A nonlinear multi-scale model for blood circulation in a realistic vascular system

Figures

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  Figure 1.. The quasi-three-dimensional numerical model for a two-dimensional spatial problem. The xy (vertical) axis represents the two-dimensional computational domain in space and the third axis (horizontal) is the model axis. Blood flows from the roots of the arterial network through the arterial network (red), then to the continuous domains of the capillary compartments (light red for arterial and light blue for venous compartment) and finally to the venous network structure (blue) and to the venous roots at the bottom of the venous network. The arterial and venous terminal nodes are connected to their respective continuous capillary domains in the capillary bed (red solid and blue dashed dotted lines). The green dotted lines represent the pixel-wise bridge between capillary compartments, which is modelled as the blood perfusion in §2.4.

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  Figure 2.. Illustration of pressure drop at junctions, see (2.3). The arrow represents the direction of blood flow.

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  Figure 3.. An anatomical frog tongue image from a classical textbook [23], with arterial (red) and venous (blue) vascular network structures. The networks are obtained by segmenting the anatomical vessel structures. Our capillary domain is the region inside the tongue’s boundary. Vessel A connects two arterial networks, while Vessel B connects two vein networks.

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  Figure 4.. The static blood pressure distribution from our simulations has good estimation across the whole vascular system. The computed pressures (asterisks) follow the effective pressure from experimental data (brown solid line), the light brown area representing the variation from several experimental data measurements [24]. Our simulations produce some overestimation in the small arteriole (label a) and venule (label b) regions. The inaccurate simulation pressure is likely caused by vessel diameters in the range of the pixel size, so that the rounding of the vessel radius in the segmentation is in mismatch to the true value.

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  Figure 5.. The pressure distribution for the baseline model: the segment thickness in the vascular graphs is proportional to the flow rate passing through it and the local pressure distribution in the Darcy domain. (a) The arterial network structure and the arterial compartment with pressure ranging from 7.6 to 11 mmHg (8.6 ± 0.7 mmHg). The right arterial vessel input provides 16% more blood flow compared to the left vessel, causing higher pressure in the right area. (b) The vein network structure and venous compartment with pressure ranging from 7.4 to 10.8 mmHg (8.5 ± 0.8 mmHg). The right vein vessel output has 4.3 times more blood flow than the left vessel, playing a vital role in blood distribution in the vein compartment.

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  Figure 6.. The vascular system resistance R as a function of the occlusion rate in vessels: (a) Resistance estimates for occlusions, baseline model. (b) Resistance estimates for occlusions, linear model. The occlusion rate (x-axis) in a vessel varies from 5% to 100% of the original radius for each simulation. Arteries 1 and 2 refer to a segment in the right and left big vessel of the arterial structure. Veins 1 and 2 refer to a segment in the right and left big vessel of the venous structure. The artery occlusions cause an increment of the macroscopic flow resistance in the whole system for both the baseline and linear models. For the vein 1 occlusion, the entire vascular resistance is almost constant (±3% of the original total resistance). By contrast, the increment of the total resistance in vein 2 occlusion shows that the vein 2 plays vital role in draining blood from the organ tissue. This effect is only observed in the nonlinear simulation, while the linear model failed to capture this anomaly.

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  Figure 7.. Pressure drop distribution in the capillary compartments with simulated occlusions (marked by black circle) computed with the baseline model. (a,b) Pressure drop distribution for artery occlusions are not mirroring each other due to the asymmetric structure of the arterial network with artery 1 occlusion generated bigger pressure drop. (c) Even with a negligible change in the macroscopic pressure drop (table 2), the pressure drop distribution in vein 1 experiment shows a notable pressure drop distribution for both compartments. (d) A higher pressure gradient between several terminal nodes in the bottom right corner enhances blood flow through the venous vessel rather than in the venous capillary. This microscopic alteration can occur with small change in the macroscopic flow.