Fractal geometric analysis of colorectal polyps

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.

On the morphological stability of multicellular tumour spheroids growing in porous media

Multicellular tumour spheroids (MCTSs) are extensively used as in vitro system models for investigating the avascular growth phase of solid tumours. In this work, we propose a continuous growth model of heterogeneous MCTSs within a porous material, taking into account a diffusing nutrient from the surrounding material directing both the proliferation rate and the mobility of tumour cells. At the time scale of interest, the MCTS behaves as an incompressible viscous fluid expanding inside a porous medium. The cell motion and proliferation rate are modelled using a non-convective chemotactic mass flux, driving the cell expansion in the direction of the external nutrients' source. At the early stages, the growth dynamics is derived by solving the quasi-stationary problem, obtaining an initial exponential growth followed by an almost linear regime, in accordance with experimental observations. We also perform a linear-stability analysis of the quasi-static solution in order to investigate the morphological stability of the radially symmetric growth pattern. We show that mechano-biological cues, as well as geometric effects related to the size of the MCTS subdomains with respect to the diffusion length of the nutrient, can drive a morphological transition to fingered structures, thus triggering the formation of complex shapes that might promote tumour invasiveness. The results also point out the formation of a retrograde flow in the MCTS close to the regions where protrusions form, that could describe the initial dynamics of metastasis detachment from the in vivo tumour mass. In conclusion, the results of the proposed model demonstrate that the integration of mathematical tools in biological research could be crucial for better understanding the tumour's ability to invade its host environment.

Geometric complexity identifies platelet activation in familial hypercholesterolemic patients

Familial hypercholesterolemia (FH), a genetic disease, is associated with a severe incidence of athero-thrombotic events, related, also, to platelet hyperreactivity. A plethora of methods have been proposed to identify those activated circulating platelets, none of these has proved really effective. We need efficient methods to identify the circulating platelet status in order to follow the patients after therapeutic procedures. We propose the use of computerized fractal analysis for an objective characterization of the complexity of circulating platelet shapes observed by means of transmission electron microscopy in order to characterize the in vivo hyperactivated platelets of familial hypercholesterolemic patients, distinguishing them from the in vivo resting platelets of healthy individuals. Platelet boundaries were extracted by means of automatically image analysis. Geometric complexity (fractal dimension, D) by box counting was automatically calculated. The platelet boundary observed by electron microscopy is fractal, the shape of the circulating platelets is more complex in FH (n = 6) than healthy subjects (n = 5, P < 0.01), with 100% correct classification in selected individuals. In vitro activated platelets from healthy subjects show an analogous increase of D. The observed high D in the platelet boundary in FH originates from the in vivo platelet activation. Computerized fractal analysis of platelet shape observed by transmission electron microscopy can provide accurate, quantitative data to study platelet activation in familial hypercholesterolemia and after administration of drugs or other therapeutic procedures.

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.

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.

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.

Mathematical modelling of tumour acidity

Acid-mediated tumour invasion is receiving increasing experimental and clinical attention. Previous models proposed to describe this phenomenon failed to capture key properties of the system, such as the existence of the benign steady state, or predicted incorrectly the size of the inter-tissue gap. Here we show that taking proper account of quiescence ameliorates these drawbacks as well as revealing novel behaviour. The simplicity of the model allows us to fully identify the key parameters controlling different aspects of behaviour.

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.

The role of acidity in solid tumour growth and invasion

Acidic pH is a common characteristic of human tumours. It has a significant impact on tumour progression and response to therapies. In this paper, we develop a simple model of three-dimensional tumour growth to examine the role of acidosis in the interaction between normal and tumour cell populations. Both vascular and avascular tumour dynamics are investigated, and a number of different behaviours are observed. Whilst an avascular tumour always proceeds to a benign steady state, a vascular tumour may display either benign or invasive dynamics, depending on the value of a critical parameter. Analysis of the model allows us to assess novel therapies directed towards changing the level of acidity within the tumour.

Fractal dynamics in the differential diagnosis of thyroideal follicular neoplasms

In this study, we investigated the sensitivity, specificity, and predictive values of morphometric parameters of thyroideal follicular neoplasms based on concepts of fractal geometry. Thirty-seven follicular adenomas and 36 well-differentiated follicular carcinomas were assessed morphometrically. The nuclear area, nuclear area fraction, nuclear regularity factor, nuclear elongation factor, and slope setting (representing the ratio between the nuclear perimeter and nuclear regularity factor) were subjected to fractal dimensions analysis. By univariate analysis, the nuclear area, nuclear area fraction, nuclear regularity factor and slope values discriminate between adenomas and carcinomas. By multivariate analysis, the nuclear area, nuclear area fraction and slope values possess significant discriminatory powers in distinguishing between adenomas and carcinomas. Incorporating the nuclear area, nuclear area fraction, and slope values leads to a discriminatory power with 92% specificity and 83% sensitivity. The reciprocal relationships between the nuclear area, nuclear perimeter, and nuclear regularity factor of the cells of thyroideal adenomas and carcinomas may be expressed by fractal dimensions. Analysis limited to one parameter provides incomplete data. Expressing variations of the nuclear perimeter as a function of the nuclear regularity factor, the slope values constitute an independent attribute that significantly differentiates thyroideal adenomas from carcinomas.

A survey on histological image analysis-based assessment of three major biological factors influencing radiotherapy: proliferation, hypoxia and vasculature

Image analysis is a rapidly evolving field with growing applications in science and engineering. In cancer research, it has played a key role in advancing techniques of major diagnostic importance, minimising human intervention and providing vital clinical information. Especially in the field of tissue microscopy, the use of computers for the automated analysis of histological sections is becoming increasingly important. This paper presents an overview of various image analysis methodologies and summarises developments in this field, with great emphasis given on the assessment of three major biological factors known to influence the outcome of radiotherapy: proliferation, vasculature and hypoxia. A brief introduction followed by a survey is provided in each of these areas.

Architectural analysis of oral cancer, dysplastic, and normal epithelia

BACKGROUND

We present a novel, automated, and quantitative approach to evaluate local epithelial tissue architecture based on mathematical graph theory.

METHODS

Four hundred forty-one images of three diagnostic classes of oral epithelium (normal, dysplastic, and neoplastic) were analysed. The epithelial compartment was partitioned into exclusive areas associated with each nucleus to approach the theoretical cell extents. The spatial arrangement of cells in neighbourhoods of two sizes was characterised by constructing graph networks based on the cell centroids and recording 29 statistical properties. We analysed 104,627 and 67,590 neighbourhoods of diameters 37.5 and 75 microm, respectively.

RESULTS

The discrimination power of the architectural descriptors was evaluated by using discriminant analysis. The best neighbourhood discrimination rate was 75% for normal versus carcinoma. For the pooled data, discrimination into three classes based on largest number of neighbourhoods associated with each class was 100% correct. Case-wise, discrimination rates were 67%, 100%, and 80% correct for normal, premalignant, and malignant. When considering two classes, discrimination rates was 89% (normal) and 100% (malignant) correct, with 71% of premalignant cases assigned to the malignant class.

CONCLUSIONS

The results indicate that unbiased and reproducible quantification of tissue architectural features is possible and may provide valuable morphological information for diagnostic purposes.

Fractal dimensions of breast lesions on cytology smears

In the present study, box-counting fractal dimensions of benign and malignant cells of breast tumors on cytology material were measured and compared. We selected fine-needle aspiration cytology smears of 14 cases of histopathology-proven infiltrating duct carcinomas and 7 cases of fibroadenoma of the breast. Five cells were randomly selected from each case. Box-counting fractal dimensions of all cells were measured with the help of an image cytometer (Leica, Cambridge, UK), using Quantimet 600 software (Leica). In total, 70 malignant cells and 35 benign cells were studied. The mean fractal dimensions of malignant cells and benign cells were 0.9571 +/- 0.1265 and 0.8354 +/- 0.1367, respectively. There was significant difference in the fractal dimension of malignant and benign cells (P = 0.006, Mann-Whitney nonparametric test). The measurement of fractal dimension may be helpful in discriminating malignant from benign cells. This may be another discriminating feature of malignant cells, along with classic image morphometry based on Euclidean geometry.

Fractal tumor growth of ovarian cancer: sonographic evaluation

OBJECTIVE

The objective of this study was to determine whether sonographically depicted ovarian tumor growth is fractal, and the mean fractal dimension differs according to stages of the disease and histologic types.

METHODS

The fractal dimensions of outlines of sonographically depicted solid components in 160 ovarian tumors were measured using a box-counting method.

RESULTS

The mean fractal dimensions of the surface of intracystic solid components in serous, mucinous, endometrioid, and clear cell adenocarcinoma were 1.259, 1.243, 1.238, and 1.182, respectively. These values were significantly greater than the topological dimension of a line (=1). The value was significantly higher in stage I or II (1.381) than stage III or IV (1.205) in serous carcinoma (P = 0.02), but not significantly different in clear cell carcinoma (1.187 and 1.172, respectively). In stage I or II, the value of serous carcinoma (1.381) was significantly higher than that of clear cell carcinoma (1.187) (P = 0.03). The value of mucinous cystadenoma of low malignant potential was 1.337, which was also significantly greater than 1. The mean fractal dimensions of outlines of solid tumors in cases with dysgerminoma and thecoma-fibroma were 1.036 and 1.023, respectively. These values were not significantly different from 1.

CONCLUSION

This study shows that the surface of solid components in cystic epithelial ovarian cancers has a fractal structure, and the mean fractal dimension may differ according to stages of the disease and histologic types. Fractal geometry, a vocabulary of irregular shapes, can be useful for describing the pathological architecture of ovarian tumors and for yielding insights into the mechanisms of tumor growth.

Short-term rhythmic proliferation of human breast cancer cell lines: surface effects and fractal growth patterns

Kinetic studies of cell proliferation rates shed light on the growth dynamics of cancer. Most such studies are based on measurements of cell numbers that were evaluated in time intervals of about 12 h. Studies of the initial tumour growth with short measuring intervals are rare. This study was therefore designed with 1 h measuring intervals over a 24 h period. Human breast cancer cell lines (ZR-75-1, SK-BR-3, MCF-7) and a benign cell line (HBL-100) were used to study the hourly thymidine uptake as a measure of cells in synthesis. In parallel experiments, the same cell lines were also exposed to tumour necrosis factor alpha (TNF-alpha) to explore the effect of an apoptosis-inducing substance on initial tumour growth kinetics. In time-evolution plots, there was an oscillation of the labelling index of thymidine uptake for all investigated cell lines, with and without TNF-alpha. Based on the results obtained, a mathematical model was developed mimicking the real experiment. To describe the system dynamically a cellular automaton model was studied. The growth kinetics revealed by the simulation were in accordance with our experimental data. Two- and three-dimensional growth simulations of this computer model yielded objects morphologically similar to real images of human breast cancer. Almost identical fractal dimensions of the virtual and real tumours further supported this visual similarity. The cellular automata models could, therefore, be seen as a bridge towards realistic in vivo scenarios. From a clinical point of view, the results obtained may be applicable not only to primary tumours, but even to tumour cell microfoci and small metastases, which are a major concern in early metastasizing tumours such as breast cancer.

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.

Genetic analysis of multiple synchronous lesions of the colon adenoma-carcinoma sequence

The colorectal adenoma-carcinoma sequence represents a well-known paradigm for the sequential development of cancer driven by the accumulation of genomic defects. Although the colorectal adenoma-carcinoma sequence is well investigated, studies about tumours of different dignity co-existent in the same patient are seldom. In order to address the distribution of genetic alterations in different lesions of the same patient, we coincidently investigated carcinomas, adenomas and aberrant crypt foci in patients with sporadic colon cancer. By utilizing polymerase chain reaction, single-strand conformation polymorphism, heteroduplex-analysis, restriction fragment length polymorphism, protein truncation test and sequencing techniques we looked for mutations and microsatellite instability of APC, H-ras, K-ras, p53, DCC and the DNA repair genes hMLH1/hMSH2. In accordance with the suggested adenoma-carcinoma sequence of the colon, four patients reflected the progressive accumulation of genetic defects in synchronously appearing tumours during carcinogenesis. However, two patients with non-hereditary malignomas presented different genetic instabilities in different but synchronously appearing tumours suggesting non-clonal growth under almost identical conditions of the environment. Thus, sporadically manifesting multiple lesions of the colon were not necessarily driven by similar genetic mechanisms. Premalignant lesions may transform into malignant tumours starting from different types of genetic instability, which indicates independent and simultaneous tumorigenesis within the same organ.

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.

Chaodynamic loss of complexity and self-similarity in cancer

'Omnis cellula a cellula': each cell descends from another cell--and most complex organ structures are built up of multiple cells at least. Even the underlying mechanisms of physiological as well as pathological processes reflect complexity in a pronounced manner. Loss of complexity, however, has been detected in aging and apparently also in the case of a number of diseases. This paper describes the loss of complexity in both carcinogenesis and the related growth pattern of cancer. After disruption of the cell's stability as a result of mutations, a decrease in the ability of the cell to induce a self-organized response is associated with the loss of control parameters at the cellular and molecular level. Complexity, understood as a principle of order, is altered in certain malignant tumors, as is indicated by a loss of the golden mean and by the disappearance of self-similarity. The lack of self-similarity could, therefore, be an ideal practical marker of malignancy in medical image analysis.

Fractal characterization of chromatin appearance for diagnosis in breast cytology

This study explores the use of fractal analysis in the numerical description of chromatin appearance in breast cytology. Images of nuclei from fine-needle aspiration biopsies of the breast are characterized in terms of their Minkowski and spectral fractal dimensions, for 19 patients with benign epithelial cell lesions and 22 with invasive ductal carcinomas. Chromatin appearance in breast epithelial cell nuclear images is demonstrated to be fractal, suggesting that the three-dimensional chromatin structure in these cells also has fractal properties. A statistically significant difference between the mean spectral dimensions of the benign and malignant cases is demonstrated. The two fractal dimensions are very weakly correlated. A statistically significant difference between the benign and malignant cases in lacunarity, a fractal property characterizing the size of holes or gaps in a texture, is found over a wide range of scales. These differences are particularly pronounced at the smallest and largest scales, corresponding respectively to fine-scale texture, indicating whether chromatin is clumped or fine, and to large-scale structures like nucleoli. Logistic regression and artificial neural network classification models are developed to classify unknown cases on the basis of fractal measures of chromatin texture. Using leave-one-out cross-validation, the best logistic regression classifier correctly diagnoses 95.1 per cent of the cases. The best neural network model can correctly classify all of the cases, but it is unclear whether this is due to overtraining. Fractal dimensions and lacunarity are useful tools for the quantitative characterization of chromatin appearance, and can potentially be incorporated into image analysis devices to assure the quality and reproducibility of diagnosis by breast fine-needle aspiration biopsy.

Fractal properties of cancellous bone of the iliac crest in vertebral crush fracture

Fractal analysis is a method for describing complex shapes, including the cancellous structure of bone. It describes the surface texture and form of individual trabecular profiles and the overall cancellous structure. Sixty-four postmenopausal women with symptoms of back pain were referred for investigation for osteoporosis. The patients were divided into two groups for comparison: vertebral crush fracture (n = 31, mean age 68.58 +/- 6.47 years), and no vertebral crush fracture (n = 33, mean age 63.36 +/- 7.21 years). Cores of cancellous bone, 3 mm in diameter, were taken from the iliac crest and sectioned. A box-counting method implemented on an image analyzer was used to measure the fractal dimension. Three fractal dimensions describing trabecular surface texture (fractal 1), trabecular shape (fractal 2), and trabecular arrangement (fractal 3) were measured, indicating that cancellous bone has sectional self-similarity. Conventional histomorphometry was also performed on the samples. The results show that fractal 2 is significantly lower in the vertebral crush fracture group than in the nonfracture group (1.15 +/- 0.10 < 1.23 +/- 0.090, p < 0.0013). The histomorphometric analysis shows that bone surface total volume (p < 0.0002), trabecular number (p < 0.0001), and osteoid surface bone surface (p < 0.028) are significantly lower in the fracture group than the nonfracture group. Eroded surface/bone surface (p < 0.056) follows this trend, whereas trabecular separation (p < 0.001) is significantly higher in the fracture group than in the nonfracture group. Fractal 1 and fractal 3 were not significantly different between study groups. The fractal dimension detects changes in the cancellous architecture and gives information about iliac bone transformation in postmenopausal women with vertebral fracture.

Proteoglycan expression in the rat temporomandibular joint in response to unilateral bite raise

The vertebrate articular tissue consists of collagen fibers embedded in a ground substance. Collagen resists tensile forces, while proteoglycans in the ground substance provide resilience and resistance to compression. It was hypothesized that unilateral bite raise would induce increasing expression of proteoglycans in TMJ articular tissues. As a test of this hypothesis, six- and nine-week-old Sprague-Dawley rats received unilateral bite-raising appliances bonded to their right upper molars for 4 wks. A group of nine-week-old rats was housed for an additional 4 wks after removal of the appliances they had worn for 4 wks. Proteoglycans that carry abundant chondroitin sulfate and keratan sulfate side-chains, most likely aggrecans, were detected by safranin O in the fibrocartilaginous zone of the condyle in parasagittal sections. A monoclonal antibody against a large chondroitin sulfate proteoglycan related to versican reacted strongly in the surface fibrous layer of the mandibular condyle and moderately in the discs of the treated specimens. Computer quantification for safranin O and anti-versican antibody staining revealed that the average intensities of the treated specimens were significantly higher than those of their corresponding sham-operated controls, and the average intensities of the treatment-reversal specimens had no significant differences from their corresponding sham-operated controls. Thus, unilateral bite raise appeared to have induced an increase in the expression of aggrecan in the condylar cartilage and a proteoglycan related to versican in the TMJ disc and the articular surface of the condyle. The elevated proteoglycan expression is interpreted to suggest that unilateral bite raise leads to an increase in the magnitude of compressive forces in the rat temporomandibular joint.

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.

Fractal properties of subchondral cancellous bone in severe osteoarthritis of the hip

Primary osteoarthritis of the hip results in changes to the architecture of subchondral cancellous bone. These changes in architecture occur through the action of osteoclasts and osteoblasts in selectively removing and adding bone. The quantitative description of the bone architecture helps in understanding the etiology of primary osteoarthritis. Fractal analysis is a method for describing complex shapes, which is expressed numerically as the fractal dimension. A box counting method was used, where the perimeter of binary profiles of cancellous bone samples was measured for different box sizes. The fractal dimension was the absolute value of the slope of the straight line segments from the plot of the log number of boxes versus the log box size. Cancellous bone samples from two subchondral regions, superior and inferomedial, to the fovea were analyzed from primary severe osteoarthritic specimens taken following total hip replacement surgery (n = 19, aged 51-80 years) and autopsy controls (n = 25, aged 18-90 years). There were three straight line segments identified on the log-log plot, for each subject, indicating a fractal dimension over three different ranges of scale. The results show that in the superior region there is a highly significant difference between the groups (p < 0.0001) for fractal 1 and pivot point 2. The histomorphometry shows significant differences for bone volume/total volume, bone surface/total volume, trabecular separation, and osteoid surface/total volume between groups. In the inferomedial region fractal 1 and fractal 2 are significantly different. For the histomorphometry, trabecular thickness and eroded surface/total volume are significantly different between the groups. The pivot points, i.e., the box size at which the fractal dimension changes, were of similar magnitude to the trabecular thickness and trabecular separation. These data suggest that the fractal geometry analysis of cancellous bone identifies architectural features not easily recognized by conventional bone histomorphometry. The fractal dimension is a descriptor of bone structure which simplifies the description of a complex structure and enables changes in cancellous bone architecture, due to disease, to be identified.

Three-dimensional reconstruction of non-Hodgkin's lymphoma in bone marrow trephines

The objective of this study was to analyse the location in the bone marrow of deposits of low-grade non-Hodgkin's lymphoma. This was achieved using computer-generated three-dimensional reconstruction techniques applied to serial tissue sections of five bone marrow trephines involved by lymphoma. For comparative purposes, previously published three-dimensional models of benign lymphoid aggregates in the bone marrow were need. The images generated by this new study showed that deposits of low-grade non-Hodgkin's lymphoma involving the bone marrow always assumed a paratrabecular pattern of infiltration at some point. This is in direct contrast to the pattern of bone marrow infiltration shown by benign lymphoid aggregates. It is concluded that location within the marrow space is a crucial factor in distinguishing between benign and low-grade malignant lymphoid infiltrates in the bone marrow.

How important is tumour shape? Quantification of the epithelial-connective tissue interface in oral lesions using local connected fractal dimension analysis

Quantification of the local complexity of the epithelial-connective tissue interface (ECTI) in normal mucosa, epithelial dysplasia, and squamous cell carcinoma of the floor of the mouth was investigated by estimating the local connected fractal dimension in tissue profiles from histological sections. The use of certain parameters of the distribution of the local connected fractal dimensions of the ECTI classifies the cases belonging to these three histopathological diagnoses with 85 per cent accuracy by means of linear discriminant analysis. The values of the local fractal dimension were also used to produce colour-coded dimensional images of the ECTI, to highlight locations with higher irregularity that may correlate with locally invasive 'higher-risk' areas.

The potential role of apoptosis (programmed cell death) in a chaotic determined carcinogenesis

The proposed hypothesis may interpret apoptosis as a dynamic parameter by means of chaos theory. Regular tissue renewal is determined by the quantity of mitosis and cell death (mainly apoptosis). These two components are describable as so-called 'limit cycle attractor' representing a reference cycle of the tissue system. If tissue kinetics remain within these limits, the system stays in a homeostatic equilibrium. Disruptive factors such as genetic alterations may have the ability to destabilize this homeostasis and induce the quasiperiodic development which represents one way in which a system becomes chaotic. The inhibition of apoptosis prevents the elimination of mutation-affected cells which leads to an uncoordinated proliferation of a mutated cell clone. Thus, this clone represents an independent oscillator which effects a new attractor: it is called 'torus' attractor. Additional perturbations induce the chaotic 'strange attractor': the system is bound to shift into chaos, morphologically detectable as cancer.

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.