The Science of Half-Life: Pharmacokinetics Explained
Understanding the mathematical and biological principles behind drug elimination, steady-state concentrations, and clinical pharmacokinetics.
In This Guide
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
Parameter | First-Order | Zero-Order |
---|---|---|
Rate of elimination | Proportional to concentration | Constant amount per time |
Half-life | Constant | Increases as dose decreases |
Graph (linear plot) | Curved (exponential) | Straight line |
Graph (semi-log plot) | Straight line | Curved |
Examples | Most 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:
Elimination Time Calculations
To calculate when a specific percentage of drug is eliminated:
t = -log₂(remaining fraction) × t½
% Eliminated | Fraction Remaining | Number of Half-Lives | Formula |
---|---|---|---|
50% | 0.50 | 1 | 1 × t½ |
75% | 0.25 | 2 | 2 × t½ |
90% | 0.10 | 3.32 | 3.32 × t½ |
95% | 0.05 | 4.32 | 4.32 × t½ |
99% | 0.01 | 6.64 | 6.64 × t½ |
99.9% | 0.001 | 9.97 | 9.97 × 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.