Engineering the Tissue Which Encapsulates
Subcutaneous Implants. I. Diffusion Properties
A. Adam Sharkawy, Bruce Klitzman, George A. Truskey, W. Monty Reichert
Dept. of Biomedical Engineering, Duke University
J Biomed Mater Res. 1997. 37: 401-412
Motivation
• Demonstrate that implant surface architecture impacts the mass
transfer properties of the surrounding tissue
Objectives
• Demonstrate impact of implant surface on encapsulation tissue
• Measure binary diffusion coefficient of a small-molecule analyte
through each tissue
Approach
• Implantation in subcutaneous tissue of rats
• Histology of encapsulation tissue at implant surface
• Two-chamber measurements of diffusion coefficient across tissue
Engineering the Tissue Which Encapsulates
Subcutaneous Implants. I. Diffusion Properties
A. Adam Sharkawy, Bruce Klitzman, George A. Truskey, W. Monty Reichert
Dept. of Biomedical Engineering, Duke University
J Biomed Mater Res. 1997. 37: 401-412
Motivation
• Demonstrate that implant surface architecture impacts the mass
transfer properties of the surrounding tissue
Objectives
• Demonstrate impact of implant surface on encapsulation tissue
• Measure binary diffusion coefficient of a small-molecule analyte
through each tissue
Approach
• Implantation in subcutaneous tissue of rats
• Histology of encapsulation tissue at implant surface
• Two-chamber measurements of diffusion coefficient across tissue
Engineering the Tissue That Encapsulates
Subcutaneous Implants. I. Diffusion Properties
A. Adam Sharkawy, Bruce Klitzman, George A. Truskey, W. Monty Reichert
Dept. of Biomedical Engineering, Duke University
J Biomed Mater Res. 1997. 37: 401-412
Motivation
• Demonstrate that implant surface architecture impacts the mass
transfer properties of the surrounding tissue
Objectives
• Demonstrate impact of implant surface on encapsulation tissue
• Measure binary diffusion coefficient of a small-molecule analyte
through each tissue
Approach
• Implantation in subcutaneous tissue of rats
• Histology of encapsulation tissue at implant surface
• Two-chamber measurements of diffusion coefficient across tissue
Implants in Sprague-Dawley Rats
Implant Types
Parenthetical values are length of implantation
in weeks
SQ - normal subcutaneous tissue (4)
SS - stainless steel cages (3 or 12)
PVA-skin - non-porous PVA (4)
PVA-60 - PVA sponge, 60 m pore
size (4)
PVA-350 - PVA sponge, 350 m pore
size (4)
PVA Sponge
Stainless Steel Mesh
Porosity Reduces Encapsulation
PVA-60
PVA-skin
S = A3/2
Is this an appropriate
assumption?
Fibrous Tissue Inhibits Diffusion
Comparison of ExperimentaleffD
/Do to
Concentrated
Chamber
Dilute
Chamber
0 .7
0 .6
0 .5
Deff/Do
0 .4
Membrane
Ussing-type Diffusion Chamber
0 .3
0 .2
M eas ured
M axwell Spheres
Rayleigh S pheres
Rayleigh C ylinders
0 .1
Fluorescein
0
MW 376
0 .4
0 .4 5
0 .5 5
0 .6
0 .6 5
0 .7
0 .7 5
0 .8
A rea Fract ion
PVA-350
1 cA o  2cB  ADt
 ln

2  cA o  xV
0 .5
SQ
PVA-60
Maxwell’s correlation for
composite media:
SS
PVA-skin
Deff 2(1 s )

Do
2 s
0 .8 5
Finite Difference Modeling
Step Change
Ramp
This is a Good Paper
This is a good paper
-It presented qualitative evidence that the implant surface
could be engineered to minimize the formation of fibrous scar
tissue
- It presented internally-consistent data showing that fibrous
tissue inhibited the diffusion of small molecule analytes
- The community agrees; nearly 100 citations plus 100 more
for 2 companion papers
But, this is a very difficult experiment, and it isn’t without its
flaws…
The Paper Does Have Flaws
• Absence of a control membrane that allows quantitative
comparison to other studies
•The FD model adds nothing to the paper; I got the same answer
they did in 30 seconds w/out using Matlab
• Why do experiment and theory correlate poorly in this study?
• Rats aren’t humans; subcutaneous tissue isn’t abdominal tissue these results offer a qualitative picture, not an absolute
quantitative measure
But to reiterate: This is a difficult experiment!
Supplemental Slides
Two-Chamber Diffusion
•
•
Assume membrane adjusts rapidly
to changes in concentration
DH 
ji  
c  ci, upper 
 l 
 i, lower
Vlower
Vupper
dC i, lower
dt
dC i, upper
dt
Expanding flux terms
d
ADH  1
1 
Ci, lower  Ci, upper  


C

 i, lower  Ci, upper   DCi, lower  Ci, upper 
dt
l Vlower Vupper 
•
Species balance for each tank 

•
•
Ci, lower  Ci, upper
 e Dt
o
o
C i, lower  C i, upper
  Aji
 Aji

•
Combine species balances
dCi, lower
dt

dCi, upper
dt
Integrating w/ Coi,lower-Coi,upper @ t = 0
Assuming tanks are equal volumes,
we can say Ci,lower = Coi,lower-Ci,upper


 o

1 C i,lower  2Ci,upper  ADHt
 ln


2 
C o i,lower
lV




 1
1 
 Aji 


V
V
 lower
upper 

Maxwell’s Composite Correlation
In Maxwell’s derivation, we can
consider some property, v
(temperature, concentration, etc.),
whose rate of change is governed
by a material property, Z
(diffusivity, conductivity, etc.)
We now consider an isolated sphere with property Z’ embedded
within an infinite medium with property Z. Far from the sphere,
there is a linear gradient in v along the z-axis such that v = Vz. We
want to know the disturbance in the linear gradient introduced by
the embedded sphere.
Maxwell’s Composite Correlation
We assume profiles of the form:
v  Vr cos 
B
cos
r2
v Arcos
Outside Sphere
Inside Sphere
Subject to 
the boundary conditions:
v
v
Z  Z
r
r
v = v’
for r = a, 0 ≤  ≤ 
Solving for A and B, we find:

Va 3 (Z  Z )
v  Vr cos  2
cos

r (2Z  Z )
v 

3ZVz
2Z  Z 
Maxwell’s Composite Correlation
We now consider a larger sphere of radius b with many smaller
spheres of radius a inside, such that na3 = b3, where  is the volume
fraction of small spheres in the large one. The following must be
true:
na3 (Z  Z )
v  Vz  3
Vz
r (2Z  Z )

b3 (Z  Zeff )
v  Vz  3
Vz
r (2Z  Zeff )
Equating these two expressions, we can solve for Zeff:

Z eff 
3ZZ   (2Z  Z )Z(1  )
3Z  (2Z  Z )(1  )
This expression can be written in various forms, including the
 in the paper.
one listed
Other Composite Correlations
Maxwell’s Correlation for Diffuse Spheres
 1
2
1
1 

 2 s  
Deff Ds Do
Ds Do 

 1
Do
2
1
1 

  s  
Ds Do
Ds Do 
Rayleigh’s Correlation for Densely-Packed Spheres
D
3s
eff  1
Ds  2Do 
 Ds  Do  10 / 3
Do

 s 1.569
s  ...
 Ds  Do 
3Ds  4Do 
Rayleigh’s Correlation for Long Cylinders

Deff, xx
2s
 1
Ds  Do 
Ds  Do 
Do
4
 8  ...

 s  
0.30584  0.013363
Ds  Do 
Ds  Do 
Source: BSL, 2nd Edition, p.281-282.
What are the Volume Fractions?
Comparison of ExperimentaleffD
/Do to
0 .5
0 .4 5
0 .4
0 .3 5
Deff/Do
0 .3
0 .2 5
0 .2
0 .1 5
0 .1
0 .0 5
M eas ured
M axwell Spheres
Rayleigh S pheres
Rayleigh C ylinders
0
0 .4
0 .4 5
0 .5
0 .5 5
0 .6
0 .6 5
A rea Fract ion
0 .7
0 .7 5
0 .8
0 .8 5

Other Way to Estimate the Lag Time
NA  Ct ADABxA  Ct ADAB
L
x

 A  Rt
Ct ADAB N A
Composite Resistances
Rt 
L1
L2

Ct A1DAB,1 Ct A2 DAB,2
xA
L
C
DAB,1
DAB,2
L1
L2
Other Way to Estimate the Lag Time
In Cartesian Co-ords (A1=A2):
Rt 
For DAB,1 = 2.35 and DAB,2 = 1.11:
1
1

DAB,1 DAB,2
Ro
 3.1
Rt
In Cylindrical Co-ords:

Rt 
1
1

1.5DAB,1 DAB,2

Ro
 2.8
Rt
In Spherical Co-ords:


1
1
Rt 

2.25DAB,1 DAB,2
Ro
 2.6
Rt
The Finite Difference Model
 2c  2c 
c
 D 2  2 
t
x y 
c t 1i, j  F Ct i1, j  c t i1, j c t i, j 1  c t i, j1 1 4Fc t i, j
Discretized Transient Species Balance
Transient Species Balance
Boundary Conditions:
c
0
x
F

Dt
x 2
1/F > 20 in the model to ensure stability

c
0
y
c
D  j reaction
y
where

j reaction x m 2
 100
Dm c m
Rats v. Humans
“This study reveals profound physiological differences at
material-tissue interfaces in rats and humans and highlights the
need for caution when extrapolating subcutaneous rat
biocompatibility data to humans.” - Wisniewski, et al. Am J
Physiol Endocrinol Metab. 2002.
“Despite the dichotomy between primates and rodents
regarding solid-state oncogenesis, 6-month or longer
implantation test in rats, mice and hamsters risk the accidental
induction of solid-state tumors...” - Woodward and Salthouse,
Handbook of Biomaterials Evaluation, 1987.
2-Bulb Problem
No flux @ boundaries
--> Nt = 0
As w/ our membrane, we assume that the concentrations can adjust very
rapidly in the connecting tube (pseudo steady-state). Thus, we obtain a linear
profile connecting the two bulbs:
D2 x 0
J 
ct
D x L  x R
L
Species Balance for a bulb

ct
dxi
  Ni
dt

Div.Thm.
ctV
dxi
 Ni A  Ji A
dt
2-Bulb Problem
For left bulb:
dx i L
c tVL
 J i A
dt
Substituting our expression for the molar flux and rearranging:

dx L 
dt



A
R
L
D
x

x






LV L
We can eliminate the right-side mole fraction via an equilibrium balance.
Applying and simplifying:

dx L 
A  V L 

L

1
D
x

x








dt
LV L  V R 


In a multicomponent system, we’d need to decouple these equations to
solve them analytically. For our binary system, we can solve directly:

L

xi  xi
 Dt

e
L,o

xi  xi
Sources of Error
•
•
•
•
•
•
•
•
•
•
•
•
•
1-D Assumption
Quasi-Steady State Assumption
Infinite Reservoir Assumption
Constant cross-sectional area
Constant tissue thickness
Implantation errors
Dissection errors
Image Analysis errors
Cubic volume fraction assumption
Tissue shrinkage/swelling
Stokes-Einstein estimation
Sampling errors
Dissection-triggered cell changes