1. 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 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

2. 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 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

3. 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 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

4. 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-350 - PVA sponge, 350  m pore size (4)

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

6. 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

7. Finite Difference Modeling Step Change Ramp

8. 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…

9. 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!

10. Supplemental Slides

11. 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 i,upper @ t = 0 Assuming tanks are equal volumes, we can say C i,lower = C o i,lower -C i,upper

12. 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.

13. 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

14. 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.

15. Other Composite Correlations Rayleigh’s Correlation for Densely-Packed Spheres Rayleigh’s Correlation for Long Cylinders Source: BSL, 2nd Edition, p.281-282. Maxwell’s Correlation for Diffuse Spheres

16. What are the Volume Fractions?

17. Other Way to Estimate the Lag Time Composite Resistances D AB,1 D AB,2 L1L1 L2L2 CC

18. 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 ):

19. The Finite Difference Model Transient Species Balance Discretized Transient Species Balance Boundary Conditions: where 1/F > 20 in the model to ensure stability

20. 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.

21. 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 flux @ boundaries --> N t = 0 Species Balance for a bulb Div.Thm.

22. 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:

23. 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