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32 ESSENTIAL PHARMACOLOGY
Obviously,suchdrugs must be extensively bound to tissues and/or
sequestered in fat. It should also be clear that Vd does not necessarily
correspond to an identifiable physiological compartment (see Table6.1
for representative values of volumes of distribution for drugss).
1. Clearance CL ml/min per kg
Clearance is related to the rate of elimination of a drug by any, or
by all, routes. It is derived by defining the rate of elimination of
the drug in terms of its plasma concentration:
Rate of eliminationCLC
The assumption inherent in this formulation is that the
mechanisms of drug elimination function at a rate that is proportional
to the plasma concentration of the drug with a proportionality constant
that is the clearance (CL). In the case of a drug that is eliminated by
filtration in the kidney, this seems a very intuitive assumption. Drug
molecules that are free in solution in the plasma simply flow through
the capillary fenestrations in the glomerular filter unimpeded. The
rate at which they are filtered is therefore directly related to their
concentration in the filtrate. (Figure6.3: LEFT)
Table6.1
Examples of drugs Volume of
Distribution, l/kg
Comment
plasminogen activator
space
aminoglycoside antibiotics
space
Ethanol, creatinine,
phenytoin
Dissolved in total body
water
Acetaminophen, lidocaine,
thiopental
Moderate binding to tissues
Propranolol, verapamil,
meperidine
4– 8 High binding to tissues
Chlorpromazine, desip-
10– 40 Hydrophobic drugs with
very high binding to fat
tissue
Clinical Pharmacokinetics 33
It is less obvious that other mechanisms of drug elimination
also bear this linear relationship to drug concentration. However,
recall that the biotransformation reactions that eliminate active
compounds by changing their structure are all enzyme catalyzed and
that for most drugs the enzymes are exposed to drug concentrations
well below the Michaelis constant, Km. Therefore, the metabolizing
systems are operating on the linear portion of the enzyme velocity/
substrate concentration curve, where the rate of metabolism
of the drug is directly proportional to the drug concentration
(Figure6.3: RIGHT).
The final consideration is of drugs that are eliminated by active
transport processes such as transport across the kidney tubules
into the urine, or across the choroid plexus from the cerebral
spinal fluid, or across the hepatocyte membrane into the bile.
The transport proteins that carry the drug molecules across the
cell membranes function in a manner analogous to enzymes and
thus obey Michaelis– Menton kinetics. Here too, for most drugs
the transport systems are not saturated, and thus their activity is
directly proportional to the drug concentration.
For drugs that are eliminated by several routes, the total body
clearance is simply the sum of the clearances at the different organs
Excretion
∝
C
Linear range
V
∝
C
MetabolismFiltration
V
C
Figure6.3 Clearance of a drug is proportional to plasma concentration of the
34 ESSENTIAL PHARMACOLOGY
since in each case the rate of elimination is proportional to the
plasma concentration. For example, in the case of a drug eliminated
by both the kidney and the liver:
Total rate of eliminationrate of elimination in the kidneyyrate of elimination
in the liver
CL liver CCLkidneyC CLL liver CL kidneyC
Hence, CL total=CL liver + CL kidney.
What are the units of clearance? The rate of elimination of
a drug must be in units of mass per time, e.g., mg/min. Often this
is normalized per kg of the patient’s mass to give mg/ml per kg.
The plasma concentration must be a mass per volume, e.g., mg/ml.
Therefore, dimensional analysis of the equation above indicates
clearance must have the units of volume per time, often normalized
to the patient’s mass, ml/min per kg:
Rate of eliminationmgper kg Clearance ml perkgPl/min /min aasma
concentrationmgml/
As a mnemonic, notice that the units of clearance are such that
the clearance for a given drug is numerically equal to a volume of
blood that would be completely “cleaned” of the drug in a unit of time.
Of course the body doesn’t work this way– it actually cleans up
a little of the drug in a large volume of blood during each unit of
time. Nonetheless, this is useful for remembering the units associ-
ated with clearance.
Now let’s look at some actual values of clearance for drugs in
current use, in the Tables6.2 and6.3. For comparison, it is useful
to know that the total blood flow (cardiac output) of a 70 kg man is
about 5 l of blood per minute, which is equivalent to about 40 ml of
plasma per minute per kg. The flow of plasma through the kidneys
and liver combined is about half of this.
Clinical Pharmacokinetics 35
Table6.3
Substance Clearance
ml/min per kg
Comments
—
—
Propranolol 99% of clearance occurs in the liver
the liver
Acetominophen
Phenobarbital
6.3 A Simple Pharmacokinetic Model
We can now develop a simple, but quite general model that
describes the time- dependent concentration of a drug in the plasma
of a patient. In this model we will assume that the important
Table6.2
Substance Clearance
ml/min per kg
Comments
—
Para- amino
hippurate
Creatinine
Tobramycin
Uric acid
secreted
36 ESSENTIAL PHARMACOLOGY
pharmacokinetic parameter, the clearance of the drug, is constant
in time and independent of the plasma concentration of the drug.
Because the rate of elimination of the drug is linearly dependent
on the plasma concentration in this model, it is sometimes called
the linear pharmacokinetic model, or, because the clearance is
independent of the size of the dose of drug given, it is also called the
dose- independent pharmacokinetic model. We will also assume
that the volume of distribution is a constant. This is also a fairly
good assumption; exceptions could arise if, for example, there is a
change in the percentage of body fat of the patient during therapy
so that less or more of the drug becomes trapped in adipose tissue.
The model is also known as a single compartment model, as
we will treat the whole body as one compartment. More complex,
multiple compartment models can come into play during rapid
infusion of a drug that first is distributed in the blood, and then
slowly enters tissue reservoirs.
Keeping the above assumptions in mind, let us develop the model.
Our goal is to obtain an expression for the plasma concentration of
the drug as a function of time, C(t). We begin by considering how
much drug is in the patient, and what the factors are that change
this amount. The total amount of drug in the patient is given by the
volume of distribution multiplied by the plasma concentration:
TotaldrugVdCt
Only two things can change this total amount of drug “on
board”: Administration of more drug and elimination of some of
the drug by biotransformation or excretion. The rate of change of
the total drug can be written as a differential with respect to time:
Rate of change dC Vd dt/
Administering the drug at a given dosing rate D.R. should con-
tribute positively to this differential, and the rate of elimination,
which is determined by the clearance and the plasma concentration
Clinical Pharmacokinetics 37
(CL × C(t)), should contribute negatively. Therefore, we can write
the complete differential equation
dC V
dt
DR CL C
d
..
This is a linear differential equation for C(t), another reason that
the model is called the linear pharmacokinetic model. The equation
can be integrated by using standard methods involving a variable
substitution as shown below.
Vd dC
DR CL C
dt
..
(6.1)
LetDRC
LC
..
Then
d
dC
CL dC
CL
d;
1
Substituting into(6.1):
Vd
CL
d
dt
Integration yields
ln .
CL
Vd
tConst
where const. is a constant. Taking the exponent:
e
CL
Vd
tConst.
Substituting for µ:
DR CL Ce e
CL
Vd
t
Const
..
(6.2)
At the initial time, t=0:
DR CL Ce
Const
..
0
where c
0
is the plasma concentration at time t=0.
Substituting for e
const
in(6.2):
38 ESSENTIAL PHARMACOLOGY
DR CL CDRCLC e
CL
Vd
t
.. ..
0
This can be rearranged to
C
DR
CL
eC
e
CL
Vd
t
CL
Vd
t
..
1
0
This expression, the solution of the differential equation, gives
the plasma drug concentration as a function of time. To understand
what it is telling us, let us consider some special conditions.
If no drug is administered, the plasma concentration falls as
defined by
CCe
CL
Vd
t
0
Thus the concentration of drug falls exponentially and semi- log
plots are linear (Figure 6.4):
Although this is a simple expression, it is somewhat inconven-
ient for intuitive application because we are not used to thinking
in terms of the exponential function, e. For this reason it is more
convenient to use this expression to derive a term that represents
2.0
1.5
1.0
Concentration
0.5
0.0
0510
Time, Hours
15 20 25
10
1
10
0
Concentration
10
–1
10
–2
0510
Time, Hours
15 20
25
Figure6.4 Time course of the fall of the concentration of a drug in plasma due
Clinical Pharmacokinetics 39
the time it takes for half of the drug to be eliminated, the so- called
half- time of elimination, T
1/2
(or half- life of elimination). At
what time has the plasma concentration fallen to one half of its
initial value?
CCWhen e
or e
CL
Vd
t
tT
Vd
CL
CL
Vd
t
CL
Vd
t
1
2
1
2
2
2
0 693
0
1
2
ln
.
wheere 0 693 2
.l
n
Now let us ask if this expression makes good intuitive sense.
The half- life of the drug is seen to be inversely proportional to the
clearance. This does seem to make sense since the drug will be
removed more quickly (a shorter half- life) if the mechanisms for
elimination of the drug are more active (a larger clearance). The
direct linear relationship to the volume of distribution is a little
more obscure. Remember that the volume of distribution deter-
mines how much of the drug must be given to a patient to achieve
a certain plasma concentration. If the volume of distribution is
large, then more drug must be given, and more total drug is in the
patient at a given plasma concentration. Therefore, it should take
longer (larger T
1/2
) to reduce the plasma concentration of such a
drug at a constant clearance value because more drug must be
eliminated.
We can use the half- life of a drug in a convenient rule of thumb
for the length of time it takes for a drug to be eliminated from a
patient after the last dose is administered. For practical purposes
we can say that the drug has been eliminated when over 95% is lost.
As each half- life passes, one half of the drug present is eliminated.
After one half- life 50% is lost, after two half- lifes, 75% is lost,
after 3half- lives 87.5%, after 4half- lives 93.75%, and after 5 half-
lives 96.875%.
40 ESSENTIAL PHARMACOLOGY
Therefore, we must wait 4– 5 half- lives for the drug to be
“completely” eliminated and no longer of clinical concern. For
most drugs currently in use, this ranges from hours to weeks.
Now let us analyze the situation where C
0
=0; that is, when we are
initiating therapy with a given drug, so no drug is present in the
plasma at time 0. We will assume that the dosing rate is constant
in time. This is the case clinically if, for example, a drug were being
given by intravenous infusion at a constant rate, a typical inpatient
procedure. If the drug were being given orally instead, as a pill every
few hours, then the dosing rate obviously would fluctuate with time.
We will see later that in this case our differential equation gives us
an accurate estimate of the average plasma concentration over time.
With C
0
=0, our solution to the differential equation reduces to:
A Sample Calculation
Penicillin G has a volume of distribution of 0.26l/Kg and a clear-
ance of 5.3ml/min- Kg. What is its half- life?
TVdCL
lKgm
lK
g
12
0 693
0 693 02653
/
./
..//./min
Notice that both the clearance and the volume of distribution are
given as “normalized values” relative to the weight of the patient in
Kg. Therefore, the Kg cancels out and it is not necessary to know the
actual weight of a particular patient to make this calculation. How-
ever, the Vd is given in terms of liters while the clearance involves
ml. Therefore, the l will have to be converted to mls to continue the
calculation:
0 693 260 53
34
./
./min
min
ml ml
Clinical Pharmacokinetics 41
C
DR
CL
e
DR
CL
e
CL
Vd
t
T
t
..
..
.
/
1
1
0 693
12
Note that the rate of accumulation of drug in the plasma is
governed by the same exponential term as was the washout of drug.
Therefore, the halftime for accumulation of drug to steady state is
also T
1/2
as defined above. 95% of the steady- state concentration
will be reached between four and five halftimes after instituting a
constant rate of dosing (Figure 6.5).
This can be a significant problem in the case of drugs with
long half- times. For example, if you were giving a drug with a
half- time of 48 hours, the full therapeutic benefit would not be
0510
Time, Hours
15 20
25
Concentration
2.0
1.5
1.0
0.5
0.0
Increase in plasma concentration
at the initiation of therapy
c10 = 1
cl1 = 0.2
dr1 = 0.0
c20 = 0
cl2 = 0.2
dr2 = 0.2
Figure6.5 Time course of drug accumulation in plasma during a constant rate
Compare with the bottom curve, which represents the fall in plasma concentra-