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DYNAMICS OF A DATA BASED OVARIAN CANCER GROWTH

AND TREATMENT MODEL WITH TIME DELAY

R. A. Everett1, J. D. Nagy1,2, and Y. Kuang1

1 School of Mathematical and Statistical SciencesArizona State University, Tempe, AZ 85287, USA

2 Department of BiologyScottsdale Community College, Scottsdale, AZ 85256, USA

Abstract. We present a simple model that considers ovarian tumor growth andtumor induced angiogenesis, subject to on and off anti-angiogenesis treatment.The tumor growth is governed by Droop’s cell quota model, a mathematicalexpression developed in ecology. Here, the cell quota represents intracellularconcentration of necessary nutrients provided through blood supply. We presentmathematical analysis of the model, including proving positivity of the solutionsso that they are biologically meaningful, as well as discussing stability and globallystability of the solutions. The mathematical model is able to fit both on-treatmentand off-treatment clinical data using the same biologically relevant parameters.We also state an open mathematical question.

1. Introduction

Ovarian cancer, also known as the ‘silent killer’ [13, 4], causes more deaths thanany other gynecological malignancies and is the 5th leading cause of death from non-skin cancers among women [14, 29]. The American Cancer Society estimates 21,290new cases of ovarian cancer and 14,180 deaths due to ovarian cancer in the UnitedStates in 2015 [30]. Only about 20% of ovarian cancers are detected at an earlystage [4], in part due to the lack of effective screening strategy and in part since theindications are often symptomatic of other diseases; symptoms include abdominaldiscomfort or fullness, bloating, and dyspepsia [2]. Ovarian cancer is characterizedby intraperitoneal (IP) tumors and ascitic fluid [22, 15].

While cytoreductive surgery and chemotherapy are common treatments for ovar-ian cancer, more than 70% of advanced-stage patients will develop drug resistanceand the disease recur within 5 years [4, 2]. Thus molecular targeted therapy has beenrecently researched, particularly anti-angiogenesis therapy [2], an idea proposed byFolkman more than 40 years ago [8]. Angiogenesis, the development of new bloodvessels from pre-existing vessels, is essential for tumor growth and expansion byproviding the necessary oxygen and nutrient support to the tumor [22, 9, 10]. An-giogenesis is regulated by pro-angiogenic and anti-angiogenic factors. When thesefactors become unbalanced in favor of angiogenesis, the tumor acquires angiogeneicproperties, known as the ‘angiogenic switch’ [10]. One of these pro-angiogenic fac-tors, vascular endothelial growth factor (VEGF), is expressed in most malignant

1

2 DYNAMICS OF A DATA BASED OVARIAN CANCER MODEL

tumors and is one of the most important tumor angiogenesis factors [31], althoughits role in tumor angiogenesis is still not understood completely [33]. VEGF isinvolved in cyclic growth of ovarian follicles and corpus luteum development andmaintenance [12], it is expressed higher in women with ovarian cancers compared tothose with benign tumors [14].

Several studies have investigated the relationship between VEGF and tumorgrowth. In 1998, Mesiano et al. [22] analyzed the role of VEGF in tumor growth,progression and ascites formation in ovarian cancer in tumors induced in immun-odeficient mice using the human ovarian carcinoma cell line SKOV-3. The authorsconcluded that tumor-derived VEGF is necessary for the formation of ascites, butmay not be obligatory for IP growth. In 2000, Hu et al. [15] researched the effectsof a PI3-K inhibitor, LY294002, on tumor progression and ascites formation in thesame mouse model of IP ovarian carcinoma using the OVCAR-3 ovarian cancer cellline. The authors concluded that LY294002 significantly inhibited growth and as-cites formation. While this study did not investigate VEGF directly, the authorsbelieve LY294002 may have blocked the signal transduction pathway of VEGF. Themonoclonal anti-VEGF antibody bevacizumab became approved by the FDA in 2004for first-line treatment of metastatic colorectal cancer [33]. In 2013, Ye and Chen[34] analyzed the efficacy and safety of bevacizumab in ovarian cancer treatmentusing four, phase III randomized controlled trials. They concluded that combin-ing bevacizumab to chemotherapy provided improvement in objective response rateand progression-free survival for both first-line and recurrent disease treatment, butprovided no benefits to overall survival. These results were confirmed by a 2014systematic review of bevacizumab combined with chemotherapy for ovarian cancertreatment [2]. This result is typical of treatments targeting angiogenesis in humans.Although many anticipated great benefits of anti-VEGF/VEGF receptor therapy,studies across several cancers have shown only modest results [31].

One approach to modeling angiogenesis and tumor growth is to apply ecologicalmodeling techniques to cancer modeling. Healthy and cancerous cells live in anecological system where they interact with each other, competing for resources,nutrition, and space [3, 17, 21, 28, 23, 24, 26, 5]. Kuang et al. [18], applied thetheory of ecological stoichiometry to a model of tumor angiogenesis. This theoryconsiders the balance of multiple chemical substances, or sometimes energy andmaterials, in ecological interactions and processes [32]. One of the main hypothesisfrom this theory is the growth rate hypothesis, which, “proposes that ecologicallysignificant variations in the relative requirements of an organism for C, N and P aredetermined by its mass-specific growth rate because of the heavy demand for P-richribosomal RNA under rapid growth” [18, 32]; organisms with high growth rates havehigh P:C ratios due to the increased allocation of P to RNA. Since tumor cells oftenhave high growth rates, it makes sense to apply this hypothesis to cancer biology.Elser et al. [7] tested this and determined that the growth rate hypothesis mighthold true for some cancers, but not for all cancers. Kuang et al. [18] propose amodel which considers healthy cells, tumor cells, and tumor microvessels, or maturevascular endothelial cells (VECs) in the tumor. The growth rate of these cells are

DYNAMICS OF A DATA BASED OVARIAN CANCER MODEL 3

possibly limited by the nutrient phosphorus, depending on the concentration ofextracellular phosphorus. The growth of the cancerous cells can also be limited bya lack of blood vessels, which carry important nutrients and supplies. The authorsassume a time delay τ , which represents the time it takes for the tumor vessels toform. This idea of representing the micro vessel formation process using a time delayhas also been used in previous mathematical models of tumor-induced angiogenesis[16, 1].

We present a first approximation mathematical model of tumor growth andtumor-induced angiogenesis in the simplest context, using a minimum number ofparameters. We apply the idea of nutrient limited induced angiogenesis from Kuanget al. [18] through the use of Droop’s Cell Quota model [6], a mathematical expres-sion developed in ecology and used in previous ecological and cancer models. Wealso express the processes of microvessel formation through the use of a time delay.We consider the tumor growth both on and off anti-VEGF treatment using the sameparameter set. We present our mathematical model in Section 2 and then presentthe analysis of the model in Sections 3 and 4. Section 5 contains simulations andcomparisons of the model to clinical data from Mesiano et al. [22].

2. Tumor Model

We present a simple vascularized tumor growth model where tumor growth isgoverned by the Droop equation [6]. Let y represent the vascularized tumor volumeand Q represent the intracellular concentration of necessary nutrients provided byangiogenesis, or the cell quota of some limiting nutrient from angiogenesis. Ourmodel takes the following form:

y′ = µm

(1− q

Q

)y︸︷︷︸

growth

− dy︸︷︷︸death

,(1a)

Q′ = αy(t− τ)

y(t)︸︷︷︸nutrient uptake

−µm (Q− q)︸︷︷︸dilution

.(1b)

The growth of the tumor is given by the Droop equation, where µm represents themaximum tumor growth rate and q represents the minimum cell quota, or minimumconcentration of limiting nutrient needed to sustain the cell. The tumor death rateis assumed to be constant, represented by d. Similarly to [18], we assume that ittakes τ time for the vascular endothelial cells to respond to the angiogenic signaland mature to fully functional vessels. While a mechanistic model would need totrack the blood vessels, we simplify by assuming that the nutrient uptake rate isproportional to the nutrient concentration in the interstitial fluid, which in turn isproportional to the blood vessel density τ time units in the past. The delay arisesbecause the tumor is assumed to grow into regions that are unvascularized, andit takes τ units of time for them to vascularize. Once a region is vascularized, itremains static. VEGF is assumed to be the primary signal generating new blood


vessel growth and is secreted primarily where new tumor tissue is forming, sincethese are the regions that are hypoxic. Parameter α represents both uptake rate ofthe nutrients in the interstitial fluid and resulting nutrient concentration per tumorunit. Term µm(Q−q) represents the dilution of the nutrient as the tumor grows. Weassume that when the cells die they release the nutrient, which remains in the nearbyenvironment [18]. See Table 1 for a list of the variable and parameter meanings,units, and values.

We now consider the model when an anti-VEGF treatment is applied. Duringtreatment, the blood vessel growth will be impaired due to the inhibition of VEGF,but existing vasculature is likely not to be affected by the treatment. In that casenutrient delivery no longer depends upon the tumor volume τ days ago, but remainsconstant due to the static vasculature. We assume that blood vessel sprouts thatbegan forming within τ time units before the onset of treatment will not be fullyformed and patent. Let t0 represent the time of treatment onset and y = y(t0 − τ).Then nutrient delivery is dependent upon y, and the delay differential equationmodel becomes an ordinary differential equation model:

y′ = µm

(1− q

Q

)y︸︷︷︸

growth

− dy︸︷︷︸death

(2a)

Q′ = αpy

y(t)︸︷︷︸nutrient uptake

−µm (Q− q)︸︷︷︸dilution

(2b)

When comparing the model to clinical data from Mesiano et al [22], the parameterα was too large during treatment. Therefore, the α during treatment cannot bethe same as during off-treatment. Thus, we introduced a discount parameter p toaccount for this biological observation. Since the nutrient uptake term is capturingthe processes of angiogenic signaling, angiogenesis itself, and uptake of nutrients,the parameter p could represent an additional inhibition, above the treatment effectoriginally modeled, in any of these processes. One possible explanation could be thecharacteristic blood-filled cysts that form in untreated ovarian malignancies that areoften depleted once treatment is ongoing [22]. If this explanation is correct, thenblood within these cysts must provide nutrients to the tumor.

3. Basic Analysis for the System (1)

The following section provides a mathematically basic but practically adequateanalysis of system (1), verifying the positivity of the solution in order to be biolog-ically meaningful, providing a simple condition ensuring the tumor cell populationtends to 0, and discussing a condition for approximating the solution.

Theorem 3.1. Solutions system (1) with the initial conditions QM > Q(t) > qand y(t) > 0 for t ∈ [−τ, 0] will remain in this region for all t > 0, where QM =

maxQ(s), q + α

µmedτ , s ∈ [0, τ ]

. If µm ≤ d, then limt→∞ y(t) = 0.


Table 1. Tumor model parameter ranges. In the table, vol stands for volumeunit and Ref stands for Reference.

Para. Meaning Unit Value Refy tumor volume mm3 - [22]Q cell quota (cell nutrient density) mol/vol -q minimum cell quota mol/vol 0.0021-0.0099µm maximum growth rate day−1 0.47-1.58 [27]d death rate day−1 0.28-1.43 [27]α nutrient uptake coefficient mol/vol/day 0.0084-0.70p reduction in nutrient uptake rate - 0.17-0.47τ time delay day 10 [20]

Proof. We first establish the positivity of the solutions. Assume by contradiction,there exists time t1 ∈ [0, τ ] such that a trajectory with initial conditions Q(t) > qand y(t) > 0 for t ∈ [−τ, 0] crosses a boundary Q = q or y = 0 for the first time.

Case 1. Assume the trajectory crosses the boundary Q(t1) = q first. Then fort ∈ [0, t1], y(t) > 0 and

Q′(t) =αy(t− τ)

y(t)− µm(Q(t)− q)

≥ −µm(Q(t)− q)

Then

Q′(t) + µmQ(t) ≥ µmqand so

Q(t) ≥ q + (Q(0)− q)e−µmt > q

Thus Q(t1) > q, which contradicts Q(t1) = q. Therefore a trajectory cannot crossthis boundary first.

Case 2. Assume the trajectory crosses the boundary y(t1) = 0 first or thetrajectory crosses both y(t1) = 0 andQ(t1) = q at the same time. Then for t ∈ [0, t1],Q(t) ≥ q and

y′(t) =

(µm

(1− q

Q(t)

)− d)y(t) ≥ −dy(t)

Then y(t) ≥ y0e−dt > 0. Therefore y(t1) > 0, which contradicts y(t1) = 0. Therefore

a trajectory cannot cross the boundary y = 0 first nor both boundaries at the sametime.

Now we will show Q is bounded above. Since y′ ≥ −dy, then y(t) ≥ y(t− τ)e−dτ

and so y(t−τ)y(t) ≤ e

dτ . Then for t > τ ,

Q′ ≤ αedτ − µm(Q− q)


which implies

Q(t) ≤ q +α

µmedτ +

(Q0 − q −

α

µmedτ)e−µmt

Then

lim supt→∞

Q(t) ≤ q +α

µmedτ

and

Q(t) ≤ max

Q(s), q +

α

µmedτ , s ∈ [0, τ ]

= QM .

Assume now µm ≤ d, we see that

y′ ≤ −µmqQ

y,

which implies y(t) is strictly decreasing to a constant y∗ ≥ 0. We will now show thaty′(t) is uniformly continuous. Let ε > 0 and choose δ = ε

d−µm+µmq

(QM+y(0)). Then

for y1, y2, Q1, Q2 ∈ R, |y1 − y2|+ |Q1 −Q2| < δ implies

|(y1, Q1)− f(y2, Q2)| =∣∣∣∣µm(1− q

Q1

)y1 − dy1 − µm

(1− q

Q2

)y2 + dy2

∣∣∣∣≤ |µm − d| |y1 − y2|+ µmq

∣∣∣∣ y1

Q1− y2

Q2

∣∣∣∣≤ |µm − d| |y1 − y2|+ µmq

∣∣∣∣y1Q2 − y2Q1

Q1Q2

∣∣∣∣≤ |µm − d| |y1 − y2|+

µmq|y1Q2 − y2Q1|

≤ |µm − d| |y1 − y2|+µmq|y1Q2 − y1Q1 + y1Q1 − y2Q1|

≤ |µm − d| |y1 − y2|+µmq

(y1 |Q2 −Q1|+Q1 |y1 − y2|)

< (d− µm)δ +µmq

(y1δ +Q1δ)

≤ δ(d− µm +

µmq

(y(0) +QM )

)= ε

The uniform continuous property of y′(t) ensures that (By Barbalat’s Lemma)limt→∞ y

′(t) = 0. If y∗ 6= 0, then we must have

limt→∞

µm

(1− q

Q(t)

)− d = 0.

This yields that

limt→∞

q

Q(t)= 1− d

µm≤ 0,


implying limt→∞Q(t) = ∞ if d = µm and a contradiction if d > µm. However,when d = µm and y∗ 6= 0, we can find a T > 0, such that for t > T , we havey∗/2 < y(t) < 2y∗. Hence we have for t > T + τ,

Q′(t) ≤ 4α− µm(Q(t)− q),implying that lim supt→∞Q(t) ≤ q + 4α

µm< ∞, again a contradiction. This proves

that if µm ≤ d, then limt→∞ y(t) = 0.

In the following, we assume that µm > d.The full system described in (1) is nonlinear with a time delay and so there is no

standard approach for handling the system in order to study the dynamics of thesolution. Biologically we can assume that the uptake of nutrients is on a faster timescale than the population dynamics. Thus we apply a quasi-steady state argumentby allowing Q′ = 0. Then

Q′(t) = 0 =αy(t− τ)

y(t)− µm(Q∗(t)− q)

and so

Q∗(t) =αy(t− τ)

µmy(t)+ q

Then

y′(t) = µm

(1− q

Q∗(t)

)y(t)− dy(t)

= µm

(1− qµmy(t)

αy(t− τ) + qµmy(t)

)y(t)− dy(t)

=

(µmαy(t− τ)

αy(t− τ) + qµmy(t)− d)y(t)

= f (y(t), y(t− τ))(3)

Motivated by the off-treatment tumor growth data in [22] (see Section 5), we lookfor the existence of a dominating exponential solution. Let y(t) = y0e

λt. Then

(4) eλτ =α(µm − λ− d)

µmq(λ+ d).

Note that eλτ is a monotone increasing function of lambda and α(µm−λ−d)µmq(λ+d) is a

monotone decreasing function of lambda (see Figure 1). Observe that if α(µm−d) >µmqd, we see that there exists a unique real eigenvalue, 0 < λ1 < µm−d, that satisfies(4). Then a solution to the system is

(5) (y1(t), Q∗) =

(y0e

λ1t,αe−λ1τ + µmq

µm

).

However, there are also infinitely many complex solutions to the equation (4). Thefollowing theorem provides a sufficient condition that ensures λ1 as the dominanteigenvalue, i.e. when the solution can be approximated by (5). Also, note that sincey1(t) = y0e

λ1t is a solution to the system, we know that y(t) is not bounded above.


−5 −4 −3 −2 −1 0 1 2 3 4 5−30

−20

−10

0

10

20

30

λ

λ1= 0.09972

y 1(λ ) = e λ τ

y 2(λ ) = α ( µ m− λ − d )µ mq ( λ + d )

λ 1

Figure 1. Plot of (4) with µm = .64, d = .43, q = .006, α = .05, τ = 10

Theorem 3.2. If α(µm − d) > µmqd and α < µmqeλ1τ , then λ1 ≥ supRe(λ) :

λ is any solution of (4).

Proof. Consider the characteristic equation defined in (4) and let λ1 be the uniquepositive real eigenvalue that satisfies the characteristic equation. Then∥∥∥eλ1τ

∥∥∥ =

∥∥∥∥α(µm − λ1 − d)

µmq(λ1 + d)

∥∥∥∥implies

e2λ1τ =(α(µm − λ1 − d))2

(µmq(λ1 + d))2 .

We assume, by contradiction, that there exists a λ = a + ib such that a > λ1.Then ∥∥∥e(a+ib)τ

∥∥∥ =

∥∥∥∥α(µm − a− d) + i(−αb)µmq(a+ d) + i(µmqb)

∥∥∥∥implies

(6) e2aτ =(α(µm − a− d))2 + (αb)2

(µmq(a+ d))2 + (µmqb)2 =

A+ C

B +D

where A = (α(µm − a− d))2, B = (µmq(a+ d))2, C = (αb)2, and D = (µmqb)2.

Since a > λ1 and 0 < λ1 < µm − d, we have

(7) e2aτ > e2λ1τ =(α(µm − λ1 − d))2

(µmq(λ1 + d))2 >(α(µm − a− d))2

(µmq(a+ d))2 =A

B

Let E = e2λ1τ and F = 1. Then by (7),A

B<E

F.

Sinceα

µmq< eλ1τ , then

(αb)2

(µmqb)2< e2λ1τ and so

C

D<E

F.

Note that ifA

B<E

Fand

C

D<E

F, then

A+ C

B +D<E

F.


Then by (6), e2aτ =A+ C

B +D<E

F= e2λ1τ . However, by (7), e2aτ > e2λ1τ . Thus we

have reached a contradiction. Therefore λ1 ≥ supRe(λ) : λ is any solution of (4).

4. Global Analysis for the System (2)

The following section provides a global analysis of the model in system (2), ver-ifying the boundedness and invariance of the solution and discussing the stabilityand global stability of the positive equilibrium solution.

Theorem 4.1. The solutions of the system (2) are bounded away from zero.

Proof. First we will show y is bounded away from 0.We will show by contradiction that y > L, where L is chosen to be sufficiently

small. Let tL = mint > 0 : y(t) = L, i.e., let tL be the first time y(t) reaches L.We now choose M such that L < M < y0. Let tM = maxt < tL : y(t) = M, i.e.,let tM be the last time y(t) reaches M before it reaches L. Then 0 < tM < tL andy(t) < M for t ∈ (tM , tL]. See Figure 2 for a sketch.

M

L

tM tL

y

t

Figure 2. Sketch of proof

We first choose M such that

M < min

y0,

αpy(µm − d)

qµ2m

which implies that

αpy

Mµm>

qµmµm − d

.

We now choose L such that∣∣∣∣(M − αpy

Mµm− q)e

−µmd

ln(ML )∣∣∣∣ < q,


or equivalently

L < M

q∣∣∣M − αpyMµm

− q∣∣∣ d

µm

.

Since tL is the first time y(t) reaches L and L < y0, then

y′(tL) =

(µm

(1− q

Q(tL)

)− d)y(tL) ≤ 0

and so

µm

(1− q

Q(tL)

)≤ d

or equivalently

(8) Q(tL) ≤ qµmµm − d

.

Since y < M for t ∈ (tM , tL], then for all t ∈ (tM , tL],

Q′ =pαy

y− µm(Q− q) > αpy

M− µm(Q− q)

and so

Q(t) >αpy

Mµm+ q +

(M − αpy

Mµm− q)eµm(tM−t).

Then for t = tL,

Q(tL) >αpy

Mµm+ q +

(M − αpy

Mµm− q)e−µm(tL−tM ).

If(M − αpy

Mµm− q)≥ 0, then

Q(tL) >αpy

Mµm+ q +

(M − αpy

Mµm− q)e−µm(tL−tM )

≥ αpy

Mµm+ q

>qµmµm − d

,

a contradiction to equation (8).

We now consider the case when(M − αpy

Mµm− q)≤ 0. Since y′ ≥ −dy for all

t > 0, then y(tL) ≥ y(tM )e−d(tL−tM ) or equivalently

tL − tM ≥1

dln

(M

L

).

Thene−µm(tL−tM ) ≤ e

−µmd

ln(ML )


and so(M − αpy

Mµm− q)e−µm(tL−tM ) ≥

(M − αpy

Mµm− q)e

−µmd

ln(ML ) > −q

Then

Q(tL) >αpy

Mµm+ q +

(M − αpy

Mµm− q)e−µm(tL−tM )

>αpy

Mµm

>qµmµm − d

.

However, this again contradicts (8) and therefore y > L.

Now we will show Q is bounded below by q.For t > 0,

Q′ =αpy

y− µm(Q− q)

≥ −µm(Q− q).Then

Q′ + µmQ ≥ µmqand so

Q(t) ≥ q + (Q0 − q)e−µmt > q.

Therefore Q(t) > q for all t > 0.Therefore the solutions of the system (2) are bounded away from zero.

Theorem 4.2. The solutions to the system (2) are bounded from above.

Proof. Let z = yQ and z0 = y0Q0. Then

z′ = αpy − dzwhich implies

z(t) =αpy

d+(z0 −

αpy

d

)e−dt.

Then

lim supt→∞

z(t) ≤ αpy

d

and

(9) z(t) ≤ maxz0,

αpy

d

= M.

Thus y(t)Q(t) ≤ M .

Since Q > q by Theorem 4.1, then y(t) ≤ MQ(t) <

Mq . Therefore y is bounded

above.Since y > L by Theorem 4.1, then Q(t) ≤ M

y(t) <ML . Thus Q is bounded above.

Therefore the solutions of the system (2) are bounded.


0 0.005 0.01 0.015 0.02 0.025 0.030

100

200

300

400

500

600

700

800

900

1000

Q

y

Droop model phase plane

Q nullcine

y nullcline

y nullcine

Q lower bound (q)

Figure 3. Phase plane with steady state value (y∗, Q∗) =(αpyµmqd

, qµmµm−d

)

The only steady state of the system is E1 = (y∗, Q∗) =(αpy(µm−d)

µmqd, qµmµm−d

). See

Figure 3 for the phase plane. In order for the steady state to be positive, we assumeµm − d > 0. The following theorem discusses the global stability of this positiveequilibrium point.

Theorem 4.3. The equilibrium point E1 is globally asymptotically stable.

Proof. We will first show that E1 is locally asymptotically stable. Using the Jaco-bian, we see

J(E1) =

0αpy(µm − d)

(qµm)2d−(µmqd)2

αpy(µm − d)2−µm

Then Tr(E1) = −µm < 0 and Det(E1) =

d

µm − d> 0. Thus E1 is a stable equilib-

rium point. Now we must show there are no periodic orbits. Let M be defined asin (9) and

Ω =

0 < y <

M

q, q < Q <

M

L

.

Consider the system

y′ = µm

(1− q

Q

)y − dy = F (y,Q)(10)

Q′ = αpy

y(t)− µm (Q− q) = G(y,Q)(11)


ThendF

dy+dG

dQ= µm

(1− q

Q

)− d− µm =

−µmqQ

− d < 0.

Then by the Bendixon’s negative criterion theorem, there cannot be a closed orbitcontained within Ω. Therefore, since Ω is simply connected and positively invariantand contains no orbits, by the Poincare-Bendixson Theorem, all solutions of thesystem (2) starting in Ω will converge to E1. Thus, E1 is globally asymptoticallystable.

5. Data and Simulation Results

We compare our model to clinical data from Mesiano et al. [22]. The authorsfirst studied subcutaneous (SC) tumors induced in immunodeficient mice using thehuman ovarian carcinoma cell line SKOV-3, in order to monitor the tumor growthdirectly. The data we compare to our model consists of the SC tumor volume (mm3)over time (days). Since ovarian cancer is not a subcutaneous cancer, the authorsalso studied IP tumors. However, these tumors could not be monitored directlydue to the spread within the abdomen and the results were only obtained frompostmortem examination. Thus we do not have data for the IP tumor volume overtime to compare to our model.

We performed simulations in MATLAB. After first fitting parameters by hand,we used the function fminsearchbnd, a bounded version of the built-in functionfminsearch that uses the Nelder-Mead simplex algorithm [19], to find values forthe free parameters that minimize mean square error (MSE) between data andmodel within a pre-determined bounded region. The delay differential equation (1)simulations begin on the third data point in order to avoid modeling the transitionaldynamics. Since the first three data points are fairly constant, we assume a constanty history of the average of the first three data points. We also assume a constantQ = Q0 history, which was considered a parameter and found using fminsearchbnd.The initial conditions for the ordinary differential equation (2) simulations wereassumed to be the value of the third data point for y and Q = Q0.

Figures 4-8 show the simulation results compared to the data as well as thecorresponding dynamics on the phase plane. In Figures 4-6, we are able to use thesame parameter values to model both the on-treatment and off-treatment tumorvolumes. To confirm computationally that the solution defined in equation (5)is in fact a solution to the off-treatment model, we ran the simulation with thehistory equivalent to equation (5) (Figure 5). We can see that the off-treatmentsolution is the same as y0e

λ1t and the Q value is equivalent to Q∗. As expected, theratio between the off-treatment solution and y0e

λ1t was equal to one at each timestep. We then considered a constant Q = Q0 history with the y history remainingthe exponential solution (Figure 6). The Q solution approached Q∗ and the ratiobetween the off-treatment solution and y0e

λ1t approached a constant value. Thissuggests that even when perturbing the initial condition, the solution still has thesame exponential form.


Figures 7 and 8 show solutions where the model reverses from off-treatment toon-treatment and from on-treatment to off-treatment, respectively. When reversingfrom off-treatment to on-treatment, the initial conditions of the ordinary differentialequations are last values of the off-treatment solution. When reversing from on-treatment to off-treatment, we approximate the on-treatment solution for the lastτ days with a function and use this function as the history for the delay differentialequations. These approximations are plotted in Figure 8.

0 5 10 15 20 25 30 350

100

200

300

400

500

600

700

800

Days

SC

Tu

mo

r vo

lum

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Figure 4. y vs. t (left) and phase plane (right) simulation of (1) and (2) withµm = 0.47, d = 0.28, q = 0.0064, a = 0.050, p = 0.17, Q0 = 0.014

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6. Discussion

We present a simple yet biologically meaningful model that considers ovarian tu-mor growth and tumor induced angiogenesis, subject to both on and off anti-VEGF


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treatment. The growth of the tumor is governed by the intracellular limiting nutri-ent concentration, or cell quota. We present analysis of the off-treatment model, andverify positivity of the solutions so that the solutions are biologically meaningful.Motivated by the data, we also discuss approximating the solution with an expo-nential solution. We analyze the on-treatment model, proving positivity of solutionsand the existence of a globally stable equilibrium point.

We then compare the simulation results to both on and off treatment biologicaldata. The tumor-derived VEGF activity was inhibited using the function-blockingmonoclonal antibody A4.6.1, the murine-equivalent of bevacizumab (Avastin) [11],which blocks VEGF receptors VEGFR-1(flt-1) and VEGFR-2 (KDR/flk-1). Theauthors concluded that the antibody significantly inhibited the growth of the SCtumors, by significantly inhibiting tumor vascularization, though tumor growth re-sumed once treatment stopped. From postmortem examination of the IP tumors,


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Figure 8. First row: y vs. t (left) and phase plane (right) simulation of (1)and (2) with µm = 0.67, d = 0.47, q = 0.0021, a = 0.0084, p = 0.47, Q0 = 0.0070.Second row: corresponding plot of y (left) and Q (right) history functions thatapproximate the on-treatment ODE solution in the top left figure

the authors observed partially inhibited tumor growth in the treatment group com-pared to the controls for a variety of treatment regimes. Tumor burden in treatedIP mice varied from minimal to high. Therefore, Mesiano et al. suggested that IPmetastasis might contain both angiogenesis-independent (thin layer tumor growth)and angiogenesis-dependent (large solid tumors) components. The simulations fitboth the on-treatment and off-treatment data using the same parameters as well asreversing from off- to on-treatment and vice versa, which supports using Droop’smodel and applying ecological ideas to cancer biology.

Here, we assume that the limiting nutrient is delivered to the tumor via the bloodvessels, although we do not specify the limiting nutrient. However, a future directionwould be to consider a specific limiting nutrient supplied through the blood vessels,such as oxygen, phosphorus, nitrogen, or glucose. When considering a specific nu-trient, parameter ranges can be more tightly specified, because it is likely that atleast some data regarding uptake of the specific nutrient and minimum cell quota


can be obtained. Nagy [25] suggested a “best guess” minimum intracellular phos-phorus concentration parameter value of approximately 0.01. Since our simulationssuggested a minimum cell quota less than about 0.0099, perhaps phosphorus is thelimiting nutrient provided by angiogenesis. We hope that these results will motivatebiologists to consider a limiting nutrient when collecting data in the future.

Although the model we present is quite simple, there still remain open questionsin analyzing the system. We were able to show positivity of the solutions for theoff-treatment model as well as proving the dominance of λ1 given a condition. In thesimulations presented, that given condition does not hold, yet the solutions seem totake the form y0e

λ1t. This suggests that Theorem 3.2 can be improved. We end thispaper with the following intriguing mathematical question:Is it always true that λ1 ≥ supRe(λ) : λ is any solution of (4)?

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