Algorithm 3. Correlated Beta (CoBe) Dose Optimisation Algorithm

BEGIN ALGORITHM 1. Initialisation: a. Choose in collaboration with clinicians and experts i. Total trial participants available, N ii. Sampling cohort size, c (=6 in this work) iii. Determine whether a single-administration, prime/boost, or prime/boost/second-boost paradigm is being used. iv. Determine all potential doses, $d_i$, in the discretized dosing domain, see [Discretization]). v. Choose length parameter(s) for the efficacy similarity kernel ($l=0.2$, $l_1=l_2=0.25, l_1=l_2=l_3=0.4$ in this work) vi. Choose length parameter(s) for the toxicity similarity kernel (the same as for efficacy in this work) vii. Query experts to determine any potential priors. 2. Initialization of Beta distributions - in silico a. Initialise description of efficacy probability distribution for each dose $d_i$ as $p_{i,eff}^0~Beta\left({\alpha }_{i,eff}^0,{\beta }_{i,eff}^0\right)$ b. Initialise description of toxicity probability distribution for each dose $d_i$ as $p_{i,tox}^0~Beta\left({\alpha }_{i,tox}^0,{\beta }_{i,tox}^0\right)$ 3. Thompson sampling for dose selection-in silico a. For each dose $d_i$, sample ${\widehat{p}}_{1,eff}$ and ${\widehat{p}}_{2,eff}$ from the relevant Beta distributions. b. Select for trialing dose $d_i$ such that ${\widehat{U}}_i>{\widehat{U}}_j$ for all doses $d_j$, where $U_i\left(p_{i,eff},p_{i,tox}\right)$ is the utility function to be maximised. 4.Repeat step 3 until sampling cohort is full (c repeats total) 5. Trialing and data collection – practical a. Conduct a trial of c individuals, respectively, at the c doses chosen in steps 3 and 4. This is simulated in this work but would be practical lab work in real life application. b. Record c data points consisting of {dose given, whether efficacy was observed, whether toxicity was observed} 6. Model Updating-in silico a. Update ${\alpha }_{i,eff}^{n-c},{\beta }_{i,eff}^{n-c},{\alpha }_{i,tox}^{n-c},{\beta }_{i,tox}^{n-c}$ to ${\alpha }_{i,eff}^n,{\beta }_{i,eff}^n,{\alpha }_{i,tox}^n,{\beta }_{i,tox}^n$ using: i. Update $\alpha { }_{i,eff}^{n-c},{\beta }_{i,eff}^{n-c},{\alpha }_{i,tox}^{n-c},{\beta }_{i,tox}^{n-c}$ to ${\alpha }_{i,eff}^{n-c+1},{\beta }_{i,eff}^{n-c+1},{\alpha }_{i,tox}^{n-c+1},{\beta }_{i,tox}^{n-c+1}$ using Algorithm 2 with a data point gathered in step 5. ii. Repeat for all other data points gathered in step 5 (order does not matter) 7. Prediction of optimal dose – in silico a. For each dose di, calculate the median response probabilities ${\overline{p}}_{i,eff}$ and ${\overline{p}}_{i,tox}$ b. The predicted optimal dose is $d_i$ such that $U\left({\overline{p}}_{i,eff},{\overline{p}}_{i,tox}\right)\ge U\left({\overline{p}}_{j,eff},{\overline{p}}_{j,tox}\right)$ where $U_i\left(p_{i,eff},p_{i,tox}\right)$ is the utility function to be maximised. 8.Repeat steps 3-7 until all N trial participants have been utilised. END ALGORITHM