In this work we study a bone remodeling model for the evolution of the myeloma disease. The biological problem is written as a coupled nonlinear system consisting of parabolic partial differential equations. They are written in terms of the concentrations of osteoblasts and osteoclasts, the density of the relative bone and the concentration of the tumor cells. Then, we deal with the numerical analysis of this variational problem, introducing a numerical approximation by using the finite element method and a hybrid combination of both implicit and explicit Euler schemes. We perform some a priori error estimates and show a few numerical simulations to demonstrate the accuracy of the approximation. Finally, we present the comparison with previous works and the behavior of the solution in two‐dimensional examples.

中文翻译：

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细胞动力学引起的骨髓瘤疾病的骨重塑模型的分析。

在这项工作中，我们研究了骨髓瘤疾病演变的骨重塑模型。生物学问题被写成由抛物线偏微分方程组成的耦合非线性系统。用成骨细胞和破骨细胞的浓度，相对骨骼的密度和肿瘤细胞的浓度来表示。然后，我们对这个变分问题进行了数值分析，通过使用有限元方法以及隐式和显式Euler方案的混合组合来引入数值逼近。我们执行一些先验误差估计，并显示一些数值模拟来证明近似的准确性。最后，我们在二维示例中介绍了与先前工作的比较以及解决方案的行为。