Multispecies model of cell lineages and feedback control in solid tumors
Highlights
► A model of spatiotemporal dynamics of cell lineages in solid tumors is developed. ► Soluble factors regulate cell proliferation, self-renewal and differentiation. ► Spatiotemporal heterogeneous distributions of cells and factors develop. ► Clusters of stem cells appear at the tumor boundary, consistent with experiments. ► Combining differentiation and radiation therapies is a very effective strategy.
Introduction
Tumors arise when the carefully regulated balance of cell proliferation and programmed cell death (apoptosis) that ordinarily exists in normal homeostatic tissues breaks down. In the traditional view, cancer cells are assumed to acquire, through genetic or epigenetic changes, a common set of traits (Hanahan and Weinberg, 2000): (i) self-sufficiency in growth signals, (ii) insensitivity to growth inhibitory signals, (iii) ability to evade apoptosis, (iv) limitless replicative potential, (v) ability to sustain angiogenesis and (vi) invasiveness and metastatic capability. There is an increasing body of evidence, however, that not all proliferating cells in a tumor matter equally (e.g., Visvader and Lindeman, 2008, Charafe-Jauffret et al., 2009, Alison et al., 2011).
As with cells in normal tissues, tumor cells appear to progress through lineage stages, in which the capacity for unlimited self-renewal is, at some point, lost. The existence of a small population of cells capable of initiating cancer, known as cancer initiating cells or cancer stem cells (CSCs), was first demonstrated in leukemia (Furth and Kahn, 1937, Lapidot et al., 1994, Bonnet and Dick, 1997) by showing that the transplantation of only certain types of leukemic cells consistently result in leukemia in the animal. Implantation of even one of these cancer stem cells into a mouse can cause leukemia. Later, studies have identified such cancer stem cells in solid tumors including breast (Al-Hajj et al., 2003), brain (Hemmati et al., 2003, Singh et al., 2004), prostate (Collins, 2005), melanoma (Fang, 2005, Monzani. et al., 2007), ovarian (Bapat et al., 2005), colon (O’Brien et al., 2007, Ricci-Vitiani et al., 2007), liver (Ma et al., 2007, Yin et al., 2007), lung (Ho et al., 2007), pancreas (Hermann et al., 2007, Li et al., 2007, Olempska et al., 2007) and gastric cancer (Fukuda et al., 2009, Takaishi et al., 2009). These studies have given further credence to the cancer stem cell hypothesis, which states that cancer diagnostic, prognostic and therapeutic efforts need to be focused on that population of cells—often a small minority—that undergoes long-term self-renewal. While this hypothesis acknowledges the existence of lineage progression in cancers, it does not address the role that lineages normally play in cancer biology.
A lineage is a set of progenitor–progeny relationships within which progressive changes in cell character occur. Typically, lineages are traced back to a self-perpetuating stem cell, and end with a terminally differentiated cell that is either postmitotic or divides slowly compared with its normal lifespan. In between stem and terminal cells are a number of “committed” progenitor cell stages. There is increasing evidence, however, that stem and committed progenitor cells are not necessarily cell types per se, but rather patterns of cell behavior that emerge when cells at different lineage stages find themselves in specific environments (e.g., Loeffler and Roeder, 2002, Zipori, 2004, Jones et al., 2007, Clayton et al., 2007, Chang et al., 2008, Lander et al., 2009). Thus, within a lineage, which cell stages behave as stem cells and which as committed progenitor cells may be more a matter of context than pre-determination, may change over time, and may vary with spatial location.
Every population of dividing cells at a given lineage stage can be characterized by a parameter P, that is the fraction of daughter cells resulting from cell division that remains at the same lineage stage (i.e., 1−P is the fraction of the daughter cells that progress to the next stage). When P=0.5, we are usually inclined to call this population stem cells, because they maintain constant numbers while producing differentiated progeny. When P<0.5, we tend to call this population committed progenitor cells, or transit amplifying cells, because their lineages self-extinguish after several rounds of division (the lower the P, the sooner the extinction). Note that this characterization makes no reference to cell division symmetry. From the population standpoint it does not matter whether a value of P=0.5 is achieved by having all cells divide asymmetrically or having some divide symmetrically to generate two of themselves and an equal number divide symmetrically to generate two cells of the next stage.
It has long been argued that tissue growth must be controlled by feedback (e.g., Bullough, 1965). Tissue-specific signals affect the behaviors of stem, committed progenitor and also possibly terminally differentiated cells. For instance, McPherron et al. (1997) showed that when growth and differentiation factor 8 (GDF-8)/myostatin, a protein belonging to the transforming growth factor-beta (TGFβ) superfamily, is genetically eliminated in mice, this results in the production of an excessive number of terminally differentiated cells (myocytes) and increase in muscle mass. Wu et al. (2003) and Gokoffski et al. (2011) showed that other closely related members of the TGFβ superfamily, activin B and GDF11, control the number of stem and committed progenitor cells in the mouse olfactory epithelium. Control of cell numbers through the regulation of self-renewal also occurs during hematopoiesis (e.g., Kirouac et al., 2009, Marciniak-Czochra et al., 2009). In all cases, control of cell populations involves negative feedback loops that reduce not only mitosis rates, but also the self-renewal fractions, i.e. P.
Other TGFβ superfamily members have been found to decrease self-renewal and differentiation rates of stem cells both in normal tissues and in cancer (e.g., Watabe and Miyazono, 2009, Anido et al., 2010, Meulmeester and Ten Dijke, 2011). Some members of the TGFβ family may also increase tumor invasiveness in the later stages of tumor progression. Many other factors, such as Wnts, Notch, Sonic Hedgehog (Shh), and fibroblast growth factor (FGF) have been found to upregulate stem and committed progenitor cell renewal and proliferation rates in normal tissues and cancer (e.g., Dontu et al., 2004, Lie et al., 2005, Katoh and Katoh, 2007, Bailey et al., 2007, Klaus and Birchmeier., 2008, Kalani et al., 2008, Bisson and Prowse, 2009, Pannuti et al., 2010, Turner and Grose, 2010). A number of these signaling factors, together with their inhibitors such as Dickkopf (Dkk) and secreted frizzled proteins (SFRPs) which inhibit Wnt singaling, are also found to promote the development of invasive cancer (e.g., González-Sancho et al., 2005, Guo et al., 2007, Klaus and Birchmeier., 2008, Bovolenta et al., 2008, Takahashi et al., 2010, Li and Zhou, 2011, Meulmeester and Ten Dijke, 2011). Further, experiments reveal considerable spatial heterogeneity in signaling factors and in the distributions of stem and non-stem cells; see for example Fig. 3, Fig. 4 and the accompanying description in Section 4.1 below.
Using a mathematical model, Lander et al. (2009) and Lo et al. (2009) demonstrated that feedback regulation of the P-values of the cell stages by more differentiated cells in the lineage forms the basis of a powerful integral control strategy that can explain many features of homeostasis, such as insensitivity of tissue size to stochastic fluctuations (e.g., in proliferation, renewal and differentiation rates) and the rapid regeneration of tissues in response to injury. Such feedback can also drive the spatial stratification of epithelia (Chou et al., 2010). Moreover, such studies show that in multistage lineages, the relative strengths of the different feedback loops determine which cell stage adopts stem or committed progenitor cell behaviors and suggest that feedback is the reason why stem and committed progenitor cell behaviors emerge in tissues.
The fact that lineages are also apparently present in cancer, suggests therefore that feedback regulation is operating in tumors although not necessarily normally. As evidence for this hypothesis we note that recent research shows that there may be several types of stem and committed progenitor cell subpopulations in solid breast tumors (e.g., Hwang-Verslues et al., 2009). Further, by implanting BRCA1/p53 breast tumor cells in mice, Shafee et al. (2008) demonstrated that the fraction of cells displaying normal mammary stem cell markers in the fully developed tumors varies little from tumor to tumor (roughly 3–8% of all cells) regardless of the stem cell fraction initially implanted. Yet perturbations of the tumor and its microenvironment can dramatically change the stem cell fractions. For example, repeated treatment by cisplatin can cause the stem cell fraction in the BRCA1/p53 breast tumors to dramatically increase, which can lead to chemoresistance and enhanced invasiveness (Shafee et al., 2008). Analogously, stem cell fractions may increase during fractionated radiotherapy, which can result in an accelerated repopulation of the tumor and increased invasiveness (e.g., Kim and Tannock, 2005, Pajonk et al., 2010). Hypoxia in the microenvironment, however, can act as a radiosensitizer and protects cells from radiation damage (e.g., Pajonk et al., 2010). In addition, hypoxia may also increase stem cell fractions and invasiveness by promoting reprogramming cells to a cancer stem cell phenotype (e.g., Heddleston et al., 2009). In general, feedback processes in tumors may create new ways for tumor progression and invasion to occur.
There have been many mathematical models of tumor growth developed in recent years. See, for example, the recent reviews by Roose et al. (2007), Harpold et al. (2007), Anderson and Quaranta (2008), Tracqui (2009), Attolini and Michor (2009), Preziosi and Tosin (2009), Lowengrub et al. (2010), Byrne (2010), Edelman et al. (2010), Rejniak and Anderson (2011) and Frieboes et al. (2011). Increasingly, mathematical models incorporating stem cell dynamics have been developed. Much of this work has dealt with hematopoietic cancers such as leukemia and their treatment. See, for example, the review by Michor (2008) and the references therein. In solid tumors, much work has focused on studies of colorectal cancer including stochastic and deterministic models of intestinal crypts that incorporated limited feedback loops among the cell types as well as extracellular sources of signaling factors such as Wnt (e.g., D'Onofrio and Tomlinson, 2007, Johnston et al., 2007a, Johnston et al., 2007b, Johnston et al., 2010, van Leeuwen et al., 2006, van Leeuwen et al., 2009). General stochastic spatiotemporal discrete models have been recently developed to simulate the dynamics of stem and differentiated cells in tumor clusters that were not specific to a particular type of cancer (e.g., Galle et al., 2009, Enderling et al., 2009a, Enderling et al., 2009b, Enderling et al., 2009c, Enderling et al., 2010a, Enderling et al., 2010b, Sottoriva et al., 2010a, Sottoriva et al., 2010b). A general ODE-based cell compartment model was developed earlier (Ganguly and Puri, 2006). None of these models, however, explicitly accounted for spatiotemporally varying cell signaling and feedback among tumor cells at the different lineage stages.
In this paper, we present a general model that accounts for spatiotemporally heterogeneous signaling factors produced by cells in the lineages and nutrients supplied by the microenvironment. Together, these regulate the rates of proliferation, self-renewal and differentiation of the cells within the lineages and control the cell population sizes and distributions. In particular, terminally differentiated cells release proteins (e.g., from the TGFβ superfamily) that feedback upon less differentiated cells in the lineage and promote differentiation and decrease rates of proliferation (and self-renewal). Stem cells release a short-range feedback factor that promotes self-renewal (e.g., representative of Wnt signaling factors), as well as a long-range inhibitor of this factor (e.g., representative of Wnt inhibitors such as Dickkopf (Dkk) and secreted frizzled proteins (SFRPs)). Generally speaking, the results of modeling such feedback are generic—i.e. they do not depend on the type of molecule that implements feedback—and therefore should also be relevant to processes such as Notch, BMP, Shh, FGF mediated signaling, “contact inhibition”, mechanical forces or even indirect feedback through depletion of nutrients, growth factors or spatial limitations.
The outline of the paper is as follows. In Section 2, we present the mathematical model. In Section 3, we nondimensionalize and simplify the equations. In Section 4, results are presented where we investigate tumor progression and the response to treatment under various feedback and treatment scenarios. In Section 5, we present conclusions, comparisons with previous work and discussions of future work. In the Appendix, we present a nondimensionalization of the model. Additional details are provided in the Supplementary Material.
Section snippets
Multispecies tumor model
We develop a spatial model for lineage dynamics by adapting the multispecies tumor mixture model from Wise et al. (2008) and Frieboes et al. (2010) to account for cell lineages. In Fig. 1, a schematic is shown of a cell lineage, which is composed of cancer stem cells (CSC), committed progenitor cells (CP), terminal cells (TC) and dead cells (DC). Differentiation and feedback processes, described below, link the cells in the lineage through the self-renewal fractions and mitosis rates of the CSC
Model simplification and nondimensional equations
We next simplify the model and consider only two types of viable tumor cells: TCs and non-TCs. In particular, the non-TC population contains the CSCs and CPs. For simplicity, we refer to the non-TC population as CSCs in the remainder of the paper, although we expect that the CPs dominate the population of the non-TC compartment. Consequently, the proliferation rate for the combined population is taken to be that for the CPs (e.g., on the order of 1 day). Preliminary results (not shown) from the
Results
In this section, we present numerical results in 2D and 3D for tumor progression with varying degrees of response to feedback signaling, shape perturbations and therapy application. To solve the governing equations efficiently, an adaptive finite difference-nonlinear multigrid method is developed following previous work by Wise et al., 2008, Wise et al., 2011. The details of the method and numerical implementation are briefly described in the Supplementary Material (Section S1).
Unless otherwise
Summary
In this paper, we have developed and simulated a multispecies continuum model of the dynamics of cell lineages in solid tumors. We have also suggested a number of experiments that could be performed to test the model predictions. The model accounted for spatiotemporally varying cell proliferation and death mediated by the heterogeneous distribution of oxygen and factors with varying solubilities that regulated the self-renewal and differentiation of the different cells within the lineages.
Acknowledgment
The authors thank H. Enderling, T. Hillen, C. Ladagec, F. Pajonk and E. Vlashi for valuable discussions. We especially thank C. Ladagec and F. Pajonk for performing new experiments to generate the large in vitro tumor spheroid shown in Fig. 3(b) to test our predictions, and for kindly providing the visualizations of the ZsGreen-ODC cells, which are thought to be cancer stem cells, in the U87MG-derived tumors shown in Fig. 3. We also thank the reviewers for their helpful suggestions to improve
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2021, Journal of Theoretical BiologyCitation Excerpt :The present study provides a solid theoretical basis for implicating the presence of feedback regulatory loops as a determinant of responses to cancer therapy. This adds to the mathematical literature quantifying the role of feedback regulation for tissue and tumor dynamics (Arino and Kimmel, 1986; Komarova, 2013; Komarova and van den Driessche, 2018; Konstorum et al., 2016; Kunche et al., 2016; Lander et al., 2009; Rodriguez-Brenes et al., 2011, 2013b, 2017; Stiehl et al., 2018; Yang et al., 2015; Youssefpour et al., 2012), and builds upon the wider mathematical literature concerned with the dynamics of hierarchically structured cell populations, e.g. (Enderling et al., 2013; Glauche et al., 2007; Marciniak-Czochra et al., 2009; Michor, 2008; Roeder and Loeffler, 2002; Stiehl and Marciniak-Czochra, 2011; Werner et al., 2011) and stem cell fractions (Enderling, 2014). Two major approaches can be mentioned as most relevant in the present context.
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2019, Physica A: Statistical Mechanics and its ApplicationsActivation of the HGF/c-Met axis in the tumor microenvironment: A multispecies model
2018, Journal of Theoretical BiologyCitation Excerpt :Despite the prevalence of tumor and tumor-microenvironment models, based on our current knowledge, no tissue-level models of CAF-tumor interactions have been developed that specifically addresses the HGF/c-Met and tumor-derived growth-factor signaling pathway dynamics. Using, as a starting point, a spatiotemporal, multispecies model of tumor growth that accounts for feedback signaling between CSCs and non-CSCs (Yan et al., 2017; 2016; Youssefpour et al., 2012), we investigate how the development and spread of a tumor is impacted by a dynamic interaction between tumor-derived growth factors and CAF-derived HGF, and the physiological effect of therapies directed at reducing the strength of this feedback mechanism. A multispecies continuum model of tumor growth with lineage dynamics and feedback regulation was developed by Youssefpour et al. (2012), who investigated two-stage lineages primarily in two dimensions and Yan et al. (2016), who investigated three-stage lineages in three dimensions.
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2017, Journal of Theoretical BiologyCitation Excerpt :Using these criteria, a selection algorithm was devised that identified three of the 32 possible control networks as most likely the ones corresponding to the regulation of homeostasis of human colon crypts. This paper contributes to the growing literature on the theory of stem cells, which ranges from ODE modeling (Nakata et al., 2012; Stiehl and Marciniak-Czochra, 2011) to stochastic modeling (Dingli et al., 2007; Enderling et al., 2009a, 2007, 2009b, 2009c), and includes research of stem cells in the context of feedback mechanisms (Konstorum et al., 2016; Kunche et al., 2016; Lander et al., 2009; Youssefpour et al., 2012), carcinogenesis (Ashkenazi et al., 2008, 2007; Boman et al., 2008; Enderling and Hahnfeldt, 2011; Ganguly and Puri, 2006, 2007; Hardy and Stark, 2002; Johnston et al., 2007; Yatabe et al., 2001), modeling hematopoietic SC dynamics (Foo et al., 2009; Glauche et al., 2007; Marciniak-Czochra et al., 2009; Stiehl and Marciniak-Czochra, 2012), and cancer stem cells (Dingli and Michor, 2006; Enderling, 2015; Enderling and Hahnfeldt, 2011; Hillen et al., 2013; Johnston et al., 2010; Scott et al., 2014). We have measured the number and location of dividing cells (Ki-67 positively stained cells) and non-dividing cells (Ki-67 non-stained cells) in 49 colon crypts in human biopsy specimens.