Anti-Angiogenic and Chemotherapeutic Drug

8/4/2019 Anti-Angiogenic and Chemotherapeutic Drug

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Mathematical Modelling of Flow in 2D and 3D Vascular

Networks: Applications to Anti-angiogenic and

Chemotherapeutic Drug Strategies*

A. Stphanou1

, S.R. McDougall1

, A.R.A. Anderson2

, M.A.J. Chaplain2

, and J.A Sherratt3

1 Institute of Petroleum Engineering, Heriot-Watt University,Edinburgh EH14 4AS, Scotland

2 The SIMBIOS Centre, Department of Mathematics, University of Dundee,

Dundee DD1 4HN, Scotland.3 School of Mathematical and Computer Sciences, Heriot-Watt University,

Edinburgh EH14 4AS, Scotland

Abstract

The aim of this paper is to investigate the conditions required to optimize the amount of chemotherapeutic

drug absorbed by a solid tumour through a network of blood vessels. This work is based on a study of

vascular networks generated from a discrete mathematical model of tumour angiogenesis (Anderson andChaplain, 1998), which describes the formation of a capillary network in response to chemical stimuli

released by a solid tumour. Simulations of blood flow in the vasculature connecting the parent vessel to the

solid tumour are then performed by adapting modelling techniques from the field of petroleum engineering

to this biomedical application (McDougall et al., 2002).We begin with a qualitative, comparative study relating to the efficiency of drug delivery in 2D

and 3D tumour-induced vasculatures and evaluate the influence of key parameters (mean capillary radius,

blood viscosity and delivery regime) upon uptake by the tumour. We then go on to examine the impact of

the vascular architecture upon nutrient (e.g. oxygen) and drug delivery by comparing the efficiency of threevasculatures characterised by different spatial distributions of branching order and anastomosis density (i.e.

the number of fused loops or connections). We identify the main criteria required of a tumour-induced

vascular network for optimised delivery of nutrients and/or cytotoxic agents.We conclude by focussing on a particular vascular network and investigate how capillary

pruning (i.e. network re-modelling) modifies the network connectivity and associated blood flow

distribution. We demonstrate how random removal of vessels may lead to a significant increase in the

amount of drug delivered to the tumour. Selective removal of vessels characterised by low flow is seen to

speed up delivery, whilst the targeting of high flow capillaries leads to a rapid shut down of the entirecapillary bed. These results allow us to propose the possibility of optimised cancer treatment therapies,

based upon a coupled anti-angiogenic/chemotherapy strategy the anti-angiogenesis treatment would be

used to optimise network efficiency, thereby maximising drug uptake during subsequent chemotherapy

treatments.

1 Introduction

Angiogenesis is the process by which new blood vessels develop from an existing vasculature,

through endothelial cell sprouting, proliferation and fusion (Risau, 1997). Adult endothelial cells

are normally quiescent and, apart from certain developmental processes (e.g. embryogenesis), and

wound healing, angiogenesis is generally a pathological process implicated in arthritis (Walsh,1999), some eye diseases and tumour development, invasion and metastasis (Folkman, 1995).

* Article submitted to the Journal of Mathematical and Computer Modelling

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Tumour angiogenesis is believed to occur when a small avascular tumour exceeds some critical

diameter (~2 mm), above which normal tissue vasculature is no longer able to support its growth

(Folkman, 1971). At this stage, the tumour cells lacking nutrients and oxygen become hypoxic.

This is assumed to trigger cellular release of tumour angiogenic factors (TAFs) (Folkman and

Klagsbrun, 1987), which start to diffuse into the surrounding tissue and approach the endothelial

cells of nearby blood vessels. Endothelial cells subsequently respond to the TAF concentration

gradient by forming sprouts, dividing and migrating towards the tumour (Schoefl, 1963;Ausprunk and Folkman, 1977; Sholley et al. 1984). It takes approximately 10 to 21 days for the

growing network to link the tumour to the parent vessel (Gimbrone et al., 1974; Ausprunk andFolkman, 1977; Muthukkaruppan et al., 1982), and this vascular connection subsequently

provides all the nutrients and oxygen required for continued tumour growth.

At this stage in tumour development, chemotherapy treatments can be administered and tumour

cells can be specifically targeted via the newly developed vasculature. However, although most ofthe drugs used to date have proven to be successful on small animals (e.g. mice), their efficiency

in humans remains highly variable from one patient to another. One of the main reasons

suggested to explain such variability is the issue of drug delivery to the tumour it is thought

possible that most of the drug can bypass large areas of the target (Jain, 1987; Jain, 1988). Based

on this fact, more theoretical models are being developed to try to better understand vasculararchitecture (Secomb, 1995; Godde, 2001; Krenz, 2002) and solve this issue (Baish, 1996; El-

Kareh, 1997; Jackson, 1999; Quarteroni, 2000).

McDougall et al. (2002) were among the first to specifically model flow in a tumour-induced

vascular network. This was achieved by adapting computational techniques that have been

extensively developed in the context of the petroleum engineering industry to model flow through

porous media. These methods were successfully applied in a biomedical setting, in particular to

assess the efficiency of the drug delivery process to different types of vasculature. Indeed,

vascular networks share some topological similarities with interconnected pore networks. This

earlier study examined the relative importance of a variety of associated parameters either

related to the vascular architecture itself (vessel radii) or to the blood properties (viscosity), which

might influence the efficacy of the drug delivery process. It was thus possible to understand thecircumstances under which bypassing effects might occur and to predict, for a given vasculature,

the success of chemotherapy treatment.

In this paper, we extend the earlier work considerably in a number of important ways:

(i) 3D networks are generated and drug-uptake results compared with those from previous

2D simulations. We show that both the vascular connectivity and the system dimensionality affect

drug delivery to the tumour.

(ii) a number of different vascular architectures are examined and the spatial distribution of

the branching order and the anastomosis density (i.e. the number of fused connections) are

quantified. These distributions are then used to infer treatment efficiency.

(iii) capillary pruning of a flowing vasculature is studied, whereby certain capillary elementsare removed, either randomly or according to flow criteria. We then discuss how the results could

help to define new cancer treatmenttherapies, such as a coupled treatment strategy that uses anti-

angiogenic and chemotherapeutic agents sequentially.

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2 A mathematical model for tumour-induced angiogenesis

2.1 A discretestochastic description of network growth

All of the vasculatures considered in this paper were generated on the basis of the tumour-

induced angiogenesis model proposed by Anderson and Chaplain (1998), Chaplain and Anderson

(1999). This model describes how endothelial cells emerging from a parent vessel, respond and

migrate through (i) random motility, (ii) chemotaxis via gradients of tumour angiogenic factor

(TAF) released by the tumour and (iii) haptotaxis via fibronectin gradients in the extracellular

matrix. We denote the endothelial cell density per unit area n, the TAF concentration c and thefibronectin concentration f. The nondimensional system of partial differential equationsdescribing the vascular growth process is thus given by:

( ) ( )

}

}

.

,

,)(

deg

2

uptake

radationproduction

haptotaxischemotaxisrandom

cnt

c

fnnt

f

fncncnDt

n

=

=

=

876

484764484476876

(1)

The chemotactic migration is characterized by the function (c)=/(1+c) which reflects thedecrease in chemotactic sensitivity with increased TAF concentration. The coefficientsD, and characterize the random, haptotactic and chemotactic cell migration respectively. and are

coefficients describing the rates of TAF uptake and fibronectin degradation by the endothelial

cells respectively, and is the rate at which fibronectin is produced by migrating endothelial

cells.

In order to capture the salient features of network growth, such as branching and anastomosis

(loop formation), we use the 3D discretized form of the system of partial differential equations(see McDougall et al., 2002 for the 2D implementation). The discrete model allows us to track

the motion of individual endothelial cells located at the capillarysprout tips and the subsequent

formation of capillaries. The discretized 3D system of equations is obtained by using the standardEuler finite difference approximation (Mitchell and Griffiths, 1980):

[ ][ ].1

,1

,

,,,,

1

,,

,,,,,,

1

,,

61,,51,,4,1,3,1,2,,11,,10,,

1

,,

q

wml

q

wml

q

wml

q

wml

q

wml

q

wml

q

wml

q

wml

q

wml

q

wml

q

wml

q

wml

q

wml

q

wml

q

wml

nkcc

nknkff

PnPnPnPnPnPnPnn

=

+=

++++++=

+

+

++++

(2)

where l, m, w and q are positive parameters which specify the location on the grid and the time

step i.e.,x=lx, y=my, z=wzand t=qt. The migration of an individual endothelial cell locatedat the tip of a sprout is determined by the set of coefficientsP0 toP6which are proportional to theprobabilities of the endothelial cell being stationary (P0), or moving left (P1), right (P2), up (P3),

down (P4), out of the plane (P5), or into the plane (P6). These coefficients incorporate the effects

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of random, chemotactic and haptotactic movement and strongly depend upon the local chemical

environment (fibronectin and TAF concentrations).

The processes of branching (formation of new sprouts from existing sprout tips) and anastomosis

(formation of loops by fusion of two colliding capillary sprouts) are incorporated into the

discretized form of the model by assuming the following:

for branching:

the probability of an existing sprout branching increases with the local TAF concentration, a sprout must reach a certain level of maturation before it becomes capable of branching.

for anastomosis:

if two sprouts collide as they grow, only one of them is allowed to keep growing (the choice of

which is random), if a sprout tip meets another sprout, they fuse to form a loop (Paweletz and Knierim, 1989).

2.2 Simulation details of the discretestochastic model

Simulations are performed by calculating the coefficients P0 to P6 at each time step and theseprobabilities define the potential displacement directions for each endothelial cell located at each

sprout tip. Probability ranges are then computed by summing the coefficients to produce five

ranges:

00 =R to 0P

=

=1

0

j

i

ij PR to wherej=1 to 6.=

j

i

iP0

We assume that the endothelial cells cannot move back onto the sprouts from which they emerge.

Therefore each cell can either remain stationary or move in five possible directions (in 3D) (seeFigure 1 for a 2D schematic). A random number between 0 and 1 is then generated. Depending

on the range in which the number falls, the endothelial cell moves accordingly. The larger a

particular range, the greater the probability that the corresponding coefficient will be selected.

2.3 Network growth simulation

Figure 2 presents a 2D simulation for vascular network growth obtained using the model outlined

above. This simulation was carried out on a 70x70 grid, with a space step x=y. In order to

contain the growth within the limits of the domain, zero flux conditions were imposed on the

boundaries of the grid. The initial distribution of fibronectin and TAF concentrations used in the

simulation are the same as those discussed in Anderson and Chaplain (1998). For presentationpurposes, we have arbitrarily chosen five positions, evenly distributed along the parent vessel, as

our initial sprouting sites.

Unless otherwise indicated, the dimensionless parameter values used for the simulations

presented in this paper are as follows:

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D=0.00035 =0.6 =0.38 =0.22

=0.05 =0.1 =0.1

Details of the parameter normalization are given in Anderson and Chaplain (1998). The time

parameter was normalized as:

t

t =

~

with =L2/Dc, whereL=2 mm is the length of the domain andDc=2.9 x 10-7 cm2s-1 is taken as the

diffusion coefficient for TAF (Sherratt and Murray, 1990; Bray, 1992). The simulation of

vascular growth presented in Figure 2 shows the random character of each of the five sprout

trajectories, which progress towards the tumour (the surface of which occupies the lower

boundary of the domain). At time =2 (corresponding to t=3 days), we observe that branchingand anastomosis have already occurred. The local TAF concentration increases as the sprouts

approach the tumour, the branching process is significantly amplified, and the sprout density and

anastomosis density consequently increase.

The resulting vasculature can now be directly used to perform flow simulations in order toinvestigate the efficiency of the network for drug delivery (McDougall et al., 2002).

3 Modelling flow through a network

This area of research has been extensively studied in the context of petroleum engineering, where

the main focus has been to investigate the flow of water, oil, and gas through the interstices of a

porous rock (see, for example, McDougall and Sorbie, 1997 for an overview). Adaptation of fluid

modelling techniques from the oil industry to biomedical applications has previouslybeen

performed by McDougall et al. (2002)

Network flow calculations are based upon principles similar to those characterising Kirchoff'slaws in electrical circuit theory here, however, fluid mass and not electrical charge is the

conserved variable. A lattice (2D) or network (3D) is constructed using interconnected capillaryelements (of known geometry) and pertinent boundary conditions are imposed upon the system

boundaries (usually, but not exclusively, these take the form of imposed pressure profiles). Before

ensemble flows can be calculated, however, a local relationship between pressure gradient and

flow must be assumed at the scale of a single capillary element. Poiseuille's law for flow in a

rigid, impermeable, circular cylinder has been assumed here, where the flow rate is given by:

L

PRQ

8

4= (3)

where is the fluid viscosity, Pis the pressure drop across the vessel,R andL are the radius andlength of the vessel respectively. Clearly, this assumption is somewhat idealistic: at the capillary

scale, it is rather radical to assume that blood can be modelled as a continuum of constant

viscosity. In reality, capillary beds are characterized by unsteady-state flow dynamics, with some

capillaries periodically stagnant. However, more complex flow/pressure-drop relationships

(including. haematocrit-dependent viscosity, and R(P) relationships) can easily be incorporatedinto the formulation at a later date. For now, we restrict ourselves to Poiseuille-type flow and will

take the term "viscosity" as a cipher for "apparent viscosity" (see Fung, 1993, for a fuller

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treatment). Assuming mass conservation and incompressible flow at each junction where

capillary elements meet (a node), we can write for each node:

=

=

=Nk

k

kjiQ1

),,( 0 (4)

where the index krefers to adjacent nodes andN=4 in 2D and 6in 3D. This procedure leads to a

set of linear equations for the nodal pressures (P i) which can be solved numerically using any of a

number of different algorithms (e.g. Successive Over-Relaxation (SOR), Choleski conjugate

gradient method, Lanczos method).

Once nodal pressures are known, equation (3) can be used to calculate the flow in each capillary

element in turn. When simulating drug infusions, these elemental flows are used to update

capillary drug concentrations at each time step (with complete mixing assumed at each node). A

fuller account of this procedure can be found in McDougall et al(2002).

4 Qualitative comparison of drug delivery in 2D and 3D vasculatures

In this section we compare drug delivery in 2D and 3D vasculatures the two vasculaturesconsidered in these simulations are displayed in Figure 3, where the 2D vasculature correspondsto the network whose growth was presented earlier in Figure 2. The dimensions of the grid for the

2D network were 70x70 and the 3D vasculature was grown (i.e. simulated) on a 30x30x30domain. In both cases, a continuous infusion treatment regime was simulated by fixing the input

drug concentration to Cmax at the inlet of the parent vessel (left-hand side in all figures). The timewas normalized using the "filling time" of the parent vessel (volume/flow) as follows:

=

pvpv

pv

PR

L

tt

2

2

*

8

The values of the parent vessel variables are given in Table 1. These are in line with experimental

data and physiological observations(see Levick, 2000).

Parameter Value

(cp)Ppv (Pa)Rpv (m)L pv(mm)

4 x 10-3(cp)

800(Pa)10(m)2(mm)

Table 1:physiological input parameters

Simulations were continued until the mass of drug in each vasculature tended towards a steady

state (corresponding to t*= 1000 in 2D and 5000 in 3D, Figure 4). Figure 5 shows the drugevolution for the 2D case, whilst the 3D concentrations are given in Figure 6. At a normalizedtime of t*=150, it can be seen from the 2D simulations that some drug has reached the tumour

although some bypassing and return flow to the parent vessel has already begun (via the 4 th

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branch along the parent vessel). In the 3D case, we observe that drug reaches the tumour at asimilar time and that bypassing of the drug is also occurring.

At this stage, our aim is not to provide a detailed quantitative comparison of drug delivery

efficiency for2D and 3D networks but rather to evaluate, on a qualitative basis, the influence of

various parameters (e.g. blood viscosity and vessel radii) upon drug uptake by the tumour. Theaim here is to extend the 2D results published in a previous paper by McDougall et al. (2002) to

3D networks. Figure 7 presents, in parallel, the results obtained for the 2D and 3D vasculaturesup to steady-state (left and right columns respectively). In both cases, three series of simulations

have been performed. The first series (Figure 7-a,b) examines the influence of blood viscosity

(=1, =4 and=8 cp), the second (Figure 7-c,d) the influence of mean capillary radius (=2,

=4, and =6m), and the third (Figure 7-e,f) the influence of the regime of delivery(continuous or bolus injection of the drug). We observe that the sensitivities are similar in both

2D and 3D an increase in blood viscosity and/or a decrease in mean capillary radius contributeto an increase in circulatory resistance and this considerably slows down the delivery of the drug

to the tumour.

Figure 8 highlights a direct comparison of drug delivery to the tumour between 2D and 3Dvasculatures (left and right columns show sensitivity to blood viscosity and mean capillary radius,

respectively). When observed over a timescale t*1000, we see that delivery via the 3D networkis initially far slower: the three-dimensional vasculature has very different percolation properties

associated with it (Stauffer and Aharony, 1992) and any treatment must negotiate highly tortuous

pathways before reaching the tumour. These results clearly demonstrate that it is the combined

effects of branching/anastomosis density, dimensionality of the vasculature, and duration of the

infusion itself that ultimately determine how much drug reaches the target site.

The comparisons above are somewhat qualitative in nature. In order to make more meaningful

quantitative comparisons between the 2D and 3D simulation results, we normalize drug delivery

data using the capacity of each vasculature. The capacity of the vasculature is defined as the

maximum mass of drug that the vasculature can potentially contain (i.e. if it was completely filledwith drug at maximum concentration). Table 2 shows a number of normalized measures for 2D

simulations (t*1000) and Table 3 the corresponding results for3D simulations (t*5000). Inall cases, simulations continued until the mass of drug in each vasculature tended towards a

steady state.

In both 2D and 3D networks, it is clear that the steady-state concentration distribution does notcorrespond to a network that is fully saturated with drug. In 2D, the capillary bed only reachesbetween 65.4% and 80.9% saturation, whilst the 3D system only attains 57-63.2%. This is due to

the numerous dead-end and re-circulating capillaries that contribute little to the flowing

backbone of the vasculature. Extraneous recirculation can be more dominant in the 3D case.Perhaps the most striking results relate to the percentage of injected drug that has actually reached

the tumour at steady state. In both 2D and 3D simulations, a staggering 97 to 99.9% of theinjected treatment bypassed the tumour completely. These results clearly show that some measure

of vessel connectivity and anastomosis density should be taken into account when planning

chemotherapy treatment regimes. These issues will be examined more fully in the next section.

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(r,) Volume Total Mass In Networkas % of Capacity

Uptakeas % ofCapacity

% of inputdelivered totumour

% of inputremainingin network

% of inputbypassingtumour

(4,1) 4.33x10-12

80.9 55.5 0.614 0.896 98.5(4,4) 4.33x10-12 78.3 9.22 0.102 0.867 99

(4,8) 4.33x10-12

70.0 2.15 0.0238 0.775 99.2

(2,4) 1.56x10-12 65.4 0.0554 0.000221 0.261 99.7

(6,4) 8.94x10-12 78.7 30.3 0.693 1.8 97.5

Table 2: Influence of the parameters and rupon drug uptake and bypassing in the 2D vasculature. Totaldrug input (continuous infusion) = 3.91x10-9 at time t*=1000.

(r,) Volume Total Mass In Networkas % of Capacity

Uptakeas % of

Capacity

% of inputdelivered to

tumour

% of inputremaining

in network

% of inputbypassing

tumour

(4,1) 1.04x10-11 63.2 287 1.86 0.410 97.7

(4,4) 1.04x10

-11

60.2 61.3 0.397 0.391 99.2(4,8) 1.04x10-11 57.0 24.1 0.156 0.370 99.5

(2,4) 3.19x10-12 59.3 5.43 0.0108 0.118 99.9

(6,4) 2.25x10-11 61.0 156 2.19 0.853 97

Table 3: Influence of the parameters and r upon drug uptake and bypassing in the 3D vasculature. Total

drug input (continuous infusion) = 1.61 x10-8 at time t*= 5000.

5 Network connectivity and efficiency of drug delivery to the tumour

5.1 Comparison of drug delivery via 3 different vasculatures

The vascular networks generated from our model agree with the observation that there exists a

wide range of heterogeneity in tumour-induced capillary architecture. This depends upon many

parameters, including; the initial tumour size, its distance from the parent vessel, the nature of the

surrounding tissues, and the local chemical environment. In this section, we consider the

treatment efficacy of three different vascular architectures (Figure 9-a,c,e) characterized by

different spatial distributions of branching coefficients and anastomosis densities. Note, that we

use the term "branching coefficient" to refer to local vascular connectivity, where the connectivity

at a given point in the network corresponds to the number of elementary vessels (capillary

elements) meeting at a given node (junction). Hence, for a 2D vasculature, the branching

coefficient at each node can either be 4, 3, 2 or 1 (identified as z4, z3, z2, z1, see Figure 10). Inorder to further map (characterise)the global architecture of a network, we scan the capillary bed

parallel to the parent vessel and calculate, at each cross sectional ordinate (y), the sums:

ztoti = total number of nodes having i capillary elements connected to it (i=1,2,3,4)

This procedure allows us to construct a two-part "fingerprint" for each of the different network

configurations:

Part (i) a unique branching map can be constructed by using different colours to show the spatial

distribution of each branching coefficientzi (Figure 9-a,c,e.), and

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Part (ii) a graphical signature can be constructed from thezitot(defined above)and plotted (Figure

9-b,d,f.).

The three networks that we chose to consider are characterised by significantly different

signatures. The first network (Figure 9-a,b) exhibits a gradual increase in anastomosis density as

the vessels approach the tumour. The connectivity of the network is dominated by z2 near theparent vessel but evolves towards a more homogeneous representation of the connectivities z4, z3

andz2 in the lower part of the vasculature (i.e. near the tumour). The second network (Figure 9-c,d) was generated by considering a decrease in the haptotactic coefficient. The vascular growth

is then dominated by chemotactic effects, which reduce the lateral migration of the endothelial

cells. This leads to longer main vessels which start to branch only when they become very close

to the tumour surface. The signature of the network reflects the observed differences. The final

network was obtained by considering a small circular tumour (located in the middle of the righthand boundary) as the source of tumour angiogenesis factor (TAF), leading to a radially

symmetric initial TAF concentration profile. In this particular case, the network is characterized

by a low density of poorly connected vessels and the contribution of nodes with connectivityz4 isnegligible.

Simulations of blood flow in each vasculature were performed and, in each case, identical bolus

injections were input to the parent vessel. The results are shown in Figure 11. Figure 11-a shows

how the total mass of drug in the vasculature varies with time observe the sharp decrease in

drug concentration at time t*=150, corresponding to the end of the bolus injection and subsequent

post-flush with clean blood. Despite the same amount of drug being injected into each of the three

corresponding parent vessels, we observe that the total amount of drug remaining at the end of the

injection period varies from case to case. This is due to the fact that in some vasculatures, notably

2 and 3, some drug leaves the network while the parent vessel is still delivering the treatment.

Figure 11-b shows the corresponding plots of drug uptake by the tumour. We see that, at time

t*=150, only vasculature 2 has delivered drug to the tumour. No uptake has yet occurred viavasculature 1, although most of the drug injected is still dispersed in the network. For vasculature

3, the drug has bypassed the tumour completely and has simply recirculated to the parent vessel.

Visualization of the distribution of capillary blood flow in each of the three vasculatures (Figure

12) is extremely useful in interpreting these results. In network 3 (Figure 12-c), for example, flow

is high only in the two main vessels, which form a loop causing the drug to bypass the entire

system. The rest of the network is dominated by poorly perfused capillaries. In vasculatures 1 and

2 (Figures 12-a and 12-b), flow is sufficiently high in the distal region of the capillary bed (close

to the tumour) to expedite drug delivery. The average capillary flow in vasculature 2 is

significantly higher than that in vasculature 1 due to its lower anastomosis density and this

explains why vasculature 2 delivers drug far more efficiently.

5.2 Influence of a posteriori changes in network connectivity and anastomosis

density upon drug uptake

In this section, we investigate more closely the delivery characteristics of vasculature 1. The aim

is to determine how modifications to the network connectivity might influence the efficiency of

drug delivery to the tumour. The motivation for this comes from the possibility of targeting

particular areas of a growing vasculature with cytotoxic compounds (e.g. anti-angiogenesis

drugs). The current study focuses upon a posteriori capillary pruning (i.e. capillary removal from

a static pre-existing capillary bed). However, this should be envisaged as a first step towards

dynamic vascular remodelling, where capillary architecture is modified as the bed itself is still

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growing and forming. At present, modifications are achieved by removing a fraction of vessels in

the lower part of the vasculature (from ordinatesy=40 to y=70).

Three pruning algorithms have been considered: (i) vessels are removed in a totally random

fashion; (ii) vessels are removed if their flow lies below a given threshold; (iii) vessels are

removed if their flow exceeds a given threshold. These three approaches seek to cover the

spectrum of possible targeting strategies for anti-angiogenic treatments the first relates to a broad-based indiscriminate cytotoxic drug, the second relates to an anti-angiogenic drug that

preferentially targets immature, poorly perfused capillaries, and the third relates to the possibletargeting of capillaries characterized by high wall stresses. In all simulations, a bolus injection at

concentration Cmaxand duration t*=150 was considered.

5.2.1 Random vessel removal

Figure 13 presents the variation in drug uptake at time t*=1000 as we gradually increase the

fraction (FR) of vessels randomly removed from the initial vasculature. Note, that t*=1000corresponds to the time at which drug uptake via the unpruned vasculature (i.e. FR=0.0) ceased(see Figure 11). We observe from Figure 13 that the total amount of drug uptake increases to a

maximum forFR=0.06. A sharp decrease is then observed, corresponding to poorly connectedarchitectures that are unable deliver drug effectively.

Visualization of the flow distribution in the optimized vasculature is given in Figure 14, together

with the corresponding branching/anastomosis signature of the network. We observe an increase

in flow in the distal part of the capillary bed (close to the tumour) due to a decrease in the z4branching coefficient and increases inz3 andz2. This redistribution increases the efficacy of drugdelivery to the tumour by 130% (Figure 14-d). These simulations clearly show that drug deliveryto a tumour becomes optimized when (i) highly interconnected regions of the vasculature are

removed (i.e. z4 is decreased); and (ii) a high capillary density can be preserved close to the

tumour surface. If an anti-angiogenic treatment could achieve this architecture, then the efficacy

of subsequent chemotherapy treatments would be much improved. However, we should also point

out that if an anti-angiogenic agent were to be used in isolation (i.e. without chemotherapeuticfollow-up), then the effect may be to simply optimize nutrient supply and this could actually

encourage tumour growth.

5.2.2 Flow-dependent vessel removal: low flow rate threshold

As we saw in previous simulations, the process of random removal of vessels can lead to drastic

changes in drug delivery to a tumour and these variations can be either positive or negative. In

this section, motivated by the possibility of targeting drugs at poorly perfused areas of the

vasculature, we selectively remove some of the vessels whose flow falls below a given threshold

value.

Figure 15 shows a case where vessels having flow less than 1% of the maximum capillary flow(Qmax) were removed (the vessels removed are highlighted in Figure 15-c) the fractionremoved represents 40% of the total capillary bed. We observe that the flow distribution remainsessentially unchanged when compared with the unmodified vasculature and a comparison of drug

uptake in the two systems shows that maximum uptakes are also similar. We note, however, that

the modified network initially delivers the drug more quickly: the "treated" vasculature has been

optimized and drug delivery is accelerated by approximately 30%.

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5.2.3 Flow-dependent vessel removal: high flow rate threshold

The final approach to capillary pruning relates to the possible targeting of "bottleneck" capillaries

characterized by high wall stresses. In this scenario, capillaries with flow exceeding 5% of themaximum network value were removed this corresponded to a removed fraction FR=0.05.

Figure 16 shows the resulting flow distribution, branching/anastomosis signature, and uptake

profile. We observe that the network has essentially shut down in this case, with no drug deliveryto the tumour. Hence, if an anti-angiogenic treatment could be developed to target these fast-

flowing "bottleneck" capillaries, it would be a highly efficient way of slowing tumour growth.

The results presented in this section have shown that drug delivery to a tumour is highly sensitive

to the associated capillary architecture. Moreover, we have also shown that the main flowing

backbone plays a dual role: it helps to carry the blood towards the tumour but also contributes to a

lot of bypassing. It is the management of this flowing backbone that anti-angiogenic therapiesshould address.

6 Discussion and Conclusion

The main aim of this paper has been to examine the issue of capillary blood flow in tumour-

induced vasculatures. In particular we have identified the conditions under which drug delivery to

a tumour is optimised. 3D networks have been generated and drug-uptake results compared with

those from previous 2D simulations. We have shown that both vascular connectivity and system

dimensionality affect delivery to the tumour in particular, extraneous recirculation seems to be

a dominant effect in 3D networks. In both 2D and 3D simulations, 97 to 99.9% of the injectedtreatment drug bypassed the tumour completely and these results clearly show that some measure

of vessel connectivity and anastomosis density should be taken into account when planning

treatment regimes.

The second part of the paper has focussed upon a number of different vascular architectures and

the spatial distribution of branching order and anastomosis density have been quantified. These

distributions have subsequently been used to infer treatment efficiency. We have shown that themost favourable conditions for drug uptake are those where the density of capillaries is high in

the distal region of the vasculature (near the tumour surface), together with a small number of

anastomoses in the region proximal to the parent vessel. The combination of these two conditions

significantly reduces bypassing effects.

Having compared different capillary networks characterized by different connectivity/

anastomosis signatures, we then focused on one particular vasculature and investigated how

vessel removal (capillary pruning) affected the distribution of blood flow in the system. Capillary

pruning was done either randomly or according to flow criteria. Random removal of vessels

showed that it was possible to significantly increase uptake (by around 130%). This was due toslight modifications in the backbone structure of the vasculature, which allowed redirection of the

flow preferentially towards the tumour. Random removal also preserved a high capillary densitynear the tumour surface. In the case of low-flow capillary pruning, 40% of the vessels with thelowest flow magnitude (< 1% of the maximum flow) were removed. Whilst drug uptake was not

improved (as the backbone structure remained essentially unmodified), the speed of delivery was

accelerated by 30% due to the removal of extraneous dead-end vessels. In the case of high-flowcapillary pruning, where vessels with flow > 5% of the maximum flow were removed, the mainvessels feeding the tumour became ineffective and delivery to the tumour was completely shut

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down. If an anti-angiogenic treatment could be developed to target these fast-flowing

"bottleneck" capillaries, it would be a highly efficient way of slowing tumour growth.

Experimentally, alteration of the vasculature is made possible by the use of antiangiogenic factors

(Kerbel, 2000; Hidalgo and Eckhardt 2001; Ruhrberg, 2001). These factors are mainly used at an

early stage of tumour development in order to prevent the formation of the vasculature, which

otherwise will feed the tumour and allow it to grow and spread (via dissemination of tumourcells). However, at a later stage of tumour development i.e. when the tumour is already

vascularized, antiangiogenic factors can still be used to disturb the constant reorganization of thecapillary bed (Keshet and Ben-Sasson, 1999). Antiangiogenic factors are usually efficient in

destroying younger immature vessels, i.e. those that have not yet stabilized their structure with

periendothelial cell coating (Benjamin et al., 1999; Darland and DAmore, 1999) and thesevessels mainly correspond to small capillaries characterized by low-flow magnitudes. This would

justify our choice of removing low-flow vessels in some of our simulations.

The fact that the drug uptake can be increased by the random removal of vessels suggests the

possibility of coupling an antiangiogenic treatment (to preliminarily optimize the vasculature)

with chemotherapy treatment to ensure maximum uptake. One possible way to test the efficiency

of the antiangiogenesis treatment and its effect on flow distribution could be to monitor theintensity of a fluorescent probe injected before and after the treatment.

Whilst the work presented here has dealt with individual vascular beds emerging from a single

parent vessel, we should note that, in vivo, the tumour may be fed by a number of different

vascular networks emerging from different parent vessels (Figure 17). Subtle differences in

architecture mean that each of these vasculatures could respond differently to an anti-

angiogenesis treatment some may become totally inefficient for drug delivery whilst others

could become optimized. This would result in a highly heterogeneous distribution of drug to the

tumour, with some regions being killed by high concentrations of cytotoxic drugs and other

regions becoming hypoxic and triggering the release of additional angiogenic factors. Several

cycles of coupled anti-angiogenic and chemotherapy treatments should then be considered as a

potential solution to contain tumour growth.

Acknowledgments

The authors gratefully acknowledge the support of the following grants: EPSRC grant

GR/R74789 (AS); EPSRC/MRC Discipline Hopping Award G0100660 (SMcD); EU RTN Grant

HPRN-CT-2000-00105 Using Mathematical Modelling and Computer Simulation to ImproveCancer Therapy(MC); SHEFC RDG 099 The SIMBIOS Centre (MC); The Royal Society ofEdinburgh Personal Research Fellowship (ARAA).

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References

Anderson, A.R.A. and M.A.J. Chaplain (1998) Continuous and discrete mathematical model of tumor-

induced angiogenesis.Bull. Math. Biol. 60, 857-899.

Ausprunk, D.H. and J. Folkman (1977) Migration and proliferationof endothelial cells in preformed and

newly formed blood vessels during tumour angiogenesis. Microvasc. Res. 14, 53-65.

Baish, J.W., Y. Gazit, D.A. Berk, M. Nozue, L.T. Baxter and R.K. Jain (1996) Role of tumor vascular

architecture in nutrient and drug delivery: an invasion percolation-based network model. Microvasc. Res.

51, 327-346.

Benjamin, L.E., D. Golijanin, A. Itin, D. Pode and E. Keshet (1999) Selective ablation of immature blood

vessels in established human tumors follows vascular endothelial growth factor withdrawal. J. Clin. Invest.

103(2), 159-165.

Bray, D. (1992) Cell Movements,New-York: Garland Publishing.

Chaplain, M.A.J. and A.R.A. Anderson (1999) Modelling the growth and form of capillary networks, pp.

225-250, in On Growth and Form: Spatio-temporal Pattern Formation in Biology, M.A.J. Chaplain, G.D.Singh and J.C. McLachlan (Eds.), Wiley: Chichester.

Darland, D.C. and P. DAmore (1999) Blood vessel maturation: vascular development comes of age. J.

Clin. Invest.103(2), 157-158.

El-Kareh, A.W. and T.W. Secomb (1997) Theoretical models for drug delivery to solid tumours. Crit. Rev.

Biomed. Eng. 25(6), 503-571.

Folkman, J. (1971) Tumor angiogenesis: therapeutic implications.N. Engl. J. Med. 285, 1182-1186.

Folkman J. and M. Klagsbrun (1987) Angiogenic factors. Science. 235, 442-447.

Folkman, J. (1995) Angiogenesis in cancer, vascular, rheumatoid and other disease.Nat. Med. 1, 21-31.

Fung, Y.C. (1993) Biomechanics, Mechanical properties of living tissues, Springer-Verlag, New York.

Gimbrone, M.A., R.S. Cotran, S.B. Leapman and J. Folkman (1974) Tumor growth and neovascularization:

an experimental model using the rabbit cornea.J. Natl. Cancer Inst. 52, 413-427.

Godde, R. and H. Kurz (2001) Structural and biophysical simulation of angiogenesis and vascular

remodelling.Developmental Dynamics . 220, 387-401.

Hidalgo, M. and S.G. Eckhardt (2001) Development of matrix metalloproteinase inhibitors in cancer

therapy.J. Natl. Cancer Inst.93(3), 178-193.

Jackson, T.L., S.R. Lubkin and J.D. Murray (1999) Theoretical analysis of conjugate localization in two-

step cancer chemotherapy.J. Math. Biol. 39, 353-376.

Jain, R.K. (1987) Transport of molecules in the tumor interstitium: a review. Cancer Res.47(12), 3039-

3051.

Jain, R.K. (1988) Determinants of tumour blood flow: a review. Cancer Res.48(10), 2641-2658.

Kerbel, R.S. (2000) Tumor angiogenesis: past, present and the near future. Carcinogenesis.21(3), 505-515.

13


14/31

Keshet, E. and S.A. Ben-Sasson (1999) Anticancer drug targets: approaching angiogenesis.J. Clin. Invest.

104(11), 1497-1501.

Krenz, G.S. and C.A. Dawson (2002) Vessel distenbility and flow distribution in vascular trees. J. Math.

Biol. 44, 360-374.

Levick, J.R. (2000)An Introduction to Cardiovascular Physiology, 3rd Edition, London, Arnold.

McDougall, S.R. and K. Sorbie (1997) The application of network modelling techniques to multiphase flow

in porous media.Petroleum Geosci . 3, 161-169.

McDougall, S.R., A.R.A. Anderson, M.A.J. Chaplain and J.A. Sherratt (2002) Mathematical modelling offlow through vascular networks: implications for tumour-induced angiogenesis and chemotherapy

strategies.Bull. Math. Biol. 64(4), 673-702.

Mitchell, A.R. and D.F. Griffiths (1980) The finite difference method in partial differential equations.Wiley: Chichester.

Muthukkaruppan, V.R., L. Kubai and R. Auerbach (1982) Tumor-induced neovascularization in the mouseeye.J. Natl. Cancer Inst. 69, 699-705.

Paweletz, N and M. Knierim (1989) Tumor-related angiogenesis. Crit. Rev. Oncol. Hematol. 9, 197-242.

Quarteroni, A., M. Tuveri and A. Veneziani (2000) Computational vascular fluid dynamics: problems,

models and methods. Comput. Visual. Sci.2, 163-197.

Risau, W. (1997) Mechanisms of angiogenesis.Nature386, 671-674.

Ruhrberg C. (2001) Endogenous inhibitors of angiogenesis.J. Cell Sci. 114, 3215-3216.

Secomb, T.W.; A.R. Pries and P. Gaehtgens (1995) Architecture and hemodynamics of microvascularnetworks.Biological Flow, Edited by MY. Jaffrin and C. Caro, Plenum Press, New York. 9, 159-176.

Sherratt J.A. and J.D. Murray (1990) Models of epidermal wound healing.Proc. Roy. Soc. Lond. B241, 29-36.

Shoefl, G.I. (1963) Studies of inflammation III. Growing capillaries: Their structure and permeability.Virchows Arch. Path. Anat. 337, 97-141.

Sholley, M.M., G.P. Ferguson, H.R. Seibel, J.L. Montour and J.D. Wilson (1984) Mechanisms of

neovascularization. Vascular sprouting can occur without proliferation of endothelial cells.Lab. Invest. 51,

624-634.

Stauffer, D. and A. Aharony (1992) Introduction to percolation theory.Ed. Taylor and Francis, London.

Walsh, D.A. (1999) Angiogenesis and arthritis.Rheumatology. 38, 103-112.

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Figure 1: Schematic diagram of a section of the 2D-grid used in the simulation procedure

illustrating how the capillary sprout growth process is taken into account in the simulations. Ateach node, the sprout can grow in 3 possible directions in 2D (and 5 possible directions in 3D).Each growing sprout is automatically assigned a radius in order to directly use the resulting

network for flow simulations.

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=1 =2 =3

=4 =5 =14.5

Figure 2: Simulation of the growth of a vascular network from five initial capillary sprouts

emerging from the parent vessel. The tumour, which is located on the lower boundary of the

domain, secretes TAF which diffuses into the tissue, and which then contributes to the orientation

of the endothelial cell migration towards the tumour. The angiogenic process was completed in

this simulation in 22 days (=14.5).

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(a) (b)

Figure 3: (a) the simulated 2D vasculature (70x70 grid) and (b) the simulated 3D vasculature

(30x30x30 grid) used for flow simulations. These two vascular networks were used in order to

compare delivery efficacy for various capillary and fluid parameters.

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Figure 4: Comparison of the total mass of drug in the 2D and 3D vasculatures [see Figures 3a,

3b](normalized by the total mass in the parent vessel) up to their respective stationary states for

the continuous drug delivery process.

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t* = 5 t* = 30 t* = 50

t* = 100 t* = 150 t* = 200

t* = 300 t* = 500 t* = 1000

Figure 5: Snapshots in normalized time of the drug concentration during continuous infusion

from the parent vessel into the 2D vascular network. The various colours correspond to differentnormalized drug concentrations C(x,y,t*)/Cmax as shown in the colour scale.

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t* = 2 t* = 20 t* = 40

t* = 100 t* = 200 t* = 400

t* = 600 t* = 800 t* = 5000

Figure 6: Snapshots in normalized time of the drug concentration during continuous infusionfrom the parent vessel into the 3D vascular network. The various colours correspond to differentnormalized drug concentrations C(x,y,t*)/Cmax as shown in the colour scale.

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(a) (b)

(c) (d)

(e) (f)

Figure 7: Plots(a,c,e) correspond to the simulation results associated with the 2D vasculature andplots (b,d,f) to the simulation results associated with the 3D vasculature. The three series of

simulations investigate the influence of the viscosity parameter (a,b), the mean radius of the

vessels (c,d) and the regime of infusion (e,f) respectively.

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(a) (b)

(c) (d)

(e) (f)

Figure 8: Comparisons of the initial 2D and 3D drug uptakes, (a,c,e) comparison for changes inthe viscosity parameter, (b,d,f) comparison for changes in the mean radius parameter.

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(a) vasculature 1

(b)

(c) vasculature 2

(d)

(e) vasculature 3

(f)

Figure 9: Plots (a,c,e) showvascular networks with spatial connectivity distributions coloured

using the accompanying scale. Plots(b,d,f) give graphical signatures for each network involving

the densities of branching coefficients (z4,.z3,z2,z1)

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Figure 10: Representation of the 4 possible connectivities encountered at a junction (node) of a

2D vascular grid.

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(a) (b)

Figure 11: Comparison of the total mass of drug in the network (a) and the total drug uptake by

the tumour (b) for the 3 different vasculatures of Figure 9.

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(a) (b) (c)

Figure 12: Visualization of the flow intensity in the 3 different vasculatures (Figure 9). The

colours code used represent the normalized flow intensity Q(x,y,t)/Qmax.

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Figure 13: Plot of the total amount of drug received by the tumour at t*=1000 vs. fraction ofvessels randomly removed from the capillary bed (the fraction removed, FR, is varied from 0.0 to

0.18).

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(a) FR=0.0(b)

(c) FR=0.06(d)

Figure 14: (a) Plot showing the blood flow distribution in the original vasculature (i.e. no vessels

removed, FR=0.0); (b) Plot showing thedensities of branching coefficients (z4,.z3,z2,z1) after afraction FR=0.06 of capillary elements have been randomly removed; (c) Visualization of the

vessels randomly removed (in black) and the subsequent modified flow distribution; (d) Plotshowing the comparison of the drug uptake between the FR=0.06 vasculature and the original

FR=0.0 case.

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(a) FR=0.0(b)

(c) FR=0.4(d)

Figure 15: (a) Plot showing theblood flow distribution in the original vasculature with no vessels

removed i.e.FR=0.0; (b) Plot showing the densities of branching coefficients (z4,.z3,z2,z1) after a

fraction FR=0.4 of capillary elements have been randomly removed (those whose flow magnitudeis 1% of the maximum capillary flow); (c) Visualization of the vessels randomly removed (inblack) and the subsequent modified flow distribution; (d) Plot showing the comparison of the

drug uptake between the FR=0.4 vasculature and the original FR=0.0 case.

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(a) FR=0.0 (b)

(c)FR=0.05 (d)

Figure 16: (a) Plot showing theblood flow distribution in the original FR=0.0 vasculature i.e. no

vessels removed; (b) Plot showing the densities of branching coefficients (z4,.z3,z2,z1) after a

fraction FR=0.05 of capillary elements have been randomly removed (those whose flowmagnitude is 5% of the maximum capillary flow; (c) Visualization of the vessels randomly

removed (in black) and the subsequent modified flow distribution; (d) Plot showing the

comparison of the drug uptake between the FR=0.05 vasculature and the originalFR=0.0 case.

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Figure 17: Schematic representation of a more complex tumour-induced angiogenesis scenario,where several vascular networks eventually connect the tumour to the circulation via a number of

different parent vessels.

Anti-Angiogenic and Chemotherapeutic Drug

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