Post on 06-May-2023
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Journal of Advanced & Applied Sciences (JAAS) Volume 03, Issue 02, Pages 46-50, 2015
ISSN: 2289-6260
Analytical expression for compartment mathematical model of liver
V.Ananthaswamy1,*, S.Uma Maheswari2
1Department of Mathematics, The Madura College, Madurai, Tamil Nadu, India. 2M.Phil., Mathematics, The Madura College, Madurai, Tamil Nadu, India.
* Corresponding author. Tel.:8903550705;
E-mail address: ananthu9777@rediffmail.com
A b s t r a c t
Keywords:
Pharmacokinetics;
Compartment model; liver;
System of ordinary
differential equations;
Homotopy perturbation
method. .
In this research article, we discuss the mathematical model of pharmacokinetics is a growing
area in drug development. Pharmacokinetics concerns with the distribution of drugs,
chemicals tracers as a fluid among the different compartment of human body. The system of
ordinary differential equations is solved by using the Homotopy perturbation method (HPM).
Our analytical results are compared with the previous work and a satisfactory agreement is
noted. This method can be easily extended to solve the other non-liner initial and boundary
value problems in physical, chemical in engineering and sciences.
Accepted:18 April2015 © Academic Research Online Publisher. All rights reserved.
1. Introduction
A mathematical model is a description of a system
using mathematical concepts and language. The
process of developing a mathematical model is
termed mathematical modeling .A model may help
to explain a system and to study the effects of
different components and to make predictions
about behavior. Mathematical models can take
many forms, including but not limited to dynamical
system, statistical models, differential equations, or
game theoretic models.
The liver is a vital organ of vertibrates and
some other animals. In the human it is located in
the upper right quadrant of the abdomen, below the
diaphragam. The liver is a gland and plays a major
role in metabolism.
Pharmacokinetics concerns with the
distribution of drugs, chemicals tracers by a fluid
among the different compartment of human body
[2]. The present research of this three compartment
model related to the liver in the human body [3-5].
The liver carries a substance thyroxin, when it is
injected into the blood stream. Thyroxin is
transformed to iodine by the liver, iodine is
immersed into the bile [6]. Though, the
transformation or the immersion would occur
simultaneously. Thyroxin would be re circulated
when they are entered into the blood stream. The
three compartment mathematical models are given
below:
Compartment 1 represents the blood vessels,
Compartment 2 represents the liver, Compartment
3 represents the bile are shown in the following
figure;
V. Ananthaswamy et al. / Journal of Advanced & Applied Sciences (JAAS) 3 (2): 46-50, 2015
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K12
K23 K21 K23
Fig.1: Mathematical model for liver
Where tX 1 is the quantity of thyroxin in blood
vessel at the instant time ‘ t ’, tX 2 is the quantity
of iodine in the liver, tX 3 is the quantity of
iodine absorbed in to the bile, 12K is the rate of
conversion of thyroxin into iodine, 21K is the rate
of quantity of unabsorbed thyroxin sent out for
recycling from compartment 2 , 23K is the rate of
absorption of iodine from compartment 2 into bile
compartment3. 2010 , XX and 30X are initial
values of 21 , XX and 3X respectively and the
rate of constants 2112, KK and 23K are positive
values.
It is assumed that the rate tx1 which
thyroxin is converted to iodine as it is transferred
from compartment 1 to compartment 2 is
proportional to concentration tX 1 of thyroxin in
the compartment 1.
2. Model block diagrams and model equations:
= = +
From above diagram we get the equation,
1012211121 0, XXXKXK
dt
dX (1)
= - -
-
From above diagram we get the equation,
Compartment 3
(X3)
Bile
Compartment 2
(X2)
Liver
Compartment 1
(X1)
Blood vessels
The rate of change of
thyroxin in compartment1
x1 (t)
The rate of conversion of
thyroxin into Iodine from
compartment 1 to
compartment2
(K12 X1)
(K12X1
The rate of reentry of
uncovered thyroxin from
compartment2 to
compartment 1
(K21 X2)
The rate of
absorption of
Iodine
compartment2
X2(t)
The rate of entry of
thyroxin into
compartment2 from
compartment1
(K12 X1)
The rate of reentry
from compartment2 to
compartment 1 for
recycling
(K21 X2)
The rate of transfer
of iodine from
compartment2 to
compartment3
(K23 X2)
V. Ananthaswamy et al. / Journal of Advanced & Applied Sciences (JAAS) 3 (2): 46-50, 2015
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202223211212 0, XXXKKXK
dt
dX (2)
=
From above diagram we get the following equation,
.3032233 0, XXXK
dt
dX
(3)
When there are no diffusion terms
Then the eqn.(1) to (3) becomes,
02211121 XKXK
dt
dX (4)
0223211122 XKKXK
dt
dX (5)
02233 XK
dt
dX (6)
Where 1X represent the quantity of thyroxin in the
blood vessel at the instant time t , 2X represent the
quantity of iodine in the liver, 3X represent the
quantity of iodine absorbed in to the bile, 12K
represent the rate of conversion of thyroxin into
iodine, 21K represent the rate of the quantity of
unabsorbed thyroxin sent out for recycling from
compartment 2 to compartment 1, 23K represent
the rate of absorption of iodine from compartment2
into bile compartment3.
Let the initial conditions be,
00 ii XX ,where ,....3,2,1i (7)
4. Approximate analytical solution of the initial
value problem using the Homotopy perturbation
method (HPM)
Recently, many authors have applied the
Homotopy perturbation method (HPM) to solve the
non-linear problem in physics and engineering
sciences [7-14]. This method is also used to solve
some of the non-linear problem in physical
sciences [15-17]. This method is a combination of
Homotopy in topology and classic perturbation
techniques. Ji-Huan [11-13] He used to solve the
Lighthill equation, the Duffing equation and the
Blasius equation [9-12]. The HPM is unique in its
applicability, accuracy and efficiency. The HPM
uses the imbedding parameter p as a small
parameter, and only a few iterations are needed to
search for an asymptotic solution. Using this
method [14-16], we can obtain solution to the eqns.
(1)-(3)asfollows:
tKKtK
tK
ee
KKK
XK
eX
tX
232112
12
122321
2021
10
1 (8)
tKKtK
tKK
ee
KKK
XK
eX
tX
232112
2321
122321
1012
20
2 (9)
tXtXXXXtX 213020103
(10)
The rate of change of material in
compartment 3 to compartment 2
X3(t)
The rate of transfer
compartment2 to
compartment3
(K23 X2)
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4. Results and discussion
Fig .2: The variation of thyroxin 1X , iodine 2X , bile 3X verses the dimensionless time t . The variations 321 ,, XXX
are computed by using eqns. (8-10) in some fixed coefficient values 232112 ,, KKK , when .1000t
Fig .3: The variation of thyroxin 1X , iodine 2X , bile 3X verses the dimensionless time t . The variations 321 ,, XXX
are computed by using eqns. (8-10) in some fixed coefficient values 232112 ,, KKK , when .1000t
Fig .4: The variation of thyroxin 1X , iodine 2X , bile 3X verses the dimensionless time t . The variations 321 ,, XXX
are computed by using eqns. (8-10) in some fixed coefficient values 232112 ,, KKK , when .1000t
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Fig .5: The variation of thyroxin 1X , iodine 2X , bile 3X verses the dimensionless time t . The variations 321 ,, XXX
are computed by using eqns. (8-10) in some fixed coefficient values 232112 ,, KKK , when .500 t
Fig .6: The variation of thyroxin 1X , iodine 2X , bile 3X verses the dimensionless time t . The variations 321 ,, XXX
are computed by using eqns. (8-10) in some fixed coefficient values 232112 ,, KKK , when .500 t
Fig .7: The variation of thyroxin 1X , iodine 2X , bile 3X verses the dimensionless time t . The variations 321 ,, XXX
are computed by using eqns. (8-10) in some fixed coefficient values 232112 ,, KKK , when .500 t
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Fig .8: The variation of thyroxin 1X , iodine 2X , bile 3X verses the dimensionless time t . The variations 321 ,, XXX
are computed by using eqns. (8-10) in some fixed coefficient values 232112 ,, KKK , when .500 t
Fig .9: The variation of thyroxin 1X verses the dimensionless time t . The variation of 1X are computed by using eqns. (8-
10) in some fixed coefficient values of 2112, KK and different values of 23K , when .1000t
Fig. 10: The variation of iodine 2X verses the dimensionless time t . The variation of 2X are computed by using eqns.
(8-10) in some fixed coefficient values of 2112, KK and different values of 23K , when .1000t
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Fig. 11: The variation of bile 3X verses the dimensionless time t . The variations 3X are computed by using eqns. (8-10)
in some fixed coefficient values 2112, KK and different values of 23K , when .1000t
Figs.(2)-(8) represents the variation of thyroxin
1X , iodine 2X , bile 3X verses the
dimensionless time t . The variations of 321 ,, XXX
are computed by using the eqns. (8-10) in some
fixed values of the dimensionless parameters
232112 and, KKK . From the Figs.(2)-(4), it is
clear that when t increases the variables 1X and
2X decreases and the variables 3X increases in
some fixed values of the dimensionless parameters
in the range of .1000t Fig.(5)-(8) represents
thyroxin 1X , iodine 2X , bile 3X verses the
dimensionless time t respectively. From Figs.(5)-
(8), it is observed that, when the time increases the
variables 1X and 2X decreases and the variables
3X increases in some fixed values of the
dimensionless parameters in the range of
.500t Fig.(9) represents thyroxin 1X versus
the dimensionless time t . From Fig.(9) it is noted
that when time increases the variable 1X decreases
in some fixed values of the dimensionless
parameters. Fig.(10) denotes the iodine 2X
versus the dimensionless time .t From Fig.(10) it is
clear that when the time increases the variable 2X
decreases in some fixed values of the
dimensionless parameters. Fig.(11) indicates the
bile 3X versus the dimensionless time t . From
Fig.(11), we observe that when the time increases,
the variable 3X increases in some fixed values of
the dimensionless parameters.
5. Conclusion
The system of linear ordinary differential
equations for a three compartment mathematical
model of liver has been solved analytically. The
variation of thyroxin tX1 , iodine tX 2 , bile
tX 3 where obtained by using the Homotopy
perturbation method (HPM). Our analytical results
are compared with previous work and found to be
in good agreement. The primary result of this work
is simple and approximate expressions for all
values of the dimensionless parameters.
Acknowledgement
The authors are thankful to Shri. S.
Natanagopal, Secretary, The Madura College
Board, Dr. R. Murali, The Principal and Mr. S.
Muthukumar, Head of the Department of
Mathematics, The Madura College (Autonomous),
Madurai, Tamil Nadu, India for their constant
encouragement.
V. Ananthaswamy et al. / Journal of Advanced & Applied Sciences (JAAS) 3 (2): 46-50, 2015
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Appendix A
Basic concept of the Homotopy perturbation
method (HPM) [10-12]
To explain this method, let us consider the
following function:
r ,0)()( rfuDo (A.1)
with the boundary conditions of
r ,0) ,(
n
uuBo (A.2)
where oD is a general differential operator, oB is
a boundary operator, )(rf is a known analytical
function and is the boundary of the domain .
In general, the operator oD can be divided into a
linear part L and a non-linear part N . The eqn.
(A.1) can therefore be written as
0)()()( rfuNuL (A.3)
By the Homotopy technique, we construct a
Homotopy ]1,0[:),( prv that satisfies
.0)]()([
)]()()[1(),(
0
rfvDp
uLvLppvH
o
(A.4)
.0)]()([)(
)()(),(
0
0
rfvNpupL
uLvLpvH (A.5)
where p[0, 1] is an embedding parameter, and
0u is an initial approximation of eqn. (A.1) that
satisfies the boundary conditions. From eqns. (A.4)
and (A.5), we have
0)()()0,( 0 uLvLvH (A.6)
0)()()1,( rfvDvH o (A.7)
When p=0, the eqns. (A.4) and (A.5) become linear
equations. When p =1, they become non-linear
equations. The process of changing p from zero to
unity is that of 0)()( 0 uLvL to
0)()( rfvDo . We first use the embedding
parameter p as a “small parameter” and assume
that the solutions of eqns. (A.4) and (A.5) can be
written as a power series in p :
...22
10 vppvvv (A.8)
Setting 1p results in the approximate solution
of the eqn. (A.1):
...lim 2101
vvvvup
(A.9)
This is the basic idea of the HPM.
Appendix B
Solution of the boundary value problem eqns.
(1)-(3) using the Homotopy perturbation method
[7-17]
In this appendix we indicate the equations (4) – (7)
are derived in this paper. To find the solution of the
equations (1) – (3), we construct a Homotopy as
follows:
0
1
2211121
1121
XKXKdt
dXp
XKdt
dXp
(B.1)
0
1
223211122
223212
XKKXKdt
dXp
XKKdt
dXp
(B.2)
01 22333
XK
dt
dXp
dt
dXp (B.3)
The analytical solutions of (B.1),(B.2) and (B.3) is,
........210 1
2111 XppXXX (B.4)
........210
22
222 XppXXX (B.5)
........210
32
333 XppXXX (B.6)
Substituting the eqns. (B.4), (B.5) and (B.6) in
(B.1), (B.2) and (B.3) respectively we get,
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0
.....
.....
.....
.....
.....
1
2
10
2
10
210
2
10
210
22
22
21
12
11
12
12
11
12
11
12
12
11
Xp
pXXK
Xp
pXXK
dt
XppXXd
p
Xp
pXXK
dt
XppXXd
p
(B.7)
0
.....
.....
)(
.....
.....
)
.....
1
2
10
2
10
2
10
2
10
210
12
11
12
22
22
2321
22
22
22
22
2321
22
22
Xp
pXXK
Xp
pXX
KK
dt
Xp
pXXd
p
Xp
pXX
KK
dt
XppXXd
p
(B.8)
0
.....
.....
.....1
2
102
210
210
22
2
23
32
33
32
33
Xp
pXX
K
dt
XppXXd
p
dt
XppXXdp
(B.9)
Comparing the coefficients like powers of p in
(B.7), (B.8) and (B.9) we get,
.0:0
0
112
10 XKdt
dXp (B.10)
.0:0
0
22321
20 XKKdt
dXp (B.11)
.0: 030 dt
dXp (B.12)
01
1
221112
11 : XKXKdt
dXp (B.13)
.:01
1
11222321
21 XKXKKdt
dXp (B.14)
.:0
1
223
31 XKdt
dXp (B.15)
The initial approximations are as follows:
303202101 )0(;)0(;0 XXXXXX (B.16)
......,3,2,1;00 iX i (B.17)
Solving the eqns. (B.10) to (B.15) and using the
initial conditions (B.16) and (B.17), we obtain the
following results:
tX01 tK
eX 1210
(B.18)
tX11
.
232112
122321
2021
tKKtKee
KKK
XK
(B.19)
tX02
tKKeX 2321
20
(B.20)
tX12
tKKtKee
KKK
XK
232112
122321
1012
(B.21)
tXtXXXXtX 213020103 (B.22)
According to the HPM, we can conclude that
)(lim 111
11 10
XXtXXp
(B.23)
)(lim10 222
12 XXtXX
p
(B.24)
)(lim 333
13 10
XXtXXp
(B.25)
After putting the eqns. (B.18) and (B.19) into an
eqn. (B.23) and (B.20) and (B.21) into an eqn. eqn.
(B.24) and (B.22) into an eqn. (B.25) we obtain the
solutions in the text eqns. (8)-(10).
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Appendix: C
Nomenclature
Symbol Meaning
tX1 The quantity of thyroxin in the blood vessels at the instant time t
tX 2 The quantity of iodine in the liver.
tX 3 The quantity of iodine absorbed in to the bile.
12K The rate of conversion of thyroxin into iodine.
21K The rate of the quantity of un absorbed thyroxin sent out for recycling from
compartment 2 to compartment 1.
23K Rate of absorption of iodine from compartment 2 into bile compartment 3.
tx1 The rate of change of thyroxin into iodine from compartment 1 to compartment 2is proportional to
concentration of thyroxin in the compartment 1.
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