Unit Cell Characterization, Representation, and Assembly of 3D Porous Scaffolds Connie Gomez, M. Fatih Demirci, Craig Schroeder Drexel University 4/19/05

Outline Overview of Project Overview of Project Earth Mover’s Distance (EMD) Earth Mover’s Distance (EMD) Summary Summary

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

Overview of Approach Compare Unit Cells from Library to the current scaffold Initial Assembly Using a Given/Reference: Unit Cell(s) based on Application Heterogeneous Scaffold and Implant Design Preprocessing Unit Cell Characterization Application Requirements Given Cell Library Growing the Scaffold + Unit Cell Rotation + Scaffold Analysis and Property Update

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

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

Assembly Given the volume (anatomical geometry) Given the volume (anatomical geometry) Bottom Up Approach Bottom Up Approach Starts from a unit cell at a given location within the volume with only a primary direction Starts from a unit cell at a given location within the volume with only a primary direction Top Down Approach Top Down Approach Starts with two unit cells and a path to optimize. Starts with two unit cells and a path to optimize.

Alignment Approaches Bottom-Up Approach Bottom-Up Approach Assemble the unit cells along the primary direction 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 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 Grow plane of unit cells in the third direction, starting from the reference unit cell Top-Down Approach Top-Down Approach Construct the path, filling in cells along the path so as to minimize total discontinuity between cells 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 Fill in the rest of the scaffold, choosing cells that minimize discontinuity

Outline Overview of Project Overview of Project Earth Mover’s Distance (EMD) Earth Mover’s Distance (EMD) Summary Summary

Earth Mover’s Distance (EMD) Measure of dissimilarity between two sets of elements Measure of dissimilarity between two sets of elements How much “work” is required to turn the first set into the second set How much “work” is required to turn the first set into the second set Known to be suitable in other domains Known to be suitable in other domains E.g., Optimization E.g., Optimization

Motivating Problem Ensure optimal alignment between two unit cells Ensure optimal alignment between two unit cells Optimization based on application Optimization based on application One unit cell is source One unit cell is source Other is destination Other is destination Velocity in terms of EMD Velocity in terms of EMD Source Unit Cell 1 Destination Unit Cell 2

Why EMD? Allows partial matches in a natural way Allows partial matches in a natural way Two skeletons might not match up exactly Two skeletons might not match up exactly Partial matches make the comparison more flexible Partial matches make the comparison more flexible What else goes here? What else goes here?

Computing EMD EMD can be expressed in terms of the transportation problem EMD can be expressed in terms of the transportation problem Source (elements of skeleton characterizing void of one skeleton) Source (elements of skeleton characterizing void of one skeleton) Destination (elements of skeleton) Destination (elements of skeleton) Transport “properties” from one skeleton to the other Transport “properties” from one skeleton to the other Source Unit Cell 1 Destination Unit Cell 2

Earth Mover’s Distance (EMD) EMD is given EMD is given Two skeletons, with their properties and geometry Two skeletons, with their properties and geometry A function to determine how dissimilar two skeleton points are A function to determine how dissimilar two skeleton points are Taking into account geometry Taking into account geometry Called ground distance Called ground distance Eg, Euclidean distance between the skeleton points Eg, Euclidean distance between the skeleton points EMD’s goal EMD’s goal A measure of how dissimilar the two skeletons are A measure of how dissimilar the two skeletons are Optimal rotation of skeletons Optimal rotation of skeletons (and therefore unit cells)

Example Property: Velocity Property: velocity Property: velocity Speed of flow at any point Speed of flow at any point Skeleton points hold the average flow for the areas surrounding them Skeleton points hold the average flow for the areas surrounding them r=6.7

Velocity in Terms of EMD At skeleton points on the boundary, the velocity is a component of flux At skeleton points on the boundary, the velocity is a component of flux EMD pairs up skeleton points in one unit cell with skeleton points in the other EMD pairs up skeleton points in one unit cell with skeleton points in the other Flow can pass across the boundary between them Flow can pass across the boundary between them EMD favors closer matches, thus improving connections EMD favors closer matches, thus improving connections

Total Flow and EMD The total velocity of one skeleton boundary is entirely matched with the other skeleton boundary The total velocity of one skeleton boundary is entirely matched with the other skeleton boundary The maximum amount of flow for one of the two unit cells is computed The maximum amount of flow for one of the two unit cells is computed The total flow is limited by the amount either unit cell can transmit The total flow is limited by the amount either unit cell can transmit Flow can be divided up Flow can be divided up The flow out of one skeleton point may enter the other unit cell through multiple skeleton points The flow out of one skeleton point may enter the other unit cell through multiple skeleton points

Scope of Problem Skeletons instead of exact representation Skeletons instead of exact representation Reduced complexity of operations Reduced complexity of operations Averaging – loses information Averaging – loses information

Ground Distance in EMD EMD needs some way to determine geometric distance between two skeleton points EMD needs some way to determine geometric distance between two skeleton points Skeleton points have positions Skeleton points have positions A pos, B pos A pos, B pos GroundDistance(A, B) = Distance(A pos, B pos ) GroundDistance(A, B) = Distance(A pos, B pos ) Properties are not taken into account yet! Properties are not taken into account yet!

EMD Usage An association between two skeleton points An association between two skeleton points A dissimilarity: GroundDistance(A, B) A dissimilarity: GroundDistance(A, B) An amount of property being associated: a An amount of property being associated: a Cost of the association: GroundDistance(A, B) * a Cost of the association: GroundDistance(A, B) * a An alignment between two skeletons An alignment between two skeletons A set of associations between skeleton points A set of associations between skeleton points All of the property of skeleton has been associated with skeleton points from the other skeleton All of the property of skeleton has been associated with skeleton points from the other skeleton Total cost of alignment is minimal Total cost of alignment is minimal

Scope of Problem (cont’d) Turbulence ignored across boundary Turbulence ignored across boundary What we are not ignoring What we are not ignoring Disparity between the skeletons is not ignored Disparity between the skeletons is not ignored Boundary discontinuity Boundary discontinuity Flow capacity is being used in alignment Flow capacity is being used in alignment Boundary flow discontinuity Boundary flow discontinuity

Example Two skeletons to compare Two skeletons to compare Circle Rectangle

One Match One match between skeleton points One match between skeleton points

(68.0, 82.2) (80.7, 80.0) (57.7, 80.0) (76.9, 76.0) (64.0, 80.0) (76.9, 76.0)(64.0, 80.0) (57.7, 80.0)(64.0, 80.0) (57.7, 80.0)(68.0, 82.2) (80.7, 80.0) (68.0, 82.2) Point 1 Point 2 DistAttrContribution

Attributes and Work Each arrow represents an association between skeleton points Each arrow represents an association between skeleton points Total attribute amount for a set of skeleton points is divided among the arrows attached to them (*) Total attribute amount for a set of skeleton points is divided among the arrows attached to them (*) * Remember that one skeleton is not entirely matched.

Not One-to-One: Multiple Matches Some points have multiple arrows touching them Some points have multiple arrows touching them These points have more of the attribute than should be contributed These points have more of the attribute than should be contributed Multiple arrows, total, contain the full attribute contribution of that point Multiple arrows, total, contain the full attribute contribution of that point * Remember that one skeleton is not entirely matched

Not One-to-One: No Match Some skeleton points have no arrows touching them Some skeleton points have no arrows touching them There is more of this attribute in the red skeleton There is more of this attribute in the red skeleton Not all of the red points can be associated with green Not all of the red points can be associated with green All of green points match with one or more red points All of green points match with one or more red points

Computing EMD EMD

Reference Unit Cell Skeleton with Properties Skeleton Alignment Using Multiple Parameters Characterized Reference Unit Cell Unit Cell Regrowth in Scaffold with and without Rotation Heterogeneous Scaffold

Future Work: Use all properties simultaneously to formulate this problem as one optimization problem Use all properties simultaneously to formulate this problem as one optimization problem Pareto optimization with multiple objectives Pareto optimization with multiple objectives Find the optimal rotation, regrow the void inside the scaffold. Find the optimal rotation, regrow the void inside the scaffold.

Questions

EMD Equations Amount of property sent is always positive Capacity of source not exceeded Capacity of destination not exceeded Work is “force” times “distance” EMD is normalized by the total amount of flow to deal with partial matches