J Dyn Diff Equat (2016) 28:1393–1414DOI 10.1007/s10884-015-9498-y

Dynamics of a Data Based Ovarian Cancer Growth andTreatment Model with Time Delay

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

Received: 31 January 2015 / Revised: 7 July 2015 / Published online: 30 September 2015© Springer Science+Business Media New York 2015

Abstract Wepresent a simplemodel that describes ovarian tumor growth and tumor inducedangiogenesis, subject to on and off anti-angiogenesis treatment. The tumor growth is governedby Droop’s cell quota model, a mathematical expression developed in ecology. Here, thecell quota represents the intracellular concentration of necessary nutrients provided throughblood supply. We present mathematical analysis of the model, including proving positivityof the solutions so that they are biologically meaningful, as well as discussing local andglobal stability. The mathematical model can be employed to fit both on-treatment and off-treatment preclinical data using the same biologically relevant parameters. We also state anopen mathematical question.

Keywords Stability · Delay equation · Droop model · Ovarian tumor model · Datavalidation

1 Introduction

Ovarian cancer, also known as the ‘silent killer’ [4,14], causes more deaths than any othergynecological malignancies and is the 5th leading cause of death from non-skin cancersamong women [15,31]. The American Cancer Society estimates 21,290 new cases of ovariancancer and 14,180 deaths due to ovarian cancer in the United States in 2015 [32]. Only about20% of ovarian cancers are detected at an early stage [4], in part due to the lack of effectivescreening strategy and in part since the indications are often symptomatic of other diseases;symptoms include abdominal discomfort or fullness, bloating, and dyspepsia [2]. Ovariancancer is characterized by intraperitoneal (IP) tumors and ascitic fluid [16,24].

B Y. Kuangkuang@asu.edu

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

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

123

1394 J Dyn Diff Equat (2016) 28:1393–1414

While cytoreductive surgery and chemotherapy are common treatments for ovarian cancer,more than 70% of advanced-stage patients will develop drug resistance and the disease recurwithin 5 years [2,4]. Thus molecular targeted therapy has been recently researched, particu-larly anti-angiogenesis therapy [2], an idea proposed by Folkman more than 40 years ago [9].Angiogenesis, the development of new blood vessels from pre-existing vessels, is essentialfor tumor growth and expansion by providing the necessary oxygen and nutrient support to thetumor [10,11,24]. Angiogenesis is regulated by pro-angiogenic and anti-angiogenic factors.When these factors become unbalanced in favor of angiogenesis, the tumor acquires angio-genic properties, known as the ‘angiogenic switch’ [11]. One of these pro-angiogenic factors,vascular endothelial growth factor (VEGF), is expressed in most malignant tumors and isone of the most important tumor angiogenesis factors [33], although its role in tumor angio-genesis is still not understood completely [35]. Although VEGF is involved in cyclic growthof ovarian follicles and corpus luteum development and maintenance [13], it is expressedhigher in women with ovarian cancers compared to those with benign tumors [15].

Several studies have investigated the relationship between VEGF and tumor growth. In1998, Mesiano et al. [24] analyzed the role of VEGF in tumor growth, progression andascites formation in ovarian cancer in tumors induced in immunodeficient mice using thehuman ovarian carcinoma cell line SKOV-3. The authors believed that tumor-derived VEGFis necessary for the formation of ascites, but may not be obligatory for IP growth. In 2000,Hu et al. [16] researched the effects of a PI3-K inhibitor, LY294002, on tumor progressionand ascites formation in the same mouse model of IP ovarian carcinoma using the OVCAR-3ovarian cancer cell line. The authors believed that LY294002 significantly inhibited growthand ascites formation.While this study did not investigate VEGF directly, the authors believeLY294002may have blocked the signal transduction pathway ofVEGF. Themonoclonal anti-VEGF antibody bevacizumab became approved by the FDA in 2004 for first-line treatmentof metastatic colorectal cancer [35]. In 2013, Ye and Chen [36] analyzed the efficacy andsafety of bevacizumab in ovarian cancer treatment using four phase III randomized controlledtrials. They concluded that combining bevacizumab to chemotherapy provided improvementin objective response rate andprogression-free survival for bothfirst-line and recurrent diseasetreatment, but provided no benefits to overall survival. These results were confirmed by a2014 systematic review of bevacizumab combined with chemotherapy for ovarian cancertreatment [2]. This result is typical of treatments targeting angiogenesis in humans. Althoughmany anticipated great benefits of anti-VEGF/VEGF receptor therapy, studies across severalcancers have shown only modest results [33].

One approach to model angiogenesis and tumor growth is to apply ecological modelingtechniques to cancer modeling. Healthy and cancerous cells live in an ecological systemwhere they interact with each other, competing for resources, nutrition, and space [3,6,18,23,25,26,28,30]. Kuang et al. [20], applied the theory of ecological stoichiometry to a modelof tumor angiogenesis. This theory considers the balance of multiple chemical substances,or sometimes energy and materials, in ecological interactions and processes [34]. One of themain hypothesis from this theory is about the growth rate hypothesis, which, “proposes thatecologically significant variations in the relative requirements of an organism for C, N andP are determined by its mass-specific growth rate because of the heavy demand for P-richribosomal RNA under rapid growth” [20,34]; organisms with high growth rates have highP:C ratios due to the increased allocation of P to RNA. Since tumor cells often have highgrowth rates, it makes sense to apply this hypothesis to cancer biology. Elser et al. [8] testedthis and determined that the growth rate hypothesis might hold true for some cancers, butnot for all cancers. Kuang et al. [20] proposed a model that consisted of healthy cells, tumorcells, and tumor microvessels, or mature vascular endothelial cells (VECs) in the tumor.

123

J Dyn Diff Equat (2016) 28:1393–1414 1395

The growth rate of these cells are possibly limited by the nutrient phosphorus, dependingon the concentration of extracellular phosphorus. The growth of the cancerous cells can alsobe limited by the lack of blood vessels, which carry important nutrients and supplies. Theauthors assumed 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 delay hasalso been used in previous mathematical models of tumor-induced angiogenesis [1,17].

We present a first approximation mathematical model of tumor growth and tumor-inducedangiogenesis in the simplest context, using a minimum number of parameters. We apply theidea of nutrient limited induced angiogenesis from Kuang et al. [20] through the use ofDroop’s Cell Quota model [7], a mathematical expression developed in ecology and usedin previous ecological and cancer models. We also express the processes of microvesselformation by a time delay.We consider the tumor growth both on and off anti-VEGF treatmentusing the same parameter set. We present our mathematical model in Sect. 2 and then presentthe analysis of the model in Sects. 3 and 4. Section 5 contains simulations and comparisonsof the model to preclinical data from Mesiano et al. [24].

2 Tumor Model

We present a simple vascularized tumor growth model where tumor growth is governed bythe Droop equation [7]. Let y represent the vascularized tumor volume and Q represent theintracellular concentration of necessary nutrients provided by angiogenesis, or the cell quotaof some limiting nutrient from angiogenesis. Our model 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)

Thegrowthof the tumor is givenby theDroop equation,whereμm represents themaximumtumor growth rate and q represents the minimum cell quota, or minimum concentration oflimiting nutrient needed to sustain the cell. The tumor death rate is assumed to be constant,represented by d . Similar to [20], we assume that it takes τ units of time for the vascularendothelial cells to respond to the angiogenic signal and mature to fully functional vessels.While a mechanistic model would need to track the blood vessels, we simplify by assumingthat the nutrient uptake rate is proportional to the nutrient concentration in the interstitialfluid, which in turn is proportional to the blood vessel density τ time units in the past. Thedelay arises because 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, it remainsstatic. VEGF is assumed to be the primary signal generating new blood vessel growth andis secreted primarily where new tumor tissue is forming, since these are the regions that arehypoxic. Parameter α represents both uptake rate of the nutrients in the interstitial fluid andresulting nutrient concentration per tumor unit. The term μm(Q − q) represents dilution ofthe nutrient as the tumor grows. We assume that when the cells die they release the nutrient,which remains in the nearby environment [20]. See Table 1 for a list of the variable andparameter meanings, units, and values.

123

1396 J Dyn Diff Equat (2016) 28:1393–1414

Table 1 Tumor model parameter ranges

Para. Meaning Unit Value Ref

y Tumor volume mm3 – [24]

Q Cell quota (cell nutrient density) mol/vol –

q Minimum cell quota mol/vol 0.0021–0.0099

μm Maximum growth rate day−1 0.41–1.58 [29]

d Death rate day−1 0.28–1.43 [29]

α Nutrient uptake coefficient mol/(volday) 0.0084–0.70

p Reduction in nutrient uptake rate – 0.17–0.47

τ Time delay day 10 [22]

In the table, vol volume unit and Ref reference

Wenowconsider themodelwhen an anti-VEGF treatment is applied.During treatment, theblood vessel growth will be impaired due to the inhibition of VEGF, but existing vasculatureis likely not to be affected by the treatment. In that case nutrient delivery no longer dependsupon the tumor volume τ days ago, but remains constant due to the static vasculature. Weassume that blood vessel sprouts that began forming within τ time units before the onset oftreatment will not be fully formed and patent. Let t0 represent the time of treatment onsetand y = y(t0 − τ). Then nutrient delivery is dependent upon y, and the delay differentialequation model 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 preclinical data from Mesiano et al. [24], the parameterα was too large during treatment. Therefore, the α during treatment cannot be the sameas during off-treatment. Thus, we introduced a discount parameter p to account for thisbiological observation. Since the nutrient uptake term is capturing the processes of angiogenicsignaling, angiogenesis itself, and uptake of nutrients, the parameter p could represent anadditional inhibition, above the treatment effect originallymodeled, in any of these processes.One possible explanation could be the characteristic blood-filled cysts that form in untreatedovarianmalignancies that are often depleted once treatment is ongoing [24]. If this explanationis correct, then blood 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 adequate analysis ofsystem (1), verifying the positivity of the solution in order to be biologically meaningful,providing a simple condition ensuring the tumor cell population tends to 0, and discussing acondition for approximating the solution. Note that system (1) has no equilibrium points.

123

J Dyn Diff Equat (2016) 28:1393–1414 1397

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

{Q(s), q + α

μmedτ , s ∈ [0, τ ]

}. If μm ≤ d, then limt→∞ y(t) = 0.

Proof We first establish the positivity of the solutions. Assume by contradiction, there existstime t1 ∈ [0, τ ] such that a trajectory with initial conditions Q(t) > q and y(t) > 0 fort ∈ [−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 for t ∈ [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) ≥ μmq

and so

Q(t) ≥ q + (Q(0) − q) e−μmt > q

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

Case 2.Assume the trajectory crosses the boundary y(t1) = 0 first or the trajectory crossesboth y(t1) = 0 and Q(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 atrajectory cannot cross the boundary y = 0 first nor both boundaries at the same time.

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

y(t) ≤ edτ . 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′ ≤ −μmq

Qy ≤ −μmq

QMy

123

1398 J Dyn Diff Equat (2016) 28:1393–1414

and so

y′(t) ≤ y(0)e− μmq

QMt.

Therefore as t→∞, 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 standardapproach for handling the system in order to study the dynamics of the solution. Biologicallywe can assume that the uptake of nutrients is on a faster time scale than the populationdynamics. Thus we apply a quasi-steady state argument by allowing Q′ = 0. Then

Q′(t) = 0 = αy(t − τ)

y(t)− μm

(Q∗(t) − q

)

and so

Q∗(t) = αy(t − τ)

μm y(t)+ q

Then

y′(t) = μm

(1 − q

Q∗(t)

)y(t) − dy(t)

= μm

(1 − qμm y(t)

αy(t − τ) + qμm y(t)

)y(t) − dy(t)

=(

μmαy(t − τ)

αy(t − τ) + qμm y(t)− d

)y(t)

= f (y(t), y(t − τ)) (3)

Note that y∗ = 0 is the only equilibrium point for the limiting system (3), assumingμmα

α+qμm�= d . The following proposition presents a simple result on the stability of this steady

state.

Proposition 3.1 If d >μmα

α+qμm, then the solutions of the limiting system (3) tend to y = 0.

Proof We prove this proposition by applying a Rezumikhin type argument [19, part (ii) of

Theorem 2.6.1] with p(s) = ρ2s for some ρ > 1. Since d >μmα

α + qμm, we see there is a

ρ > 1 such that d >μmαρ

αρ + qμm. Let V (y(t)) = y(t)2. If V (y(t + θ)) < p (V (y(t))) , for

θ ∈ [−τ, 0], then y(t + θ) < ρy(t) for θ ∈ [−τ, 0] anddV (y(t))

dt≤

(μmαρy(t)

αρy(t) + qμm y(t)− d

)y(t) =

(μmαρ

αρ + qμm− d

)y(t).

Let u(s) = v(s) = s2 and w(s) =(d − μmαρ

αρ + qμm

)s, we see that the part (ii) of Theorem

2.6.1 in [19] can be applied here with p(s) = ρ2s which implies that the solutions of thelimiting system (3) tend to y = 0. �

Motivated by the off-treatment tumor growth data in [24] (see Sect. 5), we look for theexistence of a dominating exponential solution. Let y(t) = y0eλt . Then

eλτ = α(μm − λ − d)

μmq(λ + d). (4)

123

J Dyn Diff Equat (2016) 28:1393–1414 1399

−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

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

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

is amonotone decreasingfunction of λ (see Fig. 1). Observe that if α(μm − d) > μmqd, we see that there exists aunique real eigenvalue, 0 < λ1 < μm − d, that satisfies (4). Then a solution to the system is

(y1(t), Q∗) =

(y0e

λ1t ,αe−λ1τ + μmq

μm

). (5)

However, there are also infinitely many complex solutions to the equation (4). The followingtheorem provides a sufficient condition that ensures λ1 as the dominant eigenvalue, i.e. whenthe solution can be approximated by (5). Also, note that since y1(t) = y0eλ1t is a solution tothe system, we know that y(t) is not bounded above.

Theorem 3.2 If α(μm − d) > μmqd and α < μmqeλ1τ , then λ1 ≥ sup{Re(λ) : λ is anysolution of (4)}.

Proof Consider the characteristic equation defined in (4) and let λ1 be the unique positivereal 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

e2aτ = (α(μm − a − d))2 + (αb)2

(μmq(a + d))2 + (μmqb)2= A + C

B + D(6)

123

1400 J Dyn Diff Equat (2016) 28:1393–1414

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

e2aτ > e2λ1τ = (α(μm − λ1 − d))2

(μmq(λ1 + d))2>

(α(μm − a − d))2

(μmq(a + d))2= A

B(7)

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 ≥ sup{Re(λ) : λ is any solution of (4)}. �The following proposition presents a simple result on the instability of the solution y1(t).

Proposition 3.2 If μmα(αe−2λ1τ +qμm )

(αe−λ1τ +qμm )2< d <

μmαα+qμm

, then the solution y1 = y0eλ1t is

unstable.

Proof First we must linearize around the solution y1(t) = y0eλ1t , where λ1 satisfies thefollowing equation

eλ1τ = α(μm − λ1 − d)

μmq(λ1 + d).

We let u(t) = y(t) − y1(t). Then

u′ = fy(t) (y1(t), y1(t − τ)) u(t) + fy(t−τ) (y1(t), y1(t − τ)) u(t − τ)

where

fy(t) (y1(t), y1(t − τ)) = μmα2e−2λ1τ(αe−λ1τ + qμm

)2 − d

fy(t−τ) (y1(t), y1(t − τ)) = qμ2mα(

αe−λ1τ + qμm)2

.

Then

u′ = −β1u(t) + γ1u(t − τ), (8)

where β1 = d −(

μmα2e−2λ1τ

(αe−λ1τ + qμm )2

)and γ1 =

(qμ2

mα

(αe−λ1τ + qμm )2

).

By Theorem 2.2.1 in [19], if β1 > |γ1|, then there are no stability changes as τ changes.To determine the stability, we can consider τ = 0. Then

u′(t) =(

μmα2

(α + qμm)2− d + qμ2

mα

(α + qμm)2

)u(t)

=(

μmα

α + qμm− d

)u(t)

Therefore if d <μmα

α+qμm, then u(t) will diverge, and thus y(t) will move away from y1(t).

In other words, y1(t) is unstable. �

123

J Dyn Diff Equat (2016) 28:1393–1414 1401

Observe that the equation (8) is a linear equation with parameters β1 and γ1 as analyticfunctions of the time delay. Although the geometric stability result of Beretta and Kuang in2002 [5] is applicable, the actual application is very tedious and we leave this as a futureproject.

4 Global Analysis for the System (2)

The following section provides a global analysis of the model in system (2), verifying theboundedness and invariance of the solutions and discussing the stability and global stabilityof the positive equilibrium solution.

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

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

tL = min{t > 0 : y(t) = L}, i.e., let tL be the first time y(t) reaches L . We now choose Msuch that L < M < y0. Let tM = max{t < tL : y(t) = M}, i.e., let tM be the last time y(t)reaches M before it reaches L . Then 0 < tM < tL and y(t) < M for t ∈ (tM , tL ]. See Fig. 2for a sketch.

We first choose M such that

M < min

{y0,

αpy(μm − d)

qμ2m

}

which implies that

αpy

Mμm>

qμm

μm − d.

We now choose L such that∣∣∣∣(M − αpy

Mμm− q

)e

−μmd ln

(ML

)∣∣∣∣ < q,

Fig. 2 Sketch of proof forTheorem 4.1

M

L

tM tL

y

t

123

1402 J Dyn Diff Equat (2016) 28:1393–1414

or equivalently

L < M

⎛⎝ q∣∣∣M − αpy

Mμm− q

∣∣∣

⎞⎠

dμm

.

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

Q(tL ) ≤ qμm

μm − d. (9)

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

Q′ = pα y

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

>qμm

μm − d,

a contradiction to equation (9).

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

).

Then

e−μm (tL − tM ) ≤ e−μmd ln

(ML

)

123

J Dyn Diff Equat (2016) 28:1393–1414 1403

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

>qμm

μm − d.

However, this again contradicts (9) 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 ≥ μmq

and 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 − dz

which implies

z(t) = αpy

d+

(z0 − αpy

d

)e−dt .

Then

lim supt→∞

z(t) ≤ αpy

d

and

z(t) ≤ max

{z0,

αpy

d

}= M . (10)

Thus y(t)Q(t) ≤ M .Since Q > q by Theorem 4.1, then y(t) ≤ M

Q(t) < Mq . Therefore y is bounded above.

Since y > L byTheorem4.1, then Q(t) ≤ My(t) < M

L . ThusQ is bounded above. Thereforethe solutions of the system (2) are bounded. �

123

1404 J Dyn Diff Equat (2016) 28:1393–1414

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 nullciney nullcliney nullcineQ lower bound (q)

Fig. 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

,qμm

μm−d

). See Fig. 3

for the phase plane. In order for the steady state to be positive, we assume μm − d > 0. Thefollowing theorem discusses the global stability of this positive equilibrium point.

Theorem 4.3 The equilibrium point E1 is globally asymptotically stable.

Proof Wewill first show that E1 is locally asymptotically stable. Using the Jacobian, 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 equilibrium

point. Now we must show there are no periodic orbits. Let M be defined as in (10) and

={0 < y <

M

q, q < Q <

M

L

}.

Consider the system

y′ = μm

(1 − q

Q

)y − dy = F(y, Q) (11)

Q′ = αpy

y(t)− μm (Q − q) = G(y, Q) (12)

Then

dF

dy+ dG

dQ= μm

(1 − q

Q

)− d − μm = −μmq

Q− d < 0.

Then by the Bendixon’s negative criterion theorem, there cannot be a closed orbit containedwithin . Therefore, since is simply connected and positively invariant and contains no

123

J Dyn Diff Equat (2016) 28:1393–1414 1405

orbits, by the Poincaré-Bendixson Theorem, all solutions of the system (2) starting in willconverge to E1. Thus, E1 is globally asymptotically stable. �

5 Data and Simulation Results

We compare our model to preclinical data fromMesiano et al. [24]. The authors first studiedsubcutaneous (SC) tumors induced in immunodeficient mice using the human ovarian carci-noma cell line SKOV-3, in order to monitor the tumor growth directly. The data we compareto our model consists of the SC tumor volume (mm3) over time (days). Since ovarian canceris not a subcutaneous cancer, the authors also studied IP tumors. However, these tumors couldnot be monitored directly due to the spread within the abdomen and the results were onlyobtained from postmortem examination. Thus we do not have data for the IP tumor volumeover time to compare to our model.

We performed simulations in MATLAB. After first fitting parameters by hand, we usedthe function fminsearchbnd, a bounded version of the built-in function fminsearch that usesthe Nelder-Mead simplex algorithm [21], to find values for the free parameters that minimizemean square error (MSE) between data and model within a pre-determined bounded region.The delay differential equation (1) simulations begin on the third data point in order to avoidmodeling the transitional dynamics. Since the first three data points are fairly constant, weassume a constant y history of the average of the first three data points. We also assume aconstant Q = Q0 history, whichwas considered a parameter and found using fminsearchbnd.The initial conditions for the ordinary differential equation (2) simulations were assumed tobe the value of the third data point for y and Q = Q0.

Figures 4, 5, 6, 7, and 8 show the simulation results compared to the data as well as thecorresponding dynamics on the phase plane. In Figs. 4, 5, and 6, we are able to use thesame parameter values to model both the on-treatment and off-treatment tumor volumes. Toconfirm computationally that the solution defined in equation (5) is in fact a solution to theoff-treatmentmodel, we ran the simulationwith the history equivalent to equation (5) (Fig. 5).We can see that the off-treatment solution is the same as y0eλ1t and the Q value is equivalentto Q∗. As expected, the ratio between the off-treatment solution and y0eλ1t was equal to oneat each time step.We then considered a constant Q = Q0 historywith the y history remainingthe exponential solution (Fig. 6). The Q solution approached Q∗ and the ratio between theoff-treatment solution and y0eλ1t approached a constant value. This suggests that even whenperturbing the initial condition, the solution still has the same exponential form.

Figures 7 and 8 show solutions where the model reverses from off-treatment to on-treatment and from on-treatment to off-treatment, respectively. When reversing fromoff-treatment to on-treatment, the initial conditions of the ordinary differential equations arelast values of the off-treatment solution. When reversing from on-treatment to off-treatment,we approximate the on-treatment solution for the last τ days with a function and use thisfunction as the history for the delay differential equations. These approximations are plottedin Fig. 8.

We also performed simulations in MATLAB to determine how the solutions changefor given changes in parameters (Fig. 9). We first produced bifurcation diagrams for theon-treatment ODE system (2) for the given parameters. Since the steady state is globallyasymptotically stable, the bifurcation diagrams show how the steady state y∗ = αpy(μm−d)

μmqdchanges with respect to changes in the parameters. The off-treatment DDE system givenin (1) has no equilibrium points and a solution to the reduced system (3) is an exponential

123

1406 J Dyn Diff Equat (2016) 28:1393–1414

05

1015

2025

3035

0

100

200

300

400

500

600

700

800

Day

s

SC Tumor volume (mm3)

off-t

reat

men

tso

lnon

-tre

atm

ent

soln

Dru

gA

4.6.

1C

ontr

olP

BS

DD

Eh

isto

ry

00.

010.

020.

030.

040.

050.

060.

070.

080.

090.

10

200

400

600

800

1000

1200

1400

1600

1800

2000

Q

y

off-t

reat

men

tso

lnon

-tre

atm

ent

soln

yn

ull

clin

eQ

on-t

reat

.n

ull

.Q

ineq

n(5

)q

Fig.4

yversus

t(left)andph

aseplane(right)simulationof

(1)and(2)with

μm

=0.41

,d

=0.28

,q

=0.00

64,α

=0.05

0,p

=0.17

,Q0

=0.01

4

123

J Dyn Diff Equat (2016) 28:1393–1414 1407

05

1015

2025

3035

0

100

200

300

400

500

600

700

800

Day

s

SC Tumor volume (mm3)

off-t

reat

men

tso

lnon

-tre

atm

ent

soln

Dru

gA

4.6.

1C

ontr

olP

BS

DD

Eh

isto

ryy

=e

λ1t

on-t

reat

.st

ead

yst

ate

00.

020.

040.

060.

080.

10.

120.

140.

160.

180

200

400

600

800

1000

1200

1400

1600

1800

2000

Q

y

off-t

reat

men

tso

lnon

-tre

atm

ent

soln

yn

ull

clin

eQ

on-t

reat

.n

ull

.Q

ineq

n(5

)q

Fig.5

yversus

t(left)andph

aseplane(right)simulationof

(1)and(2)with

μm

=0.87

,d

=0.73

,q

=0.00

99,α

=0.36

,p

=0.18

,Q0

=0.01

4

123

1408 J Dyn Diff Equat (2016) 28:1393–1414

05

1015

2025

3035

0

100

200

300

400

500

600

700

800

Day

s

SC Tumor volume (mm3)

off-t

reat

men

tso

lnon

-tre

atm

ent

soln

Dru

gA

4.6.

1C

ontr

olP

BS

DD

Eh

isto

ryy

=e

λ1t

on-t

reat

.st

ead

yst

ate

00.

005

0.01

0.01

50.

020.

025

0.03

0.03

50.

040.

045

0

200

400

600

800

1000

1200

1400

1600

1800

2000

Q

y

off-t

reat

men

tso

lnon

-tre

atm

ent

soln

yn

ull

clin

eQ

on-t

reat

.n

ull

.Q

ineq

n(5

)q

Fig.6

yversus

t(left)andph

aseplane(right)simulationof

(1)and(2)with

μm

=0.64

,d

=0.43

,q

=0.00

63,α

=0.04

8,p

=0.23

,Q0

=0.01

1

123

J Dyn Diff Equat (2016) 28:1393–1414 1409

05

1015

2025

3035

400

200

400

600

800

1000

1200

Day

s

SC Tumor volume (mm3)

off-t

reat

men

tso

lnon

-tre

atm

ent

soln

Dru

gA

4.6.

1C

ontr

olP

BS

DD

Eh

isto

ryT

reat

men

tR

ever

se

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0

200

400

600

800

1000

1200

Q

y

off-t

reat

men

tso

lnon

-tre

atm

ent

soln

yn

ull

clin

eQ

on-t

reat

.n

ull

.

Qin

eqn

(5)

q

Fig.7

yversus

t(left)andph

aseplane(right)simulationof

(1)and(2)with

μm

=1.58

,d

=1.43

,q

=0.00

53,α

=0.70

,p

=0.23

,Q0

=0.00

60

123

1410 J Dyn Diff Equat (2016) 28:1393–1414

05

1015

2025

3035

4045

50050100

150

200

250

300

350

400

450

500

Day

s

SC Tumor volume (mm3)

off-t

reat

men

tso

lnon

-tre

atm

ent

soln

Dru

gA

4.6.

1C

ontr

olP

BS

DD

Eh

isto

ryT

reat

men

tR

ever

se

24

68

1012

x 10

−3

050100

150

200

250

300

350

400

450

500

Q

y

off-t

reat

men

tso

lnon

-tre

atm

ent

soln

yn

ull

clin

eQ

on-t

reat

.n

ull

.Q

ineq

n(5

)q

510

1520

2530

76788082848688909294

Day

s

y

y O

DE

sol

n

y hi

stor

y fu

nctio

n

510

1520

2530

7

7.1

7.2

7.3

7.4

7.5

7.6

7.7

7.8

x 10

−3

Day

s

Q

Q O

DE

sol

n

Q h

isto

ry fu

nctio

n

Fig.8

Firstrow

yversus

t(left)andph

aseplane(right)simulationof

(1)and(2)with

μm

=0.67

,d

=0.47

,q

=0.00

21,α

=0.00

84,p

=0.47

,Q0

=0.00

70.S

econ

drow

correspo

ndingplot

ofy(left)andQ

(right)historyfunctio

nsthatapprox

imatetheon

-treatmentO

DEsolutio

nin

thetopleftfig

ure

123

J Dyn Diff Equat (2016) 28:1393–1414 1411

α0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Tum

orvo

lum

e(y

)

0

500

1000

1500

2000

2500

3000

3500

4000

y

α0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

Tum

orgr

owth

rate

( λ1)

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

λ1

α(um − d) = umqd

q0 0.005 0.01 0.015 0.02 0.025

Tum

orvo

lum

e( y

)

0

2000

4000

6000

8000

10000

12000

14000y

q0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Tum

orgr

owth

rate

(λ1)

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2λ1

α(um − d) = umqd

d0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Tum

orvo

lum

e(y

)

0

2000

4000

6000

8000

10000

12000

14000

16000

18000y

d = μm

d0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Tum

orgr

owth

rate

(λ1)

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3λ1

α(um − d) = umqd

μm

0 0.5 1 1.5 2 2.5 3

Tum

orvo

lum

e(y

)

0

100

200

300

400

500

600

700

yy = apy

qd

um = d

um

0 0.5 1 1.5 2 2.5 3

Tum

orgr

owth

rate

(λ1)

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

λ1

α(um − d) = umqd

eλ1t = αq(λ1+d)

p0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Tum

orvo

lum

e( y

)

0

200

400

600

800

1000

1200

y

τ0 10 20 30 40 50 60 70 80 90 100

Tum

orgr

owth

rate

(λ1)

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11λ1

Fig. 9 On-treatment tumor volume versus parameters (left) and off-treatment growth rate versus parameters(right) for α (first row), q (second row), d (third row), μm (fourth row), and p, τ (last row). μm = 0.41, d =0.28, q = 0.0064, α = 0.050, p = 0.17, Q0 = 0.014 for the parameters not given

123

1412 J Dyn Diff Equat (2016) 28:1393–1414

function. Thus for the DDE system, we decided to look at how the growth rate λ1 (the realsolution of the characteristic equation eλτ = α(μm−λ−d)

μmq(λ+d)) changes with respect to the given

parameters.We can see that the solutions make sense biologically. As the nutrient uptake α increases

(first row), the on-treatment tumor volume also increases (left). Similarly, the off-treatmentsolution y1(t) = y0eλ1t grows at a faster rate since λ1 also increases with α (right). Notethat when α = 0, the on-treatment solution y = 0 and when α(μm − d) = μmqd , givenby the dashed red line, λ1 = 0. As the minimum intracellular nutrient concentration qincreases or the death rate d increases (second and third rows), the on-treatment solution yand off-treatment growth rate λ1 decrease. As the maximum growth rateμm increases (fourthrow), the on-treatment solution y and λ1 also increase accordingly. Note that ifμm = d , theny = 0 and the off-treatment solution remains constant. However, if we consider a proportionalrelationship between the death rate and maximum growth rate by letting d = kμm wherek < 1, then the on-treatment steady state y∗ = αpy(1−k)

qkμm. Assuming this relationship between

μm and d , then we can see that y∗ is inversely proportional to μm . This suggests that whileon-treatment and assuming a proportional relationship between μm and d , a faster growingovarian tumor may have a smaller equilibrium size. The discount parameter p (last row, left)represents an additional inhibition in the process of angiogenic signaling, angiogenesis itself,and/or uptake of nutrients when on-treatment. As p increases to maximum of 1, meaningthere is no additional inhibition, the on-treatment solution increases. Note that when p = 0,meaning there is complete inhibition, the on-treatment tumor volume y = 0. As the delayτ increases, the growth rate λ1 decreases. Thus, the fastest off-treatment growth rate occurswhen τ = 0, meaning the vascular endothelial cells respond to the angiogenic signal andmature to fully functional vessels instantaneously.

6 Discussion

We presented a simple yet biologically meaningful model that considered ovarian tumorgrowth and tumor induced angiogenesis, subject to both on and off anti-VEGF treatment.The growth of the tumor was governed by the intracellular limiting nutrient concentration,or cell quota. We presented analysis of the off-treatment model, and verified positivity ofthe solutions so that the solutions are biologically meaningful. Motivated by the data, wealso discussed approximating the solution with an exponential solution. We analyzed theon-treatment model, proving positivity of solutions and the existence of a globally stableequilibrium point.

We then compared the simulation results to both on and off treatment biological data.The tumor-derived VEGF activity was inhibited using the function-blocking monoclonalantibody A4.6.1, the murine-equivalent of bevacizumab (Avastin) [12], which blocks VEGFreceptors VEGFR-1(flt-1) and VEGFR-2 (KDR/flk-1). The authors believed that the anti-body significantly inhibited the growth of the SC tumors, by significantly inhibiting tumorvascularization, though tumor growth resumed once treatment stopped. From postmortemexamination of the IP tumors, the authors observed partially inhibited tumor growth in thetreatment group compared to the controls for a variety of treatment regimes. Tumor burdenin treated IP mice varied from minimal to high. Therefore, Mesiano et al. suggested thatIP metastasis might contain both angiogenesis-independent (thin layer tumor growth) andangiogenesis-dependent (large solid tumors) components. The simulations fit both the on-treatment and off-treatment data using the same parameters as well as reversing from off- to

123

J Dyn Diff Equat (2016) 28:1393–1414 1413

on-treatment and vice versa, which supports using Droop’s model and applying ecologicalideas to cancer biology.

Here, we assumed that the limiting nutrient is delivered to the tumor via the blood vessels,although we did not specify the limiting nutrient. However, a future direction would be toconsider a specific limiting nutrient supplied through the blood vessels, such as oxygen,phosphorus, nitrogen, or glucose. When considering a specific nutrient, parameter rangescan be more tightly specified, because it is likely that at least some data regarding uptake ofthe specific nutrient and minimum cell quota can be obtained. Nagy [27] suggested a “bestguess” minimum intracellular phosphorus concentration parameter value of approximately0.01. Since our simulations suggested a minimum cell quota less than about 0.0099, perhapsphosphorus is the limiting nutrient provided by angiogenesis. We hope that these results willmotivate biologists to consider a limiting nutrient when collecting data in the future.

Although the model we presented is quite simple, there still remain open questions inanalyzing the system. We were able to show positivity of the solutions for the off-treatmentmodel as well as proving the dominance of λ1 given a condition. In the simulations presented,that given condition does not hold, yet the solutions seem to take the form y0eλ1t . Thissuggests that Theorem 3.2 can be improved. We end this paper with the following intriguingmathematical question:Is it always true that λ1 ≥ sup{Re(λ) : λ is any solution of (4)}?

Acknowledgments We would like to thank the referee for many helpful suggestions. This work is partiallysupported by an ARCS scholarship to Rebecca Everett and by NSF Grants DMS 1148771 and DMS-1518529.

References

1. Agur, Z., Arakelyan, L., Daugulis, P., Ginosar, Y.: Hopf point analysis for angiogenesis models. Discret.Contin. Dyn. Syst. 4(1), 29–38 (2004)

2. Aravantinos, G., Pectasides, D.: Bevacizumab in combination with chemotherapy for the treatment ofadvanced ovarian cancer: a systematic review. J. Ovarian Res. 7(57), 1–13 (2014)

3. Basanta, D., Anderson, A.R.A.: Exploiting ecological principles to better understand cancer progressionand treatment. Interface Focus 3, 20130020 (2013)

4. Bast Jr, R.C., Hennessy, B., Mills, G.B.: The biology of ovarian cancer. Nat. Rev. Cancer 9, 415–528(2009)

5. Beretta, E., Kuang, Y.: Geometric stability switch criteria in delay differential dystems with delay depen-dent parameters. SIAM J. Math. Anal. 33, 1144–1165 (2002)

6. Bickel, S.T., Juliano, J.D., Nagy, J.D.: Evolution of proliferation and the angiogenic switch in tumorswith high clonal diversity. PLoS One 9(4), e91992 (2014)

7. Droop, M.R.: Vitamin B12 and marine ecology, iv: the kinetics of uptake, growth and inhibition inmonochrysis lutheri. J. Mar. Biol. Assoc. UK 48(3), 689–733 (1968)

8. Elser, J.J., Kyle, M.M., Smith, M.S., Nagy, J.D.: Biological stoichiometry in human cancer. PLoS One10, e1028 (2007)

9. Folkman, J.: Tumor angiogenesis: therapeutic implications. N. Engl. J. Med. 118, 1182–1186 (1971)10. Folkman, J.: What is the evidence that tumors are angiogenesis dependent? J. Natl. Cancer Inst. 82(1),

4–6 (1990)11. Folkman, J.: Role of angiogenesis in tumor growth and metastasis. Semin. Oncol. 29(6), 15–18 (2002)12. Gerber, H.-P., Ferrara, N.: Pharmacology and pharmacodynamics of bevacizumab as monotherapy or in

combination with cytotoxic therapy in preclinical studies. Cancer Res. 65(3), 671–680 (2005)13. Geva, E., Jaffe, R.B.: Role of vascular endothelial growth factor in ovarian physiology and pathology.

Fertil. Steril. 74(3), 429–438 (2000)14. Goff, B.A., Mandel, L., Muntz, H.G., Melancon, C.H.: Ovarian carinoma diagnosis. Cancer 89(10),

2068–2075 (2000)15. Gómez-Raposo, C., Mendiola, M., Barriuso, J., Casado, E., Hardisson, D., Redondo, A.: Angiogenesis

and ovarian cancer. Clin. Transl. Oncol. 11, 564–571 (2009)

123

1414 J Dyn Diff Equat (2016) 28:1393–1414

16. Limin, H., Zaloudek, C., Mills, G.B., Gray, J., Jaffe, Robert B.: In Vivo and in Vitro ovarian carcinomagrowth inhibition by a phosphatidylinositol 3-kinase inhibitor (LY294002). Clin. Cancer Res. 6, 880–886(2000)

17. Jain, H.V., Nör, J.E., Jackson, T.L.: Modeling the VEGF-Bcl-2-CXCL8 pathway in intratumoral angio-genesis. Bull. Math. Biol. 70, 89–117 (2008)

18. Korolev, K.S., Xavier, J.B., Gore, J.: Turning ecology and evolution against cancer. Nat. Rev. Cancer 14,371–380 (2014)

19. Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press Inc,San Diego (1993)

20. Kuang, Y., Nagy, J.D., Elser, J.J.: Biological stoichiometry of tumor dynamics: mathematical models andanalysis. Discret. Cont. Dyn. Syst. 4(1), 221–240 (2004)

21. Lagarias, J.C., Reeds, J.A., Wright, M.H., Wright, P.E.: Convergence properties of the nelder-meadsimplex method in low dimensions. SIAM J. Optim. 9(1), 112–147 (1998)

22. Leunig, M., Yuan, F., Menger, M.D., Boucher, Y., Goetz, A.E., Messmer, K., Jain, R.K.: Angiogenesis,microvascular architecture, microhemodynamics, and interstitial fluid pressure during early growth ofhuman adenocarcinoma LS174T in SCID mice. Cancer Res. 52, 6553–6560 (1992)

23. Lauren, L.M.F., Pepper, J.W., Reid, B.J., Maley, C.C.: Cancer as an evolutionary and ecological process.Nat. Rev. Cancer 6, 924–935 (2006)

24. Mesiano, S., Ferrara, N., Jaffe, R.B.: Role of vascular endothelial growth factor in ovarian cancer: inhi-bition of ascites formation by immunoneutralization. Am. J. Pathol. 153(4), 1249–1256 (1998)

25. Nagy, J.D.: Competition and natural selection in a mathematical model of cancer. Bull. Math. Biol. 66,663–687 (2004)

26. Nagy, J.D.: The ecology and evolutionary biology of cancer: a review of mathematical models of necrosisand tumor cell diversity. Math. Biosci. Eng. 2(2), 381–418 (2005)

27. Nagy, J.D.: Hypertumors in cancer can be caused by tumor phosphorus demand. Proc. Appl. Math. Mech.7, 1121703–1121704 (2007)

28. Nagy, J.D., Armbruster, D.: Evolution of uncontrolled proliferation and the angiogenic switch in cancer.Math. Biosci. Eng. 9(4), 843–876 (2012)

29. Panetta, J.C.: A mathematical model of breast and ovarian cancer treated with paclitaxel. Math. Biosci.146, 89–113 (1997)

30. Pienta, K.J., McGregor, N., Axelrod, R., Axelrod, D.E.: Ecological therapy for cancer: defining tumorsusing an ecosystem paradigm suggests new opportunities for novel cancer treatments. Transl. Oncol. 1(4),158–164 (2008)

31. Siegel, R., Ma, J., Zou, Z., Jemal, A.: Cancer statistics, 2014. Cancer J. Clin. 64(1), 9–29 (2014)32. Siegel, R., Miller, K.D., Jemal, A.: Cancer statistics, 2015. Cancer J. Clin. 65(1), 5–29 (2015)33. Sitochy, B., Nagy, J.A., Dvorak, H.F.: Anti-VEGF/VEGFR therapy for cancer: reassessing the target.

Cancer Res. 72(8), 1909–1914 (2012)34. Sterner, R.W., Elser, J.J.: Ecological Stoichiometry The Biology of Elements from Molecules to the

Biosphere. Princeton University Press, Princeton (2002)35. Waldner, M.J., Neurath, M.F.: Targeting the VEGF signaling pathway in cancer therapy. Expert Opin.

Ther. Targets 16(1), 5–13 (2012)36. Ye, Q., Chen, H.-L.: Bevacizumab in the treatment of ovarian cancer: a meta-analysis from four phase iii

randomized controlled trials. Arch. Gynecol. Obstet. 288(3), 655–666 (2013)

123