Designing optimal allocations for cancer screening using queuing network models

Abstract
Cancer is one of the leading causes of death, but mortality can be reduced by detecting tumors earlier so that treatment is initiated at a less aggressive stage. The tradeoff between costs associated with screening and its benefit makes the decision of whom to screen and when a challenge. To enable comparisons across screening strategies for any cancer type, we demonstrate a mathematical modeling platform based on the theory of queuing networks designed for quantifying the benefits of screening strategies. Our methodology can be used to design optimal screening protocols and to estimate their benefits for specific patient populations. Our method is amenable to exact analysis, thus circumventing the need for simulations, and is capable of exactly quantifying outcomes given variability in the age of diagnosis, rate of progression, and screening sensitivity and intervention outcomes. We demonstrate the power of this methodology by applying it to data from the Surveillance, Epidemiology and End Results (SEER) program. Our approach estimates the benefits that various novel screening programs would confer to different patient populations, thus enabling us to formulate an optimal screening allocation and quantify its potential effects for any cancer type and intervention.

Introduction
Cancer is a potentially fatal disease with a large annual incidence worldwide [1]. Since it is the result of the gradual accumulation of genetic and/or epigenetic changes [2] that eventually lead to uncontrolled proliferation and dissemination of cells, its stage at diagnosis has a large impact on a patient’s prognosis [3]. Therefore, diagnosing cancer early through screening can result in substantially reduced mortality and treatment-associated morbidity [4]. For most cancer types, sensitive screens remain unavailable [5] and even in cases when screening technology exists, screens take time, are expensive, and often lead to psychological distress [6], particularly regarding false positives and possibly overtreatment [7]. In some cases, screening has not been demonstrated to prolong survival, for instance with PSA screening for prostate cancer [8]. These tradeoffs lead to considerations regarding the costs and benefits of different screening programs. The advent of novel diagnostic tools that can detect signatures of circulating tumor DNA (ctDNA) in plasma heralds a revolution in early cancer detection [9],[10],[11],[12],[13]. Using these assays, mutations or epigenetic states of interest can be characterized without the need for an invasive biopsy. Innovations such as these advances might make previously unviable cancer screening programs soon worth pursuing on a more widespread basis, motivating the development of mathematical models of such potential screening programs and their optimization based on incidence and survival data.

Materials and methods
Mathematical background
Our mathematical modeling framework is underpinned by the theory of queuing networks (S1 Appendix). Queuing theory is the formal, mathematical study of networks of waiting lines. The length of a queue is represented as a non-negative, integer-valued stochastic process. Formally a queue is described by detailing an arrival process, a service time distribution, and the number of servers operating at the head of the queue. This approach is succinctly summarized by Kendall’s X/Y/Z notation, where X specifies the arrival process, Y the service time distribution, and Z is the number of servers [36]. For instance, the M/G/∞ queue has Markovian arrivals (a Poisson point process), general (arbitrary) service times, and an infinite number of servers (meaning all customers are served in parallel). The equilibrium length of the M/G/∞ queue with arrival rate λ and mean service time 1/μ has a Poisson distribution with mean λ/μ [21]. Here we focus on infinite server queues, but in S1 Appendix we discuss examples that go beyond this paradigm.

Results
Assessing the benefits of pancreatic cancer screening
Pancreatic ductal adenocarcinoma is a particularly deadly form of cancer with limited treatment options and low overall survival [46],[47],[2]. By the time it is detected, it has often progressed to metastatic disease with poor prognosis [48], [49]. Currently no widespread pancreatic cancer screen is available, but several approaches are under investigation, for instance a cell free DNA-based screen for early diagnosis [50]. With early detection, pancreatic cancer patients may receive potentially curative treatment [51], and evidence from genomic sequencing indicates a 15-year period of genetic progression from disease initiation to the metastatic stage, suggesting a sizable window during which screening would be beneficial [52]. Screening for PDAC is not standard amongst the general population because of its low incidence and lack of a highly sensitive and specific test [53]. However, high-risk individuals, i.e. those with a family history, genetic predisposition or chronic disease, generally have access to screening modalities [44]. Previous approaches investigate the effectiveness of endoscopic screening of high-risk individuals [54] and the potential benefits of biannual MRI scans [55].

Discussion
Here we describe how a queuing-theoretic framework can be used as a versatile computational method to generate simple stochastic models to quantify the benefits of screening for cancer and to design optimal allocation and screening strategies. We illustrate the versatility of this modeling approach by discussing example queuing network models that cover a range of medical applications. We demonstrate how the queuing approach permits generalizations that go significantly beyond deterministic compartmental models and Markov chain models, while also providing more detailed answers. The exact results we obtain circumvent the need for simulations entirely and offer a transparent relationship between inputs and outputs of the model. Basic performance analysis of the queuing network models also yields natural and explicit analytical quantifications of the benefits of screening. This finding suggests simple rules for developing optimal screening strategies when resources are scarce and for extending our methodology to factor in cost (as in Fig 2B), which is particularly important in the setting of differential incidence and targeted screening. We apply this modeling framework to datasets from the SEER cancer incidence database with a particular focus on pancreatic cancer.

Citation: Dean J, Goldberg E, Michor F (2022) Designing optimal allocations for cancer screening using queuing network models. PLoS Comput Biol 18(5): e1010179. https://doi.org/10.1371/journal.pcbi.1010179

Editor: Attila Csikász-Nagy, Pázmány Péter Catholic University: Pazmany Peter Katolikus Egyetem, HUNGARY

Received: October 1, 2021; Accepted: May 7, 2022; Published: May 27, 2022

Copyright: © 2022 Dean et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All relevant code is within the manuscript and its Supporting Information files and on our GitHub page: https://github.com/evanhsph/Dean_et_al Publicly available data used can be found at https://seer.cancer.gov/data https://minorityhealth.hhs.gov/omh/browse.aspx?lvl=3&lvlid=63 https://www.census.gov/data/tables/time-series/demo/popest/2010s-national-detail.html.

Funding: This work was supported by the Dana-Farber Cancer Institute’s Physical Sciences-Oncology Center (U54CA193461 to F.M.). http://psoc.dfci.harvard.edu/ The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Competing interests: I have read the journal’s policy and the authors of this manuscript have the following competing interests: F.M. is a co-founder and consultant of Harbinger Health and is a consultant for Red Cell Partners and Zephyr AI. No other conflicts to declare.

https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1010179#abstract0