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Review: Mathematical Modeling of Prostate Cancer and Clinical Application

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We review and synthesize key findings and limitations of mathematical models for prostate cancer, both from theoretical work and data-validated approaches, especially concerning clinical applications. Our focus is on models of prostate cancer dynamics under treatment, particularly with a view toward optimizing hormone-based treatment schedules and estimating the onset of treatment resistance under various assumptions. Population models suggest that intermittent or adaptive therapy is more beneficial to delay cancer relapse as compared to the standard continuous therapy if treatment resistance comes at a competitive cost for cancer cells. Another consensus among existing work is that the standard biomarker for cancer growth, prostate-specific antigen, may not always correlate well with cancer progression. Instead, its doubling rate appears to be a better indicator of tumor growth. Much of the existing work utilizes simple ordinary differential equations due to difficulty in collecting spatial data and due to the early success of using prostate-specific antigen in mathematical modeling. However, a shift toward more complex and realistic models is taking place, which leaves many of the theoretical and mathematical questions unexplored. Furthermore, as adaptive therapy displays better potential than existing treatment protocols, an increasing number of studies incorporate this treatment into modeling efforts. Although existing modeling work has explored and yielded useful insights on the treatment of prostate cancer, the road to clinical application is still elusive. Among the pertinent issues needed to be addressed to bridge the gap from modeling work to clinical application are (1) real-time data validation and model identification, (2) sensitivity analysis and uncertainty quantification for model prediction, and (3) optimal treatment/schedule while considering drug properties, interactions, and toxicity. To address these issues, we suggest in-depth studies on various aspects of the parameters in dynamical models such as the evolution of parameters over time. We hope this review will assist future attempts at studying prostate cancer.
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applied
sciences
Review
Review: Mathematical Modeling of Prostate Cancer
and Clinical Application
Tin Phan 1,*, Sharon M. Crook 1, Alan H. Bryce 2, Carlo C. Maley 3, Eric J. Kostelich 1
and Yang Kuang 1
1School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85281, USA
2Division of Hematology and Medical Oncology, Mayo Clinic, Phoenix, AZ 85054, USA
3Arizona Cancer Evolution Center, Arizona State University, Tempe, AZ 85287, USA
*Correspondence: tin.t.phan@asu.edu
Received: 15 February 2020; Accepted: 8 April 2020; Published: 15 April 2020


Abstract:
We review and synthesize key findings and limitations of mathematical models for prostate
cancer, both from theoretical work and data-validated approaches, especially concerning clinical
applications. Our focus is on models of prostate cancer dynamics under treatment, particularly with
a view toward optimizing hormone-based treatment schedules and estimating the onset of treatment
resistance under various assumptions. Population models suggest that intermittent or adaptive
therapy is more beneficial to delay cancer relapse as compared to the standard continuous therapy if
treatment resistance comes at a competitive cost for cancer cells. Another consensus among existing
work is that the standard biomarker for cancer growth, prostate-specific antigen, may not always
correlate well with cancer progression. Instead, its doubling rate appears to be a better indicator
of tumor growth. Much of the existing work utilizes simple ordinary differential equations due to
difficulty in collecting spatial data and due to the early success of using prostate-specific antigen
in mathematical modeling. However, a shift toward more complex and realistic models is taking
place, which leaves many of the theoretical and mathematical questions unexplored. Furthermore, as
adaptive therapy displays better potential than existing treatment protocols, an increasing number
of studies incorporate this treatment into modeling efforts. Although existing modeling work has
explored and yielded useful insights on the treatment of prostate cancer, the road to clinical application
is still elusive. Among the pertinent issues needed to be addressed to bridge the gap from modeling
work to clinical application are (1) real-time data validation and model identification, (2) sensitivity
analysis and uncertainty quantification for model prediction, and (3) optimal treatment/schedule
while considering drug properties, interactions, and toxicity. To address these issues, we suggest
in-depth studies on various aspects of the parameters in dynamical models such as the evolution of
parameters over time. We hope this review will assist future attempts at studying prostate cancer.
Keywords:
prostate cancer; mathematical models; clinical application; mathematical oncology;
parameter evolution; personalized medicine; adaptive therapy; optimal schedule; parameter
identification; uncertainty quantification
1. Introduction
The past 20 years have seen an accelerating development of mathematical models for prostate
cancer. One reason is the success of earlier work to explain and replicate experimental and clinical
data. Another reason is the availability of experimental and clinical data due to collaborations with
physicians and advancements in technology. Thus far, modeling efforts have utilized a broad range
of model types and methods to uncover some fundamentals about prostate cancer progression and
the efficacy of its treatments. However, there has not been a systematic and comprehensive synthesis
Appl. Sci. 2020,10, 2721; doi:10.3390/app10082721 www.mdpi.com/journal/applsci
Appl. Sci. 2020,10, 2721 2 of 29
of these results, perhaps due to each modeling effort being somewhat independent of the others.
In this article, we systematically synthesize existing results and summarize general limitations from
mathematical modeling work relating to prostate cancer.
1.1. Prostate Cancer as a Public Health Problem
Prostate cancer (PCa) is the leading cause of cancer in US men (behind non-melanoma skin
cancers), as well as the second leading cause of cancer-related mortality in men. The American
Cancer Society estimates that in 2020, 191,930 Americans will be diagnosed with prostate cancer,
with approximately 21.5% of all new cancer cases in men. The average five-year survival rate for all
cases of prostate cancer is about 99%, which is a 31% point higher than in the 1970s [
1
]. However,
once cancer metastasizes, the five-year survival rate drops to 30%, an increase of 10 percentage-point
compared to the 1970s [
2
]. While the incidence of prostate cancer in the US is decreasing (see Figure 1),
one possible cause is doctors’ discouragement of early PCa diagnosis due to the fear of overtreating
cancer. Globally, the incidence of prostate cancer is increasing, especially among developing nations [
3
].
Figure 1. Statistics of prostate cancer in the US—data obtained from [4].
It is difficult to state exactly what contributes to the development of prostate cancer; however,
age, race, and hereditary factors are the most firmly supported risk factors for prostate cancer [
5
].
While prostate cancer is most commonly diagnosed in men over 50, a benign tumor can start its
development decades earlier. The slow progression rate of prostate cancer means that often, patients
diagnosed with prostate cancer will not die because of it. Furthermore, the side effects of prostate
cancer treatment can be significant, so active surveillance without treatment is the recommended
management for some cases of low-risk disease [6].
1.2. Prostate Cancer: Physiology and Treatments
In this section, we aim to give a summary of the biological and clinical details of prostate cancer
and its treatments.
The prostate is a walnut-sized gland that is part of the endocrine system and the male reproductive
system. Its main job is to secrete seminal fluids that are used to nourish and transport sperm.
The function, proliferation, and atrophy of healthy and cancerous prostate cells depend highly on
the circulating level of androgen, particularly testosterone [
7
]. Androgens are steroid hormones that
regulate the development and maintenance of male characteristics by binding to androgen receptors
Appl. Sci. 2020,10, 2721 3 of 29
(AR), which is a cytoplasmic receptor that is activated by binding to androgens, which induce the AR
to migrate to the nucleus and bind to the Androgen Response Elements (ARE) within the DNA [
8
].
The testicular–hypothalamic–pituitary axis, also known as the hypothalamus–pituitary–gonadal
axis—where gonads indicate the reproductive system—produces androgen in a negative feedback
loop. This means that the higher the androgen level in the body, the lower its production rate
becomes [
9
]. Approximately 90%–95% of the body’s androgen is produced by the testes and the
remainder by the adrenal gland [
10
]. Once androgens, or testosterones, enter the prostate, about 90%
is converted to the more potent form 5
α
-Dihydrotestosterone (DHT) by the 5
α
-reductase enzyme [
11
].
DHT has been observed to be a better agonist to the androgen receptors, with about 2–3 and 15–30
times higher affinity than testosterone and adrenal androgens, respectively [
12
,
13
]. Additionally,
the unbinding rate of DHT and AR is roughly five times slower than that of testosterone and AR [
14
].
Once androgens bind to the AR, they are transported to the nucleus to bind to the ARE. This process
results in the transcription of androgen-regulated genes, which signals the proliferation and inhibition
of apoptosis in both healthy and cancerous prostate cells. The proliferation of prostate cells produces a
prostate-specific antigen (PSA). Most of this byproduct of prostate cell growth is normally contained
within the prostate wall; however, in the presence of prostate cancer cells, the cell wall is disrupted,
which allows for an increased leakage of PSA into the bloodstream. For this reason, a simple blood test
can be used to measure the level of PSA, which acts as a biomarker for prostate cancer and is often
used as a proxy for tumor volume in mathematical models [
15
]. A simplified schematic of prostate
cancer dynamics is described in Figure 2.
Figure 2.
A simplified schematic of the dynamics of androgen and prostate-specific antigen in prostate
cancer-adapted from [
16
]. (1) Serum androgen flows into the tissue of prostatic epithelial cells. (2) It is
then converted to Dihydrotestosterone (DHT) by 5-
al pha
reductase. (3) Both testosterones and DHT
bind to androgen receptors. (4) This further activates the proliferative and survival signals. (5) Cancer
activity results in the production of prostate-specific antigen (PSA), which is leaked out of the cells
(6). Note that the gray compartments indicate measurable concentrations that are relatively easier
to obtain.
Appl. Sci. 2020,10, 2721 4 of 29
The diagnosis of prostate cancer usually starts with a blood test to measure the PSA level.
To verify the presence of a prostate tumor, a digital rectal exam, a transrectal ultrasound, and/or a
prostate biopsy may be carried out. If cancer is confirmed, clinicians also assign the patient a Gleason
score, which measures the resemblance of biopsy cancer tissues to healthy tissues. The Gleason
score is often used as part of the clinical process to determine which treatment would be best for
the patient [
17
], but it is not often utilized in mathematical models. For localized prostate cancer,
the main options are: watchful waiting and active surveillance, surgical removal of the prostate,
and radiation. If the cancer has metastasized, hormonal therapy is the standard treatment. Initially,
hormonal therapy was administered as a continuous treatment, also known as continuous androgen
suppression therapy (CAS). However, hormonal therapy cuts off the production of male hormones,
so it often associates with a high level of toxicity. Some of the common side effects are impotence,
depression, bone demineralization, and dementia [
1
]. Furthermore, cancer cells eventually develop
resistance to hormonal therapy. Therefore, instead of CAS, intermittent androgen suppression therapy
(IAS) was introduced as an alternative to reduce the side effects of CAS and potentially delay the time
to treatment failure [
18
]. In contrast with CAS where the patient is put on treatment until it becomes
ineffective, during IAS, patients go on and off of therapy according to either a fixed interval of time or
threshold PSA level. The on-off cycles are repeated until the treatment is ineffective. While IAS gives
patients a better quality of life, it is controversial whether it is better than CAS for delaying the onset of
treatment resistance [
19
]. Recently, the concept of adaptive therapy was introduced into clinical studies
of prostate cancer, which has generated some exciting results and opportunities [
20
]. On the other
hand, while hormonal therapy is the focus of many modeling efforts, immunotherapy and chemical
therapies can also be used to treat prostate cancer—usually after the cancer has developed treatment
resistance to hormonal therapy [21,22].
1.3. Mathematical Models for Prostate Cancer
Due to a lack of detailed knowledge of the complex mechanisms of PCa and the effects of
numerous therapies, some researchers have begun to use mathematical models to better explain
the observations from experimental and clinical studies [
15
]. Whereas statistical models can help to
establish the relationships between variables in a study, mathematical models pursue the dynamics
of the interaction of those variables. One could argue that the studies using mathematical models
should come after the observed relationships of the variables. Additionally, mathematical models can
be developed using known physical, biological, or empirical laws and incorporate control effects (such
as drugs), making them suitable for a variety of purposes, such as testing a hypothesis or finding an
optimal combination of drugs. This means that mathematical models often contain assumptions on
the nature of the phenomenon being studied. Furthermore, mathematical models can range wildly in
their complexity, which may result in either highly unrealistic or mathematically and computationally
untractable models. There have been numerous reviews of the growing literature on mathematical
oncology in the past 15 years [
15
,
23
26
]. However, none of these reviews exclusively and extensively
cover mathematical models for prostate cancer. This review aims to address the current lack of an
extensive review of the rapidly-developing area of mathematical modeling for prostate cancer. To do
so, we exhaustively collect existing mathematical models developed specifically for prostate cancer
and synthesize their results. First, we present conclusions that are independent of the details of the
models. This is carried out by categorizing the models based on some important characteristics and
then finding the common conclusion of models that belong in the same category. Secondly, we provide
a summary of key findings and their implications for each model. Finally, we point out relevant but
lesser explored concepts related to prostate cancer and its treatment that may be addressed by future
research. Before carrying out our analysis, we first introduce the terminology and key characteristics
of existing models.
Appl. Sci. 2020,10, 2721 5 of 29
2. Elements of Mathematical Models for Clinical Applications
In this section, we introduce and discuss the elements of mathematical models that are important
for transitioning from theoretical studies to clinical applications. While we do not consider all aspects of
mathematical models, the chosen categories should serve as a solid foundation for a fruitful discussion.
2.1. Population Structure and Dynamics
To have a complete understanding of the structure of prostate cancer and how the subpopulations
of cells interact, with or without treatment, would be similar to having solved the problem of cancer
from a theoretical perspective. A system-biology approach treats cancer as a unified population of cells
that is part of a larger ecosystem. On the other hand, cancer can also be looked at like a society with
many interacting sub-types of cancer cells. While it would perhaps be ideal to understand the details
of each aspect of prostate cancer, it is impossible to do so. Instead, each model has to make certain
simplifying assumptions that aim to answer a specific set of questions. Thus, we will examine the
underlying assumptions of the population structure and dynamics on which each model was built and
analyzed. Specifically, we aim to explore the following features of mathematical models in more detail.
Tumor heterogeneity and evolution: treatment failure due to cancer cells developing resistance is
an unsolved problem in treating advanced prostate cancer. The most likely explanation is that over
the course of treatment, selective pressure allows selection for resistant cell types. Thus, all existing
population-type prostate cancer models incorporate some mechanisms that describe this heterogeneity
and the evolution of the cancer population during treatment. For this feature, we study the existing
literature to obtain conclusions that explore the heterogeneity and evolution of a tumor and its effects.
Tracking the progression of tumor size: having accurate and consistent measurements for the growth
of the tumor that can be incorporated into mathematical models is crucial for clinical applicability and
model assessment. While imaging data is perhaps best suited when it comes to measuring the changes
in the tumor, the challenges of collecting high-quality spatial data hinder this possibility. Instead,
time-serial PSA data has often been used as a proxy for the progression of prostate cancer. In this
regard, we synthesize suggestions from existing work to provide an assessment of PSA as a biomarker
for prostate cancer and explore other alternatives.
Model types and dynamics: some of the most successful prostate cancer models make use of the
characteristics of the cancer progression or the observed data, while others construct more mechanistic
models that make use of either fundamental or physical laws, like conservation laws, and/or the
biochemical networks of cancers. In essence, mathematical modeling of prostate cancer is at the
interface between simple classical mathematical models and complex system-biology type models,
which offers rich opportunities to study interesting properties of these models from a mathematical,
biological, or computational perspective. Additionally, while modeling with ordinary differential
equations still dominates existing studies, the potential for using other types of dynamics has also
been explored. For this reason, we compare and contrast different types of models and how they may
offer insights for treating prostate cancer.
2.2. Applicability in Clinical Settings
While theoretical work offers insights into the fundamentals of prostate cancer, the complex
interactions of various components in a real situation blur the implications from theories. Moreover,
in the case of mathematical models, the predictions of the progression of the tumor must be validated
against real-world clinical data, which often means fitting and forecasting using incomplete sets of
data. Aside from these theoretical issues, clinical applications come with another set of questions
that are often case-specific and not explored in depth in theoretical settings. For example, how does
one predict whether a certain treatment will be effective for a specific patient? If so, how long will
it take for the treatment to be effective? If not completely, then for how long would such treatment
be effective? What is the dose level of certain drugs that should be used to achieve optimal results?
Appl. Sci. 2020,10, 2721 6 of 29
What is the best combination of drugs to optimize effectiveness? Cancer treatments can be invasive
and expensive, which can harm patients physically and financially [
1
,
6
]. Although treating cancer is
a nonlinear process, this is not the case in practice; different adjustments to the treatment should be
made based on the specific situation. Existing modeling studies attempt to give some hints to possible
treatments. However, other features of treatment, such as dosage, combinations of drugs and therapies,
can also be optimized to achieve better results. For these reasons, we discuss several important gaps
between theoretical work and the clinical application of mathematical models.
Real-time estimability: in theoretical studies, researchers often have the luxury of knowing the
complete set of a patient’s data. However, in the clinical setting, researchers must make do with
a smaller but increasing set of a patient’s data. This means the parameters for the model cannot
be identified with high certainty at the earlier stage, rendering implications from mathematical
models unreliable.
Uncertainty, identifiability, and sensitivity: another concern that arises with models of high
complexity is the sensitivity of the model outputs to variations in the parameters. If the variation is
small, it is usually referred to as sensitivity analysis, whereas uncertainty quantification refers to larger
variation in the parameters, perhaps due to the large uncertainty in the data. Since measurements
and models are imperfect, uncertainty quantification allows physicians to put confidence bounds
on a model’s predictions. On the other hand, sensitivity analysis can help pinpoint the important
parameters in a model, which can aid with decision making during treatment.
In close connection with sensitivity and uncertainty, the term identifiability refers to the ability to
determine a unique set of model parameters with respect to the given data. This concept is further
broken down into structural identifiability and practical identifiability. Structural identifiability refers
to the relationship between the inherent structure of the model and perfect data. In essence, structural
identifiability asks whether all of the parameters in a model can be uniquely identified given some
perfect set of data on the variables in the model. On the contrary, when the data contains errors,
the same question refers to the practical identifiability of the model. While these are important topics,
there have been few attempts to study these closely related aspects in the case of prostate cancer.
Optimal schedule, optimal treatment, and patient classification: while the previous issues deal with
the validation of mathematical models in clinical settings, optimal schedule, optimal treatment, and
patient classification are important contributions of mathematical models. For instance, patients
can be categorized in advance based on whether they would react positively toward a certain type
of treatment using mathematical models. Furthermore, to fully draw out the effect of a treatment,
a mathematical model can be used to study how to best administer the treatment as pertaining to
treatment schedule or in combination with other treatments. Additionally, other factors, such as cost,
could also be taken into account for an optimal treatment study.
3. Implications from Mathematical Models
3.1. Population-Based Models of Tumor Relapse
In this section, we discuss the implications from mathematical models. A synthesis of our findings
is presented in Table 1.
Table 1. Aspects of mathematical modeling: summary of findings and future exploration.
Aspects of Summary of Findings Future Exploration
mathematical Modeling
Tumor heterogeneity Treatment resistance appears to
come at a competitive cost for
cancer, which implies that
intermittent and adaptive therapy
would be superior to continuous
therapy.
Quantitative measures of the
evolutionary cost and when it
does occur would be useful.
Furthermore, the competition
rates between sub-types of cancer
cells should be quantified.
and evolution
Sections 3.13.4
Appl. Sci. 2020,10, 2721 7 of 29
Table 1. Cont.
Aspects of Summary of Findings Future Exploration
Mathematical Modeling
Tracking the The most commonly used
biomarker for prostate cancer
growth (PSA) is useful but can be
unreliable. Instead, other
measurements can be used in
place or in concurrence with PSA
to track tumor progression.
The accuracy of tracking the
progression of tumors using
multiple biomarkers needs to be
examined. While accuracy is key,
the availability of such biomarkers
should also be taken into account.
progression of tumor
Section 3.5
Model types
Ordinary differential equations
build the foundation for studying
prostate cancer. However, the lack
of various modes of modeling
implies many aspects of prostate
cancer are left unexplored.
As more data becomes available,
especially imaging data, spatial,
memory-based, and stochastic
models will become useful in
capturing spatial patterns in
cancer progression and
interaction, specifically,
the metastatic processes.
and dynamics
Section 3.6
The highly heterogeneous nature of prostate cancer, especially after it has metastasized, is
one important reason why population-type models take up a large percentage of existing work.
This includes simple models that describe only the interactions among sub-populations of cells, or more
complex models that also account for external factors, such as environmental changes due to treatment.
The population-type models, in general, focus on the interactions of the cancer sub-populations with
or without external effects. On the other hand, kinetic-type models tend to view cancer as a whole and
how cancer interacts within the human physiological network, often via biochemical pathways.
Yorke et al. (1993)
. The earliest approach to model prostate cancer progression mathematically
is credited to Yorke et al. [
27
]. In their work, they developed a simple kinetic model based on
Gompertzian growth that aims to explain the progression of prostate cancer, its metastasis process,
and how the metastasis process affects local treatments (surgical and radiation therapy). By comparing
their model simulations with the patient’s survival data in Fuk et al. [
28
], Yorke and colleagues
suggested that local relapse is associated with more aggressive cancer phenotypes. Furthermore, due
to the aggressive cancer phenotypes observed in local relapse, they suggest that local control (e.g., the
complete eradication of the primary tumor and lymph node metastasis) can significantly improve the
disease-free survival and metastasis-free survival time. However, after the cancer has metastasized,
standard treatments show early positive results but fail to contain tumor relapse.
Jackson (2004)
. The relapse following positive responses to hormonal castration has long been
hypothesized to be a result of prostate cancer being made up of different subpopulations of cells,
each with its own level of dependency on the extracellular androgen. This concept is first demonstrated
with mathematical modeling in 2001 by Jackson [
29
]. Jackson balances proliferation, apoptosis,
and fluxes of two types of cells under radial symmetry to construct a system of partial differential
equations for androgen deprivation therapy in prostate cancer, as shown in Equations (1)–(2).
dR
dt =u(R(t),t). (1)
Dpp
r(R,t)u(R,t)p(R,t) = 0, Dqq
r(R,t)u(R,t)q(R,t) = 0. (2)
In this model,
u(R
,
t)
is the net rate of collective cellular motion,
R(t)
is the tumor radius and
p(R
,
t)
,
q(R
,
t)
are the volume fractions of androgen dependent (AD) and androgen independent (AI)
prostate cancer cells, respectively. The model assumes that a high level of androgen increases the
proliferation rate of androgen dependent cells but does not affect the proliferation rate of androgen
independent cells. On the other hand, a low level of androgen decreases the apoptosis rate of
Appl. Sci. 2020,10, 2721 8 of 29
androgen dependent cells and increases the apoptosis rate of androgen independent cells. With these
assumptions, the models were able to capture the qualitative dynamics observed from experimental
data of LuCaP 23 in mice [
30
]. Thus, Jackson concludes that the model supports the notion that cancer
relapse occurs because AI cells have a lower apoptosis rate, rather than a higher proliferation rate,
a conclusion that agrees with some experimental data [
31
]. This conclusion supports the hypothesis
that prostate cancer cells pay a cost to evolve resistance to treatment—thus having a disadvantage
against AD cells when no treatment is applied [20].
Additionally, Jackson shows that androgen deprivation therapy is only effective for a small region
of parameters—which agrees with clinical data showing that androgen deprivation therapy is not
curative, even in early stage disease (see Figure 3). As with all models, Jackson’s modeling effort
has certain limitations. First, the assumption of spherical symmetry in the model formulation limits
the application of the model to the early stage, where the cancer can be approximated with a sphere.
Secondly, the assumption that increasing the level of androgen increases the apoptosis rate of androgen
independent cells is arguable because androgen independent cells may still be dependent on AR,
which is known as the outlaw pathway to resistance [
11
,
15
,
32
]. Regardless, Jackson’s work has paved
the way for future modeling studies.
Figure 3.
Treatment effectiveness in Jackson’s model—figure reproduced from Jackson [
29
] with
permission, which is distributed under a Creative Commons Attribution (CC BY) license. The vertical
axis represents the proliferation rate of AD cells post treatment. The horizontal axis represents the
steady state androgen concentration post treatment. The striped region indicates the tumor radius
decreasing in the first and second order approximations. (
a
) The figure shows a small striped region in
the parameter space where partial blockage of androgen leads to successful containment of the cancer.
(
b
) Often times, successful treatment is not possible. Further analysis in the case of a complete blockage
of androgen shows a small expansion of the successful region.
Ideta et al. (2008)
. Inspired by Jackson’s work, Ideta et al. [
33
] studied a system of ordinary
differential equations describing the interaction of androgen dependent and independent cells,
while explicitly treating the androgen level as a dynamical variable.
da
dt =γa+γa0(1u)(3)
dxi
dt =αipi(a)xiβiqi(a)xi+ (1)im(a)x1(4)
Appl. Sci. 2020,10, 2721 9 of 29
Here,
a
represents the androgen level, the index
i
runs between 1 and 2, which represents the
androgen dependent cells and androgen independent cells, respectively.
pi(a)
,
qi(a)
, and
m(a)
are
functions that represent the effects of androgen on the proliferation, death, and mutation rates of the
respective cells. The authors consider three different assumptions of how androgen affects the growth
and death rates of the androgen independent population (
p2(a)
is different for each assumption).
Furthermore, the model incorporates the idea that androgen dependent cancer cells can transform into
androgen independent cancer cells due to survival pressure.
Ideta et al. used this model to compare the effects of intermittent androgen deprivation therapy
verses continuous androgen deprivation therapy, making it the first mathematical modeling work
to carry out such a comparison in the decade-old debate of CAS and IAS. They observed that
relapse can be averted only under the assumption that the AI cell population decreases when the
androgen level is normal. Their comparison concludes that if androgen dependent cancer cells have a
competitive advantage over androgen independent cancer cells without treatment, then IAS is superior
in prolonging the onset of treatment resistance than CAS. It is noteworthy to point out that both
Jackson and Ideta make the assumption that AI cells do not have a competitive advantage over AD
cells. This assumption is partially justified due to the rare occurrence of AI cells in early PCa, yet AI
cells are quite common after the cancer has metastasized.
While the mechanisms leading to the occurrence of AI relapse are still somewhat of a mystery,
the study by Zhang and colleagues [
20
] shows strong evidence to support the assumption that AI cells
are out-competed by AD cells in an androgen rich environment. In addition, Zhang et al. hypothesizes
that in order for cancer cells to obtain treatment resistance, they must expend part of their system to
develop some mechanisms to do so, which makes them less fit than the cells that do not have such
mechanisms. However, Jackson’s and Ideta et al. models do not directly incorporate the competitive
effect of the two populations.
Shimada and Aihara (2008), Yang et al. (2016)
. To address this limitation, Shimada and
Aihara [
34
] and Yang et al. [
35
] proposed models that directly account for competition between
the two cancer phenotypes. The competition model of Shimada and Aihara introduces a competition
term with a constant rate, whereas the competition terms in Yang et al. depend on the androgen
level and are asymmetrical. Furthermore, Yang et al. also introduce intraspecific competition, which
addresses the theoretical issue of having an environment that can support an infinite amount of cancer
cells in Ideta’s work. Numerical simulations for both models show the potential successes of IAS,
which increases with higher competitive advantages for AD cells.
Guo et al. (2008), Tao et al. (2010)
. Aside from approaches using ordinary differential equations,
Jackson’s model inspired spatial models by Guo et al. [
36
] and Tao et al. [
37
]. The work by Guo et al.
is a spatial extension of the Ideta et al. model using some assumptions from Jackson’s formulation.
Their results give similar conclusions to Ideta’s. Tao et al., on the other hand, analyzed the case where
a mutation inhibitor is used as part of the regime for androgen deprivation therapy. They conclude
that CAS can neither control nor cure PCa, even with a mutation inhibitor. However, their simulations
suggest that mutation inhibition may delay the relapse of cancer.
Friedman and Jain (2013), Lorenzo et al. (2016)
In addition to these studies, Friedman and
Jain [
38
] construct a spatial model of prostate cancer based on a similar framework and prove the
existence and uniqueness of solutions for the model. Taking on a more computational approach,
the phase-field method, which accounts for the dynamics and co-existence of healthy and cancerous
cells, has also been applied in Lorenzo et al. [
39
], where the authors simulated several observed
patterns of growth in prostate tumor (in vitro) with their model.
3.2. Data-Validated Models
Hirata et al. (2010)
. These early approaches suffer from a common problem: they are observation
driven but not data validated, meaning the model is not tested against real-world data quantitatively.
This brings us to the work by Hirata, Bruchovsky, and Aihara (HBA) [
40
]. In 2010, Hirata et al. took
Appl. Sci. 2020,10, 2721 10 of 29
a different approach to develop a model for prostate cancer. Hirata and colleagues observed that
when treatment starts, the PSA level first plummets to a certain level, then decreases more slowly to
an undetectable level. Hirata et al. proposed that the decay trend of PSA is most naturally modeled
by a three-population model with interacting populations via linear rates, which results in a matrix
representation. To account for the effect of treatment, Hirata et al. use a discrete switching mechanism.
When the patient is on treatment, the HBA model takes the form:
d
dt
x1(t)
x2(t)
x3(t)
=
w1
1,1 0 0
w1
1,2 w1
2,2 0
w1
1,3 w1
2,3 w1
3,3
x1(t)
x2(t)
x3(t)
, (5)
and during the off-treatment period,
d
dt
x1(t)
x2(t)
x3(t)
=
w0
1,1 w0
2,1 0
0w0
2,2 0
0 0 w0
3,3
x1(t)
x2(t)
x3(t)
. (6)
The parameters
wij
are the transformation rates from
xi
to
xj
and for simplicity, the PSA level is
taken as the sum of
x1
,
x2
,
x3
. The model is tested against different sets of clinical data [
40
,
41
]. While
the model is phenomenologically based, it has successfully reproduced the highly nonlinear trends in
PSA data in many instances [
23
,
24
]; see Figure 4. Hirata and colleagues also use the model to explore
the classification of cancer patients to determine which treatment (IAS or CAS) may benefit a patient
the most. This is done by looking at the numerically estimated parameters for each patient based
on their PSA data. Furthermore, the optimal treatment schedule has also been studied using their
model [42].
Figure 4.
The fitting and forecasting capability of the Hirata, Bruchovsky, and Aihara (HBA)
model—figures reproduced from [
40
] with permission, which is distributed under a Creative Commons
Attribution (CC BY) license. (
a
) An example of data fitting using the HBA model. The circles represent
patient data. The black curve represents the model result after fitting to the patient’s data during
IAS. The grey curve represents the simulated PSA level in the hypothetical situation when continuous
androgen suppression therapy (CAS) is carried out. (
b
) An example of forecasting using the HBA
model. The vertical dashed line represents the cut-off between fitting and forecasting.
Portz et al. (2012)
. While purely phenomenological models, such as the HBA model, are useful
in establishing the core structure of cancer dynamics, they are perhaps better viewed as a building
block to a more comprehensive and biologically tractable model. In this regard, and Portz, Kuang,
and Nagy (PKN) [
43
] formulated a mechanistic model of prostate cancer in 2012. The model utilizes
the idea of cell quota as previously studied by Droop in a marine ecology context [
44
] and has gained
traction since. The PKN model assumes only two subpopulations of cancer cells that can mutually
transform from one to the other. The model assumes that androgen is the limiting nutrient (or driving
Appl. Sci. 2020,10, 2721 11 of 29
force) in the proliferation and death of cancer cells and incorporates it as a cell quota. Specifically,
the model presumes that there is a minimum quantity of androgen required for the growth of both AD
and AI cells.
dxi
dt =µm1qi
Qixi
| {z }
growth
dixi
|{z}
death
λi(Qi)xi
| {z }
cell transformation
+λj(Qj)xj
| {z }
cell transformation
, (7)
dQi
dt =νmqmQ
qmqi
A
A+νh
| {z }
androgen influx to cells
µm(Qiqi)
| {z }
uptake
bQi
|{z}
clearance
, (8)
dP
dt =
i=1,2 σ0+σi
Qm
i
Qm
i+ρm
ixi
| {z }
PSA production
δP
|{z}
PSA clearance
. (9)
In this model,
x1
and
x2
are the AD and AI cancer cells, respectively.
Qi
is the intracellular
androgen level, in contrast to the serum androgen level
A
. PSA is represented by the variable
P
. While
the most commonly presented form of the PKN model is in Equation (7), the cell-quota assumption in
the model means that the growth term takes a more accurate form of
µmmax{
0, 1
qi
Qi}xi
. In this form,
the biological meaning of the model is preserved when
Qi
dips below
qi
. This should be noted in both
mathematical analysis and data fitting to avoid negative population simulations [
45
]. The model is first
validated using the data from Akakura [
46
]. The PKN model requires the input of serum androgen
data, so an exponential model is used to interpolate serum androgen levels. Similar to the HBA model,
the PKN model also reproduces the time series observed in clinical data (see Figure 5). Additionally,
a comparison of the HBA and the PKN models shows that they are equally capable of fitting and
forecasting clinical PSA data [23,24].
Figure 5.
The fitting capability of Portz et al. model—figure reproduced from Portz et al. [
43
] with
permission, which is distributed under a Creative Commons Attribution (CC BY) license. (
a
) The red
circles followed by doted dash lines are patient’s data. The blue curve represents the fitting of the
model. The dashed black curve and dotted dashed curves represent the relative level of androgen
independent (AI) and androgen dependent (AD) population as predicted by the model. (
b
) The plot
shows the level of intracellular androgen for AD and AI cells (
Q1
and
Q2
, respectively) relative to the
minimum cell quotas for AD and AI cells (q1and q2, respectively).
Baez and Kuang (2016)
. The work by Portz et al. has inspired a series of modifications to address
various limitations of the model. First, the PKN model requires interpolation of androgen data,
so an assumption of the future androgen level is required, which makes accurate forecasts difficult to
obtain. Thus, Baez and Kuang (BK) [
47
] introduced an improved version of the model where the main
Appl. Sci. 2020,10, 2721 12 of 29
difference is the direct incorporation of androgen as a variable in the model, similar to the Ideta et al.
model. This allows for the simultaneous fitting of androgen and PSA data. Furthermore, by having
androgen as a variable, forecasts can be compared directly to clinical data.
Aside from addressing the limitation of forecasting in the PKN model, the BK model also implicitly
addresses another limitation, which is pointed out by Hatano et al. [
23
]. Using approximation,
Hatano et al. showed that the PKN model is not able to reproduce PSA relapse under continuous
androgen deprivation therapy. This was missed by the original effort, most likely because their
attention was focused on intermittent therapy. Phan et al. [
48
] show that the main reason for
this biological limitation is due to the PKN model having a reversible transformation between the
two subpopulations. On the other hand, by having only a transformation from AD to AI cells,
which inversely depends on the serum androgen level, the BK model avoids this biological limitation
and can produce relapse for CAS. This leads to the suggestion that the simplest way to ensure this
biological aspect is by formulating the model with a unidirectional transformation from AD to AI cells.
Furthermore, their conclusion shows that the success of cancer treatment is often heavily influenced
by the structure of the model. Specifically, if a model assumes reversible transformation between the
cancer sub-populations, then the control of the tumor tends to be simpler. Whereas a unidirectional
transformation to a resistant population almost guarantees resistance will take place.
Phan et al. (2019)
. Phan et al. [
48
] also compared a two subpopulation version of the BK model
and a three subpopulation version with a similar structure to the HBA model. They observed that the
limitation of data types and data points hinders significant improvements in using a more complex
model. This concept is further analyzed using a Fisher information matrix and profile likelihood in
Wu et al. [
49
], where a similar conclusion is reached. Additionally, in the work by Baez and Kuang,
upon off treatment, the level of androgen grows very rapidly—causing the model to overshoot in
its forecasts of androgen and PSA levels. Phan et at. [
50
] pointed out that this is perhaps due to two
main reasons. First, the parameter associated with androgen production seems to be much larger
than previous estimates. Second, the BK model assumes instantaneous diffusion of serum androgen
to intracellular androgen. Thus, a new model is constructed by adding a compartment for serum
androgen. By addressing these two features, Phan and colleagues show an improvement in the fitting
and forecasting of androgen and PSA using clinical data. This suggests that separating intracellular
and serum androgen is an effective and natural method to improve model fitting.
3.3. Models of Cellular Kinetics
Barton and Andersen (1998), Potter et al. (2006), Reckell et al. (2020), Cerasuolo et al. (2020)
.
As hormonal therapy is the standard of care for metastatic prostate cancer, it is crucial to obtain a good
understanding of how the drugs used in hormonal therapy affects the androgen level and consequently
the prostate cancer cells. To this end, the work by Barton and Andersen in 1998 [
9
] has paved the
way for an initial framework of androgen regulation of prostate growth. Its extension and validation
using rat data was later carried out in Potter et al. [
51
]. Recently, Reckell et al. [
52
] also formulated a
pharmacokinetics model that incorporates the specific properties of a drug on androgen production in
a prostate cancer model and tests it with clinical patient data. While Reckell et al. focus on the effects
of combination hormonal drugs (Cyproterone acetate and Leuprolide acetate) on a macroscopic level
using phenomenological functional responses, the stochastic differential equation phamacokinetics
model by Cerasuolo et al. [
53
] attempts to capture the molecular interactions (oxygen consumption
and protons production of cancer cells) between the cells and a newer hormonal drug (enzalutamide)
using mice data. In essence, these works are successful in their attempts to replicate the drug effects on
the dynamics of androgen and prostate growth; however, the complexity of these models often make
them mathematically untractable and highly difficult to fit with high certainty. To address this issue,
in both pharmacokinetics attempts, the parameter estimations are carried out in a multi-level process.
Eikenberry et al. (2010)
Utilizing the intracellular kinetics framework of the androgen receptor,
testosterones, and 5
α
-dihydrotestosterones in the work of Potter et al., Eikenberry et al. [
54
] built
Appl. Sci. 2020,10, 2721 13 of 29
a multi-scale model to study the effect of androgen in the evolutionary process from the benign to
treatment stage of prostate cancer. The model couples the AR kinetics model with a state-transition
model, where 100 states included with each state representing a different strain of prostate cells
with varying AR expression. This reflects the observation that AR up-regulation is one of the most
important pathways in which cancer cells becomes AI. Their results demonstrate how a heterogeneous
population of prostate cancer cells can be skewed to select for androgen independent phenotypes in a
low androgen environment. This suggests that while low levels of androgen may delay the appearance
of malignant cancer cells, it may increase the chance of more aggressive cancer phenotypes. This result
offers a reasonable explanation to the observation that finasteride, a 5-
α
reductase inhibitor, can reduce
the overall rate of prostate cancer but may increase the rate of high-grade prostate cancer [55].
Jain et al. (2011), Jain and Friedman (2013)
. While Eikenberry et al. considered the role of
androgen in the evolutionary dynamics of prostate cells using hundred of cell types, Jain et al. [
16
]
applied the framework into a two cell types model under androgen deprivation therapy (both
intermittent and continuous). Since their model is rather complex, they carry out multi-level fitting to
obtain the estimates for the parameter values separately. By fitting average patient data reported in
Goldenberg et al. [
56
], they show that their model not only contains patient specific parameters but is
also capable of reproducing a variety of clinically observed dynamics of cancer progression. The model
predicts similar conclusions as previous work: intermittent androgen suppression therapy is superior
compared to continuous therapy when the AD cells have a competitive advantage over the AI cells and
vice versa. Due to the complexity of the model, not much analytical work can be done. Furthermore,
the large number of parameters presented in the model means clinical parameter estimation for each
patient is difficult. Therefore, in subsequent work, Jain and Friedman [
57
] simplify their model and
carry out mathematical analysis on the simplified version. By defining rigorously the definition of
treatment viability and failure, they were able to compare the efficacy of continuous vs. intermittent
therapy, which yields a similar conclusion to previous work. Additionally, they discover that even if it
is possible to control a tumor with an optimal schedule of intermittent therapy, a sub-optimal one may
still lead to the emergence of treatment resistance.
Zhang et al. (2017)
. In the same spirit of using a simple phenomenological model, but in contrast
to Hirata et al. where the focus is to examine the mutation process that governs the dynamics of
the cancer progression, Zhang et al. [
20
] utilized a three population Lotka–Volterra type model to
investigate solely the competition aspect of cancer sub-populations. The model takes the form:
dxi
dt =rixi 13
j=1aij xj
Ki!(10)
and
dP
dt =
3
i=1
xi0.5P. (11)
Here,
aij
is the competition matrix, where
x1
,
x2
, and
x3
represent AD cells, treatment-sensitive AI cells,
and treatment-resistant AI cells, respectively. The PSA level is given by the variable
P
. The model is
used in concurrence with their clinical study. Their work comes at a time where there is no conclusive
evidence of whether IAS is better than CAS in terms of prolonging the time to the relapse of cancer.
They argue that the result of previous clinical comparisons of intermittent and continuous treatments
that study whether intermittent therapy gives any benefit in terms of delaying the onset of treatment
resistance [
58
,
59
] are not well supported. Zhang and colleagues were able to offer clinical evidence
that the previous protocol for intermittent therapy, in essence, has the same effect as continuous
therapy. Furthermore, they show that by changing the standard protocol, they were able to obtain
significant improvement in the delay of treatment resistance for intermittent therapy with abiraterone
over previous results. Their results show that AD cells are likely to have a significant competitive
advantage over AI cells when treatment is not applied. In the same work, they support an alternative
Appl. Sci. 2020,10, 2721 14 of 29
form of therapy, namely adaptive therapy. The main idea behind adaptive therapy is not to cure
cancer but to obtain a stable tumor burden and maintain it indefinitely [
60
]. Compared to standard
treatment where a stable tolerable dose is used, adaptive therapy uses varying drug dosages that
change, depending on the response of the tumor to the drug, resulting in a lower overall drug usage
than the standard treatment over the same treatment period. By using a smaller dosage, the treatment
does not wipe out the drug-sensitive population. Hence, the treatment can be said to rely on the
sensitive cancer population to control the resistance population due to their competitive advantage.
West et al. (2018, 2019)
. Subsequently, the work by Zhang et al. was extended further in
West et al. [
61
], where an additional compartment was added in order to implement multi-drug
therapy. By noting the limitation of parameter identification, West et al. put forward an assumption
that aids the parameter fitting process: parameters corresponding to cancer cells are similar across
different patients. What distinguishes the cancer progression between different patients is the initial
relative size of the cancer subpopulations. Using this assumption, West and colleagues parametrized
the parameters and fixed them across patients. They then carry out the data fitting process only on
the initial size of each cancer subpopulation. They argue that the identifiability of their parameter
estimation process is supported by the theorem in Sontag’s paper [
62
]. Their work shows that
the incorporation of an additional drug (adding of docetaxel to abiraterone) may further delay the
onset of treatment failure. A similar concept is demonstrated in the work by West et al. [
63
] but for
chemotherapy in prostate cancer.
3.4. Models Of Immunology
Peng et al. (2016)
. While many of the mathematical modeling efforts for metastasized prostate
cancer focus on hormonal therapy, several researchers have also taken note of the recent development
of immunotherapy (vaccination) for prostate cancer. One prominent example is Provenge (or
sipuleucel-T), which has been approved by the Food and Drug Administration for treating prostate
cancer. A notable example is the work by Peng et al. [
64
] in 2016. In this work, Peng and colleagues
construct a system of differential equations consisting of castrate resistant and sensitive tumor cells and
the immune micro-environment. After parametrizing the models using mouse data, they study the
treatment efficacy of combination therapy between four different treatments (castration, vaccination,
cytokine interleukin-2/IL2 neutralization and regulatory T cells/Treg depletion). The concurrent
use of castration and vaccination is motivated by the review of Ching et al. [
65
], providing evidence
that ADT can increase the efficacy of immunotherapy. On the other hand, the incorporation of IL2
neutralization and Treg depletion is motivated by a study [
66
], showing that ADT is followed by the
activation of cytotoxic T lymphocytes (killer T-cells/CTLs); however, the production of IL-2 and Treg
may inhibit the activity of CTLs in the prostate lymph nodes. Their work shows the potential of using
system biology type models to address complex multi-drug approaches for prostate cancer. While
their study shows the potential of a system biology type approach in the modeling of prostate cancer,
their limitation lies in the lack of a sufficient source of data for validation of their complex model.
Portz and Kuang (2013), Rutter and Kuang (2017), Kronik et al. (2010)
. Portz and Kuang [
67
]
propose an alternative approach to modeling immunotherapy motivated by the work of Kirschner and
Panetta [
68
]. Aside from the AD and AI cells, the model also considers the number of activated T-cells,
the concentration of cytokines, the concentration of androgen, and the number of dendritic cells. In
computational studies, they show a small advantage of combining IAS with immunotherapy over CAS
with immunotherapy. Subsequently, Rutter and Kuang [
69
] extend their model to study the effects that
drug dosage amounts and frequencies of administration on the time to relapse. Their computational
and mathematical analysis of the model show the possibility of tailoring the vaccine dosage to the
patient’s effective immune system to maximize the effectiveness of treatment. In another work,
Kronik et al. [
70
] formulated a simple mathematical model that describes the immune response in
prostate cancer patients receiving immunotherapy. Kronik and colleagues fit the model to individual
Appl. Sci. 2020,10, 2721 15 of 29
patient data and carry out the forecasting using the estimated parameters. Their model produces
robust predictions with regards to the data set of 15 patients used for validation.
Elishmereni et al. (2016), Stura et al. (2016)
. The idea of predicting treatment failure time is also
presented in the work of Elishmereni et al. [
71
], where a simple model of cancer growth that includes
three types of cancer cells with testosterone is used for hormonal therapy. The model is fitted to all
patient data. Then the parameters for an individual patient are obtained using Markov Chain Monte
Carlo (MCMC) from the distribution obtained from the parameter estimates of the full cohort. Using
the individual parameters, 1000 simulations are carried out to predict the biochemical failure, or the
onset of castrate resistance, timing for each patient. The model shows high accuracy in predicting the
timing of biochemical failure for ADT. In preference of a simpler model for predicting time to cancer
relapse in prostatectomized patients, Stura et al. [
72
] used a form of the generalized Von Bertalanffy
growth law to model prostate cancer growth. Using statistical analysis, they highlight the importance
of the growth parameter for PSA as a means to predict cancer relapse. Interestingly, they also note that
this growth parameter is larger in the case without androgen deprivation therapy as opposed to with
androgen deprivation therapy—a conclusion that is supported by other research.
3.5. Limitations of Psa as a Proxy for Tumor Size
Swanson et al. (2001)
. Mathematical models of prostate cancer rely on clinical and experimental
data for their validation. For clinical purposes, a model is often validated using the byproduct of
prostate cell activity, PSA. However, at the patient level, the correlation between PSA level and cancer
volume is questionable. Motivated by this observation, Swanson et al. [
73
] examined how well PSA
represents the prostate (both healthy and cancerous) volume.
dy
dt =βhVh+βcVc(t)ky(t)(12)
This simple differential equation assumes linear production rates of PSA based on the proportion
of cancerous/healthy cells coupled with a linear degradation rate.
y(t)
represents the serum PSA
with linear production rates
βh
and
βc
from the healthy and cancerous cancer cells, respectively.
The volume of healthy cells is assumed to be constant (
Vh
is constant), while the volume of cancerous
cells is governed by exponential growth (
Vc=V0eρt
). The parameter values are estimated using
human-derived mouse xenograft LuCaP 23 published in Ellis et al. [
30
]. Although there are limitations
in the biological basis of the model, it offers a valuable explanation for the abnormality in the PSA
representation of prostate tumor volume. They conclude this abnormality can be explained by the
ratio of the PSA degradation rate and tumor growth rate.
Vollmer et al. (2002), Vollmer and Humphrey (2003)
. On the other hand, by adjusting the
parameter values in Swanson et al. to account for differences in mouse and human, Kuang et al. [
15
]
(section 5.3.1) argue that while variation in cancer growth rates can explain some of the poor
correlation, the key parameters are the PSA production rates by cancer cells and PSA degradation
rates. This conclusion is further supported by the work of Vollmer and Humphrey in 2003 [
74
]. Due to
the limitation of PSA as a biomarker of prostate cancer, further investigations have been carried out to
explore alternative measurements to PSA in predicting cancer progression. For example, the work by
Vollmer et al. [
75
] in 2001 on the dynamics of PSA during watchful waiting show that PSA amplitude
and relative velocity are better predictors of cancer progression—a conclusion supported by clinical
studies [76].
Dimonte (2010), Dimonte et al. (2012)
. Existing modeling studies sometimes use additional
clinical measurement besides PSA as either input or validation for the model. One such example is
the work by Dimonte [
77
], where the author constructed a cell kinetics model to track prostate cancer
progression from diagnosis to final outcome. In this case, the patient’s Gleason score is used to obtain
the transition rate of cells. The model was used to study the author ’s and another patient’s data. In a
subsequent work, Dimonte and colleagues [
78
] simplify the model and use it to explain the variability
Appl. Sci. 2020,10, 2721 16 of 29
in the recurrence time of prostate cancer patients. They reach a similar conclusion to use PSA doubling
time to improve predictive power. There are various potential biomarkers for prostate cancer that are
being studied clinically. For a list of potential biomarkers for prostate cancer, the readers are referred
to [
79
]. Modeling work aiming to incorporate multiple biomarkers can potentially result in increasing
the accuracy of the model’s predictive power.
3.6. Other Approaches to Mathematical Modeling of Prostate Cancer
Lorenzo et al. (2019), Farhat et al. (2017), Liu et al. (2015)
. Aside from the aforementioned
work that focuses on hormonal therapy, there are other works that utilize mathematical models
to study other aspects of prostate cancer. For instance, Lorenzo et al. [
80
] use mathematical
models to study personalized treatment with radiation therapy, where they suggest several potential
prognostic measurements that can be obtained from the model. A recent study by Farhat et al. [
81
]
formulates a mathematical model of metastatic prostate cancer while taking into account the bone
micro-environment to investigate several possible therapeutic strategies. Both modeling approaches
are novel; however, their work lacks substantial validation to support their findings. Taking on a
more computational approach, Liu et al. [
82
] use a hybrid automaton model to find a personalized
therapeutic strategy.
Tanaka et al. (2010), Zazoua and Wang (2019), Baez (2017), Mizrak et al. (2020)
. Furthermore,
while existing modeling work for prostate cancer relies strongly on deterministic models, stochasticity
in the model has also been considered numerically by Tanaka et al. [
83
] and Cerasuolo et al. [
53
].
Furthermore, analytical consideration of stochastic modeling has been carried out by Zazoua and
Wang [
84
]. Their results show that the stochastic models can capture the statistical components of
the dynamics of PSA time serial data. Additionally, extensions using delay differential equations and
fractional differential equations for prostate cancer modeling have been carried out by Baez [
85
] and
Mizrak et al. [
45
], which show some improvement in data fitting. Although these approaches show
early promise, follow-up studies are needed to verify and expand their implications.
4. Mathematical Models in Clinical Settings
Many modeling efforts for prostate cancer diverge from one another, where previous results are
often not utilized. In this section, we discuss the limitations of current work in a clinical setting and
how to potentially address them. A summary of our discussion is presented in Table 2.
4.1. Real-Time Estimability
Mathematical models need to be validated against data prior to any application. In theoretical
studies, the complete set of patient data is often known to the researchers and is used for model
validation purposes. This makes sense because existing mathematical models often contain a large
number of parameters, so having more data gives a better chance of estimating the parameters. A study
of model identification methods by Hirata et al. [
86
] shows that 1.5 cycles, where a cycle refers to an on-
and off-treatment period in IAS, of data is the minimum requirement for model identification in most
cases. Yet in clinical settings, a model should be useful even when a limited amount of information is
available; however, this is often not the case. If just a few data points are used to parametrize a model,
its predictions would be highly unreliable [
49
]. On the other hand, if researchers wait for more data
as the treatment progresses, it may be too late, perhaps because the cancer becomes resistant and the
patient must switch to a different treatment. Furthermore, this issue affects even the classification
systems that are introduced along with some mathematical models [40,87].
One might assume that obtaining data more quickly, perhaps by way of a self-measuring device
or enticing the patient to visit the hospital more regularly, would resolve this issue; however, this
is deceptive. The shape of the patient’s data trend (PSA or androgen) is unique. Thus, forcing a
collection of data quickly does not expose completely the shape of this trend; instead, this would only
oversample certain sections leading to biases.
Appl. Sci. 2020,10, 2721 17 of 29
Table 2. Aspects of clinical applications: summary of findings and future exploration.
Aspects of Summary of Findings Future Exploration
Clinical Applications
Real-time estimability The estimation of parameters in
mathematical models often
require a large quantity of data.
However, the nature of data
collection in real-time means that
reliable estimation of parameters
for patients may not be possible at
the early stages of treatment.
Some parameters share similar
values across patients, while
others are more patient-specific.
This distinction should be studied
in detail. Utilizing multiple data
sets is another possibility to allow
early estimates of parameters.
Finally, parameter evolution can
be accounted for to address the
limitation of data availability.
Section 4.1
Uncertainty, sensitivity, Due to the large number of
parameters and the heavy reliance
on parameter fitting, model
predictions can be unreliable.
Furthermore, the issue of
parameter identifiability is often
ignored, which can lead to wildly
different predictions for a specific
patient.
Local sensitivity analysis and
uncertainty quantification should
be studied for each patient. Clear
links between each parameter and
its physical interpretation should
be established, which potentially
allows for laboratory
estimates/bounds to resolve
identifiability.
and identifiability
Section 4.2
Optimal schedule, Studies on optimal schedule and
treatment yield useful information
on how intermittent, adaptive,
and combination therapies should
be carried out. Patient
classification based on treatment
effectiveness can be done using
model parameters. However, both
aspects are affected heavily by the
estimability of the parameters and
the uncertainty in the model’s
forecasts.
The usefulness of optimal studies
and classification hinges on how
well the uncertainty in the model
can be quantified, which relates to
previous issues. In addition,
the objective of optimal studies
may be extended to include drugs’
properties, cost, and important
features of each treatment.
optimal treatment, and
patient classification
Section 4.3
There are two proposed ways to address the issue of data limitation in practice. West et al. [
61
]
noticed the issue of the high ratio of the number of parameters to data points, so they assumed that the
parameters in the model are the same across patients and the only difference is the initial population
of cancer cells. If this assumption holds, the number of parameters needed to be estimated for existing
models would decrease dramatically—allowing mathematical models to be useful in clinical studies.
However, this is generally not the case. Various studies on personalized medicine, or the idea that each
patient should have his own unique set of parameters, show that the parameters vary significantly
across patients [
88
]. This variability between patients makes intuitive sense. The characteristics of a
tumor and how the patient reacts to certain drugs should depend on the specificity of the patient’s
physiology and the tumor’s composition, metastatic site, etc. However, it may be possible to show that
their assumption holds for patients within a certain group, for example, patients who share certain
physical traits. Additionally, some parameters may in fact be relatively constant among patients. Of
course, if such categorization is possible, it could potentially be used to resolve this issue.
The second possibility is not as clear cut as the first; however, it is inspired by classical physical
studies. In this approach, we could consider models that contain mostly (if not only) parameters
that can be measured or estimated from laboratory testing. Thus, most of the parameters would not
come from data fitting but from actual physical testing of the patients. However, this method has
its own set of problems. Perhaps the biggest problem is the construction of the model. One cannot
mindlessly add all known mechanisms/factors into the model, There are many limitations by doing
so, such as increasing the complexity of the model, existence of unknown biological details, not well
tested biological details that may turn out to be incorrect in the future, etc. In order to create such a
model, the model formulation rests on the expertise in both mathematical modeling and biological
Appl. Sci. 2020,10, 2721 18 of 29
knowledge. One compromise using this approach is to carry out studies that focus on multi-level
fitting as done in previous work. This would allow for more accurate estimations of certain parameter
values. However, one must carefully isolate the parameters with respect to the data to avoid ambiguity
in the biological interpretations of the parameters.
An alternative approach to resolve the limitation of data in clinical settings is to account for the
evolution of parameters as the treatment progresses, instead of relying extensively on data fitting.
The rationale is that the parameters in dynamical models are often stand-ins for more complex
processes, which means their values represent the average values over a certain time interval of
the underlying processes. Hence, as time goes on, there would be changes accumulated from the
underlying processes of the parameters. The effect of the underlying process may be minimal in
many cases, but it could also be substantial. This is a complex subject that is akin to a time-scale
analysis. Sometimes, fast changing processes can be accounted for using an average as long as the
interval of time that the average is taken over is sufficiently long enough for the effect of the fast
changing process to be negligible. On the other hand, other processes take a long time for their effects
to be noticeable. When it comes to parameter estimations, there should be consideration for the time
interval used for data fitting. For example, the extended version of the BK model (Phan et al. [
50
])
contains a parameter that stands in for the maximal level of serum androgen. While they take this
as constant, the observed maximal level of androgen as treatment goes on tends to decrease with
each cycle, see Figure 6. The reason for this phenomenon is not known; however, it is suggested
to be due to an accumulated damage from drugs or the patient’s behavior changes over time after
learning of the cancer. Furthermore, Phan et al. [
50
] demonstrate that by focusing on the more recent
data using a time-weighted objective, significant improvements on the fitting and forecasts can be
obtained. This observation supports the implementation of parameter evolution within dynamical
models. The first study to directly incorporate this approach into the prostate cancer model treats the
parameter associated with the resistance of cancer as a dynamic variable [
47
]. This not only simplifies
the model but also results in better fitting compared to more complex models. From a computational
perspective, the evolution of parameters can be thought of as updating the parameter values as new
data becomes available by using Kalman filters as shown in Wu et al. [
49
]. The concept of parameter
evolution over time recently was incorporated in Brady et al. [
89
] and is explored conceptually in a
more general ecological context by Loladze [90].
Figure 6.
A representative example of time serial androgen data under intermittent androgen
suppression (IAS). The red circles represent the recorded data. The data is taken from
Bruchovsky et al. [
91
]. Note that the trend of the maximal level of androgen goes down over the
course of treatment.
Appl. Sci. 2020,10, 2721 19 of 29
4.2. Uncertainty, Identifiability, And Sensitivity
Bootstrapping and the ensemble Kalman filter are two standard methods that have been used to
study the uncertainty in prostate cancer models [
41
,
49
]. As shown in those works, existing models
often contain large uncertainty in their prediction, especially when longer forecasts are needed to
make a decision on optimal schedule, see Figure 7a. Other variance-based uncertainty quantification
methods can also be applied for similar purposes, such as the work by Elishmereni et al. [
71
]. In
the case of sensitivity analysis, a standard approach is to vary the parameters one by one by some
percentage and evaluate the corresponding changes at a fixed time point [
47
]. This method can also
be extended to study the sensitivity over the entire treatment [
50
,
92
]. The advantage of studying
the sensitivity of a parameter over time is evident in intermittent and adaptive therapy. Since there
are orders of magnitudes difference for certain variables (such as PSA level) for different phases of
the treatment, the sensitivity of the parameter is also affected, see Figure 8. Thus, having a better
understanding of how the sensitivity of each parameter changes as the treatment progresses can
provide a better tool to optimize treatment. Additionally, sampling-based methods to account for
simultaneous effects of all parameters is also possible; however, this should be done in a relatively
small range after the parameters have been identified for a specific patient.
There are a variety of methods to estimate the parameters of a model. However, recently, when
the identifiability of some prostate cancer models were examined, Wu et al. [
49
] show that these
models are not practically identifiable. They further show that for an unidentifiable model, different
sets of parameters may yield an indistinguishable fitting but result in vastly different forecasting, see
Figure 7b. Such a result is troublesome as it could undermine the applicability of mathematical models
in a clinical setting. To take into account these issues, some researchers rely on the 2n + 1 law by
Sontag [
62
]. However, this law requires models to be structurally identifiable—a condition that is
often not examined when applying the law. Alternatively, if one carries out the sensitivity analysis of a
model before hand, then information revealing the least sensitive parameters could be used to enhance
the identifiability of the model. The issue of identifiability ties back to the complexity of the model and
the limited data in clinical settings. It essentially requires a sufficient number of data points and data
types, to uniquely determine the parameters of a model. In Wu et al. [
49
], the authors suggest the use
of an observer experiment to quantify the minimum required data that would allow the model to be
identifiable. The suggested technique is to use the Fisher information matrix because of its ability to test
different combinations of data sets easily. While the process is tedious, such information can provide
insight into the types of data needed for a model to be useful. Additionally, if the functional forms of
some parameters are available (either through study of parameter evolution or study of the nonlinear
relationships between parameters), it may directly address the issue of model identifiability [93].
4.3. Optimal Schedule and Patient Classification
Due to the lack of a gold standard when it comes to determining a treatment schedule for IAS,
its full benefits may not be realized [
20
]. Furthermore, in some instances, patients may benefit more
from CAS. To tackle this issue, Hirata and colleagues studied patient classification based on whether a
patient would benefit more or less from IAS as compared to CAS [
40
]. Another classification system
was introduced by Morken et al. [
87
], where the type of treatment resistance is studied based on cell
death rate analysis. In treating prostate cancer, mathematical models have been used to study potential
optimal treatment schedules that may give the best chance for patients. While various studies have
been done on this topic, the goal of an optimal schedule has varied among them. For instance, Suzuki
and Aihara [
94
] choose the objective to be the minimization of the amount of time a patient is on
treatment while still keeping IAS effective in controlling the cancer progression. Hirata et al. [
42
]
instead focus on delaying the relapse of cancer as long as possible, and Cunningham et al. [
95
] study
three potential objectives: minimizing average tumor volume, tumor mass variance, or average density
of androgen independent cells. While these are obvious objectives to be minimized for prostate cancer
Appl. Sci. 2020,10, 2721 20 of 29
(or cancer in general), other creative alternatives such as minimizing cancer activity at any given time,
or PSA doubling rate, may provide better control of cancer.
Figure 7.
(
a
) Figure reproduced from Hirata et al. [
41
], with permission, which is distributed
under a Creative Commons Attribution (CC BY) license. The spread denotes the 80% confidence
interval using the bootstrap method, where the red crosses represent a patient’s data. (
b
) Figure
reproduced from Wu et al. [
49
] with permission, which is distributed under a Creative Commons
Attribution (CC BY) license. In the fitting portion, the fittings using five different sets of parameters
are nearly indistinguishable. However, in the forecasting portion, only one set of parameters provides
accurate forecasting.
Figure 8.
Example of varying sensitivity for parameter during IAS-figure reproduced from
Voth et al. [92] with permission, which is distributed under a Creative Commons Attribution (CC BY)
license. (
a
) The sensitivity index of the mutation parameter
m1
with respect to androgen independent
cancer cells
x2
for intermittent treatment. (
b
) The sensitivity index of the mutation parameter
m1
with
respect to androgen dependent cancer cells
x1
for intermittent treatment. Note that while the dynamical
behaviors of the sensitivity indexes can be non-trivial, they should be explainable using biological
understanding of the system.
The study of drug combinations can be considered as a subset of the optimal schedule. Various
studies have been done on finding the best combinations of drugs for certain cases [
52
,
61
]. While these
Appl. Sci. 2020,10, 2721 21 of 29
studies hold promises, their implications hinge on how well the mathematical models represent the
underlying biological system, which means drawing definitive conclusions is difficult due to the
aforementioned problems with model validation. Furthermore, finding an optimal time (free terminal
time) for a treatment may aid the design of a treatment, especially in the case of adaptive therapy.
A comprehensive collection of tutorials for optimal control application in bio-sciences are presented by
Lenhart and Workman [96].
5. Data, Parameter Ranges, and a Framework for Clinical Application
5.1. Data
Various relevant data sets exist, ranging from experimental studies of cancer cells to clinical
studies of patients undergoing hormonal therapy. In this section, we summarize some of these data.
Perhaps the most used data set for prostate cancer model validation at the patient level comes from
a clinical trial at Vancouver Prostate Center. The study admitted patients who showed a rising serum
PSA level after undergoing radiotherapy without evidence of distant metastasis or being previously
subjected to hormonal therapy, with the exception of less than three months of neoadjuvant hormonal
therapy [
91
]. Additionally, all patients exhibit high serum PSA levels (
6
µ
g/L) prior to the study.
Another set of data comes from a clinical study [
46
], where three stage C and four stage D prostate
cancer patients were treated with intermittent hormonal therapy for 21 to 47 months (two to four
cycles). All patients show decreasing PSA levels during the course of the study. Moreover, there are
various clinical studies that have been used in mathematical modeling. For instance, patients’ data
sets were used in the study by Draghi et al. [
88
], which can also be extracted directly from their paper.
Another study [
41
] also utilized additional data sets from different clinical studies in the United States
and Japan [
97
100
]. Additionally, we would like to point out a clinical trial [
20
,
61
] that was the first
for prostate cancer that utilizes adaptive therapy to treat metastatic castrate-resistant prostate cancer.
However, as this is ongoing, the data may not be readily available. For a review of clinical studies that
focus more on the statistical correlation of different variables for prostate cancer, we refer the readers
to the work by Dimonte [77].
5.2. Parameter Ranges
One of the main issues that is common to existing work is determining the values of model
parameters. Here, we also provide references with regard to some important parameters.
Cancer maximum proliferation rates: by making the assumption that measurements of cell growth
and death for hormonally untreated patients to be that of AD cells, the study of tumor doubling
time at various stages in Berges et al. [
101
] can be used to suggest that the range for the proliferation
rates of AD cells is (0.004–0.081)
[day]1
. Similarly, we can find a range of (0.001–0.046)
[day]1
for
AI cells. Note that these ranges fall within the expectation that AD cells out-compete AI cells in
an androgen-rich environment. Additionally, growth rates for specific cell types can be obtained
in vitro [
20
,
61
]. However, appropriate scaling or fitting is necessary to account for differences in tumor
environment. Since the environment in a laboratory is ideal for the growth of cancer cells, their growth
rate should be lower in practice. Thus, we consider these ranges to be for maximum proliferation rates.
Cancer cell death rates: similar to cancer growth rates, we estimate the ranges of cancer death rates
for AD and AI cells to be (0.001–0.0525)[day]1and (0.015–0.0775)[day]1, respectively.
Cancer cell maximal transformation rate: Robust estimate of cancer transformation rates
from experimental data is lacking. However, numerical experiments and fitting show
that (10
5
–10
4
)
[day]1
is an appropriate range for the maximal transformation rate [
24
,
33
].
The transformation rate encompasses the mutation rate. While there are several estimates of the
PCa cell mutation rate, mathematical models often use transformation rates instead of the more specific
mutation rate.
Appl. Sci. 2020,10, 2721 22 of 29
PSA clearance rate and production rates by healthy and cancerous cells: initial estimates [
73
] were
for human-derived mouse xenograft sublines LuCaP 23.1, 23.8, and 23.12. These values were later
extrapolated for human prostate cancer in Section 5.3.1 [
15
] using the study of Berges et al. [
101
]
and Vesely et al. [
102
]. The PSA production rate from healthy cells is (2.870
×
10
5
–1.354
×
10
4
)
[ng][ml]1[mm]3[day]1
. On the other hand, estimating PSA production rate is perhaps best
carried out using specific cell sublines. For LuCaP 23.1, 23.8, and 23.12, this rate is estimated to be
1.7210, 2.1841, and 6.9722
[ng][ml]1[mm]3[day]1
, respectively [
73
]. Additionally, the PSA clearance
rate is within the range of (0.1754–0.4030)[day]1[103]. We summarize these in Table 3.
Table 3.
Ranges for some commonly used parameter values in mathematical models for prostate
cancers. Note that for more accurate ranges, the specificity of the situation needs to be taken into
account, for example, sublines of cancer cells.
Description Range Unit Source
Max proliferation rate (AD) 4.00×103- 8.10×102[day]1[101]
Max proliferation rate (AI) 1.00×103- 4.60×102[day]1[101]
Death rate (AD) 1.00×103- 5.25×102[day]1[101]
Death rate (AI) 1.50×102- 7.75×102[day]1[101]
Max transformation rate 1.00 ×105- 1.00 ×104[day]1[24]
PSA clearance rate 1.75×101- 4.03×101[day]1[103]
PSA production rate (healthy) 2.87 ×105- 1.35 ×104[ng][ml]1[mm]3[day]1[15]
PSA production rate (cancer) 1.72×100- 6.97×100[ng][ml]1[mm]3[day]1[73]
Some parameters are difficult to estimate robustly, for instance, the competition rates of cancer
cells. In the case of difficulty in estimating certain parameters from laboratory data, researchers also
rely on experts’ opinions or fix the parameters in an ad hoc way [
20
,
43
]. Alternatively, the model can
also be derived from physical laws (e.g., conservation laws), which introduce competition rates without
adding additional explicit parameters for competition between cells [
47
,
48
,
50
]. For readers interested
in how some parameters are estimated in existing models, we refer to studies [
16
,
51
], where binding
rates are estimated from multi-level processes.
6. Conclusions
Mathematical models play a vital role in the study of prostate cancer dynamics. Over the past
two decades, various models have been developed to examine different aspects of prostate cancer,
including treatment options and schedules. In this review, we collect and synthesize the results of
existing studies to share conclusions that are well agreed upon and raise questions that are in need of
further investigation.
The study of prostate cancer at the multi-cell level gives insights on the treatment efficacy with
respect to the competition among the cells. Specifically, the consensus among existing work is that IAS
is superior to CAS at delaying the relapse of metastasized cancer, if AD cells have some competitive
advantages over AI cells in an androgen-rich environment. While some studies have shown that AD
cells can indeed out-compete AI cells, there is no extensive evidence that this is true in general. Instead
of suggesting that the competition advantage is an intrinsic property of AD cells, this property needs
to be examined on a case-by-case basis because many factors such as the physiology of the cancer or
the specific cancer phenotype can affect the relative competition between cancer cells. Additionally, we
find that lowering the androgen level as a preventive measure of prostate cancer may result in selection
for aggressive androgen independent phenotypes, if the tumor forms. On the other hand, studies
that focus on tools used to track prostate cancer progression, such as PSA, show that while PSA is an
overall good biomarker for cancer growth, it can display poor correlation in many cases due to the
patient’s specific PSA production rates by cancer cells and PSA degradation rates. Instead, the relative
velocity of PSA appears to be a better alternative for monitoring tumor progression. Furthermore,
Appl. Sci. 2020,10, 2721 23 of 29
to tackle the problem of identifiability of models, multiple biomarkers should be used to enhance the
accuracy of models.
Thus far, there exists mathematical models for almost all clinical stages of prostate cancer. Some
models are even flexible enough to be used across many stages. However, the application of each
model is still limited to certain stages. Having a comprehensive model that can describe tumor
development as a whole may be useful for clinical practice. Treatment effectiveness is one of the
most examined issues using mathematical models with a focus on hormonal therapy. This is often
coupled with the study of optimal schedule, especially for IAS and adaptive therapies. While the
studies so far show potential, they lack key components for use in clinical settings. First, the drugs
being considered in a mathematical model are often generalized—meaning the specific properties of
the drug are not accounted for. Secondly, for combinations of drugs, the actual dosages and toxicity are
not considered. This is perhaps due to the fact that previous studies are dominated by the traditional
way of administering hormonal treatment (fixed dosage). However, as adaptive therapy is becoming
more prominent, the actual dosages and toxicity of multi-therapy should be accounted for in modeling
efforts. Furthermore, in the case of optimal schedule, the best objective is often not clear, which is
perhaps due to the lack of a credible set of biomarkers that can track cancer growth.
Perhaps the most urgent issue with existing efforts in modeling prostate cancer is the validation
of mathematical models. In the early stage of development, models are validated using observations
of clinical trends, which can only give qualitative descriptions of the process. As the field progresses,
there have been more attempts to validate models against clinical data in a quantitative way. While
this is an improvement, efforts are lacking and parameter identification becomes a big problem for
the validity of many mathematical models. This issue is further evident when mathematical models
attempt to reach clinical application because of the lack of retrospective data sets. The identification of
model parameters is coupled with issues of uncertainty, sensitivity, and stochasticity in making any
conclusions or forecasts from mathematical models, while useful in providing an indication of likely
scenarios, problematic. As many of the existing models are phenomenological, the issue of model
identification is further emphasized. Instead, we suggest the use of mechanistic models, which tend
to be more complex, but parameters can often be linked to experimental data. The formulation of
mechanistic models should be based on well-tested structures from phenomenological models, while
acknowledging the biological details. In this way, previous results can aid future studies.
As optimal treatments require model predictions far into the future, it severely limits the usage
of mathematical models due to increasing uncertainty, especially for long term treatments such as
hormonal therapy. To address this issue, we observe that parameters in dynamical systems are often
an average of an underlying process over a period of time (e.g., the period of time used in parameter
estimation). Thus, the parameters in dynamical models should be updated regularly. There are
several means to incorporate parameter evolution in dynamical models, such as applying the Kalman
filter [
49
], incorporating the dynamics of the parameters into the model [
47
], using weighted fitting
to emphasize more recent data [
50
], or studying the trends of parameter evolution [
89
]. If the trends
of certain parameters can be established for a specific treatment, it can allow for better prediction
without higher certainty. An in-depth exploration of parameter evolution could significantly bridge
the gap between mathematical models and clinical practices. Furthermore, it could provide a new set
of interesting dynamical questions [104].
In summary, mathematical modeling efforts for prostate cancer so far have shown that (1) IAS or
adaptive therapy is better than CAS at delaying cancer relapse, if AD cells hold competitive advantages
over AI cells, and (2) cancer progression is better associated with the relative velocity of PSA due to
variation in PSA production by the tumor and its degradation rate. Moving forward, to close the gap
between modeling work and clinical application, the following questions need to be addressed: (1) how
to obtain the individualized model parameters without extensive data, (2) what are the most important
parameters that affect cancer growth and how to use this knowledge in clinical applications, (3) what is
the most appropriate objective for an optimal treatment/schedule study, and (4) how to best quantify
Appl. Sci. 2020,10, 2721 24 of 29
the uncertainty in model prediction? Many physicians have found value in mathematical models for
improving our understanding of prostate cancer progression and creating better treatment for patients.
However, many issues stand in the way of a complete theory and a clinically applicable model. In this
paper, we have highlighted the general consensus among existing studies and limitations that should
be addressed in the future. We hope that this review can aid future attempts at studying prostate
cancer. Moreover, we note that while the findings and issues emphasized in this paper focus on
various aspects of mathematical modeling of prostate cancer, they are also relevant to other cancers
and their respective treatments. For instance, breast cancer shares many similarities with prostate
cancer. The issue of modeling treatments in the case of co-occurring cancers is not well explored and
we are not aware of any attempt to model treatment of prostate cancer in the presence of another
cancer. However, since prostate cancer is more prevalent in older men (55+), when cancer is more
likely and detectable, modeling attempts in the case of co-occurrence of cancers can be beneficial. Thus,
we hope this review can provide insights into future modeling work in other areas of mathematical
oncology.
Author Contributions:
Conceptualization, T.P. and Y.K.; investigation, T.P., S.M.C., A.H.B., C.C.M., E.J.K., and Y.K.;
data curation, T.P.; writing—original draft preparation, T.P.; writing—review and editing, T.P., S.M.C., A.H.B.,
C.C.M., E.J.K., and Y.K.; visualization, T.P.; supervision, E.J.K. and Y.K.; project administration, T.P.; funding
acquisition, E.J.K. and Y.K. All authors have read and agree to the published version of the manuscript.
Funding:
This research was partially supported by a grant from Arizona Biomedical Research Commission.
Research of YK is partially supported by NSF grants DMS-1615879, DEB-1930728 and an NIH grant
5R01GM131405-02.
Acknowledgments:
The authors would like to thank Kyle Nguyen, Trevor Reckell, and Penny Wu for
helpful discussions.
Conflicts of Interest: The authors declare no conflict of interest.
Abbreviations
The following abbreviations are used in this manuscript:
PCa Prostate cancer
AR Androgen receptors
ARE Androgen Response Elements
DHT 5al pha-Dihydrotestosterone
PSA Prostate-specific antigen
CAS Continuous androgen suppression (therapy)
IAS Intermittent androgen suppression (therapy)
AD Androgen-dependent (cancer cells)
AI Androgen-independent (cancer cells)
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... If this unique set of parameters can be determined for a particular patient, then the model is presumed to be useful in predicting the most appropriate treatment for that patient. Thus, the concept of mathematical modeling coincides with the central idea behind precision medicine, where treatment should be formulated from the characteristics of each individual [18,19]. ...
... Many mathematical models for prostate cancer have been developed in the past two decades [1,18,47] with many recent studies focused on immuno-and chemo-treatments of prostate cancer [48][49][50][51][52][53][54][55][56]. Here, we divert from this trend and instead use two simpler mechanistic models to demonstrate the concept of model identifiability in practice. ...
... Mathematical models not only contribute to the foundation of cancer theory but can also be integrated to provide a better prognostic tool for clinicians in clinical settings. For example, one may apply mathematical models to better understand cancer progression dynamics and make predictions of treatment outcomes based on a patient's characteristics [18]. Yet, the issue of practical identifiability remains a major obstacle to realizing the clinical potential of mathematical models. ...
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Mathematical models are a core component in the foundation of cancer theory and have been developed as clinical tools in precision medicine. Modeling studies for clinical applications often assume an individual’s characteristics can be represented as parameters in a model and are used to explain, predict, and optimize treatment outcomes. However, this approach relies on the identifiability of the underlying mathematical models. In this study, we build on the framework of an observing-system simulation experiment to study the identifiability of several models of cancer growth, focusing on the prognostic parameters of each model. Our results demonstrate that the frequency of data collection, the types of data, such as cancer proxy, and the accuracy of measurements all play crucial roles in determining the identifiability of the model. We also found that highly accurate data can allow for reasonably accurate estimates of some parameters, which may be the key to achieving model identifiability in practice. As more complex models required more data for identification, our results support the idea of using models with a clear mechanism that tracks disease progression in clinical settings. For such a model, the subset of model parameters associated with disease progression naturally minimizes the required data for model identifiability.
... Treatment for prostate cancer has advanced considerably over the past several decades [2], but treatment resistance remains a significant aspects of prostate cancer and its treatments [12,13,16,[23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41]. Most of these results have been reviewed and synthesized previously [5,42,43]. ...
... In general, existing prostate cancer models are built on the assumption that the rate of PSA production is a linear function of the amount of tumor cells [5]. However, PSA production is intrinsically linked to cancer proliferation via the same transcription factor: bound androgen receptors ( Figure 1a). ...
... Table 1 contains a summary of model parameters and ranges, and additional information on the model formulation and its parameters can be found in the supplemental section. [5,26]. The range column indicates the upper and lower bounds within which an error-minimizing function may establish an optimal value with respect to a concrete set of patient data. ...
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Prostate cancer is a serious public health concern in the United States. The primary obstacle to effective long-term management for prostate cancer patients is the eventual development of treatment resistance. Due to the uniquely chaotic nature of the neoplastic genome, it is difficult to determine the evolution of tumor composition over the course of treatment. Hence, a drug is often applied continuously past the point of effectiveness, thereby losing any potential treatment combination with that drug permanently to resistance. If a clinician is aware of the timing of resistance to a particular drug, then they may have a crucial opportunity to adjust the treatment to retain the drug’s usefulness in a potential treatment combination or strategy. In this study, we investigate new methods of predicting treatment failure due to treatment resistance using a novel mechanistic model built on an evolutionary interpretation of Droop cell quota theory. We analyze our proposed methods using patient PSA and androgen data from a clinical trial of intermittent treatment with androgen deprivation therapy. Our results produce two indicators of treatment failure. The first indicator, proposed from the evolutionary nature of the cancer population, is calculated using our mathematical model with a predictive accuracy of 87.3% (sensitivity: 96.1%, specificity: 65%). The second indicator, conjectured from the implication of the first indicator, is calculated directly from serum androgen and PSA data with a predictive accuracy of 88.7% (sensitivity: 90.2%, specificity: 85%). Our results demonstrate the potential and feasibility of using an evolutionary tumor dynamics model in combination with the appropriate data to aid in the adaptive management of prostate cancer.
... Furthermore, applying any therapy to the point of failure only serves to further fortify the existing resistance. If it were possible to identify an incipient resistance then a clinician would have an opportunity to change treatment strategy to potentially reach a more favorable outcome of the overall regimen (15,23,24). ...
... Since then, there has been an array of models developed to study different aspects of prostate cancer and its treatments (5,(8)(9)(10)(11)(12)(13)17,25,(32)(33)(34)(35)(36)(37)(38)(39)(40)(41)(42)(43)50). Most of these results have been reviewed and synthesized previously (15,44,45). ...
... In general, existing prostate cancer models are built on the assumption that the rate of PSA production is a linear function of the amount of tumor cells (15). However, PSA production is intrinsically linked to cancer proliferation via the same transcription factor: bound androgen receptors (Fig 1-a). ...
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Prostate Cancer is a serious public health concern in the United States. The primary obstacle to effective long-term management for prostate cancer patients is the eventual development of treatment resistance. Due to the uniquely chaotic nature of the neoplastic genome, it is difficult to determine the evolution of tumor composition over the course of treatment. Hence, a drug is often applied continuously past the point of effectiveness, thereby losing any potential treatment combination with that drug permanently to resistance. If a clinician is aware of the timing of resistance to a particular drug, then they may have a crucial opportunity to adjust the treatment to retain the drug usefulness in potential treatment combination or strategy. In this study, we investigate new methods of predicting treatment failure due to treatment resistance using a novel mechanistic model built on an evolutionary interpretation of Droop cell quota theory. We analyze our proposed methods using patient PSA and androgen data from a clinical trial of intermittent treatment with androgen deprivation therapy. Our results produce two indicators of treatment failure. The first indicator is calculated using our mathematical model with a predictive accuracy of 87.3% (sensitivity: 96.1%, specificity: 65%). The second indicator is calculated directly from serum androgen and PSA data with a predictive accuracy of 88.7% (sensitivity: 90.2%, specificity: 85%). The high sensitivity of the first indicator and the high specificity of the second indicator means they can complement one another in clinical settings. Our results demonstrate the potential and feasibility of using evolutionary tumor dynamics models in combination with the appropriate data to aid in the adaptive management of prostate cancer.
... We have employed scaled versions of the parameters' values, with a conversion value of ten to the fifth power. All of the parameters have no dimensions (due to the fact that their values are scaled using various conversion factors) [42] [43]. We have used the clinically relevant initial values and desired values, which are as follows: ...
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... Monotherapy with anti-CTLA-4 or anti-PD-1 in clinical trials did not improve tumor growth in most cases. Mathematical models of prostate cancer that consider treatment with androgen deprivation are reviewed in a number of papers (e.g., [50][51][52]); models with intermittent androgen ablation strategies aimed to reduce androgen resistance were developed in [50,53], where additional references are given. ...
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Metastatic castration resistant prostate cancer (mCRPC) is commonly treated by androgen deprivation therapy (ADT) in combination with chemotherapy. Immune therapy by checkpoint inhibitors, has become a powerful new tool in the treatment of melanoma and lung cancer, and it is currently being used in clinical trials in other cancers, including mCRPC. However, so far, clinical trials with PD-1 and CTLA-4 inhibitors have been disappointing. In the present paper we develop a mathematical model to assess the efficacy of any combination of ADT with cancer vaccine, PD-1 inhibitor, and CTLA-4 inhibitor. The model is represented by a system of partial differential equations (PDEs) for cells, cytokines and drugs whose density/concentration evolves in time within the tumor. Efficacy of treatment is determined by the reduction in tumor volume at the endpoint of treatment. In mice experiments with ADT and various combinations of PD-1 and CTLA-4 inhibitors, tumor volume at day 30 was always larger than the initial tumor. Our model, however, shows that we can decrease tumor volume with large enough dose; for example, with 10 fold increase in the dose of anti-PD-1, initial tumor volume will decrease by 60%. Although the treatment with ADT in combination with PD-1 inhibitor or CTLA-4 inhibitor has been disappointing in clinical trials, our simulations suggest that, disregarding negative effects, combinations of ADT with checkpoint inhibitors can be effective in reducing tumor volume if larger doses are used. This points to the need for determining the optimal combination and amounts of dose for individual patients.
... The goal is to detect the single model (or the group of models) that best represent the information in the considered dataset and, therefore, if possible, the most promising biological frame representing them. A general and historical review of the available prostate cancer models can be found elsewhere (Phan et al. 2020). ...
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The prostate is an exocrine gland of the male reproductive system dependent on androgens (testosterone and dihydrotestosterone) for development and maintenance. First-line therapy for prostate cancer includes androgen deprivation therapy (ADT), depriving both the normal and malignant prostate cells of androgens required for proliferation and survival. A significant problem with continuous ADT at the maximum tolerable dose is the insurgence of cancer cell resistance. In recent years, intermittent ADT has been proposed as an alternative to continuous ADT, limiting toxicities and delaying time-to-progression. Several mathematical models with different biological resistance mechanisms have been considered to simulate intermittent ADT response dynamics. We present a comparison between 13 of these intermittent dynamical models and assess their ability to describe prostate-specific antigen (PSA) dynamics. The models are calibrated to longitudinal PSA data from the Canadian Prospective Phase II Trial of intermittent ADT for locally advanced prostate cancer. We perform Bayesian inference and model analysis over the models’ space of parameters on- and off-treatment to determine each model’s strength and weakness in describing the patient-specific PSA dynamics. Additionally, we carry out a classical Bayesian model comparison on the models’ evidence to determine the models with the highest likelihood to simulate the clinically observed dynamics. Our analysis identifies several models with critical abilities to disentangle between relapsing and not relapsing patients, together with parameter intervals where the critical points’ basin of attraction might be exploited for clinical purposes. Finally, within the Bayesian model comparison framework, we identify the most compelling models in the description of the clinical data.
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Prostate cancer results from enlargement of the prostate gland in the male reproductive system. It results in the leakage of a protein called prostate-specific antigen into the primary bloodstream. Several treatments are available for prostate cancer, with androgen deprivation, or hormone therapy, being one of them. This therapy uses drugs that destroy the hormones that accelerate the growth of cancer cells. Numerous mathematical models exist in the literature that consider cancer dynamics under hormone therapy. However, in most of the studies, the cancer cells develop resistance to the therapy, and therefore, there is a need for proper drug scheduling. To achieve this purpose, four different nonlinear controllers (sliding-mode control, super-twisting, integral sliding mode control, and integral-based super-twisting control) are designed using the hormone therapy model. The objective is to bring down the androgen levels that help in the proliferation of cancer cells. The suppression of androgen hormones reduces prostate cancer cells to zero. The asymptotic stability of the controllers is analysed using the Lyapunov stability criteria. The derived control laws provide researchers with information for accurate drug scheduling. All the results are validated using the MATLAB/SIMULINK platform.
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In this paper we present a new version of a mathematical model of Elishmereni et al. describing androgen deprivation therapy (ADT) for hormone sensitive prostate cancer patients (HSPC). We first focus on the detail description of the model, and then we present mathematical analysis of the proposed model, starting from the simplified model without resistance and ending on the full model with two resistance mechanisms present. We make a step towards personalization proposing an underlying tumor growth law base on a cohort of patients from Mayo hospital. We conclude that the model is able to reflect reality, that is in clinical scenarios the level of testosterone in HSPC patients inevitably rises leading to the failure of ADT.
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The past two decades have seen the development of numerous mathematical models to study various aspects of prostate cancer in clinical settings. These models often contain large sets of parameters and rely on limited data sets for validation. The quantitative analysis of the dynamics of prostate cancer under treatment may be hindered by the lack of identifiability of the parameters from the available data, which limits the predictive ability of the model. Using three ordinary differential equation models as case studies, we carry out a numerical investigation of the identifiability and uncertainty quantification of the model parameters. In most cases, the parameters are not identifiable from time series of prostate-specific antigen, which is used as a clinical proxy for tumor progression. It may not be possible to define a finite confidence bound on an unidentifiable parameter, and the relative uncertainties in even identifiable parameters may be large in some cases. The Fisher information matrix may be used to determine identifiable parameter subsets for a given model. The use of biological constraints and additional types of measurements, should they become available, may reduce these uncertainties. Ensemble Kalman filtering may provide clinically useful, short-term predictions of patient outcomes from imperfect models, though care must be taken when estimating “patient-specific” parameters. Our results demonstrate the importance of parameter identifiability in the validation and predictive ability of mathematical models of prostate tumor treatment. Observing-system simulation experiments, widely used in meteorology, may prove useful in the development of biomathematical models intended for future clinical application.
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