Pharmacology & Pharmacy, 2010, 1, 42-52
doi:10.4236/pp.2010.12007 Published Online October 2010 (http://www.SciRP.org/journal/pp)
Rational Drug Delineation: A Global Sensitivity
Approach Based on Therapeutic Tolerability to
Deviations in Execution
Denis Goue Gohore1, Frédérique Fenneteau2, Olivier Barrière1, Jun Li1,3, Fahima Nekka1,3
1Faculté de Pharmacie, Université de Montréal, Montréal, Canada; 2Pharsight, A certara company, Montréal, Canada; 3Centre de
Recherches Mathématiques, Université de Montréal, Montréal, Canada.
Received August 25th, 2010; revised September 10th, 2010; accepted September 20th, 2010.
ABSTRACT
Noncompliance to therapeutic regimen is a real public health problem with tremendous socioeconomic consequences.
Instead of direct intervention to patients, which can add extra burden to the already overloaded health system, alterna-
tive strategies oriented to drugs’ own properties turns to be more appealing. The aim of this study was establish a ra-
tional way to delineate drugs in terms of their “forgiveness”, based on drugs PK/PD properties. A global sensitivity
analysis has been performed to identify the most sensitive parameters to dose omissions. A Comparative Drug For-
giveness Index (CDFI), to rank the drugs in terms of their tolerability to non compliance, has been proposed. The index
was applied to a number of calcium channel blockers, namely benidipine, nivaldipine, manidipine and felodipine. Using
the calculation, benedipine and manidipine showed the best performance among those considered. This result is in ac-
cordance with what has been previously reported. The classification method developed here proved to be a powerful
quantitative way to delineate drugs in terms of their forgiveness and provides a complementary decision rule for clini-
cal and experimental studies.
Keywords: Compliance, Drug Forgiveness, Global Sensitivity Analysis, Comparative Drug Forgiveness Index,
Monte-Carlo
1. Introduction
Compliance has been referred to as a dimensionless,
blanket concept encompassing the extent to which pa-
tients’ drug dosing histories conform, or not, to pre-
scribed drug dosing regimen, in terms of both persistence
and quality of execution compliance [1]. As a human
behaviour, the patient compliance has an intrinsic com-
plex nature which is in part responsible for the gap ob-
served between the abundance of descriptive research
and the shortage of quantitative tools. The tendency of
the public health care system towards reducing hospi-
talization costs along with the increase in more powerful
self-administered drugs, call for efficient evaluation
methods to capture the multidimensional character of
compliance and evaluate its clinical impact [2]. When
dealing with adherence-related problems, the most spread
practice is to enhance patients’ adherence through inter-
vention programs. This interactive approach, when suc-
cessful, has proved to be beneficial for all the involved
parts, including the patient, health care givers as well as
the pharmaceutical industry. However, this individual-
ized approach can easily become a burden for the health
system, with too many aspects involved in the manage-
ment of the patient’s adherence [2] Recent efforts are
more focused on the development of objective ways for
compliance control and improvement. Remarkable ad-
vances in this important therapeutic-related area have
been achieved, as reported in the review paper of Düsing
[3]. In fact, work on compliance can be viewed from
different angles. One can address the quantitative rela-
tionship of drug intake with its therapeutic outcomes, or
alternatively looks for solutions to reduce the negative
impact of poor compliance. The underline of the former
aspect relies on the direct link of compliance to therapy.
The latter however considers minimizing the impact of
poor compliance upstream, putting emphasis on drugs
and their pharmacokinetic and pharmacodynamic (PK/

Rational Drug Delineation: A Global Sensitivity Approach Based on Therapeutic Tolerability to Deviations in Execution 43
PD) properties, with the intention to compare drug toler-
ability to changes in drug execution. This has led to the
concept of “drug forgiveness” which is formally defined
as the drugs post-dose duration of action minus the pre-
scribed dosing interval [4]. An early molecule screening
procedure, targeted to prioritize flexible drugs in terms of
their forgiveness, is important in drug research and de-
velopment for enhancement of the quality of therapy and
reduction in costs. Indeed, this procedure could add a
(market) value to those drugs having the least sensitive
profile to irregular drug intake. The implication of com-
pliance in pharmaceutical value has been previously
highlighted by Urquhart [4].
Modeling and simulation approaches have become an
integral part of the biopharmaceutical research, encom-
passing all aspects of the critical path of drug develop-
ment and evaluation, including adherence studies. Many
papers have focused on modeling human behaviour in
relation to treatment recommendations [5-11]. Others
have tried to understand the complex relationship be-
tween adherence, exposure and therapeutic response to a
treatment [8,12-14]. In this paper, a modeling and simu-
lation strategy based on sensitivity analysis to classify
drugs according to their degrees of forgiveness was de-
veloped. It is based on the control of uncertainties in
drug-related information that aims at ranking drugs in
terms of their tolerance to dose omissions. The design of
this classification procedure uses a recently developed
global sensitivity analysis strategy, involving the Partial
Ranked Correlation Coefficient (PRCC) method [15-19].
In this method, a family of calcium channel blockers was
used in which four of them were chosen as drug models
exhibiting a large spectrum of PK/PD properties, namely
benedipine, nivaldipine, manidipine and felodipine.
This paper is organized as follows. In Materials and
Methods, we describe our modeling approach and the
global sensitivity analysis that will be used here and we
define the comparative drug forgiveness index and ex-
plain how it can be used to classify drugs in terms of
drug forgiveness. In Results, we present the results of
drug classification in terms of their PK and PD properties
and analyse its robustness for various compliance mod-
els.
2. Materials and Methods
2.1. The General Approach
In this study, the approach relied on building a combined
model composed of three sub-models describing one-by
one and in a chronological way, drug intake, drug dispo-
sition through the pharmacokinetics and the relationship
between pharmacokinetics and pharmacodynamics as
shown in Figure 1. The sensitivity analysis for the model
parameters that were likely to carry out the most infor-
mation on therapeutic effect in response to dose omis-
sions for different dosing regimens was performed. For
this, two clinical compliance indices that translate the
impact of patient compliance in therapeutic outcomes
were introduced. A global sensitivity analysis was per-
formed to determine the coefficient of correlation (CC)
between the PK and PD parameters for a given compli-
ance index. These CCs were then converted into transi-
tory scores that were used to estimate Comparative Drug
Forgiveness Index (CDFI) that were used to classify
drugs having similar pharmacological mechanisms. As
an application of this approach, four long acting calcium
channel blockers with various compliance situations
were studied. Compliance scenarios were generated
through three modeling approaches, namely, 1) Markov
chain, 2) fixed percentages of taken doses and 3) differ-
ent cases of drug holidays.
2.2. Model Components
2.2.1. Compliance Model
Several modeling approaches were used to simulate pa-
tients’ drug intake. They include Markov Chain compli-
ance model, drug holidays compliance model and Fixed
Figure 1. Conceptual model describing the three linked
drug intake-PK-PD components, with the receptor-binding
models describing the drug effect.


1
a
Kh
is the absorp-
tion rate constant,


1
12
Kh
 and

1
21
Kh
 are the trans-
fer rate constants from the central compartment to the pe-
ripheral compartment, and from the peripheral compart-
ment to the central compartment, respectively;


1
e
Kh
 is
the constant of elimination of drug from the central com-
partment,



1
on
Kngh

 is the second-order association
rate constant and


1
off
Kh
 is the first-order dissociation
rate constant.

Rational Drug Delineation: A Global Sensitivity Approach Based on Therapeutic Tolerability to Deviations in Execution 44
percentage of taken doses compliance model. The Mar-
kov chain-based approach is the most reported one that is
used to simulate compliance from real data [20].
2.2.1.1. Markov Chain Compliance Model
Markov chain is a mathematical tool used to predict fu-
ture states from the current ones. In the context of com-
pliance, it was assumed that there are three possible dose
states at nominal times: omitted dose (0), one taken dose
(1) or a double dose (2). Transitions between these states,
from one nominal time to the next, were represented by a
3 × 3 transition matrix P, with each (i,j) element, noted
ij
p, corresponding to the transition probability from dose
state i to dose state j. Hence, if we use a 3-dimensional
vector

012
,,ppp  , where i p are probabilities for
the dose states i, i = 0, 1, 2 with 012 1ppp  to note
the current state, then the next dose state probabilities are
expressed by:

0001 02
012 101112
2021 22
,, .
ppp
ppppp p
ppp






P (1)
In this study, transition matrix estimated by Sun et al.
from data collected from 177 patients following an HIV
clinical trial study was used. [21]. This matrix is:
0.23 0.58 0.19
0.12 0.81 0.07.
0.14 0.75 0.11





P (2)
To mimic realistic compliance scenarios, dosing inter-
vals with normal distributions are used, where average
dosing intervals and standard deviation are set to μ = 24
h and σ = 12 h, respectively. Using the approach de-
scribed in [6], the generated negative values are truncated
and replaced with an arbitrary chosen small time length
(0.01 h in our case) and are assigned to double doses. A
typical dosing history is illustrated in Figure 2.
2.2.1.2. Drug Holidays Compliance Model
‘Drug holidays’ have been proved to be relevant to
therapeutic outcomes. Defined as drug omissions over
three successive days or more [9], they are reported to
occur during weekends and in special events such as
travel periods.
2.2.1.3. Compliance Model with Fixed Percentages of
Taken Doses
The percentage of taken doses is the traditional cut-off
used to classify patient compliance. It is commonly ac-
cepted that a patient who has taken at least 80% of pre-
scribed doses is a ‘perfect’ compliant.
2.2.2. Pharmacokinetic Model
A two-compartmental PK model with first-order absorp-
Figure 2. Illustrative example of dosing history. A single
circle represents one taken dose, a double circle is for a
double dose while a cross symbol is for an omitted dose.
tion and elimination for the calcium channel blockers
was considered in the disposition model [22-24]. The
disposition model was then linked to the compliance
model through the gastro-intestinal tract as illustrated in
Figure 1. The PK model was given by the following sys-
tem of equations:
D
aD
dA QK A
dt

 (3)

12122
12 1
11
aD
e
KFA dCKC V
K
KC
dt VV
 

 (4)
21211
21 2
2
dCKC V
K
C
dt V


 (5)
where
D
A
(mg) was the absorbable amount of drug in
the gastro-intestinal tract, Q(mg) was a time-dependant
function of drug intake determined by compliance model ,
1
C (mg/L) and 2 C (mg/L) were the central and periph-
eral concentrations, respectively, 1 V (L/kg) and 2 V
(L/kg) were the apparent central and peripheral volumes
of distribution, respectively. The other parameters were
as in Figure 1. For sake of simplicity, bioavailability F
was assumed to be 1.
2.2.3. PK/PD Model
2.2.3.1. Drug Class
Calcium channel antagonists are largely used for the ma-
nagement of various cardiovascular diseases including
hypertension. It has been reported that there is a direct
link between blood pressure and compliance to these
drugs, where over 37% of hypertensive patients that un-
dergo treatment, are stated non compliant [25].

Rational Drug Delineation: A Global Sensitivity Approach Based on Therapeutic Tolerability to Deviations in Execution 45
2.2.3.2. Pharmacodynamic Model
The calcium channel blockers bind to calcium channels
to limit the entry of calcium into the vascular and cardiac
smooth muscles thus preventing muscle contractility. The
intensity and duration of their action depend on their
ability to dissociate at the target site. The indirect PK/PD
model to characterize the effect of these drugs was used
[22]. The effect E(mHg) can be modeled as:

maxon off
dE
K
EEKE
dt   (6)
where

max
EmHg
, Kon and Koff are the maximum
effect, the association and dissociation rate constants,
respectively.
2.3. Sensitivity Analysis of PK/PD Parameters
A global sensitivity analysis (GSA) of the combined mo-
del was applied to identify input parameters suspected to
have determinant role on compliance indices under in-
vestigation. GSA is a probabilistic approach used to de-
termine the sensitivity of the model outcomes to the
variation of input parameters [26]. Using this approach,
possible input parameter values were simulated and sta-
tistically analysed according to their distribution func-
tions and possible correlations. GSA has recently been
introduced to analyse the physiological based pharma-
cokinetic models [15,16].
The following terms were used in this work: the ‘input
parameter ,
i
X
refers to one of the involved pharma-
cokinetic and pharmacodynamic parameters, and the
‘output variable ,
j
Y to the model response.
2.4. Input Parameters
Nine parameters were investigated using GSA, namely:
a
K
, e
K
12
K
, 21
K
, 1 V, 2 V, on
K
, of f
K
and max E.
Based on the statistical description of input parameters
for calcium channel inhibitors given in Table 1 [22,27,28]
Monte Carlo approach was used to generate a large
number (N = 1000) of drugs, each having a specific vec-
tor composed of m = 9 input parameters (i.e., a matrix of
m × 1000). These input parameters were used to generate
the corresponding output parameters.
2.5. Output Parameters
Two important compliance indices relevant to anti-hyper-
tensive therapy were chosen:
2.5.1. Number of Subtherapeutic Days (SD)
A patient was considered to have subtherapeutic plasma
drug concentrations if the systolic blood pressure devi-
ated by ΔE = 20% from the expected value if the patient
was a perfect compliant. Using this well accepted clinical
criterion [32], the number of SD following the calcula-
Table 1. Statistical description of input parameters assumed
to be log-normally distributed.
Parameters Mean Std IC


1
a
K
h 0.80 0.37 0.30-3.20


1
e
K
h 0.36 0.13 0.10-1.20


1
12
K
h 0.14 0.12 0.02-1.00


1
21
K
h 0.10 0.10 0.008-0.82


1
VL 992.20 550.90 359.10-2,739


2
VL 5,758 3,770 2,788-18,563


max
EmHg 27.00 8.25 15.00-48.00


1
.
on
K
ng h0.70 0.90 0.05-3.00


1
off
K
h 0.36 0.50 0.01-5.00
tion of ΔE for different compliance scenarios and input
parameters was estimated. A higher SD indicates that the
treatment success can be jeopardized.
2.5.2. Smoothness Index (SI)
SI is used to assess the fluctuation in blood pressure
driven by the drug or treatment regimen. Clinically, this
index indicates the homogeneity of blood pressure reduc-
tion induced by antihypertensive drug treatment over the
24 hours. A large variation in blood pressure (low SI) is
likely to trigger organ damage, in comparison to a higher
SI that indicates a smooth blood pressure [29-34]. Hence,
an SI decrease can raise therapeutic concerns.
The smoothness index was obtained as:
D
H
D
H
m
SI

 (7)
where
D
H
m and
D
H

are the mean and standard de-
viation of systolic blood pressure calculated for a same
individual, respectively.
2.6. Input-Output Correlation
In order to identify the important parameters and quan-
tify their influence on model outcomes, the correlation
(CC), rank correlation (RCC), partial correlation (PCC)
or partial rank correlation (PRCC) coefficients were cal-
culated according to the linearity or monotonicity prop-
erties of the input-output relationship, as well as to the
correlation between input parameters [16]. In this study,
nonlinear but monotonous relationships were observed
between some input and output parameters, justifying
thus the use of RCC. To take into account the possible
correlation between input parameters, the partial rank
coefficients of correlation (PRCC) between an input pa-

Rational Drug Delineation: A Global Sensitivity Approach Based on Therapeutic Tolerability to Deviations in Execution 46
rameter i
X
and an output parameter Y was calculated
as follows:

1
11
,iY
i
ii YY
C
PRCC YX
CC


 (8)
where 1
C is the inverse matrix of C:
121 1
212 2
12
12
1
1
1
1
1
NY
NY
T
NN NY
YY YN
rrr
rrr
A
B
CB
rr r
rr r

















(9)
where A was the input parameters correlation matrix with
elements ij rRCC and B was the input-output
correlation vector with elements
j
Y
r. A positive PRCC
value indicates that the output parameter increases with
the input parameter, and vice versa.
To understand the relative determinant roles of input
parameters on the effect of dose omission, the score (i SC )
was defined from the input-output PRCC value as fol-
lows:



1
,
,
i
im
i
i
PRCC YX
SC
PRCC YX


 (10)
where i = 1,2,...m.
Once the estimated PRCC values and scores were ob-
tained using N=1000 simulated drugs, the results were
used for the classification of n chosen drugs in terms of
their forgiveness. The classification process can be direct
if a single parameter emerges as the most sensitive one.
However, it is possible that more than one parameter
were identified as important, for which case a more deli-
cate criterion, based on the scores, to delineate drugs
forgiveness was developed.
2.7. Drug Forgiveness Estimation: Comparative
Drug Forgiveness Index (CDFI)
The defined scores to compare n drugs in terms of their
forgiveness were illustrated. For this, the concept of
Comparative Drug Forgiveness Index (CDFI) was intro-
duced and calculated for the n considered drugs from the
class of calcium channel blockers.
For each drug, CDFI was directly computed from its
PK and PD parameters and the predetermined scores of
the corresponding pharmacological class. This made
CDFI an accessible method easily applicable in practice
for drug forgiveness classification.
2.8. Calculation of CDFI
Assume m PK/PD parameters were used for each of the n
considered drugs; each parameter was represented by a
vector


12
,,
iii in
X
XXX, i = 1, 2, …, m, with each
component corresponding to the i-th PK/PD parameter of
one drug. Y was the n-ry vector of the corresponding
compliance index, namely SD and SI. i SC is the score
of i X defined by Equation 10.
Depending on


,i PRCC Y X values and considering
that an increase of a given compliance index positively or
negatively influences the therapeutic outcome, the for-
giveness index ij
F
was defined to measure the relative
performance of the i-th parameter ij
X
of the j-th drug in
terms of drug forgiveness.
To calculate ij
F
, the drug index 0 j for which 0
ij
X
indicates, in terms of drug forgiveness, the worst per-
formance among ij
X
, j = 1, 2, …, n was first determined.
The different cases are summarized as follows:
1) An increase in compliance index negatively influ-
ences the therapeutic outcome (e.g., an increase in SD
negatively influences blood pressure control)
a) If


,0
i
PRCC YX,

0
j
ij
jargmaxX and we
let 0
ij i
F
SC
b) If


,0
i
PRCC Y

X,

0
j
ij
jargminX and we
let 0
ij i
F
SC
2) An increase in compliance index positively
influences the therapeutic outcome (e.g., an increase in SI
positively influences blood pressure homogeneity)
a) If


,0
i
PRCC YX,

0
j
ij
jargminX and we
let 0
ij i
F
SC
b) If


,0
i
PRCC Y

X,

0
j
ij
jargmaxX and we
let 0
ij i
F
SC
Hence the forgiveness index ij
F
for the i-th parame-
ter of the j-th drug was defined as follows:
0
0
0
0
0
if
if
ij
ij ij
ij
ij
F
jj
XF Fjj
X






(11)
Table 2 illustrates a simplified diagram for the com-
putation of the forgiveness index.
Table 2. A simplified diagram of the forgivness index calculation


 indicates an increase (decrease).
Terapeutic OutcomeY Terapeutic OutcomeY
(,) 0
i
PRCC YX


/min
ijijij ij
F
XSC X



/max
ijijij ij
F
XSC X
(,) 0
i
PRCC YX 


/max
ijijij ij
F
XSC X


/min
ijijij ij
F
XSC X

Rational Drug Delineation: A Global Sensitivity Approach Based on Therapeutic Tolerability to Deviations in Execution 47
3) For the j-th drug, j = 1, 2, …, n, the Comparative
Drug Forgiveness Index (CDFI) was defined by
summing its individual forgiveness indices ij
F
:

1
m
ij
i
CDFI jF



(12)
4) Finally, a ranking of drugs was based one their
CDFI values, where a higher CDFI indicates a better
drug forgiveness.
2.9. Application of CDFI
To evaluate the relevance and robustness of the approach
defined here for the evaluation of drugs in terms of their
forgiveness, CDFI for four long-action calcium channel
blockers, namely benidipine, nivaldipine, manidipine and
felodipine were calculated. Concentration and blood pre-
ssure data used in the study were obtained from lit-
erature [22,27,28] and were used for each of these drugs.
The two compartment model generally adopted for these
drugs in literature was used to estime the relevant PK
and PD parameters using WinNonlin software package
(Pharsight Corporation, Mountain View, CA, USA). The
estimated values of PK and PD parameters were sum-
marized in Table 3.
As observed in Table 3, large differences in PK and
PD properties exist between the four calcium channel
inhibitors investigated, assuring thus the robustness of
the approach. Upon these disposition and effect models,
the previously mentioned three compliance models were
applied to these drugs to generate data for the assessment
of their forgiveness.
3. Results
3.1. Exploratory Analysis of Input-Output
Relationships
To check for the monotonicity in the input-output rela-
tionships, scatter plots representing the input parameters
vs the model output SI were displayed in Figure 3. For
each of the m = 9 input parameters N = 1000 copies were
simulated.
Linear trends as well as nonlinear ones were displayed
by these pairs. The monotonicity exhibited by these lin-
ear and nonlinear relationships justified our use of the
rank coefficient of correlation (RCC) approach or the
partial rank coefficient of correlation (PRCC). The latter
was identified as the most appropriate and powerful
method when parameters were correlated. In Figures 4
and 5, results of the RCC and PRCC of each input pa-
rameters with outcome parameters, SD and SI, when
neglecting or not the correlations between various input
parameters, respectively, are shown. Difference parame-
Table 3. PK and PD parameters of the four long-action
calcium channel inhibitors; Data from Shimada and al.,
Kirsten and al. [22,27,28].
ParametersBenedipineNivaldipine Manidipine Felodipine
PK


1
a
K
h 1.33 0.67 0.63 0.51


1
e
K
h 0.50 0.42 0.32 0.20


1
12
K
h1.07 0.11 0.20 0.27


1
21
K
h0.73 0.01 0.16 0.013


1
VL 1323.20 562.23 1449.90 465.20


2
VL 3045.70 6044.50 1833.10 9402.40
PD


max
E mHg23.40 36.92 17.63 29.62


1
.
on
K
ng h1.26 0.143 0.32 0.54


1
off
K
h0.012 0.37 0.13 0.21
Figure 3. Correlation between PK and PD input parameters
and SI.
ters rankings were obtained using these two approaches.
For example, RCC identified e
K
, 12
K
, off
K
as the
most important parameters for SD, whereas off
K
, on
K
and 21
K
were those identified by PRCC. The result
rationalizes the choice for the PRCC approach in this
study.
The PRCC values in Figure 4 indicate that both output
parameters, SD and SI, were sensitive to off
K
, and at a
less extent to on
K
and 21
K
. The latter parameter be-
longs to the PK model while the two others to the PD
model. While the ranking of these three parameters was
preserved, the sign of correlation was reversed as ex-

Rational Drug Delineation: A Global Sensitivity Approach Based on Therapeutic Tolerability to Deviations in Execution 48
Figure 4. Coefficients of correlation between input and out-
put parameters when possible input correlations are ac-
counted for.
Figure 5. Coefficients of correlation between input and out-
put parameters when input correlations are ignored.
pected as a consequence of the SD and SI definitions.
From a clinical point of view, given two or more calcium
channel blockers, this implied that drugs with smaller
off
K
, 21
K
, and larger on
K
were preferred in terms of
drug forgiveness. However, these conditions could be
rarely satisfied for one drug at the same time, which led
us to the development of a criterion for more general
cases.
3.2. Choice of the Most Forgiving Drug Based on
the Sensitivity of PK or PD Parameters
Three input parameters were identified as the most im-
portant ones, on
K
and off
K
were PD related while
21
K
was PK related. In terms of dose omission, ni-
valdipine was classified as the most forgiving drug if the
delineation procedure is solely PK-based, while benedi-
pine was the least forgiving one, as shown in Table 3.
The conclusion may be reversed if the procedure was
PD-based. Thus, a sensitivity analysis solely based on the
PK properties while ignoring the PD component (and
vice versa) could lead to erroneous classification of drugs.
This result confirms the need to take into account as
much PK/PD properties as possible for a proper charac-
terization of drug tolerability to dose omissions.
3.3. Use of CDFI to Test Tolerability to Dose
Omissions
The CDFI approach was applied in this study to classify
the four calcium channel blockers in terms of their for-
giveness to dose omissions. The PK/PD parameters of
these four drugs (Table 1) were in the range of the Monte
Carlo generated PK/PD parameters (Table 1), validating
thus the use of PRCC method and consequently CDFI.
Since a sensitivity analysis was based only on PK or
PD parameters, it cannot fulfill the task of classification,
and therefore the CDFI approach was performed on these
drugs as shown in Table 4.
For both output parameters, benidipine showed the
highest CDFI, which means it holds for the longest effec-
tive therapeutic period and causes the least inhomogene-
ity in blood pressure. It was followed by manidipine for
SD and by felodipine for SI.
3.4. CDFI Classification versus Direct
Classification
Compared to direct classification approaches based on
therapeutic markers, which require specific simulations
for each drug, the advantage of CDFI was obvious.
Moreover, CDFI was computed with the same compli-
ance scenario to classify a whole pharmacological class,
having a wide range of PK and PD properties. However,
it was important to ensure the robustness of CDFI classi-
fication by considering different compliance patterns
using direct classification approaches.
In this study, two therapeutic markers, namely SD or
SI, can be used to study the performance of CDFI for the
four chosen blockers by considering the three compliance
scenarios above.
3.5. Compliance Scenario Using Markov Chain
Model
In this study, 500 drug intake profiles were simulated for
each drug using Markov chain approach to analyse the
impact of drug intake irregularity on the therapeutic out-
come. Table 5 shows the values of several therapeutic
markers that we calculated or extracted from the literature.

Rational Drug Delineation: A Global Sensitivity Approach Based on Therapeutic Tolerability to Deviations in Execution 49
Table 4. Comparative drug forgiveness index for four long-
action calcium channel drugs.
Benedipine Nivaldipine Manidipine Felodipine
CDFI(SD) −10 −47 −27 −58
CDFI(SI) 51 −3.8 10 16
Table 5. Values of compliance markers obtained after simu-
lation and experimental data.
Therapeutic
marker Benedipine Nivaldipine Manidipine Felodipine
SD (h) 1.3 70.6 17.5 80.45
SI 8 0.13 0.59 1.13
Experimen-
tal SI 2a - 0.6b 1
c
aData from Nakajima and al. [32]; bData from Roca-Cusachs and al. [34];
cData from Mancia and al. [31]
In terms of SD, benedipine, with the least therapeutic
time of 1.3 h, had the best forgiveness. Manidipine was
ranked second with SD = 17.5 h, while felodipine had the
worst forgiveness with an SD of 80.45 h. When it comes
to SI, benidipine and felodipine with SI value of 8 and
1.3 respectively, manifest their fine quality in the control
of harmful fluctuations in blood pressure compared to the
other two drugs. These results were consistent with those
found using CDFI (Table 4). Moreover, simulated SI
was in accordance with experimental SI reported in lit-
erature from real data, which was a strong indication of
the suitability of this study.
3.6. Noncompliance Based on Drug Holidays
Different scenarios based on the number of drug holidays,
going from 1 to 15 times, each lasting exactly three days
were explored. In this method, the perfect compliance
was simulated into which a number of occasional drug
holidays were used. Figure 6 shows the relationship be-
tween SD, i.e., number of subtherapeutic days, and num-
ber of drug holidays for different fixed systolic blood
pressure deviations ΔE. With nivaldipine, the percentage
of subtherapeutic days exceeds 10% after five drug holi-
days for ΔE = 0.2. However, when ΔE = 0.1, only
benedipine showed an SD under 10%. It was noted that
benedipine showed a better tolerance for drug holidays
compared to the other drugs.
In Figure 7, the evolution of SI for each drug versus
the number of drug holidays was shown. Benedipine had
the largest SI decreasing ratio compared to other drugs.
However, SI for nivaldipine and felodipine were almost
not altered by drug holidays. This indicates that drug
Figure 6. Number of days that a patient is inefficiently
treated vs. number of drug holidays. ∗ = Benedipine; ⋄ =
Manidipine; × = Felodipine and ○ = Nivaldipine.
Figure 7. Profile of smoothness index vs. number of drug
holidays. ∗ = Benedipine; ⋄ = Manidipine; × = Felodipine
and ○ = Nivaldipine.
omission had almost no influence on blood pressure
fluctuation. Therefore, the difference between perfect
and poor compliers in terms of organ damage induced by
drug holidays can be neglected during hypertension
treatment. These results were consistent with the CDFI
classification (Table 4).
3.7. Noncompliance Based on Percentage of
Taken Doses
For a fixed total dose, scenarios were simulated with an
increasing percentage of taken doses, ranging from 10%
to 100%. For each percentage of taken doses, percentage
of subtherapeutic days (SD) was calculated; the results
are shown in Figure 8. Similar to the compliance model

Rational Drug Delineation: A Global Sensitivity Approach Based on Therapeutic Tolerability to Deviations in Execution 50
based on drug holidays, benedipine had a better forgive-
ness for dose omission, followed by manidipine. The
other two drugs showed very poor forgiveness.
However, in terms of SI, benedipine changed more
rapidly against percentage of taken doses than other
drugs (Figure 9), which suggests a high risk of organ
damages for non compliant patients during hypertension
treatment. The above results were again consistent with
the CDFI classification (Table 4).
4. Discussion
Many therapeutic strategies consider the issue of drug
compliance as crucial for a treatment to be efficient. In
Figure 8. Number of days, the patient is inefficiently treated
vs. percentage of taken doses. ∗ = Benedipine; ⋄ = Manidi-
pine; × = Felodipine and ○ = Nivaldipine.
Figure 9. Profile of smoothness vs. percentage of dose taken.
∗ = Benedipine; ⋄ = Manidipine; × = Felodipine and ○ =
Nivaldipine.
this context, two options can be put forward under dif-
ferent philosophies, one is centered around the patient
while the other targets the patient drug use. The first
builds on the interactive synergy between the health
caregiver and the patient, with the ultimate goal of the
patient to be an integral part of drug execution. Since this
approach heavily depends on the patient willingness and
collaboration, it can be time consuming and provides no
guarantee of improvement in patient drug intake. This
has led to consider alternative drug-based strategies, with
attempt to favor drugs that are more tolerable to the ir-
regular drug intake, thus reducing the risk for therapeutic
failure. This is particularly relevant for specific popula-
tions where compliance to medication has proved to be
poor. The drug forgiveness issue has been raised by
Urquhart [35], and suggested as an additive criterion in
the drug evaluation. In the drug selection process, the
characterization of PK/PD properties is crucial. Restric-
ted by clinical and ethical conditions, a M&S approach can
play a major role for this purpose. A step towards this di-
rection has been made by Nony and coworkers [36].
In this study, a global sensitivity analysis (GSA) ap-
proach was adopted, that considers a whole package of
PK/PD parameters and quantifies their roles on therapeu-
tic outcomes in terms of compliance. For this, a previ-
ously developed GSA method was used and aimed at
identifying important input parameters and quantify their
influence on drug distribution in different tissues [15].
GSA incorporates the correlations between input pa-
rameters in the quantification of their influence on the
model outcomes.
This approach for a PK and PD model was related to a
compliance model. As a case-study, four calcium channel
blockers with different PK/PD parameters have been
tested. Using those PK and PD parameters identified as
important by GSA, the drug classification in terms of
forgiveness can be different whether the PK and/or PD
models have been included or not. This indicates that
considering the PK/PD parameters as independent vari-
ables can lead to erroneous conclusions. For instance, for
drugs acting through direct effect model, the one having
a very long elimination half-life is considered more for-
giving. However, for drugs acting through indirect model,
the one with the longest elimination half-life cannot be
automatically judged as the most forgiving since the dis-
sociation rate constant may also have an influence on the
length of drug effect.
For the calcium channel blockers considered in this
paper, the GSA approach reveals that the high degree of
benedipine forgiveness is in part related to PD properties
(off
K
and on
K
), while it can be more linked to PK
properties (21
K
) for the case of felodipine.

Rational Drug Delineation: A Global Sensitivity Approach Based on Therapeutic Tolerability to Deviations in Execution 51
For the same pharmacology class drugs, it is difficult
to classify drugs in terms of their forgiveness when sev-
eral parameters simultaneously influence the drug re-
sponse expressed through compliance indices. Our study
raises the issue of the validation of those studies involv-
ing the sole knowledge of PK or PD components without
consideration of the whole drug intake-PK/PD process.
In this work, the GSA-based CDFI approach, which
takes into account the unavoidable and complex rela-
tionship between these three components, can be used as
a reasonable tool for the classification of drugs in terms
of their forgiveness.
The results obtained in this study are reassuring and
support the relevance of CDFI approach. Indeed, the
drug forgiveness classification is consistent with the
clinical results, which confirm the efficacy and long-
action effect of the benedipine and manidipine compared
to other calcium channel antagonists [37,23,34]. More-
over, simulated SI are also close to clinical data (see Ta-
ble 5), indicating that benedipine induces high homoge-
neity of blood pressure, followed by felodipine and mani-
dipine [30-34].
This work, which uses for the first time the global sen-
sitivity analysis to compare drugs in terms of their for-
giveness is a step forward towards a strategy that favor
drugs that are more tolerable to deviations in drug execu-
tions.
5. Conclusions
In this work, a global sensitivity analysis has been per-
formed to identify the most sensitive parameters to dose
omissions. A Comparative Drug Forgiveness Index
(CDFI), designed to rank drugs in terms of their toler-
ability to non compliance, has been proposed. The classi-
fication results are in accordance with what has been
previously reported for the calcium channel blockers.
The classification method developed here proved to be a
powerful quantitative way to delineate drugs in terms of
their forgiveness and provides a complementary decision
rule for clinical and experimental studies.
6. Acknowledgements
This work has been supported by the NSERC and FQRNT
grants held by Dr. Fahima Nekka. The Mathematical
Centre of Excellence (MITACS) and Ivory Coast Repub-
lic are also acknowledged for their support.
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