Two-compartment model

Mohammad Issa Saleh

Typical plasma concentration (Cp) versus time profiles for a drug that obeys a two-compartment model following

intravenous bolus administration

y axis: normal scale y axis: logarithmic scale

A schematic representation of three types of two-compartment models consisting of a central and a peripheral compartment.

Please note the difference in each type is reflected in the placement of an organ responsible for the elimination of the drug from the body. K12, K21, transfer rate constants; K10, K20, elimination rate constants.

X1 X2

X1 X2

X1 X2

Assumptions of the model

• Upon drug absorption there is instantaneous distribution of drug throughout the central compartment (sampling compartment) having a volume V1 (Vc)

• Transfer of drug from the central compartment to the peripheral compartment is by a first-order process

• Transfer of drug from of drug from the peripheral compartment to the central compartment is by a first-order process

One Compartment Two Compartments

Rapid or prompt equilibrium is attained.

Distribution equilibrium is slow (takes finite time).

There is a single disposition phase

Distribution and post-distribution are two distinct phases.

Linear: drug elimination follows first order kinetics

Linear: distribution and

elimination both follow first order

A. We start with virtually no drug in the second compartment, but re-equilibration moves drug in – levels rise

B. A brief equilibrium - no net movement – at the peak of the curve, levels are neither rising nor falling

C. Re-equilibration moves into reverse and drug leaves the second compartment – levels fall

Drug concentrations in the two compartments following a single i.v. bolus injection

X1 X2

K12

K21

K10

1101122211 XKXKXK

dt

dX

2211122 XKXK

dt

dX

Distribution rate from X1 to X2 = 112XK

221XK

110XK

Distribution rate from X2 to X1 =

Elimination rate =

tt eKX

eKX

X

)()( 210210

1

)e(eβ)(α

XoKX αtβt

122

t

C

t

C

eV

KXe

V

KXC

)(

)(

)(

)( 2102101

VC is the volume of the central compartment

Amount in the central compartment

Conc in the central compartment

Amount in the peripheral compartment

tt BeAeC 1

)(

)( 21

Vc

KXoA

)(

)( 21

Vc

KXoB

Determination of the postdistributionrate constant (β) and the

coefficient (B)

• Postdistribution phase to determine:

1. Determine β from the graph by using the slope

2. The y-axis intercept of the extrapolated line is B

Determination of thedistribution rate constant (α) and

the coefficient (A)• Method of residuals: The

difference between measured concentrations and those obtained by extrapolation of the post-distribution line is plotted vs time

1. Determine α from the graph by using the slope

2. The y-axis intercept of the extrapolated line is A

Determination of micro rate constants: the inter-compartmental rate constants (K21 and K12) and

the pure elimination rate constant (K10)

2110 K

αβ K

102112 - Kβ-KαK

B)(A

BαAβK

21

Volume of distribution of the central compartment (VC)

• Volume of distribution of the central compartment (VC). This is a proportionality constant that relates the amount of drug and the plasma concentration immediately (i.e. at t=0) following the administration of a drug.

BA

XoVc

Volume of distribution during the terminal phase (Vb or Vβ)

• This is a proportionality constant that relates the plasma concentration and the amount of drug remaining in the body at a time following the attainment of distribution equilibrium, or at a time on the terminal linear portion of the plasma concentration time data

VcKV

10

Volume of distribution at steady state (Vss)

• This is a proportionality constant that relates the plasma concentration and the amount of drug remaining in the body at a time, following the attainment of practical steady state. This volume of distribution is independent of elimination parameters such as K10 or drug clearance.

VcK

KK

Css

XssVss

21

1221 )(

The area under the plasmaconcentrationtime curve (AUC)

• Model independent: Trapezoid method

• Model dependent:

BAAUC

dtBeAedttCAUC tt

00.).(

Example: The pharmacokinetics of amrinone after a single IV bolus injection (75 mg) in 14 healthy adult male volunteers followed a two-compartment open model and fit the following parameters:A = 4.62 ± 12.0 µg/mLB = 0.64 ± 0.17 µg/mL = 8.94 ± 13 hr–1

= 0.19 ± 0.06 hr–1

From these data, calculate:a. The volume of the central compartmentb. The volume of the tissue compartmentc. The transfer constants k12 and k21

d. The elimination rate constant from the central compartmente. The elimination half-life of amrinone after the drug has equilibrated with the tissue compartment

X1 X2

K12

K21

K10

1101122211 XKXKXKXK

dt

dXaa

2211122 XKXK

dt

dX

Xa

Ka

Two Compartment Extravascular

X1 X2

K12

K21

K10

Xa

Ka

ttt CeBeAeC 1

CBA

Two Compartment Extravascular