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. organelles. I. Cytological organization of human exocrine epithelia. J. Electr. Microsc. Tech., 7:167–175. Hemon, D., Bourgeois, C.A., and Bouteille, M. (1981) Analysis of the spatial organization of the cell: A statistical method for revealing the non-random location of an organelle. J. Microsc., 121:29–37. Letourneau, P.C., and Wire, J.P. (1995) Three-dimensional organization of stable microtubules and the Golgi apparatus in the somata of developing chick sensory neurons. J. Neurocytol., 24:207–223. Mattfeldt, T., and Mall, G. (1984) Estimation of surface and length of anisotropic capillaries. J. Microsc., 135:181–190. Mayhew, T.M. (1992) A review of recent advances in stereology for quantifying neural structure. J. Neurocytol., 21:313–328. Nikonenko, A.G. (1996) The Cell Sectioning Model. Nuclear/cytoplasmic ratio studied by computer simulation. Anal. Quant. Cytol. Histol., 18:23–34. Pentcheff, N.D., and Bolender, R.P. (1992) Computer assisted data collection for stereology: Rationale and description of point counting stereology (PCS) software. Microsc. Res.Tech., 21:347–354. Rashbass, J., Stekel, D., and Williams, E.D. (1996) The use of a computer model to simulate epithelial pathologies. J. Pathol., 179:333–339. Reverter, J.-C., Feliu, E., Climent, C., Rozman, M., Berga, L., and Rozman, C. (1993) Stereological study of human bone marrow adipocytes. A comparison of four methods for estimating size distributions. Path. Res. Pract., 189:1215–1220. Rigaut, J.P., Carvajal-Gonzalez, S., and Vassy, J. (1992) 3-D image cytometry. In: Visualization in Biomedical Microscopies. A. Kriete, ed. VCH, Weinheim, pp. 205–248. Verbeek, F.J., Huijsmans, D.P., Baeten, RJ., Schoutsen, NJ., and Lamers, WH. (1995) Design and implementation of a database and program for 3D reconstruction from serial sections: A data-driven approach. Microsc. Res. Tech., 30: 496–512. 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. REFERENCES Abrams, D.-C., Facer, P., Bishop, A.E., Polak, J.M. (1994) A computerassisted stereological quantification program: OpenStereo. Microsc. Res. Tech., 29: 240–247. Baudhuin, P., Leroy-Houyet, M.A., Quintart, J., and Berthet, P. (1979) Application of cluster analysis for characterization of spatial distribution of particles by stereological methods. J. Microsc., 115:1–17. Black, V.H., and Rosen, S. (1996) COSAS 2.0: A Macintosh-based stereological analysis system. J. Struct. Biol., 116:176–180. Burkhardt, J.K. (1996) In search of membrane receptors for microtubule-based motors: Is kinektin a kinesin receptor ? Trends Cell Biol., 6:127–131. Clem, C.J., Konig, D., and Rigaut, J.P. (1997) A three-dimensional dynamic simulation model of epithelial tissue renewal. Anal.Quant. Cytol.Histol.,19:174–184. Cruz-Orive, L.M. (1976) Quantifying ‘‘pattern’’: A stereological approach. J. Microsc., 107: 1–12. Diaz, G., Cossu, M., and Riva, A. (1987) Quantitative ultrastructural approach to the study of the spatial relationships among cell 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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