When a drug that distributes rapidly in the body is given in the form of a rapid intravenous injection (i.e. i.v. bolus or slug), it takes about one to three minutes for complete circulation and therefore the rate of absorption is neglected in calculations.

__One-Compartment Open Model__

**Intravenous Bolus Administration**

When a drug that distributes rapidly in the body is
given in the form of a rapid intravenous injection (i.e. i.v. bolus or slug),
it takes about one to three minutes for complete circulation and therefore the
rate of absorption is neglected in calculations. The model can be depicted as
follows:

The general expression for **rate of drug presentation** to the body is:

dX/dt = Rate in (availability) - Rate out
(elimination) (9.1)

Since **rate
in** or absorption is absent, the equation becomes:

dX / dt = - Rate out (9.2)

If the **rate
out** or elimination follows first-order kinetics, then:

dX/dt = K_{Ε} X (9.3)

where, K_{E} = first-order elimination rate
constant, and

X = amount of drug in the body at any time t
remaining to be eliminated.

Negative sign indicates that the drug is being lost
from the body.

__Estimation of Pharmacokinetic Parameters__

For a drug that follows one-compartment kinetics
and administered as rapid i.v. injection, the decline in plasma drug
concentration is only due to elimination of drug from the body (and not due to
distribution), the phase being called as elimination phase. **Elimination phase** can be characterized
by 3 parameters—

1. Elimination rate constant

2. Elimination half-life

3. Clearance.

**Elimination Rate Constant**:
Integration of equation 9.3 yields:

ln X = ln X_{o} – K_{E} t (9.4)

where, X_{o} = amount of drug at time t =
zero i.e. the initial amount of drug injected.

Equation 9.4 can also be written in the exponential
form as:

X = X_{o} e^{–KEt }(9.5)

The above equation shows that *disposition of a drug that follows one-compartment* *kinetics is ***monoexponential**.

Transforming equation 9.4 into common logarithms
(log base 10), we get:

Since it is difficult to determine directly the
amount of drug in the body X, advantage is taken of the fact that a constant
relationship exists between drug concentration in plasma C (easily measurable)
and X; thus:

X = V_{d} C (9.7)

where, V_{d} = proportionality constant
popularly known as the *apparent volume of
distribution*. It is a pharmacokinetic parameter that permits the use of
plasma drug concentration in place of amount of drug in the body. The equation
9.6 therefore becomes:

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

Equation 9.8 is that of a straight line and
indicates that a semilogarithmic plot of log C versus t will be linear with *Y*-intercept log C_{o}. The
elimination rate constant is directly obtained from the slope of the line (Fig.
9.2*b*). It has units of min^{–1}.
Thus, a linear plot is easier to handle mathematically than a curve which in
this case will be obtained from a plot of C versus t on regular (Cartesian)
graph paper (Fig. 9.2*a*).

**Fig. 9.2 ***(a) Cartesian plot of a drug that follows one-compartment kinetics and
given by rapid i.v. injection, and
(b) Semilogarithmic plot for the rate of elimination in a one-compartment
model.*

Thus, C_{o}, K_{E} (and t_{½})
can be readily obtained from log C versus t graph. The elimination or removal
of the drug from the body is the sum of urinary excretion, metabolism, biliary
excretion, pulmonary excretion, and other mechanisms involved therein. Thus, K_{E}
is an additive property of rate constants for each of these processes and
better called as **overall** **elimination rate constant**.

K_{E} = K_{e} + K_{m} + K_{b}
+ K_{l} + ...... (9.9)

The fraction of drug eliminated by a particular
route can be evaluated if the number of rate constants involved and their
values are known. For example, if a drug is eliminated by urinary excretion and
metabolism only, then, the fraction of drug excreted unchanged in urine F_{e}
and fraction of drug metabolized F_{m} can be given as:

**Elimination Half-Life: **Also
called as** biological half-life**, it
is the oldest and the best** **known of
all pharmacokinetic parameters and was once considered as the most important
characteristic of a drug. *It is defined
as the time taken for the amount of drug in the body as* *well as plasma concentration to decline by
one-half or 50% its initial value*. It is expressed* *in hours or minutes. Half-life is related to elimination rate
constant by the following equation:

Elimination half-life can be readily obtained from the graph of log C versus t as shown in Fig 9.2.

Today, increased physiologic understanding of
pharmacokinetics shows that *half-life is
a* *secondary parameter that depends
upon the primary parameters *clearance and apparent* *volume of distribution according to following equation:

**Apparent Volume of Distribution: **The two separate
and independent pharmacokinetic** **characteristics
of a drug are –

1. Apparent volume of
distribution, and

2. Clearance.

*Since these parameters are closely related with the physiologic
mechanisms in the body, they are called as ***primary parameters**.

Modification of equation 9.7 defines apparent
volume of distribution:

V_{d} is a measure of the extent of
distribution of drug and is expressed in liters. The best and the simplest way
of estimating V_{d} of a drug is administering it by rapid i.v.
injection, determining the resulting plasma concentration immediately and using
the following equation:

Equation 9.14 can only be used for drugs that obey
one-compartment kinetics. This is because the V_{d} can only be estimated
when distribution equilibrium is achieved between drug in plasma and that in
tissues and such equilibrium is established instantaneously for a drug that
follows one-compartment kinetics. A more general, more useful noncompartmental
method that can be applied to many compartment models for estimating the V_{d}
is:

For drugs given as i.v. bolus,

For drugs administered extravascularly (e.v.),

where, X_{o} = dose administered, and F =
fraction of drug absorbed into the systemic circulation. F is equal to *one* i.e. complete availability when the
drug is administered intravenously.

**Clearance: **Difficulties arise when one
applies elimination rate constant and half-life as** **pharmacokinetic parameters in an anatomical/physiological context
and as a measure of drug elimination mechanisms. A much more valuable
alternative approach for such applications is use of clearance parameters to
characterize drug disposition. *Clearance
is the most important* *parameter in
clinical drug applications and is useful in evaluating the mechanism by which a
drug is eliminated by the whole organism or by a particular organ*.

Just as V_{d} is needed to relate plasma
drug concentration with amount of drug in the body, clearance is a parameter to
relate plasma drug concentration with the rate of drug elimination according to
following equation:

**Clearance ***is defined as the theoretical volume of body fluid containing drug*** **(i.e. that** **fraction of
apparent volume of distribution) *from
which the drug is completely removed in a* *given period of time*. It is expressed in ml/min or liters/hour.
Clearance is usually further* *defined
as **blood clearance** (Cl_{b}),
**plasma clearance** (Cl_{p}) or
clearance based on unbound or free drug concentration (Cl_{u})
depending upon the concentration C measured for the right side of the equation
9.17.

**Total Body Clearance: **Elimination
of a drug from the body involves processes occurring** **in kidney, liver, lungs and other eliminating organs. *Clearance at an individual organ level* *is called as ***organ clearance**. It can be estimated by dividing the rate of
elimination by each* *organ with the
concentration of drug presented to it. Thus,

*The ***total body clearance**, Cl_{T},* also called as ***total systemic clearance**,* is
an additive property of individual organ clearances*. Hence,

*Total Systemic Clearance *

Cl_{T} = Cl_{R} + Cl_{H} + Cl_{Others}
(9.18d)

Because of the additivity of clearance, the
relative contribution by any organ in eliminating a drug can be easily
calculated. *Clearance by all organs other
than kidney is* *sometimes known as ***nonrenal clearance*** *Cl_{NR}. It is the difference between total clearance* *and renal clearance.

According to an earlier definition (equation 9.17),

Substituting dX/dt = K_{E}X from equation
9.3 in above equation, we get:

Since X/C = V_{d} (from equation 9.13), the
equation 9.19 can be written as:

Cl_{T }= K _{E} V_{d} (9.20a)

Parallel equations can be written for renal and
hepatic clearances as:

Cl_{R = }K _{e} V_{d } (9.20b)

Cl_{H = }K _{m} V_{dm } (9.20c)

Since K_{E} = 0.693/t_{½} (from
equation 9.11), clearance can be related to half-life by the following
equation:

Identical equations can be written for Cl_{R}
and Cl_{H} in which cases the t_{½} will be urinary excretion
half-life for unchanged drug and metabolism half-life respectively. Equation
9.21. shows that as Cl_{T} decreases, as in renal insufficiency, t_{½}
of the drug increases. As the Cl_{T }takes into account V_{d},
changes in V_{d} as in obesity or oedematous condition will reflect
changes in Cl_{T}._{}

The noncompartmental method of computing total clearance for a drug that follows one-compartment kinetics is:

For a drug given by i.v. bolus, the renal clearance
Cl_{R} may be estimated by determining the total amount of unchanged
drug excreted in urine, X_{u}^{∞} and AUC.

**Organ Clearance: **The best way of understanding
clearance is at individual organ level.** **Such
a physiologic approach is advantageous in predicting and evaluating the
influence of pathology, blood flow, P-D binding, enzyme activity, etc. on drug
elimination. At an organ level, the rate of elimination can be written as:

Substitution of equations 9.25 and 9.26 in equation
9.24 yields:

**Rate of elimination = **Q C_{in
}- Q C_{out}

(also called as **Rate of extraction**) = Q (C_{in }-_{ }^{C}out
^{) }(9.27)

Division of above equation by concentration of drug
that enters the organ of elimination C_{in} yields an expression for
clearance of drug by the organ under consideration. Thus:

where, ER = (C_{in} – C_{out})/C_{in}
is called as **extraction ratio**. It
has no units and its value ranges from zero (no elimination) to one (complete
elimination). Based on ER values, drugs can be classified into 3 groups:

1. Drugs with **high
ER **(above 0.7),

2. Drugs with **intermediate
ER** (between 0.7 to 0.3), and

3. Drugs with **low
ER** (below 0.3).

**ER ***is an index of how efficiently the eliminating organ clears the blood
flowing through it*** ***of drug*. For example, an ER of 0.6 tells that 60% of the
blood flowing through the organ will* *be
completely cleared of drug. The fraction of drug that *escapes removal* by the organ is expressed as:

F=1-ER (9.29)

where, **F**
= **systemic availability** when the
eliminating organ is liver.

**Hepatic Clearance: **For certain drugs, the nonrenal
clearance can be assumed as equal to** **hepatic
clearance Cl_{H}. It is given as:

Cl_{H }=
Cl_{T} - Cl_{R} (9.30)

An equation parallel to equation 9.28 can also be
written for hepatic clearance:

Cl_{H }= Q_{H} ER _{H }(9.31)

where,

Q_{H} = hepatic blood
flow (about 1.5 liters/min), and

ER_{H} = hepatic
extraction ratio.

The hepatic clearance of drugs can be divided into
two groups:

1. Drugs with hepatic blood flow
rate-limited clearance, and

2. Drugs with intrinsic
capacity-limited clearance.

**1. Hepatic Blood Flow: **When ER_{H}** **is one, Cl_{H}** **approaches its maximum value i.e.** **hepatic blood flow. In such a
situation, hepatic clearance is said to be **perfusion
rate-limited** or **flow-dependent**.
Alteration in hepatic blood flow significantly affects the elimination of drugs
with high ER_{H} e.g. propranolol, lidocaine, etc. Such drugs are
removed from the blood as rapidly as they are presented to the liver (high
first-pass hepatic metabolism). Indocyanine green is so rapidly eliminated by
the human liver that its clearance is often used as an indicator of hepatic
blood flow rate. First-pass hepatic extraction is suspected when there is lack
of unchanged drug in systemic circulation after oral administration. **Maximum oral** **availability F **for such drugs can be computed from equation 9.29.
An extension of the same** **equation is
the noncompartmental method of estimating F:

**TABLE 9.1**

On the contrary, hepatic blood flow has very little
or no effect on drugs with low ER_{H} e.g. theophylline. For such
drugs, whatever concentration of drug present in the blood perfuses liver, is
more than what the liver can eliminate (low first-pass hepatic metabolism).
Similar discussion can be extended to the influence of blood flow on renal
clearance of drugs. This is illustrated in Table 9.1. Hepatic clearance of a
drug with high ER is independent of protein binding.

**2. Intrinsic Capacity Clearance: ***Denoted as*** Cl _{int}**,

The hepatic and renal extraction ratios of some
drugs and metabolites are given in Table 9.2.

**TABLE 9.2**

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