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March 27, 2000 11:34 WSPC/103-M3AS 0049

Mathematical Models and Methods in Applied SciencesVol. 10, No. 3 (2000) 379–407c© World Scientific Publishing Company

ADVECTION-DIFFUSION MODELS FOR SOLID TUMOUR

EVOLUTION IN VIVO AND RELATED FREE

BOUNDARY PROBLEM

E. DE ANGELIS∗ and L. PREZIOSI†

Department of Mathematics, Politecnico di Torino, Corso Duca degli Abruzzi 24,10129 Torino, Italy

Communicated by N. Bellomo

Received 7 October 1998Revised 25 January 1999

This paper proposes a multicell model to describe the evolution of tumour growth fromthe avascular stage to the vascular one through the angiogenic process. The model isable to predict the formation of necrotic regions, the control of mitosis by the presenceof an inhibitory factor, the angiogenesis process with proliferation of capillaries justoutside the tumour surface with penetration of capillary sprouts inside the tumour,the regression of the capillary network induced by the tumour when angiogenesis iscontrolled or inhibited, say as an effect of angiostatins, and finally the regression of thetumour size. The three-dimensional model is deduced both in a continuum mechanicsframework and by a lattice scheme in order to put in evidence the relation betweenmicroscopic phenomena and macroscopic parameters. The evolution problem can bewritten as a free-boundary problem of mixed hyperbolic–parabolic type coupled with aninitial-boundary value problem in a fixed domain.

1. Introduction

In the last decade the deduction of mathematical models addressed towards the

study of the evolution of biological systems related to tumour growth has become

one of the relevant fields of research activity in bio-mathematics. The type of cells

involved in this description, the interactions between cells and the effects to consider

make the problem very complex.

As can be recovered in the recently published book by Adams and Bellomo,1 and

in the special issue edited by Chaplain,2 the attempts of deducing mathematical

models to simulate different aspects of tumour evolution can be divided in those

paying more attention to phenomena occurring at the cellular scale and those aimed

at describing the macroscopic growth of tumours.

∗E-mail: [email protected]†E-mail: [email protected]

379

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380 E. De Angelis & L. Preziosi

The first approach refers to the early stage of the evolution of the tumour,

when tumour cells are not condensed yet in a macroscopically observable quasi-

spherical mass and when the competition with the immune system is important.

It requires working in the framework of mathematical methods that are typical

of nonequilibrium statistical mechanics and phenomenological kinetic theory. This

approach has been used by Bellomo and coworkers (see Refs. 8 and 18 for a recent

review).

The second approach aims at describing the evolution of condensed tumour

masses by using continuum mechanics tools. Reference prior to 1995 can be re-

covered in Refs. 3 and 14. Referring to the most recent papers, it is possible to

distinguish those focusing on the avascular phase, e.g. Refs. 6, 7, 9–11, 30, 31, 33

and 34 and those focusing on angiogenesis and the vascular phase, e.g. Refs. 4, 5,

12–14, 25–27.

In the latter case, attention is focused on what happens outside the tumor.

Hence, the evolving domain is the environment and the tumour is seen as the

entity which from outside the domain is responsible for the distribution of tumour

angiogenesis factor in the time-independent domain of integration.

In the former case, attention is on the region occupied by the tumor with the

interaction with the outer environment restricted to the boundary conditions. This

means, for instance, that the chemical factors produced by the tumour remain

inside the tumour and neither TAF nor endothelial cells are considered. These

papers usually consider the tumour as an incompressible material, without giving

it a cellular structure. The distinction between dead, quiscent and replicating cells

is done by identifying the level of nutrient reaching the regions.

In addition, most of them work under the assumption of spherical symmetry, so

that from an integral mass balance it is possible to obtain the evolution equation

determining tumour growth.

Dropping this assumption would require determining the velocity at which cells

move. In Refs. 7, 30 and 31, this is done by assuming that the “incompressible

fluid” moves according to Darcy’s law.

Ward and King,33,34 instead, consider the tumour as made up of two con-

stituents, living and dead cells. This allows to consider the death of cells as a

gradual process and not as an instantaneous response to concentration of nutrient

below that needed by the cells. They assume that both cells move with the same

velocity and as other papers working in spherical symmetry and assuming that

cells occupy the whole space can avoid giving a phenomenological law for the cell

velocity.

Starting from this cultural background, this paper develops a multicell modelling

framework which can describe the smooth passage from the avascular phase to

the vascular one through angiogenesis. All the different phases of tumor growth

are described in a unique three-dimensional model. Actually it is the evolution

itself which allows to identify periods which are dominated by the different pheno-

mena. In addition, the model also includes the possibility of considering the effects

March 27, 2000 11:34 WSPC/103-M3AS 0049

FBP for Solid Tumour Evolution 381

of angiostatins with regression of the capillary network and then of the tumour

itself.

Another novelty point of the paper with respect to the existing literature is

the particular attention paid to the interactions between the growing tumour mass

and the evolving environment it is embedded in. In this respect, the model involves

some state variables defined in the tumour only, and some defined in the whole

environment and the mathematical problem can be written as a free boundary

problem.

There are many interactions between tumour and environment which were not

considered before in an evolving framework, e.g. continuous emission of chemical

factors from a growing tumour, diffusion of chemical factors inside and outside the

tumour yielding a time-dependent distribution, penetration of capillary sprouts into

the tumour, dependence of tumour growth rate on the vascularization, regression

of the induced capillary network in the presence of proteins, such as angiostatins,

inhibiting endothelial cell proliferation, tumour shrinking as a result of the reduction

of available nutrient.

In addition, the model takes into account the phenomena which were already

included in the previously cited papers, such as the diffusion of nutrient through the

surface of the tumour and the capillary network and its difficulty in reaching the

central region of the tumour with the formation of necrotic regions as a result of lack

of nutrient, the existence of a proliferating outer layer of nearly constant width16,21

and of a quiescent layer, again of nearly constant width, which can activate when the

amount of nutrient present in the environment increases,20 e.g. after the formation

of the induced capillary network; the fact that the passage to the necrotic region

is not sharp, as noted in Ref. 32; the control of tumour mitosis by the presence of

inhibitory factors; the existence of a limit radius of tumor growth in the avascular

phase;19,22–24 the angiogenic process with proliferation of capillaries just outside

the tumour surface directing towards the tumour.

The general modelling process develops using both a lattice schematization and

a continuum mechanics approach. This is done in order to keep in mind and put in

evidence the relation between microscopic phenomena and macroscopic variables, as

many phenomenological observations are done at a cellular scale and these knowl-

edges have to be transferred at a macroscopic scale. Each parameter characterizing

the model refers to a specific phenomenon and could in principle be identified by

suitable phenomenological observations.

The problem consists in a set of five parabolic equations, describing the evolution

of the densities of living tumour cells, nutrients, capillaries, growth inhibitory factor

and tumour angiogenesis factor, and a hyperbolic equation, describing the evolution

of the density of death cells. Three of them are valid in the whole domain, i.e. both

inside and outside the tumour, and three inside the tumour, which is a time varying

domain. Hence the problem can be written as a free-boundary problem, the free

boundary being the tumour surface. This domain is defined by a material interface

which moves with tumour cells and therefore has the same velocity appearing in

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the equation describing the evolution of living tumour cells. An important role is

played by the relation between nutrient supply and capillary density just outside

the tumour surface and by the diffusion of angiogenesis factors produced by the

tumour outside it.

In more detail, the paper develops as follows. Section 2 explains the biological

background obtained by phenomenological observation which is taken as the stand-

ing block of the modelling procedure. Section 3 considers the general modelling

frameworks, and more precisely lattice schemes in Sec. 3.1 and continuum mecha-

nics approach in Sec. 3.2. In Sec. 4, after introducing the specific assumptions, the

model is deduced. Section 5 deals with the formulation of the free boundary pro-

blem while Sec. 6 provides the description of some qualitative behaviours of the

model. Finally, the last section draws some conclusions and suggests some possible

developments of the model.

A description of the conceivable scenarios and their dependence on the para-

meters is given in Ref. 17. The simulations therein show the qualitative behaviors

which are described in Sec. 2 and an agreement on the qualitative characteristics

deduced in Sec. 6.

2. Phenomenological Observation of the Biological System

Three overlapping phases of growth are usually identified in the stage of growth

of tumour cells condensed into a compact form: avascular, angiogenic and vascular

phase.

In the avascular phase, tumour cells are aggregated in the form of multicell

spheroids and feed on oxygen and nutrients present in the environment. These

nutrients filtrate through the surface of the spheroid and diffuse in the intracellular

space. Consumption of nutrients and cell proliferation are characterised by strong

nonuniformities. After the early stages of growth, the spheroids give an inner zone

of dead necrotic cells for lack of nutrients and a thin outer zone of living cells.

This last zone can be further divided into a layer with prevalence of quiescent cells

(a) (b) (c)

Fig. 1. Movement of tumour cells. (a) Configuration before a mitosis, (b) mitosis and pressureexerted on neighbouring cells, (c) cell movement and propagation of the force field generated bymitosis.

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and a layer with prevalence of proliferating cells, though dead cells are also found

adjacent to both quiescent and proliferating cells.32

When a tumour cell undergoes mitosis, the new-born cell presses the cells nearby

to make space for itself. This generates a displacements of the neighbouring cells,

as shown in Fig. 1, to eventually reach a configuration in which each cell has all

the space it needs. In particular, this leads to tumour growth.

As at this stage the tumour is not surrounded yet by capillaries, this phase can

be observed and studied in laboratory by culturing cancer cells. In particular, it

has been observed (see, for instance, Refs. 19, 23 and 24) that the tumours tend

to assume a limit radius corresponding to a situation with balance between mitosis

and disintegration of tumour cells into waste products, mainly water.

In addition, during their life tumour cells produce many chemical factors which

diffuse both in the tumour and in the surrounding tissue in a spatially-dependent

way. Two important factors are the so-called growth inhibitory factor (GIF) and

tumour angiogenesis factor (TAF), respectively. The former influences the mitotic

rate of tumour cells, the latter stimulates the proliferation of endothelial cells.

With the diffusion of TAF the beginning of a new phase, called tumour angiogen-

esis, can be identified. This is characterised by the fact that the endothelial cells

reached by TAF release some enzymes which degrade their basement membrane

causing proliferation of endothelial cells with invasion of the extracellular matrix

and migration towards the source of angiogenic stimulus, that is the tumour cells.

Capillary sprouts then form by accumulation of endothelial cells. They at last reach

the tumour surface and penetrate the tumour bringing it much more nutrient. The

tumour cells then become very aggressive and their mitotic rate increases leading

to a much faster growth. This process is schematised in Fig. 2.

Finally, the vascular phase is characterised by the presence of a dense network of

capillaries around the tumour which is then able to provide for itself large amounts

of nutrients and to increase considerably its mitotic rate. Finally, it invades the

surrounding tissue metastasising to different parts of the body.

Geometrically one can then identify two regions: the outer environment and the

region occupied by the tumour mass. This is a time-dependent domain denoted by

T (t) included in a much larger fixed domain D which is called environment. The

region D−T (t) is called outer environment. According to the previous observations,

tumour cells are confined in T (t) and it is expected that the evolution leads to the

formation of a necrotic region inside T (t), that is a region in which the number of

dead cells is larger (or much larger) than living cells. On the other hand, capillaries

initially exist only outside T (t), i.e. in the outer environment, but because of an-

giogenesis they proliferate and penetrate the tumour. Therefore it is expected that

the evolution leads to the formation of capillary sprouts near the tumour surface

∂T (t) and within T (t). GIFs and TAFs are produced inside T (t) but can diffuse in

the outer environment.

As we aim at describing from a macroscopic viewpoint the evolution of the

tumour mass and of the state variables affecting it, the first step of the modelling

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Fig. 2. The angiogenesis process. (a) tumour releases angiogenic factors (TAF), (b) endothelialcells are stimulated to proliferate and invade the extracellular matrix migrating towards the sourceof angiogenic stimulus, (c) new capillary sprouts are formed and supply the tumour with morenutrient and oxygen.

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Fig. 3. Nutrients (dark circles) and chemical factors (stars) diffusing in the tumor and outsideit. Clear circles are tumor cells.

process consists in defining the set of dependent variables in charge of describing

the physical state of the system. In this paper, they are taken to be:

— the density uT of living tumour cells;

— the density uD of dead tumour cells;

— the density uC of new capillaries;

— the density uI of GIF;

— the density uA of TAF;

— the density uN of nutrient.

The first three variables refer to cells, the last three to macro-molecules. Between

these two classes there is a profound difference as cells are much larger than chemi-

cal factors and macromolecules. They are delimited by a membrane and cannot

penetrate each other. Cells, then, occupy part of the available space. On the con-

trary, chemical factors are molecules which diffuse in the intercellular space, attach

to the cell membrane, or pass through it either to feed the cell or as a cell product,

so that they actually do not occupy space (see Fig. 3).

3. General Frameworks for the Derivation of the

Mathematical Model

The derivation of the mathematical model, which consists in an evolution equation

for the variable u = {uT , uD, uC , uI , uA, uN}, will be done in the following sub-

sections on the basis both of a lattice schematisation and of continuum mechanics

procedures, mainly consisting in mass balances.

In fact, when specialising the procedure to the different interacting “popula-

tions”, bearing the two procedures in mind can be helpful in determining the rela-

tive importance of the various phenomena and parameters and in identifying proper

phenomenological observations performed at a cellular scale to evaluate the para-

meters appearing in the model.

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3.1. Lattice scheme

Consider, for simplicity, but without loss of generality, a plane square lattice and

identify each point by the pair of integers (i, j). The elementary volume centred in

the node (i, j) is denoted by Vij and its volume by ∆Vij . Finally, all cells in Vij are

considered concentrated in the node (i, j).

We want to study the evolution of the number of a certain type of cells (or

chemical factors) Nij(t) found in the node (i, j) at time t, which is related to the

number density u(t,x) by

Nij(t) =

∫Vij

u(t; x) dx . (3.1)

We assume that there exist two mechanisms which govern the movement of cells

(a) A diffusive-like phenomenon governed by a Brownian motion with probability

which depends on the direction of the movement. We indicate, for instance,

with Qji,i+1 the transition probability density per unit time that a cell diffuses

from (i, j) to (i+ 1, j).

(b) A transport-like phenomenon, which generates a drift of cells with a probability

which again depends on the direction of motion, e.g. P ji,i+1 is the transition

probability density per unit time that a cell is transported along x from (i, j)

to (i + 1, j), and Rj,j+1i that it is transported along y from (i, j) to (i, j + 1).

We will assume that the drift velocity has positive components.

One then has the following scheme

Number of

cells in

(i, j) at

time t+ dt

=

Number of

cells present

in (i, j)

at time t

+

Number of

cells drifting

in along x from

(i− 1, j) and

along y from

(i, j − 1)

−

Number of

cells drifting

away along x

to (i+ 1; j) and

along y to

(i; j + 1)

+

Number

of cells

diffusing

from

neighbour-

ing nodes

−

Number

of cells

diffusing

away to

neighbour-

ing nodes

+

Genera-

tion of

new cells

− Death

of cells

(3.2)

which leads to the following finite difference equation

Nij(t+ ∆t) = Nij(t) + ∆t[P ji−1,iNi−1,j(t) +Rj−1,ji Ni,j−1(t)

−P ji,i+1Nij(t)−Rj,j+1i Nij(t)

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+Qji−1,iNi−1,j(t) +Qji+1,iNi+1,j(t)

+Qj−1,ji Ni,j−1(t) +Qj+1,j

i Ni,j+1(t)

− (Qji,i−1 +Qji,i+1 +Qj,j−1i +Qj,j+1

i )Nij(t)

+ Γij − LijNij(t)] , (3.3)

where Γij and Lij are the generation and death rates, respectively.

Notice that if at time t the cells occupy a finite number of nodes of the infinite

lattice, they will still occupy a finite number of nodes at any time. In addition, if

Γij = LijNij summing over all possible (i, j), the total number of cells is preserved.

Assuming symmetry of the transition probability density, e.g.

Qji−1,i = Qji,i−1 , (3.4)

one can rewrite Eq. (3.3) as

Nij(t+ ∆t)−Nij(t)∆t

= [P ji−1,iNi−1,j(t)− P ji,i+1Nij(t)]

+ [Rj−1,ji Ni,j−1(t)−Rj,j+1

i Nij(t)]

+ [Qji−1,iNi−1,j(t)− (Qji−1,i +Qji,i+1)Nij(t)

+Qji,i+1Ni+1,j(t)]

+ [Qj−1,ji Ni,j−1(t)− (Qj−1,j

i +Qj,j+1i )Nij(t)

+Qj,j+1i Ni,j+1(t)] + Γij − LijNij(t) . (3.5)

Assuming now that the transition probability densities of the diffusion process

depend on some properties evaluated in the mean point of the path, e.g.

Qji−1,i = Qji− 1

2

, (3.6)

and that the transition probability densities of the transport phenomenon depend

on the starting point, e.g.

P ji−1,i = P ji−1 , (3.7)

one has the scheme

Nij(t+ ∆t)−Nij(t)∆t

= [P ji−1Ni−1,j(t)− P ji Nij(t)] + [Rj−1i Ni,j−1(t)− RjiNij(t)]

+ [Qji− 1

2

Ni−1,j(t)− (Qji− 1

2

+ Qji+ 1

2

)Nij(t)

+ Qji+ 1

2

Ni+1,j(t)]

+ [Qj− 1

2i Ni,j−1(t)− (Q

j− 12

i + Qj+ 1

2i )Nij(t)

+ Qj+ 1

2i Ni,j+1(t)] + Γij − LijNij(t) . (3.8)

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Under suitable regularity assumptions one can expand N, P , Q and R, use

Nij(t) ≈ u(t, xi, yj)∆Vij , and write

∂u

∂t= −∂(Pu)

∂x− ∂(Ru)

∂y+

∂

∂x

(Q∂u

∂x

)+

∂

∂y

(Q∂u

∂y

)+ Γ− Lu , (3.9)

with Γ(t, xi, yj) = Γij(t)/∆Vij and where the indices (i, j) have been substituted

with the dependence of u and of all coefficients on the space variable. Equivalently,

one can write

∂u

∂t+∇ · (Wu) = ∇ · (Q∇u) + Γ− Lu , (3.10)

where, in two dimensions, W = (P,R).

This result could have been expected as (3.8) represents an explicit finite differ-

ence scheme of (3.9) in which the diffusive term is approximated with a second-order

central scheme and the convective term with a first-order upwind scheme.

3.2. Continuum mechanics approach

The same result can be obtained in the framework of continuum mechanics consid-

ering the number N(t) of a certain type of cells (or chemical factors) contained in

a volume V fixed in space

N(t) =

∫Vu(t,x) dx . (3.11)

This quantity may change due to the following reasons:

(i) Advective flux through the boundary ∂V of V related to cells moving with

velocity W;

(ii) Diffusive flux through the boundary ∂V of V related to random motion of cells;

(iii) Generation of cells;

(iv) Destruction of cells.

The above-mentioned can be schematised in the following diagram

Rate of

change of

number of

cells in a

volume

= −

Advective

outflux

through

boundary

−

Random

outflux

through

boundary

+Generation

of cells− Death

of cells

(3.12)

According to these hypotheses, one can then write the following mass balance

equation in integral form

d

dt

∫Vu dV = −

∫∂VuW · n dσ −

∫∂V

J · n dσ +

∫V

Γ dV −∫VLudV , (3.13)

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where n is the outward normal to V, Γ is the proliferation term and L is the death

coefficient.

Using Gauss’ theorem and under suitable regularity assumptions, one can then

write, as usual in continuum mechanics, the following balance law in local form

∂u

∂t= −∇ · (Wu)−∇ · J + Γ− Lu , (3.14)

which, if J = −Q∇u, reduces to (3.10).

The general advection-diffusion model (3.10) requires the specification of the

drift, diffusion, proliferation and death coefficients, i.e. the terms W, Q, Γ, L, and,

in particular of their dependence of the state variables.

This can be done on the basis of phenomenological arguments after having fixed

(3.10) to the various state variables.

As already mentioned, looking at the process both from a lattice viewpoint and

a continuum viewpoint and comparing the results can constitute a link between

microscopic phenomena and macroscopic description. In addition, this can help in

identifying the type of phenomenological observation which can be used to evaluate

the coefficients also from the quantitative viewpoint, or at least to have an idea of

their order of magnitude.

4. Specific Modelling Assumptions

We need to consider, in order to specify the form of the drift, diffusion, gain and loss

terms, in more detail the specific phenomena generating them. The model deduced

in this section is based on the following assumptions:

1) Assumptions on the terms influencing tumour evolution:

1.1) Mitosis of tumour cells occurs only if the cells receive a quantity of nutrient

uN , which can be strictly larger than the amount uN necessary for its

survival.

1.2) Proliferation is affected by GIF, which inhibits mitosis and by the amount

of nutrient which promotes it.

1.3) Tumour cells die only if the nutrient is not sufficient to feed all cells.

1.4) There exists a threshold density u, which is here named close packing overall

density, characterised by the fact that if the total density of all cells in a

point x is above it, then tumour cells are pressed by their neighbours and

tend to migrate towards a region characterised by a lower total density (see

Fig. 1).

1.5) Dead tumour cells do not move.

1.6) Dead tumour cells naturally disintegrate into waste products, mainly water.

1.7) Dead tumour cells outside the tumour are eaten by macrophages.

2) Assumptions on the terms influencing the diffusion of chemical factors:

2.1) Living tumour cells constantly produce the chemical factors (GIF and

TAF).

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2.2) The chemical factors diffuse both in the region T (t) occupied by the tumour

and in the tumour-free region (see Fig. 2a).

2.3) The diffusion mechanism of the chemical factors is the same and may de-

pend on the effective overall density, as cells occupy part of the space where

the factors diffuse. In particular, it may be drastically different inside and

outside the tumour.

2.4) Chemical factors naturally degrade.

3) Assumptions on the terms influencing angiogenesis:

3.1) When stimulated by TAF, endothelial cells proliferate at a rate propor-

tional to the concentration of TAF (see Fig. 2b). In addition, proliferation

decreases with the density of new capillaries. In particular, it stops if the

density of endothelial cells is higher than a threshold value.

3.2) Newborn endothelial cells move randomly and migrate toward the source

of angiogenic stimulus with formation of capillary sprouts by accumulation

of endothelial cells (see Fig. 2c).

3.3) Newborn endothelial cells undergo natural death, while old capillaries are

constantly replaced so that their distribution is constant in time.

3.4) Proteins like angiostatins are modelled as having the ability of stopping the

proliferation of endothelial cells, e.g. by drastically reducing their sensitivity

to the presence of TAF.

4) Assumptions on the terms influencing nutrient diffusion:

4.1) Nutrients are mainly carried by the capillary network, though some nu-

trients diffuse through the environment outside the capillary network. In

particular, in the absence of new capillaries the region outside the tumour

is kept at a constant amount of nutrient. With the formation of capillaries,

the nutrient reaching the tumour surface increases proportionally to the

capillary density.

4.2) Nutrient diffusion in the tumour is promoted by the presence of capillaries.

4.3) Nutrient is absorbed by living tumour cells.

4.1. Tumour evolution

Assumption 1.4 states that the movement of cell located in x is related to the total

density u of cells around it and on its local gradient.

In determining u, we assume that all cells, living tumour cells (uT ), dead tu-

mour cells (uD), new capillaries (uC), and possibly the pre-existing capillaries (uC),

contribute in the same way. Then

u = uT + uD + uC + uC . (4.1)

If the undeformed configuration of all cells were a sphere of the same radius,

the threshold density u introduced in assumption 1.4 would correspond to what is

called in the multiphase flows literature the close packing density of a monodisperse

March 27, 2000 11:34 WSPC/103-M3AS 0049


(a) (b)

Fig. 4. Effective overall density. Living tumour cells are white and dead cells are grey. Limitcases (a) ε1 = ε2 = 0, (b) ε1 = ε2 = 1.

mixture of spherulites. If, instead, radii are different, then u would correspond to a

polidisperse mixtures of sperulites.

Remark 4.1. The assumptions that all cells equally contribute to the total density

is an approximation of reality. Dead cells do not occupy the same space as living cells

and there is a geometrical difference between tumour cells and endothelial cells. The

former may be taken as deformable spheres, the latter are aggregated in cylindrical-

like shapes. In detail, one could consider the topological and geometrical microscopic

configuration of compact tumours and investigate to which extent capillaries profit

of the space found between tumour cells. One can then consider the overall density

to be given by u = uT +ε1uD+ε2(uC+ uC). For instance, the limit case ε1 = ε2 = 0

corresponds to saying that tumour cells only feel the “pressure” induced by the

presence of other living tumour cells nearby, while dead cells become so small to

stay in the space left free by the living tumour cells and capillaries sneak through

them without disturbing their distribution, as shown in Fig. 4a. However, capillaries

are larger than tumour cells and therefore it is more likely that tumour cells tend

to wrap over them adapting to the shape of the capillaries, as shown in Fig. 4b.

Going back to the movement of cells, when a new tumour cell is generated, if

there are other cells nearby, it will force them to make room for itself (see Fig. 1).

The pressure then generates a motion of the neighbouring cells and so on in a

spherically symmetric fashion. In the absence of other phenomena, the new steady

configuration will again have u = u everywhere.

The pressure P exerted can be taken to be a function of the local density of the

cells (see Fig. 5), so that one can take the following phenomenological relation

hWT = −∇P ,

which resembles Darcy’s law, or

WT = −P′(u)

h∇u . (4.2)

The function P (u) has to vanish for u = u, corresponding to the fact that no

pressure is felt by a cell when the total density u is equal to the close packing overall

March 27, 2000 11:34 WSPC/103-M3AS 0049


Fig. 5. Pressure-overall density relation determining cell motion.

density u. It is positive for u > u, corresponding to repulsive forces in compression,

and negative for u < u, corresponding to the attractive forces among the cells

which tend to stay aggregate in the form of a ulticell spheroid. In addition, it is

also characterised by the presence of an asyntote at u = uM , where uM represents

the maximum density that can be achieved by compressing the multicell spheroid.

The simplest relation having the mentioned properties is

P (u) = p(uM − u)u− uuM − u

, (4.3)

where p is the slope of P (u) in u = u.

Since in practical situations u does not depart much from u, as a first approxi-

mation one can also use the linearization

P (u) = p(u− u)

so that (4.2) can be rewritten as

WT = −wT∇u (4.4)

with wT = p/h.

It can be noted that if, independently from the value of the total density,∇u = 0,

i.e. the effective overall density has a stationary point, then again WT = 0 as the

tumour cell does not know which direction to go.

As far as the production term is concerned, recalling Assumptions 1.1 and 1.2,

a cell will proliferate if it receives a quantity of nutrient uN above a threshold value

uN , i.e. if uN > uNuT , it increases with the available nutrient and decreases with

the presence of GIFs. In addition, as proliferation is by mitosis of living tumour

cells, the growth term is proportional to the density of living tumour cells

ΓT = GTuT . (4.5)

One can then model GT as

GT =γTuN

ε+ αuIH(uN − uNuT ) , (4.6)

March 27, 2000 11:34 WSPC/103-M3AS 0049


where ε is the amount of nutrient present in the environment in the avascular case,

and (if α or uI vanishes) γT is the growth rate coefficient of tumour cells living in

an environment with an amount of nutrient equal to ε.

If, instead, the nutrient uN is not sufficient to feed the tumour, i.e. if uN < uNuTwith uN ≤ uN , then tumour cells will die, so that one has a death coefficient

LT = δTH(uNuT − uN) . (4.7)

In conclusion, the evolution equation for the living tumour cells can be written

as

∂uT

∂t+∇ · (uTWT ) =

[γTuN

ε+ αuIH(uN − uNuT )− δTH(uNuT − uN)

]uT . (4.8)

Remark 4.2. It should be mentioned that in the case of lack of nutrient, tumour

cells will die. This will decrease the number of cells which feeds upon the nutrient

arriving in the region and will increase their probability of survival. The death of

cells stops when a sufficient number of tumour cells has died so that the nutrient

reaching the region is sufficient to feed all the remaining cells. This will give rise

to a smooth transition into the necrotic region. This is different, for instance, from

Refs. 6 and 7 which assume the existence of a sharp front delimiting the necrotic

region characterised by the amount of nutrient. This would mean that when tumour

cells do not receive enough nutrient, they will suddenly die.

Remark 4.3. Allowing uN > uN corresponds to saying that in order to proliferate

tumour cells need a surplus of nutrient, as mentioned in assumption 1.1. This corres-

ponds to the existence of quiescent tumour cells which live but do not proliferate.

Of course, the death of tumour cells increases the number of dead cells, so that

the same term appearing in (4.8) as destruction term will appear as a “proliferation”

term in the evolution equation for uD. Recalling then assumption 1.5, one has

∂uD

∂t= δTH(uNuT − uN )uT − δDuD . (4.9)

4.2. Chemical factor evolution

As already stated in Sec. 2 among all chemical factors a relevant role seems to be

played by GIF and by TAF which are produced by active tumour cells at rates γIand γA, and naturally decay at rates δI and δA, respectively.

As mentioned in assumption 2.1, once produced, these chemical factors move

according to a random motion in the region T (t) occupied by the tumour, but can

also diffuse in the tumour-free region. Of course, GIF will have no effect there, as it

will not encounter tumour cells, but, as we shall see in the next section, the diffusion

of TAF plays the crucial role of inducing the generation of capillaries just outside

the tumour to bring more nutrient to the solid tumour.

March 27, 2000 11:34 WSPC/103-M3AS 0049


Having assumed in assumption 2.2 that the diffusion mechanism of the two

factors is the same, then the diffusion coefficient is the same. Of course, it may

depend on the effective overall density, as cells occupy part of the space where the

two factors diffuse. In particular, it may differ dramatically inside and outside the

tumour.

The evolution of uI and uA can then be written as

∂uI

∂t= ∇ · [KF (u)∇uI ] + γIχT uT − δIuI , (4.10)

∂uA

∂t= ∇ · [KF (u)∇uA] + γAχT uT − δAuA . (4.11)

where χT = χT (x, t) is the characteristic function of T (t), or, in the particular case

of constant diffusion coefficient,

∂uI

∂t= kF∇2uI + γIχT uT − δIuI , (4.12)

∂uA

∂t= kF∇2uA + γAχT uT − δAuA . (4.13)

4.3. Angiogenesis

Tumour angiogenesis is promoted by the presence of TAF diffusing from the solid

tumour. When the endothelial cells interact with this factor, they proliferate and

the new-born cells tend to migrate towards the source of angiogenic stimulus with

formation of capillary sprouts by accumulation of endothelial cells. Initially, the

only cells which can be stimulated are those belonging to the pre-existing vascu-

lar network. These are then the origin of new capillary sprouts, which are then

stimulated to duplicate.

The drift term WC is then directed towards the source of angiogenic stimulus

and can be modelled as

WC = WC(u)∇uA , (4.14)

giving rise to a chemotactic term. In addition, according to assumption 3.1, the

growth term can be modelled as

ΓC = γCuA(uC − uC)+(uC + uC) , (4.15)

where

(f)+ =

{f if f > 0 ;

0 otherwise ;(4.16)

is the positive part of f .

Assuming that the diffusion and drift terms are constant, the balance equation

(3.10) specialises to

∂uC

∂t+ wC∇ · (uC∇uA)

= kC∇2uC + γCuA(uC − uC)+(uC + uC)− δCuC , (4.17)

March 27, 2000 11:34 WSPC/103-M3AS 0049


Fig. 6. Geometric configuration. Dark circle are dead tumour cells and white circles are livingcells.

where uC = uC(x) is the density of pre-existing capillaries, which, according to

assumption 3.3, is time-independent, fixed in space and naturally space dependent.

This equation applies both within the solid tumour and outside it. In particular,

as TAF may diffuse outside the tumour, near the tumour surface it models the

generation of capillary sprouts in part diffusing as a result of random motion and

in part directing themselves toward the active tumour cells. The relative importance

of these two behaviours depends on the relative magnitude of kC and wC .

It can be observed that, in agreement with assumption 3.3, in the absence of

angiogenic stimulus the dead cells belonging to the new capillaries are not replaced,

so that the newly formed network eventually disappears leaving the pre-existing

vascular system unaffected.

4.4. Nutrient evolution

Assumption 4.1 states that the diffusion of nutrient inside the tumour is promoted

by the presence of capillaries. This means that the diffusion coefficient increases

with uC . Taking into account the uptake from the living tumour cells, the balance

equation can be written as

∂uN

∂t= ∇ · [(kE + kN (uC + uC))∇uN ]− δNuTuN , (4.18)

where kE is the diffusion coefficient in the absence of capillaries and kN measures

the dependence of the diffusion coefficient on the presence of capillaries.

This evolution equation has to be applied only in T (t), as it is assumed that

outside T (t) the capillary network can constantly keep the amount of nutrient in a

quantity which increases linearly with its density.

In particular, this is the amount of nutrient constantly brought from the outside

to the surface of the solid tumour. Hence,

uN = ε+ β(uC + uC) on ∂T (t) , (4.19)

represents the proper interface condition for (4.18).

March 27, 2000 11:34 WSPC/103-M3AS 0049


The nutrient that enters the solid tumour is first consumed by the outer shells

and diffuses toward the inner shells, which are reached by decreasing amounts of

nutrient. As already stated, when the nutrient is no longer enough, tumour cells

start dying generating the necrotic region.

4.5. The model

Summing up the model deduced in this section in charge of describing the evolution

of the tumour in the presence of angiogenesis is the following

in D :

∂uI

∂t= kF∇2uI + γIχT uT − δIuI ,

∂uA

∂t= kF∇2uA + ΓAχT uT − δAuA ,

∂uC

∂t+ wC∇ · (uC∇uA)

= kC∇2uC + γCuA(uC − uC)+(uC + uC)− δCuC ,

(4.20a)

in T (t) :

∂uT

∂t= wT∇ · (uT∇u) +

[γTuN

ε+ αuIH(uN − uNuT )

− δTH(uNuT − uN )

]uT ,

∂uD

∂t= δTH(uNuT − uN)uT − δDuD ,

∂uN

∂t= ∇ · [(kE + kN (uC + uC))∇uN ]− δNuTuN ,

(4.20b)

where, to be more specific, it has been set WT = −wT∇u.

In formulating the dependence of the coefficients on the state variables, we tried

to keep the number of parameters to a minimum. Despite this attempt, because

of the numerous phenomena which have been considered, the number of param-

eters is not so small, so we tried to distinguish production coefficients (γ), death

coefficients (δ), diffusion coefficients (k), transport coefficients (w) and reference

densitied (ui, ui). The index then refers to the state variable on which the constant

has its effect. In more details,

Growth coefficients:

• α refers to the inhibited proliferative effect due to the presence of GIF;

• γI is the production coefficient of GIF from living tumour cells;

• γA is the production coefficient of TAF from living tumour cells;

• γC is the proliferative coefficient of capillary sprouts;

• γT is the proliferative coefficient of tumour cells uT .

Destruction coefficients

• δI is the degradation coefficient of GIF;

March 27, 2000 11:34 WSPC/103-M3AS 0049


• δA is the degradation coefficient of TAF;

• δC is the natural death coefficient of endothelial cells;

• δT is the death coefficient of tumour cells when the nutrient is not sufficient to

feed all cells;

• δD is the disintegration coefficient into waste products of dead tumour cells;

• δN is the absorption coefficient of nutrient from active tumour cells.

Reference and threshold densities:

• uN is the amount of nutrient needed to feed a tumour cell;

• uN is the amount of nutrient needed per tumour cell to start proliferating;

• u is the threshold density, here called close packing density, above which living

tumour cells are pushed toward regions with smaller effective overall density;

• uC is the threshold overall density above which capillaries are not generated;

• uC is the density of pre-existing capillaries from which new capillaries start being

generated. This is a space dependent function.

Diffusion coefficients:

• kF is the diffusion coefficient of chemical factors;

• kC is the diffusion coefficient of capillary sprouts;

• kE is the diffusion coefficient of nutrient outside the capillary network inside the

tumour;

• kN is the diffusion coefficient of nutrient through the capillary network inside the

tumour.

Drift velocities

• wC is the migration coefficient of capillary sprouts towards the source of angio-

genic stimulus;

• wT is the migration coefficient of active tumour cells toward less compressed

regions.

However, each parameter listed above has a specific meaning, refers to a well-

defined phenomenon and has a particular effect on a specific cell population. Hence,

suitable phenomenological observations hopefully accompanied by ad hoc experi-

ments can lead to their evaluation, at least in order of magnitude or in comparison

with other effects.

To be more specific, growth and destruction coefficients, and reference and

threshold densities can be evaluated by looking at the system at the cellular scale.

In more details, growth coefficients can be evaluated by looking at the mitotic rate

of cells, and death coefficient by looking at the rate at which cells die in the absence

of mitosis. Degradation coefficients relative to chemical factors and the absorption

coefficient of the nutrient by the tumour cells can be evaluated by measuring their

evolution when they are no longer produced by the cells or by the environment, res-

pectively. The amount of nutrient which promotes proliferation or that cause death

can be evaluated by looking at the behaviour of the cells for decreasing amounts of

March 27, 2000 11:34 WSPC/103-M3AS 0049


nutrient. The close packing reference density can be evaluated by “counting” the

number of cells in a stationary configuration. It is slightly more difficult is to obtain

direct measurements of diffusion and drift coefficients, because they might involve

observations at a sub-cellular scale and might be affected by strong random fluctua-

tions. However, experiments are usually performed to measure diffusion coefficients

which turn out to be of order 10−10 m2/sec.

We finally notice that in the model presented, the action of the immune system

is not taken into account, as we assumed that tumour growth is at a stage such

that the competition with the immune system has little effect on its growth. Still

in the framework of continuous models this problem was studied in Refs. 28 and

29.

5. The Free-Boundary Value Problem

Some of the state variables in (4.20), more precisely uI , uA and uC , are defined in

the whole environment D, while others, namely uT , uD and uN , are defined in the

tumour only, T (t).

As this region depends on time (e.g. the tumour grows) one has to give an

evolution equation which is responsible for determining the position x = xT (t) of

the border ∂T (t) of the tumour. Referring to Fig. 1, it can be realised that this

interface is a material interface fixed on the tumour and therefore has to move with

the tumour cells at the free surface, i.e.

nTdxTdt

= nTWT (xT ) , (5.1)

or referring more specifically to (4.20)

nTdxTdt

= −wTnT∇u(xT ) . (5.2)

The mathematical problem describing the evolution of the tumour in the pres-

ence of angiogenesis is then written as a free-boundary value problem. The math-

ematical structure of the model is rather complex. It consists of three parabolic

equations valid in the whole domain D. Those referring to the chemical factors are

diffusion equations, while that referring to the formation of capillary sprouts is an

advection-diffusion equation which can be advection dominated, i.e. dominated by

chemotaxis. Within the time-dependent domain referring to the tumour, one has

parabolic equations for the nutrient and for the living tumour cells and a hyperbolic

equation for the dead cells.

As far as the boundary conditions are concerned, the domain D can be taken

large enough to assume that its boundary is not reached by the chemical factors or

by the new capillaries. This gives the boundary conditions

on ∂D :

uI = 0 ,

uA = 0 ,

uC = 0 .

(5.3)

March 27, 2000 11:34 WSPC/103-M3AS 0049


In posing the proper boundary conditions for the variables defined in the tumour

we will first look at growing tumours. In this case the characteristics related to the

hyperbolic equation (4.9) enter the expanding domain T (t), from the tumour surface

∂T (t), and one has to prescribe there a condition on uD. As the motion is due to

the generation and motion of living tumour cells, it is natural to impose that there

are no dead cells on the surface

uD(xT (t), t) = 0 . (5.4)

Furthermore, as the tumour surface is stress-free, the cells are not compressed there

and the overall density is equal to the close packing one, so that one can write

uT (xT (t), t) = u− uC(xT (t), t)− uC(xT (t)) , (5.5)

where Eq. (4.1) has been recalled.

If, instead, the tumour is imploding (i.e. WT · nT ≤ 0 where nT is the normal

to ∂T ), then no condition has to be imposed on uD at the tumour surface as char-

acteristics are exiting the domain. This corresponds to saying that if the implosion

is fast, in principle, one could find dead cells left beyond the retiring surface. Ac-

cording to assumption 1.7, they are, however, eaten by macrophages, so that uDcan still be considered as defined in T (t).

In summary, on ∂T the boundary conditions are

on ∂T (t) :

uT = u− uD − uC − uC ,uN = ε+ β(uC + uC) ,

uD = 0 , if WT · nT > 0 ,

(5.4′)

where the last condition is imposed only for expanding tumours.

The free boundary value problem can be solved by starting from the following

initial conditions

in D : uI = uA = uC = 0 , (5.5′)

in T (t = 0) : uT = u , uD = 0 , uN = ε , (5.6)

xT (0) = x0 , (5.7)

as uC(x) is taken with support outside T (t = 0), which mean implanting some

living tumour cells in an environment initially full of nutrient and in the absence

of chemical factors (which is natural if there where no tumour cells before the

implant).

The free-boundary value problem can be put in dimensionless form, normalizing

space by√kE/δN u, time by γT , cell densities (i.e. uT , uD, uC , uC , and u) by u,

nutrient density by ε and chemical factor densities by γiu/γT , (i = A, I). Denoting

by a star the dimensionless quantities, the free boundary problem can be written

March 27, 2000 11:34 WSPC/103-M3AS 0049


as

in D :

∂u∗I∂t∗

= k∗F∇2u∗I + χT u∗T − δ∗Iu∗I ,

∂u∗A∂t∗

= k∗F∇2u∗A + χT u∗T − δ∗Au∗A ,

∂u∗C∂t∗

+ w∗C∇ · (u∗C∇u∗A)

= k∗C∇2u∗C + γ∗Cu∗A(u∗C − u∗C)+(u∗C + u∗C)− δ∗Cu∗C ,

(5.8a)

in T (t∗) :

∂u∗T∂t∗

= w∗T∇ · (u∗T∇u∗) +u∗Tu

∗N

1 + α∗u∗IH(u∗N − u∗Nu∗T )

− δ∗TH(u∗Nu∗T − u∗N)u∗T ,

∂u∗D∂t∗

= δ∗TH(u∗Nu∗T − u∗N )u∗T − δ∗Du∗D ,

∂u∗N∂t∗

= δ∗N {∇ · [(1 + k∗N (u∗C + u∗C))∇u∗N ]− u∗Tu∗N} ,

(5.8b)

with boundary conditions

on ∂T (t∗) :

u∗T = 1− u∗D − u∗C − u∗C ,u∗N = 1 + β∗(u∗C + u∗C) ,

u∗D = 0 , if W∗T · nT > 0 ,

(5.8c)

on ∂D : u∗I = u∗A = u∗C = 0 , (5.8d)

and initial conditions

in D : uI = uA = uC = 0 , (5.8e)

in T (t∗ = 0) : u∗T = u∗N = 1 , u∗D = 0 , (5.8f)

x∗T (0) = x∗0 , (5.8g)

where

δ∗N =δN u

γT, δ∗i =

δ∗iγT

, i = A,C,D, I, T ,

k∗F =kF δN u

kEγT, k∗N =

kN u

kE, k∗C =

kCδN u

kEγT,

α∗ =αγI u

γT ε, β∗ =

βu

ε, γ∗C =

γCγAu2

γ2T

,

w∗C =wCδNγAu

2

kEγ2T

, w∗T =w∗T δN u

2

kEγT,

u∗N =uuN

ε, u∗N =

uuN

ε.

(5.9)

March 27, 2000 11:34 WSPC/103-M3AS 0049


Useful dimensionless numbers can also be

κ∗C =kC

kE=k∗Cδ∗C

, κ∗F =kF

kE=k∗Fδ∗C

, (5.10)

which are the ratio of the diffusion coefficients, and

ω∗C =w∗Ck∗C

=wCγAu

kCγT, ω∗T =

w∗Tw∗C

=wTγT

wCγA, (5.11)

which, respectively, give an indication of the relative importance of the diffusion

term versus the chemotactic one in the equation for the capillaries and of the ratio

of “tumor growth acceleration” over “chemotactic acceleration” in the equation for

the living tumor cells.

6. Qualitative Behaviours

In order to understand more on the effects produced by the different terms charac-

terising the model, we will examine the qualitative behaviour of the solution for

the one-dimensional problem. In particular, we will first consider the avascular case

with wT = 0, so that the tumour free surface does not move. In this case the

equation for uT also becomes hyperbolic and does not need boundary conditions.

1) As the nutrient evolution is governed by a diffusion equation with a sink term,

uN will attain its maximum at the tumour surface and will decrease with

increasing distance from the surface.

Fig. 7. Identification of the interfaces delimiting the proliferating, the quiescent and the regionwith dead cells.

March 27, 2000 11:34 WSPC/103-M3AS 0049


2) If u∗T = 1 were constant, then the nutrient would decrease exponentially in

time with a dimensionless characteristic time t∗ ≈ 1/δ∗N to the stationary

distribution

u∗N =coshx∗

coshx∗T. (6.1)

Hence, in this case one has that the nutrient distribution within the tumor

drops exponentially in a (dimensional) depth of order√kE/δN u. In particular,

the amount of nutrient reaching the centre of the tumour is 1/ cosh x∗T . In the

simulation presented in Ref. 17, this can also be observed for evolving u∗T , as

long as δ∗T � δ∗N . Referring to Fig. 7 and if

1 > u∗N > u∗N >1

coshx∗T,

one has two points x∗Q such that u∗N (x∗Q) = u∗N and x∗N such that u∗N (x∗N ) = u∗N .

Still assuming that u∗T = 1, these points are given by

x∗Q = cosh−1(u∗N coshx∗T ) , x∗N = cosh−1(u∗N coshx∗T ) . (6.2)

In the region between x∗T and x∗Q in the evolution equation for u∗T only the gain

term is present. This is then the proliferating region. In the region between x∗Qand x∗N (or, in the absence of x∗N , below x∗Q) neither the gain, nor the loss

term are present. This is then a quiescent region. It can be noticed from Fig. 8

that the depth of the proliferating layer rapidly becomes independent of the

tumor radius (say, for x∗T > log 10/u∗N) and nearly given by − log u∗N . In fact,

the given level of nutrient is achieved at a distance from the tumour surface

which rapidly tends to a constant. A similar thing occurs for the quiescent

layer with x∗Q − x∗N ≈ log u∗N/u∗N . This means that (if u∗T = 1) while evolving

Fig. 8. Location of given nutrient levels from the tumour surface as a function of the dimensionof the tumour. From below the curves refer to u∗N = 0.9, 0.8, 0.7, 0.6, 0.5, 0.4, 0.3, 0.2, 0.1. Thevertical line refers to a tumour of dimensionless size equal to 3 and represents the thickness of theproliferating layer (thicker line) if u∗N = 0.8 and of the quiescent layer (middle line) if u∗N = 0.5.

March 27, 2000 11:34 WSPC/103-M3AS 0049


the depths of the proliferating and of the quiescent layers remain constant, as

experimentally confirmed by the observations reported in Refs. 16 and 20. In

Fig. 8 this is shown considering u∗N = 0.8 and u∗N = 0.5. As x∗T grows, the

proliferating layer indicated with the thicker line tends to a constant, as the

quiescent layer. Consequently, the necrotic region will grow.

3) As already stated, if u∗N < 1, i.e. the amount of nutrient arriving at the tumour

surface is larger than that needed by all cells living there to proliferate, there

is a region near the tumour surface of proliferating cells. Mitosis stops when

u∗T = u∗N/u∗N (see Fig. 9(a)) as at this point the nutrient does not furnish to

the cells that surplus necessary to promote proliferation. This occurs in a time

of order

t ≈ (1 + α∗u∗I)

(u∗N

u∗Nu∗T (0)

− 1

). (6.3)

It should be mentioned that, if the assumption w∗T = 0 is dropped, this pro-

liferation will probably generate tumour expansion as u∗ > 1 near the surface

(see point 10).

4) If u∗N < 1 < u∗N , then at the tumour surface there are no proliferating cells (see

Fig. 9b). If u∗N > 1, then already on the tumour surface there is copresence

of quiescent and dead tumour cells (see Fig. 9c). More precisely, the steady

configuration will present a percentage of 1/u∗N living cells on the tumour

surface.

5) In the region below x∗N tumour cells do not have enough nutrient to survive

and start to die. However, when u∗T (t∗, x∗) has decreased to u∗N (t∗, x∗)/u∗N ,

tumour cells will stop dying. There is then a boundary layer with copresence of

living and dead tumour cells near the quiescent interface x∗N , which represents

the transition to the necrotic region, in agreement with the description given

Fig. 9. Proliferation and death of cells for (a) u∗N < u∗N < 1, (b) u∗N < 1 < u∗N , (c) 1 < u∗N < u∗N .

March 27, 2000 11:34 WSPC/103-M3AS 0049


in Ref. 32. The steady state u∗T = u∗N/u∗N is reached in a time of order

t∗ ≈ 1

δ∗T(6.4)

and the dimensional width of the layer is ∆ =√

6kE/δN u. In fact, if x∗N � 1,

the solution near x∗N can be approximated by

u∗N ≈u∗N[

1 +x∗N−x∗√

6

]2 , u∗T ≈1[

1 +x∗N−x∗√

6

]2 . (6.4′)

One can then say that the necrotic region becomes evident for depths of

order xT − xN +√

6kE/δN u.

6) Living and quiescent cells produce chemical factors, which also diffuse outside

the tumour. If uT were constant, the stationary configuration of TAF and

GIF diffusion would be characterised by diffusion layers of dimensional width√kF/δA and

√kF /δI , respectively.

Referring now in more detail to tumour expansion and angiogenesis we can

observe the following:

7) The expansion of the tumor always tries to keep the total cell density equal to

one. Hence the observation done in point 2 has a larger validity than what can

be originally expected.

8) The formation of capillary sprouts increases the amount of nutrient reaching the

tumour to u∗N(x∗T ) = 1 +β∗(u∗C + u∗C)(x∗T ) ≡ ε∗. Besides increasing the mitotic

rate of the already active cells, this moves down the interface between active

and quiescent cells waking up some tumour cells which will start proliferating

and contributing to the growth of the tumour. For instance, (6.1) and (6.2) can

be rewritten as

u∗N = ε∗coshx∗

coshx∗T, x∗Q = cosh−1

(u∗Nε∗

coshx∗T

). (6.5)

9) As a result of disintegration of dead cells in waste products, the overall density

in the necrotic region decreases below the close packing density. This induces

quiescent cells to move there, finding less nutrient and probable death. In the

same way some living cells move in the quiescent region. If, however, prolifera-

tion of living cells is strong enough, then the tumour will still grow. When the

proliferation of cells near the tumour surface balances the destruction of dead

cells in the necrotic region, the tumour will not expand. If on the other hand,

the proliferation of cells is not sufficient to balance the destruction of dead

cells, then the tumour will implode to the steady state above. This certainly

occurs in the situations discussed in point 4 above. However, it can be a result

of a decrease of available nutrient related, for instance, to the destruction of

the new capillary network.

10) Setting γ∗C to zero (i.e. with the injection of angiostations) will generate a

regression of the new capillary network in a time of order t∗ ≈ 1/δ∗C.

March 27, 2000 11:34 WSPC/103-M3AS 0049


In Ref. 17, all the above-mentioned features are studied in more details. In

particular, focusing on the avascular phase, the paper analyzes the sensitivity of

the model to the values of u∗N and u∗N and the dependence of the stationary value

of tumour size in the avascular phase on the parameters. Moreover, in dealing with

angiogenesis, it analyzes the dependence of the evolution on γ∗C and the effect of

angiostatins.

7. Summary and Discussion

The paper proposes a three-dimensional mathematical model to describe tumour

growth form the avascular to the vascular phase through the angiogenic process.

The different phases are not separated from one another, but identified during

the continuous evolution of the system. Special attention is also paid to the inter-

actions between the growing tumour and the evolving environment. As some state

variables are defined in the tumour only and others also outside, the mathematical

problem is formulated as a free-boundary problem.

The model is able to predict

• The diffusion of nutrient through the surface of the tumour and the capillary

network and its difficulty in reaching the central region of the tumour;

• The formation of necrotic regions as a result of lack of nutrient, even starting

from a situation with all living cells in an environment full of nutrient;

• The existence of an outer proliferating layer in condition of sufficient nutrient,

which for large tumour size has nearly a constant depth.

• The existence of a layer of quiescent cells which can activate when the amount

of nutrient present in the environment increases, e.g. after the formation of the

induced capillary network;

• The existence of a limit tumour radius in the avascular phase depending on the

amount of nutrient present in the environment;

• Tumour expansion due to the birth of new cells;

• The control of tumour mitosis by the presence of inhibitory factors;

• The angiogenic process with proliferation of capillaries just outside the tumour

surface and penetration of capillary sprouts inside the tumour;

• The increase of tumour growth rate with the formation of capillary sprouts;

• The regression of the new capillary network induced by the tumour as an effect

of proteins inhibiting angiogenesis as angiostatins.

• The shrinking of tumour radius when the induced capillary network is destroyed.

Some a priori estimates of the main qualitative characteristics of tumour

evolution, as the characteristic time for the formation of the necrotic region or

the regression of the capillary network and the width of the layer of living cells at

the tumour surface are also given as a function of the parameters characterising the

model.

The model could be improved by describing some phenomena in more detail, or

including several other phenomena, such as

March 27, 2000 11:34 WSPC/103-M3AS 0049


• Cell-to-cell interaction, e.g. adhesion, response to compression;

• Formation and diffusion of metastasis;

• Competition with the immune system;

• Inclusion of the extracellular liquid as the phase in which tumour cells live;

• Inclusion of other chemical factors, e.g. growth promoting factors, growth in-

hibitory factors for endothelial cells.

These topics, however, need a more careful analysis starting from the observation

of the phenomena involved. The authors are well aware of the great difficulty in

dealing with the topics studied in this paper or with the developments mentioned

in this section, and above all on the need of a continuous relation and feedback that

should link mathematical models and phenomenological descriptions. Nevertheless,

we hope that developing models and related computational schemes can be regarded

as a contribution, hopefully useful, to reduce or even more properly address an

experimental activity, which, as is well known, is expensive and time consuming.

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