LESSON 10
`Nonlinear Processes
`
`OBJECTIVES
`
`After completing Lesson 10, you should be able to:
`
`1. Describe the relationship of both drug concentration and area under the plasma drug
`concentration versus time curve (AUG) to the dose for a nonlinear, zero-order process.
`
`2. Explain the various biopharmaceutic processes that can result in nonlinear
`pharmacokinetics.
`
`3. Describe how hepatic enzyme saturation can result in nonlinear pharmacokinetics.
`
`4. Use the Michaelis-Menten model for describing nonlinear pharmacokinetics.
`
`5. Describe Vmax and Km.
`6. Use the Michaelis- Menten model to predict plasma drug concentrations.
`7. Use the t90% equation to estimate the time required for 90% of the steady-state
`concentration to be reached .
`
`Until now, we have used a major assumption in constructing models for drug pharmaco(cid:173)
`kinetics: drug clearance remains constant with any size dose. This is the case
`only when drug elimination processes are first order (as described in previous
`lessons). With a first-order elimination process, as the dose of drug increases, the
`plasma concentrations observed and the AUC increase proportionally. That is, if
`the dose is doubled, the plasma concentration and AUC also double (Figure 10-1).
`Because the increase in plasma concentration and AUC is linear with drug
`dose in first-order processes, this concept is referred to as linear pharmaco(cid:173)
`kinetics. When these linear relationships are present, they are used to predict drug
`dosage. For example, if a 100-mg daily dose of a drug produces a steady-state peak
`plasma concentration of 10 mg/L, we know that a 200-mg daily dose will result in
`a steady-state plasma concentration of 20 mg/L. (Note that linear does not refer to
`the plot of natural log of plasma concentration versus time.)
`With some drugs (e.g., phenytoin and aspirin), however, the relationships
`of drug dose to plasma concentrations and AUC are not linear. As the drug dose
`increases, the peak concentration and the resulting AUC do not increase propor(cid:173)
`tionally (Figure 10-2). Therefore, such drugs are said to follow nonlinear,
`zero-order, or dose-dependent pharmacokinetics (i.e., the pharmacokinetics
`change with the dose given). Just as with drugs following linear pharmacokinetics,
`it is important to predict the plasma drug concentrations of drugs following zero(cid:173)
`order pharmacokinetics. In this lesson, we discuss methods to characterize drugs
`that follow nonlinear pharmacokinetics.
`
`149
`
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`Concepts in Clinical Pharmacokinetics
`
`150
`
`FIGURE 10-1.
`Relationship of AUG to drug dose with first-order elimination,
`where clearance is not influenced by dose.
`
`FIGURE 10-2.
`Relationship of AUG to drug dose with dose-dependent
`pharmacokinetics.
`
`Nonlinear pharmacokinetics may refer to several
`different processes, including absorption, distribu(cid:173)
`tion, and renal or hepatic elimination (Table 10-1).
`For example, with nonlinear absorption, the frac(cid:173)
`tion of drug in the gastrointestinal (GI) tract that is
`absorbed per minute changes with the amount of drug
`present. Even though absorption and distribution can
`be nonlinear; the term nonlinear pharmacokinetics
`usually refers to the processes of drug elimination.
`When a drug exhibits nonlinear pharmaco(cid:173)
`kinetics, usually the processes responsible for drug
`elimination are saturable at therapeutic concen(cid:173)
`trations. These elimination processes may include
`renal tubular secretion (as seen with penicillins) and
`hepatic enzyme metabolism (as seen with phenytoin).
`When an elimination process is saturated, any
`increase in drug dose results in a disproportionate
`increase in the plasma concentrations achieved
`because the amount of drug that can be eliminated
`over time cannot increase. This situation is contrary
`to first-order linear processes, in which an increase
`in drug dosage results in an increase in the amount
`of drug eliminated over any given period.
`
`Of course, most elimination processes are
`capable of being saturated if enough drug is
`administered. However, for most drugs, the doses
`administered do not cause the elimination processes
`to approach their limitations.
`
`Clinical Correlate
`
`Many drugs exhibit mixed-order pharmaco(cid:173)
`kinetics, displaying first-order pharmacokinetics
`at low drug concentrations and zero-order
`pharmacokinetics at high concentrations. It is
`important to know the drug concentration at
`which a drug order switches from first to zero.
`Phenytoin is an example of a drug that switches
`order at therapeutic concentrations, whereas
`theophylline does not switch until concentrations
`reach the toxic range.
`
`For a typical drug having dose-dependent
`pharmacokinetics, with saturable elimination, the
`plasma drug concentration versus time plot after a
`dose may appear as shown in Figure 10-3.
`
`TABLE 10-1. Drugs Having Dose- or Time-Dependent Pharmacokinetics
`
`Agent
`
`Mechanism
`
`Riboflavin, methotrexate, gabapentin
`Penicillins
`
`Saturable gut wall transport
`Saturable decomposition in Gl tract
`
`Saturable transport into and out of tissues
`Saturable protein binding
`
`Active tubular secretion
`Active reabsorption
`
`Enzyme induction
`Saturable metabolism
`
`Process
`
`Absorption
`
`Distribution
`
`Renal elimination
`
`Methotrexate
`Salicylates
`
`Penicillin G
`Ascorbic acid
`
`Extrarenal elimination
`
`Garbamazepine
`Theophylline, phenytoin
`
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`

`Lesson 10 I Nonlinear Processes
`
`151
`
`100
`
`Cll:
`2~
`c~
`<II'E
`E Q) 10
`UlO
`til 1: -o
`Q.(.)
`
`FIGURE 10-3.
`Dose-dependent clearance of enzyme-saturable drugs.
`
`Time
`
`After a large dose is administered, an initial slow
`elimination phase (clearance decreases with higher
`plasma concentration) is followed by a much more
`rapid elimination at lower concentrations (curve A).
`However, when a small dose is administered (curve B),
`the capacity of the elimination process is not reached,
`and the elimination rate remains constant. At high
`concentrations, the elimination rate approaches that
`of a zero-order process (i.e., the amount of drug elim(cid:173)
`inated over a given period remains constant, but the
`fraction eliminated changes). At low concentrations,
`the elimination rate approaches that of a first-order
`process (i.e., the amount of drug eliminated over a
`given time changes, but the fraction of drug elimi(cid:173)
`nated remains constant).
`A model that has been used extensively in
`biochemistry to describe the kinetics of saturable
`enzyme systems is known as Michaelis-Menten
`kinetics (for its developers). This system describes
`the relationship of an enzyme to the substrate (in
`this case, the drug molecule). In clinical pharmaco(cid:173)
`kinetics, it allows prediction of plasma drug concen(cid:173)
`trations resulting from administration of drugs with
`saturable elimination (e.g., phenytoin).
`The equation used to describe Michaelis(cid:173)
`Menten pharmacokinetics is:
`
`lt' C
`-dC
`drug elimination rate = - - = K max C
`m +
`dt
`
`where -dCjdt is the rate of drug concentration
`decline at time t and is determined by v max• the theo(cid:173)
`retical maximum rate of the elimination process. Km
`is the drug concentration when the rate of elimi(cid:173)
`nation is half the maximum rate, and C is the total
`plasma drug concentration.
`
`max
`V
`is expressed in units of amount per unit of
`time (e.g., milligrams per day) and represents the
`maximum amount of drug that can be eliminated
`in the given time period. For drugs metabolized by
`the liver, vmax can be determined by the quantity or
`efficiency of metabolizing enzymes. This param(cid:173)
`eter will vary, depending on the drug and individual
`patient.
`the Michaelis constant, is expressed in
`K
`m•
`units of concentration (e.g., mg/L) and is the drug
`concentration at which the rate of elimination is
`half the maximum rate CVmaxl In simplified terms,
`K is the concentration above which saturation of
`m
`drug metabolism is likely.
`Vmax and Km are related to the plasma drug
`concentration and the rate of drug elimination
`as shown in Figure 10-4. When the plasma drug
`concentration is less than Km, the rate of drug
`elimination follows first-order pharmacokinetics.
`In other words, the amount of drug eliminated
`per hour directly increases with the plasma drug
`concentration. When the plasma drug concentra(cid:173)
`tion is much less than Km, the first-order elimination
`rate constant (K) for drugs with nonlinear pharmaco(cid:173)
`kinetics is approximated by vmax ; therefore, as vmax
`increases (e.g., by hepatic enzyme induction), K
`increases.
`With drugs having saturable elimination, as
`plasma drug concentrations increase, drug elimi(cid:173)
`nation approaches its maximum rate. When the
`plasma concentration is much greater than Km, the
`rate of drug elimination is approximated by vmax•
`and elimination proceeds at close to a zero-order
`process.
`
`Vmax ___________ ....,.
`
`(ll
`
`Ole: 2 0 o:;::
`-c:: o ._
`C11 E m=
`a:w
`
`Plasma Drug
`Concentration
`
`FIGURE 10-4.
`Relationship of drug elimination rate to plasma drug
`concentration with saturable elimination.
`
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`Concepts in Clinical Pharmacokinetics
`
`152
`
`Next, we consider how Vmax and Km can be calcu(cid:173)
`lated and how these determinations may be used to
`predict plasma drug concentrations in patients.
`
`Then:
`
`Calculation of Vmax' Km, and Plasma
`Concentration and Dose
`
`For drugs that have saturable elimination at the
`plasma concentrations readily achieved with thera(cid:173)
`peutic doses (e.g., phenytoin), prediction of the plasma
`concentrations achieved by a given dose is important.
`For these predictions, it is necessary to estimate Vmax
`and K111 • Therefore, we must apply the Michaelis(cid:173)
`Menten equation presented earlier in this lesson:
`
`-dC
`dt
`
`VmaxC
`=
`Km +C
`
`The change in drug concentration over time is
`related to the Michaelis-Menten parameters Vmax•
`K111, and the plasma drug concentration (C). We
`know that at steady state (after multiple drug doses)
`the rate of drug loss from the body (milligrams
`removed per day) is equal to the amount of drug
`being administered (daily dose). In the Michaelis(cid:173)
`Menten equation, -dCjdt indicates the rate of drug
`loss from the body; therefore, at steady state:
`V. C
`-dC
`= daily drug dose = max c
`-
`Km +
`dt
`Now we have an equation that relates Vmax• Km,
`plasma drug concentration, and daily dose (at steady
`state). To use this relationship, it is first helpful to
`transform the equation to a straight-line form:
`v. c
`max c
`Km+
`
`.
`dally dose=
`
`daily dose (K m +C) = VmaxC
`
`daily dose (K m) +daily dose (C) = VmaxC
`
`daily dose= -Km(daily dose/C)+ Vmax
`
`Y (slope) = mX + b (intercept)
`
`So the relationship of the Michaelis-Menten param(cid:173)
`eters, C, and dose can be expressed as a straight line
`(Figure 10-5). If the straight line can be defined,
`then V
`and Km can be determined; if Vmax and
`max
`Km are known, then the plasma concentrations at
`steady state resulting from any given dose can be
`estimated.
`To define the line, it is necessary to know the
`steady-state concentrations achieved at a minimum
`of two different doses. For example, a patient
`receiving 300 mg of phenytoin per day achieved
`a steady-state concentration (trough) of 9 mg/L;
`when the daily dose was increased to 400 mgjday, a
`steady-state concentration of16 mg/L was achieved.
`The data for this patient can be plotted as shown
`in Figure 10-6. Then a line is drawn between the
`two points, intersecting they-axis. They-intercept
`equals vmax (observed to be 700 mgjday), and the
`slope of the line equals -Km.
`
`Calculating Km
`
`K
`I
`s ope=- m =
`
`doseinitial - dose increased
`dose/ cinitial - dose/ cincreased
`
`300 mg/day- 400 mg/day
`= ( 300 mg/day _ 400 mg/day J
`9 mg/L
`16 mg/L
`
`-100 mg/day
`= - - - - -= - - - ' - -
`33.3 Uday-25 Uday
`
`= - 12.0 mg/L
`
`daily dose (C) = VmaxC- daily dose (K m)
`
`So K"' equals 12 mg/L.
`
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`

`Lesson 10 I Nonlinear Processes
`
`153
`
`Slope = -Km
`
`Q)
`Ill
`
`0 c
`~
`"' c
`
`Daily Dose/CSteady State
`
`FIGURE 10-5.
`Linear plot of the Michaelis- Menten equation.
`
`Calculating Dose
`Knowing Vmax and Km, we can then predict the dose
`necessary to achieve a given steady-state concen(cid:173)
`tration or the concentration resulting from a given
`dose. If we wish to increase the steady-state plasma
`concentration to 20 mg/L, we can use the Michaelis(cid:173)
`Menten equation to predict the necessary dose:
`
`(700 mg/day)(20 mg/L)
`=
`12 mg/L + 20 mg/L
`
`14,000 mg2 /(day x L)
`= _;____--=._____:___:____:_
`32 mg/L
`
`= 437 mg/day
`
`Note how units cancel out to yield mgjday.
`
`Calculati ng Steady-State
`Concentration from This Km and Dose
`If we wish to predict the steady-state plasma concen(cid:173)
`tration that would result if the dose is increased
`to 500 mgjday, we can rearrange the Michaelis(cid:173)
`Menten equation and solve for C:
`
`C = Km(daily dose)
`vmax - daily dose
`
`12 mg/L (500 mg/day)
`=
`700 mg/day- 500 mg/day
`
`12 mg/L (500 mg/day)
`=
`200 mg/day
`
`= 30 mg/L
`
`700
`600
`.s
`Cl 500
`Q) 400
`Ill
`0 c 300
`:?.-·;;; 200
`c
`100
`
`10
`20
`30
`Daily Dose /Csteady State
`
`40
`
`FIGURE 10-6.
`Plot of patient data using two steady-state plasma phenytoin
`concentrations at two dose levels.
`
`See Lesson 15 for examples of how these calcu(cid:173)
`lations are applied.
`
`Clinical Correlate
`
`When performing this calculation using sodium
`phenytoin or fosphenytoin, be sure to convert
`doses to their phenytoin free-acid equivalent
`before substituting these values into the equation.
`To convert, multiply the daily dose by 0.92 (92%
`free phenytoin). Fosphenytoin injection, although
`containing only 66% phenytoin free acid, is
`actually labeled in phenytoin sodium equivalents
`such that the 0.92 factor also applies to this
`product.
`
`The preceding example demonstrates how
`plasma drug concentrations and drug dose can be
`predicted. However, it also shows that for drugs
`like phenytoin, with saturable elimination, when
`plasma concentrations are above Kw small dose
`increases can result in large increases in the steady(cid:173)
`state plasma concentration.
`When clearance changes with plasma concen(cid:173)
`tration, there is no true half-life as with first-order
`elimination. As clearance changes, the elimination
`rate changes as does the time to reach steady state.
`With high doses and high plasma concentrations
`(and resulting lower clearance), the time to reach
`steady state is much longer than with low doses and
`low plasma concentrations (Figure 10-7). Theoret-
`
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`

`When the dose is increased to 400 mgjday:
`
`t 0 =
`12 mg/L (50 L)
`(700 mg/day- 300 mg/day)2
`govo
`
`[2.3(700 mg/day)-
`
`0.9(400 mg/day)]
`
`600 mg
`=
`2 [161 0 mg/day- 360 mg/day]
`(300 mg/day)
`
`= (0.0067 day2 /mg)(1250 mg/day)
`
`= 8.38 days
`
`We can see that as the dose is increased, it takes
`a longer time to reach steady state, drug continues
`to accumulate, and the plasma drug concentration
`continues to rise. When this occurs with a drug
`such as phenytoin, toxic effects (e.g., ataxia and
`nystagmus) probably will be observed if the high
`dosage is given on a regular basis.
`
`Clinical Correlate
`
`The ~o% equation will provide only a rough
`estimate of when 90% of steady state has been
`reached, and its accuracy is dependent on the
`Km value used. Other ways to check to see if a
`patient is at steady state are to examine two
`levels drawn approximately a week apart. If
`these levels are ± 1 0% of each other, then you
`can assume steady state. Additionally, it is
`safe to wait at least 2 weeks (and preferably 4
`weeks) after beginning or changing a dose before
`obtaining new steady-state levels.
`
`Concepts in Clinical Pharmacokinetics
`
`154
`
`Dose and Time Dependencies
`
`Plateau
`Rate of
`Administration Concentration
`(mg/day)
`
`425
`
`25
`
`20
`
`Q)
`
`c
`0
`~
`'E
`0 c
`8~15
`·=:a,
`o E
`~~10
`Q)
`.c
`Q.
`as
`E
`Ill as
`ii:
`
`5
`
`0 ~--~~--~--~--~~
`5
`10
`15
`20
`25
`0
`
`IX>
`
`Days
`
`FIGURE 10-7.
`Time to reach t90o;, (represented by arrows) at different daily
`dosages.
`
`ically, if the dose is greater than vmax• steady state
`will never be reached.
`Because clearance and half-life are concen(cid:173)
`tration-dependent factors, a traditional time to
`steady-state value cannot be calculated. Instead, the
`Michaelis-Menten equation can be rearranged to
`provide an equation that estimates the time required
`(in days) for 90% of the steady-state concentration
`to be reached ( t 90%), as shown below for phenytoin
`(where the dose equals the daily dose):
`Km(V)
`t90o;. =
`.
`2 [2.3Vmax -0.9 dose]
`(Vmax - daily dose)
`
`o
`
`From the previous example, when dose= 300 mgjday;
`Vmax = 700 mgjday, and Km = 12 mg/L, volume of
`distribution (V) can be estimated as 0.65 L/kg body
`weight, or (0.65 x 77 kg body weight) = 50 L.
`
`t 0 =
`12 mg/L (50 L)
`(700 mg/day- 300 mg/day)2
`govo
`
`[2.3(700 mg/day) -
`
`0.9(300 mg/day)]
`
`600 mg
`------=-2 [161 0 mg/day- 270 mg/day]
`( 400 mg/day)
`
`= (0.00375 day2 /mg)(1340 mg/day)
`= 5.0 days
`
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`

`Lesson 10 I Nonlinear Processes
`
`'I
`
`155
`
`REVIEW QUESTIONS
`
`10-1. Which drug pairs demonstrate nonlinear
`pharmacokinetics?
`A. phenytoin and aspirin
`B. penicillin G and gentamicin
`C. acetaminophen and sulfonamides
`D. A and B
`
`10-2. Linear pharmacokinetics means that the
`plot of plasma drug concentration versus
`time after a dose is a straight line.
`A. True
`B. False
`
`10-3. When hepatic metabolism becomes satu(cid:173)
`rated, any increase in drug dose will lead
`to a proportionate increase in the plasma
`concentration achieved.
`A. True
`B. False
`
`10-4. When the rate of drug elimination proceeds
`at half the maximum rate, the drug concen(cid:173)
`tration is known as:
`A.
`vmax
`B. Km
`c. ~Vmax
`D. CVmax)(C)
`
`10-5. At very high concentrations-concentrations
`much higher than the drug's Km-drugs are
`more likely to exhibit first-order elimination.
`A. True
`B. False
`
`10-6. Which of the equations below describes the
`form of the Michaelis-Menten equation that
`relates daily drug dose to V max• Km, and the
`steady-state plasma drug concentration?
`A. daily dose = -Km( daily dose/ C) (V max)
`B. daily dose= -Km(daily dose/C)+ Vmax
`c. daily dose= -Km(daily dose X C)+ vmax
`D. daily dose = -Km- (daily dose/ C) + Vmax
`
`The following information is for Questions 10-7
`to 10-11. A patient, JH, is administered phenytoin
`free acid, 300 mgjday for 2 months (assume steady
`state is achieved), and a plasma concentration
`determined just before a dose is 10 mg/L. The
`phenytoin dose is then changed to 400 mgjday;
`2 months after the dose change, the plasma
`concentration determined just before a dose is
`18 mgjL. Assume that the volume of distribution of
`phenytoin is 45 L.
`
`10-7. Calculate Km for this patient.
`A. 12.5 mg/L
`B. 25 mg/L
`C. 37.5 mg/L
`D. 10 mg/L
`
`10-8. For the same patient, JH, determine Vmax·
`A. 123 mgjday
`B. 900 mgjday
`C. 500 mgjday
`D. 678 mgjday
`
`10-9. For the case of JH above, plot both concen(cid:173)
`trations on a daily dose/C versus Vmax plot
`and then determine this patient's vmax•
`A. approximately 550 mgjday
`B. approximately 400 mgjday
`C. approximately 675 mgjday
`D. approximately 800 mgjday
`
`10-10. After the dose of 400 mgjday is begun, how
`long will it take to reach 90% of the steady(cid:173)
`state plasma concentration?
`
`A. approximately 14 days
`B. approximately 9 days
`C. approximately 30 days
`D. approximately 90 days
`
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`

`Concepts in Clinical Pharmacokinetics
`
`156
`
`10-11. If the patient, JH, misunderstood the dosage
`instructions and consumed 500 mgjday
`of phenytoin, what steady-state plasma
`concentration would result?
`A. 29.4 mg/L
`B. 36.8 mg/L
`c. 27.2 mg/L
`D. 19.6 mg/L
`
`10-5. A.
`Incorrect answer
`B. CORRECT ANSWER. At very low concen(cid:173)
`trations, drugs are more likely to exhibit
`first-order kinetics because hepatic
`enzymes are usually not yet saturated,
`whereas at higher concentrations,
`enzymes saturate, making zero-order
`kinetics more likely.
`
`ANSWERS
`
`10-1. A. CORRECT ANSWER
`B, C, D. Incorrect answers
`
`10-2. A.
`Incorrect answer
`B. CORRECT ANSWER. Linear pharmaco(cid:173)
`kinetics means that the AUC and plasma
`concentrations achieved are directly
`related to the size of the dose adminis(cid:173)
`tered. Drugs with linear pharmacokinetics
`may exhibit plasma concentrations
`versus time plots that are not straight
`lines, as with multicompartment drugs.
`
`10-3. A.
`Incorrect answer
`B. CORRECT ANSWER. There will be a
`disproportionate increase in the plasma
`concentration achieved because
`the
`amount of drug that can be eliminated
`over time cannot increase.
`
`10-4. A.
`
`Incorrect answer. Vmax is the maximum
`rate of hepatic metabolism.
`B. CORRECT ANSWER
`C.
`Incorrect answer. ~ Vmax is only one-half
`of the maximum hepatic metabolism
`and does not relate K"' to Vmax·
`Incorrect answer. CVmaxJCC) is only the
`numerator of the Michaelis-Menten
`equation.
`
`D.
`
`10-6.
`
`A, C, D. Incorrect answers
`B. CORRECT ANSWER
`
`10-7. A. CORRECT ANSWER. The Km is calculated
`from the slope of the line above:
`
`dose1 - dose2
`s ope = - m = ------''-------=--
`K
`I
`dose1 I C1 - dose 2 I C2
`
`300 mglday- 400 mglday
`300 mglday _ 400 mglday)
`(
`1 0 mgll
`18 mgll
`
`-100 mglday
`30 Uday- 22 Uday
`
`-100 mglday
`=
`8 Uday
`
`=-12.5mgll
`
`So Km equals 12.5 mg/L.
`
`B, C, D. Incorrect answers. Use dose pairs of
`300 and 400 and concentration pairs of
`10 and 18 to calculate Km.
`
`10-8. A, C. Incorrect answers. Try again; you prob(cid:173)
`ably made a math error.
`Incorrect answer. Try again, and use
`either set of dose and concentration
`pairs (i.e., 300 and 10 or 400 and 18).
`
`B.
`
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`

`

`Lesson 10 I Nonlinear Processes
`
`157
`
`10-11. A
`
`Incorrect answer. Perhaps you used a
`400-mg dose instead of a 500-mg dose.
`B. CORRECT ANSWER. The steady-state
`plasma concentration resulting from
`a daily dose of 500 mg would be esti(cid:173)
`mated from the line equation as follows:
`
`daily dose = -K m (dose/C) + Vmax
`
`500 mg/day)
`C
`500 mg/day = -12.5 mg/L
`+ 670 mg/day
`(
`
`Rearranging gives:
`
`-170 mg/day
`-12.5 mg/L
`
`500 mg/day
`= - - - -
`C
`
`13.6 Uday = 500 mg/day
`c
`
`1
`13.6 Uday
`- - -= -=
`500 mg/day C
`
`1
`0.0272 Umg = C
`
`C = 36.8 mg/L
`
`C, D. Incorrect answers. You may have made
`a simple math error.
`
`D. CORRECT ANSWER.
`
`daily dose= -Km(daily dose/C)+ Vmax
`
`400 = (-12.5)(400/18) + vmax
`
`400 = (-12.5)(22.22) + vmax
`
`400 = -277.77 + vmax
`
`677.77 = vmax
`
`10-9. A, B, D. Incorrect answers
`C. CORRECT ANSWER. See Figure 10-8
`for a plot of the daily dose versus daily
`dose/C.
`
`10-10. A, C, D. Incorrect answers
`B. CORRECT ANSWER. The time to reach
`steady state is calculated by:
`
`f90% =
`
`Km(V)
`2 [2.3Vmax -0.9 dose]
`{Vmax -dose)
`
`12.5 mg/L (45 L)
`=
`(670 mg/day- 400 mg/day)2
`
`[2.3(670 mg/day)-
`
`0.9(400 mg/day)]
`
`562.5 mg
`----=-~2 [1541 mg/day - 360 mg/day]
`(270 mg/day)
`
`= (0.00772 day2 /mg)(1181 mg/day)
`
`= 9.11 days
`
`700
`600
`Cl 500
`g
`5: 400
`0
`0 300
`~
`~ 200
`
`100
`
`20
`10
`30
`Daily Dose /CSteady State
`
`40
`
`FIGURE 1 0-8.
`Daily dose versus daily dose divided by steady-state
`concentration.
`
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`

`

`Concepts in Clinical Pharmacokinetics
`
`158
`
`Discussion Points
`
`W When using the Michaelis-Menten equa(cid:173)
`tion, examine what happens when daily
`dose is much lower than vmax• and when it
`exceeds v max·
`I!B When using the t90% equation, examine what
`happens to t90% when dose greatly exceeds
`vmax·
`
`I!J[g Using two steady-state plasma drug concen(cid:173)
`trations and two doses to solve for a new Kw
`Vmax• and dose using the Michaelis-Menten
`equation, examine the values of Km and
`Vmax obtained using this process. Are these
`values close to the actual patient population
`parameters?
`1!191 Discuss several practical methods to deter(cid:173)
`mine when a nonlinear drug has reached
`steady state.
`1!J1i1 Examine the time to 90% equation and note
`the value of Km that is used in this equation.
`Substitute several different phenytoin Km
`values based on a range of population values
`(i.e., from approximately 1 to 15 mg/L) and
`describe the effect this has on your answer.
`Based on this observation, what value of Km
`would you use when trying to approximate
`the t90% for a newly begun dose of phenytoin?
`
`I!KiJ Discuss
`the patient variables that can
`affect the pharmacokinetic calculation of a
`nonlinear drug when using two plasma drug
`concentrations obtained from two different
`doses.
`!!lll Examine the package insert for Cerebyx®
`(fosphenytoin) and answer the following
`questions:
`
`A What salt is this product?
`
`B. What percent phenytoin sodium is it?
`
`C. What percent phenytoin free acid is it?
`
`D. How many milligrams of Cerebyx® is
`equal to 100 mg of sodium phenytoin
`injection?
`
`E. What therapeutic advantage does this
`product offer?
`
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`
`

`

`LESSON 11
`Pharmacokinetic Variation
`and Model-Independent
`Relationships
`
`OBJECTIVES
`
`After completing Lesson 11, you should be able to:
`
`1.
`
`Identify the various sources of pharmacokinetic variation.
`
`2. Explain how the various sources of pharmacokinetic variation affect pharmacokinetic
`parameters.
`
`3. Describe how to apply pharmacokinetic variation in a clinical setting.
`
`4. Name the potential sources of error in the collection and assay of drug samples.
`
`5. Explain the clinical importance of correct sample collection, storage, and assay.
`
`6. Describe ways to avoid or minimize errors in the collection and assay of drug samples.
`
`7. Explain the basic concepts and calculations of the model-independent
`pharmacokinetic parameters of total body clearance, mean residence time (MRT},
`volume of distribution at steady state, and formation clearance.
`
`Sources of Pharmacokinetic Variation
`
`An important reason for pharmacokinetic drug monitoring is that a drug's effect
`may vary considerably among individuals given the same dose. These differences
`in drug effect are sometimes related to differences in pharmacokinetics. Some
`factors that may affect drug pharmacokinetics are discussed below. However, irre(cid:173)
`spective of pharmacokinetics, drug effects may vary among individuals because of
`differences in drug sensitivity.
`
`Age
`At extremes of age, major organ functions may be considerably reduced compared
`with those of healthy young adults. In neonates (particularly if premature) and
`the elderly, renal function and the capacity for renal drug excretion may be greatly
`reduced. Neonates and the elderly are also more likely to have reduced hepatic
`function. Renal function declines at a rate of approximately 1 mLjminutejyear
`after the age of 40 years. In the neonate, renal function rapidly progresses in
`infancy to equal or exceed that of adults. Pediatric patients may have an increased
`rate of clearance because a child's drug metabolism rate is increased compared to
`adults. When dosing a drug for a child, the drug may need to be administered more
`frequently.
`
`159
`
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`

`Concepts in Clinical Pharmacokinetics
`
`160
`
`Other changes also occur with aging. Compared
`with adults, the neonate has a higher proportion of
`body mass made up of water and a lower proportion
`of body fat. The elderly are likely to have a lower
`proportion of body water and lean tissue (Figure
`11-1). Both of these changes-organ function and
`body makeup-affect the disposition of drugs and
`how they are used. Reduced function of the organs
`of drug elimination generally requires that doses
`of drugs eliminated by the affected organ be given
`less frequently. With alterations in body water or fat
`content, the dose of drugs that distribute into those
`tissues must be altered. For drugs that distribute
`into body water, the neonatal dose may be larger
`per kilogram of body weight than in an adult.
`
`Disease States
`Drug disposition is altered in many disease states,
`but the most common examples involve the kidneys
`and liver, as they are the major organs of drug elimi(cid:173)
`nation. In patients with major organ dysfunction,
`drug clearance decreases and, subsequently, drug
`half-life lengthens. Some diseases, such as renal
`failure or cirrhosis, may even result in fluid reten(cid:173)
`tion and an increased volume of drug distribution.
`Alterations in drug clearance and volume
`of distribution require adjustments in the dose
`administered and/ or the dosing interval. For most
`drugs, when clearance is decreased but the volume
`of distribution is relatively unchanged, the dose
`administered may be similar to that in a healthy
`
`100
`
`90
`
`Neonate
`
`Adult
`25yo
`75yo
`
`.E 80
`Ol
`~ 70
`>
`'tl 60
`0 m
`iij 50
`
`0
`
`~ - 40
`.... c: 30
`Q) u ... Q)
`
`Fat
`
`Intracellular Water
`
`D..
`
`20
`
`10
`
`Extracellular Water
`
`FIGURE 11-1.
`Effect of age on body composition.
`
`person although the dosing interval may need to
`be increased. Alternatively, smaller doses could be
`administered over a shorter dosing interval. When
`the volume of distribution is altered, the dosing
`interval can often remain the same but the dose
`administered should change in proportion to the
`change in volume of distribution.
`
`Clinical Correlate
`
`When adjusting a dose of a drug that follows
`first-order elimination, if you do not change
`the dosing interval, then the new dose can
`be calculated using various simple ratio and
`proportion techniques. For example, if gentamicin
`peak and trough serum drug concentrations (in a
`patient receiving 120 mg every 12 hours) were 9
`and 2.3 mcg/ml, respectively, then a new dose
`can be calculated : "if 120 mg gives a peak of 9,
`then X mg will give a desired peak of 6," yielding
`an answer of 80 mg every 12 hours. Likewise,
`one can check to see if this trough would be
`acceptable with this new dose: "if 120 mg gives a
`trough of 2.3, then 80 mg will give a trough of X,"
`yielding an answer of 1.5 mcg/ml.
`
`EXAM P LE
`
`Effect of Volume of Distribution and
`Impaired Renal/Hepatic Function on
`Drug Dose
`A 23-year-old male experienced a major
`traumatic injury from a motor vehicle acci(cid:173)
`dent. On the third day after injury, his renal
`function is determined to be good ( creati(cid:173)
`nine clearance = 120 mL/minute), and his
`weight has increased from 63 kg on admis(cid:173)
`sion to 83 kg. Note that fluid accumulation
`(as evidenced by weight gain) is an expected
`result of traumatic injury. He is treated with
`gentamicin for gram-negative bacteremia.
`An initial gentamicin dose of 100 mg is
`given over 1 hour, and a peak concentra(cid:173)
`tion of 2.5 mg/L is determined. Four hours
`after the peak, the plasma concentration is
`determined to be 0.6 mgjL, and the elimina-
`
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`
`

`

`Lesson 11
`
`I Pharmacokinetic Variation & Model-Independent Relationships
`
`161
`
`tion rate constant and the volume of distri(cid:173)
`bution are determined to be 0.36 hr-1 and
`33.6 L, respectively. This volume of 33.6 L
`equals 0.40 L/kg compared to a typical V of
`0.2-0.3 L/kg. In this case, the patient's genta(cid:173)
`micin elimination rate constant is similar to
`that found in people with normal renal func(cid:173)
`tion, but the volume of distribution is much
`greater. To maintain a peak plasma genta(cid:173)
`micin concentration of 6-8 mgjL, a much
`larger dose would have to be administered at
`a dosing interval of 6 or 8 hours. Using the
`multiple-dose infusion equation from Lesson
`5 (see Equation 5-1), we would find that a
`dose as high as 220 mg given every 6 hours
`would be necessary to achieve the desired
`plasma concentrations.
`On the other hand,