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Mathematical modelling reveals cellular dynamics within tumour spheroids

Joshua A. Bull, Franziska Mech, Tom Quaiser, Sarah L. Waters, Helen M. Byrne


Tumour spheroids are widely used as an in vitro assay for characterising the dynamics and response to treatment of different cancer cell lines. Their popularity is largely due to the reproducible manner in which spheroids grow: the diffusion of nutrients and oxygen from the surrounding culture medium, and their consumption by tumour cells, causes proliferation to be localised at the spheroid boundary. As the spheroid grows, cells at the spheroid centre may become hypoxic and die, forming a necrotic core. The pressure created by the localisation of tumour cell proliferation and death generates an cellular flow of tumour cells from the spheroid rim towards its core.


By the time tumours are clinically detectable in vivo they are typically highly heterogeneous in terms of their spatial composition [1]. Tumours contain multiple cell types, including stromal cells (e.g., fibroblasts) and immune cells (e.g., macrophages, T cells) and their growth is sustained by an irregular network of tortuous and immature blood vessels which deliver vital nutrients such as oxygen to the tumour cells. When characterising tumour cell lines or testing new cancer treatments it is important to have a reproducible experimental assay. In such situations, tumour spheroids are widely used due to the predictable manner in which they grow [2].

Materials and methods

Model overview

We develop a hybrid agent-based mathematical model to describe the in vitro growth of tumour spheroids in response to an externally-supplied nutrient, here taken to be oxygen. We use the model to simulate spheroid infiltration by inert microbeads and 3H-labelled cells. Our model is implemented within Chaste (Cancer, Heart and Soft Tissue Environment, available at, an open source simulation package designed to solve computationally demanding, multiscale problems that arise in biology and medicine [35, 36]. We choose this framework because it provides an efficient means of implementing off-lattice ABMs, and has previously been used to simulate multicellular spheroids [35, 54]. The extensions to the framework described in this paper will be made available in a subsequent release of the Chaste software.


We have developed a hybrid, off-lattice ABM for oxygen-limited spheroid growth. Our model reproduces the sigmoidal growth dynamics seen in in vitro spheroids as well as their spatial structure, consisting of an outer rim of proliferating cells and a central necrotic core (Fig 2). Our simulations reveal that it is possible to generate synthetic spheroids with similar growth dynamics but different spatial compositions of proliferating, quiescent, hypoxic and necrotic cells (Fig 4). These changes in spheroid composition are driven by variations in cell-scale behaviours such as the average length of the cell cycle or the oxygen thresholds at which cells are affected by hypoxia.

Citation: Bull JA, Mech F, Quaiser T, Waters SL, Byrne HM (2020) Mathematical modelling reveals cellular dynamics within tumour spheroids. PLoS Comput Biol 16(8): e1007961.

Editor: Stacey Finley, University of Southern California, UNITED STATES

Received: December 12, 2019; Accepted: May 18, 2020; Published: August 18, 2020

Copyright: © 2020 Bull 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 data are within the manuscript and its Supporting Information files.

Funding: JAB thanks the EPSRC/MRC Centre for Doctoral Training in Systems Approaches to Biomedical Science (Grant Ref: EP/G037280/1) and the EPSRC Impact Acceleration Account (Grant Ref: EP/R511742/1). This work was supported by Cancer Research UK (CR-UK) grant number C5255/A18085, through the Cancer Research UK Oxford Centre. SLW gratefully acknowledges funding from the Royal Society in the form of a Royal Society Leverhulme Trust Senior Research Fellowship. 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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