150
8 Theoretical Determination the Function of Size Distribution for Blood Cells
The Tikhonov condition [12, 13] follows from condition (8.12):
(A
∗
h A h + αC)u
α
= A
∗
h f
Here, A h is the operator from W
1
2 [a, b] L 2 [c, d], A
∗
h is the conjugate operator
with respect to A h ,A
∗
h is the operator from L 2 [c, d] W
1
2 [a, b], and C is the operator
the matrix of which is determined in [12, 13].
In the above formulation, we consider operator A h of the original integral equation
that acts from L 2 [a, b] to L 2 [c, d] (i.e., the information regarding the smoothness of
the exact solution is missing). Then, the smoothing functional is written as
M
α
[u] = ||A h u − f δ ||
2
L 2
+ α||u||
2
L 2
→ min
and the Tikhonov equation is represented as
(A
∗
h A h + α E)u
α
= A
∗
h f,
where E is the unity operator.
Note that function u
α that minimizes functional (8.11) or (8.12) depends on regularization parameter α. To determine the regularization parameter, we employ the
method of relative residual
||Au
α
− f ||
f
= δ
(8.13)
Thus, expression (8.13) makes it possible to automatically determine the regularization parameter.
8.4 Numerical Calculations for a Model Medium
and Conclusions
Let us consider a model medium with the following characteristics. Typical layer
thicknesses are equal to d 2 = 65 · 10
−6 , d 3 = 565 · 10
−6 , d 4 = 90 · 10
−6 , n
◦
1 = 1,
χ 1 = 0, χ 2 = χ 3 = χ 4 = χ 5 = 10
−5 , refractive indices of the layers are n
◦
2 = 1.50,
n
◦
3 = 1.40, n
◦
4 = 1.35, n
◦
5 = 1.40 and the following values of parameters are a 1 =
−0.0024, b 1 = 0.020, a 2 = 0.021, b 2 = 0.030, a 3 = 0.041, b 3 = 0.051, c 1 = c 2 =
c 3 = 10
−2 . The values of parameters for the interfaces between the layers are chosen
so that the shape of the surface is maximally close to the shape of the boundary of the
corresponding layer in the structure of the normal human dermis, and wavelength is
λ = 0.63 µm (center of the line of a He−Ne laser).
Since the erythrocyte contains no cell organelles, its cellular membrane is very thin
and does not noticeably affect the scattering of light; consequently, the erythrocyte
can be treated as a homogeneous scatterer. Thus, our computations were performed
for monolayer spherulated particles simulating erythrocytes; the number of particles
8 Theoretical Determination the Function of Size Distribution for Blood Cells
The Tikhonov condition [12, 13] follows from condition (8.12):
(A
∗
h A h + αC)u
α
= A
∗
h f
Here, A h is the operator from W
1
2 [a, b] L 2 [c, d], A
∗
h is the conjugate operator
with respect to A h ,A
∗
h is the operator from L 2 [c, d] W
1
2 [a, b], and C is the operator
the matrix of which is determined in [12, 13].
In the above formulation, we consider operator A h of the original integral equation
that acts from L 2 [a, b] to L 2 [c, d] (i.e., the information regarding the smoothness of
the exact solution is missing). Then, the smoothing functional is written as
M
α
[u] = ||A h u − f δ ||
2
L 2
+ α||u||
2
L 2
→ min
and the Tikhonov equation is represented as
(A
∗
h A h + α E)u
α
= A
∗
h f,
where E is the unity operator.
Note that function u
α that minimizes functional (8.11) or (8.12) depends on regularization parameter α. To determine the regularization parameter, we employ the
method of relative residual
||Au
α
− f ||
f
= δ
(8.13)
Thus, expression (8.13) makes it possible to automatically determine the regularization parameter.
8.4 Numerical Calculations for a Model Medium
and Conclusions
Let us consider a model medium with the following characteristics. Typical layer
thicknesses are equal to d 2 = 65 · 10
−6 , d 3 = 565 · 10
−6 , d 4 = 90 · 10
−6 , n
◦
1 = 1,
χ 1 = 0, χ 2 = χ 3 = χ 4 = χ 5 = 10
−5 , refractive indices of the layers are n
◦
2 = 1.50,
n
◦
3 = 1.40, n
◦
4 = 1.35, n
◦
5 = 1.40 and the following values of parameters are a 1 =
−0.0024, b 1 = 0.020, a 2 = 0.021, b 2 = 0.030, a 3 = 0.041, b 3 = 0.051, c 1 = c 2 =
c 3 = 10
−2 . The values of parameters for the interfaces between the layers are chosen
so that the shape of the surface is maximally close to the shape of the boundary of the
corresponding layer in the structure of the normal human dermis, and wavelength is
λ = 0.63 µm (center of the line of a He−Ne laser).
Since the erythrocyte contains no cell organelles, its cellular membrane is very thin
and does not noticeably affect the scattering of light; consequently, the erythrocyte
can be treated as a homogeneous scatterer. Thus, our computations were performed
for monolayer spherulated particles simulating erythrocytes; the number of particles