5.1.

Beer-Lambert Law

The Beer-Lambert law is a simple deterministic technique which has been applied to estimate the reflectance ($R$) from a skin model. Although empirically derived,³² this is essentially a solution to the application of RTT to the interaction of light with a static homogeneous absorbing (nonscattering) medium.

The Beer-Lambert law uses an estimate of the attenuation of light ($A$) obtained by measurements of on-axis transmission through thin samples of skin, and the optical path length ($l$):

5.2.

Diffusion Approximation

The diffusion approximation to RTT is by far the most widely used deterministic approach in biomedical optics, with Farrell et al.’s method³⁶ accumulating over 500 citations alone.³⁷

The diffusion approximation is described in detail elsewhere.^28,38 In summary, it has a low demand for computing power and thus provides fast convergence for a wide range of applications. For example, it has been applied successfully to real-time light dosimetry during photodynamic therapy (PDT) of superficial skin cancers,^39,40 differentiation between pigmented skin lesions (including malignant melanoma),⁴¹ and to determine blood oxygen saturation levels faster, and over a larger area than existing methods with equivalent reliability.⁴² However, it suffers from a number of limitations, in part because it relies upon approximate solutions to an equation that itself represents an approximation to the equation of RTT.⁴³ For example, there is a requirement that scattering is dominant over absorption. This may be appropriate in clinically normal pale skin types, as the probability of scattering within the skin may be as high as twenty times that of absorption.^10,44 However, absorption is much greater in the epidermis of darker skin types, the blood vessel plexi of the normal dermis, and within PWS lesions. Thus, the validity of this approximation is limited in such cases.^{34,43,45–47}

These drawbacks have limited the application of the diffusion approximation to investigations of PWS skin. For example, Verkruysse et al.⁴⁵ reported that data obtained from their two-layered skin model could be obtained quickly but compared poorly to measured spectra, resulting in an overestimation of blood volume fraction, oxygenation levels and melanin concentration in port wine stain (PWS) skin. Zhang et al.⁴⁸ found similar problems when applying a genetic algorithm minimization procedure to their diffusion approximation results, as did Lakmaker et al.⁴⁹ when investigating a method of predicting the maximal treatment depth response required for complete clearance of PWS lesions. Svaasand et al.⁵⁰ applied a diffusion approximation to their model, predicting the effects on skin color resulting from changes in the depth and thickness of a PWS lesion. However, simulated spectral reflectance curves compared poorly to their single example of a measured dataset, suggesting that these predictions of skin color may not be applicable on an individual basis.

5.3.

Monte Carlo Method

Monte Carlo simulations used in biomedical optics are simply ray-tracing procedures, where a statistical analysis is used to calculate, step-by-step, the movements of simulated photons or light beams through a mathematical skin model⁵¹ (see Fig. 3). This has been described as a discrete version of the radiative transport equation.⁵² The Monte Carlo method was first introduced to the field of skin optics by Wilson and Adam in 1983⁵³ and was developed further by a number of groups over the next decade.^54–58 Although its use at this time was limited due to high demands in computing power, Monte Carlo methods are widely regarded as the most accurate simulations of light transport through skin,⁴⁸ and, as a result of recent advances in computing hardware, they are now routinely used for the interpretation of spectral data.¹

Fig. 2

Change in simulated spectral reflection with varying dermal blood concentration (left) and vessel diameter (right). With kind permission from Ref. 61.

Fig. 3

Example of ray traces from 100 beams at a wavelength of 590 nm, simulated using a Monte Carlo program developed by the first author (T. Lister). Beams are initiated on the x-y plane (bottom of image). Reduction in beam ‘weight’ along each path is represented as a transition from light to dark green (color online). Dimensions in mm.

In 1989, Keijzer et al. published their Monte Carlo simulation, analyzing the effectiveness of vascular laser treatments.⁵⁹ This program employs a two-layer skin model, and was developed for investigating PWS skin, in particular for determining the energy deposition within a single vessel during laser treatment. A pencil beam directed perpendicular to the surface of the skin model is initiated at a position randomly allocated within a circular region. The beam propagates through the skin model, undergoing absorption and scatter at events separated by a distance ($s$), calculated, as follows:

Keijzer et al.’s program has the advantage of easy insertion of a sphere, ellipsoid or cylinder into the skin model, to simulate blood vessels for example, but lacks versatility as it uses a predetermined set of optical parameters determined from phantom measurements.

Keijzer et al.’s program was used by Lucassen et al.⁶⁰ to study the effects of laser wavelength on the response of PWS skin to laser therapy. The simulation calculated the absorbed energy within a single straight blood vessel of varying diameters, a curved blood vessel and multiple straight blood vessels. A 585-nm laser beam was predicted to be more effective than a 577-nm beam. This is in agreement with the results found in clinical practice, as a transition of standard practice between these two wavelengths followed worldwide (prior to the shift from 585 to 595 nm), as did the introduction of epidermal cooling during laser therapy, also recommended in Lucassen et al.’s study. Verkruysse et al.⁶¹ also used Keijzer et al.’s simulation to study the influence of PWS anatomy on skin color. They created five models consisting of varying blood layers containing homogenous distributions of blood, with a correction factor to account for reduced light absorption when blood is contained within vessels. Verkruysse et al. commented that their simple skin model failed to take into account the effects of high epidermal scattering, even though they were able to model a diffuse irradiance of photons. Predicted changes of skin color with the removal of superficial vessels did not correspond with the reported clinical response.

In 1992, Wang and Jacques (the latter of which was listed as an author on the previously discussed paper by Keizer et al.) published a Monte Carlo simulation of steady-state light transport in multi-layered tissue using the ANSI Standard C computing language.^55,62 This program is freely available on the internet,⁵⁵ and allows users to create a two-dimensional skin model by inputting custom values of absorption coefficients, scattering coefficients, refractive indices and (Henyey-Greenstein) anisotropy factors for any number of layers. Like the program produced by Keijzer et al., the simulation begins with an infinitely narrow beam incident normal to the skin surface whose path through the skin is governed by Eqs. (5) and (6). A convolution algorithm may then be applied retrospectively to simulate a light source of finite size, such as a laser beam. This effectively repeats the results from the infinitely narrow beam over a finite area without performing any further simulations.

Mantis and Zonios⁶³ developed a two-layer mathematical skin model to estimate the thickness and absorption coefficient of a superficial absorbing layer using Wang and Jacques’ Monte Carlo program. They inputted reflectance values obtained from a fiber optic spectrophotometer setup and compared them to the results simulated from an infinitesimally narrow incident laser beam. Nishidate et al.⁶⁴ applied the same program to a skin model consisting of an epidermis, dermis and local blood region. The depth and thickness of the simulated local blood region was adjusted until the simulated reflectance was in adequate agreement with a measured spectrum, forming an estimate of these parameters in the measured sample. They inputted measurements from a tissue phantom using a diffuse illuminant and a charge coupled device camera setup. Both studies reported absorption and reduced scattering coefficients with errors of around 10% compared to values expected from their skin phantoms. Considering the addition of uncertainty and optical inhomogeneity in a genuine skin sample, these errors are considerable. In fact, the study by Nishidate et al.⁶⁴ used the same method to estimate the depth of in vivo human veins. They reported errors of up to 0.6 mm for the depth and 0.2 mm for the thickness of veins when compared to ultrasound measurements. Again, these errors are large when considering normal skin thickness.⁶⁵

Other research groups have developed Monte Carlo simulations for use in biomedical optics. Graaf et al. produced a condensed Monte Carlo simulation based upon the same governing equations [Eqs. (5) and (6)], but utilizing the derived relationship between simulated reflectance and albedo. In detail, the same simulated diffuse reflectance may be obtained by increasing the scattering coefficient and simultaneously decreasing the absorption coefficient, maintaining a fixed value of albedo (a dimensionless coefficient defined as the ratio of scattering coefficient to the sum of absorption and scattering coefficients).⁶⁶ To demonstrate the effects of varying absorption and scattering coefficients whilst maintaining a fixed albedo, we applied the following parameters to a simulation of 1 million photon packets, carried out using Wang and Jacques’ program.⁵⁵ The skin model consisted of a single layer with an effectively infinite depth ($1\times {10}^8\mathrm{cm}$) (Table 1).

Table 1

Results from a simulation of 1 million photons. Values are presented for a simulation carried out using the Wang and Jacques Monte Carlo program (MCML)62 with a simple, single-layered skin model of 1×108  cm depth; values of g=0.9, refractive index=1.3, and pencil beam irradiation used throughout (simulated by the first author, T. Lister).

This technique vastly improved simulation time for a uniform skin model and was shown to determine skin optical coefficients consistent with other studies.^10,67 This work was later extended by Wang et al.⁶⁸ who considered the effects of varying the diameter of the source. Wang et al. also demonstrated superior calculation times whilst maintaining good agreement with a standard Monte Carlo simulation.

Meglinski and Matcher’s program^47,69 involved a seven layered skin model to simulate the reflectance spectrum of human skin, as measured using a fiber optic probe setup. Path lengths were calculated according to Eq. (7), and absorption coefficients were applied separately [Eq. (8)]:

Meglinski and Matcher found good agreement between their calculated reflectance and a single example of measured reflectance between 450 and 600 nm wavelengths, although the skin properties used outside of this wavelength range did not appear to provide a good agreement. This shortcoming may have been as a result of the constant scattering properties applied to the skin model over the entire wavelength range investigated. When applying the coefficients used by Meglinski and Matcher to Wang and Jacques’ Monte Carlo program, Maeda et al.⁷² reported a “rather strange spectral curve that had much worse agreement with measured results than (another set of coefficients),” demonstrating a clear discrepancy between the two programs.

In summary, Monte Carlo simulations have contributed substantially to the field of skin optics since their introduction in 1983.⁵³ Initial development by Keijzer et al.⁵⁹ and later by Wang and Jacques⁵⁵ facilitated a number of studies into the color of PWS skin. More recent work has demonstrated gradual development in Monte Carlo techniques for the simulation of PWS skin color and with continuing advancements in computing power available to the researcher, and an ever increasing interest in skin optics, such advances are likely to continue into the foreseeable future.