Application of the Fractional Calculus in Pharmacokinetic Compartmental Modeling

In this study, we present the application of fractional calculus (FC) in biomedicine. We present three different integer order pharmacokinetic models which are widely used in cancer therapy with two and three compartments and we solve them numerically and analytically to demonstrate the absorption, distribution, metabolism, and excretion (ADME) of drug in different tissues. Since tumor cells interactions are systems with memory, the fractional-order framework is a better approach to model the cancer phenomena rather than ordinary and delay differential equations. Therefore, the nonstandard finite difference analysis or NSFD method following the Grunwald-Letinkov discretization may be applied to discretize the model and obtain the fractional-order form to describe the fractal processes of drug movement in body. It will be of great significance to implement a simple and efficient numerical method to solve these fractional-order models. Therefore, numerical methods using finite difference scheme has been carried out to derive the numerical solution of fractional-order two and tri-compartmental pharmacokinetic models for oral drug administration. This study shows that the fractional-order modeling extends the capabilities of the integer order model into the generalized domain of fractional calculus. In addition, the fractional-order modeling gives more power to control the dynamical behaviors of (ADME) process in different tissues because the order of fractional derivative may be used as a new control parameter to extract the variety of governing classes on the non local behaviors of a model, however, the integer order operator only deals with the local and integer order domain. As a matter of fact, NSFD may be used as an effective and very easy method to implement for this type application, and it provides a convenient framework for solving the proposed fractional-order models

Mathematical modeling with applications in biological systems, physiology, and neuroscience

Doctor of PhilosophyDepartment of MathematicsBacim AlaliDynamical systems modeling is used to describe different biological and physical systems as well as to predict the interactions between multiple components of a system over time. A dynamical system describes the evolution of a given system over time using a set of mathematical laws, typically described by differential equations. There are two main methods to model the dynamical behaviors of a system: continuous time modeling and discrete-time modeling. When the time between two measurements is negligible, the continuous time modeling governs the evolution of the system, however, when there is a gap between any two consecutive measurements, discrete-time system modeling comes into play. Differential equations are used to model continuous systems and iterated maps represent the generations in discrete-time systems. In this dissertation, we study some dynamical systems and present their applications to different problems in biological systems, physiology, and neuroscience. In chapter one, we study the local dynamics of some interesting systems and show the local stable behavior of the system around its critical points. Moreover, we investigate the local dynamical behavior of different systems including the Hénon-Heiles system, the Duffing oscillator, and the Van der Pol equation. Furthermore, we discuss about the chaotic behavior of Hamiltonian systems using two different and new examples. In chapter two, we consider some models in computational neuroscience. Due to the complexity of nerve systems, linear modeling methods are not sufficient to understand the various phenomena in neuroscience. We use nonlinear methods and models, which aim at capturing certain properties of the neurons and their complex dynamics. Specifically, we explore the interesting phenomenon of firing spikes and complex dynamics of the Morris-Lecar model. We consider a set of parameters such that the model exhibits a wide range of phenomenon. We investigate the influences of injected current and temperature on the spiking dynamics of Morris-Lecar model. In addition, we study bifurcations, and computational properties of this neuron model. Moreover, we provide a bound for the membrane potential and a certain voltage value or threshold for firing the spikes. Studying the two co-dimension bifurcations demonstrates more complicated behaviors for this single neuron model. Furthermore, we describe the phenomenon of neural bursting and investigate the dynamics of Morris-Lecar model as a square-wave burster, elliptic burster and parabolic burster. Pharmacokinetic models are mathematical models, which provide insights into the interaction of chemicals with certain biological processes. In chapter three, we consider the process of drug and nanoparticle (NPs) distribution throughout the body. We use a tricompartmental model to study the perfusion of NPs in tissues and a six-compartmental model to study drug distribution in different body organs. We perform global sensitivity analysis by LHS Monte Carlo method using Partial Rank Correlation Coefficient (PRCC). We identify the key parameters that contribute most significantly to the absorption and distribution of drugs and NPs in different organs in the body. In chapter four, we study two infectious disease models and use nonlinear optimization and optimal control theory to help in identifying strategies for transmission control and forecasting the spread of infectious diseases. We analyze the effect of vaccination on the disease transmission in these models. Moreover, we perform global sensitivity analysis to investigate the key parameters in these models. In chapter five, we investigate the complex dynamics of two-species Ricker-type discrete-time competitive model. We perform local stability analysis for the fixed points of the system and discuss about its persistence for boundary fixed points. This system inherits properties of the dynamics of a one-dimensional Ricker model such as the cascade of period-doubling bifurcation, periodic windows, and chaos. We explore the existence of chaos for the equilibrium points for a specific case of this system using Marotto theorem and show the existence of snap-back repeller. In chapter six, we study the problem of chaos synchronization in certain discrete-time dynamical systems. We introduce a drive-response discrete-time dynamical system, which is coupled using convex link function. We investigate a synchronization threshold, after which, the drive-response system uncouples and loses its synchronized behaviors. We apply this method to the synchronized cycles of the Ricker model and show that this model displays a rich cascade of complex dynamics from a stable fixed point and cascade of period-doubling bifurcation to chaos. We numerically verify the effectiveness of the proposed scheme and demonstrate how this type of coupling affects the synchronization of the system. In chapter seven, we study the synchronized cycles of a generalized Nicholson-Bailey model. This model demonstrates a rich cascade of complex dynamics from a stable fixed point to periodic orbits, quasi periodic orbits and chaos. We introduce a coupling of these two chaotic systems with different initial conditions and show how they synchronize over a short time. We investigate the qualitative behavior of Generalized Nicholson-Bailey model and its synchronized model using time series analysis and its long-time dynamics by using its bifurcation diagram

Altered Topological Structure of the Brain White Matter in Maltreated Children through Topological Data Analysis

Childhood maltreatment may adversely affect brain development and consequently influence behavioral, emotional, and psychological patterns during adulthood. In this study, we propose an analytical pipeline for modeling the altered topological structure of brain white matter in maltreated and typically developing children. We perform topological data analysis (TDA) to assess the alteration in the global topology of the brain white-matter structural covariance network among children. We use persistent homology, an algebraic technique in TDA, to analyze topological features in the brain covariance networks constructed from structural magnetic resonance imaging (MRI) and diffusion tensor imaging (DTI). We develop a novel framework for statistical inference based on the Wasserstein distance to assess the significance of the observed topological differences. Using these methods in comparing maltreated children to a typically developing control group, we find that maltreatment may increase homogeneity in white matter structures and thus induce higher correlations in the structural covariance; this is reflected in the topological profile. Our findings strongly suggest that TDA can be a valuable framework to model altered topological structures of the brain. The MATLAB codes and processed data used in this study can be found at https://github.com/laplcebeltrami/maltreated

Time varying analysis of dynamic resting-state functional brain network to unfold memory function

Recent advances in brain network analysis are largely based on graph theory methods to assess brain network organization, function, and malfunction. Although, functional magnetic resonance imaging (fMRI) has been frequently used to study brain activity, however, the nonlinear dynamics in resting-state (fMRI) data have not been extensively characterized. In this work, we aim to model the dynamics of resting-state (fMRI) and characterize the dynamical patterns in resting-state (fMRI) time series data in left and right hippocampus and inferior frontal gyrus. We use Sliding Window Embedding (SWE) method to reconstruct the phase space of resting-state (fMRI) data from left and right hippocampus and orbital part of inferior frontal gyrus. The complexity of resting-state MRI data is examined using fractal analysis. The main purpose of the current study is to explore the topological organization of hippocampus and frontal gyrus and consequently, memory. By constructing resting-state functional network from resting-state (fMRI) time series data, we are able to draw a big picture of how brain functions and step forward to classify brain activity between normal control people and patients with different brain disorders

Noninteger Dimension of Seasonal Land Surface Temperature (LST)

During the few last years, climate change, including global warming, which is attributed to human activities, and its long-term adverse effects on the planet’s functions have been identified as the most challenging discussion topics and have provoked significant concern and effort to find possible solutions. Since the warmth arising from the Earth’s landscapes affects the world’s weather and climate patterns, we decided to study the changes in Land Surface Temperature (LST) patterns in different seasons through nonlinear methods. Here, we particularly wanted to estimate the noninteger dimension and fractal structure of the Land Surface Temperature. For this study, the LST data were obtained during the daytime by a Moderate Resolution Imaging Spectroradiometer (MODIS) on NASA’s Terra satellite. Depending on the time of the year data were collected, temperatures changed in different ranges. Since equatorial regions remain warm, and Antarctica and Greenland remain cold, and also because altitude affects temperature, we selected Riley County in the US state of Kansas, which does not belong to any of these location types, and we observed the seasonal changes in temperature in this county. According to our fractal analysis, the fractal dimension may provide a complexity index to characterize different LST datasets. The multifractal analysis confirmed that the LST data may define a self-organizing system that produces fractal patterns in the structure of data. Thus, the LST data may not only have a wide range of fractal dimensions, but also they are fractal. The results of the present study show that the Land Surface Temperature (LST) belongs to the class of fractal processes with a noninteger dimension. Moreover, self-organized behavior governing the structure of LST data may provide an underlying principle that might be a general outcome of human activities and may shape the Earth’s surface temperature. We explicitly acknowledge the important role of fractal geometry when analyzing and tracing settlement patterns and urbanization dynamics at various scales toward purposeful planning in the development of human settlement patterns

Measuring fractal dynamics of FECG signals to determine the complexity of fetal heart rate

In this research, we study the fetal heart rate from abdominal signals using multi-fractal spectra and fractal analysis. We use the Abdominal and Direct Fetal Electrocardiogram Database contains multichannel fetal electrocardiogram (FECG) recordings obtained from 5 different women in labor, between 38 and 41 weeks of gestation. We apply autocorrelation or power spectral densities (PSD) analysis on these five FECG recordings to estimate the exponent from realizations of these processes and to find out if the signal of interest exhibits a power-law PSD. We perform multi-fractal analysis to discover whether some type of power-law scaling exists for various statistical moments at different scales of these FECG signals. We plot the multi-fractal spectra of this database to compare the width of the scaling exponent for each spectrum. A quantitative analysis commonly known as the Fractal Dimension (FD) using the Higuchi algorithm has been carried out to illustrate the fractal complexity of input signals. Our finding shows that the fractal geometry can be used as a mathematical model and computational framework to further analysis and classification of clinical database. Moreover, it can be considered as a framework to compare the complexity of FECG signals and a useful tool to differentiate between their patterns