1. Connie Gomez, M. Fatih Demirci, Craig Schroeder
Unit Cell Characterization, Representation, and Assembly of 3D Porous Scaffolds
Connie Gomez, M. Fatih Demirci, Craig Schroeder
Drexel University
4/11/05

2. Outline
Quick Summary of Project
Earth Mover’s Distance (EMD)

3. Problem Statement
Develop a framework to assemble biocompatible unit cell structures that mimic tissue properties to serve as a scaffold. ?
Unit Cell Structures

4. Design Considerations
Informatics
Possible Design Solutions
1) Mechanical requirements:
scaffold structural integrity
internal architectural stability
scaffold strength and stiffness
E & EEff
G & GEff ν α ρ φS
dpore
Apore
biomaterial selection
internal architecture
porosity and pore distribution
fabrication method
2) Biological requirements:
cell loading, distribution, and nutrition
cell attachment and growth
cell-tissue aggregation and formation
dpore
Apore
θpore φS
φFC
layout
pore size and interconnectivity;
vasculature
3) Geometrical requirements:
anatomical fitting L w h l φS
θpore
dpore
Apore
scaffold external geometry
interconnectivity
permeability selection φS
φFC
Apore k P V T μ ρ D Re
4) Transport requirements:
nutrient and oxygen delivery
waste removal
drug delivery

5. Overview of Approach Preprocessing Initial Assembly
Unit Cell Characterization
Application Requirements
Initial Assembly
Using a Given/Reference
Unit Cell(s)
Aligning Current Assembly with the Database Unit Cells
Adding to the Assembly +
Unit Cell Rotation
Vector Update
Heterogeneous Scaffold and Implant Design

6. Informatics: Mechanical
Scaffold Material Properties
Effective Young’s Modulus (EEff)
Effective Shear Modulus (GEff)
Poisson’s Ratio (ν)
Coefficient of Expansion (α)
Contour/Fluid Properties
Diffusion Constant (D)
Viscosity (μ)
Density (ρ)
Permeability (k)

7. Informatics: Structural
Porosity/Volume Fraction
Scaffold Material(φS)
Contour/Fluid Material(φF)
Effective/Open Porosity(φFC)
Dead/Closed Porosity (φFD)
Pore Size (dpore)
Pore Area (Apore)
Pore Angle (θpore)

8. Informatics: Transport
Mass/Fluid Flux normal to surface (m)
Velocity (u, v, w)
Pressure (P)
Geometric Tortuosity (T)

9. Model for Transport Characterization
Create model of pore space
Unit Cell Length (L)
Length of pore (l)
Height of pore (h)
Width of pore (w)
Diameter of pore (dpore)
Area of pore (Apore)
Pore angle (θpore)

10. Transport Characterization Input
Define fluid properties
Density (ρ)
Viscosity (μ)
Specific Heat
Define flow parameters
Velocity Magnitude
Velocity Direction
Turbulent or Laminar
Check Re
Define boundary conditions
Inlets
Outlets
Walls

11. Results for the given properties and flow parameters
CFD Analysis
Mesh Model
Run Analysis
Nodal Information
Node Coordinate and number
Velocity (u, v, w)
Pressure
Surface Meshing: 2570 shells
Volume Meshing:18442 cells
Results for the given properties and flow parameters

12. Unit Cell Representation
Common Engineering Representations
CAD
STL
IGES
Disadvantages:
Not suitable for computing unit cell connectivity
Complexity of optimization increases as the size of the scaffold increases
We can not associate properties using these representations.
Any operations that we do using skeletons cost much less than these common engineering representations. Rotation for example.

13. Skeletonization Skeleton:
An intuitive representation of shape and can be easily understood by the user, providing more control in the alignment process.
Captures the topology of an object in both two and three dimensions.

14. Skeletonization – Example:
(x, y, radius)

17. Skeletonization – Example:

19. Skeletonization – Example:

20. Modified 2D Skeletonization
Set of skeleton points
Coordinates and the radius of the circle used to find the skeleton point (x, y, z, r)
Layering
Three directions

21. Skeleton Point and Nodal Point Superposition
Skeletal Points
Coordinates
Radius
Nodal Points
Property Values at nodal point
Velocity (u, v, w)
Pressure

22. Property Calculation r=6.7
The nodal points can be associated with different cross sections.

23. Property Assignment Expansion of the skeleton representation
Original Skeleton Point
Unit Cell 1: [ x, y, z, r]
Layered in three directions for rotation
Expanded Skeleton Point of the Pore Material
Unit Cell 1: [x, y, z, r, p1, p2, …pn]
Properties (normal to the surface)
P1: Density
P2: Velocity Magnitude
P3: Positive/Negative Direction
P4: Pressure
P5: Flow Rate
P6: Permeability

24. Overview of Approach Preprocessing Preprocessing Initial Assembly
Unit Cell Characterization
Application Requirements
Preprocessing
Unit Cell Characterization
Application Requirements
Initial Assembly
Using a Given/Reference
Unit Cell(s)
Aligning Current Assembly with the Database Unit Cells
Adding to the Assembly +
Unit Cell Rotation
Update Vectors
Heterogeneous Scaffold and Implant Design

25. Unit Cell Assembly (Alignment)
Framework to provide a structural and/or contour connectivity between unit cells
The goal :
To develop an approach that will assemble characterized unit cell structures into a larger heterogeneous scaffold

26. Assembly Given the volume (anatomical geometry) Bottom Up Approach
Starts from a unit cell at a given location within the volume with only a primary direction
Top Down Approach
Starts with two unit cells and a path to optimize.
Goal is to provide transport connectivity for one of the parameters on this path.

27. Bottom-Up Approach Assemble a scaffold given constraints
A reference unit cell
Outer scaffold geometry
Direction for flow
Three step assembly
Assemble the unit cells along the primary direction
Grow the line of unit cells in a second direction to form plane, starting from the reference unit cell
Grow plane of unit cells in the third direction, starting from the reference unit cell

28. Bottom-up Approach
Radius and Velocity.

29. Bottom-Up Assembly

30. Top-down Approach Assemble a scaffold given constraints
A path along which optimal flow is desired
Fixed connections at either end of the path
Outer scaffold geometry
Two step assembly
Construct the path, filling in cells along the path so as to minimize total discontinuity between cells
Fill in the rest of the scaffold, choosing cells that minimize discontinuity

31. Top-down: Path Filling
Choose cubes and orientations for each cell in the path
Minimize total discontinuity between adjacent cells along the path
Illustration: filled path as part of a scaffold

32. Top-down: Scaffold Filling
Choose cubes and orientations for each remaining cell
Fill from the path outwards
Choose best fit
Illustration: filled scaffold with path highlighted

33. Bottom-Up Approach using area

34. Top Down: Example using area

35. Top Down: Example using pressure

36. Summary
Unit cell informatics, necessary for unit cell alignment, have been set forth.
Unit cell characterization approaches have been outlined.
The unit cell’s geometry has been reduced to a skeletal representation to reduce complexity.
Top-down and bottom-up approaches have been used to create an assembly inside a 3D volume.

37. Earth Mover’s Distance (EMD)
Measure of dissimilarity between two sets of elements
How much “work” is required to turn the first set into the second set

38. Earth Mover’s Distance (EMD)
Given
Two sets of elements
Some way to determining how dissimilar two elements are
Result
A measure of how dissimilar the two sets are

39. EMD Usage EMD takes two sets of elements
The elements are the skeleton points
The elements include the properties attached to the skeleton points
The sets of elements are the skeletons

40. EMD Usage
EMD needs some way to determine how dissimilar two elements are
Skeleton points differ by position
Apos, Bpos
Dissimilarity(A, B) = Distance(Apos, Bpos)
Attributes are taken into account later
This is called Ground Distance; EMD uses this

41. EMD Usage A match between two skeleton points
A dissimilarity: Distance(Apos, Bpos)
An amount of attribute being matched: a
Cost of match: Distance(Apos, Bpos) * a
A match between two skeletons
A set of matches between skeleton points
All of the attribute of skeleton has been associated with skeleton points from the other skeleton
Total cost of all matches is minimal

43. EMD for Alignment

44. Computing the best arrangement

46. Example
Two skeletons to compare
Circle
Rectangle

47. One Match
One match between skeleton points

48. Attributes and Work All matches
Attribute of each node spread across the arrows
Cost of each arrow is the length of the arrow times the amount of attribute assigned to it
“Force” required to move that much attribute
This distance times “force” is where the notion of “work” comes from
Total “work” is minimized

49. Not One-to-One
Note that some skeleton points have no arrows touching them
There is more of this attribute in the red skeleton
Not all of the red points have room to match with green
All of green points match with one or more red points
Note that others have multiple arrows touching them
Green or red nodes
These nodes have more of the attribute than what they are paired with
Multiple arrows, total, contain the full amount of the attribute of that node

50. Why EMD? Known to be suitable in other domains
Object Recognition
Who else uses this?
Allows partial matches in a natural way
Two skeletons might not match up exactly
Partial matches make the comparison more flexible
What else goes here?

51. Simple Example
Do we need some sort of illustration(s) that show a small example of EMD step by step?
Better idea of what is going on
EMD process hard to visualize – linear programming problem
Simplex method?
Example
How many nodes?
Should they be complete scaffolds?
Should they be labeled with attributes?
Should they be labeled with coordinates?
Should costs be computed and shown per edge?
Should the cost be computed and shown for the whole thing?
Should the optimal and a suboptimal match be shown for comparison?
Does he need to see this example through all stages of the pipeline?
Probably a lot of work!
He seems to be okay with the other parts of the process
Questionable benefit in understanding EMD

52. Questions