engineering the tissue which encapsulates subcutaneous implants. i. diffusion properties a. adam...
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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
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
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
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)
Porosity Reduces Encapsulation
PVA-60 PVA-skin
S = A3/2
Is this an appropriate assumption?
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
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
• 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
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 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
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.
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
⎛
⎝ ⎜
⎞
⎠ ⎟
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
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
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):
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
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
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.
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
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
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