Post on 16-Dec-2015
Response Optimization in Oncology In Vivo Studies:a Multiobjective Modeling Approach
Maksim Pashkevich, PhD(Early Phase Oncology Statistics)
Joint work with
Philip Iversen, PhD(Pre-Clinical Oncology Statistics)
Harold Brooks, PhD(Growth and Translational Genetics)
Eli Lilly and Company
MBSW’2009, May 18-20, Muncie, IN
2
Outline
• Problem overview
– In vivo studies in oncology drug development
– Efficacy and toxicity measures for in vivo studies
– Optimal regimen as balance between efficacy / toxicity
• Models for efficacy and toxicity
– Modified Simeoni model of tumor growth inhibition
– Animal body weight loss model to describe toxicity
– Statistical estimation of model parameters in Matlab
• Optimal regimen simulation
– Multiobjective representation of simulation results
– Pareto-optimal set of optimal dosing regimens
3
Motivation
In vivo studies in oncology• Typical way to assess cancer compound activity• Cancer tumors are implanted in mice or rats• Tumor size and animal weight are measured over time
Efficacy and toxicity measures• Tumor growth delay is a standard efficacy measure• Body weight loss is a typical surrogate for toxicity
Optimal dosing regimen is unknown• Goal is to achieve balance between efficacy and toxicity• Number of possible dosing regimens is very significant• Modeling should help to select promising regimens
4
Example of Efficacy DataTumor Weight vs Time
(mean +/- SE)
0
500
1000
1500
2000
2500
0 10 20 30 40 50 60Day
Tu
mo
r W
eig
ht,
mg
Vehicle, 0.2 ml, IV, TID7dx4 <n Vehicle, 0.2 ml, IV, TID7dx4Drug A, 15 mg/kg, IV, q7dx4 <n Drug A, 15 mg/kg, IV, q7dx4
Drug A, 30 mg/kg, IV, q7dx4 <n Drug A, 30 mg/kg, IV, q7dx4Drug A, 60 mg/kg, IV, q7dx1 <n Drug A, 60 mg/kg, IV, q7dx1
Drug A, 15 mg/kg, IV, BID7dx4 <n Drug A, 15 mg/kg, IV, BID7dx4Drug A, 30 mg/kg, IV, BID7dx4 Drug A, 20 mg/kg, IV, TID7dx4
Max wt. Sac, DeadLoss / Total -10 10, 0 / 10 -7 10, 0 / 10 -8 9, 1 / 10 -21 5, 5 / 10 -9 0, 0 / 10 -11 0, 0 / 10 -10 0, 0 / 10
5
Example of Toxicity DataBody Weight vs Time
(mean +/- SE)
19
20
21
22
23
24
25
26
27
28
0 10 20 30 40 50 60
Day
Bo
dy
We
igh
t, g
Vehicle, 0.2 ml, IV, TID7dx4 <n Vehicle, 0.2 ml, IV, TID7dx4Drug A, 15 mg/kg, IV, q7dx4 <n Drug A, 15 mg/kg, IV, q7dx4Drug A, 30 mg/kg, IV, q7dx4 <n Drug A, 30 mg/kg, IV, q7dx4Drug A, 60 mg/kg, IV, q7dx1 <n Drug A, 60 mg/kg, IV, q7dx1Drug A, 15 mg/kg, IV, BID7dx4 <n Drug A, 15 mg/kg, IV, BID7dx4Drug A, 30 mg/kg, IV, BID7dx4 Drug A, 20 mg/kg, IV, TID7dx4
7
Model Extension
)(2 tck n
P
x1
x2
Nonlineardrug effect
1P
cytotoxic
cytostatic
x5
x3 x41k 1k
1k
Cell death
Modifications to get adequate model• Drug effect depends on exposure in a non-linear way
• Drug has both cytotoxic and cytostatic effect
• Rationale is based on cell-cycle effect of the compound
Cell growth
8
Developed Efficacy Model
)()()1()(
)()()(
)()()(
),()()()(
)()()(1)()(
125
4314
3213
21122
121
10101
),(
)(
)( )(
txtckdt
tdxtxtxk
dt
tdx
txtxkdt
tdxtxktxtck
dt
tdx
txtcktwtxdt
tdx
n
n
n
),0(~),()()()()()(
0)()()()(),exp()(
254321
05040302001
uNutxtxtxtxtxtw
txtxtxtxuwtx
Dynamic model: system of ordinary differential equations
Initial conditions: with random effect for initial tumor weight
90 10 20 30 40 50 60
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
Time (days)
log(
Tum
or w
eigh
t)
0 10 20 30 40 50 60-2.5
-2
-1.5
-1
-0.5
0
0.5
1
Time (days)
log(
Tum
or w
eigh
t)
5 10 15 20 25 30 35 40-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
Time (days)
log(
Tum
or w
eigh
t)
5 10 15 20 25 30 35 40 45-2.5
-2
-1.5
-1
-0.5
0
0.5
1
Time (days)
log(
Tum
or w
eigh
t)
5 10 15 20 25 30 35 40-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
Time (days)
log(
Tum
or w
eigh
t)
5 10 15 20 25 30 35 40-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
Time (days)
log(
Tum
or w
eigh
t)
5 10 15 20 25 30 35 40-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Time (days)
log(
Tum
or w
eigh
t)Modeled vs. Observed for Groups
Model adequacy assessmentIndividual profiles vs. mean modeled tumor growth curves for each group
Control 15 mg/kg QD 30 mg/kg QD
60 mg/kg QD 15 mg/kg BID 30 mg/kg BID
20 mg/kg TID
10
Efficacy Model Results
0 10 20 30 40 50 600
0.5
1
1.5
2
2.5
Time (days)
Tum
or
wei
ght
Control15 mg/kg q7dx4
30 mg/kg q7dx4
60 mg/kg q7dx1
15 mg/kg BID7dx4
30 mg/kg BID7dx420 mg/kg TID7dx4
Modeled population-average tumor growth curves for each dose group
115 10 15 20 25 30 35 40
0.96
0.965
0.97
0.975
0.98
0.985
0.99
0.995
1
1.005
1.01
Time (days)
Rel
ativ
e bo
dy w
eigh
tBody Weight Loss
drug
drug
Hypothetical example: two dosing cycles at days 7 and 17
Body weight is initially in steady state
Drug exposure causes weight loss
Body weights starts to recover
Next dose causes more weight loss
Slow recovery phase: body weight growth based on Gompertz model
Maximum body weight lossis roughly 3.25%
12
Developed Toxicity Model
)(),(
)(
)()()(
)()()(
)()()()(
)()()(ln)()(
4314
3213
21122
121101
txtxkdt
tdxtxtxk
dt
tdx
txktxtckdt
tdx
txtcktwtxdt
tdx
n
n
),0(~),()()()()(
0)()()(),exp()(
24321
040302001
uNutxtxtxtxtw
txtxtxuwtx
Dynamic model: system of ordinary differential equations
Initial conditions: with random effect for initial body weight
135 10 15 20 25 30 35 40
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
1.25
Time (days)
Rel
ativ
e bo
dy w
eigh
t
5 10 15 20 25 30 35 400.75
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
1.25
Time (days)
Rel
ativ
e bo
dy w
eigh
t
5 10 15 20 25 30 35 400.75
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
1.25
Time (days)
Rel
ativ
e bo
dy w
eigh
t
5 10 15 20 25 30 35 400.75
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
1.25
Time (days)
Rel
ativ
e bo
dy w
eigh
t
5 10 15 20 25 30 35 400.75
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
1.25
Time (days)
Rel
ativ
e bo
dy w
eigh
t
5 10 15 20 25 30 35 400.75
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
1.25
Time (days)
Rel
ativ
e bo
dy w
eigh
t
5 10 15 20 25 30 35 400.75
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
1.25
Time (days)
Rel
ativ
e bo
dy w
eigh
tModeled vs. Observed for Groups
Model adequacy assessmentIndividual profiles vs. mean modeled body weight curves for each group
Control 15 mg/kg QD 30 mg/kg QD
60 mg/kg QD 15 mg/kg BID 30 mg/kg BID
20 mg/kg TID
14
Toxicity Model Results
5 10 15 20 25 30 35 400.75
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
1.25
Time (days)
Mic
e b
ody
wei
ght
(g)
Control15 mg/kg q7dx4
30 mg/kg q7dx4
60 mg/kg q7dx1
15 mg/kg BID7dx4
30 mg/kg BID7dx420 mg/kg TID7dx4
Modeled population-average animal weight curves for each dose group
15
ML Parameter Estimation
Computationally hard problem• Numerical solution of system of differential equations• Numerical integration due to random effects• Numerical optimization of resulting likelihood function• Three “heavy ”numerical problems nested in one another
Implementation in Matlab• Relying on standard functions is unacceptably slow• Special problem-specific method was developed for ODE
system solution and random effects integration • Numerical optimization was done by Matlab function
16
Regimens Simulation
Simulation settings• Dosing was performed until day 28 as in original study
• Doses from 1 to 30 mg/kg (QD, BID, TID) were used
• Dosing interval was varied between 1 and 14 days
Regimen evaluation• Efficacy and toxicity were computed for each regimen
• Efficacy was defined as overall tumor burden reduction
• Toxicity was defined as maximum relative weight loss
• Efficacy was plotted vs. toxicity for each simulation run
• Pareto-optimal solutions were identified for QD, BID, TID
17
Tumor Burden
0 10 20 30 40 50 600
0.5
1
1.5
2
2.5
Time (days)
Tum
or
wei
ght,
kg
Control
60 mg/kg q7dx4
20 mg/kg TID7dx4
Area under thetumor growth curve
18
Efficacy-Toxicity PlotRed – QD, blue – BID, green – TID
0 2 4 6 8 10 12 14 16 18 2050
55
60
65
70
75
80
85
90
95
100
30 mg/kg BID7dx4
20 mg/kg TID7dx4
Toxicity (%)
Eff
icac
y (%
)
19
Pareto-Optimal SolutionsRed – QD, blue – BID, green – TID
0 2 4 6 8 10 12 14 16 18 2050
55
60
65
70
75
80
85
90
95
100
Toxicity (%)
Eff
icac
y (%
)
200 2 4 6 8 10 12 14 16 18 20
50
55
60
65
70
75
80
85
90
95
100
Toxicity (%)
Eff
icac
y (%
)
Pareto-Optimal SolutionsRed – QD, blue – BID, green – TID
Zoomingthis part …
21
Pareto-Optimal SolutionsRed – QD, blue – BID, green – TID
0 1 2 3 4 5 680
85
90
95
14,1
15,1
16,1
17,1
18,1
8,1
9,1
10,1
5,1
6,1
7,1
10,2
12,2
Toxicity (%)
Eff
icac
y (%
)
Notation: dose in mg/kg, interval in days
220 1 2 3 4 5 6
80
85
90
95
14,1
15,1
16,1
17,1
18,1
8,1
9,1
10,1
5,1
6,1
7,1
10,2
12,2
Toxicity (%)
Eff
icac
y (%
)
Pareto-Optimal Solutions
Optimal regimens(QD, BID, TID)
Red – QD, blue – BID, green – TID
Notation: dose in mg/kg, interval in days
23
Optimal Regimens
0 10 20 30 40 50 60-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Time
log(
Tum
or w
eigh
t)
Control
QD 16 mg/kg every day
BID 9 mg/kg every day
TID 6 mg/kg every day
24
Prediction Accuracy
Regimen: TID 6 mg/kg every day
0.75 0.80 0.85 0.90 0.95
05
10
15
20
25
30
Efficacy probability distribution
Efficacy
De
nsi
ty
0.00 0.01 0.02 0.03 0.04
02
04
06
08
0
Toxicity probability distribution
Toxicity
De
nsi
ty
Methodology• Fisher’s information matrix computed numerically • Variance-covariance matrix for ML parameter estimates• Simulations performed to quantify prediction uncertainty
25
Closer Look at QD Administration
0 1 2 3 4 5 6 7 8 9 1050
55
60
65
70
75
80
85
90
95
100
10,1
11,1
12,1
13,1
14,1
15,1 16,1
17,1 18,1
18,2
19,1
19,2
20,2
21,2
22,2
23,2
24,2 25,2
26,2 27,2
28,2 29,2 30,2
Toxicity (%)
Eff
icac
y (%
)
Notation: dose in mg/kg, interval in days
26
In Vivo Study: Dosing Until Day 28
0 10 20 30 40 50 60-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Time
log(
Tum
or w
eigh
t)
Control
16 mg/kg QD until day 28
25 mg/kg Q2D until day 2835 mg/kg Q3D until day 28
30 mg/kg BID7D x 4
27
Summary
Methodological contribution• New multiobjective method for optimal regimen selection• Novel dynamic model for cancer tumor growth inhibition• Novel dynamic model for animal body weight loss
Practical contribution• More efficacious and less toxic in vivo dosing regimens• Better understanding of compound potential pre-clinically
Validation• Application of modeling results to in vivo study in progress