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We used Dr. M. L. Lapidus's Fractal Zeta Functions to analyze the complex fractal dimensions of 100 different modifications of the Sierpinski Carpet fractal construction. We will showcase the theorems that made calculations easier, as well as Desmos tools that helped in classifying the different fractals and computing their complex dimensions. We will also showcase all 100 of the Sierpinski Carpet modifications and their complex dimensions.

京都大学 / 新制・課程博士 / 博士(医学) / 甲第23077号 / 医博第4704号 / 新制||医||1049(附属図書館) / 京都大学大学院医学研究科医学専攻 / (主査)教授 中本 裕士, 教授 溝脇 尚志, 教授 羽賀 博典 / 学位規則第4条第1項該当 / Doctor of Medical Science / Kyoto University / DFAM

<p>Chronic kidney disease (CKD) has an annual mortality rate of 22% and can cause secondary complications including hypertension, anemia, secondary hyperparathyroidism, and malnutrition. Currently, clinical diagnosis and evaluation of CKD involves blood and urine testing and biopsy. MRI is not currently used to image CKD, but there is an interest in developing MRI techniques to test kidney function. Usually, renal functional MRI refers to single images reflecting tissue oxygenation. Using time series information may offer additional information about changes in kidneys as a result of disease. In this thesis, blood oxygen level-dependent (BOLD) MRI and diffusion weighted imaging (DWI) were used to investigate the effects of breath holding and water loading on kidneys. First, BOLD MRI was used to measure effects of breath holding on BOLD signal intensity. DWI and fractal analysis were used to measure changes in diffusion, perfusion and microcirculation shortly after water loading. Breath holding results showed no effect on temporal BOLD signal intensity in young, healthy subjects. A significant decrease in signal intensity was measured in the kidney of a single subject with impaired renal function. Although the renal BOLD signal was found to have fractal characteristics, no changes were measured using this technique between pre- and post-water loading scans during the time period examined. Because the signal appears to behave fractally, this technique may be a good candidate for similar kidney function studies in the future. DWI also remained unchanged as a result of water loading during the post-water loading time period examined.</p> / Master of Applied Science (MASc)

Assessment of diffuse brain disorders, where the brain may appear normal, has proven difficult to translate into personalized treatments. Previous methods based on brain magnetic resonance imaging (MRI) resting state blood oxygen level dependent (rs-BOLD) signal routinely rely on group analysis where large data sets are assessed using region-of interest (ROI) or probabilistic independent component analysis (PICA) to identify temporal synchrony or desynchrony among regions of the brain. Brain connectivity occurs in a complex, multilevel and multi-temporal manner, driving the fluctuations observed in local oxygen demand. These fluctuations have previously been characterized as fractal, as they auto-correlate at different time scales. In this study we propose a model-free complexity analysis based on the fractal dimension of the rs-BOLD signal, acquired with MRI. The fractal dimension can be interpreted as a measure of signal complexity and connectivity. Previous studies have suggested that reduction in signal complexity can be associated with disease. Therefore, we hypothesized that a detectable differences in rs-BOLD signal complexity could be observed between patients with diffuse or heterogeneous brain disorders and healthy controls. In this study, we obtained anatomical and functional data from patients with brain disorders where traditional methods have been insufficient to fully assess the condition. More specifically, we tested our method on mild traumatic brain injury, autism spectrum disorder, chemotherapy-induced cognitive impairment and chronic fatigue syndrome patients. Three major databases from the Neuroimaging Informatics Tools and Resources Clearinghouse (NITRC) project were used to acquire large numbers of age matched healthy controls. Healthy control data was downloaded from the the Autism Brain Imaging Data Exchange (ABIDE), the Alzheimer's Disease Neuroimaging Initiative (ADNI) and the Human Connectome Project specifically matching our experimental design. In all of our studies, the voxel-wise rs-BOLD signal fractal dimension was calculated following a procedure described by Eke and Herman et al. 2000. This method was previously used to assess brain rs-BOLD signal in small mammals and humans. The method consists of estimating the Hurst exponent in the frequency domain using a power spectral density approach and refining the estimation in the time domain with de-trended fluctuation analysis and signal summation conversion methods. Voxel-wise fractal dimension (FD) was then calculated for every subject in the control and patient groups to create ROI-based Z-scores for each individual patient. Voxel-wise validation of FD normality across controls was studied and non-Gaussian voxels, determined using kurtosis and skewness calculations, were eliminated from subsequent analysis. To maintain a 95 % confidence level, only regions where Z-score values were at least 2 standard deviations away from the mean were included in the analysis. In the case of chronic fatigue patients and chemotherapy induced cognitive impairment, DTI analysis was added to also determine whether white matter abnormalities were also relevent. Similar Z-score analysis on DTI metrics was also performed. Brain microscopic networks, modeled as complex systems, become affected in diffuse brain disorders. Z-scoring of the fractal rs-BOLD frequency domain delineated patient-specific regional brain anomalies which correlated with patient-specific symptoms. This technique can be used alone, or in combination with DTI Z-scoring, to characterize a single patient without any need for group analysis, making it ideal for personalized diagnostics. / Thesis / Doctor of Philosophy (PhD)