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# TOPICS GAME DEVELOPMENT CS 5890

Utah State University

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Advances in Complex Systems Vol 5 Nos 2 amp 3 2002 2477267 World Scienti c Publishing Company CELLULAR AUTOMATON MODELS OF TUMOR DEVELOPMENT A CRITICAL REVIEW JOANA MOREIRA and ANDREAS DEUTSCHl Zentmm fiir Hochleistungsrechnen Technische Universitiit Dresden Dmsden Germany momimzhrtuid7esden e ldeutsch zhr tuidresden de Received 17 June 2002 Revised 30 August 2002 Cancer development can be viewed as an example of spatiotemporal pattern forma tion Several attempts have been made to model and predict malignant tumor behavior an also to account for immune system response and the impact of possible clinical treatments Modeling started from a macroscopic perspective and developed towards cellbased approaches from which cellular automaton CA models are an exam le In thi article we rst 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 cellbased modeling H 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 nonmalignant 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 nondeterministic 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 preneoangiogenic 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 247 248 J Mmeim and A Dwtsch 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 spatiotemporal 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 rst 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 Reviews are provided by Marusic et al 34 Adam and Bellomo 2 and Chaplain 12 More recent works try to explain spatiotemporal 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 cellbased models to simulate particularly cancer development The other important advantage is the possibility to incorporate cellbased data in such models Currently a standard cellbased model does not exist In many variants of inter acting particle systems particles cells can alternatively move through conti nuous space or a xed lattice Examples of latticefree models are those by Drasdo 17 lori et al 29 and Stott et al 47 Lattice models are frequently called cellular automata and we shall review them in this article We rst 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 2 The Cellular Automaton Concept Cellular automata CA are de ned 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 selforganizing 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 threedimensional regular lattice de ning lattice cells3 see Fig 1 ii discrete states each site takes one of a nite number of states aOr lattice sites as we shall refer to in this article to avoid confusion with biological cells Cellular Automaton Models of Tumor Development 249 a von Neumann neighborhood b Moore neighborhood Fig 1 Example of a regular twodimensional lattice i a square lattice 7 showing two possible neighborhoods iii discrete dynamics time is discrete at each time step every site updates its state simultaneously according to the transition rules iv local rules the transition rules that de ne the site state dynamics depend only on the site s spatial neighborhood con guration Fig l and can be de terministic or probabilistic v homogeneity all sites are equivalent since the lattice is regular and the transition rules for each of them are the same for all times The above features can be extended giving rise to several variants of this basic CA system These are particularly Coupledmap lattices these are CA where the constraint that cells can only assume discrete states is withdrawn ie the state space is a continuous set Asynchronous CA in such systems the restriction of simultaneous update of all the sites7 states is revoked allowing asynchronous updates Nonhomogeneous CA this generalization allows transition rules to vary from site to site andor in time 3 Cellular Automaton Models of Tumor Development A huge variety of CA models related to tumor development can be found in the literature Different authors have developed characteristic models which have several applications Here we do not present an extensive list of existing models but only one example from each characteristic model Firstly we consider models of avascular tumor growth then vascular growth and later the morphology of invasive growth patterns Finally a model of angiogenesis is reviewed 31 Avascular tumor growth models The following models address various aspects of pre neoangiogenic tumor develop ment The term avascular growth model77 indicates that these models do not 250 J Momma and A Dwtsch implement vascular tissue explicitly7 although they may take into account nutri ent supply through diffusion 311 Pioneer model Some of the earliest models were proposed by W Duchting 197 20 with the goal to design a model to study the regulation of disturbed cell renewal7 through the analysis of two competing populations of cells The model has the following features Lattice o A twodimensional regular 10 X 10 square lattice is used 0 A von Neumann neighborhood is considered States 0 Discrete states 7 each lattice site corresponds to a biological cell if a cell dies the lattice site becomes empty Rules 0 Transition rules are deterministic and local A cell can survive7 divide or die 0 The update is asynchronous The model simulates the number and con guration of the cells within the lattice Two clones of cells with normal and fast growth can coexist in competition Simulations of normal cells7 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 Furthermore7 the only biological processes considered are mitosis and death7 which are implemented through transition rules that do not express biological interactions An update to this model is presented in Ref 217 in which the lattice is en larged to 100 X 100 sites and extended rules are introduced In Ref 22 a three dimensional model of 40 X 40 X 40 cells is proposed A nutrient medium is considered and cellcycle phases are taken into account7 with durations based on cellkinetic 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 Cellular Automaton Models of Tumor Development 251 312 Tumorimmune 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 effect7 and can be characterized as follows Lattice o A twodimensional regular square lattice represents the tissue 0 A von Neumann neighborhood is considered States 0 Four discrete states 7 two corresponding to tumor cells7 alive or dead7 one to the normal cells and another to complexes an immune cell interacting with a cancer cell Rules 0 Rules are probabilistic Cells can proliferate7 interact with the immune system and dissolve Transition rules are nonhomogeneous7 nonlocal and not constant in time The proliferation probability decreases as the total number of tumor cells increases to simulate nutrient consumption 0 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 rst 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 t the Gompertz curve The rules of this model do not respect CA locality 7 the proliferation rate depends on the total number of malignant cells7 in a way that the model is biased to reach a saturation with respect to the number of cells7 resembling the Gompertz curve Additionally7 the dissolution of cells does not mimic real biological behavior7 since dead tumor cells tend to accumulate forming a necrotic core The Gompertz model is a phenomenological law It gives the volume of the tumor V with respect to time v weave mx where V V0 is the initial volume and A and B are constants that can be tted to agree with experimental data 252 J Moreim and A Deutsch C d Fig 2 Sections of a twodimensional space tiled into Voronoi cells a and c show lattice cells in c the varying density of sites is observable b shows both lattice cells and the Delaunay tesselation In d darkened lattice cells represent the tumor from Ref 30 courtesy of A R Kansal 313 Bramtumor growth More lately Kansal et al 30 published a model of brain tumor growth in order to reproduce the macroscopic structure of a tumor arising from microscopic processesC Its main characteristics are Lattice o A three dimensional Delaunay lattice represents the tissue This is generated from a Vorronoi network 7 a collection of polyhedra whose centres are randomly distributed and that completely ll the space 0 The density of the lattice ie the number of sites per unit volume is higher near the centre of the lattice Fig 2 o A site s neighborhood comprises all the lattice sites polyhedra that share a common surface with that site COther publications by the same authors are in Refs 31 and 32 Cellular Automaton Models of Tumor Development 253 States 0 There are three discrete states corresponding to possible types of malignant cells proliferating cells7 quiescent cells and necrotic cells Noncancerous cells corre spond to empty sites of the lattice 0 Each lattice site corresponds to several biological cells Rules 0 Rules are probabilistic The contents of lattice sites can proliferate7 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 quantity7 depending on the overall tumor radius 0 The update is synchronous The structure of the tumor is a priori introduced into the model through a choice of input parameters that quantitatively t experimental situations The simulations presented show that the outputs of the model7 namely the average overall tumor radius7 the proliferating rim thickness as well as the necrotic frac tion agree with experimental and clinical data taken from Refs 5 107 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 varies7 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 diffusing7 the proposed gradient would only last a short time after the cells have become necrotic 314 Multicellular spheroid growth Recently7 Dormann and Deutsch 16 proposed a model of avascular tumor growth as a selforganized system with the following characteristics Lattice o A twodimensional regular square lattice with 200 X 200 sites represents the tissue 0 A von Neumann neighborhood is considered States 0 Ten discrete states 7 each lattice site can accommodate two types of cells7 namely tumor or necrotic cells that have an orientation expressed by four velocity chan nels and one resting channel Fig 3 254 J Morequ and A Dwtsch 026 1 0 0550 PC Fi Example of a cell con guration at a lattice site 7 The dark gray lled circle and the light gray lled circle denote the presence of a tumor and a necrotic cell respectively from Ref 16 courtesy of A Deutsch 0 Each channel can be occupied by at most one cell exclusion principle 0 Two continuous elds are considered a chemotactic signal emitted by necrotic cells and nutrients Rules 0 Rules are probabilistic Cells can be quiescent7 proliferate7 die or become necrotic with given probabilities 0 Transition rules are local The probabilities of mitosis7 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 o 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 multicellular 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 system7 cellular heterogeneity and additional signalling molecules as growth and suppression factors 32 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 ef cacy of pre neoangiogenic tumors was proposed by Patel et al 39 Its main characteristics are Cellular Automaton Models of Tumor Development 255 5days 11 days d12mm d2mm 15 days d2mm d2mm 3 quiescent tumor cells 5d2a smm 0d2aglsmm necrotic cell material Fig 4 Simulation of tumor growth with a CA Starting from a small number of tumor cells 44 a layered tumor forms comprised of necrotic material quiescent cells and proliferating tumor cells from Ref 16 courtesy of A Deutsch Lattice o A twodimensional regular N X N square lattice represents the tissue In the simulations N is either 100 or 200 o A von Neumann neighborhood is considered States 0 The state space is fourdimensional The rst component of the state vector can assume four different states corresponding to the type of biological entity normal cell tumor cell microvessel or a dead cell vacancy The second and third components are local concentrations of H4r and glucose respectively the state vector is continuous in these components The fourth component is used Whenever the site is a vessel stores four values for both glucose and HJr elds at the vessel s four walls providing boundary conditions 0 Each site corresponds to one biological cell or vessel 256 J Monaim andl Demtsch Rules c 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 c Transition rules are oc c 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 ie a fraction f of the total cell number With f 01 Then the equilibrium diffusion equations for glucose and HJr 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 ie HJr production and vascular density on tumor evolution and morphology are performed and statistically analyzed e results can reproduce clinically observed tumor morphologies Fig 5 28 and give i ieer nt tumor morphologies obtained with the simulations by vanying the malignant cells phenotype and their vascular en ironment Fou t mors tha have been growing for the sa im each starting from the same initial size are depicmd In a and b the vascular de ity and metabolism are the same however in a tum quiescence is ad tted nd in b it is suppressed Note that a growth te en anoeme as been obtained by the tumor in b at th e pense of necrosis su ered throughout In c the vesse density and aci rod ction rate h n lowered whil tu or quiescence su ressed resulting in necrosis co 5 ed to n a tu or that initially gro s central cores In d the vascularity is lowered further resultin but soon selfrpoisons survivi g 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 courmsy of A Paml 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 1 337 387 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 elds which can allow one to simulate therapy The absence of cell motility forces mitosis to be strongly dependent of site vacancy7 and also adhesion properties of malignant cells have not been considered in the present model 33 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 speci c functional cell properties The model has the following characteristics Lattice o A twodimensional regular 100 X 300 square lattice represents the tissue 0 A Moore neighborhood is considered States 0 There are two discrete states 7 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 xed 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 cells7 which are considered as sources The chemicals factors7 concentrations depend on the den sity of sources in the local surroundings of a cell7 however their effective in uence is randomly modulated at each step Simulations underline the importance of division7 migration and death proba bilities and that of the different chemical factors in the formation of distinct tumor patterns The model is not timescaled 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 speci ed since the action of the chemical factors introduced does not express biological processes 258 J Momma and A Dwtsch Turner and Sherratt 50 proposed an extension of the Potts modeld as a model of malignant invasion to address the problem of how the malignant cells7 phenotype specially their adhesion properties in uences the morphology and invasiveness of cancer The model can be characterized as follows Lattice o A twodimensional regular 200 X 200 square lattice represents the tissue 0 A Moore neighborhood is considered States 0 Discrete states 7 each biological cell represents a state Cells can occupy a variable number of adjacent lattice sites accordingly they have a certain two dimensional volume and form The number of possible states corresponds to the number of biological cells and can vary with time The extracellular matrix ECM is modeled by protein concentration a param eter assigned to each lattice site Rules 0 Rules are probabilistic Lattice sites can change their state with a certain prob ability calculated by a stochastic energy minimization technique Transition rules are local 7 at each time step the change in the overall energy due to a local transition is evaluated in order to calculate the probability of that event Since the transition is strictly local the change in the energy is also exclusively dependent on neighboring states Hence the rules are local The update is asynchronous The energy function mentioned above includes a term of surface interaction energy that depends on cellcell and cellECM adhesion forces and a mechanical energy term representing elastic deformation and growth of cells Haptotaxise connected to secretion of proteolytic enzymes by the malignant cellsf is taken into dThe Potts model 47 uses a method of energy minimization that was originally developed to simulate surface energy driven diffusion in nonbiological systems The model simulates a pure material composed by cells with the same surface interaction energy Each biological cell occupies several sites of the lattice At each step a site is randomly selected and a local positional change of a cell occurs with a probability depending on the energy change it induces This dynamics allows cells to move slowly adjusting their boundary positions eHaptotaxis designates the directed movement of cells due to gradients of xed substrates in the extracellular medium It differs from chemotaxis whilst in the latter the substances promoting the movement are diffusible fThe secretion of proteolytic enzymes is essential for invasion of the extracellular matrix ECM by the malignant cells These enzymes degrade the ECM providing a space into which cells can move and also generating gradients in ECM proteins that constitute chemotactic and haptotactic signals to direct malignant cells Cellulm Automaton Models of Tmmm DEVElopmem 259 I 500 z 3000 Fig 6 Early and nal steps ln a slmulatlon of the morphology ofa tumor advancing front The shape of individual cells can be observed from Ref 50 courtesy of s Turner account Haptotaxis is accompllshed through an energy term dependlng on the l39 39 39 varies in elm 439 4 ll interactions Mitosis can optlonally be included in the model Simulations ofthe invasion process are conducted Fig 6 in which the dlfferent parameters related to cellecell adhesion cell M adheslon and haptotaxis vary The maximum depth ofinvasion is used to quantify the effect of these parameters in 39 39 39 l P l l l 39 39 9 a cleventual interaction between them ost of the simulations presented do not include cell prollferatlon Since one of the maln features of cancer cells is their high mitotic potentlal it ls arguable lfthe model can falthfully reproduce the biological process While the simulations that lnclude prollferatlon are a s ep ahead only the following activities are conslde motility and adhesion division growth and haptotaxis The model does not take into account lmportant features like nutrlent consumptlon death slgnalllng or dlf ferentla 39on 7 the addition of such posslhle biological states and behaviors that mallgnant cells can in fact adopt ls yet a question to be addressed 39s model ls motivated hy the fact that minimization of the energy assoclated with cell surface lnteractlons drives reorganization of cell aggregates 18 35 as con rmed by experimental observations 44746 34 Angiogenesis A model of angiogenesis based on the mechanism of endothelial cells EC migration in uenced by gradients of bronectin and tumor angiogenesis factors TAFg 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 zen J Momm andA 17mm Ldtttoe A twddimensional regular lattioe with 200 X 200 sites represents the tisue on Ne mznn neighborhood is considered States o discrete states corresponding to presence or absenoe ofan EC sprout Continuous elds of EC density bronectin and TAF concentrations are cone sidered Rules Rules that govern migration are probabilistic Branching or anastomosish by ts an occur and are controlled by deterministic rules local Branching depends on neighboring space occupancy ns ty relative to a threshold level ofT and a minimum time interval be fore new branching occurs Anastomosis can happen below a oertain level ofTAF Th a iii noquot Mir n tin ml ycoupl equations The discretized equation for the EC diffusion is solved on a square H are ewluated and the probability of migration is chosen proportional to these coemcients T e update is synchronous Some parameters of the model are scaled according to biological data imu tions to observe the mechanism of EC migration under the in uenoe of bronectin and TAF gradients are performed Fig 7 The results can reproduoe experiments of solid tumor implants in the cornea of animals 27 36 i3 u Fig 7 cumideted in z 1 32511511111 and anastumusls is visible hum Ref 3 courtesy uiA Andaman hAnastDmDsls is the immatmn oi capillay loops by newly immirg spxuuts Cellular Automaton Models of Tumor Development 261 4 Discussion We haVe reviewed models of tumor development which are called cellular automa ton models in the literature It turns out that the proposed models can be grouped into classical7 latticegas and manysite CA models Whenever a model considers simultaneous existence of discrete cells and continuous elds7 it is called a hybrid model By classical models we mean those which respect conditions to V de scribed in Sec 2 207 41 and may haVe asynchronous update 43 The models by Patel et al 39 and Chaplain and Anderson 3 are examples of classical hybrid CA Latticegas models also respect to V and haVe a speci c state space that considers velocity channels associated with each site7 as in the hybrid model by Dormann and Deutsch 16 Manysite models respect to V with the exten sion of asynchronous update and each biological cell may occupy seVeral latice sites 50 The threedimensional model by Kansal et al 30 cannot be considered a CA model7 since its lattice is not regular and rules are neither local nor homogeneous A comparison of the models is dif cult Each model does not consider the same biological processes altogether7 but analyzes different aspects of tumor deVelopment and7 therefore7 the results cannot be compared When constructing a model7 it is important to understand which biological functions are interconnected and which can be separately modeled Below7 we describe how biological processes and properties are incorporated in the reViewed models 0 Mitosis The transition rules that control cell proliferation in the model can depend on site Vacancy7 macroscopic parameters of the system or biological conditions In particular7 the most simple and deterministic condition is the existence of a Vacant neighboring site 20 In some models7 in addition to the preVious condition7 the probability of mitosis is arbitrarily chosen considering neighboring conditions 43 or is related to the total number of cells 307 41 Mitosis can also be implemented by using biological parameters related to the aVerage time of cell cycle phases7 adding stochasticity7 either depending on Vacancy of sites 227 considering nutrient leVel 16 or eVen adhesiVe properties of cells 50 Finally7 a deterministic condition de ning thresholds for pH and nutrient leVels can be used 39 0 Cell death Deterministic rules depending on neighboring sites occupancy control cell death 20 or depending on nutrients and pH leVels control apoptosis and necrosis 39 Probabilistic local rules 417 43 or global rules 30 determine necrosis Apoptosis and necrosis are in a way for cell cycles7 signalling and nutrients in Ref 16 262 J Moreim and A Dwtsch o Nutrients uptake A xed 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 0 Cellular motility 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 cell7 and in uenced by cell adhesion7 pressure and chemotaxis 16 o Adhesion This property is explicitly simulated in Ref 50 through the energy associated With cellcell surface interaction In Ref 16 adhesion is indirectly considered through the particular choice of occupation rules Which determine the con guration of the velocity channels in each site since tumor cell adhesion decreases When cells are in contact With necrotic material7 the necrotic cells are preferably placed at the resting channels and tumor cells at the remaining velocity channels 0 Pressure The effect of pressure is implemented through a nonlocal rule in Ref 41 In Ref 16 pressure is considered to be proportional to local cell density Which directly in uences occupation rules Another implementation is through a mechanical energy term Which considers cell deformation due to expansion or compression 50 0 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 de nition of biologically based intervals of pH level inside of Which cells survive I C39hemotamis and haptotamis Chemotaxis is explicitly modeled in Ref 16 through discretization of the diffusion equation for the chemotactic signal concentration The signal7 emitted by necrotic cells7 in uences the distribution of cells in the velocity channels ie it provides a cue for directed active movement Haptotaxis is simulated in Ref 50 through a discrete eld of extracellular matrix protein concentration that changes in time depending on the presence of malignant cells Whenever a cell moves throughout the lattice7 it is assumed that there is a linear relationship between the protein gradient it experiences and local energy change 7 thus7 by requiring energy minimization7 directed motility is induced Cellulm Automaton Models of Tumo39r DMElopmem 263 a 50 days 60 days 100 days b 50 days 60 days 70 days v c 50 days 60 days 100 days lg 3 Simulation of various treatments a Aaer 50 days one half of the tumor is cut the tumor recovers from this surgery b After 50 days the cellrcell adhesion is lowered c Aaer 50 days the necrosis rate is magni ed by a mtor 10 tumor cells still survive from Ref 16 courtesy of A Deutsch 0 Tissue vascular39ity 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 elds must be ful lled 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 he 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 t pe depends on the features one wants to include in the model a hybrid model is necessary 264 J Moreira and A Dwtsch whenever diffusing or mechanical elds are considered a latticegas CA allows the incorporation of cell motion and manysites models allow the simulation of varying cell shapes Second whenever a model includes nondeterministic processes sta tistical analysis which is not standard in existing models is required to compare results There should also be a wellde ned relation between experimental data and the parameters introduced in the model Concerning the biological processes a careful analysis should be performed in order to determine which of them are essential to the model and which can be optionally included As was already mentioned this also leads to the problem of which biological functions are interconnected and thus cannot be separately modeled It would also be desirable to construct transition rules that express even tual physical processes occurring on the system As a cellbased model a CA allows manipulation of single cells and its environ ment which enables the simulation of clinical treatments Fig 8 In comparison with cellbased latticefree approaches the CA models possess some advantages Concerning implementation parallelization of the algorithm is particularly straight forward for simultaneous update simulations are fast and allow the follow up of large cell numbers In simpli ed cell interaction models stability analysis can be performed Caution has to be taken with respect to the spatiotemporal scale Cell size and the fastest biological process that shall be described by the model determine the resolution of the system Furthermore the potential and versatility of CA models along with recent exper imental data provide a very promising approach to understand tumor development as well as selforganizing spatiotemporal pattern formation systems in general References 1 Abrams H L Spira R and Goldstein N Metastases in carcinoma Analysis of 1000 autopsied cases Cancer 3 74785 1997 Adam J A and Bellomo N A Survey of Models for TumourImmune System Dynamics Birkhauser Boston 1997 Anderson A R A and Chaplain M A J Continuous and discrete mathematical models of tumourinduced angiogenesis Bull Math Biology 60 8577899 1998 Boon J P Dab D Kapral R and Lawniczak A Latticegas automata for reactive systems Phys Rep 273 557147 1996 Burgess P K Kulesa P M Murray 1 D and Alvord E C The interaction of growth rates and diffusion coe icient in a threedimensional mathematical model of gliomas J Neuropathol Exp Neurol 56 7047713 1997 Burks A W Essays on Cellular Automata University of Illinois Press Illinois 1968 Bussemaker H Deutsch A and Geigant E Mean eld analysis of a dynamical phase transition in a cellular automaton model for collective motion Phys Rev Lett 78 501875021 1997 Byrne H M and Chaplain M A J Necrosis and apoptosis Distinct cell loss mechanisms in a mathematical model of tumour growth J Theor Medicine 1 2237 235 1998 EEE E E E E H E H H E H d H OH HSQ m E m E m m 3 10 2 10 m g Cellular Automaton Models of Tumor Development 265 Carlsson J Tumour models in uitro A study of proliferation and growth in cellular spheroids Acta Univ Ups 31 52333 1978 Carlsson J and Acker H Relations between pH oxygen partial pressure and growth in cultured cell spheroids Int J Cancer 466 Uppsala 1988 Chaplain M A J Avascular growth angiogenesis and Vascular growth in solid tumours The mathematical modelling of the stages of tumour development Math Comput Modelling 23 47787 1996 Chaplain M A J Mathematical modelling of angiogenesis J Neurooncology 50 37751 2000 Chopard B and Droz M Cellular Automata Modelling of Physical Systems Cam bridge University Press New York 1998 Deutsch A Dormann S Cellular Automaton Modelling of Biological Pattern For mation Birkhauser Boston 2003 to appear Doolen G D LatticeGas Methods for Partial Di erential Equations Frisch U Hasslacher 13 Orszag S and Wolfram S eds AddisonWesley RedwoodCity 1990 Dormann S and Deutsch A Modelling of selforganized aVascular tumour growth with a 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