I am interested in all areas of applied mathematics, but my current research is in numerical methods and mathematical modeling of biological systems. A list of publications with links can be found here.

Chemotactic Models

My dissertation research involved a generalized version of the Keller-Segel model, a highly nonlinear system of partial differential equations that describes the fascinating biological phenomenon of chemotaxis. Along with my advisors Jay Gopalakrishnan and Patrick De Leenheer, I have developed theoretical results on a generalized Keller-Segel model that suggest chemotaxis as a means of initiation of pattern formation. A paper on the results can be found here. I also programmed finite element method solutions (both standard and non-standard discretizations) to find numerical stationary solutions for the Keller-Segel model. Time simulations indicate that these stationary solutions are stable, and therefore would be expressed in nature. Some steady state solutions and time simulations are shown below, and results have been published here.






Another exciting result of my dissertation research is the development of a novel computational algorithm for identifying potential patterns that may be seen in solutions to nonlinear partial differential equations. Some of the patterns visualized using our method of spectral bands can be seen below.
Potential patterns


Bone Growth and Remodeling
The bone in our bodies is constantly being destroyed and rebuilt in order to repair microfractures and keep our bones strong. Some of my more recent research has involved questions surrounding the mathematical modeling of this process. One question I have been investigating is pattern formation in the bones of patients with the rare bone disorder idiopathic hyperphosphatasia. Patients with this disorder have bones which take on a paralell plate pattern, as opposed to healthy children which have more of a honey comb pattern (see Pivonka and Komarova). Preliminary results of the application of the method of spectral bands indicate the the production level of the protein osteoprotegerin (OPG) may affect the patterning of bone tissue in this manner as seen below. Each row shows the patterning of the bone at a different level of OPG production, and each column corresponds to a different point in time.
Role of OPG in bone pattern formation