## Modeling optical properties of human skin using Mie theory for particles with different size distributions and refractive indices |

Optics Express, Vol. 19, Issue 15, pp. 14549-14567 (2011)

http://dx.doi.org/10.1364/OE.19.014549

Acrobat PDF (884 KB)

### Abstract

We used size distributions of volume equivalent spherical particles with complex refractive index to model the inherent optical properties (IOPs) in four different layers of human skin at ten different wavelengths in the visible and near-infrared spectral bands. For each layer, we first computed the size-averaged absorption coefficient, scattering coefficient, and asymmetry factor for the collection of particles in a host medium using Mie theory and compared these IOPs in each layer with those obtained from a bio-optical model (BOM). This procedure was repeated, using an optimization scheme, until satisfactory agreement was obtained between the IOPs obtained from the particle size distribution and those given by the BOM. The size distribution as well as the complex refractive index of the particles, obtained from this modeling exercise, can be used to compute the phase matrix, which is an essential input to model polarized light transport in human skin tissue.

© 2011 OSA

## 1. Introduction

1. P. A. Payne, “Measurement of properties and function of skin,” Clin. Phys. Physiol. Meas. **12**, 105–129 (1991). [CrossRef] [PubMed]

*epidermis*, the

*dermis*, and the

*sub-cutis*, each being a heterogeneous medium for transport of light. The epidermis (≈100

*μ*m), which is thinner than the dermis (≈1 mm) and the sub-cutis (≈3 mm), may be further divided into sub-layers [2], namely the

*stratum corneum*, the

*stratum lucidum*, the

*stratum granulosum*, the

*stratum spinosum*, and the

*stratum germinativum*. The stratum corneum which is 10–20

*μ*m thick, is the outermost layer of the epidermis, and is composed of non-living

*corneocyte*cells glued together with

*keratin*to form a membrane-like structure.

3. B. L. Diffey, “A Mathematical model for ultraviolet optics in skin,” Phys. Med. Biol. **28**, 647–657 (1983). [CrossRef] [PubMed]

4. K. P. Nielsen, L. Zhao, P. Juzenas, J. J. Stamnes, K. Stamnes, and J. Moan, “Reflectance spectra of pigmented and non-pigmented skin in the UV spectral region,” Photochem. Photobiol. **80**, 450–455 (2004). [PubMed]

5. A. R. Young, “Chromophores in Human Skin,” Phys. Med. Biol. **42**, 789–802 (1997). [CrossRef] [PubMed]

6. D. J. Faber, M. C. G. Aalders, E. G. Mik, B. A. Hooper, M. J. C. van Gemert, and T. G. van Leeuwen, “Oxygen saturation-dependent absorption and scattering of blood,” Phys. Rev. Lett. **93**, 1–4 (2004). [CrossRef]

7. J. Sandby-Møller, T. Paulsen, and H. C. Wulf, “Epidermal thickness at different body sites: relationship to age, gender, pigmentation, blood content, skin type and smoking,” Acta Derm. Venereol. **83**, 410–413 (2003). [CrossRef] [PubMed]

8. A. N. Bashkatov, E. A. Genina, V. I. Kochubey, and V. V. Tuchin, “Optical properties of human skin, subcutaneous and mucous tissues in the wavelength range from 400 to 2000 nm,” J. Phys. D: Appl. Phys. **38**, 2543–2555 (2005). [CrossRef]

9. B. Farina, C. Bartoli, A. Bono, A. Colombo, M. Lualdi, G. Tragni, and R. Marchesini, “Multispectral imaging approach in the diagnosis of cutaneous melanoma: potentiality and limits,” Phys. Med. Biol. **45**, 1243–1254 (2000). [CrossRef] [PubMed]

10. M. Moncrieff, S. Cotton, E. Claridge, and P. Hall, “Spectrophotometric intracutaneous analysis - a new technique for imaging pigmented skin lesions,” Br. J. Dermatol. **146**, 448–457 (2002). [CrossRef] [PubMed]

*in vivo*.

*λ*, the absorption coefficient

*μ*(

_{a}*λ*) [mm

^{−1}] of a particle embedded in a base fluid or host medium can be expressed by where

*f*is the volume fraction occupied by the absorbing particle, and where

_{a}*μ*(

_{a,p}*λ*) and

*μ*(

_{a,h}*λ*) are the absorption coefficients of the particle and host medium, respectively. If the absorption by each particle is assumed to be independent of the absorption by all other particles, the net absorption coefficient for a collection of

*N*particles becomes

*N*times

*μ*(

_{a,p}*λ*).

11. M. J. C. Van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, and W. M. Star, “Skin Optics,” IEEE Trans. Biomed. Eng. **36**, 1146–1154 (1989). [CrossRef] [PubMed]

12. S. L. Jacques, “Role of tissue optics and pulse duration on tissue effects during high-power laser irradiation,” Appl. Opt. **32**, 2447–2454 (1993). [CrossRef] [PubMed]

*in vitro*investigations of tissues of e.g. arteries, liver, and kidney. However, the attenuation of VIS and NIR radiation in human skin tissue due to absorption is much less than that due to scattering [11

11. M. J. C. Van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, and W. M. Star, “Skin Optics,” IEEE Trans. Biomed. Eng. **36**, 1146–1154 (1989). [CrossRef] [PubMed]

13. R. R. Anderson and J. A. Parrish, “The optics of human skin,” J. Invest. Dermat. **77**, 13–19 (1981). [CrossRef]

14. A. Yodh and B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today **48**, 34–40 (1995). [CrossRef]

*therapeutic window*[15].

16. J. R Mourant, J. P. Freyer, A. H. Hielscher, A. A. Eick, D. Shen, and T. M. Johnson, “Mechanisms of light scattering from biological cells relevant to non-invasive optical-tissue diagnostics,” Appl. Opt. **37**, 3586–3593 (1998). [CrossRef]

*μ*m, while Jacques [17

17. S. L. Jacques, “Optical assessment of tissue heterogeneity in biomaterial and implants,” Proc. SPIE **3914**, 576–580 (2000), doi:. [CrossRef]

*μ*m. According to Drezek et al. [18

18. R. Drezek, A. Dunn, and R. Richards-Kortum, “Light scattering from cells: finite-difference, time-domain simulations and goniometric measurements,” Appl. Opt. **38**, 3651–3661 (1999). [CrossRef]

*μ*m. Similarly, the shape of scattering particles varies. Keratin cells in the stratum corneum are flat, and melanosome particles in the lower epidermis (stratum germanitivum) are fairly round with a diameter of about 1

*μ*m [3

3. B. L. Diffey, “A Mathematical model for ultraviolet optics in skin,” Phys. Med. Biol. **28**, 647–657 (1983). [CrossRef] [PubMed]

19. K. P. Nielsen, L. Zhao, J. J. Stamnes, K. Stamnes, and J. Moan, “Importance of the depth distribution of melanin in skin for DNA protection and other photobiological processes,” J. Photochem. Photobiol. **82**, 194–198 (2006). [CrossRef]

20. I. S. Saidi, S. L. Jacques, and F. K. Tittel, “Mie and Rayleigh modelling of visible light scattering in neonatal skin,” Appl. Opt. **34**, 7410–7418 (1995). [CrossRef] [PubMed]

*μ*m and 100 nm, respectively, in skin samples of newborn infants.

18. R. Drezek, A. Dunn, and R. Richards-Kortum, “Light scattering from cells: finite-difference, time-domain simulations and goniometric measurements,” Appl. Opt. **38**, 3651–3661 (1999). [CrossRef]

21. G. J. Tearney, M. E. Brezinski, J. F. Southern, B. E. Bouma, M. R. Hee, and J. G. Fujimoto, “Determination of the refractive index of highly scattering human tissue by optical coherence tomography,” Opt. Lett. **20**, 2258 –2260 (1995). [CrossRef] [PubMed]

*Caucasian*and

*African American*human skin epidermis and dermis

*in vitro*were measured by Ding et al. [22

22. H. Ding, J. Q. Lu, W. A. Wooden, P. J. Kragel, and X.-H. Hu, “Refractive indices of human skin tissues at eight wavelengths and estimated dispersion relations between 300 and 1600 nm,” Phys. Med. Biol. **51**, 1479–1489, (2006). [CrossRef] [PubMed]

24. Q. Fu and W. Sung, “Mie theory for light scattering by a spherical particle in an absorbing medium,” Appl. Opt. **40**, 1354–1361 (2001). [CrossRef]

25. P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D **3**, 825–839 (1971). [CrossRef]

28. D. Petrov, E. Synelnyk, Y. Shkuratov, and G. Videen, “The T-matrix technique for calculations of scattering properties of ensembles of randomly oriented particles with different size,” J. Quant. Spectr. Radiat. Transfer **102**, 85–110 (2006). [CrossRef]

*μ*(

_{s}*λ*) [mm

^{−1}] due to combined contributions from both Mie and Rayleigh scattering can then be expressed by where

*f*is the fraction of scattering due to large particles contributing to Mie scattering, (1 –

_{s}*f*) is the fraction of scattering due to small particles contributing to Rayleigh scattering, and

_{s}*b*indicates the wavelength dependence of Mie scattering.

*r*may be expressed in terms of a dimensionless

*size*parameter

*x*= 2

*πr*/

*λ*, where

*λ*is the wavelength of light in the medium. Particles with

*x*≪ 1, pertain to Rayleigh scattering [29

29. A. J. Cox, A. J. Deweerd, and J. Linden, “An experiment to measure Mie and Rayleigh total scattering cross sections,” Am. J. Phys. **70**, 620–625 (2002). [CrossRef]

*λ*

^{−4}, and with the angular distribution of the scattering being equally probable in the forward and the backward direction. Rayleigh scattering is independent of the particle shape [31] as long as the condition

*x*≪ 1 is fulfilled.

### 1.1. Mie Scattering

*λ*

^{−b}. The exponent

*b*varies with particle size, and has the value

*b*= 4 for particles that are very small compared to the wavelength, in which case Mie scattering reduces to Rayleigh scattering. Recently, Jacques [30

30. S. L. Jacques, “Optical assessment of cutaneous blood volume depends on the vessel size distribution: a computer simulation study,” J. Biophoton. **3**, 75–81 (2010). DOI. [CrossRef]

*b*for skin epidermis and dermis tissues is equal to 0.838.

*E*

_{||s}and

*E*

_{⊥s}are considered at observation points sufficiently far (

*kr*≫ 1) from the sphere so that the radial component of the scattered electric field becomes negligible. If the scattering particle is a homogeneous and isotropic sphere, then

*S*

_{3}=

*S*

_{4}= 0, so that the scattering matrix

*S*(Θ) of such a sphere is given by

*S*

_{1}(Θ) and

*S*

_{2}(Θ), which provide the perpendicular and parallel polarization components of the scattered electric field vector, are given by [31] where

*n*is a summation index and

*a*and

_{n}*b*are coefficients that depend on the size parameter

_{n}*x*and the refractive index of the spherical particle relative to that of the host medium, which is assumed to be homogeneous, isotropic, and non-absorbing. The functions

*π*(

_{n}*μ*) and

*τ*(

_{n}*μ*) with

*μ*= cosΘ determine the angular dependence of the scattered field and are given in terms of associated Legendre polynomials [34].

*a*and

_{n}*b*in Eqs. (5a) and (5b) have an infinite number of terms, but in practical computations the summation must be terminated after a certain number of terms. According to Bohren and Huffman [31], the maximum number of terms required for convergence is given by

_{n}*n*=

*x*+ 4

*x*

^{1/3}+ 2.

_{1}and S

_{2}become identical, indicating that scattering by spheres in the forward direction (Θ = 0) is independent of the polarization state of the incident field. The total power extinction in the forward direction is given by [35

35. M. I. Mishchenko, “The electromagnetic optical theorem revisited,” J. Quant. Spectr. Radiat. Transfer **101**, 404–410 (2006). [CrossRef]

*is the extinction efficiency and ℜ denotes the real part. As expected, Q*

_{ext}*vanishes as*

_{ext}*x*→ 0. Q

*increases to 4 for*

_{ext}*x*≈ 1, and deceases to 2 as the particle becomes very large (

*x*→ ∞). Mie theory is generally applicable to scattering by homogeneous spheres of any size. For large values of

*x*, the total extinction Q

*tends to 2, implying that the extinction cross section becomes twice as large as the geometrical cross section, a result known as the extinction paradox [32].*

_{ext}*and scattering efficiency Q*

_{ext}*of a spherical particle are given by [31, 32] where*

_{sca}*m*is the relative refractive index of the particle. If the extinction does not involve absorption, the single-scattering albedo, which is the ratio of the scattering coefficient

*μ*to the extinction coefficient

_{s}*μ*=

_{e}*μ*+

_{s}*μ*, is equal to unity. However, in skin tissue, the extinction generally includes absorption, so that the absorption efficiency Q

_{a}*can be obtained from*

_{abs}### 1.2. Scattering Phase Function

*i.e.*where cosΘ =

*μ*, and d

*ω*is the differential solid angle centered around the direction of scattering. The scattering phase function

*p*(

*μ*) can be expanded in a series of Legendre polynomials such that where

*τ*is the optical depth and

*χ*(

_{ℓ}*τ*) is

*ℓ*th expansion coefficient given by and

*P*is the

_{ℓ}*ℓ*th Legendre polynomial. The first expansion coefficient or moment

*χ*

_{1}in Eq. (11) is called the

*asymmetry factor g*, and is given by where

*g*, which is confined to the range −1 ≤

*g*≤ +1, is dimensionless and characterizes the direction of a single-scattering. For particles that are large relative to the wavelength of the incident radiation, the scattering phase function peaks towards the forward direction (Θ = 0°), and then

*g*tends to +1. When

*g*tends to −1, the scattering is in the backward direction (Θ = 180°). If

*g*= 0, the scattering is symmetric around Θ =

*π*/2. The asymmetry factor

*g*for Rayleigh scattering is zero because the scattering phase function is proportional to 1 + cos

^{2}Θ, and hence is symmetric around Θ =

*π*/2. The parameters

*μ*,

_{a}*μ*, and

_{s}*g*depend on the wavelength and the particle’s size and relative refractive index.

### 1.3. Collection of Particles

*μ*for a single particle, provided the scattering is independent. The human skin is an inhomogeneous, turbid optical medium in which the actual particle size distribution and refractive index vary with location. Such an inhomogeneous scattering medium can be decomposed into elementary volumes in each of which the particle distribution can be considered as homogenous [37] so that the extinction strength in each elementary volume is given by the product of the single particle extinction and the number of particles.

_{s}38. E. Limpert, W. A. Stahel, and M. Abbt, “Log-normal distributions across the science: keys and clues,” BioScience **51**, 341–351 (2001). [CrossRef]

*n*(

*r*) consisting of 20 evenly spaced size intervals

*dr*within the range from

*r*

_{1}=

*r̄*– 3

*σ*to

*r*

_{2}=

*r̄*+ 3

*σ*such that where

*r̄*and

*σ*are the mean and the standard deviation of the natural logarithm of the variable

*r*. The number

*N*which represents particles of all sizes in a unit volume is given by

*(*μ ˜

_{e}*λ*) =

*μ*

*̃*(

_{a}*λ*) +

*μ*

*̃*(

_{s}*λ*) and the ensemble-averaged scattering coefficient

*(*μ ˜

_{s}*λ*) for a collection of spherical particles can be approximated as a weighted average of the contributions from individual particles as follows [39

39. B. Hamre, J. Winther, S. Gerland, J. J. Stamnes, and K. Stamnes, “Modelled and measured optical transmittance of snow-covered first-year sea ice in Kongfjorden, Svalbard,” J. Geophys. Res. **109**, 1–14 (2004). [CrossRef]

*i.e.*and the ensemble-averaged absorption coefficient for the collection then follows from

*(*μ ˜

_{a}*λ*) =

*(*μ ˜

_{e}*λ*) –

*(*μ ˜

_{s}*λ*).

*g̃*for a collection of particles is given by [23, 40

40. H. J. van Staveren, C. J. M. Moes, J. van Marie, S. A. Prahl, and M. J. C. van Gemert, “Light scattering in intralipid-10% in the wavelength range of 400–1100 nm,” Appl. Opt. **30**, 4507–4514, (1991). [CrossRef] [PubMed]

*C*=

_{sca}*πr*

^{2}

*Q*is the scattering cross section of a particle with radius

_{sca}*r*.

*includes contributions due to Rayleigh scattering*μ ˜

_{s}*K*

*λ*

^{−4}, where the constant

*K*represents the strength of Rayleigh scattering. With contributions from both types of scattering, the ensemble-averaged asymmetry factor

*g̃*becomes where the last result follows from the fact that

*g̃*= 0, since, as mentioned previously, the Rayleigh scattering phase function is symmetric around Θ =

^{R}*π*/2.

## 2. Method

### 2.1. Bio-optical Model

39. B. Hamre, J. Winther, S. Gerland, J. J. Stamnes, and K. Stamnes, “Modelled and measured optical transmittance of snow-covered first-year sea ice in Kongfjorden, Svalbard,” J. Geophys. Res. **109**, 1–14 (2004). [CrossRef]

44. K. Zhang, W. Li, H. Eide, and K. Stamnes, “A bio-optical model suitable for use in forward and inverse coupled atmosphere-ocean radiative transfer models,” J. Quant. Spectr. Radiat. Transfer **103**, 411–4233 (2007). [CrossRef]

*a*, colored dissolved organic matter (CDOM), and suspended matter). However, BOMs that describe the optics of living tissue, such as human skin tissue, are rarely found in the open literature. A simple BOM for human skin was presented by Nielsen et al. [4

4. K. P. Nielsen, L. Zhao, P. Juzenas, J. J. Stamnes, K. Stamnes, and J. Moan, “Reflectance spectra of pigmented and non-pigmented skin in the UV spectral region,” Photochem. Photobiol. **80**, 450–455 (2004). [PubMed]

*μ*,

_{a}*μ*, and

_{s}*g*) for melanosome particles from Mie theory in the UV spectral region (250–310 nm). They considered the real part of the refractive index of the melanosome particles to be constant, whereas the refractive index of human skin is actually wavelength dependent [22

22. H. Ding, J. Q. Lu, W. A. Wooden, P. J. Kragel, and X.-H. Hu, “Refractive indices of human skin tissues at eight wavelengths and estimated dispersion relations between 300 and 1600 nm,” Phys. Med. Biol. **51**, 1479–1489, (2006). [CrossRef] [PubMed]

45. A. N. Bashkatov, E. A. Genina, V. I. Kochubey, and V. V. Tuchin, “Estimation of wavelength dependence of refractive index of collagen fibers of scleral tissue,” Proc. SPIE **4162**, 265–267 (2000). [CrossRef]

4. K. P. Nielsen, L. Zhao, P. Juzenas, J. J. Stamnes, K. Stamnes, and J. Moan, “Reflectance spectra of pigmented and non-pigmented skin in the UV spectral region,” Photochem. Photobiol. **80**, 450–455 (2004). [PubMed]

46. D. L. Swanson, S. D. Laman, M. Biryulina, K. P. Nielsen, G. Ryzhikov, J. J. Stamnes, B. Hamre, L. Zhao, E. Sommersten, F. S. Castellana, and K. Stamnes, “Optical transfer diagnosis of pigmented lesions,” Dermatol. Surg. **36**, 1–8 (2010). DOI: [CrossRef]

*τ*, single-scattering albedo

_{B}*ω*, and scattering phase function asymmetry factor

_{B}*g*) for each layer of the skin from layer thickness and particle concentrations or particle volume fractions. The Balter BOM is also based on additional inputs from the literature, such as fat absorption coefficient [47], water absorption coefficient [48], and blood absorption coefficient [49

_{B}49. S. Prahl, “Tabulated Molar Extinction Coefficient for Hemoglobin in Water,” http://omlc.ogi.edu/spectra/hemoglobin/takatani.html.

*upper epidermis, lower epidermis, dermis*, and

*sub-cutis*. For each of these four layers, the layer thickness

*dz*, the volume fraction of particles [%], and the major types of particles used as input to the Balter BOM are summarized in Table 1.

*dz*, the Balter BOM generates the differential optical thickness

*dτ*(

_{B}*λ*), the single-scattering albedo

*ω*(

_{B}*λ*), and the asymmetry factor

*g*(

_{B}*λ*), which can be expressed in terms of the ensemble-averaged absorption coefficient

*(*μ ˜

_{a}*λ*), scattering coefficient

*(*μ ˜

_{s}*λ*), and asymmetry factor

*g̃*for Mie scattering derived in section 1.2,

^{M}*i.e.*

*(*μ ˜

_{e}*λ*),

*(*μ ˜

_{s}*λ*), and

*(*μ ˜

_{a}*λ*) on the right side obtained from Rayleigh and Mie scattering theory. Similarly, for each layer, we compared the BOM-generated asymmetry factor

*g*on the left side of Eq. (22) with the quantity on the right side obtained from Rayleigh and Mie scattering theory. These comparisons were performed using least-squares fitting in MATLAB

_{B}*by varying the complex refractive index and the size distribution of the particles until satisfactory agreement was obtained between the BOM-generated IOPs and those obtained from Rayleigh and Mie scattering theory.*

^{TM}### 2.2. Least-squares Fitting

*fminsearch*to match the BOM-generated IOPs with those obtained from Rayleigh and Mie scattering theory. This direct search method [50

50. R. M. Lewis, V. Torczon, and M. W. Trosset, “Direct search methods: then and now,” J. Compt. Appl. Math. **124**, 191–207, (2000). [CrossRef]

51. J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J: Optimization **9**, 112–147 (1998). [CrossRef]

*n*real variables, each iteration in this algorithm includes the evaluation of the objective function at each of a set of

*n*+ 1 points or

*vertices*in

*n*dimensional space (

*simplex*) through reflection, expansion, and contraction [52]. The iteration starts with initial guess values of the variables and continues until specified stopping criteria are reached.

*μ*(

_{a,B}*λ*),

*μ*(

_{s,B}*λ*), and

*g*(

_{B}*λ*) and the corresponding quantities

*μ*

*̃*(

_{a}*λ*),

*(*μ ˜

_{s}*λ*), and

*g̃*(

*λ*) obtained from Rayleigh and Mie scattering theory, is given by

*f*(

*m,x*) is the sum of the

*least-squares*differences between the IOPs at ten different wavelengths. These ten wavelengths, which are unevenly distributed in the range from the near UV to the NIR, are used in measurements performed with a skin scanner designed and built by Balter Medical AS, Norway [53].

*m′*,

*m″*,

*r̄*,

*σ*, and

*K*, where

*m*=

*m′*+

*im″*is the complex refractive index of the particles,

*r̄*and

*σ*are the mean and the standard deviation of the particle size distribution, and

*K*is the strength of the Rayleigh scattering. In the simulations,

*(*μ ˜

_{a}*λ*) was sensitive to the imaginary part

*m″*of the refractive index, whereas

*(*μ ˜

_{s}*λ*) and the

*g̃*(

*λ*) were sensitive to the real part

*m′*of the refractive index as well as to

*r̄*and

*σ*. The parameters

*(*μ ˜

_{a}*λ*),

_{j}*(*μ ˜

_{s}*λ*), and

_{j}*g̃*(

*λ*) appearing in our objective function Eq. (23) were obtained from computer programs published by Bohren and Huffmann [31, appendix A], which were implemented in MATLAB functions by M

_{j}*ä*tzler [54

54. C. Mätzler, “MATLAB functions for Mie scattering and absorption,” Inst. Appl. Phys., University of Bern (2002), http://arrc.ou.edu/~rockee/NRA_2007_website/Mie-scattering-Matlab.pdf.

## 3. Results and Discussion

**80**, 450–455 (2004). [PubMed]

*λ*≤ 300 nm. Magnain et al. [56

56. C. Magnain, M. Elias, and J. Frigerio, “Skin color modelling using the radiative transfer equation solved by the auxiliary function method,” J. Opt. Soc. Am. A **24**, 2196–2203 (2007). [CrossRef]

*dz*as given in Table 1, and assumed spherical particles to be present in each layer with a complex refractive index value to be determined at each of ten different wavelengths in the VIS and NIR spectral ranges. In each layer, the particle size distribution and complex refractive index were determined at each of the ten wavelengths by fitting the modeled and calculated spectral values of the absorption coefficient

*μ*, the scattering coefficient

_{a}*μ*(Mie + Rayleigh), and the asymmetry factor

_{s}*g*. The Rayleigh scattering coefficient was assumed to be the same in each layer, given by

*μ*=

_{s,Ray}*K*(

*λ*)

*λ*

^{−4}, where the strength factor

*K*was allowed to vary with

*λ*. Because of the

*λ*

^{−4}dependence, Rayleigh scattering decreases rapidly as the wavelength increases.

### 3.1. Stratum Corneum or Upper epidermis

**80**, 450–455 (2004). [PubMed]

*upper epidermis*, which is assumed to represent the stratum corneum. It is the outermost layer of the skin that protects the underlying skin tissue from the external environment. The upper epidermis layer consists of nonviable or dead keratinocyte cells mixed with melanin dust. Thus, we considered most particles in the upper epidermis layer to consist of keratin and melanin, whose volume fractions and layer thickness used as input to the Balter BOM are given in the Table 1. In the upper epidermis, part of the scattering takes place at the air-tissue surface, whereas in the subsurface layers scattering is due only to particles that are embedded in the host medium. In this respect, the scattering process in the upper epidermis can be distinguished from that in an underlying layer.

*r̄*= 128 nm. The absorption coefficient

*μ*, the scattering coefficient

_{a}*μ*, and the imaginary part

_{s}*m″*of the refractive index were found to decrease with increasing

*λ*[cf. Figs. 1(d)–(f)], whereas the real part

*m′*of the refractive index was found to increase slightly [cf. Fig. 1 (c)]. Since the variation in the absolute value of

*m′*between

*λ*= 368 nm and

*λ*= 880 nm is fairly small (0.19%), it is considered to have the constant value of

*m′*≃ 1.56 in this layer. The melanosomes are assumed to become smaller due to fragmentation as they are transported upwards from the lower epidermis and to end up as melanin dust in the upper epidermis. The melanin dust both absorbs and scatters light, whereas keratin is assumed to act only as an absorber. We considered these particles to be mixed in a host or base medium with real part of the refractive index of

*m′*= 1.36. Thus, in the upper epidermis there are two types of particles, which scatter light in accordance with Mie and Rayleigh scattering theory. Rayleigh scattering, which contributes only about 6% to the total scattering at

*λ*= 368 nm, decreases with increasing

*λ*and contributes less than 1% at

*λ*= 880 nm. The asymmetry factor

*g*depends on the size distribution of the scattering particles. The value of

*g*obtained from Mie computations was found to agree well with that provided by the Balter BOM for wavelengths in the range from

*λ*= 368 nm to

*λ*= 632 nm. However at

*λ*= 880 nm, the value of

*g*was found to be 4% lower than that obtained from the Balter BOM.

### 3.2. Lower epidermis

**80**, 450–455 (2004). [PubMed]

*upper epidermis*. The remaining five sub-layers constitute the

*lower epidermis*which is assumed to have a thickness equal to four times that of the upper epidermis. The lower epidermis is a bloodless layer, in which the most common cells are keratinocytes, melanocytes, and langerhans. These cells are immersed into the base fluid, which is slightly more dense than water. As in the upper epidermis, keratin in the lower epidermis is considered to only absorb light, whereas melanin is considered both to absorb and scatter light. Similarly to scattering in the upper epidermis, scattering in the lower epidermis depends on the amount of melanin in the base medium, whereas the absorption is determined by the amounts of both keratin and melanin.

*r̄*is equal to 206 nm which is found to be 1.6 times larger than that in the upper epidermis. This result is consistent with the prediction that the size of melanosome particles decreases as they move upward from the lower epidermis to the upper epidermis [19

19. K. P. Nielsen, L. Zhao, J. J. Stamnes, K. Stamnes, and J. Moan, “Importance of the depth distribution of melanin in skin for DNA protection and other photobiological processes,” J. Photochem. Photobiol. **82**, 194–198 (2006). [CrossRef]

*m′*varies smoothly by 0.39% when the wavelength varies from

*λ*= 368 nm to

*λ*= 880 nm, which is twice as large as the corresponding variation in the upper epidermis. This trend is opposite to that found in published results [22

22. H. Ding, J. Q. Lu, W. A. Wooden, P. J. Kragel, and X.-H. Hu, “Refractive indices of human skin tissues at eight wavelengths and estimated dispersion relations between 300 and 1600 nm,” Phys. Med. Biol. **51**, 1479–1489, (2006). [CrossRef] [PubMed]

57. A. N. Bashkatov, E. A. Genina, and V. V. Tuchin, “Optical properties of skin, subcutaneous and mucous tissues: A review,” J. Innov. Opt. Health Sci. Appl. Phys. **4**, 9–38 (2011). [CrossRef]

*m′*in the lower epidermis is fairly small, we may regard it to have a constant value, i.e.,

*m′*≃ 1.52 for wavelengths in the range

*λ*= 368 – 880 nm.

*g*and the scattering coefficient

*μ*in the lower epidermis are found to be larger than those in the upper epidermis, while absorption and Rayleigh scattering coefficients are seen to be the same in both epidermis layers. The contribution from Rayleigh scattering to the total scattering is 4% at

_{s}*λ*= 368 nm, and decreases with increasing

*λ*to less than 1% at

*λ*= 880 nm.

### 3.3. Dermis

*μ*m with log-normal mean radius of

*r̄*= 449 nm. The particle size distribution, refractive-index variation with wavelength, and associated IOPs of the dermis, obtained from the optimization, are given in Fig. 3. In the VIS spectral region, the wavelength-dependent absorption in the dermis is due to

*de-oxygenated*(Hb) and

*oxygenated*(HbO

_{2}) hemoglobin. Fig. 3(f) shows the dermis absorption coefficient

*μ*, which has peak values at 403 and 540 nm. At these wavelengths, the imaginary part

_{a}*m″*of the refractive index also has peak values [cf. Figs 3 (e)–(f)]. But both

*m″*and

*μ*are seen to have very low values at wavelengths above 600 nm.

_{a}*μ*of the dermis layer decreases with increasing

_{s}*λ*. The contribution from Rayleigh scattering to the total scattering coefficient is less than 5% at

*λ*= 880 nm, but increases to 40% at

*λ*= 368 nm. Scattering by hemoglobin in the VIS and NIR spectral regions was discussed by Faber et al. [6

6. D. J. Faber, M. C. G. Aalders, E. G. Mik, B. A. Hooper, M. J. C. van Gemert, and T. G. van Leeuwen, “Oxygen saturation-dependent absorption and scattering of blood,” Phys. Rev. Lett. **93**, 1–4 (2004). [CrossRef]

_{2}particles at

*λ*> 600 nm, but that HbO

_{2}particles scatter slightly more than Hb particles at

*λ*< 400 nm. Besides scattering by hemoglobin, dermis scattering is due primarily to collagen particles, whose sizes vary from micrometer [45

45. A. N. Bashkatov, E. A. Genina, V. I. Kochubey, and V. V. Tuchin, “Estimation of wavelength dependence of refractive index of collagen fibers of scleral tissue,” Proc. SPIE **4162**, 265–267 (2000). [CrossRef]

### 3.4. Sub-cutis

*r̄*= 182 nm, which is larger than that in the upper epidermis, but smaller than that in the lower epidermis or in the dermis. The IOPs are shown in Fig. 4(b), (d), and (f). The contribution from Rayleigh scattering to the total scattering is less than 1% within the entire range of wavelengths. Thus, in the sub-cutis layer, Mie scattering by particles having size parameters in the range 1.3 ≤

*x*≤ 5.7 dominates. In this layer, the absorption coefficient

*μ*, the scattering coefficient

_{a}*μ*, and the asymmetry factor

_{s}*g*are found to decrease with increasing wavelength, and the scattering coefficient

*μ*is larger than in any other layer. The imaginary part

_{s}*m″*of the refractive index is seen to decrease with increasing wavelength, whereas the real part

*m′*stays fairly constant with an average value of 1.57 for wavelengths above 500 nm. At wavelengths below 500 nm,

*m″*[Fig. 4(e)], the absorption coefficient [Fig. 4(f)],

*m′*[Fig. 4(c)] and the scattering coefficient [Fig. 4(d)], all have relatively high values. Therefore, the sub cutis layer is opaque to incident light at wavelengths below 500 nm.

### 3.5. Discussion

*r̄*and the variance

*v*for each layer are given in Table 2, which shows that the mean particle diameter varies from 256 nm in the upper epidermis to 898 nm in the dermis. The larger particles, which are about 3.5

*μ*m in diameter, are most probably non-oxygenated and oxygenated hemoglobin particles and collagen fibers in the dermis layer. In the dermis, the cellular structures of small-scaled collagen fibers have sizes in the range from 60 to 100 nm [8

8. A. N. Bashkatov, E. A. Genina, V. I. Kochubey, and V. V. Tuchin, “Optical properties of human skin, subcutaneous and mucous tissues in the wavelength range from 400 to 2000 nm,” J. Phys. D: Appl. Phys. **38**, 2543–2555 (2005). [CrossRef]

*μ*m. Other particles contained in the dermis may originate from fat and water with sizes smaller than those of collagen and blood particles. The smallest particles with diameters of about 80 nm, which are found in the upper epidermis, may be melanosome dust produced by the fragmentation of melanin. The diameters of melanosome particles [3

3. B. L. Diffey, “A Mathematical model for ultraviolet optics in skin,” Phys. Med. Biol. **28**, 647–657 (1983). [CrossRef] [PubMed]

**80**, 450–455 (2004). [PubMed]

*m*=

*m′*+

*im″*in the four different skin layers are shown in Table 2 for different wavelengths, and these results are plotted in Figs. 1–4(c) for the real part

*m′*and in Figs. 1–4(e) for the imaginary part

*m″*. The highest value of

*m′*is found in the sub-cutis layer, and the lowest value in the dermis. The opposite is true for the imaginary part

*m″*of the refractive index,

*i.e. m″*has its lowest values in the sub-cutis and its highest values in the dermis. Therefore, the dermis is the most light-absorbing layer, whereas the sub-cutis is the most light-scattering layer. Both

*m′*and

*m″*are slightly higher in the upper epidermis layer than in the lower epidermis layer. The refractive-index values in the four skin layers are such that

*sc*,

*de*,

*le*, and

*ue*, stand for sub-cutis, dermis, lower epidermis, and upper epidermis, respectively. Note that the lower epidermis is four times thicker than the upper epidermis in the Balter BOM.

58. V. V. Tuchin, I. L. Maksinova, D. A. Zimnyakov, I. L. Kon, A. H. Mavlutov, and A. A. Mishin, “Light propagation in tissues with controlled optical properties,” J. Biomed. Opt. **2**, 401–417 (1997). [CrossRef]

*m′*= 1.474 at

*λ*= 589 nm. In our result,

*m′*= 1.488 at

*λ*= 579 nm for the dermis, where the scattering is most likely dominated by collagen particles. Ding et al. [22

**51**, 1479–1489, (2006). [CrossRef] [PubMed]

*m*for Caucasian abdomen skin and found its real part to be higher for the epidermis than for the dermis. The data published by Ding et al. [22

**51**, 1479–1489, (2006). [CrossRef] [PubMed]

*m′*values are lower than ours both for the epidermis and the dermis, and the difference is not the same at each wavelength. The complex refractive index for

*in vitro*porcine tissues, including that of dermis tissues, was recently reported by Lai et al. [59

59. J. C. Lai, Y. Y. Zhang, Z. Li, H. Jiang, and A. He, “Complex refractive index measurement of biological tissues by attenuated total ellipsometry,” Appl. Opt. **49**, 3235–3237 (2010). [CrossRef] [PubMed]

*m*= 1.3818 +

*i*0.0049 at 632.8 nm for the dermis. This result for the real part of the refractive index is comparable to that found by Ding et al. [22

**51**, 1479–1489, (2006). [CrossRef] [PubMed]

59. J. C. Lai, Y. Y. Zhang, Z. Li, H. Jiang, and A. He, “Complex refractive index measurement of biological tissues by attenuated total ellipsometry,” Appl. Opt. **49**, 3235–3237 (2010). [CrossRef] [PubMed]

**51**, 1479–1489, (2006). [CrossRef] [PubMed]

**51**, 1479–1489, (2006). [CrossRef] [PubMed]

45. A. N. Bashkatov, E. A. Genina, V. I. Kochubey, and V. V. Tuchin, “Estimation of wavelength dependence of refractive index of collagen fibers of scleral tissue,” Proc. SPIE **4162**, 265–267 (2000). [CrossRef]

58. V. V. Tuchin, I. L. Maksinova, D. A. Zimnyakov, I. L. Kon, A. H. Mavlutov, and A. A. Mishin, “Light propagation in tissues with controlled optical properties,” J. Biomed. Opt. **2**, 401–417 (1997). [CrossRef]

59. J. C. Lai, Y. Y. Zhang, Z. Li, H. Jiang, and A. He, “Complex refractive index measurement of biological tissues by attenuated total ellipsometry,” Appl. Opt. **49**, 3235–3237 (2010). [CrossRef] [PubMed]

*f*(

*m,x*) in Eq. (23). For the input parameters to the BOM in Table 1, the sum of the least square differences

*f*(

*m,x*) between the BOM-generated IOPs and corresponding IOPs obtained from Rayleigh and Mie scattering theory is found to be different for the four skin layers such that

*f*(

*m,x*) equals to 0.013 in the upper epidermis, 0.007 in the lower epidermis, 0.043 in the dermis, and 0.0074 in the sub-cutis. Refractive-index data accessible in the literature usually do not cover a wide range of wavelengths. Moreover, available refractive-index data are mostly for the real part only, whereas a complete description of light transport in a medium such as skin tissue, requires knowledge of the complex refractive index for a wide range of different wavelengths, such as shown in Table 2.

*m″*of the refractive index, while the real part

*m′*determines the scattering, which occurs due to the mismatch between the refractive index of the scattering particles and that of the host medium. Also, the mismatch of the refractive index at the air-skin interface causes a part of the incident light to be reflected. At normal incidence, the reflected light intensity will be ≃5% of the incident light intensity if

*m′*= 1.56 (see Table 2 for the upper epidermis). The portion of the light that penetrates into the skin tissue will be absorbed and scattered depending on the IOPs of the skin tissue. The variation in the IOPs between different body sites of a given individual is larger than the variation in the IOPs of a given body site between different individuals [7

7. J. Sandby-Møller, T. Paulsen, and H. C. Wulf, “Epidermal thickness at different body sites: relationship to age, gender, pigmentation, blood content, skin type and smoking,” Acta Derm. Venereol. **83**, 410–413 (2003). [CrossRef] [PubMed]

*μ*and

_{a}*μ*vary strongly with the amount of chromophores (due to skin pigments and blood content), modeled and measured IOPs can be matched by changing the values of these physiological parameters in the BOM that produces input to the radiative transfer simulations. Here it should be noted that the tabulated data used in the Balter BOM are measured, wavelength-dependent IOPs [4

_{s}**80**, 450–455 (2004). [PubMed]

49. S. Prahl, “Tabulated Molar Extinction Coefficient for Hemoglobin in Water,” http://omlc.ogi.edu/spectra/hemoglobin/takatani.html.

57. A. N. Bashkatov, E. A. Genina, and V. V. Tuchin, “Optical properties of skin, subcutaneous and mucous tissues: A review,” J. Innov. Opt. Health Sci. Appl. Phys. **4**, 9–38 (2011). [CrossRef]

*g*, which is the average of the cosine of the scattering angle, is found to be different for different skin layers. According to Gemert et al. [11

11. M. J. C. Van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, and W. M. Star, “Skin Optics,” IEEE Trans. Biomed. Eng. **36**, 1146–1154 (1989). [CrossRef] [PubMed]

*g*in the epidermis can be considered nearly identical to that in the dermis,

*i.e.*where

*λ*is in nm. The values of

*g*calculated from Eq. (24) varies from 0.73 to 0.87 when

*λ*varies from 368 to 880 nm. For the same range of wavelengths, the values of

*g*from our optimizations vary from 0.82 to 0.59 in the upper epidermis, from 0.9 to 0.83 in the lower epidermis, from 0.59 to 0.90 in the dermis, and from 0.88 to 0.59 in the sub-cutis. These

*g*values indicate that light scattering from skin layers always is peaked in the forward direction. According to Eq. (18), the size-averaged asymmetry factor

*g̃*depends on the strengths of both Rayleigh scattering and Mie scattering. As stated previously, the averaged

*g*for Rayleigh scattering is zero. Therefore, the size-averaged asymmetry factor will be reduced when the strength of Rayleigh scattering increases, as seen in panels (b) and (d) of Fig. 3 for the dermis layer.

## 4. Conclusions

*i.e.*absorption coefficients, scattering coefficients, and scattering phase functions, as input. Since there is an abrupt change in the refractive across the air-tissue interface, solutions to the radiative transfer equation must take this change into account [36, 55

55. K. Hestenes, K. P. Nielsen, L. Zhao, J. J. Stamnes, and K. Stamnes, “Monte Carlo and discrete-ordinate simulations of spectral radiances in a coupled air-tissue system,” Appl. Opt. **46**, 2333–2350 (2007). [CrossRef] [PubMed]

41. K. P. Nielsen, L. Zhao, G. A. Ryzhikov, M. S. Biryulina, E. R. Sommersten, J. J. Stamnes, K. Stamnes, and J. Moan, “Retrieval of the physiological state of human skin from UV-VIS reflectance spectra - A feasibility study,” J. Photochem. Photobiol. B **93**, 23–31 (2008). [CrossRef] [PubMed]

55. K. Hestenes, K. P. Nielsen, L. Zhao, J. J. Stamnes, and K. Stamnes, “Monte Carlo and discrete-ordinate simulations of spectral radiances in a coupled air-tissue system,” Appl. Opt. **46**, 2333–2350 (2007). [CrossRef] [PubMed]

43. D. L. Swanson, S. D. Laman, M. Biryulina, K. P. Nielsen, G. Ryzhikov, J. J. Stamnes, B. Hamre, L. Zhao, F. S. Castellana, and K. Stamnes, “Optical transfer diagnosis of pigmented lesions: a pilot study,” Skin Res. Technol. **15**, 330–337 (2009). doi: [CrossRef] [PubMed]

46. D. L. Swanson, S. D. Laman, M. Biryulina, K. P. Nielsen, G. Ryzhikov, J. J. Stamnes, B. Hamre, L. Zhao, E. Sommersten, F. S. Castellana, and K. Stamnes, “Optical transfer diagnosis of pigmented lesions,” Dermatol. Surg. **36**, 1–8 (2010). DOI: [CrossRef]

61. E. J. Dennis, G. J. Dolmans, R. K. Jain, and D. Fukumura, “Photodynamic therapy for cancer,” Nature Reviews Cancer **3**, 380–387 (2003). [CrossRef]

## Acknowledgments

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**OCIS Codes**

(000.1430) General : Biology and medicine

(170.3660) Medical optics and biotechnology : Light propagation in tissues

(290.5850) Scattering : Scattering, particles

**ToC Category:**

Scattering

**History**

Original Manuscript: April 28, 2011

Revised Manuscript: June 22, 2011

Manuscript Accepted: June 23, 2011

Published: July 14, 2011

**Virtual Issues**

Vol. 6, Iss. 8 *Virtual Journal for Biomedical Optics*

**Citation**

A. Bhandari, B. Hamre, Ø. Frette, K. Stamnes, and J. J. Stamnes, "Modeling optical properties of human skin using Mie theory for particles with different size distributions and refractive indices," Opt. Express **19**, 14549-14567 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-15-14549

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### References

- P. A. Payne, “Measurement of properties and function of skin,” Clin. Phys. Physiol. Meas. 12, 105–129 (1991). [CrossRef] [PubMed]
- M. Gillison, A History of the Body Tissues (Williams and Wilkins Co., Baltimore, Maryland, 1962).
- B. L. Diffey, “A Mathematical model for ultraviolet optics in skin,” Phys. Med. Biol. 28, 647–657 (1983). [CrossRef] [PubMed]
- K. P. Nielsen, L. Zhao, P. Juzenas, J. J. Stamnes, K. Stamnes, and J. Moan, “Reflectance spectra of pigmented and non-pigmented skin in the UV spectral region,” Photochem. Photobiol. 80, 450–455 (2004). [PubMed]
- A. R. Young, “Chromophores in Human Skin,” Phys. Med. Biol. 42, 789–802 (1997). [CrossRef] [PubMed]
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