COPIA SIN
TÍTULO DÉCIMO QUINTO
The ICS Model is derived from the proportional hazards assumption as shown in Collett [15]. It has traditionally been used with analysing time-to-event outcomes in order to reduce bias associated with the uncertainty of the exact timings of events. For example, in the analysis of progression free survival (PFS), the event of disease progression is usually determined at scheduled visits and assessments. The exact time of progression is therefore unknown and assigned to the next scheduled assessment. When symptoms associated with progression arise, usually for control treatments, unscheduled visits and assessments occur which implies that progression can be detected earlier and the PFS is therefore obtained accurately. If a control treatment estimates the PFS accurately, the hazard associated with the control treatment is also estimated accurately. The occurrence of symptoms associated with progression may be suppressed for the experimental treatment and will therefore not be observed. The occurrence of progression may then not be observed until the scheduled assessment, which overestimates the PFS, which in turn, underestimates the hazard associated with the experimental treatment. The overall estimated hazard ratio (HR) will then be underestimated and the experimental treatment may look to perform better than it actually does. By allowing inclusion of observations at each interval, the uncertainty of the exact time of progression is localised to one specific period. This therefore
reduces the bias observed when unscheduled assessments occur intermittently with scheduled assessments.
The idea of observing events occurring in different intervals can be directly related to the design of Phase I dose-finding trials. Traditionally, dose-finding studies observe the occurrence of a patient’s first DLT during one fixed period of time. Patients should remain on therapy after the first cycle of treatment, but only observations from the first cycle will be used for analysis. The observations are binary, either they had a DLT in cycle 1 or they did not, and it is therefore very easy to analyse these events over a fixed period of time, either with a model-based analysis or rule-based. The first cycle of therapy is also the cycle for which the first DLT for a patient is expected to occur with highest frequency, although they can be expected to occur at a decreasing rate of frequency over time. By only using one cycle of therapy for the analysis, the trials are often very short in time. To observe for multiple cycles o f therapy would increase the duration of the trial and when observing binary responses, the same issue arises as described for analysing PFS. The time of DLT would not necessarily be captured if multiple cycles were combined to one fixed period of time, and the probability of DLT (P(DLT)) would be assumed to be constant for the whole time period. This may underestimate a patient’s chance of experiencing a DLT early in the treatment phase and overestimate it later on.
The ICS model now becomes attractive since it can look at a larger fixed period of time, and break it down into intervals, in this case cycles. The binary endpoint of whether a patient’s first DLT occurs or not is still utilised, however the endpoint is now whether a DLT occurred in a given cycle, and the occurrence of DLTs will now be dependent on the cycles that occur previously and a DLT not occurring. A patient will contribute information to the analysis for all cycles of therapy they complete, up
to and including the cycle during which they have a DLT. By allowing patients to contribute information for every completed cycle of therapy, this model allows for non-informative withdrawal by including cycles up to the first DLT or withdrawal. Although dropouts before the occurrence of a DLT are not particularly expected in this phase of development, it is still an issue that should be considered since cancer patients may experience progression so therefore may be withdrawn, and since the sample size for Phase I trials is so small already.
For Phase I trials, interest lies in modelling the probability of a patient having their first DLT, on dose level d{j), j = 1 denoted by p (j). Traditionally, this is the probability of having a DLT during the first cycle of therapy. Interest may lie in assessing the probability of a DLT over s cycles o f therapy, where each cycle of therapy 1,1 - l,...,s , with s being the maximum number of cycles, begins at time cM and finishes at time c, with c0 = 0. Let p {]), be the probability of a patient
experiencing their first DLT on dose level d(j) during cycle / and p (/)f be the probability that no DLT occurs during the first s cycles, i.e. the complement of p (j)s. The cumulative probability of an event occurring during the first / cycles for a patient on dose level d(j) can then be defined as:
/
m - \
Here, p (j)m for m = l,...,/are probabilities relating to mutually exclusive events. Therefore the sum from m = \,...,s + \ is equal to 1.
The ICS model is based on the probability of a DLT occurring during a given cycle conditional on the fact that there has been no DLT in previous cycles. The conditional
probability of having an event in cycle / (after time cM) given that there has been no
where T{J) is the true time at which a DLT occurs on dose level d(J). The conditional probabilities can then be combined to calculate the unconditional probabilities of DLT
p {j), for a given cycle / and dose level d(j) as follows:
It can be noted that;r0 ) i + 1 = 1 since it is assumed that if the patient remained on
treatment after surviving the firsts cycles, at some point in the interval (c,,oo) a DLT would occur. This then explains why the unconditional probability of an event occurring in the interval (cv,oo) reduces as above. This arrangement of probabilities means that the likelihood can be constructed in terms of the conditional probabilities. DLT in any interval prior to cycle I is defined as n (j)l = P(°t- 1 < T(J) < c, | T(J) > cM )
- n ( j ) l [ l P { C I - 2 K T ( j ) - C l - 1 I T ( J ) > C l - 2 ) ] P { P ( i ) > C I - 2 )
etc.
This can then be generalised to the following:
PU)i n U)i
( i - n a)1) ( l - n U)2) ... ( l -
( ! - 1Tu n ) i 1 ~ n u)2) - C1 - 7ro ),/-2 )(i - n u i i - 1)
I = 1
= 2 ,..., s I > s.
k .v+1 k .v+1
flfl
A/)/nfl[0
k (j\M/-2))",0 ^(yy-O^uv] , / = l / = 1 7 = 1 / = 1 = f i > w iw i r i o - ^(7)/ r w <4-1 > 7 = 1 /= 1 £ 5 1u■)/ = n n - o , / " o -* < ,* ) 7 = 1 / = lHere /(/)/ is equal to the number of toxicities observed on dose level d(J) during cycle / and qU)l is equal to the number of patients who have completed I cycles of therapy
without experiencing a DLT {n(j)l - f (y)/) ■
This is a Binomial likelihood so a generalised linear model can be used to model these probabilities. The link function for this generalised linear model can be defined from the proportional hazards assumption via the following mechanism (as seen in Collett [15]).
Redefining the conditional probabilities in terms of survival probabilities is shown below;
71 a)i = p{ci-i < P(j) - ci I > c>-\
)
_ SU)(cl_y) - S u)(cl)*5(7) (C/_,) ’
Where S{j)(c,) is the survival probability (i.e. the probability of ‘surviving’ the cycle without experiencing an event) associated with dose level ( j ) by the end of cycle /. This can be simplified to:
SU){c,.x)
Where r/{j) is the linear predictor of covariates associated with dose level d(j) such as
rj{j) - 6\og{d{j)) , and S0(c,) is the baseline survival function at the end of cycle I , i.e.
the survival probability associated with dose level </(0)at the end of cycle /. When the dose is transformed, this transformation of this dose level d{0) will be equal to 0, i.e. to
apply a log transformation to d(Q), d(0) will be equal to 1 such that once transformed is
equal to 0. Applying the proportional hazards assumption to the above rearrangement gives;
This link function is a complementary log-log link function, which includes a term that is dependent solely on the interval during which the event occurred. This is a factor with s levels which therefore allows separate intercept terms to be estimated for each cycle and therefore can allow for a differing dose-response relationship with time. The intercepts and the log(dose)-coefficient will all be log-hazard ratios comparing dose level d(J) to d{0).
Interest lies in estimating the dose that corresponds to a pre-defined probability of toxicity after 5 cycles of therapy. This can be defined as;
l o g ( —l o g ( l - / T ( / ) / ) ) = 770 ) + l o g - l o g A M (4.2)
= \ n + ri
V u ) ( - Cs )
( n U ) i 5 = 1
By putting in a rearrangement of the link function (in terms of;r(/)v) such that
tjU) = 0\og p U)(cs) in terms of the parameters can be found to be;
Pu)(cs ) = ^ - Qx v { du ) \ r en ~ en
which can then be rearranged in terms of the dose;
d(j) =exp
log lo g (l-/?0)( 0 )
1 1 1 *1 1
e (4.3)
On analysing the responses with the model described in equation (4.2), parameter estimates will be obtained. By replacing p U){cs) in equation (4.3) with the TTL and including all the parameter estimates, the estimate of the TD associated with the TTL will be obtained. The derivation of rearrangement (4.3) is shown in Appendix 1.