`
`IUNCIAMENTAIS
`
`CIINICAI
`pI-IARMACOkINETICS
`
`by jOI-IN q WAQNER
`
`th.B., B.S.P., B.A., Ph.D
`
`Professor of Pharmacy, College of Pharmacy, and
`
`Staff Member of the Upjohn Center for Clinical
`
`Pharmacology, Medical School, The University of
`
`Michigan, Ann Arbor, Michigan 48104, U.S.A.
`
`FIRST EDITION 1975
`
`DRUG INTELLIGENCE PUBLICATIONS,
`
`|NC., HAMILTON, ILLINOIS 62341
`
`_—J
`
`AUROBINDO EX. 1019, 1
`
`AUROBINDO EX. 1019, 1
`
`

`

`COPYRIGHT © 1975 BY DRUG INTELLIGENCE PUBLICATIONS, INC.
`
`All rights reserved
`No part of this book may be reproduced
`in any form without written
`permission of the copyright holder
`
`Library of Congress Catalog Number 75-5443
`ISBN 0—914678-20—4
`
`AUROBINDO EX. 1019, 2
`
`

`

`of a in column 2 is the ratio of the dosage interval to the half—life. When
`6 is small, the doses are given close together, and when 6 is large, the
`doses are given far apart. Since, even when 6 = 5, C223" 2 1.032 C0,
`then for all dosage regimens listed in table 3—1, drug accumulation exists
`according to the concentration build-up concept. However, in the last
`column of table 3—1, are listed the drug accumulation indices calculated
`by means of equation 3—2. Only when E > 1 is RA > 1. Hence, by the
`“amount criterion” drug accumulation only occurs for this model when
`the dosage interval is less than 1.443 times the half—life of elimination.
`Dosage regimen calculations can also be made then based on: ( 1) predic-
`tion of average amounts of drug in the “body” or a particular compart-
`ment (such as the central compartment of the two compartment open
`model), or (2) prediction that, say, a patient with poor renal function
`will have the same average steady state amount of drug in the body as
`a patient with normal renal function. Both approaches are considered
`in later sections.
`
`There is a considerable difference in the levels of sophistication
`which can be applied to dosage regimen calculations. The levels vary
`all the way from calculations which can be performed quickly by the
`human brain,
`to those which can be performed readily with a pencil
`and paper, to those which can be performed readily with an electronic
`calculator, and finally to those which require the use of a large digital
`computer. All of these types will be considered in subsequent sections.
`According to Kriiger-Thiemer,3 a dosage regimen consists of the
`following quantities: (a) a dosage interval (T); (b) a dose ratio : initial
`(loading) dose/maintenance dose; and (c)
`the maintenance dose. One
`criterion of acceptance of a dosage regimen is the ratio of the maximum
`to the minimum concentration of drug in plasma or the maximum and
`minimum amounts of drug in the “body” at steady state compared with
`the average amount.
`
`3.2 SIMPLE DOSAGE REGIMEN
`CALCULATIONS
`
`3.2.1 Based on the Elimination
`Half-Life. The half-life of elimination of
`
`a drug (or the corresponding first order rate
`constant) is
`the most
`important pharmaco—
`kinetic parameter for dosage regimen evalua—
`tion. This is the half-life estimated from termi—
`
`nal blood (serum or plasma) concentration
`data, and hence, is equal to 0.693/K for the
`one compartment open model and 0.693/[3 for
`the two compartment open model.
`
`If nothing else is known about a drug
`except
`its half—life of elimination,
`following dosage regimen “rule” is useful:
`Make the dosage interval equal to the half—life
`of elimination, make the loading or initial
`dose equal to twice the maintenance dose, and
`make the maintenance dose equal to the mini—
`mum amount of drug in the “body” necessary
`for effective therapy.6 This “rule” is based
`on consideration of the one compartment
`open model with first order absorption and the
`case when the rate constant for absorption is
`much larger than the rate constant for elimi—
`
`AUROBINDO EX. 1019, 3
`
`—
`
`AUROBINDO EX. 1019, 3
`
`

`

`nation (Le. k > K). Data for the one compart-
`ment open model with bolus intravenous in-
`jection in table 3-1 indicates that under this
`condition that 'T = t1/2 and e : 1, the ratio
`GEM/C231“ = 2, and the drug accumulation
`is equal
`to 1.443 (or 1/1112) :
`1/().693). Interestingly enough, clinical expe—
`rience and practical reasons have lead physi—
`cians in the past
`to use this rule without
`knowing its theoretical foundation.6
`
`3.2.2 Based on Average Steady-
`State Blood Levels. The equation of
`\Vagner et (1].,12 shown as equation 3-4, is most
`useful in dosage regimen calculations.
`
`_ FD
`F
`_
`: _.
`VKT
`VCI’T
`
`3—4
`
`E .
`q (
`
`)
`
`Cw
`
`In equation 3-4, 6,0 is the average (not the
`minimum) steady-state whole blood, plasma or
`serum concentration, F is the fraction of the
`dose which is absorbed, D is the dose, V is
`the volume of distribution, K is the elimina-
`
`gon rate constant, ’T is the dosage interval, and
`V0] is the blood (serum or plasma) clearance
`(sometimes called the "body” clearance). The
`middle term of equation 3-4 indicates how the
`equation was originally written,12 but
`the
`term on the far right is a more generalized
`way of writing the equation. Note that (5,,
`is
`defined by equation 3-5, and is an application
`of the central limit theorem of calculus. As
`
`a result of the equivalency of areas shown in
`figure 3-1, and
`
`t2
`
`_ a _ l.
`
`1'
`
`Cm(t)dt
`
`then F/VCl (the reciprocal of the apparent
`clearance) may be estimated from equation
`3-6 by nlaking a ratio of the area to the dose
`(Le. F/VCl : Ag/D : Afi/D). After
`single
`doses the area A; may be estimated by means
`of equations 10-2 through 10-6 given in chap-
`ter 10.
`
`It is really more desirable to estimate the
`value of F/VCl by measuring plasma concen-
`trations after several different doses of the
`
`drug and plotting A3“ versus D. It is also more
`desirable to express the dose, D,
`in mg/kg
`body weight so that the abscissa scale consti-
`tutes a distribution for each fixed D value. In
`
`this case, least squares regression techniques
`applicable to a bivariate normal distribution
`may be used to calculate the regression line.
`The slope of the regresion line for such a plot
`
`0—» o
`
`xhrsl
`fl ml 2 o
`
`|20
`
`IOO
`
`80
`
`60
`
`40
`
`CURVE0-.ml
`Y=AREAUNDERLINCOMYCINSERUMCONCENTRATION
`
`
`
`
`
`where ’T = t2 — t1
`
`Eq- (3-5)
`
`equations 3—4 and 3—5, we may also write
`equation 3-6. If concentrations
`
`Eq. (3-6)
`
`V01
`
`are measured at a sufficient number of differ—
`
`ent times following oral or intranniscular ad—
`
`0
`
`2
`
`8
`6
`4
`X = DOSE OF LlNCOMYClN (mg lkql
`
`IO
`
`l2
`
`Fig. 3-2. Plot of A3 under single dose serum
`concentration curve versus mg/kg dose of linco—
`mycin hydrochloride following intramuscular ad—
`ministration. Each point corresponds to a diEerent
`subject. The three arrays correspond to doses of
`100, 200 and 600mg. of the antibiotic. The slope
`of 12.05 represents a suitable value of F/VCl to use
`
`AUROBINDO EX. 1019, 4
`
`

`

`dictions as indicated below. An example of
`such a plot is shown in figure 3-2 where the
`area under the serum concentration curve
`
`administration of
`following intramuscular
`lincomycin hydrochloride is plotted against
`the mg/kg dose of the antibiotic.14 The re-
`gression is highly significant (P < .001). The
`1001‘2 value is 74 percent, indicating that 74
`percent of
`the variability of
`the area is
`accounted for by differences in the lug/kg
`doses. Since the intercept was not significantly
`different
`from zero the least squares line
`forced through the origin was calculated (i.c.,
`y : 12.05x where y is the estimated value of
`A3 and x is the mg/kg dose) and is the line
`drawn in the figure. The “zero intercept" is
`indicated by the theory emb_odied in equation
`3-6. Thus,
`in this case, F/VCl : 12.05 kg/L.
`(The units were obtained as follows:
`
`,ug/ml
`__ g kg
`mg/kg _ ml
`103 pg
`dosage regimen question which may be an—
`swered with such information is
`indicated
`below.
`
`:kg/L). A typical
`
`Question: What would be the expected aver—
`age steady state serum concentration if a
`60 kg person were administered 200mg of
`lincomycin hydrochloride every 8 hours?
`
`Answer:
`
`(—3,, = (—l) (2)
`
`V01
`
`7'
`200 60
`
`= (12.05) (T/) = 5 pg/ml
`
`It should be noted that the answer was ob-
`
`tained by simply separating the factors in the
`right hand side of equation 3-4, and remem-
`bering that when F/v01 has units of kg/L, D
`must be in mg/kg, and (—3,, will have dimen-
`sions of ,LLg/ml. It must also be remembered
`that this answer was obtained using a least
`squares slope value of 12.05 and because of
`the scatter of points about the line in figure
`3-2 the answer of 5 ,ug/ml is a type of “aver-
`age” expected value and may not apply ex-
`actly to an individual patient. One could use
`the 95 percent confidence interval of the least
`squares slope for the data in figure 3—2 to make
`
`not forced through the origin was 11.42 with
`the 95 percent confidence interval of 9.64 to
`13.4. If the latter values are substituted for
`F/VCl in the above equation instead of 12.05,
`one obtains 4.0 and 5.6 [Lg/1111, respectively.
`It should be noted that application of
`equations 3—4 through 3—6 above are inde—
`pendent of whether the one or two compart—
`ment open model applies to the data being
`evaluated since the mean clearance is used.
`
`In terms of the one compartment open model,
`the mean clearance is equal to VK, but for
`the two compartment open model, the mean
`clearance is equal to Vlkel 0r Vd area [3.
`Another type of question which may be
`answered with the above information is indi—
`cated below.
`
`Question: What dosage could be employed for
`a 50 kg woman to provide an average steady
`state level of lincomycin hydrochloride of
`3 pg/ml
`if
`the drug is administered intra-
`muscularly?
`Answer: First rearrange equation 3-4 and sub—
`stitute the known values as follows:
`
`2 = (Can/<70.) = <3><1/12.05) : 1/4
`’T
`
`Hence, any combination of D and 7 which will
`give a ratio of D/T equal to 1/4 will be an
`answer. You have to remember, again,
`since the clearance has units of L/kg, and its
`reciprocal, kg/L,
`that D must be in units
`of mg/kg.
`One
`answer
`100 rug/50 kg : 2 mg/kg and 7 : 8 hours
`since D/T : 13/12 = 1/4.
`Such
`
`are “ball—park” answers and very useful in the
`clinical situation.
`
`Orr et (11.13 published a method for esti-
`mating individual drug—dosage regimens, but
`what
`they call ”o_ccupancy/ml” is exactly
`equivalent
`to F/VCl = Ag/D : Ag/D and
`really does not require a new name. Their
`method is equivalent to that discussed above.
`The emphasis was, however, on measuring the
`serum or plasma concentrations in the same
`patient for which the predictions were to be
`made. Of course, this is highly desirable, but
`
`AUROBINDO EX. 1019, 5
`
`*4
`
`AUROBINDO EX. 1019, 5
`
`

`

`not always feasible. They also discussed an
`easy way to apply the trapezoidal rule to
`estimate the area. If a smooth curve is drawn
`
`through the plasma or serum concentration
`data, and the concentrations at hourly inter-
`vals are estimated from the smooth line, the
`in this case,
`is simply the sum of the
`estimated concentrations. This actually gives
`the value of Ag of equation 10-6, and the con—
`tribution of the area from T to infinity to the
`total area should be assessed from equation
`10—5, and the total area used in any predictions.
`
`3.2.3 Based on Minimum Steady-
`State Blood Levels. Equation 3—4
`is
`often inappropriately applied to minimum
`steady—state blood, plasma, 01' serum concen—
`trations, which are those measured at ’1' hours
`
`after a dose, or equivalently, just before the
`next dose at steady-state. The appropriate
`equation to apply in such a case is equa—
`
`this is not actually the case at all. To apply
`equation 3-7 in the clinical situation one can
`assume linearity tentatively, and look at
`everything in the square bracket in equation
`3—7 as just one proportionality constant relat—
`ing minimum steady—state concentrations to
`the maintenance dose, Dm. If you have some
`way of estimating [3, such as from a correla-
`tion of [3’ with endogenous creatinine clear-
`ance or a calculated clearance of creatinine,
`then one can reduce variance by using only
`[pF/Vd] as the proportionality constant.
`A model—independent version of equation
`3-7 may be written as equation 3—8.
`~
`F
`
`C211“ : _iv.)
`
`Fraction remaining at
`
`end of dosage interval
`Fraction lost during
`dosage interval
`
`m
`
`E .' 3-8
`q (
`
`)
`
`e‘/37
`F
`m = — — E .
`
`[p (Vd)(1 — e‘BT)iDm
`
`q (
`
`3—
`
`7)
`
`Note that when written this way the “p” has
`disappeared from the equation. By matching
`terms in equations 3—7 and 3—8 we can see
`that:
`
`In equation 3-7, C231“ is the minimum steady-
`state concentration, p is a function of rate
`constants for a particular model, F is the frac—
`tion of the dose, D, which is absorbed, Vd is
`the volume of distribution, Dm is the mainte—
`nance dose given every 7 hours, and [5 is the
`apparent elimination rate constant. For the
`one compartment open model with intrave-
`nous administration p = 1 and F = 1 (com—
`pare equation 3-7 to equation 2—54b in chap-
`ter 2). For the one compartment open model
`with first order absorption p : (k/k — K) :
`(k/k — [3); in this case compare equation 3—7
`with equation 3—67 and assume that e‘l” 2 0,
`and that CO : FDm /Vd. The reason that equa—
`tion 3-7 may often be used is that even though
`the concentration, time curve is described by
`a polyexponential equation, all terms but the
`one containing [3, assumed to be the smallest
`rate constant in the system, have gone to zero
`at 7' hours after the dose at steady—state. It is
`often said that if equation .‘3-7 is applied that
`
`Fraction remaining at end of dosage interval
`Fraction lost during dosage interval
`e’flT
`
`= p (CW) Eq. (34))
`It should be noted that when written as equa—
`tion 3-7' the value of “p” is model—dependent,
`while the “p” does not appear in equation 3—8.
`In the case of the two compartment open
`model the “Vd” in equations 3-7 and 3-8 is
`actually V1.
`
`3.2.4 Superposition or Overlaying
`Principle.
`This
`principle
`provides
`a
`method of predicting multiple dose blood lev—
`els in linear systems and a simple method of
`estimating a loading dose. The method is ap—
`plicable to data obeying any linear pharmaco-
`kinetic model, and requires only a pencil and
`paper. Unlike the method outlined in section
`3.2.2, this method provides estimates of multi—
`ple dose blood levels at any desired sampling
`
`AUROBINDO EX. 1019, 6
`
`

`

`Dose
`Time
`
`Number
`(Hours)
`Dose 1
`Dose 2
`Dose 3
`Dose 4
`Dose 5
`Dose 6
`1
`0
`0
`0.5
`38. 84
`1
`58.63
`2
`69. 86
`3
`65.93
`4
`57.94
`5
`49.57
`6
`41.96
`
`2
`
`0
`
`6.5
`7
`8
`9
`
`10
`11
`12
`
`(38.53)
`35.36
`29. 74
`25.00
`
`21.01
`17.65
`14.83
`
`38.84
`58.63
`69.86
`65.93
`
`57. 94
`49.57
`41.96
`
`0
`
`3
`
`4
`
`38.84
`38.53
`(13.61)
`12.5
`58. 63
`35.36
`(12.48)
`13
`69.86
`29. 74
`(10.49)
`14
`65. 93
`25.00
`(8. 82)
`15
`57.94
`21.01
`(7.41)
`16
`49.57
`17.65
`(6.23)
`17
`0
`41.96
`14.83
`(5.24)
`18
`38.84
`38.53
`13.61
`(4.81)
`18.5
`58.63
`35.36
`12.48
`(4.41)
`19
`69. 86
`29. 74
`10.49
`(3.70)
`20
`65.93
`25.00
`8.82
`(3.11)
`21
`57. 94
`21.01
`7.41
`(2.62)
`22
`49.57
`17. 65
`6.23
`(2.20)
`23
`0
`41.96
`14.83
`5.24
`(1.85)
`24
`38.84
`38.53
`13.61
`4.81
`(1.70)
`24.5
`58.63
`35.36
`12.48
`4.41
`(1.56)
`25
`69.86
`29.74
`10.49
`3.70
`(1.31)
`26
`65.93
`25.00
`8. 82
`3.11
`(1.10)
`27
`57.94
`21.01
`7.41
`2.62
`(0.92)
`28
`49.57
`17. 65
`6.23
`2.20
`(0.78)
`29
`0
`41.96
`14.83
`5. 24
`1.85
`(0.65)
`30
`38. 84
`38.53
`13.61
`4.81
`1.70
`(0.60)
`30.5
`58.63
`35.36
`12.48
`4.41
`1.56
`(0.55)
`31
`69.86
`29.74
`10. 49
`3. 70
`1.31
`(0.46)
`32
`65.93
`25. 00
`8.82
`3.11
`1.10
`(0.39)
`33
`57. 94
`21.01
`7.41
`2.62
`0.92
`(0.33)
`34
`49.57
`17.65
`6.23
`2. 20
`0.78
`(0.27)
`35
`
`36 64.76 (0.23) 0.65 1.85 5. 24 14.83 41.96
`
`
`
`
`
`
`
`5
`
`6
`
`_
`
`_
`
`_
`
`AUROBINDO EX. 1019, 7
`
`AUROBINDO EX. 1019, 7
`
`

`

`The Superposition Method in Tabular Form
`
`This method can be applied to raw data
`directly without resorting to pharmacokinetic
`analysis. It will give the correct answer for
`a linear system providing everything remains
`constant when multiple doses are adminis—
`
`To illustrate the method simulated blood
`
`levels, C(t), following a single dose were gen—
`erated with the equation:
`
`
`em = co(k EK)[e—m _ 81.]
`
`Eq.
`
`(3—10)
`
`using C0 = 100, k : 1.0455 hr‘l, and K :
`0.17425 hr‘l. Thus,
`the
`absorption half-
`time was 0.693/1.0455 : 0.663 hr, and the
`elimination half-life was 0.693/0.l7425 =
`
`the trend. It was also found that for this par—
`ticular set of data a semilogarithmic plot of
`blood level versus time was linear in the 6 to
`
`In
`12 hour period as shown in figure 3—4.
`actual practice one could fit a line by sight
`through the points shown in figure 3-4, and
`then extrapolate the line to be able to predict
`the concentrations out to 36 hours (equivalent
`to six doses as used in this example). The au-
`thor was a little fancier and fitted the least
`
`squares line through the points.
`Applying the method of least squares to
`the natural logarithms of the “blood levels”
`under the column headed “Dose 1” in the
`
`table starting with the value at 6 hours,
`namely 41.96 and ending with the value 0.23
`at 36 hours, based on equation 3-12,
`
`In C = ln C0 — Kt
`
`Eq.
`
`(3—12)
`
`Substitution of the above constants into
`
`one obtains the equation shown as equation
`3-13.
`
`equation 3-10 gave equation 3-11.
`
`ca) : 120[e—0.17425t _
`
`6 —1.04551]
`
`Eq. (3—11)
`
`Data generated with equation 3-11 are listed
`under the third column, labeled "Dose 1” in
`table 3—2. These numbers are not bracketed.
`
`ln (3 = 4.7862 — 0.17421t
`
`Eq.
`
`(3—13)
`
`The correlation coefficient was r = 1.000000
`
`showing an excellent fit. Equation 3-13 may
`be written as equation 3—14.
`
`c = 119.85e’0-17421‘
`
`Eq.
`
`(3—14)
`
`These C(t) values are plotted in figure 3—3 and
`the points are joined by straight lines to show
`
`Using equation 3-14 the bracketed values
`of C(t), listed Lmder Dose 1 in table 3—2, were
`
`LEVEL
`
`BLOOD
`
`
`
`TIME
`
`IN HOURS
`
`AUROBINDO EX. 1019, 8
`
`

`

`6
`
`8
`
`10
`
`12
`
`I4
`
`16
`
`TIME
`
`18
`
`IN
`
`2O
`
`22
`
`24
`
`26
`
`28
`
`HOURS
`
`30
`
`20
`
`E»
`§‘3a
`:7
`:6
`5
`
`:14>
`
`1‘53
`
`2
`
`So
`
`_l
`m
`
`.099.0.0.05‘UIOVOOH
`
`Fig. 3-4. Semilogarithmic plot of the blood levels in the six to twelve hour range with extrapolated
`values based on equation 3-14.
`
`calculated. These corresponded to the extra—
`polated line in figure 3-4. To apply the
`method one carries out the operations indi-
`cated in the table—relisting the values under
`“Dose 1” but starting anew at
`intervals of
`6 hours. When the table is complete for the
`number of doses one wishes to predict (exam—
`ple has 6 doses)—then one merely adds all the
`ordinate values across the rows to obtain the
`
`the predicted
`to Cn(t) or
`total equivalent
`blood level for the multiple dose regimen after
`the nth dose.
`
`The Cn(t) values in the last column of
`table 3-2 are plotted in figure 3-5. A dotted
`line has been drawn by just joining the points
`to show the trend. Hence, figure 3-5 illustrates
`the multiple dose blood levels predicted from
`the single dose blood levels by means of the
`superposition or overlaying principle.
`
`Estimation of the Loading Dose
`
`Analogous to the superposition or over-
`laying principle to estimate multiple dose
`blood levels, we can estimate the loading dose
`reasonably accurately by making a ratio of the
`peak blood level at the equilibrium state to
`the peak blood level after the first dose. Ap-
`plying this to the data in table 3—2 one obtains:
`
`Loading dose
`Maintenance dose
`
`_ Peak blood level after 6th dose
`_ Peak blood level after lst dose
`
`115.5
`= __ = 1.65
`69.86
`
`This provides an answer very close to the 1.54
`calculated with the equation of Kruger—
`
`AUROBINDO EX. 1019, 9
`
`AUROBINDO EX. 1019, 9
`
`

`

`
`Table 3-3. Predicted Blood Levels \Vith Loading Dose
`
`Time
`
`TOTAL
`
`
`(Hours)
`Dose 1
`Dose 2
`Dose 3
`Dose 4
`Dose 5
`Dose 6
`:Cn(t)
`
`0
`0
`0
`64.09
`64.09
`0.5
`96.74
`96.74
`1
`115.3
`115.3
`2
`108.8
`108.8
`3
`95. 60
`95. 60
`4
`81.79
`81.79
`5
`69.23
`0
`69.23
`6
`102.4
`38.84
`63.57
`6.5
`117.0
`58.63
`58.34
`7
`118.9
`69.86
`49.07
`8
`107.2
`65.93
`41.25
`9
`92.61
`57.94
`34.67
`10
`78.69
`49.57
`29.12
`11
`66.43
`0
`41.96
`24.47
`12
`99.83
`38.84
`38.53
`22.46
`12.5
`114.6
`58.63
`35.36
`20.59
`13
`116.9
`69.86
`29.74
`17.31
`14
`105.5
`65.93
`25.00
`14.55
`15
`91.18
`57.94
`21.01
`12.23
`16
`77.50
`49.57
`17.65
`10.28
`17
`65.44
`0
`41.96
`14.83
`8.65
`18
`98.92
`38.84
`38.53
`13.61
`7.94
`18.5
`113.8
`58.63
`35.36
`12.48
`7.28
`19
`116.2
`69.86
`29.74
`10.49
`6.11
`20
`104.9
`65. 93
`25. 00
`8. 82
`5.13
`21
`90.68
`57.94
`21.01
`7.41
`4.32
`22
`77.08
`49.57
`17.65
`6.23
`3.63
`23
`65.08
`0
`41.96
`14.83
`5.24
`3.05
`24
`98.60
`38.84
`38.53
`13.61
`4.81
`2.81
`24.5
`113.5
`58.63
`35.36
`12.48
`4.41
`2.57
`25
`115.9
`69.86
`29. 74
`10.49
`3. 70
`2.16
`26
`104. 7
`65.93
`25.00
`8.82
`3.11
`1. 82
`27
`90.50
`57.94
`21.01
`7.41
`2.62
`1.52
`28
`76.94
`49.57
`17.65
`6.23
`2.20
`1.29
`29
`65.94
`0
`41.96
`14.83
`5.24
`1.85
`1.07
`30
`96.48
`38.84
`38.53
`13.61
`4.81
`1.70
`0.99
`30.5
`113.4
`58.63
`35.36
`12.48
`4.41
`1.56
`0.91
`31
`115.9
`69.86
`29.74
`10.49
`3.70
`1.31
`0.76
`32
`104.6
`65. 93
`25.00
`8.82
`3.11
`1.10
`6. 64
`33
`90.44
`57.94
`21.01
`7.41
`2.62
`0.92
`0.54
`34
`76.88
`49.57
`17.65
`6.23
`2.20
`0.78
`0.45
`35
`
`
`
`
`
`
`
`0.38 0.65 1.85 5.24 14.83 41.9636 64.91
`
`AUROBINDO EX. 1019, 10
`
`

`

`100
`
`90
`
`80
`
`7O
`
`LEVEL 60
`50
`
`BLUOD
`
`4O
`
`30
`
`20
`
`10
`
`0
`
`
`
`
`
`
`26
`28
`30
`32
`18
`20
`22
`24
`2
`4
`6
`8
`10
`12
`14
`16
`TIME IN HOURS
`l
`
`l DOSES
`
`i
`
`l
`
`l
`
`l
`
`Fig. 3-5. Predicted multiple dose blood levels from last column of table 3-2.
`
`Thiemer which is given in a later section (see
`equation 3—30 and related text).
`To check out the answer, all blood levels
`under “Dose 1" of table 3-2 were multiplied
`by 1.65 and the blood levels due to the previ-
`ous maintenance doses used to complete the
`table as before. The new table is shown as table
`3-3. The Cn(t) values are plotted in figure 3-6.
`One can see that with this dosage regimen the
`steady state is reached at once and main-
`tained.
`
`Pitfalls of the Superposition Method
`
`The superposition method is valid only
`when the pharmacokinetics are linear and
`elimination occurs from the body according
`to first order kinetics. For accurate predictions
`one also has to extrapolate the true terminal
`log—linear decline of blood levels on semiloga—
`rithmic graph paper. The points chosen must
`be in the post—absorptive, post—tissue distribu-
`
`tion phases. Hence, blood level measurements
`must be made long enough to establish the
`log-linear line and provide enough points in
`that phase to establish the line. A reasonable
`general rule is that the blood levels must be
`followed long enough that the last blood level
`measured is between one-fifth and one-
`twentieth of the peak blood level. The lower
`one can measure, the more assured one is that
`the appropriate data are being used to make
`the extrapolation.
`An example showing Where an error
`could be made is shown in figure 3-7. The
`points in this figure were generated with
`equation 3—16, using the parameter values
`A1 = 50,
`A2 = 50,
`A3 = 100,
`[3 z 0.1, and k5‘ = 1.5.
`
`C(t) = Ale’at + Age’fit — A3e'kat
`
`Using the blood levels in the 12. to 28 hour
`
`AUROBINDO EX. 1019, 11
`
`AUROBINDO EX. 1019, 11
`
`

`

`4
`
`6
`
`l
`
`8
`
`10
`
`12
`
`I4
`
`16
`2O
`18
`TIME IN HOURS
`
`22
`
`24
`
`26
`
`28
`
`30
`
`32
`
`34
`
`36
`
`l
`
`l
`
`l
`
`1
`
`
`llWD[[VH
`
`range (solid line in figure 3-7) a rate constant
`of 0.1005 hours‘1 was obtained, which is very
`close to the smallest rate constant in equation
`3-7, namely [3 = 0.1 hours’l. However, if one
`used the blood levels in the 2 t0 8 hour range
`(dotted line in figure 3-7) a rate constant of
`0.1411 hours’1 is obtained. Extrapolation of
`the solid line would give correct values to
`apply the superposition principle, whereas
`extrapolation of the dotted line would give
`incorrect values and poor prediction of multi-
`ple dose blood levels. It has become common
`practice by many not well acquainted with
`pharmacokinetics to perform the operations
`shown in figure 3-7 and report two rate con-
`stants or their corresponding half—lives. This
`practice has no theoretical foundation and is
`extremely misleading. The rate constant and
`
`Fig. 3-6. Predicted multiple dose blood levels from the last column of table 33. These are the levels
`which are predicted when a loading dose equal to 1.65 times the maintenance dose was administered
`initially. Then maintenance doses of the same size as before were administered every six hours.
`
`A
`
`I
`
`la
`
`n
`"M!
`
`u
`II
`
`I6
`"UNIS
`
`n
`
`20
`
`21
`
`24 u u
`
`Fig. 3-7. Solid dots are simulated blood levels
`generated with equation 3-7. Solid line in the 12
`to 28 hour region gives correct estimate of elimi-
`nation rate constant and half—life, while dotted
`
`AUROBINDO EX. 1019, 12
`
`

`

`If a drug’s elimination obeys Michaelis—
`Menten kinetics, then the superposition prin—
`ciple will provide an underestimate of the
`multiple—dose blood levels.
`
`3.2.5 The Superposition Principle
`in Mathematical Form
`
`Let t = time measured from administration
`of first dose
`time measured from administration
`of nth dose
`11 2 dose number
`
`t!
`
`1' = dosage interval
`
`Then:
`
`t’=t—(n—1)T
`
`\Vhen n = 1, t’ : t; when n : 2, t’ = t — T;
`
`and in general t = t’ + (n — 1)1'.
`
`Let C1(t’), C2(t’), ——————— , Cn(t’) be the
`blood level at time t’ after the first, second,
`
`———————— , nth dose, respectively.
`
`If the B—phase" (loglinear phase) is established
`at
`1' hours after the first dose, V’Vestlake17
`showed that:
`
`Eq. (3-17)
`
`Cnv=qm+B
`(1 + e’fiT + 642/37 + _____ + e—/3(n—2)T)eifit’
`
`Eq.
`
`(3—19)
`
`Let S 21+ 6437 + 6—527
`
`Cn(tl) : C1“) + B
`
`1 fa]€“quw
`[1 _ e—(n—l)/3-r
`
`Equation 3-24 provides a method of predict-
`ing multiple—dose blood levels. To use it we
`must go back to the fitted line in figure 3—4
`and the corresponding equations 3-13 and
`3-14. From these equations the estimated
`value of [3 is 0.17421 hours—1. Now, “B” in
`equation 3-24 is
`the value of C(t) when
`t = T = 6 hours.” Substituting t = 6 into
`equation 3—14 gives B = 42.14. Hence, for this
`example, equation 3-24 may be written as
`equation 3-25.
`
`Cn(t’) = C1(t) + 42.14
`
`[1 _ e—(n—l)(0.17421)(6)
`1 _ e—(0.17421)(6)
`
`]e
`
`70.174211’
`
`When n : 6, substitution into equation 3-25
`gives:
`
`Cn(t’) = C,(t) + 4214 [9%]6—0174211’
`= C,(t) + 64646—017421"
`
`0.64840
`
`Substituting t’ = 0, 1, 2, 3, 4, 5, 6, into equa—
`tion 3—26 gives the values shown in table 3-4.
`It should be noted that the values of Cn(t’)
`
`”To apply equation 3-24 ’1' must be chosen so that it is
`2 than the time when the B-plrase is established after
`the single dose.
`
`+ ———— + e‘mflmf Eq. (3-20)
`
`Table 3-4. Values of Cn(t’) Generated with
`
`Equation 3-26 for n = 6
`
`Then e'fiTS : e‘37 + e‘fiZT
`+ ________________ + e—(n—1)Br
`
`Eq.
`
`(3—21)
`
`8(1 _ e’fiT) : 1 _ ei(1171)/3-r
`by subtraction Eq.
`
`(3—22)
`
`Hence, S =
`
`1 _ e-(n~1)BT
`1 — e‘BT
`
`t’
`
`0
`1
`2
`3
`4
`5
`6
`
`t
`
`30
`31
`32
`33
`34
`35
`36
`
`CG(t’)
`
`64.64
`112.9
`115.5
`104.3
`90.14
`76.62
`64.69
`
`°Note that in this section [3 replaces K in the previous
`section.
`
`at :t’+(n —1)T=t’+(5)(6)=t'+30
`
`AUROBINDO EX. 1019, 13
`
`AUROBINDO EX. 1019, 13
`
`

`

`in table 3—4 are essentially the same as the
`values of Cn(t) after the sixth dose shown in
`
`Table 3-5. Values of Cn(t’) Calculated by
`Means of Equations 3—28 and 3-29 for u 2 2
`and n : 6
`
`3.2.6 Use of Model Equations to
`Predict Multiple Dose
`Blood
`Levels. Let us assume you had the data
`shown in figure 3-3 and fitted these data by
`means of a digital computer and a nonlinear
`least squares program to equation 3- 10. Again
`let us assume that this fitting gave the exact
`values of the parameters, namely C0 = 100,
`k = 1.0455 hours‘1 and K : 0.17425 hours—1.
`
`It is readily shown that equation 3—27 is
`the multiple dose equation which corresponds
`with the single dose equation 3-10.
`
`Equation 328
`
`Equation 3—29
`
` t’ ta C2(t’) t" C6(t’)
`
`
`
`
`0
`6
`42.0
`30
`64.5
`1
`7
`94.0
`31
`112.9
`2
`8
`99.62
`32
`115.5
`3
`9
`90.95
`33
`104.3
`4
`10
`78.95
`34
`90.6
`5
`11
`67.22
`35
`76.64
`
`12 56. 79 366 64.70
`
`
`
`
`at=t'+(n—1)T=t'+'r
`bt:t'+(n~1)1’=t’+57
`
`Estimation of the Loading Dose
`
`For the particular model embodied by equa—
`tions 3—10 and 3-27 (i.e. the one compartment
`open model with first order absorption) the
`exact loading dose is given by equation 3—30.
`
`Loading dose 2
`
`Maintenance dose
`
`(1 __ CMKTXI _ e—k-r)
`
`Eq. (3-30)
`
`However, equation 3—30 can only be applied
`when the single dose data have been fitted
`well
`to equation 3—10.
`If we
`substitute
`k : 1.0455, K = 0.17425, and ’l' : 6 hours
`
`into equation 3-30 we find the loading dose
`should be 1.54 times the maintenance dose.
`
`Thus, this exact equation of Kriiger—Thiemer
`gives an estimate very close to the value of
`1.65 estimated by equation 3—15 after apply-
`ing the tabular superposition method.
`
`Generalization of the Multiple Dosing
`Functions as in Equation 3—27.
`
`Multiple dosing functions may be added
`to single dose blood level polyexponential
`equations without deriving the actual numeri—
`cal values of the microscopic rate constants
`of the model.
`
`For example, suppose the single dose
`blood level data were fitted to a tri-expo—
`nential equation in the form of equation 3—31.
`
`*uK'r
`
`I
`
`_ *nK'r
`
`I
`
`[(113240 )e_Kt _ (11 —ee"” )eiml
`
`Eq. (3-27)
`
`where, as before, t’ is elapsed time since the
`nth dose was given.
`As before, let CO : 100, k = 1.0455 h1"1,
`K : 0.17425 hr—1 and 7 = 6 hours.
`When n = 2, substitution of the constants into
`
`equation 3-27 yields equation 3-28.
`
`C2(tl) : 162.2 6‘017425t' _ 1202 ee1.0455t'
`Eq.
`(3—28)
`
`When n : 6, substitution of the constants into
`
`equation 3-27 yields equation 3—29.
`
`C6(t’) = 184.7 67017425" — 120.2 e“1'0455tl
`Eq. (32.9)
`
`Substituting values of t’ : 0, ———, 6 into
`equations 3-28 and 3-29 yields the values
`shown in table 3—5.
`
`The values shown in table 3—5 are actually
`the “exact” multiple dose blood levels for the
`simulation employed in sections 3.2.3 and
`3.2.4. Comparison of these values with the
`corresponding values in table 3-2 indicates
`that the tabular superposition method is very
`accurate if properly performed and that equa-
`
`AUROBINDO EX. 1019, 14
`
`

`

`Cn(t’) : A1{ 1 _ eTElT ]e’E1I' + A2
`[
`e EZI. + A3 ___— 6 Eat
`
`1 _ e—DEZT
`1 _ eeEz-r
`
`_
`
`I
`
`1 _ e—nE37
`1 _ e‘EaT
`
`7‘
`
`1 _ e—nEl'r
`
`t’ is the time after the nth
`In equation 3—32,
`dose and ’T is the dosing interval.
`
`Eq.
`
`(3—32)
`
`Summary
`
`The tabular superposition method, out-
`lined in section 3.2.3,
`is applicable to blood
`(serum or plasma) concentrations which obey
`any linear pharmacokinetic model. So also is
`the mathematical superposition method out—
`lined in section 3.2.4.
`
`If the raw data are not very smooth, or
`were collected with a very irregular sampling
`scheme,
`they may best be applied by first
`plotting the raw data on rectalinear (cartesian
`coordinate) graph paper
`and drawing a
`smooth line through the points by sight, then
`interpolating points from the smooth line.
`Usually, such interpolated points are taken at
`uniform time intervals in order to apply the
`methods.
`The method outlined in section 3.2.5 is
`
`model—dependent. The appropriate multiple
`dosing equation for the particular model must
`be used. However, it also can be made model—
`
`independent by just fitting the data to the ‘
`appropriate polyexponential equation,
`then
`forming the multiple-dosing equation as
`shown.
`
`3.3 PREDICTED MAXIMUM
`AND MINIMUM BLOOD
`LEVELS AFTER THE nTH
`DOSE AND AT STEADY
`STATE
`
`It is not necessary to determine values of
`microscopic rate constants, or assume a par—
`ticular model applies to data, to make predic—
`tions of maximum and minimum blood levels
`
`after the nth dose or at steady state. All that
`need be done is to fit the single dose blood
`
`below.
`
`,
`
`3.3.1 Blood Level Data Following
`Oral or
`Intramuscular Adminis-
`
`tration Which is Fitted by a Differ-
`ence of Two Exponentials. When
`there is a lag time, to, equation 3—10 is written
`as equation 3—33.
`
`
`C(t) : C0 (k E K)[e~K(t—t0) _ eeKa—ty]
`
`Equation 3—33 may be written as equa—
`tion 3-34.
`
`C(t) = Ae‘K‘ — Be-kt Eq. (3-34)
`
`where
`
`
`
`K]e+K‘0 Eq. (3.35)
`: C0[—T]e+k‘0 Eq. (3-36)
`
`When t0 : 0, thenA : B : C0 [fl]
`
`as in equation 3—10. Equation 3—34 would also
`be a general form for the model:
`
`'
`.
`.
`Absoi pt1on _
`
`Site
`
`k
`
`1
`k,,
`
`r
`k
`
`C
`V
`
`but the k and K in the equation would be 011
`a and /3 which would be complex square root
`functions of the k1, lg1 and K in the scheme.
`Let us assume you “stripped” a set of
`data by the method outlined in section 2.1.2
`and obtained an equation of the form of equa—
`tion 3—34. You could use the values of the
`
`coefficients and exponents obtained by the
`stripping procedure; or, better, use them as
`preliminary estimates, and obtain at
`squares fit of the original data to equation 13—34
`using a nonlinear regression program, such as
`NONLIN, and a digital computer—then use
`the least squares estimates of A, B, k and K
`to make predictions. The latter parameters
`would have less bias.
`
`By adding multiple dosing functions to
`
`AUROBINDO EX. 1019, 15
`
`AUROBINDO EX. 1019, 15
`
`