therapeutic dose. In the case of interstitial irradiation one has: tth > NthεpPa⫺1 = PDthεp(PDpPa)⫺1,
3.1
2.3.3 Cylindrical light source in depth of a tissue. Let L be the source length and rs be the radius of the PDp saturation area. At L < rs, eqn. (17) provides a rather accurate description of the PDp distribution. Otherwise, see Section 2.2.3. 2.4
Permissible energy of a light pulse
An increase in the pulse energy entails an increase in the thickness of a tissue layer for which the TSS condition is valid. However, the pulse energy is limited by the permissible heating of tissue. Other effects (for example photochemical reactions of native tissue pigments) are insignificant as it is shown by practice of application of the powerful pulsed lasers in cosmetology. Let us estimate the extent of tissue heating by a light pulse at different irradiation regimes. The relation between the concentration of photons and the rate of their absorption by the tissue can be determined from the non-stationary photon diffusion equation: ∂U/∂t = ⫺ β 2DU
(18)
The increase, ∆T , in the tissue temperature near the irradiated surface can be determined by integration of eqn. (18) 362
P h o t o c h e m . P h o t o b i o l . S c i . , 2 0 0 4 , 3, 3 6 0 – 3 6 5
(22)
where Pa is the maximum permissible average power which can be brought into the optical waveguide during irradiation.
3
Fig. 3 PDp distribution around a spherical light source at εp = 10⫺3 (1), 10⫺2 (2), 10⫺1 (3), and 1 J (4). Other parameters are the same as in Fig. 2.
(21)
Estimation of parameters and discussion Kinetics parameters
The rate constant k1 can be estimated from the ratio k1 = σγv/2, where σ is the cross-section of light absorption by the PS molecule and γ is the quantum yield of triplet states. The k1 value can vary from 10⫺8 up to 5 × 10⫺6 cm3 s⫺1. Photosensitizers characterized by high k1 values seem to be more appropriate for the TSS mode of PDT. Here we used k1 = 2 × 10⫺6 cm3 s⫺1. Typical k2 and k3 values are k2 ≅ 109 M⫺1 s⫺1 and k3 ranging from 103 to 104 s⫺1. The θ value lies between 0.5 and 1.32 The mean PS concentration in the tumor during PDT session is usually of the order of 10⫺7 to 10⫺6 M. The oxygen concentration, [O2]0, can vary from 10⫺4 M in the cells near blood vessels up to 10⫺6 M in the poorly oxygenated tumor cells. Thus, above-mentioned condition θn
