Chapter 9. Nonlinear Pharmacokinetics

Previous chapters discussed linear pharmacokinetic models using simple first-order kinetics to describe the course of drug disposition and action. These linear models assumed that the pharmacokinetic parameters for a drug would not change when different doses or multiple doses of a drug were given. With some drugs, increased doses or chronic medication can cause deviations from the linear pharmacokinetic profile previously observed with single low doses of the same drug. This nonlinear pharmacokinetic behavior is also termed dose-dependentpharmacokinetics.

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. As shown in Table 9-1, various causes of nonlinear pharmacokinetic behavior are theoretically possible. Besides saturation of plasma protein-binding or carrier-mediated systems, drugs may demonstrate nonlinear pharmacokinetics due to a pathologic alteration in drug absorption, distribution, and elimination. For example, aminoglycosides may cause renal nephrotoxicity, thereby altering renal drug excretion. In addition, gallstone obstruction of the bile duct will alter biliary drug excretion. In most cases, the main pharmacokinetic outcome is a change in the apparent elimination rate constant.

A number of drugs demonstrate saturation or capacity-limitedmetabolism in humans. Examples of these saturable metabolic processes include glycine conjugation of salicylate, sulfate conjugation of salicylamide, acetylation of p-aminobenzoic acid, and the elimination of phenytoin (Tozer et al, 1981). 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 (ie, competition effects).

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

Because these drugs have a changing apparent elimination constant with larger doses, prediction of drug concentration in the blood based on a single small dose is difficult. Drug concentrations in the blood can increase rapidly once an elimination process is saturated. In general, metabolism (biotransformation) and active tubular secretion of drugs by the kidney are the processes most usually saturated. Figure 9-1 shows plasma level–time curves for a drug that exhibits saturable kinetics. When a large dose is given, a curve is obtained with an initial slow elimination phase followed by a much more rapid elimination at lower blood concentrations (curve A). With a small dose of the drug, apparent first-order kinetics is observed, because no saturation kinetics occurs (curve B). If the pharmacokinetic data were estimated only from the blood levels described by curve B, then a twofold increase in the dose would give the blood profile presented in curve C, which considerably underestimates the drug concentration as well as the duration of action.

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 (Fig. 9-2).

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

where Vmax is the maximum elimination rate and KM is the Michaelis constant that reflects the capacity of the enzyme system. It is important to note that KM 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.

The elimination rate of a hypothetical drug with a KM of 0.1 μg/mL and a Vmax of 0.5 μg/mL per hour is calculated in Table 9-2 by means of Equation 9.1. Because the ratio of the elimination rate to drug concentration changes as the drug concentration changes (ie, dCp/dt is not constant, Equation 9.1), the rate of drug elimination also changes and is not a first-order or linear process. In contrast, a first-order elimination process would yield the same elimination rate constant at all plasma drug concentrations. At drug concentrations of 0.4 to 10 μg/mL, the enzyme system is not saturated and the rate of elimination is a mixed or nonlinear process (Table 9-2). At higher drug concentrations, 11.2 μg/mL and above, the elimination rate approaches the maximum velocity (Vmax) of approximately 0.5 μg/mL per hour. At Vmax, the elimination rate is a constant and is considered a zero-order process.

Equation 9.1 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 9.1 becomes:

Using the hypothetical drug considered in Table 9-2 (Vmax = 0.5 μg/mL per hour, KM = 0.1 μg/mL), how long would it take for the plasma drug concentration to decrease from 20 to 12 μg/mL?

Solution

Because 12 μg/mL is above the saturable level, as indicated in Table 9-2, elimination occurs at a zero-order rate of approximately 0.5 μg/mL per hour.

A saturable process can also exhibit linear elimination when drug concentrations are much less than enzyme concentrations. When the drug concentration Cp is small in relation to the KM, the rate of drug elimination becomes a first-order process. The data generated from Equation 9.2 (Cp ≤ 0.05 μg/mL, Table 9-3) using KM = 0.8 μg/mL and Vmax = 0.9 μg/mL per hour shows that enzymatic drug elimination can change from a nonlinear to a linear process over a restricted concentration range. This is evident because the rate constant (or elimination rate/drug concentration) values are constant. At drug concentrations below 0.05 μg/mL, the ratio of elimination rate to drug concentration has a constant value of 1.1 h−1. Mathematically, when Cp is much smaller than KM, Cp in the denominator is negligible and the elimination rate becomes first order.

The first-order rate constant for a saturable process, k′, can be calculated from Equation 9.3:

This calculation confirms the data in Table 9-3, because enzymatic drug elimination at drug concentrations below 0.05 μg/mL is a first-order rate process with a rate constant of 1.1 h−1. Therefore, the t1/2 due to enzymatic elimination can be calculated:

How long would it take for the plasma concentration of the drug in Table 9-3 to decline from 0.05 to 0.005 μg/mL?

Solution

Because drug elimination is a first-order process for the specified concentrations,

Because , and ,

When given in therapeutic doses, most drugs produce plasma drug concentrations well below KM for all carrier-mediated enzyme systems affecting the pharmacokinetics of the drug. Therefore, most drugs at normal therapeutic concentrations follow first-order rate processes. Only a few drugs, such as salicylate and phenytoin, tend to saturate the hepatic mixed-function oxidases at higher therapeutic doses. With these drugs, elimination kinetics is first-order with very small doses, mixed order at higher doses, and may approach zero-order with very high therapeutic doses.

The rate of elimination of a drug that follows capacity-limited pharmacokinetics is governed by the Vmax and KM of the drug. Equation 9.1 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 9.1 described by

Alternatively, the amount of drug in the body after an IV bolus injection may be calculated by the following relationship. Equation 9.5 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.

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 9.5 must be rearranged and solved for time t:

The relationship of KM and Vmax to the time for an IV bolus injection of drug to decline to a given amount of drug in the body is illustrated in Figs. 9-3 and 9-4. Using Equation 9.6, the time for a single 400-mg dose given by IV bolus injection to decline to 20 mg was calculated for a drug with a KM of 38 mg/L and a Vmax that varied from 200 to 100 mg/h (Table 9-4). With a Vmax of 200 mg/h, the time for the 400-mg dose to decline to 20 mg in the body is 2.46 hours, whereas when the Vmax is decreased to 100 mg/h, the time for the 400-mg dose to decrease to 20 mg is increased to 4.93 hours (see Fig. 9-3). Thus, there is an inverse relationship between the time for the dose to decline to a certain amount of drug in the body and the Vmax as shown in Equation 9.6.

Using a similar example, the effect of KM on the time for a single 400-mg dose given by IV bolus injection to decline to 20 mg in the body is described in Table 9-5 and Fig. 9-4. Assuming Vmax is constant at 200 mg/h, the time for the drug to decline from 400 to 20 mg is 2.46 hours when KM is 38 mg/L, whereas when KM is 76 mg/L, the time for the drug dose to decline to 20 mg is 3.03 hours. Thus, an increase in KM (with no change in Vmax) will increase the time for the drug to be eliminated from the body.

The one-compartment open model with capacity-limited elimination pharmacokinetics adequately describes the plasma drug concentration–time profiles for some drugs. The mathematics needed to describe nonlinear pharmacokinetic behavior of drugs that follow two-compartment models and/or have both combined capacity-limited and first-order kinetic profiles are very complex and have little practical application for dosage calculations and therapeutic drug monitoring.

1. A drug eliminated from the body by capacity-limited pharmacokinetics has a KM of 100 mg/L and a Vmax of 50 mg/h. If 400 mg of the drug is given to a patient by IV bolus injection, calculate the time for the drug to be 50% eliminated. If 320 mg of the drug is to be given by IV bolus injection, calculate the time for 50% of the dose to be eliminated. Explain why there is a difference in the time for 50% elimination of a 400-mg dose compared to a 320-mg dose.

   Solution

   Use Equation 9.6 to calculate the time for the dose to decline to a given amount of drug in the body. For this problem, Dt is equal to 50% of the dose D0.

   If the dose is 400 mg,

   

   If the dose is 320 mg,

   

   For capacity-limited elimination, the elimination half-life is dose dependent, because the drug elimination process is partially saturated. Therefore, small changes in the dose will produce large differences in the time for 50% drug elimination. The parameters KM and Vmax determine when the dose is saturated.

2. Using the same drug as in Problem 1, calculate the time for 50% elimination of the dose when the doses are 10 and 5 mg. Explain why the times for 50% drug elimination are similar even though the dose is reduced by one-half.

   Solution

   As in Practice Problem 1, use Equation 9.6 to calculate the time for the amount of drug in the body at zero time (D0) to decline 50%.

   If the dose is 10 mg,

   

   If the dose is 5 mg,

   

   Whether the patient is given a 10- or a 5-mg dose by IV bolus injection, the times for the amount of drug to decline 50% are approximately the same. For 10- and 5-mg doses the amount of drug in the body is much less than the KM of 100 mg. Therefore, the amount of drug in the body is well below saturation of the elimination process and the drug declines at a first-order rate.

Determination of KM and Vmax

Equation 9.1 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 9.7). When an experiment is performed with solutions of various concentration of drug C, a series of reaction rates (v) may be measured for each concentration. Special plots may then be used to determine KM and Vmax (see also Chapter 11).

Equation 9.7 may be rearranged into Equation 9.8.

Equation 9.8 is a linear equation when 1/v is plotted against 1/C. The y intercept for the line is 1/Vmax, and the slope is KM/Vmax. An example of a drug reacting enzymatically with rate (v) at various concentrations C is shown in Table 9-6 and Fig. 9-5. A plot of 1/v versus 1/C is shown in Fig. 9-6. A plot of 1/v versus 1/C is linear with an intercept of 0.33 mmol. Therefore,

because the slope = 1.65 = KM/Vmax = KM/3 or KM = 3 × 1.65 μmol/mL = 5 μmol/mL. Alternatively, KM may be found from the x intercept, where −1/KM is equal to the x intercept. (This may be seen by extending the graph to intercept the x axis in the negative region.)

With this plot (Fig. 9-6), the points are clustered. Other methods are available that may spread the points more evenly. These methods are derived from rearranging Equation 9.8 into Equations 9.9 and 9.10.

A plot of C/v versus C would yield a straight line with 1/Vmax as slope and KM/Vmax as intercept (Equation 9.9). A plot of v versus v/C would yield a slope of −KM and an intercept of Vmax (Equation 9.10).

The necessary calculations for making the above plots are shown in Table 9-7. The plots are shown in Figs. 9-7 and 9-8. It should be noted that the data are spread out better by the two latter plots. Calculations from the slope show that the same KM and Vmax are obtained as in Fig. 9-6. When the data are more scattered, one method may be more accurate than the other. A simple approach is to graph the data and examine the linearity of the graphs. The same basic type of plot is used in the clinical literature to determine KM and Vmax for individual patients for drugs that undergo capacity-limited kinetics.

Determination of KM and Vmax in Patients

Equation 9.7 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 (see Chapter 11). The body may be regarded as a single 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.

An example for the determination of KM and Vmax is given for the drug phenytoin. Phenytoin undergoes capacity-limited kinetics at therapeutic drug concentrations in the body. To determine KM and Vmax, two different dose regimens are given at different times, until steady state is reached. The steady-state drug concentrations are then measured by assay. 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 9.11 may be written for drug metabolism in the body similar to the way drugs are metabolized in vitro (Eq. 9.7). 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.

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 of phenytoin 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 9.11) with C1, R1, C2, and R2. The equations contain two unknowns, KM and Vmax, and may be solved easily.

Combining the two equations yields Equation 9.15.

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. To calculate KM and Vmax, use Equation 9.15 with the values C1 = 8.6 mg/L, C2 = 25.1 mg/L, R1 = 150 mg/d, and R2 = 300 mg/d. The results are

Substitute KM into either of the two simultaneous equations to solve for Vmax.

Interpretation of KM and Vmax

An understanding of Michaelis–Menten kinetics provides insight into the nonlinear kinetics and helps avoid dosing a drug at a concentration near enzyme saturation. For example, in the above phenytoin dosing example, since KM occurs at 0.5Vmax, KM = 27.3 mg/L, the implication is that at a plasma concentration of 27.3 mg/L, enzymes responsible for phenytoin metabolism are eliminating the drug at 50% Vmax, ie, 0.5 × 626 mg/day or 313 mg/day. When the subject is receiving 300 mg of phenytoin per day, the plasma drug concentration of phenytoin is 8.6 mg/L, which is considerably below the KM of 27.3 mg/L. In practice, the KM in patients can range from 1 to 15 mg/L, Vmax can range from 100 to 1000 mg/d. Patients with a low KM tend to have greater changes in plasma concentrations during dosing adjustments. Patients with a smaller KM (same Vmax) will show a greater change in the rate of elimination when plasma drug concentration changes compared to subjects with a higher KM. A subject with the same Vmax, but different KM, is shown in Fig. 9-12. (For another example, see the slopes of the two curves generated in Fig. 9-4.)

Dependence of Elimination Half-Life on Dose

For drugs that follow linear kinetics, the elimination half-life is constant and does not change with dose or drug concentration. 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. The relationship between elimination half-life and drug concentration is shown in Equation 9.16. The elimination half-life is dependent on the Michaelis–Menten parameters and concentration.

Some pharmacokineticists prefer not to calculate the elimination half-life of a nonlinear drug because the elimination half-life is not constant. Clinically, if the half-life is increasing as plasma concentration increases, and there is no apparent change in metabolic or renal function, then there is a good possibility that the drug may be metabolized by nonlinear kinetics.

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. Within a certain drug concentration range, an average or mean clearance (Clav) may be determined. Because the drug follows Michaelis–Menten kinetics, Clav is dose dependent. Clav may be estimated from the area under the curve and the dose given (Wagner et al, 1985).

According to the Michaelis–Menten equation,

Inverting Equation 9.17 and rearranging yields

The area under the curve, is obtained by integration of Equation 9.18 (ie,

where V′max is the maximum velocity for metabolism. Units for V′max are mass/compartment volume per unit time. V′max = Vmax/VD; Wagner et al (1985) used Vmax in Equation 9.20 as mass/time to be consistent with biochemistry literature, which considers the initial mass of the substrate reacting with the enzyme.

Integration of Equation 9.18 from time 0 to infinity gives Equation 9.20.

where VD is the apparent volume of distribution.

Because the dose Equation 9.20 may be expressed as

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

Alternatively, dividing Equation 9.17 by Cp gives Equation 9.24, which shows that the clearance of a drug that follows nonlinear pharmacokinetics is dependent on the plasma drug concentration Cp, KM, and Vmax.

Equation 9.22 or 9.23 calculates the average clearance Clav for the drug after a single IV bolus dose over the entire time course of the drug in the body. For any time period, clearance may be calculated (see Chapters 6 and 11) as

In Chapter 11, the physiologic model based on blood flow and intrinsic clearance is used to describe drug metabolism. The extraction ratios of many drugs are listed in the literature. Actually, extraction ratios are dependent on dose, enzymatic system, and blood flow, and for practical purposes, they are often assumed to be constant at normal doses.

Except for phenytoin, there is a paucity of KM and Vmax data defining the nature of nonlinear drug elimination in patients. However, abundant information is available supporting variable metabolism due to genetic polymorphism (Chapter 11). The clearance (apparent) of many of these drugs in patients who are slow metabolizers changes with dose, although these drugs may exhibit linear kinetics in subjects with the “normal” phenotype. Metoprolol and many β-adrenergic antagonists are extensively metabolized. The plasma levels of metoprolol in slow metabolizers (Lennard et al, 1986) were much greater than other patients, and the AUC, after equal doses, is several times greater among slow metabolizers of metoprolol (Fig. 9-13). A similar picture is observed with another β-adrenergic antagonist, timolol. These drugs have smaller clearance than normal.

The dose-dependent pharmacokinetics of sodium valproate (VPA) was studied in guinea pigs at 20, 200, and 600 mg/kg by rapid intravenous infusion. The area under the plasma concentration–time curve increased out of proportion at the 600-mg/kg dose level in all groups (Yu et al, 1987). The total clearance (ClT) was significantly decreased and the beta elimination half-life (t1/2) was significantly increased at the 600-mg/kg dose level. The dose-dependent kinetics of VPA were due to saturation of metabolism. Metabolic capacity was greatly reduced in young guinea pigs.

Clinically, similar enzymatic saturation may be observed in infants and in special patient populations, whereas drug metabolism may be linear with dose in normal subjects. These patients have lower Vmax and longer elimination half-life. Variability in drug metabolism is described in Chapters 11 and 12.

Paroxetine hydrochloride (Paxil) is an orally administered psychotropic drug. Paroxetine is extensively metabolized and the metabolites are considered to be inactive. Nonlinearity in pharmacokinetics is observed with increasing doses. Paroxetine exhibits autoinhibition. The major pathway for paroxetine metabolism is by CYP2D6. The elimination half-life is about 21 hours. Saturation of this enzyme at clinical doses appears to account for the nonlinearity of paroxetine kinetics with increasing dose and increasing duration of treatment. The role of this enzyme in paroxetine metabolism also suggests potential drug–drug interactions. Clinical drug interaction studies have been performed with substrates of CYP2D6 and show that paroxetine can inhibit the metabolism of drugs metabolized by CYP2D6 including itself, desipramine, risperidone, and atomoxetine.

Paroxetine hydrochloride is known to inhibit metabolism of selective serotonin reuptake inhibitors (SSRIs) and monoamine oxidase inhibitors (MAOIs) producing “serotonin syndrome” (hyperthermia, muscle rigidity, and rapid changes in vital signs). Three cases of accidental overdosing with paroxetine hydrochloride were reported (Vermeulen, 1998). In the case of overdose, high liver drug concentrations and an extensive tissue distribution (large VD) made the drug difficult to remove. Vermeulen (1998) reported that saturation of CYP2D6 could result in a disproportionally higher plasma level than could be expected from an increase in dosage. These high plasma drug concentrations may be outside the range of 20 to 50 mg normally recommended. Since publication of this article, more is known about genotype CYP2D6*10 (Yoon, 2000), which may contribute to inter-subject variability in metabolism of this drug (see also Chapter 12).

The equations presented thus far in this chapter have been for drugs given by IV bolus, distributed as a one-compartment model, and eliminated only by nonlinear pharmacokinetics. The following are useful equations describing other possible routes of drug administration and including mixed drug elimination, by which the drug may be eliminated by both nonlinear (Michaelis–Menten) and linear (first-order) processes.

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. For example, sodium salicylate is metabolized to both a glucuronide and a glycine conjugate (hippurate). The rate of formation of the glycine conjugate is limited by the amount of glycine available. Thus, the rate of formation of the glucuronide continues as a first-order process, whereas the rate of conjugation with glycine is capacity limited.

The equation that describes a drug that is eliminated by both first-order and Michaelis–Menten kinetics after IV bolus injection is given by

where k is the first-order rate constant representing the sum of all first-order elimination processes, while the second term of Equation 9.26 represents the saturable process. V′max is simply Vmax expressed as concentration by dividing by VD.

The pharmacokinetic profile of niacin is complicated due to extensive first-pass metabolism that is dosing-rate specific. In humans, one metabolic pathway is through a conjugation step with glycine to form nicotinuric acid (NUA). NUA is excreted in the urine, although there may be a small amount of reversible metabolism back to niacin. The other metabolic pathway results in the formation of nicotinamide adenine dinucleotide (NAD). It is unclear whether nicotinamide is formed as a precursor to, or following the synthesis of, NAD. Nicotinamide is further metabolized to at least N-methylnicotinamide (MNA) and nicotinamide-N-oxide (NNO). MNA is further metabolized to two other compounds, N-methyl-2-pyridone-5-carboxamide (2PY) and N-methyl-4-pyridone-5-carboxamide (4PY). The formation of 2PY appears to predominate over 4PY in humans. At doses used to treat hyperlipidemia, these metabolic pathways are saturable, which explains the nonlinear relationship between niacin dose and plasma drug concentrations following multiple-doses of Niaspan (niacin) extended-release tablets (Niaspan, FDA-approved label, 2009).

Zero-Order Input and Nonlinear Elimination

The usual example of zero-order input is constant IV infusion. 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:

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 (eg, 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.

where ka is the first-order absorption rate constant.

If the drug is eliminated by parallel pathways consisting of both linear and nonlinear pharmacokinetics, Equation 9.28 may be extended to Equation 9.29.

where k is the first-order elimination rate constant.

Two-Compartment Model with Nonlinear Elimination

RhG-CSF is a glycoprotein hormone (recombinant human granulocyte-colony stimulating factors, rhG-CSF, MW about 20,000) that stimulates the growth of neutropoietic cells and activates mature neutrophils. The drug is used in neutropenia occurring during chemotherapy or radiotherapy. Similar to many biotechnological drugs, RhG-CSF is administered by injection. The drug is administered subcutaneously and the drug is absorbed into the blood from the dermis site. This drug follows a two-compartment model with two elimination processes: (1) a saturable process of receptor-mediated elimination in the bone marrow and (2) a nonsaturable process of elimination. The model is described by two differential equations as shown below:

where k12 and k21 are first-order transfer constants between the central and peripheral comparments; k is the first-order elimination constant from the central compartment; V1 is the volume of the central compartment and the steady-state volume of distribution is Vss; X2 is the amount in the peripheral compartment; C1 is the drug concentration in the central compartments at time t; and Vmax, KM are Michaelis–Menten parameters that describe the saturable elimination.

The pharmacokinetics of this drug was described by Hayashi et al (2001). Here, α is a function of dose with no dimensions and granulocyte colony-stimulating factor (G-CSF) takes a value from 0 to 1. When the dose approaches 0, α = 1; when the dose approaches ∞, α = 0

According to Hayashi et al. (2001), the drug clearance may be considered as two parts as shown below:

where Clint is intrinsic clearance for the saturable pathway; Cln is nonsaturable clearance; and C is serum concentration.

Chronopharmacokinetics broadly refers to a temporal change in the rate process (such as absorption or elimination) of a drug. The temporal changes in drug absorption or elimination can be cyclical over a constant period (eg, 24-hour interval), or they may be noncyclical, in which drug absorption or elimination changes over a longer period of time. Chronopharmacokinetics is an important consideration during drug therapy.

Time-dependent pharmacokinetics generally refers to a noncyclical change in the drug absorption or drug elimination rate process over a period of time. Time-dependent pharmacokinetics leads to nonlinear pharmacokinetics. Unlike dose-dependent pharmacokinetics, which involves a change in the rate process when the dose is changed, time-dependent pharmacokinetics may be the result of alteration in the physiology or biochemistry in an organ or a region in the body that influences drug disposition (Levy, 1983).

Time-dependent pharmacokinetics may be due to auto-induction or auto-inhibition of biotransformation enzymes. For example, Pitlick and Levy (1977) have shown that repeated doses of carbamazepine induce the enzymes responsible for its elimination (ie, auto-induction), thereby increasing the clearance of the drug. Auto-inhibition may occur during the course of metabolism of certain drugs (Perrier et al, 1973). In this case, the metabolites formed increase in concentration and further inhibit metabolism of the parent drug. In biochemistry, this phenomenon is known as product inhibition. Drugs undergoing time-dependent pharmacokinetics have variable clearance and elimination half-lives. The steady-state concentration of a drug that causes auto-induction may be due to increased clearance over time. Some anticancer drugs are better tolerated at certain times of the day; for example, the antimetabolite drug fluorouracil (FU) was least toxic when given in the morning to rodents (Von Roemeling, 1991). A list of drugs that demonstrate time dependence is shown in Table 9-8.

In pharmacokinetics, it is important to recognize that many isozymes (CYPs) are involved in drug metabolisms. A drug may competitively influence the metabolism of another drug within the same CYP subfamily. Sometimes, an unrecognized effect from the presence of another drug may be misjudged as a time-dependent pharmacokinetics. Drug metabolism and pharmacogenetics are discussed more extensively in Chapter 12.

Circadian Rhythms and Influence on Drug Response

Circadian rhythms are rhythmic or cyclical changes in plasma drug concentrations that may occur daily, due to normal changes in body functions. Some rhythmic changes that influence body functions and drug response are controlled by genes and subject to modification by environmental factors. The mammalian circadian clock is a self-sustaining oscillator, usually within a period of ∼24 hours, that cyclically controls many physiological and behavioral systems. The biological clock attempts to synchronize and respond to changes in length of the daylight cycle and optimize body functions.

Circadian rhythms are regulated through periodic activation of transcription by a set of clock genes. For example, melatonin onset is associated with onset of the quiescent period of cortisol secretion that regulates many functions. Some well-known circadian physiologic parameters are core body temperature (CBT), heart rate (HR), and other cardiovascular parameters. These fundamental physiologic factors can affect disease states, as well as toxicity and therapeutic response to drug therapy. The toxic dose of a drug may vary as much as several-fold, depending on the time of drug administration–during either sleep or wake cycle.

For example, the effects of timing of aminoglycoside administration on serum aminoglycoside levels and the incidence of nephrotoxicity were studied in 221 patients (Prins et al, 1997). Each patient received an IV injection of 2 to 4 mg/kg gentamicin or tobramycin once daily at: (1) between midnight and 7:30 AM, (2) between 8 AM and 3:30 PM, or (3) between 4 PM and 11:30 PM. In this study, no statistically significant differences in drug trough levels (0−4.2 mg/L) or peak drug levels (3.6−26.8 mg/L) were found for the three time periods of drug administration. However, nephrotoxicity occurred significantly more frequently when the aminoglycosides were given during the rest period (midnight–7:30 AM). Many factors contributing to nephrotoxicity were discussed; the time of administration was considered to be an independent risk factor in the multivariate statistical analysis. Time-dependent pharmacokinetics/pharmacodynamics is important, but it may be be difficult to detect the clinical difference in drug concentrations due to multivariates.

Another example of circadian changes on drug response involves observations with chronic obstructive pulmonary disease (COPD) patients. Symptoms of hypoxemia may be aggravated in some COPD patients due to changes in respiration during the sleep cycle. Circadian variations have been reported involving the incidence of acute myocardial infarction, sudden cardiac death, and stroke. Platelet aggregation favoring coagulation is increased after arising in the early morning hours, coincident with the peak incidence of these cardiovascular events, although much remains to be elucidated.

Time-dependent pharmacokinetics and physiologic functions are important considerations in the treatment of certain hypertensive subjects, in whom early-morning rise in blood pressure may increase the risk of stroke or hypertensive crisis. Verapamil is a commonly used antihypertensive. The diurnal pattern of forearm vascular resistance (FVR) between hypertensive and normotensive volunteers was studied at 9 PM on 24-hour ambulatory blood pressure monitoring, and the early-morning blood pressure rise was studied in 23 untreated hypertensives and 10 matched, normotensive controls. The diurnal pattern of FVR differed between hypertensives and normotensives, with normotensives exhibiting an FVR decline between 2 PM and 9 PM, while FVR rose at 9 PM in hypertensives. Verapamil appeared to minimize the diurnal variation in FVR in hypertensives, although there were no significant differences at any single time point. Verapamil effectively reduced ambulatory blood pressure throughout the 24-hour period, but it did not blunt the early-morning rate of blood pressure rise despite peak S-verapamil concentrations in the early morning (Nguyen et al, 2000).

Hypertensive patients are sometimes characterized as “dippers” if their nocturnal blood pressure drops below their daytime pressure. Non-dipping patients appear to be at an increased risk of cardiovascular morbidity. Blood pressure and cardiovascular events have a diurnal rhythm, with a peak of both in the morning hours, and a decrease during the night. The circadian variation of blood pressure provides assistance in predicting cardiovascular outcome (de la Sierra et al, 2009).

The pharmacokinetics of many cardiovasucular acting drugs have a circadian phase-dependency (Lemmer, 2006). Examples include β-blockers, calcium channel blockers, oral nitrates, and ACE inhibitors. There is clinical evidence that antihypertensive drugs should be dosed in the early morning in patients who are hypertensive “dippers,” whereas for patients who are non-dippers, it may be necessary to add an evening dose or even to use a single evening dose not only to reduce high blood pressure (BP) but also to normalize a disturbed non-dipping 24-hour BP profile. However, for practical purposes, some investigators found diurnal BP monitoring in many individuals too variable to distinguish between dippers and non-dippers (Lemmer, 2006)

The issue of time-dependent pharmacokinetics/pharmacodynamics (PK/PD) may be an important issue in some antihypertensive care. Pharmacists should recognize drugs that exhibit this type of time-dependant PK/PD.

Another example of time-dependent pharmacokinetics involves ciprofloxacin. Circadian variation in the urinary excretion of ciprofloxacin was investigated in a crossover study in 12 healthy male volunteers, ages 19 to 32 years. A significant decrease in the rate and extent of the urinary excretion of ciprofloxacin was observed following administrations at 2200 versus 1000 hours, indicating that the rate of excretion during the night time was slower (Sarveshwer Rao et al, 1997).

Clinical and Adverse Toxicity Due to Nonlinear Pharmacokinetics

The presence of nonlinear or dose-dependent pharmacokinetics, whether due to saturation of a process involving absorption, first-pass metabolism, binding, or renal excretion, can have significant clinical consequences. However, nonlinear pharmacokinetics may not be noticed in drug studies that use a narrow dose range in patients. In this case, dose estimation may result in disproportionate increases in adverse reactions but insufficient therapeutic benefits. Nonlinear pharmacokinetics can occur anywhere above, within, or below the therapeutic window.

The problem of a nonlinear dose relationship in population pharmacokinetics analysis has been investigated using simulations (Hashimoto et al, 1994, 1995; Jonsson et al, 2000). For example, nonlinear fluvoxamine pharmacokinetics was reported (Jonsson et al, 2000) to be present even at subtherapeutic doses. By using simulated data and applying nonlinear mixed-effects models using NONMEM, the authors also demonstrated that use of nonlinear mixed-effect models in population pharmacokinetics has an important application in the detection and characterization of nonlinear processes (pharmacokinetic and pharmacodynamic). Both first-order (FO) and FO conditional estimation (FOCE) algorithms were used for the population analyses. Population pharmacokinetics is discussed further in Chapter 22.

The bioavailability of drugs that follow nonlinear pharmacokinetics is difficult to estimate accurately. As shown in Table 9-1, each process of drug absorption, distribution, and elimination is potentially saturable. Drugs that follow linear pharmacokinetics follow the principle of superposition (Chapter 8). The assumption in applying the rule of superposition is that each dose of drug superimposes on the previous dose. Consequently, the bioavailability of subsequent doses is predictable and not affected by the previous dose. In the presence of a saturable pathway for drug absorption, distribution, or elimination, drug bioavailability will change within a single dose or with subsequent (multiple) doses. An example of a drug with dose-dependent absorption is chlorothiazide (Hsu et al, 1987).

The extent of bioavailability is generally estimated using If drug absorption is saturation limited in the gastrointestinal tract, then a smaller fraction of drug is absorbed systemically when the gastrointestinal drug concentration is high. A drug with a saturable elimination pathway may also have a concentration-dependent AUC affected by the magnitude of KM and Vmax of the enzymes involved in drug elimination (Equation 9.21). At low Cp, the rate of elimination is first order, even at the beginning of drug absorption from the gastrointestinal tract. As more drug is absorbed, either from a single dose or after multiple doses, systemic drug concentrations increase to levels that saturate the enzymes involved in drug elimination. The body drug clearance changes and the AUC increases disproportionately to the increase in dose (see Fig. 9-2).

Protein binding may prolong the elimination half-life of a drug. Drugs that are protein bound must first dissociate into the free or nonbound form to be eliminated by glomerular filtration. The nature and extent of drug–protein binding affects the magnitude of the deviation from normal linear or first-order elimination rate process.

For example, consider the plasma level–time curves of two hypothetical drugs given intravenously in equal doses (Fig. 9-14). One drug is 90% protein bound, whereas the other drug does not bind plasma protein. Both drugs are eliminated solely by glomerular filtration through the kidney.

The plasma curves in Fig. 9-14 demonstrate that the protein-bound drug is more concentrated in the plasma than a drug that is not protein bound, and the protein-bound drug is eliminated at a slower, nonlinear rate. Because the two drugs are eliminated by identical mechanisms, the characteristically slower elimination rate for the protein-bound drug is due to the fact that less free drug is available for glomerular filtration in the course of renal excretion.

The concentration of free drug, Cf, can be calculated at any time, as follows.

For any protein-bound drug, the free drug concentration (Cf) will always be less than the total drug concentration (Cp).

A careful examination of Fig. 9-14 shows that the slope of the bound drug decreases gradually as the drug concentration decreases. This indicates that the percent of drug bound is not constant. In vivo, the percent of drug bound usually increases as the plasma drug concentration decreases (see Chapter 10). Since protein binding of drug can cause nonlinear elimination rates, pharmacokinetic fitting of protein-bound drug data to a simple one-compartment model without accounting for binding results in erroneous estimates of the volume of distribution and elimination half-life. Sometimes plasma drug data for drugs that are highly protein bound have been inappropriately fitted to two-compartment models.

Valproic acid (Depakene) shows nonlinear pharmacokinetics that may be due partially to nonlinear protein binding. The free fraction of valproic acid is 10% at a plasma drug concentration of 40 μg/mL and 18.5% at a plasma drug level of 130 μg/mL. In addition, higher-than-expected plasma drug concentrations occur in the elderly, hyperlidemic patients, and in patients with hepatic or renal disease.

One-Compartment Model Drug with Protein Binding

The process of elimination of a drug distributed in a single compartment with protein binding is illustrated in Fig. 9-15. The one compartment contains both free drug and bound drug, which are dynamically interconverted with rate constants k1 and k2. Elimination of drug occurs only with the free drug, at a first-order rate. The bound drug is not eliminated. Assuming a saturable and instantly reversible drug-binding process, where P = protein concentration in plasma, Cf = plasma concentration of free drug, kd = k2/k1 = dissociation constant of the protein drug complex, Cp = total plasma drug concentration, and Cb = plasma concentration of bound drug,

This equation can be rearranged as follows:

Solving for Cf,

Because the rate of drug elimination is dCp/dt,

This differential equation describes the relationship of changing plasma drug concentrations during elimination. The equation is not easily integrated but can be solved using a numerical method. Figure 9-16 shows the plasma drug concentration curves for a one-compartment protein-bound drug having a volume of distribution of 50 mL/kg and an elimination half-life of 30 minutes. The protein concentration is 4.4% and the molecular weight of the protein is 67,000 Da. At various doses, the pharmacokinetics of elimination of the drug, as shown by the plasma curves, ranges from linear to nonlinear, depending on the total plasma drug concentration.

Nonlinear drug elimination pharmacokinetics occurs at higher doses. Because more free drug is available at higher doses, initial drug elimination occurs more rapidly. For drugs demonstrating nonlinear pharmacokinetics, the free drug concentration may increase slowly at first, but when the dose of drug is raised beyond the protein-bound saturation point, free plasma drug concentrations may rise abruptly. Therefore, the concentration of free drug should always be calculated to make sure the patient receives a proper dose.

Determination Linearity in Data Analysis

During new drug development, the pharmacokinetics of the drug is examined for linear or nonlinear pharmacokinetics. A common approach is to give several graded doses to human volunteers and obtain plasma drug concentration curves for each dose. From these data, a graph of AUC versus dose is generated as shown in Fig. 9-2. The drug is considered to follow linear kinetics if AUC versus dose for various doses is proportional (ie, linear relationship). In practice, the experimental data presented may not be very clear, especially when oral drug administration data are presented and there is considerable variability in the data. For example, the AUC versus three-graded doses of a new drug is shown in Figure 9-17. A linear regression line was drawn through the three data points. The conclusion is that the drug follows dose-independent (linear) kinetics based upon a linear regession line through the data and a correlation coefficient, R2 = 0.97.

• Do you agree with this conclusion after inspecting the graph?

The conclusion for linear pharmacokinetics in Fig. 9-17 seems reasonable based on the estimated regression line drawn through the data points.

However, another pharmacokineticist noticed that the regression line in Fig. 9-17 does not pass through the origin point (0,0). This pharmacokineticist considered the following questions:

• Are the patients in the study receiving the drug doses well separated by a washout period during the trial such that no residual drug remained in the body and carried to the present dose when plasma samples are collected?
• Is the method for assaying the samples validated? Could a high sample blank or interfering material be artificially adding to elevate 0 time drug concentrations?
• How does the trend line look if the point (0,0) is included?

When the third AUC point is above the trend line, it is risky to draw a conclusion. One should verify that the high AUC is not due to a lower elimination or clearance due to saturation.

In Fig. 9-18, a regression line was obtained by forcing the same data through point (0,0). The linear regression analysis and estimated R2 appears to show that the drug followed nonlinear pharmacokinetics. The line appears to have a curvature upward and the possibility of some saturation at higher doses. This pharmacokineticist recommends additional study by adding a higher dose to more clearly check for dose dependency.

• What is your conclusion?
  • Considerations
• The experimental data is composed of three different drug doses.
  • The regression line shows that the drug follows linear pharmacokinetics from the low dose to the high dose.
  • The use of a (0.0) value may provide additional information concerning the linearity of the pharmacokinetics. However, extrapolation of curves beyond the actual experimental data can be misleading.
  • The conclusion in using the (0.0) time point shows that the pharmacokinetics is nonlinear below the lowest drug dose. This may occur after oral dosing because at very low drug doses some of the drug is decomposed in the gastrointestinal tract or metabolized prior to systemic absorption. With higher doses, the small amount of drug loss is not observed systemically.

Note if VD of the drug is known, determining k from the terminal slope of the oral data provides another way of calculating Cl (Cl = VDk) to check whether clearance has changed at higher doses due to saturation. Some common issues during data analysis for linearity are listed below.

Note: In some cases, with certain drugs, the oral absorption mechanism is quite unique and drug clearance by the oral route may involve absorption site-specific enzymes or transporters located on the brush border. Extrapolating pharmacokinetic information from IV dosing dose data should be done cautiously only after a careful consideration of these factors. It is helpful to know whether nonlinearity is caused by distribution, or absorption factors.

Unsuspected nonlinear drug disposition is one of the biggest issues concerning drug safety. Although pharmacokinetic tools are useful, nonlinearity can be easily missed during data analysis when there are outliners or extreme data scattering due to individual patient factors such as genetics, age, sex, and other unknown factors in special populations. While statistical analysis can help minimize this, it is extremely helpful to survey for problems (eg, epidemiological surveillance) and have a good understanding of how drugs are disposed in various parts of the body in the target populations.

1. Nonlinearity caused by membrane resident transporters

2. Nonlinearity caused by membrane CYPs

3. Nonlinearity caused by cellular proteins

4. Nonlinearity caused by transporter proteins at the GI tract

5. Nonlinearity caused by bile acid transport (Apical/bile canaliculus)

1Source: Evaluation of hepatotoxic potential of drugs using transporter-based assays. Jasminder Sahi AAPS Transporter Meeting, 2005 at Parsipanny, New Jersey.

Nonlinear pharmacokinetics refers to kinetic processes that result in disproportional changes in plasma drug concentrations when the dose is changed. This is also referred to as dose-dependent pharmacokinetics or saturation pharmacokinetics. Clearance and half-life are usually not constant with dose-dependent pharmaokinetics. Carrier-mediated processes and processes that depend on the binding of the drug to a macromolecule resulting in drug metabolism, protein binding, active absorption, and some transporter-mediated processes can potentially exhibit dose-dependent kinetics, especially at higher doses. The Michaelis–Menten kinetic equation may be applied in vitro or in vivo to describe drug disposition, eg, phenytoin.

An approach to determine nonlinear pharmacokinetics is to plot AUC versus doses and observe for nonlinearity curving. A common cause of overdosing in clinical practice is undetected saturation of a metabolic enzyme due to genotype difference in a subject, eg, CYP2D6. A second common cause of overdosing in clinical practice is undetected saturation of a metabolic enzyme due to coadministration of a second drug/agent that alters the original linear elimination process.

1. Define nonlinear pharmacokinetics. How do drugs that follow nonlinear pharmacokinetics differ from drugs that follow linear pharmacokinetics?

   1. What is the rate of change in the plasma drug concentration with respect to time, dCp/dt, when Cp « KM?

   2. What is the rate of change in the plasma drug concentration with respect to time, dCp/dt, when Cp » KM?

2. What processes of drug absorption, distribution, and elimination may be considered “capacity limited,” “saturated,” or “dose dependent”?

3. Drugs, such as phenytoin and salicylates, have been reported to follow dose-dependent elimination kinetics. What changes in pharmacokinetic parameters, including t1/2, VD, AUC, and Cp, could be predicted if the amounts of these drugs administered were increased from low pharmacologic doses to high therapeutic doses?

4. 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.

   1. What is the reaction order for the metabolism of this drug when given in a single intravenous dose of 10 mg/kg?

   2. How much time is necessary for the drug to be 50% metabolized?

5. How would induction or inhibition of the hepatic enzymes involved in drug biotransformation theoretically affect the pharmacokinetics of a drug that demonstrates nonlinear pharmacokinetics due to saturation of its hepatic elimination pathway?

6. Assume that both the active parent drug and its inactive metabolites are excreted by active tubular secretion. What might be the consequences of increasing the dosage of the drug on its elimination half-life?

7. 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? (Hint: see Fig. 9-4.)

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

9. The Vmax for metabolizing a drug is 10 mmol/h. The rate of metabolism (v) is 5 μmol/h when drug concentration is 4 μmol. Which of the following statements is/are true?

   1. KM is 5 μmol for this drug.

   2. KM cannot be determined from the information given.

   3. KM is 4 μmol for this drug.

10. Which of the following statements is/are true regarding the pharmacokinetics of diazepam (98% protein bound) and propranolol (87% protein bound)?

    1. Diazepam has a long elimination half-life because it is difficult to be metabolized due to extensive plasma–protein binding.

    2. Propranolol is an example of a drug with high protein binding but unrestricted (unaffected) metabolic clearance.

    3. Diazepam is an example of a drug with low hepatic extraction.

    4. All of the above.

    5. a and c.

    6. b and c.

11. Which of the following statements describe(s) correctly the properties of a drug that follows nonlinear or capacity-limited pharmacokinetics?

    1. The elimination half-life will remain constant when the dose changes.

    2. The area under the plasma curve (AUC) will increase proportionally as dose increases.

    3. The rate of drug elimination = Cp × KM.

    4. All of the above.

    5. a and b.

    6. None of the above.

12. The hepatic intrinsic clearances of two drugs are

    drug A: 1300 mL/min

    drug B: 26 mL/min

    Which drug is likely to show the greatest increase in hepatic clearance when hepatic blood flow is increased from 1 L/min to 1.5 L/min?

    1. Drug A

    2. Drug B

    3. No change for both drugs