The application of fractal geometric analysis to microscopic images

A Self-Similarity Logic May Shape the Organization of the Nervous System

From the morphological point of view, the nervous system exhibits a fractal, self-similar geometry at various levels of observations, from single cells up to cell networks. From the functional point of view, it is characterized by a hierarchical organization in which self-similar structures (networks) of different miniaturizations are nested within each other. In particular, neuronal networks, interconnected to form neuronal systems, are formed by neurons, which operate thanks to their molecular networks, mainly having proteins as components that via protein-protein interactions can be assembled in multimeric complexes working as micro-devices. On this basis, the term "self-similarity logic" was introduced to describe a nested organization where, at the various levels, almost the same rules (logic) to perform operations are used. Self-similarity and self-similarity logic both appear to be intimately linked to the biophysical evidence for the nervous system being a pattern-forming system that can flexibly switch from one coherent state to another. Thus, they can represent the key concepts to describe its complexity and its concerted, holistic behavior.

The Fractal Geometry of the Brain: AnOverview

The first chapter of this book introduces some history, philosophy, and basic concepts of fractal geometry and discusses how the neurosciences can benefit from applying computational fractal-based analysis. Further, it compares fractal with Euclidean approaches to analyzing and quantifying the brain in its entire physiopathological spectrum and presents an overview of the first section of this book as well.

Fractal Analysis in Clinical Neurosciences: An Overview

Over the last years, fractals have entered into the realms of clinical neurosciences. The whole brain and its components (i.e., neurons and astrocytes) have been studied as fractal objects, and even more relevant, the fractal-based quantification of the geometrical complexity of histopathological and neuroradiological images as well as neurophysiopathological time series has suggested the existence of a gradient in the pattern representation of neurological diseases. Computational fractal-based parameters have been suggested as potential diagnostic and prognostic biomarkers in different brain diseases, including brain tumors, neurodegeneration, epilepsy, demyelinating diseases, cerebrovascular malformations, and psychiatric disorders as well. This chapter and the entire third section of this book are focused on practical applications of computational fractal-based analysis into the clinical neurosciences, namely, neurology and neuropsychiatry, neuroradiology and neurosurgery, neuropathology, neuro-oncology and neurorehabilitation, neuro-ophthalmology, and cognitive neurosciences, with special emphasis on the translation of the fractal dimension and other fractal parameters as clinical biomarkers useful from bench to bedside.

Computational Fractal-Based Analysis of Brain Tumor Microvascular Networks

Brain parenchyma microvasculature is set in disarray in the presence of tumors, and malignant brain tumors are among the most vascularized neoplasms in humans. As microvessels can be easily identified in histologic specimens, quantification of microvascularity can be used alone or in combination with other histological features to increase the understanding of the dynamic behavior, diagnosis, and prognosis of brain tumors. Different brain tumors, and even subtypes of the same tumor, show specific microvascular patterns, as a kind of "microvascular fingerprint," which is particular to each histotype. Reliable morphometric parameters are required for the qualitative and quantitative characterization of the neoplastic angioarchitecture, although the lack of standardization of a technique able to quantify the microvascular patterns in an objective way has limited the "morphometric approach" in neuro-oncology.In this chapter, we focus on the importance of computational-based morphometrics, for the objective description of tumoral microvascular fingerprinting. By also introducing the concept of "angio-space," which is the tumoral space occupied by the microvessels, we here present fractal analysis as the most reliable computational tool able to offer objective parameters for the description of the microvascular networks.The spectrum of different angioarchitectural configurations can be quantified by means of Euclidean and fractal-based parameters in a multiparametric analysis, aimed to offer surrogate biomarkers of cancer. Such parameters are here described from the methodological point of view (i.e., feature extraction) as well as from the clinical perspective (i.e., relation to underlying physiology), in order to offer new computational parameters to the clinicians with the final goal of improving diagnostic and prognostic power of patients affected by brain tumors.

Hypertrophic cardiomyopathy or athlete's heart? A systematic review of novel cardiovascular magnetic resonance imaging parameters

Hypertrophic cardiomyopathy (HCM) is a common cause of sudden cardiac death in athletes. Cardiac Magnetic Resonance (CMR) imaging is considered an excellent tool to differentiate between HCM and athlete's heart. The aim of this systematic review was to highlight the novel CMR-derived parameters with significant discriminative capacity between the two conditions. A systematic search in the MEDLINE, EMBASE and Cochrane Reviews databases was performed. Eligible studies were considered the ones comparing novel CMR-derived parameters on athletes and HCM patients. Therefore, studies that only examined Cine-derived volumetric parameters were excluded. Particular attention was given to binary classification results from multi-variate regression models and ROC curve analyses. Bias assessment was performed with the Quality Assessment on Diagnostic Accuracy Studies. Five (5) studies were included in the systematic review, with a total of 284 athletes and 373 HCM patients. Several novel indices displayed discriminatory potential, such as native T1 mapping and T2 values, LV global longitudinal strain, late gadolinium enhancement and whole-LV fractal dimension. Diffusion tensor imaging enabled quantification of the secondary eigenvalue angle and fractional anisotropy in one study, which also proved capable of reliably detecting HCM in a mixed athlete/patient sample. Several novel CMR-derived parameters, most of which are currently under development, show promising results in discerning between athlete's heart and HCM. Prospective studies examining the discriminatory capacity of all promising modalities side-by-side will yield definitive answers on their relative importance; diagnostic models can incorporate the best performing variables for optimal results.

Machine learning-enabled cancer diagnostics with widefield polarimetric second-harmonic generation microscopy

The extracellular matrix (ECM) collagen undergoes major remodeling during tumorigenesis. However, alterations to the ECM are not widely considered in cancer diagnostics, due to mostly uniform appearance of collagen fibers in white light images of hematoxylin and eosin-stained (H&E) tissue sections. Polarimetric second-harmonic generation (P-SHG) microscopy enables label-free visualization and ultrastructural investigation of non-centrosymmetric molecules, which, when combined with texture analysis, provides multiparameter characterization of tissue collagen. This paper demonstrates whole slide imaging of breast tissue microarrays using high-throughput widefield P-SHG microscopy. The resulting P-SHG parameters are used in classification to differentiate tumor from normal tissue, resulting in 94.2% for both accuracy and F1-score, and 6.3% false discovery rate. Subsequently, the trained classifier is employed to predict tumor tissue with 91.3% accuracy, 90.7% F1-score, and 13.8% false omission rate. As such, we show that widefield P-SHG microscopy reveals collagen ultrastructure over large tissue regions and can be utilized as a sensitive biomarker for cancer diagnostics and prognostics studies.

Advanced micromorphology study of microbial films grown on Kefir loaded with Açaí extract

In this work, an advanced analysis of the 3D surface microtexture of the microbial films grown on Kefir loaded with Açaí extract was performed. Atomic force microscopy was used to characterize the 3D surface microtexture data in correlation with the stereometric analyses to allow a better understanding of the surface micromorphology consistent with ISO 25178-2: 2012. Two new parameters, fractal succolarity and fractal lacunarity, have been inserted for a quantitative approach to microtexture. The results revealed that the morphology was affected by the increase of the Açaí concentration in biofilms, as well as the fractal succolarity and lacunarity. Adhesive bacteria of the genus Lactobacillus were observed for the lowest concentrations of Açaí. Moreover, it was found that the surface of the biofilms has shown saturation when the concentration has changed from 4 to 6 % of Açaí. These results are of great interest in the characterization of surfaces with promising application like skin dressing.

Fractal analysis of concurrently prepared latex rubber casts of the bronchial and vascular systems of the human lung

Fractal geometry (FG) is a branch of mathematics that instructively characterizes structural complexity. Branched structures are ubiquitous in both the physical and the biological realms. Fractility has therefore been termed nature's design. The fractal properties of the bronchial (airway) system, the pulmonary artery and the pulmonary vein of the human lung generates large respiratory surface area that is crammed in the lung. Also, it permits the inhaled air to intimately approximate the pulmonary capillary blood across a very thin blood-gas barrier through which gas exchange to occur by diffusion. Here, the bronchial (airway) and vascular systems were simultaneously cast with latex rubber. After corrosion, the bronchial and vascular system casts were physically separated and cleared to expose the branches. The morphogenetic (Weibel's) ordering method was used to categorize the branches on which the diameters and the lengths, as well as the angles of bifurcation, were measured. The fractal dimensions (DF) were determined by plotting the total branch measurements against the mean branch diameters on double logarithmic coordinates (axes). The diameter-determined DF values were 2.714 for the bronchial system, 2.882 for the pulmonary artery and 2.334 for the pulmonary vein while the respective values from lengths were 3.098, 3.916 and 4.041. The diameters yielded DF values that were consistent with the properties of fractal structures (i.e. self-similarity and space-filling). The data obtained here compellingly suggest that the design of the bronchial system, the pulmonary artery and the pulmonary vein of the human lung functionally comply with the Hess-Murray law or 'the principle of minimum work'.

Hyperoxia Alters Ultrastructure and Induces Apoptosis in Leukemia Cell Lines

Oxygenation conditions are crucial for growth and tumor progression. Recent data suggests a decrease in cancer cell proliferation occurring after exposure to normobaric hyperoxia. Those changes are associated with fractal dimension. The purpose of this research was to study the impact of hyperoxia on apoptosis and morphology of leukemia cell lines. Two hematopoietic lymphoid cancer cell lines (a T-lymphoblastoid line, JURKAT and a B lymphoid line, CCRF-SB) were tested under conditions of normobaric hyperoxia (FiO₂ > 60%, ± 18h) and compared to a standard group (FiO₂ = 21%). We tested for apoptosis using a caspase-3 assay. Cell morphology was evaluated by cytospin, microphotography after coloration, and analysis by a fractal dimension calculation software. Our results showed that exposure of cell cultures to transient normobaric hyperoxia induced apoptosis (elevated caspase-3) as well as significant and precocious modifications in cell complexity, as highlighted by increased fractal dimensions in both cell lines. These features are associated with changes in structure (pycnotic nucleus and apoptosis) recorded by microscopic analysis. Such morphological alterations could be due to several molecular mechanisms and rearrangements in the cancer cell, leading to cell cycle inhibition and apoptosis as shown by caspase-3 activity. T cells seem less resistant to hyperoxia than B cells.

New Features for Neuron Classification

This paper addresses the problem of obtaining new neuron features capable of improving results of neuron classification. Most studies on neuron classification using morphological features have been based on Euclidean geometry. Here three one-dimensional (1D) time series are derived from the three-dimensional (3D) structure of neuron instead, and afterwards a spatial time series is finally constructed from which the features are calculated. Digitally reconstructed neurons were separated into control and pathological sets, which are related to three categories of alterations caused by epilepsy, Alzheimer's disease (long and local projections), and ischemia. These neuron sets were then subjected to supervised classification and the results were compared considering three sets of features: morphological, features obtained from the time series and a combination of both. The best results were obtained using features from the time series, which outperformed the classification using only morphological features, showing higher correct classification rates with differences of 5.15, 3.75, 5.33% for epilepsy and Alzheimer's disease (long and local projections) respectively. The morphological features were better for the ischemia set with a difference of 3.05%. Features like variance, Spearman auto-correlation, partial auto-correlation, mutual information, local minima and maxima, all related to the time series, exhibited the best performance. Also we compared different evaluators, among which ReliefF was the best ranked.

In vivo erosion of orthopedic screws prepared from nacre (mother of pearl)

BACKGROUND

Biodegradable biomaterials have been proposed to prepare orthopedic devices. Nacre is a natural aragonitic material made of calcium carbonate and is bioerodible.

WORKING HYPOTHESIS

We postulated that nacre is biodegradable without provoking bone erosion and favors bone apposition.

MATERIAL AND METHODS

We prepared orthopedic screws from nacre of the giant oyster Pinctada maxima. Threaded screws (3.5mm diameter) were implanted in 6 ewes in the upper tibial metaphysis (3 to 4 screws per animal). Their trajectory was transcortical and intramedullary to the opposite cortex. Animals were kept for 3months (n=2) and 6 months (n=4). They did not develop local inflammation. Before euthanasia, they received a double calcein labeling. Bone samples were analyzed by X-ray nanotomography and histology after embedding in poly(methyl methacrylate). The fractal dimension of the screw profiles (measured by the box-counting method) was used to quantify surface erosion.

RESULTS

3D nanotomography showed a gradual erosion of the threads, which was confirmed by a decreased fractal dimension. Histologically, multinucleated cells (non-osteoclastic appearance) were visible at the surface of the screws. No ruffled border was seen in these cells but they had extensions creeping in the organic matter between the aragonite tablets. Bone apposition was noted in the transcortical path of the screws with limited osteoconduction at the endosteum. Mineralization rate was increased in these zones composed of woven bone in contact with the nacre.

DISCUSSION AND CONCLUSION

Screws prepared from nacre have the advantage of an in vivo resorbability by macrophage-derived cells and an osteoconductive apposition in contact with the material without triggering a local inflammatory reaction.

Spatial Heterogeneity in the Tumor Microenvironment

Recent developments in studies of tumor heterogeneity have provoked new thoughts on cancer management. There is a desperate need to understand influence of the tumor microenvironment on cancer development and evolution. Applying principles and quantitative methods from ecology can suggest novel solutions to fulfil this need. We discuss spatial heterogeneity as a fundamental biological feature of the microenvironment, which has been largely ignored. Histological samples can provide spatial context of diverse cell types coexisting within the microenvironment. Advanced computer-vision techniques have been developed for spatial mapping of cells in histological samples. This has enabled the applications of experimental and analytical tools from ecology to cancer research, generating system-level knowledge of microenvironmental spatial heterogeneity. We focus on studies of immune infiltrate and tumor resource distribution, and highlight statistical approaches for addressing the emerging challenges based on these new approaches.

Fractal frontiers in cardiovascular magnetic resonance: towards clinical implementation

Many of the structures and parameters that are detected, measured and reported in cardiovascular magnetic resonance (CMR) have at least some properties that are fractal, meaning complex and self-similar at different scales. To date however, there has been little use of fractal geometry in CMR; by comparison, many more applications of fractal analysis have been published in MR imaging of the brain.This review explains the fundamental principles of fractal geometry, places the fractal dimension into a meaningful context within the realms of Euclidean and topological space, and defines its role in digital image processing. It summarises the basic mathematics, highlights strengths and potential limitations of its application to biomedical imaging, shows key current examples and suggests a simple route for its successful clinical implementation by the CMR community.By simplifying some of the more abstract concepts of deterministic fractals, this review invites CMR scientists (clinicians, technologists, physicists) to experiment with fractal analysis as a means of developing the next generation of intelligent quantitative cardiac imaging tools.

Digital differentiation of non-small cell carcinomas of the lung by the fractal dimension of their epithelial architecture

INTRODUCTION

In recent years, differences have emerged in the treatment of squamous and non-squamous non-small cell lung carcinomas (NSCLCs). This highlights the importance of accurate histopathologic classification. However, there remains inter-observer disagreement when making diagnoses based on histology. Fractal dimension (FD) is a mathematical measure of irregularity and complexity of shape. We hypothesize that the FD of carcinoma epithelial architecture can assist in differentiating adenocarcinoma (ADC) from squamous cell carcinoma (SCC) of the lung.

METHODS

134 resected (88 ADC and 46 SCC) cases of resected early-stage NSCLC were analyzed. Tissue micro arrays were generated from formalin-fixed paraffin-embedded tissue, stained with pan-cytokeratin, and digitally imaged and the FD of the epithelial structure calculated. Mean FD of ADC and SCC were compared using the independent t-test, partial correlations, and receiver operating characteristic (ROC) analyses.

RESULTS

A statistically significant difference (p<0.001) between the mean FD of ADC (M=1.70, SD=0.07) and SCC (M=1.78, SD=0.07) was found. Significance remained (p<0.001) when controlling for several possible confounders. ROC analysis demonstrated an area-under-the-curve of 0.81 (p<0.001).

CONCLUSIONS

The epithelial structure FD of NSCLC has potential as a reproducible and automated measure to help subtype NSCLCs into ADC and SCC. With further image analysis algorithm improvements, fractal analysis may be a component in computerized histomorphological assessments of lung cancer and may provide an adjunct test in differentiating NSCLCs.

Microvascular morphometrics of the hypophysis and pituitary tumors: from bench to operating theatre

The idea that microvasculature might be a histopathological biomarker in the prognosis and treatment of tumors is garnering even more attention in the scientific community. The roles of neovascularity in tumor progression and metastasis, have become a hot-topic of investigation in cancer research. A number of methods of quantitatively analyzing pituitary adenoma microvasculature have been applied, and fractal analysis is emerging as a potential effective model for this aim. Additionally, new and more specific immunological techniques have been developed for the detection of microvessels. CD105 (Endoglin) has been proposed as a valuable antigen that marks only newly formed vessels, rather than the entire tumor microvascular system. The combination of different types of immunostaining techniques for the detection of microvessels in pituitary adenomas with fractal analysis as an objective and computer-aided technique to quantify and describe morphological aspects of microvessels has potential implications in future clinical and surgical applications. Tumor treatments, such as anti-angiogenic therapy, as well as intraoperative tools, stand to be enhanced by increasing advances in microvascular research. We here review the methods used for the quantitative analysis of microvessels of the pituitary in its physiopathological states, with the aim to show the pituitary adenoma as a model for the study of neoplastic angioarchitecture and the importance of the introduction of new techniques for the study of angiogenesis, with the relative scientific, medical and surgical implications.

Application of multifractal analysis on microscopic images in the classification of metastatic bone disease

The paper considers the method, based on multifractal (MF) analysis, for classifying the shape of tissue cells from microscopis images, identifying the primary cancer in cases of metastasis bone disease. Diagnosis of primary cancer is of great importance, because further treatment depends on how successful and accurate that diagnosis is. This method can be applied as an additional and objective tool in primary cancer diagnosis, as well as in decreasing of the subjective factor and error probability. The method is tested over a large number (1050) of clinical cases from the Institute of Pathology, University of Belgrade. The results of computer-aided analysis of images have been presented and discussed.

Fractal properties of macrophage membrane studied by AFM

Complexity of cell membrane poses difficulties to quantify corresponding morphology changes during cell proliferation and damage. We suggest using fractal dimension of the cell membrane to quantify its complexity and track changes produced by various treatments. Glutaraldehyde fixed mouse RAW 264.7 macrophage membranes were chosen as model system and imaged in PeakForce QNM (quantitative nanomechanics) mode of AFM (atomic force microscope). The morphology of the membranes was characterized by fractal dimension. The parameter was calculated for set of AFM images by three different methods. The same calculations were done for the AFM images of macrophages treated with colchicine, an inhibitor of the microtubule polymerization, and microtubule stabilizing agent taxol. We conclude that fractal dimension can be additional and useful parameter to characterize the cell membrane complexity and track the morphology changes produced by different treatments.

Fractal analysis for classification of breast carcinoma in optical coherence tomography

The accurate and rapid assessment of tumor margins during breast cancer resection using optical coherence tomography (OCT) has the potential to reduce patient risk. However, it is difficult to subjectively distinguish cancer from normal fibroglandular stromal tissues in OCT images, and an objective measure is needed. In this initial study, we investigate the potential of a one-dimensional fractal box-counting method for cancer classification in OCT. We computed the fractal dimension, a measure of the self-similarity of an object, along the depth axis of 44 ultrahigh-resolution OCT images of human breast tissues obtained from 4 cancer patients. Correlative histology was employed to identify distinct regions of adipose, stroma, and cancer in the OCT images. We report that the fractal dimension of stroma is significantly higher than that of cancer (P < 10(-5), t-test). Furthermore, by adjusting the cutoff values of fractal dimension between cancer, stroma, and adipose tissues, sensitivities and specificities of either 82.4% and 88.9%, or 88.2% and 81.5%, are obtained, respectively, for cancer classification. The use of fractal analysis with OCT could potentially provide automated identification of tumor margins during breast-sparing surgery.

Atomic force microscopy comes of age

AFM (atomic force microscopy) analysis, both of fixed cells, and live cells in physiological environments, is set to offer a step change in the research of cellular function. With the ability to map cell topography and morphology, provide structural details of surface proteins and their expression patterns and to detect pico-Newton force interactions, AFM represents an exciting addition to the arsenal of the cell biologist. With the explosion of new applications, and the advent of combined instrumentation such as AFM-confocal systems, the biological application of AFM has come of age. The use of AFM in the area of biomedical research has been proposed for some time, and is one where a significant impact could be made. Fixed cell analysis provides qualitative and quantitative subcellular and surface data capable of revealing new biomarkers in medical pathologies. Image height and contrast, surface roughness, fractal, volume and force analysis provide a platform for the multiparameter analysis of cell and protein functions. Here, we review the current status of AFM in the field and discuss the important contribution AFM is poised to make in the understanding of biological systems.

Roundness variation in JPEG images affects the automated process of nuclear immunohistochemical quantification: correction with a linear regression model

The volume of digital image (DI) storage continues to be an important problem in computer-assisted pathology. DI compression enables the size of files to be reduced but with the disadvantage of loss of quality. Previous results indicated that the efficiency of computer-assisted quantification of immunohistochemically stained cell nuclei may be significantly reduced when compressed DIs are used. This study attempts to show, with respect to immunohistochemically stained nuclei, which morphometric parameters may be altered by the different levels of JPEG compression, and the implications of these alterations for automated nuclear counts, and further, develops a method for correcting this discrepancy in the nuclear count. For this purpose, 47 DIs from different tissues were captured in uncompressed TIFF format and converted to 1:3, 1:23 and 1:46 compression JPEG images. Sixty-five positive objects were selected from these images, and six morphological parameters were measured and compared for each object in TIFF images and those of the different compression levels using a set of previously developed and tested macros. Roundness proved to be the only morphological parameter that was significantly affected by image compression. Factors to correct the discrepancy in the roundness estimate were derived from linear regression models for each compression level, thereby eliminating the statistically significant differences between measurements in the equivalent images. These correction factors were incorporated in the automated macros, where they reduced the nuclear quantification differences arising from image compression. Our results demonstrate that it is possible to carry out unbiased automated immunohistochemical nuclear quantification in compressed DIs with a methodology that could be easily incorporated in different systems of digital image analysis.

Quantifying the architectural complexity of microscopic images of histology specimens

Tumour grade (a measure of the degree of cellular differentiation of malignant neoplasm) is an important prognostic factor in many types of cancer. In general, poorly differentiated tumours are characterized by a higher degree of architectural irregularity and complexity of histological structures. Fractal dimension is a useful parameter for characterizing complex irregular structures. However, one of the difficulties of estimating the fractal dimension from microscopic images is the segmentation of pathologically relevant structures for analysis. A commonly used technique to segment structures of interest is to apply a pixel intensity threshold to convert the original image to binary and extract pixel outline structures from the binary representation. The difficulty with this approach is that the value of the threshold required to segment the histological structures is highly dependent on the staining technique chosen and imaging conditions (i.e., illumination time, intensity, and uniformity) of the microscopic system. In this work, we present a method for finding the optimal intensity threshold by maximizing the corresponding fractal dimension. This method results in the segmentation of histological structures and the estimation of their fractal dimension (independent of imaging conditions). We applied our technique to 164 prostate histology sections from 82 prostate core biopsy specimens (two serial sections from each of the 63 benign prostate tissues and 19 high grade prostate carcinoma). We stained one of the serial sections with conventional hemotoxylin and eosin (H&E) and the other with pan-keratin, and found that the difference in mean fractal dimension between the two groups was statistically significant (p<0.0001) for both stains. However, using receiver operating characteristics (ROC) analysis, we conclude that our fractal dimension method applied to the images of pan-keratin stained sections provides greater classification performance (benign versus high grade) than with those stained with H&E when compared to the original histological diagnosis. The sensitivity and specificity achieved with the pan-keratin images were 89.5% and 90.5%, respectively.

Quantitative imaging of musculoskeletal tissue

Quantitative imaging of musculoskeletal tissue, including radiography, computed tomography (CT), and magnetic resonance imaging (MRI), has become the essential methodology in clinical practice for diagnosis and monitoring of various musculoskeletal conditions. Furthermore, quantitative imaging technologies have become indispensable for research and development in diseases of the human skeleton. Standardized methods of image analysis have been developed through the years to quantify measurements on bone and cartilage with high precision and accuracy. Key areas of musculoskeletal disease where quantitative imaging is currently employed are osteoporosis and arthritis.

Fractal dimension as a quantitator of the microvasculature of normal and adenomatous pituitary tissue

It is well known that angiogenesis is a complex process that accompanies neoplastic growth, but pituitary tumours are less vascularized than normal pituitary glands. Several analytical methods aimed at quantifying the vascular system in two-dimensional histological sections have been proposed, with very discordant results. In this study we investigated the non-Euclidean geometrical complexity of the two-dimensional microvasculature of normal pituitary glands and pituitary adenomas by quantifying the surface fractal dimension that measures its space-filling property. We found a statistical significant difference between the mean vascular surface fractal dimension estimated in normal versus adenomatous tissues (P = 0.01), normal versus secreting adenomatous tissues (P = 0.0003), and normal versus non-secreting adenomatous tissues (P = 0.047), whereas the difference between the secreting and non-secreting adenomatous tissues was not statistically significant. This study provides the first demonstration that fractal dimension is an objective and valid quantitator of the two-dimensional geometrical complexity of the pituitary gland microvascular network in physiological and pathological states. Further studies are needed to compare the vascular surface fractal dimension estimates in different subtypes of pituitary tumours and correlate them with clinical parameters in order to evaluate whether the distribution pattern of vascular growth is related to a particular state of the pituitary gland.

Evaluation of surface roughness of hydrogels by fractal texture analysis during swelling

The surface of a biomaterial reacts in contact with biological fluids. Hydrogels are used to prepare biomaterials. The surface roughness of materials can be explored by several techniques. However, when considering hydrogels, the surface examined in the dry state does not reflect the final conformation. How the surface roughness is affected by swelling has been little explored by quantitative methods. We have evaluated the surface roughness of poly(2-hydroxyethyl methacrylate) (i.e., pHEMA) by image analysis. Images of disks, prepared from linear pHEMA, were obtained on a light microscope after various incubation times in saline. Fractal texture analysis was done on images to determine the fractal dimension D. In this study, D exhibited a significant decrease during swelling and was highly correlated with the swelling ratio (r2 = 0.994, p < 0.00001). Water uptake by the surface of the polymer affected the surface roughness. Image analysis using fractal algorithms appears to be the most interesting technique for the quantitative exploration of surfaces of hydrated materials that cannot be measured by conventional methods.

Nuclear patterns of human breast cancer cells during apoptosis: characterisation by fractal dimension and co-occurrence matrix statistics

An analytical strategy combining fractal geometry and grey-level co-occurrence matrix (GLCM) statistics was devised to investigate ultrastructural changes in oestrogen-insensitive SK-BR3 human breast cancer cells undergoing apoptosis in vitro. Apoptosis was induced by 1 microM calcimycin (A23187 Ca(2+) ionophore) and assessed by measuring conventional cellular parameters during the culture period. SK-BR3 cells entered the early stage of apoptosis within 24 h of treatment with calcimycin, which induced detectable changes in nuclear components, as documented by increased values of most GLCM parameters and by the general reduction of the fractal dimensions. In these affected cells, morphonuclear traits were accompanied by the reduction of distinct gangliosides and loss of unidentifiable glycolipid molecules at the cell surface. All these changes were shown to be involved in apoptosis before the detection of conventional markers, which were only measurable during the active phases of apoptotic cell death. In overtly apoptotic cells treated with 1 microM calcimycin for 72 h, most nuclear components underwent dramatic ultrastructural changes, including marginalisation and condensation of chromatin, as reflected in a significant reduction of their fractal dimensions. Hence, both fractal and GLCM analyses confirm that the morphological reorganisation of nuclei, attributable to a loss of structural complexity, occurs early in apoptosis.

Effects of organelle shape on fluorescence recovery after photobleaching

The determination of diffusion coefficients from fluorescence recovery data is often complicated by geometric constraints imposed by the complex shapes of intracellular compartments. To address this issue, diffusion of proteins in the lumen of the endoplasmic reticulum (ER) is studied using cell biological and computational methods. Fluorescence recovery after photobleaching (FRAP) experiments are performed in tissue culture cells expressing GFP-KDEL, a soluble, fluorescent protein, in the ER lumen. The three-dimensional (3D) shape of the ER is determined by confocal microscopy and computationally reconstructed. Within these ER geometries diffusion of solutes is simulated using the method of particle strength exchange. The simulations are compared to experimental FRAP curves of GFP-KDEL in the same ER region. Comparisons of simulations in the 3D ER shapes to simulations in open 3D space show that the constraints imposed by the spatial confinement result in two- to fourfold underestimation of the molecular diffusion constant in the ER if the geometry is not taken into account. Using the same molecular diffusion constant in different simulations, the observed speed of fluorescence recovery varies by a factor of 2.5, depending on the particular ER geometry and the location of the bleached area. Organelle shape considerably influences diffusive transport and must be taken into account when relating experimental photobleaching data to molecular diffusion coefficients. This novel methodology combines experimental FRAP curves with high accuracy computer simulations of diffusion in the same ER geometry to determine the molecular diffusion constant of the solute in the particular ER lumen.

Vascularity and Fractal Dimension of the Dermo-Epidermal Interface in Guttate and Plaque-Type Psoriasis

BACKGROUND

Histological structures of the skin are often irregular in size and shape. Euclidean geometry and fractal analysis are complementary for assessing distinct aspects of their dimensions.

OBJECTIVE

To determine and compare the variations in shape of the dermo-epidermal junction and the size of the superficial vessels in psoriatic lesions.

METHOD

The relative microvasculature area and the fractal dimension D of the dermo-epidermal interface were measured inside and outside growth-stunted guttate lesions (n = 22) and expanding plaques (n = 37) in psoriasis of the trunk.

RESULTS

The median D values of the dermo-epidermal interface were significantly larger (p < 0.01) in psoriatic plaques (D = 1.15) than in guttate lesions (D = 1.08), and these D values on lesional skin were significantly larger (p < 0.01) than in the uninvolved skin (D = 1.03). The microvasculature was significantly (p < 0.01) more developed in lesional (plaque: 13%, guttate: 8.20%) than in uninvolved skin (3.60 and 3.85%). No correlations were found between the relative microvasculature areas and the D values of the dermo-epidermal interface, both in the uninvolved and lesional skins of each psoriatic type.

CONCLUSION

The absence of a relationship between modulations of the dermo-epidermal junction and vascular hyperplasia, both in expanding and stable psoriasis lesions, suggests that these events are regulated by different mechanisms and do not depend on each other.

Image analysis of irregularity of cluster shape in cytological diagnosis of breast tumors: Cluster analysis with 2D-fractal dimension

To establish diagnostic criteria using comparison of cell cluster shapes, between benign and malignant tumors, breast tumors demonstrating weak cellular atypia in low grade invasive ductal carcinoma (IDC) were compared. Fine-needle aspiration (FNA) specimens of breast tumors were obtained from 37 patients. Among these, 16 were histologically diagnosed as IDC low-grade and the other 21 as benign fibroadenoma (FA). For evaluation, we examined 740 clusters from these 37 FNA specimens. Nine image morphometric parameters were studied, including the cluster area, circumference, maximal length, maximal breadth, ratio of length to breadth, cluster roundness, cluster size, and the edge and distribution image fractal dimensions for cluster analysis. We evaluated the irregularity in cell cluster shape using fractal dimension analysis, and determined the correlation to cluster size. The irregularity in the IDC cluster shape was higher than that in the FA cluster shape. However, six cases (28.5%) of 21 FA clusters showed high fractal dimensions similar to those for IDC. The clusters were classified by cluster analysis into three types: IDC clusters, FA with irregular cluster shape, and FA with no irregular clusters. The average cell cluster area of the FA with irregular shape was found to be about three times larger than that of IDC clusters. When the differential diagnosis between IDC and FA is difficult, it is important to focus on irregularities in the shape and on overall size of the cell clusters. For accurate diagnosis, the cell cluster shape is as important as the individual cellular atypia.

Determination of tumour prognosis based on angiogenesis-related vascular patterns measured by fractal and syntactic structure analysis

AIMS

Intratumoural micro-vessel density (IMD) has recently been shown to be a valuable prognostic tool in many tumours. Yet, IMD does not take into account the spatial arrangement of the vessels, therefore only partly reflecting the angiogenic situation. In order to describe contextual vascular relationships more accurately, we have used fractal and syntactic structure analysis (SSA) based on computerised image processing to quantify micro-vascular hot spots.

MATERIALS AND METHODS

The parametric performance in prediction of patients' outcome was evaluated by univariate analysis and compared with manually obtained IMDs, whereas an automated K-nearest-neighbour (KNN) classifier searched most discriminative parametric combinations. The method is based on analysis of vascular 'hot-spots' of paraffin-embedded tissue sections of invasive cervical carcinoma, colorectal carcinoma and malignant mesothelioma.

RESULTS

For all three cancers, prediction of prognosis based on SSA yielded in general much higher recognition scores compared with IMD or fractal dimension. Survival of cervical carcinoma was mostly correlated with clinical data, with the vascular permeation being the only parameter with independent value. Prognosis of colorectal carcinoma is best described by SSA, completed with IMD, indicating an inverse correlation of survival time with a more irregular pattern and a slight increase in vessel number. For mesothelioma, we found a strong correlation with SSA and patients' outcome, with two SSA-parameters having independent prognostic value.

CONCLUSIONS

The more accurate angiogenic description obtained with SSA may be useful for further exploitation as a prognosticator in a general diagnostic pathology service.

Morphological development of microglia in the postnatal rat brain

Morphological transformation of lectin-positive microglia/macrophages in the developing rat cerebral hemisphere was analysed using quantitative methods. During the first postnatal month, the cells showed increases in their size and fractal dimension accompanied by a simultaneous decrease in their solidity. Regional variations in dynamics of the process indicated the existence of spatio-temporal developmental gradients within the cerebral hemisphere wall which might correspond with regional patterns of neuronal differentiation. Results of the present study prove that the quantitative methods can be the source of reliable data replacing subjective cell typologies.

Fractal dimension can distinguish models and pharmacologic changes in liver fibrosis in rats

Fractal analysis measures the complexity of geometric structures. The aim of this study was to evaluate the feasibility and accuracy of fractal analysis in liver fibrosis. A total of 77 rats were included: 10 sham, 46 with fibrosis secondary to bile duct ligation (BDL), and 21 with fibrosis due to CCl(4) intoxication. Measurements included the fractal dimension of Kolmogorov (D(k)), histologic lesions, the area of fibrosis by image analysis, liver hydroxyproline content, messenger RNA fibronectin, serum hyaluronate level, and portal pressure. Fibrotic rats were given placebo, octreotide, or O(2)-vinyl 1-(pyrrolidin-1-yl)diazen-1-ium-1,2-diolate (V-PYRRO/NO). Intraobserver agreement of D(k) was excellent with the intraclass (ic) correlation coefficient r(ic) = 0.91 (P <.0001) as well as the interobserver agreement with r(ic) = 0.88 (P <.001). D(k) was correlated with other measurements or markers of fibrosis: the area of fibrosis (r = 0.75; P <.0001), hydroxyproline content (r = 0.51; P <.001), serum hyaluronate level (r = 0.52; P <.001), and portal pressure (r = 0.52; P <.01). D(k) was significantly different between the 2 models of fibrosis (P <.0001), unlike the area of fibrosis, and this relationship was independent of other histologic lesions. The significant decrease in fibrosis observed with octreotide or V-PYRRO/NO was similarly reflected by D(k) or the area of fibrosis. The diagnostic accuracy for the fibrosis model was 97% with the 5 main measurements or markers of fibrosis studied, with D(k) isolated at the first step by stepwise analysis. In conclusion, fractal analysis is suitable for analyzing liver fibrosis and has excellent reproducibility. This is the only quantitative morphometric method that can discriminate among the models of fibrosis and is sensitive enough to detect pharmacologically induced changes in liver fibrosis.

Fractal dimension of trabecular bone: comparison of three histomorphometric computed techniques for measuring the architectural two-dimensional complexity

Trabecular bone has been reported as having two-dimensional (2-D) fractal characteristics at the histological level, a finding correlated with biomechanical properties. However, several fractal dimensions (D) are known and computational ways to obtain them vary considerably. This study compared three algorithms on the same series of bone biopsies, to obtain the Kolmogorov, Minkowski-Bouligand, and mass-radius fractal dimensions. The relationships with histomorphometric descriptors of the 2-D trabecular architecture were investigated. Bone biopsies were obtained from 148 osteoporotic male patients. Bone volume (BV/TV), trabecular characteristics (Tb.N, Tb.Sp, Tb.Th), strut analysis, star volumes (marrow spaces and trabeculae), inter-connectivity index, and Euler-Poincaré number were computed. The box-counting method was used to obtain the Kolmogorov dimension (D(k)), the dilatation method for the Minkowski-Bouligand dimension (D(MB)), and the sandbox for the mass-radius dimension (D(MR)) and lacunarity (L). Logarithmic relationships were observed between BV/TV and the fractal dimensions. The best correlation was obtained with D(MR) and the lowest with D(MB). Lacunarity was correlated with descriptors of the marrow cavities (ICI, star volume, Tb.Sp). Linear relationships were observed among the three fractal techniques which appeared highly correlated. A cluster analysis of all histomorphometric parameters provided a tree with three groups of descriptors: for trabeculae (Tb.Th, strut); for marrow cavities (Euler, ICI, Tb.Sp, star volume, L); and for the complexity of the network (Tb.N and the three D's). A sole fractal dimension cannot be used instead of the classic 2-D descriptors of architecture; D rather reflects the complexity of branching trabeculae. Computation time is also an important determinant when choosing one of these methods.

Use of fractal theory in neuroscience: methods, advantages, and potential problems

Fractal analysis has already found widespread application in the field of neuroscience and is being used in many other areas. Applications are many and include ion channel kinetics of biological membranes and classification of neurons according to their branching characteristics. In this article we review some practical methods that are now available to allow the determination of the complexity and scaling relationships of anatomical and physiological patterns. The problems of describing fractal dimensions are discussed and the concept of fractal dimensionality is introduced. Several related methodological considerations, such as preparation of the image and estimation of the fractal dimensions from the data points, as well as the advantages and problems of fractal geometric analysis, are discussed.

The roughness of the supranasal region--a morphological sex trait

The supranasal region often attracts attention by a remarkable rough and jagged quality of the bony surface caused by an irregular supranasal suture and additional holes and pores. Some authors point out that there is a positive relation between the supranasal suture, the superciliar arches, and the forehead contour. For this a relation to sex is conceivable. This present study was done to prove the value of this morphological trait for sexing skulls.A total of 80 human skulls of known sex (40 females, 40 males) were collected from autopsy material used in anatomy teaching classes and from forensic cases. The mean age of the female sample was 70.98 years (minimum 38, maximum 93), that of the male sample was 74.10 years (minimum 57, maximum 99). To quantify the roughness of the supranasal region the calculation of the box-counting dimension was used. The results were normally distributed in both, the male and female group. The male dimension values were well grouped (maximum 1.51111, minimum 0.98765, mean 1.26159, S.D. 0.12268, 95% CI 1.22236-1.26604) whereas the female showed a wide range (maximum 1.46744, minimum 0.44755, mean 1.15052, S.D. 0.21388, 95% CI 1.08212-1.21892), widely overlapping the male range. Statistical analysis showed that there was a less than 1% probability that the female box-counting dimension was lower than the male by chance (P-value 0.00593). For this results the admission of the trait 'quality of the supranasal region' into a catalogue of features regarding morphognostic sex determination following the scheme: hyperfemininity: very smooth and regular--femininity: more smooth and regular--indifferent--masculinity: more rough and irregular--hypermasculinity: very rough and irregular, seems to be justified.

Quantitative analysis of three-dimensional complexity and connectivity changes in trabecular microarchitecture in relation to aging, menopause, and inflammation

There are several types of bone loss besides that associated with normal aging, eg, that associated with the menopause, and that associated with chronic inflammation, and these are considered to be caused by different mechanisms. The microarchitecture that results from these different bone-loss mechanisms would not be the same. The purpose of this study was to investigate differences in the three-dimensional trabecular microarchitecture in various types of osteopenia, using microcomputed tomography (Micro-CT). Thirty-five Fisher 344 rats were divided into five groups (control, young, senile, ovariectomized [OVX], and inflammation-mediated osteopenia [IMO]) and distal femoral metaphysis was scanned by Micro-CT to nondestructively acquire a 3-D CT stack consisting of 50 consecutive slices at a spatial resolution of 26 microm. The volume of interest, consisting of the secondary spongiosa, was prepared to analyze the 3-D trabecular microarchitecture. A parametric analysis was carried out using bone volume fractions, fractal dimensions, and the first Betti number in order to quantitatively express the mass, complexity, and connectivity of the trabecular microarchitecture. Complexity tended to decrease with age, and decreased significantly in estrogen deficiency-induced and inflammation-mediated osteopenia. Connectivity did not appear to change with aging, but was significantly decreased in estrogen deficiency-induced and inflammation-mediated osteopenia. There was no significant difference between the OVX and the IMO groups.

Are neurons multifractals?

In the last few years, fractal analysis has found widespread application in the field of neuroscience and some investigators are starting to use multifractals as a methodology that may provide information about the distribution of fractal dimensions in biological structures. This is so, despite of the technical difficulties of multifractal analysis. In this paper, we investigate the theoretical and practical aspects of studying and measuring the multifractal dimensions of neurons. Patterns were analysed by means of the standard box-counting method and a generalised sand-box method. Our results show that odd behaviours of Dq reported in the literature are a consequence of numerical deficiencies of the box-counting method and cannot be associated to peculiar geometrical characteristics of neurons. Instead the sand-box method gives a Dq which monotonically decreases with q. Although this result may indicate that neurons are multifractals, it is argued that size effects may in fact be the origin of this apparent multifractality.

Comparison of eight histomorphometric methods for measuring trabecular bone architecture by image analysis on histological sections

Osteoporosis is defined as a disease characterized by low bone mass and microarchitectural deterioration of trabecular bone leading to enhanced bone fragility. Various histomorphometric methods have been described to measure bone architecture on histological sections. However, not all of the methods are strictly equivalent and some of them appear able to detect differences earlier in the course of the disease. We have compared 8 histomorphometric methods known to characterize the architecture of trabecular bone in 154 male osteoporotic patients. Measurements were done on transiliac bone biopsies: Trabecular number, thickness, and separation (Tb.N, Tb.Th, Tb.Sp); Trabecular Bone Pattern Factor (TBPf); Euler-Poincare's number (E); Interconnectivity Index (ICI); strut analysis of the trabecular network with the ratio of nodes/free-end (N/F); star volume of the bone marrow (V*m.space) and trabeculae (V*Tb) and the Kolmogorov fractal dimension of the trabecular boundaries (D). Relationships between the various architectural parameters were studied by hierarchical cluster analysis. Linear, hyperbolic, and exponential correlations were found between trabecular bone volume (BV/TV) and architectural parameters. Cluster analysis demonstrates the link between these architectural parameters. ICI, E, and TBPf, which reflect the amount of open/closed marrow cavities clustered together and appeared related to Tb.Sp, V*m.space which are indicators of the mean size of marrow cavities. Tb.Th, V*Tb and N/F flocked together as they reflect the trabecular size. Tb.N and D segregated together and seemed to best describe the trabecular network complexity. These histomorphometric techniques are correlated but correlations may be linear or nonlinear. Several histomorphometric techniques need to be used in parallel to appreciate the pathophysiological mechanisms of osteoporotic states.

Neurons and fractals: how reliable and useful are calculations of fractal dimensions?

In the past 15 years it has become possible to determine the fractal dimension (Df) of complex objects, including neurons, by automated image analysis methods. However, there are many unresolved issues that need to be addressed. In this paper we discuss how the Df calculated by different methods may vary and how fractal analysis may be of use for retinal ganglion cell characterization. The goal of this work was to acknowledge inherent sources of variation during measurement and evaluate current fractal analysis methods for describing structure. Our results show that different algorithms and even the same algorithm performed by different computer programs and/or experimenters may give different but consistent numerical values. All described methods demonstrated their suitability for classifying cat retinal ganglion cells into distinct groups. Our results reinforce the idea that comparison of measurements of different profiles using the same measurement method may be useful and valid even if an exact numeric value of the dimension is not realised in practice.

Estimating fractal dimension with fractal interpolation function models

Fractal dimension (fd) is a feature which is widely used to characterize medical images. Previously, researchers have shown that fd separates important classes of images and provides distinctive information about texture. We analyze limitations of two principal methods of estimating fd: box-counting (BC) and power spectrum (PS). BC is ineffective when applied to data-limited, low-resolution images; PS is based on a fractional Brownian motion (fBm) model-a model which is not universally applicable. We also present background information on the use of fractal interpolation function (FIF) models to estimate fd of data which can be represented in the form of a function. We present a new method of estimating fd in which multiple FIF models are constructed. The mean of the fd's of the FIF models is taken as the estimate of the fd of the original data. The standard deviation of the fd's of the FIF models is used as a confidence measure of the estimate. We demonstrate how the new method can be used to characterize fractal texture of medical images. In a pilot study, we generated plots of curvature values around the perimeters of images of red blood cells from normal and sickle cell subjects. The new method showed improved separation of the image classes when compared to BC and PS methods.

Fractals in pathology

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.

Morphometric applications in anatomic pathology

The components of the cell and tissue changes in many diseases are variable and can therefore be quantified. Characterization of these quantitative changes provides data that is useful not only for making a definitive, cell- and tissue-based diagnosis of disease, but also for predicting the course of disease. The spectrum of changes found in malignant tumors, ie, cell grade, architecture, cellularity, extent of invasion, nature and extent of inflammatory reaction, exemplify this range of quantifiable features. The diagnosis and prognosis of nonneoplastic diseases, ie, myopathy and metabolic bone disease, can also be determined by quantitating tissue changes. Morphometry is the quantification of changes in the "objects" of tissues, ie, cells and organelles, and their organization, using quantitative evaluation tools. The principles of morphometry have been known for a century. With the increasing availability of affordable, powerful computer systems and increasingly flexible and user-friendly software has come easier ability to measure these changes. This article discusses the principles of morphometry with illustrations of types of analysis (ie, area fraction, object counting, shape and size analyses, and mutliparametric analyses) using examples of these applications with discussions of error sources and limitations of morphometry.

Fractal geometric analysis of the renal arterial tree in infants and fetuses

Fractal geometry is a useful method of quantitating the space-filling properties of complex objects and has a particular advantage in pediatric pathology because it is independent of organ size. The fractal dimensions of angiographic images of 44 renal arterial trees from 23 consent pediatric autopsies were measured by the box-counting method. The mean fractal dimension was 1.64 and all values were greater than the topological dimension (one), indicating that the renal arterial tree in fetuses and infants has a fractal element to its structure. There was no significant association with size of the kidneys, confirming the size-independent nature of the fractal dimension. There was no significant association with age of the subject, and the mean value was not significantly different from values obtained in studies of adult kidneys, suggesting that the degree of branching, at a lobar and lobular level, does not increase after about the 21st week of gestation. The results are compatible with a diffusion-limited aggregation model of development.