complex dynamics of tumors: modeling an emerging brain tumor system with coupled...

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Physica A 327 (2003) 501–524www.elsevier.com/locate/physa

Complex dynamics of tumors: modeling anemerging brain tumor system with coupled

reaction–di'usion equationsSalman Habiba, Carmen Molina-Par./sb;c;∗,

Thomas S. Deisboeckd;eaTheoretical Division T-8, MS B285, Los Alamos National Laboratory,

Los Alamos, NM 87545, USAbMathematics Institute, University of Warwick, Coventry CV4 7AL, UK

cDepartment of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UKdComplex Biosystems Modeling Laboratory, Harvard-MIT (HST) Athinoula A. Martinos Center for

Biomedical Imaging, Harvard Medical School, Charlestown, MA 02129, USAeMolecular Neuro-Oncology Laboratory, Massachusetts General Hospital, Harvard Medical School,

Charlestown, MA 02129, USA

Received 16 October 2002

Abstract

One of the hallmarks of malignant brain tumors is their extensive tissue invasion, which rep-resents a major obstacle for e'ective treatment. In this paper we speci8cally model the invasivebehavior of such tumors viewed as complex dynamic biosystems. Based on the spatio-temporalpatterns seen in an experimental setting for multicellular brain tumor spheroids we propose aninvasion-guiding, dynamical pro7le of heterotype and homotype attractor substances. We presenta novel theoretical and numerical framework for a mathematical tumor model composed of aset of coupled reaction–di'usion equations describing chemotactic and haptotactic cell behav-ior. In particular, our continuum model simulates tumor cell motility guided by the principle ofleast resistance, most permission and highest attraction. Preliminary numerical results indicatethat the computational algorithm is capable of reproducing patterns similar to the experimentallyobserved behavior.c© 2003 Elsevier B.V. All rights reserved.

∗ Corresponding author.E-mail address: [email protected] (C. Molina-Par./s).

Abbreviations: ECM, extracellular matrix; EGF, epidermal growth factor; EGFR, epidermal growth fac-tor receptor; HGF/SF, hepatocyte growth factor/scatter factor; MTS, multicellular tumor spheroid; TGF�,transforming growth factor-alpha.

0378-4371/03/$ - see front matter c© 2003 Elsevier B.V. All rights reserved.doi:10.1016/S0378-4371(03)00391-1

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502 S. Habib et al. / Physica A 327 (2003) 501–524

1. Introduction

The poor prognosis of highly malignant neuroepithelial tumors (i.e., malignantgliomas) with a median survival time (for patients with glioblastoma multiforme) of 8months is primarily caused by rapid growth, cellular heterogeneity and extensive inva-sion of the surrounding parenchyma, rendering pure cytoreductive therapy ine'ective[1–4]. In fact, single invasive cells have been found at a distance greater than 4 cmfrom the gross tumor [5].Brain tumor invasion is a multi-step process, which includes single cell detachment

from the bulk of the tumor, receptor-mediated extracellular matrix (ECM) adhesion,followed by enzymatic matrix degradation and active cell locomotion [6,7]. Adhesion ismediated by integrin receptors, which are transmembrane heterodimer proteins involvednot only in cell cycle regulation and growth [8], but also in actin remodeling, which isa determinant of cell movement. Brain tumor cells produce a variety of proteinases, andan imbalance with their tissue inhibitors is thought to be responsible for the digestionof adjacent matrix and thus for the creation of space for the cells to move into. Celllocomotion is distinguished into active and passive movement. Active cell motility canbe driven by the gradient of a soluble factor, called chemotaxis, or can be facilitatedon a permissive solid substrate, called haptotaxis. As Thorsen et al. [9] have described,in brain tumors passive cell movement can be caused by cerebrospinal Guid dynamics.For other cell types mechanical stimuli have been discussed as a possible trigger forcell migration by Thomas et al. [10]; however, at this point, it is not clear if thisso-called contact-stimulated migration is relevant for in vivo brain tumor cell motilityas well. The main routes for brain tumor invasion in vivo are along the white 8bertracts and along the basement membranes of blood vessels or the glial limitans externa[7]. It has been suggested that those pathways not only represent speci8c (ECM-coated)permissive routes but also least resistance paths [7,11]. Finally, the complex interplaybetween proliferation and invasion reveals patterns of self-organization [12]. Moreover,using a migration assay Berens et al. [13] found that tumor cells exhibit either a highlyproliferative or a highly migratory pattern, leading to the hypothesis of a dichotomybetween both key features as presented by Giese et al. [14]. It is believed that suchsingle invasive cells can eventually turn into a proliferative mode, giving rise to distantsecondary tumor masses through local aggregation and on site division as suggestedby Bernstein et al. [15] and Suh et al. [16].

Each of the mechanisms leading to forward motion of cells is quite complicated andgiven the suggested cross-activity of some of these pathways their complex interplay islargely unknown. Most of the growth factors such as epidermal growth factor (EGF)or transforming growth factor alpha (TGF�), which trigger directed cell movementthrough receptor pathways, are also expressed by the tumor cells themselves, thussupporting the concept of autocrine and/or paracrine pathways [17,18]. At least invitro, brain tumor cells also produce ECM proteins and express integrins, thus creatinga more permissive local environment for migration and invasion [19,20]. In addition,cross-talks between these motility pathways have been suggested after recent workrevealed interconnections between the EGFR expression and migration stimulated bythe ECM protein laminin, which harbors EGF-like domains [21]. We propose that these

S. Habib et al. / Physica A 327 (2003) 501–524 503

tumors behave as complex dynamic self-organizing biosystems. We further argue thattumor invasion is a multi-step process directed by a variety of alternating intrinsic(i.e., homotype) and extrinsic (i.e., heterotype) factors, i.e., guided by a non-linear,temporally and spatially ?uctuating pro7le of these soluble and solid attractor andinhibitor substances. To validate these concepts one has to develop a new generationof computational tumor models capable of including concepts of non-linear dynamics,as well as able to generate patterns seen in experimental settings.

2. Previous work

Keller and Segel introduced the 8rst model for chemotaxis de8ning it as the “abilityof an organism of microscopic dimensions to sense and respond to macroscopic chem-ical gradients” [22]. Based on an analogy with Brownian motion they obtained arelation between the chemotactic sensitivity �c(C) and the cell di'usion �c of thetype

�c(C) = Ad�c(C)dC

; (1)

with A being a constant and C the cell density. This equation was deduced under somerestrictive assumptions and in a follow-up paper the authors [23] therefore introduceda chemotactic sensitivity of the simple form

�c(C) =�C; (2)

with � being the chemotactic coeLcient and a constant di'usion parameter for thecells, which is not of the form obtained in Eq. (1). Yet, this model seems “to be inreasonable accord with observation”. Lau'enburger and Aris [24] then used the Keller–Segel equations (with constant �c) to model migration assays and to obtain the rela-tionship between experimental cell migration distance data and the constant coeLcients�c and �c. Ford and Lau'enburger [25] applied the mathematical model, which had8rst been presented by Rivero and collaborators [26], to relate individual cell properties(i.e., “swimming speed” and “tumbling frequency”) to population parameters such asrandom motility coeLcient (i.e., di'usion coeLcient) and the chemotactic sensitivitycoeLcient. Maini et al. considered a chemotactic model, which yields pattern forma-tion on two-dimensional domains [27], whereas Nagai and Ikeda [28] employed thealgorithm introduced by Keller and Segel to show the existence of traveling waves formodels that include di'usion of the substrate. Perumpanani et al. [29] modeled a systemcomprising normal, non-invasive, and invasive tumor cells. They included both chemo-tactic and haptotactic gradients and obtained a traveling wave solution both for thevelocity of invasion and the pro8le of the invasive cells. Their one-dimensional model,however, required a large number of auxiliary parameters (more than 20). Sherratt andcoauthors have developed a model, which incorporates the receptor-based mechanismunderlying chemical regulation of cell motion [30]. The same authors concentrated,in a second paper [31], on the study of eukaryotic cell movement and considered an


approximate model based on that discussed in their earlier paper. The approximationin that model is of the Keller–Segel form and yields functional expressions for thetransport coeLcients and the chemical degradation rate. Furthermore, they were ableto relate these parameters to experimentally measurable quantities. Pettet and coauthors[32] considered the dynamics of a tumor cell population with di'usion and chemo-taxis due to the presence of nutrient gradients, based however on the steady-statestructure of the spheroid. More recently, Ferreira et al. [33] introduced a reaction–di'usion model for the growth of avascular tumors. Their cancer growth model takesinto account cell division, migration and cell death, as well as di'usion of growthfactors, yet it does not consider the e'ects of chemotaxis. It is well known, however,that cancerous cells secrete chemo-attractors into the surrounding tissue [34]. In thispaper Painter and collaborators addressed the issue of multiple chemotactic signalsfrom a macroscopic phenomenological approach, yet in particular considered bacterialchemotaxis.Discrete approaches such as the ones presented by Qi et al. [35], as well as by

Duechting et al. [36], have used cellular automata, which can properly describe the dis-crete nature of actual cells and thus model distinct cellular sub-populations as presentedby Kansal et al. [37,38]. Also, the regional e'ects of environmental heterogeneity suchas con8nement pressure can be readily simulated with such techniques. Conversely, theimpact of alternating gradients of growth factors and nutrition sources on the tumorsystem should be better suited for a continuum approach. There exists some work onmathematical brain tumor modeling predicting in vivo behavior. For example, Swansonet al. [39] recently presented a mathematical model to quantify the spatio-temporalexpansion of macroscopic gliomas in three dimensions. They speci8cally model thedi'erential motility of gliomas in the brain by assuming a larger di'usion coeLcientin the white matter.In this manuscript, we introduce a novel reaction–di'usion model to describe an

emerging brain tumor that includes both haptotactic and chemotactic e'ects. In the cur-rent iteration, we primarily study the dynamic interplay of di'erent chemotactic signalsfrom both homotype and heterotype chemo-attractors. The key di'erences between ourapproach and previous mathematical models are: (i) we consider cell di'usion, chemo-taxis and haptotaxis in one model, with the minimum number of parameters possible,in such a way that most of them can be determined experimentally or from availabledata in the literature; (ii) we consider the invasion phase and its dynamics explicitly,therefore avoiding restriction to the steady-state of the tumor spheroid; (iii) we solve(in two space dimensions) the coupled system of reaction–di'usion equations for thetime evolution of the mean density of tumor cells, homotype- and heterotype-solublechemo-attractors, and not simply the velocity pro8le of the tumor spheroidsurface.In the following section we will 8rst describe the motivation of the mathematical

model by brieGy introducing a novel experimental setting for studying expanding braintumor spheroid systems. The second part explains the oncology concept developed fromthese experimental 8ndings and therefore describes the background for the mathematicalmodel, which follows thereafter.


3. Experimental �ndings and oncology concept

3.1. Experimental 7ndings

We have previously developed a novel sandwich assay, placing multicellular tumorspheroids (MTS) between two layers of an extracellular matrix gel without addinga medium super-layer. Such spheroids behave in many ways similar to solid (butavascular) microscopic tumors [40]. The novel assay is described elsewhere in detail[12]. BrieGy, the human U87MGmEGFR glioma cells (kind gift from Dr. W. Cavenee(Ludwig Institute for Cancer Research, San Diego, CA)) were cultured in a humid-i8ed atmosphere (5% CO2 at 37◦C). These cells are stable-transfected to co-express(with the wild-type receptor) the mutant epidermal growth factor receptor mEGFR,which confers enhanced tumorigenicity in vivo [41]. U87MGmEGFR cells rapidlyform MTS in culture after reaching mono-layer conGuence and detach at a size rangeof about 500–700 �m in diameter (0.7–1:0 × 104 U87MGmEGFR cells). The MTSwere then placed between two layers of medium-supplemented, growth-factor-reduced(GFR) matrix, MatrigelR (BIOCATR, Becton Dickinson, Franklin Lakes, NJ), whichforms a reconstituted basement membrane at room temperature. It has been shown thatsuch basement membrane gels consist of interconnected networks of thin sheets ofECM proteins [42]. The total GFR-M/OPTI-MEMR [(serum-free) GIBCO Invitrogen,Carlsbad, CA] volume per well was 200 �l [48-well Gat-bottom tissue-culture-treatedMultiwellTM plates (FALCONR, Fisher Scienti8c, Pittsburgh, PA)]. The growth dy-namics were recorded daily by using a Nikon inverted light microscope and analyzedby an online image-analysis system (n= 30).As already previously reported in Deisboeck et al. [12], after a rapid growth phase,

the MTS showed decelerating growth dynamics. In parallel, an invasive networkemerged predominantly in two dimensions, i.e., in between the two layers of the gel(see Fig. 1). Between 24 h (t = 24 h) and t = 120 h, the MTS core increased from0:125 mm3 (±0:008 SEM) to 0:356 mm3 (±0:016 SEM) and the invasive area ex-panded from 0:147 mm2 (±0:013 SEM) to 1:394 mm2 (±0:104 SEM). The invasiveedge velocity showed a maximum of 4:5 �m=h (at t = 96 h), which is in agreementwith in vivo results reported by Chicoine and Silbergeld [43]. The experiments werecontinued up to 144 h post-placement of the MTS into the gel (t = 0 h).

3.2. Oncology concept

The MTS growth follows decelerating dynamics. As in macroscopic tumors, decel-erating dynamics in MTS are most likely due to increasing mechanical con8nement(corresponds to a loss of Guid over time in the gel-assay), lack of extrinsic nutritionand the onset of central apoptosis and necrosis [44,45]. These growth-limiting e'ectsare supposedly even more pronounced in our avascular microscopic tumor growth as-say than in vivo, where (neo)-vascularization can occur. From the proliferative surfaceof the tumor, cells are continually shed [46]. As can be seen in our in vitro systemthese single cells are capable of invading the surrounding environment (i.e., in vivo,the brain parenchyma).


Fig. 1. MTS assay. The central darkened spherical area depicts the proliferative MTS core. Note the extensivebranching system, which rapidly expands into the surrounding extracellular matrix gel. These branches consistof multiple invasive cells (t = 72 h; original magni8cation: ×40).

(1) We suggest that this single cell invasion follows a principle of “least resistance,most permission and highest attraction”. According to this concept, tumor cells areguided by heterogeneous properties of the gel in terms of regional mechanical resistance(consistency of the gel), permissive ECM proteins and local concentration of nutritivefactors. A similar assumption can be made for the real brain parenchyma, which consistsof neuroanatomically distinct regions, i.e., shows structural di'erences (such as neural8ber tracts and blood vessels). In particular, we assume that tumor invasion tends tobe directed towards metabolically attractive regions, thus largely avoiding hostile areas,i.e., inhibitors with high local mechanical resistance and/or toxic environment (suchas acidic or nutrition-depleted areas). Consequently, we introduce in the following ascenario in which tumor cells are attracted towards a strong gradient of a soluble factor,termed heterotype chemo-attractor, which di'uses into the gel from a distant nutritionsource. Naturally, such a source can be replenished (e.g., in the case of blood vessels)or non-replenished (as in our assay and mathematical model). In moving along thisattractive gradient (chemotaxis) cells also seek solid permissive substrates (such asECM proteins), which facilitate motility in addition (i.e., haptotaxis).


(2) Moreover, based on 8ndings in experimental brain tumor research, we furtherpropose the secretion of chemo- and haptotactic substances by the tumor cells, whichinduce the so-called “homotype (permission and) attraction”. We argue that both het-erotype and homotype factors are responsible for creating the single cell branchesseen in Fig. 1. If the “8rst” cell detects an attractive trace and moves forward tothis site choosing a least resistance and most permission path, the “choice” of thesecond cell leaving the tumor surface towards the same direction is already more de-termined. Following the preformed path, this “second” cell would move faster dueto the now slightly reduced mechanical resistance, as well as due to the enhanced(paracrine), ECM-mediated, permission. In addition, this following cell would be at-tracted by chemotactically acting soluble substances, (paracrine) produced by the 8rstcell within the “channel”. The more cells would follow, the more cells would be at-tracted until thresholds in receptor mechanisms are reached, and until raising mechanicalcon8nement pressure limits the expansion of the branch width and thus forces the cellsto either move faster within the channel or to expand the channel width beyond themechanical constraints. In this context, the initially mentioned concept of mechanicallytriggered, contact-stimulated migration could still then have relevance for 3D brain tu-mor invasion [10]. Much like the “8rst” invasive tumor cell, the following cells wouldalso move forward “choosing” a least resistance, most permission and highest attractionpath, yet then increasingly triggered by homotype, paracrine substances. Note that thiswould also ensure directed forward movement of cells shed at a later stage into a pre-sumably increasingly hostile microenvironment surrounding the main tumor related tothe necrotic tumor center, the highly metabolizing tumor surface and the nutrient con-sumption of the preceding invasive cell population. Most importantly, these proposedmechanisms would maintain a high sensitivity for heterotype attraction speci8cally inthe border zone of the invasive system. This would render the cells at the tips ofthe branches Gexible enough to rapidly approach extrinsic nutrient sources, crucial forboth, recharging the depleted “energy storage” of invasive cells as well as satisfyingthe metabolic needs for potential on-site proliferation. Overall, invasiveness should de-crease the consistency of the tissue surrounding the tumor, thus the local mechanicalcon8nement, and as such can prolong the volumetric growth phase of the main tumor(in addition to neovascularization, in vivo). This, in turn, would promote cell shedding,thus increasing the total number of invasive cells. This concept is further supportedby recently reported signs of feedback between tumor proliferation and invasion [12].In summary, our concept describes a self-sustaining invasive system, which closelyfollows the heterogeneity of the environment, resulting in an imprint of “successful”branches, i.e., cell pathways which led to attractor sites. Such a dynamic tumor systemwould be highly dependent on its cellular signal transmission, reception and processingcapability (termed “oncocybernetics”). Finally, our concepts are based on the view ofcells as opportunistic systems, which seek to limit their energy expenditure. Randommovement (i.e., chemokinesis) is therefore considered to be rather small, yet includedin the mathematical model as well.This oncology concept is based on the phenomenon seen in the experimental

setting but is also supported by extensive published data from cancer research.Drawn from this work we now introduce several candidate substances for the


proposed attractors:

1. Heterotype chemo-attractor: The original murine ECM matrix used to produceGFR-MatrigelR is 8ltered against the commonly used cell culture medium DMEM.This medium contains a variety of inorganic salts as well as amino acids, vitaminsand glucose (2:5 g=l). The maximum glucose consumption rate for brain tumor cellshas been described in Ref. [47] (8:7 fmol=cell=min), and Jain [48] has reported adi'usion coeLcient for D-glucose in the brain of 6:7 × 10−7 cm2=s. Furthermore,the ECM gel itself has structure [42]. We propose that the non-replenished “source”in the mathematical model represents a Guid-8lled void (or lesser consistency site)in the gel 8lled with a combination of DMEM components and components of theOPTI-MEMR medium used to prepare the speci8c composition of the gel.

2. Homotype chemo-attractor: Hepatocyte growth factor/scatter factor, a pleiotropicpolypeptide, is known to stimulate chemotactic cell motility in brain tumor cellsover a tyrosine kinase receptor [49]. Both this c-Met receptor and HGF/SF (90 kD)are co-expressed in brain tumor cells, leading to the hypothesis of an autocrinepathway involved in tumorigenesis. Since HGF/SF also stimulates angiogenesis inbrain tumors, a paracrine pathway has been suggested as well [49–53]. Moreover,HGF/SF seems to enhance brain tumor cell invasiveness by up-regulation of the pro-teinase activity, required for ECM degradation [54]. Rosen et al. [53] determinedthat U87MG, which is the parental cell line of the one used in our experiments,indeed produces SF (6:3 ng=106 cells=48 h). According to the vendor (Becton Dick-inson, Bedford, MA) the optimal HGF/SF amount to stimulate cells in culture isreported to be 5–50 ng=ml.The polypeptide growth factors, epidermal growth factor, and transforming growth

factor-alpha are two additional candidates. In glioma cells TGF� is endogenouslyproduced; the EGF=TGF�=EGFR system is involved in growth and motility in gliomacells [55,56] and should be active in the wtEGFR co-expressing U87MGmEGFRcell line used in our experimental model [41,57]. The receptor for both ligands,EGFR, is a trans-membrane glycoprotein with intrinsic tyrosine kinase activity. TheEGFR gene is ampli8ed and over-expressed in a majority of malignant gliomasand frequent gene rearrangements such as the mEGFR have been reported as well[58,59]. Most importantly, both EGF and TGF� are expressed in malignant braintumors and thus autocrine and/or paracrine pathways seem very likely [17]. More-over, the GFR-MatrigelR also harbors traces of growth factors (such as the epider-mal growth factor EGF), which could attract these tumor cells. In fact, by usinga modi8ed radial dish assay Chicoine et al. [60] could demonstrate the chemo-attraction of glioma cells when establishing a gradient of non-replenished EGFin agar.

3. Heterotype hapto-attractor: GFR-MatrigelR comprises the extracellular matrix pro-teins, laminin (61%), collagen IV (30%) and entactin (7%). Especially laminin andcollagen IV are known to enhance brain tumor cell haptotactic movement in vitrothrough integrin activation [13].

4. Homotype hapto-attractor: Glioma cell lines especially in 3D-settings have beenshown to produce permissive ECM proteins [61]. Enam et al. [62] reported that


U87MG (which, again, is the parental cell line of the one used in our experimentalsetting) indeed produces ECM proteins in both mono-layer and spheroid cultures,especially collagen IV, 8bronectin and tenascin.It is important to emphasize that the experimental measurement of the parameters,

which are considered in the following mathematical model, will be of considerablediLculty. This is in part because of the speci8c experimental setting necessary butalso due to the expected relatively small amounts of the various suggested attrac-tor substances within the invasive area, especially on the single cell level. Thus,we will 8rst use statistical sensitivity analysis of the parameter sets starting withbiologically reasonable approximations. Yet, in parallel we will explore speci8c ex-perimental approaches: con8ning mechanical pressure exhibited by the gel can beapproximated by experiments determining the empirical parameters of a strain energyfunction [63]. Furthermore, Stickle et al. [64] already described a method to deter-mine the concentration pro8le of a chemoattractant and its di'usion coeLcient inan under-agarose assay. Measurement of the heterotype chemo-attractor distributionrelies also on the knowledge of the di'usion-determining properties of the gel, i.e.,requires micro-structural analysis of the gel-medium composition. These measure-ments will then also help determining the scale dependence of the di'usion dynam-ics, e.g., of the 90 kD HGF/SF (candidate for the homotype chemo-attractor) versusthe 850 kD ECM protein laminin (candidate for the heterotype hapto-attractor). De-tecting the homotype factors involved as well as their regional concentration pro8lewill be especially challenging, however. First, regions of the gel can be enzymat-ically digested and quantitative bioassays (ELISA) for the determination of somefactors may be possible depending on the sensitivity of the available tests. Al-though the medical literature describes the paracrine production of the (otherwiseintensely investigated) growth factors and matrix proteins mostly qualitatively, somedata about production rates are already available, such as the one for HGF/SF with6:3 ng=106 U87MG cells=48 h [53]. Quantitative immune-histochemistry on 8xedparaLn sections, to search for example for the enhanced expression of speci8c re-ceptors or degrading enzymes, may be an option as well. Another possibility is thesearch for regional distinct gene-expression patterns with microarrays, which wouldrequire harvesting of cell populations from regions of interest at various time points.Based on this oncology concept, we will now describe the theoretical frameworkfor the mathematical model in more detail.

4. Mathematical model

In this section we present the basic model and equations that can provide one de-scription of the growth and invasion of the aforementioned microscopic brain tumors.We will address the processes that take place and the various simplifying assumptionsand approximations needed in order to make the equations mathematically tractable.The aim is to have a model Gexible enough to be generalized in the future, yet sim-ple enough so that it can be applied in a straightforward manner to the experimentsdescribed above.


The experimental setting we have in mind is the following: growth of, and invasionfrom, an initial (spherical) tumor (a set of cells of a given type, with U (x; t) describ-ing the average cell density as a function of space and time) on a given substrate[GFR-MatrigelR described in turn by its average density M (x; t)]. Our analysis will befor the evolution of ensemble averaged quantities, in other words, for averages overmany repetitions of the same experiment (of course, this does not mean that one ac-tually has to repeat the experiment many times to compare it with the theory). Thus,even though the GFR-MatrigelR is in principle microscopically inhomogeneous, viewedin an ensemble picture, its density is a smooth function in space and time. For now,we will assume that natural decay of the GFR-MatrigelR (and of any of the othercomponents introduced below) can be neglected, as a 8rst approximation.The original tumor is incorporated as an initial condition in the U (x; t) density 8eld.

Tumor cells are able to actively move in the experimental setting. In mathematicalterms, they “di'use” into the GFR-MatrigelR, with di'usion coeLcient �U [we beginby taking this di'usion coeLcient to be constant but allow for the later inclusion of adependence on the gel density, i.e., �U = �U (M)]. Furthermore, the GFR-MatrigelR iscontinuously digested by the tumor cells. Therefore, there is a separate dynamical equa-tion for M (x; t) which takes into account the fact that the gel degradation is a functionof the local cell density U (x; t). It is reasonable to assume then that this “sink” termfor 9M=9t is proportional to the density of cells U with constant of proportionality �M ,which is the degradation rate. As a 8rst approximation we take �M to be a constant, butallow for the possibility of more general dependencies. (For instance, we are neglectingthe e'ects of space and time variations in the metabolic rate of the tumor cells.)We assume that the nutritive heterotype chemo-attractor component di'uses from

a local source and that tumor cells metabolize this soluble attractor. At the initialtime we imagine a given quantity of this chemical to be present somewhere in space(line, surface, or volume). The heterotype chemo-attractor has average density Q(x; t),and di'uses with di'usion coeLcient �Q. Again, as a 8rst step we take this di'usioncoeLcient to be constant. The heterotype chemo-attractor density is also degraded bythe tumor cells. Based on the assumption that there is a minimum concentration ofthe heterotype chemo-attractor (required in order to trigger cell attraction at all), aswell as a maximum threshold for consumption, i.e., beyond which the metabolism doesnot increase further, we consider the degradation of the heterotype chemo-attractor asfollows. First, there is a minimum density threshold below which degradation doesnot take place (as a simpli8cation we set this threshold to zero, but this can easilybe relaxed). Second, there is a saturation density of the heterotype chemo-attractorabove which the rate of consumption by tumor cells does not further increase with thechemo-attractor concentration.The heterotype attractor-induced chemotactic behavior of the tumor cells is described

in stochastic language by a drift term, with drift coeLcient �Q. Chemotaxis is di-rected motion (of the cells U ) in response to a chemical gradient (of the heterotypechemo-attractor Q). In addition, tumor cells produce both a homotype chemo-attractorC1 (e.g., protein growth factors) and a homotype hapto-attractor C2 (e.g., ECM pro-teins). These homotype chemicals induce chemotaxis and haptotaxis, respectively. Letus 8rst consider the homotype chemo-attractor C1. It di'uses in the GFR-MatrigelR


with di'usion constant �C1 , which we take here to be a constant. There are two char-acteristic densities: a minimum critical density (which will be taken to be zero) and amaximum critical density, above which the intensity of signaling does not increase withthe concentration of homotype chemo-attractor. The rationale for the lower thresholdis a 8nite receptor sensitivity, whereas for the upper threshold it is the 8nite num-ber of cell-surface receptors available for a speci8c signaling path. This homotypechemo-attractor induces chemotactic behavior described by a drift term in the equationfor U , with drift coeLcient �C1 (directed motion of the U cells due to a chemicalgradient of C1). Unlike the heterotype chemo-attractor, there is a source term for thishomotype chemo-attractor C1, as it is continuously produced by the tumor cells. Theproduction of this homotype attractor depends on the metabolic rate of the tumor cells,which in turn depends on the local heterotype chemo-attractor density Q (metabolicrate increases with Q). However, there does exist a saturation point in the produc-tion of C1 as a function of Q. These two features are included in the model (seethe equations below). Finally, signal attenuation includes receptor down-regulation andligand depletion by endocytic internalization and intracellular degradation. In order toaccount for such ligand depletion, we will add a loss term to Eq. (4). Speci8cally, theadditional 8rst term on the right-hand side describes the reduction in the C1 densitydue to the intracellular depletion and includes threshold e'ects (controlled by �C1 , aC1 ,and bC1—only two parameters are actually needed, but the present form is used forconvenience).Turning now to haptotaxis, we note that while this e'ect leads to enhanced motion on

the substrate it is not directed in the sense of preferential transport in a given directiondue to a chemical gradient as in chemotaxis. However, in individual realizations ofexperiments haptotaxis can produce enhanced motion on arbitrarily shaped domains,which might give the impression of directed motion (hence termed “attractor” in both,concept and model). Nevertheless, on averaging over realizations, it seems reasonableto assume that haptotaxis can be viewed mathematically as another form of di'usion.Thus haptotaxis can be represented as an enhancement of di'usion over the substrate:�U has one component due to true Brownian motion (Fickian), �0, and a haptotacticcomponent that depends on both C2 and H (�2 and �H , respectively), where H is theaverage density of the heterotype hapto-attractor (e.g., laminin or collagen IV).The inclusion of the homotype hapto-attractor C2 into the picture dictates a dy-

namical equation for its average density. This equation will involve a di'usion termand a source term, as C2 is produced by the tumor cells (again, with minimum andmaximum saturation thresholds), yet, other than C1, C2 is not consumed by the cells.Finally, we also require a dynamical equation for the average density of the heterotypehapto-attractor H ; without loss of generality we will assume that it only di'uses. Muchlike C2, H is not consumed by the tumor cells; however, unlike C2, H is not pro-duced, and therefore there is no source term for the heterotype hapto-attractor in itsdynamical equation. Following the integrin–receptor concept, the current model alreadyincludes lower and upper thresholds for the homotype hapto-attractor. A more detaileddescription of the sensitivity thresholds of the receptors for C2, as well as for H , isdesirable, yet beyond the scope of this manuscript and thus will be dealt with in thefuture.


4.1. One-dimensional model

The mathematical model sketched above consists of the following equations (re-stricted to the case of one space dimension for convenience; two- and three-dimensionalequations are given in the Appendix A):

9M (x; t)9t =−�MU (x; t) ; (3)

9C1(x; t)9t =−�C1U (x; t)

aC1C1(x; t)bC1 + C1(x; t)

+ �C1

99x

[9C1(x; t)9x

]

+�C1 (Q)U (x; t)�C1 + U (x; t)

; (4)

9C2(x; t)9t = �C2

99x

[9C2(x; t)9x

]+�C2 (Q)U (x; t)�C2 + U (x; t)

; (5)

9Q(x; t)9t = �Q

99x

[9Q(x; t)9x

]− �QU (x; t)

aQQ(x; t)bQ + Q(x; t)

; (6)

9H (x; t)9t = �H

99x

[9H (x; t)9x

]; (7)

9U (x; t)9t = k1U (|x|; t)[k2 − k3U (|x|; t)] + 9

9x

[(�0 + �2 + �H )

9U (x; t)9x

]

− �C1

99x

[U (x; t)

9C1(x; t)9x

]− �Q 9

9x

[U (x; t)

9Q(x; t)9x

]: (8)

Eq. (3) accounts for the degradation of the gel by the tumor cells, with the cor-responding rate constant �M . Eq. (4) is the dynamical equation for the homotypechemo-attractor density C1(x; t); the 8rst term takes into account the consumption of C1

by the tumor cells, the second term describes di'usion with di'usion coeLcient �C1 ,and the third contribution is a source term, which includes thresholds (set by �C1 and�C1 ) and accounts for the metabolic rate dependence on the heterotype chemo-attractorby making �C1 a function of the heterotype chemo-attractor density Q(x; t). Eq. (5)describes the evolution of the homotype haptotatic component C2(x; t) via an equationof precisely the same form as (4), yet without the loss term (reGecting the fact that C1

is consumed by the tumor cells and C2 is not). Eq. (6) controls the evolution of theheterotype chemo-attractor component Q(x; t). In this case, there is a reduction termsimilar to that in Eq. (4) and a di'usion term but no source (as nutrient replenish-ment is not being considered here). Eq. (7) describes the evolution of the heterotypehapto-attractor H (x; t) by means of di'usion. The last equation describes the evolutionof the tumor cell density U (x; t). The 8rst term on the right-hand side is a simple lo-gistic term (more complicated models can be incorporated if desired) to model tumor


growth (proliferation), 1 the second term models ordinary and haptotactically enhanceddi'usion (homo- and heterotype), and the next two terms model chemotaxis by thehomo- and heterotype components, respectively.

5. Numerical simulations

The six dynamical Eqs. (3)–(8) underlying our model comprise a set of coupledreaction–di'usion equations. In order to solve the initial value problem for this set ofinteracting partial di'erential equations we have implemented an explicit integrationalgorithm (second order in space and fourth order in time), which runs on parallelsupercomputers (T3E NERSC). The code is capable of handling up to three spacedimensions. We display some preliminary results from a simpli8ed two-dimensionalsimulation below. More detailed results from three-dimensional simulations will betreated elsewhere.Without loss of generality we have neglected the e'ects of the gel M and haptotaxis

due to both C2 and H in order to focus, in this 8rst paper, on the chemotactic e'ectsfrom both Q and C1. Our initial conditions are as follows: at time t0 = 0 we startwith an initial tumor spheroid (see Fig. 2a), an initial spherical source of heterotypechemo-attractor Q (see Fig. 4a) and a zero initial density of homotype chemo-attractorC1 (see Fig. 3a), assuming that it requires some time post-placement into the geluntil the cells would restart producing it. These initial conditions allow us to focuson the evolution of the invasion phase characterized by the interplay between chemo-tactic attraction and the dynamics of secretion and degradation of soluble attractorsubstances.This simpli8ed model is described by the following equations (restricted to one-

dimensional space for convenience; two- and three-dimensional equations are given inAppendix A):

9C1(x; t)9t = �C1

99x

[9C1(x; t)9x

]+

�C1U (x; t)�C1 + U (x; t)

; (9)

9Q(x; t)9t = �Q

99x

[9Q(x; t)9x

]− U (x; t)


; (10)

9U (x; t)9t =

99x

[�09U (x; t)9x

]− �C1

99x

[U (x; t)

9C1(x; t)9x

]

− �Q 99x

[U (x; t)

9Q(x; t)9x

]: (11)

1 Note that the logistic term describes spherical proliferation of the tumor [for a review see, e.g., Marusicet al. [65]] and the absolute value of x, |x| is required in the equation.


020

4060

80100

120 0

20

40

60

80

100

120

00.5

11.5

22.5

33.5

44.5

5

020

4060

80100

120 0

20

40

60

80

100

120

00.5

11.5

22.5

33.5

44.5

5

020

4060

80100

120 0

20

40

60

80

100

120

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

(a) (b)

(c)

Fig. 2. (a) Initial tumor cell density U (x; y; t0); (b) intermediate tumor cell density U (x; y; t1); and (c) 8naltumor cell density U (x; y; t2).

For simplicity we have set �Q = 1 and have neglected the degradation of the homo-type chemo-attractor C1 and the dependence of �C1 on the heterotype chemo-attractordensity Q, so that �C1 is a constant coeLcient. The parameters of the model are:�C1 ; �C1 ; �C1 ; �Q; aQ; bQ; �0; �C1 , and �Q. The values of these constants have been cho-sen as shown in Table 1. 2

The choice of parameter values has been guided by the oncology concept(Section 3.2): we have set the di'usion coeLcient of the homotype chemo-attractor tozero so that the dynamics of C1 is due to the secretion term of Eq. (9); �Q is 100 timeslarger than �0 as it is reasonable to expect glucose to di'use faster than tumor cells; theheterotype chemo-attractor is metabolized by the tumor cells (second term of Eq. (10))and we therefore set �Q larger than �C1 ; the rest of the parameters have been chosenso that the numerical simulation clearly shows the dynamical relevance of the terms

2 The unit of time corresponds to the total time run by the code, the unit of distance corresponds to thelattice size of the code, and the unit of concentration is relative to the initial concentration.


020

4060

80100

120 0

20

40

60

80

100

120

-1

-0.5

0

0.5

1

020

4060

80100

120 0

20

40

60

80

100

120

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

020

4060

80100

120 0

20

40

60

80

100

120

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

(a) (b)

(c)

Fig. 3. (a) Initial homotype chemo-attractor density C1(x; y; t0); (b) intermediate homotype chemo-attractordensity C1(x; y; t1); and (c) 8nal homotype chemo-attractor density C1(x; y; t2).

Table 1Parameter values for the numerical simulations

�0 = 0:01 �C1 = 0:0 �Q = 1:0�C1 = 2:0 �C1 = 2:0 aQ = 0:1bQ = 0:3 �C1 = 0:5 �Q = 0:8

in which they appear (second term in both Eqs. (9) and (10)). This approach is ap-propriate as the purpose of these simpli8ed simulations is not to assess the model’spredictive power in detail, but to provide a qualitative picture of the kind of behaviorthat can be expected from our model, as well as to provide evidence that this model


020

4060

80100

120 0

20

40

60

80

100

120

0

1

2

3

4

5

6

7

8

9

10

020

4060

80100

120 0

20

40

60

80

100

120

0

1

2

3

4

5

6

7

8

020

4060

80100

120 0

20

40

60

80

100

120

0

1

2

3

4

5

6

7

(a) (b)

(c)

Fig. 4. (a) Initial heterotype chemo-attractor density Q(x; y; t0); (b) intermediate heterotype chemo-attractordensity Q(x; y; t1); and (c) 8nal heterotype chemo-attractor density Q(x; y; t2).

can indeed reproduce tumor growth patterns similar to the experimental ones describedin Section 3.1.Figs. 2(a), 3(a), and 4(a) show the initial conditions at time t0=0 for the tumor cell

density U (x; y; t0), the homotype chemo-attractor density C1(x; y; t0), and the heterotypechemo-attractor density Q(x; y; t0) on a 120×120 (two-dimensional) grid. The boundaryconditions enforce all densities to be zero at the grid edges.As discussed above, to focus on invasion, we have assumed that the radius of the

initial tumor spheroid is 8xed (k1 = 0; however this condition can be easily relaxed infuture iterations). In addition, in order to further simplify the setting, we have ignoredthe potential enhancement of cell di'usion due to homo- and heterotype hapto-attraction


(i.e., we have set �2 = 0 and �H = 0 as we are neglecting haptotactic e'ects in thispreliminary study) and also ignored the degradation of the matrigel by the tumor cells(�M = 0).Figs. 2(b), 3(b), and 4(b) show the density pro8les at an intermediate stage of

the simulation t1, i.e., after 130 time steps. Fig. 2(b) denotes the tumor cell den-sity U (x; y; t1), Fig. 3(b) depicts the homotype chemo-attractor density C1(x; y; t1), andFig. 4(b) shows the heterotype chemo-attractor density Q(x; y; t1). At this intermedi-ate stage the plots describe the following scenario: (a) the tumor cell pro8le developsa second smaller peak towards the location of the initial heterotype chemo-attractorsource (due to transport by means of the chemotactic drift in Eq. (11)); (b) the homo-type chemo-attractor density C1 pro8le (initially zero) has become non-zero where thetumor cell density is increasing (due to the production term in Eq. (9)) and hence itspro8le is not spherical; (c) the heterotype chemo-attractor density Q decreased fromits initial value as it is di'using, continually degraded by the metabolizing tumor cellsand generally, not replenished (Eq. (10)).Figs. 2(c), 3(c), and 4(c) show the distributions (number density) after a 8nite

time evolution t2 of 200 time steps. With the present choices of parameters and overthe timescale of the simulation we 8nd that due to chemotactic migratory behavior(a) a second tumor cell peak centered on the Q(x; y; t2) pro8le (due to chemotaxis anddi'usion) has substantially increased in size, and in fact is larger than the initial one(Fig. 2(c)); (b) the initial tumor cell population has decreased in size due to di'usion,chemotactic movement, and the fact that we are currently not considering tumor cellproliferation (Eq. (11)); (c) the homotype chemo-attractor density C1 pro8le (Fig. 3(c))has increased and shows an asymmetry towards the heterotype chemo-attractor, wherethe secondary tumor cell population is increasing (due to the production term inEq. (9)); (d) 8nally, the heterotype chemo-attractor density (Fig. 4(c)) has decreasedeven more as it is di'using, continually degraded by the metabolizing tumor cells andagain, not replenished (Eq. (10)).

6. Discussion

With this computational model we plan to investigate if distinct homo- and het-erotype attractor substances can indeed guide a multicellular invasive network. Ourcontinuum model consists of several coupled reaction–di'usion equations designed toinvestigate two intriguing concepts of tumor invasion: the principle of least resistance,most permission and highest attraction as well as the concept of homotype (per-mission and) attraction. These concepts propose the development of a self-sustaininginvasive system, which although intrinsically driven, closely follows the heterogeneityof the environment resulting in an imprinting of branches, which reached suLcientattractor sites. The experimentally observed patterns should therefore be the result ofa dynamically alternating pro8le of various intrinsic and extrinsic parameters.The proposed continuum model already incorporates several of these parameters

such as the metabolized heterotype chemo-attractor, the homotype chemo- and hapto-attractors as well as the heterotype hapto-attractor. It is a substantial progress from


earlier work in that it includes the dynamics (temporal and spatial) of all the substancesconsidered, namely the matrix M , the homotype chemo-attractor C1, the homotypehapto-attractor C2, the heterotype chemo-attractor Q and the heterotype hapto-attractorH , as well as the tumor cell density U . We allow for degradation of the matrixgel and include the e'ects of haptotaxis as an enhanced di'usion constant. For thechemotactic terms we adopt the earlier form described in Keller et al. [22] but assumingthat the chemotactic coeLcients are constant. For the degradation of the heterotypechemo-attractor we introduce a lower and a saturation threshold, and assume constantdegradation rates. Regarding the possibility of describing the receptor-based mechanismunderlying chemical regulation of cell motion [30], we have not considered it in itsfull generality but only argued that the saturation threshold takes into account thephysical limitations due to a 8nite number of surface receptors per tumor cell. It willbe interesting to see if our numerical results (work in progress) derived from theseequations yield solutions of the kind of traveling waves (of moving tumor cells) foundin earlier work [28,29].The preliminary two-dimensional numerical results already support some of the on-

cology concepts described above, in particular with regard to highest heterotype at-traction and homotype (intra-branch) attraction. The simpli8ed model that has beensolved numerically [see Eqs. (9)–(11)] includes the di'usion of tumor cells and het-erotype chemo-attractor, the metabolization of the heterotype chemo-attractor Q, theproduction of the homotype chemo-attractor C1 by the tumor cells, and chemotaxis ofthe tumor cells due to gradients in Q and C1. In order to focus on chemotactic andhomotype attraction e'ects we have started at time t0 with initial density pro8les forU and Q centered at di'erent locations in a two-dimensional 120× 120 grid. We havestarted with zero initial density for C1 to focus on its production by the tumor cells. Atan intermediate time of the simulation t1 the e'ects of heterotype chemotaxis can beseen, as the tumor cell density develops a second peak towards the initial source of Q.At time t1 production of C1 has already taken place and the pro8le of C1 is prominentwhere the tumor cell density U is non-zero. By then the heterotype chemo-attractordensity Q has slightly decreased as it is a non-replenished source (reGecting the exper-imental setting), and it is being di'used and metabolized by the tumor cells. After 200time steps in the evolution, i.e., at t2, these e'ects are even more dramatic. The secondpeak of U is larger than the initial one, the density of the homotype chemo-attractorC1 is almost twice its value at t1 and follows the density pro8le of the tumor cells,and the density of heterotype chemo-attractor has become smaller due to the e'ects ofdi'usion and consumption. These preliminary results therefore already show that bothhetero- and homotype chemo-attraction have to be considered as potential mechanismsnot only for directed motility per se but also for the development of speci8c structuralpatterns in such 3D heterogeneous biological media.The parameter values have been chosen to reGect the underlying tumor biology con-

cepts. Nevertheless, the next step requires detailed analysis of the simulation resultsfor a wider range of parameters and initial conditions. The dynamics that can be en-coded in Eqs. (3)–(8) is very rich and di'erent ranges of non-linear behavior are to beexpected as parameter values are varied. If the model yields comparable dynamics andshows similar structural pattern as the experimental model it is based on, it will further


support our oncology concept and thus may provide important insights into the com-plexity of tumor invasiveness if closer connected to the experimental setting. Thus, ourongoing interdisciplinary work will include adding soluble attractor substances into spe-ci8c locations of the ECM gel. By comparing the microscopic online analysis with thesimulation results the mathematical model will be continually improved. Conversely,statistical analysis of the mathematical model output will help determine the parame-ters with the highest impact, which then need to be examined more carefully in theexperimental model.Therefore, the next iterations of the model will likely have to consider even more

biological details such as the heterogeneity of the gel (resistance pro8le) and the decayof both gel as well as attractor substances. For example, the analysis of the microstruc-tural properties of the gel will allow us to carefully investigate the dependence of thedi'usion coeLcient, �U , on the gel density and will therefore enable us to properlysimulate the concept of “least resistance”. The incorporation of speci8c inhibitor sub-stances in both the experimental setting, as well as the mathematical model, will beanother challenging extension of our work. Furthermore, speci8cations in the proposedreceptor systems will make the model richer. For example, taking EGF or TGF� ascandidate substances for the homotype chemo-attractors, their well-described intracel-lular signaling mechanism could be incorporated to explore its impact on the emergingtumor system. In brief, mitogenic signaling paths for both ligands consist of bindingto their common surface receptor (EGFR), possible receptor dimerization and cyto-plasmatic activation of tyrosine kinase activity. After internalization and traLcking ofthe EGF (TGF�)/EGFR complexes recycling to the cell surface is possible. Recep-tor down-regulation and recycling as well as ligand depletion are important cellularmechanisms to balance growth control in non-cancerous cells. Cancer cells suppos-edly evade some of these mechanisms by over-expression of receptor genes, auto- orparacrine growth factor secretion and failed receptor down-regulation as in the caseof the mEGFR. Thus, modeling e'orts like ours may very well contribute to a bet-ter understanding of these cellular processes and their impact on emergent behaviorin tumors. As such, our multicellular model could be combined with the intracellu-lar model presented by Starbuck and Lau'enburger [66] who simulated wild-type andtruncated EGF/EGFR signaling pathways with coupled non-linear ordinary di'usionequations. As stated earlier, the characteristics of the cell line used in our experimentsinclude the expression of a truncated mEGF-receptor, which lacks signal attenuationdue to a failed down-regulation [57]. In this case, the thresholds in the model forthe ligand-binding capacity of the homotype chemo-attractor only refer to the regula-tion of the ligand-dependent, co-expressed wtEGFR, whereas the mEGFR supposedlycannot bind any homotype chemo-attractor at all. However, the tumor-speci8c ampli-8cation and over-expression of the wtEGFR itself will have to be taken into accountin our model by increasing the upper receptor-capacity-related threshold for C1. An-other interesting aspect is to implement the adaptive potential of the lower thresholdfor both Q and C1. An increase in signal sensitivity (by lowering this threshold (cur-rently kept constant at zero only for computational purposes)) could be related to achange in environmental conditions or triggered by intrinsic events. This step wouldtherefore lead to yet another important extension of the model, i.e., incorporating


multiple cellular subpopulations, another hallmark of these highly malignant braintumors.Admittedly, our model is based on numerous simplifying assumptions and thus by

no means can claim completeness. However, the equations can be generalized quiteeasily so that working in higher than one dimension will be possible (see appendix).This will be helpful in order to combine this model with our discrete 3D modelinge'orts (e.g., Ref. [37]). Only such hybrid models, i.e., a combination of discrete andcontinuum approaches will be capable of simulating to a certain extent the complexinterplay of a critical number of required tumor features and their dependence on en-vironmental conditions. In summary, the model presented in this paper is a signi8cantstep towards even more advanced interdisciplinary attempts at modeling malignant tu-mors as complex dynamic, self-organizing multicellular systems. E'orts such as thisone are necessary to advance our understanding of these tumors, which is essential inorder to be able to develop novel diagnostic tools and treatment strategies and thusultimately improve the grim outcome of patients su'ering from this devastating disease.

Acknowledgements

This work was supported in part by Grant CA69246 from the National Institutes ofHealth. The authors thank Drs. James P. Freyer (Los Alamos National Laboratory),Michael E. Berens (Barrow Neurological Institute), Leonard M. Sander (University ofMichigan) and A. Perumpanani for inspiring discussions and Dr. E. Antonio Chiocca(Harvard Medical School) for his support of the Tumor Complexity Modeling Project(TCMP). Numerical simulations were carried out at the National Energy ResearchScienti8c Computing Center (NERSC).

Appendix A. Two- and three-dimensional model

The previous equations can be generalized quite easily to the case of higher thanone dimension. We only need to replace x by a two- or three-dimensional vector andworking in Cartesian coordinates, we have in two dimensions

x = (x1; x2) ; (A.1)

and in three dimensions 3

x = (x1; x2; x3) ; (A.2)

where all the components can range over the set of real numbers. Keeping the abovevectorial notation, the generalized equations become 4

9M (x; t)9t =−�MU (x; t) ; (A.3)

3 For the sake of simplicity we only present the equations in three dimensions as the two-dimensionalones can be easily derived from Eqs. (A.3)–(A.12).

4 The quantity r in the dynamical equation for U (x; t) is the radius of the tumor spheroid.


9C1(x; t)9t = �C1∇ · ∇C1(x; t) +

�C1 (Q)U (x; t)�C1 + U (x; t)

− �C1U (x; t)aC1C1(x; t)bC1 + C1(x; t)

;

(A.4)

9C2(x; t)9t = �C2∇ · ∇C2(x; t) +

�C2 (Q)U (x; t)�C2 + U (x; t)

; (A.5)

9Q(x; t)9t = �Q∇ · ∇Q(x; t)− �QU (x; t)


; (A.6)

9H (x; t)9t = �H ∇ · ∇H (x; t) ; (A.7)

9U (x; t)9t = k1U (r; t)[k2 − k3U (r; t)] + ∇ · [(�0 + �2 + �H )∇U (x; t)]

− �C1∇ · [U (x; t)∇C1(x; t)]− �Q∇ · [U (x; t)∇Q(x; t)] ; (A.8)

where r =√x21 + x

22 + x

23 and

∇ ≡ e1 99x1

+ e299x2

+ e399x3

(A.9)

and e i, for i = 1; 2; 3, is the unit vector in the ith direction. The product ∇ · ∇ actingon a function f is given by

(∇ · ∇)f =99x1

(9f9x1

)+

99x2

(9f9x2

)+

99x3

(9f9x3

); (A.10)

and the calculation of a term such as ∇ · (f∇g), where f and g are two functions, isgiven by

∇ · (f∇g) = (∇f) · (∇g) + f(∇ · ∇)g (A.11)

=(9f9x1

)(9g9x1

)+(9f9x2

)(9g9x2

)+(9f9x3

)(9g9x3

)

+f[99x1

(9g9x1

)+

99x2

(9g9x2

)+

99x3

(9g9x3

)]; (A.12)

where we have made use of the chain rule in the 8rst line of this derivation, and alsothe previous result of Eq. (A.10) in the second line.

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