SFU: Surface-Free Utility-Based Design for Dose Optimization in Cancer Drug Combination Trials

Abstract

Precision oncology has demonstrated the potential of drug combinations in effectively enhancing anti-tumor efficiency and controlling disease progression. Nonetheless, dose optimization in early-phase drug combination trials presents various challenges and is considerably more complex than single-agent dose optimization. To address this, we propose a surface-free design for exploring the optimal doses of combination therapy within the phase I–II framework. Rather than relying on parametric models to define the shape of toxicity and efficacy surfaces, our approach centers on characterizing dose-toxicity and dose-efficacy relationships between adjacent dose combinations using surface-free models. The proposed design encompasses a run-in phase, facilitating a swift exploration of the dose space, followed by a main phase where the dose-finding rule relies on the proposed surface-free model. Through extensive simulation studies, we have thoroughly examined the operating characteristics of this innovative design. The results demonstrate that our method exhibits desirable operating characteristics across a wide range of dose-toxicity and dose-efficacy relationships.

Appendix: Extending the Proposed Method to Include Three-Dose Combinations

The proposed design can be straightforwardly generalized to the combination of three or more drugs. This appendix elucidates the models and dose optimization steps when three drugs are involved. Regarding the toxicity model, we assume $\theta ^T=1-p_{111}$. Following the “equal-jump” assumption, the parameterization of $\theta _j^T$, $\tau _k^T$, and $\nu _m^T$ is as follows:

$$\begin{aligned} \theta _j^T= & {} \frac{1-p_{jkm}}{1-p_{j-1,k,m}},j=2,\cdots ,J,k=1,\cdots ,K,m=1,\cdots ,M, \\ \tau _k^T= & {} \frac{1-p_{jkm}}{1-p_{j,k-1,m}},j=1,\cdots ,J,k=2,\cdots ,K,m=1,\cdots ,M, \\ \nu _m^T= & {} \frac{1-p_{jkm}}{1-p_{j,k,m-1}},j=1,\cdots ,J,k=1,\cdots ,K,m=2,\cdots ,M. \end{aligned}$$

The DLT rate of dose combination (j, k, m) takes the form

$$\begin{aligned} p_{jkm}={\left\{ \begin{array}{ll} 1-\theta ^{T}, &{} j=1,k=1,\text { and }m=1,\\ 1-\theta ^{T}\theta _{2}^{T}\cdots \theta _{j}^{T}, &{} j>1,k=1,\text { and }m=1,\\ 1-\theta ^{T}\tau _{2}^{T}\cdots \tau _{k}^{T}, &{} j=1,k>1,\text { and }m=1,\\ 1-\theta ^{T}\nu _{2}^{T}\cdots \nu _{m}^{T}, &{} j=1,k=1,\text { and }m>1,\\ 1-\theta ^{T}\theta _{2}^{T}\cdots \theta _{j}^{T}\tau _{2}^{T}\cdots \tau _{k}^{T}, &{} j>1,k>1,\text { and }m=1,\\ 1-\theta ^{T}\theta _{2}^{T}\cdots \theta _{j}^{T}\nu _{2}^{T}\cdots \nu _{m}^{T}, &{} j>1,k=1,\text { and }m>1,\\ 1-\theta ^{T}\tau _{2}^{T}\cdots \tau _{k}^{T}\nu _{2}^{T}\cdots \nu _{m}^{T}, &{} j=1,k>1,\text { and }m>1,\\ 1-\theta ^{T}\theta _{2}^{T}\cdots \theta _{j}^{T}\tau _{2}^{T}\cdots \tau _{k}^{T}\nu _{2}^{T}\cdots \nu _{m}^{T}, &{} j>1,k>1,\text { and }m>1. \end{array}\right. } \end{aligned}$$

The efficacy probability of dose combination (j, k, m) can be derived similarly. Let $\theta ^E=\textrm{logit}(q_{111})$. Following the “equal-jump” assumption, $\theta _j^E$,$\tau _k^E$, and $\nu _m^E$ are parameterized as follows:

$$\begin{aligned} q_{jkm}={\left\{ \begin{array}{ll} \textrm{expit}(\theta ^{E}), &{} j=1,k=1,\text { and }m=1,\\ \textrm{expit}(\theta ^{E}+\theta _{2}^{E}+\cdots +\theta _{j}^{E}), &{} j>1,k=1,\text { and }m=1,\\ \textrm{expit}(\theta ^{E}+\tau _{2}^{E}+\cdots +\tau _{k}^{E}), &{} j=1,k>1,\text { and }m=1,\\ \textrm{expit}(\theta ^{E}+\nu _{2}^{E}+\cdots +\nu _{m}^{E}), &{} j=1,k=1,\text { and }m>1,\\ \textrm{expit}(\theta ^{E}+\theta _{2}^{E}+\cdots +\theta _{j}^{E}+\tau _{2}^{E}+\cdots +\tau _{k}^{E}), &{} j>1,k>1,\text { and }m=1,\\ \textrm{expit}(\theta ^{E}+\theta _{2}^{E}+\cdots +\theta _{j}^{E}+\nu _{2}^{E}+\cdots +\nu _{m}^{E}), &{} j>1,k=1,\text { and }m>1,\\ \textrm{expit}(\theta ^{E}+\tau _{2}^{E}+\cdots +\tau _{k}^{E}+\nu _{2}^{E}+\cdots +\nu _{m}^{E}), &{} j=1,k>1,\text { and }m>1,\\ \textrm{expit}(\theta ^{E}+\theta _{2}^{E}+\cdots +\theta _{j}^{E}+\tau _{2}^{E}+\cdots +\tau _{k}^{E}\\ \qquad \ \ \nu _{2}^{E}+\cdots +\nu _{m}^{E}), &{} j>1,k>1,\text { and }m>1. \end{array}\right. } \end{aligned}$$

The MTDC for each specific axis, $j_{km}^*$, $k_{jm}^*$, $m_{jk}^*$, can be elicited by extending (7) of the manuscript:

$$\begin{aligned} j_{km}^{*}= & {} {\mathop {\mathrm{arg\,min}}\limits _{\begin{array}{c} k^{'}=1,\ldots ,K^*,\\ m^{'}=1,\ldots ,M^* \end{array}}}~|{\hat{p}}_{jk^{'}m^{'}}-\phi _{T}|, \\ k_{jm}^{*}= & {} {\mathop {\mathrm{arg\,min}}\limits _{\begin{array}{c} j^{'}=1,\ldots ,J^*,\\ m^{'}=1,\ldots ,M^* \end{array}}}~|{\hat{p}}_{j^{'}km^{'}}-\phi _{T}|, \\ m_{jk}^{*}= & {} {\mathop {\mathrm{arg\,min}}\limits _{\begin{array}{c} j^{'}=1,\ldots ,J^*,\\ k^{'}=1,\ldots ,K^* \end{array}}}~|{\hat{p}}_{j^{'}k^{'}m}-\phi _{T}|. \end{aligned}$$

${\mathcal {S}}$ is updated as:

• (A) If $j_{km}^*>j$, $k_{jm}^*>k$, and $m_{jk}^*>m$, ${\mathcal{S}} = {\{} (j-1,k-1,m), (j-1,k,m-1), (j,k-1,m-1), (j-1,k,m), (j,k-1,m), (j,k,m-1), (j,k,m), (j+1,k,m), (j,k+1,m), (j,k,m+1), (j+1,k+1,m), (j+1,k,m+1), (j,k+1,m+1){\}}$;

• (B) If $j_{km}^*=j$, $k_{jm}^*>k$, and $m_{jk}^*>m$, ${\mathcal {S}}=\{(j-1,k-1,m),\ (j-1,k,m-1),\ (j,k-1,m-1),\ (j-1,k,m),\ (j,k-1,m),\ (j,k,m-1),\ (j,k,m),\ (j,k+1,m),\ (j,k,m+1),\ (j,k+1,m+1)\}$;

• (C) If $j_{km}^*>j$, $k_{jm}^*=k$, and $m_{jk}^*>m$, ${\mathcal {S}}=\{(j-1,k-1,m),\ (j-1,k,m-1),\ (j,k-1,m-1),\ (j-1,k,m),\ (j,k-1,m),\ (j,k,m-1),\ (j,k,m),\ (j+1,k,m),\ (j,k,m+1),\ (j+1,k,m+1)\}$;

• (D) If $j_{km}^*>j$, $k_{jm}^*>k$, and $m_{jk}^*=m$, ${\mathcal {S}}=\{(j-1,k-1,m),\ (j-1,k,m-1),\ (j,k-1,m-1),\ (j-1,k,m),\ (j,k-1,m),\ (j,k,m-1),\ (j,k,m),\ (j+1,k,m),\ (j,k+1,m),\ (j+1,k+1,m)\}$;

• (E) If $j_{km}^*=j$, $k_{jm}^*=k$, and $m_{jk}^*>m$, ${\mathcal {S}}=\{(j-1,k-1,m),\ (j-1,k,m-1),\ (j,k-1,m-1),\ (j-1,k,m),\ (j,k-1,m),\ (j,k,m-1),\ (j,k,m),\ (j,k,m+1)\}$;

• (F) If $j_{km}^*=j$, $k_{jm}^*>k$, and $m_{jk}^*=m$, ${\mathcal {S}}=\{(j-1,k-1,m),\ (j-1,k,m-1),\ (j,k-1,m-1),\ (j-1,k,m),\ (j,k-1,m),\ (j,k,m-1),\ (j,k,m),\ (j,k+1,m)\}$;

• (G) If $j_{km}^*>j$, $k_{jm}^*=k$, and $m_{jk}^*=m$, ${\mathcal {S}}=\{(j-1,k-1,m),\ (j-1,k,m-1),\ (j,k-1,m-1),\ (j-1,k,m),\ (j,k-1,m),\ (j,k,m-1),\ (j,k,m),\ (j+1,k,m)\}$;

• (H) If we have three $"="$, or two $"="$ and one $"<"$, or one $"="$ and two $"<"$, ${\mathcal {S}}=\{(j-1,k-1,m),\ (j-1,k,m-1),\ (j,k-1,m-1),\ (j-1,k,m),\ (j,k-1,m),\ (j,k,m-1),\ (j,k,m)\}$;

• (I) If $j_{km}^*<j$, $k_{jm}^*<k$, and $m_{jk}^*<m$, ${\mathcal {S}}=\{(j-1,k-1,m),\ (j-1,k,m-1),\ (j,k-1,m-1),\ (j-1,k,m),\ (j,k-1,m),\ (j,k,m-1)\}$.

As per the definition of ${\mathcal {S}}$ provided earlier, the dose exploration of three drugs can be seamlessly integrated into the proposed dose-finding algorithm.