A mixture of piperacillin and tazobactam (Zosyn, Pfizer) is indicated for certain patients with moderate to severe bacterial infections. The piperacillin is administered at a constant infusion rate of 400 mg/min for 30 min every 6 hours. For your calculations, it may also be helpful to know that in a separate experiment, piperacillin was injected as an intravenous bolus and the elimination half-life was found to be approximately 1 h and its effective volume of distribution in the body was 0.243 L/kg (you can assume a 70 kg adult patient).

(a) Write the set of differential equations for input (i.e., absorption rate, dA/dt), accumulation in the body (dB/dt), and output/elimination (dE/dt) for constant infusion of piperacillin. Then, solve them to produce an expression for piperacillin concentration in the body as a function of time (t), infusion rate constant (k0), and rate constant for elimination from the body (k1). Show your calculations.

(b) Using the data given, determine the numerical value of k1. Then, make a graph of piperacillin concentration in the body versus time over a 6 hour period consisting of a 30 min delivery period followed by a 5.5 hour wash out.