Role of Dynamic Loading on Early Stage of Bone Fracture Healing

Abstract

After fracture, mesenchymal stem cells (MSCs) and growth factors migrate into the fracture callus to exert their biological actions. Previous studies have indicated that dynamic loading induced tissue deformation and interstitial fluid flow could produce a biomechanical environment which significantly affects the healing outcomes. However, the fundamental relationship between the various loading regimes and different healing outcomes has not still been fully understood. In this study, we present an integrated computational model to investigate the effect of dynamic loading on early stage of bone fracture healing. The model takes into account cell and growth factor transport under dynamic loading, and mechanical stimuli mediated MSC differentiation and tissue production. The developed model was firstly validated by the available experimental data, and then implemented to identify the loading regimes that produce the optimal healing outcomes. Our results demonstrated that dynamic loading enhances MSC and growth factor transport in a spatially dependent manner. For example, compared to free diffusion, dynamic loading could significantly increase MSCs concentration in endosteal zone; and chondrogenic growth factors in both cortical and periosteal zones in callus. Furthermore, there could be an optimal dynamic loading regime (e.g. 10% strain at 1 Hz) which could potentially significant enhance endochondral ossification.

Appendix A

Appendix A.1: Volume Fraction of Solid, Fluid and Solute

The volume fraction of solid, fluid and solute phase of the porous callus domain can be represented by $\phi^{\text{s}}$, $\phi^{\text{f}}$ and $\phi^{\text{w}}$, respectively. As the volume of cells, growth factors and other nutrients are ignorable at tissue level compared to the volume of solid and fluid phase of the callus,27 we obtain:

$$ \phi^{\text{f}} + \phi^{\text{s}} \cong 1 $$

(A1.a)

The relationship between the medium volume based solute concentration ($\bar{c}^{\text{w}}$) and solvent volume based solute concentration ($c^{\text{w}}$) can be expressed as:

$$ \bar{c}^{\text{w}} = \phi^{\text{f}} c^{\text{w}} $$

(A1.b)

Similarly, the medium volume based tissue concentration ($\bar{c}^{\text{s}}$) can be expressed as:

$$ \bar{c}^{\text{s}} = (1 - \phi^{\text{f}})c^{\text{s}} $$

where $c^{\text{s}}$ is solvent volume based tissue concentration. (A1.c)

Appendix A.2: MSC Diffusion and Differentiation

The diffusion coefficient of MSCs for the callus matrix23 as follows:

$$ D^{\text{m}} = \frac{{D^{\text{hm}} m}}{{(K^{\text{hm}})^{2} + m^{2}}} $$

(A2.a)

where $D^{\text{hm}}$ and $K^{\text{hm}}$ are coefficients of cell migration depending on the total matrix density (m) of the callus.

The chemical stimuli mediated differentiation rates of MSCs into chondrocyte ($k_{\text{d}}^{c}$) and osteoblast ($k_{\text{d}}^{\text{b}}$) can be expressed by using Hill function3 as follows:

$$ k_{\text{d}}^{\text{b}} = \frac{{Y_{1} g^{\text{b}}}}{{H_{1} + g^{\text{b}}}} $$

(A2.b)

$$ k_{\text{d}}^{c} = \frac{{Y_{2} g^{c}}}{{H_{2} + g^{c}}} $$

(A2.c)

where H₁, H₂, Y₁ and Y₂ are parameters representing growth factor mediated MSC differentiation. $k_{\text{d}}^{\text{fb}}$ is assumed to be constant.54

Appendix A.3: Transport Equation for Cells and Growth Factors

The transport equations of fibroblasts ($c^{\text{fb}}$), chondrocytes ($c^{c}$), osteoblasts ($c^{\text{b}}$) can be expressed as:

$$ \frac{{\partial \bar{c}^{\alpha}}}{\partial t} = - \nabla \left({- D^{\alpha} \nabla \bar{c}^{\alpha} + {\mathbf{v}}^{\varvec{\text{f}}} \bar{c}^{\alpha}} \right) + S_{\text{p}}^{\alpha} - S_{\text{d}}^{\alpha} $$

(A3.a)

$$ S_{\text{p}}^{\alpha} = (k_{\text{d}}^{\alpha} + \lambda_{\text{d}}^{\alpha})c^{\text{m}} $$

(A3.b)

where, α = fb (fibroblasts), α = c (chondrocytes) and α = b (osteoblasts). $S_{\text{p}}^{\alpha}$ is the production rate of α cells which is assumed to be the sum of chemical stimuli mediated production rate ($k_{\text{d}}^{\alpha})$ and mechanical stimuli mediated production rate ($\lambda_{\text{d}}^{\alpha}$). $S_{\text{d}}^{\alpha}$ is the degradation rate of α cells and $D^{\alpha}$ is the diffusion coefficient of α cells in fracture callus.

Chondrogenic growth factors ($g^{c}$) and osteogenic growth factors ($g^{\text{b}}$)

The mass balance equation for chondrogenic ($g^{\text{c}}$) and osteogenic growth factor ($g^{\text{b}}$) is given by

$$ \frac{{\partial \bar{g}^{\beta}}}{\partial t} = - \nabla \cdot \left({- D^{\beta} \nabla \bar{g} ^{\beta} + {\mathbf{v}}^{\varvec{\text{f}}} \bar{g}^{\beta}} \right) + S_{\text{p}}^{\beta} - S_{\text{d}}^{\beta} $$

(A3.c)

where β = c (chondrogenic growth factors) and β = b (osteogenic growth factors). $S_{\text{p}}^{\beta}$ and $S_{\text{d}}^{\beta}$ are the production rate and degradation rate of growth factors, respectively. $D^{\beta}$ is the diffusion coefficient of the growth factors in fracture callus.

Appendix A.4: Stimuli Index (S)

Stimuli Index S is obtained from interstitial fluid phase velocity $({\mathbf{v}}^{\text{f}})$ and octahedral shear strain $(\tau)$ within the callus.33

$$ S = \frac{\tau}{0.0375} + \frac{{{\mathbf{v}}^{\text{f}}}}{{3 \times 10^{- 6} (m/s)}} $$

(A4)

It is assumed that, during early stage of healing, low magnitude of stimuli index (S < 1) results in differentiation of osteoblast, moderate magnitude of stimuli index (1 < S < 3) favours differentiation of chondrocytes while high magnitude of stimuli index (S < 1) leads to fibroblast differentiation.51

Appendix A.5: Production of Tissues Due to Mechanical Stimuli

Similar to the cell differentiation pattern shown in Fig. 4, the production rate of fibrous tissue, cartilage and bone due to mechanical stimuli also depends on dynamic loading as follows:

$$ P_{\text{pm}}^{\gamma} = \left\{\begin{array}{*{20}l}P_{\text{pm}}^{\text{bs}} = \lambda_{\text{p}}^{\text{bs}}\bar{c}^{\text{b}}, & P_{\text{pm}}^{\text{cs}} = P_{\text{pm}}^{\text{fs}} = 0, & {\text{for}}\;S < 1 & ({\text{bone}}\;{\text{formation)}} \\ P_{\text{pm}}^{\text{cs}} = \lambda_{\text{p}}^{\text{cs}} \bar{c}^{c}, & P_{\text{pm}}^{\text{bs}} = P_{\text{pm}}^{\text{fs}} = 0, & {{\text{for}}\; 1 < S < 3} & ({\text{cartilage}}\;{\text{formation}}) \\ P_{\text{pm}}^{\text{fs}} = \lambda_{\text{p}}^{\text{fs}} \bar{c}^{\text{fb}}, & P_{\text{pm}}^{\text{bs}} = P_{\text{pm}}^{\text{cs}} = 0, & {\text{for}}\;S > 3 & ({\text{fibrous}}\;{\text{tissue}}\;{\text{formation}}) \end{array}\right.$$

(A5)

Appendix A.6: Balance of Linear Momentum

Assuming callus as homogeneous, isotropic and linear elastic mixture having no body and inertial forces, for an infinitesimal strain, the balance of linear momentum can be expressed as73:

$$ - \nabla p + \left({\lambda_{\text{s}} + 2\mu_{\text{s}}} \right)\nabla \left({\nabla {\mathbf{u}}^{\varvec{\text{s}}}} \right) + \mu_{\text{s}} \nabla^{2} {\mathbf{u}}^{\varvec{\text{s}}} = 0 $$

(A6.a)

where $p$ is the interstitial fluid pressure; λ_s and μ_s are Lame’s constants; and ${\mathbf{u}}^{\varvec{\text{s}}}$ is the solid phase displacement vector.

Conservation of Mass of Solid and Fluid Phases

From the mass balance equation for solid and fluid phase and Darcy’s law, the interstitial fluid motion within the callus ^{71, 74} can be expressed as

$$ \nabla \left({{\mathbf{v}}^{\varvec{\text{s}}} - \varvec{k}\nabla p} \right) = 0 $$

(A6.b)

where ${\mathbf{v}}^{\varvec{\text{s}}}$ is the solid phase velocity, $\varvec{k}$ is the hydraulic permeability tensor and $p$ is the interstitial fluid pressure.

Appendix A.7: Cell and Growth Factor Uptake Ratio

The average normalised uptake ratio for cells and growth factors in each zone is computed as follows70:

$$ \bar{c}_{avg}^{\text{w}} = \frac{{\mathop \smallint \nolimits_{0}^{{V_{0}}} \bar{c}^{\text{w}} dV}}{{c_{0}^{\text{w}} V_{0}}} $$

(A7)

$$ \bar{g}_{avg}^{\text{w}} = \frac{{\mathop \smallint \nolimits_{0}^{{V_{0}}} \bar{g}^{\text{w}} dV}}{{g_{0}^{\text{w}} V_{0}}} $$

where $c^{\text{w}} = c^{\text{m}}, c^{\text{fb}}, c^{\text{c}} and c^{\text{b}}$ for MSCs, fibroblasts, chondrocytes and osteoblasts concentration, $g^{\text{w}} = g^{\text{b}} {\text{and}} g^{\text{c}}$ for osteogenic and chondrogenic growth factor concentration, respectively. $c_{0}^{\text{w}}$ is the saturated concentration of particular cell over callus volume $V_{0}$. The uptake of osteogenic and chondrogenic growth factors are normalised to their respective boundary conditions $\left({g_{0}^{\text{w}} = g^{\text{b0}} \;{\text{and}}\;g^{\text{c0}}} \right)$.