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PubMed1993

Analyzing and modeling fractal intensity point processes.

Fractal intensity point processes--doubly stochastic point processes with a fractal waveform intensity process--are required to describe the discharge patterns recorded from the auditory and visual systems. The Fano factor--the ratio of the variance of the number of events in an interval to the mean of this number--captures the self-similar characteristics of the intensity via two quantities: fractal dimension and fractal time. The fractal dimension is the exponent of the asymptotic power law behavior of the Fano factor with interval duration. The fractal time delineates long-term fractal behavior from short-term characteristics of the data. The average rate and self-similarity parameter of the intensity process, absolute and relative refractory effects, and serial dependence all modify the fractal time. To generate fractal intensity point processes, stochastic fractal processes are derived by applying memoryless, nonlinear transformations to fractional Gaussian noise. The intensity's amplitude distribution in combination with the Fano factor form criteria to choose the transformation that best describes data.

Auditory Perception↗

PubMedDate not supplied

Analyte-receptor binding kinetics for biosensor applications. An analysis of the influence of the fractal dimension on the binding rate coefficient.

The diffusion-limited binding kinetics of antigen (analyte), in solution with antibody (receptor) immobilized on a biosensor surface, is analyzed within a fractal framework. Most of the data presented is adequately described by a single-fractal analysis. This was indicated by the regression analysis provided by Sigmaplot. A single example of a dual-fractal analysis is also presented. It is of interest to note that the binding-rate coefficient (k) and the fractal dimension (Df) both exhibit changes in the same and in the reverse direction for the antigen-antibody systems analyzed. Binding-rate coefficient expressions, as a function of the Df developed for the antigen-antibody binding systems, indicate the high sensitivity of the k on the Df when both a single- and a dual-fractal analysis are used. For example, for a single-fractal analysis, and for the binding of antibody Mab 0.5 beta in solution to gp120 peptide immobilized on a BIAcore biosensor, the order of dependence on the Df was 4.0926. For a dual-fractal analysis, and for the binding of 25-100 ng/mL TRITC-LPS (lipopolysaccharide) in solution with polymyxin B immobilized on a fiberoptic biosensor, the order of dependence of the binding-rate coefficients, k1 and k2, on the fractal dimensions, Df1 and Df2, were 7.6335 and -11.55, respectively. The fractional order of dependence of the k(s) on the Df(s) further reinforces the fractal nature of the system. The k(s) expressions developed as a function of the Df(s) are of particular value, since they provide a means to better control biosensor performance, by linking it to the heterogeneity on the surface, and further emphasize, in a quantitative sense, the importance of the nature of the surface in biosensor performance.

Antibodies, Monoclonal↗

PubMed1997

Generation of fractals from incursive automata, digital diffusion and wave equation systems.

This paper describes modelling tools for formal systems design in the fields of information and physical systems. The concept and method of incursion and hyperincursion are first applied to the fractal machine, an hyperincursive cellular automata with sequential computations with exclusive or where time plays a central role. Simulations show the generation of fractal patterns. The computation is incursive, for inclusive recursion, in the sense that an automaton is computed at future time t + 1 as a function of its neighbouring automata at the present and/or past time steps but also at future time t + 1. The hyperincursion is an incursion when several values can be generated for each time step. External incursive inputs cannot be transformed to recursion. This is really a practical example of the final cause of Aristotle. Internal incursive inputs defined at the future time can be transformed to recursive inputs by self-reference defining then a self-referential system. A particular case of self-reference with the fractal machine shows a non deterministic hyperincursive field. The concepts of incursion and hyperincursion can be related to the theory of hypersets where a set includes itself. Secondly, the incursion is applied to generate fractals with different scaling symmetries. This is used to generate the same fractal at different scales like the box counting method for computing a fractal dimension. The simulation of fractals with an initial condition given by pictures is shown to be a process similar to a hologram. Interference of the pictures with some symmetry gives rise to complex patterns. This method is also used to generate fractal interlacing. Thirdly, it is shown that fractals can also be generated from digital diffusion and wave equations, that is to say from the modulo N of their finite difference equations with integer coefficients.

Computer Simulation↗

PubMedDate not supplied

Development of fractal kinetic theory for enzyme-catalysed reactions and implications for the design of biochemical pathways.

Recent evidence has shown that elementary bimolecular reactions under dimensionally-restricted conditions, such as those that might occur within cells when reactions are confined to two-dimensional membranes and one-dimensional channels, do not follow traditional mass-action kinetics, but fractal kinetics. The power-law formalism, which provides the context for examining the kinetics under these conditions, is used here to examine the implications of fractal kinetics in a simple pathway of reversible reactions. Starting with elementary chemical kinetics, we proceed to characterise the equilibrium behaviour of a simple bimolecular reaction, derive a generalised set of conditions for microscopic reversibility, and develop the fractal kinetic rate law for a reversible Michaelis-Menten mechanism. Having established this fractal kinetic framework, we go on to analyse the steady-state behaviour and temporal response of a pathway characterised by both the fundamental and quasi-steady-state equations. These results are contrasted with those for the fundamental and quasi-steady-state equations based on traditional mass-action kinetics. Finally, we compare the accuracy of three local representations based on both fractal and mass-action kinetics. The results with fractal kinetics show that the equilibrium ratio is a function of the amount of material in a closed system, and that the principle of microscopic reversibility has a more general manifestation that imposes new constraints on the set of fractal kinetic orders. Fractal kinetics in a biochemical pathway allow an increase in flux to occur with less accumulation of pathway intermediates and a faster temporal response than is the case with traditional kinetics. These conclusions are obtained regardless of the level of representation considered. Thus, fractal kinetics provide a novel means to achieve important features of pathway design.

Catalysis↗

PubMed1994

Fractal analysis of renal cortical perfusion.

RATIONALE AND OBJECTIVES: Contrast-enhanced computed tomography (CT) images of canine renal cortex were analyzed to determine if the heterogeneous pixel intensity patterns met the requirements for fractal analysis and if the heterogeneity could be quantified by a fractal dimension, Df. METHODS: Contrast-enhanced CT images were obtained after injection of iodipamide ethyl ester (IDE), a vascular marker, or iohexol, a freely filtered interstitial marker, into the catheterized renal artery of an anesthetized dog. Images were mounted on a graphics workstation for analysis. A computer program was written to determine the fractal dimension of the pixel-intensity pattern within selected regions. RESULTS: All regions of renal cortex examined met the requirements for fractal analysis. Three seconds after injection of IDE, the mean fractal dimension decreased significantly from 1.21 +/- 0.05 to 1.12 +/- 0.06 (P < .05). Although the mean fractal dimensions were not significantly different, the variation in the fractal dimension around the renal cortex was significantly different with IDE as compared with iohexol at 3 seconds (P < .05). Differences in the change in fractal dimension over time were also observed with IDE as compared with iohexol. CONCLUSIONS: Fractal dimension measurement provides a new means to examine the in-vivo organization of renal vascular perfusion by quantifying pixel heterogeneity in contrast-enhanced CT. This may prove useful in understanding and quantifying the pathophysiologic changes in renal disease.

Animals↗

PubMed1996

Estimation of fractal dimension in radiographs.

In the last decade, the fractal dimension has become a popular parameter to characterize image textures. Also in radiographs, various procedures have been used to estimate the fractal dimension. However, certain characteristics of the radiographic process, e.g., noise and blurring, interfere with the straightforward application of these estimation methods. In this study, the influence of quantum noise and image blur on several estimation methods was quantified by simulating the effect of quantum noise and the effect of modulation transfer functions, corresponding with different screen-film combinations, on computer generated fractal images. The results are extrapolated to explain the effect of film-grain noise on fractal dimension estimation. The effect of noise is that, irrespective of the noise source, the fractal dimension is overestimated, especially for lower fractal dimensions. On the other hand, blurring results in an underestimation of the dimensions. The effect of blurring is dependent on the estimation method used; the dimension estimates by the power spectrum method are lowered with a constant value, whereas the underestimation by the methods working in the spatial domain is dependent on the given dimension. The influence of the MTF and noise on fractal dimension estimation seriously limits the comparability of fractal dimensions estimated from radiographs which differ in noise content or MTF. Only when the power spectrum method is used, it is possible to correct for the influence of different MTFs of screen-film combinations. It is concluded that only when using the same object-focus distance, the same exposure conditions, the same digitizer at the same resolution, can fractal dimensions as estimated in radiographs be reliably compared.

Biophysical Phenomena↗

PubMed1996

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.

Animals↗

PubMed2006

Fractal analysis of the maternal surface of the placenta: preliminary report.

AIMS: The objective of this study was to determine whether the maternal surface of the placenta is fractal, and whether the mean fractal dimension differs according to the gestational age and clinically or pathologically different conditions. METHODS: Using digitized images of the maternal surface of 75 placentas, fractal dimensions were measured with a fractal analysis software. RESULTS: The mean fractal dimension of the maternal surface of the placentas significantly exceeded the topological dimension of a surface (= 2). This means that the morphological pattern of the maternal surface fulfills the mathematical definition of fractal structures. Among the three different groups of gestational age, the mean fractal dimension in 22-29 weeks was significantly lower than that in 30-36 and 37-41 weeks (p = 0.022 and 0.014, respectively). Although not significantly different (p = 0.149), in 30-36 weeks the mean fractal dimension of the placentas complicated by pregnancy-induced hypertension (PIH) was greater than that without PIH. CONCLUSION: Fractal geometry, a vocabulary of irregular shapes, can be useful for describing quantitatively the architecture of the maternal surface of the placenta and become a useful tool for analyzing physiological and pathological placental formation mathematically.

Adolescent↗

PubMed2006

Technical factors in fractal analysis of periapical radiographs.

OBJECTIVES: Fractal analysis quantifies complex geometric structures by generating a fractal dimension, which can measure trabecular bone density. The use of non-standardized radiographic techniques potentially limits the reliability of fractal analysis. The objective of this study was to determine how variations in radiographic technique affect fractal dimension. METHODS: Periapical radiographs of maxillary incisors taken on eight dry human skulls at varying angulation, tube potential and impulse settings were subjected to Fourier transform fractal analysis. RESULTS: A significant (p < 0.05) difference in fractal dimension, ranging from 2.5 to 2.9, was found between the skulls, potentially reflecting variations in bone trabeculation. However, variations in angulation, tube potential or impulse did not have a statistically significant effect (p > 0.05) on fractal dimension, with the standard deviation of the fractal dimension ranging from +/- 0.005 to +/- 0.062 at various radiographic machine settings. CONCLUSIONS: This study indicates that non-standardized periapical radiographs may provide a reliable method for determining fractal dimensions, which could be useful in analysing changes in alveolar bone density in various bone disorders.

Alveolar Process↗

PubMed1998

Quantitative analysis of pulmonary emphysema: three-dimensional fractal analysis of single-photon emission computed tomography images obtained with a carbon particle radioaerosol.

OBJECTIVE: The purpose of this study was to quantify the heterogeneous distribution of a carbon particle radioaerosol in the lungs of patients with pulmonary emphysema using single-photon emission computed tomography (SPECT) and fractal analysis. SUBJECTS AND METHODS: Ventilation SPECT was performed on 19 patients with pulmonary emphysema, seven patients with suspected pulmonary emphysema, and 14 healthy volunteers. We delineated the lungs using five cutoff levels (15%, 20%, 25%, 30%, and 35% of the maximal pixel radioactivity) and measured the total number of pixels in the areas surrounded by the contours obtained with each cutoff level. We calculated fractal dimensions from the relationship between the total number of pixels and cutoff levels transformed into logarithms. RESULTS: The fractal dimensions increased with the severity of pulmonary emphysema. The fractal dimensions for patients with pulmonary emphysema, patients with suspected pulmonary emphysema, and healthy volunteers were 2.03+/-0.61 (mean +/- SD), 0.77+/-0.09, and 0.59+/-0.06, respectively. A significant difference (p < .0005) in the fractal dimension was found when any of these three groups was compared with any other group of subjects. CONCLUSION: Three-dimensional quantitative evaluation of the emphysematous lung was possible using SPECT and fractal analysis. Our results suggest that fractal analysis can quantify the heterogeneity of the distribution of a carbon particle radioaerosol in the lung. Fractal analysis appears to be applicable for routine clinical use because it is simple and the fractal dimensions can be obtained easily without using special software.

Aged↗

PubMed1996

Fractal tumours: their real and virtual images.

Autonomous and uncoordinated proliferation of epithelia leads to various well-known growth patterns such as expansive (cauliflower-like), radiating infiltrative, polycyclic, and roundish-ovoid figures. All attempts to describe such natural growth patterns graphically by Euclidean geometry have failed and remain no more than works of art. However, fractal geometry is a new tool for the characterization of irregularly-shaped and complex figures. Moreover, behind a fractal structure there is a basic power-law which provides the opportunity to simulate these forms artificially. A prerequisite for achieving this goal of simulating tumour growth by computer is to establish whether typical tumour growth patterns are fractal. Hence, an investigation was undertaken of 20 tumours (malignant, metastases or benign) exhibiting the above-mentioned typical patterns. If tumour outlines are fractal they have to possess a fractal non-integer dimension which significantly exceeds the integer Euclidean dimension. The fractal dimension of tumour outlines was determined using the box-counting method. Almost all tumours presented a fractal dimension and virtual tumour images were created by utilizing available fractal software. In conclusion, the determination of the fractal dimension of solid neoplasms may be an additional morphometric parameter for growth assessment and it probably provides further opportunity to simulate cancer growth and infiltration by computer animation.

Cell Division↗

PubMed1998

Fractal dimension and approximate entropy of heart period and heart rate: awake versus sleep differences and methodological issues.

1. Investigations that assess cardiac autonomic function include non-linear techniques such as fractal dimension and approximate entropy in addition to the common time and frequency domain measures of both heart period and heart rate. This article evaluates the differences in using heart rate versus heart period to estimate fractal dimensions and approximate entropies of these time series.2. Twenty-four-hour ECG was recorded in 23 normal subjects using Holter records. Time series of heart rate and heart period were analysed using fractal dimensions, approximate entropies and spectral analysis for the quantification of absolute and relative heart period variability in bands of ultra low (<0.0033 Hz), very low (0. 0033-0.04 Hz), low (0.04-0.15 Hz) and high (0.15-0.5 Hz) frequency.3. Linear detrending of the time series did not significantly change the fractal dimension or approximate entropy values. We found significant differences in the analyses using heart rate versus heart period between waking up and sleep conditions for fractal dimensions, approximate entropies and absolute spectral powers, especially for the power in the band of 0.0033-0.5 Hz. Log transformation of the data revealed identical fractal dimension values for both heart rate and heart period. Mean heart period correlated significantly better with fractal dimensions and approximate entropies of heart period than did corresponding heart rate measures.4. Studies using heart period measures should take the effect of mean heart period into account even for the analyses of fractal dimension and approximate entropy. As the sleep-awake differences in fractal dimensions and approximate entropies are different between heart rate and heart period, the results should be interpreted accordingly.

Adult↗

PubMed1991

Analysis of texture in macroradiographs of osteoarthritic knees using the fractal signature.

Texture of regions of macroradiographs (x5) of six normal and five osteoarthritic knee joints, taken on a high resolution microfocal x-ray unit, are examined using mathematical morphology. Radiographs of bones are two-dimensional projections of attenuation coefficient through the three-dimensional (3D) joint structure. Visible texture represents the summation of the attenuation from numerous thin plates of bone. Where there is no organization in the trabeculae, resultant radiographs approximate a fractal surface. Varying structuring element size in mathematical morphology allows estimation of fractal dimension over a range of resolution. Variation of fractal dimension with resolution, the fractal signature, indicates how images deviate from fractal surfaces. By correct choice of structuring element, a texture analysis method using the fractal signature has been developed, tolerant to changes in image acquisition and digitization. Texture in regions of radiographs of normal tibia approximates a fractal surface with dimension 2.8 as does vertical structure in arthritic patients. In osteoarthritic knee joints, horizontal tibial trabeculae thicken. Horizontal structure in the tibia on radiographs of arthritic patients deviates from the fractal model. This is indicated by peaks in the fractal signature whose height and position match a visual assessment of the degree of arthritic change.

Humans↗

PubMed1995

Fractal clustering of ventricular ectopy correlates with sympathetic tone preceding ectopic beats.

BACKGROUND: Fractal geometric analysis of ventricular ectopy yields a fractal dimension, which can range from zero to one and is inversely related to clustering of ventricular premature contractions (VPCs). Low values of this fractal dimension, which reflect significantly nonuniform distributions of ventricular ectopy, are found in patients with life-threatening ventricular arrhythmias and predict adverse outcomes in selected patients with congestive heart failure and with mitral regurgitation. However, the physiological mechanism and correlates of the fractal dimension are unknown. METHODS AND RESULTS: To explore the physiological correlates of clustered ventricular ectopy, we studied 30 patients with a history of sustained ventricular tachycardia or ventricular fibrillation who had inducible sustained monomorphic ventricular tachycardia during electrophysiological study and also underwent drug-free 24-hour ambulatory ECG. In addition to fractal dimension (determined by use of our previously described algorithm), we measured the mean RR interval (+/- SD) for all sinus beats preceding a sinus beat and for all sinus beats preceding a single VPC and the mean root-mean-square difference (RMSSD) of all windows of 15 successive RR intervals (excluding ectopic beats) preceding a sinus beat and preceding a single VPC. Based on the directional changes of mean RR (a measure of both sympathetic and parasympathetic tone) and of RMSSD (a measure of parasympathetic tone), each patient's inferred relative sympathetic tone preceding ventricular ectopy was classified as increased, unchanged, or decreased. If these values changed concordantly, relative sympathetic tone was indeterminate. Fractal dimension did not correlate with the mean RR interval, SD of the RR interval, or RMSSD preceding sinus beats or preceding VPCs (all P > .10). In 20 patients, fractal dimension was significantly lower among those with increased relative sympathetic tone (n = 14) than those with unchanged or decreased sympathetic tone (n = 6, P = .008). Ten patients had indeterminate relative sympathetic tone. CONCLUSIONS: Clustering of ventricular ectopy, as measured by the fractal dimension, is observed in patients at increased risk of sudden cardiac death. A low fractal dimension (clustered ventricular ectopy) is related to changes in heart rate and heart rate variability that are consistent with transient increases in cardiac sympathetic tone.

Aged↗

PubMed1993

[New horizons in medicine. Fractals].

In these last few years, a vertiginous growth in the application of informatics in the field of science has generated a new branch of geometry: the geometry of irregularity whose paradigmatical expression is fractals. We can identify in the fractal an object of complex structure which, when gradually enlarged, shows up details which repeat themselves as the scale of enlargement is progressively run through. It is possible, therefore, by using fractals to study and quantify in a much more concise way and much less approximatively those structures which were until recently described as granulous, tentacular, ramified, etc. Fractals bridge the gap between mathematics and visual art and allow us to explore the rapport between determinism and apparent chaos in the study of non-linear biological systems. Fractal dimension is an intermediary one between linear and surface dimensions. The applications of fractals images in the study of certain problematics which concern the cardiovascular system are referred to synthetically; particularly those concerning the sectorial differences of myocardial blood flow and the branches of small coronary vessels, both in physiological and pathological conditions. Fractal images permit us to analyse the regional flow of the pulmonary parenchyma, and to quantify pulmonary damage from thromboembolism. Other applications of fractal studies regard the digestive system, the nervous system and mammary pathology. In specialistic fields, fractals have been studied in both ophthalmology and odontology, and also in biochemistry to study protein surfaces and fluids turbulence.

Cardiovascular System↗

PubMed1996

Fractal dimension and architecture of trabecular bone.

The fractal dimension of trabecular bone was determined for biopsies from the proximal femur of 25 subjects undergoing hip arthroplasty. The average age was 67.7 years. A binary profile of the trabecular bone in the biopsy was obtained from a digitized image. A program written for the Quantimet 520 performed the fractal analysis. The fractal dimension was calculated for each specimen, using boxes whose sides ranged from 65 to 1000 microns in length. The mean fractal dimension for the 25 subjects was 1.195 +/- 0.064 and shows that in Euclidean terms the surface extent of trabecular bone is indeterminate. The Quantimet 520 was also used to perform bone histomorphometric measurements. These were bone volume/total volume (BV/TV) (per cent) = 11.05 +/- 4.38, bone surface/total volume (BS/TV) (mm2/mm3) = 1.90 +/- 0.51, trabecular thickness (Tb.Th) (mm) = 0.12 +/- 0.03, trabecular spacing (Tb.Sp) (mm) = 1.03 +/- 0.36, and trabecular number (Tb.N) (number/mm) = 0.95 +/- 0.25. Pearsons' correlation coefficients showed a statistically significant relationship between the fractal dimension and all the histomorphometric parameters, with BV/TV (r = 0.85, P < 0.0001), BS/TV (r = 0.74, P < 0.0001), Tb.Th (r = 0.50, P < 0.02), Tb.Sp (r = -0.81, P < 0.0001), and Tb.N (r = 0.76, P < 0.0001). This method for calculating fractal dimension shows that trabecular bone exhibits fractal properties over a defined box size, which is within the dimensions of a structural unit for trabecular bone. Therefore, the fractal dimension of trabecular bone provides a measure which does not rely on Euclidean descriptors in order to describe a complex geometry.

Adult↗

PubMed1998

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.

Biopsy, Needle↗

PubMed1994

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.

Adenoma↗