BASIC PHARMACOKINETICS - CHAPTER 13: Non-linear kinetics

Non-Linear Pharmacokinetics Non-linear pharmacokinetic models, as opposed to linear pharmacokinetic models differ in some very critical areas. When the concentration that results from the dose is proportional to that dose and the rate of elimination of the drug is proportional to the concentration, the drug is said to exhibit linear pharmacokinetics as shown below.

Figure 13-1-1

0

5

10

15

20

25

30

0 5 10 15Dose (mg)

Cp

(ng/

mL)

Figure 13-1-2

0

25

50

75

100

0 5 10 15 20 25 30

Time (hr)

Cp

(ng/

mL)

DoseCp DoseVolume

∝ = (13.1.1)

Cp CpTime∂

− ∝∂

(13.1.2)

Integrating equation (13.1.2) results in the first order equation: 0

KTCp Cp e−= (13.1.3) which can be rewritten to 0( ) ( )LN Cp LN Cp KT= − (13.1.4) A graph of which is linear with time as shown below.

Figure 13-1-3

1

10

100

0 5 10 15 20 25 30

Time (hr)

Cp

(ng/

mL)

When concentration that results from the dose is not proportional to that dose and/or the rate of elimination of the drug is not proportional to the concentration, the drug is said to exhibit non-linear kinetics as shown below.

Figure 13-1-4

0

10

20

30

40

50

60

70

80

90

100

0 100 200 300 400 500

Dose/day

Cp

(ng/

mL)

ave

Figure 13-1-5

0

100

200

300

400

500

600

700

800

900

1000

0 10 20 30 40

Time (hours)

Cp

(ng/

mL)

If we were to plot LN(Cp) vs Time as in figure (13.1.4), we would observe the following:

Figure 13-1-6

1

10

100

1000

0 10 20 30 40

Time (hours)

Cp

(ng/

mL)

Compare the terminal portion of the curve (time>25 hours) of figures 5 and 6 with figures 2 and 3. Looks similar, doesn’t it. Well, there’s a reason for it. In many cases, the rate of elimination of the drug is defined by the enzymatic metabolism. From biochemistry, we (should) remember the Michaelis-Menten equation that describes the rate of substrate metabolism.

( )

max

m

V CCt K C

∂= −

∂ + (13.1.5)

Where Vmax = the maximum velocity capable by the enzymes Km = drug concentration that is metabolized at half the maximum velocity

Drugs that exhibit this kind of behavior are said to exhibit non-linear pharmacokinetics. Let’s look at the various portions of the curve. There are three distinct portions of this curve:

1) The initial straight portion of the figure 13-1-5 (Cp >> Km) 2) The middle curved portion of figure 13-1-5 and 13-1-6 (Cp ≈ Km) 3) The terminal straight portion of figure 13-1-6 (Cp << Km)

If (Cp >> Km) then equation (13.1.5) becomes:

( )

0maxmax

V CC V KCt C

∂− = − = −

∂ (13.1.6)

What we see is that the enzymatic activity is maxed out and the rate of elimination of the drug is constant or, in math speak, the elimination rate is proportional to C to the zero power. (Math review A0=1, Chapter 2.) Thus 0KC K− = − . Drugs that exhibit this kind of behavior are said to exhibit zero order behavior. The concentration is changed at a constant rate, X mg/hour are removed by the metabolizing enzymes. If (Cp << Km) then equation (13.1.5) becomes:

( )

1max

m

V CC KCt K

∂= − = −

∂ (13.1.7)

where ( )

max

m

V KK

=

Drugs that exhibit this kind of behavior are said to exhibit first order behavior. The concentration is changed at a rate proportional to the concentration, X%/hour is removed by the metabolizing enzymes. Note equation (13.1.7) is identical to equation (13.1.2). Thus, what we see for a majority of drugs is that equations (13.1.3) and (13.1.4) describe their pharmacokinetics because the enzymes in the body are very efficient in the drug’s removal and the therapeutic concentrations are well below the drug’s Km. If (Cp ≈ Km) then the whole equation must be used and not just the limits as shown above. Why is this important? Because for those drugs where this is occurring, unlike drugs that exhibit linear kinetics, where a change in daily dose results in a proportional change in plasma concentration, as in the initial portion of figure 13-1-4 (< 200 mg/day), a small change in daily dose could result in a LARGE change in plasma concentration, as shown in the terminal portion of figure 13-1-4 (> 400 mg/day). In other words something like a 10% change in dose would not yield a 10% change in concentration but could yield a 100% (or greater) change in concentration. This could lead to toxicity.

At steady state, the rate of drug being put into the body equals the rate of drug being removed from the body by the enzymes, thus:

( )

maxfDRateIn = DailyDose = ss

m ss

V Cp RateOutK Cpτ

= =+

(13.1.8)

where tau = one day. Multiplying (13.1.8) by (Km+Cpss) ss max ssDD Km+DD Cp =V Cp⋅ ⋅ (13.1.9) Where DD = Daily Dose Divide (13.1.9) by Cpss

maxmss

DD K DD VCp

+ = (13.1.10)

Rearanging (13.1.10)

max mss

DDDD V KCp

= − (13.1.11)

And

ss

mgDD day Liter ClearanceDaymgCp

Liter= = = (13.1.12)

So: max mDD V Clearance K= − ⋅ (13.1.13) Thus, the daily dose is proportional to the clearance, and a graph of Daily Dose vs. Clearance will result in a straight line with an intercept of Vmax and a slope of - Km. Example: Your patient is 90 Kg male and the doctor would like to achieve a Cpss of 18 mg/L of Phenytoin. Previously you patient received a dose of 400 mg/day Dilantin Kapseals and attained a Cpss of 7.7 mg/L and a dose of 600 mg/day to attain a Cpss of 15.3 mg/L. Find Vmax, Km and the daily dose necessary to accieve the desired blood level.

First of all Dilantin Kapseals are Sodium Phenytoin and must be converted to Phenytion equivalents:

Molecular Weight of Sodium Phenytoin is 274.25 g/mole and the molecular weight of Phenytoin is 252.27 g/mole. So, the daily dose of Phenytoin in each case is calculated by:

MW Phenytoin Mg Phenytoin = mg Phenytoin Sodium * MW Phenytoin Sodium

(13.1.14)

Where the Molecular Weight of Sodium Phenytoin is 274.25 g/mole and the molecular weight of Phenytoin is 252.27 g/mole. So: Converting Sodium Phenytoin to phenytoin using (13.1.15) and Calculating clearance using (13.1.12) yields the following data:

Dosage Regimen

Sodium Phenytoin (mg/day)

Phenytoin (mg/day)

Cpss (mg/Liter)

Clearance (L/day)

1 400 367.94 7.7 47.78 2 600 551.91 15.3 36.07

which can be graphically represented by:

Figure 13-1-7

y = -15.711x + 1118.6

0

100

200

300

400

500

600

0 20 40 60

Clearance (L/ay)

Dai

ly D

ose

(mg)

The trend line yields a Km of 15.7 mg/L and a Vmax of 1118.6 mg/day. Plugging in the values of Km and Vmax into equation (13.1.8)

( )

max 1118.6 18 DailyDose = 15.7 18

ss

m ss

V Cp mg day mg LiterK Cp mg Liter mg Liter

⋅=

+ + (13.1.16)

we find that the daily dose needed to attain a Cpss of 18 mg/L of Phenytion is:

Daily Dose (mg Phenytoin)

Daily Dose (mg Sodium Phenytoin)

597.47 649.53 Whereas if the drug exhibitd linear kinetics and we wanted to increase the plasma concentration from 15.3 mg/Liter to 18 mg/Liter the dose of Sodium Phenytion would be

= 705.88 mg. If we did dose our patient at this daily dose a further rearrangement of equation (13.1.9) yields:

max

mss

DD KCpV DD

⋅=

− (13.1.17)

Converting 705.88 mg of Sodium Phenitoin to Phenytoin yields 651.76 mg/Phenytoin. Plugging that into equation (13.1.17) yields a plasma concentration of 21.92 mg/Liter instead of the intended 18 mg/Liter. In other words, an 18% increase in dose (652/552=1.18) yields a 44% increase in plasma concentration (22/15.3 = 1.44.) This clearly could be a problem.

Problems: Cefadroxil

Sanchez-Pico, A., et al, "Nonlinear intestinal abosrption kinetics of cefadroxil in the rat", Journal of Pharmacy Pharmacology, Vol.41, (1989), p. 179 - 185.

Cefadroxil is a cephalosporin antibiotic which is commonly used to treat various infections. It is usually given orally. This study looks at the pharmacokinetics and bioavailability of cefadroxil in the rat.

Dose Cpss

500 mg 7.31 µgmL

1000 mg 14.67 µgmL

1. Find km . 2. Find the maximum clearance,vmax , for this patient. 3. What would be the dose needed to acheive a steady-state concentration of

10 µgmL

?

4. You recommend changing the patient's dosage regimen to 300 mg/day.

What would be your patient's steady state plasma concentration?

CD4 Qian, M., et al., "Pharmacokinetic evaluation of drug interactions with anti-human immunodeficiency virus drugs: V. effect of soluble CD4 on 2',3'-dideoxycytidine

kinetics in monkeys", Drug Metabolism and Disposition, Vol. 20, (1992), p. 396 - 400. 2',3'-dideoxycytidine in combination with recombinant ST4 has been shown to be effective against HIV (human immunodeficiency virus) in vitro. This study examines whether or not the pharmacokinetics of 2',3'-dideoxycytidine are affected by administration of CD4 (an immunoglobulin). Doses of each drug were given to male adult monkeys weighing an average of 4.45 kg. The following data is for ST4 (soluble CD4).

Dose Cpss

1.1 mg/kg 10.27 µgmL

2.2 mg/kg 22.23 µgmL

Weight of Monkey = 4.45 kg


15 µgmL

?

4. You recommend changing the patient's dosage regimen to 1.5 mg/ kg.

What would be your patient's plasma concentration?

Methylprednisone Haughey, D, and Jusko W.., "Bioavailability and nonlinear dispositionof methylprednisolone and methylprednisone in the rat", Journal of Pharmaceutical Sceicnes, Vol. 81, (1992), p. 117 - 121.

Methylprednisone is a corticosteroid which is commonly used in the treatment of medical emergencies such as cardiovascular shock, asthma, and cerebral edema. The following data was obtained for two methylprednisolone doses. The plasma concentration measurement given for each dose below is that for the central compartment.

Dose Cpss

10 mg 6834 ngmL

50 mg 71519 nmolL


10,000 ngmL

?

4. You recommend changing the patient's dosage regimen to 30 mg. What

would be your patient's plasma concentration?

Mezlocillin Jungbluth, G. and Jusko, W., "Dose-dependent pharmacokinetics of mezlocillin in

rats", Antimicrobial Agents and Chemotherapy, Vol. 33, (1989), p. 839 - 843. Mezlocillin is an antibiotic used to treat various types of infection. It is usually given by the intravenous route and exhibits dose-dependent (nonlinear) pharmacokinetics. This article compares two intravenous bolus doses, one of 20 mg/kg and one of 200 mg/kg in rats. The following data was calculated from the results of this study.

Dose Cpss

20 mg/kg 158.6 µgmL

200 mg/kg 294.1 µgmL

Rat weight = 425 g


200 µgmL

?

4. You recommend changing the patient's dosage regimen to 150 mg/ kg.


Naphthol Redegeld, A., Hofman, G., and Noordhoek, J., "Conjugative clearance of 1-naphthol and

disposition of its glucuronide and sulfate conjugates in the isolated perfused rat", Journal of Harmacology and Experimental Therapeutics, Vol. 244, (1988), p. 263 - 267.

1-naphthol is a small phenolic compound which is extensively metabolized by conjugation. This study looks at the pharmacokinetics of naphthol in a rat.

Dose Cpss

30 µmol 6.79 µM 40 µmol 8.63 µM


7.7 µM? 4. You recommend changing the patient's dosage regimen to 35 µmol. What


Paroxetine Sindrug, S., Brosen, K, and Gram, L.., "Pharmacokinetics of the selective serotonin reuptake

inhibitor paroxetine: nonlinearity and relation to the sprateine oxidation polymorphism", Clinical Pharmacology and Therapeutics, Vol. 51, (1992), p. 288 - 295.

Paroxetine hydrochloride (Paxil) is a selective serotonin reuptake inhibitor which is used in the treatment of depression. Paroxetine is metabolized both by oxidation and conjugation with the conjugated metabolites excreted in the urine. Paroxetine exhibits dose-dependent (nonlinear) pharmacokinetics. The following data is for a male diabetic patient who was concurrently taking insulin.

Dose Cpss

10 mg daily 1.65 ngmL








50 µgmL

?



Phenytoin Levine, M., et al., "Evaluation of serum phanyton monitoring in an acute care setting", Therapeutic Drug Monitoring, Vol. 10, (1988)., p. 50 - 57.

Phenytoin is an agent which is commonly used in the treatment of epilepsy. This drug exhibits nonlinear kinetics. Phenytoin is mainly eliminated from the body by hepatic cytochrome P-450 metabolism. Several doses of phenytoin were studied in patients and the data is summarized below:

Dose Day Cpss

300 mg at bedtime (started on day 1)

2 5.0 mgL

200 mg BID (started on day 6)

12 11.4 mgL

200 mg BID (started on day 6)

49 21.5 mgL


15 mgL

?



Phenytoin in the Critically Ill Boucher, B. et al., "Phenytoin pharmacokinetics in critically ill trauma patients",

Clinical Pharmacology and Therapeutics, Vol. 44, (1988)., p. 675 - 683. Phenytoin is an agent which is commonly used in the treatment of epilepsy. This drug exhibits nonlinear kinetics. This study looks at several doses of phenytoin in severely ill trauma patients. The data given below is that obtained for one male, 25 year-old, patient who weighed 85 kg.

Dose Cpss

615 mg/ day 10 mgL

588 mg/day 8.5 mgL


12 mgL

?

4. You recommend changing the patient's dosage regimen to 450 mg/ day.

What would be your patient's steady-state plasma concentration?

Phenytoin in Pediatrics Bauer, L. and Blouin, R., "Phenytoin Michaelis-Menten pharmacokinetics in caucasian

paediatric patients", Clinical Pharmacokinetics, Vol. 8, (1989)., p. 545 - 549. Phenytoin is an agent which is commonly used in the treatment of epilepsy. This drug exhibits nonlinear kinetics. This study looks at several doses of phenytoin in pediatric patients of several ages. The data for the 4 to 6 year old patients is given below.

Dose Cpss

7.5 mg/ kg/ day 15 µgmL

6.5 mg/ kg/day 10 µgmL


12 mgL

?

4. You recommend changing the patient's dosage regimen to 4.5 mg/ kg/ day.

What would be your patient's steady-state plasma concentration?

Quinalapril Elliott, H., et al., "Dose responses and pharmaockinetics for the angiotensin converting enzyme

inhibitor, quinapril", Clinical Pharmacology and Therapeutics, Vol. 52, (1992), p. 260 - 265. Quinalapril is an angiotensin converting enzyme (ACE) inhibitor which is used in the treatment of hypertension and heart failure. The optimal dosage regimen for the ACE inhibitors is controversial and this study further investigates quinalapril's pharmacokinetics at various doses ranging from 0.5 to 20 mg. Quinalapril is a prodrug which is metabolized to its active form, quinalaprilat.

Dose (of quinalapril)

Cpss

(of quinalaprilat)

2.5 mg daily 47.5 ngmL

5.0 mg daily 98.1 ngmL


75 ngmL

?

4. You recommend changing the patient's dosage regimen to 3.0 mg/day.


Vanoxerine Ingwersen, S., et al., "Nonlinear multiple-dose pharmacokinetis of the dopamine reuptake

inhibitor vanoxerine", Journal of Pharmaceutical Sciences, Vol. 82, (1993)., p. 1164 - 1166. Vanoxerine is a pre-synaptic dopamine reuptake inhibitor which may be useful as an antidepressant. The bioavailability of vanoxerine is changed by food intake. The bioavailability after fasting is increased 76% by a low-fat meal and 255% by a high-fat meal. In this study, the volunteers were given doses of vanoxerine after eating a standard breakfast of one bowl of cereal with milk, two slices of toast with sunflower margarine and jam, and one cup of tea.

Dose Cpss

25 mg 3.4 nmolL

75 mg 15.1 nmolL

125 mg 46.5 nmolL


30.0 nmolL

?

4. You recommend changing the patient's dosage regimen to 100 mg. What


Nonlinear Equations

The following equations were used to solve the questions following each "nonlinear" scenario. Three scenerios have been completed for you. The answers have been provided for the remainder. 1.

k D DD

CDC

D

C

m

pss

pss

pss

=−

−

=

=

1 21 2

1 2

Where: Dose

Steady - state plasma concentration

2.

( )V

D k C

C

Vk

maxm p

p

max

m

ss

ss

=+

Where: = maximum clearance

= Michaelis - Menten Rate Constant

3. D

V Ck Cmax p

m p

ss

ss

=⋅

+

4. C D k

V Dpm

maxss=

⋅−

Phenytoin in the Critically Ill

1. k D DD

CDC

mg mgmgmgL

mgmgL

m

pss

pss

=−

−=

−

−=1 2

1 2

1 2

615 588615

10

588

8 5

.

3.517 mgL

2. ( )

VD k C

C

mg mgL

mgL

mgL

maxm p

p

ss

ss

=+

=+

=

615 3 517 10

10

.831.3 mg

3. DV Ck C

m mgL

mgL

mgL

max p

m p

ss

ss

=⋅

+=

⋅

+=

8313 10

3 517 10

.

.642.88 mg

day

g

4. C D kV D

mg mgL

mg mgpm

maxss=

⋅−

=⋅

−=

450 3 517

8313 450

.

.4.15 mg

L

_____________________________________________________________________

CD4

1. The monkeys had an average weight of 4.45 kg.

D mgkg

kg mg

D mgkg

kg mg

C gmL

mgL

C gmL

mgL

pss

pss

1

1

2 2 4 45 9 79

11 4 45 4 895

22 23 22 23

10 27 10 27

1

2

= • =

= • =

= =

= =

. . .

. . .

. .

. .

µ

µ

k D DD

CDC

mg mgmgmgL

mgmgL

m

pss

pss

=−

−=

−

−=1 2

1 2

1 2

9 4 8959

22.23

4

10.27

.79 .79 .895 . 135.09 mg

L

2. ( )

VD k C

C

mg mgL

mgL

mgL

maxm p

p

ss

ss

=+

=+

=

9 135.09 22 23

22.23

.79

.69.28 mg

3. C g

mLmgLpss

= =15 15 µ

DV Ck C

m mgL

mgL

mgL

max p

m p

ss

ss

=⋅

+=

⋅

+=

69.28 15

135.09 15

g

6.92 mgday

4. D mgkg

kg mg= • =15 4 45 6 675. . .

C D kV D

mg mgL

mg mgpm

maxss=

⋅−

=⋅

−=

6 135 09

69.28 9 79

.675

.

.14.40 mg

L

_____________________________________________________________________

Cefadroxil

1. C gmL

mgLp

ss1

14 67 14 67= =. .µ

C gmL

mgL

k D DD

CDC

mg mgmgmgL

mgmgL

pss

m

pss

pss

2

1 2

7 31 7 31

1000 5001000

14.67

500

7

1 21 2

= =

=−

−=

−

−=

. .

2144.75 mgL

µ

.31

2. ( )

VD k C

C

mg mgL

mgL

mgL

maxm p

p

ss

ss

=+

=+

= =

500 2144.75 7 31

7

.147200 mg 147.2 g

.31

3. C gmL

mgLpss

= =10 10 µ

DV Ck C

m mgL

mgL

mgL

max p

m p

ss

ss

=⋅

+=

⋅

+=

147200 10

2144.75 10

g

683.14 mgday

4. C D kV D

mg mgL

mg mgpm

maxss=

⋅−

=⋅

−=

300 2144 75

147200 300

.4.38 mg

L

Answers Cefadroxil

1. 2144.75 µgm L

2. 147.2 g/day 3. 683.14 mg 4. 4.38 µg

m L

CD4 1. 135.09 µg

m L

2. 69.28 mg/day 3. 6.92 mg/day 4. 14.4 µg

m L

Methylprednisone 1. 52345.27 ng

m L

2. 86.60 mg/day 3. 13.89 mg/day 4. 27747.1 ng

m L

Mezlocillin 1. 324.95 µg

m L

2. 25.92 mg/day 3. 68.23 mg/day 4. 1676.04 µg

m L

Naphthol 1. 46.14 µM 2. 233.86 µmol 3. 25.61 µmol 4. 43.5 µM

Paroxetine 1. 4.125 ng

m L

2. 45 mg/day 3. 41.6 mg/day 4. 16.5 ng

m L

Phenytoin 1. 4.014 µg

m L

2. 540.85 mg/day 3. 426.7 mg/day 4. 5 µg

m L

Phenytoin in the Critically Ill

1. 3.517 µgm L

2. 831.3 mg/day 3. 642.88 mg/day 4. 4.15 µg

m L

Phenytoin in Pediatrics

1. 6.67 µgm L

2. 162.5 mg/day 3. 104.46 mg/day 4. 4.74 µg

m L

Quinalapril 1. 1503.15 ng

m L

2. 81.61 mg/day 3. 3.88 mg/day 4. 57.36 ng

m L

Vanoxerine 1. 20.96 nm ol

L

2. 179.08 mg/day 3. 105.4 mg/day 4. 26.5 nm ol

L

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BASIC PHARMACOKINETICS - CHAPTER 13: Non-linear kinetics