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1.  Fractal Physiology and the Fractional Calculus: A Perspective 
This paper presents a restricted overview of Fractal Physiology focusing on the complexity of the human body and the characterization of that complexity through fractal measures and their dynamics, with fractal dynamics being described by the fractional calculus. Not only are anatomical structures (Grizzi and Chiriva-Internati, 2005), such as the convoluted surface of the brain, the lining of the bowel, neural networks and placenta, fractal, but the output of dynamical physiologic networks are fractal as well (Bassingthwaighte et al., 1994). The time series for the inter-beat intervals of the heart, inter-breath intervals and inter-stride intervals have all been shown to be fractal and/or multifractal statistical phenomena. Consequently, the fractal dimension turns out to be a significantly better indicator of organismic functions in health and disease than the traditional average measures, such as heart rate, breathing rate, and stride rate. The observation that human physiology is primarily fractal was first made in the 1980s, based on the analysis of a limited number of datasets. We review some of these phenomena herein by applying an allometric aggregation approach to the processing of physiologic time series. This straight forward method establishes the scaling behavior of complex physiologic networks and some dynamic models capable of generating such scaling are reviewed. These models include simple and fractional random walks, which describe how the scaling of correlation functions and probability densities are related to time series data. Subsequently, it is suggested that a proper methodology for describing the dynamics of fractal time series may well be the fractional calculus, either through the fractional Langevin equation or the fractional diffusion equation. A fractional operator (derivative or integral) acting on a fractal function, yields another fractal function, allowing us to construct a fractional Langevin equation to describe the evolution of a fractal statistical process. Control of physiologic complexity is one of the goals of medicine, in particular, understanding and controlling physiological networks in order to ensure their proper operation. We emphasize the difference between homeostatic and allometric control mechanisms. Homeostatic control has a negative feedback character, which is both local and rapid. Allometric control, on the other hand, is a relatively new concept that takes into account long-time memory, correlations that are inverse power law in time, as well as long-range interactions in complex phenomena as manifest by inverse power-law distributions in the network variable. We hypothesize that allometric control maintains the fractal character of erratic physiologic time series to enhance the robustness of physiological networks. Moreover, allometric control can often be described using the fractional calculus to capture the dynamics of complex physiologic networks.
PMCID: PMC3059975  PMID: 21423355
fractals; fractional calculus; physiology; allometric control; power-law statistics
2.  Fractal Analysis of Contours of Breast Masses in Mammograms 
Journal of Digital Imaging  2006;20(3):223-237.
Fractal analysis has been shown to be useful in image processing for characterizing shape and gray-scale complexity. Breast masses present shape and gray-scale characteristics that vary between benign masses and malignant tumors in mammograms. Limited studies have been conducted on the application of fractal analysis specifically for classifying breast masses based on shape. The fractal dimension of the contour of a mass may be computed either directly from the 2-dimensional (2D) contour or from a 1-dimensional (1D) signature derived from the contour. We present a study of four methods to compute the fractal dimension of the contours of breast masses, including the ruler method and the box counting method applied to 1D and 2D representations of the contours. The methods were applied to a data set of 111 contours of breast masses. Receiver operating characteristics (ROC) analysis was performed to assess and compare the performance of fractal dimension and four previously developed shape factors in the classification of breast masses as benign or malignant. Fractal dimension was observed to complement the other shape factors, in particular fractional concavity, in the representation of the complexity of the contours. The combination of fractal dimension with fractional concavity yielded the highest area (Az) under the ROC curve of 0.93; the two measures, on their own, resulted in Az values of 0.89 and 0.88, respectively.
PMCID: PMC3043900  PMID: 17021926
Box counting method; breast cancer; breast masses; breast tumors; contour analysis; fractal analysis; fractal dimension; ruler method; shape analysis; signatures of contours
3.  Fractal-based analysis of optical coherence tomography data to quantify retinal tissue damage 
BMC Bioinformatics  2014;15(1):295.
The sensitivity of Optical Coherence Tomography (OCT) images to identify retinal tissue morphology characterized by early neural loss from normal healthy eyes is tested by calculating structural information and fractal dimension. OCT data from 74 healthy eyes and 43 eyes with type 1 diabetes mellitus with mild diabetic retinopathy (MDR) on biomicroscopy was analyzed using a custom-built algorithm (OCTRIMA) to measure locally the intraretinal layer thickness. A power spectrum method was used to calculate the fractal dimension in intraretinal regions of interest identified in the images. ANOVA followed by Newman-Keuls post-hoc analyses were used to test for differences between pathological and normal groups. A modified p value of <0.001 was considered statistically significant. Receiver operating characteristic (ROC) curves were constructed to describe the ability of each parameter to discriminate between eyes of pathological patients and normal healthy eyes.
Fractal dimension was higher for all the layers (except the GCL + IPL and INL) in MDR eyes compared to normal healthy eyes. When comparing MDR with normal healthy eyes, the highest AUROC values estimated for the fractal dimension were observed for GCL + IPL and INL. The maximum discrimination value for fractal dimension of 0.96 (standard error =0.025) for the GCL + IPL complex was obtained at a FD ≤ 1.66 (cut off point, asymptotic 95% Confidence Interval: lower-upper bound = 0.905-1.002). Moreover, the highest AUROC values estimated for the thickness measurements were observed for the OPL, GCL + IPL and OS. Particularly, when comparing MDR eyes with control healthy eyes, we found that the fractal dimension of the GCL + IPL complex was significantly better at diagnosing early DR, compared to the standard thickness measurement.
Our results suggest that the GCL + IPL complex, OPL and OS are more susceptible to initial damage when comparing MDR with control healthy eyes. Fractal analysis provided a better sensitivity, offering a potential diagnostic predictor for detecting early neurodegeneration in the retina.
PMCID: PMC4261615  PMID: 25178846
Optical coherence tomography; Fractal analysis; Fractal dimension; Wavelet algorithm; Diabetic retinopathy; Ophthalmology
4.  Microarchitecture Parameters Describe Bone Structure and Its Strength Better Than BMD 
The Scientific World Journal  2012;2012:502781.
Introduction and Hypothesis. Some papers have shown that bone mineral density (BMD) may not be accurate in predicting fracture risk. Recently microarchitecture parameters have been reported to give information on bone characteristics. The aim of this study was to find out if the values of volume, fractal dimension, and bone mineral density are correlated with bone strength. Methods. Forty-two human bone samples harvested during total hip replacement surgery were cut to cylindrical samples. The geometrical mesh of layers of bone mass obtained from microCT investigation and the volumes of each layer and fractal dimension were calculated. The finite element method was applied to calculate the compression force F causing ε = 0.8% strain. Results. There were stronger correlations for microarchitecture parameters with strength than those for bone mineral density. The values of determination coefficient R2 for mean volume and force were 0.88 and 0.90 for mean fractal dimension and force, while for BMD and force the value was 0.53. The samples with bigger mean bone volume of layers and bigger mean fractal dimension of layers (more complex structure) presented higher strength. Conclusion. The volumetric and fractal dimension parameters better describe bone structure and strength than BMD.
PMCID: PMC3361288  PMID: 22654618
5.  Automated classifiers for early detection and diagnosis of retinopathy in diabetic eyes 
BMC Bioinformatics  2014;15:106.
Artificial neural networks (ANNs) have been used to classify eye diseases, such as diabetic retinopathy (DR) and glaucoma. DR is the leading cause of blindness in working-age adults in the developed world. The implementation of DR diagnostic routines could be feasibly improved by the integration of structural and optical property test measurements of the retinal structure that provide important and complementary information for reaching a diagnosis. In this study, we evaluate the capability of several structural and optical features (thickness, total reflectance and fractal dimension) of various intraretinal layers extracted from optical coherence tomography images to train a Bayesian ANN to discriminate between healthy and diabetic eyes with and with no mild retinopathy.
When exploring the probability as to whether the subject’s eye was healthy (diagnostic condition, Test 1), we found that the structural and optical property features of the outer plexiform layer (OPL) and the complex formed by the ganglion cell and inner plexiform layers (GCL + IPL) provided the highest probability (positive predictive value (PPV) of 91% and 89%, respectively) for the proportion of patients with positive test results (healthy condition) who were correctly diagnosed (Test 1). The true negative, TP and PPV values remained stable despite the different sizes of training data sets (Test 2). The sensitivity, specificity and PPV were greater or close to 0.70 for the retinal nerve fiber layer’s features, photoreceptor outer segments and retinal pigment epithelium when 23 diabetic eyes with mild retinopathy were mixed with 38 diabetic eyes with no retinopathy (Test 3).
A Bayesian ANN trained on structural and optical features from optical coherence tomography data can successfully discriminate between healthy and diabetic eyes with and with no retinopathy. The fractal dimension of the OPL and the GCL + IPL complex predicted by the Bayesian radial basis function network provides better diagnostic utility to classify diabetic eyes with mild retinopathy. Moreover, the thickness and fractal dimension parameters of the retinal nerve fiber layer, photoreceptor outer segments and retinal pigment epithelium show promise for the diagnostic classification between diabetic eyes with and with no mild retinopathy.
PMCID: PMC3996190  PMID: 24725911
6.  Quantification of the fractal nature of mycelial aggregation in Aspergillus niger submerged cultures 
Fractal geometry estimates have proven useful in studying the growth strategies of fungi in response to different environments on soil or on agar substrates, but their use in mycelia grown submerged is still rare. In the present study, the effects of certain important fermentation parameters, such as the spore inoculum level, phosphate and manganese concentrations in the medium, on mycelial morphology of the citric acid producer Aspergillus niger were determined by fractal geometry. The value of employing fractal geometry to describe mycelial structures was examined in comparison with information from other descriptors including classic morphological parameters derived from image analysis.
Fractal analysis of distinct morphological forms produced by fermentation conditions that influence fungal morphology and acid production, showed that the two fractal dimensions DBS (box surface dimension) and DBM (box mass dimension) are very sensitive indexes, capable of describing morphological differences. The two box-counting methods applied (one applied to the whole mass of the mycelial particles and the other applied to their surface only) enabled evaluation of fractal dimensions for mycelial particles in this analysis in the region of DBS = 1.20–1.70 and DBM = 1.20–2.70. The global structure of sufficiently branched mycelia was described by a single fractal dimension D, which did not exceed 1.30. Such simple structures are true mass fractals (DBS = DBM = D) and they could be young mycelia or dispersed forms of growth produced by very dense spore inocula (108–109 spores/ml) or by addition of manganese in the medium. Mycelial clumps and pellets were effectively discriminated by fractal analysis. Fractal dimension values were plotted together with classic morphological parameters derived from image analysis for comparisons. Their sensitivity to treatment was analogous to the sensitivity of classic morphological parameters suggesting that they could be equally used as morphological descriptors.
Starting from a spore, the mycelium develops as a mass fractal and, depending on culture conditions, it either turns to a surface fractal or remains a mass fractal. Since fractal dimensions give a measure of the degree of complexity and the mass filling properties of an object, it may be possible that a large number of morphological parameters which contribute to the overall complexity of the particles, could be replaced by these indexes effectively.
PMCID: PMC1382250  PMID: 16472407
7.  Pitfalls in Fractal Time Series Analysis: fMRI BOLD as an Exemplary Case 
This article will be positioned on our previous work demonstrating the importance of adhering to a carefully selected set of criteria when choosing the suitable method from those available ensuring its adequate performance when applied to real temporal signals, such as fMRI BOLD, to evaluate one important facet of their behavior, fractality. Earlier, we have reviewed on a range of monofractal tools and evaluated their performance. Given the advance in the fractal field, in this article we will discuss the most widely used implementations of multifractal analyses, too. Our recommended flowchart for the fractal characterization of spontaneous, low frequency fluctuations in fMRI BOLD will be used as the framework for this article to make certain that it will provide a hands-on experience for the reader in handling the perplexed issues of fractal analysis. The reason why this particular signal modality and its fractal analysis has been chosen was due to its high impact on today’s neuroscience given it had powerfully emerged as a new way of interpreting the complex functioning of the brain (see “intrinsic activity”). The reader will first be presented with the basic concepts of mono and multifractal time series analyses, followed by some of the most relevant implementations, characterization by numerical approaches. The notion of the dichotomy of fractional Gaussian noise and fractional Brownian motion signal classes and their impact on fractal time series analyses will be thoroughly discussed as the central theme of our application strategy. Sources of pitfalls and way how to avoid them will be identified followed by a demonstration on fractal studies of fMRI BOLD taken from the literature and that of our own in an attempt to consolidate the best practice in fractal analysis of empirical fMRI BOLD signals mapped throughout the brain as an exemplary case of potentially wide interest.
PMCID: PMC3513686  PMID: 23227008
fractals; monofractals; multifractals; time series analysis; numerical testing; fMRI BOLD; brain
8.  Fractal analysis of the structural complexity of the connective tissue in human carotid bodies 
The carotid body (CB) may undergo different structural changes during perinatal development, aging, or in response to environmental stimuli. In the previous literature, morphometric approaches to evaluate these changes have considered quantitative first order parameters, such as volumes or densities, while changes in spatial disposition and/or complexity of structural components have not yet been considered. In the present study, different strategies for addressing morphological complexity of CB, apart from the overall amount of each tissue component, were evaluated and compared. In particular, we considered the spatial distribution of connective tissue in the carotid bodies of young control subjects, young opiate-related deaths and aged subjects, through analysis of dispersion (Morisita's index), gray level co-occurrence matrix (entropy, angular second moment, variance, correlation), and fractal analysis (fractal dimension, lacunarity). Opiate-related deaths and aged subjects showed a comparable increase in connective tissue with respect to young controls. However, the Morisita's index (p < 0.05), angular second moment (p < 0.05), fractal dimension (p < 0.01), and lacunarity (p < 0.01) permitted to identify significant differences in the disposition of the connective tissue between these two series. A receiver operating characteristic (ROC) curve was also calculated to evaluate the efficiency of each parameter. The fractal dimension and lacunarity, with areas under the ROC curve of 0.9651 (excellent accuracy) and 0.8835 (good accuracy), respectively, showed the highest discriminatory power. They evidenced higher level of structural complexity in the carotid bodies of opiate-related deaths than old controls, due to more complex branching of intralobular connective tissue. Further analyses will have to consider the suitability of these approaches to address other morphological features of the CB, such as different cell populations, vascularization, and innervation.
PMCID: PMC4220644  PMID: 25414672
morphometry; fractal parameters; co-occurrence matrix; Morisita's index; carotid body; drug-related death; heroin; aging
9.  Intronic Alternative Splicing Regulators Identified by Comparative Genomics in Nematodes 
PLoS Computational Biology  2006;2(7):e86.
Many alternative splicing events are regulated by pentameric and hexameric intronic sequences that serve as binding sites for splicing regulatory factors. We hypothesized that intronic elements that regulate alternative splicing are under selective pressure for evolutionary conservation. Using a Wobble Aware Bulk Aligner genomic alignment of Caenorhabditis elegans and Caenorhabditis briggsae, we identified 147 alternatively spliced cassette exons that exhibit short regions of high nucleotide conservation in the introns flanking the alternative exon. In vivo experiments on the alternatively spliced let-2 gene confirm that these conserved regions can be important for alternative splicing regulation. Conserved intronic element sequences were collected into a dataset and the occurrence of each pentamer and hexamer motif was counted. We compared the frequency of pentamers and hexamers in the conserved intronic elements to a dataset of all C. elegans intron sequences in order to identify short intronic motifs that are more likely to be associated with alternative splicing. High-scoring motifs were examined for upstream or downstream preferences in introns surrounding alternative exons. Many of the high- scoring nematode pentamer and hexamer motifs correspond to known mammalian splicing regulatory sequences, such as (T)GCATG, indicating that the mechanism of alternative splicing regulation is well conserved in metazoans. A comparison of the analysis of the conserved intronic elements, and analysis of the entire introns flanking these same exons, reveals that focusing on intronic conservation can increase the sensitivity of detecting putative splicing regulatory motifs. This approach also identified novel sequences whose role in splicing is under investigation and has allowed us to take a step forward in defining a catalog of splicing regulatory elements for an organism. In vivo experiments confirm that one novel high-scoring sequence from our analysis, (T)CTATC, is important for alternative splicing regulation of the unc-52 gene.
Alternative splicing of precursor messenger RNA is a process by which multiple protein isoforms are generated from a single gene. As many as 60% of human genes are processed in this manner, creating tissue-specific isoforms of proteins that may be a key factor in regulating the complexity of our physiology. One of the major challenges to understanding this process is to identify the sequences on the precursor messenger RNA responsible for splicing regulation. Some of these regulatory sequences occur in regions that are spliced out (called introns). This study tested the hypothesis that there should be evolutionary pressure to maintain these intronic regulatory sequences, even though intron sequence is non-coding and rapidly diverges between species. The authors employed a genomic alignment of two roundworms, Caenorhabditis elegans and Caenorhabditis briggsae, to investigate the regulation of alternative splicing. By examining evolutionarily conserved stretches of introns flanking alternatively spliced exons, the authors identified and functionally confirmed splicing regulatory sequences. Many of the top scoring sequences match known mammalian regulators, suggesting the alternative splicing regulatory mechanism is conserved across all metazoans. Other sequences were not previously identified in mammals and may represent new alternative splicing regulatory elements in higher organisms or ones that may be specific to worms.
PMCID: PMC1500816  PMID: 16839192
10.  Analyzing Self-Similar and Fractal Properties of the C. elegans Neural Network 
PLoS ONE  2012;7(10):e40483.
The brain is one of the most studied and highly complex systems in the biological world. While much research has concentrated on studying the brain directly, our focus is the structure of the brain itself: at its core an interconnected network of nodes (neurons). A better understanding of the structural connectivity of the brain should elucidate some of its functional properties. In this paper we analyze the connectome of the nematode Caenorhabditis elegans. Consisting of only 302 neurons, it is one of the better-understood neural networks. Using a Laplacian Matrix of the 279-neuron “giant component” of the network, we use an eigenvalue counting function to look for fractal-like self similarity. This matrix representation is also used to plot visualizations of the neural network in eigenfunction coordinates. Small-world properties of the system are examined, including average path length and clustering coefficient. We test for localization of eigenfunctions, using graph energy and spacial variance on these functions. To better understand results, all calculations are also performed on random networks, branching trees, and known fractals, as well as fractals which have been “rewired” to have small-world properties. We propose algorithms for generating Laplacian matrices of each of these graphs.
PMCID: PMC3465333  PMID: 23071485
11.  The Genome Sequence of Caenorhabditis briggsae: A Platform for Comparative Genomics 
PLoS Biology  2003;1(2):e45.
The soil nematodes Caenorhabditis briggsae and Caenorhabditis elegans diverged from a common ancestor roughly 100 million years ago and yet are almost indistinguishable by eye. They have the same chromosome number and genome sizes, and they occupy the same ecological niche. To explore the basis for this striking conservation of structure and function, we have sequenced the C. briggsae genome to a high-quality draft stage and compared it to the finished C. elegans sequence. We predict approximately 19,500 protein-coding genes in the C. briggsae genome, roughly the same as in C. elegans. Of these, 12,200 have clear C. elegans orthologs, a further 6,500 have one or more clearly detectable C. elegans homologs, and approximately 800 C. briggsae genes have no detectable matches in C. elegans. Almost all of the noncoding RNAs (ncRNAs) known are shared between the two species. The two genomes exhibit extensive colinearity, and the rate of divergence appears to be higher in the chromosomal arms than in the centers. Operons, a distinctive feature of C. elegans, are highly conserved in C. briggsae, with the arrangement of genes being preserved in 96% of cases. The difference in size between the C. briggsae (estimated at approximately 104 Mbp) and C. elegans (100.3 Mbp) genomes is almost entirely due to repetitive sequence, which accounts for 22.4% of the C. briggsae genome in contrast to 16.5% of the C. elegans genome. Few, if any, repeat families are shared, suggesting that most were acquired after the two species diverged or are undergoing rapid evolution. Coclustering the C. elegans and C. briggsae proteins reveals 2,169 protein families of two or more members. Most of these are shared between the two species, but some appear to be expanding or contracting, and there seem to be as many as several hundred novel C. briggsae gene families. The C. briggsae draft sequence will greatly improve the annotation of the C. elegans genome. Based on similarity to C. briggsae, we found strong evidence for 1,300 new C. elegans genes. In addition, comparisons of the two genomes will help to understand the evolutionary forces that mold nematode genomes.
With the Caenorhabditis briggsae genome now in hand, C. elegans biologists have a powerful new research tool to refine their knowledge of gene function in C. elegans and to study the path of genome evolution
PMCID: PMC261899  PMID: 14624247
12.  Morphologic complexity of epithelial architecture for predicting invasive breast cancer survival 
Precise criteria for optimal patient selection for adjuvant chemotherapy remain controversial and include subjective components such as tumour morphometry (pathological grade). There is a need to replace subjective criteria with objective measurements to improve risk assessment and therapeutic decisions. We assessed the prognostic value of fractal dimension (an objective measure of morphologic complexity) for invasive ductal carcinoma of the breast.
We applied fractal analysis to pan-cytokeratin stained tissue microarray (TMA) cores derived from 379 patients. Patients were categorized according to low (<1.56, N = 141), intermediate (1.56-1.75, N = 148), and high (>1.75, N = 90) fractal dimension. Cox proportional-hazards regression was used to assess the relationship between disease-specific and overall survival and fractal dimension, tumour size, grade, nodal status, estrogen receptor status, and HER-2/neu status.
Patients with higher fractal score had significantly lower disease-specific 10-year survival (25.0%, 56.4%, and 69.4% for high, intermediate, and low fractal dimension, respectively, p < 0.001). Overall 10-year survival showed a similar association. Fractal dimension, nodal status, and grade were the only significant (P < 0.05) independent predictors for both disease-specific and overall survival. Among all of the prognosticators, the fractal dimension hazard ratio for disease-specific survival, 2.6 (95% confidence interval (CI) = 1.4,4.8; P = 0.002), was second only to the slightly higher hazard ratio of 3.1 (95% CI = 1.9,5.1; P < 0.001) for nodal status. As for overall survival, fractal dimension had the highest hazard ratio, 2.7 (95% CI = 1.6,4.7); P < 0.001). Split-sample cross-validation analysis suggests these results are generalizable.
Except for nodal status, morphologic complexity of breast epithelium as measured quantitatively by fractal dimension was more strongly and significantly associated with disease-specific and overall survival than standard prognosticators.
PMCID: PMC3024250  PMID: 21194459
13.  Robust estimation of fractal measures for characterizing the structural complexity of the human brain: optimization and reproducibility 
NeuroImage  2013;83:646-657.
High-resolution isotropic three-dimensional reconstructions of human brain gray and white matter structures can be characterized to quantify aspects of their shape, volume and topological complexity. In particular, methods based on fractal analysis have been applied in neuroimaging studies to quantify the structural complexity of the brain in both healthy and impaired conditions. The usefulness of such measures for characterizing individual differences in brain structure critically depends on their within-subject reproducibility in order to allow the robust detection of between-subject differences. This study analyzes key analytic parameters of three fractal-based methods that rely on the box-counting algorithm with the aim to maximize within-subject reproducibility of the fractal characterizations of different brain objects, including the pial surface, the cortical ribbon volume, the white matter volume and the grey matter/white matter boundary. Two separate datasets originating from different imaging centers were analyzed, comprising, 50 subjects with three and 24 subjects with four successive scanning sessions per subject, respectively. The reproducibility of fractal measures was statistically assessed by computing their intra-class correlations. Results reveal differences between different fractal estimators and allow the identification of several parameters that are critical for high reproducibility. Highest reproducibility with intra-class correlations in the range of 0.9–0.95 is achieved with the correlation dimension. Further analyses of the fractal dimensions of parcellated cortical and subcortical gray matter regions suggest robustly estimated and region-specific patterns of individual variability. These results are valuable for defining appropriate parameter configurations when studying changes in fractal descriptors of human brain structure, for instance in studies of neurological diseases that do not allow repeated measurements or for disease-course longitudinal studies.
PMCID: PMC3897251  PMID: 23831414
14.  Higuchi Dimension of Digital Images 
PLoS ONE  2011;6(9):e24796.
There exist several methods for calculating the fractal dimension of objects represented as 2D digital images. For example, Box counting, Minkowski dilation or Fourier analysis can be employed. However, there appear to be some limitations. It is not possible to calculate only the fractal dimension of an irregular region of interest in an image or to perform the calculations in a particular direction along a line on an arbitrary angle through the image. The calculations must be made for the whole image. In this paper, a new method to overcome these limitations is proposed. 2D images are appropriately prepared in order to apply 1D signal analyses, originally developed to investigate nonlinear time series. The Higuchi dimension of these 1D signals is calculated using Higuchi's algorithm, and it is shown that both regions of interests and directional dependencies can be evaluated independently of the whole picture. A thorough validation of the proposed technique and a comparison of the new method to the Fourier dimension, a common two dimensional method for digital images, are given. The main result is that Higuchi's algorithm allows a direction dependent as well as direction independent analysis. Actual values for the fractal dimensions are reliable and an effective treatment of regions of interests is possible. Moreover, the proposed method is not restricted to Higuchi's algorithm, as any 1D method of analysis, can be applied.
PMCID: PMC3172302  PMID: 21931854
15.  Automated Image Analysis for the Detection of Benthic Crustaceans and Bacterial Mat Coverage Using the VENUS Undersea Cabled Network 
Sensors (Basel, Switzerland)  2011;11(11):10534-10556.
The development and deployment of sensors for undersea cabled observatories is presently biased toward the measurement of habitat variables, while sensor technologies for biological community characterization through species identification and individual counting are less common. The VENUS cabled multisensory network (Vancouver Island, Canada) deploys seafloor camera systems at several sites. Our objective in this study was to implement new automated image analysis protocols for the recognition and counting of benthic decapods (i.e., the galatheid squat lobster, Munida quadrispina), as well as for the evaluation of changes in bacterial mat coverage (i.e., Beggiatoa spp.), using a camera deployed in Saanich Inlet (103 m depth). For the counting of Munida we remotely acquired 100 digital photos at hourly intervals from 2 to 6 December 2009. In the case of bacterial mat coverage estimation, images were taken from 2 to 8 December 2009 at the same time frequency. The automated image analysis protocols for both study cases were created in MatLab 7.1. Automation for Munida counting incorporated the combination of both filtering and background correction (Median- and Top-Hat Filters) with Euclidean Distances (ED) on Red-Green-Blue (RGB) channels. The Scale-Invariant Feature Transform (SIFT) features and Fourier Descriptors (FD) of tracked objects were then extracted. Animal classifications were carried out with the tools of morphometric multivariate statistic (i.e., Partial Least Square Discriminant Analysis; PLSDA) on Mean RGB (RGBv) value for each object and Fourier Descriptors (RGBv+FD) matrices plus SIFT and ED. The SIFT approach returned the better results. Higher percentages of images were correctly classified and lower misclassification errors (an animal is present but not detected) occurred. In contrast, RGBv+FD and ED resulted in a high incidence of records being generated for non-present animals. Bacterial mat coverage was estimated in terms of Percent Coverage and Fractal Dimension. A constant Region of Interest (ROI) was defined and background extraction by a Gaussian Blurring Filter was performed. Image subtraction within ROI was followed by the sum of the RGB channels matrices. Percent Coverage was calculated on the resulting image. Fractal Dimension was estimated using the box-counting method. The images were then resized to a dimension in pixels equal to a power of 2, allowing subdivision into sub-multiple quadrants. In comparisons of manual and automated Percent Coverage and Fractal Dimension estimates, the former showed an overestimation tendency for both parameters. The primary limitations on the automatic analysis of benthic images were habitat variations in sediment texture and water column turbidity. The application of filters for background corrections is a required preliminary step for the efficient recognition of animals and bacterial mat patches.
PMCID: PMC3274299  PMID: 22346657
cabled observatory; automated image analysis; squat lobster (Munida quadrispina); bacterial mat (Beggiatoa spp.); Scale-Invariant Feature Transform (SIFT); Fourier Descriptors (FD); Partial Least Square Discriminant Analysis (PLSDA); percentage of coverage, fractal dimension
16.  Computerized texture analysis of atypical immature myeloid precursors in patients with myelodysplastic syndromes: an entity between blasts and promyelocytes 
Diagnostic Pathology  2011;6:93.
Bone marrow (BM) blast count is an essential parameter for classification and prognosis of myelodysplastic syndromes (MDS). However, a high degree of cell atypias in bone marrow hemopoietic cells may be found in this group of clonal disorders, making it difficult to quantify precisely myeloblasts, and to distinguish them from promyelocytes and atypical immature myeloid precursors. Our aim was to investigate whether computerized image analysis of routine cytology would help to characterize these cells.
In May-Grünwald-Giemsa stained BM smears of 30 newly diagnosed MDS patients and 19 cases of normal BM, nuclei of blasts and promyelocytes were digitalized and interactively segmented. The morphological classification of the cells was done by consensus of two observers. Immature granulocytic precursors, which could not be clearly classified either as blasts or promyelocytes, were called "atypic myeloid precursors". Nuclear morphometry and texture features derived from the co-occurrence matrix and fractal dimension (FD) were calculated.
In normal BM, when compared to myeloblasts, nuclei of promyelocytes showed significant increase in perimeter and local texture homogeneity and a decrease in form factor, chromatin gray levels, Haralick's entropy, inertia, energy, contrast, diagonal moment, cluster prominence, the fractal dimension according to Minkowski and its goodness-of-fit. Compared to normal myeloblast nuclei, the chromatin texture of MDS myeloblasts revealed higher local homogeneity and goodness-of-fit of the FD, but lower values of entropy, contrast, diagonal moment, and fractal dimension. The same differences were found between nuclei of normal promyelocytes and those of MDS. Nuclei of atypical myeloid precursors showed intermediate characteristics between those of blasts and promyelocytes according to the quantitative features (perimeter, form factor, gray level and its standard deviation), but were similar to promyelocytes according to the texture variables inertia, energy, contrast, diagonal moment, cluster prominence, and Minkowski's fractal dimension.
BM atypical immature myeloid precursors are difficult to be correctly classified in routine cytology. Although their cytoplasm is more similar to that of myeloblasts, computerized texture analysis indicates a nuclear chromatin remodeling more close to the promyelocyte, thus indicating an asynchronous intermediate maturation stage between blast and promyelocyte.
PMCID: PMC3195698  PMID: 21958512
myelodysplastic syndromes; bone marrow; nuclear texture; cell atypias; karyometry; morphometry; fractal; chromatin; co-occurrence matrix; computerized pathology
17.  Examination of exhaustive cloning attempts reveals that C. elegans piRNAs, transposons, and repeat sequences are efficiently cloned in yeast, but not in bacteria 
Frontiers in Genetics  2014;5:275.
Genome sequencing requires insertion of random fragments of the sequenced organism’s DNA into a unicellular host, most often Escherichia coli bacteria. This manipulation was found in the past to be analogous to naturally occurring horizontal gene transfer, and moreover has proved valuable to understanding toxicity of foreign genetic elements to E. coli. Sequencing of the Caenorhabditis elegans genome was similarly achieved via DNA transformation into E. coli. However, numerous attempts have proven a significant percentage of the genome unclonable using bacteria, although clonable via yeast. We examined the genomic segments that were not clonable in bacteria but were clonable in yeast, and observed that, in line with previous hypotheses, such sequences are more repetitive on average compared with the entire C. elegans genome. In addition, we found that these gap-sequences encode significantly more for DNA transposons. Surprisingly, we discovered that although the vast majority of the C. elegans genome is clonable in bacteria (77.5%), almost all the thousands of sequences that encode for PIWI-interacting small RNAs, or 21U-RNAs (91.6%) were only clonable in yeast. These results might help understanding why most piRNAs in C. elegans are physically clustered on particular loci on chromosome IV. In worms and in a large number of other organisms, piRNAs serve to distinguish “Self” from “Non-Self” sequences, and thus to protect the integrity of the genome against foreign genetic elements, such as transposons. We discuss the possible implications of these discoveries.
PMCID: PMC4148067  PMID: 25221566
C. elegans; bacteria; lateral gene transfer; piRNAs; PIWI-interacting small RNAs; repetitive elements
18.  Efficient Physical Embedding of Topologically Complex Information Processing Networks in Brains and Computer Circuits 
PLoS Computational Biology  2010;6(4):e1000748.
Nervous systems are information processing networks that evolved by natural selection, whereas very large scale integrated (VLSI) computer circuits have evolved by commercially driven technology development. Here we follow historic intuition that all physical information processing systems will share key organizational properties, such as modularity, that generally confer adaptivity of function. It has long been observed that modular VLSI circuits demonstrate an isometric scaling relationship between the number of processing elements and the number of connections, known as Rent's rule, which is related to the dimensionality of the circuit's interconnect topology and its logical capacity. We show that human brain structural networks, and the nervous system of the nematode C. elegans, also obey Rent's rule, and exhibit some degree of hierarchical modularity. We further show that the estimated Rent exponent of human brain networks, derived from MRI data, can explain the allometric scaling relations between gray and white matter volumes across a wide range of mammalian species, again suggesting that these principles of nervous system design are highly conserved. For each of these fractal modular networks, the dimensionality of the interconnect topology was greater than the 2 or 3 Euclidean dimensions of the space in which it was embedded. This relatively high complexity entailed extra cost in physical wiring: although all networks were economically or cost-efficiently wired they did not strictly minimize wiring costs. Artificial and biological information processing systems both may evolve to optimize a trade-off between physical cost and topological complexity, resulting in the emergence of homologous principles of economical, fractal and modular design across many different kinds of nervous and computational networks.
Author Summary
Brains are often compared to computers but, apart from the trivial fact that both process information using a complex physical pattern of connections, it has been unclear whether this is more than just a metaphor. In our work, we rigorously uncover novel quantitative organizational principles that underlie the network organization of the human brain, high performance computer circuits, and the nervous system of the nematode C. elegans. We show through a topological and physical analysis of connectivity data that each of these systems is cost-efficiently embedded in physical space; they are organized as economical modular networks, paying a modest premium in wiring cost for the functional advantages of high dimensional topology. We also show that the fractal properties of human brain network connectivity can be used to explain allometric scaling relations between grey and white matter volumes in the brains of a wide range of differently sized mammals—from mouse opossum to sea lion—further suggesting that these principles of nervous system design are highly conserved across species. We propose that market-driven human invention and natural selection have negotiated trade-offs between cost and complexity in design of information processing networks and convergently come to similar conclusions.
PMCID: PMC2858671  PMID: 20421990
19.  Assessment of Performance Improvement in Content-based Medical Image Retrieval Schemes using Fractal Dimension 
Academic radiology  2009;16(10):1171-1178.
Rationale and Objectives
To investigate whether using fractal dimension as an objective index (quantitative measure) to assess and control the “visual” or “texture” similarity of reference image regions selected by a CBIR (content-based image retrieval) scheme will (or will not) affect the performance of the scheme in classification between image regions depicting suspicious breast masses.
Materials and Methods
An image dataset depicting 1500 verified mass regions and 1500 false-positive mass regions was used. We computed 14 morphological and intensity distribution based features and a fractal dimension. A CBIR scheme using a k-nearest neighbor classifier was applied and two experiments were conducted. In the first experiment, we evaluated our CBIR scheme using all 15 features. In the second experiment, we used the fractal dimension as a prescreening feature to guide the CBIR scheme to search for the most similar reference images that have similar measure in the fractal dimension.
The CBIR scheme achieved classification performance with area under ROC curve (AZ) of 0.857 with 95% confidence interval (CI) of [0.844, 0.870] using 14 features and 0.866 with 95% CI of [0.853, 0.879] after adding fractal dimension. The p-value of two classification results was 0. 005. After using fractal dimension as a prescreening feature, the CBIR scheme achieved AZ = 0.851 with 95% CI of [0.837, 0.864] without significant difference as comparing with the previous result using the original 14 features (p = 0.120). The difference of fractal dimension values between the selected similar reference images was reduced by 56.7% indicating the improvement of image texture similarity. In addition, more than half of references were early discarded without similarity comparison indicating the improvement of searching efficiency.
This study demonstrated the feasibility of applying the fractal dimension as an objective (quantitative) and efficient search index to assess and maintain texture similarity of reference mass regions selected by the CBIR schemes without reducing the scheme performance in classifying between suspicious breast masses.
PMCID: PMC2779846  PMID: 19524455
Content-based image retrieval (CBIR); Mammograms; Fractal analysis; Computer-aided diagnosis (CAD); Visual similarity; Receiver operating characteristics (ROC) analysis
20.  Generating Signals with Multiscale Time Irreversibility: The Asymmetric Weierstrass Function 
Complexity  2011;16(4):29-38.
Time irreversibility (asymmetry with respect to time reversal) is an important property of many time series derived from processes in nature. Some time series (e.g., healthy heart rate dynamics) demonstrate even more complex, multiscale irreversibility, such that not only the original but also coarse-grained time series are asymmetric over a wide range of scales. Several indices to quantify multiscale asymmetry have been introduced. However, there has been no simple generator of model time series with “tunable” multiscale asymmetry to test such indices. We introduce an asymmetric Weierstrass function WA (constructed from asymmetric sawtooth functions instead of cosine waves) that can be used to construct time series with any given value of the multiscale asymmetry. We show that multiscale asymmetry appears to be independent of other multiscale complexity indices, such as fractal dimension and multiscale entropy. We further generalize the concept of multiscale asymmetry by introducing time-dependent (local) multiscale asymmetry and provide examples of such time series. The WA function combines two essential features of complex fluctuations, namely fractality (self-similarity) and irreversibility (multiscale time asymmetry); moreover, each of these features can be tuned independently. The proposed family of functions can be used to compare and refine multiscale measures of time series asymmetry.
PMCID: PMC3398483  PMID: 22815622
multiscale time asymmetry; irreversibility; arrow of time; sawtooth function; Weierstrass function; multiscale analysis of time series
21.  Evaluation of the Dispersional Analysis Method for Fractal Time Series 
Annals of biomedical engineering  1995;23(4):491-505.
Fractal signals can be characterized by their fractal dimension plus some measure of their variance at a given level of resolution. The Hurst exponent, H, is <0.5 for rough anticorrelated series, >0.5 for positively correlated series, and =0.5 for random, white noise series. Several methods are available: dispersional analysis, Hurst rescaled range analysis, autocorrelation measures, and power special analysis. Short data sets are notoriously difficult to characterize; research to define the limitations of the various methods is incomplete. This numerical study of fractional Brownian noise focuses on determining the limitations of the dispersional analysis method, in particular, assessing the effects of signal length and of added noise on the estimate of the Hurst coefficient, H, (which ranges from 0 to 1 and is 2 − D, where D is the fractal dimension). There are three general conclusions: (i) pure fractal signals of length greater than 256 points give estimates of H that are biased but have standard deviations less than 0.1; (ii) the estimates of H tend to be biased toward H = 0.5 at both high H (>0.8) and low H (<0.5), and biases are greater for short time series than for long; and (iii) the addition of Gaussian noise (H = 0.5) degrades the signals: for those with negative correlation (H < 0.5) the degradation is great, the noise has only mild degrading effects on signals with H > 0.6, and the method is particularly robust for signals with high H and long series, where even 100% noise added has only a few percent effect on the estimate of H. Dispersional analysis can be regarded as a strong method for characterizing biological or natural time series, which generally show long-range positive correlation.
PMCID: PMC3756095  PMID: 7486356
Time series analysis; Autocovariance; Gaussian and fractional Brownian noise; Correlation; Hurst coefficient; Fractal dimension; Statistics
22.  Functional Genomic Analysis of the let-7 Regulatory Network in Caenorhabditis elegans 
PLoS Genetics  2013;9(3):e1003353.
The let-7 microRNA (miRNA) regulates cellular differentiation across many animal species. Loss of let-7 activity causes abnormal development in Caenorhabditis elegans and unchecked cellular proliferation in human cells, which contributes to tumorigenesis. These defects are due to improper expression of protein-coding genes normally under let-7 regulation. While some direct targets of let-7 have been identified, the genome-wide effect of let-7 insufficiency in a developing animal has not been fully investigated. Here we report the results of molecular and genetic assays aimed at determining the global network of genes regulated by let-7 in C. elegans. By screening for mis-regulated genes that also contribute to let-7 mutant phenotypes, we derived a list of physiologically relevant potential targets of let-7 regulation. Twenty new suppressors of the rupturing vulva or extra seam cell division phenotypes characteristic of let-7 mutants emerged. Three of these genes, opt-2, prmt-1, and T27D12.1, were found to associate with Argonaute in a let-7–dependent manner and are likely novel direct targets of this miRNA. Overall, a complex network of genes with various activities is subject to let-7 regulation to coordinate developmental timing across tissues during worm development.
Author Summary
In the past decade, microRNAs (miRNAs) have become recognized as key regulators of gene expression in many biological pathways. These small, non-coding RNAs target specific protein-coding genes for repression. The specificity is mediated by partial base-pairing interactions between the 22 nucleotide miRNA and sequences in the target messenger RNA (mRNA). The use of imperfect base-pairing means that a single miRNA can regulate many different mRNAs, but it also means that identifying these targets is not straightforward. One of the first discovered miRNAs, let-7, generally promotes cellular differentiation pathways through a repertoire of targets that is yet to be fully described. Here we utilized molecular and genetic approaches to identify biologically relevant targets of the let-7 miRNA in Caenorhabditis elegans. Our analyses indicate that let-7 regulates a large cast of genes, both directly and indirectly. Loss of let-7 activity in C. elegans results in multiple developmental abnormalities and, ultimately, death. We uncovered new targets of let-7 that contribute to these phenotypes when they fail to be properly regulated. Given the highly conserved nature of let-7 from worms to humans, our studies highlight new genes and pathways potentially under let-7 regulation across species.
PMCID: PMC3597506  PMID: 23516374
23.  Sequence space coverage, entropy of genomes and the potential to detect non-human DNA in human samples 
BMC Genomics  2008;9:509.
Genomes store information for building and maintaining organisms. Complete sequencing of many genomes provides the opportunity to study and compare global information properties of those genomes.
We have analyzed aspects of the information content of Homo sapiens, Mus musculus, Drosophila melanogaster, Caenorhabditis elegans, Arabidopsis thaliana, Saccharomyces cerevisiae, and Escherichia coli (K-12) genomes. Virtually all possible (> 98%) 12 bp oligomers appear in vertebrate genomes while < 2% of 19 bp oligomers are present. Other species showed different ranges of > 98% to < 2% of possible oligomers in D. melanogaster (12–17 bp), C. elegans (11–17 bp), A. thaliana (11–17 bp), S. cerevisiae (10–16 bp) and E. coli (9–15 bp). Frequencies of unique oligomers in the genomes follow similar patterns. We identified a set of 2.6 M 15-mers that are more than 1 nucleotide different from all 15-mers in the human genome and so could be used as probes to detect microbes in human samples. In a human sample, these probes would detect 100% of the 433 currently fully sequenced prokaryotes and 75% of the 3065 fully sequenced viruses. The human genome is significantly more compact in sequence space than a random genome. We identified the most frequent 5- to 20-mers in the human genome, which may prove useful as PCR primers. We also identified a bacterium, Anaeromyxobacter dehalogenans, which has an exceptionally low diversity of oligomers given the size of its genome and its GC content. The entropy of coding regions in the human genome is significantly higher than non-coding regions and chromosomes. However chromosomes 1, 2, 9, 12 and 14 have a relatively high proportion of coding DNA without high entropy, and chromosome 20 is the opposite with a low frequency of coding regions but relatively high entropy.
Measures of the frequency of oligomers are useful for designing PCR assays and for identifying chromosomes and organisms with hidden structure that had not been previously recognized. This information may be used to detect novel microbes in human tissues.
PMCID: PMC2628393  PMID: 18973670
24.  Improving the Caenorhabditis elegans Genome Annotation Using Machine Learning 
PLoS Computational Biology  2007;3(2):e20.
For modern biology, precise genome annotations are of prime importance, as they allow the accurate definition of genic regions. We employ state-of-the-art machine learning methods to assay and improve the accuracy of the genome annotation of the nematode Caenorhabditis elegans. The proposed machine learning system is trained to recognize exons and introns on the unspliced mRNA, utilizing recent advances in support vector machines and label sequence learning. In 87% (coding and untranslated regions) and 95% (coding regions only) of all genes tested in several out-of-sample evaluations, our method correctly identified all exons and introns. Notably, only 37% and 50%, respectively, of the presently unconfirmed genes in the C. elegans genome annotation agree with our predictions, thus we hypothesize that a sizable fraction of those genes are not correctly annotated. A retrospective evaluation of the Wormbase WS120 annotation [1] of C. elegans reveals that splice form predictions on unconfirmed genes in WS120 are inaccurate in about 18% of the considered cases, while our predictions deviate from the truth only in 10%–13%. We experimentally analyzed 20 controversial genes on which our system and the annotation disagree, confirming the superiority of our predictions. While our method correctly predicted 75% of those cases, the standard annotation was never completely correct. The accuracy of our system is further corroborated by a comparison with two other recently proposed systems that can be used for splice form prediction: SNAP and ExonHunter. We conclude that the genome annotation of C. elegans and other organisms can be greatly enhanced using modern machine learning technology.
Author Summary
Eukaryotic genes contain introns, which are intervening sequences that are excised from a gene transcript with the concomitant ligation of flanking segments called exons. The process of removing introns is called splicing. It involves biochemical mechanisms that to date are too complex to be modeled comprehensively and accurately. However, abundant sequencing results can serve as a blueprint database exemplifying what this process accomplishes. Using this database, we employ discriminative machine learning techniques to predict the mature mRNA given the unspliced pre-mRNA. Our method utilizes support vector machines and recent advances in label sequence learning, originally developed for natural language processing. The system, called mSplicer, was trained and evaluated on the genome of the nematode C. elegans, a well-studied model organism. We were able to show that mSplicer correctly predicts the splice form in most cases. Surprisingly, our predictions on currently unconfirmed genes deviate considerably from the public genome annotation. It is hypothesized that a sizable fraction of those genes are not correctly annotated. A retrospective evaluation and additional sequencing results show the superiority of mSplicer's predictions. It is concluded that the annotation of nematode and other genomes can be greatly enhanced using modern machine learning.
PMCID: PMC1808025  PMID: 17319737
25.  Comparison of C. elegans and C. briggsae Genome Sequences Reveals Extensive Conservation of Chromosome Organization and Synteny 
PLoS Biology  2007;5(7):e167.
To determine whether the distinctive features of Caenorhabditis elegans chromosomal organization are shared with the C. briggsae genome, we constructed a single nucleotide polymorphism–based genetic map to order and orient the whole genome shotgun assembly along the six C. briggsae chromosomes. Although these species are of the same genus, their most recent common ancestor existed 80–110 million years ago, and thus they are more evolutionarily distant than, for example, human and mouse. We found that, like C. elegans chromosomes, C. briggsae chromosomes exhibit high levels of recombination on the arms along with higher repeat density, a higher fraction of intronic sequence, and a lower fraction of exonic sequence compared with chromosome centers. Despite extensive intrachromosomal rearrangements, 1:1 orthologs tend to remain in the same region of the chromosome, and colinear blocks of orthologs tend to be longer in chromosome centers compared with arms. More strikingly, the two species show an almost complete conservation of synteny, with 1:1 orthologs present on a single chromosome in one species also found on a single chromosome in the other. The conservation of both chromosomal organization and synteny between these two distantly related species suggests roles for chromosome organization in the fitness of an organism that are only poorly understood presently.
Author Summary
The importance of chromosomal organization in the fitness of a species is only poorly understood. The publication of the C. elegans genome sequence in 1998 revealed features of higher level organization that suggested its chromosomes were organized into distinct domains. Chromosome arms were accumulating changes more rapidly than the centers of chromosomes. In this paper, we have compared the organization of the nematode C. briggsae genome with that of C. elegans. By building a genetic map based on DNA variations between two strains of C. briggsae, and by using that map to organize the draft genome sequence of C. briggsae published in 2003, we found the following: (1) Intrachromosomal rearrangements are frequent within and even between arms but are less common within central regions and between arms and centers. (2) Genes have remained overwhelmingly on the same chromosomes. (3) The distinctive features that distinguish C. elegans arms from centers also are seen in C. briggsae chromosomes. The conservation of these features between these two species, despite the approximately 100 million years since their most recent common ancestor, provides clear evidence of the selective advantages of the domain architecture of chromosomes. The continuing association of genes on the same chromosomes suggests that this may also be advantageous.
The conservation of both chromosomal organization and synteny between two distantly related species suggests roles for chromosome organization in the fitness of an organism.
PMCID: PMC1914384  PMID: 17608563

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