The `escalation`

package by Kristian Brock. Documentation
is hosted at https://brockk.github.io/escalation/

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

`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:

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:

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.

In Table 1 of their publication, Ji, Li, and
Bekele (2007) list some model recommendations conditional on
hypothesised numbers of toxicities in cohorts of varying size. We can
use the `get_dose_paths`

function, for instance, to calculate
exhaustive model recommendations after a single cohort of three is
evaluated at dose 2:

```
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
```

This table confirms the advice following a cohort of three to
de-escalate if 2 or 3 toxicities are seen, to escalate if no toxicity is
seen, otherwise to remain. Note that the recommendations would actually
have been the same if `next_dose = 3`

or
`next_dose = 4`

. In this five-dose setting, they would
naturally have been slightly different if `next_dose = 1`

or
`next_dose = 5`

because we cannot de-escalate below dose 1 or
escalate above dose 5.

We can easily visualise those paths with:

`graph_paths(paths)`

For more information on working with dose-paths, refer to the dose-paths vignette.

Ji, Li, and Bekele (2007) present
simulations in their Table 2, comparing the performance of their TPI
method to other designs. We can use the `simulate_trials`

function to reproduce the operating characteristics.

Their example concerns a clinical trial of eight doses that targets
25% toxicity. We must respecify the `model`

object to reflect
this. They also elect to limit the trial to a sample size of \(n=30\):

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

For the sake of speed, we will run just fifty iterations:

`num_sims <- 50`

In real life, however, we would naturally run many thousands of iterations. 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)`

at which the simulated behaviour is:

```
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
```

This reproduces their finding that dose 2 is overwhelmingly likely to be recommended, and that the sample size is virtually guaranteed to be 30, i.e. early stopping is unlikely.

For more information on running dose-finding simulations, refer to the simulation vignette.

Ji, Yuan, Yisheng Li, and B. Nebiyou Bekele. 2007. “Dose-finding in phase I clinical trials based on toxicity
probability intervals.” *Clinical Trials* 4 (3):
235–44. https://doi.org/10.1177/1740774507079442.

Ji, Yuan, and Shengjie Yang. 2017. “On the
Interval-Based Dose-Finding Designs,” 1–26. https://arxiv.org/abs/1706.03277.