New phase I methods in escalation

Introduction

This week I updated the escalation (Brock 2020) package on CRAN with new methods to implement several phase I and phase I/II dose-finding designs. This post describes the addition of the phase I methods TPI, mTPI, and the logistic model method of Neuenschwander, Branson, and Gsponer (2008), and also the addition of MCMC-based CRM methods.

TPI

The toxicity probability interval (TPI) design was introduced by Ji, Li, and Bekele (2007). It is one of a series of dose-finding trial designs that works by partitioning the probability of toxicity into a set of intervals. These designs make dose-selection decisions that are determined by the interval in which the probability of toxicity for the current dose is believed to reside.

Core to this design is a beta-binomial Bayesian conjugate model. For hyperparameters \(\alpha\) and \(\beta\), let the probability of toxicity at dose \(i\) be \(p_i\), with prior distribution

\[p_i \sim Beta(\alpha, \beta).\]

If \(n_i\) patients have been treated at dose \(i\), yielding \(x_i\) toxicity events, the posterior distribution is

\[ p_i | data \sim Beta(\alpha + x_{i}, \beta + n_{i} - x_{i}).\]

Using this distribution, let the standard deviation of \(p_i\) be denoted by \(\sigma_i\). The design seeks a dose with probability of toxicity close to some pre-specified target level, \(p_T\). The entire range of possible values for \(p_i\) can be broken up into the following intervals:

• The underdosing interval (UI), defined as \((0, p_{T} - K_{2} \sigma_{i})\);
• The equivalence interval (EI), defined as \((p_{T} - K_{2} \sigma_{i}, p_{T} + K_{1} \sigma_{i})\);
• The overdosing interval (OI), defined as \((p_{T} + K_{1} \sigma_{i}, 1)\);

for pre-specified model constants, \(K_1, K_2\). These intervals are mutally-exclusive and mutually-exhaustive, meaning that every possible probability belongs to precisely one of them. In other words, these intervals form a partition of the probability space, \((0, 1)\).

Using the posterior distribution, we can calculate the three probabilities

\[p_{UI} = Pr(p_i \in \text{UI}), \enspace p_{EI} = Pr(p_i \in \text{EI}), \enspace p_{OI} = Pr(p_i \in \text{OI}).\]

By definition, \(p_{UI} + p_{EI} + p_{OI} = 1\). The logical action in the dose-finding trial depends on which of these three probabilities is the greatest. If \(p_{UI} > p_{EI}, p_{OI}\), then the current dose is likely an underdose, so our desire should be to escalate dose to \(i+1\). In contrast, if \(p_{OI} > p_{UI}, p_{EI}\), then the current dose is likely an overdose and we will want to de-escalate dose to \(i-1\) for the next patient. If \(p_{EI} > p_{UI}, p_{OI}\), then the current dose is deemed sufficiently close to \(p_T\) and we will want to stay at dose-level \(i\).

Further to these rules regarding dose-selection, the following rule is used to avoid recommending dangerous doses. A dose is deemed inadmissible for being excessively toxic if

\[ Pr(p_{i} > p_{T} | data) > \xi,\]

for a certainty threshold, \(\xi\). If a dose is excluded by this rule, it should not be recommended by the model. Irrespective the probabilities \(p_{UI}, p_{EI}, p_{OI}\), the design will recommend to stay at dose \(i\) rather than escalate to a dose previously identified as being inadmissible. Furthermore, the design will advocate stopping if the lowest dose is inferred to be inadmissible.

In their paper, the authors demonstrate acceptable operating performance using \(\alpha = \beta = 0.005\), \(K_{1} = 1\), \(K_{2} = 1.5\) and \(\xi = 0.95\). See Ji, Li, and Bekele (2007) for full details.

library(escalation)

model <- get_tpi(num_doses = 5, target = 0.3, alpha = 0.005, beta = 0.005, 
                 k1 = 1, k2 = 1.5, exclusion_certainty = 0.95)

fit <- model %>% fit('1NNT') 

fit %>% recommended_dose()

## [1] 1

paths <- model %>% 
  get_dose_paths(cohort_sizes = 6, next_dose = 2)

library(dplyr)
as_tibble(paths) %>% 
  select(outcomes, next_dose) %>% 
  knitr::kable()

mTPI

The modified toxicity probability interval (mTPI) design was introduced by Ji et al. (2010). As the name suggests, it is a modification of the earlier TPI design. mTPI is very similar to TPI, again using a method that partitions the probability of toxicity into a set of intervals.

This design uses a beta-binomial Bayesian conjugate model. For hyperparameters \(\alpha\) and \(\beta\), let the probability of toxicity at dose \(i\) be \(p_i\), with prior distribution

\[p_i \sim Beta(\alpha, \beta).\]

If \(n_i\) patients have been treated at dose \(i\), yielding \(x_i\) toxicity events, the posterior distribution is

\[ p_i | data \sim Beta(\alpha + x_{i}, \beta + n_{i} - x_{i}).\]

The design seeks a dose with probability of toxicity close to some pre-specified target level, \(p_T\). The entire range of possible values for \(p_i\) can be broken up into the following intervals:

• The underdosing interval (UI), defined as \((0, p_{T} - \epsilon_{1})\);
• The equivalence interval (EI), defined as \((p_{T} - \epsilon_{1}, p_{T} + \epsilon_{2})\);
• The overdosing interval (OI), defined as \((p_{T} + \epsilon_{2}, 1)\);

for pre-specified model constants, \(\epsilon_{1}, \epsilon_{2}\). These intervals are mutally-exclusive and mutually-exhaustive, meaning that every possible probability belongs to precisely one of them. In other words, these intervals form a partition of the probability space, \((0, 1)\).

For a continuous random variable \(X\) with cumulative probability mass function \(F(x)\) (i.e. \(Pr(X < x) = F(x)\)), the authors define the unit probability mass (UPM) for an interval \((a, b)\) to be \((F(b) - F(a)) / (b - a)\). That is, the UPM is the probability mass in an interval divided by the width of the interval, and can be interpreted as the average probability density of the interval.

Then, using the posterior distribution identified above, we calculate the three UPMs

\[UPM_{UI} = Pr(p_i \in \text{UI}) / (p_{T} - \epsilon_{1}),\] \[UPM_{EI} = Pr(p_i \in \text{EI}) / (\epsilon_{1} + \epsilon_{2}),\]

and

\[UPM_{OI} = Pr(p_i \in \text{OI}) / (1 - p_{T} + \epsilon_{2}).\]

The logical action in the dose-finding trial depends on which of these three quantities is the greatest. If \(UPM_{UI} > UPM_{EI}, UPM_{OI}\), then the current dose is likely an underdose, so our desire should be to escalate dose to \(i+1\). In contrast, if \(UPM_{OI} > UPM_{UI}, UPM_{EI}\), then the current dose is likely an overdose and we will want to de-escalate dose to \(i-1\) for the next patient. If \(UPM_{EI} > UPM_{UI}, UPM_{OI}\), then the current dose is deemed sufficiently close to \(p_T\) and we will want to stay at dose-level \(i\).

Further to these rules regarding dose-selection, the following rule is used to avoid recommending dangerous doses. A dose is deemed inadmissible for being excessively toxic if

\[ Pr(p_{i} > p_{T} | data) > \xi,\]

for a certainty threshold, \(\xi\). If a dose is excluded by this rule, it should not be recommended by the model. Irrespective the values of \(UPM_{UI}, UPM_{EI}\) and \(UPM_{OI}\), the design will recommend to stay at dose \(i\) rather than escalate to a dose previously identified as being inadmissible. Furthermore, the design will advocate stopping if the lowest dose is inferred to be inadmissible.

In their paper, the authors demonstrate acceptable operating performance using \(\alpha = \beta = 1\), \(\epsilon_{1} = 0.05\), \(\epsilon_{2} = 0.05\) and \(\xi = 0.95\). See Ji et al. (2010) for full details.

library(escalation)

model <- get_mtpi(num_doses = 5, target = 0.3, alpha = 1, beta = 1, 
                  epsilon1 = 0.05, epsilon2 = 0.05, exclusion_certainty = 0.95)

fit <- model %>% fit('1NNT') 
fit %>% recommended_dose()

## [1] 1

paths <- model %>% 
  get_dose_paths(cohort_sizes = 6, next_dose = 2)

as_tibble(paths) %>% 
  select(outcomes, next_dose) %>% 
  knitr::kable()

Neuenschwander et al.

Neuenschwander, Branson, and Gsponer (2008) (NBG) introduced a derivative of the CRM for dose-escalation clinical trials using the model:

\[ \text{logit} p_i = \alpha + \exp{(\beta)} \log{(x_i / d^*)}, \]

where \(p_i\) is the probability of toxicity at the \(i\)th dose, \(x_i\), and \(d^*\) is a reference dose. Here \(\alpha\) and \(\beta\) are model parameters on which the authors place a bivariate normal prior. This model is very similar to the two-parameter logistic CRM. However, a notable difference is that the dose, \(x_i\), enters the model as a covariate. This dispenses with the toxicity skeleton that is used in the CRM.

library(escalation)

dose <- c(1, 2.5, 5, 10, 15, 20, 25, 30, 40, 50, 75, 100, 150, 200, 250)
outcomes <- '1NNN 2NNNN 3NNNN 4NNNN 7TT'

model <- get_trialr_nbg(real_doses = dose, d_star = 250, target = 0.3,
                        alpha_mean = 2.15, alpha_sd = 0.84,
                        beta_mean = 0.52, beta_sd = 0.8,
                        seed = 2020)

fit <- model %>% fit(outcomes)
fit

## Patient-level data:
## # A tibble: 17 x 4
##    Patient Cohort  Dose   Tox
##      <int>  <int> <int> <int>
##  1       1      1     1     0
##  2       2      1     1     0
##  3       3      1     1     0
##  4       4      2     2     0
##  5       5      2     2     0
##  6       6      2     2     0
##  7       7      2     2     0
##  8       8      3     3     0
##  9       9      3     3     0
## 10      10      3     3     0
## 11      11      3     3     0
## 12      12      4     4     0
## 13      13      4     4     0
## 14      14      4     4     0
## 15      15      4     4     0
## 16      16      5     7     1
## 17      17      5     7     1
## 
## Dose-level data:
## # A tibble: 15 x 7
##    RealDose  dose   tox     n empiric_tox_rate mean_prob_tox median_prob_tox
##       <dbl> <int> <int> <int>            <dbl>         <dbl>           <dbl>
##  1      1       1     0     3                0        0.0117         0.00511
##  2      2.5     2     0     4                0        0.0306         0.0186 
##  3      5       3     0     4                0        0.0644         0.0468 
##  4     10       4     0     4                0        0.134          0.114  
##  5     15       5     0     0              NaN        0.202          0.185  
##  6     20       6     0     0              NaN        0.264          0.251  
##  7     25       7     2     2                1        0.322          0.314  
##  8     30       8     0     0              NaN        0.374          0.370  
##  9     40       9     0     0              NaN        0.463          0.466  
## 10     50      10     0     0              NaN        0.535          0.543  
## 11     75      11     0     0              NaN        0.662          0.678  
## 12    100      12     0     0              NaN        0.741          0.760  
## 13    150      13     0     0              NaN        0.829          0.849  
## 14    200      14     0     0              NaN        0.875          0.894  
## 15    250      15     0     0              NaN        0.902          0.920  
## 
## The model targets a toxicity level of 0.3.
## The model advocates continuing at dose 7.

MCMC CRM

escalation already provides one-parameter CRM models via the dfcrm package (Cheung 2013). It now also provides MCMC implementations of those same one-parameter models:

skeleton <- c(0.05, 0.12, 0.25, 0.40, 0.55)
target <- 0.25

model1 <- get_trialr_crm(skeleton = skeleton, target = target, 
                         model = 'empiric', beta_sd = 1)
model2 <- get_trialr_crm(skeleton = skeleton, target = target, 
                         model = 'logistic', a0 = 3, 
                         beta_mean = 0, beta_sd = 1)

and also adds a two-parameter logistic model:

model3 <- get_trialr_crm(skeleton = skeleton, target = target, 
                         model = 'logistic2', 
                         alpha_mean = 0, alpha_sd = 2, 
                         beta_mean = 0, beta_sd = 1)

For more on the parameterisation of these methods, check out the CRM vignette in the escalation.

Simulations and dose-paths

As with all methods in escalation, simulations are supported right out of the box.

For illustration, Ji, Li, and Bekele (2007) present a simulation study to compare the performance of their TPI design to some alternatives. They specify a design that uses a maximum sample size of thirty patients:

model <- get_tpi(num_doses = 8, target = 0.25, k1 = 1, k2 = 1.5,
                 exclusion_certainty = 0.95) %>%
  stop_at_n(n = 30)

and their scenario 1 assumes true probability of toxicity:

sc1 <- c(0.05, 0.25, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95)

We can then reproduce their simulated characteristics by running:

set.seed(123)
sims <- model %>%
  simulate_trials(num_sims = 50, true_prob_tox = sc1, next_dose = 1)

sims

## Number of iterations: 50 
## 
## Number of doses: 8 
## 
## True probability of toxicity:
##    1    2    3    4    5    6    7    8 
## 0.05 0.25 0.50 0.60 0.70 0.80 0.90 0.95 
## 
## Probability of recommendation:
## NoDose      1      2      3      4      5      6      7      8 
##   0.00   0.14   0.76   0.10   0.00   0.00   0.00   0.00   0.00 
## 
## Probability of administration:
##     1     2     3     4     5     6     7     8 
## 0.226 0.606 0.160 0.008 0.000 0.000 0.000 0.000 
## 
## Sample size:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##      30      30      30      30      30      30 
## 
## Total toxicities:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    4.00    6.00    7.00    7.44    9.00   11.00 
## 
## Trial duration:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   15.30   27.50   30.75   30.65   34.90   42.30

For the sake of speed, we ran just fifty iterations but in a real situation, many thousands of iterations would be more appropriate.

Conclusion

Amongst pure phase I designs, escalation now supports:

• CRM
• TPI
• mTPI
• BOIN
• NBG

It also supports some designs that select dose by co-primary efficacy and toxicity outcomes, the so-called seamless phase I/II designs. Those will be the subect of the next post on this blog. Til then, good day.