BioSystems 94 (2008) 276–281 Contents lists available at ScienceDirect BioSystems journal homepage: www.elsevier.com/locate/biosystems Lacunarity as a novel measure of cancer cells behavior Przemyslaw Borys a,∗ , Monika Krasowska a , Zbigniew J. Grzywna a , Mustafa B.A. Djamgoz b , Maria E. Mycielska b a Department of Physical Chemistry and Technology of Polymers, Section of Physics and Applied Mathematics, Silesian University of Technology, 44-100 Gliwice, Ks. M. Strzody 9, Poland b Imperial College London, Division of Cell and Molecular Biology, Neuroscience Solutions to Cancer Research Group, Sir Alexander Fleming Building, South Kensington Campus, London SW7 2AZ, UK article info Keywords: Texture Lacunarity Gliding box method abstract An important goal in many branches of science, especially in molecular biology and medicine is the quan- titative analysis of the structures and their morphology. The morphology can be analyzed in many ways, in particular by the fractal analysis. Apart from the fractal dimension, an important part of the fractal analysis is the lacunarity measurement which, roughly speaking, characterizes the distribution of gaps in the fractal: a fractal with high lacunarity has large gaps. In this paper, we present an extension of the lacu- narity measure to objects with nonregular shapes that enables us to provide a successful discrimination of cancer cell lines. The cell lines differ in the shape of vacuole (the gaps in their body) which is perfectly suited for the lacunarity analysis. © 2008 Elsevier Ireland Ltd. All rights reserved. 1. Introduction Classical statistics can be used for a description of different pat- terns. The geometry of the structure of sets can be characterized by many parameters. Among them we can name the total area, mean area, etc. Very simple and powerful way to describe the spatial distribu- tion of structures are the spatial moments, provided they have a tendency to cluster around same particular value. The moments include: the total mass, the location of gravity center, the vari- ance around the gravity center, radius of gyration (deGennes, 1979; Dewey, 1997), the skewness (asymmetry) and flatness. Another way of description is use of fractal measures such as fractal dimension (Mandelbrot, 1983; Bassingthwaighte et al., 1994) and generalized fractal dimension. In this approach we divide the investigated structure image into N boxes of side length ε and calculate a (generalized) fractal dimension by the formula: D q = 1 1 - q lim ε→0 log N  i=1  q i log 1/ε (1) ∗ Corresponding author. E-mail address: Przemyslaw.Borys@polsl.pl (P. Borys). where D q , the generalized fractal dimension; q, dimension index (q = 0 for the generic fractal dimension);  i , the mass located within the box. To supplement the fractal analysis, we can also use lacunarity. Lacunarity is a term, coined by Mandelbrot (1983) that reflects the impression of gaps that appear in analyzed image (Kaye, 1989): the higher the lacunarity, the larger gaps in the mass distribution. Lacu- narity was originally introduced to distinguish between fractals with the same fractal dimension (Mandelbrot, 1983). In microscopy image analysis, such gaps can be the vacuoles of cells. The research (Mycielska et al., 2003; Krasowska et al., 2004b) shows that they may change their shape significantly if they are cancer cells. The first application of fractal measures (Krasowska et al., 2004b) has shown the fractal dimension description to be successful while lacunarity description has failed (we did not publish these results). This should not be the case since  and D q are related as will be shown in Section 3. In this paper we investigate this problem and show how to successfully describe cancer cell lines with help of . The traditional approach (Plotnick et al., 1993, 1996; Allain and Cloitre, 1991) to lacunarity, using gliding box method, was unac- ceptable for such kind of analysis, because the measure of both cell lines overlapped (See Fig. 6 and take a projection on  A axis). To overcome the difficulties, we have modified this method by chang- ing the domain of lacunarity. The results are acceptable and can serve as a discriminating factor. 0303-2647/$ – see front matter © 2008 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.biosystems.2008.05.036