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.
d. After how many minutes does the drug mass reach 90% of its steady-state level? 

Verified step by step guidance

1. Identify the variables and parameters: let \(Q(t)\) be the mass of the drug in the blood at time \(t\) (in minutes). The volume \(V\) of the blood compartment is 4 liters, the inflow concentration \(C_{in}\) is 500 mg/L, and the inflow rate \(r\) is 0.06 L/min. The initial mass is \(Q(0) = 0\) mg.
2. Write the differential equation for the mass of the drug in the blood. Since the blood is well-mixed and volume is constant, the rate of change of drug mass is inflow minus outflow: \[\frac{dQ}{dt} = r \times C_{in} - r \times \frac{Q(t)}{V}\] Here, \(\frac{Q(t)}{V}\) is the concentration in the blood at time \(t\).
3. Find the steady-state mass \(Q_{ss}\) by setting \(\frac{dQ}{dt} = 0\): \[0 = r \times C_{in} - r \times \frac{Q_{ss}}{V} \implies Q_{ss} = V \times C_{in}\] This represents the mass when the system reaches equilibrium.
4. Solve the differential equation for \(Q(t)\) with initial condition \(Q(0) = 0\). This is a first-order linear ODE with solution: \[Q(t) = Q_{ss} \left(1 - e^{-\frac{r}{V} t}\right)\] This shows how the drug mass approaches the steady-state over time.
5. To find the time \(t\) when \(Q(t)\) reaches 90% of \(Q_{ss}\), set \[Q(t) = 0.9 \times Q_{ss}\] Substitute the solution and solve for \(t\): \[0.9 = 1 - e^{-\frac{r}{V} t} \implies e^{-\frac{r}{V} t} = 0.1 \implies t = -\frac{V}{r} \ln(0.1)\] This expression gives the time to reach 90% of the steady-state drug mass.

Key Concepts

First-Order Linear Differential Equations

This problem involves modeling the change in drug concentration over time using a first-order linear differential equation. The rate of change depends on the inflow and outflow rates and the concentration difference, allowing us to describe how the drug mass evolves until it reaches steady state.

Steady-State Concentration

Steady state occurs when the drug mass in the blood no longer changes with time, meaning the inflow and outflow rates balance. Calculating the steady-state concentration helps determine the maximum drug mass achievable under constant infusion conditions.

Exponential Approach to Steady State and Time Constants

The drug mass approaches steady state exponentially, characterized by a time constant related to the volume and flow rate. Understanding this exponential behavior allows us to calculate the time required to reach a certain percentage (e.g., 90%) of the steady-state drug mass.