A general pharmacodynamic
interaction (GPDI) model
Sebastian Wicha, Chunli Chen, Oskar Clewe, Ulrika Simonsson
Pharmacometrics Research Group
Uppsala Universitet
PAGE Meeting, Lisbon, 2016-06-09
1
Combination therapy –
Drug interactions?
Combination therapy is prevalent in many therapeutic areas
– Anti-infectives
– Chemotherapy
– Antiepileptics
– Anaesthesia
What are our current modelling options and their limitations
for characterization of pharmacodynamic (PD) drug
interactions (DDI)?
2
PD DDI modelling
current options
Mechanistic approaches
• Receptor theory
– Competitive interaction models
– Noncompetitive/uncompetitive models
• Systems biology/pharmacology
Additivity-based empirical models
• Loewe Additivity
– Greco model, …
– Combination indices
• Bliss Independence
– Empiric Bliss model(s)
3
R
CA
CB
R*
R*
E
no E
Challenges with current
modelling options
Mechanistic approaches
• Lack of knowledge to parameterize mechanistic interaction models
Additivity-based empirical models
4
5
alpha = 0 Loewe Additivity
alpha > 0 Synergy
alpha < 0 Antagonism
W.R. Greco et al. Cancer Res., 50: 5318–27 (1990).
Greco model
1 =𝐶𝐴
𝐸𝐶50𝐴 ×𝐸
𝐸𝑚𝑎𝑥 − 𝐸
1/𝐻𝐴+
𝐶𝐵
𝐸𝐶50𝐵 ×𝐸
𝐸𝑚𝑎𝑥 − 𝐸
1/𝐻𝐵
+𝛼 × 𝐶𝐴 × 𝐶𝐵
𝐸𝐶50𝐴 × 𝐸𝐶50𝐵 ×𝐸
𝐸𝑚𝑎𝑥 − 𝐸
12𝐻𝐴
+1
2𝐻𝐵
Loewe Additivity
Interaction
term
Challenges with current
modelling options
Mechanistic approaches
• Lack of knowledge to parameterize mechanistic interaction models
Additivity-based empirical models
• Interaction parameters are not quantitatively interpretable
• Single underlying additivity concept
• Mono-dimensional (‘symmetric’) interactions
• Limitation to two drugs
1 =𝐶𝐴
𝐸𝐶50𝐴 ×𝐸
𝐸𝑚𝑎𝑥 − 𝐸
1/𝐻𝐴+
𝐶𝐵
𝐸𝐶50𝐵 ×𝐸
𝐸𝑚𝑎𝑥 − 𝐸
1/𝐻𝐵
+𝛼 × 𝐶𝐴 × 𝐶𝐵
𝐸𝐶50𝐴 × 𝐸𝐶50𝐵 ×𝐸
𝐸𝑚𝑎𝑥 − 𝐸
12𝐻𝐴
+1
2𝐻𝐵
Greco model
W.R. Greco et al. Cancer Res., 50: 5318–27 (1990).
Towards a general PD
interaction model
• Receptor-based mechanistic approaches
– Competitive interaction models
– Noncompetitive/uncompetitive models
• Additivity-based empirical models
– Loewe Additivity
• Greco model
– Bliss Independence
• Empiric Bliss model(s)
7
General PD interaction
model
Basis: additivity criterion
Mechanistic elements
The Emax model for
quantification of drug effects
8
0 1 2 3 4 5
0.0
0.2
0.4
0.6
0.8
1.0
Drug concentration C [fraction of EC50]
Dru
g effect
E [
frac
tion o
f Em
ax]
𝐸 =𝐸𝑚𝑎𝑥 × 𝐶𝐻
𝐸𝐶50𝐻 + 𝐶𝐻
GPDI model
implementation on EC50
• 𝐸𝐴 =𝐸𝑚𝑎𝑥𝐴×𝐶
𝐴𝐻𝐴
𝐸𝐶50𝐴
× 1+𝐼𝑛𝑡
𝐴𝐵×𝐶𝐵
𝐸𝐶50𝐼𝑁𝑇
,𝐴𝐵
+𝐶𝐵
𝐻𝐴+𝐶𝐴
𝐻𝐴
• 𝐸𝐵 =𝐸𝑚𝑎𝑥𝐵×𝐶
𝐵𝐻𝐵
𝐸𝐶50𝐵
× 1+𝐼𝑛𝑡
𝐵𝐴×𝐶𝐴
𝐸𝐶50𝐼𝑁𝑇
,𝐵𝐴
+𝐶𝐴
𝐻𝐵+𝐶𝐵
𝐻𝐵
• 4 parameter interaction model
– IntAB, IntBA (-1 < Int < Inf) (0: Additivity, <0: Synergy, >0: Antagonism)
– EC50INT,AB , EC50INT,BA
• Simplifications:
– IntAB = IntBA (joint INT parameter)
– EC50Int,AB = EC50B and EC50Int,BA = EC50A
9
0 1 2 3 4 5
0.0
0.2
0.4
0.6
0.8
1.0
Drug concentration C [fraction of EC50]
Dru
g effect
E [
frac
tion o
f Em
ax]
GPDI model
implementation on Emax
• 𝐸𝐴 =𝐸𝑚𝑎𝑥𝐴× 1+
𝐼𝑛𝑡𝐴𝐵
×𝐶𝐵𝐸𝐶50
𝐼𝑁𝑇,𝐴𝐵
+𝐶𝐵×𝐶
𝐴𝐻𝐴
𝐸𝐶50𝐴𝐻𝐴+𝐶𝐴
𝐻𝐴
• 𝐸𝐵 =𝐸𝑚𝑎𝑥𝐵× 1+
𝐼𝑛𝑡𝐵𝐴
×𝐶𝐴𝐸𝐶50
𝐼𝑁𝑇,𝐵𝐴
+𝐶𝐴×𝐶
𝐵𝐻𝐵
𝐸𝐶50𝐵𝐻𝐵+𝐶𝐵
𝐻𝐵
• 4 parameter interaction model
– IntAB, IntBA
– EC50INT,AB , EC50INT,BA
• Simplifications:
– IntAB = IntBA (joint INT parameter)
– EC50Int,AB = EC50B and EC50Int,BA = EC50A
10
0 1 2 3 4 5
0.0
0.2
0.4
0.6
0.8
1.0
Drug concentration C [fraction of EC50]
Dru
g effect
E [
frac
tion o
f Em
ax]
The GPDI model is compatible with
several additivity criteria
11
𝐸𝐴 =𝐸𝑚𝑎𝑥𝐴 × 𝐶𝐴
𝐻𝐴
𝐸𝐶50𝐴 × 1 +𝐼𝑛𝑡𝐴𝐵 × 𝐶𝐵
𝐸𝐶50𝐼𝑁𝑇, 𝐴𝐵 + 𝐶𝐵
𝐻𝐴
+ 𝐶𝐴𝐻𝐴
𝐸𝐵 =𝐸𝑚𝑎𝑥𝐵 × 𝐶𝐵
𝐻𝐵
𝐸𝐶50𝐵 × 1 +𝐼𝑛𝑡𝐵𝐴 × 𝐶𝐴
𝐸𝐶50𝐼𝑁𝑇, 𝐵𝐴 + 𝐶𝐴
𝐻𝐵
+ 𝐶𝐵𝐻𝐵
Loewe Additivity
Bliss Independence
Effect summation
Highest single agent
…
GPDI Model on EC50
Two Drugs
12
Bliss Independence
Bidirectional Synergy
Joint decrease in EC50
Bidirectional Antagonism
Joint increase in EC50
Asymmetric interaction
A increases EC50 of B
B decreases EC50 of A
GPDI Model
Three Drugs
Bidirectional interactions between three drugs A, B and C:
• 𝐸𝐴 =𝐸𝑚𝑎𝑥𝐴×𝐶𝐴
𝐻𝐴
𝐸𝐶50𝐴 × 1+𝐼𝑁𝑇𝐴𝐵×𝐶𝐵
𝐸𝐶50𝐼𝑁𝑇,𝐴𝐵+𝐶𝐵× 1+
𝐼𝑁𝑇𝐴𝐶×𝐶𝐶𝐸𝐶50𝐼𝑁𝑇,𝐴𝐶+𝐶𝐶
𝐻𝐴+𝐶𝐴𝐻𝐴
Drug C modulates interaction between A and B:
• 𝐸𝐴 =𝐸𝑚𝑎𝑥𝐴×𝐶𝐴
𝐻𝐴
𝐸𝐶50𝐴 × 1+𝐼𝑁𝑇𝐴𝐵× 1+
𝐼𝑁𝑇𝐴𝐵|𝐶×𝐶𝐶𝐸𝐶50𝐼𝑁𝑇,𝐴𝐵|𝐶+𝐶𝐶
×𝐶𝐵
𝐸𝐶50𝐼𝑁𝑇,𝐴𝐵+𝐶𝐵× 1+
𝐼𝑁𝑇𝐴𝐶×𝐶𝐶𝐸𝐶50𝐼𝑁𝑇,𝐴𝐶+𝐶𝐶
𝐻𝐴
+𝐶𝐴𝐻𝐴
13
Application of the GPDI Model
• Dataset:
Cokol et al. Mol Syst Biol, 7: 544 (2011).
– Target organism:
S. cerevisiae BY4741
– 200 combinations of anti-fungal and
non-antifungal drugs.
• Inhibition of growth model
• GPDI model
– Loewe additivity
– Bliss Independence
• Comparison to Greco model
14
200x
Mono A
Mono B
Fungal lo
ad (
OD
)
Time (h)
Ind. Prediction of Loewe Additivity
15
Observation
Prediction
Ax Bx (fold MIC)
Individual predictions using
Greco (left) or GPDI model (right)
16
Observation Prediction Ax Bx (fold MIC)
INT values at EC50
17
SYN
ANT
AN
T +
SY
N
ANT + SYN
0 2 4 6 8 10
0
2
4
6
8
10
INTAB at EC50A
INT
BA a
t EC
50
B
INT values at EC50 of
sham combinations
18
SYN
ANT
AN
T +
SY
N
ANT + SYN
0 2 4 6 8 10
0
2
4
6
8
10
INTAB at EC50A
INT
BA a
t EC
50
B
INT values at EC50 of
sham combinations
19
SYN
ANT
AN
T +
SY
N
ANT + SYN
ADD
0 2 4 6 8 10
0
2
4
6
8
10
INTAB at EC50A
INT
BA a
t EC
50
B
188/200 PD DDI supported
estimation of full GPDI model*
20
SYN
ANT
AN
T +
SY
N
ANT + SYN
ADD
0 2 4 6 8 10
0
2
4
6
8
10
INTAB at EC50A
INT
BA a
t EC
50
B
* i.e. significant separate INT
value for each drug vs.
joint INT parameter
PD DDI are often misclassified
by conventional analyses!
21
SYN
ANT
AN
T +
SY
N
ANT + SYN
Result of Greco analysis:
• Antagonism
• Additivity
• Synergy
ADD
0 2 4 6 8 10
0
2
4
6
8
10
INTAB at EC50A
INT
BA a
t EC
50
B
The choice of additivity criterion
affects the determined interaction
22
0 2 4 6 8 10
0
2
4
6
8
10
INTAB at EC50A
INT
BA a
t EC
50
B
0 2 4 6 8 10
0
2
4
6
8
10
GPDI – Loewe Additivity GPDI – Bliss Independence
76
35 35
54
27
88
27
58
Conclusions
• The GPDI model combined elements from mechanism-based
interaction models with empirical additivity concepts.
• Advantages of the GPDI model over conventional
approaches are:
– interpretable parameters
– Applicability for > 2 interacting drugs
– Compatibility with multiple additivity criteria
– More-dimensional interactions
– Scalability to adapt to complexity of the data
– Applicability in both concentration-effect and longitudinal
modelling
23
GPDI model application studies
at PAGE 2016
24
II-50 II-44
In vitro and in vivo (acute mouse infection model) drug effects of
rifampicin, isoniazid, ethambutol and pyrazinamide against
M. tuberculosis
Acknowledgements
PMx research group at Uppsala University
• Elin Svensson
• Anders Kristofferson
PreDiCT-TB consortium
25
The research leading to these results has received funding from the Innovative
Medicines Initiative Joint Undertaking (www.imi.europe.eu) under grant agreement
n°115337, resources of which are composed of financial contribution from the
European Union’s Seventh Framework Programme (FP7/2007-2013) and EFPIA
companies’ in kind contribution.
Appendix
26
Implementation of Combined
Effects on Inhibition of Growth
GPDI within Bliss Independence:
𝐸 = 𝐸𝐴 + 𝐸𝐵 − 𝐸𝐴 × 𝐸𝐵
• 𝐸𝐴 =𝐸𝑚𝑎𝑥𝐴×𝐶𝐴
𝐻𝐴
𝐸𝐶50𝐴 × 1+𝐼𝑛𝑡
𝐴𝐵×𝐶𝐵
𝐸𝐶50𝐼𝑁𝑇
,𝐴𝐵
+𝐶𝐵
𝐻𝐴+𝐶𝐴
𝐻𝐴
• 𝐸𝐵 =𝐸𝑚𝑎𝑥𝐵×𝐶𝐵
𝐻𝐵
𝐸𝐶50𝐵 × 1+𝐼𝑛𝑡
𝐵𝐴×𝐶𝐴
𝐸𝐶50𝐼𝑁𝑇
,𝐵𝐴
+𝐶𝐴
𝐻𝐵+𝐶𝐵
𝐻𝐵
GPDI within Loewe additivity:
27
𝐸𝐶50𝐴 = 𝐸𝐶50𝐴 × 1 +𝐼𝑛𝑡𝐴𝐵 × 𝐶𝐵
𝐸𝐶50𝐼𝑁𝑇, 𝐴𝐵 + 𝐶𝐵
𝐸𝐶50𝐵 = 𝐸𝐶50𝐵 × 1 +𝐼𝑛𝑡𝐵𝐴 × 𝐶𝐴
𝐸𝐶50𝐼𝑁𝑇, 𝐵𝐴 + 𝐶𝐴
*
*
* *
N
kgrowth·(1-E)
[1] W.J. Jusko, J. Pharm. Sci. 60 (1971).
[1]
The full GPDI Model is identifiable
on standard checkerboard designs
28
8x8 checkerboard
• 1000 scenarios assessed
• n=3 replicates per scenario
• max. growth: 10 log10 CFU/mL
• 𝜎add = 0.3 log10 CFU/mL
Emax ∈ {0.5,1}
EC50 ∈ {0. 5, 2}
H ∈ {1, 4}
Int ∈ {-0.9, -0.2} ∪ {2, 20}
EC50_Int ∈ {0.1, 1}
Em
ax
A
EC
50
A
HA
Em
ax
B
EC
50
B
HB
Int A
B
Int B
A
EC
50
Int,
AB
EC
50
Int,
BA
0.5
1.0
2.0
5.0
10.0
20.0
50.0
100.0
200.0
exp
ecte
d R
SE
%
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