Research in Mathematical Biology is extremely diverse in the Department of Mathematical Sciences, with faculty members working on a wide variety of modern problems ranging from the molecular scale to the organism level. Current research projects include the fluid mechanics of tear films and the effect of the blink cycle, models of atherosclerotic plaque, transport in bone and osteoporosis, imaging of bone marrow and the collective dynamics of ants.
The Department is an active partner in a Howard Hughes Medical Institute grant, which has also been awarded to the Departments of Chemistry and Biochemistry and Biological Sciences. The grant has successfully overseen the creation of a new Quantitative Biology major, housed within Mathematical Sciences, which engages students in an integrative and interdisciplinary study program.
Many of our faculty members are affiliated with the Delaware Biotechnology Institute (www.dbi.udel.edu) which maintains active programs and seminars related to the biological sciences.
Unless explicitly stated, personnel listed below are affiliated with the Department of Mathematical Sciences at the University of Delaware. The name of the primary contact is underlined in each case.
Mathematical Modeling of Pattern Formation in Ant Foraging Trails and Swarms
Lou Rossi, Drew Amis, Gloria Amakobe, Chien-Chung Shen, (Computer and Information Sciences, University of Delaware), Iain Couzin (Princeton University)
While most people know that ants are expert problem-solvers capable of finding food meters away from their nest, few people realize that there are over ten thousand distinct species of ants, that constitute between 15 and 20% of the terrestrial biomass on Earth. In short, ants have won the battle for the planet Earth, and the best we can do is study their successful strategy. Clearly, their success is not a result of size, strength or speed. In fact, individual ants are rather inept. Most ant species are nearly blind, but their behavior is partially regulated by the secretion and detection of chemicals called pheromones. This chemical communication drives a complex social structure that gives ants a tremendous advantage. Undergraduate Katie Johnson and Professor Rossi have reconciled observed traveling wave patterns in ant foraging trails with a new mathematical model (Journal of Theoretical Biology 241 (2), pp 360-369 (2006)). Undergraduates Drew Amis and Gloria Amakobe are extending this work in a number of directions. Finally, Professors Rossi and Shen are working to apply some of these methods to routing problems in mobile ad-hoc networks.
Biochemical Reactions in Thin Reaction Zones
David Edwards, Linsey Norris
Many biochemical reactions occur in thin reaction zones (along the surface of a cell, for instance). In order to better understand these systems, mathematical models are needed to include both the reaction processes and the transport near these layers. Dr. Edwards' work focuses on the study of the BIAcore, which is an experimental device used to measure rate constants of such reactions.
Mathematical Models of Atherosclerotic Plaque
Pak-Wing Fok, Brooks Emerick, Ulhas Naik (Biological Sciences, University of Delaware), William Weintraub (Christiana Care Heath System)
Plaques are fatty deposits that grow mainly in arteries and develop as a result of a chronic inflammatory response. Dr. Fok’s group studies mathematical models that describe and predict the growth of plaques. For example, graduate student Brooks Emerick is investigating the coupling of fluid flow to the Leukocyte Adhesion Cascade. The group is also developing new methods to interpret data obtained from imaging modalities such as Intravascular Ultrasound (IVUS).
Stochastic models of gene networks
Chetan Pahlajani
Ion Channel Distributions in Olfactory Cilia
David Edwards, Don French, (University of Cincinnati)
When odorant molecules enter the nose, they react with certain receptors. These chemical signals are turned into electrical signals via ion channels in olfactory cilia membranes. Although the number and position of these channels are of great interest, the only measurable quantity in practice is the current of the electrical signal. The system poses an ill-conditioned inverse problem which is studied asymptotically and numerically by Dr. Edwards’ group.
Modeling of Human Tear Films using adaptive Radial Basis Functions (RBFs) and Moving Overset Grids (MOG)
Richard Braun, L. Pamela Cook, Tobin Driscoll, Alfa Heryudono, Kara Maki, P. Ewen King-Smith, (College of Optometry, Ohio State University), Petri Fast (Lawrence, Livermore National Laboratory)
RBF and MOG are two powerful numerical methods that are being developed by Dr. Braun’s group for high order nonlinear PDEs that model the human tear film during a blink. These equations frequently include challenging and realistic boundary conditions. The models are being compared to in vivo measurements of the tear film.
Methods for inferring mechanical and microstructural properties of cancellous bone
Robert Gilbert, Philippe Guyenne, Yvonne Ou, Mathew Lewis, (University of Texas Medical School)
The aim of this project is to provide a mathematical background for understanding the use of ultrasound methodology for osteoporosis diagnosis and to begin the investigations of the dynamics of osteoporosis. To understand exactly what mechanical information of cancellous bone can be extracted from ultrasound measurement, especially in the lower frequency range (< 100 kHz) where effective theory applies, this project proposes to 1) develop accurate both low (< 100 kHz) and high (1-2 MHz) frequency ultrasound models for the isonification of cancellous bone, 2) test these models by solving the inverse acoustic model for the effective bone parameters and compare the results with experiments and 3) correlate microscopic bone parameters with macroscopic parameters using the methods of dehomogenization. Two different mathematical approaches are applied in order to go beyond periodic microstructure: one by variant of Tartar's method of oscillating test functions, the other by stochastic-two-scale homogenization.
