From the Pathology and Laboratory Medicine Service,
Veterans Affairs Maryland Health Care System, Baltimore, Maryland [1];
Department of Pathology, University of Maryland Medical System,
Baltimore, Maryland [2]; and
Department of Pathology, The Johns Hopkins Medical Institutions,
Baltimore, Maryland [3].
Send comments and correspondence to:
George.Moore4@va.gov
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ABSTRACT
Studies showing the clonal identity of various tumors have led
to the belief that most tumors originate from a single cell.
It is shown by Monte Carlo computer simulations that monoclonality
can evolve from minor differences either in cell cycle time or in the
probability of cell death in a polyclonal `founder' population.
If cells divide continuously without cell death (exponential clonal growth),
a tri-clonal population with three starting cells (cell cycle times 0.9 days,
1 day, and 1.1 days) converges to near-monoclonality in 100 generations.
For cell cycle times of 0.9 days, 1.1 days, and 1.1 days, and cell death
probabilities of 0.45 and 0.46, populations tend toward monoclonality
while the tumor is still small (<3 mm3).
Key words: monoclonal, tumor origin, polyclonal, Monte Carlo, cell death
INTRODUCTION
Although little is understood about the early cellular events in carcinogenesis, many observers have concluded that most tumors arise from a single cell. Since it is impossible to observe a tumor grow from the single cell stage to a clinically evident tumor mass, our understanding of the origins of tumors have been inferential. Many tumors have a clonal phenotype (e.g., cells of an antibody-producing myeloma produce monotypic antibodies; neoplastic cells of chronic myelogenous leukemia may all have the identical abl-bcr tumor marker). When all cells in a tumor have the same genetic characteristics, one concludes that all cells in the tumor are descended from a single cell (7,16). However, it cannot be concluded that only one neoplastic cell was present at the tumor's origin. Any number of non-clonal neoplastic cells may have been present (and even necessary) during the evolution of the tumor, only to be lost through selection as the tumor enlarged. A polyclonal theory for the origin of tumors has been proposed by Alexander (1) and others (12,17).
We propose that over generations, polyclonal populations tend toward monoclonality when there is variation in the growth properties of the original cells in the original population. We employ the Monte Carlo method to simulate the growth of cell clones with different probabilities of cell death (2,4,11). The Monte Carlo method obtains a single outcome for a function containing a probability value by selecting a pseudorandom number, comparing it to the probability value, and substituting 0 or 1 depending on whether the selected number exceeds the probability value. By repeating the simulation many times, a distribution of outcomes is obtained. Monte Carlo simulations are particularly useful in predicting outcomes where a simple set of initial conditions results in a large number of possible outcomes, from which it is too difficult or cumbersome to derive analytic solutions. By performing repeated trials and observing the distribution of outcomes that evolve through time, one can predict the behavior of complex systems.
MATERIALS AND METHODS
Cell proliferation models were programmed on an IBM PC/AT compatible computer (COMTEX, 30368 microprocessor, 25MHz, 330 Mb Priam hard disk), using American National Standard MUMPS (MGlobal, Inc., Houston, TX), and the public-domain File Manager (FileMan) database management system of the United States Department of Veterans Affairs (5). We assume that tumors begin as populations of genetically distinct and independently dividing cells, each with the potential for unbounded growth with a constant cell cycle time per generation, t, and a constant, inherited death probability, p. For the case in which there is no cell death (deterministic model), the number of cells at the nth generation is 2 to n.
In a Monte Carlo simulation, a pseudorandom number is substituted for the probability value to obtain a single outcome. The calculation is then repeated many times to obtain a distribution of outcomes. For example, in a clone beginning as a single founder cell with a death probability of 0.45, the cell divides, and each of the two daughter cells is assigned an independent pseudorandom number between 1 and 100, inclusive. If a daughter cell obtains a pseudorandom number at most 45, then it dies without dividing. If a daughter cell obtains a pseudorandom number at least 46, then it is capable of at least one additional mitosis. The process continues until no daughter cells can divide further (extinction), or until the experiment is arbitrarily terminated (in this report, at the 100th generation).
Table 1 shows the source code in American National Standard MUMPS for calculating a cell growth simulation from a single founder cell. The output is arbitrarily terminated after generation 100. In this report, the program was executed 500 times apiece at each cell death probability and each cell cycle time shown in subsequent tables and figures, yielding a distribution of outcomes for that cell death probability and cell cycle time. A single Monte Carlo simulation using this program is shown in Figure 1. Although the outcome of each single simulation may vary widely, the outcomes of 500 such simulations can be regarded as a statistical sample of simulations.
RESULTS
Figure 2 shows the relative contributions to a population that began with three cells in a deterministic model (i.e., zero cell death probability), each dividing with cell cycle times of 0.9 days (solid), 1 day (checkered), or 1.1 days (crosshatched). Even with only this 10% difference in cell cycle time, the population at 60 days is composed almost entirely of descendants of the fastest growing cell. At day 40, the cells derived from the slowest growing cell contributed only 0.3% of the total population.
Table 2 shows the results of a Monte Carlo simulation of 1500 founder cells, all growing with a constant 0.47 probability of cell death per generation. Five hundred starting cells apiece have a cell generation time of 1.1 days, 1 day, and 0.9 days. At the end of 100 days, most founder clones have become extinct. Only 251 (17%) of the original 1500 clones have survived (84 dividing every 1.1 days, 88 dividing every day, and 79 dividing every 0.9 days). The mean size of clones surviving at 100 days is 2053 cells for a 0.9 day cell cycle time, 31% higher than the mean clone size of 1569 cells for a 1 day cell cycle time, and 159% higher than the mean clone size of 792 cells for a 1.1 day cell cycle time. The number of clones surviving to 100 days is a fairly constant feature of the cell death probability, whereas the mean clonal size at 100 days varies according to cell cycle time. At cell death probabilities below 0.50, a certain proportion of clones, once established, tend to persist, with larger clone sizes at 100 days resulting from shorter cell cycle times.
Table 3 shows the results of a Monte Carlo simulation of the growth of clones of cells all having the same cell cycle time (1 day) but varying in the probability of cell death. The growth of 500 clones was simulated for each of ten probabilities of cell death, ranging from 0.44 to 0.53. Clones with probability of cell death above 0.50 all terminated before the 100th day of growth. In clones with lower probabilities of cell death, there was some clonal survival, but in all cases the majority of clones terminated before the 100th day. When the probability of cell death per generation was 0.44, 67.8% of the clones became extinct. The remaining clones grew to a mean size of 156,921 cells. When the probability of cell death was only one percent higher (0.45), the mean size of the surviving clones was 25,023 cells. In all simulations with small increments in the probability of cell death per generation, there were large differences in survival rates and clonal growth.
Among 500 initial clones with a cell death probability less than 0.50, after 100 generations the surviving clones tend to persist and behave in a near-deterministic manner. Thus it is reasonable to speak of a `cell doubling time'. For example, with a cell death probability of 0.49, starting with 500 cells, 22 clones survived to 100 generations, with a total of 2,288 cells; at 55 generations, the total number of cells was 965. In other words, it required 55 generations to double its population from 500 to nearly 1000 cells. In a Monte Carlo simulation, the time of the first doubling (over 55 generations) is longer than the time of the second doubling (less than 100 generations), due to random extinction of small clones. Thus cell doubling time is only an approximate concept in Monte Carlo simulations. The estimated doubling time for each simulation group is shown in the rightmost column of Table 3, as calculated from the equation N = N at 0 x 2 to the g/t, solved for t. N is the cell population size at the 100th generation for a particular cell death probability, N at 0 is the number of cells at the 0th generation, here 500, and g is the number of generations, here 100. Solution is given by the expression t = g/(log2 to the N - log2 to the N at 0).
Table 4 shows the growth of clones that differ by small increments both in probability of cell death at each generation and in cell generation time. The growth of 500 founder clones having a generation time of 0.9 days and a probability of cell death of 0.45 was compared with the growth of 500 founder clones having a generation time of 1.1 days and a probability of cell death of 0.46. At 100 days, the clones having a shorter generation time and lower probability of cell death had an average clonal size of 67,869 cells, compared to an average clonal size of 2,623 cells for the other group. Thus in a biclonal founder population that expands into a lesion at 100 days of growth, the average lesion would consist of approximately 7.0 x 10 to the 4 cells, 96% of which would derive from a single clone (i.e., the lesion would have converged to near-monoclonality). Such a lesion would measure approximately
2.1 mm3.
DISCUSSION
In a deterministic model of exponential cell growth, cells grow continuously with a fixed cell cycle time and a zero probability of death in any cell cycle. Under these conditions, minor differences in cell cycle time produce major changes in the clonal composition of populations with polyclonal origins. Experimentally, monoclonality is usually defined by the presence of a single clone of cells occupying 95% of a population (7). A proportion less than 100% is necesssary because all tumor preparations are contaminated by non-neoplastic stromal cells that would not be expected to belong to a (clonal) neoplastic population. In addition, techniques that measure clonality all have an inherent inaccuracies produced by the limitations of measurement. After 60 days of deterministic growth, a triclonal population of cells can easily achieve monoclonality with more than 99% of cells derived from the fastest growing founder. This model, however, does not accurately depict tumor cell growth, as it does not account for cell death. In fact, when a single cell grows exponentially for 60 days, it attains a cell mass of 2 to the 60 cells. This number of cells, at 30,000 micron3 per cell (6), would have a mass of 34,588 cubic meters! In fact, tumors grow slowly, often over many years, and although tumors have high proliferative indices, this growth is counterbalanced by cell death.
In this report, we assume that cells in a tumor all have a non-zero probability of cell death, and that this probability of cell death is stable and characteristic for a given clone. This assumption is based on observations of a relatively constant growth fraction in tumors, although the doubling times can vary greatly among various types of tumors (8,13,14). A more general model, with two-event carcinogenesis and a variable death probability, has been described (9,10). However, the stochastic form of this model is mathematically intractable, and the deterministic form is subject to asymptotic approximations. The present model requires no approximations to obtain a solution, as is required for an analytic mathematical solution. The only limitation in Monte Carlo methods is the number of repetitions required to obtain a stable solution (15).
Probably the fastest growing human tumor is Burkitt's lymphoma, a tumor endemic in African children. This tumor can double its size every three days. Other tumors, such as breast cancers, may grow very slowly, many with doubling times of about six months. The simulations shown in Table 3 result in tumor doubling times similar to the range observed in human tumors. When the tumor cell death probability is 0.49, tumor doubling time is about one-and-a-half months. When the tumor cell death probability is 0.44, the tumor doubling time is 6.4 days. It is not feasible to simulate growth when the cell death probability drops below 0.44, as the sample sizes increase dramatically, and iterative calculations for each cell require excessive computer time. Probability rates lower than 0.43 would probably result in tumors growing at rates unobserved in clinical experience.
Do the cell death probabilities chosen for our simulations correspond to cell death rates observed in human tumors? It is recognized that tumors grow at rates much slower than the potential doubling time (determined by fraction of cycling cells and their average cell cycle time). This disparity between the observed rate of tumor growth and the potential rate of tumor growth is reconciled by a cell loss factor (3,13), defined as the rate of cell loss divided by the rate of growth. In steady state conditions, such as normal skin or normal gastric mucosa, the cell loss factor is always 1.0 (cell loss rate equals cell growth rate) (13). In human tumors, the cell loss rates varies, between a low of 0.70 to a high of about 0.95 (13). Anything higher than 1.0 would produce no net growth. In the simulations shown in Table 3, probabilities for cell death that apply to a single cell can be converted to cell loss probabilities for aggregate tumor populations. For instance, a cell death probability of 0.50 results in a rate of cell loss equal to the rate of cell growth, hence a cell loss factor of 0.50/0.50 = 1.0. A cell death probability of 0.43 implies that when the cell loss rate is 0.43, the cell proliferation rate must be 0.57, producing a cell loss factor of 0.43/0.57 = 0.75. A cell loss factor of 0.75 is observed for Burkitt's lymphoma, the fastest-growing human tumor. Consequently, cell death probabilities between about 0.43 and 0.49 can account for the entire range of tumor doubling times as well as for cell loss factors observable in human tumors.
We assume that cells and their descendants have a constant cell cycle time and that this is about one day. This assumption is based on cell growth rates of tumor cells growing logarithmically in culture, and may have little bearing on the cell cycle times of cells in tumors. The one day cell cycle time is used largely as a convenience, to avoid having fractional time-variables in the simulation. However, many tumors have tumor cell cycle times between 1 and 4 days (13). The predicted sizes of lesions after any given time should thus be considered as approximations.
In summary, it is shown that with only minor variations in cell cycle time and probability of cell death per generation, cells can have large differences in average clonal growth rate (Table 4). This phenomenon might account for clonality occurring in tumors that arise from polyclonal populations. Furthermore, clonality can be attained while the lesion is still small (<3 mm3).
REFERENCES
1. Alexander P. (1985) Do cancers arise from a single transformed cell or is monoclonality of tumours a late event in carcinogenesis? Br J Cancer, 51, 453-457.
2. Berman J. J. and Moore G. W. (1990) Why do most initiated cells fail to produce early (preneoplastic) lesions? Prediction by Monte Carlo simulation of growth. Lab Invest, 62,9A.
3. Day R. S. (1987) Exploring large tumor model spaces: drawing sturdy conclusions. In: Cancer Modelling, pp. 365-386. Editors: J.R. Thompson, B.W. Brown. Marcel Dekker, New York.
4. Diggle P. J. and Gratton R. J. (1984) Monte Carlo methods of inference for implicit statistical models. J Royal Statist Soc B, 46, 193-227.
5. Davis R. G. (1987) FileMan: A User Manual. National Association of VA Physicians, Bethesda.
6. Elias H. and Sherrick J. C. (1969) Morphology of the liver, p. 13. Academic Press, New York.
7. Fialkow P. J. (1979) Clonal origin of human tumors. Ann Rev Med, 30, 135-143.
8. Laird A. K. (1969) Dynamics of growth in tumors and in normal organisms. Natl Cancer Inst Monogr, 30, 15.
9. Mookgavkar S. H. and Knudson A. G. jr. (1981) Mutation and cancer: A model for human carcinogenesis. J Natl Cancer Inst 66, 1037-1052.
10. Mookgavkar S. H. and Luebeck G. (1990) Two-event model for carcinogenesis: biological, mathematical, and statistical considerations. Risk Anal 10:323-341.
11. Moore G. W., Berman J. J. (1990) Cell growth simulations predicting polyclonal origins for "monoclonal" tumors. Lab Invest, 62, 69A.
12. Nowell P. C. (1976) Clonal evolution of tumor cell subpopulations. Science, 194, 23-28.
13. Schiffer L. M. (1987) Cellular proliferation in tumor and in normal tissues. In: Principles and Practice of Radiation Oncology, pp. 56-66. Editors: C. A. Perez, L. W. Brady. J. B. Lippincott, Philadelphia.
14. Steel G. G. (1982) Cytokinetics of Neoplasia. In: Cancer Medicine, pp. 177-189. Editors: J. F. Holland and E. Frei. Lea and Febiger, Philadelphia.
15. Thompson J. R., Atkinson E. N., Brown B. W. (1987) SIMEST: An algorithm for simulation-based estimation of parameters characterizing a stochastic process. In: Cancer Modelling, pp. 387-415. Editors: J. R. Thompson, B. W. Brown. Marcel Dekker, New York.
16. Wainscoat J. S. and Fey M. F. (1990) Assessment of Clonality in Human Tumors: A Review. Cancer Research, 50, 1355-1360.
17. Woodruff M. F. A., Ansell J. D., Forbes G. M., Gordon J. C., Burton D. I. and Micklem H. S. (1972) Clonal interaction in tumours. Nature, 299, 822-824.
Table 1. American National Standard MUMPS source program for calculating a single cell growth simulation. Program was executed 500 times apiece at each cell death probability and each generation time shown in subsequent tables and figures.
1 CLONGRTH ; MONTE CARLO CLONAL GROWTH MODEL;
2 ENTRY S FRSZ=1,DTHR=45,EOUT=100,CLSZ=FRSZ,NXCL=0 ;
3 W !,"FOUNDER CLONE: ",FRSZ," CELLS"," DEATH RATE: ",DTHR,"%",!!
4 NXCL W $J(CLSZ,7) S NXCL=NXCL+1 W:((NXCL#10)=0) ! ;
5 S DBSZ=CLSZ*2,CLSZ=0 G:(NXCL>EOUT) EOUT ;
6 F FCL=1:1:DBSZ S RND=$R(100)+1 S:(RND>DTHR) CLSZ=CLSZ+1 ;
7 G:CLSZ NXCL W !,"EXTINCTION" G EXIT ;
8 EOUT W !,"OUTPUT TERMINATED" ;
10 EXIT W !!,"EXECUTION COMPLETE" ;
Table 2. Monte Carlo simulation of polyclonal cell populations with a constant 0.47 probability of cell death, and 500 initial clones for each cell cycle time.
Cell cycle # Clones at Clone sizes
time 100 days at 100 days
Mean S.D.
1.1 DAYS 84 792 749
1 DAY 88 1569 1755
0.9 DAYS 79 2053 2398
Table 3. Monte Carlo simulation of 500 founder clones apiece, with constant 1 day cycle time and different probabilities of cell death. Cell doubling time estimated by the equation N = N at 0 x 2 to the g/t, solved for t.
Probability Percent surv- Clone Size at 100 Days Estimated
of cell death ing clones at Total Mean St. Dev. No. Doubling
per cycle 100 days Time, days
0.44 32.2% 2.53x10 7 156,921 174,568 161 6.39
0.45 25.4% 3.18x10 6 25,023 24,087 127 7.90
0.46 21.8% 6.25x10 5 5,733 7,167 109 9.70
0.47 14.4% 1.04x10 5 1,445 1,843 72 12.95
0.48 8.6% 1.34x10 4 311 411 43 21.00
0.49 4.4% 2.29x10 3 104 121 22 45.22
0.50 0.2% 415 42 47 10 INFIN
0.51 0% 0 0 INFIN
0.52 0% 0 0 INFIN
0.53 0% 0 0 INFIN
Table 4. Monte Carlo simulation of 500 founder clones apiece, that differ in growth rate by 10% and in probability of cell death by 0.01
POPULATION AT 100 GENERATIONS
Surviving Clone Size Average %
Clones at 100 days Population
Mean S.D.
0.45 probability of death
generation time 0.9 28% 67,869 79,423 96%
0.46 probability of death
generation time 1.1 26% 2,623 2,893 4%
Figure Legends.
Figure 1. Single Monte Carlo experiment, first forty cell cycles, using the computer source code in Table 1. Cell death probability is 0.45. Cell cycle number (number of generations) versus clone size (number of cells). A complete simulation consists of assembling the results of 500 individual experiments.
Figure 2. Deterministic model (i.e., zero cell death probability) showing monoclonal convergence of a triclonal tumor at sixty days, with cell cycle times of 0.9 days (solid), 1 day (checkered), and 1.1 days (crosshatched).
Last updated: 1/26/2008, by
G. William Moore, MD, PhD.