J. Math. Biol. (1995) 33:744-770
deurnol of
Mathematical
Blolo9y
© Springer-Verlag 1995
A mathematical analysis of a model for
tumour angiogenesis
M.A.J. Chaplain 1, Susan M. Giles z, B.D. Sleeman 2, R.J. Jarvis 2
1School of Mathematical Sciences, University of Bath, Bath BA2 7AY, England
e-mail: [email protected], Fax: 01225 826492
2Department of Mathematics and Computer Science, University of Dundee,
Dundee DD1 4HN, Scotland
Received 9 May 1994; received in revised form 21 December 1994
Abstract. In order to accomplish the transition from avascular to vascular
growth, solid tumours secrete a diffusible substance known as tumour angiogenesis factor (TAF) into the surrounding tissue. Neighbouring endothelial
cells respond to this chemotactic stimulus in a well-ordered sequence of events
comprising, at minimum, of a degradation of their basement membrane,
migration and proliferation. A mathematical model is presented which takes
into account two of the most important events associated with the endothelial
cells as they form capillary sprouts and make their way towards the tumour
i.e. cell migration and proliferation. The numerical simulations of the model
compare very well with the actual experimental observations. We subsequently investigate the model analytically by making some relevant biological simplifications. The mathematical analysis helps to clarify the particular contributions to the model of the two independent processes of endothelial
cell migration and proliferation.
Key words: Angiogenesis - Endothelial cells - Proliferation - Migration
1 Introduction
Solid tumours are known to progress through two distinct phases of
growth - the avascular phase and the vascular phase (Folkman 1974, 1976).
The transition from the dormant avascular state to the vascular state, wherein
the tumour possesses the ability to invade surrounding tissue and metastasise
to distant parts of the body, depends upon its ability to induce new blood
vessels from the surrounding tissue to sprout towards and then gradually
penetrate the tumour, thus providing it with an adequate blood supply and
microcirculation. In order to accomplish this neovascularization, it is now
a well-established fact that tumours secrete a diffusible chemical compound
known as tumour angiogenesis factor (TAF) into the surrounding tissue and
Tumour angiogenesis model
745
extracellular matrix. Much work has been carried out into the nature of TAF
and its effect on endothelial cells since initial research began in the early 1970's
with Folkman, culminating quite recently in the purification of several angiogenic factors, the determination of their amino acid sequences and the
cloning of their genes (Strydom et al., 1985; Folkman and Klagsbrun, 1987;
Deshpande & Shetna, 1989). The extensive current literature on the subject is
testimony to its importance in our understanding of the mechanisms by which
solid tumours develop and grow (see, for example, the reviews of Folkman and
Klagsbrun, 1987, and Paweletz and Knierim, 1989).
The first events of angiogenesis are rearrangements and migration of
endothelial cells situated in nearby vessels. The main function of endothelial
cells is in the lining of the different types of vessels such as venules and veins,
arterioles and arteries, small lymphatic vessels and the thoracic duct. They
form a single layer of flattened and extended cells and the intercellular
contacts are very tight. Large intercellular spaces are not visible and any easy
penetration of the established layer of cells is impossible. Special processes
must take place for the intra- and extravasation of different cellular elements
of the blood or the lymphatic fluids and tumour cells. Even intravascular
tumour cells have to induce the formation of gaps in the single layer of
endothelial cells in order to leave the respective vessels. These cells are the
principal characters in the drama of angiogenesis and are always centre stage
(Paweletz and Knierim, 1989).
In response to the angiogenic stimulus, endothelial cells in the neighbouring normal capillaries which do not possess a muscular sheath are activated to
stimulate proteases and collagenases. The endothelial cells destroy their own
basal lamina and start to migrate into the extracellular matrix. Small capillary
sprouts are formed by accumulation of endothelial cells which are recruited
from the parent vessel. The sprouts grow in length by migration of the
endothelial cells (Cliff, 1963; Schoefl, 1963; Warren, 1966; Sholley et al., 1984).
The experimental results of Sholley et al. (1984) demonstrated that endothelial
cells are continually redistributed among sprouts, moving from one sprout to
another. This permits the significant outgrowth of a network of sprouts even
when cell proliferation is prevented (Sholley et al., 1984). At some distance
from the tip of the sprout the endothelial cells divide and proliferate to
contribute to the number of migrating endothelial cells. The mitotic figures
are only observed once the sprout is already growing out and cell division is
largely confined to a region just behind the sprout tip. Solid strands of
endothelial cells are formed in the extracellular matrix. Lumina develop
within these strands and mitosis continues.
Initially the sprouts arising from the parent vessel grow in a more or less
parallel way to each other. They tend to incline toward each other at a definite
distance from the origin when neighbouring sprouts run into one another and
fuse to form loops or a n a s t o m o s e s . Both tip-tip and tip-branch anastomosis
occur and the first signs of circulation can be recognised. From the primary
loops, new buds and sprouts emerge and the process continues until the
tumour is eventually penetrated.
746
M.A.J. Chaplain et al.
Thus, as is clear from the above brief description, there are three main
events concerning the endothelial cells which go to make up the process of
angiogenesis after the release of TAF by the tumour cells, namely:
1. degradation of the basement membrane by enzymes secreted by the cells,
2. migration of the endothelial cells,
3. proliferation of the endothelial cells.
(a detailed description of the above events can be found in the extensive review
of Paweletz and Knierim, 1989).
We note that the second and third of these stages - endothelial cell
migration and endothelial cell proliferation- are not linked together. They are
distinct events and different types of stimuli are necessary for each of them.
Indeed, the first steps of angiogenesis can be performed without any cell
division at all (Sholley et al., 1984), and it is well known that mitotic figures
can only be found once the sprouts have already started to grow. Under
normal conditions, once they have developed, the walls of both blood and
lymphatic vessels tend to be stable structures, with the endothelial cells there
only rarely undergoing mitosis. Thus cell division is a sine qua non event for
the successful completion of angiogenesis. Endothelial cell migration together
with endothelial cell proliferation are crucial to neovascularization. Angiogenic factors must therefore induce all of the above three events in a wellordered sequence.
In this paper we present a mathematical analysis of a model developed by
Chaplain and Stuart (1993) which describes the response of the endothelial
cells to the chemotactic stimulus of tumour angiogenesis factor. In the following two sections, we give a brief resum6 of this model and present the main
results arising from the numerical simulations of the model. In Sect. 4 we make
a relevant biological simplification to the model regarding the concentration
profile of the TAF in the surrounding tissue. This reduces the model to one
single partial differential equation describing the endothelial cell density and
permits a certain amount of mathematical analysis to be carried out. We
analyse both the time-dependent problem, which is relevant for describing the
motion of the capillary sprouts as they move towards the tumour, and also the
time-independent problem, which is shown to have some relevance to the
modelling of the interactions taking place between the endothelial cells and
the tumour cells in the immediate, initial stages of vascularization. The
concluding section discusses the analysis with regard to the numerical simulations and also, more importantly, with regard to known experimental results
concerning various aspects of angiogenesis.
2 The mathematical model
Chaplain and Stuart (1993) develop a model for the response of the endothelial cells to the chemotactic stimulus of the TAF which consists of two
equations, one for the endothelial cell density per unit area of capillary sprout
Tumour angiogenesis model
747
(n) and the other for the concentration (c) of TAF in the host tissue. Using
conservation principles, the system of equations considered takes on the
general form
rate of increase
of T A F
diffusion
of TAF
rate of increase
of cell density
cell
migration
loss due to
cells
mitotic
generation
decay of
chemical
cell
loss
We now discuss each of the above equations in turn, deriving the mathematical representation of each term of each equation.
2.1 Tumour angiogenesis factor
From experimental work carried out on the cornea of test animals it is known
that there is a critical distance between the tumour implant and the neighbouring vessels. We assume therefore that the tumour implant has been placed
sufficiently close to the vessels of, for example, the corneal limbus (cf.
Gimbrone et al., 1974; Muthukkaruppan et al., 1982) so as to be within the
critical threshold distance observed by Gimbrone et al. (1974) (in these
experiments it was found that no vascularisation of the tumour occurred, or
that the time for vascularisation was substantially increased, when the tumour
implant was placed at a distance of more than 2.5 mm from the limbal vessels).
Tumour angiogenesis factor having concentration c (x, t) is secreted by the
solid turnout and diffuses into the surrounding tissue with diffusion coefficient
De, a constant (the assumption of linear Fickian diffusion).
Upon reaching neighbouring endothelial cells situated in, for example, the
limbal vessels, the TAF stimulates the release of enzymes by the endothelial
cells which degrade their basement membrane. As described in the introduction, after degradation of the basement membrane has taken place, the initial
response of the endothelial cells is to begin to migrate towards the source of
angiogenic stimulus. Capillary sprouts are formed and cells subsequently
begin to proliferate at a later stage. Once the capillary sprouts have formed,
mitosis is largely confined to a region a short distance behind the sprout tips
(Ausprunk and Folkman, 1977; Sholley et al., 1984; Paweletz and Knierim,
1989; Stokes and Lauffenburger, 1991). Ausprunk and Folkman (1977) hypothesised that the reason for this proliferation was that these cells or vessels
at the sprout tips were actin9 as sinks for the TAF. Balding and McElwain
(1985) also suggested that a sink term could be included in their model of
capillary growth. Following Chaplain and Stuart (1991, 1993), we thus incorporate a sink term, f(c), say, for the TAF in addition to a natural decay term
for the TAF and we assume that the local rate of uptake of TAF by the
endothelial cells is governed by Michaelis-Menten kinetics (cf. Lin, 1976;
McElwain, 1978; Hiltman and Lory, 1983; Chaplain and Stuart, 1991, 1993).
We also assume that this uptake rate depends on the cell density through
748
M.A.J. Chaplain et al.
some function g (n), say, i.e. the greater the density of endothelial cells, the
more TAF will be removed by the cells acting as sinks (cf. Ausprunk and
Folkman, 1977; Chaplain and Stuart, 1991, 1993). In general, the function #(n)
can therefore be chosen to be some strictly increasing function to account for
this. For simplicity the actual function used in the model is given by
#(n) = n/no, a simple linear function. More complicated choices are also
possible however and do not affect the model qualitatively.
We also assume that the decay of TAF with time is governed by first-order
kinetics, a standard assumption (cf. Sherratt and Murray, 1990), hence a term
- d c is included. This leads to the following equation for the TAF in the
external tissue
Oc
Qcn
at = D~V2c (Km + c)no dc
(2.1)
The initial condition is given by
c(x, O) = Co(X),
(2.2)
where Co(x) is a prescribed function chosen to describe qualitatively the profile
of TAF in the external tissue when it reaches the limbal vessels (cf. Chaplain
and Stuart, 1991, 1993). The TAF is assumed to have a constant value cb on
the boundary of the tumour and to have decayed to zero at the limbus (cf.
Chaplain and Stuart, 1991, 1993) giving the boundary conditions as
c(O, t) = cb,
c(L, t) = 0 .
(2.3)
2.2 Endothelial cell population balance equation
We follow the route of the endothelial cells from their origin in their
parent vessel (e.g. the limbus), their crossing of the extracellular matrix and
other material in the surrounding host tissue, to their destination within the
tumour.
The main events we model are the mi9ration and the proliferation of the
endothelial cells (the processes of anastomosis and budding will be accounted
for implicitly in the model). We note that the migration and replication of
endothelial cells are not linked together. Different types of stimuli are necessary for these two processes and we take this important fact into account in
our model. The general conservation equation for the endothelial cell density
n(x, t) may be written (Chaplain and Stuart, 1993)
an
a--~+ V . J = F(n)G(c) + H(n)
(2.4)
where J i s the cell flux, F(n) and H(n) are functions representing a normalized
growth term and a loss term respectively for the endothelial cells. We assume
that mitosis is governed by logistic type growth and that cell loss is a first
Tumour angiogenesis model
749
order process (cf. Stokes and Lauffenburger, 1991). Thus
F(n) = rn ( 1 - --~o) '
(2.5)
H(n) = -kpn ,
(2.6)
where r is a positive constant related to the maximum mitotic rate and kp is
the proliferation rate constant which is taken to be the reciprocal of the
endothelial cell doubling time (cf. Sherratt and Murray, 1990; Stokes and
Lauffenburger, 1991). We note that (2.5) contains a second order loss term
while (2.6) is a first order loss term. Balding and McElwain (1985) considered
cell loss due to anastomosis (both tip-tip and tip-branch) as essentially
a second order process, while Stokes and Lauffenburger (1991) modelled
endothelial cell loss due to budding as a first order loss term. Moreover they
assumed that the probability of budding was uniform in all sprouts for all
positions and all times. Thus these two terms implicitly account for endothelial cell loss due to anastomosis and budding respectively. Further we assume
that the endothelial cell proliferation is controlled in some way by the TAF
(Paweletz and Knierim, 1989) and this is reflected by the inclusion of the
function G(c) which is assumed to be nondecreasing. As stated previously, the
initial response of endothelial cells to the angiogenic stimulus is one of
migration (Paweletz and Knierim, 1989). Proliferation is a crucial but secondary response. In order to account for this through the function G(c) we assume
that there is a threshold concentration level of TAF below which proliferation
does not occur. Thus in the present model we chose G(c) to be of the form
0,
c<c*
G(c) = c-- c*
- - 7
Cb
(2.7)
C* < C
where c* < Cb.
There is substantial evidence that the response of the endothelial cells to
the presence of the TAF is a chemotactic one (Ausprunk and Folkman, 1977;
Terranova et al., 1985; Balding and McElwain, 1985; Stokes et al., 1990;
Stokes and Lauffenburger, 1992) and following Balding and McElwain (1985)
we assume that the flux 3" of endothelial cells consists of two parts, one
representing random motion and the other chemotactic motion of the cells.
Thus, under the assumption of linear diffusion, the flux J can be written
J = -D,Vn + nzoVc.
(2.8)
where D, is the diffusion coefficient of the endothelial cells and Zo the
(constant) chemotactic coefficient (we note that other, perhaps more biologically relevant, functional forms for Z are possible and that this particular
form has been chosen for mathematical simplicity, cf. Chaplain and Stuart,
1993).
750
M.A.J. Chaplain et al.
With the above assumptions we thus have the following population
diffusion-chemotaxis equation for the endothelial cells
3n
Ot
-
-
=
D,V2n _ zoV.(nVc) + rn (1 __~o ) G(c) _ kp n
(2.9)
where G(c) is given by (2.7). We assume that initially the endothelial cell
density at the limbus is a constant no and zero elsewhere, giving initial
condition
fno,
n(x, O) = [~0 ,
x = L
x < L
(2.10)
We assume that throughout the subsequent motion, the cell density remains
constant at the limbus and hence the boundary condition here becomes
n(L, r) = no.
(2.11)
The boundary condition for the cell density at the site of the tumour
implant is taken to be
ti.Vn=0
atx=0.
(2.12)
It is easily seen from (2.8) that while n remains small this boundary
condition approximates a zero flux condition at x = 0. Biologically, this
means that the endothelial cells remain within the capillary sprouts. Thus as
long as n remains approximately zero the phase of the sprout moving across
the ECM is modelled. However, upon reaching the tumour the value of n will
become non-zero and hence there will be a flux J = n)~oC~ < 0 into the region
x < 0. During this phase, the model is a crude first approximation to the
interaction between the endothelial cells and the tumour cells which is important during the vascular phase of growth.
Following Chaplain & Stuart (1991, 1993) we normalize the equations
using the following reference variables
1. reference TAF concentration: cb, the value of the TAF concentration at the
tumour boundary,
2. reference cell density, no, the value of the endothelial cell density at the
limbus,
3. reference length, L, the distance from the tumour boundary to the limbal
vessels,
4. reference time unit, z = L2/D.
We thus define new variables:
F = C/Cb,
fl = n/no,
:,2 = x/L,
~ = t/z .
Dropping the tildes and specialising to a one dimensional geometry (cf. Liotta
et al., 1977; Balding and McElwain, 1985; Sherratt and Murray, 1990;
Tumour angiogenesis model
751
Chaplain and Sleeman, 1990; Chaplain and Stuart, 1991, 1993) the equations
now become
Oc 8%
omc
2c,
(2.13)
at Ox2 ~ + c
On
021"1
ot=D~x2--X~x
8 (tiC)
n-~x
+pn(1--n)O(c)--,sn,
(2.14)
where
G(c) = )
L2Q
DcCb'
CbZo
~ = D---7-'
O,
c < c*
C - - C*,
C* < C
K,,
=-Y cb '
2=
L 2r
L2d
Dc '
(2.15)
D=
L2kp
P = De'
D,
-~'
(2.16)
,8= Dc
The initial and boundary conditions become respectively
c(x, O) = Co(X),
n(x,O)=
c(O, t) = 1,
n(1, t ) = l ,
1, x = 1
O, x < l
c(1, t) = 0
8n
--=0
8x
atx=O.
(2.17)
(2.18)
(2.19)
(2.20)
Using comparison principles for parabolic partial differential equations it is
straightforward to show that the TAF concentration c in (2.13) must satisfy
the condition 0 < c _< 1 and also that the endothelial cell density n in (2.14) is
positive i.e. n > 0. By standard existence theory n must also be bounded
(Protter and Weinberger, 1967; Jones and Sleeman, 1983). More precisely,
using the result of Lu and Sleeman (1993) (Theorem 5.2), it can also be shown
that n < 1.
As we have mentioned previously, angiogenic factors must be able to
provoke three main activities of the endothelial cells, namely (1) production
and secretion of enzymes capable of digesting extracellular matrix, (2) the
initiation of migration , and (3) cell proliferation. It is clear from experimental
evidence (Sholley et al., 1977; Sholley et al., 1984; Reidy and Schwartz, 1981)
that different types of stimuli are necessary for (2) and (3) and we note that this
is accounted for in the model i.e. migration is governed by chemotaxis while
proliferation is governed by the function G(c) which essentially only depends
on the TAF concentration. As stated previously the processes of anastomosis
and budding are also (implicitly) accounted for through the second and first
order loss terms respectively in (2.14). Moreover, the form of the function G(c)
ensures that the second order loss term implicitly modelling anastomosis will
only take effect after the sprouts have reached a certain distance into the
752
M.A.J. Chaplain et al.
extracellular matrix which is precisely what is observed experimentally (cf.
Paweletz and Knierim, 1989).
Parameter values
As far as possible parameter values are chosen to correspond to available
experimental data. Unfortunately, data are not available for all parameters, in
particular those which relate to the concentration of TAF. However, most of
the parameters used in (2.16) can be estimated from actual experimental data.
Diffusion coefficient D
For a reference length we choose L = 2 mm, an average distance between
a tumour implant and the limbal vessels, and the value for z is taken to be 14
days, an average time for neovascularization to occur (cf. Balding and
McElwain, 1985; Chaplain and Stuart, 1991, 1993; Stokes and Lauffenburger,
1991). According to the nondimensionalisation in (2.16), this gives a value
for Dc of 3.3 × 10 -8 cm2s -1. Sherratt and Murray (1991) found values of
3.1 x 10-7 cm2s -1 and 5.9 x 10 -6 cm2s -1 as estimates for diffusion coefficients of chemicals. Correspondingly, they estimated the diffusion coefficients
Dn of the epithelial cells under consideration in their model as
3.5 x 10- lO cm 2 s - 1 and 6.9 x 10- ~1 cm 2 s - ~. Using these values the ratio
DJD~ thus varies between 1.1 x 10 -3 and 1.2 x 10 -5. In accordance with this
range we choose D to be 10-3.
Chemotactic parameter ~c
Sokes et al. (1990) have measured the value for the chemotaxis coefficients Xo
of endothelial cells migrating in a medium containing unpurified acidic
fibroblast growth factor (aFGF) as 2600 cm z s- 1 M - 1. As above, the value for
Dc is estimated at 3.3 x 10 -8 cm 2 s- ~. In the numerical simulations carried out
the (nonzero) values for x varied between 0.3 and 1.0 which from (2.16) gives
a value of Cb of approximately 10- ~~ M which is not unreasonable.
Cell proliferation parameter i~
F r o m equation (2.9) we assume that the parameter r is a positive constant
related to the maximum mitotic rate. Using data based on epidermal wound
healing (Winter, 1972), Sherratt and Murray (1990) estimated that the maximum value rm~x for this parameter was 10 times the proliferation rate
constant. Based on in vitro experiments on endothelial cell p r o l i f e r a t i o n
(Williams, 1987) Stokes and Lauffenburger (1991) estimated the proliferation
rate constant kp of endothelial cells to be 0.056 h r - ~ under the assumption
that all cells proliferate. However cell mitosis in the sprouts is mainly confined
to a region close to the tips. To compensate for this, in most of their
simulations Stokes and Lauffenburger reduced the value of kp to 0.02 h r - 1
and assumed that all cells in a sprout may proliferate. Thus in the following
numerical simulations we assume a range for rmax of 0.2 hr-1-0.56 hr-1.
Under this assumption this gives a range of values for the cell proliferation
parameter # of approximately 70-190.
Tumour angiogenesis model
753
Cell loss parameter fl
The parameter kp is the reciprocal of the endothelial cell doubling time (cf.
Sherratt and Murray, 1990). As seen above Stokes and Lauffenburger (1991)
estimated this to lie within the range 0.02 h r - 1-0.056 h r - 1. Also, from experimental data on epidermal cells (Wright, 1983) Sherratt and Murray (1991)
estimated this to be 0.01 h r - 1. This gives a range of values for the cell loss
parameter fl of approximately 3-18.
3 Results
In this section we present the results from the numerical simulation of
(2.13)-(2.20) with appropriate boundary and initial conditions.
3.1 Initial conditions
The initial condition (2.17) was taken to be c(x, O) --- Co(X) = 1 - x 2 which is
of the correct qualitative shape for the TAF profile in the external tissue i.e.
a constant value of 1 at the tumour edge decaying away to 0 at the limbus.
Figure l(a), (b) show the profiles of the TAF concentration and the
endothelial cell density respectively in the external host tissue at various
different times. The time taken for the endothelial cells at the sprout tip to first
reach the tumour is approximately t = 0.7 which corresponds to a real time of
approximately 11 days which is within the experimentally observed time scale
(cf. Balding and McElwain, 1985). By varying the parameters/~ and x the time
taken for the endothelial cells (and hence the capillary sprouts) to reach the
tumour can also be varied. Figure l(b) also demonstrates that the initial
response of the endothelial cells is essentially one of migration with proliferation of the cells occurring at a later time (cf. Paweletz and Knierim, 1989).
The form of the function G(c) ensures that there is always a region within the
sprouts where there is zero cell proliferation and also that once cells have
started to proliferate the proliferation is mainly confined to a region a short
distance behind the sprout tips. Thus the model distinguishes, as far as is
possible within its limitations, between proliferating cells near the sprout tip
and non-proliferating cells within the rest of the sprout. Also once the cells
have started to proliferate the endothelial cell density is greatest (locally)
a short distance behind the sprout tips which is what is observed experimentally (Ausprunk and Folkman, 1977; Sholley et al., 1984; Paweletz and
Knierim, 1989). Similar results were obtained for different functions G(c)
which were of the same qualitative form as (2.7).
For all numerical simulations carried out we took c* = 0.2. Qualitatively
similar results were obtained for various other values of c* between 0.0
and 0.4.
Figure 2 shows the steady-state profile achieved. This was reached when
t ~ 1 i.e. a time scale of O(1). As stated above the boundary condition used at
754
M . A . J . Chaplain et al.
0.9
0.8 ¸
0.7
t=0
~E 0.6
d~
O
,~o.5
0.4
0.3
0.2
0.1
00
i
0.2
0.4
0.6
Distance from tumour
0.8
Fig. la. Numerical simulation of full system (2.13), (2.14). Profile of the T A F concentration
in the external host tissue at times t = 0, 0.1, 0.3, 0.5, 0.7 showing changing gradient profile;
parameter values ~ = 10, 7 --- 1, 2 = 1, D = 0.001, x = 0.75, p = 100, fl = 4, c* = 0.2
1
•
0.9
=,0.8
0.7
!
"O
0.6
.¢) o.s
r-.
"6 0.4
"O
c-
w 0.3
0.2
0.1
%
01
0.2
0.4
0.6
Distance from tumour
0.8
1
Fig. lb. Profile of the endothelial cell density in the external host tissue at times t = 0.1, 0.3,
0.5, 0.7. The boundary condition imposed at x = 0 is nx = 0. Shortly after t = 0.7 the
endothelial cells reach the t u m o u r and interactions between the endothelial cells and the
tumour cells become important; parameter values ~ = 10, 7 = 1, 2 = 1, D = 0.001, x = 0.75,
# = 100, fl = 4, c* = 0.2
x = 0 is n~ = 0 w h i c h a p p r o x i m a t e s a z e r o flux b o u n d a r y c o n d i t i o n w h i l e
n ~ 0 i.e. t h e e n d o t h e l i a l cells a r e w i t h i n t h e c a p i l l a r y s p r o u t s u n t i l t h e y first
r e a c h t h e t u m o u r a t x = 0. H o w e v e r w h e n s p r o u t t i p s r e a c h t u m o u r i n t e r a c t i o n s b e t w e e n t h e e n d o t h e l i a l cells a n d t h e t u m o u r cells b e c o m e i m p o r t a n t .
Tumour angiogenesis model
755
0.9
~0.8
"~ 0.7
N 0.a
o
•~- 0.5
r-
"~ 0.4
fLU
0.3
0.2
0.1
°0
6.2
0'.4
o'.6
Distance from tumour
6.8
Fig. 2. Steady state profile of the endothelial cell density in the external host tissue. This
was achieved after t ~ 1 i.e. on a time scale of 0(1). This profile models in a certain way the
interactions which will occur between EC and tumour cells once vascularization of the
turnout has taken place; parameter values c~= 10, 7 = 1, 2 = 1, D = 0.001, ~ = 0.75,
/t = 100, ] / = 4; c* = 0.2
W h e n the endothelial cells reach the t u m o u r the cell density at x = 0 becomes
n o n - z e r o and there is a flux of cells J = n g o c x < 0 into the region x < 0. Thus
during the time interval 0.7 _< t < 1.0 the model is a crude, first a p p r o x i m a t i o n
to the modelling of the interaction of E C with t u m o u r cells. Thus the
transition to the steady-state, during the time 0.7 < t < 1.0, can be justified
biologically.
4 Model simplification
As is a p p a r e n t from Fig. l(a) the T A F concentration profile in the external
host tissue, according to the above model, does n o t vary drastically t h r o u g h out the complete process. This is perhaps not unexpected since as we have
seen from the parameter estimations of the previous section, the T A F is
estimated to diffuse very m u c h faster than the cells and can be reasonably
expected to be in some kind of steady state. In order to simplify the complete
system a n d thus m a k e it amenable to mathematical analysis we m a k e the
assumption that the T A F concentration in the external host tissue has
a profile which does n o t depend u p o n time i.e. steady-state profile (cf. Stokes
and Lauffenburger, 1991). W e thus seek a solution of the equation
d2c
dx---2 - 2c = O ,
(4.1)
756
M.A.J. Chaplain et al.
subject to
c(O) = 1,
(4.2)
c(1) = O.
T h e solution to the above is easily seen to be
c(x) =
sinh [(1 - x)w/2 ]
sinh x//~
(4.3)
With 2 = 1 in the above equation, this is a very close approximation to the
profile c ( x ) = 1 - x which is also seen to be a reasonable assumption from
Fig. l(a). Thus, in order to simplify the mathematical analysis slightly, henceforth we assume that the T A F concentration profile is of the (time-independent) form
c(x)=(1-x),
0<x<l.
(4.4)
This profile m a y also be t h o u g h t of as being an average (in a temporal sense)
of the profile of Fig. l(a). This has the effect of reducing (2.13), (2.14) to a single
equation:
On
a2n
an
a--t = D ~-2xZ+ X~xx + #(1 -- x)n(1 - - n) - f i n ,
(4.5)
with initial condition and b o u n d a r y conditions as before.
Before proceeding with an analysis of the above equation, we first of all
solve it numerically. As can be seen from Figs. 3(a), 3(b) the numerical
1
. 0 . 9 :..
>,0.8
g 0.7
"o
0.6
~ 0.5
0
0.4
t-
t.u 0.3
0.2
0.1
O,
/
6.2
-.
0
Distance from tumour
1
Fig. 3a. Numerical simulation of caricature model (4.5). Profile of the endothelial cell
density in the external host tissue at time t = 0.1, 0.3, 0.5, 0.7; The figure is qualitatively the
same as Fig. l(b) with all the salient features being captured; parameter values D = 0.001,
x = 0.65,/~ = 75, fl = 10, c* = 0.0
Tumour angiogenesis model
757
1
0.9
steady state profile
,~0.8
~0.7
O
0.6
~0.5
t-
"6 0.4
"O
C
w 0.3
0.2
0.1
6.2
0'.4
0'.6
Distance from tumour
0'.8
Fig. 3b. Caricature model (4.5). Steady state profile of the endothelial cell density in the
external host tissue. This was achieved after t ~ 1 i.e. on a time scale of 0(1). This profile
models in an approximate way the interactions between EC and tumour cells once
vascularization has taken place; parameter values D = 0.001, x = 0.65, p = 75, fl = 10;
c* = 0.0
simulations from this caricature model are qualitatively the same as those
from the full system, with all salient features of the model being reproduced.
W e thus begin an analysis of the above equation and will attempt to draw
conclusions from this caricature model for the full system (2.13), (2.14).
5 Mathematical analysis of time-dependent problem
W e note firstly t h a t a straightforward application of the c o m p a r i s o n principle
for parabolic equations shows that if/)(x, t) is the solution of
0/)
021)
at -
D 7x
10/)
-
= 0,
subject to
v(x, 0 ) =
1,
0,
x = 1
x<l
(5.2)
and
vx = 0,
x = 0,
v = 1,
x = 1,
(5.3)
v(x, t) is a lower b o u n d for our solution n(x, t). An explicit expression for
v(x, t) is f o u n d by writing
then
v(x, t) = Vo(X) + V(x, t)
(5.4)
758
M.A.J. Chaplain et al.
where Vo(X) is the time-independent solution. A p p r o p r i a t e initial and b o u n d ary conditions for b o t h parts of the solution are then
V(x, o)= -Vo(X)
v'o (0) = O,
OV
Ox
v•(1) = 1,
O,
(5.5)
x = O,
V(1, t) = 0 .
(5.6)
(5.7)
After s o m e straightforward but s o m e w h a t tedious algebra, using standard
techniques, we have
v(x, t) =
e a(1 -~)[cosh (bx) + ~ sinh (bx)]
cosh b + -~sinh b
+~c~,e-"~(~cos(m,x)+sin(m,x))e
(5.8)
where a = x / 2 0 , b = ~/I~ 2 -q- 4Dfl/2D, 2 = - f l -- m2, D -- ~2/4D < 0, and mn is
a solution of tan(re,) = - m , / a , m o = O.
5.1 Travelling w a v e solutions
Since for 0 < x < 1, we also have 0 < (1 - x) < 1 and n(1 - n) > 0 then it
follows that
nt
--
Dnx~ -- rcnx + fin =/~(1 -- x)n(1 - n) __</~n(1 - n)
and h e n c e that
nt - Dnxx - xnx - p n ( 1 - n) + fin <=0 .
Then, by the c o m p a r i s o n principle for parabolic equations we m a y take as an
u p p e r function the solution w of
wt = D w ~ + ~cw~ + #w(1 - w) - flw
(5.9)
with the s a m e initial a n d b o u n d a r y conditions as (4.5). Thus we have n < w in
[0, 1] × [0, T ] . Rescaling the a b o v e equation in the following manner:
yields
u, = Dux:, + tcu:, + (~ -- fl)u(1 -- u).
(5.10)
T h e a b o v e e q u a t i o n is thus seen to be of a Fisher type equation with an extra
convection-like term, u~. W e thus seek a travelling wave solution to the above
equation. In the usual w a y write
u ( x , t ) = U(z),
z = x + ct .
(5.11)
Tumour angiogenesis model
759
Such a solution if it exists represents a wave travelling in the negative
x-direction with wave speed ¢( > 0, constant). Thus this would represent
a wave of endothelial cells moving towards the tumour across the extracellular
matrix. Substitution of the above into (5.11) yields
DU" = (c -- K) U' -- (# -- fl)U(1 - U)
(5.12)
Thus we seek to determine the value, or values, of c such that a nonnegative solution U exists which satisfies
lim U (z) = 0,
Z"* oO
lim U (z) = 1.
(5.13)
Z--*O
Standard phase plane analysis shows that there exists a minimum wave
speed given by
Cmi n = /~ "]-
2x/D(/~ - fi)
(5.14)
It is thus not unreasonable to hypothesise that the true solution to (4.5) will
evolve into a travelling wave with minimum speed c*i. less than this minimum
wave speed 5mln of the upper solution i.e.
:~ < Cmin
Cmin
(5.15)
i.e. the true solution trails the upper solution. This analysis would seem to
agree with the numerical results of the previous section where the concentration profile for n does indeed seem to develop into a travelling wavefront type
solution.
5.2 Zero-diffusion analysis
In order to make some progress in analysing our model explicitly, we note
that the diffusion coefficient is very small (D = 10-3) in comparison with the
other parameters and hence we neglect this term. The equation now becomes
On
~n
0-7 =/C~x + p(1 - x)n(l -- n) - fin.
(5.16)
This is now a first order equation and we shall treat this simply as an initial
value problem with initial condition n(x, O) = no(x). This has the advantage of
permitting us to view the model as focussing exclusively and explicitly upon
the endothelial cells at the capillary tips since there are no boundary conditions to satisfy. This equation can be solved using standard techniques to yield
the exact solution
n°(x + ~ct)exp [ - -llKt2
- ~ + (#(1
n(x,
t) -
-
x) -
,
fl)t
]
,
(5.17)
[1 + no(x + tct)fo ~teS(P)f(p)dp]
where g(p) = I~xp2/2 + [p(1 - x - xt) - fl]p and f(p) = xp + (1 - x - xt).
760
M.A.J. Chaplain et al.
This explicit solution clearly shows the individual contributions made by
endothelial cell migration and endothelial cell proliferation. The key term in
the above equation is the exponential term
exp - - - - ~
+ (#(1-- x) -- fl)t ,
0_<t_<l.
(5.18)
We restrict attention here 0 _ t _< 1 since our numerical simulations indicate
that angiogenesis, if it is to succeed given the chosen parameter values, should
occur on this timescale. This term is essentially an amplification factor for the
"propagating wave of initial conditions" no (x + xt). If the endothelial cells are
to grow and cross the matrix then the exponent must be positive. If the
exponent is negative then the endothelial cell density will decay to zero. Thus
for successful completion of angiogenesis we require at least, that
-- fl~t 2
h(t) =
2
+ (#(1 - x) - fi)t > O,
t > O.
(5.19)
A straightforward calculation shows that h(t) is positive for
0 _< t _ 2 ~ ( 1 - x ) - 2 9
(5.20)
0 < x < 1 - -
(5.21)
provided
#
Thus we have a "window of opportunity" (in time) for the endothelial cell
density to increase in the region 0 =< x < x* = 1 - fl//~, while for x* < x =< 1,
h(t) < 0 (t ~ 0) and hence the solution will always decay in this region. The
point x = x* = 1 - fl//~ is a critical point. It is clear that for angiogenesis to
succeed, the chemotactic term x must be large enough to ensure that the
endothelial cells migrate quickly enough so as to reach the region of growth.
This will be facilitated if the ratio of the cell proliferation parameter/~ to cell
death parameter fl is small enough. On the contrary, if the cell proliferation
rate is too small (or the cell loss rate too large) then successful completion
becomes less likely for a fixed value of x. We note that by varying the
parameters in the model, the location of the critical point x* can be changed.
Thus x*, and hence in this caricature model, the TAF concentration
c* = 1 - x*, is a bifurcation parameter. It is therefore reasonable to assume
that in the full model the TAF concentration also plays the role of a bifurcation parameter. This may well be a signal for events such as anastomosis to
take place.
6 Mathematical analysis of time-independent problem
We note that a time-independent lower bound for the solution is given by
Vo(X) calculated previously. Also a time-independent upper bound is easily
Tumour angiogenesis model
761
seen to be a solution of
0 = Du:,x + lcux -- flu + ¼ p(1 - x)
(6.1)
This 2nd order linear differential equation with b o u n d a r y conditions
u(1) = 1,
u'(0) = 0
(6.2)
can easily be solved to yield
u(x) = A e ~lx + Be ~2x + # ( f l - to)
----------T4fl
#x
4fl
(6.3)
where 2a, 22 are the roots o f ( -~: + x/to 2 + 4Dfl/2D) and
A
(1
- - 0~ - - y ) 2 2 +
=
22 e~.l
-
-
o~e '1"2
2~eX ~
-~
#(/~
c~ = 4 f l '
Y=
,
-
(~x + y - - 1 ) 2 2 - -
B =
)~2 ezz -
c~ez~
21 e z~
~)
4fl 2
6.1 Singular perturbation analysis of steady state
As we have seen in a previous section, the numerical solution of the caricature
model (4.5) indicates that the solution converges to a time-independent
steady-state whose profile is given in Fig. 3(b). We now study this steady-state
problem analytically i.e. we analyse the following equation:
O2n
On
0 = D~-~x2 + ~0xx + p(1 - x)n(1 - n) - fin,
(6.4)
which can be written
D 02n
I¢ O n
u
0 = ~ Ox--5 + ~ ~x + ~ (1 - x)n(1 - n) - n .
(6.5)
F r o m the range of values estimated for the various parameters associated with
the model (see previous section), it is clear that D/fl can be treated as small
parameter of the system. T o this end we define e = x / ~
and ~:* = x/x/rD-~
and then (6.5) becomes
0 = e2n " + etc*n' + ~ (1 - x)n(1 - n) - n
(6.6)
with b o u n d a r y conditions
n=l
atx=l,
n'=0
atx=0
(6.7)
This can be treated as a singular perturbation problem in the small
parameter e. We thus seek a solution to (6.6) of the form
n(x; e) = no(x) + eni(x) + e2nz(x) + " .
(6.8)
762
M.A.J. Chaplain et al.
Substituting this into (6.6) and equating the coefficients of powers of e yields
no=l
/,l 1 =
# ( l -flx ) '
(6.9)
flt¢*
(6.10)
This gives as an outer solution
n=l
/~(1 1 x)
- -
(1-x)[1-~U(1-x)]
(6.11)
This expansion is not uniformly valid because of the difficulties associated
with the behaviour near the left boundary (i.e. the boundary condition at
x = 0 is violated) and because of the singularities at the points
x = x* = 1 - fl/# and x = 1. Thus separate asymptotic expansions, valid in
the regions, must be constructed by introducing suitable transformations in
x in the above areas. This singular part of the solution is referred to as the
inner solution.
Correct "matching" of the inner and outer solutions in some suitable
domain of overlap should permit us to obtain a solution which is uniformly
valid over the whole domain under consideration.
Approximation
n e a r x = O:
We introduce the transformation
X=6~,
which enables us to focus on the immediate neighhourhood of x = 0. This
transformation is such that any smll neighbourhood ofx = 0 is m a p p e d onto
a large domain in the ~ plane. W e note that ~ = 0 w h e n x = O, and for any
fixed x > 0, lim~-~o~ = oo. Thus we have the following transformations
d =e-' d
.(x) -,
dx
"
This transforms (6.6) into
0 = u" + x * u ' + ~ (1 -- e~)u(1 - u) -- u
(6.12)
where the prime now denotes differentiation with respect to ~. Once again we
construct a solution of the form u(~; e) = Uo(~) + eul(~) + - • • . Equating
powers of the expansion, we have, to 0(1),
U"o + ~ * U' o + ~#
U o ( 1 - - U o ) - - Uo = 0
(6.13)
Tumour angiogenesis model
763
We take the constant solution to this equation given by
uo
The 0(e) term yields the following equation, to be solved for ul:
ul+~c ul+
1--
ul=
1-
{,
(6.15)
with boundary condition u~ (0) = 0. The solution of the above equation can be
found straightforwardly and is easily calculated to be
B
ul = (A + B ~ ) - - T e x p ( 2 ~ ) ,
where
A=
_~,f12
B=
_fl
,
2=
-x*--~//x*2+4(~
-1)
<0.
Hence we have
B
u=(l-~)+Bx+~[A--£exp(2x/e)].
(6.16)
The expansions for u and n must match in the sense that lim~_,~u({)=
lim,-.o n(x). A straightforward calculation shows this to be the case with both
limits giving
Approximation
n e a r x = 1:
We introduce the transformation x = 1 - er/, which enables us to focus
on the immediate neighbourhood of x = 1. Thus we have the following
transformations
d
_~-1 d
n (x) -+ v (~),
dx =
d-~
This transforms (6.6) into
0 = v" - tc*v' + ~# (et/)v(1 - v) -- v
(6.17)
where the prime now denotes differentiation with respect to r/. Once again we
construct a solution of the form v(t/; e) = Vo(t/) + evl (r/) + . . . . Substituting
the above expression into (6.6) and collecting terms independent of e gives
tt
Vo -
l~
!
I~ V o - v o
= O.
764
M.A.J. Chaplain et al.
0.9
•
approximation to steady state profile
~o.8
"0
rJ
0.7
0.6
.~o.5
~
,~
•
e--
"(:1
e--
uJ 0.3
0.2
~
+
0.1
~
+++
+++÷+
%
o'.2
0.4
6.6
+
0.8
Distance from tumour
Fig. 4. Approximation to steady state profile using matched asymptotic expansions
Once again the solution of the above equation can be found straightforwardly
and is easily calculated to be
Vo = exp(2t/) = exp(2(1 - x)/e),
where 2 = (~c* - ~
+ 4)/2 < 0.
Because of the severity of the singularity at x = x* = 1 - / ? / # it is very
difficult to obtain a uniformly valid solution in the region around the singular
point x*. Thus Fig. 4. shows the approximation to the steady-state profile
which is obtained from the matched asymptotic expansions given above. It
can be seen that the fit is a good one except in a region around x*. From the
numerical simulations shown previously in Figs. l(b), 3(a) it is clear that some
kind of transition does indeed occur at a specific point where the proliferation
of the EC becomes important. This is what is observed experimentally i.e.
there is a transition from motion of the EC due to migration alone to motion
due to proliferation as well as migration. It has also been observed experimentally that anastomosis always occurs at a critical distance from the
parental vessel e.g. the limbus (Paweletz and Knierim, 1989). In the following
section we perform a phase plane analysis of equation (6.6) and analyse the
behaviour of the solution about this critical point in more detail.
6.2 Phase plane analysis
We recall that the steady state caricature model is given by
0 = eZn" -t- eK*n' d- #(1 - x)n(1 - n ) - n
(6.18)
Tumour angiogenesis model
765
In the usual way, we consider writing this as the following system of ordinary
differential equations:
~nx = p ,
(6.19)
~Px = - ~ * p - ~ ( 1
wx = l,
- w)n(1 - n) + n,
(i.e. w = x).
(6.20)
(6.21)
From inspection of the above system, it is evident from the fact that x changes
slowly (order 1) compared to n and p which change more quickly (order e- 1).
We can thus treat x as a constant and analyse the equilibrium points of the
following 2-dimensional system
enx = p
= f * (n, p),
8Px = - ~ * p - ~ (1 - x)n(1 - n) + n = 9* (n, p),
(6.22)
(6.23)
as x varies between 0 and 1. Since we wish to follow the path of the EC from
the limbus (situated at x = 1) to the turnout implant (situated at x = 0) then it
is convenient to introduce the variable y = 1 - x . The above system then
transforms to
e n r = - - p = f (n, p) ,
(6.24)
#
~Pr = t¢*p + -~ y n ( 1 - - n) - n = 9 ( n , p) .
(6.25)
The equilibrium points of the above system (i.e. points satisfying ny = Pr = 0)
are located at (0, 0) and (n*, 0) where
n* = 1
/~
/ly
The stability matrix A (n, p) for the above system is given by
I
0
-1 1
-l+~y(1-2n)
~c*
(6.26)
and the stability of the critical points may be determined in the usual way by
examining the eigenvalues of A. The first critical point (0, 0) gives rise to
a stability matrix
0
-l+~y
-1 1
(6.27)
~c*
It is a straightforward task to show that the following situations arise:
1. (0, 0) is an unstable node when i l l # < y < 1 i.e. 0 < x < 1 - (fi/#) = x * ,
766
M.A.J. Chaplain et al.
2. the nature of(0, 0) is undetermined when x = 1 - fl/# = x* since one of the
eigenvalues is zero,
3. (0, 0) is a saddle point when 0 < y < fl/# i.e. x* = 1 - fl/# < x < 1.
Similarly, for the critical point (n*, 0) we have the stability matrix given by
E
0
1 -~y
-1 1 ,
~c*
(6.28)
and once again it is straightforward to show that
1. (n*, O) is a saddle point when///# < y < 1 i.e. 0 < x < 1 - / ~ / # = x*,
2. the nature of (n*, O) is undetermined when x = x* since one of the eigenvalues is zero,
3. (n*, 0) is an unstable node when 0 < y < 2/15 i.e. x* < x < 1. We note that
in the region x* < x < 1, n* < 0, which is not relevant from a biological
point of view.
We now study the phase portraits of the above system as y increases from 0
to 1 i.e. as x decreases from 1 to 0 and endeavour to determine the path which
has been taken by the EC as they have crossed the extracellular matrix from
the limbus (x = 1) to the tumour implant (x = 0). The following phase plots
were obtained using the O D E solver ODE23 within the MATLAB package.
The plots are given for n ~ [0, 1] since this is the range of n which is relevant
biologically.
At x = 1, we have, from the boundary condition, n = 1, and the only
equilibrium point is (0, 0) which is a saddle point.
From Fig. 5(a), which shows the qualitative form of the steady state
solution at a general point in the region x* < x < 1, we see that n decreases
from 1 towards the equilibrium point at (0, 0), for p > 0. Similar phase
portraits are obtained while x is in the region 0.8667 < x < 1. We note that
there is another equilibrium point which is not biologically relevant, i.e.
n* < 0 and (0, 0) is always a saddle point.
At x = x* = 13/15, (n*, 0) = (0, 0) and the two equilibrium points coalesce.
In this case one of the eigenvalues is zero and so the nature of the fixed point is
undetermined. The behaviour is irregular at this point and the problem is
non-standard. This would seem to confirm the difficulty in trying to find
a solution which was uniformly valid in this region using the techniques of the
previous section and also emphasises the fact that a critical bifurcation point
exists.
As x enters the region 0 < x < 13/15 the fixed point (n*, 0) enters the
region [-0, 1] and is a saddle point, whilst (0, 0) changes and becomes an
unstable node. Figure 5(b) illustrates the phase portrait for a typical point in
this region.
Thus it is evident from the phase portraits that the point x = x* = 13/15 is
a transition point which marks a change in the nature of the solution. This
Tumour angiogenesis model
767
0.4
0.3
0.2
0.1
n
0
-0.1
-0.,¢
-0.,,."
-O.d
0h
N
66
t
Fig. 5a. Phase plane portrait for x = 0.9. The fixed point at (0, 0) is a saddle point
0.4
0.3
0.2
0.1
o.. 0
-0.1
-05
-0.,"
-0. /
J.
0.2
1].4
N
0.6
0.8
Fig. 5b. Phase plane portrait for x = 0.6. The fixed point at (0, 0) is now an unstable node
while (2/3, 0) is a saddle point
reiterates the fact that the behaviour of n at this point is crucial. W e note that
this is in agreement with the numerical solutions for the full system whereby
there was a c h a n g e to a travelling wave type solution at some critical spatial
point. Thus it appears t h a t our space variable x is indeed a bifurcation
parameter, a n d hence, given the assumption of steady-state T A F concentration profile, so is the T A F concentration. This transition behaviour can also
768
M.A.J. Chaplain et al.
be given some biological meaning. It is known from experimental observations that anastomosis always seems to occur at some definite distance from
the parent vessel e.g. limbus. It is also known that EC proliferation does not
occur until the growing capillaries are already some way out into the extracellular matrix. Both of these processes will results in an increase in endothelial
cell density beginning at some critical distance into the extracellular matrix.
Thus the mathematical analysis may well correspond to observed experimental results.
7 Conclusions
The paper of Chaplain and Stuart (1993) developed a mathematical model of
angiogenesis which focusses attention upon the roles of the endothelial cells
and the TAF concentration profile. In this paper we made a certain relevant
and biologically justifiable simplification to this model regarding the TAF
concentration profile and in doing so have constructed a simpler caricature
model. We have shown that this simpler model, consisting of a single nonlinear pde for the endothelial cell density, captures all of the salient features of the
original model. Once again, as far as is possible all parameter values used in
the numerical simulations are based on those from experimental data. In
addition, this simplification has permitted some mathematical analysis to be
performed and we have shown that there is a good correlation between the
numerical results, the mathematical analysis and most importantly, the experimental results. This has thus permitted a genuine biological interpretation
of the results to be made. More precisely, the analysis performed on the
caricature model with zero diffusion has permitted us to focus exclusively
upon the endothelial cells at the capillary tips. These cells effectively drive the
capillary sprouts across the tissue towards the tumour (Stokes and
Lauffenburger, 1991). The analysis of this section and of Sects. 6.1 and 6.2 also
clearly indicates that the solution takes on different characteristics in two
distinct regions of the domain, the critical point x* -- 1 - fl/# marking the
transition from one region to the next. In x* < x =< 1 the solution appears to
simply propagate into the region driven mainly by the chemotactic term, while
in the region 0 ___<x < x* travelling wave-like solutions appear possible. The
analysis clearly shows that in order for successful completion of angiogenesis
to take place, both migration and proliferation are essential. These results,
both analytical and numerical, correspond very well with the actual experimental observations. Initially, the endothelial cells simply migrate outwards,
enabling the capillary sprouts to grow into the external tissue. At first the
sprouts are aligned in a parallel fashion but at some critical distance anastomosis takes place. Capillary loops are formed and then secondary sprouts
form from the loops. The secondary sprouts progress towards the tumour
with continuing anastomosis. A so-called "brush-border" of endothelial cells
is often observed and the capillary network becomes very dense. Although the
morphology of the capillary loops is well known, many questions regarding
Tumour angiogenesis model
769
anastomosis remain unanswered. In particular, why does anastomosis occur
at a definite distance from the parental vessel? F r o m our analysis of the model,
we can perhaps venture a tentative answer to this question. Since we have
(reasonably) assumed a steady-state concentration profile for the TAF, it
follows that the critical distance x* also corresponds to a critical T A F
concentration, given by (in our particular case) 1 - x* (we note that perhaps
a m o r e general expression for the critical concentration is given by
c(x*) =
sinh [(1 - x * ) x / ~ ] / s i n h x//2, cf. (4.3)). Perhaps then a critical concentration of T A F is involved in initiating the process of anastomosis. The
numerical simulations of the caricature model has also demonstrated that the
effect of chemotaxis can be modelled using a steady state T A F concentration
profile. This will have i m p o r t a n t consequences when considering numerical
simulations of the model in 2 spatial dimensions which is w o r k currently being
undertaken.
References
Ausprunk, D. H., Folkman, J.: Migration and proliferation of endothelial cells in preformed
and newly formed blood vessels during tumour angiogenesis. Microvasc. Res. 14, 53-65
(1977)
Balding, D., McElwain, D. L. S.: A mathematical model of tumour-induced capillary
growth. J. theor. Biol. 114, 53-73 (1985)
Chaplain, M. A. J., Sleeman, B. D.: A mathematical model for the production and secretion of
tumour angiogenesis factor in tumours. IMA J. Math. Appl. Med. Biol. 7, 93-108 (1990)
Chaplain, M. A. J., Stuart, A. M.: A mathematical model for the diffusion of tumour
angiogenesis factor into the surrounding host tissue. IMA J. Math. Appl. Med. Biol. 8,
191-220 (1991)
Chaplain, M. A. J., Stuart, A. M.: A model mechanism for the chemotactic response of
endothelial cells to tumour angiogenesis factor. IMA J. Math. Appl. Med. Biol. 10,
149-168 (1993)
Cliff, W. J.: Observations on healing tissue: A combined light and electron microscopic
investigation. Trans. Roy. Soc. London B246, 305-325 (1963)
Deshpande, R. G., Shetna, Y. I.: Isolation and characterization of tumour angiogenesis
factor from solid tumours and body fluids from cancer patients. Indian J. Med. Res. 90,
241-247 (1989)
Folkman, J.: Tumor angiogenesis. Adv. Cancer Res. 19, 331-358 (1974)
Folkman, J.: The vascularization of tumors. Sci. Am. 234, 58-73 (1976)
Folkman, J., Klagsbrun, M.: Angiogenic factors. Science 235, 442-447 (1987)
Gimbrone, M. A., Cotran, R. S., Leapman, S. B., Folkman, J.: Tumor growth and neovascularization: An experimental model using the rabbit cornea. J. Natn. Cancer Inst. 52,
413-427 (1974)
Hiltman, P., Lory, P.: On oxygen diffusion in a spherical cell with Michaelis-Menten oxygen
uptake kinetics. Bullet. Math. Biol. 45, 661-664 (1983)
Jones, D. S., Sleeman, B. D.: Differential Equations and Mathematical Biology, London,
George Allen & Unwin 1983
Lin, S. H.: Oxygen diffusion in a spherical cell with nonlinear uptake kinetics. J. theor. Biol.
60, 449-457 (1976)
Liotta, L. A., Saidel, G. M., Kleinerman, J.: Diffusion model of tumor vascularization and
growth. Bull. Math. Biol. 39, 117-129 (1977)
Lu, G., Sleeman, B. D.: Maximum principles and comparison theorems for semilinear
parabolic systems and their applications. Proc. R. Soc. Edinb. 123A, 857-885 (1993)
McElwain, D. L. S.: A re-examination of oxygen diffusion in a spherical cell with MichaelisMenten oxygen uptake kinetics. J. theor. Biol. 71, 255-263 (1978)
770
M.A.J. Chaplain et al.
Maini, P. K., Myerscough, M. R., Winters, K. H., Murray, J. D.: Bifurcating spatially
heterogeneous solutions in a chemotaxis model for biological pattern generation. Bull.
Math. Biol. 53, 701-719 (1991)
Murray, J. D.: Mathematical Biology. Bedim Springer-Verlag 1989
Muthukkaruppan, V. R., Kubai, L., Auerbach, R.: Tumor-induced neovascularization in the
mouse eye. J. Natn. Cancer Inst. 69, 699-705 (1982)
Paweletz, N., Knierim, M.: Tumor-related angiogenesis. Crit. Rev. Oncol. Hematol. 9,
197-242 (1989)
Protter, M. H. and Weinberger, H. F.: Maximum principles in Differential Equations.
Englewood Cliffs, NJ: Prentice-Hall 1967
Reidy, M. A., Schwartz, S. M.: Endothelial regeneration. III. Time course of intimal changes
after small defined injury to rat aortic endothelium. Lab. Invest. 44, 301-312 (1981)
Schoefl, G. I.: Studies on inflammation III. Growing capillaries: Their structure and
permeability. Virchows Arch. Pathol. Anat. 337, 97-141 (1963)
Sherratt, J. A., Murray, J. D.: Models of epidermal wound healing. Proc. R. Soc. Lond.
B 241, 29-36 (1990)
Sholley, M. M., Gimbrone, M. A., Cotran, R. S.: Cellular migration and replication in
endothelial regeneration. Lab. Invest. 36, 18-27 (1977)
Sholley, M. M., Ferguson, G. P., Seibel, H. R., Montour, J. L., Wilson, J. D.: Mechanisms of
neovascu!arization. Vascular sprouting can occur without proliferation of endothelial
cells. Lab. Invest. 51, 624-634 (1984)
Stokes, C. L., Rupnick, M. A., Williams, S. K., Lauffenburger, D. A.: Chemotaxis of human
microvessel endothelial cells in response to acidic fibroblast growth factor. Lab. Invest.
63, 657-668 (1990)
Stokes, C. L., Lauffenburger, D. A.: Analysis of the roles of microvessel endothelial cell
random motility and chemotaxis in angiogenesis. J. theor. Biol. 152, 377-403 (1991)
Strydom, D. J., Fett, J. W., Lobb, L. R., Alderman, E. M., Bethune, J. L., Riordan, J. F.,
Vallee, B. L.: Amino acid sequence of human tumour derived angiogenin. Biochemistry
24, 5486-5494 (1985)
Terranova, V. P., Diflorio, R., Lyall, R. M., Hic, S., Friesel, R., Maciag, T.: Human
endothelial cells are chemotactic to endothelial cell growth factor and heparin. J. Cell
Biol. 101, 2330-2334 (1985)
Warren, B. A.: The growth of the blood supply to melanoma transplants in the hamster
cheek pouch. Lab. Invest. 15, 464-473 (1966)
Williams, S. K.: Isolation and culture of microvessel and large-vessel endothelial cells; their
use in transport and clinical studies. In: Microvascular Perfusion and Transport in
Health and Disease (Ed. McDonagh, P.) pp. 204-245. Basel: Karger 1987
Winter, G. D.: Epidermal regeneration studied in the domestic pig. In: Epidermal Wound
Healing (Eds. Maibach, H. I. and Royce, D. T.) pp. 71-112. Chicago: Year Book Med.
Publ 1972
Wright, N. A.: Cell proliferation kinetics of the epidermis. In: Biochemistry and Physiology
of the Skin (Ed. Goldsmith, L. A.) pp. 203-229. Oxford University Press 1983