| Description |
Failure to stick to a well-planned drug taking protocol may lead to drug inefficiency during medical treatments. In this dissertation, a stochastic model of the drug delivery process is studied, where the elimination of the drug between each dosage interval is modeled by a single or a set of ordinary differential equations (ODEs), and whether a new dosage of drug is taken or not is modeled by a stochastic impulsive condition. We first derive several important algebraic equations that can be used to determine steady state probability distribution of the drug concentration, as well as some other statistics and relations to help understand this probability distribution. We then show numerical results of this distribution using two different methods for different sets of parameter values. Both linear and nonlinear drug elimination rates are studied and discussed. We next examine the dynamics of this random drug taking process by considering the first exit time problems associated with this random iteration. Specifically, we study the mean of the first exit time the drug concentration passes an effective level. We show that this mean exit time is finite and we give bounds for these estimations. We then discuss the issue of whether the patient should take a single or double dose if a dose is missed on the previous day. We show, by constructing a stochastic model that incorporates both an effective level and a toxicity level, that for a fixed toxicity level, when the effective level is high, then taking a double dose is the better strategy than if only one dose is taken and vice versa. Finally, assuming drugs are injected into the body through extravascular routes on time at each scheduled time, i.e, assuming no randomness in the timing of drug intake, we apply singular perturbation techniques to obtain critical conditions under which there is a stable periodic solution of the model equations, assuming nonlinear elimination kinetics. We also construct composite expansions with this asymptotic solution and calculate typical important biomarkers that cannot be obtained otherwise. |
