Physical Sciences and Mathematics Commons^™

A Comparison Of Computational Perfusion Imaging Techniques, Shaharina Shoha

Masters Theses & Specialist Projects

Dynamic contrast agent magnetic resonance perfusion imaging plays a vital role in various medical applications, including tumor grading, distinguishing between tumor types, guiding procedures, and evaluating treatment efficacy. Extracting essential biological parameters, such as cerebral blood flow (CBF), cerebral blood volume (CBV), and mean transit time (MTT), from acquired imaging data is crucial for making critical treatment decisions. However, the accuracy of these parameters can be compromised by the inherent noise and artifacts present in the source images.

This thesis focuses on addressing the challenges associated with parameter estimation in dynamic contrast agent magnetic resonance perfusion imaging. Specifically, we aim …

Robust Sensor Design For The Novel Reduced Models Of The Mead-Marcus Sandwich Beam Equation, Ahmet Aydin

Masters Theses & Specialist Projects

Novel space-discretized Finite Differences-based model reductions are proposed for the partial differential equations (PDE) model of a multi-layer Mead-Marcus-type beam with (i) hinged-hinged and (ii) clamped-free boundary conditions. The PDE model describes transverse vibrations for a sandwich beam whose alternating outer elastic layers constrain viscoelastic core layers, which allow transverse shear. The major goal of this project is to design a single boundary sensor, placed at the tip of the beam, to control the overall dynamics on the beam.

For (i), it is first shown that the PDE model is exactly observable by the so-called nonharmonic Fourier series approach. However, …

Analysis Of Boundary Observability Of Strongly Coupled One-Dimensional Wave Equations With Mixed Boundary Conditions, Wilson Dennis Horner

Masters Theses & Specialist Projects

*see note below

In control theory, the time it takes to receive a signal after it is sent is referred to as the observation time. For certain types of materials, the observation time to receive a wave signal differs depending on a variety of factors, such as material density, flexibility, speed of the wave propagation, etc. Suppose we have a strongly coupled system of two wave equations describing the longitudinal vibrations on a piezoelectric beam of length L. These two wave equations have non-identical wave propagation speeds c1 and c2. First, we prove the exact observability inequality with the optimal …

H-Discrete Fractional Model Of Tumor Growth And Anticancer Effects Of Mono And Combination Therapies, Kamala Dadashova

Masters Theses & Specialist Projects

In this thesis, we focus on h–discrete and h–discrete fractional representation of a pharmacokinetics-pharmacodynamics (PK-PD) model which describes tumor growth considering time on hNa, where h>0. First, we introduce some definitions, lemmas and theorems on both h–discrete and h–discrete fractional calculus in the preliminary section. In Chapter 3, we work on the PD model with delay by exam ining nabla h–discrete equations and nabla h–discrete fractional equations as well as variation of constants formulas, accordingly. We introduce our model and solve it using theorems we proved in the last section of the indicated chapter. When we do simulation for …

Mathematical Modeling Of Diabetic Foot Ulcers Using Optimal Design And Mixed-Modeling Techniques, Michael Belcher

Mahurin Honors College Capstone Experience/Thesis Projects

A mathematical model for the healing response of diabetic foot ulcers was developed using averaged data (Krishna et al., 2015). The model contains four major factors in the healing of wounds using four separate differential equations with 12 parameters. The four differential equations describe the interactions between matrix metalloproteinases (MMP-1), tissue inhibitors of matrix metalloproteinases (TIMP-1), the extracellular matrix (ECM) of the skin, and the fibroblasts, which produce these proteins. Recently, our research group obtained the individual patient data that comprised the averaged data. The research group has since taken several approaches to analyze the model with the individual …

Parameter Estimation And Optimal Design Techniques To Analyze A Mathematical Model In Wound Healing, Nigar Karimli

Masters Theses & Specialist Projects

For this project, we use a modified version of a previously developed mathematical model, which describes the relationships among matrix metalloproteinases (MMPs), their tissue inhibitors (TIMPs), and extracellular matrix (ECM). Our ultimate goal is to quantify and understand differences in parameter estimates between patients in order to predict future responses and individualize treatment for each patient. By analyzing parameter confidence intervals and confidence and prediction intervals for the state variables, we develop a parameter space reduction algorithm that results in better future response predictions for each individual patient. Moreover, use of another subset selection method, namely Structured Covariance Analysis, that …

Score Test And Likelihood Ratio Test For Zero-Inflated Binomial Distribution And Geometric Distribution, Xiaogang Dai

Masters Theses & Specialist Projects

The main purpose of this thesis is to compare the performance of the score test and the likelihood ratio test by computing type I errors and type II errors when the tests are applied to the geometric distribution and inflated binomial distribution. We first derive test statistics of the score test and the likelihood ratio test for both distributions. We then use the software package R to perform a simulation to study the behavior of the two tests. We derive the R codes to calculate the two types of error for each distribution. We create lots of samples to approximate …

Controllability And Observability Of The Discrete Fractional Linear State-Space Model, Duc M. Nguyen

Masters Theses & Specialist Projects

This thesis aims to investigate the controllability and observability of the discrete fractional linear time-invariant state-space model. First, we will establish key concepts and properties which are the tools necessary for our task. In the third chapter, we will discuss the discrete state-space model and set up the criteria for these two properties. Then, in the fourth chapter, we will attempt to apply these criteria to the discrete fractional model. The general flow of our objectives is as follows: we start with the first-order linear difference equation, move on to the discrete system, then the fractional difference equation, and finally …

Runs Of Identical Outcomes In A Sequence Of Bernoulli Trials, Matthew Riggle

Masters Theses & Specialist Projects

The Bernoulli distribution is a basic, well-studied distribution in probability. In this thesis, we will consider repeated Bernoulli trials in order to study runs of identical outcomes. More formally, for t ∈ N, we let Xt ∼ Bernoulli(p), where p is the probability of success, q = 1 − p is the probability of failure, and all Xt are independent. Then Xt gives the outcome of the tth trial, which is 1 for success or 0 for failure. For n, m ∈ N, we define Tn to be the number of trials needed to first observe n …

Iterative Methods To Solve Systems Of Nonlinear Algebraic Equations, Md Shafiful Alam

Masters Theses & Specialist Projects

Iterative methods have been a very important area of study in numerical analysis since the inception of computational science. Their use ranges from solving algebraic equations to systems of differential equations and many more. In this thesis, we discuss several iterative methods, however our main focus is Newton's method. We present a detailed study of Newton's method, its order of convergence and the asymptotic error constant when solving problems of various types as well as analyze several pitfalls, which can affect convergence. We also pose some necessary and sufficient conditions on the function f for higher order of convergence. Different …

Height Transitions, Shape Evolution, And Coarsening Of Equilibrating Quantum Nanoislands, Mikhail Khenner

Mathematics Faculty Publications

No abstract provided.

Analysis And Implementation Of Numerical Methods For Solving Ordinary Differential Equations, Muhammad Sohel Rana

Masters Theses & Specialist Projects

Numerical methods to solve initial value problems of differential equations progressed quite a bit in the last century. We give a brief summary of how useful numerical methods are for ordinary differential equations of first and higher order. In this thesis both computational and theoretical discussion of the application of numerical methods on differential equations takes place. The thesis consists of an investigation of various categories of numerical methods for the solution of ordinary differential equations including the numerical solution of ordinary differential equations from a number of practical fields such as equations arising in population dynamics and astrophysics. It …

Interplay Of Quantum Size Effect, Anisotropy And Surface Stress Shapes The Instability Of Thin Metal Films, Mikhail Khenner

Mathematics Faculty Publications

Morphological instability of a planar surface ([111], [011], or [001]) of an ultra-thin metal film is studied in a parameter space formed by three major effects (the quantum size effect, the surface energy anisotropy and the surface stress) that influence a film dewetting. The analysis is based on the extended Mullins equation, where the effects are cast as functions of the film thickness. The formulation of the quantum size effect (Z. Zhang et al., PRL 80, 5381 (1998)) includes the oscillation of the surface energy with thickness caused by electrons confinement. By systematically comparing the effects, their contributions into the …

An Investigation Of The Accuracy Of Parallel Analysis For Determining The Number Of Factors In A Factor Analysis, Mandy Matsumoto

Mahurin Honors College Capstone Experience/Thesis Projects

Exploratory factor analysis is an analytic technique used to determine the number of factors in a set of data (usually items on a questionnaire) for which the factor structure has not been previously analyzed. Parallel analysis (PA) is a technique used to determine the number of factors in a factor analysis. There are a number of factors that affect the results of a PA: the choice of the eigenvalue percentile, the strength of the factor loadings, the number of variables, and the sample size of the study. Although PA is the most accurate method to date to determine which factors …

Numerically Solving A System Of Pdes Modeling Chronic Wounds Treated With Oxygen Therapy, Stefan Stryker

Mahurin Honors College Capstone Experience/Thesis Projects

Chronic wounds such as diabetic foot ulcers are the leading cause of non-traumatic amputations in developed countries. For researchers to better understand the physiology of these wounds, a mathematical model describing oxygen levels at the wound site can be used to help predict healing responses. The model utilizes equations that are modified from work by Guffey (2015) that consists of four variables – oxygen, bacteria, neutrophils, and chemoattractant within a system of partial differential equations. Our research focuses on numerically solving these partial differential equations using a finite volume approach. This numerical solver will be important for future research in …

Discrete Fractional Hermite-Hadamard Inequality, Aykut Arslan

Masters Theses & Specialist Projects

This thesis is comprised of three main parts: The Hermite-Hadamard inequality on discrete time scales, the fractional Hermite-Hadamard inequality, and Karush-Kuhn- Tucker conditions on higher dimensional discrete domains. In the first part of the thesis, Chapters 2 & 3, we define a convex function on a special time scale T where all the time points are not uniformly distributed on a time line. With the use of the substitution rules of integration we prove the Hermite-Hadamard inequality for convex functions defined on T. In the fourth chapter, we introduce fractional order Hermite-Hadamard inequality and characterize convexity in terms of this …

Stability Of Linear Difference Systems In Discrete And Fractional Calculus, Aynur Er

Masters Theses & Specialist Projects

The main purpose of this thesis is to define the stability of a system of linear difference equations of the form,

∇y(t) = Ay(t),

and to analyze the stability theory for such a system using the eigenvalues of the corresponding matrix A in nabla discrete calculus and nabla fractional discrete calculus. Discrete exponential functions and the Putzer algorithms are studied to examine the stability theorem.

This thesis consists of five chapters and is organized as follows. In the first chapter, the Gamma function and its properties are studied. Additionally, basic definitions, properties and some main theorem of discrete calculus are …

Density-Dependent Leslie Matrix Modeling For Logistic Populations With Steady-State Distribution Control, Bruce Kessler, Andrew Davis

Mathematics Faculty Publications

The Leslie matrix model allows for the discrete modeling of population age-groups whose total population grows exponentially. Many attempts have been made to adapt this model to a logistic model with a carrying capacity (see [1], [2], [4], [5], and [6]), with mixed results. In this paper we provide a new model for logistic populations that tracks age-group populations with repeated multiplication of a density-dependent matrix constructed from an original Leslie matrix, the chosen carrying capacity of the model, and the desired steady-state age-group distribution. The total populations from the model converge to a discrete logistic model with the same …

Model For Computing Kinetics Of The Graphene Edge Epitaxial Growth On Copper, Mikhail Khenner

Mathematics Faculty Publications

A basic kinetic model that incorporates a coupled dynamics of the carbon atoms and dimers ona copper surface is used to compute growth of a single-layer graphene island. The speed of theisland's edge advancement on Cu[111] and Cu[100] surfaces is computed as a function of the growthtemperature and pressure. Spatially resolved concentration pro les of the atoms and dimers aredetermined, and the contributions provided by these species to the growth speed are discussed.Island growth in the conditions of a thermal cycling is studied.

Applications Of Latin Hypercube Sampling Scheme And Partial Rank Correlation Coefficient Analysis To Mathematical Models On Wound Healing, Hannah M. Pennington

Mahurin Honors College Capstone Experience/Thesis Projects

Latin hypercube sampling and Partial Rank Correlation Coefficient procedure (LHS/PRCC) can be used in combination to perform a sensitivity analysis that assesses a model over a global parameter space. Through this analysis, the uncertainty of the parameters and therefore the variability of the model output in response to this uncertainty can be observed. Latin hypercube sampling divides the parameter space into equiprobable regions and sample without replacement, producing a global, unbiased selection of parameter values. For montonic, non-linear relationships, the correlation between the outputs and parameters can be understood by performing a Partial Rank Correlation Coefficient procedure. This sensitivity analysis …

Analysis Of Discrete Fractional Operators And Discrete Fractional Rheological Models, Meltem Uyanik

Masters Theses & Specialist Projects

This thesis is comprised of two main parts: Monotonicity results on discrete fractional operators and discrete fractional rheological constitutive equations. In the first part of the thesis, we introduce and prove new monotonicity concepts in discrete fractional calculus. In the remainder, we carry previous results about fractional rheological models to the discrete fractional case. The discrete method is expected to provide a better understanding of the concept than the continuous case as this has been the case in the past. In the first chapter, we give brief information about the main results. In the second chapter, we present some fundamental …

Application Of A Numerical Method And Optimal Control Theory To A Partial Differential Equation Model For A Bacterial Infection In A Chronic Wound, Stephen Guffey

Masters Theses & Specialist Projects

In this work, we study the application both of optimal control techniques and a numerical method to a system of partial differential equations arising from a problem in wound healing. Optimal control theory is a generalization of calculus of variations, as well as the method of Lagrange Multipliers. Both of these techniques have seen prevalent use in the modern theories of Physics, Economics, as well as in the study of Partial Differential Equations. The numerical method we consider is the method of lines, a prominent method for solving partial differential equations. This method uses finite difference schemes to discretize the …

Boundary Problems For One And Two Dimensional Random Walks, Miky Wright

Masters Theses & Specialist Projects

This thesis provides a study of various boundary problems for one and two dimensional random walks. We first consider a one-dimensional random walk that starts at integer-valued height k > 0, with a lower boundary being the x-axis, and on each step moving downward with probability q being greater than or equal to the probability of going upward p. We derive the variance and the standard deviation of the number of steps T needed for the height to reach 0 from k, by first deriving the moment generating function of T. We then study two types of two-dimensional random walks with …

A Coupled Pde Model For The Morphological Instability Of A Multi-Component Thin Film During Surface Electromigration, Mahdi Bandegi

Masters Theses & Specialist Projects

In this thesis a model involving two coupled nonlinear PDEs is developed to study instability of a two-component metal film due to horizontal electric field and in a high-temperature environment similar to operational conditions of integrated circuits. The proposed model assumes the anisotropies of the diffusional mobilities for two atomic species, and negligible stresses in the film. The purpose of the modeling is to describe and understand the time-evolution of the shape of the film surface. Toward this end, the linear stability analysis (LSA) of the initially planar film surface with respect to small shape perturbations is performed. Such characteristics …

Analysis Of A Partial Differential Equation Model Of Surface Electromigration, Selahittin Cinar

Masters Theses & Specialist Projects

A Partial Differential Equation (PDE) based model combining surface electromigration and wetting is developed for the analysis of the morphological instability of mono-crystalline metal films in a high temperature environment typical to operational conditions of microelectronic interconnects. The atomic mobility and surface energy of such films are anisotropic, and the model accounts for these material properties. The goal of modeling is to describe and understand the time-evolution of the shape of film surface. I will present the formulation of a nonlinear parabolic PDE problem for the height function h(x,t) of the film in the horizontal …

Green's Functions Of Discrete Fractional Calculus Boundary Value Problems And An Application Of Discrete Fractional Calculus To A Pharmacokinetic Model, Sutthirut Charoenphon

Masters Theses & Specialist Projects

Fractional calculus has been used as a research tool in the fields of pharmacology, biology, chemistry, and other areas [3]. The main purpose of this thesis is to calculate Green's functions of fractional difference equations, and to model problems in pharmacokinetics. We claim that the discrete fractional calculus yields the best prediction performance compared to the continuous fractional calculus in the application of a one-compartmental model of drug concentration. In Chapter 1, the Gamma function and its properties are discussed to establish a theoretical basis. Additionally, the basics of discrete fractional calculus are discussed using particular examples for further calculations. …

Ogden College Of Science & Engineering Newsletter (Spring 2014), Cheryl Stevens, Dean

Ogden College of Science & Engineering Publications

No abstract provided.

Leslie Matrices For Logistic Population Modeling, Bruce Kessler

Mathematics Faculty Publications

Leslie matrices are taught as a method of modeling populations in a discrete-time fashion with more detail in the tracking of age groups within the population. Leslie matrices have limited use in the actual modeling of populations, since when the age groups are summed, it is basically equivalent to discrete-time modeling assuming exponential population growth. The logistic model of population growth is more realistic, since it takes into account a carrying capacity for the environment of the population. This talk will describe an adjustment to the Leslie matrix approach for population modeling that is both takes into account the carrying …

Leslie Matrices For Logistic Population Modeling, Bruce Kessler

Bruce Kessler

Leslie matrices are taught as a method of modeling populations in a discrete-time fashion with more detail in the tracking of age groups within the population. Leslie matrices have limited use in the actual modeling of populations, since when the age groups are summed, it is basically equivalent to discrete-time modeling assuming exponential population growth. The logistic model of population growth is more realistic, since it takes into account a carrying capacity for the environment of the population. This talk will describe an adjustment to the Leslie matrix approach for population modeling that is both takes into account the carrying …

Peaklet Analysis: Software For Spectrum Analysis, Bruce Kessler

Mathematics Faculty Publications

This is the presentation I was invited to give at the Kentucky Innovation and Entrepreneurship Conference, regarding the software that I have developed and worked at commercializing with the help of Kentucky Science and Technology Corporation.