Application of fractal and euclidean methods to differentiate normal and neoplastic thyroid cells

Context: The differentiated papillary and follicular thyroid neoplasms can be characterized from the notions of fractal and Euclidean geometry to overcome the challenges faced by the pathologist. This method was previously used in differentiating preinvasive lesions of cervical cancer. Aims: to characterize the irregularity of histologic samples of normal thyroid cells as well as benign and malignant thyroid papillary and follicular carcinomas, through the box-counting method using the principles of fractal and Euclidian geometry. Settings and Design: This is a retrospective study involving the measurement of thyroid cells through pixels in photographs, applying geometric methods. Subjects and Methods: Photographs of histological samples from normal and neoplastic biopsy samples were taken and processed by a software in order to delimit the borders of the nucleus and cytoplasm. Then, the box-counting method was applied by superimposing grids of 5 and 10 pixels to measure the fractal dimension and the occupied spaces of the cellular surface. Results: The set of papillary and follicular cells evaluated from the occupied spaces from the borders and surfaces of the nucleus and cytoplasm in the 5-pixel grid showed that normal cells are included within a range of values, while the neoplastic variations are differentiable from this range. Conclusions: Fractal and Euclidean geometries can differentiate normality from some benign and malignant thyroid lesions, which opens a path to develop methodologies that characterize more precisely distinctive features between normal and neoplastic cells independent of qualitative criteria from traditional pathology and histology.