Dynamics of oxygen unloading from sickle erythrocytes.

The objective of this work is to theoretically model oxygen unloading in sickle red cells. This has been done by combining into a single model diffusive transport mechanisms, which have been well-studied for normal red cells, and the hemoglobin polymerization process, which has been previously been studied for deoxyhemoglobin-S solutions and sickle cells in near-equilibrium situations. The resulting model equations allow us to study the important processes of oxygen delivery and polymerization simultaneously. The equations have been solved numerically by a finite-difference technique. The oxygen unloading curve for sickle erythrocytes is biphasic in nature. The rate of unloading depends in a complicated way on (a) the kinetics of hemoglobin S polymerization, (b) the kinetics of hemoglobin deoxygenation, and (c) the diffusive transport of both free oxygen and oxy-hemoglobin. These processes interact. For example, the hemoglobin S polymer interferes with the transport of both free oxygen and unpolymerized oxy-hemoglobin, and this is accounted for in the model by diffusivities which depend on the polymer and solution hemoglobin concentration. Other parameters which influence the interaction of these processes are the concentration of 2,3-diphosphoglycerate and total hemoglobin concentration. By comparing our model predictions for oxygen unloading with simpler predictions based on equilibrium oxygen affinities, we conclude that the relative rate of oxygen unloading of cells with different physical properties cannot be correctly predicted from the equilibrium affinities. To describe the unloading process, a kinetic calculation of the sort we give here is required.     

Effect of intracapillary resistance to oxygen transport on the diffusional shunting between capillaries

The effect of intracapillary resistance to oxygen transport on distribution of oxygen tension in the tissue and along the capillary was investigated by means of a mathematical model. In some cases resistance significantly affects the diffusive interaction between neighbouring capillaries thus aggravating the dificiency of oxygen supply around capillaries with low oxygen tension.     

Analysis of oxygen exchange between arterioles and surrounding capillary-perfused tissue.

A theoretical model is used to analyze oxygen transport in a three-dimensional tissue region containing an arteriole surrounded by an array of capillaries in planes perpendicular to the arteriole. Convective removal of oxygen from the vicinity of the arteriole by nearby capillaries is shown to increase diffusive oxygen loss from the arteriole. This effect depends on the locations of the capillaries, particularly those nearest to the arteriole. The arteriolar oxygen efflux is comparable to that predicted by a previous model which used a continuum approach, but the efflux does not increase with increasing perfusion as rapidly as predicted by the continuum model. Even a small capillary flow rate strongly influences the oxygen field surrounding the arteriole.     

Estimation of capillary density in human skeletal muscle based on maximal oxygen consumption rates

A previously developed Krogh-type theoretical model was used to estimate capillary density in human skeletal muscle based on published measurements of oxygen consumption, arterial partial pressure of oxygen, and blood flow during maximal exercise. The model assumes that oxygen consumption in maximal exercise is limited by the ability of capillaries to deliver oxygen to tissue and is therefore strongly dependent on capillary density, defined as the number of capillaries per unit cross-sectional area of muscle. Based on an analysis of oxygen transport processes occurring at the microvascular level, the model allows estimation of the minimum number of straight, evenly spaced capillaries required to achieve a given oxygen consumption rate. Estimated capillary density values were determined from measurements of maximal oxygen consumption during knee extensor exercise and during whole body cycling, and they range from 459 to 1,468 capillaries/mm2. Measured capillary densities, obtained with either histochemical staining techniques or electron microscopy on quadriceps muscle biopsies from healthy subjects, are generally lower, ranging from 123 to 515 capillaries/mm2. This discrepancy is partly accounted for by the fact that capillary density decreases with muscle contraction and muscle biopsy samples typically are strongly contracted. The results imply that estimates of maximal oxygen transport rates based on capillary density values obtained from biopsy samples do not fully reflect the oxygen transport capacity of the capillaries in skeletal muscle.     

Effect of unstirred layers on binding and reaction kinetics at a membrane surface

The effects of solution unstirred layers on the time course of chemical reactions and transport processes at a membrane surface are determined. A set of equations which describes non-steady-state diffusion through an unstirred layer coupled with chemical reaction at a membrane surface or transport through a membrane is developed. A numerical solution to the equations is obtained by uncoupling diffusive and chemical processes in an iterative manner. The diffusive process is solved by the Crank-Nicolson method; the chemical process is solved by integrating the differential equations describing the kinetics. Diffusive processes in one dimension, in three dimensions, and in the presence of an arbitrary potential near the membrane surface are solved. General characteristics of the calculated reaction time course are discussed using surface binding and membrane transport examples. Small, neglected, unstirred layers are shown to sometimes yield erroneous values of rate parameters for a surface reaction and to simulate competitive reaction kinetics. Experimental approaches for measuring unstirred layer thickness are reviewed.     

A singular perturbation approach to diffusion reaction equations containing a point source,with application to the hemolytic plaque assay

Summary Many cells secrete factors which diffuse and bind to receptors on neighboring cells. These processes can be described by a nonlinear diffusion equation with a point source and a spatially distributed binding reaction. We show via perturbation analysis how approximate solutions can be obtained for such equations when the binding reaction is fast compared to diffusive transport. We base our analysis on an example which is of great practical importance in immunology, the hemolytic plaque technique.     

Red Blood Cells: Centerpiece in the Evolution of the Vertebrate Circulatory System

All vertebrates except cold-water ice fish transport oxygenvia hemoglobin packaged in red blood cells (RBCs). VertebrateRBCs vary in size by thirtyfold. Differences in RBC size havebeen known for over a century, but the functional significanceof RBC size remains unknown. One hypothesis is that large RBCsare a primitive character. Agnathans have larger RBCs than domammals. However, the largest RBCs are found in urodele amphibianswhich is inconsistent with the hypothesis that large RBCs areprimitive. Another possibility is that small RBCs increase bloodoxygen transport capacity. Blood hemoglobin concentration ([Hb])and mean RBC hemoglobin concentration (MCHC) increase from Agnathato birds and mammals. However, the changes in [Hb] and MCHCdo not parallel changes in RBC size. In addition, RBC size doesnot affect blood viscosity. Thus, there is no clear link betweenRBC size and oxygen transport capacity. We hypothesize thatRBC size attends changes in capillary diameter. This hypothesisis based on the following observations. First, RBC width averages25% larger than capillary diameter which insures cell deformationduring capillary flow. Functionally, RBC deformation minimizesdiffusion limitations to gas exchange. Second, smaller capillariesare associated with increased potential for diffusive gas exchange.However, smaller capillaries result in higher resistances toblood flow which requires higher blood pressures. We proposethat the large capillary diameters and large RBCs in urodelesreflect the evolutionary development of a pulmonary vascularsupply. The large capillaries reduced systemic vascular resistancesenabling a single ventricular heart to supply blood to two vascularcircuits, systemic and pulmonary, without developing high pressureson the pulmonary side. The large RBCs preserved diffusive gasexchange efficiency in the large capillaries.     

Some factors affecting oxygen uptake by red blood cells in the pulmonary capillaries

In this study we investigate the equations governing the transport of oxygen in pulmonary capillaries. We use a mathematical model consisting of a red blood cell completely surrounded by plasma within a cylindrical pulmonary capillary. This model takes account of convection and diffusion of oxygen through plasma, diffusion of oxygen through the red blood cell, and the reaction between oxygen and haemoglobin molecules. The velocity field within the plasma is calculated by solving the slow flow equations. We investigate the effect on the solution of the governing equations of: (i) mixed-venous blood oxygen partial pressure (the initial conditions); (ii) alveolar gas oxygen partial pressure (the boundary conditions); (iii) neglecting the convection term; and (iv) assuming an instantaneous reaction between the oxygen and haemoglobin molecules. It is found that: (a) equilibrium is reached much more rapidly for high values of mixed-venous blood and alveolar gas oxygen partial pressure; (b) the convection term has a negligible effect on the time taken to reach a prescribed degree of equilibrium; and (c) an instantaneous reaction may be assumed. Explanations are given for each of these results.     

Mathematical analysis of oxygen concentration in a two dimensional array of capillaries

 An approach is presented for modeling transport and exchange in skeletal muscle that can be used to analyze vascular beds consisting of a large number of interacting capillaries. First the oxygen concentration is determined in a functional unit consisting of a single capillary surrounded by a region of tissue in which a flux is prescribed on the outer boundary of the region. This flux, which is a result of the interaction among all of the capillaries comprising the vascular bed, is then found by matching the concentration along the borders between adjacent units. This leads to a system of ordinary differential equations for the oxygen concentration in the capillaries coupled with a system of algebraic equations for the fluxes. The method is illustrated by obtaining the oxygen concentration within an array of capillaries for the case when each capillary has a different initial concentration and for the case when each capillary has a different flow rate. Received: 12 June 2001 / Revised version: 18 April 2002 / Published online: 17 January 2003 Key words or phrases: Skeletal muscle – Transport – Microcirculation     

Diffusive oxygen shunting between vessels in the preglomerular renal vasculature: anatomic observations and computational modeling

To understand how geometric factors affect arterial-to-venous (AV) oxygen shunting, a mathematical model of diffusive oxygen transport in the renal cortex was developed. Preglomerular vascular geometry was investigated using light microscopy (providing vein shape, AV separation, and capillary density near arteries) and published micro-computed tomography (CT) data (providing vessel size and AV separation; Nordsletten DA, Blackett S, Bentley MD, Ritman EL, Smith NP. IUPS Physiome Project. http://www.physiome.org.nz/publications/nordsletten_blackett_ritman_bentley_smith_2005/folder_contents). A "U-shaped" relationship was observed between the arterial radius and the distance between the arterial and venous lumens. Veins were found to partially wrap around the artery more consistently for larger rather than smaller arteries. Intrarenal arteries were surrounded by an area of fibrous tissue, lacking capillaries, the thickness of which increased from ～5 μm for the smallest arteries (<16-μm diameter) to ～20 μm for the largest arteries (>200-μm diameter). Capillary density was greater near smaller arteries than larger arteries. No capillaries were observed between wrapped AV vessel pairs. The computational model comprised a single AV pair in cross section. Geometric parameters critical in renal oxygen transport were altered according to variations observed by CT and light microscopy. Lumen separation and wrapping of the vein around the artery were found to be the critical geometric factors determining the amount of oxygen shunted between AV pairs. AV oxygen shunting increases both as lumen separation decreases and as the degree of wrapping increases. The model also predicts that capillaries not only deliver oxygen, but can also remove oxygen from the cortical parenchyma close to an AV pair. Thus the presence of oxygen sinks (capillaries or tubules) near arteries would reduce the effectiveness of AV oxygen shunting. Collectively, these data suggest that AV oxygen shunting would be favored in larger vessels common to the cortical and medullary circulations (i.e., arcuate and proximal interlobular arteries) rather than the smaller vessels specific to the cortical circulation (distal interlobular arteries and afferent arterioles).     

Lysosome Transport as a Function of Lysosome Diameter

Lysosomes are membrane-bound organelles responsible for the transport and degradation of intracellular and extracellular cargo. The intracellular motion of lysosomes is both diffusive and active, mediated by motor proteins moving lysosomes along microtubules. We sought to determine how lysosome diameter influences lysosome transport. We used osmotic swelling to double the diameter of lysosomes, creating a population of enlarged lysosomes. This allowed us to directly examine the intracellular transport of the same organelle as a function of diameter. Lysosome transport was measured using live cell fluorescence microscopy and single particle tracking. We find, as expected, the diffusive component of intracellular transport is decreased proportional to the increased lysosome diameter. Active transport of the enlarged lysosomes is not affected by the increased lysosome diameter.     

A theoretical model for oxygen transport in skeletal muscle under conditions of high oxygen demand.

Oxygen transport from capillaries to exercising skeletal muscle is studied by use of a Krogh-type cylinder model. The goal is to predict oxygen consumption under conditions of high demand, on the basis of a consideration of transport processes occurring at the microvascular level. Effects of the decline in oxygen content of blood flowing along capillaries, intravascular resistance to oxygen diffusion, and myoglobin-facilitated diffusion are included. Parameter values are based on human skeletal muscle. The dependence of oxygen consumption on oxygen demand, perfusion, and capillary density are examined. When demand is moderate, the tissue is well oxygenated and consumption is slightly less than demand. When demand is high, capillary oxygen content declines rapidly with axial distance and radial oxygen transport is limited by diffusion resistance within the capillary and the tissue. Under these conditions, much of the tissue is hypoxic, consumption is substantially less than demand, and consumption is strongly dependent on capillary density. Predicted consumption rates are comparable with experimentally observed maximal rates of oxygen consumption.     

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.     

Mathematical modelling of pulmonary gas transport

 Equations governing the transport of the gases oxygen and carbon dioxide inside the pulmonary capillaries are written down. By analysing these equations it is predicted that there will be negligible limitation to the transport of oxygen when oxygen concentration takes a normal physiological or higher value. For low values of oxygen concentration, there may be limitation to oxygen transport. It is predicted further that the quantity of carbon dioxide excreted from blood into alveolar gas is dependent on oxygen concentration, with low oxygen concentrations inhibiting the carbon dioxide transport process. The relatively slow reaction involving carbon dioxide in plasma also inhibits the excretion of carbon dioxide. These predictions are verified by solving the whole system of governing equations numerically. Received: 1 May 2002 / Revised version: 20 October 2002 / Published online: 19 March 2003 JPW was supported by a grant from the Engineering and Physical Sciences Research Council of Great Britain. Key words or phrases: Pulmonary gas transport – Haemoglobin – Saturation     

The flow of sickle-cell blood in the capillaries.

Oxygen tension levels and red cell velocities for the flow of sickle-cell blood in the capillaries are determined by using the Krogh model for oxygen transport and lubrication theory for the cell motion. The coupling and interaction between these arises from the red cell compliance, which is assumed to vary with the oxygen concentration. Microsieving data is used to establish an upper bound for this relationship. Calculations are carried out for a range of capillary sizes, taking into account the rightward shift of the oxyhemoglobin dissociation curve and the reduced hematocrit of sickle-cell blood, and are compared to, as a base case, the flow of normal blood under normal pressure gradient. The results indicate that under normal pressure gradients the oxygen tensions and cell velocities for sickle blood are considerably higher than for normal blood, thus acting against the tendency for cells to sickle, or significantly change their rheological properties, in the capillaries. Under reduced pressure gradients, however, the concentrations and velocities drop dramatically, adding to the likelihood of such shape or flow property changes.     

Structure and function of lymphatic postcapillaries (coordinating mechanism of the processes of interstitial transport and lymphatic resorption

Investigation performed on topography, structure and resorptive function of lymphatic microvessels in the serous membranes gives possibilities to work out a model on formation and transport of lymph. Lymphatic postcapillaries (LP) - thin-walled endothelial channels with valves make an important link of the lymphatic system. The LP structure is similar to that of capillaries. These microvessels are situated in the interstitial space area, that are characterized with their high content of plasma proteins and water. Interrelation in concentrations of protein in the LP lumen and in the tissue ensures the existence of the osmotic gradient pressure through the wall and contributes liquor resorption into the lumen of microvessels. Peritoneal lymphatic capillaries are situated in the areas of a higher hydrotation level of the interstitial space. They can control the rate of the liquor filtration from plasma into tissue and regulate the resorption level in the whole lymphatic network. The model provides a differentiated participation of the LP and capillaries in performing resorption of proteins and liquor from the interstitium. The resorption mechanisms are closely connected with processes of the lymph movement along the vessels.     

Electron microscopic research on the manifestations of transport-metabolic interactions in intraocular nerve tissue grafts

Electron microscopic study of intraocular grafts of the septal and hippocampal nervous tissue revealed intensive transport and metabolic interactions between various types of the cells. High level of transport exists between the recipient's blood and intraocular fluid, on the one hand, and the graft, on the other one; the abundance of pinocytotic vesicles in the endothelium, pericytes and the glial end-feet of the capillaries, and the presence of microvilli and cilia on the graft limiting glia. Signs of active communications between interstitial glial cells, as well as between gliocytes and neurons, such as pinocytosis and gap junctions were observed. Similar interactions exist between various parts of nerve cells. Besides, there is an evidence of microphagocytic interactions, particularly between pre- and postsynaptic elements, as shown by the presence of so-called spinules. An unusual fact was observed of internalization of cytoplasmic fragments of the degenerating neuron by contacting synaptic boutons. It is suggested that the high level of transport and metabolic processes may reflect adaptive and compensatory processes in the nervous tissue developing under strict limitations of the neural and neurotrophic influences.