cancer trials. reading instructions 6.1: introduction 6.2: general considerations 6.3: single stage...

Cancer Trials

Reading instructions

• 6.1: Introduction• 6.2: General Considerations• 6.3: Single stage phase I designs• 6.4: Two stage phase I designs• 6.5: Continual reassessment• 6.6: Optimal/flexible multi stage designs• 6.7: Randomized phase III designs

What is so special about cancer?

•Many cancers are life-threatening.•Many cancers neither curable or controlable.•Malignant disease implies limited life expectancy.

•Narrow therapeutic window.•Many drug severely toxic even at low doses.•Serious or fatal adverse drug reactions at high doses.•Difficulty to get acceptance for randomization

The disease

The drugs

Ethics

?

Some ways to do it

•No healty volunteers.•Terminal cancer patients with short life expectancy.•Minimize exposure to experimental drug.•Efficient selection of acceptable drug.

The cancer programmePhase I: Find the Maximum Tolerable Dose (MTD)

The dose with probability of dose limiting toxicity less than p0 0:max pDLTPd d

Dd

DLT=Dose Limiting Toxicity

Phase II: Investigate anti tumour actividy at MTD using e.g. tumour shrinkage as outcome.

Phase III:Investigate effect on survival

Sufficient anti tumour activity

0poften between 0.1 and 0.4Doses Iidi ,,1 ,

Phase I cancer trialsObjective: Find the Maximum Tolerable Dose (MTD)

0:max pDLTPd dDd

pP DLT xp

p

1

ln

Use maximum likelihood to estimate and

ˆ

ˆ1ln

0

0

pp

xm

Phase I cancer trials

Design A

Start with a group of 3 patients at the initial

dose level

No toxicityNext group of 3 patients

at the next higher dose level

Next group of 3 patientsat the same dose level

Toxicity in at mostone patient

Next group of 3 patientsat the next higher dose level

Trial stops

Yes

Yes

No

No

If id is the highest dose

then 1id is the estimated MTD

•Only escalation possible.•Start at the lowest dose.•Many patients on too low dose.

Phase I cancer trials

•Escalation and deescaltion possible.•No need to start with the lowest dose.

ˆ

ˆ1ln

0

0

pp

xmMTD:

Design BStart with a single

patient at the initial dose level

No toxicityNext patient at the

same dose level

Next patient at the next lower dose level

No Toxicity in two consequtive patients

Next patient at the next higher dose level

Trial stops

Yes

No

No

Toxicity in two consequtive patients

Yes

No

Next patient at the next lower dose level

Yes

Phase I cancer trials

Design DStart with a group of 3 patients at the initial

dose level

Next group of 3 patients at the same dose level

Next group of 3 patients at the next lower dose level

Toxicity in onepatient

Next group of 3 patients atthe next higher dose level

Yes

No

Yes

No

Repeat the process untilexhaustion of all dose levelsor max sample size reached

Toxicity in morethan one patient

•Escalation and deescaltion possible.•No need to start with the lowest dose.

ˆ

ˆ1ln

0

0

pp

xmMTD:

Phase I cancer trials

Design BD

Run design Buntil it stops.

DLT in last patientRun design D

starting at the next lower dose level.

Run design D starting at same

dose level.

Phase I cancer trialsContinual reassessment designs

0pAcceptable probability of DLT

00 :max pDLTPdd dDd

MTD

Dose response model: ii xx ,Plogit

Assume fixed.

Let g be the prior distribution for the slope parameter.

Phase I cancer trialsOnce the response, DLT or no DLT, is available from the current patient at dose 1ix the estimated slope is update as:

dfE iii 11 || where

dzzgzq

gqf

i

ii

1

11|

where

jj

i

yj

i

j

yj xPxPq

11

1

,1,1

is the likelihood function, and

11111 ,,,, iii yxyx is the cumulative data up to the i-1 patient.

Phase I cancer trials

The next dose level is given by minimizing 0,P px ii

MTD is estimated as the dose xm for the hypothetical n+1 patient.

The probability of DLT can be estimated as mmx ,P

•CRM is slower than designs A, B, D and BD.•Estimates updated for each patient.•CRM can be improved by increasing cohort size

Phase II cancer trialsObjective: Investigate effect on tumor of MTD.

Response: Sufficient tumour shrinkage.

•Stop developing ineffective drug quickly.•Identify promising drug quickly.

Two important things:

Progression free survival.

Phase II cancer trialsOptimal 2 stage designs.

First stage: n1 patients:

Second stage:n2 patients:

0pUnacceptable response rate:

1pAcceptable response rate: 10 pp

Test: 00 : ppH 01 : ppH vs.

Stop and reject the drug if at most r1 successes

Stop and reject the drug if at most r successes

error I TypeP error II TypeP

Phase II cancer trialsHow to select n1 and n2 ?

Minimize expected sample size under H0: 21 01 nPETnNE p

011001

1 ,;11

0pnrBpp

i

nPET

r

ip

where is the

probability of early termination.

Given p0, p1, and , select n1, n2, r1 and r such that

21 1 nPETnNE is minimized. Nice discrete problem.

),min(

101101011

11

1

0,;,;,;)drugreject (

rn

rxp pnxrBpnxbpnrBP

Phase II cancer trialsAssume specific values of p0, p1, and

For each value of the total sample size n, n1[1,n-1] and r1[0,n1]

Find the largest value of r that gives the correct error II TypeP

Check if the combination: n1, n2, r1 and r satisfies error I TypeP

If it does, compare E[N] for this design with previous feasible designs.

Start the search at

2

01

112101

21

2

pp

zzpppp

!: not unimodal

Phase II cancer trials

Efficacy hypotheses Reject drug if p0 p1 r1/n1 r/n E[N] PET 0.05 0.25 0/9 2/17 12.0 0.63 0.30 0.50 5/15 18/46 23.6 0.72 0.70 0.90 4/6 22/27 14.8 0.58

20.0error II Type P 05.0error I Type POptimal 2 stage designs with:

Efficacy hypotheses Reject drug if p0 p1 r1/n1 r/n E[N] PET 0.05 0.25 0/12 2/16 13.8 0.54 0.30 0.50 6/19 16/39 25.7 0.48 0.70 0.90 19/23 21/26 23.2 0.95

Corresponding designs with minimal maximal sample size

Phase II cancer trialsOptimal flexible 2 stage designs.

In practise it might be difficult to get the sample sizes n1 and n2 exactly at their prespecified values.

Solution: let N1{n1, …n1+k} with P(N1=n1j)=1/k, j=1,…k and N2{n2, …n2+k} with P(N2=n2j)=1/k , j=1,

…k.

P(N1=n1j ,N2=n2j)=1/k2 , j=1,…k.

N1 and N2 independent, n1+k< n2.

Total samplesize N=N1+N2

Phase II cancer trials

jnPETinNE p 21 01

For a given combination of n1 +i and n2 +j:

011001

1 ,;11

0pinrBpp

i

inPET

r

ip

where

Minimize the average E[N]

(Average over all possisble stopping points)

Phase II cancer trials

Efficacy hypotheses Reject drug if p0 p1 r1/n1 r/n E[N] PET 0.05 0.25 0/5-10, 1/11-12 2/17-21, 3/23-24 11.8 0.73 0.30 0.50 3/11, 4/12-14

5/15-16, 6/17-18 16/40-41, 17/42-44 18/45-46, 19/47

24.0 0.68

0.70 0.90 4/6, 5/7, 6/8, 7/9, 8/10-11, 9/12, 10/13

22/27, 23/28-29, 24/30, 25/31, 26/32-33, 27/34

15.2 0.74

Flexible designs with 8 consucutive values of n1 and n2.

20.0error II Type P 05.0error I Type P

Phase II cancer trialsOptimal three stage designs

The optimal 2 stage design does not stop it there is a ”long” initial sequence of consecutive failures.

First stage: n1 patients:Second stage: n2 patients:

Stop and reject the drug if no successes

Stop and reject the drug if at most r2 successesThird stage: n3

patients:Stop and reject the drug if at most r3 successes

For each n1 such that:

11 |reject 1 1 pHPp an

Determine n2, r2, n3, r3 that minimizes the expected sample size.

More?

Phase II cancer trials

Efficacy hypotheses

Reject drug if at least Stage 1 Overall

p0 p1 r1/n1 r2/n2 r3/n3 E[N] PET PET 0.05 0.25 0/7 1/15 3/26 10.9 0.70 0.87 0.30 0.50 0/5 5/15 19/49 22.5 0.17 0.73 0.70 0.90 0/5 4/6 22/27 14.8 0.00 0.58

Optimal 3 stage design with n1 at least 5 and

20.0error II Type P 05.0error I Type P

Example:

Phase II cancer trialsMultiple-arm phase II designs

Say that we have 2 treatments with P(tumour response)=p1 and p2

Select treatment i for further development if

ji pp ˆˆ

Assume p2>p1. The probability of correct secection is

2122 ,|ˆˆ ppppPPCorr

Ambiguous if ji pp ˆˆ

n

x

n

y

ynyxnxnyx pppp

y

n

x

nI

0 01122 11

Phase II cancer trials

n

x

n

y

ynyxnxnyx pppp

y

n

x

nI

0 01122 11

The probability of ambiguity is

2122 ,|ˆˆ ppppPPAmb

Ambiguous if ji pp ˆˆ

Phase II cancer trials

n P1 P2 PCorr PAmb PCorr+0.5PAmb 50 0.25 0.35 0.71 0.24 0.83 50 0.20 0.35 0.87 0.12 0.93 75 0.25 0.35 0.76 0.21 0.87 75 0.20 0.35 0.92 0.07 0.96 100 0.25 0.35 0.76 0.23 0.87 100 0.20 0.35 0.94 0.06 0.97

Probability of outcomes for different sample sizes (=0.05)

Select n such that: AmbCorr PP

Phase II cancer trialsSample size can be calculated approximately by using

ZPPCorr

ZPZPPAmb

Where 12 pp 2211 111

ppppn

1,0~ NZ

The power of the test of 211210 : vs.: ppHppH is given by

12

2/12

2/11pp

Zpp

Z

2/Z is the upper /2 quantile of the standard normal distribution

Phase II cancer trials

Letting AmbCorr PP it can be showed that:

2/1 Z

Sample size can be calulated for a given value of .

=0 =0.5 P1 P2 =0.90 =0.80 =0.90 0.05 0.20 32 13 16 0.10 0.25 38 15 27 0.15 0.30 53 17 31 0.20 0.35 57 19 34 0.25 0.40 71 31 36 0.30 0.45 73 32 38 0.35 0.50 75 32 46 0.40 0.55 76 33 47

Phase II cancer trialsMany phase II cancer trials not randomized

Treatment effect can not be estimated due to variations in:

•Patient selection•Response criteria•Inter observer variability•Protocol complience•Reporting procedure????•Sample size (?)

Phase III cancer trial

It’s all about survival! Diagnosis

Treatment

Progression

Death from the cancer

Death fromother causes

•Progression free survival•Cause specific survival•All cause survival

The competing risks model

Diagnosed with D

Death from other cause

Death cused by D)(tD

)(tD

)()()( ttt DDtot

The aim is to estimate the cause specific survival function for death caused by D.)(tSD

The usual way)(tSDThe cause specific survival, , is usually estimated

using the cause of death information and standard methods such as Kaplan-Meier or life tables, censoring for causes of death other than D.

Problem: The actual cause of death is not always equal to the registered cause of death.

)()()( ttt DDtot

)(*)()( tStStS DDtot

t

duutS0

.. )(exp)(

The model :

can be formulated using the corresponding survival functions as:

using

)(ˆ/)(ˆ)(ˆ tStStS DtotD Estimate:

Estimation)(tStot can be estimated directly from data.

)(tSD relating to deaths from causes other than D can be estimated using data from a population registry if:

)(tSD : the “expected” survival given age, sex and calender year

D is a ‘rare’ cause of death in the population.

The study population has the same risk of dyingfrom other causes as the background population.

The intuitive way (no formulas)

• We have the annual survival probability given age, sex and calender year.

• Multiply to get the probability of surviving k years for each individual

• Average to get the expected survival.

Converting intuition into formulas

Individuals i=1 …n, time intervals j=1 to k

For each individual we have the “expected” probability )( ji tPof surviving time interval j.

Now

n

i

j

hhij

ID tP

ntS

1 1

)(1

)(ˆ

is called the Ederer I estimate of the expected survial

Problemo

t

91958882857774727273706663

atriskat tj

tj tj+1

All inividuals contributes to

)(ˆjtot tSOnly individuals at risk at tj contributes to

)(ˆj

ID tS

age

Solution:Let only individuals at risk contribute to the expected survival.

j

h

n

ihitIi

hj

IID tP

tntS

h1 1

)}({ )(1)(

1)(ˆ

where )(tn is then number of individuals at risk at time t.

)(tIand is the index set of individuals at risk at time t

The Ederer II estimate

Expected survival for a group pf patients diagnosed with prostate

cancer 1992

Ederer I and Ederer II expected survival

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9 10

Time from diagnosis (years)

Exp

ecte

d su

rviv

al

Ederer I

Ederer II

Estimated cause specific survival of patients diagnosed

with prostate cancer 1992

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8

Time from diagnosis (years)

caus

e sp

ecif

ic s

urvi

val

Ederer II

Life table

Continuous time, expected hazard

)(* ti : ‘expected’ morality (hazard) from the population for individual i.

)(tYi : at risk indicator for individual i at time t.

i

i tYtY )()( : number of individuals at risk

The expected integrated hazard is now given by

t n

i

ii du

uY

uYutA

0 1

**

)(

)()()(

Cont. time relative survival

Rewriting the model: )()()( ttt DDtot

using integrated hazards we can estimate t

D duu0

)( using

)()(

)(ˆ * tAXY

DtA

tX i

iD

i

where,..., 21 XX =event times

iD = # events at time iX

Now the continuous time relative survival is given by:

)(ˆexp

)(ˆ)(ˆ

tA

tStS

D

totD

Illustrated

t

)(*1 t

)(*2 t

)(*3 t

)(*4 t

)(*1 t

Illustrated

t

)(*1 t

)(*2 t

)(*3 t

)(*4 t

)(*1 t

)(* t

Example

Example

Population based trials

In many countries there are cancer registers where data on all cases of cancer diagnoses are collected.

Many countries also have a cause of death registry

Intervension Incidence Death

Incidence Intervension Death

Often observational studies i.e. no randomization.