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

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

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 ?

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

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

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

Informatics: Structural Porosity/Volume Fraction Porosity/Volume Fraction Scaffold Material(φ S ) Scaffold Material(φ S ) Contour/Fluid Material(φ F ) Contour/Fluid Material(φ F ) Effective/Open Porosity(φ FC ) Effective/Open Porosity(φ FC ) Dead/Closed Porosity (φ FD ) Dead/Closed Porosity (φ FD ) Pore Size (d pore ) Pore Size (d pore ) Pore Area (A pore ) Pore Area (A pore ) Pore Angle (θ pore ) Pore Angle (θ pore )

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

Model for Transport Characterization Create model of pore space Create model of pore space Unit Cell Length (L) Unit Cell Length (L) Length of pore (l) Length of pore (l) Height of pore (h) Height of pore (h) Width of pore (w) Width of pore (w) Diameter of pore (d pore ) Diameter of pore (d pore ) Area of pore (A pore ) Area of pore (A pore ) Pore angle (θ pore ) Pore angle (θ pore )

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

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

Unit Cell Representation Common Engineering Representations Common Engineering Representations CAD CAD STL STL IGES IGES Disadvantages: Disadvantages: Not suitable for computing unit cell connectivity Not suitable for computing unit cell connectivity Complexity of optimization increases as the size of the scaffold increases Complexity of optimization increases as the size of the scaffold increases

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.

Skeletonization – Example: (x, y, radius)

Skeletonization – Example:

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

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

Property Calculation r=6.7

Property Assignment Expansion of the skeleton representation Expansion of the skeleton representation Original Skeleton Point Original Skeleton Point Unit Cell 1: [ x, y, z, r] Unit Cell 1: [ x, y, z, r] Layered in three directions for rotation Layered in three directions for rotation Expanded Skeleton Point of the Pore Material Expanded Skeleton Point of the Pore Material Unit Cell 1: [x, y, z, r, p 1, p 2, …p n ] Unit Cell 1: [x, y, z, r, p 1, p 2, …p n ] Properties (normal to the surface) Properties (normal to the surface) P1: Density P1: Density P2: Velocity Magnitude P2: Velocity Magnitude P3: Positive/Negative Direction P3: Positive/Negative Direction P4: Pressure P4: Pressure P5: Flow Rate P5: Flow Rate P6: Permeability P6: Permeability

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

Unit Cell Assembly (Alignment) 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.

Bottom-Up Approach Assemble a scaffold given constraints Assemble a scaffold given constraints A reference unit cell A reference unit cell Outer scaffold geometry Outer scaffold geometry Direction for flow Direction for flow Three step assembly Three step assembly 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

Bottom-up Approach

Bottom-Up Assembly

Top-down Approach Assemble a scaffold given constraints Assemble a scaffold given constraints A path along which optimal flow is desired A path along which optimal flow is desired Fixed connections at either end of the path Fixed connections at either end of the path Outer scaffold geometry Outer scaffold geometry Two step assembly Two step assembly 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

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

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

Bottom-Up Approach using area

Top Down: Example using area

Top Down: Example using pressure

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

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

Earth Mover’s Distance (EMD) Given Given Two skeletons, with their properties and geometry Two skeletons, with their properties and geometry Some function to determine how dissimilar two skeleton points are Some 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 Goal Goal A measure of how dissimilar the two skeletons are A measure of how dissimilar the two skeletons are Optimal rotation of unit cells Optimal rotation of unit cells

Ground Distance in EMD EMD needs some way to determine how dissimilar two skeleton points are EMD needs some way to determine how dissimilar two skeleton points are Skeleton points differ by position Skeleton points differ by position A pos, B pos A pos, B pos Dissimilarity(A, B) = Distance(A pos, B pos ) Dissimilarity(A, B) = Distance(A pos, B pos ) Properties are taken into account later Properties are taken into account later

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

Computing EMD EMD can be expressed in terms of the transportation problem [1] EMD can be expressed in terms of the transportation problem [1] Suppliers (nodes of one skeleton) Suppliers (nodes of one skeleton) Consumers (nodes of other skeleton) Consumers (nodes of other skeleton) Transport “properties” from one node to another Transport “properties” from one node to another [1] F. L. Hitchcock. The distribution of a product from several sources to numerous localities. J. Math. Phys., 20: , 1941.

EMD Equations Add them here… Add them here…

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

Velocity in Terms of EMD At skeleton points on the boundary, the velocity is an indicator of flux At skeleton points on the boundary, the velocity is an indicator of flux EMD tries to pair up skeleton points in one unit cell with skeleton points in the other unit cell such that flow can pass across the boundary between them EMD tries to pair up skeleton points in one unit cell with skeleton points in the other unit cell such that flow can pass across the boundary between them EMD favors closer matches, thus improving connections EMD favors closer matches, thus improving connections

Velocity in Terms of EMD The total velocity of one skeleton boundary is entirely matched with skeleton points from the boundary of the other skeleton The total velocity of one skeleton boundary is entirely matched with skeleton points from the boundary of the other skeleton The maximum amount of flow is always matched The maximum amount of flow is always matched One skeleton may not be entirely matched, indicating that it admits more flow than the other One skeleton may not be entirely matched, indicating that it admits more flow than the other 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

Velocity in Terms of EMD Simplifications Simplifications Skeletons instead of exact representation Skeletons instead of exact representation Reduced complexity of operations Reduced complexity of operations Averaging – loses information Averaging – loses information Velocity assumed to be normal to boundary Velocity assumed to be normal to boundary Velocity corresponds to flux Velocity corresponds to flux Approximation Approximation Since the error because of this will be experienced by both unit cells, this error should be reduced Since the error because of this will be experienced by both unit cells, this error should be reduced

Velocity in Terms of EMD Simplifications Simplifications Fluid properties (e.g., viscosity) are ignored Fluid properties (e.g., viscosity) are ignored 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 matching Flow capacity is being used in matching Boundary flow discontinuity Boundary flow discontinuity

EMD for Alignment

Computing the best arrangement

Example Two skeletons to compare Two skeletons to compare Circle Rectangle

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

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

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

Why EMD? Known to be suitable in other domains Known to be suitable in other domains Object Recognition Object Recognition Who else uses this? Who else uses this? 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?

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

Questions