[CANCER RESEARCH 49, 3344-3354, June 15, 1989]
Model for the Genetic Evolution of Human Solid Tumors1
Stanley E. Shackney,2 Charles A. Smith, Beverly W. Miller, Dennis R. Burholt, Kevin Murtha, HarÃ-anR. Giles,
Deborah M. Ketterer, and Agnese A. Pollice
Cancer Cell Biology and Genetics Laboratory, Allegheny-Singer Research Institute, Pittsburgh, Pennsylvania 15212 [S. E. S., C. A. S., B. W. M., D. R. B., K. M., A. A.
P.], and Department of Obstetrics/Gynecology, Allegheny General Hospital, Pittsburgh, Pennsylvania 15212 [H. R. G., D. M. KJ
ABSTRACT
A conceptual model is proposed for the genetic evolution of many
human solid tumors that is based on the observations that cancer cells
may spontaneously double their chromosome number; that cells with
excessive chromosome numbers may be cytogenetically unstable, both
losing chromosomes randomly during subsequent cell divisions, and often
developing structural abnormalities in the chromosomes that are retained;
and that some structural chromosome abnormalities may activate growthpromoting genes. The sequence of tetraploidization with chromosome
loss can occur repeatedly in a given tumor. The available evidence
supporting the model is reviewed. A computer simulation system that
embodies these concepts is described and the model is used to generate
distributions of chromosome number/cell under various simulated condi
tions and in a variety of simulated biological settings. A simulation of the
time course of changes in chromosome number per cell that accompany
the spontaneous neoplastic transformation of mouse fibroblasts in vitro
is described. The best fit to the data was obtained when provision was
made for the activation of at least two growth-promoting genes. The
conditions for generating discrete aneuploid peaks in cytogenetic and flow
cytometric studies were explored; our modeling studies suggest that the
activation of a growth promoting gene is required in order to produce a
discrete aneuploid peak. Our modeling studies suggest that the overrepresentation of individual oncogene-bearing chromosomes in aneuploid cell
lines may require the activation of gene dose-dependent growth-promoting
genes and is not likely to occur in cell lines in which at least two copies
of each normal chromosome are required for cell survival. Overall, the
results obtained using the model are consistent with a wide variety of
flow cytometric and cytogenetic studies in human solid tumors.
INTRODUCTION
The development and progression of cancer are thought to
be a multistep process, in which sequential genetic abnormali
ties in individual tumors confer incremental proliferative ad
vantages upon successive clonal overgrowths (1,2). The nature
of these genetic abnormalities and the mechanisms underlying
their development have been studied extensively. In recent
years, increased attention has been focused on the development
of structural chromosomal abnormalities, particularly translo
cations, deletions, and inversions, that may be involved in
oncogene activation (3, 4).
Human leukemias have lent themselves readily to cytogenetic
analysis, since they tend to be diploid or near diploid (5-7) and
often exhibit single, distinctive nonrandom clonal structural
chromosomal abnormalities (4,8-10). In contrast, human solid
tumors often exhibit extensive numerical chromosomal abnor
malities (11-23), with considerable variability in chromosome
number from cell to cell in the same tumor. Although distinc
tive, tumor-specific structural chromosomal abnormalities have
been identified in some solid tumors (23), multiple, random
structural chromosomal abnormalities are much more common.
Received 8/8/88; revised 3/6/89; accepted 3/22/89.
The costs of publication of this article were defrayed in pan by the payment
of page charges. This article must therefore be hereby marked advertisement in
accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
1Supported by Allegheny-Singer Research Institute.
1To whom requests for reprints should be addressed, at Cancer Cell Biology
and Genetics Laboratory, Allegheny-Singer Research Institute, 320 E. North
Avenue, Pittsburgh, PA 15212.
The striking cytogenetic differences between the leukemias and
human solid tumors may reflect early vcrsux late stages in a
common genetic evolutionary sequence, or they may reflect
fundamentally different mechanisms of genetic evolution.
Recent flow cytometry studies in a variety of human solid
tumors have demonstrated that the presence of aneuploidy is
correlated with clinically aggressive disease and shortened pa
tient survival (24-37). Since aneuploidy by flow cytometry
almost invariably reflects the presence of cell subpopulations
with numerical chromosomal abnormalities (38), these studies
implicate numerical chromosomal abnormalities in the expres
sion of biological aggressiveness of human solid tumors.
In this paper we wish to explore a model for the biological
and clinical progression of human solid tumors that would
account for both the numerical chromosomal abnormalities and
the structural chromosomal abnormalities that are commonly
observed in these malignancies.
MATERIALS AND METHODS
Conceptual Basis for the Model
There are several well known cytogenetic mechanisms for generating
excessive numbers of chromosomes in cells. One mechanism is nondisjunction of individual chromosomes. Another is endoreduplication,
which results in the abrupt, spontaneous doubling of chromosome
number per cell. A failure of cytokinesis and cell fusion can also result
in the spontaneous doubling of chromosome number per cell. For
convenience, we shall refer to the process of spontaneous doubling of
chromosome number per cell as tetraploidization.
In early cytogenetic studies of rodent ascites tumors, both endore
duplication and tetraploidization were commonly observed (39, 40).
The development of tetraploidy or hypotetraploidy has been found to
accompany spontaneous neoplastic transformation in a variety of ro
dent cell lines in vitro (41-48) and has been observed after SV40induced transformation of normal fibroblasts (49-51).
Diploid or near diploid human tumor cells may undergo spontaneous
tetraploidization with chromosome loss when they are cultivated in
vitro. When this occurs, a concomitant increase in the rate of cell
proliferation is often observed, as documented in cultured human
glioma cells (22), in renal cell carcinomas (52), and in our own studies
of a cultured human undifferentiated large cell carcinoma of the lung
(53).
There are several lines of evidence which suggest that the sequence
of tetraploidization and chromosome loss occurs commonly in human
solid tumors in vivo and that it is associated with tumor progression.
Cytogenetic and flow cytometric studies have shown that low-grade
bladder cancers are usually diploid, whereas tetraploidy and hypotetra
ploidy are generally found in high grade and/or advanced stage bladder
cancers (17, 18, 54-58). Aneuploid bladder cancers have been found to
exhibit larger fractions of cells in S phase (54) than diploid low-grade
tumors. Cytogenetic studies in cervical cancer have shown diploid
chromosome numbers in cervical dysplasia, predominantly near-tetraploid values in carcinoma in situ, and hypcrdiploidy to hypotetraploidy
in invasive carcinoma (16). The lowest frequency of aneuploidy by flow
cytometry is commonly observed in low-grade and/or early stage cancer
of the lung (32-33), colon (25-27), and ovary (35-37); a higher fre
quency of aneuploidy is observed with increasing grade and/or stage in
each of these cancers. The presence of aneuploidy by flow cytometry
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GENETIC EVOLUTION OF HUMAN SOLID TUMORS
has been shown to be associated with shortened survival in these tumors
(24-28, 32-37).
Sequential clinical sampling studies in individual patients with mul
tiple myeloma (59), Sezary syndrome (60, 61), and brain tumors (62)
have documented the development of tetraploidy or hypotetraploidy in
association with neoplastic transformation, disease recurrence, or dis
ease progression.
Thus, the available evidence suggests that the sequence of tetraploidization and chromosome loss occurs commonly in human solid tumors,
both in vitro and in vivo, and tetraploidization with chromosome loss
is often associated with increased tumor growth rate //; vitro and
increased clinical aggressiveness of disease in vivo. By what mecha
nism^) might tetraploidization with chromosome loss contribute to
tumor aggressiveness? Explanations that have been suggested in the
past include increased dosage of growth-promoting genes (63), the
greater genetic vigor that may accompany increased chromosomal
heterogeneity (64, 65), and the loss of growth-suppressing genes (66).
Our own cytogenetic studies in a human undifferentiated large cell
lung cancer cell line (53) have demonstrated repeated rounds of tetra
ploidization during the course of 40 serial passages. In these studies,
several chromosomes with distinctive structural chromosomal abnor
malities were consistently over-represented, supporting the concept that
repeated rounds of tetraploidization increase the intracellular dose of
growth-promoting genes. In these studies, the hypodiploid cells present
in early passages often exhibited the loss of both copies of each of
several normal chromosomes. This hypodiploid cell subline eventually
died out, whereas the triploid to hypertetraploid cells that persisted in
late passages commonly had at least one copy of each normal chro
mosome. These findings suggest that tetraploidization also provides an
added reserve of normal chromosomal material that might be required
for cell survival in the face of subsequent chromosome loss.
Our studies also support the role of random chromosome loss in
generating diversity among cell progeny, with respect to both chromo
some number and chromosome type, permitting the selection and
overgrowth of cells with multiple copies of chromosomes that might
bear activated growth-promoting genes.
Finally, our experimental studies suggest a higher rate of develop
ment of random structural chromosomal abnormalities following tetra
ploidization (53, 67). Shapiro and Shapiro (22) have also noted an
increase in the frequency of structural chromosomal abnormalities in
association with tetraploidization and chromosomal loss in human
(¿liornasin tissue culture. Similarly, in the transplanted rat Dunning
prostatic adenocarcinoma, the appearance of multiple structural chro
mosomal abnormalities was found to be associated with the develop
ment of tetraploidy/hypotetraploidy
in rapidly growing cell sublines
that emerged in a single passage from a slowly growing diploid parent
cell line (68).
These considerations have prompted a conceptual model for the
genetic evolution of many human solid tumors that is shown in Fig. 1.
This model features repeated rounds of tetraploidization and chromo
some loss, with an increased rate of development of random structural
chromosomal abnormalities among aneuploid cells. Some of the struc
tural abnormalities that develop may result in the activation of growthpromoting genes. Cells containing chromosomes with growth-promot
ing structural abnormalities might be expected to overgrow and emerge
as the dominant cell line.
Despite the apparent simplicity of the model shown in Fig. 1, there
are many aspects of its dynamic behavior that cannot be anticipated on
intuitive grounds alone. For example, one cannot be sure if this model
is capable of generating discrete aneuploid peaks such as those that are
commonly found in flow cytometric studies. Therefore, we have devel
oped a computer simulation system that embodies the concepts of the
model, in order to explore the consequences of the model more rigor
ously.
Description of the Computer Model
Our computer model uses Monte Carlo simulation techniques to
represent cell growth, cell division, tetraploidization, chromosome loss,
and the development of growth-promoting structural chromosomal
abnormalities stochastically. The model was written in FORTRAN and
implemented on a VAX 11/750 computer (Digital Equipment Corp.,
Mayneard, MA).
A program flow diagram is given in Fig. 2. To simulate cell growth
and progression through the cell cycle, each cell progresses sequentially
through an array of ten cell cycle compartments. The movement of a
cell from one compartment to the next is governed by a transition
probability, /Vow, that is applied with each simulated A/. Thus, /V0«
varies inversely with the cell cycle time and has a baseline value that is
subject to upward revision, depending on the number of growth-pro
moting structural chromosomal abnormalities that are present within
a given cell and the relative growth advantage that is conferred by each
of these abnormalities. As a practical matter, simulated A/ is taken to
be 1 h, so that the minimum possible cell cycle time for any cell in the
population is 10 h.
It is assumed that cells with chromosome numbers exceeding the
diploid complement are at increased risk for the development of struc
tural chromosomal abnormalities. For each cell, there are associated
arrays to store the number of copies of each chromosome and, if present,
the number and type of chromosomal abnormalities per chromosome
copy.
The probability that a particular chromosome will develop a struc
tural abnormality between cell divisions is generated by a power func
tion that increases from values of less than 1 x 10~'°for a cell with 46
chromosomes or fewer, to an asymptotic value, Pstnu:<,
in cells with more
than 46 chromosomes. The shape of this curve is subject to user
modification. In all of the simulations to be described here, the curve
describing P5UMas a function of chromosome number per cell ap
proaches its asymptote rapidly, so that cells with 50 chromosomes or
more all have the same value for /'„„,„,.
Simulated structural abnormal
ities are intended to encompass chromosomal translocations, inver
sions, and deletions. However, in the present model, each structural
abnormality was assigned to a single chromosome for the sake of
simplicity, and there was no attempt to track interchromosomal ex
changes. Of course, the model does not distinguish between structural
DIPLOIDY^
abnormalities that are detectable by light microscopy and those abnor
malities that are submicroscopic. For cells that leave the last, i.e.,
premitotic, compartment each chromosome is subjected to PstnK,.It is
assumed that there is an upper limit for the number of structural
PSEUDODIPLOIDY/ HYPEHDIPLOIDY -* TETRAPLOIDY
chromosomal abnormalities per cell, above which cell viability is lost.
In the studies presented here, this limit was set arbitrarily at 66.
One or more specific chromosomes can be designated as bearing
regions that are subject to the development of structural abnormalities
with growth-promoting properties, with the probability of /Viv per cell
•¿
CHROMOSOME LOSS *division, which varies as a function of chromosome number per cell,
much like Psmc,. Growth promotion is effected by increasing the value
for /Vow by a specified fraction in the cell containing the growthpromoting structural abnormality. It is assumed that a given cell can
INCREASED
DEVELOPMENT OF
have multiple growth-promoting structural chromosomal abnormali
•¿â€¢STRUCTURAL
CHROMOSOME *
GROWTH »
ONCOGENES
ties, the growth-promoting effect of each abnormality being additive
RATE
ABNORMALITIES
for a given cell.
Fig. 1. Schematic representation of a model for the genetic evolution of human
solid tumors. For discussion, see text.
Once P,tnK,has been applied to all of the chromosomes in a given
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GENETIC EVOLUTION OF HUMAN SOLID TUMORS
THAT
MS LOST
CONTAIN*
STRUCTURAI
ABNORMALITY
THAT
CONFERRED
A
OROWTH
ADWNTAGF
Fig. 2. Program flow diagram for computer simulation system that implements the conceptual model for the genetic evolution of human solid tumors shown in
Fig. 1.
It is assumed that certain chromosomes or combinations of chro
cell and one P.dv has been applied to those chromosomes designated as
bearing potential growth-promoting abnormalities, with adjustments
mosomes may be essential for cell survival. The model provides for the
to /'„„,»
as appropriate, then the number of chromosomes in the cell is death and removal of cells in which random chromosome loss has
doubled and the probability of tetraploidization, /'„.,„,
is applied. If the
resulted in the deficiency of a prescribed numerical chromosome com
plement required for cell survival. If, after all chromosomes have been
cell divides successfully, a new daughter cell is established in addition
subjected to /\,„„
the cell retains at least the minimum complement of
to the existing cell, and the chromosomes are divided equally between
chromosomes required for survival, then the cell is assigned to the First
the two cells. If the cell does not divide successfully, all chromosomes
cell cycle age compartment, and the cell growth cycle begins again.
are retained in the existing cell. Cells with chromosome numbers
Because of physical limitations on computational resources, no more
exceeding 200 are considered to be nonviable and are cleared from the
than 5000 cells can be followed simultaneously. When this number is
population.
exceeded, the population is thinned by random cell selection to 1200
Cells with chromosome numbers exceeding the diploid number are
cells and cell population growth continues. This limitation on simulated
assumed to be cytogenetically unstable. The probability of chromosome
loss during a given cell division is represented by /*!„„.
Like PSUva,Pi,** population size can have an effect on the results of the simulation
studies, particularly on estimates of PKm, P\<m,P«™*,
and P*, that are
is assumed to vary with chromosome number per cell but is assumed
obtained by fitting output of the model to actual time-dependent data.
to be the same for each chromosome in a given cell and independent of This problem can be obviated by the use of appropriate scaling factors
the loss of other chromosomes in that cell during the course of a given that take into account the difference between simulated and object
cell division. For the parent cell and for the daughter cell (if cell division
population size. Of course, conclusions regarding population charac
was successful), each chromosome is subjected to /',„.If a chromosome
teristics that depend on the relative rather than the absolute magnitudes
with a growth-promoting structural abnormality is lost, Pt„„
is adjusted
of Pu«*,floss, fl.dv, should not be affected by the application of such
accordingly.
scaling factors.
3346
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GENETIC EVOLUTION OF HUMAN SOLID TUMORS
In the simulation shown in Fig. 3Ä,fl|05Sis higher in relation
RESULTS
General Features of Model Behavior: The Effects of /'„•«-.,
and to fletra than in the previous example (fletra = 0.05; floss = 0.03).
In this case cells with chromosome numbers ranging between
/"loss-We first consider fleira, the probability of tetraploidization,
and /'loss, the probability of chromosome loss, and their effects diploid and tetraploid are present, and they persist over time.
When PUU, is less than floss, as in Fig. 3C, there is no net
on the behavior of the model. Fig. 3A shows the results of a
accumulation of cells throughout the diploid to tetraploid re
simulation where the probability of tetraploidization for any
gion,
and cells that had previously undergone tetraploidization
given cell division is very high (flelra = 0.1), and the probability
with chromosome loss come to rest in the diploid and nearof chromosome loss is low (floss = 0.0001). Under these con
diploid region. The karyotypes of these cells betray their history
ditions, the population of cells that emerges is predominantly
by the presence of multiple tetrasomies, trisomies, and/or montetraploid. Since tetraploid cells are, themselves, subject to a
osomies in the same cell and, in the absence of extensive
doubling of chromosome number when Putr, is relatively high,
constraints on cell viability, by the loss of both copies of one or
they produce octaploid cells in turn. This effect is observed
more chromosomes (data not shown).
whenever Ptan is high in relation to floss. That is, the hallmark
In the simulations shown in Fig. 3, no chromosomes were
of high flew»in relation to floss is the presence of cells with designated as having the potential for developing growth-pro
ploidy values exceeding 4N. Hypertetraploid cells with chro
moting structural abnormalities. Under these conditions, stable,
mosome numbers often approximately twice those of the pre
discrete aneuploid peaks were never observed.
dominant hyperdiploid to hypotetraploid cells have been ob
Effects of /'a,i, and Fractional Growth Advantage. The time of
served in experimental tumors (45, 46). High />,„„
may also
appearance of a new population that has acquired a proliferative
account for the occasional observation of hypertetraploid DNA
advantage as a result of the development of a growth-promoting
indices or hypertetraploid modal chromosome numbers in hu
structural chromosomal abnormality depends on several fac
man tumors.
tors, including (a) the baseline doubling time of the parent
0.6-0.4-0.2-o-V1
population, (b) the value of fladv,the probability per cell division
of developing a specific growth-promoting structural abnor
mality, (c) the number of potential growth-promoting genetic
abnormalities, and (<l) the relative increase in cell growth rate
resulting from the occurrence of each growth-promoting genetic
abnormality.
As might be expected, as fl,dvdecreases, the emergence of the
earliest progeny of the advantaged population is delayed. If the
fractional growth advantage rate is high, the advantaged popu
1^0
'
1
'
i230
lation will overgrow rapidly. If the fractional growth advantage
46
92
138
184A1
is low, the advantaged population will overgrow more slowly
HI
(data not shown).
Simulation of Neoplastic Transformation in Vitro. In order to
0.6-1
test the capability of the model to describe the behavior of a
B
real experimental system, we undertook the simulation of the
changes in chromosome numbers per cell during the neoplastic
0.4-0.2-n-JLáál
transformation of mouse embryo fibroblasts in vitro.
A summary of published data from several different labora
tories (43-46) is given in Fig. 4,A\,Bt, and C\. The diploid cell
fraction began to decrease within 10 days of initiation of mouse
embryo fibroblast cultures (Fig. 4/d). There was an initial
,,,
, L,.
,,
decline to values in the range of 0.5 to 0.8 between days 20 and
90 in most cell lines. After a brief plateau phase, a relatively
138
184
230
92
46
rapid decrease in the fraction of diploid cells occurred between
50 and 150 days; in most cell lines, the diploid cell fraction was
0.6-0.4-0.2-0less than 0.4 by 100 days and less than 0.1 by 150 days. Overall,
there was considerable variability in the data. In three cell lines,
the diploid cell fraction fell sharply between 60 and 90 days
after culture initiation (43). In two cell lines, diploid cells were
present up to and beyond 300 days (Fig. 4A¡,
); this may
have been related to adverse effects of the fetal calf serum in
which these two cell lines were grown (44). For the present
purposes, we considered these two curves to be nonrepresentative.
A rise in the tetraploid cell fraction was observed within 10
6
92
138
184
230
I)
4Ci'l'i
days of cell culture initiation (Fig. 4B\). The proportion of
tetraploid metaphases peaked in the range of 0.1 to 0.3 at 2070 days and then declined progressively, approaching zero at
150 to 300 days. It is clear from the published histograms of
CHROMOSOME NUMBER/CELL
chromosome number per cell (46) that at 20 and 40 days the
Fig. 3. Simulated distributions of chromosome number per cell, varying f,.„
overwhelming majority of metaphases were either diploid or
and /•,„„.
In all instances, P*, = 0. A, Pm,. = 0.1, /",„„
= 0.0001. B, />«,„
= 0.05,
A».= 0.03. C, P,m = 0.01, /V«= 0.03. For discussion, see text.
tetraploid, with very few nondiploid, nontetraploid cells
3
ü
I
u_
O
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GENETIC EVOLUTION OF HUMAN SOLID TUMORS
M
too
200
10°-
300
200
200
300
300
CO
100-,
KX>-
80-
80-
60
60
60
40-
40-
40-
20
20-
20-
100
80
0
0
100
200
300
0
100
200
300
100
200
300
300
100
200
30O
TIME, DAYS
Fig. 4. Time course of numerical and structural chromosomal abnormalities associated with the transformation of mouse fibroblasts in vitro, based on Refs. 4346 (.!("). 11.fraction of diploid cells as a function of time in culture; //,. fraction of tetraploid cells as a function of time in culture; C'., composite representation of
the data in A, and /<,: hatched area I, time of cell culture immortalization; hatched area II, time of development of neoplastic transformation. Results of simulations
providing for two growth-promoting structural abnormalities are given in .-I?, H:, and C:. A,, ten simulations of the fraction of diploid cells as a function of time in
culture; /'-. fraction of tetraploid cells as a function of time; C¡,simulation runs with the earliest, median, and latest times of intersection of the curves of the diploid
cell fraction and nondiploid, nontetraploid cell fraction. Corresponding curves of the tetraploid cell fraction are also shown. Results of simulations providing for only
one growth-promoting structural abnormality are given in A,, B}, and C ,. Panels correspond to .-I..,H;. and ( '..,respectively. For discussion, see text.
present. Metaphases with chromosome numbers in the triploid
to hypotetraploid range became increasingly more common at
100 days; triploid to hypotetraploid metaphases and hypertetraploid metaphases predominated during the period from 100
to 400 days (46).
The salient features of the data in Fig. 4, A\ and B¡,are
represented schematically in Fig. 4Ct, and their relation to the
reported times of appearance of cell line immortalization and
the times of appearance of tumorigenicity in vivo (hatched
areas) are also shown. The brief early plateau in the falling
diploid cell fraction occurred at about the time of cell culture
immortalization (4Ci, Area I). The tetraploid cell fraction
peaked early and fell gradually. The nondiploid, nontetraploid
cell fraction was calculated by summing the diploid and tetra
ploid fraction at each time point and subtracting the sum from
1. The point of intersection between the diploid curve and the
nondiploid, nontetraploid curve proved to be a useful landmark
for modeling purposes. We estimate that this occurred between
75 and 100 days after initiation of most cultures. Also shown
in Fig. 4Ci is a schematic representation of the reported accu
mulation over time of cells with minute chromosomes and
marker chromosomes. These data indicate that the accumula
tion of gross structural chromosomal abnormalities was a late
phenomenon (43-46). Since the studies shown in Fig. 4, A \-C\
were performed before the advent of chromosome-banding tech
niques, we did not attempt to model the time course of devel
opment of structural chromosomal abnormalities in the simu
lations reported here. Indeed, our simulations suggest that
growth-promoting structural chromosomal abnormalities de
veloped long before gross abnormal marker chromosomes ap
peared (See Fig. 6 and associated discussion.).
It is apparent from the studies of Todaro and Green (43) that
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GENETIC EVOLUTION OF HUMAN SOLID TUMORS
cell population-doubling times of the primary mouse embryo
ity can be appreciated more fully by examining simulations
fibroblast cultures increased progressively from under 20 h using Padvvalues that straddle the value used in Fig. 4C3. When
shortly after the time of inoculation to approximately 100 h 45 Padvwas 0.0002, the diploid cell fraction fell very rapidly, with
to 75 days later. We have made no attempt here to model this no significant early plateau (Fig. 5/ii); the tetraploid cell frac
crisis phase in detail. Rather, we have simply assigned a rela
tion rose to about 0.2 but then fell somewhat more rapidly than
tively long representative value of 48 h for the baseline cell in the actual data (Fig. 5B,). With values of Padvdecreasing to
cycle of the primary cell culture. In published studies with cell 0.00005, the slope of the early portion of the diploid cell
culture transformation, population-doubling times fell to values fraction decreased appropriately; however, the brief plateaus
in the range of 14-24 h (43). Allowing for the fact that cell loss that developed with low values of Padvwere too low and occurred
lengthens population-doubling time in relation to cell cycle much later than in the actual data (Fig. 5, A2-/Ã-4).With decreas
time, in our modeling studies provision was made for minimum
ing Padv,the simulated tetraploid curves fell more slowly, but
cell cycle times that may have been as short as 10 h.
their crests were also shifted to longer times after simulated
More recent studies have shown that the activation of at least culture initiation, resulting in poor overall fits to the actual data
two oncogenes is required for full neoplastic transformation of (Fig. 5, 52-A4).
rodent fibroblasts (69, 70). Therefore, in our modeling studies
In all of the simulations shown in Fig. 5, the fractional growth
we compared simulations in which provision was made for the advantage was 0.1. When the fractional growth advantage was
potential activation of only one oncogene with simulations in increased to 0.2, simulations using a value of 0.00005 for Pad,
which provision was made for the potential activation of two exhibited a more rapid late fall in the diploid cell fraction than
oncogenes on separate chromosomes, each of which would be that shown in Fig. 5A4, resulting in a brief plateau that was
subject to Padv.
comparable to that of the actual data; however, the tetraploid
In our initial simulations of the data given in Fig. 4, A¡-Ct, fraction rose higher than that of Fig. 5/44 between 50 and 100
it quickly became apparent that Ptnn must be relatively high in days, diverging further from the real data. Overall, we were
order to fit the early rise in the tetraploid cell fraction (Fig. unable to find a combination of values that produced a better
4Bi). Estimates of P|OSSwere then constrained by the require
fit to the data than that shown in Fig. 4Ci_3 when provision
ments that the chromosome loss rate be sufficiently high to was made for only one growth-promoting structural abnormal
provide for a limited rise in the simulated tetraploid cell fraction
ity.
When two growth-promoting structural abnormalities were
to no more than 0.3 (Fig. 4H,) and at the same time that it be
sufficiently low in relation to Pietrato permit the development
permitted, a value of Padvof 0.0001 produced a subpopulation
of cells with a single growth-promoting structural abnormality
of simulated hypertetraploidy, as in the actual published data
(43-46) (see Fig. 3A and associated discussion). Values of PKu* that began to grow approximately 40 to 70 days after simulated
in the range of 0.035 and values of P\ox in the range of 0.0006
initiation of the primary culture (Fig. 6A), at a time that
were found to produce simulated histogram patterns of chro
corresponded with the development of cell culture immortali
zation (Fig. 6A, shaded region). Between 80 and 130 days, a
mosome number per cell that corresponded closely with those
actually observed during the first 50 days of cell culture. In our second growth-promoting structural abnormality developed in
simulations, as in the actual data, early histograms exhibited
the cells that already had one such abnormality (Fig. 6B). The
predominant diploidy and tetraploidy.
time of overgrowth of these cells corresponded with the time of
We found that simulations which permitted the development
first appearance of tumorigenicity in the real data (Fig. 6B,
of two separate growth-promoting structural chromosomal ab
shaded region).
normalities fitted the actual data at and beyond 50 days better
In these simulations, as in the real data, aneuploid chromo
than simulations which permitted only one growth-promoting
some numbers per cell were broadly distributed, with no discrete
abnormality. When two growth-promoting abnormalities were aneuploid peaks in evidence.
Simulation of Discrete Aneuploid Peaks. We found that dis
permitted, and when the values chosen for Padvand the frac
crete aneuploid peaks developed in our simulations when Padv
tional growth advantage were 0.0001 and 0.1, respectively,
multiple simulation runs demonstrated an early brief plateau in is greater than zero and when the simulated distribution of
the falling diploid cell fraction that was in the range of 0.5 to chromosome numbers per cell exhibits relatively few cells in
0.8 between 40 and 80 days after culture initiation (Fig. 4A2), the hyperdiploid to tetraploid region under baseline conditions
as in the actual data (Fig. 4A,). There was an early rise in the (i.e., in the presence of low absolute values of P|OSSand Ptelraor
tetraploid cell cycle fraction to values in the range of 0.15 to high Pio»in relation to PI«™;
see Fig. 3 C and associated discus
0.2 in most simulation runs and a gradual decline between 50 sion). When Pio»is relatively large, the aneuploid peak drifts
and 200 days (Fig. 4B2), as in the actual data (Fig. 4B¡).The down relatively quickly toward the diploid/hyperdiploid region.
median simulated point of intersection of the diploid curve and Although growth-promoting abnormalities did not always pro
the nondiploid, nontetraploid curve occurred between 75 and duce discrete aneuploid peaks, we have never observed a discrete
persistent aneuploid peak without the prior development of a
100 days (Fig. 4C2).
In contrast, when only one growth-promoting structural chro
growth-promoting abnormality.
To illustrate this aspect of the model, we have chosen to
mosomal abnormality was permitted, the simulations which
fitted the actual data best were obtained with values of Pa,u of simulate certain features of a human non-small cell lung cancer
0.000125. However, these simulations did not produce a strik
cell line that we have studied serially in tissue culture. This cell
ing early plateau in the falling diploid cell fraction (Fig. 4/i ,)• line exhibited two new rounds of tetraploidization with chro
mosome loss during 40 serial passages over the course of 1 year
In multiple simulation runs, the tetraploid cell fraction often
tended to remain elevated longer than in the actual data (Fig. in culture. An increase in the rate of development of nonclonal
4/?j), even though the diploid cell fractions fell somewhat more structural chromosomal abnormalities was observed after the
rapidly than in the actual data (compare Fig. 4,4., and 4.1,). The second round of tetraploidization, and one of these became
relatively poor fit of simulations based on the assumption of established as a clonal abnormality over the course of seven
only one growth-promoting structural chromosomal abnormal
weekly passages. The actual data are presented in detail in Ref.
3349
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GENETIC EVOLUTION OF HUMAN SOLID TUMORS
Fig. 5. Simulations of mouse fibroblast
transformation which provide for only one
growth-promoting chromosomal abnormality.
Ten simulation runs are shown in each pair of
panels. .1, and /',. diploid cell fraction and
tetraploid cell fraction as a function of time,
using values for /'„,i.
of 0.0002, and a fractional
growth advantage of 0.1. A2 and B2, compa
rable to A, and B\, respectively, with P^, =
0.00015. A, and B,, />„,,
= 0.0001. A. and Bt,
/>„,,
= 0.00005.
TIME, DAYS
S3. Additional data at our disposal including doubling times in
the range of 1-3 days after the cell line had become well
established in tissue culture.3
LU
CJ
100
200
Our specific modeling objectives were to simulate conditions
under which one or more rounds of tetraploidization with
chromosome loss occurred over the course of a simulated year
of growth in tissue culture, each with a discrete, clearly identi
fied aneuploid peak the position of which shifted downward
from the tetraploid/hypotetraploid
region toward the diploid
region over a period of several weeks to several months in
culture. Under these conditions, one growth-promoting struc
tural abnormality should develop and at least partially overgrow
at some time during the course of a simulated year of observa
tion. It should be noted that the experimental cell line that we
studied was already highly evolved cytogenetically at the time
of initiation of cell culture in vitro, with 11 stable chromosome
markers already present (53). We made no attempt here to
model these preexisting chromosomal abnormalities.
We found that simulations with input values of 0.004 for
Pietra, 0.004 for P,oss, 0.0001 for /»„„v,
0.3 for the fractional
300
B
100
200
300
TIME, DAYS
Fig. 6. Ten simulated runs that provide for two growth-promoting structural
chromosomal abnormalities in murine fìbroblastsgrown in vitro. A, simulated
patterns of appearance and overgrowth of cell subpopulations with the first
growth-promoting structural chromosomal abnormality. B, simulated patterns of
appearance and overgrowth of cell subpopulations with two growth-promoting
structural abnormalities per cell. Simulated conditions identical to those of Fig.
4, /(;- f ...Shaded regions, time of immortalization and time of neoplastic trans
formation, as in Fig. 4( ,.
growth, and 20 h for the minimum cell cycle time satisfied our
modeling objectives. The results of a typical simulation run are
shown in Fig. 7. Here, a growth-promoting abnormality devel
oped in a near tetraploid cell at 80 days and a hypotetraploid
subpopulation bearing the chromosome with this structural
abnormality rapidly overgrew, representing over 80% of the
cells by 180 days. As this rapidly growing subpopulation came
to predominate, a discrete aneuploid peak was observed, the
position of which drifted downward toward the diploid value
over time, as a result of random chromosome loss. By the time
this subpopulation comprised more than 45% of the total at
210 days, random chromosome loss had produced a reduction
in overall chromosome number per cell, including the average
number of copies per cell of the chromosome with the potential
for producing the growth-promoting abnormality. The average
number of growth promoting structural abnormalities per cell
did not fall, however, reflecting the selective pressure favoring
the overgrowth of cells with this abnormality. Indeed, once the
overall population consisted almost entirely of cells with one
growth-promoting abnormality, nearly all subsequent cells
undergoing tetraploidization acquired two copies of the chro
mosome with this abnormality. Since the growth-promoting
effect was assumed to be additive in these simulation studies,
cells with two copies of the aberrant chromosome had a selective
3 Unpublished observations.
3350
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GENETIC EVOLUTION OF HUMAN SOLID TUMORS
DAY:
120
150
180
6
92
0
«
270
240
210
92
O
U
92
O
46
300
92
I
46
92
Fig. 7. Stimulation of a discrete, unstable
aneuploid peak that first appears in the neartetraploid region and migrates toward the dip
loid region over time. P,„„
= 0.004; P¡aa=
0.004; Pa, = 0.0001; relative growth advan
tage = 0.3. Also shown are population-dou
bling time as a function of time, overgrowth of
subpopulation with the growth-promoting ab
normality, the average number of copies per
cell of the chromosome bearing the potential
structural abnormality as a function of time,
and the average number of growth-promoting
abnormalities per cell as a function of time.
a
i
co
3
DAYS
growth advantage over cells with fewer copies; by the end of 1 ment for cell viability (Fig. 8/4), the chromosome bearing the
abnormality becomes increasingly overyear, nearly all copies of the chromosome with the potential for growth-promoting
producing a growth-promoting abnormality that were present
represented over time, while some of the other chromosomes
in the population did, in fact, contain the abnormality.
are lost completely. When at least one copy of each chromo
The results of multiple simulation runs are summarized in some is required for cell viability, the Overrepresentation of the
Table 1. The effects of different values for />,„„,
P¡OK,
Padv,and chromosome with the growth-promoting abnormality is less
the fractional growth advantage are also summarized in Table
pronounced (Fig. 8Ä).When at least two copies of each chro
1.
mosome are required for cell viability, the chromosome with
Selection and Overrepresentation of Individual Chromosomes
the growth promoting abnormality is no longer overrepresented
with Growth-promoting Structural Abnormalities. The process
(Fig. 8C). In these simulations, it would appear that the strin
of tetraploidization with random chromosome loss provides the gent chromosome complement requirements for cell viability
potential for uncoupling the fates of individual chromosomes
played a role in determining cell survival that competed with
and outweighed the additive growth-promoting effects of overfrom those of other chromosomes in the same cell. If chromo
representation of chromosomes with growth-promoting abnor
some loss is truly random, then cells which retain chromosomes
with growth-promoting
structural abnormalities purely by malities. The distribution of chromosome number per cell pro
chance might overgrow those cells that do not retain such vides some insight into the interaction between these two fac
chromosomes. Furthermore, if the growth-promoting effects of tors. In the simulations shown in Fig. 8, A and B, the number
a given structural chromosomal abnormality were more pro
of chromosomes per cell tended to become pseudodiploid or
nounced with increasing gene dose, then there might be a near-diploid over time. In the simulation shown in Fig. 8C, the
selection pressure favoring cells bearing multiple copies of distribution of chromosome numbers per cell ranged predomi
chromosomes with the same growth-promoting structural ab
nantly between diploid and hypertriploid. This would suggest
normality, as suggested by the simulation run shown in Fig. 7. that, under stringent viability requirements, as a tetraploid cell
While the present model supports the premise that the over- loses more and more chromosomes and approaches the diploid
representation of chromosomes in aneuploid cells may reflect number, it becomes increasingly likely that further random
the presence of growth-promoting structural abnormalities on chromosome loss will result in a failure to meet the minimum
such chromosomes, it also suggests that there are factors which chromosome complement requirement for cell viability, and the
could easily dampen this phenomenon. One such factor is the cell will be lost.
minimum chromosome complement required for cell viability,
The true minimum chromosome requirements for the viabil
as shown in Fig. 8. In Fig. 8, the average number of copies per ity of normal human cells are not known in detail, nor is it
cell of each chromosome is plotted as a function of time. The known to what extent tumor cells must conform to these
solid curve represents one chromosome designated as poten
requirements. It is reasonable to suppose that at least one copy
tially bearing a growth-promoting structural abnormality. All of most individual chromosomes and at least two copies of
the other chromosomes are represented by dashed curves. When
some chromosomes must be necessary for the survival of most
there is no requirement of a minimum chromosome comple
human cells. Some human tumors have been reported with
3351
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GENETIC EVOLUTION OF HUMAN SOLID TUMORS
Table 1 Summary of multiple simulation runs of discrete, unstable aneuploid peaks under varying conditions
Fractional
growth
advantage
Results
A growth-promoting abnormality developed
between SOand 290 days in 10 simula
tion runs, accompanied by a round of to
traploidization and chromosomal loss. In
3 of 10 runs, there were 2 rounds of tetraploidization with chromosome loss.
No hypertetraploidy was observed.
Earlier appearance of the growth-promot
ing abnormality. This occurred between
10 and 65 days in 5 simulation runs. Sig
nificant hypertetraploidy was noted in 3
of 5 runs.
Growth-promoting abnormalities and mi
grating hypotetraploid peaks occurred no
more than once and occurred relatively
late, at 100 to 200 days in 4 of 5 simula
tion runs; neither growth-promoting ab
normalities nor aneuploidy occurred in 1
of 5 simulation runs.
Significant octaploidy occurred in 4 of 5
simulation runs.
0.004
0.004
0.0001
0.3
0.004
0.004
0.001°
0.3
0.004
0.004
0.00001
0.3
0.004
0.001
0.0001
0.3
0.001
0.004
0.0001
0.3
Neither growth-promoting abnormalities
nor aneuploidy were observed in 3 of S of
simulation runs.
0.004
0.004
0.0001
0.1
A population with a growth-promoting ab
normality began to overgrow in 1 of S
simulation runs but remained at less than
10% of the total by 360 days. No aneu
ploid peaks were observed in S of S simu
lation runs.
Comments
The simulations satisfied
modeling objectives.
Hypertetraploidy was not a
prominent feature of the
real data set.
Multiple rounds of tetraploidization and chro
mosome loss were ob
served in the real data.
Octaploidy was not ob
served in the real data
set.
The simulated rate of de
velopment of tetraploid
cells was too low to per
mit the development of
growth-promoting ab
normalities during the
observation periods.
Growth-promoting abnor
malities occurred but
were lost during simu
lated cell dilution during
' Italicized values indicate deviation from values given in the top line.
8 -
Ill
Fig. 8. Patterns of average chromosome
copy numbers per cell over time in relation to
minimum chromosome complement require
ments for cell viability.
, the one chromo
some designated as potentially bearing a
growth-promoting structural abnormality. All
other chromosomes are represented by dashed
lines. For all three simulations, /"«,„
= 0.02;
P** = 0.005; /Vi, = 0.01; simulated baseline
cell cycle time = 48 h; relative growth advan
tage = 0.25. .!. no minimum chromosome
complement requirement for cell viability; B,
at least one copy of each chromosome required
for cell viability; C, at least two copies of each
chromosome required for cell viability. For
discussion, see text.
LU
CL
6-
6 -
2-
2-
O
ü
LU
O
co
O
O
oc
I
o
¡
100
150 200
O
SO
100
I
150
¡
I
200
O
50
'• !
100
'
150 200
TIME, DAYS
modal chromosome numbers per cell of 40 or less (71). Our
own studies in human undifferentiated carcinoma of the lung
indicate that at least one copy of each chromosome is required
for cell survival (53). These studies also demonstrate moderate
overrepresentation (an excess of 1-3 copies/cell) of certain
chromosomes with structural abnormalities, in keeping with
the simulation results (Fig. 8/i). In the simulations of mouse
fibroblast transformation and in the simulations for human
lung cancer cell line reported in this paper, we assumed a
minimum requirement for viability of one copy per cell of each
normal chromosome.
DISCUSSION
We have proposed a conceptual model for the development
and progression of human solid tumors that is based on the
observations that cancer cells can spontaneously double their
chromosome number; that cells with excessive chromosome
numbers are cytogenetically unstable, both losing chromosomes
randomly over the course of successive cell divisions and often
developing structural abnormalities in the chromosomes that
are retained; and that some of the structural chromosome
abnormalities that develop may activate growth-promoting
genes.
The supporting evidence for the model is of two types. The
first is a large body of cytogenetic and flow cytometric evidence
that documents the widespread occurrence of numerical chro
mosomal abnormalities that are compatible with tetraploidization and chromosome loss in a variety of animal and human
tumor cell lines in vitro, and in many different human solid
tumors /// vivo. When tetraploidization with chromosome loss
3352
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GENETIC EVOLUTION OF HUMAN SOLID TUMORS
occurs, it is often associated with an increase in growth rate
and/or neoplastic transformation in vitro (21, 41-53) and with
tumor progression and/or shortened host survival in vivo (2437, 59-62).
The second type of evidence consists of banded karyotype
studies which suggest that there is an increase in the rate of
development of structural chromosomal abnormalities in cells
that had undergone recent tetraploidization (22, 53, 67, 68).
Our model represents a synthesis of the concepts supported by
these two lines of evidence.
Our overall modeling goal was to explore the consequences
of developing growth-promoting structural chromosomal ab
normalities under the conditions of constantly changing chro
mosome numbers per cell that are associated with tetraploidi
zation and chromosome loss. To meet this goal, we set three
specific performance objectives.
The output of the model should, at the minimum, conform
with a wide range of relevant existing real-world observations.
If the model is robust, the number of variables required will be
relatively small, and the effects of each variable on simulated
outcomes would be fairly distinctive. Our model did in fact
simulate the gamut of observed distributions of chromosome
number per cell, and specific outcomes proved to be highly
dependent on the conditions under which these outcomes were
generated. Thus, for example, hypertetraploidy and pseudodiploidy were generated under very different conditions (see Fig.
3 and associated discussion), and the occurrence of a simulated
discrete aneuploid peak was never observed without the prior
development of a growth-promoting structural abnormality (see
Fig. 7 and associated discussion.)
A more challenging test of the performance of the model
would be to determine how well it excludes from its repertory
of outcomes those which are unlikely to occur in the real world.
This is exemplified by the simulations of time-dependent
changes in the distributions of chromosome number per cell
during the course of spontaneous transformation of mouse
fibroblast in tissue culture (Figs. 4-6 and associated discussion).
We found that the data were fitted best when provision was
made for the development of at least two growth-promoting
structural chromosomal abnormalities in the same cells, as in
the case in real rodent fibroblasts (69,70). When provision was
made for the development of only one growth-promoting struc
tural abnormality, the simulated results did not correspond
with the actual data as well.
A still more challenging test of the performance of a model
is the determination of how well its predictions are borne out.
Our model predicted that the occurrence and degree of overrepresentation of individual chromosomes would be dependent on
the nature of the minimum chromosome complement required
for cell viability (see Fig. 8 and associated discussion). Specifi
cally, the model predicted that if at least one normal chromo
some of each type were required for cell viability, then chro
mosomes with gene dose-dependent growth-promoting struc
tural abnormalities would be only moderately overrepresented,
with an excess of approximately 1-3 chromosomes/cell (Fig.
8 A). This prediction has been borne out in our own recent
cytogenetic studies of human lung cancer cell lines in tissue
culture (53, 67), where at least one chromosome of each type
was preserved in the cell population over time. In both cell
lines, multiple stable markers were observed; most were repre
sented, on the average, by one copy per diploid cell, and some
were consistently represented by 2 copies per diploid cell.
The usefulness of this model may lie in its ability to provide
new insights into the origins and biological significance of
aneuploidy in human solid tumors. This, in turn, may provide
new leads for future experimental studies. In this regard, the
simulated observations that discrete aneuploid peaks are never
seen without the prior development of growth-promoting struc
tural abnormalities are especially relevant, as are the simulated
observations that gene dose-dependent
growth-promoting
structural chromosomal abnormalities can, under appropriate
conditions, produce chromosomal overrepresentation in the
presence of repeated rounds of tetraploidization and chromo
some loss. The model would suggest that in human solid tumors
that are aneuploid and which also contain overrepresented
chromosomes with structural abnormalities that are apparent
cytogenetically, one might focus on the sites of these structural
abnormalities as possible loci of activated gene dose-dependent
growth-promoting genes that may be directly associated with
clinical tumor aggressiveness.
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Downloaded from cancerres.aacrjournals.org on June 16, 2017. © 1989 American Association for Cancer Research.
Model for the Genetic Evolution of Human Solid Tumors
Stanley E. Shackney, Charles A. Smith, Beverly W. Miller, et al.
Cancer Res 1989;49:3344-3354.
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