Quantitating the subtleties of microglial morphology with fractal analysis

Abstract

It is well established that microglial form and function are inextricably linked. In recent years, the traditional view that microglial form ranges between "ramified resting" and "activated amoeboid" has been emphasized through advancing imaging techniques that point to microglial form being highly dynamic even within the currently accepted morphological categories. Moreover, microglia adopt meaningful intermediate forms between categories, with considerable crossover in function and varying morphologies as they cycle, migrate, wave, phagocytose, and extend and retract fine and gross processes. From a quantitative perspective, it is problematic to measure such variability using traditional methods, but one way of quantitating such detail is through fractal analysis. The techniques of fractal analysis have been used for quantitating microglial morphology, to categorize gross differences but also to differentiate subtle differences (e.g., amongst ramified cells). Multifractal analysis in particular is one technique of fractal analysis that may be useful for identifying intermediate forms. Here we review current trends and methods of fractal analysis, focusing on box counting analysis, including lacunarity and multifractal analysis, as applied to microglial morphology.

Keywords: box counting; cell shape; fractals; image interpretation: computer-assisted; lacunarity; microglia; models: biological; multifractal analysis.

Figure 1

Microglial morphology in adult human CNS. Microglia are morphologically and functionally dynamic cells able to change form from highly ramified to completely lacking processes. The transition can be very rapid or microglia can remain in a form for years (Colton et al., 2000). The forms illustrated here represent snapshots of a transformation that is reversible at every point, with variation within each form shown. The figures are not shown to scale; they are adjusted to compare detail.

Figure 2

Self-similarity and scale invariance. (A) Thirty-two-segment quadric fractal contour illustrating exact but limited self-similarity. The pattern always resolves into 32 new parts each 1/8 of the previous size so the fractal dimension is ln 32/ln 8 = 1.67. The theoretical pattern is infinitely self-similar, but the image is limited by the smallest possible line size that can be used to draw it. [Image generated with ImageJ (Schneider et al.), fractal generator plugin. http://imagej.nih.gov/ij/plugins/fractal-generator.html. Modified from http://rsb.info.nih.gov/ij/plugins/fraclac/FLHelp/Fractals.htm#32seg.] Silhouettes of (B) unramified and (C) ramified microglia. [Note that the rightmost figure in (C) could be classified as intermediate.] (B) Showing membrane detail, and (C) showing branching detail, depict scaling that is not exactly self-similar and referred to as scale invariant. Moreover, scaling is only detectable within physical limits and the limits of the methods and media used to reveal morphology including staining and recording methods as well as magnification and resolution (both optical and digital).

Figure 3

Increasing complexity of branching patterns. Branching structures generated by computer based modeling, according to known rules. At each iteration, the ratio of branch length to parent length was changed to illustrate how branching features influence theoretical D_Fs and calculated D_Bs. For both series, the direction a branch was allowed to grow in was random. The only difference between the two series was in the amount of curving allowed in processes, which was greater in the images on the bottom. The values for the D_B are averages for sets of models rather than only the particular model shown for each series. Modeled with MicroMod, free, open-source modeling software (Karperien, ; Jelinek et al., 2002).

Figure 4

Lacunarity. The images on the bottom row are 90° rotations of the images on the top row. Images (A) and (C) look similar to their originals when rotated, but (B) and (D) are more affected by rotation; they are more heterogeneous or less rotationally invariant and this is captured and quantified by Λ [e.g., (B) has greater lacunarity than (A)]. (A) and (B) were generated with ImageJ fractal generator plugin http://imagej.nih.gov/ij/plugins/fractal-generator.html; (C) and (D) are from Figure 1.

Figure 5

In silico microglia illustrate the relationship between the D_B and branching features. The D_B decreased with scale—that is, it increased nonlinearly with the relative length of sub-branches Modified from Karperien (2004); models generated with MicroMod (Karperien, ; Jelinek et al., 2002).

Figure 6

Drawings of representative microglia grown in the laboratory. Cells grown in culture tend to appear different than microglia found in live animals. Art by Thomas R. Roy.

Figure 7

Criteria for morphological categories of microglia.