<p>Mathematical and computational modeling allow for the rationalization of complex phenomenon observed in our reality. Through the careful selection of assumptions, the intractable task of simulating reality can be reduced to the simulation of a practical system whose behavior can be replicated. The development of computational models allow for the full comprehension of the defined system, and the model itself can be used to evaluate the results of thousands of simulate experiments to aid in the rational design process.</p> <p>Biomedical engineering is the application of engineering principles to the field of medicine and biology. This discipline is composed of numerous diverse subdisciplines that span from genetic engineering to biomechanics. Each of these subdisciplines is concerned with its own complex and seemingly chaotic systems, whose behavior is difficult to characterize. The development and application of computational modeling to rationalize these systems is often necessary in this field and will be the focus of this thesis.</p> <p>This thesis is centered on the development and application of mathematical and computational modeling in three diverse systems in biomedical engineering. First, computational modeling is employed to investigate the behavior of key proteins in the post-synapse centered around learning and memory. Second, computational modeling is utilized to characterize the drug release rate from implantable drug delivery depots, and produce a tool to aid in the tailoring of the release rate. Finally, computational modeling is utilized to understand the motion of particles through an inertial focusing microfluidics chip and optimize the size selective capture efficiency.</p> <p> </p>