Edited by: Joseph L. Greenstein, Johns Hopkins University, United States
Reviewed by: Ranjan K. Dash, Medical College of Wisconsin, United States; Daniel Goldman, University of Western Ontario, Canada; Leif Østergaard, Center of Functionally Integrative Neuroscience and MINDLab, Denmark; David Robert Grimes, Queen's University Belfast, United Kingdom
This article was submitted to Computational Physiology and Medicine, a section of the journal Frontiers in Physiology
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Capillary dysfunction impairs oxygen supply to parenchymal cells and often occurs in Alzheimer's disease, diabetes and aging. Disturbed capillary flow patterns have been shown to limit the efficacy of oxygen extraction and can be quantified using capillary transit time heterogeneity (CTH). However, the transit time of red blood cells (RBCs) through the microvasculature is not a direct measure of their capacity for oxygen delivery. Here we examine the relation between CTH and capillary outflow saturation heterogeneity (COSH), which is the heterogeneity of blood oxygen content at the venous end of capillaries. Models for the evolution of hemoglobin saturation heterogeneity (HSH) in capillary networks were developed and validated using a computational model with moving RBCs. Two representative situations were selected: a Krogh cylinder geometry with heterogeneous hemoglobin saturation (HS) at the inflow, and a parallel array of four capillaries. The heterogeneity of HS after converging capillary bifurcations was found to exponentially decrease with a time scale of 0.15–0.21 s due to diffusive interaction between RBCs. Similarly, the HS difference between parallel capillaries also drops exponentially with a time scale of 0.12–0.19 s. These decay times are substantially smaller than measured RBC transit times and only weakly depend on the distance between microvessels. This work shows that diffusive interaction strongly reduces COSH on a small spatial scale. Therefore, we conclude that CTH influences COSH yet does not determine it. The second part of this study will focus on simulations in microvascular networks from the rodent cerebral cortex. Actual estimates of COSH and CTH will then be given.
Microvessels are the primary site of gas exchange in the vertebrate microvascular system due to their large surface area. Energy metabolism is largely dependent on a continuous oxygen supply from the microcirculation which is actively regulated by the microvasculature. For instance, in the cerebral cortex, dilations of pial arteries (Chen et al.,
Malfunctions in the microvasculature occur in many diseases and conditions. Cerebral small vessel disease plays a crucial role in stroke, dementia and aging (Pantoni,
Capillary dysfunction can be quantified by capillary transit time heterogeneity (CTH) which is the standard deviation of the transit time distribution. In their seminal study, Jespersen and Østergaard (
In the study by Jespersen and Østergaard (
The distribution of P
In previous modeling works (Jespersen and Østergaard,
To compute COSH, models that describe the evolution of HSH in single and multiple capillaries are developed. Diffusive oxygen transfer among RBCs is shown to be the main physical mechanism for the reduction of HSH. The diffusive interaction between RBCs in single capillaries and between parallel capillaries is modeled based on ordinary differential equations. These interaction models are validated for a large range of physiological parameters using a computational model with individual moving RBCs (Lücker et al.,
Models for the diffusive interaction between RBCs in single capillaries and between multiple capillaries were developed based on ordinary differential equations that extend those used in Lücker et al. (
Based on the observation that the RBC transit time is only one of multiple parameters that determine HS (Lücker et al.,
Diffusive interaction models are derived for these two representative situations. These simplified models directly highlight the variables that influence most the reduction of HSH. Then, the employed computational model (Lücker et al.,
The interaction models developed in this study are all formulated in an axisymmetric geometry with four distinct regions: RBCs, plasma, capillary endothelium and tissue, denoted by the indices
Schematics for RBC diffusive interaction.
In capillaries, RBCs flow in a single file with velocity
The equilibrium curve for hemoglobin and oxygen is modeled using the Hill equation
where
will often be required. Since the Hill equation is known to be inaccurate at low P
The models developed here are based on the neglect of axial diffusion and the use of steadystate equations. These common assumptions (Hellums,
where
as illustrated in Figure
Oxygen in the blood is present bound to hemoglobin in RBCs and dissolved in both plasma and RBCs. The total convective flux is given by
where
Values of
We can now summarize the above equations to obtain the evolution equation for HS. For brevity, we define total oxygen convective capacity as
Equations (4), (6) and (8) result in
When the tissue domain is a straight cylinder, the last term vanishes. If the tissue radius is not constant, this term is nonzero but was found to be negligible. Therefore, it will be omitted in further derivations. However, this term was included in all numerical computations for completeness. Equation (10) can be recast in terms of the capillary transit time τ and integrated as
Thus, HS on the distal side is influenced by the RBC transit time, the oxygen consumption per unit length, hematocrit and vessel diameter. This description of distal blood oxygen concentration is more complete than the previously used BohrKetyCroneRenkin equation (Jespersen and Østergaard,
where
Given the sink term
In the modeling of diffusive integration between parallel capillaries, knowledge of the P
Capillary networks in the cerebral microvasculature form a meshlike structure (Lorthois and Cassot,
To describe this fluctuation, the HS is treated as a random variable
where
The averaging of Equation (14) combined with Equation (15) yields an expression for
The oxygen partial pressure
By a suitable linearization, the nonlinear Equation (17) can be further simplified to an evolution equation for the standard deviation of
Using the above assumptions, the standard deviation of HS σ_{S} satisfies the differential equation
This equation can be solved numerically given the average HS
The resistance coefficient
where Δ
This quantity can be obtained from the results of the computational model. We will show that Δ
Having examined the diffusive interaction between heterogeneously saturated RBCs in the same capillary, we now consider the diffusive interaction between capillaries with different saturation levels. For our analysis, four parallel capillaries with concurrent blood flow are considered where both pairs of diagonally opposed capillaries will be denoted by the indices ϕ and ψ, respectively (Figure
Schematics for capillary diffusive interaction.
Given different HS values
Using the intravascular resistance coefficient and the Krogh model (Equation 13), the continuity of tissue P
where Δ
Given
A slight simplification in Equation (23) provides an explicit expression for the oxygen flux out of both model capillaries. Under the assumption that both capillaries have the same geometry and linear density, the intravascular resistance coefficient
The assumption that
where
and a similar expression for
By using the average oxygen outflux
This model will be referred to as explicit Kroghbased model. To derive this equation, the respective linear densities in both capillaries were assumed to be equal. However, the respective RBC velocities were still allowed to be different. Under the assumption that
This third model will be referred to as linearized capillary interaction model. This equation leads to the definition of the characteristic length scale
Similarly, the characteristic time scale τ_{CI} is defined as
Thus it is independent from the RBC velocity and depends on linear density, the average HS
The results of the models for RBC and capillary diffusive interactions were compared with numerical solutions to the advectiondiffusionreaction equations for oxygen and hemoglobin. The reaction rates between both quantities are coupled based on Clark et al. (
where
This was chosen instead of the commonly used MichaelisMenten kinetics to facilitate the comparison between the interaction models and the computational model. The oxygen transport equation is given by
where
where
For simulations in a single capillary or parallel capillaries, these equations were solved using the finitevolume method with moving RBCs (Lücker et al.,
The heterogeneity of HS was investigated in different computational domains. The physiological parameters were chosen to match the mouse cerebral cortex. The diffusive interaction between RBCs was studied in a twodimensional cylindrical domain with radius
The diffusive interaction between capillaries was investigated in an array with four parallel capillaries with radius
The metabolic rate of oxygen consumption was set to 10^{−3} μm^{3} O_{2} μm^{−3} s^{−1}, which is within the range of values measured in the anesthetized rodent cerebral cortex (Zhu et al.,
Parameter values.
α_{rbc}  O_{2} solubility in RBCs  3.38 × 10^{−5}  ml O_{2} mmHg^{−1} cm^{−3}  Altman and Dittmer, 
α_{p}  O_{2} solubility in the plasma  2.82 × 10^{−5}  ml O_{2} mmHg^{−1} cm^{−3}  Christoforides et al., 
α_{w}  O_{2} solubility in the capillary wall  3.89 × 10^{−5}  ml O_{2} mmHg^{−1} cm^{−3}  α_{t} 
α_{t}  O_{2} solubility in the tissue  3.89 × 10^{−5}  ml O_{2} mmHg^{−1} cm^{−3}  Mahler et al., 
O_{2} diffusivity in RBCs  9.5 × 10^{−6}  cm^{2} s^{−1}  Clark et al., 

O_{2} diffusivity in the plasma  2.18 × 10^{−5}  cm^{2} s^{−1}  Goldstick et al., 

O_{2} diffusivity in the capillary wall  8.73 × 10^{−6}  cm^{2} s^{−1}  Liu et al., 

O_{2} diffusivity in the tissue  2.41 × 10^{−5}  cm^{2} s^{−1}  Bentley et al., 

Hemoglobin diffusivity in RBCs  1.44 × 10^{−7}  cm^{2} s^{−1}  Clark et al., 

Dissociation rate constant  44  s^{−1}  Clark et al., 

Hill exponent  2.64  –  Fitted from Watanabe et al. ( 

Total heme density  2.03 × 10^{−5}  mol cm^{−3}  Clark et al., 

P 
47.9  mmHg  Fitted from Watanabe et al. ( 

Radius of capillary lumen  2.0  μm  Tsai et al., 

O_{2} molar volume at 36.9°C  2.54 × 10^{4}  ml O_{2} mol^{−1}  Ideal gas law  
RBC volume  59.0  μ 
Shirasawa, 
Equations (37) and (38) were solved using a custom written extension of the opensource computational fluid dynamics library OpenFOAM 2.3.0 (Weller et al.,
The evolution of HSH was simulated in the geometries shown in Figure
The diffusive interaction between RBCs with different HS was investigated in a cylindrical tissue domain (Figure
Hemoglobin saturation profiles with alternating inlet values in the cylindrical geometry. μ_{LD} = 0.3;
To reduce the computational effort in further parameter studies, we compared the results obtained with domain lengths of 100 μm (
We now examine the influence of model parameters such as linear density, RBC velocity, oxygen consumption rate and HS difference at the inlet on the results. Figure
Decay time scale τ_{RI} for RBC diffusive interaction. The time scale τ_{RI} was obtained with an exponential fit of the standard deviation of HS from simulations across a range of parameters.
The above results show that the RBC interaction models agree closely with numerical simulations when using fitted values of
Model coefficient
The capillary diffusive interaction models are now compared to our computational model for oxygen transport. Numerical simulations were run in an array of four straight, parallel capillaries (Figure
Figure
Hemoglobin saturation profiles in parallel capillaries. μ_{LD} = 0.3;
To show model robustness, several input parameters were varied and the predicted drop in HS difference Δ
The capillary interaction models rely on a single model parameter
Decay time scale τ_{CI} for capillary diffusive interaction.
The previous results all assumed the same RBC velocity, flow direction and hematocrit in each capillary. These assumptions are now dropped to further examine model robustness. First, simulations with countercurrent flow instead of concurrent flow were run. Namely, the flows in both pairs of diagonally opposite capillary were set to opposite directions with the same RBC velocity. The HS difference between the venous capillary ends turned out to be practically the same as with concurrent flow (Figure
Capillary diffusive interaction with different RBC velocities (
Capillary diffusive interaction with different linear densities (μ_{LD} = 0.6 and 0.2). Solid lines: numerical model; dashdotted lines: nonlinear Kroghbased model; dotted lines: equal oxygen flux assumption.
We identified two diffusive interaction mechanisms that cause a large reduction of HSH in capillary networks, developed associated interaction models and validated them using a computational model with individual moving RBCs. The interaction models provide explicit formulas for the reduction of HSH and the associated decay exponents, which gives more insight than a purely computational approach. This work shows that CTH only partially reflects the actual heterogeneity of blood oxygen content and that estimating HSH solely based on CTH may lead to considerable overestimation.
Diffusive interaction between RBCs in a single capillary occurs when two branches with different HS levels converge. This phenomenon is therefore more prevalent in the presence of multiple converging bifurcations along RBC paths. In the mouse cerebral cortex, Sakadžić et al. (
While the standard deviation of HS in a single capillary generally decreases, our results show that the average value of
Diffusive interaction between capillaries is the second reduction mechanism of HSH that was investigated here. While RBC diffusive interaction primarily occurs downstream of converging bifurcations, capillary diffusive interaction is a more general phenomenon since it does not require the presence of branchings. Our results qualitatively agree with the computations by Popel et al. (
The range of distances between capillaries (20 to 60 μm) that was examined in our simulations in parallel capillary arrays corresponds to Krogh cylinder radii between 11.3 and 33.8 μm. This includes the mean Krogh radii of the reconstructed MVNs in Fraser et al. (
Having shown the importance of diffusive interaction mechanisms, it is natural to ask up to which length scale they can act. While RBC interaction is confined to single capillaries, hence very local, it is not evident how far reaching capillary interaction can be. The weak dependence of the decay time scale τ_{CI} on capillary spacing (Equation (34) and Figure
The models for RBC and capillary diffusive interactions enable the computation of mean HS and its heterogeneity in single and parallel vessels, respectively. Previously, the relation between CTH and oxygen extraction fraction was studied by Jespersen and Østergaard (
The diffusive interaction models consist in ordinary differential equations that can be easily integrated. The simplifications done in their derivation give rise to slight inaccuracies with respect to the computational model. The errors of the RBC and capillary diffusive interaction models are ≤ 4% (Figure
The limitations to our diffusive interaction models include the oxygenindependent metabolic consumption term
This study extensively investigates HSH in a Krogh cylinder geometry and parallel capillary arrays. Thus, our conclusions are currently limited to tissues with approximately parallel and straight capillaries such as striated muscles. The next step is to verify whether our theoretical predictions hold when blood vessels are interconnected, tortuous and have variable spacings. In a followup article, we are going to present simulations of oxygen transport in microvascular networks from the mouse somatosensory cortex. The distribution of capillary transit times will be compared to that of outflow HS. Then, the interaction models will be used to quantify how much diffusive interaction reduces COSH. The spatial scale up to which diffusive interaction acts also requires further investigation. This could be performed using parallel capillary arrays with more vessels or large realistic microvascular networks. In addition to numerical simulations, experimental data are needed to confirm our theoretical predictions. To achieve this, measurements of CTH based on bolus tracking (GutiérrezJiménez et al.,
In conclusion, this study lays the theoretical basis for the analysis of HSH in MVNs. It is a substantial improvement over previous approaches in the brain that were limited to independent, identical capillaries without branchings. Models for RBC and capillary diffusive interactions were developed and successfully validated using a detailed computational model in simplified geometries. The following conclusions can be drawn: (1) diffusive interaction leads to a strong reduction of smallscale HSH caused by CTH or other factors; (2) HSH can arise in the absence of CTH, for instance due to differences in hematocrit or supplied tissue volume; (3) CTH influences COSH, but does not determine it. Thus, this modeling work is a major step to better understand the actual effects of CTH on blood oxygen content. This has potential implications in the study of all conditions where capillary dysfunction and CTH are thought to be involved, such as Alzheimer's disease (Østergaard et al.,
AL conceived of the study, developed the theoretical models, implemented the algorithms, ran the simulations, interpreted the data and drafted the manuscript. TS contributed the initial idea for the study. BW and PJ conceived of the study and participated in its design.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
The authors are grateful for the valuable discussions with Franca Schmid.
The Supplementary Material for this article can be found online at:
capillary outflow saturation heterogeneity
capillary transit time heterogeneity
hemoglobin saturation
hemoglobin saturation heterogeneity
red blood cell.