Norris (2020b) proposed a universal schematic for evaluating the safety of the 3+3 dose-escalation design, and conjectured that the *dose transition pathways* (DTP) of Yap et al. (2017) would enable a similar treatment of model-based designs. This vignette explores that conjecture in relation to the continual reassessment method (CRM), using the `dtpcrm`

package (Yap et al. 2019).

Dose-escalation trial designs commonly operate with a fixed set of \(D\) prespecified doses, enrolling participants in small cohorts of size \(C\), at doses selected sequentially according to observed counts of binary dose-limiting toxicities (DLTs). Under such designs, the observations at any point in the trial may be recorded in a \(C \times D\) matrix of toxicity counts \(\{0, 1, ..., C, -\}\), with ‘\(-\)’ being used where a cohort has not been enrolled. Indeed, for deterministic designs in which dose-escalation decisions depend strictly on the history of observed DLT counts, such matrices suffice to account for the full dose-escalation sequence.1 To see this, consider that one may ‘read’ such a matrix, starting from the (deterministic) starting dose, and applying the (deterministic) design rules at each step. That is, a deterministic design *by definition* imposes a unique sequence on the entries in such a matrix. For designs with upper bounds on enrollment, \(C\) may be fixed *ex ante*, and all possible paths comprehensively enumerated.

In Norris (2020b), for example, paths through the 3+3 design could be represented as \(2 \times D\) matrices because the 3+3 design enrolls at most 2 cohorts at any given dose. The mathematical treatment offered there for the path matrices \(T_{c,d}^j\) was independent of the 3+3 rules, however, and carries forward generally:

Denoting the prespecified doses by \((X_d)\) and the cumulative distribution function of \(\mathrm{MTD}_i\) by \(P\), we can write the vector \((\pi^j)\) of path probabilities:2 Products or sums over \(c\) or pairs \((c,d)\) are understood to be taken over the *non-empty* cohorts thus indexed. In R, this convention is easily applied by using `NA`

to represent ‘\(-\)’, and performing aggregate operations with option `na.rm=TRUE`

.

\[ \begin{align} p_d &= P(\mathrm{MTD}_i< X_d)\quad\mbox{(dose-wise toxicity probabilities)}\\ q_d &= 1 - p_d\\ \pi^j &= \prod_{c,d} {n \choose T_{c,d}^j} p_d^{T_{c,d}^j} q_d^{(n-T_{c,d}^j)}\quad\mbox{(path probabilities)} \tag{1} \end{align} \]

Again as in Norris (2020b), taking logs in (1) we find:

\[
\log \boldsymbol{\pi} = \sum_{c,d} \log {n \choose T_{c,d}} + \sum_c [T_{c,d},n-T_{c,d}]\left[{ \log \mathbf{p} \atop \log \mathbf{q}} \right] = \mathbf{b} + U\left[{ \log \mathbf{p} \atop \log \mathbf{q}} \right], \tag{2}
\]
where the \(J \times 2D\) matrix \(U\) and the \(J\)-vector \(\mathbf{b}\) are characteristic *constants* of the given dose-escalation design.

Finally, we may as previously introduce the log-therapeutic index \(\kappa\), enabling us to write the fatal (grade-5) fraction \(f_d\) of DLTs as:

\[ f_d = \frac{P(e^{2\kappa} \mathrm{MTD}_i< X_d)}{P(\mathrm{MTD}_i< X_d)}, \tag{3} \]

in terms of which the expected number of fatal toxicities is:

\[ \boldsymbol{\pi}^\intercal \mathrm{Y} \mathbf{f}, \tag{4} \]

where \(Y = \sum_c T_{c,d}^j\) denotes the \((J \times D)\) left half of \(U\).

The DTP idea apparently was introduced primarily as a means to reconcile model-based formulations of dose-escalation trials with the habitually rule-based thinking of clinicians. Specifically, Yap et al. (2017) cites 3 specific “perceived challenges” in making the transition from rule-based to model-based designs:

- The “flexibility” of model-based designs (which I take to mean the additional free parameters they introduce into the design space) creates choices which cannot readily be appreciated in the rule-based terms familiar to clinical investigators.
- Benefits of model-based designs are likewise not apparent to clinical investigators, against the background of “rule-based designs perceived to be ‘successful’ for decades”.
- From the trialist’s perspective, the ‘black-box’ recommendations of model-based designs “contrast unfavorably with the transparent, simple rules of a rule-based design.”

To meet these challenges, DTP effectively transforms a model-based design into a rule-based design *locally* — so that its immediate operation several steps ahead can be ‘eyeballed’ by trialists as a small set of (e.g., \(4^4 = 64\)) distinct paths. The exhaustive enumeration employed here as in Norris (2020b) differs only in its *global* reach, and its intent to support *computation* over the paths as opposed to cursory inspection.

We will adopt the same parameters as in the `dtpcrm`

package vignette:3 See Table 1 in Craddock et al. (2019).

```
7
number.doses <- 3
start.dose.level <- 21
max.sample.size <- 0.2
target.DLT <- 3 cohort.size <-
```

Note that the sample size limit allows us to set \(C=7\).

```
c(0.03, 0.07, 0.12, 0.20, 0.30, 0.40, 0.52)
prior.DLT <- 0.75 prior.var <-
```

```
function(x) {
stop_func <- stop_for_excess_toxicity_empiric(x,
y <-tox_lim = target.DLT + 0.1,
prob_cert = 0.72,
dose = 1,
nsamps = 100000)
if(y$stop){
y
x <-else {
} stop_for_consensus_reached(x, req_at_mtd = 12)
x <-
} }
```

```
proc.time()
t0 <- calculate_dtps(next_dose = start.dose.level,
viola_dtp <-cohort_sizes = rep(cohort.size, 7),
dose_func = applied_crm,
prior = prior.DLT,
target = target.DLT,
stop_func = stop_func,
scale = sqrt(prior.var),
no_skip_esc = TRUE,
no_skip_deesc = FALSE,
global_coherent_esc = TRUE
)save(viola_dtp, file="../data/viola_dtp.rda")
proc.time() - t0 # ~17 minutes on a 2.6GHz i7
```

With each of (up to) 7 cohorts having 4 possible outcomes (0, 1, 2 or 3 toxicities), the DTP tabulation lists a total of \(4^7 = 16384\) paths. These are not all distinct paths, however, since early stopping results in path degeneracy. For example, we can see that paths 1005–1008 all terminate at the 6th cohort, resulting in a \(4\times\) degeneracy:

`::kable(viola_dtp[1000:1010,]) knitr`

D0 | T1 | D1 | T2 | D2 | T3 | D3 | T4 | D4 | T5 | D5 | T6 | D6 | T7 | D7 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1000 | 3 | 0 | 4 | 0 | 5 | 3 | 3 | 3 | 1 | 2 | 1 | 1 | 1 | 3 | NA |

1001 | 3 | 0 | 4 | 0 | 5 | 3 | 3 | 3 | 1 | 2 | 1 | 2 | 1 | 0 | 1 |

1002 | 3 | 0 | 4 | 0 | 5 | 3 | 3 | 3 | 1 | 2 | 1 | 2 | 1 | 1 | 1 |

1003 | 3 | 0 | 4 | 0 | 5 | 3 | 3 | 3 | 1 | 2 | 1 | 2 | 1 | 2 | NA |

1004 | 3 | 0 | 4 | 0 | 5 | 3 | 3 | 3 | 1 | 2 | 1 | 2 | 1 | 3 | NA |

1005 | 3 | 0 | 4 | 0 | 5 | 3 | 3 | 3 | 1 | 2 | 1 | 3 | NA | NA | NA |

1006 | 3 | 0 | 4 | 0 | 5 | 3 | 3 | 3 | 1 | 2 | 1 | 3 | NA | NA | NA |

1007 | 3 | 0 | 4 | 0 | 5 | 3 | 3 | 3 | 1 | 2 | 1 | 3 | NA | NA | NA |

1008 | 3 | 0 | 4 | 0 | 5 | 3 | 3 | 3 | 1 | 2 | 1 | 3 | NA | NA | NA |

1009 | 3 | 0 | 4 | 0 | 5 | 3 | 3 | 3 | 1 | 3 | 1 | 0 | 1 | 0 | 1 |

1010 | 3 | 0 | 4 | 0 | 5 | 3 | 3 | 3 | 1 | 3 | 1 | 0 | 1 | 1 | 1 |

Paths terminating at the 5th cohort would be listed 16 times, and those terminating at the 4th cohort 64 times, etc. The net degeneracy indeed proves to be quite substantial, inflating the DTP table by a factor of `3.5`

:

```
as.matrix(unique(viola_dtp))
viola_paths <-nrow(viola_paths)
```

`## [1] 4689`

To distinguish it from the degenerate DTP table returned by `dtpcrm::calculate_dtps`

, we will refer to `viola_paths`

as the DTP *matrix*.

```
outer(viola_paths[,paste0("D",0:6)], 1:7, FUN = "==")
I <- 1
I[I] <-!I] <- NA
I[# Now I[j,c,d] is an indicator array that we may use
# in the manner of a bitmask, to select the toxicities
# into their proper positions within T[j,c,d]:
I * outer(viola_paths[,paste0("T",1:7)], rep(1,7))
T <-dimnames(T)[[2]] <- paste0("C",1:7) # best labeled as Cohorts
dimnames(T)[[3]] <- paste0("X",1:7) # label as doses X1..X7
dim(T)
```

`## [1] 4689 7 7`

```
apply(log(choose(3, T)), MARGIN = 1, FUN = sum, na.rm = TRUE)
b <-length(b)
```

`## [1] 4689`

```
apply(T, MARGIN = c(1,3), FUN = sum, na.rm = TRUE)
Y <- apply(3-T, MARGIN = c(1,3), FUN = sum, na.rm = TRUE)
Z <- cbind(Y, Z)
U <-dim(U)
```

`## [1] 4689 14`

```
log(prior.DLT)
log_p <- log(1 - prior.DLT)
log_q <- b + U %*% c(log_p, log_q)
log_pi <-sum(exp(log_pi)) # check probabilities sum to 1
```

`## [1] 1`

Whereas the analysis of Norris (2020b) was carried forth under a thoroughly logarithmic scaling of *dose*, the dosing intervals employed in the VIOLA trial are suggestive of an approximately square-root dose scaling—at least for the nonzero doses:

```
viola_dosing <- data.frame(Label = -2:4 # VIOLA dose designations -2,...,4
Level = 1:7 # We use integer dose indexes here
,Dose_mg = c(0, 2.5, 5, 10, 15, 25, 35)
,skeleton = prior.DLT
,
) viola_dosing %>%
viola_dosing <- mutate(log_dose = log(Dose_mg)
sqrt_dose = sqrt(Dose_mg)
,
)
plot(Level ~ sqrt_dose, data = viola_dosing, type = 'b'
ylab = "Dose Level"
, xlab = expression(sqrt(Dose[mg]))
, las = 1
, )
```

Indeed, the CRM skeleton looks quite close to a normal distribution under this scaling:

```
plot(skeleton ~ sqrt_dose, data = viola_dosing
type = 'b'
, xlab = expression(sqrt(Dose[mg]))
, ylim = c(0, 0.55)
, las = 1)
, seq(0, 6, 0.1)
x <- pnorm(x, mean = sqrt(33), sd = 3)
y <-lines(y ~ x, lty = 2)
```

Nevertheless, the treatment of zero dose under this scaling is pharmacologically problematic, if not outright homeopathic, so we will proceed instead using a lognormal \(\mathrm{MTD}_i\) distribution fitted to the CRM skeleton.

```
plot(skeleton ~ log_dose, data = subset(viola_dosing, is.finite(log_dose))
type = 'b'
, xlab = expression(log(Dose[mg]))
, xlim = c(0.5, 3.6)
, ylim = c(0, 0.55)
, las = 1)
,
function(mu_sigma, dose = viola_dosing$Dose_mg){
sse <- mu_sigma[1]
mu <- mu_sigma[2]
sigma <- plnorm(dose, meanlog = mu, sdlog = sigma)
lnorm.DLT <- sum((lnorm.DLT - prior.DLT)[dose > 0]^2)
sse <-
} optim(par = c(mu=log(38), sigma=1.8), fn = sse)
fit <- log(round(exp(fit$par['mu']), 1))
mu_opt <- round(fit$par['sigma'], 1)
sigma_opt <- seq(0.5, 3.55, 0.05)
x <- plnorm(exp(x), meanlog = mu_opt, sdlog = sigma_opt)
y <-lines(y ~ x, lty = 2)
```

Taking this CRM skeleton fit as a modal expectation of the \(\mathrm{MTD}_i\) distribution, we may employ Equations (3)–(4) to obtain an expected number of fatal toxicities, as a function of \(\kappa\):

```
function(kappa, mu=mu_opt, sigma=sigma_opt, X=viola_dosing$Dose_mg) {
f <-# For ease of use, this function is VECTORIZED over kappa,
# generally returning a length(X) x length(kappa) MATRIX
# which can validly right-multiply Y in t(pi) %*% Y %*% f.
# (In the special case length(kappa)==1, a vector is returned.)
plnorm(outer(X, exp(-2*kappa)), meanlog = mu, sdlog = sigma)
gr5 <- plnorm(outer(X, exp( 0*kappa)), meanlog = mu, sdlog = sigma)
dlt <- gr5/dlt
ff <-is.nan(ff)] <- 0 # replace NaN caused by X[1]==0
ff[dimnames(ff) <- list(
dose = paste0("X", seq_along(X)),
kappa = round(kappa, 3)
)if (length(kappa)==1)
return(as.vector(ff))
ff
}
seq(0.2, 1.2, 0.02)
kappa <- f(kappa = kappa)
F <- t(exp(log_pi)) %*% Y %*% F
Ef <- t(Ef) # transpose to a column-vector
Ef <-plot(Ef ~ kappa, type = 'l'
xlab = expression(kappa)
, ylab = "Expected Number of Fatal DLTs"
, las=1) ,
```

The development of CRM methods has left the status of the *skeleton* somewhat ambiguous. While evading its realistic interpretation as an \(\mathrm{MTD}_i\) distribution,4 For a related perspective on interpretational issues, see Norris (2020a). CRM proponents often fixate on the skeleton as a natural starting point for simulation studies. Braun (2020), for example, develops an approach which “hinges on the assumption that the true DLT probabilities examined are consistent with the selected skeleton.”

Nevertheless, it is incumbent on trial methodologists to examine the safety characteristics of their designs under conditions of model misspecification. The technique of Figure 3 in Norris (2020b) provides one approach.

To enable parameter departures from our lognormal skeleton fit, we require \(\log \boldsymbol{\pi}\) as a *function* of \((\mu, \sigma)\):

```
function(mu, sigma){
log_pi <- plnorm(viola_dosing$Dose_mg, meanlog = mu, sdlog = sigma)
p <- c(log(p), log(1-p))
log_pq <- b + U %*% pmax(log_pq, -500) # clamping -Inf to -500 avoids NaN's
log_pi = }
```

This enables us to compute expected fatalities as a scalar field on a 3-dimensional space defined by parameters \((\mu, \sigma, \kappa)\). But again as in Norris (2020b) we employ a minimax framing of our safety question, to achieve a dimension reduction:

` log(viola_dosing$Dose_mg)[3] # dose 3 is 2nd-highest *nonzero* dose mu_minimax <-`

The irregularly spaced doses of VIOLA don’t yield a unique delta, but the \(2\times\) mulipliers either side of dose level 3 (our `mu_minimax`

) allow us to select \(\delta = \log 2\) as a local approximation:

` mean(diff(log(viola_dosing$Dose_mg)[2:4])) delta <-`

We may now generate a grid of values, and plot contours:

```
CJ(mu = mu_minimax
focustab <-K = seq(0.5, 1.65 , 0.01) # K = kappa / sigma
, sigma = delta/seq(0.4, 2.2, 0.01)
,
):= K * sigma]
focustab[, kappa := t(exp(log_pi(mu,sigma))) %*%
focustab[, value Y %*% f(kappa=kappa, mu=mu, sigma=sigma)
.(mu,sigma)] , by =
```

Although initially put forward with different intent, the *dose transition pathways* of Yap et al. (2017) render the enumerative approach of Norris (2020b) immediately applicable to the CRM. Importantly, the analysis is found to be computationally feasible in the setting of an actual trial. Moreover, the only nontrivial computation time involved in this approach manifests an ‘embarrassing’ degree of parallelism that might enable a 10\(\times\) speedup even on a single modern CPU or even effectively zero out the computation time in a massively parallel GPU implementation.

The approach taken here exemplifies a widely applicable scheme for the safety analysis of dose-escalation trials generally, rooted in the exhaustive enumeration of all possible trial paths.

Braun, Thomas M. 2020. “A Simulation‐free Approach to Assessing the Performance of the Continual Reassessment Method.” *Statistics in Medicine*, September, sim.8746. https://doi.org/10.1002/sim.8746.

Craddock, Charles, Daniel Slade, Carmela De Santo, Rachel Wheat, Paul Ferguson, Andrea Hodgkinson, Kristian Brock, et al. 2019. “Combination Lenalidomide and Azacitidine: A Novel Salvage Therapy in Patients Who Relapse After Allogeneic Stem-Cell Transplantation for Acute Myeloid Leukemia.” *Journal of Clinical Oncology: Official Journal of the American Society of Clinical Oncology*, January, JCO1800889. https://doi.org/10.1200/JCO.18.00889.

Norris, David C. 2020a. “Comment on Wages et Al, Coherence Principles in Interval-Based Dose Finding. Pharmaceutical Statistics 2019, Doi:10.1002/Pst.1974.” *Pharmaceutical Statistics*, March. https://doi.org/10.1002/pst.2016.

———. 2020b. “What Were They Thinking? Pharmacologic Priors Implicit in a Choice of 3+3 Dose-Escalation Design.” *arXiv:2012.05301 [stat.ME]*, December. https://arxiv.org/abs/2012.05301.

Yap, Christina, Lucinda J. Billingham, Ying Kuen Cheung, Charlie Craddock, and John O’Quigley. 2017. “Dose Transition Pathways: The Missing Link Between Complex Dose-Finding Designs and Simple Decision-Making.” *Clinical Cancer Research: An Official Journal of the American Association for Cancer Research* 23 (24): 7440–7. https://doi.org/10.1158/1078-0432.CCR-17-0582.

Yap, Christina, Daniel Slade, Kristian Brock, and Yi Pan. 2019. *Dtpcrm: Dose Transition Pathways for Continual Reassessment Method*. https://CRAN.R-project.org/package=dtpcrm.