quantumbradford2025 / Clinical Trial Optimization

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Assigning subjects to treatment groups is is pivotal in experimental clinical trials for novel drugs. A critical component involves identifying distinct subject cohorts suitable for specific treatments. These cohorts should exhibit maximum similarity based on measurable covariates (or attributes), which influence individual responses to treatment, thus enabling the differentiation of treatment effects from confounding factors.

The objective of the assignment is to minimize the discrepancy between any two groups in the weighted sum of the first and second moments of the covariates.

Each patient is associated with $r=3$ covariates $\vec{w}_i = (w_{i1}, w_{i2}, w_{i3})$ that are relevant for predicting the patient's outcome. There are $n$ patients in total. The decision maker knows the total number of subjects of the experiment $n$ and the respective covariates $\vec{w}=(\vec{w}_1, ⋯, \vec{w}_n)$. The decision maker will assign $k:=n/m$ subjects to each of $m ≥ 2$ treatment groups. In this case, we will consider the scenario where only two groups are used ($m=2$).
The binary decision variables are $\vec{x} = \{x_{ip}| i=1,\cdots, n, p=1,2\}$, where $x_{ip}=1$ if patient $i$ is assigned to group $p$, and $x_{ip}=0$ otherwise.

The objective of the assignment is to minimize the discrepancy between any two groups in the weighted sum of the first and second moments of the covariates.
That is, the discrepancy is calculated as follows:

where

and

The parameter $\rho$ regulates the relative weight of the first and the second moments. In this case, we will set $\rho=0.5$.

There are several constraints needed to ensure an appropriate patient stratification:

Putting it all together, this problem can be formulated as follows:

This corresponds to a mixed-integer optimization problem with 10 continuous variables ($\{d,\Delta\mu_1, \Delta\mu_2, \Delta\mu_3, \Delta\sigma_{11}, \Delta\sigma_{22}, \Delta\sigma_{33}, \Delta\sigma_{12}, \Delta\sigma_{13},\Delta\sigma_{23} \}$) and $2n-1$ binary variables $(\{x_{11}, \cdots , x_{n1}, x_{22}, \cdots, x_{n2}\})$.

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