Bone remodeling is regulated by numerous signaling pathways being transforming growth factor-$\beta$ (TGF$\beta$) and Wnt crucial for coupling bone formation and elimination activities. Bone diseases, such as osteoporosis and bone metastasis, develop when the bone cells cross-talk is corrupted which causes unbalanced bone mass production. In this work, we explore the effects of TGF$\beta$ and Wnt in bone dynamics based on a proposed mathematical model, and thereby study different disease control strategies. In particular, we find steady-states of proposed mathematical models, and show bifurcation diagrams that indicate nonlinear dynamical behavior, such as Hopf bifurcations and hysteresis. The implications of these findings on dynamical behavior are then discussed in the context of bone diseases. Finally, using control theory, we also address the question about how input or inhibition combination therapies may help in the control of osteoporosis and cancer-induced bone disease. Numerical simulations are performed to simulate the dynamical behavior of the bone microenvironment and to analyze different treatment combinations (conventional and novel). A discussion of the results is carried out.

骨骼重塑受许多正在转化生长因子的信号转导通路的调控。$\beta$ （TGF$\beta$）和Wnt对于耦合骨骼形成和消除活动至关重要。当骨细胞的串扰受损导致骨量产生不平衡时，就会发生骨病，例如骨质疏松症和骨转移。在这项工作中，我们探讨了TGF的作用$\beta$Wnt和Wnt在骨骼动力学方面的基础上提出了数学模型，从而研究了不同的疾病控制策略。特别是，我们找到了所提出数学模型的稳态，并显示了表示非线性动力学行为的分支图，例如Hopf分支和滞后现象。然后在骨骼疾病的背景下讨论这些发现对动力学行为的影响。最后，使用控制理论，我们还解决了有关输入或抑制联合疗法如何有助于控制骨质疏松症和癌症引起的骨病的问题。进行数值模拟以模拟骨骼微环境的动力学行为并分析不同的治疗组合（常规和新颖）。对结果进行了讨论。