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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 . 1997 . 37: 401-412. Motivation

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engineering the tissue which encapsulates subcutaneous implants i diffusion properties
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 properties1
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
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
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

porosity reduces encapsulation
Porosity Reduces Encapsulation

PVA-skin

PVA-60

S = A3/2

Is this an appropriate assumption?

fibrous tissue inhibits diffusion
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
  • 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
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!

two chamber diffusion
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
maxwell s composite correlation

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 correlation1

We assume profiles of the form:

Maxwell’s Composite Correlation

Outside Sphere

Inside Sphere

Subject to the boundary conditions:

v = v’

for r = a, 0 ≤  ≤ 

Solving for A and B, we find:

maxwell s composite correlation2

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:

Maxwell’s Composite Correlation

Equating these two expressions, we can solve for Zeff:

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

other composite correlations

Maxwell’s Correlation for Diffuse Spheres

Other Composite Correlations

Rayleigh’s Correlation for Densely-Packed Spheres

Rayleigh’s Correlation for Long Cylinders

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

other way to estimate the lag time

Other Way to Estimate the Lag Time

C

Composite Resistances

DAB,1

DAB,2

L1

L2

other way to estimate the lag time1

Other Way to Estimate the Lag Time

In Cartesian Co-ords (A1=A2):

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

In Cylindrical Co-ords:

In Spherical Co-ords:

the finite difference model

The Finite Difference Model

Discretized Transient Species Balance

Transient Species Balance

Boundary Conditions:

1/F > 20 in the model to ensure stability

where

rats v humans

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

2-Bulb Problem

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.

2 bulb problem1

For left bulb:

2-Bulb Problem

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
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