My Easy Science
Michaelis-Menten Elimination
Phenytoin-style saturable elimination. A small dose increase can double the level, and past Vmax there is no steady state at all.
Model: one-compartment, first-order elimination only at low levels; elimination is saturable (Michaelis–Menten).
Set Vmax, Km and the dose, then tap Show the steady level.
For learning only. Not for patient care decisions. Always check with a qualified professional and the product label.
Results
| Value | Result | Unit |
|---|---|---|
| Starting level | — | mg/L |
| Starting elimination rate | — | mg/day |
| Steady-state level (Css) | — | mg/L |
| Daily dose as % of Vmax | — | % |
| Effective clearance at steady state | — | L/day |
| Time to 90% of Css | — | days |
Curve data
| Dose rate (mg/day) | Css (mg/L) |
|---|
| Time (days) | Concentration (mg/L) |
|---|
Values from the drawn curves.
Starting values
Worked example: Vmax 500 mg/day, Km 4 mg/L, daily dose 300 mg/day. The level settles at Css = (300 × 4) / (500 − 300) = 6 mg/L. At 400 mg/day it is 16 mg/L, and at 450 mg/day it is 36 mg/L.
How it works
Checked against three hand-worked examples (see the page source comments).
Results are shown to 2 decimal places; doses and rates are whole numbers.
Most drugs are cleared by enzymes that never fill up, so the rate is a fixed fraction of the level and the half-life stays the same. Phenytoin is different: its main enzyme can be saturated. The rate follows Michaelis-Menten, Vmax × C / (Km + C). When the level is far above Km the enzyme is working flat out and the rate is stuck near Vmax, so the level rises by a fixed amount each day, not a fixed fraction. When the level drops near Km the enzyme frees up and the fall speeds up. That is why the curve bends and why the half-life is not a single number here.
Take Vmax 500 mg/day and Km 4 mg/L. Give 300 mg every day and the level settles where the daily dose equals the daily removal: Css = (300 × 4) / (500 − 300) = 6 mg/L. Raise the daily dose by a third, to 400 mg, and the level does not rise by a third: Css = (400 × 4) / (500 − 400) = 16 mg/L, nearly three times higher. At 450 mg a day the level is 36 mg/L, six times the 300 mg level for only half as much drug again. At 500 mg a day the dose equals the fastest the enzyme can ever work, so there is no steady state and the level keeps climbing. That is the phenytoin trap, and it is why its dose is guided by blood levels, not by a fixed table.
The second graph shows the climb itself: from the starting level, the concentration rises toward the steady state, faster at first and then ever more slowly as the enzyme fills up. The dotted line is what a linear drug with the same starting clearance would do, a plain exponential approach. The gap between the two lines is the whole lesson: near saturation, small dose changes stop behaving like small changes.