AMS subject classification: 65C20,68U20,65D25,
65M06,65M12.
An analysis of antibody penetration into a pre-
vascular tumor nodule embedded in normal tis-
sue is presented in [1]. According to the mathe-
matical model presented in a scoentific modeling
text in the French external agregation competition
(public 2008), a numerical method is proposed for
calculating antibody and antigen concentrations
when reaction speed is moderate. Although the
proposed mathematical model describes well the
transport-diffusion reaction of antibodies in a tu-
mor and their interactions with antigens, it has
a significant limitation. In fact, the system has
been shown to be unstable and requires modifi-
cation. An analysis of the stability and consis-
tency is proposed, and the theoretical results are
validated by numerical tests after increasing the
reaction factor. The proposed work provides a de-
tailed analysis of a modified scheme, the effects of
the reaction factor, and the behavior of the new
scheme at infinity. We seek solutions to the sys-
tem in the form of progressive waves of the ”front”
type.
We suppose :
The liquid carrying the antibodies occupies all
the inert spaces in the medium.
The antigens are fixed to the internal walls of the
inertial cells.
The porosity ratio w=volume liquid
total volume ∈]0.1[is known.
The process takes place in a fairly thin tube, of
section Aand the flow occurs through sec-
tion wA. (ie We can confuse the dimension
of the tube with a one-dimensional medium
in space.)
Notations :
The concentration of the antibody c(x,t) =
number o f antibodies
volume o f f luid .
The concentration of the antigen s(x,t) =
number o f antigen
volume total .
The flow of antibodies q(x,t) =
number o f antibodies passed in x
time ×sur f ace which is a func-
tion of cand s.
The antibody-antigen reaction
function f[c(x,t),s(x,t)] =
number o f antibodies retained by antigens
time ×volume .
Finite difference scheme for transport-diffusion-reaction of antibodies
in a tumor : Analysis of consistency and stability
AHMED KANBER
Informatique
CRMEF MARRAKECH
Rue Mozdalifa 40000 Marrakech
MOROCCO
Abstract: According to the mathematical model presented in a scientificité modeling text in the French external
agregation competition (public 2008), antibodies are transported between tumors and are reacted with antigens
in a transport-diffusion reaction. As a result of the proposed model, an unstable system is produced. Under
certain conditions, we propose modifications that result in a new system that is stable and consistent. In this
paper, a detailed study of the stability and consistency of this new system is presented, with demonstrations and
proofs that are validated numerically.
Key-Words: Finite difference scheme, antibodies, tumor, antigens, consistency, stability, simulation.
Received: March 14, 2024. Revised: September 2, 2024. Accepted: September 25, 2024. Published: November 19, 2024.
1. Introduction
2. Mathematical Model of simultaneous of
antibody-antigen reaction in a tumor
2.1 Notations
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162
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For a time ∆tand in a volume A.∆x, as a result of
the principle of mass conservation, we have:
∆c(x,t).∆x.Aw =−∆q(x,t).∆t.A−f[c(x,t),s(x,t)].∆t.∆x.A
Thus, dividing by ∆t.∆x.Aand passing on to par-
tial derivatives we obtain:
w
∂
c(x,t)
∂
t+
∂
q(x,t)
∂
x=−f[c(x,t),s(x,t)] (1)
There are two components to the flow q=qa+qd:
A transport flow is defined as qa(x,t) = u.c(x,t)
where urepresents the transportation speed
as assumed here to be constant.
The diffusion flux qd(x,t) = −v.
∂
c(x,t)
∂
xwhere vis
the diffusion factor assumed to be constant.
As the antigens are fixed, only the time t, the
antibody-antigen reaction function fand the va-
lency pof the antibodies determine their number.
C+pS →SpC(2)
A variation in the number of antigens equals −p×
(the number of antibodies) retained by the anti-
gens, we have:
∆s(x,t).∆x.A=−p f [c(x,t),s(x,t)].∆t.∆x.A
By dividing by ∆t.∆x.Aand passing to partial
derivatives, we obtain:
∂
s(x,t)
∂
t=−p.f[c(x,t),s(x,t)] (3)
We obtain
∂
s(x,t)
∂
t=−p.k.c(x,t).s(x,t), with kas
the factor of reaction assumed to be constant and
fas the form: f=k.c.s(4)
Based on the framework previously defined, we
search for cand sthat are defined on the [0,L]×
[0,T],L,T∈R∗+. Then we give ourselves: w∈
]0,1[,u,v,k,s0∈R∗+,p∈N∗and a regular function
cdon [0,T]. Therefore, the problem is as follows:
w
∂
c
∂
t+u
∂
c
∂
x−v
∂
2c
∂
x2+k.c.s=0(5)
∂
s
∂
t+p.k.c.s=0(6)
With initial and boundary conditions:
c(0,t) = cd(t)c(L,t) = 0t∈[0,T]
c(x,0) = 0s(x,0) = s0x∈[0,L](7)
An explicit scheme:
Let M,N∈N∗and the spatial and temporal dis-
cretization steps: ∆x=L/Mand ∆t=T/N. We
consider the progressive scheme in time and cen-
tered in space :
∂
c(xj,tn)
∂
t≈cn+1
j−cn
j
∆t
∂
s(xj,tn)
∂
t≈sn+1
j−sn
j
∆t
∂
c(xj,tn)
∂
x≈cn
j−cn
j−1
∆x
∂
2c
∂
x2≈cn
j+1−2cn
j+cn
j−1
∆x2
Where (xj,tn) = ( j.∆x,n.∆t),cn
j=c(xj,tn)and
sn
j=s(xj,tn). Which give:
wcn+1
j−cn
j
∆t+ucn
j−cn
j−1
∆x−vcn
j+1−2cn
j+cn
j−1
∆x2+k.cn
j.sn
j=0(8)
sn+1
j−sn
j
∆t+p.k.cn
j.sn
j=0(9)
With initial and boundary conditions:
c0
j=0j∈ |[0,M]|(10)
cn
0=cd(tn)n∈ |[1,N]|(11)
s0
j=s0j∈ |[0,M]|(12)
* Numerical simulation of the explicit scheme :
with function cd(t) = 1and values:
w u v s0 M dt T L k p
0.9 0.1 0.003 2 100 0.01 20 1 1e4 3
2.2 Basic Equation
2.3 System setup
2.3.1 The mass conservation of antigens
2.3.2 The system of partial differential Equations
3. Numerical resolution
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The scheme (8−9)is consistent with the equation
(5−6)and it is order 1 accurate in time and space
According to Taylor’s developments we have:
cn+1
j−cn
j=∆t
∂
c(xj,tn)
∂
t+O(∆t2)
cn
j+1−cn
j=∆x
∂
c(xj,tn)
∂
x+∆x2
2
∂
2c(xj,tn)
∂
x2+O(∆x3)
cn
j−1−cn
j=−∆x
∂
c(xj,tn)
∂
x+∆x2
2
∂
2c(xj,tn)
∂
x2+O(∆x3)
cn
j+1−2cn
j+cj−1=∆x2
∂
2c(xj,tn)
∂
x2+O(∆x4)
sn+1
j−sn
j=∆t
∂
s(xj,tn)
∂
t+O(∆t2)
And by replacing the obtained equations in the
3.1 Consistency analysis and stability
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first members of (8−9)we obtain the errors :
Ec=w
∂
c(xj,tn)
∂
t+O(∆t) + u
∂
c(xj,tn)
∂
x
+∆x
2
∂
2c(xj,tn)
∂
x2+O(∆x2)−v
∂
2c(xj,tn)
∂
x2
+O(∆x2) + k.cn
j.sn
j
=O(∆x,∆t)
Es=
∂
s(xj,tn)
∂
t
+O(∆t) + p.k.cn
j.sn
j
=O(∆x,∆t)
Scheme (8-9) is stable with equation (5-6) un-
der the condition:
∆t≤min{w
u
∆x+2v
∆x2+ks0
,1
kpK }W here K =max
1≤n≤Ncd(tn)
by induction :
We have 0≤c0
j≤Ket 0≤s0
j≤s0
Suppose 0≤cn
j≤Kand 0≤sn
j≤s0for a n∈N∗.
From (8−9)we have:
cn+1
j=cn
j+u∆t
w∆x(cn
j−1−cn
j)
+v∆t
w∆x2(cn
j+1−2cn
j+cn
j−1)−k∆t
wcn
jsn
j
= ( u
w∆x+v
w∆x2)∆tcn
j−1+ (1−(u
w∆x
+2v
w∆x2+k
wsn
j)∆t)cn
j+v
w∆x2∆tcn
j+1
sn+1
j=sn
j−pk∆t
wcn
jsn
j
Finaly we obtain : 0≤cn+1
j≤K
0≤sn+1
j≤s0
We consider the scheme (15 −16)by replacing the
terms cn
jsn
jby cn+1
jsn+1
j, which can be written again
in the form:
X+A1XY +B1=0(15′)
Y+A2XY +B2=0(16′)where A1=k∆t
w,
A2=pk∆t
wand
B1=−cn
j−u∆t
w∆x(cn
j−1−cn
j)−v∆t
w∆x2(cn
j+1−2cn
j+cn
j−1)
= ( u∆t
w∆x+2v∆t
w∆x2−1)cn
j−(u∆t
w∆x+v∆t
w∆x2)cn
j−1−v∆t
w∆x2cn
j+1
B2=−sn
j,X=cn+1
jand Y=sn+1
j.
The system (15′−16′)is equivalent to:
A2X2+ (1+A2B1−A1B2)X+B1=0(15′′)
Y+A2XY +B2=0(16′)
With equation (5−6), the scheme (15 −16)is:
i) consisting and it’s order 1 accurate in time and
space .
ii) stable under the condition:
∆t≤w
u
∆x+2v
∆x2
and therefore in this case it’s stable indepen-
dently of k.
We proceed also by induction.
We should demonstrate that (cn+1
j,sn+1
j)exists and
0≤cn+1
j,0≤sn+1
j;(h).
The equation (15′′)is of 2nd degree, with discrim-
inant ∆= (1+A2B1−A1B2)2−4B1A2and admits a
unique positive solution because: u∆t
w∆x+2v∆t
w∆x2−1≤
0so B1≤0and ∆≥0. As √∆≥|1+A2B1−A1B2|,
the equation (15′′)admits 2 solutions X1≥0and
X2≤0(not necessarily distinct. and the equation
(16′)shows that Y≥0as soon as X≥0(because
−B2=sn
j≥0).
Let us now show that cn+1
j≤K,sn+1
j≤s0:
From (15 and 16)we have:
cn+1
j= ( u
w∆x+v
w∆x2)∆tcn
j−1+ (1−(u
w∆x+2v
w∆x2)∆t)cn
j+v
w∆x2∆tc n
j+1−k
w∆tcn+1
jsn+1
j
sn+1
j=sn
j−pk∆t
wcn+1
jsn+1
j
And according to induction hypotheses we
have:
cn+1
j≤(u
w∆x+v
w∆x2)∆tK + (1−(u
w∆x+2v
w∆x2)∆t)K+v
w∆x2∆tK ≤K
sn+1
j≤sn
j≤s0
Numerical simulation of the modified scheme :
with function cd(t) = 1and values:
w u v s0 M dt T L k p
0.9 0.1 0.003 2 100 0.01 20 1 1e4 3
4. Behavior for large k
4.1 Modified scheme of the method
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To get an idea of the appearance of cand sfor
large k, we consider the solution (ck,sk)of the
system (5−6)for a certain kand we make k→+∞:
w
∂
ck
∂
t+u
∂
ck
∂
x−v
∂
2ck
∂
x2+kcksk=0(5)
1
−pk
∂
sk
∂
t=cksk(6)⇔
∂
(wck−sk/p)
∂
t+u
∂
ck
∂
x−v
∂
2ck
∂
x2=0
1
−pk
∂
sk
∂
t=cksk
we make k→+∞we obtain :
∂
(wc∞−s∞/p)
∂
t+u
∂
c∞
∂
x−v
∂
2c∞
∂
x2=0(13)
If we assume that
∂
sk
∂
tis bounded with respect to
the values of k∈R∗+,then c∞s∞=0.
We concluded that :
c∞>0and s∞=0
Or
c∞=0and s∞=s0
(14)
4.2 Behavior when k → + ∞
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With a large reaction factor, we have at a position
x:
If there are antibodies (c∞>0) then all antigens
react (s∞=0).
If there are no antibodies (c∞=0) then there is
no reaction (s∞=s0)
We shall now seek a solution of the following type:
c(x,t) = C(z)s(x,t) = S(z),z=x−
σ
tand C,Sare
functions that describe wave profiles propagating
at constant speed
σ
. In order to achieve a progres-
sive (non-stationary) framework, we may replace
the domain of study with R×R+and the bound-
ary conditions with:
lim
z→−∞C(z) = cd>0 lim
z→+∞C(z) = 0
lim
z→−∞S(z) = 0 lim
z→+∞S(z) = s0>0
The Cand Sprofiles verify the ODE:
vC′′ + (w
σ
−u)C′−
σ
pS′=0(17)
where
σ
=u
w+s0
pcd
(18)
We have:
∂
c(x,t)
∂
t=
∂
c(x,t)
∂
z
∂
z
∂
t
=−
σ
C′(z)and
∂
c(x,t)
∂
x=
∂
c(x,t)
∂
z
∂
z
∂
x
=C′(z)
we obtain
∂
2c(x,t)
∂
x2=C′′(z)and
∂
s(x,t)
∂
t=−
σ
S′(z)Accordingly, we concluded:
−w
σ
C′+uC′−vC′′ =−kCS
−
σ
S′=−kpCS and then vC′′ + (w
σ
−u)C′−
σ
pS′=0
σ
S′=kpCS
We integrate in an interval [−z,z],z>0:
z
−z
[vC′′+(w
σ
−u)C′−
σ
pS′]dz =0⇒v[C′]z
−z+(w
σ
−u)[C]z
−z−
σ
p[S]z
−z=0
Taking into account the boundary conditions,
Cadmits 2 horizontal asymptotes: y=cdin
−∞and y=0in +∞we find by passing to the
limit z→+∞:
−(w
σ
−u)cd−
σ
ps0=0
In the case of large kand in the context of (14),
C∞(z) = cd(1−eu−w
σ
vz) (19)
is solution of (17)
For large kand within the context of (14); the
solution (C∞,S∞) verify: S∞=0i f C∞>0
S∞=s0i f C∞=0
Such :
S∞(x−
σ
t) = 0si x <
σ
tthe wave has exceeded x
S∞(x−
σ
t) = s0si x ≥
σ
tthe wave has not yet passed x
So we can consider S∞constant then
vC′′
∞+ (w
σ
−u)C′
∞=0on ]−∞,0] (17′)
C∞=0on [0,+∞[ (17′′).
And like u−w
σ
=u−wu
w+s0
pcd
=u(1−1
1+s0
wpcd
)>
0, the equation (17′)admits as solution: z→
α
+
β
eu−w
σ
vz. And applying the conditions:
C∞(0) = 0
C∞(z)
z→−∞
=cd⇒
α
+
β
=0
α
=cd⇒
β
=−cd
α
=cd
From where:
C∞(z) = cd(1−eu−w
σ
vz)z∈]−∞,0]
C∞(z) = 0z∈[0,+∞[
with function cd(t) = 1and values:
w u v s0 cd Nx NT p
0.9 0.1 0.003 2 1 100 5 3
4.2.1 Interpretation
5. Travelling waves
5.1 Simulation
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Initially, the analysis relied on a mathematical
model proposed in a scientificité modeling text in
the French external agregation competition (pub-
lic 2008), that calculates antibody and antigen
concentrations when reaction speed is moderate.
Despite this, the inherent instability of the sys-
tem presented a significant challenge that had to
be carefully considered.
We proposed modifications in our study that
transformed an unstable system into one that is
stable and consistent under certain conditions.
This paper provides detailed proofs of the stability
and consistency of the newly devised system, sup-
ported by both theoretical and numerical analyses.
As a robust method of evaluating the effectiveness
and reliability of the modified scheme, numerical
tests were included, particularly after augmenting
the reaction factor.
[1] R.K. Banerjee, I. Dilber, W.W. van Osdol, C.
Sung, P.M. Numerical simulation of antibody
penetration in a solid tumor nodule using fi-
nite element. ASME, Bioeng., BED-Vol. 39,
(1998) 117–118..
[2] K. Banerjee, W. van Osdol, M. Bungay, Cyn-
thia Sung , L. Dedrick Finite element model
of antibody penetration in a prevascular tu-
mor nodule embedded in normal tissue. Jour-
nal of Controlled Release, 74 (2001) 193–202.
6. Conclusion
References
Contribution of Individual Authors to the
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Policy)
The author contributed in the present research, at all
stages from the formulation of the problem to the
final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The author has no conflict of interest to declare that
is relevant to the content of this article.
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