Bayesian Continual Reassessment Method for Phase I Dose-Escalation Trials

Description

Implements a wide variety of Bayesian CRM designs, including 1-parameter, 2-parameter and Escalation With Overdose Control (EWOC) designs. The program can run interactively, allowing the user to enter outcomes after each cohort has been recruited, or via simulation to assess operating characteristics.

Usage

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bcrm(stop = list(nmax = NULL, nmtd = NULL, precision = NULL, nmin = NULL), 
     data = NULL, p.tox0 = NULL, sdose = NULL, dose = NULL, 
     ff, prior.alpha, cohort = 3, target.tox, constrain = TRUE,
     sdose.calculate = "mean", pointest = "plugin", tox.cutpoints = NULL,
     loss = NULL, start = NULL, simulate = FALSE, nsims = 1, truep = NULL, 
     threep3 = FALSE, method = "exact", burnin.itr = 2000, production.itr = 2000,
     bugs.directory = "c:/Program Files/WinBUGS14/", plot = FALSE, seed = NULL,
	quietly = FALSE, file = NULL, N, tox, notox)

Arguments

stop

A list of stopping rules for the trial. One or more of the following options should be specified

nmax

The maximum sample size of the trial

nmtd

The maximum number to be treated at final maximum tolerated dose (MTD) estimate, i.e. if the next recommended dose has already been administered to nmtd patients then the trial will stop

precision

A vector of the lower and upper percentage points that the MTD 95% credible intervals for the risk of toxicity should lie within

nmin

The minimum sample size of the trial. To be used in conjunction with nmtd or precision

data

A named data frame giving information about dose and toxicity from previously recruited patients. If missing, then it is assumed that no data have thus far been collected. Contains the following variables:

patient

Recruited patient numbers, 1,...,n

dose

Dose levels of recruited patients, ranging from 1,...,k

tox

An indicator variable for each patient (1=toxicity, 0=no toxicity)

p.tox0

A vector of length k listing the prior probabilities of experiencing the outcome at each dose level 1,...k. The standardised dose levels (CRM "skeleton") are formed from these probabilities using the inverse of the functional form, with a plug-in estimate for the prior mean or median of alpha, as specified in ff, prior.alpha and sdose.calculate. Alternatively standardised dose levels can be given directly using sdose.

sdose

A vector of length k listing the standardised doses to be used in the CRM model. Only required if p.tox0 is missing.

dose

Optional vector of length k of actual doses for plotting purposes

ff

A string indicating the functional form of the dose-response curve. Options are

ht

1-parameter hyperbolic tangent

logit1

1-parameter logistic

power

1-parameter power

logit2

2-parameter logistic

prior.alpha

A list of length 3 containing the distributional information for the prior. The first element is a number from 1-4 specifying the type of distribution. Options are

  1. Gamma(a,b), where a=shape, b=scale: mean=a*b, variance=a*b*b

  2. Uniform(a,b), where a=min, b=max

  3. Lognormal(a,b), where a=mean on the log scale, b=variance on the log scale

  4. Bivariate Lognormal(a,b), where a=mean vector on the log scale, b=Variance-covariance matrix on the log scale. This prior should be used only in conjunction with a two-parameter logistic model.

The second and third elements of the list are the parameters a and b, respectively.

cohort

The size of each cohort of patients that are sequentially recruited to the trial. Defaults to 3

target.tox

The target toxicity probability. Defaults to 1/3.

constrain

Should a dose-skipping constraint be placed on the escalation procedure, as imposed by a modified CRM? Defaults to TRUE.

sdose.calculate

What plug-in estimate of the prior alpha should be used to calculate the standardised doses? Options are "mean" (default) or "median". Only required if sdose is missing.

pointest

Which summary estimate of the posterior distribution should be used to choose the next dose. Options are "plugin" (default) where the posterior mean of the model parameter(s) is plugged into the function form to obtain estimates of toxicity, or "mean" where the posterior mean probabilities of toxicity are directly used. Alternatively, a number between 0 and 1 can be specified representing the quantile of the maximum tolerated dose (MTD) posterior distribution (e.g. 0.5 specifies the posteior median). This produces an Escalation With Overdose Control (EWOC) design if the quantile is less than 0.5 (see details). Currently, EWOC designs must be fit using MCMC methods.

tox.cutpoints

A vector of cutpoints for toxicity intervals if these are to be used to choose next dose. Defaults to NULL. For example Underdosing [0,0.2], Target dosing (0.2, 0.35], Excessive toxicity (0.35, 0.60], Unacceptable toxicity (0.60, 1.00] set tox.cutpoints=c(0.2,0.35,0.60).

loss

A vector of length length(tox.cutpoints)+1 specifying the losses associated with each toxicity interval, e.g. Underdosing = 1, Target dosing =0, Excessive toxicity=1, Unacceptable toxicity=2

start

Dose level used at the beginning of the trial. Required if constrain=TRUE.

simulate

Should a simulation be conducted to assess operating characteristics? Defaults to TRUE. If FALSE, a single CRM trial is run interactively, allowing the user to input outcomes after each cohort is recruited.

nsims

Number of simulations to perform if simulate==TRUE (defaults to 1).

truep

A vector of length k giving the true probabilities of the outcome (toxicity) at each dose level 1,...,k in order to simulate data. Only required if simulate=TRUE

threep3

Should operating characteristics of a standard 3+3 rule-based design be calculated, for comparison with bcrm design? Defaults to FALSE. Only used in a simulation study, i.e. when simulate=TRUE.

method

Optimisation method: options are "exact" (the default), "rjags", "BRugs", or "R2WinBUGS".

burnin.itr

Number of burn-in iterations (default 2000).

production.itr

Number of production iterations (default 2000).

bugs.directory

Directory that contains the WinBUGS executable if method="R2WinBUGS". Defaults to "C:/Program Files/WinBUGS14/".

plot

Should the dose-response curve be plotted after each cohort has been entered? Defaults to FALSE.

seed

Integer defining the state of the random number generator to allow reproducibility of results. The default is to not specify a seed.

quietly

Should the simulation number output be suppressed when running bcrm? Defaults to FALSE.

file

File name where the dose-response plots are stored, in a pdf format. The program will ammend the current sample size to the end of the file name.

N

Final sample size (deprecated). To be replaced with stop in future versions.

tox

(Deprecated). A vector of length k listing the number of patients who have experienced the outcome (toxicity) at each dose level 1,...,k.

notox

(Deprecated). A vector of length k listing the number of patients who have not experienced the outcome (toxicity) at each dose level 1,...,k.

Details

bcrm implements a Bayesian continual reassessment method (CRM) (O'Quigley et al., 1990); an adaptive design in which cohorts of patients are sequentially recruited into a Phase I trial. A binary toxicity outcome is assumed (e.g. Dose Limiting Toxicity / No Dose Limiting Toxicity). The current cohort are given a dose "closest" to the specified target toxicity level, as estimated from the posterior distributions of toxicity at each dose level from the patients thus far recruited. If pointest="mean" then the posterior mean probability of toxicity is used to choose the next dose. If pointest="plugin", however, the posterior mean of the model parameter(s) is plugged-into the functional form of the dose-toxicity model. To implement an EWOC design (Babb et al., 1998), pointest should be a quantile, q, between 0 and 0.5. The posterior distribution of the MTD (the dose in which the probability of toxicity is equal to the target toxicity) is then calculated and the next patient is given dose closest to the qth quantile of the MTD distribution.

Alternatively, escalation can be based on intervals of toxicity from the posterior distribution using a loss function, see Neuenschwander et al., 2008. To implement this approach, the user should specify the cutpoints of the toxicity intervals using tox.cutpoints and the associated losses using loss.

The possible choice of dose-toxicity model can be specified using ff, and includes the 1-parameter hyberbolic tangent, logistic or power "working models", and the 2-parameter logistic model as follows:

Hyperbolic Tangent

p(Tox|d*)=[(tanh(d*)+1)/2]^α

Logistic (1-parameter)

p(Tox|d*)=exp(3+α d*)/(1+exp(3+α d*))

Power

p(Tox|d*)=d*^α

Logistic (2-parameter)

p(Tox|d*)=exp(log(α_1)+α_2 d*)/(1+exp(log(α_1)+α_2 d*))

where α>0 is the single positive-valued parameter for the 1-parameter models, and log(α_1) and α_2>0 are the intercept and slope parameters of the 2-parameter model.

The standardised doses, d*, are specified by the user using sdose, or alternatively the prior probability of toxicity at each dose level is specified using p.tox0. If the latter is used, then the standardised doses are calculated using the inverse of the functional form and a plug-in estimate of the prior mean or median, as specified in sdose.calculate, as follows

d* = f^{-1}(p.tox0,α= a)

where f^{-1} is the the inverse of the chosen functional form, and the parameter(s) of the model are set equal to a, either the prior mean or median of α.

Data that have already been accrued can be entered using the data argument. A constrained CRM design can be implemented using constrain=TRUE, in which case dose-skipping is prohibited (i.e. the next cohort can only be dosed up to one dose level above the current cohort). If a constrained model is used then the starting dose must be specified using start. Alternatively, if data have already been accrued, then the dose level of the last recruited patient determines the constraint for the next patient.

The prior is set using prior.alpha. For example prior.alpha=list(1,1,1) specifies a Gamma prior with shape and scale parameters both equal to one (i.e. an Exponential(1) distribution), whilst prior.alpha=list(2,0,10) specifies a Uniform(0,10) prior.

To specify a fixed maximum sample size of size m use stop=list(nmax=m). Alternatively, the trial can stop after m2 patients have been treated at the current MTD estimate, by setting stop=list(nmtd=m2). To stop the trial when the MTD estimate is within a certain level of precision, use stop=list(precision=c(l,u)), where l and u are the lower and upper percentage points that the MTD 95% credible intervals for the risk of toxicity should lie within. Finally, to prevent the trial stopping too early using these rules, the argument stop=list(nmin=m3) can be used to ensure the sample size is greater than or equal to m3. Stopping rules can be used on their own or in combination.

The trial can be run interactively using simulate=FALSE, where the user enters the outcomes for each new cohort, or as a simulation study when simulate=TRUE.

The default calculations use exact methods (method="exact") to calculate the mean and quantiles for the posterior distributions. There are three choices for MCMC calculations: method="rjags", method="BRugs" or method="R2WinBUGS". The first uses the JAGS software, the second uses OpenBUGS, whilst the latter uses WinBUGS. To implement these methods, users require one or more of these packages to be installed on their system.

A simulated bcrm design can be compared with the standard 3+3 rule-based method, see threep3 for more details.

Value

bcrm returns an object of class "bcrm" or "bcrm.sim"; the latter occuring when a simulation has been conducted (simulate=TRUE). The function print (i.e. print.bcrm or print.bcrm.sim) can be used to obtain summary information about the design used, the data observed, current posterior estimates of toxicity, and the next recommended dose level.

An object of class "bcrm" is a list with the following components:

dose

Range of doses

sdose

Standardised doses

tox

A vector of length k listing the number of patients who have experienced the outcome (toxicity) at each dose level 1,...,k

notox

A vector of length k listing the number of patients who have not experienced the outcome (toxicity) at each dose level 1,...,k

ndose

A list of lists containing for each cohort the components ndose, the dose level recommended for the next patient, est, the estimated probabilitieis of toxicity using the chosen metric (e.g. plugin, mean, quantile), mean, the posterior mean probability of toxicity at each dose, sd, the posterior standard deviation for probability of toxicity at each dose, quantiles, the posterior quantiles for probability of toxicity at each dose. This information is only provided for cohorts recruited subsequent to any data given using tox and notox. The first component relates to the prior information.

constrain

Whether a constrained CRM design was used

start

The starting dose for the latest run of the model if constrain=TRUE

target.tox

The target toxicity level

ff

A number from 1-4 identifying the functional form, 1 = Hyperbolic tangent, 2 = 1-parameter logistic, 3 = Power, 4 = 2-parameter logistic

method

The calculation method used

pointest

The summary estimate used to choose the next dose, see pointest

prior.alpha

Information about the prior used for alpha, see prior.alpha

data

A data frame with variables ‘patient’, ‘dose’ and ‘tox’ listing the doses and outcomes of all patients in the trial

An object of class "bcrm.sim" is a list of length nsims. Each component is itself a list with components similar to those obtained from a "bcrm" object. The print function, print.bcrm.sim should be used to obtain operating characteristics from the simulation.

Note

Currently, the re-parameterisation of the two-parameter model proposed by (Babb et al., 1998) is not implemented. Therefore, users wishing to implement an EWOC design should check whether their choice of prior for the model parameter(s) translates to a sensible prior for the MTD distribution before they implement the design. For example

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	prior.alpha<-list(1,1,1)
	ff<-"ht"
	target.tox<-0.2
	samples.alpha<-getprior(prior.alpha,2000)
	mtd<-find.x(ff,target.tox,alpha=samples.alpha)
	hist(mtd)

One-parameter models are designed as working models only, and should not be used with an escalation strategy based on intervals of the posterior probabilities of toxicity.

Author(s)

Michael Sweeting mjs212@medschl.cam.ac.uk (University of Cambridge, UK), drawing on code originally developed by J. Jack Lee and Nan Chen, Department of Biostatistics, the University of Texas M. D. Anderson Cancer Center

References

Sweeting M., Mander A., Sabin T. bcrm: Bayesian Continual Reassessment Method Designs for Phase I Dose-Finding Trials. Journal of Statistical Software (2013) 54: 1–26. http://www.jstatsoft.org/article/view/v054i13

O'Quigley J., Pepe M., Fisher L. Continual reassessment method: a practical design for phase I clinical trials in cancer. Biometrics (1990) 46: 33–48.

Babb J., Rogatko A., Zacks S. Cancer phase I clinical trials: efficient dose escalation with overdose control. Statistics in Medicine (1998) 17: 1103–1120.

Neuenschwander B., Branson M., Gsponer T. Critical aspects of the Bayesian approach to phase I cancer trials. Statistics in Medicine (2008) 27: 2420–2439.

See Also

print.bcrm, print.bcrm.sim, plot.bcrm, plot.bcrm.sim, threep3

Examples

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## Dose-escalation cancer trial example as described in Neuenschwander et al 2008.
## Pre-defined doses
dose<-c(1,2.5,5,10,15,20,25,30,40,50,75,100,150,200,250)
## Pre-specified probabilities of toxicity
## [dose levels 11-15 not specified in the paper, and are for illustration only]
p.tox0<-c(0.010,0.015,0.020,0.025,0.030,0.040,0.050,0.100,0.170,0.300,0.400,0.500,0.650
  ,0.800,0.900)
## Data from the first 5 cohorts of 18 patients
data<-data.frame(patient=1:18,dose=rep(c(1:4,7),c(3,4,5,4,2)),tox=rep(0:1,c(16,2)))
## Target toxicity level
target.tox<-0.30

## A 1-parameter power model is used, with standardised doses calculated using 
## the plug-in prior median
## Prior for alpha is lognormal with mean 0 (on log scale) 
## and standard deviation 1.34 (on log scale)
## The recommended dose for the next cohort if posterior mean is used
Power.LN.bcrm<-bcrm(stop=list(nmax=18),data=data,p.tox0=p.tox0,dose=dose
  ,ff="power",prior.alpha=list(3,0,1.34^2),target.tox=target.tox,constrain=FALSE
  ,sdose.calculate="median",pointest="mean")
print(Power.LN.bcrm)
plot(Power.LN.bcrm)

## Simulate 10 replicate trials of size 36 (cohort size 3) using this design 
## with constraint (i.e. no dose-skipping) and starting at lowest dose
## True probabilities of toxicity are set to pre-specified probabilities (p.tox0) 
Power.LN.bcrm.sim<-bcrm(stop=list(nmax=36),p.tox0=p.tox0,dose=dose,ff="power"
  ,prior.alpha=list(3,0,1.34^2),target.tox=target.tox,constrain=TRUE
  ,sdose.calculate="median",pointest="mean",start=1,simulate=TRUE,nsims=10,truep=p.tox0)
print(Power.LN.bcrm.sim)
plot(Power.LN.bcrm.sim)

## Comparing this CRM design with the standard 3+3 design 
## (only considering the first 12 dose levels)
## Not run: 
Power.LN.bcrm.compare.sim<-bcrm(stop=list(nmax=36),p.tox0=p.tox0[1:12],dose=dose[1:12]
  ,ff="power",prior.alpha=list(3,0,1.34^2),target.tox=target.tox,constrain=TRUE
  ,sdose.calculate="median",pointest="mean",start=1,simulate=TRUE,nsims=50
  ,truep=p.tox0[1:12],threep3=TRUE)
print(Power.LN.bcrm.compare.sim,threep3=TRUE)
plot(Power.LN.bcrm.compare.sim,threep3=TRUE)

## End(Not run)

## A 2-parameter model, using priors as specified in Neuenschwander et al 2008.
## Posterior mean used to choose the next dose
## Standardised doses using reference dose, 250mg
sdose<-log(dose/250)
## Bivariate lognormal prior for two parameters
mu<-c(2.15,0.52)
Sigma<-rbind(c(0.84^2,0.134),c(0.134,0.80^2))
## Using rjags (requires JAGS to be installed)
TwoPLogistic.mean.bcrm<-bcrm(stop=list(nmax=18),data=data,sdose=sdose
  ,dose=dose,ff="logit2",prior.alpha=list(4,mu,Sigma),target.tox=target.tox
  ,constrain=FALSE,pointest="mean",method="rjags")
print(TwoPLogistic.mean.bcrm)
plot(TwoPLogistic.mean.bcrm)

## A 2-parameter model, using an EWOC design with feasibility bound (MTD quantile) 
## of 0.25 to choose the next dose
## Using rjags (requires JAGS to be installed)
## Not run: 
TwoPLogistic.EWOC0.25.bcrm<-bcrm(stop=list(nmax=18),data=data,sdose=sdose,dose=dose
	  ,ff="logit2",prior.alpha=list(4,mu,Sigma),target.tox=target.tox,constrain=FALSE
	  ,pointest=0.25,method="rjags")
print(TwoPLogistic.EWOC0.25.bcrm)
plot(TwoPLogistic.EWOC0.25.bcrm)

## End(Not run)

## A 2-parameter model, using a loss function based on intervals of toxicity to choose
## the next dose
## Using rjags (requires JAGS to be installed)
## Not run: 
## Toxicity cut-points
tox.cutpoints<-c(0.2,0.35,0.6)
## Losses associated with toxicity intervals 
## [0,0.2]=1, (0.2,0.35]=0, (0.35,0.6]=1, (0.6,1]=2
loss<-c(1,0,1,2)
TwoPLogistic.tox.intervals.bcrm<-bcrm(stop=list(nmax=18),data=data,sdose=sdose
  ,dose=dose,ff="logit2",prior.alpha=list(4,mu,Sigma),target.tox=target.tox
  ,constrain=FALSE,tox.cutpoints=tox.cutpoints,loss=loss,method="rjags")
print(TwoPLogistic.tox.intervals.bcrm)
plot(TwoPLogistic.tox.intervals.bcrm)
## Greater loss associated with overdosing and unacceptable toxicity
## [0,0.2]=1, (0.2,0.35]=0, (0.35,0.6]=2, (0.6,1]=4
loss2<-c(1,0,2,4)
TwoPLogistic.tox.intervals.2.bcrm<-bcrm(stop=list(nmax=18),data=data,sdose=sdose
  ,dose=dose,ff="logit2",prior.alpha=list(4,mu,Sigma),target.tox=target.tox
  ,constrain=FALSE,tox.cutpoints=tox.cutpoints,loss=loss2,method="rjags")
print(TwoPLogistic.tox.intervals.2.bcrm)
plot(TwoPLogistic.tox.intervals.2.bcrm)

## End(Not run)