This calculator helps pharmacokinetics students, researchers, and clinicians calculate the Elimination Rate Constant (ke) and the corresponding Biological Half-life (t1/2). By inputting two drug concentration measurements taken at different times during the elimination phase, this tool computes the slope of the elimination curve.
Calculate ke from Two Points
mg/L or µg/mL
Hours (post-dose)
mg/L or µg/mL
Hours (post-dose)
Elimination Rate Constant (ke):—
Half-Life (t1/2):—
Time Interval (Δt):—
Interpretation: Approximately —% of the drug is eliminated per hour.
function calculateElimination() {
var c1 = document.getElementById('conc1').value;
var t1 = document.getElementById('time1').value;
var c2 = document.getElementById('conc2').value;
var t2 = document.getElementById('time2').value;
var errorDiv = document.getElementById('error-message');
var resultDiv = document.getElementById('result-box');
// Clear previous results/errors
errorDiv.style.display = 'none';
errorDiv.innerHTML = ";
resultDiv.style.display = 'none';
// Validation
if (c1 === " || t1 === " || c2 === " || t2 === ") {
errorDiv.innerHTML = 'Please fill in all four fields.';
errorDiv.style.display = 'block';
return;
}
c1 = parseFloat(c1);
t1 = parseFloat(t1);
c2 = parseFloat(c2);
t2 = parseFloat(t2);
if (c1 <= 0 || c2 <= 0) {
errorDiv.innerHTML = 'Concentrations must be greater than zero.';
errorDiv.style.display = 'block';
return;
}
if (t2 = c1) {
errorDiv.innerHTML = 'Note: C2 should be less than C1 for elimination. (Unless calculating accumulation phase)';
// We proceed but warn, or just stop. For pure elimination calc, C2 < C1.
// Let's allow it but the result will be negative, implying absorption.
}
// Calculation: ke = (ln(C1) – ln(C2)) / (t2 – t1)
// Using natural log (Math.log)
var deltaT = t2 – t1;
var logC1 = Math.log(c1);
var logC2 = Math.log(c2);
var ke = (logC1 – logC2) / deltaT;
// Half Life: t1/2 = 0.693 / ke
var halfLife = 0.693 / ke;
// Percent loss per hour = (1 – e^-ke) * 100
var percentLoss = (1 – Math.exp(-ke)) * 100;
// Display Logic
if (ke < 0) {
document.getElementById('result-ke').innerHTML = ke.toFixed(4) + " h-1 (Absorption/Accumulation)";
document.getElementById('result-thalf').innerHTML = "N/A";
document.getElementById('percent-loss').innerHTML = "0";
} else {
document.getElementById('result-ke').innerHTML = ke.toFixed(4) + " h-1";
document.getElementById('result-thalf').innerHTML = halfLife.toFixed(2) + " hours";
document.getElementById('result-dt').innerHTML = deltaT.toFixed(2) + " hours";
document.getElementById('percent-loss').innerHTML = percentLoss.toFixed(2);
}
resultDiv.style.display = 'block';
}
Understanding the Elimination Rate Constant (ke)
The elimination rate constant, denoted as ke or kel, is a fundamental parameter in pharmacokinetics. It describes the fraction of a drug that is removed from the body per unit of time. Unlike the clearance rate (which is a volume term), ke is a fractional rate expressed in inverse time units (e.g., h-1 or min-1).
The Calculation Formulas
Assuming first-order kinetics (which applies to the vast majority of drugs at therapeutic doses), the concentration of a drug decreases exponentially over time. We can calculate ke using two plasma concentration measurements:
ke = [ ln(C1) – ln(C2) ] / ( t2 – t1 )
Where:
ln = Natural logarithm
C1 = Concentration at time t1
C2 = Concentration at time t2
Relationship with Half-Life
Once the elimination rate constant is known, the biological half-life (t1/2) can be easily derived. The half-life is the time required for the drug concentration to reduce by 50%.
t1/2 = 0.693 / ke
Real-World Calculation Example
Consider a patient administered an IV antibiotic. We draw blood at two different times to determine how fast the kidneys are clearing the drug.
Measurement 1: At 2 hours post-dose, plasma concentration is 35 mg/L.
Measurement 2: At 8 hours post-dose, plasma concentration is 10 mg/L.
This means that roughly 20.8% of the remaining drug is eliminated every hour, and the concentration drops by half every 3.32 hours.
Frequently Asked Questions
Why is the natural logarithm (ln) used instead of log base 10?
Pharmacokinetic elimination typically follows an exponential decay model ($C = C_0 \cdot e^{-kt}$). To linearize this equation for easy calculation of the slope (which represents the rate constant), we take the natural logarithm of both sides. While log base 10 can be used, the constant 0.693 in the half-life equation would change to 0.301, and the math becomes slightly more complex.
What are the units for the elimination rate constant?
The unit is always inverse time. Common examples include:
Per hour (h-1)
Per minute (min-1)
Per day (day-1)
Does ke change with dose?
In first-order kinetics, ke is independent of the dose. Whether you give 100mg or 1000mg, the fraction eliminated per hour remains the same. However, in zero-order kinetics (saturation kinetics, e.g., Ethanol or high-dose Phenytoin), the rate of elimination is constant in amount (mg/h), not fraction, meaning ke is not applicable in the same way.
What factors affect the elimination rate constant?
Physiological factors that impact clearance ($Cl$) and volume of distribution ($V_d$) will affect ke, as $k_e = Cl / V_d$. These factors include:
Renal Function: Reduced kidney function decreases ke.
Hepatic Function: Liver disease can reduce metabolic elimination.
Age: Elderly patients often have reduced clearance.
Drug Interactions: Inducers or inhibitors of enzymes can alter metabolism speed.