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 13-1-2

Figure 13-1-1 30

100

75

20

Cp (ng/mL)

Cp (ng/mL)

25

15 10

50

25

5 0

0

0

5

10

0

15

5

10

20

25

30

Time (hr)

Dose (mg)

Cp ∝ Dose = −

Dose Volume

(13.1.1)

∂Cp ∝ Cp ∂Time

(13.1.2)

Integrating equation (13.1.2) results in the first order equation: Cp = Cp0 e − KT which can be rewritten to LN (Cp) = LN (Cp0 ) − KT A graph of which is linear with time as shown below. Figure 13-1-3 100

Cp (ng/mL)

15

10

1 0

5

10

15

Time (hr)

20

25

30

(13.1.3) (13.1.4)

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 13-1-5

Figure 13-1-4 1000

100

900

80

800

70

700

Cp (ng/mL)

Cp (ng/mL) ave

90

60 50 40 30 20

600 500 400 300 200

10

100

0

0

100

200

300

400

0

500

0

10

Dose/day

20

30

40

Time (hours)

If we were to plot LN(Cp) vs Time as in figure (13.1.4), we would observe the following: Figure 13-1-6

Cp (ng/mL)

1000

100

10

1 0

10

20

30

40

Time (hours)

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. V C ∂C (13.1.5) = − max ∂t ( Km + C ) 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: V C ∂C (13.1.6)  − max = −Vmax = − KC 0 ∂t (C ) 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 − KC 0 = − 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: V C ∂C (13.1.7) = − max = − KC1 ∂t ( Km ) Vmax where =K ( Km ) 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: RateIn = DailyDose =

where

fD

τ

=

Vmax Cpss = RateOut ( K m + Cpss )

(13.1.8)

tau = one day.

Multiplying (13.1.8) by (Km+Cpss) DD ⋅ Km+DD ⋅ Cpss =Vmax Cpss Where DD = Daily Dose

(13.1.9)

Divide (13.1.9) by Cpss

DD K m + DD = Vmax Cpss

(13.1.10)

DD Km Cpss

(13.1.11)

Rearanging (13.1.10)

DD = Vmax − And

mg DD day Liter = = = Clearance Day mg Cpss Liter

(13.1.12)

DD = Vmax − Clearance ⋅ K m

(13.1.13)

So:

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 (13.1.14) MW Phenytoin Sodium Where the Molecular Weight of Sodium Phenytoin is 274.25 g/mole and the molecular weight of Phenytoin is 252.27 g/mole. Mg Phenytoin = mg Phenytoin Sodium *

So: Converting Sodium Phenytoin to phenytoin using (13.1.15) and Calculating clearance using (13.1.12) yields the following data: Dosage Regimen 1 2

Sodium Phenytoin (mg/day) 400 600

Phenytoin (mg/day) 367.94 551.91

Cpss (mg/Liter) 7.7 15.3

Clearance (L/day) 47.78 36.07

which can be graphically represented by: Figure 13-1-7 y = -15.711x + 1118.6

Daily Dose (mg)

600 500 400 300 200 100 0 0

20

40

60

Clearance (L/ay)

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) DailyDose =

Vmax Cpss 1118.6 mg day ⋅18 mg Liter = ( K m + Cpss ) 15.7 mg Liter + 18 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) 597.47

Daily Dose (mg Sodium Phenytoin) 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: DD ⋅ K m Cpss = (13.1.17) Vmax − DD 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 500 mg 1000 mg

C pss

µg mL µg 14.67 mL 7.31

1.

Find km .

2.

Find the maximum clearance, v max , for this patient.

3.

What would be the dose needed to acheive a steady-state concentration of µg 10 ? mL

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 1.1 mg/kg 2.2 mg/kg

C pss

µg mL µg 22.23 mL 10.27

Weight of Monkey = 4.45 kg

1.

Find km .

2.

Find the maximum clearance, v max , for this patient.

3.

What would be the dose needed to acheive a steady-state concentration of µg 15 ? mL

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 10 mg 50 mg

C pss

ng mL nmol 71519 L 6834

1.

Find km .

2.

Find the maximum clearance, v max , for this patient.

3.

What would be the dose needed to acheive a steady-state concentration of ng ? 10,000 mL

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. C pss

Dose 20 mg/kg 200 mg/kg

µg mL µg 294.1 mL 158.6

Rat weight = 425 g 1.

Find km .

2.

Find the maximum clearance, v max , for this patient.

3.

What would be the dose needed to acheive a steady-state concentration of µg ? 200 mL

4.

You recommend changing the patient's dosage regimen to 150 mg/ kg. What would be your patient's plasma concentration?

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

C pss

30 µmol 40 µmol

6.79 µM 8.63 µM

1.

Find km .

2.

Find the maximum clearance, v max , for this patient.

3.

What would be the dose needed to acheive a steady-state concentration of 7.7 µM?

4.

You recommend changing the patient's dosage regimen to 35 µmol. What would be your patient's plasma concentration?

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 10 mg daily 20 mg daily 30 mg daily 40 mg daily 50 mg daily 60 mg daily 70 mg daily

C pss

ng mL ng 3.30 mL ng 8.25 mL ng 13.20 mL ng 26.40 mL ng 39.60 mL ng 66.00 mL 1.65

1.

Find km .

2.

Find the maximum clearance, v max , for this patient.

3.

What would be the dose needed to acheive a steady-state concentration of µg 50 ? mL

4.

You recommend changing the patient's dosage regimen to 36 mg/day. What would be your patient's plasma concentration?

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 300 mg at bedtime (started on day 1) 200 mg BID (started on day 6) 200 mg BID (started on day 6)

Day 2 12 49

C pss

mg L mg 11.4 L mg 21.5 L 5.0

1.

Find km .

2.

Find the maximum clearance, v max , for this patient.

3.

What would be the dose needed to acheive a steady-state concentration of mg 15 ? L

4.

You recommend changing the patient's dosage regimen to 300 mg/day. What would be your patient's plasma concentration?

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 615 mg/ day 588 mg/day

C pss

mg L mg 8.5 L

10

1. Find km .

2. Find the maximum clearance, v max , for this patient.

3. What would be the dose needed to acheive a steady-state concentration of mg ? 12 L

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 7.5 mg/ kg/ day 6.5 mg/ kg/day

C pss

µg mL µg 10 mL

15

1.

Find km .

2.

Find the maximum clearance, v max , for this patient.

3.

What would be the dose needed to acheive a steady-state concentration of mg ? 12 L

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) 2.5 mg daily 5.0 mg daily

C pss

(of quinalaprilat) ng 47.5 mL ng 98.1 mL

1.

Find km .

2.

Find the maximum clearance, v max , for this patient.

3.

What would be the dose needed to acheive a steady-state concentration of ng 75 ? mL

4.

You recommend changing the patient's dosage regimen to 3.0 mg/day. What would be your patient's plasma concentration?

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 25 mg 75 mg 125 mg

C pss

nmol L nmol 15.1 L nmol 46.5 L 3.4

1.

Find km .

2.

Find the maximum clearance, v max , for this patient.

3.

What would be the dose needed to acheive a steady-state concentration of nmol ? 30.0 L

4.

You recommend changing the patient's dosage regimen to 100 mg. What would be your patient's plasma concentration?

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. km =

D1 − D2 D1 D2 − ss ss Cp Cp 1

2

Where: D = Dose Cpss = Steady - state plasma concentration

2. Vmax =

(

D km + Cpss Cpss

)

Where: Vmax = maximum clearance km = Michaelis - Menten Rate Constant

Vmax ⋅ Cpss

3.

D=

4.

Cpss =

km + Cpss

D ⋅ km Vmax − D

Phenytoin in the Critically Ill 1.

2.

km =

D1 − D2 615 mg − 588 mg mg = = 3.517 D1 D2 615 mg 588 mg L − ss − ss mg mg Cp Cp 10 8.5 1 2 L L

Vmax =

(

D km + Cpss Cpss

Vmax ⋅ Cpss



mg mg  + 10  L L  = 831.3 mg mg 10 L

) = 615 mg  3.517

mg L = 642.88 mg = mg mg day 3.517 + 10 L L 8313 . mg ⋅ 10

3.

D=

4.

mg 450 mg ⋅ 3.517 D ⋅ km L = 4.15 mg Cpss = = Vmax − D 8313 . mg − 450 mg L

km + Cpss

________________________________________________________________ _____

CD4 1.

The monkeys had an average weight of 4.45 kg. mg D1 = 2.2 • 4.45 kg = 9.79 mg kg mg D1 = 11 • 4.45 kg = 4.895 mg . kg µg mg Cpss = 22.23 = 22.23 1 mL L µg mg ss Cp = 10.27 = 10.27 2 mL L mg 9.79 mg − 4.895 mg D − D2 km = 1 = = 135.09 9.79 mg 4.895 mg D1 D2 L − ss − ss mg mg Cp Cp 22.23 10.27 1 2 L L

2.

mg mg   9.79 mg  135.09 + 22.23  D km + Cpss  L L  = 69.28 mg Vmax = = mg Cpss 22.23 L

3.

Cpss = 15

(

D=

4.

)

µg mL

Vmax ⋅ Cpss km + Cpss

D = 15 .

= 15

mg L

mg mg L = = 6.92 mg mg day 135.09 + 15 L L 69.28 mg ⋅ 15

mg • 4.45 kg = 6.675 mg kg

mg 6.675 mg ⋅ 135.09 D ⋅ km L = 14.40 mg Cpss = = Vmax − D 69.28 mg − 9.79 mg L _____________________________________________________________________

Cefadroxil 1.

2.

3.

= 14.67

(

Cpss = 10 D=

4.

µg

mg 1 mL L g mg µ Cpss = 7.31 = 7.31 2 mL L D1 − D2 mg 1000 mg − 500 mg km = = = 2144.75 D1 D2 1000 mg 500 mg L − ss − ss mg mg Cp Cp 14.67 7.31 1 2 L L mg mg   500 mg  2144.75 + 7.31  D km + Cpss  L L  Vmax = = = 147200 mg = 147.2 g mg Cpss 7.31 L Cpss = 14.67

)

µg mL

Vmax ⋅ Cpss km + Cpss

Cpss

= 10

mg L

mg mg L = 683.14 = mg mg day 2144.75 + 10 L L 147200 mg ⋅ 10

mg 300 mg ⋅ 2144.75 D ⋅ km L = 4.38 mg = = Vmax − D 147200 mg − 300 mg L

Answers Cefadroxil 1.

2144.75 µ g

2. 3.

147.2 g/day 683.14 mg

4.

4.38 µ g

mL

mL

Phenytoin 1.

4.014 µ g

2. 3.

540.85 mg/day 426.7 mg/day

4.

5 µg

CD4

mL

mL

Phenytoin in the Critically Ill

1.

135.09 µ g mL

2. 3.

69.28 mg/day 6.92 mg/day

1.

3.517 µ g

4.

14.4 µ g

2. 3.

831.3 mg/day 642.88 mg/day

4.

4.15 µ g

mL

Methylprednisone 1.

52345.27 ng

2. 3.

86.60 mg/day 13.89 mg/day

4.

27747.1 ng

Mezlocillin 1.

324.95 µ g mL

2. 3.

25.92 mg/day 68.23 mg/day

4.

1676.04 µ g

mL

Naphthol 1. 2. 3. 4.

46.14 µM 233.86 µmol 25.61 µmol 43.5 µM

Paroxetine 1.

4.125 ng

2. 3.

45 mg/day 41.6 mg/day

4.

16.5 ng

mL

mL

mL

Phenytoin in Pediatrics

mL

mL

mL

1.

6.67 µ g

2. 3.

162.5 mg/day 104.46 mg/day

4.

4.74 µ g

mL

mL

Quinalapril 1.

1503.15 ng

2. 3.

81.61 mg/day 3.88 mg/day

4.

57.36 ng

mL

mL

Vanoxerine 1.

20.96 nm ol

2. 3.

179.08 mg/day 105.4 mg/day

4.

26.5 nm ol

L

L