# Pharmacokinetic three-compartment model

Pharmacokinetics refers to the rate and extent of distribution of a drug to different tissues, and the rate of elimination of the drug. Pharmacokinetics can be reduced to mathematical equations, which describe the transit of the drug throughout the body, a net balance sheet from absorption and distribution to metabolism and excretion.

Pharmacokinetic three-compartment model divided the body into *central* compartment and
two *peripheral* compartments. The central compartment (compartment 1) consists of
the plasma and tissues where the distribution of the drug is
practically instantaneous. The peripheral compartments (compartments 2 and 3) consist of tissues
where the distribution of the drug is slower compared to compartment 1.

Compartments could represent drug amounts instead of concentrations. Drug amounts in
the compartments equal to the concentrations multiplied by volumes:
*A _{1}=C_{1}×V_{1}* and

*A*.

_{2}=C_{2}×V_{2}*A*. Drug concentration in the central compartment is equal to the concentration in the plasma (or blood):

_{3}=C_{3}×V_{3}*C*. Clearance (in units L/h) is often used instead of the fractional rate constants (in units h

_{P}=C_{1}^{-1}); in pharmacokinetics the distribution volume is given in volume units (L), and rate constants can be represented as the ratio of clearance and distribution volume,

*k=CL/V*.

In case of intravenous bolus infusion or oral administration of the drug, at time *t=0*
all concentrations are zero.
Drug concentrations in the central and peripheral compartments can be calculated with
differential equations:

In case of bolus injection of the drug, at time *t=0* the central compartment is assumed
to contain all of the injected drug, and *I(t)=0*.

## See also:

- PK two-compartment model
- PK one-compartment model
- Whole-body physiology-based pharmacokinetic model
- Plasma pharmacokinetics in PET
- Receptor occupancy
- Enzyme inhibition
- Binding potential
- Compartmental model ODEs in PET
- Whole-body model for [
^{15}O]H_{2}O

## References:

Bailey JM, Shafer SL. A simple analytical solution to the three-compartment pharmacokinetic model
suitable for computer-controlled infusion pumps. *IEEE Trans Biomed Eng.* 1991; 38(6):
522-525. doi: 10.1109/10.81576.

Bergström M, Långström B. Pharmacokinetic studies with PET. *Progr Drug Res.* 2005; 62:
280-317. doi: 10.1007/3-7643-7426-8_8.

Bourne DWA: *Mathematical Modeling of Pharmacokinetic Data*.
CRC Press, 1995. ISBN 1-56676-204-9.

Fischman AJ, Alpert NM, Rubin RH. Pharmacokinetic imaging - a noninvasive method for determining
drug distribution and action. *Clin Pharmacokinet.* 2002; 41(8): 581-602. doi:
10.2165/00003088-200241080-00003.

Jann MW, Penzak SR, Cohen LJ (eds.): *Applied Clinical Pharmacokinetics and Pharmacodynamics
of Psychopharmacological Agents*. Adis, Springer, 2016.
doi: 10.1007/978-3-319-27883-4.

Kuepfer L, Niederalt C, Wendl T, Schlender JF, Willmann S, Lippert J, Block M, Eissing T,
Teutonico D. Applied concepts in PBPK modeling: how to build a PBPK/PD model.
*CPT Pharmacometrics Syst Pharmacol.* 2016; 5(10): 516-531.
doi: 10.1002/psp4.12134.

Rescigno A. Compartmental analysis revisited. *Pharmacol Res.* 1999; 39(6): 471-478.
doi: 10.1006/phrs.1999.0467.

Rosenbaum S (ed.): *Basic Pharmacokinetics and Pharmacodynamics - An Integrated Textbook and
Computer Simulations.* 2nd ed., Wiley, 2017. ISBN 9781119143154.

Schafer SL, Siegel LC, Cooke JE, Scott JC. Testing computer-controlled infusion pumps by
simulation. *Anesthesiol.* 1988; 68: 261-266. doi:
10.1097/00000542-198802000-00013.

Zamuner S, Di Iorio VL, Nyberg J, Gunn RN, Cunningham VJ, Gomeni R, Hooker AC. Adaptive-optimal
design in PET occupancy studies. *Clin Pharmacol Ther.* 2010; 87(5): 563-571.
doi: 10.1038/clpt.2010.9.

Tags: Pharmacokinetics, Drug development, Plasma, Clearance

Updated at: 2019-01-06

Created at: 2019-01-06

Written by: Vesa Oikonen