Browsing by Subject "Fractals"

Now showing 1 - 4 of 4
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    The Dynamics of Twisted Tent Maps
    (2013-07-12) Chamblee, Stephen Joseph; Misiurewicz, Michał, 1948-; Roeder, Roland; Geller, William; Eremenko, Alexandre; Mukhin, Evgeny
    This paper is a study of the dynamics of a new family of maps from the complex plane to itself, which we call twisted tent maps. A twisted tent map is a complex generalization of a real tent map. The action of this map can be visualized as the complex scaling of the plane followed by folding the plane once. Most of the time, scaling by a complex number will \twist" the plane, hence the name. The "folding" both breaks analyticity (and even smoothness) and leads to interesting dynamics ranging from easily understood and highly geometric behavior to chaotic behavior and fractals.
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    Fractal Analysis to Detect Surface Roughness Changes After Fatique Loading in the Canine Mandibular Incisor - A Pilot Study
    (2006) Hayes, Andrew M.; Katona, Thomas R.; Baldwin, James J.; Hohlt, William F.; Platt, Jeffrey A.; Shanks, James C.
    A common occurrence resulting from orthodontic treatment is iatrogenic root resorption. Experimentally, histological sections are usually employed to study root resorption utilizing special staining techniques. Although this has been shown effective for decades, results are limited to extracted teeth. Clinically, root resorption is detected and evaluated with standard radiography. This technique is limited to evaluating the 3-dimensional phenomenon of root resorption with 2-dimensional films. Quantifying these results has proven to be cumbersome if not impossible. Recent advances in computed tomography are allowing researchers to evaluate changes in the skeleton, including dental movements in three dimensions. CT reconstructions could be used to evaluate the changes in tooth composition through the volumetric data obtained. Fractal analysis techniques have become irreplaceable in the understanding of many phenomena in various scientific fields such as astrophysics, economy, geology, ecology and agriculture. Recently, a significant amount of literature has been published describing the expansion of this technique into the fields of bio-medicine, including dentistry. The logic behind fractals is derived by observing an object through successive scales. Fractal analysis can be used to measure an object's space filling properties. Simply put, the more space the object occupies, the higher the fractal dimension. The purpose of this study is to utilize fractal analysis to quantify morphological changes in ex vivo root structure from fatigue loaded mandibular central incisors of canine (hound) specimens. After fatigue loading experimental teeth, microcomputed tomography, was used to obtain 3-dimensional data of both the experimental and control teeth. Computer software was used to develop 3-dimensional models of each specimen and fractal analysis was performed on specific regions of each tooth. To obtain the specific regions, the root was divided into 5 planes or slices horizontally, beginning with the most cervical portion to the apical portion, then further divided into a buccal and lingual portion. These can be systematically evaluated using a coordinate system of numbering from 1-10, starting with the cervical-buccal region of the root (zone 1) to apical-buccal (zone 5). The numbering then follows to the opposite side of the tooth with the cervical-lingual region (zone 10). Changes in surface roughness were detected through the fractal analysis and compared between the control and experimental animals. Statistical analysis showed that the mean roughness of the experimental group was significantly smaller than in the control group for two of the ten zones tested in each tooth.
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    A perpetual switching system in pulmonary capillaries
    (American Physiological Society, 2019-02-01) Wagner, Wiltz W., Jr.; Jaryszak, Eric M.; Peterson, Amanda J.; Doerschuk, Claire M.; Bohlen, H. Glenn; King, Judy A. C.; Tanner, Judith A.; Crockett, Edward S.; Glenny, Robb W.; Presson, Robert G., Jr.; Anesthesia, School of Medicine
    Of the 300 billion capillaries in the human lung, a small fraction meet normal oxygen requirements at rest, with the remainder forming a large reserve. The maximum oxygen demands of the acute stress response require that the reserve capillaries are rapidly recruited. To remain primed for emergencies, the normal cardiac output must be parceled throughout the capillary bed to maintain low opening pressures. The flow-distributing system requires complex switching. Because the pulmonary microcirculation contains contractile machinery, one hypothesis posits an active switching system. The opposing hypothesis is based on passive switching that requires no regulation. Both hypotheses were tested ex vivo in canine lung lobes. The lobes were perfused first with autologous blood, and capillary switching patterns were recorded by videomicroscopy. Next, the vasculature of the lobes was saline flushed, fixed by glutaraldehyde perfusion, flushed again, and then reperfused with the original, unfixed blood. Flow patterns through the same capillaries were recorded again. The 16-min-long videos were divided into 4-s increments. Each capillary segment was recorded as being perfused if at least one red blood cell crossed the entire segment. Otherwise it was recorded as unperfused. These binary measurements were made manually for each segment during every 4 s throughout the 16-min recordings of the fresh and fixed capillaries (>60,000 measurements). Unexpectedly, the switching patterns did not change after fixation. We conclude that the pulmonary capillaries can remain primed for emergencies without requiring regulation: no detectors, no feedback loops, and no effectors-a rare system in biology. NEW & NOTEWORTHY The fluctuating flow patterns of red blood cells within the pulmonary capillary networks have been assumed to be actively controlled within the pulmonary microcirculation. Here we show that the capillary flow switching patterns in the same network are the same whether the lungs are fresh or fixed. This unexpected observation can be successfully explained by a new model of pulmonary capillary flow based on chaos theory and fractal mathematics.
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    Robust estimation of fractal measures for characterizing the structural complexity of the human brain: optimization and reproducibility
    (Elsevier, 2013-12) Goñi, Joaquín; Sporns, Olaf; Cheng, Hu; Aznárez-Sanado, Maite; Wang, Yang; Josa, Santiago; Arrondo, Gonzalo; Mathews, Vincent P; Hummer, Tom A; Kronenberger, William G; Avena-Koenigsberger, Andrea; Saykin, Andrew J.; Pastor, María A.; Department of Radiology and Imaging Sciences, IU School of Medicine
    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.