A pharmaceutical research team is developing a new medication and needs to understand how the drug is processed in the human body. The concentration of the drug in the bloodstream over time follows complex mathematical models that describe how the medication is absorbed, distributed, and eliminated through different pathways. To optimize dosing schedules and ensure patient safety, the team must break down these complex expressions into simpler components.

The Pharmacokinetics Challenge

After a patient receives a single dose of the experimental medication, the drug concentration in the blood (measured in mg/L) follows the function:

[latex]C(t) = \frac{90t + 300}{(t + 2)(t + 5)}[/latex]

where [latex]t[/latex] represents time in hours after administration. This expression combines multiple biological processes: the drug’s absorption into the bloodstream and its elimination through the liver and kidneys.

The research team studies a second medication with a more complex elimination pattern:

[latex]D(t) = \frac{2t^2 + 14t + 32}{t(t + 4)^2}[/latex]

This represents a drug that is eliminated through three processes: immediate clearance, primary organ elimination, and secondary metabolic pathway with the same organ showing saturation effects.