Predicting red blood cell transport and capillary hemodynamics in angiogenic and tumor vascular networks in silico

Figure 1. (a–c) Ex vivo/in vivo images and (d–f) corresponding in silico models (identified in the text as H1, A1, and T, respectively). (a) Healthy vasculature and (b) corresponding angiogenic vascular networks grown ex vivo (44). (c) Tumor microvasculature in vivo grown in a mouse dorsal window chamber (45). Yellow arrows in (d)–(f) indicate flow directions in selected vessels, and white arrows indicate vessels mostly devoid of RBCs. Animations are provided in Videos S1, S2, and S3 and Fig. S1, a–c. Higher-resolution images of all six in silico networks modeled (H1, A1, T, and H2 [healthy], H3 [healthy], and A2) are provided in Fig. S1, d and e. Ex vivo/in vivo images from the literature are reproduced with permission. The visualization of the simulated models (d–f) given here and in the supporting material shows the full spatial extent of the simulated networks. All inlets/outlets with applied pressure values are marked in Fig. S1, a–f.

Figure 2. (a) Details of in silico networks. Diameter information refers to vasculature-wide values. (b) Vessel diameter (D) distribution. (c–e) Examples of vascular abnormalities included in silico: (c) vessels showing constricted (black arrow) and dilated (white arrow) segments, (d) junctions with multiple vessels, and (e) elliptical section for partially collapsed vessels. Transition from single to multifile flow within the same vessel can be seen in (c). (f) Time series of flow rate (Q, black) and hematocrit (H, red) in two vessels. (g and h) Highly asymmetric velocity distribution (mm/s). (i) Asymmetric RBC distribution (white arrow) caused by merging of a plasma vessel. (j) RBC coin-stacking pattern (white arrow) in a dilated segment transforming from single file in upstream constricted region (black arrow). (k) RBC length versus vessel diameter. The horizontal axis is in log scale. In all plots, red arrows indicate flow direction. Note that (c)–(e), (i), and (j) only show parts of the entire networks.

Figure 3. Time average flow rate (a), and whole blood velocity (b) in each vessel in H, A, and T networks. Inset in (a) shows data for vessels with a <0.5 nL/s flow rate. Table (c) gives the vasculature-wide mean and SD for flow rate (Q, nL/s), blood velocity (V, mm/s), and hematocrit (H, %) for each group. (d and e) Cumulative distribution function (CDF) of Q and V for each group. Inset of (e) highlights the overlapping profiles. (f and g) Difference of whole blood velocity and cell-free plasma velocity (ΔV) for each vessel and its CDF for each group. (h–j) Hematocrit versus vessel diameter, flow rate, and CDF of hematocrit. (k) The Fahraeus effect in terms of blood velocity/RBC velocity ratio. The inset shows bin-averaged variations.

Figure 4. (a) Time-average RBC partitioning. Data represent one bifurcation (blue square: H network, red circle: A, and black triangles: T networks). (b) Dispersion of time-dependent partitioning; same symbols as in (a) are used. Inset: time-dependent partition in an example bifurcation. (c and d) Examples of time-dependent partitioning in T and A networks, respectively, with large variations. Each red dot represents partitioning over a small time window (65 ms); the black triangle is the average value. (e and f) RBC dynamics corresponding to (c) and (d). The bifurcation of interest is marked by a box, and the apex indicating RBC lingering is marked by a green arrow. Black arrows indicate flow directions, and daughter vessels are marked by 1 and 2.

Figure 5. Relative viscosity per vessel in H (a) and A and T (b) vasculatures. The dash-dot curve represents empirical relative viscosity at a discharge hematocrit of 0.45 (66). (c) Cumulative distribution function (CDF); (inset) probability density function (PDF). (d and e) ΔPrbc/ΔPpl and Qpl/Qrbc in each vessel in H and A+T networks. (f) An example vessel where geometry and RBC dynamics affect the pressure distribution, leading to higher μrel. Note that in (f), the left figure is 3D, but the right two figures are on a plane. The vessel near the bottom moves out of the plane.

Figure 6. (a) Average CFL per vessel. (b) CFL variation averaged per vessel; same symbols are used as in (a). (c and d) CFL fluctuations (relative to the average CFL) along a flow path in the H and A networks. Solid and dashed lines represent fluctuations in circumferential and axial directions, respectively. (e–g) CFL variations averaged per vessel as function of tortuosity (k), vessel compression (ε), and diameter variation (Dʹ). The inset shows examples of RBC flows for each case. (h) CFL asymmetry caused by the merging of a plasma vessel.

Figure 7. (a–c) Predicted 3D WSS (dyn/cm2) distribution in healthy, angiogenic, and tumor networks. (d) Average WSS per vessel as a function of diameter (blue square: healthy; red circle: angiogenic; black triangle: tumor). The vasculature-wide mean WSS and c.v. for each network type are given in the table. (e) Average WSS gradient in three regions of interest—bifurcations, mergers, and vessels—for different vasculatures.