Construction of the virtual liver lobule

One representation of the liver is a PIPE virtual liver lobule, described in Section 1.3 in S1 Text and in our previous work [1]. In this study, we focus on describing a more realistic model representation of the liver—a NET virtual liver lobule. We constructed a virtual mouse liver lobule (Fig 3B and 3C) based on the analysis of rodent liver lobules by us (Fig 3A and Section 1.2 in S1 Text) and others [25]. A key observation is that in thick tissue sections individual hepatocytes are in contact with at least two sinusoids. In addition, we and others [25] have experimentally determined quantitative descriptions of rodent liver lobules including the volume fraction of sinusoids, average sinusoid-parenchyma interfacial area per unit parenchyma volume, branching angle and inter-branch segment length (Section 1.2 in S1 Text). These architectural quantities are criteria with which we compare our virtual liver lobules to real lobules.

(A) Micrograph of sinusoids in a deep section of a rat liver. A central vein is located near the upper right. The scale bar is 100μm and the width of the individual sinusoids is ≈ 8μm. (B) Two-dimensional view of the virtual mouse liver lobule showing the sinusoid network. Hepatocytes, sinusoids, central vein and portal triads are colored green, red, yellow and blue, respectively. (C) Partial cut-away three-dimensional view of the virtual mouse liver lobule. (D) Schematics and equations for transport and metabolism. For more details on the advection model see Section 1.5 in S1 Text.

The NET virtual liver lobule is a regular hexagon with CV at the center, six PTs at the vertices, and a dense network of sinusoids connecting the PTs and the CV. We used a set of simple rules to create the virtual sinusoid network as described in Section 1.4 in S1 Text. Briefly, the network is a collection of nodes (representing sinusoid junctions) and connecting edges (representing sinusoid segments). The first step was to determine the locations of nodes. We selected the center of the lobule hexagon as the root node (the location of the CV) and then placed concentric rings encircling the root node with radial spacing between each pair of rings equal to one hepatocyte width. The radius of the area enclosed by the first concentric ring equaled one hepatocyte size plus the radius of CV. We then divide the individual concentric rings along their perimeters every hepatocyte width placing nodes on each ring. The location of the first node on each ring was chosen randomly. We repeated this process of adding rings and subdividing out to the perimeter of the simulated lobule. Next, we connected nodes on each ring to the nearest node on the next ring outward to form an edge (a sinusoid segment). The root node connected to all nodes placed on the innermost ring, while the nodes on the outermost ring connected to the nearest nodes on the perimeter of the hexagon. Nodes did not connect to other nodes on the same ring. We then transformed the topological network into a spatial network in CompuCell3D (CC3D) [34, 35]. Along each edge, we created a sinusoid diameter cylinder to represent a sinusoid segment. To position the hepatocytes, we chose the voxel between two adjacent nodes on the same ring as the center of a hepatocyte and expanded this voxel to the proper cell volume, taking into account the cell-cell and cell-vessel interactions, in CC3D (see Fig 3B and 3C).

The constructed sinusoid network acquires a “radiating” anastomotic pattern from the root node (CV). Qualitatively, every hepatocyte is in contact with at least two sinusoids (hepatocyte at the periphery can contact more than two), and the spacing between adjacent pseudo-parallel sinusoids equals one hepatocyte width. The characteristics of the simulated liver lobule are given in (Table 1). The virtual liver lobule is three dimensional with a thickness of h_lob = 20 μm (one hepatocyte) and the bounding lobule hexagon edge length of a_lob = 200 μm, which is also the hexagon’s major radius. The constructed sinusoid network is planar and constrained to the center of this slab. The diameter of sinusoid segments is d_sin = 8μm everywhere in the simulated lobule. The complete virtual liver lobule slice contains 217 hepatocytes with mean hepatocyte volume of V_hep = 8030 μm³. Since the lobule’s sinusoid network is created with a stochastic algorithm, we examined a set of networks built with the same method (see Section 2.2 in S1 Text).

^† [25] gives this value as the radius but examination of the paper’s figures suggests that the value is the diameter.

^‡ [25] and [36] appear to be giving the half-angle.

The virtual liver lobule slice represents a minimal functional unit of a mouse liver across which a xenobiotic distributes. In a larger simulated tissue section there would be six additional units next to the simulated lobule as well as units stacked above and below the simulated lobule slice. In this study, the simulated liver lobule is isolated and independent such that no exchange of the xenobiotic occurs between the simulated liver lobule and the eight neighboring lobule slices.

One qualitative feature of the constructed virtual lobule is the uniform patterning of sinusoids, which is similar to what is see in mouse liver sections. We quantitatively compared structural properties of our simulated liver lobule with mouse data (Table 1). The volume fraction of sinusoids in the virtual lobule, excluding the CV and PT, is η_sin = 12.9%, comparable to experimentally measured values of 15% [36], 11% [37] and 10.4% ± 1.1% in our measurements in ex vivo rat lobules. Ex vivo sinusoid networks show a small variation in sinusoids diameters, with the largest diameters occurring near the PT and CV and smaller diameters in the mid-lobular region (see Section 1.2 in S1 Text). In addition, the ex vivo samples have a somewhat more densely connected network in three dimensions and it is common to observe nodes with four or more connections.

The average sinusoid-parenchyma interfacial area per unit parenchymal volume S_int, which affects xenobiotic uptake by hepatocytes, is 0.122 μm²/μm³ in our virtual liver lobule versus 0.163 μm²/μm³ in [36] and 0.143 ± 0.019 μm²/μm³ from our measurements in ex vivo rat lobules.

Branching angle θ_branch, which defines the smallest of the three inter-segment angles at bifurcations in the sinusoid network, is 66.9° ± 12.8° in the virtual liver lobule. Values measured by us are 67.4° ± 27.6° for surface section of mouse liver and 78.8° ± 22.6° for deep section of rat liver. Hoehme et al. [25] and Hammad et al. [36] reported a smaller value of 32.5° ± 11.2°, but appear to be referring to the branching half angle.

The average inter-branch sinusoid segment length L_seg is 51.0 ± 37.2 μm in our virtual liver lobule, larger than our measured value of 26.3 ± 17.4 μm and reported values, 43.1 ± 18.9 μm in [25] and 23.9 ± 5.9 μm in [36]. Our model assumption that the two-dimensional virtual sinusoid network has no inter-plane connections may be the source of the longer segment lengths in our virtual liver lobule versus real lobules. Overall, the virtual liver lobule matches many of the architectural aspects of the rodent liver summarized in Table 1.

The virtual liver lobule contains two types of computational objects; sinusoid segments (SINs) and individual hepatocytes (HEPs). Biologically, a SIN represents a volume of blood. The sinusoidal endothelial cells and the Space of Disse, which represent relatively little volume, are not explicitly modeled. We treat the blood as homogeneous fluid and do not explicitly treat, or differentiate, between serum and blood cells such as erythrocytes (red blood cells). In addition, we do not explicitly model the separate arterial and venous blood inflows at the PT, the two blood flows are considered to have already mixed. Computationally, a SIN is a container of the xenobiotic dissolved in blood and is modeled as a CC3D pseudo-cell. HEPs are CC3D cells representing individual hepatocytes.