The resistance to oxygen transport in the capillaries relative to that in the surrounding tissue

Oxygen transport across tank-treading red blood cell: Individual and joint roles of flow convection and oxygen-hemoglobin reaction

Gas, especially oxygen, transport in the microcirculation is a complex phenomenon, however, of critical importance for maintaining normal biological functions, and the cytoplasm fluid in red blood cells (RBCs) is the major vehicle for transporting oxygen from lungs to tissues via the circulatory system. Existing theoretical and numerical studies have neglected the cytoplasm convection effect by treating RBCs as rigid particles undergoing a constant translation velocity. As a consequence, the influence and mechanism of the cytoplasm flow on oxygen transport are still not clear in microcirculation research. In this study, we consider a tank-treading capsule in shear flow, which is generated with two parallel plates moving in opposite directions: the top plate of a higher oxygen pressure (P_O2) representing the RBC core in the central region of a microvessel and the bottom plate of a lower P_O2 representing the microvessel wall. Numerical simulations are conducted to investigate the individual and combined effects of cytoplasm convection and oxygen-hemoglobin (O₂-Hb) reaction on the oxygen transport efficiency across the tank-treading capsule, and different P_O2 situations and shear rates are also tested. Due to the lower oxygen diffusivity in cytoplasm, the presence of the capsule reduces the oxygen transfer flux across the gap by 7.34 % in the pure diffusion system where the flow convection and O₂-Hb reaction are both neglected. Including the flow convection or the O₂-Hb reaction has little influence on the oxygen flux; however, when they act together as in real microcirculation situations, the enhancement in oxygen transport could be significant, especially in the low P_O2 and high shear rate situations. In particular, with the respective P_O2 at 60 and 30 mmHg on the top and bottom plates and a 400 s^-1 shear rate, the oxygen flux reduction is only 0.02 %, suggesting that the cytoplasm convection can improve the oxygen transport across RBCs considerably. The simulation results are scrutinized to explore the underlying mechanism for the enhancement, and a new nondimensional parameter is introduced to characterize the importance of cytoplasm convection in oxygen transport. These simulation results, discussion and analysis could be helpful for a better understanding of the complex oxygen transport process and therefor valuable for relevant studies.

The coronary capillary bed and its role in blood flow and oxygen delivery: A review

The assumption that the coronary capillary blood flow is exclusively regulated by precapillary vessels is not supported by recent data. Rather, the complex coronary capillary bed has unique structural and geometric characteristics that invalidate many assumptions regarding red blood cell (RBC) transport, for example, data based on a single capillary or that increases in flow are the result of capillary recruitment. It is now recognized that all coronary capillaries are open and that their variations in flow are due to structural differences, local O₂ demand and delivery, and variations in hematocrit. Recent data reveal that local mechanisms within the capillary bed regulate flow via signaling mechanisms involving RBC signaling and endothelial-associated pericytes that contract and relax in response to humoral and neural signaling. The discovery that pericytes respond to vasoactive signals (e.g., nitric oxide, phenylephrine, and adenosine) underscores the role of these cells in regulating capillary diameter and consequently RBC flux and oxygen delivery. RBCs also affect blood flow by sensing <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>P</mml:mi> <mml:msub><mml:mi>O</mml:mi> <mml:mn>2</mml:mn></mml:msub> </mml:msub> </mml:math> and releasing nitric oxide to facilitate relaxation of pericytes and a consequential capillary dilation. New data indicate that these signaling mechanisms allow control of blood flow in specific coronary capillaries according to their oxygen requirements. In conclusion, mechanisms in the coronary capillary bed facilitate RBC density and transit time, hematocrit, blood flow and O₂ delivery, factors that decrease capillary heterogeneity. These findings have important clinical implications for myocardial ischemia and infarction, as well as other vascular diseases.

Origins of 1/f-like tissue oxygenation fluctuations in the murine cortex

The concentration of oxygen in the brain spontaneously fluctuates, and the distribution of power in these fluctuations has a 1/f-like spectra, where the power present at low frequencies of the power spectrum is orders of magnitude higher than at higher frequencies. Though these oscillations have been interpreted as being driven by neural activity, the origin of these 1/f-like oscillations is not well understood. Here, to gain insight of the origin of the 1/f-like oxygen fluctuations, we investigated the dynamics of tissue oxygenation and neural activity in awake behaving mice. We found that oxygen signal recorded from the cortex of mice had 1/f-like spectra. However, band-limited power in the local field potential did not show corresponding 1/f-like fluctuations. When local neural activity was suppressed, the 1/f-like fluctuations in oxygen concentration persisted. Two-photon measurements of erythrocyte spacing fluctuations and mathematical modeling show that stochastic fluctuations in erythrocyte flow could underlie 1/f-like dynamics in oxygenation. These results suggest that the discrete nature of erythrocytes and their irregular flow, rather than fluctuations in neural activity, could drive 1/f-like fluctuations in tissue oxygenation.

Computer modeling of pulsatile blood flow in elastic artery using a software program for application in biomedical engineering

BACKGROUND AND OBJECTIVE

Atherosclerosis-a condition in which an artery is constricted-alters blood flow in the artery, that can exacerbate the condition. Focusing on previous studies, it can be seen that the k-ε model has been used in the simulation. Therefore, the reverse flow on the back of stenosis is not well represented. In this study, the simulated results are much closer to clinical results, relying on the use of physiological pulses, and considering elasticity of the vessel wall, and the applying k-ω model. It can therefore be claimed that a much more accurate prediction will be made regarding the formation, development and progression of the disease.

METHODS

Modeling biological systems usually contain many parameters, which cannot be calculated experimentally, or are too costly and time consuming. In addition, it is occasionally required to examine the influence of different physical variables, which, given the complexity of the governing equations, make analytical methods feasible (or very limited). The present study is an attempt to investigate the turbulent pulsatile blood flow in an elastic artery with single and double stenoses using a finite element software program, ADINA 8.8.

RESULTS

According to the results, the k - ω turbulence model predicted a larger reverse flow in the post-stenotic region and between the two stenoses in comparison with the k - ε model. In other words, the k - ω model results suggest that a larger region is prone to atherosclerosis. In addition, that the k - ε model predicted a greater maximum shear stress at the throat and a shorter reverse flow region (Mean WSS < 0) in both stenosis scenarios. In other words, relative to the k - ε model, the k - ω model underestimated the damage to the plaque and the risk of its rupture though it predicted new stenosis developing behind the previous one. It was observed that the presence of a double stenosis causes the upstream pressure to reach the critical value in less time. Velocity profiles revealed that in the stenosis throat, the maximum velocity exceeds the normal biological state, which may cause disorders in the blood circulation.

CONCLUSIONS

The artery wall displacement results are suggestive of the greater difference between the two turbulence models in the case with double stenosis compared with single stenosis. Moreover, the difference between the two turbulence models in double stenosis is minimized in both post-stenotic and pre-stenotic regions.

The mass transfer coefficient for oxygen transport from blood to tissue in cerebral cortex

The functioning of cerebral cortex depends on adequate tissue oxygenation. MRI-based techniques allow estimation of blood oxygen levels, tissue perfusion, and oxygen consumption rate (CMRO₂), but do not directly measure partial pressure of oxygen (PO₂) in tissue. To address the estimation of tissue PO₂, the oxygen mass transfer coefficient (KTO₂) is here defined as the CMRO₂ divided by the difference in spatially averaged PO₂ between blood and tissue, and is estimated by analyzing Krogh-cylinder type models. Resistance to radial diffusion of oxygen from microvessels to tissue is distributed within vessels and in the extravascular tissue. The value of KTO₂ is shown to depend strongly on vascular length density and also on microvessel tube hematocrits and diameters, but to be insensitive to blood flow rate and to transient changes in flow or oxygen consumption. Estimated values of KTO₂ are higher than implied by previous studies, implying smaller declines in PO₂ from blood to tissue. Average tissue PO₂ can be estimated from MRI-based measurements as average blood PO₂ minus the product of KTO₂ and CMRO₂. For oxygen consumption rates and vascular densities typical of mouse cortex, the predicted difference between average blood and tissue PO₂ is about 10 mmHg.

Mini-review: Perfluorocarbons, Oxygen Transport, and Microcirculation in Low Flow States: in Vivo and in Vitro Studies

The in vivo study of microvascular oxygen transport requires accurate and challenging measurements of several mass transfer parameters. Although recommended, blood flow and oxygenation are typically not measured in many studies where treatments for ischemia are tested. Therefore, the aim of this communication is to briefly review cardinal aspects of oxygen transport, and the effects of perfluorocarbon (PFC) treatment on blood flow and oxygenation based mostly on studies performed in our laboratory. As physiologically relevant events in oxygen transport take place at the microvascular level, we implemented the phosphorescence quenching technique coupled with noninvasive intravital videomicroscopy for quantitative evaluation of these events in vivo. Rodent experimental models and various approaches have been used to induce ischemia, including hemorrhage, micro- and macroembolism, and microvessel occlusion. Measurements show decrease in microvascular blood flow as well as intravascular and tissue oxygen partial pressure (PO2) after these procedures. To minimize or reverse the effects of ischemia and hypoxia, artificial oxygen carriers such as different PFCs were tested. Well-defined endpoints such as blood flow and tissue PO2 were measured because they have significant effect on tissue survival and outcome. In several cases, enhancement of flow and oxygenation could be demonstrated. Similar results were found in vitro: PFC emulsion mixed with blood (from healthy donors and sickle cell disease patients) enhanced oxygen transport. In summary, PFCs may provide beneficial effects in these models by mechanisms at the microvascular level including facilitated diffusion and bubble reabsorption leading to improved blood flow and oxygenation.

More homogeneous capillary flow and oxygenation in deeper cortical layers correlate with increased oxygen extraction

Our understanding of how capillary blood flow and oxygen distribute across cortical layers to meet the local metabolic demand is incomplete. We addressed this question by using two-photon imaging of resting-state microvascular oxygen partial pressure (PO₂) and flow in the whisker barrel cortex in awake mice. Our measurements in layers I-V show that the capillary red-blood-cell flux and oxygenation heterogeneity, and the intracapillary resistance to oxygen delivery, all decrease with depth, reaching a minimum around layer IV, while the depth-dependent oxygen extraction fraction is increased in layer IV, where oxygen demand is presumably the highest. Our findings suggest that more homogeneous distribution of the physiological observables relevant to oxygen transport to tissue is an important part of the microvascular network adaptation to local brain metabolism. These results will inform the biophysical models of layer-specific cerebral oxygen delivery and consumption and improve our understanding of the diseases that affect cerebral microcirculation.

Defining Physiological Normoxia for Improved Translation of Cell Physiology to Animal Models and Humans

The extensive oxygen gradient between the air we breathe (Po₂ ~21 kPa) and its ultimate distribution within mitochondria (as low as ~0.5-1 kPa) is testament to the efforts expended in limiting its inherent toxicity. It has long been recognized that cell culture undertaken under room air conditions falls short of replicating this protection in vitro. Despite this, difficulty in accurately determining the appropriate O₂ levels in which to culture cells, coupled with a lack of the technology to replicate and maintain a physiological O₂ environment in vitro, has hindered addressing this issue thus far. In this review, we aim to address the current understanding of tissue Po₂ distribution in vivo and summarize the attempts made to replicate these conditions in vitro. The state-of-the-art techniques employed to accurately determine O₂ levels, as well as the issues associated with reproducing physiological O₂ levels in vitro, are also critically reviewed. We aim to provide the framework for researchers to undertake cell culture under O₂ levels relevant to specific tissues and organs. We envisage that this review will facilitate a paradigm shift, enabling translation of findings under physiological conditions in vitro to disease pathology and the design of novel therapeutics.

On the Physics Underlying Longitudinal Capillary Recruitment

Numerous researchers have found that capillary vessel haematocrit depends on the vasodilatory state of the arterioles. At rest, vessel haematocrit is down to 15 %, suggesting a red blood cell velocity three times higher than the plasma velocity. This finding is analysed in the context of present understanding of propulsion of red blood cells (RBCs) and plasma by means of the arteriovenous pressure gradient. Interfacial forces between the red blood cells and the plasma are proposed as a rational explanation of the observed red blood cell velocities. While the arteriovenous pressure gradient across the capillaries propels the red blood cell and the plasma jointly, interfacial forces along the red blood cell membrane can propel RBCs at the cost of the plasma. Different options are explored for the physical origin of these interfacial forces and oxygen gradients are found to be the most probable source.

The Relation Between Capillary Transit Times and Hemoglobin Saturation Heterogeneity. Part 1: Theoretical Models

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.

The relative influence of hematocrit and red blood cell velocity on oxygen transport from capillaries to tissue

OBJECTIVE

Oxygen transport to parenchymal cells occurs mainly at the microvascular level and depends on convective RBC flux, which is proportional in an individual capillary to the product of capillary hematocrit and RBC velocity. This study investigates the relative influence of these two factors on tissue PO₂ .

METHODS

A simple analytical model is used to quantify the respective influences of hematocrit, RBC velocity, and RBC flow on tissue oxygenation around capillaries. Predicted tissue PO₂ levels are compared with a detailed computational model.

RESULTS

Hematocrit is shown to have a larger influence on tissue PO₂ than RBC velocity. The effect of RBC velocity increases with distance from the arterioles. Good agreement between analytical and numerical results is obtained, and the discrepancies are explained. Significant dependence of MTCs on RBC velocity at low hematocrit is demonstrated.

CONCLUSIONS

For a given RBC flux in a capillary, the PO₂ in the surrounding tissue increases with increasing hematocrit, as a consequence of decreasing IVR to diffusive oxygen transport from RBCs to tissue. These results contribute to understanding the effects of blood flow changes on oxygen transport, such as those that occur in functional hyperemia in the brain.

Insights into cerebral haemodynamics and oxygenation utilising in vivo mural cell imaging and mathematical modelling

The neurovascular mechanisms underpinning the local regulation of cerebral blood flow (CBF) and oxygen transport remain elusive. In this study we have combined novel in vivo imaging of cortical microvascular and mural cell architecture with mathematical modelling of blood flow and oxygen transport, to provide new insights into CBF regulation that would be inaccessible in a conventional experimental context. Our study indicates that vasoconstriction of smooth muscle actin-covered vessels, rather than pericyte-covered capillaries, induces stable reductions in downstream intravascular capillary and tissue oxygenation. We also propose that seemingly paradoxical observations in the literature around reduced blood velocity in response to arteriolar constrictions might be caused by a propagation of constrictions to upstream penetrating arterioles. We provide support for pericytes acting as signalling conduits for upstream smooth muscle activation, and erythrocyte deformation as a complementary regulatory mechanism. Finally, we caution against the use of blood velocity as a proxy measurement for flow. Our combined imaging-modelling platform complements conventional experimentation allowing cerebrovascular physiology to be probed in unprecedented detail.

Vascular density and distribution in neocortex

An amazingly wide range of complex behavior emerges from the cerebral cortex. Much of the information processing that leads to these behaviors is performed in neocortical circuits that span throughout the six layers of the cortex. Maintaining this circuit activity requires substantial quantities of oxygen and energy substrates, which are delivered by the complex yet well-organized and tightly-regulated vascular system. In this review, we provide a detailed characterization of the most relevant anatomical and functional features of the cortical vasculature. This includes a compilation of the available data on laminar variation of vascular density and the topological aspects of the microvascular system. We also review the spatio-temporal dynamics of cortical blood flow regulation and oxygenation, many aspects of which remain poorly understood. Finally, we discuss some of the important implications of vascular density, distribution, oxygenation and blood flow regulation for (laminar) fMRI.

Development of a mathematical model to estimate intra-tumor oxygen concentrations through multi-parametric imaging

Background

Tumor hypoxia is involved in every stage of solid tumor development: formation, progression, metastasis, and apoptosis. Two types of hypoxia exist in tumors—chronic hypoxia and acute hypoxia. Recent studies indicate that the regional hypoxia kinetics is closely linked to metastasis and therapeutic responses, but regional hypoxia kinetics is hard to measure. We propose a novel approach to determine the local pO₂ by fusing the parameters obtained from in vivo functional imaging through the use of a modified multivariate Krogh model.

Methods

To test our idea and its potential to translate into an in vivo setting through the use of existing imaging techniques, simulation studies were performed comparing the local partial oxygen pressure (pO₂) from the proposed multivariate image fusion model to the referenced pO₂ derived by Green’s function, which considers the contribution from every vessel segment of an entire three-dimensional tumor vasculature to profile tumor oxygen with high spatial resolution.

Results

pO₂ derived from our fusion approach were close to the referenced pO₂ with regression slope near 1.0 and an r² higher than 0.8 if the voxel size (or the spatial resolution set by functional imaging modality) was less than 200 μm. The simulation also showed that the metabolic rate, blood perfusion, and hemoglobin concentration were dominant factors in tissue oxygenation. The impact of the measurement error of functional imaging to the pO₂ precision and accuracy was simulated. A Gaussian error function with FWHM equal to 20 % of blood perfusion or fractional vascular volume measurement contributed to average 7 % statistical error in pO₂.

Conclusion

The simulation results indicate that the fusion of multiple parametric maps through the biophysically derived mathematical models can monitor the intra-tumor spatial variations of hypoxia in tumors with existing imaging methods, and the potential to further investigate different forms of hypoxia, such as chronic and acute hypoxia, in response to cancer therapies.

Electronic supplementary material

The online version of this article (doi:10.1186/s12938-016-0235-5) contains supplementary material, which is available to authorized users.

INFLUENCE OF BIFURCATION DIAMETER ON THE VERTEBRAL ARTERY ORIGIN STENOSIS: A SIMULATION STUDY OF HEMODYNAMICS

The stenosis at the beginning segment of the vertebral artery accounts for the first risk of stroke in the posterior circulation. The extracranial vertebral arteries, especially the proximal ends, have been considered to be the predilection sites of stenosis or occlusion. From the perspective of hemodynamics, the mechanics of vertebral arteries stenosis is still unclear. In this paper, the formation of atherosclerosis in proximal end was concerned from the aspects of the effect of bifurcation diameter. Different models represent different bifurcation diameter. In order to find correlation between bifurcation diameter and WSS we build different models. Three idealized models with the vertebral artery diameter of [Formula: see text][Formula: see text]m (Model A1), [Formula: see text][Formula: see text]m (Model A2) and [Formula: see text][Formula: see text]m (Model A3) respectively and seven realistic models were analyzed by using computational fluid dynamics tools. The area of low wall shear stress (WSS, [Formula: see text] 1.5[Formula: see text]Pa) in the proximal end of vertebral artery extracted at the peak systole in the idealized models were 2.25[Formula: see text]e-7, 8.55[Formula: see text]e-7 and 1.61[Formula: see text]e-6[Formula: see text]m2, respectively. The area of low WSS on the vertebral artery origin of realistic models extracted at the peak systole were 0, 1.18[Formula: see text]e-09, 3.91[Formula: see text]e-07, 1.68[Formula: see text]e-07, 5.46[Formula: see text]e-06, 1.16[Formula: see text]e-06 and 2.25[Formula: see text]e-06[Formula: see text]m2, respectively. Moreover, the time-averaged WSSs of the three idealized models were 3.95, 3.56 and 3.19, respectively. The time-averaged WSSs of the realistic models were 6.28, 6.36, 4.48, 4.71, 3.59, 3.59 and 3.31[Formula: see text]Pa, respectively. With the increase of bifurcation diameter, the risk of endothelial dysfunction increases, and the same is to intimal hyperplasia.

Modeling of Cerebral Oxygen Transport Based on In vivo Microscopic Imaging of Microvascular Network Structure, Blood Flow, and Oxygenation

Oxygen is delivered to brain tissue by a dense network of microvessels, which actively control cerebral blood flow (CBF) through vasodilation and contraction in response to changing levels of neural activity. Understanding these network-level processes is immediately relevant for (1) interpretation of functional Magnetic Resonance Imaging (fMRI) signals, and (2) investigation of neurological diseases in which a deterioration of neurovascular and neuro-metabolic physiology contributes to motor and cognitive decline. Experimental data on the structure, flow and oxygen levels of microvascular networks are needed, together with theoretical methods to integrate this information and predict physiologically relevant properties that are not directly measurable. Recent progress in optical imaging technologies for high-resolution in vivo measurement of the cerebral microvascular architecture, blood flow, and oxygenation enables construction of detailed computational models of cerebral hemodynamics and oxygen transport based on realistic three-dimensional microvascular networks. In this article, we review state-of-the-art optical microscopy technologies for quantitative in vivo imaging of cerebral microvascular structure, blood flow and oxygenation, and theoretical methods that utilize such data to generate spatially resolved models for blood flow and oxygen transport. These “bottom-up” models are essential for the understanding of the processes governing brain oxygenation in normal and disease states and for eventual translation of the lessons learned from animal studies to humans.

Erythrocyte aggregation may promote uneven spatial distribution of NO/O2 in the downstream vessel of arteriolar bifurcations

This study examined the effect of red blood cell (RBC) aggregation on nitric oxide (NO) and oxygen (O2) distributions in the downstream vessels of arteriolar bifurcations. Particular attention was paid to the inherent formation of asymmetric cell-free layer (CFL) widths in the downstream vessels and its consequential impact on the NO/O2 bioavailability after the bifurcations. A microscopic image-based two-dimensional transient model was used to predict the NO/O2 distribution by utilizing the in vivo CFL width data obtained under non-, normal- and hyper-aggregating conditions at the pseudoshear rate of 15.6±2.0s(-1). In vivo experimental result showed that the asymmetry of CFL widths was enhanced by the elevation in RBC aggregation level. The model demonstrated that NO bioavailability was regulated by the dynamic fluctuation of the local CFL widths, which is corollary to its modulation of wall shear stress. Accordingly, the uneven distribution of NO/O2 was prominent at opposite sides of the arterioles up to six vessel-diameter (6D) away from the bifurcating point, and this was further enhanced by increasing the levels of RBC aggregation. Our findings suggested that RBC aggregation potentially augments both the formation of asymmetric CFL widths and its influence on the uneven distribution of NO/O2 in the downstream flow of an arteriolar bifurcation. The extended heterogeneity of NO/O2 downstream (2D-6D) also implied its potential propagation throughout the entire arteriolar microvasculature.

MATHEMATICAL MODEL FOR THE TRANSPORT OF OXYGEN IN THE LIVING TISSUE THROUGH CAPILLARY BED

Oxygen is essential for the survival of living tissues in the human body. The mechanism of oxygen transport in the human body is a subject of great concern. In the conditions like hypoxia and hypothermia, the amount of oxygen supply in the biological tissue loose homeostasis, thereby the concentration of O 2 and the liberation of CO 2 in the human body demands a special attention. The present study based on finite element method employed to the mass diffusion equation with suitable conditions has been established. The main objective of this work is to understand the behavior of O 2 through various compartments of the capillary bed. The concentration of O 2 at plasma and capillary layers has been estimated which in turn leads to understand the situation of oxygen transport during various situations.

Integrative models of vascular remodeling during tumor growth

Malignant solid tumors recruit the blood vessel network of the host tissue for nutrient supply, continuous growth, and gain of metastatic potential. Angiogenesis (the formation of new blood vessels), vessel cooption (the integration of existing blood vessels into the tumor vasculature), and vessel regression remodel the healthy vascular network into a tumor-specific vasculature that is in many respects different from the hierarchically organized arterio-venous blood vessel network of the host tissues. Integrative models based on detailed experimental data and physical laws implement in silico the complex interplay of molecular pathways, cell proliferation, migration, and death, tissue microenvironment, mechanical and hydrodynamic forces, and the fine structure of the host tissue vasculature. With the help of computer simulations high-precision information about blood flow patterns, interstitial fluid flow, drug distribution, oxygen and nutrient distribution can be obtained and a plethora of therapeutic protocols can be tested before clinical trials. In this review, we give an overview over the current status of integrative models describing tumor growth, vascular remodeling, blood and interstitial fluid flow, drug delivery, and concomitant transformations of the microenvironment. WIREs Syst Biol Med 2015, 7:113-129. doi: 10.1002/wsbm.1295 For further resources related to this article, please visit the WIREs website.

CONFLICT OF INTEREST

The authors have declared no conflicts of interest for this article.

THE INFLUENCE OF HEMODYNAMICS ON THE ULCERATION PLAQUES OF CAROTID ARTERY STENOSIS

The further rupture of atherosclerotic ulceration plaque is one of the main triggers of the carotid ischemic stroke. However, the abnormal hemodynamics is not well addressed yet. A lesion-based computational fluid dynamic (CFD) analysis is proposed to investigate the complex hemodynamic change of the ulceration plaque that prevails in patients. The 3D models including eight groups of ulcerations (six groups with single ulceration and two groups with two consecutive ulcerations), were reconstructed based on the computer tomography (CT) images, and the tetrahedral grid was taken to mesh the models with the appropriate numbers. After setting the boundary conditions, numerical simulation was carried out to analyze the pulsatile blood flow in the models. The complex flow in the vicinity of the ulcerations directly leads to a significant effect on the distribution of the wall shear stress (WSS). WSS is respectively from 3.29 to 35.41 Pa at the upstream, from 11.90 to 41.85 Pa at the downstream ulceration, and 18.60 and 30.60 Pa in the area between the two consecutive ulcerations. The rupture from these regions could cause the further rupture of ulceration plaques, particularly at the downstream ulceration and the area between the two consecutive ulcerations. The twisting and the curling of the flow at the ulcerations can lead to thrombosis which may break free later and go through the downstream stenosis by the effect of the flow. The different degrees of WSS in downstream and upstream ulcerations will damage the ulceration on the plaque because of pulling and stretching forces at the ulcerations. Furthermore, high wall shear stress gradient (WSSG) also increases the risk of the further rupture. Our study gives a better understanding in the further rupture mechanism of ulceration plaques and provides the information of the location of thrombosis after aggravated rupturing, which can be referred by surgeons to improve the surgical planning.

A dynamic model of oxygen transport from capillaries to tissue with moving red blood cells

Most oxygen required to support the energy needs of vertebrate tissues is delivered by diffusion from microvessels. The presence of red blood cells (RBCs) makes blood flow in the microcirculation highly heterogeneous. Additionally, flow regulation mechanisms dynamically respond to changes in tissue energy demand. These spatiotemporal variations directly affect the supply of oxygen to parenchymal cells. Due to various limiting assumptions, current models of oxygen transport cannot fully capture the consequences of complex hemodynamic effects on tissue oxygenation and are often not suitable for studying unsteady phenomena. With our new approach based on moving RBCs, the impact of blood flow heterogeneity on oxygen partial pressure (Po2) in the tissue can be quantified. Oxygen transport was simulated using parachute-shaped solid RBCs flowing through a capillary. With the use of a conical tissue domain with radii 19 and 13 μm, respectively, our computations indicate that Po2 at the RBC membrane exceeds Po2 between RBCs by 30 mmHg on average and that the mean plasma Po2 decreases by 9 mmHg over 50 μm. These results reproduce well recent intravascular Po2 measurements in the rodent brain. We also demonstrate that instantaneous variations of capillary hematocrit cause associated fluctuations of tissue Po2. Furthermore, our results suggest that homogeneous tissue oxygenation requires capillary networks to be denser on venular side than on arteriolar side. Our new model for oxygen transport will make it possible to quantify in detail the effects of blood flow heterogeneity on tissue oxygenation in realistic capillary networks.

Theoretical analysis of the determinants of lung oxygen diffusing capacity

The process of pulmonary oxygen uptake is analyzed to obtain an explicit equation for lung oxygen diffusing capacity in terms of hematocrit and pulmonary capillary diameter. An axisymmetric model with discrete cylindrical erythrocytes is used to represent radial diffusion of oxygen from alveoli through the alveolar-capillary membrane into pulmonary capillaries, through the plasma, and into erythrocytes. Analysis of unsteady diffusion due to the passage of the erythrocytes shows that transport of oxygen through the alveolar-capillary membrane occurs mainly in the regions adjacent to erythrocytes, and that oxygen transport through regions adjacent to plasma gaps can be neglected. The model leads to an explicit formula for diffusing capacity as a function of geometric and oxygen transport parameters. For normal hematocrit and a capillary diameter of 6.75 μm, the predicted diffusing capacity is 102 ml O₂ min⁻¹ mmHg⁻¹. This value is 30-40% lower than values estimated previously by the morphometric method, which considers the total membrane area and the specific uptake rate of erythrocytes. Diffusing capacity is shown to increase with increasing hematocrit and decrease with increasing capillary diameter and increasing thickness of the membrane. Simulations of pulmonary oxygen uptake in humans under conditions of exercise or hypoxia based show closer agreement with experimental data than previous models, but still overestimate oxygen uptake. The remaining discrepancy may reflect effects of heterogeneity of perfusion and ventilation in the lung.

Oxygen transport in the microcirculation and its regulation

OBJECTIVE

Cells require energy to carry out their functions and they typically use oxidative phosphorylation to generate the needed ATP. Thus, cells have a continuous need for oxygen, which they receive by diffusion from the blood through the interstitial fluid. The circulatory system pumps oxygen-rich blood through a network of increasingly minute vessels, the microcirculation. The structure of the microcirculation is such that all cells have at least one nearby capillary for diffusive exchange of oxygen and red blood cells release the oxygen bound to hemoglobin as they traverse capillaries.

METHODS

This review focuses first on the historical development of techniques to measure oxygen at various sites in the microcirculation, including the blood, interstitium, and cells.

RESULTS

Next, approaches are described as to how these techniques have been employed to make discoveries about different aspects of oxygen transport. Finally, ways in which oxygen might participate in the regulation of blood flow toward matching oxygen supply to oxygen demand is discussed.

CONCLUSIONS

Overall, the transport of oxygen to the cells of the body is one of the most critical functions of the cardiovascular system and it is in the microcirculation where the final local determinants of oxygen supply, oxygen demand, and their regulation are decided.

Imaging local neuronal activity by monitoring PO₂ transients in capillaries

Two-photon phosphorescence lifetime microscopy (2PLM) has been used recently for depth measurements of oxygen partial pressure (PO(2)) in the rodent brain. In capillaries of olfactory bulb glomeruli, 2PLM has also allowed simultaneous measurements of PO(2) and blood flow and revealed the presence of erythrocyte-associated transients (EATs), which are PO(2) gradients that are associated with individual erythrocytes. We investigated the extent to which EAT properties in capillaries report local neuronal activity. We find that at rest, PO(2) at EAT peaks overestimates the mean PO(2) by 35 mm Hg. PO(2) between two EAT peaks is at equilibrium with, and thus reports, PO(2) in the neuropil. During odor stimulation, there is a small PO(2) decrease before functional hyperemia, showing that the initial dip in PO(2) is present at the level of capillaries. We conclude that imaging oxygen dynamics in capillaries provides a unique and noninvasive approach to map neuronal activity.

Oxygen: the poison is in the dose

Cell-free hemoglobin (Hb) has been blamed for a spectrum of problems, including vasoconstriction pancreatitis, myocardial infarction, and pulmonary hypertension in hemolytic anemia, malaria, and sickle cell anemia, and from Hb-based oxygen carriers (HBOCs). Toxicities have been attributed to scavenging of nitric oxide (NO). However, while NO scavenging may explain many in vitro effects, and some effects in animal models and clinical trials with HBOCs, key inconsistencies in the theory require alternative explanations. This review considers the hypothesis that cell-free Hb oversupplies oxygen to tissues, leading to oxygen-related toxicity, possibly through formation of reactive oxygen species and local destruction of NO. Evidence for this hypothesis comes from various sources, establishing that tissue oxygen levels are maintained over very narrow (and low) levels, even at high oxygen consumption. Tissue is normally protected from excessive oxygen by its extremely low solubility in plasma, but introduction of cell-free Hb, even at low concentration, greatly augments oxygen supply, engaging protective mechanisms that include vasoconstriction and ischemia. The requirement to limit oxygen supply by cell-free Hb suggests novel ways to modify it to overcome vasoconstriction, independent of the intrinsic reaction of Hb with NO. This control is essential to the design of a safe and effective cell-free HBOC.

Modelling and detecting tumour oxygenation levels

Tumours that are low in oxygen (hypoxic) tend to be more aggressive and respond less well to treatment. Knowing the spatial distribution of oxygen within a tumour could therefore play an important role in treatment planning, enabling treatment to be targeted in such a way that higher doses of radiation are given to the more radioresistant tissue. Mapping the spatial distribution of oxygen in vivo is difficult. Radioactive tracers that are sensitive to different levels of oxygen are under development and in the early stages of clinical use. The concentration of these tracer chemicals can be detected via positron emission tomography resulting in a time dependent concentration profile known as a tissue activity curve (TAC). Pharmaco-kinetic models have then been used to deduce oxygen concentration from TACs. Some such models have included the fact that the spatial distribution of oxygen is often highly inhomogeneous and some have not. We show that the oxygen distribution has little impact on the form of a TAC; it is only the mean oxygen concentration that matters. This has significant consequences both in terms of the computational power needed, and in the amount of information that can be deduced from TACs.

Effects of cell-free layer formation on NO/O2 bioavailability in small arterioles

We developed a new time-dependent computational model for coupled NO/O(2) transport in small arterioles that incorporates potential physiological responses (temporal changes in NO scavenging rate and O(2) partial pressure in blood lumen and NO production rate in endothelium) to the temporal cell-free layer width variations. Two relations between wall shear stress (WSS) and NO production rate based on the linear and sigmoidal functions were considered in this simulation study. The cell-free layer data used for the simulation were acquired from arteriolar flows (D=48.3 ± 1.9 μm) in the rat cremaster muscles under normal flow conditions (WSS=3.4-5.6 Pa). For both cases of linear and sigmoidal relations, temporal layer width variations were found to be capable of significantly enhancing NO bioavailability and this effect was more pronounced in the latter (P<0.0005) than the former (P<0.005). In contrast, O(2) bioavailability in the arteriolar wall was not considerably altered by the temporal layer width variations, irrespective of the relation. Prominent enhancement (P<0.005) of soluble guanylyl cyclase (sGC) activation in the smooth muscle by the temporal layer width variations were predicted for both relations. The extent of sGC activation was generally lower (P<0.01) in the case of the sigmoidal relation than that of the linear relation, suggesting a lesser tendency for arterioles to dilate with the former.

Oxygen gradients in the microcirculation

Early in the last century August Krogh embarked on a series of seminal studies to understand the connection between tissue metabolism and mechanisms by which the cardiovascular system supplied oxygen to meet those needs. Krogh recognized that oxygen was supplied from blood to the tissues by passive diffusion and that the most likely site for oxygen exchange was the capillary network. Studies of tissue oxygen consumption and diffusion coefficient, coupled with anatomical studies of capillarity in various tissues, led him to formulate a model of oxygen diffusion from a single capillary. Fifty years after the publication of this work, new methods were developed which allowed the direct measurement of oxygen in and around microvessels. These direct measurements have confirmed the predictions by Krogh and have led to extensions of his ideas resulting in our current understanding of oxygenation within the microcirculation. Developments during the last 40 years are reviewed, including studies of oxygen gradients in arterioles, capillaries, venules, microvessel wall and surrounding tissue. These measurements were made possible by the development and use of new methods to investigate oxygen in the microcirculation, so mention is made of oxygen microelectrodes, microspectrophotometry of haemoglobin and phosphorescence quenching microscopy. Our understanding of oxygen transport from the perspective of the microcirculation has gone from a consideration of oxygen gradients in capillaries and tissue to the realization that oxygen has the ability to diffuse from any microvessel to another location under the conditions that there exists a large enough PO(2) gradient and that the permeability for oxygen along the intervening pathway is sufficient.

Simultaneous two-photon imaging of oxygen and blood flow in deep cerebral vessels

Uncovering principles that regulate energy metabolism in the brain requires mapping of partial pressure of oxygen (PO(2)) and blood flow with high spatial and temporal resolution. Using two-photon phosphorescence lifetime microscopy (2PLM) and the oxygen probe PtP-C343, we show that PO(2) can be accurately measured in the brain at depths up to 300 μm with micron-scale resolution. In addition, 2PLM allowed simultaneous measurements of blood flow and of PO(2) in capillaries with less than one-second temporal resolution. Using this approach, we detected erythrocyte-associated transients (EATs) in oxygen in the rat olfactory bulb and showed the existence of diffusion-based arterio-venous shunts. Sensory stimulation evoked functional hyperemia, accompanied by an increase in PO(2) in capillaries and by a biphasic PO(2) response in the neuropil, consisting of an 'initial dip' and a rebound. 2PLM of PO(2) opens new avenues for studies of brain metabolism and blood flow regulation.

Integration of cardiovascular regulation by the blood/endothelium cell-free layer

The cell-free layer (CFL) width separating red blood cells in flowing blood from the endothelial cell membrane is shown to be a regulator of the balance between nitric oxide (NO) production by the endothelium and NO scavenging by blood hemoglobin. The CFL width is determined by hematocrit (Hct) and the vessel wall flow velocity gradient. These factors and blood and plasma viscosity determine vessel wall shear stress which regulates the production of NO in the vascular wall. Mathematical modeling and experimental findings show that vessel wall NO concentration is a strong nonlinear function of Hct and that small Hct variations have comparatively large effects on blood pressure regulation. Furthermore, NO concentration is a regulator of inflammation and oxygen metabolism. Therefore, small, sustained perturbations of Hct may have long-term effects that can promote pro-hypertensive and pro-inflammatory conditions. In this context, Hct and its variability are directly related to vascular tone, peripheral vascular resistance, oxygen transport and delivery, and inflammation. These effects are relevant to the analysis and understanding of blood pressure regulation, as NO bioavailability regulates the contractile state of blood vessels. Furthermore, regulation of the CFL is a direct function of blood composition therefore understanding of its physiology relates to the design and management of fluid resuscitation fluids. From a medical perspective, these studies propose that it should be of clinical interest to note small variations in patient's Hct levels given their importance in modulating the CFL width and therefore NO bioavailability. WIREs Syst Biol Med 2011 3 458-470 DOI: 10.1002/wsbm.150

Measurement of oxygen in the microcirculation using phosphorescence quenching microscopy

According to the classical concept of Krogh, O(2) is delivered to the tissues solely by capillaries and intra-capillary resistance to O(2) diffusion is negligible. Over the past three decades longitudinal PO(2) and SO(2) gradients in arterioles have been observed with a transmural PO(2) gradient in small arterioles of only 1-2 mmHg. Application of phosphorescence quenching microscopy to measurements of PO(2) in arterioles of the rat mesentery by Tsai et al. (1998) found a large transmural PO(2) in these arterioles. That led to the provocative conclusion that the arteriolar wall is the major sink for O(2) in the microcirculation. Our studies indicate that many of these results can be explained by photo-activated O(2) consumption following phosphor excitation, combined with a large excitation area and high frequency of flash excitation. We have developed the basic principles for phosphorescence quenching microscopy including the need to use a small excitation area, a low excitation frequency and a scanning excitation for stationary samples.

Oxygen Partial Pressure in Outer Layers of Skin: Simulation Using Three-Dimensional Multilayered Models

OBJECTIVE

Papillary loops in dermal papillae are believed to facilitate oxygen delivery to the germinal layer in the epidermis. The study aims to simulate the distribution of the oxygen partial pressure, pO2, in superficial layers of skin and to quantify oxygen supply from different sources.

METHODS

A theoretical model that considered heterogeneity in tissue properties was developed based on the anatomical arrangement of the upper skin. Distribution of pO2 in tissues and in blood was solved numerically. Effects of the blood flow rate and the skin surface condition were investigated.

RESULTS

Under normal blood flow rate and skin surface conditions, approximately one-quarter of the oxygen consumed in the upper skin came from papillary loops. Subpapillary plexus was the main supplier, which accounted for more than 60% of the total oxygen supply. Oxygen diffusion into the skin from the air was restricted to superficial regions of the germinal layer and accounted for approximately 10% of the total oxygen uptake.

CONCLUSIONS

Papillary loops in dermal papillae facilitate oxygen delivery to the germinal layer from the circulation. The flow rate in papillary loops affects pO2 distribution in dermal papillae. In the epidermis, however, pO2 is strongly affected by skin surface conditions.

Functional optical imaging at the microscopic level

Functional microscopic imaging of in vivo tissues aims at characterizing parameters at the level of the unitary cellular components under normal conditions, in the presence of blood flow, to understand and monitor phenomena that lead to maintaining homeostatic balance. Of principal interest are the setting of shear stress on the endothelium; formation of the plasma layer, where the balance between nitric oxide production and scavenging is established; and formation of the oxygen gradients that determine the distribution of oxygen from blood into the tissue. Optical techniques that enable the analysis of functional microvascular processes are the measurement of blood vessel dimensions by image shearing, the photometric analysis of the extent of the plasma layer, the dual-slit methodology for measuring blood flow velocity, and the direct measurement of oxygen concentration in blood and tissue. Each of these technologies includes the development of paired, related mathematical approaches that enable characterizing the transport properties of the blood tissue system. While the technology has been successful in analyzing the living tissue in experimental conditions, deployment to clinical settings remains an elusive goal, due to the difficulty of obtaining optical access to the depth of the tissue.

A mathematical model of O2 transport in the rat outer medulla. I. Model formulation and baseline results

The mammalian kidney is particularly vulnerable to hypoperfusion, because the O(2) supply to the renal medulla barely exceeds its O(2) requirements. In this study, we examined the impact of the complex structural organization of the rat outer medulla (OM) on O(2) distribution. We extended the region-based mathematical model of the rat OM developed by Layton and Layton (Am J Physiol Renal Physiol 289: F1346-F1366, 2005) to incorporate the transport of RBCs, Hb, and O(2). We considered basal cellular O(2) consumption and O(2) consumption for active transport of NaCl across medullary thick ascending limb epithelia. Our model predicts that the structural organization of the OM results in significant Po(2) gradients in the axial and radial directions. The segregation of descending vasa recta, the main supply of O(2), at the center and immediate periphery of the vascular bundles gives rise to large radial differences in Po(2) between regions, limits O(2) reabsorption from long descending vasa recta, and helps preserve O(2) delivery to the inner medulla. Under baseline conditions, significantly more O(2) is transferred radially between regions by capillary flow, i.e., advection, than by diffusion. In agreement with experimental observations, our results suggest that 79% of the O(2) supplied to the medulla is consumed in the OM and that medullary thick ascending limbs operate on the brink of hypoxia.

Perfluorocarbon emulsions as a promising technology: a review of tissue and vascular gas dynamics

Perfluorocarbon (PFC) emulsions are halogen-substituted carbon nonpolar oils with resultant enhanced dissolved respiratory gas (O(2), N(2), CO(2), nitric oxide) capabilities. In the first demonstration of enhanced O(2) solubility, inhaled PFC could sustain rat metabolism. Intravenous emulsions were then trialed as "blood substitutes." In the last 10 yr, biocomputational modeling has enhanced our mechanistic understanding of PFCs. Contemporary research is now taking advantage of these physiological discoveries and applying PFCs as "oxygen therapeutics," as well as ways to enhance other gas movements. One particularly promising area of research is the treatment of gas embolism (arterial and venous emboli/decompression sickness). An expansive understanding of PFC-enhanced diffusive gas movements through tissue and vasculature may have analogous applications for O(2) or other respiratory gases and should provide a revolution in medicine. This review will stress the fundamental knowledge we now have regarding how respiratory gas movements are changed when intravenous PFC is present.

Direct numerical simulation of transitional flow in a stenosed carotid bifurcation

The blood flow dynamics of a stenosed, subject-specific, carotid bifurcation were numerically simulated using the spectral element method. Pulsatile inlet conditions were based on in vivo color Doppler ultrasound measurements of blood velocity. The results demonstrated the transitional or weakly turbulent state of the blood flow, which featured rapid velocity and pressure fluctuations in the post-stenotic region of the internal carotid artery (ICA) during systole and laminar flow during diastole. High-frequency vortex shedding was greatest downstream of the stenosis during the deceleration phase of systole. Velocity fluctuations had a frequency within the audible range of 100-300Hz. Instantaneous wall shear stress (WSS) within the stenosis was relatively high during systole ( approximately 25-45Pa) compared to that in a healthy carotid. In addition, high spatial gradients of WSS were present due to flow separation on the inner wall. Oscillatory flow reversal and low pressure were observed distal to the stenosis in the ICA. This study predicts the complex flow field, the turbulence levels and the distribution of the biomechanical stresses present in vivo within a stenosed carotid artery.

Time-dependant oxygen partial pressure in capillaries and tissue in the hamster window chamber model

The possibility of a plasma oxygen diffusion barrier implies a significant resistance to oxygen diffusion and the existence of capillary erythrocyte-associated transients of oxygen. This effect was analyzed by measuring intracapillary blood and tissue pO(2) in the hamster chamber window model using a noninvasive intravital microscopy palladium porphyrin phosphorescence decay technique for two set light excitations (high and low). Using high light excitation, intracapillary blood pO(2) was 13.7 +/- 6.1 mm Hg, and increased to 18.0 +/- 4.5 mm Hg for low light excitation. For high light excitation, intracapillary blood pO(2) peaks were in the range of 25-30 mm Hg, and the lowest values were in the range of 5-10 mm Hg. Reducing the excitation provided a more uniform pO(2) ranging 15-25 mm Hg. With temporal reduction in blood capillary pO(2), levels were correlated to the increase in phosphorescent amplitude that corresponded to plasma gaps. Tissue pO(2) measured at low light excitation in the proximity of capillaries was 23.1 +/- 1.8 mm Hg. In conclusion, low intracapillary blood pO(2) measurements at full hematocrit are an artifact, only observed when oxygen consumption by the measurement technique was excessive and/or absorption of the excitation light was increased by the absence of RBCs. These findings suggest that resistance to oxygen diffusion in plasma is a minor factor in tissue oxygenation by capillaries in the hamster model.

Erythrocyte-associated transients in capillary PO2: an isovolemic hemodilution study in the rat spinotrapezius muscle

Mathematical simulations of oxygen delivery to tissue from capillaries that take into account the particulate nature of blood flow predict the existence of oxygen tension (Po(2)) gradients between erythrocytes (RBCs). As RBCs and plasma alternately pass an observation point, these gradients are manifested as rapid fluctuations in Po(2), also known as erythrocyte-associated transients (EATs). The impact of hemodilution on EATs and oxygen delivery at the capillary level of the microcirculation has yet to be elucidated. Therefore, in the present study, phosphorescence quenching microscopy was used to measure EATs and Po(2) in capillaries of the rat spinotrapezius muscle at the following systemic hematocrits (Hct(sys)): normal (39%) and after moderate (HES1; 27%) or severe (HES2; 15%) isovolemic hemodilution using a 6% hetastarch solution. A 532-nm laser, generating 10-micros pulses concentrated onto a 0.9-microm spot, was used to obtain plasma Po(2) values 100 times/s at points along surface capillaries of the muscle. Mean capillary Po(2) (Pc(O(2)); means +/- SE) significantly decreased between conditions (normal: 56 +/- 2 mmHg, n = 45; HES1: 47 +/- 2 mmHg, n = 62; HES2: 27 +/- 2 mmHg, n = 52, where n = capillary number). In addition, the magnitude of Po(2) transients (DeltaPo(2)) significantly decreased with hemodilution (normal: 19 +/- 1 mmHg, n = 45; HES1: 11 +/- 1 mmHg, n = 62; HES2: 6 +/- 1 mmHg, n = 52). Results suggest that the decrease in Pc(O(2)) and DeltaPo(2) with hemodilution is primarily dependent on Hct(sys) and subsequent microvascular compensations.

Simultaneous blood-tissue exchange of oxygen, carbon dioxide, bicarbonate, and hydrogen ion

A detailed nonlinear four-region (red blood cell, plasma, interstitial fluid, and parenchymal cell) axially distributed convection-diffusion-permeation-reaction-binding computational model is developed to study the simultaneous transport and exchange of oxygen (O2) and carbon dioxide (CO2) in the blood-tissue exchange system of the heart. Since the pH variation in blood and tissue influences the transport and exchange of O2 and CO2 (Bohr and Haldane effects), and since most CO2 is transported as HCO3(-) (bicarbonate) via the CO2 hydration (buffering) reaction, the transport and exchange of HCO3(-) and H+ are also simulated along with that of O2 and CO2. Furthermore, the model accounts for the competitive nonlinear binding of O2 and CO2 with the hemoglobin inside the red blood cells (nonlinear O2-CO2 interactions, Bohr and Haldane effects), and myoglobin-facilitated transport of O2 inside the parenchymal cells. The consumption of O2 through cytochrome-c oxidase reaction inside the parenchymal cells is based on Michaelis-Menten kinetics. The corresponding production of CO2 is determined by respiratory quotient (RQ), depending on the relative consumption of carbohydrate, protein, and fat. The model gives a physiologically realistic description of O2 transport and metabolism in the microcirculation of the heart. Furthermore, because model solutions for tracer transients and steady states can be computed highly efficiently, this model may be the preferred vehicle for routine data analysis where repetitive solutions and parameter optimization are required, as is the case in PET imaging for estimating myocardial O2 consumption.

Oxygen transport and exchange in the microcirculation

The cardiovascular system is responsible for maintaining an adequate convective delivery of oxygen to the smallest branches of the network of blood vessels-the microcirculation-from which oxygen passes to the parenchymal cells by passive diffusion. The aim of this brief review is to trace the development of the study of oxygen transport from the point of view of the microcirculation. August Krogh performed measurements that allowed him to use his keen insight to draw conclusions about oxygen transport that remained the foundations of this field for decades. After an extended period of neglect, Duling rekindled interest in the field of oxygen transport by discovering that substantial amounts of oxygen diffused from the arteriolar network. Subsequent investigations confirmed this finding ill various vascular beds and extended these studies to capillaries and venules. The important contributions of computational modeling and new techniques in intravital microscopy continue to lead to more advances in our understanding of the role of the microcirculation in the supply of oxygen to tissues. Current work is applying the concepts and principles learned in normal tissues to pathophysiological situations, as well as increasing our understanding of artificial oxygen carriers, oxygen sensing, and the connections between nitric oxide and oxygen transport.

Erythrocyte-associated transients in PO2 revealed in capillaries of rat mesentery

Mathematical models have predicted the existence of Po(2) gradients between erythrocytes in capillaries in the usual case where plasma contributes substantial resistance to oxygen diffusion. According to theoretical predictions, these gradients could be detected as rapid Po(2) fluctuations (erythrocyte-associated transients, EATs) along the capillary. However, verification of a model and correct choice of its parameters can be made only on the basis of direct experimental measurements. We used phosphorescence quenching microscopy to measure Po(2) in 52 capillaries of rat mesentery to obtain plasma Po(2) values 100 times/s at a given point along a capillary. A 532-nm laser generated 10-micros pulses of light, concentrated by a x100 objective, onto a spot 0.9 microm in diameter. The presence of erythrocytes in the excitation region was detected on the basis of phosphorescence amplitude (PA), proportional to the amount of plasma encountered by the laser beam, and on the basis of the intensity of transmitted laser light (LT), detected by a photodiode placed under the capillary. The data revealed correlated waveforms in PA, LT, and Po(2) in capillaries. The magnitude of the Po(2) gradients between erythrocytes and plasma was correlated with average capillary Po(2). EATs in Po(2) were more readily detected in capillaries with relatively low oxygenation. The correlation coefficients between PA and Po(2) for the half of the capillaries (n = 26) below the median Po(2) (mean Po(2) = 17 mmHg; R = -0.72) was higher than that for the other half (mean Po(2) = 39 mmHg; R = -0.38). These results support the theoretical predictions of EATs and plasma Po(2) gradients in capillaries.

Effect of Blood Flow on Gas Transport in a Pulmonary Capillary

The effects of blood velocity on gas transport within the alveolar region of lungs, and on the lung diffusing capacity DL have for many years been regarded as negligible. The present work reports on a preliminary, two-dimensional investigation of CO convection-diffusion phenomenon within a pulmonary capillary. Numerical simulations were performed using realistic clinical and morphological parameter values, with discrete circular red blood cells (RBCs) moving with plasma in a single capillary. Steady-state simulations with stationary blood (RBCs and plasma) were performed to validate the model by comparison with published data. Results for RBCs moving at speeds varying from 1.0mm∕s to 10mm∕s, and for capillary hematocrit (Ht) from 5% to 55%, revealed an increase of up to 60% in DL, as compared to the stationary blood case. The increase in DL is more pronounced at low Ht (less than 25%) and high RBC speed and it seems to be caused primarily by the presence of plasma. The results also indicate that capillary blood convection affects DL not only by improving the plasma mixing in the capillary bed but also by replenishing the capillary with fresh (zero concentration) plasma, providing an additional reservoir for the consumption of CO. Our findings cast doubt on the current belief that an increase in the lung diffusing capacity of humans (for instance, during exercising), with fixed hematocrit, can only be accomplished by an increase in the lung volume effectively active in the respiration process.

Oxygen gradients in the microcirculation

As arterialized blood transits from the central circulation to the periphery, oxygen exits through the vessel walls driven by radial oxygen gradients that extend from the red blood cell column, through the plasma, the vessel wall, and the parenchymal tissue. This exit determines a longitudinal gradient of blood oxygen saturation whose extent is inversely related to the level of metabolic activity of the tissue, being small for the brain and considerable for skeletal muscle at rest where hemoglobin is only half-saturated with oxygen when blood arrives to the capillaries. Data obtained by a variety of methods show that the oxygen loss is too great to be explained by diffusion alone, and oxygen gradients measured in the arteriolar wall provide evidence that this structure in vivo is a very large oxygen sink, and suggests a rate of oxygen consumption two orders of magnitude greater than seen in in vitro studies. Longitudinal gradients in the capillary network and radial gradients in surrounding tissue also show a dependence on the metabolic rate of the tissue, being more pronounced in brain than in resting skeletal muscle and mesentery. Mean PO2 values increase from the postcapillary venules to the distal vessels of this network while radial gradients indicate additional oxygen loss. This circumstance may be due to pathways with higher flow having higher oxygen content than low flow pathways as well as possible oxygen uptake from adjacent arterioles. Taken together, these newer findings on oxygen gradients in the microcirculation require a reexamination of existing concepts of oxygen delivery to tissue and the role of the capillaries in this process.

The Pharmacology of Tissue Oxygenation by Biopure’s Hemoglobin-Based Oxygen Carrier, Hemopure® (HBOC-201)

Biopure's hemoglobin-based oxygen carrier, HBOC-201 (Hemopure), enhances oxygen transport by promoting both the convective and diffusive components of transport in the microcirculation. Convective transport is modified by HBOC-201 in three ways; (i) volume expansion promotes organ and tissue perfusion, (ii) the low viscosity of HBOC-201 improves flow to tissues, and (iii) oxygen delivery by HBOC Hb in the plasma is relatively insensitive to mechanisms regulating RBC distribution in the microcirculation. Diffusive oxygen transport is increased by the higher P50 compared with native RBC Hb which increases the off-loading of oxygen to tissues. Oxygen transport is also increased by reducing the diffusional barrier to oxygen transport associated with the plasma, in which oxygen is sparingly soluble. Biopure's HBOC solutions have been shown in vitro and in vivo to take up and off-load oxygen more efficiently than RBC Hb, and when added to blood can increase the efficiency of RBC oxygen transport.