Melanin and blood concentration in a human skin model studied by multiple regression analysis: assessment by Monte Carlo simulation

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1 INSTITUTE OF PHYSICSPUBLISHING Phys. Med. Biol. 46 (21) PHYSICS INMEDICINE AND BIOLOGY PII: S (1) Melanin and blood concentration in a human skin model studied by multiple regression analysis: assessment by Monte Carlo simulation M Shimada 1, Y Yamada 2,3,MItoh 4 and T Yatagai 4 1 Department of Integrated Neuroscience, Tokyo Institute of Psychiatry, Kamikitazawa 2-1-8, Setagaya, Tokyo, , Japan 2 Department of Mechanical Engineering and Intelligent Systems, University of Electro- Communications, Chofugaoka 1-5-1, Chofu, Tokyo, , Japan 3 Institute of Human Science and Biomedical Engineering, National Institute of Advanced Industrial Science and Technology, Namiki 1-2-1, Tsukuba, Ibaraki, , Japan 4 Institute of Applied Physics, University of Tsukuba, Ten-noh-dai 1-1-1, Tsukuba, Ibaraki, , Japan shimada@prit.go.jp Received 27 April 21 Published 22 August 21 Online at stacks.iop.org/pmb/46/2397 Abstract Measurement of melanin and blood concentration in human skin is needed in the medical and the cosmetic fields because human skin colour is mainly determined by the colours of melanin and blood. It is difficult to measure these concentrations in human skin because skin has a multi-layered structure and scatters light strongly throughout the visible spectrum. The Monte Carlo simulation currently used for the analysis of skin colour requires long calculation times and knowledge of the specific optical properties of each skin layer. A regression analysis based on the modified Beer Lambert law is presented as a method of measuring melanin and blood concentration in human skin in a shorter period of time and with fewer calculations. The accuracy of this method is assessed using Monte Carlo simulations. 1. Introduction Information on the amounts of melanin and capillary blood vessel in human skin is useful for the diagnosis of and therapy for nevus, melanoma and erythema. Control of skin colour by laser and medical tattooing also requires knowledge of light propagation in the skin over the whole visible spectrum. Many researchers have reported the optical coefficients of each layer of human skin and the molar absorption coefficients spectra of chromophores such as melanin and blood (Anderson et al 198, Anderson and Parrish 1981, Firbank et al 1993, /1/ $3. 21 IOP Publishing Ltd Printed in the UK 2397

2 2398 M Shimada et al Graaff et al 1993). However, a simplified relationship between skin colour and chromophores has not been determined (Feather et al 1988) Strong scattering and its dependence on wavelength make skin colour analysis difficult. The Kubelka Munk model considering only two opposite fluxes, has been used for colour analysis of scattering media. (Wan et al 1981). This model is not accurate for forward-directed scattering and complicated geometric media such as a biological tissue because this model assumes skin to be a slab of isotropic scattering medium. The Monte Carlo method, which is a statistical technique, provides a more accurate solution for various scattering directions and geometries of objects. This method is widely used to calculate the reflectance of skin models (Barton et al 1998 Kienle et al 1996, Lu et al 2, Verkruysse et al 1999); however, computing machines are burdened with the long calculation times required. It is necessary to adopt a new algorithm if we are to obtain the concentration of the chromophores from a reflectance spectrum. To calculate rapidly, the adding doubling method (Prahl et al 1993), which takes into account only four or eight fluxes, and several equations for solving the reflectance of the multi-layered model (Martelli et al 1997, Patterson et al 1989) have been reported. It is difficult to determine the optical coefficients because the constitution of blood in the capillary vessel is unknown and the scattering coefficient of the dermis depends on the direction of the collagen fibres (Nickell et al 2, Saidi et al 1995). Statistical methods widely used for the analysis of reflectance spectra (Imai et al 1996) and independent component analysis were applied to measure the amount of melanin and blood in in vivo skin (Tsumura et al 2). We have proposed a method of measuring chromophores in human skin by regression analysis based on the modified Beer Lambert law (Shimada et al 2, 21). This method assumes that mean path length is independent of the concentration of the chromophores. In this paper, the Monte Carlo simulation was conducted in order to assess the proposed method. 2. Method 2.1. Structure of human skin Human skin consists of three layers from the surface: epidermis, dermis and subcutaneous fat (Anderson et al 198, Anderson and Parrish 1981). The stratum corneum is produced by the transformation of epidermis and, for reasons of simplication, the optical coefficients of the epidermis and stratum corneum are regarded as the same. The scattering coefficient of the epidermis is about half that of the dermis (Simpson et al 1998, Cheong et al 199). The absorbance of epidermis and dermis depends mainly on the amount of melanin and capillary blood present, respectively. The effect of carotene in the epidermis is neglected because the absorption peak specific to carotene does not appear in normal skin. Individual, time and area variations in scattering coefficient are assumed to be small. On the other hand, the variations of absorption coefficients are larger than scattering coefficients because absorption coefficients vary with the number of chromophores. For example, the absorption coefficient changes after exposure to UV light or bathing in hot water. To simplify, the subcutaneous fat is assumed to diffuse all visible light because there are no remarkable chromophores in subcutaneous fat Monte Carlo simulation and inverse Monte Carlo simulation The Monte Carlo method can demonstrate light propagation in a scattering medium if the scattering coefficient µ s, the absorption coefficient µ a, the thickness d, the refractive index n and phase function p(θ) are known. µ s and µ a are defined as the inverse of the mean path

3 Measurement of melanin and blood concentration in human skin 2399 length until the next scattering and absorption, respectively. p(θ), describing the direction of scattering, can convert to an anisotropic parameter g, whichis 1, or 1 depending on whether the scattering direction is completely forward, isotropic, or backwards, respectively. The g of each layer of human skin is about.9 (Simpson et al 1998). To calculate the reflectance and transmittance of a multi-layered model, we used the Monte Carlo code developed by Wang and Jacques (1992). In this code a Henyey Greenstein function defined by g was used instead of p(θ). The code was arranged to correspond to our measurement system. Photons reaching the subcutaneous fat were assumed to reflect in all directions with equal probability. To obtain µ s and µ a, several algorithms for the inverse Monte Carlo method were explored (Dam et al 2). In this study, the Monte Carlo simulation was iterated with varying µ s and µ a until the differences between the calculated reflectance/transmittances and measured versions fell below the thresholds. The photon number was 1, and the thresholds were.1 times reflectance and transmittance The modified Beer Lambert law with average optical path length in scattering and absorbing media The absorbance, A, is defined from the reflectance, R, of the skin of the face or arm which is considered to be a semi-infinite medium A = log 1 R. (1) The absorbance A of a homogeneous scattering medium, including absorption material whose molar absorption and molar concentration are ε and C, is shown by equation (2) A = εc l(c) + G (2) where G and l(c) are scattering loss and mean path length, respectively. This formula is the modified Beer Lambert law. G depends on µ s, g, the geometry of the object, and the measurement system, but not on µ a. l(c)gets shorter as the concentration of the chromophores increases, because a greater proportion of photons with longer path lengths is absorbed than that with shorter path lengths. The relationship between C and A is illustrated schematically in figure 1. A/ C is easy to change when the scattering coefficient is high. A/ C is independent of small C (Matcher et al 1993, Delpy et al 1988): A C = ε l(c) = ε l. (3) We named this formula the limited modified Beer Lambert law, in which A is proportional to C. IfC is near zero, A is expressed using l which is the mean path length of the non-absorbent medium A = εc l + G. (4) Equation (4) is equivalent to (a) in figure 1 and A = G at C =. The gradient for a large concentration of chromophores is small as shown in (b). Equation (5) is satisfied by small C A = ε l. (5) C C In this paper, the mean path length, l,isassumedtobeequalto l.

4 24 M Shimada et al 1.2 Absorbance A Concentration C Figure 1. Relationship between absorbance of scattering medium and concentration of chromophore is linear within small change in concentration. At large concentration, the slope decreases while the scattering loss increases. The gradient of A/ C is large at strong scattering medium. lm ε 1, C1 ε m, Cm Epidermis l1 ε l2 2, C2 lb ε b, Cb Dermis Subcutaneous fat (BaSO4) (a) (b) Figure 2. (a) Path length for each chromophore in scattering media including some kinds of chromophores. (b) Schematic of a three-layered skin phantom composed of epidermis, dermis and subcutaneous fat from the surface. Absorption chromophores of epidermis and dermis are melanin (squid ink) and blood, respectively Measurement of melanin and blood concentrations by the multiple linear regression analysis The A(λ) of an inhomogeneous scattering medium, including multiple chromophores, is m m A(λ) = A i (λ) + G(λ) = ε i (λ)c i l i (C 1,...,C m,λ)+ G(λ) (6) i=1 i=1 where the subscript, i, refers to the ith chromophore. l i (λ) is the path length in the area in which the ith chromophore is distributed. l i (λ) depends on not only C i but also C 1,...,C m, but the effects of C j (j i) are small. The schema is shown in figure 2(a). The path length in

5 Measurement of melanin and blood concentration in human skin 241 the area where plural chromophores exist is considered as path length of both chromophores. l i (C i,λ)and G(λ) vary with wavelength λ because the scattering coefficients are different at each λ. For reflectance of a semi-infinite medium measured with an integrating sphere, l(c) becomes short and G becomes long at high scattering values. A simplified model of human skin is schematized in figure 2(b), showing the light propagation of a typical photon in the skin model in which subscript m and b are melanin and blood, respectively. If the C m and C b are so small as to satisfy equation (5), l i (i = m, b) and A is independent of the concentration of chromophores. The absorbance spectrum of human skin A skin (λ) can be expressed as A skin (λ) = ε m (λ)c m l m (λ) + ε b C b l b (λ) + A (λ) (7) A (λ) = A (λ) + G(λ) where A (λ) is the absorbance of chromophores other than melanin and blood. When there is no change in the optical coefficient of human skin except C m (λ) and C b (λ), equations (8) and (9) are given as follows: C m A skin (λ) = A m (λ) + C b A b (λ) + A (λ) (8) C m C b A m (λ) = ε m (λ) C m l m (λ), A b (λ) = ε b (λ) C b l b (λ). (9) If A m (λ), A b (λ) and A (λ) are known, multiple regression analysis makes it possible to estimate C m / C m and C b / C b for an arbitrary A skin (λ). To satisfy equation (8), l i (i = m, b) in equation (9) are assumed to be equal to those in equation (8). Because the scattering and absorption coefficients depend on wavelength, the gradient of (b) and G in figure 1 varies with the wavelength. Estimated C m / C m and C b / C b were designated as Ĉ m / C m and Ĉ b / C b. 3. Results and discussion We made 25 three-layered skin phantoms. How to fabricate skin phantoms is described by Shimada et al (21). Reflectance and transmittance spectra of the epidermal and dermal phantoms were measured to calculate the scattering and absorption coefficient spectra of the epidermal and dermal phantoms. The input and detected area for transmittance measurement were 5 12 mm and 12 2 mm,respectively. The five epidermal phantoms were named E(i) phantoms (i =, 1, 2, 3, 4) in which i refers to one of five levels of concentration of melanin. In the same manner, the five dermal phantoms were named D( j) phantoms (j =, 1, 2, 3, 4) in which j refers one of the five levels of concentration of blood. The thickness of epidermal and dermal phantoms was 2.5 and 6.3 mm, respectively. Figure 3 shows µ s (λ) of epidermal and dermal phantoms calculated from measured reflectance/transmittance and the inverse Monte Carlo method in which g is.9. The absorption coefficient spectra of the epidermal phantoms µ a (m)(λ) and those of the dermal phantoms µ a (b)(λ) are shown in figure 4. (m) and (b) show melanin and blood, respectively. Figure 5 shows µ a (m)(λ) and µ a (b)(λ) caused by an increasing concentration of chromophores. The shapes of the absorbance spectra of the melanin and blood solution in figure 5 are similar to µ a (m)(λ) and µ a (b)(λ). We calculated the coefficient of variation C v of µ a (m) and µ a (b) at each wavelength. The mean C v of µ a (m) is.387 while the mean C v of µ a (b) is.87. The C v of µ a (b) averaged from 4 to 58 nm, because µ a (b) is almost zero at a longer wavelength. This means that the change in µ a (b)(λ) is more linear than that of µ a (m)(λ).

6 242 M Shimada et al s Epidermis Dermis Figure 3. Average of µ s of epidermal and dermal phantom. Error bar is the standard deviation of the five phantoms..2.5 a (mm ) melanin increase a(mm ) blood increase (a) (b) Figure 4. Change in absorption spectra of skin phantoms whose concentration of chromophores increases by a degree. (a) melanin (b) blood. The three-layered skin phantoms which consist of an E(i) phantom, a D(j) phantom and a subcutaneous fat phantom are named E(i)D(j) phantom. A E(i)D(j) (λ) is the absorbance spectrum of an E(i)D(j) phantom. A m (λ) and A b (λ) caused by increasing the concentration of chromophores by a degree are defined as equations (13) and (14) in Shimada et al (21). If A E(i)D(j) (λ)(i, j =, 1,...,4) wereassumedtobesatisfiedinequation(8) by substituting A E()D() (λ) for A (λ), it is possible to estimate C m / C m and C b / C b by regression analysis. Comparisons of estimated concentrations with measured concentrations are shown in figure 6 in Shimada et al (21). The approximate equations were given by multiple regression analysis. Although the C v of µ a (b) is smaller than µ a (m), the average the coefficient of correlation for Ĉ b is.948 while that for Ĉ m is.996.

7 Measurement of melanin and blood concentration in human skin 243 a and absorbance blood/3 melanin/5 µa(blood) µa(melanin) Figure 5. Change in µ a (m) and µ a (b) spectra by a degree shown in figure 4, compared with absorbance spectra of a melanin and blood solution. Scattering of the solutions is neglected. We compared the results from actual measurements with those generated by the Monte Carlo simulation. For the Monte Carlo simulation, the µ s spectra of each layer are defined as the average shown in figure 3. The µ a spectra are defined as µ a (E(i)) = µ a (E()) + i µ a (m) (1) µ a (D(j)) = µ a (D()) + j µ a (b) (11) where µ a (E()) and µ a (D()) are the µ a spectra shown in figure 4. The thickness of epidermal and dermal phantoms has measured values. The refractive index of all the phantoms was 1.4. The input and detected versions are the same as the measurement system. The skin phantoms are assumed to be slabs and light that leaked from the sides of the phantoms was neglected. The changes in absorbance caused by increase of melanin: A E(i+1)D() (λ) A E(i)D() (λ) are shown in figure 6(a). Although the change in µ a (m) is the same at any concentration of melanin, the change in absorbance is small at large concentrations because of changes in l m. Figure 6(b) shows the change in absorbance caused by an increased presence of blood: A E()D(j+1) (λ) A E()D(j) (λ) whose spectra are distorted as blood levels increase. The change in l b at shorter wavelength is greater than at the longer wavelengths because of strong scattering and large change in absorption. Although the blood concentration of E(i)D(4)(i =, 1, 2, 3, 4) is the same, A E(i+1)D(4) (λ) A E(i)D(4) (λ) in figure 6(c) shows blood absorption peaks. For the multiple regression analysis, A m (λ) and A b (λ) were defined in the same manner. A comparison Ĉ m and Ĉ b with C m and C m is shown in figure 7(a). The average of the coefficient of correlation for Ĉ b is.893 while for Ĉ m is.997. The non-linear relationship is remarkable for Ĉ b because the µ a of blood at the shorter wavelength is larger than that for melanin, and the µ s of the dermis is stronger than that for the epidermis. Figure 7(b) indicates good approximation by quadratic equations; the average of the coefficient of correlation is.996. The gradient of Ĉ m is independent of the concentration of the blood. The first and second terms of Ĉ b, however, decrease at large concentrations of melanin because absorption of melanin shortens l b.

8 244 M Shimada et al E(i+1)D() E(i)D() melanin increase (a) E()D(i+1) E()D(i) blood increase (b).5 E(i+1)D(4) E(i)D(4) melanin increase (c) Figure 6. Change in absorbance spectra of skin phantoms against increase of chromophore (a) A E(i+1)D() (λ) A E(i)D() (λ), (b)a E()D(j+1) (λ) A E()D(j) (λ), (c)a E(i+1)D(4) (λ) A E(i)D(4) (λ). 4. Conclusions Although C v of µ a of epidermal phantoms is greater than for dermis phantoms, the values of the coefficient of correlation for dermis phantoms were lower than for epidermal phantoms. Our approximation by the limited modified Beer Lambert law is not sufficiently accurate for dermal phantoms due to a strong absorption of blood at shorter wavelengths and a strong scattering of the dermal phantom itself. The absorption of the chromophores affects the measurement of the concentration of other chromophores. The melanin concentration prevents precise measurement of blood concentration. This means that it is impossible to neglect the dependency of l b on C m. However, spectrum analysis by multiple regression analysis incorporating A m (λ) and A b (λ) is useful for making rapid estimates of the concentration of chromophores in skin phantoms. To measure concentrations with high accuracy, it is important to take into account the decrease in the mean path lengths caused by a greater density of the chromophores.

9 Measurement of melanin and blood concentration in human skin 245 Predicted concentration D D2 D4 Predicted concentration E E2 E Measured concentration (a) Measured concentration (b) Figure 7. Estimated concentration of the melanin from calculated absorbance spectra by the Monte Carlo method: (a) Estimated C m is linear to settled C m. The mean the coefficient of correlation is The mean gradient is 1.6 and its standard deviation is.266. (b) Ĉ b of E()D(i), E(2)D(i), E(4)D(i) (i =, 1, 2, 3, 4). Ĉ b is well approximated by a quadratic equation. The first and second terms of the approximate equation decrease at large concentrations of melanin. References Anderson R R, Hu J and Parrish J A 198 Optical radiation transfer in the human skin and applications in in vivo remittance spectroscopy Bioengineering and Skin ed R Marks and P A Payne (London: MTP Press) pp Anderson R R and Parrish J A 1981 The optics of human skin J. Invest. Dermatol Barton J K, Pfefer T J and Welch A J 1998 Optical Monte Carlo modeling of a true port wine stain anatomy Opt. Exp Cheong W F, Prahl S A and Welch A J 199 A review of the optical properties of biological tissues IEEE J. Quantum Electron Dam J S, Dalgaard T, Fabricius P E and Andersson-Engels S 2 Multiple polynomial regression method for determination of biomedical optical properties from integrating sphere measurements Appl. Opt Delpy D T, Cope M, van der Zee P, Arridge S, Wray S and Wyatt J 1988 Estimation of optical pathlength through tissue from direct time of flight measurement Phys. Med. Biol Feather J W, Ellis D J and Leslie G 1988 A portable reflectrometer for the rapid quantification of cutaneous heamoglobin and melanin Phys. Med. Biol Firbank M, Hiraoka M, Essenpreis M and Delpy D T 1993 Measurement of the optical properties of the skull in the wavelength range nm Phys. Med. Biol Graaff R, Dassel A C M, Koelink M H, de Mul F F M, Aarnoudse J G and Zijlstra W G 1993 Optical properties of human dermis in vitro and in vivo Appl. Opt Imai F H, Tsumura N, Haneishi H and Miyake Y 1996 Principal component analysis of skin color and its application to colorimetric color reproduction on CRT display and hardcopy J. Imaging Sci. Technol Kienle A, Lilge L, Vitkin I A, Patterson M S, Wilson B C, Hibst R and Steiner R 1996 Why do veins appear blue? A new look at an old question Appl. Opt Lu J Q, Hu X H and Dong K 2 Modeling of the rough-interface effect on a converging light beam propagating in a skin tissue phantom Appl. Opt Martelli F, Contini D, Taddeucci A and Zaccanti G 1997 Photon migration through a turbid slab described by a model based on diffusion approximation. II. Comparison with Monte Carlo results Appl. Opt Matcher S J, Cope M and Delpy D T 1993 Use of the water absorption spectrum to quantify tissue chromophore concentration changes in near-infrared spectroscopy Phys. Med. Biol Nickell S, Hermann M, Essenpreis M, Farrell T J, Krämer U and Patterson M S 2 Anisotropy of light propagation in human skin Phys. Med. Biol Patterson M, Chance B and Wilson B C 1989 Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties Appl. Opt

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