Quantitative Pharmacokinetics
Nonlinear Pharmacokinetics
Dr. Chalet Tan
1
Learning Objectives
 profiles of nonlinear kinetics
 sources and effects of dose dependency on ADME
 Michaelis-Menten equation and parameters (Vmax, Km) for
capacity-limited metabolism
2
Case Study
An epileptic patient who has not responded to
phenytoin after 2 weeks on 300 mg/day is observed to have a
plasma concentration of 4 mg/ml. Twenty days after the daily
dose is subsequently increased to 500 mg/day, the patient
develops severe toxicities. The plasma concentrations of
phenotoin is now 36 mg/L.
Review of Linear Pharmacokinetics
 ADME all obey first-order kinetics
 Pharmacokinetic parameters, e.g. elimination half-life (t1/2),
the elimination rate constant (k), the apparent volume of
distribution (V) and the clearance (CL) remain constant.
 Plasma drug concentration at a given time and AUC are
directly proportional to the dose.
 Concentrations of drug in plasma and tissues are below
protein binding saturation , i.e. fu and fuT remain constant.
4
Review of Linear Pharmacokinetics
i.v. bolus
i.v. bolus
normalized
10 mM
by dose
1 mM
1 mg
1h
10 mg
time
Log C
Log C
100 mM
100 mg
1 mg
1h
time
 Drug plasma concentrations are proportional to the dose.
 Drug plasma concentration-time profiles are superimposable
when normalized to the dose.
Review of Linear Pharmacokinetics
p. o.
2.5 mM
normalized
0.5 mM
by dose
25 mg
0.1 mM
1 mg
tmax
time
5 mg
Log C
Log C
p. o.
0.1 mM
1 mg
tmax
time
 Drug plasma concentrations are proportional to the dose.
 tmax remains unchanged.
 Drug plasma concentration-time profiles are superimposable
when normalized to the dose.
Cp or AUC
VD or CL or t1/2
Review of Linear Pharmacokinetics
Dose
Dose
7
Nonlinear Pharmacokinetics
i.v. bolus
i.v. bolus
800 mM
normalized
20 mM
10 mg
1 mM
1 mg
1h
time
by dose
Log C
Log C
100 mg
8 mM
2 mM
1 mM
1 mg 10 mg
1h
100 mg
time
 Drug plasma concentrations are not proportional to the dose.
 Drug plasma concentration-time profiles are not superimposable
8
when normalized to the dose.
Nonlinear Pharmacokinetics
p. o.
p. o.
5 mM
normalized
1 mM
by dose
100 mg
10 mg
1 mg
Log C
Log C
10 mM
1 mM
0.5 mM
0.1 mM
1 mg
10 mg
100 mg
time
time
 Drug plasma concentrations are not proportional to the dose.
 tmax may or may not change.
 Drug plasma concentrations are not superimposable when
normalized to the dose.
F, V, CL or t1/2
Cp or AUC
Nonlinear Pharmacokinetics
linear
Dose
linear
Dose
10
Common Sources for Nonlinear Pharmacokinetics
11
Linear vs. Nonlinear Pharmacokinetics
Linear
Nonlinear
(dose-independent)
(dose-dependent)
 ADME all obey first-order
kinetics.
 at least one of the ADME
processes is saturable.
 PK parameters (CL, V, F, Ka,  ≥1 PK parameters are dosedependent.
and t1/2) are constant.
 AUC is directly proportional
to the dose.
 AUC is disproportional to the
dose.
 Concentration vs. time
profile is superimposable for
all doses.
 Concentration vs. time profile is
not superimposable for different
doses.
12
Most Common Sources for Nonlinear Pharmacokinetics
 Capacity-limited oral absorption (F)
 Capacity-limited metabolism (CLH )
 Saturable protein binding (CLH, CLR, V )
 Capacity-limited excretion (CLR )
13
Capacity-Limited Oral Absorption (F)
 limited dissolution/solubility as the oral dose increases
 saturable transport across the intestinal mucosa as the oral
dose increases
 saturable first-pass metabolism in the intestinal epithelium (gut
wall) and/or liver as the oral dose increases
14
e. g.
- limited dissolution/solubility in the GI tract
normalized
to the dose
- Griseofulvin is poorly
water-soluble (10 mg/L).
- Less proportion of the drug
is being dissolved and
absorbed with the higher
dose.
- F decreases as the dose
increases.
- tmax remains the same.
15
e. g.
- Saturable transport across the intestinal epithelium
375 mg
750 mg
1500 mg
3000 mg
- Amoxicillin is actively transported
by peptide transporter in the small
intestine.
- The active transport becomes
saturated as the dose increases.
- F decreases as the dose
increases.
- tmax remains the same.
16
e. g.
- Saturable first-pass metabolism
- Nicardipine is metabolized by CYP3A4 in the intestinal
epithelium and hepatocytes.
- First-pass metabolism is saturated as the dose increases.
- F increases as the dose increases.
17
e. g.
- Saturable first-pass metabolism
18
Saturable Drug-Plasma Protein binding (CL,V)
 Drug-plasma protein binding is saturable
The saturation drug concentrations for binding with plasma
albumin and a1-acid glycoprotein are ~ 600 mM and 15 mM,
respectively.
 May increase CLH and/or CLR
 May increase V
 May be difficult to identify due to effect on both V and CL
19
- saturable plasma protein binding
- AUC and Cp of
trandolaprilat do not
increase proportionally
with D; Cp does not
accumulate with multiple
doses.
- As the dose increases,
binding to ACE
(angiotensin-converting
enzyme) in plasma is
saturated.
-Trandolaprilat is
elminated by glomerular
filtration
2 g/day
CLR= fu GFR
- As fu increases with
20
higher Cp, CLR increases.
Capacity-Limited Excretion (CLR)
 Active secretion and active reabsorption are saturable
processes.
 Saturated tubular secretion decreases CLR
Saturated tubular reabsorption increases CLR
CLR = fu GFR + (CLsecretion – CLreabsoption)
rateof tubular secretion
CLsecretion 
Cp
rateof reabsorption
CLreabsorption 
Cp
21
e. g.
- capacity-limited renal excretion
CLinulin
= GFR
When concentration is about lower than
mg/L, it could be linear. Since clearance
linear. But once it gets above 7, the
clearance rises, which makes it non-line
p.o. 30-80 mg
When Cp
above 10
mg/L starts to
saturate renal
reabs of Vit C.
i.v. 1.5-6 g
- Vitamin C is reabsorbed from urine by active transporter.
- Tubular reabsorption becomes saturated as Cp increases, i.e. as Cp
increases, CLreabsorption (= Ratereabsorption /Cp) decreases.
- ClR (=fu GFR –CLreabsorption) approaches GFR (fu=1) as Cp increases.
22
Capacity-Limited Metabolism (CLH ,F)
 Enzymatic reactions are saturable.
Vmax[ S ]
v
K m  [S ]
 Saturated hepatic metabolism decreases CLH.
 Saturated first-pass metabolism increases F.
23
e. g.
- capacity-limited metabolism
- Phenytoin is eliminated by
hepatic metabolism only.
- As the dosing rate
increases, Cp increases
disproportionally.
- As the dosing rate
increases, hepatic
metabolism is saturated
and CL decreases.
FD

- As the dosing rate
increases, it takes longer
time to reach steady state.
 CL Css
24
Michaelis-Menten Kinetics Applied to Metabolism
k 1
k 2
[ E ]  [ S ] [ ES] [ E ]  [ P]
k 1
[ E ][S ] k1
Km 

[ ES]
k1
Vmax  [ ET ]k2
Vmax[ S ]
v
K m  [S ]
n:
rate of metabolism
Vmax : maximum rate of metabolism
Km :
Michaelis constant, disassociation constant of ES
[S]:
drug concentration
25
Michaelis-Menten kinetics
Rate of Metabolism is NOT ALWAYS proportional to
drug concentrations
n
Vmax[ S ]
v
K m  [S ]
- When [S] = Km , n=1/2 nmax
Zero order
Non linear
Km is the drug concentration at
which half of the active sites on
enzymes are occupied.
Vmax[ S ]
- When [S] <<< Km , v 
Km
- When [S]
>>> Km ,
v  Vmax
First order
Km
[S]
26
Michaelis-Menten Kinetics Applied to CLM
Rate of elimination = CL x Cp
rateof elimination
CL 
Cp
Rate of metabolic elimination =
VmaxCp
v
K m  Cp
Vmax
metabolicclearance(CLM ) 
K m  Cp
27
Michaelis-Menten Kinetics Applied to Metabolism
VmaxCp
v
 CLM Cp
K m  Cp
When Cp << Km , linear PK
zero-order
nmax n
Vmax
CLM 
K m  Cp
Vmax
CLM 
Km
non-linear
VmaxCp
v
 CLM Cp
Km
When Cp >> Km ,
v  V max
first-order
Km
[Drug]
CLM  0
28
Linear vs. Saturable Metabolism
nonlinear
linear
Vmax
CLM 
K m  Cp
CL
CL
Clearance is independent of Cp
Cp
Cp
29
Michaelis-Menten Kinetics Applied to Metabolism
Rate of metabolic elimination =
Vmax[ S ]
v
K m  [S ]
At the steady-state following multiple dosing ,
FD
V max Css
Rate of metabolic eliminatio n  Dosing Rate (
)

Km  Css
V max
Metabolic clearance (CLM ) 
Km  Css
Km D / 
CSS 
V max / F  D / 
D
(V max/ F ) Css


Km  Css
30
Linear vs. Saturable Metabolism
nonlinear
CSS
CSS
linear
D/
D/
FD

 CL Css
31
Most Common Sources for Nonlinear Pharmacokinetics
 Capacity-limited oral absorption (F)
 Capacity-limited metabolism (CLH )
 Saturable protein binding (CLH, CLR, V )
 Capacity-limited excretion (CLR )
32
Case Study
At a daily intake of 75 mg of ascorbic acid (vitamin C), the
steady-state plasma concentration is 9 mg/L, whereas at a daily
dose of 10,000 mg, the steady-state concentration is about 19
mg/L in a healthy volunteer. The renal clearance of ascorbic
acid is less than 0.5 ml/min at the plasma concentration of 9
mg/ml, whereas the renal clearance is 21 ml/min at 19 mg/L.
Vitamin is absorbed by passive facilitated diffusion in the
small intestine, and undergoes tubular reabsorption in the
kidney.
Maintenance Dose Selection for Phenytoin
- Phenytoin is eliminated by
hepatic metabolism
(CYP2C9) only.
-Variability in Vmax and Km
values in patients causes a
wide range in the effective
doses needed to achieve
therapeutic levels.
therapeutic range= 10-20 mg/ml
= 10-20 mg/L
34
Michaelis-Menten Kinetics Applied to Metabolism
Rate of metabolic elimination =
Vmax[ S ]
v
K m  [S ]
At the steady-state following multiple dosing ,
FD
V max Css
Rate of metabolic eliminatio n  Dosing Rate (
)

Km  Css
V max
Metabolic clearance (CLM ) 
Km  Css
Km D / 
CSS 
V max / F  D / 
D
(V max/ F ) Css


Km  Css
35
How to Obtain Vmax/F and Km
D
(V max/ F ) Css


Km  Css
y= m x - b
-Km
*
* Slope = Vmax /F
Slope = m
= y2-y1
x2-x1
y
Css
(V max / F ) Css
Css 
 Km
D /
Css / Dose rate
-b
*
x
*
The Direct Linear Plot
Vmax/F
dosing rate 2
dosing rate 1
-C2
-C1
Biochem J, 139:715-20 (1974)
Km
37
Maintenance Dose Selection for Phenytoin
A patient has been taking phenytoin (PHE) 150 mg b.i.d for 4
months. His plasma levels of PHE averaged 5 mg/L on this
dose. Adjustment in dose to 250 mg b.i.d eventually led to a
new plateau level of 20 mg/L. Assuming true steady state,
strict patient compliance and that the measured plasma
concentrations represent average levels over the dosing
interval.
a) use a graphical method to estimate the patient's operative
Vmax/F and Km values;
b) estimate a daily dose which should provide a steady-state
plasma level of 12 mg/L.
38
Drug-Protein Binding
Nonlinear Pharmacokinetics
fu
V  Vp  VT
fu T
Rate of metabolic elimination =
CLM 
Clearance Concepts
EH 
V max
Km  Cp
Rat e of Met abolism Dosing Rat e (
fuCL int
QH  fuCL int
D

(V
Css 
(V

fu CL int
CL H  Q H EH  Q H (
)
Q H  fuCL int
fu CL int
QH

Q H  fu CL int Q H  fu CL int
Css
F  1 E  1
VmaxCp
K m  Cp
FD

)
V max Css
Km  Css
/ F ) Css
Km  Css
max
/ F ) Css
 Km
D /
max
*
*
Slope=Vmax /F
Css / Dose rate
Vmax/F
-Km
dosing rate 2
dosing rate 1
-C2
-C1
Km
40
41
42
43
44