146
8 Theoretical Determination the Function of Size Distribution for Blood Cells
main indicators of the functional and structural state of the cell [5]. Normally, a
cell’s volume regulation is performed by a number of interrelated physiological and
biochemical processes [5, 6]. It is known that in some pathological conditions this
regulation is disturbed [7–9]. In this chapter, a mathematical model has been developed to determine the distribution of blood cells by size. Normal human erythrocytes
were chosen as the experimental model system for determining average sizes. From
a mathematical point of view, the problem of reconstructing the globule distribution
function in size and shape reduces to solving Fredholm integral equations of the first
order. Note that in this approach we use the Tikhonov regularization method [10] and
the use of a priori information on smoothness, non-negativity, and finiteness of the
solution of the inverse problem. Most of the work on mathematical modeling in this
area is devoted to solving an integral equation that relates an unknown distribution
and a diffraction pattern. The kernel of this equation is a function describing the
diffraction pattern corresponding to a single particle of a given shape. As the shape
of a single particle, we will use a sphere or a cylinder [11]. For medical applications
related to the operational diagnosis of erythrocytes, it is important that rapid analysis
can be performed in the shortest possible time. Therefore, it is relevant to search for
fairly simple but at the same time informative models in combination with analytical
estimates of the basic parameters of unknown distributions that will allow us to apply
new mathematical approaches to modeling without involving resource-intensive calculations.
In this chapter, we analyze geometrical characteristics of particles simulating
erythrocytes in the upper layer of the dermis.
The problem consists of several steps. At the first stage, it is necessary to find the
coefficient of reflection of a plane wave from a smoothly irregular layer simulating a
given biological structure which consist of two continuous layers and the third layer
containing inhomogeneous inclusions simulating blood cells with different refractive
indices.
At the second stage, it is necessary to solve the problem of reflection of a with
an arbitrary cross section for the above conditions (see Chap. 4). The construction of
these parts is auxiliary.
At the third stage, we solve the problem of detection the of form for blood cells.
8.2 Reflection of a Plane Wave from a Layer
with a Slowly Varying Thickness
For detection the the function of size distribution of form of blood, it is necessary to
find the reflected field in the layer consisting irregularly shaped particles of various
sizes and coefficients refraction.
It will be determined as follows: E blood = E re f − E skin , where E re f is reflected
field from all the simulated optical system E skin is reflected field of layers: the
epidermis, the upper layer of the dermis.
8 Theoretical Determination the Function of Size Distribution for Blood Cells
main indicators of the functional and structural state of the cell [5]. Normally, a
cell’s volume regulation is performed by a number of interrelated physiological and
biochemical processes [5, 6]. It is known that in some pathological conditions this
regulation is disturbed [7–9]. In this chapter, a mathematical model has been developed to determine the distribution of blood cells by size. Normal human erythrocytes
were chosen as the experimental model system for determining average sizes. From
a mathematical point of view, the problem of reconstructing the globule distribution
function in size and shape reduces to solving Fredholm integral equations of the first
order. Note that in this approach we use the Tikhonov regularization method [10] and
the use of a priori information on smoothness, non-negativity, and finiteness of the
solution of the inverse problem. Most of the work on mathematical modeling in this
area is devoted to solving an integral equation that relates an unknown distribution
and a diffraction pattern. The kernel of this equation is a function describing the
diffraction pattern corresponding to a single particle of a given shape. As the shape
of a single particle, we will use a sphere or a cylinder [11]. For medical applications
related to the operational diagnosis of erythrocytes, it is important that rapid analysis
can be performed in the shortest possible time. Therefore, it is relevant to search for
fairly simple but at the same time informative models in combination with analytical
estimates of the basic parameters of unknown distributions that will allow us to apply
new mathematical approaches to modeling without involving resource-intensive calculations.
In this chapter, we analyze geometrical characteristics of particles simulating
erythrocytes in the upper layer of the dermis.
The problem consists of several steps. At the first stage, it is necessary to find the
coefficient of reflection of a plane wave from a smoothly irregular layer simulating a
given biological structure which consist of two continuous layers and the third layer
containing inhomogeneous inclusions simulating blood cells with different refractive
indices.
At the second stage, it is necessary to solve the problem of reflection of a with
an arbitrary cross section for the above conditions (see Chap. 4). The construction of
these parts is auxiliary.
At the third stage, we solve the problem of detection the of form for blood cells.
8.2 Reflection of a Plane Wave from a Layer
with a Slowly Varying Thickness
For detection the the function of size distribution of form of blood, it is necessary to
find the reflected field in the layer consisting irregularly shaped particles of various
sizes and coefficients refraction.
It will be determined as follows: E blood = E re f − E skin , where E re f is reflected
field from all the simulated optical system E skin is reflected field of layers: the
epidermis, the upper layer of the dermis.