Drug Dynamics, Exponential Functions and Differential Equations
Differential equations are useful in studying the dynamics of a drug in the body. The study of such dynamics is called “pharmacokinetics.” Why should we care about such dynamics? Some drugs have undesirable, or even dangerous, side effects if their concentration is too high. At the same time those drugs must be above a certain concentration to be effective. As the drug is eliminated from the body, doses need to be given periodically in order to maintain the threshold level for effectiveness, yet doses cannot be too frequent or too large or the concentration will exceed a dangerous level. Students will have the opportunity to work on projects involving the main steps in mathematical modeling: setting up the differential equations, solving the differential equations, and testing the model against real data. Students may also work on projects using differential equations to model populations of competing species.
Suppose a drug is introduced into the blood stream. The injection rapidly mixes with the whole blood supply and produces a high concentration of the drug everywhere in the blood. Several tissues will readily absorb the drug when its concentration is higher in the blood than in the tissue, so the drug moves into this second tissue "compartment." The time rate of change of the concentrations in blood and tissue can be described by differential equations. If this were the whole story, the concentration would eventually balance out so that the concentration in blood and tissue were both equal to the total amount of drug divided by the total volume. However, this usually is not the end of the story. The drug could be metabolized or excreted. Linear models work well for drugs that are not metabolized. The kidneys remove the drug from the blood at a rate proportional to the blood concentration. This causes the blood concentration to drop, and eventually it drops below the tissue concentration. At that point, the drug flows from tissue back into the blood and is continually eliminated from the blood by the kidneys. In the long term, the drug concentration tends to zero in both blood and tissue. The speeds with which these various things happen is studied in the following projects. Derivation of the differential equations are here. The solution of the differential equations are combinations of a pair of exponential functions whose coefficients satisfy a linear algebraic equation. Such pairs of exponentials are studied here. Measurement of the empirical part of the three linear projects on drug dynamics is contained here. One can actually measure the parameters of the differential equations for a real patient in order to tell how well the kidneys are working. These measurements can then be used to calculate doses of a dangerous drug. Nonlinear Models Beer, cocaine, and vitamin C are all metabolized by enzyme mediated reactions. This more complicated chemistry leads to nonlinear differential equations to describe the concentrations of the drug in the blood. Different drugs have radically different behavior, for example, sobering up from ethanol is well modeled by a linear graph, whereas blood CO2 is better approximated by an exponential. See example here. Competition and Cooperation between Species Biologically, we say that two species are competing when the presence of each detracts from the growth of the other. Say rabbits and sparrows both eat grass and bushes, but the sparrows prefer bushes and the rabbits prefer grass. The niches of rabbits and sparrows overlap by the extent to which they cannot choose their separate preferences in food and shelter. According to biologists, no two species may share the same niche, as one will inevitably be a superior competitor, driving the other to extinction or to develop a new niche. However, niches may partially overlap, with one or more common resources. This project studies competition between such species and develops a criterion for stable co-existence. The stability amounts to relating "eigenvalues" of a linear system to the ecology.