Introduction 11 in vitro ex vivo 40 2 2 3 in vitro 13 39 19 27 30 16 23 25 p 2 Materials and methods 1 1 1 1 1 1 2 1 1 1 1 1 Figure 1. Longitudinal (left) and transverse (right) view of the internal geometry of the AMC-BAL with a – gas outlet; b – gas inlet; c – plasma inlet port; d – plasma outlet port; e – first mat segment; f – interspace; g – second mat segment; h – polyurethane potting to separate gas and fluid compartment; i – inner core; j – inflow zone; k – outflow zone; l – polycarbonate housing; m – gas capillaries; n – inter-capillary space through which plasma flows. The inline and triangular micro models are designated.  Computer Model 1 2 1 1 2 2 1 Figure 2. AMC-BAL micro models. Upper left, inline micro model; upper right, triangular micro model (c – capillary wall, M – non-woven matrix mat, f – inter-capillary space); lower left, inline micro model with double number of capillaries; lower right, triangular micro model with double number of capillaries.  Both models were created in the modeling software Gambit 2 (Fluent Inc., Sheffield, UK). All dimensions were derived from a laboratory-scale AMC-BAL or were supplied by the manufacturer. The curvature of the mat was neglected and the inflow and outflow zones before and after the mat segments in the AMC-BAL were not included in the micro models. Each standard micro model contains the equivalent capillary wall surface of 1 whole capillary. Since the entire laboratory-scale AMC-BAL contains 300 capillaries, each micro model can thus be considered as 1/300th part of AMC-BAL. Consequently, one AMC-BAL can be regarded as a combination of 300 separate micro models in parallel. Modeling Fluid Flow Theoretical Model in vitro 3 3 3 26 31 Resistance to Flow of the Non-woven Polyester Mat 10 −2 Boundary Conditions We assumed that total flow rate (15 ml/min) inside the research scale AMC-BAL is homogenously distributed. In this way, each micro model has the same flow rate of 0.05 ml/min, i.e. 1/300th of the total flow rate. Pressure inlet boundary conditions were used; outlet boundary condition was a zero pressure outflow; the capillary walls were ‘no-slip’ walls and symmetry boundary conditions were used at the side walls of the model. Modeling Oxygen Transport and Consumption Theoretical Model 1 1 \documentclass[12pt]{minimal}
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$$ \frac{\partial } {{\partial x_i }}(\rho u_i \phi - \rho D\frac{{\partial \phi }} {{\partial x_i }}) = S_\phi $$\end{document} p 2 p 2 −5 2 −5 2 D −9 2 −9 2 2 17 33 44 S φ 2 \documentclass[12pt]{minimal}
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$$ S_\phi = - \rho V_{\text{M}} \rho _{{\text{cell}}} \frac{\phi } {{\phi + \alpha K_{\text{M}} }} $$\end{document} 2 2 2 cell 2 et al 1 2 V M 6 K M 2 2 30 2 4 19 23 25 Hepatocyte Distribution in the Micro Models in vitro cell 6 14 cell 6 cell 6 19 20 in vitro 15 8 21 37 The complete AMC-BAL cannot be modeled by using only one cell distribution or only one micro model. Therefore, each cell distribution was applied to both micro models, leading to four basic configurations in total. All four micro model configurations were assessed independently. With a combination of these micro models, the entire AMC-BAL can be modeled in future studies. 2 2  2 2 p 2 3 2 2 3 D mat 9 2 2 \documentclass[12pt]{minimal}
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$$ \frac{{1 - {{solid}}\;{{fraction}}}} {{1 + {{solid}}\;{{fraction}}}} = 0.835 $$\end{document} 29 32 34 Figure 3. 2 p 2 p 2  2 D eff 2 2 et al 35 3 D eff D 0 D cell 3 \documentclass[12pt]{minimal}
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$$ \frac{{D_{{\text{eff}}} }} {{D_0 }} = 1 - (1 - \frac{{D_{{\text{cell}}} }} {{D_0 }})(1.727\phi - 0.8177\phi ^2 + 0.09075\phi ^3 ) $$\end{document} D cell −9 2 2 22 9 D 0 2 1 TABLE 1. 2 Region in the micro model D 0 −9 2 cell 6 φ \documentclass[12pt]{minimal}
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$$ \frac{{D_{{\text{eff}}} }} {{D_0 }} $$\end{document} D eff −9 2 Cult. med.  Plasma Cult. med. Plasma No hepatocytes     Inter-capillary space 2.92 2.18 0 0 1 2.92 2.18     In mat 2.48 1.85 0 0 1 2.48 1.85 Cell distribution 1     In mat 2.48 1.85 53.7 0.172 0.75 1.87 1.41 Cell distribution 2     In mat 2.48 1.85 31.7 0.102 0.85 2.11 1.59     In hepatocyte cell layer around capillaries – part in mat zone 2.48 1.85 81.7 0.262 0.64 1.59 1.22     In hepatocyte cell layer around capillaries – part in inter-capillary space 2.92  2.18 81.7 0.262 0.64 1.86 1.41 D eff D 0 D 0 3 D eff 6 16 18 41 43 Boundary Conditions p 2 p 2 2 2 5 2 p 2 p 2 in vitro 2 Grid Dependency One micro model mesh contained approximately 3.75 million finite volume mesh elements. Further increase in the number of cells rendered identical simulation results. Simulations Overview and Assessment 2 TABLE 2. Overview of simulations. Case Fluid p 2 gas p 2 medium Q medium # capillaries K M Reference case     (1) Standard boundary conditions Culture medium 150 146.5 0.05 1 2 Effect of the internal oxygenator p 2 gas Culture medium 0 146.5 0.05 1 2 p 2 gas p 2 medium Culture medium 0 293 0.05 1 2 Increasing oxygen availability p 2 gas Culture medium 300 146.5 0.05 1 2 p 2 gas Culture medium 722 146.5 0.05 1 2 p 2 medium Culture medium 150 293 0.05 1 2 p 2 medium Culture medium 150 722 0.05 1 2 Q medium Culture medium 150 146.5 0.10 1 2 Q medium Culture medium 150 146.5 0.50 1 2     (10) No. capill.  × 2 Culture medium 150 146.5 0.05 2 2 p 2 gas Culture medium 300 146.5 0.05 2 2 Clinical versus experimental setting     (12) Plasma Plasma 150 146.5 0.05 1 2 p 2 gas Plasma 300 146.5 0.05 1 2 2 K M Culture medium 150 146.5 0.05 1 4.75 K M Culture medium 150 146.5 0.05 1 7.5 Each case is applied to the four basic micro model configurations (inline and triangular micro model each with cell distribution 1 or 2). p 2 p 2 gas 2 p 2 p 2 2 p 2 gas K M K M et al 1 K M V M V M p 2 p 2 effective hepatocyte utilization ratio V ratio 4 30 4 \documentclass[12pt]{minimal}
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$$ V_{{\text{ratio}}} = \frac{\phi } {{\phi + \alpha K_{\text{M}} }} $$\end{document} V ratio V M V ratio 30 p 2 K M V ratio Results Fluid Flow and Shear Stress Distribution 4 4 4 Figure 4. Colorimetric contour plot of velocity magnitudes (m/s, left legend, upper part 1) and shear stress levels (Pa, right legend, lower part 2) in a transverse plane midway through the first mat segment, in the reference case (case 1) of an inline micro model with hepatocyte distribution 1 (A1 and A2 resp.) and in a triangular micro model with hepatocyte distribution 2 and with double number of capillaries (case 10–11) (B1 and B2 resp.).  Figure 5. p 2 V ratio p 2  4 4 4 4 4 4 4 3 TABLE 3. Overview of the maximum velocity in the inter-capillary space, the uniform velocity in the mat/hepatocyte layer zone, the static pressure loss over the entire model and the maximal shear stress for all simulation cases and for different hepatocyte distributions. Cases Hepatocyte distr. 1 Hepatocyte distr. 2 Velocity in mat/hepatocyte cell layer (mm/s) – Maximum velocity in inter-capillary space (mm/s)     (1,2,3,4,5,6,7,14,15) Standard flow rate 0.0085 – 3.59  0.011 – 4.60  Q medium 0.017 – 7.17 0.022 – 9.20 Q medium 0.084 – 35.7 0.11 – 45.0     (10,11) No capillaries × 2 0.0195 – 5.69 0.037 – 7.75     (12,13) Plasma 0.0085 – 3.59 0.011 – 4.60 Static pressure loss over micro model (Pa)     (1,2,3,4,5,6,7,14,15) Standard flow rate 15.7 20.6 Q medium 31.4 41.1 Q medium 155.7 203.3     (10,11) No capillaries × 2 36.1 69.3     (12,13) Plasma 30.0 39.3 Maximum shear stress (Pa)     (1,2,3,4,5,6,7,14,15) Standard flow rate 0.032 0.041 Q medium 0.064 0.083 Q medium 0.31 0.40     (10,11) No capillaries × 2 0.056 0.083     (12,13) Plasma 0.057 0.073 Values for the inline and triangular micro model are identical. Importantly, both the inline and triangular micro model had the same flow field, static pressure loss and shear stress distributions within a simulation case with certain boundary conditions and hepatocyte distribution (comparison not shown). Oxygen Transport and Consumption p 2 V ratio 5 4 V ratio TABLE 4. V ratio V ratio Case Hepatocyte distribution 1 Hepatocyte distribution 2 Inline (%) Triangular (%) Inline (%) Triangular (%) Mat (%) Capill. (%) Mat (%) Capill. (%) Reference case (1) Standard boundary conditions 15.7 15.8 28.8 28.6 3.3 54.1 3.3 53.5 The effect of the internal oxygenator p 2 gas 1.7 1.7 1.7 1.7 3.1 0.4 3.1 0.4 p 2 gas p 2 medium 6.2 6.2 6.3 5.9 11.0 1.7 10.3 1.7 Increasing oxygen availability p 2 gas 30.4 30.2 50.3 50.3 5.1 95.1 5.0 94.9 p 2 gas 62.4 64.5 80.7 88.3 61.2 100.0 76.3 100.0 p 2 medium 20.1 20.2 34.5 33.9 11.8 57.0 11.0 56.5 p 2 medium 35.5 35.1 54.2 53.4 40.6 67.7 37.5 69.1 Q medium 17.9 18.1 31.6 31.4 6.6 56.4 6.5 56.0 Q medium 28.4 28.6 45.7 45.6 22.2 68.9 22.1 68.9     (10) No capillaries × 2 31.8 31.8 56.7 56.8 14.1 100.0 14.3 100.0 p 2 gas 61.0 59.9 87.5 84.8 75.3 100.0 69.9 100.0 Clinical versus experimental setting     (12) Plasma 12.0 12.0 22.9 23.0 2.6 43.0 2.6 43.0 p 2 gas 22.8 22.9 40.8 40.9 2.9 78.4 2.9 78.3 2 K M 8.7 8.7 18.9 18.9 1.1 36.4 1.1 36.4 K M 5.0 5.1 12.9 13.0 0.3 25.3 0.3 25.5 In the case of hepatocyte distribution 2, distinction is made between the percentage of the total number of hepatocytes in the mat and the percentage of total number of hepatocytes in the cell layers around the capillaries. Reference Case Simulations (Case 1) 5 p 2 p 2 p 2 p 2 p 2 5 p 2 V ratio 5 p 2 V ratio V ratio V ratio 4 p 2 p 2 V ratio V ratio p 2 V ratio p 2 The Effect of the Internal Oxygenator (Case 2,3) p 2 gas V ratio p 2 V ratio Increasing Oxygen Availability (Case 4–11) p 2 V ratio 4 p 2 gas V ratio p 2 V ratio p 2 medium V ratio V ratio V ratio V ratio p 2 gas 5 p 2 p 2 p 2 p 2 5 V ratio p 2 5 V ratio V ratio 4 Clinical Versus Experimental Setting (Case 12,13) V ratio p 2 gas in vitro V ratio 2 K M p 2 p 2 V ratio Discussion Fluid Flow and Shear Stress Distribution Perfusion of a micro model was largely influenced by the presence of the non-woven mat. Although the cross-sectional area of the inter-capillary space and the non-woven mat are roughly the same size, fluid flow in the mat zone was generally two orders of magnitude smaller as compared to the flow in the inter-capillary space. This effect is caused by the higher resistance to flow of the non-woven mat, forcing the majority of fluid flow through the inter-capillary spaces, which have a negligible flow resistance. This also causes the overall pressure loss to be minimal. Since the same hydraulic permeability is used for the hepatocyte layers, fluid velocities there are also in the range of micrometer per second. Consequently, apart from the incorporation of additional gas capillaries (case 10–11), also the presence of hepatocyte layers around the capillaries (cell distribution 2) increased flow velocities and pressure loss in the model as the free cross-sectional area of the inter-capillary space is reduced. Fluid flow simulations for the inline and triangular variant of the different case studies render identical results for velocity profiles, pressure loss and shear stress distributions. This was expected as the micro models consist of the identical geometrical entities, which are only changed in location relative to each other. From a fluid dynamical point of view, we conclude that the change in capillary arrangement along the course of the spiral mat in the AMC-BAL does not influence fluid flow, pressure drop or shear stress distribution. 1 in vivo 7 28 Shear stress was also assessed as it is a possible determinant of cellular damage and reduced metabolic function. Shear stress is directly proportional to the local velocity gradient and the fluid viscosity. Consequently, shear stresses are generally more elevated in cases with higher velocity magnitudes in the inter-capillary space (e.g. in case of increased flow rate, doubled number of capillaries, cell distribution 2) and where fluid viscosity is increased (e.g. when plasma was used instead of culture medium – ‘clinical setting’ – case 12–13). Results show that only hepatocytes located at the side of the mat and at the border of the hepatocyte layers with the inter-capillary space are subjected to a certain level of shear stress. In vivo \documentclass[12pt]{minimal}
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$$ \tau  = {\raise0.7ex\hbox{${\Delta p \cdot D}$} \!\mathord{\left/ {\vphantom {{\Delta p \cdot D} {4L}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${4L}$}}$$\end{document} p D L 7 28 24 27 in vivo et al 38 in vitro 38 2 2 Oxygen Transport and Consumption We assessed the oxygen availability in different case studies and in different micro model configurations. In this paragraph, the results are discussed per case. Reference Case Simulations (Case 1) p 2 2 2 2 V ratio p 2 V ratio p 2 Since regions which are oxygenated by one gas capillary in particular do not overlap or influence each other in the reference case simulations, diffusive oxygen supply by the capillaries is independent of relative capillary location. Convective oxygen transport is also identical as flow distribution is irrespective of capillary placement (section “Fluid flow and shear stress distribution”). Consequently, inline and triangular capillary pattern give identical results concerning oxygen transport and consumption in these reference cases when the same hepatocyte distribution is used, as is confirmed by simulation results. 2 V ratio p 2 p 2 42 p 2 10 p 2 36 V ratio V M K M in vitro V M K M 2 The Effect of the Internal Oxygenator (Case 2,3) 2 V M 6 6 V ratio p 2 p 2 V ratio p 2 Increasing Oxygen Availability (Case 4–11) V ratio V ratio p 2 V ratio p 2 p 2 p 2 medium p 2 p 2 V ratio p 2 V ratio p 2 V ratio p 2 12 p 2 p 2 4 V ratio V ratio p 2 V ratio p 2 V ratio V ratio p 2 gas V ratio V ratio V ratio p 2 2 V ratio in vitro Clinical Versus Experimental Setting (Case 12,13) V ratio p 2 V ratio V ratio 2 K M K M p 2 2 V ratio V ratio p 2 4 p 2 p 2 V ratio p 2 V ratio K M p 2 V ratio p 2 Conclusions V ratio p 2 2 in vitro 2 in vitro in vitro