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42 ESSENTIAL PHARMACOLOGY
achieved for 8 or 9 days, which would likely not be acceptable in
treating a serious, acute illness. In addition, possible undesirable
side effects might not be manifest until after a week, at which
time the association with the initiation of medication might not
be apparent to the patient. In order to avoid these problems, it is
typical to administer a larger dose of the drug initially, called a
loading dose, in order to raise the plasma concentration to the
target level as soon as possible, and then provide the maintenance
dosage that will prevent the plasma concentration from falling
significantly. The loading dose might be administered, for example,
as an injection in the doctor’s office, and then the maintenance
dose supplied in the form of pills to be taken orally one or several
times a day. The appropriate loading dose can be calculated from
the relationship between dose, the plasma concentration, and the
volume of distribution of the drug:
loading doseV target concentrationV
dC
ds
s
where C
ss
is the desired, steady- state concentration.
The steady- state solution of the differential equation governing the
pharmacokinetic model occurs when the derivative of the plasma
concentration is equal to 0. This occurs when
D.R.= CL × C (refer to the equation above). Using C
ss
to stand
for the steady state concentration, D.R. = CL × C
ss
. But remember
that CL × C
ss
is defined as the rate of elimination. Therefore, dosing
rate= rate of elimination at the steady state, which is the funda-
mental equation defining the steady state that we started with. Rear-
ranging this equation as
CDRCL
ss
../
we see we have a very simple and useful relationship that governs
the steady- state plasma concentration of a drug.
Clinical Pharmacokinetics 43
A Sample Calculation
Isradipine, a calcium channel blocker, is given to a patient at a
dosing rate of 2 mg every eight hours. What is the steady- state drug
concentration in the plasma?
CDRCLmg hour
sC
L
ss
..///28
You will need to know the clearance of isradipine, which is 10 ml/
min- Kg. However, you will also need to know the weight of the patient
so you can use a “total body” clearance in this equation since the dos-
ing rate applies to the whole patient. Suppose the patient weighs 75 Kg:
DR CL mg hoursml-Kg Kg..////min28 10 75
You will also need to convert the hours in the dosing rate into min-
utes since the clearance term is given in terms of minutes:
DR CL mg ml
mg ml
..//min/ / min
./min //min
2860 750
0 0042 750
0 0000056 56././mg ml ng ml ng nanogram
Now let’s “run” the model for a few other conditions. Let the target
therapeutic concentration of drug in the plasma be 1 μg/ml.
Assume the volume of distribution is 60 ml/Kg and the clearance
is 0.2 ml/min- Kg.
1. Compare 2 dosing rates (Figure 6.6)
C
0
=0 μg/ml, CL=0.2 ml/min- Kg
D.R.=0.2 μg/min- Kg (curve A, bottom curve)
D.R.=0.4 μg/min- Kg (curve B, top curve)
2. Compare 2 clearance rates (Figure 6.7)
C
0
=0 μg/ml, D.R.=0.2 μg/min- Kg
CL = 0.2 ml/min- Kg (curve A, bottom curve)
CL=0.1 ml/min- Kg (curve B, top curve)
44 ESSENTIAL PHARMACOLOGY
0510
Time, Hours
15 20
25
Concentration
2.0
1.5
1.0
0.5
0.0
Comparing two dosing rates
c10 = 0
cl1 = 0.2
dr1 = 0.2
c20 = 0
cl2 = 0.2
dr2 = 0.4
Figure6.6
0510
Time, Hours
15 20
25
Concentration
2.0
1.5
1.0
0.5
0.0
Comparing two values of clearance
c10 = 0
cl1 = 0.2
dr1 = 0.2
c20 = 0
cl2 = 0.1
dr2 = 0.2
Figure6.7
Clinical Pharmacokinetics 45
3. Apply a loading dose (Figure 6.8)
CL=0.2 ml/min- Kg, D.R.=0.2 μg/min- Kg
C
0
=0 μg/ml (curve A, bottom curve)
C
0
=1 μg/ml (curve B, top curve (straight line)
4. Change dosing rate (Figure 6.9)
CL=0.2 ml/min- Kg
C
0
=0 μg/ml, D.R.=0.2 μg/min- Kg (curve A)
C
0
=1 μg/ml, D.R.=0.3 μg/min- Kg (curve B)
Note in Figure6.9 that if the dosage is changed (i.e., from 0.2
to 0.3 μg/min- kg) the new steady state is approached with the same
characteristic T1/2 (31/2hours in this case).
This illustrates a very general rule that systems following lin-
ear pharmacokinetics obey. When changing either the dosing
rate or the clearance (increasing or decreasing, large or small
changes), the plasma concentration goes from what it was to
0510
Time, Hours
15 20
25
Concentration
2.0
1.5
1.0
0.5
0.0
Application of a loading dose
at the initiation of therapy
c10 = 1
cl1 = 0.2
dr1 = 0.0
c20 = 1
cl2 = 0.2
dr2 = 0.2
Figure6.8
plasma concentration to achieve the target concentration immediately (depending
-
46 ESSENTIAL PHARMACOLOGY
what if will be at the new steady state with the same T1/2 char-
acteristic of the system. This is one of the most useful features
of the linear model for making calculations and predictions!
Rapid reduction of plasma concentration. Now let us consider
a situation in which it is found that the plasma concentration is too high
due to a dosing rate that is too high, leading to undesirable side effects.
What should you do? It probably would be best to reduce the dosing
rate. You could do this in two ways. Recalculate the correct dosing rate
to give the desired steady state. Or, you could go “cold turkey” – stop
administering the drug until the plasma concentration falls to an appro-
priate level, then reintroduce the drug at a more appropriate rate.
This situation is illustrated in Figure6.10. Suppose the thera-
peutic target is 1.0 μg/ml, but the patient’s plasma concentration
is measured to be 2.0 μg/ml. Since you want to reduce the plasma
concentration by 50%, you could reduce the dosing rate from 4.0 to
2.0 μg/min- Kg (a 50% decrease) (top curve). This looks good. As seen
in the figure, the plasma concentration falls to 1.0 μg/ml as desired.
0510
Time, Hours
15 20
25
Concentration
2.0
1.5
1.0
0.5
0.0
Effects of a change in dosing rate
c10 = 0
cl1 = 0.2
dr1 = 0.2
c20 = 1
cl2 = 0.2
dr2 = 0.3
Figure6.9
μ
μ
Clinical Pharmacokinetics 47
However, consider that you will have to wait 4– 5half- lives for this
change to be completed. This might not be acceptable if the higher
plasma concentration is having serious side effects. A wiser choice
would be to stop administering the drug. Then in just one half- life the
desired plasma concentration would be achieved (bottom curve). At
that point in time dosing could be resumed at the desired lower rate.
Goal: You wish to reduce the plasma level from 2.0 to 1.0.
CL=0.2 ml/min- Kg,
C
0
=2 μg/ml, D.R.=0.2 μg/min- Kg (top curve)
C
0
=2 μg/ml, D.R.=0 μg/min- Kg (bottom curve)
6.4 A “Dose- Dependent” (or “Non- Linear”)
Pharmacokinetic Model
Although the behavior of most drugs in current use follows the linear
pharmacokinetic model we have just described, a number of agents are
used at levels that begin to saturate the mechanisms of drug clearance.
0510
Time, Hours
15 20
25
Concentration
2.0
1.5
1.0
0.5
0.0
Strategies for reducing plasma concentration
c10 = 1
cl1 = 0.2
dr1 = 0.2
c20 = 2
cl2 = 0.2
dr2 = 0.0
Figure6.10
-
centration to the desired target level more rapidly than reducing the dosage to the
48 ESSENTIAL PHARMACOLOGY
Some examples include theophylline, used for treatment of asthma, and
phenytoin, used for treatment of epilepsy, as well as aspirin and ethanol,
both readily available to patients. Here we will develop a pharmacoki-
netic model for drugs that behave this way. We will learn that a risk inher-
ent in using drugs with this behavior is that plasma concentrations
can rise to very high, potentially toxic levels if dosing is not carefully
controlled. This is because the ability of the body to remove or destroy
these drugs may not be able to keep up with the rate at which they are
being administered (remember, in contrast, that in the linear model the
body can always keep up because elimination gets faster as the plasma
concentration becomes greater (Rate of elimination=Cl × C)).
In this model we will make the realistic assumption that
drug elimination is due to the action of an enzyme that follows
classical Michaelis- Menton kinetics. Thus the rate of elimina-
tion is given by the expression for enzyme velocity according to
Michaelis- Menton theory:
Rate of eliminationVmC Km C
/.
We may also write this expression as Vm/(Km + C) × C, which
emphasizes that the important pharmacokinetic parameter,
clearance, that relates the rate of elimination to the plasma con-
centration is now Vm/(Km + C). That is to say, the clearance is
no longer a constant, but is a function of plasma concentration.
Indeed, the clearance decreases as plasma drug concentration
increases. That is why this model is sometimes referred to as a
“dose dependent model,” because the clearance will depend on
the size of the dose given (which in turn determines the magni-
tude of the plasma concentration). It is also called a “nonlinear
model” because the rate of drug elimination is no longer linearly
related to the plasma concentration.
The differential equation for the rate of change of the total amount
of drug in the patient is now:
dC V
dt
DR
VC
KC
d
m
m
..
Clinical Pharmacokinetics 49
This is a nonlinear differential equation without an analytical
solution. However, it can be solved numerically, most conveniently
on a computer. In the examples shown below the Euler method for
numerical analysis has been used to provide solutions. Let us con-
sider the plasma concentration at which the steady state is achieved:
DR
VC
KC
C
DR K
VDR
ms
s
ms
s
ss
m
m
..
..
..
You can see from the expression above that we have real trouble
when the dosing rate (D.R.) approaches the V
m
for the clearance pro-
cesses and the predicted steady- state concentration approaches infin-
ity! This simply means that if Vm equals D.R there is no steady state.
The same conclusion would follow if the D.R. were larger than the Vm
because then the expression for the steady- state concentration would
yield a negative number.
Now let’s “run” a few examples with this model:
Figure 6.11. Curve A: The dose- independent model with
D.R.=0.2 μg/min- Kg,
CL ml Kg Cugml02 0
0
./min ,/.
Curve B: The dose- dependent model with D.R.=0.2μg/min- Kg,
KugmlV ug -KgC ug ml
mm
10
20
0
/, ./min
,/
.
Note that in this case the dose- dependent model predicts
that the plasma concentration will not come to a steady state.
This is the major concern when using drugs that do not follow
linear pharmacokinetics.
Can we find a dosing rate that is low enough that a stable steady
state can be obtained? We showed above that at a steady state
50 ESSENTIAL PHARMACOLOGY
DR
VC
KC
ms
s
mss
..
So if we put in 1 μg/ml for Css, as desired, and 1 μg/ml for Km
and 0.2 μg/min- Kg for Vm as before, then D.R. = 0.1 μg/min- Kg.
Using these values, the graphs below are obtained (Figure 6.12)
Curve A: The dose- independent model, as in Figure6.11.
Curve B: The dose- dependent model with D.R.=0.1μg/min- Kg,
K
m
=1 μg/ml, V
m
=0.2 μg/min- Kg, C
0
=0 μg/ml.
Note that a steady state can be achieved in this model if the dos-
ing rate is adjusted properly, although the approach to the steady
state is somewhat slower because of the lower dosing rate. If the slow
approach to the steady state is problematic in a therapeutic setting, it
can be overcome by using a loading dose as in Figure6.13.
Note that with the dose- dependent model, as seen before with
the dose- independent model, the introduction of a loading dose
assures a constant plasma drug concentration from the initiation of
therapy at time 0.
2.0
1.5
1.0
Concentration
0.5
0.0
0510
Time, Hours
A
B
15 20
25
Figure6.11 Comparison of the increase in plasma concentration in the lin-
0
m
V
m
0
Clinical Pharmacokinetics 51
Now let’s consider the “washout” of a drug that follows nonlin-
ear kinetics (Figure 6.14). We take as an example the “recreational”
drug ethanol. In this case we’ll set C
0
=2 μg/ml, D.R.=0, K
m
=1 μg/
ml, and V
m
=0.3 μg/min- Kg.
2.0
1.5
1.0
Concentration
0.5
0.0
0510
Time, Hours
A
B
15 20
25
Figure6.12 Comparison of the increase in plasma concentration in the lin-
ear model and the nonlinear model when the dosing rate is reduced to avoid
m
m
0
2.0
1.5
1.0
Concentration
0.5
0.0
0510
Time, Hours
A
B
15 20
25
Figure6.13 Addition of a loading dose to the nonlinear model overcomes the
C
0
m
m