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Abstract

and size not provided, image not shown), when analyzed by the boxplot method, provided a set of points that were not aligned in the full domain of analysis. As mentioned in the discussion of our report, natural objects differ from purely mathematical ones because they are not self-similar but exhibit self-affinity in a limited domain. This is true for liver fibrosis composed of fibrils, grouped in fibers, aggregated in collagen and noncollagen buddles and finally in more or less large fibrotic sheats. The slight concavity of the log-log plots is characteristic of natural objects observed in the "real life" instead of ideal mathematical objects. This was stressed by several authors cited in our report (e.g., Cross 1,2 ); other authors have also attempted to describe subdomains limited by break points on the log-log plots, 3,4 whereas others refer to asymptotic fractal to describe this phenomenon. [5][6][7] It is why the selection of a "domain of interest" may appear crucial before performing the box-plot analysis. In the present study, the microscopic magnification and the size of the boxes were chosen according to criteria extensively described by others in the biological literature and described in the material and methods. Dr. Wu et al. also claim that the box-plot method would also reflect the "pore size" rather than the object analyzed (i.e., the fibrosis). This assumption is derived from analysis of artificial mathematical objects. In other studies, we have compared Kolmogorov, Minkowski-Bouligand, and mass radius fractal dimensions with other well-known descriptors of the architecture of cancellous bone (a tissue composed of trabeculae encircling, more or less, large and round marrow cavities). When using a cluster tree discriminant analysis, we clearly confirmed that fractal dimensions were related to the trabeculae and not to the marrow pores. 8,9 As reported by Dollinger et al., "natural fractals show some smallest length scale terminating the scale-invariance." 7 In our report, we never claimed that liver fibrosis was a fractal object in the purely mathematical sense, but we reasonably used fractal geometry to show that various treated/ untreated rats presented self-affinity over an adequately chosen microscopic range, based on the study of collagen fibers, that allowed a correct and powerful classification. Finally, whether fractal dimension is a direct or an indirect marker of liver fibrosis is not relevant for medical applications in validated conditions. Thus, prothrombin time has not yet a biological plausibility in liver fibrosis but we have recently shown that its diagnostic accuracy was as high as that of the most accurate direct blood marker of liver fibrosis. 10