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Mathematics

Mathematical Modeling of Dendritic Cell Immuno-Theraphy for Cancer

Megan Hunter ('10); Chris DeBoever ('10 HMC); Helen Wu ('10 HMC); Ami Radunskaya; Lisette de Pillis (HMC)

One of the most promising immune therapies for cancer is dendritic cell (DC) treatment. We extend an earlier delay-differential equations model of dendritic cell trafficking to include a cancer compartment. Experimental tumor growth data with varying levels of DC injections are used to derive maximum likelihood estimates of parameters relevant to the tumor-immune system interaction. Sensitivity and bifurcation analyses are used to identify clinically useful parameters, and to determine equilibria and their stability. In the estimated parameter regime, there are two equilibria: one zero tumor (“healthy”) equilibrium which is unstable and a high tumor equilibrium (“diseased”) which is stable. This implies that, without treatment, the system must move towards the stable, high-tumor equilibrium. We suggest an effective immunological treatment that manipulates critical parameters in order to bring the system past a bifurcation point into a regime in which a new stable low-tumor equilibrium exists.
Funding provided by: Pomona College SURP; HMC Summer Research Program

Commuting Power Series in a P-Adic Ring

Minsoo Kim ('10); Emily Ognacevic ('10 St. Louis University); Jessica Olsen ('10 University of Puget Sound); Samuel Schiavone ('10 Amherst College); Joel Specter ('10 Wesleyan College); Ghassan Sarkis

Using a new notion of absolute value that measures the divisibility of a number by a given prime, we developed the idea of Zp, the ring of p-adic integers. Examining questions of when power series would commute over this ring led us to Lubin's conjecture, which states that when two such power series exist, they must be related to the endomorphisms of a formal group Zp as well. This poster presents two possible directions to follow Lubin's conjecture: on one hand, creating counter examples with "condensation" in order to refine the statement of the conjecture, while on the other hand, attempting to show the conjecture as true in a special case, namely when we havae a non-invertible, invertible non torsion, and a p-1 torsion power series all in Zp that commute with each other.
Funding provided by: The Paul K. Richter and Evalyn E. Cook Richter Award (MK); National Science Foundation, Claremont Colleges

Research at Pomona