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