Continuous Intravenous Drug Infusion (Biomedical Engineering)

In biomedical engineering, clinical pharmacokinetics, and the firmware design of automated drug infusion pumps (such as smart insulin delivery systems or intensive-care vasodilators), clinicians model how therapeutic substances are cleared from the vascular bloodstream over time.

When a programmable infusion pump delivers medication through $n$ discrete micro-boluses per hour, the systemic concentration $C_n(t)$ remaining in the patient's bloodstream after elapsed time $t$ (in hours) is modeled by the discrete clearance relation:

C_n(t) = C_0 \left(1 + \frac{k t}{n}\right)^{-n}

where

C_0 = 50.0\text{ mg/L}

represents the initial post-loading therapeutic concentration, and

k = 0.150\text{ hr}^{-1}

is the patient's biological renal/hepatic clearance rate constant.

To minimize physiological shock, prevent sharp concentration ripples, and simulate true continuous physiological perfusion, biomedical engineers program the micro-pump's stepper motor to increase pulse frequency toward the continuous limit ( $n \to \infty$ ).

Evaluating the limit as $n \to \infty$ yields an indeterminate exponential power:

\lim_{n \to \infty} C_n(t) = C_0 \left(1 + \frac{kt}{\infty}\right)^{-\infty} = C_0 (1)^{-\infty} = 1^\infty

Because the base approaches $1$ while the exponent approaches $-\infty$ , the limiting concentration cannot be determined by naive substitution. Biomedical engineers must resolve this limit rigorously to program safe delivery thresholds into the pump's microcontroller.

Classify the indeterminate form of $\lim_{n \to \infty} C_n(t)$ , and execute the 4-step Logarithmic Transformation Algorithm using L'Hôpital's Rule to prove that the continuous infusion concentration converges to: $C(t) = \lim_{n \to \infty} C_n(t) = C_0 e^{-k t}$
Compute the exact systemic concentration remaining after an $8.0\text{-hour}$ overnight shift ( $t = 8.0\text{ hr}$ ) under continuous infusion ( $n \to \infty$ ). Compare this with a low-frequency discrete dosing protocol that delivers only $n = 2$ boluses per hour ( $C_2(8.0)$ ).
Explain why discrete bolus delivery creates saw-tooth concentration spikes that risk toxicity or therapeutic failure, and describe how verifying the mathematical limit $\lim_{n \to \infty} C_n(t) = C_0 e^{-kt}$ enables biomedical software engineers to validate automated closed-loop control algorithms in FDA-regulated medical devices.