Lecture 3 - Mono and Multi-Exponential Pharmacokinetic Models
Week Two: Mono and Multi-Exponential Pharmacokinetic Models *NOTE : Equations in slides
Review of Previous Lecture
Discussion of foundations of pharmacokinetics.
Key concepts covered:
ADME (Absorption, Distribution, Metabolism, Excretion)
Clearance
Half-life
Importance of pharmacokinetics in clinical decision-making.
Learning Objectives for Today
Understand the relationship between drug concentration and time.
Distinguish mono-exponential from multi-exponential decline in pharmacokinetics.
Define and calculate various pharmacokinetic parameters:
Clearance
Volume of distribution
Elimination rate constant
Area under curve (AUC)
Half-life
Interpret pharmacokinetic data to predict drug behavior.
Definition and Importance of Pharmacokinetic Models
Pharmacokinetic Models:
Mathematical representations of how drug concentrations change over time.
Important due to complex body dynamics; simplifies drug behavior into compartments (not real anatomical spaces).
Mono-Exponential Pharmacokinetics
Mono-Exponential Model:
Assumes the body as a single compartment, with:
First-order kinetics (elimination rate proportional to drug concentration).
Instantaneous absorption (immediate availability in systemic circulation).
Linear pharmacokinetics (parameters constant with dose/time).
Elimination occurs from the central compartment only; no saturation.
Graphical Representation:
Initial concentration decreases exponentially over time.
Requires plot on a logarithmic scale for analysis (results in a straight line).
Key equation:
Ct=C0e−kt
where:
C_t = concentration at time t
C_0 = initial concentration
k = elimination rate constant
t = time.
After applying logarithm,
ext{log}(C) = ext{log}(C_0) - rac{k}{2.303} t
where the slope equals -k.
Pharmacokinetic Parameters
Elimination Rate Constant (k)
Represents fraction of drug eliminated per unit time (units: time⁻¹).
Determines how quickly a drug is eliminated:
High k = rapid elimination
Low k = slow elimination.
Calculation Methods:
Log-linear plot (slope method) to determine k directly from the slope.
Two-point method (using two concentrations):
k = rac{ ext{ln}(C1) - ext{ln}(C2)}{T2 - T1}
Example with values: C1 = 18 ext{ mg/L}, C2 = 6 ext{ mg/L}, T1 = 3 ext{ hours}, T2 = 8 ext{ hours}.
Half-life method:
k = rac{0.693}{T_{1/2}}
where T_{1/2} is the half-life.
Clearance and volume of distribution method using density parameters.
Half-life (T_{1/2})
Time required for the concentration to decrease by 50%.
In the mono-exponential model, half-life is constant.
Formula:
T_{1/2} = rac{0.693}{k} .
Important rule:
Drug reaches steady state in 5-7 half-lives.
Time for washout is also 5-7 half-lives.
Volume of Distribution (VD)
Apparent volume into which the drug distributes; indicates degree of drug affinity.
Units: ext{mL} or ext{L/kg}.
High VD suggests widespread tissue distribution; low VD indicates confinement in plasma.
Formula:
V_D = rac{ ext{Dose}}{ ext{Concentration}}.
Determines loading dose:
ext{Loading Dose} = VD imes CT.
Clearance (Cl)
Volume of plasma cleared of drug per unit time; describes efficiency of elimination.
Units: ext{L/hour}.
Critical for determining maintenance dose:
ext{Maintenance Dose} = ext{Clearance} imes C_{ss} (steady state concentration).
Calculation using different equations for situational clarity:
Cl = k imes V_D,
Cl = rac{ ext{Dose}}{ ext{AUC}}.
Area Under Curve (AUC)
Reflects total drug exposure over time when plasma concentration is plotted against time.
High AUC indicates greater bioavailability.
Calculation Methods:
Analytical formulas for IV bolus:
AUC = rac{C_0}{k}
Trapezoidal method for graphical data:
Divide into trapezoids for calculation.
Total the areas for comprehensive AUC.
Recap on Mono-Exponential Model
Summary:
Assumes single compartment; characterized by first-order elimination and single half-life.
Primary utility in rapidly distributing drugs.
Introduction to Multi-Exponential Models
Multi-Exponential Pharmacokinetics:
Used when drug eliminates according to more than one exponential decline (i.e., non-homogeneous distribution).
Model subdivided into:
Distribution phase (alpha phase) - rapid drop in concentration as drug distributes.
Elimination phase (beta phase) - slower decline as drug is eliminated.
Two Compartment Model:
Assumes central compartment (blood) and peripheral compartments (tissues).
Drug moves rapidly from the central to peripheral compartments before reaching equilibrium.
Utilizes two distinct rate constants:
Alpha: Distribution rate constant.
Beta: Elimination rate constant.
Equations:
C_t = A e^{- ext{alpha} t} + B e^{- ext{beta} t} .
Key Parameters in multi-compartment modeling:
Alpha: Rapid distribution; determined by K{12} and K{21}.
Beta: Terminal elimination; clinically relevant for determining half-life.
Mathematical Concepts and Methods in Multi-Exponential Models
Method of Residuals:
Used to separate distribution and elimination curves for clearer interpretation.
Allows calculation of distribution parameters.
Conclusion
The understanding of pharmacokinetic models facilitates informed decisions in clinical settings regarding dosages, potential toxicity risks, and adjustments based on physiological changes or disease states.
Mono-exponential models provide a framework for understanding rapidly distributing drugs while multi-exponential models serve for those that exhibit more complex behaviors.
Next Lecture
Focus on IV infusion and steady state.
Prepare questions for Q&A sessions following today's lecture!