Cross S S
Department of Pathology, University of Sheffield Medical School, U.K.
J Pathol. 1997 May;182(1):1-8. doi: 10.1002/(SICI)1096-9896(199705)182:1<1::AID-PATH808>3.0.CO;2-B.
Many natural objects, including most objects studied in pathology, have complex structural characteristics and the complexity of their structures, for example the degree of branching of vessels or the irregularity of a tumour boundary, remains at a constant level over a wide range of magnifications. These structures also have patterns that repeat themselves at different magnifications, a property known as scaling self-similarity. This has important implications for measurement of parameters such as length and area, since Euclidean measurements of these may be invalid. The fractal system of geometry overcomes the limitations of the Euclidean geometry for such objects and measurement of the fractal dimension gives an index of their space-filling properties. The fractal dimension may be measured using image analysis systems and the box-counting, divider (perimeter-stepping) and pixel dilation methods have all been described in the published literature. Fractal analysis has found applications in the detection of coding of coding regions in DNA and measurement of the space-filling properties of tumours, blood vessels and neurones. Fractal concepts have also been usefully incorporated into models of biological processes, including epithelial cell growth, blood vessel growth, periodontal disease and viral infections.
许多自然物体，包括病理学中研究的大多数物体，都具有复杂的结构特征，并且它们结构的复杂性，例如血管分支程度或肿瘤边界的不规则性，在很宽的放大倍数范围内都保持在恒定水平。这些结构还具有在不同放大倍数下重复出现的模式，这种特性称为尺度自相似性。这对于长度和面积等参数的测量具有重要意义，因为对这些参数进行欧几里得测量可能是无效的。分形几何系统克服了欧几里得几何对于此类物体的局限性，分形维数的测量给出了它们空间填充特性的一个指标。分形维数可以使用图像分析系统进行测量，文献中已经描述了盒计数法、分割器（周长步长法）和像素扩张法。分形分析已应用于DNA编码区域的检测以及肿瘤、血管和神经元空间填充特性的测量。分形概念也已被有效地纳入生物过程模型，包括上皮细胞生长、血管生长、牙周病和病毒感染。
