Applied Mathematics, 2011, 2, 1417-1423
doi:10.4236/am.2011.211200 Published Online November 2011 (http://www.SciRP.org/journal/am)
Effect of Deformation of Red Cell on Nutritional Transport
in Capillary-Tissue Exchange System
Rekha Bali1, Swati Mishra1, Mamta Mishra2*
1Department of Mathematics, Harcourt ButlerTechnological Institute, Kanpur, India
2Department of Mathematics, School of Applied Sciences, NSIT, New Delhi, India
Received October 27, 2010; revised August 9, 2011; accepted August 16, 2011
Abstract
A mathematical analysis of a model for nutritional exchange in a capillary-tissue exchange system is pre-
sented in this paper. This model consists of a single file flow of red blood cell in capillary when diameter of
red blood cell is greater than tube diameter. In this case, the cell must be deformed. Due to concentration
gradients, the dissolved nutrient in substrate diffuses into surrounding tissue. Introducing approximations of
the lubrication theory, squeezing flow of plasma in between the gap between cell and capillary wall have
been solved with the help of approximate mathematical techniques. The computational results for concentra-
tion of dissolved nutrients, diffusive flux and normal component of velocity have been presented and dis-
cussed through graphs. We have also shown the effect of deformation parameter and permeability on these
results.
Keywords: Capillary-Tissue-Exchange, Red Blood Cell, Nutrients, Diffusion, Diffusive Flux
1. Introduction
A major difficulty in studying the microcirculation is the
small dimensions of the blood vessels. Experimental data
on Pressure, velocity, flow, shear stress, mass transfer,
etc. are difficult to obtain in vivo. Hence, a model in vi-
tro experiment that obeys geometric and dynamic simi-
larity has been very useful. Model experiments, however,
are sometimes impractical, tedious or too difficult to
carry out. As such, mathematical modeling is an attrac-
tive alternative. The most physiologically important fun-
ction of the circulation of blood through capillaries is to
supply nutrients to every living cell of the organism and
also to remove various waste products from every cell.
This function is fulfilled by the transport of the various
components of blood across a capillary wall into the sur-
rounding tissue. Nutrients, dissolved in plasma, enter the
tissue from capillary wall. The material is transported by
convection and diffusion in the capillary, whereas in the
tissue the material is transported through diffusion only
as the convection velocity in the tissue is very small.
As blood flows through the capillary, the dissolved
materials exchange between plasma and surrounding
tissue. These materials (glucose, albumin and lactoge) pre-
sent in dissolved state. Renkin [1] has considered that the
solute, in plasma enters at a constant rate into the capil-
lary from the arterial end, escapes by diffusion as the
plasma in the gap between cell and wall. He has calcu-
lated the reduced concentration of the solute at the ve-
nous end and suggested that the transport of the sub-
stances across the wall is affected by the plasma flow at
the interface of capillary and tissue as well as the capil-
lary surface area, permeability and partition coefficient.
Some authors [2-4] proposed various representative
models for blood in narrow capillaries. Two, three region
flow models [5,6] have also been developed. Tandon et
al. [7] have developed a model consisting of the viscous
fluid representation which is identically same as the sus-
pending medium of the blood. Other models have also
been proposed to discuss the nutrition transport in capil-
laries, but no work has been undertaken except Tandon
et al. [8] in very narrow capillary i.e. when capillary is of
radius less than diameter of red cell including the study
of mass transfer which also constitute a very important
aspect to analyze nutritional transport from plasma to the
capillary wall [9]. Only Flow problems in very narrow
capillary have been discussed till now by some authors
[10-15]. While the blood flow in microvessels constitute
attractive problems for mathematical analysis. As capil-
lary size decreases, most of the cells are seemed to move

R. BALI ET AL. 1418
like piston and some time quite a few of them move to-
gether, like stacked coins. In smaller capillaries, the cells
enter edge wise and deform into the bullet shape with a
convex leading edge. As the capillary size further de-
creases the sheared core flow changes to the axial train
configuration. The red blood cells then move in a single
file surrounded by thin plasmatic layer near the capillary
wall when blood flows through the capillary whose di-
ameter is less than that of a red cell, For such cases
Lighthill assumed undeformed cell shape near the wall to
be parabolic and cell deformation is depend on local
pressure. Barnard et al. has assumed the cell as flexible
circular sheet deformed into hollow thimble shape with
isotropic tension acting on the membrane. Zarda et al.
[12] considered red blood cell with axisymmetric shape
for analyzing the flows in capillary at low velocity. Lin
et al. [13] has done same work as Bernard but the cell
was represented as a solid bullet like shape with isotropic
tension acting in the cell membrane. A theoretical model
is used to investigate the effect of flow velocity on mo-
tion and axisymmetric deformation of red blood cell in a
capillary with an endothelial surface layer [15,16] A
theoretical model is also proposed by Secomb et al. [17]
to analyze the effects of the glycocalyx layer on hema-
tocrit and resistance to blood flow in capillaries. They
have considered the glycocalyx as a porous layer that
resist penetration by red blood cells.
In all the above mentioned work, the problem of nutri-
tional transport in very narrow capillaries has not been
discussed. Therefore, in this paper our aim is to study the
nutritional transport through the plasma, in between the
cell and capillary wall, into the tissue. Lubrication theory
is used to describe the squeezing flow of plasma in be-
tween the cell and tissue wall. A detailed analysis of nu-
tritional transport has been discussed in this paper. The
overall system has been modeled as two region flow and
diffusion models: squeezing flow of purely viscous fluid
in between cell and capillary wall and Darcy’s flow of
filtered plasma at the capillary tissue barrier into the po-
rous tissue.
2. Formulation of the Problem
We have interoduced two dimensional cartesian geome-
try (Figure 1). The red cell is modeled as axisymmetric
containing an incompressible fluid. Single file flow of
red blood cell is considered and cell to cell interactions
are neglected. The fluid film thickness of the plasma
between the cell and the tissue wall is represented by
h


2
04
x
ha 


  (1)
where is the pressure in fluid film region, is the
focal length of the initially assumed shape of parabola
Pa
Figure 1. Diagram of single red blood cell in capillary sur-
rounded by tissue.
and


0






 represent the further deformation
due to increased pressure in wedge formed in between
the parabola and the capillary, U0 is the velocity of cell,
0 is the velocity of the fluid.
 and  are parameters
representing coefficients of small deformation in cell and
wall,
V
0


is the reference pressure.
Flow region is divided into two Sub-regions:
1) Fluid Film region;
2) Porous tissue region.
2.1. Governing Equations
2.1.1. Fluid Film Region
Using the following assumptions
1) Thickness of the fluid layer between the cell and the
wall is assumed to be sufficiently small so that stoke’s
equation can be reduced to the Reynold’s equation using
lubrication theory;
2) The suspending fluid is assumed to be in com-
pressible and Newtonian.
The governing equation of motion for the flow of
plasma in between the cell and capillary tissue wall are
given by
2
2
u
xy

  

  (2)
0
y
  

 (3)
but due to small leakage in to the porous wall we retain
the continuity equation as:
0
uv
xy




 (4)
where u

and v

are the velocity component along
axial and transverse directions and

is the viscosity of
plasma in the capillary.
2.1.2. Porous Tissue Region
Following Darcy’s law the axial and normal components

R. BALI ET AL.1419
of velocity in tissue region are given by
,
KK
uv
x
y 

 

 
 



(5)
where u and v are the axial and transverse veloci-
ties of the fluid in the porous matrix, K is the permeabil-
ity of tissue.
Using these velocities in the equation of continuity, we
get the Laplace’s Equation for the pressure distribution
in the porous tissue region as given below:
22
22
0
xy

 



 . (6)
2.2. Nutritional Transport
2.2.1. Fluid Film Region
The concentration of the dissolved nutrients has been
assumed to be uniform due to the mixing action of vortex
ring type plasma flow in between the two cells and for
very small wedge in between cell and the wall.
2.2.2. Porous Tissue Region
Under the admissible assumptions, the approximate dif-
fusion equation in porous matrix is given by:
2
2
dd
dd
cc
vD
yy

m

 
 (7)
where the concentration of dissolved nutrients in the
tissue region, Dis the diffusion coefficient and m
c


is

the rate of production or consumption of the nutrient
with in the tissue.
2.3. Boundary and Matching Conditions
0atuU yh
 
at 0
u
uy
y



 
 


0atx
x
  


0
0atx
x

 


0
0atx

 
0at x
   

,0
x
 
0at 0cc y



d0at
d
cyH
y



2.4. Non-Dimensional Scheme
2
0 0
;; ;
x
yu
xy u
H
V V 



 


;
00
02
0 0
;;Re
VH v
vV
V






 ;
0
UH
eD 

;
00
mH
MVc


where H is the thickness of porous matrix, 0 is the
uniform concentration of the nutrient in the capillary,
c

is the Partition coefficient.
2.5. Governing Equation in Dimensionless Form
Using the non-dimensional scheme, the Equations (2) to
(7) are transformed as given below:
2
0
1hx


 




 (8)
2
2
Re u
x
x



 (9)
0
Hu v
xy



 (10)
Re Re
;
KK
uv
H
xHy


 


; (11)
22 2
22 2
0
Hu v
xy


  (12)
2
2
dd
dd
cc
v
yyM

. (13)
2.6. Boundary and Matching Conditions in
Dimensionless Form
0
0
at
U
uy
Vh


at 0
u
uy
y





0atvyh



0at0x
xx
 



0at x

 
0atx






,0at 0xx y



R. BALI ET AL. 1420
at 0cy


d0at
d
cy
y1
where
H





2
00
V
H





2
4aH
 .
2.7. Solution of the Problem
2.7.1. Velocity Distribution in Capillary Region
We have solved the equation of motion using the bound-
ary conditions 0
0
U
uV  at and yhu uy 



at
0y. we get the solution for axial velocity as given
below:





2
20
0
Re hy Uy uy
xh Vh




 

 

 


. (14)
2.7.2. Pressure Distribution in Capillary Region
Considering velocity distribution in the equation of con-
tinuity, the pressure distribution in capillary region is
obtained as
  
056
2
3K
PFxF
H 
 x (15)
where






 
513
2
0
cos cos
2sinh nn
n
nn
FxF F
x
EnH y












 



26
4
62
n0
1
1
2sinh 12 60











n
n
x
x
FxFEnH y




2.7.3. Pressure Distribution in Tissue Region
Solving Laplace Equation (6) using method of separation
of variables and boundary and matching condition the
expression for pressure distribution in tissue region is
obtained as:



n0
cosh 1coscos 


 
nn
Enyx x


n
(16)
where

21
2l




n
n
  
04
4
2cos hH0.5 

 
nn
I
En Fn
132
361
4 4





 


F
2432
36
44
 



 


F
332
31
4 4





 


F
  


1
413223
22
2
4
132 4
2
I 3II
Icos 1
12 60















 
 





n
nn
n
K
FnF FF
H
FFF I
1
1sin21
22










n
In


n
n
235
32
1
1( 1)1( 1)
2
12 ()
31
4 4

 


  





 
 




 



 


n
Inn
n
 




 
nn
2
33
57
11
11
60 2
61 12112
n
nn
Inn
nnn


7


















 
 




4
11










n
In.
2.7.4. Concentration Di st ri bution
Solving Equation (13) we have obtained the expression
for concentration distribution of the nutrients in tissue
region as given below.
P
2e
P


 
n
Vey
n n
M
cy
v ve
M
(17)
where


2
2
0
21
sinhsin 2 


 

 


 nnn
n
n
KRe
vE
H.
2.7.5. Diffusive Flux on the Capillary Tissue Interface
Diffusive Flux on the wall is given as
d
d

F
c
DD
y (18)

R. BALI ET AL.1421
and expression is given as below

P1
2P
F
n
DD e
ve
 
.
n
Vey
n
MM
v




(19)
3. Results and Discussions
nd permeability of the
elocity, deffusive flux
Parameter Description Value
Effect of deformation of the cell a
issue on normal component of vt
on the interface and concentration distribution in the tis-
sue has been shown through the graphs. The following
values of parameter are used in the model (Table 1).
Figure 2(a) represents the variation of normal com-
ponent of velocity with axial distance for different values
of deformation parameter ε. Normal component of veloc-
ity decreases with axial distance increases. As deforma-
tion parameter increases the normal component of veloc-
ity also increases. These deformations may be due to
viscous stress or velocity of the fluid or due to the in-
creased pressure developed in the gap.
Figure 2(b) shows the variation of diffusive flux with
penetration depth for different value of permeability.
Diffusive flux on the wall decreases as the permeability
of the tissue decreases. As deformation increases, the
normal component of velocity increases. Therefore dif-
fusive flux also increases. The increase in diffusive flux
is beneficial for the health of the tissue due to fact that
due to increased diffusive flux nutrition supply to the
deeper region also increase, therefore they get proper
nutrition.
Figure 3(a) represents the normal velocity at the wall
with axial distance for different values of permeability.
As permeability increases normal velocity also increases.
This is due to the fact that as permeability increases more
fluid enters into tissue region and cells of the tissue get
more nutrition.
Figure 3(b) represents the variation of diffusive flux
in penetration depth for different values of deformation
parameter. As deformation parameter decreases, the flux
Table 1. List of parameter used in the model.
K Prmeability 10–12 m2
ε
(a)
(b)
Figure 2. (a) Variation of Normal component of velocit
with axial distance for diffe values of deformation pa-
s in penetra-
y
rent
rameter; (b) Variation of Diffusive flux with penetration
depth for different values of permeability.
of dissolved material is hindered so the diffusive flux
ith in the tissue decreases. It also decreasew
tion depth.
Figures 4(a) and (b) describe the concentration dis-
tribution in tissue region with penetration depth for dif-
ferent values of permeability and deformation parameters
respectively. Concentration decreases towards the no
flux boundary of the tissue. When the permeability of the
tissue increases the concentration increases. In this case,
the cells of the tissue of the deeper region also get nutri-
tion. Due to increase in permeability of the tissue more
fluid enters into tissue so that more dissolved nutrients
enter into the tissue along with the fluid. As deformation
parameter increases concentration in the tissue region
also increases. This is due to the fact that due to defor-
mation, normal velocity and diffusive flux increase
therefore more solute diffuses into the tissue region
along with the fluid. The results are similar to those ob-
tained by Tandon et al. [17,18].
Defo eter 0.00015
Radial f tube 0.b
Ra ll
ρ 1.0 3
Va 0.01 dyc/cm2
rmation Param5,0.01,0.
γ compliance o001 μm/m
β dial compliance of the ce0.06 μm/mb
 Length of the capillary 30 μm
σ Slip Parameter 0.05
Density of blood 5 gm/cm
μ iscosity of plasmne se
Re Reynold Number 0.25

R. BALI ET AL. 1422
(a)
(b)
Figure 3. (a) Variation of Normal component of velocity
with axial distance for different values of permeability; (b
Variation of diffusive flux w penetration depth for dif-
this paper, we have developed a model for nutritional
hrough microvessels. Our results for
e deformation of the cell are useful for continuous flow
he authors are greatly indebted to Prof P. N. Tandon
rtment of Mathematics, of
.B.T.I. Kanpur) for their invaluable suggestions and
)
ith
ferent values deformation parameter.
4. Conclusions
In
transport to tissue t
th
of blood in capillary, so that the tissue gets proper nutri-
ents. The studies introduce the geometry of the deformed
cell in the vicinity of the porous tube wall is most im-
portant in determining the pressure drop whereas as the
shape of the rest of the cell and particle spacing had mi-
nor influence on the results.
5. Acknowledgements
T
(Retd. Prof. and Head, Depa
H
(a)
(b)
Figure 4. (a) Variation of concentration with penetration
depth for different values of deformation parameter; (b)
Variation of concentration penetration depth for dif-
] E. M. Renkin, “Transport of Potassium—42 from Blood
ated Mammalian Skeletal Muscles,” Ame-
Physiology, Vol. 197, 1959, p. 1205.
59,
ress, London.
with
ferent values of permeability.
comments. UGC major research project F No. 36-320/2008
(SR) is greatfully acknowledged.
6. References
[1
to Tissue in Isol
rican Journal of
[2] N. Casson, “A Flow Equation for Pigment-Oil Suspen-
sions of the Printing Ink Type,” In: C. C. Mill, Ed., Rhe-
ology of Disperse Systems, Pergamon Press, Oxford, 19
pp 84-104.
[3] S. E. Charm and G. S. Kurland, “Blood Rheology and
Cardiovascular,” In: D. H. Bergel, Ed., Fluid Dynamics,
Academic P
[4] E. C. Eringen, “Theory of Micropolar Fluids,” Journal of

R. BALI ET AL. 1423
and H. M. Jaffrin, “A Three
Mathematical Fluid Mechanics, Vol. 16, 1966, pp. 1-18.
[5] B. B. Gupta, K. K. Nigam
Layer Semi Empirical Model for Flow of Blood and
Other Particulate Suspension through Narrow Tubes,”
Journal of Biomechanical Engineering, Vol. 104, No. 2,
1982, pp. 129-135. doi:10.1115/1.3138326
[6] H. R. Haynes, “Physical Basis of the Dependence of
Blood Viscosity on Tube Radius,” American Journal of
Physiology, Vol. 198, No. 6, 1966, pp. 1193-2000.
1.
,
[7] P. N. Tandon and R. Agarwal, “A Study of Nutritional
Transport in Synovial Joints,” Computers & Mathematics
with Applications, Vol. 17, No. 7, 1989, pp. 1131-114
[8] P. N. Tandon, M. Mishra and A. Chaurasia, “A Model for
Nutritional Transport in Capillary-Tissue Exchange Sys-
tem,” International Journal of Bio-Medical Computing
Vol. 37, No. 1, 1994, pp. 19-28.
doi:10.1016/0020-7101(94)90068-X
[9] H. S. Lew and Y. C. Fung, “The Motion of the Plasma
between Red Cells in the Bolus Flow,”
6, 1969, pp. 109-119
Biorheology, Vol.
o. 1, 1968, pp. 113-143.
[10] M. J. Lighthill, “Pressure Forcing of Tightly Fitting
Pellets along Fluid Filled Elastic Tubes,” Journal of Fluid
Mechanics, Vol. 34, N
doi:10.1017/S0022112068001795
[11] A. C. Barnard, L. Lopez and J. D. Hellums, “Basic The-
ory of Blood Flow in Capillaries,” Microvascu
search, Vol. 1, No. 1, 1968, pp. 23-
lar
24.
Re-
doi:10.1016/0026-2862(68)90004-6
[12] P. R. Zarda, S. Chien and R. Skalak, “Interaction of Vis-
cous Incompressible Fluid with an Elastic Body,” T. Be-
lystschko and T. L. Geers, Eds., Computational Methods
for Fluid-Solid Interaction Problems, American Society
of Mechanical Engineers, New York, 1977, pp. 65-82.
[13] K. L. Lin, L. Lopez and J. D. Hellums, “Blood Flow in
Capillaries,” Microvascular Research, Vol. 5, No. 1, 1973,
pp. 7-19. doi:10.1016/S0026-2862(73)80003-2
[14] T. W. Secomb and R. Skalak, “A Two Dimensional
Model for the Capillary Flow of an Axisymmetric Cell,”
Microvascular Research, Vol. 24, No. 2, 1982, pp 194-
203. doi:10.1016/0026-2862(82)90056-5
[15] T. W. Secomb, R. Skalak, N. Ozakaya and J. F. Gross,
“Flow of Axisymmetric Red Blood Cells in Narrow Cap-
illaries,” Journal of Fluid Mechanics, Vol. 163, 1986, pp.
405-423. doi:10.1017/S0022112086002355
[16] T. W. Secomb, R. Hsu and A. R. Pries, “Motion of Red
. Pries, “A Model for Red
,
Blood Cell in Acapillary with an Endothelial Surface
Layer: Effect of Flow Velocity,” American Journal of
Physiology: Heart and Circulatory Physiology, Vol. 28,
No. 2, 2001, pp. H629-H636.
[17] T. W. Secomb, R. Hsu and A. R
Blood Cell Motion in Glycocalyx-Lined Capillaries,”
American Journal of Physiology: Heart and Circulatory
Physiology, Vol. 274, No. 3, 1998, pp. H1016-H1022.
[18] P. N. Tandon, P. Nirmala, M. Tiwari and U. V. Rana
“Analysis of Nutritional Transport through a Capillary—
Normal and Stenosed,” Computers & Mathematics with
Applications, Vol. 22, No. 12, 1998, pp. 3-13.
doi:10.1016/0898-1221(91)90142-Q