(CANCER RESEARCH 51. 2542-2547. May 15. 1991]

Population Ecology Issues in Tumor Growth

Robert A. GatenbyDepartment of Diagnostic Imaginx, Fox Chase Cancer Center, Philadelphia, Pennsylvania IVI 11

ABSTRACT

Mathematical models developed from population ecology are appliedto tumor-host interactions and demonstrate the importance of increasedefficiency in substrate absorption as a mechanism enabling tumor cellsto (a) proliferate despite inefficient energy production and (b) competesuccessfully for resources with the numerically superior host cells. Aswith many biological invasions observed in nature, success of the invaderscan be enhanced by disruption of the local ecology if the disruption resultsin decreased viability of the native populations reducing their ability toinhibit tumor cell growth either directly through an immunological response or indirectly by competition with the tumor cells for availableresources.

Following successful invasion and displacement of normal cells from avolume of tissue, tumor cells achieve new equilibrium states with theenvironment based on their efficiency of consumption and the vascularityof the tissue.

Tumor therapy may be enhanced by reducing available resources belowa level which will support growth of the tumor cells or above a thresholdwhich will allow repopulation by normal cells.

INTRODUCTION

The principles of population ecology provide a novel conceptual framework for the examination of tumor-host interaction.Every discrete volume of tissue within an organism can berepresented as an ecological domain populated by species ofnormal epithelial and mesenchymal cells in equilibrium withresources which enter through diffusion or blood vessels. Tumors begin when a single or small number of individuals froma new (tumor) species enter or arise within this domain. Although the causes and manifestation of oncogenesis must beunderstood on a molecular level, the formation of tumor occursonly when individual malignant cells compete successfully withthe host cells allowing them to obtain adequate levels of substrate for proliferation despite the numerical advantage of thepreexisting population and the "predatory" effect of the im

mune system. Interaction of the invading tumor cells with thenative host cells may result in three general outcomes: (a)extinction of the original population, (b) achievement of a newstable equilibrium in which the tumor cells coexist with normalcells, or (c) extinction of the invading population.

Mathematical models from population ecology, thus, may beapplied to tumor-host interaction. Although this approach requires oversimplification of complex biological processes, itmay provide useful insight into the basic mechanisms of tumorgrowth and should make predictions which can be tested bothin vivo and in vitro. Previous attempts to model tumor growthhave emphasized fitting tumor volume data to Gompertzian ( 1)or exponential (2) curves. Michaelson el al. (3) applied population ecology principles to competition between tumor sub-populations, but these models have not previously been used toanalyze tumor-host interaction.

Received 12/6/90; accepted 2/22/91.The costs of publication of ihis article were defrayed in part by the payment

of page charges. This article must therefore be hereby marked advertisement inaccordance with 18 U.S.C. Section 1734 solely to indicate this fact.

RESOURCE ASSIMILATION AND UTILIZATION

Each available resource (R) is absorbed and used within thecell for maintenance and reproduction:

c, = m-,+ p;, i = I, 2 K (A)

where c, = consumed resource/individual cell/unit time, m, =resource utilized for maintenance/cell/unit time, and p, = resource available for reproduction/cell/unit time for K resources.

Consumption (uptake) of each resource will vary dependingon the amount of the resource available and the capacity of thecell to acquire it. Thus,

ci=MR,) (B)

This analysis will use glucose as a model resource because itis a critical substrate for energy production and thus essentialfor both maintenance and reproduction. Furthermore, themechanisms of uptake via glucose transport polypeptides (4, 5)and utilization for energy production (6, 7) are well established.

Holling (8) distinguished three typical shapes for the consumption curve c =/(/?) as shown in Fig. 1. In this study, theHolling type III sigmoid response curve will be used as areasonable approximation of the acquisition of glucose by cells,although the actual shape of the consumption curve does notultimately affect the conclusions of the analysis.

Returning to Equation A, the total resource absorbed andutilized by a population of ¿Vtumor cells per unit time will be

AT = N m + Np. (C)

when c = average consumed resource/cell, m = average resourceutilization for maintenance/cell, and p = average resource available for reproduction/cell. For the tumor cell population togrow (dN/dt > 0), p > 0 and thus c - m > 0. That is, resource

consumption must exceed resource utilization for maintenanceif the cell population is to increase.

The relationship of population kinetics (dN/dt) to consumption and metabolism can assume many forms but may beexpressed as

(D)

where /Vmavis the maximum number of cells that may occupy agiven volume of tissue if resources are unlimited and should berelated to such constraints as cell volume and cell-cell contactinhibition. D is the intrinsic rate of growth (e.g., doubling time)if resources are unlimited and is dependent on a variety offactors such as the tumor cell genetics, level of growth factorspresent, and the number of receptors available for growthfactors on cell surfaces, y is a conversion factor (9) for thetransformation of resources into new individuals. In the remainder of this work, D/y will be expressed as a single constantB.

For a Holling type III functional response, Equation D canbe restated as

ER2-m R02+ R2 (E)

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TUMOR GROWTH MATHEMATICAL MODELS

TUMOR CELL INTERACTION WITH ENVIRONMENT

In the absence of consumers, the quantity of a resource inany given volume of tissue can be expressed

Typei

Type III

Type II

RFig. 1. Possible consumption curves as described by Moiling. All three curves

assume a saturation level. Types II and III approach the saturation curve gradually. Type I exhibits an abrupt saturation. The type III consumer is inefficient atlow resource levels.

dR R(F)

where r = the rate of resource delivery and K = the carryingcapacity of the environment. When R = K, dR/dt = 0 becausethe resource lost because of diffusion to adjacent tissues oroutflow in the vasculature is equal to the inflow of the resource.Clearly, K and r will decrease substantially if the tissue ishypovascular or if the tumor destroys the vasculature. Thedashed curves of Fig. 3 demonstrate two possible resourcecurves for a volume of tissue with (top line) and without (bottomline) vasculature.

When consumption by a population of cells is considered,the resource kinetics can be expressed as

(G)

For the Moiling type III consumption response this becomes:

dRdt

R2R02 (H)

where E is the saturation level of c and /?„is the half-saturationdensity of the resource (i.e., the value of R at which c is "Aof

E).From Equations D and E, it can be seen that when available

resources are less than the requirements for maintenance (m >c) dN/dt < 0 and the population will decline. When c = 0, dN/dt will approach -NBm and thus the term -Bin can be inter

preted as the death rate in the absence of the resource. Fig. 2examines consumption curves for cells with Holling type IIIfunctional response with varying R0 and E and graphicallydemonstrates the effects of these parameters on the ability ofcell populations to compete for a resource.

From Equation D, it is clear that, for a population to growwhen N < jVmax,it must minimize the resource commitment formaintenance (—m)and maximize resource absorption ( c") for

the ranges of R within the volume of tissue making up thedomain. When this principle is applied to glucose as a modelresource, it is clear that the preferential use of aerobic glycolysis(6, 7) poses a problem for tumor cells because it increasesglucose requirements for maintenance. Increased absorption( c ) of glucose is therefore required if tumor cells are to haveadequate energy for proliferation. Furthermore, as this modelwill demonstrate, this necessary enhancement of glucose absorption will ultimately be critical in allowing tumor cells tosuccessfully compete with normal cells for available substrate.

Finally, in Equations D and E, doubling time (D) has norelevance to tumor population survival. That is, dN/dt is positive or negative depending only on the (c —ni) term. If dN/dt

>0, however, ßwillsignificantly effect the actual rate of growthof the population.

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This is illustrated by the solid lines in Fig. 3 which demonstrate the loss of R because of the presence of varying densitiesof cells with a Holling type III consumption response. For anylevel of R and A', dR/dt will be the result of subtracting the

solid line from the dashed line. Equilibrium is achieved whendR/dt = 0, which is graphically represented by the intersection

of a solid and dashed line.

E3,E2 -

E1-

PopulatJon 2

Population 3

Population 1

R0,(3)

Fig. 2. Three possible Holling type III consumption responses. Both £3andE2 are larger than E\ and. as a result, consumers II and III absorb more resourcesat high R. However, because Ä.XUis less than /?ooj. consumer 1 will absorb moreresource at low R. Because RM2,is less than RM>.consumer 2 will absorb moreresources at low R than consumer III, even though the saturation point is thesame for both.

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TI MOR GROWTH MATHEMATICAL MODELS

fi

dtr2-

High N

Intermediate N

Low N

K2

Fig. 3. Dashed lines, resource delivery in the absence of consumers in vascular-i/ed (A]) and nonvasculari/ed (A'2) tissue; solid lines, consumption curves for apopulation at different densities A': intersection of the lines, equilibrium state (dR/

dt = 0).

Population equilibrium will be achieved when dN/dt = 0.From Equation E, it is clear that dN/dt = 0 when N = yvmaxifthe population is not resource limited. However, if the population is resource limited and Nc<tis </Vmax,dN/dt = 0 when:

R<,2

or (I)

Rea E -ni

where /?t.g= the value of R at equilibrium.The relationship of the equilibrium values of R and N are

shown in Fig. 4. The solid lines are the equilibrium values of Rfor any given N depending on the presence (topcun'e) or absence(bottom cun'e) of tumor vasculature and are derived from Equation H for dR/dt = 0 so that

1 +

R0- - ÄfK

R-(J)

The equilibrium values of R (/f„,when dN/dt = 0 from Equations E and I) for a resource-limited population with threelevels of consumption are shown by the dashed horizontal lines.The entire system is in equilibrium when dN/dt = dR/dt = 0and is shown as the intersection of the dashed lines with thesolid lines.

If tumor expansion into a volume of tissue results in destruction of the vasculature in that volume. A', and r, in Equation Hwill be replaced by A': and r?. Since A': < A', and r^< r,, dR/dt

will initially be <0 (because R/K, < /?/A%), reducing resourceswithin the domain and resulting in dN/dt < 0 (from EquationE). A new equilibrium will then be achieved with R returning

to the initial /?cqbut with a new lower value of A'(from Equation

J and graphically demonstrated in Fig. 4), resulting in the deathof some fraction of the original cell population. This wouldresult in areas of necrosis within the tumor.

Thus, the interaction of the tumor cell population with thehost environment can result in multiple possible equilibriumstates depending on the carrying capacity (vasculature) of theenvironment and the capacity of the tumor cells to absorb theavailable resources. As demonstrated in Fig. 4, vascularizedregions of tumor have varying cell densities (/Vc.g)and resourcelevels (/?„,).Devascularized regions of tumor are only able tosupport cell densities lower than that seen in vascularized areas,resulting in cell death and thus partial or even complete (if Rcq< A'2)necrosis.

TUMOR INTERACTION WITH HOST CELLS

Until now, we have treated tumors as a species entering anunoccupied environment. In most situations, however, the tumor species enters an ecological domain completely filled byexisting species of normal cells at equilibrium with the availableresources. Thus, individual tumor cells must compete successfully with normal cells for these resources in order to gaincontrol of this volume of tissue.

Using an analysis similar to Armstrong and McGehee (10)consider two species (Fig. 5) competing for a limited resourcewith a Holling type III functional response. One populationconsists of tumor cells (W,), while the other consists of the hostcells normally present (N2). In this analysis, the heterogeneity(11) of tumor cell societies and the varied cell types present in

NFig. 4. Solid lines, equilibrium states (dR/dt = 0) for any value of N for

vascularized (A'i) and nonvascularized (A'2) tissue and redrawn for computer-

generated solutions of Equation J; dashed horizontal lines, equilibrium states dN/dt = 0 expressed in terms of K (from Equation I) for consumers of high. low. andintermediate efficiency of absorption. Note the change in the equilibrium valueof A'in vascularized and nonvascularized tissue demonstrated by the intersection

of the dashed line with the two different solid curves.

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TUMOR GROWTH MATHEMATICAL MODELS

dt

E2-

Population 1

Population 2

T, T2 K

RFig. 5. Dashed line, resource levels within the volume of tissue when no

consumer is present; solid curves, two populations with Hollings type III consumption responses which differ in that the E of population 1 (tumor) is greaterthan the E of population 2 (normal), while the Ä„of population 1 is less thanthat of population 2. Intersection of the consumption line of each population withthe logistics curve of the resource, equilibrium state for each population. Clearly.population 1 with a higher £value achieves a equilibrium state at a lower R (T,)value than population 2 (r2).

normal tissue are simplified by "averaging" the various subpop-

ulations. A more realistic approach would expand the numberof competing populations into several tumor subpopulations(./Vu, Nib, 7Vlc)and normal cell subpopulations (A^2a,A^b, N2c,etc.). This, however, markedly increases the complexity of themathematical analysis and will not be pursued further at thistime.

In the two species analysis, growth for the tumor populationW) ¡s:

—mi+£,*2

and for the normal population (N2):

—mi +

+ R'

E2R2

(K)

(L)

resulting in resource kinetics of:

dt K

£,A22

E2R2(M)

Before the tumor cells enter, normal cells will have achievedan equilibrium in which

dN2 dR—r- = ——= 0 at some valuesdt dt

N2 = 72 and R2 = r2

Where 72 = A72(m„)or, if the population is resource limited,

is the population at which

«v+ (T2)2= 0

and

E2(r2f

If a small number of tumor cells (/V, «:jYi(max))enters or arisesin this volume of normal tissue, the tumor population will growif ¿/Vi/</i>Oand thus

-m, +Rod, + (T2)2

If dN\ldt > 0, then A/i will continue to increase until a newsteady state is reached (dNJdt = 0) at equilibrium values of Nt= 7, and R = T,.

This is termed invasability (10) and it is clear from the abovethat the tumor population (/V,) can invade the normal cellpopulation if T: > T¡.As shown in Fig. 5, T, will be less than r2if the population TV,is more efficient at resource absorption inthe range of R in the domain. The invasability of the normalcells into the tumor cells can be analyzed similarly. If tumorcells are at equilibrium at R = T, and

(r,)2R 2

0(2)

<0

then dN2/dt < 0 when TV,and R are at equilibrium and 7V2cannot invade or even coexist with A'i. Thus, if tumor cells can

take up resources more efficiently than normal cells and T, >T2, then the tumor population can invade normal tissue andlocally drive the normal cells to extinction.

In summary, this analysis predicts that the ability of tumorcells to successfully invade the host is critically linked to theircapacity to take up glucose and other resources more effectivelythan normal cells. This is consistent with and adds significanceto experimental observations that the transport of glucose (12-14) and other substrate (15-17) across the cell membrane issignificantly increased in transformed cells.

The above model is a laissez-faire (10) analysis because it

assumes that the two species do not interfere with one anotherdirectly but simply compete for available resources. This isprobably not realistic because it neglects the inhibitory effectsof the immune system on the invading tumor cells. Mathematical expression of this poorly defined phenomenon is difficultand could undoubtedly assume many forms. One possible general expression is:

(N)= fi.A', - (m, + c,) l - -) - DA',

where O is a probability of mortality/unit time caused by thehost response. It seems likely that this host interference willdepend on the number of host cells present (N2) and theresources available since the vasculature delivers both components of the immune system and the substrate (i.e., oxygen,complement, and antibodies) necessary for the immunologicalcells to act.

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TUMOR GROWTH MATHEMATICAL MODELS

Thus,

D=f(N2,R) (O)

The mortality function will not be further defined in thisanalysis except to argue that D will probably decrease as both/V2and R decrease. Furthermore (from Equation N), in earlytumor growth (/V, «KyV|max),dN,/dt will be >0 and D will notaffect the outcome of the tumor-normal cell competition

provided

Ai -m, +

or, expressed in terms of resource R:

> D

D\-I + B¡m[

R>

Thus, minimizing R0 and maximizing E may be adequate toallow tumor survival even in the face of significant predatoryactivity by the immune system.

Interference in the normal cell populations by tumor cells isalso possible. This will not be considered here since the mainconcern of the analysis is invasability of N2 by Nt. When N, isvery small, this interfering term should be negligible.

As with biological invasions observed in nature (18, 19), themodels predict that disruption of the local ecology may favorthe invader. For example, if local tissue injury, inflammation,or devascularization reduces the resource levels for a prolongedduration, R could decline below r2 resulting in a decrease in thepopulation (dN2/dt < 0) and low levels of resources for theremaining cells. If a tumor population arises in this volume oftissue and if R is >T¡,the tumor cells can proliferate uncheckedby any competition from the already declining normal hostpopulation. Furthermore, from Equation O, limited resourcesshould reduce the mortality function (D) associated with theimmune response.

SUMMARY OF TUMOR-HOST INTERACTION

Tumor and adjacent host tissue can be represented as multiple regions in differing states of equilibrium or transition. Asdemonstrated in Fig. 6, the host tissue surrounding the tumorcontains normal cells in equilibrium with available resources.The tumor-host interface is a site of transition in which a small

number of tumor cells invade and compete with the normalcells for resources. The tumor rim is populated solely by tumorcells which are in a new equilibrium with the resources previously used by the normal cells. The tumor core may be identicalto the tumor rim if the vasculature remains intact within thetumor. However, if the tumor destroys the host vasculature andfails to promote new vessels, the available resources in thetumor core will decline leading to a new equilibrium state inwhich the density of tumor cells is less than that in the vascu-

larized rim resulting in varying degrees of necrosis.As the tumor size increases with time, each volume adjacent

to the tumor will experience a transition from normal tissue tocompetition to tumor rim to tumor core as the "wave" of tumor

cells passes through.

Time T

Normal Tissue

N, -0N2-Y2

R -T2N, >0

N2-»0

R-T,

N2-0

Normal Tissue

Tumor-Host Interface

Tumor Edge

Tumor Core

Time T + AT

Normal Tissuej

N, >0

N, -Y,N2-»0

N2-0

Ri T,

N, S Y,N2-0

Rit,

Tumor-Host Interface

Tumor Edge

Tumor Core

Tumor Core

Fig. 6. a, graphic representation of areas of equilibrium and transition in tumorand adjacent normal tissue as described in the text. In b. with increasing time,the tumor radius increases and the tumor passes through the adjacent normaltissue and the tumor core expands.

IMPLICATIONS FOR THERAPY

If tumor cells are killed or damaged by therapy, Fig. 4demonstrates that, starting from any equilibrium state, R willincrease when dN,/dt < 0. Using Equations L and M, we findthat during treatment, as long as N¡< 7Veq,R will be >T,. Whentherapy is withdrawn, the resources available in the environment will allow dN¡/dt > 0 provided N¡> 0. Thus, therapydirected at cancer cells alone will ultimately be ineffective unlessN, can be reduced to 0.

Therapy, however, could be enhanced by the following: (a)reduction in R below T, will maintain dN}/dt < 0 and add tothe negative growth effects of therapy. This would have to beaccomplished by destruction of the blood vessels in and adjacentto the tumor; (b) increase R above r2 allowing dN2/dt > 0 andthus proliferation of normal cells (assuming they are not killedor damaged by the tumor therapy) to compete with the remaining tumor cells. Returning to Fig. 4, it is clear that, in vascular-ized tumors with 7Vcqsmall and /?eq large, this process couldoccur because even a small percentage kill of tumor cells wouldallow the R value to increase to high levels (presumably aboveT2>.

For equilibrium states in vascularized tumors in which N} islarge and A«,is small, an equivalent percentage of reduction inN, will result in a much lower R, probably preventing anyrepopulation by normal cells.

In portions of the tumor which have become devascularized,it is possible that even a maximum reduction in N, will notallow R to increase above r2. In these regions, therapy will beeffective only if TV,is reduced to 0 or if resources are increasedby some therapeutic intervention.

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TUMOR GROWTH MATHEMATICAL MODELS

CONCLUSION

Population ecology models may be applied to the phenomenon of tumor growth and yield interesting insights into tumor-host interaction. Specifically, the models make the followingpredictions:

1. Success of tumor cells in gaining dominance is determinedsubstantially by their ability to absorb resources more efficientlythan normal cells. In the specific case of glucose used in thisanalysis, increased uptake provides two advantages: (a) it allowsadequate energy for proliferation despite the use of inefficientaerobic glycolysis by tumor cells, and (b) it allows tumor cellsto compete successfully with normal cells for available resourcesenabling them to invade and ultimately destroy the normal cellsin a given volume of tissue. This prediction is consistent withexperimental evidence demonstrating a significant increase inmembrane transport of glucose and other substrate in transformed cells.

In the competition between normal and tumor cells, doublingtime of the cells is not critical to the survival of the population(i.e., dN/dt > or < 0) and contributes to growth rate only ifdN/dt > 0.

2. Low R (i.e., TI < R < r2) states will favor tumor growth bymaintaining dN2/dt < 0 while dN\jdt > 0. This could be a localprocess such as with injured, devascularized, inflamed tissue ora generalized phenomenon such as host starvation. Tumor-mediated suppression of host vascularity and metabolism locally or through the induction of cachexia, thus, could enhancetumor growth.

3. The interaction of tumor cells and host environment canassume several clearly defined patterns with predictable finalequilibrium states which depend on the degree of vasculariza-tion and the efficiency of resource assimilation by the cells.

4. If tumor therapy does not effect the resource level withinthe tumor, dN,/dt will always be >0 at the end of treatmentunless N, = 0. Treatment will be favored, however, if the Rstate of the tumor region is decreased below levels which cansupport the tumor population or if R increases sufficiently toallow repopulation by normal cells which may then competesuccessfully with remaining tumor cells. The latter suggests thattumors at equilibrium at moderate N and high R will be mostresponsive to cytotoxic agents. Tumors which are devascularized or vascularized but at an extremely low R and high A'equilibrium will be difficult to treat because destruction of thetumor cells may not allow a new equilibrium value of R highenough to allow proliferation of competing normal cells.

ACKNOWLEDGMENTS

The author wishes to thank Joan Dillon for her patient and meticulous preparation of the manuscript and multiple reviewers for theirhelpful comments, particularly Drs. Lawrence Coia, William E. King,and Samuel Litwin.

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1991;51:2542-2547. Cancer Res   Robert A. Gatenby  Population Ecology Issues in Tumor Growth

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