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Tumor-immune agent-based modeling: drawing insights from learned spaces
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Content
Tumor-immune agent-based modeling: drawing insights from learned spaces
by
Colin Cess
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
BIOMEDICAL ENGINEERING
May 2023
ii
Dedication
I would like to dedicate this dissertation to everyone who helped make it possible.
iii
Acknowledgements
Throughout my time in graduate school, I was fortunate to have the support of many people.
I have never been one for words and I refuse to start now, so I will keep this brief. First and
foremost is my professor, Dr. Stacey Finley, for helping me develop into the researcher that
I am today. Second is my parents, who taught me to truly appreciate the value of an
education. Third is my lab mates, for helpful discussions throughout the years. Lastly, my
friends, for providing needed distractions.
iv
Table of Contents
Dedication .................................................................................................................................................................... ii
Acknowledgements ................................................................................................................................................ iii
List of Tables .............................................................................................................................................................. vi
List of Figures .......................................................................................................................................................... vii
Abbreviations ............................................................................................................................................................ ix
Abstract .................................................................................................................................................................... x
Chapter 1: Introduction ......................................................................................................................................... 1
References ........................................................................................................................................................... 5
Chapter 2: Computational model development .......................................................................................... 9
Introduction ....................................................................................................................................................... 9
Methods ............................................................................................................................................................. 10
Results ............................................................................................................................................................... 16
Discussion ........................................................................................................................................................ 27
References ........................................................................................................................................................ 32
Chapter 3: Representation learning for quantitative comparisons between model
simulations and image data ............................................................................................................................... 35
Introduction ......................................................................................................................................................... 35
Chapter 3a: Representation learning for a generalized, quantitative comparison of
complex model outputs .................................................................................................................................. 36
Introduction .................................................................................................................................................... 36
Results ............................................................................................................................................................... 37
Discussion ........................................................................................................................................................ 49
Methods ............................................................................................................................................................. 51
References ........................................................................................................................................................ 54
Chapter 3b: Calibrating agent-based models to tumor images using representation
learning .................................................................................................................................................................. 60
Introduction .................................................................................................................................................... 60
Methods ............................................................................................................................................................. 61
Results ............................................................................................................................................................... 67
Discussion ........................................................................................................................................................ 71
v
References ........................................................................................................................................................ 74
Conclusion ............................................................................................................................................................ 76
Chapter 4: Model parameter estimation and analysis methods ........................................................ 77
Introduction .................................................................................................................................................... 77
Methods ............................................................................................................................................................. 78
Results ............................................................................................................................................................... 85
Discussion ........................................................................................................................................................ 96
References ........................................................................................................................................................ 98
Chapter 5: Conclusion .......................................................................................................................................... 99
References ..................................................................................................................................................... 101
Appendices ............................................................................................................................................................ 102
Data-driven analysis of a mechanistic model of CAR T cell signaling predicts effects of
cell-to-cell heterogeneity ............................................................................................................................ 103
Multi-scale modeling of macrophage – T cell interactions within the tumor
microenvironment ......................................................................................................................................... 112
Multiscale modeling of tumor adaptation and invasion follow anti-angiogenic therapy 147
vi
List of Tables
Table 2.1: Force parameters………………………………………………………………………………....... 28
Table 2.2: Cancer cell parameters…………………………………………………………………………… 29
Table 2.3: Macrophage parameters………………………………………………………………………… 29
Table 2.4: CD4 parameters…………………………………………………………………………………...... 30
Table 2.5: CD8 parameters…………………………………………………………………………………...... 30
Table 2.6: Environment parameters……………………………………………………………………...... 31
Table 3b.1: Top 10 best fits from fitting an ABM to model-generated data for
Example 1………………………………………………………………………………………………………….......
69
Table 3b.2: Top 10 best fits from fitting an ABM to tumor imaging data for
Example 2……………………………………………………………………………………………………………...
71
Table 4.1: Fluorescence markers that correspond to each cell type………………………….. 81
Table 4.2: Cell ratios for each cell type. Used to constrain parameter search…………….. 86
Table 4.3: Estimated mean and coefficient of variation for each parameter…………….... 86
vii
List of Figures
Figure 2.1: Schematic of the simulation loop…………………………………………………………… 16
Figure 2.2: Colors for each cell type in the figures showing spatial layouts……………….. 17
Figure 2.3: Time-courses for simulations with only cancer cells and CD8 cells…………. 18
Figure 2.4: Spatial layout for one recruitment rate of CD8 cells, comparing PD-L1- and
PD-L1+ tumors………………………………………………………………………………………………………
19
Figure 2.5: Time-courses showing impact of M2 macrophage inhibition………………….. 20
Figure 2.6: Spatial layout for one recruitment rate of CD8 cells, comparing the impact
of M2 inhibition……………………………………………………………………………………….....................
21
Figure 2.7: Spatial layouts for varying rates of CD8 recruitment and CD4
differentiation………………………………………………………………………………………………………..
22
Figure 2.8: Time-courses for extreme cases of CD8 recruitment and CD4
differentiation………………………………………………………………………………………………………..
23
Figure 2.9: Fractions of each cell type for extreme values of CD8 recruitment and CD4
differentiation………………………………………………………………………………………………………..
24
Figure 2.10: Spatial layouts for varying rates of CD4 recruitment and CD4
differentiation………………………………………………………………………………………………………..
25
Figure 2.11: Time-courses for extreme cases of CD4 recruitment and CD4
differentiation………………………………………………………………………………………………………..
26
Figure 2.12: Fractions of each cell type for extreme values of CD4 recruitment and
CD4 differentiation………………………………………………………………………………………………...
27
Figure 3a.1: Schematic displaying how two simulations are compared via projection
to learned space……………………………………………………………………………………………………..
38
Figure 3a.2: Test projected datasets……………………………………………………………………….. 40
Figure 3a.3: Consensus scores for the three test models………………………………………….. 42
Figure 3a.4: Change in model state when changing lower bound for test model 1…….. 43
Figure 3a.5: Knockout results from the flux model………………………………………………….. 44
Figure 3a.6: Shifting the value of a single parameter for test model 2………………………. 45
Figure 3a.7: Local sensitivity analysis for test model 2…………………………………………….. 46
Figure 3a.8: Shifting the value of a single parameter for test model 3………………………. 47
Figure 3a.9: Clustering Monte Carlo simulations for test model 3…………………………….. 49
Figure 3b.1: Schematic of using the distance in projected space as an objective
function…………………………………………………………………………………………………………………
63
Figure 3b.2: Schematic of data-processing……………………………………………………………… 66
Figure 3b.3: Schematic of initial parameter range estimation………………………………….. 67
Figure 3b.4: Fitting results for Example 1……………………………………………………………….. 69
Figure 3b.5: Fitting results for Example 2……………………………………………………………….. 72
Figure 4.1: Schematic of Chapter 4………………………………………………………………………… 80
Figure 4.2: Example fluorescence tumor image……………………………………………………….. 81
Figure 4.3: Schematic showing that adjacent time points should be closer to each
other in projected space…………………………………………………………………………………………
85
viii
Figure 4.4: Processed versions of each fluorescence image……………………………………… 88
Figure 4.5: Box plots for each estimated parameter…………………………………………………. 89
Figure 4.6: Top 5 best fits for one of the images………………………………………………………. 92
Figure 4.7: Average training loss vs training epoch (training step) for the neural
networks……………………………………………………………………………………………………………….
93
Figure 4.8: Projection for a single simulation………………………………………………………….. 95
Figure 4.9 Projections for all 700 simulations…………………………………………………………. 95
Figure 4.10: Projections separated by clustering the final time point……………………….. 96
Figure 4.11: Bar graphs showing mean parameter values for each cluster……………….. 97
Figure 4.12: Projections of the final time points, colored by cell counts……………………. 98
ix
Abbreviations
ABM – agent-based model
TAM – tumor-associated macrophage
M0 – naïve macrophage
M1 – macrophage in the M1 state
M2 – macrophage in the M2 state
FasL – Fas ligand
ODE – ordinary differential equation
PDE – partial differential equation
IFN-gamma – interferon gamma
PD-L1 – programmed death-ligand 1
Treg – regulatory T cell
IL-2 – interleukin 2
IL-10 – interleukin 10
IL-12 – interleukin 12
TGF-beta – transforming growth factor beta
GA – genetic algorithm
HER2 – human epidermal growth factor receptor 2
DAPI - 4′,6-diamidino-2-phenylindole
CK5 – keratin 5
Epcam - Epithelial cell adhesion molecule
FOXP3 – forkhead box P3
F4/80 - EGF-like module-containing mucin-like hormone receptor-like 1
NN – neural network
x
Abstract
The interactions between a tumor and the immune system have long been known to have
important clinical impacts. However, immunotherapies are not always successful, due to the
complex network of interactions between various immune cells and cancer cells. One way to
study these interactions is with computational modeling, allowing researchers to simulate
how the tumor behaves under different conditions. A common way of modeling tumor
interactions is with agent-based models, where cells are modeled as discrete individuals that
interact with each other in a rule-based manner, capturing complex spatial structures and
population dynamics. These models, however, suffer from difficulties in parameter
estimation, due to the difficulties in comparing simulations to tumor images in a quantitative
way. Additionally, besides extracting simple spatial metrics, it is difficult to compare model
simulations to each other. Therefore, in this thesis, I present a novel application of
representation learning, using neural networks to project images to low-dimensional points,
as a way of quantitatively comparing agent-based model simulations to tumor images
without the need to manually calculate complex metrics. This comparison facilitates a
rigorous parameter estimation for these types of models, better ensuring the accuracy of
parameter values. Using tumor images from a mouse study, I show how parameters can be
estimated for an agent-based model containing several different immune cell types, along
with how representation learning can be used to analyze model simulations. This work
provides a foundation for methods that can be used to bridge the gap between model
simulations and experimental images.
1
Chapter 1: Introduction
1.1 Tumor-immune ecosystem
Cells, the basic building block of multicellular organisms, take on a wide range of functions
that, together, sustain the organism’s life. Much like social creatures, it is through
cooperation and different roles of the parts that the whole can survive. However, sometimes
cells can malfunction and, instead of undergoing proper self-directed death, they become
harmful to the organism. The dysfunctional cells undergo uncontrolled growth and spread
to distal locations, eventually leading to the organism’s death. This uncontrolled growth is
known as a cancerous tumor, and tumors have become prevalent in human patients. Rather
than growing in isolation, a tumor has extensive interactions with the rest of the body.
1,2
One
key interaction is with the immune system, and it is these interactions that form what is
termed the “tumor-immune microenvironment”.
3–7
The tumor-immune microenvironment can be divided into two components: cancer cells and
the immune system. Cancer cells differ from normal, healthy cells in a few key areas. The
main difference is that under normal conditions, cell growth and division is constrained, with
cells growing, dividing, and dying (via apoptosis) in response to certain signals, maintaining
homeostasis within the tissue. Cancer cells ignore these signals, achieving sustained and
rapid proliferation and often failing to undergo self-directed apoptosis. What separates a
cancerous, or malignant, growth from a benign (generally harmless) tumor is that cancer
cells divide at a more rapid pace and proceed to invade into the surrounding tissue, later
metastasizing to distal sites. Cancerous tumors exert further changes to their local
environment, co-opting normally regulated processes in order to sustain their uncontrolled
growth. For example, angiogenesis, the formation of new blood vessels, is normally reserved
for processes such as wound healing. However, tumors often secrete pro-angiogenic factors
in order to increase their blood, and thus nutrient, supply to sustain their growth.
1,2
The other facet of the tumor-immune microenvironment is the immune system. In the most
basic of terms, the role of the immune system in the body is to differentiate between “self”
and “not-self,” eliminating things that fall into the latter category, often viruses and
bacteria.
8–10
A key feature of cancer cells is that they mutate, expressing what are termed
neoantigens, thus allowing them to be detected by the immune system as not-self.
11–13
The
immune system is made up of two main parts: innate
14
and adaptive
15
, with the former being
a rapid, first line of defense, while the latter is a slower, but specific response to a threat.
Combined, there are many types of immune cells that comprise the innate and adaptive
immune responses. I, however, will focus on two types of cells that are present in large
2
numbers in the tumor and play significant roles in tumor development and the anti-tumor
immune response: macrophages and T cells.
Macrophages are part of the innate immune system and have many roles within the tumor
microenvironment. Tumor-associated macrophages (TAMs) take on a range of states but are
often thought of as M1-like or M2-like.
16
Although this is a major simplification, it makes it
easy to understand their different functions. M1 macrophages are considered to be
“classically” activated, and they exhibit anti-tumor behavior through several direct and
indirect mechanisms. Directly, M1 macrophages secrete nitric oxide and other reactive
oxygen species, which have direct tumor killing effects.
16–21
They can also kill tumor cells via
antibody-dependent cell-mediated cytotoxicity
16
. Indirectly, M1 macrophages can promote
the adaptive immune response through antigen presentation and the secretion of cytokines,
which promote anti-tumor T cell functions.
17–23
Overall, a high presence of M1-like TAMs is
associated with higher T cell infiltration and function and a more clinically favorable
microenvironment.
M2 macrophages, “alternatively” activated, exhibit a wound-healing state and are heavily
tumor-promoting, both increasing cancer cell survival and suppressing T cell responses.
Most TAMs exist in an M2-like state. M2 macrophages are known to promote angiogenesis,
increasing blood supply to the growing tumor. Additionally, these cells secrete factors that
promote tumor growth. M2 macrophages also increase tumor metastasis by breaking down
the extracellular matrix and facilitating the epithelial-to-mesenchymal transition of cancer
cells.
16,17,19–25
Besides promoting the functions of cancer cells, M2 macrophages inhibit the
cytotoxic response of the immune system. Cytokine secretion prompts more macrophages
to differentiate to the M2 state, continuing their tumor-promoting functions. M2
macrophages secrete cytokines that reduce CD8 T cell cytotoxicity and proliferation and
prompt CD4 T cells to differentiate to a regulatory state. M2 macrophages also express
immune checkpoints, which directly suppress CD8 T cells, preventing them from killing
cancer cells.
16–19,21,24–27
Studies have found that the elimination of TAMs or inducing an M1-
like state leads to greater survival.
18,23,24,28,29
The other main type of immune cell in the tumor microenvironment is the T cell, a part of
the adaptive immune response. T cells can be broken down into two main groups: CD8
cytotoxic T cells and CD4 helper T cells. CD8 T cells are the main killing arm of the immune
response to a tumor, with a higher number of CD8 cells being a favorable prognostic marker
for immunotherapy.
30–33
These cells learn to recognize tumor neoantigens and exert a direct
cytotoxic effect to kill cancer cells via two pathways: granule exocytosis and Fas ligand
(FasL). Granule exocytosis involves first the secretion of perforin, which punctures holes in
the cancer cell membrane, followed by the secretion of granzymes, which enter these holes
and induce apoptosis. FasL is secreted by T cells and binds to its receptor on cancer cells,
3
prompting intracellular signaling that induces apoptosis.
33
Additionally, CD8 cells secrete
IFN-gamma, which promotes macrophage differentiation to the M1 state.
31
CD4 T cells, on the other hand, generally act to augment CD8 function. There are many known
CD4 subtypes that exert various effects, including Th1, Th2, Th17, and Treg.
34
Here, I will
discuss the CD4 subtypes simply in terms of a helper state and a regulatory state. As the
name suggests, the former acts to help CD8 function, promoting CD8 proliferation, survival,
and cytotoxicity. This is done via the secretion of IL-2, which increases CD8 proliferation,
survival, and production of granzyme B.
35,36
In fact, studies have found that IL-2 from CD4
helper cells is necessary for a successful antitumor CD8 response.
37
CD4 helper cells also
secrete IFN-gamma, which both promotes CD8 recruitment and macrophage differentiation
to the M1 state.
34
Regulatory cells instead act to limit CD8 function. This is part of a normal immune response,
which functions to prevent autoimmunity. However, regulatory T cell recruitment is
increased to the tumor site, along with conversion of CD4 helper cells to the regulatory
state.
38
These cells suppress the effects of IL-2 on CD8 T cells by acting as a sink, taking up
IL-2 so that it cannot be used by CD8 cells.
37
Additionally, T regulatory cells express PD-L1
and CTLA-4, which are immune checkpoints that suppress CD8 cells.
39
These cells also
secrete cytokines such as TGF-beta, which inhibit CD8 function. Lastly, regulatory T cells
promote the differentiation of macrophages to the M2 state via the secretion of IL-10, which
can go on to influence the conversion of more CD4 cells into the regulatory state.
38–40
1.2 Tools for studying the tumor-immune ecosystem
To explore how tumors interact with the immune system, three domains of research exist:
in vivo, in vitro, and in silico. While none of these domains faithfully reproduce tumors in
human patients, each has certain advantages. In vivo experiments take place in a living
organism, usually mice, and therefore provide the most realistic tumor behavior. The main
disadvantage of these models is that data collection is much harder, with the main temporal
data that is collected being simply tumor volume over time.
41–43
While biopsies can be taken
over time, the bulk of the tumor is only analyzed at the final time point, when the tumor is
removed from the organism.
44
Additionally, because of the systemic response, specific
interactions become difficult to analyze due to the complex nature of the immune response.
45
The second domain is in vitro, where cells are grown in cell culture. This allows for examining
specific interactions between cell types or responses to various stimulants in a controlled
setting, facilitating the understanding of many different cell properties.
45
These studies are
performed much more often than in vivo experiments because they are generally much faster
and less expensive to perform. Additionally, specific interactions can be studied without the
muddling influence of other interactions.
45
However, these experiments examine
4
interactions in a very controlled manner and in isolation from other interactions. Therefore,
some cellular behaviors that are observed in vitro may not occur in vivo due to the presence
of other interactions. Furthermore, while some in vitro systems can replicate basic spatial
behavior, these studies cannot truly replicate the complex structure of the tumor or provide
a systemic response.
45
Because of these limitations, insights generated from these
experiments do not always translate well to in vivo biology.
46
The last domain is in silico, meaning computer models. These models often leverage
information from both in vivo and in vitro experiments in order to bridge the gap between
them and make predictions about how a tumor would behave under different conditions.
47,48
The main advantages of in silico models are that they are cheaper and faster to run than in
vivo and in vitro experiments, and the models can make predictions about interactions that
are difficult to measure experimentally. Generally, experimental data is used to develop an
in silico model, whose predictions are then used to inform further experiments. The main
disadvantage of in silico modeling is that it cannot fully account for all aspects of tumor
biology, with some features of the system modeled in great detail and others treated very
simply. As with in vivo and in vitro systems, this disadvantage is an unfortunate necessity in
order to use the advantages.
47,48
1.3 Leveraging in silico modeling and computational analyses to study tumor growth
This thesis rests in the domain of in silico modeling, aiming to develop novel methods to
better model the tumor-immune microenvironment and to better analyze these models.
Additionally, this thesis aims to develop methods for bridging the gap between in silico
models and in vivo experiments to better inform model development. In Chapter 2, I present
an agent-based model of interactions between cancer cells, macrophages, CD4 T cells, and
CD8 T cells, creating a generic model that can be adjusted for many different studies.
Additionally, this model makes key developments to improve computational efficiency while
still maintaining a high degree of biological accuracy. In Chapter 3, I describe a novel
application of representation learning as a way of both analyzing computational models
(agnostic to the type of model being analyzed) and as a way of quantitatively comparing
agent-based models directly to tumor images for use in parameter estimation. Finally, in
Chapter 4, I apply the methods developed in Chapter 3 to estimate the parameters for the
model developed in Chapter 2 using fluorescent tumor images taken from a mouse study. I
then apply representation learning to analyze the calibrated model. Together, this work
provides a bridge between computational models that replicate spatial behavior and tumor
images, paving the way for models that are truly informed from imaging data and can thus
better replicate specific tumor behavior.
5
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9
Chapter 2: Computational model development
Introduction
To properly investigate a system computationally, several considerations must be made first
in determining the model framework and then in the implementation of that framework. In
the field of mathematical oncology, three main types of models exist: ordinary differential
equations (ODE), partial differential equations (PDE), and agent-based models (ABM).
1
Each
has distinct advantages and disadvantages, with their employment based on modeling goals.
ODE models are often the simplest type, describing how the sizes of cell populations, and
sometimes the overall concentrations of diffusible factors, change with time. These models
are fast to simulate and relatively easy to analyze, either mathematically or numerically.
Their main disadvantage is that each cell population is treated as a homogeneous population
and that the tumor is well-mixed. PDEs offer an improvement by including the spatial
dimension, particularly for predicting the distributions of diffusible factors; however, PDE
models often retain the homogenous phenotype assumption for cells. These models are
much more computationally expensive to simulate, thus leading them to have fewer
equations, simplifying the model.
ABMs are unique in relation to the previous two types of models in that instead of using
equations to describe cellular behavior, ABMS model cells as discrete entities that interact
with each other in a rule-based manner. It is from relatively simple interactions between
cells that complex population dynamics can be simulated, which are often too complex to be
captured with equation-based models. In tumor modeling, ABMs often include the spatial
domain, either in two or three dimensions. More importantly, by modeling cells discretely,
heterogeneities between cells of the same type can be captured and more realistic spatial
interactions are modeled.
2
Many tumor ABMs have been developed to examine different aspects of the tumor
microenvironment, ranging from angiogenesis to interactions with immune cells to effects
of extracellular matrix fibers. These models contain a wide range of detail and ways of
implementing similar interactions. Amongst models dealing with tumor-immune
interactions, there is still a wide variety of models, due to the complex nature of the immune
system, which contains many different cell types. Some models focus only on a specific
interaction between cancer cells and one type of immune cell, while other models focus on
broader interactions between different types of immune cells and how those interactions
impact tumor growth.
3
The goal of this chapter is to develop a model that accounts for the complex interactions
between three major immune cell types present in the microenvironment: macrophages,
10
CD4 T cells, and CD8 T cells. These cells have very complex functions, but can be described
in brief as follows. Macrophages take on a range of different states, but are often described
simply as being in an M1 or M2 state, which have opposing functions. M1 is the classical pro-
immune state, acting to eliminate a threat, in this case the tumor, through a combination of
direct killing and promotion of a cytotoxic T cell response.
4–7
M2, the alternative state, is
thought of as a wound-healing state, promoting tumor survival and suppressing other
immune responses.
6,8
CD4 T cells have a wide variety of known subtypes, however, for ease,
they can be thought of as taking a helper state, which promotes the immune response, or a
regulatory state, which suppresses the immune response.
9,10
Lastly, CD8 T cells are the main
cytotoxic cell for tumor elimination, acting to identify and kill cancer cells. However, they are
often suppressed in the tumor microenvironment.
11,12
This model aims to phenotypically capture the interactions between these cells and provide
a generic model of the tumor-immune microenvironment, where specific interactions can
either be removed or made more detailed, based on the goals of future studies. Specifically,
I aim to simplify certain phenomena in order to reduce the number of parameters that need
to be estimated and to improve the computational efficiency of the model.
Methods
Model Overview
This model is based on my previous work that examined the interactions between cancer
cells, T cells, and macrophages.
13
Here, besides increasing the biophysical detail of the
modeling framework, as I did in another previous model,
14
the model is expanded to include
additional cell types and interactions. Besides providing a framework for detailed
simulations of these cells within the tumor microenvironment, this model introduces
multiple novel representations of complex phenomena in simplified forms, allowing the
model to capture complex behaviors while reducing the number of parameters and
improving computational efficiency.
Cell forces
As in a previous model that I developed, cells are modeled using a center-based approach
that represents cells as a point and a radius, using these to calculate physical forces between
cells.
14,15
This approach provides a greater level of biological realism than a cellular-
automata method, which constrains cells to a regular grid. At the same time, the center-based
approach is not as complex or computationally expensive as modeling frameworks that
account for more detailed cell shapes, such as a vertex model.
15
Forces between cells are
calculated via the following equation.
11
!
!"
(#)
=
⎩
⎪
⎪
⎨
⎪
⎪
⎧
µ
!"
+
!"
(#),-
!"
(#)log11+
4,
!"
(#)4−+
!"
(#)
+
!"
(#)
6,89: 4,
!"
(#)4<+
!"
(#),
µ
!"
=4,
!"
(#)4−+
!"
(#)>,-
!"
(#)exp1−B
#
4,
!"
(#)4−+
!"
(#)
+
!"
(#)
6,89: +
!"
(#)≤4,
!"
(#)4≤:
$%&
,
0,89: 4,
!"
(#)4>:
$%&
In this equation, µ
!"
is the spring constant, ,
!"
(#) is the vector between cells i and j at time t,
,-
!"
(#) is the unit vector, B
'
is the decay of the attractive force, +
!"
(#) is the sum of the radii of
cells i and j, and ,
$%&
is the maximum interaction distance. For all cells, I assume the same µ
and B
'
. This equation is solved numerically for each cell, as displayed in the following
equation.
,
!
(#+∆#)= ,
!
(#)+
∆#
G
H !
!"
(#)
"∈)
!
(+)
In this equation, G is the drag coefficient and I
!
is the set of cells within ,
$%&
of cell i. To
preserve accuracy, this equation must be solved at a small timestep (J#), which I set as 0.005
hours.
When simulating the model, it is computationally expensive to determine I
!
(#) for each cell,
as the distances between each cell need to be calculated. To improve computational
efficiency, I use a larger simulation timestep of 1 hour. At the beginning of each simulation
step, I define a cell neighborhood for each cell. This is a list of cells with a 10,
$%&
distance of
cell i. I then determine I
!
(#) only from cells in that neighborhood and proceed to solve the
above equation at the small timestep. This hastens computational time while preserving the
accuracy of the force calculations.
Proliferation and cell death
To simplify proliferation and cell death, these processes are modeled as probabilities of
occurring at each time point. In this model, only cancer cells and CD8 T cells can undergo
proliferation. When a cell proliferates, the daughter cell is placed the distance of the cell
radius away from the mother cell at a randomly selected angle. When a cell dies, it is removed
from the simulation.
Cell proliferation is also influenced by the physical presence of other cells, making the
assumption that cell proliferation is inhibited by excess physical forces. If the total overlap
with other cells is above a threshold, a cell’s proliferation is inhibited until this overlap
decreases.
14
12
Immune cell recruitment
Immune cells (macrophages and T cells) are recruited to the environment at a rate
proportional to the number of cancer cells. They are recruited at a random angle around the
tumor center, under the assumption that the area surrounding the tumor is well
vascularized. When recruited, these cells are placed a random distance away from the tumor
edge, sampled from a uniform distribution.
Immune cells migration
Immune cells migrate towards the tumor center, mimicking chemotaxis due to tumor-
secreted chemokines.
16–18
Cells migrate with a specified amount of bias, which is modeled as
the probability at each migration step of moving in the direction of the tumor center versus
a random direction. This represents the impact of the extracellular matrix surrounding the
tumor, along with immune cell migration behaviors.
19–23
However, there are many barriers
to immune cell infiltration into the tumor, leading to three general tumor-immune states:
ignored (immune cells cannot detect the tumor), excluded (immune cells are restricted to
the outside of the tumor), and inflamed (immune cells can infiltrate the tumor).
24
Some of
these are due to interactions between immune cells, while others are due to physical barriers
of the tumor. To capture this phenomenon in a simplistic manner with as few additional
parameters as possible, I assume that both migration speed and bias decrease upon entry
into the tumor. By adjusting these decreases, this model can capture different phenotypic
tumor behaviors. Migration for all immune cell types is modeled in the same way, however
each type has its own bias and speed parameters.
Diffusion
Within the TME, cells can communicate via diffusible factors, such as cytokines.
18
These are
often modeled using PDEs, with cells acting as point sources/sinks.
25–27
These, however,
require a great amount of computational time to solve, especially as environment size and
the number of factors increases. Additionally, cells secrete several different cytokines, many
with overlapping effects. This complexity makes it difficult to model the explicit biological
effects of all of the cytokines. Often, the effects of these factors are modeled either as a
probability or a threshold value, instead of explicit modeling of downstream intracellular
signaling. Because of this, in some studies, researchers pooled different cytokines into
generic factors.
27
In other studies, researchers have replaced diffusible factors altogether
with a distance threshold, where an effect occurs if one cell is within a specified distance
from another cell.
28
Here, I extend the latter approach to better account for the gradient nature of diffusion and
the impact of multiple secreting cells. I refer to this as the “cell influence,” where the closer
cell i is to cytokine-secreting cell j, the greater the effect of cell j’s cytokines on cell i. This is
modeled by an exponential decay displayed in the following equations:
13
K
!"
(#)= exp (−L4,
!"
(#)4)
L = −log
-
(M
+.
)
0.693
,
+.
The use of an exponential decay is a reasonable approximation of the effects of diffusible
factors, as the response to a cytokine-secreting cell has been experimentally found to
resemble an exponential decay.
29,30
In these equations, 4,
!"
(#)4 is the distance between cell
centers, ,
+.
is the soft threshold for the maximum influence distance that can be thought of
as the diffusion limit, and M
+.
is the probability of an effect occurring at distance ,
+.
. By
setting M
+.
and ,
+.
, I can calculate L. The total influence on cell i is calculated via
K
!,0
(#)= 1− R 1−K
!"
(#)
"∈0(+)
In this equation, K
!,0
(#) is the total influence, from 0 to 1, on cell i from all cells of cell type S.
Thus, each cell records a separate influence for each cell type in the simulation. These
influences are used to determine downstream effects. These downstream effects are
probabilities, which are then multiplied by the relevant influence. Thus, the cell influence
acts as a scaling factor for the probabilities of effects occurring. If two cell types can lead to
the same effect (such as both cancer cells and M2 macrophages promoting M0 differentiation
into M2), their influences are combined as K(#)= 1−(1−K
0
"
)(1−K
0
#
).
Modeling this way means that the cloud of diffusible factors follows each cell as it moves
through the simulation environment. This follows the usual assumption that diffusion is
resolved on a much faster timescale than cellular processes.
Cancer cells
In this model, cancer cells have three main actions: proliferation, death, and expression of
PD-L1. Proliferation and death occur as described above. Cancer cells gain PD-L1 at a
probability proportional to the influence from CD4 helper and CD8 cells (via T cell-secreted
IFN-gamma, which promotes PD-L1 expression).
25,31
PD-L1 expression is represented as a
probability of suppressing CD8 cells upon direct contact.
25
After proliferation, the daughter
cells inherit the PD-L1 expression of the mother.
Macrophages
Macrophages enter the simulation in a naïve (M0) state and differentiate into either M1 (via
IFN-gamma secretion by CD4 helper and CD8 T cells) or M2 (via IL-4 and IL-10 secreted by
cancer cells and Treg cells).
6
Additionally, because macrophage state is plastic and
influenced by environmental conditions, macrophages have a probability of redifferentiating
at each timestep.
7
Following other modeling efforts, macrophages have a probability of
remaining naïve (static value) or differentiating into M1 or M2 (influenced by environmental
14
conditions), which are then scaled to sum to 1.
26
This is displayed in the following equations,
which utilize the cell influence described above.
T
1
= U
1,V8 W
1
0,V8 X9# W
1
T
2
= B
32
(1−(1−K
'45
)(1−K
'46
))
T
-
= B
3-
(1−Y1−K
3
#
Z(1−K
#%7#89
)=1−K
0
$%&
>)
These values are then divided by their sum to get the probability of differentiating into each
state.
M1 cells increase the killing capability of CD8 T cells (via cytokines such as IL-12)
6,32
and are
able to kill nearby cancer cells (via nitric oxide secretion).
5,6,8
M2 cells reduce the killing
capability of CD8 T cells and proliferation rate of CD8 T cells (via IL-10 and TGF-beta),
promote the differentiation of CD4 T cells into the regulatory state (via IL-10 and TGF-beta),
and promote M0 differentiation into the M2 state (via TGF-beta).
5,6,8
I model all of these
effects via the cell influence function described above. M2 cells also express PD-L1,
represented as a probability of suppressing a CD8 cell upon direct contact.
CD4 T cells
While CD4 T cells can take on a variety of states, for simplicity, I only model two: helper cells
and regulatory cells. In this model, CD4 cells enter the simulation in the helper state and can
be converted into regulatory cells based on influence from M2 macrophages and cancer
cells.
5,6,8,33
The probability of differentiation is proportional to the combined influence of M2
macrophages and cancer cells, as shown in the following equations.
K
:!;;
= 1−(1−K
3
#
)(1−K
#%7#89
)
T
:!;;
= T
:!;;
'
K
:!;;
In the helper state, CD4 cells promote the killing capability and proliferation of CD8 T cells
(via IL-2),
10,34,35
and promote M0 differentiation to the M1 state (via IFN-gamma).
9
As
regulatory cells, they express CTLA-4 (which has the same function as PD-L1),
36
decrease the
killing and proliferative capabilities of CD8 T cells (via acting as an IL-2 sink),
10
and promote
M0 differentiation into the M2 state (via IL-10).
33
CD8 T cell
CD8 T cells perform the main function of killing cancer cells. This is modeled as a probability
of killing a cancer cell upon direct contact. When a CD8 cell comes into contact with a PD-L1
expressing cell, the CD8 cell becomes suppressed based on the probability of PD-L1.
25
Suppressed CD8 cells lose all functions (including migration) besides cell death. CD8 killing
is influenced by the other immune cells, as shown in the following equations, where B
<#%=!7>
is the maximum factor of increase/decrease.
15
K
?@<
= 1−(1−K
3
"
)(1−K
0.
)
K
78>
= 1−(1−K
3
"
)(1−K
0
$%&
)
T
A!==
= T
A!==
'
(B
<#%=!7>
4
()*
B4
+%&
)
Additionally, CD8 proliferation is influenced by nearby cells. Studies have shown that CD8 T
cells require IL-2 from CD4 helper cells in order to proliferate, while Treg cells and M2
macrophages inhibit CD8 proliferation via acting as an IL-2 sink and suppressive cytokine
secretion.
5,8,10
CD8 proliferation rate is then calculated via the following equations:
K
?@<
= K
0.
K
78>
= 1−(1−K
3
#
)(1−K
0
$%&
)
T
'45?9@=
= T
'45?9@=
'
(K
?@<
−K
78>
)
I note that negative values for this equation are equivalent to a probability of zero.
Simulation Loop
A simulation proceeds as follows:
1. Immune cells are recruited to the environment
2. Cell neighborhoods are determined
3. Influence from diffusible factors is calculated and direct contact effects
(killing, PD-L1 suppression) are accounted for
4. Forces between cells are solved at a smaller timestep
5. Proliferation and spontaneous cell death
Simulation proceeds until either the tumor is eliminated or the maximum simulation time is
reached.
16
Figure 2.1: Schematic of the simulation loop.
Results
The purpose of this Chapter is to develop a general tumor-immune ABM, which I use in later
Chapters for analysis. Here, I display several sets of simulations to show that it captures
realistic tumor behaviors under different conditions. Simulations ran for 24 days (duration
of the mouse study in Chapter 4). Parameters were set based on the mouse study in Chapter
4. These parameter values are shown in tables at the end of this Chapter. I ran several sets
of simulations to display a variety of tumor behaviors, exploring some of the key interactions
in the model. However, this is not an exhaustive interrogation of all possible model
behaviors. For figures showing spatial layouts, Figure 2.2 shows the colors of each cell type.
17
Figure 2.2: Colors for each cell type in the figures showing spatial layouts.
CD8 only: impact of cancer PD-L1
The first set of simulations were performed with only cancer cells and CD8 T cells, comparing
the effects of CD8 recruitment rate and cancer cell PD-L1 expression. In Figure 2.3, I show
the effects of increasing CD8 recruitment for both PD-L1- and PD-L1+ tumors. As expected,
tumor growth is slowed, or even tumor regression, as CD8 recruitment increases. This is easy
to conceptualize based on model rules, as sufficient numbers of CD8 cells are needed to
overcome the cancer cell proliferation rate. With the addition of PD-L1, tumor elimination is
slowed, preventing the even tumor regression.
Figure 2.4 shows the final state for two simulations with the same CD8 recruitment rate, one
with PD-L1 and one without. Without PD-L1, the tumor is much smaller and there is
increased CD8 cell infiltration into the tumor. With PD-L1, CD8 cells are suppressed along
the outside, reducing their infiltration into the tumor, allowing the tumor to continue to
grow.
18
Figure 2.3: Time-courses for simulations with only cancer cells and CD8 cells. Tumors
were either PD-L1- or PD-L1+. Simulations were performed at varying rates of CD8 recruitment
(rates shown in the legend).
19
Figure 2.4: Spatial layout for one recruitment rate of CD8 cells, comparing PD-L1- to PD-
L1+ tumors.
All immune cells: impact of M1 macrophages
With the addition of all cells, and excluding cancer PD-L1 expression, I test model behavior
using the baseline parameter values versus a scenario where macrophages can only
differentiate into the M1 state. With the inclusion of all immune cells, interestingly, the tumor
growth rate is slightly slower than for simulations with the same CD8 recruitment rate and
no PD-L1 (Figure 2.5). While M1 macrophages can exert some killing effect, there do not
appear to be enough M1 macrophages to cause this much of a decrease in growth (Figure
2.5). It is instead likely due to the presence of large numbers of immune cells which cause
spatial suppression of tumor growth. Differentiation only to the M1 state does promote
tumor removal, as M1 macrophages promote CD8 function, even without affecting the
overall numbers of CD8 cells. With the inhibition of M2 macrophages, there is much more
CD8 infiltration into the tumor (Figure 2.6)
20
Figure 2.5: Time-courses showing the impact of M2 macrophage inhibition. Simulations
were performed at various recruitment rates of CD8.
21
Figure 2.6: Spatial layout for one recruitment rate of CD8 cells, comparing the impact of
M2 inhibition.
All immune cells: Varying CD8 recruitment and CD4 differentiation
In the model, CD4 cells promote CD8 function while T regulatory cells inhibit it. In order to
explore how these interactions impact tumor growth, I varied both CD8 recruitment rate and
CD4 differentiation to the regulatory state (Figure 2.7) with all immune cells present. As
more CD4 cells differentiate to the regulatory state, the tumor becomes slightly larger.
However, visually, CD8 recruitment plays a much larger role in these simulations.
In Figure 2.8, I show time courses for the most extreme cases: lowest CD8 recruitment and
lowest CD4 differentiation (black), lowest CD8 recruitment and highest CD4 differentiation
(blue), highest CD8 recruitment and lowest CD4 differentiation (red), and highest CD8
recruitment and highest CD4 differentiation (magenta). As seen from the spatial layouts of
the end states, CD8 recruitment rate had a much larger impact than CD4 differentiation,
though increased differentiation did yield a slightly larger tumor. There there is a large
number of M1 macrophages with increased CD8 recruitment, with this number decreasing
slightly as CD4 differentiation increases, due to a shift in M1 influence from CD4 and CD8
cells to an M2 influence from regulatory T cells. An important result to be aware of is that
while the temporal behavior is very similar to that in Figures 1.3 and 1.5, the spatial layout
is much different, showing a key advantage that ABMs have over equation-based models.
Lastly, in Figure 2.9, I show, for the extreme cases, the fraction of cells in the tumor at the
final state. To determine this, I first find the tumor radius, calculated as the furthest cancer
cell from the center. Any cells that fall within that radius are counted for these fractions.
22
These ratios show similar behavior as the time-courses, where CD8 recruitment had a much
more noticeable effect than CD4 differentiation. As expected, a much larger fraction of CD8
cells, both active and suppressed, at the higher rates of CD8 recruitment, with slightly few
CD8 cells at higher CD4 differentiation. Notably, there is also a much larger fraction of M1
macrophages at the higher rate of CD8 recruitment, likely due to the influence of CD8 cells
on macrophage differentiation.
Figure 2.7: Spatial layouts for varying rates of CD8 recruitment and CD4 differentiation.
Boxes around the simulations in the four corners are colored to correspond to the time-courses
in the following figure.
23
Figure 2.8: Time-courses for the extreme cases of CD8 recruitment and CD4
differentiation.
24
Figure 2.9: Fractions of each cell type for extreme values of CD8 recruitment and CD4
differentiation.
All immune cells: Varying CD4 recruitment and CD4 differentiation
I next explored how varying CD4 recruitment rate impacted tumor growth (Figure 2.10). At
all levels of differentiation, increasing the recruitment of CD4 cells increases overall immune
cell infiltration into the tumor. However, at low differentiation, this infiltration is mainly a
mixture of CD4 cells, M1 macrophages, and M2 macrophages. At a high differentiation, this
infiltration is mostly regulatory T cells and M2 macrophages.
Interestingly, despite CD4 cells having no killing effect of their own and not causing a great
increase in the number of CD8 cells (Figure 2.11), tumor size decreases with increased
recruitment. As this persists even with high differentiation, where most CD8 cells will be
suppressed, I assume that this is, like in Figure 2.5, due to the spatial inhibition from large
numbers of immune cells. Here, despite low differentiation leading to a greater number of
M1 macrophages, changing differentiation had no noticeable impact on the tumor growth
curve at either recruitment rate.
For cell fractions, as expected, increasing the CD4 recruitment rate yields a higher fraction
of CD4 cells, with a higher differentiation rate shifting these to the regulatory state. There is
a slight increase in the fraction of CD8 cells within the tumor, along with an increase in the
fraction of M1 macrophages, due to the influence from CD4 cells.
25
Figure 2.10: Spatial layouts for varying rates of CD4 recruitment and CD4
differentiation. Boxes around the simulations in the four corners are colored to correspond to
the time-courses in the following figure.
26
Figure 2.11: Time-courses for the extreme cases of CD4 recruitment and CD4
differentiation.
27
Figure 2.12: Fractions of each cell type for extreme values of CD4 recruitment and CD4
differentiation.
Discussion
In this Chapter, I present an ABM of the tumor-immune microenvironment, accounting for
the interactions between cancer cells, macrophages, CD4 cells, and CD8 cells. This model
provides a framework to systematically explore the effects of cell-cell interactions, with the
major interactions between these cells modeled in a simplistic manner that captures known
phenomenological behavior. As shown by the simulation results, the model produces
expected behavior based on biological observations.
Visually, the model can capture a broad range of tumor behaviors, such as immune excluded
and inflamed tumors, which are based on immune cell infiltration into the tumor.
24
Changing
just one or two parameter values is enough to drastically alter the course of tumor growth
and the spatial layout of the immune cells. Simulations showed that setting parameters to
more anti-tumor values did indeed shrink or even eliminate the tumor, while increasing
immunosuppression yielded larger tumors. The model displays appropriate responses to
changes in parameters that do not directly affect the killing of cancer cells, such as CD4
differentiation to the regulatory state and macrophage differentiation to the M1 state.
Overall, this model is a suitable framework for interrogating many different parts of the
tumor-immune microenvironment.
One of the main advances of this model is the treatment of diffusion. Specifically, I replace
PDEs or simple distance thresholds with a distance-based exponential decay function
29,30
28
that accounts for the cumulative effects that cells have on one another, representing the
effects of different cytokines on a scale from 0 to 1. This preserves the gradient nature of
PDEs without actually needing to solve them. As many ABMs that include PDEs use either
diffusible factor thresholds or treat concentration as a probability,
25–27
instead of explicitly
modeling binding and downstream signaling promoted by diffusible factors, this approach
is a reasonable simplification that greatly improves computational time. This also makes it
easy to add additional cell interactions that are based on cytokines, such as the effects of M2
macrophages on cancer cell proliferation, without having to explicitly model the additional
cytokines. This approach can be applied not only to diffusible factors secreted by cells but
also to processes such as nutrient diffusion from blood vessels and to the tumor as a whole
to represent the penetration of therapeutics into the tumor.
From the various simulations performed, the model displays a wide range of spatial
behaviors. Indeed, while many of the tumor growth curves are very similar to each other, the
corresponding spatial layouts are very different. Outside of simple measurements, such as
tumor radius and average immune cell infiltration, it is a difficult task to devise a range of
spatial statistics to compare large numbers of simulations without having to visually
compare them. Additionally, it is very difficult to compare simulations to tumor images in a
quantitative manner, which is needed to perform a rigorous parameter estimation. With this
challenge in mind, I will display in Chapter 3 a novel method for comparing model
simulations without the need to specify comparison metrics. I demonstrate how this method
can be applied to compare model simulations to images. I then apply this novel method in
Chapter 4 to the model developed in this Chapter to estimate model parameters from
fluorescence images and then analyze the model.
Model Parameters
* indicates the parameter was estimated in Chapter 4. Value set to the average of the best fits
in that Chapter.
Table 2.1: Force parameters
Parameter (in text) Description Value Reference
kc Decay of attractive
force
10
14
µ Spring constant 50
14,15
Drag coefficient Drag coefficient 1
15
xmax Maximum
interaction distance
1.5 x cell radius
15
29
Table 2.2: Cancer cell parameters
Parameter (in text) Description Value Reference
- Cell radius 10 µm
25
- Cell division
probability
0.04 hr
-1
*
- Cell death
probability
0.0053 hr
-1
*
xth Influence limit 25 µm
30
, scaled based on
Chapter 4
- PD-L1 probability,
when expressed
0 Set manually
- Probability of
gaining PD-L1
0 Set manually
Table 2.3: Macrophage parameters
Parameter (in text) Description Value Reference
- Cell radius 7.5 µm
37
- Death probability 0.015 hr
-1
*
- Migration speed 20 µm/hr
20
, scaled based on
Chapter 4
- Migration speed
within tumor
13.3 µm/hr *
kM1 Probability of
differentiating to M1
0.187 *
kM2 Probability of
differentiating to M2
0.433 *
- Influence limit 25 µm
30
, scaled based on
Chapter 4
- Migration bias 0.123 *
- Migration bias in
tumor
0.067 *
30
- PD-L1 probability
when M2
0.264 hr
-1
*
- Probability of
redifferentiation
0.400 hr
-1
*
- Probability of killing
cancer cell (as M1)
0.044 hr
-1
*
Table 2.4: CD4 parameters
Parameter (in text) Description Value Reference
- Cell radius 5 µm
25
- Cell death
probability
0.039 hr
-1
*
- Migration speed 90 µm/hr
38
, scaled based on
Chapter 4
- Migration speed in
tumor
60.2 µm/hr *
pdiff,0 Probability of
differentiating
0.13 hr
-1
*
- Influence limit 25 hr
-1
30
, scaled based on
Chapter 4
- Migration bias 0.074 *
- Migration bias in
tumor
0.068 *
- CTLA-4 probability
when Treg
0.139 hr
-1
*
Table 2.5: CD8 parameters
Parameter (in text) Description Value Reference
- Cell radius 5 µm
25
- Cell death
probability
0.014 hr
-1
*
31
- Migration Speed 90 µm/hr
38
, scaled based on
Chapter 4
- Migration speed in
tumor
51.0 µm/hr *
pkill,0 Kill probability 0.041 hr
-1
*
kscaling Scaling factor for kill
probability
5.76 *
- Influence limit 25 µm
30
, scaled based on
Chapter 4
- Migration bias 0.076 *
- Migration bias in
tumor
0.022 *
- Division probability 0.053 hr
-1
*
Table 2.6: Environment parameters
Parameter (in text) Description Value Reference
- Macrophage
recruitment rate
6.22x10
-3
cells/(hr*n_cancer)
*
- CD4 recruitment
rate
1.03x10
-3
cells/(hr*n_cancer)
*
- CD8 recruitment
rate
1.87x10
-4
cells/(hr*n_cancer)
*
- Maximum
recruitment
distance from tumor
edge
200 µm Set manually
- Recruitment delay 2.86 days *
32
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35
Chapter 3: Representation learning for quantitative
comparisons between model simulations and image
data
Introduction
In order to be successfully applied, a computational model requires two things: parameters
that accurately replicate biological behavior and the proper tools to analyze the model. For
tumor agent-based models, parameter estimation is difficult as there is no way to compare
model simulations to tumor images in a quantitative way without having to calculate
complex spatial metrics, which still do not provide an all-encompassing measure of the
differences between simulation and image. Additionally, this lack of a comparison method
inhibits the comparison of simulations to each other. Generally, only a small dimension of
the possible outputs is compared between simulations, failing to account for the simulation
as a whole.
In this Chapter, made up of two publication preprints, I make two novel advances in model
analysis by using representation learning. Representation learning is the use of neural
networks to project complex inputs into low-dimensional space. The first preprint presents
a generalized method for comparing model simulations, regardless of the type of model, as a
single continuous value, which simply answers the question, “how different are two
simulations?” The second preprint applies this method not to compare simulations to each
other, but rather to compare ABM simulations to tumor images, using the distance between
them in low-dimensional space as an objective function for parameter estimation.
36
Chapter 3a: Representation learning for a generalized, quantitative
comparison of complex model outputs
Adapted from Cess and Finley, 2022. arxiv.org
(https://doi.org/10.48550/arXiv.2208.06530)
Introduction
As the modern century marches forward, computing power continues to increase. At the
same time, advances in experimental techniques enable one to capture enormous amounts
of data
1–5
. By taking advantage of these powerful computing resources and large
experimental datasets, scientists can construct larger and more complex computational
models of various systems
6–13
. These models allow for the interrogation of a system in
extensive detail, predicting how the system responds to various perturbations. In some
cases, models enable researchers to examine pieces of the system that cannot yet be
experimentally measured. In addition, models permit researchers to perform large-scale
simulations to produce in silico data that would either take too much time or be too costly to
generate experimentally.
To understand the system being modeled, these model simulations must be properly
analyzed. Many methods currently exist for analyzing computational models and in silico
data. Some methods, such as sensitivity analyses or parameter sweeps, are designed
specifically to examine a model
14–21
. Other methods, such as partial least squares and
clustering, are typically used to analyze data
22–25
but can easily be used to analyze models as
well
26
. However, a current drawback of most model analysis methods is that they generally
examine only a piece of the system, rather than the whole set of simulated data
27–31
. For
example, performing a sensitivity analysis requires one to explicitly define the output of
interest, and the output selected influences the analysis itself. Consider a system that
oscillates with time. For one state variable, outputs such as the amplitude and frequency of
peaks are easily calculated. But as the number of state variables increases, to get a complete
view of the system, one would have to account for how the amplitudes and frequencies for
each variable relate to those of every other variable. To be complete, one must also consider
the possibility that a state variable may not oscillate at all given the parameter values being
simulated. Additionally, one would have to scale the different comparisons to eliminate
potentially unjust influence based on output values. This could introduce bias from the
researcher, who would have to manually determine these comparison metrics and could
easily fail to account for every possible comparison. This example highlights limitations in
obtaining a holistic and unbiased comparison of model outputs.
37
To address these limitations, in this study, I present a generalized, model-agnostic approach
for comparing complex model outputs as a single, continuous value by comparing learned
representations of the outputs. Representation learning, in general, uses neural networks to
project inputs, such as images and text, into lower-dimensional feature-vectors, which can
then be used for various downstream tasks
32
. One specific use of representation learning is
Siamese neural networks. Siamese networks are a pair of identical neural networks that
project a pair of inputs into individual, low-dimensional points. By taking the distance
between these points, the similarity between the inputs can be determined
33,34
. This
implementation has been used to detect signature forgeries
35
, to visualize single-cell data
36
,
for face verification
37
, and as a measure of continuous disease progression
38
. It has also been
also been shown to work very well for dimensionality reduction
39
and for clustering mass
spectra
40
.
Here, I use representation learning with a Siamese implementation to compare the outputs
of complex computational models. By projecting outputs to low-dimensional space and
taking the distance between points, I can determine how different two simulations are. The
distance between the points provides a single, holistic value that accounts for the complex
relationships between model output features that would be otherwise difficult to calculate
manually. This approach is unique in that I am applying a method traditionally used to
analyze real-world data to analyze computational models. Other works have applied deep-
learning methods to model-generated data; however, that was to better analyze
experimental data, not to analyze a model itself
41
. Some methods exist that compare
simulated time series data
42,43
; however, these are not applicable to other types of model
outputs. I display this approach on several example models that have very different types of
outputs to show its broad applicability.
Results
I first clarify some terminology to eliminate potential confusion throughout the paper. The
term “model” is used to refer to any sort of computational model, such as systems of
differential equations, that can produce a “model output”, any simulation or calculation
performed by a model. These two terms, model and model output, specifically refer to the
computational models that are being analyzed, and not to the neural networks that are used
to perform the analysis. The terms “projected space” and “projections” is used to refer to the
learned representations of model outputs that are used to perform analyses.
Here, I describe this approach and provide three example models that produce very different
types of outputs. The goal with this study is to display the utility of the approach, potential
ways that it can be used to analyze a variety of models, and areas where it could be improved.
38
I briefly discuss each model and the analysis results so that the implementation of the
approach can be understood in the context of the model. However, I avoid discussing the
implications of the results on the modeled systems, as the aim is to demonstrate that this
approach can be applied to disparate models and not to draw novel insights about the
modeled systems.
Representation learning: overview and training
This comparison approach makes use of neural networks to reduce the dimensionality of
model outputs and then calculate the distance between the projected points to determine
how different they are from each other
33,34
. A schematic of this is displayed in Figure 3a.1.
There exist many different ways to train a neural network for representation learning. I use
a modified version of SimCLR
44
. SimCLR uses an encoder to convert inputs into
representations, followed by a projection head, with loss calculated on the projections. After
training, the projection head is removed, and the representations are used to train a linear
classifier. The key aspect of training is that, in each training batch, each input is augmented
twice, sent through the network. The projected augmentations are compared to each other
and the augmentations from all other inputs using normalized temperature-scaled cross
entropy loss (NT-Xent) with cosine similarity. As expanded upon in the Methods, I make two
changes from this approach. Because the downstream task is the use of the distance between
representations, I remove the projection head and calculate loss directly on the
representations. I also replace cosine similarity with 1/(1 + Euclidean distance), as Euclidean
distance is the main measurement.
Figure 3a.1: Schematic displaying how two simulations are compared via projection to
learned space.
39
In practice, neural networks are often used in ensembles. For the results presented below, I
report the mean of the distances projected by an ensemble of 50 neural networks, along with
their standard deviations.
Reconstruction of 2-dimensional data
Before applying this method to high-dimensional, model-generated data, I first perform a
simple test to confirm that the proposed method works as intended. Following Szubert et
al
36
, I generate two sets of two-dimensional data (Figure 3a.2, top row) (x, y), which are then
transformed into nine-dimensional space (x+y, x-y, xy, x
2
, y
2
, x
2
y, xy
2
, x
3
, y
3
). I then use this
approach to learn two-dimensional representations of this transformed dataset, aiming to
reconstruct the original structure (Figure 3a.2, middle row). I find that this approach
performs a reasonable reconstruction of the original data, meaning that the learned
representations have useful information.
Consensus score
Because neural networks are stochastically initialized and trained, different networks can
potentially yield different representations. Additionally, unlike the test set in Figure 3a.2, it
is not known beforehand what the learned representations should look like. Therefore, I
developed the “consensus score” metric to evaluate how different the learned
representations for multiple neural networks are from each other. The higher the consensus
score, the more similar the learned representations. As the analyses that I wish to perform
are based on the distance between points in learned space, a higher consensus between
40
networks is favorable, as it means that there is higher agreement between the networks in
an ensemble, increasing confidence in the results.
Figure 3a.2: Test projected datasets. Top row: original data. Middle row: learned
representations from 9-d transformation of original data. Bottom row: consensus scores for a
range of neighborhood sizes.
This consensus score (expanded upon in the methods) compares the local spatial layouts of
two sets of projections. It is calculated using the fraction of n nearest neighbors to a point
that are shared for projections from two networks. For example, if the three nearest
neighbors of point p are points [2, 3, 5] for network A, and points [2, 3, 4] for network B, then
the consensus for that point for those two networks is 0.66. I perform this calculation for
each point and average the results to get the overall consensus between two networks. I then
average all of the pairwise consensus scores between each network in an ensemble to get
41
the overall consensus for that neural network structure for a particular set of simulations.
The consensus scores for both of the above test datasets at several values of n are shown in
Figure 3a.2, bottom row. In general I would expect the consensus score to increase as n
increases. However, the consensus for these two examples already display very high scores
and are varying only over a range of 0.04. Thus, I believe I have developed a robust metric
for characterizing the similarity between results from distinct neural networks.
Overview of test models
I test this approach on three disparate computational models: a constraint-based metabolic
flux model, an ordinary differential equation (ODE) model, and a spatial agent-based model
(ABM). These models (detailed in the following sections) were chosen because they produce
outputs in different formats, which are, respectively: a vector of fluxes for each reaction (1D
vector), time-series for each ODE species (2D matrix), and a spatial layout for each agent type
(3D matrix). Because of the different output formats, I use different types of neural networks
for each model.
For analyzing each model, I use ensembles of 50 neural networks, each trained on 10,000
simulations. For all three test models, I project to 16 dimensions, with the final layer of
neurons having a linear activation. The consensus for each model is shown in Figure 3a.3. As
expected, the consensus was lower than it was for the datasets used in Figure 3a.2 (as model
outputs were not originally in a low dimension) and increases as the number of neighbors
used to calculate score increases. Interestingly, there was a lower consensus between the
neural networks used to represent the flux model, compared to the other two models. This
result was unexpected, as this model has the simplest output format. Further work is needed
to explore this specific model type and find methods that can produce greater consistency
between representations. However, as the primary goal is to demonstrate the use of this
method to represent complex model outputs to enable standard analyses, I present example
analyses that may be performed using the results from this approach.
In the following three sections, I detail each model and show example analyses that can be
performed using this approach to better understand model behavior. I display two analyses
for each model. The first is the same for each, and that is shifting the value of one parameter
to see how the distance from the base parameter set changes. I use this same analysis to
show how this approach translates to different model types. The second test is different for
each model and is one that could be of interest to that specific model type. However, these
specific tests can also be used for other model types.
42
Figure 3a.3: Consensus scores for the three test models.
Flux model
The first example uses a model of metabolic reactions for E. coli, containing 2,477 reactions
organized into 44 subsystems
68
. Here, the model parameters are the upper and lower flux
bounds for each reaction. These bounds are fixed, and the model estimates the reaction
fluxes within the bounds needed to optimize a specified objective function
69
. Positive fluxes
represent flow in the forward direction, while negative fluxes are flow in the reverse
direction. The output for this model is a one-dimensional vector of the flux through each
reaction for the optimized state. Because the output is a simple vector, I use a basic
feedforward neural network to analyze this model.
For the first test, I shift the lower bound for uptake of uridine diphosphate glucose (a nutrient
for the organism) from 0 to -1,000 (in nutrient uptake, negative flux corresponds to flow into
the cell) and compute the distance from the optimized flux state to that of when the lower
bound is 0. This shows how the overall metabolic state changes as the cell becomes able to
take up more of this nutrient. I find that there is a steep increase in the distance from the
base state once the organism becomes able to uptake this nutrient (Figure 3a.4), meaning
that this nutrient causes a distinct change in the metabolic pathways utilized, compared to
the base state. This difference continues to increase as E. coli can take up more of the
nutrient, until it finally levels off, with increased potential uptake no longer changing the
metabolic state.
The second test I perform, which is very specific to this type of model, is a series of reaction
knockouts
70–72
. Here, I set the upper and lower bounds for the flux through a specific reaction
43
equal to zero before solving the model for fluxes through the rest of the reactions (Figure
3a.5). This means the organism cannot utilize that particular reaction, and it has to direct
flux through alternative pathways. Because of the large number of reactions in this model, it
would be difficult to compare the effect of each individual knockout on the estimated flux
through each metabolic reaction. Instead, I examine the average change for knockouts in
each subsystem. For each subsystem, I knockout each reaction individually, compute the
distance to the base state, and then average the distances for each reaction in the subsystem.
This shows which subsystems contribute the most to the base state and thus could be subject
to further analysis.
Figure 3a.4: Change in model state when changing lower bound for test model 1.
Calculated as the distance in projected space. Line – mean; shaded area – standard deviation.
44
Figure 3a.5: Knockout results from the flux model. Bars show the average distance from the
base state for each subsystem. Error bars show standard deviation.
Ordinary differential equation model
The second example is a Lotka-Volterra model, which is comprised of ordinary differential
equations. This model has been used in many fields, such as ecology, chemistry, and
economics
45–47
. I chose this model because of its broad usage and because, depending on
parameter values, it can reach a steady state or produce oscillations. Specifically, I use a four-
species model, setting the base parameters to a set listed in Vano et al., who examined chaotic
behavior in this system
48
. I increase three of the parameter values from zero to 0.01 (shown
in equation (3) of Vano et al.), which maintains the oscillatory behavior, but allows me to
sample these parameters above and below the base value of 0.01 when producing the
training dataset. The output of this model is a vector of values at each time point for each
species. I organize the model outputs as a two-dimensional matrix, #V[\T9VX#+×+T\^V\+ .
The neural network I use for this type of model is a one-dimensional convolutional neural
network, which has been used to classify time-series data
49,50
.
The first test that I perform is shifting the value of a parameter; here, from two-fold below to
two-fold above the base value. I calculate the distance between the output from each
parameter value and the base output (Figure 3a.6). As expected, the further the parameter
value is shifted from the base value, the larger the predicted distance is. While this is a simple
test, it shows how this approach can be used to easy perform a holistic evaluation of how
model behavior changes as a parameter value is changed.
45
Figure 3a.6: Shifting the value of a single parameter for test model 2. Values are shifted
over a two-fold range (x-axis) and calculating the distance from the base model output (red
point) parameter (y-axis). Line – mean; shaded area – standard deviation.
The second test for this model is a local sensitivity analysis (Figure 3a.7), which is used to
determine how sensitive a model output is to changes in a parameter. A local sensitivity
analysis changes one parameter at a time from a base value (here, increasing by 10%) and
records the corresponding change in a specified output. I consider two cases: (1) using the
distance in projected space to holistically compare the model outputs and (2) specifying a
single model
46
Figure 3a.7: Local sensitivity analysis for the test model 2. Compares the most sensitive
parameters when using a Siamese network as the output (top) to using the final value of each
variable as the output (bottom). Values are normalized to the maximum for each sensitivity
output. Error bars (top) show standard deviation. Inset in the top right shows the base
simulation. The other insets show model simulations following a 10% increase for parameter 6
(left) and parameter 19 (right).
output. In projected space, I calculate the distance between the output from the perturbed
parameter and the base output to get an overall comparison between the two. For the
specified output, I use the average change of all four final values of the differential equations
in the model. This output is only capturing one moment in time, and fails to account for
temporal behaviors, such as the oscillations that this model is able to produce. One common
usage for a sensitivity analysis is to determine which parameters most strongly impact the
output, and thus are most important for parameter estimation. With this in mind, I averaged
the sensitivity results for increasing each parameter value for each case of sensitivity
analysis (representations and specified output) and ranked the parameters by the most
sensitive. Because of the difference in how I compare perturbed simulations to the baseline,
I normalize the sensitivities to between 0 and 1 so that I can compare them qualitatively.
Comparing the two cases, I find that some parameters (3, 11, 16) have similar rankings,
whereas other parameters have very different rankings. Specifically, I draw attention to
parameters 6 and 19. In Figure 3a.7, the box in the top right shows the base simulation, while
the other insets show the simulations for a 10% increase in parameters 6 (left) and 19
47
(right). The specified output ranks parameter 19 as slightly more influential while using
learned representations ranks parameter 6 as having a much higher influence on the model
output. From a visual comparison to the base simulation, one would likely argue that
perturbing parameter 6 yields a larger overall change in the model output than parameter
19. With this, I show how this approach displays how perturbing a parameter impacts model
behavior as a whole, instead of looking at a small region of parameter space.
As with the previous examples, I first performed a single parameter shift. I varied the tumor
cell proliferation rate across 21 different values (Figure 3a.8), with the base simulation being
the middle value. Because this is a stochastic model, I performed each simulation with ten
replicates. When taking the distance between simulations for two different parameter sets,
I compared each replicate to every other replicate and then took the average distance
between them. I again find that the distance between model outputs increases as I shift the
parameter value.
Figure 3a.8: Shifting the value of a single parameter for test model 3. Compares the
distance calculated in projected space to the middle parameter value (red point). Line – mean;
shaded area – standard deviation.
The second test I performed was clustering the model Monte Carlo simulations that
produced the training dataset (Figure 3a.9). I use the distances in projected space to group
the 10,000 simulations via hierarchical clustering
67
. From these clusters, I examined the
distributions for model parameters that produced the simulations (top two rows). I looked
48
at different numbers of clusters, with two producing clear separation in the parameter
distributions. I see clear differences in the distributions for the parameters, allowing for the
identification of how different model end-states compare to each other based on model
parameter values. Additionally selected specific model outputs can then be looked at
(bottom row). This gives insight into some of the systemic characteristics of each cluster. For
example, in the orange cluster, low numbers of cancer cells are clustered with high numbers
of T cells, which kill cancer cells. Furthermore, these two outputs are clustered with a low
macrophage recruitment rate, which makes sense as, in this model, macrophages suppress
T cell activity. This example demonstrates that this approach can be used to cluster model
simulations and examine how overall model output is linked to model parameters and
specific outputs.
49
Figure 3a.9: Clustering Monte Carlo simulations for test model 3. Distributions for four
sampled parameters are shown for each of the two clusters in the white plots (top two rows)
and for two selected model outputs in the gray plots (bottom row).
Discussion
With this study, I display how representation learning can be used to compare complex
model simulations as a single quantitative value. I show how the same method can be applied
to three very different types of models, only needing to change the neural network structure
to accommodate the format of the model output. By using a neural network to reduce the
50
model output to a low-dimensional point and taking the distance between two points, a
holistic view of how different two simulations are from each other can be obtained, without
the need to manually calculate complex comparison metrics.
To display the utility of this approach, I provided a total of six tests performed across three
example models, showing how the approach allows for the comparison of model simulations
in an unbiased manner. The first test for each model was the same, where I shift the value of
a single parameter across a range of values. This simple test displays how model outputs
change overall as a parameter is varied. This analysis can be used to determine areas of
parameter space where the model output changes more drastically, or the identify potential
regions where biphasic behavior is exhibited. For the first example model, I also displayed
how this approach can be applied to reaction knockout simulations. Often, these simulations
are used to identify perturbations that optimize a specific biological process
70–72
. This
approach can extend these analyses to determine how different the new metabolic states are
from the original or from other knockout states. The second example model is a time series
differential equation model used in a variety of fields. I show how this approach can be used
to aid in sensitivity analyses
18
, which can then identify important parameters for parameter
fitting, model reduction, and model expansion. Lastly, I used this approach to cluster
simulations from an agent-based model. From these clusters, it is possible to examine how
different parameter values tend to create different simulation end-states, along with how
specific model outputs are linked to the different end-states. In the context of the tumor
model that I examined, this approach would allow the researcher to better predict how
different tumor properties yield different final tumor states.
I only displayed simple examples of analyses so that this approach could be easily
understood; however, it can be extended to more complex analyses. For example, the
sensitivity analysis that I showed was a simple, local analysis. However, this only captures a
small region of parameter space, whereas global sensitivity analyses yield a better
exploration of the model
21,73,74
. Instead of comparing to a base parameter set, each
dimension of the projected points can be treated as an output for a global sensitivity analysis,
providing a more exhaustive description of how sensitive the model as a whole is to each
parameter. This approach could also be used to aid in uncertainty and robustness analyses
75–
78
. By accounting for model behavior as a whole instead of only focusing on a small part of
the output space, this approach provides a better characteristic of the system. Another
potential use is for optimizing a perturbation to a system that would produce a desired
change in an output of interest while minimizing the overall change in the system.
I note that I did not explore the many other ways to perform representation learning.
Potentially other methods can improve the consistency of the learned representations,
which would improve confidence in the calculated distance between representations. For
51
example, one could consider implementing different training methods, variational
autoencoders, or supervised pre-training, where one would first predict model parameters
from the model outputs
79–84
. The purpose of this work, however, is to present a novel
application of representation learning, and I leave extensive analyses to future work, where
this approach can be optimized in more case-specific settings.
The main limitation with this approach is that it can be computationally expensive. One can
quickly generate large numbers of training samples for small models; however, as model
complexity increases, the time required to perform Monte Carlo simulations increases
greatly. Additionally, it takes time to determine an optimal neural network structure,
especially when more complex neural networks need to be implemented, based on the
format of the model outputs. However, once a neural network is trained, it can be used for
many different analyses. Another limitation is the nature of neural networks. Because they
are black-box methods, it is difficult to interpret how model outputs are being projected into
low-dimensional space. This approach simply determines how different two simulations are,
not why they are different. However, as I show with the third example, combining this
approach with specific model outputs helps yield further insight into the system.
Despite these limitations, I present a useful approach to compare model outputs via
representation learning. This work addresses the limitations of commonly used model
analysis methods; namely, a narrowly focused and potentially biased comparison. I show
how these neural networks can be trained on a set of model-generated data and how the
trained networks can be applied to analyze a range of model outputs. I demonstrate that this
approach provides an additional way to interrogate models beyond current methods.
Methods
The approach I detail here uses representation learning to train neural networks to project
the outputs of computational models into low-dimensional space, where the neural
networks can then be applied in a Siamese fashion to compare model outputs as a single,
continuous value. In the following sub-sections, I describe: (1) an overview of representation
learning, (2) the specific training approach, (3) generation of training data and specific
neural network construction, (4) calculation of a “consensus score,” and (5) implementation
for analyzing model outputs.
Overview of representation learning
Representation learning is a form of self-supervised learning, where a series of low-
dimensional features are learned from much more complex inputs
32
. These learned
representations are then used to perform more specific downstream tasks. Ultimately,
52
representation learning achieves a dimensionality reduction, where complex inputs are
projected into lower-dimensional feature vectors. In the case of Siamese networks, the
distance between a pair of projected inputs is used as a measure of similarity
33
.
Training approach
Many methods exist for training neural networks for representation learning
32
. Here, I use
the training method proposed in SimCLR
44
. This method aims to maximize the similarity of
the projections for two augmentations of an input in comparison to their similarity to the
other inputs. I chose this method because it does not require inputs to be labeled as similar
or dissimilar to each other, whereas training methods such as triplet loss require explicitly
labeling training samples as similar or dissimilar
85,86
.
I make two changes to this approach. SimCLR, like many representation learning
frameworks, makes use of a projection head that follows the learned representations, with
loss being calculated based on the cosine similarity between the outputs of this projection
head. The use of a projection head has been found to improve accuracy when training a linear
classifier on the learned representations of images
44
. However, the aim is to instead compare
distances in projected space. Therefore, like many studies using Siamese networks, I do not
use a projection head and instead calculate loss directly on the learned representations.
Additionally, I use Euclidean similarity instead of cosine similarity, as shown in Equation 1,
where _(T
2
,T
-
) is the Euclidean distance between points in projected space. The reason for
this is that simulations are projected to a low number of dimensions, where the distance
between simulations is used as a measure of comparison. This comparison should be
unbounded, hence the use of Euclidean distance instead of cosine.
+V[V`a:V#b=
1
1+_(T
2
,T
-
)
Generation of training data and neural network structure
I take a straightforward approach to generate training data. I simply perform Monte Carlo
simulations with each computational model, sampling parameters from a uniform
distribution across the entire range of parameter space that may over. This produces a wide
range of model outputs for training the neural network. Each model simulation is treated as
a unique individual, whether or not the model is stochastic. In this study, I perform 10,000
simulations. Sampling ranges were chosen based on the specific model being analyzed.
The main part of this approach that is model-specific is the neural network structure, which
must be chosen based on the format of the model output. For temporal models, a one-
dimensional convolutional neural network is often sufficient, whereas a two-dimensional
53
convolutional network is suitable for most spatial outputs. More complex networks may be
suitable based on the nature and scope of the model’s output.
Consensus Score
One of the issues with neural networks is that they have large numbers of parameters and
are stochastically initialized, meaning that each time a neural network is trained, the values
of neuron weights are different. Because of this, different representations may be learned
each time a network is trained. Therefore, I developed what I term the “consensus score”
between two trained neural networks, which quantifies how similar the projections are to
each other. I make the assumption that if multiple stochastically initialized neural networks
converge to similar projections, then there is greater confidence in those representations.
This score is calculated in the following way. For each neural network in an ensemble, I
project the training data into low-dimensional space. For each projected point, I find its
nearest n neighbors, referring to this as the point’s neighborhood. To determine the
consensus between neural networks A and B, for each point, I calculate the fraction of the
neighborhoods that are the same based on their projections from A and B. I average this
across all points. This then gives the consensus between networks A and B.
I calculate the pairwise consensus score between each neural network in an ensemble and
average them to get the overall consensus within the ensemble. I select neural network
structures that maximize this score.
Implementation for analyzing model outputs
By using the trained neural networks in a Siamese implementation, the distance between
projected points can be used as a measure of how different they are, with closer points being
more similar to each other than they are to distant points. The main advantage of this
approach is that, by using a neural network to learn representations of model outputs, there
is no need to manually calculate a difference metric. I note that the distance between two
projected points can be influenced by the neural network structure and training, and thus
has little meaning by itself. Instead, comparisons between multiple outputs are needed to
give it meaning. For example, for outputs O1, O2, and O3, the distance between O1 and O2 alone
provides little information, however if that distance is smaller than the distance between O1
and O3, I can conclude that O1 is more similar to O2 than it is to O3.
For all three test models, I project to 16 dimensions. This final layer of neurons uses a linear
activation, as I are performing analyses directly on this layer. Other neural network
hyperparameters were adjusted to maximize the consensus score.
54
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60
Chapter 3b: Calibrating agent-based models to tumor images using
representation learning
Adapted from Cess and Finley, 2023. biorxiv.org
(https://doi.org/10.1101/2023.01.12.523847)
Introduction
Agent-based models (ABMs) of cellular systems have been used to explore various facets of
physiology and disease
1,2
. Specifically in the field of oncology, ABMs have been used to
explore the many different ways in which the tumor is influenced by the microenvironment,
such as through hypoxia, angiogenesis, invasion, and interactions with the immune system
3–
5
. While ABMs are powerful tools that can simulate many different tumor properties and
predict emergent behaviors that occur on the spatial level, they are limited in the realm of
parameterization. In general, fitting spatial models of cell populations to spatial data is a
difficult task, where specific features of the spatial state have to be extracted in order to
perform comparisons
6
. For ABMs, key parameters, such as proliferation rates, cell lifespans,
and migration rates, can often be experimentally measured outside of the scope of the
modeled system and are commonly found in the literature. However, this sometimes leaves
specific parameters with unknown values. Many of these parameters are often phenotypic
summaries of complex biological mechanisms, reducing phenomena such as the secretion of
cytokines and subsequent activation of intracellular signaling pathways to simple
interactions governed by a single parameter
7–11
. Therefore, an ABM needs to be compared
to data in order to properly estimate the unknown parameter values.
The spatially-resolved and stochastic nature of ABMs makes parameter estimation more
difficult than in equation-based models. For an equation-based model, while parameter
estimation is not a trivial task, it is straightforward, requiring a quantitative comparison of
the model outputs and experimental data. In most cases, the experimental data are
represented by a single, continuous value, which is the same format as the model output.
This makes it easy to calculate the distance between experimental data and model
simulation
12
. One can then use parameter estimation algorithms that minimize the distance
between the two
13–16
. For ABMs, however, this is more complicated. While a comparison to
quantitative data, such as tumor volume over time, is possible, this type of data loses the
spatial aspect that is a defining feature of an ABM. Therefore, in order to better estimate
unknown parameters, a comparison to tumor images, which provide spatial information,
would be beneficial.
Comparison to tumor images poses a challenge in that it is very difficult to compare imaging
data to model simulations. For ABMs of tumors, simple qualitative comparisons are often
61
performed to determine how well the model represents the spatial biology. In some cases,
simple metrics are extracted and used for comparison
17–20
. However, this fails to capture the
full extent of the spatial properties being modeled. Furthermore, selecting comparison
metrics introduces bias to the parameter estimation, becomes unwieldy as model complexity
increases, and can potentially miss complex spatial interactions that are hard for the
researcher to detect.
To this end, I endeavored to develop a method for quantitatively comparing experimental
tumor images to ABM simulations as a scalar metric. I have previously described an approach
for learning low-dimensional representations of model simulations using neural networks,
which allows for the comparison of complex model simulations in terms of a single distance
metric
21
. Here, I extend that approach to comparing ABM simulations to tumor images. The
closer the projected points for two inputs to the neural network are, the more similar the
inputs are. This distance between a tumor image and ABM simulation can be used as the
objective function for parameter estimation. Neural networks are widely used in various
forms of image analysis and have previously been shown to work better than methods such
as PCA for image dimensionality reduction
22
.
Here, I display how I process both fluorescent images and the spatial output from ABM
simulations into a similar format so that they can be compared with the neural network
method. I detail how I train a neural network to learn low-dimensional representations of
this type of model output and how this can be used as an objective function for parameter
estimation. I present two examples to demonstrate the application of this method. In the first
test case, I fit a model to its own simulations to display method functionality in a controlled
setting. In the second example, I fit an ABM to a fluorescence image from an in vitro device
that recapitulates a hypoxic tumor microenvironment, displaying how this method works on
real data and enables fitting an ABM across spatial scales.
Methods
This fitting approach is based on the previously described method for comparing complex
model simulations
21
. In brief, that method uses representation learning to train a neural
network on Monte Carlo simulations of a model to project model simulations into low-
dimensional space. As a result, the difference between simulations is given by the distance
in low-dimensional space. The method enables a holistic comparison of model simulations
without the need to manually calculate comparison metrics. Here, I train a neural network
on model-generated data from an ABM and use it to compare model simulations to a tumor
image. This method is very similar to the previous one, with the main distinction here being
the pre-processing of simulations and tumor images so that they are in a comparable format.
62
After projecting the processed model simulation and tumor image into low-dimensional
space, the distance between projected points can be used as an objective function. This
permits the use of a parameter estimation algorithm to minimize the distance between
projections, thus fitting the ABM to the tumor image. I display a schematic of this process in
Figure 3b.1.
Figure 3b.4: Schematic of using the distance in projected space as an objective
function. Two inputs, a tumor image and the spatial output from an ABM simulation, are
inputted into the same neural network and projected to low-dimensional space. The
Euclidean distance between these projected points is used as the objective function to be
minimized with a parameter estimation algorithm.
Data processing
I start by describing the data-processing step, as it is a key piece of this study. I note that the
processing I describe here is specific to this use-case with ABMs and tumor images. However,
using a different data-processing step would allow this method to be used with other types
of models and imaging data. Additionally, this is only one potential way to process images
and simulations, and other methods may be able to yield similar effects. The exact
implementation of data-processing is based on the model framework and available data. In
their original formats, ABM simulations and fluorescence images have little in common.
ABMs output a list of cells with their coordinates and properties, while a fluorescence image
is a picture with several color-channels, one for each fluorescence stain. The first step of
processing is to extract the cell coordinates from the fluorescence image, which can be done
using readily available software. Here, I use the Analyze Particles function in ImageJ
23
. Based
on the fluorescence stains, I now have converted the image to a list of cells with their
coordinates and properties, the same as the ABM simulations. After this, the cell lists for the
simulations are pruned to only contain the same cell types and properties as that from the
image.
63
The second step of data-processing involves converting the cell lists back into an image
format, which I term the “simplified images.” This is done to both the cell lists from tumor
images and from model simulations. By converting the cell lists into images, convolutional
neural networks can be used, which are used extensively for image analysis. Additionally, by
converting from cell lists, I generate images that are qualitatively comparable and in the
same format. I form the simplified images by discretizing cell coordinates from their
continuous values to a grid, where one grid space is the size of one cell diameter, akin to the
format of an on-lattice ABM. There is one grid per cell type or property, yielding a three-
dimensional matrix for each image or simulation, with the first two dimensions
corresponding to the x,y-dimensions and the remaining dimension for cell types and
properties. Each cell type or property is represented as a grid that corresponds to a color-
channel, and each site within the grid corresponds to a pixel in the simplified image. Grids
for cell types take on values of zero or one, indicating whether or not that cell type is present.
Grids for other properties, such as ligand expression, have continuous values, and I scale
these so that each grid has a maximum value of one, since values in the model do not directly
equal that of fluorescence intensity.
The final step resizes the simplified images to a smaller, uniform size. This serves two
purposes. The first is so that it produces the inputs that can be used in convolutional neural
networks, which require images of the same size. The second is that it aggregates discrete
cell locations into regions of cell density, allowing a comparison across spatial scales. This is
important since tumor images are generally of a larger size than is computationally feasible
to simulate with an ABM. Before resizing, I crop the simplified images to be bound to the
dimensions of the tumor. Then, to resize the images, I use the resize function from the
OpenCV Python library, using area interpolation
24
. I then rescale each grid such the values
for all grid sites are between zero and one. A schematic of this final processing step is shown
in Figure 3b.2A, with an example from the first test model (described in a later section)
shown in Figure 3b.2B, where it is separated into four grids, with three for cell locations and
one for ligand expression. I note that this step brings with it some considerations for
parameter estimation, which I cover in the Discussion.
Overall, this processing step serves to convert both model simulations and tumor images
into coarse-grained images showing cell densities. While this does lose some of the more
detailed spatial features, it is necessary in order to have both simulations and images in a
comparable format.
Generation of training data
Following establishment of an image processing protocol, the next step of this method is to
generate the model simulations (to which the processing protocol is applied). These
simulations are needed to train the neural network. I randomly sample each parameter to be
64
fitted over the widest range of potential values, generating 10,000 Monte Carlo simulations.
Because agent-based models are stochastic, I perform simulation replicates to capture
differences in model behavior. Once simulations are completed, I process them using the
above method. To simplify method development, here I only vary the model parameters.
However, especially in cases where the initial conditions are unknown, it could be useful to
also vary the initial conditions.
Training of the neural network
I then use the processed dataset to learn representations of the model output using the
approach outlined in SimCLR, which I applied in previously. This approach requires
augmentations of the training inputs, which I perform by randomly mirroring and rotating
each simplified image. The goal of this training approach is to move the representations of
two augmentations of an input closer together in projected space than they are to
augmentations of other inputs. Full details of this method can be found in previous
works
21,25
.
I train the neural network to model-generated data. The reason for using simulated data
instead of training to biological images is that there are generally very few images obtained
in a specific experimental study. In comparison, a model can generate a large amount of
training data relatively quickly. This means that the neural network can better learn to
generate representations of a broader range of model behaviors, thus improving the
accuracy of the comparisons to image data.
Here, I project to two-dimensional space. It is important to keep the number of projected
dimensions low, as Euclidean distance becomes less accurate in high dimensions. I
recommend using the lowest dimension that yields meaningful results. I train an ensemble
of 50 neural networks on 10,000 simulations and average their predicted distances when
applying them to parameter fitting.
65
Figure 3b.5: Schematic of data-processing. Conceptual example (A) and example
simulation (B). From the continuous spatial layout, cells are separated by property and
discretized to a series of grids. The grids are then reduced to a smaller size, converting
discrete locations to densities. In (B, left), pink dots are tumor cells (darker = higher PD-L1),
teal dots are active T cells, and white are suppressed T cells. In (B, right), cell densities range
from light pink to purple. For ease of visualization, I show the same size figures in the center
and right panels, though I note there are fewer grid spaces in the right panel, compared to
center.
Initial estimation of parameter ranges
Now that I have an ensemble of neural networks that are trained to create representations
of the model, I can use it to estimate the model parameters. I first determine the general
region of parameter space that the tumor image sits in, to avoid having to search the entire
parameter space, thus reducing the computational time needed for fitting. To do this, I use
the neural network to project both the processed tumor image and the training simulations
into low-dimensional space. I then take the n closest simulations to the image and examine
66
their parameter values, using the minimum and maximum parameter values as the lower
and upper bounds for parameter estimation. A schematic of this is shown in Figure 3b.3.
Figure 3b.6: Schematic displaying the initial parameter range estimation. The tumor image
and the Monte Carlo simulations used for training the neural network are projected into low-
dimensional space. Parameter values for the n closest simulations to the tumor image are
compared and used to set the upper and lower bounds for parameter estimation.
Example models
Now that I have described the approach, I will detail the models that I use to test it. I note
that I am using these models simply to test this method, and not to produce biological insight
regarding tumor growth. In Example 1, I fit an ABM to its own simulation results. This model
simulates the interactions between T cells and tumor cells. This model is a center-based
model, meaning that each cell is represented by a point and a radius
26
. Tumor cells
proliferate at a set rate and gain expression of the immune checkpoint inhibitor PD-L1 at a
set rate in the presence of T cells
20
. The effect of PD-L1 is represented as a probability of the
tumor cells suppressing a nearby T cell at each timestep. T cells are recruited around the
tumor and migrate towards the tumor center up to a specified distance. T cells kill nearby
tumor cells at a set probability. T cells suppressed by PD-L1 no longer migrate or kill tumor
cells. I fit four parameters, based on how strongly they influence model simulations: T cell
67
killing probability, T cell infiltration distance, maximum PD-L1 probability, and rate of PD-
L1 increase.
The model for Example 2 was designed to be fitted to a fluorescence image taken from Ando
et al. (Figure 5a of their paper), which stained for living and dead cells
27
. The authors of that
study examine CAR T cell interactions with tumors via a tumor-on-a-chip model. The tumor
is represented by a thin region of tumor cells, and T cells enter from the edge and migrate
into the tumor. Their system also accounts for hypoxia in the tumor center. To model this, I
make two modifications to the previously described model used in Example 1. The first is
that I add a hypoxic region in the tumor center, where any cells within the region have an
increased probability of spontaneous death. The second is that I do not remove dead tumor
cells from the environment. I note that these simple extensions do not account for all of the
complex biological mechanisms occurring in the in vitro system. However, the purpose of
this paper is to display how a quantitative metric can be created that can be used in an
objective function for fitting ABMs to images. Thus, I kept this model as simple as possible.
Here, I fit the following five parameters, which are context-dependent and should be fit to
data: T cell killing probability, T cell infiltration distance, basal tumor cell death probability,
size of the hypoxic area, and probability of dying from hypoxia.
Results
Here, I show fitting examples for the two test models, displaying the functionality of this
approach. I only show the results of fitting and do not infer any biological implications of the
results, as the focus of this study is to display the use of representation learning for specifying
objective functions. In addition, I keep the fitting process simple overall, as the focus is on
the objective function and not parameter estimation as a whole.
Example 1: Fitting to model-generated data
With the first test model, I produced a base simulation by manually setting the parameters
that would be fitted. The purpose of this is to test the method in a controlled scenario, where
I know what the fitted parameters should be. Thus, a strong demonstration of the method
will produce fitted parameters that are close to those of the base simulation.
I fit four parameters that strongly influence the number and spatial distribution of cells in
the ABM using the approach described in the Methods. The base simulation was projected to
low-dimensional space and the nearest 100 training simulations were used to set the upper
and lower bounds for parameter estimation. This number can be adjusted based on the
model being fit. Parameter estimation was then performed using a genetic algorithm (GA)
consisting of 300 individuals, with the objective function being the distance in low-
68
dimensional space between the base simulation and the fitting simulations. A genetic
algorithm was chosen simply because of easy of implementation on the computing cluster.
However, other estimation algorithms can be used with the objective function produced in
this method. I only perform one simulation replicate for each parameter set in the GA, in
order to improve computational time. The GA converges to similar parameter sets which
produce the best fit, thus creating simulation replicates that account for the inherent
stochasticity of ABMs.
The results of this fitting are shown in Figure 3b.4 and Table 3b.1. The average and best fits
quickly level off after only a small number of fitting steps (Figure 3b.4A). In Figure 3b.4B, I
display the end-point of the base simulation together with the best fit, providing a visual
confirmation that the fitted model captures the baseline simulation. The best-fit parameters
are very close to the nominal parameters (Table 3b.1), and the maximum, mean, and
minimum for the top 10 fits are fairly constrained around the nominal parameters. These
results show that learned representations have potential for use as objective functions to
compare ABM simulations to images.
Figure 3b.7: Fitting results for Example 1. (A) Fitting with a GA. Black dots – individuals
in the GA. Red line – average fit. Blue line – best fit. (B) Visual comparison of the spatial
layouts of the base simulation (top) and the best-fit simulation (bottom).
Table 3b.1. Top 10 best fits from fitting an ABM to model-generated data for Example
1.
T cell killing
probability
T cell
infiltration
Maximum
PD-L1
Rate of PD-
L1 increase
69
Nominal
Parameters
0.02 0.8 0.01 5x10
-5
Best Fit 0.0186 0.853 0.0102 4.57´10
-5
Maximum 0.0199 0.875 0.0106 5.21´10
-5
Minimum 0.0186 0.787 0.0068 3.62´10
-5
Mean 0.0192 0.827 0.0097 4.52´10
-5
Coefficient of
Variation
0.0198
0.0438
0.1062
0.0810
Example 2: Fitting to in vitro fluorescence image
Next, I test the approach using real-world data. Here, I apply the same method as in the
previous section, with the exception of using 400 individuals in the GA given the slightly
more complex model. The fluorescence image that I fit the model to only stained for living
cells and dead cells, and I decided to fit only to the locations of the dead cells for simplicity.
In the imaging data, there was some overlap between living and dead cells that was not
captured with the model. A more complex model, with more parameters, is needed to fully
reproduce the behavior shown in the experimental images. However, to test the method and
demonstrate its utility, I only used dead cells for fitting. A key difference that separates this
test from the previous, besides the fact that I am now comparing model simulations to real
data, is that the image is spatially much larger than the tumor that I am simulating. The
diameter of the image is roughly 5,500 microns, while I use the ABM to simulate a tumor of
approximately 3,000 microns in diameter (150 cells). In terms of cell numbers, simulating a
tumor of the actual size would involve roughly 3.5 times more cells than the simulation that
I performed. This would greatly increase the computational time needed to run each
simulation and perform parameter estimation. Additionally, tumors taken from mouse
models and patients are even larger, and, in most situations, it is prohibitive to simulate at a
true biological scale. Therefore, while it would have been feasible to simulate a tumor that is
the same size as this image, I deliberately simulate a smaller tumor to test how this approach
functions when comparing across scales. Thus, I specifically aim to demonstrate that this
approach works well even when the sizes of the tumors simulated by ABMs are different
from the size of the tumor that was imaged.
Parameter fitting was performed in a similar manner as the first example. In addition to
narrowing parameter bounds using the distance between tumor and simulated images based
on the low-dimensional representation, I also adjusted some parameter bounds based on
fitting results. I recognize that this may insert some bias into the fitting and could potentially
be avoided by performing a larger number of initial simulations. However, the goal with this
70
work is primarily to display the use of representation learning for calculating an objective
function, while balancing the use of available computational resources.
The final fit is shown in Figure 3b.5A. As in the first example, not many fitting steps (< 10)
were required before the best and average fits leveled off. Visually comparing the processed
image from experimental data that is input to the neural network (Figure 3b.5B, top) and the
actual spatial layouts of the cells (Figure 3b.5B, bottom), I see that the best-fit simulation
bears strong resemblance to the tumor image, confirming the fitting result. Although the
actual values of the fitted parameters are not the focus of this study, I demonstrate that best-
fit parameters are tightly constrained and identifiable (Table 3b.2). Overall, this example
not only displays that this approach can be used to compare ABM simulations to real-world
images, but that it can be used to fit parameters across spatial scales.
Table 3b.2. Top 10 best fits from fitting an ABM to tumor imaging data for Example 2.
T cell killing
probability
T cell
infiltration
Cancer cell
death
probability
Radius of
hypoxic area
Death
probability in
hypoxic area
Best Fit 3.99´10
-3
0.386 9.98´10
-4
641
5.00´10
-3
Maximum 3.99´10
-3
0.398 9.99´10
-4
698 5.00´10
-3
Minimum 3.95´10
-3
0.382 9.86´10
-4
636 4.81´10
-3
Mean 3.98´10
-3
0.391 9.96´10
-4
647 4.94´10
-3
Coefficient of
Variation
3.94´10
-3
1.33´10
-2
3.98´10
-3
2.98´10
-2
1.21´10
-2
71
Figure 3b.8: Fitting results for Example 2. (A) Fitting with a GA. Black dots – individuals
in the GA. Red line – average fit. Blue line – best fit. (B) Visual comparison of the processed
image for the tumor image (top) and the best-fit simulation (bottom).
Discussion
With this study, I present a method for performing a quantitative comparison between
tumor images and ABM simulations and fitting model parameters. To my knowledge, this is
the first method for truly performing such a comparison, accounting for spatial layouts
without having to manually calculate differences between tumor images and model
simulations for user-defined comparison metrics. I compare the images and model
simulations using representation learning to project model simulations and tumor images
into low-dimensional space and then calculate the distance between the two. Using this
distance as an objective function, it is possible to fit to both model-generated data and actual
biological data. Importantly, I show that with the data-processing step, it is possible to fit
across spatial scales. This is vital for tumors simulated using ABMs, as they are generally
much smaller than actual tumors.
I believe this method provides researchers a way to more accurately set tumor ABM
parameters based on actual image data. This would allow for a stronger bridge between
computational modeling and clinical and experimental research by ensuring that ABMs more
accurately represent tumor images. Additionally, this could have broader impacts beyond
ABMs used for mathematical oncology, as it can be applied to fit ABMs to spatially resolved
data for a range of applications. Specifically, by modifying the data-processing step, this
72
method could be applied to any type of spatial model that requires image data where an
objective function would otherwise have to be designed based on specific metrics extracted
from the image.
One aspect with the data-processing step that must be considered is the scaling of the
simplified images. First, I note that this step can be ignored if the model is the same spatial
scale as image data. However, when ABM simulations and data span different scales, two
considerations are important. The first is that many parameters, such as diffusion rates,
would have to be re-estimated due to the difference in spatial scale between the model and
actual biology. In other studies, researchers scaled temporal parameters to account for this
difference in scale
28
, however other parameters may also be need to be re-estimated. This
means that model parameters become specific to the difference in scale between the model
simulations and the image used for parameter estimation, making it more difficult to transfer
the parameters to a different model. The second consideration is that by cropping each
simplified image to the tumor border, each simulation during parameter estimation is on a
different scale. In this way, the scaling between the tumor image and the simulation becomes
an additional parameter that is estimated.
This method is of course not without limitations. The major limitation is that tumor images
represent a single timepoint, meaning that this method is not temporally resolved. To
improve this, an ABM should be fit simultaneously to an image along with temporal data. For
example, the measured tumor volume over time can be used to further constrain parameter
estimation. An additional limitation is related to the imaging data that can be acquired. If
whole-tumor images are not available, the model simulations should be cropped to the same
region as the image. Lastly, as with many parameter estimation approaches, this method is
computationally expensive, as many simulations are needed in order to fit the neural
network and to perform subsequent parameter estimation. There are strategies for
potentially overcoming the computational resources required. In Example 2, I manually
adjusted the bounds of some parameters and limited the size of the GA population. In
addition, I note that the simulations required for initial estimation of the parameter ranges
can be run in parallel, reducing the computational expensive. Finally, I acknowledge that the
analyses of the parameter estimation results are limited. However, I chose not to include a
critical evaluation of the estimated parameters and their biological implications in order to
maintain focus of this study on implementation of representation learning as an objective
function for tumor ABMs. Future work can provide an in-depth description of critical aspects
of model calibration, including optimizing the cell types and interactions included,
performing parameter sensitivity analysis, and analyzing the fitted parameter values.
I developed a novel method for fitting ABM simulations to tumor images by using the
distance between images and simulations in projected space as the objective function for
73
parameter estimation. I show that this approach is successful, using both model-generated
data and actual tumor images. Overall, this method provides a new way to determine ABM
parameters beyond a visual comparison or simple user-defined metrics.
74
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Conclusion
This Chapter detailed the development and application of a novel use of representation
learning for computational model analysis and parameter estimation of ABMs to tumor
images. The former provides an additional analysis method for computational models,
allowing for a holistic comparison of model simulations. The latter provides, the first method
for directly comparing ABM simulations and tumor images without specifying any metrics,
and shows that this method can be used as an objective function for parameter estimation.
In Chapter 4, I will demonstrate how the methods developed here can be applied to the
model developed in Chapter 2 in order to estimate model parameters from fluorescence
images from a mouse study and to analyze the model.
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Chapter 4: Model parameter estimation and analysis
methods
Introduction
In order to accurately portray biological processes, computational model parameters need
to be estimated based on biological data. Often, parameters will be set based on values from
the literature in order to reduce the number of parameters that need to be estimated. This is
especially the case with tumor ABMs, which have a modular construction and are a challenge
to fit to biological data. However, even for patients with the same cancer type, cancer stage,
and similar tumor markers, it is the general view that each tumor is unique.
1
Therefore, by
fitting to tumor-specific data, a model becomes more capable of predictions that are not only
realistic, but also more relevant.
As discussed in Chapter 3, the challenge of parameter estimation for tumor ABMs is their
spatial dimension. As seen in Chapter 2, simulations that produce similar temporal behaviors
in regard to simply the numbers of cells can have very different spatial behaviors. That
provided the motivation for Chapter 3, where I developed a method to compare the spatial
layout of ABM simulations to image data, displaying its capability first on model-generated
data and then on a simple fluorescence image from an in vitro tumor system. Those examples,
however, simply provided an easy test-bed for the method. Therefore, in this Chapter, I apply
that method to the model developed in Chapter 2 in order to estimate model parameters
from fluorescence images taken from an in vivo mouse study. This Chapter displays that the
method is indeed capable of comparing ABM simulations to in vivo images, and focuses on
considerations and challenges associated with this task.
The first part of this Chapter describes parameter estimation for the model in Chapter 2,
describing how I modify the approach from Chapter 3 in order to account for the more
complex image data and larger numbers of parameters in the model that need to be
estimated. I discuss the challenges, along with its limitations and ways to potentially improve
parameter estimation, both on the computational side and on the experimental side.
Furthermore, I show how the method from Chapter 3 can be used as a way to analyze tumor
ABMs. In that Chapter, I discussed how projecting simulations into low-dimensional space
provides an easy way to compare them simply by taking the distance between them in
projected space. As I already discussed how that can be applied, albeit using different
computational models, I will not discuss the use of the distance metric here. Instead, I extend
that approach to show how using representation learning to project model simulations into
78
low-dimensional space allows for the visualization of how the spatial state develops with
time, for many simulations at once.
Due to limitations that I will discuss in parameter identifiability, the focus of this Chapter is
to present how the methods from Chapter 3 can be applied to in vivo data and then potential
analyses that can be performed, instead of discussing biological insights from this model and
data. This Chapter therefore provides a necessary computational foundation for future
studies.
Methods
Overview
This Chapter applies the model analysis methods developed in Chapter 3 to the model
constructed in Chapter 2 in order to show how they can be utilized in a real-world setting.
First, I use a modification of the Chapter 3 methods to estimate the parameters for the
Chapter 2 model from fluorescent tumor images taken from a mouse study. After parameter
estimation, I show how representation learning can be applied specifically to tumor ABMs in
order to visualize and compare spatial trajectories across large numbers of simulations, and
to cluster simulations based on their spatial states.
I first describe the image data and how I process it and simulations, along with modifications
to the model based on characteristics of the images. I then detail the parameter estimation
that I performed, detailing modifications from what was utilized in Chapter 3 and the
reasonings for those modifications. Lastly, I describe how I use representation learning to
analyze model spatial development, detailing how I train a NN for this and how it is
implemented.
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Figure 4.1: Schematic of Chapter 4. After processing both the fluorescence images and model
simulations, I constrain the parameter space via Monte Carlo simulations and basic
comparisons to the images. I then use the distance between simulations and images in projected
space to estimate model parameters. Lastly, I train neural networks to project spatial layouts
based on time points, using this to analyze spatial trajectories via projected space.
Imaging data
The fluorescence images used in this Chapter were taken from mouse studies done by
collaborators at the Keck School of Medicine. Injected HER2+ breast cancer cells grew for 23
or 24 days before mice were sacrificed, the tumors extracted, and tumor sections imaged.
The images show the entire tumor slice, with 7 of the tumors being roughly circular (an
example shown in Figure 4.2), allowing for an easy comparison to model simulations. Images
were stained with fluorescence markers for DAPI, CK5, CD4, HER2/NEU, CD8a, Epcam,
FOXP3, PD-L1, and F4/80. Using QuPath software,
2
cell coordinates were extracted based on
DAPI staining, and immune cells were identified based on expression of markers shown in
Table 4.1. Fluorescence thresholds were set based on a visual inspection of the images. Cells
not identified as immune cells are assumed to be cancer cells.
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Figure 4.2: Example fluorescence tumor image. Circle shows an example region of the tumor
following the “Cell Detection” function in QuPath, with cells outlined in red. Fluorescence for
CD4 and CD8a are shown, highlighting one each of those respective cell types.
Table 4.1: Fluorescence markers that correspond to each cell type.
Cell Type Markers
Macrophage F4/80
CD4 T cell CD4
T regulatory cell CD4, FOXP3
CD8 T cell CD8a
Model modification
The tumors from the mouse study show large necrotic regions roughly in their centers.
3,4
To
capture this with the model, I add a necrosis module. A circular necrotic region grows from
the center of the tumor outward at a rate proportional to the number of cancer cells. This
region grows up to a set limit, which is the distance between the necrotic region and the
tumor edge, representing the diffusion limit of oxygen and factors such as angiogenesis.
5
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Therefore, if the tumor shrinks at any point, the necrotic region shrinks as well. I assume the
necrotic region exerts a minor outward force (due to the buildup of cell debris) and any cells
that enter the necrotic region or are overcome by it are killed. While this is not as detailed as
the necrotic regions produced by explicitly modeling cancer cell metabolism and
vasculature,
6
it produces a necrotic region using a minimum number of parameters,
facilitating parameter estimation.
Image and simulation processing
Image and simulation scaling
With agent-based modeling of tumors, it is often the case that model simulations are
noticeably smaller than biological tumors. In addition, simulation times are shorter than
biological times, as human tumors often take many years to develop. This reduction in space
and time is due to limits in computational power and is an unfortunate necessity when
modeling. Some models are able to achieve large-scale simulations, however the
computational time needed prohibits large numbers of simulations.
7
Therefore, in order to
compare simulations to images, these differences in scale must be accounted for.
8
In this
study, I only need to account for spatial scaling, as the time duration of the mouse studies is
within a feasible computational time.
To determine the spatial scaling factor, I first determine the size of the final mouse tumors.
Using QuPath, I take the vertical and horizontal radii of each tumor and average them to get
an approximate radius of each tumor. Taking the average for all tumors, which were all of
similar sizes, I get an estimate of how large the tumor should be at its final time point. I then
divide this by a simulation radius that would fall within a reasonable computational time,
which gives the scaling factor between simulations and images. Adjusting this to an integer
value for ease of implementation, I obtain a scaling factor of 4. To then determine the initial
tumor size for the simulations, I calculate the initial size of the injected tumors. Mice were
injected with either 5x10
4
or 1x10
5
cancer cells (I note this did not appear to impact final
tumor size). Using the volume of one cell (assuming a 20µm cell diameter), I calculate the
initial tumor volume and, assuming a spherical tumor, I calculate the initial tumor radius and
divide by the scaling factor. I determine that, based on injected tumor size, the initial
simulation radius should be either 4.6 or 5.8 cells. For simplicity, I use a radius of 5 cells for
all simulations.
Image and simulation formatting
As described above, cell coordinates are extracted from stained images, based on DAPI
staining, giving a list of cells with their locations and types. This is the same format as the
ABM output, and simplified images were constructed following the method established in
Chapter 3, yielding images 50 pixels x 50 pixels in size with five color channels (cancer cells,
macrophages, CD4, Treg, CD8). While the ABM differentiates between macrophage subtypes
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and active/suppressed CD8 cells, this information is not present in the images. Therefore,
when processing model simulations, macrophage subtypes are binned together, as are CD8
states. As described later, this reduction of information inhibits the estimation of some
parameters.
When creating the simplified images, I account for tumor size instead of binding the
simplified images to the tumor edge, like I do in Chapter 3, as I already know how large the
simulated tumor should be. The size of the grid (number of grid spaces multiplied by the size
of one grid space) prior to shrinking to 50x50 pixels is set constant for all simulations and
tumor images that is 1.1 times the largest tumor size (from the images), and cells are
discretized so that the tumor is centered in the grid. Based on the size of the simulated tumor,
the tumor may extend past the edges of the grid (if the tumor is very large) or there could be
a lot of empty space between the tumor edge and the edges of the grid (if the tumor is very
small). I do this to achieve a more accurate size comparison when comparing simulations to
the images, thus increasing the accuracy of the estimated parameters.
Parameter estimation
Overview
In general, parameter estimation is a difficult task, one that becomes harder and harder as
the number of parameters being estimated grows, since different regions of parameter space
may produce similar behaviors and not all parameters may be identifiable.
9,10
It is also
important to note that while the model is on the same temporal scale as the data, they are on
different spatial scales. This means that some parameters that would otherwise be taken
from the literature need to be either scaled or estimated in order to account for the spatial
scaling. While some studies have scaled certain parameters based on differences in temporal
scales, the difference in spatial scale is usually ignored.
8
In order to properly account for
spatial differences, I include parameters such as cell division rates and lifespan in the
estimation. To reduce some of the parameters that need to be estimated, cell migration rates
and diffusion limits were taken from the literature and then scaled using the scaling factor
calculated above, as these are direct spatial parameters.
As will be discussed in the following sections, parameter estimation for this model is an
extremely challenging task. Many parameters can contribute to similar effects and many are
also temporally influenced. Because of the relatively low amount of information contained
in the images (macrophage subtypes are binned, and images are only for the final timepoint),
I cannot confidently estimate a single set of best parameters. Indeed, even parameters that
provide the best fit may not be entirely accurate due to the lack of temporal information and
macrophage state. Therefore, instead of using a computationally expensive genetic algorithm
as in Chapter 3 to perform an exhaustive parameter estimation, which would take
significantly longer than in Chapter 3 due to the large number of parameters that need to be
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estimated, I perform a much simpler estimation to obtain several sets of reasonable
parameters, which I use to demonstrate later analyses.
Initial parameter range estimation
In order to narrow down the search ranges for each parameter, I first performed 100,000
Monte Carlo simulations. Parameter ranges were set based on the nature of each parameter.
After these simulations were completed, I parsed parameter sets based on if the final
simulation state was both within the scaled diameter bounds and contained cell type ratios
similar to those calculated from the images. The parameter sets that met these criteria were
used to then set parameter ranges for the next step described below, starting with a new set
of 10,000 Monte Carlo simulations from this new range.
Parameter estimation via learned representations
In Chapter 3, I introduced the concept of using neural networks to project images and
simulations into low-dimensional space, using the distance between them as an objective
function for a genetic algorithm. However, from the initial parameter estimation, I find that
a wide range of parameter values will produce simulations that meet the size and cell ratio
requirements. To proceed from this, I take the top 100 closest parameter sets for each image
(after parsing based on tumor size and cell ratios), as calculated by an ensemble of 50 neural
networks. From these, I generate 10 new parameter sets per parameter set by randomly
sampling from 1.25 times above and below each parameter, therefore exploring the local
parameter space around each set. I perform this several times, taking the top 10 best fits per
image from the final set of simulations as the final parameters. The neural networks used for
this section were trained in the same manner as in Chapter 3 using only the tumor state at
the final time point, and distance was calculated as an average of an ensemble of 50
networks.
Simulation analysis
Overview
One of the largest advantages of in silico modeling is that the system can be interrogated at
every time point. However, with ABMs, it is difficult to compare the spatial state of model
simulations over time for large numbers of simulations. Therefore, I use representation
learning to project spatial layouts for multiple time points into two-dimensional space,
allowing for the visualization of how the simulations change with time, facilitating easy
comparisons between simulations.
I describe first how I train neural networks for this analysis, extending the training method
described in Chapter 3, followed by the simulated data used for training. I then describe the
simulations used to showcase this approach as an analysis tool.
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Neural network training for trajectory analysis
I train a neural network for representation learning via the method introduced in Chapter 3,
with one key modification. Because I want to use this neural network to project different
time points, I make the assumption that subsequent time points for a simulation should be
near each other in projected space (Figure 4.3). While it is possible that the neural network
will simply learn this on its own, I decided instead to enforce this while training. Therefore,
instead of training pairs consisting of two augmentations of the same input, training pairs
consist of augmentations of subsequent time points. This influences the learned
representations so that their projections will be near each other.
Figure 4.3: Schematic showing that adjacent time points should be closer to each other
in projected space.
Besides modifying the training method, I increase the amount of information contained in
the inputted simulations. Because I are no longer comparing to tumor images from the
mouse model of breast tumor growth, but rather using only ABM simulations, I can include
macrophage subtype and CD8 state (active or suppressed) in this analysis, so that each
simplified image contains 8 channels (cancer, M0, M1, M2, CD4, Treg, CD8, CD8 suppressed).
Neural network training data
I generate a set of training data via Monte Carlo sampling. The reason for using a different
set of training data, which sampled over a wider parameter range, is that I found it led to an
overall lower training loss, which, upon an inspection of the projected trajectories, led to
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better projections (described in detail in the results). For each of the top 10 parameters for
each image, I sample 10 parameter sets, randomly selecting each parameter from 2-fold
below to 2-fold above the value of the best fit. This gives a total of 700 simulations for
training, each containing 25 time points. Starting at the first time point, I pair subsequent
points so that each time point only exists as part of one pair, meaning that it is only paired
with one of its neighbors. I found that this did not impact the training results, however, it
greatly improves training time. This gives a total of 8,400 training pairs.
Simulations for analysis
To display how tumor trajectories can be analyzed using representation learning, I perform
10 simulation replicates with each of the 10 best parameter fits for each image. From this,
stochasticity between simulation replicates can be compared, along with how different
parameter sets lead to different spatiotemporal trajectories.
Results
Processing of tumor images
In Table 4.2, I show the ratios of each cell type following extraction of cell coordinates and
determination of cell type based on fluorescence ratios. From these, I set cutoff ratios for
simulations when performing parameter estimation, which were slightly larger than the
ranges identified from the images. Following the described discretization process, I create
50x50 pixel simplified images with 5 color channels (cancer, macrophage, CD4, T regulatory,
CD8), which are shown for each image in Figure 4.4. An import note is that there are very
low numbers of T cells present in these images, especially CD8 cells. As seen in Chapter 2,
large numbers of CD8 cells are needed to overcome tumor growth. Because of the very low
number of CD8 cells present here, they would have very little impact on tumor growth
regardless of immunosuppression. Therefore, the values of many of the parameters relating
to CD8 suppression are likely unidentifiable, which is why I focus only on the applications of
computational methods here and not on biological insight.
Table 4.2: Cell ratios for each cell type. Used to constrain parameter search.
Cell Type Fraction in images (min, mean,
max)
Fraction for parameter
estimation (min, max)
Cancer (0.746, 0.889, 0.975) (0.65, 1.0)
Macrophage (0.015, 0.100, 0.240) (0.01, 0.3)
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CD4 (0.004, 0.008, 0.011) (0.0035, 0.03)
T regulatory (0.0004, 0.001, 0.002) (0.00035, 0.005)
CD8 (0.0002, 0.003, 0.008) (0.00015, 0.01)
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Figure 4.4: Processed versions of each fluorescence image.
Parameter estimation
From parameter estimation, I produce 10 best parameter sets for each image, for a total of
70 parameters. After dividing each parameter distribution by its mean in order to eliminate
88
the effect of magnitude, I show in Figure 4.5 boxplots for each parameter. Table 4.3 shows
the means and coefficients of variation for each parameter. The mean estimated value for
cancer division probability yielded an interesting result. This is one of the few parameters in
the model that has direct biological meaning, with its inverse yielding cell cycle duration. I
chose to estimate this parameter instead of using literature values due to the fact that I are
modeling on a different spatial scale, however the mean estimated cell cycle was around 24
hours, which is a biologically reasonable value.
11–13
This means that cell cycle rates can
indeed be taken from literature values for ABMs, even when operating on a smaller spatial
scale. In future studies, I can then fix this parameter to lower the number of parameters that
need to be estimated.
Figure 4.5: Box plots for each estimated parameter. Parameter distributions were first
divided by their means in order to facilitate comparison, due to differences in magnitude.
Table 4.3: Estimated mean and coefficient of variation for each parameter.
Name Description Mean CoV
cancerDiv Cancer proliferation probability 0.040 0.070
cancerDeath Cancer death probability 0.005 0.214
cancerPDL1_m Cancer PD-L1 probability, when expressed 0.738 0.179
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cancerPDL1_g Cancer probability of gaining PD-L1 0.5213 0.224
cancerM1Death Cancer death probability from M1 0.044 0.346
cd4Death CD4 death probability 0.038 0.106
cd4MigSpeed_inf CD4 migration speed in tumor 60.1 0.217
cd4Diff CD4 differentiation probability 0.132 0.513
cd4MigBias CD4 migration bias 0.147 0.203
cd4MigBias_inf CD4 migration bias in tumor 0.137 0.183
cd4CTLA4 CD4 CTLA-4 probability when Treg 0.139 1.04
cd8Death CD8 death probability 0.013 0.546
cd8MigSpeed_inf CD8 migration speed in tumor 51.0 0.146
cd8KillProb CD8 kill probability 0.041 0.198
cd8InfScale CD8 killing scale 5.75 0.192
cd8MigBias CD8 migration bias 0.151 0.217
cd8MigBias_inf CD8 migration bias in tumor 0.022 2.75
cd8Div CD8 proliferation probability 0.053 0.175
macDeath Macrophage death probability 0.014 0.371
macMigSpeed_inf Macrophage migration speed in tumor 13.3 0.128
macM1 Macrophage differentiation probability to M1 0.186 0.326
macM2 Macrophage differentiation probability to M2 0.432 0.242
macMigBias Macrophage migration bias 0.245 0.040
macMigBias_inf Macrophage migration bias in tumor 0.134 0.236
macPDL1 Macrophage PD-L1 probability as M2 0.264 0.195
macPlasticity Macrophage probability of redifferentiation 0.400 0.237
90
cd8RecRate CD8 recruitment rate 0.0001 0.220
macRecRate Macrophage recruitment rate 0.006 0.257
cd4RecRate CD4 recruitment rate 0.001 0.305
recDelay Recruitment delay 2.85 0.152
necroticGrowth Necrosis growth rate 0.008 0.307
necrosisLimit Necrotic region limit 940.3 0.196
In Figure 4.6, I show the processed simulations for the top five best fits for one of the images.
One thing to note is that the model is radially symmetric, whereas the actual tumor is not, as
displayed by the shape of the necrotic region and the areas where immune cells are
infiltrating. Interestingly, across all of the images, macrophages tended to cluster on one side
of the tumor, while the T cells showed more radially consistent infiltration. However,
increasing the detail of the model to capture this behavior would have added more
parameters to estimate, making overall parameter estimation more difficult.
91
Figure 4.6: Top 5 best fits for one of the images.
Using representation learning to analyze tumor trajectories
I tried several approaches to training the neural networks for analyzing simulations. In later
sections, I analyze 700 simulations: 10 simulation replicates for the top 10 best fit
parameters for each of the 7 images. Initially, I trained the neural networks solely on the
92
simulations that I were analyzing, which yielded, in projected space, trajectories that circled
back towards the initial time point. This, however, did not match with actual simulation
behavior, based on a visual inspection. I noticed that neural networks that achieved lower
training losses yielded projections that better matched simulation behavior, based on the
visual inspection. I then generated a separate set of training data by sampling parameters
from 2 times below to 2 times above the values for the simulations I aimed to analyze. I found
that this yielded lower training loss and produced better projections of the analysis
simulations (Figure 4.7).
Figure 4.7: Average training loss vs training epoch (training step) for the neural
networks. Extended dataset - randomly sampling parameters two-fold above and below the
estimated values. Projected dataset - training the neural network on the simulations
(generated using only the estimated parameter values) that I project and analyze in later
sections.
In Figure 4.8, I show an example projection for a single simulation, along with the spatial
layouts for several time points. I see that, as expected from the spatial layouts, the projected
tumor moves farther and farther away from the initial state as the simulation progresses.
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This displays how a complex spatial trajectory can be easily visualized in two dimensions
using representation learning.
In Figure 4.9, I show projected trajectories for 10 simulation replicates for the top 10
parameters for each image (a total of 700 simulations). I trained 25 neural networks,
showing the projections for the 10 with the lowest training loss, in order from lowest loss to
highest. The green point is the starting point, which is the same for all simulations, the red
points are the end-states, and the black lines the rest of the time points for each simulation.
I see that, although the trajectories are fairly wide, most simulations tend to move in the
same direction. This is expected, as, from a visual perspective, the seven tumor images were
all similar to each other. Therefore, the best-fit parameters should yield similar simulations.
Additionally, while the orientations of the projections are not consistent between neural
networks, the overall shapes of each projection are very similar. This is expected, as neural
networks are stochastically initialized and trained. The consistency also provides confidence
to the projections, meaning that the neural network is learning the same features each time
it is trained.
94
Figure 4.8: Projection for a single simulation. Each point represents one day, with the
spatial layouts shown for days 0, 8, 16, and 24.
95
Figure 4.9: Projections for all 700 simulations. Each subplot is the projections for one neural
network. The green dot represents day 0, which is the same for all simulations. The red dots are
the final states at day 24. The black lines are the rest of the time points for each simulation.
Clustering final simulated tumor representations
For the following analysis, I only use the neural network that trained with the lowest loss.
Clustering analysis reveals that the projected points for the final time points separate into
two very distinct clusters (Figure 4.10), coloring the entire trajectories in order to visualize
how the simulations change with time. I see that, from the onset, the clusters develop
following different paths. However, while clearly distinct from each other, these two paths
have similar overall shapes, originally going down before moving to the left. From these two
clusters, I can examine the differences in parameters for each cluster. In Figure 4.11, I first
divide all of the parameters by their means (from the 700 simulations) before separating
them by cluster and plotting the mean of each parameter for each cluster. The largest
differences in parameter means for the two clusters are in CD4 differentiation, regulatory T
cell CTLA-4 expression, CD8 migration bias in the tumor, and the growth of the necrotic
region. Lastly, I colored the projected points by their cell counts (Figure 4.12), which is an
easy metric to extract from the simulations. I see that the red cluster has more cancer cells
and regulatory T cells, but fewer of the other cell types, than the blue cluster, providing some
insight into the differences between the two clusters.
Figure 4.10: Projections separated by clustering the final time point.
96
Figure 4.11: Bar graphs showing mean parameter values for each cluster. Error bars
show the standard deviation. Bars are colored to correspond to the previous figure.
Figure 4.12: Projections of the final time points, colored by cell counts. Purple is low, while
yellow is high. In each subplot, the top-left cluster corresponds to the red cluster in Figure 4.9
while the bottom right cluster corresponds to the blue cluster.
Discussion
In this Chapter, I apply the methods developed in Chapter 3 to the model developed in
Chapter 2 and show how these methods work in an actual biological setting. I display how
these methods are best applied to work with complex imaging data from an in vivo study.
Despite making use of imaging data and a complex biological model, the focus of this Chapter
is on the application of the methods. Because of this, along with some parameter limitations
discussed below, this Chapter does not focus heavily on the biological implications of the
fitted model.
97
Based on the nature of the images and the number of parameters to estimate, I perform a
simple estimation, instead of the rigorous method described in Chapter 3. This method
combined simple general statistics extracted from the images, such as the relative ratios of
cell types present, to filter out unsuitable parameter sets with the distance metric calculated
via projection to learned space. I implement a simple parameter search method where the
closest projected simulations explore their local region in parameter space, keeping the best
performing parameter sets after each step. Estimation could be improved in several ways,
including investing more time and computational power into the fitting, using different
image staining to increase the information contained in the images, incorporating temporal
measurements (such as tumor volume) along with the images, or simplifying the model to
reduce the number of parameters that need to be estimated. One important result is that,
despite the difference in spatial scale, the estimated cancer division rate is a biologically
realistic value. This means that this and other parameters can either be set to literature
values to eliminate the need to estimate them, or their ranges can be much smaller. Other
parameters can also be fixed based on researcher intuition and biological understanding of
the system. However, I chose to avoid this, simply to observe how the method behaved.
The main limitation in this Chapter, which makes biological analysis difficult, is that many
parameters are essentially non-identifiable. For some, such as macrophage differentiation
probabilities, this is due to the low information contained in the image data. By only having
the locations of all macrophages, and not their distinct states, macrophage differentiation is
essentially impossible to estimate confidently. This second reason for why some parameters
are difficult to analyze is that their value makes no impact on the simulation. CD8 killing
probability, for example, makes very little difference due to CD8 cells being a very small
percent of the population. Even if this probability was set to 1, there would be very little
effect, simply because there are so many cancer cells in comparison. As I saw in Chapter 2, a
large number of CD8 cells is needed to overcome the tumor. Therefore, other parameters,
such as PD-L1 expression and parameters related to the functions of immune cells, are
difficult to accurately estimate.
Overall, this Chapter provides a necessary foundation for fitting tumor ABMs to imaging data
and then analyzing model simulations using representation learning. I show that it is
possible to fit directly to fluorescence images taken from in vivo studies after minor image
processing, then display how the spatial states of simulations can be analyzed temporally to
compare tumor development between simulations. I acknowledge some important points to
consider when fitting a large ABM. However, I demonstrate that this novel approach can
provide meaningful parameter estimates and can form the basis of relevant model analyses.
98
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3. Yee, P. P. & Li, W. Tumor necrosis: A synergistic consequence of metabolic stress and
inflammation. BioEssays 43, 2100029 (2021).
4. Bredholt, G. et al. Tumor necrosis is an important hallmark of aggressive
endometrial cancer and associates with hypoxia, angiogenesis and inflammation responses.
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5. Lugano, R., Ramachandran, M. & Dimberg, A. Tumor angiogenesis: causes,
consequences, challenges and opportunities. Cellular and Molecular Life Sciences 77, 1745–
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6. Cess, C. G. & Finley, S. D. Multiscale modeling of tumor adaption and invasion
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7. Ghaffarizadeh, A., Heiland, R., Friedman, S. H., Mumenthaler, S. M. & Macklin, P.
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8. Alfonso, J. C. et al. Tumor-immune ecosystem dynamics define an individual
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9. Lillacci, G. & Khammash, M. Parameter estimation and model selection in
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11. Gong, C. et al. A computational multiscale agent-based model for simulating spatio-
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Chapter 5: Conclusion
This thesis aimed to develop methods to improve the utility and analysis of agent-based
models of tumor growth. The focus was on the development and implementation of novel
model-analysis methods, rather than on insights into tumor biology. While the work here
focused on tumor interactions with the immune system, it can be easily applied to other
aspects of the tumor microenvironment, bridging the gap between spatial models and in vivo
images.
The purpose of Chapter 2 was to present a generalized model of the interactions between
cancer cells, macrophages, CD4 T cells, and CD8 T cells. This model compiles complex
networks of interactions into phenomenological summaries in order to capture population-
level behavior. It provides a framework that can be easily modified based on specific
modeling objectives. One of the major novel pieces of this model is the replacement of
diffusion with a distance function. This function assumes that the maximal effect of a
diffusible factor is at the center of the secreting cell and decreases exponentially as the
distance increases.
1,2
The output of this function is treated as a scaling factor for whatever
effect it induces, with the value of the parameter related to the maximum effect possible. The
effects of multiple cells that secrete the same factor are treated as probabilities, so that the
maximum value that can be reached is 1. This is similar to how other studies have treated
diffusion,
3–5
without the need to numerically solve PDEs, greatly improving computational
time. Though I only show this concept with factors secreted by cells, it can easily be extended
to other diffusible processes, such as oxygen diffusion out of blood vessels or diffusion of
drug into a tumor, depending of course on the level of detail required for the model.
Chapter 3 was a novel application of representation learning to analysis of computational
models. Representation learning involves the use of neural networks to learn a low-
dimensional representation of a complex input in order to facilitate downstream tasks.
6,7
Instead of applying this to actual data, I instead apply it to model simulations as a form of
analysis. By training using Euclidean distance, model simulations can be projected into low-
dimensional space with the trained neural network. The distance between simulations in
low-dimensional space is then used as a measure of their difference, with the assumption
that simulations that are closer in space are more similar to each other. This works
regardless of the type of simulation and avoids the need for complex comparison metrics. Of
course, this approach does not negate the need for more specific comparisons between
simulations. However, it provides an additional method that holistically compares
simulations and also facilitates clustering without introducing bias from the researcher. I
then proceed to extend this approach to compare tumor ABM simulations to images in order
to create an objective function for parameter fitting that does not require specific
calculations to compare images and simulations, providing the first method that creates an
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actual objective function for fitting spatial models to image data without heavy extraction of
statistical metrics.
8
I show the validity of this approach first by fitting an ABM to its own
simulation, where I know what the estimated parameters should be, then by fitting to images
from an in vitro tumor system.
Finally, Chapter 4 combines Chapters 2 and 3. The parameters for the model from Chapter 2
are estimated from fluorescence tumor images taken from an in vivo mouse study using the
methods from Chapter 3. The purpose of this Chapter was to show that parameters can be
estimated from in vivo images and how the model can then be analyzed. I show that the fitting
method from Chapter 3 is indeed applicable to more complex models and images. I also
display how representation learning can be applied to analyze tumor ABMs, specifically
demonstrating how it allows for plotting spatial trajectories over time. In this Chapter, I
discussed some of the challenges of estimating large numbers of parameters from image data
and potential ways this could be improved, both on the modeling side and experimental side.
I then discussed how a neural network can be trained to explicitly learn spatial trajectories.
Lastly, I use the approach to project simulations into two-dimensional space for easy analysis
of tumor development.
In conclusion, this thesis provides the necessary foundation for future studies of tumor
growth that will now be able to take better advantage of imaging data. Combined with other
measurements such as tumor volume, models can be easily calibrated to mouse studies,
providing a computational playground for testing many different aspects of the tumor
microenvironment in order to inform further experimental studies. This thesis also provides
a way to easily compare spatial states for large numbers of simulations. Besides building
models from mouse studies, this work is foundational to more complex studies that will be
able to calibrate models to patient data, allowing for personalized predictions.
101
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Appendices
C.G. Cess and S.D. Finley. Data-driven analysis of a mechanistic model of CAR T cell signaling
predicts effects of cell-to-cell heterogeneity. Journal of Theoretical Biology. 2020.
https://doi.org/10.1016/j.jtbi.2019.110125
C.G. Cess and S.D. Finley. Multi-scale modeling of macrophage – T cell interactions within the
tumor microenvironment. PLOS Computational Biology. 2020.
https://doi.org/10.1371/journal.pcbi.1008519
C.G. Cess and S.D. Finley. Multiscale modeling of tumor adaption and invasion following anti-
angiogenic therapy. Computational and Systems Oncology. 2022.
https://doi.org/10.1002/cso2.1032
Journal of Theoretical Biology 489 (2020) 110125
Contents lists available at ScienceDirect
Journal of Theoretical Biology
journal homepage: www.elsevier.com/locate/jtb
Data-driven analysis of a mechanistic model of CAR T cell signaling
predicts effects of cell-to-cell heterogeneity
Colin G. Cess
a
, Stacey D. Finley
a , b , c , ∗
a
Department of Biomedical Engineering, University of Southern California, Los Angeles, CA, United States
b
Mork Family Department of Chemical Engineering and Materials Science, University of Southern California, Los Angeles, CA, United States
c
Department of Biological Sciences, University of Southern California, Los Angeles, CA, United States
a r t i c l e i n f o
Article history:
Received 18 October 2019
Revised 13 December 2019
Accepted 18 December 2019
Available online 19 December 2019
Keywords:
Intracellular signaling
Computational modeling
Partial least-squares
a b s t r a c t
Due to the variability of protein expression, cells of the same population can exhibit different responses
to stimuli. It is important to understand this heterogeneity at the individual level, as population averages
mask these underlying differences. Using computational modeling, we can interrogate a system much
more precisely than by using experiments alone, in order to learn how the expression of each protein af-
fects a biological system. Here, we examine a mechanistic model of CAR T cell signaling, which connects
receptor-antigen binding to MAPK activation, to determine intracellular modulations that can increase
cellular response. CAR T cell cancer therapy involves removing a patient’s T cells, modifying them to ex-
press engineered receptors that can bind to tumor-associated antigens to promote tumor cell killing, and
then injecting the cells back into the patient. This population of cells, like all cell populations, would
have heterogeneous protein expression, which could affect the efficacy of treatment. Thus, it is impor-
tant to examine the effects of cell-to-cell heterogeneity. We first generated a dataset of simulated cell
responses via Monte Carlo simulations of the mechanistic model, where the initial protein concentrations
were randomly sampled. We analyzed the dataset using partial least-squares modeling to determine the
relationships between protein expression and ERK phosphorylation, the output of the mechanistic model.
Using this data-driven analysis, we found that only the expressions of proteins relating directly to the
receptor and the MAPK cascade, the beginning and end of the network, respectively, are relevant to the
cells’ response. We also found, surprisingly, that increasing the amount of receptor present can actually
inhibit the cell’s ability to respond due to increasing the strength of negative feedback from phosphatases.
Overall, we have combined data-driven and mechanistic modeling to generate detailed insight into CAR
T cell signaling.
©2019 The Author(s). Published by Elsevier Ltd.
This is an open access article under the CC BY-NC-ND license.
( http://creativecommons.org/licenses/by-nc-nd/4.0/ )
1. Introduction
Even among cells of the same type, phenotypic differences can
arise due to variations in protein abundance, which are caused by
the stochastic nature of gene expression ( Fraser and Kaern, 2009 ;
Mantzaris, 2007 ; Niepel et al., 2009 ). While the response of a sig-
naling network may be robust to variations in the expressions of
some proteins, the expressions of other proteins may be highly
influential, causing variances that compound into significant dif-
ferences at the cellular level ( Altschuler and Wu, 2010 ). Although
methods such as flow cytometry can measure protein expression
∗
Corresponding author at: 1042 Downey Way, DRB 140, Los Angeles, CA 90089,
United States.
E-mail address: sfinley@usc.edu (S.D. Finley).
in individual cells, it is still difficult to examine experimentally all
of the proteins in a network and how they relate to network re-
sponse. Using computational modeling, it is possible to determine
how variations in protein expression affect phenotypic outcome by
precisely controlling protein amounts and simulating their effects.
Computational mechanistic models comprised of ordinary dif-
ferential equations can be analyzed using various approaches.
Some methods of analyzing these models, such as sensitivity anal-
ysis, require a large computational cost to be performed in all di-
mensions. The computational resources required to analyze such
models can be prohibitive, especially as model size and complex-
ity increases. As an alternative, data-driven methods can be used
as a way of analyzing a model in all possible dimensions at once.
While data-driven methods are unable to model actual biological
interactions, they are able to generalize the relationships between
https://doi.org/10.1016/j.jtbi.2019.110125
0022-5193/© 2019 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license.
( http://creativecommons.org/licenses/by-nc-nd/4.0/ )
2 C.G. Cess and S.D. Finley / Journal of Theoretical Biology 489 (2020) 110125
model inputs and outputs, providing information on how each in-
put acts across all dimensions. For example, Hua et al. used a de-
cision tree to analyze the effects of differences in protein expres-
sion in a model of Fas-mediated caspase-3 activation. This allowed
the authors to see how individual proteins worked together to in-
fluence the response of the system ( Hua et al., 2006 ). However, a
major drawback to this approach is that as the number of proteins
in the network increases, the decision tree must expand as well,
containing many more nodes and branches until it becomes very
difficult to analyze. We propose here to use partial least-squares
(PLS) as an alternative way to analyze a mechanistic model.
PLS provides information on how the inputs of a system, in this
case the initial protein expressions, relate to the system’s outputs
( Wold et al., 2001 ; Kreeger 2013 ). Although PLS does not give in-
formation about specific relationships between proteins, it does tell
how the expression of each protein generally affects the response,
providing information on the population as a whole. It is also easy
to analyze, with multiple quantitative metrics providing informa-
tion on the influence of the inputs.
In this study, we apply PLS to a mechanistic model of chimeric
antigen receptor (CAR) T cell signaling. The model connects
receptor-antigen binding to the MAPK cascade, resulting in the
phosphorylation of ERK, which is one characteristic of T cell ac-
tivation. CAR T cells are a type of cell-based immunotherapy in
which T cells that have been taken from a cancer patient are mod-
ified to express the CAR on the cell surface, such that the cells can
directly bind to the tumor-associated antigen that the CAR recog-
nizes. Once put back into the patient, these engineered T cells can
be directly activated by tumor cells, allowing the CAR T cells to
kill the diseased cells without targeting other cells in the body
( Androulla and Lefkothea, 2018 ; Cho et al., 2018 ). However, this
therapeutic approach has some limitations; for example, there are
many instances of no response from the patient. Many different
types of CARs have been developed to try to increase tumor killing.
Most of those modifications focus on the CAR itself (i.e., creat-
ing a new receptor with different signaling domains) and not the
downstream signaling network. Using PLS, we focus on intracellu-
lar variations of proteins involved in a signaling network that in-
fluences how a CAR T cell responds to stimulation. We use Monte
Carlo simulations to generate a dataset with protein expressions
as the inputs and ability to respond to stimulation, based on ERK
phosphorylation, as the output. We find that only a small subset
of proteins in the network, those relating to the receptor and to
the MAPK cascade, have expressions that significantly influence the
network’s response.
2. Methods
2.1. Mechanistic model of T cell signaling
This study employed the use of a mechanistic model of CAR
T cell signaling that was previously developed ( Rohrs et al.,
2019 ). This model was constructed using a modular approach with
smaller models that account for lymphocyte-specific protein ty-
rosine kinase (LCK) regulation, CAR phosphorylation, LAT signalo-
some formation, CD45 phosphatase activity, mitogen-activated pro-
tein kinase (MAPK) activation, and feedback from the phosphatase
SHP1. The model structure is shown in Fig. 1 . In our model, the
kinase LCK initiates signaling once the antigen CD19 is bound. LCK
can undergo autophosphorylation. It can also catalyze phosphory-
lation of various sites on the CAR, which consists of the CD3 ζ do-
main and the CD28 co-stimulatory domain. The CD3 ζ domain is
comprised of three immunoreceptor tyrosine-based activation mo-
tifs (ITAMs), each having two phosphorylation sites. These six sites
on CD3 ζ are phosphorylated by LCK independently, in a random
order, and with distinct kinetics ( Rohrs et al., 2018 ). Once two sites
on an ITAM are phosphorylated, ZAP70 can bind. ZAP70 can also be
phosphorylated by LCK to become catalytically active. Active ZAP70
promotes the formation of the multi-protein complex called the
“LAT signalosome”, including LAT, SLP76, ITK, SOS, and other pro-
teins. The LAT signalosome promotes signaling through the MAPK
pathway, a three-layer cascade of phosphorylation reactions involv-
ing Ras, Raf, MEK, and ERK.
Using this model, we are able to simulate T cell signaling ini-
tiated by antigen binding to the extracellular domain of the CAR
and culminating with activation of the MAPK pathway to produce
doubly phosphorylated ERK (ppERK). Here, we consider the con-
centration of ppERK as the primary model prediction, and it is the
focus of all model simulations.
In total, the model consists of 245 total species, 23 of which
have non-zero initial conditions, and 159 parameters. The species
in the model represent molecular biochemical species that can in-
teract with one another through binding and catalyzing phospho-
rylation and dephosphorylation reactions. Thus, the set of species
consists of unphosphorylated proteins (such as ERK), proteins with
varying levels of phosphorylation (for example, singly and doubly
phosphorylated ERK), free proteins (such as unbound ZAP70) and
various protein complexes (such as ZAP70 bound to a CAR where
ITAM A is doubly phosphorylated). The parameters characterize the
rates of phosphorylation and dephosphorylation, protein binding
rates, and enzyme catalytic activities.
The model was constructed using BioNetGen ( Harris et al.,
2016 ). BioNetGen is a rule-based approach for model construction
that produces the set of nonlinear, coupled ordinary differential
equations (ODEs) to describe how the molecular species’ concen-
trations evolve over time. The parameters of the model were pre-
viously fit to quantitative membrane reconstitution experiments
( Hui et al., 2017 ; Hui and Vale, 2014 ; Rohrs et al., 2018 ) and val-
idated using in vitro cellular experiments ( Rohrs et al., 2019 ). The
initial protein concentrations were taken from previous literature
( Rohrs et al., 2019 ). Here, we simulated the model using MATLAB
(MathWorks, Inc.).
2.2. Simulating cell-to-cell heterogeneity
In order to explore how heterogeneity in protein expression im-
pacts CAR T cell activation, we performed Monte Carlo simulations
to create a population of 10 0,0 0 0 cells. This population size was
deemed large enough for the data-driven analysis described in the
following sections to infer the relationships between model inputs
and outputs. For each of the simulated cells, the initial protein con-
centrations were sampled from a log-uniform distribution over a
range of 10-fold above and below the baseline protein expression
used in the original model. Such a range was chosen to encom-
pass high and low values of protein expression to determine how
cells would behave at more extreme values away from the mean.
Each cell was then simulated (i.e., the model was run with each
combination of the initial protein concentrations), given the same
amount of antigen. The antigen concentration was set to be high
enough so that it would always saturate the amount of receptor.
The reason for this is that here, we examine the intracellular com-
ponents of the network. Varying the antigen concentration would
inevitably affect the model output, which is not the focus of the
present analysis.
The model was simulated for a duration of 15 min. This du-
ration was chosen because we are interested in factors affecting
a rapid response to stimulation. We repeated the simulations and
subsequent analysis for longer durations, 30 and 60 min, to see if
the influential proteins change with longer stimulation.
C.G. Cess and S.D. Finley / Journal of Theoretical Biology 489 (2020) 110125 3
Fig. 1. Schematic of the mechanistic model. Module I: LCK regulation, autophosphorylation, and phosphorylation. Module II: Phosphatase activity of CD45 and SHP1. Module
III: Formation of the LAT signalosome and its downstream signaling. Module IV: MAPK signaling and ERK-mediated negative feedback. Arrows and bars indicate activating
and inhibitory interactions, respectively. Dashed lines denote the same species in multiple Modules. Reprinted with permission from Rohrs et al. (2019) .
2.3. Characterizing cellular response
We used the concentration of double phosphorylated ERK
(ppERK) as a way to characterize the cells’ response to antigen
stimulation. Once the time course for ppERK was simulated for
each cell, the final value was compared to the initial total amount
of ERK in the cell. If the relative amount of ppERK reached 50%
or greater, the cell was considered to have responded to stimula-
tion and was placed into a group called “high ERK response.” Cells
that failed to reach 50% of ERK phosphorylation were considered
to be “low ERK response.” This threshold was chosen based on
its usage in previous experimental studies ( Altan-Bonnet and Ger-
main, 2005 ). This classification is the primary output of the model.
2.4. Partial least squares analysis
In brief, partial least-squares (PLS) allows for the formation of
a predictive model of a system’s outputs given any number of in-
puts. However, unlike a mechanistic model, which describes a sys-
tem based on its biological interactions, the parameters found by a
PLS model do not correspond to actual biological functions. Rather,
the parameters are chosen based on their ability to relate the in-
puts to the outputs. A PLS model can be used generalize the re-
lationship between the inputs and the outputs, providing valuable
multivariate information that would be more difficult to get out of
a mechanistic model.
A PLS model is a type of multivariate regression that relates
input variables to output variables by maximizing the correlation
between the variables. In PLS, both the inputs and the outputs are
transformed into a new dimensional space comprised of principal
components (linear combinations of the inputs), and a linear re-
gression is performed between these new variables before trans-
forming them back to the original dimensions. When transforming
the inputs into the principal components space, the dimensionality
of the inputs is reduced. This allows PLS to handle noisy data and
collinear inputs. Additionally, this analysis calculates how much
each input contributes to the transformed variables, indicated by
the weight of each input. The weights can then be examined to
learn general relationships between the inputs and the outputs,
and determine which inputs are most influential. For these rea-
sons, PLS is an attractive tool for multivariate analysis of large net-
works ( Cosgrove et al., 2010 ; Loiben et al., 2017 ; Wold et al., 2001 ;
Wu et al., 2008 ).
In this study, the initial protein concentrations were used as
inputs to predict which group, “high ERK response” or “low ERK
response”, the cell would belong to. For this analysis, the non-
linear iterative partial least-squares (NIPALS) algorithm ( Geladi and
Kowalski, 1986 ) was used for fitting the PLS model. For a detailed
description of the process, see Geladi and Kowalski (1986) . The in-
puts were taken as the log-value of the initial concentrations of the
23 proteins that have a non-zero starting value. Prior to perform-
ing the analysis, these input values were scaled by subtracting the
mean of the training set and then dividing by its standard devia-
tion. The result of this is that each input had a mean of zero and a
standard deviation of one. Such scaling is important so as to elim-
inate the effects of highly varying input ranges on the model.
The model was trained on simulations from two-thirds of the
cells, with the remaining one-third left for validating the PLS
model. To determine the robustness of the PLS model, training and
validation was performed 100 times, randomly splitting the popu-
4 C.G. Cess and S.D. Finley / Journal of Theoretical Biology 489 (2020) 110125
lation that was created in Section 2.2 into training and validation
sets each time. This was performed for each possible number of
principal components that the model could have. The final num-
ber of principal components was chosen as the lowest number of
components where the addition of more components failed to im-
prove the accuracy of the model.
2.5. Identification of influential proteins
The primary way of identifying the most influential inputs to a
PLS model is by calculating the variable importance of projection
(VIP) scores. VIP scores are also used in variable selection in order
to determine which variables to keep for model reduction when
dealing with large numbers of inputs. VIP scores are calculated us-
ing the weights from the inputs to each component, along with the
amount of output variance explained by each component. A higher
VIP score indicates that the input is more influential to the out-
puts. Traditionally, an input is considered to be highly influential
and chosen during variable selection if its VIP score is greater than
one ( Akarachantachote et al., 2014 ).
To further determine how each protein’s expression influences
the response, the components of the PLS model and their rela-
tions to each protein were examined. First, looking at the compo-
nents of the PLS model, we can see if high or low values of the
component are associated with a particular group (“high ERK re-
sponse” or “low ERK response”). Second, the absolute value of the
weight for each input that makes up the component indicates how
much influence that input has on a group. Finally, the sign of the
weight indicates in which direction each input influences the value
of the component. As an example, if high values of a component
correspond to the “high ERK response” group, then inputs (initial
protein concentrations) with positive weights in that component
are positively associated with that “high ERK response” group. This
means that increasing the values of those proteins’ initial concen-
trations will increase the number of cells with high ppERK levels.
This examination of the PLS components and the weights of the
inputs that make up the components provides a straightforward
method for determining how protein expressions influence the sys-
tem at the population level.
3. Results
3.1. Mechanistic model predicts heterogeneous response in ERK
activation
The simulations for the population of 10 0,0 0 0 cells with the
mechanistic model show that there is a large range of responses
within the population. This is not unexpected, as the signaling
responses of cells directly depend on the initial protein levels,
which we explicitly varied. Both the time at which ERK activa-
tion occurs and amount of activation vary widely. Fig. 2 shows the
time courses for 100 randomly selected cells as a representation of
the population. Approximately half of the cells in the population
(42.3%) reached a level of ERK activation high enough to be clas-
sified as “high ERK response”. While there were some cells that
achieved intermediate levels of ERK activation, most of the cells
were at the extremes, with either almost complete activation or
almost no activation. In total, approximately 91% of the cells ex-
perienced ERK activation in which ppERK was either greater than
90% or less than 10% of the cell’s initial amount of ERK. The distri-
butions of the final relative ppERK values for all 10 0,0 0 0 cells are
shown in Figure S1A. From this, the clear “all-or-nothing” phospho-
rylation of ERK that is characteristic of T cells can be seen ( Altan-
Bonnet and Germain, 2005 ; Birtwistle et al., 2012 ). The relative
ppERK response resembles a biomodal distribution, with two very
sharp peaks at either end. Due to this, we concluded that group-
ing the cells into “high” and “low” response (that is, a discrete
classification) was most appropriate for our analysis. We did at-
tempt to use a continuous output (data not shown); however, this
was largely unsuccessful due to the extreme grouping of ppERK
responses. Figure S1B shows the initial concentrations of ERK that
lead to “high ERK response” and “low ERK response.” From this, it
is clear that the initial expression of ERK has little influence on its
phosphorylation.
3.2. PLS model identifies influential proteins
A PLS model was developed to determine how variations in
protein expression influence the ability of a CAR T cell to respond
to stimulation. The inputs to the PLS model were the initial protein
concentrations for each cell (see Methods), and the output was the
cell’s classification as a “high ERK response” or “low ERK response.”
We considered different PLS models where the number of compo-
nents ranged from two to 23, which was the maximum number of
components possible (the total number of inputs). The final model
consisted of two components, as adding more components failed
to improve accuracy in predicting the classification of the cells and
unnecessary components increases the chance of overfitting. Using
100 randomized sets of training and validation data, the model was
able to achieve an average accuracy of 86.9% in predicting which
group a simulated cell with particular initial concentrations would
belong to. Here, the training set was two-thirds of the 10 0,0 0 0
simulated cells, and the validation set was the remainder of the
simulated data. Considering “high ERK response” to be positive, the
true positive prediction accuracy was 85.1%. The true negative pre-
diction accuracy was 88.2%. These values are close to each other
and to the overall prediction accuracy, meaning that the PLS model
is not biased towards either group. This is high accuracy, consid-
ering that PLS is a linear method, while the mechanistic model
that was used to produce the data is very complex and inevitably
has many nonlinearities. To see how the number of training sam-
ples affected the ability of the PLS model to fit the mechanistic
model, we trained the model on smaller sets of samples. Even us-
ing only 1% of the simulated population as the training set yielded
similar accuracy. However, with fewer training samples, the VIP
scores were less consistent between randomized batches, and the
results heavily depended on which samples were used for train-
ing. Because of these differences, we used a large amount of train-
ing data when calculating the final VIP scores to make sure that
there was no bias based on which samples were chosen for train-
ing. While it is possible to overfit a PLS model, we have avoided
overfitting given the low number of components that we used and
because training performed using a much smaller number of sam-
ples still provided similar accuracy as training with large amounts
of samples. Overall, we established a PLS model that is able to pre-
dict which CAR-engineered T cells respond to antigen stimulation,
given the initial concentrations of the intracellular signaling pro-
teins.
Next, we used the predictive PLS model to evaluate the impor-
tance of each protein on ERK activation in the simulated CAR T
cells. In order to determine which proteins hold the most influ-
ence over the response, the VIP score for each protein was calcu-
lated ( Fig. 3 ). Out of the 23 proteins in the network whose ini-
tial concentrations were varied, six achieved a VIP score of greater
than one and were thus identified as being highly influential: LCK,
CD3 ζ, Ras, Raf, MEK, and SHP1. These proteins were then exam-
ined in detail to determine how they are influential to the system.
In some instances, when training with fewer numbers of samples,
the VIP score for CD3 ζ would drop below one, and the score for
ZAP70 would rise above one. Although ZAP70 does not meet the
VIP score cutoff for being influential with the full training set, we
C.G. Cess and S.D. Finley / Journal of Theoretical Biology 489 (2020) 110125 5
Fig. 2. Relative ppERK time courses for a representative set of 100 cells. The red line at 50% shows the level of phosphorylation needed to be considered as a “high ERK
response” following stimulation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. VIP scores when grouping “high ERK response” and “low ERK response”.
Proteins that achieved a VIP score greater than one are colored orange, indicating
that they significantly influence the PLS model’s classification of a cell being a “high
ERK response” or “low ERK response”. (For interpretation of the references to color
in this figure legend, the reader is referred to the web version of this article.)
still discuss it in the following section due to it being identified as
an influential input when training with small amounts of data.
As a way of validating these proteins as being influential, we
investigated how the initial concentration of the 23 proteins that
have non-zero starting amounts influences the percentage of “high
ERK responses”. We utilized the mechanistic model to simulate a
new population consisting of 10 0 0 cells. In these simulated cells,
we set the initial concentration of a single protein to be constant
across the population, with the initial amounts of the other 22 pro-
teins with non-zero starting values randomly sampled from their
log-uniform distribution as before. We simulated the population to
determine the percentage of cells that achieved a “high ERK re-
sponse” at that protein level. We ran simulations in which the ini-
tial concentration of one of the proteins with a non-zero starting
value was varied from 10-fold below the baseline level to 10-fold
above its baseline value. In total, we performed 230,0 0 0 simula-
tions: varying the initial concentration of each of the 23 proteins
with non-zero starting values, at 10 different levels, 10 0 0 times for
each protein level.
We found that as we increased the initial amount of a protein,
the percentage of “high ERK responses” monotonically increased,
monotonically decreased, or remained constant. Figure S2 shows
the absolute values of the change in the percentage of cells with
“high ERK responses” for the lowest concentration (10 times be-
low the baseline value) of each protein compared to the highest
concentration (10 times above the baseline value). We see that the
greatest difference is for the proteins identified as being influen-
tial by the PLS model, further confirming the importance of those
proteins.
3.3. PLS model characterizes the role of the influential proteins
The VIP score for each input indicates whether that input sig-
nificantly influences the model output. However, it does not tell
whether varying the input increases or decreases the output. In
order to determine how each of the six proteins with a VIP score
greater than one influenced activation, the values of transformed
inputs for the full model were examined. Since the optimal PLS
model consisted of two components, and thus each original set of
inputs was condensed into two variables, the values of the trans-
formed inputs were viewed as a scatter plot in two dimensions
( Fig. 4 ). Each transformed input was colored based on whether it
corresponded to a cell with high or low ERK response. This plot al-
lowed for determining visually how the value of each component
6 C.G. Cess and S.D. Finley / Journal of Theoretical Biology 489 (2020) 110125
Fig. 4. Transformed input values for each PLS model component. Blue points cor-
respond to cells classified as “high ERK response” based on the model inputs. Red
points correspond cells classified as “low ERK response” based on the model in-
puts. (For interpretation of the references to color in this figure legend, the reader
is referred to the web version of this article.)
corresponded to the response of the cell. As shown, while there is
some overlap between the two groups, there is a clear separation
of “high ERK response” and “low ERK response” when considering
the first component. The second component, however, provides lit-
tle additional information for classifying the cells. From the PLS
model, we found that the first component captures more than half
(53.3%) of the variance in the output, while the second compo-
nent captures very little of the variance (0.01%). Therefore, we only
used the first component to determine which groups the inputs re-
late to. High values for the first component were associated with
“low ERK response” while low values were associated with “high
ERK response”. Therefore, if a protein had a positive weight for
transformation into the first component, increased expression of
that protein would make the cell less likely to have high levels of
ppERK. The opposite is true for proteins that had negative weights.
If the value of a weight was large in magnitude, then changes in
the expression of its corresponding protein would have a larger ef-
fect compared to a protein whose weight for the first component
is smaller in magnitude.
With this insight, the weights of the first component ( Fig. 5 )
show how each of the influential proteins affects cell activation.
The analysis indicates that increasing the initial amount of LCK,
Ras, Raf, or MEK positively influences the cell and promotes phos-
phorylation of ERK. Conversely, increasing CD3 ζ or SHP1 negatively
influences the system, inhibiting ERK phosphorylation. We further
examined the network to understand where these points of con-
trol are placed in the signaling pathway. CD3 ζ, LCK, and SHP1 can
all be grouped as proteins involved in the initiation of the signal,
while Ras, Raf, and MEK, which make up the MAPK pathway, are
all involved in conversion of the signal to a digital output. It must
be noted that together, these proteins are positioned at the begin-
ning and end of the signaling pathway.
To determine the biological reason for the influence of these six
proteins, we looked into the roles that have been established in the
literature. Upon binding to the antigen, specific tyrosine residues in
the ITAMs on the CD3 ζ chain, which is part of the CAR, are phos-
phorylated by LCK. These phosphorylated ITAMs can then proceed
to activate downstream signaling proteins, initiating signal trans-
duction via ZAP-70 ( Simeoni, 2017 ). ZAP70 serves as a bridge be-
tween the receptor and downstream signaling by phosphorylat-
Fig. 5. Weights from the inputs to the first component of the PLS model. Blue bars
represent proteins that lower the value of the first component, thus influencing the
cell to have “high ERK response”. Red bars represent proteins that increase the value
of the first component, thus influencing the cell to have “low ERK response”. The
proteins found to be influential have darker bars. (For interpretation of the refer-
ences to color in this figure legend, the reader is referred to the web version of this
article.)
ing LAT and SLP-76, which then recruit other signaling proteins
( Wang et al., 2010 ). Logically, it makes sense that increasing levels
of LCK increases the cell’s ability to induce intracellular signaling,
since more phosphorylation of CD3 ζ would lead to a stronger sig-
nal transduction. Similarly, increasing ZAP70 leads to an increase
in ppERK. On the other hand, upon receptor stimulation, SHP1 can
bind to CD3 ζ, become activated by LCK, and then proceed to de-
phosphorylate CD3 ζ, LCK, and ZAP-70. This provides a form of neg-
ative feedback to prevent noise from accidentally leading to ERK
activation ( Altan-Bonnet and Germain, 2005 ). Therefore, increas-
ing levels of SHP1 increases the strength of the negative feedback
and prevents the signal from the receptor from being transmit-
ted downstream. Finally, our analysis reveals that increasing CD3 ζ,
the last protein involved in signal initiation, inhibits cell activa-
tion. This result is discussed in detail in the following section as
the conclusion drawn from the PLS model is particularly interest-
ing and unexpected.
Ras, Raf, and MEK are three proteins at the end of the CAR-
mediated signaling network and comprise the MAPK pathway, and
their activation is important for the digital interpretation of a sig-
nal ( Das et al., 2009 ). MAPK signaling leads to the rapid phospho-
rylation of ERK in an all-or-nothing fashion ( Birtwistle et al., 2012 ).
These actions allow the cell to make important decisions based on
signals from the external environment ( Shaul and Seger, 2007 ). As
expected, analysis of the weights in the PLS model show that hav-
ing increased amounts of the proteins that make up this pathway
leads to more phosphorylation of ERK.
3.4. High levels of CD3 ζ can increase the ERK response time
Of interest is the fact that the PLS model found that increas-
ing the expression of CD3 ζ would negatively influence the cell,
causing it to not respond to stimulation. At first, this seems coun-
terintuitive, since CD3 ζ binds to the antigen, becomes phospho-
rylated, and initiates downstream signaling. Logically, having more
CD3 ζ should lead to a greater ability to initiate signaling. To de-
termine the cause for this result, we performed a series of sim-
ulations with the mechanistic model in which the initial concen-
trations of all of the proteins were set to their average (baseline)
values, except for CD3 ζ, which was varied 10-fold above and be-
C.G. Cess and S.D. Finley / Journal of Theoretical Biology 489 (2020) 110125 7
Fig. 6. Predicted dynamics of signaling species. Time courses for phosphorylated ERK (A) and SHP1 (B) for different initial values of CD3 ζ.
low its mean value. These simulations showed that the time of ERK
activation was delayed as CD3 ζ concentration increased ( Fig. 6 A).
Upon studying the predicted time courses for other proteins, we
found that activated SHP1 reached higher levels when CD3 ζ con-
centration was increased ( Fig. 6 B). Examining the interactions in-
volving CD3 ζ and SHP1 provided an explanation for these results.
Inactive SHP1 binds to CD3 ζ, where it becomes activated by LCK
and is then able to inactivate other molecules in the pathway.
Thus, increased levels of CD3 ζ allow SHP1 to become activated
faster, which then inhibits downstream signaling. This analysis re-
veals that together, the PLS model and the details of the signaling
network produce relevant insight into the cells’ response.
3.5. Fewer proteins are influential as simulation time increases
Finally, we aimed to determine whether the influential proteins
change when the system is simulated for longer times. We re-
peated our analysis after simulating the system for 30 min and
60 min. As simulation duration increases, more cells eventually
reach a “high ERK response”, with 54.1% at 30 min and 62.4%
at 60 min. We also found that as simulation duration increased,
the number of influential proteins decreased, with their VIP scores
dropping below one. At 30 min, CD3 ζ ceased being influential,
while the other five proteins retained their influence. At 60 min,
LCK, RAF, and MEK were the only proteins found to still be influ-
ential. These results indicate that the key modulators of the ppERK
level vary with time, and that certain proteins are only influential
in mediating a rapid response to antigen stimulation.
4. Discussion
In this work, we applied partial least-squares to analyze predic-
tions from a detailed mechanistic model of CAR T cell signaling.
In particular, we study how heterogeneity in protein expression af-
fects cell behavior characterized by MAPK signaling and phospho-
rylation of ERK. Through this analysis, we determined the influence
of the levels of individual proteins on the ability of CAR T cells to
respond to stimulation. Although the network that mediates sig-
nal transduction is fairly large and complex, we found that the ex-
pressions of only a handful of proteins play an influential role in
the response. The influential proteins are positioned at the begin-
ning and the end of the signaling pathway. The analysis was able
to determine how each protein influenced the response, and while
the effects of most of these proteins made sense based on their
known functions, the influence of CD3 ζ was found to go against
what might be intuitively assumed.
Previous modeling work has also explored the role of heteroge-
neous protein expression on influencing cellular signaling. A study
by Birtwistle et al. examined the response of the MAPK cascade
( Birtwistle et al., 2012 ). They compared the result of stochastic
simulation using the Gillespie algorithm to the effect of randomly
sampling initial protein concentrations and found that the latter
matched their experimental measurements much better than the
former. This provides support to our analysis presented here. Our
work expands on the general conclusion that initial protein con-
centrations affect cellular response and uses analysis of the mecha-
nistic model to identify which proteins cause the observed hetero-
geneity. As such, our computational analysis provides novel insight
that would be much more difficult to obtain experimentally.
An earlier study by Feinerman et al. also examined heterogene-
ity in T cell activation using flow cytometry to explore the influ-
ence of the expressions of CD8, ERK, and SHP1 ( Feinerman et al.,
2008 ). Similar to the results we present here, they found that in-
creasing the initial ERK has little effect on the ability of the pop-
ulation to become activated, while increasing expression of SHP1
lowered the percent of the population that could respond. With
our analysis, we were also able to identify additional proteins in
the system that influence the response.
By gaining an understanding of which proteins in the network
contribute heavily to the response to an input we can determine
how an intracellular signaling network can be modulated to induce
a desired response. Ideally, one would be able to modulate each
cell individually to achieve optimal response; however, that is not
likely feasible. The analysis done here is useful as the VIP scores,
which determine how influential a protein is, and the weights of
the inputs, which determine the direction in which an input influ-
ences the output, are calculated based on all of the sets of inputs.
This means that the PLS model tells how the population in gen-
eral would respond to an increase or decrease of a specific protein
and provides information on which proteins could be modulated
at the population level to attain a desired response. Moreover, by
performing the data-driven analysis for different durations in the
8 C.G. Cess and S.D. Finley / Journal of Theoretical Biology 489 (2020) 110125
mechanistic model, it is possible identify time-based strategies for
altering the cells’ responses.
There are a few limitations to using PLS for the analysis done
here. The primary issue is that PLS assumes linear relationships
between the inputs and outputs. It is unlikely that a system of
this size is perfectly linear. However, as shown by the high accu-
racy of the PLS model we developed, linear relationships can be a
reasonable approximation. The use of nonlinear methods, such as
neural networks, would provide a higher prediction accuracy, but
at the cost of being more difficult to analyze. Another limitation is
that all of the initial protein concentrations were sampled from the
same range. Due to the variety of different methods of gene regu-
lation, some proteins could have wider or narrower distributions. It
is possible that changing the distribution could influence the pre-
dicted numbers of each cell phenotype (i.e., high or low ERK re-
sponse). However, we do not believe that this significantly impacts
our analysis as we are interested in determining the effects of each
protein even at higher or lower levels compared to the mean value.
As more quantitative measurements for the single-cell concentra-
tions and distributions of proteins become available, we can incor-
porate that information into our model.
Encoding the outputs of a mechanistic model as a data-driven
model, which reduces the computational time, has multiple uses.
As described in this work, the data-driven model can then be used
as an analysis tool. Here, the data-driven model enabled a better
understanding of the important relationships between model vari-
ables and cellular response. Those relationships can inform how
to engineer the cell for a desired purpose. Another potential ap-
plication is using the data-driven model inside of an agent-based
model (ABM). While some ABMs do use simple ODE models as
a way of making cellular decisions ( Hendrata and Sudiono, 2016 ;
Wang et al., 2007 ; Zhang et al., 2009 ), most use discrete or prob-
abilistic rules to govern how each cell behaves, as that is much
more computationally efficient. Using a data-driven model, ODE
networks could potentially be simplified, allowing ABMs to become
more biologically detailed without a significant increase in compu-
tational cost.
The results from this study show that data-driven analyses pro-
vide insight into large mechanistic models. Using a relatively fast
analysis, we were able to determine which proteins were the most
influential in determining the response of the system, and whether
each protein had a positive or a negative influence. Using PLS pro-
vides information on how to push the population as a whole to-
wards a specific response. In the context of CAR T cell signaling,
we found that the system was most sensitive to proteins at the
very beginning or very end of the network. While most of the pro-
teins (LCK, SHP1, Ras, RAF, and MEK) influenced the system in a
way that is expected based on their biological functions, we found
that CD3 ζ can actually influence the system towards no response,
despite being part of the receptor that initiates signaling. This re-
sult was explained by examining the specific interactions in the
mechanistic model. Overall, we find that data-driven methods are
capable of analyzing detailed signaling networks, rather than just
being used in cases where forming a mechanistic model is not fea-
sible.
5. Conclusion
Here we used a data-driven analysis of a mechanistic model to
study how variations in protein expression influence the ability of
a CAR T cell to respond to stimulation and promote ERK phos-
phorylation. We identified six proteins relating to either the re-
ceptor or the MAPK cascade that strongly influenced the output of
the system. We also found the counterintuitive result that increas-
ing the amount of receptor in the system can actually hinder ERK
phosphorylation, as it increases the level of active phosphatase in
the system. By combining data-driven and mechanistic modeling,
we gain useful insight into cell signaling.
CRediT authorship contribution statement
Colin G. Cess: Formal analysis, Investigation, Software, Visu-
alization, Writing - original draft, Writing - review & editing.
Stacey D. Finley: Conceptualization, Supervision, Resources, Writ-
ing - review & editing.
Acknowledgments
The authors acknowledge members of the Finley research group
for constructive feedback and Ms. Lauren Slowskei for generating
initial simulation results.
Funding
This work was supported by the USC Viterbi/Graduate School
Merit Fellowship (to CGC).
Supplementary materials
Supplementary material associated with this article can be
found, in the online version, at doi: 10.1016/j.jtbi.2019.110125 .
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RESEARCHARTICLE
Multi-scalemodelingofmacrophage—Tcell
interactionswithinthetumor
microenvironment
ColinG.Cess
ID
1
,StaceyD.Finley
ID
1,2,3
1 DepartmentofBiomedicalEngineering,UniversityofSouthernCalifornia,LosAngeles,California,United
StatesofAmerica,2 DepartmentofQuantitativeandBiologicalSciences,UniversityofSouthernCalifornia,
LosAngeles,California,UnitedStatesofAmerica,3 MorkFamilyDepartmentofChemicalEngineeringand
MaterialsScience,UniversityofSouthernCalifornia,LosAngeles,California,UnitedStatesofAmerica
*sfinley@usc.edu
Abstract
Withinthetumormicroenvironment,macrophagesexistinanimmunosuppressivestate,
preventingTcellsfromeliminatingthetumor.Duetothis,researchisfocusingonimmuno-
therapiesthatspecificallytargetmacrophagesinordertoreducetheirimmunosuppressive
capabilitiesandpromoteTcellfunction.Inthisstudy,wedevelopanagent-basedmodel
consistingoftheinteractionsbetweenmacrophages,Tcells,andtumorcellstodetermine
howtheimmuneresponsechangesduetothreemacrophage-basedimmunotherapeutic
strategies:macrophagedepletion,recruitmentinhibition,andmacrophagereeducation.We
findthatreeducation,whichconvertsthemacrophagesintoanimmune-promotingpheno-
type,isthemosteffectivestrategyandthatthemacrophagerecruitmentrateandtumorpro-
liferationrate(tumor-specificproperties)havelargeimpactsontherapyefficacy.Wealso
employanovelmethodofusinganeuralnetworktoreducethecomputationalcomplexityof
anintracellularsignalingmechanisticmodel.
Authorsummary
Wepresentamulti-scaleagent-basedmodelofmacrophagesandTcellswithinthetumor
microenvironment.Toincreasethebiologicaldetail,weincludeanintracellularmecha-
nisticmodelinthemacrophages,employingamethodofusingneuralnetworkstoreduce
themechanisticmodelintoasimpleinput/outputmodel.Withthemechanisticmodel,we
areabletopredicttheeffectsofspecificallyinhibitingapartoftheintracellularsignaling
pathway,asopposedtojustmakingphenotypicpredictions.Usingtheintegratedmodel-
ingframework,weareabletopredicttheimpactsofimmunosuppressivemacrophageson
Tcellfunctionandpredicthowmacrophage-basedimmunotherapiescanreduceimmu-
nosuppression.Altogether,wepresentausefulframeworkforstudyingcell-cellinterac-
tionsinthetumormicroenvironmentandtheeffectsofimmunecell-targetingtherapies.
PLOS COMPUTATIONAL BIOLOGY
PLOSComputationalBiology|https://doi.org/10.1371/journal.pcbi.1008519 December23,2020 1/35
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Citation:CessCG,FinleySD(2020)Multi-scale
modelingofmacrophage—Tcellinteractions
withinthetumormicroenvironment.PLoSComput
Biol16(12):e1008519.https://doi.org/10.1371/
journal.pcbi.1008519
Editor:KathrynMiller-Jensen,YaleUniversity,
UNITEDSTATES
Received:August3,2020
Accepted:November11,2020
Published:December23,2020
PeerReviewHistory:PLOSrecognizesthe
benefitsoftransparencyinthepeerreview
process;therefore,weenablethepublicationof
allofthecontentofpeerreviewandauthor
responsesalongsidefinal,publishedarticles.The
editorialhistoryofthisarticleisavailablehere:
https://doi.org/10.1371/journal.pcbi.1008519
Copyright: 2020Cess,Finley.Thisisanopen
accessarticledistributedunderthetermsofthe
CreativeCommonsAttributionLicense,which
permitsunrestricteduse,distribution,and
reproductioninanymedium,providedtheoriginal
authorandsourcearecredited.
DataAvailabilityStatement:Github:https://
github.com/FinleyLabUSC/Early-TME-ABM-PLOS-
Comp-Bio.
Funding:ThisworkwassupportedbytheUSC
Viterbi/GraduateSchoolMeritFellowshiptoCGC
Introduction
Akeyfeatureofthetumormicroenvironment(TME)isthatthenormalimmuneresponse,
whichshouldbeabletotargetandkillmalignantcells,isdysfunctional[1,2].Specifically,some
immunecellsaresuppressedandunabletocarryouttheirfunctions,whileotherimmunecells
arecorruptedintoapro-tumorstateandactivelyworktoincreasetumorgrowth.Thedysfunc-
tionalimmuneresponseisacommonfeatureacrosstumortypes,anditisthoughtthattumors
developonlyafterevadingtheimmunesystem[3].Recently,immunotherapieshavebeen
developedinattemptstoreactivatetheimmunesystemsothatitcancarryoutitsnormalfunc-
tionandremovethetumor[4–6].Althoughtheefficacyofimmunotherapyhasincreasedin
recentyears,theuseofimmunotherapeuticstrategiestotreatsolidtumorshasbeenlargely
unsuccessful[7–9].Therefore,furtherexaminationoftheimmunesuppressingmechanisms
withintheTMEisneededinordertodevelopmoreeffectiveimmunotherapies.
Oneofthemostcommon,andmostinfluential,typesofimmunecellintheTMEisthe
tumor-associatedmacrophage(TAM)[10,11].Macrophageshavevariousroleswithinthenor-
malimmuneresponse,havingcytotoxiccapabilityandtheabilitytopresentantigenstoTcells
[10].Dependingonenvironmentalsignals,macrophagescandisplayavarietyofphenotypes,
rangingfrompro-inflammatoryandimmune-supportingtoimmunosuppressivewithwound-
healingproperties.Whilemacrophagephenotypeexistsonaspectrum[12],forconceptual
purposesitisdividedintotwomainstates:M1(immune-promoting)andM2(immunosup-
pressive).Duetoinfluencefromthetumor,mostTAMsareinanM2-likestateandfurther
promotetumorgrowth.TAMsareabletopromotetumorcellproliferation,induceangiogene-
sis,enabletumorcellmigrationandmetastasis,andsuppressthefunctionofanti-tumor
immunecells[10,11,13,14].Duetotheirimportantrolesintumorgrowth,TAMshavebecome
thesubjectofvariousimmunotherapies[9,15,16].Thesetreatmentstrategiesaimtoeither
reducethenumberofTAMswithintheTME,whichwouldlimittheirsuppressionofTcells,
orconvertTAMsintoanM1-likestate,whichwouldenhanceTcellfunction.
Tcellsareconsideredtobethemainportionoftheadaptiveimmunesystemforeliminat-
ingtumors,beingabletodetecttumor-associatedantigensandthenkilltumorcells[17–19].It
ishypothesizedthatmosttumorsareeliminatedbyTcellsearlyonandthatonlyasmallnum-
beroftumorsmanagetoescapeandgoontohaveclinicalsignificance[3].Astumorsgrow
further,Tcellfunctionissuppressed,diminishingtheimmuneresponse[17–20].CytotoxicT
cells(CTLs)canbecomeexhausted,havinglimitedproliferativeandcytotoxicfunction,dueto
excessivestimulationandtheexpressionofcheckpointproteinssuchasCTLA-4andPD-L1
onM2macrophagesandtumorcells[21].Thetumoralsoincreasestheexcretionofchemo-
kinesthatattractregulatoryTcells(Tregs)andcytokinesfromthatpromotetheconversionof
Thelper(Th)cellsintoTregs,whichassistinsuppressingCTLfunction[22].M2macrophages
canalsoproducetheseTreg-promotingcytokines.
Variousimmunotherapieshavebeendevelopedtoreactivatetheimmunesystemandpro-
motetumorremoval.CheckpointinhibitionaimstoblockligandssuchasCTLA-4andPD-L1
ontumorsandM2macrophages[4–7,20].Blockingtheseligandshasbeenshowntoincrease
Tcellexpansionatthetumorsiteandpromotethekillingoftumorcells,thoughitisunclear
whetherthisisduetotherestorationofthefunctionsofTcellsalreadyatthetumorsiteordue
totheinfiltrationofnewTcells,asevidencehasbeenfoundforboth[20,23].Anothermethod
ofimmunotherapyisadoptiveTcelltherapy,inwhichTcellsareremovedfromthepatient,
expanded ex vivo,andthengivenbacktothepatientinanattempttoboosttheimmune
response[5,6,24].SometimestheTcellsaremodifiedinordertobetterdetecttumorcells,asis
thecasewithchimericantigenreceptor(CAR)-engineeredTcells.Whileimmunotherapyis
successfulinsomecases,particularlyinhematologicalmalignancies,itoftenfailsinsolid
PLOS COMPUTATIONAL BIOLOGY Modelofmacrophage-Tcellinteractionsinthetumor
PLOSComputationalBiology|https://doi.org/10.1371/journal.pcbi.1008519 December23,2020 2/35
(http://graduateschool.usc.edu)andtheUSC
CenterforComputationalModelingofCancer
(http://modelingcancer.usc.edu).Thefundershad
noroleinstudydesign,datacollectionand
analysis,decisiontopublish,orpreparationofthe
manuscript.”
Competinginterests:Theauthorshavedeclared
thatnocompetinginterestsexist.
tumors.Thislackofsuccessindicatesthatthereisacomplexinterplaywithinthecellsresiding
intheTMEthatpreventsimmunefunction.
Manycomputationalmodelshavebeendevelopedtobetterunderstandinteractionsbetween
tumorandimmunecells.Werecentlyreviewedmathematicalmodelsoftumor-immuneinter-
actionsacrossvariousscales[25].Thesemodelsaimtobothexaminegeneraltumor-immune
behaviorandtotesttheeffectsofimmunotherapy.Significantmodelingeffortshavefocusedon
Tcell-mediatedkilling.Forexample,Gongetal.developedamodelofhowPD-L1expression
impactsTcellresponse[26].Theyalsoexaminedhowtumormutationalburdenandantigen
strength,whichimpactthestrengthoftheTcellresponse,influencetumorgrowth.Katheretal.
examinedTcellresponseinrelationtotumorstroma,whichphysicallyinhibitsbothimmune
cellinfiltrationandtumorgrowth[27].Theydeterminedthatahighlevelofstromaslows
tumorgrowthwhenthenumberofTcellsislowbutpreventsimmunecell-mediatedelimina-
tionofthetumorwhenthereisahighnumberofTcells.Therefore,combiningtherapiesthat
increaseTcellcountwiththerapiesthatreducetumorstromacouldincreaseTcellinfiltration
andtumorremoval.Otherstudieshavefocusedonlymphnodedynamicsandtheeffectsof
tumorproliferation,antigenicity,andTcellrecruitmentontumorremoval[28].
Mathematicalmodelinghasalsobeenappliedtostudytheroleofmacrophagesintumor
elimination.SomeofthesemodelshavefocusedontheinterplaybetweenM1andM2macro-
phagesattheearlystagesoftumorgrowth.BothWellsetal.[29]andMalbacheretal.[30]
developedmodelswheremacrophagescandifferentiateintoeitheranM1oranM2phenotype
basedonfactorssecretedbythegrowingtumor.Thesemacrophagestheneitherinhibitorpro-
motetumorgrowth,dependingontheirphenotype.El-kenawietal.focusedontheeffectsof
tumor-inducedacidityonmacrophagedifferentiation,allowingmacrophagestodifferentiate
onaspectrumasopposedtodiscretephenotypes,tomoreaccuratelyrepresentmacrophage
state[31].Othermodelshavefocusedoninteractionsbetweenmacrophagesandtumorscells
thatleadtoincreasedtumormigrationandmetastaticpotentialduetoparacrinesignaling
loopsbetweenthetwocelltypes[32–34].
Whilethereareexamplesofmathematicalmodelsthatconsidermultipletypesofimmune
cellsinthelocaltissuemicroenvironment[35],mostmodelsdonotconsidermultipletypesof
immunecellsintheTME.However,itisimportanttoaccountforthevariousimmunecell
populationssinceimmunecellsinteractwitheachotherdirectlyandviadiffusiblesignaling
factors.Therefore,inthisstudy,wefocusontheinteractionsbetweenmacrophagesandT
cells.Usinganagent-basedmodel(ABM),weexaminethegrowthofamicrometastasisand
howmacrophagedifferentiationaffectstheabilityoftheTcellstoeliminatethetumor.We
alsomodeltheeffectsofthreemacrophage-basedimmunotherapies(macrophagedepletion,
recruitmentinhibition,andmacrophagereeducation)andinvestigatehowdifferingratesof
tumorproliferationandmacrophagerecruitmentaffecttheefficacyofeachtreatment.To
modelmacrophagedifferentiation,weemployamechanisticintracellularsignalingmodel
withinthemacrophages.Inordertoimprovecomputationaltime,weuseaneuralnetworkto
predictthemechanisticmodeloutputsbasedonsignalinginputsfromtheTME.Wefindthat
macrophagereeducationisthemostpowerfulofthethreeimmunotherapiessimulateddueto
thepromotionofTcellfunction,andthattumorproliferationrateandmacrophagerecruit-
mentratecanhavelargeimpactsontheefficacyoftherapy.
Results
Modelconstruction
Asexplainedindetailinthemethodssection,wehaveconstructedamulti-scaleagent-based
modeloftheinteractionsbetweenmacrophages,Tcells,andtumorcells,alongwiththe
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cytokinesIL-4andIFN- .Themodelrepresentsa2Dtissueslice,andcellsareconstrainedtoa
lattice,withonecellperlatticesite.Briefly,macrophagesareinitiallypresentwithinthetissue
andmorearerecruitedtothetumorsiteassimulationprogresses.Theyareabletodifferentiate
basedoncytokineconcentrationintoeitheranM1-likeorM2-likestate.Differentiationis
basedonanintracellularsignalingmodel,whichwasthenreducedtoasimpleinput/output
modelusinganeuralnetworktorelatecytokineconcentrationstodifferentiation.Tcellsare
recruitedbasedontumorcelldeath,andacttokilltumorcells.Theybecomefullyactiveupon
antigencontact,allowingthemtokilltumorcellsandsecreteIFN- .Activationispromoted
byM1macrophagesandinhibitedbyM2macrophages.Tumorcellsproliferateandsecrete
IL-4.Withthismodel,wepredicttheeffectsofthreemacrophage-basedimmunotherapies
(macrophagedepletion,recruitmentinhibition,andmacrophagereeducationviaPI3Kinhibi-
tion)andhowtheyimpacttheabilityofTcellstoremovethetumor.Amodelschematicalong
withanexamplespatialdistributionofcellsisshowninFig1.Fromthespatialdistribution
(Fig1B),whichshowsasamplesimulationwithouttreatment,weseethatmacrophagessur-
roundthetumor,primarilyintheM2state,preventingsomeTcellsfromreachingthetumor
andinhibitingTcellsthatarerecruitedtothetumorsite.ThereareactiveTcellsimmediately
adjacenttotumorcells;however,therearenotenoughtoeliminatethetumor.
Modelbehaviorwithouttreatment
Priortosimulatingmacrophage-basedinterventions,weransimulationswithoutanytreat-
menttohaveareferenceforcomparingtheefficacyofdifferenttreatmentstrategies.Tounder-
standhowvariousparametersrelatingtotheimmuneresponseaffectmodelbehavior,we
sampledoverawiderangeofvaluesaboveandbelowourbaseparametervaluesusingLatin
HypercubeSampling(LHS),generatingatotalof500parametersets.Duetothestochastic
behaviorofthemodel,eachparametersetwassimulated100timesinordertoobtainaverage
behavior.WenotethatLHSgeneratesaseriesofparametersetswhereeachparametervalue
onlyappearsonce.Therefore,thesharpspikesseenintheresultsareduetoeachparameter
Fig1. Modelschematicandrepresentativesimulationresult.(A)Modelschematic.TcellssecreteIFN- ,whichpromotes
M1differentiation.TumorcellsandM2macrophagessecreteIL-4,whichpromotesM2differentiation.Tcellskilltumor
cells.M1macrophagespromoteTcellfunction,whileM2macrophagesinhibitit.(B)Representativespatialdistributionof
cellsoncethetumorreachestheequilibriumstate.Tumorcells(black),Tcells(yellow),activeTcells(red),M0macrophages
(blue),M2macrophages(purple).TherearenoM1macrophagespresentattheendofthissimulation.
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valuebeingsimulatedonce,andnottheaverageresponseattheparametervalue.Despitethis,
westillseetrendsforcertainparameters.Wefoundthatmacrophagerecruitmentratehadthe
onlynoticeableeffectonthefractionoftumorsremovedbytheimmunesystem.Specifically,
higherrecruitmentrates,andthusmoremacrophagesinthesystem,leadtoalowerfractionof
tumorsthatwereremoved(S1Fig).Wealsofoundthatmacrophagerecruitmentrateandlife-
spanwerecorrelatedtothefinalnumberoftumorcellsattheendofsimulation(S2Fig).That
is,havingmoremacrophages,whichwouldbeintheM2state(discussedindetailbelow),pres-
entintheenvironment,duetoeitherincreasedrecruitmentorlongerlifespan,leadstomore
tumorcellsattheendofsimulation.Theseresultssupportthefindingthathighnumbersof
TAMshaveaworseclinicaloutcome[15].Thisanalysisalsosupportsourfocusonpredicting
theeffectsofmacrophage-basedtherapies.Thefinalvaluesofthemodelparameterswereset
sothatfewtumorswereremovedbytheimmunesystemwithouttreatment,sothattumorkill-
inginsubsequentsimulationswouldbedueprimarilytotreatment,andtomatchthoseused
insimilarmodels.
Weusedthebasemodeltoexplorethedynamicsofeachcelltypeanddiffusiblefactorpres-
entintheTME.ShowninFig2arethetimecoursesfor100simulationsover200dayswithout
treatment.Forallsimulationswheretheimmunesystemfailedtoeliminatethetumor(99%),
eachcelltypereachedanequilibriumstate.Thisstatecanbeconsideredthe“immunecontrol”
phaseoftumorgrowth,whichisthoughttotakeplaceoverseveralyearsandinvolvestheselec-
tionoftumorcellsthatareresistanttotheimmunesystem[3].Weconcludethatthisequilib-
riumstateisindeedduetothecytotoxicfunctionoftheTcellsandnotspatialinhibition,with
S3Figcomparingtumorgrowthcurvesintheabsenceofimmunecellsandwithimmunecells
presentbutwithoutfunction,findingthesetwocurvestobealmostidentical.Afteraninitial
dropduetotheintroductionofTcellstotheenvironment,thetumorcellpopulationevens
outataround350cells(Fig2A).Thenaivemacrophagepopulationdropsimmediatelyand
staysatalownumberduetocontinuousrecruitmenttotheTMEandsubsequentdifferentia-
tion(Fig2B).Inthissetofsimulations,weseenomacrophagedifferentiationtotheM1phe-
notype(Fig2C)whereasthenumberofM2macrophagesreachesahighlevel(Fig2D).The
timecoursesforthetotalnumberofTcells(Fig2E)resemblesadelayedversionofthosefor
tumorcells.ThetimecoursesforactiveTcellsfollowsthis,thoughatverylowlevels(Fig2F).
Duetothenumberoftumorcells,averageandmaximumIL-4levelsarerelativelyhigh(Fig
2Gand2H)whileIFN- levels(Fig2Iand2J)arelowduetolowTcellactivation.Thisdiffer-
encecausestheabsenceofM0differentiationintotheM1phenotype.Wealsobrieflyexam-
inedhowinitialmacrophagedensityimpactstumordynamics(S4Fig).Wefoundthatwhile
thisimpactsinitialtumordynamics,long-termdynamicswerenotimpactedbyinitialmacro-
phagecount.
Effectsofcontinuoustreatment
Afterevaluatingtumorgrowthintheabsenceoftreatment,weevaluatedtheefficacyofthree
macrophage-basedtherapies:(1)macrophagedepletion,(2)inhibitionofmacrophagerecruit-
ment,and(3)re-educationofmacrophagesviaPI3Kinhibition.Toinitiallyexplorehoweach
treatmentstrategyaffectstheimmuneresponse,wesimulatedeachtreatmentattendifferent
strengths.Treatmentwasstarted100daysaftertumorinitiation,whichwasafterthesystem
hadreachedequilibrium.Thetreatmentcontinueduntileitherthetumorwasremovedor
until200daysofsimulationtimewasreached.Werepeatedeachsimulation100timesand
averagedtheresultstoobtainthegeneraleffectofeachtreatment.Aprimaryoutputofthese
simulationsisthefractionoftumorsremoved—thenumberoftumorseliminatedaftertreat-
mentwasstarted.Wealsocalculatedthetimefromstartoftherapytotumoreliminationand
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themaximumnumbersofM1macrophages,Tcells,andactiveTcells.Becausetreatment
startsonceequilibriumisreached,thesystemisalreadyatthemaximumnumberoftumor
cellsandM2macrophages,sowedonotrecordtheirmaximumvaluesduringtreatment.
First,weinvestigatedtheeffectsofmacrophagedepletion(Fig3A).The“DepletionProba-
bility”displayedonthex-axisistheprobabilitythatamacrophagehasofbeingremovedat
eachtimestep,withanequalprobabilityforeachmacrophagephenotype.Fig3A-ishowsthe
percentoftumorseliminatedaftertreatmentisstarted.Weseethatthereisadrasticjump
fromalmostnoeffectivenessataprobabilityof0.001toalmostcompletetumorremovalata
probabilityof0.002,whichthenincreasesslightlyandstaysatcompletetumorremovalasthe
probabilityisincreased.Itisinterestingthatthereissuchasharpincreaseineffectivenesswith-
outacorrespondingincreaseinthemaximumnumberofM1macrophagesoractiveTcells.
Webelievethatthisincreaseisduetoremovalofenoughmacrophagesaroundthetumorto
allowTcellsnearthetumortoinfiltratebetterandremovethetumor.Followingthis,theaver-
agenumberofdaysneededtoremovethetumordecreasesandthenremainsstatic(Fig3A-ii).
WeseethatthemaximumnumberofM1macrophagesincreaseswithdepletionprobabilityto
apoint,thenbeginstodecreaseatthehighestdepletionprobabilities(Fig3A-iii).Webelieve
thattheincreaseisduetothedecreaseinIL-4astumorcellsandM2macrophagesare
removed,allowingnewlyrecruitedmacrophagestodifferentiateintotheM1state.InS5Fig
weshowthetimecoursesatadepletionprobabilityof0.006,whichshowsadecreaseinIL-4
andandincreaseinIFNjustpriortotheincreaseinM1macrophages.Asdepletionprobability
furtherincreases,thesenewmacrophagesareremovedfastenoughtodecreasethemaximum
numberofM1macrophages.However,duetothelownumbersofM1macrophagesandlarge
errorbars,wecannotmakeanydefinitiveinferencesfromthis.ThetotalnumberofTcells
increasesveryslightlywithdepletionprobability(Fig3A-iv)whileTcellactivationseesamore
drasticincrease(Fig3A-v)correspondingwiththenumberofM1macrophages.
Examiningthetimecourseforadepletionprobabilityof0.002(S6Fig),wefindthatinsim-
ulationsthatfailtoremovethetumor,thetumorcellpopulationreachesanewequilibrium
statethatislowerthantheoriginal(S6AFig).Asthetumorisremoved,thereisanincreasein
thenumberofnaïvemacrophages(S6BFig)duetolessofaninfluencefromthetumorfor
macrophagedifferentiation.Atthislevelofdepletion,noM1macrophagesariseduringsimu-
lation(S6CFig),whilethenumberofM2macrophagesdecreases(S6DFig).Wealsoseelittle
differenceinTcellandactiveTcelldynamics(S6EandS6FFig)betweensimulationsthat
removedthetumorandthosethatdidnot,presumablyduetothelackofM1macrophages,
whichpromoteTcellactivation.
Thesecondtreatmentstrategyweemployedwasinhibitionofmacrophagerecruitment,
whichissimilartomacrophagedepletioninthatiteliminatesmacrophagesregardlessofphe-
notype.Onthex-axesforFig3Bisthe“InhibitionStrength,”whichisthefractionthatthe
parameterforrecruitmentratewasreducedby.Weseethatbelow0.6inhibition,treatmentis
ineffectiveinremovingthetumor.At0.6thereisveryminoreffectiveness,whichthenrapidly
increaseswithinhibitionstrength(Fig3B-i).Asinhibitionstrengthincreases,weseeaslight
decreaseintheaveragetimeneededtoremovethetumor(Fig3B-ii),howeveritdoestakelon-
gertoremovethetumorthanwithmacrophagedepletionprobabilitiesthatachievedsimilar
tumorremoval(compareFig3A-iiandFig3B-ii).Whereasmacrophagedepletionshowed
Fig2. Timecourseswithbaselineparametersandnotreatment.Timecoursesfortumorsthatwerenotremovedby
theimmunesystemareshowninblack;thosefortumorsthatwereremovedareshowninred.Thereisonlyone
simulationherethatledtotumorremoval.(A)Cancercells,(B)M0cells,(C)M1cells,(D)M2cells,(E)totalTcells,
(F)activeTcells,(G)averageIL-4,(H)maximumIL-4,(I)averageIFN- ,(J)MaximumIFN- .(G)–(J):unitsare
numberofmolecules.
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someincreaseinthenumberofM1macrophages,theyareessentiallynonexistentwithrecruit-
mentinhibition(Fig3B-iii).Webelievethisisduetothefactthatmacrophagedepletion
removesmacrophagesfromtheenvironmentwhilerecruitmentinhibitionpreventsnewmac-
rophagesfromentering.Whilebothmethodslowerthetotalnumberofmacrophages,the
macrohpagesthatremainevenwithrecruitmentinhibitionareclosertothetumor,whereIL-4
levelsarehighest,whichkeepsthemintheM2state.Withmacrophagedepletion,newmacro-
phagesentertheenvironmentfartherfromthetumorwhereIL-4levelsarelower,allowing
someofthemtodifferentiatetotheM1state.Withthis,weseethatthereisnochangeinthe
numbersoftotaloractiveTcells(Fig3B-iv,v),withTcellactivationbeingfairlylow.
Examiningthetimecoursesforarecruitmentinhibitionof0.7,whicheliminatedroughly
halfofthetumors,wefindthatthetimecoursesfortumorsthatwereremovedareverysimilar
tothosethatwerenot(S7Fig).Thedynamicsforthetumorcellsareverysimilarfortheinitial
decrease,withtumorsthatwereeliminatedcontinuingtodecrease,whiletumorsthatwerenot
suddenlylevelingofftoanewequilibrium(S7AFig).Webelievethisisduetoinherentran-
domnessandM2macrophagespreventingTcellsfromadvancingtthetumor.Wedosee
someincreaseinnaivemacrophagesintumorsthatwereeliminated(S7BFig).Likethesimu-
lationswithouttreatment,noM1macrophagesappear(S7CFig).ThenumbersofM2macro-
phages,totalTcells,andactiveTcellsexhibitsimilarbehaviorasthetumorcells(S7D–S7F
Fig).IL-4andIFN- concentrationsmirrortumorcellandactiveTcelldynamics,respectively
(S7G–S7JFig).
Thefinaltreatmentstrategy,PI3Kinhibition,yieldedsimilarefficaciesasrecruitmentinhi-
bition,thoughwithasteepershiftfromnon-effectivenesstoeffectiveness(Fig3Ci).However,
PI3Kinhibitiondisplayedthefastesttumorremovaltime(Fig3Cii)duetoincreasesinM1
macropahgesandthusTcellfunction.Asintended,becausePI3Kinhibitionaimstoreeducate
themacrophages,thenumberofM1macrophagesismuchhigherwiththisstrategy(Fig
3Ciii).ThisincreasecorrelateswithanincreaseinthenumberoftotalandactiveTcells(Fig
3Civ,v).WealsoexaminedthetimecoursesforaPI3Kinhibitionof0.8(S8Fig).Thereisthe
expecteddecreaseintumorcells(S8AFig)and,sincethistreatmentdoesnotremovemacro-
phages,thereisanincreaseinnaivemacrophages(S8BFig)duetoadecreasedpressuretodif-
ferentiateasthenumberofcancercellsisreduced.ThenumberofM1macrophagesrapidly
increasesimmediatelyaftertreatment,whilethenumberofM2macrophagesrapidlydecreases
(S8CandS8DFig).BothtotalandactiveTcellsincreasegreatlywiththeincreaseinM1mac-
rophages(S8EandS8FFig)duetotheinfluenceoftheM1macrophages.Aswithothertreat-
ments,IL-4andIFN- levelsfollowtumorandTcelldynamics(S8G–S8JFig).
Overall,wefindthat,atthebaselineparameters,treatmentefficacyforallthreestrategies
steeplyincreasesfromineffectivetoveryeffectivewhensimulatedcontinuously.PI3Kinhibi-
tion,whichconvertsthemacrophagestotheM1state,leadstohigherlevelsofTcellactivation
andremovesthetumoratafasterratethantheothertwotreatments.Interestingly,although
bothofthesetreatmentsreducethenumberofmacrophagesintheTME,weseethatmacro-
phagedepletionleadstoaslightincreaseinM1macrophageswhilerecruitmentinhibition
doesnot.
Fig3. Effectsofcontinuousimmunotherapystartedat100daysofsimulation.(A)Macrophagedepletion,(B)recruitmentinhibition,
and(C)PI3Kinhibition.(i)fractionoftumorsremovedafterstartingtherapy.(ii)time(days)fromstartingtreatmenttotumorremoval.Itis
averagedoverthe100simulationsandisequaltozeroifnotumorswereremovedatthattreatmentlevel.(iii)maximumnumberofM1
macrophages.(iv)maximumnumberoftotalTcells.(v)maximumnumberofactiveTcells.Notethedifferencesiny-axisscalesacross
treatmentstrategies.Asteriskssignifythataresultisstatisticallysignificant(p<0.01)fromtheresultofthelowesttreatmentstrength.We
notethatforthetimeneededtoremovethetumor(ii),weplotthetimeaveragedoveronlysimulationswherethetumorwasremoved.
Therefore,whilesomebarsmayappearmuchhigherthanthatofthelowesttreatmentstrength,theyonlyrepresentasmallnumberof
simulationsoutof100andthuswerenotfoundtobestatisticallysignificant.
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Treatmentcycles
Whileitisidealtobeabletogiveacontinuoustreatment,thisisunrealisticduetovariousrea-
sons,suchaspatientcompliance,inabilitytogivetreatmentcontinuouslyduetopharmaceuti-
calimplementation,orthedesiretogiveaslittletreatmentaspossibleinordertoavoid
potentialside-effects.Therefore,weranseveralsetsofsimulationswheretreatmentwascycled
onandoff.Wevaryboththetotallengthofthetreatmentcycleandtheamountoftimeduring
thecyclethattreatmentisgiven.Weperformedthesesimulationsonlyforhighertreatment
strengths,asitcanbereasonedthatcyclingtreatmentwouldnotbeaseffectiveasconstant
treatment.
Formacrophagedepletion,weimplementtreatmentbyremovingthefractionofmacro-
phagesequalto“DepletionStrength”atthebeginningofeachtreatmentcycle(Fig4A),shown
onthey-axis.Thedurationofeachcycleisshownonthex-axis.Evenwithcyclingtreatment
onandoff,wefindthatthereisasteepincreaseinthetreatmentbeingineffectivetoveryeffec-
tiveasdepletionstrengthisincreasedorcycledurationisdecreased(Fig4A-i).Thetime
neededforthetreatmenttoremovethetumoralsodecreasesslightlyasdepletionstrengthis
increasedandcycledurationdecreased(Fig4A-ii).Surprisingly,weseeaslightincreaseinM1
macrophages,totalTcells,andactiveTcells(Fig4A-iiito4A-v),especiallyatthehighest
depletionstrengthandslightlylongercycledurations,toahigherlevelthanwithconstant
treatment.ThisismostlikelyduetothedecreaseinIL-4astumorcellsandM2macrophages
areremoved,allowingsomeofthenewmacrophagestodifferentiateintotheM1state.
Whereasmacrophagedepletioncontinuestobeeffectivewhentreatmentiscycled,recruit-
mentinhibitionbecomesratherineffective,evenatcompleteinhibitionofmacrophage
recruitment(Fig4B).Whencyclingrecruitmentinhibition,treatmentwasturnedonforthe
numberofdaysshownonthey-axisandturnedofffortheremainderofthecycle(x-axis).
Whentreatmentisgivenforalmosttheentirecycle,thereisstrongtumorremoval,however
thisquicklyfallswhentreatmentisnotgivenforaslongorwhencycledurationincreases(Fig
4B-i).Thetimeneededtoremovethetumorisfairlyconsistentincaseswherethereistumor
removal(Fig4B-ii).WhiletherearelittletonoM1macrophagesforalmostallofthetreatment
combinations,thereisoneinstancewhereisalargenumberofM1macrophages(Fig4B-iii).
However,becausethatcombinationsawverylittletumorremoval,weassumeitisaveryrare
stochasticoccurrence.ThenumbersofTcellsandactiveTcellsremainlowandconstant
acrosscombinations,excepttheinstancewheretherewasanincreaseinM1macrophages(Fig
4B-iv,v).
Interestingly,aPI3Kinhibitionof0.8,whichisthelowestsuccessfulinhibitionforcontinu-
oustreatment,isstillverysuccessfulwhencyclingtreatment,evenwhentreatmentisgiven
brieflyforlongcycledurations(Fig4C-i).Forthebulkofthetreatmentcombinations,the
timeneededtoremovethetumorisfairlyconstant(Fig4C-ii).M1macrophages,Tcells,and
activeTcellsareallattheirmaximumvalues(Fig4C-iiito4C-v).TodeterminewhyPI3K
inhibitioncontinuestobesuccessfulevenwhengivenforshortdurations,weexaminedthe
timecourseswhentreatmentwasgivenfor2daysoutofa25-daycycle(Fig5).Interestingly,
treatmentcyclesdonotalwaysremovethetumor;however,whentumoreliminationdoeshap-
pen,itoccursveryswiftly(Fig5A).Itisunclearwhythisoccurs,thoughwedoseethaton
cyclesthatdonotleadtoremoval,thereislittlechangeinthenumberofM2macrophages.
BecausethesustainedresponsethatleadstotumorremovalisduetoTcellproducedIFN- ,
webelievethatnotenoughTcellsarebecomingactivatedduringthesecyclestocausethesus-
tainedresponse.Wehavehighlightedasingletimecourseinredtomakeiteasiertovisualize.
Immediatelyfollowingasuccessfulcycleoftreatment,weseeagradualriseinnaivemacro-
phagesasthetumoriseliminated(Fig5B),andarapidincreaseinM1macrophagesfollowed
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byagradualdecrease(Fig5C).M2macrophages,ontheotherhand,rapidlydecreasesince
theyarebeingconvertedintheM1state(Fig5D).CorrespondingwithM1behavior,thenum-
bersofTcellsandactiveTcellsrapidlyincreaseduetoinfluencefromtheM1macrophages
(Fig5Eand5F).IL-4andIFN- levelscorrelatewithtumorandTcelldynamics(Fig5G–5J).
WeseethatduetosustainedTcellactivationthereissustainedIFN- ,whichisresponsiblefor
promotingM1differentiation.ThiscreatesafeedbackloopbetweentheM1macrophagesand
TcellsthatsustainsM1differentiationandtumorremovalevenaftertreatmentisremoved.
Overall,whencyclingtreatmentonandoff,wefindthatmacrophagedepletionyields
expectedresults,withhigherdepletionstrengthsandshortertreatmentcyclesleadingtoa
strongertumorremoval.Interestingly,recruitmentinhibitionbecomesveryineffectivewhen
cycled,unlesstreatmentisgivenforalmosttheentiretyofsimulation.PI3Kinhibition,how-
ever,isveryeffectiveforalmosteverycombinationoftimeonandtimeoff.Wefindthatthisis
becauseconvertingthemacrophagestotheM1phenotypepromotesTcellactivationandIFN-
secretion,whichsustainstheM1phenotypeaftertreatmentisremoved.
Changingtumorproliferationrateandmacrophagerecruitmentrate
Ofinterestishowtumorproliferationrateandmacrophagerecruitmentrateaffectthe
immuneresponsewithandwithouttreatment.Presumablythesetwoparameterswouldlead
toadifferentequilibriumstate,whichwouldchangetheeffectivenessofeachtreatment.We
repeatedtheaboveanalysisforthreesetsoftumors:increasedproliferationrate,increased
macrophagerecruitmentrate,andboth.Wefirstconsidertumorgrowthwithouttreatment,
andthenimplementthethreemacrophage-basedtreatmentstrategies.
Comparedtothebaselineparameters,weseeonlyaslightincreaseintheequilibriumnum-
beroftumorcells(Fig6A)whentumorproliferationrateisincreasedfrom0.8perdayto1.2
perday.Whatismostinterestingisthatatrandompointsthroughouttheequilibriumstate,
theimmunesystemwillsuddenlyremovethetumor,aphenomenonnotseenwiththebaseline
parameters.Aswithprevioussimulations,adecreaseintumorcellpopulationisfollowedby
anincreaseinnaivemacrophagepopulation(Fig6B).Forsimulationsthatremovedthe
tumor,weseearapidincreaseinM1macrophageswithacorrespondingdecreaseinM2mac-
rophages(Fig6Cand6D).ThedynamicsoftotalTcellsandactiveTcellsfollowsthatofM1
macrophages(Fig6Eand6F).IL-4andIFN- dynamicsfollowtumorcellsandTcells,respec-
tively(Fig6G–6J).
Attheincreasedtumorproliferationrate,weseeamoregradualincreaseintheeffective-
nessofconstantPI3Kinhibition(Fig7A).However,theseresultsareconflatedduetopropen-
sityoftheimmunesystemtospontaneouslyremovethetumorduringtheequilibriumstate,as
evidentbythelargeerrorbarsinFig7B.Acrossinhibitionstrengths,thereisalargeamountof
M1differentiation,Tcellnumbers,andTcellactivation(Fig7C–7E).CyclingPI3Kinhibition
issuccessfulthroughoutthedifferentcycledurations,evenwiththeincreasedtumorprolifera-
tionrate(S9CFig).Macrophagedepletionandrecruitmentinhibitiondonotshowaclear
Fig4. Effectsofcycledimmunotherapystartedat100daysofsimulation.Formacrophagedepletion(A),thefraction
ofmacrophagesremovedatthebeginningofeachcycleisgivenas“DepletionStrength”andthelengthofeachcycleis
“CycleDuration.”Forrecruitmentinhibition(B)andPI3Kinhibition(C),thenumberofdaysinthecyclethattreatment
isonforisgivenas“DaysTreatmentisOn.”Recruitmentinhibitionissimulatedatastrengthof1.0(completeinhibition)
andPI3Kinhibitionissimulatedatastrengthof0.8.ForrecruitmentinhibitionandPI3Kinhibition,spacesmarkedwith
anXarethosewheretreatment-ontimeisequalorgreatertothecycleduration,thuswerenotsimulated.(i)fractionof
tumorsremovedafterstartingtherapy.(ii)time(days)fromstartingtreatmenttotumorremoval.Itisaveragedoverthe
100simulationsandisequaltozeroifnotumorswereremovedatthattreatmentlevel.(iii)maximumnumberofM1
macrophages.(iv)maximumnumberoftotalTcells.(v)maximumnumberofactiveTcells.
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trendandtheresultsareagainconflatedduetotheinherenttumorremovalfoundwithout
treatment(S9AandS9BFig,S10FigandS11Fig).
Whereasincreasedtumorproliferationonlycausedaslightincreaseintheequilibrium
tumorpopulation,doublingtherateofmacrophagerecruitmentgreatlyincreasedtheequilib-
riumtumorpopulationtoroughlytwiceofthebaseline(Fig8A).NaiveandM1macrophage
dynamics(Fig8Band8C)areverysimilartothebaselinewhiledynamicsfortheremaining
cellpopulationsandcytokinesreachahigherequilibriumstate(Fig8D–8J).
Atthehighermacrophagerecruitmentrate,macrophagedepletiontherapybearssimilar
effectivenessasthebaselinerecruitmentrate(Fig9A),howevereffectivenessappearsata
depletionprobabilityof0.003ratherthan0.002.Thetimeneededtoremovethetumoriscon-
stantacrossdepletionprobabilities(Fig9B)andisabithigherthanatthebaselinerecruitment
rate.ThereisalsoalargeincreaseinthenumberofM1macrophages(Fig9C),andthepeakat
amoderatedepletionprobabilitybecomesmoreclear.Thisresponseismirroredwiththe
numbersofTcellsandactiveTcells(Fig9Dand9E),thoughwithamoregradualtail.The
othertherapiesresemblethebaselinecase(S12,S13andS14Figs).
Increasingbothtumorproliferationrateandmacrophagerecruitmentrateleadstothe
highestequilibriumtumorpopulation(Fig10A).Thedynamicsoftheothercellsandcytokines
behavesimilarlytotheprevioussimulationswithouttreatment(Fig10B–10J).
Interestingly,lowertreatmentstrengthsofcontinuousPI3Kinhibitionaremoreeffective
herethanatthebaselineparameters,despitethehigherequilibriumstateofthetumor(Fig
11A),andefficacyincreasesgraduallywithinhibitionstrength.However,atthehigherinhibi-
tionstrengths,treatmentisnotaseffectiveasitwasatthebaselineparameters(Fig3Ci).The
timeneededtoremovethetumorisfairlyconstantacrossinhibitionstrengths(Fig11B).The
numbersofM1macrophages,totalTcells,andactiveTcellsallincreasegraduallyfollowing
treatmentefficacy,however,duetothewidestandarddeviations,itdoesnotappeartobetoo
significantofanincrease(Fig11C–11E).Neithercontinuousmacrophagedepletionnor
recruitmentinhibitionhaveasignificantabilitytoremovethetumor(S15andS16Figs).
WhilecyclingPI3Kinhibitionremainssimilartothebaselinecase(S17Fig),weseeavery
differentbehaviorwithmacrophagedepletionandrecruitmentinhibitionwhentumorprolif-
erationandmacrophagerecruitmentratesareincreased,comparedtobaselinecase.InFig12
weshowtheeffectsofcyclingmacrophagedepletionwhiletheeffectsofrecruitmentinhibi-
tion,whichareverysimilar,areshowninS18Fig.Interestingly,moderatedepletionstrengths
andtreatmentcyclelengthsorhighdepletionstrengthsatlongtreatmentcycleswerethemost
effectiveatremovingthetumor(Fig12A).Thisresultisverydifferentfromprevioussimula-
tions,whereefficacycorrelatedtotheamountoftreatmentgiven.Thisshowsthatmoremod-
eratetreatmentcouldpotentiallybemoreeffective.Whiletheefficacyherestillisnotvery
high,itishigherthancontinuoustreatmentforthesametumorparameters.Timeneededto
removethetumorisfairlyconstantforcaseswheretherewastumorremoval(Fig12B).The
levelsofM1macrophagesandTcellsmirrorthebehavioroftumorelimination(Fig12C–
12E).
Overall,wefindthattherateofmacrophagerecruitmentismoreimpactfulontheequilib-
riumstatethanthetumorproliferationrate.Wealsonotethat,surprisingly,increasingthe
tumorproliferationrateleadstospontaneoustumorremovalduringtheequilibriumstate.As
Fig5. PI3Kinhibitionof0.8forcycledimmunotherapy.IndividualtimecoursesforPI3Kinhibitionof0.8atacycle
durationof25dayswithtreatmentgivenfor2dayspercycle.Tumorsthatsurvivedtotheendofsimulationareshown
inblack.Tumorsthatwereeliminatedareshowningreen.Onetimecourseisshowninredforeaseofunderstanding.
(A)Cancercells,(B)M0macrophages,(C)M1macrophages,(D)M2macrophages,(E)Tcells,(F)ActiveTcells,(G)
AverageIL-4,(H)MaximumIL-4,(I)AverageIFN- ,(J)MaximumIFN- .
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Fig6. Timecourseswithincreasedtumorproliferationrateandnotreatment.Thetumorproliferationrateis
increasedto1.2/day.Timecoursesfortumorsthatwerenotremovedbytheimmunesystemareshowninblack;those
fortumorsthatwereremovedareshowninred.(A)Cancercells,(B)M0cells,(C)M1cells,(D)M2cells,(E)TotalT
cells,(F)ActiveTcells,(G)AverageIL-4,(H)MaximumIL-4,(I)AverageIFN- ,(J)MaximumIFN- .
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Fig7. ConstantPI3Kinhibitionattumorproliferationrateof1.2/day.(A)fractionoftumorsremovedafterstartingtherapy,(B)averagetime
neededtoremovethetumor,(C)themaximumnumberofM1macrophages,(D)themaximumnumberoftotalTcells,(E)themaximumnumber
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canbeexpected,treatmentefficacydecreasesastheequilibriumstateincreases,thoughPI3K
inhibitionretainedstrongefficacyathighertreatmentstrengths.Themostinterestingresultis
fromthesimulationsatincreasedtumorproliferationandmacrophagerecruitment.Here,
whencyclingmacrophagedepletionandrecruitmentinhibition,wefindthatmoderatetreat-
mentsareactuallymoreeffective,causingaslightincreaseinM1macrophagesandtumor
removal,howevertheireffectivenessiswellbelowthatofPI3Kinhibition.
Discussion
Inthisstudy,wepresentanABMexaminingmacrophage-Tcellinteractionsandhowmacro-
phage-basedimmunotherapiescaninfluencetumorgrowthandtheimmuneresponse.While
macrophageshaveanumberofeffectsontheTME,includingimmunosuppression,angiogen-
esis,andtumorcellinvasion[10,11,13,14],wefocusontheirimmune-relatedinteractionsand
howtheycanpromoteorinhibittheTcellresponse.Byusinganagent-basedmodel,weare
abletoexploretheemergentbehaviorthatarisesfromcell-to-cellinteractionsthatwouldoth-
erwisebeverydifficulttocapturewithdeterministicequations.Tobetterexploremacrophage-
basedimmunotherapies,weutilizeamechanisticmodelofintracellularmacrophagepheno-
typemarkersinresponsetotwotypicalM1andM2relatedcytokines,respectivelyIFN- and
IL-4.
Wefind,consistentwithexperimentalobservations,thatalmostallofthemacrophagesin
thesystemdisplayanM2phenotypewhennotreatmentisgiven,whichisindicativeofapoor
clinicaloutcome[36].Interestingly,wefindthatthesystemreachesanequilibriumwherethe
Tcellsareabletofunctionenoughtopreventtumoroutgrowthbutareunabletoremovethe
tumor.Thiscanbeconsideredthe“immunecontrol”phaseoftumorgrowthandisverydiffi-
culttoexploreexperimentallyas in vitromethodsareunabletocaptureimmunecellrecruit-
mentorthelongtimescaleoverwhichimmunecontroloccurs.Additionally,itisdifficultto
findthisphase in vivoasthisphaseiscompletedbythetimeatumorcanbedetected.Atour
baselineparameters,wefindthat,atahighenoughstrength,eachtreatmenthasastrongeffi-
cacyforremovingthetumorwhengivencontinuously.Whencyclingtreatment,macrophage
depletionretainsitsefficacyathigherstrengthsandshortercycleswhilerecruitmentinhibition
becomeslargelyineffective.PI3Kinhibitionretainsastrongefficacy,evenwhengivenfora
shortamountoftimeoveralongcycle,duetoapositivefeedbackloopbetweentheM1macro-
phagesandTcells,highlightingtheimportanceoftheseinteractions.
Atincreasedtumorproliferationratesandmacrophagerecruitmentrates,whichincrease
theequilibriumtumorpopulation,macrophagedepletionandrecruitmentinhibitionbecome
lesseffectivewhilePI3Kinhibitionretainsefficacy.Whatisnotablehereisthat,whencycling
macrophagedepletionandrecruitmentinhibition,thereisaslightincreaseintumorremoval
andthenumberofM1macrophagesatmoderatetreatmentcycles,whichmeansthatastron-
gertreatmentmaynotalwaysbethemosteffectiveinthecaseofloweringthenumberofmac-
rophagesintheTME.WebelievethisisbecausethesetreatmentsremoveM2macrophages
fromthesystem,whichdecreasesIL-4levelsandallowsTcellstoactivate,increasingIFN- levels.Thisthenallowsmacrophagesnewlyrecruitedtothetumorsite,whichenterthesimula-
tionatadistancefromthetumorandthusfarfrompeakIL-4levels,todifferentiatetoM1and
increasetheTcellresponse.Atmoderatelevelsoftreatment,enoughmacrophagesarestill
ofactiveTcells.Notethedifferencesiny-axisscalesacrosstreatmentstrategies.Asteriskssignifythataresultisstatisticallysignificant(p<0.01)
fromtheresultofthelowesttreatmentstrength.Wenotethatforthetimeneededtoremovethetumor(B),plottedisthetimeaveragedoveronly
simulationswherethetumorwasremoved.Therefore,whilesomebarsmayappearmuchhigherthanthatofthelowesttreatmentstrength,they
onlyrepresentasmallnumberofsimulationsoutof100andthuswerenotfoundtobestatisticallysignificant.
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enteringthesystemtodifferentiateintoM1macrophages,whereashigherlevelsofthesetreat-
mentspreventnewmacrophagesfromenteringanddifferentiating.Theseresults,alongwith
theefficacyofPI3Kinhibition,highlighttheimportanceofM1macrophagesandtheirinterac-
tionswithTcellswithintheTME.Thus,havingamodelingframeworkthatexplicitlyaccounts
fortheseinteractionsisparticularlyuseful.
Whileitisdifficulttoextensivelymatchoursimulationresultsto in vivodata,wehave
foundsomestudiesthatqualitativelysupportourresults.Nyweningetal.[37]presentaphase
1bclinicaltrialofanoral,small-moleculeCCR2inhibitor,whichdecreasesmacrophage
recruitmenttothetumor.TheyfoundthatthislowersthenumberofTAMsinthetumorand
improvestheanti-tumorimmuneresponse.Thissupportsourexplanationofthemodelpre-
dictionfortheeffectofreducingmacrophagerecruitment–thatthisstrategyreducesthenum-
berofM2macrophages(S4Fig).Germanoetal.[38]performedapre-clinicalstudyexamining
amacrophage-depletingdrugthatwasabletoeliminatemacrophagesfromthetumorsite
withoutaffectingthenumberofTcells,whichimprovedTcellremovalofthetumor,whichis
inlinewiththemodelpredictions(Fig3A-iv).Kanedaetal.[39]examinedPI3Kinhibitionin
miceandfoundthatitstimulatesananti-tumorimmuneresponse.Theyfoundthat, in vitro,
PI3Kinhibitioncausesadecreaseinimmunosuppressivemacrophagemarkersandanincrease
inimmunesupportingmarkers.ThisissimilartothemodelpredictionsforhowPI3Kinhibi-
tionaffectsIL-4andIFN- (S5Fig).Theyalsofoundthat,inmice,PI3Kinhibitionledtoan
increaseinIFN- expressionbyTcellsandtumorremoval.
Anovelpieceofourworkisthereductionofcomplexmechanisticmodelsintosimpleneu-
ralnetworksfortheirinclusionintheindividualagents.Thoughthereareseveralstudiesthat
incorporatemechanisticmodelsintoABMs,thisbringswithitagreatcomputationalcost,
whichiswhymostmodelsonlyusesimplediscrete/stochasticrules.Whilesimplisticrulescan
stillbeusedtodrawgreatinsightaboutthesystem,sinceABMsaregenerallyconcernedabout
behavioratthemulticellularscale,addingthisadditionalbiologicalscaleallowsustobetter
understandthemechanisticmodelandgivesusinsightintohowchangesattheintracellular
levelcancompoundintochangesatthemulticellularlevel.Usingthemechanisticmodelout-
sideoftheABMtotrainaneuralnetworkgreatlyimprovesthecomputationalspeed,allowing
ustorunmoresimulationsandexploreadditionalaspectsoftumorgrowth.Whilesomeinfor-
mationisinevitablylost,withtheneuralnetworkonlypredictingcategoricalbehavior,we
believeittobeanacceptabletrade-off.Wedemonstratethatnotonlycanweusetheneural
networktopredicttheresultsofdifferingcytokinelevels,butwecanalsoincludedifferent
kineticparameterssothatwecansimulatetheeffectsofspecifictargets.Thismethodprovides
ampleopportunityforfuturesimulationswhereweexploreintercellularheterogeneityby
includinginitialproteinconcentrationsintheMonteCarlosimulationsandhavingthesepro-
teinconcentrationsasneuralnetworkinputsaswell.
Weacknowledgesomelimitationsofourmodel.Themainlimitationisthatmacrophage
phenotypeandinteractionswithTcellsaresimplified.Macrophages,inreality,displayarange
ofpropertiesandcanexistasmixedphenotypes.However,simplifyingdifferentiationintodis-
cretephenotypes,achoicemadebysimilarmodels,capturesenoughofthemacrophagebehav-
iortobesufficientforthisstudy.Also,theinteractionsbetweenmacrophagesandTcellsare
mediatedbymanydifferentcytokines,makingtheirinteractionsmuchmorecomplexthan
howwemodeledthem.However,addinginmorecytokinesandligandexpressionswould
Fig8. Timecourseswithmacrophagerecruitmentratedoubledandnotreatment.(A)Cancercells,(B)M0cells,
(C)M1cells,(D)M2cells,(E)TotalTcells,(F)ActiveTcells,(G)AverageIL-4,(H)MaximumIL-4,(I)AverageIFN-
,(J)MaximumIFN- .
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havemadethemodelmuchmorecomplex,andwedonotbelievethisadditionallayerofcom-
plexitywouldhavesignificantlycontributedtotheobjectiveofthecurrentstudy.Another
modellimitationisthatwedonotaccountfornutrientuptakeorhypoxia,whichpromotes
M2differentiationandTcellsuppression[31].Wechosetonotincludetheseeffectsbecause
wewantedtofocussolelyontheinteractionsbetweenmacrophagesandTcells.Wecan
accountfortheeffectsofnutrientsinlatermodels.Inaddition,inthiswork,wedonotstudy
theeffectsofchangingtheTcellnumbersorbehaviorsinordertoisolatehowmacrophage-
basedstrategiesinfluencetumorgrowth.ExpandingthestudytovaryTcelldynamicscanbe
anotherfocusoffuturework.Wealsoaimtoexploreamoredetailedspatialanalysisofthe
modeltobetterunderstandtheobservedphenomena.
Overall,ourmodelcapturestheoverarchinginteractionsbetweenmacrophagesandTcells
withintheTMEandpredictshowthreemainmacrophage-basedimmunotherapiesimpactthe
immuneresponsetothetumor.WehighlighttheimportanceoftheinteractionsbetweenM1
macrophagesandTcellsforpromotingarobustanti-tumorresponse.Wealsointroducea
methodforreducingthecomputationalcostofincorporatingmechanisticmodelsintoan
ABMbytraininganeuralnetworkonthecalculatedmechanisticmodelresponse.
Methods
Here,wepresentamulti-scale,hybrid,ABMoftheTME,consistingofseveralcelltypesand
diffusiblefactors.Ourmodelislattice-based,witheachcelltakinguponelatticesite.Eachcell
typehascertainbehaviors:cellscanproliferateintoemptylatticesites,migratetoanewlattice
site,andinteractwithothercellsanddiffusiblefactors.Usingthismodel,weexaminevarious
immunotherapiesandtheirimpactontheTME.
Overviewofmodel
Themodelrepresentsearlytumorgrowthorasmallinitialmetastasis,andwesimulatethe
tumorin2D,representingatissueslice.Themodelconsistsofthreecelltypes:Tcells,macro-
phages,andcancercells,representedasdiscreteagents.Interactionsbetweencelltypesand
cytokinesareshowninFig1A.Thelatticeisa100x100gridrepresentinga1.5x1.5mmtissue
slice,witheachsitebeinga15-micrometersquare,thesizeofonecelldiameter[40].Assuch,
onlyonecellcanoccupyasiteatatime.Whiledifferentcelltypesdohavedifferentsizes,the
“onecellpersite”assumptionisanecessarylimitationofon-latticemodels,withmanysimilar
modelsmakingthesameassumption[27,29,30].Off-latticemodelsareabletoeasilyaccount
fordifferentcellsizes,howeverthesemodelsaremorecomputationallyexpensive.Fig1Bdis-
playsanexamplesimulationshowingthespatialdistributionofdifferentcelltypes.Tomodel
theproductionanddiffusionofdiffusiblefactors,wehavealayerofpartialdifferentialequa-
tions(PDEs).Parametersareeithertakenfrompreviousmodelingeffortsorsetbasedon
experimentalobservations.Ourmodeldoesnotrepresentaspecifictumortype.Instead,itis
meanttoexaminegeneralizedtumorbehavior.Wedescribethemodelindetailbelowandthe
parametervaluesarelistedinS1Table,alongwithsupportingreferences.
Fig9. Constantmacrophagedepletionstartedat100dayswithmacrophagerecruitmentratedoubled.(A)fractionoftumorsremovedafter
startingsimulation,(B)averagetimeneededtoremovethetumor,(C)themaximumnumberofM1macrophages,(D)themaximumnumber
oftotalTcells,(E)themaximumnumberofactiveTcells.Notethedifferencesiny-axisscalesacrosstreatmentstrategies.Asteriskssignifythat
aresultisstatisticallysignificant(p<0.01)fromtheresultofthelowesttreatmentstrength.Wenotethatforthetimeneededtoremovethe
tumor(B),plottedisthetimeaveragedoveronlysimulationswherethetumorwasremoved.Therefore,whilesomebarsmayappearmuch
higherthanthatofthelowesttreatmentstrength,theyonlyrepresentasmallnumberofsimulationsoutof100andthuswerenotfoundtobe
statisticallysignificant.
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Diffusiblefactors
Therearethreediffusiblefactorspresent:atumor-secretedfactorthatactivatesmacrophages
andtwocytokines,IL-4andIFN- .Thefirstisreferredtoasthemacrophageactivationfactor,
whichalertsthemacrophagesofthetumorandcausesthemtodifferentiate.Thisfactoris
secretedbythetumorcellsandprimarilyrepresentshighmobilitygroupbox1protein
(HMG-B1)[29].Alsosecretedbythetumor,andbyM2macrophages,isIL-4,whichisan
immunosuppressivecytokinethatpromotesmacrophagedifferentiationintotheM2tumor-
promotingphenotype[41].ThefinaldiffusiblefactorisIFN- whichissecretedbyactivatedT
cellsandisapartoftheTh1response[42].Thisimmuneresponsepromotesmacrophagedif-
ferentiationintotheM1immune-promotingphenotype.IL-4andIFN- werechosenbecause
theyaretypicalpro-tumorandanti-tumorcytokines,respectively.Also,theyareusedasthe
inputstothemechanisticmodelusedtodeterminemacrophagedifferentiation,describedin
detailinlatersections.
ThediffusionandsecretionofthesefactorswasmodeledusingPDEsshowninEq1.
@C
i
@t
à Dr
2
C
i
á k
sec;i
Cells x;y Ö Ü Ö 1Ü Here, C
i
istheconcentrationofdiffusiblefactor i, k
sec,i
isthesecretionrateofthefactor,and
Cells(x,y)arethecoordinatesofcellsthatsecretethefactor.Tosolvetheequations,weusea
finitedifferencemethodtodiscretizethem.
Macrophages
Macrophagesareinitiallypresentinthetissue,andmorearerecruitedtotheTMEduetothe
secretionofvariouschemokines,suchasCCL2[15,16,43].Tosimulatemacrophageinfiltration
ofthetumor,wesetarateofmacrophagerecruitment[29].MacrophagesentertheTMEin
thenaïvestate(M0).Differentiationoccursonceasufficientlevelofactivatingfactorispres-
ent.Inourmodel,amacrophagechecksthelevelsofIL-4andIFN- thatarepresentintheir
localenvironment,andthendifferentiatesaccordingtotheintracellularmodeldescribedin
thefollowingsection.Macrophagesmigratetowardsthetumor,mimickingchemotaxis,and
theyhaveafinitelifespan.
Macrophagedifferentiationmodel
Toincreasethelevelofbiologicaldetailinthemodel,weincorporateamechanisticordinary
differentialequation(ODE)modelofmacrophageintracellularsignalinginresponsetoIL-4
andIFN- ,whichwasdevelopedbyZhaoetal[41].Althoughmacrophagephenotypeis
showntobeonacontinuousspectrum[31],weusediscretephenotypesformodelingsimplic-
ity,asotherpapershavedone[29,30].FollowingthepaperbyZhaoetal.,wefirsttakethe
modeloutputs,whicharethetimecoursesforiNOS,TNF- ,CXCL9,CXCL10,andIL-12,
whicharecharacteristicofanM1phenotype,andIL-10,Arg-1,andVEGF,whicharetypical
ofanM2phenotype,normalizedtotheirstartingvalues.Theproductofthetimecoursesof
theM1outputsisdividedbytheproductofthetimecoursesoftheM2outputstoobtaina
timecourseofthe“M1/M2Score.”Whiletherearemanypossiblewaysthatmacrophage
Fig10. Timecourseswithincreasedtumorproliferationandmacrophagerecruitmentrates,withouttreatment.
Themacrophagerecruitmentrateisdoubled,andtumorproliferationrateincreasedto1.2/day.Timecoursesfor
tumorsthatwerenotremovedbytheimmunesystemareshowninblack;thosefortumorsthatwereremovedare
showninred.(A)Cancercells,(B)M0cells,(C)M1cells,(D)M2cells,(E)TotalTcells,(F)ActiveTcells,(G)Average
IL-4,(H)MaximumIL-4,(I)AverageIFN- ,(J)MaximumIFN- .
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Fig11. ConstantPI3Kinhibitionatincreasedtumorproliferationandmacrophagerecruitmentrates.(A)fractionoftumorsremovedafter
startingtherapy,(B)averagetimeneededtoremovethetumor,(C)themaximumnumberofM1macrophages,(D)themaximumnumberoftotal
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differentiationcanbedeterminedfromhere,forourpurposes,wecalculatethisratioovera
24-hoursimulationandthentaketheaveragevalue.Iftheaveragevalueisgreaterthanone,we
assumethatthereisagreaterM1-promotingsignaling,andthemacrophagedifferentiatesinto
theM1phenotype.Ifthevalueislessthanone,themacrophagedifferentiatesintotheM2phe-
notype.Macrophagesarealsoshowntobeplastic,changingtheirphenotypebasedonchang-
ingenvironmentalconditions[44].Assuch,weallowmacrophagestoreevaluatetheirlocal
microenvironment,processtheinputsignalsviatheintracellularsignalingmodel,andredif-
ferentiateevery24hours.Weassumethatmacrophagescanredifferentiateindefinitely.
WhileODEmodelsareabletoprovideanincreasedlevelofbiologicaldetailtotheABM,
theygreatlyincreasethecomputationalburdenofsimulation.Therefore,toimprovecomputa-
tionaltime,wereplacedthemechanisticmodelwithadata-drivenmodelthattakesthecytokine
concentrationsasinputsandoutputsthemacrophagephenotype.Wehavealreadyshownthat
data-drivenmodelsareabletopredictmechanisticmodeloutputs[45].Here,weuseaneural
networktopredictmacrophagephenotype.Briefly,aneuralnetworkisamachine-learning
methodthatpredictsoutputsgivenasetofinputs,evenwithacomplexnon-linearrelationship
betweenthetwo.Itistrainedusinglargesetsofinput-outputdata.Aneuralnetworkcancon-
tainseverallayers,eachwithmanyneurons.Asingleneurontakesasitsinputthelinearcombi-
nationofthevaluesofeitherthemodelinputsortheoutputsofthepreviouslayer.Theneuron
thentransformsthisinputwithan“activationfunction,”suchasasigmoidfunction.Theoutput
ofthisfunctionbecomestheneuron’soutputandisusedaninputtothenextlayer.
Totraintheneuralnetwork,weran100,000MonteCarlosimulationsofthemechanistic
model,randomlysamplingcytokineconcentrationsovertherangethattheywouldbepresent
intheABM.Wethendeterminedthephenotypeasdescribedabove,creatingadatasetof
inputsandoutputs.TheneuralnetworkwasthentrainedinPythonusingTensorFlow[46].
Thefinalneuralnetworkconsistedofonehiddenlayerwithfourneuronsusingasigmoidacti-
vationfunction.Wetrainedthenetworkseveraltimestodetermineitsabilitytocapturethe
ODEmodeloutputs,eachtimerandomlysplittingthedatasetintotrainingandtestingsets.
Witheachtestingset,theneuralnetworkachievedapredictionaccuracyof>98%.Becauseof
thisveryhighaccuracyinpredictingtheoutputoftheODEmodel,wedeterminedthatthe
neuralnetworkisanefficientwaytosimplifytheODEmodelintoasimpleinput-output
model.
Tcells
Inordertoeliminatethetumor,TcellshavetoberecruitedtotheTME.Uponcancercell
death,tumorantigenisbroughttothelymphnodes,initiatinganimmuneresponse.Tomodel
thisprocess,weimplementaTcellrecruitmentrateforeachtimestep,calculatedusingthefol-
lowingequations,whichweretakenfromGongetal[26].TherateatwhichTcellsare
recruitedisthenmultipliedbythedurationofeachtimesteptogetthenumberofTcellsthat
arerecruitedatthatiteration,andthenthatmanyTcellsarerandomlyplacedinavailablesites
onthelattice,sinceweassumethereissufficientvascularizationforimmunecellrecruitment.
r t Ö Ü à k
a
N
c;death
Ö t t
delay
Ü r
1
1
k
i
á N
c;death
t t
delay
⇣ ⌘ Ö 2Ü Tcells,(E)themaximumnumberofactiveTcells.Notethedifferencesiny-axisscalesacrosstreatmentstrategies.Asteriskssignifythataresultis
statisticallysignificant(p<0.01)fromtheresultofthelowesttreatmentstrength.Wenotethatforthetimeneededtoremovethetumor(B),plotted
isthetimeaveragedoveronlysimulationswherethetumorwasremoved.Therefore,whilesomebarsmayappearmuchhigherthanthatofthe
lowesttreatmentstrength,theyonlyrepresentasmallnumberofsimulationsoutof100andthuswerenotfoundtobestatisticallysignificant.
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N
c;death
Ö tÜ à P
tá 0:5⇤t
window
sà t0:5⇤t
window
n
c;death
Ö sÜ Ö 3Ü Therate,r(t),ofTcellrecruitmentiscalculatedusingthemutationalburden,k
a
,whichis
theextentoftumorcellmutation,thebasalrecruitmentrate,r
1
,theneoantigenstrength,k
i
,
andthenumberofcancercelldeathsoveraninterval,N
c,death
.Whileantigencharacteristics
canvarywidelyacrosstumors,weleavethemconstantasitisnotthefocusofthisstudy.The
t
delay
parameteraccountsforTcellprimingandtraffickingtothetumorsite,leadingtoa
delayedresponse.Thet
window
parametersetsatimerangecenteredaroundthecurrentsimula-
tiontimeminust
delay
fromwhichtoaccumulatethenumberofdeadcancercells,whichinflu-
encestherateofTcellrecruitment.
Tcellscandooneoffouractions:migratetowardsthetumor(representingchemotaxis),
becomefullyactiveuponcontactwithantigenonacancercell,proliferateiffullyactive,and
killanearbycancercell.FullyactiveTcellsarealsoabletosecreteIFN- toinfluenceM1
polarization.InteractionsbetweenTcellsandmacrophagesaredescribedinmoredetail
below.
Tcell–macrophageinteractions
TherearenumerouswaysthatmacrophagesareabletomodulateTcellbehavior,including
cytokineexcretion,antigenpresentation,andinhibitoryligandexpression.TomodelTcell-
macrophageinteractions,wecondensethesebehaviorsintohavingmacrophagespromoteor
inhibitTcellactivationbasedontheirphenotype[9,15,20,47–49].M1macrophagesareableto
activateneighboringTcells,whileM2macrophagesareabletopreventneighboringTcells
frombecomingactive.Thisyieldsthefollowingequationthatdeterminestheprobabilityofa
Tcellbecomingfullyactive:
P
act
à antigenPresence⇥
1
1á e
k antigenPresence
numM2
numM1á 1
Ö Ü s Ö Ü Ö 4Ü whereantigenPresenceequals1ifthereisatumorcelloranM1macrophagepresenttoactivate
theTcell.numM1andnumM2arethenumberofM1andM2macrophages,respectively,
neighboringtheTcell.Theparameterskandsarescalingparameters.Weformulatedthis
equation,andhand-tunedparameterskands,togiveareasonablerangeofprobabilitiesbased
onneighboringmacrophages.
Cancercells
Cancercellsinthemodelsimplyproliferate.Afteracancercell’sinternalclockhasreachedthe
specifiedproliferationtime,thecancercellwillproliferate,unlessthereisnoroomforitto
proliferateorithasreachedthespecifiedlifespan.Ifthereisnotsufficientroomforanewcan-
cercelltobeplaced,itwillbecomequiescent.
Tcellkillingofcancercells
WhenanactiveTcellisneighboringacancercell,theTcellcanrecognizetheantigenonthe
cancercellandbegintokillit.Duetothetimeittakestokillacancercell,boththeTcelland
Fig12. Cyclingmacrophagedepletionwithincreasedtumorproliferationandmacrophagerecruitmentrates.Themacrophage
recruitmentrateisdoubled,andtumorproliferationrateincreasedto1.2/day.(A)fractionoftumorsremovedafterstarting
simulation,(B)averagetimeneededtoremovethetumor,(C)themaximumnumberofM1macrophages,(D)themaximumnumber
oftotalTcells,(E)themaximumnumberofactiveTcells.
https://doi.org/10.1371/journal.pcbi.1008519.g012
PLOS COMPUTATIONAL BIOLOGY Modelofmacrophage-Tcellinteractionsinthetumor
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thecancercellareconsideredtobeengagedanddonotundergootherprocessesforthedura-
tionofkilling[27].Oncethecancercelldies,itisremovedfromsimulation,andtheTcellis
freetocontinuekilling,untilitreachesthemaximumnumberofcancercellsthatitcankill,at
whichpointtheTcellbecomesexhausted[27].
Initializationofsimulation
Simulationstartswith25cancercellsplacedinthecenterofthelattice,representingtheearly
growthofatumororamicrometastasis.Apopulationoftissue-residentmacrophagesisran-
domlyscatteredthroughouttheremaininglatticesites.Weconsidertheareasurroundingthe
tumortobevascularizedenoughtoallowmacrophagesandTcellstoberecruitedthere.
Modelsimulationstepsandimplementation
Ateachtimestep,weproceedthroughthefollowingsteps.First,wecalculatethediffusionand
secretionofdiffusiblefactorsforthedurationofthetimestep.Then,werecruitmoremacro-
phagestotheenvironmentandproceedtoiteratethrougheachmacrophageintheenviron-
mentinarandomorderandallowthemtocarryouttheirvariousfunctions.Wethenrepeat
thisforTcells.Afterthis,weiteratethroughthecancercellsandallowthemtoproceedwith
theirfunctions.Lastly,weremovealldeadcellsfromtheenvironment.Themodelwasimple-
mentedinC++.Thefullmodelisavailableat:https://github.com/FinleyLabUSC/Early-
TME-ABM-PLOS-Comp-Bio.
Treatment
AsthefocusofthisstudyisoninteractionsbetweenmacrophagesandTcells,weexaminesev-
eralmacrophage-basedtherapiestoseehowtheyimpacttheabilityoftheTcellstoremovethe
tumor.Foreachtherapy,wevarytheeffectiveness,whichwasimplementedasthefractionthat
thetargetparameterwasreducedby.Treatmentswerechosenbasedontargetsinthelitera-
ture,withthetwomainstrategiesbeingtoreducethenumberofmacrophagesintheTMEand
toreeducateM2macrophagestoanM1phenotype.Thethreetherapiesexploredherearemac-
rophagedepletion,recruitmentinhibition,andPI3Kinhibition.
Treatmentwasimplementedintwoforms.Thefirstwastosimulatetreatmentcontinuously
forthedurationofthesimulation.Thesecondwastocyclethetreatment,withtreatmenton
forseveraldays,thenoffforseveraldays.
Recruitmentinhibition
Asdescribedabove,macrophagesarerecruitedtotheTMEbychemokinessuchasCCL2.As
manymacrophagesintheTMEdisplayanimmunosuppressiveM2phenotype,itisthought
thatpreventingmacrophagerecruitmentwouldallowTcellstobetterbeabletoremovethe
tumor[15,16].Weimplementtargetingmacrophage-recruitingchemokinesbyreducingthe
rateatwhichmacrophagesarerecruitedtotheenvironment.
Macrophagedepletion
Similartorecruitmentinhibition,depletionofmacrophagescanbeusedtoreducethenumber
ofmacrophagesintheTME[15,16].Itistobenotedthatbothofthesestrategiesfailtodis-
criminatebetweenM1andM2macrophages,thusalsoinhibitingtheimmune-promoting
propertiesofM1macrophages.Wemodelmacrophagedepletionbygivingthemacrophagesa
probabilityofundergoingapoptosisateachtimestepthattreatmentison.Whensimulating
continuoustreatment,wedecreasedtherangeofdepletionprobabilitiesuntilwereachedthe
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pointwherethelowestprobabilityfailstoremovealargenumberoftumorssothatwecould
compareineffectivelevelsofdepletiontoeffectivelevels.
Macrophagereeducation
Inordertopreventtheeliminationofimmune-promotingM1macrophages,reeducationof
M2macrophagesintoanM1phenotypeisapotentialtherapy.Onenotedtargetisinhibition
ofPI3K[15,16].Asthisispresentinthemechanisticmodelthatwasutilizedformacrophage
differentiation,weincludevariationsinPI3KactivityintheMonteCarlosimulationsandhave
PI3Kactivityasaninputtotheneuralnetwork.Targetingthisparameterallowsustorediffer-
entiatethemacrophagesintoanM1phenotype.Wesimulatethistreatmentbyreducingthe
valueofthePI3Kactivityparameter.
Supportinginformation
S1Fig.ResultsfromtheLatinHypercubeSampling–tumorremoval.Thefractionoftumors
removedasafractionofrelevanttumormicroenvironmentparameters:macrophagerecruit-
mentrate,tumorIL-4secretionrate,TcellIFN- secretionrate,tumorproliferationrate,M2
IL-4secretionrate,andmacrophagelifespan.WithLatinHypercubeSampling,parametersets
aresampledsothateachparametervalueappearsonlyonce.Thisisdoneforthesakeof
computationalefficiency.Becauseonlyoneparametersetwassimulatedforeachparameter
value,theplotsareverydiscontinuousanddonotshowtheaveragemodelbehaviorforeach
parametervalue.Despitethis,acleartrendisvisibleformacrophagerecruitmentrate.
(TIF)
S2Fig.ResultsfromtheLatinHypercubeSampling–finaltumorcellcount.Thefinalnum-
beroftumorcellsisonthey-axisandtherelevantparametersareonthex-axes(macrophage
recruitmentrate,tumorIL-4secretionrate,TCellIFN- secretionrate,tumorproliferation
rate,M2IL-4secretionrate,andmacrophagelifespan).Becauseonlyoneparametersetwas
simulatedforeachparametervalue,theplotsareverydiscontinuous.However,acleartrendis
visibleformacrophagerecruitmentrateandmacrophagelifespan.Themaximumnumberof
tumorcellshereis5,000becausewechosetoendsimulationwheneithermaximumsimula-
tiontimewasreached,thetumorwaseliminated,ortumorcellcountreached5,000.Thisis
becauseatthatnumberoftumorcells,thereismuchlessspaceavailableforrecruitedimmune
cells,sotheirrecruitmentinherentlydecreasesduetothenatureofthemodel.
(TIF)
S3Fig.Tumorgrowthcurveswithoutimmunecellsandwithoutimmunefunction.Com-
parisonoftumorgrowthcurveswhentherearenoimmunecellspresentinthesimulationand
whenimmunecellsarepresentbutlackfunction.Becausethesecurvesareverysimilar,we
concludethatspatialinhibitionisnotthecauseoftheequilibriumstateseenwithouttreat-
ment.Simulationsweredoneinreplicatesof10.
(TIF)
S4Fig.Impactofinitialmacrophagedensity.Comparisonoftumorcurvesandtotalmacro-
phagecurvesfordifferingstartingmacrophagedensities:2x10
-4
(blue),2x10
-3
(red,valueused
forrestofthesimulations),and2x10
-2
(green)cellspersite.Eachdensitywassimulatedinrep-
licatesof10.Plottedaretheaveragetimecoursesforthosereplicates.
(TIF)
S5Fig.Individualtimecoursesformacrophagedepletionprobabilityof0.006pertime-
step.Tumorsthatwereeliminatedareshowninorange.(A)Cancercells,(B)M0
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macrophages,(C)M1macrophages,(D)M2macrophages,(E)Tcells,(F)ActiveTcells,(G)
AverageIL-4,(H)maximumIL-4,(I)averageIFN- ,(J)MaximumIFN- .
(TIF)
S6Fig.Individualtimecoursesformacrophagedepletionprobabilityof0.002pertime-
step.Tumorsthatsurvivedtotheendofsimulationareshowninblack.Tumorsthatwere
eliminatedareshowninorange.(A)Cancercells,(B)M0macrophages,(C)M1macrophages,
(D)M2macrophages,(E)Tcells,(F)ActiveTcells,(G)AverageIL-4,(H)maximumIL-4,(I)
averageIFN- ,(J)MaximumIFN- .
(TIF)
S7Fig.Individualtimecoursesforrecruitmentinhibitionof0.7.Tumorsthatsurvivedto
theendofsimulationareshowninblack.Tumorsthatwereeliminatedareshowninblue.(A)
Cancercells,(B)M0macrophages,(C)M1macrophages,(D)M2macrophages,(E)Tcells,(F)
activeTcells,(G)AverageIL-4,(H)MaximumIL-4,(I)AverageIFN- ,(J)MaximumIFN- .
(TIF)
S8Fig.IndividualtimecoursesforPI3Kinhibitionof0.8.Tumorsthatsurvivedtotheend
ofsimulationareshowninblack.Tumorsthatwereeliminatedareshowningreen.(A)cancer
cells,(B)M0macrophages,(C)M1macrophages,(D)M2macrophages,(E)Tcells,(F)active
Tcells,(G)AverageIL-4,(H)MaximumIL-4,(I)AverageIFN- ,(J)MaximumIFN- .
(TIF)
S9Fig.Effectsofcycledimmunotherapystartedat100daysofsimulationathighertumor
proliferation.Formacrophagedepletion(A),thefractionofmacrophagesremovedatthe
beginningofeachcycleisgivenas“DepletionStrength”andthelengthofeachcycleis“Cycle
Duration.”Forrecruitmentinhibition(B)andPI3Kinhibition(C),thenumberofdaysinthe
cyclethattreatmentisonforisgivenas“DaysTreatmentisOn.”Recruitmentinhibitionis
simulatedatastrengthof1.0(completeinhibition)andPI3Kinhibitionissimulatedata
strengthof0.8.ForrecruitmentinhibitionandPI3Kinhibition,spacesmarkedwithanXare
thosewheretreatment-ontimeisequalorgreatertothecycleduration,thuswerenotsimu-
lated.(i)fractionoftumorsremovedafterstartingtherapy.(ii)time(days)fromstartingtreat-
menttotumorremoval.Itisaveragedoverthe100simulationsandisequaltozeroifno
tumorswereremovedatthattreatmentlevel.(iii)maximumnumberofM1macrophages.(iv)
maximumnumberoftotalTcells.(v)maximumnumberofactiveTcells.
(TIF)
S10Fig.Constantmacrophagedepletionstartedat100dayswithincreasedtumorprolifer-
ationrate.(A)fractionoftumorsremovedafterstartingsimulation,(B)averagetimeneeded
toremovethetumor,(C)themaximumnumberofM1macrophages,(D)themaximumnum-
beroftotalTcells,(E)themaximumnumberofactiveTcells.Notethedifferencesiny-axis
scalesacrosstreatmentstrategies.Asteriskssignifythataresultisstatisticallysignificant
(p<0.01)fromtheresultofthelowesttreatmentstrength.Wenotethatforthetimeneededto
removethetumor(B),plottedisthetimeaveragedoveronlysimulationswherethetumorwas
removed.Therefore,whilesomebarsmayappearmuchhigherthanthatofthelowesttreat-
mentstrength,theyonlyrepresentasmallnumberofsimulationsoutof100andthuswerenot
foundtobestatisticallysignificant.
(TIF)
S11Fig.Constantrecruitmentinhibitionstartedat100dayswithincreasedtumorprolif-
erationrate.(A)fractionoftumorsremovedafterstartingsimulation,(B)averagetime
neededtoremovethetumor,(C)themaximumnumberofM1macrophages,(D)the
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maximumnumberoftotalTcells,(E)themaximumnumberofactiveTcells.Notethediffer-
encesiny-axisscalesacrosstreatmentstrategies.Asteriskssignifythataresultisstatistically
significant(p<0.01)fromtheresultofthelowesttreatmentstrength.Wenotethatforthetime
neededtoremovethetumor(B),plottedisthetimeaveragedoveronlysimulationswherethe
tumorwasremoved.Therefore,whilesomebarsmayappearmuchhigherthanthatofthelow-
esttreatmentstrength,theyonlyrepresentasmallnumberofsimulationsoutof100andthus
werenotfoundtobestatisticallysignificant.
(TIF)
S12Fig.Constantrecruitmentinhibitionstartedat100dayswithmacrophagerecruitment
ratedoubled.(A)fractionoftumorsremovedafterstartingsimulation,(B)averagetime
neededtoremovethetumor,(C)themaximumnumberofM1macrophages,(D)themaxi-
mumnumberoftotalTcells,(E)themaximumnumberofactiveTcells.Notethedifferences
iny-axisscalesacrosstreatmentstrategies.Asteriskssignifythataresultisstatisticallysignifi-
cant(p<0.01)fromtheresultofthelowesttreatmentstrength.Wenotethatforthetime
neededtoremovethetumor(B),plottedisthetimeaveragedoveronlysimulationswherethe
tumorwasremoved.Therefore,whilesomebarsmayappearmuchhigherthanthatofthelow-
esttreatmentstrength,theyonlyrepresentasmallnumberofsimulationsoutof100andthus
werenotfoundtobestatisticallysignificant.
(TIF)
S13Fig.ConstantPI3Kinhibitionstartedat100dayswithmacrophagerecruitmentrate
doubled.(A)fractionoftumorsremovedafterstartingsimulation,(B)averagetimeneededto
removethetumor,(C)themaximumnumberofM1macrophages,(D)themaximumnumber
oftotalTcells,(E)themaximumnumberofactiveTcells.Notethedifferencesiny-axisscales
acrosstreatmentstrategies.Asteriskssignifythataresultisstatisticallysignificant(p<0.01)
fromtheresultofthelowesttreatmentstrength.Wenotethatforthetimeneededtoremove
thetumor(B),plottedisthetimeaveragedoveronlysimulationswherethetumorwas
removed.Therefore,whilesomebarsmayappearmuchhigherthanthatofthelowesttreat-
mentstrength,theyonlyrepresentasmallnumberofsimulationsoutof100andthuswerenot
foundtobestatisticallysignificant.
(TIF)
S14Fig.Effectsofcycledimmunotherapystartedat100daysofsimulationatmacrophage
recruitmentratedoubled.Formacrophagedepletion(A),thefractionofmacrophages
removedatthebeginningofeachcycleisgivenas“DepletionStrength”andthelengthofeach
cycleis“CycleDuration.”Forrecruitmentinhibition(B)andPI3Kinhibition(C),thenumber
ofdaysinthecyclethattreatmentisonforisgivenas“DaysTreatmentisOn.”Recruitment
inhibitionissimulatedatastrengthof1.0(completeinhibition)andPI3Kinhibitionissimu-
latedatastrengthof0.8.ForrecruitmentinhibitionandPI3Kinhibition,spacesmarkedwith
anXarethosewheretreatment-ontimeisequalorgreatertothecycleduration,thuswerenot
simulated.(i)fractionoftumorsremovedafterstartingtherapy.(ii)time(days)fromstarting
treatmenttotumorremoval.Itisaveragedoverthe100simulationsandisequaltozeroifno
tumorswereremovedatthattreatmentlevel.(iii)maximumnumberofM1macrophages.(iv)
maximumnumberoftotalTcells.(v)maximumnumberofactiveTcells.
(TIF)
S15Fig.Constantmacrophagedepletionstartedat100dayswithincreasedtumorprolifer-
ationandmacrophagerecruitmentrates.(A)fractionoftumorsremovedafterstartingsimu-
lation,(B)averagetimeneededtoremovethetumor,(C)themaximumnumberofM1
macrophages,(D)themaximumnumberoftotalTcells,(E)themaximumnumberofactiveT
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cells.Notethedifferencesiny-axisscalesacrosstreatmentstrategies.Asteriskssignifythata
resultisstatisticallysignificant(p<0.01)fromtheresultofthelowesttreatmentstrength.We
notethatforthetimeneededtoremovethetumor(B),plottedisthetimeaveragedoveronly
simulationswherethetumorwasremoved.Therefore,whilesomebarsmayappearmuch
higherthanthatofthelowesttreatmentstrength,theyonlyrepresentasmallnumberofsimu-
lationsoutof100andthuswerenotfoundtobestatisticallysignificant.
(TIF)
S16Fig.Constantrecruitmentinhibitionstartedat100dayswithincreasedtumorprolif-
erationandmacrophagerecruitmentrates.(A)fractionoftumorsremovedafterstarting
simulation,(B)averagetimeneededtoremovethetumor,(C)themaximumnumberofM1
macrophages,(D)themaximumnumberoftotalTcells,(E)themaximumnumberofactiveT
cells.Notethedifferencesiny-axisscalesacrosstreatmentstrategies.Asteriskssignifythata
resultisstatisticallysignificant(p<0.01)fromtheresultofthelowesttreatmentstrength.We
notethatforthetimeneededtoremovethetumor(B),plottedisthetimeaveragedoveronly
simulationswherethetumorwasremoved.Therefore,whilesomebarsmayappearmuch
higherthanthatofthelowesttreatmentstrength,theyonlyrepresentasmallnumberofsimu-
lationsoutof100andthuswerenotfoundtobestatisticallysignificant.
(TIF)
S17Fig.CyclingPI3Kinhibitionwithincreasedtumorproliferationandmacrophage
recruitmentrates.Themacrophagerecruitmentrateisdoubled,andtumorproliferationrate
increasedto1.2/day.(A)fractionoftumorsremovedafterstartingsimulation,(B)average
timeneededtoremovethetumor,(C)themaximumnumberofM1macrophages,(D)the
maximumnumberoftotalTcells,(E)themaximumnumberofactiveTcells
(TIF)
S18Fig.Cyclingrecruitmentinhibitionwithincreasedtumorproliferationandmacro-
phagerecruitmentrates.Themacrophagerecruitmentrateisdoubled,andtumorprolifera-
tionrateincreasedto1.2/day.(A)fractionoftumorsremovedafterstartingsimulation,(B)
averagetimeneededtoremovethetumor,(C)themaximumnumberofM1macrophages,
(D)themaximumnumberoftotalTcells,(E)themaximumnumberofactiveTcells
(TIF)
S1Table.ModelParameters.Parametervalues,withsupportingreferences.Forthetumor
divisiontimeandmacrophagerecruitmentrateparameters,valuesshowninparenthesesare
theincreasedvaluesusedforcertainsimulations,asdescribedintheResults.
(DOCX)
Acknowledgments
TheauthorsthankmembersoftheFinleyresearchgroupforcriticalcommentsand
suggestions.
AuthorContributions
Conceptualization:ColinG.Cess,StaceyD.Finley.
Formalanalysis:ColinG.Cess.
Investigation:ColinG.Cess.
Resources:StaceyD.Finley.
PLOS COMPUTATIONAL BIOLOGY Modelofmacrophage-Tcellinteractionsinthetumor
PLOSComputationalBiology|https://doi.org/10.1371/journal.pcbi.1008519 December23,2020 32/35
Supervision: StaceyD.Finley.
Writing–originaldraft:Colin G.Cess.
Writing–review&editing: ColinG.Cess, StaceyD.Finley.
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PLOS COMPUTATIONAL BIOLOGY Modelofmacrophage-Tcellinteractionsinthetumor
PLOSComputationalBiology|https://doi.org/10.1371/journal.pcbi.1008519 December23,2020 35/35
Received:!"August !#!$ Revised: $%January !#!! Accepted: $&January !#!!
DOI: $#.$##!/cso!.$#"!
ORIGINAL ARTICLE
Multiscalemodelingoftumoradaptionandinvasion
followinganti-angiogenictherapy
ColinG.Cess
!
StaceyD.Finley
!,",#
$
DepartmentofBiomedicalEngineering,
UniversityofSouthernCalifornia,
LosAngeles,California,USA
!
DepartmentofQuantitativeand
ComputationalBiology,Universityof
SouthernCalifornia,LosAngeles,
California,USA
"
MorkFamilyDepartmentofChemical
EngineeringandMaterialsScience,
UniversityofSouthernCalifornia,
LosAngeles,California,USA
Correspondence
StaceyD.Finley,DepartmentofBiomedi-
calEngineering,DepartmentofQuantita-
tiveandComputationalBiology;andMork
FamilyDepartmentofChemicalEngineer-
ingandMaterialsScience,Universityof
SouthernCalifornia,CA,USA.
Email:sfinley@usc.edu
Fundinginformation
AmericanCancerSociety,Grant/Award
Number:$"#%"!-RSG-$&-$""-#$-CSM;USC
Provost’sOffice
Abstract
In order to promote continued growth, a tumor must recruit new blood ves-
sels, a process known as tumor angiogenesis. Many therapies have been tested
that aim to inhibit tumor angiogenesis, with the goal of starving the tumor of
nutrientsandpreventingtumorgrowth.However,manyofthesetherapieshave
beenunsuccessfulandcanparadoxicallyfurthertumordevelopmentbyleading
to increased local tumor invasion and metastasis. In this study, we use agent-
based modeling to examine how hypoxic and acidic conditions following anti-
angiogenic therapy can influence tumor development. Under these conditions,
we find that cancer cells experience a phenotypic shift to a state of higher sur-
vivalandinvasivecapability,spreadingfurtherawayfromthetumorintothesur-
rounding tissue. Although anti-angiogenic therapy alone promotes tumor cell
adaptationandinvasiveness,wefindthataugmentingchemotherapywithanti-
angiogenic therapy improves chemotherapeutic response and delays the time
it takes for the tumor to regrow. Overall, we use computational modeling to
explain the behavior of tumor cells in response to anti-angiogenic treatment in
thedynamictumormicroenvironment.
KEYWORDS
agent-basedmodel,anti-angiogenictreatment,heterogeneity,tumorcellevolution
! INTRODUCTION
It has long been known that angiogenesis, the forma-
tion of new blood vessels, is a feature common to almost
everysolidtumor.Asthetumorgrows,itdepletesitslocal
environmentofoxygen(called“hypoxia”),andthesubse-
quent response to hypoxia is the secretion of various pro-
angiogenic factors, such as vascular endothelial growth
factor (VEGF). These factors recruit blood vessels to the
tumor, resupplying nutrients and removing waste prod-
ucts. It has also been found that angiogenesis promotes
ThisisanopenaccessarticleunderthetermsoftheCreativeCommonsAttributionLicense,whichpermitsuse,distributionandreproductioninanymedium,providedthe
originalworkisproperlycited.
© !#!!TheAuthors.ComputationalandSystemsOncologypublishedbyWileyPeriodicalsLLC
tumormetastasistodistalsites[$,!].Therefore,beingable
tomodulatetumorangiogenesisisseenasawaytopoten-
tiallyprovidesometherapeuticbenefit.Sincethediscovery
oftumorangiogenesis,varioustherapieshavebeendevel-
opedthatprimarilytargettheeffectsofpro-angiogenicfac-
tors,withtheaimofpreventingvascularrecruitment[",%].
These therapies work by starving the tumor of nutrients,
preventingcontinuedgrowthandkillingthetumor.How-
ever, the therapies provide limited actual clinical benefit
[',(]. In addition, clinical and mouse studies have found
that inhibiting angiogenesis can not only fail to remove
ComputSystOncol. !#!!;!:e$#"!. wileyonlinelibrary.com/journal/cso! !of!"
https://doi.org/$#.$##!/cso!.$#"!
!of"! CESS!"#FINLEY
the tumor but also lead to a more invasive and metastatic
tumor[$–%].
To understand why invasion and metastasis occur fol-
lowing anti-angiogenic therapy, we must look at how the
tumor responds to the harsh microenvironmental con-
ditions (primarily hypoxia and lowered pH) that occur
when its blood supply is cut off. Hypoxia has been shown
to lead to other adaptive changes in cancer cells besides
angiogenic factor secretion, notably an upregulation of
glucose transporters, which contributes to the Warburg
effect [&'–&(] and ultimately leads to elevated glucose
metabolism and increased production of lactic acid as
a by-product. This lactic acid secretion helps the can-
cer cells invade nearby tissue by killing normal cells and
degrading the extracellular matrix [&),&*]. Hypoxia has
also been found to promote cancer cell migration and
epithelial-mesenchymaltransition[&+,&$],furtherincreas-
ing the cells’ invasive capabilities. In addition to hypoxia-
mediatedeffects,cancercellsadapttotheacidicconditions
producedbyincreasedglycolysis,allowingthemtosurvive
atapHthatwouldkillnormalcells[&),&,].Theseadapta-
tions, which are a result of lowered blood supply, explain
theinvivoandclinicalobservationthattumorinvasionis
advancedbyanti-angiogenictreatment.
In order to explore the tumor in a more detailed set-
ting, computational models have been used to simulate
tumor angiogenesis. These models allow for many differ-
ent perturbations to be tested and a vast amount of infor-
mation examined. These studies would otherwise be very
prohibitivetoperforminvitroorinvivo.Whilemanydif-
ferent types of models have been developed to examine
angiogenesis, ranging from intracellular signaling models
towholetumormodels[&%–-)],wewillfocusheresolelyon
a type of model known as the agent-based model (ABM).
ABMs model cells as discrete individuals, capturing com-
plex cell-to-cell interactions and spatial morphologies. To
date, a plethora of ABMs has been developed to exam-
ine various aspects of tumor vasculature [-*–()]. These
models focus on a variety of different aspects of angio-
genesis,vascularproperties,andmicroenvironmentalcon-
ditions. Depending on the model focus, vasculature has
been treated as simply as point sources or as complexly
as explicit vessel structure with detailed blood rheology.
Together, these models have advanced understanding of
tumor-vasculature dynamics by allowing researchers to
explore in great detail how modulating different interac-
tionsleadstodifferencesinthetemporalandspatialdevel-
opmentofthetumorecosystem.
Inthisstudy,weaimtouseagent-basedmodelingtobet-
ter understand the mechanisms that contribute to more
invasive and metastatic tumors following anti-angiogenic
treatment. While previous models on this subject focus
mostly on the interplay between tumor and vasculature,
they often neglect how cancer cells change with time in
responsetotheirdynamicenvironment,whichisthefocus
ofourworkhere.Usingasimpleapproachtosimulatethe
vasculature within an ABM,we focuson how tumor cells
adapt to harsh microenvironmental conditions and how
reducing angiogenesis leads to changes in tumor pheno-
type and spatial morphology. In addition, we focus on an
effectofanti-angiogenictreatmentknownasvascularnor-
malizationandhowitaugmentstheeffectsofchemother-
apy by transiently increasing blood flow. Overall, we find
that anti-angiogenic therapy leads to tumor adaptation
and a phenotypic shift toward more invasive properties:
increased glycolysis, increased resistance to acidic pH,
and increased migration, leading to invasion of the sur-
rounding tissue. However, we also find that augmenting
chemotherapy with anti-angiogenic therapy improves the
chemotherapy’sabilitytosuppresstumorgrowth.
! METHODS
!." Modeloverview
Here, we provide a brief overview of model construc-
tion. A detailed description can be found in the sup-
plementary methods. Our model consists of a tumor
growing in a -Dtissue slice with vessels represented
as discrete point sources. Cells uptake glucose and oxy-
gen, secreting acid (represented as H+) as a metabolic
by-product. An overview of the model is displayed
in Figure &.The model is constructed in C++ and
can be found at: https://github.com/FinleyLabUSC/Anti-
angiogenic-treatment-ABM-model.
-.&.& Cellinteractions
We model cells using an overlapping sphere method [(*].
With this method, cells are represented by a point and
a radius. We calculate attractive and repulsive forces
between cells to determine how they physically interact.
Allcellsarethesamesize(sameradius).
-.&.- Diffusion
We model three groups of diffusible factors: metabolism
(oxygen,glucose,andH+),angiogenicfactor(VEGF),and
agenericchemotherapeuticagent.Thediffusionequations
aresolvedusingafinitedifferencemethod.
-.&.( Cellfunctions
Cells migrate up the oxygen gradient. At a set probabil-
ity, they stop migrating and enter the cell cycle until they
26899655, 2022, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cso2.1032, Wiley Online Library on [26/03/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
CESS!"#FINLEY !of"#
FIGURE " Modelschematic.Cancercellsstartatthebasephenotype(gray)andadaptphenotypicallytoenvironmentalconditions.
Aciditycausesanincreaseinacidresistance(blue),whilehypoxiacausesincreasesinglycolysis(purple)andmigration(yellow).Hypoxiaalso
causesvascularendothelialgrowthfactorsecretionbycancercells,whichpromotesangiogenesisandlimitshypoxiaandacidity
proliferate, at which point they continue migrating until
entering the cell cycle again. Cells also become necrotic
based on the local pH and threshold for acid resistance.
Necroticcellsshrinkbutremaininthesimulation.
$.%.& Celladaptation
Cellphenotypeshiftsinresponsetoenvironmentaleffects.
Hypoxiacausesashifttowardincreasedglucoseconsump-
tionandadecreaseintheproliferativeprobability,leading
tomoremigratorycells.LowpHcausesanincreaseinacid
resistance of cells. These shifts occur at a constant rate in
response to the presence of the respective environmental
conditions,uptoamaximumorminimumvalue.Cellsdo
notreverttheirphenotype.Phenotypesarepasseddownto
daughtercells.
$.%.' Vasculature
Vessels are represented as point sources that are fixed in
place, providing nutrients and removing acid from the
environment. If cells exert sufficient force upon a vessel,
the vessel will collapse and is removed from the simu-
lation. Vessels are recruited at a rate proportional to the
VEGF concentration in a location, which is secreted by
hypoxic cells. Because tumor-recruited vessels are aber-
rant,weassumethattheydelivernutrientsataslowerrate
andrequirelessforcetocollapse.
$.%.( Anti-angiogenictherapy
We model anti-angiogenic therapy as simply turning off
vessel recruitment. Therapy also normalizes the vascula-
ture, restoring recruited vessel properties to those of non-
recruitedvessels.
$.%.) Chemotherapy
Chemotherapy has been employed along with anti-
angiogenic therapy in order to utilize the vascular-
normalization effect of anti-angiogenic therapy and
improve the delivery of chemotherapy. We model
chemotherapy after Pérez-Velázquez and Rejniak [*(],
where the effect of chemotherapy is a trade-off between
theaccumulationofcelldamageandcellresistance.
! RESULTS
!." Higherrateofangiogenesis
preservesprotectiveoutertumorlayer
We varied the rate of angiogenesis and the health of
recruited vessels to see how these parameters impacted
tumor growth and development. Simulations were run
until the tumor reached a diameter that corresponded
to a volume of $ mm
*
. We note that tumor diameter is
calculated here as the largest distance between any two
26899655, 2022, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cso2.1032, Wiley Online Library on [26/03/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
! of"# CESS!"#FINLEY
26899655, 2022, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cso2.1032, Wiley Online Library on [26/03/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
CESS!"#FINLEY !of"#
cancer cells with a proliferative probability greater than
$.%.InFigure &, we show representative images of the
final tumor state, with black cells as necrotic, gray cells
representing the base phenotype, and the colored cells
representingtheshifttowardincreasedglycolysis(purple),
increased acid resistance (blue), and increased migratory
potential (yellow), with a more vibrant color representing
a furthershift. At the lower rate of vessel recruitment, we
see that an increase in vascular health, while preventing
some dissemination into the surrounding tissue, still has
a large necrotic region with cells beginning to spread
outward. Additionally, many of the cells in these tumors
adapted to harsher conditions, making the tumor more
invasive. At a higher rate of vessel recruitment, tumors
retain an outer layer of cells in the initial phenotype.
Here, the outer layer of tumor cells was not exposed to
harsher environmental conditions, and thus there was
no pressure to adapt. With this layer acting as a barrier
and the sufficient supply of oxygen into the center of the
tumor by recruited vessels (Figure S%), the more invasive
cellsareretainedwithinthetumorcoreandfailtospread
outward. The higher rate of angiogenesis also leads to
muchsmallerregionsofnecrosis.
$.# Anti-angiogenictreatment
promotestumorevolutionandlocal
invasion
Wethenlookedathowanti-angiogenictreatmentimpacts
tumor shape and the distribution of tumor cells with the
threepropertiesofinterest(glycolyticshift,acidresistance,
and migratory potential). Here, and for the rest of the
study, we focused our simulations on one combination of
vesselrecruitmentrateandvascularhealth,higherrecruit-
ment (3.16×10
−6
(ℎ$×%m
2
)
−1
)andlowerhealth($.&').
Figure ( shows a comparison of representative tumor
simulations without treatment (left) and with treatment
(right).Here,wesimulatedtheno-treatmentcaseuntilthe
tumorreachedadiametercorrespondingtoavolume of &
mm
(
asdescribedabove.Wethensimulatedthetreatment
case for the same time duration as the no-treatment case.
Treatment was started when the tumor reached % mm
(
.
While anti-angiogenic treatment succeeded in greatly
reducingthenumberofcancercells,weseemuchfurther
dissemination into the surrounding tissue along with a
large necrotic region in the tumor core. This is consistent
withclinicalandinvivoobservationsthatanti-angiogenic
treatment promotes local invasion as cancer cells are
forced to search for a more habitable environment [)–*].
Wealsoseethatmuchofthevasculatureinthesimulation
environment was collapsed by the invading tumor. To
complement these figures showing the final tumor state,
ittcn Figure S&, we provide several snapshots of tumor
development following therapy. The model also predicts
heterogeneity in the patterning of the cells, including
areaswithhighcelldensityandareaswithfewcells.Some
regionsoflessdensenecrosis(whiteareas)areduetolive
cells migrating away from the core. Others are because
the radius of necrotic cells shrinks over time. The denser
regions are caused from the growth of live cells nearby,
whichpushthenecroticcellstogether;regionswithlower
densityexperiencedlessofthiscompression.
In Figure +, we show frequency distributions of cel-
lular properties for '$ simulation replicates, comparing
simulations with anti-angiogenic treatment (red) and the
no-treatment case (black). These graphs only account for
living cells present at the end of the simulation. The dis-
tributions provide quantitative evidence of the qualitative
observations from the spatial layout. There is a clear shift
incelllocationawayfromthecenterofthesimulationenvi-
ronmentduetoanti-angiogenictreatment(Figure+A).In
addition, cells display increased glycolysis (Figure +B), a
lowerpHthresholdforcelldeath(Figure+C),andalower
probability of committing to the cell cycle, thus being
more migratory (Figure +D). Altogether, the simulations
indicate the way in which anti-angiogenic therapy pro-
motes the phenotypic shift of tumor cells, making them
moretoleranttoharshermicroenvironmentalconditions.
$.$ Anti-angiogenictherapylimits
tumorgrowthafterchemotherapy
We next examined how augmenting chemotherapy with
anti-angiogenictherapycanchangephenotypicevolution.
We simulated a chemotherapeutic agent that diffuses out
of the blood vessels. Cancer cells died once accumulat-
ing sufficient damage, based on the concentration of the
chemotherapeutic agent. In addition, cells acquired toler-
ance to chemotherapy at a constant rate upon sufficient
exposuretothedrug.Boththerapieswerestartedoncethe
FIGURE # Predictedtumorcellpropertieswithouttreatment.Spatiallayoutofrepresentativefinaltumorstatesatdifferentratesof
vesselrecruitment(1.58×10
−6
and3.16×10
−6
(ℎ$×%m
2
)
−1
)anddifferentlevelsofrecruitedvesselhealth($.&'and$.').Phenotypic
propertiesareshowninthethreecolumns,withgraycellsrepresentingthebasephenotypeandcoloredcellsrepresentingincreasesin
glycolysis(purple),acidresistance(blue),andmigratorypotential(yellow)asthelog
%$
oftheprobabilityofcommittingtothecellcycle.
Necroticcellsareshowninblack.Vesselsareshowninred.Colorbarsatthebottomofeachcolumnshowtherangeofphenotypicproperties
forthisfigureandtheotherspatialfiguresthatwepresent
26899655, 2022, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cso2.1032, Wiley Online Library on [26/03/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
!of"# CESS!"#FINLEY
FIGURE $ Tumor cells’ response to anti-angiogenic treatment. Spatial layout of sample final tumor states without treatment (left) and
withanti-angiogenictreatment(right).Cancercellphenotypeisshowninthethreerows,withblackcellsrepresentingthebasephenotype
andcoloredcellsrepresentingincreasesinglycolysis(purple),acidresistance(blue),andmigratorypotential(yellow).Necroticcellsare
showninblackandvesselsareshowninred
tumor diameter corresponded to $ mm
%
, and simulation
ended either when the diameter corresponded to & mm
%
or simulation time reached '( days, whichever occurred
first.Anti-angiogenictherapywassimulatedasconstantly
on,whereaschemotherapywascycledonfor)daysandoff
for $( days until the simulation ended. This combination
of continuous anti-angiogenic therapy and intermittent
chemotherapyhasbeenusedinpre-clinicalstudies[&&].
Time courses for living cancer cell numbers with
chemotherapyalone(blue)andchemotherapyaugmented
26899655, 2022, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cso2.1032, Wiley Online Library on [26/03/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
CESS!"#FINLEY !of"#
FIGURE $ Distributionsofcellproperties.Frequencydistributionsfor$%simulationreplicatesshowingcelldistancefromthetumor
centerinmicrometers(A),increaseinthevalueof!
"
(B),decreaseinthepHvaluethatinducescelldeath(C),anddecreaseintheprobability
ofcommittingtothecellcycle,meaninganincreaseincellmigration(D).Distributionswithouttreatmentareshowninblack,while
distributionswithanti-angiogenictreatmentareinred.Notethatatlowvaluesofthex-axisforpanel(D),theredandblackcurvesoverlap
by anti-angiogenic therapy (red) are shown in Figure $.
To investigate how the response to combination therapy
changesasafunctionofdrugresistance,wevariedtherate
at which cancer cells became resistant to the chemother-
apeutic agent. At a lower rate of acquired resistance
(Figure $Ai), we see that the number of tumor cells is
greatly reduced and maintained at a low number, with
simulationsendingat&%daysinsteadofmeetingthestop-
ping criteria based on tumor diameter, indicating that the
cancercellshavenotinvadedintothesurroundingtissue.
Atahigherrateofacquiringresistance(Figure$Bi),wesee
that the tumor begins to regrow at a faster rate. However,
the addition of anti-angiogenic therapy slows regrowth,
compared to chemotherapy alone. At the highest rate of
resistance,weseearapidregrowthofthesimulationswith
chemotherapy alone (Figure $Ci); however, the regrowth
is much smaller with combination therapy. Despite the
lower cell number, we do see that combination therapy
leadstoasmallincreaseinmigrationintothesurrounding
tissue.ThisisshowninFigure$A-C,ii,whereasresistance
increases, the distribution of cell distance from the center
increases for combination therapy. Interestingly, even at
thehighestrateofresistance,thedistancedistributionfor
combination therapy is not as extreme as that for anti-
angiogenictherapyalone(dottedblackline,Figure$Ciii).
This is likely due to the continued killing effect of
chemotherapyasmigratorycellsapproachbloodvessels.
Similar to comparing no treatment to only anti-
angiogenic therapy, we see that the addition of anti-
angiogenic therapy to chemotherapy yields a small
increase in the distribution of glycolytic shift (Fig-
ure $A-C,iii).Wealsoseethatthereisagreaterportionof
cells with combination therapy that experience a greater
shift in resistance to acidic pH and increased migration
(Figure$A-C,iv,$A-C,v).Fromthis,weseethatwhilecom-
binationtherapyleadstoasmallercancercellpopulation,
itstillinducesinvasivepropertiesinthatpopulation.Inter-
estingly, for migratory potential, the cells exposed to only
anti-angiogenic therapy experienced a greater shift than
thosewithcombinationtherapy.
In Figure ', we show representative final tumor states
for each rate of resistance from Figure $,comparingthe
effectsofchemotherapywithoutandwithanti-angiogenic
treatment. We note that in this figure, we only show cell
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!of"# CESS!"#FINLEY
FIGURE $ Effectofacquiredtoleranceandsingle-agentorcombinedtherapyontumorproperties.Time-courses(i)fordifferentratesof
acquiringtolerancetochemotherapy:$%
–&
(A),$%
–'.&
(B),and$%
–'
(C).Frequencydistributionsforthedistancefromtumorcenter(ii),
increasein!
"
(iii),pHvaluethatinducescelldeath(iv),andprobabilityofcommittingtothecellcycle(v).Simulationsareshownfor&%
replicateswherethetumorwasnotremovedbytherapyforchemotherapyalone(blue),chemotherapyaugmentedwithanti-angiogenic
therapy(red),andanti-angiogenictherapy(blackdottedline,fromFigure')
26899655, 2022, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cso2.1032, Wiley Online Library on [26/03/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
CESS!"#FINLEY !of"#
FIGURE $ Effectofacquiredtoleranceandsingle-agentorcombinedtherapyonfinaltumorstates.Spatiallayoutforrepresentative
finaltumorstateswithchemotherapyalone(left)andchemotherapywithanti-angiogenictherapy(right),fordifferentratesofacquiring
resistance.Vesselsareshowninred,livingcancercellsareshowningreen,andnecroticcellsinblack
26899655, 2022, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cso2.1032, Wiley Online Library on [26/03/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
!"of!# CESS!"#FINLEY
location,withgreencellsbeinglivingcellsandblackcells
beingnecroticcells,andphenotypicpropertiesareshown
inFiguresS$andS%.Atalowlevelofresistance,regrowth
ofthetumorissimilarbetweenchemotherapyandcombi-
nationtherapy,withthetumorhavingregrownfromoneor
twolocations.Asresistanceincreases,theregrowntumors
begin to resemble those without chemotherapy. We show
inFiguresS&andS',examplesoftumorgrowthovertime
forchemotherapyaloneandcombinationtherapy.Inaddi-
tion,themodelagainpredictsthatsomeareasofthetumor
will have high cell density, while other areas contain few
cells.Thisspatialheterogeneityisinfluencedbycellmigra-
tionandshrinkageofnecroticcells.
$ DISCUSSION
Inthisstudy,wehighlighthowtheuseofanti-angiogenic
therapies can lead to a more invasive tumor that is better
adapted to harsher microenvironmental conditions. Our
results agree with previous in vivo and clinical studies
thatfindthatlocalinvasionisincreasedbyanti-angiogenic
therapy [(–)]. By using a computational model, we are
able to further explore this phenomenon, examining how
the tumor not only evolves spatially but adapts pheno-
typically to the environmental pressure exerted on it by
treatment.We have extendedprevious modelingworks to
includeexplicitphenotypicchangesbasedonenvironmen-
tal changes and the impact of cell migration on tumor
growth. The metabolic changes predicted by the model,
increased glycolysis and resistance to acidic pH, are asso-
ciatedwithmoreaggressivetumors,allowingthesecellsto
betterinvadeintothesurroundingtissue.
Here, we found that anti-angiogenic therapy alone
prompts cancer cells that have developed in the harsh
microenvironment of the tumor core to migrate out into
thesurroundingtissueinsearchofmorehabitableregions.
Compared to simulations without any treatment, we find
that preventing angiogenesis causes further phenotypic
shifts toward properties that are induced by hypoxia and
low pH. From this, we see that while anti-angiogenesis
alone can reduce the number of cancer cells, it also has
thepotentialtoadvancetumordevelopmentbycreatinga
microenvironment that selects for more robust and inva-
sive cancer cells. This highlights how care must be taken
toavoidDarwinianselectionofcancercellsthataremore
fitforharsherenvironments[$(].
Despite these negative effects of introducing anti-
angiogenic treatment, the model predicts that anti-
angiogenic treatment provides some benefits when
combined with chemotherapy. The combination of
these two strategies has been shown to be effective in
some cancers [*,$,$+]. Indeed, we found that including
anti-angiogenic therapy is able to slow tumor regrowth
following chemotherapy; however, there was little differ-
enceinthephenotypicshiftbetweenchemotherapyalone
andtheadditionofanti-angiogenictherapy.
Our model is limited in some ways, which can be
addressedinfuturestudies.Here,wekepttheratesofphe-
notypic shift constant, choosing instead to focus on the
effectsoftherapy.Wecanexpandthemodelbyexamining
howtheseratesimpacttumordevelopmentwithandwith-
out the addition of anti-angiogenic therapy. We can also
includeheterogeneityincellsizesothatcellsstartwithdif-
ferent radii. Additionally, our treatment of normal tissue
is limited, represented only by an increase in the damp-
ing coefficient. We can extend our modeling of normal
tissue to include more explicit tissue forces and account
for regrowth of the tumor when appropriate. We can also
expand this section of the model to include destruction of
normalcellsandextracellularmatrixbyloweredpH,thus
increasingthemodel’sdetailonhowacidityenablestumor
invasion.Inaddition,here,wehavefocusedontheimpacts
of therapy on tumor phenotypic development rather than
adetailedanalysisofallmodelparameters.Suchasensitiv-
ityanalysiscanbeperformedinfutureworkasweexplore
themodelingreaterdepth.
% CONCLUSION
Here, we find that anti-angiogenic therapy promotes
tumor invasion by exposing the tumor to a harsher
microenvironment that forces the cancer cells to adapt
in order to survive. These cells exhibited further shifts in
aerobic glycolysis, acid resistance, and migratory poten-
tial, compared to simulations without any treatment.
Additionally, we find that augmenting chemotherapy
withanti-angiogenictherapycanprolongtumorregrowth
following the acquisition of resistance. Overall, our
model generates quantitative and mechanistic insights
into experimental and clinical observations following
anti-angiogenictreatment.
ACKNOWLEDGMENTS
The authors thank members of the Finley research group
forcriticalcommentsandsuggestions.Thisworkwassup-
ported by the American Cancer Society (,$-%$*-RSG-,(-
,$$--,-CSMtoSDF)andtheUSCProvost’sPhDFellowship
(CGC).
AUTHOR CONTRIBUTION
Conceptualization,formalanalysis,investigation,method-
ology, writing–original draft, writing–review and editing:
Colin G. Cess. Conceptualization, resources, writing-
reviewandediting:StaceyD.Finley.
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CESS!"#FINLEY !!of!"
DATA AVAILABILITY STATEMENT
Thedatathatsupportthefindingsofthisstudyareopenly
availableinGitHubathttps://github.com/FinleyLabUSC/
Anti-angiogenic-treatment-ABM-model and in article
supplementarymaterial.
ORCID
ColinG.Cess https://orcid.org/$$$$-$$$%-&'()-%)*+
StaceyD.Finley https://orcid.org/$$$$-$$$&-+($&-'+(%
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SUPPORTING INFORMATION
Additional supporting information may be found in the
onlineversionofthearticleatthepublisher’swebsite.
Howtocitethisarticle: C.G.Cess,andS.D.
Finley,Multiscalemodelingoftumoradaptionand
invasionfollowinganti-angiogenictherapy.Comp.
Sys.Onco."(%&%%),e'&$%.
https://doi.org/'&.'&&%/cso%.'&$%
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Abstract (if available)
Abstract
The interactions between a tumor and the immune system have long been known to have important clinical impacts. However, immunotherapies are not always successful, due to the complex network of interactions between various immune cells and cancer cells. One way to study these interactions is with computational modeling, allowing researchers to simulate how the tumor behaves under different conditions. A common way of modeling tumor interactions is with agent-based models, where cells are modeled as discrete individuals that interact with each other in a rule-based manner, capturing complex spatial structures and population dynamics. These models, however, suffer from difficulties in parameter estimation, due to the difficulties in comparing simulations to tumor images in a quantitative way. Additionally, besides extracting simple spatial metrics, it is difficult to compare model simulations to each other. Therefore, in this thesis, I present a novel application of representation learning, using neural networks to project images to low-dimensional points, as a way of quantitatively comparing agent-based model simulations to tumor images without the need to manually calculate complex metrics. This comparison facilitates a rigorous parameter estimation for these types of models, better ensuring the accuracy of parameter values. Using tumor images from a mouse study, I show how parameters can be estimated for an agent-based model containing several different immune cell types, along with how representation learning can be used to analyze model simulations. This work provides a foundation for methods that can be used to bridge the gap between model simulations and experimental images.
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Asset Metadata
Creator
Cess, Colin Graham
(author)
Core Title
Tumor-immune agent-based modeling: drawing insights from learned spaces
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Biomedical Engineering
Degree Conferral Date
2023-05
Publication Date
04/27/2023
Defense Date
03/31/2023
Publisher
University of Southern California
(original),
University of Southern California. Libraries
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Tag
agent-based modeling,Immune System,OAI-PMH Harvest,oncology,representation learning,tumor microenvironment
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theses
(aat)
Language
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Electronically uploaded by the author
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Finley, Stacey (
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)
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cess@usc.edu,colin.cess@gmail.com
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