Modelling and analysis of 3‐D arrangements of particles by point processes with examples of application to biological data obtained by confocal scanning light microscopy

Within the concept of point processes, a review is presented of quantities which can be used in studies of three‐dimensional (3‐D) aggregates of particles. Suitable characteristics and estimators are given for both unmarked and marked point processes. To demonstrate the feasibility of such quantitative approaches, an application in histology, dealing with 3‐D arrangements of cell nuclei in rat liver, is described. Using a confocal scanning light microscope, 3‐D images are recorded and image analysis used to obtain the coordinates of the centroid, together with the volume and DNA content, of each cell nucleus. Examples of results are given, using both unmarked and marked point processes. In the latter case, cell type, nuclear volume and ploidy group are suitable marks.

[1]  O. Kallenberg.  Random Measures , 1983 .

[2]  Tom W. Berrie,et al.  Queues and Point Processes , 1983 .

[3]  A. Baddeley,et al.  A cautionary example on the use of second-order methods for analyzing point patterns , 1984 .

[4]  G. J. G. Upton,et al.  Spatial data Analysis by Example , 1985 .

[5]  P. Diggle,et al.  Monte Carlo Methods of Inference for Implicit Statistical Models , 1984 .

[6]  D. Stoyan,et al.  The pair correlation function for point and fibre systems and its stereological determination by planar sections * , 1985, Journal of microscopy.

[7]  B. Ripley.  Statistical inference for spatial processes , 1990 .

[9]  W. D. Ray.  Infinitely Divisible Point Processes , 1979 .

[10]  K. Hanisch,et al.  Some remarks on estimators of the distribution function of nearest neighbour distance in stationary spatial point processes , 1984 .

[11]  Brian D. Ripley,et al.  Stochastic Simulation , 2005 .

[12]  J. Vassy,et al.  High-resolution three-dimensional images from confocal scanning laser microscopy. Quantitative study and mathematical correction of the effects from bleaching and fluorescence attenuation in depth. , 1991, Analytical and quantitative cytology and histology.

[13]  Dieter König.  Probabilistic Characterization of the Inner Order of Random Structures , 1986 .

[14]  P. J. Green,et al.  Density Estimation for Statistics and Data Analysis , 1987 .

[15]  Bernard W. Silverman,et al.  Methods for Analysing Spatial Processes of Several Types of Points , 1982 .

[16]  R. E. Miles.  The Sampling, By Quadrats, of Planar Aggregates , 1978 .

[17]  Thomas Fiksel,et al.  Edge-corrected density estimators for point processes , 1988 .

[18]  G. J. Brakenhoff,et al.  Three-dimensional chromatin distribution in neuroblastoma nuclei shown by confocal scanning laser microscopy , 1985, Nature.

[19]  B. Ripley.  Modelling Spatial Patterns , 1977 .

[20]  K. Mardia.  Statistics of Directional Data , 1972 .

[21]  J. G. White,et al.  Confocal microscopy comes of age , 1987, Nature.

[22]  David A. Agard,et al.  Three-dimensional architecture of a polytene nucleus , 1983, Nature.

[23]  Alan Boyde,et al.  Three-dimensional analysis of the spatial distribution of particles using the tandem-scanning reflected light microscope , 1987 .

[24]  B. Ripley.  The Second-Order Analysis of Stationary Point Processes , 1976 .

[25]  V. A. Epanechnikov.  Non-Parametric Estimation of a Multivariate Probability Density , 1969 .

[26]  A. Boyde,et al.  Unbiased estimation of particle density in the tandem scanning reflected light microscope , 1985, Journal of microscopy.

[27]  Dietrich Stoyan,et al.  Correlations of the Marks of Marked Point Processes - Statistical Inference and Simple Models , 1984, J. Inf. Process. Cybern..

[28]  H. J. G. Gundersen,et al.  The impact of recent stereological advances on quantitative studies of the nervous system , 1986, Journal of Neuroscience Methods.

[29]  Ewald R. Weibel,et al.  Stereological Analysis of Anisotropic Structures using Directional Statistics , 1985 .