## Cancer Vaccine Phase I Design with Simultaneous Evaluation of Immunogenecity and Toxicity

#### Description

This package contains the functions for implementing the visit design for Phase I cancer vaccine trials.

#### Background

Phase I clinical trials are the first step in drug development to apply a new drug or drug combination on humans. Typical designs of Phase I trials use toxicity as the primary endpoint and aim to find the maximum tolerable dosage. However, these designs are generally inapplicable for the development of cancer vaccines because the primary objectives of a cancer vaccine Phase I trial often include determining whether the vaccine shows biologic activity.

The visit design allows dose escalation to simultaneously account for immunogenicity and toxicity. It uses lower dose levels as the reference for determining if the current dose level is optimal in terms of immune response. It also ensures subject safety by capping the toxicity rate with a given upper bound. These two criteria are simultaneously evaluated using an intuitive decision region that avoids complicated safety and immunogenicity trade-off elicitation from physicians.

There are several considerations that are clinically necessary for developing the design algorithm. First, we assume that there is a non-decreasing relationship that exists between toxicity and dosage, i.e., the toxicity risk does not decrease as dose level increases. Second, the immune response rate may reach a plateau or even start to decline as the dose level increases.

#### Notation

For subject \(s\), let \(D_s=l\) (\(l=1,…,L\)) denote the received dose level, \(T_s=1\) if any DLT event is observed from the subject and \(0\) otherwise, \(R_s=1\) if immune response is achieved for the subject and \(0\) otherwise.

Let \(θ^{(l)}_{ij}=P(T=i, R=j|D=l)\) for \(i,j=0,1\), \(θ^{(l)}=\{θ_{ij}^{(l)}:i,j=0,1\}\) and \(Θ = \{θ^{(l)}: l=1,…,L\}\). Furthermore, for dose level \(l\), let \(p^{(l)}=P(T=1|D=l)=θ_{10}^{(l)}+θ_{11}^{(l)}\) be the DLT risk, \(q^{(l)}=P(R=1|D=l)=θ_{01}^{(l)}+θ_{11}^{(l)}\) be the immune response probability, and \(r^{(l)}=θ_{00}^{(l)}θ_{11}^{(l)}/θ_{01}^{(l)}θ_{10}^{(l)}\) be the odds ratio. Let \(n_{ij}^{(l)}\) be the observed number of subjects with \(T=i\) and \(R=j\) at dose level \(l\), \(n^{(l)}=\{n_{ij}^{(l)}:i,j=0,1\}\) and \(H\) denote all the data observed by the time the current analysis is conducted.

#### Dose escalation algorithm

The dose escalation algorithm is based on the posterior probability distribution of \(π(p^{(l)}, q^{(l)}|H)\), where \(p^{(l)}\) and \(q^{(l)}\) represent the DLT risk and immune response rate, respectively, of the current dose level \(l\), and \(H\) denotes the cumulative data at the time of interim analysis.

Let \(p_l\) denote the lower boundary of DLT risk below which the dose is considered absolutely safe, \(p_u\) denote the upper boundary of DLT risk above which the dose is considered toxic. visit implements a sequential identification approach based on conditional probabilities derived from \(π(p^{(l)}, q^{(l)}|H)\). Let \(C_1, C_2, C_3\) be fixed cut-off values in \([0,1]\). The steps are as follows:

Step 1.

  If \(Prob(p^{(l)} > p_U|H) > C_1\), then the current dose level is considered to be too toxic. The trial should be stopped and the next lower dose level should be reported as the recommended dose.

Step 2.

  \(Prob(q^{(l)} ≤q q_L| p^{(l)} ≤q p_U, H) > C_2\), then the current dose level is considered to be no more effective than its lower dose levels. The trial should be stopped and the next lower dose level should be reported as the recommended dose.

Step 3.

  If \(Prob(p^{(l)} ≤q p_L| p^{(l)} ≤q p_U, q^{(l)} > q_L, H) > C_3\), then the current dose level is considered to be safe and effective. The trial will escalate to dose level \(l+1\).

Step 4.

  The current dose level is considered to be uncertain. The trial should continue to treat more patients at dose level \(l\).

The values of should be chosen \(C_1, C_2, C_3\) prior to study initiation and reflect the considerations of the investigators and patients. These thresholds should also give reasonable overall study operating characteristics.

We can see that, based on the posterior distribution of \(π(p^{(l)}, q^{(l)}|H)\), the currently dose level is in one of the four regions: 1: too toxic, 2: no more effective than its lower dose, 3: safe and effective, and 4: uncertain. These regions are termed as a Decision Map.

#### Probability models

visit provides several options for the probability models that can be considered for Bayesian inference. The models are non-decreasing with respect to the dose-toxicity relationship and avoid monotonic assumptions for the dose-immune response curve.

### Non-parametric model

As one of the simplest models, we posit no assumptions on the dose-toxicity or dose-immune response relationships and assume the outcome data \(n_{00}, n_{01}, n_{10}, n_{11}\) follow a multinomial distribution.

### Non-parametric+ model

This is the simplified non-parametric model with the odds ratios \(r=1\).

### Partially parametric model

Compared to non-parametric models, a parametric model may allow the incorporation of dose-toxicity, dose-efficacy, and toxicity-efficacy relationships in dose escalation. In the context of evaluating cancer vaccines, however, it is difficult to posit assumptions on the dose-efficacy relationship, since the immune response rate may even decrease as the dose level increases. On the other hand, it remains reasonable to assume that the dose-toxicity curve is non-decreasing. Therefore, we propose a partially parametric model that only makes assumptions about dose-toxicities but leaves the dose-immune response relationship unspecified.

Specifically, we construct the dose-toxicity model as:

\( \log p^{(l)}= e^α \log τ^{(l)}. \)

The \(τ^{(l)}\)'s are deterministic design parameters reflecting the expectation of the DLT risk at dose level \(l\) with \(τ^{(l)} > τ^{(l')}\) for \(l> l'\).

For the immune response and the odds ratio, we assume \(q^{(l)}\) and \(r^{(l)}\) at different dose levels are independent a priori.

### Partially parametric+ model

This is the simplified partially parametric model with the odds ratios \(r=1\).

#### References

Wang, C., Rosner, G. L., & Roden, R. B. (2019). A Bayesian design for phase I cancer therapeutic vaccine trials. Statistics in medicine, 38(7), 1170-1189.