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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 : 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 Motivation Demonstrate that implant surface architecture impacts the mass transfer properties of the surrounding tissue

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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 : 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 Motivation Demonstrate that implant surface architecture impacts the mass transfer properties of the surrounding tissue

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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 : 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 Motivation Demonstrate that implant surface architecture impacts the mass transfer properties of the surrounding tissue

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Implants in Sprague-Dawley Rats PVA Sponge Stainless Steel Mesh 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 PVA sponge, 350 m pore size (4)

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Porosity Reduces Encapsulation PVA-60 PVA-skin S = A 3/2 Is this an appropriate assumption?

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Fibrous Tissue Inhibits Diffusion Ussing-type Diffusion Chamber Maxwell’s correlation for composite media: Fluorescein MW 376 PVA-350 SQ PVA-60 SS PVA-skin Concentrated Chamber Dilute Chamber Membrane

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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!

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Supplemental Slides

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Two-Chamber Diffusion Assume membrane adjusts rapidly to changes in concentration Species balance for each tank Combine species balances Expanding flux terms Integrating w/ C o i,lower -C o t = 0 Assuming tanks are equal volumes, we can say C i,lower = C o i,lower -C i,upper

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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 Correlation We assume profiles of the form: Subject to the boundary conditions: v = v’ for r = a, 0 ≤ ≤ Solving for A and B, we find: Outside Sphere Inside Sphere

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Maxwell’s Composite Correlation We now consider a larger sphere of radius b with many smaller spheres of radius a inside, such that na 3 = b 3, where is the volume fraction of small spheres in the large one. The following must be true: Equating these two expressions, we can solve for Z eff : This expression can be written in various forms, including the one listed in the paper.

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Other Composite Correlations Rayleigh’s Correlation for Densely-Packed Spheres Rayleigh’s Correlation for Long Cylinders Source: BSL, 2nd Edition, p Maxwell’s Correlation for Diffuse Spheres

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What are the Volume Fractions?

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Other Way to Estimate the Lag Time Composite Resistances D AB,1 D AB,2 L1L1 L2L2 CC

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Other Way to Estimate the Lag Time For D AB,1 = 2.35 and D AB,2 = 1.11: In Cylindrical Co-ords: In Spherical Co-ords: In Cartesian Co-ords (A 1 =A 2 ):

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The Finite Difference Model Transient Species Balance Discretized Transient Species Balance Boundary Conditions: where 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 “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: No boundaries --> N t = 0 Species Balance for a bulb Div.Thm.

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2-Bulb Problem For left bulb: We can eliminate the right-side mole fraction via an equilibrium balance. Applying and simplifying: Substituting our expression for the molar flux and rearranging: In a multicomponent system, we’d need to decouple these equations to solve them analytically. For our binary system, we can solve directly:

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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

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