A method for approximating and controlling the distributed parameter model of a MIMO chemical diffusion system

Abstract

Chemical distribution is an important factor in many biological systems. Chemotaxis, the directed movement of an organism in response to chemical in its environment, is known to be a prominent driver in the directed growth of new blood vessels from pre-existing ones, a phenomenon called angiogesis. In order to properly study the effects of various chemical inputs to an angiogenic assay, it is crucial to have strict control over the delivery of these chemicals, which are carried to the sprouting stie via diffusion. More specifically, we use, as a model system, a microfluidic, in vitro assay in which a cell scaffold is bounded between two microchannels. The scaffold is taken to be a porous region through which diffusion occurs, in one dimension, from one channel to the other. In this system, we can specify the chemical concentration and gradient within the region by changing the concentrations in the channels that bound it. In control terms, this is a multi-input, multi-output (MIMO) system, with two inputs and two outputs. However, the dynamics of diffusion are governed by a partial differential equation, meaning that the plant in question is a distributed parameter system, not immediately amenable to controller design. Thus, in this thesis, we present a method for transforming the diffusion equation (Fick's Law) into a finitely-approximated, MIMO, state-space system, by first deriving a matrix of infinite transfer functions. With this state-space system, it is shown that classical control techniques can be applied to manipulate the dynamics as well as track a reference step input with zero steady-state error.