Week 5

• Compartmental analysis


  A mathematical model is developed to describe the shape of a drug's pharmacokinetic profile
• Compartmental analysis


  • Assumes there is some kind of underlying physiological process that underpins the pharmacokinetics of the drug
  • Involves the development of a mathematical model to describe the concentration-time curve
  • The body is divided into a series of linked homogenous compartments that represent the disposition of the drug
  • Predictive as well as descriptive
• Compartmental analysis


  Mass balance - what goes in is what comes out
• Convolution


  The concentration-time course of a drug can be considered to comprise of input and disposition phases
• Convolution


  1. Differentiation of input into disintegration, dissolution and absorption

  2. Disposition = distribution and elimination

  3. Combining different input and disposition models to create the best complete model that characterizes the shape of the concentration-time curve
• Typically, about 10 input and 9 disposition models describe the usual range of PK behaviours for most drugs, providing approximately 90 PK "input-output" models
• Zero-order absorption


  Rate of drug absorption is constant and independent of amount of drug remaining to be absorbed from the GIT
• First-order absorption


  Rate of drug absorption is dependent on the amount of drug remaining to be absorbed from the GIT
• Input models


  • Zero-order absorption
  • First-order absorption
  • Zero-order and first-order input + lag time
  • Zero-order and first-order
  • Double first-order and double first-order with lag
• First order elimination (iv bolus)


  Rate of drug elimination is proportional to the concentration of drug in the plasma
• Zero-order elimination (iv bolus)


  Amount of drug eliminated per unit time remains constant, no matter what the drug concentration
• Disposition models


  • First order elimination (iv bolus)
  • Zero-order elimination (iv bolus)
  • One compartment distribution (first order elimination)
  • Two compartment distribution (first order elimination)
  • Three compartmental distribution (first order elimination)
• Drawing a pharmacokinetic model


  1. Representing the body as a single compartment

  2. Representing the body as the gut and one body compartment

  3. Representing the body as the gut and two body compartments
• Mass balance


  The amount of drug in a particular compartment at a particular time (t) may be calculated by the difference given by the rate of drug entering the compartment and the rate of drug leaving the compartment
• Ordinary differential equation (ODE)


  The basis for defining a pharmacokinetic model, representing the rate of change in the amount of drug in a compartment
• First-order reaction


  The rate of loss from the compartment is proportional to the current value of the amount of drug
• Zero-order reaction


  The amount of drug eliminated per unit time remains constant, no matter what the drug concentration
• Developing pharmacokinetic models


  1. Conceptualise the model as a series of linked boxes

  2. Define the rate processes that link the boxes to write the ODEs

  3. Transform the ODEs into an explicit equation or leave them as ODEs
• Non-linear pharmacokinetics


  When the dose of a drug is increased, the concentration at steady-state (Css) will not increase proportionally
• Non-linear pharmacokinetic behaviours


  • The plasma concentration changes either more or less than would be expected from a change in the dose rate
  • Can cause problems when adjusting drug doses, especially when a drug has a narrow safety margin
• Linear or first-order pharmacokinetics


  The drug concentration achieved is proportional to the dose given at all times
• Non-linear pharmacokinetic behaviour


  The pharmacokinetic response (drug exposure) changes disproportionally with dose
• Causes of non-linear pharmacokinetics


  • Saturation of drug absorption, distribution, metabolism and/or excretion processes
  • Limitation in the ability of biology to process drug movement in the body
• Saturable drug absorption


  Some drugs require energy dependent transporters to carry otherwise "unabsorbable" drugs from the gastrointestinal tract to the portal circulation
• Saturable drug dissolution


  As the dose of a poorly water soluble drug increases, relatively less of the dose can dissolve sufficiently to be absorbed
• Saturable first-pass metabolism


  Drugs which are metabolised by a first-pass effect may display nonlinear absorption as the enzymes responsible become saturated leading to higher systemic bioavailability
• Drug binding to body tissues does not appear to show any non-linearity
• Saturable drug distribution


  Occurs almost exclusively with drugs that have a high-affinity for plasma proteins and bind to low-capacity proteins (limited binding sites per mol)
• Saturable drug excretion


  Tubular secretion involves movement of a drug from the blood into the proximal renal tubule via transporter proteins, which is saturable
• Saturable drug metabolism


  The most important aspect of non-linearity in therapeutics, reflected in saturation of enzyme mediated processes, especially metabolism (capacity-limited metabolism of specific liver enzymes)
• Michaelis-Menten model


  Describes the interaction between a drug (D) and an enzyme (E) to form a metabolite (M), where the availability of E is limited with respect to the amount of D to be metabolised
• Michaelis-Menten equation


  Specifies the rate (v) of an enzyme reaction as a function of the substrate concentration (C), where Vmax is the maximum possible reaction rate and Km is the value of C at which half the Vmax is achieved
• Michaelis-Menten pharmacokinetics


  As the concentration of a drug in blood increases, the rate of metabolism increases until it approaches an asymptote (plateau) at which point the enzyme is saturated
• Enzyme induction


  Leads to a larger amount of enzyme and, therefore larger Vmax (enzyme induction does not affect the basic enzymatic activity, thus Km does not change)
• Competitive inhibition


  Increases Km because the activity is lessened (Vmax is unchanged because at high C the maximum metabolic rate is still achieved)
• Phenytoin pharmacokinetics


  • Phenytoin disposition can be described by the Michaelis-Menten equation
  • Phenytoin exhibits marked saturation of metabolism at concentrations in its therapeutic target range
  • Small dose increase -> large (and variable) changes in the phenytoin plasma concentration at steady-state
  • Drug "half-life" changes from 12 hours at low drug doses to around 1 week at higher drug doses
• Compartmental analysis


  Development of a mathematical model to describe the concentration-time profile of a drug
• Compartmental analysis


  • Different input and disposition models can be combined to create the best complete pharmacokinetic model
• Developing a pharmacokinetic model


  1. Create a diagram

  2. Write a set of ordinary differential equations describing drug movement across the compartments that make up the diagram
• Amount of drug in a particular compartment at a particular time


  Calculated by the difference given by the rate of drug entering the compartment and the rate of drug leaving the compartment