Malignant cancers that lead to fatal outcomes for patients may remain dormant for very long periods of time. Although individual mechanisms such as cellular dormancy, angiogenic dormancy and immunosurveillance have been proposed, a comprehensive understanding of cancer dormancy and the “switch” from a dormant to a proliferative state still needs to be strengthened from both a basic and clinical point of view. Computational modeling enables one to explore a variety of scenarios for possible but realistic microscopic dormancy mechanisms and their predicted outcomes. The aim of this paper is to devise such a predictive computational model of dormancy with an emergent “switch” behavior. Specifically, we generalize a previous cellular automaton (CA) model for proliferative growth of solid tumor that now incorporates a variety of cell-level tumor-host interactions and different mechanisms for tumor dormancy, for example the effects of the immune system. Our new CA rules induce a natural “competition” between the tumor and tumor suppression factors in the microenvironment. This competition either results in a “stalemate” for a period of time in which the tumor either eventually wins (spontaneously emerges) or is eradicated; or it leads to a situation in which the tumor is eradicated before such a “stalemate” could ever develop. We also predict that if the number of actively dividing cells within the proliferative rim of the tumor reaches a critical, yet low level, the dormant tumor has a high probability to resume rapid growth. Our findings may shed light on the fundamental understanding of cancer dormancy.
Citation: Chen D, Jiao Y, Torquato S (2014) A Cellular Automaton Model for Tumor Dormancy: Emergence of a Proliferative Switch. PLoS ONE 9(10): e109934. https://doi.org/10.1371/journal.pone.0109934
Editor: Daotai Nie, Southern Illinois University School of Medicine, United States of America
Received: May 9, 2014; Accepted: September 12, 2014; Published: October 16, 2014
Copyright: © 2014 Chen 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 authors confirm that all data underlying the findings are fully available without restriction. All relevant data are within the paper and its Supporting Information files.
Funding: The research described was supported by the National Cancer Institute under Award NO. U54CA143803 (http://www.nih.gov/). 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.
Cancer dormancy, the phenomena that the tumor's volume or the number of tumor cells stays at a very low level for a certain period of time before the tumor begins to grow rapidly, has been an outstanding issue in cancer research for many years , . Currently, the mechanisms responsible for the “switch” from a dormant state to a rapid growth state for different tumors are not well understood, although it is well known that such a “switch” in secondary metastatic tumors can be triggered by the removal of the primary tumor. This could eventually lead to failure of tumor treatment and fatal outcomes for the patient. Therefore, a comprehensive understanding of the “switch” from a dormant to a proliferative state is crucial to our fundamental understanding of cancer progression and recurrence and might lead to the development of novel treatments for cancer.
Dormancy has been observed in many types of cancer. This includes tumor dormancy before any metastases take place and the latency of cancer recurrence after therapy. In some cases of pancreatic cancer, the tumor can remain in a benign dormant state for about 20 years . During this time, it is undetectable by conventional clinical methods, and it is only afterwards that the tumor becomes highly malignant and grows aggressively with highly fatal outcomes after about a year. In the cases of breast and prostate cancer, it is reported that 20%–45% of patients will relapse years or decades later after the resection of the primary tumor –. In addition, recurrence has been observed in brain tumors, which indicates the existence of a large number of micrometastases that are dormant in the presence of the primary tumor , .
Extensive studies over years have revealed three major cancer dormancy mechanisms: cellular dormancy, angiogenic dormancy and immunosurveillance , . On the cellular level, a tumor cell could be arrested at a certain stage of the cell cycle and unable to complete the cell division process successfully, resulting in a dormant solitary cell –. On the cell population level, when the population does not gain enough ability to recruit blood vessels and promote neovascularization, the tumor cannot obtain sufficient nutrients necessary for its proliferation and as a result, angiogenic dormancy occurs , . On other hand, immunosurveillance operates when the immune system suppresses the proliferation of tumor cell population and leads to the dormancy of the tumor –. Figure 1(a) shows an image of tumor tissue surrounded by immune cells. Figure 1(b) compares the morphology and vascular structure of dormant and fast-growing tumors.
Nuclei appear blue (DAPI). Image courtesy of Michael Graham Espey, PhD, National Cancer Institute, NIH (private communication). (b) Representative pictures of dormant and fast-growing tumors and their vascular structure. Reprinted from Cancer Letters, 294, Almog N, Molecular mechanisms underlying tumor dormancy, 139–146, Copyright (2010), with permission from Elsevier.
A comprehensive understanding of cancer dormancy and the “switch” from a dormant to a proliferative state still needs to be strengthened. This is mainly due to the fact that efficient and accurate experimental or clinical approaches to track the states of individual cells in a dormant tumor in vivo throughout the entire dormancy period are still under development –.
Given the current need for further understanding of dormancy, computational modeling provides a powerful means to probe various scenarios for the underlying mechanisms. Specifically, modeling enables one to probe a variety of different dormancy scenarios by examining different combinations of mechanisms in order to see which ones provide possible explanations for experimental and clinical observations. Over the past few decades, computational modeling has played an important role in the study of the progression of solid tumors ; a variety of models based on different mathematical schemes have been developed, including continuum models –, discrete cell models ,  and hybrid models , . Various models have been used to investigate cancer dormancy caused by cancer-immune interactions and other mechanisms, including ordinary differential equation-based models , , stochastic differential equation-based models , models based on kinetic theory for active particles –, and cellular automaton models . However, the aforementioned studies neither explicitly demonstrated how the dynamic process of active proliferation after a certain period of dormancy emerges from various microscopic mechanisms nor showed the associated growth dynamics of the “switch” phenomenon. Therefore, predictive computational models that incorporate cellular-level microscopic mechanisms are needed to address these important issues.
In this paper, we generalize a two-dimensional (2D) cellular automaton (CA) model that we have devised to study proliferative growth of avascular solid tumors – in order to investigate tumor dormancy. Our goal is to formulate a dynamical model in which the “switch” to a proliferative state spontaneously emerges by incorporating additional interactions between the tumor and the microenvironment, for example the effects of immune system, which were not included in our previous CA model. The new rules of our CA model induce a “competition” between the tumor's propensity to proliferate and the microenvironmental factors that suppress its growth. In our model, a fraction of the dormant cells undergo phenotypic transformations triggered by intracellular factors or external stimulus and acquire the ability to actively proliferate. Subsequently, those microenvironmental factors act to suppress the growth of these transformed tumor cells either by killing some of these cells or turning these actively dividing proliferative cells back into dormant cells.
The “competition” between the tumor and the microenvironmental suppression factors either results in a “stalemate” for a period of time in which the tumor either eventually wins (spontaneously emerges) or is eradicated; or it leads to a situation in which the tumor is eradicated before such a “stalemate” could ever develop. Since we are mainly interested in the situations in which tumor growth involves a period of dormancy, we will henceforth focus on those situations in which a “stalemate” between the tumor and the microenvironmental suppression factors develops. Our model demonstrates that a variety of parameters characterizing the tumor-host interactions may greatly alter the growth dynamics of the tumor. These parameters include the rate of phenotypic transformation, by which the tumor cells gain the ability to proliferate against those suppression factors, the suppression rate imposed by suppression factors on individual tumor cells, and the mechanical rigidity of the microenvironment. The growth dynamics influenced by these parameters include the existence of a dormant period in tumor's growth, the length of the dormant period (if there exists one) and the existence of a sudden “switch” to a highly proliferative state. We also demonstrate that if the number of actively dividing cells within the proliferative rim reaches a critical, yet low level, the tumor has a high probability to begin rapid proliferation. While we study a 2D CA model for simplicity in this paper, our model can be easily generalized to three dimensions (3D).
Materials and Methods
We divide the two-dimensional square simulation box into different polygonal units (i.e., automaton cells). Our model is coarse-grained, allowing us to grow the tumor from a very small size with a cross section of roughly 1000 real cells through to a fully developed tumor with a cross section consisting of 2.0×106 cells. Specifically, the innermost automaton cells represent roughly 100 real tumor cells or or a region of host microenvironment of similar size, while the outermost automaton cells represent roughly 104 real tumor cells or a region of host microenvironment of similar size. To generate the automaton cells in the simulation box, we first fill the simulation box with non-overlapping circular disks (or spheres in 3D) using random-sequential-addition packing method  until there is no void space left for additional circular disks (or spheres in 3D). Periodic boundary conditions are used for generating the packing. Then we divide the simulation box into polygons (or polyhedra in 3D), each polygon (or polyhedron in 3D) associated with a particle center, such that any point within a polygon (or polyhedron in 3D) [i.e., a Voronoi polygon (or polyhedron in 3D)] is closer to its associated particle center than to any other particle centers. The resulting Voronoi polygons (or polyhedra in 3D) are referred to as automaton cells. In this paper, we focus on the two-dimensional case, but our model should be readily generalized to three dimensions.
The microenvironment surrounding a tumor is mainly composed of stroma cells and extracellular matrix (ECM). In the current model, we explicitly take into account the effects of the ECM macromolecule density, ECM degradation by the proliferative cells, and the pressure built up due to the ECM deformation by tumor growth. The effects of the stroma cells are not explicitly considered in our current model. Henceforth, we will refer to the regions of microenvironment as ECM-associated cells for simplicity. In addition, we consider the interactions between the tumor and the various suppression factors in the microenvironment, for example the immune system. Since we consider development of primary tumor or local recurrences of micrometastases under microenvironmental suppression, invasive tumor growth is not a mechanism relevant for our purposes and hence is not included in our dormancy model.
In our model of noninvasive proliferative tumor growth, tumor cells can be in one of the three possible states: proliferative, quiescent or necrotic, depending on their nutrient supply. Proliferative cells are tumor cells that have enough nutrients and possess the ability to divide. Quiescent (or arrested) cells are tumor cells that are alive, but do not have enough nutrient supply to support cell division. Quiescent cells can eventually become inert, necrotic (dead) cells due to an insufficient nutrient supply. In this paper, we focus on avascular tumor growth and assume that there is no explicit angiogenesis during the growth process (although this assumption can be relaxed). The nutrients available to tumor cells are those that diffuse into the tumor region through tumor edge. As the tumor grows, the amount of nutrient supply, which is proportional to the perimeter of the tumor interface (or surface area of the tumor interface in 3D), cannot meet the needs of all of the tumor cells. As a result, quiescent and necrotic regions emerge near the center of the tumor. The state of a tumor cell is determined by its distance to the tumor edge (i.e., the source of nutrients). We assume that proliferative cells more than δp away from the tumor edge become quiescent and quiescent cells more than δn away from the tumor edge become necrotic (see details below).
In this section, we will introduce our CA dormancy model, which modifies our previous basic CA models of tumor growth , ,  by introducing several additional parameters to incorporate the interactions between tumor cells and the microenvironmental suppression factors. This dynamical model is capable of producing situations in which a “switch” from a dormant state to a proliferative state spontaneously emerges.
Noninvasive proliferative tumor growth
We now specify the cellular automaton rules used in our model for noninvasive proliferative tumor growth. Each ECM-associated automaton cell is assigned a specific density ρECM, representing the density of the ECM molecules within the automaton cell. If a proliferative cell divides, its daughter cell occupies a nearby ECM-associated cell. The daughter cell pushes away or degrades the ECM within the ECM-associated cell it occupies. Initially, a tumor is introduced by designating several automaton cells at the center of the growth permitting region as proliferative tumor cells. Then time is discretized into units, with each time step representing one day. At each time step, the tumor grows according to the following cellular automaton rules.
• Quiescent cells more than a certain distance δn from the tumor's edge are turned necrotic. The tumor's edge, which is assumed to be the source of nutrients, consists of all ECM-associated cells that border the tumor. The critical distance δn for quiescent cells to turn necrotic is computed as follows:(1)where a is the necrotic thickness controlled by nutritional needs, d is the Euclidean spatial dimension and Lt is the distance between the geometric centroid xc of the tumor (i.e., , where N is the total number of cells in the tumor) and the tumor edge cell that is closest to the quiescent cell under consideration.
• Proliferative cells more than a certain distance δp from the tumor's edge are turned quiescent. The critical distance δp is given by(2)where b is the proliferative thickness controlled by nutritional needs, d is the spatial dimension and Lt is the distance between the geometric tumor centroid xc and the tumor edge cell that is closest to the proliferative cell under consideration.
• The probability of division for a proliferative cell used in our model is(3)where p0 = 0.192 is the base probability of division linked to cell-doubling time, ρECM is the local ECM density, is a parameter taking into account the effect of pressure, is the ratio of current average ECM density over the initial density, and and are, respectively, the length and width of local protrusion tips.
Interactions between the tumor and the microenvironmental suppression factors
Here, we specify the additional interaction rules between the tumor and the microenvironmental suppression factors beyond the aforementioned ones for noninvasive proliferative growth, which were not included in our previous CA models. We assume that there are two possible states of proliferative cells, dormant or actively dividing, depending on their interactions with the microenvironmental suppression factors.
• Initially, we assume that all proliferative cells are kept in dormant states by the microenvironmental suppression factors, which means that they are not able to divide.
• At each day, beyond the aforementioned CA rules for proliferative noninvasive growth, each dormant proliferative cell has a certain probability to change in their phenotypes due to intracellular factors or external stimulus. The cell with phenotype change gains different degrees of resistance to the suppression factors in the microenvironment, depending on the specific phenotype change the cell undergoes. For example, mutated leukaemic cells in acute myeloid leukaemia acquire resistance to cytotoxic T lymphocytes-mediated cell lysis, whose degree is related to the level of the cell's expression of B7-H1 or B7.1 . For simplicity, we divide the phenotypic changes into two different types: weak changes and strong changes with respect to their resistance to the suppression factors in the microenvironment (i.e. their ability to actively proliferate). Henceforth, we will refer to these phenotypic changes as “transformations” and the cells that undergo these changes as “transformed” cells for simplicity. Strong-type “transformed” cells gain a larger competition advantage and thus have a greater ability to divide actively. The quantities xW and xS are the fractions of weak-type “transformations” and strong-type “transformations”. Henceforth, we set xW 0.99 and xS 0.01.
• At each subsequent day, the microenvironmental suppression factors will counteract the weak-type “transformed” and strong-type “transformed” cells with probabilities αW and αS. The suppression factors in the microenvironment will either kill the “transformed” cells or turn them back into dormant cells , .
• When the number of tumor cells reaches a certain threshold NT, strong reactions of the microenvironmental factors are triggered and those factors start to kill the “transformed” cells. The parameter NT is introduced to ensure that the tumor is not completely removed by the microenvironmental suppression factors. Note that the particular choice of NT barely has any effect on the simulation results within a relatively wide range of NT values. In this work NT is set to be 50, a sufficiently small value that leads to biophysically realistic outcomes. As the tumor grows, the microenvironmental factors are weakened by the tumor, resulting in weaker suppression of the tumor cells , . Therefore, when the microenvironmental factors counteract the “transformed” cells, the fraction of the cells that are killed can be coupled with the growth rate of the tumor by(4)where k0 is a constant characterizing the strength of the suppression factors in the microenvironment, dA/dt is the daily area change of the tumor (i.e. the growth rate of the tumor), and is the critical value of the tumor's growth rate. In this work, is chosen as half of the tumor's maximum growth rate under suppression, but our numerical tests have revealed that the simulation results are insensitive to the choice of as long as is smaller than the tumor's maximum growth rate. When the growth rate of the tumor reaches this critical value, the suppression factors become too weak to kill any actively dividing tumor cells and k is set to be 0 , .
• Due to the “competition” between the tumor and the suppression factors in the microenvironment, the ratio of the number of actively dividing proliferative cells over the total number of proliferative cells changes with time. The larger is this ratio , the larger is the amount of nutrients the tumor tissue consumes. As a result, the nutrient concentration around the tumor depends on the ratio . Therefore, we make the necrotic thickness a and proliferative thickness b functions of :(5)(6)where a0 = 0.58 mm1/2 and b0 = 0.30 mm1/2 are base necrotic thickness and base proliferative thickness respectively, q = 1.6, s = 2.0 are parameters determining the ranges of necrotic thickness and proliferative thickness as changes.
The aforementioned additional parameters associated with the new rules that we employ for dormancy (beyond the ones for noninvasive proliferative growth) are summarized in Table 1. These parameters are sufficient to formulate a model in which the transition from “dormant” to proliferative state emerges spontaneously. Note that unlike other parameters listed in Table 1, the two critical threshold parameters themselves do not incorporate any additional CA rules. Instead, the critical threshold parameters determine when the microenvironmental suppression factors are able to kill the proliferative cells. Also, it is noteworthy that we map the complicated tumor-host interactions onto a number of “effective” parameters. The values of these parameters could differ for different tumors in different microenvironments. It is noteworthy that currently due to a lack of detailed in-vivo or in-vitro data for the growth dynamics of a dormant tumor, we are not able to determine the values of the parameters in our model for a specific real system. Instead, we have done a full parametric study to probe different outcomes corresponding to different parameter values in the subsequent sections. However, once we obtain the statistics of a dormant tumor as a function of time from the initiation of the tumor, we should be able to extract the parameter values for the tumor by fitting the statistics. At this stage, the extracted parameter values could be applied to other tumors of similar type.
Noninvasive proliferative tumor growth under suppression
Here, we specify how the additional interaction rules are coupled together with the original CA rules for noninvasive proliferative tumor growth, resulting in noninvasive proliferative tumor growth under suppression.
• As mentioned above, proliferative cells in the dormant state do not divide. Only proliferative cells in actively dividing states actually proliferate.
• At each day, each dormant proliferate cell is checked to see if it enters the active state according to the interaction rules. Once it begins to actively divide, it proliferates according to the CA rules for proliferative tumor growth.
• At each day, each active proliferative cell is checked to see if it is killed or turned back into dormant cell according to the interaction rules.
• Quiescent cells and necrotic cells act according to CA rules for proliferative tumor growth. However, the values of parameters a and b determining the transitions from necrotic cells to quiescent cells and from proliferative cells to quiescent cells respectively are influenced by interaction rules, as mentioned above.
Note that our CA dormancy model should be readily generalized to angiogenic dormancy by explicitly considering the angiogenic process and vascular tumor growth. This is beyond the scope of this work and will be addressed in future work.
In this section, we apply our CA model and show that it produces a dormancy period of the tumor that can lead to a subsequent emergent “switch” behavior to a proliferative state. A homogeneous distribution of ECM density is used for simplicity . A circular growth permitting region containing ∼2×104 automaton cells is employed. Simulating the growth of a 2D tumor from several cells (representing roughly 1000 real cells) to a macroscopic-size tumor (a cross section of 5 cm2 consisting of ∼2×106 real cells) with a period of dormancy up to a person's life (∼80 years) generally takes no more than a few minutes on a standard Dell Workstation (Precision T3400).
Initially, a few automaton cells at the center of the growth-permitting region are designated as proliferative cells. Then the initial tumor is allowed to grow according to our CA model incorporating the additional interaction rules between the tumor and the suppression factors in the microenvironment. Certain geometrical characteristics of the tumor (e.g., tumor area, areas of different tumor cell populations) and its morphology (e.g., the geometrical positions of the tumor cells) are collected every days. We set , αW = 0.75, αS = 0.15, k0 = 0.8 and use these parameter values throughout this paper, except where otherwise stated.
Statistics of tumor growth
Here we consider the growth of a proliferative tumor in a confined space with . As shown in Figure 2(a), with the interactions between the tumor and the microenvironmental suppression factors incorporated, there exists a period of dormancy in the tumor's growth. Specifically, for the initial approximate 900 days, the tumor stays in a dormant state. Suddenly at approximately day 900, the tumor switches its behavior and begins rapid proliferation. The virtual patient would die 100 days after this critical point in time. Figure 2(b) shows the areas A of different populations normalized by the area of the growth-permitting area A0. For purposes of comparison, Figure 2(c) and Figure 2(d) show the statistics of the tumor growth without the suppression of microenvironmental factors. Moreover, by comparing Figure 2(a) and Figure 2(c), it is seen that the interactions between the tumor and the microenvironmental suppression factors lead to the existence of a dormancy period and a subsequent emergent “switch” behavior of the tumor from a dormant state to a proliferative state. Also, from the comparison of Figure 2(b) and Figure 2(d), one can see that the additional interaction rules alter the fractions of necrotic cell population and proliferative cell population within the tumor. When suppression of the tumor growth is present, the necrotic region decreases and the proliferative region increases relatively; the area of the quiescent region remains almost unchanged.
(a) Tumor area AT normalized by the area A0 of the growth permitting region. (b) Areas of different cell populations normalized by the area A0 of the growth permitting region. Lower panel: statistics of a simulated noninvasive tumor growing in the ECM with without suppression. (c) Tumor area AT normalized by the area A0 of the growth permitting region. (d) Areas of different cell populations normalized by the area A0 of the growth permitting region.
Figure 3 shows snapshots of the simulated 2D tumor. It can be clearly seen that the tumor develops a highly aspherical morphology due to the interactions between the tumor and the microenvironmental factors. Figure 3 also demonstrates that the tumor hardly grows during the period of dormancy, but once the “switch” occurs, the tumor expands very rapidly. Henceforth, we will use the “CA dormancy model” to investigate the effects of the various parameters characterizing the tumor-host interactions on the growth dynamics of the tumor. These parameters include the rate of phenotypic transformation, by which the tumor cells gain the ability to proliferate against those suppression factors, the suppression rate imposed by suppression factors on individual tumor cells, and the mechanical rigidity of the microenvironment.
Suppression rate vs transformation rate
Here we investigate growth dynamics of the tumor under different suppression rates α and phenotypic transformation rates . The suppression rate α is defined as the following weighted average:(7)where xW = 0.99 and xS = 0.01 are the fractions of weak-type “transformations” and strong-type “transformations”, and αW and αS are the suppression rates of the weak-type “transformed” cells and strong-type “transformed” cells by the microenvironmental factors. It is found that increasing α and decreasing generally increases the length of the dormancy period and delays the “switch” from a dormant state to a proliferative state, as demonstrated in Figure 4(a). Within some regimes of α and , the tumor could lie dormant for a period equal to or longer than a person's life (∼80 years).
A schematic phase diagram that characterizes the growth dynamics of a noninvasive tumor growing in the ECM with under different α and (b).
Based on our simulation results, we construct a “phase diagram” to characterize the tumor's growth dynamics in terms of α and , as shown in Figure 4(b). There are two regions in this phase diagram: proliferative and dormant regions. By “proliferative”, we mean that the tumor resumes rapid proliferation after a period of dormancy and the length of the dormancy period is less than a virtual patient's life; by “dormant”, we mean that the tumor remains in a dormant state during the whole life of a person and undetected by conventional clinical methods (usually clinicians call such tumors “benign” –). The solid line separates the two regions, and crossing this boundary line is associated with a “phase transition”.
Rigidity of the microenvironment
Various mechanical cues in the microenvironment could influence the growth dynamics of the tumor . Here we only consider the effects of the ECM macromolecule density, ECM degradation by the proliferative cells, and the pressure built up due to the ECM deformation by tumor growth. As shown in Figure 5, when ECM rigidity increases, the time at which the switch occurs gets delayed significantly and the final size of the tumor when it plateaus appreciably decreases. For example, with all the other parameters fixed, when the tumor grows in a soft ECM with , the “switch” point occurs approximately on day 250. This is to be contrasted with growth in a rigid ECM with where the dormancy period could last for 2,000 days before a “switch” to a proliferative state occurs. Also, the plateau size of the tumor growing in a soft ECM with is five times as large as that of one growing in a rigid ECM with . In other words, when the tumor grows in a harsher microenvironment, it's harder for the tumor to break out of a dormant state and potential proliferative growth is largely suppressed. Note that the rigidity of the microenvironment could also affect tumor growth via various intracellular signaling processes . Those mechanotransduction effects will be incorporated into our CA dormancy model in future work, which could result in different scenarios from those reported here .
Strength of the suppression factors
Here we investigate how tumor growth dynamics changes with the strength of the microenvironmental suppression factors. As shown in Figure 6, increasing the fraction of actively dividing tumor cells that are killed [i.e., increasing k0 in the equation (4)] when the microenvironmental suppression factors (which we recall could either kill the “transformed” cells or turn them back into dormant cells) delays the “switch” point from a dormant state to a rapid proliferative state and decreases the final tumor size. However, relatively speaking, the simulated tumor growth statistics are insensitive to k0 compared to the influences of the aforementioned other factors. Note that even when the suppression factors can only turn the “transformed” cells back into dormant cells and do not kill any “transformed” cells (i.e. k0 = 0), a “switch” behavior from a dormant state to a rapid proliferative state can still emerge. This indicates that turning the active proliferative cells back into dormant cells could also be a possible independent mechanism leading to a dormancy period and a subsequent “switch” to a proliferative state.
In this paper, we generalized a two-dimensional cellular automaton (CA) model previously developed for proliferative growth of avascular solid tumors to investigate tumor dormancy and evasion from dormancy to proliferation. Our CA dormancy model incorporates a variety of cell-level tumor-host interactions, including those between the tumor and the suppression factors in the microenvironment, for example the immune system. Our CA dormancy model induces a “competition” between the tumor's propensity to proliferate and the microenvironmental factors that suppress its growth. Our CA dormancy model predicts a dramatic emergent “switch” behavior from a dormant state to a rapidly proliferative state. Our results show that under the suppression of microenvironmental factors, the tumor develops a highly aspherical morphology with an larger proliferative region and a smaller necrotic region than those of a tumor that grows without the presence of suppression factors. We also predict that if the number of actively dividing cells within the proliferative rim of tumor reaches a critical, yet low level, the tumor has a large probability to resume rapid regrowth and exit dormancy. In addition, we demonstrate that a variety of different factors could greatly alter tumor growth dynamics, including the rate of phenotypic transformations, the suppression rate by the microenvironmental factors, the mechanical rigidity of the microenvironment, and the strength of the suppression factors. However, relatively speaking, the tumor growth is insensitive to the strength of the suppression factors in terms of killing active proliferative cells. We inferred from our simulation results a qualitative phase diagram to characterize the growth dynamics of the tumor under the suppression of microenvironmental factors in terms of the phenotypic transformation rate and the suppression rate. In this paper we focused on the two-dimensional case, but our model should be easily generalized to three dimensions.
At the cellular level, the origin of the “stalemate” between the tumor and microenvironmental suppression factors remains unclear. This “stalemate” may come from cell proliferation balanced by cell death, which could be the case for a dividing cancer stem cell . Arrested tumor cell proliferation imposed by microenvironmental factors could also result in the “stalemate” between the tumor and the microenvironmental suppression factors. Both scenarios could account for the case of differentiated cancer cells, since our CA dormancy model is coarse-grained and therefore considers the effective behavior of the tumor.
Our CA dormancy model may shed light on the fundamental understanding of cancer dormancy phenomenon. Specifically, our CA dormancy model proposes possible scenarios for cancer dormancy that during the dormancy period the great majority of proliferative cells stay in a dormant state, while only a small portion of proliferative cells, i.e., “transformed” cells are actively dividing, and the microenvironmental suppression factors counteract these “transformed” cells by either killing them or turning them back into dormant cells. As a result, the tumor cell population is barely expanding during the dormancy period. It is noteworthy that our CA dormancy model predicts that the tumor either eventually spontaneously emerges or is eradicated after a period of a “stalemate” between the tumor and the microenvironmental suppression factors; or the tumor is eradicated before such a “stalemate” could ever develop. These predicted scenarios arising from the interaction between the tumor and the microenvironmental suppression factors in our simulation qualitatively match the experimental observations of the cancer immunoediting process, by which the immune system controls the tumor growth and necessarily leads to tumor escape or elimination . The predictions of our CA dormancy model can be further verified by comparing the macroscopic geometrical and dynamical properties of our simulated tumor in different microenvironments  to those obtained by experimental data from future animal studies. In future work we plan on incorporating recently discovered mechanisms for cancer dormancy via the clinical trials and experiments ,  to better inform our computational model. These results together could aid in answering the important fundamental question of whether the majority of cancer cells in a dormant tumor are arrested at a certain stage of the cell cycle or not. Furthermore, they will have significant treatment implications in terms of what stage of the cell cycle the therapies should target .
Besides the aforementioned influences, our findings informed by clinical data might be able to provide further insights to novel early cancer detection and therapy. For example, a new cancer drug that suppresses the emission of CD47 by the tumor tissues, which helps the tumor cells evade attack by the immune system, has been discovered . It was shown via in vitro experiments that this drug is able to kill a variety of cancer cell types. Thus, an effective clinical application of this drug depends upon the ability to identify different tumor cell populations while they are dormant. Our work may serve to provide insights to the application of this new drug as well by contributing to the development of new early detection methods. In addition, our work may shed light on why the immune system may not always be able to prevent tumor progression. Specifically, our work shows that even if the immune system maintains its strength throughout the tumor growth process, there is still a high possibility that the immune system could eventually fail, which is to be contrasted with the simple explanations that it becomes weaker as the tumor develops –. Also, for a tumor of a specific type, we can extract the parameter values in our model by fitting our simulation results to the statistics of a real in-vitro or in-vivo tumor of this type. Then we can utilize our model to explore optimal treatment strategies for the tumors of this specific type. In addition, once we determine the effects of a specific microenvironmental factor (e.g., specific integrins ) on the parameter values in our model, we could then study the effects of this microenvironmental factor on the tumor growth dynamics.
Our current CA dormancy model is still preliminary, and to achieve our ultimate goal of understanding cancer dormancy and progression, we need to develop robust models that incorporate appropriate cell-level tumor-host interactions that are informed by experiments. For example, by explicitly considering angiogenesis and using more realistic distribution of “transformed” tumor cells' resistance to microenvironmental suppression factors (currently we just divide the “transformed” cells into two types with respect to their resistance to microenvironmental suppression factors: weak and strong), our model might be able to yield more realistic results and improve our understanding of cancer dormancy and progression. Also, the effects of tumor cell competition, cooperation and the microenvironmental changes caused by tumor cell activities could be incorporated to further strengthen our CA dormancy model . It is noteworthy that although we employ interaction rules based on a discrete cell model to describe “competition” between the tumor and the microenvironmental suppression factors, alternatives such as evolutionary game theory implemented by partial-differential equations are also available to address the interplay between the tumor and the microenvironment .
S. T. is grateful to Paul Davies for bringing his attention to the cancer dormancy problem.
Conceived and designed the experiments: DC YJ ST. Performed the experiments: DC YJ ST. Analyzed the data: DC YJ ST. Contributed reagents/materials/analysis tools: DC YJ ST. Wrote the paper: DC YJ ST.
- 1. Almog N (2010) Molecular mechanisms underlying tumor dormancy. Cancer Lett 294: 139–146.
- 2. Aguirre-Ghiso JA (2007) Models, mechanisms and clinical evidence for cancer dormancy. Nat Rev Cancer 7: 834–846.
- 3. Yachida S, Jones S, Bozic I, Antal T, Leary R, et al. (2010) Distant metastasis occurs late during the genetic evolution of pancreatic cancer. Nature 467: 1114–1117.
- 4. Karrison TG, Ferguson DJ, Meier P (1999) Dormancy of mammary carcinoma after mastectomy. J Natl Cancer I 91: 80–85.
- 5. Pfitzenmaier J, Ellis WJ, Arfman EW, Hawley S, Mclaughlin PO, et al. (2006) Telomerase activity in disseminated prostate cancer cells. BJU Int 97: 1309–1313.
- 6. Weckermann D, Müller P, Wawroschek F, Harzmann R, Riethmüller G, et al. (2001) Disseminated cytokeratin positive tumor cells in the bone marrow of patients with prostate cancer: detection and prognostic value. J Urol 166: 699–704.
- 7. Aguirre-Ghiso JA, Kovalski K, Ossowski L (1999) Tumor dormancy induced by downregulation of urokinase receptor in human carcinoma involves integrin and mapk signaling. J Cell Biol 147: 89–104.
- 8. Naumov GN, Bender E, Zurakowski D, Kang SY, Sampson D, et al. (2006) A model of human tumor dormancy: an angiogenic switch from the nonangiogenic phenotype. J Natl Cancer I 98: 316–325.
- 9. Naumov GN, Townson JL, MacDonald IC, Wilson SM, Bramwell VH, et al. (2003) Ineffectiveness of doxorubicin treatment on solitary dormant mammary carcinoma cells or late-developing metastases. Breast Cancer Res Tr 82: 199–206.
- 10. Aguirre-Ghiso JA, Ossowski L, Rosenbaum SK (2004) Green fluorescent protein tagging of extracellular signal-regulated kinase and p38 pathways reveals novel dynamics of pathway activation during primary and metastatic growth. Cancer Res 64: 7336–7345.
- 11. Kusumbe AP, Bapat SA (2009) Cancer stem cells and aneuploid populations within developing tumors are the major determinants of tumor dormancy. Cancer Res 69: 9245–9253.
- 12. Hanahan D, Folkman J (1996) Patterns and emerging mechanisms of the angiogenic switch during tumorigenesis. Cell 86: 353.
- 13. Baeriswyl V, Christofori G (2009) The angiogenic switch in carcinogenesis. Semin Cancer Biol 19: 329–337.
- 14. Krahenbuhl JL, Lambert Jr LH, Remington JS (1976) The effects of activated macrophages on tumor target cells: escape from cytostasis. Cell Immunol 25: 279–293.
- 15. Weinhold KJ, Goldstein LT, Wheelock E (1979) The tumor dormant state. Quantitation of l5178y cells and host immune responses during the establishment and course of dormancy in syngeneic dba/2 mice. J Exp Med 149: 732–744.
- 16. Matsuzawa A, Takeda Y, Narita M, Ozawa H (1991) Survival of leukemic cells in a dormant state following cyclophosphamide-induced cure of strongly immunogenic mouse leukemia (dl811). Int J Cancer 49: 303–309.
- 17. Zou W (2005) Immunosuppressive networks in the tumour environment and their therapeutic relevance. Nat Rev Cancer 5: 263–274.
- 18. Finn OJ (2006) Human tumor antigens, immunosurveillance, and cancer vaccines. Immunol Res 36: 73–82.
- 19. Uhr JW, Scheuermann RH, Street NE, Vitetta ES (1997) Cancer dormancy: opportunities for new therapeutic approaches. Nat Med 3: 505–509.
- 20. Marches R, Scheuermann R, Uhr J (2006) Cancer dormancy from mice to man: A review. Cell Cycle 5: 1772–1778.
- 21. Quesnel B (2008) Dormant tumor cells as a therapeutic target? Cancer Lett 267: 10–17.
- 22. Byrne HM (2010) Dissecting cancer through mathematics: from the cell to the animal model. Nat Rev Cancer 10: 221–230.
- 23. Burton AC (1966) Rate of growth of solid tumours as a problem of diffusion. Growth 30: 157–176.
- 24. Greenspan H (1972) Models for the growth of a solid tumor by diffusion. Stud Appl Math 51: 317–340.
- 25. Owen MR, Byrne HM, Lewis CE (2004) Mathematical modelling of the use of macrophages as vehicles for drug delivery to hypoxic tumour sites. J Theor Biol 226: 377–391.
- 26. Smallbone K, Gavaghan DJ, Gatenby RA, Maini PK (2005) The role of acidity in solid tumour growth and invasion. J Theor Biol 235: 476–484.
- 27. Jiang Y, Pjesivac-Grbovic J, Cantrell C, Freyer JP (2005) A multiscale model for avascular tumor growth. Biophys J 89: 3884–3894.
- 28. Quaranta V, Rejniak KA, Gerlee P, Anderson AR (2008) Invasion emerges from cancer cell adaptation to competitive microenvironments: quantitative predictions from multiscale mathematical models. Semin. Cancer Biol 18: 338–348.
- 29. Anderson AR (2005) A hybrid mathematical model of solid tumour invasion: the importance of cell adhesion. Math Med Biol 22: 163–186.
- 30. Kim Y, Stolarska MA, Othmer HG (2007) A hybrid model for tumor spheroid growth in vitro i: theoretical development and early results. Math Mod Meth Appl S 17: 1773–1798.
- 31. Page K, Uhr J (2005) Mathematical models of cancer dormancy. Leukemia Lymphoma 46: 313–327.
- 32. Kuznetsov VA, Knott GD (2001) Modeling tumor regrowth and immunotherapy. Math Comput Model 33: 1275–1287.
- 33. Lefever R, Horsthemke W (1979) Bistability in fluctuating environments. implications in tumor immunology. Bull Math Biol 41: 469–490.
- 34. De Angelis E, Lods B (2008) On the kinetic theory for active particles: A model for tumor–immune system competition. Math Comput Model 47: 196–209.
- 35. Brazzoli I, De Angelis E, Jabin PE (2010) A mathematical model of immune competition related to cancer dynamics. Math Methods Appl Sci 33: 733–750.
- 36. Bellomo N, Delitala M (2008) From the mathematical kinetic, and stochastic game theory to modelling mutations, onset, progression and immune competition of cancer cells. Phys Life Rev 5: 183–206.
- 37. Takayanagi T, Kawamura H, Ohuchi A (2006) Cellular automaton model of a tumor tissue consisting of tumor cells, cytotoxic t lymphocytes (ctls), and cytokine produced by ctls. Inf Media Technol 1: 780–786.
- 38. Kansal A, Torquato S, Harsh Iv G, Chiocca E, Deisboeck T (2000) Cellular automaton of idealized brain tumor growth dynamics. Biosystems 55: 119–127.
- 39. Schmitz JE, Kansal AR, Torquato S (2002) A cellular automaton model of brain tumor treatment and resistance. Comput Math Methods M 4: 223–239.
- 40. Torquato S (2011) Toward an ising model of cancer and beyond. Phys Biol 8: 015017.
- 41. Jiao Y, Torquato S (2011) Emergent behaviors from a cellular automaton model for invasive tumor growth in heterogeneous microenvironments. PLoS Comput Biol 7: e1002314.
- 42. Jiao Y, Torquato S (2012) Diversity of dynamics and morphologies of invasive solid tumors. AIP Adv 2: 011003–011003.
- 43. Torquato S (2002) Random heterogeneous materials: microstructure and macroscopic properties: Springer-Verlag.
- 44. Jiao Y, Torquato S (2013) Evolution and morphology of microenvironment-enhanced malignancy of three-dimensional invasive solid tumors. Phys Rev E 87: 052707.
- 45. Saudemont A, Quesnel B (2004) In a model of tumor dormancy, long-term persistent leukemic cells have increased B7-H1 and B7.1 expression and resist CTL-mediated lysis. Blood 104: 2124–2133.
- 46. Bos PD, Rudensky AY (2012) Treg cells in cancer: a case of multiple personality disorder. Sci Transl Med 4: 164fs44–164fs44.
- 47. Blatner NR, Mulcahy MF, Dennis KL, Scholtens D, Bentrem DJ, et al. (2012) Expression of rorγt marks a pathogenic regulatory t cell subset in human colon cancer. Sci Transl Med 4: 164ra159–164ra159.
- 48. Hoffman HJ, Becker L, Craven M (1980) A clinically and pathologically distinct group of benign brain stem gliomas. Neurosurgery 7: 243–248.
- 49. Mahaley Jr MS, Mettlin C, Natarajan N, Laws Jr ER, Peace BB (1989) National survey of patterns of care for brain-tumor patients. J Neurosurg 71: 826–836.
- 50. England DM, Hochholzer L, McCarthy MJ (1989) Localized benign and malignant fibrous tumors of the pleura: a clinicopathologic review of 223 cases. Am J Surg Pathol 13: 640–658.
- 51. Seewaldt V (2014) ECM stiffness paves the way for tumor cells. Nat Med 20: 332–333.
- 52. Kleffel S, Schatton T (2013) Tumor dormancy and cancer stem cells: two sides of the same coin? Adv Exp Med Biol 734: 145–179.
- 53. Matsushita H, Vesely MD, Koboldt DC, Rickert CG, Uppaluri R, et al. (2012) Cancer exome analysis reveals a T-cell-dependent mechanism of cancer immunoediting. Nature 482: 400–404.
- 54. Kloxin AM, Tibbitt MW, Kasko AM, Fairbairn JA, Anseth KS (2010) Tunable hydrogels through controlled photodegradation for external manipulation of the cell microenvironment. Adv Mater 22: 61–66.
- 55. Barkan D, Green JE (2011) An in vitro system to study tumor dormancy and the switch to metastatic growth. J Vis Exp 54: e2914.
- 56. Chaterjee M, van Golen KL (2011) Breast cancer stem cells survive periods of farnesyl-transferase inhibitor-induced dormancy by undergoing autophagy. Bone Marrow Res 2011: 362938.
- 57. Wells A, Griffith L, Wells JZ, Taylor DP (2013) The dormancy dilemma: quiescence versus balanced proliferation. Cancer Res 73: 1–6.
- 58. Willingham SB, Volkmer JP, Gentles AJ, Sahoo D, Dalerba P, et al. (2012) The cd47-signal regulatory protein alpha (sirpa) interaction is a therapeutic target for human solid tumors. Proc Natl Acad Sci U.S.A. 109: 6662–6667.
- 59. Barkan D, El Touny LH, Michalowski AM, Smith JA, Chu I, et al. (2010) Metastatic growth from dormant cells induced by a col-I-enriched fibrotic environment. Cancer Res 70: 5706–5716.
- 60. Klein CA (2013) Selection and adaptation during metastatic cancer progression. Nature 501: 365–372.
- 61. Wu A, Liao D, Tlsty TD, Sturm JC, Austin RH (2014) Game theory in the death galaxy: interaction of cancer and stromal cells in tumour microenvironment. Interface Focus 4: 20140028.