1 Nonlinear Pharmacokinetics Dr. Chalet Tan Quantitative Pharmacokinetics

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Nonlinear Pharmacokinetics

Dr. Chalet Tan

Quantitative Pharmacokinetics

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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

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.

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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.


Drug plasma concentrations are proportional to the dose.

Drug plasma concentration-time profiles are superimposable when normalized to the dose.

normalized

by dose

time

i.v. bolus

100 mg1

mg

10 mg

Log

C

100M

10M

1M

1 h

i.v. bolus

1 mg

timeLo

g C

1 h


Drug plasma concentrations are proportional to the dose.

tmax remains unchanged.

normalized

by dose

p. o.

25 mg

1 mg

5 mg

Log

C

timetmax

0.5M

0.1M

2.5M

Drug plasma concentration-time profiles are superimposable when normalized to the dose.

p. o.

1 mg

Log

C

timetmax

0.1M

7


Dose

Cp

or A

UC

Dose

VD o

r C

L or

t1

/2

8


Drug plasma concentrations are not proportional to the dose.

Drug plasma concentration-time profiles are not superimposable when normalized to the dose.

normalized

by dose

i.v. bolus

100 mg

1 mg

10 mg

time

Log

C

20M

1M

800M

1 htime

i.v. bolus

100 mg1 mg 10 mg

Log

C

1 h

1M2M

8M


p. o.

100 mg10 mg

Log

C

time

1 mg

M

1M

10M

normalized

by dose

Drug plasma concentrations are not superimposable when normalized to the dose.

Drug plasma concentrations are not proportional to the dose.

tmax may or may not change.

p. o.

100 mg

10 mg

Log

C

time

1 mg

1M

0.5M

0.1M

10


linear

linear

Dose

Cp

or A

UC

Dose

F,

V,

CL

or t

1/2

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Common Sources for Nonlinear Pharmacokinetics

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Linear vs. Nonlinear Pharmacokinetics

ADME all obey first-order kinetics.

PK parameters (CL, V, F, Ka, and t1/2) are constant.

AUC is directly proportional to the dose.

Concentration vs. time profile is superimposable for all doses.

Linear Nonlinear

at least one of the ADME processes is saturable.

≥1 PK parameters are dose-dependent.

AUC is disproportional to the dose.

Concentration vs. time profile is not superimposable for different doses.

(dose-dependent)(dose-independent)

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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 )

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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

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- 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.

e. g.

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- 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.

e. g.

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- 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.

e. g.

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- Saturable first-pass metabolism

e. g.

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Saturable Drug-Plasma Protein binding (CL,V)

Drug-plasma protein binding is saturable

The saturation drug concentrations for binding with plasma albumin and 1-acid glycoprotein are ~ 600 M and 15 M, respectively.

May increase CLH and/or CLR

May increase V

May be difficult to identify due to effect on both V and CL

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- 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

CLR= fu GFR

- As fu increases with higher Cp, CLR increases.

2 g/day

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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)

pCCL

secretion tubular of ratesecretion

pCCL

onreabsorpti of rateonreabsorpti

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- capacity-limited renal excretion

- 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.

CLinulin = GFR

i.v. 1.5-6 g

p.o. 30-80 mg

When Cp above 10 mg/L starts to saturate renal reabs of Vit C.

e. g. When concentration is about lower than 7 mg/L, it could be linear. Since clearance is linear. But once it gets above 7, the clearance rises, which makes it non-linear.

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Enzymatic reactions are saturable.

Saturated hepatic metabolism decreases CLH.

Saturated first-pass metabolism increases F.

Capacity-Limited Metabolism (CLH ,F)

][

][max

SK

SVv

m

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- capacity-limited metabolism

e. g.

- 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.

- As the dosing rate increases, it takes longer time to reach steady state.

ssCCLFD

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Michaelis-Menten Kinetics Applied to Metabolism

][

][max

SK

SVv

m

1

1

][

]][[

k

k

ES

SEKm

][ 2max kEV T

][][][][][21

1

PEESSEkk

k

: rate of metabolism

Vmax : maximum rate of metabolism

Km : Michaelis constant, disassociation constant of ES

[S]: drug concentration

mK

SVv

][max

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Michaelis-Menten kinetics

Rate of Metabolism is NOT ALWAYS proportional to drug concentrations

][

][max

SK

SVv

m

- When [S] = Km , =1/2 max

Km is the drug concentration at which half of the active sites on enzymes are occupied.

- When [S] <<< Km ,

- When [S] >>> Km , maxVv

[S]Km

Zero order

First order

Non linear

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Michaelis-Menten Kinetics Applied to CLM

Rate of elimination = CL x Cp

pm

p

CK

CVv

max

Rate of metabolic elimination =

pCCL

neliminatio of rate

pm

MCK

VCL

max)( clearance metabolic

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When Cp << Km , linear PK

pMpm

pCCL

CK

CVv max

pM

m

pCCL

K

CVv max

When Cp >> Km ,

maxVv

pm

MCK

VCL

max

m

MK

VCL max

[Drug]

max

Km

zero-order

first-order

non-linear

0MCL

29Cp

CL

Linear vs. Saturable Metabolism

Clearance is independent of Cp

linear nonlinear

CL

pm

MCK

VCL

max

Cp

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At the steady-state following multiple dosing ,

ssmM

CK

VCL

max )( clearance Metabolic

ssm

ss

CK

CFVD

)/( max

ssm

ss

CK

CVFD

max)( Rate Dosing neliminatio metabolic of Rate

][

][max

SK

SVv

m Rate of metabolic elimination =

//

/

max

DFV

DKC

mSS

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Linear vs. Saturable Metabolism

linear nonlinear

D/

CS

S

D/

CS

S

ssCCLFD

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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 )

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.

Case Study

Vitamin is absorbed by passive facilitated diffusion in the

small intestine, and undergoes tubular reabsorption in the

kidney.

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Maintenance Dose Selection for Phenytoin

therapeutic range= 10-20 g/ml = 10-20 mg/L

- 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.

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At the steady-state following multiple dosing ,

ssmM

CK

VCL

max )( clearance Metabolic

ssm

ss

CK

CFVD

)/( max

ssm

ss

CK

CVFD

max)( Rate Dosing neliminatio metabolic of Rate

][

][max

SK

SVv

m Rate of metabolic elimination =

//

/

max

DFV

DKC

mSS

How to Obtain Vmax/F and Km

ssm

ss

CK

CFVD

)/( max

mss

ss KD

CFVC

/ )/ ( max

**

Css / Dose rate

Css

Slope = Vmax /F

-Km

y= m x - b

Slope = m

= y2-y1

x2-x1

**

x

y

-b

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The Direct Linear Plot

Vmax/F

Km-C1-C2

dosing rate 1

dosing rate 2

Biochem J, 139:715-20 (1974)

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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.

)

(

int

int

CLfQ

CLfQEQCL

uH

uHHHH

int int

int

11

CLfQ

Q

CLfQ

CLfEF

uH

H

uH

u

Tfu

fuVVV Tp

**

Slope=Vmax /F

-Km

Css / Dose rate

Css

Nonlinear PharmacokineticsDrug-Protein Binding

int

int

CLfQ

CLfE

uH

uH

Vmax/F

Km-C1-C2

dosing rate 1

dosing rate 2

ssm

ss

CK

CVFD

max)( Rate Dosing Metabolism of Rate

ssm

ss

CK

CFVD

)/( max

Clearance Concepts

pmM

CK

VCL

max

pm

p

CK

CV

max

Rate of metabolic elimination =

mss

ss KD

CFVC

/ )/ ( max

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Documents

1 Nonlinear Pharmacokinetics Dr. Chalet Tan Quantitative Pharmacokinetics