Simulation Study on the Effects of Environment and Structure on Bone Tissue Scaffold Flow Properties

Abstract

One of the hottest topics in current research is the creation of scaffolds for bone tissue restoration that are both biocompatible and tissue inducible. The aim of this work is to develop a numerical model to study the effects of temperature, velocity, and scaffold structure on flow and biomechanical properties, as well as to optimize design parameters to improve tissue engineering outcomes. The results show that the fluid transport properties of cylindrical unit cell architectures are superior. For effective mass transfer, pore diameters > 4 mm and porosity > 60% are ideal design parameters. With important clinical and financial implications, these discoveries offer theoretical direction and economical methods for developing bone tissue engineering.

1. Introduction

In tissue engineering and regenerative medicine, bone repair is still a crucial but difficult area of study. According to Mohammed [1], bone is a stiff connective structure that facilitates mobility; shields internal organs; and controls blood pH, hematopoiesis, and mineral homeostasis. Collagen and calcium phosphate crystals make up bones; the organic portion of the bone is mainly constituted of collagen, while the inorganic portion is made up of calcium phosphate crystals found in the collagen matrix [2]. Bone is the second most transplanted tissue. Natural regeneration capability is frequently exceeded by severe bone abnormalities brought on by trauma, cancer, osteoporosis, or rheumatoid arthritis [3,4], which severely reduce patient quality of life and present serious clinical issues [5]. The problem of bone defects can be partially addressed by traditional bone repair techniques like bone grafting [6], which include autologous bone grafting [7] and allogeneic bone grafting [8]. Unfortunately, each of these techniques has drawbacks [9], including poor donor availability, immunological rejection, and discrepancies between material degradation and bone regeneration [10,11]. For effective and secure defect healing, it is crucial to create the best bone repair scaffolds that replicate the structure and function of natural bone.

Bone Tissue Engineering (BTE), which focuses on repairing tissues that have lost their functionality as a result of illness or trauma, can heal bone abnormalities [12,13]. The utilization of artificial scaffolds (such as bioceramics) as materials for bone tissue healing is currently the main trend in the development of BTE [14,15]. These scaffolds can assist the growth of new bone by imitating the extracellular matrix (ECM) found in animals. Artificial scaffolds can reduce the risk of infection or immunogenicity, shorten surgical procedures, eliminate the risk of disease transmission, and enable customized scaffold shape and function design based on patient and case characteristics, all of which improve treatment outcomes. It has been shown that bone scaffolds are a great way to address flaws or anomalies in the bone [16].

Since artificial scaffolds are biomaterials that resemble bone, they must be biocompatible, biodegradable, and possess mechanical and thermal properties similar to those of genuine bone [17,18]. Cell survival and tissue growth are significantly influenced by the structural characteristics of bone tissue repair scaffolds, such as pore size, porosity, geometric shape, and interconnected pore architecture [19,20]. Sobral et al. demonstrated that the distribution of channels or holes encourages fluid flow, whereas interconnected pores help move waste and nutrients. And the form and geometry of interior pores are also structural features that affect bone conduction and mechanical qualities [21]. Pore size affects the adhesion, migration, and proliferation of cells within pores. Mukasheva et al. reported findings across tissue types that Larger pores (e.g., 40–100 μm) encourage the creation of vascular structures, moderate-sized pores (e.g., 2–12 μm) support skin migration, and small pores (e.g., 1–2 μm) aid epidermal cell adhesion for skin regeneration. Multilayer scaffolds in bone tissue engineering that have bigger pores (200–400 μm) improve vascularization and nutrient diffusion, whereas smaller pores (50–100 μm) encourage cellular adhesion [20]. Higher porosity increases surface area and spatial availability while keeping the scaffold volume constant. This improves scaffold integration and contact with adjacent tissues, which promotes cell migration, adhesion, and proliferation [22]. A range of 60% to 90% is often ideal for bone tissue repair scaffolds [23,24,25], since numerous studies have demonstrated that the choice of porosity is influenced by various parameters, including bone type and scaffold material [26]. The cellular behavior and structural stability of scaffolds are greatly influenced by their geometric design. Various scaffold geometries can be divided into lattice frameworks, truss-based architectures, basic unit cell configurations [27], and additional types. A growing variety of bone tissue repair scaffold architectures is being investigated. These scaffolds must provide a microenvironment conducive to cell growth and possess characteristics that promote the differentiation and growth of target cells. These characteristics can be determined by analyzing fluid flow through the scaffolds.

Given the structural complexity and experimental feasibility challenges in bone scaffold development, computational modeling enables pre-fabrication design optimization and performance prediction, significantly reducing development cycles and costs. Established computational approaches in engineering, particularly Computational Fluid Dynamics (CFD) and Finite Element Analysis (FEA), have been increasingly adapted for bone tissue engineering applications. These simulation tools offer distinct advantages, including non-invasive investigation, multi-scale analysis capability, parametric visualization, and cost-effective precision modeling, making them indispensable for modern scaffold design workflows.

Recent years have witnessed a surge in bone scaffold applications. For instance, Bini et al. established a modeling framework to simulate diffusion in trabecular bone reconstructed from micro-CT scans [28], laying a critical methodological foundation for this study. Ouyang et al. numerically analyzed permeability, mass flow rate, fluid velocity, and wall shear stress (WSS) in porous titanium scaffolds. Their study compared CAD models with three CT-reconstructed models, validating simulations against cell seeding experiments in bioreactors [29]. Rahbari et al. conducted experimental and numerical comparisons of permeability in unit cell-based bone scaffolds. They demonstrated a $y=1.297x$ linear relationship between experimental and simulated permeability values, confirming the validity of numerical methods for scaffold permeability assessment [30]. Scaffold performance is primarily governed by porosity, pore dimensions, geometric configuration, and microstructural features. Current research prioritizes advancing novel biomaterials and fabrication techniques to develop high-performance bone repair scaffolds. While existing optimization strategies predominantly focus on single-physics domains, they lack a systematic analysis of thermo-fluid–solid multiphysics coupling. This work innovatively integrates numerical simulation and thermodynamic regulation to investigate scaffold thermomechanical behavior, develop coupled numerical methods, and analyze how flow velocity, temperature, and structural parameters influence cell proliferation and differentiation during perfusion culture. Through optimizing scaffold architecture and cultivation conditions, we aim to propose a bone repair scaffold with superior thermomass transfer efficiency and osteogenic potential. This study injects innovative vitality into resolving long-standing challenges in bone tissue engineering, accelerates interdisciplinary development, and holds significant theoretical and practical value.

2. Materials and Methods

2.1. Structural Construction

2.1.1. Three-Dimensional Modeling Design

1.

Pre-Processing

The current study is based on a regular-structured (cellular) scaffold that is frequently utilized in BTE studies. This scaffold is made up of several repeating cellular units that have apertures running through each surface of the cell to form flow channels inside the cell. The pores are connected, and the cells are arranged in a linear pattern that fits each other closely. This study employed SolidWorks 2021, a CAD software, to construct a three-dimensional (3D) model of the scaffold for bone tissue regeneration. Since the workstation can no longer perform calculations, the number of grid cells in the 5 cm × 5 cm × 5 cm model of the cellular structural bone tissue repair scaffold, which satisfies the minimum grid quality requirements, has surpassed 50 million, so the full structure of the scaffold was not simulated in this paper due to computer performance limitations. The Representative Volume Elements (RVE) method can be used to concentrate the numerical simulation’s scope on one or more relatively small representative volume elements, which reduces computation time and the amount of computation because cell-structured scaffolds have a symmetric, cyclic structure. The results of pre-simulation of five simplified stent models with varying numbers of cells (composed of 3 × 3 × 5, 3 × 3 × 10, 5 × 5 × 5, 5 × 5 × 10, and 7 × 7 × 10 cells, respectively) demonstrated that the two simplified stents made up of 5 × 5 × 10 and 7 × 7 × 10 cells could meet the simulation requirements and could accurately respond to the necessary parameters (temperature, pressure, velocity, and WSS) of the trends, with a deviation of less than 0.05% between the two sets of simulation results. The numerical simulation experiments are conducted using a simplified scaffold consisting of 5 × 5 × 10 cells, taking into account the simulation experiments’ efficiency.

This study has so far established the structure of the scaffold for bone tissue restoration and created various pore diameters and pore distributions. The CAD geometric model of the scaffold with a square pore cell structure and the fluid domain is shown in Figure 1. Following geometric modeling completion, export files in STL or STEP (Standard for the Exchange of Product Model Data) format for import into ANSYS fluent 2021 R1 Workbench pre-processing.

This study uses ANSYS Workbench (Wb), the project management platform of ANSYS. Wb integrates a variety of simulation software and modeling and data-processing tools, which can significantly improve work efficiency. In order to ensure the accuracy of the simulation results, structured grids are used for models with regular shapes, and tetrahedral meshes are generated using the patch conforming method for models with complex shapes, and mesh encryption is performed in local areas with complex shapes in order to ensure the quality of the mesh. Figure 2 shows some of the mesh models generated in this paper.

2.

Fluent Solver Setup

Confirm that the mesh quality in the cell mass, skewness, and orthogonal mass meet the requirements of the actual analysis of the experiment, and then define the model boundaries, including the flow inlet, flow outlet, different wall surfaces, and so on. Open Fluent, turn on the display mesh option, and check whether the inlet and outlet conditions and wall settings are read correctly and whether the model size is scaled. In this experiment, the simulation of incompressible flow with low velocity flow is used, and the liquid velocity is used as the main variable, so the type of pressure-based solver is selected, and the momentum equations are solved sequentially to calculate the velocity field, and the mass conservation equations are solved to calculate the pressure field until the convergence condition is reached to get a stable solution of the flow field. Set the Y-axis gravitational acceleration to −9.81 m/s².

Additionally, since this experiment requires modeling heat transfer, the energy equation is activated. The flow regime is determined via the Reynolds number, expressed as shown in Equation (1):

$$\mathrm{Re}={\displaystyle \frac{\rho \nu L}{\mu }},$$

(1)

Here, Re represents the Reynolds number; $\rho$ is the fluid density in kg·m⁻³; $\nu$ is the characteristic flow velocity in m·s⁻¹; $L$ is the characteristic length in m; and $\mu$ is the dynamic viscosity of the fluid in Pa·s.

After pre-calculation, the Reynolds number of the flow model is much less than the critical value of 2000, and the flow is laminar, so the laminar model is selected in the viscosity option. DMEN (Dulbecco’s Modified Eagle Medium) was selected as the simulation liquid material, and the physical properties of DMEN were set as density 1036 kg/m³, specific heat 4182 J/kg∙k, thermal conductivity 0.6 w/m∙k, and viscosity 0.00145 kg/m∙s. When regulating the temperature, the change in physical properties with temperature needs to be considered, and the physical properties need to be set to piecewise-linear form, the solid material of the stent is selected as TC4 (Ti-6Al-4V), and the physical properties are set to a density of 4430 kg/m³, a specific heat of 613 J/kg·K, and a thermal conductivity of 6.93 W/m·K. The surrounding fluid surface is set to symmetric condition. The fluid inlet is set as a velocity inlet, different inlet velocities are set according to the demand of simulation and analysis, the temperature is set as 310 K, and the outlet is set as a 0 Pa pressure outlet. The solid and fluid interfaces were coupled to each other to simulate thermal mass exchange.

Considering that the flow model is a low Reynolds number laminar flow, but the flow region is relatively complex, in order to meet the accuracy requirements and time and cost considerations, the coupled solver (Coupled) is selected in Fluent, the gradient term is selected based on the cell-based least-squares method, the discretization of the pressure term is selected in the PREssure STaggering Option (PRESTO!) format, and a second-order upwind scheme is used for the momentum and energy terms.

The relaxation factor (ranging from 0 to 1) in the solution control can adjust the preference of convergence and convergence speed when solving, decreasing the relaxation factor makes the convergence speed slower while the convergence improves; increasing the relaxation factor will speed up the convergence speed and decrease the convergence. The relaxation factor should be adjusted according to the simulation when simulating. The convergence criterion of the residuals is set to 0.00001 to ensure the accuracy of the results, and the calculations are run after mixing initialization.

3.

Post-Processing

• Mesh Irrelevance Study

Grids with different cell counts were generated for the square-pore unit cell-type scaffold model, and simulations were performed under conditions of a fluid inlet velocity of 1 mm/s and a temperature of 310 K. Computations diverged when the grid cell count was below 800,000. Due to workstation hardware limitations, the maximum grid cell count was 7,590,000 cells. Four groups of valid data were extracted, with grid sizes of 996,728, 2,219,115, 3,721,514, and 7,591,866 cells. Figure 3 shows the changes in average pressure of the bone tissue repair scaffold model at different heights for the four grid groups.

As depicted in the data graphs, the disparity in the average pressure at varying heights across the four groups of models is minimal. When comparing the average pressure at different heights between two groups of models with adjacent grid cell counts, the most pronounced data divergence occurs between the grid with 990,000 cells and the one with 2.22 million cells, reaching a maximum divergence of 12.46%. The maximum divergence between the group with 2.22 million cells and the group with 3.72 million cells is 1.43%, and that between the group with 3.72 million cells and the group with 7.69 million cells is 1.95%.

Consequently, it can be inferred that the simulation results start to stabilize once the number of grid cells surpasses 2.22 million. Taking into account the restricted computational resources, the numerical simulation results corresponding to a grid cell count of 3.72 million are employed for the analysis in this instance.

• Model Validity Test

Simulations were carried out at different flow velocities (1 mm/s, 0.5 mm/s, 0.3 mm/s, and 0.1 mm/s) for the square-pore unit cell-type scaffold model to obtain the pressure drop of the fluid flow through the bone tissue repair scaffolds, and the permeability of the scaffolds of this structure at different flow velocities was calculated according to Equation (3). The results are shown in Figure 4.

As shown, for the same scaffold structure, the permeability of the scaffold is independent of the velocity when the fluid physical properties are kept constant, which is in accordance with the definition of permeability, and the slight fluctuation of the permeability with the velocity (with a maximum difference of 0.11%) may be due to the entrance and exit effects or the accuracy of the simulation experiments. The permeability of this bone tissue repair scaffold was in the range of the permeability of bone trabeculae from 0.02 to 20 × 10⁻⁹ m², which is in the same order of magnitude as data in the literature for scaffolds of similar structure, proving the validity of the model.

2.1.2. Unit Cell Structure Design

(1)

Pore Size

This part uses a holder composed of several unit cells with a side length of 1 mm. Every unit cell has square holes on six sides, with a pore size of 0.7 mm, and its porosity is calculated to be 79.4%. Through pre-simulation experiments, the simplified model consists of 5 × 5 × 10 unit cells, and this scaffold is named S-0.7. On the basis of the S-0.7-type scaffold, the pore sizes of the cells are adjusted to be 0.4 mm (named S-0.4) and 0.62 mm (named S-0.62), respectively. The cell geometries of S-0.4 and S-0.62 are shown in Figure 5, and the porosities are, respectively, 35.2% and 67.7%.

(2)

Unit Cell Size

In order to investigate the modulation of unit cell size on bone tissue repair scaffolds, scaffolds with cell lengths of 0.8 mm and 1.2 mm were designed for comparative studies with scaffolds with cell lengths of 1 mm. To ensure the same porosity and pore shape, the scaffold with a cell length of 0.8 mm had a square pore diameter of 0.56 mm (78.3% porosity) and was named S-0.8-0.56; the scaffold with a cell length of 1.2 mm had a square pore diameter of 0.84 mm (78.4% porosity) and was named S-1.2-0.84.

(3)

Unit Cell Shape

Three different cell structures were designed in this experiment, namely circular pore cell, rhombic pore cell, and columnar cell, which have the same cell size and are all square cells with 1 mm side length. Two pore sizes were considered for the circular pore cell: the pore size of 0.7 mm with 66.9% porosity, with the scaffold of this cell structure named C-0.7; and the pore size of 0.8 mm with 79.4% porosity, with the scaffold of this cell structure named C-0.8. The rhombic pore cell had a pore size of 0.62 mm with 64.8% porosity, named D-0.62. The cylindrical cell has a pore diameter of 0.6 mm and a porosity of 71.38% and is named Z-0.6. The CAD models of several cell types are shown in Figure 6.

2.1.3. Design of Environmental Variables

(1)

Velocity

Through the summary of existing research in the literature, various simulation experiments and laboratory experiments were conducted to study the velocity of the bioreactor in the range of 0.01 mm/s to 1 mm/s. This part of the experiment controls the other variables using S-0.7. Set up four groups of simulation experiments with inlet flow velocities of 0.3 mm/s, 0.5 mm/s, 1 mm/s, and 1.2 mm/s, respectively, and calculate their respective flow inlet and outlet pressure difference, permeability, and WSS for comparison.

(2)

Temperature

Heat therapy is a therapeutic means of using heat to treat diseases or promote tissue repair. After studying the relevant literature, it was determined that the temperature range of heat therapy in the application of bone tissue repair is 38~45 °C, in which the 40~42 °C interval can achieve the best results. On the temperature part, an attempt will be made to use numerical simulation to explore the role of temperature in bone tissue repair in the regulation of in-scaffold flow and cell induction.

When heat therapy is applied to the bone tissue repair scaffold cell culture system in vitro or in vivo, the heat is transferred to the inside of the scaffold through heat conduction and thermal convection, so this part of the simulation model used in this paper is adapted to simulate the fluid domain at the periphery of the scaffold by setting it to be 37 °C, 39 °C, 41 °C, and 43 °C, respectively, and simplifying the liquid material to water by piecewise-linear. The density of water at 37 °C~43 °C is 991.04~993.33 kg/m³, the specific heat is 4.174 kJ/kg∙K, the thermal conductivity is 0.63 W/m∙K, and the viscosity is 6.22~6.97 × 10⁻⁴ kg/m∙s. Define the properties of water to simulate its changes under different temperature conditions.

2.2. Biomechanical Property Studies

While techniques such as Particle Tracking Velocimetry (PTV) are available, numerical analysis has emerged as a predominant approach for analyzing tissue repair scaffolds due to its efficiency, cost-effectiveness, and visualizability. Important indicators for assessing the biocompatibility of scaffolds for bone tissue healing and encouraging cell growth and development include permeability [31] and WSS [32]. The ability of a fluid to flow through a porous material is measured by its permeability, which is a crucial factor in characterizing the fluid transport characteristics of scaffolds used for bone tissue healing. WSS is the tangential force that results from friction between the fluid and the medium wall when the fluid passes through a porous material. Among these, the bone tissue repair scaffold’s permeability should not be less than 5.13 × 10⁻⁹ m², the wall shear force is best suited for bone differentiation within the range of 0.1–10 mPa, with 10–30 mPa potentially inducing MSC differentiation into cartilage, 30–60 mPa causing the formation of fibrous tissues, and below 0.1 mPa or above 60 mPa causing bone resorption or cell death.

2.2.1. WSS

This study investigated the effect of different pore sizes, cell sizes, and cell shapes of the unit cell structure as well as different flow velocities and temperatures on the WSS within the scaffold. WSS is an important factor to consider when designing bone tissue engineering scaffolds due to the fluid flow inside the scaffold as it affects the cellular processes involved in new tissue formation [32]. It has been shown that different WSS leads to different biosignals affecting mesenchymal stromal cell differentiation and ECM mineralization, leading to differences in tissue repair outcomes. WSS also affects the self-assembly and directional alignment of collagen fibers, which is critical for the mechanical properties of bone tissue.

The formula for studying WSS is Equation (2).

$$\mathrm{WSS}=\eta {\left.\left({\displaystyle \frac{\partial v_i}{\partial x_j}}+{\displaystyle \frac{\partial v_j}{\partial x_i}}=\right)\right|}_{x_i\in {\Gamma }_s},$$

(2)

where $v_i$ and $v_j$ are velocity vectors in different directions, while $x_i$ and $x_j$ are the i-th (or j-th) space coordinates.

The expression of WSS shows that the fluid velocity and viscosity are the main causes of WSS. Numerous studies have documented the influence of WSS on the differentiation of bone mesenchymal stem cells (MSCs). Current recommendations suggest applying mechanical stimulation with WSS levels in the range of 0.1 to 10 mPa to promote the osteogenic differentiation of MSCs. Conversely, WSS in the range of 10 to 30 mPa has been shown to enhance chondrogenesis [25]. Furthermore, in vitro studies have demonstrated that WSS ranging from 0.05 to 25 mPa can effectively facilitate tissue growth [33]. Vetsch et al. investigated the impact of different flow velocities on mineralized tissue growth generated by human mesenchymal stromal cells (hMSCs) cultured on scaffolds. By integrating WSS distributions obtained from CFD simulations with three-dimensional micro-computed tomography data, they demonstrated that shear stress below 0.39 mPa correlates with increased DNA content, while shear stress between 0.55 mPa and 24 mPa promotes osteogenic differentiation [34]. Porter et al. reported that an average surface shear stress of 5 × 10⁻⁵ Pa (0.05 mPa) is associated with high cell viability and osteoblast proliferation, whereas a peak shear stress of 5.7 × 10⁻² Pa (57 mPa) leads to cell death [6]. The observed discrepancies among these studies likely stem from variations in experimental parameters, including scaffold structure, dimensions, material types, and specific conditions (such as flow velocity and frequency within bioreactors). Additionally, local variations in surface chemistry and morphology can significantly impact cell-biomaterial interactions, thereby influencing cellular responses to mechanical stimuli. Consequently, accurately predicting fluid shear stress and flow paths via CFD simulations is critical for optimizing the design parameters of scaffolds and bioreactor systems.

2.2.2. Permeability

Permeability ($K$) is expressed by Darcy’s Law, meaning the ability of a fluid per unit area to pass through a porous medium per unit time, and is derived from Darcy’s Law with the expression (3).

$$K=-Q{\displaystyle \frac{\Delta L}{\Delta P}},$$

(3)

where $K$ is the permeability in m²; $Q$ is the volume velocity through the porous medium in m³/s; $\Delta L$ is the length of the path of the fluid through the porous medium in m; and $\Delta P$ is the pressure difference of the fluid on the flow path $\Delta L$ in Pa.

Darcy’s law with the fluid velocity and fluid viscosity of the expression for (4):

$$K=-\eta {\displaystyle \frac{v}{\Delta P}},$$

(4)

where $\eta$ is the fluid viscosity in Pa·s and $v$ is the velocity at the inlet of the porous medium in m/s.

The use of permeability as a design parameter for bone tissue repair scaffolds lies in the fact that it encompasses the influence of multiple factors on the mass transport properties of the scaffold, such as scaffold geometry structure, pore size, and porosity. Permeability is critical for cell growth, and scaffolds with high permeability produce favorable conditions for cell growth [23]. However, too high permeability can also negatively affect the scaffold performance, e.g., leading to lower overall mechanical properties, reduced cell–scaffold interactions, and unfavorable cell attachment to the scaffold [24]. In different articles, the permeability of scaffolds with different structures ranges from 0.018 to 50 (×10⁻⁹ m²), and permeability increases with increasing porosity and scaffold cell size. Permeability may also be related to specific surface area. Research by Jung et al. demonstrated that the fluid permeability of TPMS-structured scaffolds is inversely proportional to the specific surface area of these structures [35]. The existing literature reports that the threshold permeability required for vascularization and mineralization within porous implants is approximately 3 × 10⁻¹¹ m² [36]. Human trabecular bone exhibits permeability ranging from 0.02 to 20 × 10⁻⁹ m², with an average value of 5.13 × 10⁻⁹ m² [37,38]. Consequently, the designed permeability of bone tissue repair scaffolds should be no less than 5.13 × 10⁻⁹ m².

It is worth explaining that the nutrients required and metabolites discharged by cells per unit of time are limited, so there is a threshold for the increase in the permeability of bone tissue repair scaffolds, and after exceeding the threshold, the increase in permeability will no longer promote the growth and differentiation of cells. For scaffolds of the same structure, the permeability is determined by the pore diameter and pore spacing. When the pore diameter increases to a certain value, the supply of nutrients reaches saturation, and then, according to the “curvature-driven mechanism”, scaffolds with smaller pore diameters provide higher mean curvature, thus promoting cell proliferation.

3. Results and Discussions

3.1. Velocity Effects on Unit Cell-Structured Bone Tissue Repair Scaffolds

Firstly, the analysis was carried out for the flow of 1 mm/s bioreactor inlet velocity in the bone tissue repair scaffolds with unit cell structure. The pressure distribution of S-0.7 is shown in Figure 7a.

With the inflow of fluid from the inlet (Y = 0.01 m), the pressure decreases gradually with the flow direction, and the overall pressure distribution is uniform. Using Equation (2), the permeability was determined to be 1.23 × 10⁻⁸ m², which exceeds the reported average permeability of human trabecular bone and can satisfy cellular nutrient supply requirements. Figure 7b presents the velocity vector field within the scaffold.

Fluid flow primarily occurred along Y-axis aligned pores of the scaffold, forming multiple primary flow streams with the highest velocity at the pore center and a gradual velocity decay toward the wall. Flow velocities within X- and Z-axis pores were significantly lower, resulting in a periodically regular but non-uniform velocity distribution. This non-uniformity may compromise nutrient supply and cellular metabolism in low-velocity regions of the scaffold. The WSS distribution of the S-0.7 scaffold at 1 mm/s for the scaffold was plotted in Figure 7c. As illustrated in the figure, WSS distribution exhibits similar trends to the velocity field, with high WSS primarily concentrated on the scaffold walls aligned with the main flow direction (Y-axis), while X- and Z-axis pore walls experience lower magnitudes. Statistical analysis of WSS values revealed a mean of 15.77 mPa and a maximum of 59.69 mPa. The relative frequency distribution of WSS across different intervals is presented in Figure 7d.

As illustrated, at an inlet velocity of 1 mm/s, the WSS distribution along the S-0.7 scaffold walls was non-uniform. The 0–10 mPa WSS range accounted for 53% of the total distribution, and this wall region promotes osteogenic differentiation of MSCs [25]. In contrast, the 10–30 mPa WSS region induces ECM mineralization or cartilaginous tissue formation, while the 30–60 mPa region stimulates fibrous tissue development. Combining these results with the WSS contour plot, the non-uniform WSS distribution can be attributed to heterogeneous fluid flow patterns within the scaffold pores.

If all the cells on the scaffold wall are differentiating towards osteoblasts, the scaffold structure or the velocity of the bioreactor needs to be modulated. The velocity was first changed to modulate the scaffold WSS distribution. Numerical simulations were performed for the S-0.7 scaffold at different inlet velocities with the rest of the settings kept constant. The pressure distribution cloud plots and WSS distribution contour plots at three inlet flow velocities of 1.2 mm/s, 1.0 mm/s, and 0.5 mm/s were selected for comparison after comparative analysis of the simulation results.

Pressure and WSS contour plots for the S-0.7 scaffold under varying inlet velocities are presented in Figure 8.

Through comparison, it is found that the pressure distribution on the surface of the scaffold with different flow velocities all show the same trend, decreasing layer by layer with the direction of flow inlet (Y-axis direction), with the maximum pressure in the inlet layer, the minimum pressure in the outlet layer, and the uniform pressure distribution between each layer. With the increase in inlet velocity, the pressure difference between the upper and lower layers between different groups increases. And the WSS distribution on the surface of the S-0.7 stent shows the same trend at different flow velocities, with the high WSS region distributed on the wall close to the main flow, while the pore wall parallel to the main flow direction has a lower WSS value. Unlike the pressure distribution, the WSS distribution between the cells of each layer does not show a decreasing trend layer by layer, and the WSS distribution between the cells of each layer is relatively uniform; it can be hypothesized that the velocity between different layers also shows a uniform distribution.

The flow entrance and exit pressure differences were calculated for four sets of simulations with inlet flow velocities of 0.3 mm/s, 0.5 mm/s, 1 mm/s, and 1.2 mm/s, respectively, and the permeability was derived to obtain Figure 9a. As well as the average flow rates between the cell layers of different heights of the S-0.7 stent were statistically calculated for different inlet flow velocities and plotted in Figure 9b.

With the increase in velocity, the differential pressure of fluid at the entrance and exit of the S-0.7 stent is linear, while the permeability remains constant, a trend that is consistent with the definition of permeability and indicates the validity of the simulation model. In the control group with different flow velocities, the average flow velocities between the cell layers were not much affected by the height of the stent, and they were all slightly lower than the inlet velocity, and the distribution of the flow velocities between the cells in each layer was similar, which was consistent with the previous speculation and reflected the reasonableness of the modulation of the distribution of the WSS on the wall of the bone tissue repair stent by the velocity. The WSS frequency distribution of S-0.7 scaffolds with different velocity control groups is plotted in Figure 10.

Comparison of the control groups with inlet velocities of 1.0 mm/s and 1.2 mm/s, respectively, revealed that the frequency distribution of WSS on the stents had a similar trend, with WSS distributed most between 0 and 10 mPa, followed by 10–20 mPa, and the frequency distribution between 20 and 30 mPa appeared to have a very small point before increasing slightly. The overall WSS frequency distribution showed an equiproportional scaling trend with decreasing velocity. The frequency difference of WSS in the 10–30 mPa interval was not significant in the four control scaffolds, indicating that the S-0.7 scaffolds had strong potential for cartilage differentiation. At an inlet velocity of 1.2 mm/s, 10% of the scaffolds of the S-0.7 scaffolds showed a WSS of more than 50 mPa, which was close to the threshold for cell death and, therefore, the S-0.7 scaffolds were used for the differentiation of cartilage, and the WSS of these scaffolds was not significant. Therefore, the velocity should be controlled not to be higher than 1.2 mm/s when using S-0.7 scaffolds for perfusion culture. When the velocity was 1.0 mm/s, only 0.49% of the WSS exceeded 50 mPa, but it should be noted that at this time, the fine [22] cells will be differentiated towards bone, cartilage, and fibrous tissues. When the inlet velocity is 0.5 mm/s and less, the distribution of WSS on the scaffolds is almost always less than 30 mPa, and at this time the MSCs will not differentiate towards fibrous tissue. To increase the proportion of cells differentiating toward osteogenesis, the velocity of the perfused bioreactor should be adjusted within 0.1–0.3 mm/s.

3.2. Influence of Scaffold Structure on Bone Tissue Repair Scaffolds and Modulation

3.2.1. Aperture Size

Numerical simulations of S-0.4 and S-0.62 scaffolds were conducted at an inlet velocity of 1 mm/s, with all other parameters identical to the baseline setup. Pressure contour plots were generated and compared against the S-0.7 scaffold, yielding the results presented in Figure 11.

From the pressure distribution comparison chart, it can be seen that the pressure distribution trend of stents with different aperture diameters is the same, decreasing layer by layer along the direction of inlet velocity. The smaller the pore diameter, the larger the pressure difference between the inlet and outlet, and the more drastic the change in pressure with the height of the stent. The pressure difference between the inlet and outlet of the S-0.62 stent is 1.88 Pa, which is 1.6 times that of the inlet and outlet of the S-0.7 stent, and the pressure difference between the inlet and outlet of the S-0.4 stent is 10.48 Pa, which is 8.9 times that of the inlet and outlet of the S-0.7 stent. When the pore diameter is larger, the liquid molecules can pass through the pore more easily, and the flow resistance is smaller, while when the pore diameter is reduced, due to the viscous effect of the liquid, it makes the friction between the molecules inside the liquid and the shear force between the liquid and the wall increase, which leads to a greater resistance to flow, and therefore the pressure difference increases.

The change in permeability and the change in porosity for the three types of scaffolds are plotted in Figure 12.

It can be seen that the porosity of the square pore unit cell-type scaffolds varies linearly with the cell pore diameter; the permeability increases with the increase in the pore diameter, and the larger the pore diameter, the greater the rate of change in the permeability, indicating that the permeability is very sensitive to the change in the cell pore diameter. When the permeability is low, the mass transfer inside the scaffold is obstructed, which will affect the cell nutrient supply and metabolic waste discharge demand.

The velocity vector of the S-0.4 scaffold at an inlet velocity of 1 mm/s is plotted in Figure 13.

From the velocity vector diagram, it can be seen that the mainstream direction of the S-0.4-type stent is consistent with the flow direction at the inlet and runs through the whole stent along the Y-axis direction, with the most intense flow at the center of the pore in the Y-axis direction and very few velocity vectors in the X-axis direction and Z-axis direction and slow flow in the lateral direction. Compared with the S-0.7-type stent, the velocity distribution inside the S-0.4-type stent was more inhomogeneous. Due to the decrease in pore diameter and the increase in wall thickness, the volume of the low-velocity flow zone between the pores on the side of the stent increased, and a large amount of non-permeable volume existed, which was unfavorable to the growth and development of cells.

The frequency distribution of WSS at an inlet velocity of 1 mm/s was statistically plotted in Figure 14 for the S-0.4-type stent and the S-0.62-type stent.

From the figure, it can be seen that the WSS distribution of square pore cell type scaffolds with different pore sizes are not uniform, and there is a frequency concentration phenomenon, with the largest proportion of the relatively lowest WSS region, and the WSS distribution trend of other values between groups is slightly different, which is due to the flow inhomogeneity. The WSS distribution trend of S-0.4 scaffolds is similar to that of S-0.7, and it is similar to that of S-0.62 scaffolds, which is a concave distribution. The S-0.62 scaffold had a similar trend of WSS distribution as S-0.7, with a concave distribution. Under the inlet velocity of 1 mm/s, 49.4% of the area of WSS was between 0 and 10 mPa, and the cells in this part of the area would be differentiated to osteoblasts; 24.2% of the area of WSS was between 10 and 30 mPa, which would induce the cells to be differentiated to cartilage; 25.2% of the area of WSS was between 30 and 60 mPa, which would induce the cells to be differentiated to fibrous tissues; only 1.1% of the area was unfavorable for cell culture.

The inlet velocity was adjusted to 0.5 mm/s for simulation, and the frequency distribution of WSS was counted and is plotted in Figure 15.

Comparative analysis of Figure 14 and Figure 15 shows that for square crystalline cell-structured scaffolds with different pore diameters, the lower velocity decreases the overall WSS level, but the trend of the WSS distribution remains consistent, which is in keeping with the conclusions of Section 3.1. At an inlet velocity of 0.5 mm/s, 17% of the area of the S-0.4 scaffold was still unsuitable for cell growth and differentiation, indicating that the small pore size of the crystalline cell structure bone tissue repair scaffolds is unsuitable for cell culture, and that the scaffolds should be avoided in designing smaller pore sizes. At an inlet velocity of 0.5 mm/s, the S-0.62 scaffold will induce cell differentiation in two directions: 62.5% of the area will be induced to form osteoblasts, and the remaining area will be induced to form chondrocytes. If it is desired that all cells in the region differentiate into osteoblasts, it is necessary to continue to reduce the inlet velocity of the perfusion bioreactor. After simulation experiments, it was found that for the S-0.4-type scaffold, at an inlet velocity of 0.2 mm/s, the WSS were all distributed within 0–10 mPa, which could meet the biostimulation requirements for cell differentiation to osteoblasts.

3.2.2. Cell Size

The S-0.8-0.56 and S-1.2-0.84 stents were simulated at an inlet velocity of 0.5 mm/s, and the pressure distributions were plotted in Figure 16.

As illustrated, S-0.8-0.56 and S-1.2-0.84 scaffolds exhibit consistent pressure distribution trends at identical inlet velocities, with pressure decreasing gradually with decreasing Y-axis height. Quantitatively, smaller-aperture unit cells demonstrate larger inlet–outlet pressure differences and steeper pressure gradients along the scaffold height, attributed to increased flow resistance from the smaller pore diameters in S-0.8-0.56 scaffolds.

Permeability calculations for S-0.8-0.56 and S-1.2-0.84 scaffolds, compared against S-0.7 scaffolds, are presented in Figure 17.

It can be seen that the permeability of the three groups of scaffolds increased with the increase in the unit cell length, and the permeability increased by 37% when the cell length increased from 0.8 mm to 1.0 mm; when the cell length increased from 0.8 mm to 1.2 mm, the permeability increased by a factor of 1.1, and this trend was similar to the change in permeability with the pore size. Since the porosities of the three groups of scaffolds were almost the same (maximum difference of 1.4%), it can be hypothesized that for the square-pore shaped cell-type scaffolds, the pore size is the main influencing factor of the permeability, and the change in porosity does not aptly reflect the change in the permeability of the scaffolds.

Statistical analysis of WSS for S-0.8-0.56 and S-1.2-0.84 scaffolds at an inlet velocity of 0.5 mm/s is presented in Figure 18.

As shown in the figure, the two scaffolds with different cell sizes had significantly different WSS distribution trends at the same inlet velocity, but the distribution intervals were similar, and both scaffolds would induce cellular differentiation toward bone and cartilage at an inlet velocity of 0.5 mm/s. The WSS distribution of the S-0.8-0.56 scaffold with a cell length of 0.8 mm was between 0 and 20 mPa, with 74% of the WSS in the 0–10 mPa region, which would induce bone differentiation, and the rest of the region would induce chondrogenic differentiation. The WSS of S-1.2-0.84 scaffolds with a cell length of 1.2 mm was distributed between 0 and 30 mPa, with 61% of this region inducing bone differentiation and the rest of the region inducing chondrogenic differentiation. The average WSS of S-0.8-0.56 scaffold was calculated to be 9.067 mPa and that of S-1.2-0.84 scaffold was 6.009 mPa, with a difference of 50.77%, which shows that the larger the cell size, the smaller the scaffold WSS at the same inlet velocity under the same scaffold porosity, and the rate of change in the WSS is smaller than that of the permeability.

3.2.3. Unit Cell Shapes

Several sets of numerical simulations were performed to analyze the scaffolds with different cell shapes, which will be discussed in separate cases. Firstly, the pressure was analyzed. C-0.7 and C-0.8 differed only in pore size, which has been discussed in Section 3.2.1. Therefore, the pressure distribution plots of C-0.7, D-0.62, and Z-0.6 stents at an inlet velocity of 1 mm/s are compared here, as shown in Figure 19.

At identical inlet velocities, scaffolds with different unit cell shapes exhibit highly similar pressure distributions, all decreasing from the upper to lower layers along the inlet flow direction. Surface integral calculations of inlet–outlet pressure differences yielded values of 1.8433 Pa (C-0.7), 1.0132 Pa (C-0.8), 1.9351 Pa (D-0.62), and 1.6386 Pa (Z-0.6), from which permeability values were calculated and compared against square-pore scaffolds in Figure 20.

The six scaffolds have the same cell size, with differences in cell shape and pore size. C-0.8 and S-0.7 scaffolds have the same porosity, and C-0.8 scaffolds have a larger pore size and higher permeability. The pore size has a greater effect on the permeability than the cell shape for the same porosity. C-0.8 and S-0.7 scaffolds have the same pore size, and C-0.8 scaffolds have a larger pore size and higher permeability.

Comparison of S-0.7 and C-0.7 stents shows that the square pore cell increases the permeability through higher porosity. Comparison of D-0.62, Z-0.6, and S-0.6 reveals that the stents with columnar cell structure have the largest porosity and the highest permeability when the pore sizes are the same, and the difference between the diamond-shaped pore stents and the square pore stents is not obvious in terms of porosity and permeability. It can be seen that for the cell-type scaffolds, the columnar cell is superior to the rhombic pore cell and the square pore cell from the perspective of increasing permeability, and the round pore cell is the least effective. In addition, the permeability of these six types of cells was higher than the value of human trabecular bone permeability (5.13 × 10⁻⁹ m²), and their overall mass transfer capacity met the demand of osteoblast growth and development.

The flow vector in Figure 21 of the Z-0.6 scaffolds was plotted to analyze the effect of cell shape on flow within the scaffolds.

Velocity vector fields indicate that columnar unit cell scaffolds exhibit axial flow dominance along the inlet direction (perpendicular to the XZ-plane), with peak velocities at XZ-plane pore centers and radial velocity decay toward pore walls. Z-0.6 scaffolds demonstrate superior internal flow homogeneity due to fluid–wall viscous coupling, where wall-adjacent fluid migrates laterally, enhancing inter-pore flow mixing and reducing flow stagnation zones. This uniform flow architecture creates favorable microenvironments for cell growth and differentiation.

WSS contour plots for Z-0.6, D-0.62, and C-0.7 scaffolds are presented in Figure 22.

The cylindrical structure of the Z-0.6 scaffold has the highest WSS in the region closest to the center of the main stream, decreasing radially in the direction away from the main stream, with an ellipsoidal WSS isosurface, and the lowest WSS in the region centered on the unit cell sidewall. The D-0.62 scaffold has high WSS regions distributed on walls close to the main stream, with the lowest WSS at the inter-unit–cell wall junctions. The C-0.7 scaffold exhibits a polarized WSS distribution, with higher WSS on mainstream-aligned walls and lower WSS on lateral pore walls. The transition zone between high and low WSS is indistinct.

WSS values of Z-0.6, D-0.62, and C-0.7 scaffolds were statistically analyzed and plotted in Figure 23.

The frequency distribution graph of WSS was consistent with the performance of the cloud data. The WSS frequency distribution plot aligned with the contour plot observations. Z-0.6 scaffolds exhibited the broadest WSS distribution, with frequency decreasing monotonically with increasing WSS; however, 8.5% of the area exceeded 60 mPa (detrimental to cell growth/differentiation), necessitating perfusion bioreactor velocities < 1 mm/s for cell culture. D-0.62 scaffolds showed slightly higher WSS inhomogeneity than Z-0.6: 34% of the area (0–10 mPa) induced osteogenesis, 36% (10–30 mPa) promoted chondrogenesis, 27.5% (30–60 mPa) favored fibrosis, and only 2.2% exceeded 60 mPa. C-0.7 scaffolds had the most uneven distribution: 48.2% < 10 mPa, 20.3% (10–20 mPa), and 31.3% (30–60 mPa), with the smallest excessive WSS region, indicating advantages at high inlet velocities.

3.3. Effects and Regulation of Temperature on Unit Cell Structure Bone Tissue Repair Scaffolds

Numerical simulations were carried out for the S-0.7 scaffold inlet velocity of 1 mm/s at different temperatures to derive the pressure difference between the inlet and outlet of the stent under each temperature condition. The permeability was calculated separately and the results are shown in

Table 1. Pressure difference and permeability change.

Temperature (°C)	Pressure Difference Between the Inlet and Outlet (Pa)	Permeability (×10−8 m2)
37	0.5916	1.19064
39	0.5687	1.18523
41	0.5479	1.18379
43	0.5326	1.17839

.

As temperature increases, the inlet–outlet pressure difference of the scaffold and its permeability decrease, attributed to reduced water viscosity and temperature-driven natural convection. A velocity vector plot of the scaffold interior at 43 °C is presented in Figure 24 to analyze internal flow.

Compared with Figure 7b, it can be seen that the flow inhomogeneity of the S-0.7 scaffold is reduced under the heating condition, and the gradient of velocity change is significantly reduced, although there is still a main flow stream that follows the inlet flow direction consistently. The highest flow velocity is at the center of each layer of stent aperture in the Y-axis direction, which is caused by the narrowing of the flow channel by the constraint of the stent wall. According to the law of conservation of mass, which requires that the mass of fluid passing through the channel with smaller cross-sectional area must be equal to the mass of fluid passing through the channel with larger cross-sectional area per unit time, the flow velocity increases as the fluid passes through the aperture of each layer of the wall surface, and spreads out rapidly after passing through the wall surface. Calculate the average flow rate of fluid through the bracket at different temperatures. The average velocity within the scaffold gradually increased with temperature elevation, ranging from 1.245 to 1.248 mm/s. enhancing intra-scaffold mass transfer and optimizing cellular growth conditions on the scaffold. Concurrently, results indicate that permeability-based measurement of scaffold mass transfer capacity becomes less precise under changing fluid viscosities.

A temperature distribution contour plot is presented in Figure 25 to analyze the scaffold and internal fluid temperature fields.

As illustrated, the S-0.7 scaffold and its internal fluid domain exhibit nearly uniform temperature distribution, with slight cooling at the inlet region, indicating that the regular scaffold architecture promotes temperature field uniformity and excellent heat transfer performance.

WSS distributions of the S-0.7 scaffold at different temperatures were quantified. Given WSS frequencies in 30–60 mPa < 0.5% across all datasets, only 0–30 mPa data were analyzed, and frequency distributions are presented in Figure 26.

From the frequency distribution plot, as temperature increases, the frequency of 0–5 mPa WSS rises, 20–30 mPa WSS gradually decreases, and S-0.7 scaffold WSS exhibits a downward trend with minimal variation in overall distribution. Combined with the velocity–temperature trend, WSS reduction is attributed to decreased liquid viscosity. Thermal regulation of WSS distribution is less effective than velocity-based regulation.

4. Conclusions

Changing the inlet flow velocity of a bioreactor is an effective method for regulating WSS in bone tissue repair scaffolds. By adjusting the flow velocity, scaffolds of different structures can achieve proportional scaling (up or down) of the WSS distribution, thereby promoting cell differentiation along different orientations. However, flow velocity regulation cannot alter the pattern of the WSS distribution. During cell perfusion culture, altering the flow orientation enables cells to differentiate along the same direction. Heating can improve the heterogeneity of flow distribution within bone tissue repair scaffolds and enhance their mass transfer efficiency. Due to decreased viscosity of the liquid, heating results in a slight reduction of WSS within the scaffolds.

Comparative assessment of several scaffold architectures reveals that cylindrical unit cell structures exhibit the highest permeability and superior internal flow uniformity, eliminating WSS concentration. When designing scaffolds, curvilinear geometries should be prioritized to generate multidirectional flow paths, thereby enhancing intra-scaffold flow homogeneity. Crucially, linear collinear pore channels must be avoided, as they create centralized high-velocity flow cores, causing heterogeneous WSS distribution, which may induce cell detachment or death. Among structural parameters, pore diameter (>0.4 mm) represents a primary determinant of permeability, while porosity should exceed 60%. Strategic architectural modification serves as an effective approach for optimizing scaffold-wide WSS distribution.

This study conducts a comprehensive investigation into the optimization of bone tissue repair scaffolds, analyzing the regulatory mechanisms of unit cell-based scaffolds and proposing a multi-parameter control methodology integrating flow velocity, temperature, and structural design to achieve scaffold optimization. The approach significantly enhances design efficiency while reducing experimental costs, demonstrating substantial practical and economic value. However, the research relies on static cellular response assumptions, whereas actual bone regeneration involves dynamic macrophage-osteoblast-endothelial cell interactions, neglecting impacts of cellular proliferation and mineralization on scaffold structural evolution, which introduces limitations. Future studies should refine computational models by incorporating dynamic cellular feedback mechanisms to improve accuracy and explore combinatorial arrangements of heterogeneous scaffold architectures beyond the current homogeneous designs to identify novel optimization pathways.