Cel/ Tissue Kinet. (1973) 6, 69-79.
COMPUTER SIMULATION OF GROWING CELL POPULATIONS A.-J. V ALLERON
AND
E.
FRINDEL
Unité de Recherches Statistiques et Institut de Radiobiologie clinique de l'Institut National de la Santé et de la Recherche médicale, Villejuif, France (ReceiL'ed 12 July 1972; revisioll received 25 August 1972)
ABSTRACT
A new technique of computer simulation of growing population of cells is presented. This method can be used to study various biological problems the examples of which are discussed in this paper. INTRODUCTION Mathematical models have been used for many years to study ceIl kinetics. Most of the time, models have been devised to help with the interpretation of specific experimental data; for instance, many mathematical models have been developed to obtain maximal information from the percentage labelled mitoses (PLM) experiment (Barrett, 1966; MacDonald, 1970; Takahashi, 1968; Trucco & Brockwell, 1968). Models have also been used to formalize hypotheses about growing populations of cells; for instance, an alternative to the model of Mendelsohn (1962) was presented by Burns & Tannock (1970) and one allowing estimation of cellloss was developed by Steel (1967). Only a few papers deal with a mathematical representation of populations of cells without specifying, a priori, a particular application. Such a general procedure has been presented by Hahn (1970). The aim of this paper is to present a computer method of representing growing populations of cells which closely reflects the specific biological events assumed to occur under various hypotheses, particularly when variability is involved. THE METHOD
Computer representation of the descendence of a ce/l In this first step, we represent the family tree originating from a single cell born at time O. In order to specify the tree we have first to find a system of identification which allows the definition of the relationship between any two celIs, second to be able to store and to retrieve the pertinent data about events in the life of each ceIl (for example, the number, identity and position in the ceIl cycle of the ce Ils alive at time t may be required to be Correspondence: Dr A.-J. Valleron, Unité de Recherches Statistiques de l'I.N.S.E.R.M., 16 bis, avenue Paul Vaillant Couturier, 94-Villejuif, France.
69
A.-J. Valleron and E. Frindel
70
calculated). In Fig. l, such a tree covering four generations is represented assuming that there is neither ceII death nor ceII loss. Each branch of this graph represents a ceII and the four segments of each branch represent the four phases of the cycle. The numbers shown above each ceII are used to identify these cells. From these numbers one can immediately find the solution to the first problem posed; if k generations are studied these numbers range from 1 to 2k - 1 (here, k = 4, the numbers of identification range from 1 to 24 - 1 = 15). The procedure of identification is based on the fact that if J is the number associated with a specific cell and JI the number associated with the mother of that cell, the relationship JI = Int(Jj2) holds (where 'Int' means 'integer part of'). For instance, the mother of cell 13 is cell6 because 6 = Int(6·5) = Int(13j2). Thus, it is possible to compute the relationship of any two cells.
2
Time 0
Generation
FIG. 1. Family tree originating from cell 1. The identification number is shown above each cell. The length of the thick line indicates Y(13,2).
To store the events of the life of each cell of the tree, we define a two-dimensional array Y(J,K) where Y(J,K) is the time elapsed between time 0 and the end of the Kth phase of cell J. For instance, Y(1 3, 2) is shown on Fig. 1 (thick line). * To fill the array Y(J, K), t4e following recurrence equations are used: Y(J, 1) = Y(J,2) = Y(J, 3) = Y(J, 4) =
Y(J1>4) + GI(J) Y(J, 1) + S(J) Y(J, 2) + Gz(J) Y(J, 3) + M(J)
with JI
=
Int(Jj2)
(1)
GI(J), S(J), G 2 (J) and M(J) are the durations of the phases of the cycle for cell J. The four equations are obvious; for instance, the first expresses the fact that between the end • If g generations are simulated, the dimensions of this array will be respectively 2" - 1 and 4.
Computer simulation of growing cel! populations
71
of mitosis of cell JI and the end of G l for cell J, the time elapsed is equal to the duration of G l for cell J. In most of our applications, Gl(J), S(J), GiJ) and M(J) are lognormally and independently distributed. The procedure described above makes it possible to compute easily the number of cells of the family tree present at time t, present in a specified phase at time t (for instance while simulating simple labelling, it is important to know which cells are in S), or present in a specified phase at both time t and t' (as in double labelling): Cell J is present at time
f
if Y(Jl ,4) < t Y(J,4) > t
JI
=
(2)
Int(J/2)
The first equation expresses that, at time t, cell J has been formed; the second one expresses that cell J is still in the population. Cell J is present in S at time t, if Y(J, 1) < t Y(J,2) > t
(3)
Cell J is present in S at both time t and t' (l' < t) Y(J, 1) < f' Y(J,2) > t
(4)
Death and cell loss The case described here is the one where a proportion p of the cells may leave the population as a consequence of death, differentiation or migration at the end of a given phase. A new array D(J) is defined with D(J) = 1 if cell J is present and D(J) = if J is lost to the population studied. For each cell J, a random variable r, uniformly distributed on (0,1), is compared with p. If r < p, we let D(J) = 0 and, if r > p, D(J) = 1. The following relations are also considered:
°
D(Jl ) = 0
--+
D(J) = 0
and JI = Int(J/2)
(5)
The relation expresses the fact if cell JI is dead its daughter cannot exist. In this case, the array Y(J, K) is not filled.
Simulation of P-cells and Q-eells , Suppose that, as in Fig. 2, a population is divided into two compartments, one filled with proliferating cells (P-cells), the other with non-proliferating cells (Q-ceIls) and that at birth, a cell may enter the P-compartment with a probability p. The growth fraction GF (Mendelsohn, 1962) is related to p by the equation GF = 2p - 1. Thus, in order to represent a population with a growth fractionGF, a variable h uniformly distributed in (0,1) is, for each ceIl, compared to p; if h > p, Gl(J) is made to be very large (for instance equal to thirty durations ofP-cell cycle); if h - LAB(J) = 1 and Y(J, 1) < fi Y(J,2) > t i
JI = Int(Jj2)
cell Jin S LAB(J) = 1 at t = t l i= 1, 2, ... , N
(t>
(6a) fi)
(7)
Grain counts may also be easily simulated by modifying the equation (7): when cell J is in S, LAB(J) is taken to be equal to a variable simulating the number of grains after exposure; the equation (6a) is modified as foIlows: LAB(J)
=
LAB(JJ/2
(6b)
In equation (6b) it is assumed that the number of grains in cell J is equal to half the number of grains in cell JI'
Computer representation of the M'hole population If we study a family tree during g generations, we have to store in any problem at least \ 5(29 - 1) numbers which represent Y(J, K) and D(J). Th'us, if we want to study simul-
Computer simulation of growing cel! populations
73
taneously a large number of family trees, we can only do this for a few generations. To circumvent this limitation, calculations are made on consecutive family trees as summarized . in Fig. 3. Special computation procedures are necessary to shorten the total computing time and to reduce the storage required in the central memory of the computer. For instance, rather than having ail cells synchronized at birth at time 0 (as indicated on Fig. 1), the age of the
Building of Nth progeny d uring generations occording to Fig.1
K
Count of cells of the Nth progeny in Ihe differenl phases al differenl limes
No
FIG.
3. Diagram of computing procedure.
first cell of each family tree at time 0 is a variable distributed in the asymptotic age distribution. However, the simulations of experiments or counts should be do ne at a time greater than three generations when the age distribution becomes stationary. The number of generations simulated in each family tree never exceeds 13, this limitation being due to the size of the central memory of the computer used (UNIV AC 1107). The number of family trees needed may be great in some cases: for example, if one wishes to represent the evolution of the percentage of labelled mitoses after a 3H-thymidine injection. Two hundred mitoses per point may be needed in order not to have a high noise level, if the
74
A.-J. Valleron and E. Frindel
simulation of the experiment begins at the fourth generation, and if the mitotic index is 2 % about 1250 family trees will be required. SOME ILLUSTRATIONS OF THE METHOD Four examples are presented to show the ability of the method to simulate various biological experiments on growth kinetics. We wiII set aside aIl discussion of the biological aspect of the examples which wiIl only be I}sed in order to demonstrate our technique. The data used below is from a study of the NCTC 2472 fibrosarcoma (Frindel et al., 1969) grown in an ascitic form. The analysis was made on the fourth day following an intraperitoneal transplantation of 10 6 ceIls. Sequential labelling technique This technique has been described in a previous paper (VaIleron, Frindel & Tubiana, 1968): a weak dose (0·3 pCi) of 3H-thymidine is injected as a first label and, 1 hr later, a heavy dose (70 pCi) of 3H-thymidine is injected intraperitoneaIly. Thus, on the autoradiography one can distinguish three populations of ceIls; an unlabeIled population (UL), a weakly labeIIed population (WL) and a heavily labelled population (HL); Fig. 4 shows
- - - - WeoK label - - Heovy label
40
3 5 10
20, 30
40
50
>60
No. of groins
FIG. 4. Histogram of grain counts in a sequential two dose 3H-thymidine labelling experiment. After 3 weeks of exposure.
Computer simulation of growing cel! populations G 2 window First injection
, =-1 hr
Weak:0'3pCi
o
,= - l'
Second injection
f = 0 hr
Heavy 70pCi
f = 0 hr
t
=
x hr
No variability
Variability
FIG. 5. Representation of event.s after two sequential doses of 3H-thymidine labelling. t = -1' is the time immediately before the second injection.
20
*u
......
:;;; 10
+
+
\~+~+ + ----+ +
o o
5
10
15
Time (hr)
FIG. 6. Ratio of weakly labelled mitotic figures to weakly labelled cells as a function of time after the second label. + indicates experimental points; the line refers to the computed curve where G 2 is assurned to he lognormally distributed with a mean value of J.lG2 = 1·6 hr and a standard deviatiori of UG2 = 1'0 hr.
75
76
A.-J. Valleron and E. Frindel
the histograms of grain counts for WL and HL populations. The principle of the method is illustrated in Fig. 5: the rate of progression through the cell cycle of the WL population (G 2 window, the first label being weak) can be followed by determining the percentage of WL mitotic figures among the WL cells as a function of time after the second injection. The results are shown in Fig. 6. The time course variation of the mitotic index among the WL population depends on the distribution of T G2 also of S, provided we are able to assume that the duration of synthesis is larger than 1·0 hr for any cell. If the first injection would have been the heavy one, the mitotic index would depend on the joint distribution of TG2 and Ts . Two thousand family trees similar to the one shown in Fig. 1 were used to simulate the experiment; the curve has been smoothed by hand. The best fit with a lognormal distribution has been obtained with a O 2 duration of 1·6 hr and a standard deviation of 1·0 hr. The slight peàk observed on the experimental curve 5 hr after the second labelling might be explained by a bimodality of O 2 but, until we have further experimental evidence of this, we have not attempted to take this into account in fitting the data. This type of experiment is interesting from the biological point of view because it permits a direct study of the flux of cells from S to O 2 and a direct measurement of T G2 . However, without the help of our simulation technique the experimental data would be difficult to interpret.
Percentage labelled mitoses cun'e The percentage of labelled mitoses observed after different time intervals after injection of 3H-thymidine are indicated in Fig. 7.
oF>
5(
>2 Ë "0 ~
0; .0
50
.2
ëCl> u
&' 25
J ~
10
20
30
40
Time alter labelling (hr)
FIG. 7.
+ indicates experimental points. The solid line refers to the computed FLM curve
ÛlGI = 3·9 hr, aGI = 2'4 hr; Ils = 10'2 hr, as = 2·7 hr; Ilm = 1·6 hr.
aGZ
= 1·0 hr).
Computer simulation of growing cel! populations
77
The cell cycle parameters havè been estimated in the following ways: the G 2 parameters are known from the sequential labelling procedure presented above; the other cell cycle parameters have been computed from the experimental data by using a procedure similar to that of Barrett (1966) assuming a steady state population. These parameters are used in the simulation of the PLM experiment on an exponentially growing population. The corresponding curve is shown in Fig. 7. It has been obtained here too, after a manual smoothing. One thousand family trees were used in the computation. One can note that the fit of experimental points is good. Thus, the parameters computed by using the Barrett procedure provide, in this case, a good fit of experimental points although the simulation is performed on an exponentially growing population. Moreover, our technique às compared to other methods has the advantage of making it possible to simulate the PLM experiment on populations with any age distribution (even
- ------::--====---------==---=======-----
100
,0°1 ')":>1 / " ')01
•
GF
75
5
10
15
Time (hr)
FIG. 8. Multiple labelling experiment. Per cent labelled cells: experimental points (e) and computed curves (--).
non-stationary as the one resulting from a temporary block after treatment) and, although computer time consuming, may be needed for the interpretation of sorne experiments. Multiple labelling The percentage of labelled cells were determined after repeated injections of 10 jlCi 3H-TdR every 5 hr for 40 hr. This is analogous to a continuous labelIing since the mean duration of S is 10·2 hr. The results of the experiment are shown in Fig. 8. The continuous labelling curves assuming growth fractions of 100 %, 95 %and 90 %were calculated using our technique. Two hundred family trees were used in the simulations. Comparison with experimental data shows that the hypothesis of an almost completely proIiferating population is not incompatible. Here, the advantages of the technique are the same as those mentioned for the simulation of the PLM experiment. Quiescent cells We have shown in a previous paper (Frindel et al., 1969) that the method described by Mendelsohn (1962) for the evaluation of the growth-fraction including the correction due
A.-J. Valleron and E. Frindel
78
to growth could not be used for the NCTC 4 days ascites tumour. When applied, this method gives an absurd value of 700 % for the growth fraction. One explanation may be that the Mendelsohn's method assumes that there is a one way flow from the proliferative (P) compartment to the quiescent one (Q). An alternative might assume a flow from Q to P. The time variation in the number of grains can be simulated for either model. Experimentally, we have measured the grain tounts of mitotic cells at 2 hr, and at 120 hr, after the labelling. When studying the histograms of grains found on mitotic cells as compared to intermitotic cells, we are sure that the cells studied are the proliferating ones. In Fig. 9 the experimental cumulative histograms of grain counts are shown. Probit scale for ordinate and logarithmic scale for abscissa have been chosen, because the distribution of grain +
99 98
90
'"
80
0' 0
~ 70
e
1i 60
/ /
+ /
/
+/ /
/
/
/
"
/
+
• •••
/
/ /
•
/+
'13 u
/
/
•
20
5
•
•
30
10
/
/
'l! 50
~ 40
+//
• • 10
100
No. of groins (log scale)
FIG. 9. Experimental cumulative histograms of grain counts in labelled mitotic figures: 7 hr (e) and 5 days (+) after 3H-thymidine injection. If there was no back flow from Q to P compartment, the theoretical 60 hr and 5 day curves would be respectively (---) and (--).
counts is close to the lognormal one (Lazar & Gerard-Marchant, 1965). The lines drawn in Fig. 9 for 60 and 120 hr are the theoretical cumulative histograms of grain counts for mitotfc figures which are obtained by simulation from the experimental 7 hr histogram assuming that there is no return from Q to P. One can note the discrepancy between the experimental and simulated 120 hr histograms; in the no Q --+ P flow hypothesis, the grain distribution which is observed at 120 hr would have been attained at 60 hr. Further investigations are being undertaken to provide better interpretations of the data. This example demonstrates the advantages of our method which permit simulation of the time course variation of the grain cou nt distribution for any model of tumour growth.
Computer simulation of growing cel! populations
79
CONCLUSION We have presented a si!11ple computer method of simulation of growing populations of cells. This method could be of interest to those who wish to compute the consequences of different hypotheses on various biological situations where variability is involved. In the examples which we have examined, we have mainly shown the resuIts on the simulation of autoradiographie experiments. Moreover this method can also be used to interpret effects of radiations or of chemotherapy. ACKNOWLEDG MENTS
We wish to thank Professor P. Tubiana for helpful suggestions, criticisms, and constant interest in our work. REFERENCES BARRETT, J.c. (1966) A mathematical model of the mitotic cycle and its application to the interpretation of percentage labelled mitoses data. J. nat. Cancer /nst. 37, 443. BURNs, F.J. & TANNOCK, 1. (1970) On the existence of a Go-phase in the cell cycle. Cel! Tissue Kinet. 3, 321. FRINDEL, E., VALLERON, A.-J., VASSORT, F. & TUBIANA, M. (1969) Proliferation kinetics of an experimental ascites tumour. Cell Tissue Kinet. 2, 51. HAHN, G.M. (1970) A formalism describing the kinetics of sorne mammalian cell populations. Math. Biosci. 6, 295. LAZAR, P. & GERARD-MARCHANT, R. (1965) Une technique de représentation graphique des numérations de grains sur autohistoradiographie. Ann. Histochim. 10, 43. MACDONALD, P.D.M. (1970) Statistical inference from the fraction labelled curve. Biometrika, 57, 489. MENDELSOHN, M.L. (1962) Autoradiographie analysis of cell proliferation in spontaneous breast cancer of C3H mouse. III. The growth fraction. J. nat. Cancer /nst. 28, 1005. STEEL, G. (1967) Cellloss as a factor in the growth rate of human tumours. Europ. J. Cancer, 3,381. , TAKAHASHI, M. (1968) Theoretical basis for cell cycle analysis. II. Further studies on labelled mitosis wave method. J. theoret. Biol. 18, 195. TRUCCO, E. & BRocKwELL, P.J. (1968) Percentage labelled mitosis curve in exponentially growing cell population. J. theoret. Biol. 20, 321. VALLERON, A.-J., FRINDEL, E. & TUBIANA, M. (1968) Méthode de mesure in vivo de la distribution des durées des phases du cycle cellulaire. C.r. Acad. Sei. (Paris), 277, 2189.
