1. Cell Adhesion for Improved Prosthetics:
Determining Spatial Parameters to Allow
Cell Spreading to Reach New Heights
Gene B. Merewether
Submitted in Partial Fulfillment
of the Requirements of the
Degree of
Bachelor of Arts
To the Department of Chemistry
of
Princeton University
April 22, 2013
4. iv
Acknowledgments
I would like to thank Professor Jeffrey Schwartz for helping me tailor my project to my
interests. Professor, your sharp wit and wry humor were much appreciated. Your insights
and patience guided me to the successful completion of this project, and more importantly,
taught me how to carry out such a research project.
To Stephen Bandini and Patrick Donnelly, thank you for all your assistance with refin-
ing the design of experiments and carrying them out. You pushed me to refine my critical
analysis skills, and were always willing to hear out my latest crazy idea. Steve and Pat,
best of luck next year and onward! To Steven Lowe, in the Jadwin Machine Shop, thank
you for teaching me the satisfying and practical skill of machining. To Gerald Poirier, at
the PRISM Imaging & Analysis Center, your help with imaging techniques was crucial.
Steve, Pat, Steve, and Jerry, I couldn’t have done it without your easygoing generosity and
willingness to point me in the right direction.
To my friends on the Princeton Sailing Team, with PFARS, and in Tower, thank you for
making the last four years the best of my life. You guys constantly inspire me with your
passions and drive. Thanks for putting me back on track when I hit life’s speed bumps.
Mom and Dad, you inspired my interest in science and engineering, and taught me to
design and build. These are among the most important lessons I have learned. Dad, you
were a lifesaver on those late nights where you answered the phone for long-distance help
on what seemed like everything! To my whole family, you’re the best I ever could have
asked for. Your unconditional love and support have taught me self-confidence, and I strive
to follow your example as role-models. I love you.
5. v
Abstract
Currently, there exists a mismatch between titanium bone replacement implants and
native bone. Polyetheretherketone (PEEK) is favored as a replacement, but cells do not
adhere well to untreated PEEK. This work attempts to combine 3D printing, materials
properties, surface treatments, and control of structure geometries to test:
1. Is 3D printing a viable route to medical devices that promote cell in- and on-growth?
2. What dimensions allow cells to climb and perfuse on and in an entire engineered
structure?
To answer these questions, a porous PEEK scaffold with 800µm channels was con-
structed by selective laser sintering (SLS) and inspected visually and by scanning electron
microscopy (SEM) for proper formation. The results indicate that these dimensions are
near the minimum viable dimensions for channels in SLS PEEK, while still building suc-
cessfully. A thin zirconium oxide layer was formed on the PEEK surface by chemical vapor
deposition (CVD), followed by formation of a self-assembled monolayer (SAM) of diphos-
phonic acid. X-ray photoelectron spectroscopy (XPS) analysis of surfaces of sections of
the device indicated treatment success in the device interior. Computer simulations of the
discrepancy between the design of 3D parts and the finished SLS products were carried
out. To assay the tolerance of cells to microstructure of devices, PEEK cell migration de-
vices were constructed using standard machining techniques. Cell adhesion and migration
were assayed on plain PEEK, zirconated PEEK, and phosphonated PEEK. Cells adhered
the farthest from the center of the devices on the phosphonated surface, second farthest on
the zirconated surface, and least far on plain PEEK.
This work serves to characterize the parameters of manufacturing methods viable for
producing bone replacement prosthetics. The influence of surface characteristics and cell-
adhesive surface treatments on cell adhesion and migration are examined, hopefully leading
to improved implant fixation and customizable shape.
8. viii
List of Acronyms
AFM Atomic Force Microscopy
CT Computerized Tomography
CVD Chemical Vapor Deposition
DCM Dichloromethane
DMEM Dulbecco’s Modified Eagle Medium
MRI Magnetic Resonance Imaging
PBS Phosphate-Buffered Saline
PEEK Polyetheretherketone
RPM Revolutions per Minute
SAM Self-Assembled Monolayer
SBF Simulated Body Fluid
SEM Scanning Electron Microscopy
SLS Selective Laser Sintering
TMS Tetramethylsilane
11. 2
1.1 Background and Significance1
1.1.1 Medical Implant Technology
As average life expectancy increases due to medical advances, aging populations re-
quire prosthetic devices as replacements for failing bones and joints. Implant techniques
and materials from these prosthetics extend to a wide range of hard implants, including
fracture stabilization plates and dental implants, that benefit many segments of the popula-
tion [10]. The current state-of-the-art material, titanium, benefits from a 50-year history of
research, production techniques, and testing. Titanium exhibits a high strength-to-weight
ratio, and is compatible with high-temperature sterilization. Proteins adhere to the surface
of titanium, adhering to hydroxyl and oxide functional groups [14].
However, titanium exhibits disadvantages that have led to a search for a new material.
Lack of cell adhesion to uncoated titanium causes implant failure in certain circumstances.
For example, if the gap between an implant surface and native bone is greater than 0.1mm,
the implant fails due to relative motion of the implant and the bone [17]. Additionally,
bone tissue requires continual application of stress in order to survive. This presents a
difficulty because the titanium alloy commonly used in bone implants has a tensile modulus
of 110GPa, while compact bone has a tensile modulus of only 18GPa [10]. This mismatch
causes loss of bone tissue in a phenomenon known as stress shielding [7]. Finally, titanium
is opaque under x-ray imaging, complicating studies of implant effectiveness. The metal
also interferes with computer tomography (CT) scanning and magnetic resonance imaging
(MRI) [8]. Radiolucent hard implants that induce cell adhesion will increase quality of life
for many segments of the population.
1.1.2 Standard Alternative to Titanium
Research on a replacement for titanium focuses on polyetheretherketone (PEEK) (Fig-
ure 1.1 on the following page), a high-temperature and high-strength thermoplastic. PEEK
composites have appropriate tensile modulus values, and are radiolucent and biocompatible
[10].
1Parts of this section were adapted from the author’s Junior Paper, submitted in May 2012.
12. 3
Figure 1.1: Polyetheretherketone (PEEK) repeat unit
The tensile modulus of PEEK carbon fiber composites can be tuned from four GPa with
no carbon fiber to over 100GPa, allowing the modulus to match that of bone [21]. The FDA
has approved PEEK-OPTIMA® and a carbon fiber reinforced variety, manufactured by In-
vibio®, for long-term implantation in humans [10]. Also, PEEK is radiolucent and does
not interfere with CT or MRI, allowing for medical imaging [9]. Due to its chemical inert-
ness, PEEK strongly resists degradation in vivo, and does not cause a significant adverse
reaction [21]. In work carried out by Williams, et al., material properties of PEEK were
maintained during a six month implantation of test samples in rabbits. Williams replicated
that result in vitro in simulated body fluid (SBF) [21].
Unfortunately, osteoblasts and other cells do not adhere well to unmodified PEEK sur-
faces [12][10]. Successful implants require attachment of native tissue for enhanced struc-
tural attachment, so PEEK surfaces must be modified to improve cell adhesion.
1.1.3 Bioadhesive Surface Treatment of PEEK
The Schwartz lab took a novel approach to PEEK surface treatment for improved cell
adhesion: formation of a thin zirconium oxide layer on PEEK by chemical vapor deposition
(CVD), followed by formation of a self-assembled monolayer (SAM) of diphosphonic acid
[3].
The zirconium oxide layer deposits from vapor of zirconium tetra(tert-butoxide) un-
der modest vacuum. The PEEK ether and ketone functional groups ligate the zirconium.
Then, controlled heating cleaves the tert-butoxl ligands, leaving hydroxyl groups to form
oxide bonds between adjacent zirconium atoms, stabilizing the structure. Alkoxide ligands
can remain ligated at the edge of the zirconium thin film. These alkoxide ligands can be
13. 4
replaced by hydroxyl ligands by soaking in water overnight, giving the surface shown in
Figure 1.2.
Figure 1.2: Zirconated PEEK repeat unit
The SAM is formed on the ZrO2 surface by slow evaporation of a solution of phos-
phonic acids in ethanol, giving surfaces of the type shown in Figure 1.3. As the solvent
evaporates, a monolayer of phosphonic acids bonds to the zirconium oxide surface. Phos-
phonic acids can be chosen that exhibit cell-adhesive properties, or further chemical steps
can be followed to attach cell-adhesive molecules.
Figure 1.3: Phosphonated PEEK repeat unit
The thickness of the ZrO2 film deposited by this treatment ranges in thickness from
one nm after five minutes of deposition to eight nm after one hour. Therefore, the interface
14. 5
experiences lower mechanical stresses during flexing, and is less likely to fracture and
flake off than hydroxyapatite, an alternative coating material [4]. Additionally, the shear
strength of the layer nearly matches the strength of a hydroxyapatite layer on titanium.
The number of adherent osteoblasts nearly doubled for PEEK treated with zirconium and
dodecylbisphosphonic acid compared with untreated PEEK [3]. Finally, the process will
likely be compatible with interior porosity and microscale features. The vapor should easily
permeate a device with interior porosity, and the thin deposited film should not occlude
engineered surface features of the underlying PEEK.
1.1.4 Common Issues Encountered During Imaging and Manufactur-
ing of PEEK
Ideal implants would have complex, patient-specific shapes, exhibiting interior porosity
and microfeatures. Currently, none of the three major manufacturing methods (injection
molding, standard machining, and rapid prototyping) can satisfy all of these requirements.
Injection molding, the standard method for producing plastic parts in complex and con-
tinuous geometries, has high startup costs associated with the production of molds for the
process. The molds function only for one shape, and therefore do not allow for patient-
specific implants.
Standard machining techniques allow for customized parts, but these techniques cannot
easily generate interior features such as channels and pores. Custom parts have high costs
associated with the labor of skilled machinists who produce them. Also, when machining
PEEK, tools need to be very sharp to prevent melting of the polymer surface. Therefore,
carbide tools are most appropriate, but they are more expensive than tools made from stan-
dard tool steel.
Rapid prototyping of PEEK offers a high degree of control over the shape of an implant,
and makes single production runs of a particular shape economically feasible. Rapid pro-
totyped PEEK parts are produced by selective laser sintering (SLS), a process where small
pellets of material are melted together by a moving laser spot. The computer model of the
part to be made is separated into layers of fixed size. The build bed of pellets is heated to
near its melting point, and the laser spot traces the first layer of the part, melting pellets in
15. 6
those areas together. Then, a blade scrapes another layer of pellets across the build surface.
The process continues with the next layer. The nature of the process causes errors in the
finished part. First, the minimum size of the pellets is limited to 20-80µm. Below this
size, static electricity forces dominate pellet-pellet and pellet-build platform interactions,
preventing even coverage of the build platform by the scraping blade. Also, during the
SLS process, pellets outside the intended sintering volume are sintered to the finished part
because they stick to the liquid volume under the laser spot. Features such as pores and
wells collapse if they are too small, because the region currently being sintered can flow
into a neighboring pore or well. Finally, during additive manufacturing processes, contin-
uous surfaces must be approximated by the 3D equivalent of pixels, commonly known as
voxels. For example, in the case of an inclined plane, the printed part will have a series of
steps whose corners lie along the intended angled plane. When designing an implant to fit
tightly to a patient’s geometry, these errors must be taken into account.
An additional difficulty arises when imaging cells on PEEK. One workhorse technique
of cell analysis is fluorescence microscopy, where cell components are tagged with dyes
that absorb laser light at one wavelength and fluoresce at another. PEEK autofluoresces
strongly, interfering with this technique [1].
1.1.5 Gaps in Present Understanding
After implant installation in the body, successful implants must induce natural healing
processes. Additionally, for secure implant fixation, cells of the native tissue must migrate
onto and into the device, especially if the device has interior porosity or engineered vascular
channels. If a device is to be pre-seeded with cells, these cells must still migrate through
the device during the seeding process. Thus, research must address the following:
• How do cell-adhesive surface treatments influence the migration of cells?
• How do these treatments affect the ability of cells to migrate against gravity?
• How do these treatments affect the ability of cells to migrate across a sharp angle
between two planes?
16. 7
• How can the roughness of SLS PEEK surfaces be reduced?
• How can pellet size be reduced?
• Can post-sintering treatments be applied to smooth the finished part?
• What are the effects of cell-scale roughness on cell viability, adhesion, and migra-
tion?
1.1.6 Concluding Statement
I hypothesize that the bioadhesive surface treatments studied in the lab of Professor
Schwartz are compatible with 3D porous structures. I also hypothesize that these treatments
will increase fibroblast adhesion and migration ability on machined surfaces. These results
may lead to improved prosthetic outcome.
17. 8
1.2 Specific Aims
• Question: Can CVD and ligand exchange (LE) permeate a porous PEEK scaffold?2
– Hypothesis: CVD and LE will translate from 2D PEEK films to 3D porous
PEEK structures, leading to constant zirconium and phosphorus signals with
section depth as measured by x-ray photoelectron spectroscopy (XPS).
– Experiment: Deposit zirconium on PEEK scaffold, carry out LE, section the
samples, and analyze with XPS.
• Question: What is the most appropriate common machining technique for producing
smooth PEEK planar surfaces that intersect at a sharp angle?
– Hypothesis: Chamfer mill cutters will be the most appropriate tool.
– Experiment: Machine the desired PEEK features with chamfer mill cutters,
carbide end mills in two cuts, and custom-ground flycutter tools. Analyze with
SEM and atomic force microscopy (AFM).
• Question: How do cell-adhesive surface treatments affect cell migration against
gravity? How do cell-adhesive surface treatments affect cell migration across a sharp
angle between two planes?
– Hypothesis: Cells will migrate slowest on plain PEEK, at intermediate speed
on zirconated surfaces, and fastest on phosphonated surfaces. Fewest cells will
migrate past an angle on plain PEEK, an intermediate number on zirconated
PEEK, and the most on phosphonated PEEK.
– Experiment: Plate cells on plain, zirconated, and phosphonated surfaces. Al-
low to grow and spread. Analyze with SEM and fluorescence microscopy.
2This experiment represents the successful conclusion of work initiated in the author’s Junior Paper, sub-
mitted in May 2012.
18. 9
1.3 Experimental Approach
1.3.1 Surface Treatment and Analysis
PEEK samples (films, SLS structures, high heels, and conical cups) were cleaned and
attached to a metal base plate inside the CVD chamber. The metal base plate served to
transmit external thermal conditions into the chamber and distribute them over the surface
of the PEEK. This helps compensate for the low thermal conductivity of PEEK, which is
only 0.29 W
m⇤K [18], compared to steel (around 40 W
m⇤K ) or titanium (around 30 W
m⇤K ). The de-
position chamber was cooled to help favor the deposited phase over the vapor phase. After
exposure to vapor under vacuum, the chamber was sealed off from external evacuation, and
the chamber was heated to cross-link zirconium atoms. Finally, samples were immersed in
the LE solution of phosphonic acid in ethanol.
The success of surface treatments was determined by analysis of XPS spectra, both
survey scans and detail scans of the zirconium 3d, phosphorus 2p, and carbon 1s peaks.
XPS was chosen because fine spatial detail was not required and the region of interest was
limited to the surface of the material. XPS can distinguish between zirconium and phos-
phorus signals, unlike energy-dispersive X-ray spectroscopy, another common technique
for surface analysis. These results determine the applicability of these surface treatments
to interior porosity of laser sintered samples as well as to PEEK prepared by traditional
machining techniques.
1.3.2 SLS PEEK
A balance must be struck between large channel size for nutrient, oxygen, and waste
exchange, and small channel size for increased surface area for cell adhesion. Melchels,
et al., explored cell viability in salt-leached versus 3D printed microstructures, discovering
limits on the size and configuration of channels [11]. After seeding mesenchymal stem
cells onto the surface of the assay devices, the devices were sectioned at multiple time
points and cells were stained with methylene blue. Although cells remained viable in the
interior of the device after five days, by 20 days the surface of the scaffold was occluded
by cells and matrix, causing death of interior cells. Based on their results, they designed
19. 10
an improved architecture with 600µm-wide channels for nutrient and waste exchange con-
nected to 250µm-wide channels for cell adhesion and matrix deposition [11]. Ashman, et
al., investigated tissue ingrowth into porous polymethylmethacrylate scaffolds, and found
that connective tissue grew into pores larger than 100µm, and osseous tissue grew into
pores larger than 450µm [2].
An approximately 30mm-long SLS PEEK device (Figure 2.3 on page 16) was designed
with intersecting channels in the {x,y,z} axes to determine the penetration of CVD and LE
treatments into interior porosity. Communications with engineers at SolidConcepts indi-
cated that ~800µm was the minimum viable width for such channels. Surface treatments
were applied and characterized as described above. The SLS part was visually inspected
to confirm proper channel manufacture, and qualitative SEM analysis of the microstructure
was carried out. These results provide data on the viability of channels of this size, data that
is currently not available. Except for one notable example [15], commercial manufacturers
do not publish data on the minimum size features that will build during SLS processes.
The example in question does not provide data on pores or channels. SLS samples were
sectioned through the interior paralleling the device faces, and XPS was carried out on the
exterior of the device and on the sections.
The following gaps in knowledge are not addressed in this research:
• How can the roughness of SLS PEEK surfaces be reduced?
• How can pellet size be reduced?
• Can post-sintering treatments be applied to smooth the finished part?
Instead, this research focuses on determining parameters of cell migration on a surface
of much lower roughness. Once the parameters of acceptable roughness and maximum
traversable sharp angle are known, goals for the aforementioned unanswered questions can
be set.
This work also follows the methods outlined by Patil, et al., in order to simulate the error
introduced from a different source: the voxelization inherent in 3D printing methods [13].
Triangles in 3D space represent the boundaries of objects in an STL file, a common format
for the exchange of model data. The intersections of all such triangles in a model with
20. 11
lines paralleling the x-axis are calculated, and the regions between intersections are filled
with voxels of the specified resolution. Then, the voxelized model is displayed. Knowing
the extent of the voxelization effect will help prosthetic designers determine if the errors of
standard SLS PEEK fall within desired tolerances.
1.3.3 Machining
The tradeoffs of several methods of machining were explored. Milling in multiple
steps using standard end mills employs tools most likely to be available in machine shops.
Chamfer end mills (Figure 2.6a on page 19) and flycutters (Figure 2.6b on page 19) are
more likely to make a sharp critical angle. The angle is milled at the same time as the
smooth base plane and the smooth inclined plane. Two-step milling, on the other hand,
relies on precise alignment of the edge of the end mill on the first cut with the edge of
the end mill on the second cut, made after reclamping the part. Chamfer mill angles were
chosen based on the commonly available tool angles.
Flycutters can be easily made in most machine shops, which have grinding wheels and
high strength steel stock bars for grinding custom tools. The flycutter tools were hand-
ground to determine if that method of fabrication would suffice, as opposed to the time-
consuming approach of clamping tool steel precisely to a surface grinder. One drawback
of flycutter tools is their removal of material through a wide arc. They are not compatible
with attempting to mill other features in the area they sweep out during cutting. Chamfer
end mills do not suffer from this restriction, but they are not commercially sold in forms
that would make the desired critical angles. Custom chamfer end mills, which could be
produced for the desired angles, are expensive and would be specific to each critical angle.
For milling conical cups, lathe tools were chosen because the circular symmetry of
lathe machining facilitates building the conical shape. A lathe tool was ground with cutting
edges on the radially interior and exterior sides to increase the versatility of the tool. The
compound angle setting of the lathe bed was used to advance the lathe tool radially inward,
in order that cutting could proceed in one sequence without tool removal or switching.
AFM test samples were fabricated by cutting flat surfaces using the flycutter tool and
lathe tool respectively. Flat samples were chosen rather than samples inclined at the base
21. 12
angle due to the height limitations of AFM. AFM was chosen because it provides quanti-
tative surface height data. Quantitative surface height data obtained by AFM was compli-
mented by qualitative data provided by SEM. During SEM, samples were placed on their
sides, with the critical angle oriented nearly vertically. This provided images of the profiles
of surface features and of the critical angle.
1.3.4 Migration Experiment
The literature does not address cell migration and adhesion on surfaces manufactured
using standard machining techniques. This work addresses the following gaps in knowl-
edge:
• How do cell-adhesive surface treatments influence migration of cells against gravity,
and across a critical angle?
• How do cells adhere and migrate on a machined surface?
• What is the effect of micron-scale, random roughness on cell viability, adhesion, and
migration?
Cell assays were devised with a lower section tilted above the horizontal by a base angle.
This lower region intersected another region tilted more sharply above the horizontal. This
intersection is referred to as the critical angle. The base angle was engineered in the cell
assay devices to prevent the possibility of plating cells above the critical angle on a device
with a flat base angle. The first type of cell assay, the high heel device (Figure 2.5 on
page 19), had two planar regions demarcated by a line in 3D space along the critical angle.
The second type, the conical cup device (Figure 2.11a on page 22), presented two conical
regions demarcated by a circular critical angle.
After cell assays on the high heels showed no cell presence, the conical cup cell assay
was devised to physically constrain the cell media during the plating step. Channels were
added to the conical cups to provide for nutrient and waste exchange. The base angle was
increased in the conical cup devices to provide additional physical separation between the
rim of the droplet of media containing cells and the critical angle. The media was main-
tained in a static position as opposed to stirring to prevent disturbing the liquid droplets.
22. 13
The 30 base angle was chosen based on an migration experiment conducted with chicken
heart fibroblasts that traversed an obtuse angle ground on optical glass. In that experiment,
a 16 angle caused individual cells to deflect, but migrating cell fronts successfully tra-
versed the angle. A 32 angle caused the layer of cells to deflect [5]. The 30 base angle
was chosen to test the limits of cell adhesion with cell adhesive surface treatments.
24. 15
2.1 Chemical Surface Treatments
This CVD procedure is based on that developed by Joseph Dennes in our lab [3]. Sam-
ples were cleaned by sonication in DCM for 10 min and rinsed twice in DCM, then once
in ethanol. Samples were blown dry with N2, then secured on a copper plate with con-
ductive tape. The copper plate was placed inside the deposition chamber (Figure 2.1a),
which was wrapped in dry ice in aluminum foil. Samples were exposed to zirconium(IV)
tert-butoxide vapor under vacuum (10 3 torr) for 5 min. Then, the deposition chamber was
closed to vacuum, and the chamber was heated to 70 C as measured by thermocouple for
15 min. The chamber was backfilled with N2, then opened to air.
Figure 2.1: Deposition chamber and vacuum system
(a) Deposition chamber and Zr bulb (b) Vacuum supply system
This ligand exchange procedure was provided by Professor Jeffrey Schwartz. Ligand
exchange solution was made by dissolving butane-1,4-diphosphonic acid (Figure 2.2) in
ethanol at 0.25mg/ml, and sonicating the solution for 10 min. Samples were suspended in
ligand exchange solution for 24 hr, vigorously rinsed in ethanol, and air dried.
Figure 2.2: Butane-1,4-diphosphonic acid
25. 16
2.2 “Porous” PEEK Scaffolds
PEEK scaffolds were designed using AutoCAD. First, a repeat cube (Figure 2.3a) was
constructed by subtracting channels of 0.762mm in width oriented along {x,y,z} from a
cube with sides of 1.778mm in length. Then, this cube was tiled (Figure 2.3b) by (8,8,16)
repeat units along {x,y,z}.
Figure 2.3: Porous PEEK scaffold
(a) Scaffold repeat cube (dimensions in mm) (b) Unit cube tiled by (8,8,16)
Models were ordered from SolidConcepts1, where they were built using an EOSINT
P 800 PEEK LS Laser Sintering System [16]. Sections were cut from the scaffold using
a Buehler IsoMet® low speed saw, with water as coolant and lubricant. A custom fixture
provided by the Jadwin Machine Shop was used to clamp the samples. Samples were
sectioned at 1.3mm and 5.1mm, and XPS was carried out on the exterior of the device and
on the sections.
2.3 Simulation of Voxelization
STL files are packaged by AutoCAD in binary format for enhanced compression as
compared with text [22]. All numeric values occupy four bytes of space each. After an 80-
1https://www.solidconcepts.com/
26. 17
byte header, the number of triangles is recorded. Then, for each triangle, triplets of floating
point numbers specify the normal vector and each of the three vertices of that triangle.
The main script (E.1) reads a specified binary STL file, calculates the minimum and
maximum specified values of {x,y,z}, and uses the desired number of voxels in each di-
mension to voxelize and display the model. The voxelize function (E.2) converts a list
of triangles representing the boundary of the object into a three-dimensional array where
values of one represent filled voxels. This function, based on the work of Patil, et al., op-
erates by generating rays parallel to the x-axis with {y,z} coordinates at the center of each
possible voxel within the calculated bounding box. For each line, each triangle is checked
for intersections with that line. The list of intersections is sorted, and the voxels that lie
between each odd-numbered and the following even-numbered intersection are filled [13].
The triangle class (E.3) represents a triangle in STL format. It also provides a method that
returns the x-value of the intersection of a line parallel to the x-axis with the plane con-
taining the triangle. Another method computes areas of subtriangles using cross products
to determine if the intersection point falls within the boundaries of the triangle. A third
method draws the triangle as a filled polygon. The tRead function (E.4) reads and returns
the next triangle from the given filestream. The inArray function (E.5) avoids generating
duplicate intersections by searching a given array for a value within a specified tolerance.
For example, this duplication otherwise occurs when an intersection-finding line strikes the
boundary between two STL triangles. The plotVoxel function (E.6) plots each face of the
given voxel based on the coordinates at the center, the bounding box, and the voxel resolu-
tion. The plotVoxels function (E.7) calls plotVoxel on each voxel to plot the entire model.
The toReal (E.8) and toVoxel (E.9) functions convert between real coordinates and voxel
indices.
2.4 Cell Studies and Sample Preparation
Cell seeding on “high heel” and conical cup devices was done by Stephen Bandini and
Patrick Donnelly in our group, in the lab of Professor Jean Schwarzbauer. Samples were
placed in individual wells of 24- or 6-well plates and rinsed twice with phosphate-buffered
27. 18
saline (PBS). NIH 3T3 fibroblasts were plated at 30,000 cells per well on the high heel
substrates in serum-free Dulbecco’s Modified Eagle Medium (DMEM) and were allowed
to attach at 37 °C for 3 hr, with media only covering half of the base region of the device
(Figure 2.4a) to prevent plating cells on or above the critical angle. After the media was
suctioned off and the devices were rinsed twice with PBS, the medium was changed to
DMEM with 10% calf serum. The wells were filled until the devices were completely
submerged to allow any cells that attached during the plating step to migrate and spread for
an additional 3 days (Figure 2.4b). With conical cup substrates, 10,000 cells were plated
in 150µl DMEM with serum in a single drop at the center of the cup. The space in the well
around the device was filled with a predetermined volume of medium until the two liquids
communicated, as measured before the experiment using food coloring to assay effective
communication. Thereafter, the same procedure was followed as for the high heel devices.
Figure 2.4: Cell migration assay steps
(a) Initial plating step (b) Migration step
Fixing and permeabilization were done by Stephen Bandini and Patrick Donnelly. Cells
were fixed using 3.7% formaldehyde in PBS for 15 min, and permeabilized with 0.5%
NP-40 detergent in PBS for 15 min at room temperature. Next, cell samples were dried
successively in 10,30,50,70,90% ethanol solutions in deionized water, then in 1:1 tetram-
ethylsilane (TMS):ethanol and pure TMS, for 5 min each. Samples were air dried from
TMS solution in a fume hood. After drying, high heel device samples were coated with
gold to 40 Å in thickness, using an ion beam sputterer. Samples were rotated and tilted to
ensure even coverage. Conical cup samples were not coated. Progress of cell migration
was determined by SEM, by visually locating the furthest cells from the center of the cup.
28. 19
Fluorescence staining was done by Stephen Bandini and Patrick Donnelly. After fixing and
permeabilization, high heel device samples were stained with rhodamine-phalloidin stain
for actin and DAPI stain for DNA for cell shape and orientation studies.
2.5 “High Heel” Devices
Figure 2.5: Design of the “high heel” cell migration assay device
This cell migration assay device was designed using AutoCAD (Figure 2.5). The base
angle remained 15 across all samples, while the critical angle varied from 0 to 40 .
Figure 2.6: Single-step critical angle milling devices
(a) 45 degree chamfer end mill (b) Custom flycutter tools, 10 40
Test devices were created using each of the following methods for milling a precise
critical angle: chamfer end mills, two-step milling, and custom flycutter tools. The base
29. 20
angle for the SEM test pieces was 0 , and critical angles varied as follows: 60 for chamfer
end mills, 15 for two-step milling, and 10 and 20 for flycutter tools. For AFM, flat-
surfaced samples were milled using each technique, without a critical angle feature.
Figure 2.7: Chamfer end mill cutting volume and critical angle
Figure 2.8: Two-step critical angle milling
High heels with a 60 critical angle were milled using a chamfer end mill in one pass
(Figure 2.7). During two-step milling, the lower section of the device was milled with
an 1/8 inch four flute carbide end mill, leaving a rectangular step (Figure 2.8). Then,
the device was clamped at an angle, and the slanted upper section was milled from the
remaining rectangular step. Custom flycutter tools (Figure 2.9 on the next page) were
ground for each desired critical angle. The 20 angle of the flycutter body was compensated
for in the grinding of the angle on the flycutter bit.
For the cell assay, flycutter machining was chosen. 1” PEEK round stock was clamped
in a collet block on top of 15 angle blocks. The flycutter was spun in a counterclockwise
direction at 1750 revolutions per minute (RPM). The flycutter was moved in the positive x
30. 21
Figure 2.9: Flycutter assembly and cutting surface
(a) High heel machining with flycutter (b) Flycutter body
and negative z directions until the high heels took shape (Figure 2.10). For the final critical
angle cut, the flycutter was moved approximately 25 microns in the z direction and 100
microns in the positive x direction while not cutting. Then, to make the final cut, the x-
axis of the table was locked, and the flycutter was moved across the surface from the y
side of the table at about 1mm/sec. The segment of the rod containing the devices was
separated from the stock with a bandsaw. Then, the segment was inserted upside down into
the collet block with no angle blocks below the collet block, and the base of the devices
was machined flat with the flycutter. Individual devices were cut from the row of devices
with the bandsaw, and edges were filed smooth.
Figure 2.10: Setup and manufacture of high-heels
31. 22
2.6 Conical Cup Devices
This cell migration assay device (Figure 2.11a) and control device (Figure 2.11b) were
designed using AutoCAD. The pitch of the first inclined surface was 25 , and the critical
angle was 30 . The central cup was designed to hold cells during the initial plating step.
Radial channels were cut into the devices to allow passage of nutrients and waste in and
out of the central cup.
Figure 2.11: Conical cup migration assay devices
(a) 30 Conical cup assay device (b) Control assay device
1” PEEK round stock was held in a South Bend lathe. A custom lathe tool with two
cutting edges at 90 from each other was rotated to produce the desired critical angle (Fig-
ure 2.12 on the next page). The lathe compound slide, a table axis set at an adjustable
angle relative to the spinning axis of the lathe, was set to manufacture the base angle, 25 .
After advancing the lathe tool parallel to the axis of the rod to the desired depth, the tool
was advanced with the compound slide until the tip of the lathe tool reached the center of
the round stock. The part was spun at 628 RPM for heavy cuts and 940 RPM for light,
finishing cuts. Then, the conical cup was separated from the round stock using a parting
tool. Channels were cut using a Craftsman Variable Speed Rotary Tool, using the Dremel
545 Diamond Wheel and 420 Cutoff Wheel.
32. 23
Figure 2.12: Lathe setup for conical cup devices
2.7 Imaging
AFM was done on a Veeco Dimension NanoMan in tapping mode, with Bruker RTESP
tips. SEM was carried out on a FEI Quanta 200 ESEM operating in low vacuum mode with
pressures of 0.65-0.80 torr of water vapor, a beam voltage of 15.00 kV, and a spot size of
3. Fluorescence microscopy was done by Joe Goodhouse, Confocal Core Lab Manager in
the Department of Molecular Biology, on a Nikon A1 confocal microscopy system. For
XPS characterization, PEEK coupons from Goodfellow were used as reference samples to
compare against SLS PEEK and machined PEEK samples.
34. 25
3.1 SLS PEEK Devices
Figure 3.1: SLS PEEK scaffold
(a) Engineered channel, sectioned between
junctions
(b) Engineered channel, sectioned through
junction
(c) Sectioned through junctions, viewed at
45 (d) Incompletely-formed channel walls
A clear path could be visualized through all channels during visual inspection of the
SLS PEEK printed scaffold, showing that 800µm channels formed at these dimensions.
The minimum size for a pore of this sort in SLS PEEK is 0.03”, or 762µm (unpublished;
personal communication from John Thiell, Senior Project Engineer at SolidConcepts). This
is borne out by the successful construction of these pores. Some channel walls did not
build properly (Figure 3.1d), indicating that these dimensions are near the minimum size
35. 26
for channels of this type. Solid regions appear to be completely sintered together, with full
melting and joining of PEEK pellets. However, neighboring regions (Figure 3.1b on the
previous page) show significant unintended sintering of particles due to proximity to the
laser spot. During sectioning of the device, sections often fragmented from gentle physical
handling. This indicates reduced strength from the thin channels. This high degree of
porosity is likely to be inviable in a functioning implant for strength reasons. However, an
implant could be designed with a solid core, with increasing porosity towards the exterior
surface. SLS allows building such objects while maintaining the ability to customize the
overall shape and design on a patient-by-patient basis. SEM images (Figure 3.2) of an SLS
printed PEEK high heel device illustrate the limitations of the technique for designing cell
migration assays. For a cell, this material presents an extremely convoluted surface. Both
surfaces show roughness on the same scale as an unspread cell. This roughness dwarfs
a well-spread cell, making cell adhesion and migration experiments very difficult on this
material.
Figure 3.2: PEEK 20 high heel manufactured by SLS
(a) Step-shaped roughness from layer size (1
mm scale bar)
(b) Step roughness (tilted at 30 for imaging,
500µm scale bar)
Commercial printing services do not always release their layer size. However, the layer
size can be approximated by the following method: Measure the horizontal step using the
ImageJ1 distance measurement tool. Compare the length in pixels of a step versus a scale
1http://rsb.info.nih.gov/ij/
36. 27
bar. Use trigonometry to compute the vertical step height from the horizontal step and
angle of the plane specified in the 3D model. The calculated average layer size (Figure A
on page A1) is about 100µm. However, the layer size can easily be changed with one
setting on most SLS machines. These results provide experimental confirmation of an
anecdotal report of the minimum pore size buildable in SLS PEEK. Computer simulations
require this information and the layer size in order to correctly model the SLS process. The
next step in this research would be to attempt to smooth the final sintered part through heat
treatment. If SLS PEEK were heated to near the melting point, surface tension might draw
partially-molten pellets together without compromising overall design features of a device.
XPS analysis of sections at 1.3mm and 5.1mm of the SLS device after CVD and LE
(Figure C.1 on page C2) shows deposition at these depths. When viewed on the survey
scan, phosphorus and zirconium signal strength are constant with depth relative to the car-
bon peak. In contrast to other surface treatments such as plasma spray, this treatment is
compatible with channels of this dimension. The ability of this treatment to permeate a
porous structure will cause the entire surface of such a device to exhibit cell-adhesive prop-
erties.
3.2 “High Heel” Devices
In a side-by-side comparison of two-step milling, chamfer mill cutting, and flycutter
machining (Figure 3.3 on the next page), disadvantages of two-step milling can be clearly
seen. Incompletely-removed chips from the milling process remain attached to the surface
near the critical angle. Also, the region where the two cuts of the mill meet along the
critical angle can be seen to significantly alter the critical angle feature. Flycutter-milled
surfaces appear as smooth as chamfer milled surfaces.
Commercial chamfer mills are typically sold in 60 and 90 included angles (Figure 3.4
on page 29). Custom chamfer mills may be ordered, but they are specific to each combi-
nation of base angle and critical angle. Also, ordering these tools introduces significant lag
time in manufacturing or experimentation, on the order of several weeks. Based on these
tradeoffs, flycutter mills were chosen to build the high heel cell migration device.
37. 28
Figure 3.3: SEM image of critical angles, tilted at 85 from vertical
(a) Two-step milling (b) 60 chamfer mill
(c) 20 angle cut with flycutter
Height profiles (Figure B.1 on page B2) from AFM of test surfaces milled with the
flycutter bits indicated that surface features remained within a 1µm height range. Von
Wilmonsky, et al., found no differences in growth of fibrous tissue on SLS bone implant
surfaces with roughness varying from 0.3 to 7.5µm [19]. The roughness of flycutter milled
surfaces as determined by AFM and visual inspection falls within this range. The rough-
ness parameter was calculated to be 0.185µm. Huang, et al., found that osteoblast-like
cells adhered most strongly to titanium surfaces with roughness of RA = 0.15µm [6]. On
the other hand, Boyan, et al., found that osteoblast-like cells deposited optimal matrix on
rougher surfaces, with RA ⇡ 4µm. For a migration and adhesion experiment, this rough-
38. 29
Figure 3.4: 90 and 60 included-angle chamfer mills
ness parameter is in the correct range. For a finished device, though, the roughness might
actually need to be increased.
At the conclusion of the cell plating experiment, high heel devices were imaged as
described in Section 2.4 in order to visualize cell adhesion and migration. SEM analysis
(Figure 3.5) of the high heels found no evidence of fibroblast adhesion. Cells most likely
slipped off of the base angle before adhering, during the initial 3 hr plating step. To avoid
this possibility, the experiment was redesigned with conical cups to hold the cells during the
plating step. Fluorescence microscopy was inconclusive because PEEK autofluorescence
swamped any potential signal from DAPI and rhodamine-phalloidin stains.
Figure 3.5: SEM of high heel devices after cell assay experiment
39. 30
3.3 Conical Cup Devices
Height profiles (Figure B.2 on page B3) from AFM of test surfaces cut with the tip of
the lathe tool indicated that surface features remained within a several micron height range.
The surface roughness is 0.587µm, so as before with the high heel devices, this value fa-
vors cell adhesion and migration. Walboomers et al. found that fibroblasts tend to align to
microgrooves in a surface [20]. Therefore, the occurrence of radial microgrooves must be
minimized as much as possible to prevent interference with migration. There is no apparent
difference in cell microstructure (Figure 3.6 on the following page) on the three different
surface chemistries. Cells appear equally well-spread, indicating that the roughness and
features of the plain, zirconated, and phosphonated surfaces are similar. This helps show
that the oxide layer is surface-conforming, as the cells adhere to the underlying surface
grooves in the same way. Cell migration fronts (Figure D.1 on page D2) were observed
at the following distances: 3.0mm on the 30 plain PEEK device, 3.3mm on the 30 zir-
conated PEEK device, and 4.8mm on the 0 phosphonated PEEK device. Cells were not
observed on the remaining devices. This may be due to irregularities in cell concentration
in the plating volume. Alternatively, when the cell plating volume communicated with the
media surrounding the conical cup, cells may have been drawn out with fluid flow through
the channel. The migration front distance result is interpreted with care: during the initial
plating, the droplets spread differently on the surfaces according to their wetting properties.
The droplet even reached the critical angle on the 30 phosphonated PEEK device during
the initial plating step.
To improve this cell assay experiment, the next step in imaging would be to stain the
samples with a dye of longer wavelength, such as 647nm. This would serve two purposes:
enhanced microstructure analysis and more rapid counting of adherent cells. By selecting
stains for various cytoskeletal components, the nature of skeletal stresses and focal adhe-
sions could be investigated on each surface type. With clear distinction between the device
surface and the fluorescence of stained cells, these cells could be counted in a large field
of view. Alternatively, future work could explore Nylon 12 as an alternative material for
these experiments, as it does not autofluoresce. XPS of a Nylon 12 coupon treated by CVD
and LE (Figure C.2 on page C2) indicated zirconium and phosphorus on the surface. This
40. 31
Figure 3.6: SEM of cell microstructure on conical cup devices
(a) Plain PEEK 30 device (b) Zirconated 30 device
(c) Phosphonated 0 control
material is attractive as it has favorable materials properties: though it is not as strong as
PEEK, it still exhibits a Young’s modulus of about 2GPa.
The next refinements in cell adhesion steps would be to plate cells at a higher density
for a longer initial period of time, in a smaller volume of media. Also, the migration
experiment would be run for a longer period of time after the removal of unadhered cells.
These modifications would prevent the initial cell plating volume from reaching the critical
angle, while favoring a cell migration front farther from the center of the device, ideally
reaching and traversing the critical angle. Also, these modifications would lead to a more
radially uniform cell migration front.
41. 32
3.4 Conclusion
In this work, 3D printing, computer simulations, materials properties, surface treat-
ments, and control of structure geometries were integrated to examine viable routes to
medical devices that promote cell in- and on-growth. A PEEK scaffold with 800 micron
channels was obtained, illustrating that this dimension is near the minimum feasible for
SLS PEEK. XPS data indicated that CVD and LE permeate the porous structure at this
dimension, leading to even coating with depth.
In spite of these results, 3D printing is not yet mature for bone prosthetic devices. Sur-
face roughness from voxelization and unintentionally sintered particles of PEEK act as
major obstacles to cell adhesion and migration, as well as to imaging techniques. Future
work exploring post-manufacturing modifications is necessary to reduce the surface rough-
ness and voxelization of sintered parts. Also, PEEK particle size, laser spot size, and layer
size must be optimized during the manufacturing process. In the meantime, computer sim-
ulations to determine error introduced by the SLS process can be carried out. The code in
AppendixE approximates the effects of voxelization on the intended part model.
In order to determine the acceptable parameters for cell adhesion and migration, PEEK
surfaces with micron-scale roughness were machined. A cell adhesion and migration assay
device was developed. During the conical cup cell adhesion and migration assay, cells
adhered the farthest from the center of the devices on the phosphonated surface, second
farthest on the zirconated surface, and least far on plain PEEK. This result is interpreted
carefully due to difficulties encountered in standardizing and controlling the experiment, as
well as with visualizing cells. Further work will focus on increasing the robustness of the
assay.
Additionally, the conical cup style of cell migration assay device can be manufactured
with different materials, machining techniques, and surface features. These variations will
help advance polymers with cell-adhesive surface treatments as alternative to titanium for
hard implants.
42. 33
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48. B2
Figure B.1: AFM of surface milled by flycutter
(a) False-color image with profile line indicated (b) 3D height depiction of surface
(c) Height along profile
49. B3
Figure B.2: AFM of surface cut by lathe tool point
(a) False-color image with profile line indicated (b) 3D height depiction of surface
(c) Height along profile
51. C2
Figure C.1: XPS data for two sections through SLS PEEK device
(a) Phosphorus 2p peak on slice at 1.3mm (b) Zirconium 3d peak on slice at 1.3mm
(c) Phosphorus 2p peak on slice at 5.1mm (d) Zirconium 3d peak on slice at 5.1mm
Figure C.2: XPS of Nylon 12 surface treated with CVD and LE
(a) Phosphorus 2p peak (b) Zirconium 3d peak
55. E2
Listing E.1: Main script that reads the STL file and calls the voxelization function
1 % Main script that reads the STL file and calls the voxelization
function
2
3 clear;
4
5 res = [10 10 10]’; % number of voxels in x,y,z
6
7 fname = uigetfile(’*.stl’,’Select the STL model file ’);
8
9 fid = fopen(fname);
10
11 % skip 80 bytes of STL header
12 fread(fid , 80);
13
14 % read count of triangles
15 count = fread(fid , 1, ’uint32 ’);
16 T = triangle.empty(count , 0);
17
18 figure;
19 hold on
20 for i = 1: count % read and draw each triangle
21 T(i) = tRead(fid);
22 T(i).draw ();
23 end
24 hold off
25
26 fclose(fid);
27
28 % find min and max values of x,y,z
29 pts = [T.v1 T.v2 T.v3]’;
30 mn = min(pts) ’;
31 mx = max(pts) ’;
32
33 figure;
34 V = voxelize(T, mn , mx , res);
57. E4
Listing E.2: This function converts a triangulated surface representation of a model to a
voxelized version
1 function V = voxelize(T, mn , mx , res)
2 % This function converts a triangulated surface representation of
a model
3 % to a voxelized version.
4 %
5 % This function accepts the following arguments:
6 % T: an array of Triangles
7 % mn: a 3x1 array for the minimum x, y, and z values of the
model
8 % mx: a 3x1 array for the maximum x, y, and z values of the
model
9 % res: a 3x1 array for the number of voxels in x, y, and z
10 %
11 % and returns an array of voxels where 0 represents unfilled and
1
12 % represents filled.
13
14 DUP_TOL = 1e-4; % tolerance for duplicate intersections
15
16 V = zeros(res ’);
17
18 for j = 1:res (2) % generate crossing rays in y
19 for k = 1:res (3) % and in z
20 xOns = zeros(size(T)) ’;
21 count = 1;
22 % find y and z true coordinates of ray
23 r = toReal(mn , mx , res , [0 j k]’);
24 % for each triangle:
25 for i = 1: size(T,2)
26 % find x coordinate of intersection of ray and plane
27 x = T(i).xOn(r(2), r(3));
28 p = [x r(2) r(3)]’;
29 % confirm that intersection is in triangle and not yet
recorded
58. E5
30 if (T(i).isIn(p) && ~inArray(x, xOns , DUP_TOL))
31 xOns(count) = x;
32 count = count + 1;
33 end
34 end
35 % sort intersections in order of occurrence
36 xOns = sort(xOns (1: count));
37 if (mod(size(xOns , 2), 2) ~= 0)
38 % if there are an odd number of intersections , discard
the
39 % last
40 xOns = xOns (1:end -1);
41 end
42 % fill pixels between odd and subsequent even
intersections
43 for h = 0: size(xOns ,1)/2-1
44 % convert to voxel coordinates
45 x1 = toVoxel(mn ,mx ,res ,[ xOns(h*2+1) ,r(2),r(3)]’);
46 x1 = x1(1);
47 x2 = toVoxel(mn ,mx ,res ,[ xOns(h*2+2) ,r(2),r(3)]’);
48 x2 = x2(1);
49 V(x1:x2 , j, k) = 1;
50 end
51 end
52 end
53
54 end
59. E6
Listing E.3: This class stores a triangle in 3D
1 classdef triangle
2 % This class stores a triangle in 3D. It also provides ray
3 % intersection finding , point inclusion tests , and drawing
methods.
4
5 properties
6 n % vector normal to this triangle
7 v1 % first vertex of triangle
8 v2 % second vertex of triangle
9 v3 % third vertex of triangle
10 i % intercept of plane equation
11 end
12
13 properties (Constant)
14 AREA_TOL = 1e-5; % tolerance for comparing areas in
isIn method
15 end
16
17 methods
18 function t = triangle(n, v1 , v2 , v3)
19 t.n = n;
20 t.v1 = v1;
21 t.v2 = v2;
22 t.v3 = v3;
23 t.i = dot(n, v1);
24 end
25
26 function in = isIn(obj , p)
27 % compute areas of subtriangles
28 a1 = norm(cross(obj.v1 - p, obj.v2 - p));
29 a2 = norm(cross(obj.v2 - p, obj.v3 - p));
30 a3 = norm(cross(obj.v3 - p, obj.v1 - p));
31 aT = norm(cross(obj.v2 - obj.v1 , obj.v3 - obj.v1));
32 % compare areas; point not included if difference
greater than
60. E7
33 % tolerance
34 in = (abs(aT - a1 - a2 - a3) <= obj.AREA_TOL);
35 end
36
37
38 function x = xOn(obj , y, z)
39 % calculate x coordinate on plane with given (y, z)
40 x = (obj.i - obj.n(2) * y - obj.n(3) * z) / obj.n(1);
41 end
42
43 function draw(obj)
44 % draw the triangle
45 p = [obj.v1 obj.v2 obj.v3];
46 fill3(p(1,:), p(2,:), p(3,:), ’b’);
47 end
48 end
49
50 end
61. E8
Listing E.4: This function reads the next Triangle from the given file stream
1 function t = tRead(fid)
2 % This function reads the next Triangle from the given file stream
.
3 %
4 % This function accepts the following argument:
5 % fid: a file identifier obtained from fopen
6 %
7 % and returns a Triangle created from the data read.
8
9 % read three values formatted as 32-bit floating point numbers for
each of:
10 % normal vector , and the three vertices
11 n = fread(fid , 3, ’float32 ’);
12 v1 = fread(fid , 3, ’float32 ’);
13 v2 = fread(fid , 3, ’float32 ’);
14 v3 = fread(fid , 3, ’float32 ’);
15 fread(fid , 1, ’uint16 ’); % this value is not needed
16 t = triangle(n, v1 , v2 , v3);
17
18 end
62. E9
Listing E.5: This function searches for duplicates in an array with a specified tolerance
1 function exists = inArray(n, A, tol)
2 % This function searches for duplicates in an array with a
specified
3 % tolerance.
4 %
5 % This function accepts the following parameters:
6 % n: the value to search for
7 % A: the array to search
8 % tol: the tolerance for value matching
9 %
10 % and returns 1 for present , and 0 for absent.
11
12 for i = 1: size(A,1)
13 if (abs(A(i) - n) <= tol)
14 exists = 1;
15 return;
16 end
17 end
18
19 exists = 0;
20
21 end
63. E10
Listing E.6: This function plots a 3D box around the center point of a voxel
1 function plotVoxel(r, mn , mx , res)
2 % This function plots a 3D box around the center point of a voxel.
3 %
4 % This function accepts the following arguments:
5 % r: the real coordinates for the center of the voxel
6 % mn: a 3x1 array for the minimum x, y, and z values of the
model
7 % mx: a 3x1 array for the maximum x, y, and z values of the
model
8 % res: a 3x1 array for the number of voxels in x, y, and z
9 %
10 % This function operates by plotting each face of the box in
turn.
11
12 hold on
13
14 half = (mx - mn) ./ res / 2;
15 X = zeros (5);
16 Y = zeros (5);
17 Z = zeros (5);
18
19 % top
20
21 Z(1:5) = r(3) + half (3);
22
23 X(1:2) = r(1) + half (1);
24 X(3:4) = r(1) - half (1);
25 X(5) = X(1);
26
27 Y(1) = r(2) + half (2);
28 Y(2) = r(2) - half (2);
29 Y(3) = r(2) - half (2);
30 Y(4:5) = Y(1);
31
32 fill3(X,Y,Z,’b’);
66. E13
Listing E.7: This function plots all of the voxels in the voxel array
1 function plotVoxels(V, mn , mx , res)
2 % This function plots all of the voxels in the voxel array.
3 %
4 % This function accepts the following arguments:
5 % V: the 3D voxel array , where 1 indicates filled
6 % mn: a 3x1 array for the minimum x, y, and z values of the
model
7 % mx: a 3x1 array for the maximum x, y, and z values of the
model
8 % res: a 3x1 array for the number of voxels in x, y, and z
9
10 hold on
11 for i = 1:res (1)
12 for j = 1:res (2)
13 for k = 1:res (3)
14 if V(i, j, k)
15 % convert from voxel index to real coordinates
16 r = toReal(mn , mx , res , [i j k]’);
17 plotVoxel(r, mn , mx , res);
18 end
19 end
20 end
21 end
22 hold off
23
24 end
67. E14
Listing E.8: This function converts from voxel index to real coordinates
1 function p = toReal(mn , mx , res , v)
2 % This function converts from voxel index to real coordinates.
3 %
4 % This function accepts the following arguments:
5 % v: the voxel indices in x,y,z
6 % mn: a 3x1 array for the minimum x, y, and z values of the
model
7 % mx: a 3x1 array for the maximum x, y, and z values of the
model
8 % res: a 3x1 array for the number of voxels in x, y, and z
9 %
10 % and returns the real coordinates at the center of the voxel
11
12 p = (mx - mn) .* (v - 0.5) ./ res;
13
14 end
68. E15
Listing E.9: This function converts from real coordinates to voxel indices
1 function v = toVoxel(mn , mx , res , p)
2 % This function converts from real coordinates to voxel indices.
3 %
4 % This function accepts the following arguments:
5 % p: the real coordinates in x,y,z
6 % mn: a 3x1 array for the minimum x, y, and z values of the
model
7 % mx: a 3x1 array for the maximum x, y, and z values of the
model
8 % res: a 3x1 array for the number of voxels in x, y, and z
9 %
10 % and returns the voxel indices.
11
12 v = round(res .* (p - mn) ./ (mx - mn)) + 1;
13
14 end
