vascular growth paper

8/8/2019 Vascular Growth Paper

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Mathematical modelling of vascular tumour

growth and implications for therapy

Jasmina Panovska1, Helen M Byrne2, and Philip K Maini3

1 Chemical Engineering, Heriot-Watt University, Riccarton Campus, Edinburgh,EH14 4AS [email protected] Centre for Mathematical Medicine, Division of Applied Mathematics, School of

Mathematical Sciences, University of Nottingham, Nottingham NG7 [email protected] Centre for Mathematical Biology, Mathematical Institute, Oxford University,

Oxford OX1 3LB [email protected]

Summary. In this paper we briefly discuss the results of a mathematical model for-mulated in [22] that incorporates many processes associated with tumour growth.The deterministic model, a system of coupled non-linear partial differential equa-tions, is a combination of two previous models that describe the tumour-host in-teractions in the initial stages of growth [11] and tumour angiogenic process [6].Combining these models enables us to address combination cancer therapies thattarget different aspects of tumour growth. Numerical simulations show that themodel captures both the avascular and vascular growth phases. Furthermore we re-cover a number of characteristic features of vascular tumour growth such as the rateof growth of the tumour and invasion speed. We also show how our model can beused to investigate the effect of different anti-cancer therapies.

Key words: vascular tumours, angiogenesis, hypoxia, anti-cancer therapy

1.1 Introduction

Tumour growth is a complex process that involves a sequence of well orches-trated events. These characterise the initial avascular phase of growth, theangiogenesis that enables the tumour to become vascularised and the vascu-lar phase of growth. During the early stages of growth, oxygen is deliveredto the tumour cells via diffusion from nearby blood vessels and the tumourcells proliferate rapidly and consume more oxygen than the host cells [7]. Dueto diffusion limited supply of oxygen such growth is limited in size [31]. Togrow larger the tumour must undergo a cascade of processes that include thesecretion of tumour angiogenic factors, such as Vascular Endothelial GrowthFactor (VEGF). VEGF stimulates the formation of a tumour-specific vascu-


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lar network from the host vessels. Upon successful vascularisation oxygen israpidly supplied to the tumour and it can grow larger.

Within the last three decades a number of mathematical models for tu-mour growth have been developed as part of the quest to understand tumourgrowth dynamics. Most of these models focus on one particular aspect, for ex-ample, avascular growth (e.g. [27, 32]), tissue-tissue interactions (e.g [11, 28]),angiogenesis (e.g. [1, 6, 21]) or vascular tumour growth (e.g. [4, 5, 14, 18]).However, if we wish to compare and contrast the effectiveness of different

treatment protocols via mathematical modelling, we need a model that inte-grates several key processes that occur during tumour growth. A first attemptat deriving such a model was by de Angelis and Preziosi [8]. This paper pro-posed a multicell model to describe the evolution of tumour growth from theavascular stage to the vascular stage through the angiogenic process. Themodel was able to predict the formation of necrotic regions, the control ofmitosis by the presence of an inhibitory factor, the angiogenesis process withproliferation of capillaries just outside the tumour surface as well as the re-gression of the tumour and the angiogenic capillaries when angiogenesis wascontrolled or inhibited. Here we briefly describe an extended model to theone in [8] by including the density of the healthy host cells in the systemand also we model two distinct components of the vasculature, distinguish-ing between capillary tips and blood vessels. We refer the reader to [22] for

full details. In section 1.2, we present the model equations. In section 1.3 weillustrate the types of behaviour that the model yields when formulated ona one-dimensional spatial domain. The potential use of our model for test-ing anti-tumour drug protocols is illustrated in section 1.4. We present ourconclusions and comment on future research directions in section 1.5.

1.2 Model Formulation

The model we develop comprises a system of non-linear partial differentialequations and aims to reproduce the animal chamber experiments of Gimbroneet al. [12] and Muthukkaruppan et al. [17]. Thus we consider a small solidtumour implanted in the cornea of a test animal close to the limbal vessels.

Angiogenesis is quick (14-21 days) and tumour growth evolves continuouslyfrom the avascular phase, through angiogenesis to the vascular phase.

A novel feature of our model, compared to previous models, is that tumour-host interactions are active in the region while the new capillary network isforming during angiogenesis. This enables us to investigate how the couplingof tumour-host dynamics and angiogenesis influences tumour growth. To ourknowledge this has not been considered in existing models, which have tendedto focus on a single specific aspect of tumour development.

Tumour growth, via invasion of the surrounding host cells, and angiogen-esis is a multi-dimensional process. By averaging the dependent variables in a


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1 Mathematical Modelling of vascular tumour growth etc. 5

where k represents the rate of tip decrease at the limbus. The blood vessels atthe limbus supply the region with oxygen and also remove the excess VEGF .For the tumour density we impose a no-flux boundary condition at x = 0. Weassume symmetry of the tumour about its centre and hence impose no fluxboundary conditions for n2,a ,c and n3 at x = 1. We assume that initiallysome tumour cells are located at x = 1, the rest of the domain is filled withnormal cells and that the vasculature is only present near the limbus. Initiallythe region is well oxygenated and no VEGF is present.

1.3 Model Simulations

We investigate, using numerical computation, the behaviour of the model invarious parameter regimes. We use the NAG library routine DO3PCF whichdiscretises the system of equations using finite differences and solves the re-sulting system of ordinary differential equations using backward differentiation[26]. We find that, by changing parameter values, the model can simulate agrowing tumour before and after vascularisation, as well as the clearance ofthe tumour due to interactions with the host tissue. Qualitatively we can cap-ture avascular tumour growth with invasion of the host cells, the migration ofthe neovasculature during the angiogenic process as well as vascular tumour

growth characterised by the tumour growing larger and invading the host cellsmore rapidly than its avascular counterpart.

In Figure 1.1(a)-(e) we present numerical solutions of the equations (1.1)-(1.10) for different parameter values. We observe avascular tumour growth andtumour invasion of the host cells (see Figure 1.1(a)); successful angiogenesisand tumour invasion of the host cells (see Figure 1.1(b)); avascular tumourgrowth and tumour coexistence with the host cells (see Figure 1.1(c)); success-ful angiogenesis and tumour-host coexistence (see Figure 1.1(d)); and tumourregression during avascular growth only (see Figure 1.1(e)). We note thatchanges in key model parameters (competition parameters c1, c2 and oxygenconsumption 2 as well as the chemotactic parameters , ) allow us to switchfrom one type of behaviour to another. We illustrate this in the bifurcation di-agrams in Figure 1.2(a)-(c) where the parameter space is divided into distinct

regions depending on the outcomes of the simulations.Our results suggest that the success of the angiogenic process depends on

the strength of tumour-host competition: only when the tumour can competewith the host cells will angiogenesis be completed (Regions A and B in Figure1.2(a)-(b)). Thus we predict that angiogenesis must follow invasive avasculartumour growth and it is not possible for a tumour that initially regresses tothen undergo angiogenesis and invade the host cells. This is because in ourmodel the tumour cells are the main source of VEGF. Hence when the normalcells are dominant the tumour recedes and VEGF secretion decreases (seeFigure 1.1(e)). Tumour vascularisation is quicker when the tumour consumeslarger amounts of oxygen (i.e. as 2 increases; see Figure 1.3(a)). Equally,


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0 10

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Fig. 1.1. Series of plots illustrating the types of behaviour that emerge from equa-tions (1.1)-(1.10). The normal n1() and the tumour cell density n2() propagateas waves of normal cell regression and tumour invasion before and after successfulvascularisation (a)-(d); or when the normal cells are better competitors the tumourdensity regresses (e) in which case angiogenesis is unsuccessful. Following successfulvascularisation the tumour grows larger ((b) and (d)). During avascular tumour in-vasion in (a) and (c) the capillary tips n3(.) propagate from the limbus towardsthe tumour with increasing speed and increased maximum density. Post angiogenesistip profiles propagate with constant speed and either increase to a maximum valuewithin the tumour mass (b) or decrease towards the tumour centre (d). The resultsare shown at dimensionless t = 5, 10, 15 in (a), (c) and (e) and t = 20, 25 and 30 in (b)and (d). Parameter values: r1 = 4, 1 = 8, R1 = 1, r2 = 10, 2 = 15, R2 = 2, dn2 =0.0007, h = 10,1 = 0.1, r3 = 0.1, r4 = 10, p1 = 10, dc = 0.28, = 1, dn3 = 0.0001, = 0.8, = 1.5, p2 = 50, 1 = 10, s1 = 1, = 0.25, k = 30, p3 = 10, 2 =

250, a2 = 0

.9, 3 =

4 = 250

, a3 =

a4 = 0 and (a)

c1 = 10

, c2 = 5

, 2 = 50 in(b) c1 = 1, c2 = 5, 2 = 0.5 and in (c) c1 = 1, c2 = 25, 2 = 50.

increasing 2, and making the region hypoxic, in Figures 1.2(a)-(b), increasesthe size of the region B thus making it more likely for the tumour to invade thehost cells. This suggests that tumour invasion is stronger in hypoxic conditionsand this is a new prediction of our model. Combining these results we predictthat hypoxic conditions, brought about by large oxygen consumption by thetumour cells, render the tumour more invasive and able to vascularise morequickly.


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Region II

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Fig. 1.2. (a)-(b) Diagram showing how the competition (c1, c2) parameter spacecan be decomposed into distinct regions depending on the long time behaviour ofthe model solutions, for two different values of 2. In (a) 2 = 5 and in (b) 2 = 50.In regions A and B the tumour is a similar or better competitor than the normalcells, the numerical solutions evolve as travelling waves of tumour invasion of thehost cells during the avascular phase followed by successful angiogenesis and vasculartumour growth. In region C the normal cells dominate, the tumour regresses andfails to become vascularised. (c) Diagram showing the existence of a numericallycalculated region where the tumour grows as avascular or vascular depending on thevalues of the chemotactic parameters and . This parameter space is determinedwith detailed numerical simulation. The rest of the parameter values are as perFigure 1.1(a) with tumour being the better competitor and (c1, c2) B from (a)-(b). Qualitatively the results are the same when (c1, c2) A from (a)-(b).

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Fig. 1.3. (a) Numerically calculated decrease in the time when the tumour becomesvascularised as a function of the parameter 2 that controls oxygen consumption by

the tumour cells. (b)-(c) Diagrams illustrating how the (1,2) and (2, 2) param-eter spaces can be decomposed into distinct regions depending on the effectivenessof an anti-proliferative therapy. In (b) in the long term, only tumour cells are killed(II), only normal cells are killed (IV), both cell types are killed (III) by the ther-apy, or it has no effect on tumour growth (I). In (c) the therapy is not effective inregion N; the tumour regresses and is removed in region R and the tumour reachessaturated growth in region S.

Successful angiogenesis, in Figures 1.4(c)-(d), is characterised by vascu-lar profiles that propagate from the limbal vessels towards the tumour, with


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increasing speed and increasing maximum density (see Figure 1.4(c)). Inaddition the capillary tip profiles precede the vessel profiles (compare Fig-ures 1.4(c) and 1.4(d)). These are features of the so-called brush-border ef-fect associated with successful tumour vascularisation in the experiments byMuthukkaruppan [17]. By tracking the position of a point in the wave frontover time, we estimate the speed of propagation of the tips to be approxi-mately 0.03 in dimensionless units (or 0.11 mm day1 in dimensional units)near the limbus and 0.13 in dimensionless units (or 0.4 mm day1) near the

tumour in Figure 1.4(c). This agrees with experimental measurements show-ing the vascular speed increasing from 0.1-0.2 mm day1 near the limbus to0.3-0.8 mm day1 near the tumour [10]. Once the tumour is vascularised thespeed of the vascular front becomes constant and approximately that near thelimbus prior to angiogenesis. Therefore once the vessels penetrate the tumourtheir rate of propagation becomes constant. Furthermore the maximum den-sity of the capillary tips either reaches a maximum value within the tumourmass (see Figure 1.1(b)) or decreases towards the tumour centre once the tipshave penetrated the tumour (see Figure 1.1(d)). The former case occurs whenthe tumour is a better competitor than the host cells, whereas the latter caseoccurs when the tumour cells coexist with the host cells. Therefore we pre-dict that the outcome of the tumour-host interaction affects the behaviourof the vasculature during vascular growth. A large vascular density (capillary

tips in Figure 1.1(b)) during vascular tumour growth is, in our simulations,present during successful invasion of the host tissue by the tumour cells. Asmall vascular density (capillary tip density in Figure 1.1(d)) is associatedwith tumour-host coexistence during vascular growth.

The model also shows that angiogenesis enhances the ability of the tumourcells to invade the host tissue. For example, for the profiles depicted in Figure1.1(c) the tumour invasion speed increases from 0.09 (or 0.27 mm day1) to0.105 (or 0.315 mm day1) following successful angiogenesis. In addition thetumour density is much larger following angiogenesis (compare Figures 1.1(a)and 1.1(b)). These results suggest that in the later stages of tumour growth,following vascularisation, the tumour grows much larger and is more invasivethan during the initial avascular stages of tumour growth. These observationsagree with experimental observations of tumour growth in vivo [10].

1.4 Applications of the model

The model (1.1)-(1.10) can be used to investigate the effect of different anti-cancer treatments. We now extend our model to include an equation for ablood-borne drug with different modes of action. For example, model sim-ulations suggest that a tumour which in the absence of therapy invades theadjacent host tissue, when treated continuously with an anti-proliferative drugcan saturate in growth or can regress. The effect of the drug in our simulationsis thus similar to the effect of chemotherapeutic drugs such as doxorubicin [3]


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Fig. 1.4. Series of plots illustrating the profiles of the capillary tips and the vesseldensity during unsuccessful angiogenesis (a)-(b) and during successful angiogene-sis with brush border (c)-(d). The parameters are as in Figure 1.1(a) apart from = 19, = 50 for (a)-(b) and = 1, = 1.5 for (c)-(d). We plot the profiles atdimensionless t = 5, 10, 15, 20.

which target rapidly proliferating cells. In Figure 1.3(b)-(c) we depict the bi-furcation diagrams for the parameters that control the effectiveness of theanti-proliferative therapy: 1 and 2 associated with the potency of the drugon the healthy and the tumour cells and 2 representing the rate of drug up-take by the tumour cells. We predict that the therapy is most effective whenthe potency on the tumour cells and the uptake of the drug by the tumourcells are large, whereas the potency on the normal cells is small.

Alternatively we study administration of a drug that destroys the vascularnetwork. Such a drug may fall into two categories: one that targets the angio-genic stimuli (e.g. VEGF) via inhibitors such as endostatin [19] or angiostatin[20]; or a drug such as combretastatin (CAP4) [33] that directly targets theimmature blood vessels. Within our model we are able to incorporate thesedifferent modes of action of the anti-vascular drug and compare the outcomes.Qualitatively the results are the same. Our simulations predict that upon ad-ministration of anti-vascular drug tumour vascularisation can be preventedbut tumour invasion of the host cells continues in this case. The tumour den-sity resembles an avascular mass that invades with a constant speed. This isunrealistic, as we know that avascular tumours cannot grow indefinitely, and


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occurs because the model [11] does not properly include necrosis. Recently,an extension of the model in [11], has been proposed and shown to exhibitgrowth saturation [30]. A future extension of the present integrated modelwould be to include this new model.

When a combination of anti-proliferative and anti-vascular therapy areintroduced into our model, the qualitative outcome is similar to that whenonly anti-proliferative therapy is applied. Tumour invasion into the host cellscan either be halted and the tumour reaches a saturated growth, or tumour

invasion is reversed and the tumour regresses. For more details of these andother therapeutic applications see [22] and [24].

1.5 Conclusions

We believe that the mathematical model presented here enables us to betterunderstand the complex interactions that govern tumour growth. The contin-uum approach adopted in [22] allows us to make analytical predictions, forexample, of wavespeed of invasion. Recently many cellular automata (CA) ap-proaches have been developed to describe different aspects of tumour growth(see [16]). CA allows one to consider properties of individual cells but thereis little in the way of mathematical theory developed for such models. Our

model was the first deterministic model to study how tumour cells, host cellsand host blood vessels interact. With our results we can capture avascularfollowed by vascular tumour growth, as well as tumour elimination due to in-teractions with the host tissue. We can confirm that a vascular tumour is moreaggressive and grows larger than its avascular counterpart. This agrees within vivo observations of tumour growth and the modelling results presented in[4]. Furthermore we predict that it is not possible for a tumour that initiallyregresses to undergo angiogenesis and then invade the host cells. This may bea consequence of the fact that we do not allow for genetic mutations of thetumour cells. Our simulations also suggest that, during vascular growth, themaximum density of immature vessels within the tumour mass stays constantor decreases towards the tumour centre. In practise the situation which arisesdepends on the nature of the tumour-host interactions.

In terms of novel therapies, our numerical results suggest that targeting avascular tumour with a highly potent anti-proliferative drug in combinationwith reducing the VEGF influence in the region (and thus preventing angio-genesis) is the most effective treatment. When the therapy only destroys thevascular network, we predict that angiogenesis can be prevented but tumourinvasion will continue unaffected. We note that in our model there is con-tinuous infusion of the drug. Questions remain as to whether such a therapyis feasible and, of course, our modelling framework does not account for theissue of side-effects. A full critique of this modelling approach, together withpossibilities for further research are presented in [24].


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