Browsing by Subject "Mathematical modeling"
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Item A Systems Biology Approach to Develop Models of Signal Transduction Pathways(2011-10-21) Huang, ZuyiMathematical models of signal transduction pathways are characterized by a large number of proteins and uncertain parameters, yet only a limited amount of quantitative data is available. The dissertation addresses this problem using two different approaches: the first approach deals with a model simplification procedure for signaling pathways that reduces the model size but retains the physical interpretation of the remaining states, while the second approach deals with creating rich data sets by computing transcription factor profiles from fluorescent images of green-fluorescent-protein (GFP) reporter cells. For the first approach a model simplification procedure for signaling pathway models is presented. The technique makes use of sensitivity and observability analysis to select the retained proteins for the simplified model. The presented technique is applied to an IL-6 signaling pathway model. It is found that the model size can be significantly reduced and the simplified model is able to adequately predict the dynamics of key proteins of the signaling pathway. An approach for quantitatively determining transcription factor profiles from GFP reporter data is developed as the second major contribution of this work. The procedure analyzes fluorescent images to determine fluorescence intensity profiles using principal component analysis and K-means clustering, and then computes the transcription factor concentration from the fluorescence intensity profiles by solving an inverse problem involving a model describing transcription, translation, and activation of green fluorescent proteins. Activation profiles of the transcription factors NF-?B, nuclear STAT3, and C/EBP? are obtained using the presented approach. The data for NF-?B is used to develop a model for TNF-? signal transduction while the data for nuclear STAT3 and C/EBP? is used to verify the simplified IL-6 model. Finally, an approach is developed to compute the distribution of transcription factor profiles among a population of cells. This approach consists of an algorithm for identifying individual fluorescent cells from fluorescent images, and an algorithm to compute the distribution of transcription factor profiles from the fluorescence intensity distribution by solving an inverse problem. The technique is applied to experimental data to derive the distribution of NF-?B concentrations from fluorescent images of a NF-?B GFP reporter system.Item Mathematical modeling of flow through vegetated regions(2013-08) Mattis, Steven Andrew; Dawson, Clinton N.Understanding flow processes of sea and fresh water through complex coastal regions is of utmost importance for a number of applications of interest to the scientific and engineering community, including wetland health and restoration, inland flooding due to tropical storms and hurricanes, and navigation through coastal waters. In such regions, the existence of vegetation increases flow resistance, which is a major factor in determining velocity and water level distribution in wetlands and inland. Commonly, the momentum loss due to vegetation is included in a bottom friction term in the model equations; however, such models may oversimplify the complex resistance characteristics of such a system. With recent increases in computational capabilities, it is now feasible to develop and implement more intricate resistance models that more accurately capture these characteristics. We present two methods for modeling flow through vegetated regions. With the first method, we employ mathematical and computational upscaling techniques from the study of subsurface flow to parametrize drag in a complex heterogeneous region. These parameterizations vary greatly depending on Reynolds number. For the coastal flows in which we are interested the Reynolds number at different locations in the domain may vary from order 1 to order 1000, so we must consider laminar and fully turbulent flows. Large eddy simulation (LES) is used to model the effects of turbulence. The geometry of a periodic cell of vegetative obstacles is completely resolved in the fluid mesh with a standard no-slip boundary condition imposed on the fluid-vegetation boundaries. The corresponding drag coefficient is calculated and upscaling laws from the study of inertial flow through porous media are used to parametrize the drag coefficient over a large range of Reynolds numbers. Simulations are performed using a locally conservative, stabilized continuous Galerkin finite element method on highly-resolved, unstructured 2D and 3D meshes. The second method we present is an immersed structure approach. In this method, separate meshes are used for the fluid domain and vegetative obstacles. Taking techniques from immersed boundary finite element methods, the effects of the fluid on the vegetative structures and vice versa are calculated using integral transforms. This method allows us to model flow over much larger scales and containing much more complicated obstacle geometry. Using a simple elastic structure model we can incorporate bending and moving obstacles which would be extremely computationally expensive for the first method. We model flexible vegetation as thin, elastic, inextensible cantilever beams. We present two numerical methods for modeling the beam motion and analyze their computational expense, stability, and accuracy. Using the immersed structure approach, a fully coupled steady-state fluid-vegetation interaction model is developed as well as a dynamic interaction model assuming dynamic fluid flow and quasi-static beam bending. This method is verified using channel flow and wave tank test problems. We calculate the bulk drag coefficient in these flow scenarios and analyze their trends with changing model parameters including stem population density and flow Reynolds number. These results are compared to well-respected experimental results. We model real-life beds of Spartina alterniflora grass with representative beds of flexible beams and perform similar comparisons.Item Mathematical modeling, analysis and simulation of the productivity index for non-linear flow in porous media, with applications in reservoir engineering(2007-08) Eburi Losoha, Simeon; Heinze, Lloyd R.; Seshaiyer, Padmanabhan; Aulisa, Eugenio; Ibragimov, AkifFluid flow in porous media has been often modeled using Darcy’s Law. Because it has been observed that Darcy’s Law is valid only for low velocities, there have been several attempts to solve this problem by using modifications of Darcy’s equation. Forchheimer modified Darcy’s equation by adding a new term to account for inertia caused by high velocity flow initially thought to occur only in gas reservoir but later confirmed that some oil reservoirs may also exhibit non-Darcy flow behavior. This paper takes a look into high rate oil wells to determine if they may experience nonlinearity due to non-Darcy flow and to examine the discrepancy between the differential equation derived from the traditional Darcy’s Law and a new approach using a non-linear Darcy-Forchheimer equation. Mathematical modeling of Darcy and Darcy-Forchheimer diffusivity equations will be performed to simulate high velocity flow. Finite differences methods will be used to approximate the solution of the partial differential equations. In this paper, both explicit and implicit methods will be used to evaluate the differential equations. Implicit methods are more commonly used for domain discretization because they are unconditionally stable. Stability analysis will be conducted to determine the condition for stability of the explicit method used. The results will then be compared to results from finite element method software. The productivity index from the Darcy-Forchheimer diffusivity equation will be calculated with different boundary conditions to evaluate the effect of nonlinear flow on well potential, pressure gradient, and velocity.Item Model-based Biomarker Detection and Systematic Analysis in Translational Science(2012-07-16) Sun, YoutingThis dissertation is concerned with the application of mathematical modeling and statistical signal processing into the rapidly expanding fields of proteomics and genomics. The research is guided by a translational goal which drives the problem formalization and experimental design, and leads to optimization, prediction and control of the underlying system. The dissertation is comprised of three interconnected subjects. In the first part of the dissertation, two Bayesian peptide detection algorithms are proposed to optimize the feature extraction step, which is the most fundamental step in mass spectrometry-based proteomics. The algorithms are designed to tackle data processing challenges that are not satisfactorily addressed by existing methods. In contrast to most existing methods, the proposed algorithms perform deisotoping and deconvolution of mass spectra simultaneously, which enables better identification of weak peptide signals. Unlike greedy template-matching algorithms, the proposed methods have the capability to handle complex spectra where features overlap. The proposed methods achieve better sensitivity and accuracy compared to many popular software packages such as msInspect. In the second part of the dissertation, we consider modeling and assessing the entire mass spectrometry-based proteomic data analysis pipeline. Different modules are identified and analyzed, resulting in a framework that captures key factors in system performance. The effects of various model parameters on protein identification rates and quantification errors, differential expression results, and classification performance are examined. The proposed pipeline model can be used to aid experimental design, pinpoint critical bottlenecks, optimize the work flow, and predict biomarker discovery results. Finally, the same system methodology is extended to analyze the work flow in DNA microarray experiments. A model-based approach is developed to explore the relationship among microarray data properties, missing value imputation, and sample classification in a complicated data analysis pipeline. The situations when it is suitable to apply missing value imputation are identified and recommendations regarding imputation are provided. In addition, a missing value rate-related peaking phenomenon is uncovered.Item The dynamics of mathematical models for machupo viral infection in a rodent population(2007-12) Banerjee, Chandrani; Allen, Linda J. S.; Allen, Edward J.; Paige, Robert; Salazar-Bravo, JorgeMachupo virus is a zoonotic disease that is spread by wild rodents. In humans this disease is known as Bolivian hemorrhagic fever as it was first identified in an outbreak in Bolivia. Humans are exposed to this disease through urine or feces or saliva of infected animals. The mortality rate in humans is approximately 30\%. Thus it is very important to study the spread of the disease in rodent populations so that it's spread to the human population can be prevented. We begin by giving some background on Machupo viral infection in rodents. Machupo virus is transmitted horizontally, vertically, and sexually. In addition, rodents respond differently to infection depending on various conditions. Either rodents develop immunity and recover (referred to as immunocompetent) or they do not develop immunity and remain infected (referred to as immunotolerant). We use this information to formulate a general deterministic model for male and female rodents consisting of eight differential equations, four for females and four for males. The four states in the differential equation are susceptible, immunocompetent, immunotolerant and recovered, denoted as $S$, $I^t$, $I^c$ and $R$, respectively. We compute the disease-free equilibrium (DFE) and study the dynamics for this model near the DFE. A basic reproduction number ${\mathcal R}_0 $ is computed and it is shown that the DFE is locally asymptotically stable if ${\mathcal R}_0<1$. The basic reproduction number shows important relationships among the various model parameters that determine whether an outbreak will occur (${\mathcal R}_0>1)$. Since the general model with eight differential equations is difficult to analyze, we consider special cases of this model. We study $SI^t$ and $SI^cR^c$ models. In the first model $SI^t$ all infected individuals are assumed immunotolerant and in the second model $SI^cR^c$ it is assumed that all infected individuals are immunocompetent. For these models, we compute a basic reproduction number ${\mathcal R}_0$ and show the DFE is locally asymptotically stable if ${\mathcal R}_0<1$. For a simple $SI$ model similar to the $SI^t$, model but which is not differentiated by the sex of the rodent, the dynamics are completely understood. There exists a DFE and possibly two endemic equilibria. Finally, we formulate stochastic differential equations for the general model and all the other special cases. For the stochastic models, the extinction state is an absorbing state. Therefore, if population sizes become very small, it is possible for complete population extinction, even though this may not be the case for the deterministic models. We illustrate some of the analytical results with numerical examples. Numerical approximations for the solution of the differential equations are graphed and compared to a sample path of the stochastic differential equations. Our future goals are to obtain better estimates for the model parameters and to use these models to predict when outbreaks in rodent populations are likely to occur.Item The Effects of Exploratory Learning Environments on Students' Mathematics Achievements(2013-07-05) Sokolowski, AndrzejThe objective of this dissertation was to advance the knowledge about mathematics instruction regarding the use of exploratory graphical embodiments in Pre-K to College levels. More specifically, the study sought to find out which graphical representations generate the highest learning effect sizes as well as which teaching method is the most supportive when graphical representations are applied. The dissertation is organized into three coherent research studies that correspond to different schooling levels. The primary method of data analysis in this study was meta-analysis supported by synthesis of qualitative and comparative studies. A total of 73 primary studies (N = 9055) from 22 countries conducted over the past 13 years met the inclusion criteria. Out of this pool, 45 studies (N = 7293) were meta-analyzed. The remaining 28 studies (N = 1762) of qualitative or mixed method designs where scrutinized for common themes. The results support the proposed hypothesis that visualization aids mathematics learning. At the primary level, the mean effect size for using exploratory environment was ES = 0.53 (SE = 0.05, 95% CI: 0.42-0.63), the mean effect size for using computerized programs at the grade levels 1-8 was ES = 0.60 (SE = 0.03, 95% CI: 0.53-0.66), and the results of applying congruent research techniques at the high school and college levels revealed an effect size of ES = 0.69 (SE = 0.05, 95% CI: 0.59?0.79). At each of the teaching level, implementing an exploratory environment generated a moderate effect size when compared to traditional teaching methods. These findings support a need for a broader implementation of exploratory learning media to mathematics school practice and provide evidence to formulate a theoretical instructional framework.Item Variation in tick host preference and its epidemiological impact(2014-12) Pierce, Kelly Anne; Meyers, Lauren Ancel; Sarkar, Sahotra; Bolnick, Daniel; Leibold, Mathew; Miller, Jennifer; Williamson, PhillipTick-borne pathogens pose a significant health risk to humans and wildlife. The complex interactions between ticks and their hosts make management of tick-borne pathogens particularly challenging. Many of the most common species of ticks feed on a wide variety of hosts, but transmit pathogens that are only capable of infecting a narrow range of susceptible host species. Prior research has focused on understanding which tick hosts are capable of serving as pathogen reservoir hosts by carrying and transmitting tick- borne pathogens. However, relatively little attention has been given to studying how ticks choose their hosts. Host choice is of particular importance to the epidemiology of tick- borne pathogens when not all hosts are pathogen reservoirs. My dissertation research investigates the nature of host choice and its impact on disease prevalence in two tick species with similar life histories and host ranges: the lone star tick (Amblyomma americanum) and the American dog tick (Dermacentor variabilis). I conducted an experiment to demonstrate that lone star ticks can respond to host scent. Certain host scents, including those from some individual opossums and raccoons, are attractive to ticks. Proximity to scent also influences tick movement. I also looked for evidence that American dog tick populations are genetically structured by host species identity, and found that certain tick genotypes correlate with host species. This suggests that these ticks may have heritable host preferences that influence their feeding behaviors. Finally, I used a mathematical model to predict disease transmission probability and lone star tick preference for reservoir hosts. I considered hypothetical wildlife communities with different reservoir host relative abundances, and found that changes in relative abundance influence both disease transmission probability and tick host preference estimates. The model also suggests that lone star ticks must parasitize reservoir hosts more frequently when those hosts are less common. These results highlight the importance of host choice and host community composition as determinants of tick-borne disease prevalence.