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MICROSCOPY RESEARCH AND TECHNIQUE 42:451–458 (1998)
Computer Simulation as a Tool in Evaluating Intracellular
Spatial Arrangement of an Organelle Using
Random Section Data
ALEXANDER G. NIKONENKO*
Laboratory of Quantitative Morphology, Cascade Medical Ltd., 252042, Kiev, Ukraine
KEY WORDS
mathematical model; microscopy; microtomy; stereology
ABSTRACT
A random profile of a cell gives a highly biased presentation of the location of
various organelles within an intracellular space. The technique combining mathematical modeling
and computer simulation presented here is aimed to overcome this bias and interpret intracellular
spatial arrangement of an organelle using two-dimensional (2-D) observations from random
sections. It allows to simulate random sectioning of a cell whose shape approximates to an ellipsoid
of rotation, and to obtain the coordinates for the center of an organelle profile located within a cell
profile. The pilot study was performed to investigate the influence of different three-dimensional
(3-D) scattering patterns of an organelle on the coordinates of an organelle profile’s center. Computer
tests were carried out on a personal computer using the original software written in Pascal. It was
ascertained that statistical properties of a sample of organelle profile’s center coordinates allow for a
quantative estimation of some 3-D features, including the position of an organelle with respect to a
cell center and specific characteristics of an organelle position (e.g., its fixity or randomness). J.
Microsc. Res. Tech. 42:451–458, 1998. r 1998 Wiley-Liss, Inc.
INTRODUCTION
In a living cell the intracellular position of some
organelles (e.g., mitochondria or lysosomes) is changing. There is some evidence that these organelles
interact with a cytoskeleton and move within a cell in a
regulated fashion (Burkhardt, 1996; Letourneau and
Wire, 1995). Preliminary reasoning prompts that various cell function alterations associated with the reorganization of a cytoskeleton may, at the same time,
influence the spatial arrangement of an organelle. So,
this arrangement can be a marker for some kinds of cell
activity and/or pathology. The analysis of 3-D arrangement of an organelle can be accomplished by reconstructing a cell from serial sections or by visualizing intracellular space by confocal laser microscopy (Rigaut et al.,
1992; Verbeek et al., 1995). In random sections, spatial
pattern of an organelle is usually assessed rather
qualitatively, whereas the quantitative investigations
are scanty. The main reason for such a tendency lies in
the method of cell sectioning, which puts a bias on
two-dimensional (2-D) observations of an organelle
position and often makes their adequate interpretation
difficult (Cruz-Orive, 1976). In order to overcome these
obstacles and interpret such 2-D observations adequately, the stereology-based computer simulation
technique is proposed. In a pilot study the validity of
this technique has been tested.
MATERIALS AND METHODS
Cell Sectioning Model
The previously described Cell Sectioning Model (Nikonenko, 1996) has been modified to simulate the
coordinates for the center of an organelle profile located
within the profile of a randomly sectioned cell. The
model is based on the following assumptions: (1) Cell
r 1998 WILEY-LISS, INC.
and organelle shapes are limited by an ellipsoid of
rotation. In this study the organelle approximates to a
sphere. (2) The organelle position inside a cell may vary.
(3) The cell is randomly sectioned. (4) Cell sections have
no thickness.
The location of a cell profile made by the cutting
plane ⍀ , is governed by length and spatial orientation
of a line segment, OM, which is perpendicular to the
sectioning plane (Fig.1). The spatial orientation of the
line segment OM is determined by the angles ␸ and ␪
enclosed by this segment and the x- and z-axis, respectively. It is defined that: 0 ⱕ ␸ ⱕ 2␲ and 0 ⱕ ␪ ⱕ ␲. The
length of the segment OM (d) takes values between 0
and an upper limit, which depends on the angle ␪ value,
so that
dmax ⫽ 冑c2 · cos2 ␪ ⫹ a2 · sin2 ␪
(1)
Any section of the rotational ellipsoid representing a
cell, cut by the plane ⍀, is an ellipse. When ⍀ intersects
with an organelle, the coordinates of the organelle
profile’s center, Q, in ⍀ (Fig. 2A) will be given by:
uQ ⫽ x · cos ␸ · cos ␪ ⫹ y · sin ␸ · cos ␪ ⫺ z
· sin ␪ ⫹ d ·
sin ␪ · cos ␪ · (c2 ⫺ a2)
c2 · cos2 ␪ ⫹ a2 · sin2 ␪
(2)
vQ ⫽ y · cos ␸ ⫺ x · sin ␸
*Correspondence to: Dr. A.G. Nikonenko, Laboratory of Quantitative Morphology, Cascade Medical Ltd., blvd. Druzhby Narodov 19, 252001, P.O.Box B-51,
Kiev, Ukraine. E-mail: agn@serv.biph.kiev.ua
Received 3 January 1998; accepted in revised form 4 June 1998
452
A.G. NIKONENKO
Fig. 1.
Schematic representation of cell and organelle sectioning.
where c and a denote the ellipsoid’s semiaxes (c semiaxis of rotation); x, y, z - are running coordinates in
the Cartesian system indicated in Figure 1.
In this study, relative values of the coordinates of the
organelle profile’s center in ⍀ were used. They were
obtained by normalizing uQ and vQ values with respect
to the length of the corresponding semiaxis. Hence,
␰⫽
uQ
U
⫽
a·
冑
uQ
,
12
d2 1
1⫺
·
c
H
and
(3)
H
␩⫽
vQ
V
⫽
a·
冑
vQ
1⫺
12
d2 1
·
c
H
,
in which U, V are the ellipse’s semiaxes and H equals
cos2 ␪ ⫹ (a/c)2 · sin2 ␪. The values of ␨ - and ␩ range from
0 to 1.
In order to design a random sectioning scheme, the
value of d was generated at random within the interval:
0 ⱕ d ⱕ dmax. Isotropic vectorial directions of the
segment OM in space were simulated according to
Mattfeldt and Mall (1984) by insertion of independent
random numbers (x1, x2) uniformly distributed within
the interval [0,1] into the:
␸ ⫽ 2␲ · x1
(4)
␪ ⫽ arccos (1 ⫺ 2x2)
(5)
The mathematical details for some procedures mentioned above can be found in the Appendix.
453
SPATIAL ARRANGEMENT OF AN ORGANELLE
selected randomly. In the latter case, the coordinates
were statistically dependent, maintaining a minimum
distance between the cell and the organelle surface
(⌬R) of 0.1 unit. In the tests of the first series, the real
values for ␨ - and ␩ -coordinates were generated. The
tests of the second and third series were carried out by
generating the absolute values for both coordinates of
an organelle profile’s center. The tests of the first and
second series were performed with fixed sample size
(N ⫽ 1,000) and applied all three schemes for generation of organelle center’s coordinates (1–2). The tests of
the third series were carried out applying two schemes
for generation of the coordinates (2–3) with sample size
changing from 10 to 300 in step of 10. A random
sectioning scheme was used in all cases. The statistical
analysis of the samples included evaluation of mean
(M), variance (V), skewness (Sk), and kurtosis (K).
Kurtosis values were calculated using the formula:
n
兺 (x ⫺ x)
i
K⫽
i⫽1
n · ␴4
4
⫺ 3.
The Kolmogorov-Smirnov two-sample test was used to
examine the statistical significance of differences between samples of organelle profile’s center coordinates.
Fig. 2. Schematic representation of a cell profile showing an
organelle profile (A) and two scatter plots for the coordinates of an
organelle profile’s center. Organelle center’s coordinates are fixed
outside the cell center (B) and random (C).
Simulations
The original software to simulate the coordinates of
an organelle profile’s center ( ␨ - and ␩-coordinates) was
written in Turbo Pascal 6.0 (Borland International Inc.,
Scotts Valley, CA). The simulation study was performed
on the IBM AT486 DX4 computer (Hyundai, South
Korea) running the MS-DOS operating system.
Three series of tests were carried out with different
schemes for generation of organelle center’s coordinates (x, y, z). The cell ellipsoid was given a long
semiaxis, c, of 10 units, and a short semiaxis, a, of 5
units, and the organelle sphere was given a radius, R,
of 1 unit. The schemes for generation of organelle
center’s coordinates included the following variations:
(1) the coordinates were fixed at the cell center
(x⫽y⫽z⫽0) ; (2) the coordinates were fixed outside the
cell center (x⫽y⫽2; z⫽6), and (3) the coordinates were
RESULTS
The data indicate that when the organelle center’s
coordinates were fixed at the cell center, the ␩ -coordinate was equal to zero, and the ␨ -coordinate value
fluctuated within the range: -0.09974 ⱕ ␨ ⱕ ⫹0.09906.
The histogram’s mode took the central position corresponding to the center of a cell section (Fig. 3A), and the
sample was characterized by very small values of
variance and skewness: 0.002 and ⫹0.010, respectively.
The negative kurtosis was equal to - 0.591.
The analysis of the case when the organelle center’s
coordinates were fixed outside the cell center has shown
that the ␨ -coordinate value varied within significantly
wider limits than in the previous case: -0.92375 ⱕ ␨ ⱕ
⫹0.80928. The histogram shifted to smaller values (Fig.
3B). The sample’s variance and skewness were equal to
0.140 and ⫹1.126, correspondingly, being significantly
larger than in the previous case. The positive kurtosis
equaled ⫹ 0.572. The ␩ -coordinate value changed
within the following limits: -0.77446 ⱕ ␩ ⱕ ⫹0.78663.
The histogram of the sample had two modes that were
symmetrical with respect to the histogram’s center
(Fig. 3C). The variance of the ␩-coordinate sample,
being 0.228, exceeded that of the ␨ -coordinate sample.
The skewness value was near zero (- 0.023), but negative kurtosis was equal to -1.495, being the maximum
among all cases considered. The scatter plot of the
coordinates of an organelle profile’s center for the case
when the organelle center’s coordinates were fixed
outside the cell center is shown in Figure 2B.
When random organelle center’s coordinates were
applied, the ␨ -coordinate value varied within the range:
-0.87088 ⱕ ␨ ⱕ ⫹0.91140. The sample’s histogram had
no explicit mode (Fig. 3D). The ␨ -coordinate sample’s
variance was equal to 0.171. The positive skewness
equaled ⫹ 0.088, and kurtosis was equal to -0.927. The
␩-coordinate value changed within the following limits:
-0.80541 ⱕ ␩ ⱕ ⫹0.78544. The histogram and statisti-
454
A.G. NIKONENKO
Fig. 3. Histograms illustrating that changes in an organelle position inside a cell influence
characteristics of the ␨ - (A,B,D) and ␩- (C,E) coordinate samples. Organelle center’s coordinates are fixed
at the cell center (A), fixed outside the cell center (B,C) and random (D,E).
cal parameters of the sample were similar to those of
the ␨ -coordinate sample (Fig. 3E). The variance was
equal to 0.195, the skewness being ⫹ 0.088, and the
kurtosis being -1.212. The scatter plot of the coordinates of an organelle profile’s center for the case with
random organelle center’s coordinates is shown in
Figure 2C.
There is no possibility to sign a coordinate within the
real cell sections. While generating the absolute values
for the coordinates of an organelle profile’s center
instead of real values, and thus simulating the situation similar to that in the real sections, it was ascertained that when the organelle center’s coordinates
were fixed at the cell center, the absolute value for the
␩-coordinate (abs( ␩)-coordinate) was equal to zero. The
abs( ␨) -coordinate value varied within the following
limits: 0.00002 ⱕ ␨ ⱕ 0.09972. The mean of the abs( ␨)-coordinate sample equaled 0.036, the variance
being 0.001. The histogram’s mode took the extremely
left position (Fig. 4A), and the sample was characterized with the moderate values of positive skewness
⫹0.463 and negative kurtosis: - 0.914.
When the organelle center’s coordinates were fixed
outside the cell center, the abs( ␨) -coordinate value
varied within the range: 0.00173 ⱕ ␨ ⱕ 0.92073. The
mean and variance of the abs( ␨)-coordinate sample
were significantly larger than in the previous case,
being 0.507 and 0.053, respectively. The sample’s histogram shifted to the larger values (Fig. 4B). The skewness equaled -0.485, the kurtosis being -0.787. The
abs-( ␩) -coordinate value varied within the limits:
-0.00008 ⱕ ␨ ⱕ ⫹0.79150. The mean and variance of the
abs( ␩) -coordinate sample equaled 0.433 and 0.041,
correspondingly. The sample was characterized by negative skewness (-0.480) and kurtosis (-0.881) (Fig. 4C).
When random organelle center’s coordinates were
applied, the abs( ␨) -coordinate value varied within the
following range: 0.00299 ⱕ ␨ ⱕ ⫹0.92169. The mean
and variance of the abs( ␨)-coordinate sample were
equal to 0.352 and 0.049, respectively. There was no
explicit mode in the sample’s histogram (Fig. 4D). The
skewness was positive, being ⫹0.284, and the negative
kurtosis was equal to -0.990. The abs( ␩)-coordinate
value varied within the limits: 0.00001 ⱕ ␨ ⱕ ⫹0.79094.
SPATIAL ARRANGEMENT OF AN ORGANELLE
455
Fig. 4. Histograms illustrating that changes in an organelle position inside a cell influence
characteristics of the abs( ␨ )- (A,B,D) and abs( ␩)- (C,E) coordinate samples. Organelle center’s
coordinates are fixed at the cell center (A), fixed outside the cell center (B,C), and random (D,E).
The mean and variance of the abs( ␩)-coordinate sample were equal to 0.381 and 0.047, respectively. The
sample’s histogram was very similar to that of the abs( ␨)-coordinate sample (Fig. 4E), its skewness being
⫹0.017, and kurtosis being -1.139.
The analysis of results obtained in the third series of
tests has shown that statistical characteristics of samples of organelle profile’s center coordinates allow for
distinguishing between the cases with random organelle center’s coordinates and those fixed outside the cell
center. To tell the cases with abs( ␨)-coordinate from one
another, the sample size needed to be not less than 40 for P ⬍ 0.01 level of statistical significance, and not less
than 70 - for P ⬍ 0.001 level. In contrast, the differences
between the abs( ␩)-coordinate samples became statistically significant only when N reached 110 (P ⬍ 0.05),
acquiring the higher levels of statistical significance
when N attained 200 (P ⬍ 0.01) and 290 (P ⬍ 0.001).
DISCUSSION
To evaluate in cell sections the topographical relationships between various subcellular structures and to
characterize the space accessible to them, some researchers applied cluster analysis tools. It was ascertained, for example, that in hepatoma cells lysosomes
gain access only to a limited (less than 10%) portion of
the cell volume (Baudhuin et al., 1979). The analysis of
associations calculated for pairs of cell structures and
the pattern of multiple objects’ associations obtained by
a clustering technique is assumed to be a useful description of spatial relationships between various subcellular components (Diaz et al., 1987).
To evaluate spatial pattern for an organelle, Hemon
et al. (1981) applied a method that uses the distance
between an organelle profile and a point of reference.
This method is not restricted with a shape and size
heterogeneity of the objects analyzed, but it allows for
determining whether the organelle’s location is random.
Combining computer technologies and stereological
methods seems today to be a very natural way. Several
software systems have been developed, which increase
the efficiency of stereological estimation (Abrams et al.,
1994; Black and Rosen, 1996; Pentcheff and Bolender,
456
A.G. NIKONENKO
1992). Reverter et al. (1993) used computer simulation
to test the validity of several stereological techniques
for estimating size distribution of bone marrow adipocytes. Computer modeling was used to simulate the
organization of cells in a tissue, as well as processes
that may occur in a variety of pathological conditions
(Clem et al., 1997; Rashbass et al., 1996).
The particular application of the Cell Sectioning
Model described in this paper is aimed at adequate
interpretation of 2-D observations of 3-D spatial arrangement of an organelle. Within the model, an organelle intracellular position (i.e., point P) is expressed via
the organelle center’s coordinates (x, y, and z) in a
three-coordinate system, the axes of which coincide
with cell ellipsoid axes. The position of an organelle
profile within a cell profile (i.e., point Q) is expressed via
the normalized coordinates of an organelle profile’s
center ( ␨, ␩) in a two-coordinate system, the axes of
which coincide with those of the cell profile ellipse.
When the position of a simulated organelle satisfies the
sectioning criterion, the old coordinates (x, y, and z) are
being transformed into the new ones ( ␨, ␩). This paper
deals with the general description of the method and
doesn’t consider all its possible applications. In order to
demonstrate this method, we examine the simplest
case when the organelle is given the shape of a sphere.
It simplifies the algorithm, because the projection of the
sphere’s center always coincides with the center of any
given profile in the same plane. However, the model can
be easily applied to the case when the shape of an
organelle is approximated with an ellipsoid of rotation.
In the real cell sections one cannot sign a coordinate. To
simulate a situation similar to the real one, the series of
tests was carried out generating absolute values for
normalized coordinates of an organelle profile’s center.
The data suggest that at least some features concerning the spatial pattern for an organelle can be estimated using 2-D measurements made in random sections. It is worth noting that the criteria for such an
estimation do not involve individual observations, but
only statistical properties of 2-D measurement samples.
The ␨ - and ␩-coordinates are the 2-D parameters of
interest. The absolute values of these coordinates can
be evaluated in real cell sections as a displacement of
an organelle profile’s center from a cell profile center in
the direction of the corresponding axis of a profile
ellipse.
In this pilot study, it was found that if organelle
center’s coordinates are fixed at the cell center, it leads
to a zero value for the ␩-coordinate, and to grouping of
the ␨ -coordinate values around zero. The latter feature
can be easily revealed in both the ␨ - and abs( ␨)coordinate samples. It must be stressed that the use of
fixed organelle center’s coordinates is an oversimplification of reality, because an organelle position in a living
cell usually alters. However, the fixation of organelle
center’s coordinates allows us to demonstrate more
clearly the influence of the sectioning process on 2-D
measurement observations.
If organelle center’s coordinates are fixed and asymmetrical in respect to the cell center, the pattern for the
coordinates of an organelle profile’s center is quite
different from the previous case. Previously seen sym-
metry in the ␨ -coordinate pattern disappears, which is
demonstrated explicitly in the sample’s histogram (Fig.
3B). The same conditions cause shifting of the histogram for the abs( ␨)-coordinate sample to larger values.
It is obvious that the larger absolute value for the x, y,
and, especially, z organelle center’s coordinates will be
(i.e., the farther the organelle is from the cell center),
the larger values of the abs( ␨)-coordinate the histogram’s mode will correspond to. The ␩-coordinate pattern obtained under the same conditions is, in contrast
to that of the ␨ -coordinate, symmetrical. The sample’s
histogram has two modes that take marginal positions
on equal distances from the histogram’s center (which
corresponds to ␩ ⫽ 0) (Fig. 3C). At the same time, the
abs( ␩)-coordinate sample’s histogram has single mode,
witnessing the grouping of the distribution around the
larger values. The same rule, apparently, can be applied to the features of both abs( ␨)- and abs( ␩)coordinates: the larger the values for the corresponding
organelle center’s coordinates will be, the larger the
values the mode of the distribution of organelle profile’s
center coordinates will correspond to. It is worth noting,
however, that while the value for an ␨ -coordinate
depends on the values of all three organelle center’s
coordinates (x, y, and z), the value for a ␩-coordinate
depends on the values of only two of them (x and y).
When random organelle center’s coordinates are used,
both coordinates of an organelle profile’s center are
distributed very similarly to one another, with their
sample’s histograms lacking explicit maximums. Changing real values for the coordinates of an organelle
profile’s center into absolute ones results in a shifting of
both samples to smaller values. The latter feature for
the abs( ␨)-coordinate sample is, however, less pronounced than in the case of the centrally-fixed organelle, so these cases can easily be distinguished. The
results of the tests performed with random organelle
center’s coordinates point to one interesting aspect.
They suggest that, in contrast to conclusions drawn by
Hemon et al. (1981), the 3-D randomness of an organelle intracellular position does not imply the 2-D randomness of its profile location within a cell profile (Fig.
3D,E). Preliminary data provide evidence that the
shape of a randomly sectioned cell influences the 2-D
spatial representation of organelle profiles.
On practice, a researcher analyzing random section
data often needs to determine whether the organelle is
definitely localized in a particular part of a cell. It was
shown that characteristics of a sample of organelle
profile’s center coordinates allow for distinguishing
between the cases of different localization of an organelle. Since distributions under consideration are not
Gaussian, the non-parametrical Kolmogorov-Smirnov
test was used to determine the statistical significance of
differences between samples of organelle profile’s center coordinates generated according to two definitely
different schemes for organelle center’s coordinates. It
was shown that, to distinguish the two cases, with the
organelle center’s coordinates fixed outside the cell
center and randomly selected, one needs to estimate the
position of 40 (P ⬍ 0.01) or 70 (P ⬍ 0.001) organelle
profiles using abs( ␨)-coordinate values, and 110 (P ⬍
0.05), 200 (P ⬍ 0.01), or 290 (P ⬍ 0.001) organelle
profiles, using abs( ␩)-coordinates values.
457
SPATIAL ARRANGEMENT OF AN ORGANELLE
The purpose of the simulations was to demonstrate
the possibilities for application of the Cell Sectioning
Model to the evaluation of an organelle intracellular
arrangement using random section data. This paper
does not deal with all possible variants of a scheme for
generation of organelle center’s coordinates. We are
going to discuss them in forthcoming publications.
The basic limitations of the Cell Sectioning Model
were discussed earlier (Nikonenko, 1996). They include
the approximation of the shape of a simulated object to
an ellipsoid of rotation and cell sections having no
thickness. The method proposed in this paper is an
assumption-based one, and doesn’t fit the design-based
approach, which allows for an estimation of various
parameters of cells without any assumption made
about the shape of an object under study. It is well
known that design-based methods are efficient in extrapolating 3-D volume, area, length, and number
quantities from simple counts made on 2-D sections
(Mayhew, 1992), but the situation is different for the
quantitative estimation of an organelle intracellular
spatial arrangement. The rare publications dealing
with this subject illustrate it clearly. In this context, the
application of the model to interpret 2-D measurement
observations from thin and semi-thin sections in cells of
particular types seems to be valuable.
Thus, intracellular spatial arrangement of an organelle can be revealed in random sections using computer
simulation modeling as an interpretational procedure.
The statistical properties of a sample of organelle
profile8s center coordinates depend on the pattern of
spatial scattering of an organelle in the cytoplasm of a
cell. This allows, using 2-D observations from random
sections, to interpret some 3-D spatial regularities,
including the following: the position of an organelle in
respect to a cell center and the fixity or randomness of
the intracellular position of an organelle. The technique
proposed can be used as a tool for computerized morphometry and/or as an algorithm of stereological software for educational purposes.
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ACKNOWLEDGMENTS
1. first rotation is made around the z-axis on the angle
␸;
2. second rotation is made around the y’-axis on the
angle ␪ ;
3. the coordinate system is displaced along the z’’-axis
(the OM segment) for a distance d (the length of the
OM segment ) (Fig.1), we have:
The author is indebted to Dr. Richard I. Moisia,
Ph.D., for his valuable advice. He also thanks Mr. Oleg
N. Nelip for his assistance in performing this research.
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APPENDIX
Within the Cell Sectioning Model, the shape of a
simulated cell (and/or an organelle) is approximated by
an ellipsoid of rotation. The equation of a rotational
ellipsoid is:
x2⫹ y 2
a2
⫹
z2
c2
⫽ 1,
(A.1)
where a (⫽b) ⬍ c - semiaxes.
Expressing the equation (A.1) in coordinates transformed in the following way:
1
x⫹d·
3
2
E2
D
1
a2 · c2 ⫺ d2 · F ⫺
D
E2
D
24
(A.2)
2
⫹
3
y
1
a ·c ⫺d · F⫺
2
2
2
c2
E2
D
4
2
⫽ 1,
458
A.G. NIKONENKO
where
D ⫽ c2 · cos2 ␪ ⫹ a2 · sin2 ␪,
E ⫽ sin ␪ · cos ␪ · (c2 ⫺ a2),
F ⫽ c2 · sin2 ␪ ⫹ a2 · cos2 ␪.
This equation differs from the canonical one only by
the displacement of an ellipse’s center on value d ·
(E/D). The latter value equals the distance on which an
ellipsoid center projection on a profile ( the point M) can
be displaced in respect to a profile center (the point M1 ,
Figs. 1, 2A). It is obvious from the equation (A.2) that
this displacement ( the length of the MM1 segment) can
occur only along the longer axis of a cell profile ellipse.
The value of such displacement is used in the fourth
step of coordinate transformation procedure to define
the center of a cell profile, i.e., the beginning of the new
coordinate system (see the formulae (2) in the text of
the paper). The more detailed description of the basic
formulae derivation has been published earlier (Nikonenko, 1996).
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