In Silico Biology of Bone Regeneration Inside Calcium Phosphate Scaffolds

Abstract

Bone tissue engineering plays a key role in finding better solutions for the healing of large bone defects and non-unions. Despite extensive experimental research, many of the mechanisms of the bone regeneration process still remain to be elucidated. As such, mathematical modeling is a useful tool to further investigate the different influential factors and their interactions in silico. This chapter starts with a description of the biological processes that take place during bone regeneration in calcium phosphate (CaP) scaffolds. The second section gives an overview of the most recent mathematical models of bone regeneration in (CaP) scaffolds. One model is explained in more detail and used to illustrate the potential of mathematical modeling in the bone tissue engineering field. Finally, the drawbacks of the current modeling techniques and the need for more quantitative experimental research, together with possible solutions are presented.

Appendix

The equations contain the following model parameters:

$$\begin{aligned} J(t) = J_{in}.\frac{\mathit{Ca}(t)}{H_{\mathit{Ca}4} + \mathit{Ca}(t)} \end{aligned}$$

$$\begin{aligned} A_{m}(t) = \frac{A_{m0}.m(t)}{K_{m}^{2} + m(t)^{2}} \end{aligned}$$

$$\begin{aligned} \beta_{cm}(t) = \frac{a_{cm}}{c_{cm}}.\exp \biggl( - \frac{1}{2}. \biggl( \frac{\mathit{Ca}(t) - b_{cm}}{c_{cm}} \biggr)^{2} \biggr) \end{aligned}$$

$$\begin{aligned} F_{1}(t) = \frac{Y_{11}.g_{b}(t)^{6}}{H_{11}^{6} + g_{b}(t)^{6}}.F_{11}.\exp \biggl( - \frac{1}{2}. \bigl( \mathit{Ca}(t) - F_{12} \bigr)^{2} \biggr) \end{aligned}$$

$$\begin{aligned} A_{b}(t) = \frac{A_{b0}.m(t)}{K_{b}^{2} + m(t)^{2}} \end{aligned}$$

$$\begin{aligned} \beta_{cb}(t) = \frac{a_{cb}}{c_{cb}}.\exp \biggl( - \frac{1}{2}. \biggl( \frac{\mathit{Ca}(t) - b_{cb}}{c_{cb}} \biggr)^{2} \biggr) \end{aligned}$$

$$\begin{aligned} E_{gb}(t) = \frac{G{}_{gb}.g_{b}(t)}{H_{gb} + g_{b}(t)} \end{aligned}$$

$$\begin{aligned} R(t) = \bigl \vert c{}_{m}(t) - c_{m}(t - t_{3}) \bigr \vert .\frac{G_{\mathit{con}}.g_{b}(t)}{H_{\mathit{con}} + g_{b}(t)} \end{aligned}$$

The following scaling factors were chosen for the non-dimensionalization of the model variables:

$$\begin{aligned} \widetilde{t} = \frac{t}{T}, \qquad \widetilde{c_{m}} = \frac{c_{m}}{c_{0}}, \qquad \widetilde{c_{b}} = \frac{c_{b}}{c_{0}}, \qquad \widetilde{m} = \frac{m}{m_{0}}& \\ \widetilde{b} = \frac{b}{m_{0}}, \qquad \widetilde{g_{b}} = \frac{g_{b}}{g_{0}}, \qquad \widetilde{\mathit{Ca}} = \frac{\mathit{Ca}}{\mathit{Ca}_{0}}& \end{aligned}$$

The time T=1 day was considered to be a representative unit time for the process under study (similar to fracture healing models e.g. Geris et al. [50]). Representative concentrations for the collagen content (m ₀=0.1 g/ml) and growth factors (g ₀=100 ng/ml) are adopted from Geris et al. [50]. A typical value for the cell density (c ₀=10⁶ cells/ml) is derived from Bailón-Plaza and van der Meulen [48]. The scaling factor for the calcium concentration was assumed to be equal to the extracellular calcium concentration (Ca ₀=1 mM). An overview of the model parameter values and the initial variable values is given in Table 1 and Table 2 respectively.

The model parameters were non-dimensionalized as follows (the tildes represent the non-dimensional parameters):

$$\begin{aligned} \widetilde{P_{bs}} = \frac{P_{bs}.c_{0}.T}{m_{0}}, \qquad \widetilde{\kappa_{b}} = \kappa_{b}.m_{0}, \qquad \widetilde{A_{m0}} = \frac{A_{m0}.T}{m_{0}},& \\ \widetilde{K_{m}} = \frac{K_{m}}{m_{0}}, \qquad \widetilde{a_{cm}} = \frac{a_{cm}}{\mathit{Ca}_{0}}, \qquad \widetilde{b_{cm}} = \frac{b_{cm}}{\mathit{Ca}_{0}},& \\ \widetilde{c_{cm}} = \frac{c_{cm}}{\mathit{Ca}_{0}}, \qquad \widetilde{\alpha_{m}} = \alpha_{m}.c_{0}, \qquad \widetilde{H_{11}} = \frac{H_{11}}{g_{0}},& \\ \widetilde{Y_{11}} =Y_{11}.T, \qquad \widetilde{G_{gb}} =\frac{G_{gb}.T.c_{0}}{g_{0}}, \qquad \widetilde{H_{gb}} =\frac{H_{gb}}{g_{0}},& \\ \widetilde{d_{gb}} = d_{gb}.T, \qquad \widetilde{A_{b0}} = \frac{A_{b0}.T}{m_{0}}, \qquad \widetilde{K_{b}} = \frac{K_{b}}{m_{0}}, \qquad \widetilde{a_{cb}} = \frac{a_{cb}}{\mathit{Ca}_{0}},& \\ \widetilde{b_{cb}} = \frac{b_{cb}}{\mathit{Ca}_{0}}, \qquad \widetilde{c_{cb}} = \frac{c_{cb}}{\mathit{Ca}_{0}}, \qquad \widetilde{\alpha_{b}} =\alpha_{b}.c_{0},& \\ \widetilde{d_{b}} = d_{b}.T, \qquad \widetilde{P_{bb}} = \frac{P_{bb}.c_{0}.T}{m_{0}}, \qquad \widetilde{\kappa_{bb}} =\kappa _{bb}.m_{0},& \\ \widetilde{\sigma} =\sigma.T, \qquad \widetilde{\mathit{Ca}_{\infty}} = \frac{\mathit{Ca}_{\infty}}{\mathit{Ca}_{0}}, \qquad \widetilde{J_{\mathit{leaky}}} = \frac{J_{\mathit{leaky}}.T.c_{0}}{\mathit{Ca}_{0}},& \\ \widetilde{H_{\mathit{ca}4}} = \frac{H_{\mathit{ca}4}}{\mathit{Ca}_{0}}, \qquad \widetilde{d_{Ca}} =d_{\mathit{Ca}}.T.c_{0}, \qquad \widetilde{F_{11}} =F_{11}, \qquad \widetilde{F_{12}} =F_{12},& \\ \widetilde{G_{\mathit{con}}} =G_{\mathit{con}}.c_{0}, \qquad \widetilde{H_{\mathit{con}}} =\frac{H_{\mathit{con}}}{g_{0}}, \qquad \widetilde{d_{cm}} = d_{cm}.T.m_{0}& \end{aligned}$$