A MATHEMATICAL MODEL OF COMPETITION FOR NUTRIENTS BETWEEN

MALIGNANT AND SECONDARY TUMORS

by

Jonathan Alexander Winkler

An Honors Thesis Presented in Partial Fulfillment of Honors College Graduation Requirements (Department of Mathematics and Statistics)

ARIZONA STATE UNIVERSITY

MAY 2000

Abstract

Metastasis is a primary reason for cancer treatment failure. While a great deal of

experimental work has been directed toward metastasis, much of the basic theory remains

obscure. In particular, the question of how tumor cells survive to initiate secondary

growth at a site distant from their origin despite competition for nutrients from cells in

the primary tumor is still open. This thesis addresses this question using mathematical

modeling. I developed a system of four ordinary differential equations that expresses the

competition between a primary tumor and one of its metastatic offspring. In particular,

the model addresses how a secondary tumor can survive with the primary tumor in an

environment where nutrients are limited. The masses of both a primary and secondary

tumor are modeled along with the dynamics of two nutrients: carbon and nitrogen. The

parameters of the model are estimated from biological data whenever possible. The

model predicts that one characteristic of importance with regards to tumor growth and

competition is a given tumor’s optimal carbon to nitrogen (C:N) nutrient ratio. In

general, the outcome of competition between a primary and secondary tumor can be

predicted based on which tumor has a higher optimal C:N ratio, meaning the tumor can

grow with less nitrogen per unit carbon. Comparisons of the results of the model are

made to results of other models that consider nutrient dynamics, and experiments are

suggested which could test the model’s predictions.

I. Introduction

The National Cancer Institute (NCI) estimates that there were over 1.3 million

new incidences of cancer in the United States in 2004 [National Cancer Institute 2004].

The NCI also estimates that over 500,000 Americans died from cancer that year. Despite

these statistics, it is important to note that cancer incidence and death rates actually

decreased slightly between 1991 and 2001, primarily because of improvements in the

general understanding of this complex disease and the corresponding improvements in

healthcare [National Cancer Institute 2004]. Indeed, the development of new

biotechnologies such as microarray analysis and the completion of the Human Genome

Project have enabled a formal characterization of the traits required for cancer

development and progression.

According to the most widely accepted theory, there are six unique traits that a

cell must acquire in order to become cancerous [Hanahan and Weinberg 2000].

Specifically, malignant cells must produce their own growth signals, become insensitive

to antigrowth signaling, evade programmed cell death, sustain angiogenesis, acquire the

ability to replicate an unlimited number of times, and invade neighboring tissues and

organs to initiate secondary growth. While different combinations of these traits, called

the “Hallmarks of Cancer” by Hanahan and Weinberg, are sufficient for abnormal

cellular behavior, it is generally believed that all six are required to induce full-blown

malignancy.

In particular, metastasis, or the initiation of secondary growth, is known to be the

main reason for cancer treatment failure, primarily because treatment, often involving

invasive surgery followed by radiation and chemotherapy, becomes much less effective

after tumor cells have spread to secondary sites. One main difficulty is that it is often

difficult to determine exactly where a given metastatic tumor cell has moved to in the

body. Although studies have characterized the most common metastatic sites for various

malignant tumors [Hanahan and Weinberg 2000, Nicolson 1986, Ziljstra et al. 2002],

mysteries still remain. For example, while it is known that tumor cells most frequently

metastasize to the organs first encountered while traveling through the blood stream,

many examples exist of tumor cells disseminating to organs distant from the initial site

[Nicolson 1986]. One example of this phenomenon is the frequent spread of breast

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adenocarcinoma cells to bone, brain, and adrenal sites, along with the expected

dissemination to sites in the lung [Nicolson 1986]. Indeed, this example also indicates

how remarkably different the environment of a secondary site can be compared to the

environment of the primary site. Despite the large quantity of experimental work in this

area, exactly how a metastasizing cell survives the circulation and initiates metastatic

growth in a completely new environment is still an open question. Similarly, how

metastasizing cells are capable of competing for nutrients with cells from the primary

tumor is also unknown.

What is known is that a metastasizing cell must go through a series of steps in

order to break away from the primary tumor and spread to a new site in the body [Liotta

1985, Saidel et al. 1976, Ziljstra et al. 2002]. In particular, a cell must leave the primary

tumor mass, invade a nearby blood or lymph vessel, avoid immune attack in the

circulation, adhere to the capillary endothelial wall at a new site, exit the circulation and

enter the interstitium, and finally initiate secondary growth [Liotta 1985, Saidel et al.

1976]. Each of these steps requires its own complex series of molecular events. For

example, for a cell to break away from the primary tumor mass, it must somehow

dissociate itself from other cells and the extracellular matrix (ECM). [Hanahan and

Weinberg 2000]. This process necessarily involves the disruption of cell-cell adhesion

proteins and alterations to molecules such as integrins, which bind cells to the ECM

[Hanahan and Weinberg 2000]. Such changes might occur through activation of certain

proteases that cleave cell-ECM and cell-cell binding proteins [Hanahan and Weinberg

2000, Liotta 1985]. As a second example, penetration of the target organ also requires

activation of specific protease enzymes, such as collagenases [Liotta 1985]. Throughout

this entire process, tumor cells must evade immune system responses.

Recent experimental research has attempted to characterize expression of the

various genes involved in controlling the metastasis process. However, information

gained thus far has been limited due to the large number of different genes that show

altered expression in metastasizing cells [Hanahan and Weinberg 2000]. Indeed,

characterizing integrin expression alone has been difficult, as there are a large number of

integrin genes, and integrin itself consists of multiple subunits that allow for the

expression of over 20 different versions of the protein in metastasizing cells [Hanahan

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and Weinberg 2000]. Further difficulties arise because a single metastatic cell expresses

different genes in the different environments through which it travels [Hanahan and

Weinberg 2000]. For example, because of environmental differences, a metastatic cell

moving from the kidney to the liver will express different metastasis-associated genes

than would a cell traveling from a primary tumor in the skin to the brain. Such variations

in gene expression make it difficult to fully characterize the set of genetic changes

associated with metastasis.

Perhaps one technique to overcome some limitations facing experimental

metastasis research could be the use of mathematical modeling. There is a long and rich

history of mathematical models being used to understand cancer [Araujo and McElwain

2004]. Indeed, models have been developed to investigate many phenomena associated

with cancer, from cell-cycle regulation [Hatzimanikatis et al. 1999] to tumor necrosis and

pleomorphism [Nagy 2005]. The general idea behind most models has been to create

equations that realistically represent cancer development and progression. These

equations are based upon known biological principles and often contain parameters with

values estimated from the experimental literature, although rarely can one characterize all

parameters in a realistic model with precision. Nevertheless, such an approach serves

scientific exploration in three ways. First, mathematical models can be used to make

experimentally testable predictions about biological systems or phenomena. In this way,

the models essentially give direction for experimental investigation, making the scientific

process more efficient. Second, the models can help one develop an overall context for

biological data gathered experimentally. This quality has become especially important

with the advent of microarray and other high throughput technologies that have produced

massive amounts of biological data requiring a context in order to be useful. Finally,

mathematical models can be used to study biological systems or phenomena that are

difficult to study experimentally. Complex problems that involve multiple variables or

steps rarely lend themselves to straightforward experimental analysis. Mathematical

models help one systematically analyze such problems quickly and efficiently, without

loss of biological relevance.

One such thorny problem is metastasis, which has been the subject of various

mathematical models in the past [Araujo and McElwain 2004]. In a classic paper, Saidel,

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et al. [1976] developed a model that encapsulates all of the steps of metastasis, including

tumor vascularization, intravasation, transport through the circulatory system, arrest at a

target organ, and initiation of secondary growth. The model consisted of a system of

ordinary differential equations, and its predictions matched well with experimental

observations. In particular, the model examined the effects of tumor trauma and

resection on initiation of metastasis and the model’s behavior was compared to

experimental data obtained using mechanical massage to induce tumor trauma in addition

to amputation of tumor-bearing limbs of model organisms to experimentally simulate

tumor resection [Saidel et al. 1976]. The model and experimental results matched well,

showing that showers of circulating tumor cells appear not long after tumor trauma and

that tumor amputation does not necessarily limit the formation of metastases [Saidel et al.

1976]. Overall, the model offered a way to examine different aspects of the metastatic

mechanism. Since the work of Saidel et al. [1976], however, more has been learned about

the roles of proteases and extracellular matrix anchors such as integrins in metastasis, and

Saidel et al.’s [1976] model does not include any of this new biological information

[Araujo and McElwain 2004].

While mathematical models have offered valuable biological insight into

metastasis, many questions remain unanswered. For example, no model fully explains

why certain tumor cells spread to specific organs or how they survive in environments

that are completely different from their initial environments. Nor do the models truly

answer why it is beneficial for tumor cells to spread away from the primary site in the

first place. One hypothesis suggests that cells metastasize because of oxygen constraints

in the primary tumor [Hanahan and Weinberg 2000]. However, there are problems with

this hypothesis. For example, small cell lung carcinoma most commonly metastasizes to

the brain [Nicolson 1986], but if oxygen depletion is the most important inducer of

metastatic behavior, why would a tumor cell ever leave the highly-oxygenated

environment of the lung? Clearly there may be other factors besides hypoxia that can

stimulate metastasis.

Indeed, other hypotheses are beginning to arise in the literature. For example,

Pescarmona, et al. (1999) hypothesize that iron is the key limiting nutrient driving

metastasis, and they studied this hypothesis with a series of mathematical models. They

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argue that oxygen is not generally a limiting factor for cellular proliferation, which is

apparently a crucial process characterizing cancerous cells. Iron, on the other hand, may

be limiting because it is involved in DNA synthesis and ATP production and therefore

may be a key nutrient in tumor development and proliferation [Pescarmona et al. 1999].

This conclusion is supported by two other pieces of evidence. First, the concentration of

iron is low at the surface of the earth. Second, cancer cells have higher numbers of

receptors for siderophores such as transferrin, which facilitate iron uptake into cells. This

increase in the number of receptors for iron uptake is taken to be an indication that cancer

cells require more iron than typical healthy cells do in order to support the uncontrolled

growth and proliferation characteristic of malignancy.

Pescarmona et al. [1999] developed a model consisting of a system of discrete,

non-linear difference equations that center on a section of tissue containing the tumor and

assumes a single limiting nutrient influencing cancer cell growth [Pescarmona et al.

1999]. Precise rules governing iron uptake by cells, iron consumption, cellular birth and

death processes, and diffusion were defined to complete the model. The model produced

two-dimensional images representing the growing tumor that were then compared with

actual images of tumor growth. The authors conclude that despite its simplicity, the

model describes many different possible behaviors for tumor growth [Pescarmona et al.

1999]. For example, the model predicted that low iron availability and high rate of tumor

cell growth could promote metastasis because cells may begin to seek out nutrients in

other areas as resources become limited. Essentially, high consumption of nutrient within

the tumor leads to local nutrient depletion, which in turn can lead to metastasis if

nutrients are available elsewhere. Metastasis becomes unlikely, however, if nutrients are

not available elsewhere, in which case the model predicts that a benign tumor will result,

as there is not enough iron in the environment to support tumor growth [Pescarmona et

al. 1999].

While this model does indeed offer a method of visualizing various possible

behaviors of tumor growth with respect to nutrient limitations, there are some problems

overall. First, the model does not include any boundary effects that might arise because

of anatomical limitations, and is therefore only applicable to tumors in soft tissues such

as the brain [Pescarmona et al. 1999]. Also, the authors assume that iron is the only

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limiting nutrient, and while it is possible that iron alone may limit the growth rate, such a

situation is not very plausible. Elser, et al. [2003] note, for example, that phosphorus

may in fact be a growth-rate limiting molecule in tumors. Their argument is based upon

principles from the field of biological stoichiometry, which involves the study of how

changes in the balance of energy and in the ratios of chemical elements such as carbon,

nitrogen, and phosphorous can have large scale effects on biological systems [Elser et al.

2003, Kuang et al. 2004, Sterner and Elser 2002]. In particular, Elser and colleagues

argue that because tumor cells proliferate at higher than normal rates, the cells require

more molecular machinery for cellular division, especially ribosomes. Ribosomes,

consisting largely of ribonucleic acids, are rich in phosphorus, and therefore tumor cells

may have a great need for this element.

To investigate this hypothesis, Kuang et al. [2004] developed a mathematical

model examining how phosphorus limitation could affect tumor growth dynamics. In

particular the authors examined whether dietary restrictions on phosphorus intake could

inhibit tumor growth, and therefore be used as a new type of non-invasive cancer

treatment. What they found was that limiting the amount of phosphorus did indeed limit

the growth of tumor cells, but such a limitation also damaged healthy tissue; therefore,

limiting dietary phosphorus would not be an effective treatment. However, the model

suggested that limiting the amount of phosphorus sequestered by tumor cells while

leaving healthy cells unaffected would be effective at limiting tumor growth.

Interestingly, the model also predicted that over time, in a phosphorus-limited

environment, tumor cells with a slower growth rate, and consequently fewer phosphorus-

rich ribosomes, will tend to dominate the tumor eventually [Kuang et al. 2004]. Kuang et

al. [2004] speculate that more aggressive tumor cell types may therefore be induced to

metastasize to a secondary environment where the phosphorous concentration is not as

limited.

The goal of this thesis is to investigate this idea more fully using mathematical

modeling. Instead of phosphorous, however, the model developed here examines another

element that may also limit the growth rate of tumors. Nitrogen is particularly important

in protein synthesis, as it is found in all amino acids as part of the amino group and in

many of the variable groups. Indeed, the average nitrogen content of the 20 amino acids

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in proteins is roughly 17% [Sterner and Elser 2002]. Cellular division requires that cells

essentially double the number of proteins and other molecules before division can occur.

Therefore, the hypothesis examined here is that tumor cells, which exhibit uninhibited

cellular division, have higher requirements for nitrogen than healthy cells because they

must produce the requisite number of amino acids and other nitrogen-containing

molecules necessary for protein synthesis every time they divide. In order to get a better

understanding of these cellular requirements, the ratio of carbon (C) to nitrogen (N) is

examined. The reason for looking at the ratio of carbon to nitrogen (C:N) is that the

tumor cells require both elements in order to survive. For example, suppose one were to

give a cell a massive amount of carbon and no nitrogen at all. The cell would die because

it would not have the nitrogen it needs to make proteins. Giving the cell a small amount

of carbon would improve the cell’s chances of surviving, but because there is so much

carbon for every nitrogen molecule, the cell must waste a large amount of energy

excreting all of the excess carbon. In other words, the cell must act inefficiently, which

in turn affects its overall ability to reproduce. In general, examining the C:N ratio helps

to capture this type of behavior which can influence cell growth. Also, taking this

approach may offer information about what C:N ratio is optimal for tumor growth and

metastasis.

The other goal of this thesis is to examine the question of how metastasizing cells

are able to survive in new environments despite competition for nutrients from cells in

the primary tumor. To address this question, the model developed looks at both a

primary and secondary tumor in competition for two nutrients, namely, carbon and

nitrogen. The hypothesis is that cells from the primary and secondary tumor are able to

coexist due to genotypic differences that influence their relationship to the nutrients in the

environment. The logic is that differences on the genetic level between cells from the

primary and secondary tumor may make it so that the cells have completely dissimilar

nutrient requirements. Experimental evidence suggests that this may be true, as even

cells found within the same tumor can show remarkably different genotypes [Hanahan

and Weinberg 2000]. The results of the model are then compared to those of Pescarmona

et al.’s [1999], which considered iron as a sole limiting nutrient. Order of magnitude

ranges for parameters in the model are estimated from experimental data whenever

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possible. Using parameter values obtained from experimental data helps to ensure that the

predictions of the model are biologically relevant.

II. The Model

The model developed consists of the following four ordinary differential

equations:

11 max,1

,1

( ) ( )( )d

Vdp c tV e d p tdt K c t

σ δ1−β⎡ ⎤⎛ ⎞

= α − − −⎢ ⎥⎜ ⎟+⎢ ⎥⎝ ⎠⎣ ⎦

(1)

22 max,2

,2

( ) ( )( )d

Vds c tV e d s tdt K c t

σ δ2−β⎡ ⎤⎛ ⎞

= α − − −⎢ ⎥⎜ ⎟+⎢ ⎥⎝ ⎠⎣ ⎦

, (2)

[ ] ( ) (1 21 1 2 2( ) ( ) ( )V V

c Bdc C c t V p t e V s t edt

βλ γ α γ α−= − − − )β− , (3)

[ ] 1 2( ) ( ) ( ) ( ) ( )N Bdn N n t p t n t s t n tdt

λ ω ω= − − − , (4)

( )( )c tV

n t=

+ ε. (5)

Table 1 lists the dependent variables of the model, their biological interpretation,

and their biological units; Table 2 lists the model’s parameters, along with their

biological interpretation and units.

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Table 1: The model’s dependent variables, their biological interpretation, and their

units.

Dependent Variable Biological Interpretation Units

p(t) Mass of the primary tumor grams (g)

s(t) Mass of the secondary tumor g

c(t) Carbon (C) concentration mol/liter (M)

n(t) Nitrogen (N) concentration mol/liter (M)

Note: t refers to the independent variable time, which has the units of days in this model.

Table 2: The model’s parameters, their biological interpretation, and their units.

Parameter Biological Interpretation Units

iα Per capita growth rate for tumor i (day)-1

iβ Sensitivity of tumor i’s growth rate to variations in the C:N

ratio

Unitless

max,id Maximum rate of death due to nutrient limitation for tumor

i

(day)-1

iσ Per capita death rate due to nutrient limitation for tumor i (day)-1

,d iK Concentration of nutrient at half the maximum death rate

for tumor i

mol/liter (M)

δ Death rate from causes not related to lack of nutrient (day)-1

Cλ Rate of carbon influx into the environment (day)-1

λΝ Rate of nitrogen influx into the environment (day)-1

BC Interstitial carbon concentration mol/liter (M)

BN Interstitial nitrogen concentration mol/liter (M)

iγ Rate of carbon sequestration for tumor i mol/(g·liter)

iω Rate of nitrogen sequestration for tumor i (g·day)-1

Note: i = 1 or 2, representing the primary or secondary tumor, respectively.

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The model looks at the mass of both a primary (p(t)) and a secondary tumor (s(t))

in a resource-dependent context. Equations (1) and (2) represent the growth rates of the

primary and secondary tumors, respectively. The maximal per capita growth rate of

tumor type i is , and growth is directly affected by the ratio of carbon concentration to

nitrogen concentration in the environment, which is signified by V. To avoid a

singularity in this quantity when the nitrogen concentration is zero, V is modified slightly

by adding ε to the denominator. Since growth rates of primary and secondary tumors are

generally different [Saidel et al. 1976], the per capita growth rates ( ) for the two

tumors will also generally be different. The parameter

iα

iα

iβ represents the sensitivity of

growth rates to changes in V. Varying this parameter is akin to changing the reciprocal of

the optimal C:N ratio for tumor growth. A larger iβ implies a smaller optimal C:N ratio

and a smaller iβ implies a larger optimal C:N ratio.

In this model, tumor cell death arises from two sources. First, there is a natural

death rate, represented by δ , which is assumed to have the same value for both the

primary and secondary tumors. This parameter represents cell death due to a variety of

factors, essentially anything not related to nutrient status. The second cause of death for

tumor cells is assumed to be nutrient-related. The term max,

,

( )( )

ii

d i

c tdK c t

σ⎛ ⎞−⎜ ⎟

+⎝ ⎠ represents the

overall tumor cell death due to nutrient limitations. The mortality rate from carbon

limitation is assumed to never go above a finite maximum, represented by . The

death rate from carbon limitations is assumed to decrease as the concentration of carbon

increases because with more carbon cells are more capable of producing sufficient ATP

for growth. However, as the concentration of carbon goes to infinity, the level of survival

only goes to

max,id

iσ , and therefore the death rate in an infinitely-rich carbon environment is

max,id iσ− . Because of this, must necessarily be greater than or equal to max,id iσ ,

because otherwise the overall rate of death would be positive, which would mean that

new cancer cells are arising from nowhere. Essentially, the biological assumption being

made is that tumor cell survival is positively related to the concentration of carbon,

primarily from carbohydrates, in the environment. Survival in this case is assumed to

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follow a Michaelis-Menten-like form, following a common convention seen in many

previous mathematical models [Araujo and McElwain 2004].

Equations (3) and (4) represent the dynamics of carbon and nitrogen

concentrations, respectively. While both equations contain similar terms for nutrient

influx and natural depletion in the overall environment ( [ ]( )C BC c tλ − and

[ ]( )N BN n tλ − ), the terms for nutrient sequestration by the tumor cells are different for

the different nutrients, representing critical biological differences between how those

nutrients are used. It is assumed that the body attempts to maintain constant carbon and

nitrogen concentrations in the blood at all times, with set points and , respectively.

Deviations from that set point are corrected at a rate proportional to the difference

between the set point and blood nutrient concentrations, with rate constants

BC BN

Cλ and λΝ .

Tumor cell sequestration of carbon is directly related to the growth rates of the tumors,

whereas sequestration of nitrogen is directly related to the overall size of the tumors.

Biologically, these assumptions arise because reduced carbon is assumed to be the main

energy source powering cell proliferation, whereas nitrogen is used primarily for

structural components of cells. Furthermore, I assume that the carbon released when cells

die is not reusable by other cells, because it has been completely oxidized to carbon

dioxide or some other unusable product. Also, in general, the per-capita uptake of

nitrogen is not assumed to vary over the course of time for each tumor, because the

nitrogen content of cells is assumed to be invariant throughout the tumor.

All parameters in the model can be varied so as to simulate different possible

situations that might arise in terms of competition between the primary and secondary

tumor. The model assumes that the tumors are essentially in found in the same

environment with regards to resources. In other words, both tumors “see” the same

amounts of carbon and nitrogen. However, the genotypes of the cells that compose the

tumors can be modified to reflect different abilities to use the nutrients in the

environment. For example, it is possible to examine how tumor growth is affected if

cells in the primary and secondary tumor have different genotypes with regards to

nutrient sequestration by maintaining the per capita growth rate iα and varying the

carbon and nitrogen sequestration rates, iγ and iω , respectively. Specifically, by varying

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iω , the effects of the tumors having different nitrogen sequestration capabilities due to

different genotypes can be simulated. Biologically, such a situation arises when cells

from the primary and secondary tumors differ in terms of the number of nitrogen

receptors they form on their membranes. Similarly, varying iγ allows for a study of how

the growth dynamics are affected when the tumors have different abilities to sequester

carbon.

Another parameter that will be varied between the different tumors is iβ . By

changing the value of iβ , the situation where the tumors have different optimal ratios of

C:N can be simulated. Essentially, it is possible that the tumors could have genotypic

differences which allow them to grow differently based on the ratio of carbon to nitrogen

in the environment. For example, different growth dynamics will arise if one tumor has a

higher optimal C:N ratio than the other. A tumor with a high optimal C:N ratio will grow

the best when the amount of carbon is much greater than the amount of nitrogen in the

environment. This tumor will generally grow more efficiently at low nitrogen

concentrations, because it can effectively use all of the nitrogen that is present in the

environment. However, if the C:N ratio becomes too low, which could occur if the

nitrogen concentration becomes too high, then the tumor cells waste a great deal of

energy excreting excess nitrogen that they cannot use, and tumor growth will potentially

be affected.

Overall, by altering different parameters while keeping others constant, it may be

possible to identify those parameters that are most important with regards to tumor

growth and competition dynamics. The biological characteristics that these parameters

represent can then be concentrated on experimentally.

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Table 3: Fixed parameters and their values

Parameter 1α 2α max,1d max,2d 1σ 2σ ,1dK ,2dK δ Cλ λΝ ε

Value 0.05 0.05 1.00 1.00 1.00 1.00 0.50 0.50 0.01 19.0 6.40 0.00001

III. Parameterization

In the numerical investigation of this model, certain values were maintained

constant throughout (Table 3). The initial size of the primary tumor is assumed to be .5

g, while the initial size of the secondary tumor is assumed to be .4 g, following

[Pescarmona et al. 1999] and [Nagy 2004]. The smaller initial size for the secondary

tumor is assumed because I imagine that this tumor began to develop at a later time than

the primary tumor, although it is not uncommon for secondary tumors to be detected

clinically before the primary tumor is [Nicolson 1986]. The initial carbon and nitrogen

concentrations are set such that the ratio of carbon to nitrogen is roughly 13:1, based on

evidence presented by Sterner and Elser (2002) that indicates that the human body has a

stoichiometric ratio of carbon to nitrogen of 85,700,000 : 6,430,000. iβ , the reciprocal of

the optimal C:N ratio for growth, is also based on this ratio initially and is 0.077. The per

capita growth rates of the two tumors were initially taken to be the same value of 0.05,

based on information found in Pescarmona et al. [1999], although these values are

changed later in the numerical investigation based on evidence from [Nicolson 1986],

which points out the possibility that the primary and secondary tumors grow at different

rates. The rate of carbon influx into the body ( Cλ ) was assumed to be three-fold greater

than the rate of influx of nitrogen ( λΝ ), based on Sterner and Elser (2002), which states

that proteins are roughly 57% carbon and 17% nitrogen. It was speculated that an

average person takes in about 230 g of carbon in a day, which is roughly 19 moles of

carbon per day. Therefore Cλ = 19.0 and λΝ is assumed to be about one-third of that,

which is 6.40. The values of the rest of the parameters were speculated to be within an

order of magnitude of the true values based on information in Kuang et al. [2004], Nagy

[2004], and Pescarmona et al. [1999].

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Figure 1: Competition between tumors in which the homeostatic C:N ratio is 3:1. ( = 9.0, = 3.0).

(a) Tumor growth dynamics. (b) Carbon dynamics. (c) Nitrogen Dynamics. BC BN

1β = = 0.077, 2β 1γ = 2γ =

0.60, = = 0.20. All other parameter values can be found in Table 3. 1ω 2ω

IV. Results

The general approach chosen to investigate the dynamics of the model was to

vary the values of different parameters to determine how they affect tumor dynamics

overall. While examining the overall effects of a specific parameter value, other

parameter values were kept constant. The purpose of this approach was to try to

determine those parameter values that are most important with regards to tumor growth

and competition dynamics.

Variation of the interstitial nutrient concentrations

The first part of the numerical investigation involved varying the interstitial

nutrient concentrations to illustrate why it is useful to look at the ratio of carbon to

nitrogen as opposed to either nutrient alone. Figs. 1-3 show the system’s response to

variations in interstitial concentrations of carbon and nitrogen. In Fig. 1, the body’s

homeostatic C:N ratio is 3; in Figs. 2 and 3 it is 1 and 1/3, respectively. Fig. 1

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Figure 2: Competition between tumors in which the homeostatic C:N ratio is 1:1. ( = 9.0, = 9.0). (a) Tumor growth dynamics. (b) Carbon dynamics. (remains steady around 9.0) (c) Nitrogen Dynamics.

= = 0.077, = = 0.60, =

BC BN

1β 2β 1γ 2γ 1ω 2ω = 0.20. All other parameter values can be found in Table 3.

shows that neither the primary nor secondary tumor can grow until the nitrogen

concentration falls below some critical value. While the nitrogen concentration is high,

both tumors must inefficiently excrete all of the excess carbon that they cannot use, and

doing so wasting their energy in that manner affects their overall growth. As evidence

for this, notice that the carbon concentration remains roughly steady as the nitrogen

concentration decreases. Tumor growth is delayed because the ratio of carbon to nitrogen

is such that neither tumor can grow until the nitrogen in the environment decreases. Fig. 2

shows that if the carbon and nitrogen concentrations in the environment are the same,

then the tumor’s overall use of nitrogen becomes very inefficient as the tumor cells use

up a lot of energy while excreting all of the excess nitrogen that they cannot use.

Because of this inefficient behavior, the tumors rapidly die. This can be concluded

because the ratio of carbon to nitrogen in the environment is 1.0, whereas the optimal

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Figure 3: Competition between tumors in which the homeostatic C:N ratio is 1:3. ( = 3.0, = 3.0). (a) Tumor growth dynamics. (b) Carbon dynamics. (remains steady around 9.0) (c) Nitrogen Dynamics.

= = 0.077, = = 0.60, =

BC BN

1β 2β 1γ 2γ 1ω 2ω = 0.20. All other parameter values can be found in Table 3. C:N ratio for both tumors is much greater than 1.0 because iβ is small. Similarly, in Fig.

3, both tumors die because while their optimal C:N ratio for growth is much greater than

1.0, the actual C:N ratio in the environment is much less than 1.0.

Interestingly, Fig. 4 shows that the tumors can grow even though the interstitial

concentrations of carbon and nitrogen are equal, which is in contrast to the behavior

shown in Fig. 2. The difference is that the interstitial concentrations of carbon and

nitrogen are both very high at 30.0 M. In Fig. 2, at equally low interstitial concentrations

of carbon and nitrogen, the tumors rapidly die as a result of inefficient use of resources

caused by a non-optimal C:N ratio in the environment. The nitrogen in the environment

actually rises up to a steady-state level because the tumor cells cannot use it efficiently

with respect to carbon for metabolism maintenance. In contrast, in Fig. 4, the amount of

nitrogen decreases until the C:N ratio is optimal for tumor growth. When the C:N ratio

reaches the optimal level for growth, the tumors are able to proliferate up to a steady-state

level, with the primary tumor reaching a mass of roughly 1200 g and the secondary tumor

reaching a mass of 1000 g.

16

Figure 4: Competition between tumors in which the homeostatic C:N ratio is 1:1, but at much higher interstitial concentrations. ( = 30.0, = 30.0). (a) Tumor growth dynamics. (b) Carbon dynamics.

(remains steady around 9.0) (c) Nitrogen Dynamics. BC BN

1β = 2β = 0.077, 1γ = 2γ = 0.60, = = 0.20. All other parameter values can be found in Table 3.

1ω 2ω

Figure 5: Competition between tumors in which the primary tumor sequesters nitrogen at a faster rate than the secondary tumor does. ( = 0.60,1ω 2ω = 0.20). (a) Tumor growth dynamics. (b) Carbon dynamics. (c)

Nitrogen Dynamics. = 9.0, = 3.0,BC BN 1β = 2β = 0.077, 1γ = 2γ = 0.60, All other parameter values can be found in Table 3.

17

Figure 6: Competition between tumors when the secondary tumor sequesters nitrogen at a faster rate than the primary tumor does. ( = 0.20,1ω 2ω = 0.60). (a) Tumor growth dynamics. (b) Carbon dynamics. (c)

Nitrogen Dynamics. = 9.0, = 3.0,BC BN 1β = 2β = 0.077, 1γ = 2γ = 0.60, All other parameter values can be found in Table 3.

Examining the effects of changing nutrient sequestration abilities

By varying and , the genotypes of the cells that constitute the two tumors

could be varied in terms of the cells’ abilities to sequester carbon and nitrogen,

respectively. Initially, I fixed the rate at which the tumors sequestered carbon and varied

their rates of nitrogen sequestration. I also assumed that the interstitial concentration of

carbon was three-fold greater than the interstitial concentration of nitrogen ( = 9.0,

= 3.0). Interestingly, it appears that nitrogen sequestration abilities only affect how

large the tumors actually become but does not seem to alter which tumor is favored by

selection. Three situations are examined: one where the two tumors sequester nitrogen

equally well, one where the primary tumor sequesters nitrogen better than the secondary

tumor, and one where the secondary tumor sequesters nitrogen better than the primary

tumor. Fig. 1 actually shows the situation that occurs when the tumors sequester nitrogen

equally well. In this case the primary tumor grows to a steady size of 300 g and the

secondary tumor grows to a size of 250 g. Once again, tumor growth is delayed by the

iγ iω

BC

BN

18

Figure 7: Competition between tumors in which the rates of nitrogen and carbon sequestration for both tumors are all the same. ( = 0.20,1ω 2ω = 0.20, 1γ = 2γ = 0.20 ). (a) Tumor growth dynamics. (b) Carbon

dynamics. (c) Nitrogen Dynamics. = 9.0, = 3.0 BC BN 1β = 2β = 0.077. All other parameter values can be found in Table 3.

fact that initially there is so much nitrogen in the environment that the ratio of C:N is

much smaller than the optimal ratio that the tumors require to grow. As the nitrogen

concentration decreases, however, the optimal ratio is reached, and tumor growth to a

steady-state mass occurs. Fig. 5 shows the situation where the primary tumor sequesters

nitrogen at a three-fold faster rate than the secondary tumor does. The tumors grow to

the steady-state mass when the optimal C:N ratio in the environment is reached, but this

time the tumors grow to sizes that are different than when the rates of nitrogen

sequestration were equal. In this case the primary tumor grows to a size of 125 g and the

secondary tumor grows to a size of 100 g. Similar results are seen when the secondary

tumor sequesters nitrogen at a faster rate than the primary tumor does (Fig. 6). The

reason the tumors grow to different sizes when their nitrogen sequestration abilities are

different is because the tumors’ per capita growth rates remain the same. Essentially,

when one of the tumors nitrogen sequestration rates is increased, it is forced to

inefficiently sequester nitrogen that it cannot use for metabolism maintenance and

19

Figure 8: Competition between tumors in which the primary tumor sequesters carbon at a faster rate than the secondary tumor does. ( = 1.8,1γ 2γ = 0.60). (a) Tumor growth dynamics. (b) Carbon dynamics. (c)

Nitrogen Dynamics. = 9.0, = 3.0,BC BN 1β = 2β = 0.077, 1ω = 2ω = 0.20. All other parameter values can be found in Table 3.

growth. This nitrogen gets excreted in an unusable form, which in turn affects how well

the other tumor can grow, because there is less usable nitrogen in the system

The next situation simulated was when both tumors sequester carbon and nitrogen

at equal rates (Fig. 7). In this case, the tumors grow to steady-state sizes of 280 g and

220 g for the primary and secondary tumor, respectively. The low sequestration rates of

the nutrients enables the tumor cells to avoid sequestering more carbon and nitrogen than

they can use for metabolism. The cells are growing at a very efficient level, and therefore

they are more capable of reaching a relatively high steady-state mass. Interestingly, as

the tumors start to rapidly grow, the amount of carbon in the environment dips

drastically. Up to this point the nitrogen in the system has been gradually decreasing so

that the C:N ratio approaches the optimal level for tumor growth. When the ratio gets

close to the optimal level, the tumors begin to grow and start using a lot of carbon (in the

form of glucose) to support that growth. As the optimal C:N ratio is reached and the

tumors reach a steady-state mass, the amount of carbon begins to rise up

20

Figure 9: Competition between tumors in which the secondary tumor sequesters carbon at a faster rate than the primary tumor does. ( = 0.60,1γ 2γ = 1.80). (a) Tumor growth dynamics. (b) Carbon dynamics. (c)

Nitrogen Dynamics. = 9.0, = 3.0,BC BN 1β = 2β = 0.077, 1ω = 2ω = 0.20. All other parameter values can be found in Table 3. again because the tumors are sequestering nutrients efficiently and are only taking in

carbon that they need to support the level of growth they have reached.

Next, carbon sequestration ( iγ ) was then changed to simulate three different

situations: when the primary and secondary tumors sequester carbon at equal rates, when

the primary tumor sequesters carbon faster than the secondary tumor, and when the

secondary tumor sequesters carbon better than the primary tumor. Fig. 1 once again

shows what occurs when the primary and secondary tumors sequester carbon equally well

( 1 2γ γ= = .6). The optimal nutrient ratio is initially much higher than the nutrient ratio

that is actually found in the environment, and therefore initially the tumor cannot grow.

The amount of nitrogen is gradually depleted from the environment until the C:N ratio

begins to approach the optimal level, and both tumors begin to rapidly grow. Fig. 8

shows the situation that occurs when the primary tumor sequesters carbon at a faster rate

than the secondary tumor does. In this case the tumors grow to sizes of roughly 280 g for

21

the primary tumor and 220 g for the secondary tumor. Figure 9 shows that the opposite

situation, where the secondary tumor sequesters carbon at a rate that is three-fold greater

than the primary tumor does, the tumors overall end up at larger sizes of 300 g (primary

tumor) and 250 g (secondary tumor).

The preceding numerical investigations offer evidence for the value of studying

the C:N ratio as opposed to only studying carbon or nitrogen individually. By studying

the nutrient ratio it is possible to describe how effectively the tumor cells are using the

carbon and nitrogen in the environment for their growth and proliferation. However, the

parameter conditions thus far have only resulted in either death for both the primary and

secondary tumors or the formation of benign tumors that maintain a steady-state mass.

More information needs to obtained about the conditions necessary for one tumor to grow

while the other tumor does not. Because thus far the determining factor with regards to

tumor growth appears to be the C:N ratio in the environment, varying the parameter iβ ,

which is a measure of the reciprocal of the optimal C:N ratio for growth for each tumor,

may offer other valuable information.

Varying the optimal C:N ratio for tumor growth

Three situations were examined: one where the optimal C:N ratio for tumor

growth is the same for both tumors, one where the primary tumor has a higher optimal

C:N ratio (lower iβ ) , and one where the secondary tumor has a higher optimal C:N ratio.

As before, the interstitial carbon concentration is assumed to be three-fold greater than

the interstitial nitrogen concentration, and so Fig. 1 shows an example of the model’s

behavior when the tumors have equal optimal C:N nutrient ratios. Both tumors rapidly

grow following a delay to a steady state level, with the primary tumor growing to 300 g

and the secondary tumor growing to 220 g. Once again, the delay is caused by the fact

that the C:N ratio in the environment lower than the optimal ratio for growth for both

tumors. In contrast, Fig. 10 shows that increasing the optimal C:N ratio (decreasing iβ )

for growth for the primary tumor by only 10% causes the primary tumor to out-compete

the secondary tumor. The primary tumor grows to a size of about 600 g following a

delay whereas the secondary tumor grows for a short period of time to almost 200 g and

22

Figure 10: Competition between tumors in which the primary tumor has a 10% higher optimal C:N ratio than the secondary tumor does ( = .0693 1β 2β = 0.077) (a) Tumor growth dynamics. (b) Carbon dynamics.

(c) Nitrogen Dynamics. = 9.0, = 3.0,BC BN 1γ = 2γ = 0.60, 1ω = 2ω = 0.20. All other parameter values can be found in Table 3.

then gradually dies out after roughly 100 days because it cannot survive at the C:N ratio

that has been reached in the environment. The delay seen before the primary and

secondary tumors begin to grow is due to the fact that the C:N ratio in the environment is

too low to support tumor growth. However, nitrogen depletion from the environment

raises the C:N ratio towards the optimal levels for tumor growth for both tumors. The

primary tumor grows faster because it can use more carbon for growth per nitrogen

molecule than the secondary tumor can. The amount of carbon in the environment

depletes as the tumors grow more and more, and the ratio of carbon to nitrogen goes to a

level where the primary tumor can maintain a steady level of growth while the secondary

tumor gradually dies off. In general, the primary tumor is more efficient than the

secondary tumor at using its resources for growth.

The exact opposite situation is shown in Fig. 11. In this case, the secondary

tumor’s optimal C:N ratio is 10% greater than that of the primary tumor, and the

secondary tumor ends up out-competing the primary tumor. The secondary tumor grows

to a steady-state mass of roughly 600 g while the primary tumor gradually dies out over a

23

Figure 11: Competition between tumors in which the secondary tumor has a 10% higher optimal C:N ratio than the primary tumor does ( = .077 1β 2β = 0.0693) (a) Primary tumor growth dynamics. (b) Secondary

tumor growth dynamics (c) Carbon dynamics. (d) Nitrogen Dynamics. = 9.0, = 3.0, = BC BN 1γ 2γ =

0.60, = = 0.20. All other parameter values can be found in Table 3. 1ω 2ω

period of roughly 100 days. This situation is interesting because it is an example of how

a secondary tumor might be able to out-compete the primary tumor that spawned it.

Clinically, this may also explain why some secondary tumors are detected before the

primary tumors are. It is possible that the primary tumor could experience enough

necrosis to become undetectable clinically because of competition for resources from the

secondary tumor.

Varying both growth rates and nutrient ratio sensitivity

To examine the relationship between the tumor growth rates and their optimal

C:N nutrient ratios, I decided to vary both iα and iβ simultaneously. Fig. 12 shows the

situation where the primary tumor has both a slower growth rate and a lower optimal C:N

nutrient ratio. The secondary tumor was expected to out-compete the primary tumor in

this case because the primary tumor grew more slowly and in general had higher

24

Figure 12: Competition between tumors in which the secondary tumor has both a 10% higher optimal C:N ratio than the primary tumor does ( = .077 1β 2β = 0.0693) and a faster growth rate ( = .05 > = .04) (a) Primary tumor growth dynamics. (b) Secondary tumor growth dynamics (c) Carbon dynamics.

1α 2α

(d) Nitrogen Dynamics. = 9.0, = 3.0,BC BN 1γ = 2γ = 0.60, 1ω = 2ω = 0.20. All other parameter values can be found in Table 3.

requirements for nitrogen per carbon molecule than the secondary tumor did. Also, in

general, with a higher optimal nutrient ratio coupled with a faster growth rate, the

secondary tumor can efficiently use the carbon and nitrogen supplied in the environment

to support itself as it grows. As expected, the secondary tumor did indeed out-compete

the primary tumor. The primary tumor showed a small amount of growth up to a mass of

about 30 g before it died off due to differences between its optimal nutrient ratio and the

C:N ratio in the environment. The secondary tumor, in contrast, grew up to about 600 g

and maintained that level of growth. Essentially, the secondary tumor was more capable

of using the nutrients provided in the environment for its growth, because its growth rate

was higher in conjunction with a higher optimal C:N ratio.As expected, the exact

opposite results were seen when the primary tumor was given a higher optimal C:N ratio

and a faster growth rate (Fig. 13). In this case the secondary tumor grew to roughly 27 g,

which is smaller than the mass that the primary tumor grew to previously, but this can

most likely be explained by the fact that the secondary tumor starts off smaller than the

25

Figure 13: Competition between tumors in which the secondary tumor has both a 10% higher optimal C:N ratio than the primary tumor does ( 1β = .0693 2β = 0.077) and a faster growth rate ( = .04 < 1α 2α = .05) (a) Primary tumor growth dynamics. (b) Secondary tumor growth dynamics (c) Carbon dynamics. (d) Nitrogen Dynamics. = 9.0, = 3.0,BC BN 1γ = 2γ = 0.60, 1ω = 2ω = 0.20. All other parameter values can be found in Table 3.

primary tumor does. In general, the symmetry of the model allowed for this result to be

predicted beforehand.

Fig. 14 shows an interesting result that occurs when the primary tumor is made to

have a slower growth rate and a higher optimal C:N ratio when compared to the

secondary tumor. In other words, the tumor grows slower, but it can grow more

efficiently with less nitrogen per unit of carbon. As can be seen from the figure, initially

the secondary tumor grows rapidly up to about 400 g, whereas the primary tumor grows

relatively slowly, reaching less than 100 g at the same point. With a faster growth rate,

the secondary tumor is able to more effectively use the nutrients in the environment when

the C:N ratio is relatively small. However, the tumor reaches its peak at 400 g and begins

to die off after that point, whereas the primary tumor grows steadily up to roughly 580 g

and maintains its mass at that level. The secondary tumor dies off because the ratio of

C:N has become too large with respect to the tumor’s optimal C:N ratio: there is not

enough nitrogen in the environment for the tumor to grow. The primary tumor, on the

26

Figure 14: Competition between tumors in which the primary tumor has a 10% higher optimal C:N ratio but a 20% slower growth rate than the secondary tumor has ( 1β = .0693, 2β = 0.077, = .04 < = .05) (a) Primary tumor growth dynamics. (b) Secondary tumor growth dynamics (c) Carbon dynamics.

1α 2α

(d) Nitrogen Dynamics. = 9.0, = 3.0,BC BN 1γ = 2γ = 0.60, 1ω = 2ω = 0.20. All other parameter values can be found in Table 3.

hand, has a higher optimal C:N ratio, and can therefore survive better when the amount of

nitrogen is low with respect to carbon. Because of the model’s symmetry, the exact

opposite results are seen when the growth rate of the secondary tumor is less than that of

the primary tumor but its optimal nutrient ratio is higher. In general, these results suggest

that a tumor’s per capita growth rate may not be as important as its optimal C:N ratio

with regards to its survival.

V. Discussion

This thesis presents a mathematical model consisting of four ordinary differential

equations that describe both primary and secondary tumor growth dynamics following

metastasis. In general, the model attempts to address two major questions. First, can

nitrogen limitations in the environment (as measured with respect to carbon) influence

tumor growth dynamics and survival? Second, how is it possible for metastasizing cells

27

to compete for nutrients with cells from the primary tumor? In other words, what

conditions dictate that a secondary tumor will be able to survive in a nutrient-limited

environment?

The first part of this thesis involved determining the overall effect that the

interstitial carbon and nitrogen concentrations had on tumor growth dynamics. By

varying the interstitial carbon and nitrogen concentrations, it was possible to get a sense

of how important the carbon to nitrogen ratio is with regards to tumor growth and

competition. When the homeostatic C:N ratio was high, both tumors grew to a large

steady-state size because they could efficiently use the nutrients supplied in the

environment for growth. Making the C:N ratio 1, with low overall concentrations of

carbon and nitrogen, resulted in both of the tumors immediately dying out, primarily

because the cells of both tumors ended up wasting metabolic energy to excrete excess

nitrogen which they could not use due to a lack of carbon. Tumor death also resulted

when the C:N ratio was 1/3, once again because the cells ended up wasting a lot of

energy sequestering and excreting nitrogen that they could not use without more carbon.

Finally, making the concentrations of carbon and nitrogen relatively high while still

maintaining a C:N ratio of 1 did not result in tumor death. Rather, the overall nitrogen

concentration depleted enough so that the environmental C:N ratio approached the

optimal ratio for the tumors, and both tumors grew to a steady-state mass as a result.

In general, varying the carbon and nitrogen interstitial concentrations indicated

how important the C:N nutrient ratio is. A tumor that sequesters too much nitrogen must

then expend energy to excrete the excess nitrogen that it cannot use. Such inefficient

behavior negatively affects tumor growth and may even result in tumor necrosis. If too

little nitrogen exists in the system, however, then tumor cells would no longer be able to

make all proteins needed for proliferation or, indeed, survival. Using the ratio of carbon

to nitrogen captures the essence of this behavior and therefore more accurately represents

the dynamics that occur when the tumor cells compete for the limiting nutrients.

The next part of the thesis involved changing the genotypes of the tumor cells

with regards to their nutrient sequestration capabilities to see the overall effects on

growth. Essentially, by altering the carbon and nitrogen sequestration rates, which is

biologically akin to altering the number of cellular membrane nutrient receptors, the

28

effects of nutrient sequestration on tumor growth and competition could be observed.

Those cells with a higher nitrogen sequestration rate, for example, were assumed to have

more cellular receptors for nitrogen, and therefore they could use more of the nitrogen

from the general environment. One assumption made was that improving the tumor

cells’ abilities to sequester a given nutrient would result in greater growth for the overall

tumor, which in turn might affect the outcome of competition between the primary and

secondary tumors. In reality, making the nutrient sequestration rates of the two tumors

different had little effect on the overall outcome of competition. Both the primary and

secondary tumors grew to a steady-state mass no matter which one had the greater ability

to sequester one of the given nutrients. The reason for this is even though the tumors

could sequester nitrogen at a faster rate, that sequestration did not translate into overall

tumor growth because the per capita growth rate was not directly linked to nutrient

sequestration per se. Essentially, without a corresponding increase in their rates of

growth, the tumors ended up sequestering nutrients that that they could not use to

contribute to their biomass to support unlimited growth. This explains why the tumors

rose to steady-state levels of size instead of continuing to grow: their abilities to sequester

nutrients could only take them so far without a requisite increase in their ability to

contribute those nutrients towards their biomass. Overall, the key point is that the

tumors’ nutrient sequestration abilities probably are not the most important characteristics

to concentrate on with respect to tumor growth and competition, because in general the

growth dynamics of the model were insensitive to changes in these parameters.

A parameter that does seem to be important with regards to tumor growth and

competition is iβ , which represents the reciprocal of the optimal C:N ratio for growth.

Varying this parameter directly affected the outcome of competition between the primary

and secondary tumors. In general, the tumor with the higher optimal C:N ratio (lower

iβ ) would be the one that would out-compete the other. Having a higher optimal C:N

ratio essentially means that the tumor is capable of surviving with less nitrogen per

carbon molecule. In other words, a tumor with a high optimal ratio can survive and grow

in an environment that has less nitrogen with respect to carbon. Conversely, a tumor with

a low optimal C:N ratio requires more nitrogen per unit of carbon, and therefore faces

more selection pressure when nitrogen becomes limiting in the environment.

29

The final part of the numerical investigation involved examining the interaction

between the per capita growth rates ( iα ) of the tumors and their optimal nutrient ratios

( iβ ). In general, the tumor with the higher optimal C:N nutrient ratio and higher growth

rate was capable of out-competing the other tumor, because having a higher optimal

nutrient ratio in conjunction with a higher growth rate means that the tumor can more

effectively use the nutrients in the environment to support an increase in biomass.

Essentially, not only is such a tumor more capable of surviving as nitrogen becomes

limiting, it is also capable of growing at a faster rate when nutrients are not limiting.

Interesting dynamics arose when one tumor was given a faster growth rate while the other

tumor was given a higher optimal C:N nutrient ratio. The outcome was that the tumor

with the higher optimal C:N ratio would end up out-competing the other tumor, most

likely because a higher optimal nutrient ratio prevents the tumor cells from dying when

nitrogen becomes limiting, whereas a faster growth rate does not prevent such death and

,indeed, may even make it more likely. The tumor with the faster growth rate will

initially grow faster because it is more capable of using the nutrients in the environment

to increase its biomass, especially at a lower C:N environmental ratio. However, as

nitrogen is depleted from the environment and the ratio of C:N becomes larger, the tumor

with the higher optimal C:N ratio will inevitably out-compete the other tumor, because

this tumor cannot survive in an environment where nitrogen is limited with respect to

carbon. The key biological point of this part of the investigation is that it describes

conditions whereby a secondary tumor can initially outgrow a primary tumor, only to be

out-competed by the primary tumor later on. This may explain why clinically the

secondary tumors are often found before the primary tumors are discovered.

Overall, this model predicts that in a competition for nutrients between a primary

and secondary tumor, that tumor which has the higher optimal C:N nutrient ratio will out-

compete the other. It may be possible to test this hypothesis in an experimental setting.

For example, it is certainly plausible that cells with a high optimal C:N nutrient ratio

could be selected for in vitro by using defined media that presents an environment where

the ratio of carbon to nitrogen is relatively large. These cells could then be used in a

model metastasis system such as the chick embryo to see the overall effects on tumor

growth dynamics [Ziljstra et al. 2002].

30

Comparing these results to the model by Pescarmona et al. [1999], who claimed

that iron was the primary limiting nutrient, one can see some fundamental similarities and

differences. Both models agree that a metastatic tumor’s interaction with its

environmental conditions critically influences its ability to survive in competition with

the primary tumor. Pescarmona et al. [1999] claim that low nutrient (iron) availability

and higher cellular growth rates stimulate metastasis. In contrast, the results of my model

indicate that, while nutrient availability in the form of nitrogen may be important, cellular

growth rates are far less important than the carbon to nitrogen nutrient ratio. Also,

Pescarmona et al. [1999] predict that high nutrient availability and high nutrient

consumption rates can also favor metastatic behavior. Herein lies a fundamental

difference between Pescarmona et al.’s [1999] model and my own: while their model

concentrates on a single limiting nutrient (iron), my model looks at a ratio of two

nutrients (carbon and nitrogen). The results from my model indicate that high nutrient

availability may not lead to metastasis or even support tumor growth. In fact, having a

large amount of carbon and nitrogen can lead to tumor necrosis. Considering a nutrient

ratio might more accurately capture realistic biological dynamics because nutrients like

carbon and nitrogen are often found in the same molecules such as proteins. So while it

is certainly possible that iron and other elements such as phosphorus may in fact be

limiting, in order to truly develop an understanding of how exactly these nutrients affect

the complex metastatic process, it is necessary to model them together using nutrient

ratios as has been done in this thesis.

One of the benefits of the model presented in this thesis is that it is general

enough to allow for modification in order to attempt to answer one of the most important

questions regarding metastasis: how is it that a cell that comes from a primary tumor in

an environment that offers specific nutrients is capable of surviving and initiating

secondary growth in another environment where the nutrients offered may be completely

different? Answering this question would enable clinicians to identify the most common

secondary sites for any given primary tumor and adjust therapies accordingly. As it

stands now, the model presented in this thesis does not answer this question. Rather, it

answers the question of how a secondary tumor is capable of surviving the competition

31

for nutrients that exists with the primary tumor. However, the model could be changed to

address the question of how cells are able to survive in completely different

environments. In order to do so, it would be necessary to add two more equations to the

model. Three of the equations would deal with the interstitial nutrient concentrations and

the growth of the primary tumor in one environment, while the other three would deal

with the interstitial nutrient concentrations and the growth of the secondary tumor in a

different environment. This new model, in conjunction with the model presented in this

thesis, would more fully describe much of the tumor growth dynamics that ensue during

the complex process of metastasis.

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