A physiological model A common assumption in modeling drug assimilation is that the blood volume in a person is a single compartment that behaves like a stirred tank. Suppose the blood volume is a four-liter tank that initially has a zero concentration of a particular drug. At time t = 0, an intravenous line is inserted into a vein (into the tank) that carries a drug solution with a concentration of 500 mg/L. The inflow rate is 0.06 L/min. Assume the drug is quickly mixed thoroughly in the blood and that the volume of blood remains constant.
a. Write an initial value problem that models the mass of the drug in the blood, for t ≥ 0.

Verified step by step guidance

1. Define the variable: let \(M(t)\) represent the mass of the drug (in mg) in the blood at time \(t\) (in minutes).
2. Identify the rate of change of the mass of the drug in the blood, \(\frac{dM}{dt}\), which depends on the inflow and outflow of the drug.
3. Since the blood volume is constant at 4 liters and the drug is well mixed, the concentration of the drug in the blood at time \(t\) is \(\frac{M(t)}{4}\) mg/L.
4. The inflow rate of the drug mass is the product of the inflow volume rate and the concentration of the incoming drug solution: \$0.06 \text{ L/min} \times 500 \text{ mg/L} = 30$ mg/min.
5. The outflow rate of the drug mass is the product of the outflow volume rate (same as inflow, 0.06 L/min) and the concentration in the blood: \$0.06 \times \frac{M(t)}{4} = 0.015 M(t)\( mg/min. Therefore, the differential equation modeling the mass is: \)\frac{dM}{dt} = 30 - 0.015 M(t)\(, with the initial condition \)M(0) = 0$ since initially there is no drug in the blood.

Key Concepts

Formulating Differential Equations from Physical Models

This involves translating a real-world scenario into a mathematical equation that describes how a quantity changes over time. In this problem, the rate of change of the drug mass in the blood depends on the inflow concentration and rate, and the mixing assumption allows us to model it with a first-order differential equation.

Initial Value Problems (IVP)

An initial value problem consists of a differential equation paired with an initial condition that specifies the state of the system at the start time. Here, the initial drug mass is zero at time t = 0, which helps uniquely determine the solution to the differential equation modeling the drug concentration.

Conservation of Mass and Mixing Assumptions

The conservation of mass principle states that the change in drug mass in the blood equals the mass entering minus the mass leaving. Assuming the blood is a well-mixed compartment means the drug concentration is uniform throughout, simplifying the model to depend only on total mass and inflow parameters.