I
DRI
TDFigure 2. Light interaction with turbid media, when illuminated by a collimated light beamI0. IF depicts the intensity of Fresnel reflection from the boundary of the media, ISS represents single scattered light beam, IDR is the diffuse re-flected and multiple scattered light beam. IT B andIT D shows transmission, either ballistic or diffuse intensity. The light, which is not reflected nor trans-mitted, is absorbed.
Although, the reflectance and transmission in turbid media can be easily measured, the determination of the intrinsic optical properties, absorption and scattering, is still hard in general cases, since scattering and absorption are intermixed in a complicated way.
Usually some kind of model is used in solving the optical properties of the medium based on reflectance and transmission measurements.
2.2 Optical modelling of turbid media
The absorption of a substance is usually expressed as an absorption coefficient, µa, which is the molar absorptivity, or extinction coefficient of the chromophore,ε [1/cm/-mol], multiplied by the concentration of the chromophore, c [mol]. The molar ex-tinction coefficient of a substance usually depends on the wavelength, λ. Therefore the absorption coefficient is (Ishimaru 1977; van Gemert, Jacques, Sterenborg & Star 1989):
(4) µa(λ) =cε(λ) [1/cm]
The inverse of the absorption coefficient is the mean free pathlength,pmf,a, of a photon between absorption events.
The scattering of light in a substance is often described using a scattering coefficient, µs(λ). The scattering coefficient depends on the density of the scattering particles,ρs [1/cm3], and the scattering cross section of the particles,σs[cm2]. Therefore, according to (Ishimaru 1977; van Gemertet al.1989):
(5) µs(λ) =ρsσs(λ) [1/cm]
The scattering coefficient is the inverse of the mean free pathlength, pmf,s, between the scattering events.
Scattering in tissue can be modelled using the Mie and Rayleigh scattering modes (Saidi, Jacques & Tittel 1995). Mie scattering occurs from large tissue structures, such as collagen fibers. Mie scattering in tissue is anisotropic, biased towards forward scat-tering. The scattering events from particles smaller than wavelengths, such as from various small skin organelles, can be modelled as Rayleigh scattering, which leads to scattering oriented almost equally to all directions (Saidi et al. 1995). This kind of scattering is called isotropic. The scattering angle is stochastic, but the tendency for forward or backward scattering is an expectation value of the cosine of the phase func-tion, normalized (Jacques, Alter & Prahl 1987; Gandjbakhche 2001):
(6) g=hcosθi=2π
Z π
0
p(θ)cos(θ)sin(θ)dθ
Many estimates of the tissue phase function are used. The most common phase func-tions are probably the delta–Eddington (Joseph, Wiscombe & Weinman 1976) and Henyey–Greenstein phase functions (Jacques et al. 1987; Gandjbakhche 2001). The Henyey–Greenstein phase function was first used in modelling interstellar scattering, but it is shown to be suitable also in describing scattering in skin and other biological tissues. The Henyey–Greenstein phase function is the following:
(7) p(θ) = 1
4π
1−g2
(1+g2−2g cosθ)3/2
The scattering of a photon is depicted in Figure 3. The Henyey–Greenstein phase function and the scattering angle are also shown.
Photon after
scattering
Photon
Figure 3. The scattering of the photon from the scatterer. The scattering angle, θ is the angle between the original direction of the photon and the direction after the scattering event. The larger ellipse depicts the probability density of the scattering angle, calculated using the Henyey–Greenstein phase function, when the anisotropy,g=0.8, and the smaller ellipse, wheng=0.7.
If the anisotropy factorg=0, the scattering is fully isotropic. In pure forward scattering media,g=1 andg=−1 in the case of pure backward scatter. The average anisotropy of the skin is often: g∈[0.7,0.95](Tuchin 2007; Gandjbakhche 2001).
When a collimated beam of light enters into a strongly scattering substance, it will become isotropic when undergoing enough scattering events, even if the substance is heavily forward scattering. Therefore, it is common to replace the scattering coefficient and the anisotropy with reduced scattering coefficient:
(8) µ0s=µs(1−g) [1/cm]
The reduced scattering coefficient describes the medium in which the scattering is purely isotropic, but weaker than in the original medium. If the path of the light beam in the medium is long enough to make the light isotropic, the net result of medium having scatteringµsand anisotropygis approximately the same as the medium withµ0s andg=0. Therefore,µ0s is usually used instead of andµs andg.
By combining the effects of scattering and absorption, a total interaction coefficient,µt, is obtained.
(9) µt=µa+µs [1/cm]
2.2.1 Radiative transport equation
The propagation of electromagnetic radiation is often described by Maxwell equations.
However, the skin may be too complicated a medium for Maxwell equations, due to the inhomogeneity and complex micro structures. For more than twenty years, the light transport equation (RTE) has been more popular in tissue optics than the Maxwell equations. The RTE model assumes that the light follows purely the particle model, there is no interaction between photons, nor interference. The motivation in RTE mod-elling is to predict the energy transport in turbid media. For this purpose, the RTE models the time and space change of radiance, the flux density, in the tissue. Radiance, L[W/m2/sr], is a radiometric measure which indicates how much radiation originat-ing from a particular area is transmitted to the given solid angle. The light transport in tissue can be modelled by examining how the radiance changes when it passes through an infinitely small volume. The RTE is shown in Equation (10) (Chandrasekhar 1960;
Ostermeyer 1999; Thompson 2004):
(10)
1 v
∂
∂tL(~r,~s,t) =−~s·∇L(~r,~s,t)
| {z }
net flow
−µtL(~r,~s,t)
| {z }
losses
+
µs Z
4π
L(~r,~s,t)p(~s,~s0)dω0
| {z }
scattering gain
+Q(~r,~s,t)
| {z }
sources
,
where L is the radiance of light at location~r moving towards~s. In volume element in location,~r, the radiation is scattered to a new direction,~s0. The scattering angle is determined by the scattering phase function,p(~s,~s0).
The left hand side of RTE is the time derivative of radianceL, divided by the speed of light in the medium, v, to represent the change of radiance per distance travelled.
This change is caused by the four additive terms on the right hand side. The first of these terms is the net flow through the volume element. If the spatial derivative of radiance parallel to~s is nonzero, the net flow through the volume element equals the negative derivative along~s. The second term accounts for scattering and absorption losses within the volume. All absorption and scattering is counted. The third term describes the increase of radiance, due to the scattering into the direction of~s. The last term represents the net source within the volume element, dV.
In steady state condition, without sources, the RTE can be written as (Cheong, Prahl &
Welch 1990; Ostermeyer 1999):
(11) ~s·∇L(~r,~s) =−µtL(~r,~s) +µs Z
4π
L(~r,~s0)p(~s,~s0)dω0,
Even Equation (11) is too complicated to be used in the practical modelling of light transport in skin. Therefore, the RTE is usually approximated in order to obtain a more tangible model. The most common analytical approximations of RTE include Beer–Lambert–Bouguer law, Adding-Doubling method (Prahl, Van Gemert & Welch 1993), Kubelka-Munk theory (Kubelka & Munk 1931), and diffusion approximation (Ishimaru 1977). In addition to analytical approximations, the numerical approxima-tion by stochastic Monte Carlo simulaapproxima-tion (Prahl, Keijzer, Jacques & Welch 1989) is also common.