Quantification of fractal dimension and Shannon's entropy in histological diagnosis of prostate cancer

Prostate histological investigation in rats exposed to bisphenol a and phytochemicals during the perinatal period and subjected to hormonal stimulus in adulthood

Bisphenol A (BPA) is an environmentally dispersed chemical associated with tumor development. Phytochemicals such as indole-3-carbinol (I3C) and genistein (GEN) have chemoprotective effects on tumor cells. Thus, this study aimed to evaluate the prostatic morphological aspects of rats exposed to BPA, GEN, and I3C during the perinatal period and submitted to hormonal stimulus in adulthood. Blood was collected to obtain hormone concentrations. Slides stained with hematoxylin & eosin, and picrosirius were subjected to fractal, stereological, morphometric, and collagen quantification analysis. I3C decreased the plasma dihydrotestosterone levels, and both phytochemicals increased the plasma estrogen levels. Unlike phytochemicals, BPA did not alter any of the parameters evaluated. GEN reduced the epithelial height, while I3C increased the fractal dimension and stromal collagen. Although BPA did not alter the prostate morphology, the phytochemicals provided beneficial effects for the prostate histological organization in adult animals subjected to hormonal stimulus.

Long exposure to a mixture of endocrine disruptors prediposes the ventral prostate of rats to preneoplastic lesions

Endocrine disruptors (ED) are compounds dispersed in the environment that modify hormone biosynthesis, affecting hormone-dependent organs such as the prostate. Studies have only focused on evaluating the effects of ED alone or in small groups and short intervals and have not adequately portrayed human exposure. Therefore, we characterized the prostate histoarchitecture of rats exposed to an ED mixture (ED Mix) mimicking human exposure. Pregnant females of the Sprague-Dawley strain were randomly distributed into two experimental groups: Control group (vehicle: corn oil, by gavage) and ED Mix group: received 32.11 mg/kg/day of the ED mixture diluted in corn oil (2 ml/kg), by gavage, from gestational day 7 (DG7) to post-natal day 21 (DPN21). After weaning at DPN22, the male pups continued to receive the complete DE mixture until they were 220 days old when they were euthanized. The ED Mix decreased the epithelial compartment, increased the fractal dimension, and decreased glandular dilation. In addition, low-grade prostatic intraepithelial neoplasia was observed in addition to regions of epithelial atrophy in the group exposed to the ED Mix. Exposure to the mixture decreased both types I and III collagen area in the stroma. We concluded that the ED Mix was able to cause alterations in the prostatic histoarchitecture and induce the appearance of preneoplastic lesions.

Fractal analysis is a useful tool for evaluating prostate tissue remodeling caused by ethanol consumption and androgen therapy

Alcohol has been widely consumed for centuries and is linked to the aggravation of diseases. Several studies have shown that excessive consumption of ethanol results in morphophysiological changes in the male reproductive system. One of the effects of ethanol is the decrease in testosterone concentration and hormonal therapies are an alternative to minimize the changes resulting from chronic alcoholism. Qualitative studies were commonly carried out to evaluate the male histopathological alterations resulting from ethanol consumption, being necessary quantitative and non-subjective techniques. This study analyzes the importance of fractal analysis as a useful tool to identify and quantify tissue remodeling in rats submitted to ethanol consumption and hormone therapy with testosterone. Prostate of animals submitted to chronic ethanol consumption showed tissue disorganization, which was confirmed by an increasing of fractal dimension. Regarding the prostatic stroma, collagen fractal dimension and quantification revealed lower values in animals that were only submitted to androgen therapy. Thus, we can conclude that the fractal analysis was a useful tool to quantify tissue changes caused by ethanol consumption and androgen therapy.

Evaluation of entropy and fractal dimension as biomarkers for tumor growth and treatment response using cellular automata

Cell-based models provide a helpful approach for simulating complex systems that exhibit adaptive, resilient qualities, such as cancer. Their focus on individual cell interactions makes them a particularly appropriate strategy to study cancer therapies' effects, which are often designed to disrupt single-cell dynamics. In this work, we propose them as viable methods for studying the time evolution of cancer imaging biomarkers (IBM). We propose a cellular automata model for tumor growth and three different therapies: chemotherapy, radiotherapy, and immunotherapy, following well-established modeling procedures documented in the literature. The model generates a sequence of tumor images, from which a time series of two biomarkers: entropy and fractal dimension, is obtained. Our model shows that the fractal dimension increased faster at the onset of cancer cell dissemination. At the same time, entropy was more responsive to changes induced in the tumor by the different therapy modalities. These observations suggest that the prognostic value of the proposed biomarkers could vary considerably with time. Thus, it is essential to assess their use at different stages of cancer and for different imaging modalities. Another observation derived from the results was that both biomarkers varied slowly when the applied therapy attacked cancer cells scattered along the automatons' area, leaving multiple independent clusters of cells at the end of the treatment. Thus, patterns of change of simulated biomarkers time series could reflect on essential qualities of the spatial action of a given cancer intervention.

Demystifying neuroblastoma malignancy through fractal dimension, entropy, and lacunarity

PURPOSE

Neuroblastoma is a pediatric solid tumor with a prognosis associated with histology and age of the patient, which are the parameters of the well-established current classification (Shimada classification). Despite the development of new treatment options, the prognosis of high-risk neuroblastoma patients is still poor. Therefore, there is a continuous need to stratify the children suffering from this tumor. A mathematical and computational approach is proposed to enable automatic and precise cancer diagnosis on the histological slide.

METHODS

We targeted the complexity of neuroblastoma by calculating its image entropy (S), fractal dimension (FD), and lacunarity (λ) in a combined mathematical code. First, we tested the proposed method for patient-derived glioma images. It allowed distinguishing between normal brain tissue, grade II, and grade III glioma, which harbor different outcomes.

RESULTS

In neuroblastoma, our analysis of image's FD, S, and λ combined with a machine learning algorithm automatically predicted tumor malignancy with a receiver operating characteristic curve of 0.82. FD, S, and λ distinguish between neuroblastoma and ganglioneuroma, but they only partially differentiate between the normal samples and the other classes. Ganglioneuroma, the most differentiated form, and poorly-differentiated neuroblastoma display different values of FD, S, and λ.

CONCLUSIONS

FD, S, and λ of imaging recognize groups in neuroblastic tumors. We suggest that future studies including these features may challenge the current Shimada classification of neuroblastoma with categories of favorable and unfavorable histology. It is expected that this methodology could trigger multicenter studies and potentially find practical use in the clinical setting of children's hospitals worldwide.

Evaluating the effects of orthodontic treatment with clear aligners and conventional brackets on mandibular condyle bone quality using fractal dimension analysis of panoramic radiographs

OBJECTIVE

Orthodontic treatment carried out with clear aligners has become a method that is preferred by patients. In this study, the effect following orthodontic treatment with clear aligners and with conventional brackets, on the mandibular condyle trabecular bone formation is aimed to be examined.

MATERIALS AND METHODS

Thirty two individuals (16 females and 16 males) with a mean age of 16.80 ± 3,10 years who received orthodontic treatment with clear aligners and conventional brackets were included in the study. Individuals included in the study from the archive data were divided into two groups according to treatment types. The mandibular condyle bone quality in the panoramic images taken from the patients before orthodontic treatment (T0) and after orthodontic treatment (T1) has been compared retrospectively. This examination is carried out via the fractal analysis method.

RESULT

The mandibular condyle bone density in individuals that have undergone orthodontic treatment with clear aligners, has been reduced following orthodontic treatment. In individuals that have undergone orthodontic treatment with conventional brackets however the mandibular condyle bone density has increased following treatment (T0) in comparison to before being treated (T1). The mandibular condyle bone density change according to different treatment groups has been found to be statistically different.

CONCLUSION

Orthodontic treatment with clear aligners provides several advantages to patients. However, besides determining the effectiveness of the treatment, complications must also be examined. Application of clear aligners in orthodontic treatment may cause a decrease in bone density in the mandibular condyle. When evaluating orthodontic treatments carried out with clear aligners, the change of the mandibular condyle head bone should be taken into consideration.

CGAT: Cell Graph ATtention Network for Grading of Pancreatic Disease Histology Images

Early detection of Pancreatic Ductal Adenocarcinoma (PDAC), one of the most aggressive malignancies of the pancreas, is crucial to avoid metastatic spread to other body regions. Detection of pancreatic cancer is typically carried out by assessing the distribution and arrangement of tumor and immune cells in histology images. This is further complicated due to morphological similarities with chronic pancreatitis (CP), and the co-occurrence of precursor lesions in the same tissue. Most of the current automated methods for grading pancreatic cancers rely on extensive feature engineering involving accurate identification of cell features or utilising single number spatially informed indices for grading purposes. Moreover, sophisticated methods involving black-box approaches, such as neural networks, do not offer insights into the model's ability to accurately identify the correct disease grade. In this paper, we develop a novel cell-graph based Cell-Graph Attention (CGAT) network for the precise classification of pancreatic cancer and its precursors from multiplexed immunofluorescence histology images into the six different types of pancreatic diseases. The issue of class imbalance is addressed through bootstrapping multiple CGAT-nets, while the self-attention mechanism facilitates visualization of cell-cell features that are likely responsible for the predictive capabilities of the model. It is also shown that the model significantly outperforms the decision tree classifiers built using spatially informed metric, such as the Morisita-Horn (MH) indices.

Fractal dimension analysis as an easy computational approach to improve breast cancer histopathological diagnosis

Histopathology is a well-established standard diagnosis employed for the majority of malignancies, including breast cancer. Nevertheless, despite training and standardization, it is considered operator-dependent and errors are still a concern. Fractal dimension analysis is a computational image processing technique that allows assessing the degree of complexity in patterns. We aimed here at providing a robust and easily attainable method for introducing computer-assisted techniques to histopathology laboratories. Slides from two databases were used: A) Breast Cancer Histopathological; and B) Grand Challenge on Breast Cancer Histology. Set A contained 2480 images from 24 patients with benign alterations, and 5429 images from 58 patients with breast cancer. Set B comprised 100 images of each type: normal tissue, benign alterations, in situ carcinoma, and invasive carcinoma. All images were analyzed with the FracLac algorithm in the ImageJ computational environment to yield the box count fractal dimension (Db) results. Images on set A on 40x magnification were statistically different (p = 0.0003), whereas images on 400x did not present differences in their means. On set B, the mean Db values presented promissing statistical differences when comparing. Normal and/or benign images to in situ and/or invasive carcinoma (all p < 0.0001). Interestingly, there was no difference when comparing normal tissue to benign alterations. These data corroborate with previous work in which fractal analysis allowed differentiating malignancies. Computer-aided diagnosis algorithms may beneficiate from using Db data; specific Db cut-off values may yield ~ 99% specificity in diagnosing breast cancer. Furthermore, the fact that it allows assessing tissue complexity, this tool may be used to understand the progression of the histological alterations in cancer.

Quantitative phase and polarization imaging through an optical fiber applied to detection of early esophageal tumorigenesis

Phase and polarization of coherent light are highly perturbed by interaction with microstructural changes in premalignant tissue, holding promise for label-free detection of early tumors in endoscopically accessible tissues such as the gastrointestinal tract. Flexible optical multicore fiber (MCF) bundles used in conventional diagnostic endoscopy and endomicroscopy scramble phase and polarization, restricting clinicians instead to low-contrast amplitude-only imaging. We apply a transmission matrix characterization approach to produce full-field en-face images of amplitude, quantitative phase, and resolved polarimetric properties through an MCF. We first demonstrate imaging and quantification of biologically relevant amounts of optical scattering and birefringence in tissue-mimicking phantoms. We present an entropy metric that enables imaging of phase heterogeneity, indicative of disordered tissue microstructure associated with early tumors. Finally, we demonstrate that the spatial distribution of phase and polarization information enables label-free visualization of early tumors in esophageal mouse tissues, which are not identifiable using conventional amplitude-only information.

Light localization properties of weakly disordered optical media using confocal microscopy: application to cancer detection

We have developed a novel technique to quantify submicron scale mass density fluctuations in weakly disordered heterogeneous optical media using confocal fluorescence microscopy. Our method is based on the numerical evaluation of the light localization properties of an 'optical lattice' constructed from the pixel intensity distributions of images obtained with confocal fluorescence microscopy. Here we demonstrate that the technique reveals differences in the mass density fluctuations of the fluorescently labeled molecules between normal and cancer cells, and that it has the potential to quantify the degree of malignancy of cancer cells. Potential applications of the technique to other disease situations or characterizing disordered samples are also discussed.

Is cancer a pure growth curve or does it follow a kinetics of dynamical structural transformation?

BACKGROUND

Unperturbed tumor growth kinetics is one of the more studied cancer topics; however, it is poorly understood. Mathematical modeling is a useful tool to elucidate new mechanisms involved in tumor growth kinetics, which can be relevant to understand cancer genesis and select the most suitable treatment.

METHODS

The classical Kolmogorov-Johnson-Mehl-Avrami as well as the modified Kolmogorov-Johnson-Mehl-Avrami models to describe unperturbed fibrosarcoma Sa-37 tumor growth are used and compared with the Gompertz modified and Logistic models. Viable tumor cells (1×10⁵) are inoculated to 28 BALB/c male mice.

RESULTS

Modified Gompertz, Logistic, Kolmogorov-Johnson-Mehl-Avrami classical and modified Kolmogorov-Johnson-Mehl-Avrami models fit well to the experimental data and agree with one another. A jump in the time behaviors of the instantaneous slopes of classical and modified Kolmogorov-Johnson-Mehl-Avrami models and high values of these instantaneous slopes at very early stages of tumor growth kinetics are observed.

CONCLUSIONS

The modified Kolmogorov-Johnson-Mehl-Avrami equation can be used to describe unperturbed fibrosarcoma Sa-37 tumor growth. It reveals that diffusion-controlled nucleation/growth and impingement mechanisms are involved in tumor growth kinetics. On the other hand, tumor development kinetics reveals dynamical structural transformations rather than a pure growth curve. Tumor fractal property prevails during entire TGK.

Fractal Dimension in Quantifying Experimental-Pulmonary-Hypertension-Induced Cardiac Dysfunction in Rats

BACKGROUND

Right-sided heart failure has high morbidity and mortality, and may be caused by pulmonary arterial hypertension. Fractal dimension is a differentiated and innovative method used in histological evaluations that allows the characterization of irregular and complex structures and the quantification of structural tissue changes.

OBJECTIVE

To assess the use of fractal dimension in cardiomyocytes of rats with monocrotaline-induced pulmonary arterial hypertension, in addition to providing histological and functional analysis.

METHODS

Male Wistar rats were divided into 2 groups: control (C; n = 8) and monocrotaline-induced pulmonary arterial hypertension (M; n = 8). Five weeks after pulmonary arterial hypertension induction with monocrotaline, echocardiography was performed and the animals were euthanized. The heart was dissected, the ventricles weighed to assess anatomical parameters, and histological slides were prepared and stained with hematoxylin/eosin for fractal dimension analysis, performed using box-counting method. Data normality was tested (Shapiro-Wilk test), and the groups were compared with non-paired Student t test or Mann Whitney test (p < 0.05).

RESULTS

Higher fractal dimension values were observed in group M as compared to group C (1.39 ± 0.05 vs. 1.37 ± 0.04; p < 0.05). Echocardiography showed lower pulmonary artery flow velocity, pulmonary acceleration time and ejection time values in group M, suggesting function worsening in those animals.

CONCLUSION

The changes observed confirm pulmonary-arterial-hypertension-induced cardiac dysfunction, and point to fractal dimension as an effective method to evaluate cardiac morphological changes induced by ventricular dysfunction.

Fractal analysis and the diagnostic usefulness of silver staining nucleolar organizer regions in prostate adenocarcinoma

Pathological diagnosis of prostate adenocarcinoma often requires complementary methods. On prostate biopsy tissue from 39 patients including benign nodular hyperplasia (BNH), atypical adenomatous hyperplasia (AAH), and adenocarcinomas, we have performed combined histochemical-immunohistochemical stainings for argyrophilic nucleolar organizer regions (AgNORs) and glandular basal cells. After ascertaining the pathology, we have analyzed the number, roundness, area, and fractal dimension of individual AgNORs or of their skeleton-filtered maps. We have optimized here for the first time a combination of AgNOR morphological denominators that would reflect best the differences between these pathologies. The analysis of AgNORs' roundness, averaged from large composite images, revealed clear-cut lower values in adenocarcinomas compared to benign and atypical lesions but with no differences between different Gleason scores. Fractal dimension (FD) of AgNOR silhouettes not only revealed significant lower values for global cancer images compared to AAH and BNH images, but was also able to differentiate between Gleason pattern 2 and Gleason patterns 3–5 adenocarcinomas. Plotting the frequency distribution of the FDs for different pathologies showed clear differences between all Gleason patterns and BNH. Together with existing morphological classifiers, AgNOR analysis might contribute to a faster and more reliable machine-assisted screening of prostatic adenocarcinoma, as an essential aid for pathologists.

[Objective grading of prostate carcinoma based on fractal dimensions: Gleason 3 + 4= 7a ≠ Gleason 4 + 3 =7b]

BACKGROUND

Significant intra- and interobserver variability ranging between 40 and 80% is observed in tumor grading of prostate carcinoma. By combining geometric and statistical methods, an objective system of grading can be designed.

MATERIAL AND METHODS

The distributions of cell nuclei in two-dimensional patterns of prostate cancer classified subjectively as Gleason score 3+3, 3+4, 4+3, 4+4, 4+5, 5+4, and 5+5 were analyzed with algorithms measuring the global fractal dimensions of the Rényi family and with the algorithm for the local connected fractal dimension (LCFD).

RESULTS

The dimensions for global fractal capacity, information, and correlation (standard deviation) were 1.470 (045), 1.528 (046), and 1.582 (099) for homogenous Gleason grade 3 (n = 16), 1.642 (034), 1.678 (041), and 1.673 (084) for homogenous Gleason grade 4 (n=18), and 1.797 (042), 1.791 (026), and 1.854 (031) for homogenous Gleason grade 5 (n=12), respectively. The LCFD algorithm can be used to distinguish both qualitatively and quantitatively between mixed and heterogeneous patterns, such as Gleason score 3+4=7a (intermediate risk cancer) and Gleason score 4+3=7b (high-risk cancer). Sensitivity of the method is 89.3%, and specificity 84.3%.

CONCLUSION

The method of fractal geometry enables both an objective and quantitative grading of prostate cancer.

On the relationship between tumor structure and complexity of the spatial distribution of cancer cell nuclei: a fractal geometrical model of prostate carcinoma

BACKGROUND

A risk of the prostate cancer patient is defined by both the objective and subjective criteria, that is, PSA concentration, Gleason score, and pTNM-stage. The subjectivity of tumor grading influences the risk assessment owing to a large inter- and intra-observer variability. Pathologists propose a central prostate pathology review as a remedy for this problem; yet, the review cannot eliminate the subjectivity from the diagnostic algorithm. The spatial distribution of cancer cell nuclei changes during tumor progression. It implies changes in complexity measured by the capacity dimension D0, the information dimension D1, and the correlation dimension D2.

METHODS

The cornerstone of the approach is a model of prostate carcinomas composed of the circular fractals CF(4), CF(6 + 0), and CF(6 + 1). This model is both geometrical and analytical, that is, its structure is well-defined, the capacity fractal dimension D0 can be calculated for the infinite circular fractals, and the dimensions D0, D1, D2 can be computed for their finite counterparts representing distribution of cell nuclei. The model enabled both the calibration of the software and the validation of the measurements in 124 prostate carcinomas. The ROC analysis defined the cut-off D0 values for seven classes of complexity.

RESULTS

The Gleason classification matched in part with the classification based on the D0 values. The mean ROC sensitivity was 81.3% and the mean ROC specificity 75.2%. Prostate carcinomas were re-stratified into seven classes of complexity according to their D0 values. This increased both the mean ROC sensitivity and the mean ROC specificity to 100%. All homogeneous Gleason patterns were subordinated to the class C1, C4, or C7. D0 = 1.5820 was the cut-off D0 value between the complexity class C2 and C3 representing low-risk cancers and intermediate-risk cancers, respectively.

CONCLUSIONS

The global fractal dimensions eliminate the subjectivity in the diagnostic algorithm of prostate cancer. Those complexity measures enable the objective subordination of carcinomas to the well-defined complexity classes, and define subgroups of carcinomas with very low malignant potential (complexity class C1) or at a large risk of progression (complexity ass C7).

Comparison of fractal dimension and Shannon entropy in myocytes from rats treated with histidine-tryptophan-glutamate and histidine-tryptophan cetoglutarate

INTRODUCTION

Solutions that cause elective cardiac arrest are constantly evolving, but the ideal compound has not yet been found. The authors compare a new cardioplegic solution with histidine-tryptophan-glutamate (Group 2) and other one with histidine-tryptophan-cetoglutarate (Group 1) in a model of isolated rat heart.

OBJECTIVE

To quantify the fractal dimension and Shannon entropy in rat myocytes subjected to cardioplegia solution using histidine-tryptophan with glutamate in an experimental model, considering the caspase markers, IL-8 and KI-67.

METHODS

Twenty male Wistar rats were anesthetized and heparinized. The chest was opened, the heart was withdrawn and 40 ml/kg of cardioplegia (with histidine-tryptophan-cetoglutarate or histidine-tryptophan-glutamate solution) was infused. The hearts were kept for 2 hours at 4ºC in the same solution, and thereafter placed in the Langendorff apparatus for 30 min with Ringer-Locke solution. Analyzes were performed for immunohistochemical caspase, IL-8 and KI-67.

RESULTS

The fractal dimension and Shannon entropy were not different between groups histidine-tryptophan-glutamate and histidine-tryptophan-acetoglutarate.

CONCLUSION

The amount of information measured by Shannon entropy and the distribution thereof (given by fractal dimension) of the slices treated with histidine-tryptophan-cetoglutarate and histidine-tryptophan-glutamate were not different, showing that the histidine-tryptophan-glutamate solution is as good as histidine-tryptophan-acetoglutarate to preserve myocytes in isolated rat heart.

Proliferation, apoptosis, and fractal dimension analysis for the quantification of intestinal trophism in sole (Solea solea) fed mussel meal diets

Background

The evaluation of intestinal trophism, mainly the mucosal layer, is an important issue in various conditions associated with injury, atrophy, recovery, and healing of the gut. The aim of the present study was to evaluate the kinetics of the proliferation and apoptosis of enterocytes by immunohistochemistry and to assess the complexity of intestinal mucosa by fractal dimension (FD) analysis in Solea solea fed different experimental diets.

Results

Histomorphological evaluation of all intestinal segments did not show signs of degeneration or inflammation. Cell proliferation index and FD were significantly reduced with a diet high in mussel meal (MM; p = 0.0034 and p = 0.01063, respectively), while apoptotic index did not show any significant difference for the same comparison (p = 0.3859). Linear regression analysis between apoptotic index (independent variable) and FD (dependent variable) showed a statistically significant inverse relationship (p = 0.002528). Linear regression analysis between cell proliferation index (independent variable) and FD (dependent variable) did not show any significant correlation (p = 0.131582).

Conclusions

The results demonstrated that diets containing increasing levels of mussel meal in substitution of fishmeal did not incite a hyperplastic response of the intestinal mucosa. The mussel meal, which is derived from molluscs, could mimic the characteristics of the sole’s natural prey, being readily digestible, even without increasing the absorptive surface of intestinal mucosa. Interestingly, from this study emerged that FD could be used as a numeric indicator complementary to in situ quantification methods to measure intestinal trophism, in conjunction with functional parameters.

Stratification of gene coexpression patterns and GO function mining for a RNA-Seq data series

RNA-Seq is emerging as an increasingly important tool in biological research, and it provides the most direct evidence of the relationship between the physiological state and molecular changes in cells. A large amount of RNA-Seq data across diverse experimental conditions have been generated and deposited in public databases. However, most developed approaches for coexpression analyses focus on the coexpression pattern mining of the transcriptome, thereby ignoring the magnitude of gene differences in one pattern. Furthermore, the functional relationships of genes in one pattern, and notably among patterns, were not always recognized. In this study, we developed an integrated strategy to identify differential coexpression patterns of genes and probed the functional mechanisms of the modules. Two real datasets were used to validate the method and allow comparisons with other methods. One of the datasets was selected to illustrate the flow of a typical analysis. In summary, we present an approach to robustly detect coexpression patterns in transcriptomes and to stratify patterns according to their relative differences. Furthermore, a global relationship between patterns and biological functions was constructed. In addition, a freely accessible web toolkit “coexpression pattern mining and GO functional analysis” (COGO) was developed.

Marker-free phenotyping of tumor cells by fractal analysis of reflection interference contrast microscopy images

Phenotyping of tumor cells by marker-free quantification is important for cancer diagnostics. For the first time, fractal analysis of reflection interference contrast microscopy images of single living cells was employed as a new method to distinguish between different nanoscopic membrane features of tumor cells. Since tumor progression correlates with a higher degree of chaos within the cell, it can be quantified mathematically by fractality. Our results show a high accuracy in identifying malignant cells with a failure chance of 3%, which is far better than today's applied methods.