How well mixed is inert gas in tissues?

The washout of inert gas from tissues typically follows multiexponential curves rather than monoexponential curves as would be expected from homogeneous, well-mixed compartment. This implies that the ratio for the square root of the variance of the distribution of transit times to the mean (relative dispersion) must be greater than 1. Among the possible explanations offered for multiexponential curves are heterogeneous capillary flow, uneven capillary spacing, and countercurrent exchange in small veins and arteries. By means of computer simulations of the random walk of gas molecules across capillary beds with parameters of skeletal muscle, we find that heterogeneity involving adjacent capillaries does not suffice to give a relative dispersion greater than one. Neither heterogeneous flow, nor variations in spacing, nor countercurrent exchange between capillaries can account for the multiexponential character of experimental tissue washout curves or the large relative dispersions that have been measured. Simple diffusion calculations are used to show that many gas molecules can wander up to several millimeters away from their entry point during an average transit through a tissue bed. Analytical calculations indicate that an inert gas molecule in an arterial vessel will usually make its first vascular exit from a vessel larger than 20 micron and will wander in and out of tissue and microvessels many times before finally returning to the central circulation. The final exit from tissue will nearly always be into a vessel larger than 20 micron. We propose the hypothesis that the multiexponential character of skeletal muscle tissue inert gas washout curves must be almost entirely due to heterogeneity between tissue regions separated by 3 mm or more, or to countercurrent exchanges in vessels larger than 20 micron diam.     

Vasomotion in critically perfused muscle protects adjacent tissues from capillary perfusion failure

We analyzed the incidence and interaction of arteriolar vasomotion and capillary flow motion during critical perfusion conditions in neighboring peripheral tissues using intravital fluorescence microscopy. The gracilis and semitendinosus muscles and adjacent periosteum, subcutis, and skin of the left hindlimb of Sprague-Dawley rats were isolated at the femoral vessels. Critical perfusion conditions, achieved by stepwise reduction of femoral artery blood flow, induced capillary flow motion in muscle, but not in the periosteum, subcutis, and skin. Strikingly, blood flow within individual capillaries was decreased (P < 0.05) in muscle but was not affected in the periosteum, subcutis, and skin. However, despite the flow motion-induced reduction of muscle capillary blood flow during the critical perfusion conditions, functional capillary density remained preserved in all tissues analyzed, including the skeletal muscle. Abrogation of vasomotion in the muscle arterioles by the calcium channel blocker felodipine resulted in a redistribution of blood flow within individual capillaries from cutaneous, subcutaneous, and periosteal tissues toward skeletal muscle. As a consequence, shutdown of perfusion of individual capillaries was observed that resulted in a significant reduction (P < 0.05) of capillary density not only in the neighboring tissues but also in the muscle itself. We conclude that during critical perfusion conditions, vasomotion and flow motion in skeletal muscle preserve nutritive perfusion (functional capillary density) not only in the muscle itself but also in the neighboring tissues, which are not capable of developing this protective regulatory mechanism by themselves.     

Numerical Modeling of Interstitial Fluid Flow Coupled with Blood Flow through a Remodeled Solid Tumor Microvascular Network

Modeling of interstitial fluid flow involves processes such as fluid diffusion, convective transport in extracellular matrix, and extravasation from blood vessels. To date, majority of microvascular flow modeling has been done at different levels and scales mostly on simple tumor shapes with their capillaries. However, with our proposed numerical model, more complex and realistic tumor shapes and capillary networks can be studied. Both blood flow through a capillary network, which is induced by a solid tumor, and fluid flow in tumor’s surrounding tissue are formulated. First, governing equations of angiogenesis are implemented to specify the different domains for the network and interstitium. Then, governing equations for flow modeling are introduced for different domains. The conservation laws for mass and momentum (including continuity equation, Darcy’s law for tissue, and simplified Navier–Stokes equation for blood flow through capillaries) are used for simulating interstitial and intravascular flows and Starling’s law is used for closing this system of equations and coupling the intravascular and extravascular flows. This is the first study of flow modeling in solid tumors to naturalistically couple intravascular and extravascular flow through a network. This network is generated by sprouting angiogenesis and consisting of one parent vessel connected to the network while taking into account the non-continuous behavior of blood, adaptability of capillary diameter to hemodynamics and metabolic stimuli, non-Newtonian blood flow, and phase separation of blood flow in capillary bifurcation. The incorporation of the outlined components beyond the previous models provides a more realistic prediction of interstitial fluid flow pattern in solid tumors and surrounding tissues. Results predict higher interstitial pressure, almost two times, for realistic model compared to the simplified model.     

Effect of sepsis on skeletal muscle oxygen consumption and tissue oxygenation: interpreting capillary oxygen transport data using a mathematical model

Inherent in the inflammatory response to sepsis is abnormal microvascular perfusion. Maldistribution of capillary red blood cell (RBC) flow in rat skeletal muscle has been characterized by increased 1) stopped-flow capillaries, 2) capillary oxygen extraction, and 3) ratio of fast-flow to normal-flow capillaries. On the basis of experimental data for functional capillary density (FCD), RBC velocity, and hemoglobin O2 saturation during sepsis, a mathematical model was used to calculate tissue O2 consumption (Vo2), tissue Po2 (Pt) profiles, and O2 delivery by fast-flow capillaries, which could not be measured experimentally. The model describes coupled capillary and tissue O2 transport using realistic blood and tissue biophysics and three-dimensional arrays of heterogeneously spaced capillaries and was solved numerically using a previously validated scheme. While total blood flow was maintained, capillary flow distribution was varied from 60/30/10% (normal/fast/stopped) in control to 33/33/33% (normal/fast/stopped) in average sepsis (AS) and 25/25/50% (normal/fast/stopped) in extreme sepsis (ES). Simulations found approximately two- and fourfold increases in tissue Vo2 in AS and ES, respectively. Average (minimum) Pt decreased from 43 (40) mmHg in control to 34 (27) and 26 (15) mmHg in AS and ES, respectively, and clustering fast-flow capillaries (increased flow heterogeneity) reduced minimum Pt to 14.5 mmHg. Thus, although fast capillaries prevented tissue dysoxia, they did not prevent increased hypoxia as the degree of microvascular injury increased. The model predicts that decreased FCD, increased fast flow, and increased Vo2 in sepsis expose skeletal muscle to significant regions of hypoxia, which could affect local cellular and organ function.     

A model of oxygen exchange between an arteriole or venule and the surrounding tissue

A mathematical model is developed to study the effect of capillary convection on oxygen transport around segments of arterioles and venules that are surrounded by capillaries. These capillaries carry unidirectional flow perpendicular to the vessel. The discrete capillary structure is distributed in a manner determined by the capillary blood flow and capillary density. A nonlinear oxyhemoglobin dissociation curve described by the Hill equation is used in the analysis. Oxygen flux from the vessel is expressed as a relationship between Sherwood and Peclet numbers, as well as other dimensionless combinations involving parameters of the capillary bed. A numerical solution is obtained with a finite difference method. The numerical results obtained within the physiological range of parameters allow the prediction of longitudinal gradients of hemoglobin-oxygen saturation along the arterioles and venules.     

A mechanism for erythrocyte-mediated elevation of apparent viscosity by leukocytes in vivo without adhesion to the endothelium

In spite of the relatively small number of leukocytes in the circulation, they have a significant influence on the perfusion of such organs as skeletal muscle or kidney. However, the underlying mechanisms are incompletely understood. In the current study a combined in vivo and computational approach is presented in which the interaction of individual freely flowing leukocytes with erythrocytes and its effect on apparent blood viscosity are explored. The skeletal muscle microcirculation was perfused with different cell suspensions with and without leukocytes or erythrocytes. We examined a three-dimensional numerical model of low Reynolds number flow in a capillary with a train of erythrocytes (small spheres) in off-axis positions and single larger leukocytes in axisymmetric positions. The results indicate that in order to match the slower axial velocity of leukocytes in capillaries, erythrocytes need to position themselves into an off-axis position in the capillary. In such off-axis positions at constant mean capillary velocity, erythrocyte axial velocity matches on average the axial velocity of the leukocytes, but the apparent viscosity is elevated, in agreement with the whole organ perfusion observations. Thus, leukocytes influence the whole organ resistance in skeletal muscle to a significant degree only in the presence of erythrocytes.     

Dynamic oscillatory circuit of regulation of capillary hemodynamics

We used laser Doppler flowmetry with wavelet analysis of blood flow oscillations, computer capillaroscopy, and thermometry of the nail bed in 30 subjects to show an important role of the oscillatory circuit in the regulation of capillary hemodynamics, number of functioning capillaries, and linear and volumetric velocity of blood flow. The number of functioning capillaries is regulated by oscillations of myogenic and sensory peptidergic origin. The appearance of sensory oscillations, especially high-amplitude oscillations, is an adaptive neurotrophic mechanism that significantly increases the number of functioning capillaries and intensity of blood flow from arterioles to capillaries. The linear velocity of blood flow depends on both the tone of microvessels and changes in the dynamic component of blood pressure. Under conditions of skin hypoperfusion, the mean linear velocity of capillary blood flow may be inversely related to the extracapillary perfusion, including the amplitude of heart rate (A _h) and oscillations of the tone of precapillary sphincters, whereas under conditions of vasodilation and increased skin perfusion, it may be inversely related to the amplitude of arteriolar oscillations of endothelial or neurogenic sympathetic origin (A _maxe + n) and the shunting index. The A _h affects the linear velocity of blood flow in the arterial part of capillaries, whereas the A _maxe + n influences the same factor in the venous part. The contribution of oscillations to the regulation of the linear velocity varies depending on the perfusion and skin temperature. The resultant tone of distributing microvessels is determined by the competition between the stationary and oscillatory components. In addition to changes in the amplitude, the frequency of vasomotions may also be important. The regulatory importance of the oscillatory circuit is increased with a decrease in the skin blood flow.     

A heterogeneous tissue model for measurement of regional blood perfusion in the myocardium using inert gas isotopes

Inert gas isotopes are finding increasing application in the measurement of blood perfusion in the capillary beds of muscle, especially the myocardium. When measuring blood perfusion of the myocardium, washout curves are first produced by precordial monitoring of isotope activity following intracoronary artery injection of an inert gas isotope dissolved in saline. The washout curve data are then applied to a mathematical model to yield blood perfusion rate. Present models for this purpose either ignore any diffusive effects of gas movement (Kety-Schmidt model), or diffusive effects are accounted for by weighting the calculated perfusion value (Zierler's height-over-area technique). A new model is described here for convective and diffusive movement of an inert, nonpolar gas in myocardial tissue. A digital computer simulation of the model equations is used both to simply the model and to show agreement between the model response and experimental ¹³³Xe washout curves from normal and infracted canine hearts. The model assumes that the tail of the washout curves (portion after roughly 1.5 minutes) is caused by a heterogeneous, diffusion-limited tissue structure. The model provides two parameters which can be adjusted to washout curve data using model-matching techniques. These are perfusion rate, and a parameter which is an index of the diffusive nature of the particular myocardial area under study.     

Current views on oxygen transport from blood to tissues]

During the recent 25-30 years, sophisticated experiments and mathematical simulation significantly changed the generally accepted theory of oxygen transport in tissue, which was based on two major postulates, namely: 1) Blood flows in capillaries continuously at uniform velocity, 2) Gas circulation between blood and tissue takes place exclusively in capillaries. As was shown by modern research techniques, blood flow in microvessels has irregular sharp velocity fluctuations in very short time intervals (seconds). In addition, mean velocity of blood flow in microvessels of the same caliber and the same micro-region of tissue may differ several times. Therefore, efficiency of microcirculation reactions may be assessed exclusively witH mean blood velocity in capillaries of the whole micro-region, and with complicated changes of the histogram of mean velocity distribution in capillaries. It was shown that arteriolas and venulas of inactive muscles and brain account for 30 to 50% of gas circulation between blood and tissue. This resulted in fundamental change of the previous postulates in the area of tissue gas circulation physiology, and, in effect, in replacement of oxygen transport paradigm created by A. Krog. This study is an attempt to present a new modern concept of oxygen transport in tissue, to show its research significance, and possible applications.     

Modelling inert gas exchange in tissue and mixed-venous blood return to the lungs

Inert gas exchange in tissue has been almost exclusively modelled by using an ordinary differential equation. The mathematical model that is used to derive this ordinary differential equation assumes that the partial pressure of an inert gas (which is proportional to the content of that gas) is a function only of time. This mathematical model does not allow for spatial variations in inert gas partial pressure. This model is also dependent only on the ratio of blood flow to tissue volume, and so does not take account of the shape of the body compartment or of the density of the capillaries that supply blood to this tissue. The partial pressure of a given inert gas in mixed-venous blood flowing back to the lungs is calculated from this ordinary differential equation. In this study, we write down the partial differential equations that allow for spatial as well as temporal variations in inert gas partial pressure in tissue. We then solve these partial differential equations and compare them to the solution of the ordinary differential equations described above. It is found that the solution of the ordinary differential equation is very different from the solution of the partial differential equation, and so the ordinary differential equation should not be used if an accurate calculation of inert gas transport to tissue is required. Further, the solution of the PDE is dependent on the shape of the body compartment and on the density of the capillaries that supply blood to this tissue. As a result, techniques that are based on the ordinary differential equation to calculate the mixed-venous blood partial pressure may be in error.     

The interdependent contributions of gravitational and structural features to perfusion distribution in a multiscale model of the pulmonary circulation

Recent experimental and imaging studies suggest that the influence of gravity on the measured distribution of blood flow in the lung is largely through deformation of the parenchymal tissue. To study the contribution of hydrostatic effects to regional perfusion in the presence of tissue deformation, we have developed an anatomically structured computational model of the pulmonary circulation (arteries, capillaries, veins), coupled to a continuum model of tissue deformation, and including scale-appropriate fluid dynamics for blood flow in each vessel type. The model demonstrates that both structural and the multiple effects of gravity on the pulmonary circulation make a distinct contribution to the distribution of blood. It shows that postural differences in perfusion gradients can be explained by the combined effect of tissue deformation and extra-acinar blood vessel resistance to flow in the dependent tissue. However, gravitational perfusion gradients persist when the effect of tissue deformation is eliminated, highlighting the importance of the hydrostatic effects of gravity on blood distribution in the pulmonary circulation. Coupling of large- and small-scale models reveals variation in microcirculatory driving pressures within isogravitational planes due to extra-acinar vessel resistance. Variation in driving pressures is due to heterogeneous large-vessel resistance as a consequence of geometric asymmetry in the vascular trees and is amplified by the complex balance of pressures, distension, and flow at the microcirculatory level.     

A computational study of the effect of capillary network anastomoses and tortuosity on oxygen transport

The objective of this study was to investigate the effects of capillary network anastomoses and tortuosity on oxygen transport in skeletal muscle, as well as the importance of muscle fibers in determining the arrangement of parallel capillaries. Countercurrent flow and random capillary blockage (e.g. by white blood cells) were also studied. A general computational model was constructed to simulate oxygen transport from a network of blood vessels within a rectangular volume of tissue. A geometric model of the capillary network structure, based on hexagonally packed muscle fibers, was constructed to produce networks of straight unbranched capillaries, capillaries with anastomoses, and capillaries with tortuosity, in order to examine the effects of these geometric properties. Quantities examined included the tissue oxygen tension and the capillary oxyhemoglobin saturation. The computational model included a two-phase simulation of blood flow. Appropriate parameters were chosen for working hamster cheek-pouch retractor muscle. Our calculations showed that the muscle-fiber geometry was important in reducing oxygen transport heterogeneity, as was countercurrent flow. Tortuosity was found to increase tissue oxygenation, especially when combined with anastomoses. In the absence of tortuosity, anastomoses had little effect on oxygen transport under normal conditions, but significantly improved transport when vessel blockages were present.     

Multiscale modeling of lymphatic drainage from tissues using homogenization theory

Lymphatic capillary drainage of interstitial fluid under both steady-state and inflammatory conditions is important for tissue fluid balance, cancer metastasis, and immunity. Lymphatic drainage function is critically coupled to the fluid mechanical properties of the interstitium, yet this coupling is poorly understood. Here we sought to effectively model the lymphatic-interstitial fluid coupling and ask why the lymphatic capillary network often appears with roughly a hexagonal architecture. We use homogenization method, which allows tissue-scale lymph flow to be integrated with the microstructural details of the lymphatic capillaries, thus gaining insight into the functionality of lymphatic anatomy. We first describe flow in lymphatic capillaries using the Navier-Stokes equations and flow through the interstitium using Darcy's law. We then use multiscale homogenization to derive macroscale equations describing lymphatic drainage, with the mouse tail skin as a basis. We find that the limiting resistance for fluid drainage is that from the interstitium into the capillaries rather than within the capillaries. We also find that between hexagonal, square, and parallel tube configurations of lymphatic capillary networks, the hexagonal structure is the most efficient architecture for coupled interstitial and capillary fluid transport; that is, it clears the most interstitial fluid for a given network density and baseline interstitial fluid pressure. Thus, using homogenization theory, one can assess how vessel microstructure influences the macroscale fluid drainage by the lymphatics and demonstrate why the hexagonal network of dermal lymphatic capillaries is optimal for interstitial tissue fluid clearance.     

Apparent Diffusivity and Taylor Dispersion of Water and Solutes in Capillary Beds

A physical theory explaining the anisotropic dispersion of water and solutes in biological tissues is introduced based on the phenomena of Taylor dispersion, in which highly diffusive solutes cycle between flowing and stagnant regions in the tissue, enhancing dispersion in the direction of microvascular flow. An effective diffusion equation is derived, for which the coefficient of dispersion in the axial direction (direction of capillary orientation) depends on the molecular diffusion coefficient, tissue perfusion, and vessel density. This analysis provides a homogenization that represents three-dimensional transport in capillary beds as an effectively one-dimensional phenomenon. The derived dispersion equation may be used to simulate the transport of solutes in tissues, such as in pharmacokinetic modeling. In addition, the analysis provides a physically based hypothesis for explaining dispersion anisotropy observed in diffusion-weighted imaging (DWI) and diffusion-tensor magnetic resonance imaging (DTMRI) and suggests the means of obtaining quantitative functional information on capillary vessel density from measurements of dispersion coefficients. It is shown that a failure to account for flow-mediated dispersion in vascular tissues may lead to misinterpretations of imaging data and significant overestimates of directional bias in molecular diffusivity in biological tissues. Measurement of the ratio of axial to transverse diffusivity may be combined with an independent measurement of perfusion to provide an estimate of capillary vessel density in the tissue.     

Mathematical model of gas bubble evolution in a straight tube.

Deep sea divers suffer from decompression sickness (DCS) when their rate of ascent to the surface is too rapid. When the ambient pressure drops, inert gas bubbles may form in blood vessels and tissues. The evolution of a gas bubble in a rigid tube filled with slowly moving fluid, intended to simulate a bubble in a blood vessel, is studied by solving a coupled system of fluid-flow and gas transport equations. The governing equations for the fluid motion are solved using two techniques: an analytical method appropriate for small nondeformable spherical bubbles, and the boundary element method for deformable bubbles of arbitrary size, given an applied steady flow rate. A steady convection-diffusion equation is then solved numerically to determine the concentration of gas. The bubble volume, or equivalently the gas mass inside the bubble for a constant bubble pressure, is adjusted over time according to the mass flux at the bubble surface. Using a quasi-steady approximation, the evolution of a gas bubble in a tube is obtained. Results show that convection increases the gas pressure gradient at the bubble surface, hence increasing the rate of bubble evolution. Comparing with the result for a single gas bubble in an infinite tissue, the rate of evolution in a tube is approximately twice as fast. Surface tension is also shown to have a significant effect. These findings may have important implications for our understanding of the mechanisms of inert gas bubbles in the circulation underlying decompression sickness.     

Capillary perfusion patterns in single alveolar walls.

We studied capillary perfusion patterns in single alveolar walls through a transparent thoracic window implanted in pentobarbital-anesthetized dogs. The capillaries were maximally opened by brief inflation of a balloon in the left atrium to raise pressure. After the balloon was deflated and pulmonary hemodynamics returned to zone 2 baseline conditions, the capillaries that remained perfused in the observed field were videotaped with the use of in vivo microscopy. The cycle of elevated pressure and baseline observation was repeated three times. Perfusion of different capillaries during each of the observations would imply that the capillaries had characteristics that permitted flow to switch between segments. Perfusion of a specific set of pathways through the network each time would demonstrate that flowing blood sought a unique and repeatable combination of segments, presumably with the least total pathway resistance. We found that the same capillary segments were perfused 79% of the time, a strong indication that a reproducible combination of individual segmental resistances determined the predominant pattern of pulmonary capillary perfusion.     

Microvascular and capillary perfusion following glycocalyx degradation.

Systemic parameters and microvascular and capillary hemodynamics were studied in the hamster window chamber model before and after hyaluronan degradation by intravenous injection of Streptomyces hyaluronidase (100 units, 40-50 U/ml plasma). Glycocalyx permeation was estimated using fluorescent markers of different molecular size (40, 70, and 2,000 kDa), and electrical charge. Systemic parameters (blood pressure, heart rate, blood gases) and microhemodynamics (vascular tone, velocity, and blood flow) remained statistically unchanged after injection of hyaluronidase, compared with inactivated hyaluronidase. Conversely, capillary hemodynamics were drastically affected. Functional capillary density, the capillaries perfused with red blood cells (RBCs), decreased by 35%, capillary Hct of the remaining functional capillaries increased from 16 to 27%, and penetration of 70-kDa fluorescent marker increased. Furthermore, plasma-only perfused capillaries statistically increased 30 min after hyaluronidase. The decrease in functional capillary density accounted for an increased RBC flux in the remainder of the capillaries, since the same number of RBCs had to traverse a reduced number of capillaries. Flux balances showed a reduction from baseline of 11% for the RBC flux and 20% for the plasma flux after treatment. These discrepancies are within the margin of error of the techniques used and could be explained by accounting for RBC over-velocity compared with plasma. These findings suggest that the decrease in the glycocalyx leads to capillary perfusion impairments.     

Theoretical simulation of K(+)-based mechanisms for regulation of capillary perfusion in skeletal muscle

Muscle fibers release K(+) into the interstitial space upon recruitment. Increased local interstitial K(+) concentration ([K(+)]) can cause dilation of terminal arterioles, leading to perfusion of downstream capillaries. The possibility that capillary perfusion can be regulated by vascular responses to [K(+)] was examined using a theoretical model. The model takes into account the spatial relationship between functional units of muscle fiber recruitment and capillary perfusion. Diffusion of K(+) in the interstitial space was simulated. Two hypothetical mechanisms for vascular sensing of interstitial [K(+)] were considered: direct sensing by arterioles and sensing by capillaries with stimulation of feeding arterioles via conducted responses. Control by arteriolar sensing led to poor tissue oxygenation at high levels of muscle activation. With control by capillary sensing, increases in perfusion matched increases in oxygen demand. The time course of perfusion after sudden muscle activation was considered. Predicted capillary perfusion increased rapidly within the first 5 s of muscle fiber activation. The reuptake of K(+) by muscle fibers had a minor effect on the increase of interstitial [K(+)]. Uptake by perfused capillaries was primarily responsible for limiting the increase in [K(+)] in the interstitial space at the onset of fiber activation. Vascular responses to increasing interstitial [K(+)] may contribute to the rapid increase in blood flow that is observed to occur after the onset of muscle contraction.