Inert gas diffusion in heterogeneous tissue II: With perfusion

In this paper, a tractable mathematical model is proposed to describe transient inert gas diffusion in heterogeneous tissue with perfusion controlling gas input to the cellular region. The corresponding solution of overall mass uptake of the inert gas is derived exactly and should be useful in interpreting washout curves from particular tissue zones, whether there is any interaction with cellular diffusion or not. It is shown that the solution contains effectively nearly all models hitherto proposed to describe gas uptake in tissue. However some indication is given of a possible situation where perfusion, extra-cellular and cellular diffusion will need to be treated separately.     

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.     

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.     

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 Kinetics of Distribution of the Fat-Soluble Inert Gas Cyclopropane in the Body

A kinetic analysis is made of the experimentally measured time course of respiratory uptake of the highly fat-soluble, inert gas cyclopropane by normal human subjects. The analysis is based on the well-known perfusion-limited model in which a number of body compartments are arranged in parallel with the lungs via the circulating blood. Three distinct body compartments are derived from the data. These are tentatively identified as: (a) adipose tissue (b) fat-poor tissue of low perfusion such as resting muscle, skin, and connective tissue (c) fat-poor tissue of high perfusion such as brain, heart, gut, liver, and kidney. Blood flow rates to the several compartments are also derived from the data. The rates to compartments (a) and (b) are each approximately 10 per cent of the estimated total cardiac output. The derived perfusion (blood flow rate/compartment weight) of the three compartments are in the range, respectively, (a) 2 to 4, (b) 1 to 2.5, (c) 25 to 75 ml/min/100 gm. Uncertainties arising from the experimental data and from simplifications of the model (neglect of lung fill-up phase of uptake and gross diffusion of cyclopropane from one tissue into another) are discussed. The present type of uptake experiment is significant for the problems of total body fat determination, of gross body composition in relation to weight change, of gross shunting of blood flow from one compartment to another, of anesthesia by fat-soluble substances, and of decompression sickness.     

Diffusion in lung gas and across alveolar membrane in mammalian lungs

The relative importance of stratified inhomogeneity in the mammalian lung has been considered both theoretically and experimentally. Calculations show that alveolar-capillary diffusion resistance is not limiting for inert gas exchange. Previous model calculations have concluded that stratified inhomogeneity is not likely to be an important mechanism in limiting gas exchange. Experimental data with inspired boluses of inert gas show the presence of stratified inhomogeneity. However, another possible mechanism, related to interaction of diffusion and convection, remains as a possible explanation for these findings. Stratified inhomogeneity appears to play a role in gas exchange in the normal rat, but the role of stratified inhomogeneity in humans and other animals is yet to be determined.     

Factors determining temporal pattern of isobaric supersaturation

It is possible to produce a transient supersaturation or undersaturation in tissues and blood by sequentially breathing gases with different equilibration rates. If the ambient gas pressure is sufficiently high, the induced supersaturation can produce vascular bubbles. By means of the classical perfusion-dependent model of inert gas elimination, which assumes that the effects of diffusion are minimal, the magnitude of the total inert gas pressure can be predicted. If, however, the effects of diffusion cannot be ignored, the supersaturation could be substantially larger. This paper estimates the effects of diffusion in a Krogh cylinder on the supersaturation produced by suddenly changing the inert gas partial pressure in the blood. The results of these estimates indicate that diffusion plays a role in this transient supersaturation only in long Krogh cylinders with high blood flows. The effects of diffusion are further reduced by the finite time necessary to switch the inert gases in arterial blood. The conclusions are supported by experiments that measure vascular bubble production after a switch of the inert portion of the inspired gas. These experiments further show that the formation of vascular bubbles after such a switch cannot be entirely explained by the different diffusion constants of the gases used.     

Modeling and analysis of diffusion and reaction in legume nodules

Diffusion of gases through legume nodules is important for nitrogen fixation. A mathematical model is presented for diffusion and enzymatic reaction for legume nodules with a reactive core and an inert shell. The transient model is solved numerically for spherical geometry for acetylene reduction by nitrogenase enzyme. The results are used to estimate the diffusivities of acetylene and ethylene in the nodules by comparing predicted and experimental lag times. The experimental results are also analyzed using an effectiveness factor plot for spherical nodules with inert shells and reactive cores. The results show that the diffusivities are slightly higher than those for acetylene and ethylene in water because of some contribution of gas phase diffusion. Applications to oxygen diffusion through nodule tissue are suggested.     

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 Diffusion of Oxygen, Carbon Dioxide, and Inert Gas in Flowing Blood

Measurements were made of exchange rates of oxygen, carbon dioxide, and krypton-85 with blood at 37.5°C. Gas transfer took place across a 1 mil silicone rubber membrane. The blood was in a rotating disk boundary layer flow, and the controlling resistance to transfer was the concentration boundary layer. Measured rates were compared with rates predicted from the equation of convective diffusion using velocities derived from the Navier-Stokes equations and diffusivities calculated from the theory for conduction in a heterogeneous medium. The measured absorption rate of krypton-85 was closely predicted by this model. Significant deposition of material onto the membrane surface, resulting in an increased transfer resistance, occurred in one experiment with blood previously used in a nonmembrane type artificial lung. The desorption rate of oxygen from blood at low P_o2¹ was up to four times the corresponding transfer rate of inert gas. This effect is described somewhat conservatively by a local equilibrium form of the convective diffusion equation. The carbon dioxide transfer rate in blood near venous conditions was about twice that of inert gas, a rate significantly greater than predicted by the local equilibrium theory. It should be possible to apply these theoretical methods to predict exchange rates with blood flowing in systems of other geometries.     

The secretion of inert gas into the swim-bladder of fish

The composition of the gas mixture secreted into the swim-bladders of several species of fish has been determined in the mass spectrometer. The secreted gas differed greatly from the gas mixture breathed by the fish in the relative proportions of the chemically inert gases, argon, neon, helium, and nitrogen. Relative to nitrogen the proportion of the very soluble argon was increased and the proportions of the much less soluble neon and helium decreased. The composition of the secreted gas approaches the composition of the gas mixture dissolved in the tissue fluid. A theory of inert gas secretion is proposed. It is suggested that oxygen gas is actively secreted and evolved in the form of minute bubbles, that inert gases diffuse into these bubbles, and that the bubbles are passed into the swim-bladder carrying with them inert gases. Coupled to a preferential reabsorption of oxygen from the swim-bladder this mechanism can achieve high tensions of inert gas in the swim-bladder. The accumulation of nearly pure nitrogen in the swim-bladder of goldfish (Carassius auratus) is accomplished by the secretion of an oxygen-rich gas mixture followed by the reabsorption of oxygen.     

Conducting airway gas exchange: diffusion-related differences in inert gas elimination.

We studied CO2 and inert gas elimination in the isolated in situ trachea as a model of conducting airway gas exchange. Six inert gases with various solubilities and molecular weights (MW) were infused into the left atria of six pentobarbital-anesthetized dogs (group 1). The unidirectionally ventilated trachea behaved as a high ventilation-perfusion unit (ratio = 60) with no appreciable dead space. Excretion of higher-MW gases appeared to be depressed, suggesting a MW dependence to inert gas exchange. This was further explored in another six dogs (group 2) with three gases of nearly equal solubility but widely divergent MWs (acetylene, 26; Freon-22, 86.5; isoflurane, 184.5). Isoflurane and Freon-22 excretions were depressed 47 and 30%, respectively, relative to acetylene. In a theoretical model of airway gas exchange, neither a tissue nor a gas phase diffusion resistance predicted our results better than the standard equation for steady-state alveolar inert gas elimination. However, addition of a simple ln (MW) term reduced the remaining residual sum of squares by 40% in group 1 and by 83% in group 2. Despite this significant MW influence on tracheal gas exchange, we calculate that the quantitative gas exchange capacity of the conducting airways in total can account for less than or equal to 16% of any MW-dependent differences observed in pulmonary inert gas elimination.     

On the theory of blood-tissue exchanges: II. Applications

The application of a previously developed theory of inert gas absorption is here outlined. Interpretation of uptake curves and the method of obtaining tissue constants from such curves is discussed, together with illustrations from actual experiment. A critique of previous analytic procedures is included. The material in this article should be construed only as the personal opinion of the writers and not as representing the opinion of the Navy Department officially.     

Spatial interaction between tissue pressure and skeletal muscle perfusion during contraction

The vascular waterfall theory attributes decreased muscle perfusion during contraction to increased intramuscular pressure (P(IM)) and concomitant increase in venous resistance. Although P(IM) is distributed during contractions, this theory does not account for heterogeneity. This study hypothesises that pressure heterogeneity could affect the interaction between P(IM) rise and perfusion. Regional tissue perfusion during submaximum (100kPa) tetanic contraction is studied, using a finite element model of perfused contracting skeletal muscle. Capillary flow in muscles with one proximal artery and vein (SIM(1)) and with an additional distal artery and vein (SIM(2)) is compared. Blood flow and pressures at rest and P(IM) during contraction ( approximately 25kPa maximally) are similar between simulations, but capillary flow and venous pressure differ. In SIM(2), venous pressure and capillary flow correspond to P(IM) distribution, whereas capillary flow in SIM(1) is less than 10% of flow in SIM(2), in the muscle half without draining vein. This difference is caused by a high central P(IM), followed by central venous pressure rise, in agreement with the waterfall theory. The high central pressure (SIM(1)), obstructs outflow from the distal veins. Distal venous pressure rises until central blood pressure is reached, although local P(IM) is low. Adding a distal vein (SIM(2)) restores the perfusion. It is concluded that regional effects contribute to the interaction between P(IM) and perfusion during contraction. Unlike stated by the vascular waterfall theory, venous pressure may locally exceed P(IM). Although this can be explained by the principles of this theory, the theory does not include this phenomenon as such.     

A probabilistic approach to some problems in blood-tissue exchange

The problem of exchange of substances between circulating blood and tissue is discussed in terms of a model with continuously distributed tissue characteristics. Consideration is given to the distribution of perfusion rates, the distribution of metabolism coefficients for a substance consumed at a rate proportional to its concentration, and the distribution of permeability coefficients for a slowly penetrating inert substance. Equations are derived for obtaining crude estimates of the moments of these distributions from exchange data.     

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.     

Correction of inert gas washin or washout for gas solubility in blood

We show that when an inert gas is washed into the lungs its retention in the blood during any one breath is approximately proportional to its solubility. This relationship makes possible the correction of washin or washout data for blood uptake or release, provided that two gases of different solubility are used simultaneously. The method automatically allows for the characteristics of an individual washin or washout and for the occurrence of recirculation within a fairly short washin or washout period. It has been tested in models with nonuniform ventilation and perfusion and closely approximates the behavior of a truly insoluble gas. In the derived ventilation distribution, gas solubility appears as ventilation to units of low turnover. In the case of N2 this effect is small but causes appreciable overestimation of lung volume. The recovered dead space and main alveolar distribution are insignificantly affected.     

Importance of oscillations in alveolar gas concentrations in the analysis of rebreathing data

Cyclic rebreathing of a soluble inert gas can be used to estimate lung tissue volume (Vt) and pulmonary blood flow (Qc). A recently proposed method for analyzing such cyclic data (Respir. Physiol. 48: 255-279, 1982) mathematically assumes that ventilation is a continuous process. However, neglecting the cyclic nature of ventilation may prevent the accurate estimation of Vt and Qc. We evaluated this possibility by simulating the uptake of soluble inert gases during rebreathing using a cyclic model of gas exchange. Under cyclic uptake conditions alveolar gases follow an oscillating time course, because gas concentrations tend to increase during inspiration and to decrease during expiration. We found that neglecting these alveolar gas oscillations leads to the underestimation of soluble gas uptake by blood, particularly during the early rebreathing breaths. When continuous ventilation is assumed Vt and Qc are overestimated unless rapid rebreathing rates, large tidal volumes, and gases of moderately low solubility are used. Under these conditions the amplitude of the cyclic oscillations is minimized, the alveolar time course more closely resembles that expected from continuous ventilation, and the resulting errors are minimized. Alternatively, when the effect of oscillating alveolar gas concentrations on mass transfer are considered, these estimation errors can be eliminated without restricting rebreathing rate or gas solubility. We conclude that failure to consider the effect of cyclic rebreathing on the time course of alveolar gas concentrations may result in significant errors when evaluating rebreathing data for Vt and Qc.     

Revision of the theory of tracer transport and the convolution model of dynamic contrast enhanced magnetic resonance imaging

Counterexamples are used to motivate the revision of the established theory of tracer transport. Then dynamic contrast enhanced magnetic resonance imaging in particular is conceptualized in terms of a fully distributed convection–diffusion model from which a widely used convolution model is derived using, alternatively, compartmental discretizations or semigroup theory. On this basis, applications and limitations of the convolution model are identified. For instance, it is proved that perfusion and tissue exchange states cannot be identified on the basis of a single convolution equation alone. Yet under certain assumptions, particularly that flux is purely convective at the boundary of a tissue region, physiological parameters such as mean transit time, effective volume fraction, and volumetric flow rate per unit tissue volume can be deduced from the kernel.      

A model of extravascular bubble evolution: effect of changes in breathing gas composition.

Observations of bubble evolution in rats after decompression from air dives (O. Hyldegaard and J. Madsen. Undersea Biomed. Res. 16: 185-193, 1989; O. Hyldegaard and J. Madsen. Undersea Hyperbaric Med. 21: 413-424, 1994; O. Hyldegaard, M. Moller, and J. Madsen. Undersea Biomed. Res. 18: 361-371, 1991) suggest that bubbles may resolve more safely when the breathing gas is a heliox mixture than when it is pure O(2). This is due to a transient period of bubble growth seen during switches to O(2) breathing. In an attempt to understand these experimental results, we have developed a multigas-multipressure mathematical model of bubble evolution, which consists of a bubble in a well-stirred liquid. The liquid exchanges gas with the bubble via diffusion, and the exchange between liquid and blood is described by a single-exponential time constant for each inert gas. The model indicates that bubbles resolve most rapidly in spinal tissue, in adipose tissue, and in aqueous tissues when the breathing gas is switched to O(2) after surfacing. In addition, the model suggests that switching to heliox breathing may prolong the existence of the bubble relative to breathing air for bubbles in spinal and adipose tissues. Some possible explanations for the discrepancy between model and experiment are discussed.