If a picture is worth a thousand words, says William Schiesser, then the right mathematical equation, in today’s Information Age, can be worth a thousand pictures.
This modern twist on an old saying is making the rounds among the mathematicians and engineers who use numerical models to gain a clearer, more detailed picture of the physical world.
These researchers, says Schiesser, who is the R.L. McCann Professor Emeritus of Engineering and Mathematics in the department of chemical engineering, are making critical contributions to the field of biomedical engineering.
CT (computed tomography) scans and other computer imaging tools like magnetic resonance imaging and optical coherence tomography have improved the scope and accuracy of medical diagnosis. The huge quantities of data they generate are providing new insights into the physiology of the human body by, among other things, illuminating the distinctions between healthy tissue and cancerous tissue.
But analyzing endless streams of data can be a daunting task for human beings, he says.
“How do we interpret data when it can be overwhelming? We use mathematical models based on differential equations. With the right software, computers can analyze huge streams of data and detect patterns and other things that human beings can’t.”Modeling for time and for space
Schiesser is the author of Partial Differential Equation Analysis in Biomedical Engineering: Case Studies with Matlab
, a textbook published in December 2012 by Cambridge University Press.Two more books
, Schiesser’s 12th and 13th, will be published in March 2014 by John Wiley & Sons Inc. Their titles are nearly identical: Differential Equation Analysis in Biomedical Science and Engineering: Ordinary Differential Equation Applications with R
and Differential Equation Analysis in Biomedical Science and Engineering: Partial Differential Equation Applications with R
The three books represent the culmination of Schiesser’s lifelong efforts to help his students and his peers use the computer to solve ordinary and partial differential equations, or ODEs and PDEs. The two types of equations show how phenomena vary over time (ODEs) and over time and space (PDEs).
ODEs and PDEs are ideally suited, Schiesser says, for modeling the physiological phenomena he analyzes in PDE Analysis in Biomedical Engineering
. The book devotes one chapter each to the kinetics of antibody binding, the growth of tumors, the transport of oxygen in the retina, the dynamics of kidney dialysis, the healing of wounds and the distribution of encapsulated drugs.
Differential equations were co-invented more than three centuries ago by Isaac Newton and Gottfried Wilhelm Leibniz and have since been used by other scientists, including Albert Einstein.
“Computers have revolutionized the field by routinely and quickly solving differential equations,” says Schiesser. “Our emphasis is to put sets of differential equations on the computer, which can solve more complicated problems than a human being with a pen and a piece of paper.”Tackling macular degeneration
As a teacher, Schiesser often used examples from biomedicine to show his students how to solve differential equations on the computer. He created a library of routines for solving PDEs on the computer and made them available in Matlab, a commercial programming language, and later in R, an open-source programming language. More than a thousand academic and industrial organizations have requested his software program
in the past six years and have used the program to solve problems
in medicine, geology, physics and a host of other subjects. One of the most recent requests, from the department of ophthalmology at Harvard Medical School’s Children’s Hospital, is to use the retinal oxygen transport model presented in PDE Analysis in Biomedical Engineering
In retirement, Schiesser has joined research teams at the Perelman School of Medicine at the University of Pennsylvania and at the University of Michigan’s Medical School.
“Almost the same day I stepped down, I got a phone call from Penn’s medical school inviting me to join them as a researcher part-time. It took me 10 seconds to decide. They had been using my computer routines for solving PDEs for a long time.”
At Penn, Schiesser works with ophthalmologists to mathematically model the diffusion of proteins in the retina
, the light-sensitive layer of tissue behind the inner eye. He discussed this project in A Compendium of PDE Models: Methods of Lines Analysis with Matlab
, a book he coauthored in 2009.
“The diffusion of proteins in the retina is very important in macular degeneration, which is the leading cause of blindness in older people,” he says. “As the population of baby boomers ages, we’re looking at 10 million cases of macular degeneration in the U.S. alone.
“But the disease is not very well understood. The standard treatment is to give people injections of proteins in the eye. We’ve got to do better than that.”
At the University of Michigan, Schiesser is working with medical doctors to model the diffusion of anesthesia in different kinds of tissues, a phenomenon that has much in common with the manner in which encapsulated drugs are released from a polymer matrix and distributed into the body.
“When anesthesiologists administer anesthesia,” he says, “they need to have some kind of idea where that anesthesia is going and in what quantities. We hope our PDE models can give us this idea.
“Drug distribution from a polymer matrix is similar. An encapsulated drug has to leave the capsule and enter the bloodstream. This has to happen at a prescribed rate—slowly over time. The drug is very powerful, and if too much is released at one time, it can do more harm than good.
“Also, the movement of the drug through tissue can vary depending on the type of tissue or organ where it is released. So we are trying to determine what the right rates of release are. This ties in with my work with the University of Michigan and the spread of anesthesia through tissue and where to administer the anesthesia.”
Schiesser, who also taught courses in Lehigh’s department of mathematics, has no plans to retire completely anytime soon.
“Many people who retire,” he says, “close their doors, and we never see them again.
“I feel very fortunate to be able to continue to contribute.”