Abstract: The homotopy analysis method is used to obtain semi-analytic solutions for the mathematical model describing a solid tumor growth in the initial a vascular stage of growth. During a vascular tumor growth, the balance between cell proliferation and cell loss determines whether the colony expands or progress. We focus on the chemical inhibition of mitosis within multicell spheroids. The main assumption of modeling the diffusion of a growth inhibitory factor (GIF) within a multicell spheroid and its possible effects on cell mitosis and proliferation

N H Sweilam

S A Ghwail

CiteSeerX

Math. Sci. Lett.4, No. 1, 39-43 (2015) 39
Mathematical Sciences Letters
An International Journal
http://dx.doi.org/10.12785/msl/040109
Semi-Analytical Solution for the Multicell Spheroid
model for Vascular Tumor Growth
N. H. Sweilam1 and S. A.Ghwail2,∗
1 Department of Mathematics, Faculty of Science, Cairo Univers ty, Giza, Egypt
2 Department of Mathematics, Faculty of Science, Suez Canal Uiversity, Ismailia, Egypt
Received: 47 Jul. 2014, Revised: 23 Sep. 2014, Accepted: 5 Oct. 2014
Published online: 1 Jan. 2015
Abstract: The homotopy analysis method is used to obtain semi-analytic solutions for the mathematical model describing a solid
tumor growth in the initial a vascular stage of growth. During a vascular tumor growth, the balance between cell proliferation and cell
loss determines whether the colony expands or progress. We focus on the chemical inhibition of mitosis within multicellspheroids. The
main assumption of modeling the diffusion of a growth inhibitory factor (GIF) within a multicell spheroid and its possible effects on
cell mitosis and proliferation.
Keywords: Homotopy analysis method, Tumor growth
1 Introduction
A vahcular tumor growth models studied extensively in
the last three decade, see for example, ([5], 7], [8]).
Mathematical model for a vascular tumor growth and
development that spans three distinc scales; cellular level,
subcellular level, extracellular level, for more details see
([1]-[4]). Although, almost all studies reach similar
conclusions that a vascular tumor can only grow up to a
limited size, the saturation mechanisms that are assured in
different models are not same ([5]-[6]), depending on
nutrients concentration tumor cells are supposed to be in
one of the three stages; proliferating, resting or dead.
While the tumor expands, the nutrient concentration at the
center falls below critical level. The cell proliferation rate
will be decrease which causes a slow growth rate.
Eventually, these interior cells can die off, creating what
is known as a necrotic core. Although a significant
progress in modeling tumor has been achieved by now,
most of these models are based on numerical approach.
Hence, an approach other than numerical approach such
as semi-exact solutions is needed
Recently, some principles of the physics and It is
known that approximate analytical method such as
Homotopy Analysis Method(HAM) which advised by
Shi-Jun Liao in ([15]-[18]) has been successfully applied
to solve many types of nonlinear problems in science and
engineering by many authors, see ([1], [13], [14], and the
references cited therein). The aim in this work is to
employ HAM for establish an infinite series solutions for
the mathematical model describing a solid tumor growth
in the initial a vascular stage, for more details on the
HAM method for linear and nonlinear differential
equations, see([11]-[12]), [15], ([27]-[28]), and the
references cited therein. We can pointed out here that ,
HAM contains the auxiliary parameterh̄, which provides
us with a simple way to adjust and control the
convergence region of solution series[18], moreover, the
approximate solutions can be obtained using a few
number of iterations ([27]-[28]).
2 Basic idea of HAM
Consider the following general differential equation:
F(u(t)) = 0, (1)
where F is a nonlinear operator,u(t) is an unknown
function.For simplicity, we ignore all boundary and initial
conditions, which can be treated in the similar ways in
[12], [15], [17].
∗ Corresponding author e-mail:suzy.future@gmail.com
c© 2015 NSP
Natural Sciences Publishing Cor.
40 N. H. Sweilam et. al.: Semi-Analytical Solution for the Multicell...
2.1 Zeroth-order deformation equation
Liao ([15]-[18]), constructed the so-called zeroth-order
deformation equation
(1− q)L[Φ(t;q)− u0(t)] = qh̄F [Φ(t;q)], (2)
whereL is an auxiliary linear operator,u0(t) is an initial
guess,h̄ is an auxiliary parameter andq ∈ [0,1] is the
embedding parameter. Obviously, whenq = 0 andq = 1,
it holds, respectively
Φ(t;0) = u0(t), Φ(t;1) = u(t). (3)
Thus, asq increasing from 0 to 1, the solutionΦ(t;q)
various fromu0(t) to u(t). ExpandingΦ(t;q) in Taylor
series with respect to the embedding parameterq, one has
Φ(t;q) = u0(t)+
∞
∑
m=1
um(t)q
m, (4)
where
um(t) =
1
m!
∂ mΦ(t;q)
∂qm
∣
∣
∣
q=0
. (5)
Assume that the auxiliary linear operator, the initial guess
and the auxiliary parameterh̄ are selected such that the
series (4) is convergent atq = 1, then atq = 1 and by (20),
the series (4) becomes
u(t) = u0(t)+
∞
∑
m=1
um(t). (6)
2.2 The mth-order deformation equation
Define the vector
un(t) = [u0(t),u1(t), ...,un(t)]. (7)
Differentiating equation (2) m times with respect to the
embedding parameterq, then settingq = 0 and dividing
them bym!, finally using (5), we have the so-calledmth-
order deformation equations
L[um(t)− δmum−1(t)] = h̄ℜm(um−1), (8)
where
ℜm(um−1) =
1
(m−1)!
∂ m−1F[Φ(t;q)]
∂qm−1
∣
∣
∣
q=0
, (9)
and
δm =
{
0, m ≤ 1;
1, m > 1. (10)
3 Mathematical formulation
We formulate this model in mathematical equation which
describe the diffusion, production and degradation of the
growth inhibitory factor GIF ([6], [7]) within the spheroid,
It can be written as,
∂C
∂ t
= D∇2C+ f (C)+λ S(r) (11)
whereC = C(r, t) is the concentration of GIF within
the spheroid occupying the regionR3 and λ is the
inhibitor production rate (molecules per unit volume per
second). Shymko and Glass [22], and Adam [2] use the
function f (C) = −γC, whereγ is the depletion rate or
decay constant, various forms for the source functionS(r)
have been used. In the original model in [22], the GIF is
assumed to be produced at a constant rate throughout the
tissue, yielding the uniform source function[7]:
S(r) =
{
1− r
2
R2
, 0≤ r ≤ R;
0, r > R.
(12)
Then the model will be,
∂C
∂ t
= D∇2C− γC+λ S(r), r ∈ Ω , (13)
∂C
∂ r
= 0, r = 0, (14)
D(r)
∂C
∂ r
+PC = 0 on ∂Ω , P > 0, (15)
C(r,0) = 0, r ∈ Ω . (16)
Initially, we assume that production of GIF is via the
uniform source function S(r). Considering the spherical
geometry described in the introduction and assuming
radial symmetry, the above system reduces to
∂C
∂ t
=
1
r2
∂
∂ r
(r2D
∂C
∂ r
)− γC+λ S(r), r ≤ R, (17)
∂C
∂ r
= 0, r = 0, (18)
D(r)
∂C
∂ r
+PC = 0 on r = R, P > 0, (19)
C(r,0) = 0, r = R3. (20)
Before continuing with an analysis of the above system, it
is appropriate to recast them in terms of dimensionless
variables. Denoting byR andt, the radius of the spheroid
GIF concentration and as reference time. The system now
becomes, upon dropping the tildes for notational
convenience,
∂C
∂ t
=
1
r2
∂
∂ r
(r2D
∂C
∂ r
)−B2C+ aB2(1− r2), r ≤ R,
(21)
c© 2015 NSP
Natural Sciences Publishing Cor.
Math. Sci. Lett.4, No. 1, 39-43 (2015) /www.naturalspublishing.com/Journals.asp 41
∂C
∂ r
= 0, r = 0, (22)
D(r)
∂C
∂ r
+PC = 0 on r = R, P > 0, (23)
C(r,0) = 0, r = R3, (24)
whereB = kR, k2 = γD ,a =
λ
γθ andη =
γD1/2
P . With this
non-dimensionlisation, we see that once the parameters
for a particular spheroid are determined, the only
undetermined parameter is the radiusR. the solutions to
the above equation can therefore be monitored for
different size of spheroidsR. Let r = x and using the
transformation
u(x, t) = xC(x, t). (25)
Then,
∂u
∂x
= C+ x
∂C
∂x
,
∂ 2u
∂x2
= 2
∂C
∂x
+ x
∂ 2C
∂x2
, (26)
∂u
∂ t
= x
∂C
∂x
.
Substitute (25)-(26) into(21), we claim
∂u
∂ t
= D
∂ 2u
∂x2
−B2u+ aB2x(1− x2), (27)
subject to the initial conditionsu(x,0) = 0.
4 Method of solution
In this section, we apply HAM to obtain the approximate
solution to the problem of vascular tumor growth.
Consider above eqn(27).
∂u
∂ t
= D
∂ 2u
∂x2
−B2u+ aB2x(1− x2),
with initial conditionsu(x,0) = 0. By means of homotopy
analysis method we choose the linear operator
L[ϕ(x, t;q)] =
∂ϕ(x, t;q)
∂ t
, (28)
where the operatorL satisfies the relationL[c] = 0 for some
arbitrary constantc also we define the non-linear operator
as
N(ϕ(x, t;q)) =
∂ϕ(x, t;q)
∂ t
−D
∂ 2ϕ(x, t;q)
∂x2
(29)
+ B2ϕ(x, t;q)− aB2x(1− x2), (30)
we can construct the zeroth-order deformation
equation in the form
(1− q)L[ϕ(x, t;q)− u0(x, t)] = qh̄N(ϕ(x, t;q)). (31)
Obviously whenq = 0 andq = 1, we obtain
ϕ(x, t;0) = u0(x, t), ϕ(x, t;1) = u(x, t).
Therefore, as the embedding parameterq increases
from 0 to 1,ϕ(x, t;q) varies from the initial guessu0(x, t)
to the solution u(x, t). Expanding ϕ(x, t;q) in Taylor
series with respect toq
ϕ(x, t;q) = u0(x, t)+
∞
∑
m=1
qmum(x, t), (32)
where
um(x, t) =
1
m!
∂ mϕ(x, t;q)
∂qm
|q=0. (33)
Now, we define the vectoru(x, t) = [u0(x, t),u1(x, t), ...].
Themth-order deformation equation is
L[um(x, t)− δmum−1(x, t)] = h̄Rm(−→u m−1), (34)
with initial conditionsum(x,0) = 0, where
Rm(
−→u m−1) =
∂um−1(x, t)
∂ t
−D
∂ 2um−1(x, t)
∂x2
(35)
+ B2um−1(x, t)− aB
2x(1− x2). (36)
Now, the solution of themth-order deformation equation
becomes
um(x, t) = δmum−1(x, t)+ h̄
∫ t
0
Rm(
−→u m−1)dt, (37)
then the approximate solution will take the following
u(x, t) = u0(x, t)+ u1(x, t)+ ... (38)
5 Existence and Convergence of HAM
To investigate the influence ofh̄ on the convergence of the
solution series given by HAM, we first plot the so-called
h̄-curve ofu(x, t). According to theh̄-curve, it is easy to
discover the valid region of̄h at−1.5≤ h̄ ≤ 2 (see figure
1).
We use four terms in evaluating the approximate
solution,
u0(x, t) = aB
2h̄x(1− x2)t, (39)
u1(x, t) = −aB
2h̄2x(3D+
B2
2
(1− x2))t2− (1− x2)t, (40)
u2(x, t) = aB
4h̄3x(2D+
B2
6
(1− x2))t3, (41)
u3(x, t) = −a
B6
4
h̄4(3D+
B2
6
(1− x2))t4. (42)
It is noted that our approximate solutions converge at
(−2 ≤ h̄ ≤ 2) see figure 1. Figures 2 and 3 show the
results for HAM for various values of R, which show the
development of a spheroid from its early stages of growth
c© 2015 NSP
Natural Sciences Publishing Cor.
42 N. H. Sweilam et. al.: Semi-Analytical Solution for the Multicell...
Fig. 1: h̄ curve of GIF att = 10, a = 76.764, D = 5× 10−7,
R = 0.02,γ = 5×10−5 .
Fig. 2: Approximate solution of GIF for̄h = −1.5, t = 10, a =
76.764,D = 5×10−7, R = 0.02,γ = 5×10−5.
1
2
3 0
5
10
15
−5
0
5
10
15
20
tx
u(
x,
t)
Fig. 3: Approximate solution of GIF in 3D for̄h = −1.5, a =
76.764,D = 5×10−7, R = 0.02,γ = 5×10−5.
to its diffusion - limited size of a stable radius of 0.2 cm.
Our results in agreement with [7] where the threshold
value for the GIF concentration isC = 1. Thus if the
concentration of GIF is greater than 1 in any region
within the spheroid, then mitosis will be inhibited
(necrotic core) to be distinguished. Fig 1 shows that the
model predicts that the onset of necrosis occurs in the
center of spheroids are (C = 1 atr = 0).
6 Conclusions
In this work, HAM is adapted to obtain series solutions
with a high degree of accuracy for the nonlinear
mathematical model describing a solid tumor growth in
the initial a vascular stage of growth. No discretizations,
linearization , small perturbations or restrictive
assumption are needed when we apply this method.
It may be concluded that this methodology is very
powerful and efficient technique in finding semi-exact
solutions for wide classes of problems. It is also worth
noting to point out that the advantage of this methodology
shows a fast convergence of the solutions by means of the
auxiliary parameter̄h. We can conclude that HAM is easy
to apply for both linear or nonlinear differential systems.
References
[1] S. Abbasbandy, Homotopy analysis method for heat radiation
equations, International and Communication of Heat Mass
Transfer, 34, 380-387 (2007).
[2] J. A. Adam, A simplified mathematical model of tumor
growth, Math. Biosci. 86:183-211 (1987).
[3] J. A. Adam, A mathematical model of tumor growth, I.
Effects of geometry and Spatial non uniformity on stability,
Math. Biosci. 86:183-211 (1987).
[4] J. A. Adam, A mathematical model of tumor growth.-
III. Comparison with experiment, Math. Biosci. 86:213-227
(1987).
[5] J.A. Adam, Corrigendum: A mathematical model of tumor
growth by diffusion, Math. Biosci.94:155 (1989).
[6] A. C. Burton, Rate of growth of solid tumors as a problem of
diffusion, Growth 3:157-176 (1966).
[7] M. A. J. Chaplain and N. F. Britton, On the concentration
profile of a growth inhibitory factor in multicell spheroids,
Math. Biosci.115:233-245 (1993).
[8] M. A. J. Chaplain and A. M. Stuart, 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).
[9] H. P. Greenspan, Models for the growth of a solid tumor by
diffusion, Stud.A&. Math.51:317-340 (1972).
[10] H. P. Greenspan, On the growth and stability of cell cultures
and solid tumours, J. Theor. Biol.56:229-242 (1976).
[11] I. Hashim, O. Abdulaziz and S. Momani, Homotopy
analysis method for fractional IVPs, Communications in
Nonlinear Science and Numerical Simulation, 14, 674-
684(2009).
c© 2015 NSP
Natural Sciences Publishing Cor.
Math. Sci. Lett.4, No. 1, 39-43 (2015) /www.naturalspublishing.com/Journals.asp 43
[12] J. Hilton, An Introduction to Homotopy Theory, Cambridge
University Press, Cambridge, (1953).
[13] M. Inc, On exact solution of Laplace equation with Dirichlet
and Neumann boundary conditions by the homotopy analysis
method, Physics Letters A, 365, 412-415 (2007).
[14] H. Jafari and S. Seifi, Solving a system of nonlinear
fractional partial differential equations using homotopy
analysis method, Communications in Nonlinear Science and
Numerical Simulation, 14, 1962-1969 (2009).
[15] S. J. Liao, Beyond Perturbation: Introduction to the
Homotopy Analysis Method, Champan& Hall/CRC Press,
Boca Raton, (2003).
[16] S. J. Liao, On the homotopy analysis method for non linear
problems, Applied Mathematics and Computation, 147,499-
513 (2004).
[17] S. J. Liao, The Proposed Homotopy Analysis
Techniquefor the Solution of Nonlinear Problems, PhD
dissertation,Shanghai Jiao Tong University, (1992).
[18] S. J. Liao, Notes on the homotopy analysis method-
Some definitions and theorems, Commun. Nonlinear
Sci.Numer.Simulat., 14, 983-997 (2009).
[19] J. Landry, J. P. Freyer, and R. M. Sutherl and, A model for
the growth of multicell spheroids, Cell Tissue Kinet. 15585-
594 (1982).
[20] D. L. S. McElwain and P. J. Ponzo, A model for the growth
of a solid tumor with non-uniform oxygen consumption,
Math. Biosci. 35:267-279 (1977).
[21] S. A. Maggelakis and J. A. Adam, Mathematical model of
prevascular growth of a spherical carcinoma, Math. Comp.
Modelling 13:23-38 (1990).
[22] R. M. Shymko and L. Glass, Cellular and geometric control
of tissue growth and mitotic instability, J. Theor. Biol.
63:355-374 (1976).
[23] R. M. Sutherland, Cell and environment interactions in
tumor microregions: The multicell spheroid model, Science
240:177-184 (1988).
[24] R. M. Sutherland and R. E. Durand, Growth and cellular
characteristics of multicell spheroids, Recent Results Cancer
Rex 95:24-49 (1984).
[25] R. M. Sutherland, J. A. McCredie and W. R. Inch, Growth
of multicell spheroids as a model of nodular carcinomas, J.
Nutl.Cancer Inst. 46:113-120 (1971).
[26] G. W. Swan, The diffusion of an inhibitor in a spherical
tumor, Math. Biosci. 108:75-79 (1992).
[27] N. H. Sweilam and M. M. Khader, Semi exact solutions for
the bi-harmonic equation using homotopy analysis method,
World Applied Sciences Journal, 13, 1-7 (2011).
[28] M. M. Khader, N. H. Sweilam, Z. I. EL-Sehrawy and
S.A.Ghwail, Analytical study for the nonlinear vibrations
of multiwalled carbon nanotubes using homotopy analysis
method, Appl. Math. Inf. Sci. 8, 1-10 (2014).
Nasser H. Sweilam,
professor of numerical
analysis at the Department
of Mathematics, Faculty
of Science, Cairo University.
He was a channel system
Ph.D. student between
Cairo University, Egypt, and
TU-Munich, Germany. He
received his Ph.D. in ”Optimal Control of Variational
Inequalities, the Dam Problem”. He is the Head of the
Department of Mathematics, Faculty of Science, Cairo
University, since May 2012. He is referee and editor of
several international journals, in the frame of pure and
applied Mathematics. His main research interests are
numerical analysis, optimal control of differential
equations, fractional and variable order calculus,
bio-informatics and cluster computing and ill-posed
problems.
S. A. Ghwail received the
Msc degree in Mathematics
Department, Suez Canal
University (Ismailia) -(2011).
Her research interests
are in the areas of Applied
Mathematics including
the mathematical methods
and numerical techniques for
solving mathematical models in (Nano-technology,
solitary waves,...).
c© 2015 NSP
Natural Sciences Publishing Cor.


Semi-Analytical Solution for the Multicell Spheroid model for Vascular Tumor Growth

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Semi-Analytical Solution for the Multicell Spheroid model for Vascular Tumor Growth

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