The kinetics of capacity-limited or saturable processes is best described by Michaelis.

**MICHAELIS MENTEN EQUATION**

The kinetics of capacity-limited or saturable
processes is best described by Michaelis-

Menten equation:

Where,

–dC/dt = rate of decline of drug
concentration with time,

V_{max} = theoretical
maximum rate of the process, and

K_{m} = Michaelis
constant.

Three situations can now be considered depending
upon the values of K_{m} and C:

__1. When K _{m} = C__

Under this situation, the equation 10.1 reduces to:

i.e. the rate of process is equal to one-half its
maximum rate (Fig. 10.1).

*Fig. 10.1 **A plot of Michaelis-Menten
equation (elimination rate dC/dt versus concentration
C). Initially, the rate increases linearly (first-order) with concentration,
becomes mixed-order at higher concentration and then reaches maximum (Vmax)
beyond which it proceeds at a constant rate (zero-order).*

__2. When K _{m} >> C__

Here, K_{m} + C ≡ K_{m} and the equation 10.1 reduces to:

The above equation is identical to the one that
describes first-order elimination of a drug where V_{max}/K_{m}
= K_{E}. This means that the drug concentration in the body that results
from usual dosage regimens of most drugs is well below the K_{m} of the
elimination process with certain exceptions such as phenytoin and alcohol.

__3. When K _{m} << C__

Under this condition, K_{m} + C ≡ C and the equation 10.1 will become:

The above equation is identical to the one that
describes a zero-order process i.e. the rate process occurs at a constant rate
V_{max} and is independent of drug concentration e.g. metabolism of
ethanol.

The parameters of capacity-limited processes like
metabolism, renal tubular secretion and biliary excretion can be easily defined
by assuming one-compartment kinetics for the drug and that elimination involves
only a single capacity-limited process.

The parameters K_{m} and V_{max}
can be assessed from the plasma concentration-time data collected after i.v.
bolus administration of a drug with nonlinear elimination characteristics.

Rewriting equation 10.1.

Integration of above equation followed by
conversion to log base 10 yields:

A semilog plot of C versus t yields a curve with a
terminal linear portion having slope –V_{max}/2.303K_{m} and
when back extrapolated to time zero gives *Y*-intercept
log __Bar__*C*_{0} () (*see* Fig. 10.2). The equation that
describes this line is:

**Fig. 10.2 ***Semilog plot of a drug given as i.v. bolus with nonlinear elimination
and that fits one-compartment
kinetics.*

At low plasma concentrations, equations 10.5 and
10.6 are identical. Equating the two and simplifying further, we get:

K_{m} can thus be obtained from above
equation. V_{max} can be computed by substituting the value of K_{m}
in the slope value.

An alternative approach of estimating V_{max}
and K_{m} is determining the rate of change of plasma drug
concentration at different times and using the reciprocal of the equation 10.1.
Thus:

where C_{m} = plasma concentration at
midpoint of the sampling interval. A double reciprocal plot or the **Lineweaver-Burke plot** of 1/(dC/dt)
versus 1/C_{m} of the above equation yields a straight line with slope
= K_{m}/V_{max} and y-intercept = 1/V_{max}.

A *disadvantage*
of Lineweaver-Burke plot is that the points are clustered. More reliable plots
in which the points are uniformly scattered are **Hanes-Woolf plot** (equation 10.9) and **Woolf-Augustinsson-Hofstee plot** (equation 10.10).

The above equations are rearrangements of equation
10.8. Equation 10.9 is used to plot C_{m}/(dC/dt) versus C_{m}
and equation 10.10 to plot dC/dt versus (dC/dt)/C _{m}. The parameters
K_{m} and V_{max} can be computed from the slopes and
y-intercepts of the two plots.

When a drug is administered as a constant rate i.v.
infusion or in a multiple dose regimen, the steady-state concentration C_{ss}
is given in terms of **dosing rate** DR
as:

DR = C_{ss}Cl_{T }(10.11)

where DR = R_{o} when the drug is administered
as zero-order i.v. infusion and it is equal to FX_{o}/τ when administered as multiple oral dosage regimen (F is fraction
bioavailable, X_{o} is oral dose and is dosing interval).

At steady-state, the dosing rate equals rate of
decline in plasma drug concentration and if the decline (elimination) is due to
a single capacity-limited process (for e.g. metabolism), then;

A plot of C_{ss} versus DR yields a typical
*hockey-stick shaped curve* as shown in
Fig. 10.3.

**Fig. 10.3 ***Curve for a drug with nonlinear kinetics obtained by plotting the
steady-state concentration versus
dosing rates.*

To define the characteristics of the curve with a
reasonable degree of accuracy, several measurements must be made at steady-state
during dosage with different doses.

Practically, one can graphically compute Km and
Vmax in 3 ways:

Taking reciprocal of equation 10.12, we get:

Equation 10.13 is identical to equation 10.8 given
earlier. A plot of 1/DR versus 1/C_{ss} yields a straight line with
slope K_{m}/V_{max} and y-intercept 1/V_{max }

**Fig. 10.4 **

Here, the graph is considered as shown in Fig.
10.5. A pair of C_{ss} viz. C_{ss,1} and C_{ss,2}
obtained with two different dosing rates DR_{1} and DR_{2} is
plotted. The points C_{ss,1 }and DR_{1} are joined to form a
line and a second line is obtained similarly by joining C_{ss,2} and DR_{2}.
The point where these two lines intersect each other is extrapolated on DR axis
to obtain V_{max} and on x-axis to get K_{m}.

**Fig. 10.5** *Direct linear plot for estimation of K _{m} and V_{max}
at steady-state concentrations of a drug given at different dosing rates.*

** 3. The third graphical method **of
estimating K

A plot of DR versus DR/C_{ss} yields a
straight line with slope -K_{m} and *Y*-intercept
V_{max}^{.}

K_{m} and V_{max} can also be
calculated numerically by setting up simultaneous equations as shown below:

Combination of the above two equations yields:

After having computed K_{m}, its subsequent
substitution in any one of the two simultaneous equations will yield V_{max}.

It has been observed that K_{m} is much
less variable than V_{max}. Hence, if mean K_{m} for a drug is
known from an earlier study, then instead of two, a single measurement of C_{ss}
at any given dosing rate is sufficient to compute V_{max}.

There are several limitations of K_{m} and
V_{max} estimated by assuming one-compartment system and a single
capacity-limited process. More complex equations will result and the computed K_{m}
and V_{max} will usually be larger when:

1. The drug is eliminated by more
than one capacity-limited process.

2. The drug exhibits parallel
capacity-limited and first-order elimination processes.

3. The drug follows
multicompartment kinetics.

However, K_{m} and V_{max} obtained
under such circumstances have little practical applications in dosage
calculations.

Drugs that behave nonlinearly within the
therapeutic range (for example, phenytoin shows saturable metabolism) yield
less predictable results in drug therapy and possess greater potential in
precipitating toxic effects.

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