Stereological analysis and modelling of gradient structures

PRELIMINARY INVESTIGATIONS OF TRANSPORT IN HETEROGENEOUS RANDOM MEDIA

Random media emerge in several applications in reactor physics and safety analysis. Most often, models of stochastic media assume spatial homogeneity, whereas real-world complex materials, such as fuel chunks resulting from core degradation, typically display apparent heterogeneities. In a series of previous works, we have shown that stochastic tessellations can be successfully used in order to describe the material properties of several classes of random media. In this paper we extend these results to the case of heterogeneous random media by using Voronoi tessellations with space-dependent seed distributions, allowing for spatial gradients.

Statistical methods for mechanical characterization of randomly reinforced media

Advanced materials with heterogeneous microstructure attract extensive interest of researchers and engineers due to combination of unique properties and ability to create materials that are most suitable for each specific application. One of the challenging tasks is development of models of mechanical behavior for such materials since precision of the obtained numerical results highly depends on level of consideration of features of their heterogeneous microstructure. In most cases, numerical modeling of composite structures is based on multiscale approaches that require special techniques for establishing connection between parameters at different scales. This work offers a review of instruments of the statistics and the probability theory that are used for mechanical characterization of heterogeneous media with random positions of reinforcements. Such statistical descriptors are involved in assessment of correlations between the microstructural components and are parts of mechanical theories which require formalization of the information about microstructural morphology. Particularly, the paper addresses application of the instruments of statistics for geometry description and media reconstruction as well as their utilization in homogenization methods and local stochastic stress and strain field analysis.

Densities of mixed volumes for Boolean models

In generalization of the well-known formulae for quermass densities of stationary and isotropic Boolean models, we prove corresponding results for densities of mixed volumes in the stationary situation and show how they can be used to determine the intensity of non-isotropic Boolean models Z in d-dimensional space for d = 2, 3, 4. We then consider non-stationary Boolean models and extend results of Fallert on quermass densities to densities of mixed volumes. In particular, we present explicit formulae for a planar inhomogeneous Boolean model with circular grains.

Nonstationary Poisson hyperplanes and their induced tessellations

Results about stationary Poisson hyperplane processes and the induced hyperplane mosaics are extended to the case where, instead of stationarity, it is only assumed that the intensity measure has a (possibly continuous) density with respect to some translation-invariant measure. Intensities and quermass densities, which are constant in the stationary case, are then replaced by functions. In a similar way, the associated zonoid (Matheron's Steiner convex set) is generalized.

Nonstationary Poisson hyperplanes and their induced tessellations

Results about stationary Poisson hyperplane processes and the induced hyperplane mosaics are extended to the case where, instead of stationarity, it is only assumed that the intensity measure has a (possibly continuous) density with respect to some translation-invariant measure. Intensities and quermass densities, which are constant in the stationary case, are then replaced by functions. In a similar way, the associated zonoid (Matheron's Steiner convex set) is generalized.

Inhomogeneous spatial point processes by location-dependent scaling

A new class of models for inhomogeneous spatial point processes is introduced. These locally scaled point processes are modifications of homogeneous template point processes, having the property that regions with different intensities differ only by a scale factor. This is achieved by replacing volume measures used in the density with locally scaled analogues defined by a location-dependent scaling function. The new approach is particularly appealing for modelling inhomogeneous Markov point processes. Distance-interaction and shot noise weighted Markov point processes are discussed in detail. It is shown that the locally scaled versions are again Markov and that locally the Papangelou conditional intensity of the new process behaves like that of a global scaling of the homogeneous process. Approximations are suggested that simplify calculation of the density, for example, in simulation. For sequential point processes, an alternative and simpler definition of local scaling is proposed.

Inhomogeneous spatial point processes by location-dependent scaling

A new class of models for inhomogeneous spatial point processes is introduced. These locally scaled point processes are modifications of homogeneous template point processes, having the property that regions with different intensities differ only by a scale factor. This is achieved by replacing volume measures used in the density with locally scaled analogues defined by a location-dependent scaling function. The new approach is particularly appealing for modelling inhomogeneous Markov point processes. Distance-interaction and shot noise weighted Markov point processes are discussed in detail. It is shown that the locally scaled versions are again Markov and that locally the Papangelou conditional intensity of the new process behaves like that of a global scaling of the homogeneous process. Approximations are suggested that simplify calculation of the density, for example, in simulation. For sequential point processes, an alternative and simpler definition of local scaling is proposed.

Block bootstrap methods for the estimation of the intensity of a spatial point process with confidence bounds

This paper deals with the estimation of the intensity of a planar point process on the basis of a single point pattern, observed in a rectangular window. If the model assumptions of stationarity and isotropy hold, the method of block bootstrapping can be used to estimate the intensity of the process with confidence bounds. The results of two variants of block bootstrapping are compared with a parametric approximation based on the assumption of a Gaussian distribution of the numbers of points in deterministic subwindows of the original pattern. The studies were performed on patterns obtained by simulation of well-known point process models (Poisson process, two Matérn cluster processes, Matérn hardcore process, Strauss hardcore process). They were also performed on real histopathological data (point patterns of capillary profiles of 12 cases of prostatic cancer). The methods are presented as worked examples on two cases, where we illustrate their use as a check on stationarity (homogeneity) of a point process with respect to different fields of vision. The paper concludes with various methodological discussions and suggests possible extensions of the block bootstrap approach to other fields of spatial statistics.

Stereological estimation of cell wall density of DR12 tomato mutant using three-dimensional confocal imaging

BACKGROUND AND AIMS

The cellular structure of fleshy fruits is of interest to study fruit shape, size, mechanical behaviour or sensory texture. The cellular structure is usually not observed in the whole fruit but, instead, in a sample of limited size and volume. It is therefore difficult to extend measurements to the whole fruit and/or to a specific genotype, or to describe the cellular structure heterogeneity within the fruit.

METHODS

An integrated method is presented to describe the cellular structure of the whole fruit from partial three-dimensional (3D) observations, involving the following steps: (1) fruit sampling, (2) 3D image acquisition and processing and (3) measurement and estimation of relevant 3D morphological parameters. This method was applied to characterize DR12 mutant and wild-type tomatoes (Solanum lycopersicum).

KEY RESULTS

The cellular structure was described using the total volume of the pericarp, the surface area of the cell walls and the ratio of cell-wall surface area to pericarp volume, referred to as the cell-wall surface density. The heterogeneity of cellular structure within the fruit was investigated by estimating variations in the cell-wall surface density with distance to the epidermis.

CONCLUSIONS

The DR12 mutant presents a greater pericarp volume and an increase of cell-wall surface density under the epidermis.

Modelling tree roots in mixed forest stands by inhomogeneous marked Gibbs point processes

The aim of the paper is to apply point processes to root data modelling. We propose a new approach to parametric inference when the data are inhomogeneous replicated marked point patterns. We generalize Geyer's saturation point process to a model, which combines inhomogeneity, marks and interaction between the marked points. Furthermore, the inhomogeneity influences the definition of the neighbourhood of points. Using the maximum pseudolikelihood method, this model is then fitted to root data from mixed stands of Norway spruce (Picea abies (L.) Karst.) and European beech (Fagus sylvatica L.) to quantify the degree of root aggregation in such mixed stands. According to the analysis there is no evidence that the two root systems are not independent.

Neuronal distribution in the neocortex of schizophrenic patients

It has been postulated that the prefrontal cortices of schizophrenic patients have significant alterations in their neuropil space. However, previous results have been contradictory and inconclusive, reporting both decreases and increases in the prefrontal neuropil. The present study re-examines these findings based on measurements of cell density, and inter-cellular distances within and between cell minicolumns. The results indicate alterations in the neuropil of schizophrenic patients according to both the lamina and cortical area examined. Alterations were present in all cortical areas studied. The findings suggest an alteration in the modulatory systems innervating the cell minicolumn. Furthermore, the lack of variation in core columnarity parameters argues in favor of a defect post-dating the formation of the cell minicolumn.

Stochastic geometric models, and related statistical issues in tumour-induced angiogenesis

In the modelling and statistical analysis of tumor-driven angiogenesis it is of great importance to handle random closed sets of different (though integer) Hausdorff dimensions, usually smaller than the full dimension of the relevant space. Here an original approach is reported, based on random generalized densities (distributions) á la Dirac-Schwartz, and corresponding mean generalized densities. The above approach also suggests methods for the statistical estimation of geometric densities of the stochastic fibre system that characterize the morphology of a real vascular system. A quantitative description of the evolution of tumor-driven angiogenesis requires the mathematical modelling of a strongly coupled system of a stochastic branching-and-growth process of fibres, modelling the network of blood vessels, and a family of underlying fields, modelling biochemical signals. Methods for reducing complexity include homogenization at mesoscales, thus leading to hybrid models (deterministic at the larger scale, and stochastic at lower scales); in tumor-driven angiogenesis the two scales can be bridged by introducing a mesoscale at which one locally averages the microscopic branching-and-growth process, in presence of a sufficiently large number of vessels (fibers).

Stereological methods to assess tissue response for tissue-engineered scaffolds

The stereological approach can provide an objective unbiased assessment of structural change in biological systems. In this review, we elucidate the basic principles of stereology and their implementation in the analysis of tissue response to tissue-engineering scaffolds. A brief outline of tissue response parameters that can be estimated using stereological approach is included. The focus is on frequently quantified parameters in tissue response, such as host tissue infiltration, inflammatory cell numbers, angiogenesis, fibrous tissue thickness, areas of calcification, and/or necrosis, among others. Special consideration is given to sampling techniques and how these techniques can influence the reliability of the obtained results as well as minimizing potential sources of bias. These basic principles are illustrated with practical examples, where measurements are performed and estimations calculated using conventional stereological techniques. As the next generation of biomaterials continue to be developed, it is essential that researchers develop a rigorous and unbiased method of performance quantification.