R package precautionary

Escalation vs Titration

• We escalate between or across cohorts …
• and titrate within individuals.

Senn, Statistical Issues in Drug Development 2ed. (2007), p.318

• NB: On this view, ‘intra-patient dose escalation’ is a misnomer!

Rhetorical role of ‘IPDE’

Structure of the debate

Difficulties of mounting an ‘external’ challenge

Departmental Robert Frost

Unraveling one-size-fits-all dose finding ‘from within’

Refactoring by Martin Fowler (1999)

Dose-finding simulations rife with rbinom(size=1):

library(crmPack); getMethod(simulate, "TDDesign")

function (object, nsim = 1L, seed = NULL, ...) 
{
    .local <- function (object, nsim = 1L, seed = NULL, truth, args = NULL, firstSeparate = FALSE,
        parallel = FALSE, nCores = min(parallel::detectCores(), 5), ...) 
    {
        ...
        runSim <- function(iterSim) {
            ...
            thisDose <- object@startingDose
            while (!stopit) { # Note the abundant 'rbinom(size=1)' Bernoulli draws
                ...
                if (firstSeparate && (thisSize > 1L)) {
                  thisDLTs <- rbinom(n = 1L, size = 1L, prob = thisProb)
                  if (thisData@placebo && (thisSize.PL > 0L)) 
                    thisDLTs.PL <- rbinom(n = 1L, size = 1L, prob = thisProb.PL)
                  if (thisDLTs == 0) {
                    thisDLTs <- c(thisDLTs, rbinom(n = thisSize - 1L, size = 1L, prob = thisProb))
                    if (thisData@placebo && (thisSize.PL > 0L)) 
                      thisDLTs.PL <- c(thisDLTs.PL, rbinom(n = thisSize.PL, size = 1L, prob = thisProb.PL))
                  }
                }
                else {
                  thisDLTs <- rbinom(n = thisSize, size = 1L, prob = thisProb)
                  if (thisData@placebo && (thisSize.PL > 0L)) 
                    thisDLTs.PL <- rbinom(n = thisSize.PL, size = 1L, prob = thisProb.PL)
                }
                ...
            }
            ...
            return(thisResult)
        }
        ...
        return(ret)
    }
    .local(object, nsim, seed, ...)
}

rbinom internals

double rbinom(double nin, double pp)
{
  /* EXCERPTED & ANNOTATED to focus n=1 (Bernoulli) case */

  if (np < 30.0) { /* inverse cdf logic for mean less than 30 */
    qn = R_pow_di(q, n);
    goto L_np_small;
  }

L_np_small:
  /*---------------------- np = n*p < 30 : ------------------------- */

  repeat {
    ix = 0;
    f = qn; /* NB: n=1 ==> f = q^1 = (1-p) */
    u = unif_rand();
    repeat {
      if (u < f) /* i.e., if (1-u) > p ... */
        goto finis;
        /* ... */
    }
  }
finis:
  /* ... */
  return (double)ix;
}

• \(u \sim \mathscr{U}[0,1] \implies I(u < p) \sim \mathscr{B}\mathit{ern}(p)\)

\(u_i \sim \mathscr{U}[0,1]\) : 2 decades of fleeting appearances

To each subject \(j\) we can associate a numerical value \(v_j\) indicating his/her toxicity tolerance threshold to the treatment under study. Without loss of generality, the \(v_j\) can be generated from a uniform distribution.

…

Of course, in any practical setting, the observations are not complete, we are unable to independently replicate experimentation for the same subject at all levels, and so such an optimal method is not available to us.

— O’Quigley, Paoletti & Maccario (2002) doi:10.1093/biostatistics/3.1.51

Let \(S\) be a random variable that denotes the lowest dose at which a subject has a DLT. Given value \(s_j\) for patient \(j\) and supposing an increasing dose-toxicity relation, the probability of toxicity at any level is either zero or one.

…

We can notice that it exactly corresponds to what we are used to doing when we computationally simulate the response of a patient at a given dose level. A random number is generated over (0,1) and compared with the probability of toxicity. S is not far from the notion of a latent variable.

— Paoletti & Kramar (2009) doi:10.1002/sim.3682

In simulating the data, the tolerance of each patient can be considered a uniformly distributed random variable on the interval \([0, 1]\), which we term a patient’s latent toxicity tolerance and denote \(u_i\) for the \(i\)th entered patient. At the dose \((d_j)\) assigned to patient \(i\), if the tolerance is less than or equal to its true dose-limiting toxicity probability (ie, \(u_i \le \pi_j\)), then patient \(i\) has a dose-limiting toxicity; otherwise the patient has a non–dose-limiting toxicity outcome. Of course, in a real trial, it is impossible to observe a patient’s latent tolerance, but it is a useful tool in simulation …

— Wages, Iasonos, O’Quigley & Conaway (2019) doi:10.1002/pst.1974

Unfolding this ‘hidden program’ in package escalation

Compare https://refactoring.com/catalog/extractVariable.html
See also https://pubpeer.com/publications/4CEB4434EA94CC0AA3BCAF63617A78#1

git diff

Foreshadowing some consequences of \(u_i\)

• It’s nothing less than a principium individuationis

• It lets us inject ordinal toxicities into trial simulations,

• including grade-5 (fatal) toxicities,

• thus providing an index into questions of trial safety

• … about which no one can say, “that’s somebody else’s department.”

Breaking the ‘fourth wall’ : Therapeutic Index (TI)

knowledge about potential toxicities needs to be interpreted in relation to the desired pharmacology of the drug candidate, as well as the intended clinical indication, the associated unmet medical need and the competitive environment.
…
the therapeutic index (TI) of drug candidates … is a quantitative relationship between their efficacy (pharmacology) and safety (toxicology) that can be calculated using various pairs of pharmacological and toxicological end points.

— Muller & Milton (2012) doi:10.1038/nrd3801

• Outside oncology: \[TI = \frac{TD_{50}}{ED_{50}} \gg 1\]

• For oncology indications—esp. with on-target toxicities: \[ \mathrm{MTD}_i \cong \text{Paoletti & Kramar's}\ S \\ \mathrm{MTD}_i^g := \max_x \{ x\ |\ y_i(x) < g \}, g \in \{1,...,5\} \\ y_i : dose \mapsto Y_i \in \{0,1,2,3,4,5\}\;\; \text{(left-continuous step function)}\; \\ TI := \frac{\mathrm{MTD}_i^{g=4}}{\mathrm{MTD}_i^{g=3}} = \frac{\text{Gr4 tox dose threshold}}{\text{Gr3 tox dose threshold}} \]

To recap the story up to this point …

• Dose-finding simulations use Bernoulli random draws
• We unfolded sim code to reveal embedded \(u_i \sim \mathscr{U}[0,1]\)
  • We applied Fowler’s ‘extract variable’ refactoring,
  • in a sense, reversing the ‘inline variable’ factoring that had swept \(u_i\) under the rug.
• We’re not the first to notice \(u_i\), only the first to interpret it realistically:
  • \(u_i\) is the CDF of a ‘latent’ \(\mathrm{MTD}_i\)
  • ‘Latent’ is not the same as ‘nebulous’!
  • \(\mathrm{MTD}_i\) is realizable via titration.
• Introduced an oncology-adapted TI concept …
  • providing counterfactual linkage from binary DLTs to ordinal toxicities,
  • enabling dose-escalation trial simulations to predict fatalities.

21 CFR §312.42(b)(1)(i)

Title 21: Food and Drugs
PART 312—INVESTIGATIONAL NEW DRUG APPLICATION
Subpart C—Administrative Actions

§312.42 Clinical holds and requests for modification.
…
(b) Grounds for imposition of clinical hold–(1) Clinical hold of a Phase 1 study under an IND. FDA may place a proposed or ongoing Phase 1 investigation on clinical hold if it finds that:
(i) Human subjects are or would be exposed to an unreasonable and significant risk of illness or injury;

An opening for deeply adaptive dose-finding

Weakly vs strongly adaptive dose finding

The method is fully adaptive, makes use of all the information available at the time of each dose assignment, and directly addresses the ethical need to control the probability of overdosing.

— Babb, Rogatko & Zacks (1998) “Cancer phase I clinical trials: efficient dose escalation with overdose control” Stat Med. 1998 May 30;17(10):1103-20.

• Weak adaptiveness of CRM &al.
  • Benchmark is 3+3 ‘bogeyman’ w/ forgetful last-cohort-only dependence
  • CRM &c. incorporate all previous observed toxic responses
  • Adaptation occurs across cohorts — in the ‘N-dimension’ of enrollment
  • Adaptation is dose-centered
  • Patient ≈ Bernoulli random variate
• ‘Strong’ adaptiveness of DTAT
  • Adaptation primarily within individuals — by a dose titration algorithm (DTA)
  • …but the DTA is subject to adaptation as well — DTA + Tuning = DTAT
  • Thus DTAT involves a nested (‘deep’) adaptation
  • Adaptation is patient-centered
  • Patient ≈ stochastic state-space model

Precautionary Coherence

See: Norris DC. Precautionary Coherence Unravels Dose Escalation Designs. bioRxiv. Dec 29, 2017. doi:10.1101/240846

Extension to Phase I/II

Forthcoming in Mayo Clinic Proceedings

DTAT Bibliography

1. Norris DC. Dose Titration Algorithm Tuning (DTAT) should supersede ‘the’ Maximum Tolerated Dose (MTD) in oncology dose-finding trials. F1000Research. 2017;6:112. doi:10.12688/f1000research.10624.3. lay explainer
2. –––. Dose Titration Algorithm Tuning (DTAT) should supplant ‘the’ MTD. May 2017. [podium presentation] Symposium on Dose Selection for Cancer Treatment Drugs, Stanford Center for Innovative Study Design (CISD) May 12, 2017. doi:10.7490/f1000research.1114209.1.
3. –––. Costing ‘the’ MTD. bioRxiv. August 2017:150821. doi:10.1101/150821. lay explainer ; 2-minute video
4. –––. Costing ‘the’ MTD: What Is the Economic and Human Cost of 1-Size-Fits-All Dose Finding in Oncology? [poster] Presented at 8th American Conference on Pharmacometrics (ACoP8), October 16, 2017. doi:10.7490/f1000research.1114988.1.
5. –––. One-size-fits-all dosing in oncology wastes money, innovation and lives. Drug Discovery Today. 2018;23(1):4-6. doi:10.1016/j.drudis.2017.11.008. Shiny app
6. –––. Precautionary Coherence Unravels Dose Escalation Designs. bioRxiv. December 2017:240846. doi:10.1101/240846. lay explainer ; Shiny+D3 app
7. –––. Costing ‘the’ MTD … in 2-D. bioRxiv. July 2018:370817. doi:10.1101/370817 lay explainer
8. –––. Ethical Review and Methodologic Innovation in Phase 1 Cancer Trials. JAMA Pediatrics. April 2019. doi:10.1001/jamapediatrics.2019.0811 2-minute video
9. —. Comment on Wages et al., Coherence principles in interval-based dose finding. Pharmaceutical Statistics 2019, DOI: 10.1002/pst.1974. Pharmaceutical Statistics. March 2020. doi:10.1002/pst.2016
10. –––. Retrospective analysis of a fatal dose-finding trial. arXiv:2004.12755 [stat.ME]. April 2020. http://arxiv.org/abs/2004.12755 Tweetorial
11. Norris DC, Sen S, Groisberg R, Subbiah V. Patient Centered, Physician Investigator Friendly, Pragmatic Phase I/II Trial Designs: The 4P Model. Mayo Clinic Proceedings. Forthcoming 2020.