Experimental-based mechanobiological modeling of the anabolic and catabolic effects of breast cancer on bone remodeling

Abstract

Bone is a biological tissue characterized by its hierarchical organization. This material has the ability to be continually renewed, which makes it highly adaptative to external loadings. Bone renewing is managed by a dynamic biological process called bone remodeling (BR), where continuous resorption of old bone and formation of new bone permits to change the bone composition and microstructure. Unfortunately, because of several factors, such as age, hormonal imbalance, and a variety of pathologies including cancer metastases, this process can be disturbed leading to various bone diseases. In this study, we have investigated the effect of breast cancer (BC) metastases causing osteolytic bone loss. BC has the ability to affect bone quantity in different ways in each of its primary and secondary stages. Based on a BR mathematical model, we modeled the BC cells’ interaction with bone cells to assess their effect on bone volume fraction (BV/TV) evolution during the remodeling process. Some of the parameters used in our model have been determined experimentally using the enzyme-linked immune-sorbent assay (ELISA) and the MTT assay. Our numerical simulations show that primary BC plays a significant role in enhancing bone-forming cells’ activity leading to a 6.22% increase in BV/TV over 1 year. On the other hand, secondary BC causes a noticeable decrease in BV/TV reaching 15.74% over 2 years.

Appendices

Appendix

Bone remodeling general model

The general mathematical model formulation of bone cell behavior is presented as follows, where the bone cells involved are: Osteoblast precursors (OBp), active osteoblast (OBa), and active osteoclasts (OCa) (Pivonka et al. 2008):

$$ \left\{ \begin{gathered} \frac{{dC_{{{\text{OBp}}}} \left( t \right)}}{dt} = D_{{{\text{OBu}}}} \pi_{{{\text{act}}}}^{{{\text{OBu}} \to {\text{OBp}}}} C_{{{\text{OBu}}}} + {\mathcal{P}}_{{{\text{OBp}}}} \Pi_{{\text{act,OBp}}}^{{{\text{mech}}}} C_{{{\text{OBp}}}} - D_{{{\text{OBp}}}} \pi_{{{\text{rep,TGF}}\beta }}^{{{\text{OBp}} \to {\text{OBa}}}} C_{{{\text{OBp}}}} \hfill \\ \frac{{dC_{{{\text{OBa}}}} \left( t \right)}}{dt} = D_{{{\text{OBp}}}} \pi_{{{\text{rep,TGF}}\beta }}^{{{\text{OBp}} \to {\text{OBa}}}} C_{{{\text{OBp}}}} - A_{{{\text{OBa}}}} C_{{{\text{OBa}}}} \hfill \\ \frac{{dC_{{{\text{OCa}}}} \left( t \right)}}{dt} = D_{{{\text{OCp}}}} \pi_{{{\text{act}},\left[ {{\text{RANK.RANKL}}} \right]}}^{{{\text{OCp}} \to {\text{OCa}}}} C_{{{\text{OCp}}}} - A_{{{\text{OCa}}}} \pi_{{{\text{act,TGF}}\beta }}^{{{\text{OCa}} \to + }} C_{{{\text{OCa}}}} \hfill \\ \end{gathered} \right. $$

(A1-A3)

${C}_{\mathrm{OBu}}$, ${C}_{\mathrm{OBp}}$, ${C}_{\mathrm{OBa}}$, ${C}_{\mathrm{OCp}}$, ${C}_{\mathrm{OCa}}$ represent, respectively, OBu concentration, OBp concentration, OBa concentration, OCp concentration, and OCa concentration. ${D}_{\mathrm{OBu}}$, ${D}_{\mathrm{OBp}}$ and ${D}_{\mathrm{OCp}}$ are, respectively, differentiation rates of OBu, OBp, and OCp. ${\mathcal{P}}_{\mathrm{OBp}}$ is the proliferation rate of the OBp and ${A}_{\mathrm{OBa}}$ and ${A}_{\mathrm{OCa}}$ represent, respectively, the apoptosis rates of OBa and OCa.

In Eqs. 1, 2, ${\pi }_{\mathrm{act}}^{\mathrm{OBu}\to \mathrm{OBp}}$, ${\pi }_{\mathrm{rep},\mathrm{TGF}\beta }^{\mathrm{OBp}\to \mathrm{OBa}}$, and ${\Pi }_{\mathrm{act},\mathrm{OBp}}^{\mathrm{mech}}$ represent, respectively, the ability of TGF_β and Wnt to stimulate the natural differentiation of OBu into OBp, the ability of TGF_β to inhibit the natural differentiation of OBp into OBa, and the ability of mechanical strains to promote preosteoblasts’ proliferation.

In Eq. 3, ${\pi }_{\mathrm{act},\mathrm{TGF}\beta }^{\mathrm{OCa}\to +}$. ${\pi }_{\mathrm{act},[\mathrm{RANK}.\mathrm{RANKL}]}^{\mathrm{OCp}\to \mathrm{OCa}}$ and ${\pi }_{\mathrm{act},\mathrm{TGF}\beta }^{\mathrm{OCa}\to +}$ represent, respectively, the ability of RANK/RANKL binding to promote preosteoblasts’ differentiation and the ability of the TGF_β to stimulate active osteoclasts’ apoptosis.

The fraction of extravascular bone matrix $BV/TV$ behavior is determined by Eq. 4. $BV/TV$ depends on active osteoblasts and osteoclasts’ concentrations, where ${k}_{\mathrm{form}}$ and ${k}_{\mathrm{res}}$ represent, respectively, the daily volume of bone matrix formed by osteoblast and the daily volume of bone matrix resorbed by osteoclast.

$$ \frac{{d{\text{BV}}/{\text{TV}}\left( t \right)}}{dt} = \left( {k_{{{\text{form}}}} C_{{{\text{OBa}}}} - k_{{{\text{res}}}} C_{{{\text{OCa}}}} } \right) $$

(A4)

Seeking to represent the cellular response to ligand stimulation, Hill function has been used. The Hill activation and repression functions used in the model are expressed as follows

$$ \begin{gathered} \pi_{{{\text{act}}}} = \frac{{C_{X} }}{{K_{{{\text{act}}}} + C_{X} }} \hfill \\ \pi_{{{\text{rep}}}} = \frac{{K_{{{\text{rep}}}} }}{{K_{{{\text{rep}}}} + C_{X} }} \hfill \\ \end{gathered} $$

(A5)

where ${C}_{X}$ is the concentration of the ligand X governing the cellular response, and ${K}_{\mathrm{act}}$ and ${K}_{\mathrm{rep}}$ are, respectively, the activation and repression constants.

The cellular response to different ligands of the model parameters are grouped in Table

Table 11 Description of the biochemical and mechanical factors’ integration into the normal BR model

11.