Fractal dimension and vessel complexity in patients with cerebral arteriovenous malformations

The fractal dimension (FD) can be used as a measure for morphological complexity in biological systems. The aim of this study was to test the usefulness of this quantitative parameter in the context of cerebral vascular complexity. Fractal analysis was applied on ten patients with cerebral arteriovenous malformations (AVM) and ten healthy controls. Maximum intensity projections from Time-of-Flight MRI scans were analyzed using different measurements of FD, the Box-counting dimension, the Minkowski dimension and generalized dimensions evaluated by means of multifractal analysis. The physiological significance of this parameter was investigated by comparing values of FD first, with the maximum slope of contrast media transit obtained from dynamic contrast-enhanced MRI data and second, with the nidus size obtained from X-ray angiography data. We found that for all methods, the Box-counting dimension, the Minkowski dimension and the generalized dimensions FD was significantly higher in the hemisphere with AVM compared to the hemisphere without AVM indicating that FD is a sensitive parameter to capture vascular complexity. Furthermore we found a high correlation between FD and the maximum slope of contrast media transit and between FD and the size of the central nidus pointing out the physiological relevance of FD. The proposed method may therefore serve as an additional objective parameter, which can be assessed automatically and might assist in the complex workup of AVMs.     

桂花叶片的分形特征

提供了一种叶片分形维数的测定方法,并用其测定了桂花(Osmanthus fragrans Lour)的叶片分形维数。通过分析发现:(1)桂花叶片具有典型的分形特征并且这种分形特征可以用分形理论的一般方法加以研究;(2)同一株桂花树的成熟叶片在统计意义上具有相同的分形维数;(3)对于桂花这个物种的成熟叶片在统计意义上具有相同的分形维数。由此推测同一种植物的成熟叶片(或者其他构件)具有相同分形维数。最后提出创建植物分形分类学的设想。     

水生植物叶片的分形特征研究

应用分形几何理论对保安湖部分水生植物叶的分形特征进行研究 ,以寻找其中的自相似性特征 ,计算其分形维数 ,并力图了解各个分维的生态学含义。计算得到各物种分维平均值 ,马来眼子菜为 1 441(1 3 2 2 -1 52 0 ,sd =0 0 547) ,菹草为 1 472 (1 3 81-1 565,sd =0 0 489) ,大茨藻为 1 3 2 5(1 2 0 9-1 40 6,sd =0 0 470 ) ,轮叶黑藻为 1 10 3(1 0 0 1-1 2 17,sd =0 0 456)。按分维值排序 ,菹草 >金鱼藻 >马来眼子菜 >茨藻 >黑藻。叶片分维的大小 ,说明了不同种类水生植物叶片占据空间的程度和利用生态空间的能力有差异。通过对五种植物叶的空间格局的定量化描述 ,更好地理解水生植物对水环境的适应机理 ,为进一步对更大尺度下的水生植物空间格局研究提供基础信息     

桂花叶片的分形特征

提供了一种叶片分形维数的测定方法,并用其测定了桂花(Osmanthus fragrans Lour)的叶片分形维数。通过分析发现：(1)桂花叶片具有典型的分形特征并且这种分形特征可以用分形理论的一般方法加以研究;(2)同一株桂花树的成熟叶片在统计意义上具有相同的分形维数;(3)对于桂花这个物种的成熟叶片在统计意义上具有相同的分形维数。由此推测同一种植物的成熟叶片(或者其他构件)具有相同分形维数。最后提出创建植物分形分类学的设想。     

Fractal geometry of bean root systems: correlations between spatial and fractal dimension

An obstacle to the study of root architecture is the difficulty of measuring and quantifying the three-dimensional configuration of roots in soil. The objective of this work was to determine if fractal geometry might be useful in estimating the three-dimensional complexity of root architecture from more accessible measurements. A set of results called projection theorems predict that the fractal dimension (FD) of a projection of a root system should be identical to the FD of roots in three-dimensional space (three-dimensional FD). To test this prediction we employed SimRoot, an explicit geometric simulation model of root growth derived from empirical measurements of common bean (Phaseolus vulgaris L.). We computed the three-dimensional FD, FD of horizontal plane intercepts (planar FD), FD of vertical line intercepts (linear FD), and FD of orthogonal projections onto planes (projected FD). Three-dimensional FD was found to differ from corresponding projected FD, suggesting that the analysis of roots grown in a narrow space or excavated and flattened prior to analysis is problematic. A log-linear relationship was found between FD of roots and spatial dimension. This log-linear relationship suggests that the three-dimensional FD of root systems may be accurately estimated from excavations and tracing of root intersections on exposed planes.     

Age-Related Changes in Electroencephalographic Signal Complexity

The study of active and healthy aging is a primary focus for social and neuroscientific communities. Here, we move a step forward in assessing electrophysiological neuronal activity changes in the brain with healthy aging. To this end, electroencephalographic (EEG) resting state activity was acquired in 40 healthy subjects (age 16–85). We evaluated Fractal Dimension (FD) according to the Higuchi algorithm, a measure which quantifies the presence of statistical similarity at different scales in temporal fluctuations of EEG signals. Our results showed that FD increases from age twenty to age fifty and then decreases. The curve that best fits the changes in FD values across age over the whole sample is a parabola, with the vertex located around age fifty. Moreover, FD changes are site specific, with interhemispheric FD asymmetry being pronounced in elderly individuals in the frontal and central regions. The present results indicate that fractal dimension well describes the modulations of brain activity with age. Since fractal dimension has been proposed to be related to the complexity of the signal dynamics, our data demonstrate that the complexity of neuronal electric activity changes across the life span of an individual, with a steady increase during young adulthood and a decrease in the elderly population.     

On the Fractal Geometry of DNA by the Binary Image Analysis

The multifractal analysis of binary images of DNA is studied in order to define a methodological approach to the classification of DNA sequences. This method is based on the computation of some multifractality parameters on a suitable binary image of DNA, which takes into account the nucleotide distribution. The binary image of DNA is obtained by a dot-plot (recurrence plot) of the indicator matrix. The fractal geometry of these images is characterized by fractal dimension (FD), lacunarity, and succolarity. These parameters are compared with some other coefficients such as complexity and Shannon information entropy. It will be shown that the complexity parameters are more or less equivalent to FD, while the parameters of multifractality have different values in the sense that sequences with higher FD might have lower lacunarity and/or succolarity. In particular, the genome of Drosophila melanogaster has been considered by focusing on the chromosome 3r, which shows the highest fractality with a corresponding higher level of complexity. We will single out some results on the nucleotide distribution in 3r with respect to complexity and fractality. In particular, we will show that sequences with higher FD also have a higher frequency distribution of guanine, while low FD is characterized by the higher presence of adenine.     

Fractal dimension in health and heart failure.

BACKGROUND: Non-linear analysis of heart rate variability (HRV) can give additional information about autonomic control of the heart rate. This study applied the fractal dimension (FD) in a congestive heart failure (CHF) population. METHODS: FD and HRV were evaluated in a healthy population (n=21) and an end-stage heart failure population (n=21) using 1-h segments during the day and night from Holter recordings. RESULTS: CHF patients presented a loss of circadian variation in both FD and conventional time- and frequency-domain HRV indices. FD was higher in the CHF population both during the day and night. In the CHF population the correlation between FD and high-frequency power of HRV was lost. CONCLUSION: Day-night variations of heart rate fluctuations are lost in heart failure. Changes in FD reflecting physiological and pathophysiological changes were observed.     

Fractal nature of protein surface roughness: a note on quantification of change of surface roughness in active sites,before and after binding

Year 2010 marked the 25th year since we came to know that roughness of a protein surface has fractal symmetry. Ever since the publication of Lewis and Rees' paper, hundreds of works from a spectrum of perspectives have established that fractal dimension (FD) can be considered as a reliable marker that describes roughness of protein surface objectively. In this article, we introduce readers to the fundamentals of fractals and present categorical biophysical and geometrical reasons as to why FD‐based constructs can describe protein surface roughness more accurately. We then review the commonality (and the lack of it) between numerous approaches that have attempted to investigate protein surface with fractal measures, before exploring the patterns in the results that they have produced. Apart from presenting the genealogy of approaches and results, we present an analysis that quantifies the difference in surface roughness in stretches of protein surface containing the active site, before and after binding to ligands, to underline the utility of FD‐based measures further. It has been found that surface stretches containing the active site, in general, undergo a significant increment in its roughness after binding. After presenting the entire repertoire of FD‐based surface roughness studies, we talk about two yet‐unexplored problems where application of FD‐based techniques can help in deciphering underlying patterns of surface interactions. Finally, we list the limitations of FD‐based constructs and put down several precautions that one must take while working with them. Copyright © 2013 John Wiley & Sons, Ltd.     

Ultrastructural complexity of nuclear components during early apoptotic phases in breast cancer cells.

Fractal morphometry was used to investigate the ultrastructural features of the plasma membrane, perinuclear membrane and nuclear chromatin in SK-BR-3 human breast cancer cells undergoing apoptosis. Cells were incubated with 1 microM calcimycin (A23187) for 24 h. Cells in the early stage of apoptosis had fractal dimension (FD) values indicating that their plasma membranes were less rough (lower FD) than those of control cells, while their perinuclear membranes were unaffected. Changes of the chromatin texture within the entire nucleus and in selected nuclear domains were more pronounced in treated cells. This confirms that the morphological reorganization imputable to a loss of structural complexity (reduced FD) occurs in the early stage of apoptosis, is accompanied by the inhibition of distinct enzymatic events and precedes the onset of conventional cellular markers, which can only be detected during the active phases of the apoptotic process.     

Plant light interception can be explained via computed tomography scanning: demonstration with pyramidal cedar (Thuja occidentalis, Fastigiata)

BACKGROUND AND AIMS: Light interception by the leaf canopy is a key aspect of plant photosynthesis, which helps mitigate the greenhouse effect via atmospheric CO(2) recycling. The relationship between plant light interception and leaf area was traditionally modelled with the Beer-Lambert law, until the spatial distribution of leaves was incorporated through the fractal dimension of leafless plant structure photographed from the side allowing maximum appearance of branches and petioles. However, photographs of leafless plants are two-dimensional projections of three-dimensional structures, and sampled plants were cut at the stem base before leaf blades were detached manually, so canopy development could not be followed for individual plants. Therefore, a new measurement and modelling approach were developed to explain plant light interception more completely and precisely, based on appropriate processing of computed tomography (CT) scanning data collected for developing canopies. METHODS: Three-dimensional images of canopies were constructed from CT scanning data. Leaf volumes (LV) were evaluated from complete canopy images, and fractal dimensions (FD) were estimated from skeletonized leafless images. The experimental plant species is pyramidal cedar (Thuja occidentalis, Fastigiata). KEY RESULTS: The three-dimensional version of the Beer-Lambert law based on FD alone provided a much better explanation of plant light interception (R(2) = 0.858) than those using the product LV*FD (0.589) or LV alone (0.548). While values of all three regressors were found to increase over time, FD in the Beer-Lambert law followed the increase in light interception the most closely. The delayed increase of LV reflected the appearance of new leaves only after branches had lengthened and ramified. CONCLUSIONS: The very strong correlation obtained with FD demonstrates that CT scanning data contain fundamental information about the canopy architecture geometry. The model can be used to identify crops and plantation trees with improved light interception and productivity.     

Fractal analysis on root systems of rice plants in response to drought stress

A fractal analytical method was used to examine the developmental responses of root systems in upland rice genotype CT9993-5-10-1-M (japonica) and lowland genotype IR62266-42-6-2 (indica) (abbreviated as CT9993 and IR62266, respectively) to soil water stress. The root systems were grown for one month in root boxes with 25 cm in length, 2 cm in width and 40 cm in depth, which were filled with soil. The root systems were sampled by following the needle-pinboard method, and then spread on the transparent plastic films with nets after carefully washing out the soils. The two-dimensional images of root systems were digitized by using a scanner. The digitized images were used for analysis based on fractal geometry with the box-counting method. The reductions in shoot dry weight, photosynthesis rate and transpiration rate of IR62266 by soil drought were greater than those of CT9993. The change of fractal parameters in response to soil moisture conditions differed between the two rice genotypes. The values of fractal abundance (FA) and fractal dimension (FD) in well-watered IR62266 plants were larger than in CT9993. The value of FA of IR62266 was decreased more by drought stress than that of CT9993, indicating that the volume of soils explored by the whole root systems of CT9993 was maintained or less decreased under drought stress in comparison to IR62266. Moreover, the values of FD tended to increase in CT9993 while it tended to decrease in IR62266 in response to drought. These root responses detected by the fractal analysis in CT9993 may be advantageous for its extracting more water from drying soils, which explains its better growth under drought-stressed condition.     

Fluctuations of the fractal dimension of the electroencephalogram during periodic breathing in heart failure patients

The physiological mechanisms responsible for periodic breathing (PB) in heart failure (HF) patients are still debated. A role for rhythmic shifts in the level of wakefulness has been suggested, but their existence has never been proven. In this study we investigated the existence of an oscillation in EEG activity during PB in these patients and assessed its relationship with the ventilatory oscillation. EEG activity was measured by the fractal dimension (FD) and by a spectral technique (weighted mean frequency, WMF) in 17 stable HF patients (mean age ± SD: 57±10 yrs, NYHA class: 2.6 ± 0.4, LVEF: 24 ± 6%), with sustained PB during supine rest. The relationship between minute ventilation (MV) signal and FD and WMF was assessed by coherence analysis. Most patients (10/17) showed a well defined oscillation in FD and WMF at the frequency of PB closely linked (coherence > 0.7) with the oscillation of MV. In the remaining patients, neither FD nor WMF showed a clear oscillatory pattern synchronous with MV. Overall, the two EEG-derived parameters showed the same coherence with the ventilatory oscillation (mean coherence ± SD: 0.65 ± 0.25 vs 0.66 ± 0.23, for FD and WMF respectively, p = 0.44). Our results provide evidence that during PB in HF patients, EEG activity often, but not always, fluctuates synchronously with the ventilatory oscillation. These fluctuations can be effectively detected by the fractal dimension, but classical spectral methods provide substantially the same information. Other mechanisms, particularly chemical instability in the respiratory control system, are likely to play a role in the genesis of PB.     

植被格局的分形模型：植被格局的分形特征

 目前植被格局研究存在一些问题,分形模型是解决这些问题的有力工具。选择从分枝到景观的各个等级层次上若干有代表性的研究实例进行介绍,给出了尺度、自相似性和分形维数在植被格局研究中的应用方法,说明了分形模型描述植被格局的实用性和优越性。在此基础上提出植被格局的分形机制,认为植被格局是具有自组织特征的分形体。最后对植被格局的分形模型进行了评价,提出研究中需要注意的若干问题以及未来的发展方向。     

植被格局的分形特征

目前植被格局研究存在一些问题,分形模型是解决这些问题的有力工具。选择从分枝到景观的各个等级层次上若干有代表性的研究实例进行介绍。给出了尺度,自相似性和分形维数在植被格局研究中的应用方法,说明了分形模型描述植被格局的实用性和优越性。在此基础上提出了植被格局的分形机制,认为植被格局是具有自组织特征的分形体。最后对植被格局的分形模型进行了评价,提出了研究中需要注意的若干问题以及未来的发展方向。     

Modelling applicability of fractal analysis to efficiency of soil exploration by roots

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.     

Fractal characteristics of May-Grünwald-Giemsa stained chromatin are independent prognostic factors for survival in multiple myeloma

Background

The use of computerized image analysis for the study of nuclear texture features has provided important prognostic information for several neoplasias. Recently fractal characteristics of the chromatin structure in routinely stained smears have shown to be independent prognostic factors in acute leukemia. In the present study we investigated the influence of the fractal dimension (FD) of chromatin on survival of patients with multiple myeloma.

Methodology

We analyzed 67 newly diagnosed patients from our Institution treated in the Brazilian Multiple Myeloma Study Group. Diagnostic work-up consisted of peripheral blood counts, bone marrow cytology, bone radiograms, serum biochemistry and cytogenetics. The International Staging System (ISS) was used. In every patient, at least 40 digital nuclear images from diagnostic May-Grünwald-Giemsa stained bone marrow smears were acquired and transformed into pseudo-3D images. FD was determined by the Minkowski-Bouligand method extended to three dimensions. Goodness-of-fit of FD was estimated by the R² values in the log-log plots. The influence of diagnostic features on overall survival was analyzed in Cox regressions. Patients that underwent autologous bone marrow transplantation were censored at the day of transplantation.

Principal Findings

Median age was 56 years. According to ISS, 14% of the patients were stage I, 39% were stage II and 47% were stage III. Additional features of a bad prognosis were observed in 46% of the cases. When stratifying for ISS, both FD and its goodness-of-fit were significant prognostic factors in univariate analyses. Patients with higher FD values or lower goodness-of-fit showed a worse outcome. In the multivariate Cox-regression, FD, R², and ISS stage entered the final model, which showed to be stable in a bootstrap resampling study.

Conclusions

Fractal characteristics of the chromatin texture in routine cytological preparations revealed relevant prognostic information in patients with multiple myeloma.     

Histomorphometric parameters and fractal complexity of the equine placenta from healthy and sick foals

Computer-based digital image analysis of tissue samples shows promise both to reduce the subjectivity of traditional manual tissue assessments and potentially to shorten the time required to analyze each sample. The present study used digital image analysis to investigate the histomorphometric parameters and fractal complexity of the equine placenta from healthy and sick foals. We hypothesized that the placentas of sick foals could have a different growth pattern and complexity that could be objectively estimated by their fractal dimension (FD). Fourteen placentas from 30 mares were selected in the 2013 breeding season and divided into two groups: seven mares with normal pregnancy, eutocic delivery, and healthy foals (group 1) and seven mares with normal or high-risk pregnancy, eutocic delivery and sick foals (group 2). Four mares in group 2 were classified as having a high-risk pregnancy on the basis of anamnesis and/or ultrasound findings. Clinical diagnosis of group 2 foals included perinatal asphyxia syndrome (n = 4), prematurity/dysmaturity (n = 2), and both diagnoses (n = 1). Seven out of fourteen placentas showed diffuse gross abnormalities. Grossly abnormal placentas were observed in one out of seven (14.28%) animals in group 1 and in six out of seven (85.72%) animals in group 2. Digital image analysis proved to be reliable and efficient in segmentation, calculation, outline extraction of villi as also resulted in sampled test images. The placentas of group 1 foals displayed a uniform and homogeneous villi development, as revealed by geometric parameters and FD. These results can be interpreted as a harmonic growth pattern of microcotyledons throughout the placenta in healthy foals. By contrast, the placentas of group 2 foals showed a nonuniform growth pattern and complexity with more villi and more developed villi in pregnant horn (PH) and nonpregnant horn (NPH) compared with body (B) and higher FD in NPH than in the other areas. This finding can be interpreted as a compensatory growth with increased complexity. Our results show that morphometric analysis, particularly FD measurement, can be proposed as an ancillary histological tool for equine placenta evaluation. Chorionic villi tend to have greater branching and complexity in sick than in healthy foals, particularly in the NPH. This could represent an attempt to increase the exchange area between fetal and maternal compartments of the equine placenta, and merits further investigation.     

The ability of freeze-dried bull spermatozoa to induce calcium oscillations and resumption of meiosis

The objective was to investigate the ability of freeze-dried (FD) bull spermatozoa to induce calcium oscillations in mouse oocytes and meiosis resumption in in vitro-matured bovine oocytes after intracytoplasmic sperm injection (ICSI). Bull spermatozoa were freeze-dried and stored for 1 y at +25, +4, or -196 degrees C. In the first experiment, rehydrated sperm heads were microinseminated into hybrid mouse oocytes loaded with fluo-3/AM, and the kinetics of intracellular calcium concentration was monitored for 1h. Repetitive increases of intracellular calcium concentration were recorded in the majority of injected oocytes, with exception of a few oocytes injected with FD sperm heads stored at +4 degrees C (11%) and +25 degrees C (8%) that exhibited a single increase or no response (non-oscillated). The proportion of oocytes that oscillated with high frequency (>or=10 spikes/h) was higher in the non-dried control group (79%; P<0.05) than in the FD groups (58, 55, and 54% for storage at -196, +4, and +25 degrees C, respectively). In the second experiment, control and FD spermatozoa were microinseminated into in vitro-matured, denuded bovine oocytes. The oocytes were fixed and stained 12h after ICSI. A higher proportion of bovine oocytes injected with control spermatozoa (70%; P<0.05) resumed meiosis than those injected with +25, +4 and -196 degrees C stored FD spermatozoa (53, 48, and 57%, respectively). The proportion of ICSI oocytes that developed to the pronuclear stage (complete activation) was higher in the control group (64%; P<0.05) than those in all the FD groups (34, 27, and 28% for storage at -196, +4, and +25 degrees C, respectively). Thus, the ability of bull spermatozoa to induce frequent intracellular calcium spikes in mouse oocytes was impaired by the process of freeze-drying, without differences among storage at +25, +4 or -196 degrees C, probably resulting in a lower proportion of bovine oocytes that resumed meiosis and/or developed to the pronuclear stage.