Saddles, arrows, and spirals: deterministic trajectories in cyclic competition of four species.

Population dynamics in systems composed of cyclically competing species has been of increasing interest recently. Here we investigate a system with four or more species. Using mean field theory, we study in detail the trajectories in configuration space of the population fractions. We discover a variety of orbits, shaped like saddles, spirals, and straight lines. Many of their properties are found explicitly. Most remarkably, we identify a collective variable that evolves simply as an exponential: Q ∝ e(λt), where λ is a function of the reaction rates. It provides information on the state of the system for late times (as well as for t→-∞). We discuss implications of these results for the evolution of a finite, stochastic system. A generalization to an arbitrary number of cyclically competing species yields valuable insights into universal properties of such systems.

Calculation of electric fields induced in the human knee by a coil applicator.

Calculations are presented of the induced electric fields and current densities in the cartilage of the knee produced by a coil applicator developed for applying pulsed magnetic fields to osteoarthritic knees. This applicator produces a sawtooth-like magnetic field waveform composed of a series of 260-micros pulses with a peak to peak magnitude of approximately 0.12 mT in the cartilage region. The simulations were performed using a recently developed 3 dimensional finite difference frequency domain technique for solving Maxwell's equations with an equivalent circuit model. The tissue model was obtained from the anatomically segmented human body model of Gandhi. The temporal peak electric field magnitude was found to be -153 mV/m, averaged within the medial cartilage of the knee for the typical dB/dt excitation levels of this coil. The technique can be extended to analyze other excitation waveforms and applicator designs.

Magnetic resonance behavior of normal and diseased lungs: spherical shell model simulations.

The alveolar air-tissue interface affects the lung NMR signal, because it results in a susceptibility-induced magnetic field inhomogeneity. The air-tissue interface effect can be detected and quantified by measuring the difference signal (Delta) from a pair of NMR images obtained using temporally symmetric and asymmetric spin-echo sequences. The present study describes a multicompartment alveolar model (consisting of a collection of noninteracting spherical water shells) that simulates the behavior of Delta as a function of the level of lung inflation and can be used to predict the NMR response to various types of lung injury. The model was used to predict Delta as a function of the inflation level (with the assumption of sequential alveolar recruitment, partly parallel to distension) and to simulate pulmonary edema by deriving equations that describe Delta for a collection of spherical shells representing combinations of collapsed, flooded, and inflated alveoli. Our theoretical data were compared with those provided by other models and with experimental data obtained from the literature. Our results suggest that NMR Delta measurements can be used to study the mechanisms underlying the lung pressure-volume behavior, to characterize lung injury, and to assess the contributions of alveolar recruitment and distension to the lung volume changes in response to the application of positive airway pressure (e.g., positive end-expiratory pressure).

Effects of endotoxin lung injury on NMR T2 relaxation.

The effects of endotoxin injury on lung NMR relaxation times (T1, CPMG T2, and Hahn decay constant (Hahn T2)) were studied in excised unperfused rat lungs. Blinded histologic examination showed no clear-cut separation between endotoxin and control lungs. Morphometric lung tissue volume density and gravimetric lung water content did not differ significantly between the two groups. In contrast, the values of the fast, intermediate, and slow T2 components, obtained by multiexponential analysis of the CPMG decay curve, increased markedly after endotoxin administration, with minimal overlap between endotoxin and control values. The response of Hahn T2 was, in general, in the same direction as that of CPMG T2; however, Hahn T2 may be more affected by measurement errors and may be less sensitive to the presence of lung injury. T1 showed minimal changes after injury. The present data suggest that CPMG T2 measurements can consistently detect the presence of lung injury even when conventional histologic, morphometric, and gravimetric studies provide negative or equivocal results, and that the CMPG T2 method is superior, in this respect, to the Hahn decay method. T1 does not appear to be sensitive to lung injury in the absence of significant lung water accumulation.

Calculation of electric fields induced near metal implants by magnetic resonance imaging switched-gradient magnetic fields.

Electric (E) fields induced near metal implants by MRI switched-gradient magnetic fields are calculated by a new equivalent-circuit numerical technique. Induced E-field results are found for a metallic spinal-fusion implant consisting of two thin wires connected to the metallic case of a current generator as well as for its subsections: a bare U-shaped wire, an insulated U-shaped wire, a cut insulated wire, and a generator. The presence of the metallic implants perturbs the E field significantly. Near the ends of the bare U-shaped wire, the E field is 89.7 times larger than in the absence of the wire. The greatest E field concentration occurs near the ends of the cut insulated wire, where the E field is 196.7 times greater than in the absence of the wire. In all cases, the perturbation of the induced E field by the implanted wire is highly localized within a few diameters of the wire.

The effects of weak extremely low frequency magnetic fields on calcium/calmodulin interactions.

Mechanisms by which weak electromagnetic fields may affect biological systems are of current interest because of their potential health effects. Lednev has proposed an ion parametric resonance hypothesis (Lednev, 1991, Bioelectromagnetics, 12:71-75), which predicts that when the ac, frequency of a combined dc-ac magnetic field equals the cyclotron frequency of calcium, the affinity of calcium for calcium-binding proteins such as calmodulin will be markedly affected. The present study evaluated Lednev's theory using two independent systems, each sensitive to changes in the affinity of calcium for calmodulin. One of the systems used was the calcium/calmodulin-dependent activation of myosin light chain kinase, a system similar to that previously used by Lednev. The other system monitored optical changes in the binding of a fluorescent peptide to the calcium/calmodulin complex. Each system was exposed to a 20.9 microT static field superimposed on a 20.9 microT sinusoidal field over a narrow frequency range centered at 16 Hz, the cyclotron frequency of the unhydrated calcium ion. In contrast to Lednev's predictions, no significant effect of combined dc-ac magnetic fields on calcium/calmodulin interactions was indicated in either experimental system.

Comparison of calculated and experimental NMR spectral broadening for lung tissue.

NMR lineshapes were calculated for a model of lung, and NMR proton spectra were measured for individual voxels in an excised inflated rat lung. NMR lines for parenchymal lung regions containing alveoli, alveolar ducts, and capillaries were calculated using a computer simulation of the NMR signal from a three-dimensional honeycomb-like structure, a collection of modified Wigner-Seitz cells. These cells were modified by rounding the corners and increasing the thickness of the boundaries to model various degrees of lung inflation and lung water. NMR lineshapes were also calculated for the central or nonparenchymal lung regions containing bronchi and large blood vessels. A comparison of theoretical lineshapes with those measured in individual voxels both in the parenchymal and in the central (largely nonparenchymal) regions in excised rat lungs at an inflation pressure of 30 cm of water shows excellent agreement. These results indicate that the NMR lineshape reflects the underlying lung geometry. This research constitutes the first calculations and measurements of NMR lineshapes in lung. The appendix describes a new method for calculating the magnetic field inside a weakly diamagnetic material of arbitrary shape placed in an otherwise uniform external magnetic field. This new method does not require either solution of simultaneous equations or evaluation of integral expressions.

Lung water measurement by nuclear magnetic resonance: correlation with morphometry.

Estimates of lung water content obtained from nuclear magnetic resonance (NMR) and morphometric and gravimetric measurements were compared in normal and experimentally injured rats. Average lung water density (rho H2O) was measured by an NMR technique in excised unperfused rat lungs (20 normal lungs and 12 lungs with oleic acid-induced edema) at 0 (full passive deflation) and 30 cmH2O lung inflation pressure and in vivo (4 normal rats and 8 rats with lung injury induced by oleic acid or rapid saline infusion). The rho H2O values were compared with morphometric measurements of lung tissue volume density (Vv) obtained from the same lungs fixed at corresponding liquid-instillation pressures. A close correlation was observed between rho H2O and Vv in normal and injured excised lungs [correlation coefficient (r) = 0.910, P < 0.01]. In vivo rho H2O was also closely correlated with Vv (r = 0.897, P < 0.01). The correlation coefficients between rho H2O and gravimetric lung water content (LWGr) were lower in the excised lung group (r = 0.663 and 0.692, respectively, for rho H2O at 0 and 30 cmH2O lung inflation pressure, P < 0.01) than in the in vivo study (r = 0.857, P < 0.01). Our results indicate that NMR techniques, which are noninvasive and nondestructive, provide reliable estimates of lung water density and that the influence of lung inflation on rho H2O is important (compared with the effect of lung water accumulation in lung injury) only in the presence of deliberately induced very large variations in the lung inflation level.

Alveolar air/tissue interface and nuclear magnetic resonance behavior of normal and edematous lungs.

The alveolar air/tissue interface markedly affects the NMR properties of lungs by causing an NMR signal loss as a result of internal (tissue-induced) magnetic field inhomogeneity. The signal loss can be measured as the difference in NMR signal intensity (difference signal delta) between a pair of images obtained using temporally symmetric and asymmetric spin-echo sequences. Previous data indicate that the difference signal measured at an asymmetry time of 6 ms (delta 6ms) is very low in degassed lungs and increases markedly with alveolar opening. Theoretically, the NMR behavior of edematous lungs is expected to differ from that of normal nondegassed lungs because alveolar flooding and collapse are equivalent to partial (regional) degassing. To test this prediction, we measured delta 6ms in normal and edematous (oleic acid-injured) excised unperfused rat lungs at 5, 10, 20, 30, and 0 (full passive deflation) cm H2O inflation pressure (PL). Lung volume changes were estimated from NMR lung water density (pH2O) measurements. In normal lungs, delta 6ms did not vary with PL. In edematous lungs delta 6ms was, as predicted, significantly lower than normal at 5 and 10 cm H2O PL but rose markedly (to about normal) as PL was further increased. Upon subsequent deflation from 30 to 0 cm H2O PL, delta 6ms did not vary significantly or decreased. On the basis of our theoretical models, the data could be interpreted as reflecting the loss of alveolar air/tissue interface as a result of alveolar flooding and the relative contributions of airspace recruitment and distension to the lung volume changes. Histologic and morphometric data obtained from the same lungs supported this interpretation.(ABSTRACT TRUNCATED AT 250 WORDS)

Comparison of in vivo and in vitro Hahn T2 measurements in rat lung.

We compared in vivo and in vitro Hahn echo T2 measurements in rat lungs in both imaging and nonimaging modes. All measurements could be characterized by multiexponential functions consisting of either two or three exponentials. Essentially the same values of the time constants were observed for spontaneously breathing rats and for excised lungs.

Investigation of AC-DC magnetic field effects in planar phospholipid bilayers.

Observations recently reported by others indicate that a combination of a weak dc magnetic field and extremely-low-frequency ac magnetic field can produce resonant effects in biological systems. We report measurements of the effects of combined dc and ac magnetic fields on the dc current through channel-free planar phospholipid membranes. The combined dc-ac magnetic fields did affect the dc current through planar phospholipid membranes, but not in every membrane, and not consistently at the same values of magnetic flux density and frequency. None of our measurements showed resonant response akin to the cyclotron-like resonance reported in diatoms [Smith et al., 1987] and lymphocytes [Liboff et al., 1987].

Calcium binding to metallochromic dyes and calmodulin in the presence of combined, AC-DC magnetic fields.

The possibility that weak, ac and dc magnetic fields in combination may affect binding equilibria of calcium-ions (Ca2+) was investigated with two metallochromic dyes as calcium-binding molecules: murexide and arsenazo III. Calcium-dye equilibria were followed by measuring solution absorbances with a fiber-optic spectrophotometer. A Ca(2+)-arsenazo solution was also used indirectly to monitor the binding of Ca2+ to calmodulin. Parallel, ac and dc magnetic fields were applied to each preparation. The ac magnetic field was held constant during each of a series of experiments at a frequency in the range between 50 and 120 Hz (sine wave) or at 50 pps (square wave) and at an rms flux density in the range between 65 and 156 microT. The dc magnetic field was then varied from 0 to 299 microT at 1.3 microT increments. The magnetic fields did not measurably affect equilibria in the binding of metallochromic dyes or calmodulin to Ca2+.

Computer-aided design of two-dimensional electric-type hyperthermia applicators using the finite-difference time-domain method.

A hyperthermia applicator design tool consisting of a finite-difference time-domain (FDTD) technique in combination with a graphical display of electric fields and normalized linear temperature rise is described. This technique calculates, rather than assumes, antenna current distributions; it includes mutual interactions between the body and the applicator, and it calculates driving-point impedance and power delivered to the applicator. Results show that the fundamental limitation of 2-D electric-type applicators is overheating of the fat by normal components of the electric field, which exist because of near fields and capacitive coupling with the muscle. Two factors which contribute to the capacitance are the muscle conductivity and the small antenna size in air. Two examples of applicators designed to avoid fat overheating are described: a 27-MHz segmented dipole for heating large tumors to 7 cm depth, and a 100-MHz dipole for small tumors to 5 cm depth. The first uses a water bolus, and the second uses a water bolus with low-permittivity strips to reduce normal fields at the antenna ends. The results of this study describe fundamental limitations of electric field applicators, and illustrate the use of a powerful applicator design tool that allows rapid evaluation of a wide range of ideas for applicators which would require months and years to test experimentally.

Alveolar air-tissue interface and nuclear magnetic resonance behavior of lung.

Inflated lungs are characterized by a short nuclear magnetic resonance (NMR) free induction decay (rapid disappearance of NMR signal), likely due to internal (tissue-induced) magnetic field inhomogeneity produced by the alveolar air-tissue interface. This phenomenon can also be detected using temporally symmetric and asymmetric NMR spin-echo sequences; these sequences generate a pair of NMR images from which a difference signal (delta) is obtained (reflecting the signal from lung water experiencing the air-tissue interface effect). We measured delta in normal excised rat lungs at inflation pressures of 0-30 cmH2O for asymmetry times (a) of 1-6 ms. Delta was low in degassed lungs and increased markedly with alveolar opening when measured at a = 6 ms (delta 6 ms); delta 6 ms varied little during the rest of the inflation-deflation cycle. Delta 1 ms (a = 1 ms) did not vary significantly on inflation and deflation. Measurements of delta at a = 3 and 5 ms generally lay between those of delta 1 ms and delta 6 ms. These findings, which are consistent with theoretical predictions, suggest that measurements of delta at appropriate asymmetry times are particularly sensitive to alveolar opening and may provide a means of distinguishing alveolar recruitment from alveolar distension in the pressure-volume behavior of the lung.

Electromagnetic pulsed-wave radiation in spherical models of dispersive biological substances.

In analytical studies, we investigated induced-field patterns and SAR distributions in a lossy, dispersive, homogeneous, dielectric sphere typical of muscle tissue as irradiated by a plane-wave pulse train consisting of a pulse-modulated sinusoidal carrier wave. Calculations were made for carrier frequencies of 1, 3, and 15 GHz, pulse widths of 0.333, 2.0 and 4 ns, and pulse repetition rates of 1.11 x 10(6), 100 x 10(6), and 181.18 x 10(6) pps. The classical Mie solution was modified for a train of incident pulses that was represented by a Fourier series, and the fast-Fourier transform was used to sum the series. Computationally, the technique proved to be feasible and less expensive than we expected. The calculated field patterns show that the sphere's physical dimensions and the internal wavelength of the carrier greatly influence the nature of pulse-train propagation in the sphere. Harmonics having internal wavelengths nearly equal to the radius of the sphere produce most of the absorption; other harmonics produce little absorption. An intense hot spot is observed in spheres with radii that match the carriers' wavelengths.

Regional effects of repetition time on NMR quantitation of water in normal and edematous lungs.

It is well known that pulmonary edema is, in general, spatially nonuniform. Since the NMR spin-lattice relaxation time (T1) is increased by lung edema, the spatial distribution of T1 will be nonuniform. When the repetition time (TR) is short relative to the T1 of edematous lung, lung water content will be underestimated and this underestimation will be spatially nonuniform as well. Therefore, technical artifacts which are a complex function of lung edema and its spatial distribution are expected. We compared overall and regional (topographic) lung water density measurements obtained from living rats (with normal or edematous lungs) using repetition times of 2.0 and 6.2 s (at a magnetic field of 1 T), to quantify this uneven T1 effect for normal and edematous lungs. NMR measurements at TR = 2.0 s underestimated whole lung water density (-rho H2O) TR = 6.2 s) by an average of 7.2% in normal rats and 22.5% in rats with pulmonary edema. Regional -rho H2O underestimation (%delta-rho H2O) varied from 2.2 to 8.8% (groups means) in normal lungs and from 7.3 to 30.8% in edematous lungs. As a result, the interquartile range (of the voxel distribution as a function of rho H2O) underestimated the spatial nonuniformity of lung water density by 28.0% in edematous lungs, likely because of greater loss of NMR signal from high-water-density, long-T1 lung regions. Both %delta-rho H2O and T1 were significantly correlated with -rho H2O at TR = 6.2 s.(ABSTRACT TRUNCATED AT 250 WORDS)