Optimal Decision Criteria for the Study Design and Sample Size of a Biomarker-Driven Phase III Trial

Abstract

Background

The design and sample size of a phase III study for new medical technologies were historically determined within the framework of frequentist hypothesis testing. Recently, drug development using predictive biomarkers, which can predict efficacy based on the status of biomarkers, has attracted attention, and various study designs using predictive biomarkers have been suggested. Additionally, when choosing a study design, considering economic factors, such as the risk of development, expected revenue, and cost, is important.

Methods

Here, we propose a method to determine the optimal phase III design and sample size and judge whether the phase III study will be conducted using the expected net present value (eNPV). The eNPV is defined using the probability of success of the study calculated based on historical data, the revenue that will be obtained after the success of the phase III study, and the cost of the study. Decision procedures of the optimal phase III design and sample size considering historical data obtained up to the start of the phase III study were considered using numerical examples.

Results

Based on the numerical examples, the optimal study design and sample size depend on the mean treatment effect in the biomarker-positive and biomarker-negative populations obtained from historical data, the between-trial variance of response, the prevalence of the biomarker-positive population, and the threshold value of probability of success required to go to phase III study.

Conclusions

Thus, the design and sample size of a biomarker-driven phase III study can be appropriately determined based on the eNPV.

Appendices

Appendix 1: Formulation of PoS

The PoS of the traditional and enriched designs is formulated as follows, using the predicted distribution of treatment effect in each population of equation (1).

$$\begin{aligned} Po{\text{S}}_{{T_{n} }} & = \mathop \int \limits_{ - \infty }^{ + \infty } P\left( {\hat{\Delta }_{F} \ge z_{\alpha } \sqrt {\frac{{2\sigma^{2} }}{n}} |\Delta_{F}^{\ast} } \right)f\left( {\Delta_{F}^{\ast} |\Delta_{P}^{1} , \ldots ,\Delta_{P}^{H} ,\Delta_{N}^{1} , \ldots ,\Delta_{N}^{H} ,\tau^{2} } \right)d\Delta_{F}^{\ast} \\ & = 1 - {{\Upphi }}\left[ {\frac{{z_{\alpha } \sqrt {\frac{{2\sigma^{2} }}{n}} - \left( {\lambda \frac{{\sum w_{P,i} \hat{\Delta }_{P,i} }}{{\sum w_{P,i} }} + \left( {1 - \lambda } \right)\frac{{\sum w_{N,i} \hat{\Delta }_{N,i} }}{{\sum w_{N,i} }}} \right)}}{{\sqrt {\lambda^{2} \left( {\frac{1}{{\sum w_{P,i} }} + 2\tau^{2} } \right) + \left( {1 - \lambda } \right)^{2} \left( {\frac{1}{{\sum w_{N,i} }} + 2\tau^{2} } \right)} }}} \right], \\ Po{\text{S}}_{{E_{n} }} & = \mathop \smallint \limits_{ - \infty }^{ + \infty } P\left( {\hat{\Delta }_{P} \ge z_{\alpha } \sqrt {\frac{{2\sigma^{2} }}{n}} |\Delta_{P}^{\ast} } \right)f\left( {\Delta_{P}^{\ast} |\Delta_{P}^{1} , \ldots ,\Delta_{P}^{H} ,\tau^{2} } \right)d\Delta_{P}^{\ast} \\ & = 1 - {{\Upphi }}\left[ {\frac{{z_{\alpha } \sqrt {\frac{{2\sigma^{2} }}{n}} - \frac{{\sum w_{P,i} \hat{\Delta }_{P,i} }}{{\sum w_{P,i} }}}}{{\sqrt {\frac{1}{{\sum w_{P,i} }} + 2\tau^{2} } }}} \right]. \\ \end{aligned}$$

The PoS of the stratified and adaptive enrichment designs is calculated by simulations. The PoS in these study designs is represented as follows based on the study success criteria in each design mentioned above.

$$\begin{aligned} PoS_{{S_{n} }} & = P\left( {S_{S,1} |\Delta_{P}^{\ast} ,\Delta_{N}^{\ast} } \right)\left( { = PoS_{{S_{n} ,1}} } \right) + P\left( {S_{S,2} only|\Delta_{P}^{\ast} } \right)\left( { = PoS_{{S_{n} ,2}} } \right), \\ PoS_{{A_{n} }} & = P\left( {S_{A1,1} |\Delta_{P}^{\ast} ,\Delta_{N}^{\ast} } \right)\left( { = PoS_{{A1_{n} ,1}} } \right) + P\left( {S_{A2,1} only|\Delta_{P}^{\ast} } \right)\left( { = PoS_{{A2_{n} ,1}} } \right) \\ & \quad + P\left( {S_{A,2} |\Delta_{P}^{\ast} ,\Delta_{N}^{\ast} } \right)\left( { = PoS_{{A_{n} ,2}} } \right) + P\left( {S_{A,3} |\Delta_{P}^{\ast} } \right)\left( { = PoS_{{A_{n} ,3}} } \right). \\ \end{aligned}$$

Appendix 2: Expected discounted total revenue

In the case of success in enriched design, the expected value ($E_{E} )$ and variance ($V_{E} )$ of efficacy are calculated in the same manner as that in the traditional design. The expected discounted total revenue of enriched design ($R_{{E_{n} }}$) is given by:

$$R_{{E_{n} }} = PoS_{{E_{n} }} \cdot \lambda M \cdot m\left( {E_{{E_{n} }} ,V_{{E_{n} }} } \right) = \mathop \int \limits_{{T_{E}^{\ast} = s + \frac{2n}{\lambda r} + f + e}}^{T} e^{{ - \frac{\rho }{12}t}} \text{d}t.$$

The expected value of efficacy and its variance when the stratified and adaptive enrichment designs are successful are calculated by simulation. The expected discounted total revenue for sample size n in each arm of the stratified design ($R_{{S_{n} }}$) is given by

$$R_{{S_{n} }} = PoS_{{S_{n} ,1}} \cdot M \cdot m\left( {E_{{S_{n} ,1}} ,V_{{S_{n} ,1}} } \right)\mathop \int \limits_{{T_{S}^{\ast} = s + \frac{2n}{r} + f + e}}^{T} e^{{ - \frac{\rho }{12}t}} \text{d}t + PoS_{{S_{n} ,2}} \cdot \lambda M \cdot m\left( {E_{{S_{n} ,2}} ,V_{{S_{n} ,2}} } \right)\mathop \int \limits_{{T_{S}^{\ast} = s + \frac{2n}{r} + f + e}}^{T} e^{{ - \frac{\rho }{12}t}} \text{d}t,$$

where $E_{{S_{n} ,1}}$ and $E_{{S_{n} ,2}}$ are the expected values of efficacy in the case of $S_{S,1}$ and $S_{S,2},$ respectively, and $V_{{S_{n} ,1}}$ and $V_{{S_{n} ,2}}$ are their variances. Additionally, the expected discounted total revenue for sample size n in each arm of the adaptive enrichment design ($R_{{A_{n} }}$) is given by

$$R_{{A_{n} }} = PoS_{{A1_{n} ,1}} \cdot M \cdot m\left( {E_{{A1_{n} ,1}} ,V_{{A1_{n} ,1}} } \right)\mathop \smallint \limits_{{T_{A}^{\ast} = s + \frac{2n}{r} + f + e}}^{T} e^{{ - \frac{\rho }{12}t}} \text{d}t + PoS_{{A2_{n} ,1}} \cdot \lambda M \cdot m\left( {E_{{A2_{n} ,1}} ,V_{{A2_{n} ,1}} } \right)\mathop \smallint \limits_{{T_{A}^{\ast} = s + \frac{2n}{r} + f + e}}^{T} e^{{ - \frac{\rho }{12}t}} \text{d}t + PoS_{{A_{n} ,2}} \cdot M \cdot m\left( {E_{{A_{n} ,2}} ,V_{{A_{n} ,2}} } \right)\mathop \smallint \limits_{{T_{A}^{\ast} = s + \frac{2n}{r} + f + e}}^{T} e^{{ - \frac{\rho }{12}t}} \text{d}t + PoS_{{A_{n} ,3}} \cdot \lambda M \cdot m\left( {E_{{A_{n} ,3}} ,V_{{A_{n} ,3}} } \right) \cdot \mathop \smallint \limits_{{T_{A}^{\ast} = s + \left( {\frac{2nt}{r} + \frac{2n - 2nt}{\lambda r}} \right) + f + e}}^{T} e^{{ - \frac{\rho }{12}t}} \text{d}t,$$

where $E_{{A1_{n} ,1}} ,E_{{A2_{n} ,1}} ,E_{{A_{n} ,2}}$ and $E_{{A_{n} ,3}}$ are the expected values of efficacy in the case of $S_{{A1_{n} ,1}} ,S_{{A2_{n} ,1}} ,S_{{A_{n} ,2}}$ and $S_{{A_{n} ,3}},$ respectively, and $V_{{A1_{n} ,1}} ,V_{{A2_{n} ,1}} ,V_{{A_{n} ,2}}$ and $V_{{A_{n} ,3}}$ are their variances.

Appendix 3: Expected phase III cost

The discounted phase III cost in the enriched and stratified designs is given below:

$$\begin{aligned} C_{{E_{n} }} & = r\lambda \left( {c_{\rm sub} + c_{\rm bio} } \right)\mathop \int \limits_{s}^{{s + \frac{2n}{r\lambda }}} e^{{ - \frac{\rho }{12}t}} \text{d}t + c_{\rm trial} \mathop \int \limits_{s}^{{s + \frac{2n}{r\lambda } + f}} e^{{ - \frac{\rho }{12}t}} \text{d}t + c_{\rm CDx}, \\ C_{{S_{n} }} & = r\left( {c_{\rm sub} + c_{\rm bio} } \right)\mathop \int \limits_{s}^{{s + \frac{2n}{r}}} e^{{ - \frac{\rho }{12}t}} \text{d}t \quad + c_{\rm trial} \mathop \int \limits_{s}^{{s + \frac{2n}{r} + f}} e^{{ - \frac{\rho }{12}t}} \text{d}t + c_{\rm CDx}. \\ \end{aligned}$$

Additionally, let $p_{k}$ ( $k = 1, \ldots ,4$) denote the probability of selecting case 1 to case 4 in Table 5 and $C_{{A_{n} ,k}}$ denote the cost, excluding the interim analysis cost required in each case. The discounted phase III cost of conducting the adaptive enrichment design ($C_{{A_{n} }}$) is given by the following equation:

$$C_{{A_{n} }} = \mathop \sum \limits_{k = 1}^{4} p_{k} C_{{A_{n} ,k}} + c_{\rm interim} e^{{ - \frac{\rho }{12}\left( {s + \frac{{2nt^{\prime}}}{r} + f} \right)}} + c_{\rm CDx} .$$

Note that

$$\begin{aligned} C_{{A_{n} ,1}} & = C_{{A_{n} ,2}} = r\left( {c_{\rm sub} + c_{\rm bio} } \right)\mathop \int \limits_{s}^{{s + \frac{2n}{r}}} e^{{ - \frac{\rho }{12}t}} \text{d}t + c_{\rm trial} \mathop \int \limits_{s}^{{s + \frac{2n}{r} + f}} e^{{ - \frac{\rho }{12}t}} \text{d}t, \\ C_{{A_{n} ,3}} & = \left( {c_{\rm sub} + c_{\rm bio} } \right)\left\{ {r\mathop \int \limits_{s}^{{s + \frac{{2nt^{\prime}}}{r}}} e^{{ - \frac{\rho }{12}t}} \text{d}t + r\lambda \mathop \int \limits_{{s + \frac{{2nt^{\prime}}}{r}}}^{{s + \frac{{2nt^{\prime}}}{r} + \frac{{2n - 2nt^{\prime}}}{r\lambda }}} e^{{ - \frac{\rho }{12}t}} \text{d}t} \right\} \\ & + c_{\rm trial} \mathop \int \limits_{s}^{{s + \frac{{2nt^{\prime}}}{r} + \frac{{2n - 2nt^{\prime}}}{r\lambda } + f}} e^{{ - \frac{\rho }{12}t}} \text{d}t, \\ C_{{A_{n} ,4}} & = r\left( {c_{\rm sub} + c_{\rm bio} } \right)\mathop \int \limits_{s}^{{s + \frac{{2nt^{\prime}}}{r}}} e^{{ - \frac{\rho }{12}t}} \text{d}t + c_{\rm trial} \mathop \int \limits_{s}^{{s + \frac{{2nt^{\prime}}}{r} + f}} e^{{ - \frac{\rho }{12}t}} \text{d}t. \\ \end{aligned}$$

Table 5. The Optimal Sample Size, PoS, and eNPV in the Motivating Example.

Appendix 4: The value of parameters used in the numerical example and the motivating example

See Table A1, A2 and A3.

Table A1. The Value of Variable Parameters Used in the Numerical Example.

Table A2. The Values of Fixed Parameters Used in the Numerical Example.

Table A3. The Values of Parameters Used in the Motivating Example.