Simulation of sunscreen performance

Sun protection factors from UV transmittance

The basic principle of sun protection factor calculations is the calculation of the factor by which the intensity of the UV radiation is reduced due to the presence of a sunscreen. This factor is given by the inverse of the UV transmittance of the absorbing film, 1/T. At a certain wavelength λ, 1/T(λ) is also designated as monochromatic protection factor (MPF). As the spectral range relevant for the formation of erythema is between 290 and 400 nm, the monochromatic protection factors should be averaged over this range. In order to obtain the SPF, this average must be weighted with the intensity of the light source, S_s(λ) and the erythemal action spectrum, s_er(λ), leading to eq. (1) contributed by Sayre et al. [19]:

Data for S_s(λ) and s_er(λ) are available [10, 20], but T(λ) has to be determined for the respective sunscreen either by in vitro measurement or by calculation. S_s(λ) is given in units of W m⁻² nm⁻¹, s_er(λ) without units. The product of S_s(λ) and s_er(λ) is called erythemal effectiveness spectrum or erythemally effective irradiance, E_er(λ), and again has units of W m⁻² nm⁻¹. The quantities S_s(λ) and s_er(λ) are plotted in Fig. 1. In Fig. 2 the erythemally effective irradiance in the case of no protection, E_er(λ), is compared to situations of protection, E_er(λ)·T(λ), with sunscreens of SPF = 10 and SPF = 20.

Fig. 1:

Erythemal action spectrum and spectrum of standard sun irradiance.

Fig. 2:

Erythemal effectiveness in absence and presence of sunscreens.

Overall UV spectrum of a UV filter composition in a sunscreen

Sunscreens in most cases contain more than one UV filter. In order to obtain the overall absorbance of a UV filter composition via Beer-Lambert’s law, the overall molar concentration c̅ (from the average molecular weight M̅) and the average molar extinction coefficient ε̅(λ) are calculated via eq. (2),

where M_i is the molar mass, β_i the concentration in percent (w/v), and ε_i(λ) the molar extinction coefficient of an individual UV absorber [16, 17].

In order to calculate transmittance via Beer–Lambert’s law, the only quantity still missing is the thickness d of the sunscreen film on the skin:

In the in vivo method for SPF determination, 2 mg/cm² sunscreen are applied on human skin, approximately corresponding to 2 μL/cm². This results in a film of 20 μm (0.002 cm) thickness. However, this film is not homogenous, but irregular. As water evaporates, the film will get thinner, but at the same time the concentrations of the UV-filters will increase accordingly, such that the product c·d will stay constant.

The above calculation of the film thickness is based on the volume of the sunscreen spread on certain skin area. The error made here taking the volume instead of the weight cancels with the error assuming the filter concentrations given in weight per volume, as in practice formulations are prepared in weight per weight.

Accounting for film irregularity

When measuring UV transmittance in vitro, in some way the irregular surface structure of the human skin has to be modeled by using appropriate substrates with rough surface. This is important because the optical transmittance of an absorbing film of uniform thickness always is lower compared to that of a corresponding irregular film of the same average thickness. This fact is illustrated in Fig. 3.

Fig. 3:

A homogeneous film always absorbs light better than an irregular film.

The effect can be quite dramatic, and several models describing sunscreen film irregularity have been published [21–24]. The model of Ferrero and coworkers [23], which is employed in the simulations reported here, uses an approach common in surface metrology. In this case, the film profile becomes equivalent to the bearing area curve of Abbot and Firestone [25, 26] and is constructed based on a cumulative distribution function F(h) containing the film height h as a random variable. Among possible probability functions, the authors chose a gamma law representing an asymmetrical distribution, and f(h) is the associated probability density function:

where h is the random variable “relative height,” c is the shape parameter to be adjusted, b is needed for normalization and Γ(c) is the value of the gamma function at c. The cumulative height distribution F(h) is obtained by integration of f(h) from 0 to ∞:

In order to construct the film thickness profile, h is deduced from its cumulative distribution F(h), ranging from 0 to 1. As the area under the film profile is normalized to 1, the following relationship must hold:

The transmittance of the film can then be calculated:

The transmittance calculated via eq. (7) can be fed into eq. (1), resulting in an SPF value. The shape parameter c is the screw by which the calculated result can be adjusted to fit the in vivo SPF. In Fig. 4, the density function given in eq. (4) and its cumulative version indicated in eq. (5) are depicted for a shape parameter of c = 1.5. Figure 5 shows the respective film profile defined in eq. (6). In Fig. 6 the influence of the shape parameter is demonstrated for a sunscreen composition with in vivo SPF of 15.5. The larger the shape parameter, the higher the calculated SPF value. Obviously, the value of the SPF is largely determined by the thickness of the thinner parts of the film. In Fig. 5 the film profile of the irregular film is drawn as well as the profile of the corresponding regular film, also called parent film. When plotting the irregular film extinction (E, synonymous with absorbance) against parent film extinction, a non-linear relationship is observed, shown in Fig. 7 for different shape parameters and the composition of the P3 standard formulation [6], but without treatment of photoinstabilities.

Fig. 4:

Probability density distribution f(h) of film thickness h and the corresponding cumulative distribution F(h).

Fig. 5:

Construction of irregular film profile by interpreting F(h) as film extension.

Fig. 6:

Structure of irregular film with different values of shape parameter c; corresponding SPF data calculated for the P3 standard sunscreen (composition: 2.78 % PBSA, 3 % EHMC, 0.5 % BMDBM).

Fig. 7:

Absorbance of irregular film as function of parent film absorbance, calculated for P3 standard sunscreen (composition: 2.78 % PBSA, 3 % EHMC, 0.5 % BMDBM) with different values of shape parameter c.

Changes of UV transmittance due to photoinstabilities

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:

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:

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]:

,

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:

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 Table 1 values of k_i are given for the UV filters used in this work. Table 2 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.

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:

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:

where

and

The integrated absorbance can then be obtained from integrated transmittance:

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.

Calibration of the model

As already mentioned, the screw by which the calculated SPF can be adjusted to SPF in vivo data is the shape parameter in eq. (4). At first sight, the model has only this single parameter. However, it is not possible to get satisfactory correlation with compositions covering a brought range of sun protection factors using just one value of the shape parameter. So the shape parameter has to vary as function of the filter composition, and there are two observations indicating, how this could be explained: First, presence of a higher amount of emollient leads to an increase of the SPF at constant filter composition, and second, there is a synergistic effect on the SPF when sunscreens contain water- and oil-soluble UV filters rather than having the UV filters only in one phase of the preparation. As the only input for the calculations is the UV filter composition, the first point can be handled by scaling the shape parameter with the overall filter concentration (OFC) which for most sunscreen compositions is an indicator of the amount of oil used. The overall UV filter concentration for filters i to n is given as:

For the second point, a relevant parameter is the relative erythema active extinction in the oil phase, REAE. For a single filter i with concentration β_i the erythema active extinction EAE_i is calculated via eq. (18):

where E₁₁(i, λ) is the specific extinction of filter i at wavelength λ, the other quantities have the same meaning as in eqs. (1) and (2). The relative erythema active extinction in the oil phase is given via eq. (19):

where EAE_k(w) is the erythema active extinction of filter k in the water-phase. For the variation of the shape parameter as function of filter composition, the following function has been found:

The values of the parameters have been adjusted to p1 = 0.33, p2 = 0.69, p3 = 0.89, p4 = 1.95, and p5 = 0.5. Thus, the distribution of the UV filters in the water and the oil phase is taken into account as well as the overall filter content, which in a way can be interpreted as formulation influences.

The calibration of the parameters in eq. (20) was achieved by using in vivo data of ring tests where several laboratories had been involved, or our own data, where in vivo studies had been repeated several times on the same sunscreen formulations (P1, P2 and P3 [6], S2 [30, 31], S1, C1, C2 and P4 [17, 32]). Table 3 lists the UV filter compositions and corresponding data, and the correlation of simulated and in vivo data is shown in Fig. 10. The slope of the correlation is 1.015 and the correlation coefficient is r² = 0.9423.

Fig. 10:

Correlation of calculated SPF and in vivo SPF of formulations used for calibration of the model parameters (data in Table 3).

Protection against UVA radiation

The in vivo method used for UVA protection assessment is persistent pigment darkening (PPD). PPD is the official method for assessment of the UVA-protection in Japan [32]. In analogy to the erythemal response expressed by the SPF, the UVA-protection factor (UVAPF) values are determined by the biological endpoint of spontaneous pigmentation after irradiation with a UVA light source.

The UVAPF is simulated in a similar way like the SPF from calculated transmittance T(λ) taking photodegradation of the filter molecules into account, but now employing the spectrum of an UVA lamp, S_UVA(λ), and the PPD action spectrum s_PPD(λ) in eq. (8), and the time t_MPD after which a minimal pigmenting dose (1 MPD) is transmitted:

For the calculation of UV transmittance in eq. (21), the same principles are applied as previously described for the calculation of the SPF. However, the overall UV dose needed to achieve 1 MPD is higher than that to achieve 1 MED. Thus, irradiation time with PPD experiments is about five times longer. Considering that the UVB is filtered off, still a factor 3.2 has to be taken into account for the calculation of UV filter photodegradation.

An in vitro method for determination of the UVA-PF has been developed [13] and is meanwhile standardized [14]. In order to simulate this method, the film irregularity model is not used, but a relationship between parent film extinction and extinction of the film on the substrate, based on in vitro data, leading to similar curves as shown in Fig. 7:

Assuming the thickness of the homogenous parent film to be 0.00075 cm, the parameters were determined to a = 0.5072 and b = 0.5466. In that way in vitro extinction and transmittance spectra are obtained resulting in a simulated in vitro SPF, the value of which may differ from the simulated in vivo SPF. The corresponding simulated in vitro transmission spectrum is then transformed, such that the simulated in vitro SPF matches the simulated in vivo SPF. This is achieved by adjusting a parameter C as an exponent to the transmittance T(λ):

The irradiation step involved in this method [14], which scales with the UVAPF obtained without irradiation, is also taken into account.

Other quantities characterizing the UVA protection such as UVA/UVB-ratio, critical wavelength and others [15] can also be deduced from the calculated absorbance or transmittance spectra. The UVA/UVB-ratio is the average absorbance in the UVA-range divided by the average absorbance in the UVB-range. The critical wavelength (λ_c) is given as the wavelength at which 90 % of the total area under the absorbance curve is achieved.

Variation of application amount

The question how the sun protection factor depends on application amount is of interest, since people tend to apply much less sunscreen as is used in the in vivo SPF test performed on volunteers. A multi-center study showed, that there is a linear dependence of the SPF on application amount [33], although some authors claim an exponential relationship [34]. From eq. (1) it can be deduced that an exponential relationship can only be expected if the transmittance would be approximately the same at all wavelength between 290 and 400 nm. As there is also significant erythemal efficacy in the UVA range, the SPF of a pure UVB sunscreen will even show saturation-like behavior when its application amount is increased.

Figures 11 and 12 show calculations for two extreme examples, comparing a broad-spectrum sunscreen (5 % BEMT and 5 % MBBT, UVA/UVB-ratio = 0.92) and a sunscreen containing only UVB filters (5 % EHT and 5 % PBSA) UVA/UVB-ratio = 0.16), both with overall filter concentrations of 10 %. Indeed, the SPF of the broad-spectrum composition shows slightly exponential behavior, whereas that of the UVB sunscreen approaches saturation when the application amount is increased.

Fig. 11:

Calculated sunscreen spectra of irregular films of composition 5 % EHT + 5 % PBSA (UVA/UVB-ratio = 0.16) and 5 % BEMT + 5 % MBBT (UVA/UVB-ratio = 0.92).

Fig. 12:

Calculated SPF as function of application amount for two sunscreens with different UVA/UVB-ratios (0.16 and 0.92).

In [33] three sunscreen products were investigated in a ring study. All products showed linear dependence of the SPF on application amount. In order to simulate the behavior of these products, the filter contents of products B and C were analyzed. Product B with label SPF 20 contained 10 % OCR, 2.7 % BMDBM, 1.8 % TiO₂, 1.0 % DTS, 1.0 % EHT, and 0.5 % TDSA (abbreviations are explained in the experimental section). Product C with label SPF 25 contained 4.2 % EHT, 3.5 % TiO₂, and 2.0 % BEMT. Knowing the UV filter concentrations it is possible to calculate the SPF as function of application amount. The simulations shown in Fig. 13 are in good agreement with the experimental data [33] and also represent approximately the linear dependence of the SPF on application amount. The UVA/UVB-ratios are in between those of the extreme examples shown in Fig. 12 (0.72 for product B, and 0.53 for product C).

Fig. 13:

SPF as function of application amount for two sunscreens investigated in reference [33].

Variation of light source spectrum

For determination of SPF in vivo, a light source is used with a spectrum deviating in the UVA-I range strongly from solar spectra, as the visible light is filtered off. The question how much this affects the protection under natural solar conditions, has already been addressed in an experimental study [35]. It was observed that the SPF is lowered by about one third when UVA-I is not filtered. It was also demonstrated, that with different types of solar simulators, SPF results may vary by more than a factor of two [36]. Figure 14 shows a comparison of a standard solar spectrum [10], the spectrum defined in the International SPF method [8] and the latter one multiplied with the transmission spectrum of the WG335 filter [37].

Fig. 14:

Different light source spectra.

In Fig. 11 the irregular film absorbances of two quite different UV filter compositions are shown for the usual application amount of 2 mg/cm². Considering the SPF of the two compositions, the case with UVA/UVB-ratio = 0.16 gives an SPF of 13.7 when calculated with the laboratory light source spectrum. Taking the spectrum of the standard sun, the SPF is only 10.6, about 23 % lower. In comparison, the SPF of the composition with UVA/UVB-ratio = 0.92 is 26.3 with the laboratory lamp and 25.0 when taking the standard sun spectrum, which is only 5 % lower. That means, the more uniform a sunscreen absorbs in the UVB and the UVA range, the less its performance will depend on the spectrum of the light source.

Development of new sunscreens

The sunscreen simulator model described here [38] is widely used in the cosmetic industry for the design of new sunscreen formulations. An example where it has been successfully applied was the development of a premix concept for a simple preparation procedure of sunscreens [39, 40]. The concentrate consists of a mixture of four UV absorbers, 20 % EHMC, 20 % DHHB, 6 % EHT, and 5 % BEMT, and cosmetic oils and emulsifiers added to 100 %. Upon dilution with water an emulsion forms spontaneously. In that way sunscreen formulations can be easily prepared. With the chosen filter composition the resulting emulsions comply with the requirements for UVA protection, such as UVAPF/SPF ≥ 1/3 and λ_c ≥ 370 nm. In Table 4 filter concentrations, in vivo SPF and calculated SPF data are given for different dilutions of the concentrate. The correlation is shown in Fig. 15. The model prediction of the SPF (slope = 0.944) and the correlation of in vivo and simulated SPF data (r² = 0.9680) are excellent.

Fig. 15:

Correlation of calculated SPF and in vivo SPF of formulations based on a UV filter concentrate (data in Table 4).