Mechanobiological osteocyte feedback drives mechanostat regulation of bone in a multiscale computational model

Abstract

Significant progress has been made to identify the cells and signaling molecules involved in the mechanobiological regulation of bone remodeling. It is now well accepted that osteocytes act as mechanosensory cells in bone expressing several signaling molecules such as nitric oxide (NO) and sclerostin (Scl) which are able to control bone remodeling responses. In this paper, we present a comprehensive multiscale computational model of bone remodeling which incorporates biochemical osteocyte feedback. The mechanostat theory is quantitatively incorporated into the model using mechanical feedback to control expression levels of NO and Scl. The catabolic signaling pathway RANK–RANKL–OPG is co-regulated via (continuous) PTH and NO, while the anabolic Wnt signaling pathway is described via competitive binding reactions between Wnt, Scl and the Wnt receptors LRP5/6. Using this novel model of bone remodeling, we investigate the effects of changes in the mechanical loading and hormonal environment on bone balance. Our numerical simulations show that we can calibrate the mechanostat anabolic and catabolic regulatory mechanisms so that they are mutually exclusive. This is consistent with previous models that use a Wolff-type law to regulate bone resorption and formation separately. Furthermore, mechanical feedback provides an effective mechanism to obtain physiological bone loss responses due to mechanical disuse and/or osteoporosis.

Appendices

Appendix 1: Formulation of the evolution of the osteocytes population

As written in Eq. (4), we chose here to represent the change in osteocytes concentration ${\mathsf {Ot}}$ as a function of bone matrix fraction $f_{\mathsf {bm}}$. It is well accepted that osteocytes are differentiated osteoblastic cells buried in the bone matrix. As a result, one could argue that the evolution of the osteocytes population could be written as follows:

$$\begin{aligned} \dfrac{{\mathrm{d}}{\mathsf {Ot}}}{{\mathrm{d}}t}=D_{{\mathsf {Ob}}_{\mathrm{a}}} {\mathsf {Ob}}_{\mathrm{a}}- R^{{\mathsf {Oc}}_{\mathrm{a}}}_{{\mathsf {Ot}}} {\mathsf {Oc}}_{\mathrm{a}}, \end{aligned}$$

(29)

where one can note that the sink term is proportional to the concentration of osteoclasts ${\mathsf {Oc}}_{\mathrm{a}}$. Osteoclasts resorb bone and trigger osteocyte apoptosis by means of a local acidification. This action is quantified through the factor $R^{{\mathsf {Oc}}_{\mathrm{a}}}_{{\mathsf {Ot}}}$ in Eq. (29). If we set the osteoblasts differentiation coefficient to $D_{{\mathsf {Ob}}_{\mathrm{a}}} =\eta k_{\mathsf {form}}$ and the resorption coefficient to $R^{{\mathsf {Oc}}_{\mathrm{a}}}_{{\mathsf {Ot}}}=\eta k_{\mathsf {res}}$, we obtain readily the formulation given in Eq. (4) in Sect. 2.1.

Appendix 2: Formulation of the RANKL production

We made the assumption that RANKL is expressed by osteocytes and osteoblasts precursors, following experimental evidence (Nakashima et al. 2011; Xiong et al. 2015). Therefore, the body production term for RANKL reads as follows:

$$\begin{aligned}&P_{{\mathsf {RANKL}},b}= \beta _{{\mathsf {RANKL}},{\mathsf {Ot}}} {\mathsf {Ot}}\left( 1- \frac{[{\mathsf {RANKL}}]_{\mathsf {tot}}}{[{\mathsf {RANKL}}]_{\mathsf {max}}} \right) \\& \quad + \beta _{{\mathsf {RANKL}},{\mathsf {Ob}}_{\mathrm{p}}}\pi _{{\mathsf {act/rep}},{\mathsf {RANKL}}}^{{\mathsf {PTH}},{\mathsf {NO}}} {\mathsf {Ob}}_{\mathrm{p}}\left( 1- \frac{[{\mathsf {RANKL}}]_{\mathsf {tot}}}{[{\mathsf {RANKL}}]_{\mathsf {max}}}\right) , \end{aligned}$$

(30)

where the total concentration of RANKL (bound and free) $[{\mathsf {RANKL}}]_{\mathsf {tot}}$ is defined as follows:

$$\begin{aligned}&[{\mathsf {RANKL}}]_{\mathsf {tot}}= [{\mathsf {RANKL}}] \\&\quad \left( 1+\frac{[\mathsf {RANK}]}{K_{D}^{\mathsf {RANK}-{\mathsf {RANKL}}}}+\frac{[{\mathsf {OPG}}]}{K_{D}^{{\mathsf {OPG}}-{\mathsf {RANKL}}}}\right) . \end{aligned}$$

(31)

The parameters used in Eqs. (30), (31) are displayed in Table 3. When knocking out numerically the expression of RANKL by osteocytes ($\beta _{{\mathsf {RANKL}},{\mathsf {Ot}}}=0$), we obtain the osteopetrotic phenotype observed experimentally (Nakashima et al. 2011). This result is illustrated in Fig. 12, where the steady state (dashed line) is compared to the RANKL-deficient state (solid line).

Fig. 12

Influence of osteocytes’ RANKL production on bone homeostasis: we compare the steady state (dashed–dotted line) to the RANKL-deficient state (solid line)

Appendix 3: Properties of matrix and fluid in the micro-mechanical model

The stiffness tensor of the bone matrix reads as follows (Kelvin notation):

$$\begin{aligned} \mathbb {c}_{\mathsf {bm}}= \begin{pmatrix} 18.5 &{}\quad 10.3 &{}\quad 10.4 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 10.3 &{}\quad 20.8 &{}\quad 11.0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 10.4 &{}\quad 11.0 &{}\quad 28.4 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 12.9 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 11.5 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 9.3 \\ \end{pmatrix} \mathrm {GPa} \end{aligned}$$

(32)

The stiffness tensor of the saturating fluid is:

$$\begin{aligned} \mathbb {c}_{\mathsf {vas}}= k_{\mathsf {H_2 O}} \mathbb {J} + \mu _{\mathsf {H_2 O}} \mathbb {K}, \end{aligned}$$

(33)

where the bulk modulus and the shear modulus are, respectively, $k_{\mathsf {H_2 O}}=2.3\ \mathrm {GPa}$ and $\mu _{\mathsf {H_2 O}}=0\ \mathrm {GPa}$, and $\mathbb {J}$ is the volumetric part of the fourth-order unit tensor $\mathbb {I}$, and $\mathbb {K}$ is its deviatoric part, $\mathbb {K}=\mathbb {I}-\mathbb {J}$.

Appendix 4: Derivation of free sclerostin concentration

The following equation describes sclerostin balance:

$$\begin{aligned}&P_{{\mathsf {Scl}},b}+P_{{\mathsf {Scl}},d}=\tilde{D}_{\mathsf {Scl}}[{\mathsf {Scl}}] \\&\quad +D_{{\mathsf {Scl}}\ - {\mathsf {LRP5/6}}}[{\mathsf {Scl}}\ - {\mathsf {LRP5/6}}] \end{aligned}$$

(34)

with

$$\begin{aligned}&[{\mathsf {Scl}}- {\mathsf {LRP5/6}}]=\frac{[{\mathsf {LRP5/6}}][{\mathsf {Scl}}]}{K_{D}^{{\mathsf {LRP5/6}}\ - {\mathsf {Scl}}}} \\&\quad =\frac{[{\mathsf {Scl}}][{\mathsf {LRP5/6}}]_{\mathsf {tot}}}{K_{D}^{{\mathsf {LRP5/6}}\ - {\mathsf {Scl}}}\, (1+\frac{[{\mathsf {Wnt}}]}{K_{D}^{{\mathsf {Wnt}},{\mathsf {LRP5/6}}}}+\frac{[{\mathsf {Scl}}]}{K_{D}^{{\mathsf {Scl}},{\mathsf {LRP5/6}}}})}. \end{aligned}$$

(35)

After inserting Eqs. (18) and (35) into Eq. (34), we uncover a second-order polynomial equation on the concentration of free sclerostin $[{\mathsf {Scl}}]$:

$$\begin{aligned} A[{\mathsf {Scl}}]^2+B[{\mathsf {Scl}}]+C=0, \end{aligned}$$

(36)

where

$$\begin{aligned} A= \, & {} \tilde{D}_{\mathsf {Scl}}+ \frac{\beta _{\mathsf {Scl}}[{\mathsf {Ot}}] \pi ^{{\varPsi _{\mathsf {bm}}}}_{{\mathsf {rep}},{\mathsf {Scl}}}}{[{\mathsf {Scl}}]_{\mathsf {max}}} \\ B= \, & {} K_{D}^{{\mathsf {Scl}}{-}{\mathsf {LRP5/6}}} (\tilde{D}_{\mathsf {Scl}}\\&+\,(\beta _{\mathsf {Scl}}[{\mathsf {Ot}}] \pi ^{{\varPsi _{\mathsf {bm}}}}_{{\mathsf {rep}},{\mathsf {Scl}}})/[{\mathsf {Scl}}]_{\mathsf {max}} )\\&(1+[{\mathsf {Wnt}}]/K_{D}^{{\mathsf {Wnt}}{-}{\mathsf {LRP5/6}}} )\\&+\,\tilde{D}_{{\mathsf {Scl}}{-}{\mathsf {LRP5/6}}} [{\mathsf {LRP5/6}}]_{\mathsf {tot}}-P_{{\mathsf {Scl}},d} \beta _{{\mathsf {Scl}}} [{\mathsf {Ot}}] \pi ^{{\varPsi _{\mathsf {bm}}}}_{{\mathsf {rep}},{\mathsf {Scl}}} \\ C= \, & {} -P_{{\mathsf {Scl}},d} \beta _{\mathsf {Scl}}[{\mathsf {Ot}}] \pi ^{{\varPsi _{\mathsf {bm}}}}_{{\mathsf {rep}},{\mathsf {Scl}}} (1+[{\mathsf {Wnt}}]/K_{D}^{{\mathsf {Wnt}}{-}{\mathsf {LRP5/6}}} ) \end{aligned}$$

The term A is a sum of positive quantities, one of them being strictly positive, and the term C is the opposite of the product of strictly positive quantities except for the concentration of osteocytes that we can assume to be non-null. From linear algebra, we know the product of the solutions is equal to $\frac{C}{A}<0$. Thus, provided solutions are in the real space, there are exactly two roots of opposed signs satisfying this equation. Hence, as the solution of the equation is a concentration, the only acceptable root is:

$$\begin{aligned} {[}{\mathsf {Scl}}]=\frac{-B+\sqrt{B^2-4AC}}{2A} \end{aligned}$$

Appendix 5: Nomenclature

The abbreviations used in the present paper are summarized in Table 5.

Table 5 Nomenclature