(Pharmacokinetics). Differential equations can be used in pharmacology to model the amount of a drug in the body, from the initial uptake through its elimination via metabolism and excretion. When a drug is introduced into the body, it often spreads differently through body plasma (i.e., blood and other fluids) than it does through the denser tissues that it is intended to target. Therefore, it is appropriate to model the amounts of the drug in the blood and tissue separately, which leads naturally to a system of differential equations. Suppose that a dose of the drug dylar, consisting of 0.5 mg per kilogram of body mass, is administered intravenously at a constant rate over the course of 24 hours. The drug spreads quickly through the body plasma, the volume of which (in liters) we estimate to be 16% of the body’s total mass (in kilograms). The dylar spreads more slowly in the brain tissue that it targets, and we estimate the volume of this tissue (in L) to be 11% of the body’s mass (in kg). For simplicity, we assume the dylar is uniformly distributed in both the plasma and brain tissue at any given time. Assume that water carrying dylar enters the brain at a rate of 0.4 L/hour and leaves the brain at a rate of 0.6 L/hour. The dylar in the bloodstream is metabolized by the liver and excreted from the body, with a half-life of 1.72 hours.
(a) Suppose dylar is administered to a patient weighing 100 kg. We let QP (t) denote the amount of dylar (in mg) present in the patient’s body plasma after t hours and QB(t) the amount of dylar in the brain. Write down a system of differential equations for QP and QB.
(b) Determine the amount of dylar in both the patient’s blood plasma and brain after t hours.