Equation (1) refers to a static value of the SPF without considering any changes of UV transmittance due to photoinstabilities, which may occur in the course of the irradiation process involved with SPF determination in vivo. In order to take this into account when performing in vitro SPF measurements, eq. (1) was modified by Stanfield et al. [27], with *t*_{MED} referring to the time, after which one minimal erythemal dose (1 MED) is transmitted through the sunscreen film:

$$SPF\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{{\displaystyle {\sum}_{0}^{{t}_{\text{MED}}}{\displaystyle {\sum}_{290}^{400}{s}_{er}}}\mathrm{(}\lambda \mathrm{)}\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}{S}_{s}\mathrm{(}\lambda \mathrm{)}}{{\displaystyle {\sum}_{0}^{{t}_{\text{MED}}}{\displaystyle {\sum}_{290}^{400}{s}_{er}}}\mathrm{(}\lambda \mathrm{)}\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}{S}_{s}\mathrm{(}\lambda \mathrm{)}\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}T\mathrm{(}t,\lambda \mathrm{)}}\text{\hspace{1em}(8)}$$(8)

This approach mimics the conditions of SPF determination in vivo. A similar and equivalent method was described by Wloka et al. [28]. These procedures were developed for in vitro SPF measurements, but can equally be applied when UV transmittance is calculated, under the prerequisite that decay constants of the photodegradation of the UV filters are known [17, 18]. The kinetics of UV filter photodegradation is approximated with a first-order law, where *c*_{i} is the molar concentration of UV filter *i* (*c*_{i,0} without irradiation, *c*_{i,UVdose} after irradiation with a certain UV dose), and *k*_{i} is its kinetic constant of photodegradation:

$${c}_{i,\text{\hspace{0.17em}}\text{UV\hspace{0.17em}dose}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{c}_{i,0}\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\mathrm{exp}\mathrm{(}-{k}_{i}\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}UVdose\mathrm{)}\text{\hspace{1em}(9)}$$(9)

For practical reasons, instead of time, UV dose is written in eq. (9), which is proportional to irradiation time. There can be stabilizing or destabilizing interactions between different UV filters. This is accounted for by formalisms which decrease the *k*_{i} in case of stabilization and increase them in case of destabilization. For stabilization the following equation was found [18]:

$${k}_{i,j}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\mathrm{(}1/{k}_{i,0}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{q}_{j}{\beta}_{j}\mathrm{)}}^{-1},\text{\hspace{1em}(10)}$$(10),

where *k*_{i,0} is the degradation constant without stabilization, *q*_{j} is the stabilizer constant of stabilizer *j*, and *β*_{j} is its concentration in percent (w/v). For destabilization eq. (11) is employed:

$${k}_{i,m}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{k}_{i,0}\mathrm{(}1\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{r}_{m}{\beta}_{m}\mathrm{)},\text{\hspace{1em}(11)}$$(11)

where *k*_{i,0} is the degradation constant without destabilization, *r*_{m} is the destabilization constant of destabilizer *m*, and *β*_{m} is its concentration in percent (w/v). In values of *k*_{i} are given for the UV filters used in this work. lists stabilization and destabilization constants. By determining the interaction factors *k*_{i,j}/*k*_{i,0} and *k*_{i,m}/*k*_{i,0}, any number of interactions can be taken into account by multiplying *k*_{i,0} with all existing interaction factors.

Table 1 Photodegradation constants of UV filters used in this work.

Table 2 Stabilizing constants *q*_{j} and destabilizing constants *r*_{m}.

In addition, the photostability of photounstable UV filters can be improved when UV light is strongly absorbed in their environment. The following relationship was established empirically in order to implement this effect:

$${k}_{i,OD}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\mathrm{(}\frac{O{I}_{i}}{O{I}_{sf}}\mathrm{)}}^{0.8304},\text{\hspace{1em}(12)}$$(12)

where *OI*_{i} = overlap integral of the absorbance spectrum of 2 % (w/v) of the respective UV filter at an optical thickness of *d* = 0.002 cm and the irradiance spectrum of the lamp used for the photostability assessment [29], and *OI*_{sf} = overlap integral of the absorbance spectrum of the irregular sunscreen film and the irradiance spectrum of the lamp used for photostability measurements. If *k*_{i,OD} < 1, there is a protective effect due to the overall optical density. Only then *k*_{i,OD} is considered by multiplying it with the decay constant of the respective filter.

For the SPF calculation with consideration of photoinstabilities the approach of Wloka et al. [28] is used. For that purpose UV dose dependent concentrations of the UV filters are calculated using eq. (9) and with that, UV dose dependent SPF values. The inverse of the UV dose-dependent SPF is plotted against the irradiation dose given in minimal erythemal doses (MEDs). The area under the curve can be interpreted as the erythemally effective irradiation dose. When this area becomes unity, 1 MED has been transmitted through the sunscreen film, what exactly corresponds to the principle of the in vivo SPF measurement. Thus the SPF can be read at this point from the respective UV-dose given on the abscissa. An example with a resulting SPF of 10 is shown in Fig. 8.

Fig. 8: Evaluation of the calculated SPF with photounstable sunscreens.

With photounstable UV filters, the spectrum of the absorbance of the sunscreen film varies with time. According to Stanfield et al. [27] it is possible to determine the effective UV absorbance spectrum, regarding the period from beginning of irradiation up to the point where 1 MED is transmitted. To this end, the transmitted UV-doses with (*D*_{p}) and without protection (*D*_{u}) are integrated over time, and transmittance, now called integrated transmittance, is calculated for each wavelength *λ*:

$${T}_{\mathrm{int}}\mathrm{(}\lambda \mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{D}_{p}\mathrm{(}\lambda \mathrm{)}/{D}_{u}\mathrm{(}\lambda \mathrm{)},\text{\hspace{1em}(13)}$$(13)

where

$${D}_{u}={\displaystyle {\int}_{0}^{{t}_{\text{MED}}}{S}_{s}}\mathrm{(}\lambda \mathrm{)}dt\text{\hspace{1em}(14)}$$(14)

and

$${D}_{p}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\displaystyle {\int}_{0}^{{t}_{\text{MED}}}{S}_{s}}\mathrm{(}\lambda \mathrm{)}\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}T\mathrm{(}\lambda ,t\mathrm{)}dt.\text{\hspace{1em}(15)}$$(15)

The integrated absorbance can then be obtained from integrated transmittance:

$${E}_{\text{int}}\mathrm{(}\lambda \mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\mathrm{lg}\mathrm{(}1/{T}_{\mathrm{int}}\mathrm{(}\lambda \mathrm{)}\mathrm{)}.\text{\hspace{1em}(16)}$$(16)

Figure 9 illustrates the concept of integrated transmittance and absorbance with an example of a photounstable composition (5 % EHMC, 4 % BMDBM, 3 % OCR) with calculated SPF of 17.5.

Fig. 9: Integrated spectra calculated for a sunscreen composition of 5 % EHMC, 4 % BMDBM, and 3 % OCR.

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