SIAM Undergraduate Research Online

Volume 2, Issue 2


A Mathematical Model of the Immune System's Role in Obesity-Related Chronic Inflammation

Published electronically August 10, 2009
DOI: 10.1137/08S010323

Authors: Pablo Díaz, Michael Gillespie, Justin Krueger, José Pérez, Alex Radebaugh, Toby Shearman, Garret Vo, and Christine Wheatley (University of North Carolina at Greensboro, St. Augustine's College, Miami University, University of Puerto Rico, Mayagüez Campus, Bucknell University, Virginia Tech, Montana State University, Alma College)
Sponsors: J. Bassaganya-Riera, J. Borggaard , J. Burns , E. Cliff , A. Guri, S. Faulkner, R. Hontecillas-Magarzo, A. Jarrah, C. Koelling, R. Laubenbacher, H. Mortveit, L. Zietsman (Virginia Bioinformatics Institute at Virginia Tech and Interdisciplinary Center for Applied Mathematics at Virginia Tech)

Abstract: Obesity is quickly becoming a pandemic. The low-grade chronic inflammation associated with obesity leads to health risks such as cancer, heart disease, and type 2 diabetes mellitus. To better understand the progression of obesity-related chronic inflammation, mice were fed either a high-fat or low-fat diet over 140 days. At Days 0, 35, 70, and 140, the percentages of macrophage subsets, CD4+ T cells, and regulatory T cells infiltrating the intra-abdominal white adipose tissue (WAT) were examined. Monocyte chemoattractant protein-1 (MCP-1) mRNA expression in WAT was also quantified. Additionally, glucose-normalizing ability was examined by administering peritoneal glucose tolerance tests. A system of ordinary differential equations models this system. The model consists of 8 differential equations, has 25 parameters, and has 1 forcing function. Tools used to characterize the model include parameter estimation, sensitivity analysis, and stability analysis. Based on the data provided, the system describes the growth of adipocyte size and chronic inflammation over 105 days beginning at Day 35, which is approximately when the adipose cells become hypertrophic, or too large to function normally. The model shows that without intervention, chronic inflammation escalates and the related health problems persist.

Misclassification Rates in Hypertension Diagnosis due to Measurement Errors

Published electronically August 10, 2009
DOI: 10.1137/08S010311

Authors: Camila Friedman-Gerlicz and Isaiah Lilly (Claremont McKenna College, California State University at Sacramento)
Sponsor: Xianggui Qu (Oakland University)

Abstract: Using a mixture of two normal distributions, we estimate the false positive and false negative errors in the diagnosis of hypertension. Parameters in the mixture are estimated by the expectation-maximization (EM) algorithm. It is shown that both errors depend on cutoff points. Repeated measurements reduce both errors dramatically. The number of repeated measurements is recommended through a simulation study.

A Numerical Study of Generalized Multiquadric Radial Basis Function Interpolation

Published electronically October 16, 2009
DOI: 10.1137/09S01040X

Author: Maggie E. Chenoweth (Marshall University)
Sponsor: Scott A. Sarra (Marshall University)

Abstract: This work focuses on the generalized multiquadric (GMQ) radial basis function. The GMQ is derived from the multiquadric (MQ), which is used in radial basis function (RBF) interpolation. This is a relatively new field of research, and many properties of the GMQ are still unknown. Numerical experiments will be performed involving the GMQ, and results will be analyzed to gain further understanding into this type of function.

Finding the Maximum Modulus of a Polynomial on the Polydisk Using a Generalization of Stečkin's Lemma

Published electronically October 28, 2009
DOI: 10.1137/09S010460

Author: Gabriel De La Chevrotière (McGill University)
Sponsor: Stephen Drury (McGill University)

Abstract: This paper is a generalization of the work of J.J. Green in Calculating the maximum modulus of a polynomial using Stečkin's Lemma. This lemma is generalized to higher dimensions and is used in an algorithm to locate the absolute global max of a polynomial on the polydisk. How to apply this algorithm to the real sphere and the complex ball is also explained.