Title: Physiologically Based Pharmacokinetic Modeling of MTBE: Role of Metabolism
Mentor: Dr. Marina Villafane Evans (US Environmental Protection Agency)
Physiologically based pharmacokinetic (PBPK) models are computational tools used to convert external exposure into target or individual organ doses. These PBPK models can account for differences in physiology, metabolism, and absorption across different species and routes of exposure. These models are based on the principle of mass balance, and make use of ordinary differential equations to describe transport in different compartments representing the organs needed for calculating the chemical's distribution. Metabolism is usually associated with health effects for many volatile compounds. The students this year will use published data on MTBE (methyl tert butyl ether), a volatile and water soluble environmental contaminant. PBPK models are used to study different metabolic hypotheses, and to estimate potential risk to humans . Results will be communicated to the National Center for Environmental Assessment within the US EPA.
Title: Computerized Design of Klystrons
Mentor: Dr. Lawrence Ives (Calabazas Creek Research, Inc.) and
Hien Tran (NC State University)
Klystrons are high power RF sources used for communications, radar, cancer therapy, industrial heating and high energy accelerators. An electron beam interacts with a series of cavities to convert electrical power into RF power. Klystrons provide power levels from 10s of watts to more than 50 MW at frequencies from a few hundred megawatts to more than 30 gigahertz. These RF sources are critical for medical applications, scientific research and national defense.
Although klystrons were first invented more than fifty years ago, their design is still a manual process. Advance simulation codes can predict the performance, but iteration of design parameters is a time consuming, labor intensive process. Typical klystrons contain six to seven cavities, each with several design parameters. The parameter values depend on several inter-related factors and the required klystron performance. Klystrons performance parameters include output power, gain, bandwidth, and efficiency. Cavity parameters include frequency, stored energy, and impedance. With the large number of parameters, klystron design often requires several months to complete.
This program will leverage computer optimization to accelerate the design process and improve klystron performance. This will require generation of goal functions to achieve the desired performance while varying several interrelated design variables with restricted value ranges. In many cases, tradeoffs will be required to optimize conflicting performance specifications.
Given the large number of design variables and the complexity of simulation codes, this will be a challenging project. There is, however, a significant opportunity to dramatically reduce the cost of klystron design while improving the performance of these critically important RF sources.
Title: Portfolio Optimization with Conditional Value at Risk (CVaR)
Faculty Mentor: Prof. Tao Pang (NC State University)
The traditional mean-variance portfolio optimization framework usually
assumes asset returns follow normal distribution. However, more and
more evidence has been found that financial asset returns usually
follow distributions with heavier tail than normal distribution.
Therefore, the optimal strategy may not be truly optimal in the real
world. In this project, students will learn how to use heavy tail
distributions to model the financial asset return data and how to
derive the optimal portfolio under certain constraints. In addition, a
very popular risk measure, conditional value at risk (CVaR) will be
introduced to replace the variance in the mean-variance portfolio
optimization framework. We will investigate the effects of higher
moments, such as skewness and kurtosis, in the CVaR optimization.
Further, we will compare the mean-CVaR optimization results to the
traditional mean-variance optimization results under normal market
condition, as well as stressed market condition, such as market during
the period of 2008 financial crisis.
Students should have solid background in calculus, linear algebra and
probability theory. Background in linear and nonlinear programming is
also desired, but not required.
Title: Mathematical Description of the Mechanical Behavior of Arterial Wall: Changes Between Species
Mentors: Dr Adam Mahdi and Prof. Mette Olufsen (NC State University)
Students will work on developing models predicting
mechanical behavior of the arterial wall. They will study differences
between species (rats vs. sheep and humans), and integrate these
models with models of baroreflex regulation. Models will be validated
using data from rat, sheep, and human arteries.
Prerequisites: Differential equations; Modeling (desired); Interest in
biology; Experience with programming in Matlab.
Title: Prediction and Real-Time Object Classification
Mentors: Drs. David Padgett and Lori Layne (MIT Lincoln Laboratory), and
Hien Tran (NC State University)
Pattern recognition and the classification of data is a mathematically rich problem with many practical applications in areas such as engineering, finance, and medicine. Several algorithms exist to classify objects based on features which are distilled from measurements, but algorithm performance can vary depending on many factors, such as type of data and feature selection. In a previous REU, students implemented and assessed the accuracy of various classification schemes. This summer, we will explore how incorporating prediction and opponent modeling into classification algorithms can help us identify objects in real time. Using a time-dependent sequence of object features, we will identify factors which can assist in the correct classification of objects and prediction of subsequent objects in the sequence. Time permitting, we will also explore real-time classifier training. That is, if a previously unknown object is introduced into the sequence, can we train our classifier as the unknown objects come in to recognize and anticipate the new objects? There are time-dependent feature databases that will be suggested for use by the students and the students may identify and use their own feature databases.
Title: Development of Computational Models to Estimate Maternal and Fetal
Tissue Levels of Selected Hazardous Chemicals
Mentor: Dr. Hisham El-Masri (U.S. Environmental Protection Agency)
The US Environmental Protection Agency (USEPA) is engaged in several research projects to identify health hazards to people exposed to environmental contaminants. For instance, USEPA has established a research program (Virtual Embryo) to investigate the chain of biochemical events impacting the health of a developing fetus during pregnancy. One key component of this research program is the development of an exposure to tissue dose to toxic effect relationship based on understanding of the toxicokinetics of the chemical(s) in question. The purpose of this project is to develop mathematical computational models to predict maternal and fetal tissue levels of chemicals that are considered hazardous to the mother, fetus and developing infant health. These computational models can include simple compartmental models for mother and fetus or a more physiologically based pharmacokinetic (PBPK) modeling for mother/fetus and neonate. Simple models use less complex mathematical formulations but carry the important factors that impact tissue dosimetry. PBPK models include more mathematical descriptions of physiological factors that may impact the absorption, distribution, metabolism and excretion (ADME) of chemicals. Students participating in this project will learn 1) how experimental and literature information is used to identify hazardous chemicals, 2) what physiological and biochemical factors impact the disposition of hazardous chemicals in maternal and fetal tissues, and 3) how to estimate parameters for these factors, and 4) how to incorporate knowledge from 1-3 above to develop, parameterize and simulate mathematical computational models related to developmental toxicity.
Title: Numerical Analysis in high dimensions
Faculty Mentor: Prof. Pierre Gremaud (NC State University)
Many applications demand the resolution of high dimensional problems, from data analysis to complex engineering models. Unfortunately, the computational complexity of most algorithms in dimension d grows exponentially in d. This curse of dimensionality is a very significant impediment to scientific progress.
Fortunately, the curse can sometimes be broken or at least somewhat circumvented... Statisticians have long relied on analysis of variance (ANOVA) representations of functions. In numerical linear algebra, low rank approximations do wonders. More generally, several promising theoretical results have surfaced such as Kolmogorov's representation theorem. Numerical methods that can deal efficiently with high dimensional problems are slowly emerging.
In this project, we will start by considering low rank approximations and characterizing mathematical properties of scalar functions in high dimensions. We will then analyze and implement separation techniques. Finally, we will explore to what extend these techniques can be adapted to the numerical resolution of high dimensional Fokker-Planck equations.
Prerequisites: differential equations, some linear algebra, interest and taste in analysis and numeric.
Title: Cell Arrangements and Rearrangements in Tissues
Faculty Mentor: Prof. Sharon Lubkin (NC State University)
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.
Title: Mathematical Models for Illicit Drug Use
Faculty Mentor: Prof. Alun Lloyd (NC State University)
In this project, differential equation models will be developed to describe the dynamics of illicit drugs in a community and how these change in response to law enforcement efforts. Drug use will be modeled like a disease moving through the population. Of particular interest will be the dynamical interaction between two (or more) illicit drugs and how drug switching might occur in response to anti-drug crackdowns. Model parameters will be estimated using real-world data, and conditions under which bifurcations (such as moving from a methamphetamine community to a heroin community) occur will be explored. Optimal control theory will be used to identify drug control strategies to minimize overall negative impacts on communities. Finally, if time permits, we shall explore more detailed network-based models to understand dynamics at a finer spatial scale, with the aim of identifying what parts of illicit drug networks should be targeted to have the biggest impact.
Previous exposure to differential equations would be helpful. During the early stages of the project we will cover the basics of disease modeling, parameter estimation techniques, topics from bifurcation theory and optimal control.