STEP FILM MODEL FOR SPFs 17 (1-f)-d ,:_:_:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.1.:.:.:.:.:.:.1. ............................... ,:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:., •.:.:.:.:.:.:.:.:.:.:-:.:.:.:-:.:.:.:.:.:.:.:.:.:.:.:.•.:.:. .:. ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: •.:.1.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:. . ::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::: •••:•:•:•:•:•:•:•:•••:•••••:.:•:••:•:•:.•••••••••••••••••••.••:•••••••••••••:•••••••••••:•••:.• ••.:.:.:.:.:.:.:.:.:.•.:.:.:.:.:..:.:.:.:.•.:.:.•.:.•.:.•.•.•.•.•.•.•.:.:••.:.:.•.•.:.:.:.:.:.•. ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: -K g (l-g) Figure 5. Illustration of O'Neill's step film model explanations in the text. = ' +1 (1) 1-g From the Lambert-Beer law the transmission T(X) of light with wavelength X is given as T(X) = 10 -•(x) (2) where E(X) is the extinction that is a function of the molar decadic extinction coefficient e(X), the molar concentration is c, and the optical pathlength is d: E(X) = e(X).c ß d (3) The transmission of a step film TOO can then be written as the sum of the transmissions through the two fractions of the film: T(X) = g' 10 -•(x'•"/'{•-/} + (1 - g)' 10 -•(x)'c'd'ts'f/(1-g)+l} (4) where d is the average thickness of the step film. Since the in vivo methods for SPF measurements define an amount of sunscreen of 2 mg/cm 2 = 2 pl/cm 2 that is applied on the skin, the average thickness of the film is d = 20 pm. Again, e(X) is the molar extinction coefficient and c is the molar concentration of the UV absorber. With cosmetic sunscreens, in most cases several UV absorbers are combined. Consid- ering a mixture of n different UV absorbers, the concentrations of the individual ab- sorbers are most conveniently given as percentages •i (weight per volume). The corre- sponding molar extinction coefficients are the e00i, and the molecular weights ]Vl i. The -- average molecular weight M, the molar concentration based on the average molecular weight ?, and the average molar absorption coefficient e(X) of the mixture can be calculated using the following formulae:

18 JOURNAL OF COSMETIC SCIENCE In the case of mixtures of several UV absorbers, in equation 4, e(}t) has to be replaced by e(}t) and c by ?. Figure 6 shows e(}t) for the case of standard P3. CALCULATION OF SUN PROTECTION FACTORS FROM TRANSMISSION VALUES Using the step film model, the transmissions TOO of sunscreens can be calculated, and for certain values of the parameters g andf the result may be close to the situation in vivo. The inverse of the transmission at a certain wavelength }t equals the factor of attenuation of the corresponding radiation. Therefore, l/T00 has the meaning of a monochromatic protection factor, MPF(k): 1/T(M = MPF(?t) (6) The spectral range of solar radiation that is regarded as potentially dangerous for the skin comprises the wavelengths between 290 nm and 400 nm. In order to calculate the sun protection factor, the monochromatic protection factors in this spectral range must be averaged. This average has to be weighted by the intensities of solar radiation S(}t) and by the skin sensitivity towards UV light and the erythemal action spectrum EA(?t), which both are functions of wavelength. The resulting formula for the calculation of the weighted average of monochromatic protection factors was first given by Sayre et al. (10): 4OO s(x). X=290 SPF= 400 (7) s(x). EA(X). X=290 S0•) and of EA(h.) are available from the literature. Values for S0•) used in this work are taken from Diffey and Robson (5) and values for EA(h.) from CIE (11). The T0•) data are calculated using the step film model as indicated above. CALIBRATION OF THE STEP FILM PARAMETERS Figure 6 shows the UV spectrum of the SPF standard P3 in terms of the average extinction coefficients in the range between 290 and 400 nm as calculated from the contributions of the individual filters according to equation 5c. Taking into account these extinction coefficients e(}t), the optical pathlength (d -- 20 pm), and the molar concentration ? based on the average molecular weight (? = 0.221 mol/1 for P3), the transmission TOO of a step film can be calculated according to equation 4 for different values of the step film parameters g and f, which both can vary between 0 and 1. For each pair of step film parameters, a spectrum of transmissions TOO in the range between 290 and 400 nm is calculated, which is transformed via equation 7 into an SPF value. In that way the SPF of a formulation with a certain set of UV absorbers can be obtained for every pair of step film parameters. This is shown in Figure 7 as a three-dimensional plot of the SPF as function ofg andf for the P3 standard. From the plot it is obvious that there are different sets of parameters g and f that reproduce exactly the in vivo SPF value of standard P3 of 15.5. Corresponding calculations for the other two standards, P1 and P4, were performed in