Inert gas diffusion in heterogeneous tissue I: Without perfusion

A mathematical model is proposed to describe transient gas diffusion into a block of heterogenous tissue placed on an impermeable base. The corresponding asymptotic sultion of mass uptake of the gas is derived on the assumption that the diffusion constant is very much smaller in the cellular phase. It is expected that this will be useful in evaluating the diffusion constant in cellular material, and the volume fraction of extracellular fluid, providing the partition coefficient is known. The phenomenon of mutual interaction and multiple feedback between cellular and extracellular fluid is clearly seen in the overall response of the tissue. In this regard it is shown that the extraction of the two least dominant time constants, by backward projection of the experimental data curve of gas uptake, is likely to confuse the numerical evaluation of the physical parameters of the system. In an appendix, the problem of diffusion straight through a tissue slice is solved at the asymptotic stage, before steady state is reached. The resulting expression predicts the by-passing of cells by the diffusing gas and shows how the parameters cannot reliably be, determined.     

Linear bulk diffusion into heterogeneous tissue

In this paper an expression is derived which describes the transient overall uptake of an inert solute by a section of tissue excised with parallel faces and placed upon an impermeable base. The approach diverges from the conventional analyses for perfused tissue (Morales and Smith,Bull. Math. Biophysics,6, 125–141, 1944;7, 47–99, 1945) because the extravascular zone is regarded as a heterogeneous diffusion medium. Account for this is taken by regarding tissue as effectively composed of two phases—a continuous (extracellular) phase similar to water, and a dispersed phase comprising cells of irregular profile. In both phases the relevant mode of uptake is taken as bulk diffusion rather than surface permeation, thus emphasizing the influence of the internal geometry of the tissue upon its overall exchange response.     

Radial bulk diffusion into heterogeneous tissue

An expression is derived which describes the transient distribution of solute diffusing into heterogeneous tissue from a fully-stirred cylindrical region in which there has been a step change in solute concentration. This is envisaged as a model for the uptake of drugs by vessel walls, although the same approach has been extended for estimating the mode of total uptake of substances by annular specimens of tissue. Tissue is regarded as effectively composed of two phases—an extracellular (continuous) phase, similar to water, and a dispersed phase comprising cells of irregular profile. In each phase the relevant mode of uptake is taken as bulk diffusion rather than the permeation of a surface membrane.     

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.     

The interaction of diffusion and perfusion in homogeneous tissue

A mathematical model is proposed to examine the interaction between blood perfusion and gas diffusion in the uptake of inert gases in tissue. The standard Haldane perfusion model is contrasted with the Hills radial bulk diffusion model in a variety of homogeneous tissue types used in decompression theory. It is the intention of the present analysis to fix ideas on the role of diffusion, perfusion and axial concentration and quantitative studies are given and seem to show that Haldane's perfusion theory is at best a poor approximation even at asymptotic times. It is shown that a strong interaction exists between diffusion and perfusion in muscle tissue and neither approach adequately describes the actual uptake half-time of an inert gas.     

Inert gas clearance from tissue by co-currently and counter-currently arranged microvessels

To elucidate the clearance of dissolved inert gas from tissues, we have developed numerical models of gas transport in a cylindrical block of tissue supplied by one or two capillaries. With two capillaries, attention is given to the effects of co-current and counter-current flow on tissue gas clearance. Clearance by counter-current flow is compared with clearance by a single capillary or by two co-currently arranged capillaries. Effects of the blood velocity, solubility, and diffusivity of the gas in the tissue are investigated using parameters with physiological values. It is found that under the conditions investigated, almost identical clearances are achieved by a single capillary as by a co-current pair when the total flow per tissue volume in each unit is the same (i.e., flow velocity in the single capillary is twice that in each co-current vessel). For both co-current and counter-current arrangements, approximate linear relations exist between the tissue gas clearance rate and tissue blood perfusion rate. However, the counter-current arrangement of capillaries results in less-efficient clearance of the inert gas from tissues. Furthermore, this difference in efficiency increases at higher blood flow rates. At a given blood flow, the simple conduction-capacitance model, which has been used to estimate tissue blood perfusion rate from inert gas clearance, underestimates gas clearance rates predicted by the numerical models for single vessel or for two vessels with co-current flow. This difference is accounted for in discussion, which also considers the choice of parameters and possible effects of microvascular architecture on the interpretation of tissue inert gas clearance.     

Dynamic studies of proton diffusion in mesoscopic heterogeneous matrix: II. The interbilayer space between phospholipid membranes

The thin water layer, as found in chloroplast or mitochondria, is confined between low dielectric amphypathic surfaces a few nm apart.

The physical properties of this mesoscopic space, and how its dimensions affect the rate of chemical reactions proceeding in it, is the subject for this study.

The method selected for this purpose is time resolved fluorometry which can monitor the reversible dissociation of a proton from excited molecule of pyranine (8 hydroxy pyrene 1,3,6 tri sulfonate) trapped in thin water layers of a multilamellar vesicle made of neutral or slightly charged phospholipids.

The results were analyzed by a computer program of N. Agmon (Pines, E., D. Huppert, and N. Agmon. 1988. J. Am. Chem. Soc. 88:5620-5630) that simulates the diffusion of a proton, subjected to electrostatic attraction, in a thin water layer enclosed between low affinity, proton binding surfaces. The analysis determines the diffusion coefficient of the proton, the effective dielectric constant of the water and the water accessibility of the phosphomoieties of the lipids.

These parameters were measured for various lipids [egg-phosphatidylcholine (egg PC), dipalmitoyl phosphatidylcholine (DPPC), cholesterol + DPPC (1:1) and egg PC plus phosphatidyl serine (9:1)] and under varying osmotic pressure which reduces the width of the water layer down to ~10 ~ across.

We found that: (a) The effective dielectric constant of the aqueous layer, depending on the lipid composition, is ~40. (b) The diffusion coefficient of the proton in the thin layer (30-10 ~ across) is that measured in bulk water D = 9.3 10^-5 cm²/s, indicating that the water retains its normal liquid state even on contact with the membrane. (c) The reactivity of the phosphomoiety, quantitated by rate of its reaction with proton, diminishes under lateral pressure which reduces the surface area per lipid.

We find no evidence for abnormal dynamics of proton transfer at the lipid water interface which, by any mechanism, accelerates its diffusion.

     

Kinetics of gas diffusion in mammalian cell culture systems. II. Theory

Mathematical models have been constructed which relate the depth of the culture fluid overlay to the oxygen available to mammalian cells cultured under static conditions. These models suggest that the maintenance of a given rate of oxygen utilization by some culture systems may be critically depended on this fluid depth and on the solubility and rate of diffusion of oxygen in the culture fluid. The importance of these concepts as applied to the isolation and growth of differentiated cells representative of the tissue of origin are noted.     

External and internal diffusion in heterogeneous enzymes systems

The intrusion of diffusion in heterogeneous enzyme reactions, which follow. Michaelis-Menten kinetics, is quantitatively characterized by dimensionless parameters that are independent of the substrate concentration. The effects of these parameters on the overall rate of reaction is illustrated on plots commonly employed in enzyme kinetics. The departure from Michaelis-Menten kinetics due to diffusion limitations can be best assessed by using Hofstee plots which are also suitable to distinguish between internal and external transport effects. A graphical method is described for the evaluation of the reaction rate as a function of the surface concentration of the substrate from measured data.     

Oxygen diffusion and consumption in active aerobic granules of heterogeneous structure

The interior structure of aerobic granules is highly heterogeneous, hence, affecting the transport and reaction processes in the granules. The granule structure and the dissolved oxygen profiles were probed at the same granule in the current work for possible estimation of transport and kinetic parameters in the granule. With the tested granules fed by phenol or acetate as carbon source, most inflow oxygen was consumed by an active layer thickness of less than 125 μm on the granule surface. The confocal laser scanning microscopy scans also revealed a surface layer thickness of approximately 100 μm consisting of cells. The diffusivities of oxygen transport and the kinetic constant of oxygen consumption in the active layers only were evaluated. The theoretical models adopted in literature that ignored the contributions of the layered structure of aerobic granule could have overlooked the possible limitations on oxygen transport.     

Homogeneous transport in a heterogeneous membrane: water diffusion across human stratum corneum in vivo.

The objective of this study was to determine whether a structurally heterogeneous biomembrane, human stratum corneum (SC), behaved as a homogeneous barrier to water transport. The question is relevant because the principal function of the SC in vivo is to provide a barrier to the insensible loss of tissue water across the skin. Impedance spectra (IS) of the skin and measurements of the rate of transepidermal water loss (TEWL) were recorded sequentially in vivo in human subjects as layers of the SC were progressively removed by the serial application of adhesive tape strips. The low-frequency (< or = 100 rad s-1) impedance of skin was much more significantly affected by tape stripping than the higher frequency values; removal of the outermost SC layer had the largest effect. In contrast, TEWL changed little as the outer SC layers were stripped off, but increased dramatically when 6-8 microns of the tissue had been removed. It follows that the two noninvasive techniques probe SC barrier integrity in somewhat different ways. After SC removal, recovery of barrier function, as assessed by increasing values of the low-frequency impedance, apparently proceeded faster than TEWL decreased to the prestripping control. The variation of TEWL as a function of SC removal behaved in a manner entirely consistent with a homogeneous barrier, thereby permitting the apparent SC diffusivity of water to be found. Skin impedance (low frequency) was correlated with the relative concentration of water within the SC, thus providing an in vivo probe for skin hydration. Finally, the SC permeability coefficient to water, as a function of SC thickness, was calculated and correlated with the corresponding values of skin admittance derived from IS.     

Effects of diffusion on total biomass in heterogeneous continuous and discrete-patch systems

Theoretical models of populations on a system of two connected patches previously have shown that when the two patches differ in maximum growth rate and carrying capacity, and in the limit of high diffusion, conditions exist for which the total population size at equilibrium exceeds that of the ideal free distribution, which predicts that the total population would equal the total carrying capacity of the two patches. However, this result has only been shown for the Pearl-Verhulst growth function on two patches and for a single-parameter growth function in continuous space. Here, we provide a general criterion for total population size to exceed total carrying capacity for three commonly used population growth rates for both heterogeneous continuous and multi-patch heterogeneous landscapes with high population diffusion. We show that a sufficient condition for this situation is that there is a convex positive relationship between the maximum growth rate and the parameter that, by itself or together with the maximum growth rate, determines the carrying capacity, as both vary across a spatial region. This relationship occurs in some biological populations, though not in others, so the result has ecological implications.     

Convection and diffusion in tissues and tissue cultures

The relative importance of transport by diffusion and convection in permeable tissues is investigated in three-dimensional structures. The transport of a solute takes place from a number of sources embedded in the tissue, to a number of sinks similarly embedded. An enhancement factor E is defined to be the ratio of the transport rate in the presence of a pressure difference between the sources and sinks, to the transport rate without a pressure difference. It is shown that E is a unique function of a parameter W, which characterizes the properties of the tissue and the pressure difference. This relation is independent of the number or the geometries of the sources and sinks.This relation is compared with experimental data obtained in a hollow fiber tissue culture device, with two sets of hollow fibers kept at different pressures. This relation is also used to estimate the importance of convection in vivo for a wide range of mammalian tissues and solute molecules.