Pär Karlsson, AstraZeneca FMS vårmöte med årsmöte 23 Mars 2011 Örebro Universitet A simple model of phase II-III clinical programs.

Pär Karlsson, AstraZeneca

FMS vårmöte med årsmöte 23 Mars 2011 Örebro Universitet

A simple model of phase II-III clinical programs

Model

Phases of development

3 Pär Karlsson, FMS 20110323

Phase 2a Phase 2b

Stop

Phase 3

Phase 3

Stop Stop

Launch

Time of Ph 2a Time of Ph 2b Time of Ph 3 Time of sales

Probabilities of• Continue• Discontinuedevelopment after phase 2a

Probabilities of• Continue• Discontinuedevelopment after phase 2b

Probabilities of• Submit• Not submitNDA/MAA after phase 3

Transition probabilities - efficacy


Phase 2a Phase 2b

Stop

Probabilities of• Continue• Discontinuedevelopment after phase 2a

These probabilities are described by the power function

)Power(1)ntinue,Prob(Disco

)Power()nue,Prob(Conti

error] II Typefor 1 anderror I Typefor 0 [

efficacy True

yprobabiliterror II Type

yprobabiliterror I Type

functionon distributi normal Standard

)))()(()(()Power( 111

truetrue

truetrue

truetrue

true

truetrue

ee

ee

ee

e

α

ee

Similar transition probabilities after phase 2b. As Phase 3 consists of 2 independent studies the total power is the square of the power for one study.Independent of assumed effect and variability, but these will affect the size of the studies (costs)

Transition probabilities - safety

• There is always the possibility that an unforeseen safety finding is detected in a clinical study

• This is implemented as an independent Bernoulli variable in each phase. The probability of a safety finding might depend on the phase

- Probability to continue = Power * (1 - Prob(safety) )

Pär Karlsson, FMS 201103235

Value


• If the true efficacy is e then the Gain (sales-costs of sales) during one year is G(e)- In the simple model G(e) is constant above one critical value, and 0 below it.

Another assumption is that the Gain is constant during the period of marketing. Also the Gain is positive only until the expiry of the patent, after the expiry the Gain is 0.

• The total gain after Launch is G(e)*(Time of sales)• The value (=gain-cost) at the beginning of phase 3 is

G(e)*(Time of sales)*Power(Ph3,e) - Cost(Ph3)• The value at the beginning of phase 2b is

(G(e)*(Time of sales)*Power(Ph3,e) - Cost(Ph3))*Power(Ph2b,e) - Cost(Ph2b)

• The value at the beginning of phase 2a is((G(e)*(Time of sales)*Power(Ph3,e) - Cost(Ph3))*Power(Ph2b,e) - Cost(Ph2b))*Power(Ph2a,e) - Cost(Ph2a)

Apriori distribution of the true efficacy

• In order to get the value prior to phase 2a, we must integrate the value function over the apriori distribution of e

• In the simple model a beta-distribution is used• The support of the beta-distribution is [0,1]

• Simplification- same efficacy in all phases

• The integration over the apriori distribution is done by numeric integration (implemented in both Excel and SAS)


Input / Output to the model


Phase 2a Phase 2b Phase 3

Cost

Time

Type I error

Type II error

Probability of safety finding

Apriori distribution

Beta lower parameter

Beta upper parameter

Launch

End time of sales (from start of phase 2a)

Effect where sales begin to increase from zero

Effect where sales reach peak

Peak amount of sale

Output from the model

Expected cost

Expected value

Expected value / Expected cost

Results

Basic model

Input / Output to the model



Cost 10 25 100

Time 1 2 3

Type I error 0.05 0.05 0.025

Type II error 0.2 0.1 0.1


0.2 0.1 0


Beta lower parameter 0

Beta upper parameter 0

Launch


12


0.9


0.9

Sales per year 1000

Output from the model

Expected cost 34.3

Expected value 184.4

Expected value / Expected cost 5.38

Power for the different studies

Phase 2bPhase 2a

Phase 3

Power for the different studies, adjusted for safety findings

Phase 2bPhase 2a

Phase 3

Power for the different studies, adjusted for safety findings, adding program power

Phase 2bPhase 2a

Phase 3

Phase 2

Phase 2 and 3

What type I error rate in the Ph2a study is optimal?


Type I error

Type II error

0.01 0.44

0.03 0.27

0.05 0.20

0.10 0.11

0.15 0.07

0.20 0.05


Expected cost Expected value


Expected value / Expected costalpha beta E(cost) E(value)

E(value)/E(cost)

0.005 0.536 20.0 99.4 4.965

0.010 0.436 23.3 124.4 5.329

0.015 0.376 25.6 139.7 5.448

0.020 0.333 27.5 150.7 5.488

0.025 0.299 29.0 159.2 5.494

0.030 0.272 30.3 166.0 5.484

0.035 0.250 31.4 171.7 5.464

0.040 0.231 32.5 176.6 5.439

0.045 0.214 33.4 180.8 5.411

0.050 0.200 34.3 184.4 5.382

0.055 0.187 35.1 187.6 5.351

0.060 0.176 35.8 190.5 5.321

0.065 0.165 36.5 193.1 5.290

0.070 0.156 37.2 195.4 5.260

0.075 0.148 37.8 197.6 5.231

0.080 0.140 38.4 199.5 5.202

0.085 0.133 38.9 201.3 5.173

0.090 0.126 39.4 202.9 5.145

0.095 0.120 39.9 204.4 5.118

0.100 0.114 40.4 205.8 5.091

Is the optimal ph 2a type I error independent of apriori distribution?



E = 0.333

E = 0.5

E = 0.667

apriori distributions


alpha beta E =

0.333 E = 0.5E =

0.6670.005 0.536 -0.215 4.965 7.7590.010 0.436 -0.129 5.329 8.1080.015 0.376 -0.095 5.448 8.2040.020 0.333 -0.079 5.488 8.2250.025 0.299 -0.073 5.494 8.2160.030 0.272 -0.071 5.484 8.1930.035 0.250 -0.072 5.464 8.1630.040 0.231 -0.075 5.439 8.1290.045 0.214 -0.079 5.411 8.0940.050 0.200 -0.084 5.382 8.0590.055 0.187 -0.089 5.351 8.0230.060 0.176 -0.094 5.321 7.9880.065 0.165 -0.100 5.290 7.9540.070 0.156 -0.105 5.260 7.9210.075 0.148 -0.111 5.231 7.8880.080 0.140 -0.117 5.202 7.8570.085 0.133 -0.122 5.173 7.8260.090 0.126 -0.128 5.145 7.7960.095 0.120 -0.134 5.118 7.7670.100 0.114 -0.139 5.091 7.739

E = 0.333

E = 0.5

E = 0.667


Is a phase 2a study a waste of money?


Phase 2a Phase 2b

Phase 3

Phase 3

Stop Stop Stop

Launch

Phase 2b

Phase 3

Phase 3

Stop Stop

Launch

Alternative with a phase 2a study

Alternative without a phase 2a study


Input to the model



Cost 10 or 0 25 100

Time 1 or 0 2 3

Type I error 0.05 or 1 0.05 0.025

Type II error 0.2 or 0 0.1 0.1


0.2 0.1 0


Beta lower parameter 1 to 0

Beta upper parameter 0 to 1

Launch


12


0.9


0.9

Sales per year 1000

Apriori distributions

Lower parameter

Upper parameter

Expected value

0 1 0.3330.2 1 0.3750.5 1 0.4291 1 0.5001 0.5 0.5711 0.2 0.6251 0 0.667






No ! not with these assumptions

How high sales are necessary?

Input to the model



Cost 10 25 100

Time 1 2 3

Type I error 0.05 0.05 0.025

Type II error 0.2 0.1 0.1


0.2 0.1 0


Beta lower parameter 1 to 0

Beta upper parameter 0 to 1

Launch


12


0.9


0.9

Sales per year 500, 750, 1000


Lower parameter

Upper parameter

Expected value

0 1 0.3330.2 1 0.3750.5 1 0.4291 1 0.5001 0.5 0.5711 0.2 0.6251 0 0.667




The cost is same, as only the gain is different

The lower the gain the higher apriori confidence is needed



Note, that for this question it is enough to look at the expected value, as the important point is when the lines cross zero

Summary

Summary

• A simple model of the drug development has been developed

• Some important questions can be addressed

• As the model is simple, a more complete/complicated model should be developed

• The main purpose is to inspire statisticians to improve the development processes of drug development


Pär Karlsson, AstraZeneca FMS vårmöte med årsmöte 23 Mars 2011 Örebro Universitet A simple model of phase II-III clinical programs.

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