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.