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)
CA
R*
E
R*
no E
R
CB
3
Challenges with current
modelling options
Mechanistic approaches
• Lack of knowledge to parameterize mechanistic interaction models
Additivity-based empirical models
4
Greco model
Loewe Additivity
1=
𝐶𝐴
𝐸
𝐸𝐶50𝐴 ×
𝐸𝑚𝑎𝑥 − 𝐸
1/𝐻𝐴
+
𝐶𝐵
𝐸
𝐸𝐶50𝐵 ×
𝐸𝑚𝑎𝑥 − 𝐸
𝛼 × 𝐶𝐴 × 𝐶𝐵
+
𝐸𝐶50𝐴 × 𝐸𝐶50𝐵 ×
𝐸
𝐸𝑚𝑎𝑥 − 𝐸
1
1
+
2𝐻𝐴 2𝐻𝐵
1/𝐻𝐵
Interaction
term
alpha = 0  Loewe Additivity
alpha > 0  Synergy
alpha < 0  Antagonism
W.R. Greco et al. Cancer Res., 50: 5318–27 (1990).
5
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
Greco model
• Limitation to two drugs
1=
𝐶𝐴
𝐸
𝐸𝐶50𝐴 ×
𝐸𝑚𝑎𝑥 − 𝐸
+
W.R. Greco et al. Cancer Res., 50: 5318–27 (1990).
1/𝐻𝐴
+
𝐶𝐵
𝐸
𝐸𝐶50𝐵 ×
𝐸𝑚𝑎𝑥 − 𝐸
𝛼 × 𝐶𝐴 × 𝐶𝐵
𝐸
𝐸𝐶50𝐴 × 𝐸𝐶50𝐵 ×
𝐸𝑚𝑎𝑥 − 𝐸
1
1
+
2𝐻𝐴 2𝐻𝐵
1/𝐻𝐵
Towards a general PD
interaction model
• Receptor-based mechanistic approaches
– Competitive interaction models
– Noncompetitive/uncompetitive models
Mechanistic elements
• Additivity-based empirical models
– Loewe Additivity
• Greco model
– Bliss Independence
• Empiric Bliss model(s)
General PD interaction
model
Basis: additivity criterion
7
Drug effect E [fraction of Emax]
0.0 0.2 0.4 0.6 0.8 1.0
The Emax model for
quantification of drug effects
𝐸𝑚𝑎𝑥 × 𝐶𝐻
𝐸=
𝐸𝐶50𝐻 + 𝐶𝐻
0
1
2
3
4
5
Drug concentration C [fraction of EC50]
8
𝐻𝐴
+𝐶𝐴 𝐻𝐴
,
𝐸𝑚𝑎𝑥𝐵 ×𝐶𝐵𝐻𝐵
• 𝐸𝐵 =
𝐸𝐶50𝐵 ×
𝐼𝑛𝑡 ×𝐶𝐴
1+𝐸𝐶50 𝐵𝐴 +𝐶
𝐴
𝐼𝑁𝑇 𝐵𝐴
,
𝐻𝐵
+𝐶𝐵 𝐻𝐵
0.4 0.6 0.8 1.0
𝐸𝐶50𝐴 ×
𝐼𝑛𝑡 ×𝐶𝐵
1+𝐸𝐶50 𝐴𝐵 +𝐶
𝐵
𝐼𝑁𝑇 𝐴𝐵
0.2
• 𝐸𝐴 =
0.0
𝐸𝑚𝑎𝑥𝐴 ×𝐶𝐴𝐻𝐴
Drug effect E [fraction of Emax]
GPDI model
implementation on EC50
0
•
•
1
2
3
4
5
Drug concentration C [fraction of EC50]
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
𝐼𝑛𝑡 ×𝐶
0.4 0.6 0.8 1.0
• 𝐸𝐵 =
𝐴
𝐸𝑚𝑎𝑥𝐵 × 1+𝐸𝐶50 𝐵𝐴 +𝐶
×𝐶𝐵𝐻𝐵
𝐴
𝐼𝑁𝑇 𝐵𝐴
𝐻𝐵 , 𝐻𝐵
𝐸𝐶50𝐵 +𝐶𝐵
0.2
• 𝐸𝐴 =
0.0
𝐼𝑛𝑡 ×𝐶
𝐵
𝐸𝑚𝑎𝑥𝐴 × 1+𝐸𝐶50 𝐴𝐵 +𝐶
×𝐶𝐴𝐻𝐴
𝐵
𝐼𝑁𝑇 𝐴𝐵
𝐻𝐴 , 𝐻𝐴
𝐸𝐶50𝐴 +𝐶𝐴
Drug effect E [fraction of Emax]
GPDI model
implementation on Emax
0
•
•
4 parameter interaction model
– IntAB, IntBA
– EC50INT,AB , EC50INT,BA
Simplifications:
– IntAB = IntBA (joint INT parameter)
– EC50Int,AB = EC50B and EC50Int,BA = EC50A
1
2
3
4
5
Drug concentration C [fraction of EC50]
10
The GPDI model is compatible with
several additivity criteria
Loewe Additivity
𝐸𝐴 =
𝐸𝐵 =
𝐸𝑚𝑎𝑥𝐴 × 𝐶𝐴𝐻𝐴
𝐼𝑛𝑡𝐴𝐵 × 𝐶𝐵
𝐸𝐶50𝐴 × 1 +
𝐸𝐶50𝐼𝑁𝑇, 𝐴𝐵 + 𝐶𝐵
𝐸𝑚𝑎𝑥𝐵 × 𝐶𝐵𝐻𝐵
𝐼𝑛𝑡𝐵𝐴 × 𝐶𝐴
𝐸𝐶50𝐵 × 1 +
𝐸𝐶50𝐼𝑁𝑇, 𝐵𝐴 + 𝐶𝐴
𝐻𝐴
𝐻𝐵
+ 𝐶𝐴 𝐻𝐴
Bliss Independence
+ 𝐶𝐵 𝐻𝐵
Effect summation
Highest single agent
…
11
GPDI Model on EC50
Two Drugs
Bliss Independence
Bidirectional Antagonism
Joint increase in EC50
Bidirectional Synergy
Joint decrease in EC50
Asymmetric interaction
A increases EC50 of B
B decreases EC50 of A
12
GPDI Model
Three Drugs
Bidirectional interactions between three drugs A, B and C:
𝐻
𝐸𝑚𝑎𝑥𝐴 ×𝐶𝐴 𝐴
• 𝐸𝐴 =
𝐸𝐶50𝐴 ×
𝐼𝑁𝑇
×𝐶𝐵
1+𝐸𝐶50 𝐴𝐵 +𝐶
𝐼𝑁𝑇,𝐴𝐵 𝐵
×
𝐼𝑁𝑇 ×𝐶𝐶
1+𝐸𝐶50 𝐴𝐶 +𝐶
𝐼𝑁𝑇,𝐴𝐶 𝐶
𝐻𝐴
+𝐶𝐴 𝐻𝐴
Drug C modulates interaction between A and B:
• 𝐸𝐴 =
𝐻
𝐸𝑚𝑎𝑥𝐴 ×𝐶𝐴 𝐴
𝐸𝐶50𝐴 × 1+
𝐼𝑁𝑇𝐴𝐵|𝐶 ×𝐶𝐶
𝐼𝑁𝑇𝐴𝐵 × 1+𝐸𝐶50
×𝐶𝐵
𝐼𝑁𝑇,𝐴𝐵|𝐶 +𝐶𝐶
𝐸𝐶50𝐼𝑁𝑇,𝐴𝐵 +𝐶𝐵
𝐻𝐴
𝐼𝑁𝑇𝐴𝐶 ×𝐶𝐶
𝐸𝐶50𝐼𝑁𝑇,𝐴𝐶 +𝐶𝐶
× 1+
+𝐶𝐴 𝐻𝐴
13
Application of the GPDI Model
200x
Mono B
Mono A
Fungal load (OD)
• 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
Time (h)
14
Ind. Prediction of Loewe Additivity
Observation
Prediction
Ax Bx (fold MIC)
15
Individual predictions using
Greco (left) or GPDI model (right)
Observation
Prediction
Ax Bx (fold MIC)
16
ANT + SYN
6
2
4
ANT
0
INTBA at EC50B
8
10
INT values at EC50
ANT + SYN
SYN
0
2
4
6
8
10
INTAB at EC50A
17
ANT + SYN
6
2
4
ANT
0
INTBA at EC50B
8
10
INT values at EC50 of
sham combinations
ANT + SYN
SYN
0
2
4
6
8
10
INTAB at EC50A
18
ANT + SYN
6
2
4
ANT
0
INTBA at EC50B
8
10
INT values at EC50 of
sham combinations
ANT + SYN
SYN
ADD
0
2
4
6
8
10
INTAB at EC50A
19
ANT + SYN
6
2
4
ANT
0
INTBA at EC50B
8
10
188/200 PD DDI supported
estimation of full GPDI model*
ANT + SYN
SYN
ADD
0
2
4
6
8
10
* i.e. significant separate INT
value for each drug vs.
joint INT parameter
INTAB at EC50A
20
ANT + SYN
ANT
2
4
6
Result of Greco analysis:
• Antagonism
• Additivity
• Synergy
0
INTBA at EC50B
8
10
PD DDI are often misclassified
by conventional analyses!
ANT + SYN
SYN
ADD
0
2
4
6
8
10
INTAB at EC50A
21
The choice of additivity criterion
affects the determined interaction
GPDI – Bliss Independence
10
6
6
4
27
0
2
2
4
76
0
INTBA at EC50B
27
8
54
8
10
GPDI – Loewe Additivity
35
35
0
2
4
6
8
10
88
58
0
2
4
6
8
10
INTAB at EC50A
22
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
In vitro and in vivo (acute mouse infection model) drug effects of
rifampicin, isoniazid, ethambutol and pyrazinamide against
M. tuberculosis
II-50
II-44
24
Acknowledgements
PMx research group at Uppsala University
• Elin Svensson
• Anders Kristofferson
PreDiCT-TB consortium
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.
25
Appendix
26
Implementation of Combined
Effects on Inhibition of Growth
GPDI within Bliss Independence:
𝐸 = 𝐸𝐴 + 𝐸𝐵 − 𝐸𝐴 × 𝐸𝐵
•
𝐸𝑚𝑎𝑥𝐴 ×𝐶𝐴 𝐻𝐴
𝐸𝐴 =
𝐸𝐶50𝐴 ×
𝐼𝑛𝑡𝐴𝐵×𝐶𝐵
1+
𝐸𝐶50𝐼𝑁𝑇 𝐴𝐵+𝐶𝐵
𝐻𝐴
+𝐶𝐴 𝐻𝐴
,
•
kgrowth·(1-E)
N
𝐸𝑚𝑎𝑥𝐵 ×𝐶𝐵 𝐻𝐵
𝐸𝐵 =
𝐸𝐶50𝐵 ×
𝐼𝑛𝑡𝐵𝐴×𝐶𝐴
1+
𝐸𝐶50𝐼𝑁𝑇 𝐵𝐴+𝐶𝐴
𝐻𝐵
+𝐶𝐵 𝐻𝐵
[1]
,
GPDI within Loewe additivity:
𝐼𝑛𝑡𝐴𝐵 × 𝐶𝐵
𝐸𝐶50𝐼𝑁𝑇, 𝐴𝐵 + 𝐶𝐵
𝐼𝑛𝑡𝐵𝐴 × 𝐶𝐴
*
𝐸𝐶50𝐵 = 𝐸𝐶50𝐵 × 1 +
𝐸𝐶50𝐼𝑁𝑇, 𝐵𝐴 + 𝐶𝐴
𝐸𝐶50𝐴* = 𝐸𝐶50𝐴 × 1 +
*
[1] W.J. Jusko, J. Pharm. Sci. 60 (1971).
*
27
The full GPDI Model is identifiable
on standard checkerboard designs
EC50Int,BA
EC50Int,AB
IntBA
IntAB
HB
EC50B
EmaxB
HA
2.0
1.0
0.5
EC50A
∈ {0.5,1}
∈ {0. 5, 2}
∈ {1, 4}
∈ {-0.9, -0.2} ∪ {2, 20}
∈ {0.1, 1}
20.0
10.0
5.0
EmaxA
Emax
EC50
H
Int
EC50_Int
200.0
100.0
50.0
expected RSE %
8x8 checkerboard
• 1000 scenarios assessed
• n=3 replicates per scenario
• max. growth: 10 log10 CFU/mL
• 𝜎add = 0.3 log10 CFU/mL
28