Development of a Model to Estimate the Association Between Delay in Cancer Treatment and Local Tumor Control and Risk of Metastases

Key Points Question What are the potential risks of decreased local tumor control and increased metastatic spread for patients with cancer who experience treatment delays due to the coronavirus disease 2019 (COVID-19) pandemic? Findings In this decision analytical model, a simplified mathematical model of tumor growth, metastasis formation, and tumor control by radiotherapy was developed and applied to 3 cancers with different doubling times and propensities to metastasize. Estimated detrimental risks were largest for fast-growing and intermediate-growing tumors and for longer treatment delays. Meaning This model provided quantitative risk estimates that may help to guide physicians and patients with treatment decision-making during the COVID-19 pandemic.


Introduction
The effects of the coronavirus disease 2019 (COVID- 19) pandemic have altered medical care globally.
From March 2020 to the present, hospital systems across the United States and globally have altered their cancer care practices and functions to provide the necessary resources to handle the large number of patients with COVID-19. 1,2 Routine procedures, including cancer screening, diagnosis, and treatments (ie, surgery, chemotherapy, and radiotherapy), have been postponed or delayed to accommodate the current situation. [3][4][5][6] In addition, even after the peak of the pandemic, patients have continued to avoid or postpone medical center visits and treatments because of concerns about the risk of COVID-19 exposure. 4 In parallel, sharp increases in the unemployment rate nationally will likely lead to a significant loss in access to medical care. 7,8 It is not clear when established standards of cancer care will routinely resume, and the effects of these delays on patients with cancer remain understudied.
In response to the rapidly changing medical landscape, guidelines have been published and distributed to help oncologists determine which patients are appropriate for treatment delays. 2,6 These global measures to contain the spread of COVID-19 have in turn focused recent attention on measuring the indirect effect of the pandemic on cancer treatment. 1,3,[9][10][11][12] In particular, there have been several recent modeling studies and expert commentaries reporting on the potential impact of cancer treatment delays due to the COVID-19 pandemic and associated mortality risks. [1][2][3][4][5][8][9][10][11][12][13] There is general consensus on the critical importance of more high-quality data and quantitative analysis on the potential impact of treatment delays on cancer outcomes.
We propose a mathematical model, built on previously published tumor growth and control models and informed by published literature on the association of delayed treatment with cancer upstaging, to determine the potential association of cancer treatment delays with local tumor control probability (TCP) and the formation of new metastases. 14,15 Such estimated risks may help to shape guidelines on selecting which patients should be treated without delay and those who could more safely chose to postpone treatments if necessary. We modeled 3 examples of malignant neoplasms that can grow rapidly, for which there were quality published data on tumor kinetics and for which timing of treatment initiation likely determines survival, ie, head and neck cancers, colorectal cancers, and non-small cell lung cancers. 9

Methods
This study was deemed exempt from institutional review board approval and the requirement for informed consent by Weill Cornell Medicine. This study followed the Consolidated Health Economic Evaluation Reporting Standards (CHEERS) reporting guideline.

Mathematical Model of Tumor Growth and Metastasis
To illustrate the proof of principle of how substantial delays in cancer treatment could affect a patient's risk of local tumor progression and development of new metastases, we developed a simple mathematical model of tumor growth and metastasis. The model was calibrated using data from the literature published from 2001 to 2020 and applied to an example of a fast-growing tumor (head and neck), an intermediate-growing tumor (colorectal), and a slow-growing tumor (non-small cell lung).
The literature suggests that the probability of developing new metastases after delayed treatment was higher in head and neck malignant neoplasms than in lung malignant neoplasms. 16,17 The propensity to metastasize appeared to be even higher for colorectal tumors, although there is some uncertainty behind these estimates. 18,19 The model assumed that (1) the number of primary tumor cells as function of treatment delay time since diagnosis, P(t), grows exponentially and (2) the hazard function for metastasis formation at any given time, M(t), is proportional to the number of primary tumor cells at that time. These assumptions have clear limitations, but they provide a simplified approximation for the processes of interest with a minimal number of adjustable parameters. They are represented by the following system of differential equations, where g is the primary tumor growth rate and k is the proportionality constant for metastasis formation. Equation 1 is as follows: This system of equations has the following solution (Equation 2), where P 0 is the initial number of primary tumor cells at t = 0 (ie, at cancer diagnosis) and M 0 is the initial metastasis hazard at t = 0: The local TCP for a tumor treated at time t is described by the Equation 4, based on an assumed Poisson distribution of tumor cells that survive treatment, where S is their surviving fraction: Equations 1 to 4 assume that all tumors grow at the same rate g. To improve the model's realism, we added an assumption that the tumor growth rate is a normally distributed random variable with mean G and standard deviation σ. Then the probability of any given tumor having a growth rate g is p(g), described in Equation 5 as follows: This approach accounts in a simplified way for observations that some patients have aggressive fastgrowing tumors (g > G), and others have dormant ones (g = 0) or even spontaneous tumor regression (g < 0). It allows the probability distributions of the variables of interest, such as primary tumor size (P dist ), metastasis probability (PM dist ), and TCP (TCP dist ), to be calculated as function of time (Equation 6):

Statistical Analysis
The model contains 5 adjustable parameters: P 0 , S, G, σ, and k. The initial number of primary tumor cells P 0 was estimated based on the following assumptions: (1) the tumor cell density is 10 8 cells/cm −3 for all studied cancers; (2) tumors are spherical in geometry; and (3) the initial diameter for studied lung tumors is 0.5 cm (stage I); for head and neck tumors, 3.0 cm (stage II); and for colon tumors, 5.0 cm (stage II). These values were chosen based on availability of published tumor growth rate and metastasis formation data from published peer-reviewed literature from 2001 to 2020. [16][17][18][19][20][21] Consequently, P 0 was set to 6.54 × 10 6 cells for lung tumors, 1.41 × 10 9 cells for head and neck tumors, and 6.54 × 10 9 cells for colon tumors. The initial metastasis probability M 0 was assumed to be 0 for each cancer type, representing the situation with no metastases at primary tumor diagnosis.
The fraction of tumor cells S that survive treatment was estimated for each cancer using Equation 4, substituting 0 for t, 0.9 for TCP (which represents the clinically plausible scenario of 90% local tumor control over 5 years for tumors treated without delay), and the cancer-specific P 0 value, and solving for S. In other words, S was estimated based on a TCP of 90% at cancer diagnosis and on the previously estimated initial tumor size. S was 1.61 × 10 −8 for lung cancer, 7.45 × 10 −11 for head and neck cancer, and 1.61 × 10 −11 for colon cancer.
The average tumor growth rate G and its standard deviation σ were estimated from the literature on tumor doubling times. The metastasis formation parameter k was estimated for each cancer based on a literature-reported metastasis probability at a specified time, given the other already described parameters. [16][17][18][19][20] All risk estimates were calculated with normal 95% CIs. There were no inferential statistical tests used because this was not an inferential data analysis but rather a theoretical model risk estimate.
The software used was Maple 2020 (Maplesoft).

Results
Head and neck cancers had the fastest reported rates of proliferation, with an estimated median   The quantitative estimates from our study show that, as a proof of principle, delays in cancer treatment by several months are associated with reduced treatment effectiveness. In some patients, these associations can be great, as shown by the range of potential outcomes in the Figure. Rapidly growing tumors, such as head and neck cancers, and rapidly metastasizing tumors, such as colorectal cancers, are likely to have worse estimated outcomes after treatment delay compared with slower growing tumors, such as lung cancers.

Discussion
Patient population heterogeneity and tumor heterogeneity could lead to a range of potential outcomes from delaying treatment, but the results of this study would estimate a significant loss in locoregional control and a substantial risk of increased incidence of metastatic diagnoses, with significantly higher associated risks with rapidly doubling tumors, such as head and neck malignant neoplasms. Many other cancer types, such as cervical cancers, breast cancers, and anal cancers, have tumor kinetics estimated to lie somewhere along this spectrum of possible outcomes between the head and neck and non-small cell lung cancers modeled here. [22][23][24]29,30,34,35 The risks associated with treatment delays appear to increase exponentially with any treatment delay interval and become particularly substantial for delays longer than 3 months. The underlying basis behind this exponential showed that even treatment delay intervals as short as 4 weeks were associated with an increase in the risk of death. In that meta-analysis, 9 the authors noted that the association was even stronger for some radiotherapy and systemic indications, with a 9% and 13% increased risk of death for definitive head and neck radiotherapy and adjuvant systemic treatment for colorectal cancer, respectively. The findings from this meta-analysis are concordant with the findings from our study that the fasterproliferating tumors are associated with greater risks. Another cancer modeling study 23 estimated a decrease of 8% in local TCP with a 4-week delay in treatment for tumors with a doubling time of 90 days. A systematic review of head and neck cancer management 25 found an association with treatment delay and worse survival. Other studies have found that time from surgery to the initiation of radiotherapy within 6 weeks was associated with improved recurrence-free or overall survival. [26][27][28] For patients with cervical cancer, treatment delays greater than 7 weeks from diagnosis of locally advanced disease may be associated with overall survival. 29 Another study indicated that treatment delays of 4 months or longer for cervical cancer may be associated with a 2-fold or greater increase in the risk of death. 30 For patients with lung cancer, some studies demonstrate worse survival in patients with delayed diagnosis and treatment, including 2 studies that included patients identified through population-based mass screening. 31,32 Finally, a 2020 modeling study on the association of COVID-19 with breast and colorectal cancer outcomes estimated a potential 10 000 excess deaths (or 1% increase in deaths) for breast and colorectal cancer during the next decade. 1

Limitations
There are limitations and simplifications behind our modeling study. The primary purpose of this study was to establish conceptual estimates to guide clinical decision-making rather than to generate rigorously precise estimates. Tumor growth and metastasis formation are complex biological processes affected by many variables. Numerous modeling strategies for these phenomena, with varying degrees of mechanistic detail, have been proposed and incorporated in the development of our mathematical model. [36][37][38][39] The situation we considered here-delays in cancer treatmentinvolves a limited period of a few months, which corresponds to a few tumor doubling times. We believe that during this limited time, a maximally simplified modeling approach with a minimum of parameters is adequate for describing the main patterns of the expected responses to treatment delays in terms of tumor volume and metastasis formation. Consequently, we assumed such a simplified approximation, which consists of exponential tumor growth with normally distributed variability in growth rates between tumors, and a proportional dependence between metastasis hazard and primary tumor cell number. However, more detailed models (eg, Gompertz or power law with metastasis emission proportional to the tumor surface area) would be needed to describe tumor development and metastatic risks over longer periods, such as several years.
In applying our simplified model of cancer treatment delays, we could not adjust for real-world cancer treatment countermeasures implemented in response to the initial COVID-19 risk containment measures. While surgeries, radiation treatments, and chemotherapy treatments are often being delayed, patients with cancer are also receiving alternative strategies, such as selected systemic therapies, as bridging therapeutic strategies. Furthermore, over time, hospitals and clinics have put safety measures in place to allow for safe encounters and to reduce potential treatment delays. Nevertheless, as additional COVID-19 waves continue to affect hospital systems nationally and globally, treatment delays for cancer patients remain relevant.
Third, while the exponential nature of tumor doubling is expected to be the dominant effect in clinical oncology, it is well established that tumor kinetics are also affected by the microenvironment. 40 It is known that tumors eventually outstrip their blood supply, leading to necrosis and slowing down their exponential growth. Metastatic development in patients occurs through a long, complex series of physiologic processes, including selection and adaptation through invasion, transportation, seeding, and growth into distal sites. Aside from time to treatment, many other variables can influence metastatic progression. The nuances of such complex processes are simplified and approximated in this model. Future clinical studies and more sophisticated models with better estimates of tumor kinetics will allow us to refine and improve the accuracy of these risk estimates, broadening the applicability of quantitative models of risks associated with cancer treatment delays.