Fang

Eﬃcient Design and Analysis of
Combination Experiments to Improve
Early Stage Clinical Development

Hong-Bin Fang, Ph.D.

Division of Biostatistics
University of Maryland Greenebaum Cancer Center
and Department of Epidemiology and Preventive Medicine
Baltimore, MD 21201, USA
Email: hfang@som.umaryland.edu

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Outline
• Introduction
• Existing Methods for Combination Study Design
• Maximal Power Experimental Design
• Existing Analysis Methods for Synergy
• Statistical Analysis of Interaction Index
• Conclusion and Further Research


Introduction: Combination Therapy
• Reduce the frequency of acquired resistance
• Achieve greater eﬃcacy with lower doses and reduced toxicity
• Achieve enhanced potency (or sensitization) exploring synergistic ac-
tivities
• Provide a ﬁrmer basis for potential clinical trials

Joint action is divided into three types:
1. Independent joint action
2. Simple similar (additive) action
3. Synergistic/antagonistic action

Refs.: Berenbaum MC. (1989). Pharmacological Reviews 41: 93-141.
Greco WR, et al. (1995). Pharmacological Reviews 47, 331-385.
Fitzgerald et al.(2006). Nature Chemical Biology 2, 458 - 466


Interaction Index
To assess the interaction of two drugs at the combination of (xA, xB ),
Berenbaum (1977) deﬁned an interaction index (τ ),
xA xB
τ= +
XA XB
XA and XB are the doses of drugs A and B that when administered alone
yield the same eﬀect as does the combination (xA, xB ).
• τ = 1, A and B are additive at (xA, xB );
Loewe Independence (Additivity)
• τ < 1, A and B are synergistic at (xA, xB );
• τ > 1, A and B are antagonistic at (xA, xB ).

Refs.: Berenbaum MC. (1977). Clin. Exp. Immunol. 28: 1-18.
Berenbaum MC. (1989). Pharmacological Reviews 41: 93-141.


Bliss Independence (Additivity)
• The joint eﬀect of inhibitors A and B is the product of the eﬀect of
each
EAB = EA ∗ EB
• Assumption: Inhibitors can bind simultaneously and mutually nonex-
clusively through distinct mechanism


Fitzgerald et al. Nature 2006

Statistical Approaches for Design
• Statistical (instead of mechanistic) evaluation is very valuable in
clinical trials because it is impractical for a measure of success (such
as synergism between two drugs) to change with every biochemical
advance.
• Variation: the administration of precisely the same dose to aliquots
(or virtually genetically identical animals) may result in diﬀerent levels
of dose eﬀect.
• Sample Size: how many samples are needed, namely, how to identify
doses in the combinations and how many replicates at each combina-
tion and how to analyze the data produced


Current Methods of Design
• Equal regression lines (Finney, 1971)
• An optimal design by ﬁxing the total dose for
speciﬁc models (Abdelbasit and Plackett, 1982)
• Fixed ratio/ray design (Tallarida et al., 1992)
• Checkerboard (Lattice) design (Martinez-Irujo et
al., 1996)
• A D-optimal design for in vitro combination
studies in linear models (Greco et al., 1995) but
n too large
Refs.: Finney DJ (1971). Probit Analysis. Cambridge University Press.
Abdelbasit KM, Plackett RL (1982). Biometrics 38, 171-179.
Greco WR, et al. (1995). Pharmcological Reviews 47, 331-385.
Martinez-Irujo, et al.(1997). Biochemical Pharmacology 51, 635-644.
Tallarida RJ, et al.(1997). Life Science 61, 417-425.

Fixed Ratio/Ray Design
• Assume that two drugs are synergistic, additive or antagonistic for all
doses of a ﬁxed ratio;
• Suboptimal dose allocations result in false synergistic combinations or
miss an apparent interaction at a particular combination;
• Statistical power to detect additivity is undermined substantially.


Maximal Power Design
Statistical Model
Single dose eﬀect: y = fA(XA), y = fB (XB )
−1
Potency of B relative to A: ρ(XB ) = fA fB (XB )/XB
The joint model at the combination x is assumed
y = fA(xA + ρ(XB )xB ) + g(xA, xB ) + ε

• g(xA, xB ) = 0, A and B are additive at x;
• g(xA, xB ) > 0, A and B are synergistic at x;
• g(xA, xB ) < 0, A and B are antagonistic at x.

Experimental design is based on testing the additive
action of drugs A and B H0 : g = 0 versus H1 :
g = 0.

Statistical Inference
After transformation, the additive model
y = fA(xA + ρ(XB )xB ) = (or ≈)g(z1) + g(z2)
(i) (i)
Let z(i) = (z1 , z2 )T , i = 1, 2, . . . , m,
y = (y11, · · · , y1k , · · · , ymk )T ,
(i) (i)
Z: m × 2 matrix, its ith row: (g1(z1 ), g2(z2 ))
Then, under H0,
yT (J − V )y/(m − 2)
F = T ∼ Fm−2,m(k−1)
y (I − J)y/(mk − m)
where U = I 1 k , V = U Z(Z T U T U Z)−1Z T U T , J = U (U T U )−1U T


Statistical Inference
If z(1), . . . , z(m) are uniformly scattered in S, under
H1, the F statistic has a noncentral F -distribution
with degrees of freedom m − 2 and m(k − 1) and
the noncentrality parameter,
mk
δ= 2 g 2(z)dz,
σ S
is maximized. Thus, the power for detecting de-
partures from additivity of two drugs is maximized
when m mixtures z(1), . . . , z(m) are uniformly scat-
tered in S.
Refs.: Wiens (1991). Statistics & Probability Letters, 12: 217-221.
Tan et al.(2003). Statistics in Medicine, 22: 2091-221.


Sample Size Determination
Given the type I error rate: α, power: 1 − β
the smallest meaningful diﬀerence: η
the measurement variation: σ 2
the sample sizes can be obtained from the noncen-
tral F -distribution function,
∞
e−δ/2(δ/2)k (m − 2)x
P (F ≤ x) = P Fm−2+2k,n−m ≤ ,
k! m − 2 + 2k
k=0

δ = nη 2/σ 2.


Sample Size Calculation
α = 0.05, 1 − β = 0.8,
d = η 2/σ 2 # of Replications n0
1 2 3 4 5 6
d = 0.1 − − 139(556) 87(435) 61(366) 46(322)
d = 0.2 − 78(234) 42(168) 27(135) 19(114) 14(98)
d = 0.3 107(214) 40(120) 21(84) 14(70) 10(60) 7(49)
d = 0.4 68(136) 25(75) 14(56) 9(45) 3(18) 3 (21)
d = 0.5 48(96) 18(54) 10(40) 6(30) 3(18) 3(21)
d = 0.8 24(48) 9(27) 4(16) 3(15) 3(18) 3(21)
d=1 18(36) 6(18) 3(12) 3(15) 3(18) 3(21)


Experimental Design I
The sample sizes are based on the F-test to detect
departures from additive action with 80% power at
a significance level of 0.05
• Choose the dose range of significance: e.g., ED20-
ED80 (based on pharmacology)
• Choose the meaningful difference in the dose-
effect outcome to be detected: η
• Choose the number of replicates at each combi-
nation
• The variance is estimated based on the pooled
variations from the two single drug experiments.

Experimental Design II
• The additive model depends on the individual
dose-response;
• Diﬀerent individual dose-responses result in dif-
ferent experimental designs;

We have considered three classes of drugs classiﬁed
by the shape of individual dose-response curves:
• both log-linear;
• both linear;
• linear + log-linear.


Log-linear + Log-linear
Let the dose-response of drugs A and B be
y(xA) = αA+βA log xA, y(xB ) = αB +βB log xB
The potency ρ of B relative to A is
βB /βA−1
ρ(xB ) = ρ0xB , ρ0 = exp[(αB − αA)/βA].
The additive model at combination (xA, xB ):
y(xA, xB ) = αA + βA log(xA + ρxB )
(βB −βA)/βA
ρ = ρ0 ρ−1x A + xB .


Log-linear + Log-linear
Ray Design: Using lattice points undermines the power to detect the
additivity;
Maximum Power Design: Using uniformly scattered points achieves
maximum power to detect departures from additivity.


Results of Simulation Studies
Number of combinations 9 16
Maximum Discrepancy (CL2) 0.003583 0.001190
Power Type I error 0.0501 0.0493
Design Power 0.7949 0.8345
Discrepancy (CL2) 0.022719 0.017986
Ray/Lattice Type I error 0.0532 0.0501
Design Power 0.6540 0.4062
Discrepancy (CL2) average: 0.035943 0.020812
SD: 0.0173668 0.0147279
Monte Type I error average: 0.0503 0.0502
Carlo SD: 0.00311 0.00227
method Power average: 0.5330 0.7439
SD: 0.31456 0.26184
y = 20 log(z1 ) + 70 log(z2 ) + g(z1 , z2 ) + ε, g(z1 , z2 ) = 50(z1 − 2) sin(z1 ) cos(z2 ), ε ∼ N (0, σ 2 ) and
σ 2 = 250.
η 2 = 100, with 10,000 replications


Example: SAHA + Ara-C against K562

y = 43.69 − 10.93 log(S), y is the viability

y = 35.98 − 8.25 log(C)
Shiozawa et al. (2009). CCR. 15:1698-1707

Mixtures SAHA and Ara-C for combination experiment

Exper. SAHA Ara-C Exper. SAHA Ara-C
# (µM) (µM) # (µM) (µM)
1 0.706 0.101 7 1.427 3.193
2 3.035 0.099 8 3.402 2.681
3 2.160 2.030 9 1.008 0.924
4 0.635 5.122 10 3.139 0.781
5 0.393 0.727 11 4.524 0.724
6 0.118 2.684

dose range: ED20-ED80; η = 15%(viability); 5 replicates; σ 2 = 804.564
ˆ


Analysis of Synergy
Estimation of the Interaction Index Surface
(i) (i)
yij : the jth response at (xA , xB ),
(i) (i)
With the single dose-response curves, the interaction indexes at (xA , xB )
are
(i) (i)
xA xB
τij = +
exp{(yij − αA)/βA} exp{(yij − αB )/βB }
j = 1, . . . , k, i = 1, . . . , m.

The method of two-dimensional B-splines (thin plate splines) is employed
to estimate the interaction index surface τ = h(xA, xB ),

Ref.: Fang, et al. Stat. Medicine 2008


SAHA: Estimation of relative potency
0.842
The potency of Vorinostat relative to Etoposide is ρ(XB ) = 0.368XB ,
which is non-constant and depends on dose.
The predicted additive model is

y(xA, xB ) = 41.52 − 13.02 log(xA + ψ(xA, xB )xB ),

where ψ(xA, xB ) is determined by ψ(xA, xB ) = 0.368(ψ −1(xA, xB )xA +
xB )0.842.


Dose-response surface

Observations: 66(11 combinations with 5 replicates at each combination)
Maximum viability: 82.81%; Minimum viability: 17.72%; Mean: 30.67%;
Standard deviation: 13.40.
Output: F9,55 = 8.14, p-value< 0.0001

Interaction Index Surface


Linear + Log-linear
Let
z1 = xA, z2 = βAξ(xA, xB )/[βB ψ(xA, xB )xB ],
then, the additive model becomes
y(xA, xB ) = αA + βAz1 + βB z2.
The m experimental points should be uniformly
scattered in the tetragon,
z1
: a < αA + βAz1 + βB z2 < b, z1 > 0, z2 > 0
z2
for given a and b.

Example: LY-168 with Sorafnib against WM164

y = 111.85 − 9.56xA, y is the viability

y = 101.91 − 31.17xB

dose range: ED20-ED80; η = 15%(viability); 6 replicates; σ 2 = 1352.724
ˆ


Example: LY-168 + Sorafnib against WM164
Dose-response surface

Observations: 133(19 combinations with 6 replicates at each
combination) Maximum viability: 93.58%; Minimum viability: 3.53%;
Mean: 30.45%; Standard deviation: 25.347.
Output: F17,114 = 162.9696, p-value< 0.0001

Example: LY-168 + Sorafnib against WM164
Interaction Index Surface


Follow-up in Early Clinical Stage Trials
Three cases classiﬁed by the shape of individual
dose-response curves considered:
• both log-linear (SAHA + Ara-C; Clini-
cal trial ongoing);
• both linear (potentially resurrect a
promising drug combination) ;
• linear + log-linear (In vivo experiment
ongoing).


Conclusion
• Maximal power design produces data eﬃciently.
It is optimal in that statistical power to detect
departures from additivity is maximized.
• The F test can be used to test departures from
additivity and the thin plate splines to estimate
the interaction index surface eﬀectively with data
generated by the MP design.
• SYNSTAT R package at
www.umgcc.org/research/biostatistics.htm


Acknowledgments
Ming Tan, PhD, UMGCC
Guo-Liang Tian, PhD, University of Hong Kong
Dr. Douglas D. Ross’s Lab, U Maryland School of Medicine
Dr. Wei Li’s Lab, U Tennessee College of Pharmacy
Dr. Pei Feng’s Lab, U Maryland Dental School
Dr. Peter Houghton, St Jude Children’s Research Hospital


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