The Science of Half-Life: Pharmacokinetics Explained

Understanding the mathematical and biological principles behind drug elimination, steady-state concentrations, and clinical pharmacokinetics.

Basic Concepts & Definitions

What is Half-Life?

Half-life (t½) is the time required for the concentration of a substance in the body to decrease by 50%. It's a fundamental parameter in pharmacokinetics that determines:

  • How long a drug remains active in the body
  • Optimal dosing intervals
  • Time to reach steady-state
  • Duration needed for complete elimination

Core Pharmacokinetic Parameters

Volume of Distribution (Vd)

The theoretical volume needed to contain the total amount of drug at the same concentration as in the blood. Lipophilic drugs have larger Vd as they distribute into tissues.

Vd = Dose / C₀

Clearance (CL)

The volume of plasma from which a drug is completely removed per unit time. Combines renal, hepatic, and other elimination pathways.

CL = k × Vd = 0.693 × Vd / t½

Bioavailability (F)

The fraction of administered dose that reaches systemic circulation. IV drugs have F=1, while oral drugs typically have F<1 due to first-pass metabolism.

F = AUC(oral) / AUC(IV)

Elimination Rate Constant (k)

The fraction of drug eliminated per unit time. Directly related to half-life and determines the slope of elimination curve.

k = 0.693 / t½

First-Order Elimination Kinetics

The Exponential Decay Model

Most drugs follow first-order kinetics, where the rate of elimination is proportional to the drug concentration. This creates an exponential decay pattern:

C(t) = C₀ × e^(-kt)

or equivalently: C(t) = C₀ × (1/2)^(t/t½)

Characteristics:

  • • Constant fraction eliminated per unit time
  • • Half-life independent of dose
  • • Linear on semi-log plot
  • • Clearance remains constant

Clinical Implications:

  • • Predictable elimination pattern
  • • Dose proportionality
  • • Simple dose adjustments
  • • Standard dosing intervals

Zero-Order vs First-Order Kinetics

ParameterFirst-OrderZero-Order
Rate of eliminationProportional to concentrationConstant amount per time
Half-lifeConstantIncreases as dose decreases
Graph (linear plot)Curved (exponential)Straight line
Graph (semi-log plot)Straight lineCurved
ExamplesMost drugs (>95%)Alcohol, phenytoin (at high doses)

Steady-State Concentrations

Understanding Steady State

Steady state occurs when the rate of drug input equals the rate of drug elimination. At this point, plasma concentrations fluctuate predictably between doses but maintain consistent peak and trough levels.

Average Steady-State Concentration:

CSS,avg = (F × Dose) / (CL × τ)

Where τ = dosing interval

Peak Concentration (CSS,max):

CSS,max = Dose / (Vd × (1 - e^(-kτ)))

Trough Concentration (CSS,min):

CSS,min = CSS,max × e^(-kτ)

Accumulation Factor

The accumulation factor (R) predicts how much drug builds up with repeated dosing:

R = 1 / (1 - e^(-kτ))

τ = t½

R = 2

2x accumulation

τ = 2×t½

R = 1.33

33% accumulation

τ = 3×t½

R = 1.14

14% accumulation

Mathematical Models & Calculations

One-Compartment Model

The simplest pharmacokinetic model assumes the body is a single, well-mixed compartment:

IV Bolus Administration

C(t) = (Dose/Vd) × e^(-kt)

Oral Administration

C(t) = (F×Dose×ka)/(Vd×(ka-k)) × (e^(-kt) - e^(-ka×t))

Where ka = absorption rate constant

IV Infusion

C(t) = (R₀/CL) × (1 - e^(-kt))

Where R₀ = infusion rate

Loading Dose Calculations

A loading dose rapidly achieves therapeutic concentrations without waiting for steady state:

Loading Dose = CSS,target × Vd / F

Clinical Examples:

Digoxin (Vd = 7 L/kg, target = 1.5 ng/mL)LD = 10.5 mcg/kg
Phenytoin (Vd = 0.7 L/kg, target = 15 mg/L)LD = 15-20 mg/kg
Vancomycin (Vd = 0.7 L/kg, target = 20 mg/L)LD = 25-30 mg/kg

Elimination Time Calculations

To calculate when a specific percentage of drug is eliminated:

t = -log₂(remaining fraction) × t½

% EliminatedFraction RemainingNumber of Half-LivesFormula
50%0.5011 × t½
75%0.2522 × t½
90%0.103.323.32 × t½
95%0.054.324.32 × t½
99%0.016.646.64 × t½
99.9%0.0019.979.97 × t½

Clinical Applications

Therapeutic Drug Monitoring (TDM)

Drugs with narrow therapeutic windows require monitoring to maintain safe and effective levels:

Common TDM Drugs:

  • • Aminoglycosides (gentamicin, tobramycin)
  • • Antiepileptics (phenytoin, valproic acid)
  • • Cardiac glycosides (digoxin)
  • • Immunosuppressants (tacrolimus, cyclosporine)
  • • Lithium
  • • Vancomycin

Monitoring Principles:

  • • Draw levels at steady state (5 half-lives)
  • • Time samples appropriately (peak/trough)
  • • Consider protein binding changes
  • • Account for drug interactions
  • • Adjust for organ dysfunction
  • • Monitor clinical response

Dosing Interval Optimization

Choosing the optimal dosing interval balances efficacy, toxicity, and convenience:

τ < t½

Continuous effect

Minimal fluctuation, higher accumulation, used for drugs requiring constant levels

τ = t½

Balanced approach

50% fluctuation, 2× accumulation, good efficacy-safety balance

τ > 2×t½

Intermittent dosing

Large fluctuation, minimal accumulation, for concentration-dependent killing

Dose Adjustments in Organ Dysfunction

Renal Impairment:

Adjusted Dose = Normal Dose × (Patient CrCl / Normal CrCl)

Alternative: Extend interval while maintaining dose

Hepatic Impairment:

More complex due to multiple factors:

  • Reduced drug metabolism (↑ t½)
  • Decreased protein binding (↑ free drug)
  • Altered volume of distribution
  • Portosystemic shunting

Child-Pugh Score guides dose reductions: Class B (25-50% reduction), Class C (>50% reduction)

Special Populations

Pediatric Considerations

Neonates (0-1 month):

  • • Immature hepatic enzymes (↑ t½)
  • • Reduced renal function
  • • Higher body water content
  • • Lower protein binding
  • • Immature blood-brain barrier

Children (1 month - 12 years):

  • • Faster metabolism (↓ t½)
  • • Higher clearance per kg
  • • Different Vd ratios
  • • Rapid developmental changes
  • • Weight-based dosing essential

Geriatric Considerations

Age-related changes significantly affect drug pharmacokinetics:

Decreased Clearance

Reduced hepatic blood flow, enzyme activity, and renal function increase half-life

Altered Distribution

Increased fat, decreased muscle and water affect lipophilic and hydrophilic drugs differently

Increased Sensitivity

Enhanced pharmacodynamic response, especially to CNS-active drugs

Dosing Strategy:

"Start low and go slow" - Begin with 50-75% of adult dose

Pregnancy & Lactation

Pregnancy Changes:

  • • ↑ Cardiac output (↑ hepatic flow)
  • • ↑ GFR (↑ renal clearance)
  • • ↑ Volume of distribution
  • • ↓ Protein binding
  • • Altered GI absorption
  • • Placental metabolism
  • • Fetal drug exposure
  • • Changed enzyme activity

Lactation Considerations:

Drug transfer to breast milk depends on:

Milk/Plasma Ratio = f(MW, pKa, lipophilicity, protein binding)

Lower MW, higher lipophilicity, and lower protein binding increase transfer

Advanced Topics

Nonlinear Pharmacokinetics

Some drugs exhibit saturable (Michaelis-Menten) kinetics at therapeutic doses:

dC/dt = -(Vmax × C) / (Km + C)

Where Vmax = maximum elimination rate, Km = Michaelis constant

Clinical Examples:

Phenytoin

Small dose increases cause disproportionate concentration rises near saturation

Alcohol

Zero-order kinetics at typical consumption levels (~7-10 g/hour elimination)

Aspirin

Transitions from first-order to zero-order at anti-inflammatory doses

Population Pharmacokinetics

Modern approaches use statistical models to understand variability in drug response:

Sources of Variability:

  • • Inter-individual variability (genetics, demographics)
  • • Intra-individual variability (circadian rhythms, compliance)
  • • Residual unexplained variability

Covariate Effects:

CL = θCL × (Weight/70)^0.75 × (CrCl/100) × e^η

Where θ = population parameter, η = random effect

Practical Tools & Resources

Quick Reference Formulas

Half-life from k:

t½ = 0.693 / k

Time to steady state:

tss = 5 × t½

Loading dose:

LD = Css × Vd / F

Clearance:

CL = k × Vd

Maintenance dose:

MD = Css × CL × τ / F

Accumulation factor:

R = 1 / (1 - e^(-k×τ))

Clinical Pearls

Apply These Concepts

Use our calculator to see these principles in action

Now that you understand the science behind half-life calculations, try our interactive calculator to visualize drug concentrations, steady-state achievement, and elimination patterns.