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VIGRE seminar, spring 2000: Soft Tissue Modeling

Jay Walton, David Dobson, Jeff Morgan, Patrick Wilber
Students enrolled
Heather Gollmer, Vincent Lemoine, Autumn Moody, Nicholas Neumann, Matthew Riddle, Melinda Williams (undergraduate mathematics students); Nandini Duraiswamy, Yuan Xu (graduate bio-engineering students); Guido David, Sami Hamid, Damali Moore, John Ryan, Ediz Tufekcioglu, David Zeigler (graduate mathematics students)
The general topic involved soft-tissue modeling. Specifically, we worked with Jay Humphrey from the Bioengineering Department. The problem concerned modeling the mechanical response of arteries. The clinical issue involves the important phenomenon of mechanically induced chemical changes in living cells. For example, it is conjectured that mechanical stresses and strains play an important role in the initial formation and subsequent growth of atherosclerotic plaques on the inner surfaces of arteries. Humphrey's students are conducting mechanical tests on arteries trying to model their mechanical response to inflation, elongation and torsion. They had been trying to fit certain standard models due to Y. C. Fung from nonlinear elasticity to their data with very limited success. They had difficulty in matching their experimental data and also in trying to solve boundary value problems numerically. One of Humphrey's students who was actually collecting data on arteries was a student in the class. So we had access to the best data available. The mathematics was a very heavy go for the undergraduates, but we did begin to develop some new models. Pat Wilber and Jay Walton decided to check the standard models the engineers were using for certain mathematical properties, namely strong ellipticity and other ellipticity or convexity properties, that have a strong bearing on the qualitative predictions they make and their numerical stability. These investigations led to two papers by Walton and Wilber in which they show that the standard models fail to satisfy strong ellipticity for ANY choice of parameters (to say the least, this surprised the experienced practitioners) and developed some new classes of models for which they gave convenient tests for when strong ellipticity holds and when it doesn't.
One of the main challenges for the modeling and analysis of arterial tissue is that it is highly nonhomogeneous and anisotropic The students found that taking isotropic models or models with spatially averaged fibre orientation and stiffnesses do not do a very satisfactory job of modeling the experimental data. The class was divided into groups who worked with different modeling approaches on the same data. They then wrote reports and gave presentations on their results comparing how each approach fared. The new approaches got reasonable agreement with some of the test data, but not so good with the other. Some of this work carried over to future VIGRE courses.
Two papers were started by Walton and Wilber. One of them has now been accepted by the International Journal of Non-Linear Mechanics.