MECHANICAL INVESTIGATION OF NOVEL BIOMIMETIC CELLS WITH HYBRID POROSITIES FOR ORTHOPEDIC APPLICATIONS

1. Introduction

Orthopedic diseases and their treatment methods are crucial parts of modern healthcare, since the demands of extreme life and job conditions, especially for the elderly population, increases the number of chronic orthopedic injuries. Despite recent advances and improvements in treatment methods and equipment, over 1.7 million injuries and 83000 deaths were suspected to be from implanted medical devices in over 10 years, from 2008 to 2018, in the U.S. alone.^1,2 In many cases, a second surgery is needed to either treat the damage, or to remove the implant. Based on reports and a recent study, about 42% of hip and knee replacements fail in a course of 25 years.³⁴ For hard tissue replacements, such as orthopedics applications, the implant materials should have several characteristics. They have to be non-toxic, non-allergic, non-carcinogenic, non-irritant. In addition to the aforementioned, they have to have adequate porosity and pores so as to provide sufficient osteointegration and bone ingrowth from the host bone to the implant surface because the presence of pores significantly increases the internal surface and surface-to-body ratio of the structure.⁵

Implants with numerous materials of metals,⁶ polymers,⁷ and even ceramics⁸ are recently being tested. Co-Cr alloys and Stainless Steel (316L) are two of metallic biomaterial groups which provide high elastic modulus and high fatigue strength.^9,10 However, these materials are not as biocompatible as Titanium alloys for bone replacements, as Cr and Co are reported to have toxic effects on human body.¹¹ Furthermore, Titanium-based implants can reduce stress shielding effect because of their lower elastic modulus (110 to 140GPa),^12,13 and enhanced corrosion resistance,¹⁴ which makes them a better choice for orthopedic applications. Nonetheless, their elastic modulus stays much higher than that of bone, so, geometrical configuration is needed to achieve a better structure to prevent the stress shielding effect of the implants. Therefore, various porous configurations and formations have been proposed in this regard in numerous research studies.

The porosity and its morphology also alter and affect the mechanical behavior of the structure.¹⁵ Many attempts have been made to mimic the bone structure, which led to a better mechanical behavior of the structure, similar to the bone itself. Sogutlu et al.¹⁶ developed a method to create porous structures with stochastic connected pores, which replicated bone geometry. Nevertheless, these structures have later been proven to deform improperly due to their stochastic pore connections inside the structure.¹⁷ Moreover, it has been observed that structures with non-stochastic porosities have higher mechanical capabilities.¹⁸ Conjoined with the pores, the struts of the structure are also of importance in defining the mechanical characteristics of the structure. Many studies have been done on size, orientation, connection of the struts in a porous material. There have also been some studies on topology optimization of the pores and strut shapes to reach a balance in energy absorption, bending elastic modulus, compressive strength of the porous structures.¹⁹ Additionally, Volker et al.²⁰ studied the effect of the orientation of porous structure and its strut angles on mechanical behavior of additive manufactured parts.

One of the key features of porous biomaterials fabricated by additive manufacturing (AM) technology is the customizability of their structure which can be modified to mimic the properties of trabecular or even cortical bone. This feature is highly valuable in contrast to solid metallic biomaterials which are about 15 times stiffer than bone.^15,21,22,23 Customizing topological orders and features make it certain to reach a fully interconnected porous structure which can improve osseointegration and increase bone regeneration rate. Plus, increased oxygenation and nutrition delivery to the cells migrated into the porous structure cause this performance to improve.¹⁵

Biofunctionalization of the implant can greatly benefit from the increased surface resulting from the pores, by means of pharmaceutics or craving nanopatterns. Drug delivery devices can also benefit from the large porous space in AM biomaterials.^24,25 The small-scale topological design of AM porous biomaterials determines their large-scale properties.¹⁵ Their porous structure decreases their elastic modulus, which helps decrease the stress shielding effect of the load-bearing implant, prevent the implant from loosening. Proper pore size and morphology can also increase the bone ingrowth rate.³ Furthermore, the pore size should be between 100 to 700$\mu$m to avoid the structure from obstruction and to provide enough surface area for cell adhesion.^26,27,28 However, the literature lacks the study on structures having both micro and macro pores in a single unit cell structure, fabricated by additive manufacturing.

Lately, studies regarding porous structures for orthopedic implants have been focused on cellular, non-stochastic open cell porous structures with the help of additive manufacturing techniques.^{8,29,30,31,32} Many unit cells which are mostly CAD-based geometries, have been developed and studied for orthopedic applications, such as simple hollow cube,^31,33,34 diamond,^{22,31,35,36,37} tetrahedron,²⁶ truncated cube,²² truncated cuboctahedron,^22,31 rhombicuboctahedron,²² and octet truss.²⁶ Recently, triply periodic minimal surfaces (TPMS), which have a zero mean curvature and the closest architecture to internal trabecular bone structure have been of inclined interest despite being notably weaker and having a lower elastic modulus when it comes to load-bearing capabilities.^38,39,40 In this paper, we tried to achieve a higher surface-to-volume ratio in unit cells by incorporating basic hollow cube which has a proper mechanical strength, creating new unit cells by adding micropores in the struts to increase the total internal surface area to further assist bone integration and cell proliferation.

The hierarchical morphology of the human bone was the inspiration of this research. To mimic the bone structure which consists of pores with varying sizes and morphologies, a relatively simple geometry, Hollow Cube, was chosen as the base of this study. To increase the porosity of unit cell, micropores were added to the Hollow Cube’s struts which helped to lighten the weight of the implant as well as increase the surface-to-volume ratio of the structure that improves the bone integrity of the implant. The micropores were studied in three different families, specifically, cubic, diamond, circular micropores. Ti6Al4V alloy with an elastic modulus of 113GPa, which is relatively lower compared to other metallic alloys used in orthopedic biomaterials, was chosen as the material for this study. Buckling and failure analyses were then performed to see if thin features in the struts with new structures could withstand the physiological loads under normal conditions. Finally, experimental compression tests were performed to validate the numerical procedure and its results.

2. Materials and Methods

2.1. Modeling and characterization of unit cells with hybrid micropores

To create the porous structure with hybrid pores, the Hollow Cube was used as the basis for placing the micropores into the struts. The chosen micropores varied in size and shape, however, they were categorized into three families of cubic, circular and diamond. In whole, the simplicity of the cubic unit cells allowed us to only focus on the effects of the micropores implemented in the struts, automatically omit the complications and other parameters which may have appeared as a result of a more complex unit cells with angled struts. All the micropores had a fixed height of 200$\mu$m and a variable dimension, which was width for cubic, diameters for diamond and circular families. These variable dimensions were changed gradually throughout the models, ranging from 50 to 400$\mu$m for cubic, from 100 to 400$\mu$m for circular and diamond pores. This led to a total of twelve unit cells with different micropores generated. As for the micropore types, cubic was chosen to demonstrate cube hierarchy, diamond to display another orientation of the cube micropores and finally, circular to evaluate micropores without any sharp edges. Apart from the aforementioned, these simple micropore geometries made the fabrication process more plausible. The Hollow Cube itself and the micropore placements (blue regions) as well as the three micropores along with their variable dimensions are presented in Fig. 1.

The generated geometries are named after their family and their variable dimension. As an example, Cu150 refers to the unit cell with Cubic micropores with 150mm widths. Two of the models not only have micropores along their struts, but also in their corners. These variants were designed to inspect the possible effect of further porosity in the corner regions of the unit cell. These two models have a ($+$) sign next to their names. The micropores in their corners differ from the ones in the struts and have 200mm width and height. Figure 2 shows the generated unit cells and their respective names. The volume of a $3\times 3\times 3$mm non-porous cube was considered to calculate the porosity of the specimens. The volume ratio was reported as the volume of the porous unit cell to that of the non-porous cube.

2.2. Material selection

One of the most common biocompatible metals used in orthopedic implants and tissue engineering scaffolds is Titanium and its alloys. However, pure Titanium is not a proper choice for load-bearing implants, as it has low yield stress (less than 480MPa) and low tensile strength (less than 550MPa). Hence, its alloys are preferred due to their improved mechanical performance. Ti6Al4V has been known for its high yield strength and high load-bearing capability which makes it a preferred material for osteosynthesis and orthopedic implants. Ti6Al4V is the material used for finite element analysis with an elastic modulus of 113GPa, Poisson’s ratio of 0.34, density of 4430kg/m³. Furthermore, for experimental tests and evaluation of numerical computations, polymeric replicas were fabricated with ABS from thermoplastic polymers family.

2.3. Numerical implementation

All hybrid cubic unit cells have been created using SOLIDWORKS^TM and then exported to ANSYS^TM 19.2 Student edition for computational studies in 3D and static structural setup. To perform finite elements analysis, tetrahedral elements were implemented to properly fill the geometries. Mesh size was considered small enough to give us relatively precise results near sharp corners. Generated mesh for a simple geometry, as well as the boundary conditions of the simulation can be seen in Fig. 3. The blue upper surface is under arbitrary 200Pa pressure and the bottom surface is fixed in all directions. Although 200Pa is an arbitrary value, however, it is in the elastic range of the material and follows Hook’s Law (Eq. (1)).

Hook’s law was implemented to evaluate the elastic modulus of the unit cells in the linear region. Hook’s law, elastic modulus (E), strain ($\epsilon$), initial length (L), displacement (u) are presented in Eqs. (1)–(3). Also Eq. (4) was used to evaluate the porosity percentage of models in which $V_p$ is the volume with porosities and $V_t$ is the volume of the unit cells without any porosity. To evaluate their mechanical characteristics such as elastic modulus, they were placed on a fixed surface and put under 200Pa compressive pressure on their upper face. In the computations, P was equal to the pressure applied to the top surface (200Pa). Axial displacement (u) proportional to the pressure was read from the results of the numerical computations which was then used to evaluate the strain of the unit cell based on Eq. (2). Elastic modulus then was obtained by Eq. (3).

Creating micropores in already porous structure causes the unit cells to be more vulnerable to the effects of compressive loadings, buckling, stress concentration in their thin and corner regions. Hence, after confirming that the elastic modulus of the unit cells is safely in the human bone range, the buckling test was performed to obtain the critical loads required for the unit cells to fail by buckling. To observe the initiating failure points in the simulation, safety factor was used. Using the parametric study of ANSYS, the safety factor was set to 1 the load was gradually increased. The process could have been stopped the instant the safety factor reached 1, nonetheless, to see other possible starting failure regions, the parametric study continued even after reaching initial failure. This gave us a few points in the structure suggesting the failure during compressive loads.

2.4. Fabrication and test

To validate the numerical results, three Cu200 unit cell samples were fabricated using Digital Light Processing method (DLP) with ABS material and Zmorph printer. Figure 4(a) shows one of the fabricated samples and to put scale to the infinitesimal sample size, in Fig. 4(b) a sample can be seen alongside a coin. Thereafter, compressive test was performed following ISO 13314:2011 instructions (mechanical testing of metals’ compression test for the porous and cellular metals) which indicates a constant 1 mm/min compressive displacement. SANTAM DBBP-200 instrument was used to perform the compression test. The force and their respective displacements were obtained from the tests which will be discussed in the results section.

3. Results and Discussion

3.1. Mesh study and convergency

To mesh the geometries, tetrahedral elements were used and, to reach mesh independency as well as solution convergence, mesh refinement was performed. Nine cases were investigated regarding refining the mesh from 5117 to 353762 number of elements in which the values of displacements on upper surface aligned with the direction of pressure were measured.

For the convergency analysis, the number of elements and element displacements were obtained. The data have been presented in Table 1 and plotted in Fig. 5. Observing the convergency curves, the quadratic element type entered a convergent bound of $\pm 0.0005$$\mu$m, from 124616 elements. On the other hand, the linear element demonstrated larger displacement gap between each case. However, it was observed that even with 353762 elements, the linear mesh did not reach sensible convergence. Considering the high computational cost of higher element numbers, quadratic elements were decided to be used for all the cases.

3.2. Mechanical properties

3.2.1. Static analysis and mechanical properties

The elastic modulus of the unit cells was calculated from both the results of numerical computations and Eqs. (1)–(3), in Numerical Method section. Figure 6 shows key properties of materials used for orthopedic implants: elastic modulus, porosity, internal surface, surface-to-volume ratio. Also, simple Hollow Cube (HC) is placed at the first position for comparison with other models. Other unit cells are ordered by their elastic modulus in an ascending manner. Additionally, 2D corner view of each unit cell has been placed under their mechanical properties bar for better understanding. The exact derived values of these properties are presented below the chart.

Compared to the Hollow Cube, all new unit cells indicated lower elastic modulus, but remained in the range of bone’s elastic modulus. Furthermore, all new unit cells had significant increase in their internal surface area, surface-to-volume ratio, porosity. The more surface is available to the adjacent bone, the better cell proliferation and bone integration will happen.⁵ In addition, higher porosity helps the bone cells to grow more inside the implant. Together, the surface area and the porosity provide a proper understanding of how a porous structure behaves regarding bone integration which is essential for orthopedic implants. To help achieve such understanding, the internal surface area-to-volume ratio (SV) of the unit cells should be considered. Higher SV happens under one of the following conditions: (a) significant increase of the internal surface area, (b) notable porosity increase (higher porosity leads to lesser volume) and, (c) a combination of both of the aforementioned occurring together, to some notable extent.

To investigate morphologies, the specimens with the same micropore heights were chosen from each family. Cu400, D400, C400 were chosen for this cause. In Fig. 6, it can be seen that the diamond family had higher elastic modulus and tends to be stronger when bearing loads. It is also notable that cells like Cu400$+$ and C200$+$ which include extra micropores in their corners, had notably higher surface-to-volume ratio which suggests better bone integration. Despite their mechanical characteristics, it is better to choose the unit cells depending on the region which needs implantation. Proper porosity of porous orthopedic implant designs ranges from 50% to 80%.⁴¹ Furthermore, human cortical bone has an elastic modulus from roughly 3 to 30GPa.⁴¹ Similar studies in the recent years on the subject have reported much lower elastic modulus, commonly ranging between roughly 0.5 to 5GPa^26,42 which is acceptable, however they provide such stiffness with relatively lower porosities. A unit cell designed in the shape of dodecahedron with 75% porosity was reported to possess an elastic modulus of 1.9GPa,²⁶ while the Cu150 model in this study with a porosity almost equal to 68.5% showed an elastic modulus of 9.4GPa which is notably higher. Lowering the porosity of the unit cells as low as 67% in the D100 model, provided an elastic modulus of 11GPa, while unit cells with much lower porosity of 50% have been reported to be as stiff as 4.6GPa.²⁶ It can be claimed that an overall increase in porosity while maintaining a high elastic modulus is the outcome of the current approach to include hierarchical micropores in the struts of the already porous unit cells.

Many studies attempt to propose new unit cell topologies for bone implants. Although it is needed to keep the stiffness in the range of human bone, lower elastic modulus helps further reduce the stress shielding phenomenon. The three nominated cases from each family of micropores in this study, namely Cu200, C200, D200 showed 70.37%, 71.93% and 69.63% porosities, respectively, which were 4 to 7GPa lower than those of unit cells with similar porosity investigated in a recent study.⁴³ In another study, for a unit cell with 70% porosity they reported an elastic modulus of 15GPa whereas in this paper, D400 with the same porosity demonstrated the module of 10.83GPa.⁴⁴ D100 with a relative density of 33% had an elastic modulus of 10.93GPa, which was about 2GPa less than CFC-50 investigated in another study with the same porosity.³⁴ Overall, what this research achieved is notably higher surface-to-volume ratio and internal surface area, while maintaining a low and sufficient elastic modulus along with relatively high porosity.

3.2.2. Buckling analysis of thin features

Decreasing the elastic modulus can guarantee preventing stress shielding phenomena, although the chance of buckling in thin struts may increase. To investigate whether or not the cells fail under extreme physiological conditions, buckling numerical analysis was performed and the results are shown in Fig. 7. The magnitude of the critical buckling loads suggest that cells can withstand normal physiological conditions and will not fail due to their thin features. Nevertheless, large and sudden loads like impacts and heavy sport should be considered while using such structures, as this test does not focus on abnormal loading conditions. There is an overall reverse relation between the porosity and the buckling force, though this statement is not absolute. Inspecting the morphologies suggests the effect of micropore shape in the aforementioned relation. For example, Cu200 has lower porosity compared to Cu400 and C400, but it’s buckling load lies in between the other two. A similar situation can also be seen for Cu150 and D400, which have a relatively lower buckling load compared to their previous and next unit cells on the graph, but their porosity lies between the other two.

Furthermore, some unit cells displayed very close elastic modulus, but their buckling load differed significantly. Cu400, Cu400+, Cu50, Cu150 exhibited extremely close elastic modulus, unlike Cu400 and Cu400+ which failed sooner due to the buckling of their thin features. Thus, the elastic modulus alone is not a good basis for choosing load-bearing implants. Investigating the unit cells with equal micropore widths like Cu200, D200, C200, suggests that diamond family has higher elastic modulus but, they buckle quicker than cubic and circular families. As well, the hybrid unit cells with micropores in their corners such as Cu400+ and C200+ behaved much weaker in terms of buckling, compared to their counterparts with no micropores in their corners. Hence, the increase in their internal surface area comes at the cost of their strength under buckling loads.

3.2.3. Initiating failure regions

Failure due to heavy compressive loads usually happens in thin features, corners, the junctions of a structure. This issue was expected to be seen in the new suggested porous unit cells as well. The parametric study was performed to obtain possible regions of failure. Figure 8 shows the possible initial failure regions under axial compression. These regions have the highest chance in initiating failure process of the unit cells based on the distortion energy theory. They are marked with red paint in the magnified regions.

An interesting observation is that the diamond family and the circular family unit cells start to fail from their corner regions and inside the median area of strut micropores. Cubic family unit cells start failure only from corners and the micropores still can withstand compressive stress. To obtain the exact value of failure stress, the parametric study on the unit cells was performed and the results are presented in Fig. 9.

As mentioned before, the diamond family unit cells had relatively higher elastic modulus and were stronger under compression loads. This can be seen in the slope of the dot line of the diamond family which is above cubic and circular families. Regardless, the structure failed sooner under axial compression and had a lower yield point. This matter alongside the buckling results, suggested their overall weaker load-bearing ability. These leave engineers with the need to precisely understand the region and the type of the unit cell they are going to use. Elastic modulus, buckling load, failure load all together have to be considered when dealing with load-bearing unit cells in implants.

3.2.4. Experimental test results

The force-displacement diagram for three samples of Cu200 micropores was generated from results obtained from the axial compression test. Figure 10 shows the linear portion (elastic region) of the results (dotted line) as well as the numerical values (triangle dots).

The diagram indicates approximately 9% error between the simulation and experimental results. The error was obtained by comparing the slope of the results in Fig. 10. Usually, this error could be attributed to geometric mismatch between the CAD file and the fabricated model in the form of micro scale defects. This ensures the reliability of the numerical results obtained in this research with a reasonable accuracy.

4. Conclusions

In this research, we designed new unit cells which simultaneously included both macropores and micropores to mimic the hierarchical structure of the bone. The micropores included three geometry families of cubic, circular and diamond and this approach provided increased internal surface area and up to 87% porosity. Using the Ti6Al4V alloy, these unit cells showed stiffness ranging from 7.9GPa to 11GPa, which is in the range of the stiffness of the bone. The diamond family tended to be slightly stiffer, but under compressive load it reached the elastic limit quicker than the circular and cubic families. Thus, it is suggested to choose the appropriate structure for orthopedic application accordingly.

Continuation of this research would benefit from further studying diverse micropore topologies and arrangements, as this project was only an initial step, only taking three fundamental topologies into account. Considering different materials suitable for orthopedic applications can also help deepen the understanding of the effects of having such hybrid porosities on the mechanical performance of the porous unit cells. Lastly, it is suggested to incorporate more advanced techniques of additive manufacturing to fabricate these structures, as the current state of the models lacked the utmost desired quality with the equipment at hand.

ORCID

ALI ABOUEI MEHRIZI  https://orcid.org/0000-0002-7515-9855