The inflammatory phase is an important event in the skin wound healing process. The deposition of granulation tissue in the wound bed and the rebuilding of the vascular network occur as inflammation diminishes. An angiogenic component in the formation of granulation tissue is the secretion of vascular endothelial growth factor, which assists in the chemotaxis, proliferation, and replication of fibroblasts. In this paper, we develop a mathematical model of skin wound healing angiogenic factors based on inflammatory cells (macrophages and neutrophils) and mediators (interleukin 6 and interleukin 10). We highlight the importance of this process in vascular endothelial growth factor release and in the formation of new capillary tips. We used a mathematical model of partial differential equations based on the reaction-diffusion-advection equations. In order to calibrate the parameters, we considered an in vivo model composed by four treatments: hydroalcoholic extract and oil-resin of Copaifera langsdorffii at 10% concentration, collagenase, and Lanette cream. Using the laboratory data for the wound edge, our mathematical model estimated the values of vascular endothelial growth factor concentration, and tips density in the center of the wound with a maximum error of 2.9%, and predicted healing time required for each treatment. The region of viability for the parameters, in the proposed model, was found through numerical simulations from the Interleukin 6 and 10 dysregulation and we obtained that, among the parameters analyzed, the greatest influencer in the dynamics of the system is the one, which represents the production of Interleukin 10 during phagocytosis.

中文翻译：

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大鼠皮肤伤口愈合模型的细胞、分子和血管生成参数的数学和数值分析


炎症阶段是皮肤伤口愈合过程中的重要事件。当炎症消退时，肉芽组织在伤口床中沉积并重建血管网络。肉芽组织形成中的血管生成成分是血管内皮生长因子的分泌，其有助于成纤维细胞的趋化、增殖和复制。在本文中，我们开发了基于炎症细胞（巨噬细胞和中性粒细胞）和介质（白细胞介素 6 和白细胞介素 10）的皮肤伤口愈合血管生成因子的数学模型。我们强调这一过程在血管内皮生长因子释放和新毛细血管尖端形成中的重要性。我们使用基于反应扩散平流方程的偏微分方程数学模型。为了校准参数，我们考虑了由四种处理组成的体内模型：浓度为 10% 的 Copaifera langsdorffii 的水醇提取物和油树脂、胶原酶和 Lanette 霜。利用伤口边缘的实验室数据，我们的数学模型估计了伤口中心的血管内皮生长因子浓度和尖端密度值，最大误差为2.9%，并预测每次治疗所需的愈合时间。通过对白细胞介素 6 和 10 失调的数值模拟，发现了所提出的模型中参数的可行性区域，我们发现，在分析的参数中，对系统动力学影响最大的一个是，它代表吞噬过程中白细胞介素 10 的产生。