ELSEVIER
`
`An Interactive Computer Program for Determining Areas Bounded by
`Drug Concentration Curves Using Lagrange Interpolation
`C. Ediss and Y.K. Tam
`
`Faculty of Pharmacy
`
`and Pharmaceutical
`
`Sciences, Universig of Alberta, Edmonton, Alberta, Canada
`
`The LAGRAN method of Rocci and Jusko for determining the area of plasma concentration
`curves has been implemented
`in a user-friendly form. Drug concentration versus time data
`may be entered using the keyboard or imported
`in the form of simple text files from
`spreadsheets or other software. The MSDOS program allows prompt graphic observation of
`the data. The effect of selecting different fitting modes for each segment of the curve may be
`viewed interactively using this graphic display. Pharmacokinetic parameters that are provided
`by the program include mean residence time, variance of the residence time, plasma clearance,
`and steady-state volume of distribution.
`
`Key Words: AUC; Computer program; Lagrange; Parameter estimation; Pharmacokinetics
`-
`
`Introduction
`Useful pharmacokinetic
`parameters can be obtained
`by determining
`the area
`(AUC)
`under drug plasma
`concentration
`versus
`time curves. Rocci and Jusko
`(1983) described a technique using lagrange polynomials
`to interpolate
`concentration
`levels during
`the intervals
`between measurements. The advantage of such polyno-
`mials is that
`the fitted curves pass exactly through
`the
`measured data points. Storey and Davies (1986) imple-
`mented a program using a combination
`of spline fitting
`and Simpson’s rule to calculate AUC. The advantage of
`their method
`is that it results in a simple program, but
`there is no graphic display to readily confirm
`the quality
`of the fit. Borsi et al. (1988) introduced
`the PharmCalc
`program
`to automate
`the calculation of AUC
`for meth-
`otrexate
`infusions using the trapezoidal method. Pharm-
`Calc produces a graphic printout of the data. Wijnand
`(1992) has described
`the SIMF&KA
`and ESTF&KA
`programs employing
`the regression method of truncated
`areas for linear pharmacokinetics. Results are provided
`in printed
`form. Another
`approach has been
`to use
`programs such as NONLIN
`(Weiner, 1986) to determine
`the value of parameters
`that give the best fit of the data
`to some proposed model. These parameters can then in
`
`to Mr. Chris
`University
`
`of Pharmacy
`Faculty
`Ediss,
`of Alberta,
`Edmonton,
`T6G
`
`requests
`reprint
`Address
`Sciences,
`and Pharmaceutical
`2N8, Alberta,
`Canada.
`Received
`May
`5, 1995;
`
`accepted
`
`June
`
`22, 1995.
`
`to determine AUC. For ex-
`instances be used
`some
`travascular drug input or constant
`infusions, NONLIN
`(V4.2) uses the linear trapezoidal method
`for determin-
`ing AUC.
`imple-
`The current program provides an interactive
`mentation
`of Rocci and Jusko’s
`lagrangian method,
`which with
`linear and logarithmic
`trapezoidal options
`allows broad
`flexibility. The
`lagrangian
`fitting mode
`is
`most suitable for rapidly changing areas of considerable
`curvature. The linear or logarithmic
`trapezoidal modes
`describe
`the gradually diminishing
`regions effectively
`and avoid the possibility of oscillatory
`lagrangian solu-
`tions (Yeh and Kwan, 1978). Ready access to a graphic
`display of the data allows prompt user feedback during
`the fitting procedure.
`(in Turbo C, Borland
`The program has been written
`International)
`so that it is compatible with the broadest
`range of MSDOS computers as possible. The program
`will run on 8088 (or better) machines and requires no
`co-processor.
`The program
`generates monochrome
`graphics and is compatible with display adapters ranging
`from CGA
`to SVGA.
`
`Methods
`Each interval between data points is fitted separately.
`A cubic lagrangian
`is generated
`from a total of four data
`points,
`the
`two boundary
`points of
`the
`interval
`in
`question and their
`immediate neighbors. The first and
`
`of Phamacological
`Journal
`Inc.
`0 1995 Elsevier Science
`655 Avenue
`of
`the Americas,
`
`and Toxicological
`
`Methods
`
`34, 165-168
`
`(1995)
`
`New York, NY 10010
`
`1056-8719/95/$9.50
`1056-8719(95)00062-M
`
`SSDI
`
`AMN1076
`Amneal Pharmaceuticals LLC v. Alkermes Pharma Ireland Limited
`IPR2018-00943
`
`

`

`166
`
`the original
`
`- C.Ediss
`Program LAGRAN Vl.OD
`Data
`Input
`paper
`from
`- ID: Test data
`Fl
`by Rocci and Jusko
`# Time
`Cp
`F2 - Bolus dose
`(500*****)
`1 o*****
`o*******
`406.31**
`the
`F3 - Number of points
`in
`2 0.5***
`3 1*****
`494.96**
`(5*)
`terminal
`phase
`4 2*****
`406.69**
`5 4*****
`206.46**
`the
`F4 - Decay constant
`of
`6 6*****
`115.04**
`terminal
`phase
`(0.25723*)
`7 a*****
`68.16***
`F5 - &d
`em
`file
`(********************)
`a IO****
`41.11***
`9 12****
`24.9****
`F6 - Load
`.LGN file
`(C:\TC\DATA.lGN******)
`10 24****
`1.24****
`**
`o*****
`********
`**
`******
`**A.*****
`Control/F9
`- Erase all data
`**
`******
`********
`*x ******
`*******+I
`FlO - Return
`to
`the main menu
`Tab -5 Cp; Up/Down/Right
`arrow keys.
`mter/Retum
`-5 Next Time;
`Enter a time value greater
`than 24
`
`30th September
`
`Fitting
`cp
`t.cp
`lag
`lag
`lag
`lag
`lag
`lag
`lag
`lag
`lag
`lag
`log
`lag
`log
`lag
`log
`lag
`log
`lag
`
`JPM Vol. 34, No. 3
`November
`1995:165-168
`
`I994
`
`I
`!
`
`modes
`t.t.q
`lag
`lag
`lag
`lag
`lag
`lag
`lag
`lag
`lag
`
`Figure 1. The Data Input screen.
`
`lagrangian
`last intervals are characterized by a quadratic
`obtained
`from
`three data points,
`the
`two boundary
`points and the only available neighbor.
`long enough
`Data are seldom collected over periods
`to ensure that the drug has been completely eliminated
`from
`the system. Thus, in order
`to obtain
`the complete
`area under the curve, it is necessary to have some means
`of estimating
`the area for the remaining
`time from
`the
`last data collection point
`to infinity.
`It is assumed that
`the drug concentration
`decays exponentially during
`this
`period. The existence of such a monoexponential
`phase
`is one of the properties most often used in non-compart-
`mental methods
`(Gillespie, 1991). The decay constant
`can be entered manually or determined by the method
`of weighted
`least squares applied
`to a selected number
`of final data points. Detailed algorithms are described by
`Rocci and Jusko (1983).
`The area bounded by three functions are obtained by
`the program:
`the area of drug concentration
`versus time
`(AUC),
`the product of time and concentration
`versus
`time (AUTC),
`and the product of (time)* and concen-
`tration versus time (AUT2C).
`(assuming a bolus
`The pharmacokinetic
`parameters
`intravenous dose)
`that can be determined
`from
`these
`areas include:
`time, MRT = AUTC/AUC
`Mean
`residence
`Variance of the residence
`time, VRT = (AUT2C/
`AUC)
`-
`(AUTC/AUC)*
`Plasma clearance, CZp = Dose/AUC
`Steady state volume of distribution, V,S’ = (Dose X
`AUTC)/(AUC*)
`
`func-
`The user interacts with the program by making
`tion key selections
`from a hierarchal system of menus.
`From
`the main menu,
`the Data
`Input area (F3) is used
`to provide
`the program with experimental
`values. Then
`the Display Graphics area (F5) may be viewed
`to see
`how well the data has been fitted. The user may alter-
`nate between
`the Data
`Input screen and the Display
`Graphics screen to tailor
`the fitting modes and other
`input parameters
`to
`improve
`the
`fit. This
`interactive
`capability expedites prompt convergence
`toward optimal
`conditions. A report of the results may be displayed (F6)
`or printed
`(F7). Input data and parameters may be saved
`(F4) for subsequent
`re-examination.
`
`CP
`
`Program
`
`LAGRAN
`
`Vl.OD
`
`- CEdiss
`
`30th September
`
`1994
`
`0
`F2 - t.Cp;
`
`10
`F3 - t.t.Cp;
`
`L - Linear;
`
`20
`X - Expand;
`
`30
`- Exit;
`
`FIO
`
`Figure 2. The Graphics Display screen.
`
`AMN1076
`Amneal Pharmaceuticals LLC v. Alkermes Pharma Ireland Limited
`IPR2018-00943
`
`

`

`C. EDISS AND Y.K. TAM
`LAGRANGE
`DETERMINATION
`
`OF AUC
`
`167
`
`CP
`
`Program
`
`LAGRAN
`
`Vl.OD
`
`- CEdiss
`
`30th September
`
`1994
`
`0
`F2 - t.Cp;
`
`10
`F3 - t.t.Cp;
`
`L - Linear;
`
`20
`X - Expand;
`
`30
`- Exit;
`
`FIO
`
`3. An example of ill-fitting data displayed on the
`Figure
`Graphics Display screen.
`
`The main focus of the Data Input screen (Figure 1) is
`a scrolling
`template
`into which
`time (t) and concentra-
`tion (C,) values may be entered. By default each interval
`
`(lug).
`to be fitted by the lagrangian method
`is flagged
`Occasionally
`the graphic display demonstrates
`that the
`lagrangian solution
`for a particular
`interval can result in
`widely oscillatory
`interpolations.
`The visual impact of
`the graphic display provides a useful tool for identifying
`these troublesome areas and in these instances trapezoi-
`dal
`(lin) or exponential
`(log)
`fitting modes may be
`(lug, Zin, or log) may be as-
`selected. Separate modes
`signed for each segment of each function C,, t * CP, or
`t2 - c,.
`
`Results and Discussion
`The data set employed
`in Rocci and Jusko’s original
`description of the method
`is shown
`in Figure 1. Note
`that the last four
`intervals have been flagged
`for expo-
`nential
`fitting. The corresponding Graphics Display
`screen is shown in Figure 2. The effect of fitting
`the last
`four
`intervals by the
`lagrangian method
`is shown
`in
`Figure 3. The graphic display readily exposes trouble-
`some areas and results in swifter correction
`than would
`be obtained
`from
`results viewed
`in tabular
`form. A
`typical printout of the results produced by the program
`is shown in Figure 4.
`
`30th September
`and Jusko
`
`by Rocci
`
`4’
`1994
`
`'C:\DATA.IGN'
`file
`the
`for
`from
`the
`original
`paper
`points
`= 10
`points
`in
`the
`of
`the
`terminal
`beginning
`of
`Area CC@)
`(lag)
`(lag)
`(lag)
`(lag)
`(lag)
`(log)
`(log)
`
`Report
`Lagran
`ID
`: Test
`data
`IGumber of data
`Number of data
`Decay
`constant
`Cp value
`at
`the
`Time
`cp
`n
`
`0
`
`0.5
`1
`
`2
`
`4
`
`6
`
`8
`10
`
`12
`
`24
`
`406.31
`494.96
`
`406.69
`
`206.46
`
`115.04
`
`68.16
`41.11
`
`24.90
`
`1.24
`
`114.81
`
`234.97
`472.09
`
`609.05
`
`308.72
`179.13
`
`107.00
`
`Dose
`= 500
`Bolus
`phase
`terminal
`= 5
`= 0.25723
`phase
`the
`terminal
`phase
`Area(t.Q)
`47.09
`
`173.60
`
`680.17
`
`1732.34
`1529.20
`1235.40
`953.66
`
`= 2.6946E
`
`5
`
`life
`Half
`= 1.12015
`Area(t.t.Cp)
`
`13.24
`139.74
`1038.13
`5064.29
`
`7566.12
`8594.46
`
`8535.42
`
`7733.31
`
`25317.26
`
`(lag)
`(lag)
`(lag)
`(lag)
`(lag)
`(lag)
`(lag)
`(lag)
`(lag
`
`4. A typical printed report
`Figure
`produced by the program.
`
`(lag)
`(lag)
`(lag)
`(lag)
`(lag)
`(lag)
`(lag)
`(lag)
`(lag)
`
`64.66
`
`94.65
`
`(log)
`
`(log)
`
`706.33
`
`1277.49
`
`2185.09,
`=
`AUC
`~ Partial
`8335.29,
`=
`AUTC
`Partial
`64001.96,
`AUT2C =
`Partial
`= 0.228368
`Plasma Clearance
`= 0.882073
`Steady
`State
`Volume
`of Distribution
`Mean Residence
`Time = 3.862500
`Variance
`of
`the Residence
`Time = 15.890034
`
`= 2189.44
`AUC
`= 8456.73
`AUTC
`ATJTZC = 67454.47
`
`Total
`Total
`Total
`
`AMN1076
`Amneal Pharmaceuticals LLC v. Alkermes Pharma Ireland Limited
`IPR2018-00943
`
`

`

`168
`
`to
`least squares
`the method of weighted
`By using
`the terminal phase, the resulting exponen-
`characterize
`tial does not in general pass through
`the last data point.
`Thus, although
`this method does give a good estimate of
`the exponential decay constant,
`it introduces a disconti-
`nuity in the curve (Figure 2). The program may force the
`trailing exponential
`to pass through
`this last data point in
`the
`following way: First, a reasonable number of the
`final data points are selected to characterize
`the expo-
`nential
`tail (5 in the example shown), and the program
`determines
`the corresponding decay constant. Then the
`Data Entry area may be revisited, and the number of
`data points in the terminal phase changed
`to zero. The
`previous decay constant
`is retained, but the exponential
`decay is now anchored
`to the last data point. Thus the
`usual displacement
`at
`the beginning
`of
`the
`trailing
`exponential may be avoided.
`The program has been used for its original purpose in
`the evaluation of the pharmacokinetics
`of Hydralazine
`(Semple et al., 1990) and Methyldopa
`(Skerjanec et al.,
`1995). It has also proved useful in the field of radiation
`dosimetry. The integration of activity versus time data is
`used
`to produce
`cumulated
`activity and, ultimately,
`radiation dose (McQuarrie
`et al., 1994).
`readily permits
`The modular nature of the program
`the
`inclusion of improvements.
`For
`instance a fourth
`fitting mode in addition
`to Zag, Liz, or log could readily be
`accommodated without
`requiring an extensive redesign
`of the data input area. Also the statistical selection of the
`number of data points
`in the
`terminal phase
`imple-
`mented by Kowalski
`(1994) could be included as an
`option. Other suggested
`improvements
`include
`the cal-
`culation of AUC within user-selected arbitrary
`time
`boundaries
`and
`the accommodation
`of infusions and
`multiple dosing.
`
`JPM Vol. 34, No. 3
`November
`1995:165-168
`
`Availability
`The program may be obtained directly by mail from
`the authors upon
`receipt of a formatted
`disk of the
`desired format. Alternatively
`the authors will attempt
`to
`respond to e-mail requests (directed to cediss@pharmacy.
`ualberta.ca), provided the desired electronic delivery mode
`is clearly specified.
`
`for
`Program
`of methotrex-
`
`the
`
`An algorithm
`studies.
`
`for estimating
`Comput
`Methods
`
`the
`
`terminal
`Programs
`
`half-life
`Biomed
`
`A,
`RP, Niesen
`of
`comparison
`174H.64
`mono-
`
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`(1988)
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`LAGRAN
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`
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`
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`Biomed
`
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`Hydralazine
`(1990)
`An evaluation
`of
`the dog as an animal
`
`in
`
`calcu-
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`
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`
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`
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`lagrange
`6:79-98.
`
`integrating
`of numerical
`spline
`approximation.
`
`J
`
`and
`
`AMN1076
`Amneal Pharmaceuticals LLC v. Alkermes Pharma Ireland Limited
`IPR2018-00943
`
`