`
`[UNCAMENTAIS
`
`clinical
`pHARMACOKiNETICS
`
`by john G WAGNER
`
`Phm.B., B.S.P., B.A., Ph.D
`
`Professor of Pharmacy, College of Pharmacy, and
`Staff Memberof the Upjohn Centerfor Clinical
`Pharmacology, Medical School, The University of
`Michigan, Ann Arbor, Michigan 48104, U.S.A.
`
`FIRST EDITION 1975
`
`DRUG INTELLIGENCE PUBLICATIONS, INC., HAMILTON, ILLINOIS 62341
`
`AUROBINDO EX. 1019, 1
`
`eeeee
`
`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
`
`

`

`in column2 is the ratio of the dosage interval to the half-life. When
`of €
`« is small, the doses are given close together, and whene« is large, the
`doses are given far apart. Since, even when ¢ = 5, CR** = 1.032 Co,
`thenforall dosage regimenslisted in table 3-1, drug accumulationexists
`according to the concentration build-up concept. However, in the last
`columnof table 3-1, are listed the drug accumulation indices calculated
`by means of equation 3-2. Only when ¢« > 1 is Ry > 1. Hence, by the
`“amountcriterion” drug accumulation only occurs for this model when
`the dosage interval is less than 1.443 times the half-life of elimination.
`Dosage regimencalculations can also be made thenbasedon:(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 performedreadily with a pencil
`and paper, to those which can be performedreadily 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 subsequentsections.
`According to Kriiger-Thiemer,® a dosage regimen consists of the
`following quantities: (a) a dosage interval (7); (b) a dose ratio = initial
`(loading) dose/maintenance dose; and (c)
`the maintenance dose. One
`criterion of acceptance of a dosage regimenis the ratio of the maximum
`to the minimumconcentration 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 parameterfor 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 and0.693/f for
`the two compartment open model.
`
`If nothing else is known about a drug
`except
`its half-life of elimination,
`following dosage regimen “Tule” is useful:
`Makethe dosageinterval equal to the half-life
`of elimination, make the loading orinitial
`dose equal to twice the maintenance dose, and
`make the maintenance dose equal to the mini-
`mum amountof drug in the “body” necessary
`for effective therapy.® This “rule” is based
`on consideration of the one compartment
`open modelwithfirst order absorption and the
`case whenthe rate constant for absorptionis
`muchlarger than the rate constant for elimi-
`
`(acaaaa
`
`AUROBINDOEX. 1019, 3
`
`AUROBINDO EX. 1019, 3
`
`

`

`nation(i.e. k > K). Data for the one compart-
`ment open model with bolus intravenous in-
`jection in table 3-1 indicates that underthis
`condition that + = t,,. and € = 1, the ratio
`Cmax/Cmin = 2, and the drug accumulation
`is equal
`to 1.443 (or 1/1n2) =
`1/0.693). Interestingly enough, clinical expe-
`rience andpractical reasons have lead physi-
`cians in the past
`to use this rule without
`knowing its theoretical foundation.®
`
`3.2.2 Based on Average Steady-
`State Blood Levels. The equation of
`Wagneret al.,'” shownas equation 3-4, is most
`useful in dosage regimencalculations.
`
`=D mes
`“VKr Vat
`In equation 3-4, C.,
`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-
`tionrate constant, 7 is the dosage interval, and
`Vo is the blood (serum or plasma) clearance
`(sometimescalled the “body”clearance). The
`middle term of equation 3-4 indicates how the
`equation was originally written,!? but
`the
`term on the far right is a more generalized
`way of writing the equation. Note that C,, 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 shownin
`figure 3-1, and
`
`ty
`
`f C.(t)dt
`CG _ AG _ ty
`epg
`where 7 = t, — t,
`
`Eq. (3-5)
`
`equations 3-4 and 3-5, we may also write
`equation 3-6. If concentrations
`
`Kq. (3-6)
`
`are measured at a sufficient numberof differ-
`ent times following oral or intramuscular ad-
`
`then F/Vo, (the reciprocal of the apparent
`clearance) may be estimated from equation
`3-6 by making a ratio of the area to the dose
`(ie. F/Vq = A/D = AG/D). After
`single
`doses the area Aj 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/Vor by measuring plasma concen-
`trations after several different doses of the
`drug and plotting Aj 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 regressionline.
`Theslope of the regresion line for such a plot
`
`a °
`
`xhrs)
`29 ml > °
`
`~ °
`
`100
`
`a o
`
`a o
`
`bp Oo
`
`20
`
`Y= 0
`AREAUNDERLINCOMYCINSERUMCONCENTRATIONCURVE0-@(
`
`
`
`
`
`2
`
`8
`6
`4
`X= DOSE OF LINCOMYCIN (mg/kg)
`
`10
`
`12
`
`Fig. 3-2. Plot of Aj under single dose serum
`concentration curve versus mg/kg dose of linco-
`mycin hydrochloride following intramuscular ad-
`ministration. Each point correspondsto a different
`subject. The three arrays correspond to doses of
`100, 200 and 600 mg. of the antibiotic. The slope
`of 12.05 represents a suitable value of F/V,, to use
`
`AUROBINDO EX. 1019, 4
`
`

`

`dictions as indicated below. An example of
`not forced through the origin was 11.42 with
`the 95 percent confidence interval of 9.64 to
`such a plot is shownin figure 3-2 where the
`area under the serum concentration curve
`13.4. If the latter values are substituted for
`F'/V¢ in the above equation instead of 12.05,
`following intramuscular
`administration of
`lincomycin hydrochloride is plotted against
`one obtains 4.0 and 5,6 g/ml, respectively.
`the mg/kg dose of the antibiotic.'* The re-
`It should be noted that application of
`gression is highly significant (P < .001). The
`equations 3-4 through 3-6 above are inde-
`100r? value is 74 percent, indicating that 74
`pendent of whetherthe one or two compart-
`percent of
`the variability of
`the area is
`ment open model applies to the data being
`accounted for by differences in the mg/kg
`evaluated since the meanclearance is used.
`doses. Since the intercept was notsignificantly
`In termsof the one compartment open model,
`different
`from zero the least squares line
`the mean clearance is equal to VK, but for
`forced through the origin wascalculated(i.e.,
`the two compartment open model, the mean
`y = 12.05x where¥is the estimated value of
`clearance is equal to V,k. or Vg area B-
`Ag and x is the mg/kg dose) andis the line
`Another type of question which may be
`answered with the above information is indi-
`drawn in the figure. The “zero intercept” is
`cated below.
`indicated by the theory embodied in equation
`3-6. Thus,
`in this case, F/Vo = 12.05 kg/L.
`Question: What dosage could be employedfor
`(The units were obtained as follows:
`a 50 kg womanto provide an average steady
`
`ug/ml wg
`kg
`state level of lincomycin hydrochloride of
`= kg/L).
`A_
`typical
`mg/kg ml**103 ng
`3 pg/ml
`if
`the drug is administered intra-
`muscularly?
`dosage regimen question which may be an-
`Answer: First rearrange equation 3-4 and sub-
`swered with such information is
`indicated
`stitute the knownvalues as follows:
`below.
`Question: What would be the expected aver-
`age steady state serum concentration if a
`60 kg person were administered 200 mg of
`lincomycin hydrochloride every 8 hours?
`
`P = (G.)(1/Vo) = (8)(1/12.08) ~ 1/4
`T
`
`Answer:
`
`CG. = (=) (2)
`= (12.05) (a) = 5 pg/ml
`
`\t
`200/60
`
`Vo’
`
`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 FVoy has units of kg/L, D
`must be in mg/kg, and C,, will have dimen-
`sions of pg/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 answerof 5 pg/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 intervalofthe least
`squaresslopeforthe datain figure 3-2 to make
`
`Hence, any combinationof D and 7 which will
`give a ratio of D/7 equal to 1/4 will be an
`answer. You have to remember, again,
`since the clearance has units of L/kg, andits
`reciprocal, kg/L,
`that D must be in units
`of mg/kg.
`One
`answer
`100 mg/50 kg = 2mg/kg and 7 = 8 hours
`since D/r = 3/12 = 1/4.
`Such
`are “ball-park” answers andvery useful in the
`clinical situation.
`Orret al.13 published a methodforesti-
`mating individual drug-dosage regimens, but
`what
`they call “occupancy/ml” is exactly
`equivalent
`to F/Vo = Aj/D = Aj/D and
`really does not require a new name. Their
`methodis 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
`
`OL,EE|
`
`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 curveis drawn
`through the plasma or serum concentration
`data, and the concentrations at hourly inter-
`vals are estimated from the smoothline, the
`in this case,
`is simply the sum of the
`estimated concentrations. This actually gives
`the value of Aj 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, or serum concen-
`trations, which are those measured at 7 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-
`
`e 87
`F
`min —|p(E)\(_e* _\|p
`
`[P (VG — | m
`
`zg.
`
`Ba 8)
`
`(3-
`
`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 minimumsteady-state concentrations to
`the maintenancedose, D,,. If you have some
`wayof estimating £, such as from a correla-
`tion of B with endogenous creatinine clear-
`ance or a calculated clearance of creatinine,
`then one can reduce variance by using only
`[pF/V,] as the proportionality constant.
`A model-independentversion of equation
`3-7 may be written as equation 3-8.
`i
`F
`
`Cmin a.(¥,)
`
`Fraction remaining at
`end of dosage interval
`Fraction lost during
`dosage interval
`Note that when written this way the “p”has
`disappeared from the equation. By matching
`terms in equations 3-7 and 3-8 we cansee
`that:
`
`Ed’(3-8
`
`(3-8)
`
`D.
`
`m
`
`eq:
`
`In equation 3-7, C™™is the minimumsteady-
`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, V, is
`the volumeofdistribution, D,, is the mainte-
`nance dose given every 7 hours, and f 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
`withfirst order absorption p = (k/k — K) =
`(k/k — £); in this case compare equation 3-7
`with equation 3-67 and assume that e~*7 ~ 0,
`and that Cy = FD,,/V,. The reasonthat 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 /, 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 remainingat endof dosageinterval
`Fraction lost during dosage interval
`e Ar
`
`=p (—] Eq. (3-9)
`It should be noted that when written as equa-
`tion 3-7the value of “p” is model-dependent,
`while the “p” does not appearin equation3-8,
`In the case of the two compartment open
`model the “V,” in equations 3-7 and 3-8 is
`actually V,.
`
`3.2.4 Superposition or Overlaying
`Principle.
`This
`principle
`provides
`a
`methodof predicting multiple dose blood lev-
`els in linear systems and a simple method of
`estimating a loading dose. The methodis ap-
`plicable to data obeying any linear pharmaco-
`kinetic model, and requires only a pencil and
`paper. Unlike the methodoutlined insection
`3.2.2, this method provides estimates of multi-
`ple dose bloodlevels at any desired sampling
`
`AUROBINDO EX. 1019, 6
`
`

`

`After Multiple Doses in a Linear System
`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
`4.4]
`(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
`4.4]
`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
`
`=
`
`—_
`
`AUROBINDOEx. 1019, 7
`
`AUROBINDO EX. 1019, 7
`
`

`

`The Superposition Method in Tabular Form
`
`This method can be applied to rawdata
`directly without resorting to pharmacokinetic
`analysis. It will give the correct answerfor
`a linear system providing everything remains
`constant when multiple doses are adminis-
`
`Cit) = (Efe— o]
`
`Eq. (3-10)
`
`using Cy = 100, k = 1.0455 hr7}, and K =
`0.17425 hr~!. Thus,
`the
`absorption _half-
`time was 0.693/1.0455 = 0.663 hr, and the
`elimination half-life was 0.693/0.17425 =
`
`the trend. It was also found that forthis par-
`ticular set of data a semilogarithmic plot of
`bloodlevel versus time waslinear in the 6 to
`12 hour period as shown infigure 3-4,
`In
`actual practice one couldfit 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 illustrate the method simulated blood
`to six doses as used in this example). The au-
`levels, C(t), following a single dose were gen-
`thor wasalittle fancier and fitted the least
`erated with the equation:
`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,
`
`InC =InC,) —Kt_—
`
`Eq. (3-12)
`
`Substitution of the above constants into
`equation 3-10 gave equation 3-11.
`InC = 4.7862 — 0.17421t—Eq. (3-13)
`C(t) = 120[e~0-17425t _
`e 71.0455]
`Eq. (3-11)
`
`one obtains the equation shown as equation
`3-13.
`
`Data generated with equation 3-11 are listed
`underthe third column, labeled “Dose 1” in
`table 3-2. These numbers are not bracketed.
`These C(t) values are plottedinfigure 3-3 and
`the points are joined by straight lines to show
`
`The correlation coefficient was r = 1.000000
`showing an excellent fit. Equation 3-13 may
`be written as equation 3-14.
`
`C i TTG85e-O raat
`
`Eq. (3-14)
`
`Using equation 3-14 the bracketed values
`of C(t), listed under Dose 1 in table 3-2, were
`
`
`
`TIME
`
`IN HOURS
`
`LEVEL
`BLOOD
`
`AUROBINDO EX. 1019, 8
`
`

`

`6
`
`8
`
`10
`
`12
`
`14
`
`16
`
`TIME
`
`18
`
`IN
`
`20
`
`22
`
`24
`
`26
`
`28
`
`HOURS
`
`30
`
`20
`
`98
`
`2s
`
`7=
`
`65
`
`4
`
`3
`
`2
`
`©99999>’NWOnuNeo-
`
`a
`
`> =
`
`Ss
`—_d
`Ss
`es
`
`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
`numberof doses one wishes to predict (exam-
`ple has 6 doses)—then one merely adds all the
`ordinate values across the rows to obtain the
`total equivalent
`to C,(t) or
`the predicted
`bloodlevel for the multiple dose regimenafter
`the nth dose.
`The C,(t) values in the last column of
`table 3-2 are plotted in figure 3-5. A dotted
`line has been drawnby justjoining the points
`to show thetrend. 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
`bloodlevels, we can estimate the loading dose
`reasonably accurately by making a ratio of the
`peak blood level at the equilibrium state to
`the peak bloodlevel after the first dose. Ap-
`plying this to the datain table 3-2 one obtains:
`
`Loading dose
`Maintenance dose
`
`_ Peak blood level after 6th dose
`~~ Peak blood level after Ist dose
`— 115.5 _ |6s
`~~ 69.86
`
`This provides an answervery close to the 1.54
`calculated with the equation of Kriiger-
`
`AUROBINDO Ex. 1019, 9
`
`AUROBINDO EX. 1019, 9
`
`

`

`Table 3-3. Predicted Blood Levels With Loading Dose
`
`Time
`TOTAL
`(Hours)
`Dose 1
`Dose 2
`Dose 3
`Dose 4
`Dose 5
`Dose 6
`=C,(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
`it
`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
`Il
`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.Al
`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
`131
`0.76
`32
`104.6
`65.93
`25.00
`8.82
`3.11
`110
`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
`
`

`

`24
`
`
`
` 26 =«=28 30 32
`
`
`20
`22
`
`18
`
`16
`TIME IN HOURS
`\
`\
`h
`f
`A poses
`Fig. 3-5. Predicted multiple dose blood levels from last columnof table 3-2.
`
`100
`
`90
`
`80
`
`70
`
`LEVEL 60
`BLOOD
`
`50
`
`40
`
`30
`
`0
`
`2
`
`4
`
`6
`
`8
`
`10
`
`12
`
`14
`
`\
`
`Thiemerwhichis given in a later section (see
`equation 3-30 andrelatedtext).
`To check out the answer, all blood levels
`under “Dose 1” of table 3-2 were multiplied
`by 1.65 andtheblood levels due to the previ-
`ous maintenance doses used to complete the
`table as before. The new table is shownastable
`3-3, The C,,(t) values are plotted infigure 3-6.
`Onecansee that with this dosage regimenthe
`steady state is reached at once and main-
`tained.
`
`Pitfalls of the Superposition Method
`The superposition methodis valid only
`when the pharmacokinetics are linear and
`elimination occurs from the body according
`to first orderkinetics. 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 thatthelast bloodlevel
`measured is between one-fifth and one-
`twentieth of the peak blood level. The lower
`one can measure, the moreassured oneis 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
`A, =50, A,=50, A3;= 100,
`B =0.1, andk, = 16.
`
`C(t) = A,et + A,eft — Ae a!
`
`Using the blood levels in the 12 to 28 hour
`
`AUROBINDO EX. 1019, 11
`
`AUROBINDO EX. 1019, 11
`
`

`

`16 18
`weV@od °
`
`range (solid line in figure 3-7) a rate constant
`of 0.1005 hours~! was obtained, which is very
`close to the smallest rate constant in equation
`3-7, namely 2 = 0.1 hours~!. However,if one
`used the bloodlevels in the 2 to 8 hour range
`(dottedline in figure 3-7) a rate constant of
`0.1411 hours? 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 poorprediction of multi-
`ple dose blood levels. It has become common
`practice by many not well acquainted with
`pharmacokinetics to perform the operations
`shownin 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
`
`4
`
`6
`
`8
`
`10
`
`12
`
`14
`
`20
`TIME IN HOURS
`
`22
`
`24
`
`26
`
`28
`
`30
`
`32
`
`34
`
`36
`
`A
`A
`\
`\
`A
`Fig. 3-6. Predicted multiple dose blood levels from the last columnof table 3-3. 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.
`
`6
`
`8
`
`10
`
`12
`TIME
`
`16
`14
`IN HOURS
`
`18
`
`20
`
`22
`
`24 2
`
`26
`
`Fig. 3-7. Solid dots are simulated blood levels
`generated with equation 3-7. Solid line in the 12
`to 28 hourregion gives correct estimate of elimi-
`nation rate constant andhalf-life, while dotted
`
`AUROBINDO EX. 1019, 12
`
`

`

`If a drug’s elimination obeys Michaelis-
`Mentenkinetics, then the superposition prin-
`ciple will provide an underestimate of the
`multiple-dose bloodlevels.
`
`3.2.5 The Superposition Principle
`in Mathematical Form
`
`Let t = time measured from administration
`of first dose
`t’ = time measured from administration
`of nth dose
`n = dose number
`7 = dosage interval
`
`Then:
`
`t=t—(n—1)r
`
`When n = 1, t’ = t; whenn = 2, t’ =t —7;
`and in general t = t’ + (n — I),
`
`Let C,(t’), C,(t’), ------- , C,(t’) be the
`blood level at time t’ after the first, second,
`-------- , nth dose, respectively.
`
`C,(t’) = C,(t) + B
`
`| _ ema-DBrle Je Eq. (3-24)
`
`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 B is 0.17421 hours~t. Now, “B” in
`equation 3-24 is
`the value of C(t) when
`t =7 =6 hours.*® Substituting t = 6 into
`equation 3-14 gives B = 42.14. Hence,forthis
`example, equation 3-24 may be written as
`equation 3-25,
`
`C,(t’) = C,(t) + 42.14
`
`1 — e-W.i74206)
`2 — em-10.17421)6)
`
`|e
`
`0.174210
`
`Whenn = 6,substitution into equation 3-25
`gives:
`
`If the B-phase® (loglinear phase) is established
`0.99463|_
`at + hours after the first dose, Westlake!”
`C(t’) = 42,14.|W222302|e-0.17421¢t
`
`
`showedthat:
`
`a(t’)=C(t) + |oeeass |e
`
`= C(t) + 64.64e~0-17421"
`Eq. (3-17)
`C(t’) = C,(t) + Be-At
`C,(t’) = C,(t) + Be“ft + Be-Bt' +”)
`>
`=C,(t) + Be-#"(1 + e-87) Eq, (3-18)
`
`Substituting t’ = 0, 1, 2, 3, 4, 5, 6, into equa-
`tion 3-26 gives the values shownintable 3-4.
`It should be noted that the values of C,(t’)
`
`C,(t’) = Cy(t) + B
`(1 peBr 4RF 4 wena feBln-2)7)g—BU’
`Eq.(3-19)
`
`Let S = 1 + ef? 4 e827
`+ ---- +e287 Eq. (3-20)
`
`Then e~f7S = e-f7 + e827
`4 —---------------- pS e -a-DBr
`Eq. (3-21)
`
`S(1 = e #7) =] —e-@-vpr
`by subtraction Eq. (3-22)
`_ ea-DBr
`1 —eArt
`
`Hence, S =
`
`°Note that in this section £ replaces K in the previous
`section.
`
`°°To apply equation 3-24 7 must be chosen so that it is
`> thanthe time whenthe /-phase is established after
`the single dose.
`
`Table 3-4. Values of C,(t’) Generated with
`Equation 3-26 for n = 6
`
`t’
`
`0
`1
`2
`3
`4
`5
`6
`
`t
`
`30
`3l
`32
`33
`34
`35
`36
`
`C,(t’)
`
`64.64
`112.9
`115.5
`104.3
`90,14
`76.62
`64.69
`
`ar =t’ + (n — 17 = tt’ 4 (5)(6) = t’ + 30
`
`AUROBINDO EX. 1019, 13
`
`AUROBINDO EX. 1019, 13
`
`

`

`in table 3-4 are essentially the same as the
`values of C,(t) after the sixth dose shown in
`
`3.2.6 Use of Model Equations to
`Predict Multiple Dose
`Blood
`Levels. Let us assume you had the data
`shownin figure 3-3 andfitted these data by
`means of a digital computer and a nonlinear
`least squares programto equation 3-10. Again
`let us assume that this fitting gave the exact
`values of the parameters, namely Cy = 100,
`k = 1.0455 hours"! and K = 0.17425 hours7}.
`It is readily shown that equation 3-27 is
`the multiple dose equation which corresponds
`with the single dose equation 3-10.
`
`—nKr
`
`j
`
`(4 See ews ~ (4 =et Jer"
`
`— p—nKr
`
`x
`
`Eq. (3-27)
`where, as before, t’ is elapsed time since the
`nth dose was given.
`Asbefore, let Cy = 100, k = 1.0455 hr7!,
`K = 0.17425 hr“! and + = 6 hours.
`When n= 2, substitution of the constants into
`equation 3-27 yields equation 3-28.
`
`C,(t’) = 162.2 e70-17425t — 120.2 e71.0455t"
`Eq. (3-28)
`
`When n = 6, substitution of the constants into
`equation 3-27 yields equation 3-29.
`Colt’) = 184.7 e--17425"' _ 190.9 e~L 0455"
`Eq. (3-29)
`
`Substituting values of t’ = 0, ---, 6 into
`equations 3-28 and 3-29 yields the values
`shownin table 3-5.
`The values shownin 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 methodis very
`accurate if properly performed and that equa-
`tion 3-24 (whichstates the superposition prin-
`
`Table 3-5. Values of C,(t’) Calculated by
`Means of Equations 3-28 and 3-29 for n = 2
`andn =6
`
`Equation 3-28
`
`Equation 3-29
`
` ' Cyt’) Catt’)
`
`
`0
`6
`42.0
`30
`64.5
`1
`‘k
`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
`
`
`
`
`ft=t'+(n-—)r=t’ +7
`baot+(n—-—Ir=t’ +57
`
`Estimation of the Loading Dose
`
`Forthe 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 =
`
`Maintenance dose
`(1 _ eK] _ ek)
`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 + = 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.
`
`AUROBINDO EX. 1019, 14
`
`

`

`wn
`
`] — e Tnyr
`
`Jewe + Ay
`Ci) =i [Ta
`($aJem 4 a(Laer)aw
`
`—e
`
`QT
`
`—e
`
`37
`
`t’ is the time after the nth
`In equation 3-32,
`dose and 7 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 onrectalinear(cartesian
`coordinate) graph paper
`and drawing a
`smoothline throughthe points by sight, then
`interpolating points from the smoothline.
`Usually, such interpolated points are takenat
`uniform time intervals in order to apply the
`methods.
`The methodoutlined in section 3.2.5 is
`model-dependent. The appropriate multiple
`dosing equationfor the particular model must
`be used. However,it also can be made model-
`independent byjust 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 minimumbloodlevels
`after the nth dose orat steady state. All that
`need be doneis 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,tp, equation 3-10 is written
`as equation 3-33.
`
` C(t) = Gy (, = x)lew _ eKitt]
`
`Equation 3-33 may be written as equa-
`tion 3-34.
`
`C(t) = Ae“Kt — Bet Eq, (3-34)
`
`where
`
`k-K
`
` A= Co
`Jew Eq. (3-35)
`B=C,|*—|e
`|—k_]enea,
`—~c
`Whenty, = 0, thenA = B = Cy Je]
`
`as in equation 3-10, Equation 3-34 would also
`be a general form for the model:
`
`an
`K
`
`k
`
`k
`k
`
`Cc
`Vv
`
`>)
`7°
`.
`Absor ption |_
`
`site
`but the k and K in the equation would be on
`a and f which would be complex square root
`functions of the k,, k_,; and K in the scheme.
`Let us assume you “stripped” a set of
`data by the methodoutlined in section 2.1.2
`and obtained an equationof 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
`squaresfit of the original data to equation 3-34
`using a nonlinear regression program, suchas
`NONLIN,anda digital computer—thenuse
`the least squares estimates of A, B, k and K
`to make predictions. The latter parameters
`wouldhaveless bias.
`By adding multiple dosing functions to
`
`AUROBINDO EX. 1019, 15
`
`AUROBINDO EX. 1019, 15
`
`