Unit Cell Characterization, Unit Cell Characterization, Representation, and Representation, and

Assembly of 3D Porous Assembly of 3D Porous ScaffoldsScaffolds

Connie Gomez, M. Fatih Demirci, Craig Connie Gomez, M. Fatih Demirci, Craig SchroederSchroeder

Drexel UniversityDrexel University

4/19/054/19/05

OutlineOutline

Overview of ProjectOverview of Project Earth Mover’s Distance (EMD)Earth Mover’s Distance (EMD) SummarySummary

Problem StatementProblem Statement Develop a framework to assemble Develop a framework to assemble

biocompatible unit cell structures that biocompatible unit cell structures that mimic tissue properties to serve as a mimic tissue properties to serve as a scaffold.scaffold.

Unit Cell Structures

?

Overview of ApproachOverview 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

Unit Cell AssemblyUnit Cell Assembly

Framework to provide a structural Framework to provide a structural and/or contour connectivity between and/or contour connectivity between unit cells unit cells

The goal:The goal: To develop an approach that will To develop an approach that will

assemble characterized unit cell assemble characterized unit cell structures into a larger heterogeneous structures into a larger heterogeneous scaffoldscaffold

AssemblyAssembly Given the volume (anatomical geometry)Given the volume (anatomical geometry) Bottom Up ApproachBottom Up Approach

Starts from a unit cell at a given location Starts from a unit cell at a given location within the volume with only a primary within the volume with only a primary directiondirection

Top Down ApproachTop Down Approach Starts with two unit cells and a path to Starts with two unit cells and a path to

optimize.optimize.

OutlineOutline

Overview of ProjectOverview of Project Earth Mover’s Distance (EMD)Earth Mover’s Distance (EMD) SummarySummary

Earth Mover’s Distance Earth Mover’s Distance (EMD)(EMD)

Measure of dissimilarity between two Measure of dissimilarity between two sets of elementssets of elements

How much “work” is required to turn How much “work” is required to turn the first set into the second setthe first set into the second set

Known to be suitable in other Known to be suitable in other domainsdomains E.g., OptimizationE.g., Optimization

Motivating ProblemMotivating Problem Ensure optimal Ensure optimal

alignment between alignment between two unit cellstwo unit cells Optimization based on Optimization based on

applicationapplication One unit cell is sourceOne unit cell is source Other is destinationOther is destination Velocity in terms of Velocity in terms of

EMDEMD

SourceUnit Cell 1

DestinationUnit Cell 2

Computing EMDComputing EMD

EMD can be expressed in terms of EMD can be expressed in terms of the transportation problemthe transportation problem Source (elements of skeleton Source (elements of skeleton

characterizing void of one skeleton)characterizing void of one skeleton) Destination (elements of skeleton)Destination (elements of skeleton) Transport “properties” from one Transport “properties” from one

skeleton to the otherskeleton to the other

SourceUnit Cell 1

DestinationUnit Cell 2

Earth Mover’s Distance Earth Mover’s Distance (EMD)(EMD)

EMD is givenEMD is given Two skeletons, with their properties and geometryTwo skeletons, with their properties and geometry A function to determine how dissimilar two A function to determine how dissimilar two

skeleton points areskeleton points are Taking into account geometryTaking into account geometry Called Called ground distanceground distance Eg, Euclidean distance between the skeleton pointsEg, Euclidean distance between the skeleton points

EMD’s goalEMD’s goal A measure of how dissimilar the two skeletons areA measure of how dissimilar the two skeletons are Optimal rotation of skeletonsOptimal rotation of skeletons

(and therefore unit cells)(and therefore unit cells)

Example Property: VelocityExample Property: Velocity Property: velocityProperty: velocity

Speed of flow at any pointSpeed of flow at any point Skeleton points hold the average flow Skeleton points hold the average flow

for the areas surrounding themfor the areas surrounding them

r=6.7

Velocity in Terms of EMDVelocity in Terms of EMD

At skeleton points on the boundary, At skeleton points on the boundary, the velocity is a component of fluxthe velocity is a component of flux

EMD pairs up skeleton points in one EMD pairs up skeleton points in one unit cell with skeleton points in the unit cell with skeleton points in the otherother Flow can pass across the boundary Flow can pass across the boundary

between thembetween them EMD favors closer matches, thus EMD favors closer matches, thus

improving connectionsimproving connections

Total Flow and EMDTotal Flow and EMD

The total velocity of one skeleton The total velocity of one skeleton boundary is entirely matched with the boundary is entirely matched with the other skeleton boundary other skeleton boundary The maximum amount of flow for one of the The maximum amount of flow for one of the

two unit cells is computedtwo unit cells is computed The total flow is limited by the amount either The total flow is limited by the amount either

unit cell can transmitunit cell can transmit Flow can be divided upFlow can be divided up

The flow out of one skeleton point may enter The flow out of one skeleton point may enter the other unit cell through multiple skeleton the other unit cell through multiple skeleton pointspoints

Scope of ProblemScope of Problem

Skeletons instead of exact Skeletons instead of exact representationrepresentation Reduced complexity of operationsReduced complexity of operations Averaging – loses informationAveraging – loses information

EMD UsageEMD Usage

An association between two skeleton pointsAn association between two skeleton points A dissimilarity: GroundDistance(A, B)A dissimilarity: GroundDistance(A, B) An amount of property being associated: aAn amount of property being associated: a Cost of the association: GroundDistance(A, B) * aCost of the association: GroundDistance(A, B) * a

An alignment between two skeletonsAn alignment between two skeletons A set of associations between skeleton pointsA set of associations between skeleton points All of the property of skeleton has been All of the property of skeleton has been

associated with skeleton points from the other associated with skeleton points from the other skeletonskeleton

Total cost of alignment is minimalTotal cost of alignment is minimal

ExampleExample

Two skeletons to compareTwo skeletons to compareCircle

Rectangle

6.49

8.203.65

4.46

(68.0, 82.2)

(80.7, 80.0)

(57.7, 80.0)

(76.9, 76.0)

(64.0, 80.0)

54

45

46

52

55

(76.9, 76.0) (64.0, 80.0) 13.413.499

8.208.20 110.5110.555

(57.7, 80.0) (64.0, 80.0) 6.316.31 3.653.65 23.0523.05

(57.7, 80.0) (68.0, 82.2) 10.510.566

4.464.46 47.0447.04

(80.7, 80.0)

(68.0, 82.2)

6.4912.93

83.93

Point 1Point 1 Point 2Point 2 DistDist AttrAttr ContributiContributionon

Attributes and WorkAttributes and Work Each arrow Each arrow

represents an represents an association association between between skeleton pointsskeleton points

Total attribute Total attribute amount for a amount for a set of skeleton set of skeleton points is points is divided among divided among the arrows the arrows attached to attached to them (*)them (*)

* Remember that one skeleton is not entirely matched.

Not One-to-One: Multiple Not One-to-One: Multiple MatchesMatches

Some points Some points have multiple have multiple arrows arrows touching touching themthem These points These points

have more of have more of the attribute the attribute than should than should be be contributedcontributed

Multiple Multiple arrows, total, arrows, total, contain the contain the full attribute full attribute contribution contribution of that pointof that point* Remember that one

skeleton is not entirely matched.

6.49

8.203.65

4.46

10.95

7.62

8.11

8.22

11.85

54

45

46

52

55

Not One-to-One: No MatchNot One-to-One: No Match Some skeleton Some skeleton

points have no points have no arrows touching arrows touching themthem There is more of There is more of

this attribute in this attribute in the red skeletonthe red skeleton

Not all of the Not all of the red points can red points can be associated be associated with greenwith green

All of green All of green points match points match with one or with one or more red pointsmore red points

Computing EMDComputing EMDEdge Point 1 Point 2 Dist Attr Contribution

1 (63.9, 45.8) (44.5, 24.5) 28.80 1.35 38.942 (62.8, 50.9) (48.0, 68.9) 23.33 1.84 43.003 (31.7, 50.0) (48.0, 35.6) 21.78 5.13 111.744 (96.1, 100.0) (100.0, 97.8) 4.45 3.71 16.525 (46.9, 5.8) (88.0, 6.7) 41.10 4.59 188.576 (63.7, 29.0) (100.0, 0.0) 46.47 1.35 62.837 (95.7, 95.7) (95.7, 97.5) 1.81 4.33 7.838 (92.3, 8.0) (84.0, 8.9) 8.34 3.87 32.299 (42.2, 96.0) (88.0, 93.3) 45.87 4.09 187.5810 (96.1, 4.0) (92.0, 4.4) 4.17 4.05 16.89… … … … … …56 (100.0, 0.0) (100.0, 0.0) 0.00 2.29 0.0057 ( 0.0, 36.2) (51.9, 24.7) 53.17 1.35 71.7358 (62.7, 50.0) (55.8, 22.9) 27.96 0.00 0.0359 (57.7, 20.0) (63.6, 19.3) 5.93 5.79 34.3360 (84.6, 16.0) (71.5, 15.4) 13.06 5.81 75.8661 (52.3, 12.7) (80.0, 11.1) 27.71 5.90 163.56

Total 4948.92 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:Future Work:

Use all properties simultaneously to Use all properties simultaneously to formulate this problem as one formulate this problem as one optimization problemoptimization problem Pareto optimization with multiple Pareto optimization with multiple

objectivesobjectives Find the optimal rotation, regrow the Find the optimal rotation, regrow the

void inside the scaffold.void inside the scaffold.

QuestionsQuestions

EMD EquationsEMD 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