simulated brain tumor growth dynamics using a three...

J. theor. Biol. (2000) 203, 367}382doi:10.1006/jtbi.2000.2000, available online at http://www.idealibrary.com on

Simulated Brain Tumor Growth DynamicsUsing a Three-Dimensional Cellular Automaton

A. R. KANSAL*, S. TORQUATO*-?, G. R. HARSH IVA,E. A. CHIOCCAEB** AND T. S. DEISBOECKEB**--

*Department of Chemical Engineering, Department of Chemistry, -Princeton Materials Institute,Princeton ;niversity, Princeton, NJ 08544, ;.S.A., ADepartment of Neurosurgery, Stanford ;niversity

Medical School, Stanford, CA 94305, ;.S.A., ENeurosurgical Service, BBrain ¹umor Center,**Molecular Neuro-Oncology ¸aboratory, Massachusetts General Hospital East,

Harvard Medical School, Charlestown, MA 02129, ;.S.A.

(Received on 20 August 1999, Accepted in revised form on 14 January 2000)

We have developed a novel and versatile three-dimensional cellular automaton model ofbrain tumor growth. We show that macroscopic tumor behavior can be realisticallymodeled using microscopic parameters. Using only four parameters, this model simulatesGompertzian growth for a tumor growing over nearly three orders of magnitude in radius. Italso predicts the composition and dynamics of the tumor at selected time points in agreementwith medical literature. We also demonstrate the #exibility of the model by showing theemergence, and eventual dominance, of a second tumor clone with a di!erent genotype. Themodel incorporates several important and novel features, both in the rules governing themodel and in the underlying structure of the model. Among these are a new de"nition of howto model proliferative and non-proliferative cells, an isotropic lattice, and an adaptive gridlattice.

( 2000 Academic Press

1. Introduction

The incidence of primary malignant brain tumorsis already 8/100 000 persons per year and is stillincreasing. The vast majority (80%) consists ofhigh-grade malignant neuroepithelial tumorssuch as glioblastoma multiforme (GBM) (Fig. 1)(Annegers et al., 1981; Werner et al., 1995). Inspite of aggressive conventional and advancedtreatments, the prognosis remains uniformly fatalwith a median survival time for patients with

?Author to whom correspondence should be addressed.E-mail: [email protected]

--T. S. Deisboeck, M.D. is also a$liated with the Depart-ment of Neurosurgery, University of Munich (Germany).

0022}5193/00/080367#16 $35.00/0

GBM of 8 months (Black, 1991; Whittle, 1996).The main reason for this grim outcome is notonly the rapid tumor growth but especially thefact that, long before the neoplasm can be diag-nosed, it has already grossly invaded the sur-rounding brain parenchyma, rendering surgicalremoval virtually ine!ective. The proposed se-quence of proliferation, then invasion, followedby proliferation again suggests that invasive cellsleft behind after an operation not only causeparenchyma destruction, but also eventuallytumor recurrence (Suh & Weiss, 1984; Burgeret al., 1988; Nazzaro & Neuwelt, 1990; Silbergeld& Chicoine, 1997). Anti-proliferative treatmentsfail because of poor transport across the

( 2000 Academic Press

FIG. 1. T1-contrast enhanced brain MRI-scan showinga right frontal GBM tumor. Perifocal hypointensity iscaused by signi"cant edema formation. The hyper-intense,white region (ring-enhancement) re#ects an area of extensiveblood}brain/tumor barrier leakage. Since this regionalneovascular setting provides tumor cells with su$cient nu-trition it contains the highly metabolizing e.g. dividing,tumor cells (Selker et al., 1982; Burger et al., 1983; Kellyet al., 1987; Earnest IV et al., 1988). Therefore, this areacorresponds to the outermost concentric shell of highlyproliferating neoplastic cells in our model (see Fig. 6).

368 A. R. KANSAL E¹ A¸.

blood}brain barrier, acquired treatment resist-ance, and lack of susceptibility of single invadingcells due to their minimal proliferative activity(Giese et al., 1996; Schi!er et al., 1997).

The rapid growth and resilience of tumorsmake it di$cult to believe that they behave asrandom, disorganized and di!use cell masses andsuggests instead that they are emerging, opportun-istic systems. If this hypothesis holds true, thegrowing tumor and not only the single cell(Kraus & Wolf, 1993) must be investigated andtreated as a self-organizing complex dynamicsystem. This cannot be done with currently avail-able in vitro/in vivo models or common math-ematical approaches. Therefore, there is a need

for novel computational models to simulate themechanistic complexity of solid tumor growthand invasion, combining a range of disciplinesincluding medical, engineering and statisticalphysics research.

Here we report the "rst step towards acomplex self-organizing tumor model focusingon volumetric growth of a solid tumor as thenecessary precondition for subsequent and ongo-ing invasion. We have developed a three-dimen-sional cellular automaton (CA) model whichdescribes tumor growth as a function of time. Thealgorithm takes into account that this growthstarts out from a few cells, passes through a multi-cellular tumor spheroid (MTS) stage (Fig. 2) andproceeds to the macroscopic stages at clinicallydesignated time-points for a virtual patient:detectable lesion, diagnosis and death (Suther-land, 1988; Mueller-Klieser, 1997). This simpli"-ed growth approach models macroscopic GBMtumors as an enormous idealized MTS, mathe-ematically described by a Gompertz function(Brunton & Wheldon, 1977; Vaidya & AlexandroJr., 1982; Norton, 1988; Marusic et al., 1994).Modeling the ideal tumor at every step as anoversized spheroid is especially suited for GBM,since this tumor, like a large MTS, compriseslarge areas of central necrosis surrounded bya rapidly expanding shell of viable cells (Fig. 1).In accordance with experimental data, the algo-rithm also implicitly takes into account that in-vasive cells are continually shed from the tumorsurface. Future work will treat the invasive dy-namics explicitly.

The simulation contains several features novelto the simulation of tumor growth:

z The ability of cells to divide is treated ina new manner. By rede"ning the transitionbetween dividing and non-dividing cells, it ispossible to generate medically realistic over-all tumor growth rates using a biologicallyreasonable cell-doubling time.

z Previous researchers have used the Voronoitessellation in histopathological image anal-ysis of tumors (Preston Jr. & Siderits, 1992;Kiss et al., 1995; Haroske et al., 1996). How-ever, the present work represents the "rst useof the Voronoi tessellation (described later inthe text) to study the dynamics of tumor

FIG. 2. MTS-gel assay showing a central spheroid with multiple &&chain''-like invasion pathways leading towards theboundary (magni"cation: ]200).

TUMOR DYNAMICS VIA CELLULAR AUTOMATON 369

growth in a cellular automaton. This tessel-lation is isotropic in space and hence doesnot create the arti"cial anisotropies possiblewith square or cubic lattices, which havetypically been used in tumor simulations.

z The model uses a varying density of latticesites (an adaptive grid lattice). This allowssmall tumors to be simulated with greateraccuracy, while still allowing the tumor togrow to a large size. Using this variable-den-sity lattice, tumors can be simulated overnearly three orders of magnitude in radius.

A CA model treats the discrete nature of actualcells realistically, giving it great adaptability intreating complex situations (Kau!man, 1984;Wolfram, 1984). For example, the addition ofa blood vessel or other growth promoting hetero-geneity could be studied by altering a single para-meter. Similarly, the e!ect of the surgical removalof a portion of the tumor could be readilymodeled. Most importantly, a discrete modelreadily allows the inclusion of a number of dis-tinct subpopulations, corresponding to di!erent

cell clones. In addition, this work is intended toserve as a preliminary step to modeling systemicinvasive growth dynamics, the nature of which isideally suited to study using discrete modeling.Finally, each site in a CA model can be thoughtof as a group of actual cells. This interpretationallows the model to serve as an intuitive comp-lement to the results obtained from a continuummodel.

The simulation is designed to predict clinicallyimportant criteria such as the fraction of thetumor which is able to divide (GF), the non-proliferative (G

0/G

1arrest) and necrotic frac-

tions, as well as the rate of growth (volumetricdoubling time) at given radii. The simulationresults re#ect a test case derived from the medicalliterature very well, proving the hypothesis thatmacroscopic tumor growth behavior may bemodeled with primarily microscopic data. Cur-rent limitations and potential implications of thismodel for further tumorigenesis research arediscussed. Since an approach claiming to modelmalignant tumor complexity must take into ac-count cell invasion, we will also brie#y describe


initial theoretical considerations to model tumorcell invasion.

In summary, our model is characterized byseveral biologically important features:

z The model is able to grow the tumor froma very small size of roughly 1000 real cellsthrough to a fully developed tumor with 1011cells. This allows a tumor to be grownfrom almost any starting point, through tomaturity.

z The thickness of di!erent tumor layers,i.e. the proliferative rim and the non-prolif-erative shell, are linked to the overall tumorradius by a 2/3 power relation. This re#ectsa surface-area-to-volume ratio, which can bebiologically interpreted as nutrients di!usingthrough a surface.

z Our inclusion of mechanical con"nementpressure enables us to simulate the physiolo-gical con"nement by the skull at di!erentlocations within the brain di!erently.

z The discrete nature of the model and thevariable density lattice allow us to controlthe inclusion of mutant &&hot spots'' in thetumor, i.e. consider genetic instability andemergence of clonal subpopulations. Thevariable density lattice will allow us to lookat such an area at a higher resolution.

In the following section (Section 2), we outlinesome of the earlier work that has been done in the"eld of tumor modeling. Section 3 discussesthe details of the procedure for the simulation.A summary of our results is contained in Section4. This is followed by a discussion and concludingremarks regarding our current work, as well asfuture work, in Section 5.

2. Previous Work

Signi"cant research has been done in themodeling of tumors using theoretical models andcomputer simulations. Previous modeling hasbeen done on a range of tumor behaviors, includ-ing proliferative growth of the tumor core (seereferences immediately below), invasive growth(Tracqui, 1995; Perumpanani et al., 1996) andimmune response (Sherratt & Nowak, 1992).Here we will focus on works which have explicitly

modeled the proliferative growth of the tumorcore.

Some of the earliest work in modeling of tu-mors using a three-dimensional cellular automa-ton on a cubic lattice was carried out byDuK chting & Vogelsaenger (1985) for very smalltumors. These automaton rules were designed tore#ect nutritional needs for tumor growth. Otherimportant factors, such as surrounding cellsand mechanical pressure, however, remained un-considered.

Qi et al. (1993) considered a two-dimensionalcellular automaton tumor model that reproducedidealized Gompertz results. However, cells couldonly divide if one of their nearest neighbors wasempty. This created an unrealistically small frac-tion of a tumor which may divide. Furthermore,dead tumor cells were assumed to simply dissolveaway rather than accumulating into a necroticcore, as is seen in real tumors. In addition, thetransition from dividing cells to the resting statewas handled in a purely stochastic manner,rather than the more biologically reasonablenutrient-based method used by DuK chting andVogelsaenger.

Recent work by Smolle & Stettner (1993)showed that the macroscopic behavior of atumor can be a!ected by the presence of growthfactors at the microscopic level and added theconcept of cellular migration to the behavior ofthe cells. The work done by Smolle and Stettner,however, was qualitative and designed to showthe range of behaviors obtainable from a simplemodel. Recent work has been carried out to esti-mate the model parameters needed to generatea given con"guration (Smolle, 1998). This model,as well as those designed by Qi et al. and DuK cht-ing and Vogelsaenger, relied on an underlyingsquare (or cubic) lattice. While this providesa simple method of organizing the automata, itintroduces undesirable asymmetries and otherarti"cial lattice e!ects.

Other researchers have attempted to study thegrowth patterns of tumors in a more macroscopicfashion. The work of Wasserman et al. (1996)used a "nite element analysis technique to de-scribe the macroscopic behavior of a tumorbased on stresses imposed by various factors.Wasserman's approach is similar to that of Chap-lain & Sleeman (1993), in which the authors used


nonlinear elasticity theory to model a tumor. Inthat paper, the growth of the tumor was governedby a strain-energy function.

Another approach that has been taken bya number of researchers is to create equationswhich describe the tumor phenomenologically.The best known is the Gompertz model, whichdescribes the volume, <, of a tumor vs. time, t, as

<"<0expA

AB

(1!exp (!Bt))B , (1)

where <0

is the volume at time t"0 and A andB are parameters (Steel, 1977). Qualitatively, thisequation gives exponential growth at small timeswhich then saturates at large times (deceleratinggrowth). Many tumors, however, contain morethan one clonal population. A second populationmight have di!erent division rates and nutri-tional needs, resulting in competitive e!ectswhich cannot be accounted for by the Gompertzmodel. The Jansson}Revesz equations model theinteraction of two populations in a competitivesetting (Cruywagen et al., 1995). These equationsare essentially the classical Lotka}Volterra equa-tions of logistic growth with an added term toaccount for the conversion of one species to theother. Work by Cruywagen et al. has sought toapply the Jansson}Revesz equations to tumorgrowth. In addition, they have included a dif-fusive term to each equation, to account for pass-ive cellular motion. These mathematical modelsare useful to describe the general size of a tumorunder relatively simple conditions (two popula-tions, Fickian di!usion), but have yet to be ex-tended to multiple populations or active cellularmotility. A review of several other mathematicalmodels is contained in Marusic et al. (1994).

Finally, the growth of the tumor core has beenmodeled in a continuum setting using di!erentialequations with explicit spatial dependence byseveral researchers. Adam (1986) used an ordi-nary di!erential equation, which re#ects massconservation of tumor cells, coupled with a reac-tion-di!usion equation, re#ecting the distribu-tion of nutrients within the tumor. Ward & King(1997) used nonlinear partial di!erential equa-tions to generate pro"les for growth of an avascu-lar tumor based on a nutrient distribution. More

recently, Byrne & Chaplain (1998) have proposeda model which also includes the e!ects of nutrientdi!usion and incorporates both apoptosis andnecrosis explicitly. Other models, notably thoseby Tracqui et al. (1995) and Woodward et al.(1996), attempt to model both tumor growth andthe impact of treatment.

3. Simulation Procedure

The underlying lattice for our algorithm is theDelaunay triangulation, which is the dual latticeof the Voronoi tessellation (Okabe et al., 1992).The lattice generation is accomplished by "rstchoosing a set of sites, which are simply pointsdistributed in space according to some randomprocess. Each cell in the "nal Voronoi lattice willcontain one of these sites. The cell is then de"nedby the region of space nearer to that particularsite than to any other site. In two spatial dimen-sions, a Voronoi lattice is a collection of suchpolygons, which "ll the plane. This is illustratedin Fig. 3. In the three-dimensional analog, theVoronoi cells take the form of polyhedra.

The Delaunay lattice is generated from theVoronoi network by connecting those siteswhose polyhedra (in three dimensions) share acommon face. This determines which sites arenearest neighbors of one another. All referencesto nearest neighbors in this paper refer to thosedetermined in this way. A two-dimensional ana-log of this is also depicted in Fig. 3.

For our algorithm, a prescribed number ofrandom points in space are generated. The speedand memory requirements of the subsequent pro-grams are strongly dependent on the number ofpoints chosen. In addition, however, the largerthe number of points chosen, the more accuratethe model. As such, the largest number of pointsthat can be accommodated in computer memoryand run in a reasonable length of time has beenused.

Because a real brain tumor grows over severalorders of magnitude in volume, the lattice wasdesigned with a variable grid size. In our lattice,the density of site was allowed to vary continu-ously with the radius of the tumor. The density oflattice sites near the center of the lattice wassigni"cantly higher than that at the edge. A high-er site density corresponds to less real cells per

FIG. 3. Two-dimensional space tiled into Voronoi cells.Points represent sites and lines denote boundaries betweencells. (a) and (b) depict a very small section of a lattice.(a) shows the Voronoi cells, while (b) shows both theVoronoi cells, along with the Delaunay tessellation. (c) and(d) show a more representative section of the lattice, with thevariable density of sites evident. (c) shows the entire latticesection, (d) shows the same section with the darkened cellsrepresenting a tumor.


automaton cell, and so to a higher resolution.The higher density at the center enables us tosimulate the #at small-time behavior of the Gom-pertz curve. In the current state of the model, theinnermost automaton cells represent roughly 100real cells, while the outermost automaton cellsrepresent roughly 106 real cells. The average dis-tance between lattice sites was described by thefollowing relation:

f"16

r2@3 , (2)

in which f is the average distance between latticesites and r is the radial position at which thedensity is being measured. This relation restrictsthe increase in the number of proliferating cells asthe tumor grows. Note the 2/3 appearing in theexponent, which is intended to re#ect a surface-area-to-volume-like relation. This conforms tothe di!usion of nutrients through the surface

of the tumor, which is known to be a crucialfactor in governing the tumor's growth dynamics(Folkman & Hochberg, 1973).

While an isotropic Voronoi tessellation can begenerated from any list of random points, not allpoint sets are equally reasonable. A purely ran-dom distribution of points (the Poisson distribu-tion) will have regions in which the density ofpoints is very high, corresponding to a very smallVoronoi cell, and regions with a very low density,corresponding to very large cells. While somevariation in the size and shape of cells is impor-tant to ensure isotropy, it is biologically unreas-onable to have large variations. To solve thisproblem, a technique common in statistical phys-ics known as the random sequential addition(RSA) process was used (Cooper, 1988). In thistechnique, as random points are generated, theyare tested to ensure that they are not withina given distance of any other point. This elimin-ates the possibility of high-density clusters,though they can be added later to simulate highlyproliferative tumor &&hot spots''. In addition, it iswell known that in the RSA process there isa maximum fraction of space which can be "lled.In three-dimensions, this &&jamming limit'' corres-ponds to an occupied volume fraction of 0.38. Byapproaching this density, the possibility of a largelow-density area is also eliminated. This tech-nique was adapted for our work to allow theminimum distance between points to vary, givinga list of points suitable for generating a biolo-gically reasonable adaptive lattice. The formulaused to do so was

Rs"0.146r2@3, (3)

where Rsis the minimum distance between points

at distance r from the lattice center.The list of random points in space was fed to

a program written by Ernst Mucke called detri(Mucke, 1997). The detri program generates theDelaunay triangulation in three spatial dimen-sions for a given list of points. To test the detriprogram, a second Delaunay program, del-tree3,was also downloaded (Devillers, 1996). Both pro-grams were run on the same list of points and theresults compared to ensure that they were identi-cal. Because the detri program provided a moreconvenient form of output (listing tetrahedra by

FIG. 4. An idealized tumor. A detailed description of thedi!erent regions is contained in the text.

n


the points' indices rather than the speci"c coordi-nates) it was used for the remainder of the workdone.

The detri program was run on an IBM SP2parallel computer. Due to memory limitations,the list of sites was divided into sublists. Theprogram was then run on each sublist, generatingpartial lattices. Culling these partial lattices intoa single complete lattice required some care. Inorder to avoid missing triangles across the divis-ions between sublists, a signi"cant amount ofoverlap was included in each sublist. A secondchallenge, however, are the triangles formed atthe edges of each sublist. This led to extratriangles being included in the overall lattice.Because these extra triangles are localized at thesubset boundaries, however, it is possible to elim-inate them. This is done by generating two di!er-ent sets of sublists from the same original list andthen comparing the two "nal lattices. Any tri-angles not found in both lattices are consideredto be e!ects of the division and are discarded.This produces a Delaunay lattice over the entirepoint set. In order to check the procedure fol-lowed here, the division process was done ona point set that could be run in its entirety. Thedivision method gave the same "nal lattice aswhen the entire point set was run and so wasconsidered to be accurate.

Once the lattice is generated, the proliferationalgorithm can be run. This algorithm is designedto allow a tumor consisting of a few automatoncells, representing roughly 1000 real cells, to growto a full macroscopic size. An idealized model ofa macroscopic tumor is a spherical body consist-ing of several concentric shells. The inner core,the gray region in Fig. 4, is composed of necroticcells. The necrotic region has a radius R

n, which

is a function of time, t, and is characterized by itsdistance from the proliferation rim, d

n. The next

shell, the cross-hatched region in the "gure, con-tains cells which are alive but in the G

0cell-cycle

rest state. This is termed the non-proliferativeregion and is de"ned in terms of its distance fromthe edge of the tumor, d

p. This thickness is the

maximum distance from the tumor edge witha high enough nutrient concentration to main-tain active cellular division. In real MTS tumors,however, only about one-third of the viable cellsincrease the tumor size by proliferation [which is

numerically comparable to the growth fraction inmacroscopic tumors (Hoshino & Wilson, 1975)].The others are actively dividing, but the new cellsleave the central tumor mass and supposedlytrigger invasion into the surrounding tissue(Landry et al., 1981; Freyer & Schor, 1989). Fi-nally, as discussed below, an individual cell canonly divide if free space exists within a certaindistance of it. This distance must also be d

p(as

de"ned above) to properly account for the nutri-ent gradient basis for the transition of cells be-tween the actively dividing and G

0arrested

states. This distance is depicted as a small brokencircle in Fig. 4. Both real tumors and oursimulated tumors, however, are not perfectlyspherical. As such, the values of R

tand R

nvary

over the surface of the tumor. The single valuesused in the algorithm and listed in our results areobtained by averaging the radii of all the cells atthe edge of the tumor or of the necrotic region,respectively, according to the relations:

Rt"

+Np

i/1ri

Np

, (4)

Rn"

+Nn

i/1ri

N, (5)


where Npdenotes the number of cells on the edge

of the proliferative region and Nn

denotes thenumber of cells on the edge of the necrotic core.

This de"nition of proliferative cells allows cellsthat are not on the immediate edge of the tumorto divide. In order to accommodate these divis-ions, without allowing discontinuous division, analgorithm which allows for expansive growth viaintercellular mechanical stress (IMS) is used.A one-dimensional schematic of IMS growth isdepicted in Fig. 5. In part (a) of the "gure, cell A isa non-tumorous cell. Non-tumorous cells aretreated as empty in the current model, whichsimply means that, when &&"lled'', they are con-sidered to have been forced into the surroundingregion of indistinguishable non-tumorous cells.Cells B, C, and D are tumor cells which maydivide. When cell D attempts to divide, it cannot"nd an empty space within d

pand will turn non-

proliferative. When cell C attempts to divide itcan "nd cell A and so it will divide. Note that,biologically, cell C actually cannot &&see'' cell A,but rather senses its location because of the high-er nutrient level. Division creates a cell C@ which"lls the space previously occupied by cell B,which is in turn forced to the space previously"lled by cell A. Cell A is regarded as being forced

FIG. 5. Schematic of expansive growth via intercellularmechanical stress (IMS). In (a) cell A is non-tumorous, B}Dare tumorous and able to divide. (b) Depicts the results ofattempted divisions by cells C and D, in which C@ is the newcell created by the successful division of C. A full descriptionis contained in the text.

into the surrounding tissue and disappears (fornow) from our consideration. This is the newcon"guration depicted in part (b) of Fig. 5.

In three spatial dimensions, 1.5 million latticesites are used. This has been found to be theminimum required to give adequate spatial res-olution over the entire range of tumor radii. Nat-urally, more sites would be desirable, however,practical limits dictate that as few sites as possiblebe used. The initial tumor is a few automatoncells, representing roughly 1000 real cells, locatedat the center of the lattice.

In summary, the four key quantities Rt, d

p, d

n,

and pdare functions of time calculated within the

model. To "nd them, the simulation utilizes fourmicroscopic parameters: p

0, a, b and R

max. These

parameters are linked to the cell-doubling time,the nutritional needs of growth-arrested cells, thenutritional needs of dividing cells, and the e!ectsof con"nement pressure, respectively. The quant-ities are calculated according to the followingalgorithm.

z Initial setup: The cells within a "xed initialradius of the center of the grid are designatedproliferative. All other cells are designated asnon-tumorous.

z Time is discretized and incremented, so thatat each time step:*Each cell is checked for type: non-tumor-

ous or (apoptotic and) necrotic, non-pro-liferative or proliferative tumorous cells.*Non-tumorous cells and tumorous nec-

rotic cells are inert.*Non-proliferative cells more than a cer-

tain distance, dn, from the tumor's edge

are turned necrotic. This is designed tomodel the e!ects of a nutritional gradi-ent. The edge of the tumor is taken to bethe nearest non-tumorous cell.

dn"aR2@3

t, (6)

where a, the base necrotic thickness, isa parameter with units of (length)[email protected] the 2/3 in the exponent, againindicating a surface-area}volume-typerelation.

TABLE 1Summary of time-dependent functions and input parameters

for our model

Functions within the model (time dependent)

Rt

Average overall tumor radiusdp

Proliferative rim thickness (determines growth fraction)dn

Non-proliferative thickness (determines necrotic fraction)pd

Probability of division (varies with time and position)

Parameters (constant inputs to the model)

p0

Base probability of division, linked to cell-doubling timea Base necrotic thickness, controlled by nutritional needsb Base proliferative thickness, controlled by nutritional needsR

maxMaximum tumor extent, controlled by pressure response


*Proliferative cells are checked to see ifthey will attempt to divide. This is a ran-dom process, though the probability ofdivision, p

d, is in#uenced by the location

of the dividing cell (r), re#ecting the ef-fects of mechanical con"nement pressure.This e!ect requires the use of an addi-tional parameter, the maximum tumorextent, R

max. This probability is deter-

mined by the equation:

pd"p

0A1!r

RmaxB . (7) .

*If a cell attempts to divide, it will searchfor su$cient space for the new cell begin-ning with its nearest neighbors and ex-panding outwards until either an empty(non-tumorous) space is found or noth-ing is found within the proliferationradius, d

p. The radius searched is

calculated as

dp"bR2@3

t, (8)

where b, the base proliferative thickness,is a parameter with units of (length)1@3.*If a cell attempts to divide but cannot"nd space it is turned into a non-prolif-erative cell.(The above two steps constitute the re-de"nition of the proliferative to non-

proliferative transition that is one of themost important new features of themodel. They allow a larger number ofcells to divide, since cells no longer needto be on the outermost surface of thetumor to divide. In addition, it ensuredthat cells which cannot divide are cor-rectly labeled as such.)

z After a predetermined amount of time hasbeen stepped through, the volume and radiusof the tumor can be plotted as a function oftime.

z The type of cell contained in each grid canalso be saved at given times.

Table 1 summarizes the important time-depen-dent functions calculated by the algorithm andthe constant parameters used.

4. Results

The simulation has been compared with avail-able experimental data for an untreated GBMtumor from medical literature. The parameterscompared were cell number, growth fraction,necrotic fraction and volumetric doubling time.Medically, these data are used to determine a tu-mor's malignancy and the prognosis for its futuregrowth. Because it is impossible to determine theexact time a tumor began growing, the medicaldata are listed at "xed radii. The di!erent cellfractions used were extrapolated from the sphe-roid level and compared to data published for


cell fractions at macroscopic stages. Previousresearch has shown that the expanding tumorincreases both its cell loss (through necrosis/apo-ptosis and invasion) and its quiescent cell popula-tion (G

0/G

1arrested) due to a declining gradient

of nutritional elements towards the center of therapidly growing avascular mass (Folkman &Hochberg, 1973; Durand, 1976; Landry et al., 1981;Freyer & Sutherland, 1986; Mueller-Klieser et al.,1986; Rotin et al., 1986; Freyer & Schor, 1989). Atadvanced tumor stages, volumetric growth slowsdown mostly due to the declining growth fraction(GF), an increasing cell loss and G

0-fraction

(Turner & Weiss, 1980; Bauer et al., 1982; Freyer& Sutherland, 1985) caused by a lack of nutritionas well as an increasing con"nement pressure.

Summarized in Table 2 is the comparisonbetween simulation results and data (experi-mental, as well as clinical) taken from the medicalliterature. For macroscopic glial tumors (thelatter three time points) cell numbers, tumorvolumes, and volumetric doubling time measure-ments, including growth and necrotic fractionsand survival times, were taken from Hoshino& Wilson (1975); Hoshino & Wilson (1979),Yamashita & Kuwabara (1983), Yoshii et al.(1986), Alvord (1995), Blankenberg et al. (1995),Pierallini et al. (1996) and Burgess et al. (1997).For the microscopic spheroid level, doublingtimes as well as viable rim diameter, rim cellfractions, necrotic fractions and cell shedding

TAB

Comparison of test case data and simulatsimulation data only and is taken from

theoretical start o

Spherio

Time Sim. Day 6Radius Data 0.5 mm

Sim. 0.5Cell no. Data 106

Sim. 7]10Growth fraction Data 36%

Sim. 35Necrotic fraction Data 46%

Sim. 38Volume-doubling time Data 6 days

Sim. 9

data were taken from Haji-Karim & Carlsson(1978), Carlsson et al. (1983), Carlsson & Acker(1988), Freyer & Schor (1989) and Landry et al.(1981). Since it is virtually impossible to measuretotal cell numbers in macroscopic tumor (thelatter three time points), we have used the vol-ume-doubling times to estimate the number ofcells in the tumors, including at the microscopic(MTS) level. This method, however, leads toa rather high cell number at the MTS level [ex-perimental values range from 104 to 105 cells(Freyer, 1988, 1998)]. In a few cases, glioma MTSdata were not available and thus data from otherwell-characterized cell lines were used after care-ful evaluation. On the whole, the simulation datareproduce the test case very well. The virtualpatient would die roughly 11 months after thetumor radius reached 5 mm and 3.5 months afterthe expected time of diagnosis. The fatal tumorvolume is about 65 cm3.

These data were created using a tumor whichwas grown from an initial radius of 0.1 mm. Thefollowing parameter set was used:

p0"0.192, a"0.42 mm1@3, b"0.11 mm1@3,

Rmax

"37.5 mm.

This value of p0

corresponds to a cell-doublingtime of 4.0 days, which is reasonable for high-grade glial tumors (Hoshino & Wilson, 1979;

LE 2ion results (Sim). Note that the time row isthe start of the simulation not from thef the tumor growth

d Detect. Diagnosis Deathlesion

9 Day 223 Day 454 Day 5605 mm 18.5 mm 25 mm

5 18.5 25109 5]1010 1011

5 6]108 4]1010 9]101030% 20% 9%30 18 11

49% 55% 60%53 58 63

45 days 70 days 105 days36 68 100

FIG. 7. Cross sections of a fully developed tumor(radius"25 mm). (a) The central slice. (b) and (c) are taken10 mm from the center. (d) Taken 20 mm from the center, onthe same side as (b). The dark-gray region is proliferatingcells, the light-gray region non-proliferative cells and theblack region necrotic cells. The scales are in millimeters.


Pertuiset et al., 1985). The a and b parametershave been chosen to give a growth history thatquantitatively "ts the test case. As discussed be-low, the speci"cation of these parameters corres-ponds to the speci"cation of a clonal strain. Thisis manifested in qualitative behavior that is inde-pendent of the choice of the a and b parameters,but quantative behavior that is strongly a!ectedby them. The R

maxparameter was similarly

chosen to match the test case history. In this case,however, the "t is relatively insensitive to thevalue of R

max, as long as the parameter is some-

what larger than the fatal radius in the test case.Indeed, the "t is relatively insensitive to the exactform of eqn (7) in general.

Since a three-dimensional CA image is di$cultto visualize, cross sections of the tumors areshown instead. The growth of the tumor can befollowed graphically over time in Fig. 6. It depictsthe central cross section of the tumor, with theconvention that necrotic cells are labeled withblack, non-proliferative tumorous cells with lightgray and proliferative tumor cells with dark gray.Note that the plotted points are not intendedto depict the exact shape of the Voronoi cells,but rather just their positions. Figure 7 depicts

FIG. 6. The development of the cross-central section ofa tumor in time. (a) corresponds to the tumor spheroid stage,(b) to the "rst detectable lesion, (c) to diagnosis and (d) todeath. The dark-gray outer region is comprised of proliferat-ing cells, the light-gray region is non-proliferative cells andthe black region is necrotic cells. The scales are in milli-meters.

several cross sections taken at di!erent positionsfrom within a single fully developed tumor. Asexpected in this idealized case, the tumor is essen-tially spherical, within a small degree of random-ness. The high degree of spherical symmetryensures that the central cross section is arepresentative view. The volume and radius ofthe developing tumor are shown vs. time in Fig. 8.Note that the virtual patient dies while theuntreated tumor is in the rapid growth phase.

While these results con"rm that the algorithmis able to reproduce a very idealized case, they donot illustrate its ability to model more complexsituations. As mentioned in the Introduction, onesuch complexity is the inclusion of multiple dis-tinct tumor clones. An interesting question wehave begun to address is that of the emergence ofa clonal population from a small mutated hotspot in the original cell population. To addressthis, a second strain is de"ned by the parameterset:

p0"0.384, a"0.42 mm1@3, b"0.11 mm1@3,

Rmax

"37.5 mm.

FIG. 8. Plots of the radius and volume of the tumorversus time. The lines correspond to simulation predictions,using the "rst parameter set given in the text. The plottedpoints re#ect the test case derived from the medical litera-ture. A quantitative comparison of the simulation with thetest case is given in Table 2.


The second strain is a more rapidly dividingclone than the "rst, parental strain de"ned above.The increase in p

0gives a decreased cell-doubling

time (1.7 vs. 4 days in the primary strain), chosento re#ect the degree of heterogeneity betweendi!erent clonal strains. This example corres-ponds to the emergence of a more rapidly divid-ing clone from the more slowly proliferatingmalignant parental strain and thereby models the&&progression pathway'' of GBM tumors from lessmalignant precursors (Lang et al., 1994). Thissimulation begins in the same way as the pre-vious one, with a very small tumor, composedentirely of cells of the primary strain. This tumoris then allowed to grow until it reaches aprede"ned overall radius of 3.8 mm. A mutatedhot spot is then introduced as a single, randomlyselected, CA cell (corresponding to roughly 105real cells, i.e. the size of a spheroid) changing fromthe primary strain to the secondary strain. Thisrepresents the appearance of the second genotypein roughly 0.01% of the viable tumor cells at thattime, assuming monoclonal expansion. At thistime we do not distinguish between randommutational activity (intrinsic) or environmentallycaused mutational stresses. Figure 9 depicts thedevelopment of the secondary strain (shown inblue) in the cross-sectional slice nearest the ap-pearance of the mutation, which is displaced from

the tumor center by roughly 2.5 mm in this case.To aid in distinguishing between strains, cells ofthe primary clonal strain (dark gray in Fig. 6)have been colored red, while the non-prolif-erative cells (light gray in previous "gures) havebeen colored yellow. A bulge near the initialposition of mutation appears soon after themutation occurs. This corresponds to a morerapidly advancing tumor rim due to a higherlocal proliferation rate. This higher proliferationrate also leads to additional cells being shed perunit of time, which in turn leads to a higherinvasive potential towards this area. Gradually,the secondary strain overtakes the primary strainand comes to dominate the entire tumor. Thisexample shows the spatial evolution of a second-ary tumor population. From such a simulation, itis possible to consider the e!ects of mutations onthe shape and location of a tumor. In this simula-tion, the secondary strain overtakes the primarystrain su$ciently rapidly so that the center of massof the entire tumor shifts considerably from itsoriginal location. In addition, the bulge created bythe more rapidly advancing front increases thesurface-area-to-volume ratio of the tumor, there-by increasing the growth fraction and causing thetumor growth to accelerate more than would bethe case if the tumor remained spherical.

Currently, we are exploring the survival prob-ability of a secondary strain as a function of thenumber of cells that have mutated and the com-petitive advantage it enjoys as well as the posi-tion within the tumor at which the second strainarises. Given too small an advantage, the secondstrain will be quickly overwhelmed by the muchmore common primary strain and will likely dis-appear. Preliminary results show that for a fairlysmall mutation, like the one considered here,a large competitive advantage in doubling time isrequired (of the order of a factor of two), for thesecondary population to survive. In biomedicalterms, the tumor is &&searching'' for the best suitedheterogeneic pattern, i.e. the genotypes andphenotypes (considering gene expression changes,along with mutations) that are optimal relative tothe tumor's growth rate. The current modelfocuses on the importance of growth-promotinggenetic combinations. From a biological per-spective, the model incorporates selectionpressures leading to the dominance of the clonal

FIG. 9. Cross-section of a tumor, showing the emergence and eventual dominance of a more rapidly growing secondarystrain. The red region corresponds to proliferating cells of the primary strain (darker gray in Figs 6 and 7), the blue region tothose of the secondary strain, while the yellow (light gray in previous "gures) and black regions correspond to non-proliferative and necrotic cells of either strain, respectively. The cross-sections are taken 2.5 mm from the tumor's centralplane. (a) Depicts the tumor roughly 1 month after the initial mutated CA cell was introduced in the simulation, (b) 5 monthsafter the mutation, (c) 10 months after the mutation, and (d) 20 months after the mutation. Note that the tumor origin islocated at (0, 0), but because of the mutation, the center of mass of the tumor shifts substantially from this location, beforebeing forced back by the boundary conditions.


population best suited to the tumor's environ-ment. Realistically, environmental in#uencesshow spatial heterogeneity and as such will notnecessarily lead to the dominance scenario depic-ted here, but rather may lead to a tumor withmultiple, coexisting tumor clones (i.e. the hall-mark of GBM tumors), each best suited to theirlocal environment.

5. Discussion and Conclusions

Substantial progress has been made in the vari-ous specialized areas of cancer research. Yet thecomplexity of the disease on both the single celllevel as well as the multicellular tumor stage hasled to the "rst attempts to describe tumors ascomplex, dynamic, self-organizing biosystems,rather than merely focusing on single features(Bagley et al., 1989; Schwab & Pienta, 1996;Co!ey, 1998; Waliszewski et al., 1998). To beginto understand the complexity of the proposedsystem, novel simulations must be developed, in-corporating concepts from many scienti"c areassuch as cancer research, statistical mechanics,applied mathematics and nonlinear dynamicalsystems.

To our knowledge, this is the "rst three-dimen-sional cellular automaton model of solid tumorgrowth, which realistically models the macro-scopic behavior of a malignant tumor as a func-tion of time using predominantly microscopicparameters. This four-parameter model predictsthe composition and dynamics of malignantbrain tumor proliferation at selected clinicallyrelevant time points in agreement with experi-mental and clinical data. From a modeling per-spective, this is also the "rst use of the Voronoitessellation to study tumor growth in a cellularautomaton. The use of a variable density latticeenables the simulation of tumor growth overnearly three orders of magnitude in radius.Finally, the ability of internal cells to divide rep-resents a physiologically more realistic situationand changes the proliferative dynamics from pre-vious models.

In addition, the discrete nature of our modelenables us to directly simulate more complexphysiological situations with only minor alter-ations. Examples include regional di!erences instructural con"nement and the in#uence of envir-

onmental factors, such as blood vessels. The im-pact of surgical procedures or other treatmentscan also be modeled. Further, we can currentlyinclude the hallmark of GBM tumors*hetero-geneity*by using tumorous subpopulationswith di!erent growth behaviors (Giangaspero& Burger, 1983; Burger & Kleihues, 1989; Paulus& Pfei!er, 1989). These tumor subpopulationsrepresent a clear step forward from previous&&monoclonal'' (single population) models. Envir-onmental stresses are known to exert a strongin#uence on tumor growth (Helmlinger et al.,1997). These stresses can both cause geneticvariations, and so increase the number of sub-populations, and select for the most "t tumorsubpopulation. It is this diversity in structure andfunction which enables tumors to react locallyand globally. It is important to mention that,according to this concept, systemic tumoralgrowth increases both intrinsic and extrinsicstress and therefore leads to a higher selectionpressure and mutational probability. Since thismay advance tumor progression, it clearly arguesagainst Foulds' rule III (that tumor growth andprogression are independent) (Foulds, 1954;Rubin, 1994). In the next iteration of the model,we will simulate the conditions which create re-gional genetic instability and study the e!ect ofspeci"c mutations in tumor cells on the macro-scopic growth of tumors. Such information couldallow tumor biopsies, which are currently region-ally limited, to be put into the context of anevolving system. This will lead to important in-formation about intratumoral competition andcooperation and how tumors adapt to maintaintheir "tness in the environment they are present-ed (Greenspan, 1976; Axelrod & Hamilton, 1981;Gatenby, 1996).

Another important step on our way to acomplex dynamic tumor model is the speci"cinclusion of the other key feature of tumors,namely single cell invasion. The current modelonly includes such an e!ect implicitly. Active cellmotility is a crucial feature, not only because ofits local disruptive capacity, but also because ofits signi"cance to treatment. If a solid tumor isremoved, these invasive cells, which would be leftbehind, can eventually cause recurrence of thetumor. We are currently working to theoreticallyassess the factors which may drive some of the


structural elements within the invasive network,in accordance with a proposed deterministicchaos principle (Habib et al., unpublished data).The spatial distribution of these factors will alsodepend on the heterogeneous intercellular space.While this space is not included in the current work,it will be properly accounted for in further work.

In conclusion, such complex interdisciplinarymodels are likely to lead to insights within cancerresearch and into complex biosystems in general.Such insights hold promise for increasing ourunderstanding of tumors as self-organizing sys-tems, which in turn may have signi"cant impactboth on cancer research and on clinical practice.By no means does this model claim such com-pleteness. It is instead a "rst, but promising, steptowards advanced modeling techniques treatingtumors as complex dynamic systems.

S.T. gratefully acknowledges the GuggenheimFoundation for his Guggenheim fellowship to conductthis work. This work has also been supported in partby grants CA84509 and CA69246 from the NationalInstitutes of Health. Calculations were carried out onan IBM SP2, which was kindly provided by the IBMCorporation (equipment grant to Princeton Univer-sity for the Harvard}Princeton Tumor Modeling Pro-ject). The technical assistance provided by Drs Kirk E.Jordan and Gyan V. Bhanot of the IBM T.J. WatsonLabs is gratefully acknowledged. The authors wouldalso like to thank Drs Stuart A. Kau!man of the SantaFe Institute for Complex Science, Jerome B. Posnerof Memorial Sloan Kettering Cancer Center, MichaelE. Berens of the Barrows Neurological Institute andH.J. Reulen of the University of Munich Neuro-surgical Department for valuable discussions.

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simulated brain tumor growth dynamics using a three...

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