The hepatic artery, portal vein and hepatic vein form a compact three-dimensional dendritic architecture within the liver. Each tree architecture is composed of a main trunk subdivided into smaller and smaller braches. The thinner channels form the canopy of the HA and PV trees and irrigate the lobules which behave like a porous system. Further downstream, the flow from the two inlet trees is reconstituted into a single stream through the outlet HV tree. Conceptually, the liver vascularization can be seen as 2 trees matching canopy-to-canopy bathing a porous architecture made of lobules, as presented in Fig. 3. Configurations of trees matching canopy-to-canopy were already presented by our group in the context of engineering applications^{37,38,39}. We showed that the global flow resistance decreases as the number of bathed elements connected to the trees increases.

Should the objective be to flow from the gastrointestinal tract to the inferior vena cava and the heart, then the straight channel would be the configuration that we should see. On the contrary, the blood flowing in reaches first the smallest liver units, the lobules, before flowing out through the hepatic veins and then the vena cava inferior. The assembly of lobules constitutes a functionalized porous medium which must allow the mixing of the oxygenated blood from the hepatic artery and the deoxygenated blood from the portal vein*.* The hepatic artery brings oxygenated blood. The portal vein brings blood rich in nutrients and antigens from the gastrointestinal system. Both sources of blood mix in the sinusoids. All the cells of the porous lobule-system fulfill the metabolic and filtering functions. Once mixed the blood is pushed into the hepatic vein.

### The shape of the elemental system

The lobules which constitute the designed porous medium have a highly regular design. Figure 4 depicts cross sections of them, and gives an overview of one single lobule, the elemental system. The hexagonal cross section shows 6 portal triads made of the bile duct, the portal vein and the hepatic artery. Here we do not consider the bile canal. The blood mixes along each of the sinusoids, which are mainly perpendicular to the portal triad. Blood is then pushed out when reaching the center of the lobule through the central vein parallel to the portal triad. Describing the lobules under the assumption of slices of highly vascularized hexagons represents a commonly admitted hypothesis.

Why a hexagon? If we look into Fig. 4, we see that the square image is made of about 16 hexagonal shapes of side ({L}_{h}). Calling A_{h} the area of the hexagon, we have ({A}_{h}=frac{3sqrt{3}}{2}{L}_{h}^{2}). The mass flow rate that exits one hexagon is ({dot{m}}_{h}), and therefore the total mass flow rate through the square delimited by the dashed lines would be ({16 dot{m}}_{h}). Each central vein of a hexagon receives ({dot{m}}_{h}/)6 from the 6 triads distant of ({L}_{h}) from the central vein (the hexagon is made of 6 equilateral triangles of side ({L}_{h})). Assume one main sinusoid of diameter ({d}_{h}) connects a triad to the central vein. The assembly of hexagonal shapes makes each hepatic artery and portal vein in contact with 3 lobules. For the sake of simplicity, assume that the hepatic artery and the portal vein are one single conduct of diameter (d) and length ({L}_{d}), through which the mass flow rate is hence (3{dot{m}}_{h}/6). The flow path is represented on the left hand side of Fig. 5.

The total pressure difference between the inlet and the outlet of a lobule is given by

$${Delta p}_{t,h} sim frac{{dot{m}}_{h}}{2}frac{{L}_{d}}{{d}^{4}}+frac{{dot{m}}_{h}}{6}frac{{L}_{h}}{{d}_{h}^{4}}$$

(9)

Another way to pave entirely the square domain represented in Fig. 4 would be to use squared shaped lobules of side ({L}_{S}) with ({A}_{S}={L}_{S}^{2}). We consider that ({A}_{h}={A}_{S}), which means that the hexagon is reshaped to become a square, and ({L}_{S}=frac{{3}^{3/4}}{sqrt{2}}{L}_{h}cong 1.6{L}_{h}). The total number of squares is the same as the number of hexagons, namely 16. Therefore, because the total mass flow rate is a constant, the mass flow rate per element must continue to be ({dot{m}}_{h}). Each square element receives the blood from each of its 4 corners. This means that in this configuration, the central vein would be connected to 4 radial branches of diameter ({d}_{c}) and length ({L}_{c}=sqrt{2}{L}_{S}) through which the mass flow rate is ({dot{m}}_{h}/4). As each square element is in contact with 3 other ones, the mass flow rate through the duct of diameter (d) and length ({L}_{d}) must be ({dot{m}}_{h}). Finally the overall pressure loss is

$${Delta p}_{t,c} sim {dot{m}}_{h}frac{{L}_{d}}{{d}^{4}}+frac{{dot{m}}_{h}}{4}frac{{L}_{c}}{{d}_{c}^{4}}$$

(10)

The pressure loss along the sinusoids is (Delta {p}_{h}sim {dot{m}}_{h}/6times {L}_{h}/{d}_{h}^{4}) in the case of the hexagonal shape, and (Delta {p}_{c} sim {dot{m}}_{h}/4times {L}_{h}/{d}_{c}^{4}) in the case of the square shape. The volume of blood flowing through the lobule is a constant. Therefore we write

$$V=frac{3}{2}pi {d}_{h}^{2}{L}_{h}=pi {d}_{c}^{2}{L}_{c}$$

(11)

And because ({L}_{c}=frac{{3}^{3/4}}{2}{L}_{h}), we have

$${left(frac{{d}_{h}}{{d}_{c}}right)}^{2}={3}^{-1/4}$$

(12)

Leading to the sinusoid pressure drop ratio between a square and a hexagonal lobule:

$$frac{{Delta p}_{c}}{{Delta p}_{h}}=frac{3}{2}frac{{L}_{c}}{{L}_{h}}{left(frac{{d}_{h}}{{d}_{c}}right)}^{4}$$

(13)

In other words, (Delta {p}_{h}cong Delta {p}_{c}).

Finally, we conclude that the overall pressure losses will be smaller in the case of the assembly of hexagonal lobules (Eq. 9) because the mass flow rate along the ({L}_{d}) ducts of diameter (d) (hepatic artery and portal vein) is half the one of a square assembly. This is consistent with the results provided by Siggers et al.^{40} whose finite elements modeling in 2D indicates a reduction in blood flow rate in a square lattice as opposed to a hexagonal one.

### The porous medium approach

Scanning Electron Microscope images of human liver reveal that the elemental system, the lobule, is entirely vascularized^{19}. The sinusoids tortuous network bathes the entire lobule, supplying the blood to the central vein from the 6 sources (hepatic artery + portal vein). Note that the number of 6 branches is the limit for which the radial connections exhibited in Fig. 5 is a good pattern. We demonstrated previously that beyond the value of 6 connected branches, radial networks should be replaced by tree-shaped ones with optimized diameter ratios (Eq. 3) and length ratios (Eq. 6) in order to spend less pumping power for the same fluid volume^{26}.

We note that each source is in charge of 1/6th of the hexagonal lobule cross section. From one source, the blood not only invades the sinusoids network in the broad direction of the central vein, but it must also flow along the hexagon periphery at mid-distance from the two neighboring sources. This way, what was initially a local fluid source becomes a distributed fluid source. The sector covered has an angle of π/6 from the central vein, see Fig. 6. The entire network is similar to a river delta, except that in the case of the lobule the fluid flows in the reverse direction as in the river basin. The entire volume of the lobules is fixed because the blood volume is fixed. The network that drives the flow of blood towards the central vein is not radial as the radial design does not allow minimum friction losses^{26}. At such a small scale, it seems appropriate to use a porous medium analogy^{16,41,42,43}. The overall pressure loss from the hexagon rim to the central vein is the sum of the pressure losses along the branches of the flow dendritic pathway.

In our previous works on engineered flow architectures^{26,33}, a general expression of the pressure losses was derived as a function of the fluid volume (V) and a flow resistance factor ({f}_{n}) that varies with the bifurcation level of the dendritic structures.

$$Delta p=8pi nu {dot{m}}_{h}frac{{{L}_{h}}^{3}}{{V}^{2}}{f}_{n}$$

(14)

Here, ({L}_{h}) is the distance between the lobule center and its perimeter.

The work dealt with a surface with a round cross section. Nevertheless considering that the hexagonal shape is close to the circle shape, the same expression was used in this work to predict the order of magnitude of the lobule permeability. The resistance factor ({f}_{n}) is obtained from the search of minimum overall flow resistance in a laminar dendritic structure: the diameter ratio (which follows the Hess-Muray’s law^{27,28}), the branches lengths and bifurcation angles correspond to minimum pressure losses. Its generic expression is

$${f}_{n}={n}_{0}{left[sum_{0}^{n}{2}^{i/3}widehat{{L}_{i}}right]}^{3}$$

(15)

where ({n}_{0}) is the number of sectors connected to the source (({n}_{0}=) 6 here), (widehat{{L}_{i}}={L}_{i}/{L}_{h}) the non-dimensional branch length, and n is the bifurcation level. We gathered the values of ({f}_{n}) from Wechsatol et al.^{33} They are 13.16, 16.31, 18.67, 20.5, 21.8 and 22.6 for n = 2 to 7 respectively, which means that for increasing pairing levels, ({f}_{n}) reaches an asymptote close to 25.

Considering Darcy flow through the porous elemental system (i.e. one lobule), the average velocity of the blood flowing to the central vein is

$$U=frac{K}{mu }frac{Delta p}{{L}_{h}}$$

(16)

where *K* is the intrinsic permeability of the lobule.

The mass flow rate is related to the average velocity through ({dot{m}}_{h}=6rho U{L}_{h}t), with *t* the lobule thickness. Finally, the permeability of a lobule of volume (V) is

$$Ksim frac{{V}^{2}}{48pi {{t L}_{h}}^{3}{f}_{n}}$$

(17)

which, in view of the asymptotic value of ({f}_{n}), gives

$$Kcong 2.6 1{0}^{-4}frac{{V}^{2}}{{{t L}_{h}}^{3}}$$

(18)

According to the literature^{15,44,45}, the average human liver has a volume of 1500 cm^{3}, and contains 10–20% of blood, while it possesses about 10^{6} lobules. This would give a lobule volume of 1.5 mm^{3}. On another hand, Debbaut et al.^{46} reported a value of 0.134 mm^{3} for 3 human liver lobules. The lobules dimensions are also difficult to find in the literature. They are reported to range from 500 µm up to 2.5 mm in diameter for humans^{14,42,47}. In the absence of more precise data, Eq. 18 gives a permeability K ranging between 3 10^{–10} m^{2} and 9 10^{–12} m^{2}. This result is in agreement with the literature as the radial and tangential permeability of a lobule were estimated to be about 1.5 10^{–14} m^{2} in Ref^{46}, while Ref^{11}. reports a lobule permeability of 4.8 10^{–9} m^{2}.