Investigation of mechanism of bone regeneration in a porous biodegradable calcium phosphate (CaP) scaffold by a combination of a multi-scale agent-based model and experimental optimization/validation

Abstract

Herein, we have developed a novel approach to investigate the mechanism of bone regeneration in a porous biodegradable calcium phosphate (CaP) scaffold by a combination of a multi-scale agent-based model, experimental optimization of key parameters and experimental data validation of the predictive power of the model. The advantages of this study are that the impact of mechanical stimulation on bone regeneration in a porous biodegradable CaP scaffold is considered, experimental design is used to investigate the optimal combination of growth factors loaded on the porous biodegradable CaP scaffold to promote bone regeneration and the training, testing and analysis of the model are carried out by using experimental data, a data-mining algorithm and related sensitivity analysis. The results reveal that mechanical stimulation has a great impact on bone regeneration in a porous biodegradable CaP scaffold and the optimal combination of growth factors that are encapsulated in nanospheres and loaded into porous biodegradable CaP scaffolds layer-by-layer can effectively promote bone regeneration. Furthermore, the model is robust and able to predict the development of bone regeneration under specified conditions.

1. Introduction

Bone reconstruction is a complex physiological process that exhibits nonlinear characteristics under different material and boundary conditions[2, 3]. The interaction between the external environment and internal factors makes the dynamic balance between osteoclasts and osteoblasts affect the formation of bone[4]. Recently, bone reconstruction has grown in significance in clinical studies due to its importance in human health. Nonetheless, because the microenvironment of bone growth is very complicated and involves various biological and physical processes, it is costly to perform in vivo/in vitro bone reconstruction experiments to investigate the bone formation process in detail. For this reason, this study proposes an in silico mathematics model that can not only simulate the bone reformation process within a porous biodegradable CaP scaffold under mechanical stimulation with a related experimental design, but can also predict the change in bone mass with respect to various growth factors that are encapsulated in nanospheres and loaded into porous biodegradable CaP scaffolds layer-by-layer after we optimize the key parameters and validate the predictive power of the model by experimental training and testing data.

Since the beginning of the 21st century, many biomaterials scientists have carried out many studies of bone reconstruction in different research areas such as the investigation of the relationship between osteoclasts and osteoblasts[4], osteogenic differentiation[5], mechanical stimulation of bone [6], cell mechanics[2], signalling pathways of bone cells [7, 8] and bone growth factors[9]. For example, Sun et al.[1] developed a multi-scale mathematical model, which not only reconstructed the 3D bone regeneration system and examined the effects of pore size and porosity on bone formation, but also studied the impact of growth factors on the change in bone mass. However, it did not take mechanical stimulation into consideration or use experimental design to investigate the priority of impact of bone mass-related growth factors. In addition, Checa et al.[10] developed a lattice-based mechanical platform to model tissue differentiation under conditions of angiogenesis, but it neither investigated the impact of the key growth factors on bone formation nor verified the predictive power of the model via experimental data. Recently, Sanz-Herrera et al.[11] constructed a mathematical model of bone tissue regeneration that can investigate a set of physiological processes associated with bone cells, such as porosity, mechanical properties and permeability, but it did not perform parameter analysis or validate the robustness of the model.

For the first time, this study has integrated mechanical stimulation, a data-mining algorithm and experimental design into our established 3D multi-scale model of bone reconstruction [1, 12]. Therefore, it can not only predict changes in bone mass under stimulation by different growth factors via experimental design, but also investigate the impact of mechanical stimulation on bone reformation. In particular, this study employed the currently widely used calcium phosphate (CaP) scaffolds, which are ideal materials for bone repair owing to their biocompatibility, adjustable degradation rates, and excellent bioactivity. The results demonstrate that not only can mechanical stimulation significantly promote the growth of active osteoblasts (OBa), pre-osteoblasts (OBp) and mesenchymal stem cells (MSC), but the model also has good predictive precision and robustness for the prediction of bone regeneration and reveals the priority of the impact of growth factors on bone regeneration via experimental design.

2. Methods

This study developed a 3D multi-scale agent-based model of bone regeneration with four biological and physical scales, namely, molecular, cellular, scaffold, and bone tissue scales (Fig. 1). It not only employed a set of reaction-diffusion equations to describe the diffusion of growth factors released from the porous biodegradable CaP scaffold and the transportation of nutrients by the vasculature, but also used an agent-based module[13] to simulate the activities of bone cells under mechanical stimulation and the related signaling transduction pathway. Here, the biodegradable CaP scaffolds contained both human cells (MSC, osteocytes and endothelial cells) and growth factors for bone tissue repair.

Figure 1.

Schematic of the computational flow

Fig. 1 shows that stimulated by the growth factors diffused in the scaffold scale and regulated by the nutrient transported in the bone tissue scale, the signaling pathway in the molecular scale determines cell’s phenotype switching on the cellular scale. On the contrary, cell’s phenotype switch can remodel and impact the scaffold environment and the diffusion of the growth factors on the scaffold and bone tissue scale, respectively.

2.1 Molecular scale: the signaling pathway

Runx2 and Osterix (Osx) are two crucial transcription factors in osteoblast differentiation and bone regeneration (refer to Fig. 1 in Sun et al.[12]). As reported by previous studies[4], their expression can be regulated by the release of growth factors (BMP2 and Wnt) via the activation of intracellular proteins or related transcription factors such as Smad1/5 (S1), Smad2/3 (S2) and β-catenin[14].

The molecular regulatory mechanisms involved in the intracellular signalling pathway were modelled using a system of ordinary differential equations such as Eq. 1, as detailed in our previous research[12]. Values of the key parameters of Eq. 1 are listed in [1].

$$\frac{d[S1]}{dt}=\frac{V_1·[\mathit{BMP}2]}{K_1·[\mathit{BMP}2]}·([\mathit{TotalS}1]-[S1])-d_1·[S1]$$ (1.1)

The change in concentration of phosphorylated Smad1/5 ([S1]) depends on the concentrations of BMP2 ([BMP2]) and unphosphorylated Smad1/5 ([TotalS1] − [S1]) and its dephosphorylation.

$$\frac{d[\beta \mathit{Catenin}]}{dt}=a-((\frac{[\mathit{Wnt}]+b}{c·[\mathit{Wnt}]+d})·(\frac{e}{e+[\beta \mathit{Catenin}]}+f)·[\beta \mathit{Catenin}])$$ (1.2)

where the concentrations of Wnt and β-catenin are represented by [Wnt]and [βCatenin], respectively.

$$\frac{d[\mathit{Runx}2]}{dt}=\frac{V_1·[S1]}{K_1+[S1]}+\frac{V_5·[S2]}{K_5+[S2]}+\frac{V_6·[\beta \mathit{Catenin}]}{K_6+[\beta \mathit{Catenin}]}-d_1·[\mathit{Runx}2]$$ (1.3)

Runx2[15] has been found to be a crucial transcription factor in osteogenic differentiation. Its expression and activation can be promoted by BMP2 via Smad1/5 and Smad2/3 ([S2]) and by Wnt ligands via β-catenin and other proteins. The concentration of Runx2 is represented by [Runx2].

$$\frac{d[\mathit{Osx}]}{dt}=\frac{V_7·[S1]}{K_7+[S1]}+\frac{V_8·[\mathit{Runx}2]}{K_8+[\mathit{Runx}2]}-d_5·[\mathit{Osx}]$$ (1.4)

Osx[16] is also a critical transcription factor in osteoblast differentiation, acting downstream of Runx2 and Smad1/5.

Here, the intracellular signaling pathway described by Eq. 1 is stimulated by the concentrations of the growth factors (BMP2, Wnt, Runx2 and Osx) released on the scaffold scale to determine the phenotype switch of the cell on the cellular scale.

2.2 Cellular scale: cell activities

2.2.1 Migration

MSC and OBp migrate along the gradient of the normalized concentration of growth factors, including bone morphogenetic protein 2 (BMP2), Wnt, and nutrients. The probability ( $P_i^{\mathit{mig}}$) that MSC and OBp will migrate along the i^th direction is given by Eq. 2. OBa is assumed not to migrate in this study, with reference to a previous report[17].

$$P_j^{\mathit{mig}}\propto (\nabla G_i+\nabla O_2){·\mathit{1}}_j,i=1,2,3;j=1,2,\dots ,6$$ (2)

where G_i and O₂ are the concentrations of growth factors (BMP2 and Wnt ligands) and oxygen, respectively, and l_j is the directional vector along the j^th direction. There are six directions in which a cell can migrate in our three-dimensional lattice.

2.2.2 Differentiation

Activated Runx2 and Osx play different roles in different stages of an osteoblastic lineage. Both Runx2 and Osx can promote the differentiation of MSC into OBp, whereas Runx2 can inhibit the differentiation of OBp into OBa[14, 18]. The probabilities that MSC will differentiate into OBp ( $P_{\mathit{MSC}\to \mathit{OBp}}^{\mathit{diff}}$) and OBp will differentiate into OBa ( $P_{\mathit{OBp}\to \mathit{OBa}}^{\mathit{diff}}$) are related to the levels of expression of activated Runx2 and Osx. We employ Hill functions[4] to model these situations by Eq. 3 and 4, as in previous studies[11].

$$P_{\mathit{MSC}\to \mathit{OBp}}^{\mathit{diff}}=(\frac{V_{D1,\mathit{Runx}2}·[\mathit{Runx}2]}{K_{D1,\mathit{Runx}2}+[\mathit{Runx}2]}+\frac{V_{D1,\mathit{Osx}·[\mathit{Osx}]}}{K_{D1,\mathit{Osx}}+[\mathit{Osx}]})·P_{\mathit{MSC}\to \mathit{OBp}}^{\mathit{diff}}$$ (3)

$$P_{\mathit{OBp}\to \mathit{OBa}}^{\mathit{diff}}=(\frac{1}{1+[\mathit{Runx}2]/K_{D2,\mathit{Runx}2}}+\frac{V_{D2,\mathit{Osx}}·[\mathit{Osx}]}{K_{D2,\mathit{Osx}}+[\mathit{Osx}]})·p_{\mathit{OBp}\to \mathit{OBa}}^{\mathit{diff}}$$ (4)

where $P_{\mathit{MSC}\to \mathit{OBp}}^{\mathit{diff}}$ and $P_{\mathit{OBp}\to \mathit{OBa}}^{\mathit{diff}}$ are the baseline probabilities that MSC will differentiate into OBp and OBp into OBa, respectively. Because both the concentration of oxygen and the strain[19] play major roles in the differentiation of MSC, we model this situation by the following rules (Fig. 2).

Figure 2.

(a) Workflow of the computational algorithm (b) Finite element analysis model

1. If (0.1 < S < 0.53) and O₂ is greater than Thr^O2, then mature MSC differentiate towards OBa.

2. If (0.53 < S < 1) and O₂ is greater than Thr^O2, then mature MSC differentiation towards OBp.

3. Otherwise, the direction of differentiation of MSC depends on Eq. 3 and 4, where Thr^O2 is the threshold of oxygen (O₂) concentration.

2.2.3 Proliferation

MSC, OBp and OBa can proliferate with different probabilities (p^pro). With reference to our previous research[20], we set these as constants. Here, Table 1 lists the values of the important parameters for the model.

Table 1.

Values of the important parameters for the model

Symbol	Value	Unit	Description	Reference
$p_{\mathit{MSC}\to \mathit{OBp}}^{\mathit{diff}}$	0.3	Day−1	Differentiation rate of MSC to OBp	[6, 10, 21]
$p_{\mathit{OBp}\to \mathit{OBa}}^{\mathit{diff}}$	0.3	Day−1	Differentiation rate of OBp to OBa	[6, 10, 21]
KD 1, Runx 2	500	[−]	Regulation coefficient 1 regarding Runx2	[12]
KD 1, Osx	4000	[−]	Regulation coefficient 1 regarding Osx	[12]
VD 1, Runx 2	8000	[−]	Hill regulatory factor 1 regarding Runx2	[12]
VD 1, Osx	4 × 104	[−]	Hill regulatory factor 1 regarding Osx	[12]
KD2, Runx2	0.8	[−]	Regulation coefficient 2 regarding Runx2	[12]
KD 2, Osx	100	[−]	Regulation coefficient 2 regarding Osx	[12]
ppro	0.6	Day−1	Proliferation rate of MSC and OBp	[6, 10, 21]
$p_{\mathit{MSC}}^{\mathit{apop}}$	0.05	Day−1	Apoptosis rate of MSC	[6, 10, 21]
$p_{\mathit{OBp}}^{\mathit{apop}}$	0.10	Day−1	Apoptosis rate of OBp	[6, 10, 21]
$p_{\mathit{OBa}}^{\mathit{apop}}$	0.16	Day−1	Apoptosis rate of OBa	[6, 10, 21]
ϕ	125	[−]	Positive constant	Estimated
thoxygen	0.0015	[−]	Threshold of cell apoptosis	Estimated
DC	6.7 × 10−8	cm2 s−1	Diffusivity of water	[22]
kC	0.15	[−]	Degradation rate of water	[23]
kM	0.15	[−]	Degradation rate of CaP molecular weight	[23]
D G i	1.55 × 10−5	cm2 s−1	Diffusivity of growth factor in water	[24]
r G i	0.2	[−]	Growth factor release constant	Estimated
u G i	10	ng/106 cells/day	Depletion rate of cytokine by osteoblastic cells	[25]
d G i	0.03	Day−1	Degradation rate of cytokine	[25]
KC	0.5	[−]	Michaelis constant	Estimated
α	2600	cm2s−1M−1	Chemotactic coefficient	[20]
kV	1.67 ×10−10	[−]	Positive constant controlling chemotactic sensitivity	[20]
DN	8 × 10−5	cm2s−1	Diffusivity of oxygen	[20]
qN	0.5	[−]	Vessel permeability for oxygen	[20]
Nblood	0.0025	[−]	Blood oxygen concentration	[20]
uN	6.25 ×10−4	[−]	Cell uptake rate of oxygen	[20]

2.2.4 Apoptosis

Because hyperbaric oxygen will attenuate cell apoptosis[26], Eq. 5 has been developed to describe the relation between the apoptosis rate of cells and the oxygen concentration. It is noted that if the oxygen concentration is less than the threshold (Th_oxygen), osteoblast cells will die.

$$P^{\mathit{apop}}=p_b^{\mathit{apop}}+\varphi (O_2^{\mathit{average}}-O_2)$$ (5)

where $p_b^{\mathit{apop}}$ and P^apop represent the baseline probability and probability of apoptosis of MSC, OBp and OBa, respectively, $O_2^{\mathit{average}}$ represents the normal oxygen concentration, an φ is a positive constant[11].

2.3 Scaffold scale: scaffold degradation and growth factor release

In the same way as in our previous research[12], growth factors are encapsulated in nanospheres and loaded into porous biodegradable CaP scaffolds layer-by-layer. After being implanted into defected bone, calcium phosphate can be degraded via hydration reactions and network breakage. The diffusion of the extracellular liquid and disintegration of calcium phosphate are described by Eq. 6 and 7, respectively:

$$\frac{\partial C}{\partial t}=D_C{\nabla }^{2}C-k_{C}CM$$ (6)

$$\frac{\partial M}{\partial t}=-k_{M}CM$$ (7)

where C and M are the water concentration and molecular weight of calcium phosphate, respectively, D_C is the diffusivity of water, and k_C and k_M are the degradation rates for water and calcium phosphate, respectively.

BMP2, Wnt ligands and VEGF are released from the degrading CaP scaffold and continuously diffuse within the scaffold pores. The paracrine and autocrine activity of these cytokines are ignored because we assume that the concentrations of these cytokines secreted by individual cells are quite low compared with those released from the CaP scaffold. These processes are modeled by Eq. 8 [1]:

$$\frac{\partial G_i}{\partial t}=D_{G_i}{\nabla }^2{G}_i+{\chi }_{\mathit{scaffold}}{r}_{G_i}(G_{i,max}-G_i)\frac{C}{C+K_c}-{\chi }_{\mathit{osteo}}{u}_{G_i}{G}_i-d_{G_i}{G}_i$$ (8)

where D_Gi is the diffusivity of each growth factor, G_i,max is the maximum concentration of each growth factor initially loaded into the scaffold, r_Gi is the release constant, u_Gi is the depletion rate of the cytokine, d_Gi is the degradation rate and K_c is the Michaelis constant. The time-dependent characteristic function χ_scaffold(t, x) is equal to 1 in the calcium phosphate matrix and 0 in the pores of the scaffold. The value of χ_osteo(t, x) is equal to 1 if an osteoblastic cell is present at x; otherwise, it is equal to 0. Both χ_scaffold and χ_osteo are updated at each simulation step.

2.4 Bone tissue scale: angiogenesis and oxygen transportation

2.4.1 Angiogenesis

We assume that the motion of an individual endothelial cell located at the tip of a capillary sprout governs the motion of the entire sprout, and chemotaxis in response to VEGF gradients guides the motion of the endothelial cells at the tip of the capillary sprout. Eq. 9 defines the probability of migration of endothelial cells:

$$P_j\propto \alpha \frac{k_V}{k_V+v}\nabla V·l_j,j=1,2,\cdots ,6$$ (9)

where V is the concentration of VEGF, α is the chemotactic coefficient and K_V is a positive constant[27].

2.4.2 Nutrient transportation

Oxygen can be transported by the neovasculature to osteoblasts within scaffold pores, which is described by Eq. 10.

$$\frac{\partial O_2}{\partial t}=D_{O_2}{\nabla }^2{O}_2+{\chi }_{\mathit{ves}}(t,x)q_{O_2}(O_2^{\mathit{blood}}-O_2)-{\chi }_{\mathit{osteo}}(t,x)u_{O_2}{O}_2$$ (10)

where D_O2 is the diffusivity of oxygen, q_O2 is the permeability of the vessel for oxygen, $O_2^{\mathit{blood}}$ is the blood oxygen concentration, and u_O2 is the oxygen uptake rate of the cell.

2.5 Establish computational models by Abaqus[4]

Because our research aims to investigate the mechanism of bone regeneration in a porous biodegradable calcium phosphate (CaP) scaffold[1], we have to consider the impacts of nutrient transportation, growth factors, angiogenesis, and mechanical stimulation together from the biological perspective of the system. In particular, because biodegradable material is used to repair the bone tissue scaffold, it indirectly affects mechanical stimulation via regulating the release of cytokines owing to its material properties. However, because mechanical stimulation is sufficiently complicated, this study integrates a well-established mechanical stimulation[11] module into the model to study the mechanism of bone regeneration as follows.

The time course of tissue formation and vascularization of the scaffold is modeled as an iterative process (Fig. 2a). Fig. 2b shows how to compute the number of cells in each sub-cube at each run. The densities of mature bone consisting of OBp (Density_p) and immature bone consisting of OBa (Density_a) are described by Eq. 11.1 and Eq. 11.2, respectively:

$${\mathit{Density}}_p=N_{\mathit{OBp}}/V_{sc}$$ (11.1)

$${\mathit{Density}}_a=N_{\mathit{OBa}}/V_{sc}$$ (11.2)

where N_OBp and N_OBa represent the number of OBp and OBa, respectively, and V_sc is the volume of the sub-cube.

The computation of stress for each small cube (Fig. 2b) is illustrated as follows.

1. For each sub-cube, if Density_p is greater than Density_a, then the material type of the sub-cube is set to immature bone; otherwise, it is set to mature bone.

2. Eq. 12.1 and Eq. 12.2 are employed to compute the Young’s modulus[4] of immature and mature bone, respectively:

   $$E=\mathit{Density}\times S_p$$ (12.1)

   $$E=\mathit{Density}\times S_a$$ (12.2)

   where S_p and S_a represent the speed of propagation of sound in immature and mature bone, respectively.

3. The stress (S) for each sub-cube can be computed by Eq. 13.

   $$S=E\times D$$ (13)

   where D is the strain of the sub-cube[4].

2.6 Model training and testing

To make the results of the model believable, this study optimized the key parameters and estimated the predictive capacity of the model by using experimental data (Table 2) and a data-mining algorithm. The experimental data were obtained on the designated days after treating MSCs with BMP2 on day 1 followed by Wnt on day 4. Alkaline phosphatase (ALP) and double-stranded DNA (dsDNA) are two critical biomarkers for quantifying the early differentiation rate of MSC towards OBp and the total cell number, respectively.

Table 2.

ALP and dsDNA data

	ALP		dsDNA
Day	Mean	SD	Mean	SD
1	8.643728	0.499756	0.138832	0.005635
3	160.7598	7.218334	0.348147	0.017524
7	74.13583	6.928568	0.361439	0.031959
10	73.31991	10.62254	0.491355	0.067455

A particle swarm optimization (PSO) algorithm[28] was used to train two key parameters (Thr^O2 and V_i) of the model by fitting the simulated data against the experimental data using Eq. 14. Thr^O2 and V_i represent the oxygen concentration and the speed of propagation of sound in the material, respectively:

$${\mathrm{\Theta }}^{\ast }=\text{argmin}\sum _{i=1}^N\sum _{j=1}^T\omega {(x_i^{\mathit{sim}}(t_j,\mathrm{\Theta })-x_i^{exp}(t_j))}^2$$ (14)

where ${\omega }_i=1/{(\underset{j}{max}{x}_i^{exp}(t_j))}^2$, N and T denote the number of cells and time points, respectively, and $x_i^{\mathit{sim}}(t_j,\mathrm{\Theta })$ and $x_i^{exp}(t_j)$ represent the number of cells at each time step of the simulation and experiment, respectively.

Eq. 15 was used to compute the average relative error (ARE)[17, 29, 30] for testing the predictive precision of the model:

$$AR\mathrm{E}=\sum _{i=1}^R\frac{\mid \mathit{Est}{\theta }_i-L\theta \mid }{R\times \mid L\theta \mid }\times 100\%$$ (15)

where R represents the number of repetitions for the simulation and Estθ_i and Lθ represent the simulated and experimental results, respectively.

3. Results

This section consists of the following three major results. Firstly, we investigate the effects of mechanical stimulation, angiogenesis and bone formation in detail after the 3D bone regeneration system within a porous biodegradable CaP scaffold is developed. Secondly, we compare the changes in bone mass between the presence and absence of mechanical stimulation. Thirdly, we employ training data to optimize the key parameters of the model and use testing data to validate the predictive power of the model, as well as analyzing the robustness of the model by sensitivity analysis.

3.1 Simulation of 3D vascularized bone regeneration

Based on our previous research[1], we reconstruct the system of vascularized bone regeneration within the 3D porous CaP scaffold, which consists of the coupled processes of the evolution of scaffold degradation (Eq. 6 and 7), exogenous growth factor release (Eq. 8), angiogenesis (Eq. 9), differentiation of MSCs (Eq. 3 and 4), and cell growth within the scaffold pores over time.

This research incorporates mechanical stimulation into the well-developed model of 3D bone regeneration[1] within a porous biodegradable CaP scaffold to investigate the impact of mechanical stimulation on bone regeneration. More importantly, because the dynamics of the differentiation of MSCs will cause changes in OBs as described by Eq. 3, it is directly related to the bone mass. Fig. 3 shows a 2/3D simulation of the bone regeneration process under mechanical stimulation. On day 10, blood vessels were rare and scattered on the surface of the porous CaP scaffold (Fig. 3a and d); on day 20, newly formed blood vessels grew into the pores located at the periphery of the scaffold and several blood vessels started branching to form a tree (Fig. 3b and e). On day 28, a branched vasculature was observed within the peripheral pores of the scaffold and a few blood vessels were already growing into the pores at the center of the scaffold (Fig. 3c and f).

Figure 3.

3D bone formation and 2D slices of bone formation over time. (a), (b) and (c) show the 3D model under mechanical stimulation on days 10, 20 and 28, respectively. (d), (e) and (f) show 2D slices of bone formation on days 10, 20 and 28, respectively. The porous CaP scaffold, MSC, OBa, angiogenesis and pore structure are colored cyan, yellow, blue, red and white, respectively.

Fig. 4 shows the dynamics of the various cells over time in the presence and absence of mechanical stimulation within a porous biodegradable CaP scaffold. The changes in the trend in the cell numbers for MSC, OBa and OBp are similar regardless of the presence or absence of mechanical stimulation, but Fig. 4a, b and c indicate that mechanical stimulation will significantly increase the cell numbers for these three types of cell after 28 days, especially for OBa. The numbers of MSC, OBa and OBp increased in the beginning and decreased after reaching peaks at different times. The number of MSC peaked around day 3 (Fig. 4a), that of OBa peaked around day 7 (Fig. 4b), and that of OBp peaked around day 14 (Fig. 4c).

Figure 4.

Dynamics of MSC (a), OBp (b) and OBa (c) over 28 days.

3.2 Training and testing of the model

The predictive power of the model was verified by fitting the simulated results against the experimental results. A PSO algorithm [31] was employed to train the model by setting the initial values of the key parameters and the training data. The refinement step was repeated five times to obtain optimal and stable results (Fig. 5), which show a similarity between the simulated and experimental results after the model training process. We define the important variables and equations for the testing and training of the model in Table 3.1, and list the related results in Table 3.2.

Figure 5.

Results of training (a) and testing (b)

Table 3.

(1) Definitions of the parameters for model testing; (2) comparison between experimental and simulated results

(1)
Parameter definition	Description
${CS}_1^{\mathit{MSC}\to \mathit{OBp}}=\frac{{SN}_3^{\mathit{MSC}\to \mathit{OBp}}-{SN}_1^{\mathit{MSC}\to \mathit{OBp}}}{{SN}_1^{\mathit{MSC}\to \mathit{OBp}}}$	${CS}_1^{\mathit{MSC}\to \mathit{OBp}}$ is the rate of change in the differentiation of MSC to OBp during days 1 to 3 in the simulation.
${CS}_2^{\mathit{MSC}\to \mathit{OBp}}=\frac{{SN}_7^{\mathit{MSC}\to \mathit{OBp}}-{SN}_3^{\mathit{MSC}\to \mathit{OBp}}}{{SN}_3^{\mathit{MSC}\to \mathit{OBp}}}$	${CS}_2^{\mathit{MSC}\to \mathit{OBp}}$ is the rate of change in the differentiation of MSC to OBp during days 3 to 7 in the simulation.
${CS}_3^{\mathit{MSC}\to \mathit{OBp}}=\frac{{SN}_{10}^{\mathit{MSC}\to \mathit{OBp}}-{SN}_7^{\mathit{MSC}\to \mathit{OBp}}}{{SN}_7^{\mathit{MSC}\to \mathit{OBp}}}$	${CS}_3^{\mathit{MSC}\to \mathit{OBp}}$ is the rate of change in the differentiation of MSC to OBp during days 7 to 10 in the simulation.
${CE}_1^{\mathit{MSC}\to \mathit{OBp}}=\frac{{EN}_3^{\mathit{MSC}\to \mathit{OBp}}-{EN}_1^{\mathit{MSC}\to \mathit{OBp}}}{{EN}_1^{\mathit{MSC}\to \mathit{OBp}}}$	${CE}_1^{\mathit{MSC}\to \mathit{OBp}}$ is the rate of change in the differentiation of MSC to OBp during days 1 to 3 in the experiment.
${CE}_2^{\mathit{MSC}\to \mathit{OBp}}=\frac{{EN}_7^{\mathit{MSC}\to \mathit{OBp}}-{EN}_3^{\mathit{MSC}\to \mathit{OBp}}}{{EN}_3^{\mathit{MSC}\to \mathit{OBp}}}$	${CE}_2^{\mathit{MSC}\to \mathit{OBp}}$ is the rate of change in the differentiation of MSC to OBp during days 3 to 7 in the experiment.
${CE}_3^{\mathit{MSC}\to \mathit{OBp}}=\frac{{EN}_{10}^{\mathit{MSC}\to \mathit{OBp}}-{EN}_7^{\mathit{MSC}\to \mathit{OBp}}}{{EN}_7^{\mathit{MSC}\to \mathit{OBp}}}$	${CE}_3^{\mathit{MSC}\to \mathit{OBp}}$ is the rate of change in the differentiation of MSC to OBp during days 7 to 10 in the experiment.
${CS}_1^{\mathit{Total}}=\frac{{SN}_3^{\mathit{Total}}-{SN}_1^{\mathit{Total}}}{{SN}_1^{\mathit{Total}}}$	${CS}_1^{\mathit{Total}}$ is the rate of change in the total cell number during days 1 to 3 in the simulation.
${CS}_2^{\mathit{Total}}=\frac{{SN}_7^{\mathit{Total}}-{SN}_3^{\mathit{Total}}}{{SN}_3^{\mathit{Total}}}$	${CS}_2^{\mathit{Total}}$ is the rate of change in the total cell number during days 3 to 7 in the simulation.
${CS}_3^{\mathit{Total}}=\frac{{SN}_{10}^{\mathit{Total}}-{SN}_7^{\mathit{Total}}}{{SN}_7^{\mathit{Total}}}$	${CS}_3^{\mathit{Total}}$ is the rate of change in the total cell number during days 7 to 10 in the simulation.
${CE}_1^{\mathit{Total}}=\frac{{EN}_3^{\mathit{Total}}-{EN}_1^{\mathit{Total}}}{{EN}_1^{\mathit{Total}}}$	${CE}_1^{\mathit{Total}}$ is the rate of change in the total cell number during days 1 to 3 in the experiment.
${CE}_2^{\mathit{Total}}=\frac{{EN}_7^{\mathit{Total}}-{EN}_3^{\mathit{Total}}}{{EN}_3^{\mathit{Total}}}$	${CE}_2^{\mathit{Total}}$ is the rate of change in the total cell number during days 3 to 7 in the experiment.
${CE}_3^{\mathit{Total}}=\frac{{EN}_{10}^{\mathit{Total}}-{EN}_7^{\mathit{Total}}}{{EN}_7^{\mathit{Total}}}$	${CE}_3^{\mathit{Total}}$ is the rate of change in the total cell number during days 7 to 10 in the experiment.

(2)
Differentiation of MSC to OBp				Total number of cells
${CS}_1^{\mathit{MSC}\to \mathit{OBp}}$	14.250	${CE}_1^{\mathit{MSC}\to \mathit{OBp}}$	18.598433	${CS}_1^{\mathit{Total}}$	1.595570	${CE}_1^{\mathit{Total}}$	2.507686
${CS}_2^{\mathit{MSC}\to \mathit{OBp}}$	2.202786	${CE}_2^{\mathit{MSC}\to \mathit{OBp}}$	0.467379	${CS}_2^{\mathit{Total}}$	1.797395	${CE}_2^{\mathit{Total}}$	1.038179
${CS}_3^{\mathit{MSC}\to \mathit{OBp}}$	0.498009	${CE}_3^{\mathit{MSC}\to \mathit{OBp}}$	0.9758315	${CS}_3^{\mathit{Total}}$	1.057147	${CE}_3^{\mathit{Total}}$	1.359441

In the training process, simulated data ( ${CS}_1^{\mathit{MSC}\to \mathit{OBp}}$ and ${CS}_2^{\mathit{MSC}\to \mathit{OBp}}$) were employed to optimize the key parameters of the model by fitting them against experimental data ( ${CE}_1^{\mathit{MSC}\to \mathit{OBp}}$ and ${CE}_2^{\mathit{MSC}\to \mathit{OBp}}$). In the testing process, a leave-one-out cross-validation (LOOCV) [32] algorithm was employed to validate the predictive power of the model by using three time points of the experimental data as the training data set ( ${CS}_2^{\mathit{MSC}\to \mathit{OBp}}$ and ${CS}_3^{\mathit{MSC}\to \mathit{OBp}}$) for the key parameters of the optimization of the model and the remaining time points of the experimental data as the testing data set ( ${CE}_2^{\mathit{MSC}\to \mathit{OBp}}$ and ${CE}_3^{\mathit{MSC}\to \mathit{OBp}}$) for computation of the accuracy of the model (Eq. 16). The small relative error (RE) indicates high predictive accuracy.

$$RE=\frac{{\chi }_i^{sin}(t_j,\theta )-{\chi }_i^{exp}(t_j)}{{\chi }_i^{exp}(t_j)}$$ (16)

As computed by Eq. 16, the average and standard deviation of the predictive power of the model are 62.21% and 0.019, respectively.

3.3 Parameter sensitivity analysis in our model

Sensitivity analysis [17] was employed to evaluate the impact of the parameters on the behavior of the model. The value of each parameter in Table 4.1 was varied independently over the specified range [17] about 10 times, while keeping the values of the other parameters the same as their baseline. Table 4.2 lists the Spearman rank-order correlation coefficient [33] and p value for each parameter. Also, Table 4.2 shows that the parameters V₆, K_6, d₁, S_p, and S_a are sensitive to OBp, the parameters V₆, d₁, V₈, d_s, S_p, and S_a are sensitive to OBa, and the parameters S_p and S_a are sensitive to MSC.

Table 4.

(1) Estimated key parameters of the model; (2) estimated key parameters of the intracellular pathway.

(1)
Symbol	Value	Unit	Description
V 1	1.9608	nM/hr	Maximum activation velocity of Smad1/5 by BMP2 [1]
K 1	33.7255	nM	Michaelis activation coefficient of Smad1/5 by BMP2 [1]
d 1	1.0000	hr−1	Decay rate of phosphorylated Smad1/5 [1]
V 2	40.0000	nM/hr	Maximum activation velocity of Smad2/3 by TGFβ [1]
K 2	10.0000	nM	Michaelis activation coefficient of Smad2/3 by TGFβ [9]
d 2	0.1961	hr−1	Decay rate of phosphorylated Smad2/3 [9]
V 4	0.8198	nM/hr	Maximum activation velocity of Runx2 by Smad1/5 [9]
K 4	93.4510	nM	Michaelis activation coefficient of Runx2 by Smad1/5 [9]
V 5	0.8786	nM/hr	Maximum activation velocity of Runx2 by Smad2/3 [9]
K 5	9.8039	nM	Michaelis activation coefficient of Runx2 by Smad2/3 [9]
V 6	9.8039	nM/hr	Maximum activation velocity of Runx2 by β-catenin [9]
K 6	590.5882	nM	Michaelis activation coefficient of Runx2 by β-catenin [12]
d 1	36.8634	hr−1	Degradation rate of Runx2 [12]
V 7	0.0519	nM/hr	Maximum activation velocity of Osx by Smad1/5 [12]
K 7	984.3137	nM	Michaelis activation coefficient of Osx by Smad1/5 [12]
V 8	0.0392	nM/hr	Maximum activation velocity of Osx by Runx2 [12]
K 8	139.2157	nM	Michaelis activation coefficient of Osx by Runx2 [12]
ds	0.0224	hr−1	Degradation rate of Osx [12]
Sp	450.0000	m/s	Speed of propagation of sound in immature bone [12]
Sa	720.0000	m/s	Speed of propagation of sound in mature bone [12]

(2)
Parameter	Mesenchymal stem cell		Pre-osteoblast		Active osteoblast
	Spearman ρ	p value	Spearman ρ	p value	Spearman ρ	p value
V 6	0.6512	0.7985	0.5201	5.2047×10−3	0.3719	1.5261×10−3
V 2	0.1839	2.4592	0.1459	3.6012	−0.3596	6.1427
K 6	0.4116	0.8583	0.5206	2.3935×10−2	0.1216	0.0003
K 2	−0.7195	3.0152	0.7633	4.9112	0.1769	3.5903
d 1	0.3552	0.7985	−0.0178	7.561×10−5	0.3819	4.7219×10−5
d 2	−0.1268	0.8916	0.3145	1.3927	0.1181	0.9265
V 8	0.9816	5.3071	0.9761	6.3275	0.8995	1.6231×10−6
ds	0.5545	2.7046	0.5103	1.091	0.6534	3.1659×10−4
Sp	0.8912	0.0152	0.4169	3.5916×10−3	0.6051	2.6719×10−3
Sa	0.9609	0.0429	0.9872	5.6201×10−3	0.9910	1.9602×10−3

3.4 Significance of effects of growth factors on bone formation

It is well known that using orthogonal experimental design [34] can reduce the number of experiments. Therefore, an orthogonal test table L9(34) [35, 36] was employed to investigate which single growth factor or combination of these growth factors (VEGF, BMP2 and Wnt) will most effectively promote bone formation. Therefore, three major growth factors (BMP2, VEGF and Wnt) were set as the experimental factors (columns 2, 3 and 4 in Table 5.1) for the experimental design with three levels: high (10), medium (5) and low (0), in accordance with the principles of experimental design [35, 36]. The cumulative number of OBa was set as the experimental test index [37] to represent the bone mass at the designated times (columns 5 and 6 in Table 5.1) in the experimental design, because the growth rate of bone mass is proportional to the number of OBa. Table 5.1 shows that the number of OBa under mechanical stimulation is greater than the number of OBa not under mechanical stimulation. Table 5.2 and Table 5.3 [35, 37] indicate that the optimal combination of experimental factors is always BMP2 (medium), VEGF (medium) and Wnt (high), and the order of priority of the growth factors (BMP2 > VEGF > Wnt) is statistically significant regardless of mechanical stimulation, respectively.

Table 5.

(1) Orthogonal test data; (2) optimal combination analysis; (3) analysis of variance

(1)
Factor	Wnt dose	BMP2 dose	VEGF dose	OBa number	OBa number
	A	B	C	Without mechanical stimulation	With mechanical stimulation
1	0	0	0	1791	2397
2	0	5	5	13714	15156
3	0	10	10	13679	14815
4	5	0	5	2535	2918
5	5	5	10	14156	16156
6	5	10	0	13310	14167
7	10	0	10	2486	2769
8	10	5	0	13131	13982
9	10	10	5	13773	14107

(2)
K-value	Without mechanical stimulation			With mechanical stimulation
K1	29184	6812	28232	32368	8084	30546
K2	30001	41001	30022	33241	45294	32181
K3	28390	40762	30321	30858	43089	33740
k1 = (K1/3)	9728	2270.6	9410.7	10789.3	2694.7	10182
k2 = (K2/3)	10000.3	13667	10007.3	11080.3	15098	10727
k3 = (K3/3)	9463.3	13596.3	10107	10286	14363	11246.7
Range	537	11396.4	2089	796.3	11403.3	1064.7
Optimal solution	A2	B2	C3	A2	B2	C3

4. Discussion

This study considers the impact of angiogenesis on bone growth [26]. Fig. 3a, b and c demonstrate that angiogenesis can promote the accumulation of VEGF in the center of the porous biodegradable CaP scaffold rather than on the surface, whereas new blood vessels will start growing on the surface of the porous biodegradable CaP scaffold rather than in the center because of inhibition by the wall of the porous biodegradable CaP scaffold. These phenomena imply that blood vessels will develop much faster in the loose structure of the scaffold rather than in the dense structure, and also that a more uniform distribution of VEGF in the scaffold will significantly promote the growth of blood vessels. Moreover, Fig. 3d shows that the initial porous biodegradable CaP scaffold has a pore structure, whereas Fig. 3e and f show that the porous biodegradable CaP scaffold moves into the center of the bone over time. This phenomenon demonstrates that the scaffold made from calcium phosphate is hydrolyzed during bone regeneration, as was expected.

As reported by previous research [1], because mechanical stimulation will result in the differentiation of more MSC to OBa and OBp, it will increase the bone mass density and finally change the material properties. Here, Fig. 4 indicates that mechanical stimulation plays a role in bone regeneration, because the numbers of the three kinds of cells (MSC, OBp and OBa) under mechanical stimulation are significantly greater than the number not under mechanical stimulation, while they display almost the same dynamic trend. Also, Fig. 4 indicates that the decreasing trend in the numbers of MSC, OBp and OBa slows down around day 10, because mechanical stimulation significantly changes the rate of differentiation of MSC to OBa or OBp (Fig. 2a).

After the training (Fig. 4), testing (Fig. 5) and sensitivity analysis (Table 4.2) procedures, the model is robust enough to predict bone regeneration with high accuracy.

The small p values in Table 4.2 indicate that the parameters (V₆ and d₁) that determine the expression of Runx2 are significantly related to OBp and OBa, and also that the parameters (V₈ and d_s) that determine the expression of Osx are significantly related to OBa. These results prove that Runx2 actually plays an important role in promoting the differentiation of MSCs into OBp and inhibiting the differentiation of OBp into OBa [38], and also that Osx will promote OBa via the intracellular signaling pathway [4].

Finally, experimental design was employed to investigate the impact of key growth factors such as BMP2 and Wnt on bone regeneration. By comparing the number of OBa (Table 5.1 and 5.2) in the presence and absence of mechanical stimulation, we can conclude that mechanical stimulation is positively related to bone regeneration. Also, Table 5.3 demonstrates that the priority of the impact of the key growth factors is regardless of mechanical stimulation.

In brief, this model can not only predict vascular bone regeneration by using experimental data [39] and a data-mining algorithm [40], but can also employ experimental design [41] to investigate which growth factors play major roles in bone regeneration under mechanical stimulation within the porous biodegradable calcium phosphate (CaP) scaffold. However, it does not consider the impact of sequential drug delivery or employ the experimental data to optimize the key parameters of the related signaling transduction pathway. Therefore, we are going to develop related biological experiments and mathematical models to investigate potential cytokine combinations, optimal drug doses and drug delivery sequences within the porous biodegradable CaP scaffold in the near future.