Mathematical model of macrophage-facilitated breast cancer cells invasion
Introduction
In breast cancer, the presence of macrophages at the tumor site is related to poor prognosis (Condeelis and Pollard, 2006, Lewis and Pollard, 2006). This is surprising since macrophages are a part of our immune system. However, studies have shown that macrophages play various roles in tumor development and progression, one of which is to increase tumor cell motility. Tumor cells and macrophages in close proximity communicate via a short-ranged chemical signalling loop based on epidermal growth factor, EGF, secreted by macrophages, and colony stimulating factor 1, CSF-1, secreted by tumor cells (Lin et al., 2001, Lin et al., 2002, Goswami et al., 2005, Condeelis and Pollard, 2006, Lewis and Pollard, 2006, Wyckoff et al., 2007, Beck et al., 2009, Patsialou et al., 2009). This signalling results in aggregation (van Netten et al., 1993), so that tumor cells migrate alongside macrophages towards blood vessels or surrounding tissues and organs (Wyckoff et al., 2007). Extravasation (escape out of a blood vessel) results in metastasis, the formation of secondary tumors, a primary cause of death in breast cancer patients. Hence, limiting or eliminating tumor cell motility is a crucial part of cancer treatments. Experiments have shown that when the number of macrophages is decreased at breast cancer sites, tumor progression is slower and fewer cells are able to metastasize, resulting in increased survival rates (Lin et al., 2001). Here we use mathematical modelling to examine interactions between tumor cells and macrophages in a nutrient-rich in vitro situation.
In order for the tumor cell-macrophage interactions to result in group migration, the cells must have a net tendency for aggregation. By aggregation, we mean the tendency of the system to develop a nonuniform spatial distribution where cell clusters form. The density of such clusters is generally well-elevated over background densities, though we do not claim a specific density or size in order to call such a cluster “an aggregate”. Some clues about the underlying process are provided by quantitative measurements. For example, in Wyckoff et al. (2004) it is found that the ratio of tumor cells to macrophages is 3 to 1 in experiments conducted in mice where cells were collected into micro-needles containing EGF. In Patsialou et al. (2009) a similar experiment is conducted using a human breast cancer cell line and the ratio between tumor cells and macrophages increases to 15 to 1. In this paper we explore how features of the paracrine and autocrine signalling loops contribute to this tendency. Principally, we ask the following questions:
- •
Under what conditions is the paracrine loop sufficient to produce aggregation of tumor cells and macrophages?
- •
How does the size of an aggregate depend on aspects of signalling such as rates of secretion?
- •
What signalling aspects are most easily changed by drugs to eliminate aggregation?
- •
How would treatment by drugs affect the process?
- •
What are the key differences between various cancer cell lines?
- •
What governs the observed 3 to 1 ratio between motile tumor cells and macrophages?
To address these questions, we introduce two models for interactions and motility of tumor cells and macrophages. The first, described in Section 2.1, is a continuous 1D Eulerian model, amenable to both analysis and simulations. The second (Section 2.2) is a discrete Lagrangian model where individual cells are tracked. In Section 3 we adapt methods from Luca et al. (2003) and Green et al. (2010) to perform a linear stability analysis of the Eulerian model. This leads to conditions for spontaneous aggregation of cells. Results of full simulations of the PDEs and of the discrete model are presented in Section 5. The advantages of this dual approach are that we can use analytical PDE tools to understand parameter dependence (using the continuum model) while preserving our ability to track individual cells and how they move (using the discrete simulation). We summarize our findings in Section 6 and discuss how our models can be useful for designing cancer treatments.
Macrophages are a type of white blood cell comprising approximately 5% of the body׳s white blood cell count. Macrophages originate from monocytes circulating in the bloodstream and are recruited to tumor sites by chemotactic factors such as the colony stimulating factor-1, CSF-1 (Lewis and Pollard, 2006). Up to 50% of the cell mass in breast tumors can be macrophages (Lin et al., 2002). In Table 1 we provide information on cancer cell lines of interest here.
Tumor cells manipulate innate macrophage signalling in order to migrate. The tumor cells secrete CSF-1, which can bind to CSF-1 receptors on macrophages. This activates the macrophages to chemotax towards a CSF-1 gradient and to secrete EGF. The EGF can then bind to receptors on tumor cells, continuing the chain of activation. Activated tumor cells respond by secreting more CSF-1 and chemotacting in the direction of the EGF gradient (Beck et al., 2009, Pu et al., 2007, Goswami et al., 2005, Wyckoff et al., 2004). This process results in a short-ranged chemotactic signalling loop, also called a paracrine loop, see Fig. 1.
The first indication of a macrophages role in tumor cell motility was provided by van Netten et al. (1993). In their experiment, macrophages and tumor cells plated together form multicellular aggregates within 24 h. Wyckoff et al. (2004) conducted in vivo experiments in mice to study motility and intravasation (crossing into blood vessels) of tumor cells and macrophages. They used PyMT-induced mammary tumors and a multi-photon microscope to view the process. Tumors were grown for 16–18 weeks after which the anaesthetized mice were viewed under a microscope. Collection needles containing 25 nM EGF were placed inside the tumor. In 4 h, approximately 1000 cells were collected, with 73% tumor cells and 26% macrophages. This ratio of approximately 3:1 tumor cells to macrophages was also observed when MTLn3 cells were grown in rats.
Patsialou et al. (2009) showed that, in addition to the paracrine loop, there can also be a CSF-1/CSF-1R autocrine signalling loop (tumor cells both secrete and respond to CSF-1 gradients). This appears to be the case in some human breast cancer cell lines such as MDA-MB-231, which have CSF-1 receptors in addition to EGF receptors. Results from both in vivo (human tumor cells transplanted into mice) and in vitro experiments reported in Patsialou et al. (2009) indicate that invasion of the MDA-MB-231 cell line is less dependent on the macrophages. For example, in micro-needles, only 6% of the collected cells were macrophages (compared to 25% in the experiments with MTLn3 and PyMT).
Motivated by van Netten et al. (1993), we will examine conditions necessary for aggregation of tumor cells and macrophages. In view of Wyckoff et al. (2004), we will also explore what features of the signalling loop control the ratio of the two cell types observed in such aggregates. We will investigate this in our simulations by introducing a localized source of EGF and observing how this ratio changes as model parameters are varied. Motivated by Patsialou et al. (2009), we will use our model to investigate how autocrine signalling affects the ratio of cell types in the aggregates.
Section snippets
Models
Keller and Segel first used partial differential equations to study chemotaxis in 1970 (Keller and Segel, 1971) to investigate aggregation of the slime mold Dictyostelium discoideum. In their analysis, Keller and Segel associated instability in the system with spontaneous aggregation of cells (Keller and Segel, 1970). Since then, similar models have been used to study a wide variety of systems that involve chemotaxis (Sherratt, 1994, Lapidus and Schiller, 1974, Luca et al., 2003, Green et al.,
Linear stability analysis of continuum model
An advantage of the Eulerian model is that it is amenable to analytic methods. Here, we carry out a linear stability analysis of the simplified model consisting of Eqs. (20), (21) to explore conditions for spontaneous aggregation. We ask whether a small perturbation from the spatially uniform steady state can lead to aggregation. Consequently we substitute the forms into (20), (21) to obtain the linearized system:
Parameter estimates
We used experimental literature, where available, to quantify parameters (Table 2). However, as such data is sparse, we also rely on previous estimates of some rates and values from other modelling papers, e.g. Kim and Friedman (2010), a common custom in the field of cancer modelling. We use a default cell density of cells/cm2, based on the density used in experiments (Goswami et al., 2005, Kim and Friedman, 2010, Liu et al., 2009). The rates of motility associated with the two cell
Simulations of the continuum model
We solve the system of four dimensionless PDEs, (1), (2), (3), (4), numerically (using the pdepe solver in MATLAB). We use no-flux boundary conditions, as previously noted. Recall that a given uniform initial tumor cell and macrophage distribution will give rise to a steady state uniform concentration of EGF and CSF-1. Here we start with an initial population of cells and let the chemical concentration evolve via secretion/decay. To induce instability, we impose a small perturbation of the
Discussion
Cancer is a heterogeneous disease where an interplay of multiple chemical and mechanical signals in different cell types takes place. Microscopic events occurring within a cell and mesoscopic events involving cell–cell interactions need to be considered when studying the disease. These events take place at varying time and spatial scales, motivating the need for multi-scale modelling.
Mathematical modelling has been used in cancer research for decades, and has been on the rise in recent years.
Acknowledgements
The research was supported by an NSERC Discovery Grants to L.E.K. and E.P. The continuum model and its analysis was initiated at an International Graduate Training Center in Mathematical Biology summer graduate course, sponsored by the Pacific Institute for the Mathematical Sciences. This research was enabled in part by support provided by WestGrid (www.westgrid.ca) and Compute Canada Calcul Canada (www.computecanada.ca).
References (53)
- et al.
Macrophages: obligate partners for tumor cell migration, invasion, and metastasis
Cell
(2006) - et al.
Non-local models for the formation of hepatocyte–stellate cell aggregates
Journal of theoretical biology
(2010) - et al.
Cell invasion of highly metastatic MTLn3 cancer cells is dependent on phospholipase D2 (PLD2) and Janus kinase 3 (JAK3)
J. Mol. Biol.
(2011) - et al.
Initiation of slime mold aggregation viewed as an instability
J. Theor. Biol.
(1970) - et al.
Model for chemotaxis
J. Theor. Biol.
(1971) - et al.
A mathematical model for bacterial chemotaxis
Biophys. J.
(1974) - et al.
Non-local concepts and models in biology
J. Theor. Biol.
(2001) - et al.
Chemotactic signaling, microglia, and alzheimer׳s disease senile plaquesis there a connection?
Bull. Math. Biol.
(2003) - et al.
Pattern formation and spatiotemporal irregularity in a model for macrophage–tumour interactions
J. Theor. Biol.
(1997) - et al.
Mathematical modelling of the use of macrophages as vehicles for drug delivery to hypoxic tumour sites
J. Theor. Biol.
(2004)