Supplemental Book Resources for:
Berman JJ, Moore GW.
Spontaneous regression of residual tumour burden: prediction by Monte Carlo simulation.
Anal Cell Pathol. 1992 Sep;4(5):359-368.
Spontaneous regression of early (preneoplastic) lesions is one of the
fundamental mysteries in carcinogenesis. We have hypothesized that lesions
regress for strictly probabilistic reasons. Small populations composed
of dividing cells that have a non-zero probability of death in any division
cycle may either grow or shrink as chance dictates. Complete regression
occurs when a small population shrinks to zero cells. In a computerized
Monte Carlo simulation, the cell growth of groups of 500 cells
with constant generation times and preset prob-abilities of cell death
was determined. Regression of proliferating clones (extinction of clones
before the 100th generation) was the usual outcome. When the probability
of cell death in each cell at each generation is set at 50%, 49%, 48%, 47%,
46%, and 45%, regression of clones before the 100th generation occurs
in 98%, 96%, 91%, 86%, 78%, and 75% of clones, respectively.
Simulations with probabilities of cell death greater than 50% lead
to early clonal extinction. When the probability of cell death
is appreciably lower than 45%, clones grow so rapidly that their simulation
is not consistent with observed tumor growth. Results suggest
that regression of early lesions and initiated cells may occur
purely as a probabilistic consequence of intrinsic growth conditions
in small populations (early lesions). Cell death probability
near 45% produces simulated clonal growth rates similar to empirically
observed tumor growth.
Key Words: spontaneous regression, preneoplastic lesion,
liver, Monte Carlo simulation, cell death, initiation, promotion.
Spontaneous regression of preneoplastic lesions is a fundamental
mystery of carcinogenesis. The Solt-Farber model of liver carcinogenesis
is a widely used system for the study of cell population events
that lead to cancer. In this system, initiation occurs when rats
are exposed to a carcinogen, and promotion of initiated cells
is achieved by partial hepatectomy. In the ensuing months, hundreds
of preneoplastic lesions arise. Most early lesions regress for reasons
that are entirely unknown. The observation of emerging early proliferative
lesions followed by regression of most early lesions is also observed
in rodent models of skin tumorigenesis. In humans, the regression
of dysplastic lesions of the cervix is an accepted clinical phenomenon.
We have hypothesized that lesions regress for strictly probabilistic
reasons. Small populations composed of dividing cells that have a non-zero
probability of death in any division cycle may either grow or shrink
as chance dictates. In a computerized Monte Carlo model of clonal population
growth, groups of 500 cells with constant generation times
and preset probabilities of cell death was simulated. Regression
of proliferating clones (extinction of clones before the 100th generation)
was the usual outcome. When the probability of cell death in each cell
at each generation is set at 50%, 49%, 48%, 47%, 46%, and 45%, regression
of clones before the 100th generation occurs in 98%, 96%, 91%, 86%, 78%,
and 75% of clones. Simulations with probabilities of cell death greater
than 50% lead to early clonal extinction. When the probability of cell death
is appreciably lower than 45%, clones grow so rapidly that their simulation
is inconsistent with clinically and experimentally observed rates of tumor
growth. Results suggest that regression of preneoplastic lesions and
initiated cells may occur purely as a probabilistic consequence
of intrinsic growth conditions in small populations (early lesions).
La rgression spontane des lsions prnoplastiques
est un mystre fondamental de carcinogense. Le modle de Solt-Farber
de carcinogense du foie est un systme, qui est employ de loin
pour l'tude des vnements de la population des cellules qui introduitent
le cancer. En ce systme initiation a lieu quand les rats sont exposs
carcinogen, et la promotion des cellules inities est ralise par une
hpatectomie partielle. Dans les mois suivants, les centains des lsions
prnoplastiques viennent. La plupart des lsions prnoplastiques regresse
cause des raisons inconnues. L'observation des lsions prnoplastiques
prolifratives mergentes qui suivant par rgression de la plupart
des lsions prnoplastiques est vu ainsi dans les modles de tumorogense
de la peau des rongeurs. Dans les tres humains, la rgression des lsions
dysplastiques du cervix est un phnomne clinique accept.
Nous avons hypothesis que les lsions regressent pour les raisons
seulement probabilistiques. Les petites populations des cellules
qui divisent et qui ont une probabilit de non-zero
mourir dans un cycle
de division peuvent crotre ou retrcir par hazard. Dans le modle
computeris de Monte Carlo de la croissance de population, les groupes
de cinq cents cellules avec les priodes de gnrations constantes
et les probabilits de la mort des cellules qui sont fixes en avance,
tait simul. La rgression des clones qui prolifre (l'extinction des
clones avant la centime gnration) tait de rsultat normal. Quand la
probabilit de la mort de cellule dans chaque cellule en chaque gnration
50%, 49%, 48%, 47%, 46%, et 45%, la rgression des clones
avant la centime gnration a lieu dans 98%, 96%, 91%, 86%, 78%, et 75%
des clones. Les simulations des probabilits de la mort des cellules
plus de 50% a fait d'extinction commenante des clones. Quand la probabilit
de la morte des cellules est beaucoup moins de 45%, les clones croissent
si vite qui leure simulation est inconsistent o on a remarqu
ou cliniquement ou exprimentalement les taux de croissance du tumeur.
Les rsultats suggrent que la rgression des lsions prnoplastiques
et des cellules inities puisse avoir lieu seulement comme une consquence
probabilistique des conditions de la croissance intrinsique dans les petites
populations (lsions prnoplastiques).
Mots-clefs: rgression spontane, lsions prnoplastiques,
foie, simulation de Monte Carlo, mort de la cellule, initiation, promotion.
La regresi¢n espont nea de lesiones preneopl sicas es un misterio
fundamental de carcinognesis. El modelo Solt-Farber de carcinognesis
del higado es un sistema, de amplia aceptaci¢n, usado para el estudio
de eventos que generan cancer en poblaciones celulares. En este sistema,
la iniciaci¢n ocurre cuando se exponen ratas a un carcin¢geno, y la promoci¢n
de clulas iniciadas se lleva a cabo por hepatectom¡a parcial. En los meses
siguientes, surgen cientos de lesiones preneopl sicas. La mayor¡a de las
lesiones preneopl sicas tienen una regresi¢n por razones enteramenta
desconocidas. La observaci¢n de lesiones proliferativas tempranas seguidas
par la regresi¢n de la mayor¡a de lesiones preneopl sicas es observada
tambin en modelos de tumorognesis en piel en roedores. En humanos,
la regresi¢n de lesiones displ sicas del crvix es un fen¢meno
Nuestra hip¢tesis es que las lesiones tienen una regresi¢n
por razones estrictamente de probabilidad. Peque¤as poblaciones
compuestas por clulas en divisi¢n que tienen una probabilidad nocero
de muerte en cualquier ciclo de divisi¢n pueden crecer o disminuir
de azarosamente. En el modelo computerizado Monte Carlo de poblaci¢n
de crecimiento clonal, grupos de 500 clulas con tiempos de generaci¢n
constantes y probabilidades de muerte clular fijadas previamente fueron
simulados. La regresi¢n de los clones proliferativos (extinci¢n
de los clones antes de la generaci¢n 100) fue el resultado usual.
Cuando la probabilidad de muerte celular en cada clula a cada generaci¢n
es fijada a 50%, 49%, 48%, 47%, 46%, y 45%, la regresi¢n de los clones antes
de la generaci¢n 100 ocurre en 98%, 96%, 91%, 86%, 78%, y 75% de los clones.
Simulaciones con probabilidades de muerte celular mayores de 50% llevan
a la extinci¢n temprana de los clones. Cuando la probabilidad de muerte
celular es apreciablemente menor de 45%, los clones crecen tan r pido
que su simulaci¢n es inconsistente con los indices de crecimiento tumoral
clinicalmente y experimental-mente observados. Los resultados sugieren
que la regresi¢n de las lesiones preneopl sicas y de clulas iniciadas puede
ocurrir solamente como una consequencia de probabilidad de las condiciones
intr¡nsicas de crecimiento en poblaciones peque¤as (lesiones preneopl sicas).
Palabras-claves: regresi¢n espont nea, lesiones preneopl sicas,
higado, muerte celular, iniciaci¢n, promoci¢n.
The Solt-Farber model of liver carcinogenesis is a widely used
system for the study of cell population events that lead to cancer [
3]. In this system, initiation occurs
when rats are exposed to a carcinogen (diethylnitrosamine). Promotion
of initiated cells is achieved by partially hepatectomizing the rats.
In the ensuing months, hundreds of preneoplastic lesions arise.
Most early lesions regress. The observation of emerging early
proliferative lesions followed by regression of most early lesions
is also observed in rodent models of skin tumorigenesis
In humans, the occurrence of dysplastic lesions that may regress without
resulting in carcinoma is commonplace (e.g., low-grade cervical dysplasias).
The regression of precancerous lesions is an unsolved
and fundamental mystery of carcinogenesis. We hypothesize
that early lesion regression is governed by the same principles
that limit the growth of established tumors: the intrinsic tumor
cell death probability. In experimental studies of tumors,
it has been shown that tumor cells are continuously dying,
often at a rate that matches their cell growth
Cell death in tumors has been attributed to tumor cells
that have outstripped their blood supply. However, it can be
inferred that, even in the first few months of growth, a tumor must have
a high rate of tumor cell death. For instance, if a single hepatoma
cell (30,000 µ3 ),
with a generation time of 1 day is permitted to double with a zero death
probability per generation, the ensuing tumor mass would occupy
1 cubic meter in 45 days. In 55 days, it would grow to be over
1,000 cubic meters. In fact, tumors grow slowly, often over many years,
and although tumors and preneoplastic lesions have high proliferative
indices, this growth is counter-balanced by cell death.
In this study we employ the Monte Carlo method to simulate
the growth of cell clones with varied probabilities of cell death
The Monte Carlo method obtains a distribution of outcomes
for a mathematical experiment, using a computerized pseudorandom number
generator as a substitute for the probability value in the theoretical
distribution. 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
to obtain analytic solutions. By performing repeated trials and observing
how outcomes evolve over time, one can make predictions for complex systems
that defy direct evaluation. In this report, the simulated growth of 500
initiated cells was computed using one day as the generation time.
The one day generation time was chosen because hepatocellular cell lines
and regenerating liver cells divide about every twenty-four hours. Varying
probabilities of cell death were assigned to each founder cell. Cell death
probabilities were varied between 45% and 53% at 1% intervals.
For comparison, a deterministic evaluation of cell death was obtained
for the same range of cell death probabilities.
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.
In a Monte Carlo simulation, a founder clone consisting
of a single cell in generation zero divides once to yield two daughter cells.
For each daughter cell at generation one, the simulation program
independently draws a pseudorandom number between 1 and 100 in a uniform
(i.e., equiprobable) distribution. If the pseudorandom number exceeds
the death probability, then that cell continues for at least one more cell
division; otherwise the cell divides no further. For example, in a clone
with death probability 0.40, 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 is assigned a pseudorandom number at most 40,
then it dies without dividing. If a daughter cell is assigned a pseudorandom
number at least 41, 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).
In a deterministic model, fractional cells are permitted at each
generation, so that no clone dies unless it has a death probability of one.
The number of cells at generation k+1, namely c at k+1,
equals 2ck × (1-death probability).
A sample output for a single cell with death probability 0.49
per generation is shown in Table 1.
There are 62 generations prior to extinction, in which the clone reaches
a maximum size of 111 cells in generation 12, and then becomes smaller.
Generation zero has one cell, which duplicates to form two cells;
both cells obtain pseudorandom numbers greater than 49, allowing them
to proceed to the next generation. Generation one has two cells, which
duplicate to form four cells; all four cells obtain pseudorandom numbers
greater than 49, allowing them to proceed to the next generation.
Generation two has four cells, which duplicate to form eight cells;
this time, only four of these cells obtain pseudorandom numbers greater than
49, allowing them to proceed to the next generation. Generation 62 has
two cells, which duplicate to form four cells. However, none of the four
cells obtain pseudorandom numbers greater than 49, and the clone becomes
extinct. In a comparable deterministic experiment, there are only 3.41
cells after 62 generations, but the clone never becomes extinct.
The results of the Monte Carlo simulations for groups
of 500 clones are shown in Table 2.
When 500 clones of initiated cells, all having a 24 hour generation time
and a 50% probability of any cell dying in any cell cycle, half of the cells
died within the first generation cycle. The remaining 250 cells gave rise
to lesions of varying sizes, all but ten of which terminated before
the 100th cell generation (slightly over 3 months of growth).
Among the ten clones that survived to the 100th generation,
the largest clone reached a size of only 149 cells. When the probability
of cell death was decreased, many more clones reached their
hundredth generation and clone sizes grew. In all cases,
even in the presence of large surviving clones, the extinction of clones
was the rule and clonal survival was the exception. Even when the cell death
rate was 45%, there were only 127 (out of 500 initial clones) that reached
their 100th generation. The clones that survived to the 100th generation
had an average size of 24,851 cells. The largest clone was 89,851.
A smaller number of simulations was run for a death probability of 44%,
because the numbers of surviving cells in clones became so large that
the computer time needed to predict the growth outcome of every cell
In the deterministic model, in which fractional cells
were permitted and thus no cells could become extinct, the total number
of surviving cells usually exceeded the surviving cell numbers predicted
by Monte Carlo simulation, but outcomes were all at a similar order
In tumor cell growth simulations where cells grow probabilistically,
the number of possible outcomes (distributions of descendants) increases
exponentially with each generation. Consequently, it becomes impossible
to determine all possible outcomes that might arise when the starting cell
populations grow. A practical way to achieve some understanding
of how complex populations grow is to run growth simulations and
observe the patterns that emerge.
Cell death rates (probability of cell death for each cell
in each generation) greater than 50% result in few or no clones
reaching 100 generations, and clones that grow to 100 generations are small.
For simulations run with cell death probabilities between 44% and 50%,
small decrements in death probability resulted in large differences
in clonal growth. For instance, at the 100th generation of clones
growing with a 45% probability of cell death, the surviving clones
had an average size of 24,851. If the cell death rate is 1% higher (46%),
the average size of the surviving clones drops to 5,734 cells
(more than a 4-fold difference). If the cell death rate is 1% lower
(44%), the average size of the surviving clones rises to 131,447.
For the simulation at a 44% probability of cell death (per cell
per generation), only 30% of the initial clones survived to 100 generations.
In other words, 70% of the population regressed. Regression resulted
from initial growth conditions without the intervention of any subsequent
event to precipitate the regression phenomenon. The maximum-size clone
at the 100th generation of a clone with a 44% death probability was 613,500.
Using 30,000 microns3 as the size of a hepatocyte, a mass of 613,500 cells
would have a volume of approximately 20 mm3.
The actual size of a lesion composed of 613,550 cells would be larger,
due to allowances for intercellular space and stroma. However,
the predicted clonal cell mass at 100 generations (100 days when the
generation time is 24 hours) is similar to the observed size of cells
at three months in the Solt-Farber model.
It is a popular conception that tumors consist of two populations;
a proliferating and a nonproliferating population. The non-proliferating
population is presumed to consist of differentiated cells that can no longer
divide or of dead cells. The proliferating population is regarded as the
subset of cells responsible for tumor growth. Strictly speaking,
this model is correct, in that any given tumor cell is either capable
of dividing or not. However, the assumption that there is a specific
proliferative subpopulation in a tumor is untenable. If there were a
population that proliferated and gave rise to other proliferating cells,
that popuation would soon overtake the tumor (all other cells belonging
to the non-proliferating or dead cell population). Subsequently,
the tumor would grow with enormous speed. We propose that tumors
are composed of a non-proliferating population of cells that have died
or that are post-mitotic (differentiated), as well as a population
of cells that can divide. However, the cells that can divide may also die,
and the probability that they die is a factor influencing tumor growth rate.
If the probability of cell death in the proliferating fraction is stable,
then the growth fraction of the tumor would remain constant, as the growth
rate is determined experimentally by dividing the [3H-thymidine]-labeling
index of the whole tumor cell population by the labeling index of the
proliferating subpopulation. The labeling index of the proliferating
cell population would be proportionately reduced by the death probability
for the population of dividing cells.
A one day generation time was chosen because hepatocellular cell
lines and regenerating liver cells divide about every twenty-four hours.
This generation period does not have to be exact or even close to the
actual tumor cell generation time, as clonal extinction outcomes are
independent of the tumor cell growth rate. Cell generation times
do have bearing on the size of surviving clones, however, but we consider
predictions of tumor growth to be generally inaccurate, serving only to
illustrate possible (not necessarily likely) model outcomes.
Simulations were performed with cell death probabilities near 50%.
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.
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). The death probability for a skin basal cell presumably
decreases in wound repair. The assumption that tumor cells proliferate with
a stable probability of cell death is inferred from the observed properties
of early (preneoplastic) lesion growth, tumor growth,
and normal tissue growth.
Monte Carlo simulations show that minor decrements in the rate
of cell death can produce enormous changes in clonal growth.
It is possible that a feature of the malignant phenotype is the acquisition
of a less-than-50% cell death probability (for tumor cells).
Minor additional perturbations of the tumor cell death probability
may account for dramatic increases in the rate of growth of tumors.
These Monte Carlo simulations show that early lesion regression occurs
when there is clonal growth with an intrinsic probability of cell death
We wish to thank Mme. Barbara L. Moore for the French translation
and Dr. Manuel Rivero for the Spanish translation of the abstract.
1. Solt DB, Cayama E, Tsuda H, Enomoto K, Lee G, Farber E.
Promotion of liver cancer development by brief exposure
to dietary 2-acetylaminofluorene plus partial hepatectomy
or carbon tetrachloride.
Cancer Res. 1983 Jan;43(1):188-191.
Solt DB, Cayama E, Sarma DS, Farber E.
Persistence of resistant putative preneoplastic hepatocytes
induced by N-nitrosodiethylamine or N-methyl-N-nitrosourea.
Cancer Res. 1980 Apr;40(4):1112-1118.
Ogawa K, Solt DB, Farber E.
Phenotypic diversity as an early property of putative
preneoplastic hepatocyte populations in liver carcinogenesis.
Cancer Res. 1980 Mar;40(3):725-733.
Farber E, Solt D, Cameron R, Laishes B, Ogawa K, Medline A.
Newer insights into the pathogenesis of liver cancer.
Am J Pathol. 1977 Nov;89(2):477-482.
Solt DB, Medline A, Farber E.
Rapid emergence of carcinogen-induced hyperplastic lesions
in a new model for the sequential analysis of liver carcinogenesis.
Am J Pathol. 1977 Sep;88(3):595-618.
Solt DB, Hay JB, Farber E.
Comparison of the blood supply to diethylnitrosamine-induced
hyperplastic nodules and hepatomas and to the surrounding liver.
Cancer Res. 1977 Jun;37(6):1686-1691.
2. Farber E.
Putative precursor lesions: summary and some analytical considerations.
Cancer Res 36,2703-2705, 1975.
3. Goldfarb S, Pugh TD.
Multistage rodent hepatocarcinogenesis.
In: Popper H, Schaffner F, eds. Progress in Liver Diseases.
Orlando, FL: Grune and Stratton, Inc. 1986;8:597-620.
4. Foulds L.
London: Academic Press, Inc. 1969;1:.
5. Foulds L.
London: Academic Press, Inc. 1975;2:.
6. Laird AK.
Dynamics of growth in tumors and in normal organisms.
Natl Cancer Inst Monogr 1969;30:15.
7. Elias H, Sherrick JC.
Morphology of the Liver.
New York: Academic Press. 1969;:.
8. Cashwell ED, Everett CJ.
A Practical manual on the Monte Carlo Method for Random Walk Problems.
New York: Pergamon Press. 1959;:.
9. Berman JJ, Moore GW.
Why do most initiated cells fail to produce
early (preneoplastic) lesions? Prediction by Monte Carlo simulation
of cellular growth.
Lab Invest 62:9A, 1990.
10. Steel GG.
Cytokinetics of Neoplasia.
In: Holland, JF, Frei E, eds. Cancer Medicine.
Philadelphia: Lea and Febiger. 1982;:177-189.
INITIAL FOUNDER CLONE: 1 CELL
DEATH RATE: 49%
1 2 4 4 6 12 22 32 41 80
95 95 111 64 65 52 47 45 43 53
56 65 71 55 63 60 63 56 61 59
56 57 57 51 48 45 42 49 53 60
80 81 78 87 84 85 65 64 41 26
29 25 23 25 22 15 14 12 8 5
4 4 2
Sample growth simulation of a single cell with death probability 0.49,
starting in generation 0 with a single cell. There is a total of 62
generations prior to extinction. The clone reaches a maximum size
of 111 cells in generation 12. The clone size drops to 52 in generation
15, then rises to a second maximum of 87 in generation 43. Generation 62
has two cells, which duplicate to form four cells, but none of the four cells
obtain pseudorandom numbers greater than 49, and the clone becomes extinct.
PROB. OF CLONES AVERAGE LARGEST CLONE
CELL DEATH REACHING SIZE OF TO REACH 100
(PER CELL 100 GEN- SURVIVING GENERATIONS,
PER CYCLE) ERATIONS CLONES, # OF CELLS
# OF CELLS
0.53 0 - -
0.52 1 10 10
0.51 8 19 (S.D. 16) 44
0.50 10 42 (S.D. 47) 149
O.49 22 104 (S.D. 121) 488
0.48 43 312 (S.D. 411) 2,005
0.47 72 2,336 (S.D. 8,412) 71,245
0.46 109 5,734 (S.D. 7167) 37,948
0.45 127 24,850 (S.D. 24,169 99,277
0.44 * 131,447(S.D. 138,125) 613,510
0.00 500** 1.3 x 10 to the 30
Fate of 500 clones of cells at generation 100, all having
a 24-hour-generation time and various probabilities of any cell dying
at any cell cycle. For a 53% cell death probability, no clones survive
to 100 generations. For 52% cell death probability, 1 clone of 10 cells
survives to 100 generations. For 51% cell death probability, 8 clones
(average 19 cells per clone, largest clone 44 cells) survive to 100
generations, etc. Even for 45% cell death probability, there were only 127
(out of 500 initial clones) that reached the 100th generation
(average 24,851 cells per clone, largest clone 89,851 cells).
*Calculations for p=0.44 were performed for 180 clones only,
as the calculations required excessive computer time. For 180 clones,
54 reached 100 generations.
**For cell death rate zero, Monte Carlo and deterministic solutions
key words: precancer, precancerous condition, precancerous conditions,
premalignant, preneoplastic, early cancer, early lesions, early lesion,
neoplasm classification, cancer classification, in situ neoplasia,
in situ cancer, non-invasive cancer, noninvasive cancer, pre-invasive cancer,
preinvasive cancer, preneoplasia, premalignancy
Last modified: July 2, 2008