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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 3.1–3.4

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)

Dp∂p

∂r(R,t)−u(R,t)p(R,t) = 0, Dq∂q

∂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(1−u)(3)

dxi

dt =αipi(a)xi−βiqi(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 =µm1−qi

Qixi

| {z }

growth

−dixi

|{z}

death

−λi(Qi)xi

| {z }

cell transformation

+λj(Qj)xj

| {z }

cell transformation

, (7)

dQi

dt =νmqm−Q

qm−qi

A

A+νh

| {z }

androgen inﬂux to cells

−µm(Qi−qi)

| {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 1−∑3

j=1aij xj

Ki!(10)

and

dP

dt =

3

∑

i=1

xi−0.5P. (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) 4.00×10−3- 8.10×10−2[day]−1[101]

Max proliferation rate (AI) 1.00×10−3- 4.60×10−2[day]−1[101]

Death rate (AD) 1.00×10−3- 5.25×10−2[day]−1[101]

Death rate (AI) 1.50×10−2- 7.75×10−2[day]−1[101]

Max transformation rate 1.00 ×10−5- 1.00 ×10−4[day]−1[24]

PSA clearance rate 1.75×10−1- 4.03×10−1[day]−1[103]

PSA production rate (healthy) 2.87 ×10−5- 1.35 ×10−4[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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