**Research Experience for Undergraduates:Prior Research Interests::**

Improperly posed problems, systems of reaction-diffusion equations, nonlinear wave equations.

**Current Research Interests:**

Mathematical modelling of tumor driven angiogenesis. This work was supported by the National Science Foundation Program in Applied Mathematics Grant #DMS 9803992. My coauthors were Professor Brian Sleeman of the University of Leeds, and Professor Marit Nilsen-Hamilton of the Department of Biochemistry, Biophysics and Molecular Biology at Iowa State are my principal collaborators on this project. Sleeman is supported as a consultant on the grant. The computations were done by S. Pamuk, a PhD student of mine. Together with Mike Smiley and Anna Tucker, Prof. Nilsen-Hamilton and I have extended these ideas to the so called "p53-switch" in which loss of the wild type p-53 function leads to uncontroled or excess growth factor expression. Professor Nilsen-Hamilton, K. Boushaba and I have recently modeled the control of secondary tumors by primary tumors.

** Abstract and Summary: Problems in chemotaxis and angiogenesis.**

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Check out the following plots for the progress of angiogenesis: The system of ordinary and partial differential equations consists of a one dimensional reinforced random walk pde for the EC combined with two odes arising from a Michaelis-Menten mechanism for VEGF conversion to proteolytic enzyme (PE) and one production-consumption equation for fibronectin. These equations, which describe events in the mother capillary, are coupled, via so called transmission relationships, with a two dimensional reinforced random walk pde for the EC, a two dimensional diffusion equation for the diffusion of VEGF from a tumor source to the mother capillary, an ode for proteolytic enzyme and a diffusion-production equation for fibronectin. The fibronectin is assumed to diffuse via mean curvature.

The figures below are only for the ECM. A tumor is assumed to be 25 microns from a capillary segment of 50 microns in length. The times are in hours. The subscripts on the "T"s refer to the fraction of the ECM that the daughter capillary has crossed from the mother capillary to the tumor. The numerical results, when scaled to the dimensions used by Folkman in his classic rabbit eye cornea experiments, agree very well with his experimental travel times as well as other observations such as the accelleration of the tip as it nears the tumor. The model also includes two mechanisms for angiostatin, one in which angiostatin is converted into an inhibitor by endothelial cells and a second in which angiostatin is itself an inhibitor or protease. (Mathematically, the second case is a sub case of the first.)

In the first set of four figures below, no angiostatin is present.

### Endothelial Cell Concentration.

Notice the tip proliferation of the EC density.

### Proteolytic Enzyme Concentration.

Notice the tip proliferation of the enzyme density.

### Angiogenic Growth Factor Concentration.

Notice that the growth factor is consumed only in the region of capillary formation.

### Fibronectin Concentration.

The next set of figures illustrate what happens in the presence of angiostatin. Angiostatin is introduced into the mother capillary just as the daughter capillary reaches the tumor side.

### Endothelial Cell Concentration.

Notice the drop in EC concentration.

### Active Proteolytic Enzyme Concentration.

Compare the concentration of active enzyme to total enzyme in the following figure.

### Total Proteolytic Enzyme Concentration.

### Angiogenic Growth Factor Concentration.

### Fibronectin Concentration.

Notice that it takes quite awhile for the slowly diffusing fibronectin to fill the channel.

### Angiostatin Concentration.

### Inhibitor Concentration.

I have had eight Masters students and six PhD students who have completed their degrees with me. I have one current PhD student and a post doctoral fellow who is supported by a NATO fellowship. Details can be found on pages 5 and 6 of my curriculum vita. I will be happy to send you hard copies of my early papers or either postscript or tex files of my more recent publications.

**Click on Partial Differential Equations for more information on Partial Differential Equations at Iowa State University.**