STEP FILM MODEL FOR SPFs 19 25000 20000 15000 10000 5OOO 300 320 340 360 380 400 Wavelength / nm Figure 6. UV spectrum of sunscreen standard P3 in terms of molar decadic extinction coefficients. the same way. There exists no unique set of parameters g andf which is able to reproduce exactly the i, vivo SPF results of all three standards at the same time. However, it is possible to search for the set of step film parameters that results in the closest possible simulation of the i, vivo SPF values of the three standards P1, P3, and P4. For that purpose a function Ag,f was defined that describes the overall quadratic deviation of the calculated SPF values from the i, vivo values of the sunscreen standards P1, P3, and P4, which are 4.2, 15.5, and 35.7, respectively. 4-5 + + ([SPF(P1)g•-4.2] -35 • + 4.2 + 15.5 [SPF(P3)•-15.5] + 35.7 [SPF(P4)•, (8) where SPF(P 1)•,f are the SPF values calculated according to the spectrum of standard P 1 as a function of parameters g and f and SPF(P3)g,fand SPF(P4)•,f are the correspondingly calculated SPF values for standards P3 and P4. The first three terms of equation 8 represent the squared deviations of each individual standard between the calculated values and the in vivo SPF, normalized to the corresponding in vivo SPF. The fourth term considers the square of the sum of the deviations of all three standards between calcu- lated and in vivo SPF, taking account of the fact that before their squares are computed, deviations of different terms may partly cancel.

20 JOURNAL OF COSMETIC SCIENCE 80 60 I.t. 40 20 0 0.0 0.2 0.0 0.4 1.0 1.0 Figure 7. SPF calculated on the basis of the UV spectrum and UV absorber contents of standard P3 as a function of the step film parameters g and f The task is now to find the set of parameters g and f for which Ag,f has a minimum. At this point the deviation of the calculated values compared to the in vivo values is minimal and the agreement is best. The function hg,f is visualized in Figure 8. The minimum is located at g = 0.269 and f = 0.935, which therefore can be regarded as the step film parameters calibrated from standaMs P1, P3, and P4. The calculation was carried out with a 200-times-200 matrix of g- and f-values in the range between 0 and 1 using the MathCad 6.0 software running under Microsoft Windows © NT 4.0 on a personal computer (IBM 300PL). For standard P1 the calculated SPF value is 5.0 (in vivo SPF 4.2), for P3 it is 10.9 (in vivo SPF 15.5), and for P4 it is 38.5 (in vivo SPF 35.7). In order to cover a broad range of SPF values with this step film model calibration, low, medium and high SPF standards had been chosen. RESULTS AND DISCUSSION CORRELATION OF IN VIVO SPF VALUES AND SPF VALUES OBTAINED WITH THE CALIBRATED STEP FILM MODEL In order to correlate in vivo SPF values of sunscreen products with values calculated using