Use of fractals in determining the malignancy degree of lung nodules.

Introduction: A Computer-Assisted Detection (CAD) System for classification into malignant-benign classes using CT images is proposed. Methods: Two methods that use the fractal dimension (FD) as a measure of the lung nodule contour irregularities (Box counting and Power spectrum) were implemented. The LIDC-IDRI database was used for this study. Of these, 100 slices belonging to 100 patients were analyzed with both methods. Results: The performance between both methods was similar with an accuracy higher than 90%. Little overlap was obtained between FD ranges for the different malignancy grades with both methods, being slightly better in Power spectrum. Box counting had one more false positive than Power spectrum. Discussion: Both methods are able to establish a boundary between the high and low malignancy degree. To further validate these results and enhance the performance of the CAD system, additional studies will be necessary.

A spatial randomness test based on the box-counting dimension.

Statistical modelling of a spatial point pattern often begins by testing the hypothesis of spatial randomness. Classical tests are based on quadrat counts and distance-based methods. Alternatively, we propose a new statistical test of spatial randomness based on the fractal dimension, calculated through the box-counting method providing an inferential perspective contrary to the more often descriptive use of this method. We also develop a graphical test based on the log-log plot to calculate the box-counting dimension. We evaluate the performance of our methodology by conducting a simulation study and analysing a COVID-19 dataset. The results reinforce the good performance of the method that arises as an alternative to the more classical distances-based strategies.

Fractal evaluation in retinographies of individuals with different grades of diabetic retinopathy / Avaliação fractal de retinografias de indivíduos com diferentes graus de retinopatia diabética

ABSTRACT Purpose To evaluate the retinal blood vascular network of the retinographies of patients with different grades of diabetic retinopathy. Methods Ninety Retinographies (MESSIDOR database) were used, with different grades of diabetic retinopathy divided into 4 groups: no retinopathy (n=23), grade one (n=20), grade two (n=20) and grade three (n=27) diabetic retinopathy. The grades of diabetic retinopathy were classified according to the number of microaneurysms, number of hemorrhages and the presence of neovascularization. The images were skeletonized and quantified by fractal methods: dimension of box-counting (Dbc) and information (Dinf). Results The means of Dbc values of groups were around 1.25, without statistically significant difference in the dimension values between groups for whole retina. There was also no statistical difference in Dinf values between groups, whose means ranged between 1.294 ± 0.013 (group of grade 1) and 1.3 ± 0.017 (group of grade 3). The retinographies were divided into regions of equal areas. The fractal values of some retinal regions showed statistical differences, but these differences were not enough to show the sensitivity of fractal methods in identifying diabetic retinopathy. Conclusion The fractal methods were not able to identify the different grades of diabetic retinopathy in retinographies.

RESUMO Objetivo Avaliar a rede vascular sanguínea da retina a partir de retinografias de pacientes com diferentes graus de retinopatia diabética. Métodos Foram utilizadas 90 retinografias (banco de dados MESSIDOR), com diferentes graus de retinopatia diabética divididas em quatro grupos: sem retinopatia (n=23), retinopatia diabética de grau um (n=20), grau dois (n=20) e grau três (n=27). Os graus de retinopatia foram classificados conforme o número de microaneurismas, número de hemorragias e presença de neovascularização. As imagens foram esqueletizadas e quantificadas pelos métodos fractais: dimensão da contagem de caixas e informação. Resultados As médias dos valores das dimensões de contagem de caixas para todos os grupos foram próximas a 1,25, sem diferença estatisticamente significativa nos valores das dimensões entre os grupos para retina inteira. Também não houve diferença estatística nos valores da dimensão de informação entre os grupos, cujas médias variaram entre 1,294 ± 0,013 (grupo do grau 1) e 1,3 ± 0,017 (grupo do grau 3). As imagens retinianas foram divididas em regiões de áreas iguais. Os valores fractais de algumas regiões retinais mostraram diferenças estatísticas, mas estas não foram suficientes para mostrar a sensibilidade dos métodos fractais na identificação da retinopatia diabética. Conclusão Os métodos fractais não foram capazes de identificar os diferentes graus de retinopatia diabética em retinografias.

Caracterización geométrica de los eritrocitos nucleados de tilapia roja (Oreochromis spp) / Geometric characterization of nucleated erythrocytes of red tilapia (Oreochromis spp)

RESUMEN Introducción. En hematología, el estudio de las alteraciones de la morfología eritrocitaria contribuye con el diagnóstico de la normalidad o anormalidad de estas estructuras, sin embargo, el carácter cualitativo de los criterios diagnósticos dificulta su interpretación y alcance. Objetivo. Caracterizar los eritrocitos nucleados de tilapia roja (Oreochromis spp), en el contexto de la geometría fractal y euclidiana. Metodología. Se tomaron 50 eritrocitos nucleados de 20 extendidos de sangre de tilapia roja. Posteriormente todos los contornos del núcleo y el citoplasma de los eritrocitos fueron delineados, para superponer dos rejillas, una con el doble tamaño que la otra, para calcular mediante el método de Box Counting la dimensión fractal de cada eritrocito delineado. Adicionalmente fue calculada la superficie de estas dos partes del eritrocito. Resultados: Los resultados de este estudio revelaron que los valores de la dimensión fractal no permiten hacer comparaciones entre eritrocitos nucleados. Por su parte, la superposición de rejillas de 5x5 y 10x10 píxeles permitió observar que los valores de ocupación del citoplasma y el núcleo permiten hacer comparaciones entre los eritrocitos nucleados, junto con los valores de la superficie de estas dos partes del eritrocito nucleado. Conclusión: Los eritrocitos nucleados de tilapia roja pueden ser caracterizados mediante la medición de los valores espacios ocupados por su citoplasma y el núcleo, junto con los valores de la superficie de cada una de estas dos partes del eritrocito.

ABSTRACT Introduction. In hematology, the study of erythrocyte morphology alterations contributes to the diagnosis of normality or abnormality of these structures. However, the qualitative nature of the diagnostic criteria makes their interpretation and scope difficult. Objective. Characterize the nucleated erythrocytes of red tilapia (Oreochromis spp) in the context of fractal and Euclidean geometry. Methodology. Fifty nucleated erythrocytes were taken from twenty red tilapia blood smears. Subsequently, all the contours of the nucleus and the cytoplasm of the erythrocytes were delineated to superimpose two grids, one twice the size of the other, to calculate the fractal dimension of each delineated erythrocyte using the Box Counting method. Additionally, the surface of these two parts of the erythrocyte was calculated. Results: This study revealed that the fractal dimension values do not allow comparisons between nucleated erythrocytes. The superposition of 5x5 and 10x10 pixel grids allowed us to observe that the occupancy values of the cytoplasm and the nucleus allow comparisons between the nucleated erythrocytes, together with the values of the surface of these two parts of the nucleated erythrocyte. Conclusion: Red tilapia nucleated erythrocytes can be characterized by measuring the values of the spaces occupied by their cytoplasm and nucleus, together with the values of the surface of each of these two parts of the erythrocyte.

Can the fractal dimension be applied for the early diagnosis of non-proliferative diabetic retinopathy?

The fractal dimension has been employed as a useful parameter in the diagnosis of retinal disease. Avakian et al. (Curr Eye Res 2002; 24: 274-280), comparing the vascular pattern of normal patients with mild to moderate non-proliferative diabetic retinopathy (NPDR), found a significant difference between them only in the macular region. This significant difference in the box-counting fractal dimension of the macular region between normal and mild NPDR patients has been proposed as a method of precocious diagnosis of NPDR. The aim of the present study was to determine if fractal dimensions can really be used as a parameter for the early diagnosis of NPDR. Box-counting and information fractal dimensions were used to parameterize the vascular pattern of the human retina. The two methods were applied to the whole retina and to nine anatomical regions of the retina in 5 individuals with mild NPDR and in 28 diabetic but opthalmically normal individuals (controls), with age between 31 and 86 years. All images of retina were obtained from the Digital Retinal Images for Vessel Extraction (DRIVE) database. The results showed that the fractal dimension parameter was not sensitive enough to be of use for an early diagnosis of NPDR.