Non-stem cancer cell kinetics modulate solid tumor progression
© Morton et al; licensee BioMed Central Ltd. 2011
Received: 17 October 2011
Accepted: 30 December 2011
Published: 30 December 2011
Solid tumors are heterogeneous in composition. Cancer stem cells (CSCs) are believed to drive tumor progression, but the relative frequencies of CSCs versus non-stem cancer cells span wide ranges even within tumors arising from the same tissue type. Tumor growth kinetics and composition can be studied through an agent-based cellular automaton model using minimal sets of biological assumptions and parameters. Herein we describe a pivotal role for the generational life span of non-stem cancer cells in modulating solid tumor progression in silico.
We demonstrate that although CSCs are necessary for progression, their expansion and consequently tumor growth kinetics are surprisingly modulated by the dynamics of the non-stem cancer cells. Simulations reveal that slight variations in non-stem cancer cell proliferative capacity can result in tumors with distinctly different growth kinetics. Longer generational life spans yield self-inhibited tumors, as the emerging population of non-stem cancer cells spatially impedes expansion of the CSC compartment. Conversely, shorter generational life spans yield persistence-limited tumors, with symmetric division frequency of CSCs determining tumor growth rate. We show that the CSC fraction of a tumor population can vary by multiple orders of magnitude as a function of the generational life span of the non-stem cancer cells.
Our study suggests that variability in the growth rate and CSC content of solid tumors may be, in part, attributable to the proliferative capacity of the non-stem cancer cell population that arises during asymmetric division of CSCs. In our model, intermediate proliferative capacities give rise to the fastest-growing tumors, resulting in self-metastatic expansion driven by a balance between symmetric CSC division and expansion of the non-stem cancer population. Our results highlight the importance of non-stem cancer cell dynamics in the CSC hypothesis, and may offer a novel explanation for the large variations in CSC fractions reported in vivo.
KeywordsCancer Stem Cells Agent-Based Modeling Solid Tumor Growth Kinetics
The cancer stem cell hypothesis suggests that a subset of cells in a tumor is uniquely capable of driving disease. Definitively identifying cancer stem cells is often hailed as the Holy Grail of cancer research, as they are believed to be the sole initiator and driver of tumor growth, and thus their eradication may offer targeted tumor treatment. Characterization of cancer stem cells and quantification of their frequency within solid tumors, however, are topics still in their infancy and subjects of debate. Computational modeling in the context of the cancer stem cell hypothesis is an invaluable tool for assessing the relative contributions of basic cellular kinetics to macroscopic growth dynamics and composition of solid tumors. Two tumors, presenting the same cellular population size at clinical detection, can comprise very different subsets of cell types. As a result, the prognosis downstream of presentation could be very different. Unexpectedly, although cancer stem cells are the engine of tumor progression, the proliferative capacity of non-stem cancer cells appears to be modulating cancer stem cell kinetics and thus overall tumor progression dynamics. Understanding the oft-neglected contribution of the non-stem population within a heterogeneous tumor will help explain the large variations in cancer stem cell frequency reported in the literature, and ultimately help guide the design of appropriate treatments for patients with different disease presentations.
The interplay of CSCs and CCs in tumor progression lends itself to quantitative agent-based modeling of tumor behavior using cellular automata, an established approach to analyze complex system dynamics and test systemic response to perturbation under a minimal set of rules and constraints [4–8]. Of particular intrigue is the paradoxical dependence of tumor progression on cell death within the CSC hypothesis, and the apparent disconnect between the clinical strategy of inciting cell death through chemical or radiological assault and the mathematical behavior of an untreated system that might well be self-limiting . We have shown previously that a system looking solely at the balance between cell proliferation, migration, and death can be sufficient to explain anomalous features of the growth kinetics and morphology of simulated tumors . In this model, without sufficient migration capability, a CSC may quickly become surrounded and spatially inhibited by non-stem CC progeny, and thus able to initiate only a clone of self-limiting size . The resulting tumor achieves a pseudo-steady dormant state, characterized by a balance between cell death of CC with exhausted proliferative capacity at the periphery and subsequent proliferation of previously quiescent CC. Growth beyond the initial clone is impeded by the lack of space available for the CSC to divide potentially symmetrically to seed a new clone nearby - a mechanism others and we have described as self-metastatic tumor progression [10, 12]. Herein we describe further exploration of cell-cell interactions underlying these phenomena toward describing tumor growth kinetics and CSC fraction as a function of the generational life span of the CCs. Depending on this inherited maximum proliferative capacity, CCs generate clonal population sizes that contribute to form smaller or larger tumors, respectively enabling or inhibiting CSC proliferation. Without sufficient telomerase activity, cells progressively shorten telomeres upon each division, eventually leading to terminal failure to replicate chromosomes and cell death . The number of cell divisions is likely to be dependent on the function, morphology, age and developmental history of the specific organ. However, the specificity of telomere length to tissue type and the process of telomere shortening at a measurable rate  suggests that the remaining proliferative capacity of a given cell can be quantitatively modeled as the number of sequential mitoses until death (Figure 1). Agent-based models can capture the interactive consequences of cell-intrinsic properties and tumor population dynamics while allowing for the distinction of the participating CSC and CC compartments. Through stochastic simulations, we explore how altering CC generational life span influences tumor growth kinetics and CSC prevalence. CCs are thought to be the dominant population in a tumor, yet a large CC population impairs tumor progression in silico. Therefore, non-linear modulation of tumor kinetics by the generational life span of CCs is expected. We identify optimal parameter values for tumor progression and inhibition, as predicted by the model, and discuss their biological applicability.
Materials & methods
We extend an established agent-based cellular automaton model [9, 10] of CSCs and CCs interacting in tumor growth. We assume that (i) CSCs are immortal with unlimited replicative potential, (ii) CC maximum proliferative capacity ρmax is inherited from the parent CSC following an asymmetric division, (iii) the probability of symmetric (ps) and asymmetric (1 - ps) CSC division is constant and stochastic, and (iv) cells require adjacent available space to migrate or proliferate. Individual cells are equipped with a cell cycle time of 24 hours and migration speeds of μ = 0, 50, 100, or 150 μm (i.e., 0, 5, 10, or 15 cell widths) per day . We model migration speed as a trait inherent to a cell. While there are undoubtedly tissue types that naturally demonstrate faster or slower migration rates associated with the biological function of that organ system, or that change to a faster migration regime in a context such as wound healing, these tissue-level observations can be emergent properties of an agent-based model rather than imposed behaviors. How cell migration speed and environmental chemotactic gradients modulate tumor progression has been discussed elsewhere [10, 16].
Total number of cancer cells and number of cancer stem cells are recorded at simulated seven-day intervals. Simulation run times on a Quad-core 2.3 GHz Intel Xeon Mac OS and 64-bit CentOS servers range from ten minutes to seven days. Raw data from each simulation are analyzed to generate average growth curves and CSC fraction in the emerging tumors.
Tumor growth kinetics display non-monotonic dependence on replicative potential
Summary of model tumor growth simulations across all conditions.
Symmetric division frequency
p s = 1%
p s = 10%
Migration rate (cell widths/day)
μ = 0
μ = 5
μ = 10
μ = 15
μ = 0
μ = 5
μ = 10
μ = 15
Optimum ρmax (Figs. 3-4)
1303 ± 19
115 ± 4
95 ± 5
85 ± 6
130 ± 1
23 ± 1
17 ± 1
14 ± 1
CSC Fraction at optimum ρmax
ρmax pairs(Figs. 5-6)
Sizes at tcritical
1.7 × 104
0.86 × 104
2.1 × 104
4.0 × 104
3.8 × 104
1.9 × 104
Times to 5 × 104 cells (weeks)
Similar non-monotonic behavior is also observed at higher cell migration speeds (Figure 3; Table 1). At μ = 10, tumors reach 50,000 cells fastest if ρmax = 7 (tcritical = 95 ± 5 weeks), with increasing tumor sizes from ρmax = 0 (1,977 cells) to ρmax = 6 (37,152 cells) and decreasing tumor sizes from ρmax = 8 (40,066 cells) to ρmax = 10 (7,840 cells). The higher migration rate leads to an increase in the CC generational life span that facilitates fastest tumor growth. A larger maximum proliferative capacity and the resulting increase in CC persistence are accommodated by faster cell migration that loosens the intratumoral spatial confinement. A further increase of cell migration speed to μ = 15 shifts the fastest growth to tumors with ρmax = 8 (tcritical = 85 ± 6 weeks), again with consistently increasing tumor sizes as ρmax increases from 0-8 and decreasing tumor sizes as ρmax increases further.
Tumor CSC fraction correlates with growth kinetics downstream of a critical size
When symmetric CSC division was comparatively rare (ps = 1%, Figure 5), tumors with very distinct morphologies and compositions were apparent on either side of the optimum ρmax for fastest growth. For example, at μ = 10, ρmax = 3 and ρmax = 10 tumors both reached ~0.86 × 104 cells in 95 weeks, but would require 124 or 196 weeks, respectively, to grow to 5 × 104 cells. With more frequent symmetric division (ps = 10%, Figure 6), morphologies were similar, but in some cases the differences in tumor composition and growth kinetics downstream of tcritical were significant. For example, when μ = 15, tumors with CC that inherit either ρmax = 1 or ρmax = 9 could reach ~1.9 × 104 cells in 14 weeks, but would require 18 or 21 weeks, respectively, to then progress to 5 × 104 cells. In this case, morphologies at tcritical were indistinguishable but the CSC fractions of 30% (ρmax = 1) and 0.32% (ρmax = 9) were dramatically different.
Tumor CSC fractions are dependent on non-stem cancer cell replicative potential
Cancer stem cell fraction fluctuates during early tumor growth due to the stochastic interpretation of (a)symmetric cancer stem cell division. Once a compositional pseudo equilibrium is established, tumor growth is dominated by a fractal self-similarity associated with self-metastatic progression as observed in previous modeling studies .
The high variance in frequency of cancer cells expressing stem cell biomarkers within different tumors of the same tissue type can be attributed to intrinsic tumor hierarchy. While cancer stem cells (CSCs) are the engine of tumor progression , the contribution of non-stem cancer cells (CCs) to tumor growth kinetics is often disregarded, despite the fact that in many cases, these cells constitute the majority of the tumor. We set out to investigate if and how CCs modulate CSC dynamics and thus tumor progression. We utilized an agent-based model to describe the behavior of individual cells in response to their local environment, and for simplicity, focused only on the availability of space for cells to proliferate and migrate. Higher-order mechanisms, such as mechanical forces between cells and nutrient gradients, could be included in such a framework in the future to facilitate detailed understanding of how tumors overcome more specific, environmentally-imposed barriers to growth. Here we limited our analysis to the early stages of tumor development to sizes below angiogenic limitations  and focused on interaction of cells with only their nearest neighbors, which has been previously shown to be sufficient to reveal intriguing and often counterintuitive dynamics [9, 10].
Much of the debate regarding the frequency of CSCs in various tumors derives from measurements of expression of cell surface proteins and/or gene expression profiling. While transformation at different locations on the somatic differentiation axis could lead to variable expression levels of stem cell markers following clonal expansion, our model suggests that the observed compositional heterogeneity could arise more simply from varying the generational life spans of the CC progeny, a trait inherited from asymmetric division of the CSCs driving tumor growth. As such, a transformation event yielding a CSC will confer a discrete maximum proliferative capacity to the resulting non-stem cancer cells based on a variety of factors, including tissue type  and host age [24–28]. In addition, heterogeneity of telomere lengths within the cells of a given organ at a certain age may lead to variability in tumor growth kinetics and composition following otherwise identical origin. We have shown a direct connection between the character of the CSC, the commensurate proliferative capacity of the non-stem cancer cells, and the macroscopic growth kinetics and compositional heterogeneity of the tumor.
Interestingly, simulated tumors arising from CSCs conferring intermediate maximum proliferative capacities demonstrated the most aggressive growth kinetics for non-zero migration rates, resulting in a non-monotonic dependence of the growth kinetics on the depth of the CC hierarchy. Macroscopically, this suggests an optimal balance between the freeing of space through death of cells with exhausted replicative potential and migration of CSCs to the resulting available space. The expansion of the immortal CSC compartment is critical for self-metastatic tumor progression . In tumors with non-stem proliferative capacities either lower or higher than the optimum, macroscopic growth was less efficient. In the former case, the failure of CCs to persist before inevitable cell death slowed total population expansion, and space resulting from CC death opened up more quickly than the CSC population could exploit due to the infrequency of symmetric CSC division. In the latter case, enrichment of CSCs was inhibited by crowding of the surrounding CCs, whose long generational life spans inhibit cell activity in the tumor interior [5, 9, 11, 29, 30]. This bi-modal contribution of CCs to tumor growth and progression yields a non-monotonic dependence of tumor growth kinetics on CC generational life span. Optimum proliferative capacity is dependent on the interplay of other presumably independent tumor growth kinetic parameters, such as CSC symmetric division frequency and cell migration rate.
Tumors comprising CCs with generational life spans on either side of the optimum proliferative capacity can grow to similar sizes yet harbor CSC fractions that vary by multiple orders of magnitude, eventually resulting in vastly different growth kinetics. The degree to which two presenting tumors of identical size but distinct composition may respond differently to treatment must be considered. While tumors gaining size early through rapid expansion of CCs with longer generational life spans may contain only a small number of CSCs, CCs with shorter generational life spans yield later progressing tumors that may harbor many CSCs. These CSC-rich tumors have a significantly steeper growth curve at hypothetical time of detection and thus a worse prognosis. Comparatively self-limiting tumors arising from CSCs conferring high replicative potential to their mortal progeny should have a slower progression rate and thus experience an initial shrinkage in response to cytotoxic treatment, but the space opened up through killing of the mortal CCs might lead to enrichment of the immortal, self-metastatic CSC fraction at a rate faster than would occur naturally . Conversely, tumors with already high fractions of CSCs are limited by the short life spans of the mortal progeny, such that a cytotoxic treatment would not exacerbate disease progression through selection for the immortal compartment.
The model presented herein is based on a small number of plausible, biologically motivated first-order assumptions. The interaction of cells has been restricted to their immediate neighborhood, and long-range interactions achieved through physical forces have been ignored for simplicity. Furthermore, we focused on initial tumor growth from a single cell to a small cluster that is thought to develop without nutrient deprivation or physical constraints enforced by the local host environment. These model assumptions apply to all simulations presented regardless of parameter values. As such, we believe is valuable to compare the time intervals for model tumor progression at different conditions in a qualitative sense only, as the translation from simulation times to biologically accurate growth kinetics would require more mechanistic detail.
Inclusion of extrinsic forces on tumor progression is expected to further enhance the observed behavior as modulators of the fundamental cellular level processes of proliferation, migration, quiescence, and cell death, all of which are already accounted for in the model. Further modeling efforts would be needed to explore the contribution of applied cytotoxic pressure (with varying degrees of differential susceptibility between the CSCs and mortal cancer cells) to overall growth kinetics and assess the role of stored information in the tumor microenvironment in determining decision pathway probability vectors at the cellular level.
While these model simulations begin with a single CSC, in a clinical setting, "time zero" for the patient and physician is the onset of disease presentation rather than the initiation event. Moreover, the number of human tumors that presents clinically is necessarily only the aggressive fraction of the total number of tumors existing, with a significant number likely being present in a dormant or slow-progressing state . We speculate that solid tumors in a given organ with distinct CSC fractions may demonstrate substantially different progression patterns downstream of presentation and diagnosis, and that these differences may be attributable to the compositions in the respective populations as governed by CC generational life span. We believe that further exploration of the observed in silico correlation between CSC fraction and growth kinetics in vivo may help reveal tools that augment clinical predictive power.
List of Abbreviations
c ancer s tem c ell(s)
non-stem c ancer c ell(s)
s tandard e rror of the m ean.
This project was supported by the AACR Centennial Postdoctoral Fellowship in Cancer Research 08-40-02-ENDE (H. Enderling), and the National Cancer Institute under Award Number U54CA149233. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Cancer Institute or the National Institutes of Health. The authors would also like to thank Clare Lamont for help with figure design and Lauren Frick for critical reading of the manuscript.
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