Interfaces play a key role on diseases development because they dictate the energy inflow of nutrients from the surrounding tissues. What is underestimated by existing mathematical models is the biological fact that cells are able to use different resources through nonlinear mechanisms. Among all nutrients, lactate appears to be a sensitive metabolic when talking about brain tumours or neurodegenerative diseases. Here we present a partial differential model to investigate the lactate exchanges between cells and the vascular network in the brain. By extending an existing kinetic model for lactate neuro-energetics, we first provide analytical proofs of the uniqueness and the derivation of precise bounds on the solutions of the problem including diffusion of lactate in a representative volume element comprising the interface between a capillary and cells. We further perform finite element simulations of the model in two test cases, discussing the relevant physical parameters governing the lactate dynamics.

中文翻译：

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乳酸神经能的偏微分模型：分析结果和数值模拟

界面在疾病发展中起关键作用，因为它们决定了周围组织营养物质的能量流入。现有数学模型低估了生物学事实，即细胞能够通过非线性机制使用不同的资源。在所有营养素中，当谈论脑部肿瘤或神经退行性疾病时，乳酸似乎是一种敏感的新陈代谢。在这里，我们提出了一种偏微分模型，以研究细胞与大脑中的血管网络之间的乳酸交换。通过扩展现有的乳酸神经能量动力学模型，我们首先提供解析性证明，证明该问题的解决方案具有唯一性和精确界限，包括乳酸在包括毛细管和细胞之间的界面在内的代表性体积元素中的扩散。我们在两个测试案例中进一步进行了模型的有限元模拟，讨论了控制乳酸动力学的相关物理参数。