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1.  Correcting for finite spatial scales of self-similarity when calculating fractal dimensions of real-world structures 
Fractal geometry is a potentially valuable tool for quantitatively characterizing complex structures. The fractal dimension (D) can be used as a simple, single index for summarizing properties of real and abstract structures in space and time. Applications in the fields of biology and ecology range from neurobiology to plant architecture, landscape structure, taxonomy and species diversity. However, methods to estimate the D have often been applied in an uncritical manner, violating assumptions about the nature of fractal structures. The most common error involves ignoring the fact that ideal, i.e. infinitely nested, fractal structures exhibit self-similarity over any range of scales. Unlike ideal fractals, real-world structures exhibit self-similarity only over a finite range of scales.
Here we present a new technique for quantitatively determining the scales over which real-world structures show statistical self-similarity. The new technique uses a combination of curve-fitting and tests of curvilinearity of residuals to identify the largest range of contiguous scales that exhibit statistical self-similarity. Consequently, we estimate D only over the statistically identified region of self-similarity and introduce the finite scale- corrected dimension (FSCD). We demonstrate the use of this method in two steps. First, using mathematical fractal curves with known but variable spatial scales of self-similarity (achieved by varying the iteration level used for creating the curves), we demonstrate that our method can reliably quantify the spatial scales of self-similarity. This technique therefore allows accurate empirical quantification of theoretical Ds. Secondly, we apply the technique to digital images of the rhizome systems of goldenrod (Solidago altissima). The technique significantly reduced variations in estimated fractal dimensions arising from variations in the method of preparing digital images. Overall, the revised method has the potential to significantly improve repeatability and reliability for deriving fractal dimensions of real-world branching structures.
PMCID: PMC1688706
Branching Structures Fractal Dimension Self-Simularity Solidago Altissima Spatial Scales
2.  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
3.  Quantification of left ventricular trabeculae using fractal analysis 
Left ventricular noncompaction (LVNC) is a myocardial disorder characterized by excessive left ventricular (LV) trabeculae. Current methods for quantification of LV trabeculae have limitations. The aim of this study is to describe a novel technique for quantifying LV trabeculation using cardiovascular magnetic resonance (CMR) and fractal geometry. Observing that trabeculae appear complex and irregular, we hypothesize that measuring the fractal dimension (FD) of the endocardial border provides a quantitative parameter that can be used to distinguish normal from abnormal trabecular patterns.
Fractal analysis is a method of quantifying complex geometric patterns in biological structures. The resulting FD is a unitless measure index of how completely the object fills space. FD increases with increased structural complexity. LV FD was measured using a box-counting method on CMR short-axis cine stacks. Three groups were studied: LVNC (defined by Jenni criteria), n=30(age 41±13; men, 16); healthy whites, n=75(age, 46±16; men, 36); healthy blacks, n=30(age, 40±11; men, 15).
In healthy volunteers FD varied in a characteristic pattern from base to apex along the LV. This pattern was altered in LVNC where apical FD were abnormally elevated. In healthy volunteers, blacks had higher FD than whites in the apical third of the LV (maximal apical FD: 1.253±0.005 vs. 1.235±0.004, p<0.01) (mean±s.e.m.). Comparing LVNC with healthy volunteers, maximal apical FD was higher in LVNC (1.392±0.010, p<0.00001). The fractal method was more accurate and reproducible (ICC, 0.97 and 0.96 for intra and inter-observer readings) than two other CMR criteria for LVNC (Petersen and Jacquier).
FD is higher in LVNC patients compared to healthy volunteers and is higher in healthy blacks than in whites. Fractal analysis provides a quantitative measure of trabeculation and has high reproducibility and accuracy for LVNC diagnosis when compared to current CMR criteria.
PMCID: PMC3680331  PMID: 23663522
Cardiomyopathy; Heart failure; Trabeculation
4.  Wavelet-Based 3D Reconstruction of Microcalcification Clusters from Two Mammographic Views: New Evidence That Fractal Tumors Are Malignant and Euclidean Tumors Are Benign 
PLoS ONE  2014;9(9):e107580.
The 2D Wavelet-Transform Modulus Maxima (WTMM) method was used to detect microcalcifications (MC) in human breast tissue seen in mammograms and to characterize the fractal geometry of benign and malignant MC clusters. This was done in the context of a preliminary analysis of a small dataset, via a novel way to partition the wavelet-transform space-scale skeleton. For the first time, the estimated 3D fractal structure of a breast lesion was inferred by pairing the information from two separate 2D projected mammographic views of the same breast, i.e. the cranial-caudal (CC) and mediolateral-oblique (MLO) views. As a novelty, we define the “CC-MLO fractal dimension plot”, where a “fractal zone” and “Euclidean zones” (non-fractal) are defined. 118 images (59 cases, 25 malignant and 34 benign) obtained from a digital databank of mammograms with known radiologist diagnostics were analyzed to determine which cases would be plotted in the fractal zone and which cases would fall in the Euclidean zones. 92% of malignant breast lesions studied (23 out of 25 cases) were in the fractal zone while 88% of the benign lesions were in the Euclidean zones (30 out of 34 cases). Furthermore, a Bayesian statistical analysis shows that, with 95% credibility, the probability that fractal breast lesions are malignant is between 74% and 98%. Alternatively, with 95% credibility, the probability that Euclidean breast lesions are benign is between 76% and 96%. These results support the notion that the fractal structure of malignant tumors is more likely to be associated with an invasive behavior into the surrounding tissue compared to the less invasive, Euclidean structure of benign tumors. Finally, based on indirect 3D reconstructions from the 2D views, we conjecture that all breast tumors considered in this study, benign and malignant, fractal or Euclidean, restrict their growth to 2-dimensional manifolds within the breast tissue.
PMCID: PMC4164655  PMID: 25222610
5.  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
6.  Quantitative evaluation and modeling of two-dimensional neovascular network complexity: the surface fractal dimension 
BMC Cancer  2005;5:14.
Modeling the complex development and growth of tumor angiogenesis using mathematics and biological data is a burgeoning area of cancer research. Architectural complexity is the main feature of every anatomical system, including organs, tissues, cells and sub-cellular entities. The vascular system is a complex network whose geometrical characteristics cannot be properly defined using the principles of Euclidean geometry, which is only capable of interpreting regular and smooth objects that are almost impossible to find in Nature. However, fractal geometry is a more powerful means of quantifying the spatial complexity of real objects.
This paper introduces the surface fractal dimension (Ds) as a numerical index of the two-dimensional (2-D) geometrical complexity of tumor vascular networks, and their behavior during computer-simulated changes in vessel density and distribution.
We show that Ds significantly depends on the number of vessels and their pattern of distribution. This demonstrates that the quantitative evaluation of the 2-D geometrical complexity of tumor vascular systems can be useful not only to measure its complex architecture, but also to model its development and growth.
Studying the fractal properties of neovascularity induces reflections upon the real significance of the complex form of branched anatomical structures, in an attempt to define more appropriate methods of describing them quantitatively. This knowledge can be used to predict the aggressiveness of malignant tumors and design compounds that can halt the process of angiogenesis and influence tumor growth.
PMCID: PMC549205  PMID: 15701176
7.  Fractal dimension of chromatin is an independent prognostic factor for survival in melanoma 
BMC Cancer  2010;10:260.
Prognostic factors in malignant melanoma are currently based on clinical data and morphologic examination. Other prognostic features, however, which are not yet used in daily practice, might add important information and thus improve prognosis, treatment, and survival. Therefore a search for new markers is desirable. Previous studies have demonstrated that fractal characteristics of nuclear chromatin are of prognostic importance in neoplasias. We have therefore investigated whether the fractal dimension of nuclear chromatin measured in routine histological preparations of malignant melanomas could be a prognostic factor for survival.
We examined 71 primary superficial spreading cutaneous melanoma specimens (thickness ≥ 1 mm) from patients with a minimum follow up of 5 years. Nuclear area, form factor and fractal dimension of chromatin texture were obtained from digitalized images of hematoxylin-eosin stained tissue micro array sections. Clark's level, tumor thickness and mitotic rate were also determined.
The median follow-up was 104 months. Tumor thickness, Clark's level, mitotic rate, nuclear area and fractal dimension were significant risk factors in univariate Cox regressions. In the multivariate Cox regression, stratified for the presence or absence of metastases at diagnosis, only the Clark level and fractal dimension of the nuclear chromatin were included as independent prognostic factors in the final regression model.
In general, a more aggressive behaviour is usually found in genetically unstable neoplasias with a higher number of genetic or epigenetic changes, which on the other hand, provoke a more complex chromatin rearrangement. The increased nuclear fractal dimension found in the more aggressive melanomas is the mathematical equivalent of a higher complexity of the chromatin architecture. So, there is strong evidence that the fractal dimension of the nuclear chromatin texture is a new and promising variable in prognostic models of malignant melanomas.
PMCID: PMC2902442  PMID: 20525386
8.  Quantification of echodensities in tuberculous pericardial effusion using fractal geometry: a proof of concept study 
The purpose of this study was to quantify the heterogeneous distribution of echodensities in the pericardial fluid of patients with tuberculous pericarditis using echocardiography and fractal analysis, and to determine whether there were differences in the fractal dimensions of effusive-constrictive and effusive non-constrictive disease.
We used fractal geometry to quantify the echocardiographic densities in patients who were enrolled in the Investigation of the Management of Pericarditis in Africa (IMPI Africa) Registry. Sub-costal and four chamber images were included in the analysis if a minimum of two clearly identified fibrin strands were present and the quality of the images were of a standard which allowed for accurate measurement of the fractal dimension. The fractal dimension was calculated as follows: Df = limlog N(s)/[log (l/s)], where Df is the box counting fractal dimension of the fibrin strand, s is the side length of the box and N(s) is the smallest number of boxes of side length s to cover the outline of the object being measured. We compared the fractal dimension of echocardiographic findings in patients with effusive constrictive pericarditis to effusive non-constrictive pericardial effusion using the non-parametric Mann–Whitney test.
Of the 14 echocardiographs from 14 participants that were selected for the study, 42.8% (6/14) of images were subcostal views while 57.1% (8/14) were 4-chamber views. Eight of the patients had tuberculous effusive constrictive pericarditis while 6 had tuberculous effusive non-constrictive pericarditis. The mean fractal dimension Df was 1.325 with a standard deviation (SD) of 0.146. The measured fibrin strand dimension exceeded the topological dimension in all the images over the entire range of grid scales with a correlation coefficient (r2) greater than 0.8 in the majority. The fractal dimension of echodensities was 1.359 ± 0.199 in effusive constrictive pericarditis compared to 1.330 ± 0.166 in effusive non-constrictive pericarditis (p = 0.595).
The echocardiographic densities in tuberculous pericardial effusion have a fractal geometrical dimension which is similar in pure effusive and effusive constrictive disease.
PMCID: PMC3464936  PMID: 22838492
Pericardial effusion; Tuberculosis; Fractal dimension; Effusive constrictive pericarditis; Effusive non-constrictive pericarditis
9.  Lightning and the Heart: Fractal Behavior in Cardiac Function 
Physical systems, from galactic clusters to diffusing molecules, often show fractal behavior. Likewise, living systems might often be well described by fractal algorithms. Such fractal descriptions in space and time imply that there is order in chaos, or put the other way around, chaotic dynamical systems in biology are more constrained and orderly than seen at first glance. The vascular network, the syncytium of cells, the processes of diffusion and transmembrane transport might be fractal features of the heart. These fractal features provide a basis which enables one to understand certain aspects of more global behavior such as atrial or ventricular fibrillation and perfusion heterogeneity. The heart might be regarded as a prototypical organ from these points of view. A particular example of the use of fractal geometry is in explaining myocardial flow heterogeneity via delivery of blood through an asymmetrical fractal branching network.
PMCID: PMC3175793  PMID: 21938081
10.  Quantitative Assessment of Early Diabetic Retinopathy Using Fractal Analysis  
Diabetes Care  2009;32(1):106-110.
OBJECTIVE—Fractal analysis can quantify the geometric complexity of the retinal vascular branching pattern and may therefore offer a new method to quantify early diabetic microvascular damage. In this study, we examined the relationship between retinal fractal dimension and retinopathy in young individuals with type 1 diabetes.
RESEARCH DESIGN AND METHODS—We conducted a cross-sectional study of 729 patients with type 1 diabetes (aged 12–20 years) who had seven-field stereoscopic retinal photographs taken of both eyes. From these photographs, retinopathy was graded according to the modified Airlie House classification, and fractal dimension was quantified using a computer-based program following a standardized protocol.
RESULTS—In this study, 137 patients (18.8%) had diabetic retinopathy signs; of these, 105 had mild retinopathy. Median (interquartile range) retinal fractal dimension was 1.46214 (1.45023–1.47217). After adjustment for age, sex, diabetes duration, A1C, blood pressure, and total cholesterol, increasing retinal vascular fractal dimension was significantly associated with increasing odds of retinopathy (odds ratio 3.92 [95% CI 2.02–7.61] for fourth versus first quartile of fractal dimension). In multivariate analysis, each 0.01 increase in retinal vascular fractal dimension was associated with a nearly 40% increased odds of retinopathy (1.37 [1.21–1.56]). This association remained after additional adjustment for retinal vascular caliber.
CONCLUSIONS—Greater retinal fractal dimension, representing increased geometric complexity of the retinal vasculature, is independently associated with early diabetic retinopathy signs in type 1 diabetes. Fractal analysis of fundus photographs may allow quantitative measurement of early diabetic microvascular damage.
PMCID: PMC2606840  PMID: 18835945
11.  Modelling Applicability of Fractal Analysis to Efficiency of Soil Exploration by Roots 
Annals of Botany  2004;94(1):119-128.
• Background and Aims Fractal analysis allows calculation of fractal dimension, fractal abundance and lacunarity. Fractal analysis of plant roots has revealed correlations of fractal dimension with age, topology or genotypic variation, while fractal abundance has been associated with root length. Lacunarity is associated with heterogeneity of distribution, and has yet to be utilized in analysis of roots. In this study, fractal analysis was applied to the study of root architecture and acquisition of diffusion‐limited nutrients. The hypothesis that soil depletion and root competition are more closely correlated with a combination of fractal parameters than by any one alone was tested.
• Model The geometric simulation model SimRoot was used to dynamically model roots of various architectures growing for up to 16 d in three soil types with contrasting nutrient mobility. Fractal parameters were calculated for whole roots, projections of roots and vertical slices of roots taken at 0, 2·5 and 5 cm from the root origin. Nutrient depletion volumes, competition volumes, and relative competition were regressed against fractal parameters and root length.
• Key Results Root length was correlated with depletion volume, competition volume and relative competition at all times. In analysis of three‐dimensional, projected roots and 0 cm slices, log(fractal abundance) was highly correlated with log(depletion volume) when times were pooled. Other than this, multiple regression yielded better correlations than regression with single fractal parameters. Correlations decreased with age of roots and distance of vertical slices from the root origin. Field data were also examined to see if fractal dimension, fractal abundance and lacunarity can be used to distinguish common bean genotypes in field situations. There were significant differences in fractal dimension and fractal abundance, but not in lacunarity.
• Conclusions These results suggest that applying fractal analysis to research of soil exploration by root systems should include fractal abundance, and possibly lacunarity, along with fractal dimension.
PMCID: PMC4242372  PMID: 15145791
Fractal dimension; fractal abundance; lacunarity; phosphorus; depletion; competition; Phaseolus vulgaris; SimRoot; simulation modelling
12.  Experimental study of cake formation on heat treated and membrane coated needle felts in a pilot scale pulse jet bag filter using optical in-situ cake height measurement 
Powder Technology  2011;214(3-3):388-399.
Pulse-jet bag filters are frequently employed for particle removal from off gases. Separated solids form a layer on the permeable filter media called filter cake. The cake is responsible for increasing pressure drop. Therefore, the cake has to be detached at a predefined upper pressure drop limit or at predefined time intervals. Thus the process is intrinsically semi-continuous. The cake formation and cake detachment are interdependent and may influence the performance of the filter. Therefore, understanding formation and detachment of filter cake is important. In this regard, the filter media is the key component in the system. Needle felts are the most commonly used media in bag filters. Cake formation studies with heat treated and membrane coated needle felts in pilot scale pulse jet bag filter were carried out. The data is processed according to the procedures that were published already [Powder Technology, Volume 173, Issue 2, 19 April 2007, Pages 93–106]. Pressure drop evolution, cake height distribution evolution, cake patches area distribution and their characterization using fractal analysis on different needle felts are presented here. It is observed that concavity of pressure drop curve for membrane coated needle felt is principally caused by presence of inhomogeneous cake area load whereas it is inherent for heat treated media. Presence of residual cake enhances the concavity of pressure drop at the start of filtration cycle. Patchy cleaning is observed only when jet pulse pressure is too low and unable to provide the necessary force to detach the cake. The border line is very sharp. Based on experiments with limestone dust and three types of needle felts, for the jet pulse pressure above 4 bar and filtration velocity below 50 mm/s, cake is detached completely except a thin residual layer (100–200 μm). Uniformity and smoothness of residual cake depends on the surface characteristics of the filter media. Cake height distribution of residual cake and newly formed cake during filtration prevails. The patch size analysis and fractal analysis reveal that residual cake grow in size (latterly) following regeneration initially on the base with edges smearing out, however, the cake heights are not leveled off. Fractal dimension of cake patches boundary falls in the range of 1–1.4 and depends on vertical position as well as time of filtration. Cake height measurements with Polyimide (PI) needle felts were hampered on account of its photosensitive nature.
Graphical abstract
Cake formation studies with heat treated and membrane laminated needle felts has been carried out. Pressure drop evolution, cake patches area distribution and their fractal analyses are investigated. Results reveal that filter media may affect cake formation. A strong influence of the distribution of filter cake is also observed.
► It is revealed that filter media has an influence on the evolution of pressure drop. ► Filter cakes of different heights exist on the needle felts. ► Cake formation is two dimensional (lateral and vertical) after regeneration contrary to one dimensional only (vertical).
PMCID: PMC3886367  PMID: 24415801
Cake formation; Needle felts; Bag filter; In-situ; Cake height measurements
13.  Meaningful Interpretation of Subdiffusive Measurements in Living Cells (Crowded Environment) by Fluorescence Fluctuation Microscopy 
In living cell or its nucleus, the motions of molecules are complicated due to the large crowding and expected heterogeneity of the intracellular environment. Randomness in cellular systems can be either spatial (anomalous) or temporal (heterogeneous). In order to separate both processes, we introduce anomalous random walks on fractals that represented crowded environments. We report the use of numerical simulation and experimental data of single-molecule detection by fluorescence fluctuation microscopy for detecting resolution limits of different mobile fractions in crowded environment of living cells. We simulate the time scale behavior of diffusion times τD(τ) for one component, e.g. the fast mobile fraction, and a second component, e.g. the slow mobile fraction. The less the anomalous exponent α the higher the geometric crowding of the underlying structure of motion that is quantified by the ratio of the Hausdorff dimension and the walk exponent d f /dw and specific for the type of crowding generator used. The simulated diffusion time decreases for smaller values of α ≠ 1 but increases for a larger time scale τ at a given value of α ≠ 1. The effect of translational anomalous motion is substantially greater if α differs much from 1. An α value close to 1 contributes little to the time dependence of subdiffusive motions. Thus, quantitative determination of molecular weights from measured diffusion times and apparent diffusion coefficients, respectively, in temporal auto- and crosscorrelation analyses and from time-dependent fluorescence imaging data are difficult to interpret and biased in crowded environments of living cells and their cellular compartments; anomalous dynamics on different time scales τ must be coupled with the quantitative analysis of how experimental parameters change with predictions from simulated subdiffusive dynamics of molecular motions and mechanistic models. We first demonstrate that the crowding exponent α also determines the resolution of differences in diffusion times between two components in addition to photophyscial parameters well-known for normal motion in dilute solution. The resolution limit between two different kinds of single molecule species is also analyzed under translational anomalous motion with broken ergodicity. We apply our theoretical predictions of diffusion times and lower limits for the time resolution of two components to fluorescence images in human prostate cancer cells transfected with GFP-Ago2 and GFP-Ago1. In order to mimic heterogeneous behavior in crowded environments of living cells, we need to introduce so-called continuous time random walks (CTRW). CTRWs were originally performed on regular lattice. This purely stochastic molecule behavior leads to subdiffusive motion with broken ergodicity in our simulations. For the first time, we are able to quantitatively differentiate between anomalous motion without broken ergodicity and anomalous motion with broken ergodicity in time-dependent fluorescence microscopy data sets of living cells. Since the experimental conditions to measure a selfsame molecule over an extended period of time, at which biology is taken place, in living cells or even in dilute solution are very restrictive, we need to perform the time average over a subpopulation of different single molecules of the same kind. For time averages over subpopulations of single molecules, the temporal auto- and crosscorrelation functions are first found. Knowing the crowding parameter α for the cell type and cellular compartment type, respectively, the heterogeneous parameter γ can be obtained from the measurements in the presence of the interacting reaction partner, e.g. ligand, with the same α value. The product α⋅γ=γ˜ is not a simple fitting parameter in the temporal auto- and two-color crosscorrelation functions because it is related to the proper physical models of anomalous (spatial) and heterogeneous (temporal) randomness in cellular systems. We have already derived an analytical solution for γ˜ in the special case of γ = 3/2 . In the case of two-color crosscorrelation or/and two-color fluorescence imaging (co-localization experiments), the second component is also a two-color species gr, for example a different molecular complex with an additional ligand. Here, we first show that plausible biological mechanisms from FCS/ FCCS and fluorescence imaging in living cells are highly questionable without proper quantitative physical models of subdiffusive motion and temporal randomness. At best, such quantitative FCS/ FCCS and fluorescence imaging data are difficult to interpret under crowding and heterogeneous conditions. It is challenging to translate proper physical models of anomalous (spatial) and heterogeneous (temporal) randomness in living cells and their cellular compartments like the nucleus into biological models of the cell biological process under study testable by single-molecule approaches. Otherwise, quantitative FCS/FCCS and fluorescence imaging measurements in living cells are not well described and cannot be interpreted in a meaningful way.
PMCID: PMC3583073  PMID: 20553227
Anomalous motion; broken ergodicity; Continuous Time Random Walks (CTRW); Continuous Time Random Walks (CTRW) on fractal supports; cellular crowding; Cytoplasmic Assembly of Nuclear RISC; ergodicity; FCS; FCCS; Fluorescence Fluctuation Microscopy; GFP-Ago1; GFP-Ago2; heterogeneity; living cells; meaningful interpretation of subdiffusive measurements; microRNA trafficking; physical model of crowding; physical model of heterogeneity; random walks on fractal supports; resolution limits of measured diffusion times for two components; RNA Activation (RNAa); Single Molecule; Small Activating RNA (saRNA); Temporal autocorrelation; Temporal two-color crosscorrelation; Fluorescence imaging; Time dependence of apparent diffusion coefficients.
14.  Trabecular Morphometry by Fractal Signal Analysis is a Novel Marker of Osteoarthritis Progression 
Arthritis and rheumatism  2009;60(12):3711-3722.
To evaluate the utility of subchondral bone texture from a baseline x-ray image for predicting 3-year knee osteoarthritis (OA) progression.
A total of 138 participants in the Prediction of Osteoarthritis Progression (POP) study were evaluated at baseline and 3 years. Fixed-flexion knee radiographs of the 248 non-replaced knees underwent fractal analysis of the medial subchondral tibial plateau using a commercially available software tool. OA progression was defined as a 1-grade change in joint space narrowing (JSN) or osteophyte based on a standardized knee atlas. Statistical analysis of fractal signatures was performed using a new method based on modeling the overall shape of fractal dimension versus radius curves.
Baseline fractal signature of the medial tibial plateau was predictive of medial knee JSN progression (area under the curve [AUC] of Receiver Operating Characteristic plot of 0.75), but not progression based on osteophyte or progression of the lateral compartment. The traditional covariates (age, gender, body mass index, knee pain), general bone mineral content, and baseline joint space width fared little better than random variables for predicting OA progression (AUC 0.52–0.58). The maximal predictive model combined baseline fractal signature, knee alignment, traditional covariates, and bone mineral content (AUC 0.79).
We identified a prognostic marker of OA that is readily extracted from a plain radiograph by fractal signature analysis. The global shape approach to analyzing these data is a potentially efficient means of identifying individuals at risk of knee OA progression that needs to be validated in a second cohort.
PMCID: PMC3711179  PMID: 19950282
osteoarthritis; imaging; biomarker; subchondral bone
15.  Intercellular cancer collisions generate an ejected crystal comet tail effect with fractal interface embryoid body reassembly transformation 
We have documented self-assembled geometric triangular chiral crystal complexes (GTCHC) and a framework of collagen vascular invariant geometric attractors in cancer tissues. This article shows how this system evolves in time. These structures are incorporated together and evolve in different ways. When the geometric core is stable, and the tissue architecture collapses, fragmented components emerge, which reveal a hidden interior identifying how each molecule is reassembled into the original mold, using one common connection, ie, a fractal self-similarity that guided the system from the beginning. GTCHC complexes generate ejected crystal comet tail effects and produce strange helicity states that arise in the form of spin domain interactions. As the crystal growth vibration stage progresses, biofractal echo images converge in a master-built construction of embryoid bodies with enolase-selective immunopositivity in relation to clusters of triangular chiral cell organization. In our electro-optic collision model, we were able to predict and replicate all the characteristics of this complex geometry that connects a physical phenomenon with the signal patterns that generate biologic chaos. Intrinsically, fractal geometry makes spatial correction errors embrace the chaotic system in a way that permits new structures to emerge, and as a result, an ordered self-assembly of embryoid bodies with neural differentiation at the final stage of cancer development is a predictable process. We hope that further investigation of these structures will lead not only to a new way of thinking about physics and biology, but also to a rewarding area in cancer research.
PMCID: PMC3101111  PMID: 21625398
embryoid bodies; cancer; electro-optic collision model
16.  Scale-Specific Multifractal Medical Image Analysis 
Fractal geometry has been applied widely in the analysis of medical images to characterize the irregular complex tissue structures that do not lend themselves to straightforward analysis with traditional Euclidean geometry. In this study, we treat the nonfractal behaviour of medical images over large-scale ranges by considering their box-counting fractal dimension as a scale-dependent parameter rather than a single number. We describe this approach in the context of the more generalized Rényi entropy, in which we can also compute the information and correlation dimensions of images. In addition, we describe and validate a computational improvement to box-counting fractal analysis. This improvement is based on integral images, which allows the speedup of any box-counting or similar fractal analysis algorithm, including estimation of scale-dependent dimensions. Finally, we applied our technique to images of invasive breast cancer tissue from 157 patients to show a relationship between the fractal analysis of these images over certain scale ranges and pathologic tumour grade (a standard prognosticator for breast cancer). Our approach is general and can be applied to any medical imaging application in which the complexity of pathological image structures may have clinical value.
PMCID: PMC3760300  PMID: 24023588
17.  The role of the circadian system in fractal neurophysiological control 
Many neurophysiological variables such as heart rate, motor activity, and neural activity are known to exhibit intrinsic fractal fluctuations - similar temporal fluctuation patterns at different time scales. These fractal patterns contain information about health, as many pathological conditions are accompanied by their alteration or absence. In physical systems, such fluctuations are characteristic of critical states on the border between randomness and order, frequently arising from nonlinear feedback interactions between mechanisms operating on multiple scales. Thus, the existence of fractal fluctuations in physiology challenges traditional conceptions of health and disease, suggesting that high levels of integrity and adaptability are marked by complex variability, not constancy, and are properties of a neurophysiological network, not individual components. Despite the subject's theoretical and clinical interest, the neurophysiological mechanisms underlying fractal regulation remain largely unknown. The recent discovery that the circadian pacemaker (suprachiasmatic nucleus) plays a crucial role in generating fractal patterns in motor activity and heart rate sheds an entirely new light on both fractal control networks and the function of this master circadian clock, and builds a bridge between the fields of circadian biology and fractal physiology. In this review, we sketch the emerging picture of the developing interdisciplinary field of fractal neurophysiology by examining the circadian system’s role in fractal regulation.
PMCID: PMC3766492  PMID: 23573942
fractal fluctuations; fractal physiology; circadian biology; suprachiasmatic nucleus; heart rate; spontaneous motor activity; scale-invariance; physiological control; nonlinear dynamics
18.  Fractal spatial distribution of pancreatic islets in three dimensions: a self-avoiding growth model 
Physical biology  2013;10(3):036009.
The islets of Langerhans, responsible for controlling blood glucose levels, are dispersed within the pancreas. A universal power law governing the fractal spatial distribution of islets in two-dimensional pancreatic sections has been reported. However, the fractal geometry in the actual three-dimensional pancreas volume, and the developmental process that gives rise to such a self-similar structure, have not been investigated. Here, we examined the three-dimensional spatial distribution of islets in intact mouse pancreata using optical projection tomography and found a power law with a fractal dimension, 2.1. Furthermore, based on two-dimensional pancreatic sections of human autopsies, we found that the distribution of human islets also follows a universal power law with fractal dimension 1.5 in adult pancreata, which agrees with the value previously reported in smaller mammalian pancreas sections. Finally, we developed a self-avoiding growth model for the development of the islet distribution and found that the fractal nature of the spatial islet distribution may be associated with the self-avoidance in the branching process of vascularization in the pancreas.
PMCID: PMC3767849  PMID: 23629025
fractal; power law; pancreatic islet; growth; self-avoidance; mathematical model
19.  Microbial growth patterns described by fractal geometry. 
Journal of Bacteriology  1990;172(3):1180-1185.
Fractal geometry has made important contributions to understanding the growth of inorganic systems in such processes as aggregation, cluster formation, and dendritic growth. In biology, fractal geometry was previously applied to describe, for instance, the branching system in the lung airways and the backbone structure of proteins as well as their surface irregularity. This investigation applies the fractal concept to the growth patterns of two microbial species, Streptomyces griseus and Ashbya gossypii. It is a first example showing fractal aggregates in biological systems, with a cell as the smallest aggregating unit and the colony as an aggregate. We find that the global structure of sufficiently branched mycelia can be described by a fractal dimension, D, which increases during growth up to 1.5. D is therefore a new growth parameter. Two different box-counting methods (one applied to the whole mass of the mycelium and the other applied to the surface of the system) enable us to evaluate fractal dimensions for the aggregates in this analysis in the region of D = 1.3 to 2. Comparison of both box-counting methods shows that the mycelial structure changes during growth from a mass fractal to a surface fractal.
PMCID: PMC208582  PMID: 2106504
20.  Dose Verification in Intensity Modulation Radiation Therapy: A Fractal Dimension Characteristics Study 
BioMed Research International  2013;2013:349437.
Purpose. This study describes how to identify the coincidence of desired planning isodose curves with film experimental results by using a mathematical fractal dimension characteristic method to avoid the errors caused by visual inspection in the intensity modulation radiation therapy (IMRT). Methods and Materials. The isodose curves of the films delivered by linear accelerator according to Plato treatment planning system were acquired using Osiris software to aim directly at a single interested dose curve for fractal characteristic analysis. The results were compared with the corresponding planning desired isodose curves for fractal dimension analysis in order to determine the acceptable confidence level between the planning and the measurement. Results. The film measured isodose curves and computer planning curves were deemed identical in dose distribution if their fractal dimensions are within some criteria which suggested that the fractal dimension is a unique fingerprint of a curve in checking the planning and film measurement results. The dose measured results of the film were presumed to be the same if their fractal dimension was within 1%. Conclusions. This quantitative rather than qualitative comparison done by fractal dimension numerical analysis helps to decrease the quality assurance errors in IMRT dosimetry verification.
PMCID: PMC3722795  PMID: 23956976
21.  Measuring Fractality 
When investigating fractal phenomena, the following questions are fundamental for the applied researcher: (1) What are essential statistical properties of 1/f noise? (2) Which estimators are available for measuring fractality? (3) Which measurement instruments are appropriate and how are they applied? The purpose of this article is to give clear and comprehensible answers to these questions. First, theoretical characteristics of a fractal pattern (self-similarity, long memory, power law) and the related fractal parameters (the Hurst coefficient, the scaling exponent α, the fractional differencing parameter d of the autoregressive fractionally integrated moving average methodology, the power exponent β of the spectral analysis) are discussed. Then, estimators of fractal parameters from different software packages commonly used by applied researchers (R, SAS, SPSS) are introduced and evaluated. Advantages, disadvantages, and constrains of the popular estimators (d^ML, power spectral density, detrended fluctuation analysis, signal summation conversion) are illustrated by elaborate examples. Finally, crucial steps of fractal analysis (plotting time series data, autocorrelation, and spectral functions; performing stationarity tests; choosing an adequate estimator; estimating fractal parameters; distinguishing fractal processes from short-memory patterns) are demonstrated with empirical time series.
PMCID: PMC3345945  PMID: 22586408
fractal; 1/f noise; ARFIMA; long memory
22.  Comprehensive Fractal Description of Porosity of Coal of Different Ranks 
The Scientific World Journal  2014;2014:490318.
We selected, as the objects of our research, lignite from the Beizao Mine, gas coal from the Caiyuan Mine, coking coal from the Xiqu Mine, and anthracite from the Guhanshan Mine. We used the mercury intrusion method and the low-temperature liquid nitrogen adsorption method to analyze the structure and shape of the coal pores and calculated the fractal dimensions of different aperture segments in the coal. The experimental results show that the fractal dimension of the aperture segment of lignite, gas coal, and coking coal with an aperture of greater than or equal to 10 nm, as well as the fractal dimension of the aperture segment of anthracite with an aperture of greater than or equal to 100 nm, can be calculated using the mercury intrusion method; the fractal dimension of the coal pore, with an aperture range between 2.03 nm and 361.14 nm, can be calculated using the liquid nitrogen adsorption method, of which the fractal dimensions bounded by apertures of 10 nm and 100 nm are different. Based on these findings, we defined and calculated the comprehensive fractal dimensions of the coal pores and achieved the unity of fractal dimensions for full apertures of coal pores, thereby facilitating, overall characterization for the heterogeneity of the coal pore structure.
PMCID: PMC4053310  PMID: 24955407
23.  Fractal rotation isolates mechanisms for form-dependent motion in human vision 
Biology Letters  2007;3(3):306-308.
Here, we describe a motion stimulus in which the quality of rotation is fractal. This makes its motion unavailable to the translation-based motion analysis known to underlie much of our motion perception. In contrast, normal rotation can be extracted through the aggregation of the outputs of translational mechanisms. Neural adaptation of these translation-based motion mechanisms is thought to drive the motion after-effect, a phenomenon in which prolonged viewing of motion in one direction leads to a percept of motion in the opposite direction. We measured the motion after-effects induced in static and moving stimuli by fractal rotation. The after-effects found were an order of magnitude smaller than those elicited by normal rotation. Our findings suggest that the analysis of fractal rotation involves different neural processes than those for standard translational motion. Given that the percept of motion elicited by fractal rotation is a clear example of motion derived from form analysis, we propose that the extraction of fractal rotation may reflect the operation of a general mechanism for inferring motion from changes in form.
PMCID: PMC2464696  PMID: 17360252
motion; visual perception; after-effect; psychophysics; feature tracking
24.  Analysis of fractal electrodes for efficient neural stimulation 
Planar electrodes are increasingly used in therapeutic neural stimulation techniques such as functional electrical stimulation, epidural spinal cord stimulation (ESCS), and cortical stimulation. Recently, optimized electrode geometries have been shown to increase the efficiency of neural stimulation by increasing the variation of current density on the electrode surface. In the present work, a new family of modified fractal electrode geometries is developed to enhance the efficiency of neural stimulation. It is shown that a promising approach in increasing the neural activation function is to increase the “edginess” of the electrode surface, a concept that is explained and quantified by fractal mathematics. Rigorous finite element simulations were performed to compute electric potential produced by proposed modified fractal geometries. The activation of 256 model axons positioned around the electrodes was then quantified, showing that modified fractal geometries required a 22% less input power while maintaining the same level of neural activation. Preliminary in vivo experiments investigating muscle evoked potentials due to median nerve stimulation showed encouraging results, supporting the feasibility of increasing neural stimulation efficiency using modified fractal geometries.
PMCID: PMC3709379  PMID: 23874290
neural stimulation; fractal geometry; electrodes; epidural spinal cord stimulation; cortical stimulation; deep brain stimulation (DBS)
25.  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

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