Biphasic flow dynamics and polarized mass transportation in branched hepatic sinusoids

The configuration and geometry of the calculating model are shown in Fig. 1(a). The size of the actual computational domain is 12.0 × 80.0 μm2, corresponding to a total of 120 × 800 grids in numerical calculation. No slip boundary condition is assumed on the hepatocyte surface. The inlet flow rate is set to be 8.14 × 10−5 ml min−1, with an inlet Reynolds number (Re) of 0.0012. The outlet is free to flow out, with the velocity settings as that y-velocity is zero and x-velocity adopts the mass-modified outlet boundary condition to ensure flow conservation. The plasma is assumed to be an incompressible Newtonian fluid with a viscosity of 1.2 cp and a density of 1000kg m−3. A dimensionless relaxation time Δt/τ was used in LBM equations and our programming codes, and the relaxation time in LBM is related to the value of fluid viscosity coefficient μD. The computational stability and accuracy of the model used in this work have been validated in our previous work3030. T. H. Wang, S. Lu, Y. Hao, Z. Su, M. Long, and Y. Cui, Biophys. J. 120, 4859 (2021). https://doi.org/10.1016/j.bpj.2021.09.020 and other studies.55. L. Li and Y. Liu, J. Clin. Hepatol. 35, 913 (2019). https://doi.org/10.3969/j.issn.1001-5256.2019.04.046 The permeability coefficient, k, of the endothelial layer is ranged at 10−16 m2–10−13 m.2929. Z. Z. Shi, K. Stephen, B. A. David, and C. Wu, Plos One 1, 1–39 (2016). https://doi.org/10.1371/journal.pone.0161131 The hydraulic resistivity of the macromolecular layer adjacent to the endothelium is assigned to be a value of ∼108 dyn s cm−4 in the literature,3131. T. W. Secomb, R. Hsu, and A. R. Pries, Microcirculation 9, 189 (2002). https://doi.org/10.1038/sj.mn.7800132 from which the endothelial permeability coefficient is estimated to be approximately 10−14 m2. Thus, the permeability coefficient of the hepatic microvilli and collagen layer, kr, is set on a scale between 10−16 and 10−14 m2. The higher the permeability coefficient, the greater is the porosity of the hepatocyte villi and collagen layer, indicating relatively sparse collagen filling. All calculated parameters were summarized in Table I. The convergence criterion for the steady state of the flow was given for the relative error of velocity of less than 10−6.5,305. L. Li and Y. Liu, J. Clin. Hepatol. 35, 913 (2019). https://doi.org/10.3969/j.issn.1001-5256.2019.04.04630. T. H. Wang, S. Lu, Y. Hao, Z. Su, M. Long, and Y. Cui, Biophys. J. 120, 4859 (2021). https://doi.org/10.1016/j.bpj.2021.09.020 Total 1.0 × 104–2.0 × 104 time steps were set for calculating blood flow, yielding a physical duration of 1–3h when the calculation time step is 1.67 × 10−8 s. By selecting different grid number sets of 60 × 400, 120 × 800, and 180 × 1200, it was found that this 120 × 800 set evidently meets both the requirements of computational accuracy and efficiency (data not shown).The boundary conditions for mass transport calculation are set as follows: FFA concentrations at sinusoidal inlet and outlet are CFFA = C0 and ∂CFFA/∂x=0, respectively. Inlet and outlet TG concentrations are set as periodic boundary conditions. The diffusion coefficients and parameters of FFA and VLDL are summarized in Table II. The codes were programmed using Fortran language. The computational accuracy and reliability were verified by Poiseuille pressure-driven flow, resulting in an error within 1% between analytical solution and numerical estimation.

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