THE ROLE OF CELL DEATH IN THE
GROWTH OF PRENEOPLASTIC LESIONS:
A MONTE CARLO SIMULATION MODEL.
7/8/2009.
Berman JJ, Moore GW.
http://www.netautopsy.org/celdeath.htm
Berman JJ, Moore GW.
The role of cell death in the growth of preneoplastic lesions:
a Monte Carlo simulation model.
Cell Prolif. 1992 Nov;25(6):549-557.
PMID: 1457604.
PubMed Entry
Full Text:
http://www.netautopsy.org/celdeath.htm
Last tested: July 8, 2009.
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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0. DISCLAIMER.
DISCLAIMER. United States Government Work, uncopyrighted,
public-domain, DRAFT COPY ONLY. This document does not necessarily
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This document is provided "as is", without warranty of any kind, express
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1. ABSTRACT.
A variety of experimental and clinical examples of preneoplasia
demonstrate that regression of early lesions is common. This paper examines
the hypothesis that early lesions operate under the identical growth kinetics
of "late" lesions (neoplasms), but that kinetic features favoring continuous
growth in established lesions tend to favor extinction of lesions composed
of small numbers of cells. Growth simulations of early lesions were produced
using the Monte Carlo method, a technique demanding intensive computations.
With the advent of powerful personal computers, this technique is now widely
available to biologists. Simulating growth under conditions of cell loss
similar to those observed in established tumors, the model predicts that
the great majority of initiated cell clusters are expected to reach
extinction within a few cell doubling times, and most early (promoted)
lesions would not likely progress to the size of a clinically detectable
lesion within the life span of the host organism. These Monte Carlo
simulations provide a model of initiated cell growth consistent with
the recently demonstrated role of early lesion cell death in the development
of human lymphomas and in transgenic mice expressing the bcl-2 oncogene.
The model demonstrates that small increments in the intrinsic cell loss
probability in even the earliest progenitors of malignancy can strongly
influence the subsequent development of neoplasia from initiated foci.
2. INTRODUCTION.
In most animal models, the most likely outcome for preneoplastic lesions
is regression, rather than progression. In the preoneoplastic liver model
(Solt, Medline and Farber 1977), hundreds of early
liver lesions result in only one or a few established carcinomas. A variety
of preneoplastic lesions in man, some with proven aneuploidy, have only
a small chance of ever progressing to invasive cancer. These include cervical
dysplasia, Barrett's esophagus, oral leukoplakia, low-grade papillary tumors
of bladder, lobular carcinoma in situ of breast, and borderline tumors of the
ovary. In all these cases, progression to cancer is the exception,
not the rule.
Two unproven and divergent theories addressing the phenomenon
of early lesion regression are: (1) that early lesions are detected
by a host immune surveillance system that destroys preneoplastic cells
identified by distinctive immunologic markers; (2) that early lesions
do not have all the genetic lesions necessary for sustained growth.
We propose that the growth kinetics of early lesions may favor lesion
regression, rather than early lesion growth. Recently we modeled tumor cell
growth using an intrinsic cell death probability, and found that all observed
human tumor growth could be modeled by cell death probabilities varying
between 0.43 and 0.50 (a href="#moore1991">Moore and Berman 1991,
Berman and Moore, 1992). In our model, tumors with
cell death probabilities above 0.50 cannot increase their size over time,
whereas tumors with cell death probabilities less than 0.43 grow faster than
the fastest growing clinically observed tumors in man. In this study,
we extend our model to simulations of small colonies of cells dividing with
varying cell death rate probabilities. Simulations in this study demonstrate
regression (extinction) of small colonies under the same growth kinetic
parameters that were previously shown to produce continuous population
expansion in large populations of cells
(Berman and Moore, 1992).
Experimental evidence for the role of cell death rates
in carcinogenesis was recently provided by Korsmeyer and colleagues
((Sentman et al. 1991,
McDonnell and Korsmeyer 1991).
This group has shown that the bcl-2 oncogene, associated with
malignant lymphoma in man, is an important inhibitor of programmed
cell death. Transgenic mice with deregulated bcl-2 in their
germline develop polyclonal B cell hyperplasias leading to lymphomas.
The increase in B cells in early lesions results from their extended
survival rather than from increased proliferation. Other novel oncogenes
and agents may exert their biological activity by blocking cell death
(Rotello et al. 1991). Thus evidence is mounting
that cell death in tumors is a biologically determined phenomenon
and not the simple result of noxious microenvironmental conditions
within the tumor.
3. MATERIALS AND METHODS.
Monte Carlo cell growth models were programmed on an IBM PC/AT
compatible computer (COMTEX, 80386 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
(Davis 1987).
The source code is available in the public domain.
We assumed that a cell-cluster begins as a population of genetically distinct
and independently dividing cells, each with a potential for unbounded growth,
with a constant and inherited death probability, p. For the particular
case in which there is no cell death, the number of cells nth
generation is 2n.
In a Monte Carlo simulation, a pseudorandom number generator
is substituted for the probability value in the theoretical model.
In generation zero there is a starting cell-cluster consisting of one or more
premitotic cells (founder cluster), each of which is capable of at least one
additional mitosis. The cell count thus refers only to those cells capable
of an additional mitosis; this cell count may underestimate an observed cell
count, which would include a variable number of postmitotic cells.
Each premitotic cell in generation k divides to form two daughter cells
in generation k+1. For each daughter cell, a pseudorandom number,
r, is selected independently from the uniform (equiprobable) density
function over the unit interval. If r < p, then the cell
is premitotic; if r®MDUL¯>®MDNM¯p, then the cell is incapable of further mitosis.
The process continues to extinction of all premitotic cells, or to an
arbitrary termination point (in this report, the 60th generation).
A sample output for 10 single cells, each with death probability
0.48 per generation, is shown in Table 1.
In generation 0, there is a single, premitotic cell in each of the
ten clusters. Each premitotic cell divides once, producing two daughter
cells apiece in generation 1. In clusters 2, 5, and 8, both daughter cells
draw a pseudorandom number less than 0.48, and the cluster becomes extinct.
In clusters 3, 4, 6, 7, and 10, one daughter cell draws a pseudorandom number
less than 0.48, while the other daughter cell draws a pseudorandom number
at least 0.48, to yield one premitotic cell in generation 1. In clusters
1 and 9, both daughter cells draw a pseudorandom number at least 0.48,
to yield two premitotic cells in generation 1. Clusters 4 and 10 become
extinct in generation 2. Cluster 1 becomes extinct in generation 4.
Cluster 7 becomes extinct in generation 5. Cluster 9 becomes extinct
in generation 8. Cluster 3 becomes extinct in generation 19.
Cluster 6 becomes extinct in generation 29.
In this study, we examined the behavior of 100 initial cell clusters
(Monte Carlo trials) at each of nine cell death probabilities (0.45, 0.46,
0.47, 0.48, 0.49, 0.50, 0.51, 0.52, and 0.53) and five initial cluster sizes
(1, 5, 10, 25, 50). Each cluster was arbitrarily terminated after generation
60. The mean and maximum sizes of clusters surviving at generation 60
was obtained.
4. RESULTS.
Figure 1 shows the results of 100 Monte
Carlo simulation trials after 60 generations, with an initial cluster size
of 1 cell and death probability p=0.45. Twenty initial clusters (20%)
survived, ranging in size from 43 cells (surviving cluster 4) to 5,396 cells
(surviving cluster 15), and averaging 780 cells per surviving cluster.
Eighty clusters (80%) became extinct. Tumors with a cell death probability
of 0.45 and a cell cycle time of 1 day have a tumor doubling time
of approximately 8 days (Moore et al. 1991).
Table 2
shows the distribution of 100 cell-clusters, each with initially 10 cells,
at cell death probabilities 0.53, 0.52, 0.51, 0.50, 0.49, 0.48, 0.47, 0.46,
or 0.45. After 60 generations, the 100 initial cell-clusters with cell-death
probability 0.53 had 3 surviving clusters, the 100 initial cell-clusters with
cell-death probability 0.52 had 6 surviving clusters, etc. Among
the 3 surviving clusters at cell-death probability 0.53, the largest
surviving cluster had 30 cells, and the average cluster size was 15 cells.
Among the 100 initial cell-clusters with cell-death probability 0.45, after 60 generations there were 99 surviving clusters, in which the largest surviving cluster had 9,117 cells, and the average cluster size was 2,001 cells. Clearly the three measures of cluster survival, namely percent of surviving clusters, size of largest surviving cluster, and mean surviving cluster size, all have an inverse relationship with cell death probability.
Table 3 shows the distribution of 100 cell-clusters, each with initial cluster sizes of 1, 5, 10, 25, or 50 cells, at cell death probabilities 0.53, 0.52, 0.51, 0.50, 0.49, 0.48, 0.47, 0.46, or 0.45. After 60 generations, the 100 initial cell-clusters with cell-death probability 0.53 and initial cluster size 1 had 1 surviving cluster containing 2 cells, whereas the 100 initial cell-clusters with cell-death probability 0.45 and initial cluster size 50 had 100 surviving clusters with a maximum cluster size of 28,208 cells. The number of surviving clusters and maximum cluster size after 60 generations are inversely correlated with cell death probability and directly correlated with initial cluster size.
5. DISCUSSION.
Two popular hypotheses address the phenomenon of early lesion regression:
(1) that early lesions are caught by a host immune surveillance system
that detects and destroys preneoplastic cells identified by distinctive
immunologic markers; (2) that early lesions do not have all the
genetic lesions necessary for sustained growth. Both these hypotheses have
drawbacks. The immune surveillance mechanism for early lesion regression
fails to explain the lack of heightened cancer susceptibility in immune
deficient mice (nude mice). In man, immune suppression is not associated
with increased risk of malignancy for most tumors. Rather, immune suppressed
patients seem to develop tumors whose etiology is directly dependent on the
immune deficient state for growth (i.e., lymphomas and virally-induced
neoplasms). The second argument, that early lesions must obtain additional
mutations in order to grow as fully developed neoplasms, implies that the
additional genetic changes observed in neoplastic development (such as
oncogene acquisition or suppressor gene loss) involve growth kinetics,
an unproven hypothesis. It is completely feasible that additional and
necessary genetic alterations attained during neoplastic development may
relate to phenotypic properties independent from growth kinetics (e.g.,
properties related to invasiveness and not to cell proliferation).
Evidence supporting similar growth kinetics for early lesions and
for late lesions is provided by the work of Collins and colleagues, who
compared the time needed for tumors to evolve with the time needed for tumors
to recur after unsuccessful treatment
(Collins, Loeffler and Tivey 1956,
Collins 1958).
For example, gestational chorio-carcinoma is almost always detected by the
thirteenth month following conception. When a recurrence appears, it almost
always appears within thirteen months of tumor treatment. Similar results
are observed in studies of Wilms' tumors and of Burkitt's tumors
(Bagshawe 1976,
Iverson 1974).
In the case of solid tumors in adults, recurrences may appear decades after
treatment of the tumor, but these tumors also seem to have a very long
evolution. These studies would suggest that the growth kinetics of
an initiated cell or a cell in an early preneoplastic lesion or a cell
in a developed tumor all have the same growth kinetics.
Laird has demonstrated that cell division in many tumors is accompanied by high rates of tumor cell death
(Laird 1969).
Most tumors consist of a large proportion of cells that are not capable of further division, typically in the range of 20% to 70%
(Schiffer 1987).
Many tumors have a growth fraction of only 25% and a cell loss factor of 70% or more
(Schiffer 1987).
Even normal tissue has a high rate of cell loss. In skin and gut, under
conditions of homeostasis, each dividing cell produces two daughter cells,
one that can divide and another that is post-mitotic (i.e., moribund).
The destiny of the post-mitotic cell is to be sloughed (in the case of skin)
or extruded (in the case of the gut). Normal tissue is apparently balanced
by a 50% probability of cell death (end-stage differentiation) for cell
divisions. In normal tissue that is not undergoing net growth (e.g.,
epidermis), a 50% probability of cell death (via end-stage differentiation)
is the rule. For example, when a basal cell of skin divides, it produces
a differentiated skin cell (to replace the sloughed cell of the stratum
corneum) and another basal cell (to replace itself). This is not to say that
whenever a basal cell divides, the determination of which daughter cell will
live (replicate again) and which daughter cell will die (becomes
post-mitotic) is determined in vivo on a probabilistic basis. We presume
that a biologic mechanism exists to control the proliferative destiny of the
daughter cells. However, since we know that on average, one cell dies for
each cell that lives, we can model normal epidermal growth in a probabilistic
model wherein each cell has a 50% chance of dying in any cell cycle.
Furthermore, the rate of cell death is static only under the condition
of no net growth (the normal condition of skin). During wound repair,
for example, the rate of cell growth must at least temporarily exceed the
rate of cell death. Wound repair would be modeled with a cell death
probability less than 50%.
The likelihood of a cell entering a non-dividing state, although
presumably determined by specific biological mechanisms, can be modeled
probabilistically, since tumors as a whole seem to maintain characteristic
population cell loss rates
(Schiffer 1987).
We performed computer simulations of the growth of clusters of cells, each
cell of the cluster growing with the same probability of cell death as all
other cells in the cluster. Simulations were performed to determine whether
the extinction of cell clusters initially containing only a few cells might
result under the same growth kinetic conditions that favor continuous growth
of initially large cell clusters. Additional Monte Carlo simulations were
performed for model populations of initiated cells having different constant
cell death probabilities. Repeated Monte Carlo simulations using the same
set of initial parameters yield different outcomes (a range of possible
results), permitting analysis of potentially unexpected trends and the
detection of unusual events.
We have shown previously that Monte Carlo modeling of tumor doubling times at various cell death rates can model the observed range of human tumor doubling times
(Moore et al. 1991).
A cell doubling time of a little over 6 days is modeled by a cell death
probability is 0.44 and a cell cycle time of 1 day. A cell doubling time
of about 45 days is modeled by a cell death probability of 0.49 and a cell
cycle time of 1 day.
A Monte Carlo simulation, applied to a probabilistic problem, does not
obtain an analytic solution, but rather substitutes a pseudorandom number
generator for the probability value in the theoretical distribution. Then
a series of computational experiments are performed. If an analytic solution
were obtained for cell death probability p, then the solution at the nth
generation would have 2®MDSU¯n®MDNM¯ possible outcomes. It is apparent that the analytic
solution is too unwieldy to examine the model's average behavior. In our
opinion, simulation models have some important benefits compared to analytic
models incorporating cell death probabilities
(Takahashi 1968,
Goldie 1989),
because they can be used to describe the behavior of individual components
of a system, whereas analytic systems deal with the aggregate system behavior
(Widman 1990).
Repeated Monte Carlo simulations produce a variety of possible
outcomes for a single set of initial conditions. In this case, Monte Carlo
simulation demonstrated how early lesions may reach extinction. Simulations
demonstrated cell growth trends that would not have been predicted on
a purely intuitive basis. This is particularly true in the case of the
maximal regressing cluster size. Among all outcomes, regardless of the
intrinsic cell death probability, once a cluster reaches a size of about
50 cells, it does not regress to zero(Berman and Moore, 1992).
In a deterministic model that allows growth among fractional cells,
this finding could not have been anticipated.
In our model, small differences (on the order of one percent)
in the cell death probability account for large differences in the number
of surviving clusters at the 60th generation. Although this paper does not
specifically address the biological roles of initiation and promotion,
one can speculate that promotion may exert its biological effect through
a downward adjustment of the cell death rate of initiated cells, leading
to rapid expansion of initiated clusters.
It is important to add that the rate of cell proliferation does not
change the number of surviving cell clusters that reach extinction. The rate
of proliferation only changes the speed with which clusters become extinct.
The model serves to demonstrate that simulating cell growth by conditions
assumed in this study often results in population extinction. Like all
models, ours does not provide a proof of biological mechanism. It does allow
us to see biological posibilities that are not intuitively obvious or, in the
case of Monte Carlo models, are not readily predicted by more traditional
analytic methods.
The fact that a population of persons can exhibit exponential growth
was appreciated by Leonhard Euler (1748),
who ridiculed the objections of those persons who "denied that the whole
earth could be filled in a short time with inhabitants descended from
a single man": "Quam ob causam maxime ridiculae sunt eorum incredulorum
hominum objectiones, qui negant tam brevi temporis spatio ab uno homine
universam terram incolis impleri potuisse." Furthermore,
Thomas Malthus (1817) noted that despite
a substantial probability of extinction for any particular ancestor
in that population, has been appreciated since the monograph
by Thomas Malthus (1817).
In previous studies by the authors, the growth of neoplasms was
modeled by Monte Carlo simulations. The present study extends the results
obtained for established lesions to preneoplastic lesions. We demonstrate
that small cell populations may tend to regress under the same growth
kinetics that result in rapid growth of large populations of cells
(established neoplasms). These findings provide a plausible explanation
for the regression of preneoplastic lesions (including the extinction
of initiated cells) that does not assume the existence of biologic properties
beyond those already observed experimental and human neoplastic development.
These simulations also quantitatively model the effect on proliferating
populations exerted by modulators of cell death (such as the bcl-2 oncogene).
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TABLE 1.
10 MONTE CARLO TRIALS, 1 INITIAL CELL, DEATH PROBABILITY 0.48
CLUSTER #: 1 2 3 4 5 6 7 8 9 10
GENER-
ATION:
0 1 1 1 1 1 1 1 1 1 1
1 2 1 1 1 1 2 1
2 2 2 2 1 2
3 3 3 1 2 2
4 4 1 1 2
5 3 2 1
6 4 3 2
7 3 3 1
8 3 4
9 2 5
10 1 5
11 1 8
12 2 6
13 2 9
14 2 10
15 2 9
16 2 7
17 1 3
18 1 5
19 2
20 2
21 3
22 2
23 3
24 1
25 2
26 1
27 1
28 1
EXTINCTION OF ALL CELL LINES
EXECUTION COMPLETE
TABLE 2.
DISTRIBUTION OF CLUSTERS IN 100 MONTE CARLO TRIALS, WITH 10-CELL
STARTING CLUSTERS, AND DIFFERENT CELL DEATH PROBABILITIES.
PROB. OF % 0F CLUSTERS LARGEST AVERAGE SIZE OF
CELL DEATH SURVIVING SURVIVING SURVIVING CLUSTERS,
PER CELL TO SIXTY CLUSTER NO. OF CELLS
PER GENERATION GENERATIONS
0.53 3 30 15
0.52 6 43 16
0.51 8 37 10
0.50 29 59 19
O.49 47 197 40
0.48 79 528 107
0.47 88 1,283 265
0.46 90 3,850 925
0.45 99 9,117 2,001
TABLE 3.
SURVIVAL OF CLUSTERS (TO 60 GENERATIONS),
VARYING INITIAL CLUSTER SIZES,
AND VARYING CELL DEATH PROBABILITIES.
PROB. OF TOTAL SURVIVING CLUSTERS AND (SIZE OF LARGEST CLUSTER)*
CELL DEATH
PER CELL INITIAL CLUSTER SIZE
PER CYCLE 1 5 10 25 50
0.53 1(2) 3(28) 3(30) 7(17) 13(21)
0.52 0(0) 1(6) 6(43) 23(23) 33(37)
0.51 1(14) 9(22) 8(37) 39(52) 67(94)
0.50 4(38) 11(24) 29(59) 68(114) 88(157)
0.49 8(145) 34(184) 47(197) 87(275) 93(678)
0.48 16(256) 45(228) 79(528) 93(930) 100(1,638)
0.47 15(556) 64(1,105) 88(1,283) 99(2,780) 99(4,724)
0.46 23(800) 59(2,508) 90(3,850) 99(6,252) 100(11,182)
0.45 20(5,396) 80(6,245) 99(9,117) 99(15,269) 100(28,208)
*The first number is the number of original 100 clusters that survived 60 generations. The number in parentheses is the number of cells in the largest cluster at generation 60.
FIGURE LEGEND.
FIGURE 1.
Results of 100 Monte Carlo simulation trials after 60 generations,
with an initial cluster size of 1 cell and cell death probability, p=0.45.
Twenty clusters survived, ranging in size from 43 cells (surviving cluster 4)
to 5,396 cells (surviving cluster 15), and averaging 780 cells
per surviving cluster.
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