While photodynamic therapy (PDT) has received regulatory approval for treatment of some types of cancer in a number of countries [1], its dosimetry remains relatively crude. Typically the injected or topically applied dose of photosensitizer is recorded along with the incident light fluence. However the actual drug concentration in the target volume at the time of treatment will vary from subject to subject. The optical properties of the tissue will also vary, affecting the actual light fluence delivered to critical sites within the target volume. In addition, the efficacy of PDT depends on oxygen concentration which, in turn, depends on the vascular supply to the tumor and oxygen utilization within the tumor.

One approach to PDT dosimetry is to measure all three of these variables (drug concentration, light fluence, oxygen concentration) in vivo in individual patients. These data could then be used as input to models which attempt to predict biological effect. The measurement of any one of these variables is a challenge in the clinical environment, although progress has been made in developing noninvasive techniques. A further complication is that all three variables may change during the treatment due to processes such as photobleaching, oxygen consumption by PDT and vascular shutdown. For a complete discussion of these issues, the reader is referred to the paper by Wilson et al. [2]. An alternative dosimetric approach which would compensate for this dynamic interaction is very appealing. Recently a number of authors have suggested that the photobleaching of the sensitizer itself may fulfil this role [2-4]. The relative concentration of sensitizers can be readily monitored during treatment by fluorescence spectroscopy and such measurements have been demonstrated clinically [5]. In this paper we will examine the relationship between the fluorescence signal and the yield of singlet oxygen, the putative cytotoxic product in PDT. We will demonstrate that, in principle, different features of the photobleaching process should be measured depending on the mechanism of bleaching. Experimental confirmation of these concepts is still necessary.

**Materials and Methods**
Symbols and Definitions

We list here symbols and quantities to be used in the following discussion. We have attempted to be consistent with the notation used by Georgakoudi et al. [6] and our earlier paper [2].

E light fluence rate, photosensitizer extinction coefficient [So]photosensitizer ground state concentration S) fraction of triplet quenching collisions between triplet and oxygen that yield1O2 kot bimolecular rate of triplet quenching by oxygen koa bimolecular rate of reaction between 1O2 and So F* measured fluorescence signal ft triplet yield kd monomolecular decay rate for 1O2 s photobleaching rate constant
T total treatment time [3O2] concentration of ground state triplet oxygen [1O2] concentration of excited singlet oxygen [A]concentration of acceptor molecules.

Theory

In this section we will examine the relationship between singlet oxygen yield and features of the F* (t) curve, i.e. the detected fluorescence signal as a function of time. Our first assumption is that singlet oxygen, 1O2, is the toxic agent and that treatments which produce the same amount of 1O2 are isoeffective regardless of the time course of 1O2 production. Mathematically we seek expressions for

where T is the total treatment time.

3.1 Photobleaching mediated by singlet oxygen. In this case bleaching of the sensitizer is caused by reaction with singlet oxygen, so that the relevant kinetic equation is

which, as shown by Georgakoudi et al. [6], has solution

so that the expression in (1) becomes

Assuming that the detected fluorescence signal is proportional to the photosensitizer concentration, i.e. F* (t) " [So] (t), Eq. (4) can be written as

This implies that treatments for which the fraction of drug bleached is identical should be isoeffective. This is true regardless of changes in [302] or E(t) during the treatment.

Case of “abundant” oxygen
Here we follow Forrer et al. [3] to derive an expression for (1) in terms of the fluence rate and the fluorescence signal. We start with Georgakoudi’s [6] expression for [1O2]
If there is “abundant” oxygen so that [3O2] (t) >> kp/kot and if koa [A] >> kos [So] (reasonable because of the high concentration of potential acceptors)
and further assuming that [A] (t) ~ [A], so that acceptors are not depleted during the treatment
This is equivalent to Forrer’s assumption that the instantaneous singlet oxygen concentration is proportional to the product of the fluence rate and the photosensitizer concentration. So now the damage integral
and we would expect treatments to be isoeffective if the integral
were identical.

3.3 Case of constant fluence rate
A further simplification is possible if the fluence rate is constant during the treatment. From Eq. (8)and substituting in Eq. (2)

where

Eq. (12) is equivalent to Forrer’s second order kinetic equation for bleaching. It may be solved to yield

and

so isoeffective treatments are those which conserve the product ET [So](0) or equivalently ET F*(0). Note that ET is just the total delivered fluence, so Eq. (14) suggests that one could measure the initial fluorescence and the delivered fluence in order to predict biological effect. Clinical results in support of this hypothesis have been reported by Oseroff [7].
3.4 Photobleaching not mediated by singlet oxygen
Here we consider the case where the sensitizer is bleached by reaction of the excited state with some substrate [8]. We expect the relevant kinetic equation to be first order
Note that Eq. (5) is still valid (kos would be zero) so our conclusion that isoeffective treatments would conserve the integral in (10) is still valid as long as there is abundant oxygen. As we did in Section 3, we can derive an explicit expression for the damage integral by noting that the solution of Eq. (15) is
so that Eq. (9) becomes
For the case of constant E(t) = E, this integral can be solved to yield the explicit solution
Equation (18) implies that isoeffective treatments would maintain F* (0) - F* (T) and that no explicit knowledge of the fluence is necessary.

**Discussion and Conclusion**
The preceding theoretical development involves several assumptions and simplifications, but should generate testable hypotheses regarding PDT dosimetry based on photobleaching. The following conclusions are based on this analysis:
As long as there is “abundant” oxygen the dose metric given by the integral

is applicable, whether the bleaching is mediated by singlet oxygen or not. If oxygen is significantly depleted, an independent measure of [3O2] would be required to “correct” this measurement.
If the bleaching is known to be mediated by singlet oxygen, the appropriate metric is F* (0)/F* (T). No measurement of fluence rate or oxygen is required, even if oxygen is depleted.
If the bleaching is not singlet oxygen mediated, the simplest metric is
F* (0) - F* (T). However if oxygen is depleted, this will not be adequate.

Our goal in the future is to test these hypotheses in cell suspensions and animal tumor models.

Acknowledgment

This research was supported by the National Institutes of Health PO1-CA 43892.

**References**
For recent review articles on various aspects of PDT see J. Clin. Laser Med. Surg. 14(5), 1996.

B.C. Wilson, M.S. Patterson and L. Lilge, Implicit and explicit dosimetry in photodynamic therapy: a new paradigm, Lasers Med. Sci. 12:182-199 (1997).

M. Forrer, T. Glanzmann, D. Braichotte, G. Wagnieres, H. van den Bergh, J-F. Savary and P. Monnier, In vivo measurement of fluorescence bleaching of meso-tetra hydroxyphenyl chlorin (mTHPC) in the esophagus and the oral cavity, SPIE Proc. Vol. 2627:33-39 (1996).

D.J. Robinson, H.S. de Bruijn, N. van der Veen, M.R. Stringer, S.R. Brown and W.M. Star, Fluorescence photobleaching of ALA-induced protoporphyrin IX during photodynamic therapy of normal hairless mouse skin: the effect of light dose and irradiance and the resulting biological effect, Photochem. Photobiol. 67:140-149 (1998).

D. Braichotte, J-F. Savary, T. Glanzmann, P. Westermann, S. Folli, G. Wagnieres, P. Monnier and H. van den Bergh, Clinical pharmacokinetic studies of tetra (meta hydroxyphenyl) chlorin in squamous cell carcinoma by fluorescence spectroscopy at 2 wavelengths, Int. J. Cancer 63:198-204 (1995).

I. Georgakoudi, M.G. Nichols and T.H. Foster, The mechanism of Photofrin photobleaching and its consequences for photodynamic therapy, Photochem. Photobiol. 65:135-144 (1997).

A. Oseroff, unpublished, shown in [2].

I. Georgakoudi and T.H. Foster, Singlet oxygen- versus nonsinglet oxygen-mediated mechanisms of sensitizer photobleaching and their effects on photodynamic dosimetry, Photochem. Photobiol. 67:612-625 (1998).