Monte Carlo simulations of light transport in sunscreen formulations

Abstract

Sunscreens are used for the protection of human skin against the harmful effects of solar UV radiation. Due to the low thickness of sunscreen films typically applied to the skin, it can be challenging to achieve the strong absorbance needed for good UV-protection, and most efficient sunscreen compositions are desirable. The presence of scattering particles can increase the efficacy of dissolved UV-absorbers in the oil or water phases of the formulation. As many sunscreens contain UV-absorbing particles, it is of interest how much the scattering effect of such materials contribute to the protection of the respective sunscreen. The currently available software programs for simulating sunscreen performance are based on a Beer–Lambert law approach and do not take into account such scattering effects of particles. However, Monte Carlo simulations of the UV-light transport through sunscreen films are capable to take scattering from particles into consideration. Using Monte Carlo simulations, this work shows that the efficacy of absorbance is indeed increased in the presence of scattering particles. However, this is of limited significance when the particles are UV-absorbers themselves.

Graphical abstract

1 Introduction

Sunscreens are used for the protection of human skin against the harmful effects of solar UV radiation. Due to the low thickness of sunscreen films typically applied to the skin (a few micrometers), the filtering of UV radiation must be very effective. Therefore, sunscreens often contain significant amounts of UV absorbing substances [1]. However, the concentrations of individual UV absorbers in sunscreen compositions are limited for various reasons such as low solubility in oils or water, regulatory constraints and patent restrictions [2], and thus it is desirable to explore possibilities for most efficient sunscreen compositions.

Computational models serving the simulation of sunscreen performance to estimate the protective properties of possible sunscreen compositions are freely available and extensively used since years [3, 4]. Such models are based on performance data of the different filter substances with respect to UV-absorption and photostability, using a Beer–Lambert law approach with a distribution of optical pathlengths due to the irregular sunscreen film on the skin [5]. The interaction of absorbing and scattering properties of the filter substances so far is not considered in the models.

Besides organic oil- or water-soluble UV absorbers, also inorganic or organic particulate UV-absorbers are often employed as UV filtering materials [6], like TiO₂ and ZnO, or methylene-bis-benzotriazolyl tetramethylbutylphenol (MBBT) and tris-biphenyl triazine (TBPT), respectively. MBBT and TBPT are organic UV-absorbers with extremely poor solubility in water and cosmetic oils [7] and are therefore prepared in the form of particles in aqueous dispersions. The two before mentioned inorganic oxides may be dispersible in either water or in oils via adapted coatings. In most cases particulate UV filters in sunscreens are in the size range below 100 nm [8, 9].

Often, particulate and soluble UV-absorbers are combined. Due to the scattering and absorption properties of particulate UV-absorbers a complex interplay can then be expected. The scattering of the UV radiation in different directions can lead to an increase of the effective optical pathlength of the incoming radiation and thus also to a higher probability that absorption will take place, especially when soluble UV absorbers are dissolved in the continuous phase [10]. For that reason, a certain amplification effect in terms of UV absorbance might be observed when particles are present in a sunscreen, leading to a decrease of UV transmission [11]. In the case of high particle content-dependent scattering may arise. This effect is caused by destructive or constructive interference of light scattered from different particles which are spatially correlated. An overall result of dependent scattering is a decrease in scattering and hence higher light transmission compared to independent scattering [12].

In this work, we used Monte Carlo simulations to study systems of dissolved absorber molecules in combination with particles, the latter exhibiting either both, absorption and scattering, or only scattering. With that we try to explore possible consequences of the interplay between absorption and scattering for the performance simulation of sunscreen compositions.

2 Monte Carlo simulations

The principle of the Monte Carlo simulations is shown in Fig. 1. The sample, i.e. a sunscreen film is simulated as an infinitely large film in x- and y- and finite size in z-direction. Here, I₀ is the intensity of the incident radiation. The light hits the sample where it can be either reflected from the surface or scattered form within the sample (represented by I_R), scattered or absorbed in the sample, reflected from the sunscreen–skin interface, or transmitted directly or diffusely to the skin. Such processes occur with certain probabilities depending on the optical properties of the sunscreen, like the absorption coefficient µ_a and the scattering coefficient µ_s, as well as the refractive indices of the sunscreen and the skin. The fate of a specific photon in the sample is simulated using uniformly distributed pseudo random numbers (RN) between 0 and 1. In practice we used the C++ rand() function normalized to RAND MAX with a seed rsand() depending on the system time.

Fig. 1

Simulation of sample as an infinitely large film in x- and y- with finite size in z-direction, symbols explained in the text

The Monte Carlo program is described in detail elsewhere [12]. The simulation comprises several steps. Photons are launched one after the other onto the sample. At the air-sunscreen interface, the photon can either be reflected and thus terminated, or it enters the film material. When the photon has entered the material, it will travel a distance Δs, after which it is either absorbed or scattered. µ_a and µ_s are the inverse mean free paths for absorption and scattering [12], respectively, after which the intensity I is decreased to 1/e:

$$ln\left(\frac{I}{{I}_{0}}\right)=-\Delta s\left({\mu }_{a}+{\mu }_{s}\right).$$

(1)

The ratio I/I₀ can adopt values between 0 and 1 and can be simulated by a uniformly distributed random number between 0 and 1. Δs can be calculated as:

$$\Delta s=-\frac{\text{ln}\left(RN\right)}{{\mu }_{a}+{\mu }_{s}}.$$

(2)

The probabilities for absorption and scattering after travelling the distance Δs are µ_a/(µ_a + µ_s) and µ_s/(µ_a + µ_s), respectively. In case of absorption the photon terminates, and a new photon is launched. In case of scattering (RN ≤ µ_s/(µ_a + µ_s)), the scattering angle θ_s is obtained by calculating a normalized integrated phase function, V_q(θ_s):

$${V}_{q}\left({\theta }_{s}\right)=\frac{\underset{0}{\overset{{\theta }_{s}}{\int }}f\left(\theta \right)\text{sin}\left(\theta \right)\text{d}\theta }{\underset{0}{\overset{\uppi }{\int }}f\left(\theta \right)\text{sin}\left(\theta \right)\text{d}\theta }.$$

(3)

Here, f(θ) is the scattering phase function. V_q(θ_s) is calculated once at the beginning of the MC simulation for distinct scattering angles θ_s and the results are stored in a table. The scattering angle is then obtained using again a pseudo random number between 0 and 1, and V_q(θ_s) is obtained from V_q(θ_s) = RN, and θ_s can be read from the table.

For the calculation of the phase function and scattering coefficients, Mie theory is employed, i.e. refractive indices of particles and continuous phase, absorption indices of the particles as well as the particle sizes have to be known. To account for dependent scattering, the static structure factor S(θ) has to be incorporated in to the calculation of the scattering coefficient [13]:

$${\mu }_{s,dep}=\frac{3\phi\uppi {Q}_{s}}{2a}\underset{-1}{\overset{1}{\int }}f\left(\theta \right)S\left(\theta \right)d\text{cos}\left(\theta \right),$$

(4)

where a is the particle radius, ϕ the volume fraction of the particles and Q_s the scattering efficiency, the latter obtained from Mie theory. The static structure factor is calculated using the hard sphere model in the Percus–Yevick approximation [13]. This leads typically to smaller scattering coefficients compared to independent scattering. However, dependent scattering not only manifests in smaller scattering coefficients but additionally in a change of the distribution of scattered light over the scattering angles. To account for this effect a dependent scattering phase function f_dep(θ) was employed:

$${f}_{dep}\left(\theta \right)=\frac{f\left(\theta \right)S\left(\theta \right)}{2\uppi \underset{-1}{\overset{1}{\int }}f\left(\theta \right)S\left(\theta \right)\text{dcos}\left(\theta \right)}$$

(5)

To obtain the macroscopic properties like absorbance or transmittance of the sample, a large number of photons N₀, typically between 10⁵ and 10⁶, is launched onto the sample. A photon is propagated by scattering until it has been reflected, absorbed or transmitted into the skin, and a new photon is launched until all photons are terminated. In Fig. 1, I₀ is the intensity of the incident radiation (corresponding to the total number of launched photons N₀), I_R of the intensity of reflected radiation, and I_T and I_T,D refer to the intensity of directly or diffusely transmitted radiation. Both, I_T and I_T,D (obtained from the number of transmitted photons N_T) contribute to the radiation transmitted to the skin (T). N_T is related to the number of incident photons N₀ leading to the diffuse transmittance:

$$T= {N}_{T}/{N}_{0}.$$

(6)

From this the extinction (synonymous with absorbance) A can be obtained:

$$A=ln\left({T}^{-1}\right).$$

(7)

3 Experimental

3.1 Chemicals

As a water-soluble absorber Patent Blue V (alternative name Acid Blue 3), the chemical name of which is 2-[(4-diethylaminophenyl)(4-diethylimino-2,5-cyclohexadien-1-ylidene)methyl]-4-hydroxy-1,5-benzene-disulfonate, from Sigma–Aldrich was employed.

Two oil-soluble UV absorbers were used, 2,4-Bis-{[4-(-2-ethyl-hexyloxi)-2-hydroxy]-phenyl}-6-(4-methoxyphenyl)-(1,3,5)-triazine with the INCI name (international nomenclature of cosmetic ingredients) bis-ethylhexyl-oxyphenol methoxyphenyl triazine (BEMT) and trade name Tinosorb® S (BASF), and hexyl 2-[4-(diethylamino)-2-hydroxybenzoyl]benzoate with the INCI name diethylamino hydroxybenzoyl hexyl benzoate (DHHB) and trade name Uvinul® A plus (BASF). A liquid oil-miscible UV-absorber was also employed, 3-(4-methoxyphenyl)-2-propenoic acid 2-ethylhexyl ester with the INCI name ethylhexyl methoxycinnamate (EHMC) and trade name Uvinul® MC80 (BASF).

As continuous phase served bi-distilled water and hexanedioic acid dibutyl ester with the INCI name dibutyl adipate (DBA) p.a. from Aldrich.

Two particulate organic particulate UV absorbers were used as well. Those were 2,2′-methylene-bis-(6-(2H-benzotriazole-2-yl)-4-(1,1,3,3-tetramethylbutyl)-phenol with the INCI name methylene-bis-benzotriazolyl tetramethylbutylphenol (MBBT) and trade name Tinosorb® M (BASF), and 2,4,6-tris(biphenyl-3-yl)-1,3,5-triazine with INCI name tris-biphenyl-triazine (TBPT) and trade name Tinosorb® A2B (BASF). MBBT and TBPT, both, are commercially available as 50% (w) nano-particulate aqueous dispersions [6].

3.2 Measurements

Particle size and particle size distributions of particulate UV filters were determined using FOQELS (Fibreoptic Quasi-Elastic Light Scattering). FOQELS is a dynamic light scattering technique based on the correlation analysis of the fluctuations of the back-scattered light from a particle dispersion. We typically used particle concentrations of 2% (w/w). The method is described in detail in ref. [8].

Quantitative UV/Vis spectra of diluted UV absorber solutions were measured using a Perkin Elmer Lambda 1050 spectrometer (without integration sphere) operating with the PE UV-Winlab Software and employing Hellma QS optical cells with 0.1, 0.5 and 1.0 cm optical thickness as well as sandwich-type cuvettes of 50 µm optical thickness. Stock solutions of BEMT and DHHB were prepared with Dibutyl Adipate (DBA) as solvent and further diluted with DBA to concentrations resulting in maximum absorbances of 1.6. In similar manner, quantitative UV/Vis spectra of the dye Patent Blue were obtained using bi-distilled water as solvent.

The complex refractive indices of the particulate UV-filters (MBBT and TBPT) and the oils (EHMC and DBA) were determined by ellipsometry. For ellipsometry measurements a Woollam M2000 device (J.A. Woollam Co. Instr.) with WVASE Spectroscopic Ellipsometry Data Acquisition and Analysis Software was used. For calibration of the instrument, a silicon wafer with a thermally grown silicon dioxide layer on top was employed. The samples to be measured needed to have a flat and smooth surface. For oils, this is easily feasible, by pouring an aliquot of the oil into a petri-dish-like vessel. In case of MBBT and TBPT the pure solid materials were pressed to a tablet. To get the absorber powder into this form, 0.2 to 0.4 g of the material was filled into a molding with 1.3 cm of diameter, and then pressed with an ATLAS T25 press with 10 tons. During the pressing procedure, a vacuum was applied, to avoid gas inclusions in the UV absorber tablet. For all ellipsometry measurements, an angle of incidence of 65° was used. The raw data of an ellipsometry measurement consist of an amplitude ratio and a phase difference of the incident and reflected light beams, from which the refractive index n and the absorption constant k can be evaluated. This was achieved using the WVASE Software. For evaluation of the non-absorbing parts of the spectrum a Cauchy model was applied, and for the absorbing regions an oscillator model such as the Lorentz oscillator.

4 Results and discussion

4.1 Material properties

In Figs. 2 and 3, results of particle size measurements with MBBT and TBPT particles obtained with the FOQELS technique are depicted, respectively. The diagrams show the volume weighted particle size distributions and the corresponding cumulative distributions. From the latter one the D(0.5) value can be read, also called the median, which is the diameter up to which 50% of the total volume of the particles in the particle size distribution is reached. For MBBT, a value of D(0.5) = 150 ± 10 nm was obtained and for TBPT D(0.5) = 104 ± 8 nm. Those values are averages and respective standard deviations obtained from 10 measurements. The diagrams shown in Figs. 2 and 3 are individual examples out of these measurements, and the D(0.5)-value indicated by the arrow in the diagrams may slightly differ from the average. For the Monte Carlo simulations, we used a particle size of 150 nm for MBBT and of 100 nm for TBPT, reflecting the average particle size from FOQELS measurements (for TBPT rounded from 104 to 100 nm, still in the error range of the measurement).

Fig. 2

Volume-weighted particle size distribution of MBBT particles (2% w/w in water) measured with the FOQELS technique, average particle size 150 ± 10 nm

Fig. 3

Volume-weighted particle size distribution of TBPT particles (2% w/w in water) measured with the FOQELS technique, average particle size 104 ± 8 nm

In Fig. 4, the molar absorption coefficients ɛ are drawn based on natural logarithm as function of wavelength λ, in the range from 250 to 750 nm, obtained from UV/Vis spectroscopic measurements. For Patent Blue V bi-distilled water was used as solvent and with BEMT and DHHB the medium was Dibutyl Adipate (DBA). Patent Blue V shows apart from two smaller absorption maxima in the UV range a strong absorption maximum at λ = 637 nm, DHHB at 351 nm and BEMT two maxima at 311 and 345 nm.

Fig. 4

Molar absorption coefficients of BEMT, DHHB (both in dibutyl adipate) and Patent Blue V (in water)

Figure 5 shows the absorption constants k of TBPT, MBBT and EHMC as function of wavelength λ in the same wavelength range as before obtained from ellipsometry measurements. TBPT has a maximum absorption at λ = 317 nm, EHMC at 309 nm and MBBT shows two maxima at 306 and 350 nm.

Fig. 5

Absorption constants of TBPT, MBBT and EHMC from ellipsometry measurements

In Figs. 6 and 7, the refractive indices of BEMT, EHMC, DBA, DHHB, MBBT, TBPT and water are depicted in the wavelength range from 250 to 750 nm, obtained from ellipsometry measurements, except water, the data of which were taken from literature [14].

Fig. 6

Refractive indices of BEMT, EHMC, DBA and DHHB from ellipsometry measurements

Fig. 7

Refractive indices of MBBT, TBPT and water from ellipsometry measurements

4.2 Monte Carlo studies with a model system of dye and particles

In this and the following paragraph, we study systems of visible or UV light absorbing molecules dissolved in a medium, in the presence of non-absorbing or absorbing particles dispersed in the same medium. We hypothesize that due to the scattering of the particles the average path of photons travelling in the sample is increased, leading to a higher probability for absorption of the photons by the dissolved absorber molecules. To quantify this effect, we first look at the absorbance when both, dissolved absorber and dispersed particles are present. After, the absorbance of the system containing only particles without dissolved absorbers is determined and subtracted from the first. In that way, the absorbance due to the particles is removed, and the net effect of the elongated path length of the photons in the sample is obtained in terms of an increased absorbance compared to a system without particles.

We applied the Monte Carlo simulations first to a model system, for which experimental data were available. This model system comprised an aqueous solution of the dye Patent Blue V at constant concentration in presence of various concentrations of MBBT particles. The absorption of the dye with an absorption maximum at λ = 637 nm does not interfere with the absorption of the MBBT particles in the UV-range (see Figs. 5 and 6). Experimental data for this system have been reported in ref. [11]. In the experiments, a cuvette containing an aqueous buffer solution with the dye, plus a certain concentration of MBBT particles was placed into the beam of a UV/Vis spectrometer (Perkin Elmer Lambda 20) at the entrance window of an integration sphere detector. This set-up corresponds to the situation modelled with the MC program, in which diffusely transmitted photons are counted, as illustrated in Fig. 8. In addition to the sample measurement the reference was measured, which consisted of an aqueous buffer solution without dye but with the same concentration of MBBT particles as the sample. From the transmittance of the sample dispersion obtained, and from the transmittance of the reference the absorbance of the MBBT dispersion without dye, A₆₃₇(MBBT). The difference of these two values results in the dye absorbance with the net effect of the particles on its absorbance, A₆₃₇(Dye @ MBBT):

Fig. 8

Schemes of a experimental set-up and b Monte Carlo simulation for determination of the diffuse transmittance

$${A}_{637}\left(Dye @ MBBT\right)={A}_{637}\left(Dye plus MBBT\right)-{A}_{637}\left(MBBT\right).$$

(8)

This procedure was modelled with the Monte Carlo program. First, the transmittance of the MBBT dispersion in the presence of dye was simulated (the sample), and after that the transmittance of the MBBT dispersion without dye (the reference). From the transmittances, the absorbances were calculated, and the absorbance of the reference was subtracted from the absorbance of the sample, according to Eq. (8). To illustrate the procedure, the three terms in Eq. (8) are listed in Table 1 for this case. Figure 9 displays results of experiments and MC calculations for a constant dye concentration of 0.017 mM in cuvettes of 0.1 cm optical thickness at a range of MBBT concentrations between 0 and 9%(w/w). The absorbance of the dye significantly increases with MBBT concentration in the experiment [11] as well as in the MC simulation. This effect can be explained by scattering of the light due to the presence of particles, leading to longer pathways of the photons and increased probability of absorption from the dye molecules. The agreement between experiment and simulation is good up to an MBBT concentration of about 5%. The lower absorbance in the experiments at higher MBBT concentrations is probably caused by limitations of the spectrometer in terms of dynamic range at such conditions (at 6% MBBT and 0.1 cm optical path length, the absorbance of MBBT with dye at λ = 637 nm is 2.07, which is already close to the linearity limit of the spectrometer used in [11]). With the highest MBBT concentration of 9% (w/w), 0.017 mM of the dye and 0.1 cm cuvette thickness we tested also the convergence of the Monte Carlo simulations, verifying that 10⁵ to 10⁶ photons are sufficient to obtain a stable result (Table 2). The difference between the absorbance data with 1∙10⁵ photons (A₆₃₇ = 2.549) and with 1.5∙10⁶ photons (A₆₃₇ = 2.552) is less than 0.12%.

Table 1 Monte Carlo simulation of the net effect of MBBT particles on the absorbance of the water-soluble dye Patent Blue V, A₆₃₇ (Dye @ MBBT), at the absorption maximum of the dye, a constant dye concentration of 0.017 mM and an optical path length of 0.1 cm

Fig. 9

Effect of MBBT particles on Patent Blue absorbance, optical path length d = 0.1 cm, c_{patent blue} = 0.017 mM. Error bars given for each point (but sometimes hidden by the symbols)

Table 2 Convergence of Monte Carlo simulations, investigated with respect to absorbance of 0.017 mM Patent Blue V in presence of 9% (w/w) MBBT at the absorption maximum of the dye and an optical path length of 0.1 cm with increasing number of photons N

Figure 10 shows results for the same system at a smaller optical path length of 50 µm and a constant dye concentration of 0.282 mM, with a concentration of MBBT particles ranging up to about 11%. Each experimental data point is the average of six individual measurements using 50 µm sandwich cuvettes. In Fig. 10 two sets of experimental data are depicted [15, 16]. Though experimental scatter is higher at this small optical thickness, the agreement between experiments and simulations is reasonable over the whole range of MBBT concentrations. As the concentration range of MBBT is approximately the same in Fig. 9 (0–9% w/w) and Fig. 10 (0–11% w/w) the influence of dependent scattering should be the same in both series. This indicates that the deviation of the data in Fig. 9 at high MBBT concentrations is not due to a problem with simulating dependent scattering at the higher particle contents, but most likely due to the limited linear range of the UV/Vis spectrometer used [11, 15, 16].

Fig. 10

Effect of MBBT particles on Patent Blue absorbance, optical path length d = 50 µm, c_{patent blue} = 0.282 mM, experimental data from [14] (closed squares) and from [15] (open circles)

The results in Figs. 9 and 10 make clear that the amplification of dye absorbance in the presence of scatterers depends a lot on optical thickness. In Fig. 9, with 0.1 cm optical thickness, the dye absorbance without particles equals 0.17, which at 4% MBBT is increased to 0.92, corresponding to a factor of 5.4. In Fig. 10, at 50 µm optical thickness the amplification of the dye absorbance at 4% MBBT is only increased by a factor 1.4 from 0.14 to 0.20. In terms of sunscreen films the latter case is more relevant, since after standard application conditions they have an average thickness of approximately 20 µm before evaporation of volatile constituents like water.

It should be mentioned that in this study the set-up of the MC-program did not take into account reflections at the quartz–air and quartz–liquid interfaces of the cuvettes. This was not necessary, since such effects mostly cancelled out in the experiments by subtracting the absorbance from the reference measurement.

Though the calculations were performed using extinction coefficients based on natural logarithms, the absorbance data shown in the Figures are based on decadic logarithms.

4.3 Monte Carlo simulations of films with absorbing and non-absorbing particles

In the following examples, systems of particulate and oil-soluble UV-absorbers are studied with MC simulations only, without comparing with experimental data. Although MBBT- and TBPT-particles are applied as aqueous dispersions, dibutyl adipate is employed as continuous phase. This corresponds to the situation in sunscreen films, in which water evaporates within a couple of minutes after application [17].

Like in the previous paragraph with the system of Patent Blue V and MBBT, the absorbance of the reference without BEMT but with particles in DBA is subtracted from the absorbance with BEMT in presence of the respective MBBT particle concentration in DBA. The influence of MBBT particles on the absorbance of BEMT is investigated at the maximum absorption of BEMT (λ = 345 nm) and was evaluated according to Eq. (9):

$${A}_{345}\left(BEMT @ MBBT\right)={A}_{345}\left(BEMT plus MBBT\right)-{A}_{345}\left(MBBT\right).$$

(9)

Both substances are broad-spectrum UV-absorbers (Figs. 4 and 5), covering almost the whole spectral range between 290 and 400 nm.

In Fig. 11, results at λ = 345 nm are shown at 50 µm film thickness and 0.6 mM BEMT for a range of MBBT concentrations of up to 6%(w) in dibutyl adipate. In the simulations, it is easily possible to switch off the absorption of the particles, by setting their absorption constants to zero. In that way the influence of non-absorbing and absorbing particles on BEMT absorbance can be compared. In presence of non-absorbing MBBT a significant amplification of BEMT absorbance can be observed. The BEMT absorbance without MBBT is 0.15 and increases by a factor of 2 to about 0.3 at 3% non-absorbing MBBT. The picture looks quite different with absorbing MBBT. There is only a very small increase of BEMT absorbance from 0.15 to 0.16 (by a factor of 1.07) when regarding concentrations of absorbing MBBT of 0 to 3%(w/w). Obviously, the scattering effect of UV-light is shielded by the absorption of the MBBT particles.

Fig. 11

Effect of non-absorbing and absorbing MBBT particles on BEMT absorbance, optical path length = 50 µm, c_BEMT = 0.6 mM

Similar results are obtained with a film of 10 µm thickness with a BEMT concentration kept constant at 3.0 mM and non-absorbing and absorbing MBBT in a concentration range up to 6% (Fig. 12). There is again a significant amplification of BEMT absorbance at λ = 345 nm by factor of 1.27 from absorbance 0.15 at 0% and 0.19 at 3% non-absorbing MBBT concentration. With absorbing MBBT the corresponding increase of BEMT absorbance is only by a factor of 1.03.

Fig. 12

Effect of non-absorbing and absorbing MBBT particles on BEMT absorbance, optical path length d = 10 µm, c_BEMT = 3.0 mM

Another example of this kind is depicted in Fig. 13 with the particulate UV-filter TBPT and the oil-soluble DHHB at λ = 350 nm, again with DBA as medium, with an optical thickness of 10 µm, a DHHB concentration of 3.0 mM and a range of TBPT concentrations between 0 and 6%(w). Again, the absorbance of the reference without DHHB but with TBPT particles in DBA is subtracted from the absorbance with DHHB in presence of the respective TBPT particle concentration in DBA. The influence of TBPT particles on the absorbance of DHHB is investigated at the maximum absorption of DHHB (λ = 350 nm) and was evaluated according to Eq. (10):

Fig. 13

Effect of non-absorbing and absorbing TBPT particles on DHHB absorbance, optical path length d = 10 µm, c_DHHB = 3.0 mM

$${A}_{350}\left(DHHB @ TBPT\right)={A}_{350}\left(DHHB plus TBPT\right)-{A}_{350}\left(TBPT\right).$$

(10)

At 3% non-absorbing TBPT the DHHB absorbance is increased by factor of 1.44, but with 3% absorbing TBPT only by a factor of 1.02. Again, the scattering effect of UV-light is shielded by the absorption of the TBPT particles.

4.4 Monte Carlo simulations of films with absorbing particles with and without index-matching

As the light scattered in forward direction due to the presence of particles reaches the skin, a protective effect from scattering is only relevant for back-scattered light. This can be studied for the two particulate UV-filters considered here by switching on and off their scattering ability and comparing the respective absorbances. Scattering gets down to zero, when the refractive index of the particles is equal to that of the continuous phase. Index-matching conditions were realized by exchanging the refractive indices of the particles by that of the surrounding medium. Since pure aqueous dispersions of particles were considered in that case, the particle refractive indices were exchanged by those of water. As reference, water was taken at the same optical path length. Figure 14 shows results for a dispersion of 1% (w) MBBT at an optical path length of 10 µm. Thus, the absorbance spectra shown in Fig. 14 are derived from the transmittance of dispersions of absorbing particles with and without scattering. The absorbance spectrum of the index-matched MBBT particles is indeed smaller than that of the MBBT absorbance spectrum with scattering. The difference of both spectra can be interpreted as absorbance spectrum of the back-scattered light, which is not transmitted due to scattering. The percentage of back-scattering in relation to the total absorbance varies as function of wavelength between 0.5 and 25% and is in average about 8%. Figure 15 shows similar results for TBPT particles at 0.5% (w) concentration and 10 µm optical path length. The back-scattering magnitude of TBPT is about half of that of MBBT.

Fig. 14

UV spectra of MC simulated absorbance of 1% (w/w) MBBT particles in water, optical path length d = 10 µm, with and without scattering (scattering switched off using for MBBT the refractive index of the continuous phase, water)

Fig. 15

UV spectra of MC simulated absorbance of 0.5% (w/w) TBPT particles in water, optical path length d = 10 µm, with and without scattering (scattering switched off using for TBPT the refractive index of the continuous phase, water)

4.5 Monte Carlo simulations of sunscreen films with realistic irregularity profile

The following simulations refer to protecting effects of sunscreen films on skin. Thus, we just look at the transmittance of UV-light without considering a reference. In the simulation, for the outer layer of the skin, the stratum corneum, a refractive index of 1.55 is applied [18]. The irregularity profile of the sunscreen film caused by the surface structure of the skin was represented by a modified gamma distribution of film thicknesses [19] according to Eq. (11):

$$f\left(h\right)=\frac{{\left(\frac{d+{d}_{0}}{b}\right)}^{c-1}\bullet exp\left(-\frac{d+{d}_{0}}{b}\right)}{b\cdot \Gamma \left(c\right)},$$

(11)

where d is thickness, b and c are scale and shape parameters, respectively, Γ(c) is the Gamma function with argument c and d₀ is the shift of the thickness axis to account for a finite frequency of zero thickness. In ref. [19] all three c, b and d₀ were treated as adjustable parameters and fitted to experimentally determined film thickness distributions for different types of sunscreen formulations. For the oil-in-water cream (OW-C) the parameters c = 2.26, b = 1.25 and d₀ = 0.733 had been obtained, which were used in the following simulations. Figure 16 shows the corresponding film profile obtained from the cumulative film thickness distribution. The average thickness is 2.3 µm, as indicated by the red dotted line. This value had also been obtained experimentally for the oil-in-water cream [20]. In the MC calculations, we simulate the film thickness variation as follows: In a first step a normalized cumulative function of the film thickness is calculated from Eq. (11) in the range between 0 and 24 µm in 0.3 µm steps and the results are stored in a table. During the simulation at each propagation step this function is then sampled by a random number and the respective film thickness is obtained. According to the chosen film thickness and the current position of the photon, it is decided whether the photon is within the film or outside.

Fig. 16

Film profile from cumulative film thickness distribution, Eq. (7), the red dotted line indicating the average film thickness (2.3 µm)

In Fig. 17, UV spectra of 0.02 M BEMT (corresponding to 1.26% (w) are depicted in presence of various concentrations of MBBT without absorption in the range between 0 and 10% (w), based on an irregular film according to Eq. (11) for the parameters obtained with oil-in-water creams [19]. The BEMT absorbance is increased by the presence of non-absorbing MBBT particles. Figure 18 shows the respective absorbance spectra of 0.02 mM BEMT in presence of various concentrations of absorbing MBBT.

Fig. 17

MC simulated absorbance spectra with BEMT at c_BEMT = 0.02 M (1.26%) in DBA at different MBBT particle concentrations (MBBT without absorption) employing an irregular film structure simulated with the gamma distribution profile from Eq. (7)

Fig. 18

MC simulated absorbance spectra with BEMT at c_BEMT = 0.02 M (1.26%) in DBA at different MBBT particle concentrations (MBBT with absorption) employing an irregular film structure simulated with the gamma distribution profile from Eq. (7)

The results in Fig. 18 were calculated using a film profile based on a gamma distribution as written in Eq. (11), which resulted in an average film thickness of 2.3 µm. To illustrate the effect of film irregularity on absorbance, calculations were performed with the same concentrations but using a constant film thickness of 2.3 µm (Fig. 19). Comparing Figs. 18 and 19 shows that film irregularity leads to lower absorbance, which has been discussed elsewhere in more detail [5]. For the simulation of realistic sun protection factors it is necessary to apply an irregular film model [19].

Fig. 19

MC simulated absorbance spectra with BEMT at c_BEMT = 0.02 M (1.26%) in DBA at different MBBT particle concentrations (MBBT without absorption) at fixed film thickness of 2.3 µm

The film model defined by Eq. (11) with the appropriate parameters can be used to calculate sun protection factors (SPFs), which can be derived from transmittance data. The basic principle of sun protection factor calculations is the determination 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(λ) at wavelength λ. As the spectral range relevant for the formation of erythema is between 290 and 400 nm, the inverse of the transmittance should be averaged over this range. To obtain the sun protection factor, 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. (12) [21]:

$$\text{SPF}=\frac{\sum_{290}^{400}{s}_{er}\left(\uplambda \right)\bullet {S}_{s}\left(\uplambda \right)}{\sum_{290}^{400}{s}_{er}\left(\uplambda \right)\bullet {S}_{s}\left(\uplambda \right)\bullet T\left(\uplambda \right)}.$$

(12)

Data for S_s(λ) and s_er(λ) are available from literature [22, 23], only the transmittance T(λ) has to be determined, what was carried out using the MC program for 23 sunscreens with known compositions and SPF in vivo data. Except for TBPT, those in vivo data were taken from reference [24]. All in vivo data were determined according to the method described in reference [22]. The transmittance was simulated with the MC program, using the irregular film model given by Eq. (11) with the parameters determined for the OW-C case (oil-in-water cream). Again, the step width for the film height variation for simulating the transmittance of the irregular film was set to 0.3 µm in a range from 0 to 24 µm, and the wavelengths were varied from 290 to 400 nm in 5 nm steps.

In Table 3, in vivo and simulated SPF data are listed together with the respective sunscreen compositions. The UV-filter compositions are given in weight percent, referring to the sunscreen before application. However, for the MC simulations one has to consider that water evaporates, resulting in less film material and a corresponding increase of the concentrations. From the composition of the oil-in-water cream [20] one can deduce that the residue of the sunscreen after evaporation makes 25% of the initial mass. For that reason, we have multiplied the UV-filter compositions reported in Table 1 by a factor of 4.0 in case of the MC calculations.

Table 3 Comparison of in vivo SPF data with simulations

Figure 20 shows the correlation of MC simulated sun protection factors with the corresponding in vivo data. The coefficient of determination r² is 0.920 with a slope of 1.105. In Fig. 21 a similar correlation with the same set of UV-filter compositions and in vivo data is shown, calculated with a simulation tool [19, 25] based on Beer–Lambert law without MC calculations, but using the same irregular film parameters as with the MC model. In that case, the quality of the correlation is slightly decreased with a coefficient of determination r² of 0.870 and a slope of 0.960. The larger slope of the correlation in case of the MC simulations of 1.105 indicates a slight overestimation of the SPFs in that case, which could be explained by the scattering effect of the particulate filters. 16 out of 23 compositions contained particulate UV-filters, also indicated in Figs. 20 and 21. This scattering was not taken into account with the Beer–Lambert based tool, where the slope of the correlation was 0.960, indicating a slight underestimation of the SPFs. This supports the results in Figs. 11, 12, 13, showing that the particulate UV-filters have only little effect on the enhancement of absorbance in sunscreens. In spite of the slight overestimation of the SPFs obtained here, MC simulations can be a suitable tool for the prediction of sun protection factors.

Fig. 20

Correlation of simulated SPF (MC) with in vivo SPF, slope = 1.105, coefficient of determination with MC: r² = 0.920

Fig. 21

Correlation of simulated SPF (Beer–Lambert model [18]) with in vivo SPF, slope = 0.960, coefficient of determination with BL: r² = 0.870

5 Conclusion

With Monte Carlo (MC) calculations, it is possible to simulate the absorbance and with that the sun protection factor of sunscreens when the optical properties of the constituents in the composition are known. In addition, MC simulations also allow to study the effect of particles in presence of soluble UV-absorbers. It is possible to model certain situations, which are difficult to study experimentally, like switching off absorption or scattering of the particles.

The simulations have shown that the boost of absorbance of soluble absorbers in the presence of particles due to scattering can be significant when the absorption of the particles does not overlap with that of the soluble absorbers. This was experimentally realized using a dye with an absorption maximum far beyond the absorption band of the absorbing particles. With MC simulations on the same system of dye and particles a good agreement was obtained with the experimental data.

To study the effect of particulate UV-filters on soluble UV-absorbers, only MC simulations were used. When the UV-absorption of the particles is turned off, a strong amplification of the absorbance of the soluble UV-absorbers is found. On the other hand, when the absorption of the particles is maintained, the boosting effect is very small. This could be explained by reabsorption of scattered photons from the investigated particulate UV-absorbers due to their strong absorption of in the UV-range.

These results may encourage the use of non-absorbing particles for the boosting of the sun protection factor. The amount of the effect of such particles strongly drops with decreasing film thickness. In the simulations we looked at films of 10 µm thickness, for instance with BEMT and (non-absorbing) MBBT particles (Fig. 12) as well as DHHB with (non-absorbing) TBPT particles, where an approximately 1.4-fold increase of the absorbance at 5% particle concentration was observed in both cases. Whether this would allow a corresponding boosting of the sun protection factor of sunscreens will depend on the absorbance characteristics of the specific sunscreen and should be further investigated.

When turning scattering off and on with only particulate filter dispersions, the protective effect resulting only from (back-)scattering can be studied. This is obviously quite small in comparison to the effect from absorption.

Sun protection factors (SPFs) were also studied with the MC simulations, using a gamma distribution for the consideration of film thickness variations. The results were compared with results from an SPF simulation model based on a Beer–Lambert law approach. Correlations with corresponding in vivo SPF data of a number of UV-filter compositions were satisfactory in both cases.

In the currently used sunscreen simulating models, the effects from light scattering of particulate UV-filters with respect to amplification of the absorbance or back-scattering of radiation are not taken into account. The results of this study show that this seems to be a reasonable approximation, as such scattering effects are quite small in most sunscreens, also such containing absorbing particles. However, this could be different when non-absorbing particles would be added.