JUNE 2020 | VOL 3 | 1Journal of Emerging Investigators • www.emerginginvestigators.org
Signicance of Tumor Growth Modeling in the Behavior
of Homogeneous Cancer Cell Populations: Are Tumor
Growth Models Applicable to Both Heterogeneous and
Homogeneous Populations?
SUMMARY
The ability to predict and slow the spread of cancer in
the human body is a task that medical professionals
have been trying to accomplish for many years.
Being able to give factual basis to the use of
certain growth models for application in not just
heterogeneous, but also homogeneous cancer cell
populations is imperative to treatment research as
using mathematical analysis to predict the dynamics
of tumor growth allows professionals to simulate how
tumors might behave in the human body. This study
follows the process of single-cloning and the growth
of a homogeneous cell population in a supercial
environment over the course of six weeks with the
end goal of showing which of ve tumor growth
models commonly used to predict heterogeneous
cancer cell population growth (Exponential, Logistic,
Gompertz, Linear, and Bertalanffy) would also best
exemplify that of homogeneous cell populations.
We hypothesized that the Gompertz, Linear, and
Bertalanffy models would provide the best t to the
homogeneous cancer (clonal) cell population growth
data while models such as the exponential and
logistic model, which are most commonly associated
with the growth of heterogeneous cancer cell
populations in natural environments (i.e. malignant
tumors), would veer off the growth data. It was shown
that Gompertz and Linear functions provided the best
t for this population, while exponential and Logistic
functions fell slightly behind. The data collection and
analysis for this research was performed through the
University of Michigan Research Labs and Solver by
Frontline Systems.
INTRODUCTION
Cancer has always been a burden to the health system
as malignant tumors are notorious for slowly taking over the
body due to their ability to evade apoptosis and reproduce
indenitely [1]. In 2020, the American Cancer Society
estimates that there will be 191,930 new cases of prostate
cancer and about 33,330 deaths resulting from the disease
(2). Scientists have tried to identify the “most accurate”
mathematical model to t tumor growth. However, results
Divya Reddy, Dr. Alexander Zaslavsky, Dr. Todd Morgan, and Mr. Joe Rasmus
Williamston High School, Williamston, MI
Article
are either inconclusive or vary greatly depending on the
cell population as most studies only focus on the early
stages of tumor growth. Similarly, though many studies have
determined a model that best t that set of tumor growth data,
these ndings were not applicable to other data sets as tumor
growth can vary greatly depending on the tumor environment
[3]. If a certain model ts better than others and behavioral
patterns of tumor growth predicted by mathematical analysis
can be generalized to all cell populations, scientists will have
a better idea of the timeline of tumors in the human body.
Treatment can then be developed to inhibit this behavior,
effectively slowing down the spread of cancer and the growth
of malignant tumors [3].
Major key concepts to be addressed in this research
are the application of mathematical tumor growth models
in association with homogeneous cancer cell populations,
proliferation rates of cancer cell lines (specically in the VcaP
cell line), and the cell cloning procedure as a whole. The
proliferation rate refers to the rate of growth of the cancer
cell population or the malignant tumor itself. Proliferation
could also refer to the reproduction rate of the cells. In this
research, the proliferation rates among these cell lines was
directly measured through cell counting as the increase in
cell count over the course of the experiment would signify
the overall proliferation rate of the population. Cell lines are
specic cell samples from general cancer that are purchased
for the research lab. For example, both VcaP and PC3 are cell
lines of prostate cancer that are used for general research in
urologic cancer labs. Cell cloning is the process of taking a
cell line sample and diluting it to the point where you have a
single cell isolated in its well (cell plate). Over time this cell
will then reproduce innitely and form a homogeneous cell
population [4]. In this research, the proliferation rate of one
prostate cancer homogeneous cell population was measured
over a time period of six weeks and compared to ve different
tumor growth models (Exponential, Logistic, Gompertz,
Linear, and Bertalanffy).
The exponential growth model yields a growth rate that
is proportional to the cell population and is most commonly
associated with the early stages of tumor proliferation [3].
However, this model has proven to be less accurate as tumor
growth progresses in the body due to angiogenesis and
JUNE 2020 | VOL 3 | 2Journal of Emerging Investigators • www.emerginginvestigators.org
Table 1: Five growth functions are being utilized. For each equation,
a and b serve as parameters that are relative to the data set. V
exemplies the change in tumor size over time. In this case, V will
be representing the cell population number at the given time [3, 9].
nutrition depletion. When graphing this model, the solution to
the initial equation is used [5]. The solution is given
by:
with t serving as the week the data was collected, r serving
as a growth parameter, and V
0
serving as the initial cell
population measurement. The model assumes that growth is
proportional to the surface area and that there is a decrease
in tumor volume due to cell death. The Logistic Delay growth
model yields that the growth of a cancerous cell population
is limited by some capacity relative to the tumor size. The
logistic model assumes that the cell population increases
linearly until zeroing out at carrying capacity for the said
population and embodies a standard sigmoidal curve [3]. The
Gompertz growth model is a generalization of the logistic
model and is known to be the best example of breast and
lung cancer in the human body [3]. The model had the original
intent of displaying the human mortality curve, however,
it proves to be applicable to the growth of organisms with
a sigmoidal curve that is asymmetrical with the point of
inection [6]. It also predicts that as the tumor grows in size,
its growth rate decelerates. The model shows that a tumor’s
rate of growth is the most signicant at the early stages of
growth when there are no means to detect the tumor clinically
[7]. The linear growth model assumesthat a cell population
grows at a constant rate, rst exponentially then linearly over
time. The model was also initially used to predict the growth
rate of cancer cell populations when this type of research
was moderately new [3]. The Bertalanffy growth model was
created to exemplify organism growth and is known to provide
the best example of tumor growth [3]. Each model has a
specic tumor growth formula that can be applied to unique
data sets in order to predict the future growth of a tumor or
cancer cell population (Table 1).
The hypothesis of this study is that when a homogeneous
cell population is monitored from its start in an articial setting
it will best resemble a constant rate of growth exemplied by
the Linear, Gompertz, and Bertalanffy model as opposed to
a constant growth rate given by the Exponential or Logistic
growth model. This is because these two models are more
commonly associated with early stages of heterogeneous
tumor growth in a natural environment with limited resources.
While some of the models studied here have proven to be
good examples of specic cases heterogeneous tumor
growth (i.e. the Gompertz Model is known for best predicting
breast and lung cancer) the exponential and logistic models
are known for being a good basic generalization of early
heterogeneous tumor growth of varying types [3]. Along
with that, we hypothesized that the Linear, Gompertz, and
Bertalanffy growth models would be a better t for the
experimental data generated, as based on a previous study
of supercial heterogeneous tumor growth, they seem to be
signicantly more exible than the exponential model when
it comes to tting data points generated in lab environments
[3]. These models are not commonly associated with general
heterogeneous tumor growth [3]. Similarly, the Logistic model,
though having about the same t as the Gompertz and Linear
model in the study mentioned, were still assumed to have
a lacking t since this study was focused on homogeneous
growth and the Logistic model is, as previously mentioned,
more commonly associated with the basic generalization
of natural heterogeneous tumor growth [3]. Major issues
to be addressed through this research include identifying
what growth model best ts the early stages of growth of a
homogeneous cell population in a supercial environment, as
well as determining if those models differ from those more
commonly associated with heterogeneous tumor growth
in its early stages and in a natural environment with limited
resources (i.e. exponential and logistic) [3]. In this study, growth
data from a prostate cancer single-cell clone population was
collected over a period of time starting from the formation
of the cell population and the experimental data points’ t
to the growth models being utilized (Exponential, Logistic,
Gompertz, Linear, and Bertalanffy) to determine which model
best described homogeneous cancer cell population growth
in this context. The models utilized in this study were chosen
based on their widespread use in measuring tumor growth
in the human body [3, 6]. The two other commonly used
models in tumor growth — Surface and Mendelsohn — were
not utilized in this research due to the fact that these models
commonly use tumor volume as a measurement and not cell
population count. Since the measurement unit in this study
is cell population count, it is important to only use models
that are versatile when it comes to units of measurement [2].
The purpose of this research is to provide evidence as to
which model ts best with the behavior of a homogeneous
cell population and give reasoning as to why this outcome
exists. No specic treatment is being imposed other than the
cell cloning process itself and parameter generation for each
model.
Model
Equation
Exponential
Logistic
Gompertz
Linear
Bertalanaffy
JUNE 2020 | VOL 3 | 3Journal of Emerging Investigators • www.emerginginvestigators.org
RESULTS
This experiment was conducted through rst the
procedure of single-cell cloning and then data optimization
to generate an equation for each of the tumor growth models
being studied from the original cell population data. A single
cell was rst taken from a VcaP prostate cancer cell line
sample and left to incubate over six weeks with growth media
that was changed in weekly intervals. Over this time, periodic
cell population counts were taken weekly, and these data
points were then optimized through Solver (Google Sheets)
to generate the best t equation for each tumor growth model
being studied. In order to decide which model suited the
homogeneous cell population best, the NMSE (Normalized
Mean Square Error) and SSR (Sum of Least Squares) were
taken for each model. The objective was to have a small SSR
and NMSE, as that indicates the best t. The AICc (Aikake’s
Information Criterion) for each model was also taken to help
identify possible bias within the models, with a high AICc
indicating potential bias.
We hypothesized that when a homogeneous cell
population is monitored from its start it will best resemble a
Linear, Gompertz, or Bertalanffy growth model as opposed
to an Exponential or Logistic growth model, which are more
commonly associated with heterogeneous tumor growth
in its early stages and in a natural environment with limited
resources [3]. After calculating the NMSE and AICc for each
model, we concluded that while the Bertalanffy model had an
NMSE and SSR of zero, indicating a perfect t, it also had an
abnormally high AICc pointing to potential bias having to do
with sample size and eliminating it from the being a possible
t to the data. As expected, when considering both the NMSE
and AICc, the Linear model had the lowest values, indicating
a generally good t and minimal bias. The Gompertz model
followed, with the second lowest NMSE and AICc, and the
logistic and exponential models followed after that (Table 2).
The Bertalanffy, Gompertz, and Linear models were almost
indistinguishable from each other due to their close t to the
experimental data (Figure 1B), however, the Exponential and
Logistic Models showed high variation from the experimental
data points (Figure 1A & C). It also needs to be mentioned that
due to confounding variables data was unable to be collected
for week three of this study, qualifying it as an erroneous data
point for all gures and further discussion of results.
As stated before, the SSR, NMSE, and AICc for each
model had a good amount of variation. While the Bertalanffy
model had a perfect SSR and NMSE, it also had an abnormally
high AICc, pointing to potential bias most likely having to do
with a small set of data points. The Gompertz model was the
best t for the experimental data overall, as when considering
the three forms of analysis together, the Gompertz model had
the lowest overall values proving it to be a good t with minimal
bias. The Linear model was the second-best t in terms of the
overall analysis; however, it still fell behind the Gompertz in
the sum of squared residual analysis. This is slightly surprising
as even with having the second highest SSR, the linear model
still had the best NMSE and AICc in the sense that both
Figure 1: (a) A graph of the models in comparison to the experimental
data. The Bertalanffy, Gompertz, and Linear models very closely
overlap over the experimental data, while the Exponential and
Logistic models veer off the experimental data further. (b) A visual
comparison of the experimental data points for weekly cell count
and the corresponding points for the Gompertz, Bertalanffy, and
Linear models. (c) A visual representation of each model graphed
separately in comparison to the experimental data points.
A
B
C
JUNE 2020 | VOL 3 | 4Journal of Emerging Investigators • www.emerginginvestigators.org
values were the closest to zero when compared to those of
the other models. The logistic and exponential models proved
to be the worst t for the data, both having high AICc and
NMSE in comparison to the other models. It is also surprising
that both the exponential and logistic models had the highest
AICc after the Bertalanffy model as the exponential model
only had one parameter (Table 2).
Going off that, the parameters generated for each
model had a slight amount of variation as well (Table 3).
Each model had two parameters of a and b, apart from the
exponential model which only required one parameter of r,
signifying growth rate. Though these parameters play no part
in interpreting each model’s actual t to the data they still hold
interest. While most of the parameters generated (for those
of a and b) centered around zero, those of the linear model
were substantially high which could be expected, as this is a
commonality in similar experiments [3].
DISCUSSION
We hypothesized that when a homogeneous cell
population is monitored from its start it will best resemble a
Linear, Gompertz, or Bertalanffy growth model as opposed
to an Exponential or Logistic growth model, which are more
commonly associated with heterogeneous tumor growth
in its early stages and in a natural environment with limited
resources [2]. These ndings support the original hypothesis
in that the logistic and exponential models proved to be the
worst t for a homogeneous cell population, suggesting
that there was a difference between the growth, model of a
single-cell population, and the growth of a heterogeneous cell
population. Many factors contribute to this including the fact
that the single-clone population was articially created and
existed in a controlled environment. A heterogeneous cell
population existing in the body would have different nutritional
and environmental conditions that would alter the growth of
the population overtime.
When interpreting which models exemplied a good
t to the data, it was also important to take note of the
circumstances in which each model works best and how this
could translate to potential bias. The best example of this
with this set of data is the Bertalanffy model. Though it is
always suspicious when a model has a perfect t, it could
be assumed that this model may have a “perfect” t due to
the small sample size of the study. The Bertalanffy model
assumes that growth is proportional to surface area, and
since the number of data points as well as timespan was very
small in this study, it is safe to assume that the surface area
of the cell population had minimal to no change over time
even though the cell number itself rose. This was supported
as even though the model did have a “perfect” t to the data
with an SSR and NMSE value of zero, the AICc of the model
was signicantly high. Since an AICc is meant to correct for
sample size and parameter bias, it is reasonable to assume
that the reason the Bertalanffy model had such an accurate t
to the data is that there were too few data points. Under those
circumstances and the assumption of bias, the Bertalanffy
model was rejected as a potential t to this set of data as it
would only be fair to take the equation under consideration
with a large number of data points that simulate innite time.
That leaves both the Gompertz and Linear models as good
representations of the data provided by the population. This
is unsurprising as linear models were originally used to
predict the growth of cell populations in early research, and
the Gompertz model was also used for this purpose [3]. Both
models are also quite versatile in terms of units, as they can
estimate growth from both tumor volume measurements
and quantitative cell population data. Both models also have
the ability to work well with smaller sets of data such as
that utilized in this study. This is because the Linear model
generates a slope directly from the given data, so no matter
the sample size the model will generally t the points fairly
well due to its exibility [3]. Similarly, the Gompertz model
follows the pattern of a function that is asymmetrical with the
point of inection. This makes the Gompertz function quite
exible, as it allows the function to t any amount of data quite
closely since the function does not have to be symmetrical
with the data’s point of inection like in the Logistic model [3].
Along with the Logistic model, the Exponential model is quite
inexible when it comes to smaller sets of data as the function
only has one parameter, allowing for very little change to
the general function when inputting a small number of data
points [3]. Both of these models had the highest AICc after
Bertalanffy, suggesting that they had some form of bias. This
most likely also had to do with limited sample size, since both
functions are typically used with larger sets of data due to
their inexibility when it comes to small sets of data [3, 6]. We
Table 2: The SSR, NMSE, and AIC c for each model. The ideal for
each form of analysis is having a number relatively close to zero. In
terms of the AICc positive or negative values do not hold any value
over one another.
Model SSR NMSE AICc
Exponential 1021839.2 0.085281 80.27213
Logistic 8 0 237.75 6 0.0066965 75.00594
Gompertz 105.20054 0.0000087799 35.18465
Linear 38904.237 0.0000018203 21.81120
Bertalanaffy 0 0 -170.6488
Table 3: The parameters for each model. The parameter of r comes
from the exponential solution, not the initial function listed (Table 1).
Model a b r
Exponential n/a n/a 1.193
Logistic -2.591 x10
-3
2.18 5 n/a
Gompertz 1.14 4 2.103 x10
-2
n/a
Linear 71522.726 70588.288 n/a
Bertalanaffy -5.13 x10
-11
-0.999 n/a
JUNE 2020 | VOL 3 | 5Journal of Emerging Investigators • www.emerginginvestigators.org
also suspected that the reason the logistic and exponential
models proved to not be a great t is that the cell population
being studied was not in an environment with a limited number
of resources or space.
In terms of parameters, all numbers computed seem to
t the general data trends set by previous studies of this kind.
For instance, in previous studies, the parameters of all of the
model’s used except Linear were centered around zero, while
those of the Linear model all were signicantly higher. The
same trend can be seen in this study as the parameters of a
and b in the linear model were signicantly higher than those
in the Gompertz, Bertalanffy, and Logistic models. Along with
that, in considering the Gompertz model, most other studies
dealing with the Gompertz model utilize a three-parameter
equation of a, b, and, c (Figure 2). This equation is given by:
The equation utilized in this study had two parameters
only (Table 1) [9]. While this should not have had an impact on
the t of the model itself as both are functions of Gompertz,
it would have increased the model’s AICc as it deals directly
with a higher number of parameters in correlation to bias [3].
The three-parameter equation was not utilized in this study
due to variable constraints in Solver. The parameters of a,
b, and r in each data set serve as “free number” parameters
that are generated by a data optimization function to create
an equation for each model that best ts the experimental
data. However, these parameters should hypothetically fall
into the following denitions for each growth model. For the
exponential model, r should serve as the growth constant. For
the Logistic model, a should serve as a growth constant and
b should serve as a carrying capacity [3]. For this formulation
of the Gompertz model, a and b truly serve as free parameter
values [9]. For this formulation of the Linear model, a/b should
give the initial exponential growth rate, and a should give the
later constant growth rate. For the Bertalanffy model, a should
serve as a growth constant, and b should serve as a constant
of cell death [3]. For some of these models, specically the
Logistic and Bertalanffy models, these parameter denitions
do not make sense. This again could be due to sample size
bias as they both had a very high AICc when compared to
models with an equal number of parameters such as the
Linear and Gompertz models.
Boundaries that possibly inhibited the course of this
study include time, material, and sample size. Travel to the
lab was only possible once a week, which limited the number
of observations and experiments that could be carried out
as well as limited the amount of data that could be collected
within a specic timeframe. This could provide for some
unwanted bias in some models (i.e. Bertalanffy, Exponential,
and Logistic) that rely heavily on large data counts [3]. Another
boundary that exists in terms of this research is material. This
hypothesis on more than one cell line or cell single clone
population due to the limited amount of cancer lines available
in cell culture at this given time.
In terms of uncontrollable factors in relation to this study,
time and material constraints likely had an impact on the overall
execution. Week three of data collection was considered an
erroneous data point as independent circumstances did not
allow data collection to happen; however, ideally that would
not be the case. Similarly, time constraints only allowed for
this set of data to have a small number of points, leading to
a great deal of potential bias among equations that require
a larger set of data such as the Bertalanffy, Logistic, and
Exponential models. It is also generally known that larger sets
of experimental data yield stronger results. Time constraints
also limited the set of data that could be produced. Ideally,
at least three single clones would be produced, and a large
number of data points would be collected. However, there
was only enough time to perform the study on one clonal
population after deciding which cell line to use. The material
available also played a factor in this. Along with that, ideally,
data would also be taken on a daily basis rather than a weekly
one, as that would provide more insight into the population’s
growth rate and would generate a larger set of data. This
was not possible as transportation constraints only allowed
data to be taken on a weekly basis. If these uncontrollable
factors did not exist and the situation was ideal, then the
models would probably show more accuracy, and models
such as the Bertalanffy model would not have been rejected
due to sample size bias. However, since the Gompertz and
Linear models had a low AICc for this data set and had limited
known bias under the circumstances of this experiment, we
can still assume that these models still provide the best t for
homogeneous cell populations, while exponential and logistic
models might be a better t for heterogeneous cell populations
under natural circumstances. For procedural improvements,
the single-clone procedure itself was performed under
proper circumstances and conditions, indicating that there
was no bias with the experimental data set. In terms of data
application and parameter optimization, proper methods
and functions were utilized for each model. Apart from the
minimal amount of data points, each function was applied
with the proper method as well. For future research, using a
much larger sample size both in terms of number of single-
clone cell populations and time data points collected would
be recommended, as well as the utilization of the VcaP cell
line as it provides for a stable homogeneous cell population.
By being aware of the differences in the growth between
heterogeneous and homogeneous cell populations, as well
as the growth functions that model each, cancer treatment
research can be taken to the next level as scientists would
be able to accurately predict the growth of tumors prior
to their spread and not be limited by time or material. The
ndings of this research provide useful knowledge about
the development of homogeneous cancer cell populations;
however, they still leave room for additional research since
not all cancer cell lines and growth functions were utilized.
JUNE 2020 | VOL 3 | 6Journal of Emerging Investigators • www.emerginginvestigators.org
MATERIALS AND METHODS
A VcaP prostate cancer cell line was used to create
a homogeneous single clone population, and population
counts were recorded weekly over a period of six weeks.
Afterward, the raw data was applied to ve different tumor
growth functions with the goal of providing the best t to the
population growth.
Single Clone and Cell Counting Procedure
Through the process of the single clone procedure cells
were cultured in RPMI-1640 supplemented with 10% FBS,
1% Penicillin/Streptomycin and 1% L-glutamine. Phosphate-
buffered saline, and trypsin were utilized to clean holding
asks and detach cells for counting. The cell clone procedure
was performed as follows. Growth media was aspirated out
of the original ask holding the Vcap cells, and 10 mL of PBS
was added to the ask. The PBS was aspirated, and one mL
of trypsin was added to the ask in order to detach the cell
from the ask wall. After three minutes, 5 mL of PBS was
pipetted into the ask in an up and down matter to neutralize
the trypsin creating a 6 mL cell solution within the ask. Three
mL was taken from this cell solution and put into a test tube.
The test tube was then inserted into a centrifuge and spun at
194 RCF for a total of 5 minutes in order to pellet the cells.
After the cell was spun down fully, the solution above the cell
pellet was aspirated out of the tube to just leave the pellet.
10 mL of PBS was pipetted into the tube in order to dilute
the cell pellet. A 20 µL sample of test tube solution was then
inserted into and counted using the hemocytometer. After a
cell density and dilution factor was calculated for the sample,
the test tube cell solution was diluted to the ratio of 100 cell/
mL and pipetted into two 96-well plates. The plates were then
studied under the microscope, and wells observed to only
have one cell present were identied and marked. The plates
were put in the incubator to rest. After one week, the plates
were studied again and a single well (out of the ones initially
marked) was chosen to continue the rest of the experiment.
Only one well was chosen because in this experiment only
one of the wells marked held a healthy cell population at the
rst weekly check. The cells of this well were transferred to a
bigger ask every two weeks, so the cell population was not
spatially limited in terms of cell density. Cell population counts
were taken weekly.
A hemocytometer was used to take all cell population
counts as well as provide the dilution factor for the initial single-
clone process. Ink-counting was also utilized throughout this
study for population count validation. The procedure for using
the hemocytometer was as follows. First, 10 µl was taken
from the cell suspension (post-trypsin) and pipetted into a
small test tube. Ten µL of the solution was pipetted into each
side of the hemocytometer plate. The plate was placed under
the microscope and the number of cells in each of the four
corner grids were counted and the hemocytometer formula
for measured cell density was imposed to determine the initial
dilution factor [8]. The formulas for measured cell density and
initial dilution factor are given by:
After performing this formula for each side of the
hemocytometer, the results from both sides were averaged
[8]. For the initial cloning procedure, the cell was diluted
to one-hundred cell per mL (or 1 cell per 100 uL). For the
periodic cell counts after the cloning was completed, just the
measured cell density (cell population count) formula was
used and the result was recorded after being scaled to the
proper volume. In the sample used, the initial dilution factor
was 480 mL, meaning that in order to achieve 100 cell per mL,
480 mL needed to be added to the sample. To do this, only
1 mL of the initial 10 mL sample was taken, and 48 mL was
added to that sample in order to reach the desired dilution
of 100 cells per mL. For the periodic cell counts the dilution
factor in the measured cell density formula was equal to one
as the sample was not diluted, for the initial cloning procedure
dilution calculation, it was equivalent to 10 as the cell pellet
was diluted by 10 mL.
The hemocytometer was used to count the cell population
of the VcaP single-cell population on a weekly basis for a six-
week period. Counts were taken only during transfers of the
population to larger plates and with small amounts of trypsin in
order to limit potential confounding variables. Over the course
of this experiment, the VcaP population was transferred four
times and trypsinized for a total of seven. As the cell counts
for the VcaP cell line were gathered over the experimental
period, a graph was created to visually display the growth of
the homogeneous cell population.
Throughout the course of the study, continued contact
was made with the primary investigator, the lab manager,
and the lab attendant of the research lab that housed the cell
population.
Mathematical Modeling
After population counts were collected over a period of
six weeks, the data was applied to ve mathematical growth
functions commonly used to model tumor growth in the human
body: the Exponential model, the Gompertz model, the
Logistic model, the Linear model, and the Bertalanffy model.
Each function was applied to the data provided by the VcaP
single clone cell population via optimization through Solver
(Frontline Systems, Google Sheets), and the parameters for
each differential equation was generated with the objective of
a minimum sum of squared residuals given by:
with y
i
representing the experimental data points, and y
representing the corresponding data points on the model [3].
After the parameters were tted the normalized mean square
error (NMSE) and Aikake’s information criterion (AICc) were
JUNE 2020 | VOL 3 | 7Journal of Emerging Investigators • www.emerginginvestigators.org
also computed using Google Sheets. These are given by:
with K being the number of parameters and n being the
number of data points. Since models have a different number
of parameters, the AICc was used to correct for smaller
sample sizes as well as eliminate potential bias having to
do with parameters. It is known that models with a higher
number of free parameters will ultimately be able to t the
data better than those with fewer parameters. A higher AICc
in comparison to the other models would signify that there
is a potential bias in correlation to the data for that specic
model. Models with a NMSE and/or AICc relatively close to
zero were deemed to be the best t. AICc is also meant to be
interpreted in terms of absolute value, so positive or negative
values hold no difference [3].
ACKNOWLEDGMENTS
Visuals were provided by Google Sheets. Data
generation was provided by Solver by Frontline Systems. Data
collection was made available by the Morgan/Palapattu Lab
of the University of Michigan Cancer Center. A special thank
you to Dr. Todd Morgan, Dr, Alexander Zaslavsky, and Ms.
Xiyu Cao for project assistance through the duration of the
data collection, as well as overall project aid and guidance.
Another special thank you to Mr. Joe Rasmus, and Dr. Ed
Robinson for providing thoughtful and helpful input on the
overall manuscript.
REFERENCES
1. Hanahan, Douglas, and Robert A. Weinberg. “Hallmarks
of Cancer: The Next Generation.” Cell, vol. 144, no. 5, 13
Feb. 2011, pp. 646–674., doi:10.1016/j.cell.2011.02.013.
2. “Key Statistics for Prostate Cancer: Prostate Cancer
Facts.” American Cancer Society, American Cancer
Society, Inc, 2020, www.cancer.org/cancer/prostate-
cancer/about/key-statistics.html.
3. Murphy, Hope, et al. “Differences in Predictions of ODE
Models of Tumor Growth: a Cautionary Example.” BMC
Cancer, vol. 16, no. 1, 2016, doi:10.1186/s12885-016-
2164-x.
4. Pavillon, Nicolas, and Nicholas I. Smith. “Immune Cell
Type, Cell Activation, and Single Cell Heterogeneity
Revealed by Label-Free Optical Methods.” Scientic
Reports, vol. 9, no. 1, 2019, doi:10.1038/s41598-019-
53428-3.
5. Talkington, Anne, and Rick Durrett. “Estimating Tumor
Growth Rates In Vivo.” Bulletin of Mathematical Biology,
vol. 77, no. 10, 2015, pp. 1934–1954., doi:10.1007/
s115 3 8 - 015 - 0110 - 8.
6. Gerlee, P. “The Model Muddle: In Search of Tumor
Growth Laws.” Cancer Research, vol. 73, no. 8, July
2013, pp. 2407–2411., doi:10.1158/0008-5472.can-12-
4355.
7. Crean, Daniel, and Emily Jones. “Mathematical Models
for Cancer Growth.” Chemotherapy for Cancer
Treatment RSS, chemoth.com/tumorgrowth.
8. Samuel, et al. “Hemocytometer Calculation”
Hemocytometer, 21 Feb. 2017, www.hemocytometer.org/
hemocytometer-calculation/.
9. Wang, Jue. “Modeling Cancer Growth with Differential
Equations.” STUDENT VERSION: Modeling Cancer
Growth with Differential Equations, SIMIODE, 2008, pp.
45–48.
10. Hindeya, Bayru Haftu. “Application of Growth Functions
to Describe the Dynamics of Avascular Tumor in Human
Body.” Applied and Computational Mathematics, vol. 5,
no. 2, 2016, p. 83.,doi:10.11648/j.acm.20160502.18.
11. Li, Fengzhi. “Every Single Cell Clones from Cancer Cell
Lines Growing Tumors in Vivo May Not Invalidate the
Cancer Stem Cell Concept.” Molecules and Cells , vol.
27, no. 4, 2009, pp.491–492., doi:10.1007/s10059-009-
0056-5.
Article submitted: June 20, 2020
Article accepted: July 27, 2020
Article published: June 4, 2021
Copyright: © 2021 Reddy et al. All JEI articles are distributed
under the attribution non-commercial, no derivative license
(http://creativecommons.org/licenses/by-nc-nd/3.0/). This
means that anyone is free to share, copy and distribute an
unaltered article for non-commercial purposes provided the
original author and source is credited.
