A one-compartment pharmacokinetic model with first-order oral absorption, but with an elimination rate constant that oscillates on a 24-hour circadian rhythm — because hepatic drug-metabolizing enzymes (CYP450 family) genuinely vary in abundance and activity across the day:
dA_gut/dt = −k_a · A_gut
dC/dt = k_a · A_gut / V − k_e(t) · C
k_e(t) = k_e0 · [ 1 + amp · cos( 2π(t − φ) / 24 ) ]
A_gut is the amount of drug still in the gut, C is plasma concentration, V is the volume of distribution, k_a is the fixed absorption-rate constant, k_e0 is the baseline elimination-rate constant, amp is how strongly clearance swings around that baseline, and φ is the hour at which enzyme activity — and therefore clearance — peaks. The system is integrated with RK4 across a fixed 48-hour grid (dt = 0.05 h); the single dose is injected into A_gut at the chosen administration hour.
- Dosing time — shifts when the impulse of drug enters the gut compartment relative to the circadian clearance cycle. The exact same dose can clear the toxic threshold if given near the elimination trough, or stay safely inside the therapeutic window if given near the elimination peak.
- Dose amount — the mass of drug administered.
- Rhythm amplitude — how strongly clearance swings around its baseline over 24 h; 0% collapses the model to ordinary constant-clearance PK.
- Peak-clearance hour — the time of day the liver metabolizes the drug fastest (chronotype of the enzyme system).
The left dial is a 24-hour clock: its shading is the instantaneous clearance rate ke(t) (brighter = faster clearance), the yellow handle is the dosing hour and the blue handle is the peak-clearance hour — both draggable. The right-hand chart is the resulting 48-hour plasma-concentration curve, colour-coded against the therapeutic window and toxic threshold; the strip beneath it plots ke(t) itself over the same 48 h so the cause (clearance rhythm) sits directly under the effect (concentration curve). It is exactly the reasoning behind real chronotherapy protocols — timing chemotherapy, corticosteroids or antihypertensives to the body's own metabolic clock instead of the calendar clock.