Back Calculation of the Permeability of Reported Scaffolds Using Mathematical Models Wettergreen MA, Liebschner MA Computational and Experimental Biomechanics Laboratory Rice University, Houston, TX 77098

Outline Permeability in tissue regeneration Measurement of permeability Mathematical modeling and comparison Conclusions

Permeability in tissue regeneration

Measurement of permeability

Mathematical modeling and comparison

Conclusions

Introduction Porous conduits currently the research standard for tissue regeneration Permeability is important parameter determining transfer properties and success of tissue integration in porous conduits Few studies account for permeability in design of scaffolds Modulation of void volume can change transfer properties without changing mechanical properties Modeling may aid in determining specific fluid profile required for successful tissue integration

Porous conduits currently the research standard for tissue regeneration

Permeability is important parameter determining transfer properties and success of tissue integration in porous conduits

Few studies account for permeability in design of scaffolds

Modulation of void volume can change transfer properties without changing mechanical properties

Modeling may aid in determining specific fluid profile required for successful tissue integration

Methodology Methodology Literature review of the use of porous scaffolds Application of current models for back determining permeability Evaluation of permeability of generated porous solids

Methodology

Literature review of the use of porous scaffolds

Application of current models for back determining permeability

Evaluation of permeability of generated porous solids

Literature survey Porogen Leaching Square or spherical pore geometries Porosity 60 < x < 95 Pore volumes ranged (150 µ m to 700 µ m) Depending upon tissue of interest Complex tortuosity and permeability Rapid Prototyping Uniform, orthogonal channels Porosity 50 < x < 85 Pore volumes ranged (20 µ m to 900 µ m) (Mostly) Uniform tortuosity and permeability

Porogen Leaching

Square or spherical pore geometries

Porosity 60 < x < 95

Pore volumes ranged (150 µ m to 700 µ m)

Depending upon tissue of interest

Complex tortuosity and permeability

Rapid Prototyping

Uniform, orthogonal channels

Porosity 50 < x < 85

Pore volumes ranged (20 µ m to 900 µ m)

(Mostly) Uniform tortuosity and permeability

Determination of pore architecture parameters for permeability Mercury porosimetry Non-reactive, non-wetting liquid will penetrate pores under sufficient pressure Assumes circular cross-sections Washburn equation: µCT Global scaffold characteristics Relies upon density thresholding SEM Surface analysis only Direct determination of parameters through visualization only Porogen Dimensions Measured from Imaging 0 200 400 600 800 1000 Arm Length Leg Width Leg Length Dimension (um) Designed SEM uCT

Mercury porosimetry

Non-reactive, non-wetting liquid will penetrate pores under sufficient pressure

Assumes circular cross-sections

Washburn equation:

µCT

Global scaffold characteristics

Relies upon density thresholding

SEM

Surface analysis only

Direct determination of parameters through visualization only

Washburn Equation: P is the applied pressure, D is the diameter, is the surface tension of mercury (480 dyne cm-1 and is the contact angle between mercury and the pore wall, usually near 140°. As pressure increases, the instrument senses the intrusion volume of mercury by the change in capacitance between the mercury column and a metal sheath surrounding the stem of the sample cell. As the mercury column shortens, the pressure and volume data are continuously acquired and displayed by an attached personal computer.

Modeling limitations Pore limitations Trapped pores Dead end pores Basic modeling assumptions Porosity Pore distribution No-slip conditions tortuosity

Pore limitations

Trapped pores

Dead end pores

Basic modeling assumptions

Porosity

Pore distribution

No-slip conditions

tortuosity

Capillary models Simple model accounts for n channels of uniform length and diameter Complex model accounts for variation in pore diameter and length

Simple model accounts for n channels of uniform length and diameter

Complex model accounts for variation in pore diameter and length

Hydraulic Radius Application of capillary based models Hydraulic radius is ratio of volume to surface area of global solid Works well for packed solids Kozeny-Carman relation:

Application of capillary based models

Hydraulic radius is ratio of volume to surface area of global solid

Works well for packed solids

Drag Models Opposite of Capillary model Capillaries are obstructions to flow Permeability is dependent upon flow rate Darcy substitution:

Opposite of Capillary model

Capillaries are obstructions to flow

Permeability is dependent upon flow rate

Phenomenological Models Permeation factor, K (m/s) is exponentially related to parameters Includes empirically calculated parameters

Permeation factor, K (m/s) is exponentially related to parameters

Includes empirically calculated parameters

Results – Experimental vs. Models 1.58x10 -2 3.22x10 -7 7.05x10 -8 1.10x10 -8 1.21x10 -8 5.47x10 -4 64% Y-Shape 7.64x10 -3 2.2x10 -7 3.88x10 -8 9.47x10 -9 1.04x10 -8 1.71x10 -4 55% Y-Shape Phenomenological Drag Theory Hydraulic Radius Serial Capillary Straight Capillary Measured Permeability COMPARISON OF MODELS TO EXP. RESULTS 1.70E-04 4.49E-04 9.31E-04 3.3145E-15 1.85623E-15 1.27405E-15 2.68E-08 1.00E-08 5.52E-09 7.70E-08 6.48E-08 5.57E-08 2.87E-05 2.42E-05 2.08E-05 3.33x10 -4 9.25x10 -5 2.38x10 -5 76% NaCl 64% NaCl 55% NaCl

Conclusions Improved modeling can be accomplished by taking into account architectural considerations (phenomenological) Error matches previous studies (Breysse); similar specimens may exhibit orders of magnitude difference in results Hierarchical model or network model combined with topological concepts may be optimal for the complete representation and determination of the fluid properties Current theoretical models do not take into account complex geometry, may need to use reverse engineering to develop predictive model through curve fitting

Improved modeling can be accomplished by taking into account architectural considerations (phenomenological)

Error matches previous studies (Breysse); similar specimens may exhibit orders of magnitude difference in results

Hierarchical model or network model combined with topological concepts may be optimal for the complete representation and determination of the fluid properties

Current theoretical models do not take into account complex geometry, may need to use reverse engineering to develop predictive model through curve fitting

Future Directions Complete application of models to measurement of permeability Modeling of permeability of regular architectures

Complete application of models to measurement of permeability

Modeling of permeability of regular architectures