The model can be depicted as shown below with elimination from the central compartment.

__Two-Compartment Open Model__

__Intravenous Bolus Administration__

The model can be depicted as shown below with
elimination from the central compartment.

After the i.v. bolus of a drug that follows
two-compartment kinetics, the decline in plasma concentration is biexponential
indicating the presence of *two
disposition processes viz.* *distribution
and elimination*. These two processes are not evident to the eyes in a
regular* *arithmetic plot but when a
semilog plot of C versus t is made, they can be identified (Fig. 9.12).
Initially, the concentration of drug in the central compartment *declines rapidly*; this is due to the
distribution of drug from the central compartment to the peripheral
compartment. The phase during which this occurs is therefore called as the **distributive phase**. After sometime, a *pseudo-distribution equilibrium* is
achieved between the two compartments following which the subsequent loss of
drug from the central compartment is slow and mainly due to elimination. This *second, slower rate process is called as the*
**post-distributive **or** elimination phase**. In contrast to the
central compartment, the drug** **concentration
in the peripheral compartment first increases and reaches a maximum. This
corresponds with the distribution phase. Following peak, the drug concentration
declines which corresponds to the post-distributive phase (Fig.9.12).

**Fig. 9.12. **

Let K_{12} and K_{21} be the
first-order distribution rate constants depicting drug transfer between the
central and the peripheral compartments and let subscript c and p define
central and peripheral compartment respectively. The rate of change in drug
concentration in the central compartment is given by:

Extending the relationship X = V_{d}C to
the above equation, we have

where X_{c} and X_{p} are the
amounts of drug in the central and peripheral compartments respectively and V_{c}
and V_{p} are the apparent volumes of the central and the peripheral compartment
respectively. The rate of change in drug concentration in the peripheral
compartment is given by:

Integration of equations 9.85 and 9.87 yields
equations that describe the concentration of drug in the central and peripheral
compartments at any given time t:

where X_{o} = i.v. bolus dose, α and β are **hybrid first-order constants**
for the rapid distribution phase and the slow elimination phase respectively
which depend entirely upon the first-order constants K_{12}, K_{21}
and K_{E}.

*The constants ***K _{12}**

α + β = K_{12} + K_{21} + K_{E }(9.90)

αβ = K_{21}K_{E }(9.91)

Equation 9.88 can be written in simplified form as:

C_{c} = Ae ^{–αα} + Be ^{βt }(9.92)

C_{c} = Distribution exponent Elimination
exponent

where A and B are also hybrid constants for the two
exponents and can be resolved graphically by the method of residuals.

where C_{o} = plasma drug concentration
immediately after i.v. injection.

**Method of Residuals: **The
biexponential disposition curve obtained after i.v. bolus of a** **drug that fits two compartment model
can be resolved into its individual exponents by the method of residuals.
Rewriting the equation 9.92:

C_{c} = Ae ^{-αα} + Be ^{-βt }(9.92)

As apparent from the biexponential curve given in
Fig. 9.12., the initial decline due to distribution is more rapid than the
terminal decline due to elimination i.e. the rate constant α >> ß and hence the term e^{–}^{α}^{t} approaches zero much faster than does e^{–}^{β}^{t}. Thus, equation 9.92 reduces to:

In log form, the equation

where C = back extrapolated plasma concentration
values. A semilog plot of C versus t yields the terminal linear phase of the
curve having slope –β/2.303 and when back extrapolated
to time zero, yields y-intercept log B (Fig. 9.13.).The t_{½} for the
elimination phase can be obtained from equation t_{½} = 0.693/β.

Subtraction of extrapolated plasma concentration
values of the elimination phase (equation 9.95) from the corresponding true
plasma concentration values (equation 9.92) yields a series of residual
concentration values C_{r}.

In log form, the equation becomes:

A semilog plot of C_{r} versus t yields a
straight line with slope –α/2.303 and *Y*-intercept log A (Fig. 9.13).

**Fig. 9.13. **

**Assessment of Pharmacokinetic Parameters: **All the parameters of equation 9.92 can be** **resolved by the method of residuals as described above. Other
parameters of the model viz. K_{12}, K_{21}, K_{E},
etc. can now be derived by proper substitution of these values.

C_{0 }= A + B (9.99)

It must be noted that for two-compartment model, K_{E}
is the rate constant for elimination of drug from the central compartment and β *is the rate constant for
elimination from the* *entire body*.
Overall elimination t_{½}* *should
therefore be calculated from β.

Area under the plasma concentration-time curve can
be obtained by the following equation:

The apparent volume of central compartment V_{c}
is given as:

Apparent volume of peripheral compartment can be
obtained from equation:

The apparent volume of distribution at steady-state
or equilibrium can now be defined as:

It is also given as:

Total systemic clearance is given as:

The pharmacokinetic parameters can also be
calculated by using urinary excretion data:

An equation identical to equation 9.92 can be
derived for rate of excretion of unchanged drug in urine:

The above equation can be resolved into individual
exponents by the method of residuals as described for plasma concentration-time
data.

Renal clearance is given as:

Cl_{R} K_{e} V_{c }(9.111)

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