Bench and mathematical modeling of the effects of breathing a helium/oxygen mixture on expiratory time constants in the presence of heterogeneous airway obstructions

ABSTRACT: BACKGROUND: Expiratory time constants are used to quantify emptying of the lung as a whole, and emptying of individual lung compartments. Breathing low-density helium/oxygen mixtures may modify regional time constants so as to redistribute ventilation, potentially reducing gas trapping and hyperinflation for patients with obstructive lung disease. In the present work, bench and mathematical models of the lung were used to study the influence of heterogeneous patterns of obstruction on compartmental and whole-lung time constants. METHODS: A two-compartment mechanical test lung was used with the resistance in one compartment held constant, and a series of increasing resistances placed in the opposite compartment. Measurements were made over a range of lung compliances during ventilation with air or with a 8/22% mixture of helium/oxygen. The resistance imposed by the breathing circuit was assessed for both gases. Experimental results were compared with predictions of a mathematical model applied to the test lung and breathing circuit. In addition, compartmental and whole-lung time constants were compared with those reported by the ventilator. RESULTS: Time constants were greater for larger minute ventilation, and were reduced by substituting helium/oxygen in place of air. Notably, where time constants were long due to high lung compliance (i.e. low elasticity), helium/oxygen improved expiratory flow even for a low level of resistance representative of healthy, adult airways. In such circumstances, the resistance imposed by the external breathing circuit was significant. Mathematical predictions were in agreement with experimental results. Time constants reported by the ventilator were wellcorrelated with those determined for the whole-lung and for the low-resistance compartment, but poorly correlated with time constants determined for the high-resistance compartment. CONCLUSIONS: It was concluded that breathing a low-density gas mixture, such as helium/oxygen, can improve expiratory flow from an obstructed lung compartment, but that such improvements will not necessarily affect time constants measured by the ventilator. Further research is required to determine if alternative measurements made at the ventilator level are predictive of regional changes in ventilation. It is anticipated that such efforts will be aided by continued development of mathematical models to include pertinent physiological and pathophysiological phenomena that are difficult to reproduce in mechanical test systems.     

Theory of the kinetic analysis of patch-clamp data.

This paper describes a theory of the kinetic analysis of patch-clamp data. We assume that channel gating is a Markov process that can be described by a model consisting of n kinetic states and n(n - 1) rate constants at each voltage, and that patch-clamp data describe the occupancy of x different conductance levels over time. In general, all the kinetic information in a set of patch-clamp data is found in either two-dimensional dwell time histograms describing the frequency of observation of sequential dwell times of durations tau 1 and tau 2 (Fredkin, D. R., M. Montal, and J. A. Rice, 1985, Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer, vol. 1, 269-289) or in three-point joint probability functions describing the probability that a channel is in a given conductance at time t, and at time t + tau 1, and at time t + tau 1 + tau 2. For the special case of channels with a single open state plus multiple closed states, one-dimensional analyses provide all of the kinetic information. Stationary patch-clamp data have information that can be used to determine H rate constants, where H = n(n - 1) - G and G is the number of intraconductance rate constants. Thus, to calculate H rate constants, G rate constants must be fixed. In general there are multiple sets of G rate constants that can be fixed to allow the calculation of H rate constants although not every set of G rate constants will work. Arbitrary assignment of the G intraconductance rate constants equal to zero always provides a solution and the calculation of H rate constants. Nonstationary patch-clamp data have information for the determination of H rate constants at a reference voltage plus n(n - 1) rate constants at all test voltages. Thus, nonstationary data have extra information about the voltage dependencies of rate constants that can be used to rule out kinetic models that cannot be disqualified on the basis of stationary data.     

Comparison of two-pulse sodium inactivation with reactivation in Myxicola giant axons.

Values for the time constant of reactivation of the sodium conductance following depolarization sufficient to completely inactivate GNa have been compared over a 15 mV range of membrane potential with the time constants of inactivation during a depolarization prepulse. Over this range the reactivation time constants were consistently 30-50% larger than the inactivation time constants determined simultaneously at the same potential in the same axon. The data suggests that inactivation and reactivation do not occur by identical mechanisms, and therefore implies that there are at least three kinds of experimental procedures necessary to fully characterize the sodium inactivation process in any particular system.     

Kinetic time constants independent of previous single-channel activity suggest Markov gating for a large conductance Ca-activated K channel

Models for the gating of ion channels usually assume that the rate constants for leaving any given kinetic state are independent of previous channel activity. Although such discrete Markov models have been successful in describing channel gating, there is little direct evidence for the Markov assumption of time-invariant rate constants for constant conditions. This paper tests the Markov assumption by determining whether the single-channel kinetics of the large conductance Ca-activated K channel in cultured rat skeletal muscle are independent of previous single-channel activity. The experimental approach is to examine dwell-time distributions conditional on adjacent interval durations. The time constants of the exponential components describing the distributions are found to be independent of adjacent interval duration, and hence, previous channel activity. In contrast, the areas of the different components can change. Since the observed time constants are a function of the underlying rate constants for transitions among the kinetic states, the observation of time constants independent of previous channel activity suggests that the rate constants are also independent of previous channel activity. Thus, the channel kinetics are consistent with Markov gating. An observed dependent (inverse) relationship between durations of adjacent open and shut intervals together with Markov gating indicates that there are two or more independent transition pathways connecting open and shut states. Finally, no evidence is found to suggest that gating is not at thermodynamic equilibrium: the inverse relationship was independent of the time direction of analysis.     

EXCITATION OF NERVE FIBERS IN THE SQUID (LOLIGO PEALII)

1. Strength-duration data for the giant fiber of the great stellar nerve of the squid (Loligo pealii) can be approximately described by several mathematical formulations. 2. Excitation time constants for isolated giant fibers are essentially the same as constants of the giant fibers in the intact nerve. 3. The strength-duration curves of the fibers in the intact nerve lie higher on the voltage axis than those of the isolated fibers. It is concluded that the principal effect of other fibers upon the excitation of one fiber in a nerve trunk is that of shunting the stimulating current. 4. Deterioration of the nerve shifts the curve upward and to the left, resulting in shorter time constants. 5. Decreasing interelectrode distance also shifts the curve upward and to the left. 6. Excitation time constants of the giant fibers are larger with plate electrodes than with wire or pore electrodes. 7. The strength-duration curves of the smaller fin nerve fibers lie consistently to the right of, and the time constants are longer than those of the giant fibers.     

Recovery of Ca currents from inactivation: The roles of Ca influx,membrane potential,and cellular metabolism

Ca currents were examined with regard to their recovery from inactivation. The experiments were done on isolated nerve cell bodies of Helix aspersa using a combined suction pipet , microelectrode method for voltage clamp, and internal perfusion. Ca currents were separated by suppressing K and Na currents. The time course of recovery was determined by applying a test pulse at intervals ranging from 1 msec to 20 sec after prepulses varying from 20 to 3000 msec in duration. Each pair of pulses was preceded by a control pulse to ensure that the Ca currents had recovered before the next test pair was applied. Ba and Ca currents were compared and the effects of intracellular perfusion with EGTA, ATP, and vanadate were examined. Ba currents recovered in two stages and this time course was well fit by a sum of two exponentials with amplitudes and time constants given by A1 and tau 1 for the fast component and A2 and tau 2 for the slow component. In Ba the time constants were unchanged when prepulse durations were prolonged from 70 to 700 msec, although the initial amplitudes A1 and A2, particularly A2, were increased. Comparable influxes of Ca during the prepulse caused much more inactivation, but interestingly the recovery occurred at the same rate. The time course of Ca current recovery was also fit by a sum of two exponentials, the time constants of which were both smaller than the time constants of Ba current recovery. However, the time constants of Ca current recovery were increased markedly when prepulse durations were prolonged. Increasing the extracellular Ca concentration had a similar effect. Increasing the Ba influx had no effect on the recovery time constants, and the Ba results are consistent with reversible inactivation gating of potential-dependent membrane Ca channels. The Ca results show that Ca influx enhances inactivation. Intracellular perfusion with EGTA resulted in less inactivation in the cast of Ca but it had no effect on Ba currents. Intracellular ATP increased the rate of recovery of Ca currents, and intracellular vanadate inhibited recovery. It is concluded that recovery of Ca channels depends upon both Ca influx and membrane potential and is modulated by agents which affect Ca metabolism.     

Theoretical considerations in the equilibrium binding of myosin fragments on F-actin

In a previous paper, equilibrium constants for the binding of myosin fragments onto F-actin were assumed known and the statistical problems encountered when the actin sites are occupied to an arbitrary fractional extent were analyzed. The object of the present paper is to attempt to understand the observed order of magnitude of these equilibrium constants in terms of the statistical mechanical degrees of freedom involved. That is, we examine here the equilibrium constants them- selves rather than the statistical consequences of the equilibrium constants. The treatment given amounts to a semi-quantitative sketch or outline of the problem. Structural details are much too uncertain to warrant a careful and rigorous treatment at this time. But the discussion suffices to establish the essential qualitative features of the problem. The procedure used is to exa- mine the important equilibrium constants, one at a time, in terms of the factors (partition functions) that contribute to each constant, together with numerical estimates for these factors.     

Solutions for transients in arbitrarily branching cables: I. Voltage recording with a somatic shunt.

An analytical solution is derived for voltage transients in an arbitrarily branching passive cable neurone model with a soma and somatic shunt. The response to injected currents can be represented as an infinite series of exponentially decaying components with different time constants and amplitudes. The time constants of a given model, obtained from the roots of a recursive transcendental equation, are independent of the stimulating and recording positions. Each amplitude is the product of three factors dependent on the corresponding root: one constant over the cell, one varying with the input site, and one with the recording site. The amplitudes are not altered by interchanging these sites. The solution reveals explicitly some of the parameter dependencies of the responses. An efficient recursive root-finding algorithm is described. Certain regular geometries lead to "lost" roots; difficulties associated with these can be avoided by making small changes to the lengths of affected segments. Complicated cells, such as a CA1 pyramid, produce many closely spaced time constants in the range of interest. Models with large somatic shunts and dendrites of unequal electrotonic lengths can produce large amplitude waveform components with surprisingly slow time constants. This analytic solution should complement existing passive neurone modeling techniques.     

Estimation of rate constants of drug binding to open channels of cardiac transient outward current

An improved approach to the evaluation of the rate constants of drugs binding to the open channel underlying the transient outward potassium current I(t0) is described. It is based on an analysis of a quantitative model formulated by a set of twelve differential equations. The rate constants are calculated from the time constants resulting from an approximation of the time course of apparent inactivation of the recorded I(t0) by two exponentials in the absence and by three exponentials in the presence of a blocking agent. The model study confirmed significantly higher accuracy in comparison with the existing electrophysiological method.     

The effect of holding time on nanoindentation measurements of creep in bone

Viscoelasticity may affect both the elastic and fracture characteristics of bone. Nanoindentation can be used to measure the creep behavior of bone by fitting the depth vs. time data at constant load to rheological models. However, the creep data may be influenced by latent effects arising during the loading phase of indentation. As such, the loading protocol, particularly the holding time, may affect the measured creep time constants. To characterize the effect of holding time on the measures, four cortical bone samples were prepared from four bovine femora and subjected to nanoindentation to measure the creep behavior. The creep time constants were found by fitting the indentation depth vs. time curve to three different rheological models: the standard linear solid, Burgers model, and a two-dashpot Kelvin model. All three models provided good fits to the data, which were relatively insensitive to the initial parameter estimates. The calculated creep time constants increased monotonically with increasing holding time for all three models. However, the relative differences between measurements within a single osteon, within a single sample, and between samples were maintained for creep holding times over 16s. Hence, while the creep time constants measured by nanoindentation with hold times up to 30s may not provide accurate property measurements, comparisons between samples are valid if all are assessed at the same holding time. Considering the long-term viscosity of bone tissue, Burgers model provided the best performance in terms of stability and goodness of fit, and is recommended for future studies.     

Time Constants and Electrotonic Length of Membrane Cylinders and Neurons

A theoretical basis is provided for the estimation of the electrotonic length of a membrane cylinder, or the effective electrotonic length of a whole neuron, from electrophysiological experiments. It depends upon the several time constants present in passive decay of membrane potential from an initially nonuniform distribution over the length. In addition to the well known passive membrane time constant, τ_m = R_mC_m, observed in the decay of a uniform membrane potential, there exist many smaller time constants that govern rapid equalization of membrane potential over the length. These time constants are present also in the transient response to a current step applied across the membrane at one location, such as the neuron soma. Similar time constants are derived when a lumped soma is coupled to one or more cylinders representing one or more dendritic trees. Different time constants are derived when a voltage clamp is applied at one location; the effects of both leaky and short-circuited termination are also derived. All of these time constants are demonstrated as consequences of mathematical boundary value problems. These results not only provide a basis for estimating electrotonic length, L = [unk]/λ, but also provide a new basis for estimating the steady-state ratio, ρ, of cylinder input conductance to soma membrane conductance.     

Physiological and ecological implications of a simple model of heating and cooling in reptiles

A model of heat exchange in reptiles is used to investigate the role of blood flow in controlling rates of heating/cooling in animals in complex thermal environments. The model suggests an allometry of heating and cooling time constants and of the effects of blood flow on those time constants that accords with published data. The model suggests a simple physical reason for the increased effect of blood flow on time constants in large animals. Two tools (the model and an impulse response method) are presented to allow projection of body temperatures in complex thermal habitats. Application of the model to ecologically important situations suggest that mass, blood flow, and shuttling schedules affect the rate of heating and cooling and the effect of blood flow on the range of body temperatures experienced.     

A Subsequent Fit of Time Series and Amplitude Histogram of Patch-Clamp Records Reveals Rate Constants up to 1 per Microsecond

Fast gating in time series of patch-clamp current demands powerful tools to reveal the rate constants of the adequate Hidden Markov model. Here, two approaches are presented to improve the temporal resolution of the direct fit of the time series. First, the prediction algorithm is extended to include intermediate currents between the nominal levels as caused by the anti-aliasing filter. This approach can reveal rate constants that are about 4 times higher than the corner frequency of the anti-aliasing filter. However, this approach is restricted to time series with very low noise. Second, the direct fit of the time series is combined with a beta fit, i.e., a fit of the deviations of the amplitude histogram from the Gaussian distribution. Since the “theoretical” amplitude histograms for higher-order Bessel filters cannot be calculated by analytical tools, they are generated from simulated time series. In a first approach, a simultaneous fit of the time series and of the Beta fit is tested. This simultaneous fit, however, inherits the drawbacks of both approaches, not the benefits. More successful is a subsequent fit: The fit of the time series yields a set of rate constants. The subsequent Beta fit uses the slow rate constants of the fit of the time series as fixed parameters and the optimization algorithm is restricted to the fast ones. The efficiency of this approach is illustrated by means of time series obtained from simulation and from the dominant K⁺ channel in Chara. This shows that temporal resolution can reach the microsecond range.     

Interactions of nucleic acid double helices induced by electric field pulses

Electric field pulses induce a substantial increase of the light scattering intensity of double-helical DNA. The relative change of light scattering and also the reciprocal relaxation time constants under electric field pulses increase with increasing nucleotide concentration. These observations, together with a large difference between dichroism orientation time constants and light scattering time constants under electric field pulses, demonstrate that the main part of the light scattering effect is due not to field-induced orientation but to interactions between DNA helices. From the concentration dependence of the light scattering time constants we obtain, according to an isodesmic reaction model, association rate constants in the range 3 × 10¹⁰ M^?1 helices s^?1 for DNA with approx. 300 base-pairs. These values are at the limit of a diffusion-controlled DNA association and do not show any dependence upon the field strength. The dissociation rate constants k_d decrease strongly with increasing field strength E and thus demonstrate that the interactions between the helices are induced by the electric field. This conclusion is consistent with independent measurements which do not reveal any DNA association at zero field strength. The observed linear relation between log(k_d) and E² suggests a field-induced reaction driven by dipole changes. According to this interpretation the change of dipole moment should be in the range of approx. 1400 debye. The dissociation rates for DNA helices with approx. 300 to approx. 800 base-pairs strongly increase with increasing sail concentration (measured in the range 1–5 mM ionic strength), whereas the association rate constants remain virtually unchanged. Measurements of the linear dichroism in the same range of DNA chain length demonstrate that for long field pulses of e.g., 40 μs, the amplitude approaches a maximum value and then decreases. The dichroism relaxation curves observed after long field pulses exhibit a component with a positive dichroism and an increased decay time. These observations suggest the formation of a DNA aggregate with an unusual arrangement of the bases.     

Dynamic behavior of a complete-mixing activated sludge system

The dynamic behavior of a laboratory-scale activated sludge biological waste treatment process with recycle and wasting of sludge was investigated by subjecting the system to step changes in the influent waste concentration, the recycle flow rate, or the sludge wasting rate. The dynamic behavior of the system was examined by measuring adenosine triphosphate (ATP) in addition to dissolved chemical oxygen demand (COD) and cell dry weight in the aeration tank. Cell dry weight of the recyle flow and effluent COD were also measured. Analysis of the results and estimation of time constants assuming first order responses showed that the time constants characterizing the dynamic responses of the sludge were directly related to the sludge mean residence time. The time constants estimated from dissolved COD measurements were of the same order of magnitude as the fluid residence time in the aeration tank. The ATP transient response was frequently different from that of the cell dry weight in the aeration tank.     

Influx and efflux kinetics of cationic dye binding to respiring mitochondria.

We have investigated the kinetics of interaction of cationic fluorescent lipophiles (dyes) rhodamine 123, rhodamine 6G, tetramethyl rhodamine ethyl ester, safranine O, 1,1'-diethyloxacarbocyanine, 1,1'-diethyloxadicarbocyanine, and 1,1'-diethylthiadicarbocyanine iodide with isolated respiring rat-liver mitochondria (RLM). Dye flux across the RLM inner membrane was measured by following the kinetics of fluorescence signal change after mixing of dye and RLM. The time course of fluorescence was analysed in terms of a kinetic model of the binding and transport processes involved. The rate constants of dye influx and efflux were extracted from the observed effect on the apparent time constant of fluorescence change to equilibrium intensity upon mixing dye with increasing concentrations of RLM. From the influx rate constants obtained, the apparent permeability constants for dye influx (at zero potential) across the membrane were calculated and ranged from 3 to 140 x 10(-4) cm/s. The influx rate constant was found to be linearly related to relative dye lipophilicity, as predicted by the model. As another test of the model, from the ratio of the influx and efflux rate constants, the apparent trans-membrane potential, psi, was calculated and found generally to agree with reported values, but to depend on the lipophilicity of the dye used. Not predicted by the simple model was a dissymmtry observed in the influx and efflux time constants for fluorescence change to equilibrium intensity. Inferences are made relating to the utility of these dyes as probes of psi.     

Time profile of hemin aggregation: an analysis

The status of hemin, whether monomeric or aggregated, appears to be a key factor in deciding its biological activity. The molecular basis for the tendency of hemin to aggregate and the factors that regulate it are not yet fully understood. We have investigated the time profile of aggregation of hemin in aqueous solutions and the effect of temperature on the process of aggregation. Aggregation increases with increase in temperature. The time profile data, as monitored by change in absorbance at 398 nm, fits a nonlinear equation with three time constants, suggesting a possibility of three processes. Interestingly, the variation of these three time constants with temperature are different.     

Potassium current kinetics in Myxicola axons. Effects of conditioning prepulses

In Myxicola giant axons the time constants for activation of the potassium conductance (GK) after prepulses less depolarized than a test pulse are comparable to the time constants for turn off of GK after prepulses more depolarized than the same test pulse. The absolute magnitude of the steady-state level of GK is also independent of prepulse amplitude in Myxicola. The results are contrasted with recent observations on voltage-clamped frog nodes.     

An improved method to analyze the stress relaxation of ligaments following a finite ramp time based on the quasi-linear viscoelastic theory

The quasi-linear viscoelastic (QLV) theory proposed by Fung (1972) has been frequently used to model the nonlinear time- and history-dependent viscoelastic behavior of many soft tissues. It is common to use five constants to describe the instantaneous elastic response (constants A and B) and reduced relaxation function (constants C, tau 1, and tau 2) on experiments with finite ramp times followed by stress relaxation to equilibrium. However, a limitation is that the theory is based on a step change in strain which is not possible to perform experimentally. Accounting for this limitation may result in regression algorithms that converge poorly and yield nonunique solutions with highly variable constants, especially for long ramp times (Kwan et al. 1993). The goal of the present study was to introduce an improved approach to obtain the constants for QLV theory that converges to a unique solution with minimal variability. Six goat femur-medial collateral ligament-tibia complexes were subjected to a uniaxial tension test (ramp time of 18.4 s) followed by one hour of stress relaxation. The convoluted QLV constitutive equation was simultaneously curve-fit to the ramping and relaxation portions of the data (r2 > 0.99). Confidence intervals of the constants were generated from a bootstrapping analysis and revealed that constants were distributed within 1% of their median values. For validation, the determined constants were used to predict peak stresses from a separate cyclic stress relaxation test with averaged errors across all specimens measuring less than 6.3 +/- 6.0% of the experimental values. For comparison, an analysis that assumed an instantaneous ramp time was also performed and the constants obtained for the two approaches were compared. Significant differences were observed for constants B, C, tau 1, and tau 2, with tau 1 differing by an order of magnitude. By taking into account the ramping phase of the experiment, the approach allows for viscoelastic properties to be determined independent of the strain rate applied. Thus, the results obtained from different laboratories and from different tissues may be compared.     

Excited-state dynamics of bacteriorhodopsin probed by broadband femtosecond fluorescence spectroscopy

The impact of varying excitation densities (approximately 0.3 to approximately 40 photons per molecule) on the ultrafast fluorescence dynamics of bacteriorhodopsin has been studied in a wide spectral range (630-900 nm). For low excitation densities, the fluorescence dynamics can be approximated biexponentially with time constants of <0.15 and approximately 0.45 ps. The spectrum associated with the fastest time constant peaks at 650 nm, while the 0.45 ps component is most prominent at 750 nm. Superimposed on these kinetics is a shift of the fluorescence maximum with time (dynamic Stokes shift). Higher excitation densities alter the time constants and their amplitudes. These changes are assigned to multi-photon absorptions.