16
2 Overview of Theoretical Approaches to the Analysis of Light Scattering
the scattering anisotropy factor is g ≈ 0.6 − 0.9, and in some cases, for example
blood, can reach values of 0.990–0.999 [17]. This substantially restricts applicability
of the diffusion approximation. Several papers were devoted to the study of the
accuracy of the diffusion approximation [18]. Based on the comparison of solutions
of the diffusion equation with the results of the Monte-Carlo simulation[19] (see
below) one concluded that the diffusion approximation may be a solution on some
orders from the truth. In optics the tissues simpler methods have been found for
solving the transport equation, such as the two-flux Kubelka-Munk model, three-,
four-and seven-flux models [12].
Two-and multiflux approximation. This theory is based on the model of the two
light beams propagating in the forward and backward directions. The main assumption of this theory is that the radiation intensity is diffuse. Inside the tissue diffuse flux
is divided into two parts: L 1 the flow in the direction of incident radiation and flux
scattered back L 2 . For the absorption and scattering of diffuse radiation we introduce
two Kubelka-Munk coefficients: A K M and S K M .
We have two differential equations
d L 1
dz
= −S K M L 1 − A K M L 1 + S K M L 2
d L 2
dz
= −S K M L 2 − A K M L 2 + S K M L 1 ,
where z is the average direction of the incident radiation.
Coefficients A K M and S K M values μ a and μ s are written as follows [12]: A K M =
2μ a , S K M = μ s .
The Kubelka-Munk theory is a special case of multiflux theory, where the transport
equation is transformed into a matrix differential equation which takes into account
the intensity of the radiation in the direction of many of the individual solid angles.
The two-flux theory is not applicable to describe the incident on a medium collimated beam. In this case, we use the four-flux theory. The Four-flux theory [12] takes
into account two counter diffuse flux as the Kubelka-Munk theory, as well as two
collimated laser beams, the external incident and reflected from the back surface of
the sample The Seven- flux theory is a three-dimensional representation of the incident laser beam and the scattering of radiation in a semi-infinite medium [20]. Note
that the Kubelka-Munk theory can only be applied to a one-dimensional geometry of
the system. The numerical approximation of the transport equation can be obtained
by the Monte-Carlo method.
Monte-Carlo method. The general scheme of the Monte-Carlo method is based on
the central limit theorem of probability theory. General properties of the Monte-Carlo
method:
• absolute convergence of the solution is of order
1
N
;
• independence of the error on the number of tests is of order approximate
1
√
N
• the main method reducing the error is the maximum variance reduction;
2 Overview of Theoretical Approaches to the Analysis of Light Scattering
the scattering anisotropy factor is g ≈ 0.6 − 0.9, and in some cases, for example
blood, can reach values of 0.990–0.999 [17]. This substantially restricts applicability
of the diffusion approximation. Several papers were devoted to the study of the
accuracy of the diffusion approximation [18]. Based on the comparison of solutions
of the diffusion equation with the results of the Monte-Carlo simulation[19] (see
below) one concluded that the diffusion approximation may be a solution on some
orders from the truth. In optics the tissues simpler methods have been found for
solving the transport equation, such as the two-flux Kubelka-Munk model, three-,
four-and seven-flux models [12].
Two-and multiflux approximation. This theory is based on the model of the two
light beams propagating in the forward and backward directions. The main assumption of this theory is that the radiation intensity is diffuse. Inside the tissue diffuse flux
is divided into two parts: L 1 the flow in the direction of incident radiation and flux
scattered back L 2 . For the absorption and scattering of diffuse radiation we introduce
two Kubelka-Munk coefficients: A K M and S K M .
We have two differential equations
d L 1
dz
= −S K M L 1 − A K M L 1 + S K M L 2
d L 2
dz
= −S K M L 2 − A K M L 2 + S K M L 1 ,
where z is the average direction of the incident radiation.
Coefficients A K M and S K M values μ a and μ s are written as follows [12]: A K M =
2μ a , S K M = μ s .
The Kubelka-Munk theory is a special case of multiflux theory, where the transport
equation is transformed into a matrix differential equation which takes into account
the intensity of the radiation in the direction of many of the individual solid angles.
The two-flux theory is not applicable to describe the incident on a medium collimated beam. In this case, we use the four-flux theory. The Four-flux theory [12] takes
into account two counter diffuse flux as the Kubelka-Munk theory, as well as two
collimated laser beams, the external incident and reflected from the back surface of
the sample The Seven- flux theory is a three-dimensional representation of the incident laser beam and the scattering of radiation in a semi-infinite medium [20]. Note
that the Kubelka-Munk theory can only be applied to a one-dimensional geometry of
the system. The numerical approximation of the transport equation can be obtained
by the Monte-Carlo method.
Monte-Carlo method. The general scheme of the Monte-Carlo method is based on
the central limit theorem of probability theory. General properties of the Monte-Carlo
method:
• absolute convergence of the solution is of order
1
N
;
• independence of the error on the number of tests is of order approximate
1
√
N
• the main method reducing the error is the maximum variance reduction;