# The Computed Sinusoid

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. The Hepatic Sinusoid

#### 1.2. Models of the Hepatic Sinusoids

## 2. Materials and Methods

#### 2.1. Computational Fluid Dynamics (CFD) Simulations

^{®}2021 R1 Academic software and a laptop with the features listed below (Table 3).

#### 2.2. Geometry and Mesh

- 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).
- The SoD was modeled as a 1 µm thick 2D chamber surrounding the sinusoid lumen and communicating with it via fenestrations.
- The fenestrations were modeled as 100 nm long and 150 nm high channels connecting the sinusoidal lumen with the SoD (Figure 3).

- 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).

#### 2.3. Solver Configuration

^{−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).

^{−3}) and u is the 3D velocity vector (m/s)

## 3. Results

## 4. Discussion

#### 4.1. Major Insights about Sinusoidal Pressure (P)

#### 4.2. Major Insights Regarding Flow Velocity (V)

#### 4.3. General Considerations and Limitations

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**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 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.**Schematics of the numerical models of the sinusoid. The model was designed as a half-section. 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.**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.**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.**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.**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 7.**Pressure 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 (top/bottom).

**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.**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.

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. 76.57 nm 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 (10.21–10.68 fenestrations/µm^{2} vs. 5.74–6.26 fenestrations/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) |

Komatsu, 1990 [5] | (i) Rat; (ii) in vivo fluorescence microscopy; (iii) the diameter of the sinusoid increases from zone 1 to zone 2 to zone 3; 6.4 µm–7 µm–8.3 µm; (iv) the flow rate increases along the sinusoid, 143–221–331 µm/s; (v) the interpolated values of pressure within sinusoids are as follows: zone 1, 68–50; zone 2, 50–40; and zone 3, 40–28 mmHg |

MacPhee, 1995 [4] | (i) Mouse and rat; (ii) high resolution in vivo microscopy; (iv) the flow speed is highly variable due to interactions between blood cells and the cells of the sinusoid; generally, the velocity in zone 3 is greater than in zone 1 |

Yoon, 2013 [12] | (i) Mouse; (ii) computed tomography; (iii) zone 1 features a smaller diameter (8.8 vs. 13.7 µm) than zone 3; (vi) zone 1 has a lower porosity than zone 3 |

Ryou, 2020 [13] | (v) Clinical portal hypertension has pressure above 5 mmHg (666 Pa), while normal pressure is around 3.4 mmHg (450 Pa) |

**Table 2.**The most significant studies on numerical models of the liver’s microvasculature. Ref. = Reference, Mod. Obj. = models of a liver sinusoid or lobule, Dim. = dimensions, Bound. Cond. = boundary conditions, Eval. Param. = evaluation parameter, v = velocity; FR = flow rate; WSS = wall shear stress; P = pressure; 2D = two-dimensional; 3D = three-dimensional.

Ref. | Mod. Obj. | Dim. | Origin | Bound. Cond. | Eval. Param. | Highlights |
---|---|---|---|---|---|---|

Bonfiglio (2010) [19]; Siggers (2014) [20] | Lobule | 2D | Numerical | Phys., post-resection, and lymph production | P, blood flow distribution (v), and lymph flow | An infinite lattice of hexagonal lobules, the sinusoid space as a porous medium, the resection effect, anisotropy and shear-dependent tissue deformation, and lymph production |

Debbaut (2012) [21] | Three lobules | 3D | Three human lobule casts digitized using a micro-CT scanner | Phys. | P, permeability, preferential flow pathways, and WSS | A liver circulation anisotropy estimation |

Piergiovanni (2017) [22] | Sinusoidal network | 3D | In vivo images; mouse model | Phys. | v_{mean}, FR_{mass}, and WSS | Local hemodynamics; an investigation into different degrees of occlusion |

Hu (2017) [23] | Lobule | 3D | Numerical | Phys.; path. (fibrosis; cirrhosis) | P, v_{mean}, and FR_{vol} | Porous media approach; fibrotic–cirrhotic lobule |

Processor | Intel i5-10300H |
---|---|

Clock Freq. [GHz] | 2.50 |

Core # | 8 |

Ram [GB] | 8 |

**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 diverging-radius microchannels).

Constant Radius | Divergent Radius | |||
---|---|---|---|---|

P [Pa] | V [m/s] | P [Pa] | V [m/s] | |

max | 1067.69 | 0.001 | 1066.95 | 0.0032 |

min | 800.146 | 0.0008 | 799.876 | 0.0007 |

avg | 933.5973 | 0.00085 | 871.9508 | 0.0015 |

Std.dev | 77.1903 | 1.00 × 10^{−5} | 69.201 | 0.0007 |

**Table 5.**Velocity magnitudes 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 centre line; f = fenestrations; D = Space of Disse.

Const. rad. 5% | Const. rad. Var | Const. rad 20% | |||||||
---|---|---|---|---|---|---|---|---|---|

l | f | D | l | f | D | l | f | D | |

max | 0.00087 | 0.000038 | 0.000034 | 0.0015 | 0.000033 | 0.000035 | 0.0033 | 0.000016 | 0.000035 |

min | 0.00013 | 0 | 0 | 0.00054 | 0 | 0 | 0.00085 | 0 | 0 |

avg | 0.00084 | 2.8 × 10^{−6} | 0.000029 | 0.00086 | 1.2 × 10^{−6} | 0.00003 | 0.00086 | 0.000001 | 0.000032 |

Std.dev | 0.000047 | 0.000005 | 0.000008 | 0.000044 | 2.5 × 10^{−6} | 7.7 × 10^{−6} | 0.00013 | 1.9 × 10^{−6} | 6.7 × 10^{−6} |

Div. rad. 5% | Div. rad. Var | Div. rad. 20% | |||||||

l | f | D | l | f | D | l | f | D | |

max | 0.0031 | 0.00009 | 0.000053 | 0.0032 | 0.00009 | 0.000054 | 0.019 | 0.000049 | 0.000075 |

min | 0.000022 | 0 | 0 | 0.000019 | 0 | 0 | 0.0007 | 0 | 0 |

avg | 0.0015 | 0.000004 | 0.000025 | 0.0015 | 0.000002 | 0.000026 | 0.0015 | 1.8 × 10^{−6} | 0.000028 |

Std.dev | 0.00066 | 8.5 × 10^{−6} | 0.000016 | 0.00067 | 6.63 × 10^{−6} | 0.000016 | 0.0011 | 4.3 × 10^{−6} | 0.000022 |

**Table 6.**Velocity magnitudes in sinusoids modeled with lymphatic drainage. Const. rad. = constant radius; Div. rad. = diverging radius; porosity given as %; Var = variable increasing porosity 5–20%; l = lumen centre line; f = fenestrations; D = Space of Disse.

Const. rad. 5% | Const. rad. Var | Const rad. 20% | |||||||
---|---|---|---|---|---|---|---|---|---|

l | f | D | l | f | D | l | f | D | |

max | 0.002 | 0.0013 | 0.0014 | 0.002 | 0.0014 | 0.0014 | 0.0035 | 0.0014 | 0.003 |

min | 0.00075 | 0 | 0 | 0.00065 | 0 | 0 | 0.000014 | 0 | 0 |

avg | 0.00086 | 0.000057 | 0.00012 | 0.00086 | 0.000019 | 0.00012 | 0.00085 | 0.000029 | 0.00014 |

Std.dev | 0.00024 | 0.00016 | 5.25 × 10^{−5} | 0.00025 | 0.00085 | 0.00026 | 0.0004 | 0.00012 | 0.00042 |

Div. rad. 5% | Div. rad. Var | Div. rad. 20% | |||||||

l | f | D | l | f | D | l | f | D | |

max | 0.0041 | 0.0013 | 0.0014 | 0.004 | 0.0013 | 0.0014 | 0.025 | 0.0014 | 0.003 |

min | 0.000016 | 0 | 0 | 0.000016 | 0 | 0 | 0.0007 | 0 | 0 |

avg | 0.0014 | 0.000051 | 0.00011 | 0.0014 | 0.000025 | 0.00011 | 0.0017 | 0.000031 | 0.00012 |

Std.dev | 0.0008 | 0.00015 | 0.00026 | 0.0008 | 0.00011 | 0.00025 | 0.0016 | 0.00013 | 0.00041 |

**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.

Const. rad. 5% | Const. rad. Var | Const rad. 20% | |||||||
---|---|---|---|---|---|---|---|---|---|

l | f | D | l | f | D | l | f | D | |

max | 1067 | 1054 | 1043 | 1067 | 1055 | 1044 | 1067 | 1061 | 1056 |

min | 802 | 813 | 824 | 796 | 806 | 811 | 785 | 806 | 810 |

avg | 934 | 934 | 934 | 933 | 883 | 931 | 933 | 933 | 933 |

Std.dev | 76 | 73 | 73 | 77 | 67 | 74 | 77 | 76 | 76 |

Div. rad. 5% | Div. rad. Var | Div. rad. 20% | |||||||

l | f | D | l | f | D | l | f | D | |

max | 1068 | 1031 | 1002 | 1067 | 1031 | 1001 | 1074 | 1056 | 1040 |

min | 809 | 8110 | 813 | 805 | 802 | 806 | 748 | 826 | 827 |

avg | 878 | 877 | 878 | 874 | 844 | 873 | 891 | 891 | 891 |

Std.dev | 67 | 62 | 59 | 68 | 53 | 61 | 65 | 63 | 62 |

**Table 8.**Pressure in sinusoids modeled with 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.

Const. rad. 5% | Const. rad. Var | Const rad. 20% | |||||||
---|---|---|---|---|---|---|---|---|---|

l | f | D | l | f | D | l | f | D | |

max | 1067 | 949 | 934 | 1067 | 952 | 939 | 1067 | 989 | 983 |

min | 794 | 592 | 102 | 798 | 588 | 102 | 800 | 604 | 103 |

avg | 917 | 876 | 836 | 919 | 857 | 840 | 917 | 900 | 881 |

Std.dev | 73 | 65 | 140 | 71 | 50 | 140 | 69 | 65 | 110 |

Div. rad. 5% | Div. rad. Var | Div. rad. 20% | |||||||

l | f | D | l | f | D | l | f | D | |

max | 1067 | 880 | 865 | 1067 | 883 | 868 | 1066 | 929 | 915 |

min | 805 | 570 | 102 | 806 | 586 | 102 | 656 | 600 | 105 |

avg | 869 | 828 | 791 | 870 | 823 | 792 | 850 | 834 | 816 |

Std.dev | 65 | 50 | 129 | 65 | 34 | 128 | 68 | 50 | 93 |

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**MDPI and ACS Style**

Boninsegna, M.; McCourt, P.A.G.; Holte, C.F.
The Computed Sinusoid. *Livers* **2023**, *3*, 657-673.
https://doi.org/10.3390/livers3040043

**AMA Style**

Boninsegna M, McCourt PAG, Holte CF.
The Computed Sinusoid. *Livers*. 2023; 3(4):657-673.
https://doi.org/10.3390/livers3040043

**Chicago/Turabian Style**

Boninsegna, Matteo, Peter A. G. McCourt, and Christopher Florian Holte.
2023. "The Computed Sinusoid" *Livers* 3, no. 4: 657-673.
https://doi.org/10.3390/livers3040043