Browsing by Author "Sharon Lubkin, Committee Member"

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Differential Equation Models for the Hormonal Regulation of the Menstrual Cycle
(2002-04-24) Harris, Leona Ann; James F. Selgrade, Committee Chair; Sharon Lubkin, Committee Member; John Franke, Committee Member; Paul Schlosser, Committee Member
There are growing concerns about the effects of environmental substances on the sexual endocrine system. It is believed that estrogenic substances may disrupt the sexual endocrine system by initiating or promoting such adverse effects as cancer, developmental disorders, and the reduction of fertility [17,40]. While these effects appear to be more imminent during high levels of exposure to estrogenic substances, concerns are increasing because low levels of exposure to estrogenic substances occur more frequently for longer periods of time; in our diets (phytoestrogens), in the environment (pesticides), and in contraception (spermicides and birth control pills) and hormonal therapies [17,40]. These effects might have a profound effect on the menstrual cycle. Therefore, mathematical models that accurately predict the serum levels of hormones that control the menstrual cycle would be useful tools in evaluating the effects of environmental substances. The human menstrual cycle is controlled by the pituitary hormones, luteinizing hormone (LH) and follicle-stimulating hormone (FSH), and the ovarian hormones, estradiol, progesterone, and inhibin. The pituitary hormones stimulate the growth of ovarian follicles that secrete hormones and work to produce a fertilized ovum. The mathematical models to be presented in this work predict the blood levels of these five hormones as they interact to regulate and maintain the menstrual cycle. The unmerged model has a pituitary component and an ovarian component consisting of linear systems of ordinary differential equations with time dependent coefficients. The pituitary systems describe the synthesis, release, and clearance of LH and FSH during the menstrual cycle, based on their response to estradiol, progesterone, and inhibin. Functions representing the ovarian hormones are used as inputs into these systems. The ovarian system describes the roles of FSH and LH in the development of ovarian follicles and the production of estradiol, progesterone, and inhibin during the menstrual cycle. Functions representing the pituitary hormones are used as inputs into this system. The merged model is formed by merging the pituitary and ovarian systems together. The merged system is a highly nonlinear system of delay differential equations that describes the interactions between the five hormones throughout the menstrual cycle. This model predicts reasonably accurate blood levels of these hormones observed in normally cycling women as reported in the literature. The merged system is shown to have two stable periodic solutions for the same parameter set, a large amplitude solution that fits data found in the literature for normally cycling women and a small amplitude solution arising from Hopf bifurcation in the system parameters. The small amplitude cycle possesses many similarities to the menstrual cycle disorder referred to as polycystic ovarian syndrome (PCOS). Hormonal treatments for this abnormality are simulated and the large amplitude cycle fitting the data for normally cycling women is successfully recovered. In addition, simulations of exogenous estrogen exposure show that the large amplitude cycle can be perturbed into the small amplitude cycle. Therefore, in this modeling environment, an exogenous estrogen input disrupts the normal menstrual cycle.
Equivalent Safe Response Model for Evaluating the Closed Loop Handling Characteristics of UAS to Contribute to the Safe Integration of UAS into the National Airspace System.
(2010-10-29) Southwell, John; Charles Hall, Committee Chair; Sharon Lubkin, Committee Member; Ashok Gopalarathnam, Committee Member
Finite Element Methods for Interface Problems with Locally Modified Triangulations
(2009-08-04) Xie, Hui; Kazufumi Ito, Committee Member; Xiao-Biao Lin, Committee Member; Sharon Lubkin, Committee Member; Zhilin Li, Committee Chair
Interface problems arise in many applications such as heat conduction in different materials. The partial differential equations (PDEs) that describe these applications have domains that consist of different subdomains. The different subdomains can have complicated shapes or can have different properties. For instance, different subdomains can represent different phases of the same material, such as water and ice. The coefficients of the PDEs can be discontinuous across the interfaces of the subdomains, and the source terms can be singular. Due to these irregularities, the solutions to the PDEs can be nonsmooth or even discontinuous. Here we restrict ourselves to interface problems that do not depend on time and can be expressed in terms of elliptic or elasticity PDEs. We present finite element methods (FEMs) for elliptic and elasticity problems with interfaces. The FEMs are based on body-fitted meshes with a locally modified triangulation. A FEM based on a body-fitted mesh uses a triangulation that is aligned with the interfaces. However, for complicated interfaces it can be difficult and expensive to generate such triangulations. That is why we use a locally modified triangulation based on Cartesian meshes. We first form a Cartesian mesh, then move the grid points near the interfaces to the interfaces. This leads to a locally modified triangulation. We use the standard FEM with the locally modified triangulation to solve the elliptic and elasticity problems with interfaces. By FEM theory, the method is second order accurate in the infinity norm for piecewise smooth solutions. We present some numerical examples to show the second order accuracy of the method. We also present a new second order finite difference method that does not require to compute the curvature. At points away from the interface we can approximate the PDE by using the standard 5-point scheme. At points where the interface crosses the 5-point scheme, we still use the 5-point scheme by introducing some ghost values for the grid points on the other side of interface. The price is that we need to find an equation for each ghost value. We will use the interface conditions, either the jump in Dirichlet or Neumann boundary conditions, to form the equations for the ghost values to complete the linear system. We also present some numerical examples to show the second order accuracy of the method.
Immersed-Interface Finite-Element Methods for Elliptic and Elasticity Interface Problems
(2007-07-31) Gong, Yan; Jason Osborne, Committee Member; Sharon Lubkin, Committee Member; Zhilin Li, Committee Chair; Xiao-Biao Lin, Committee Member
The purpose of the research has been to develop a class of new finite-element methods, called immersed-interface finite-element methods, to solve elliptic and elasticity interface problems with homogeneous and non-homogeneous jump conditions. Simple non-body-fitted meshes are used. Single functions that satisfy the same non-homogeneous jump conditions are constructed using a level-set representation of the interface. With such functions, the discontinuities across the interface in the solution and flux are removed; and equivalent elliptic and elasticity interface problems with homogeneous jump conditions are formulated. Special finite-element basis functions are constructed for nodal points near the interface to satisfy the homogeneous jump conditions. Error analysis and numerical tests are presented to demonstrate that such methods have an optimal convergence rate. These methods are designed as an efficient component of the finite-element level-set methodology for fast simulation of interface dynamics that does not require re-meshing. Such simulation has been a powerful numerical approach in understanding material properties, biological processes, and many other important phenomena in science and engineering.
Investigation of Prompt Gamma-Ray Neutron Activation Analysis for Determining the Phase Amounts in Multiphase Flow
(2008-06-05) Mutiso, Athanas M; Sharon Lubkin, Committee Member; Robin P. Gardner, Committee Chair; Mohamed A. Bourham, Committee Member
Mathematical Modeling of Cartilage Regeneration in Cell-Seeded Scaffolds.
(2010-07-26) Haugh, Janine; Mansoor Haider, Committee Chair; Ralph Smith, Committee Member; Sharon Lubkin, Committee Member; Farshid Guilak, Committee Member
Mathematical Models and Numerical Methods for Analysis of Mechanical and Chemical Loading in Articular Cartilage
(2005-06-27) Schugart, Richard Charles; Mansoor Haider, Committee Chair; Farshid Guilak, Committee Member; Ralph Smith, Committee Member; Sharon Lubkin, Committee Member
Articular cartilage is the primary load-bearing soft tissue in joints such as the knee, shoulder, and hip. Multiphasic continuum mixture models have been used to describe the relative contribution of effects due to solid, fluid, and ionic phases in cartilage. This research is motivated by the need to quantify differences between the normal and osteoarthritic mechanical and physico-chemical states in the tissue. In this dissertation, three studies were conducted involving the development of numerical methods and mathematical models pertaining to the cells and extracellular matrix of articular cartilage. In the first investigation, an accelerated numerical method for the continuous spectrum biphasic poroviscoelastic model of articular cartilage deformation was developed. A common constitutive law for modeling the intrinsic dissipation in cartilage extracellular matrix is the theory of quasi-linear viscoelasticity, in which the solid matrix stress depends on the strain rate via a hereditary integral with a continuous relaxation spectrum. The proposed numerical method was based on an alternate formulation of the viscoelastic law that was implemented using Gaussian quadrature time integration in combination with quadratic interpolation of the strain history. The accuracy and cost of the numerical method were evaluated and compared to a theoretical solution using a finite difference implementation of the 1-D confined compression stress-relaxation problem. Comparisons were also made between the accelerated numerical method and a discrete spectrum method, which is commonly used to overcome the cost of evaluating the hereditary stress-strain integral via an exponential series approximation of the continuous spectrum relaxation function. The second study consisted of the formulation and application of a triphasic me-chano-chemical model to analyze osmotic loading experiments for an isolated articular cartilage cell. The cell was modeled as a charged-hydrated mixture of three phases (solid, fluid, ionic). The model was formulated under the hypothesis that the cell membrane was permeable to both water and ions. Under osmotic loading, isolated cartilage cells exhibit a passive volumetric response, which is that of an ideal osmometer. The triphasic mechano-chemical model was analyzed for consistency with the Boyle-van't Hoff law, which represents the ideal response. The resulting triphasic model suggested that Donnan osmotic pressure alone was not sufficient to balance the elastic stress at equilibrium. A non-zero chemical-expansion stress, which is a measure of the charge-to-charge repulsive forces within the cell, was required to balance the elastic stresses. Since the existence of an intracellular chemical-expansion stress is not well established, it was hypothesized that the triphasic cell model should be modified to include a selectively permeable membrane. The third investigation consisted of the formulation and application of a mechano-chemical model used for analysis of osmotic loading experiments for an isolated chondron. The chondron is comprised of a cartilage cell and its encapsulating pericellular matrix (PCM). In the chondron, the cell membrane was assumed to be permeable to water, but impermeable to ions and the cell was modeled as an ideal osmometer. The PCM was modeled as a triphasic continuum mixture with a fixed charge density arising from the negatively-charged proteoglycans that are characteristic of cartilage extracellular matrix. Both parametric and asymptotic analyses were conducted to compare cell, PCM, and chondron deformation under osmotic loading.