2. Compartmental Models

Compartmental Models in Pharmacokinetics

Overview

  • Topic presented by: Trine Meldgaard Lund, Associate Professor, University of Copenhagen

  • Focus: Understanding the distribution and absorption of drugs in the body through pharmacometric models.

Learning Objectives

  • Determine the factors responsible for drug distribution in the body.

  • Calculate parameters related to the distribution phase based on plasma concentration data.

  • Calculate parameters related to the absorption phase based on plasma concentration data.

Distribution Phase

Rate of Distribution

  • Perfusion Rate Limitation
      - Increasing blood flow to different organs impacts drug distribution.
      - Particularly important for small lipophilic drugs.
      - High perfusion rates are noted in:
        - Kidneys: 0.5 - 10 mL/min/mL tissue
        - Lungs, Liver, and Brain: Similar high rates.
      - Low Perfusion Rates: Found in fat tissue and inactive muscles (~0.025 mL/min/mL tissue).

  • Permeability Rate Limitation
      - Influenced by characteristics of the membrane:
        - Types: Loose, tight, presence of transporters (Blood-Brain Barrier - BBB).
      - Influenced by characteristics of the drug:
        - Factors: Molecular size, lipophilicity, charge, pKa, and pH.
        - Critical for tighter membranes like those in the hepatocytes, kidneys, and BBB.

Extent of Distribution

  • Volume of Distribution (Vd)
      - Apparent Volume of Distribution: Quantifies how extensively a drug is distributed throughout body tissues.
      - Protein Binding: Both in blood and peripheral tissues influences distribution.

Blood Flow and Perfusion Rate

  • Formulas:
      - Blood Flow (Q) = Rate of blood flow through an organ.
      - Perfusion Rate = QVT\frac{Q}{{V_T}} (where VTV_T = volume of the tissue)
      - Equilibrium Distribution Ratio: Kp=racCTCVK_p = rac{C_T}{C_V}
      - Half-life for Distribution:
        - T12=ln2⋅Kp(QVT)T_{\frac{1}{2}}=l{{n2}\cdot}\frac{{K_{p}{}{_{}}}}{\left({}\frac{Q}{{V_{T}}}\right)}

Tissue-Specific Half-Lives

  • Calculated Half-lives:
      - Kidney:
        - T½=racextln24imesrac10.5=0.17extminT_{½} = rac{ ext{ln 2}}{4} imes rac{1}{0.5} = 0.17 ext{min}
      - Brain:
        - T½=racextln20.5imesrac11.0=1.4extminT_{½} = rac{ ext{ln 2}}{0.5} imes rac{1}{1.0} = 1.4 ext{min}
      - Fat:
        - T½=racextln20.03imesrac11.0=23extminT_{½} = rac{ ext{ln 2}}{0.03} imes rac{1}{1.0} = 23 ext{min}

Key Observations

  • Greater tendency of a drug to concentrate in a tissue can result in longer distribution times.

  • For example:
      - For a concentration ratio KpbK_{pb}:
        - Kpb = 5 (most lipophilic):
          - T½=racextln2imes50.03=115extminT_{½} = rac{ ext{ln 2} imes 5}{0.03} = 115 ext{min}
        - Kpb = 2:
          - T½=racextln2imes20.03=46extminT_{½} = rac{ ext{ln 2} imes 2}{0.03} = 46 ext{min}
        - Kpb = 1:
          - T½=racextln2imes10.03=23extminT_{½} = rac{ ext{ln 2} imes 1}{0.03} = 23 ext{min}

Distribution Profiles for Drugs

Classification of Drugs

  • Acidic Drugs:
      - Bind to Albumin (50 g/L in plasma).

  • Basic Drugs:
      - Bind to α1-acid glycoprotein and lipoproteins (1 g/L in plasma).

Volume of Distribution Based on Drug Properties

  • Volume in relation to the free fraction:
      - V=VP+VTWimesracfufuTV = V_P + V_TW imes rac{f_u}{f_{uT}}
      - For drugs with mw > 70,000 g/mol:
        - When solely in plasma: V=VP=3extLV = V_P = 3 ext{L}
      - For acidic drugs with high plasma protein binding:
        - fu≈0<br>ightarrowV=Valbumin=7.5extLf_u ≈ 0 <br>ightarrow V = V_{albumin} = 7.5 ext{L}
      - For drugs not binding to proteins:
        - fu=fuT=1f_u = f_{uT} = 1
          - V=16extLV = 16 ext{L} (extracellular fluid)
          - V=42extLV = 42 ext{L} (total body water)
      - Examples:
        - Ethanol and caffeine have a volume of distribution of V=42extLV = 42 ext{L}.

Compartment Models: Understanding Drug Distribution

One-Compartment Model

  • Characteristics: Rapid distribution (e.g., aminoglycosides).

  • Graphical Representation:
      - Plasma concentration at specified times before and after drug administration.

Two-Compartment Model

  • Characteristics: Slow distribution (e.g., vancomycin).

  • Graphical Representation:
      - Plasma concentration change before and after administration and during equilibrium distribution.

Rate Constants in Compartment Models

  • Variables:
      - k=ke=k10k = k_e = k_{10}: Elimination rate constant.
      - V1V_1: Initial distribution volume (central compartment).
      - V2V_2: Volume of peripheral compartment.
      - k12k_{12}: Rate constant from central to peripheral compartment.
      - k21k_{21}: Rate constant from peripheral to central compartment.

ADME (Absorption, Distribution, Metabolism, Excretion)

Phases in Pharmacokinetics

  • Visualization: Absorption, Distribution, Elimination (Metabolism + Excretion).

  • Often represented graphically for clarity in understanding timelines for each phase.

First-Order Processes Framework

  • Basic Equation:
      - RR=−kimesAR_{R} = -k imes A (e.g., cell division, radioactive decay, drug elimination).

  • Integrated Form:
      - Ct=C0imese−kimestC_t = C_0 imes e^{-k imes t}
      - or equivalently, Ct=extln(C0)−kimestC_t = ext{ln}(C_0) - k imes t