Quantitation of the renal arterial tree by fractal analysis

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.

Fractal nature of human gastrointestinal system: Exploring a new era

The morphological complexity of cells and tissues, whether normal or pathological, is characterized by two primary attributes: Irregularity and self-similarity across different scales. When an object exhibits self-similarity, its shape remains unchanged as the scales of measurement vary because any part of it resembles the whole. On the other hand, the size and geometric characteristics of an irregular object vary as the resolution increases, revealing more intricate details. Despite numerous attempts, a reliable and accurate method for quantifying the morphological features of gastrointestinal organs, tissues, cells, their dynamic changes, and pathological disorders has not yet been established. However, fractal geometry, which studies shapes and patterns that exhibit self-similarity, holds promise in providing a quantitative measure of the irregularly shaped morphologies and their underlying self-similar temporal behaviors. In this context, we explore the fractal nature of the gastrointestinal system and the potential of fractal geometry as a robust descriptor of its complex forms and functions. Additionally, we examine the practical applications of fractal geometry in clinical gastroenterology and hepatology practice.

Space-filling and benthic competition on coral reefs

Reef-building corals are ecosystem engineers that compete with other benthic organisms for space and resources. Corals harvest energy through their surface by photosynthesis and heterotrophic feeding, and they divert part of this energy to defend their outer colony perimeter against competitors. Here, we hypothesized that corals with a larger space-filling surface and smaller perimeters increase energy gain while reducing the exposure to competitors. This predicted an association between these two geometric properties of corals and the competitive outcome against other benthic organisms. To test the prediction, fifty coral colonies from the Caribbean island of Curaçao were rendered using digital 3D and 2D reconstructions. The surface areas, perimeters, box-counting dimensions (as a proxy of surface and perimeter space-filling), and other geometric properties were extracted and analyzed with respect to the percentage of the perimeter losing or winning against competitors based on the coral tissue apparent growth or damage. The increase in surface space-filling dimension was the only significant single indicator of coral winning outcomes, but the combination of surface space-filling dimension with perimeter length increased the statistical prediction of coral competition outcomes. Corals with larger surface space-filling dimensions (Ds > 2) and smaller perimeters displayed more winning outcomes, confirming the initial hypothesis. We propose that the space-filling property of coral surfaces complemented with other proxies of coral competitiveness, such as life history traits, will provide a more accurate quantitative characterization of coral competition outcomes on coral reefs. This framework also applies to other organisms or ecological systems that rely on complex surfaces to obtain energy for competition.

A healthy dose of chaos: Using fractal frameworks for engineering higher-fidelity biomedical systems

Optimal levels of chaos and fractality are distinctly associated with physiological health and function in natural systems. Chaos is a type of nonlinear dynamics that tends to exhibit seemingly random structures, whereas fractality is a measure of the extent of organization underlying such structures. Growing bodies of work are demonstrating both the importance of chaotic dynamics for proper function of natural systems, as well as the suitability of fractal mathematics for characterizing these systems. Here, we review how measures of fractality that quantify the dose of chaos may reflect the state of health across various biological systems, including: brain, skeletal muscle, eyes and vision, lungs, kidneys, tumours, cell regulation, skin and wound repair, bone, vasculature, and the heart. We compare how reports of either too little or too much chaos and fractal complexity can be damaging to normal biological function, and suggest that aiming for the healthy dose of chaos may be an effective strategy for various biomedical applications. We also discuss rising examples of the implementation of fractal theory in designing novel materials, biomedical devices, diagnostics, and clinical therapies. Finally, we explain important mathematical concepts of fractals and chaos, such as fractal dimension, criticality, bifurcation, and iteration, and how they are related to biology. Overall, we promote the effectiveness of fractals in characterizing natural systems, and suggest moving towards using fractal frameworks as a basis for the research and development of better tools for the future of biomedical engineering.

Fractal properties of biophysical models of pericellular brushes can be used to differentiate between cancerous and normal cervical epithelial cells

Fractal behavior is found on the topographies of pericellular brushes on the surfaces of model healthy and cancerous cells, using dissipative particle dynamics models and simulations. The influence of brush composition, chain stiffness and solvent quality on the fractal dimension is studied in detail. Since fractal dimension alone cannot guarantee that the brushes possess fractal properties, their lacunarity was obtained also, which is a measure of the space filling capability of fractal objects. Soft polydisperse brushes are found to have larger fractal dimension than soft monodisperse ones, under poor solvent conditions, in agreement with recent experiments on dried cancerous and healthy human cervical epithelial cells. Additionally, we find that image resolution is critical for the accurate assessment of differences between images from different cells. The images of the brushes on healthy model cells are found to be more textured than those of brushes on model cancerous cells, as indicated by the larger lacunarity of the former. These findings are helpful to distinguish monofractal behavior from multifractality, which has been found to be useful to discriminate between immortal, cancerous and normal cells in recent experiments.

Planar cell polarity genes frizzled4 and frizzled6 exert patterning influence on arterial vessel morphogenesis

Quantitative analysis of the vascular network anatomy is critical for the understanding of the vasculature structure and function. In this study, we have combined microcomputed tomography (microCT) and computational analysis to provide quantitative three-dimensional geometrical and topological characterization of the normal kidney vasculature, and to investigate how 2 core genes of the Wnt/planar cell polarity, Frizzled4 and Frizzled6, affect vascular network morphogenesis. Experiments were performed on frizzled4 (Fzd4^-/-) and frizzled6 (Fzd6^-/-) deleted mice and littermate controls (WT) perfused with a contrast medium after euthanasia and exsanguination. The kidneys were scanned with a high-resolution (16 μm) microCT imaging system, followed by 3D reconstruction of the arterial vasculature. Computational treatment includes decomposition of 3D networks based on Diameter-Defined Strahler Order (DDSO). We have calculated quantitative (i) Global scale parameters, such as the volume of the vasculature and its fractal dimension (ii) Structural parameters depending on the DDSO hierarchical levels such as hierarchical ordering, diameter, length and branching angles of the vessel segments, and (iii) Functional parameters such as estimated resistance to blood flow alongside the vascular tree and average density of terminal arterioles. In normal kidneys, fractal dimension was 2.07±0.11 (n = 7), and was significantly lower in Fzd4^-/- (1.71±0.04; n = 4), and Fzd6^-/- (1.54±0.09; n = 3) kidneys. The DDSO number was 5 in WT and Fzd4^-/-, and only 4 in Fzd6^-/-. Scaling characteristics such as diameter and length of vessel segments were altered in mutants, whereas bifurcation angles were not different from WT. Fzd4 and Fzd6 deletion increased vessel resistance, calculated using the Hagen-Poiseuille equation, for each DDSO, and decreased the density and the homogeneity of the distal vessel segments. Our results show that our methodology is suitable for 3D quantitative characterization of vascular networks, and that Fzd4 and Fzd6 genes have a deep patterning effect on arterial vessel morphogenesis that may determine its functional efficiency.

Fractal lacunarity of trabecular bone and magnetic resonance imaging: New perspectives for osteoporotic fracture risk assessment

Osteoporosis represents one major health condition for our growing elderly population. It accounts for severe morbidity and increased mortality in postmenopausal women and it is becoming an emerging health concern even in aging men. Screening of the population at risk for bone degeneration and treatment assessment of osteoporotic patients to prevent bone fragility fractures represent useful tools to improve quality of life in the elderly and to lighten the related socio-economic impact. Bone mineral density (BMD) estimate by means of dual-energy X-ray absorptiometry is normally used in clinical practice for osteoporosis diagnosis. Nevertheless, BMD alone does not represent a good predictor of fracture risk. From a clinical point of view, bone microarchitecture seems to be an intriguing aspect to characterize bone alteration patterns in aging and pathology. The widening into clinical practice of medical imaging techniques and the impressive advances in information technologies together with enhanced capacity of power calculation have promoted proliferation of new methods to assess changes of trabecular bone architecture (TBA) during aging and osteoporosis. Magnetic resonance imaging (MRI) has recently arisen as a useful tool to measure bone structure in vivo. In particular, high-resolution MRI techniques have introduced new perspectives for TBA characterization by non-invasive non-ionizing methods. However, texture analysis methods have not found favor with clinicians as they produce quite a few parameters whose interpretation is difficult. The introduction in biomedical field of paradigms, such as theory of complexity, chaos, and fractals, suggests new approaches and provides innovative tools to develop computerized methods that, by producing a limited number of parameters sensitive to pathology onset and progression, would speed up their application into clinical practice. Complexity of living beings and fractality of several physio-anatomic structures suggest fractal analysis as a promising approach to quantify morpho-functional changes in both aging and pathology. In this particular context, fractal lacunarity seems to be the proper tool to characterize TBA texture as it is able to describe both discontinuity of bone network and sizes of bone marrow spaces, whose changes are an index of bone fracture risk. In this paper, an original method of MRI texture analysis, based on TBA fractal lacunarity is described and discussed in the light of new perspectives for early diagnosis of osteoporotic fractures.

Beyond linear methods of data analysis: time series analysis and its applications in renal research

Analysis of temporal trends in medicine is needed to understand normal physiology and to study the evolution of disease processes. It is also useful for monitoring response to drugs and interventions, and for accountability and tracking of health care resources. In this review, we discuss what makes time series analysis unique for the purposes of renal research and its limitations. We also introduce nonlinear time series analysis methods and provide examples where these have advantages over linear methods. We review areas where these computational methods have found applications in nephrology ranging from basic physiology to health services research. Some examples include noninvasive assessment of autonomic function in patients with chronic kidney disease, dialysis-dependent renal failure and renal transplantation. Time series models and analysis methods have been utilized in the characterization of mechanisms of renal autoregulation and to identify the interaction between different rhythms of nephron pressure flow regulation. They have also been used in the study of trends in health care delivery. Time series are everywhere in nephrology and analyzing them can lead to valuable knowledge discovery. The study of time trends of vital signs, laboratory parameters and the health status of patients is inherent to our everyday clinical practice, yet formal models and methods for time series analysis are not fully utilized. With this review, we hope to familiarize the reader with these techniques in order to assist in their proper use where appropriate.

The fractal spatial distribution of pancreatic islets in three dimensions: a self-avoiding growth model

The islets of Langerhans, responsible for controlling blood glucose levels, are dispersed within the pancreas. A universal power law governing the fractal spatial distribution of islets in two-dimensional pancreatic sections has been reported. However, the fractal geometry in the actual three-dimensional pancreas volume, and the developmental process that gives rise to such a self-similar structure, has not been investigated. Here, we examined the three-dimensional spatial distribution of islets in intact mouse pancreata using optical projection tomography and found a power law with a fractal dimension of 2.1. Furthermore, based on two-dimensional pancreatic sections of human autopsies, we found that the distribution of human islets also follows a universal power law with a fractal dimension of 1.5 in adult pancreata, which agrees with the value previously reported in smaller mammalian pancreas sections. Finally, we developed a self-avoiding growth model for the development of the islet distribution and found that the fractal nature of the spatial islet distribution may be associated with the self-avoidance in the branching process of vascularization in the pancreas.

Structural markers for normal oral mucosa and oral sub-mucous fibrosis

This article presents a quantitative approach for the characterization of normal oral mucosa (NOM) in respect to thickness and textural properties of its entire epithelial layer. Histological images of oral mucosa depict that both thickness and tissue architecture at cellular and tissue level undergo change, as mucosa converts from normal to precancerous or cancerous state. In this study the thickness and fractal dimension of the mucosal epithelium of NOM and oral sub-mucous fibrosis (OSF) condition have been computed using 83 normal and 29 OSF images of oral mucosa. The result shows significant delineation between NOM and OSF in respect of both the epithelial thickness (in microm) and fractal dimensions. This quantitative characterization of oral epithelium will be of immense help for oral onco-pathologists and researchers to assess the biological nature of normal and diseased (OSF) mucosa with higher accuracy. Moreover, further differential applications may enable them to find out newer accurate quantitative diagnostic procedures to that of the usual histopathological gold standard for the assessment of malignant potentiality.

Effects of old age on vascular complexity and dispersion of the hepatic sinusoidal network

OBJECTIVES

In old age, there are marked changes in both the structure of the liver sinusoidal endothelial cell and liver perfusion. The objective of this study was to determine whether there are also aging changes in the microvascular architecture and vascular dispersion of the liver that might influence liver function.

METHODS

Vascular corrosion casts and light micrographs of young (4 months) and old (24 months) rat livers were compared. Fractal and Fourier analyses and micro-computed tomography were used. Vascular dispersion was determined from the dispersion number for sucrose and 100-nm microspheres in impulse response experiments.

RESULTS

Age did not affect sinusoidal dimensions, sinusoidal density, or dispersion number. There were changes in the geometry and complexity of the sinusoidal network as determined by fractal dimension and degree of anisotropy.

CONCLUSIONS

There are small, age-related changes in the architecture of the liver sinusoidal network, which may influence hepatic function and reflect broader aging changes in the microcirculation. However, sinusoidal dimensions and hepatic vascular dispersion are not markedly influenced by old age.

Euclidean and fractal geometry of microvascular networks in normal and neoplastic pituitary tissue

In geometrical terms, tumour vascularity is an exemplary anatomical system that irregularly fills a three-dimensional Euclidean space. This physical characteristic and the highly variable shapes of the vessels lead to considerable spatial and temporal heterogeneity in the delivery of oxygen, nutrients and drugs, and the removal of metabolites. Although these biological characteristics are well known, quantitative analyses of newly formed vessels in two-dimensional histological sections still fail to view their architecture as a non-Euclidean geometrical entity, thus leading to errors in visual interpretation and discordant results from different laboratories concerning the same tumour. We here review the literature concerning microvessel density estimates (a Euclidean-based approach quantifying vascularity in normal and neoplastic pituitary tissues) and compare the results. We also discuss the limitations of Euclidean quantitative analyses of vascularity and the helpfulness of a fractal geometry-based approach as a better means of quantifying normal and neoplastic pituitary microvasculature.

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.

Nuclear fractal dimension as a prognostic factor in oral squamous cell carcinoma

Strong theoretical reasons exist for using fractal geometry in measurements of natural objects, including most objects studied in pathology. Indeed, fractal dimension provides a more precise and theoretically more appropriate approximation of their structure properties and especially their shape complexity. The aim of our study was to evaluate the nuclear fractal dimension (FD) in tissue specimens from patients with oral cavity carcinomas in order to assess its potential value as prognostic factor. Relationships between FD and other factors including clinicopathologic characteristics were also investigated. Histological sections from 48 oral squamous cell carcinomas as well as from 17 non-malignant mucosa specimens were stained with Hematoxylin-Eosin for pathological examination and with Feulgen for nuclear complexity evaluation. The sections were evaluated by image analysis using fractal analysis software to quantify nuclear FD by the box-counting method. Carcinomas presented higher mean values of FD compared to normal mucosa. Well differentiated neoplasms had lower FD values than poorly differentiated ones. FD was significantly correlated with the nuclear size. Patients with FD lower than the median value of the sample had statistically significant higher survival rates. Within the sample of patients studied, FD was proved to be an independent prognostic factor of survival in oral cancer patients. In addition this study provides evidence that there are several statistically significant correlations between FD and other morphometric characteristics or clinicopathologic factors in oral squamous cell carcinomas.

Transit time kinetics in ordered and disordered vascular trees

Imaging modalities exploit tracer-dilution methods to measure bulk haemodynamic parameters such as blood flow and volume at the level of the microcirculation. Here, we ask the question of whether the kinetics of a tracer can reveal morphological information about the vessels through which the tracers flow. The goal is to relate the acquired time-intensity characteristic to details of the vascular structure that lies below the imaging resolution. Two fractal vascular models are developed that represent organized 'kidney-like' and disorganized 'tumour-like' structures. The models are generated using simple rules of branching and fractal geometry in two dimensions. Blood flow and tracer kinetics are simulated using fundamental laws of haemodynamics. The flow conditions are matched in the two models. The fractal box dimensions of the kidney (D(B) = 1.67 +/- 0.01) and the tumour (D(B) = 1.80 +/- 0.01) vasculatures fall in the range given in the literature (D(B) = 1.61 +/- 0.06 and D(B) = 1.84 +/- 0.04, respectively). The tracer kinetic curves of the kidney and the tumour vasculatures have the same initial slope and final asymptote, corresponding to the same flow rate and vascular volume, but have different forms. The difference in the two curves is related to the distribution function of transit times of the vascular models, and is a consequence of the randomness introduced in vessel diameter and length. In principle, the form of the tracer kinetic curve from a contrast imaging study may offer information relating not only to vascular volume and flow rate, but also to the organization of a microvascular network.

Three-dimensional reconstruction and fractal geometric analysis of serrated adenoma

Serrated adenoma (SA) is a relatively newly defined entity of colorectal neoplasm first characterized by Longacre and Fenoglio-Preiser in 1990. This lesion is characterized by a complicated serrated edge of crypts. In this study, we performed three-dimensional (3-D) reconstruction, including 3-D distribution patterns of Ki-67-positive cells and fractal dimension of SA, in order to evaluate the nature of the complicated architecture, including its possible morphogenesis. We studied nine colonoscopic polypectomy specimens including three SAs, three tubular adenomas (TAs), and three hyperplastic polyps (HPs). Sixty serial tissue sections per case were stained alternately with hematoxylin and eosin (H&E) and Ki-67 immunostain. Each serial image was then digitized for 3-D computer analysis and the distribution pattern of Ki-67-positive cells was evaluated. Ki-67-immunostained sections were also subjected to 2-D quantitative morphometric study. In addition, the fractal dimensions of images from H&E-stained sections were examined using a box-counting method. Results of the 3-D reconstruction study demonstrated that glandular budding and branching were more frequent in SA than in TA or HP. These findings were confirmed quantitatively by the results of fractal geometric analysis of these polyps (fractal dimension:1.34 +/- 0.08 for SA, 1.23 +/- 0.07 for TA, and 1.28 +/- 0.12 for HP). Ki-67-positive cells in HP were localized mainly in the bottom of crypts and those in TA were diffusely distributed, while Ki-67-positive cells in SA were mainly aggregated in the depressed sites of serrated epithelia. These findings were also confirmed quantitatively using 2-D morphometry. These distribution patterns of the proliferative zone of SA are considered to contribute to the formation of the characteristic serrated epithelia and the complicated morphological appearance of SA.

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.

Fractal analysis: an objective method for identifying atypical nuclei in dysplastic lesions of the cervix uteri

OBJECTIVES

Fractal geometry is a tool used to characterize irregularly shaped and complex figures. It can be used not only to generate biological structures (e.g., the human renal artery tree), but also to derive parameters such as the fractal dimension in order to quantify the shapes of structures. As such, it allows user-independent evaluation and does not rely on the experience level of the examiner.

METHODS

We applied a box-counting algorithm to determine the fractal dimension of atypical nuclei in dysplastic cervical epithelium. An automatic algorithm was used to determine the fractal dimension of nuclei in order to prevent errors from manual segmentation. Four groups of patients (CIN 1-3 and control) with 10 subjects each were examined. In total, the fractal dimensions of 1200 nuclei were calculated.

RESULTS

We found that the fractal dimensions of the nuclei increased as the degree of dysplasia increased. There were significant differences between control and atypical nuclei found by an analysis of variance. Atypical nuclei associated with CIN 1, CIN 2, and CIN 3 also differed significantly among these groups.

CONCLUSION

We conclude that the fractal dimension is a valuable tool for detecting irregularities in atypical nuclei of the cervix uteri and thus allows objective nuclear grading.

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 analysis and imaging of the proximal nephron cell

Cells of the S1 proximal renal tubule were examined to determine whether their peculiar shapes are a result of certain constructs of fractal mathematics. Morphometric measurements of the cell perimeter were made at several levels of cell height by measuring the intercellular boundaries that appear on electron micrographs of tubule cross sections. When the measurements were made over a range of scale lengths, the fractal dimension, D, of the cell perimeter was found to increase from 1.3 near the cell apex to 1.78 near the cell base. The length of scale was found to range between 8 and 0.4 micron and to represent the approximate dimensions of actual cell processes. Fractal patterns that conformed to the measured parameters were then constructed from a fractal generator composed of budlike formations that originated near the cell apex and that increased in number and decreased in size with cell depth according to a fractal scaling. It was found that the fractal rule of keeping a constant relative scale could be maintained between budding processes but, to obtain patterns that resemble biological structure, the processes must be positioned randomly on the cell periphery. It is shown that when the relative sizes of the buds decrease exponentially and their numbers increase geometrically, the perimeter can grow to the correct length without overlap. This suggests that patterns of the cell periphery corresponding to different levels of cell height obey a law of scale but occur randomly in a way that increases to high fractal dimension or near plane-filling values at the cell base. The fractal patterns that correspond to the measured fractal dimensions can be assembled into a three-dimensional model that closely resembles the known shape of the proximal tubule cell.

Fluctuating asymmetry and fractal dimension of the sagittal suture as indicators of inbreeding depression in dama and dorcas gazelles

The effects of inbreeding on the developmental instability of skulls of dorcas (Gazella dorcas) and dama (G. dama) gazelles were investigated. In total, 132 dorcas gazelle skulls and 74 dama gazelle skulls from the Estación Experimental de Zonas Aridas in Almera, Spain, were measured. The fluctuating asymmetry of 9 meristic characters, consisting of the numbers of foramina on the two sides of the skull and mandible, was calculated. Although only the foramen infraorbitalis showed a significant increase in asymmetry with inbreeding in dorcas gazelles, the sum of the foramina in 5 of the skull regions clearly indicates an increase in asymmetry with inbreeding in both dorcas and dama gazelles. The fractal dimension of the sagittal suture was calculated by means of the coastline method. A greater effect of inbreeding on the sagittal suture in dama than in dorcas gazelle was observed, in concordance with the more evident deleterious effects of inbreeding depression in dama than in dorcas gazelles.

Fractal patterns for cells in culture

The present study aims to understand the growth of human malignant tumours in vitro, using the geometry of fractals as a method of analysis. The fractal dimensions of HN-5 and MDCK cell growth patterns have been measured. The first results may suggest the possibility of distinct growth processes characterized by different (time-dependent) effective fractal dimensions for MDCK and HN-5 cells. If this is true, the fractal dimensions may yet prove to be a useful discriminant for comparing different diagnostic categories.

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.

Fractal geometric analysis of material from molar and non-molar pregnancies

Histological sections from 25 non-molar pregnancies, nine partial hydatidiform moles, and 16 complete hydatidiform moles were examined (diagnosis was taken as the consensus of seven experienced histopathologists) and the fractal dimension was measured using a box-counting method implemented on a microcomputer-based image analysis system. The fractal dimensions of the different diagnostic categories were normally distributed with a mean of 1.50 for non-molar pregnancies, 1.44 for partial moles, and 1.42 for complete moles. All the measured fractal dimensions were greater than the topological dimension (1), demonstrating that the specimens had a fractal element to their structure. There was a significant difference between the fractal dimensions of non-molar pregnancies and complete moles (P = 0.0005), but not between partial moles and non-molar pregnancies (P = 0.0823) or complete and partial moles (P = 0.4400). Using the fractal dimension to predict the histopathological diagnosis assigned 56 per cent of the cases to the correct category with a kappa statistic of 0.26, so the fractal dimension, used alone, is not a useful morphometric discriminant in the diagnosis of molar and non-molar pregnancy.

Fractal geometric analysis of colorectal polyps

Colorectal polyps have a subjectively self-similar structure which suggests that these structures may have fractal elements and that the fractal dimension may be a useful morphometric discriminant. The fractal dimensions of images from haematoxylin and eosin-stained sections of 359 colorectal polyps (214 tubulovillous adenomas, 41 'pure' tubular adenomas, 29 'pure' villous adenomas, 68 metaplastic polyps, and 7 inflammatory polyps) were measured using a box-counting method implemented on a microcomputer-based image analysis system. Results were assessed using polychotomous logistic regression, confusion matrices, and kappa statistics. All examined polyps were shown to have a fractal structure in the range of scales examined. The fractal dimension was significantly different between different diagnostic categories (P < 0.0001) and was a useful discriminant between these categories (kappa statistic 0.60 for the confusion matrix with size as the other variable). The fractal dimension did not shown any significant correlation with the grade of epithelial dysplasia (P > 0.05). This study shows that colorectal polyps have a fractal structure over a defined range of magnification and Euclidean morphometric measurements will be invalid outside precisely defined conditions of resolution and magnification. The fractal dimension is a better way of quantitating the polyp shape and is a useful morphometric discriminant between diagnostic categories.

The application of fractal geometric analysis to microscopic images

Fractal geometry is a relatively new tool for the quantitative microscopist that is a more valid way of measuring dimensions of complex irregular objects than the integer-dimensional geometries (such as Euclidean geometry). This review discusses the theory of fractal geometry using the classic examples of the Von Koch curve, the Cantor set and the Sierpinski gasket. The problems of describing the dimensions of these objects are discussed and the concept of fractal dimensionality is introduced. Methods for measuring fractal dimensions are discussed, including their implementation on microcomputer-based image analysis systems . The advantages and problems of fractal geometric analysis are discussed and current applications in the field of microscopy are reviewed.