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