[Faculty and Mentors] [Environment]

PROJECT: Modeling Ultrasound Imaging in Cardiovascular Tissue
Advisor: Mansoor Haider

In this project the REU group will develop computational models for  ultrasound imaging of cardiovascular tissue.  This project is motivated by a novel imaging technique called Acoustic Radiation Force Impulse (ARFI) that has the potential to non-invasively detect atherosclerotic plaques.  Model development will involve the formulation and numerical solution of boundary value problems for wave propagation in elastic and viscoelastic materials.  Solutions will be applied to analyze ultrasound data and determine the extent to which material parameter maps can enhance detection of atherosclerosis.

Prerequisites: This project will be well suited to undergraduate students with a solid foundation in vector calculus, some experience with partial differential equations or numerical analysis, and with interests in mathematical modeling.

PROJECT: Cell Rearrangements in Tissues
Advisor: Sharon Lubkin

Developmental biology is the study of how we go from a spherically symmetric egg to a complex dynamic structure with dozens of differentiated cell types in spatially complex and useful arrangements, such as lungs, hands, and brains. Part of the mechanism of development is cell rearrangements, which are due to forces generated by the cell interacting with forces applied to the cell. We will model the rearrangements of cells in morphogenesis using a stochastic simulation technique.

Some computer programming experience is necessary.

PROJECT: Who's #1? The Science Of Building Ranking Systems
Advisor: Carl Meyer

Every day our lives are touched or influenced by some kind of ranking system. For example, the heart of Google's search engine depends on their innovative PageRank concept to rank the importance of web pages.  Voting systems are ways to rank candidates running for an office. The BCS ranking systems determine who gets to play for the national championship in NCAA football. These are only a few of the innumerable places where mathematical models are created for ranking a set of competing entities. The purpose of this project is to investigate the scientific techniques used to create a variety of different ranking schemes.  The ultimate goal is to try to build some new and innovative ranking models that can outdo some of the standard ones. While the applications are not necessarily tied to sports ranking, the abundance of sports data and statistics is a good place to start, so our project will begin by examining mathematical models for ranking sports competition, and we will build from there.

Prerequisites: The tools employed are elementary probability, networks and graphs, linear algebra, and number crunching. Some elementary knowledge in these areas would be helpful.

PROJECT: Differential Inequalities & Maximum Principles
Advisor: Robert Martin

A basic differential inequality from freshman calculus asserts that if is continuous on , differentiable on and on , then is decreasing on . In particular, . We will extend this basic idea to more general inequalities and especially to inequalities involving systems of differential equations. Particular applications include positive feedback systems and competitive population models.

A basic maximum principle that can be easily established using calculus is the fact that if is continuous on , twice differentiable on , and on , then reaches its maximum or (i.e., the maximum of occurs on the boundary of ). We extend this simple idea to functions of more than one variable and then develop methods using these principles to studying, for example, the behavior of solutions to reaction-diffusion equations such as

PROJECT: Mortgage-backed Securities (MBS) Pricing and the Risks Behind
Advisor: Tao Pang

The sub-prime mortgage backed securities (MBS) was consider the trigger of the recent financial crisis. In this project, we are going to study MBS products and how to price MBS product by virtue of Monte Carlo simulation. In addition, we will investigate the risks, such as prepayment risk, interest rate risk and default risk, involved in MBS products. By looking at the risks, we will have a better understanding about the sub-prime MBS and why it caused the financial crisis. Some basic knowledge of probability theory and computer skills such as Excel, Matlab or C language are expected for the participants.

PROJECT: Canonical Gravity Quantization of Spatially Homogeneous Models
Advisor: Arkady Kheyfets

Canonical gravity quantization is an advanced topic on the frontier of modern mathematical physics. It involves very sophisticated mathematical techniques in general case. However, many issues related to the subject can be (and very often are) investigated for the case of spatially homogeneous models, called quantum cosmologies. In this case, mathematical formulation becomes simplified to an extent that it can be handled, with appropriate guidance, by advanced undergraduate students. The project is aimed at a comparative study of various techniques of quantization limited to the case of quantum cosmologies.

The necessary background for the project is mostly multivariable calculus. Understanding of ordinary differential equations and some background in physics is useful but not necessary.

PROJECT: Probabilistic Algorithms in Linear Algebra
Advisor: Ilse Ipsen

Probabilistic algorithms perform matrix operations by including random vectors or by making decisions based on probabilities. We will assess the potential of probabilistic algorithms by comparing them to the traditional "deterministic" algorithms. The algorithms under consideration include matrix approximation, least squares, condition

estimation, and subset selection. This project involves programming in Matlab, matrix analysis, and probability theory.

PROJECT: Modeling the Spread of Disease
Advisor: Alun Lloyd

We shall develop a number of different types of models for the spatial spread of a disease within and between cities and, more generally, across a country. We will examine how the spatial distribution of people affects the spread. These models will be used to test different disease control strategies and to answer the question of whether a given intervention would be able to slow or halt the spread. We shall look at data from a number of disease outbreaks in an attempt to match the behavior of our models with observed data.

A number of different techniques could be used in this project, including ordinary differential equations, partial differential equations, Markov chain models and individual-based models. We shall choose one or more approaches based on the skills and interests of the group, but we will also explore alternatives, providing an introduction to a few approaches that are less familiar to us.

PROJECT: Development of a Quantitative Model to Predict Tumor Incidence in Mice and Rats
Advisors: N.S. Luke, and H.A. El-Masri
USEPA/ORD/NHEERL/ETD/PKB, Research Triangle Park, NC

Biologically-Based Dose Response (BBDR) modeling of environmental pollutants can be utilized to inform the mode of action (MOA) by which compounds elicit adverse health effects. Chemicals that produce tumors are typically described as either genotoxic or non-genotoxic. One commonly proposed MOA for non-genotoxic carcinogens is characterized by the key events of cytotoxicity and regenerative proliferation. The increased division rate associated with such proliferation causes an increase in the probability of mutations, which can result in tumor formation. In this project, three carcinogens that are thought to induce tumors in mice through a cytotoxic mode of action (chloroform, carbon tetrachloride, and dimethyl formamide) will be quantitatively compared using a generalized BBDR quantitative model for tumor incidences. For each compound, a physiologically based pharmacokinetic (PBPK) model will be developed and linked to a pharmacodynamic model of cytolethality and cellular proliferation. The rate of proliferation is then linked to a clonal growth model which predicts tumor incidences. The BBDR model will be used to quantitatively indentify limits of cellular injury and proliferation that would result in a significant increase in tumor incidences in mice. Additionally, the BBDR model will be applied to investigate the impact of species differences in tumor incidences between rats and mice.

 

PROJECT: Design of Electron Devices Using Computer Optimization
Advisors: R.L. Ives (Calabazas Creek Research, Inc. (CCR)) and H.T. Tran (North Carolina State University)

The performance of vacuum electron
devices producing radio frequency (RF) power depends on the operational characteristics of several sub- components. For example, the efficiency of a traveling wave tube depends on the quality of an electron beam, the parameters of the RF circuit, the energy recovery characteristics of the spent beam collector, and the matching characteristics of the output waveguide system. Each subcomponent is related to one or more of the others in determining the overall performance of the device. However, current development of three-dimensional (3D) electron beam devices is placing increasing demands on design engineers. The number of parameters available for refinement becomes limited by the designers ability to manipulate 3D surfaces and visualize electric and magnetic fields in complex geometries.

Computer optimization offers a solution to this problem. Recent advances in optimization technology and its application to 2D problems demonstrates the improved performance achievable. During this program, the students will develop optimization methodologies for the optimal design of electron devices using Beam Optics Analysis (BOA) code from CCR for modeling charge particles and SolidWorks (TriMech Solutions) for modeling the complex geometries. Successful implementation of this capability will allow efficient design of new generation of RF sources, accelerators, and other charged particle beam devices.

Project: Viscoelasticity of the Arterial Wall
Advisor: Mette S. Olufsen

Blood flow in the body is transported through a network of arteries and veins and the structure of these vessels help transforming the flow from a highly pulsatile flow to a slow almost steady flow. The arterial wall is composed of tissue that is viscoelastic allowing it to dampen out some of the wave-reflections observed during the transport of the pulse wave. During this project we will formulate viscoelastic models relating blood pressure and vessel area and we will validate this model against experimental data obtained from in-vitro studies in sheep arteries. Furthermore, we will learn about blood pressure profiles and their relation to blood flow velocity through experimental study that we will conduct in our own laboratory. Students working with this project should have some experience with computer modeling, a solid understanding of calculus, and some experience with differential equations. The most important component is a curiosity and interest in learning about physiological applications.