7.1 Mathematical Modeling of Physiological Systems
3 min read•august 9, 2024
Mathematical modeling of physiological systems is a powerful tool for understanding complex biological processes. By representing the body's functions as equations and algorithms, researchers can simulate and predict how different systems interact and respond to stimuli.
This topic covers various modeling techniques, from to , and their applications in pharmacology and . It also explores model components, analysis methods, and validation approaches essential for creating accurate physiological simulations.
Modeling Techniques
Compartmental and Differential Equation Models
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Gas Exchange | Anatomy and Physiology II View original
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Compartmental models divide systems into interconnected compartments representing distinct physiological spaces
Compartments exchange substances or energy based on defined transfer rates
(ODEs) describe time-dependent changes in compartment quantities
First-order ODEs commonly used for simple pharmacokinetic models
Higher-order ODEs applied to more complex systems (cardiovascular, respiratory)
(PDEs) incorporate spatial variations in addition to temporal changes
Used for modeling processes with spatial gradients (diffusion, fluid dynamics)
Examples include oxygen diffusion in tissues, blood flow in vessels
Numerical Methods for Solving Equations
Analytical solutions often impossible for complex physiological models
provide approximate solutions through iterative calculations
offers simple first-order approximation for ODEs
Calculates next state based on current state and rate of change
Accuracy improves with smaller step sizes
provide higher-order accuracy
(RK4) widely used for balance of accuracy and efficiency
applied to solve PDEs
Discretize space and time domains into grids
Approximate derivatives using differences between adjacent grid points
Physiological Applications
Pharmacological Modeling
models drug absorption, distribution, metabolism, and excretion
One-compartment models assume rapid drug distribution throughout body
Multi-compartment models account for varying drug concentrations in different tissues
describes drug effects on the body
relate drug concentration to physiological effects
characterizes maximum effect and drug sensitivity
Physiologically-based pharmacokinetic (PBPK) models integrate anatomical and physiological data
Represent organs as compartments with specific blood flows and metabolic capacities
Predict drug concentrations in various tissues over time
Systems Biology and Homeostasis
Systems biology examines complex interactions within biological systems