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Advances in Complex Systems, Vol. 5, Nos. 2 & 3 (2002) 247– ©c World Scientific Publishing Company
CELLULAR AUTOMATON MODELS OF
TUMOR DEVELOPMENT: A CRITICAL REVIEW
JOANA MOREIRA∗^ and ANDREAS DEUTSCH† Zentrum f¨ur Hochleistungsrechnen, Technische Universit¨at Dresden, Dresden, Germany ∗[email protected] †[email protected]
Received 17 June 2002 Revised 30 August 2002
Cancer development can be viewed as an example of spatio-temporal pattern forma- tion. Several attempts have been made to model and predict malignant tumor behavior and also to account for immune system response and the impact of possible clinical treatments. Modeling started from a macroscopic perspective and developed towards cell-based approaches, from which cellular automaton (CA) models are an example. In this article, we first introduce the general concept of CA systems. Then, we review CA models of tumor development, focusing on avascular and vascular growth, tumor invasion and angiogenesis. Finally, a comparative analysis of the models as well as criteria for designing new CA models are provided and future perspectives are outlined. Keywords: Cellular automata; tumor growth; pattern formation; cell-based modeling.
- Introduction
Cancer arises from accumulation of usually somatic mutations occurring in an indi- vidual cell line. Through this process, mutated cells gain some competitive advan- tage over their non-malignant neighbors, being able to reproduce faster and invade territories normally reserved for other cells. In this perspective, cancer is the anti- thesis of embryological development, characterized by a non-deterministic sequence of events leading to disruption of the orderly multicellular organism architecture. The evolution of a malignant tumor typically follows three main phases corresponding to a sequence of different processes occurring in the organism: pre-neoangiogenic phase, neoangiogenic phase and invasion. During the pre- neoangiogenic phase, malignant cells acquire a phenotype that disables homeostatic responses usually present in healthy cells; malignant cells are metabolically very active and use native tissue vascularity, hence the tumor develops under hypoxia and in an acid medium. In the neoangiogenic phase, the tumor induces formation of new blood vessels from the pre-existing vasculature through chemical signals; the new vasculature enhances nutrient supply, allowing the cancer to grow auto- nomously. Invasion of neighboring tissues follows, and eventually some malignant
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cells are transported by the vascular system to other organs, where metastases of the primary cancer can appear. Cancer evolution, comprising these distinct phases, can be viewed as an example of spatio-temporal pattern formation. Several attempts have been made to model and predict malignant tumor behavior and also to account for immune response and the impact of possible therapies, in order to assist clinical treatments. The first steps towards this goal were taken in macroscopic approaches, by means of systems of ordinary or partial differential equations. In a macroscopic model, tissues are characterized by concentrations, since the scale of the model is large with respect to individual cells. Some examples of macroscopic tumor models focusing on different aspects of tumor development are those by Tracqui et al. [49], Chaplain [11], Woodward et al. [53], Ward and King [52], and Byrne and Chaplain [8]. Reviews are provided by Marusic et al. [34], Adam and Bellomo [2] and Chaplain [12]. More recent works try to explain spatio-temporal pattern formation starting from individual cells and their interactions. The fact that cancer arises from muta- tions in single cells is a strong motivation to use cell-based models to simulate, particularly, cancer development. The other important advantage is the possibility to incorporate cell-based data in such models. Currently, a standard cell-based model does not exist. In many variants of inter- acting particle systems, “particles” (cells) can, alternatively, move through conti- nuous space or a fixed lattice. Examples of lattice-free models are those by Drasdo [17], Iori et al. [29] and Stott et al. [47]. Lattice models are frequently called cellular automata and we shall review them in this article. We first introduce the cellular automaton concept in its basic form and possible extensions that are particularly relevant to tumor modeling. Then, we review typical examples of cellular automaton models of tumor growth, invasion and angiogenesis. Finally, a comparative discussion of the models, especially concerning biological functions and their implementation, is provided.
- The Cellular Automaton Concept
Cellular automata (CA) are defined as a class of spatially and temporally discrete, dynamical systems based on local interactions [6, 37, 54, 55]. These systems seem to incorporate many features of self-organizing complex systems and hence they have been applied numerous phenomena in physics [13, 15, 42], chemistry [4] and biology [14, 23]. The main characteristics of CA are:
(i) discrete space: the space consists of a one-, two- or three-dimensional regular lattice, defining lattice cellsa^ (see Fig. 1); (ii) discrete states: each site takes one of a finite number of states;
aOr lattice sites, as we shall refer to in this article to avoid confusion with biological cells.
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implement vascular tissue explicitly, although they may take into account nutri- ent supply through diffusion.
3.1.1. Pioneer models
Some of the earliest models were proposed by W. D¨uchting [19, 20], with the goal to design a model to study the regulation of disturbed cell renewal, through the analysis of two competing populations of cells. The model has the following features:
Lattice:
- A two-dimensional regular 10 × 10 square lattice is used.
- A von Neumann neighborhood is considered.
States:
- Discrete states — each lattice site corresponds to a biological cell; if a cell dies the lattice site becomes empty.
Rules:
- Transition rules are deterministic and local. A cell can survive, divide or die.
- The update is asynchronous.
The model simulates the number and configuration of the cells within the lattice. Two clones of cells (with normal and fast growth) can coexist in competition. Simulations of normal cells’ growth as well as simulations of both normally and rapidly proliferating (malignant) cells together are performed. Surgical removal of cells is also simulated. The results are not compared with experimental data, which were missing at the time this model was proposed. The model suffers from the small computational power existing at that time, which limited the lattice size. Since there is at most one cell per site, the number of biological cells is unrealistically small if one desires to model beyond very early stages of tumor development. Furthermore, the only biological processes considered are mitosis and death, which are implemented through transition rules that do not express biological interactions. An update to this model is presented in Ref. 21, in which the lattice is en- larged to 100 × 100 sites and extended rules are introduced. In Ref. 22, a three- dimensional model of 40× 40 ×40 cells is proposed. A nutrient medium is considered and cell-cycle phases are taken into account, with durations based on cell-kinetic data. The process of necrosis and a resting phase are also included. The results are in good agreement with experimentally observed data from tumor spheroids grown in vitro (according to Sutherland et al. [48], Folkman and Hochberg [24] and Carlsson [9]).
Cellular Automaton Models of Tumor Development 251
3.1.2. Tumor-immune system interaction and Gompertz growth
The model by Qi et al. [41] tries to explain the Gompertz growth curveb^ which characterizes the growth behavior of some tumors. It includes the immune system surveillance effect, and can be characterized as follows:
Lattice:
- A two-dimensional regular square lattice represents the tissue.
- A von Neumann neighborhood is considered.
States:
- Four discrete states — two corresponding to tumor cells, alive or dead, one to the normal cells and another to “complexes” (an immune cell interacting with a cancer cell).
Rules:
- Rules are probabilistic. Cells can proliferate, interact with the immune system and dissolve.
- Transition rules are non-homogeneous, non-local and not constant in time. The proliferation probability decreases as the total number of tumor cells increases, to simulate nutrient consumption.
- The update is synchronous.
The effect of mechanical pressure is described by introducing an anisotropy in the system; whenever a malignant cell divides, the daughter cell will always occupy the vacant adjacent site that is closer to the centre of the lattice (where the first malignant cells are placed, in the simulations); only when the density of cells is high enough may the cell occupy an outward site. The maximum number of cancer cells is arbitrarily chosen to avoid the tumor reaching the border of the lattice. The results given by the simulations fit the Gompertz curve. The rules of this model do not respect CA locality — the proliferation rate depends on the total number of malignant cells, in a way that the model is biased to reach a saturation with respect to the number of cells, resembling the Gompertz curve. Additionally, the dissolution of cells does not mimic real biological behavior, since dead tumor cells tend to accumulate, forming a necrotic core.
bThe Gompertz model is a phenomenological law. It gives the volume of the tumor, V , with respect to time, t:
V = V 0 e AB^ (1−e−Bt^ )^ ,
where V = V 0 is the initial volume and A and B are constants that can be fitted to agree with experimental data.
Cellular Automaton Models of Tumor Development 253
States:
- There are three discrete states corresponding to possible types of malignant cells: proliferating cells, quiescent cells and necrotic cells. Non-cancerous cells corre- spond to empty sites of the lattice.
- Each lattice site corresponds to several biological cells.
Rules:
- Rules are probabilistic. The contents of lattice sites can proliferate, become qui- escent or necrotic with certain probabilities.
- Transition rules are not local. The proliferation probability depends on the posi- tion of the site: only lattice cells within a given distance from the surface of the tumor are able to divide. This distance is a global quantity, depending on the overall tumor radius.
- The update is synchronous.
The structure of the tumor is a priori introduced into the model, through a choice of input parameters that quantitatively fit experimental situations. The simulations presented show that the outputs of the model, namely the average overall tumor radius, the proliferating rim thickness as well as the necrotic frac- tion agree with experimental and clinical data (taken from Refs. 5, 10, 26 and 40. Further extended references are provided in Ref. 30). This model disregards the homogeneity property of CA. The lattice is not regular and the density of sites varies, so each lattice site represents a different number of biological cells. The transition rules are neither local nor homogeneous. The authors introduce heterogeneity in order to simulate a nutrient gradient pointing from inside the tumor to the outside medium. This may not be a realistic assumption, since the necrotic core is formed by dead cells that do not consume nutrients; so if nutrients are not consumed whilst still diffusing, the proposed gradient would only last a short time after the cells have become necrotic.
3.1.4. Multicellular spheroid growth
Recently, Dormann and Deutsch [16] proposed a model of avascular tumor growth as a self-organized system, with the following characteristics:
Lattice:
- A two-dimensional regular square lattice with 200×200 sites represents the tissue.
- A von Neumann neighborhood is considered.
States:
- Ten discrete states — each lattice site can accommodate two types of cells, namely tumor or necrotic cells that have an orientation expressed by four velocity chan- nels and one resting channel (Fig. 3).
254 J. Moreira and A. Deutsch
c 4
c 3
c 1
c 5
c 2 r
Fig. 3. Example of a cell configuration at a lattice site r. The dark gray filled circle and the light gray filled circle denote the presence of a tumor and a necrotic cell, respectively (from Ref. 16, courtesy of A. Deutsch).
- Each channel can be occupied by at most one cell exclusion principle.
- Two continuous fields are considered: a chemotactic signal emitted by necrotic cells and nutrients.
Rules:
- Rules are probabilistic. Cells can be quiescent, proliferate, die or become necrotic with given probabilities.
- Transition rules are local. The probabilities of mitosis, necrosis and apoptosis depend on nutrient concentration and local cell density. Cancer cells are attracted by the chemotactic signal produced by necrotic material.
- The update is synchronous.
- The necrotic signal and nutrients diffuse.
The model is scaled using cytological kinetic parameters taken from the litera- ture. Simulations and statistical analysis of the results are presented (Fig. 4). The CA model reproduces experimental results of multi-cellular spheroid growth studies [24, 25]. In particular, the model is able to simulate the emergence of the layered tumor structure based solely on local rules. Manipulations to simulate surgical removal and changes in cell properties are also performed. Further extensions of the model could incorporate interactions with the immune system, cellular heterogeneity and additional signalling molecules as growth and suppression factors.
3.2. Vascular growth models
A model of tumor growth to examine the roles of native tissue vascularity and anaerobic tumor metabolism on the growth and invasion efficacy of pre- neoangiogenic tumors was proposed by Patel et al. [39]. Its main characteristics are:
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Rules:
- Rules are deterministic. Transition to a death state, survival in quiescent state or mitosis depends on upper and lower thresholds of pH and glucose levels at each site. Furthermore, mitosis can only occur if a neighboring cell is vacant.
- Transition rules are local.
- The update is asynchronous. To overcome the large disparity between time scales of cell proliferation and chemical diffusion, the change in cellular distribution is considered as a perturbation when solving the diffusion equations for H+^ and glucose. The update is carried out simultaneously on a randomly chosen subset of cells, i.e. a fraction, f , of the total cell number, with f = 0.1. Then, the equilibrium diffusion equations for glucose and H+^ are solved. The process is repeated 1/f times until all automaton cells have been updated.
The model is scaled according to biological data. The microvessel automaton elements are randomly distributed throughout the lattice. Simulations to observe the effects of malignant cell metabolism, i.e. H+^ production, and vascular density on tumor evolution and morphology are performed and statistically analyzed. The results can reproduce clinically observed tumor morphologies (Fig. 5) [28] and give
Fig. 5. Different tumor morphologies obtained with the simulations, by varying the malignant cells’ phenotype and their vascular environment. Four tumors that have been growing for the same time, each starting from the same initial size, are depicted. In (a) and (b) the vascular density and metabolism are the same, however in (a) tumor quiescence is admitted and in (b) it is suppressed. Note that a growth rate enhancement has been obtained by the tumor in (b) at the expense of necrosis suffered throughout. In (c) the vessel density and acid production rate have been lowered, while keeping tumor quiescence suppressed, resulting in necrosis confined to central cores. In (d) the vascularity is lowered further, resulting in a tumor that initially grows but soon self-poisons, surviving only in cords around blood vessels. Presumably, the acquisition of additional blood supply by the tumor in (d) through neoangiogenesis would restore its aggressive growth (from Ref. 39, courtesy of A. Patel).
Cellular Automaton Models of Tumor Development 257
insights concerning the effects of pH level and vascularity on tumor growth that can be related to experimental observations [1, 33, 38, 51]. These conclusions might have therapeutic implications in the future. The above model can be further developed to include oxygen diffusion, capillary destruction and creation, cell heterogeneity and additional chemical fields which can allow one to simulate therapy. The absence of cell motility forces mitosis to be strongly dependent of site vacancy, and also adhesion properties of malignant cells have not been considered in the present model.
3.3. Modeling the morphology of invasive growth patterns
Smolle and Stettner [43] published a study to address the question of how histo- logical patterns of tumors relate to specific functional cell properties. The model has the following characteristics:
Lattice:
- A two-dimensional regular 100 × 300 square lattice represents the tissue.
- A Moore neighborhood is considered.
States:
- There are two discrete states — if a site is occupied, it represents a tumor cell; if it is empty, it represents a stroma cell.
Rules:
- Transition rules are local.
- Rules are probabilistic: cells have a fixed probability to divide, migrate or die.
- The update is asynchronous.
- Neighborhood interactions are included in this model via growth, motility and death chemical factors originating from both tumor and stroma cells, which are considered as sources. The chemicals factors’ concentrations depend on the den- sity of sources in the local surroundings of a cell, however, their effective influence is randomly modulated at each step.
Simulations underline the importance of division, migration and death proba- bilities and that of the different chemical factors in the formation of distinct tumor patterns. The model is not time-scaled and the cytological kinetic probabilities are not taken from experimental data, hence the results are qualitative. The functional properties of malignant cells remain to be clearly specified, since the action of the chemical factors introduced does not express biological processes.
Cellular Automaton Models of Tumor Development 259
Fig. 6. Early and final steps in a simulation of the morphology of a tumor advancing front. The shape of individual cells can be observed (from Ref. 50, courtesy of S. Turner).
account. Haptotaxis is accomplished through an energy term depending on the gradient of protein concentration that varies in time according to cell interactions. Mitosis can (optionally) be included in the model.
Simulations of the invasion process are conducted (Fig. 6), in which the different parameters related to cell-cell adhesion, cell-ECM adhesion and haptotaxis, vary. The maximum depth of invasion is used to quantify the effect of these parameters in tumor invasiveness and to establish the relative importance and eventual interaction between them. Most of the simulations presented do not include cell proliferation. Since one of the main features of cancer cells is their high mitotic potential, it is arguable if the model can faithfully reproduce the biological process. While the simulations that include proliferation are a step ahead, only the following activities are considered: motility and adhesion, division, growth and haptotaxis. The model does not take into account important features like nutrient consumption, death, signalling or dif- ferentiation — the addition of such possible biological states and behaviors that malignant cells can, in fact, adopt is yet a question to be addressed. This model is motivated by the fact that minimization of the energy associated with cell surface interactions drives reorganization of cell aggregates [18, 35], as confirmed by experimental observations [44–46].
3.4. Angiogenesis
A model of angiogenesis based on the mechanism of endothelial cells (EC) migration influenced by gradients of fibronectin and tumor angiogenesis factors (TAF)g^ was proposed by Chaplain and Anderson [3]. Its characteristics are:
gFibronectin is a component of the ECM that enhances EC adhesion to collagen in the ECM. TAF are secreted by tumor cells and constitute a chemotactic signal for the EC.
260 J. Moreira and A. Deutsch
Lattice:
- A two-dimensional regular lattice with 200 × 200 sites represents the tissue.
- A von Neumann neighborhood is considered.
States:
- Two discrete states corresponding to presence or absence of an EC sprout.
- Continuous fields of EC density, fibronectin and TAF concentrations are con- sidered.
Rules:
- Rules that govern migration are probabilistic. Branching or anastomosish^ by capillary sprouts can occur and are controlled by deterministic rules.
- Transition rules are local. Branching depends on neighboring space occupancy, EC density relative to a threshold level of TAF and a minimum time interval be- fore new branching occurs. Anastomosis can happen below a certain level of TAF.
- The fields of EC, fibronectin and TAF are described by coupled partial differential equations. The discretized equation for the EC diffusion is solved on a square lattice — five coefficients proportional to EC density at each site and its neighbors are evaluated and the probability of migration is chosen proportional to these coefficients.
- The update is synchronous.
Some parameters of the model are scaled according to biological data. Simula- tions to observe the mechanism of EC migration under the influence of fibronectin and TAF gradients are performed (Fig. 7). The results can reproduce experiments of solid tumor implants in the cornea of animals [27, 36].
x
y
t=3.
0 0.2 0.4 0.6 0.8 1
0
1 x
y
t=7.
0 0.2 0.4 0.6 0.8 1
0
1
Fig. 7. Simulation of the spatio-temporal evolution of a capillary network. The plots show migra- tion of EC sprouts under influence of haptotaxis and chemotaxis (a linear source of tumor cells is considered in x = 1). Branching and anastomosis is visible (from Ref. 3, courtesy of A. Anderson).
hAnastomosis is the formation of capillary loops by newly forming sprouts.
262 J. Moreira and A. Deutsch
A fixed outward nutrient gradient is assumed in Ref. 30. Nutrient consumption is explicitly modeled in Refs. 16 and 39 in a discrete approximation to the diffusion equation.
Probabilistic rules are the basis for cell migration in Ref. 43. Haptotaxis along with minimization of energy related to cell shape and adhesion are used to determine cell motility in Ref. 50. More explicitly, velocity can be assigned to every cell, and influenced by cell adhesion, pressure and chemotaxis [16].
This property is explicitly simulated in Ref. 50 through the energy associated with cell-cell surface interaction. In Ref. 16 adhesion is indirectly considered through the particular choice of occupation rules which determine the configuration of the velocity channels in each site; since tumor cell adhesion decreases when cells are in contact with necrotic material, the necrotic cells are preferably placed at the resting channels and tumor cells at the remaining velocity channels.
The effect of pressure is implemented through a non-local rule in Ref. 41. In Ref. 16, pressure is considered to be proportional to local cell density which directly in- fluences occupation rules. Another implementation is through a mechanical energy term which considers cell deformation due to expansion or compression [50].
- Cellular metabolism (pH level )
The level of tumor cell metabolism is assumed to directly correspond to the pH level in Ref. 39; it determines normal and malignant cell survival through definition of biologically based intervals of pH level, inside of which cells survive.
- Chemotaxis and haptotaxis
Chemotaxis is explicitly modeled in Ref. 16 through discretization of the diffusion equation for the chemotactic signal concentration. The signal, emitted by necrotic cells, influences the distribution of cells in the velocity channels, i.e. it provides a cue for directed, active movement. Haptotaxis is simulated in Ref. 50 through a discrete field of extracellular matrix protein concentration that changes in time depending on the presence of malignant cells; whenever a cell moves throughout the lattice, it is assumed that there is a linear relationship between the protein gradient it experiences and local energy change — thus, by requiring energy minimization, directed motility is induced.
Cellular Automaton Models of Tumor Development 263
(a) 50 days 60 days 100 days
(b) 50 days 60 days 70 days
(c) 50 days 60 days 100 days
Fig. 8. Simulation of various treatments: (a) After 50 days one half of the tumor is cut; the tumor recovers from this surgery. (b) After 50 days the cell-cell adhesion is lowered. (c) After 50 days the necrosis rate is magnified by a factor 10; tumor cells still survive (from Ref. 16, courtesy of A. Deutsch).
The presence of vascular tissue is simulated in Ref. 39 by assigning some lattice sites as microvessels, where special boundary conditions, accounting for vessel per- meability and serum levels of the glucose and H+ concentration fields must be fulfilled.
Some important processes have not been included in mathematical models so far; in particular, mutations, metastasis formation, and the effects of angiogenesis in tumor growth have never been considered. The following criteria should be taken into account when building a new CA model for tumor development. First, it should be comparable with existing CA models in terms of the mathematical structure. The choice of the CA type depends on the features one wants to include in the model; a hybrid model is necessary
Cellular Automaton Models of Tumor Development 265
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