Computational Investigation of the Fluidic Properties of Triply Periodic Minimal Surface (TPMS) Structures in Tissue Engineering

: Tissue engineering, a rapidly advancing field in medicine, has made significant strides with the development of artificial tissue substitutes to meet the growing need for organ transplants. Three-dimensional (3D) porous scaffolds are widely utilized in tissue engineering, especially in orthopedic surgery. This study investigated the fluidic properties of diamond and gyroid structures with varying porosity levels (50–80%) using Computational Fluid Dynamics (CFD) analysis. The pressure and velocity distributions were analyzed, and it was observed that the pressure decreased gradually, whereas the velocity increased in the central area of the surface structures. Specifically, the pressure drop ranged from 2.079 to 0.984 Pa for the diamond structure and from 1.669 to 0.943 Pa for the gyroid structure as the porosity increased from 50% to 80%. It was also found that the permeability increased as the porosity level increased, with values ranging from 2.424 × 10 − 9 to 5.122 × 10 − 9 m 2 for the diamond structure and from 2.966 × 10 − 9 to 5.344 × 10 − 9 m 2 for the gyroid structure. The wall shear stress (WSS) was also analyzed, showing a consistent decrease with increased porosity for both types of structures, with WSS values ranging from 9.903 × 10 − 2 to 9.840 × 10 − 1 Pa for the diamond structure and from 1.150 × 10 − 1 to 7.717 × 10 − 2 Pa for the gyroid structure. Overall, this study provides insights into the fluidic properties of diamond and gyroid structures, which can be useful in various applications such as tissue engineering.


Introduction
In recent years, tissue engineering has seen substantial advancements, driven by the creation of novel artificial tissue replacements designed to address the increasing demand for organ transplants.Particularly in orthopedic surgery, three-dimensional (3D) porous scaffolds with diverse structural designs are frequently employed in tissue engineering.Advancements in manufacturing technologies, including additive manufacturing, have enabled the production of 3D scaffolds with a variety of materials and structures.Nevertheless, achieving the ideal scaffold design remains challenging, as the design parameters significantly influence the scaffold's mechanical properties and biocompatibility [1,2].For example, while a high porosity is favorable for biological compatibility, it may compromise the mechanical integrity of the scaffold.It is also necessary to properly design the mass flow characteristics of porous structures, such as permeability, to permit the passage of nutrients and oxygen into cells inside the structure [3,4].
In tissue engineering, it is a common practice to predict the permeability of scaffolds using computer models.Studies have shown that porous structures can serve as scaffolds to facilitate osteogenic differentiation, adhesion, proliferation, host integration, cell attachment, and cell viability.To facilitate nutrient diffusion, cell infiltration, and the removal of metabolic waste products, scaffolds must exhibit certain characteristics.They should have interconnected pores and a high level of porosity [5].It is difficult to develop a porous structure that meets the requirements of both mechanical and biological functions.However, the geometry and curvature of the surfaces where cells reside are recognized as crucial factors in evaluating the rate of tissue production.Recent theories suggest that, unlike flat and convex surfaces, high-curvature surfaces and concave surfaces, respectively, promote and enhance tissue regeneration [6].Additionally, surface roughness is an important quality measure as it supports cell differentiation and growth, while also potentially localizing stress.These requirements can be addressed by designing scaffolds based on Triply Periodic Minimal Surfaces (TPMS) [3,7].
A TPMS, or the surface of the minimal area, possesses a boundary delineated by a closed curve [8].TPMS scaffolds are constructed on this surface, with each unit cell created by adding thickness to the surface and subsequently combining to form a final scaffold that exhibits cubic symmetry and comprises periodically repeated interconnected pores.Therefore, a TPMS is more desirable for bone regeneration [9], and has characteristics similar to the curvature of the trabecular bone [10].Several natural geometries such as cubosomes, cell membranes, and plant prolamellar bodies can be characterized by a TPMS [7].Previous research has investigated how various pore sizes impact the bond strength between an implant and bone.Their findings indicated that pore sizes between 300 µm and 600 µm were optimal in facilitating continuous bone growth to enhance osseointegration.The largest pore size demonstrated the quickest fluid movement, indicating that increasing pore size can enhance the rate of cell migration [11].
Evaluating the characteristics related to biocompatibility, such as the permeability and effect of wall shear stress, is of the utmost importance because of their impact on the biological activity of cells within the supportive structures [12].Numerous investigations have been undertaken to measure the permeability of scaffolds and analyze the factors that impact this permeability.A scaffold that allows substances such as nutrients and gasses to pass through its pores adequately while also enabling the effective removal of waste is considered permeable [13].Pires et al. found that calculating permeability is crucial for optimizing cell cultivation and determining treatment duration, as it directly influences cell growth and proliferation within the supportive structures [14].Ma et al. revealed that, by modifying the design parameters, it is possible to adjust the permeability and mechanical characteristics of the Gyroid structure to mimic those found in trabecular bones within the human body.Furthermore, they observed that permeability had a significant impact on initial cell growth [15].When designing a scaffold, permeability is a vital consideration, as it significantly influences the transport of oxygen, nutrients, and the removal of metabolic waste.Geometric factors, including pore size, surface thickness, and structure type, also affect permeability [5,16,17].Researchers have investigated the concept of developing the microstructure of scaffolds through topology optimization.The findings demonstrated a direct relationship between computational and experimental permeability, underscoring the significance of permeability as a crucial design factor.The study illustrated that the regeneration of tissue is influenced by factors such as porosity and permeability, highlighting their potential for furthering our understanding of the impact of scaffold architecture on biological processes [18].Aqila et al. [19] conducted a study to examine how the size of pores influences the mechanical behavior and fluid permeability of bone scaffolds.Their findings indicated that larger pore sizes were associated with improved permeability and reduced wall shear stress across different flow rates, facilitating a more effective transportation of fluids and nutrients.Ntousi et al. [20] used a computational fluid dynamic (CFD) analysis to analyze two scaffold structures with a pore size of 500 µm.The results revealed that the designed scaffolds exhibited enhanced permeability and fluid flow, making them advantageous for bone tissue engineering applications.Pires et al. [14] evaluate the permeability of various TPMS scaffold geometries through a combination of experimental pressure drop tests and CFD simulations.The results suggested that Gyroid structures were the most optimal choice for applications in bone tissue engineering, as they consistently demonstrated fluid permeation, closely followed by Schwarz D. Conversely, Schwarz P configurations displayed uniform flow patterns and notable fluctuations in permeability based on porosity levels.
Another critical factor in studying the fluidic response of scaffolds is Wall Shear Stress (WSS).WSS has been recognized for its impact on cell growth [8,21].Fluid-induced WSS provides a mechanical stimulus that can be sensed by cells, leading to enhanced cell differentiation and proliferation [22].Ali et al. [23] conducted a comparative analysis using CFD simulations to assess the permeability and WSS of TPMS and lattice-based scaffolds.The analysis revealed how internal structures affect the behavior of fluids.Lesman et al. [24] conducted CFD simulations to demonstrate that when cells grow within 3D scaffolds during perfusion cell culturing, the WSS increases as a result of contraction in the channels of the scaffolds.Marin et al. [25] conducted a CFD analysis to investigate how the variation in structure among scaffolds affects the distribution and magnitude of WSS.The findings of their research revealed that the presence of geometric inconsistencies led to significant variations in velocity and WSS across the different samples.A CFD study was conducted in scaffolds with square pores of varying sizes, with a constant porosity of 63% to evaluate the permeability and WSS.The research revealed that s higher surface roughness led to a decline in both permeability and WSS.It is noteworthy that there was a notable decrease in WSS observed when comparing smooth and rough scaffolds [3].Zhao et al. [26] conducted a study that analyzed the impact of the geometric characteristics of scaffolds using a fluid-structure interaction model.The results showed that pore size significantly influences mechanical stimulation, and simultaneous fluid perfusion and mechanical load increase WSS.In addition, Lee et al. [27] utilized the Allen-Cahn equation to observe and analyze the dynamic growth behaviors of cells within various scaffold structures.Although various authors have explored the impacts of the geometric aspects of TPMS structures on mechanical performance [22,28,29], studying the fluid behavior of different TPMS structures remains a promising area of research, particularly in applications related to bone implants.
The primary objective of this research is to create TPMS structures utilizing nTopology 4.24.3 software and subsequently optimize them.Following this, the optimized scaffolds, namely Diamond and Gyroid, which have different levels of porosity, are evaluated to determine their fluidic flow characteristics.CFD analysis is employed to examine physical characteristics including pressure differentials, permeability, and WSS.These parameters provide a basis for investigating the biological behavior of these structures.Our investigation focuses on evaluating the flow characteristics of these structures.The goal is to assess their potential as bone scaffolds and contribute to the advancement of optimized bone tissue engineering applications.

Materials and Methods
Initially, a scaffold was developed leading to the creation of a three-dimensional model.Following these procedural steps, a CFD analysis was undertaken on the scaffolds, investigating the fluidic behavior of blood under varying porosity levels.

Scaffold Design
The design of scaffolds is critical in CFD studies because of the behavior of fluidic flow, which is dependent on lattice structure.Two architectural scaffolds, namely diamond and gyroid, were selected and developed utilizing the nTopology software having different porosities ranging from 50% to 80% with an increment of 10% (Figure 1).The scaffolds were subsequently exported to ANSYS SpaceClaim, where they were converted from STL to solids.The different porosities in the diamond and gyroid structures are achieved by altering the wall thickness shown in Table 1 Thicker walls reduce the void space, resulting in lower porosity, while thinner walls increase the void space, leading to higher porosity.
The cells display symmetry along each of the three orthogonal axes (x, y, and z), making them isotropic in their geometric properties.
CFD analysis was conducted using the ANSYS Fluent 2021 R2 module.To streamline computational time, each model was maintained at a size of 2 mm × 2 mm × 2 mm, taking into account the geometric complexity of the model.The 2 mm × 2 mm × 2 mm size is optimal for computational models due to its balance between efficiency and capturing the geometric intricacies of bone scaffolds [6].While not reflecting all clinical scenarios, it represents a subset for analyzing fundamental fluid behavior within computational constraints.For larger scaffold structures, we emphasize the importance of cautious evaluation and experimental validation.Our study aims to provide critical insights on fundamental fluid dynamics principles, which are essential for a wide range of clinical scaffold sizes.In Figure 1, the subscript indicates the porosity percentage, like D 50 which represents 50% porosity.The scaffolds were subsequently exported to ANSYS SpaceClaim, where they were converted from STL to solids.The different porosities in the diamond and gyroid structures are achieved by altering the wall thickness shown in Table 1 Thicker walls reduce the void space, resulting in lower porosity, while thinner walls increase the void space, leading to higher porosity.The cells display symmetry along each of the three orthogonal axes (x, y, and z), making them isotropic in their geometric properties.CFD analysis was conducted using the ANSYS Fluent 2021 R2 module.To streamline computational time, each model was maintained at a size of 2 mm × 2 mm × 2 mm, taking into account the geometric complexity of the model.The 2 mm × 2 mm × 2 mm size is optimal for computational models due to its balance between efficiency and capturing the geometric intricacies of bone scaffolds [6].While not reflecting all clinical scenarios, it represents a subset for analyzing fundamental fluid behavior within computational

Governing and Adopted CFD Equations
The following approach utilized the Navier-Stokes equation, taking into account an incompressible fluid with constant density.
where ρ, u, µ, ∇, p, and F represent the density of the fluid kg m 3 , the fluid velocity (m/s), dynamic viscosity of fluid (kg/m.s), the del operator, pressure (Pa), and other forces (gravity or centrifugal force; in this case F = 0), respectively [6].The blood density was assumed to remain constant at 1060 kg/m 3 for the CFD models.
The Carreau-Yasuda model was used to compute the fluid viscosity for the analysis, considering blood as a non-Newtonian fluid.The Carreau-Yasuda model was chosen for its ability to describe viscosity plateaus in bone scaffold applications, capturing fluid behavior under varying porosities.It proved essential in predicting pressure drop-flow rate relationships, demonstrating effectiveness in analyzing Carreau fluid flow through porous media [30].The model's accuracy in predicting fluid behavior makes it a valuable tool for understanding and optimizing blood flow through intricate scaffold structures, aiding in the design of efficient bone tissue engineering systems.
The equation utilized for Carreau fluids generalized from Darcy law is presented below: where u and L denote the inlet fluid velocity (m/s) and model length (m), respectively.The values of µ ∞ and µ 0 represent the shear rate limit viscosities when the shear rate is either infinite or zero, respectively, relaxation time constant is λ, and the power law index is n [30].
In a laminar flow system, the WSS, denoted as τω, can be mathematically expressed as the normal velocity gradient on a wall.
where m indicates the x, y, and z directions [16].
The Reynolds number for non-Newtonian fluid can be calculated by [31]: R e = ρuL µ a (6) where ρ, u, L, and µ a denote density, velocity of fluid, length, and viscosity of fluid.
The Reynolds number for this non-Newtonian fluid under the given conditions is approximately 0.0052, which indicates a laminar flow regime.The characterization of blood flow as laminar in the CFD analysis of TPMS structures is based on physiological considerations, primarily the low flow velocities within blood vessels, resulting in Reynolds numbers conducive to laminar flow.The relatively high viscosity of blood and the smallscale nature of the vessels further support the assumption of laminar flow.This modeling choice aligns with typical flow conditions in the human circulatory system, enhancing the credibility of the simulation results.

Boundary Conditions
The study employed a laminar flow of fluid with an inlet velocity of 0.0001 m/s [6,17].The direction of flow was chosen based on its potential to simplify the computational setup and boundary conditions, leading to a more accurate analysis and understanding of outcomes.Figure 2 shows the boundary conditions.To specify different fluid properties at the surface, such as velocity and pressure, a wall boundary condition was employed.In this study, a no-slip condition was imposed on the porous structure, which was assumed to be hydrophilic, consistent with typical practices in bone tissue engineering applications [25,32].This assumption accounts for wetting behavior, leading to more accurate and practical CFD simulations.Furthermore, the outlet pressure was set to zero [15], to simplify the pressure differential computation, which was a critical parameter.
at the surface, such as velocity and pressure, a wall boundary condition was employed.In this study, a no-slip condition was imposed on the porous structure, which was assumed to be hydrophilic, consistent with typical practices in bone tissue engineering applications [25,32].This assumption accounts for wetting behavior, leading to more accurate and practical CFD simulations.Furthermore, the outlet pressure was set to zero [15], to simplify the pressure differential computation, which was a critical parameter.

Numerical Schemes
In this study, the Finite Volume Method (FVM) has been employed as the numerical framework to simulate blood flow through the scaffold structures.To account for the non-Newtonian behavior of blood, the Carreau-Yasuda model has been integrated, offering a comprehensive description of viscosity variations with shear rate.The governing equations were solved using a coupled scheme, which simultaneously solves the momentum and continuity equations to enhance convergence stability and efficiency, particularly beneficial for flow interactions within the complex geometries of the scaffolds.For spatial discretization, the Green Gauss Node Based method was employed for gradient calculations due to its superior accuracy in handling unstructured meshes.The pressure interpolation was conducted using a second-order scheme to ensure a higher accuracy in capturing pressure variations within the flow domain.Momentum equations were discretized using a first-order upwind scheme, which, despite being less accurate than higher-order schemes, provides robust and stable solutions, particularly in the initial stages of simulation or in regions with high gradients and complex flow patterns.This combination of numerical schemes was chosen to balance computational efficiency and solution accuracy, ensuring reliable simulation results for the analysis of blood flow behavior through the TPMS scaffolds.

Grid Convergence
The study focused on the characterization of blood as a non-Newtonian fluid.A mesh sensitivity analysis was conducted to assess how variations in mesh element sizes influenced the results.Figure 3 shows a visual representation of the mesh in the fluid domain containing the scaffold.Mesh elements with sizes of 0.2, 0.5, and 1 mm were used to compare the permeability and WSS values of a specific model.The findings revealed that using a mesh size of 0.2 mm resulted in a minimal variation, less than 2%, in the WSS of the unit

Numerical Schemes
In this study, the Finite Volume Method (FVM) has been employed as the numerical framework to simulate blood flow through the scaffold structures.To account for the non-Newtonian behavior of blood, the Carreau-Yasuda model has been integrated, offering a comprehensive description of viscosity variations with shear rate.The governing equations were solved using a coupled scheme, which simultaneously solves the momentum and continuity equations to enhance convergence stability and efficiency, particularly beneficial for flow interactions within the complex geometries of the scaffolds.For spatial discretization, the Green Gauss Node Based method was employed for gradient calculations due to its superior accuracy in handling unstructured meshes.The pressure interpolation was conducted using a second-order scheme to ensure a higher accuracy in capturing pressure variations within the flow domain.Momentum equations were discretized using a first-order upwind scheme, which, despite being less accurate than higher-order schemes, provides robust and stable solutions, particularly in the initial stages of simulation or in regions with high gradients and complex flow patterns.This combination of numerical schemes was chosen to balance computational efficiency and solution accuracy, ensuring reliable simulation results for the analysis of blood flow behavior through the TPMS scaffolds.

Grid Convergence
The study focused on the characterization of blood as a non-Newtonian fluid.A mesh sensitivity analysis was conducted to assess how variations in mesh element sizes influenced the results.Figure 3 shows a visual representation of the mesh in the fluid domain containing the scaffold.Mesh elements with sizes of 0.2, 0.5, and 1 mm were used to compare the permeability and WSS values of a specific model.The findings revealed that using a mesh size of 0.2 mm resulted in a minimal variation, less than 2%, in the WSS of the unit cell.As a result, a size of element of 0.2 mm was selected for the ongoing investigation.Furthermore, calculations were performed for WSS, permeability, pressure, and fluid velocity.The spatial discretization is accomplished using a tetrahedral structured mesh.To ensure the model's performance sensitivity, results were brought into convergence by establishing a residual criterion value of 1 × 10 −6 .
cell.As a result, a size of element of 0.2 mm was selected for the ongoing investigation.Furthermore, calculations were performed for WSS, permeability, pressure, and fluid velocity.The spatial discretization is accomplished using a tetrahedral structured mesh.To ensure the model's performance sensitivity, results were brought into convergence by establishing a residual criterion value of 1 × 10 −6 .

Numerical Model Validation
The present numerical methodology is validated by comparing the results with the literature.In this regard, the permeability results for the Gyroid structure are compared with the Ali and Sen [16] study.They analyzed the Gyroid structure using the power law model with an inlet velocity of 0.7 mm/s.The results from the present methodology match well with those from the Ali and Sen [16] study with a reasonable accuracy.Figure 4 shows the validation of our numerical methodology for the permeability of a Gyroid structure.

Fluidic Characteristics of Scaffolds
An analysis of pressure and velocity distribution was conducted to examine the fluidic characteristics of Diamond (D) and Gyroid (G) structures with varying porosity levels ranging from 50% to 80%.The pressure distribution contours of the D surface (i.e., D , D , D , and D ) and the G surface (i.e., G , G , G , and G ) are presented in Figure

Numerical Model Validation
The present numerical methodology is validated by comparing the results with the literature.In this regard, the permeability results for the Gyroid structure are compared with the Ali and Sen [16] study.They analyzed the Gyroid structure using the power law model with an inlet velocity of 0.7 mm/s.The results from the present methodology match well with those from the Ali and Sen [16] study with a reasonable accuracy.Figure 4 shows the validation of our numerical methodology for the permeability of a Gyroid structure.
cell.As a result, a size of element of 0.2 mm was selected for the ongoing investigation.Furthermore, calculations were performed for WSS, permeability, pressure, and fluid velocity.The spatial discretization is accomplished using a tetrahedral structured mesh.To ensure the model's performance sensitivity, results were brought into convergence by establishing a residual criterion value of 1 × 10 −6 .

Numerical Model Validation
The present numerical methodology is validated by comparing the results with the literature.In this regard, the permeability results for the Gyroid structure are compared with the Ali and Sen [16] study.They analyzed the Gyroid structure using the power law model with an inlet velocity of 0.7 mm/s.The results from the present methodology match well with those from the Ali and Sen [16] study with a reasonable accuracy.Figure 4 shows the validation of our numerical methodology for the permeability of a Gyroid structure.

Fluidic Characteristics of Scaffolds
An analysis of pressure and velocity distribution was conducted to examine the fluidic characteristics of Diamond (D) and Gyroid (G) structures with varying porosity levels ranging from 50% to 80%.The pressure distribution contours of the D surface (i.e., D , D , D , and D ) and the G surface (i.e., G , G , G , and G ) are presented in Figure

Fluidic Characteristics of Scaffolds
An analysis of pressure and velocity distribution was conducted to examine the fluidic characteristics of Diamond (D) and Gyroid (G) structures with varying porosity levels ranging from 50% to 80%.The pressure distribution contours of the D surface (i.e., D 50 , D 60 , D 70 , and D 80 ) and the G surface (i.e., G 50 , G 60 , G 70 , and G 80 ) are presented in Figure 5.It is evident that there is a gradual decrease in pressure from the inlet to the outlet in all structures, regardless of the porosity level.The observed pressure variation in TPMSbased scaffolds, with the higher pressure near the inlet diminishing as the fluid progresses downwards, is a consequence of the intricate interplay between geometric complexities and fluid dynamics.The initial compression near the inlet, induced by the complex scaffold geometry, leads to regions of elevated pressure.Subsequent fluid flow through smoother paths and porous structures results in a gradual pressure dissipation.At the bottom of each Designs 2024, 8, 69 9 of 15 picture, negative values of the pressure appear.This is because, near the outlet, the speed was increasing (Figure 5) so the pressure is decreased.Also, due to the effect of gravity, the speed increases as the fluid progresses so the pressure decreased.
5. It is evident that there is a gradual decrease in pressure from the inlet to the outlet in all structures, regardless of the porosity level.The observed pressure variation in TPMSbased scaffolds, with the higher pressure near the inlet diminishing as the fluid progresses downwards, is a consequence of the intricate interplay between geometric complexities and fluid dynamics.The initial compression near the inlet, induced by the complex scaffold geometry, leads to regions of elevated pressure.Subsequent fluid flow through smoother paths and porous structures results in a gradual pressure dissipation.At the bottom of each picture, negative values of the pressure appear.This is because, near the outlet, the speed was increasing (Figure 5) so the pressure is decreased.Also, due to the effect of gravity, the speed increases as the fluid progresses so the pressure decreased.The velocity distribution within scaffolds, as illustrated in Figure 6, reveals the profound impact of internal design on flow characteristics.It is crucial to observe that the central regions of scaffold structure surface have velocities that exceed the initial inlet velocity.The increase in speed can be attributed to the complex lattice geometry, which causes a localized rise in fluid velocity.At the same time, adjacent regions demonstrate the combined influence of geometric features and fluid dynamics [15].The observed acceleration of the fluid profile, particularly in the central areas, has significant implications for the transportation of nutrients to deeper structures within the lattice.The improved fluid dynamics in these central regions suggest a potential optimization for the dispersion of nutrients, which agrees with results of the previous studies [33,34].The velocity distribution within scaffolds, as illustrated in Figure 6, reveals the profound impact of internal design on flow characteristics.It is crucial to observe that the central regions of scaffold structure surface have velocities that exceed the initial inlet velocity.The increase in speed can be attributed to the complex lattice geometry, which causes a localized rise in fluid velocity.At the same time, adjacent regions demonstrate the combined influence of geometric features and fluid dynamics [15].The observed acceleration of the fluid profile, particularly in the central areas, has significant implications for the transportation of nutrients to deeper structures within the lattice.The improved fluid dynamics in these central regions suggest a potential optimization for the dispersion of nutrients, which agrees with results of the previous studies [33,34].Figure 7 shows the pressure drop versus porosity, as by increasing the porosity, the pressure drop decreases in both the Diamond and Gyroid structures.The Diamond structure shows a higher pressure drop than the Gyroid structure due to its higher stiffness and strength, which result in a lower permeability.A study investigated the mechanical properties of TPMS-based Diamond and Gyroid Ti6Al4V scaffolds for bone implants.The findings revealed that, when comparing scaffolds with identical pore sizes, the Diamond scaffold exhibited a greater stiffness and strength than the Gyroid scaffold.As a result, the Diamond scaffold exhibited a higher pressure drop [35].This phenomenon can be attributed to the inherent differences in the geometrical characteristics and flow dynamics of the two structures.The complex geometry of the Diamond structure induces more resistance to fluid flow, resulting in a higher pressure drop, whereas the Gyroid structure, with its distinct architecture, presents a comparatively lower resistance, leading to a reduced pressure drop.Figure 7 shows that the pressure drop ranges from 2.079 Pa to 0.984 Pa for the Diamond structure and 1.669 Pa to 0.943 Pa for the Gyroid structure for porosities ranging from 50% to 80%.
The TPMS structures are made from stainless steel, which has a high Young's modulus (200 GPa) and yield strength (500 MPa), making them rigid and resistant to deformation under fluid loads.Additionally, the small dimensions (a few millimeters) and thick walls of the structures ensure structural integrity, minimizing deformation.Similar studies, such as those by Song et al. [36], have also assumed non-deformable vessels for TPMS structures due to their rigidity and structural design, supporting our assumption.Figure 7 shows the pressure drop versus porosity, as by increasing the porosity, the pressure drop decreases in both the Diamond and Gyroid structures.The Diamond structure shows a higher pressure drop than the Gyroid structure due to its higher stiffness and strength, which result in a lower permeability.A study investigated the mechanical properties of TPMS-based Diamond and Gyroid Ti6Al4V scaffolds for bone implants.The findings revealed that, when comparing scaffolds with identical pore sizes, the Diamond scaffold exhibited a greater stiffness and strength than the Gyroid scaffold.As a result, the Diamond scaffold exhibited a higher pressure drop [35].This phenomenon can be attributed to the inherent differences in the geometrical characteristics and flow dynamics of the two structures.The complex geometry of the Diamond structure induces more resistance to fluid flow, resulting in a higher pressure drop, whereas the Gyroid structure, with its distinct architecture, presents a comparatively lower resistance, leading to a reduced pressure drop.Figure 7 shows that the pressure drop ranges from 2.079 Pa to 0.984 Pa for the Diamond structure and 1.669 Pa to 0.943 Pa for the Gyroid structure for porosities ranging from 50% to 80%.
Designs 2024, 8, x FOR PEER REVIEW 12 of 17 structure shows a higher pressure drop than the Gyroid structure due to its higher stiffness and strength, which result in a lower permeability.A study investigated the mechanical properties of TPMS-based Diamond and Gyroid Ti6Al4V scaffolds for bone implants.
The findings revealed that, when comparing scaffolds with identical pore sizes, the Diamond scaffold exhibited a greater stiffness and strength than the Gyroid scaffold.As a result, the Diamond scaffold exhibited a higher pressure drop [35].This phenomenon can be attributed to the inherent differences in the geometrical characteristics and flow dynamics of the two structures.The complex geometry of the Diamond structure induces more resistance to fluid flow, resulting in a higher pressure drop, whereas the Gyroid structure, with its distinct architecture, presents a comparatively lower resistance, leading to a reduced pressure drop.Figure 7 shows that the pressure drop ranges from 2.079 Pa to 0.984 Pa for the Diamond structure and 1.669 Pa to 0.943 Pa for the Gyroid structure for porosities ranging from 50% to 80%.The TPMS structures are made from stainless steel, which has a high Young's modulus (200 GPa) and yield strength (500 MPa), making them rigid and resistant to deformation under fluid loads.Additionally, the small dimensions (a few millimeters) and thick walls of the structures ensure structural integrity, minimizing deformation.Similar studies, such as those by Song et al. [36], have also assumed non-deformable vessels for TPMS structures due to their rigidity and structural design, supporting our assumption.

Permeability
Equation ( 4) was used to calculate the permeability of the Diamond and Gyroid structures, considering the pressure difference obtained through the CFD analysis.To assess the transport efficiency of different porous structures, both permeability (k) and pressure difference (∆p) were taken into account.
For the successful occurrence of mass transfer within any porous structure, permeability is of the utmost importance.In Figure 8, the expected permeability of both the Diamond and Gyroid surfaces is depicted for the porosity range of 50% to 80%.It is noted The TPMS structures are made from stainless steel, which has a high Young's modulus (200 GPa) and yield strength (500 MPa), making them rigid and resistant to deformation under fluid loads.Additionally, the small dimensions (a few millimeters) and thick walls of the structures ensure structural integrity, minimizing deformation.Similar studies, such as those by Song et al. [36], have also assumed non-deformable vessels for TPMS structures due to their rigidity and structural design, supporting our assumption.

Permeability
Equation ( 4) was used to calculate the permeability of the Diamond and Gyroid structures, considering the pressure difference obtained through the CFD analysis.To assess the transport efficiency of different porous structures, both permeability (k) and pressure difference (∆p) were taken into account.
For the successful occurrence of mass transfer within any porous structure, permeability is of the utmost importance.In Figure 8, the expected permeability of both the Diamond and Gyroid surfaces is depicted for the porosity range of 50% to 80%.It is noted that a higher porosity results in an increased permeability because the pores in the scaffold allow for better fluid flow.Research has indicated that TPMS scaffolds with higher porosity levels have higher permeability.The Gyroid scaffold type has been determined to be more permeable than the Diamond scaffold type [37].The continuous, three-dimensional architecture of the Gyroid offers a larger surface area for the flow of fluids and facilitates a more interconnected pathway for fluid passage.These characteristics of the Gyroid, including its enhanced interconnectivity and greater void space, contribute to its higher permeability, allowing fluids to move through the Gyroid scaffold with reduced resistance compared to the Diamond scaffold.Ali et al. [23] studied Gyroid-and lattice-based architectures with porosities ranging from 65% to 90% to observe their permeability.Our results show a good agreement with the results from this study.Gómez S. et al. [38] observed the properties of 3D scaffolds for bone tissue engineering.They showed that the properties of scaffolds can be controlled in design stage to match exactly with trabecular bone.Van Bael S. [39] performed a study to observe the effect of pore size and permeability on Ti6Al4V scaffolds of three different pore shapes at 500 µm and 1000 µm.Nauman E. et al. [40] conducted a study to observe the permeability of trabecular bone for tissue engineering applications.Beaudoin et al. [41] observed the permeability of cancellous bone.
Designs 2024, 8, x FOR PEER REVIEW 13 of 17 compared to the Diamond scaffold.Ali et al. [23] studied Gyroid-and lattice-based architectures with porosities ranging from 65% to 90% to observe their permeability.Our results show a good agreement with the results from this study.Gómez S. et al. [38] observed the properties of 3D scaffolds for bone tissue engineering.They showed that the properties of scaffolds can be controlled in design stage to match exactly with trabecular bone.Van Bael S. [39] performed a study to observe the effect of pore size and permeability on Ti6Al4V scaffolds of three different pore shapes at 500 µm and 1000 µm.Nauman E. et al. [40] conducted a study to observe the permeability of trabecular bone for tissue engineering applications.Beaudoin et al. [41] observed the permeability of cancellous bone.In this study, the permeability ranged from 2.424 10 m to 5.122 10 m for the Diamond structure and 2.966 10 m to 5.344 10 m for the Gyroid structure.Table 2 shows the good agreement of the obtained results with previously reported results.In this study, the permeability ranged from 2.424 × 10 −9 m 2 to 5.122 × 10 −9 m 2 for the Diamond structure and 2.966 × 10 −9 m 2 to 5.344 × 10 −9 m 2 for the Gyroid structure.Table 2 shows the good agreement of the obtained results with previously reported results.

WSS
Equation ( 5) was utilized to determine the WSS of diamond and gyroid structures.The analysis of the WSS on the Gyroid and Diamond structures revealed a consistent trend.WSS consistently decreased with an increase in porosity for both types of structures, which attributed to the reduced surface area available for fluid interaction within the porous framework.
When comparing the WSS values between the two TPMS structures, the Diamond TPMS exhibited consistently lower WSS values compared to the Gyroid TPMS across various porosities shown in Figure 9.The results show that the Diamond TPMS is more effective in reducing WSS compared to the Gyroid TPMS, which is beneficial for improving the overall performance of the TPMS structure.This difference in WSS values between the two TPMS structures may have significant implications for their performance in different applications.Figure 9 shows that the WSS ranges from 9.903 × 10 −2 Pa to 9.840 × 10 −1 Pa for the Diamond structure and 1.150 × 10 −1 Pa to 7.717 × 10 −2 Pa for the Gyroid structure.

WSS
Equation ( 5) was utilized to determine the WSS of diamond and gyroid structures.The analysis of the WSS on the Gyroid and Diamond structures revealed a consistent trend.WSS consistently decreased with an increase in porosity for both types of structures, which attributed to the reduced surface area available for fluid interaction within the porous framework.
When comparing the WSS values between the two TPMS structures, the Diamond TPMS exhibited consistently lower WSS values compared to the Gyroid TPMS across various porosities shown in Figure 9.The results show that the Diamond TPMS is more effective in reducing WSS compared to the Gyroid TPMS, which is beneficial for improving the overall performance of the TPMS structure.This difference in WSS values between the two TPMS structures may have significant implications for their performance in different applications.Figure 9 shows that the WSS ranges from

Conclusions
In this study, the fluidic properties of Diamond and Gyroid TPMS-based scaffolds with varying porosities (50-80%) were examined through CFD analysis, focusing on their potential applications in orthopedic implants.The findings highlight significant differences in pressure distribution, permeability, and WSS between the two structures.Gyroid scaffolds consistently demonstrated superior fluidic performance compared to Diamond structures, with lower pressure drops and higher permeability across all porosity levels.These characteristics suggest that Gyroid scaffolds are more conducive to nutrient flow and waste removal, and their gradual decrease in WSS with increased porosity indicates a favorable environment for cell proliferation and tissue regeneration.Therefore, Gyroid structures are recommended as the preferred choice for orthopedic implants applications due to their balanced fluidic properties and potential to support biological processes effectively.

Conclusions
In this study, the fluidic properties of Diamond and Gyroid TPMS-based scaffolds with varying porosities (50-80%) were examined through CFD analysis, focusing on their potential applications in orthopedic implants.The findings highlight significant differences in pressure distribution, permeability, and WSS between the two structures.Gyroid scaffolds consistently demonstrated superior fluidic performance compared to Diamond structures, with lower pressure drops and higher permeability across all porosity levels.These characteristics suggest that Gyroid scaffolds are more conducive to nutrient flow and waste removal, and their gradual decrease in WSS with increased porosity indicates a favorable environment for cell proliferation and tissue regeneration.Therefore, Gyroid structures are recommended as the preferred choice for orthopedic implants applications due to their balanced fluidic properties and potential to support biological processes effectively.
Although the results were derived from a model with limited dimensions (2 mm × 2 mm × 2 mm), they provide foundational insights that could be applicable to larger scaffold samples.Future research should focus on scaling these models and validating them with experimental data to ensure their applicability to clinically relevant scaffold sizes.The necessity of experimental validation is evident; while the simulations offer a comprehensive understanding of fluidic behavior, physical experiments, such as flow perfusion bioreactor studies, are required to confirm the predicted pressure drops, permeability, and WSS values under physiological conditions.Additionally, experimental studies could explore the biological responses, such as cell adhesion and growth, within these scaffolds under dynamic flow conditions.

Figure 3 .
Figure 3. Visual representation of mesh in fluid domain containing the scaffold.

Figure 4 .
Figure 4. Validation of numerical methodology for the permeability of Gyroid structure.

Figure 3 .
Figure 3. Visual representation of mesh in fluid domain containing the scaffold.

Figure 3 .
Figure 3. Visual representation of mesh in fluid domain containing the scaffold.

Figure 4 .
Figure 4. Validation of numerical methodology for the permeability of Gyroid structure.

Figure 4 .
Figure 4. Validation of numerical methodology for the permeability of Gyroid structure [16].

Figure 5 .
Figure 5. Pressure distribution contours of diamond structure (left side) and gyroid structures (right side).

Figure 5 .
Figure 5. Pressure distribution contours of diamond structure (left side) and gyroid structures (right side).

Figure 6 .
Figure 6.Velocity streamlines the distribution of diamond structures (left side) and gyroid structures (right side).
9.903 10 Pa to 9.840 10 Pa for the Diamond structure and 1.150 10 Pa to 7.717 10 Pa for the Gyroid structure.

Table 1 .
The wall thickness of TPMS-based scaffolds.

Table 1 .
The wall thickness of TPMS-based scaffolds.

Table 2 .
Permeability of porous scaffolds and bone.

Table 2 .
Permeability of porous scaffolds and bone.