Unified Screening for Multiple Diseases

We thank the reviewer for the thorough review of our paper and constructive comments.

Limited results: In (1), we formalize the joint screening and diagnosis problem. This is distinct from (2), which focuses on the referral problem. We choose to solve (2) rather than (1) because our aim is not to propose entirely new screening and diagnosis policies—which can be difficult to justify and implement in clinical practice. Instead, we build upon existing screening policies already endorsed in clinical guidelines.

Characterizing the optimal referral policy becomes significantly more complex and far less intuitive when considering more than two diseases. One would expect the number of boundaries of interest (e.g., $5$ as shown in Figure 1(b) for $N=2$) to grow combinatorial with $N$, as one must account for boundaries that separate screening decisions across different subsets of diseases.

We adopt periodic screening policies with deterministic schedules because such approaches are widely used in clinical practice, as recommended by numerous guidelines—particularly for the early detection of chronic or progressive diseases. Common examples include annual mammograms for breast cancer screening, biennial colonoscopies for colorectal cancer, and regular HbA1c tests for diabetes monitoring.

Screening target $Y_n$: May represent a biomarker value, clinical metric, or similar quantity. Conditional independence over time is utilized in likelihood ratio and in proofs characterizing the decision boundaries. While the underlying disease state may evolve over time, this evolution can be modeled as a drift in the expectation of $Y_n$. The randomness in $Y_n$ is then primarily attributed to measurement noise or natural biological fluctuations, which we assume to be independent across time.

Why $x$ is entirely observable? One can use off-the-shelf risk prediction models to obtain disease risks for particular diseases. For instance, Gail model (for breast cancer), QRISK3 (for cardiovascular disease), (normalized) polygenic risk scores, and even AI-based models can be used to provide risk scores.

Why objective make sense: Our objective can be seen as a simplification of the standard Quality-Adjusted Life Years (QALYs) framework. Each year of life is either $1$ (acceptable quality, e.g., healthy) or $0$ (unacceptable quality, e.g., impaired, suffering). If the disease is detected before the adverse event happens, we assume issues related to the disease can be resolved or managed at a level such that the disease will not cause an unacceptable quality of life. However, after the adverse event happens, the disease causes unacceptable quality of life.

Relation to broader literature: To the best of our knowledge, there is no existing decision-making problem in the literature that yields the same structure of decision boundaries as those derived in our work. While our screening cost budget constraint bears some resemblance to the constraint in knapsack problems, the underlying objective in our setting is fundamentally different.

About empirical results: We agree that any simulation experiment may not fully capture the real-world benefits of joint screening. A definitive evaluation of our joint screening protocol versus independent screening (current standard practice as indicated by many guidelines) would require a large-scale randomized controlled trial with two arms—an undertaking that is beyond the scope of our current work. Nonetheless, we hope that our theoretical results and in-silico experiments, which demonstrate the potential benefits of joint screening in a simplified yet intuitive and representative setting, will inspire further empirical research in this area.

About an actual algorithm: Our referral problem (2) is a linear program (LP) and can be solved by utilizing any LP solver. Some coefficients in this LP are computed by Monte Carlo sampling.

Relation to policy learning: If the goal were to learn the optimal referral rule directly from a dataset of patient screening trajectories, the problem could indeed be framed as a policy learning task with an appropriately defined cost function. However, our work takes a complementary, orthogonal approach.

We begin with a well-defined optimization problem (2) and focus on analytically characterizing the structure of its optimal solution (the decision boundaries described in Figure 1(b) and formalized in Proposition 4.5). Rather than learning from data, our emphasis is on understanding the geometry and properties of the optimal referral rule under a known probabilistic model.

Our results suggest a follow-up question: can the characterized decision boundaries be leveraged to develop sample-efficient algorithms for learning optimal referral rules from limited data? We view this as an important direction for future work.

We hope that our response has addressed your concerns. Please let us know if you have any other concerns.

Thank you for the thorough review of our paper and constructive comments.

Figure 1 1(a) shows the decision boundaries for independent screening (current standard). 2(a) shows the boundaries that characterize the optimal policy for our referral problem (2). Our main contribution is to mathematically characterize these boundaries (Lemma 4.1 to Lemma 4.4, Proposition 4.5). Main message: to act optimally, screening for disease 1 should also depend on the risk of disease 2. Boundary shapes also depend on the screening budget, screening costs, adverse event times, and risk distribution (see Appendix C). Practical interpretation: Clinics can use 1(b) to decide for what a patient with risk vector $(x_1,x_2)$ should be screened.

Scaling to more diseases: $>2$ diseases does not put significant computational burden. Our referral policy optimization problem is a linear program (LP); hence, SoTA complexity bounds for LP apply to our setting.

Characterization of the optimal referral policy will be much more complex and far less intuitive for the case of $>2$ diseases. One would expect the number of boundaries of interest (e.g., $5$ as shown in Figure 1(b) for $N=2$) to grow combinatorially with $N$ as there can be boundaries separating screening of one subset of diseases from another.

Clinical validation: Any simulation study will not fully capture the real-world benefits of joint screening. A definitive evaluation of our joint screening protocol versus independent screening (current standard) would require a large-scale randomized controlled trial with two arms—an undertaking that is beyond the scope of our current work. Nonetheless, we hope that our theoretical results and in-silico experiments, which demonstrate the potential benefits of joint screening in a simplified yet intuitive and representative setting, will inspire further empirical research in this area.

Uncertain disease risks and variable screening costs: One can use off-the-shelf risk prediction models to obtain disease risks for particular diseases. For instance, Gail model (for breast cancer), QRISK3 (for cardiovascular disease), (normalized) polygenic risk scores, and even AI-based models can be used to provide risk scores. In our formulation, screening costs for different diseases can be different. In Figure 4, we also show the effects of varying screening costs.

Statistical significance: Since our experiments are on simulated data, we report exact gains obtained by solving (2). Significance tests are not necessary since we are not making claims based on limited data.

Parameter selection and sensitivity of our results: The choice of $T_0$ and $\mu_n$ is done by examining clinical literature (first paragraph of Section 5.1). While our choices of screening budget and costs are not borrowed from a real-world study, in Appendix C, we discuss in detail how the behavior and the performance of the optimal referral policy changes as we vary each of these parameters. We hope that our detailed appendix resolves the reviewer's concerns about parameters.

Limited practical applicability: There are many other examples where this methodology could be used to guide policymakers. One example can be found in sexual health services. For many settings, human immunodeficiency virus (HIV) and human papillomavirus (HPV) screening protocols are conducted at different health services and hence are independently considered (e.g., https://www.who.int/publications/i/item/9789240024168, https://pubmed.ncbi.nlm.nih.gov/38297406).

Most of these screening protocols are guided by a certain level of risk (e.g., sexual behaviors). These screening measures carry varying costs (HIV antibody is not costly, whereas HPV requires more costs associated with invasive procedures carried out by a gynecologist (for cervical neoplasia) or proctologist (for anal neoplasia)). Finally, delayed screening has implications on the time to present with severe HIV disease (for HIV) and time to develop CIN2/3 or AIN2/3, that is, cervical or anal pre-cancer (for HPV). This methodology could help define which risk levels would warrant joint screening protocols (to plead for streamlined services) or if screening protocols can remain part to minimize these times to undesirable outcomes.

Minor improvements: Compute complexity is not an issue, as the problem is LP. For $N=2$, our characterization of the decision boundaries and visualizations (e.g., Fig 1(b)) provide interpretable explanations of who should be screened. The same screening resources can be used more efficiently (no increase in screening costs for the population). We improve over independent screening in all cases (even when disease risks are independent). Reported numbers may vary based on the experimental setup, but the main message remains the same.

We hope that our response has addressed your concerns. Please let us know if you have any other concerns.

We thank the reviewer for the thorough review of our paper and the constructive comments.

Why screening of one disease depends on the screening of another: In the example of pulmonary hypertension and cardiac disease, risks are clearly contingent. Our approach is not limited to such examples, as we offer a non-trivial solution even when disease risks are independent. Due to the nature of our objective function (related to survival time) and our constraint function (screening cost budget), the optimal threshold for activating the screening of one disease depends on the risk of the other disease. This coupling distinguishes our model and solution from independent screening, which yields a suboptimal policy for our referral problem. We will further clarify this in the revised paper.

Regarding the benefit of screening: Thanks for pointing out this important review paper. We agree that screening does not always offer benefits and sometimes can even be harmful, as in the case of overdiagnosis. We will mention this review paper in the revised version and explicitly discuss the limitations of screening in line with this work. In our work, we focus on the case when screening does not hurt. In our case, the only detrimental effect of screening is a false positive rate, which is expressed as a constraint in the optimization problem. We cannot screen everyone for every disease since screening is costly and screening resources are limited. We characterize exactly how limited screening resources should be distributed over the population so that the expected benefit is maximized.

Clarification of the screening example: Consider a patient who (potentially) has both heart disease and cancer but is only screened for cancer, where the adverse event from cancer is expected to occur after the adverse event from heart disease. If we only screen for cancer, but the patient unexpectedly dies from heart disease, then the cancer screening offers no benefit to the patient in terms of lifetime gain. If the patient is screened for both diseases, heart disease will be identified earlier, the adverse event due to heart disease will be prevented, and cancer screening will result in lifetime gain by detecting cancer before the adverse event associated with it happens.

We hope that our response has addressed your concerns. Please let us know if you have any other concerns.