数理解析研究所講究録
第 828 巻 1993 年 115-121

115
A Competition Model for Tumour GrowthV.S. Manoranjan
Department of Pure and Applied Mathematics
Washington State UniversityPullman, Washington 99164-3113, U.S.A.

Abstract

A simple competition model, which couples Gompertz and logistic growth mechanisms,
is presented in order to study the growth pattern of two different types of tumour cells
growing together. It is shown that the model prediction is in fairly good agreement withthe experimental finding.
1. INTRODUCTION

Cancer research has been one of the major areas of medical research for a long time.
However, still there are many aspects of tumour growth which are not well understoodby medical scientists. In a recent experimental study [9], human tumours were grown as
xenografts in nude mice and the differences in the rates of growth at contralateral sites
were examined. It was found that the tumour pieces implanted into the dorsal left flanksof the mice grew into significantly larger tumours than the ones implanted into the dorsal
right flanks. This was true for the tumour types such as colon carcinoma Colo-205 andgastric carcinoma MKN 45. A somewhat similar finding was noted for mouse tumours
too. It was not clear why there should be such anatomical differences in growth rates.
But it was hypothesized that it could be due to the effect of morphogenetic gradients asin the case of control differentiation during ontogeny.
In some other experimental investigations controversial results have been reported inregard to the effect of interferons on human myelogeneous cell line (HL-60) establishedfrom the peripheral blood of a person with acute promyelocytic leukemia. Certain studies$[2,4]$ find that these cells are insensitive to human alpha and beta interferons. However,there are other studies [10] which show that alpha and beta interferons are effective in theinhibition of HL-60 and do not concur with data in [4]. A convincing explanation either
to justify both sets of results or to repudiate one of them is still not available.
Further, most of the research has concentrated on homogeneous tumour growth. It
is known that [1] stage A2 prostatic cancer is composed of a heterogeneous group of tumours. But there are not many pathologic studies which investigate this heterogeneity.
The question is, can one come up with mathematical models for tumour growth which
could be useful for the experimentalists? Since tumour growth experiments can produce
lots of quantitative data, it should be possible to construct such models. In this paper,

116
we look at the problem of heterogeneity in tumour growth and construct a mathematicalmodel. This model is shown to satisfy some of the experimental findings available. We
believe that experimentalists could find this simple model very useful.

2. MATHEMATICAL MODEL

One of the growth curves that has been widely used [3, 7, 11] in characterising tumour
growth is the S-shaped Gompertzian growth curve. This curve is given by the equation,(1)$N(t)=\kappa\exp[-\exp(a-bt)]$, the carrying capacity andwhere $N(t)$ is the total number of tumour cells at time ,are appropriate constants. The expression in (1) can be thought of as the solution of
the set of differential equations,
$t$

$a,$

$\kappa$$b$$\frac{dN(t)}{dt}=\gamma(t)N(t)$

(2)

,(3)

and
$\frac{d\gamma(t)}{dt}=-\alpha\gamma(t)$where is a positive constant. If the growth curve is a simple exponential growth curve
will be just a positive constant in (2).
$\alpha$$\gamma(t)$It can be argued that there is no particular reason to expect the Gompertz curve toshow any wider range of fitting power to a given data set than any other S-shaped curvewith three constants. For example, the logistic curve which can be written as,(4)$N(t)=\kappa/[1+\exp(a-bt)]$

is also S-shaped and has three constants (having the same meaning as in (1)), and. This curve can also be employed to fit any given data. But it should be noted that
there is a subtle difference between the Gompertz and the logistic curves. Although, bothcurves have points of inflection when $t=a/b$ , the ordinate that corresponds to Gompertz
is. This means, that in the cases of Gompertz
while that corresponds to logistic isand logistic the point of inflection occurs when approximately 37% and 50% respectively,of the total growth has been realized. So, it is really up to the experimentalist to decideon the best fitting growth curve for his or her experimental data by determining where
the point of inflection should be.
$\kappa$$\kappa$$a$$b$$\kappa/e$$\kappa/2$

In order to understand the experiments reported in [9] (as mentioned in the previoussection), where larger tumours in the left flank were seen than in the right flank, we tried
to fit the data obtained for gastric carcinoma MKN 45 by different types of growth curves.
It is seen (figures 1 and 2) that Gompertz type curves are the best fitting curves for thesedata (number of tumour cells are assumed to be proportional to tumour volume). Further,
the only difference between the Gompertz expression in the left flank and that of the right

117
is the constant . The values of the constants and in both cases do not change. Thisimplies that the left flank has a larger carrying capacity than the right. So, one could
argue that the anatomical differences in growth rates of any particular type of tumour
cells are essentially due to differing carrying capacities of different regions and nothing
else.
$a$

$\kappa$$b$Now, we will focus on the problem of two types of tumour cells growing together. In
this case, the two cell populations must compete for vital nutrients, growth factors, etc. A
study consisting of Ehrlich ascites cells (of different types) proliferating in the peritoneumof mice was carried out in [5]. The types of cells examined were diploid and tetraploidcells. It was found [5, 6, 8] that no matter what the initial distribution of cells were, at thesteady state, always, 96% of the total number of cells were diploid and 4% were tetraploid.
Can a mathematical model predict this finding? In order to answer this we construct thefollowing competition model:$\frac{dN_{1}}{dt}=rN_{1}(\ln\kappa-\ln N_{1})-\alpha N_{1}N_{2}$(5)$\frac{dN_{2}}{dt}=sN_{2}(L-N_{2})-\beta N_{1}N_{2}$(6)whereandare the number of diploid cells and tetraploid cells respectively at any
given time . The terms
andrelate to the competition between the cell
populations with and being positive constants. As can be noted from equations (5)and (6), we have taken the growth of diploid cells to be Gompertz growth and of tetraploidcells to be logistic growth. This is simply because the growth data of diploid cells growing
on their own and of tetraploid cells growing on their own given in [5] were found best fitted
by Gompertz and logistic curves, respectively.
$N_{2}$$N_{1}$$\alpha N_{1}N_{2}$$t$$\alpha$$\beta N_{1}N_{2}$$\beta$In order to analyse the model, let us consider the phase plane diagram for the equations(5) and (6) (figure 3). It is clear that there are four steady states with $(0,0),$and$(0, L)$ being the trivial ones. The only non-trivial steady state is given by the point ofintersection of the curves,$(\kappa, 0)$$N_{2}= \frac{r}{\alpha}\ln(\kappa/N_{1})$and$N_{2}=L- \frac{\beta}{s}N_{1}$and it can be easily checked that this steady state is asymptotically stable. So, this could
be the steady state which relates to 96% to 4%.

118$Growt1_{1}$curve of MKN45 tumor xenograft in left flank.

Growth curve of MKN45 tumor xenograft in right flank.

119Fig. 3

Phase plane diagram for the Competition model.

In [5], the growth data for diploid and tetraploid cells growing together are also given.
Using this data set along with the data set when these cell populations are growing on
their own, we are able to estimate the constants $L,$and . It should be pointed
out that for this estimation procedure only data up to 8 days are used. In the experiments the steady state is reached after day 20. This means that our estimation is doneat the initial transient phase of the growth process. Is it possible for our model to predict
what happens after day 20 using these (up to 8 day) estimates? We obtain the answer tothis by numerically solving the coupled ordinary differential equations (5) and (6) using
the estimated values for the constants. The result is presented in figure 4 and it is obviousthat the steady state is reached after day 20. But, what percentage are the diploid cellsof the total cell population at steady state? We find this to be around 91%. Compare itto the value (96%) obtained in experiments. Clearly, the model’s prediction is very good,
if one makes allowances for the experimental errors, estimation errors and so on.
$\kappa,$$r,$ $s,$$\alpha$$\beta$120
Fig. 4

Numerically simulated growth curves for diploid and tetraploid cells.

3. CONCLUDING REMARKSIn this short paper, a simple competition model is developed to describe the growthof two different types of tumour cells growing together. We demonstrated that the model
can predict the experimental finding fairly well. It should be pointed out that this model
can be easily expanded or modified if one needs to have more than two different cell
types. For example, if the tetraploid cells springing from diploid cells through endomitotic
differentiation are to be considered in a study, one can simply assume them to be a third
type of cells. Then, all we need, to modify the mathematical model are an extra growthequation (in differential form) for this third type of cell and a loss term in the growthequation for diploids. Therefore, we feel that the model presented here could be a very
valuable quantitative tool to the experimentalists.

121
References
1. W.N. Christensen, A.W. Partin, P.C. Walsh and J.I. Epstein, Pathologic findings inclinical stage A2 prostate cancer, Cancer 65, 1021 (1990).

2. S.J. Collins, R.C. Gallo and R. Gallagher, Continuous growth and differentiation ofhuman promyelocitic cell line HL-60, Exp. Cell Res. 106, 494 (1980).

3. M.A.B. Deakin, Gompertz curves, allometry and embryogenesis, Bull. Math. Biophysics 32, 445 (1970).
4. A.W. Hamburger, C.P. White, R.E. Sienbenlist, J.J. Sedmak and S.E. Crossberg,Cytotoxicity of human -interferon for differentiating leukemic HL-60 cells, Cancer
Res. 45, 5369 (1985).
$\beta$5. B. Jansson and L. R\’ev\’esz, Analysis of the growth of tumor cell populations, Math.
Biosciences 19, 131 (1974).

6. K. Kaziwara, Derivation of stable polyploid sublines from a hyperdiploid Ehrlich
ascites carcinoma, Cancer Res. 14, 795 (1954).
7. W.S. Kendal, Gompertzian growth as a consequence of tumor heterogeneity, Math.
Biosciences, 33, 103 (1985).

8. G. Klein and L. R\’ev\’esz, Qualitative studies on multiplication of neoplastic cells invivo I. Growth curves of the Ehrlich and MCIM ascites tumors, J. Nat. Cancer Inst.
14, 2 (1953).
9. M.V. Pimm and T.M. Morris, Growth rates of human tumors in nude mice, Eur. J.
Cancer, 26, 764 (1990).
10. G. Sica, G. Lama, R. Tartaglione, L. Pierelli, L. Frati, G.R.D. Cuna and P. Marchetti,
Effects of natural beta-interferon and recombinant alpha-2B-interferon on proliferation, glucocorticoid receptor content and antigen expression in cultured HL-60 cells,
Cancer 65, 920 (1990).
11. C.P. Winsor, The Gompertz curve as a growth curve, Proc. Nat. Acad. Sci. 18, 1(1932).