Biopharmaceutics Lec. 10 Dr. AA Yas

Non – Linear [Dose – Dependent] Pharmacokinetics

Introduction: -

•Many of the processes of drug absorption, distribution, biotransformation, and

excretion involve enzymes or carrier-mediated systems. For some drugs given at

therapeutic levels, one of these specialized processes may become saturated.

•Drugs that demonstrate saturation kinetics usually show the following characteristics:

1. Elimination of drug does not follow simple first-order kinetics—that is, elimination

kinetics are nonlinear.

2. The elimination half-life changes as dose is increased. Usually, the elimination half-

life increases with increased dose due to saturation of an enzyme system.

However, the elimination half-life might decrease due to ―self‖-induction of liver

biotransformation enzymes, as is observed for carbamazepine.

3. The area under the curve (AUC) is not proportional to the amount of bioavailable

drug.

4. The saturation of capacity-limited processes may be affected by other drugs that

require the same enzyme or carrier-mediated system (i.e.; competition effects).

5. The composition and/or ratio of the metabolites of a drug may be affected by a

change in the dose.

Table 1: Examples of Drugs Showing Nonlinear Kinetics

•In general, metabolism (biotransformation) and active tubular secretion of drugs by the

kidney are the processes most usually saturated. Drug concentrations in the blood can

increase rapidly once an elimination process is saturated.

Figure 1: Plasma level–time curves for a drug that exhibits

a saturable elimination process. Curves A and B represent

high and low doses of durg, respectively, given in a single

IV bolus. The terminal slopes of curves A and B are the same.

Curve C represents the normal first-order elimination of a

different drug

•In order to determine whether a drug is following dose-dependent kinetics, the drug is

given at various dosage levels and a plasma level–time curve is obtained for each dose.

The curves should exhibit parallel slopes if the drug follows dose-independent kinetics.

Alternatively, a plot of the areas under the plasma level–time curves at various doses

should be linear.

Figure 2: Area under the plasma level–time curve versus

dose for a drug that exhibits a saturable elimination process.

Curve A represents dose-dependent or saturable elimination

kinetics. Curve C represents dose-independent kinetics.

Saturable Enzymatic Elimination Processes: -

•The elimination of drug by a saturable enzymatic process is described by Michaelis–

Menten kinetics. If Cp is the concentration of drug in the plasma, then:

…..(Eq. 1)

where Vmax is the maximum elimination rate and KM is the Michaelis constant that

reflects the capacity of the enzyme system and is not an elimination constant, but is

actually a hybrid rate constant in enzyme kinetics, representing both the forward and

backward reaction rates and equal to the drug concentration or amount of drug in the

body at 0.5Vmax. The values for KM and Vmax are dependent on the nature of the drug

and the enzymatic process involved.

•Equation 1 describes a nonlinear enzyme process that encompasses a broad range of

drug concentrations. When the drug concentration Cp is large in relation to KM (Cp »

KM), saturation of the enzymes occurs and the value for KM is negligible. The rate of

elimination proceeds at a fixed or constant rate equal to Vmax. Thus, elimination of drug

becomes a zero-order process and equation 1 becomes:

…..(Eq. 2)

Drug Elimination by Capacity – Limited Pharmacokinetics: One – Compartment

Model, IV Bolus Injection: -

•The rate of elimination of a drug that follows capacity-limited pharmacokinetics is

governed by the Vmax and KM of the drug. Equation 1 describes the elimination of a

drug that distributes in the body as a single compartment and is eliminated by

Michaelis–Menten or capacity-limited pharmacokinetics. If a single IV bolus injection

of drug (D0) is given at t = 0, the drug concentration (Cp) in the plasma at any time t

may be calculated by an integrated form of equation 1 described by:

…..(Eq. 3)

•Alternatively, the amount of drug in the body after an IV bolus injection may be

calculated by the following relationship. Equation 4 may be used to simulate the decline

of drug in the body after various size doses are given, provided the KM and Vmax of drug

are known: …..(Eq. 4), where D0 is the amount of drug in the

body at t = 0.

•In order to calculate the time for the dose of the drug to decline to a certain amount of

drug in the body, equation 4 must be rearranged and solved for time t:

…..(Eq. 5)

•Determination of KM and Vmax – equation 1 relates the rate of drug biotransformation

to the concentration of the drug in the body. The same equation may be applied to

determine the rate of enzymatic reaction of a drug in vitro. Equation 6 for an

experiment is performed with solutions of various concentration of drug C, a series of

reaction rates (v) may be measured for each concentration: …..(Eq. 6)

•A rearrangement of equation 6, equation 7 is a linear

equation when 1/v is plotted against 1/C. The y intercept …..(Eq. 7)

for the line is 1/Vmax, and the slope is KM/Vmax.

•Determination of KM and Vmax in Patients – equation 6 shows that the rate of drug

metabolism (v) is dependent on the concentration of the drug (C). This same basic

concept may be applied to the rate of drug metabolism of a capacity-limited drug in the

body compartment in which the drug is dissolved. The rate of drug metabolism will

vary depending on the concentration of drug Cp as well as on the metabolic rate

constants KM and Vmax of the drug in each individual.

•At steady state, the rate of drug metabolism (v) is assumed to be the same as the rate of

drug input R (dose/day). Therefore, equation 8 may be written for drug metabolism in

the body similar to the way drugs are metabolized in vitro, equation 6. However, steady

state will not be reached if the drug input rate, R, is greater than the Vmax; instead, drug

accumulation will continue to occur without reaching a steady-state plateau.

…..(Eq. 8)

where R = dose/day or dosing rate, Css = steady-state plasma drug concentration, Vmax = maximum metabolic rate constant in the body, and KM = Michaelis–Menten constant of the drug in the body. •Determination of KM and Vmax by Direct Method - when steady-state concentrations are known at only two dose levels, there is no advantage in using the graphic method. KM and Vmax may be calculated by solving two simultaneous equations formed by substituting CSS and R, equation 8 with C1, R1, C2, and R2. The equations contain two unknowns, KM and Vmax, and may be solved easily: & . •Combining the two equations yields: …..(Eq. 9), where C1 is steady-state plasma drug concentration after dose 1, C2 is steady-state plasma drug concentration after dose 2, R1 is the first dosing rate, and R2 is the second dosing rate. •Dependence of Elimination Half-Life on Dose - For a drug that follows nonlinear kinetics, the elimination half-life and drug clearance both change with dose or drug concentration. Generally, the elimination half-life becomes longer, clearance becomes smaller, and the area under the curve becomes disproportionately larger with increasing dose. …..(Eq. 10)

•Dependence of Clearance on Dose - the total body clearance of a drug given by IV

bolus injection that follows a one-compartment model with Michaelis–Menten

elimination kinetics changes with respect to time and plasma drug concentration and is

dose dependent. To obtain mean body clearance, Clav is then calculated from the dose

and the AUC: …..(Eq. 11)

…..(Eq. 12)

•Alternatively, dividing equation by Cp gives equation 13, which shows

that the clearance of a drug that follows nonlinear pharmacokinetics is dependent on the

plasma drug concentration Cp, KM, and Vmax : …..(Eq. 13).

Drug Distributed as One – Compartment Model and Eliminated by Nonlinear

Pharmacokinetics: -

•Mixed Drug Elimination - drugs may be metabolized to several different metabolites

by parallel pathways. At low drug doses corresponding to low drug concentrations at

the site of the biotransformation enzymes, the rates of formation of metabolites are first

order. However, with higher doses of drug, more drug is absorbed and higher drug

concentrations are presented to the biotransformation enzymes. At higher drug

concentrations, the enzyme involved in metabolite formation may become saturated,

and the rate of metabolite formation becomes nonlinear and approaches zero order.

•The equation that describes a drug that is eliminated by both first-order and Michaelis–

Menten kinetics after IV bolus injection is given by: …..(Eq. 14),

where k is the first-order rate constant representing

the sum of all first-order elimination processes, while the second term of equation 14

represents the saturable process. V′max is simply Vmax expressed as concentration by

dividing by VD.

•Zero-Order Input and Nonlinear Elimination - if the drug is given by constant IV

infusion and is eliminated only by nonlinear pharmacokinetics, then the following

equation describes the rate of change of the plasma drug concentration:

…..(Eq. 15), where k0 is the infusion rate and VD is the apparent

volume of distribution.

•First-Order Absorption and Nonlinear Elimination - the relationship that describes

the rate of change in the plasma drug concentration for a drug that is given

extravascularly (e.g.; orally), absorbed by first order absorption, and eliminated only by

nonlinear pharmacokinetics, is given by the following equation. CGI is concentration in

the GI tract: …..(Eq. 16), where ka is the first-order absorption

rate constant.

•If the drug is eliminated by parallel pathways consisting of both linear and nonlinear

pharmacokinetics, then: …..(Eq. 17), where k is the first-

order elimination rate constant.

Frequently Asked Questions: -

1- What kinetic processes in the body can be considered saturable?

2- Why is it important to monitor drug levels carefully for dose dependency?

A patient with concomitant hepatic disease may have decreased biotransformation

enzyme activity. Infants and young subjects may have immature hepatic enzyme

systems. Alcoholics may have liver cirrhosis and lack certain coenzymes. Other patients

may experience enzyme saturation at normal doses due to genetic polymorphism.

Pharmacokinetics provides a simple way to identify nonlinear kinetics in these patients

and to estimate an appropriate dose. Finally, concomitant use of other drugs may cause

nonlinear pharmacokinetics at lower drug doses due to enzyme inhibition.

3- What is the Michaelis–Menten equation? How are Vmax and KM obtained? What are

the units for Vmax and KM ? What is the relevance of Vmax and KM?

4- What are the main differences in pharmacokinetic parameters between a drug that

follows linear and a drug that follows nonlinear pharmacokinetics?

A drug that follows linear pharmacokinetics generally has a constant elimination half-

life and a constant clearance with an increase in the dose. The steady-state drug

concentrations and AUC are proportional to the size of the dose. Nonlinear

pharmacokinetics results in dose-dependent Cl, t1/2, and AUC. Nonlinear

pharmacokinetics are often described in terms of Vmax and KM.

5- What is the cause of nonlinear pharmacokinetics that is not dose related?

Chronopharmacokinetics is the main cause of nonlinear pharmacokinetics that is not

dose related. The time-dependent or temporal process of drug elimination can be the

result of rhythmic changes in the body. For example, nortriptyline and theophylline

levels are higher when administered between 7 and 9 am compared to between 7 and 9

pm after the same dose. Biological rhythmic differences in clearance cause a lower

elimination rate in the morning compared to the evening. Other factors that cause

nonlinear pharmacokinetics may result from enzyme induction (eg, carbamazepine) or

enzyme inhibition after multiple doses of the drug. Furthermore, the drug or a

metabolite may accumulate following multiple dosing and affect the metabolism or

renal elimination of the drug.

6- For drugs that have several metabolic pathways, must all the metabolic pathways be

saturated for the drug to exhibit nonlinear pharmacokinetics?

7- What are the main differences between a model based on Michaelis—Menten kinetic

(Vmax and KM) and the physiologic model that describes hepatic metabolism based on

clearance?

The physiologic model based on organ drug clearance describes nonlinear drug

metabolism in terms of blood flow and intrinsic hepatic clearance. Drugs are extracted

by the liver as they are presented by blood flow. The physiologic model accounts for the

sigmoid profile with changing blood flow and extraction, whereas the Michaelis—

Menten model simulates the metabolic profile based on Vmax and KM. The

Michaelis—Menten model was applied mostly to describe in-vitro enzymatic reactions.

When Vmax and KM are estimated in patients, blood flow is not explicitly considered.

This semiempirical method was found by many clinicians to be useful in dosing

phenytoin. The organ clearance model was more useful in explaining clearance change

due to impaired blood flow. In practice, the physiologic model has limited use in dosing

patients because blood flow data for patients are not available.

Learning Questions: -

1- What processes of drug absorption, distribution, and elimination may be considered

―capacity limited,‖ ―saturated,‖ or ―dose dependent‖?

Capacity-limited processes for drugs include:

• Absorption: Active transport and Intestinal metabolism by microflora.

• Distribution: Protein binding.

• Elimination: Hepatic elimination, Biotransformation and Active biliary secretion.

• Renal excretion: Active tubular secretion and Active tubular reabsorption.

2- A given drug is metabolized by capacity-limited pharmacokinetics. Assume KM is 50

μg/mL, Vmax is 20 μg/mL per hour, and the apparent VD is 20 L/kg. a. What is the

reaction order for the metabolism of this drug when given in a single intravenous dose

of 10 mg/kg? b. How much time is necessary for the drug to be 50% metabolized?

3- The drug isoniazid was reported to interfere with the metabolism of phenytoin.

Patients taking both drugs together show higher phenytoin levels in the body. Using the

basic principles in this chapter, do you expect KM to increase or decrease in patients

taking both drugs?

When INH is coadminstered, plasma phenytoin concentration is increased due to a

reduction in metabolic rate v. Equation 9.1 shows that v and KM are inversely related

(KM in denominator). An increase in KM will be accompanied by an increase in plasma

drug concentration. Figure 9-4 shows that an increase in KM is accompanied by an

increase in amount of drug in the body at any time t. Equation 9.4 relates drug

concentration to KM, and it can be seen that the two are proportionally related, although

they are not linearly proportional to each other due to the complexity of the equation.

An actual study in the literature shows that k is increased severalfold in the presence of

INH in the body.

4- Explain why KM sometimes has units of mM/mL and sometimes mg/L.

The KM has the units of concentration. In laboratory studies, KM is expressed in moles

per liter, or micromoles per milliliter, because reactions are expressed in moles and not

milligrams. In dosing, drugs are given in milligrams and plasma drug concentrations are

expressed as milligrams per liter or micrograms per milliliter. The units of KM for

pharmacokinetic models are estimated from in vivo data. They are therefore commonly

expressed as milligrams per liter, which is preferred over micrograms per milliliter

because dose is usually expressed in milligrams.

The two terms may be shown to be equivalent and convertible. Occasionally, when

simulating amount of drug metabolized in the body as a function of time, the amount of

drug in the body has been assumed to follow Michaelis–Menten kinetics, and KM

assumes the unit of D0 (eg, mg). In this case, KM takes on a very different meaning.