Experimental and theoretical investigations of the dynamics of collective phenomena in blood II: Towards a physically more realistic model

本项目的目的是通过实验和计算机模拟研究体外施加剪切流下红细胞悬浮液，重点研究高浓度下的集体动力学和非线性响应。该项目以题为“血液中集体现象动力学的实验和理论研究I：理想囊泡/液滴模型”的同一倡议的第一部分为基础。由于上一个项目的成就（参见下面的进度报告），我们现在能够在剪切速率、细胞变形能力和体积分数/红细胞压积（细胞占据的通道体积的一部分）的参数范围内研究一个物理上更真实的红细胞模型，而不是液滴模型。新建立的模型所得到的结果是很有希望的，值得进行更细致的研究。特别是，下列开放性问题应通过实验和计算机模拟来解决。•剪切诱导的扩散率：即使在≈6%的相当低的体积分数下，我们的模拟表明（流动介导的）红细胞之间的碰撞显著地改变了它们的个体动力学。除了流动之外，红细胞似乎在足够大的时间内表现出扩散运动，近似于粒子位移的高斯分布。当监测垂直于流动方向的位移时，这一点最为明显。一个有趣的问题是，剪切引起的扩散如何随施加的剪切速率而变化，这与有效粘度对剪切速率的依赖如何比较。此外，在足够高的红细胞压积下，悬浮液开始发展成固体状特性，诸如动态屈服和剪切驱动融化等问题变得相关。在这种情况下，我们期望在一些典型的时间和长度尺度上发展出与高斯行为的强烈偏差。研究这些长度和时间尺度如何取决于剪切速率和红细胞压积将是非常有趣的。•合作运动：与上述方面密切相关的一个问题是动态关联的可能性。事实上，粒子位移的非高斯分布提供了粒子动力学中相关性存在的提示（这里是红细胞）。研究动态相关的红细胞簇是否存在以及在何种条件下存在是很有趣的。我们预计这些团簇的寿命随着施加的剪切速率而减少，而在更高的浓度/红细胞压积时增加。与上面提出的剪切诱导扩散和粘度之间关系的问题类似，研究控制动态相关簇的寿命的物理定律是否与描述扩散和粘度的物理定律相比是非常有趣的。要回答这个问题远非微不足道，而且肯定需要对这些团簇的长度尺度作为剪切速率和红细胞压积的函数进行彻底的研究：颗粒大小的距离上的扩散可能足以破坏小团簇。然而，粒子在更大尺度上的运动对于包含大量粒子的簇内的动态相关性的衰减可能是必要的。屈服应力：如前所述，高浓度的红细胞悬浮液可能会形成固体状。除了上述讨论的动态问题外，这也可能表现为屈服应力的出现（这里定义为维持流动所需的最小应力）。有限屈服应力的存在可能会产生严重的生物学后果，因为如果血管中的粘性应力低于屈服应力，则可能导致血流阻塞。通过一个简单的论证可以表明，如果血管半径低于某一值，这种情况确实可能发生在狭窄的血管中。因此，研究屈服应力及其与红细胞压积和其他系统参数的关系具有重要意义。•红细胞可变形性的影响：上述问题通常取决于红细胞的可变形性。例如，如果红细胞的可变形性降低，细胞间碰撞的影响和由此产生的有效扩散预计将在较低的体积分数下感受到。同样，随着细胞硬度的增加（如由镰状细胞性贫血或其他广泛传播的疾病，如糖尿病或疟疾引起），我们预计屈服应力可能随着红细胞压积的增加而增长得更快，从而导致血流阻塞的可能性更高。这里感兴趣的一个问题是，剪切速率和变形能力/单元刚度是否在定性上起独立的作用，或者一个单一的无量纲参数，如剪切能与弹性变形能的比率，足以描述系统在一定参数范围内的行为。

The aim of the present project is to study via experiments and computer simulations suspensions of red blood cells (RBCs) under externally imposed shear flow with a focus on collective dynamics and non-linear response at high concentrations. This project builds upon the part I of the same initiative entitled “Experimental and theoretical investigations of the dynamics of collective phenomena in blood I: Idealized vesicle/fluid droplet models”. Thanks to the achievements of the previous project (cf. progress report below), we are now able to investigate a physically more realistic –as compared to droplets– model of RBCs in the parameter range of interest in terms of shear rate, cell deformability and volume fraction/ hematocrit (the portion of the channel volume occupied by the cells). Results obtained with the newly developed model are very promising and deserve more elaborate investigations. In particular, the following open issues shall be addressed both experimentally and via computer simulations. • Shear-induced diffusivity: Even at a rather low volume fraction of ≈6%, our simulations show that (flow-mediated) collisions between red blood cells significantly alter their individual dynamics. In addition to the flow, RBCs seem to exhibit a diffusive motion at sufficiently large times with, approximately, a Gaussian distribution of particle displacements. This is best visible when monitoring displacements perpendicular to the flow direction. An interesting question here is how this shear-induced diffusion varies with imposed shear rate and how does this compare to the shear rate dependence of the effective viscosity. Furthermore, at a sufficiently high hematocrit, the suspension starts to develop a solid-like character and issues such as dynamic yielding and shear driven melting become relevant. In this case, we expect strong deviations from Gaussian behavior to develop at some typical time and length scales. It would then be very interesting to study how these length and time scales depend on shear rate and hematocrit. • Cooperative motion: An issue, closely related to the above aspect, concerns the possibility of dynamic correlations. Indeed, a non-Gaussian distribution of particle displacements provides a hint on the presence of correlations in the dynamics of particles (here red blood cells). It is interesting to investigate whether and under which conditions dynamically correlated clusters of RBCs exist. We expect the life time of these clusters to decrease with the imposed shear rate and to increase at higher concentration/hematocrit. Similar to the above raised question on the relation between shear-induced diffusion and viscosity, it is very interesting to study whether physical laws governing the life time of dynamically correlated clusters compare to those describing diffusion and viscosity. Answer to this question is far from being trivial and will certainly require a thorough study of the length scale of these clusters as a function of shear rate and hematocrit: Diffusion over a distance of a particle size may be sufficient to destroy small clusters. However, particle motion on a lager scale may be necessary for a decay of dynamic correlations within a cluster which contains a large number of particles. • Yield stress: As already mentioned, at high concentrations, suspensions of red blood cellsmay develop a solid-like character. In addition to the above discussed dynamic issues, thismay also manifest itself in the emergence of a yield stress (defined here as the minimumstress needed to maintain the flow). The presence of a finite yield stress may have severebiological consequences, since it may lead to a flow blockage if the viscous stress in theblood vessel falls below the yield stress. As can be shown by a simple argument, this may indeed occur in narrow vessels, if the vessel radius is below a certainvalue. Therefore, it is important to study the issue of yield stress and its dependence onhematocrit and other system parameters.• The effect of RBC-deformability: The above issues do in general depend on the deformability of red blood cells. For instance, the effect of cell-cell collision and the resulting effective diffusion is expected to be felt at a lower volume fraction if the deformability of red blood cells is reduced. Similarly, with increasing cell-stiffness (caused e.g. via sickle cell anemia or other widespread diseases such as diabetes mellitus or malaria) one expects that yield stress may grow faster with hematocrit and thus lead to a higher probability of flow blockage. A question of interest here is whether both shear rate and deformability/cell-stiffness play qualitatively independent roles, or a single dimensionless parameter such as the ratio of shear energy to the elastic deformation energy is sufficient to describe the system behavior in a certain parameter-range.