Drug Pharmacokinetics & Steady-State Calculator
Educational tool only. This calculator models idealised one-compartment pharmacokinetics. It must not be used to select, prescribe or adjust medication for a real patient. Clinical dosing requires approved product information, patient-specific assessment and therapeutic drug monitoring when appropriate.
Introduction to steady-state drug concentration
Repeated doses overlap because some drug remains in the body when the next dose arrives. Concentrations therefore rise over successive intervals until the amount eliminated during an interval equals the amount entering systemic circulation. The resulting peak-and-trough pattern repeats and is called steady state.
This calculator estimates that repeating pattern with a linear, one-compartment model and instantaneous absorption. It reports the concentration immediately after a dose, the concentration immediately before the next dose, the interval average, clearance, accumulation ratio and approximate time to steady state. It also estimates a loading dose that reaches the predicted steady-state peak and, when a target average is supplied, target-based loading and maintenance doses.
The peak, trough and average answer different questions. The peak represents the model’s highest concentration and may matter when toxicity is concentration-related. The trough shows how far concentration falls before another dose. The average reflects total exposure across an interval. A regimen can have an acceptable average but an unsuitable peak or trough, particularly when the dosing interval is long relative to the half-life.
How to use the steady-state pharmacokinetics calculator
Enter the maintenance dose and select milligrams, micrograms or grams. Bioavailability F is the fraction that reaches systemic circulation: use 1 for an intravenous bolus and a value between 0 and 1 for an incompletely available dose. Enter the elimination half-life and dosing interval with their correct time units. Finally, enter the apparent volume of distribution as an absolute number of litres or as a weight-normalised value.
If volume is entered in L/kg or mL/kg, body weight is used to obtain the absolute volume. The optional target field asks the model to calculate the dose associated with that average concentration. The chart length controls how many doses appear on the concentration-time graph. All reported concentrations use mg/L, which is numerically equal to µg/mL.
Check units before interpreting the result. Confusing micrograms with milligrams creates a thousand-fold error, while entering a half-life stated in minutes as hours changes both elimination and accumulation. The calculator validates positive inputs but cannot determine whether a value is clinically appropriate.
The formulas for elimination, accumulation, peak and trough
For an instantly available dose in a one-compartment system, concentration declines exponentially:
Here is bioavailability, is dose, is apparent volume of distribution and is the first-order elimination rate constant. Half-life and clearance connect through:
The calculator uses at full numerical precision. Clearance follows from . Average steady-state concentration is input rate divided by clearance:
Because is the average systemic input rate, halving both dose and interval keeps the average unchanged. It does not keep the fluctuation unchanged. The instantaneous steady-state peak is:
The trough is the peak after one interval of uninterrupted first-order decay:
The accumulation ratio compares the steady-state peak with the first-dose peak:
The peak-to-trough difference equals the concentration jump from one dose, . A loading dose aimed at a chosen concentration is:
Worked example: 500 mg every 12 hours
Consider the default educational example: 500 mg every 12 hours, bioavailability 0.8, half-life 8 hours and volume of distribution 40 L. The elimination constant is 0.693147 ÷ 8 = 0.08664 h⁻¹, giving clearance of 3.466 L/h. One dose produces a 0.8 × 500 ÷ 40 = 10.00 mg/L concentration jump.
Across 12 hours, the remaining fraction is e−0.08664 × 12 = 0.3536. The accumulation ratio is therefore 1 ÷ (1 − 0.3536) = 1.547. The steady-state peak is 10.00 × 1.547 = 15.47 mg/L, and the trough is 15.47 × 0.3536 = 5.47 mg/L. The average is 0.8 × 500 ÷ (3.466 × 12) = 9.62 mg/L.
The model reaches 90% of its final pattern after about 3.32 half-lives, or 26.6 hours here. It reaches 95% after roughly 34.6 hours and about 97% after 40 hours. A theoretical loading dose matching the steady-state peak is 500 × 1.547, or about 773 mg. These are mathematical outputs, not dosing recommendations.
Accumulation and time to steady state
The fraction of steady state reached depends on elapsed half-lives rather than dose size. After one half-life the system is 50% of the way to its eventual pattern; after two it is 75%; after three it is 87.5%; and after five it is 96.9%. Doubling a dose doubles concentrations in a linear model but does not shorten this timeline.
| Elapsed half-lives | Approximate steady state reached | Meaning |
|---|---|---|
| 1 | 50% | Substantial accumulation remains |
| 3.32 | 90% | Near the final repeating range |
| 4.32 | 95% | Common practical threshold |
| 5 | 96.9% | Conventional steady-state approximation |
Loading doses and target concentration design
A loading dose fills the apparent distribution volume without waiting several half-lives. It changes the approach to steady state, not the eventual average produced by the maintenance regimen. The target-based loading dose depends primarily on volume of distribution and bioavailability, whereas maintenance dose depends on clearance, interval and bioavailability.
When a target average is entered, this calculator estimates the loading dose as target × volume ÷ bioavailability and the maintenance dose as target × clearance × interval ÷ bioavailability. Available dosage forms, absorption rate, safety limits and patient variability are not considered.
Renal function, bioavailability and unit interpretation
Reduced renal or hepatic clearance generally lengthens half-life and raises accumulation for an unchanged regimen. Enter the half-life that applies to the situation being modelled rather than substituting a healthy-volunteer value. Changes in apparent volume can alter the loading dose and the size of each concentration jump even when average exposure is unchanged.
Bioavailability scales the amount reaching circulation. A value of 0.5 means half of the entered dose contributes to this model. Concentrations are displayed in mg/L; this is numerically identical to µg/mL, but not to ng/mL. One mg/L equals 1,000 ng/mL.
Limitations of the one-compartment steady-state estimate
The equations assume linear first-order elimination, one well-mixed compartment, constant clearance and volume, perfect adherence, fixed intervals and instantaneous absorption. Oral drugs usually reach their peak later than this model predicts. Multi-compartment distribution can produce a fast early decline followed by a slower terminal phase, so one half-life may not describe the full curve.
Saturable elimination, active metabolites, dialysis, changing organ function, drug interactions, enterohepatic recycling and concentration-dependent protein binding can invalidate the calculation. Therapeutic ranges also vary by drug, indication, sampling time and patient. Use this page to study relationships among dose, interval, half-life, clearance and distribution—not to make clinical decisions.
Reading the concentration chart and trainer game
The result chart uses superposition to show each dose adding to residual drug from earlier doses. The shaded region marks the predicted steady-state peak-to-trough band, while dashed lines identify peak, trough and average values. Increasing the number of charted doses makes convergence easier to see.
The optional Steady State trainer turns the same relationships into a five-case tuning challenge. Adjust dose and interval so the animated concentration spends as much time as possible between the minimum effective concentration and minimum toxic concentration. Later cases introduce lower bioavailability, longer half-life and narrower corridors. A complete rotation lasts about 80 seconds of active play.
Frequently asked questions about steady-state concentrations
How long does repeated dosing take to reach steady state?
A first-order drug reaches approximately 90% of steady state after 3.3 half-lives, 95% after 4.3 half-lives and 97% after five half-lives. The approach is asymptotic, so the exact final value is never reached mathematically.
Why can the average be acceptable while the peak or trough is not?
The average depends on systemic dose rate divided by clearance. Peak-to-trough fluctuation also depends on individual dose size, interval, half-life and distribution volume. Larger, less frequent doses usually create wider swings than smaller, more frequent doses with the same total daily amount.
Why is bioavailability required?
Only the systemically available fraction contributes to the modeled plasma concentration. Intravenous bioavailability is conventionally 1, while oral bioavailability may be lower because of incomplete absorption or first-pass metabolism.
Does a loading dose change the eventual steady state?
No. It changes how quickly concentration reaches the intended region. The maintenance dose and interval determine the eventual repeating exposure when clearance and bioavailability remain constant.
Sources and model basis.
- Rowland M, Tozer TN. Clinical Pharmacokinetics and Pharmacodynamics: Concepts and Applications, 5th edition, Wolters Kluwer, 2019.
- Brunton LL, Knollmann BC, editors. Goodman & Gilman’s The Pharmacological Basis of Therapeutics, 14th edition, McGraw Hill, 2023.
- US Food and Drug Administration, Population Pharmacokinetics guidance.
This educational model does not replace product labelling, clinical judgement or therapeutic drug monitoring.
Steady State: therapeutic corridor trainer
Tune each regimen so its concentration curve remains between the minimum effective concentration (MEC) and minimum toxic concentration (MTC). Change dose to move the curve and interval to control fluctuation. Five increasingly difficult cases make one compact rotation.
Case 1: normal clearance with a broad therapeutic corridor.
Ready. Start the trainer, then tune the dose and interval while the simulated course runs.
Controls: drag or tap the two sliders. On a keyboard, use ↑/↓ for dose, ←/→ for interval, L for loading dose, Space to pause and R to restart.