On the theory of blood-tissue exchanges: I. Fundamental equations

A mathematical analysis of the absorption of an inert gas by a heterogeneous system ofn phases, e.g., a limb consisting ofn tissues, is presented. The total uptake of gas, ϕ(t), up to timet is given in terms of arterial concentration, cardiac delivery, blood volume, and the volume, permeability, and partition coefficient of each tissue. The theory predicts how the uptake curve should change in shape under a variety of physiological conditions, and how from the numerical values of the constants the values of certain tissue constants, e.g. permeabilities, may be obtained. 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.     

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.     

Stochastic simulation of the gas diffusion in the air phase of the human lung

The solution of the diffusion equation in the gas phase of the human lung is very difficult because of the structure of the bronchial tree. It is shown by means of physical arguments, how one can reduce the diffusion equation to a simple one-dimensional form. The solution is then obtained by a stochastic simulation, which is easily realized on a digital computer.     

Kinetics of absorption atelectasis during anesthesia: a mathematical model.

Recent computed tomography studies show that inspired gas composition affects the development of anesthesia-related atelectasis. This suggests that gas absorption plays an important role in the genesis of the atelectasis. A mathematical model was developed that combined models of gas exchange from an ideal lung compartment, peripheral gas exchange, and gas uptake from a closed collapsible cavity. It was assumed that, initially, the lung functioned as an ideal lung compartment but that, with induction of anesthesia, the airways to dependent areas of lung closed and these areas of lung behaved as a closed collapsible cavity. The main parameter of interest was the time the unventilated area of lung took to collapse; the effects of preoxygenation and of different inspired gas mixtures during anesthesia were examined. Preoxygenation increased the rate of gas uptake from the unventilated area of lung and was the most important determinant of the time to collapse. Increasing the inspired O2 fraction during anesthesia reduced the time to collapse. Which inert gas (N2 or N2O) was breathed during anesthesia had minimal effect on the time to collapse.     

On the theory of blood-tissue exchange of inert gases: VI. Validity of approximate uptake expressions

It is shown that the equation of inert gas uptake by a distinct parallel tissue-blood arrangement coincides, under certain conditions, with two formulations which neglect the possible existence of a blood-tissue barrier. The first of these approximations is the classic von Schrötter equation in continuous form, whereas the second is the empirical one frequently used by contemporary authors. The condition for coincidence is that the product of permeability and blood-tissue exchange surface greatly exceed the rate of blood flow to the tissue. It is difficult to examine this condition at present because of a dearth of gas permeability measurement and because apparently there exist no measurements of surface and flow on the same tissue. A compilation is made of such values as are available, and it is found that on the assumption that gas permeabilities are of the order of 1×10^?3 cm sec^?1, the conditions for neglecting the blood-tissue barrier may be met in many cases and certainly not met in many others. It is concluded that under these circumstances the more exact equations, taking into account the barrier, should be employed, at least until precise independent measurements justifying the approximations become available.     

Some remarks on the mathematical theory of nutrition of fishes

V. S. Ivlev [Experimental Ecology of Nutrition of Fishes, 1955, Moscow (in Russia)] has shown that the food uptake by fishes during a fixed interval of time is an exponential function of the concentration of food. Ivlev's equation is derived here, and it is shown that it can hold only for non-stationary conditions, such as prevailed in Ivlev's experiments. For a stationary state, the rate of food uptake should tend asymptotically to a limiting value as the concentration increases, but the variation is not exponential. Different other aspects of the problem are investigated, and definite new experimental procedures suggested. The implications of Ivlev's findings on the effect of non-uniformity of food distribution upon the rate of food consumption are studied from a mathematical point of view. The conclusion is reached that whereas a fish does not, in the process of eating, move directly to an individual food particle which it perceives, it does move more or less directly to large aggregates of particles, if the latter are distributed nonuniformly.     

Soil-atmosphere CO exchanges and microbial biogeochemistry of CO transformations in a Brazilian agricultural ecosystem

Although anthropogenic land use has major impacts on the exchange of soil and atmosphere gas in general, relatively little is known about its impacts on carbon monoxide. We compared soil-atmosphere CO exchanges as a function of land use, crop type, and tillage treatment on an experimental farm in Par?na, Brazil, that is representative of regionally important agricultural ecosystems. Our results showed that cultivated soils consumed CO at rates between 3 and 6 mg of CO m(-2) day(-1), with no statistically significant effect of tillage method or crop. However, CO exchange for a pasture soil was near zero, and an unmanaged woodlot emitted CO at a rate of 9 mg of CO m(-2) day(-1). Neither nitrite, aluminum sulfate, nor methyl fluoride additions affected CO consumption by tilled or untilled soils from soybean plots, indicating that CO oxidation did not depend on ammonia oxidizers and that CO oxidation patterns differed in part from patterns reported for forest soils. The apparent K(m) for CO uptake, 5 to 11 ppm, was similar to values reported for temperate forest soils; V(max) values, approximately 1 micro g of CO g (dry weight)(-1) h(-1), were comparable for woodlot and cultivated soils in spite of the fact that the latter consumed CO under ambient conditions. Short-term (24-h) exposure to elevated levels of CO (10% CO) partially inhibited uptake at lower concentrations (i.e., 100 ppm), suggesting that the sensitivity to CO of microbial populations that are active in situ differs from that of known carboxydotrophs. Soil-free soybean and corn roots consumed CO when they were incubated with 100-ppm concentrations and produced CO when they were incubated with ambient concentrations. These results document for the first time a role for cultivated plant roots in the dynamics of CO in an agricultural ecosystem.     

A reinterpretation of the mathematical biophysics of the central nervous system in the light of neurophysiological findings

The fundamental equations for the interaction between neurons used in mathematical biophysics seem at first incompatible with the actual neurophysiological findings on the synaptic transmission. It is shown, however, that those equations may be readily interpreted in terms of accepted neurophysiological views. What has been termed “synapse” in mathematical biophysics must be regarded as a complicated network of internuncial neurons. It is shown that, under rather conditions, the number of those interneurons willstatistically vary with time according to the differential equation postulated for the excitatory and inhibitory factors. The latter are thus interpreted as the number of excitatory and inhibitory interneurons.     

The resonance of the arterial system

The arterial system is assumed to consist of two elastic chambers connected by a conducting channel. It is assumed that a current of fluid enters one chamber, whereas the other chamber is drained by a pipe with a certain peripheral resistance. The continuity of the fluid is described by a differential equation for each chamber. The inertia resistance of the conducting channel is taken into consideration. It is shown that the system may possess a resonance frequency. The latter, if it exists, as well as the damping coefficients are expressed in terms of the elastic moduli of the chambers, the conductivity of the channel, and the peripheral resistance. It is shown that with plausible values of the latter variables the resonance frequency as determined theoretically has the right order of magnitude as found experimentally.     

Pulmonary alveolar macrophages express a polyamine transport system

Polyamine transport is an important mechanism by which cells regulate their intracellular polyamine content. It is well established that the lung has a high capacity for polyamine transport, and recently the polyamine putrescine has been shown to be selectively accumulated into the type II pneumocyte of rabbit lung slices (Saunders et al.: Lab. Invest., 95:380-386, 1988). In addition, it has been suggested that there may be more than one polyamine transport system in lung tissue (Byers et al.: Am. J. Physiol., 252:C663-C669, 1987). In the present study, we have examined whether there are differences in the distribution of putrescine and spermidine uptake activities in isolated rabbit lung cells. We report that pulmonary alveolar macrophages have a greater rate of uptake of both putrescine and spermidine than the total lung cell population. Kinetic analysis of the polyamine uptake system present in macrophages showed putrescine uptake consisted of a saturable (Km = 2.1 microM) and nonsaturable component whilst spermidine uptake consisted of both a high- and a low-capacity saturable component (Km = 0.16 microM and 1.97 microM, respectively). The rate of polyamine transport was similar to those reported for many proliferative or tumor cell-lines and appears to be greater than any other major lung cell type. Inhibition studies of the transport of polyamines into pulmonary alveolar macrophages suggested that the uptake of both putrescine and spermidine was mediated by the same system, which could not be described by simple Michaelis-Menten kinetics. The transport appears to be reversible due to significant efflux. This is the first study to describe the presence of multiple polyamine transport systems in pulmonary alveolar macrophages.     

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.     

Carotenoid incorporation into natural membranes from artificial carriers: liposomes and beta-cyclodextrins

Lung cells are among the first tissues of the body to be exposed to air-borne environmental contaminants. Consequently the function of these cells may be altered before other cells are affected. As gas exchange takes place in the lungs, changes in cellular function may have serious implications for the processes of oxygen uptake and carbon dioxide elimination. In order for these processes to occur, the lung must maintain a high degree of expandability. This latter function is accomplished in part by the pulmonary surfactant which is synthesized and released by alveolar type II cells. Earlier studies have shown that exposure to gas phase materials such as smoke or organic solvents can alter the composition and function of the surfactant. The present study examines the ability of highly toxigenic mold spores to alter surfactant composition. Stachybotrys chartarum spores suspended in saline were instilled into mouse trachea as described earlier. After 24 h, the lungs were lavaged and the different processing stages of surfactant isolated by repeated centrifugation. Intracellular surfactant was isolated from the homogenized lung tissue by centrifugation on a discontinuous sucrose gradient. Samples were extracted into chloroform-methanol, dried and analyzed by Fourier-Transform infrared spectroscopy (FTIR). Exposure to S. chartarum induced an overall reduction of phospholipid among the three surfactant subfractions. The intermediate and spent surfactant fractions in particular were reduced to about half of the values observed in the saline-treated group. The relative distribution of phospholipid was also altered by spore exposure. Within the intracellular surfactant pool, higher levels of phospholipid were detected after spore exposure. In addition, changes were observed in the nature of the phospholipids. In particular strong intramolecular hydrogen bonding, together with other changes, suggested that spore exposure was associated with absence of an acyl chain esterified on the glycerol backbone, resulting in elevated levels of lysophospholipid in the samples. This study shows that mold spores and their products induce changes in regulation of both secretion and synthesis of surfactant, as well as alterations in the pattern of phospholipid targeting to the pulmonary surfactant pools.     

Contrasting modes of action of D-glucose and 3-O-methyl-D-glucose as protectors of the rat pancreatic B-cell against alloxan

Both D-glucose and its nonmetabolized analog 3-O-methyl-D-glucose are known to protect the pancreatic B-cell against the toxic action of alloxan, as if the protective action of hexoses were to involve a membrane-associated glucoreceptor site. In the present study, the protective actions of the two hexoses were found to differ from one another in several respects. Using the process of glucose-stimulated insulin release by rat pancreatic islets as an index of alloxan cytotoxicity, we observed that the protective action of D-glucose was suppressed by D-mannoheptulose and menadione, impaired by NH4Cl, and little affected by aminooxyacetate. These findings and the fact that D-glucose failed to decrease [2-14C]alloxan uptake by the islets suggest that the protective action of D-glucose depends on an increase in the generation rate of reducing equivalents (NADH and NADPH). The latter view is supported by the observation that the protective action of a noncarbohydrate nutrient, 2-ketoisocaproate, was also abolished by menadione. Incidentally, the protective action of 2-ketoisocaproate was apparently a mitochondrial phenomenon, it not being suppressed by aminooxyacetate. In contrast to that of glucose, the protective action of 3-O-methyl-D-glucose was unaffected by D-mannoheptulose, failed to be totally suppressed by menadione, and coincided with a decreased uptake of [2-14C]-alloxan by the islets. It is concluded that the protective action of D-glucose in linked to the metabolism of the sugar in islet cells, whereas that of 3-O-methyl-D-glucose results from inhibition of alloxan uptake. This conclusion reinforces our opinion that the presence in the B-cell of an alleged stereospecific membrane glucoreceptor represents a mythical concept.     

Polyamines in the lung: polyamine uptake and polyamine-linked pathological or toxicological conditions

The natural polyamines putrescine, cadaverine, spermidine, and spermine are found in all cells. These (poly)cations exert interactions with anions, e.g., DNA and RNA. This feature represents their best-known direct physiological role in cellular functions: cell growth, division, and differentiation. The lung and, more specifically, alveolar epithelial cells appear to be endowed with a much higher polyamine uptake system than any other major organ. In the lung, the active accumulation of natural polyamines in the epithelium has been studied in various mammalian species including rat, hamster, rabbit, and human. The kinetic parameters (Michaelis-Menten constant and maximal uptake) of the uptake system are the same order of magnitude regardless of the polyamine or species studied and the in vitro system used. Also, other pulmonary cells accumulate polyamines but never to the same extent as the epithelium. Although different uptake systems exist for putrescine, spermidine, and spermine in the lung, neither the nature of the carrier protein nor the reason for its existence is known. Some pulmonary toxicological and/or pathological conditions have been related to polyamine metabolism and/or polyamine content in the lung. Polyamines possess an important intrinsic toxicity. From in vitro studies with nonpulmonary cells, it has been shown that spermidine and spermine can be metabolized to hydrogen peroxide, ammonium, and acrolein, which can all cause cellular toxicity. In hyperoxia or after ozone exposure, the increased polyamine synthesis and polyamine content of the rat lung is correlated with survival of the animals. Pulmonary hypertension induced by monocrotaline or hypoxia has also been linked to the increased polyamine metabolism and polyamine content of the lung. In a small number of studies, it has been shown that polyamines can contribute to the suppression of immunologic reactions in the lung.     

Reverse Engineering of Oxygen Transport in the Lung: Adaptation to Changing Demands and Resources through Space-Filling Networks

The space-filling fractal network in the human lung creates a remarkable distribution system for gas exchange. Landmark studies have illuminated how the fractal network guarantees minimum energy dissipation, slows air down with minimum hardware, maximizes the gas- exchange surface area, and creates respiratory flexibility between rest and exercise. In this paper, we investigate how the fractal architecture affects oxygen transport and exchange under varying physiological conditions, with respect to performance metrics not previously studied.We present a renormalization treatment of the diffusion-reaction equation which describes how oxygen concentrations drop in the airways as oxygen crosses the alveolar membrane system. The treatment predicts oxygen currents across the lung at different levels of exercise which agree with measured values within a few percent. The results exhibit wide-ranging adaptation to changing process parameters, including maximum oxygen uptake rate at minimum alveolar membrane permeability, the ability to rapidly switch from a low oxygen uptake rate at rest to high rates at exercise, and the ability to maintain a constant oxygen uptake rate in the event of a change in permeability or surface area. We show that alternative, less than space-filling architectures perform sub-optimally and that optimal performance of the space-filling architecture results from a competition between underexploration and overexploration of the surface by oxygen molecules.     

Role of collateral paths in long-range diffusion in lungs.

The long-range apparent diffusion coefficient (LRADC) of (3)He gas in lungs, measured over times of several seconds and distances of 1-3 cm, probes the connections between the airways. Previous work has shown the LRADC to be small in health and substantially elevated in emphysema, reflecting tissue destruction, which is known to create collateral pathways. To better understand what controls LRADC, we report computer simulations and measurements of (3)He gas diffusion in healthy lungs. The lung is generated with a random algorithm using well-defined rules, yielding a three-dimensional set of nodes or junctions, each connected by airways to one parent node and two daughters; airway dimensions are taken from published values. Spin magnetization in the simulated lung is modulated sinusoidally, and the diffusion equation is solved to 1,000 s. The modulated magnetization decays with a time constant corresponding to an LRADC of approximately 0.001 cm(2)/s, which is smaller by a factor of approximately 20 than the values in healthy lungs measured here and previously in vivo and in explanted lungs. It appears that collateral gas pathways, not present in the simulations, are functional in healthy lungs; they provide additional and more direct routes for long-range motion than the canonical airway tree. This is surprising, inasmuch as collateral ventilation is believed to be physiologically insignificant in healthy lungs. We discuss the effect on LRADC of small collateral connections through airway walls and rule out other possible mechanisms. The role of collateral paths is supported by measurements of smaller LRADC in pigs, where collateral ventilation is known to be smaller.     

The effect of mussel size, temperature, seston volume, food quality and volume-specific toxin concentration on the uptake rate of PSP toxins by mussels (Mytilus galloprovincialis Lmk)

The accumulation of paralytic shellfish poisoning (PSP) toxins by bivalves is a serious threat to public health all over the world. However, very little is known about the uptake kinetics of these toxins and the environmental factors that may modify this process. We have studied the effect of mussel size, temperature, seston volume, food quality, and volume-specific toxin concentration (VOSTOC), on the uptake rate of paralytic shellfish poisoning (PSP) toxins by mussels (Mytilus galloprovincialis), by means of a second order factorial experiment. Over a 3-day period, the mussels were fed artificial diets containing Alexandrium minutum AL1V (a PSP toxin producer), Tetraselmis suecica, Ensiculifera sp1 and silt, to the levels required by each treatment. Mussel size, seston volume and VOSTOC were found to be statistically significant when the total toxin accumulated per weight of wet tissue was considered. Mussel size affected the uptake negatively and latter two positively. The interactions, mussel size-VOSTOC and mussel size-food quality were also significant. The response was not linear as shown by the significance of the quadratic term of mussel size. Notwithstanding, when the PSP toxins accumulation per mussel was analysed, only one factor, the VOSTOC and the interactions, food quality-mussel size and food quality-seston volume, were found to be significant. VOSTOC was the most important factor in the accumulation of toxins, in our opinion, probably due to toxin assimilation being mainly regulated by the probability of contact between the toxins and the cellular walls of the digestive system. The size of the bivalve is also especially important because toxin concentration is usually calculated per weight of bivalve tissue and because the weight-specific ingestion increases with mussel size. The food quality, which was directly related to the assimilation of organic matter, had an inverse effect on toxin assimilation. In our opinion, this is probably due to the effect of inorganic particles in enhancing the disruption of Alexandrium cells. Temperature had no effect on the uptake rate except for the accumulation of the gonyautoxin GTX1.     

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.     

Theory of the acoustic instability and behavior of the phase velocity of acoustic waves in a weakly ionized plasma

The amplification of acoustic waves due to the transfer of thermal energy from electrons to the neutral component of a glow discharge plasma is studied theoretically. It is shown that, in order for acoustic instability (sound amplification) to occur, the amount of energy transferred should exceed the threshold energy, which depends on the plasma parameters and the acoustic wave frequency. The energy balance equation for an electron gas in the positive column of a glow discharge is analyzed for conditions typical of experiments in which acoustic wave amplification has been observed. Based on this analysis, one can affirm that, first, the energy transferred to neutral gas in elastic electron-atom collisions is substantially lower than the threshold energy for acoustic wave amplification and, second, that the energy transferred from electrons to neutral gas in inelastic collisions is much higher than that transferred in elastic collisions and thus may exceed the threshold energy. It is also shown that, for amplification to occur, there should exist some heat dissipation mechanism more efficient than gas heat conduction. It is suggested that this may be convective radial mixing within a positive column due to acoustic streaming in the field of an acoustic wave. The features of the phase velocity of sound waves in the presence of acoustic instability are investigated.     

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.