Konrad Sandau

Measuring fractal dimension and complexity — an alternative approach with an application

Fractal dimension has often been applied as a parameter of complexity, related to, for example, surface roughness, or for classifying textures or line patterns. Fractal dimension can be estimated statistically, if the pattern is known to be self‐similar. However, the fractal dimension of more general patterns cannot be estimated, even though the concept may be retained to characterize complexity. We here show that the usual statistical methods, e.g. the box counting method, are not appropriate to measure complexity. A recently developed approach, the extended counting method, whose properties are closer to what fractal dimension means, is considered here in more detail. The methods are applied to geometric and to blood vessel patterns. The weak assumptions about the structure, and the lower variance of the estimate, suggest that the extended counting method has beneficial properties for comparing complexity of naturally occurring patterns.

Modelling of vascular growth processes: A stochastic biophysical approach to embryonic angiogenesis

In a computer simulation, growth of a capillary network is driven by a stochastic process on a planar hexagonal grid. Starting at a point source, the probabilities for the formation of new capillary elements depend on local biophysical knowledge. This knowledge is mainly derived from the flow theorem of Hagen–Poiseuille and the diameter exponent Δ. The hexagonal grid is visualized as being supported by a cylinder or a sphere. An arterial tree results from the adaptive diameter augmentation, and is considered to have limited fractal properties. The dimension of its border, and the time course of growth and of blood pressure are compared with biological data from the chorioallantoic membrane (CAM) of incubated chicken eggs. The model is discussed in view of mechanosensitivity and cell–matrix interactions of endothelial cells, and CAM haemodynamics.

Measuring the surface area of a cell by the method of the spatial grid with a CSLM—a demonstration

The spatial grid is a method for estimating the surface area of particles. A stack of perfectly registered sections is the essential prerequisite for its use. The confocal scanning light microscope provides such a stack by optical sectioning. The spatial grid method is briefly described and applied to an osteocyte lacuna in dry mineralized human mandible. This type of cell was chosen because of its very complex shape. The variance of the area estimate is studied and compared with the results of a simulation.

On the bifurcation of blood vessels — Wilhelm Roux's Doctoral Thesis (Jena 1878) — A seminal work for biophysical modelling in developmental biology

Wilhelm Rouxs doctoral thesis described the relationship between the angle and diameter of bifurcating blood vessels. We have re-read this work in the light of biophysics and developmental biology and found two remarkable aspects hidden among a multitude of observations, rules and exceptions to these rules. First, the author identified the major determinants involved in vascular development; genetics, cybernetics, and mechanics; moreover, he knew that he could not deal with the genetic and regulatory aspects, and could hardly treat the mechanical part adequately. Second, he was deeply convinced that the laws of physics determine the design of organisms, and that a necessity for optimality was inherent in development. We combined the analysis of diameter relationships with the requirement for optimality in a stochastic biophysical model, and concluded that a constant wall-stress condition could define a minimum wall-tissue optimum during arterial development. Hence, almost 120 years after Wilhelm Rouxs pioneering work, our model indicates one possible way in which physical laws have determined the evolution of regulatory and structural properties in vessel wall development.

The chord length transform and the segmentation of crossing fibres

Segmentation of crossing fibres is a complex problem of image processing. In the present paper, various solutions are presented basing on tools of morphological image processing. Two new image transforms are introduced – the lineal distance transform and the chord length transform. Both transforms are applied to two‐dimensional images and their results are three‐dimensional images. Thus, the segmentation problem originally formulated for crossing fibres observed in a two‐dimensional image can be reformulated as a segmentation problem in a three‐dimensional image. This can be solved by a segmentation in the three‐dimensional image. Algorithms for the lineal distance transform and the chord length transform are given and their use in image analysis is demonstrated. Furthermore, the chord length distribution function of the foreground of a binary image can efficiently be estimated via the chord length transform.

Blood Vessel Growth: Mathematical Analysis and Computer Simulation, Fractality, and Optimality

The structural complexity of the circulatory system exceeds the available genetic information. In the developmental process, therefore, self-organization on epigenetic levels can be postulated, which exploits information that is being generated during embryogenesis. We used mathematical tools to analyze patterns and complexity, and designed a computer model to predict geometrical and biophysical properties of bifurcating vessel systems. In particular, some boundary conditions during development, and the problem of optimality are addressed. We propose that the complexity of blood vessel formation in vivo and in sapio may be adequately described with a combination of various classical geometrical and physical concepts, supplemented by concepts of fractal geometry.

Some remarks on the accuracy of surface area estimation using the spatial grid

A set of three line grids in three orthogonal directions is called a spatial grid. This spatial grid can be used for surface area estimation by counting the number of intersection points of a surface with the grid lines. If direction and localization of the spatial grid are suitably randomized, the expectation of this number is proportional to the surface area of interest. The method was especially developed for cases where the surface to be measured is embedded in a medium, which is the usual case in microscopical applications, and where a stack of serial optical sections of the surface is available.

An approach based on two-dimensional graph theory for structural cluster detection and its histopathological application.

An approach based on graph theory is described for detecting clusters of cells in tissue specimens (two‐dimensional space). With a set of discrete basic elements (cell nuclei) having several measurable features (area, surface, main and minor axis of best‐fitting ellipses) a graph is defined as having attributes associated with edges. Different minimum spanning trees (MSTs) can be constructed using different weight functions on the attributes (attributed MST). Analysis of the MST and of an attributed MST by use of a decomposition function allows detection of image areas with similar local properties. These clusters, which are then clusters of the tree, describe, for example, partial growth in different directions in a case of a human fibrosarcoma assuming that tumour cell nuclei are homogeneous with respect to their configuration and size. The model allows the separation of clusters of tumour cells growing in different directions and the approximation of the different growth angles. This decomposition also allows us to create new (higher) orders of structure (cluster tree).

Spatial Fibre and Surface Processes - Stereological Estimations and Applications

The aim of this paper is to give an introduction to the treatment of spatial fibre and surface processes especially under the aspects of modelling and stereology. In the last two decades mathematical tools were developed for modelling spatial fibres and surfaces as they occur in biology or other scientific disciplines. These tools also allow to quantify characteristics of the objects like length or surface area. In quantitative microscopy actually the objects are sectioned. Therefore the spatial characteristics have to be derived from plane sections. This is the fundamental problem treated in stereology, a mathematical discipline, collecting methods to relate the measurements on sections to the characteristics of three-dimensional structures. Several examples demonstrate in which way theory and applications of this kind can be combined passing the three stages ’modelling, stereology, statistics’.

Characterization of Microstructures Using the Chord Length Transformation

Many tools of image analysis are based on methods used in quantitative microscopy and are applied since a long time. But nowadays digital image analysis offers tools which need such a huge effort, that they never would be applied without a machine. The recently developed chord length transformation (CLT) is such a tool which helps to segment structures and which also helps to find suitable features for classifying structures [1]. This contribution presents the CLT and applies it at different levels of quantitative image analysis. The CLT assigns to every pixel in the foreground of a binary image and to each angle (of a discrete set of angles) the length of the chord. This length is measured along a line through the pixel point having a slope derived from this angle. The length is the length of the connecting chord determined by the intersection of the line with the foreground. For each fixed angle the resulting data set can be depicted as a gray value image where the gray values are proportional to the chord lengths. Figure 1.a shows a binary image and Figure 1.b shows the image of chord lengths for the angle /6. From the 1 whole data set (all angles) statistical information can be derived. So maximal and minimal chord lengths can be assigned to each pixel. A visualization of maximal chord lengths is given in Figure 1.c. The local information can be used for the segmenting process as is shown in the first example, where the CLT was applied to a binary image showing lamellas of graphite in gray cast iron. A specimen, grinded and etched, was regarded using a light microscope equipped with a lens of magnification 10. Then graphite can be seen embedded in the steel phase. One typical arrangement of the graphite is an arrangement in lamellas (Figure 1.a). For further measurements connected lamellas should be segmented in line shaped parts. The segmentation has been done here using the CLT as a 3D data structure and thresholding local adaptively using the local information about maximal and minimal chord lengths. Afterwards each segment in 3D can be projected to the 2D image. In this way crossing lamellas can be measured one by one. Figure. 1.d illustrates this (the arrow in the lower left corner points at a configuration, which is segmented into three parts two are vertical and one is longitudinal arranged). A second example uses the information along the angles to get an angle distribution and to segment the image with respect to the local main directions. This is demonstrated here for a microscopical image of a perlitic microstructure (Fig. 2.a). Obviously one observes several main areas of homogeneous direction. These areas are characterized by the direction of the maximal chord length in the foreground and in the background image. Figure 2.c shows the result of the segmentation process. The smoothed regions partition the image in the main areas as expected by the human eye.