• Typically first study of drug (treatment) focused on clinical outcomes, specifically safety

• Passed pre-clinical hurdles. Drug mechanism works in cell lines, animals.
  • e.g. Targeted therapy has been shown to inhibit genetic pathway has been inhibited

• Assumption is that toxicity (and potential for efficacy) increases with dose level

• Goal is to identify "maximum tolerated dose" (MTD) : greatest dose level that induces an acceptable level of toxicity

• Common elements: focus on safety

• Points of divergence: Multiple dose levels? What are expected side effects of drug? For whom is it intended?

• Design driven by type, severity of expected toxicities, particularly dose-limiting toxicities (DLTs):

  • When DLTs are not terrible, subjects are healthy volunteers. Designs more closely resemble true experiments

  • When they are terrible, e.g. cytotoxic drugs, subjects are patients. Designs focus on ethical treatment of enrolled patients

• Often randomized, placebo-controlled:
  1. Randomize patients to (i) placebo or (ii) one of several dose levels
  2. Monitor patients for DLTs, e.g. headache, rash, cold symptoms. Typically minor and reversible
  3. MTD could be largest dose level in which estimated rate of DLT is less than X%
• These designs discussed later in the course

1. What is definition of healthy?

2. Is it safe to extrapolate from healthy to diseased?

3. How much should volunteers be paid? Does financial compensation account for risks?

• Not randomized. No placebo
• Typically dose-escalation study:
  1. Start at very low, conservative dose.
  2. Monitor for DLTs, e.g. hematologic, renal or hepatic function, cardiac, neurotoxicity
  3. Escalate dose level in next patient(s) if warranted
  4. Stop when toxicity is too large. MTD is largest dose level in which estimated rate of DLT is less than X%

1. Acceptable to purposefully induce serious or life-threatening toxicity?

2. Is truly informed consent possible?

• We'll be discussing designs for these trials

• Enrolling sick/terminally-ill patients means:

  1. Small study population \(\rightarrow\) few patients \(\rightarrow\) low precision, no adjustment for confounding

  2. Heterogenous patients \(\rightarrow\) complicated disease and many prior therapies \(\rightarrow\) limited extrapolation, high false-positive rate

  3. Toxicities occur over time \(\rightarrow\) irreversible toxicities (i.e. death) may occur

Key assumption of phase 1 designs: risk of toxicity (and therefore efficacy/response) monotonically increases with dose

1. How should first dose (\(d_1\)) be chosen?
   • Too low risks being sub-therapeutic

   • Too high may be too toxic

   • \(d_1\) is often \(0.1\times\text{MELD}_{10}\) = 1/10th of dose that kills 10% of mice

1. How should subsequent dose levels (\(d_2, \ldots, d_m\)) be selected?
   • Subsequent dose levels are fractional increases of \(d_1\) (next page)

Fibonacci sequence: \[1, 2, 3, 5, 8, 13, \ldots\]

\[ \begin{aligned} \tfrac{2}{1}, \tfrac{3}{2}, &\tfrac{5}{3}, \tfrac{8}{5}, \tfrac{13}{8}\Rightarrow\\ d_2&=2d_1, \\d_3&=1.5d_2, \\d_4&=1.67d_3, \\d_5&=1.6d_4, \\d_6&=1.63d_5 \end{aligned} \]

\[d_2=2d_1, \\d_3=1.67d_2, \\d_4=1.5d_3, \\d_5=1.33d_4, \\d_6=1.33d_5\]

Alternative is to use constant multiplier: \[d_2=c\times d_1, \\d_3=c\times d_2, \\d_4=c\times d_3, \\d_5=c\times d_4, \\d_6=c\times d_5\] equivalent to equal-spaced log-dose levels

Remaining questions (how to escalate?; how many patients to enroll?; what is MTD?) tied to choice of design. Two broad categories:

1. Algorithmic

2. Model-based

Commonly known as "3+3"

1. Enroll three at current dose (start at \(d_1\))

2. If
   1. 0/3 toxicities, escalate
   2. 1/3, enroll 3 more
      • if 1/6, escalate
      • if \(>\) 1/6, stop trial
   3. \(>\) 1/3, stop trial

3. MTD is dose below largest dose having \(>\) 1/3 or \(>\) 1/6

• Many clinical trial questions may be answered with a yes/no (binary) answer, which might be "conclude that a particular dose is the MTD" (phase 1), "determine that an agent is active" (phase 2), or "conclude that new agent is better than the standard of care" (phase 3)

• Want to know \(\Pr(C)\), but this may be analytically complex

• Monte Carlo methods allows us to estimate by sampling \(C\) from its distribution: \[ \Pr(C) \approx \sum_{i=1}^M c_m / M \]

• For example, if \(C\) corresponds to "dose 2 is found to be MTD", simulate 1000 3+3 trials and report the proportion that recommend dose 2 as estimate of \(\Pr(C)\)

• Often called an 'operating characteristic' (OC). Trials will have many OCs

• Either analytic or simulation-based derivations require specifying all parameters, e.g.
  • In a 3+3 with \(k\) dose levels, must specify \(k\) probabilities \(\pi_i = \Pr(\text{Tox} | d_i)\), \(i=1,\ldots,k\)
• So an OC is conditional on particular truth

• R is freely available at https://cran.r-project.org/

• Rstudio provides a useful way to interface with R https://www.rstudio.com/

Suppose two dose levels, with \(\pi_1=0.1\) and \(\pi_2 = 0.25\).

set.seed(1);
nsim = 2e3;
true_prob = c(0.1, 0.25);
cohort1a = rbinom(nsim, 3, true_prob[1]);
cohort1b = rbinom(nsim, 3, true_prob[1]);
cohort2a = rbinom(nsim, 3, true_prob[2]);
cohort2b = rbinom(nsim, 3, true_prob[2]);

#Recommend dose 1 as MTD
mean((cohort1a == 0 | (cohort1a + cohort1b <= 1)) 
     & (cohort2a + cohort2b > 1));

## [1] 0.435

#Recommend dose 2 as MTD
mean((cohort1a == 0 | (cohort1a + cohort1b <= 1)) 
     & (cohort2a + cohort2b  <= 1));

## [1] 0.472

source("../StorerA.R");
results = 
  sim_storerA_fancy(true_probs = c(0.05, 0.10, 0.20, 0.30, 0.50, 0.70), 
                    nsim = 2e3, seed = 1);
names(results);

## [1] "seed"           "true_probs"     "all_results"    "mtd_individual"
## [5] "enrollment"     "mtd_summary"

results$mtd_summary;#Dose selection probability

##      0      1      2      3      4      5      6 
## 0.0240 0.1085 0.2780 0.3290 0.2300 0.0290 0.0015

#Proportion of patients treated at each dose
summary(factor(results$all_results[,"dose_num"])) / nrow(results$all_results)

##      1      2      3      4      5      6 
## 0.2167 0.2544 0.2538 0.1870 0.0775 0.0107

• Will always pick dose with 0/3 or 1/6 toxicities
• No need for statisticians or software

• No de-escalation
• As few as 3 patients enrolled
• What is interpretation of MTD?
• Significant degree of uncertainty:

• Four trials, same MTD

  Dose 1	Dose 2	Dose 3
  0/3	0/3	1/3+1/1
  1/3+0/3	1/3+0/3	1/3+1/1
  1/3+0/3	0/3	1/3+1/3
  0/3	1/3+0/3	1/3+1/2

Stage 1

• Evaluate patients individually
• Escalate after each DLT-free patient
• De-escalate after first DLT, go to stage 2

Stage 2

• Escalate after two consecutive DLT-free patients
• De-escalate after each DLT
• Stop after fixed sample size

To estimate MTD, fit logistic regression, \(\text{logit}\Pr(\text{DLT at dose } k) = \alpha + \beta k\), and choose \(k\) that is closest to desired rate

• Ability to go up and down
• Fast `climbing' of dose-toxicity curve
• Easy
• Fixed sample size

• No guarantee that current patients are assigned to the best dose level
• Not using full patient information to make dose assignments

• Also up-and-down
• Always de-escalate after DLT
• Escalate with probability \(q\) if no DLT, otherwise stay at current dose level

sim_biasedcoin = function(true_probs, 
                          start = 1, 
                          q_esc = 1,
                          n = 20, 
                          nsim = 100, 
                          seed=sample(.Machine$integer.max,1)) {
#Put code in me 
}

• Phase 1 trials are heterogenous and context dependent.

• Ethical imperative of cancer and HIV/AIDS trials has motivated development of specialized designs

• Algorithmic approaches ("3+3") are simple, fairly conservative, and very prevalent (\(>98\%\) of phase 1 cancer trials (Rogatko et al., 2007)) but demonstrably inferior in terms of identifying right dose.

• Concept of patient horizon, e.g. (Simon, 1977)

• Must consider quality of the knowledge gained and ethics of enrolling future patients in additional trials (Ashcroft et al., 1997)