Drug Pharmacokinetics & Steady-State Calculator
Drug Accumulation and Steady-State Concentration in Repeated Dosing
When medications are taken repeatedly at fixed intervals, the body does not instantly eliminate each dose. Instead, drug accumulates in the blood and tissues until a dynamic equilibrium—the "steady state"—is reached, where drug eliminated per dosing interval equals the dose administered. Understanding this accumulation is critical for medication safety and efficacy. Too-frequent dosing can lead to toxic accumulation; too-infrequent dosing may leave concentrations subtherapeutic. This calculator models the pharmacokinetics of repeated dosing, predicting blood concentrations over time and the steady-state level achieved with continued dosing.
Pharmacokinetics is the study of how the body handles drugs: absorption, distribution, metabolism, and excretion. For oral or intravenous medications, the plasma concentration follows first-order kinetics: concentration decays exponentially with a characteristic half-life, the time required for the concentration to halve. Repeated doses before complete elimination lead to accumulation, with each successive dose building on residual levels from previous doses.
First-Order Kinetics and Accumulation Formula
For drugs following first-order kinetics, the concentration after a dose is absorbed is:
where is concentration at time , is the dose, is the volume of distribution, and is the elimination rate constant related to half-life by:
With repeated dosing at intervals , the accumulation factor (ratio of steady-state to single-dose peak) is:
At steady state, the peak concentration (immediately after a dose) is the single-dose peak multiplied by the accumulation factor.
Worked Example: Calculating Warfarin Steady-State Concentration
A patient takes warfarin (an anticoagulant) 5 mg once daily. Warfarin has a half-life of ~40 hours and volume of distribution of ~8 liters (simplified estimate). Calculate steady-state concentration:
Step 1: Calculate elimination rate constant – k = 0.693 ÷ 40 hours ≈ 0.0173 hr⁻¹
Step 2: Calculate single-dose peak concentration – C_peak = 5 mg ÷ 8 L = 0.625 mg/L
Step 3: Calculate dosing interval in elimination terms – τ = 24 hours; k×τ = 0.0173 × 24 ≈ 0.415
Step 4: Calculate accumulation factor – R_acc = 1 ÷ (1 − e^{-0.415}) = 1 ÷ (1 − 0.660) = 1 ÷ 0.340 ≈ 2.94
Step 5: Calculate steady-state peak – C_ss = 0.625 × 2.94 ≈ 1.84 mg/L
Step 6: Calculate steady-state trough – C_trough = 1.84 × e^{-0.415} ≈ 1.22 mg/L
Result: At steady state, warfarin concentrations fluctuate between approximately 1.22 mg/L (just before a dose) and 1.84 mg/L (just after), with an average of ~1.53 mg/L. Therapeutic concentrations typically range 0.5–2.5 mg/L, so this regimen provides good coverage without toxicity. It takes approximately 5–7 half-lives (~200–280 hours or 8–12 days) to approach steady state.
Steady-State Achievement and Time to Therapeutic Concentration
The following table shows how many half-lives are required to achieve different percentages of steady-state concentration:
| Half-Lives Elapsed | % of Steady-State Achieved | Time (if t₁/₂ = 6 hours) | Clinical Implication |
|---|---|---|---|
| 1 | 50% | 6 hours | Partial therapeutic effect begins |
| 2 | 75% | 12 hours | Most therapeutic effect present |
| 3 | 87.5% | 18 hours | Good therapeutic coverage |
| 4 | 93.75% | 24 hours | Nearly steady state |
| 5 | 96.875% | 30 hours | Practical steady state |
| 7 | 99.2% | 42 hours | True steady state |
Drugs with short half-lives (like 6 hours) reach steady state quickly (within 1–2 days); drugs with long half-lives (like 40 hours for warfarin) take much longer (1–2 weeks). This is why certain medications (like digoxin) require "loading doses" to rapidly achieve therapeutic concentration, followed by lower maintenance doses to maintain steady state.
Loading Dose Strategy
A loading dose, often larger than subsequent maintenance doses, rapidly achieves the steady-state concentration immediately without waiting days or weeks. The loading dose is calculated as:
where the ratio is the accumulation factor. A loading dose immediately achieves what would take 5–7 half-lives to reach naturally, allowing medications with critical therapeutic windows (like digoxin for heart failure, warfarin for anticoagulation) to provide immediate benefit.
Using the Calculator
Enter the single dose in milligrams, the drug's half-life in hours, the dosing interval (typically 4, 6, 8, 12, or 24 hours), and the volume of distribution in liters (obtain from drug references or estimates). The calculator computes peak and trough concentrations after each successive dose, showing accumulation over time until steady state is approached. This helps predict when therapeutic concentrations are achieved and whether the regimen will be safe (avoiding toxic levels) and effective (maintaining therapeutic levels).
Limitations and Clinical Considerations
This calculator assumes simple first-order kinetics, linear pharmacokinetics, and that the volume of distribution is constant—assumptions that hold for many common drugs but not all. Some drugs follow nonlinear kinetics (e.g., phenytoin, aspirin at high doses), where half-life changes with concentration. Individual variation in metabolism (due to genetics, liver/kidney function, drug interactions, and comorbidities) means actual concentrations vary around predicted values. The calculator provides theoretical predictions; actual clinical drug levels should be measured when therapeutic drug monitoring is available and clinically indicated (e.g., digoxin, warfarin, phenytoin, lithium). Additionally, this calculator is educational; prescribing decisions must be made by qualified healthcare providers based on individual patient factors, not calculator results alone.