Optimal Biomarker-Guided Design for Targeted Therapy with Imperfectly Measured Biomarkers

Abstract

Targeted therapy revolutionizes the way physicians treat cancer and other diseases, enabling them to adaptively select individualized treatment according to the patient’s biomarker profile. The implementation of targeted therapy requires that the biomarkers are accurately measured, which may not always be feasible in practice. In this article, we propose two optimal biomarker-guided trial designs in which the biomarkers are subject to measurement errors. The first design focuses on a patient’s individual benefit and minimizes the treatment assignment error so that each patient has the highest probability of being assigned to the treatment that matches his/her true biomarker status. The second design focuses on the group benefit, which maximizes the overall response rate for all the patients enrolled in the trial. We develop a likelihood ratio test to evaluate the subgroup treatment effects at the end of the trial. Simulation studies show that the proposed optimal designs achieve our design goal and obtain desirable operating characteristics.

Appendices

Appendix

Proof of Theorem 1

For stage II patients, the treatment assignment is solely determined by W, therefore, conditional on W, T and M are independent. It follows that the probability of misassignment for subjects assessed with the error-prone measure W is given by

$$\displaystyle\begin{array}{rcl} & & \mathrm{pr}(T\neq M\vert W) = 1 -\mathrm{ pr}(T = M = 1\vert W) -\mathrm{ pr}(T = M = 0\vert W) {}\\ & & \quad = 1 -\mathrm{ pr}(M = 1\vert W)\mathrm{pr}(T = 1\vert W) - (1 -\mathrm{ pr}(M = 1\vert W))(1 -\mathrm{ pr}(T = 1\vert W)) {}\\ & & \quad =\mathrm{ pr}(M = 1\vert W) +\mathrm{ pr}(T = 1\vert W) - 2\mathrm{pr}(M = 1\vert W)\mathrm{pr}(T = 1\vert W) {}\\ & & \quad =\mathrm{ pr}(M = 1\vert W) +\mathrm{ pr}(T = 1\vert W)(2\mathrm{pr}(M = 0\vert W) - 1). {}\\ \end{array}$$

Therefore, if 2pr(M = 0 | W) − 1 < 0, i.e., π(W) ≡ pr(M = 0 | W) ≤ 1∕2, the misassignment probability pr(T ≠ M | W) is minimized when pr(T = 1 | W) = 1, that is, assigning the patient to the treatment T = 1. Similarly, if pr(M = 0 | W) > 1∕2, pr(T ≠ M | W) is minimized when pr(T = 0 | W) = 1, that is, assigning the patient to the treatment T = 0.

Proof of Theorem 2

Let f(W) denote the density function of W, and define C = p ₀₁pr(M = 0) + p ₁₀pr(M = 1), D ₀ = p ₀₀ − p ₀₁, D ₁ = p ₁₁ − p ₁₀, ω = D ₀ + D ₁ and δ = D ₁∕D ₀. It follows that

$$\displaystyle\begin{array}{rcl} & & \mathrm{pr}(Y = 1) =\sum _{ j=0}^{1}\sum _{ k=0}^{1}\mathrm{pr}(M = j,T = k)p_{ jk} {}\\ & =& C + D_{0}\int \mathrm{pr}(M = 0\vert W)\mathrm{pr}(T = 0\vert W)f(W)dW {}\\ & & +D_{1}\int \mathrm{pr}(M = 1\vert W)\mathrm{pr}(T = 1\vert W)f(W)dW {}\\ & =& C + D_{0}\int (1 -\mathrm{ pr}(M = 1\vert W))(1 -\mathrm{ pr}(T = 1\vert W))f(W)dW {}\\ & & +D_{1}\int \mathrm{pr}(M = 1\vert W)\mathrm{pr}(T = 1\vert W)f(W)dW {}\\ & =& C +\int \left [D_{0}\left \{1 -\mathrm{ pr}(M = 1\vert W)\right \} +\mathrm{ pr}(T = 1\vert W)\left \{D_{1} -\omega \mathrm{ pr}(M = 0\vert W)\right \}\right ]f(W)dW. {}\\ \end{array}$$

As a result, when ω > 0 which indicates a positive predictive marker effect, if π(W) ≡ pr(M = 0 | W) ≤ D ₁∕ω = δ∕(1 +δ), pr(Y = 1) is maximized when pr(T = 1 | W) = 1, that is, assigning the patient to the treatment T = 1; and if π(W) > δ∕(1 +δ), pr(Y = 1) is maximized when pr(T = 1 | W) = 0, that is, assigning the patient to the treatment T = 0. Similarly, when ω ≤ 0 which indicates a negative predictive marker effect, if π(W) ≡ pr(M = 0 | W) ≤ δ∕(1 +δ), pr(Y = 1) is maximized when assigning the patient to the treatment T = 0; and if π(W) > δ∕(1 +δ), pr(Y = 1) is maximized when assigning the patient to the treatment T = 1. In general, if π(W) ≤ δ∕(1 +δ), pr(Y = 1) is maximized when assigning the patient to the treatment T = I(ω > 0); and otherwise to T = 1 − I(ω > 0).