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![Page 1: Engineering the Tissue Which Encapsulates Subcutaneous Implants. I. Diffusion Properties A. Adam Sharkawy, Bruce Klitzman, George A. Truskey, W. Monty.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649c9e5503460f9495e04f/html5/thumbnails/1.jpg)
Engineering the Tissue Which Encapsulates Subcutaneous Implants. I. Diffusion Properties
A. Adam Sharkawy, Bruce Klitzman, George A. Truskey, W. Monty ReichertDept. of Biomedical Engineering, Duke University
J Biomed Mater Res. 1997. 37: 401-412
Objectives• Demonstrate impact of implant surface on encapsulation tissue• Measure binary diffusion coefficient of a small-molecule analyte through each tissueApproach
• Implantation in subcutaneous tissue of rats
• Histology of encapsulation tissue at implant surface
• Two-chamber measurements of diffusion coefficient across tissue
Motivation
• Demonstrate that implant surface architecture impacts the mass transfer properties of the surrounding tissue
![Page 2: Engineering the Tissue Which Encapsulates Subcutaneous Implants. I. Diffusion Properties A. Adam Sharkawy, Bruce Klitzman, George A. Truskey, W. Monty.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649c9e5503460f9495e04f/html5/thumbnails/2.jpg)
Engineering the Tissue Which Encapsulates Subcutaneous Implants. I. Diffusion Properties
A. Adam Sharkawy, Bruce Klitzman, George A. Truskey, W. Monty ReichertDept. of Biomedical Engineering, Duke University
J Biomed Mater Res. 1997. 37: 401-412
Objectives• Demonstrate impact of implant surface on encapsulation tissue• Measure binary diffusion coefficient of a small-molecule analyte through each tissueApproach
• Implantation in subcutaneous tissue of rats
• Histology of encapsulation tissue at implant surface
• Two-chamber measurements of diffusion coefficient across tissue
Motivation
• Demonstrate that implant surface architecture impacts the mass transfer properties of the surrounding tissue
![Page 3: Engineering the Tissue Which Encapsulates Subcutaneous Implants. I. Diffusion Properties A. Adam Sharkawy, Bruce Klitzman, George A. Truskey, W. Monty.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649c9e5503460f9495e04f/html5/thumbnails/3.jpg)
Engineering the Tissue That Encapsulates Subcutaneous Implants. I. Diffusion Properties
A. Adam Sharkawy, Bruce Klitzman, George A. Truskey, W. Monty ReichertDept. of Biomedical Engineering, Duke University
J Biomed Mater Res. 1997. 37: 401-412
Objectives• Demonstrate impact of implant surface on encapsulation tissue• Measure binary diffusion coefficient of a small-molecule analyte through each tissueApproach
• Implantation in subcutaneous tissue of rats
• Histology of encapsulation tissue at implant surface
• Two-chamber measurements of diffusion coefficient across tissue
Motivation
• Demonstrate that implant surface architecture impacts the mass transfer properties of the surrounding tissue
![Page 4: Engineering the Tissue Which Encapsulates Subcutaneous Implants. I. Diffusion Properties A. Adam Sharkawy, Bruce Klitzman, George A. Truskey, W. Monty.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649c9e5503460f9495e04f/html5/thumbnails/4.jpg)
Implants in Sprague-Dawley Rats
PVA SpongeStainless Steel Mesh
Implant TypesParenthetical 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)
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Porosity Reduces Encapsulation
PVA-60 PVA-skin
S = A3/2
Is this an appropriate assumption?
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Fibrous Tissue Inhibits DiffusionComparison of Experimental Deff/D o to
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85
Area Fraction
Deff
/D
o
Measured
Maxwell Spheres
Rayleigh Spheres
Rayleigh Cylinders
€
−1
2ln
cAo − 2cB
cAo
⎛
⎝ ⎜
⎞
⎠ ⎟=
ADt
ΔxV
Ussing-type Diffusion Chamber
€
Deff
Do
=2(1−φs )
2+φs
Maxwell’s correlation for composite media:
Fluorescein
MW 376
PVA-350 SQ PVA-60 SS PVA-skin
Concentrated Chamber
Dilute Chamber
Membrane
![Page 7: Engineering the Tissue Which Encapsulates Subcutaneous Implants. I. Diffusion Properties A. Adam Sharkawy, Bruce Klitzman, George A. Truskey, W. Monty.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649c9e5503460f9495e04f/html5/thumbnails/7.jpg)
Finite Difference Modeling
Step Change
Ramp
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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…
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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!
![Page 10: Engineering the Tissue Which Encapsulates Subcutaneous Implants. I. Diffusion Properties A. Adam Sharkawy, Bruce Klitzman, George A. Truskey, W. Monty.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649c9e5503460f9495e04f/html5/thumbnails/10.jpg)
Supplemental Slides
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Two-Chamber Diffusion• Assume membrane adjusts rapidly
to changes in concentration
• Species balance for each tank
€
ji =DH
l
⎡
⎣ ⎢ ⎤
⎦ ⎥ ci, lower − ci, upper( )
€
Vlower
dCi, lower
dt= −Aji
€
Vupper
dCi, upper
dt= Aji
• Combine species balances
€
dCi, lower
dt−
dCi, upper
dt= −Aji
1
Vlower
+1
Vupper
⎛
⎝ ⎜
⎞
⎠ ⎟
• Expanding flux terms
€
d
dtCi, lower − Ci, upper( ) =
ADH
l
1
Vlower
+1
Vupper
⎛
⎝ ⎜
⎞
⎠ ⎟Ci, lower − Ci, upper( ) = βD Ci, lower − Ci, upper( )
• Integrating w/ Coi,lower-Co
i,upper @ t = 0
€
Ci, lower − Ci, upper
Coi, lower − Co
i, upper
= e−βDt
• Assuming tanks are equal volumes, we can say Ci,lower = Co
i,lower-Ci,upper
€
−1
2ln
Coi,lower − 2Ci,upper
Coi,lower
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
=ADHt
lV
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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.
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Maxwell’s Composite CorrelationWe assume profiles of the form:
€
v = Vr cosΘ +B
r2cosΘ
€
′ v = ArcosΘ
Subject to the boundary conditions:
v = v’
€
′ Z ∂ ′ v
∂r= Z
∂v
∂r for r = a, 0 ≤ ≤
Solving for A and B, we find:
€
v = Vr cosΘ +Va 3(Z − ′ Z )
r2(2Z + ′ Z )cosΘ
€
′ v =3ZVz
2Z + ′ Z
Outside Sphere
Inside Sphere
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Maxwell’s Composite CorrelationWe 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:
€
v = Vz +na3(Z − ′ Z )
r3(2Z + ′ Z )Vz
Equating these two expressions, we can solve for Zeff:
€
Zeff =3Z ′ Z φ + (2Z + ′ Z )Z(1− φ)
3Zφ + (2Z + ′ Z )(1− φ)€
v = Vz +b3(Z − Zeff )
r3(2Z + Zeff )Vz
This expression can be written in various forms, including the one listed in the paper.
![Page 15: Engineering the Tissue Which Encapsulates Subcutaneous Implants. I. Diffusion Properties A. Adam Sharkawy, Bruce Klitzman, George A. Truskey, W. Monty.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649c9e5503460f9495e04f/html5/thumbnails/15.jpg)
Other Composite Correlations
€
Deff
Do
= 1+3φs
Ds + 2Do
Ds − Do
⎛
⎝ ⎜
⎞
⎠ ⎟−φs +1.569
Ds − Do
3Ds − 4Do
⎛
⎝ ⎜
⎞
⎠ ⎟φs
10 / 3 +...
Rayleigh’s Correlation for Densely-Packed Spheres
Rayleigh’s Correlation for Long Cylinders
€
Deff , xx
Do
= 1+2φs
Ds + Do
Ds − Do
⎛
⎝ ⎜
⎞
⎠ ⎟−φs +
Ds − Do
Ds + Do
⎛
⎝ ⎜
⎞
⎠ ⎟ 0.30584φ 4 + 0.013363φ 8 +...( )
Source: BSL, 2nd Edition, p.281-282.
Maxwell’s Correlation for Diffuse Spheres
€
Deff
Do
=
2
Ds
+1
Do
− 2φs
1
Ds
−1
Do
⎛
⎝ ⎜
⎞
⎠ ⎟
2
Ds
+1
Do
+ φs
1
Ds
−1
Do
⎛
⎝ ⎜
⎞
⎠ ⎟
![Page 16: Engineering the Tissue Which Encapsulates Subcutaneous Implants. I. Diffusion Properties A. Adam Sharkawy, Bruce Klitzman, George A. Truskey, W. Monty.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649c9e5503460f9495e04f/html5/thumbnails/16.jpg)
What are the Volume Fractions?Comparison of Experimental Deff/D o to
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85
Area Fraction
Deff
/D
o
Measured
Maxwell Spheres
Rayleigh Spheres
Rayleigh Cylinders
![Page 17: Engineering the Tissue Which Encapsulates Subcutaneous Implants. I. Diffusion Properties A. Adam Sharkawy, Bruce Klitzman, George A. Truskey, W. Monty.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649c9e5503460f9495e04f/html5/thumbnails/17.jpg)
Other Way to Estimate the Lag Time
€
NA = −Ct ADAB∇xA ≈ −Ct ADAB
ΔxA
L
€
−L
Ct ADAB
=ΔxA
NA
= Rt
€
Rt =L1
−Ct A1DAB ,1
+L2
−Ct A2DAB ,2
Composite Resistances DAB,1 DAB,2
L1 L2
C
![Page 18: Engineering the Tissue Which Encapsulates Subcutaneous Implants. I. Diffusion Properties A. Adam Sharkawy, Bruce Klitzman, George A. Truskey, W. Monty.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649c9e5503460f9495e04f/html5/thumbnails/18.jpg)
Other Way to Estimate the Lag Time
For DAB,1 = 2.35 and DAB,2 = 1.11:
€
Rt ∝1
DAB ,1
+1
DAB ,2
In Cylindrical Co-ords:
€
Rt ∝1
1.5DAB ,1
+1
DAB ,2
In Spherical Co-ords:
€
Rt ∝1
2.25DAB ,1
+1
DAB ,2
€
Ro
Rt
= 3.1
€
Ro
Rt
= 2.8
€
Ro
Rt
= 2.6
In Cartesian Co-ords (A1=A2):
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The Finite Difference Model
€
∂c
∂t= D
∂ 2c
∂x 2+
∂ 2c
∂y 2
⎛
⎝ ⎜
⎞
⎠ ⎟
€
c t +1i, j = F C t
i+1, j + c ti−1, jc t
i, j +1 + c ti, j−1( ) + 1− 4F( )c t
i, j
€
∂c
∂x= 0
€
∂c
∂y= 0
€
D∂c
∂y= jreaction
Transient Species Balance Discretized Transient Species Balance
Boundary Conditions:
where
€
jreaction xm2
Dmcm
=100
€
F =DΔt
Δx 2
1/F > 20 in the model to ensure stability
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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.
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2-Bulb Problem
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:
€
J = −c t
LD[ ] x L − xR
No flux @ boundaries --> Nt = 0
€
D[ ] ∇ 2x( ) = 0
Species Balance for a bulb
€
c t
dx i
dt= −∇ • N i
€
c tVdx i
dt= −N iA = −JiA
Div.Thm.
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2-Bulb ProblemFor left bulb:
We can eliminate the right-side mole fraction via an equilibrium balance. Applying and simplifying:
€
c tVL
dx iL
dt= −JiA
Substituting our expression for the molar flux and rearranging:
€
d x L( )
dt=
A
LV L D[ ] xR( ) − x L
( )( )
€
d x L( )
dt=
A
LV L1+
V L
V R
⎛
⎝ ⎜
⎞
⎠ ⎟ D[ ] x∞
( ) − x L( )( )
In a multicomponent system, we’d need to decouple these equations to solve them analytically. For our binary system, we can solve directly:
€
x iL − x i
∞
x iL,o − x i
∞ = e−βDt
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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