Pharmacokinetics & Drug Delivery Systems

Modelling how drugs move through the body. Learn ADME (absorption, distribution, metabolism, excretion), one- and two-compartment models, volume of distribution, clearance and half-life, IV bolus, infusion and oral dosing, repeated dosing and steady state, therapeutic windows, controlled-release systems (diffusion, erosion, osmotic pumps), nanoparticles and targeted delivery — with Python simulating dosing regimens.

A drug only works if the right concentration reaches the target for the right time. Too little is ineffective, too much is toxic. Pharmacokinetics (PK) — what the body does to the drug — is modelled with the same compartment and differential-equation tools engineers use for tanks and reactors.

Drug-delivery engineering designs systems that shape these curves: extended-release tablets, transdermal patches, implants, inhalers, insulin pumps and nanoparticles.

An analogy

The body is like a leaky bucket: each dose pours water in, and the leak (clearance) drains a fixed fraction per hour. Dose regularly enough and the level oscillates around a steady height — the aim is to keep it between 'too low to work' and 'overflowing into toxicity'.

ADME

Absorption (from gut, skin, lungs, muscle into blood; bioavailability F), Distribution (into tissues — described by V_d), Metabolism (mainly liver enzymes, e.g. CYP450), Excretion (kidneys, bile). PK parameters summarise these processes; pharmacodynamics (PD) links concentration to effect.

Compartment models

A one-compartment model treats the body as one well-mixed tank with first-order elimination. Two-compartment models add a peripheral tissue compartment, producing a fast distribution phase and slower elimination phase. Oral dosing adds first-order absorption (Bateman function). Physiologically based PK (PBPK) models represent each organ with real blood flows and volumes.

Dosing regimens

Continuous infusion reaches C_ss = R₀/CL after ~5 half-lives; a loading dose (C_target × V_d) gets there immediately. Repeated dosing accumulates by R = 1/(1 − e^(−kτ)). Narrow-therapeutic-index drugs (digoxin, vancomycin, aminoglycosides) use therapeutic drug monitoring and model-based dose individualisation.

Controlled and targeted delivery

Engineering the release profile:

Formulas

Worked example: Half-life

Given: V_d 40 L, CL 5 L/h

Find: t½.

Answer: 5.5 h

Worked example: Infusion steady state

Given: Target 10 mg/L, CL 5 L/h

Find: Infusion rate.

Answer: 50 mg/h (loading dose 400 mg with V_d 40 L)

Worked example: Oral average level

Given: F 0.8, 500 mg every 12 h, CL 5 L/h

Find: C_ss,avg.

Answer: 6.7 mg/L

One-compartment PK: bolus and repeated dosing

import math
# One-compartment pharmacokinetics: IV bolus and repeated oral dosing
Vd, CL = 40.0, 5.0            # L, L/h
k = CL / Vd
t_half = math.log(2) / k
print(f"elimination rate {k:.3f} /h, half-life {t_half:.1f} h")
dose = 500.0                  # mg IV bolus
for t in (0, 4, 12, 24):
    print(f"t={t:2d} h: C = {dose/Vd*math.exp(-k*t):.2f} mg/L")
# Repeated dosing every 8 h: accumulation factor and steady-state trough
tau = 8.0
R = 1 / (1 - math.exp(-k * tau))
print(f"accumulation factor {R:.2f}, steady-state peak {dose/Vd*R:.1f} mg/L, trough {dose/Vd*R*math.exp(-k*tau):.1f} mg/L")
# elimination rate 0.125 /h, half-life 5.5 h
# t= 0 h: C = 12.50 mg/L
# t= 4 h: C = 7.58 mg/L
# t=12 h: C = 2.79 mg/L
# t=24 h: C = 0.62 mg/L
# accumulation factor 1.58, steady-state peak 19.8 mg/L, trough 7.3 mg/L

Steady state is reached after ~4–5 half-lives (~1 day here). Dosing every 8 h with a 5.5 h half-life gives a 2.7:1 peak-to-trough ratio; extended-release formulations or more frequent smaller doses flatten it when the therapeutic window is narrow.

Common mistakes

Applications

Key takeaways

Going further