QUICK FACTS
In the Department of Mathematics at the University of Alabama, Dr. David Halpern is using the IMSL Fortran Numerical Library to create mathematical models of physical phenomena in biomedical sciences such as pulmonary airway closure and reopening. The information uncovered by this research helps develop better medical tools and procedures for serious health issues and provides a basis for enhanced learning.

THE PROBLEM
At the University of Alabama, Ph.D. David Halpern, professor in The Department of Mathematics, focuses his research on developing theoretical models to simulate and analyze nonlinear physical phenomena primarily with applications in the biomedical sciences. Dr. Halpern works with collaborators from a number of other universities, including Tulane University, University of Nottingham, Heriot-Watt University, and the University of Michigan.

Dr. Halpern and his collaborators are currently researching the stability of core-annular flows. This is a layered arrangement of two fluids that, while not mixing together, fill a circular pipe and flow along its axis. These types of flows occur in the lungs, for example, where air passes through airways that are lined with a thin, cohesive and sticky, liquid layer. Instability of the thin film can result in airway closure by the development of a liquid plug and the subsequent collapse of the airway wall.

This instability often happens in premature infants who do not produce sufficient quantities of surfactant, a substance which helps lower surface tension in the airways. Surfactant is formed relatively late in fetal life; thus premature infants born without adequate amounts experience respiratory distress and can die if left untreated. High surface tension may cause the liquid lining to thicken causing a liquid plug to form. In some diseases, the airway walls become excessively compliant, and coupled with the surface tension instability, may cause the airway to collapse.

The goals of Dr. Halpern and the collaborating research groups have been to understand these instabilities and to propose mechanisms that may reduce the detrimental effects of such volatilities.

In addition, they have done some basic modeling of surfactant delivery to the lungs to treat surfactant deficiency in premature babies. Surfactant is delivered in the form of a spray or a liquid plug through the mouth. The aim is to spread the surfactant throughout the lungs, primarily to the furthest areas. This is where the smaller airways are located which are most susceptible to closure.

Together with his collaborators, Dr. Halpern has developed several computational models with the aim of answering questions such as, how long it takes for the surfactant to spread throughout the lungs, how much surfactant reaches the furthest areas of the lungs and should one large dose of surfactant be given or several small doses?

THE SOLUTION
Aware of the comprehensive set of mathematical and statistical functions found in the IMSL Fortran Library from Visual Numerics, Dr. Halpern knew that it would have the range of options and trusted algorithms that would be necessary for this research. The IMSL Fortran Library is the gold standard mathematical and statistical code library for Fortran programmers developing high performance computing applications. It contains highly accurate and reliable algorithms with full coverage of mathematics and statistics and complete backward compatibility.

Dr. Halpern has been using the IMSL Fortran Library in a wide variety of applications. For example, in closure and spreading problems, he must find numerical solutions to systems of differential equations that describe how the thickness of a liquid layer coats an airway, and how the surfactant concentration along the interface between the layer and the air change with time and space. The researchers rely on components from the numerical differential equations and the transforms (FFTs) areas of the IMSL Numerical Library.

To investigate the reopening of collapsed small airways, they consider the movement of a finger of air through a viscous fluid contained between two narrowly spaced compliant plates. The goal is to gain an understanding of the stability of the pressure flow response of this system, estimate pressure and forces required to reopen the small airways and determine what impact these forces will have on the cell lining of the airway.

In this case, the numerical simulation of the flow is described by boundary integral equations. Here they use components from linear algebra (solution of linear equations), numerical integration (Gaussian quadrature), function approximation (cubic spline interpolation), and nonlinear systems of equations, all from the IMSL Fortran Library.

RETURN ON INVESTMENT
"From the problems described regarding pulmonary airway closure and reopening, mathematical models are created using the IMSL Numerical Library. These models consist of complex equations that need to be solved numerically. The researchers involved in these studies believe that the reliability of the algorithms found in the IMSL Fortran Library along with its ease of use and wide range of options, makes it possible to solve these equations that are essential to their research. Dr. Halpern explains, “I work on applied mathematics and theoretical/ computational fluid dynamics related to pulmonary mechanics. By using the IMSL Fortran Library we are able to better understand the functioning of the lungs in normal health, in disease, and also in unusual environments such as outer space."