Medicine & HealthcarePharmacologyA-Level

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Drug Half-Life

Calculate drug concentration after multiple half-lives.

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C_{t} = C_{0} \times (\frac{1}{2})^{n}

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Core idea

Overview

The drug half-life equation models the exponential decay of a pharmaceutical substance's concentration in the bloodstream over time. It specifically describes first-order kinetics, where a constant percentage of the drug is eliminated per unit of time rather than a constant amount.

When to use: This formula is applied when a drug follows first-order elimination kinetics, which applies to the vast majority of medications at therapeutic doses. It is used to estimate the remaining amount of a drug in the body or to determine how many half-life intervals have passed since administration.

Why it matters: Calculating half-life is essential for clinicians to establish safe dosing schedules and avoid drug toxicity. It allows medical professionals to predict when a drug will reach a steady state or when it will be effectively cleared from a patient's system for surgery or drug testing.

Symbols

Variables

C_t = Final Concentration, C_0 = Initial Concentration, n = Number of Half-Lives

C_{t}

Final Concentration

m g

C_{0}

Initial Concentration

m g

n

Number of Half-Lives

V a r iab l e

Walkthrough

Derivation

Derivation of Drug Half-Life (First-Order Kinetics)

The half-life is the time required for plasma drug concentration to fall to half its initial value under first-order elimination.

First-order elimination: rate of removal is proportional to concentration.
Single-compartment model with rapid mixing (plasma concentration represents the compartment).
No additional dosing during the elimination phase.
Elimination rate constant k is constant over the concentration range considered.

Start with the first-order elimination solution:

For first-order kinetics, concentration decays exponentially with elimination rate constant k.

C (t) = C_{0} e^{- k t}

Apply the half-life definition:

At the half-life $t_{1/2}$ , the concentration has fallen to half the initial value.

C (t_{1/2}) = \frac{C _{0}}{2} = C_{0} e^{- k t_{1/2}}

Cancel C_0 and take natural logs:

Dividing by $C_{0}$ removes dependence on the starting dose; logging removes the exponential.

\frac{1}{2} = e^{- k t_{1/2}} \Rightarrow ln (\frac{1}{2}) = - k t_{1/2}

Rearrange for t_{1/2}:

Because $ln$ (1/2)=- $ln$ 2, the negatives cancel, giving the standard half-life expression.

t_{1/2} = \frac{ln 2}{k} \approx \frac{0.693}{k}

Note: If elimination becomes saturated (zero-order), half-life is no longer constant and this formula does not apply.

Result

t_{1/2} = \frac{ln 2}{k} \approx \frac{0.693}{k}

Source: Standard curriculum — A-Level Biology/Medicine (Pharmacokinetics)

Free formulas

Rearrangements

Solve for $C_{t}$

Make Ct the subject

C_{t} = C_{0} \times (\frac{1}{2})^{n}

Ct is already the subject of the formula.

Difficulty: 1/5

Solve for $C_{0}$

Make C0 the subject

C_{0} = C_{t} \times 2^{n}

Start from the Drug Half-Life formula. To make C0 the subject, multiply both sides by 2^n.

Difficulty: 2/5

Solve for $n$

Drug Half-Life: Solve for Number of Half-Lives

n = \frac{ln ( C _{0} / C _{t} )}{ln 2}

Start from the Drug Half-Life formula. To find the number of half-lives (`n`), first isolate the exponential term, then take natural logarithms of both sides, and finally solve for `n`.

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph displays an exponential decay curve where C_t approaches zero as n increases, creating a horizontal asymptote at the x-axis. For a healthcare student, this shape illustrates that while drug concentration drops rapidly at low n values, it persists at trace levels for much longer as n increases. The most critical feature is that the curve never reaches zero, meaning that even after many half-lives, a theoretical trace of the drug remains in the system.

Graph type: exponential

Why it behaves this way

Intuition

Imagine a smooth curve starting at a high initial drug concentration and steadily dropping by half over equal time intervals, gradually flattening out as the drug becomes nearly cleared from the system.

C_{t}

The concentration of the drug in the body at a specific time 't'

This is how much drug is still active and measurable in the patient's system at a given moment.

C_{0}

The initial concentration of the drug in the body, typically at the start of the observation period or after full absorption

This is the starting amount of drug from which the decay process begins.

1/2

The fraction of the drug concentration remaining after one half-life period

This term directly represents the core concept of 'halving' that defines a drug's half-life.

The number of half-life intervals that have elapsed since the initial concentration C_0

This value tells us how many times the drug's concentration has been reduced by half.

Signs and relationships

(1/2)^n: This entire term signifies the exponential decay. The base '1/2' means that for every increment of 'n' (each passing half-life), the remaining concentration is multiplied by one-half, leading to a progressively smaller

Free study cues

Insight

Canonical usage

This equation requires consistent units for initial and final concentrations, while the exponent term is dimensionless.

Common confusion

A common mistake is using different units for $C_{t}$ and $C_{0}$ , or misinterpreting 'n' as a time value rather than a dimensionless count of half-lives.

Dimension note

The term (1/2)^n is dimensionless because 'n' represents a count of half-life periods, ensuring that the units of $C_{t}$ and $C_{0}$ are consistent.

Unit systems

$C_{t}$ mass/volume (e.g., mg/L, μg/mL) or moles/volume (e.g., mol/L) · Concentration of the drug at time t. Must be in the same units as C_0.

$C_{0}$ mass/volume (e.g., mg/L, μg/mL) or moles/volume (e.g., mol/L) · Initial concentration of the drug. Must be in the same units as C_t.

$n$ dimensionless · The number of half-lives that have passed. This is a ratio of total elapsed time to the drug's half-life (t / t_1/2).

Ballpark figures

Quantity:

One free problem

Practice Problem

A patient is administered a 400 mg dose of a medication. If the drug follows first-order kinetics, calculate the amount of drug remaining in the patient's system after exactly 3 half-lives have elapsed.

Initial Concentration400 mg

Number of Half-Lives3

Solve for: $C t$

Hint: Use the formula Ct = C0 × (1/2)ⁿ where n is the number of half-lives.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Warfarin has 36-42hr half-life, takes days to clear.

Study smarter

Tips

Ensure the initial concentration (C0) and final concentration (Ct) use the same units of measurement.
The variable n represents the total time elapsed divided by the drug's specific half-life duration.
Most drugs are considered clinically eliminated after approximately 4 to 5 half-lives.
If solving for n manually, use logarithms to isolate the exponent if n is not a whole number.

Avoid these traps

Common Mistakes

Confusing half-life with duration of action.
Not considering accumulation.

Common questions

Frequently Asked Questions

The half-life is the time required for plasma drug concentration to fall to half its initial value under first-order elimination.

This formula is applied when a drug follows first-order elimination kinetics, which applies to the vast majority of medications at therapeutic doses. It is used to estimate the remaining amount of a drug in the body or to determine how many half-life intervals have passed since administration.

Calculating half-life is essential for clinicians to establish safe dosing schedules and avoid drug toxicity. It allows medical professionals to predict when a drug will reach a steady state or when it will be effectively cleared from a patient's system for surgery or drug testing.

Confusing half-life with duration of action. Not considering accumulation.