DOSE- AND TIME-DOSE- AND TIME-DEPENDENT DEPENDENT

PHARMACOKINETICSPHARMACOKINETICS

CAUSES OF DOSE- OR TIME-DEPENDENT KINETICSCAUSES OF DOSE- OR TIME-DEPENDENT KINETICS

PROCESSPROCESS EXAMPLEEXAMPLE PARAMETERPARAMETER

Saturable gut wall transport riboflavin FSaturable gut wall metabolism salicylamide FPoor solubility griseofulvin F

Saturable plasma protein binding disopyramide fup

Active tubular secretionActive tubular secretion penicillin Gpenicillin G CLCLRR

Active tubular reabsorptionActive tubular reabsorption ascorbic acidascorbic acid CLCLRR

Alterations in urine pHAlterations in urine pH salicylic acidsalicylic acid CLCLRR

Alterations in urine flowAlterations in urine flow theophyllinetheophylline CLCLRR

NephrotoxicityNephrotoxicity gentamicingentamicin CLCLRR

CAUSES OF DOSE- OR TIME-DEPENDENT KINETICSCAUSES OF DOSE- OR TIME-DEPENDENT KINETICS

PROCESSPROCESS EXAMPLEEXAMPLE PARAMETERPARAMETER

Capacity-limited metabolism phenytoin CLH

Autoinduction carbamazepine CLH

Co-substrate depletion acetaminophen CLH

Product (metabolite) inhibition phenylbutazone CLH

I. ABSORPTIONI. ABSORPTIONEffect of dose on riboflavin urinary recovery when given on an empty stomach. Date from: Levy G, Jusko WJ. Factors affecting the absorption of riboflavin in man. J Pharm Sci 55:285-289, 1966.

Effect of dose on ascorbic acid absorption. Data from Blanchard J et al. Am J Clin Nutr 66:1165-1171, 1997

Steady-state Vitamin C plasma concentration as a function of dose in 13 Steady-state Vitamin C plasma concentration as a function of dose in 13 female subjects receiving doses from 30 to 2,500 mg. From: Levine M, et al. female subjects receiving doses from 30 to 2,500 mg. From: Levine M, et al. A new recommended dietary allowance of vitamin C for healthy young A new recommended dietary allowance of vitamin C for healthy young women. women. Proc Natl Acad Sci USAProc Natl Acad Sci USA 98:9842-9846, 2001. 98:9842-9846, 2001.

From: Levine M, et al. A new recommended dietary allowance of vitamin C for healthy From: Levine M, et al. A new recommended dietary allowance of vitamin C for healthy young women. young women. Proc Natl Acad Sci USAProc Natl Acad Sci USA 98:9842-9846, 2001. 98:9842-9846, 2001.

Reproduced from: Rowland M, Tozet TN. Reproduced from: Rowland M, Tozet TN. Clinical Pharmacokinetics – Concepts and ApplicationsClinical Pharmacokinetics – Concepts and Applications , 3, 3rdrd edition, edition, 1995, p. 397.1995, p. 397.

Reproduced from: Rowland Reproduced from: Rowland M, Tozer TN. Ibid, p. 396.M, Tozer TN. Ibid, p. 396.

II. ELIMINATIONII. ELIMINATION A. CAPACITY-LIMITED ELIMINATIONA. CAPACITY-LIMITED ELIMINATION

1. MATHEMATICAL ANALYSIS1. MATHEMATICAL ANALYSISThese processes can be described via the These processes can be described via the Michaelis-Michaelis-MentenMenten relationship: relationship:

][][][][][21

1

PEESSEkk

kf

][][][][][21

1

PEESSEkk

kf

][][][ ESEE fT

][][]][[][

211 ESkESkSEkdt

ESdf

][][]][[][

211 ESkESkSEkdt

ESdf

0dt

d[ES] statesteady at

][][]][[ 211 ESkESkSEk f

])[(]][[ 211 ESkkSEk f

1

21

][

]][[

k

kk

ES

SE f

kk-1-1 is a dissociation process, whereasis a dissociation process, whereas k+2 k+2 requires the requires the

breaking of bonds; thus, breaking of bonds; thus, kk-1-1>>>>kk+2+2

mf K

k

k

ES

SE

1

1

][

]][[

][

]][[

ES

SEK fm

Remember that [Remember that [EEff] = [] = [EETT] – [] – [ESES]]

][

]][[

ES

SESEK Tm

]][[]][[][ SESSEESK Tm

]][[]][[][ SESESESK Tm

]][[)]([ SESKES Tm

)(

]][[][

SK

SEES

m

T

)(

]][[][

SK

SEES

m

T

The rate of formation of the product is given as:The rate of formation of the product is given as:

22 ][or ][

k

vESvESk

By implication, the maximum rate is given asBy implication, the maximum rate is given as

2

max2max ][or ][

k

VEkEV TT

][

][2

max

2 SK

SkV

k

v

m

][

][max

SK

SVv

m

][

][max

CK

CV

dt

dC

m

For most drugs, For most drugs, KKmm >> >>CC. Hence. Hence

mK

CV

dt

dC ][max

mK

CV

dt

dC ][max

Since Since VmaxVmax and and KmKm are constant for a given drug in a are constant for a given drug in agiven individual, this ratio will be constant. Eliminationgiven individual, this ratio will be constant. Eliminationwill proceed in a first-order fashion.will proceed in a first-order fashion.

Cdt

dC

K

V

m

ere whmax

Drugs for which Drugs for which KKmm << << C:C:

ethanolethanolsalicylatesalicylatephenytoinphenytoin

Numerous drugs after first-passNumerous drugs after first-pass

2. Clinical Consequences2. Clinical Consequences a. Relationship btwn dose and Ca. Relationship btwn dose and Cpp

Reproduced from: Tozer TN, Winter ME. Phenytoin, In: Evans WE, Schentag JJ, Jusko WJ, Reproduced from: Tozer TN, Winter ME. Phenytoin, In: Evans WE, Schentag JJ, Jusko WJ, Applied Applied Pharmacokinetics – Principles for Therapeutic Drug Monitoring. 3Pharmacokinetics – Principles for Therapeutic Drug Monitoring. 3 rdrd edition, 1992, edition, 1992, p. 25-12p. 25-12

b. Relationship btwn dose and time to b. Relationship btwn dose and time to steady-statesteady-state

From: Ibid.From: Ibid.

c. Relationship btwn dose and AUCc. Relationship btwn dose and AUCoo

0

1

2

3

0 100 200 300 400 500

Oral Dose (mg)

(AU

C (

mcg

-hr/

mL

)

Plasma AUC of Plasma AUC of lorcainide in a subject lorcainide in a subject as a function of dose. as a function of dose. Data from: Janchen E et al. Data from: Janchen E et al. Clin Pharmacol TherClin Pharmacol Ther 26:187, 26:187, 1979.1979.

c. Relationship btwn dose and AUCc. Relationship btwn dose and AUCoo

0

0.001

0.002

0.003

0.004

0.005

0.006

0 100 200 300 400 500

Oral Dose (mg)

AU

C/D

ose

Plasma AUC/Dose of Plasma AUC/Dose of lorcainide in a subject lorcainide in a subject as a function of dose. as a function of dose. Data from: Janchen E et al. Data from: Janchen E et al. Clin Pharmacol TherClin Pharmacol Ther 26:187, 26:187, 1979.1979.

d. Relationship btwn dose and d. Relationship btwn dose and bioavailabilitybioavailability

Bioavailability of Bioavailability of nicardipine after oral nicardipine after oral administration. administration. Data Data from: Wagner JG et al. from: Wagner JG et al. Biopharm Drug DisposBiopharm Drug Dispos 8:133- 8:133-148, 1987.148, 1987.

e. Relationship btwn Cp and timee. Relationship btwn Cp and time

3. Determination of Michaelis-Menten 3. Determination of Michaelis-Menten ParametersParameters

a. Lineweaver-Burke Expressiona. Lineweaver-Burke Expression

maxmax

max

max

11

1

VCV

K

v

CV

CK

v

CK

CVv

m

m

m

1/1/vv

1/C1/C

1/K1/Kmm

1/V1/Vmaxmax

Slope = KSlope = Kmm/V/Vmaxmax

b. b. In Vivo DeterminationIn Vivo Determination

ssm

mssss

ssssm

ssssm

ssm

ss

ssm

ss

C

KKVK

KKCVCK

CVCKKK

CVCKK

CK

CVK

K

CK

CVv

0max0

0max0

max00

max0

max0

0

max

)(

)(

rateinput If

KK00

KK00/C/Cssss

VVmaxmax

KKmm

JB is an 18 yo male receiving phenytoin for prophylaxis JB is an 18 yo male receiving phenytoin for prophylaxis of post-traumatic head injury seizures. The following of post-traumatic head injury seizures. The following steady state concentrations were obtained at the steady state concentrations were obtained at the indicated doses:indicated doses:

Dose (mg/d)Dose (mg/d) Css (mg/L)Css (mg/L) 100100 3.73.7 300300 4747

From this data, determine this patient’s From this data, determine this patient’s KKmm and V and Vmaxmax

for phenytoin.for phenytoin.

JB is an 18 yo male receiving phenytoin for prophylaxis JB is an 18 yo male receiving phenytoin for prophylaxis of post-traumatic head injury seizures. The following of post-traumatic head injury seizures. The following steady state concentrations were obtained at the steady state concentrations were obtained at the indicated doses:indicated doses:

Dose (mg/d)Dose (mg/d) Css (mg/L)Css (mg/L) Dose Rate/Css (L/d)Dose Rate/Css (L/d) 100100 3.73.7 2727 300300 4747 6.46.4

KK00 (mg/d) (mg/d)

KK00/C/Cssss (L/d) (L/d)

VVmaxmax = 362 mg/d = 362 mg/d

KKmm = 9.7 mg/L = 9.7 mg/L

What CWhat Cssss would be expected if a dose of 200 mg/d were would be expected if a dose of 200 mg/d were

given to this patient?given to this patient?

LmgC

CLmg

Cdmgdmg

CK

CVK

ss

ss

ss

ssm

ss

/ 12

)/ 7.9(

)/ 362(/ 200

max0

4. Application to Alcohol4. Application to Alcohol

EthanolAlcohol dehydrogenase

acetaldehyde

Avg Avg VVmaxmax = 10 g/hr = 10 g/hr

KKmm = 100 mg/L = 100 mg/L

Detectable pharmacologic effect: 250 mg/LDetectable pharmacologic effect: 250 mg/LLethal concentrations >7000 mg/LLethal concentrations >7000 mg/L

Note: EtOH metabolism Note: EtOH metabolism becomes zero-order. becomes zero-order. One jigger (45 mL) of 80 One jigger (45 mL) of 80 proof EtOH contains proof EtOH contains ~14 g of ethanol – which ~14 g of ethanol – which exceeds the exceeds the Vmax!Vmax!

Data from: Rowland M, Tozer TN. Ibid, p. 406.

Reproduced from: Ibid, p. 408.

Reproduced from: Ibid, p. 408.Reproduced from: Ibid, p. 408.

B. AutoinductionB. Autoinduction

C. Saturable Renal Tubular ReabsorptionC. Saturable Renal Tubular Reabsorption

Steady-state plasma ascorbic acid concentration in healthy adults Steady-state plasma ascorbic acid concentration in healthy adults receiving various regimens twice daily for 3 to 4 weeks. receiving various regimens twice daily for 3 to 4 weeks. Control Control subjects had no supplement. Estimated daily dietary intake of ascorbic acid was 50-subjects had no supplement. Estimated daily dietary intake of ascorbic acid was 50-75 mg. From: 75 mg. From: Nutr Rep InternNutr Rep Intern 30:597-601, 1984. 30:597-601, 1984.

Reproduced from: Rowland M, Tozer TN. Ibid, p. 404.Reproduced from: Rowland M, Tozer TN. Ibid, p. 404.

III. SATURABLE PROTEIN BINDINGIII. SATURABLE PROTEIN BINDING

Dose vs AUC for naproxen after single (AUCs) and Dose vs AUC for naproxen after single (AUCs) and multiple (AUCm) doses. multiple (AUCm) doses. From: From: Clin Pharmacol TherClin Pharmacol Ther 15:261-266, 1974. 15:261-266, 1974.

In vitro In vitro binding of naproxen as a function of Cp.binding of naproxen as a function of Cp.

Naproxen Plasma Concentration (mg/L)

Per

cen

t F

ree

0

1

2

3

0 100 200 300 400 500

free concentration

total concentration

ER (mg/hr)

Naproxen Concentration

fu at 2 hr postdose

0.0 0.2 0.4 0.6 0.8 1.0

CL/F

Tot

al, L

/hr

0.0

0.1

0.2

0.3

0.4

0.5

40

80

120

160

200

CL/F

Unbou

nd, L

/hr

Total Drug

Unbound Drug

Relationship between oral clearance and fraction Relationship between oral clearance and fraction unbound of oxaprozin. unbound of oxaprozin. From: From: J Clin PharmacolJ Clin Pharmacol 36:985-997, 1996. 36:985-997, 1996.