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Engineering the Tissue Which Encapsulates Subcutaneous Implants. I. Diffusion Properties

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## Engineering the Tissue Which Encapsulates Subcutaneous Implants. I. Diffusion Properties

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### Maxwell’s Composite Correlation

### Maxwell’s Composite Correlation

### Maxwell’s Composite Correlation

### Other Composite Correlations

### Other Way to Estimate the Lag Time

### The Finite Difference Model

### Rats v. Humans

### 2-Bulb Problem

### 2-Bulb Problem

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

Fibrous Tissue Inhibits Diffusion

Concentrated Chamber

Dilute Chamber

Membrane

Ussing-type Diffusion Chamber

Fluorescein

MW 376

PVA-350 SQ PVA-60 SS PVA-skin

Maxwell’s correlation for composite media:

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!

Assume membrane adjusts rapidly to changes in concentration

Species balance for each tank

Two-Chamber Diffusion- Expanding flux terms

- Integrating w/ Coi,lower-Coi,upper @ t = 0

- Assuming tanks are equal volumes, we can say Ci,lower = Coi,lower-Ci,upper

- Combine species balances

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.

We assume profiles of the form:

Outside Sphere

Inside Sphere

Subject to the boundary conditions:

v = v’

for r = a, 0 ≤ ≤

Solving for A and B, we find:

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:

Equating these two expressions, we can solve for Zeff:

This expression can be written in various forms, including the one listed in the paper.

Maxwell’s Correlation for Diffuse Spheres

Rayleigh’s Correlation for Densely-Packed Spheres

Rayleigh’s Correlation for Long Cylinders

Source: BSL, 2nd Edition, p.281-282.

In Cartesian Co-ords (A1=A2):

For DAB,1 = 2.35 and DAB,2 = 1.11:

In Cylindrical Co-ords:

In Spherical Co-ords:

Discretized Transient Species Balance

Transient Species Balance

Boundary Conditions:

1/F > 20 in the model to ensure stability

where

“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.

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:

Species Balance for a bulb

Div.Thm.

Substituting our expression for the molar flux and rearranging:

We can eliminate the right-side mole fraction via an equilibrium balance. Applying and simplifying:

In a multicomponent system, we’d need to decouple these equations to solve them analytically. For our binary system, we can solve directly:

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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