Compartmental Models

Mathematical Modeling · Axiom Academy

Understanding systems where quantities flow between interconnected compartments 1. What is a Compartmental Model? A compartmental model divides a system into distinct "compartments" or "boxes," each representing a homogeneous group or location. Material (mass, population, energy, etc.) flows between compartments according to specified rules. Compartments: Distinct boxes holding quantities (populations, concentrations, masses) Flows: Arrows showing how material moves between compartments Rates: Functions describing how fast material flows Sources/Sinks: External inflows and outflows 2. Conservation Laws and Balance Equations The fundamental principle underlying all compartmental models is conservation . The rate of change of material in a compartment equals what flows in minus what flows out. 3. Flow Rates Between Compartments How do we describe the rate at which material flows? The most common assumption is that the flow rate is proportional to the amount in the source compartment. Linear (First-order): Flow = k * (amount in source) Constant: Flow = k (fixed rate, like an IV drip) Saturating (Michaelis-Menten): Flow = V max * X / (K m + X) Mass-action: Flow = k * X * Y (depends on multiple compartments) 4. Single Compartment Models: Drug Pharmacokinetics The simplest compartmental model has just one compartment. A classic example is modeling how a drug concentration changes in the bloodstream after injection.

This is the written version of the interactive lesson above. See the full Mathematical Modeling course.