Tissue Oxygenation Around Capillaries: Effects of Hematocrit and Arteriole Oxygen Condition

Abstract

Oxygen transfer in the microvasculature is a complex phenomenon that involves multiple physical and chemical processes and multiple media. Hematocrit, the volume fraction of red blood cells (RBCs) in blood, has direct influences on the blood flow as well as the oxygen supply in the microcirculation. On the one hand, a higher hematocrit means that more RBCs present in capillaries, and thus, more oxygen is available at the source end. On the other hand, the flow resistance increases with hematocrit, and therefore, the RBC motion becomes slower, which in turn reduces the influx of oxygen-rich RBCs entering capillaries. Such double roles of hematocrit have not been investigated adequately. Moreover, the oxygen–hemoglobin dissociation rate depends on the oxygen tension and hemoglobin saturation of the cytoplasm inside RBCs, and the dissociation kinetics exhibits a nonlinear fashion at different oxygen tensions. To understand how these factors and mechanisms interplay in the oxygen transport process, computational modeling and simulations are favorite since we have a good control of the system parameters and also we can access to the detailed information during the transport process. In this study, we conduct numerical simulations for the blood flow and RBC deformation along a capillary and the oxygen transfer from RBCs to the surrounding tissue. Different values for the hematocrit, arteriole oxygen tension, tissue metabolism rate and hemoglobin concentration and affinity are considered, and the simulated spatial and temporal variations of oxygen concentration are analyzed in conjunction with the nonlinear oxygen–hemoglobin reaction kinetics. Our results show that there are two competing mechanisms for the tissue oxygenation response to a hematocrit increases: the favorite effect of the higher RBC density and the negative effect of the slower RBC motion. Moreover, in the low oxygen situations with RBC oxygen tension less than 50 mmHg at capillary inlet, the reduced RBC velocity effect dominates, resulting in a decrease in tissue oxygenation at higher hematocrit. On the opposite, for RBC oxygen tension higher than 50 mmHg when entering the capillary, a higher hematocrit is beneficial to the tissue oxygenation. More interestingly, the pivoting arteriole oxygen tension at which the two competing mechanisms switch dominance on tissue oxygenation becomes lower for higher oxygen–hemoglobin affinity and lower hemoglobin concentration. This observation has also been analyzed based on the oxygen supply from RBCs and the oxygen–hemoglobin reaction kinetics. The results and discussions presented in this article could be helpful for a better understanding of oxygen transport in microcirculation.

Appendices

Appendix A: The Pre-Conditioning Period of Simulations

Fig. 11

${\bar{J}}^w_{O_2}$–${\bar{P}}^c_{O_2}$ correlation during the simulation. The filled circles represent the starting states of the simulations, and the arrows indicate the time evolution direction. Four specific states are labeled along the 90/90/90 simulation process (thick blue curve), which correspond to the instants with ${\bar{P}}^c_{O_2}$=90 mmHg (State O, initial state), 70 mmHg (State A), 50 mmHg (State B) and 30 mmHg (State C). Two insets are also included to show more details on how the curves from different initial conditions merge together (Color figure online)

One dilemma for simulating this unsteady, nonlinear and multi-media system is that the initial $P_{\textrm{O}_2}$ and $S_{\textrm{O}_2}$ distributions are needed to start the calculation, but they are available, and any artificially assumed initial distributions will affect the calculated results in the early stage of the simulation, short or long. This is similar to the inlet condition in computational fluid dynamics (CFD) simulations. To ensure that the imposed inlet boundary condition does not affect the simulation result, sensitivity analysis can be performed by conducting numerical tests with the inlet boundary position gradually moving upstream (i.e., increase the flow distance from the inlet to the interesting region), until no apparent difference is observed in simulation results. Following this idea, we can start the oxygen transfer calculation with an initial condition with $P_{\textrm{O}_2}$ higher than what we are really interested in, and this gives the system a pre-conditioning period to digest the impact from the artificially imposed initial distributions. In principle, the simulation results at a particular oxygen state (for example, ${\bar{P}}^c_{O_2}$=50 mmHg) from different initial conditions should be very close (i.e., not affected by the artificial initial condition). This provides us a practical criterion to identify how long the pre-conditioning period is and when the simulation results can be taken for further analysis. Here we follow a similar approach as that in Vadapalli et al. (2002) and compare the system performance under different initial conditions. To demonstrate this process, we take the oxygen flux ${\bar{J}}^w_{O_2}$ as an indicator of the dynamic system behavior and plot the time course of ${\bar{J}}^w_{O_2}$–${{\bar{P}}}^c_{O_2}$ in Fig. 11 for $H_\mathrm{{ct}}=20\%$. Five initial boundary conditions are considered and they are labeled as the initial averaged $P_{\textrm{O}_2}$ values in the cytoplasm/plasma/tissue regions. For example, 70/40/22 means that we set the initial $P_{\textrm{O}_2}$ distribution as 70 mmHg in cytoplasm, 40 mmHg in plasma and 22 mmHg in tissue at the beginning of the simulation. The uniform distribution method (cases 70/70/70, 80/80/80 and 90/90/90) has been used in Vadapalli et al. (2002), and the stairwise distribution method (cases 70/40/22 and 90/40/22) has been used in Lucker et al. (2015). The initial transient period is apparent as a rapid increase in the oxygen flux ${\bar{J}}^w_{O_2}$. The curves from 90/90/90 and 80/80/80 meet at ${\bar{P}}^c_{O_2}$=72 mmHg, and they further merge with the 70/70/70 curve after ${\bar{P}}^c_{O_2}$=62 mmHg. This indicates that the results from 90/90/90 after 72 mmHg, and those from 80/80/80 after 62 mmHg, are reliable. On the other hand, for the stairwise initial distributions, the pre-conditioning periods take longer, and the results from 90/40/22 and 70/40/22 can only be taken after 30 mmHg. In this study, all simulations start with the 90/90/90 initial distribution, and results are only collected after ${\bar{P}}^c_{O_2}$ decreases to 70 mmHg for further analysis.

Appendix B: Axial $P_{\textrm{O}_2}$ Distributions

Figure 12 displays the axial $P_{\textrm{O}_2}$ distribution profiles on the capillary wall and in the tissue region. Two curves are plotted there for the capillary wall, one for the inner plasma side and one for the outer tissue side. The difference between these two curves indicates the $P_{\textrm{O}_2}$ discontinuity across the capillary wall due to its permeability (see Eq. 13). It can be seen that the axial variation is only evident on the lumen side of the capillary wall, and it quickly disappears in the tissue region. For $r>6~\mu$m, the $P_{\textrm{O}_2}$ profiles are nearly horizontal lines along the axial direction. The maximum $P_{\textrm{O}_2}$ value on the inner capillary surface is observed at $x=2$–$4~\mu$m, which is near the rear end of the RBC (see Fig. 4). In addition to the decrease in $P_{\textrm{O}_2}$ magnitude and variation amplitude from the capillary wall, the wavy variation pattern also shifts leftward. This feature is shown more clearly in Fig. 4, where the white dashed lines are plotted by connecting the peak positions of the axial $P_{\textrm{O}_2}$ profiles at different r (as those in Fig. 12). The peak lines are similar to the wake lines behind a swimming duck in a lake, and they are the combined product of the cell motion along the capillary and the oxygen diffusion outward.

Fig. 12

Axial $P_{\textrm{O}_2}$ profiles at different radial positions (labeled in a) at instants with ${\bar{P}}^c_{O_2}=70$ mmHg (a), 50 mmHg (b) and 30 mmHg (c)