Material properties for FE models were assumed to be homogenous, isotropic, and linearly elastic. For loading condition, axial load was applied to the polyhedrons to simulate the compression as shown in Figure 2. The obtained equivalent von Mises EQV stress distributions on each of the considered polyhedrons in this study were compared. The polyhedrons having EQV high stress under described loading conditions were then eliminated. The cells need certain pore size of tissue substitute to allow proliferation, and it is necessary for tissue engineering scaffolds to be a porous structure.
Geometrically, the problem due to the assembly of some polyhedrons may generate the enclosed pore. The material removal in enclosed pores after the fabrication using AM techniques is then not possible. Consequently, polyhedrons have null porosity. To be able to eliminate the problem from polyhedrons, the three-dimensional CAD models of polyhedrons that passed Criteria B were assembled and evaluated for possible enclosed pore as shown in Figure 3.
Apart from the close-cellular tissue engineering scaffolds, the wireframe of the polyhedrons can be thickened to make the open-cellular tissue engineering scaffolds as demonstrated in Figure 4. For open-cellular form of polyhedrons, the pores can be geometrically generated by two ways:.
For better understanding of the effect of beam thickness dimension on the generation of enclosed pore, it is desirable to investigate how the ratio between pore size and beam thickness PO: BT influences the geometry of open-cellular polyhedrons. BT values were 1: BT values of each open-cellular form of polyhedrons which produced enclosed pore, would be excluded. Criteria A eliminated stellations and nonconvex forms of polyhedrons. The remained polyhedrons were tetrahedron P-1 , octahedron P-2 , hexahedron P-3 , icosahedron P-4 , dodecahedron P-5 , truncated tetrahedron P-6 , truncated octahedron P-7 , truncated hexahedron P-8 , truncated icosahedrons P-9 , truncated dodecahedron P , cuboctahedron P , icosidodecahedron P , rhombicuboctahedron P , rhombicosidodecahedron P , rhombitruncated cuboctahedron P , rhombitruncated icosidodecahedron P , snub cube P , and snub dodecahedron P In Criteria B, the stress analyses based on described loading conditions were employed for 18 polyhedrons.
As shown in Table 1 , the results revealed that the polyhedrons could be classified into three groups according to the stress levels. The first group contained the polyhedrons having very high EQV stress under the load which were tetrahedron P-1 , dodecahedron P-5 , truncated tetrahedron P-6 , and truncated icosahedrons P The second group contained the polyhedrons having moderate EQV stress which were octahedrons P-2 , icosahedron P-4 , truncated dodecahedron P , icosidodecahedron P , rhombicosidodecahedron P , rhombitruncated icosidodecahedron P , and snub dodecahedron P The last group contained the polyhedrons having relatively low stress compared to the others, which were hexahedron P-3 , truncated octahedron P-7 , truncated hexahedron P-8 , cuboctahedron P , rhombicuboctahedron P , rhombitruncated cuboctahedron P , and snub cube P As the polyhedrons in the last group present the low stress level compared to the others, therefore, they were eligible passing Criteria B.
From the stress analysis, it can also be seen that the excluded polyhedrons were asymmetric as well as lack of interface for connecting itself to another one.
Most of the polyhedrons with the low stress group were isotropic symmetry, except the snub cube P The isotropic symmetric polyhedrons are normally proffered by CAD software. This is because, in the polyhedron combination process, the isotropic symmetry CAD models require low memory and short computation time. As a result, even though snub cube P presented the low EQV stress, an open-cellular library of snub cube may lack of interface for combination. Snub cube P had better to be eliminated. Criteria C-1 was employed to evaluate the polyhedrons which could be the close-cellular libraries.
Figure 6 shows the potential close-cellular scaffold libraries included truncated octahedron P-7 , rhombicuboctahedron P , and rhombitruncated cuboctahedron P Criteria C-2 was employed to evaluate the polyhedrons which could be the open-cellular libraries. The potential open-cellular scaffold libraries are shown in Figure 7.
Table 2 shows the analytical results of the influence of PO: BT ratio to the geometry of the libraries as well as the generation of enclosed pore. From the table, the completeness of geometry could be classified into three groups as follows. The porosity is determined by the relationship between the volume of scaffold material and the apparent scaffold volume bounding volume of scaffold.
The mathematical formula relating to the porosity calculation is given in the following equation:. As shown in Figure 8 , porosity of the close-cellular scaffold libraries was constant regardless of pore size. Truncated octahedron had the highest degree of porosity whereas rhombitruncated cuboctahedron had the lowest degree of porosity. For the open-cellular scaffold libraries, the PO: BT ratio in Group A and B was analyzed for the porosity. As shown in Figure 9 , the order of porosities ranged from high to low degree in all PO: From the chart, it can be seen that truncated octahedron P-7 and cuboctahedron P had the equivalent values of porosity.
BT and porosity of open-cellular scaffold libraries. BT ratio and porosity could be determined mathematically using the regression analysis. Various regression functions were trialed to observe the correlation between both parameters. Many functions could well describe the relationship since the coefficient of correlations r of these functions was high.
Nevertheless, some regression functions are rather complex and contain many constants, it is therefore difficult to be used. In this present study, Logistic Power function was used to describe the relationship which can be written in the following form:. In 2 , a , b , and c are constants whereas x and y are independent and dependent variables, respectively.
Table 3 shows the set of equations describing the relationship between PO: BT ratios and porosity of each open-cellular library. A set of equations described the relationship between PO: BT ratio and porosity. Mechanically, the high porosity scaffold has the lower strength than its null porosity solid structure. This is because the solid material is removed, thus the subject volume to the loads becomes less. The high porosity scaffold is nevertheless proper for regeneration environment, since there is a large space for fluid containing necessary substances required for cell growth to circulate into and away from the scaffold.
The designed scaffold having high porosity may not cope with the strength under physiological loads. Additionally, some organs have more than single mechanical properties, the scaffold may require differently the strength in each location. The different scaffold libraries may be selected and composing up the entire scaffold structure [ 33 ].
In order to investigate the feasibility on merging two different polyhedrons, the geometric mismatch of polyhedron interfaces and the stress exhibiting on the interface under axial loads were carried out. The analysis of geometric mismatch of polyhedron interfaces was performed by placing two of the three-dimensional CAD models of scaffold libraries contiguously. Low geometric mismatch refers to the large common interface area whereas high geometric mismatch refers to the small common interface area. The common interfaces can be measured as intersection index in percentage using the following equations.
A hundred percent common intersection index indicates the perfect match of interface of both scaffold libraries. Good example of perfect matching interface is the merging of the isometric symmetrical polyhedrons such as truncated octahedron P In addition, zero percent intersection index implies no intersection of both interfaces. According to the analysis result, the merging of two different close-cellular scaffold libraries was geometrically possible because of the high interface area. The interface area of the close-cellular scaffold library is always higher than that of the open-cellular scaffold library.
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Focusing on the analytic results of open-cellular scaffold libraries, Table 4 shows the results of geometric mismatch analysis. The results can be classified into three groups as follows.
The couples in this group could not be joined together due to lack of common faces. The couples in Group I are listed as follows:. Figure 10 a shows one of the couples in Group I. Figure 10 b shows one of the couples in Group II. The couples in this group could be properly joined together.
Most of contact surfaces of one library almost completely contact to another library, therefore the high intersection index could be observed. The couples in this group were. Figure 10 c shows one of the couples in Group III. Rigid plates were created at both ends of the FE models. In each FE model, the rigid plates were compressed by 0.
Figure 11 shows one of the FE models. Table 4 and Figure 12 show the results of stress analysis. Truncated octahedron-rhombicuboctahedron presented the highest stress whereas cuboctahedron-truncated hexahedron presented the lowest EQV stress among the couples. Additionally, level of the EQV stress was irrelevant to the intersection index. Three-dimensional tissue engineering scaffold is considered to be a key element in success of tissue regeneration.
The tissue engineering scaffold can be extracted from the natural substances or synthesized by various polymer processing techniques [ 10 , 21 ]. By means of these techniques, lack of uniformity and strength of scaffold are problems [ 22 — 25 ]. A good alternative fabrication technique is applying the digital system, that is, CAD and AM, to avoid those problems. Based on the digital system, the scaffold parameters can be controlled in the fabrication process. One way to develop the scaffold libraries, the primitive geometries such as polyhedrons are suitable to be utilized. Although, there are various available polyhedrons, not all of them are suited for using in tissue engineering applications.
Therefore, the criteria for selecting the proper polyhedrons to be used as the scaffold libraries for tissue engineering applications were developed in this present study. Three criteria were proposed for the open-cellular and the close-cellular scaffolds in terms of the limitation of fabrication devices and geometry. Each polyhedron required the orderly assessment to all criteria. For the first criteria Criteria A , feasibility of production based on AM, the complex polyhedrons were eliminated as some components may be suppressed during the fabrication due to the limitations of AM devices in fabrication small dimensional scaffold.
For example, fabrication of bone scaffold may require pore size ranges from — micron [ 36 ]. Criteria B and Criteria C were established to evaluate the geometrical limitations of the polyhedrons after assembly making up the scaffold. The evaluation was based on FE method which is widely accepted as a useful technique to evaluate or predict the biomechanical behavior of biological substitutes [ 19 ], implants [ 44 ], and organs [ 45 ] under certain loading conditions.
According to the results, it can be obviously noticed that the polyhedrons could be categorized into three groups relevant to the exhibited stress level. For the polyhedrons in the high EQV and moderate EQV stress groups, they assembled together for making up the scaffold by connecting each other by vertex or edge. This relates to the effect of interface between the polyhedron. In engineering terms, the less interfaces produce the higher EQV stress concentration around the junction between unit cells under axial loading condition. Therefore, the assembly by connecting vertex-to-vertex or edge-to-edge can be at risk.
On the other hand, the polyhedrons connect to each other by face so that they have the large interface. The large interface normally allows the force to distribute throughout the interface, the EQV stress level subsequently reduces. According to the evaluation results, the polyhedrons which are proper for using as the open-cellular and the close-cellular scaffold libraries were isotropic symmetry: The results were also compared with the ones in previous study [ 22 , 30 ]; it was found that the proposed polyhedrons for utilizing as scaffold libraries are almost similar.
However, the mechanical aspect has not been assessed in the previous study. In the previous study, the proposed polyhedrons also included prisms which are triangular prism, hexagonal prism, and octagonal prism. However, these prisms are asymmetric and may be complicated to join together using CATE automatic software system generating the scaffold.
Concept of the CATE is that the scaffold libraries are duplicated along Cartesian axis inside the specified boundary volume. Some CATE software requires the symmetry scaffold libraries to reduce the time required for software computation. In addition, the previous study suggested that Archimedean dual called rhombic dodecahedron is suitable for scaffold library [ 22 , 30 ].
In the present investigation, the Archimedean duals are beyond the scope, and the evaluation results of this present study were slightly different from that previous investigation. The porosity of the scaffold depends on the amount of apparent volume space occupied by material. The close-cellular scaffold libraries presented the constant porosity no matter of the increasing or the decreasing of pore size. The pore size of close-cellular scaffold libraries could be determined by the void among adjacent polyhedrons. Thus, the size of polyhedron influenced directly on the volume of void space.
Even though the large polyhedron produced the large pore size, unfortunately, the increase of pore size required the increase of polyhedron size by the same amount of volume. From the analysis, some polyhedrons were not included to be used as close-scaffold libraries because their assembly generated the enclosed pore.
After fabrication process by AM technique, it is subsequently impossible for material to be removed. The excessive material is trapped inside the enclose pores. The scaffold finally produces almost null porosity.
ComiXology Thousands of Digital Comics. Reverse engineering for medical, manufacturing and security applications Book 1 edition published in in English and held by 3 WorldCat member libraries worldwide. Currently available manufacturing technology, that is, additive manufacturing is essentially applied to fabricate the scaffold according to the predefined computer aided design CAD model. Parametric library and assembly program. The development of mathematical methods is quite relevant to understand cell biology and human tissues as well to model, design and fabricate optimized and smart scaffolds. Metallic scaffolds for bone regeneration. Investigation of the mechanical properties and porosity relationships in selective laser-sintered polyhedral for functionally graded scaffolds.
In order to effectively raise the porosity of the close-cellular scaffold, the interconnected pore structure IPS can be used to join between polyhedrons as shown in Figure Some libraries of the open-cellular scaffold with some PO: BT ratios could not be fabricated, because they contained the enclosed voids inside the scaffold. Exactly similar reason to the enclosed pore found in the close-cellular scaffold is that the excessive material is trapped and cannot be removed.
The libraries having higher score provided the better availability for fabrication in a wide range of PO: Hexahedron P-3 was considered to be the best library as its score was higher than those of the others. The other libraries which had the lower scores were truncated octahedron P-7 and cuboctahedron P , rhombicuboctahedron P , rhombitruncated cuboctahedron P , and truncated hexahedron P-8 , respectively.
This availability sequence can be geometrically explained that the polyhedrons having few edges and faces require only few beams to compose the scaffold. Therefore, in case of the large beam generated, there is less possibility that the beam overlaps to other beams. Diversely, the polyhedrons having many edges and faces, such as rhombitruncated cuboctahedron P , contain many beams.
The beams may overlap in which can generate enclosed pore. The polyhedron without truncated faces allows the space inside the scaffold to be maximized leaving no space between the polyhedrons. For this reason, the small pore between polyhedrons is absent. However, if the space inside the scaffold is not maximized, pore between polyhedrons is existed. The size of pore depends on the position and angle of the truncated faces. For example, the structure of truncated hexahedron P-3 is almost similar to cuboctahedron P , but the position of truncated faces is different.
The vertices of truncated faces in the cuboctahedron P are mid-edge of surface whereas the vertices of truncated faces in the truncated hexahedron P-8 are shifted toward the corner-edge of surface, as shown in Figure The vertices of truncated face located at the mid-edge of surface are optimized, that is, cuboctahedron P , the space between the polyhedrons and inside peripheral of polyhedrons.
Truncated hexahedron P-8 presents the space between polyhedrons significantly smaller than the space inside polyhedrons. The large beam for open-cellular truncated hexahedron P-8 library may generate the enclosed pore between polyhedrons. For this reason, the availability score of cuboctahedron P was higher than the availability score of truncated hexahedron P Optimized space and possible enclosed pore resulting from position of vertex of truncated face of polyhedrons. From the Figure 9 , it can be noticed that the porosity of the open-cellular scaffold library was strongly influenced by the PO: Different libraries can be assembled together, making the reinforced structure, to meet the mechanical and biological requirements of host tissue.
Despite the fact that the combination of different polyhedrons is a good choice, the compatibility in terms of mechanics and geometry needed to be under consideration. From the geometric mismatch analysis, the higher intersection index was found in the combination between library and different library geometry, but geometrically subset of another one, such as hexahedron-truncated hexahedron, and rhombicuboctahedron-rhombitruncated cuboctahedron.
Although, the geometric details are slightly different, the main components remain similar. Therefore, the common interface of polyhedrons in those couples was possible to connect to each other.
Most of the combinations in the present study yielded no common interface between polyhedrons or have only partial common interface, as the intersection index was null or low. Nevertheless, to combine the couples having without or partial common interface, the torus portion proposed by Wettergreen et al. This can be explained that the intersection index agree with the EQV stress level. However, the EQV stress depends not only on the intersection index, but also on the stiffness of the structure.
In general, the porosity corresponds inversely to the stiffness [ 46 ]. The scaffold libraries containing vertical beam are led to higher stiffness of the structure. Since rhombicuboctahedron P was in the low porosity group, it tends to stiffer than the libraries in the other groups. From the results, because of the high stiffness of rhombicuboctahedron P , the EQV stress at the junction of the couple containing rhombicuboctahedron P was high.
The possible couples for utilizing as the reinforced scaffold included hexahedron-truncated hexahedron, cuboctahedron-truncated hexahedron, and cuboctahedron-rhombitruncated cuboctahedron. Furthermore, since the anatomical geometry of some organs, for example long bone has gradient distribution of pore size, the concept of heterogeneous scaffold functionally graded scaffold, FGS has therefore been recently a new trend in tissue engineering [ 46 ].
The scaffold stiffness can be controlled by the size of pore. The lower pore size is led to higher stiffness of the structure. From the results in the present study, in order to simplify the design of FGS for long bone defect, the units block inside the scaffold may be varied. BT scaffold may be applied to the layer of cortical bone whereas the higher PO: BT scaffold may be applied to the tubercular layer. Figure 15 shows an example of the bone scaffold composing up by integrating different PO: BT ratios to make it suitable for cortical and tubercular layers.
BT scaffold generated based on the CAD model. This study presented the evaluation of polyhedrons for using as the open-cellular and the close-cellular scaffold libraries. Three proposed criteria were used to evaluate each polyhedron. According to the analysis, the proper polyhedrons for using as the close-cellular library included truncated octahedron P-7 , rhombicuboctahedron P , rhombitruncated cuboctahedron P , and snub cube P For open-cellular libraries, the proper polyhedrons included hexahedron P-3 , truncated octahedron P-7 , truncated hexahedron P-8 , cuboctahedron P , rhombicuboctahedron P , and rhombitruncated cuboctahedron P In addition, some of PO: These processes are also increasingly being used by surgeons to plan complex operations, especially in the craniofacial and maxillofacial areas.
Further, prototyping shows promise in the area of tissue engineering through the use of biomaterials including the direct manufacture of biologically active implants.
Bio-Materials and Prototyping Applications in Medicine focuses on bio-materials and prototyping applications in medical environments. The applications that are discussed integrate bio-materials, CAD, and physical prototyping techniques. Bio-Materials and Prototyping Applications in Medicine is a must have for researchers, professionals, engineers and academics seeking a comprehensive overview of this important subject and examples of medical applications using rapid prototyping, including tissue engineering, dental applications, and bone replacement. This title presents these new uses for rapid prototyping in state-of-the-art medical applications.
Design, Prototyping, and Manufacturing features fundamental discussions of all facets of materials processing and manufacturing processes across a wide range of medical devices and artificial tissues. Tissue Engineering is a multidisciplinary field involving scientists from different fields. The development of mathematical methods is quite relevant to understand cell biology and human tissues as well to model, design and fabricate optimized and smart scaffolds.
Emphasis is put on mathematical and computational modeling for scaffold design and fabrication. This particular area of tissue engineering, whose goal is to obtain substitutes for hard tissues such as bone and cartilage, is growing in importance. Advances on modeling in tissue engineering by Paulo R Fernandes 9 editions published in in English and held by WorldCat member libraries worldwide This book presents a collection of chapters describing the state of the art on computational modelling and fabrication in tissue engineering.
The chapter authors are the distinguished keynote speakers at the first Eccomas thematic conference on Tissue Engineering where the emphasis was on mathematical and computational modeling for scaffold design and fabrication. Materials, Processes and Applications brings together contributions from leaders in the field in one comprehensive reference volume, and would be invaluable for researchers and engineers working in the field.
Innovative developments in virtual and physical prototyping: A procedure for quality evaluation. Innovative developments in design and manufacturing: This title presents papers covering: