Introduction

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.

Summary of the TPI Design

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) and Ji and Yang (2017) for full details.

Implementation in escalation

To demonstrate the method, let us fit the design to a cohort of three patients treated at the first of five doses, one of whom experienced toxicity. For illustration, use the parameters chosen in Ji, Li, and Bekele (2007):

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') 

The dose recommended for the next cohort is

fit %>% recommended_dose()
#> [1] 1

Unsurprisingly, the design does not advocate escalation. Importantly, the modest toxicity seen so far is not enough to render dose 1 inadmissible:

fit %>% dose_admissible()
#> [1] TRUE TRUE TRUE TRUE TRUE

Let us imagine that we treat another two cohorts at dose 1, and see no toxicity:

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

Now, the design is happy to escalate:

fit
#> Patient-level data:
#> # A tibble: 9 × 4
#>   Patient Cohort  Dose   Tox
#>     <int>  <int> <int> <int>
#> 1       1      1     1     0
#> 2       2      1     1     0
#> 3       3      1     1     1
#> 4       4      2     1     0
#> 5       5      2     1     0
#> 6       6      2     1     0
#> 7       7      3     1     0
#> 8       8      3     1     0
#> 9       9      3     1     0
#> 
#> Dose-level data:
#> # A tibble: 6 × 8
#>   dose     tox     n empiric_tox_rate mean_prob_tox median_prob_tox admissible
#>   <ord>  <dbl> <dbl>            <dbl>         <dbl>           <dbl> <lgl>     
#> 1 NoDose     0     0            0             0                0    TRUE      
#> 2 1          1     9            0.111         0.112            0.08 TRUE      
#> 3 2          0     0          NaN             0.5             NA    TRUE      
#> 4 3          0     0          NaN             0.5             NA    TRUE      
#> 5 4          0     0          NaN             0.5             NA    TRUE      
#> 6 5          0     0          NaN             0.5             NA    TRUE      
#> # ℹ 1 more variable: recommended <lgl>
#> 
#> The model targets a toxicity level of 0.3.
#> The model advocates continuing at dose 2.

Let us imagine, however, that dose 2 is surprisingly toxic, yielding three toxicities:

fit <- model %>% fit('1NNT 1NNN 1NNN 2TTT') 

Despite the low sample size, the statistical model believes that dose 2 is excessively toxic:

fit %>% prob_tox_exceeds(threshold = 0.25)
#> [1] 0.1007690 0.9999679        NA        NA        NA

and thus inadmissible:

fit %>% dose_admissible()
#> [1]  TRUE FALSE FALSE FALSE FALSE

Note that since dose 2 is believed to be inadmissible, the assumption of monotonically increasing toxicity means that the doses higher than dose 2 are excessively toxic too.

paths <- model %>% get_dose_paths(cohort_sizes = c(3), next_dose = 2)

library(dplyr)
as_tibble(paths) %>% select(outcomes, next_dose) %>% print(n = 100)
#> # A tibble: 5 × 2
#>   outcomes next_dose
#>   <chr>        <dbl>
#> 1 ""               2
#> 2 "NNN"            3
#> 3 "NNT"            2
#> 4 "NTT"            1
#> 5 "TTT"            1

graph_paths(paths)

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

num_sims <- 50

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

set.seed(123)
sims <- model %>%
  simulate_trials(num_sims = num_sims, 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   0.20   0.72   0.06   0.02   0.00   0.00   0.00   0.00 
#> 
#> Probability of administration:
#>     1     2     3     4     5     6     7     8 
#> 0.246 0.600 0.144 0.010 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. 
#>     5.0     6.0     7.0     7.4     8.0    11.0 
#> 
#> Trial duration:
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#>   18.55   26.62   29.19   29.81   31.98   45.52