Mathematical model of prevascular growth of a spherical carcinoma

Marhl Cornput. Modehg, Vol. 13, No. 5, pp. 23-38. 1990 Printed in Great Britain. All rights reserved

Copyright

MATHEMATICAL MODEL OF PREVASCULAR OF A SPHERICAL CARCINOMA S.

Department

of Mathematics

A.

and J. A,

MAGGELAKIS

and Statistics,

Old Dominion

6

0895-7177190 $3.00 + 0.00 1990 Pergamon Press plc

GROWTH

ADAM

University,

Norfolk,

VA 23529, U.S.A.

(Received July 1989; in revised form November 1989; received for publication January 1990) Communicated

by E. Y. Rodin

Abstract-A mathematical model of prevascular tumor growth by diffusion, which is an extension of previous models, has been constructed. The effects of nonuniform nutrient consumption and nonuniform inhibitor production on the growth rate of a spherically symmetric carcinoma are examined in this model, and the complete growth history of the tissue is followed analytically. A four-layer structure in the dormant steady state is predicted, and the evolution of a multicellular spheroid from an initial form is determined by an integro-differential (growth) equation. A parameter is used to measure the degree of nonuniformity of the inhibitor production rate, and growth patterns are presented for several values of this parameter. As this nonuniformity parameter is varied frpm zero to unity, there is a considerable adius, and also in the corresponding inner relative increase in the asymptotic steady-state outer tumo necrotic core size. A detailed understanding of the math Fmattcal model presented here can form the basis for a further level of description in deterministic models of tumor growth.

1.

INTRODUCTION

Over the last decade or two, various mathematical models of prevascular tumor growth by diffusion have been developed [l-6]. A moving boundary value problem in one-dimensional slab geometry (modeling oxygen concentrations in idealized tumors) has been studied recently [7,8]. The concern in that model is to couple theoretical results from time-dependent diffusion models with tumor oxygen radiosensitivity data to determine optimum radiation dosages for radiotherapy. The effects of uniform inhibitor production and uniform nutrient consumption on the growth of onedimensional and three-dimensional spherically symmetric tissue were examined in the models developed by Greenspan [3,4]. Recently, Maggelakis and Adam [9] reconstructed the onedimensional model [4] by incorporating the effects of nonuniform inhibitor production on the growth of a tissue. A direct comparison of the results of the two models were made and a significant range of deviation over the predictions of the uniform model was discovered. These interesting and suggestive results, along with the recent renewed interest in tumor angiogenesis and diffusion models [lo-131, motivated a further study of a more sophisticated spherically symmetric growth model which is presented in this paper. Deakin [5] modeled a problem of oxygen diffusion with the effects of nonuniform oxygen consumption on the viable rim thickness of a spheroid. This was later extended by McElwain and Ponzo [6] to include the effects of nonuniform oxygen consumption on the growth rate of a solid tumor. The present paper proposes a model in which the effects of nonuniform nutrient consumption and nonuniform inhibitor production on the growth rate of a spherically symmetric multicellular spheroid are examined. The source of the growth inhibitor is assumed to be the necrotic debris which is a result of insufficient nutrients for every cell of the spheroid. Due to the combined effects of nonuniform nutrient consumption and nonuniform inhibitor production, a four-layer structure in the dormant steady state is predicted, and successive stages of growth are examined. The three-dimensional spherically symmetric diffusion equations for the nutrient consumption and inhibitor production, with the corresponding integro-differential equation for the outer radius of the tumor have been solved for each stage of growth. The time-dependent analyses of King et al. [12-141 and Liapis et al. [7,8], together with growth-diffusion models of the type presented in this paper, are complementary approaches to one another, and it is hoped that this will provide a firm theoretical base from which a deeper understanding of tumor growth and tumor characteristics will evolve. 23

24

S. A. 2.

The basic assumptions Only the most important

THE

MAGGELAKIS and J. A. ADAM

DIFFUSION

necessary to construct of these assumptions

MODEL

this model are discussed are restated here:

in detail elsewhere

[3,6].

(i) The solid tumor is a sphere and is spherically symmetric at all times such that the concentrations of the nutrient c(r) and the growth inhibitor P(r) depend only on the radial distance, r, from the center of the sphere. (ii) The transport of the nutrient and growth inhibitor that regulate the prevascular development of a tumor are governed by diffusion processes. (iii) The nutrient is consumed by living cells only, and the tumor cells die when the concentration of nutrient falls below a critical value (TV.Furthermore, the nutrient consumption rate is assumed to decrease from its normal value when g, < o(r) < 6 (8 is the critical nutrient concentration, above which the consumption of nutrient is normal). (iv) The growth inhibitor, which is produced within the tumor, inhibits cell mitosis without causing their death. This inhibition of mitosis occurs when the local concentration of the growth inhibitor exceeds a critical value pi. (v) Since the time-scale for growth is large compared with a typical diffusion time, the tumor is in a state of diffusive equilibrium at all times. (vi) A parameter 6 is introduced to account for the effects of the inhibitor concentration on the nutrient consumption rate. There is some justification for this in the paper by Mueller-Klieser and Sutherland [15], in which the effects of toxic products from the necrotic core on cellular oxygen consumption were examined (see Ref. [16] for more details). (vii) The inhibitor production rate is nonuniform throughout the necrotic core. A measure of this nonuniformity is given by the parameter b. This is justified by the fact that in reality the transition from the necrotic core to the quiescent region will not be a discontinuous one. For models in which the inhibitor is produced in the necrotic core this is a reasonable assumption (see Ref. [3] and relevant data in Ref. [17]). The parameter b is also a useful means of determining the sensitivity of the model to nonuniformity of inhibitor production, especially in the light of the results by Adam [18]. 2.1. The growth equation

n/t’

The time-development of a spherically symmetric multicellular spheroid is determined by an integro-differential equation [3,6]. Following Greenspan [3], we may write the conservation of mass or cell volume (growth equation) as follows:

(1)

A=B+C-D-E, where A = total volume B = initial

volume

of living cells at any time t, of living cells at time t = 0,

C = total volume

of cells produced

in time t 2 0,

D = total volume

of necrotic

at time t

debris

and E = total volume of cells lost in the necrotic

core in time t 2 0.

We denote by R,(t), l?(t), R,(t) and R,(t) the outer radius of the spheroid, the radius at which the concentration of nutrient a(r) = 6 (the critical concentration), the radius at which the concentration of inhibitor P(r) = pi, and the necrotic core radius, respectively. S(o, B) is a measure

25

Prevascular growth of a spherical carcinoma

of the cell proliferation rate (the mitosis “source” term), and ;I is a proportionality the volume loss term. Hence the individual terms of equation (1) become A = $r[R:(t)

constant for

- R?(t)],

B = $&;(O), , R,(f) C=4n dt s0 s max(Ri(r).

S(a, P )r2 dr, R,(r))

D = $cRj(t)

and t E=$T

3AR’(t) dt,

s0 so

that equation (1) can be written as , R:(t)=R:(O)+3

dt

s0 or, in time-differentiated

I

R3

s max(R,.

S(o, B)r2 dr - 3

AR: dt

(2)

s0

Rg)

form, RZdRo ’

_

dt

R”

s max(R,.

The functional form of the proliferation function H(x), as

S(o, /3)r2 dr - AR:.

(3)

Rs)

rate S(o, fl) is stated in terms of the Heaviside unit step

S(O, B) = S s [H(O - Oi) - H(G - 8)] + H(O - 8) (H(Bi - p)} I i (4)

= &(a(r))S&3(r)),

with s being the normal proliferation rate of cells, i.e. when a(r) > d and /3(r) < Bi for a discontinuous inhibitor switch mechanism. This proliferation rate, S(a, fi), is obtained from the solutions of the time-independent diffusion equations for a(r) and j?(r).

3.

THE

MATHEMATICAL

ANALYSIS

OF

THE

MODEL

As in Ref. [6], it is assumed that the growth of a multicellular spheroid proceeds in several stages. Figure 1 shows schematically the sequence of stages in the growth of the spheroid. 3.1. Phase I All cells obtain sufficient nutrient, i.e. cr(r) > 8 holds everywhere and mitosis proceeds normally. It is assumed that all cells consume nutrient at a uniform rate, and so the tumor grows at a normal rate until the nutrient concentration at the center of the nodule drops to the critical value 8 [i.e. until a(r) = d at r = 01. The appropriate diffusion equation is

P(r)=0

Vr,

(5)

with the boundary conditions: a(r) bounded at the origin (r = 0), a(r) = CJ, at r = R,;

(6)

S. A.

26

PHASE

I

MAGGELAKIS and J. A. ADAM

PHASE

PHASE II

I

CASE I PHASE III

PHASE II

CASE 2 PHASE IV PHASE Ill

Fig. la. The sequence of stages in the growth of a spheroid for Case I of Phase IV of the growth development.

PHASE IV

Fig. lb. The sequence of stages in the growth of a spheroid for Case II of Phase IV of the growth development.

where k is the diffusion coefficient of the nutrient, a(r), in the nodule, A is the normal nutrient consumption rate and 6, is the external nutrient concentration (r > R,). The solution to this system is a(r) = 0, + 2

(r2 - R:),

O
(7)

The growth equation during this phase is given by R0 r’S(a, fl) dr.

4n dR; Tdt=4n

s0

(8)

From equation (4), for o(r) > 8.

S(a, P) = 3 Thus, equation (8) becomes dRi

dt By integrating equation (9) and introducing 5 = BR,, where B = (A jkb)“‘, we get

-

sR3

O’

(9)

a scaled time r as r = sr and a scaled radius 5 as

r(r) = t(O)e”‘.

(10)

This phase continues until o(r) = 6 at r = 0. The corresponding outer radius r, of the nodule, which is the scaled outer radius when the nutrient a(r) has reached this critical value 8, is given by

e,=[6ff- l)jY

(11)

Prevascular growth of a spherical carcinoma

27

3.2. Phase ZZ

In this phase a(r) < 8 in the central region of the tumor, and the cell proliferation rate decreases in that region. The overall growth rate begins to slow down due to the insufficient nutrient concentration, and a two-layer structure constitutes the nodule. The outer layer consists of cells with normal proliferation rate, as in Phase 1. Inside this, a layer of viable cells exists with proliferation rate below normal. The diffusion for this phase is

p(r) =0

Vr,

(12)

with the boundary conditions: (i) a(r) bounded at the origin, (ii) a(r) = 6, at r = ri, (iii) a(r) = aoo, at r = R,, da (iv) a(r), - are continuous at r = R.

(13)

dr

The solution to this system is a(r) =

dR sinh Br r sinh BR ’

O
(144

and -$(li’-R:)

Continuity

8
(14b)

of a(r) and daldr at r = Z? yields the expression

am --l==(<=-p=)+

(15)

6

where p = Sk

The growth equation for this phase becomes 4n dR; ydt=4n

a RO

r’S(a, /?) dr

r’S(a, 8) dr +

s

1.

(16)

From equation (4),

1 3,

W, B) =

/&)

d ’

Hence, equation (16) takes the form

a(r) > B CTi


li dR3 a(r) -2 = 3s r2dr + s(Ri -Z?‘). dt O B s Integration of equation (17) with a(r), given by equation (14a), and transformation dimensionless variables gives 5’2

=f(t3-p3)+p(p

cothp - l),

(17) into

(18)

which, using equation (15), can be rewritten as (19)

S.

28

A. MAGGELAKISand J. A. ADAM

This phase continues until a(r) = ai at the center (r = 0) of the tumor, marking the onset of necrosis (inhibitor production commences at the next stage), and this occurs when p = pC, where pCcan be obtained from PC

ai -=-_

B

(20)

sinh pC

The corresponding outer radius e, of the nodule, which is the scaled outer radius when necrosis starts, is given by equation (15) with p = pC. 3.3. Phase ZZZ This is a further period of retarded growth. There are two possibilities here:

(9 The tumour may reach a dormant steady state with a three-layer structure. In

the outer shell (d < r < R,, a(r) > 6) there is normal nutrient consumption and mitosis. In the middle region (Ri < r < I?, ai < a(r) < 8) consumption of nutrient and mitosis still occur, but at a reduced level. In the inner region (r < Ri, a(r) = ai) necrotic debris exists and no nutrient consumption occurs. In this steady state, inhibitor is produced (nonuniformly) in r < Ri, but the concentration /3(r) need not exceed the critical value Bi, since volume loss due to necrosis balances the volume gain due to mitotic activity outside the necrotic core. This is, in effect the model of McElwain and Ponzo [6], exhibiting however the early stages of inhibitor production. (ii) The growth is retarded by necrosis volume loss and consumption decrease but not enough to yield a steady state. Under these circumstances the model enters a fourth stage. The diffusion of nutrient and growth inhibitor is governed by the equations

(21)

and

(22)

where K is the diffusion coefficient for the growth inhibitor, P is the inhibitor production rate and which measures the degree of nonuniformity for the inhibitor production rate. These equations are subject to the following conditions:

b is a parameter

(i) a(r), P(r) are bounded at the origin (r = 0), da dS (ii) a(r), -, p and - are continuous dr dr (iii) a(r) = ai, b(r) = fii and $

= 0 at r = Ri,

(iv) a(r) = 6 at r = R, (4

a(r) = a, and B(r) =0

across all interfaces,

at r = R,.

(23)

Prevascular

The solutions

to these systems

growth

of a spherical

29

carcinoma

are:

o(r) = 2

QR,,

O
a(r) = bir [sinh B(r - Ri) + BR; cash B(r - R,)], -$I?‘-Rfi)

c~(r)=(i,+$(r~-R;)+

” 1

Ri < r < I?,

fs,

0

i?
(24)

and

,

O

RiGr
p(r)=

Application of the conditions at I? and the use of the nondimensional results in the following system of equations:

variables

g = f [sinh( p - v]) + q cosh( p - q)]

(25)

r, p and q = BR,

(264

and

gsc-

-

6

1

=:(5’-p2)+(GP _$)(; A),

Wb)

with G =>[cosh(p Again

from equation

-q)+q

sinh(p

-yl)]-

1.

(4),

and

such that

The growth

equation

for this phase is dR’ o = s(R;(t) dt

R

- &(t))

+ 3s

c(r) r2 dr - 31R:(t). 8 s4

(27)

S. A. MAGGELAKIS and J. A.

30

Using equation (24) and integrating, r2~=i(53-p3)+~P[cosh(P

ADAM

the growth equation (27) in dimensionless form becomes

-q)+q

sinh(p -q)] - ; [sinh(p - VI)+ ‘I cosh(p - q)l-

yq3,

(28)

where y = A/s. A simpler expression for the growth equation can be obtained by using equations (26a, b) to get 2d( 5 z=fs'j+

~-1-#2-p2) [

P5 -1 5_p

yq3,

(29)

This phase continues until b(r) = /Ii at Y= Ri (inhibition of mitosis occurs in Phase IV). Inhibition is then first evident when p(r) = pi and equation (25), for 0 < r d Ri, becomes (30) Let Q2 = pi KAlPk6,

then equation (30) takes the form

Q is therefore a measure of the competing effects of nutrient consumption/diffusion and inhibitor production/diffusion. As mentioned earlier, under certain conditions a steady state is reached or a fourth stage is initiated. These cutoff conditions depend critically on the parameters Q, b and y. 3.4. Phase IV This is a period of retarded growth due to necrosis volume loss, nutrient consumption decrease and chemical inhibition of mitosis, beginning when /I(r) = Bi at r = Ri. There are two cases to be considered in this stage of growth (if it occurs). Case I. We assume that R, < i? and that the chemically inhibited cells, since they are not dead, continue to consume some nutrient. The rate of growth decreases further and the tumor eventually reaches a steady state with a four-layer structure. In the outer region (A < r < R,), the growth is normal. Inside this, there is a region (Rg < r < 8) of retarded growth of cells (due to insufficient nutrient, (Ti< a(r) < 6). Adjacent to this region there is a layer (R, c R -=cRg) of viable cells with a proliferation rate below normal due to the inhibition of mitosis. In the innermost region (0 < r < Ri) there is accumulation of necrotic debris. The diffusion equations that govern this case are

a(r) A -> B

(32)

A,

and (33)

Prevascular growth of a spherical carcinoma

with the associated

boundary

31

conditions

(i) a(r),

B(r) bounded do - =0

(ii) a(r)=ai,

at the origin

(r = 0),

at r = Ri,

dr

(iii) P(r) = p, at r = R,, (iv) a(r) = 6 at r = 1, (v) c(r)

= am and b(r)

(vi) a(r),

= 0 at r = R,,

da

- continuous dr

(vii) b(r), g A parameter concentration

(34)

at r = R, and r = A, = Ri,r = R,andr

con t’muousatr

=R.

d2 (0 < 6 < 1) is introduced to measure the effect (if any) of nonzero inhibitor /l(r) on the nutrient consumption rate. Solving the systems above we obtain

a(r) = pi)

[sinh 6B(r - R,) + 6BRi cash 6B(r - Ri)],

a(r) = s a(r) =

O
d&B

sinh B(r - RJ + pai sinh B(ff - r) r6B sinh B(R - Rg)

o(r)=o:,+G(r’-Ri)+


1

RidrdR,,

(35) R, < r d I?,

, 1

-$?2-

R;)

1 ;s,

0

Ri
J

where p = sinh 6B(R,

- R,) + 6BR, cash 6B(R,

- Ri)

O
(36) P(r)=

R, d r < R,

Application of the conditions P(r) = pi at r = R,, doldr continuous at r = R, and doldr at r = Z?, results in the following three equations (note [ = BR,):

continuous

(37)

-= ’ smh(~-r)[cosh6([-~)+6~sinh6([-~)] CL + sinh(p

- [)coth(p

- [)

[sinh6(1

--a)+s~

cosh6([

-q)]

6P

(38)

and

ox - 1 = -

-

B

?(I +ff)+:(t2-p2)+p’t -P)coth(p 3 -[)

r

_

a,cr -PI 68t

cosech(p

- i)[sinh

S(i - V) + 6~ cash S(LI - ?)I,

(39)

S. A. MAGGELAKISand J. A. ADAM

32

respectively.

The proliferation

rate S(o, p), from the equation O
0, $“(‘) d

R, d r < I?,

’

k
s, Hence,

the growth

equation

is k

dR’ z=s(R;(t)-R’(t)+; dt

By integrating equation (40), substituting variables 5, p, n and i, a dimensionless 12s

=i(43-p3)+p[p

< Riy

Ri
0, S(o, P) =

(4), becomes

coth(p

r20(r)

dr - 3ARf(t).

s% a(r) from equation (35) and using the nondimensional form of the growth equation can be obtained:

-i)--[cosech(p

-[)-

1]

+~[5coth(p-r)-pcosech(p-_)+l][sinhd(T-4)+~6cosh6(i-rl)l-y~’.

(41)

At this point some further analysis is of interest. The radius, R,, of the mitotically inactive region may move out and “pass” the radius, fi, of the region with retarded cell growth converting outer shell material into regions of inhibited growth, which is Case II of this phase. Case II. It is now supposed that R, > 8, which may never be realized, and the tumor consists of a three-layer structure. An inner shell (0 < r < Ri) which is the necrotic core, a middle shell (R, < r < RP) of mitotically inactive cells and the outer shell (R, < r < R,) in which the growth of cells is normal. The diffusion of the nutrient o(r) is described by c

O
0,

d Ri,

(42)

14

R,dr

GR,,

of inhibitor /l(r) is described by equation (33) with solutions given by while the diffusion as in Case I, apply in this case. Thus, the solution equations (36). The same boundary conditions, of system (42) above is given by a(r) = oi,

O
a(r) = $

[sinh 6B(r - Ri) + 6BR, cash 6B(r - Ri)],

o(r)=&

r2+

R,Ro (R;-R;)+g r(R, - Rg)

- r(RR,R [

+ ~ The conditions equations:

0

1

0

I2

)[o,R,-criRiCOSh6B(R,-Ri)]+ P

0

[~~ R, - pi R, cash 6B(R,

Ri < r < R,

- Ri)],

Ois~~~~~(“a)R1) 0 R,
[R, _ r]

T2

G R,.

(43)

g

P(r) = p, at r = R, and

da/dr

continues

at r = R, give the following

system

of

(44)

Prevascular growth of a spherical carcinoma

33

and

+$(& )

I [sinh S([ - q) + 61 cash S([ - ?)I.

The proliferation

rate for this case is

S(a,B)

and the appropriate

(45)

growth

equation

=

0,

O
dRi,

0,

Ri
dR,,

i s,

R,
GR,,

in dimensionless

form becomes

In all these various phases and cases, the full system of equations relating Ri(t), R,(t), 8(t) and R,(t) to each other must be solved, in essence, simultaneously. Obviously computational methods will be required, and these are discussed in the next section.

4. DISCUSSION The model studied in this paper is deterministic in that the evolution of a multicellular spheroid from an initial form is determined by an integro-differential equation, the kernel of which depends on the solutions of spherically symmetric diffusion equations for the concentration of nutrient and growth inhibitor within the tumor. The governing differential equation for the outer tumor radius is derived and solved subject to the appropriate relationships between the nutrient and inhibitor concentrations and various inner radii. These relationships are obtained by solving analytically the diffusion equations for the spatially-dependent nutrient and inhibitor concentrations. The radial dependence induced by spherical symmetry introduces considerable simplifications in the analysis, while retaining the basic physics and appropriate geometric features [3]. In this model the tumor develops into a prevascular nodule with a four-layer structure (possibly), and its growth is characterized by four phases. A full growth pattern is obtained by prescribing values of or /&, 0,/b’, y, Q, b and 6. Once these values are given, the differential equations that describe the growth pattern of each stage are solved numerically for the outer radius 5 using a fourth-order Runge-Kutta method (except for the growth equation of Phase I which is solved analytically). As mentioned earlier, these growth equations depend on the relationships between the various dimensionless inner radii p, i and v, and the concentrations of the nutrient g(r) and inhibitor P(r). Consequently, the equations that give these relationships must be solved numerically several times during each step of the Runge-Kutta method. All the results presented below are for Case I in Phase IV. The values of the parameters b and 6, when varied between zero and unity induce significant differences in the growth patterns for the spheroid. Figure 2 shows the change of the inhibitor concentration b(r) as the parameter b takes on different values, and Fig. 3 shows the change of the nutrient concentration c(r) as the values of the parameter 6 are varied. A detailed discussion of the full set of results in a more biological context will be presented elsewhere [16] (Part II of this paper), so we will content ourselves here with a qualitative description of the overall growth characteristics predicted by this mathematical model. The full growth pattern of the spheroid in terms of the changes of the different parameters that describe the system are graphically illustrated below. Figure 4 shows the effect of nonuniform inhibitor production on the outer tumor radius. As b increases, the inhibitor concentration j?(r) decreases (see Fig. 2) and hence the mitotically active region grows due to the lack of enough inhibitor to reduce mitosis. In Case I of this figure more rapid growth is implied at an early stage

S.

A.

MAGGELAKIS

and J. A. ADAM

10.00 -

b=O

6.00 - br0.5 P c.00 - brO.9

0.00

2.00

L.00

6.00

6.00

10.00

r

Fig. 2. The inhibitor concentration b(r) as a function of the radial distance r for various values of the parameter b (i.e. 0, 0.5 and 0.9).

for b = 0 and slower growth at a later stage, both compared with the b = 0.9 case. The reason for this is to be found by examining equation (31). This equation determines the end of Phase III for given Q and 6, and clearly this varies with b. For b = 0.9, Phase III lasts longer than the corresponding rapid growth stage for b = 0. Hence, for b = 0 the asymptotic approach to dormancy begins earlier, but this asymptote must lie below that for b = 0.9, because the overall inhibition is greater. This accounts for the crossover in this figure. The changes of the parameter y = A/s are shown in Fig. 5. An increase in the proportionality constant for the volume loss J or a decrease in the normal proliferation rate s results in an increase in y. This causes a decrease in the overall growth of the tumor. Recall that the parameter Q is given by

Therefore, an increase in either the diffusion coefficient K for the inhibitor p(r) or in the normal nutrient consumption rate A (or both), or a decrease in either the production rate P of the inhibitor or the diffusion coefficient k for the nutrient (or both), result in an increase in Q, which in turn generates an increase in the growth of the nodule, shown in Fig. 6. Finally, as the parameter 6 decreases, the nutrient concentration decreases in the inhibited growth region (Ri s r < Rg) and in the mitotically active region (R, < r sz R), but the flux of nutrient decreases in the inhibited growth region and increases in the region where mitosis takes place, causing the growth of the tissue to increase, which is graphically illustrated in Fig. 7. The four stages of tumor growth development for the outer radius 5 and the inner radii p, [ and 9, as the values of the parameters y, Q, b and 6 vary, are illustrated in Figs 8-11, respectively. It should be noted in these figures that the derivatives du/dz and dp/dt remain finite as q and p approach zero in this model. This can readily be established for p, in particular, from equation (18). The above results show a significant difference between the growth pattern predicted by this model and that predicted by the McElwain and Ponzo model [6]. The data used to analyze and compare existing models [3,6] with the proposed model was taken from the McElwain/Ponzo and Greenspan models. It should be noted that the different parameters that determine the growth pattern of the present model could be prescribed by clinical or experimental data [12-141. A fuller discussion of the implications of this model in the light of such data will be presented elsewhere [ 161.

10.00

b=O

ii.5

15.00

bZO.2

12.00

b=O.5 bZO.2

r

0.00 4.00 8.00 12.00 16.00

4.00

5.00

b=0.9

I

8.00

L

L.00

R9br,
0.00 0 .QQ

2.00

L.00

6.00

8.00

10.00

0.00 t 0.00

-4.00

8.00

bz0.9

Fig. 3. The nutrient concentration o(r) as function of the radial distance r and the flux of nutrient da/dr in the inhibited growth region (R, < r < /t,) and in the mitotically active region (R, < r ,< R) for various values of the parameter 6 (i.e. 6 =0.2. 0.5 and 0.9).

k dr

do r

0

12.00

16.00

t

i

5.00

2

1

10.00

/

15.00

20.00

T

30.00

b=0.9

b-0.9 b=O

Fig. 4. The effect of nonuniform inhibitor production on the growth of the tumor, for various values of the parameters b (i.e. b = 0 and b = 0.9 for both cases of Phase IV of the growth development), with l being the scaled outer tumor radius. The values for the other parameters are Q = 0.8, y = 0.4 and 6 = 0.5.

0.00

25.00

eb=O

/-

o.oo/

1

CASE

CASE

1

5.00 i

10.00

15.00

20.00

0.00

2.00

4.00

6.00

8.00

10.00

12.00

t

10.00

15.00

20.00

25.00

30.00

Fig. 5. The change of the scaled outer tumor radius l as the parameter y takes on various values (i.e. y = 0.2, 0.5 and 0.9 for Case I; and y = 0.25, 0.4 and 0.8 for Case II of phase IV of the growth development of the tumor). The values used for the other parameters are Q = 0.8, b = 0.4 and 6 = 0.5, for both cases.

5.00

1 CASE 2

0.00

0.00

20.00

0.00

1.00

8.00

12.00

16.00

t

T

10.00 15.00 20.00 25.00 30.00

Fig. 6. The change of the scaled outer tumor radius 5 as the parameter Q changes (i.e. Q = 0.2, 0.5 and 0.8 for Case I; and Q = 0.5, 0.9 and I. 15 for Case II of Phase IV of the growth development of the tumor). The values used for the other parameters are y = 0.4, b = 0.4 and 6 = 0.5, for both cases.

5.00

CASE

0.00

4.00

6.00

8.00

10.00

12.00

0.00

,.,,{

0.00

44

lO.OO-

5.00

CASE

D

15.00-

20.00-

25.00-

o.oo

5.00

10.00

CASE

2

1

10.00

/

/

5

15.00

k-l

F

20.00

25.00

30.00

b=o.9

b=0.5

bEO.2

b-0.9

b=0.5

b=0.2

Fig. 7. The change of the scaled outer tumor radius 5 as the parameter 6 changes (i.e. 6 = 0.2, 0.5 and 0.9 for both cases of Phase IV of the growth development of the tumor). The values used for the other parameters are Q = 0.8, b = 0.4 and y = 0.4, for both cases.

r.

c

15.00

20.00

4.00

7=0.2

Fig. 8. The full growth pattern of the tumor for different values of the parameter y (i.e. y = 0.2, 0.5 and 0.9). The values used for the other parameters are Q =0.8, b = 0.4 and 6 = 0.5.

0.00

2.00

4.00

6.00

8.00

10.00

0.00

2.00

6.00 a

8.00

10.00

12.00

0.00

4.00

8.00

12.00

5 z

z

z 0

? 3

16.00

Prevascuiar

HIMOUE)

growth

of a spherical

carcinoma

HlMOtlO

HlMOtlO

S. A. MAGGELAKIS and J. A. ADAM

38 AcknonYedgement-This

research

was partially

supported

by NSF Grant

No. 396141

REFERENCES model of tumor growth. Math1 Biosci. 81, 224-229 (1986). 1. J. A. Adam, A simplified mathematical 2. J. A. Adam, A mathematical model of tumor growth. II. Effects of geometry and spatial nonuniformity on stability. Math/ Biosci. 86, 1833211 (1987). 3. H. P. Greenspan, Models for the growth of a solid tumor by diffusion. Stud. uppl. Math. 52, 317-340 (1972). On the self inhibited growth of cell cultures. Growth 38, 81 (1974). 4. H. P. Greenspan, 5. A. S. Deakin, Model for the growth of a solid in vitro tumor. Growth 39, 159 (1975). oxygen consumption. 6. D. L. S. McElwain and P. J. Ponzo, A model for the growth of a solid tumor with nonuniform Math1 Biosci. 35, 267-279 (1977). 7. B. H. Arve and A. I. Liapis, Oxygen tension in tumors predicted by a diffusion with absorption model involving a moving free boundary. MathI C~~puf. Modelling 10, 159-174 (1988). diffusion in absorbing tissue. Math1 Mode&w 8. A. I. Liaois. G. G. Lioscomb and 0. K. Crosser. A model of oxygen _I 3, 83-92‘(1982). L and J. A. Adam, Note on a diffusion model of tissue growth. Appl. Mafh. Left. 3, 27-31 (1990). 9. S. A. Maggelakis 10. L. J. Old, Tumor necrosis factor. Scienr. Am. May, 59-75 (1988). II. J. Folkman and M. Klagsbrun, Angiogenic factors. Science 235, 442-447 (1987). models for describing oxygen tension profiles in human 12. W. E. King, D. S. Schultz and R. A. Gatenby, Multi-region tumors. Chem. Engng Commun. 47, 73-91 (1986). using multi-region models. 13. W. E. King, D. S. Schultz and R. A. Gatenby, An analysis of systemic tumor oxygenation Chem. Engng Commun. 64, 137-153 (1988). 14. D. S. Schultz and W. E. King, On the analysis of oxygen diffusion and reaction of biological systems. Marhl Biosci. 83, 1799190 (1987). and R. M. Sutherland, Oxygen tensions in multicell spheroids of two cell lines. Br. J. Cancer 15. W. F. Mueller-Klieser 45, 256-263 (1982). of a spherical carcinoma. Bull. math. Biol. 16. J. A. Adam and S. A. Maggelakis, Diffusion regulated growth characteristics (in press). Growth of multicell spheroids in tissue culture as a model of nodular carcinomas. J. natn. Cancer 17. R. M. Sutherland, Inst. 46, 113-120 (1971). model of tumor growth. III. Comparison with experiment. Math1 Biosci. 86, 213-227 18. J. A. Adam, A mathematical (1987). Cell and environment interactions in tumor microregions: the multicell spheroid model. Science 240, 19. R. M. Sutherland, 177-184 (1988).