About this Tool

This Elimination Half-Life from Plasma Data calculator provides a streamlined method for determining key pharmacokinetic (PK) parameters from concentration-time data. By applying log-linear regression to the terminal elimination phase, it estimates how long it takes for a substance's concentration in the body to reduce by half, a critical factor in dosing regimen design and drug safety assessment.

What This Calculator Does

The primary function is to analyze paired time and plasma concentration data points. After a user enters this data and selects the points that represent the terminal (or elimination) phase of the drug's profile, the calculator performs the following actions:

• Log-Linear Regression: It transforms the selected concentration data into their natural logarithms (ln) and performs a linear regression against time.
• Calculates Key Parameters: From the regression, it derives the elimination rate constant (kₑₗ), the terminal half-life (t½), and the coefficient of determination (R²).
• Calculates AUC: It computes the Area Under the Curve from the first to the last data point (AUC₀₋t) using the linear trapezoidal method and extrapolates this to infinity (AUC₀₋∞).
• Visualizes Data: It generates a semi-logarithmic plot of concentration versus time, showing all data points, the selected points for regression, and the calculated regression line.

When to Use It

This calculation is fundamental in pharmacokinetics and is used in various settings:

• Early Phase Clinical Trials: To characterize the basic PK profile of a new drug entity in first-in-human studies.
• Bioequivalence Studies: To compare the PK profiles of a generic drug against a reference listed drug.
• Preclinical Research: To determine the PK parameters of a compound in animal models.
• Therapeutic Drug Monitoring: To understand drug clearance in specific patient populations or individuals.
• Academic Research: For teaching and research purposes in pharmacology and pharmaceutical sciences.

Inputs Explained

• Time Units: The unit of time for your measurements (e.g., hours, minutes). This must be consistent for all time points.
• Concentration Units: The unit of concentration for your plasma samples (e.g., ng/mL, µg/L). This must be consistent for all concentration points.
• Data Points (Table): This is where you input your raw data. Each row represents a single sample.
  • Time: The time point at which the sample was collected, post-administration.
  • Concentration: The measured drug concentration in the plasma at that time point. Must be a positive number.
  • Use in Regression: A checkbox to select which data points belong to the terminal elimination phase. You should only select points after the peak concentration (Cmax) that appear to fall on a straight line on a semi-log plot.

Results Explained

• t½ (Half-Life): The time required for the plasma concentration of a drug to decrease by 50% during the elimination phase. It is a primary indicator of how long a drug will remain in the body.
• kₑₗ (Elimination Rate Constant): Also known as λz, this is the slope of the terminal log-linear phase multiplied by -1. It represents the fraction of the drug eliminated per unit of time.
• R² (R-squared): The coefficient of determination. It indicates how well the selected data points fit the calculated regression line. A value closer to 1.0 (e.g., >0.95) suggests a strong linear relationship and high confidence in the calculated kₑₗ and t½.
• AUC₀₋t: The Area Under the concentration-time Curve from time zero to the last measured time point. It is calculated using the trapezoidal rule and represents the total drug exposure over that period.
• AUC₀₋∞: The AUC extrapolated to infinity. It represents the total drug exposure after a single dose. It is calculated by adding the extrapolated area (Last Concentration / kₑₗ) to the AUC₀₋t.

Formula / Method

The calculator uses standard, well-established pharmacokinetic formulas:

1. Log Transformation: For each selected point (t, C), the concentration is transformed: ln(C).
2. Linear Regression: A best-fit line is calculated for the transformed points using the equation: ln(C) = -kₑₗ * t + ln(C₀).
   • The slope of this line is equal to -kₑₗ.
3. Elimination Rate Constant (kₑₗ): kₑₗ = -1 * slope
4. Half-Life (t½): t½ = ln(2) / kₑₗ ≈ 0.693 / kₑₗ
5. AUC (Trapezoidal Rule): AUC between two points (t₁, C₁) and (t₂, C₂) is calculated as (C₁ + C₂) / 2 * (t₂ - t₁). The AUC₀₋t is the sum of these areas across all consecutive data points.

Step-by-Step Example

Imagine you have collected the following plasma concentration data for a drug administered intravenously.

1. Enter Data: Paste or manually enter the time and concentration pairs into the data table.
2. Set Units: Ensure Time Units are set to "hours" and Concentration Units are set to "ng/mL".
3. Select Terminal Phase: Observe the data. The concentration is consistently decreasing after the 2-hour mark. The points from 4 to 24 hours appear to represent the terminal elimination phase. Tick the "Use in Regression" checkbox for the time points 4, 8, 12, and 24 hours.
4. Review Results: The calculator will automatically compute the results. You might see something like:
   • t½: ~7.5 hours
   • kₑₗ: ~0.092 1/hours
   • R²: >0.99 (indicating an excellent fit)
   • AUC₀₋t: ~1150 ng*h/mL
5. Analyze Graph: The semi-log plot will show all points, with the last four highlighted and a straight line running through them, confirming a good fit for the elimination phase.

Tips + Common Errors

• Select Enough Points: Always select at least 3 points for the regression to be statistically meaningful. More points are generally better if they all belong to the terminal phase.
• Identify the Correct Phase: Do not include points from the absorption or distribution phase in the regression. This is the most common error and will lead to an incorrect (usually shorter) half-life calculation. The terminal phase is the final, log-linear decline in concentration.
• Check for Positive Values: Concentration values must be greater than zero, as the natural logarithm of zero or a negative number is undefined.
• Consistent Units: Ensure all time points share the same unit, and all concentration points share the same unit. Mixing units will produce incorrect results.
• Poor R² Value: If your R² value is low (<0.90), it may indicate that the selected points do not form a straight line, the data has high variability, or the drug follows a more complex multi-compartment model. Re-evaluate which points constitute the terminal phase.

Frequently Asked Questions (FAQs)

1. Why is the graph on a semi-logarithmic scale?
   First-order elimination processes appear as a straight line when the concentration axis is logarithmic and the time axis is linear. This visual representation makes it easy to identify the terminal elimination phase.
2. What does a "terminal" half-life mean?
   It refers to the half-life calculated from the final, slowest rate of elimination. Some drugs have multiple phases of distribution and elimination, but the terminal half-life governs the ultimate time to clearance.
3. Can I use this calculator for oral (PO) administration data?
   Yes, but you must be careful to only select data points well after Cmax (the time of maximum concentration) that clearly represent the elimination phase, not the absorption phase.
4. Why do I need at least 3 data points for regression?
   With only two points, a straight line can always be drawn perfectly between them (R²=1.0), which provides no information about the goodness of fit. Three or more points are needed to assess linearity and calculate a meaningful R² value.
5. What if my drug follows a two-compartment model?
   This calculator determines the terminal half-life (β-phase half-life). It does not resolve the faster distribution phase (α-phase). For a full two-compartment analysis, more advanced software is required.
6. How is AUC extrapolated to infinity?
   It's calculated by assuming the drug concentration continues to decrease according to the calculated kₑₗ. The formula AUC(extrapolated) = C_last / kₑₗ gives the area from the last measured point to infinity.
7. What does a positive slope error mean?
   This error occurs if the selected concentration data is increasing over time. The elimination phase must, by definition, show decreasing concentration. Check your data entry and point selection.
8. Does the starting time point have to be zero?
   No. The calculator computes AUC using the trapezoidal rule between the points you provide. However, for a correct AUC₀₋t, your first data point should be as close to time zero as possible.
9. How accurate are the results?
   The accuracy depends entirely on the quality and richness of the input data. More frequent sampling, especially during the terminal phase, and low analytical variability will yield more accurate results.

References

• Gibaldi, M., & Perrier, D. (2007). Pharmacokinetics (2nd ed.). Informa Healthcare.
• Birkett, D. J. (2002). Pharmacokinetics Made Easy. McGraw-Hill Australia.
• Toutain, P. L., & Bousquet-Mélou, A. (2004). Plasma terminal half-life. Journal of veterinary pharmacology and therapeutics, 27(6), 427-439.
• U.S. Food and Drug Administration. (2022). Guidance for Industry: Bioavailability and Bioequivalence Studies Submitted in NDAs or INDs — General Considerations.