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Karakoç, Alp

RegionTPMS — Region based triply periodic minimal surfaces (TPMS) for 3-D printed multiphase bone scaffolds with exact porosity values

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10.1016/j.softx.2021.100835 Published: 01/12/2021

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Karakoç, A. (2021). RegionTPMS — Region based triply periodic minimal surfaces (TPMS) for 3-D printed multiphase bone scaffolds with exact porosity values. SoftwareX, 16, [100835].

https://doi.org/10.1016/j.softx.2021.100835

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Original software publication

RegionTPMS — Region based triply periodic minimal surfaces (TPMS) for 3-D printed multiphase bone scaffolds with exact porosity values

Alp Karakoç

Aalto University, Department of Communication and Networking, FI 00076, Espoo, Finland

a r t i c l e i n f o

Article history:

Received 19 April 2021

Received in revised form 8 September 2021 Accepted 20 September 2021

Keywords:

Triply periodic minimal surface (TPMS) Scaffold

Porosity

Additive manufacturing (AM) Functionally graded Multiphase

a b s t r a c t

With the advances in biocompatible materials for additive manufacturing (AM) methods, triply periodic minimal surface (TPMS) functions have been gaining grounds in design and development of bone scaffolds. Their promising results for cell proliferation, differentiation and physical characteristics in- vivo have drawn attention in both biomedical and engineering communities. In order to advance the current state-of-the-art, a region based TPMS scaffold generation algorithm is proposed, where the exact porosity values and explicitly described surface functions are used as the input parameters.

Rather than adding thickness to the minimal surfaces, as being the common practice, regions are defined within the range of local minimum and maximum of the investigated surface functions in the present study. Therefore, it is possible to design and print the 3-D scaffolds with the exact desired porosity and also define interfaces for functionally graded scaffolds. In order to demonstrate the region- based scaffold concept, first, porous scaffolds based on different mathematical functions (Schwarz-D, Schwarz-P, Gyroid, Fischer–Koch S, Fischer–Koch Y, Lidinoid, I-WP, F-RD and Split-P surface functions) with the porosity values{0.2,0.3,0.4,0.5,0.6,0.7}were demonstrated. Secondly, multi-phase and multi-porous scaffolds based on the region ranges were presented. The related editable Mathematica notebooks and sample 3D printing files in the standard triangle language (STL) format were also made available for the community use athttps://github.com/metudust/RegionTPMSunder MIT license.

Code metadata

Current code version RegionTPMS_202104

Permanent link https://github.com/ElsevierSoftwareX/SOFTX-D-21-00076

Legal Code License MIT

Required software for applications which generates scripts Mathematica≥v12

Compilation requirements, operating environments Windows

User documentation, videos, files and manual https://github.com/metudust/RegionTPMS/README.md

1. Motivation and significance

Bone (osseous tissue) scaffold regeneration have been an im- portant topic in the field of biomedical engineering for decades [1, 2]. Successful implementations of such scaffolds require ade- quate cell proliferation, colonization and mechanical stability [3, 4]. However, due to the complex structure of the bone tissue involving porosity, interconnectivity for cell and nutrient diffu- sion, the design and manufacturing processes of bone scaffolds have been challenging [5]. Especially, with the advances in non- invasive imaging techniques such as computed tomography, bone

E-mail address: alp.karakoc@alumni.aalto.fi.

structures have been characterized by various researchers in the field [6–8]. The characterization studies in the literature demonstrated that the triply minimal periodic surface (TPMS) functions are one of the most suitable candidates (among beam-based structures, foams, lattices, etc.) that can closely mimic the cancellous and cortical bone layers [9–15]. Moreover, the developments and accessibility of additive manufacturing technologies have led the bone tissue regeneration process more automatized and precise. These improvements thus enable the researchers directly use the patient-specific data to develop bone scaffolds regeneration workflows as schematized exemplified inFig. 1.

The design and manufacturing process are very crucial steps for the regeneration of bone scaffolds, especially in consideration

https://doi.org/10.1016/j.softx.2021.100835

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Alp Karakoç SoftwareX 16 (2021) 100835

Fig. 1. Patient-specific surgical framework for bone regeneration.

Fig. 2. The workflow of RegionTPMS.

to the necessity of controlled porosity and phases, which provide structural integrity, interconnectivity and cell growth [16].

Therefore, the present study provides precise design and manufacturing practices for the scaffolds with the desired porosity values (5-digit precision) and explicitly described mathematical functions, the Mathematica notebooks under MIT licenses and STL file samples for 3D printing are provided at https://github.com/

metudust/RegionTPMS(as listed in Code metadata table).

2. Software description

As schematized in Fig. 2, the workflow of RegionTPMS can be explained in three stages: input data for the porosity and TPMS function f(^x,^y,^z), process for computing the local minimum min(^f(^x,^y,^z)), local maximum max(^f(^x,^y,^z)) ^{of the} TPMS function and the upper boundξ ∈_Rcorresponding to the input parameters within the region boundaries.

In theRegionTPMS notebooks, the idea is to define the region, which corresponds to the desired porosity value, e.g.

min(f(x,^y,^z)) ≤ f(x,^y,^z) ≤ ξ, ξ ∈ _R, within the desired bounds. The presented method differs from the conventional approach of minimal surface thickening at f(^x,^y,^z) = 0 for

defining the porosity. Here,ξis obtained through numerical root- finding process, e.g. via bisection method as implemented here.

Hence, as illustrated in Fig. 3, it is both possible to generate porous and non-porous domains as distinct spaces separated by the surface function f(^x,^y,^z) and infinite number of material phases within a scaffold. For this purpose, the Mathematica notebooks include a large variety of TPMS functions, which are Schwarz-D, Schwarz-P, Gyroid, Fischer–Koch S, Fischer–Koch Y, Lidinoid, I-WP, F-RD and Split-P surface functions, are described as nodal equations as listed inTable 1.

As listed in Table 2, RegionTPMS consists of three Mathe- matica notebooks, which are RegionTPMS_ExactPorosityScaffold, RegionTPMS_MultiphaseScaffold and RegionTPMS_GradedPorosity Scaffold, respectively. RegionTPMS_ExactPorosityScaffold was designed to generate graphics object for visualization and STL file for printing for TPMS functions with the desired porosity as the inputs. The second notebook,RegionTPMS_MultiphaseScaffold, was designed to generate graphics and STL files for scaffolds with several material inclusion. For instance, soft and hard tissues within a scaffold can be thus tailored and manufactured. The last notebook, RegionTPMS_GradedPorosityScaffold, was designed for

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Fig. 3. Region based TPMS scaffold concept with different porosities (above) and implementation of multiphase materials as regions (below) with equal volumes, for which the minimal surfacef (x,y,z) = 0 was taken as the interface. Here, min(f(x,^y,^z))≤f(x,^y,^z)≤ξrefers to the region function, for whichξ∈R.

Table 1

Surface functionsf (x,y,z) representing the triply periodic minimal surfaces (TPMS) [17–19].

Function name Mathematical expressionf (x,y,z)

Schwarz D (diamond) sin(x) sin(y) sin(z)+sin(x) cos(y) cos(z)+cos(x) sin(y) cos(z)+cos(x) cos(y) sin(z) Schwarz G (gyroid) cos(x) sin(y)+sin(x) cos(z)+cos(y) sin(z)

Schwarz P cos(x)+cos(y)+cos(z)

Fischer–Koch S sin(x) cos(y) cos(2z)+cos(2x) sin(y) cos(z)+cos(x) cos(2y) sin(z) Fischer–Koch Y sin(x) sin(y) sin(z)+cos(x) cos(y) cos(z)+sin(2x) sin(y)

+cos(x) sin(2y)+sin(x) sin(2z)+sin(2x) cos(z)+sin(2y) sin(z)+cos(y) sin(2z) Lidinoid sin(x) sin(2y) cos(z)+sin(2x) cos(y) sin(z)+cos(x) sin(y) sin(2z)

−cos(2x) cos(2y)−cos(2x) cos(2z)−cos(2y) cos(2z)+0.³

Split-P −0.2(cos(2x) cos(2y)+cos(2x) cos(2z)+cos(2y) cos(2z))−0.^4(cos(2x)+cos(2y)+cos(2z)) +1.1(sin(x) sin(2y) cos(z)+sin(2x) cos(y) sin(z)+cos(x) sin(y) sin(2z))

Neovius 4 cos(x) cos(y) cos(z)+3(cos(x)+cos(y)+cos(z))

I-WP 2(cos(x) cos(y)+cos(x) cos(z)+cos(y) cos(z))−(cos(2x)+cos(2y)+cos(2z)) F-RD 4 cos(x) cos(y) cos(z)−(cos(2x) cos(2y)+cos(2x) cos(2z)+cos(2y) cos(2z))

varying porosity in the radial direction, scaffolds out of which can simply mimic the cancellous and cortical bone tissues.

These notebooks can be directly implemented or editable for specific use such as graphical visualization or 3D printing file

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Table 2

Mathematica notebooks and their outputs.

Fig. 4. Examples for user controllability over the codes: (a) resolution variations with respect toPlotPointsgraphics option in MathematicaRegionPlot3D[] function, (b) use of various boundary domains for the region.

generation with the STL format. Therefore, user has the full controllability over the codes and can further develop them based on their needs and applications. For instance, as illustrated in Fig. 4(a), user can change the resolution of the graphical objects,

hence, the surface finish of the 3D printing file, by fine-tuning thePlotPoints, which is a graphical option for theRegionPlot3D[]

function and specifies the number of initial sample points for function plotting. As also shown inFig. 4(b), user can constrain

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Table 3

Scaffolds with a variety of TPMS functions and predefined porosities.

the region bounds by defining various forms of boundary domains such as cubic, spherical or elliptical domain.

3. Illustrative examples

Three illustrative examples created withRegionTPMSare provided, the first of which focuses on the generation of scaffolds based on the predefined TPMS functions and porosity values as seen in Table 3. The related Mathematica notebook for this example isRegionTPMS_ExactPorosityScaffold,which is published in the github link. By means of this notebook, the user can have significant control over the internal pore architecture of the scaffold. Thus, design and fabrication of scaffolds with micron-level precision can be achieved with the-state-of-the-art direct writing or resin printing technologies, e.g., stereolithography (SLA) or projection microstereolithography (PµSL), etc. With the emerging additive manufacturing technologies combined with the materials science, complex 3D polymer-based scaffolds have been

mimicked and fabricated [20–22]. For instance, cell growth and mechanical support in tissue engineering should be well op- timized with discrete compartments, which can be mimicked and achieved withRegionTPMS_MultiphaseScaffold, as tabulated in Table 4. Especially, cell proliferation and differentiation should be guided for the tissue regeneration [23,24]. For instance, in order to enhance the bone tissue regeneration efficiency, graded pore architectures in the same scaffold have been investigated and promising results have been obtained [25,26]. RegionTPMS_

GradedPorosityScaffoldwas coded for the design and fabrication of such scaffolds, some of which are listed in Table 5. Here, Mathematica built-inShow[] function was used to combine the regions and form the graded architecture graphics objects and 3D printing files.

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Table 4

Multiphase (or functionally graded) scaffolds based on various TPMS functions with equally divided regions, i.e. two regions (50% filling each), three regions (33% filling each), four regions (25% filling each), five regions (20% filling each).

4. Impact

There have been several investigations on the bone scaffold characterization and regeneration, which are based on beam- based structures, foams, lattices and zero-thickness minimal surfaces [7,8,12,13,27,28]. However, to the author’s knowledge, there is no available source code as presented here that uses region based explicitly expressed functions with the desired porosity in the literature and provides 3D printing files (in the STL format) for the community use. The remarkable physical characteristics and interconnectivities of TPMS architecture have been found out to be well-mimicking the natural bone structures [29–31].

These supporting findings in the literature and the present study will have an impact on the prospective research studies and advance the current state-of-the-art in the fields of biomechanics.

In addition to its current scope, the provided Mathematica notebooks and 3D printing files can be also directly implemented in different engineering fields including aerospace and transporta- tion engineering, for which energy absorption plays a critical role; mass and heat transfer applications, for which the porous micro-feature surface structure arrays on large surface areas are needed; civil engineering applications, where the cellular cementitious structures can provide lightweight yet resilient solutions for prefabricated construction, to name a few [32–35].

5. Conclusions

In the present study, editable Mathematica notebooks and example 3D printing files in STL format, which implement region based TPMS definitions, are made available for public use. There- fore, researchers worldwide can design, manufacture, investigate and characterize bone scaffolds in timely manner so that healing conditions and periods can be enhanced for the patients suffering from bone diseases and fractures. In addition to its applicability in biomechanics, the provided source codes and 3D printing files are believed to be impactful and beneficial for the researchers whom work in various fields of engineering, materials science and biomedicine.

Declaration of competing interest

The authors declare that they have no known competing finan- cial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

AK generated the concept, codes and wrote the manuscript.

The source codes and example STL files are available athttps:

//github.com/metudust/RegionTPMS. The author gratefully ac- knowledges the funding from Academy of Finland BESIMALpro- ject (decision number 334197).

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Table 5

Scaffolds with the inner (0<r/R<0.8) and outer regions (0.8<r/R<1.0) of porosities=0.6 and 0.2, respectively.Ris the total radius andris the radial distance from the center.

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