Oxygen transport and release of adenosine triphosphate in micro-channels
Abstract
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Terry E. Moschandreou
Western University
Department of Applied Mathematics
London, Ontario
Canada
Corresponding author: Terry E. Moschandreou, Western University, Department of Applied Mathematics,
London,Canada,tmoschan@uwo.ca
The governing nonlinear steady equations for oxygen transport in a microfluidic channel are solved analytically. The Lagrange inversion theorem is
used which admits complete integrable solutions in the channel. Considering a cell-rich and cell free region with RBCs and blood plasma, we obtain
results showing clearly that there is a significant decrease in oxygen tension in the vicinity of an oxygen permeable membrane placed on the upper
channel/tube wall and to the right side of it in the downstream field. The purpose of the membrane is to cause a rapid change in oxygen saturation as
RBC’s flow through channel/tube. To the right of the membrane downstream the greatest amount of ATP is released. The method of solution is
compared to numerical results. The analytical results obtained could prove useful for the corresponding time dependent problem in future studies.
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1.Introduction
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ATP is made up of Adenine, Ribose and Phosphate groups. It is the molecule of most organisms to transfer energy. Adenine is one of the bases found
in DNA and RNA. Ribose is a sugar molecule found in RNA. There exist high energy bonds between the phosphates. Breaking the first high energy
bond in ATP releases energy and a free phosphate. This energy is used to power cellular activities. What is left with 2 phosphates and Adenosine is
called Adenine diphosphate or ADP. Energy from the breakdown of food is used to change the ADP to an ATP molecule. In cells, mitochondria
supply the major portion of ATP consumed in the cystol. ADP-ATP translocation in mitochondria is studied by Klingenberg [1]. Mitchell [2]
proposed that electron transport and ATP synthesis are related by a proton gradient across the inner mitochondrial membrane. A mechanism for
microvascular blood flow regulation that is under current investigation depends on the oxygen saturation-dependent release by red blood cells
(RBCs) of the signaling molecule ATP [3,4]. Increased RBC ATP release due to decreased O2 saturation results in a vasodilation signal that conducts
upward through the microvascular tree. This signal can originate in arterioles, capillaries or venules. Moschandreou et al. [5] have presented a
theoretical model for steady-state radial and longitudinal oxygen transport in arterioles containing plasma and red blood cells and surrounded by
living tissue. The governing equations were solved in a cell-rich and cell-free region. Using the transport equations for O2 transport in [5] together
with ATP concentration equations in [6,7], both defined in a micro-channel and then applied to representative arterioles, important information is
obtained about ATP release. 3D printed microfluidic devices have been recently used to collect ATP while simultaneously measuring the release
stimulus which is the reduced oxygen concentration [8,9]. RBC-derived ATP is measured when a luciferin luciferase mixture is applied and resulting
chemiluminescence is determined. [10,11]
To assess intraluminal resistance in arterioles, Hellums and co-workers [12,13,14] have developed a model for intraluminal oxygen transport in
arteriole-size vessels (‘large capillaries’), where the vessel is considered to be embedded in a thin film of silicone rubber in order to match an in vitro
experimental system [15, 16]. The model takes into account convection of plasma and RBCs in a cell-rich region together with a thin cell-free region
of plasma. The model contains four coupled partial differential equations describing diffusion of free O2 in the cell-free layer, in the plasma
surrounding RBCs, and in RBCs, and diffusion of hemoglobin-bound O2 in RBCs. Three of these equations are non-linear because of the slope of the
oxyhemoglobin dissociation curve. These equations have a term defined as the reaction rate, which is the rate of oxygen dissociation from
hemoglobin per unit volume of RBC. Since no hemoglobin escapes from the RBCs, the total heme concentration within the RBC remains constant.
The convective terms have plasma and RBC velocity profiles together with a radial distribution of hematocrit. In simulating the experimental results
of Boland et al. [15], transport in the silicone rubber phase is incorporated as a boundary condition at the capillary wall as described by Lemon et al.
[16].
In this paper, we utilize the simplified model of Nair et al. [12] describing the intraluminal problem of oxygen transport in an arteriole containing a
RBC-rich core with a plasma sleeve region joined to it. In this model, the four partial differential equations describing free and hemoglobin-bound
O2 transport [13] are reduced to one non-linear equation and one linear equation by: (1) assuming chemical equilibrium within the RBCs between
oxygen and oxyhemoglobin, and (2) neglecting intracellular and extracellular boundary layer resistances. Here we consider the same nonlinear partial
differential equation (PDE) used by Nair et al. [12] to describe O2 transport in the RBC-rich core (see also Hellums et al. [17]) together with a
steady-state convection-diffusion equation describing O2 transport in the cell-free plasma region.
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2.Materials and Methods
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In the present paper an analytical solution method is used to determine the oxygen tension and concentration of ATP in a microfluidic channel. The
use of the Lagrange Inversion theorem[29,30] makes it possible to integrate the governing equation associated with an RBC-rich core. The plasma
sleeve region with defined linear equation is also solved and the results are extended to plasma sleeve region. The corresponding numerical problem
is also solved.
2.1 Channel Flow and Oxygen Transport
Hellums and co-workers [12,13,14,17] have developed a model for intraluminal oxygen transport in arteriole size vessels.
We adopt this model for square channel flow where Equation (1) is used below to model oxygen transport in a micro-fluidic device.
The geometry of the present mathematical model is shown in Figure 1.
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pg. 4
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Figure 1
Geometry of mathematical model used for channel flow

K
 v p (1  H T )  v RBC H T RBC

Kp

 HbT  dSO2
1 
K RBC dPO 2

  PO 2
 2 PO 2
 
 Dp

y 2
  z
(1)
There is a core region of blood flow with RBCs and plasma and a cell-free region with only plasma flowing as shown in Figure 1. The blood plasma
velocity is v p and v RBC is the velocity of RBCs with plasma in the cell-rich region. The velocity of the RBCs is lower due to the slip between
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plasma and RBCs. [20] There is a reduction of red cell transit time through a given tube segment known as tube hematocrit H T . The discharge
hematocrit, is the hematocrit of blood entering or leaving the tube segment. Dynamic hematocrit reduction is known as the Fahraeus effect. [21 ] In
blood vessels with diameters less than 500 microns both the hematocrit decreases with decreasing vessel diameter.
The distribution of RBC`s is such that hematocrit is higher at the center of the channel and lower near the wall. [ 5] . The term
dSO2
is the slope of
dPO2
the oxyhemoglobin dissociation curve and is a nonlinear function of PO 2 . (Nair et al. [12,13,14 ]) The dissociation curve is approximated by the Hill
equation in Eq. (1a) , where N is an empirical constant and P50 is the oxygen tension that yields 50 % oxygen saturation. HbT  is the total heme
concentration which is equal to four times the hemoglobin concentration due to fact that there are four heme groups on each hemoglobin molecule.
D p is the oxygen diffusion coefficient in plasma and has units of m2 / s , and K RBC and K p are the solubilities of
O 2 in the RBCs and plasma,
respectively, and have units of M/mmHg. Differentiating the Hill equation with respect to PO2 gives Eq. (1b).
N
SO2 ( y, z ) 
PO2 ( y, z )
N
N
PO2 ( y, z )  P50
(1a)
NP50N PO 2N 1
dSO2

2
dPO 2
P50N  PO 2N

(1b)

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Generally blood flow velocity varies from zero at the wall of the channel to maximum at center. In [22 ] a quoted value of arteriole flow velocity is
in the range of 1 to 1.275 mm/s (1000 to 1275 microns/s).
Equation (2) governs oxygen transport in the cell-free plasma layer,
PO2
 2 PO2
 Dp
z
y 2
(2)
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vp
2.2 Channel Flow and ATP Model
ATP concentration given by
C ATP
is governed by the following transport equation.
C ATP
 2C ATP
vp
(1  H T )  DATP
(1  H T )  R  H T  kt C ATP (1  H T )
z
y 2
where
D ATP
is the diffusion coefficient of ATP in blood plasma with units of
(3)
m 2 / s
and
k t is a concentration rate constant with units of
s 1 .
Equations (1) and (3) are coupled partial differential equations through the nonlinear function R(y,z) [6],[24],
R( y, z)  14  SO2 ( y, z)  14
,
(4)
where,
2.7
SO2 ( y, z ) 
PO2 ( y, z )
2.7
PO2 ( y, z )  272.7
(5)
and a maximum ATP release rate of 14 M / s .[6]
Coupled equations 1 and 3 are solved using Maple 18 [18].
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2.3 Channel Inlet and Boundary Conditions for Oxygen Transport and ATP Concentration
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The inlet conditions for both oxygen tension and ATP concentration at the entrance to the channel where the oxygen porous membrane is located at
z=0 microns are given by Equation 6:
PO2 ( y, z)  P0  150 mmHg ,  L  z  0
C ATP ( y, z)  C0  0 M ,  L  z  0
(6)
The boundary conditions for both oxygen tension and ATP concentration in the channel are given by Equation 7:
PO 2
(h, z )  0
y
 L  z  0
PO 2
(0, z )  0
y
 L  z  L
Dp K p
PO2( PM )  PO2 (h, z )
PO2
(h, z )  Dm K m
y

C ATP
(h, z )  0
y
0  z  L
(7)
 L  z  L
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C ATP
(0, z )  0
y
 L  z  L
TABLE 1 Model Parameters
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Physical parameters used in oxygen transport model.
Blood plasma velocity
v p ( m / s )
1275[22]
Red Blood Cell
velocity
Slip coefficient
v RBC
1147.5 [22]
m / s
Slp
0.1 [20]
Discharge Hematocrit
HD
0.4 [21,23]
Tube Hematocrit
HT
0.28 [21,23]
O2 Diffusivity in
D p ( m 2 / s )
2750[5],[6]
Dm ( m 2 / s )
160000[6]
T( m )
1[6]
PO2(PM ) (mmHg)
10[6]
plasma
O2 Diffusivity in O2
Permeable Membrane
Plasma Layer
Thickness
O2
Permeable
Membrane
PO2

pg. 8
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start of O2 Permeable
Membrane along
channel/arteriole
Axial position for end
of O2 Permeable
Membrane along
channel/arteriole
Inlet
PO2
Microfluidic
Device/Arteriole
length
z 0 ( m )
0
L ( m )
∞
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Axial position for
P0 (mmHg)
150 [6]
L ( m )
∞
Porous Membrane
Thickness
Hill Coefficient
 ( m )
100 [6]
N
2.7[5]
O2 Solubility in
Km (
O2
Porous Membrane
M
)
mmHg
17.959[6]
M
)
mmHg
M
K RBC (
)
mmHg
1.33[5],[6]
Reaction Rate
Kt
0.1 [6]
Total Heme
Concentration
[ HbT ] ( M )
O2 Solubility in
Plasma
O2
Solubility in
RBCs
Kp(
s 1
1.47 [5],[6]
5350 [5],[6]
pg. 9
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D ATP ( m 2 / s )
475 [6]
P50 (mmHg)
27 [5],[6]
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ATP Diffusivity in
plasma
Partial Pressure at
50% Saturation
3. Analytical Methods for RBC core
3.1 Lagrange Inversion Theorem
From Eq. 1b we let ( PO 2 ) 
P50N N  [ HbT ] PO 2N 1
P
N
50
 PO 2N

2
Eq (1) can be written in compact form as:
PO 2  2 PO 2
[a  b( PO 2 )]

z
y 2
where a    
(8)
, b  
pg. 10
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and  
vp
Dp
(1  H T ) ,   v RBC  H T
1
K RBC
,  
K RBC
K p Dp
Let ( PO2 )  f ( y)  g ( z ) , and  ( fg )   1  fg  , then we obtain:
(9)

2
[a  bfg ]  ( fg )  2  ( fg )
z
y
(10)
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 1
 2 1
[a  bfg ]  ( fg )  2  ( fg )
z
y
and by applying chain rule on left side of Eq.(10 ) we obtain,
[a  bfg ]

 ( fg )   
   ( fg ) 
( fg ) z
y  y

(11)
and also applying c-r to right side of Eq.(11 ),
[a  bfg ] f  g '

 
  

   f 'g 
( fg ) y  
( fg )  
[a  bfg ] f  g '  ' ( fg )  gf ' '  ' ( fg )  g 2 f ' 2  ' ' ( fg )
[[a  bfg ] f  g ' gf ' ' ]
d
d

ln
d ( fg ) d ( fg )
g 2 f '2
 df 
dg
dg d 2 f
p
 p12 g
 2 g  k 2 g 2  
dz
dz dy
 dy 
(12)
(13)
(14)
2
(15)
pg. 11
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d 2
d
 k2
2
d ( fg )
d ( fg )
(16)
Solving Eq. (15) treating all variables as constant and k=0 and with g a function of z only, we obtain:
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
 2 qzp  qCp

 p1 e
 p W 
p



g ( z )  exp  

p








  zq  Cq 









(17)
where W is the Lambert W function, C is a constant, p  af , p1  bf 2 . Equation (15) is solved with k not equal to zero. A perturbation expansion for
g is obtained as g ( z )  g 0 ( z )  k 2 g1 ( z ) , where g 0 ( z ) is given by Eq.(17) and g1 ( z ) is solution to:
p  g ' ( z)  p12  g ( z )  g ' ( z)  q  g ( z)  k 2  r  0
(18)
where r  k 2 f ' ( y) 2 and the expansion for g 0 ( z ) is taken to be near z=0.
pg. 12
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g1 ( z) 
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C1q 2
zq 2
k 2 rp12









  p12 e  qp k 2 rp12  qp k 2 rp12  qp k 2 rp12  zq 2  C q 2  k 2 rp12
1
 zq 2    W 





 qp  k 2 rp12
 qp  k 2 rp12






1 

zq 2  C1q 2  k 2 rp12 
qp12  



 p12 e  qp k 2 rp12  zq 2  C q 2  k 2 rp12 

1
k 2 rp12  C1 q 2

  W  
2
2
2
2


 qp  k rp1 
 qp  k rp1














 pq  












(19)
Substitution of g(z) into Eq.(15) yields the following where higher order terms in r and q are assumed negligible at the start of the membrane region.
 p12
2
2 p W 
 p
 2 4
 p12
2
qk rp1  2 p W 

 p
2

 qk 2 r p14  0

(20)
Recalling the definitions of p, p1, q and r in terms of the function f we obtain:
f ( y) 


e1  e2  e3 y 2

e e 4  e5 y 2  e 6 y 4
where the constants

(21)
e i are given in Appendix B.(To be added later)
pg. 13
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To obtain a solution for z downstream in microfluidic channel, we expand left side of Eq.(15) in powers of q and retain the only term in z where
downstream higher powers of q are assumed small.
2
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 W  p12  
p 

 e    r








 p12 


 W
   p12   2




W
2


 p   1  e  p  
2
 p12 




p
1

W




2

2
  p  



p
1



2
e
z

C
r
  W  p1 

1W 
1
  p12 
  



 p12 









W 
 

p 
 W  p 
 p  




 p 
 q  
 1e


e
0
2
p

 p12  
  1  W  p1  





1

W

p
 p  
 

 p 

















(22)
We can reduce this to:
z  f ' ( y) 2  f ' ' ( y)  C1
b
 f ' ( y) 
f ( y)  f ' ' ( y)  2
0
a c1
a  f ( y)
2
(23)
pg. 14
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In the downstream region as z approaches infinity maple was used to solve Eq (22). The solution is:
f ( y)   y  C3   e C3
(24)
All of the previous solutions for f are substituted into the expression for g in Eq. (17) to obtain ( PO2 )
Solving for

requires use of the Lagrange Inversion Theorem[ 29,30]
( PO2 )  f ( y)  g ( z) can be inverted as follows due to Lagrange inversion theorem:

such that  ( )  PO2   1 ( ) . We also assume that   ( PO2 )  f ( y)  g ( z) , then:
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Let’s assume there exists a
n

 [ fg  ( PO )]n d n1 


w

PO

20
20

 
PO2  G( fg )  PO20    lim 
 n1 
n!
dw  ( w)  ( PO20 )    ,
n 1  w PO2 0 




(25)
where the expansion is taken about a known value PO20 , that is an arbitrary value of oxygen tension and fg ~ ( PO2 0 ) .We take this to be the inlet
PO 2 given in this paper as 150 mmHg..
The inversion is obtained from Eq. (16) for the oxygen tension,
PO2  C6 fg  C7
where C 6 , C 7 are constants to be determined.(To be added later)
4. Analytical Methods for plasma sleeve
Equation (2) is a simple steady state convection diffusion equation which has a well known closed form solution, [31]. We have used solution
strategy in [31] and boundary conditions (6) and (7) to solve the governing equation in the plasma sleeve region and match the solution in RBC core
to plasma sleeve. Matching solutions between plasma and core regions was done by first solving across the RBC core for analytical results and then
solving Eq. (2) analytically separately in thin plasma layer and finally verifying that the solution is continuous across boundary between RBC
core/plasma and plasma layer. The plots show complete solution in region from y=0 to y=50 microns and the results incorporate both layers. The
plasma layer is 1 micron thick.
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5. Comparison to Numerical Solution
The default method is used in Maple 18 [18] which uses a second order(in radial or y direction and axial direction z) centered, implicit finite
difference scheme to obtain the solution. The number of points in the stencil of finite difference scheme is one greater than the order of each
equation. A first order accurate boundary condition is used for a second order accurate method and still produces a solution of second order accuracy.
Derivatives in the initial/boundary conditions are specified in indexed D notation.( D p K p D1 ( PO2 )(50, x)  Dm K m
10  PO2 (50, x)
is the boundary

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condition along oxygen porous membrane.) The calling sequence in Maple is pdsolve(PDE,conditions,numeric,options). The parameters include the
type of PDE(Eqs. 1 to 5) which allows for a single z-dependent partial differential equation in one independent variable y, and the set of initial and
boundary conditions given by Eq. (6) and (7). A numeric keyword is used to invoke a centered, implicit finite difference scheme and options
specifies the spacing of the spatial points on the discrete mesh on which solution computes and defaults to 1/20 th of the spatial range of problem.
In order to solve the governing equations in both the plasma cell-free region and cell rich core it was necessary to solve Eq.(2) and (3) in cell-free
region first with boundary conditions given by Eq. (34). Since the thin plasma layer is about 1 micron in height, we used a flux of -78.5 at y=49
microns and solved a boundary value problem with Robin boundary condition at y=50 microns (Eq.34) and a flux of -78.5 at y=49 microns. The next
step was to solve the governing equation, Eq. (1) and (3) in RBC core region with a no flux condition at y=0, and matching value boundary condition
of oxygen tension at y=49 microns at the interface region between RBC core and thin plasma layer.(boundary conditions in Eq.(35), used solution
U(y,z) in plasma layer). We then stored the values of this solution in RBC core and used it as an inlet condition at z1 to the no flux region
downstream with matching values of oxygen tension at interface between thin plasma layer and RBC core downstream. The final step was to solve
Eq (2) and (3) downstream in thin plasma layer. In summary we solve numerically as in the following flow chart.
Plasma layer upstream  RBC Core upstream  RBC Core downstream  Plasma Layer downstream
Case 1
In maple the following commands are used.
pds:= pdsolve(sys,IBC,numeric)
IBC : {PO 2 ( y,7000)  150, D1 ( PO 2 )(49, z )  78.5, D1 ( PO 2 )(50, z )  785  0.1  785  0.01 PO 2 (50, z ),
C ATP ( y,7000)  0.1, D1 (C ATP )(50, z )  0, D1 (C ATP )(49, z )  0}
(26)
IBC : {PO2 ( y,7000)  150, D1 ( PO2 )(0, z)  0, u(49, z)  U (49, t ), C ATP ( y,7000)  0.1, D1 (C ATP )(49, z)  0, D1 (C ATP )(0, z)  0}
(27)
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The boundary conditions and entry conditions are such that the entry condition for blood flow is uniform oxygen tension of 150 mmHg at a length of
7000 microns from inlet of microfluidic device and a flux condition through permeable membrane shown in Figure 1, due to oxygen source at
permeable wall at length
IBC between
z  z0  7000 microns, height y=50 microns .[6] The oxygen flux through membrane was set above in definition of
z 0 and z1 in Fig(1).
The condition of oxygen impermeability between 0 and z 0 and z1 and L is set in the IBC definition in Maple for both oxygen tension and ATP
concentration. The command pdsolve was used again to solve Equation 1 using the solution in Case 1 above as an entry condition for the region
z1 and L (Fig 1) where a no flux condition or flux of zero is set in the definition of the boundary condition.
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between
That is we have the following :
Case 2.
In Maple the solution obtained from Case1 is written as:
U:=subs(pds:-value(output=listprocedure), PO2 ( y, z ) );
(28)
Proceeding in Maple language, Eq. 1 is defined as in Case 1 by defining the equation by a variable ‘PDE’.
The entry and boundary conditions become as follows using the command given by Eq. (36):
IBC : {PO2 ( y,7700)  U ( y,7700), D1 ( PO2 )(0, z)  0, D1 ( PO2 )(50, z)  0
It can be observed that the solution obtained from Case 1 is used up to the end of the oxygen permeable wall and set to be an inlet condition in z to
where the non-permeable wall condition starts to the right of the oxygen porous membrane.The advantage to store the solution in the computational
field by using (36) is to solve the partial differential equations in two steps in the entire region from z=0 to z=L as shown in Figure 1.
6. Results and Discussion
Steady State Oxygen Transport and ATP Release in Micro-Channel
In Figure 2 results show that near inlet to region with permeable wall, the oxygen tension is nearly 150 mmHg across the channel height with a
rather steep decrease towards the wall. These results are in agreement with numerical results shown in Figure 5 and 7. The results for the numerical
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solution of Eq. (33) are shown in Figure 8 for moderate value of x+c, near center of membrane. For steady-state oxygen transport in a channel of
radius 50 microns it is observed that the greatest decrease in oxygen tension is near the oxygen permeable membrane and to the right of it
downstream. Downstream the value in oxygen tension decreases to approximately 97.5 mmHg as shown by maple results shown in Fig. 7. In Fig. 5
results are shown for oxygen tension within membrane region.The largest ATP concentration occurs 3 mm downstream from the membrane.
Throughout the channel and downstream of membrane the ATP concentration flattens out at center of channel (y=0), and increases at right end of
channel height from centerline. This behavior is similar to results in [6] as shown there in Figure 2-C. It is concluded that there is a significant
decrease in oxygen tension to the right of the permeable membrane downstream. It is in this region that there is significant concentration of ATP
released (Fig.6) by RBC’s uniformly across the height of channel. The associated time dependent problem has been dealt with numerically in [24-25]
and requires use of the oxygen binding capacity of hemoglobin [26] in the time derivative coefficient. It is important to mention that a rapid decrease
in oxygen saturation is not the only means to produce ATP, it is also accomplished by means of applying shear stress on the RBC as shown in [28].
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Conclusions
Acknowledgements
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An analytical approach to solve the governing equations of oxygen transport and ATP concentration in a microfluidic channel with a section of
permeable wall subjected to a non-zero flux condition has been presented. The analytical and numerical results clearly show that significant
concentrations of ATP are released downstream. Future considerations include extension of analytical methods of solution to corresponding time
dependent problem.
I would like to acknowledge my student Keith Christian Afas, for suggesting the analytical procedure used in the first part of this paper to me and for
time spent working together on this preprint. We would like to see this work published jointly in a peer-reviewed journal in the near future. The
present preprint requires considerable more work to be ready for submission for publication.
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Appendix
The following equation governs blood plasma velocity, [3]
m 3
(h 2  y 2 ),

s
v p ( y) 
p 2
 2
2
( y i  y 2 ),
 3  p
 3   (h  y i ) 





4 w h  
 1 y i 
c


 c
 

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3 130 10
6
where w is the width of the channel which in [6] is 1500 microns,
yi  y  h
(A1)
0  y  yi
yi is 49 microns, h is 50 microns and
p
is 1.26 1 as obtained from [20] for a
c
tube of diameter 100 microns with hematocrit 0.2.
Glossary
Nomenclature
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PO 2
P50
SO2
C ATP
partial pressure of oxygen
partial pressure at 50% saturation
oxygen saturation
ATP concentration
slip velocity
vp
blood plasma velocity
v RBC
HT
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slp
red blood cell velocity
tube hematocrit
[ HbT ]
total heme concentration
Kp
O 2 solubility in plasma
K RBC
O 2 solubility in RBCs
Km
O2 Solubility in Porous Membrane
Dp
O 2 diffusivity in plasma
D ATP
Dm
diffusion coefficient of ATP
O2 Diffusivity in O2 Permeable Membrane
N
Hill coefficient
R
ATP release rate
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t
time t
normal unit vector
z
axial coordinate measured from channel/tube entrance
y
channel height coordinate or radial coordinate measured from tube axis
L
length of channel/tube
H
height of channel/tube
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n
References
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Fig. 2 Oxygen tension versus y-variation in channel height for analytical results k=1.
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Figure 2
Figure 3- Analytical results for oxygen saturation in region of permeable membrane, at z=7000 up to z=10700 microns downstream of membrane.
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150 z
versus z
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Figure 4 g ( z )  z 0
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Figure 5 Numerical results within region where permeable exists for oxygen tension(mmHg) versus y-variation in channel height.(microns)
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ATP concentration in channel at various axial locations within region where permeable membrane exits.
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Figure 6
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Figure 7 Downstream numerical results from permeable membrane region for oxygen tension (mmHg) versus y-variation in channel height(
microns).
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