Bonfiglio Andrea, Leungchavaphongse Kritsada, Repetto Rodolfo, Siggers Jennifer H
Department of Civil, Environmental and Architectural Engineering, University of Genoa, Genoa, Italy.
J Biomech Eng. 2010 Nov;132(11):111011. doi: 10.1115/1.4002563.
In this paper, we develop a mathematical model of blood circulation in the liver lobule. We aim to find the pressure and flux distributions within a liver lobule. We also investigate the effects of changes in pressure that occur following a resection of part of the liver, which often leads to high pressure in the portal vein. The liver can be divided into functional units called lobules. Each lobule has a hexagonal cross-section, and we assume that its longitudinal extent is large compared with its width. We consider an infinite lattice of identical lobules and study the two-dimensional flow in the hexagonal cross-sections. We model the sinusoidal space as a porous medium, with blood entering from the portal tracts (located at each of the vertices of the cross-section of the lobule) and exiting via the centrilobular vein (located in the center of the cross-section). We first develop and solve an idealized mathematical model, treating the porous medium as rigid and isotropic and blood as a Newtonian fluid. The pressure drop across the lobule and the flux of blood through the lobule are proportional to one another. In spite of its simplicity, the model gives insight into the real pressure and velocity distribution in the lobule. We then consider three modifications of the model that are designed to make it more realistic. In the first modification, we account for the fact that the sinusoids tend to be preferentially aligned in the direction of the centrilobular vein by considering an anisotropic porous medium. In the second, we account more accurately for the true behavior of the blood by using a shear-thinning model. We show that both these modifications have a small quantitative effect on the behavior but no qualitative effect. The motivation for the final modification is to understand what happens either after a partial resection of the liver or after an implantation of a liver of small size. In these cases, the pressure is observed to rise significantly, which could cause deformation of the tissue. We show that including the effects of tissue compliance in the model means that the total blood flow increases more than linearly as the pressure rises.
在本文中，我们建立了肝小叶内血液循环的数学模型。我们旨在找出肝小叶内的压力和通量分布。我们还研究了部分肝脏切除后压力变化的影响，部分肝脏切除常常会导致门静脉高压。肝脏可分为称为小叶的功能单位。每个小叶具有六边形横截面，并且我们假设其纵向范围与其宽度相比很大。我们考虑由相同小叶组成的无限晶格，并研究六边形横截面中的二维流动。我们将肝血窦空间建模为多孔介质，血液从门静脉分支（位于小叶横截面的每个顶点处）进入，并通过中央静脉（位于横截面中心）流出。我们首先建立并求解一个理想化的数学模型，将多孔介质视为刚性且各向同性的，将血液视为牛顿流体。跨小叶的压降和通过小叶的血液通量相互成比例。尽管该模型很简单，但它能让我们深入了解小叶内的实际压力和速度分布。然后我们考虑对该模型进行三种修改，以使它更符合实际情况。在第一次修改中，我们考虑到肝血窦倾向于优先沿中央静脉方向排列这一事实，通过考虑各向异性多孔介质来实现。在第二次修改中，我们使用剪切变稀模型更准确地描述血液的真实行为。我们表明这两种修改对行为都只有很小的定量影响，但没有定性影响。最后一次修改的目的是了解部分肝脏切除后或植入小尺寸肝脏后会发生什么情况。在这些情况下，观察到压力会显著升高，这可能会导致组织变形。我们表明在模型中纳入组织顺应性的影响意味着随着压力升高，总血流量的增加超过线性增加。
