Compartmental Modeling Tutorial

Opening Comments

Compartmental models are, in theory, instantaneously mixed throughout, which means that the outflow concentration for all solutes is equal to the concentration within the flowing fluid. This allows ordinary differential equations (ODE's) to be used, for example, to calculate the change in concentration, C, within the unit and at its outflow: dC/dt = Flow*(Cin - C)/V, where Flow = volume of flow in and out, the volume, V, is a constant and Cin is the concentration of solute in the entering flow.

Jim Bassingthwaighte jbb@bioeng.washington.edu

Further reading:

“Tracers in Physiological Systems Modeling”: Download PDF file.
Models and descriptions for figures in paper.
Basic principles in the use of tracers in physiological model development, Anderson JC and Bassingthwaighte JB, Published as Anderson JC and Bassingthwaighte JB. Tracers in physiological systems modeling. In: “Mathematical Modeling in Nutrition and Agriculture”. Proc 9th International Conf on Mathematical Modeling in Nutrition, Roanoke, VA, August 14-17, 2006, edited by Mark D. Hanigan JN and Casey L Marsteller. Virginia Polytechnic Institute and State University, Blacksburg, VA 2007, pp 125-159.

One Compartment

Comp1_Decay	This is a one compartment model with exponential decay (and also growth).
Comp1_Flow	A flow, F carries an input concentration, Cin, into a one compartment model with volume V1. It is constantly and instantaneously well mixed becoming C1, the concentration in the compartment. C1 empties out of the compartment and is designated Cout. No reactions occur in this system.
Comp1_1Reaction	A model describing the reaction of solute ‘C1’ to form solute ‘D1’.
Comp1_Flow_React	Single stirred tank, compartment model with constant flow and volume, V1, and first order consumption of solute, when G = 0, AREA_in of inflow concentration-time curve = AREA_out of outflow curve.
Comp1_Flow_1Reaction	Single stirred tank, compartment model with constant flow, F, and volume, V1, and first order consumption of solute C1 producing solute D1.
Comp1_Flow_2Reaction	Single stirred tank, compartment model with constant flow, F, and volume, V1, and first order consumption of solute C1 producing solute D1 and first order of D1 producing E1.

Two Compartments

Comp2_Flow	This is a two compartment model with flow, F, volumes V1 and V2, their respective concentrations, C1 and C2, and an exchange coefficient PS (an acronym for Permeabilty-Surface area product which allows material to move between the two compartments. A two compartment model with exchange can be thought of as two distinct volumes separated by a porous membrane.
Comp2_Flow_React	Two compartmental plasma-ISF exchange model with two reactants. Flow through the plasma region brings in reactant C, and the reversible reaction C←→B occurs in the extravascular (interstitial) region only.
Comp2_Exchange	A model describing the exchange of solute between two compartments separated by a permeable membrane.
Comp2_Flow_Exchange	A model describing the exchange of a single solute between two compartments while there is flow through one of the compartments.
Comp2_Exchange_1Reaction	A model describing solute concentrations over time in two adjacent compartments, attached together by a permeable barrier. A reaction occurs that has solute ‘a’ react to become solute ‘b’.
Comp2_Flow_Exchange_1Reaction	A model describing the concentrations of solute in two adjacent compartments separated by a permeable barrier through which solute exchange occurs. Solute ‘a’ reacts to from solute ‘b’. There is flow through one compartment.
Comp2_Flow_Exchange_MATCH_TO_DATA	A model describing the exchange of a single solute between two compartments while there is flow through one of the compartments. Flow occurs at a constant rate and solute may exchange in either direction. Sample data is given for the curve to be matched to.
Comp2_F_MMtransp	Stirred tank model for exchange between flowing instantly mixed plasma, and the stagnant, instantly mixed ISF. An external input function, Cin, enters with the flow. Reaction of solute C –> solute B (and the reverse reaction) occurs in ISF.
Comp2_MRI_Contrast	This is a tank model for infusion of GdDTPA into an inflowing stream of water and converting (by reaction) the water to water spins (so to speak), AND washing both W and WS as well as Gd out of the field of the ROI into the effluent.
Comp2_Lignocaine_Passage_from_Injection_Site_to_Heart	A model of the first pass passage of drugs from i.v. injection site to the heart based on the work of R.N. Upton [1996 British J. of Anaesth, 77, 764-772] and developed by Jinfei Yu (2006) as a final project for BIOEN 589, University of Washington.
Comp2_Osmotic_Solute-Water_Interaction	Model describing osmotic solute-water interaction in a 2 compartment system.
Comp2_Osmotic_Water_Solute_Exchange	Model describing 2 compartments with osmotic water and solute exchange (no solute water interaction).

Multiple Compartments

Comp3_Flow	Stirred tank model for exchange between plasma, ISF and parenchymal cell with an external input function, Cin.
Comp6_Recirc	A system composed of three operators in series with recirculation. Can be used to explore theories of John L.Stephenson Bull (Math Biophys 10: 117-121, 1948 and Math Biophys 22:1-17, 1960). Exchanges are allowed in operator A only. Operators B and C are each composed of two stirred tank operators in series. The input function Cin is positioned between A and B, but could be put anywhere.