We derive a cellular solids approach to the design of bone scaffolds for stiffness and pore size. Specifically, we focus on scaffolds made of stacked, alternating, orthogonal layers of hydroxyapatite rods, such as those obtained via micro-robotic deposition, and aim to determine the rod diameter, spacing and overlap required to obtain specified elastic moduli and pore size. To validate and calibrate the cellular solids model, we employ a finite element model and determine the effective scaffold moduli via numerical homogenization. In order to perform an efficient, automated execution of the numerical studies, we employ a geometry projection method so that analyses corresponding to different scaffold dimensions can be performed on a fixed, non-conforming mesh. Based on the developed model, we provide design charts to aid in the selection of rod diameter, spacing and overlap to be used in the robotic deposition to attain desired elastic moduli and pore size.

References

1.
Suzuki
,
O.
,
Imaizumi
,
H.
,
Kamakura
,
S.
, and
Katagiri
,
T.
, 2008, “
Bone Regeneration by Synthetic Octacalcium Phosphate and its Role in Biological Mineralization
,”
Curr. Med. Chem.
,
15
(
3
), pp.
305
313
.
2.
Zhang
,
F.
,
Xia
,
Y.
,
Xu
,
L.
, and
Gu
,
N.
, 2008, “
Surface Modification and Microstructure of Single-Walled Carbon Nanotubes for Dental Resin-Based Composites
”.
J. Biomed. Mater. Res., Part B: Appl. Biomater.
,
86
(
1
), pp.
90
97
.
3.
Manjubala
,
I.
,
Sastry
,
T.
, and
Kumar
,
R.
, 2005, “
Bone In-Growth Induced by Biphasic Calcium Phosphate Ceramic in Femoral Defect of Dogs
,”
J. Biomater. Appl.
,
19
(
4
), p.
341
.
4.
Cao
,
W.
, and
Hench
,
L. L.
, 1996, “
Bioactive Materials
,”
Ceram. Int.
,
22
, pp.
493
507
.
5.
U.S. Census Bureau
, 2011,
Health and Nutrition. Statistical Abstracts of the United States
,
U.S. Government Printing Office
.
6.
Jahangir
,
A.
,
Nunley
,
R.
,
Mehta
,
S.
,
Alok
,
S.
, and Washington Health Policy Fellows,, 2008, “
Bone Graft Substitutes in Orthopaedic Surgery
,”
J. Am. Acad. Orthop. Surg.
,
2
(
1
), pp.
35
37
.
7.
U.S. Census Bureau
, 2011,
Population. Statistical Abstracts of the United States
,
U.S. Government Printing Office
.
8.
Khan
,
Y.
,
Yaszemski
,
M.
,
Mikos
,
A.
, and
Laurencin
,
C.
, 2008, “
Tissue Engineering of Bone: Material and Matrix Considerations
,”
J. Bone Jt. Surg.
,
90
(
1
), p.
36
.
9.
LeGeros
,
R.
, 2008, “
Calcium Phosphate-Based Osteoinductive Materials
,”
Chem. Rev.
,
108
(
11
), pp.
4742
4753
.
10.
Habibovic
,
P.
, and
de Groot
,
K.
, 2007, “
Osteoinductive Biomaterials-Properties and Relevance in Bone Repair
,”
J. Tissue Eng. Regener. Med.
,
1
(
1
), pp.
25
32
.
11.
Mistry
,
A.
, and
Mikos
,
A.
, 2005, “
Tissue engineering strategies for bone regeneration
,”
Regenerative Medicine II: Clinical and Preclinical Applications
,
Yannis
,
I.
, ed. (Advances in Biochemical Engineering/Biotechnology).
Springer-Verlag Berlin Heidelberg
, Vol.
94
, Chap. 1, pp.
1
22
.
12.
Silber
,
J.
,
Anderson
,
D.
,
Daffner
,
S.
,
Brislin
,
B.
,
Leland
,
J.
,
Hilibrand
,
A.
,
Vaccaro
,
A.
, and
Albert
,
T.
, 2003, “
Donor Site Morbidity After Anterior Iliac Crest Bone Harvest for Single-Level Anterior Cervical Discectomy and Fusion
,”
Spine
,
28
(
2
), p.
134
.
13.
Betz
,
R.
, 2002, “
Limitations of Autograft and Allograft: New Synthetic Solutions
,”
Orthopedics
,
25
(
5
), p.
s561
.
14.
Wheeler
,
D.
, and
Enneking
,
W.
, 2005, “
Allograft Bone Decreases in Strength In Vivo Over Time
,”
Clin. Orthop. Relat. Res.
,
435
, p.
36
.
15.
Hollister
,
S.
, 2005, “
Porous Scaffold Design for Tissue Engineering
,”
Nature Mater.
,
4
(
7
), pp.
518
524
.
16.
Reichert
,
J.
, and
Hutmacher
,
D.
, 2011, “
Bone Tissue Engineering
,”
Tissue Engineering: From Lab to Clinic
,
N.
Pallua
and
C.
Suscheck
, eds.,
Springer
,
New York
.
17.
Karageorgiou
,
V.
, and
Kaplan
,
D.
, 2005, “
Porosity of 3D Biomaterial Scaffolds and Osteogenesis
,”
Biomaterials
,
26
(
27
), pp.
5474
5491
.
18.
O’Brien
,
F.
,
Harley
,
B.
,
Waller
,
M.
,
Yannas
,
I.
,
Gibson
,
L.
, and
Prendergast
,
P.
, 2007, “
The Effect of Pore Size on Permeability and Cell Attachment in Collagen Scaffolds for Tissue engineering
,”
Technol. Health Care
,
15
(
1
), pp.
3
17
.
19.
Sigmund
,
O.
, 2002, “
On the Optimality of Bone Microstructure
,”
IUTAM Symposium on Synthesis in Bio Solid Mechanics
,
Springer
,
New York
, pp.
221
234
.
20.
Hollister
,
S.
,
Liao
,
E.
,
Moffitt
,
E.
,
Jeong
,
C.
, and
Kemppainen
,
J.
, 2011, “
Defining Design Targets for Tissue Engineering Scaffolds
”.
Tissue Engineering: From Lab to Clinic
,
N.
Pallua
and
C.
Suscheck
, eds.,
Springer
,
New York
.
21.
Lewis
,
J.
, 2006, “
Direct Ink Writing of 3d Functional Materials
,”
v. Funct. Mater.
,
16
(
17
), pp.
2193
2204
.
22.
Miranda
,
P.
,
Pajares
,
A.
,
Saiz
,
E.
,
Tomsia
,
A.
, and
Guiberteau
,
F.
, 2008, “
Mechanical Properties of Calcium Phosphate Scaffolds Fabricated by Robocasting
,”
J. Biomed. Mater. Res. Part A
,
85
(
1
), pp.
218
227
.
23.
Miranda
,
P.
,
Pajares
,
A.
, and
Guiberteau
,
F.
, 2008, “
Finite Element Modeling as a Tool for Predicting the Fracture Behavior of Robocast Scaffolds
,”
Acta Biomater.
,
4
(
6
), pp.
1715
1724
.
24.
Cordell
,
J.
,
Vogl
,
M.
, and
Wagoner Johnson
,
A.
, 2009, “
The Influence of Micropore Size on the Mechanical Properties of Bulk Hydroxyapatite and Hydroxyapatite Scaffolds
,”
J. Mech. Behav. Biomed. Mater.
,
2
(
5
), pp.
560
570
.
25.
Smay
,
J. E.
,
Cesarano
III,
J.
, and
Lewis
,
J. A.
, 2002, “
Colloidal Inks for Directed Assembly of 3-D Periodic Structures
,”
Langmuir
,
18
(
14
), pp.
5429
5437
.
26.
Hoelzle
,
D.
,
Alleyne
,
A.
, and
Wagoner
,
J.
, 2008, “
Micro-Robotic Deposition Guidelines by a Design of Experiments Approach to Maximize Fabrication Reliability for the Bone Scaffold Application
,”
Acta Biomater.
,
4
(
4
), p.
897
.
27.
Lan Levengood
,
S.
,
Polak
,
S.
,
Poellmann
,
M.
,
Hoelzle
,
D.
,
Maki
,
A.
,
Clark
,
S. G.
,
Wheeler
,
M. B.
, and
Wagoner Johnson
,
A. J.
, 2010, “
The Effect of BMP-2 on Micro and Macroscale Osteointegration of Biphasic Calcium Phosphate Scaffolds with Multiscale Porosity
,”
Acta Biomater.
,
6
(
8
), pp.
3283
3291
.
28.
Lan Levengood
,
S.
,
Polak
,
S.
,
Wheeler
,
M. B.
,
Maki
,
A.
,
Clark
,
S. G.
,
Jamison
,
R.
, and
Wagoner Johnson
,
A. J.
, 2010, “
Multiscale Osteointegration as a New Paradigm for Bone Scaffold Design
,”
Biomaterials
,
31
, pp.
3552
3563
.
29.
Polak
,
S.
,
Levengood
,
S.
,
Wheeler
,
M.
,
Maki
,
A.
,
Clark
,
S.
, and
Johnson
,
A.
, 2010, “
Analysis of the Roles of Microporosity and BMP-2 on Multiple Measures of Bone Regeneration and Healing in Calcium Phosphate Scaffolds
,”
Acta Biomaterialia
,
7
(
4
), pp.
1760
1771
.
30.
Woodard
,
J.
,
Hilldore
,
A.
,
Lan
,
S.
,
Park
,
C.
,
Morgan
,
A.
,
Eurell
,
J.
,
Clark
,
S.
,
Wheeler
,
M.
,
Jamison
,
R.
, and
Wagoner Johnson
,
A.
, 2007. “
The Mechanical Properties and Osteoconductivity of Hydroxyapatite Bone Scaffolds with Multi-Scale Porosity
,”
Biomaterials
,
28
(
1
), pp.
45
54
.
31.
Schek
,
R.
,
Wilke
,
E.
,
Hollister
,
S.
, and
Krebsbach
,
P.
, 2006, “
Combined Use of Designed Scaffolds and Adenoviral Gene Therapy for Skeletal Tissue Engineering
,”
Biomaterials
,
27
(
7
), pp.
1160
1166
.
32.
LeGeros
,
R.
, 2002, “
Properties of Osteoconductive Biomaterials: Calcium Phosphates
,”
Clin. Orthop. Relat. Res.
,
395
, p.
81
.
33.
Hing
,
K.
, 2005, “
Bioceramic Bone Graft Substitutes: Influence of Porosity and Chemistry
,”
International Journal of Applied Ceramic Technology
,
2
(
3
), pp.
184
199
.
34.
Damien
,
C.
, and
Parsons
,
J.
, 1991, “
Bone Graft and Bone Graft Substitutes: A Review of Current Technology and Applications
,”
Journal of Applied Biomaterials
,
2
(
3
), pp.
187
208
.
35.
Valentini
,
P.
,
Abensur
,
D.
,
Wenz
,
B.
,
Peetz
,
M.
, and
Schenk
,
R.
, 2000, “
Sinus Grafting with Porous Bone Mineral (Bio-Oss) for Implant Placement: A 5-year study on 15 Patients
,”
Int. J. Periodontics Restorative Dent.
,
20
(
3
), p.
245
.
36.
Le Guehennec
,
L.
,
Soueidan
,
A.
,
Layrolle
,
P.
, and
Amouriq
,
Y.
, 2007, “
Surface Treatments of Titanium Dental Implants for Rapid Osseointegration
,”
Dent. Mater.
,
23
(
7
), pp.
844
854
.
37.
Cahill
,
S.
,
Lohfeld
,
S.
, and
McHugh
,
P.
, 2009, “
Finite Element Predictions Compared to Experimental Results for the Effective Modulus of Bone Tissue Engineering Scaffolds Fabricated by Selective Laser Sintering
,”
J. Mater. Sci.: Mater. Med.
,
20
(
6
), pp.
1255
1262
.
38.
Charriere
,
E.
,
Lemaitre
,
J.
, and
Zysset
,
P.
, 2003, “
Hydroxyapatite Cement Scaffolds with Controlled Macroporosity: Fabrication Protocol and Mechanical Properties
,”
Biomaterials
,
24
(
5
), pp.
809
817
.
39.
McIntosh
,
L.
,
Cordell
,
J.
, and
Wagoner
,
J.
, 2009, “
Impact of Bone Geometry on Effective Properties of Bone Scaffolds
,”
Acta Biomater.
,
5
(
2
), p.
680
.
40.
Hollister
,
S.
,
Levy
,
R.
,
Chu
,
T.
,
Halloran
,
J.
, and
Feinberg
,
S.
, 2000, “
An Image-Based Approach for Designing and Manufacturing Craniofacial Scaffolds
,”
Int. J. Oral Maxillofac Surg.
,
29
(
1
), pp.
67
71
.
41.
Williams
,
J.
,
Adewunmi
,
A.
,
Schek
,
R.
,
Flanagan
,
C.
,
Krebsbach
,
P.
,
Feinberg
,
S.
,
Hollister
,
S.
, and
Das
,
S.
, 2005, “
Bone Tissue Engineering Using Polycaprolactone Scaffolds Fabricated via Selective Laser Sintering
,”
Biomaterials
,
26
(
23
), pp.
4817
4827
.
42.
Hollister
,
S.
,
Lin
,
C.
,
Saito
,
E.
Schek
,
R.
,
Taboas
,
J.
,
Williams
,
J.
,
Partee
,
B.
,
Flanagan
,
C.
,
Diggs
,
A.
,
Wilke
,
E.
,
Van Lenthe
,
G. H.
,
Müller
,
R.
,
Wirtz
,
T.
,
Das
,
S.
,
Feinberg
,
S. E.
, and
Krebsbach
,
P. H.
, 2005, “
Engineering Craniofacial Scaffolds
,”
Orthod. Craniofacial Res.
,
8
(
3
), pp.
162
173
.
43.
Hollister
,
S.
,
Maddox
,
R.
, and
Taboas
,
J.
, 2002, “
Optimal Design and Fabrication of Scaffolds to Mimic Tissue Properties and Satisfy Biological Constraints
,”
Biomaterials
,
23
(
20
), pp.
4095
4103
.
44.
Lin
,
C.
,
Kikuchi
,
N.
, and
Hollister
,
S.
, 2004, “
A Novel Method for Biomaterial Scaffold Internal Architecture Design to Match Bone Elastic Properties with Desired Porosity
,”
J. Biomech.
,
37
(
5
), pp.
623
636
.
45.
Hollister
,
S.
, and
Lin
,
C.
, 2007, “
Computational Design of Tissue Engineering Scaffolds
,”
Comput. Methods Appl. Mech. Eng.
,
196
(
31-32
), pp.
2991
2998
.
46.
Gibson
,
L.
, and
Ashby
,
M.
, 1999,
Cellular Solids:Structure and Properties
.
Cambridge University Press
,
London
47.
Muskhelishvili
,
N.
, 1977,
Some Basic Problems of the Mathematical Theory of Elasticity
,
Springer
,
New York
.
48.
Puttock
,
M.
, and
Thwaite
,
E.
, 1969,
Elastic Compression of Spheres and Cylinders at Point and Line Contact
,
Commonwealth Scientific and Industrial Research Organization
.
49.
Beyer
,
W.
,
Fawcett
,
L.
,
Hartem
,
L.
, and
Swartz
,
B.
, 1992, “
The Volume Common to Two Congruent Circular Cylinders
,”
J. Symb. Comput.
,
13
(
2
), pp.
221
230
.
50.
Norato
,
J.
,
Haber
,
R.
,
Tortorelli
,
D.
, and
Bendsøe
,
M.
, 2004, “
A Geometry Projection Method for Shape Optimization
,”
Int. J. Numer. Methods Eng.
,
60
(
14
), pp.
2289
2312
.
51.
Bendsøe
,
M.
, and
Sigmund
,
O.
, 2003,
Topology Optimization: Theory, Methods, and Applications
,
Springer Verlag
,
Berlin
.
52.
Park
,
J.
, and
Lakes
,
R.
, 2007,
Biomaterials: An Introduction.
Springer
,
New York
.
53.
Katz
,
J.
, and
Ukraincik
,
K.
, 1971, “
On the Anisotropic Elastic Properties of Hydroxyapatite
,”
J.Biomech.
,
4
(
3
), pp.
221
227
.
54.
Prokopiev
,
O.
, and
Sevostianov
,
I.
, 2006, “
Dependence of the Mechanical Properties of Sintered Hydroxyapatite on the Sintering Temperature
,”
Mater. Sci. Eng., A
,
431
(
1-2
), pp.
218
227
.
55.
Bangerth
,
W.
,
Hartmann
,
R.
, and
Kanschat
,
G.
, 2007, “
Deal.II – A General Purpose Object Oriented Finite Element Library
,”
ACM Trans. Math. Softw.
,
33
(
4
), pp.
24/1
24/27
.
56.
Dalcin
,
L.
, 2010,
MPI for Python, Release 1.2.2
.
57.
The Scilab Consortium, 2010,
Scilab Manual, 5.3.0
. See also: http://help.scilab.org/http://help.scilab.org/.
58.
Michna
,
S.
,
Wu
,
W.
, and
Lewis
,
J.
, 2005, “
Concentrated Hydroxyapatite Inks for Direct-Write Assembly of 3-D Periodic Scaffolds
,”
Biomaterials
,
26
(
28
), pp.
5632
5639
.
59.
Nordson Engineered Fluid Dispensing, 2006,
Ultra™ fluid dispensing systems and dispensing components (Price List)
.
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