School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK.
Philos Trans A Math Phys Eng Sci. 2011 Nov 13;369(1954):4162-82. doi: 10.1098/rsta.2011.0170.
The placenta is an essential component of the life-support system for the developing foetus, enabling nutrients and waste to be exchanged between the foetal and maternal circulations. Maternal blood flows between the densely packed branches of villous trees, within which are foetal vessels. Here, we explore some of the challenges in modelling maternal haemodynamic transport using homogenization approaches. We first show how two measures can be used to estimate the minimum distance over which the distribution of villous branches appears statistically homogeneous. We then analyse a simplified model problem (solute transport by a unidirectional flow past a distribution of point sinks) to assess the accuracy of homogenization approximations as a function of governing parameters (Péclet and Damköhler numbers) and the statistical properties of the sink distribution. The difference between the leading-order homogenization approximation and the exact solute distribution is characterized by large spatial gradients at the scale of individual villi and substantial fluctuations that can be correlated over lengthscales comparable to the whole domain. This study highlights the importance of quantifying errors owing to spatial disorder in multi-scale approximations of physiological systems.
胎盘是胎儿生命支持系统的重要组成部分，使营养物质和废物在胎儿和母体循环之间进行交换。母体血液在绒毛树的密集分支之间流动，其中有胎儿血管。在这里，我们探讨了使用均匀化方法对母体血液动力学传输进行建模的一些挑战。我们首先展示了如何使用两种度量来估计绒毛分支的分布在统计学上均匀的最小距离。然后，我们分析了一个简化的模型问题（单向流动通过点汇分布的溶质传输），以评估均匀化近似的准确性作为控制参数（佩克莱特数和达姆克勒数）和汇分布的统计特性的函数。主流化近似与精确溶质分布之间的差异的特点是在单个绒毛的尺度上存在大的空间梯度，并且在与整个域相当的长度尺度上可以相关的大量波动。这项研究强调了量化由于生理系统多尺度近似中的空间无序而产生的误差的重要性。
