1 Introduction

Most fractures heal uneventfully. However, between 5% and 10% of them result in nonunion[1]. Large bone defects with a size of over 3 mm generally do not naturally heal, which presents a clinical challenge. The growing demand for bone grafts has led surgeons to investigate more advanced techniques for reconstruction of bone defects. Tissue engineering may be a promising alternative to conventional treatments. One commendable approach was to culture mesenchymal stem cells in vitro within a biomaterial scaffold to produce a bone-like layer of tissue[2–4]. Then, the engineered bone tissue is implanted into the body to reconstruct large bone defects. Although promising results have been obtained for the healing of large bone defects[5], much more research is needed before the bone engineering can enter daily clinical practice.

The cultivation of tissue constructs in vitro and static conditions, typically result in an outer shell of viable cells, while the inner core becomes necrotic due to the poor diffusive delivery of nutrients and the accumulation of wastes[6]. A bioreactor system has been developed to generate dynamic fluid perfusion within cell-seeded scaffolds, and it was found to have beneficial effects on cell function and the growth of bone tissue[7–11]. Compared to static controls, the improvement of mass transport and nutrient exchange in bioreactor systems can increase cell viability and extracellular matrix formation[12, 13]. Meanwhile, cell response was associated with the fluid flow velocity and wall shear stress in perfusion configurations[14].

Although there is a clear relationship between macroscopic mechanical loads and tissue differentiation, it is difficult to determine the local mechanical stimuli sensed by the cells at a microscopic level. Fluid flow velocity and wall shear stress distributions within a scaffold at a microscopic level can be obtained by using the computational fluid dynamics (CFD) method. Different types of mechanical stimuli have been considered for the prediction of tissue formation[15–18].

Others have used computer-aided design approaches where scaffolds were built as a repeat pattern of unit cells[11, 19]. This method allows proper actual estimation results of fluid flow velocity, wall shear stress and pressure within the model which can be used for easy manufacture. However, a more realistically geometric representation of a scaffold was obtained through micro-computed tomography (micro-CT). Combined with the CFD method, more accurate results will be achieved to indicate the natural distributions of fluid flow velocity, wall shear stress and pressure.

The main goal of this study was to build CFD models to obtain detailed information of the interstitial fluid velocity, wall shear stress and pressure within the pores of three scaffolds for several initial conditions, and to assess the microstructure and fluid simulation results of these three models with given initial conditions.

2 Materials and methods

In this study, we used a designed scaffold (DS) model and the size was 2.2 mm × 2.2 mm × 3 mm, as shown in Fig. 1 (a). The diameter of the canal was 0.2mm, and the gap of the canal was 0.2 mm. Two other scaffolds approximately 2 mm in length, 2 mm in width and 3 mm in height with different morphologies were used, as shown in Fig. 1 (b)(freeze-drying scaffold, FS) and (c) (dog femora scaffold, NS). These two samples were scanned using a MicroCT80 with a resolution of 20 µm. Computer reconstruction of these samples in three dimensions was made using Mimics, as shown in Fig. 2.

Fig. 1
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CAD models of 3 types of scaffolds

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Fig. 2
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Procedure from micro-CT images to model selection shown for natural bone scaffold

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The fluid phase was obtained by subtracting the scaffold from a chamber (DS: 2.6 mm × 2.6 mm × 3.4 mm; FS: 2.5mm × 2.5mm × 4mm; NS: 3.1mm × 3.1mm ×4mm), as shown in Fig.3.

Fig. 3
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The structure diagram of the algorithm

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Three-dimensional (3D) tetrahedral meshes for the domains include the interconnected pores and the perfusion chamber which were made in HyperMesh with an element size of 50 mm, as shown in Fig. 4. There were 1 050 000 elements made for DS, 820 000 elements made for FS and 1 150 000 elements made for NS. The size of the osteocyte adhering to the surface of the scaffold was about 50 mm. Consequently, seven boundary layers (the first minimum layer thickness was 0.002 mm) based on a growth ratio of 1.2 on the surface of the scaffold models which cell mainly perch were made to capture the distribution of fluid flow velocity and wall shear stress, as shown in Fig. 4.

Fig. 4
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Meshes for DS, FS and NS

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The analysis was carried out by using fluid. The outlet fluid pressure was set as zero, and these surfaces of the scaffolds were set as no-slip walls. Operating pressure was 101 325 Pa, while gravitational acceleration was 9.8m/s2. The culture medium was modeled as an incompressible and homogeneous Newtonian fluid with a density of 1 000 kg/m3, as shown in Fig. 5. Four levels of inlet velocity and three levels of fluid viscosity were used to make up 36 different cases, as shown in Table 1. Fluid flow velocity, wall shear stress and pressure were calculated.

Fig. 5
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Inlet velocity of DS, FS and NS

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Table 1 Levels of fluid viscosity and inlet fluid velocity used for CFD simulation
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3 Results

Fluid flow velocities for DS, FS and NS were calculated at 4 levels of inlet velocity and 3 levels of fluid viscosity. The maximal and average velocities are presented in Table 2.

Table 2 Results of fluid flow velocity
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For the same scaffold, with the viscosity increasing, the maximal and average fluid flow velocity decreased correspondingly and slightly for each value of inlet velocity. The viscosity had a limited impact on the fluid flow velocity. The fluid flow velocity increased correspondingly as the inlet velocity increased. The fluid flow velocity remained about 2 to 4 times the inlet velocity. At the same inlet velocity and viscosity, the maximal fluid flow velocity appeared for DS. The maximal fluid flow velocity in DS was nearly more than 172% of FS and NS’s velocity (initial conditions: inlet velocity was 10 m/s and viscosity was 2.1 × 10−3 Pas, DS: 42.535 µm/s, FS: 15.618 µm/s and NS: 18.868 µm/s), while the average velocity was more than 440% of FS and NS’s velocity (initial conditions: inlet velocity was 10 µm/s and viscosity was 2.1 ×10−3 Pas, DS: 13.246 µm/s, FS: 2.454 µm/s and NS: 2.867 µm/s). At the same initial conditions, the maximal and average fluid flow velocities in FS were a little more than those in NS. From this it can be considered that FS had a more reasonable microscopic structure than DS. The microscopic structure of FS was closer to that of the real scaffold (NS).

From Fig. 6, it is observed that the maximal and average fluid flow velocities in FS were closer to those in NS than in DS. The biggest difference was concentrated in the inlet velocity range between 10 µm/s and 100 µm/s. Relevant studies showed that it was quite useful for the cultivation of cell-seeded scaffold in a bioreactor. So FS was more suitable instead of natural bone than DS from the point of view of fluid flow velocity.

Fig. 6
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Fluid flow velocity for DS, FS and NSat at viscosity of 1.45 × 10−3 Pas and inletvelocity of 1 µm/s, 10 µm/s, 100 µm/s and 1000 µm/s

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Wall shear stresses for DS, FS and NS were calculated at 4 levels of inlet fluid velocity and 3 levels of fluid viscosity. The maximal and average wall shear stresses are presented in Table 3.

Table 3 Results of wall shear stress
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As the inlet velocity increased, the maximal and average wall shear stresses increased correspondingly. At the same inlet velocity and viscosity, wall shear stresses showed mainly a linear relationship, as shown in Fig. 7. Although increased viscosity reduced the fluid flow velocity, the wall shear stress increased, which indicated that the influence of viscosity on the wall shear stress was greater than that of the inlet velocity within a certain region. For the condition of inlet velocity of 100 µm/s and viscosity of 1.45 ×10−3 Pas, the maximal and average wall shear stresses were 15.675 mPas and 3.223 mPas in DS, 67.126 mPas and 5.949 mPas in FS, and 20.190 mPas and 1.629 mPas in NS. From Fig. 8, the maximal and average wall shear stresses in FS were lower than in DS except for these results obtained at an inlet velocity of 10 µm/s. The wall shear stress was not consistent in FS and NS. This was explained by the fact that the pores were so small in FS that the distribution of wall shear stress was greatly affected.

Fig. 7
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Wall shear stress for DS, FS and NS at inlet velocity 100 m/s and viscosity 1.003 ×10−3 Pas, 1.45 × 10−3 Pas and 2.1 × 10−3 Pas

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Fig. 8
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Wall shear stress for DS, FS and NS at viscosity 1.45 × 10−3 Pas and inletvelocities 1 µm/s, 10 µm/s, 100 µm/s and 1000 µm/s

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The main distribution of the wall stress was within 10% of the maximum, which is shown in Figs. 9 and 10. Both the histograms and contours show that the distribution of wall shear stress is highly matched in FS and NS. Compared to DS, the microstructure and the distribution of wall shear stress in FS and NS were more suitable for perfusion cultivation.

Fig. 9
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Scaffolds’ main wall shear stress distribution at viscosity 1.45 ×10−3 Pas and inlet velocity 100 µm/s

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Fig. 10
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Scaffolds’ main wall shearstress visualization distribution at viscosity 1.45 ×10−3 Pas and inlet velocity 100 µm/s

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Wall pressures for DS, FS and NS were calculated at 4 levels of inlet velocity and 3 levels of fluid viscosity. The maximal and average wall pressures are presented in Table 4.

Table 4 Results of wall pressure
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As the inlet velocity and viscosity increased, the maximal and average pressures in these 3 kinds of scaffold model all increased. From Fig. 11, the maximal and average wall pressures in FS were larger than those in DS and NS at the same inlet velocity and viscosity. Curves B and C remained parallel, which indicated that these distributions of pressure in FS and NS were similar. It meant that FS and NS had very similar microscopic structures.

Fig. 11
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Scaffolds’ wall shear stress distribution at viscosity 1.45 ×10−3 Pas and inlet velocity 100 µm/s

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Fig. 12
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Scaffolds’ wall pressure at viscosity of 1.45 × 10−3 Pas and inlet velocity of 1 µm/s, 10 µm/s, 100 µm/s and 1000 µm/s

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Fig. 13
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Scaffolds’ wall pressure at viscosity distribution 1.45 × 10−3 Pas and inlet velocity 100 µm/s

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Fig. 14
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Scaffolds’ wall pressure visualization distribution at viscosity 1.45 × 10−3 Pas and inlet velocity 100 µm/s

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4 Discussion

The DS, FS and NS were reconstructed in this study. These 3 samples were simulated in perfusion conditions. The distributions of fluid flow velocity, wall shear stress and pressure were obtained and analyzed. The dog femora scaffold (NS) was regarded as a reference criterion, and the other two samples were used for comparison.

The structure of DS was quite regular with homogeneous microscopic characteristics, which resulted in gradient distribution of fluid flow velocity, wall shear stress and pressure[20, 21]. The distribution of fluid flow velocity, wall shear stress and pressure in FS was multifarious resulting from the in homogeneous microstructure, which was close to that of NS and compatibly used instead of natural bone. At the same initial condition, the fluid flow velocity and wall pressure in FS were a little more than those in NS possibly due to the bad guidance quality of the microstructure.

The maximal and average wall shear stresses in FS were very different from those in NS, which may be a result of too many locally small pores and poor connectivity associated with the freeze-drying process. Compared to FS, the maximal and average wall shear stresses in DS were closer to those in NS. However, the single structure was unfavorable for the cultivation of cells[22]. From Fig. 7, with the inlet velocity increased (from 1 µm/s to 1000 µm/s), the maximal and average wall shear stresses in NS linearly increased, which indicated that the favorable microstructure retained the complexity of the formation and the stable flow field environment.

These models consisted of tetrahedral element meshes as irregular microstructures. Seven boundary layers (the first minimum layer thickness was 0.002 mm) based on a growth ratio of 1.2 on the surface of the scaffold models which cell mainly perch were made to capture the distribution of fluid flow velocity and wall shear stress. When the size of bone defects is 3 mm, the body finds it difficult to repair itself. We chose a model where one dimension was 3 mm, which can be more powerful for obtaining accurate results. As the computer resource was limited, 0.05 mm elements were decided upon and nearly 1 000 000 elements were created.

In this study, biological conditions were not included in these models. Cell migration, proliferation and differentiation were not simulated. Because of the irregular microstructure of the models, each region in the pores will have a different combination of fluid flow velocity, wall shear stress and pressure. Cells migrate into the scaffolds by means of the interstitial fluid flow. They do not reach the pores where the fluid flow does not penetrate. Abrupt changes of fluid flow velocity, wall shear stress and pressure will be perceived by cells during the migration process. Once cells are seeded in the scaffold, they can attach to specific areas of a pore, according to the mechanical stimuli of each point. However, as cells tend to differ, they synthesize an extracellular matrix and the number of cells increases, which lead to an increase in the fluid viscosity[23]. In this study, the analyzed fluid viscosity was 2.1 × 103 Pas.

A linear relationship between wall shear stress and imposed flow was found as a result of the very low Reynolds numbers causing the convective terms in the Navier-Stokes equations to be negligible[24]. A Newtonian fluid in the steady state was simulated in this study. None penetration wall model and a gravity condition were set. In this study, we reconstructed the nature of the real bone scaffold model, carried out the flow field analysis, set up 4 levels of inlet velocity, made a comprehensive analysis and compared these 3 models. From Fig. 9, considering the distribution of the wall shear stress for the conditions of 100 m/s inlet velocity and 1.45 × 10−3 Pas viscosity, the wall shear stress was within 0.015 4 Pa in DS, 0.098 5 Pa in FS and 0.023 6 Pa in NS, which were in good agreement with relevant literature[25]. The highest cell viability and proliferation were obtained at an average wall shear stress of 0.05 mPas. An upregulation of osteo-related gene messenger ribouncleic acid (mRNA) was observed as the fluid flow velocity increased from 0.005 to 0.1 mm/s leading to a wall shear stress of 5 mPas, detrimental to cell viability. These results demonstrated that cells can be stimulated by an average wall shear stress of around 1 mPas[23, 26]. From Table 3, at an inlet velocity of 100 µm/s, for 3 values of viscosity, the average wall shear stress in NS was 1.167 mPas, 1.629 mPas and 2.434 mPas, which was favorable for cell viability, while the corresponding values were 2.493 mPas, 3.223 mPas and 5.099 mPas in DS and 4.426 mPas, 5.494 mPas and 8.601 mPas in FS. The inlet velocity in DS and FS needed to be reduced to obtain an average wall shear stress of around 1 mPas at the expense of cell seeding efficiency[27].

5 Conclusions

We have transformed one 3D designed geometry and 2 geometries reversed from micro-CT to meshed models suitable for CFD simulations. This study provided detailed information about mechanical stimuli within the internal walls of these 3 biomaterial scaffolds with different morphologies. Fluid flow velocity, wall shear stress and pressure were determined for different conditions within the scaffolds according to the architecture of the samples suggesting that cells would be exposed to different stimulations. The dog femora scaffold as a reference standard, the designed scaffold and the porous and biodegradable β-TCP based on freeze-drying scaffold were evaluated in terms of microstructure and fluid simulation results. Actually, cell migration, proliferation and differentiation gradually changed the morphology of the scaffold and viscosity of the medium, which could be taken into account in future works. A dynamic perfusion system could be made and future experiments would be necessary. This methodology allowed us to obtain more insight into the complex concept of tissue engineering and will likely help in understanding and eventually improving scaffold design methods.