The Computed Sinusoid

: Hepatic sinusoids are lined with thin endothelial cells with transcellular pores, termed fen-estrations. These fenestrations are open channels that connect the sinusoidal lumen to the underlying Space of Disse (SoD) and the hepatocytes of the liver parenchyma. Fenestrations range from 0.05 to 0.35 µ m in diameter and cover 5–15% of the sinusoidal endothelial surface area, depending on their location along the sinusoids. The direct measurement of hemodynamic parameters, such as pressure and ﬂow velocity, remains challenging within the narrow sinusoids. Such knowledge would increase our understanding of the physiology of the hepatic niche and possible implications in aging or diseases in which fenestrations are reduced or lost. Few simulations of liver blood ﬂow focus on the level of the individual sinusoid, and fewer still include the transcellular pores (fenestrations) of the sinusoidal endothelium. Furthermore, none have included (i) a porosity gradient along the sinusoid wall, modeled using through-all pores rather than a porous medium, (ii) the presence of the SoD, or (iii) lymphatic drainage. Herein, computed ﬂuid dynamics (CFD) simulations were performed using a numerical model with relevant anatomical characteristics (length, diameter, porosity, inlet/outlet pressure, and lymphatic outﬂow from the portal region of the SoD). The greatest contribution to luminal velocity magnitude and pressure was the overall shape of the vessel. Divergent-radius models yielded velocity magnitudes 1.5–2 times higher than constant-radius models, and pressures were 5–8% lower in the divergent-radius models compared to the constant-radius models. Porosity only modestly contributed to luminal pressure. The luminal velocity magnitude was largely unaffected by the presence or absence of lymphatic drainage. Velocity magnitudes through fenestrations were lower in higher-porosity models (20%) vs. lower-porosity models (5%) across all models (0.4–0.55-fold lower). Velocity magnitudes through the space of Disse were increased 3–4 times via the addition of lymphatic drainage to the models, while pressures were decreased by 6–12%. The ﬂow velocity in the SoD was modiﬁed via differences in porosity, while the ﬂow velocity in the lumens of the sinusoids was largely unaffected. The overall shape of the vessel is the single most important factor in the pressure ﬂow behavior of the sinusoidal lumen. The ﬂow rate over hepatocytes and the SoD is modestly affected by the distribution of porosity along the sinusoid and greatly affected by the lymphatic drainage, parameters that would be of interest for modeling the exchange of blood with the hepatic parenchyma.

Figure 1.Schematic of the liver sinusoid: liver sinusoid endothelial cells (LSECs) form the highly specialized and fenestrated endothelium of the sinusoid.Resident macrophages (Kupffer cells (KCs)) populate the sinusoidal lumen, while hepatic stellate cells can be found within the Space of Disse (SoD), an approximately 1 µm thick region with a sparse extracellular matrix (grey bundlesproteoglycans and collagen type III) that separates the LSECs from the hepatocytes (hep).Blood rich in nutrients and oxygen flows from the portal vein (PV) and the hepatic artery (HA) toward the central vein (CV) (purple arrows).Bile is formed in the hepatocytes and flows through the bile canaliculi, which are situated between hepatic cords (dark green arrows).Lymph is largely (ca.80%) formed from the filtrate in the SoD and flows into the lymphatic vasculature (LV), which is in the Space of Mall (SoM) (light green arrows).
In rats, the mean linear flow rate of blood within the hepatic microvasculature is roughly 144 µm/s.Flow speed and fluidic resistance increase from the periportal zone to the perivenous zone [1,5,6].Conversely, the pressure along the sinusoid decreases from 70 mmHg (9333 Pa) in zone 1 to 30 mmHg (2666 Pa) in zone 3 [5].While velocities can be measured directly by tracking leukocytes or other particles in the sinusoids via in vivo microscopy, pressures across the sinusoids must be estimated from measurements of terminal portal venules and terminal hepatic venules [5].Endothelial porosity (the area of the endothelium covered with fenestrations) also varies across hepatic zones, with fewer and narrower fenestrations in the periportal zone relative to the pericentral zone(Table 1) [2,[7][8][9][10].
Table 1.Partial summary of the literature concerning sinusoidal dimensions and flow parameters.

Reference (i) Model; (ii) Method; (iii) Sinusoid Dimensions; (iv) Flow; (v) Pressure; (vi) Fenestrations
Wisse, 1983 [10] (i) Rat; (ii) SEM; (vi) porosity is higher and fenestrations have wider diameters in zone 3 than in zone 1 (97.92 vs. 76.57nm and 11.63 vs. 6.81%)Vidal-Vanaclocha and Barbera-Guillem, 1985 [8] (i) Rat; (ii) SEM; (vi) zone 3 has wider fenestrations (94-121 nm vs. 73-101 nm) and a higher frequency (10.21-10.68fenestrations/µm 2 vs. 5.74-6.26fenestrations/µm 2 ) than zone 1 and a greater number of sieve plates (1.73-fold greater) Horn, 1986 [7] (i) Human; (ii) SEM; (vi) in zone 3, fenestrations are more numerous (23.5 vs. 19.2%)than in zone 1, and porosity is higher in zone 3 than in zone 1 (9.1 vs. 7.6%) Wake, 1988 [3] (i) Rat; (ii) light and electron microscopy; (iii) centrilobular LSECs are larger (longer and wider) than periportal LSECs Henriksen and Lassen, 1988 [11] (i) Theoretical model; (iv) the shape of the sinusoid does not affect the flow profile, which is characterized by an increasing speed moving from zone 1 to zone 3; (v) in humans, the pressure drop between the portal and central veins is between 3 and 5 mmHg (450 Pa) In rats, the mean linear flow rate of blood within the hepatic microvasculature is roughly 144 µm/s.Flow speed and fluidic resistance increase from the periportal zone to the perivenous zone [1,5,6].Conversely, the pressure along the sinusoid decreases from 70 mmHg (9333 Pa) in zone 1 to 30 mmHg (2666 Pa) in zone 3 [5].While velocities can be measured directly by tracking leukocytes or other particles in the sinusoids via in vivo microscopy, pressures across the sinusoids must be estimated from measurements of terminal portal venules and terminal hepatic venules [5].Endothelial porosity (the area of the endothelium covered with fenestrations) also varies across hepatic zones, with fewer and narrower fenestrations in the periportal zone relative to the pericentral zone (Table 1) [2,[7][8][9][10].Henriksen and Lassen, 1988 [11] (i) Theoretical model; (iv) the shape of the sinusoid does not affect the flow profile, which is characterized by an increasing speed moving from zone 1 to zone 3; (v) in humans, the pressure drop between the portal and central veins is between 3 and 5 mmHg (450 Pa) The liver is the largest site of lymph production in the body, with up to 50% of the lymph that drains into the thoracic duct formed here [14].Hepatic lymph generation begins with the filtration of blood through the fenestrations of the sinusoidal lining, followed by drainage through the lymphatic vasculature beginning in the Space of Mall (SoM), a region of the portal tract situated between the outermost hepatocytes and the hepatic stroma [15][16][17]).Lymph production is correlated with hydrostatic pressure within the sinusoids, with even slight pressure changes increasing lymph production and flow [14,15].This physiological consequence, which is particularly evident in pathological conditions such as portal hypertension, is due to the high permeability of the sinusoids [18].

Models of the Hepatic Sinusoids
Given the inaccessibility of the liver sinusoids to sensors for direct measurements of hemodynamic variables, computed fluid dynamics (CFD) simulations were used to model the flow field here (Table 2).There are few models of the fluid dynamics of the liver or hepatic sinusoids that account for the presence of fenestrations [24], with most studies simulating whole lobules or larger areas of the liver [19][20][21][22][23][25][26][27][28][29][30][31][32][33] and, as such, accounting for porosity in a more general way (as porous medium) in their models.No one has, to our knowledge, added variable porosity to their models as such.Furthermore, the porosity of the liver sinusoid is reduced in several pathologies/conditions, such as cirrhosis [34], alcoholic liver disease [35], and in aging [36].We therefore believe there is value added in a model that examines the single-sinusoid level, investigating the contributions made by the overall shape and the distribution of fenestrations (the porosity) in a computational fluid dynamics model of a single sinusoid.This is especially due to the effects found in microfluidics of an increasing versus constant porosity on the fluid flow velocity through a microchannel [37].Brainerd et al. found that in a micro-channel lined with pores, the magnitude of the outflow velocity dropped significantly along its length if the porosity (% area fraction covered by open pores) was even along the length, while to achieve an even outflow from the channel, the porosity needed to increase along the length.Taken together with electron microscopic observations made on liver tissue samples, porosity was expected to contribute to fluid dynamics in the liver sinusoid.
Here, we seek to model the hemodynamics of a single liver sinusoid with a computationally inexpensive model that contains the most important ultrastructural details of the sinusoid.We aim to decipher the relative contributions of (i) the radius of the sinusoid (either as constant or expanding), (ii) the presence or absence of a periportal lymphatic drainage outlet in the space of Disse, and (iii) the distribution of fenestrations to pressure profiles and flow behavior.

Computational Fluid Dynamics (CFD) Simulations
CFD simulations were performed using Ansys ® 2021 R1 Academic software and a laptop with the features listed below (Table 3).The numerical 2D models utilized in the simulations were two half-sections of a simplified sinusoid with either a constant luminal radius (named C = constant-radius sinusoid) or a diverging luminal radius (named D = diverging sinusoid).
Each model was tested with different porosity and inlet pressure configurations (physiological vs. pathological pressure).
The effect of a lymphatic outflow in the portal tract of the SoD was explored for each model.
Linear porosity was defined as the ratio between the length given by the sum of the fenestrations and the length of the sinusoid.
Zonal linear porosity was defined as the ratio between the length given by the sum of the fenestrations in a certain zone and the length of the zone itself.
Numerical models with constant porosity were obtained by arranging fenestrations of a constant pitch along the whole length of the sinusoid.When variable porosity was applied, the fenestration pitch varied zone by zone (but the fenestrations were evenly spaced inside the zone itself).Zones 1, 2, and 3 of the sinusoid had porosities of 5%, 6%, and 20%, respectively.Thus, changes in porosity were applied by increasing the number of fenestrations rather than enlarging their diameter (Figure 2).
Numerical models with constant porosity were obtained by arranging fenestrations of a constant pitch along the whole length of the sinusoid.When variable porosity was applied, the fenestration pitch varied zone by zone (but the fenestrations were evenly spaced inside the zone itself).Zones 1, 2, and 3 of the sinusoid had porosities of 5%, 6%, and 20%, respectively.Thus, changes in porosity were applied by increasing the number of fenestrations rather than enlarging their diameter (Figure 2).

Geometry and Mesh
The numerical 2D models of the sinusoid were designed to comply with both the computational capabilities of the hardware and anatomical likelihood.

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The sinusoid was designed as a half-section measuring 275 µm long.Two half-sections were evaluated, one with a constant radius (3.5 µm) and one with a linearly increasing radius (the inlet/outlet radii were, respectively, set to 3.5 µm and 7.5 µm).

Geometry and Mesh
The numerical 2D models of the sinusoid were designed to comply with both the computational capabilities of the hardware and anatomical likelihood.

•
The sinusoid was designed as a half-section measuring 275 µm long.Two half-sections were evaluated, one with a constant radius (3.5 µm) and one with a linearly increasing radius (the inlet/outlet radii were, respectively, set to 3.5 µm and 7.5 µm).

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The SoD was modeled as a 1 µm thick 2D chamber surrounding the sinusoid lumen and communicating with it via fenestrations.

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The fenestrations were modeled as 100 nm long and 150 nm high channels connecting the sinusoidal lumen with the SoD (Figure 3).

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The SoD was modeled as a 1 µm thick 2D chamber surrounding the sinusoid lumen and communicating with it via fenestrations.

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The fenestrations were modeled as 100 nm long and 150 nm high channels connecting the sinusoidal lumen with the SoD (Figure 3).Proper geometry is essential to facilitating the meshing process (the discretization of the whole surface into tiny sub-surfaces defined by nodes for which the solver computes the solutions of the fluid dynamics equations).The main design strategies used to obtain the geometry of the sinusoid were as follows:

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The main walls (of the sinusoidal lumen and the Space of Disse lumen) were formed as two coaxial rectangles (or trapezoids when the sinusoid had a diverging section).

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Fenestrations were modeled as a linear pattern.

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The sketch was converted into a surface, and a symmetry axis was introduced (halving the model).
The finite element method (FEM) facilitates a complex system's numerical simulation.This involves the discretization of a continuous system into small elements (named cells, which are defined by nodes) over which to solve the equations.The obtained local solutions are ultimately integrated over entire domains and bodies to produce a global solution.Thus, the quality of the solutions generated by the solver strictly depend on the quality of the mesh, which defines the size, distribution, and shape of the finite elements.A reduced number of elements lead to a coarse solution with low computational costs.A high number of elements gives an accurate solution which requires time-consuming cal- Proper geometry is essential to facilitating the meshing process (the discretization of the whole surface into tiny sub-surfaces defined by nodes for which the solver computes the solutions of the fluid dynamics equations).The main design strategies used to obtain the geometry of the sinusoid were as follows:

•
The main walls (of the sinusoidal lumen and the Space of Disse lumen) were formed as two coaxial rectangles (or trapezoids when the sinusoid had a diverging section).

•
Fenestrations were modeled as a linear pattern.

•
The sketch was converted into a surface, and a symmetry axis was introduced (halving the model).
The finite element method (FEM) facilitates a complex system's numerical simulation.This involves the discretization of a continuous system into small elements (named cells, which are defined by nodes) over which to solve the equations.The obtained local solutions are ultimately integrated over entire domains and bodies to produce a global solution.Thus, the quality of the solutions generated by the solver strictly depend on the quality of the mesh, which defines the size, distribution, and shape of the finite elements.A reduced number of elements lead to a coarse solution with low computational costs.A high number of elements gives an accurate solution which requires time-consuming calculations.The mesh obtained had a good quality (Figure 3), thus ensuring accurate solutions.However, the mesh can be further improved by reducing the size of the elements (set here to 0.1 µm).
The laminar flow module of a pressure-based solver, which couples mass and momentum conservation with no-slip boundary conditions, was applied to disclose the pressure and linear speed profiles of a steady flow for an incompressible fluid (blood).

Solver Configuration
A Fluent solver (by Ansys) was utilized, setting a laminar-flow module with a no-slip condition at the boundaries (v = 0 at the walls).Blood at 37 • C was selected as the material (η = 0.0035 kg•m −1 •s −1 , ρ = 1060 kg/m 3 ).Since the computational model was based on a pressure-driven flow, the physiological pressures were set to 1067 and 800 Pa, respectively, at the inlet/outlet [13].Pathological conditions (e.g., portal hypertension) were introduced, elevating inlet pressure up to 2400 Pa [13].To simulate lymphatic drainage, a pressure outlet was added at the portal region (zone 1) of the SoD, and the selected exit pressure was set to 100 Pa [38].The equations were solved using the COUPLED algorithm (keeping default under-relaxation factors).The solutions converged after 105 iterations (which were initially set to 2000 iterations).
The physics of the numerical model can be explained with the following partial differential equations (PDEs): That is, the mass conservation equation for an incompressible fluid where ρ is the density (kg/m −3 ) and u is the 3D velocity vector (m/s) which is the equation of momentum for Newtonian fluids (constant µ) where P is the pressure (Pa) and µ is the dynamic viscosity (Pa•s).

Results
The overall shape of the vessel had the greatest effect on the measured parameters, with the velocity magnitude greater in the divergent models compared with the constantradii models.Velocity modules through the lumen were approximately 1.5-2-fold higher in the divergent models vs. the cylindrical models (Figures 4-6 and Tables 4-6), and the average pressure in the lumen was about 5-8% lower in the divergent models vs. the cylindrical models (Figures 7-9 and Tables 4, 7 and 8).The increment of zonal porosity increased the velocity in the SoD by 9-16% while decreasing the velocity through fenestrations by 40-55%; there was a modest decrease in luminal velocity in the cylindrical models, while there was a slight increase in the divergent models.Velocity magnitudes through the SoD were 3-4-fold higher in models with lymphatic drainage than in those without (Figures 4, 5, and 8; Tables 5 and 6).Velocity magnitudes through fenestrations were lower in high-porosity models (high = 20%) without lymphatic drainage, while in models with lymphatic drainage, the velocities were higher when the porosity was low (low = 5%) than when the porosity was high; additionally, velocities were lower in variable-porosity models (variable = 5, 6, 20%) than in either low-or high-porosity models (Figures 4, 5, and 8; Tables 5 and 6).
The luminal velocity magnitude was slightly higher in constant-radius models without lymphatic drainage (with high and variable levels of porosity) and unchanged in divergent-radius models.In constant-radius models with lymphatic drainage, the luminal velocity was higher in low-and variable-porosity models.The luminal velocity in divergent-radius models was higher when models had high levels of porosity and lymphatic drainage and were equal in all others (Figures 4, 5, and 8; Tables 5 and 6).
For models without lymphatic drainage and with a divergent radius, the luminal pressure was the lowest in variable-porosity models, while for constant-radius models with lymphatic drainage, variable-porosity models had slightly higher luminal pressure values.In divergent models with lymphatic drainage, variable-porosity models had the highest average pressure, with lower pressures observed in low-porosity models.The pressure was even lower in high-porosity models (Figures 6 and 7; Tables 7 and 8).constant radius (cylinder) or with a diverging radius (conical) and with or without lymphatic drainage.
The pressure across fenestrations was the lowest in the variable-porosity models compared to those with uniform porosity.In the latter, pressure across fenestrations was greatest in the high-porosity models compared to the low-porosity models (Figures 6, 7 and 9; Tables 7 and 8).Table 7. Pressure in sinusoids modeled without lymphatic drainage.Const.rad.= constant radius; Div.rad.= diverging radius; porosity given as %; Var = variable increasing porosity 5-20%; l = lumen centreline; f = fenestrations; D = Space of Disse.The increment of zonal porosity increased the velocity in the SoD by 9-16% while decreasing the velocity through fenestrations by 40-55%; there was a modest decrease in luminal velocity in the cylindrical models, while there was a slight increase in the divergent models.Velocity magnitudes through the SoD were 3-4-fold higher in models with lymphatic drainage than in those without (Figures 4-6; Tables 5 and 6).

Const
Velocity magnitudes through fenestrations were lower in high-porosity models (high = 20%) without lymphatic drainage, while in models with lymphatic drainage, the velocities were higher when the porosity was low (low = 5%) than when the porosity was high; additionally, velocities were lower in variable-porosity models (variable = 5, 6, 20%) than in either low-or high-porosity models (Figures 4-6; Tables 5 and 6).
The luminal velocity magnitude was slightly higher in constant-radius models without lymphatic drainage (with high and variable levels of porosity) and unchanged in divergentradius models.In constant-radius models with lymphatic drainage, the luminal velocity was higher in low-and variable-porosity models.The luminal velocity in divergent-radius models was higher when models had high levels of porosity and lymphatic drainage and were equal in all others (Figures 4-6; Tables 5 and 6).
For models without lymphatic drainage and with a divergent radius, the luminal pressure was the lowest in variable-porosity models, while for constant-radius models with lymphatic drainage, variable-porosity models had slightly higher luminal pressure values.In divergent models with lymphatic drainage, variable-porosity models had the highest average pressure, with lower pressures observed in low-porosity models.The pressure was even lower in high-porosity models (Figures 7 and 8; Tables 7 and 8).
The pressure across fenestrations was the lowest in the variable-porosity models compared to those with uniform porosity.In the latter, pressure across fenestrations was greatest in the high-porosity models compared to the low-porosity models (Figures 7-9; Tables 7 and 8).
In models without lymphatic drainage, the pressure in the SoD was always lower in the variable-porosity models, whereas in models with lymphatic drainage, the pressure increased from low to variable to high (Figures 7-9; Tables 7 and 8).
In general, the variable porosity model had velocity and pressure curves between low constant porosity and high constant porosity, being somewhat closer to low constant porosity (Figures 4-9).
In total, the overall shape of the vessel (Figures 4-9, Tables 4-8) and the presence or absence of lymphatic drainage in the periportal zone had the largest effects on the flow parameters, while porosity had some less-pronounced effects.
Modeling with a pathological (elevated) pressure regimen did not show any changes in pressure or flow behavior.Increasing the input/output pressures to 2400/800 Pa, as per Ryou 2020 [13], only rescaled proportionally to increases in pressure, with the same patterns as those found for physiological pressures, i.e., the differences between them were merely rescaled.
Admixing: Regarding flowlines, the addition of lymphatic drainage in the periportal zone leads to more fluid moving through the SoD and more admixing relative to models without, whereas porosity only has a modest effect when comparing models with low, high, or variable porosity (Figures S1-S3).Divergent models show the formation of stronger vortexes at the outlet compared with constant-radius models.

Discussion
Four physiologically relevant 2D models of hepatic sinusoids were generated for CFD simulations to provide hemodynamics insights.Simulations were carried out using a laminar and steady flow of blood (constant dynamic viscosity) generated via a differential pressure between the sinusoid's inlet and outlet.Velocity and pressure trends were collected for all models for physiological and pathophysiological (elevated) pressure conditions.Also, an extra outlet was added to the model to reproduce the lymphatic drainage in the portal zone of the SoD.

Major Insights about Sinusoidal Pressure (P)
Pressure decreases linearly in constant-radius models (Cmods) and exponentially in diverging-radius models (Dmods).Dmods generally have lower pressure throughout when compared with Cmods.Increased porosity in the pericentral zone implies a generalized pressure reduction, mostly borne by the SoD.Lymphatic drainage reduces with the pressure within the sinusoid, especially inside the SoD.This effect might be exaggerated by the model (see below in general considerations).
Pathological conditions, (PatCs), elevated pressures, merely re-scale the same pressure behavior obtained for physiological conditions (PhyCs).In terms of average pressure, the sinusoidal lumen and SoD have comparable pressure, while through fenestrations, pressure is generally lower.

Major Insights Regarding Flow Velocity (V)
In general, Cmods have constant luminal velocity and an almost constant velocity within the SoD.Velocity through the fenestration develops along the sinusoid with a parabolic trend, with higher values at the inlet/outlet (where the flow enters/exits the SoD).In general, Dmods demonstrate a slowly decreasing velocity along the sinusoidal lumen and along the SoD.Through the fenestrations, velocity decreases within the first 50-100µm and then follows a sinusoidal trend, with values increasing in the proximity of the outlet.Globally, Dmods produced greater velocities in all compartments of the sinusoid when compared with Cmods.Some of the models (especially but not only the Dmods) present a reverse flow at the outlet.Porosity augmentation in the perivenous zone mostly affects velocity of the SoD and through fenestrations (with no effect on the luminal speed-the flow through fenestrations and the SoD seeks to compensate for the changes in porosity ).Generally, velocity increases within the SoD, while it decreases through fenestrations.Adding lymphatic drainage to the perivenous zone generated a shift in the velocity trend toward the outlet.The average flow speed through fenestrations reaches higher values and decreases more slowly within the first half of the sinusoid (an increased flow exchange between the lumen and SoD).Similarly, the flow along the SoD is characterized by a higher average speed.PatCs merely re-scale the same velocity behavior obtained for PhyCs.

General Considerations and Limitations
The overall shape of the sinusoids had the greatest influence on the luminal pressure and velocities; this is a parameter that may be altered in disease states [12,34].When implementing a larger-scale model, this is likely the single most important parameter investigated here.Intuitively, we would have expected that porosity would have far greater effects on these luminal parameters, but this was not the case except at an extremely high porosity.
The addition of lymphatic drainage affected the flow through the fenestrations and SoD.It is therefore an important parameter to consider when modeling liver sinusoids.Lymphatic drainage in the SoD was modeled as a depressurization affecting the flow within the entire sinusoid.Since the lymphatic flow rate is estimated to be 100-500 times lower than the flow rate of blood [39], it may be necessary to adjust the pressure value at the drainage outlet.Also, lymphatic drainage is expected to be much higher under pathological conditions [14,16]; this aspect was not taken into account in our simulations.
With the current boundary conditions (PhyCs and PatCs), porosity variations seem to be fully compensated for through an exchange of flow between the lumen and SoD via fenestrations without affecting the flow velocity inside the lumen.If this model translates to the liver, then, in theory, the liver can change the flow in the SoD by changing the porosity without altering the luminal flow velocity.In the model, the elevated pressure seen in pathologies does not cause alterations to the flow pattern with unchanged geometries.While the model does not account for cellular responses to elevated pressure and flow velocity, it shows that the sinusoidal geometry must be altered for flow patterns to change.
The model employs a homogeneous variation of porosity, with evenly spaced and sized fenestrations.We believe this simplification in our model is justified for our application, but others will need to evaluate the complexity required by their inquiry.The model simplifies the sinusoid into a straight line, whereas the real case would be curved and branching.In addition, liver sinusoids are flexible and dynamic structures due to the fact they are in a soft tissue and are exposed to pulsatile flow.Our simplifications were necessary to home in on the focus of this article, namely the variable porosity and diameter in and of the sinusoid.
Luminal flow and pressure are mainly affected by the overall shape, i.e., the evolution of the vessel's diameter, with porosity mostly affecting flow within the SoD.Variable porosity, with higher porosity toward the pericentral/zone 3, modestly increases flow velocity through the SoD relative to a constant-porosity model but also decreases velocity through fenestrations significantly.In the variable-porosity models, pressures through fenestrations were lower than for models with either high or low constant porosity.Similarly, the pressure in the SoD was lower for models without lymphatic drainage, or similar (less than 1% increase) for models with lymphatic drainage, in the variable-porosity models compared with the constant-low-porosity models and always lower than in the constanthigh-porosity models.
We simulated lymphatic drainage by adding an outlet in the periportal area of the SoD; this inclusion increased flow velocity and the exchange between the lumen and SoD.However, more detailed studies of how this parameter evolves are required as it is poorly understood and it is currently not feasible to measure it directly.This is beyond the scope of the current study.Some effects may be underestimated in the model due to parameter reductions as pulsatile flow, curved geometry, tissue compressibility, and the obstruction of flow by migrating blood cells [4,40] were not incorporated.The addition of these may be feasible with better computational hardware.Adding fenestrations with realistic porosity to a larger, more detailed model, such as the one used by Piergiovanni (2017) [22], would perhaps help elucidate the distribution patterns of various solutes and colloids at the sinusoid level but would have greater requirements in terms of both time and hardware.
The model did not account for the pulsatility of the flow, as is the case for blood flow, or the elasticity of the tissue itself, which can compress in response to pressure.Lymphatic outflow was simplified to a constant, and more accurate modeling would

Figure 1 .
Figure1.Schematic of the liver sinusoid: liver sinusoid endothelial cells (LSECs) form the highly specialized and fenestrated endothelium of the sinusoid.Resident macrophages (Kupffer cells (KCs)) populate the sinusoidal lumen, while hepatic stellate cells can be found within the Space of Disse (SoD), an approximately 1 µm thick region with a sparse extracellular matrix (grey bundles-proteoglycans and collagen type III) that separates the LSECs from the hepatocytes (hep).Blood rich in nutrients and oxygen flows from the portal vein (PV) and the hepatic artery (HA) toward the central vein (CV) (purple arrows).Bile is formed in the hepatocytes and flows through the bile canaliculi, which are situated between hepatic cords (dark green arrows).Lymph is largely (ca.80%) formed from the filtrate in the SoD and flows into the lymphatic vasculature (LV), which is in the Space of Mall (SoM) (light green arrows).

Figure 2 .
Figure 2. Schematics of the numerical models of the sinusoid.The model was designed as a halfsection.Two main versions were adopted: constant-radius and diverging sinusoids a).Sizes are in microns.An example of the more complete model adopted in the simulations b): a diverging section, including a variable porosity (5%, 6%, and 20%) and an extra outlet at the portal side of the SoD to mimic lymphatic drainage (dark arrows indicate the direction of the flow).

Figure 2 .
Figure 2. Schematics of the numerical models of the sinusoid.The model was designed as a halfsection.Two main versions were adopted: constant-radius and diverging sinusoids (a).Sizes are in microns.An example of the more complete model adopted in the simulations (b) a diverging section, including a variable porosity (5%, 6%, and 20%) and an extra outlet at the portal side of the SoD to mimic lymphatic drainage (dark arrows indicate the direction of the flow).

Figure 3 .
Figure 3. Mesh highlights shown for the diverging sinusoid model.The side of each element of the mesh was set to a max size of 0.1 µm.Further, in the bottom panel, the quality spectrum for the orthogonality metric is reported.

Figure 3 .
Figure 3. Mesh highlights shown for the diverging sinusoid model.The side of each element of the mesh was set to a max size of 0.1 µm.Further, in the bottom panel, the quality spectrum for the orthogonality metric is reported.

Figure 4 .
Figure 4. Velocity magnitudes along sinusoids modeled without lymphatic drainage.Models are with a constant radius (cylinder) or with a diverging radius (conical) and low (5%) porosity or high (20%) porosity.

Figure 4 .
Figure 4. Velocity magnitudes along sinusoids modeled without lymphatic drainage.Models are with a constant radius (cylinder) or with a diverging radius (conical) and low (5%) porosity or high (20%) porosity.

Figure 5 .
Figure 5. Velocity magnitudes along sinusoids modeled with lymphatic drainage.Models are with a constant radius (cylinder) or with a diverging radius (conical) and low (5%) porosity or high (20%) porosity.

Figure 6 .
Figure 6.Velocity along sinusoids modeled with variable porosity (5, 6, 20%).Models are with a constant radius (cylinder) or with a diverging radius (conical) and with or without lymphatic drainage.

Figure 8 .
Figure 8. Pressure along sinusoids modeled without lymphatic drainage.Models are with a constant radius (cylinder) or with a diverging radius (conical) and low (5%) porosity or high (20%) porosity.

Figure 9 .
Figure 9. Pressure along sinusoids modeled with variable porosity (5, 6, 20%).Models are with a constant radius (cylinder) or with a diverging radius (conical) and with or without lymphatic drainage.

Figure 9 .
Figure 9. Pressure along sinusoids modeled with variable porosity (5, 6, 20%).Models are with a constant radius (cylinder) or with a diverging radius (conical) and with or without lymphatic drainage.

Table 1 .
Partial summary of the literature concerning sinusoidal dimensions and flow parameters.

Table 2 .
The most significant studies on numerical models of the liver's microvasculature.

Table 4 .
Quantitative evaluation of pressure (P) and velocity (V) at the axis of the simplified models of the sinusoid without fenestrations or lymphatic drainage (constant-radius and divergingradius microchannels).