ORIGINAL PAPER

A 3-D model of tumor progression based on complexautomata driven by particle dynamics

Rafał Wcisło & Witold Dzwinel & David A. Yuen &

Arkadiusz Z. Dudek

Received: 2 February 2009 /Accepted: 20 March 2009 /Published online: 23 May 2009# Springer-Verlag 2009

Abstract The dynamics of a growing tumor involvingmechanical remodeling of healthy tissue and vasculature isneglected in most of the existing tumor models. This is dueto the lack of efficient computational framework allowingfor simulation of mechanical interactions. Meanwhile, justthese interactions trigger critical changes in tumor growthdynamics and are responsible for its volumetric anddirectional progression. We describe here a novel 3-Dmodel of tumor growth, which combines particle dynamicswith cellular automata concept. The particles represent bothtissue cells and fragments of the vascular network. Theyinteract with their closest neighbors via semi-harmoniccentral forces simulating mechanical resistance of the cellwalls. The particle dynamics is governed by both theNewtonian laws of motion and the cellular automata rules.These rules can represent cell life-cycle and other biologicalinteractions involving smaller spatio-temporal scales. We

show that our complex automata, particle based model canreproduce realistic 3-D dynamics of the entire systemconsisting of the tumor, normal tissue cells, blood vesselsand blood flow. It can explain phenomena such as theinward cell motion in avascular tumor, stabilization oftumor growth by the external pressure, tumor vasculariza-tion due to the process of angiogenesis, trapping of healthycells by invading tumor, and influence of external (bound-ary) conditions on the direction of tumor progression. Weconclude that the particle model can serve as a generalframework for designing advanced multiscale models oftumor dynamics and it is very competitive to the modelingapproaches presented before.

Keywords Angiogenesis . Complex automata . Computersimulation . Discrete particle model . Tumor progression

Introduction

Despite the huge amount of resources that have beendevoted to cancer research, according to the NationalCenter for Health Statistics (http://www.cdc.gov/nchs/FASTATS/lcod.htm), cancer is the second killer (after heartdisease) in the US. The set of diseases, which arecategorized as cancer, are characterized by serious dis-ruptions in the control mechanisms regulating growth andhomeostasis in normal tissue. Up to now many aspects ofcancer remain obscure for experimentalists and clinicians,and many of the currently used therapeutic strategies arenot entirely effective.

A better understanding of the dynamics of tumorformation and its growth are expected from computermodels and simulations. The modeling could improve theoverall clinical outcome by predicting the results of specific

J Mol Model (2009) 15:1517–1539DOI 10.1007/s00894-009-0511-4

R. Wcisło :W. Dzwinel (*)Institute of Computer Science,AGH University of Science and Technology,Al. Mickiewicza 30,30-059 Krakow, Polande-mail: dzwinel@agh.edu.pl

R. Wcisłoe-mail: wcislo@agh.edu.pl

D. A. YuenMinnesota Supercomputing Institute, University of Minnesota,Minneapolis, MN 55455-0219, USAe-mail: daveyuen@gmail.com

A. Z. DudekDivision of Hematology, Oncology, and Transplantation,Department of Medicine,University of Minnesota Medical School,Minneapolis, MN 55455-0219, USAe-mail: dudek002@umn.edu

forms of treatment administered at specific time points.Moreover, it might help to optimize the existing therapeuticprocedures, design new ones, or even provide post-treatment predictions [1–4]. The degree of angiogenesis inhuman tumors varies widely and may be very low in sometypes of tumors. Therefore, tumors with a low level ofangiogenesis may not be benefited when treated withantiangiogenic agents [2, 5]. Computer modeling of tumorprogression involving its both avascular and vascularphases, could allow for answering the question if theangiogenic therapy is justified or not in these uncertaincases.

Nevertheless, we should be aware of the computermodeling limitations from which the most important areas follows:

1. The replication of living organism in silico is acomputationally irreducible problem due to impassablebarriers of the theory of computation, the theory ofchaos and technological limits. Many aspects of tumordevelopment can be investigated on a case-by-casebasis.

2. Tumor evolution is a complex process involvingprocesses occurring over a variety of time and lengthscales: from the DNA level and intracellular processes,through tumor vascularization and metastasis, to theholistic, mental and environmental factors. Up to now,no significant progress has been made in constructingconsistent and rigorous mathematical formalism andcomputational methodologies, which can exploit effi-ciently current computational resources and program-ming paradigms in modeling truly multiscalephenomena.

3. Even well known physical processes occurring in thereal tissue, such as diffusion and blood flow, which canbe described using partial differential equations (PDEs),are very difficult to simulate. This is due to the intricateboundary conditions imposed by heterogeneous envi-ronment (e.g., extracellular matrix (ECM) structure,blood vessel structure, blood rheology etc.), incompleteparameter space, and non-linear reaction terms.

Although global modeling of cancer development isimpossible, a more selective approach focusing on basiccarcinogenic processes can allow for substantial reduction ofboth many methodological problems and computationalresources. The approximate theories and numerical modelscan be used then to detect more precisely the cancer “weakpoints” - the future targets of anticancer therapeutic strategies.

We focus our attention on the process of tumor growthfrom avascular phase to angiogenic phase, i.e., the processof the blood vessel formation from a pre-existing vascula-ture. Judah Folkman published in 1971 the theory [6, 7]that angiogenesis is a principal process in tumor progres-

sion [9, 10]. Vascularized tumor invades the surroundingtissue, blood and lymphatic vascular systems and thepossibility of the cancer spreading (metastasis) increasesdramatically. Antiangiogenic targeting of the neovascula-ture within tumors is currently considered as one of themost promising strategies in the search for novel antineo-plastic therapies [2, 6, 11]. Because molecular phenotype ofimmature, angiogenic blood vessels is distinctly differentfrom that of resting blood vessels, tumor blood vessels canbe selectively targeted without affecting the normal organvasculature. The process of angiogenic signaling andformation of blood vessels can be disrupted or sloweddown with small molecules. In addition to treatmentsdirected to specific target, non specific agents can be usedto eliminate endothelial cells thus inhibiting angiogenesis.This involves numerous expensive and demanding inves-tigations both of hundreds of factors inhibiting angiogen-esis and antiangiogenic chemical species which can beconsidered in drug design process. To cut expenses, thepredictive power of mathematical modeling and computersimulation has to be employed. As shown in [12], in silicoexperiments can play the role of angiogenesis assays.

In the following section we shortly describe the mainprocesses, phenomena and growth factors contributingcancerogenesis. Then we present the most advanced modelsof tumor growth including continuum, cellular automataand hybrid models. We focus on such modeling aspects ascell cycle, tumor growth factors, angiogenesis, blood flow,and vessel remodeling. Next, we describe the model ofcomplex automata driven by particle dynamics and wediscuss its assumptions, limitations, and simulation con-ditions. The following section collects the results fromsimulation of avascular and vascular phases of tumorgrowth. We scrutinize the role of mechanical interactionsbetween swelling tumor, normal tissue, and expandingvascular network on tumor dynamics. We explain theinfluence of mechanical interactions on compartmentaliza-tion of the avascular tumor into external shell of welloxygenated cells, deeper layer of cells in hypoxia andnecrotic interior. We study the effect of mechanicalremodeling on tumor progression in vascular phase andon differentiation of microvascular density in variousregions of tumor. The role of processes involved inbranching and vessels maturation, such as (Dll4) - Noch1signaling [13, 14], is also analyzed. The results areconfronted with experimental studies. For simplicity wedo not discuss a parallel to blood vessel, lymph vesselnetwork which assist normal tissue and tumor with lymphoutflow and decrease in extracellular matrix pressures. Thisnetwork is a critical component of tumor metastasespathway. However, it is only minimally involved indynamics of tumor growth. In the last section we presentthe conclusions.

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Simplistic model of tumor growth

Generally, we can recognize a few principal phases oftumor progression, representing various spatio-temporalscales [1, 15–17].

1. Subcellular scale - involves genetic changes, distortionin the cell cycle and loss of apoptosis, absorption ofvital nutrients.

2. Cellular scale - comprises interaction at the cellularlevel with immune cells and other tumor cells,activation and inhibition of the immune system. Thecellular scale refers to the main activities of the cells:activation and proliferation of tumor cells.

3. Avascular tumor scale - includes condensation of tumorcells into cluster, macroscopic diffusion of TAF (tumorangiogenic factors) and oxygen.

4. Angiogenic phase - involves angiogenesis, sprouting,anastomosis, angiogenesis termination.

5. Vascular phase - includes blood flow, remodeling(vessel dilation and regression, mechanical reshapingof tumor and vessels), vessel maturation.

6. Detachment of metastases and invasion.

In Fig. 1 we present the scale separation map (SSM) forthis simplified model of tumor growth. The microscopicprocesses such as cell motility and cell cycle are discrete.The macroscopic scale refers to phenomena which are

typical for continuum systems such as diffusion (of oxygenand TAF), overall tumor condensation and blood flow. Inmacroscopic models, microscopic phases can be approxi-mated by coarse grained models as long as the methodol-ogy of multi-scale simulation and adequate computationalresources are lacking. The arrows in Fig. 1 show therelationships between these processes. The particle modelpresented in the following sections refers only to theprocesses shaded in blue.

Four overlapping phases of tumor growth are usuallyidentified: the avascular phase, the angiogenic phase, thevascular phase, and vessels and tumor remodeling. Thesephases go with physical processes such as nutrients andTAF diffusion, blood flow and mechanical remodelingcaused by interaction between growing tumor, vessels andthe normal tissue.

Avascular tumor

The proliferating tumor cells start to condensate into acompact cluster and interact with the external environment.Solid tumor, which is smaller than 1–2 mm in diameter(about 106–107 cells [11]), removes wastes and acquiresnutrients and oxygen through passive diffusion. Theoxygen (and nutrients) is supplied to the tumor nucleus bythe closest, mature blood vessels. It percolates through thesurface of the solid tumor and diffuses inside its mass. Due

Fig. 1 The scale separation map of all the main processes considered in modeling of tumor-induced angiogenesis. The processes controlled in ourmodel are colored blue

J Mol Model (2009) 15:1517–1539 1519

to differences in concentration of oxygen, the tumor clusterconsists of the outer region with proliferating cells, anintermediate region of cells in hypoxia and the necrotic coreof dead tumor cells. Most solid tumors, even those 1–3 mmin diameter, exhibit hypoxic fractions that may range from10 to 30% [11, 18, 19]. The avascular tumor is in dynamicequilibrium. Its critical mass has been reached, anddiffusion based transport is no longer efficient forproliferation-dominated growth. Tumor size is regulatedby proliferation speed, oxygen penetration depth and theexternal pressure [19].

Angiogenesis

As shown in Fig. 2, the tumor-induced angiogenesis isinitiated by tumor cells with the shortage of oxygen supply.The cells trigger the release of tumor angiogenic factors(TAF) [7, 18, 20, 21, 34]. Among many tumor growthfactors, vascular endothelial growth factor (VEGF) hasbeen identified as one of the key components [7]. VEGFcytokines, released from the tumor cells in hypoxia, diffusethrough the extracellular matrix (ECM) - the biologicalmaterial between tumor and existing vasculature - andproduces a chemical gradient. Once VEGF has reached avessel, it binds to the receptors located on endothelial cellslining blood vessel walls. This stimulates several enzymes(e.g., metalloproteinases) to degrade of basement mem-brane and sets off a cascade of events which triggers theoutgrowth of new vessel sprouts [9, 10, 20]. Endothelial tipcells proliferate and migrate through the ECM. The tip cellsfollow along the VEGF gradient toward regions of higher

concentration (chemotaxis) [21]. As shown in [20, 21], inaddition to the soluble isoform of VEGF, the presence ofother VEGF isoforms significantly influence morphologyof capillary network formation.

After initial sprouts have extended into the ECM forsome distance, repeated branching of the tips can beobserved. This causes numerous tip-to-sprout transitions.The newly formed vessels may form loops in a processcalled anastomosis. Along with anastomosis, the formationof lumen establishes a network that allows the bloodcirculation. The blood starts to circulate in the parts ofnetwork with nonzero blood pressure gradient.

There exist many other important factors which bothpromote and inhibit the process of angiogenesis (e.g., see[9, 10, 15, 20, 21]). Some of them will be discussed in thefollowing sections.

Vascularization

Tumor vascularization is a complex process of vascularnetwork development. It is initiated by the process ofangiogenesis and begins with the formation of a primarycapillary plexus. Once the sprouts approach the tumor, theirbranching dramatically increases until the tumor is eventu-ally penetrated by vascular network. Due to better oxygen-ation, the concentration of TAF decreases also inside thetumor. However, the newly formed vessels are subsequentlyremodeled by the growing tumor and pushed awayproducing regions of lesser concentration of oxygen, whichinitiates TAF production. This stimulates simultaneousgrowth in size of both tumor and its vasculature [2, 5, 19].

Fig. 2 The scheme of tumor-induced angiogenesis

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Tumor vessels can be segregated into three categories[5].

1. Immature - highly proliferative, nonperfused endothe-lial cell (EC) sprouts emanating from functionalvessels.

2. Intermediate - small, perfused vessels which lacksupport from mural, smooth muscle cells, and peri-cytes.

3. Mature - larger vessels, which have recruited pericytes,and smooth muscle cells with quiescent ECs and fewassociated sprouts.

Tumor vessels develop through these stages, beginningwith treads of endothelial cells sprouting from functionalblood vessels.

Remodeling

The process of tumor vascularization never becomesquiescent because the primitive vessels are continuouslyremodeled by dynamically evolving angiogenic factors,mechanical forces exerted by growing tumor, blood flow,and intraluminal thrombosis.

Local modifiers of vessel reshapining include growthfactors, pericytes, extracellular matrix (ECM), and neigh-boring cells, such as smooth muscle cells, fibroblasts andmacrophages. For example, inhibition of VEGF results insignificant increase in EC apoptosis [22] contributing tofast regression of newly formed vessel. The inadequatepericyte coverage is also the reason of vessel decay [5, 22,23]. As shown in [24, 25], the ECM density and itsstructure modify the direction of growing vessels. Theresults shown in [25] confirm different degrees of tumorperfusion or vascularization, depending on the ECMheterogeneity. The vessel wall is not always formed by ahomogenous layer of endothelial cells. Instead, it may belined with neighboring cancer cells and endothelial cells[20] changing vessel functionality.

Global vessel modifiers are: the blood flow andmechanical remodeling involved by the interactions amonggrowing tumor tissue, normal tissue and vasculature. Highpressure and shear forces of blood exerted on vessel wallcause its perfusion and dilation. Both the dilated vesselsand EC treads without circulating blood collapse. Vesselcollapsing in the interior of the tumor initiates percolationprocess which is driven toward criticality - the percolationthreshold - using a mechanism of vessel stabilization byincreased blood flow in the remaining vessels [19]. Bloodflow is also the source of both shear stress-dependent andpressure-dependent vessels reshaping [26]. However, themost robust remodeling factor is the mechanical interactionbetween the tumor cells, normal tissue cells and vascula-ture. The high pressure exerted on the vessel walls can

make them collapse. The mechanical forces may changealso the location of vessels, destabilizing the tumor growthand changing its direction [4, 19].

Computer models of tumor growth

Mathematical modeling of angiogenesis extends back anumber of years [1, 3, 4, 12, 15, 16, 27–29]. The modelingconcentrates on key events such as the response ofendothelial cells to tumor angiogenic factors secreted by asolid tumor, endothelial cell proliferation, endothelial cellinteractions with extracellular matrix macromolecules,capillary sprout branching, blood flow and vessel matura-tion. The substantial difference with respect to the physicalmodels is that the microscopic state of the cells is definednot only by mechanical variables, such as position, velocity,pressure, but also by internal biological microscopicphenomena reflecting activities of the cells.

A large bibliography about mathematical models oftumor growth driven by the process of angiogenesis can befound in two books edited by Adam and Bellomo [1] andby Preziosi [16] and in more recent overview by Mantzariset al. [15]. Three major categories of models of tumor-induced angiogenesis can be recognize: (a) continuummodels that treat the EC and chemical species densities ascontinuous variables that evolve according to a reaction-diffusion system, (b) mechanochemical models thatincorporate some of the mechanical effects of EC-ECMinteractions (c) discrete, cellular automata or agent basedmodels in which cells are treated as units which growand divide according to prescribed rules (d) hybridmultiscale models involving processes from micro-to-macroscale.

Continuum models only incorporate the chemical inter-actions between the EC and the environment. Among thecontinuum models, Greenspan proposed some of theearliest mathematical description of tumor growth [30].His models of avascular tumor growth were formulated asmoving boundary problems, in which the solid tumor growsin suspension. The continuum models do not allow forcellular heterogeneity within the tumor mass, and thetreatment of the mechanical properties of the tissue israther simplistic. Moreover, they neglect mechanical inter-actions between vasculature and its environment. Sometypes of mechanical interactions were incorporated intomechanochemical models introduced in [30–32]. However,despite continuum models can provide significant insightinto the relative role that different process play in theformation of vascular network; they cannot predict itsstructure. Within these models, it is not possible to capturesuch important events as repeated sprout branching and theoverall dendritic structure of the network.

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In contrast to continuum models, many types of discretetechniques have been presented in the literature to explainand describe the branching morphology of the vascularnetworks. There are, for example, percolation models [33],Eden models [18, 19], random walk and diffusion limitedaggregation (DLA) models [24], cell based models [34],lattice models and cellular automata [35–37]. Unlike in thecontinuum models, discrete models can follow individualcells and can reveal more details about cell dynamics andits interaction with the tissue.

Discrete models are usually hybridized with continuumapproaches in which molecular species are represented bytheir concentration while migrating EC tip cells aremimicked by particles [12, 21, 25]. The Stokes-Lauffenburger [21] model is one of the oldest hybridmodels simulating two-dimensional spatial distribution ofsprouts. It uses s classic Folkman simulation conditions [6,7]. The domain is a square (box) of surface S (volume V).The tumor is located at the top center of the domain and thecapillary at the bottom. All of the boundaries have no fluxconditions for cells and TAF. The evolution of molecularspecies is governed by reaction-diffusion equations that arediscretized on grid. The particle, which dynamics on theTAF concentration field is described by a stochasticdifferential equation, simulates the migration of EC tipcells. This approach is the source of many other hybridmodels (e.g., [12, 17, 18, 24, 25, 35, 37, 38]) which differin:

1. methodology of simulation of the process of vascular-ization and tumor growth (stochastic, deterministic,cellular automata, lattice-gas, DLA etc.),

2. modeling accuracy and its depth - defined by thenumber of factors and subprocesses included in themodel,

3. assumptions about geometrical properties of the simu-lation such as: dimensionality (2-D, 3-D), discretizationof space and time (on-grid, gridless), structure ofvascular network (rigid, structured, unstructured) etc.

One of the most popular paradigm used for modelingtumor growth are cellular automata (CA) (see the criticaloverview [36]). Cellular automata deal with the dynamicsof discrete elements populating the nodes of structural(mostly rectangular) grid. The elements take their state froma discrete (finite or infinite) space of states and evolve indiscrete space and time. The dynamics of the elements isdefined in terms of local, either deterministic or probabilis-tic, rules. Many models incorporate modifications to theclassical definition of a CA hybridizing it, like in theStokes-Lauffenburger model [21], with continuum fields ofdiffusive substances, such as nutrients or signaling sub-stances. CA models, however, typically do not address theimportant mechanical interactions between the tumor and

healthy tissue. Complex automata (CxA) are a generaliza-tion of cellular automata and represent a scalable hierarchi-cal aggregation of CA and agent-based models [39]. Thecomponents represent a sub-system operating on its typicalspatial and temporal scales. Globally, CxA can behaveeither as the classical CA nodes on a structural lattice or asinteracting particles whose dynamics is described by theNewtonian laws of motion or stochastic laws. These CxAcan be an interesting framework for the development of themultiple scale models.

The accuracy of the computational model is defined byits correspondence to the multiple processes and multiplescales involved in tumor growth (see Fig. 1). Because of thecomplexity of the entire process, many models to date focuson single key sub-processes, disregarding their interactionswith others. Many attempts assume either a static tumor andconcentrate on dynamic vascularization in the absence oftumor growth [23–26, 29] or a static network topology [32,35]. Some of them use dynamic network with bloodflowing, neglecting its interaction with concentration fieldsand tissue components (e.g., [26]).

Multiscale and multi-physics models represent the mostadvanced simulation methodologies. Multiscale modelsextend the moving boundary approach to incorporatecellular heterogeneity, intercellular phenomena and theuse of more complex mechanical laws to describe theresponse of the tissue to external forces. Advancedmultiscale models of tumor progression are presented inpapers by Bellamo et al. [1] and by Alacorn et al. [17].

The principal weaknesses of these impressive multiscalemodels are rigid geometric constrains which disablerealistic visualization and are the sources of many seriousartifacts. The reduction of one spatial dimension belongs tothe most drastic simplifications. The 2-D angiogenesismodels can adequately approximate only tumors develop-ing on a flat surface. However, even for this case, 2-Dapproaches result in unrealistic and excessive tumorcompartmentalization onto many regions separated byvessels. This compartmentalization may produce the artifactsuch as the excessive increase of the number of cells inhypoxia in separated regions and, consequently, theincrease of microvascular density (MVD). Other 2-Dartifact resulting in excessive MVD increase is illustratedin Fig. 3. The 2-D nucleus pushes the vascularization awayfrom its interior, while for 3-D case the tumor cells growbetween the vessels.

Other constrains like rigid structure of vascular network(e.g., in [17, 18]), considerably limit the role of vesselremodeling in tumor progression. In [17], for example, theprocess of angiogenesis is replaced by artificial dilation ofblood capillaries, without changes in vessel structure.Moreover, spatial anisotropy imposed by structural CAlattices (rectangular, hexagonal) falsifies both the structural

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characteristic of vascular network and macroscopic behav-ior of the growing tumor.

Disregarding all microscopic and mesoscopic biologicaland biophysical processes, in macroscopic scales tumorgrowth is a purely mechanical phenomenon. Mechanicalinteractions influences most the structure of vasculature,blood flow, tumor shape and decides about its directionalprogression. Due to the lack of adequate computationalframework any of existing computational paradigms is ableto reproduce this basic process. In our opinion, this is theprinciple obstacle in creation of a truly multiscale model oftumor growth, which not only mimics all the chemical andbiochemical inter and intracellular processes but allows forrealistic tumor progression in macroscopic scale as well.

Complex automata model driven by particle dynamics

Our complex automata (CxA) model of angiogenesisfollows the general principles of particle model describedin [40–43]. The system, representing the fragment of tissue,is made of a set ΛN={Oi: O(xi,vi,ai), i=1,…,N} of particles(agents, in terms of CxA terminology [44]), where i is theparticle index and N is the number of particles. Eachparticle is defined by three vectors: position xi, velocity vi,and attributes ai.

The attribute vector a=(tp, kd, sz, tcell, thp, cTAF, cO2,cother, p), where:

tp is particle type: ∈{tumor cell (TC), normal cell (NC),endothelial cell (EC)}

kd cell life cycle state: ∈ {newly formed, mature, inhypoxia, after hypoxia, apoptosis, necrosis}

sz cell sizetcell cell age (life clock)thp hypoxia timeck concentrations of k=TAF, O2 and other factorsp total pressure exerted on particle i from its closest

neighbors and the walls of the computational box.

In fine-grained models a particle mimics a single cell orcomponents of ECM. Nevertheless, this assumption becomesvery computationally demanding for modeling tumors ofrealistic sizes. The largest molecular dynamics (MD) simula-tion involved 2×1010 particles [45]. However, the number oftimestep of this simulation was not so impressive (only 50–100 timesteps). These computational restrictions imposecurrent limits on spatio-temporal scales simulated by ourparticle model. A tumor of 1 mm in diameter consists at leastof a million cells [11]. To simulate the proliferation of atumor of this size, located in a fragment of tissue of 8 mm3 involume in 3-D, one needs at least 1.6×107 particles simulatedin 105 timesteps. This requires approximately the samecomputational resources as the largest MD simulations.

However, particle method can also be used in its coarse-grained form (as in [42]). In particle models, such asdissipative particle dynamics (DPD) [46], fluid particledynamics (FPM) [47] or smoothed particle dynamics(SPH)[48], the particle can represent the cluster of atoms ormolecules. The clusters interact with each other via acentral potential such as in the pure MD method. This time,however, the interaction is different than for MD. It usuallyconsists of conservative (repulsive), dissipative and Brow-nian parts [46, 47]. In a similar way, we can assume that inour coarse-grained model a particle can represent afragment of tissue with additional attributes, such asconcentration of normal, tumor, EC cells and extracellularmatrix ingredients. By defining cluster size (the modelgranulation), this approach allows for simulating tumorgrowth in various spatio-temporal resolutions.

Unlike in purely fine-grained particle model, where allcells and tissue ingredients are represented by particles, inour model a particle represents a single tumor or normalcell in ECM envelope. Thus, some ECM properties, such asdensity, can be reflected in parameters of cell-cell interac-tion model.

In a truly fine-grained model, the vessel walls had to bemade of small and thin endothelial cells. Moreover,

Fig. 3 Two snapshots fromsimulation of avascular tumorgrowing in two (2-D) and three(3-D) dimensions in similarconditions. The particle basedCxA model was employed. Intwo dimensions the vasculatureis unrealistically stretched andpushed away from the interior ofgrowing nucleus while in 3-Dthe tumor is rooted by thevessels

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sprouting and bifurcation of vessel tip is a very complexbiological process involving tens of cells, many growthfactors, chemical species and ECM structure [9, 15, 21, 22].For the sake of simplicity we assume here that the vessel isconstructed of tube-like “particles” - EC-tubes - each madeof many endothelial cells. As shown in Fig. 4, we definethree types of interactions: sphere-sphere (A), sphere-tube(B), and tub-tube (C).

The forces between cells should mimic both mechanicalrepulsion from squashed cells and attraction due to celladhesiveness [17] and depletion interactions [43, 49]. Themechanical repulsion can be approximated by Hook law.We assume, for simplicity, that the attractive tail has similarcharacter but it is less rigid than the repulsive part (seeFig. 4D). The attractive forces are short-ranged. Thisheuristic potential Ω(dij) (illustrated in Fig. 4D) we defineas follows:

Ω dij� � ¼ a1d

2ij; for dij < dcut;where a ¼ a1 for dij < 0

a2 for dij � 0and a1 << a2

(a2d

2cut; for dij � dcut

8>><>>: ð1Þ

and

dij ¼ ~rij�� ��� ri þ rj

� � ð2Þwhere dij is the distance between cell walls. In moreadvanced models, tumor and normal cell interactions can berepresented employing different interaction parameters a1,2due to different mechanical properties: elasticity andadherence of normal and tumor cells [15, 17, 39]. Theinteractions between spherical cells and EC-tubes havesimilar character. We assume that despite the elongated

shape of the tube the interactions are central. The tube-tube interactions are also central. This assumption isjustified because torque will not produce remarkableeffects for slow and highly dissipative tumor dynamics.Moreover, by neglecting angular momentum, we canreduce computational load required for numerical integra-tion of equations of motion. Only the tips of interactingtubes can fuse them together. Consequently, the chain oftubes forms blood vessels such as in Fig. 4C and inFig. 8.

Fig. 4 The types of particlesand interactions used in theparticle based model

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The particle dynamics is governed by the Newtonianlaws:

midVi

dt¼ �a � rΩ dij

� �� l � Vidridt

¼ Vi

rij ¼ ri � rj� � � ri � rj

� �T ð3Þ

where mi, ri and Vi are the mass, position and velocity ofparticle i, respectively, while λ is a friction coefficient. Thisset of equations of motion is solved numerically by usingdirect leap-frog scheme (see overview [43, 49]). The totalforces acting on particles are calculated by using linked-cells method combined with the Verlet algorithm (see e.g.,in [49, 50]).

The particle system representing a growing tumor is veryunstable. The number of cells increases and/or fluctuatesbecause they can replicate or disappear. During the life-cycle, the normal and tumor cells change their states fromnew to apoptotic (or necrotic) according to their individualclock and oxygen concentration. The cells of certain ageand size and being well oxygenated undergo the process of“mitosis”. They split into two daughter cells. The newformed cell is in new state and its diameter is equal to dMIN.The diameter increases with time up to dMAX, proportionallyto the oxygen concentration. The minimum and maximumparticle diameters depend on the model granularity. Finally,after time TN, the particles undergo apoptosis, i.e.,programmed cell death. Consequently, dead cells areremoved from the system. For oxygen concentrationsmaller than a given threshold, the cell (which is not inthe necrotic state) changes its state to hypoxia. Such cellsbecome the source of TAF. The cells, which are in hypoxiafor a period of time longer than a given threshold, die andbecome necrotic. We assume that in the beginning, thediameter of a necrotic cell decreases twice and, after sometime, the cell is removed. This is contrary to apoptotic cells,which are rapidly digested by their neighbors or bymacrophages. The diagram of the cell cycle is shownbelow (Fig. 5). As was mention before, the duration ofphases of the cell cycle for normal and tumor cells differsconsiderably. Also the behavior of tumor and normal cell isdifferent. For example, the proliferation rate of tumor cells,which were in hypoxia state, can change. As shown in

Fig. 6, the cells being a certain period of time in hypoxiaproliferate faster than those evolving in normal conditions.In Fig. 7 we present two snapshots from simulation of agrowing cell cluster.

The cell cycle for EC-tubes is different than for sphericalcells. In fact, EC-tubes are clusters of endothelial cells. Thetubes grow both in length and in diameter. The vesselscollapse due to a combination of severely reduced bloodflow, the lack of VEGF, dilation, perfusion and solid stressexerted by the tumor. However, because the EC-tube is acluster of EC cells, its division onto two adjoined tubesdoes not represent the process of mitosis but is acomputational metaphor of vessel growth. Unlike, normaland tumor cells, the tubes can appear as tips of newlycreated capillaries sprouting from existing vessels. The newsprout is formed when the TAF concentration exceeds agiven threshold. Then the “vessel particle” undergoes theprocess of “mitosis” directed to the local gradient of TAFconcentration. The haptotaxis involving fibronectin [12, 15]is not included yet in the model. In Fig. 8 we show twosnapshots from simulation of spontaneously growing tubecluster.

The cells interact with each other not only by means ofmechanical forces. As shown in [17], under certain circum-stances, cancer cells can modify their environment. Partic-ularly, they are able to increase local acidity. Because tumorcells show greater resistance to acid concentration than thenormal cells, they can eliminate normal cells from tumorcells neighborhood by increasing acidity. In our model,totalistic cellular automata rules are used to simulatecompetition between tissue and tumor cells. The decisionsabout survival or death of neighboring healthy cells will betaken comparing the intracellular concentration of acid to acritical threshold.

The transport of bloodborn oxygen and TAF into thetissue is modeled by means of reaction-diffusion equations.The distribution of hematocrit is the source of oxygen,while the distribution of tumor cells in hypoxia is thesource of TAF. On the other hand, the distribution of cellsgives us the spatially distributed sink of both oxygen andTAF. We assume that the cells of any type consume oxygenand the rate of oxygen consumption depends on both celltype and its current state [51]. We assume additionally thatonly EC-tubes absorb TAF. TAF is washed out from thesystem due to blood flow. Of course, one could employmore detailed models involving VEGFb absorption in ECM[25], however, at the expense of considerable increase ofcomputational load.

Diffusion of oxygen and TAF is many orders ofmagnitude faster than the process of tumor growth. Onthe other hand, the blood circulation is slower thandiffusion but still faster than the mitosis cycle (seeFig. 1). Therefore, we can assume that both the concen-Fig. 5 The simplified diagram of cell fate

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trations (of oxygen and TAF) and hydrodynamic quantitiesare in steady state in the time scale defined by the timestepused for numerical integration of equations of motion(Eq. 3). This allows for employing fast approximationprocedures for both calculation of blood flow rates incapillaries and solving the reaction-diffusion equation.

We compute the blood flow rates employing Kirchoff’slaws (same as in [17]). We assume that only vessels withcirculating blood (anastomosing vessels) are the sources ofoxygen. Moreover, both the oxygen and hematocritconcentrations in blood do not change along capillariesand the oxygen supply is proportional to the blood flowrate.

To calculate the concentrations of oxygen and TAFwe solve the reaction-diffusion equations numericallyby using approximation theory. One can estimate afunction f at position r by using smoothing kernels W asfollows:

f rð Þ ¼Xnj¼1

mjfjrjW r� rj; h� � ð4Þ

where mj is the mass, rj is the position, ρj is the densityand fj is the quantity f for neighbor particle j, respectively.Here, n is the number of neighboring particles within cutof radius h (|r − rj| ≤ h). When r=ri, f(r) is denoted by fi.The smoothing kernel approximates a local neighborhood

r within distance h. Thus, we can estimate the density ρifor a particle i at location ri by:

ri ¼Xnj¼1

mjW ri � rj; h� � ð5Þ

where j is the index of the neighboring particle. Thekernel should be smooth, symmetric and satisfy thefollowing equation:ZΩ

W r; hð Þdr ¼ 1: ð6Þ

We used 3D poly6 kernel proposed by Muller et al. [52].

Wpoly6 r; hð Þ ¼ 315

64ph9h2 � rj j2� �3

rj j � h

0 otherwise

8<: ð7Þ

We selected this kernel due to its simplicity. As shown in[52], better kernels can be used, however, at the cost ofcomputational efficiency. The Laplacian can be approxi-mated then:

Δfi ¼Xnj¼1

mj

rjfj�ΔW ri � rj; h

� �: ð8Þ

Substituting Laplacian in the reaction-diffusion equationby Eq. (8) we got the following expression for concen-

Fig. 7 Two snapshots fromsimulation of a growing spheri-cal particle cluster due to mitosisof cells

Fig. 6 The diagrams illustrating speed of growth of mesothelioma cell line before (control) and after hypoxia (0, 24 and 48 hours)

1526 J Mol Model (2009) 15:1517–1539

trations ciK of K={oxygen, TAF} in particle i (χK - reaction

factor). When K=oxygen, then I=TAF and vice versa.

cKi ¼

cIi cIi

D � Pnj¼1;i 6¼j

mj

rjcKi �ΔW ri � rj; h

� � !

miriΔW 0; hð Þ � cKi

D

� � ð9Þ

By solving this equation iteratively in each time-step ofNewtonian equation integration, we obtained the approxi-mate concentration of TAF and oxygen in each particlelocation.

The particles are confined in the cubical computationalbox of volume V. Because the average kinetic energy in thesystem is negligible small, from the virial theorem weobtain that:

P � 1

3V�XNi< j

Fij � rij: ð10Þ

The internal pressure increases due to increasing numberof particles (cells). The increase of box volume Vcompensates the pressure increase above a given threshold.

In Fig. 9 we show basic procedures of our CxA particlemodel, which include the model initialization phase, i.e.,definition of initial and boundary conditions, and itsevolution driven by the following phenomena, which aremodeled explicitly:

a. Newtonian dynamics of interacting cells,

b. diffusion of oxygen and TAF,c. cellular life cycle modeled by CxA rules,d. vessels sprouting and growth,e. vessels remodeling due to flow and maturation.

In the following section, we describe results frommodeling of the tumor proliferation in both avascular andvascular phases by using the particle model describedabove.

Results of modeling

Assumptions and simulation parameters

In this paper we do not investigate a growth pattern of aspecific cancer histology. We focus our attention on thefunctionality and correctness of our model as a metaphor oftumor growth. Below we discuss the parameters andshortcomings we employ in our model.

The spatial scales we used in the particle model aredefined by the cell diameter, the size of computational boxand microvascular density (MVD). We have assumed thatthe tumor cell (TC) diameter is 30 μm. It is larger than intypical models (e.g., in [17–19]) (10 μm) but it is smallerthan V-79 cell described in [7]. This is a rather technicalshortcoming to decrease the computational load. The tumorof about 1 mm in diameter would consists then of 5×104

Fig. 8 The snapshots from simulation of growth of tube particle cluster due to mitosis

J Mol Model (2009) 15:1517–1539 1527

cells instead of millions. The particles (cells) are confinedin the computational box of size 1.5 mm×1.5 mm×1.5 mm(or 2 mm×2 mm×2 mm for the largest simulations). Totalnumber of cells in the box is 105 of order. We haveassumed that if the total pressure exceeds a given threshold,the stress is dissipated and the pressure in the box stabilizes.In order to keep it constant, those cells which are close tothe box walls and which are the most suppressed areremoved from the system. The initial state of the particlesystem at t=0 represents a regularly vascularized region ofa given microvascular density MVD with a small tumor in

the center (see Fig. 11). Vessels are arranged in a regulargrid with a lattice constant a=100 μm. This yields MVD=100 vessels per mm2 in 3-D. As shown in Fig. 10, thisstructure reflects well real oxygenation of modeled tissuefragment.

We have chosen the timestep of simulation assumingtypical time scales of cell life cycle and tumor progressionrates such as division time, average cell life time inhypoxia, doubling time of tumor volume (see Table 1). InEden model described in [18, 19] Δt is set to 1 h. Due tocells motion driven by cells proliferation and their death,

Fig. 9 The main procedures ofthe model

1528 J Mol Model (2009) 15:1517–1539

the time-step for numerical integration of Eq. (3) in theparticle model has to be smaller. On the other hand, becauseof high dissipation rate in the equations of motion(sufficiently large friction λ in Eq. (3)) the time-step hasnot to be so rigorously defined as for other particlemethods, e.g., such as in molecular dynamics. On the baseof our experience we assumed that in our simulations Δt=0.25 h.

Functional vessels, i.e., the vessels with flowing blood,are sources of oxygen while the cells are spatiallydistributed sinks of oxygen. For computational tractabilitywe assumed a fixed O2 secretion rate along the vessel witha given flow intensity. This assumption overestimates theO2 concentration in regions with high MVD, but this doesnot alter the model outcome significantly [18, 19]. We solvediffusion equation assuming that the diffusion coefficient isDO2=1.75×10

5 cm2/s [51, 53]. The resulting O2 concen-tration profile is fit to experimental data from Fig. 10.Given this initial configurations, we use Kirchoff’s laws tocalculate the flow intensities in newly formed anastomosesand remodeled vessels. This then provides us with initialconditions for updating the O2 concentration profiles in thefollowing simulation steps. As shown in Table 1, maximalsurvival time in hypoxia are considerably lower for normalthan for tumor cells, moreover, tumor cells can survive inmuch lower O2 concentration [39, 54]. The metabolicparameters and thresholds for cells in hypoxia are partlycopied from [39, 51, 55] and from experimental resultsobtained by one of the authors of this paper (A.Z. Dudek).Diffusion coefficient for TAF is assumed to be 10−6 cm2/s[18, 53]. The concentration profiles for TAF and oxygen areupdated simultaneously according to Eq. (9).

At the beginning of simulation, apart from the grid ofvessels, the box is filled with only normal cells. Weassumed that the normal cells do not proliferate and theirlife-time is very long. However, they can die due to necroticfactors like hypoxia or high local acidity caused by tumorcells [17]. The growing tumor pushes away the vessels,opening the gap of poorly oxygenated cells. When hypoxicregion in avascular tumor is sufficiently large to secreteenough quantity of TAF, the angiogenic phase is initiated[6, 10, 54].

The newly formed blood vessels become functionalwhen they form anastomoses allowing for blood flow dueto pressure difference on its ends. Otherwise, we assumethat the functional capillaries without pericyte supportdissolve during 70 h. Moreover, nonfunctional and imma-ture vessels without blood flow are removed from thesystem after 10 h from the moment the VEGF concentrationfalls below a given threshold [5, 57]. The values of themost of parameters are copied from existing computationalmodels, e.g., [2, 17–19].

The vessel maturation is controlled by the density ofpericytes [5, 15, 22, 23]. The varying degrees of pericyterecruitment indicate differences in the functional status ofthe tumor vasculature. In our simulations we assume a verysimplistic model of vessel maturation. The regression timedepends on the local density of EC tubes. If the density istoo high regression time is shorter. Dll4 regulates thesprouting rate. The lack of Dll4 results in excessivesprouting of newly formed vessels. As shown in Table 1,some simulation parameters such as those defining particle-particle interactions are specific for our particle model andcannot be measured experimentally. The others, likepressure limits and energy dissipation rates were setheuristically on the base of observation of the modelbehavior. The majority of measurable biological parameteris typical for similar type of computer modeling presentedin [12, 15, 17, 18]. The main parameters are collected inTable 1.

Modeling of avascular phase of tumor growth

Because the size of thecomputational box is about 1.5–2 mm, we assume that the diameter of avascular tumor doesnot exceed 0.5 mm. It means that the thresholds on vesselssprouting must be kept on such a level that TAF secretionfrom hypoxic part of the tumor is not sufficient to initiatethe process of angiogenesis. The other factor, whichcontrols the tumor growth rate, is the pressure computedfrom Eq. (10). As shown in Figs. 3; 11, the swelling tumorpushes away the capillaries increasing the space of poorlyoxygenated cells in its interior. The nodule after beingexposed to the critical stress, gradually relaxes its stressallowing the pathology to continue expanding. At the

Fig. 10 Oxygen penetration depth for mouse according to [56]. TheO2 pressure (pO2) dropped exponentially and reached plateau valuesof 3 mm Hg at around 200 μm away from the blood vessel. Assumingthat critical pO2 for healthy tissue is about 8-10 mmHg while about 2-4 mmHg for tumor cells [51, 54, 56], the average distance to thehypoxic TCs should be about 150 μm. The red curve is theexponential fit of experimental data

J Mol Model (2009) 15:1517–1539 1529

Table 1 Main simulation parameters (p.u. – program units)

1. Global physical and numerical parameters

Name Description Values Units

timestep The length of timestep 0.25 h

no_timesteps Typical number of timesteps 104 - 105

box_size The size of computational box 1-2 mm

no_cells Typical total number of cells simulated up to 3×105

diffusion_coef_TAF Diffusion coefficient of TAF 0.002 p.u.*

diffusion_coef_O2 Diffusion coefficient of O2 0.035 p.u.

force_cut_off Cut-off radius in forces calculations 30 μm

2. Biological and local physical parameters

a. Tumor cells

tc_mass Mass of cell 0.5-1 p.u.

tc_diameter Diameter of mature cell 15-30 μm

tc_mitosis_rate Doubling time for tumor 200 h

tc_nutrient_consumption O2 consumption speed in cell 0.12 p.u./h

tc_nutrient_mitosis Minimal concentration of O2 for mitosis 6 mmHg

tc_nutrient_hypoxia Min-Max concentration of O2 in hypoxia 2-5 mmHg

tc_max_hypoxia_time Maximal life time in hypoxia 50 h

tc_grow_speed Cell growth rate 0.2 μm /h

tc_min_mitosis_size Minimal cell size for mitosis 90 %

tc_taf_generation TAF generation rate 0.1 p.u./h

tc_force_in(out) Coefficient of cell-cell interactions 0.001 p.u.

tc_mitosis_pressure_limit Pressure inhibiting cell mitosis 0.3 p.u.

tc_grow_pressure_limit Pressure inhibiting cell growth 0.5 p.u.

tc_shrink_speed Speed of necrotic cell shrinking 10 μm /h

tc_dead_age Maximal time rate of dead cell removal 10 h

tc_dead_reduction Radius reduction rate for dead cell 1 μm /h

b. healthy cells

hc_mass Mass of cell 0.5-1 p.u.

hc_diameter Diameter of cell 15-30 μm

hc_max_live_age Maximal life time of cell 5000 h

hc_nutrient_consumption O2 consumption speed in cell 0.12 p.u.h-1

hc_nutrient_mitosis Minimal concentration of O2 for mitosis 10 mmHg

hc_nutrient_hypoxia Min-Max concentration of O2 in hypoxia 5-7 mmHg

hc_max_hypoxia_time Maximal life time in hypoxia 5 h

hc_force_in(out) Coefficients of cell-cell interactions 0.001 p.u.

hc_shrink_speed Speed of necrotic cell shrinking 10 μm /h

hc_dead_age Maximal time rate of dead cell removal 20 h

hc_dead_reduction Radius reduction rate for dead cell 1 μm /h

c. vessel tubes

vc_mass Mass of EC tube 2.3-5 p.u.

vc_diameter Diameter of EC tube 10-50 μm

vc_ length The length of the EC tube 70-150 μm

vc_tip_mitosis_rate Doubling time for vessel tip tubes 200 h

vc_grow_speed Growth speed 0.2 μm/h

vc_thicken_speed Thickening speed 0.002 μm/h

vc_min_mitosis_size Minimal tube diameter for division 95 %

vc_branch_taf_trigger TAF concentration allowing for sprouting 0.00005 p.u.

vc_force_in(out) Coefficient of tube-tube interactions 0.001 p.u.

vc_glue_force Coefficient of forces gluing tubes in vessels 0.005 p.u.

vc_mitosis_pressure_limit Pressure inhibiting cell mitosis 100 p.u.

vc_grow_pressure_limit Pressure inhibiting cell growth 100 p.u.

vc_max_flowless_time Regression time allowed for non-functional vessels 40 h

* p.u. - program units

1530 J Mol Model (2009) 15:1517–1539

beginning the pressure weakly constrains tumor progres-sion. However, as displayed in Fig. 12, after an initialexponential growth phase leading to tumor expansion,growth saturation is observed even in the presence of aperiodically applied nutrient supply. This observation is in a

good correspondence with experimental results [7]. Thesection of the tumor spheroid, shown in Fig. 11, displays alayered structure. A core zone composed mainly of necroticmaterial is surrounded by a thin layer of quiescent tumorcells and an outer ring of proliferating tumor cells. It iscrucial to understand the processes, which are responsiblefor the growth of a layered and saturating tumor.

With increasing size and cell number, the spheroidrequires more oxygen (more energy). Since the nutrientconcentration is the lowest in the center of the avasculartumor, cells will starve here first, and may eventually die(necrosis). The cells under necrosis swell and burst,forming a necrotic site. The necrotic center collects thenecrotic remains, and is much ‘softer’ than living cell layers[35]. The external pressure stimulates the mixing phenom-enon similar in spirit to Rayleigh-Taylor instability boostedadditionally by spherical geometry of the particle system(see [41]).

As shown in Fig. 13, the cells from the outer shell arepushed toward the necrotic center. Initially, the inward flowis slow since the necrotic core is not existing or small.Consequently, the outmoving cell population dominates, i.e., the tumor expands. Later on, if the necrotic core hasreached a critical size, the inward motion takes over whichlimits further tumor growth.

The experimental work by Dorie et al. [58] confirms thisobservation. They showed that there are not only cellsmoving toward the periphery but also there is a significantnumber of proliferative and quiescent tumor cells movingfrom the periphery toward the core area. This inward cell

Fig. 12 The plots displaying the number of cells – the total number oftumor cells, cells in hypoxia, and dead cells - with time for growingtumor (see Fig. 11) under different external pressure exerted by thetissue on the tumor mass. The filled symbols correspond to the tumordynamics inside the tissue which is 10% softer than while opensymbols

Fig. 11 The snapshots from 3-D simulation of avascular tumor.The tumor from B is stable andits size does not increase anylonger. C and D show the cross-sections through the tumor fromB. The tumor diameter is about0.35 mm. The initial distancebetween the vessels is 0.15 mm.Only tumor cells are shown.Blue particles are well oxygen-ated cells; green particles repre-sent hypoxic cells and blackones the necrotic center. Thetissue cells are invisible andwe assume that they are welloxygenated

J Mol Model (2009) 15:1517–1539 1531

motion is a necessary condition for growth saturation ofavascular tumor. If there would be no cell flowing towardthe center but only stationary cells and cells moving in thedirection of the outer shell, constant oxygen delivery wouldimply unbounded tumor growth.

This “antagonistic growth direction” was explained in[35] by the chemotaxic migration of tumor cells into thedirection of the maximum necrotic signal gradient. Accord-ingly, in [35] the authors assumed that a diffusible signalemitted by bursting tumor cells is attracting living tumorcells. In our opinion, explanation of the inward cell motionby chemotaxis directed to the necrotic center is superfluousin the context of Occam razor principle.

As shown in Fig. 12, higher external pressure producestumors of smaller volume. However, this effect is notspectacular. Considerable increase of the external pressure(two times) results in rather small (10%–15%) decrease oftumor volume and causes that also the volume fractionoccupied by quiescent cells decreases (by 20% in Fig. 12).Most of solid tumors, even small ones 1–3 mm indiameter, exhibit hypoxic fractions that may range from10 to 30% [55]. On the other hand, the size of necroticcenter cannot shrink to be too small to stop the inwardmotion of cells.

Its size should be adequate to guarantee the stabilizationof the tumor proliferation. We can conclude that highexternal pressure can decrease the hypoxic fraction ofavascular tumor below a threshold required to initiate theangiogenic processes. Such the quiescent tumor canpreserve its size for years until the drop of external

pressure. It can be caused, e.g., by tissue matrix degradationdue to destructive influence of tumor cells (see Fig. 17).

When avascular tumor is getting larger, non-uniformremodeling of capillaries caused by mechanical forces (seeFig. 3B and Fig. 14) may result in hypoxia not only in thecentral but also in many peripheral parts of the spheroid(see Fig. 14B). Deployment of additional sources of TAFcan accelerate the process of angiogenesis. The angiogen-esis in the peripheral part of the tumor is even morefeasible, because of high density of blood vessels on itsborder (see Fig. 3 and Fig. 15).

As shown in Fig. 14C, the process of angiogenesiseliminates the hypoxic spots. All the tumor cells are almostuniformly oxygenated. The angiogenic phase stops. In thefollowing section we show how the process of vasculari-zation changes the way of tumor proliferation.

Modeling of tumor vascularization and remodeling

Because of high computational load required for 3-Dsimulations we assumed that angiogenesis initiates in avery early stage of tumor growth (its diameter is about100 μm instead of a more realistic 1 mm). This assumptiondoes not change too much the scenario of tumor progres-sion at the start of vascularization process. As shown inFig. 16, increasing vascularization leads to increasingacceleration growth rate. The tumor evolution starts frompower law type of growth. The power order jumps whenavascular phase is replaced by angiogenic phase of tumorgrowth. At this stage however, we cannot estimate which

Fig. 13 The snapshots fromsimilar simulation as shown inFig. 11. The figures illustrate theinward motion of tumor cellsfrom the tumor surface to itsnecrotic interior. The threearrows of various colors showthe cells which disappear, oneby one, in the necrotic tumorcenter

1532 J Mol Model (2009) 15:1517–1539

kind of growth is expected for a larger tumor. Because oflimited size of computational box, the growth is restraineddue to increasing pressure.

The tumor evolution in bone cavity is illustrated inFig. 17. During unrestricted growth (no pressure exertedfrom healthy tissue) avascular tumor expands fast along thecapillary (Fig. 17B). In angiogenic phase, tumor cells fillthe whole cavity to the moment when external pressure ishigher than the internal pressure. Stimulation of bone lysisby tumor cells allows for local destruction of the porousbone structure. The existing and newly formed bloodvessels position themselves within bone matrix niches,

helped by high pressure exerted by the tumor on the bonewalls (see Fig. 17).

In vascular phase the growing tumor produces its ownvascular network with a specific microvascular density(MVD). MVD is used as an important diagnostic tool incancer therapy [5]. However, MVD is not uniform in tumors.Therefore, a quantitative understanding of the mechanism thatcauses compartmentalization of the tumor vasculature iscritical for more accurate assessment of influence of vasculargrowth stimulators on tumor proliferation. The mechanismleading to excessive vascularization on the peripheral of thetumor was explained in the previous section (see Figs. 3, 15).

Fig. 14 The cross-sections of avascular tumors (A, B) and tumor withvascularization (C). The tumor diameters are about 0.45 mm (A) and0.55 mm (B, C). Unlike the tumor from A, the tumor from B has a

few necrotic centers. The tissue cells are invisible and only tumor cellsare shown. Blue particles represent well oxygenated cells while greenand black cells are hypoxic and necrotic, respectively

Fig. 15 The histogram of vas-cularization around the pre-angiogenic tumor. Blue barsshow the microvascular densityin the absence of tumor

J Mol Model (2009) 15:1517–1539 1533

We showed that a high fraction of vessels close to the tumorshell come from normal tissue. These vessels are denseenough to stabilize neovasculature.

The vasculature of tumor interior is in many respectsdifferent from the regular vasculature in normal tissues.Tumor vessels are less stable than their normal counterpartsand when engulfed by the growing cancerous tissue theyundergo processes of regression. As a consequence, newly

formed immature vessels fail functionally. Arrested bloodflow in deficient blood vessels within tumor interior causestheir collapse. This stimulates both the growth of newvessels in this place and increases the blood pressure inother immature vessels. The regression-growth processesrepeat all over again. As shown in [18, 19], this process isdriven toward criticality - the percolation threshold - via amechanism of vessel stabilization by increase of blood flowin the remaining vessels.

The functional status of the neovasculature depends onmany factors. The most unstable are open vessels withoutblood flowing that dissolve quickly. Only vessels creatinganastomozes and pass up blood have a chance to survive.Nevertheless, inhibition of VEGF, e.g., in cocultures, leadto a 75% increase in EC apoptosis [22] which cases vesselregression and blood leaks. All of these factors cause that inmost cases the MVD decreases in direction from peripheryto the tumor interior [2, 5, 18].

As shown in Fig. 18 which presents the snapshots from2-D simulation of tumor growth in well oxygenated normaltissue, our particle model reproduces well this highlyinhomogeneous tumor specific capillary network. Theresulting network is very heterogeneous, composed ofdense and void regions, and has distinctly differentstructure from normal arteriovenous or normal capillarynetworks [2]. Moreover, the emerging tumor morphology ischaracterized by the compartmentalization of the tumor intoseveral regions differing in vessel density, and extent ofhypoxia [5]. This situation is clearly displayed in Fig. 18.

Fig. 16 The acceleration of tumor progression due to angiogenesisassuming two VEGF thresholds

Fig. 17 The cross-section of 3-D tumor evolution in a bonecavity. The white motionlesscells represent the bone. Thetumor fills the cavity exertingpressure on the bone walls

1534 J Mol Model (2009) 15:1517–1539

In our model we consider two corresponding factorsregulating vessels stability. First, we can mimic the activation(inhibition) of delta-like 4 (Dll4)-Notch1 signaling [13, 14]. Itregulates the formation of appropriate numbers of tip cells tocontrol vessel sprouting and branching. Inhibition of Notchsignaling promotes increased number of tip cells. Conversely,activation of Notch leads to fewer tip cells and vesselbranches. Pericytes are the second factor which regulatevessel stability [15, 22, 23]. Pericytes secrete an inhibitor thatcontrols endothelial growth at certain cell densities. Thevarying degrees of pericyte recruitment indicate differences inthe functional status of the tumor vasculature and may reflectvarying degrees of maturation of the tumor vascular bed.

Inhibition of Dll4, resulting in excessive sprouting,branching and formation of immature vessels without blood

flow, causes that pericytes concentration becomes too lowand vessels not supported by pericytes undergo regression.Otherwise, for high activation of Dll4, as in the situationshown in Fig. 19, pericyte covered vessels stabilize andbecome functional. The maturation of vessels due topericytes for high activation of (Dll4)-Notch1 signalingwere simulated using our 2-D particle model. The snap-shots from this simulation are presented in Fig. 20.

Concluding remarks

The concept of complex automata and interacting particlesrepresent a novel framework for constructing multi-scalemodels of tumor progression. We show that this approach is

Fig. 18 The snapshots from 2-D simulation of vascularizedtumor evolution. The gray cellscorrespond to normal (healthy)tissue while colored cells repre-sent tumor. The color denotesthe extent of cells oxygenation(from blue - highly oxygenatedcells, to green – cells in hypox-ia). The arrows shows thetissue cells absorbed inside thetumor mass due to mechanicalremodeling

Fig. 19 The snapshot from 2-D model of angiogenesis, showing pericyte uptake by growing vessels

J Mol Model (2009) 15:1517–1539 1535

competitive both to existing paradigms based on continuummodels and classical cellular automata. The possibility oftumor dynamics simulation, which enables mechanicalremodeling of the blood vessel system, is the greatestadvantage of our model. Moreover, realistic 3-D dynamicsof the entire system consisting of the tumor and other tissuecells, blood vessels and blood flow can be reproduced byusing the same homogeneous particle model.

Our simulation results indicated that not only redistribu-tion and uptake of angiogenic factors during angiogenesishad significant effects on the structure and functionality ofvascular networks. The results of mechanical remodelingcan suggest as follows.

1. The layered structure of tumor can be explained solelyby the self-organized growth of an initially smallnumber of tumor cells.

2. The inward motion of peripheral tumor cells to itsnecrotic interior can be explained on purely mechanicalground, without somewhat suspicious necrotic chemo-taxis suggested in [35].

3. Larger value of MVD on the tumor peripheral isinitiated due to mechanical interaction of expandingtumor borders on existing vasculature.

4. External pressure decelerates tumor proliferation. Itevolves to a quiescent state.

5. Our modeling results support controversy to Folkman’soptimistic declaration that disruption of angiogenesisyields a starving tumor, which eventually dies. As show

the results produced by our model, due to mechanicalremodeling of healthy tissue, many disparate fragmentsof tumor become very well oxygenated. Expandingtumor can push out healthy cells and use regularvasculature to support its growth without the need ofexcessive angiogenesis.

6. The functionality of neovasculature depends on thebalance between pericyte and Dll4 concentrations.

To check if the vascular networks generated in numericalsimulations are really similar to the realistic ones we plan tovalidate our model. This validation will be conducted onthe basis of the comparison between realistic images oftumor vascular networks from confocal microscopy andcomputer experiments. The comparisons will be madeemploying structural properties of tumor vascularization.The vascular networks can be described by the featurevectors with statistical and/or algebraic descriptors ofcomplex networks [59] as the feature vector components.Finally, pattern recognition methods such as clustering andfeature extraction will be used for the vector classification.

The computational complexity and modeling constraintsare the main disadvantages of our model. The first problemresults from shortcomings we used in our simulations. Weapplied our particle model to a more developed stage oftumor growth, where the tumor is likely to comprisemillions of cells rather than thousands of cells. To solvethis problem we need to define a multi-scale model andresort to parallel computing techniques. Although this

Fig. 20 The snapshots of 2-Dsimulation of vascularization intumor. The green vessels repre-sent mature, pericyte coveredvessels

1536 J Mol Model (2009) 15:1517–1539

certainly implies some degree of complication from thecomputational point of view, the particle model is not toodifficult to implement in parallel [43, 49, 60]. The mostserious modeling constraints and simplifications used in ourmodel are as follows:

1. Due to limited size of computational box and growingnumber of cells the pressure in the box increases.Computationally expensive procedures have to be usedto maintain constant pressure inside the box.

2. The cells interaction is radial, which suggests thatthey are of spherical shape. Implementation of morerealistic anisotropic interactions is very demandingcomputationally.

3. We use for simplicity rigid EC-tubes instead of ECcells, which influences the shape of vessels and mayintroduce structural artifacts.

Our model of tumor-induced angiogenesis addresses alimited number of biological processes. For example, bloodhydrodynamics have received much less attention than inother models (e.g., in [3, 4, 17, 18, 29]). We did thisintentionally to reduce the computational load. However,we definitely agree that in larger tumors, vascular remodel-ing due to blood circulation cannot be neglected. Theexisting flow models (e.g., as in [17, 18]) can be easilyadapted in the scope of our particle model. For simplicity,we neglect also other phenomena. In particular, our modelhas not addressed the effect of perfusion and differentconcentration of oxygen along the vessel. Moreover, weconsider only oxygen as the nutrient while tumor cells oftendo not require much oxygen for growth, but glucose, fat,and amino acids. Another area in which our model could beimproved is introduction of microscopic sub-models repre-senting important intra cellular phenomena such as cellproliferation in hypoxic conditions. This challenge, how-ever, can be undertaken provided that the efficient parallelversion of the model will be implemented.

In our CxA framework it is easy both to developprincipal mechanisms of tumor dynamics such as cellproliferation, vascularization, vessels remodeling and in-corporate sub-models of other important processes such asmicroscopic intracellular phenomena, blood circulation, andextracellular matrix models. The particle model can besupplemented by more precise models of reproductionmechanism, cell-life cycle, lumen growth dynamics, hapto-taxis mechanism and others. Some models of intracellularphenomena can be copied directly from the work ofAlancorn et al. [17]. Moreover, more angiogenic factorscan be considered, such as various TAF activators andinhibitors. Instead of a simplistic model of blood circula-tion, advanced blood flow algorithms from Chaplain [12]and Stephanou et al., [4, 29] can be implemented in astraightforward way.

For simplicity, our model does not reflect the influenceof the extracellular matrix composition on endothelial cellmigration and network formation [15]. These very impor-tant processes can be simulated easily employing morecomplicated (e.g., anisotropic) interaction forces betweencells. One can assume, for example, that the particlerepresent the cell with a fragment of extracellular matrix.We can then use interaction forces depending on both therange and the angle of interactions. A more realistic thoughmore computationally demanding way for simulatingprocesses occurring in ECM involves additional type ofparticles representing various ECM components.

Most of these possible improvements will requiresupercomputing power to simulate realistic tumor sizes.This is mainly due to the assumption we made that oneparticle corresponds to a single cell. This puts the upperlimit on the size of simulated 3-D system to a fewmillimeters or at most centimeters when employingnowadays high performance multiprocessor systems. How-ever, as shown in [42, 43, 49], particle models can be usedfor simulating fluid flows in various spatio-temporal scalesby using a coarse graining procedure. We postulate that asimilar procedure can be defined for modeling the tissue.Depending on the spatio-temporal scale of simulation, theparticle can represent a cell, a cell with a fragment of ECMmatrix, cluster of cells and fragments of ECM matrix, or acloud made of mixture of cells, fragments of ECM matrixand microscopic blood capillaries. If such scalability of theparticle model is possible, this would allow for simulatingtumors in various spatio-temporal scales from smallavascular clusters to large masses.

In summary, we showed that the CxA particle modeldeveloped in this study can be used as a robust modelingframework for developing more advanced tumor growthmodels. Consequently, these models can be used toevaluate the effects of various types of therapies on tumordynamics, understanding interactions between variousgrowth factors, and to generate experimentally testablehypotheses.

Acknowledgments This research is financed by the Polish Ministryof Education and Science, Project No.3 T11F 010 30, internal AGHInstitute of Computer Science grant and Vlab project of NationalScience Foundation. The exemplar movies from simulations can bedownloaded from http://www.icsr.agh.edu.pl/∼wcislo/Angiogeneza/index.html.

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