Analytical model for the prediction of permeability of triply periodic minimal surfaces

Multi-parameter design of triply periodic minimal surface scaffolds: from geometry optimization to biomechanical simulation

This study introduces a multi-parameter design methodology to create triply periodic minimal surface (TPMS) scaffolds with predefined geometric characteristics. The level-set constant and unit cell lengths are systematically correlated with targeted porosity and minimum pore sizes. Network and sheet scaffolds featuring diamond, gyroid, and primitive level-set structures are generated. Three radially graded schemes are applied to each of the six scaffold type, accommodating radial variations in porosity and pore sizes. Computer simulations are conducted to assess the biomechanical performance of 18 scaffold models. Results disclose that diamond and gyroid scaffolds exhibit more expansive design ranges than primitive counterparts. While primitive scaffolds display the highest Young's modulus and permeability, their lower yield strength and mesenchymal stem cell (MSC) adhesion render them unsuitable for bone scaffolds. Gyroid scaffolds demonstrate superior mechanical and permeability performances, albeit with slightly lower MSC adhesion than diamond scaffolds. Sheet scaffolds, characterized by more uniform material distribution, exhibit superior mechanical performance in various directions, despite slightly lower permeability. The higher specific surface area of sheet scaffolds contributes to elevated MSC adhesion. The stimulus factor analysis also revealed the superior differentiation potential of sheet scaffolds over network ones. The diamond sheet type demonstrated the optimal differentiation. Introducing radial gradations enhances axial mechanical performance at the expense of radial mechanical performance. Radially decreasing porosity displays the highest permeability, MSC adhesion, and differentiation capability, aligning with the structural characteristics of human bones. This study underscores the crucial need to balance diverse biomechanical properties of TPMS scaffolds for bone tissue engineering.

Novel 3D printed TPMS scaffolds: microstructure, characteristics and applications in bone regeneration

Bone defect disease seriously endangers human health and affects beauty and function. In the past five years, the three dimension (3D) printed radially graded triply periodic minimal surface (TPMS) porous scaffold has become a new solution for repairing bone defects. This review discusses 3D printing technologies and applications for TPMS scaffolds. To this end, the microstructural effects of 3D printed TPMS scaffolds on bone regeneration were reviewed and the structural characteristics of TPMS, which can promote bone regeneration, were introduced. Finally, the challenges and prospects of using TPMS scaffolds to treat bone defects were presented. This review is expected to stimulate the interest of bone tissue engineers in radially graded TPMS scaffolds and provide a reliable solution for the clinical treatment of personalised bone defects.

Three-Dimensional Printing of Graphene Oxide/Poly-L-Lactic Acid Scaffolds Using Fischer-Koch Modeling

Accurately printing customizable scaffolds is a challenging task because of the complexity of bone tissue composition, organization, and mechanical behavior. Graphene oxide (GO) and poly-L-lactic acid (PLLA) have drawn attention in the field of bone regeneration. However, as far as we know, the Fischer-Koch model of the GO/PLLA association for three-dimensional (3D) printing was not previously reported. This study characterizes the properties of GO/PLLA-printed scaffolds in order to achieve reproducibility of the trabecula, from virtual planning to the printed piece, as well as its response to a cell viability assay. Fourier-transform infrared and Raman spectroscopy were performed to evaluate the physicochemical properties of the nanocomposites. Cellular adhesion, proliferation, and growth on the nanocomposites were evaluated using scanning electron microscopy. Cell viability tests revealed no significant differences among different trabeculae and cell types, indicating that these nanocomposites were not cytotoxic. The Fischer Koch modeling yielded satisfactory results and can thus be used in studies directed at diverse medical applications, including bone tissue engineering and implants.

In silico assessment of the bone regeneration potential of complex porous scaffolds

Mechanical environment plays a crucial role in regulating bone regeneration in bone defects. Assessing the mechanobiological behavior of patient-specific orthopedic scaffolds in-silico could help guide optimal scaffold designs, as well as intra- and post-operative strategies to enhance bone regeneration and improve implant longevity. Additively manufactured porous scaffolds, and specifically triply periodic minimal surfaces (TPMS), have shown promising structural properties to act as bone substitutes, yet their ability to induce mechanobiologially-driven bone regeneration has not been elucidated. The aim of this study is to i) explore the bone regeneration potential of TPMS scaffolds made of different stiffness biocompatible materials, to ii) analyze the influence of pre-seeding the scaffolds and increasing the post-operative resting period, and to iii) assess the influence of patient-specific parameters, such as age and mechanosensitivity, on outcomes. To perform this study, an in silico model of a goat tibia is used. The bone ingrowth within the scaffold pores was simulated with a mechano-driven model of bone regeneration. Results showed that the scaffold's architectural properties affect cellular diffusion and strain distribution, resulting in variations in the regenerated bone volume and distribution. The softer material improved the bone ingrowth. An initial resting period improved the bone ingrowth but not enough to reach the scaffold's core. However, this was achieved with the implantation of a pre-seeded scaffold. Physiological parameters like age and health of the patient also influence the bone regeneration outcome, though to a lesser extent than the scaffold design. This analysis demonstrates the importance of the scaffold's geometry and its material, and highlights the potential of using mechanobiological patient-specific models in the design process for bone substitutes.

Mechanical Properties Directionality and Permeability of Fused Triply Periodic Minimal Surface Porous Scaffolds Fabricated by Selective Laser Melting

Titanium alloy porous scaffolds possess excellent mechanical properties and biocompatibility, making them promising for applications in bone tissue engineering. The integration of triply periodic minimal surface (TPMS) with porous scaffolds provides a structural resemblance to the trabecular and cortical bone structures of natural bone tissue, effectively reducing stress-shielding effects, enabling the scaffold to withstand complex stress environments, and facilitating nutrient transport. In this study, we designed fused porous scaffolds based on the Gyroid and Diamond units within TPMS and fabricated samples using selective laser melting technology. The effects of the rotation direction and angle of the inner-layer G unit on the elastic modulus of the fused TPMS porous scaffold were investigated through quasi-static compression experiments. Furthermore, the influence of the rotation direction and angle of the inner-layer G unit on the permeability, pressure, and flow velocity of the fused TPMS porous scaffold structure was studied using computational fluid dynamics (CFD) based on the Navier-Stokes model. The quasi-static compression experiment results demonstrated that the yield strength of the fused TPMS porous scaffold ranged from 367.741 to 419.354 MPa, and the elastic modulus ranged from 10.617 to 11.252 GPa, exhibiting stable mechanical performance in different loading directions. The CFD simulation results indicated that the permeability of the fused TPMS porous scaffold model ranged from 5.70015 × 10-8 to 6.33725 × 10-8 m2. It can be observed that the fused porous scaffold meets the requirements of the complex stress-bearing demands of skeletal structures and complies with the permeability requirements of human bone tissue.

Assessing the effect of manufacturing defects and non-Newtonian blood model on flow behaviors of additively manufactured Gyroid TPMS structures

In the field of medical engineering, Triply Periodic Minimal Surfaces (TPMS) structures have been studied widely owing to their physical attributes similar to those of human bones. Computational Fluid Dynamics (CFD) is often used to reveal the interaction between structural architectures and flow fields. Nevertheless, a comprehensive study on the effect of manufacturing defects and non-Newtonian behavior on the fluid responses in TPMS scaffolds is still lacking. Therefore, the present study fabricated Gyroid TPMS with four relative densities from 0.1 to 0.4. Non-destructive techniques were used to examine surface roughness and geometric deviation. We found that the manufacturing defects had a minor effect on fluid responses. The pressure drop comparison between defect-containing and defect-free models could be differed up to 7%. The same comparison for the average shear stress showed a difference up to 23%, in which greater deviation between both models was observed at higher relative density. On the contrary, the viscosity model played a significant role in flow prediction. By comparing the Newtonian model with Carreau-Yasuda non-Newtonian model, the resulting pressure drop and average wall shear stress from non-Newtonian viscosity could be higher than those of the Newtonian model by more than a factor of two. In addition, we matched the fluid-induced shear stress from both viscosity models with desirable ranges of shear stresses for tissue growth obtained from the literature. Up to 70% from the Newtonian model fell within the desirable range while the matching stress reduced to lower than 8% for the non-Newtonian results. Furthermore, by correlating geometric features with physical outputs, the geometric deviation was seen associated with surface curvature while the local shear stress revealed a strong correlation with inclination angle. Overall, the present work emphasized the importance of the viscosity model for CFD analysis of the scaffolds, especially when resulting fluid-induced wall shear stress is of interest. In addition, the geometric correlation has introduced the alternative consideration of structural architectures from local perspectives, which could assist the further comparison and optimization among different porous scaffolds in the future.

Roles of irregularity of pore morphology in osteogenesis of Voronoi scaffolds: From the perspectives of MSC adhesion and mechano-regulated osteoblast differentiation

Bone scaffolds designed based on the Voronoi-tessellation algorithm have been increasingly studied owing to their structural similarity with natural cancellous bone. The irregularity of pore morphology (IPM) influences the osteogenesis efficiency of Voronoi scaffolds since it may alter the static and hydromechanical microenvironments for the initial adhesion and mechano-regulated osteoblast differentiation (MrOD) of mesenchymal stem cells (MSCs). In this work, animal experiments were conducted to explore the relationship between IPM and osteogenesis efficiency in Voronoi scaffolds. A computational fluid dynamics (CFD) analysis based on discrete phase models was performed to predict the efficiency of MSC adhesion in different IPMs. Another combined finite element and CFD analysis based on the mechano-regulation algorithm was performed to predict the influence of IPM on the MrOD of the adhesive MSCs. The results showed that the osteogenesis efficiency of the Voronoi scaffolds increased as the IPM rose from low to moderate and then dropped as the IPM further rose. Same trends were also found in the MSC adhesion and MrOD, which caused by the changes of strain tensors on the strut surface and the tortuosity and fluid velocity of the fluid pathway. Moderate IPM induced the highest osteogenesis efficiency owing to its highest efficiencies of MSC adhesion and MrOD. This work identified the optimal IPM for the osteogenesis of Voronoi scaffolds and clarified its biomechanical mechanisms from the adhesion and mechano-regulated differentiation of MSCs, which is of great importance for guiding Voronoi scaffold design when it is used for bone defect repair.

Nondeterministic multiobjective optimization of 3D printed ceramic tissue scaffolds

Despite significant advances in the design optimization of bone scaffolds for enhancing their biomechanical properties, the functionality of these synthetic constructs remains suboptimal. One of the main challenges in the structural optimization of bone scaffolds is associated with the large uncertainties caused by the manufacturing process, such as variations in scaffolds' geometric features and constitutive material properties after fabrication. Unfortunately, such non-deterministic issues have not been considered in the existing optimization frameworks, thereby limiting their reliability. To address this challenge, a novel multiobjective robust optimization approach is proposed here such that the effects of uncertainties on the optimized design can be minimized. This study first conducted computational analyses of a parameterized ceramic scaffold model to determine its effective modulus, structural strength, and permeability. Then, surrogate models were constructed to formulate explicit mathematical relationships between the geometrical parameters (design variables) and mechanical and fluidic properties. The Non-Dominated Sorting Genetic Algorithm II (NSGA-II) was adopted to generate the robust Pareto solutions for an optimal set of trade-offs between the competing objective functions while ensuring the effects of the noise parameters to be minimal. Note that the nondeterministic optimization of tissue scaffold presented here is the first of its kind in open literature, which is expected to shed some light on this significant topic of scaffold design and additive manufacturing in a more realistic way.

Additively manufactured lattice structures with controlled transverse isotropy for orthopedic porous implants

Additively manufactured lattice structures enable the design of tissue scaffolds with tailored mechanical properties, which can be implemented in porous biomaterials. The adaptation of bone to physiological loads results in anisotropic bone tissue properties which are optimized for site-specific loads; therefore, some bone sites are stiffer and stronger along the principal load direction compared to other orientations. In this work, a semi-analytical model was developed for the design of transversely isotropic lattice structures that can mimic the anisotropy characteristics of different types of bone tissue. Several design possibilities were explored, and a particular unit cell, which was best suited for additive manufacturing was further analyzed. The design of the unit cell was parameterized and in-silico analysis was performed via Finite Element Analysis. The structures were manufactured additively in metal and tested under compressive loads in different orientations. Finite element analysis showed good correlation with the semi-analytical model, especially for elastic constants with low relative densities. The anisotropy measured experimentally showed a variable accuracy, highlighting the deviations from designs to additively manufactured parts. Overall, the proposed model enables to exploit the anisotropy of lattice structures to design lighter scaffolds with higher porosity and increased permeability by aligning the scaffold with the principal direction of the load.

Additively Manufactured Scaffolds with Optimized Thickness Based on Triply Periodic Minimal Surface

Triply periodic minimal surfaces (TPMS) became an effective method to design porous scaffolds in recent years due to their superior mechanical and other engineering properties. Since the advent of additive manufacturing (AM), different TPMS-based scaffolds are designed and fabricated for a wide range of applications. In this study, Schwarz Primitive triply periodic minimal surface (P-TPMS) is adopted to design a novel porous scaffold according to the distribution of the scaffold stress under a fixed load with optimized thickness to tune both the mechanical and biological properties. The designed scaffolds are then additively manufactured through selective laser melting (SLM). The micro-features of the scaffolds are studied through scanning electron microscopy (SEM) and micro-computed tomography (CT) images, and the results confirm that morphological features of printed samples are identical to the designed ones. Afterwards, the quasi-static uniaxial compression tests are carried out to observe the stress-strain curves and the deformation behavior. The results indicate that the mechanical properties of the porous scaffolds with optimized thickness were significantly improved. Since the mass transport capability is important for the transport of nutrients within the bone scaffolds, computational fluid dynamics (CFD) are used to calculate the permeability under laminar flow conditions. The results reveal that the scaffolds with optimized structures possess lower permeability due to the rougher inner surface. In summary, the proposed method is effective to tailor both the mechanical properties and permeability, and thus offers a means for the selection and design of porous scaffolds in biomedical fields.

Design and properties of graded polyamide12/hydroxyapatite scaffolds based on primitive lattices using selective laser sintering

Scaffolds with favorable biological characteristics and controlled functional gradient architectures are preferable for the repair of damaged tissues in bone tissue engineering. In this study, the triply periodic minimal surfaces (TPMS) were introduced to design functional gradient porous scaffolds based on Primitive lattices which were then manufactured by selective laser sintering (SLS) using pure polyamide12 (PA12) material and PA12/hydroxyapatite (HA) composite material. The mechanical properties and permeability of the scaffolds were then evaluated by mechanical compression experiments and computational fluid dynamics (CFD) analysis. The radial-graded scaffold was found to have superior good mechanical properties and permeability and be favorable for the subsequent growth of bone tissue. Further, the optimal PA12/HA composition was determined by analyzing the effect of the addition of HA particles on the hydrophilicity and mechanical properties of the composite scaffold. Additionally, the cytotoxicity tests were performed to evaluate the effects of PA12/HA gradient scaffold on cell growth. The obtained results demonstrate that the radial gradient scaffold with 15% HA addition exhibits a feasible combination of comprehensive performance and biological activity, indicating a great application potential in the field of bone tissue engineering.

Design procedure for triply periodic minimal surface based biomimetic scaffolds

Cellular additively manufactured metallic structures for load-bearing scaffolds in the context of bone tissue engineering (BTE) have emerged as promising candidates. Due to many advantages in terms of morphology, stiffness, strength and permeability compared to conventional truss structures, lattices based on triply periodic minimal surfaces (TPMS) have recently attracted increasing interest for this purpose. In addition, the finite element method (FEM) has been proven to be suitable for accurately predicting the deformation behavior as well as the mechanical properties of geometric structures after appropriate parameter validation based on experimental data. Numerous publications have examined many individual aspects, but conceptual design procedures that consider at least the essential requirements for cortical and trabecular bone simultaneously are still rare. Therefore, this paper presents a numerical approach to first determine the actual admissible design spaces for a choice of TPMS based lattices with respect to key parameters and then weight them with respect to further benefit parameters. The admissible design spaces are limited by pore size, strut size and volume fraction, and the subsequent weighting is based on Young's modulus, cell size and surface area. Additively manufactured beta-Ti-42Nb with a strain stiffness of 60.5GPa is assumed as material. In total, the procedure considers twelve lattice types, consisting of six different TPMS, each as network solid and as sheet solid. The method is used for concrete prediction of suitable TPMS based lattices for cortical bone and trabecular bone. For cortical bone a lattice based on the Schwarz Primitive sheet solid with 67.572μm pore size, 0.5445 volume fraction and 18.758GPa Young's modulus shows to be the best choice. For trabecular bone a lattice based on the Schoen Gyroid network solid with 401.39μm pore size, 0.3 volume fraction and 4.6835GPa Young's modulus is the identified lattice. Finally, a model for a long bone scaffold is generated from these two lattices using functional grading methods in terms of volume fraction, cell size and TPMS type. In particular, the presented procedure allows an efficient estimation for a likely suitable biometric TPMS-based scaffolds. In addition to medical applications, however, the method can also be transferred to numerous other applications in mechanical, civil and electrical engineering.