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Nonlinear pharmacokinetics
Dr Mohammad Issa Saleh
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Introduction: Linear
• Pharmacokinetic parameters, such as elimination half life
(t1/2), the elimination rate constant (K), the apparent
volume of distribution (V), and the systemic clearance
(Cl) of most drugs are not expected to change when
different doses are administered and/or when the drug is
administered via different routes as a single dose or
multiple doses
• The kinetics of these drugs is described as linear, or
dose-independent, pharmacokinetics and is
characterized by the first-order process
• The term linear simply means that plasma concentration
at a given time at steady state and the area under the
plasma concentration versus time curve (AUC) will both
be directly proportional to the dose administered
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Introduction: Linear
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Introduction: Nonlinear
• For some drugs, however, the above
situation may not apply
• For example, when the daily dose of
phenytoin is increased by 50% in a patient
from 300 mg to 450 mg, the average
steady-state plasma concentration,
(Cp)ss, may increase by as much as 10fold
• This dramatic increase in the
concentration (greater than directly
proportional) is attributed to the nonlinear
kinetics of phenytoin
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Introduction: Nonlinear
• For drugs that exhibit nonlinear or dose
dependent kinetics, the fundamental
pharmacokinetic parameters such as clearance,
the apparent volume of distribution, and the
elimination half life may vary depending on the
administered dose
• This is because one or more of the kinetic
processes (absorption, distribution and/or
elimination) of the drug may be occurring via a
mechanism other than simple first-order kinetics
• For these drugs, therefore, the relationship
between the AUC or the plasma concentration at
a given time at steady state and the
administered dose is not linear
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Introduction: Nonlinear
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Introduction: Nonlinear
Administration of different
doses of drugs with nonlinear
kinetics may not result in
parallel plasma concentration
versus time profiles expected
for drugs with linear
pharmacokinetics
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Introduction: Nonlinear
• Nonlinearity may arise at any one of the pharmacokinetic
steps, such as absorption, distribution and/or elimination
• For example, the extent of absorption of amoxicillin
decreases with an increase in dose
• For distribution, plasma protein binding of disopyramide
is saturable at the therapeutic concentration, resulting in
an increase in the volume of distribution with an increase
in dose of the drug
• As for nonlinearity in renal excretion, it has been shown
that the antibacterial agent dicloxacillin has saturable
active secretion in the kidneys, resulting in a decrease in
renal clearance as dose is increased
• Both phenytoin and ethanol have saturable metabolism,
which means that an increase in dose results in a
decrease in hepatic clearance and a more than
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proportional increase in AUC
Nonlinearity in metabolism
Capacity-limited metabolism
• Capacity-limited metabolism is also called
saturable metabolism, Michaelis–Menten
kinetics
• Nonlinearity in metabolism, is one of the
most common sources of nonlinearity
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Nonlinearity in metabolism
Capacity-limited metabolism
• The rate of metabolism, or the rate of
elimination if metabolism is the only
pathway of elimination, is defined by the
Michaelis–Menten equation:
Vmax C
Metabolism rate 
Km  C
• where Vmax is the maximum rate (unit:
amount/time) of metabolism; Km is the
Michaelis–Menten constant (unit: same as
the concentration [amount/volume]), and C
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is the drug concentration
Nonlinearity in metabolism
Capacity-limited metabolism
• Two cases:
– Km>>C
– Km<<C
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Nonlinearity in metabolism
Capacity-limited metabolism
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Estimation of Michaelis–Menten parameters
from administration of a single dose
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Estimation of Michaelis–Menten parameters
from administration of a single dose
Terminal line
(C<< Km)
Observed conc
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Estimation of Michaelis–Menten parameters
from administration of a single IV bolus dose
IV bolus
administration
(dose = X0)
Drug amount in the
Body (X)
Elimination process
Based on the assumption of nonlinear elimination process:
Vmax C
Eliminatio n rate 
Km  C
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Estimation of Michaelis–Menten parameters
from administration of a single IV bolus dose
Derivation of observed concentration equation
Vmax
Vmax C
dX
C
dC
Vd


dt
K m  C Divide by Vd
dt
Km  C
Assume that

Vmax C
dC
dt
K m  C Rearrangement
 V
Vmax  max
Vd
- Vmax

Km
dt 
dC  dC
C
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Estimation of Michaelis–Menten parameters
from administration of a single IV bolus dose
- Vmax

Km
dt 
dC  dC
C
Integration

C0  C
Vmax t
lnC 
 lnC 0 
Km
Km
Previous equation represent the observed conc
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Estimation of Michaelis–Menten parameters
from administration of a single dose
Terminal line
(C<< Km)
Observed conc

C0  C
Vmax t
lnC 
 lnC 0 
Km
Km
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Estimation of Michaelis–Menten parameters
from administration of a single IV bolus dose
Derivation of terminal concentration equation
When C>>Km: Km+C ≈ C
Vmax C
Vmax C
dX
dt
Km  C
Km

Vmax
dC
Divide by Vd

C
dt
Km
lnC  lnC 0

*
Vmax

t
Km
First order elimination
This equation represent
the terminal
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concentration equation
Estimation of Michaelis–Menten parameters
from administration of a single dose
Terminal line
(C<< Km)

V
*
lnC  lnC 0  max t
Km
Observed conc

C0  C
Vmax t
lnC 
 lnC 0 
Km
Km
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Estimation of Michaelis–Menten parameters
from administration of a single dose

Vmax
Slope(log)  
2.303  K m
K m 
C0
C *
ln 0
C0
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Estimation of Michaelis–Menten parameters
from administration of a single IV bolus dose
•
Steps:
1. Plot log(conc)-time profile
2. Get the initial conc (C0)
3. Extrapolate the terminal line to get an initial
terminal conc (C0*)
4. Calculate the slope of the terminal line using
the log
K m 
C0
C *
ln 0
C0

Vmax  2.303  K m  slope
Vd 
Dose
C0

Vmax  Vmax  Vd
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Example 1
•
The following concentration time profile was
constructed after administration of 300 mg
dose of drug A to an adult patient.
find
1.
2.
3.
4.
Vm
Km
Vd
The dose required to produce a steady-state conc
of 20 mg/L in this patient.
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Example 1
• From the figure the following were
calculated: C0=10 mg/L, C0*= 45 mg/L,
and Slope (using the log) = -0.985
Km 
C0
C 
ln  
 C0 
*
0
 10
 45 
ln  
 10 
 6.65 mg / L
Vm   slope * Km * 2.303  0.985 * 6.65 * 2.303  15.1 mg / L / hr
Dose 300
Vd 

 30 L
C0
10
Vm  Vm * Vd  15.1 * 30  453 mg / hr
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Example 1
• The dose required to produce a steadystate concentration of 20 mg/L in this
patient:
VmC SS
453 * 20
Dosing rate 

 340 mg / hr
Km  C SS 6.65  20
Daily Dose  Dosing rate * 24  8160 mg  8.16 gm
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Estimation of Michaelis–Menten parameters
from two steady-state drug concentrations
arising from two dosing rates
• At steady state:
Input rate = output rate
Dosing rate = Elimination rate
Vmax C
R
Km  C
R is the input rate that is described as:
R
FD

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Estimation of Michaelis–Menten parameters
from two steady-state drug concentrations
arising from two dosing rates
• Two dosing rates resulted in the following
steady state conc:
Dosing rate
Css
R1
Css1
R2
Css2
• Estimate Vmax and Km
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Estimation of Michaelis–Menten parameters
from two steady-state drug concentrations
arising from two dosing rates
Vmax Css1
R1 
 Vmax Css1  R1  K m  R1  Css1
K m  Css1
Vmax Css2
R2 
 Vmax Css2  R2  K m  R2  Css2
K m  Css2
Two equations with two unknowns
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Example 2
• RM is a 32 year old, 80kg male who is being seen in
the Neurology Clinic. Prior to his last visit he had
been taking 300mg of Phenytoin daily; however,
because his seizures were poorly controlled and
because his plasma concentration was only 8mg/L,
his dose was increased to 350mg daily. Now he
complains of minor CNS side effects and his
reported plasma Phenytoin concentration is 20mg/L.
Renal and hepatic function are normal. Assume that
both of the reported plasma concentrations
represent steady state and that the patient has
compiled with the prescribed dosing regimens.
Calculate RM’s apparent Vm and Km and a new daily
dose of Phenytoin that will result in a steady state
level of about 15mg/L.
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Example 2
R1* Km  R1* CSS (1)  Vm * CSS (1)
R 2 * Km  R 2 * CSS (2)  Vm * CSS (2)
R1  300 mg / day, CSS (1)  8 mg / L
R 2  350 mg / day, CSS (2)  20 mg / L
300 * Km  300 * 8  Vm * 8  37.5 * Km  300  Vm (1)
  
350 * Km  350 * 20  Vm * 20  17.5 * Km  350  Vm (2)
Eqn (1)- Eqn(2):
20 * Km  50  0
50
Km 
 2.5 mg / L
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Eqn (1):
Vm  37.5 * Km  300  37.5 * 2.5  300  393.75 mg / day
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Example 2
• Calculate RM’s a new daily dose of
Phenytoin that will result in a steady state
level of about 15mg/L
VmC SS
393.75 *15
Dosing rate 

 337.5 mg / day
Km  CSS
2.5  15
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