Pharmacokinetics

First-Order vs Zero-Order Kinetics

Simulates how a plasma concentration falls under first-order, zero-order or Michaelis–Menten (mixed) elimination.

Half-life (first-order)

6.93h

Rate proportional to concentration – constant half‑life

Inputs

Initial concentration C₀ (mg/L)

mg/L

Plasma concentration at time zero.

Elimination rate constant kₑ (h⁻¹)

h⁻¹

First-order drugs typically 0.01–0.5 h⁻¹.

Try an example

Results

Half-life = 0.693 ÷ 0.16.93h

Time to eliminate = 100 ÷ 1010.0h

Half-life uses kₑ and applies to first-order elimination; time to eliminate uses Vmax and applies to zero-order elimination.

Concentration–time profile

Simulated plasma concentration (mg/L) at 0, 1, 2, 4, 6, 8, 12 and 24 hours.

Time (h)	First-Order (mg/L)
0	100.00
1	90.48
2	81.87
4	67.03
6	54.88
8	44.93
12	30.12
24	9.07

Model comparison

Parameter	First-order	Zero-order
Rate	k·C	Vmax
Half-life	Constant	Variable
Examples	Most drugs	Ethanol, Aspirin (high dose)

First-order: C(t) = C₀ · e^(−kₑ·t) t½ = 0.693 / kₑ

Zero-order: C(t) = max(0, C₀ − Vmax·t) time to eliminate = C₀ / Vmax

Michaelis–Menten: rate = Vmax·C / (Km + C)

The mixed curve is built step by step: between each pair of sampling times the rate is worked out from the concentration at the start of the step and held constant for the whole step (simple Euler integration), and the concentration is not allowed to go below zero.

When C is much smaller than Km, the Michaelis–Menten rate is roughly (Vmax/Km)·C — first-order. When C is much larger than Km, it approaches Vmax — zero-order.

Worked example: the same starting level, two elimination models

Illustrative values. A drug reaches 80 mg/L in plasma at time zero. Compare what happens if it is eliminated by first-order kinetics with kₑ = 0.15 h⁻¹, or by zero-order kinetics at a constant 5 mg/L per hour. Choose Compare and enter C₀ = 80, kₑ = 0.15 and Vmax = 5.

Initial concentration C₀: 80 mg/L
First-order rate constant kₑ: 0.15 h⁻¹
Zero-order rate (Vmax): 5 mg/L/h

First-order half-life t½ = 0.693 / kₑ
t½ = 0.693 / 0.15 = 4.62 h
First-order concentration C = C₀ · e^(−kₑt)
t = 4 h: 80 × e^(−0.6) = 43.90 mg/L t = 8 h: 80 × e^(−1.2) = 24.10 mg/L
Zero-order concentration C = C₀ − k₀t
t = 4 h: 80 − 5 × 4 = 60.00 mg/L t = 8 h: 80 − 5 × 8 = 40.00 mg/L
Zero-order time to eliminate = C₀ / k₀
80 / 5 = 16.0 h

First-order: t½ = 4.62 h (43.90 mg/L at 4 h). Zero-order: 60.00 mg/L at 4 h, all gone by 16.0 h.

In the first 4 h the first-order drug loses 36.10 mg/L but the zero-order drug only 20 mg/L, because first-order removes a fixed fraction (high at high concentration) while zero-order removes a fixed amount. Later the order reverses: first-order slows as the level falls, so it never quite reaches zero in this model (2.19 mg/L at 24 h), while the zero-order line hits zero at 16 h. A half-life only exists for the first-order curve.

Common mistakes

Quoting a half-life for zero-order elimination. Under zero-order kinetics the time to halve the concentration depends on the starting level (t½ = C₀ / 2k₀), so it is not a constant of the drug.

Mixing time units — a kₑ in min⁻¹ entered as if it were h⁻¹ makes the half-life 60 times too short. This tool works in hours throughout.

Using log10 where the equation needs the natural log. C = C₀ · e^(−kt) gives ln C versus t a slope of −k; on a log10 plot the slope is −k / 2.303.

Confusing the units of the two rate constants: first-order k is in time⁻¹, zero-order k₀ in concentration per time (here mg/L/h).

Reading the mixed (Michaelis–Menten) curve as a prediction. Here it is stepped only at the listed sampling times, which can be up to 12 h apart, so it can fall too fast.

Assumptions and limitations

The model is one compartment after a single dose: no absorption phase, no distribution phase and no further dosing.

The half-life uses 0.693 as an approximation of ln 2 (0.6931…), which is the convention in most pharmacy texts.

The Michaelis–Menten curve uses simple Euler steps between the sampling times shown, so it is a qualitative picture rather than an exact solution.

The same zero-, first- and second-order mathematics describes drug degradation in a product, but stability shelf-life work uses % drug remaining and its own temperature conditions.

Guide: Zero-, First- and Second-Order Kinetics in Drug StabilityThe concept in full, with practice questions.Drug Stability Kinetics CalculatorDecides the order from real stability data by comparing the three linear plots.Accumulation Index CalculatorTakes a first-order half-life on to repeated dosing.

References

Ducharme MP, Shargel L, editors. Shargel and Yu's Applied Biopharmaceutics & Pharmacokinetics. 8th ed. New York: McGraw Hill; 2022.
Rowland and Tozer's Clinical Pharmacokinetics and Pharmacodynamics: Concepts and Applications. 5th ed. Philadelphia: Wolters Kluwer; 2019.
Sinko PJ, editor. Martin's Physical Pharmacy and Pharmaceutical Sciences. 8th ed. Philadelphia: Wolters Kluwer; 2023.

Common questions

Plot the log of concentration against time. First-order elimination gives a straight line; zero-order gives a straight line on the ordinary (linear) axis instead, and curves downward on the log plot.

Its usual therapeutic levels are close to its Km, so elimination is already partly saturated. A small dose increase pushes it towards zero-order, and the steady-state level can rise far more than proportionally — hence routine monitoring.

The simulation only steps between the listed sampling times, some of which are 4–12 hours apart. Over a long step the rate is overestimated, so the curve can drop too fast. Treat it as a qualitative picture, not a prediction.

Here both are concentration-based: Vmax in mg/L per hour and Km in mg/L, matching C₀ in mg/L. Clinical phenytoin dosing often quotes Vmax in mg/day instead, which is a dosing rate, not a concentration rate.

About this calculator

Most drugs are eliminated by first-order kinetics: a constant fraction is removed per hour, so half-life never changes. A few — ethanol, high-dose aspirin, phenytoin — saturate their enzymes and switch to zero-order kinetics, removing a constant amount per hour. Michaelis–Menten (mixed) kinetics describes the transition between the two.

Clinical implications

First‑order: Safe, predictable elimination; dose proportional

Zero‑order: Risk of accumulation at high doses; small dose changes can cause large concentration swings

Mixed kinetics: Requires therapeutic drug monitoring (phenytoin, theophylline)

Watch for

The mixed curve is a coarse step-by-step estimate at the sampled times, not an exact solution

Half-life is meaningful only for first-order elimination

C₀ / Vmax is meaningful only for zero-order elimination

The curves assume one compartment and no further dosing

Prepared by the PharmaWallah student team. Verify results with your textbook or instructor. Report an error · Editorial policy