Optimal cell transport in straight channels and networks

Flux of rigid or soft particles (such as drops, vesicles, red blood cells, etc.) in a channel is a complex function of particle concentration, which depends on the details of induced dissipation and suspension structure due to hydrodynamic interactions with walls or between neighboring particles. Through two-dimensional and three-dimensional simulations and a simple model that reveals the contribution of the main characteristics of the flowing suspension, we discuss the existence of an optimal volume fraction for cell transport and its dependence on the cell mechanical properties. The example of blood is explored in detail, by adopting the commonly used modeling of red blood cells dynamics. We highlight the complexity of optimization at the level of a network, due to the antagonist evolution of local volume fraction and optimal volume fraction with the channels diameter. In the case of the blood network, the most recent results on the size evolution of vessels along the circulatory network of healthy organs suggest that the red blood cell volume fraction (hematocrit) of healthy subjects is close to optimality, as far as transport only is concerned. However, the hematocrit value of patients suffering from diverse red blood cel pathologies may strongly deviate from optimality.

Figure 1

(2D simulations) Left panel: Normalized cell flow rate as a function of volume fraction for channels with different widths. Full lines are a guide for the eyes. $C_a=0.9$ and the reduced area is ${\nu }_{2\text{D}}=0.7$. Center and right panels: snapshots of the suspension for two different volume fractions in the five considered channels.

Figure 2

(2D simulations) Normalized cell flow rate as a function of volume fraction for a channel width $W=20\mu \text{m}$ and different $C_a$. Reduced area is ${\nu }_{2\text{D}}=0.7.$

Figure 3

(3D simulations) Left: The normalized cell flow rate as a function of tube volume fraction ${\mathrm{\Phi }}_t$ for different reduced volumes. Viscosity contrast is 1 and the channel width $W$ is $40\mu \mathrm{m}=14.8R$; the channel length along the flow direction is about $11.7R$, and is equal to $6.2R$ in the orthogonal direction. $C_s=0.18$. A, B, and C are snapshots of suspension configurations for the three different reduced volumes shown for the corresponding optimal volume fraction.

Figure 4

(3D simulations) Green line: Optimal flux and optimal volume fraction as a function of the actual reduced volume (3D simulations) for incompressible membranes (from Fig. 3). Violet lines: the same simulation with stretchable capsules. For each simulation the compressional elasticity ${\kappa }_{\alpha }/{\mu }_s$ has been varied in order to allow for different average reduced volume $\langle \nu \rangle$. Each capsule experiences different shear stresses within the channel, and thus has a different surface area (and thus reduced volume); the horizontal bars provide the distribution width of reduced volumes within the same suspension. $C_s=0.18$, as in Fig. 3.

Figure 5

The particle flow rate as a function of tube volume fraction for channels with different diameter calculated from our minimal model. We took $e_0=2\mu \mathrm{m}$, $\left[\eta \right]=5/2$ as for rigid spheres and for RBCs in physiological conditions [78], and ${\mathrm{\Phi }}_m$ is chosen as equal to 1 considering that the cells are deformable and can be squeezed.

Figure 6

RBC flux according to our minimal model. The reference configuration is $W=30$ µm (left) or $W=1000$ µm (right), $e_0=2$ µm, $\left[\eta \right]=5/2$, and ${\mathrm{\Phi }}_m=1$. Top panel: $e_0$ is varied; middle panel: $\left[\eta \right]$ is varied; bottom panel: ${\mathrm{\Phi }}_m$ is varied.

Figure 7

(2D simulations) Normalized cell flow rate as a function of volume fraction for channels with different widths and two different viscosity ratios $\lambda$ between the inner fluid and the external fluid. $C_a=0.9$ and reduced area is ${\nu }_{2\text{D}}=0.7$.

Figure 8

The particle flow rate as a function of tube volume fraction from our minimal model (same data as in Fig. 5). In addition, the red line indicates the ${\mathrm{\Phi }}_t$ value corresponding to a reservoir volume fraction ${\mathrm{\Phi }}_r$ of 45% for the corresponding model. Blue and green lines correspond to high ${\mathrm{\Phi }}_r$ (80%) and low ${\mathrm{\Phi }}_r$ (25%).

Figure 9

The RBC flow rate as a function of the tube hematocrit for channels of different diameters calculated from Pries empirical model [52]. The red line indicates the ${\mathrm{\Phi }}_t$ value corresponding to a reservoir hematocrit of 45%. Blue and green lines correspond to high ${\mathrm{\Phi }}_r$ (80%, polycythemia) and low ${\mathrm{\Phi }}_r$ (25%, anemia).

Figure 10

Sketch of the model network considered in Sec. 4b.

Figure 11

The RBC flow rate as a function of the reservoir hematocrit for a model network (see text) for different values of $b$ extracted from literature and empirical rheology law given in Ref. [52]. The curves are similar to those for large channels that would be obtained from Fig. 9 by considering ${\mathrm{\Phi }}_r$ instead of ${\mathrm{\Phi }}_t$. Small values of $b$ give more weight to the smallest vessels, hence an increase of the optimal hematocrit. Still, the contribution of large vessels is predominant.