Calcium has two important roles in haemodialysis. It participates in the activation of blood coagulation and calcium intakes have a major impact on patient mineral and bone metabolism. The aim of this article is to propose a mathematical model for calcium ions concentration in a dialyzer during haemodialysis using a citrate dialysate. The model is composed of two elements. The first describes the flows of blood and dialysate in a dialyzer fibre. It was obtained by asymptotic analysis and takes into account the anisotropy of the fibres forming a dialyzer. Newtonian and non-Newtonian blood rheologies were tested. The second part of the model predicts the evolution of the concentration of five chemical species present in these fluids. The fluid velocity field drives the convective part of a convection-reaction-diffusion system that models the exchange of free and complexed calcium. We performed several numerical experiments to calculate the free calcium concentration in the blood in a dialyzer using dialysates with or without citrate. The choice of blood rheology had little effect on the fluid velocity field. Our model predicts that only a citrate based dialysate without calcium can decrease free calcium concentration at the blood membrane interface low enough to inhibit blood coagulation. Moreover for a given calcium dialysate concentration, adding citrate to the dialysate decreases total calcium concentration in the blood at the dialyzer outlet. This decrease of the calcium concentration can be compensated by infusing in the dialyzed blood a quantity of calcium computed from the model.

中文翻译：

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血液透析过程中使用含有透析液的柠檬酸盐进行钙交换的数学模型。

钙在血液透析中具有两个重要作用。它参与凝血的激活，钙的摄入对患者的矿物质和骨骼代谢有重大影响。本文的目的是为使用柠檬酸盐透析液的血液透析过程中的透析器中钙离子浓度提出一个数学模型。该模型由两个元素组成。第一个描述透析器纤维中的血液和透析液的流动。它是通过渐进分析获得的，并考虑了形成透析器的纤维的各向异性。测试了牛顿和非牛顿的血液流变学。模型的第二部分预测了这些流体中存在的五种化学物质的浓度变化。流速场驱动对流-反应-扩散系统的对流部分，该系统对游离钙和复杂钙的交换进行建模。我们进行了一些数值实验，以计算使用或不使用柠檬酸盐的透析液在透析器中血液中的游离钙浓度。血液流变学的选择对流速场影响很小。我们的模型预测，仅基于柠檬酸盐的不含钙的透析液可以降低血膜界面的游离钙浓度，抑制血浆钙的浓度至足够低。此外，对于给定的透析液钙浓度，向透析液中添加柠檬酸盐会降低透析器出口处血液中的总钙浓度。