Chaotic oscillations in a simple collapsible-tube model.

A steady flow through a segment of externally pressurized, collapsible tube can become unstable to a wide variety of self-excited oscillations of the internal flow and tube walls. A simple, one-dimensional model of the conventional laboratory apparatus, which has been shown previously to predict steady flows and multiple modes of oscillation, is investigated numerically here. Large amplitude oscillations are shown to have a relaxation structure, and the nonlinear interaction between different modes is shown to give rise to quasiperiodic and apparently aperiodic behavior. These predictions are shown to compare favorably with experimental observations.     

Numerical simulation of sinusoidal flow in a straight elastic tube: effects of phase angles

Numerical simulations of flow in straight elastic (moving wall) tubes subjected to a sinusoidal pressure gradient were performed for conditions prevailing in large and medium sized arteries. The effects of varying the phase angle between the pressure gradient and the tube radius, the amplitude of wall motion, and the unsteadiness parameter (alpha) on flow rate and wall shear stress were investigated. Mean and peak flow rates and shear stresses were found to be strongly affected by the phase angle between the pressure gradient and the tube radius with greater sensitivity at higher diameter variation and higher alpha. In large artery simulations (alpha = 12), means flow rate was found to be 60% higher and peak flow rate to be 73% higher than corresponding rigid tube values for certain phase angles, while a threefold increase in mean wall shear stress and sevenfold increase in peak wall shear stress were observed in a sensitive phase angle range. Significant reversal in the wall shear stress direction occurred in the sensitive phase angle range even when there was negligible flow rate reversal. All effects were greatly diminished in simulations of medium sized vessels (alpha = 4). Some experimental evidence to support the predictions of a strong effect of phase angle on wall shear stress in large vessels is presented. Finally, physiological implications of the present work are discussed from a basis of aortic input impedance data, and a physical explanation for the extreme sensitivity of the flow field to small amplitude wall motion at high alpha is given.     

Mathematical simulation of forced expiration

Flow limitation during forced expiration is simulated by a mathematical model. This model draws on the pressure-area law obtained in the accompanying paper, and the methods of analysis for one-dimensional flow in collapsible tubes developed by Shapiro (Trans. ASME J. Biomech. Eng. 99: 126-147, 1977). These methods represent an improvement over previous models in that 1) the effects of changing lung volume and of parenchymal-bronchial interdependence are simulated; 2) a more realistic representation of collapsed airways is employed; 3) a solution is obtained mouthward of the flow-limiting site by allowing for a smooth transition from sub- to supercritical flow speeds, then matching mouth pressure by imposing an elastic jump (an abrupt transition from super- to subcritical flow speeds) at the appropriate location; and 4) the effects of levels of effort (or vacuum pressure) in excess of those required to produce incipient flow limitation are examined, including the effects of potential physiological limitation.     

Numerical simulation of aortic bifurcation flows: The effect of flow divider curvature

Two dimensional steady flow calculations in computational regions obtained from radiographs of human aortic bifurcations correlate well with unsteady measurements of wall shear in flow-through casts of the same vessels. The results suggest that wall slope may be an important factor affecting the variability of shear along the medial walls of this arterial segment. If extremes of shear stress promote atherogenesis, then variations in the curvature of the proximal iliac arteries may affect the susceptibility of these vessels to vascular disease on their medial aspect.     

Numerical simulation of streaming potentials due to deformation-induced hierarchical flows in cortical bone

Bone is a very dynamic tissue capable of modiA,fing its composition, microstructure, and overall geometry in response to the changing biomechanical needs. Streaming potential has been hypothesized as a mechanotransduction mechanism that may allow osteocytes to sense their biomechanical environment. A correct understanding of the mechanism for streaming potential will illuminate our understanding of bone remodeling, such as the remodeling associated with exercise hypertrophy, disuse atrophy, and the bone remodeling arounid implants. In the current research, a numerical model based on the finite element discretization is proposed to simulate the fluid flows through the complicated hierarchical flow system and to calculate the concomitant stress generated potential (SGP) as a result of applied mechanical loading. The lacunae-canaliculi and the matrix microporosity are modeled together as discrete one-dimensional flow channels superposed in a biphasic poroelastic matrix. The cusplike electric potential distribution surrounding the Haversian canal that was experimentallv observed and reported in the literature earlier was successfully reproduced by the current numerical calculation.     

Complex metabolic oscillations in plants forced by harmonic irradiance

Plants exposed to harmonically modulated irradiance, approximately 1 + cos(omegat), exhibit a complex periodic pattern of chlorophyll fluorescence emission that can be deconvoluted into a steady-state component, a component that is modulated with the frequency of the irradiance (omega), and into at least two upper harmonic components (2omega and 3omega). A model is proposed that accounts for the upper harmonics in fluorescence emission by nonlinear negative feedback regulation of photosynthesis. In contrast to simpler linear models, the model predicts that the steady-state fluorescence component will depend on the frequency of light modulation, and that amplitudes of all fluorescence components will exhibit resonance peak(s) when the irradiance frequency is tuned to an internal frequency of a regulatory component. The experiments confirmed that the upper harmonic components appear and exhibit distinct resonant peaks. The frequency of autonomous oscillations observed earlier upon an abrupt increase in CO(2) concentration corresponds to the sharpest of the resonant peaks of the forced oscillations. We propose that the underlying principles are general for a wide spectrum of negative-feedback regulatory mechanisms. The analysis by forced harmonic oscillations will enable us to examine internal dynamics of regulatory processes that have not been accessible to noninvasive fluorescence monitoring to date.     

Shunt effect of gas compression inside pneumotachographs during forced oscillations

Determination of the frequency response of pneumotachographs is needed whenever they are used to measure high-frequency flows, such as in the forced oscillation method. When screen and capillary pneumotachographs are calibrated using an adiabatic compression in a closed box as a reference impedance, they can be adequately described by a series of inertial-resistive elements. However, this type of reference impedance strongly differs from the actual respiratory impedance (ZL). We studied the frequency response of pneumotachographs up to 250 Hz in reference to the impedance of a compressible gas oscillating in a long tube, taken as a more generalizable model of actual ZL. We found that, with this device, the series resistance-inertance models fail to describe the frequency response of the pneumotachograph. However, when compressible effects in the pneumotachograph are taken into account by adding to the resistive models a compliance (Cpn) corresponding to the compression in half of the inner volume of the pneumotachograph, the agreement with experiments becomes satisfactory. Gas compression-related phenomena were demonstrated to be negligible only when the parameter omega Cpn magnitude of ZL is much smaller than 1 (omega pulsation). Results obtained in normal humans have shown that such a correction is required above 100 Hz. Similar correction at lower frequency might also be necessary in cases of large respiratory impedance (e.g., babies, subjects with pathological lungs, and intubated subjects).     

Numerical simulation of LDL mass transfer in a common carotid artery under pulsatile flows

Symmetrical 30-60% stenosis in carotid artery with a semi-permeable wall under steady/unsteady flows for Newtonian/non-Newtonian fluids is investigated numerically. The results show that the unsteadiness of blood flow, blood pressure rise and LDL component size increase the luminal concentration, LC, of the surface. The maximum LC occurring immediately after the separation point and the non-Newtonian fluid predicts higher LDL accumulation. LC decreased as the recirculation length is increased and reaches maximum at 40% stenosis. This process is used to estimate the time-dependent growth of the arterial wall.     

Influence of bifurcations on forced oscillations in an airway model.

Forced oscillations is a technique to determine respiratory input impedance from small amplitude sinusoidal pressure excursions introduced at the airway opening. Models used to predict respiratory input impedance typically ignore the direct effect of bifurcations on the flow, and treat airway branches as individual straight tubes placed appropriately in parallel and series. The flow within the individual tubes is assumed equivalent to that which would occur in infinitely long tubes. In this study we examined the influence of bifurcations on impedance for conditions of the forced oscillatory technique. We measured input impedance using forced oscillations in straight tubes and in an anatomically-relevant, four generation physical model of a human airway network. The input impedance measured experimentally compared well to that obtained theoretically using model predictions. The predictive scheme was based on appropriate parallel and series combinations of theoretically computed individual tube impedances, which were computed from solutions to oscillatory flow of a compressible gas in an infinitely long rigid tube. The agreement between experimental measurements and predictions indicates that bifurcations play a relatively minor direct role on the flow impedance for conditions of the forced oscillations technique. These results are explained in terms of the small tidal volumes used, whereby the axial distance traveled by a fluid particle during an oscillation cycle is appreciably smaller than branch segment lengths. Accordingly, only a small fraction of fluid particles travel through the bifurcation region, and the remainder experience an environment approaching flow in an infinite straight tube. The relevance of the study to the prediction of impedances in the human lung during forced oscillations is discussed.     

Distinguishing intrinsic limit cycles from forced oscillations in ecological time series

Ecological cycles are ubiquitous in nature and have triggered ecologists’ interests for decades. Deciding whether a cyclic ecological variable, such as population density, is part of an intrinsically emerging limit cycle or simply driven by a varying environment is still an unresolved issue, particularly when the only available information is in the form of a recorded time series. We investigate the possibility of discerning intrinsic limit cycles from oscillations forced by a cyclic environment based on a single time series. We argue that such a distinction is possible because of the fundamentally different effects that perturbations have on the focal system in these two cases. Using a set of generic mathematical models, we show that random perturbations leave characteristic signatures on the power spectrum and autocovariance that differ between limit cycles and forced oscillations. We quantify these differences through two summary variables and demonstrate their predictive power using numerical simulations. Our work demonstrates that random perturbations of ecological cycles can give valuable insight into the underlying deterministic dynamics.     

Model of forced expiratory flows and airway geometry in infants.

Early measurements of autopsied lungs from infants, children, and adults suggested that the ratio of peripheral to central airway resistance was higher in infants than older children and adults. Recent measurements of forced expiration suggest that infants have high flows relative to lung volume. We employed a computational model of forced expiratory flow along with physiological and anatomic data to evaluate whether the infant lung is a uniformly scaled-down version of the adult lung. First, we uniformly scaled an existing computational model of adult forced expiration to estimate forced expiratory flows (FEF) and density dependence for an 18-mo-old infant. The values obtained for FEF and density dependence were significantly lower than those reported for healthy 18-mo-old infants. Next, we modified the model for the infant lung to reproduce standard indexes of expiratory flow [forced expiratory volume in 0.5 s (FEV(0.5)), FEFs after exhalation of 50 and 75% forced vital capacity, FEF between 25 and 75% expired volume] for this age group. The airway sizes obtained for the infant lung model that produced accurate physiological measurements were similar to anatomic data available for this age and larger than those in the scaled model. Our findings indicate that the airways in the infant lung model differ from those in the scaled model, i.e., middle and peripheral airway sizes are larger than result from uniform downscaling of the adult lung model. We show that the infant lung model can be made to reproduce individual flow-volume curves by adjusting lumen area generation by generation.     

Nonlinearity and harmonic distortion of dog lungs measured by low-frequency forced oscillations

The nonlinearity of lung tissues and airways was studied in six anesthetized and paralyzed open-chest dogs by means of 0.1-Hz sinusoidal volume forcing at mean transpulmonary pressures (Ptp) of 5 and 10 cmH2O. Lung resistance (RL) and elastance (EL) were determined in a 32-fold range (15-460 ml) of tidal volume (VT), both by means of spectrum analysis at the fundamental frequency and with conventional time-domain techniques. Alveolar capsules were used to separate the tissue and airway properties. A very small amplitude dependence was found: with increasing VT, the frequency-domain estimates of RL decreased by 5.3 and 14%, whereas EL decreased by 20 and 22% at Ptp = 5 and 10 cmH2O, respectively. The VT dependences of the time-domain estimates of RL were higher: 10.5 and 20% at Ptp = 5 and 10 cmH2O, respectively, whereas EL remained the same. The airway resistance increased moderately with flow amplitude and was smaller at the higher Ptp level. Analysis of the harmonic distortions of airway opening pressure and the alveolar pressures indicated that nonlinear harmonic production is moderate even at the highest VT and that VT dependence is homogeneous throughout the tissues. In three other dogs it was demonstrated that VT dependences of RL and EL were similar in situ and in isolated lungs at both Ptp levels.     

Computer simulation of rapidly responding forced exponential systems

Two common problems in computer simulations are the decisions to ignore or include a particular element of a system under study in a model and the choice of an appropriate integration algorithm. To examine aspects of these problems, a simple exponential system is considered in which a large simulation error is induced by a rather small truncation error. The effect of computational precision, step size and hardware selection on this error is examined at standard and extended precisions over a range of step sizes and on a variety of computers. For this model, simulation accuracy is an exponential function of the number of bits in the mantissa of the computer word. Optimal step size is a function of accuracy required and precision used; a trade-off between truncation and round-off errors becomes important as accuracy requirements increase. Machine selection is important primarily in economic terms if the required precision is available. We conclude that the effect on a simulation of small terms such as truncation errors can be unexpectedly large, that solutions should always be checked, and that high precision and wide dynamic range are important to the successful computer simulation of models such as that examined.     

Frequency-dependent distortion of meridional intensity changes during sinusoidal length oscillations of activated skeletal muscle

Bundles of intact, tetanized skeletal muscle fibers from Rana temporaria were subjected to sinusoidal length oscillations in the frequency domain 100 Hz to 3 kHz while measuring force and sarcomere length. Simultaneously, intensity of the third-order x-ray reflection of the axial myosin unit cell (I(M3)) was measured using synchrotron radiation. At oscillation frequencies <1 kHz, I(M3) was distorted during the shortening phase of the sinusoid (i.e., where bundle length was less than rest length). Otherwise, during the stretch phase of oscillations at all frequencies, during the shortening phase of oscillations above 1 kHz, and for bundles in the rigor state, I(M3) was approximately sinusoidal in form. Mean I(M3) during oscillations was reduced by 20% compared to the isometric value, suggesting a possible change in S1 disposition during oscillations. However, the amplitude of length change required to produce distortion (estimated from the phase angle at which distortion was first evident) corresponded to that of a step release sufficient to reach the maximum I(M3), indicating a mean S1 disposition during oscillations close to that during an isometric tetanus. The mechanical properties of the bundle during oscillations were also consistent with an unaltered S1 disposition during oscillations.     

Energy stored and dissipated in skeletal muscle basement membranes during sinusoidal oscillations.

We subjected single skeletal muscle cells from frog semitendinosus to sinusoidal oscillations that simulated the strain experienced as the cells near the end of passive extension and begin active contraction in slow swimming. Other cells from which the basement membrane was removed by enzymatic and mechanical procedures were tested identically. Effectiveness of the basement membrane removal technique was evaluated by electron microscopy, by an electrophoretic and lectin-binding assay for depletion of cell surface glycoproteins, and by confirmation by means of electrophoretic and immunologic analyses that major intracellular, cytoskeletal proteins were not disrupted. Measurements of maximum stress, maximum strain, and phase lag between these maxima enabled the complex modulus (dynamic stiffness) and loss tangent (relative viscous losses to elastic energy storage) to be calculated for each mechanically tested preparation. We also calculated the amounts of energy stored and dissipated in each preparation. These calculations indicate that cells with intact basement membranes have complex moduli significantly greater than those of cells without basement membranes, and that cells with basement membrane store significantly more elastic energy than basement membrane depleted cells. However, when subjected to identical sinusoidal strains, energy dissipation in cells with intact basement membranes is over three times greater than dissipation in cells without basement membrane. The relative magnitudes of energy losses to energy storage, called the specific loss, is nearly three times greater for intact cells than for basement membrane depleted cells. Basement membranes may thereby serve as a brake for slowing passive extension of muscle before contraction begins.