17.1 Compartmental modeling of physiological processes

Compartmental modeling divides physiological systems into distinct compartments, applying mass balance principles to analyze substance flow. This approach simplifies complex biological processes, enabling the study of drug distribution, nutrient transport, and metabolic pathways.

Multi-compartment models use differential equations to describe substance transfer between compartments. Mathematical analysis techniques like Laplace transforms and eigenvalue analysis help solve these equations, providing insights into system behavior and physiological parameters.

Compartmental Modeling Principles and Applications

Principles of compartmental modeling

• Divides physiological system into distinct compartments representing homogeneous and well-mixed subsystems (organs, tissues, cell groups)
• Applies mass balance principles to each compartment
  • Change in substance amount over time equals difference between inflow and outflow rates
  • Inflow and outflow rates include transport, production, and elimination processes
• Assumes instantaneous mixing within each compartment, linear or nonlinear transfer rates between compartments, and conservation of mass across entire system

Multi-compartment model development

• Identifies relevant compartments for physiological process of interest (two-compartment model for drug distribution with plasma and tissue compartments)
• Defines transfer rates between compartments (drug absorption rate from gut to plasma compartment)
• Writes mass balance equations for each compartment
  • Example: $\frac{dC_p}{dt} = k_a C_g - k_{el} C_p - k_{pt} C_p + k_{tp} C_t$
    • $C_p$: Drug concentration in plasma compartment
    • $C_g$: Drug concentration in gut compartment
    • $C_t$: Drug concentration in tissue compartment
    • $k_a$: Absorption rate constant
    • $k_{el}$: Elimination rate constant
    • $k_{pt}$: Plasma-to-tissue transfer rate constant
    • $k_{tp}$: Tissue-to-plasma transfer rate constant
• Solves system of differential equations analytically or numerically
  1. Analytical solutions for linear systems with constant coefficients
  2. Numerical methods for nonlinear or time-varying systems (Euler's method, Runge-Kutta)

Mathematical analysis of compartmental models

• Uses Laplace transforms for solving linear systems of differential equations
  • Converts differential equations to algebraic equations in Laplace domain
  • Solves for transfer functions and applies inverse Laplace transform to obtain time-domain solutions
• Performs eigenvalue analysis for understanding system stability and response characteristics
  • Determines eigenvalues and eigenvectors of system matrix
  • Stable systems have negative real parts of eigenvalues
  • Eigenvectors represent modes of system response
• Interprets model results in terms of physiological parameters
  • Half-life, clearance, and steady-state concentrations for drug distribution models
  • Transport rates and metabolic fluxes for nutrient transport models

Limitations of compartmental modeling

• Assumes homogeneous mixing within compartments, which may not hold for heterogeneous tissues
• May not capture spatial gradients or localized effects
• Limited by availability of accurate parameter values and model validation data
• Potential extensions and improvements include:
  • Incorporating nonlinear transfer rates and saturable processes
  • Including time-varying parameters to represent circadian rhythms or adaptive responses
  • Integrating with other modeling approaches (physiologically based pharmacokinetic models, agent-based models)
  • Coupling with experimental data and parameter estimation techniques for model refinement and validation