Fractal, multifractal, and lacunarity analysis of microglia in tissue engineering.

Abstract

Tissue engineering is currently one of the most exciting fields in biology (Grayson et al., 2009). Fractal analysis is equally exciting (Di Ieva et al., 2013), as is the study of microglia, the brain’s immuno-inflammatory cell, recently shown to be of considerably more importance than previously imagined in both healthy and diseased brain (Tremblay et al., 2011). Each of these fields is developing at a pace far outstripping our capacity to integrate and translate the information gained into clinical use (Karperien et al., 2008b, 2013; Jelinek et al., 2011, 2013), and the excitement more than trebles where these fields intersect. Three elements of fractal analysis – monofractal, multifractal, and lacunarity analysis – applied to microglia may contribute significantly to the next steps forward in engineered tissues and 3D models in neuroscience. Fractal analysis and lacunarity To define “fractal analysis” would take a volume, but for this commentary, it is sufficient to understand that fractal analysis in biology assesses the scaling inherent in biological forms or events, and turns out a statistical index of complexity having no units called the “fractal dimension” (DF). This number measures not length, width, height, or density, but scale-invariant detail. For a pattern to have fractal scale-invariant detail means that the pattern repeats itself infinitely as one inspects it at closer and closer resolution (magnifies it), where that detail is not trivial. To elaborate, as one magnifies a simple line, it infinitely repeats itself quite trivially as a simple line, but as one magnifies a fractal line, one finds it never resolves into straight pieces but rather each magnified segment repeats the initial fractal pattern infinitely. A DF measures this infinite scaling, quantifying complex patterns without rendering meaningless the relative numbers of large and small measurements within them. Without getting too technical, fractal analysis of a simple line yields a DF of 1.00, and the higher the “complexity,” the higher the DF (Mandelbrot, 1983; Takayasu, 1990). Building on this so-called monofractal analysis, multifractal analysis, to summarize, is a way of finding for a single pattern a spectrum of DFs, owing to a pattern having characteristically multiple degrees of scaling, such as could be imagined for a cascading fractal phenomenon (Jestczemski and Sernetz, 1996; Falconer, 2014). The word “lacuna” is derived from the word for lake, and refers to a gap or pool. In fractal analysis, lacunarity translates to measures of gappiness or “visual texture,” such as might be seen in the patchiness of forests, for instance (Plotnick et al., 1993). It has been defined as the degree of inhomogeneity and translational and rotational invariance in an image (Plotnick et al., 1993; Smith et al., 1996), where low lacunarity implies homogeneity and that rotating the image will not change it significantly. Thus, an image having mostly similarly sized gaps and little rotational variance would be expected to have low lacunarity, and one with much heterogeneity, many differently sized gaps, and notable rotational variance, would be expected to have high lacunarity (Karperien et al., 2011a). Lacunarity is frequently assessed during fractal analysis because the data on which it is based are easily collected by the same methods. The details and calculations behind fractal analysis are beyond the scope of this commentary but user-friendly, freely available software for biologists (Karperien, 2001, 2013) and in-depth explanations are available elsewhere (Smith et al., 1996).