Supplementary Material for Microfluidics and Nanofluidics
Supplementary Material for
Mathematical Analysis of Oxygen Transfer through Polydimethylsiloxane Membrane
between Double Layers of Cell Culture Channel and Gas Chamber in Microfluidic
Oxygenator
5
Min-Cheol Kim1, 2, †,*, Raymond H. W. Lam1, 3, †, and Todd Thorsen1,* H. Harry Asada1, 2
1Department
2
BioSystem & Micromechanics IRG, Singapore MIT Alliance Research Technology, Singapore, 117543
3Department
10
of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139
of Mechanical and Biomedical Engineering, City University of Hong Kong, Hong Kong, China
† Authors contributed equally to this work.
* To whom correspondence should be addressed: mincheol@mit.edu; thorsen@mit.edu
15
20
25
Table of Contents
P.
Solution of Single Layer 1-D Convection Diffusion Equation …..……….…………………………… S2
Semi-Analytical Solution of Multiple Layer Transient 2-D Convection Diffusion Equations…….… S5
Derivation of the Dimensionless Oxygen Pressure Drop (OPD)……........................................……… S6
Sample pO2 Profiles with Different Oxygenation Configurations..…………………..………….…… S7
Figure S1……………………………………………………………………….……………………... S8
Figure S2……………………………………………………………………….……………………... S8
Figure S3……………………………………………………………………….……………………... S9
Experiment for Oxygen Diffusion Time..…………………...................................………….….........… S9
Figure S4…………………………………………………………………….……………………... S10
Figure S5…………………………………………………………………….……………………... S11
References..…………………………………………………….…...………….…..….………….…… S11
S1
Supplementary Material for Microfluidics and Nanofluidics
Solution of Single Layer 1-D Convection Diffusion Equation
The dimensionless 1-D convective diffusion equation and boundary conditions are expressed as
 i2
gi  2 gi

 mi
  2

5
gi


gi

for i 1    i and 0    1
(S1)
 Sh   Gi ( , )  g ( , , ) 
(S2)
 Sh   g ( , , )  Gi 1 ( , ) 
(S3)
  i 1
  i
 h K ( ) 
where Sh     N
 is Sherwood number (Sh) representing the ratio of length scale to the diffusive
Di 

boundary layer thickness (hN), and K ( ) is the film mass transfer coefficient at the boundary  . To solve
this non-homogeneous boundary value problem, we may let
2
gi  , ,   vi  , ,    Lij   Fij  ,  for i 1   i and 0    1,   0
10
(S4)
j 1
where
d 2 Lij ( )
d 2
 0 so that Lij ( )  Cij1  Cij2 for i 1    i and 0    1, i  1, 2,..., N , j  1, 2.
Fi1  ,   Gi  ,  and Fi 2  ,   Gi 1  ,  are unknown variables at boundaries of the ith material.
Substituting these results into Eq. S1 gives
 i2
15
2
Fij ( , )  2 vi
vi
  i2  Lij ( )

 mi .


 2
j 1
(S5)
The boundary conditions described in Eqs. S2-S3 are decomposed as (39)
Di
vi ( , i 1 , )
 hN K (i 1 )v( , i 1 , )  0

(S6)
Di
vi ( , i , )
 hN K (i )v( , i , )  0

(S7)
Di
dLi1 (i 1 )
 hN K (i 1 ) Li1 (i 1 )  hN K (i 1 )
d
(S8)
Di
20
dLi1 (i )
 hN K ( i ) Li1 ( i )  0
d
(S9)
Di
dLi 2 (i 1 )
 hN K (i 1 ) Li 2 (i 1 )  0
d
(S10)
Di
dLi 2 (i )
 hN K (i ) Li 2 (i )  hN K ( i )
d
(S11)
where Eqs. S8-S9 and S10-S11 are boundary conditions used to solve Li1 ( ) and Li 2 ( ) , respectively.
Assuming K ( i 1 ), K ( i )   , the solutions of Li1 ( ) and Li 2 ( ) are
Li1 ( ) 
25
 i 
  i 1
, Li 2 ( ) 
i  i 1
i  i 1
(S12)
with the initial condition
S2
Supplementary Material for Microfluidics and Nanofluidics
2
2
j 1
j 1
vi ( , , 0 )  gi ( , , 0 )   Lij ( ) Fij ( , 0 )  fi ( , )   Lij ( ) Fij ( , 0 )
(S13)
.
Now, we may let

vi  , ,   Vi , n  ,  i , n  
i 1    i , 0    1,   0
n 1

mi  , ,    M i , n  ,  i , n  
i 1    i , 0    1,   0
n 1
i
5
where M i , n  , 


i 1
(S14)
(S15)
mi  , ,  i , n   d
i

.
 i , n   d
2
i 1
i

Assume Lij ( )   lij , n i , n   for i 1    i , where lij , n


n 1
i 1
Lij ( ) i , n   d
i

i 1
 i , n   d
2
, and substituting Eqs.
S14 and S15 into Eq. S5:
2
i 2Vi , n  ,  i , n    i 2  lij , n
j 1
Fij  , 

i , n    M i , n  ,  i , n    i , n  Vi , n  , 
(S16)
Dividing Eq. S16 by i 2 Vi ,n  ,  i ,n   yields
2
Vi , n  ,    lij , n
j 1
10
Fij  , 

Vi , n  , 

M i , n   , 
i 2

 i , n  
i 2  i , n  
 i2,n
(S17)
where a positive constant i2,n is used to set the above equation to be negative, such that the corresponding two
ordinary differential equations are
2
Vi , n  ,   i2, nVi , n  ,    lij , n
j 1
Fij  , 

i ,n    i2,ni 2 i ,n    0
15

M i , n  , 
i 2
(S18)
(S19)
where the boundary conditions of eigenfunction in Eq. S19 are
Di i , n  i 1   hN K (i 1 )i ,n  i 1   0
(S20)
Di i , n  i   hN K (i )i ,n  i   0 .
(S21)
Hence the solution of Eq. S19 is i ,n    Ai ,n Cos(i ,n
. Substituting this solution into Eqs.
i )  Bi , n Sin(i , n
i )
S20-S21 gives a matrix form as
20
  Di i , n i Sin(i , n i i 1 )  hN K ( i 1 )Cos(i , n i i 1 ) Di i , n i Cos(i , n i i 1 )  hN K ( i 1 ) Sin(i , n i i 1 )   Ai , n   0 


    (S22)
Di i , n i Cos (i , n i i )  hN K ( i ) Sin(i , n i i )   Bi , n   0 
  Di i , n i Sin(i , n i i )  hN K ( i )Cos (i , n i i )
The orthogonality of the eigenstates, Cos(i ,n
and Sin(i,n
, lead to a condition for the eingenvalue
i )
i )
spectrum:
S3
Supplementary Material for Microfluidics and Nanofluidics
 Di i , n i Sin(i , n i i 1 )  hN K ( i 1 )Cos (i , n i i 1 ) Di i , n i Cos (i , n i  i 1 )  hN K ( i 1 ) Sin(i , n i  i 1 )
 0 (S23)
 Di i , n i Sin(i , n i i )  hN K ( i )Cos (i , n i  i )
Di i , n i Cos (i , n i  i )  hN K (  i ) Sin(i , n i  i )
.
K (hi ), K (hi 1 )   , the corresponding eigenvalues are i ,n  n 2 2  i2  i  i 1 2 . This
However, as
Sturm-Louisville problem has been solved by implementing separation of variables on the homogeneous BVP of
Eqs. S7-S8. The General solution for Eq. S18 at a simulation time  is integrated from initial time  0 that
5
yields an expression written as
Vi , n  ,   Vi , n  , 0  Exp  i2, n (   0 ) 
2

Fij ( ,  )
j 1
0

  lij , n 
Exp  i2, n (   )  d  
1
i
2


0
(S24)
M i , n  ,   Exp  i2, n (   )  d 
where Vi ,n  , 0  can be calculated with the initial condition Eq. S13:
2


 fi ( , )   Lij ( ) Fij ( , 0 )   i , n   d
i 1
j 1



i
2


d



 i,n
i
Vi , n  , 0  

i 1
i
Define i , n  fi ( ) 


i

i 1
fi ( , ) i , n   d
i

i 1
 i , n   d
2
2
  lij , n Fij ( , 0 )
.
f ( ) i , n  d
i 1
i

i 1
 i , n   d
2
in Eq. S25 and we obtain
2


Vi , n  ,    i , n  fi ( , )    lij , n Fij ( , 0 )  Exp  n 2 (   0 ) 
j 1


.
2
 Fij ( ,  )
1 
2
2
  lij , n 
Exp  i , n (   )  d  2  M i , n  ,   Exp  i , n (   )  d
0

i  0
j 1
10
By assuming
Fij ( ,  )

(S25)
j 1
(S26)
 a j and M i ,n  ,    Mi ,n  , 0  for the convolution terms in the above equation, Eq. S26
becomes
2


Vi , n  ,     i , n  fi ( , )    lij , n Fij ( , 0 )  Exp  i , n 2 (   0 ) 
j

1



1  M i , n   , 0  2

  lij , n a j  1  Exp  i , n 2 (   0 ) 

i , n 2 
i 2
j 1


where a j 
15
Fij ( , )  Fij ( , 0 )
  0
(S27)

. Updating 1   0   , where τ1 is the current simulation time and Δτ is the
time step, into Eq. S27 gives
2


Vi , n  , 1     i , n  f i ( , )    lij , n Fij ( , 0 )  Exp  i , n 2  
j 1


Fij ( , 1 )  Fij ( , 0 ) 
1  M i , n   , 0  2
2

  lij , n

 1  Exp  i , n  

i , n 2 
i 2
j 1

.

When
Fij ( ,  )


Fij ( , )  Fij ( , 0 )
  0

(S28)
and M i ,n  ,    Mi ,n  , 0  ,
S4
Supplementary Material for Microfluidics and Nanofluidics
  i , n  f i ( , )   li1, nGi ( , 0 )  li 2, nGi 1 ( , 0 )  Exp  n 2 



gi  , ,     1  M i , n  , 0 
Gi ( , )  Gi ( , 0 )
Gi 1 ( , )  Gi 1 ( , 0 ) 
2
n 1  
 li1, n
 li1, n
 1  Exp  i , n  
2 
i 2



 i , n 





 i , n  


 Li1 ( )Gi ( , )  Li 2 ( )Gi 1 ( , ).
(S29)
Semi-Analytical Solution of Multiple Layer Transient 2-D Convection Diffusion
Equations
5
To impose the dimensionless mass flux boundary conditions Eqs. 18, 20 and 22, the spatial derivative of
gi  , ,  should be calculated as
g i  , , 


10
  i , n  f i ( , )   li1, n Gi ( , 0 )  li 2, n Gi 1 ( , 0 )  Exp  i , n 2  



   1  M i , n  , 0 
Gi ( , )  Gi ( , 0 )
Gi 1 ( , )  Gi 1 ( , 0 ) 
2
n 1  
 li1, n
 li1, n
 1  Exp  i , n  
2 
2






i

 i , n 




 d  i , n  , 

d


dLi1 ( , )
dL ( , )
Gi ( , )  i 2
Gi 1 ( , )
d
d
(S30)
where  0 and  are the dimensionless priori time of iteration and time-step, respectively. Rearrange Eq. S30
as a linear combination of unknown variables and source term:
gi  , , 

 Qi  , ,   Gi ( , )  Ri  , ,   Gi 1 ( , )  Pi  , , 0 ,  
(S31)
Where Pi  , , 0 ,   is the source term, Qi  , ,   and Ri  , ,   are the coefficients of unknown
variables, which are expressed as S1
15
  i , n  f i ( , )   li1, n Gi ( , 0 )  li 2, n Gi 1 ( , 0 )  Exp  i , n 2  



Pi  , , 0 ,      1  M i , n  , 0  li1, n Gi ( , 0 )  li 2, n Gi 1 ( , 0 ) 
2
n 1  

 1  Exp  i , n  
2 

i 2

 i , n 


Qi  , ,   
Ri  , ,   
 1
dLi1 ( , ) li1, n

d

   1  Exp 
dLi 2 ( , ) li 2, n

d

   1  Exp 

n 1


n 1
2
2
i,n
i,n
 1

2
i,n
i,n


 d  i , n  , 

d


(S32)
 d i , n  , 
  
d

2

 d i , n  , 
  
d


(S33)
.
(S34)
Using the mass flux boundary conditions, Eqs. 18, 20, 22 and 23 can be converted into linear algebraic
equations written as a matrix form:
20
R1  , 0,  
 Q1  , 0,  

 1 Q  ,  ,   1 R  ,  ,    Q  ,  ,  
1
1
1
1
2
1
A     12
12

2
0
 3  2 Q2  ,  2 ,  


0
0

0
 R2  , 1 ,  
 32 2 R2  ,  2 ,    Q3  ,  2 ,  
0



0


 R3  ,  2 ,   

1

0
,
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Supplementary Material for Microfluidics and Nanofluidics
 P1  , 0, 0 ,  


 G1 ( , ) 





  1 P  ,  , ,    P  ,  , ,   
G ( , ) 
1
1
0
2
1
0
.
 and X     2
b     12
 G3 ( , ) 
 2



  3  2 P2  ,  2 , 0 ,    P3  ,  2 , 0 ,   
 G4 ( , ) 


0


(S35)
These linear equations A(ξ)·X(ξ) = b(ξ) are readily solved line-by-line along to the channel axial direction (0 ≤ ξ
≤ 1) using tridiagonal matrix algorithm (TDMA)S2.
5
Derivation of the Dimensionless Oxygen Pressure Drop (OPD)
When the mass fluxes of oxygen across each material layer are identical, it can be expressed as
q
D3 S3
2 D2 S2
2D S
Pg  P2 3  
 P23  P2b   2 2  P2b  Pcell 

H3
H2
H2
(S36)
where P2-3 is the pO2 at the interface between PDMS membrane and medium channel, P2b is the
10
bulk pO2 in medium, and Pcell is the pO2 along top cell surface. If we correlate q with the cellular
oxygen uptake rate at the equilibrium state, q can be written as
q  D1
c1
y
y  H cell

H1
0
Vmax cell c1
V  c
V * P
dy  max cell 1 H1  max cell cell
K m  c1
S1 K m  c1
K m  Pcell
(S37)
where ρ*cell (= ρcell H1) is the cell density [cells/cm2]. Equation (S36) can be converted by following
equation:
q
15
P
g
 P2 3    P2 3  P2 b    P2 b  Pcell 
.
H3
H2

D3 S3 D2 S2
(S38)
With equations (S37) and (S38), a quadratic equation with a variable Pcell can be expressed as
 H

H 
*
Pcell 2   3  2 Vmax  cell
 Km  Pg  Pcell  Pg Km  0.
 D3 S3 D2 S2 

(S39)
By dividing each term by Pg 2 , equation (S39) yields
 Pcell

 Pg
20
2
   H3
K  P
K
H V *
 2  max cell  m  cell  m  0.
  1  
Pg  Pg
Pg
   D3 S3 D2 S2  Pg
To further simplify the equation (S40), we define A 
K 'm
Pg
and B 
*
Vmax  cell
Pg
(S40)
 H3
H2 



D
S
D
2 S2 
 3 3
, and it is
simplified in following form:
 Pcell

 Pg
Solving for
2

P
  1  A  B  cell  A  0.
Pg

(S41)
Pcell
gives
Pg
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Supplementary Material for Microfluidics and Nanofluidics
Pcell 1  A  B  

Pg
1  A  B 
2
 4A
2
(S42)
.
We define the dimensionless oxygen pressure drop (OPD) between the top cell surface and gas
reservoir as
OPD 
5
Pg  Pcell
Pg
(S43)
,
and its exact solution can be simplified in following form:
OPD 
Pg  Pcell
Pg

1
1
1  A  B  
2
2
1  A  B 
2
 4 A.
(S44)
Sample pO2 Profiles with Different Oxygenation Configurations
10
Fig. S1(a-d) shows contour plots of the partial pO2 for four cases with different PDMS membrane thicknesses
(H3) 50 µm, 250 µm, 1000 µm and 6000 µm, respectively, at the steady state. For the qualitative comparison,
the cell layer thickness (H1 = 5 µm), effective channel height (H2 = 195 µm), medium flow rate (Pe = 3), cellular
oxygen uptake rate (Vmax = 4 × 10-17 mole/cell/s), surface cell coverage (ρ*cell = 3 × 104 cells/cm2) and
Michaelis–Menten constant (K’m = 0.5 mmHg) were set to be identical for the four different PDMS membrane
15
cases. The plots show that the developing length of pO2 (the length where medium starts being fully diffused, or
x for L* ~ 1) increases with the membrane thickness, which is related to the diffusive resistance.
Another parametric study was conducted to investigate how the developing length relates to the effective
channel height under the same Peclet number (Pe = 6) and same surface cell coverage (ρ*cell = 3 × 104 cells/cm2).
Fig. S2(a-d) shows contour plots of the partial pO2 in four different effective channel heights (H2) 50 µm, 100
20
µm, 250 µm and 500 µm, respectively, at the steady state with fixed cell layer thickness (H1 = 5 µm) and PDMS
thickness (H3 = 250 µm). All other parameters were the same as Fig. S1. The height of medium channel has
similar effects on the diffusion time and developing length as the PDMS thickness.
Furthermore, Fig. S3(a-e) shows contour plots of the partial pO2 in five different cell layer thickness (H1)
10 µm, 20 µm, 50 µm, 100 µm and 150 µm, respectively, under steady state conditions with a fixed channel
25
height (Hc = 200 µm) and PDMS thickness (H3 = 250 µm), and with all other parameters identical to the PDMS
thickness study illustrated in Fig. S1. The developing length of pO2 gradually decreases as the cell layer grows
in a limited culture channel (constant height: Hc = H1 + H2), inducing an increasing volumetric ratio of cell layer
to medium, because the diffusive resistance of HUVEC (H1/S1D1) is smaller than that of culture medium
(H2/S2D2).
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Supplementary Material for Microfluidics and Nanofluidics
Figure S1:
Contour plots of partial pO2 in a cross section of microfluidic device with H1 of 5 µm, H2 of 195
µm, and four values of H3, i.e. (a) 50 µm, (b) 250 µm, (c) 1000 µm and (d) 6000 µm. Each case was simulated
under identical conditions with the dimensionless parameters Pe = 3, Da = 3.93.
5
Figure S2:
Contour plots of the partial pO2 in the microfluidic devices with H1 = 5 µm, H3 = 250 µm, and
four different H2: (a) 50 µm, (b) 100 µm, (c) 250 µm and (d) 500 µm. Each case was simulated under identical
conditions with the dimensionless parameters Pe = 6 and Da = 0.98, 1.96, 4.91 and 9.83 for (a) – (d),
10
respectively.
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Supplementary Material for Microfluidics and Nanofluidics
Figure S3:
Contour plots of the partial pO2 in the microfluidic devices with Hc = 200 µm, H3 = 250 µm, and
five different H1: (a) 10 µm, (b) 20 µm, (c) 50 µm, (d) 100 µm and (e) 150 µm. Each case was simulated under
identical conditions with the dimensionless parameters Pe = 6 and Da = 7.86, 15.73, 39.32, 78.64 and 117.96 for
5
(a) – (e), respectively.
Experiment for Oxygen Diffusion Time
Chip fabrication
10
Experiments were performed to validate the simulated dissolved-oxygen (DO) concentration inside an
elastomeric PDMS channel. Microfluidic chips containing arrays of double-layer structures (as described in the
Diffusion Time of Oxygen Transfer section) with different layer thicknesses were fabricated by multilayer soft
lithographyS3 as shown in Fig. S4. Geometry of air and fluid channels was defined by SU-8/silicon molds, in
which SU-8 negative photoresist (Microchem SU-8 50) was patterned on a silicon wafer. After cleaning a 3”
15
silicon wafer (James River Semiconductors), SU-8 (thickness: 100 µm) was spin-coated (1000 rpm, 30 s) on the
wafer and patterned by photolithography (Karl Suss Mask Aligner MJB3, 2 min exposure). Afterwards the mold
was silanized with a high molecular weight trichloro-perfluorooctyl silane (Aldrich) for >5 min to reduce the
adhesion between polydimethylsiloxane (PDMS) and the Si/SU-8 surface. A mixture of 10:1 PDMS
monomer/harder (Sylgard 184, Dow Corning) was prepared and poured onto the mold with the gas channel
20
pattern before baking at 80 °C for 2 hr. Another 10:1 PDMS monomer/harder mixture was spin-coated (for H2 +
H3 ≤ 1000 μm) or poured (for H2 + H3 > 1000 μm) on the flow channel mold with the defined thickness (100 µm)
and was then baked in an oven at 80 °C for 10 min. Taken out the gas layer mold from the oven, the PDMS
substrate was cut and peeled off from the mold. After punching holes for the gas inlets and outlets, the gas layer
was aligned onto the mold with the PDMS medium layer under a dissecting scope (Olympus, Model SZX9). To
25
achieve permanent bonding and fully-cured PDMS, the whole PDMS substrate was post-baked in an oven at 80
ºC for >10 hr. The bonded PDMS was then chopped, peeled off and punched for medium inlets and outlets. On
the other hand, a glass substrate containing DO sensors was prepared from a glass slide (75 mm (l) × 25 mm (w)
× 1 mm (h)). Each sensor region on the glass slide was etched with buffered hydrofluoric acid (7 H 2O: 1 HF) for
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Supplementary Material for Microfluidics and Nanofluidics
5 min to produce a cavity (~5 μm). A polymeric sensor film was then deposited in each cavity by applying a
droplet (~1 μL) of PtOEPK/polystyrene dye solution, prepared by first dissolving PtOEPK (Frontier Scientific,
w/w 0.1 %) and polystyrene (w/w 5 %) into toluene. After toluene in the dye solution evaporates,
oxygen-permeable PtOEPK/polystyrene films (thickness: ~3 μm) were formed as DO sensors. The PDMS and
5
glass substrates were then bonded by oxygen-plasma bonding process (PLASMOD, Tegal Corporation) with
600 mTorr pressure and medium power for 45 s.
Figure S4:
10
(a) Photograph and schematic diagrams of (b) top view and (c) cross-section of a PDMS chip for
validation of diffusion time.
Experimental procedures and sample results
To measure the time-varying DO profile at the sensor layer, oxygen was continuously supplied along gas layer.
Real-time measurement was achieved by a detection method based on the Stern-Volmer relation of PtOEPK that
15
the reduction in luminescent intensity of sensing material due to oxygen quenching of the emitting excited
electronic state
S4, S5
. In the experiment, the medium channel was initially filled with fresh medium under
ambient environmental partial pressure and oxygen was then applied along the gas channel. The corresponding
numerical oxygen profiles were also calculated using the approach described in section 4-2. Comparisons of the
computational and experimental dynamic sensor responses indicate that the devices had similar pO2 profiles
20
(Fig. S5) and response/rise time as numerical solution.
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Supplementary Material for Microfluidics and Nanofluidics
5
Figure S5:
Sample sensor responses in the validation experiment of diffusion time and the corresponding
simulation results for sensor thickness (H1) 3 μm, effective medium channel height (H2) ~100 μm and membrane
thickness (H3) (a) 20 μm, (b) 100 μm, (c) 300μm , and (d) 1250 μm. The y axes were normalized by scaling the
fully saturated DO as value 1.
References
S1 G. P. Mulholland and M. H. Cobble, Int. J. Heat Mass Transfer., 1972, 15, 147–160.
10
S2 T. P. Fredman, Heat and Mass Transfer., 2003, 39, 285–295.
S3 T. Thorsen, S. J. Maerkl and S. R. Quake, Science., 2002, 298, 580–584.
S4 R. H. W. Lam, M.-C. Kim and T. Thorsen, Anal. Chem., 2009, 81, 5918–5924.
S5 A. P. Vollmer, R. F. Prostein, R. Gilbert and T. Thorsen, Lab Chip., 2005, 5, 1059–1066.
S11