Emerging Patterns in Tumor Systems: Simulating the ... - Science Direct

J. theor. Biol. (2002) 219, 343–370 doi:10.1006/yjtbi.3131, available online at http://www.idealibrary.com on

Emerging Patterns in Tumor Systems: Simulating the Dynamics of Multicellular Clusters with an Agent-based Spatial Agglomeration Model Yuri Mansuryw, Mark Kimuraz, Jose Lobozy and Thomas S. Deisboeck*Oz wHarvard-MIT Data Center, Harvard University, Cambridge, MA 02138, U.S.A. zGraduate Field of Regional Science, Cornell University, Ithaca, NY 14850, U.S.A. yBios Group, Santa Fe, NM 87501, U.S.A. OComplex Biosystems Modeling Laboratory, Harvard-MIT (HST) Athinoula A. Martinos Center for Biomedical Imaging, HST-Biomedical Engineering Center, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. and zMolecular Neuro-Oncology Laboratory, Massachusetts General Hospital, Harvard Medical School, Charlestown, MA 02129, U.S.A. (Received on 17 June 2002, Accepted in revised form on 17 July 2002)

Brain cancer cells invade early on surrounding parenchyma, which makes it impossible to surgically remove all tumor cells and thus significantly worsens the prognosis of the patient. Specific structural elements such as multicellular clusters have been seen in experimental settings to emerge within the invasive cell system and are believed to express the systems’ guidance toward nutritive sites in a heterogeneous environment. Based on these observations, we developed a novel agent-based model of spatio-temporal search and agglomeration to investigate the dynamics of cell motility and aggregation with the assumption that tumors behave as complex dynamic self-organizing biosystems. In this model, virtual cells migrate because they are attracted by higher nutrient concentrations and to avoid overpopulated areas with high levels of toxic metabolites. A specific feature of our model is the capability of cells to search both globally and locally. This concept is applied to simulate cell-surface receptor-mediated information processing of tumor cells such that a cell searching for a more growth-permissive place ‘‘learns’’ the information content of a brain tissue region within a two-dimensional lattice in two stages, processing first the global and then the local input. In both stages, differences in microenvironment characteristics define distinctions in energy expenditure for a moving cell and thus influence cell migration, proliferation, agglomeration, and cell death. Numerical results of our model show a phase transition leading to the emergence of two distinct spatio-temporal patterns depending on the dominant search mechanism. If global search is dominant, the result is a small number of large clusters exhibiting rapid spatial expansion but shorter lifetime of the tumor system. By contrast, if local search is dominant, the trade-off is many small clusters with longer lifetime but much slower velocity of expansion. Furthermore, in the case of such dominant local search, the model reveals an expansive advantage for tumor cell populations with a lower nutrientdepletion rate. Important implications of these results for cancer research are discussed. r 2002 Elsevier Science Ltd. All rights reserved. n

Corresponding author. Molecular Neuro-Oncology Laboratory, Harvard Medical School, Massachusetts General Hospital-East, Building 149, 13th Street, Charlestown, MA 02129,U.S.A. Tel.: +1-617-724-8636; fax: +1-617726-5079. E-mail address: [email protected] (T.S. Deisboeck). 0022-5193/02/$35.00/0

1. Introduction This paper proposes an agent-based search model to simulate the spatio-temporal dynamics of r 2002 Elsevier Science Ltd. All rights reserved.

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brain tumor cell invasion,* which is considered a major obstacle for effective treatment of patients suffering from this neoplasm (Nazarro & Neuwelt, 1990; Silbergeld & Chicoine, 1997; Whittle, 1996). Our framework is ‘‘agent-based’’ because the smallest unit of observation in the model is the individual tumor cell rather than the whole neoplasm. Since the migratory behavior of tumor cells is driven by their search for more viable locations to live, our framework also qualifies as a ‘‘search model.’’ We model tumor cell colonies within the invasive system as multicellular clusters in which individual cells interact among themselves and with their environment. Thus, factors governing the dynamics of the system are, first, selforganization among tumor cells themselves (intrinsic), and second, interaction between tumor cells and their environment (extrinsic). Our twodimensional (2D) model is capable of simulating brain tumor growth in sequential steps that previous works have identified: rapid proliferation and migration into neighboring brain parenchyma areas. Our approach combines local interaction of cells at the microscopic level with a macro-system perspective. This micro-macro synthesis is necessary because, on the one hand, the behavior of an individual tumor cell is not completely random and disorganized. Indeed, it can be argued that one of the key aspects of malignant disease is the ability of tumor cells for local coordination of growth and invasion (Giese et al., 1995, 1996; Deisboeck et al., 2001). On the other hand, locally interacting tumor cells may lead to self-organized networks capable of generating large-scale patterns at the macroscopic level. Thus, although focusing on the behavior of a single cell in isolation may provide valuable information about the individual cell, a multicellular analysis based on a linear extrapolation of the cells’ individual behavior does not contribute to the understanding of multicellular tumor colonies as self-organized systems (Kraus & Wolf, 1995). Such linear adding-up of individual cell behavior is invalid in the presence of the hypothesized nonlinear interactions among * We are aware of the biological difference between 3D cell invasion and 2D cell migration. In this paper, however, both tumor biology terms are used synonymously solely to describe active tumor cell locomotion.

tumor cells, and between cells and their environment. Moreover, nonlinearity would render it virtually impossible to predict the long-run dynamics of the system using a purely analytical approach. In such a case, a parallel model of computer simulation is very useful to generate predictions and to test hypotheses. In addition, since conventional in vivo imaging techniques cannot detect tumor cell colonies smaller than a few millimeters in diameter, our model aims to study the dynamics of such tumor cell colonies below the detectable limit. We will argue that such clusters of tumor colonies have significant impact in guiding overall spatiotemporal behavior of the tumor system, hence a better understanding of those microscopic foci is essential for the development of novel diagnostic tools and treatment strategies. Our model allows the density of each 2D lattice site to vary depending on how many tumor cells choose to reside in that region. Thus, in the absence of a satisfactory imaging tool to detect the spread and progression of micro-tumor foci in cancer patients, our model will prove very useful to simulate the dynamic behavior of microscopic tumor cell aggregates while still allowing the entire tumor system to grow to a macroscopic size. In the following section we will briefly describe specific experimental findings and related oncology concepts, which in turn inspired our novel computational model. 2. Experimental Findings and Modeling Concept We have developed a novel 3D in vitro assay by placing multicellular tumor spheroids (MTS) between two layers of an extracellular matrix gel. Such spheroids behave in many ways similar to non (pre)-vascularized solid microscopic tumors (Mueller-Klieser, 1997; Sutherland, 1988). The novel assay is described elsewhere in detail (Deisboeck et al., 2001). Briefly, human U87MGmEGFR glioma cells [kindly provided by Dr W. K. Cavenee (Ludwig Institute for Cancer Research, San Diego, CA); see also Nagane et al., 1996; Nishikawa et al., 1994] rapidly form MTS (500–700 mm in diameter) in culture after reaching monolayer confluence. These MTS were placed in between two layers of growth factor reduced (GFR) matrix,

EMERGING PATTERNS IN TUMOR SYSTEMS

Matrigels (BIOCATs, Becton Dickinson, Franklin Lakes, NJ), which forms a reconstituted basement membrane at room temperature. Such basement membrane gels consist of interconnected networks of extracellular matrix (ECM) proteins (Kleinman et al., 1986). The growth dynamics were recorded daily using an inverted light microscope and analysed by an online image-analysis system. Typically following a rapid growth phase, MTS showed a deceleration of volumetric growth. In parallel an emerging invasive network evolved over time. A specific pattern emerged in a pilot experiment after the addition of 10 ml conditioned medium as a (pseudo) heterotype attractor (at time t ¼ 0 hrs) into the right side of the ECM gel [symbolized by the dotted circle in Fig. 1 (left)]. Since the experimental results have been already previously reported in detail (Deisboeck et al., 2001), we will present only a brief summary here. Over 144 hrs volumetric MTS growth followed again decelerating dynamics, yet the averaged invasion area at 144 hrs of almost 6 mm2 was more than three times larger than in the ‘‘standard’’ MTS assay without additional attractor. The maximal radial extension of the invasion area toward the attractor side was more than doubled compared to the opposite control

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side. Most interestingly, specific multicellular aggregates, termed F1 and F2 clusters, emerged between 96 and 120 hrs. These clusters exhibited dynamical turnover; that is, while the MTS grew and the invasive network further advanced, F2 clusters derived spatio-temporally after F1. These F2 clusters continually increased in size in parallel to a decreasing yet still larger F1. Within the subsequent 48 hrs some cluster structures disappeared almost entirely whereas new aggregates emerged [Fig. 1 (right)]. The clusters were located within the attractor region suggesting the guiding influence of a heterogeneously diffusing single attractor site on emerging structural pattern and overall invasiveness. The average invasive edge velocity on the attractor side was 231 mm 24 hrs as compared to the velocity on the opposite control side of only 110 mm 24 hrs. The difference between attractor and control sides reached a maximum at 96 hrs, hence at the time point at which clusters began to develop. Finally, the peak velocity at the attractor side exceeded 400 mm 24 hrs, which is nearly four times higher than the peak velocity in the standard MTS assay (without attractor). Thus, in this particular experiment the nonreplenished attractor source not only guided but seemingly also accelerated the system toward its

Fig. 1. MTS assay with attractor site [10 ml conditioned medium (CM) injected into the right side of the gel (symbolized by the broken circle)] at 120 hrs (left). Note asymmetrical invasion area with elongated pathways and cluster structures. The F1 and F2 labeled arrows point toward the spatio-temporal origin of cluster sites. Within the next 48 hrs (right) the central F1 cluster has been reinforced whereas the bottom one of the two F2 clusters almost entirely disappeared at 168 hrs postplacement of the MTS. This supports the notion of a non-replenished regional attractor spot consisting of medium enriched areas within the inhomogeneous gel. However, two new clusters (arrows) emerge at sites within the F1 range, without clear indication 48 hrs earlier (original magnification: 40). (The experimental results have been previously reported in Deisboeck et al., 2001).

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site and it can be argued that multicellular clusters were involved in this process, possibly even support its dynamics. Inspired by these observations, our computational modeling framework will introduce several new features in the simulation of tumor growth and motility. First, tumor cells in our model are equipped with the intrinsic ability of performing cognitive tasks. This is not meant to be a conscious process; rather it proposes that a cell is capable of recognizing a pattern of environmental information (‘‘profile’’) and of evaluating the desirability of living in a particular location by assessing these environmental instructions (Bray, 1990). In addition, we model explicitly the interaction between tumor cells and their environment, which includes growthpromoting factors (e.g. blood vessels, nutrient supply, and oxygen), as well as growth-inhibiting factors (e.g. toxic metabolites). Growth stimuli are important because tumor cells, like other biological systems, are dissipative structures that operate far from thermodynamic equilibrium. Sources of energy thus must be present in the system to sustain further growth (Kraus & Wolf, 1995). Heterogeneity of locations due to both growth attractors and growth inhibitors also serves to capture differences in the biological microenvironment. It is well known from metastasis research that tumor cells seek to move to those locations, which have the most suitable microenvironments for their growth (Tarin, 1985). The tendency to move to favorable locations (due to e.g. the guiding principle of least resistance, most permission, and highest attraction) can be argued for locally invasive cells as well (Deisboeck et al., 2001). In this study, we model migratory behavior as a two-step search process in which locations with favorable nutrient supply and low level of toxic metabolites stimulate both cell migration to and proliferation at these sites. The two-step process consists of a global search followed by a local one. Another novel feature is the modeling of cell proliferation as a stochastic process. Proliferation is stochastic since favorable locations (with high nutrient supply and low level of toxic metabolites) are necessary but not sufficient condition for tumor cells to replicate. The higher probability of proliferation corresponds to the

higher level of nutrient supply in our model. We also model explicitly the mechanical resistance for a tumor cell to migrate toward the normal, virtual tissue on the 2D lattice since it has been shown that mechanical properties of the environment affect the pattern of both proliferation and invasion (Eaves, 1973; Helmlinger et al., 1997; Oishi et al., 1998). Finally, we model explicitly the death of tumor cells (i.e. apoptosis/ necrosis) linked with an increased level of hostile toxic metabolites per lattice site. For example, it has been demonstrated that low pH inhibits cell proliferation and contributes to cell death and necrosis within solid tumors, which in turn may induce tumor cells to migrate to more favorable locations (Tannock & Rotin, 1989; Freyer & Sutherland, 1986; Turner & Weiss, 1980). In the following section we review some relevant modeling work. 3. Previous Work A large amount of work has been devoted to the modeling of tumor growth, both using mathematical models and computer simulations. Kansal et al. (2000) simulated the growth of brain tumor using a cellular automaton model. They showed that macroscopic tumor behavior can emerge from local interactions at the microscopic level. Their model focused on volumetric growth, implicitly taking into account that invasive cells are continually shed from the tumor surface. Qi et al. (1993) also devised a cellular automaton model to simulate the growth of cancer cells in general, but they did not explicitly consider the influence of growth stimulants and inhibitors. Iwaka et al. (2000) proposed a mathematical framework to model the growth and size distribution of metastatic tumors. However, they assumed somewhat unrealistically that tumor cells live infinitely, i.e. not considering cell death. Chaplain & Sleeman (1993) developed a mathematical model of solid tumor growth employing techniques from nonlinear elasticity theory, considering the growth of tumor cells to be similar to the inflation of a balloon. A central feature of their model is the use of a strain-energy function to allow for heterogeneity of tumor cells. They specify conditions of the strain-energy function under which


invasion and proliferation of tumor cells occur. However, they did not consider location-specific factors, such as nutrient intake and toxic level, which stimulate/inhibit the growth of tumor cells. As in our paper, Delsanto et al. (2000) and Smolle & Stettner (1993) developed a mathematical model to facilitate the simulation of tumor growth and invasion. Delsanto et al. focus on the transition from tumor dormancy to latency, while Smolle & Stettner investigate the growth effects of location-specific autocrine and paracrine factors. However, they did not explicitly model how tumor cells assess the permissibility of a location. Instead, a static probability distribution is imposed exogenously to determine whether a selected cell will migrate, proliferate, or die. Such virtual cells that follow externally imposed rules cannot be ‘‘true’’ autonomous cells capable of making independent ‘‘decisions.’’ Rather, these cells obey fixed instructions that were imposed upon them externally, which is entirely different from our approach. A distinct strand of literature applies continuum mathematical models, typically based on deterministic reaction–diffusion theory, to study cell movement. For example, Tracqui (1995) formulated such a model to simulate the invasion of tumor cells, which can be used as a tool to investigate the effects of surgical resection of malignant tumors. Sherratt et al. (1993) and Sherratt (1994) modified the original Keller– Segel model of chemotaxis (i.e. movement along

Fig. 2. Initial configuration of the tumor cells.

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a chemical gradient of a soluble factor) by additionally incorporating the receptor-based mechanism, which regulates cell motion via chemical reactions. Finally, Perumpanani et al. (1996) modeled tumor invasion based on the notion that chemotaxis and haptotaxis (i.e., movement on a solid substrate) behave as traveling waves. In contrast to these continuum models, in our framework both space and time are discrete, which allows the spatio-temporal progression of brain tumors to be observed sequentially. Moreover, we avoid the shortcomings of a deterministic model by introducing stochastic elements in the behavior of tumor cells. We now describe our mathematical model in detail. 4. Mathematical Model 4.1. ALGORITHM DEVELOPMENT

The space of observation in our model F representing a virtual brain tissue slice F is a 2D lattice that contains R locations. The edges of this space lattice are joined in such a way that locations on opposite edges are connected. The population density of location j, denoted by lj, is defined as the number of tumor cells that occupy it. As the initial ‘‘seed,’’ a finite number of tumor cells is placed in a cluster located within the center of quadrant I in the square lattice (see Fig. 2). In our model the heterogeneity of locations is due to the different levels of nutrient supply,

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fj , mechanical confinement, pj, and toxicity of metabolites, tj . The initial distribution of nutrient supply, f0;j ; follows a Gaussian distribution such that the center of quadrant I has the lowest initial supply of nutrients, while the opposite center of quadrant III has the highest initial level of nutrition (see Fig. 3). The nutrient supply is non-replenished such that the nutrient level of a region will decline when tumor cells migrate toward that region and start to metabolize the limited supply. By contrast, mechanical confinement pj F representing the local tissue resistance against tumor cell movement F is dynamically fixed, but its spatial distribution resembles a Gaussian distribution with the lowest resistance to migrate at the center of quadrant I, while the center of quadrant III exhibits the highest resistance. Initiating the model in such a ‘‘least resistance’’ region is in accordance with the aforementioned tumor biology concept as it proposes that tumor cells had gathered at this lattice site at the start of the simulation. Since all tumor cells initially cluster in quadrant I, the distance between those cells and the area where nutrition is most abundant is maximal. Although tumor cells will encounter significant resistance if they try to migrate to the most growth permissive area in quadrant III, they will take that route attracted by nutrients and to avoid imminent death caused by the inevitable buildup of toxic metabolites at the initial site. At each time period, a selected cell can carry out three possible actions: (a) migration, (b) proliferation, and (c) death (i.e. apoptosis/ necrosis). The patterns of cell migration, cell proliferation, and cell death govern the dynamics of the population in a particular location. A tumor cell can migrate to another location if it is originally located on the surface of a cluster. With some positive probability, a number of tumor cells are selected to divide, while others die. The probability of a tumor cell to proliferate increases with higher levels of nutrition, fj [see eqn (10)], while the probability of cell death increases with higher level of toxic metabolites, tj [see eqn (11); see also Freyer & Sutherland, 1986]. In the non-replenished setting, both the probability of proliferation and the probability of death dynamically change depending on the

population density F and hence the pattern of migration F in the previous period (see DeHauwer et al., 1997). Figure 4 shows the emerging pattern of tumor clusters over time in a typical simulation of our model, while Fig. 5 depicts the buildup of death cells at the end of the simulation. Cell-surface receptors enable tumor cells to sense chemical gradients of ligands (e.g. nutrients, growth factor signals), which in turn can trigger cytoskeleton changes (responsible for directed motion along these gradients. See Tysnes et al., 1997). Such mechanisms enable cells to ‘‘search’’ for a more favorable location to settle. Based on the hypothesis that such signalreceptor systems with varying sensitivity are involved in guided cell movement, we propose that this search process advances in two stages: the global and the local stage. Searching in stages means that tumor cells only gradually ‘‘learn’’ the information content of a location. Cellular ‘‘learning’’ here is not meant to be a conscious mechanism, rather it is defined as information reception and processing that determine the migratory behavior of the cell in the next time step. In the global stage, a migrating tumor cell evaluates location j based on the following function: Gij ¼ f ðlj Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðr=2pÞexpðrdij2 =2Þ;

ð1Þ

where Gij is the value of location j to a tumor cell currently living in location i, lj is the number of tumor cells occupying location j, dij is the distance between locations i and j, and r represents the cost of spatial movement in terms of the required energetic-metabolic expenditure for the advancing cell. This energy expenditure is positively related to the environmental heterogeneity, which in our current model is solely represented by mechanical confinement pj such that r ¼ rr pj ;

ð2Þ

where rr is a constant that can be thought of as the intrinsic (inverse) capability of cells to migrate, which is genetically and epigenetically determined, while pj as before represents the extrinsic environmental (mechanical) resistance against migration. Low rr (i.e. high intrinsic

Fig. 3. Distribution of nutrient supplies (top) and mechanical confinement pressure profile (below). The lattice space is divided into four quadrants: frontal (south) as quadrant I, right (east) as II, distal (north) as III, and finally, left (west) as IV.

doi:10.1006 yjtbi.3131 Y. MANSURY ET AL.

Fig. 4. Cluster patterns over time. The top six snapshots derive from the simulation with rr ¼ 109, while the bottom six snapshots are from the simulation with rr ¼ 109. For each simulation, the snapshots are taken at t ¼ 0, 80, 160, 240, 320, and 400 [arbitrary units]. The figures depict the density of viable tumor cells per cluster, shown in a rainbow scale.

doi: 10.1006 yjtbi.3131 Y. MANSURY ET AL.

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capability to migrate) implies lower r (for a dynamically fixed pj) and thus lower cost of spatial movement, and vice versa. Thus, energy expenditure depends not only on mechanical resistance, but also on the individual fitness to ‘‘conquer’’ these environmental obstacles. In the future, this term will allow us to incorporate more than one cell strain, i.e. in modeling distinct cell populations, which differ in their intrinsic fitness (e.g. up/down-regulation of specific surface receptors, ‘‘downstream’’ receptor signal processing and signal amplification; production of invasive matrix-digesting enzymes). Equation (2) states that higher metabolic cost would render it more difficult for a tumor cell to move, which is an intuitive assumption given the cell is a dissipative system. In the limit, as metabolic cost approaches a very large value, no global search would occur and only local search could be performed. The underlying biological concept is that very high tissue consistency and thus high mechanical confinement, which forces migrating cells to incur high metabolic costs, also hampers the diffusion of soluble ligands, i.e. the transmission and reception of long-range signaling, leaving cells essentially with only a local information input. Global search comprises two components: the deterministicpcomponent f(lj) and the stochastic ffiffiffiffiffiffiffiffiffiffiffiffiffiffi component ðr=2pÞexpðrdij2 =2Þ: The inclusion of both a deterministic and a stochastic component implies that global search is not a pure random walk. Rather, global search is a stochastic process with a strong bias in the direction of those locations with favorable population density. In this paper, the deterministic component adopts the following form: f ðlj Þ ¼ lj clj2 :

ð3Þ

Equation (3) postulates that tumor cells prefer to aggregate with other tumor cells, but the negative coefficient of the quadratic term indicates that the tendency to cluster declines when the location is overcrowded. The parameter c thus represents the extent of competition among agglomerating tumor cells to secure a more growth-permissive location, i.e. those with higher nutrients and lower metabolic toxicity. This

notion of tumor cell agglomeration is based on the fact that tumor cells produce protein growth factors [e.g. autocrine or paracrine, e.g. transforming growth factor a (Ekstrand et al., 1991) or scatter factor (Koochekpour et al., 1997)], which in turn promote cell growth and hence may attract even more tumor cells. The stochastic component can be thought of as a Wiener Process that determines how far a tumor cell migrates. Definition. A random variable X is generated by a Wiener Process if X has stationary independent increments and X (t) B N(0,t). If we let the distance dij be a Wiener Process such that dij BNð0; 1=rÞ; then f ðdij Þ ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðr=2pÞexpðrdij2 =2Þ:

ð4Þ

Since the distance dij that a tumor cell moves is a Wiener Process, it immediately follows that t ¼ 1=r: The parameter r reflects the cost of spatial movement since larger values of r imply shorter time to migrate, and thus the scope is less for tumor cells to move to a distant location. The continuous Wiener Process serves as an approximation for the underlying discrete migratory behavior. To see this, consider a tumor cell that at any time between t and dt has the options of either stepping to the left with probability p or to the right with probability 1p. Within one unit of time, this tumor cell would take a total number of n ¼ 1=dt steps. Let n+ be the number of steps to the right, n be the number of steps to the left, and dij be the distance that this tumor cell travels in one unit of time. By definition, n++n ¼ n, and n+n ¼ dij. The probability that this tumor cell would take n+ steps to the right in one unit of time follows a binomial distribution: Pðnþ ; n; pÞ ¼

n n

þ

! þ

þ

pn ð1 pÞnn :

ð5Þ

It can be shown that (for proof, see Feller, 1957) d

Pðnþ ; n; pÞ - f ðnþ ; m; s2 Þ as n ¼ 1=dt-N;

ð6Þ

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border with it. For example, in a 2D lattice, each location is connected to its four nearest neighbors to the north, south, east, and west, as well as to its four diagonally adjacent neighbors to the northeast, northwest, southeast, and southwest.* In contrast to the global stage, ‘‘learning’’ in the second stage is local since the search space includes only those locations in the neighborhood of j, but complete. This complete set of information consists of population density and location-specific characteristics (i.e. the levels of nutrient supply, toxicity, and mechanical confinement). To a migrating tumor cell, in the local stage the value of location j depends on the characteristics of that location and its adjacent neighbors. Specifically, in the local stage a migrating tumor cell evaluates location j based on the following function:

Fig. 5. Distribution of the dead tumor cells at the end of the simulation with rr ¼ 109 (top) and with rr ¼ 109 (bottom) (corresponding to the last time point of the simulations presented in Fig. 4).

d where denotes convergence in distribution, and þ 2 ! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 n m ; f ðnþ ; m; s2 Þ ¼ ð1=2ps2 Þexp 2 s

m ¼ np; 2

s ¼ np ð1 pÞ; 2

where m and s are, respectively, the mean and variance of the probability distribution function. Since n is fixed, there is a unique pair ðnþ ; n Þ for every dij such that nþ n ¼ dij and nþ þ n ¼ n: Thus, f ðdij ; m; s2 Þ ¼ f ðnþ ; m; s2 Þ: If we further impose that m ¼ 0 (i.e. on average a tumor cell remains in the current location) and s2 ¼ 1=r; then we obtain eqn (4), which is exactly the stochastic component of eqn (1). After completing the global stage, tumor cells proceed to the local search. In the local stage, the tumor cell evaluates the neighborhood of region j to find the exact location that it will move to. The neighborhood of location j is defined as the set of all those locations that share a common

Lj ðlj ; fj ; tj ; pj Þ ¼ lj clj2 þ qf fj qt tj X qp pj þ ðli cli2 Þ;

ð7Þ

iAfj 0 s neighborhoodg

where qf 40, qt 40, and qp 40 are, respectively, the coefficients of nutrient, toxic, and mechanical resistance in the local-stage utility function. The last term of eqn (7) captures the neighborhood effect. Equation (7) postulates that tumor cells prefer abundant supply of nutrients but dislike accumulation of toxicity and higher level of resistance. Thus, the ‘‘decision’’ of tumor cells in terms of the migratory direction is influenced by both attractors and inhibitors. Tumor cells may be attracted to a location enticed by abundant nutrition and to avoid the accumulation of toxicity in its previous location. However, if the new location is not very accessible due to high mechanical confinement or if it is overpopulated, then migrating cells may decide to move elsewhere instead. The supply of nutrients is introduced as follows: supply is assumed to be non-replenished, replicating the environment in the previously described experimental in vitro setting (see Fig. 1) in which nutrients inevitably decrease as the population of tumor cells expands. The level of nutrient source at time t is modeled as the level * This type of neighborhood is called the ‘‘Moore neighborhood’’. See Hayes (1984).


of nutrient source at time tF1 less the rate of nutrient depletion, rf, multiplied by the number of cells (i.e. population density) residing in that location: ft;j ¼ ft1; j rf lt1;j :

ð8Þ

The rate of nutrient depletion represents the metabolic uptake of the tumor cells. Equation (8) implies tumor cells compete more vigorously for nutrients when the population density is higher since the depletion of nutrient sources occurs more rapidly. The rate of change of toxicity is modeled as a positive, linear function of the population density. This means that higher population density in a particular cluster region will bring about higher level of toxic metabolites in that region. Formally, tt; j ¼ tt1; j þ rt lt1; j :

ð9Þ

For proliferation to occur, two necessary conditions must be in place: (a) a tumor cell must belong to a tumor cluster and (b) the cell must be at the surface of the cluster. These conditions can be thought of as spatial restrictions on the proliferation process. The first condition mimics the experimentally proven concept that there is a dichotomy between proliferation and migration, i.e. a specific tumor cell at a given time and location exhibits only one of the key activities at maximum level, but not both (Giese et al., 1996). Thus proliferative activity should be maximal only at non-migratory (i.e. at cluster) sites. The first condition also emulates the impact of mechanical pressure F generated by an expanding cluster F that can stimulate the proliferation of tumor cells (Oishi et al., 1998). By contrast, the second condition prohibits tumor cells in a center of a cluster to proliferate. This is due to the increase of mechanical pressure inside a solid tumor that will eventually prevent the diffusion of nutrition from the surface to cells in the center of a growing cluster. For tumor spheroids, it has been shown that such nutritiondeprived cell populations halt proliferation and turn quiescent before they turn apoptotic (see Freyer & Sutherland, 1986). Locomotion is not possible for these central quiescent cells due to the fact that they are ‘‘trapped’’ within the

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interior of a compact cluster. Among the tumor cells that satisfy these necessary conditions, the probability of a tumor cell in location j to proliferate increases with higher levels of nutrition, Prproliferate;j ¼ fj =ðkf þ fj Þ;

ð10Þ

where kf is a parameter constant. Note that eqn (10) implies the probability of cell division is zero at the locations where the supply of nutrients has been completely depleted. On the other hand, the death of a cell in our model depends only on the level of local toxicity, which is more realistic for the (currently) monoclonal model than imposing a preset lifetime. Specifically, we assume that the probability of death increases when the level of toxicity tj becomes increasingly detrimental: Prdeath;j ¼ tj =ðkt þ tj Þ;

ð11Þ

where kt is a parameter constant. Note that the probability of death is zero at toxin-free locations. In other words, tumor cells always survive if their region is completely non-toxic. Equations (9) and (11) together imply that the higher the cell density of a cluster the higher the level of toxic metabolites, and therefore the more cells will eventually die within the cluster’s region (see Fig. 5). Table 1 summarizes the special (Greek) symbols that we use in our mathematical equations. In the following section, we implement our mathematical model in an operational numerical model. 4.2. ALGORITHM IMPLEMENTATION

In this section, we present the numerical technique that we employ to simulate the spatio-temporal behavior of virtual brain tumor cells. The idea is to implement the theoretical platform from the previous section with a computational tool. To that end, we built an agent-based model of virtual tumor cells with the tool of Visual C++ programming language.* As an overview, the sequence of algorithms performed in each simulation can be summarized as follows: * Microsoft Visual C++ version 5.0.

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Table 1 Summary of special symbols in the mathematical equations fj tj r m s2 qf qt qp rr rf rt kf kt

Nutrient supply at location j Level of toxicity at location j Cost of spatial cell movement Mean of probability distribution function Variance of probability distribution function Coefficients of nutrient in the local-stage utility function Coefficients of toxicity in the local-stage utility function Coefficients of mechanical resistance in the localstage utility function Intrinsic (inverse) capability of cells to migrate Rate of nutrient depletion Rate of toxic metabolite accumulation Parameter constant in the probability equation for cell proliferation Parameter constant in the probability equation for cell death

1. Initialization 2. For every period T: 2.1. Select an agent (i.e. a virtual tumor cell) randomly from the population 2.2. Determine whether this agent will die at this period 2.3. Cell death: if this agent dies, then repeat the random selection of another agent 2.4. If this agent survives, then determine whether it is eligible to proliferate 2.4.1. If this agent is eligible to proliferate, then 2.4.1.1. Proliferation (i.e. generation of offsprings or daughter cells) can occur 2.5. Determine if the agent is eligible to move 2.5.1. If this agent is eligible to move, then 2.5.1.1. Migration occurs, unless the best choice is not to move 2.6. Update the levels of nutrients and toxicity 2.7. Repeat 2.1 – 2.6 n times, where n is the number of alive agents in the population 3. End ‘‘for’’ loop It is important to note that even if an agent had been selected at period T, that agent can be selected again in the same period to proliferate and/or migrate. In other words, agents are

randomly sampled with replacement from the population pool. The details of our simulation algorithms are as follows.

4.2.1. Initialization At the beginning of every simulation, we create a torus of grid lattice (representing our space of observation), initial seeds of agents, nutrient sources, resistance to spatial movement, and toxic metabolites. The precise sequences of the initialization algorithm are as follows: (a) First, we create a 2D torus of grid lattice that contains 50 50 locations to represent a ‘‘2D’’ virtual brain tissue slice. (b) Next, we create ten agents (representing virtual tumor cells), then place them all at the center of quadrant I (see Fig. 2). (c) For each location in the grid, we create a source of nutrients, fj ; whose level is randomly distributed but weighted by the distance of that cell from the center of quadrant III. The center of the third quadrant is assigned the highest weight and thus is typically where the peak nutrient level is, but not necessarily so because of the random element. The creation of nutrient sources is implemented in four steps for every location j in the torus. First, a random number, uj, whose value is between zero and one, is generated from a uniform random-number generator. Second, the distance of that location from the center of quadrant III, dj, is computed using L-infinity metric of measurement.* Third, having computed the distance of location j from the center of the third quadrant, the weight of that location is computed as wj ¼ exp (2d2j /s2f), where sf is the parameter controlling the dispersion of nutrient level. Finally, at every location we compute the level of nutrients, fj ; as the random number generated in step one multiplied by the weight of that location: fj ¼ uj wj : (d) For each location in the grid, we initialize the mechanical resistance (i.e. confinement) * Formally, given any two points in the 2D lattice with coordinates (xi, yi) and (xj, yj), the distance between these two points is computed as dij ¼ Max(abs (xjFxi), abs (yjFyi)).


against migration to that location, pj, whose level is randomly distributed but weighted by the distance of that cell from the center of quadrant III. The center of the third quadrant has the highest weight, and thus it is typically where the peak resistance is, but again not always the case because of the stochastic element. The initialization of resistance level is implemented in four steps for every location j in the torus. Steps 1 and 2 are as in the creation of nutrition sources above. In the third step, the weight of that location is computed as wj ¼ exp (2d2j /s2p), where sp is the parameter that controls the dispersion of resistance level. Finally, the level of resistance in location j is computed as the random number generated in step one multiplied by the weight of that location: pj ¼ ujwj. (e) For each location j in the grid, the level of toxicity is initialized equal to zero: tj ¼ 0:

4.2.2. Cell Death At every period T, we select an agent randomly, and then we determine whether it dies in this period. The algorithm is as follows: (a) Select an agent, and identify the location of that agent, say location j. (b) Generate a random number, u, whose value is between 0 and 1 based on the uniform distribution function. (c) Compute the threshold probability to die in location j, Prdeath,j, using eqn (11). (d) If u happens to be less than Prdeath,j, then this agent dies. 4.2.3. Cell Proliferation

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(d) If u happens to be less than Prproliferate,j, then that agent proliferates. (e) Place the new daughter cell in the same location j where the parent agent resides. In the current algorithm implementation, we place no limit on how many daughter cells can reside in the same grid site. This implies cells can be of arbitrarily small size due to compression. 4.2.4. Cell Migration An agent decides where to migrate based on two stages of search: global and local. The search algorithm is implemented as follows: (a) The selected agent searches globally by scanning all locations in the grid torus, then determines the best location based on the globalvalue function (see eqn (1)). Denote the coordinates of the best location based on global search as (xg, yg). (b) Next, this agent searches locally by examining the Moore neighborhoods of (xg, yg), including (xg, yg) itself. This agent then selects the best location to migrate based on the local-value function (see eqn (7)). Denote the coordinates of the best location based on local search as (xn, yn). (c) The agent migrates to coordinate (xn, yn). 4.2.5. Update of Nutrients and Toxicity After an agent migrates from location a to location b, the levels of nutrients and toxic metabolites in both locations a and b are updated in the following way:

If an agent survives, then we determine whether it will proliferate using the following algorithm:

(a) Update the level of nutrients in both locations a and b using eqn (8). (b) Update the level of toxic metabolites in both locations a and b using eqn (9).

(a) Determine whether this agent is located on the surface of a cluster. (b) If the agent is on the surface of a cluster, then generate a random number, u, whose value is between 0 and 1 based on the uniform distribution function. (c) Compute the threshold probability to proliferate in that agent’s location j, Prproliferate,j, using eqn (10).

In updating the level of nutrients, it is important to recognize that the application of eqn (8) in the current algorithm implementation implies that nutrient sources are non-replenishable. That is, the level of nutrients at any given location can either decline due to cellular consumption or remain the same as in the previous period if the location harbors no cell(s) during that period. In the next section, we

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examine the results of our numerical simulations. 5. Results This section is organized into two parts. In the first part of our simulation, designated as benchmark simulation, the following parameter values, listed in Table 2, are held constant. (The values are chosen arbitrarily, which is a valid approach as the focus of this first paper is to show the rich qualitative behavior of the model and not to investigate its predictive power using specific experimental values). We ran the simulation for various values of the intrinsic (inverse) capability of a cell to migrate, rr. In the second part, we examine the robustness of the results from our benchmark simulation by varying both rr and the rate of nutrient depletion, rf [see eqn (8)]. We specifically choose rf since the metabolic uptake of tumor cells is an important parameter of interest from the experimental standpoint. Each simulation was run until all virtual tumor cells had died. Because the rates of nutrient depletion and toxic accumulation are both positive, all tumor cells will eventually die when available nutrients have been completely depleted while the environment has become increasingly toxic. Small values of rr result in lower cost of spatial movement r and thus confer higher mobility to tumor cells, making global search a worthwhile effort. By contrast, large values of rr lead to higher costs for spatial movement r, hence limit the usefulness of global search and imply a lesser intrinsic capability to move. By virtue of eqn (2), changes in mechanical confinement have similar, direct effects on the costs of spatial movement r. However, mechanical confinement is kept Table 2 Parameter values in the benchmark simulations Parameter The The The The The The

negative effect of overcrowding, c rate of nutrient depletion, rf rate of toxic accumulation, rt positive utility of nutrient, qf negative utility of toxicity, qt negative utility of resistance, qp

Equation

Value

(3) (8) (9) (7) (7) (7)

1 0.1 0.01 10 10 10

constant during all runs reported here [i.e. the distribution of mechanical confinement (shown in Fig. 3) is not affected by the behavior of the tumor cells]. In our benchmark simulation, we set rr to a very high value such that only local search occurs with no global search. The benchmark results will be compared to cases where both global and local searches can occur, and also to the case where rr is very small such that the virtual cells can migrate to any location without inhibition. Notice that global search is maximal in the latter case, which does not preclude regional search to occur. The parameter rr thus represents the extent of local search relative to global search. Larger values of rr imply greater tendency to search locally while less so globally, and vice versa. 5.1. MAXIMUM AVERAGE SIZE OF THE TUMOR CLUSTERS

The average size of tumor cell clusters is defined as the total number of viable tumor cells divided by the total number of clusters. Figure 6 plots the evolution of the maximum average size of tumor cell clusters vs. the corresponding values of rr [i.e. the (inverse) intrinsic capability of the cell to migrate). The x-axis is drawn in logscale to avoid the concentration of points at values of rr close to zero. Figure 6 suggests that the maximum average size exhibits a phase transition at rr between 0.1 and 0.3. That is, at rr smaller than 0.3 (i.e. where global search is dominant), the maximum average size is roughly constant at 800 viable tumor cells per cluster, whereas at rr larger than 0.3 (i.e. where local search is dominant), the maximum average size is stable at around 50 viable tumor cells per cluster. 5.2. TIME FOR THE FIRST TUMOR CELL TO REACH THE NUTRIENT SOURCE IN THE CENTER OF QUADRANT III

The velocity of reaching the center of quadrant III is maximal when the value of rr equals roughly 0.0001. Figure 7 shows a tendency that when rr is lower than 0.0001, increasing its value increases the speed (shortens the time) with which the first tumor cell reaches the center of


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Fig. 6. Plot of the maximum average size of tumor clusters vs. rr.

quadrant III. In sharp contrast, when rr is larger than 0.0001, raising its value lowers the speed for the first tumor cell to reach the center of quadrant III. Thus, when rr is lower than 0.0001, it would be optimal for tumor cells to raise rr (i.e. performing more local and less global search), although the resulting increase is relatively modest as the migratory velocity apparently approaches a maximum threshold at these low rr values. But when rr is large the opposite is true, namely it would be optimal for tumor cells to lower rr (i.e. performing more global and less local search); it is evident from the steep curvature that the potential velocity gain would be substantial. 5.3. LIFETIME OF THE TUMOR SYSTEM

Figure 8 plots the lifetime of the tumor system vs. the corresponding value of rr. When rr is

smaller than 0.0001, lowering its value even more will increase the lifetime of the tumor system, indicating that tumor cells survive longer by performing more global search. However, the trend is clearly positive when rr is larger than 0.0001, indicating that the tumor system can actually survive longer by performing less global search. Thus, when rr is already very small, tumor cells with even smaller rr will survive longer. Since small rr is associated with highdensity clusters (Fig. 6), such a tumor system persists longer by supporting a cell population capable of on-site proliferation as well as rapid spatial movement to the center of quadrant III (Fig. 7). By contrast, when rr is already large, it is tumor cells with larger rr (i.e. exhibiting even more local search) that will survive even longer. Since large rr is also associated with low-density clusters (see Fig. 6), such a tumor system thus prolongs its lifetime by transforming into a

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Fig. 7. Plot of the time to reach the nutrient source (located in the center of quadrant III) vs. rr.

slowly expanding type that avoids the formation of overpopulated clusters. An overpopulated cluster can be detrimental to the occupying tumor cells by exposing them to rapid buildup of toxic elements and accelerated depletion of nutrient supply. 5.4. THE NUMERATOR AND DENOMINATOR OF THE CLUSTER SIZE: TOTAL VIABLE CELLS AND CLUSTERS

Since the average size of a tumor cluster (Fig. 6) is defined as the total number of viable tumor cells divided by the total number of clusters, it is helpful to investigate the evolution of both the numerator and the denominator, and how they correlate with one another. Figure 9 plots the numerator of the maximum cluster size (the total number of viable cells) vs. the corresponding values of rr, whereas Fig. 10 plots the denominator of the maximum cluster size

(the total number of clusters) vs. the corresponding values of rr. Figure 10 shows that the maximum size of clusters undergoes the phase transition at rr between 0.1 and 0.3 due to the sharp rise in the number of clusters from 2 to 30. As a result, the maximum cluster size drops from 850 viable tumor cells per cluster to roughly 60 of these cells per cluster (see Fig. 6). If rr values were increased further, the number of viable tumor cells will fall to about 1400 at rr ¼ 105 (Fig. 9), but the number of clusters also declines to about 20, thus in fact raising the maximum cluster size slightly to 70 units per cluster in Fig. 6. Figure 11 plots the time series of the total number of clusters for rr ¼ 0.1 and 0.3 where the phase transition occurs. It confirms that if rr is increased from 0.1 to 0.3, the tumor system begins to exhibit more clusters. Figure 12 plots the time series of the total viable cells for rr ¼ 0.1


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Fig. 8. Plot of the lifetime of the tumor system vs. rr.

and 0.3. It indicates that increasing rr from 0.1 to 0.3 results in a larger number of viable tumor cells for almost the entire observation time. 5.5. RESULTS FROM VARYING THE RATE OF NUTRIENT DEPLETION, rf

5.5.1. Maximum Average size of the Tumor Clusters Figure 13 exhibits, for various values of the nutrient depletion rate rf, the evolution of the maximum average size of tumor cell clusters vs. rr. It can be seen that tumor cells with a high metabolic uptake (i.e. high rate of nutrient depletion, rf) self-organize into increasingly smaller clusters, as evident from the downward shift of the corresponding curves. In addition, the phase transition disappears as the rate of nutrient depletion is increased beyond rf of 1.5. Despite the appearance of a convergence into a

single average cluster size, if one investigates the curves beyond an rr of 1000 there still is a tendency for systems with higher metabolic uptake to exhibit smaller clusters (within the narrow range of between 9 and 12 tumor cells per cluster). Figure 14 shows the time series of the maximum average size of tumor clusters at rr ¼ 0.2 for two extreme values of the nutrientdepletion rate, i.e. for rf ¼ 0.5 (very low) and rf ¼ 10 (very high). The sharp spike (in the dotted line) indicates that under low a rate of nutrient depletion, the average size of the tumor clusters reaches its maximum peak early within the lifetime of the tumor system, but it then decays rapidly followed by the imminent extinction of the tumor system. By contrast, at a high rate of metabolic uptake the system maintains a small, yet steady and stable cluster size, seen in the flat (solid) line of rf ¼ 10. The time series for

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Fig. 9. Plot of the number of viable cells at the maximum average size of clusters vs. rr.

two extreme values, rr ¼ 106 and 106, also exhibits very similar results, indicating a behavior that is not rr dependent (data not shown here). Figure 15 depicts the time series of the total number of clusters for rf ¼ 0.5 (dotted line) and rf ¼ 10 (solid line). The pattern in Fig. 15 is very similar to that exhibited in Fig. 14: under a low rate of metabolic uptake, the number of clusters (i.e. agglomeration of tumor cells) grows rapidly early in the lifetime of the tumor system, but then declines sharply at the end. In contrast, under a high rate of nutrient depletion, tumor clusters generally maintain a smaller but stable number. As before, the time series for the two extreme values, rr ¼ 106 and 106, confirmed the results obtained with an rr of 0.2, again indicating a behavior largely independent of rr (data not shown here). Note the difference when compared with the rr-dependent behavior

observed in the time series at the transition threshold with a very low rf of 0.1 (Figs 11 and 12).

5.5.2. Time for the First Tumor Cell to Reach the Nutrient Source in the Center of Quadrant III Figure 16 shows that beyond an rr of approximately 0.01, a higher rate of metabolic uptake rf leads to slower rate of spatio-temporal expansion. It can be seen that at larger values of rr, with rising rate of cell metabolism, rf, the potential gain for implementing more global search becomes even more prominent, as evident from the increasingly steep curvature. However, since the virtual tumor cells require at least one unit of time to migrate from one location to the other, there is a limit in how fast tumor cells can


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Fig. 10. Plot of the number of clusters at the maximum average size of clusters vs. rr.

reach the center of quadrant III. This maximum velocity is reached once the graphs flatten; that is, at an rr of approximately 0.01 and lower (i.e. in areas where global search dominates) no further increase in velocity can be achieved regardless of the structural pattern (compared with Fig. 13) and the nutrient consumption of the tumor system.

5.5.3. Lifetime of the Tumor System Figure 17 shows that with an increasing rate of metabolic uptake, the lifetime of the entire tumor system increases. This can be seen from the fact that as the rate of nutrient depletion, rf, increases, the curve shifts up higher, i.e. extinction of the tumor system takes longer. Longer

lifetime of the tumor system suggests that the host (i.e. the virtual ‘‘patient’’) also survives longer, implying a generally less aggressive type of tumor cells. This is indicated in the slower spatio-temporal expansion as evident in the longer time needed to reach the center of quadrant III (Fig. 16) for tumor systems operating with high rr. Moreover, with rising metabolic uptake rf, the bifurcation becomes more prominent, as evident from the increasingly v-shaped curve. Note also that with increasing metabolic uptake the bifurcation point shifts from rr of 0.0001 (Fig. 8) to approximately 0.1. 6. Discussion and Conclusions There is growing evidence that malignant tumors behave as complex dynamic self-organizing

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Fig. 11. Time series of the total number of clusters at the phase transition.

and adaptive biosystems (Chignola et al., 1990; Coffey, 1998; Deisboeck et al., 2001; DeHauwer et al., 1997; Kraus & Wolf, 1993; Posadas et al., 1996; Schwab & Pienta, 1996). In our computational model, self-organization is manifested in the behavior of virtual brain tumor cells, which are attracted to migrate towards a distant, nonreplenished nutrient source despite the obstacle of high mechanical confinement. The migratory behavior of these cells is governed by their nutritional needs and by the tendency to leave overpopulated areas with high levels of toxic metabolites. As such, tumor cells in our model are highly adaptive to a swift change in their environment. Both self-organization and adaptation result in the spatial agglomeration or clustering of tumor cells on sites with relatively high nutrition as we kept the distribution of mechanical confinement constant (i.e. unaffected by the behavior of the tumor cells). These virtual

clusters represent actual cell aggregates observed in experimental settings using malignant brain tumors (Deisboeck et al., 2001; Tamaki et al., 1997). Here we studied specifically the effects of varying rr, the intrinsic capability of a cell to migrate on the structural dynamics of the virtual multicellular tumor system. In the following we discuss these findings specifically, relate them to the medical situation, and list possible extensions of this work (1) Our benchmark simulation suggests that the average size of the tumor clusters undergoes a phase transition at an rr around 0.1.* We argue * Delsanto et al. (2000) also found a phase transition, but in their case they vary the parameter that reflects the cell affinity for nutrient sources, which in our model is the parameter qf. By contrast, here we found a phase transition by varying the parameter rr, which reflects the intrinsic (inverse) capability of the cell to migrate, and rf, which represents the metabolic uptake of the cell.


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Fig. 12. Time series of the (total) viable cells at the phase transition.

that the phase transition bifurcates rr into two parts, each corresponds to a tumor system that can be distinguished from the other by its specific spatio-temporal expansion dynamics, caused by distinctively different emerging structural patterns. The first part represents ‘‘aggressive, highly malignant’’ tumor systems with a high ratio of global search relative to local search. Such aggressive tumor systems are characterized by a large average cluster size, large count of viable cells, small number of clusters, and fast spatial expansion. However, high velocity of spatial expansion leads to rapid depletion of (in this case, limited) nutrient sources, and thus tumor cells that belong to the same cluster must compete vigorously for nutrients to avoid premature cell death. Such systems are thus characterized by competitive behavior. The second part represents ‘‘low(er) malignant’’ tumor systems with a low ratio of global to

local search. In contrast to their highly aggressive counterparts, such lower malignant systems are characterized by a small average cluster size, low count of viable cells, large number of clusters, yet slow spatial expansion. Since these systems deplete nutrient sources at a slow pace, tumor cells belonging to the same cluster can coexist for a prolonged period of time. In short, proliferation is reduced (and thus the effect of migration is relatively increased) in the benchmark simulations for high values of rr, which in turn confers a prolonged lifetime. Interestingly, using gene expression profiling methods, Mariani et al. (2001) recently showed indeed a reduced proliferation and apoptotic activity of migrating glioma cells. We however also found that for values of rr between 0.01 and 100 (i.e. close to the system’s transition threshold), along with migration, cell proliferation appears to be a principal mechanism that promotes the

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Fig. 13. Plot of the maximum average size of tumor clusters vs. rr, for various values of rf.

expansion of the tumor system into the regions where nutrients are most abundant. This is indicated by the fact that larger clusters with less nutrient depletion move faster (Figs 13 and 16). As such, this supports the notion that, if cell metabolism comes into play within the aforementioned rr range, both key features of a tumor system, proliferation and invasion, are likely required to drive and direct spatio-temporal expansion of the tumor system (Deisboeck et al., 2001; Suh & Weiss, 1984). Furthermore, Fig. 16 suggests that if the costs for spatial movement are high and with rising rate of metabolic uptake, a tumor system would experience a significant increase in velocity gain from implementing more global search. It must be stated that by virtue of eqn (2), lowering the value of rr (and performing more global search) is not the only mechanism by

which a tumor cell could reduce the high costs of spatial movement, r, and thus optimize its spatio-temporal expansion. Although not specifically investigated here, an alternative mechanism would be for a tumor cell to retain its rr yet migrating toward areas of lower mechanical confinement pj in order to lower r, hence strictly following the ‘‘least-resistance’’ concept presented in Section 2. It is reasonable to argue for a combination of both mechanisms to operate in reality. (2) We propose the concept of global vs. local search. Whereas the latter is hardly new, the concept of a global search component influencing cellular behavior, however, is less obvious. To execute global search, which is particularly important for the aggressive, highly malignant tumors in our model, a cell requires a long-range signaling reception (and processing) capability,


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Fig. 14. Time series of the maximum average size of tumor clusters at rr ¼ 0.2, for two distinct values of rf.

e.g. through the expression of specific cellsurface receptors which enable sensing over a longer distance. An increase in (spatial) signalreception sensitivity per se correlates well with the documented overexpression of cell-surface receptors in human malignant gliomas, as described especially for the tyrosine-kinase receptors, EGF-R and HGF/SF-c Met (Sang et al., 1995; Koochekpour et al., 1997). Chemotaxis has long been shown in vitro and is presumably also important in vivo to facilitate the transmission of long-range signals.* Let us briefly visit the specific biological in vivo setting considered here. The leakage of the blood–brain barrier around brain tumors inevitably leads to * Chemotaxis has also been extensively modeled computationally; see Perumpanani et al. (1996), Sherratt et al. (1993), and Sherratt (1994).

the accumulation of tissue fluid, edema, which extends into the parenchyma, e.g. along the long white fiber tracts. Thus, soluble factors such as nutrients as well as growth promoting and inhibiting polypeptides, of which some are produced by migrating tumor cells themselves, should indeed play a substantial signaling role since they can be ‘‘transmitted’’ through the edema fluid. For the moment we are not considering the intriguing yet more complicated scenario of adaptive sensitization processes at the same receptor, e.g. that the mechanisms necessary for reception and amplification of a faint, distant signal lead to receptor desensitization and require re-sensitization during which only strong, local signals can be received. Our conceptual framework simply hypothesizes two different cell-surface receptors that guide the chemotactic movement of our virtual tumor cells

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Fig. 15. Time series of the number of tumor clusters at rr ¼ 0.2, for two distinct values of rf.

with two distinctively different lower signal detection thresholds (or alternatively, with two different intracellular amplification strengths). The lower this threshold, the higher is the cell’s sensitivity for traces of the proper diffusive signal and as such relatively distant sources can be located. Its sensitivity to an external signal likely reflects the importance of that message for the cell. In our current model, cells therefore first perform global search followed subsequently by local search, resulting in a biased random walk. A third perspective assumes chemotaxis for global search only and instead suggests a different signaling quality, e.g. mechano-transduction, to mediate local search. It is noteworthy that by presenting simulation results for very large values of rr, we also show the system’s behavior for local search alone. (3) Our work raises the contentious issue of how to define fitness in expanding tumor systems. In our model, the implications on

tumor fitness of plotting rr vs. lifetime are completely opposite to that from plotting rr vs. the time to reach the center of quadrant III (Fig. 7). As an example, when rr is large, decreasing the value of rr will increase the tumor fitness by accelerating its expansive velocity of reaching the center of quadrant III. However, decreasing the value of rr will also decrease the tumor’s fitness by shortening its lifetime (Fig. 8). Conversely, if rr were raised instead (i.e. performing more local search), the lifetime of the system would be prolonged at the cost of a considerable decrease in the velocity of spatial expansion. It is clear that ranking of the fitness of tumor systems becomes ambiguous when there is a trade-off between lifetime and the dynamics of spatial expansion. Although it is tempting to suggest that the tumor optimizes toward spatio-temporal expansion only, the definition of fitness is likely to involve a combination of several factors, including


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Fig. 16. Plot of the time to reach the nutrient source (located in the center of quadrant III) vs. rr, for various values of rf.

resistance to treatment and more generally, the tumor’s capability to adapt to changing environmental conditions. And indeed, it has been shown experimentally that multicellular tumor spheroids are capable of adapting their metabolic rates to different supply situations (Mueller-Klieser et al., 1986). Correspondingly, Fig. 13 indicates that if the proliferative activity were to increase in the presence of a limited nutrient source, the virtual tumor system adjusts by decreasing its overall metabolic uptake. (4) At the phase transition, minor differences in initial conditions can result in a tumor system exhibiting a completely distinct ratio of local to global search. It is been argued that tumors exhibit genetic instability (Lengauer et al., 1998) and as such, small genetic perturbations (at our phase transition) with sufficient impact on rr, could indeed be decisive in determining whether

a tumor system develops into a highly aggressive or a lower malignant type. Current experimental cancer research aims to determine if there are indeed genetic switches, e.g. ‘‘bifurcation point’’ genes (Waliszewski et al., 1998), which, once affected during the progression of the tumor, may alter the dynamic path of the system, much like at the bifurcation point of our numerical results. In fact, the known involvement of tumor suppressor genes (Lang et al., 1994; Louis, 1997), or more accurately their loss during tumor progression, already supports our findings. Furthermore, given the increasing prominence of the bifurcation in the lifetime of the tumor system with increasing rf (Fig. 17), cell metabolism (and thus the gene-regulatory networks controlling it) may indeed be regarded as one key characteristic distinguishing the aforementioned aggressive, highly malignant tumor

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Fig. 17. Plot of the lifetime of the tumor system vs. rr, for various values of rf.

systems from their low malignant counterparts. This notion is supported by the finding that a tumor system operating with rr of 103 progresses to an aggressive growth pattern solely by increasing its metabolic uptake, rf, as the system moved to the ‘‘left’’ side of the bifurcation (Figs 8 and 17). (5) Our findings already have some theoretical implications for current and future treatment strategies. Common treatment modalities (i.e. surgery as well as chemo- and radiotherapy) inevitably change the environmental setting within the target region. Tissue scars and radiation damage for example can make it more difficult for tumor cells to move forward, thus increase their energetic-metabolic expenditure, r. On first look, this seems to be a desirable outcome because in our simulations an increased local search translates into declining expansion

dynamics of the tumor system. However, a much more worrisome picture emerges if therapeutic efforts should lower r for advancing cells instead, for instance by increasing the perifocal edema volume, which may enable shed cells to float along least-resistance paths, thereby expending less energy. In addition, the edema fluid may facilitate long-range signaling, essential for (more) global search, employed primarily by the highly aggressive phenotype. Also, as indicated before, proliferation is reduced for high r. Referring to the concept implemented in eqn (2) anti-proliferative treatments alone will therefore be rather ineffective in tissue areas of high mechanical confinement regardless of the problem that those areas exhibit a largely intact blood–brain barrier, which represents a major delivery problem for any therapeutic agent. It is thus not surprising that the response of highly


malignant brain tumors to treatment is often frustrating, ranging from rapid re-growth to the emergence of treatment resistance and diffuse tissue infiltration (Rostomily et al., 1993). (6) (a) In the current setup the model uses arbitrary parameter values only. Its predictive power can therefore only be evaluated if more specific experimental data are available, which will then allow a more accurate tuning of the relevant parameters. (b) On the computational side several improvements are currently in progress. The feedback effect between the tumor system and its environment can be strengthened further. For example, by linking the level of resistance with cell density such that locations with low mechanical confinement will attract more cells, which in turn will lower the level of resistance even more in those locations, resulting in an adaptive grid lattice. Biologically, this is due to the fact that invading brain tumor cells produce ECM-degrading enzymes (Nakano et al., 1995) and that the formation of clusters should decrease the consistence and thus the mechanical confinement of the tissue, which will encourage the growth of even more tumor clusters. This would most likely render the low malignant case, which operates with many small clusters, ‘‘more malignant’’ as currently assumed. One way to include this feedback effect is to model the distance that a cell would travel as a function of the nutrition-to-resistance ratio. Thus, the higher this ratio, the farther a cell would travel. Most importantly, this specific extension of our work, especially in conjunction with experimental input (including genomics data), will also yield insights into a potential, dynamic relationship between the currently independent parameters rr (intrinsic capability of cells to migrate) and pj (extrinsic, mechanical resistance) in eqn (2). (c) In addition, it will be interesting to see the difference in cluster pattern and system dynamics when nutrient supply is constant. This would, for example, represent an environment in the vicinity of a blood vessel. In this scenario, the supply of nutrient is fully replenished such that it is constant in every % 0;j Þ . As such, the period with no decay ðft;j ¼ f rate of net nutrient depletion, would have to be set equal to zero, which will effectively create non-decaying nutrient sources that would

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prolong the lifetime of both tumor systems, hence also affecting the aforementioned ‘‘tradeoff’’ between lifetime and velocity. (d) It would be intriguing to investigate the effects of a hierarchical, scale-variant search ratio, i.e. the entire multicellular system optimizes toward the highly aggressive tumor type, whereas on the individual cell level local search prevails. This would imply that performing global search requires the participation of a critical number of cells rather than surface receptors on a single cell. Another possibility is to implement an adaptive search ratio for nutrients, where the dominant search mechanism self-organizes depending on the dynamic change in supply and/or demand. (e) The current algorithm assumes a monoclonal cell population, but since one key feature of malignant brain tumors is their genetic and epigenetic heterogeneity, the next logical step would be to model different tumor cell populations explicitly. Those subpopulations can differ in their migratory capability as reflected by the parameter rr, or in their susceptibility to toxic metabolites. (f ) Finally, the step from a 2D to a 3D model is eventually necessary to represent the clinical setting properly. (7) Although the current model has been developed based on observations from an experimental brain tumor model, extensions are possible to tumors that exhibit the ability to disseminate to distant organs. In this case, the current clusters would represent the emergence of secondary solid tumors, i.e. metastases. Several modifications are then required. First, lattice areas have to be specified to represent different organ-settings. Second, an additional term has to be introduced to distinguish energetic cost r for cell movement in blood vessels (e.g. for colon or liver cancer) or lymphatic pathways (e.g. for breast cancer or melanoma), needed for inter-organic spread, from r for cell expansion within solid tissue. As such the model will have to incorporate, amongst others, regional differences for example in pj and thus in the ratio of both search mechanisms. Finally, it is noteworthy that the study of chemotactic behavior is of considerable interest not only for cancer research but also for developmental biology. Examples include growth

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cone guidance at the axon tip as well as epidermal growth factor related posterior border cell migration in Drosophila oogenesis (Kalil et al., 2000; Duchek & Rorth, 2001). Most recently, Ming et al. (2002) found experimental evidence for adaptation in the chemical guidance of cultured Xenopus spinal nerve growth cones. The authors claim that this behavior allows the growth cone to re-adjust its sensitivity, which in turn is ‘‘essential’’ for what they call ‘‘long-range chemotaxis.’’ As such, applications of a similar model also in the area of developmental biology are conceivable if the computational platform is properly adjusted to the specific biological situation. In summary, the current model simulates microscopic secondary brain tumor aggregates seen in experiments, based on a concept of guiding environmental heterogeneity. The tumor cells modeled as individual agents follow a stepwise search process with regard to the direction of motility and the dynamics of spatial aggregation. As hypothesized from the experimental setting, our agent-based model showed that cluster patterns have distinct impact on the spatio-temporal dynamics of the evolving tumor system. The emergent patterns in this virtual tumor system are highly complex and show already great potential for simulating real in vitro and in vivo situations when combined with specific experimental data sets in the future. Diffuse local cell invasion in the brain as well as the secondary spread of cancers such as breast cancer, melanoma, or lung cancer to other organs are major challenges for current anticancer treatment strategies. Our innovative computational model will help us to better understand the dynamical relationship between the main tumor and its satellites as well as between the environmental setting and the tumor system itself. This is an important first step toward novel diagnostic modalities and eventually toward more efficient treatment strategies for malignant brain tumors as well as other types of cancer. This work was supported in part by grant CA69246 from the National Institutes of Health and by venture capital funding from Cornell College of Architecture, Arts, and Planning to Y.M., M.K., and J.L. The authors would like to thank especially

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