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A Mathematical Model of the Metastatic Bottleneck Predicts Patient Outcome and Response to Cancer Treatment

Ewa Szczurek, Tyll Krüger, Barbara Klink, Niko Beerenwinkel

Abstract

Metastases are the main reason for cancer-related deaths. Initiation of metastases, where newly seeded tumor cells expand into colonies, presents a tremendous bottleneck to metastasis formation. Despite its importance, a quantitative description of metastasis initiation and its clinical implications is lacking. Here, we set theoretical grounds for the metastatic bottleneck with a simple stochastic model. The model assumes that the proliferation-to-death rate ratio for the initiating metastatic cells increases when they are surrounded by more of their kind. For a total of 159,191 patients across 13 cancer types, we found that a single cell has an extremely low median probability of successful seeding of the order of 10−8. With increasing colony size, a sharp transition from very unlikely to very likely successful metastasis initiation occurs. The median metastatic bottleneck, defined as the critical colony size that marks this transition, was between 10 and 21 cells. We derived the probability of metastasis occurrence and patient outcome based on primary tumor size at diagnosis and tumor type. The model predicts that the efficacy of patient treatment depends on the primary tumor size but even more so on the severity of the metastatic bottleneck, which is estimated to largely vary between patients. We find that medical interventions aiming at tightening the bottleneck, such as immunotherapy, can be much more efficient than therapies that decrease overall tumor burden, such as chemotherapy.

Introduction

Metastases are responsible for 90% of deaths from cancer [1, 2]. The formation of metastases is a multi-step process, in which tumor cells spread from the primary site and colonize distant organs [3]. It can be divided into three phases (Fig 1a). The first phase consists of tumor cell entry into the vascular system (intravasation), transport in the blood, and exit from the vascular system to a secondary organ (extravasation). This step may or may not be preceded by the acquisition of genetic or epigenetic alterations in the primary tumor [4–6]. Experimental data suggests that the tumor cells that are shed from the primary site are already equipped with metastatic abilities [7–9]. Recent findings support that metastatic tumor cell dissemination begins in early [10, 11], rather than late stage of the disease. The first phase of metastasis formation is highly efficient, as the released tumor cells deal remarkably well with the obstacles of delivery to distant organs and their infiltration [4, 5, 12–14]. In contrast, the second phase—metastasis initiation—is extremely inefficient [4, 6, 15, 16]. Relative to the huge numbers of cells that disseminate during the long period of primary tumor growth, only very few of them successfully form distant metastases [12, 17]. This bottleneck is commonly understood as the lack of compatibility of the seeded tumor cells with the soil they encounter in the affected organ [4, 18, 19]. In mice models, the metastatic seeding potential was observed to increase with the size of tumor cell clumps [20], which was recently confirmed for human circulating tumor cell clusters [21]. In the last phase of metastasis formation, successfully initiated colonies form micrometastases and, subsequently, clinically detectable macrometastases [6].

Methods

We introduce a mathematical model of the tumor growth, extravasation, and intravasation, the metastatic bottleneck, as well as the metastasis probability and clinical outcome (Fig 1).

Mathematical model of tumor growth, extravasation, and intravasation

To describe the first and efficient phase of metastasis formation, we assume that as the primary tumor grows, it continues to release cells from its surface (Fig 1a). Primary tumor growth is modeled as an exponentially increasing spherical volume with the doubling time set to constants measured for different cancers (S1 Table). The per-cell per-year release rate is fixed to match experimental data, showing that a tumor of one gram contains about 109 cells [40] and that such a tumor sheds around 1.5 × 105 cells per day [22]. The extravasation probability is fixed to 0.8, based on experimental observations in mice [12].

Discussion

In this work, we provide a mathematical model for metastasis initiation, proposing the Allee effect as an explanation for the metastatic bottleneck. Furthermore, we estimate the severity of the bottleneck from epidemiological data for different cancers. The presented work extends current knowledge about metastasis formation in several ways. It is well known that tumor size influences prognosis, and tumor diameter is an important factor in the TNM staging system. The proposed model links tumor size to metastatic probability and provides mathematical expressions that describe the non-linear dependence of patient outcome on tumor size. We found that the size of the tumor is also predictive of how much a patient will benefit from a given treatment. Our model predictions indicate that only patients with not too large or too small tumors would benefit from additional treatments. Therefore, our model might be useful to support clinical decision making. In addition, we emphasize the importance to account for variability of metastasis initiation within and between patients. The model suggests that, apart from tumor size, also the individual bottleneck severity has a significant impact on patient response. The model predicts a high potential of treatment aiming at narrowing the metastatic bottleneck. Such bottleneck treatment would increase the chances of patient survival by reducing the chances to form a metastasis. Recent advances in immunological cancer treatment led to clinical trials of anticancer vaccines [43]. A bottleneck shift may be achieved in the future by such vaccines that strengthen the immune system against forming metastatic colonies, by reducing the number of circulating tumor cell clusters, or by such drugs that reduce cell adhesion in a cancer cell-specific manner.

Citation: Szczurek E, Krüger T, Klink B, Beerenwinkel N (2020) A mathematical model of the metastatic bottleneck predicts patient outcome and response to cancer treatment. PLoS Comput Biol 16(10): e1008056. https://doi.org/10.1371/journal.pcbi.1008056

Editor: Arne Traulsen, Max-Planck-Institute for Evolutionary Biology, GERMANY

Received: January 27, 2020; Accepted: June 15, 2020; Published: October 2, 2020

Copyright: © 2020 Szczurek 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: The raw data is available from the SEER database. Both pre-processed data and the code allowing full reproducibility of all presented results is freely available
at https://github.com/EwaSzczurek/MetastaticBottleneck.

Funding: We thank the Center for Interdisciplinary Research (ZiF), Bielefeld University, for partly funding and inspiring this work, via involvement of all authors in the ZiF Cooperation Group ”Multiscale Modeling of Tumor Initiation, Growth and Progression: From Gene Regulation to Evolutionary Dynamics” from September to December, 2016. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 766030 (to E.S. and N.B.; https://ec.europa.eu/programmes/horizon2020/en). T.K. is grateful to his Department of Control Systems and Mechatronics, Faculty of Electronics, Wroclaw University of Technology for funding several visits to Dresden, Warsaw and Basel. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Competing interests: The authors have declared that no competing interests exist.

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