Exercise 13

Deconvolution analysis vs. modelling to document the

process of drug absorption
Objectives
 To implement a WNL user model for a monocompartmental model with two
processes of absorption either with an algebraic equation or with a set of
differential equations.
 To compare two concurrent models: monocompartmental, with a single site of
absorption vs. a monocompartmental model with two sites of absorption using the
AIC criterion
 To obtain the input function of a drug using deconvolution to reveal information
about its process of absorption.

Overview

Double peaks in the plasma concentration–time profile following oral administration

have been reported for several compounds. Multiple peaks after oral administration
of a drug could have several physiological causes.
To describe a complicated drug plasma profile like a double peak, there are two
possible approaches: (i) curve fitting using a modified customary compartmental
model (ii) numerical deconvolution to identify the input rate of the drug in the
central compartment.
For the present exercise we will implement a pharmacokinetic model in WNL that
was previously used in exercise 5 (Monte Carlo simulations, Bioequivalence and
withdrawal time) incorporating absorption from two different sites. The results will be
compared with those from a deconvolution analysis.

Exercise

The goal of this experiment was to compare two formulations (A and B) of a new
drug product administered by the oral route at a dose of 100 µg/kg. Twelve (12)
animals were investigated following a cross-over design and the raw data are given
in the corresponding Excel sheet.

After importing the data set in WNL, you will inspect individual curves, you will see an
irregular initial phase with a shoulder for some animals (e.g. animal 21).

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It is clear that fitting this animal with a conventional model would most likely lead to
some misfit especially during the first hours following drug administration and even if
a fitting is apparently satisfactory, the estimated parameters could be severely
biased.
We will start to fit these different curves with a classical monocompartmental model
with a single rate constant of absorption.

The next figure shows the fitting obtained for animal 21 that looks rather good.

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However inspection of residuals is more informative and it immediately enables
departure to the model to be detected.

 Compute the different parameters associated with this monocompartmental model.

Monocompartmental model with a single rate constant of absorption, and mean
parameters for the two formulations:

Variabl Formulati N Mean SD Min Max CV% Geometric_Me

e on an
K10 A 1 0.041 0.003 0.037 0.049 9.0431 0.0416
2 8 8 7 0
K10 B 1 0.037 0.005 0.026 0.044 14.231 0.0368
2 1 3 1 7 6
V_F A 1 3.154 0.345 2.433 3.573 10.936 3.1363
2 6 0 0 4 5
V_F B 1 4.174 1.500 2.664 7.815 35.937 3.9669
2 8 3 4 9 4
K01 A 1 0.098 0.014 0.079 0.113 14.217 0.0973
2 2 0 4 3 8
K01 B 1 0.106 0.012 0.085 0.129 11.355 0.1058
2 4 1 3 4 8

Variable Formulation N Mean SD

AUC A 12 766.6596 36.7152
AUC B 12 696.0401 123.3195
CL_F A 12 0.1307 0.0062
CL_F B 12 0.1480 0.0272
K01_HL A 12 7.1911 1.0390
K01_HL B 12 6.5905 0.7583
Cmax A 12 16.8647 1.0422
Cmax B 12 14.7399 3.6567
K10_HL A 12 16.7088 1.4535
K10_HL B 12 19.0709 3.1261
Tmax A 12 15.2337 0.7055
Tmax B 12 15.2968 0.7617

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Diagnostic table for the monocompartmental model:

Formulation Item N Mean SD

A AIC 12 14.8766 27.6988
A CORR_(OBS,PRED) 12 0.9936 0.0064
A CSS 12 478.1272 64.4449
A DF 12 11.0000 0.0000
A S 12 0.6135 0.4739
A SBC 12 16.7937 27.6988
A SSR 12 6.4052 6.4875
A WCSS 12 478.1272 64.4449
A WSSR 12 6.4052 6.4875
A WT_CORR_(OBS,PRED) 12 0.9936 0.0064
B AIC 12 10.2112 27.2787
B CORR_(OBS,PRED) 12 0.9925 0.0074
B CSS 12 378.1471 174.8035
B DF 12 11.0000 0.0000
B S 12 0.5048 0.3756
B SBC 12 12.1284 27.2787
B SSR 12 4.2252 4.4506
B WCSS 12 378.1471 174.8035
B WSSR 12 4.2252 4.4506
B WT_CORR_(OBS,PRED) 12 0.9925 0.0074

Corrected sum of squared observations (CSS), weighted corrected sum of squared

observations (WCSS), sum of squared residuals (SSR), weighted sum of squared
residuals (WSSR), estimate of residual standard deviation (S) and degrees of
freedom (DF), the correlation between observed Y and predicted Y, the weighted
correlation, and two measures of goodness of fit: the Akaike Information Criterion
(AIC) and Schwarz Bayesian Criterion (SBC).

Model with two sites of absorption

We want to compare results obtained with this classical model with those of a user
model able to describe the same data but with a model incorporating two sites of
absorption.
This is the model that we previously used to simulate our data set with Crystal Ball
(exercise 5).
In this monocompartmental model, a first fraction (F1) of the administered dose is
available from site1 with a rate constant of absorption Ka1; and the second fraction
(F2) of the administered dose is available from site 2 with a rate constant Ka2. We
assumed that the availability is total and that F2=1-F1; Vc is the volume of
distribution of the central compartment;
This model can describe a double peak in plasma concentration-time curves:

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 F1 x Dose
Fraction 1 (%) Ka1

Plasma
Vc
F2 x Dose
Ka2
Fraction 2 (%)

K10

The system can be very easily described by a set of differential equations such as:
DZ(1) = -Ka1*Z(1)
DZ(2) = -Ka2*Z(2)
DZ(3) = Ka1*Z(1)+Ka2*Z(2)-K10*Z(3)
Where:
 DZ(1) = -Ka1*Z(1) is the differential equation describing the disappearing of the
drug from the first site of absorption (Z(1)) throughout Ka1, a first order rate
constant of absorption,
 DZ(2) = -Ka2*Z(2) is the differential equation describing the disappearing of the
drug from the second site of absorption (Z(2)) throughout Ka2, a first order rate
constant of absorption. For both equations, the minus sign indicates that the drug
disappears from their respective sites of administration.
 DZ(3) = Ka1*Z(1)+Ka2*Z(2)-K10*Z(3) is the differential equation describing the
arrival of drug in the central compartment (Z3) i.e. (Ka1*Z(1)+Ka2*Z(2)) minus its
disappearance from the central compartment i.e. (-K10*Z(3)) throughout K10, the
first order rate constant of elimination.
Any PK model can be described like that with a rather natural description of the drug
disposition.
The next step to describe the model is to qualify the so-call initial condition of the
system to tell the numerical solver how to proceed at time 0; here we know that a
fraction of the dose (noted FR) is in the first site of absorption and the rest of the
dose (noted 1-FR) in the other site thus:
Z(1) = FR*Dose
Z(2) = (1-FR)*Dose
Z(3) = 0
Where:
 Z(1) = FR*Dose indicates that a FR fraction of the dose (from 0 to 1) is located in
site 1, i.e. (Z1) at time 0
 Z(2) = (1-FR)*Dose, indicates that the rest of the dose should be located at time 0
in the second site of administration, i.e. Z (2)
 Z(3) = 0 indicates that there is nothing in the central compartment Z(3) at time 0
The number of parameters to estimate is 4 namely 'FR', 'Ka1', 'Ka2', and 'K10'.

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Now we have to write a set of statements to actually implement this model in WNL.

The following set of statements corresponds to the 2 site model written with
differential equations to be actually run in WNL.

A command file is made up of blocks of text, including a model block, and other
blocks specifying all or most of the information required to run the model.
Each user model must begin with the key word MODEL. It can be followed by a
number to identify the model, such as MODEL 1. The model ends with the command
EOM which indicates the “end of the model definition’.

1. Command block
The WinNonlin commands are a group of commands (in red) to define values such
as NPARAMETERS (the number of parameters). Here we have 4 parameters to
estimate.
PNAMES 'FR', 'Ka1', 'Ka2', 'K10' indicates the name of the 4 parameters to be
estimated (do not forget quotation marks to declare your parameters).
The NCONSTANTS command specifies the number of constants to be used in the
model. For our model there will be only 1 constant that is the dose; this constant
must be initialized via the WinNonlin interface as for a classical model.
The NDERIVATIVES command tells WinNonlin the number of differential equations
in the model (here n=3).
The NFUNCTIONS command specifies the number of different equations that have
observations associated with them; here we declared 3 because if I wish to simulate
my 3 compartments (2 sites of absorption plus the central compartment), WNL
should solve 3 functions.
END block allows WinNonlin commands to be associated with a model. Following the
COMMANDS statement, any WinNonlin command such as NPARAMETERS,
NCONSTANTS, PNAME, etc. may be given. The COMMAND section concludes with
an END statement.

2. Temporary variables block

Variables defined in the temporary block are general variables; that is, they may be
used in any block. Here I have only one temporary variable (Dose=CON(1))
meaning that I can replace CON(1) in my model by the dose (for example in the
initial condition block).
3. Starting values block
When a model is defined by a system of one or more differential equations (i.e.
NDERIVATIVES 3), the starting values corresponding to each differential equation in
the system, must be specified.

4. Function block
This group of statements defines the model to be fitted to a set of data. The letter N
(here 1, 2 or 3) denotes the function number. A function block must appear for each
of the NFUNCTIONS functions of the command block. The variable F should be
assigned to the function value. The function may be:
 an algebraic function (e.g. C=Dose/V*(ka1*F1*exp(-ka1*T)/(k-ka1) +
ka2*F2*exp(-ka2*T)/(k-ka2) + (ka1*F1*(ka2-k)+ka2*F2*(ka1-k))*exp(-k*T)/((ka1-
k)*(ka2-k))) for our algebraic model
or

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  a function of a differential equation or differential equations defined in one or
more differential blocks (e.g.: F=Z(1), or F= Z(1) etc)

5. Differential equation block

This set of statements is used to define a system of differential equations which
defines a model

MODEL
remark ******************************************************
remark Developer: PL Toutain
remark Model Date: 12-12-2009
remark Model Version: 1.0
remark ******************************************************
remark
remark - define model-specific commands
COMMANDS
NCONSTANT 1
NFUNCTIONS 3
NDERIVATIVES 3
NPARAMETERS 4
PNAMES 'FR', 'Ka1', 'Ka2', 'K10'
END
remark - define temporary variables
TEMPORARY
Dose=CON(1)
END
remark - define differential equations starting values
START
Z(1) = FR*Dose
Z(2) = (1-FR)*Dose
Z(3) = 0
END
remark - define differential equations
DIFFERENTIAL
DZ(1) = -Ka1*Z(1)
DZ(2) = -Ka2*Z(2)
DZ(3) = Ka1*Z(1)+Ka2*Z(2)-K10*Z(3)
END
remark - define algebraic functions
FUNCTION 1
F= Z(1)
END
FUNCTION 2
F= Z(2)
END
FUNCTION 3

F= Z(3)
END
remark - define any secondary parameters
remark - end of model
EOM

If you want to plot the 3 functions (e.g. with a simulation) you have to edit your data
set indicating the time etc. Alternatively, you can declare only a number of functions
equal to 1 i.e. only F=Z(3).

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This model using differential equations is easy to write but it is preferable, whenever
possible, to use the corresponding algebraic equations. In this case, using the
Laplace transform, the equation giving the plasma concentration is:

With:

This model does not exist in the WNL library and we will implement it.
The following set of statements corresponds to the 2 site model written with an
algebraic equation. The structure of the model is the same as for the preceding one; I
just add t=X as a temporary variable to tell WNL that the independent variable
(always X) can be written with a t in my equation:

MODEL
remark ******************************************************
remark Developer: Pl toutain
remark Model Date: 03-22-2011
remark Model Version: 1.0
remark ******************************************************
remark
remark - define model-specific commands
COMMANDS
NFUNCTIONS 1
NCON 1
NPARAMETERS 5
PNAMES 'ka1', 'ka2', 'k10', 'V', 'F1'
END
remark - define temporary variables
TEMPORARY
Dose=CON(1)
remark: t is the independent variable
t=X
END
remark - define algebraic functions
FUNCTION 1

F= (Dose/V)*(((ka1*F1*exp(-ka1*t)/(k10-ka1))+ (ka2*(1-F1)*exp(-
ka2*t)/(k10-ka2)) + (((ka1*F1*(ka2-k10))+(ka2*(1-F1)*(ka1-
k10)))*exp(-k10*t)/((ka1-k10)*(ka2-k10)))))
END
remark - define any secondary parameters
remark - end of model
EOM

Creating a new ASCII model in WNL

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Select User Model in the Model Types dialog and click Next. The ASCII Model
Selection dialog appears.

To create this new User Model, there are two options:

 The first option consists of writing your own model with the assistance of the WNL
dialog box that offers a template to edit. Two user model templates are provided
with WinNonlin: one for models that include only algebraic equations and one for
models that include differential equations.

 The second option consists of selecting one of the two options (differential or
algebraic) but, instead of writing the whole model yourself, you can copy the
present word document and paste it directly in WNL.

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Running the algebraic model corresponding to 2 sites of
absorption
 Using the algebraic model, estimate yourself the corresponding parameters for the
24 animals.
 Estimate l each individual curve; look at the different graphs including residuals
and then with the statistical tool of WNL, make the following tables.

The first table gives you mean parameters for the 24 animals and the second table is
the diagnostic table provided by WNL and that will be used to compare the merits of
the present model against those of the classical monocompartmental model.

Model with 2 sites of absorption, mean parameters for the two formulations:

Variabl Formulatio N Mean SD SE CV% Geometric_Mea

e n n
ka1 A 1 0.054 0.009 0.002 17.862 0.0535
2 3 7 8 1
ka1 B 1 0.054 0.009 0.002 17.824 0.0534
2 2 7 8 1
ka2 A 1 0.001 0.000 0.000 25.276 0.0016
2 7 4 1 2
ka2 B 1 0.001 0.000 0.000 24.944 0.0017
2 8 4 1 7
F1 A 1 0.660 0.069 0.020 10.504 0.6570
2 5 4 0 4
F1 B 1 0.580 0.180 0.052 31.100 0.5549
2 8 6 1 3
k10 A 1 0.081 0.019 0.005 23.697 0.0794
2 5 3 6 2
k10 B 1 0.081 0.019 0.005 23.843 0.0793
2 4 4 6 8
V A 1 1.185 0.206 0.059 17.433 1.1690
2 2 6 6 1
V B 1 1.194 0.221 0.063 18.499 1.1756
2 4 0 8 4

Diagnostic table for the model with 2 sites of absorption:

Formu Item N Mean SD Min Max

lation
A AIC 12 -42.3582 90.9333 -137.2456 47.9836
A CORR_(OBS,PRED) 12 0.9939 0.0064 0.9877 1.0000
A CSS 12 478.1272 64.4449 414.1760 627.5470
A DF 12 9.0000 0.0000 9.0000 9.0000
A S 12 0.5761 0.6022 0.0017 1.2943
A SBC 12 -39.1629 90.9333 -134.0503 51.1788
A SSR 12 5.9788 6.3681 0.0000 15.0761
A WCSS 12 478.1272 64.4449 414.1760 627.5470
A WSSR 12 5.9788 6.3681 0.0000 15.0761
A WT_CORR_(OBS,PRED 12 0.9939 0.0064 0.9877 1.0000
)

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B AIC 12 -45.4551 85.6979 -132.5252 44.3703
B CORR_(OBS,PRED) 12 0.9937 0.0066 0.9867 1.0000
B CSS 12 378.1471 174.8035 115.9450 670.3750
B DF 12 9.0000 0.0000 9.0000 9.0000
B S 12 0.4388 0.4728 0.0021 1.1376
B SBC 12 -42.2598 85.6979 -129.3299 47.5656
B SSR 12 3.5768 4.2168 0.0000 11.6466
B WCSS 12 378.1471 174.8035 115.9450 670.3750
B WSSR 12 3.5768 4.2168 0.0000 11.6466
B WT_CORR_(OBS,PRED 12 0.9937 0.0066 0.9867 1.0000
)
Corrected sum of squared observations (CSS), weighted corrected sum of squared
observations (WCSS), sum of squared residuals (SSR), weighted sum of squared
residuals (WSSR), estimate of residual standard deviation (S) and degrees of
freedom (DF), the correlation between observed Y and predicted Y, the weighted
correlation, and two measures of goodness of fit: the Akaike Information Criterion
(AIC) and Schwarz Bayesian Criterion (SBC).

Comparison of the two concurrent models:

 Comparison of the 2 modelling approaches (classical monocompartmental model
against our user model including two phases of absorption) can be done based on
different arguments (goodness of fitting, plot of residuals, and the consideration of
the Akaike Information Criterion (AIC)).
 AIC is a measure of goodness of fit based on maximum likelihood. When
comparing several models for a given data set, the model associated with the
smallest AIC is regarded as giving the best fit. AIC is appropriate only for
comparing models that use the same weighting scheme (no weighing here).
 AIC = -N log (WRSS) + 2P for modelling in WinNonlin. N is the number of
observations with positive weight. WRSS is the weighted residual sum of squares.
P is the number of parameters that has been estimated.

Questions:
 What are your conclusions?
 Why is the terminal half-life estimated to be 8.51h (0.693/k10) while the classical
monocompartmental model gives an estimate of 16-19 h? Is it a flip-flop
phenomenon?

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 Deconvolution analysis
Overview
Deconvolution is primarily used in PK to obtain the input function, i.e. the rate (µg/h
or µg.kg-1/h) of drug entry into the central compartment.
Deconvolution is useful to reveal in vivo drug release from a given pharmaceutical
form or in our case the delivery from the 2 sites of absorption based on data for a
known drug input (generally an IV administration).
The integral of the input function gives the bioavailability.
Deconvolution based bioavailability estimation methods are a more general approach
to estimate bioavailability than conventional methods since the former provide
estimates of both the rate of systemic uptake and the total systemic dose.
Estimated rates of bioavailability may potentially be used to:
 gain insights into the mechanisms of uptake
 enable better predictions of bioavailability under varying conditions

Technically, deconvolution is a technique that can be used to estimate an input

function, given the corresponding input-response function (our data to analyze)
and the impulse-response function (concentration profile following an IV bolus
dose) for the system.
Deconvolution is the inverse of convolution.

Literally, if we know the disposition curve from a unit IV dose (bolus) and the rate of
input of the drug into the plasma (input rate function) then we can calculate the
plasma disposition curve from that input by convolution.

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Deconvolution is the inverse problem, i.e. to determine the input function from the
plasma disposition curve given the unit disposition function, i.e. to determine the
input function f(t), given the unit impulse response Cδ(t) and the input response C(t).

Key assumptions for deconvolution are:

1) linearity: f(D1+D2) = f(D1) + f(D2)
2) time invariance: f(D) has the same shape no matter when D is given.

We will illustrate these concepts with our example:

 Open WNL with our data sheet.
 Select Deconvolution from the Tools menu:

The Deconvolution dialog appears:

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Then drag the appropriate variables to the Time and Concentration fields, drag
formulation and animals to the Sort Variables field. A separate analysis is done for
every unique combination of sort variable values.

Enter the dosing units in the Dose units field (here µg). The dosing units are used
only when the input data set has units associated with the Time and Concentration
data (ng/mL). The administered dose was 100µg/kg.

 In the Settings fields, select the number of exponential terms (N) in the unit
impulse response (here it is 1 because it is a mono-exponential model)
 Enter the values for A and alpha in the unit impulse response function (see below)

The IV bolus (unit impulse response) after a 100 µg/kg bolus dose was described by
the following equation:

With C(t) in ng/mL thus A=800 ng/mL and k10=0.1 per h

For this deconvolution, we have to enter a scaled A, i.e. A/100µg=0.8 and alpha is
k10=0.1
 Select the setting for Smoothing to be used. Automatic should be selected if you
want the program to find the optimal value for the dispersion parameter delta. This
is the default.
 Select the Initial Change in Rate is Zero check box to constrain the derivative of
the estimated input rate to be zero at the initial time (lag time).

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 Click Calculate. Deconvolution generates a new chart and workbook.
Deconvolution generates three charts for each of the sorted keys.
 The Fitted Curve output plot depicts concentration data from the input data set
plotted against time for animal #21, the curve shows a shoulder in the ascending
phase. This is the deconvoluted curve.

The next figure gives the Input Rate (µg/h) plot depicting the rate of drug input
against time for each profile. Visual inspection of this figure shows clearly the
presence of two peaks in the input rate describing a rapid and a slow absorption
process.

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The next figure is the Cumulative Input plot that displays cumulative drug input (in
µg/h) against time for animal 21. Visual inspection of the curve clearly shows that
there is a biphasic absorption explaining the initial shoulder seen on the observed
curve.

The deconvolution workbook contains one workbook with different worksheets. The
values worksheet includes values for Time, Input Rate, Cumulative Input and
Fraction Input.

Inspection of the results for animal 1 shows that the total amount of absorbed drug
was 107 µg/kg and that 50% of the drug has been absorbed after a delay of 15.84h.

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It should be stressed that all the animals do not show such a nice profile as for
animal 21.

Conclusion

Deconvolution is a powerful tool to document a process of absorption. Convolution is

also extensively used in industry for in vitro/in vivo (IVIV) extrapolations.

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