Pharmacokinetic Drug Elimination Profile (Biomedical Engineering)

In biomedical technology and pharmaceutical engineering, pharmacokinetic (PK) modeling characterizes the time-dependent bloodstream plasma concentration of therapeutic compounds following oral dosing. Under a two-compartment open model, active drug molecules are simultaneously absorbed through the gastrointestinal tract and eliminated via hepatic and renal clearance.

A biomedical device engineer is programming an automated continuous infusion pump to maintain targeted analgesic blood plasma concentrations. Following an oral tablet administration at time $t = 0$ (measured in hours, $t \ge 0$), the blood plasma concentration $C(t)$ (in $\mu\text{g/mL}$) is modeled by the biexponential curve:

1. Evaluate the initial drug concentration $C(0)$ and verify that the curve starts at the origin.
2. Determine the long-range asymptotic elimination behavior by calculating $\lim_{t \to \infty} C(t)$ using Horizontal Asymptote Principles.
3. Compute the first derivative $C'(t)$, locate the critical number $t_{\max}$, and determine the peak therapeutic plasma concentration $C_{\max}$ using the First Derivative Test.
4. Compute the second derivative $C''(t)$, and find the exact time $t_{\text{inf}}$ where the clearance curve undergoes a point of inflection.
5. Construct a complete Information Consolidation Table for $C(t)$ on $[0, \infty)$, and explain why the inflection point marks the transition from absorption-dominated kinetics to elimination-dominated clearance.