Supplementary Data
Seeding
We must specify what we mean by seeding, as to not confuse the terminology
with Enderling’s term ‘self-seeding.’ In Enderling’s models, ‘self-seeding’ referred to
stem cells that had migrated away from their initial position but still remain within
the same primary site and formed a new cluster of cells termed a ‘self-metastasis.’
When we use the term ‘seeding’ we refer to a cancer stem cell that has left the
primary site, entered the circulation and then entered a metastatic site or the
primary site again similarly to (1). In Enderling’s model, ‘seeds’ arise from the
original tumor and have migrated within the primary site, whereas in the current
model ‘seeds’ arise from a different site and are seeded in the metastatic (or
primary) site. While the seeded stem cells must have originally been migratory,
once they are seeded in the secondary site they may lose their migratory behavior
by a mesenchymal to epithelial transition (2). The current model is not specific to a
particular site or cancer type and will investigate the 3D spatial dynamics with a
focus on stem cell seeding. We hypothesized that stem cell seeding will significantly
increase the growth of the metastasis since the seeds occur on the surface on the
tumor and will not be spatially inhibited. Seeding often occurs on the surface (3, 4),
between the tumor receiving the CSC and its ‘outside’ (3), since their exit routes,
vasculature and lymph vessels, are mostly found on the periphery of the tumor (5)
and the CSC are coming from outside the mass (4) .
Parameters
Stem cell division rate, stem cell symmetric division rate, and the progenitor cell
division limit are varied based on ranges from the literature, Table 1. One iteration
of the simulation represents approximately one day. Progenitor cell division rate is
varied from 3 to 21 cell cycles based on data suggesting that epithelial cell divisions
were limited to between 3-21 cell divisions, see stochastic nonspatial sensitivity
analysis (6, 7). Stem cell symmetric division rate is varied from 0.01 to 0.09 per day
based on previous studies (8, 9) and supported by modeling (10, 11). These values
also allow the progression to cancerous growth within 1,000 days. The rate of
progenitor cell division is held constant at 0.5, which is supported by the
experimental data (9) and modeling (10, 11) and the rate of stem cell division is
held constant at 0.2 per day based on evidence that a stem cell divides 69 times per
year in humans (7). Lastly, since seeding rates of metastatic cells are hard to
measure, the stem cell seeding rate is varied from 0.002 to 0.05 to cover a wide
range of possible rates. A sensitivity analysis has been conducted for these
parameters, see supplemental data, and in other works (12-14). The simulations are
run up to 500,000 cells after which it stops. The simulation is run considering an
avascular tumor, in which we model one fourth of the growth and assume
symmetry. As such the 500,000 cell maximum is within the estimated maximum
volume of avascular tumor growth of 1-2 mm3 (15). In the model, a cell fills a 10
cubed-micron grid space with volume 1000 micron3 but the actual volume of a
sphere cell would be 523 micron3 hence about 52% of the grid space represents the
tumor cells. This is in agreement with reports calculating the breast cancer cell
percentage is around 53% (16) from estimates of vascular and interstitial space
(17).
Table 1: The Default Values for the Parameters in the Spatial Agent-based Model.
Variables
Stem Cell
Division Rate
Progenitor Cell
Division Rate
Stem Cell
Symmetric
Division Rate
Stem Cell
Asymmetric
Division Rate
Progenitor Cell
Division Limit
Seeding Rate
Symbol
r
Default Values
0.2 per day
Refereence
(7, 8)
0.5 per day
Ranges
0.14-0.33 per
day
0.5 per day
rp
rs
0.05 per day
0.026-0.125
(8, 9, 18)
1-rs
0.95
0.875-0.974
(8, 9, 18)
dmax
6 cell cycles
3-24
(7)
ps
0.002-0.05 per
day
NA
(9)
Stochastic Nonspatial Sensitivity Analysis
In order to get a better understanding of how the different parameters influence
the model results, we used a simplified nonspatial stochastic model. In this model
there are no space restrictions on growth. First, we vary the number of progenitor
cell divisions allowed before the cell is forced to apoptosis, Supplementary Figure 1.
The time until the micrometastasis becomes a macrometastasis is examined, which
is defined as reaching 500,000 cells in a quarter of the tumor, or reaching 2 mm in
diameter for the full tumor (19).
As the number of divisions before apoptosis increases, the total number of cells
is greatly increased. For 21 divisions, the total number of cells grows within the
first 100 days to over 100,000, and at a division limit of 18, it almost reaches
100,000 within the first 100 days. By 500 days, the simulation has reached 1 million
cells at a limit of 21 divisions before apoptosis. With smaller numbers of progenitor
cell divisions, such as 3 and 6, the cell population never reaches even 1,000 cells,
thus it maintains a micrometastatic size. The stem cell symmetric cell division rate
here is 0.002. It is clear from this analysis that the number of progenitor cell
divisions is a critical factor in the number of total cells in the tumor population.
Within a 10-fold difference there, the tumor cell numbers spread from less than 100
to one million after 1,000 iterations (~ 3 years).
Next, the effects of seeding, in which stem cells from the tumor primary site are
deposited into the metastatic site, on metastatic growth is examined as well as the
effects of the rate of stem cell-like symmetric division on metastatic growth. For
this analysis, we chose a progenitor division number of 6, so that the switch to
metastatic cancer would be due solely to the change in the parameter examined.
At the metastatic site, or even the primary site, there are two possible ways to
increase the number of stem cells in the population, under the assumption that
progenitor cells cannot revert back to a stem cell phenotype. The first way is for the
stem cells that are present in the metastatic site to reproduce symmetrically and
produce two stem cells, resulting in a net increase of one stem cell. The second way
is for stem cells from the primary site to leave their circulation in the blood stream
and seed the metastatic site. Each of these mechanisms is investigated separately.
In Supplementary Figure 2, stem cell symmetric division and stem cell seeding
are investigated independently. In the left panel, the stem cell symmetric division
rate is varied between 0.002 and 0.01, which demonstrates that the symmetric cell
division has a large effect on the growth of the tumor. In particular, at rates greater
than 0.006, the tumor size reaches 106 cells within the first 1,500 iterations,
whereas at a rate of 0.002 it fails to grow past 1,000 cells by the end of 1,800
iterations. When stem cell seeding is the sole method for increasing the number of
stem cells, the effects on the total number of cells is much less pronounced. There is
only about a 10-fold difference in the resulting number of cells. This is logical
because there is only a linear increase in the number of stem cells possible, when
symmetric stem cell division is not allowed.
The number of stem cells directly relates to the growth of the tumor.
Since stem cells are the only cells that can proliferate indefinitely, it makes sense
that they would be critical to the growth of the tumor. There are two ways to create
more stem cells, 1) through symmetric division, and 2) through seeding. The
symmetric division rate (rs) has a large influence on the resulting number of cells,
Supplementary Figure 3A. The mean numbers of cells, Supplementary Figure 3A,
and the mean number of stem cells, Supplementary Figure 3B, follows the same
trends but have different values. Thus the total number of cells in the metastasized
tumor is explained by the number of stem cells present at the site of metastasis.
As shown in Figure 4A, stem cell seeding increases the growth of the metastasis.
Without stem cell seeding the number of the cells in the metastasis reaches around
2,000 cells by the end of the 1,000 days of growth. With the rate of seeding set at ps
= 0.05 the mean number of cells increases to 10,000 cells after 1,000 days of growth.
Although tumor size is increased, the number of cells varies less than 10-fold by the
end of 1,000 days. While stem cell seeding (ps) increases the growth of the tumor,
the effects are less than those of symmetric division (rs) since the difference in the
number of cells in the tumor between the highest and lowest seeding rates was
about 10-fold, whereas for symmetric division it was closer to 100-fold. Thus, we
conclude in these conditions, while stem cell site seeding is important for growth,
limiting symmetric cell division would have a greater effect on reducing tumor
growth.
Our simulations indicate that generally the rate of stem cell symmetric division
has a larger effect on tumor growth than stem cell seeding when the division limit is
low. This is logical since the symmetric division rate affects all stem cells within the
tumor and governs the exponential growth of the tumor. Seeding is limited to occur
at most once every day, contributing linearly to stem cell numbers. Based on these
results, we suggest that therapeutic treatment options should be aimed more at
reducing stem cell proliferation than at restricting seeding and progenitor cell
division. This is in agreement with studies that show cancer stem cells with
decreased capacity for self-renewal have reduced ability to form mammospheres
(20, 21).
Supplementary Figure 1: Stochastic Sensitivity Analysis of the Number of Progenitor Divisions until
Death. Here we vary the number of progenitor cell divisions allowed before cell death and hold all other
parameters constant. It is clear that this parameter has a larger effect on the number of cells and after
15 divisions, the cells grow to a macrometastasis, defined here as 500,000 cells. Thus, we keep the
division limit low dmax = 6 to fully explore the other parameters.
Supplementary Figure 2: Sensitivity Analysis of Stem Cell Symmetric Division and Seeding in the
Stochastic Model. (A) The effects of cancer stem cell symmetric division rates without seeding. (B) The
effects of stem cell seeding with stem cell symmetric division. Thus all cancer stem cells reproduce to
form a cancer stem cell and a progenitor cell. The asymmetric cancer stem cell division rate is 0.3 in
both cases. Symmetric division has a much greater effect on tumor growth than cancer stem cell
seeding.
Supplementary Figure 3: Symmetric Division Rate. (A) The effects of stem cell symmetric division rates
rs. The y-axis is the mean total number of cells from 4 different runs. (B) The mean number of stem cells
from 4 different runs with different symmetric division rates rs. Both plots follow the same trends such
that the number of cells increases with increasing symmetric division rate.
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