Direct measurement of the aathway of respired gas in duck lungs

Breathing patterns and associated cardiovascular changes in intermittently breathing animals: (Partially) correcting a semantic quagmire

Many animal species do not breathe in a continuous, rhythmic fashion, but rather display a variety of breathing patterns characterized by prolonged periods between breaths (inter-breath intervals), during which the heart continues to beat. Examples of intermittent breathing abound across the animal kingdom, from crustaceans to cetaceans. With respect to human physiology, intermittent breathing-also termed 'periodic' or 'episodic' breathing-is associated with a variety of pathologies. Cardiovascular phenomena associated with intermittent breathing in diving species have been termed 'diving bradycardia', 'submersion bradycardia', 'immersion bradycardia', 'ventilation tachycardia', 'respiratory sinus arrhythmia' and so forth. An examination across the literature of terminology applied to these physiological phenomena indicates, unfortunately, no attempt at standardization. This might be viewed as an esoteric semantic problem except for the fact that many of the terms variously used by different authors carry with them implicit or explicit suggestions of underlying physiological mechanisms and even human-associated pathologies. In this article, we review several phenomena associated with diving and intermittent breathing, indicate the semantic issues arising from the use of each term, and make recommendations for best practice when applying specific terms to particular cardiorespiratory patterns. Ultimately, we emphasize that the biology-not the semantics-is what is important, but also stress that confusion surrounding underlying mechanisms can be avoided by more careful attention to terms describing physiological changes during intermittent breathing and diving.

A novel accessory respiratory muscle in the American alligator ( Alligator mississippiensis)

The muscles that effect lung ventilation are key to understanding the evolutionary constraints on animal form and function. Here, through electromyography, we demonstrate a newly discovered respiratory function for the iliocostalis muscle in the American alligator ( Alligator mississippiensis). The iliocostalis is active during expiration when breathing on land at 28°C and this activity is mediated through the uncinate processes on the vertebral ribs. There was also an increase in muscle activity during the forced expirations of alarm distress vocalizations. Interestingly, we did not find any respiratory activity in the iliocostalis when the alligators were breathing with their body submerged in water at 18°C, which resulted in a reduced breathing frequency. The iliocostalis is an accessory breathing muscle that alligators are able to recruit in to assist expiration under certain conditions.

Robust Unidirectional Airflow through Avian Lungs: New Insights from a Piecewise Linear Mathematical Model

Avian lungs are remarkably different from mammalian lungs in that air flows unidirectionally through rigid tubes in which gas exchange occurs. Experimental observations have been able to determine the pattern of gas flow in the respiratory system, but understanding how the flow pattern is generated and determining the factors contributing to the observed dynamics remains elusive. It has been hypothesized that the unidirectional flow is due to aerodynamic valving during inspiration and expiration, resulting from the anatomical structure and the fluid dynamics involved, however, theoretical studies to back up this hypothesis are lacking. We have constructed a novel mathematical model of the airflow in the avian respiratory system that can produce unidirectional flow which is robust to changes in model parameters, breathing frequency and breathing amplitude. The model consists of two piecewise linear ordinary differential equations with lumped parameters and discontinuous, flow-dependent resistances that mimic the experimental observations. Using dynamical systems techniques and numerical analysis, we show that unidirectional flow can be produced by either effective inspiratory or effective expiratory valving, but that both inspiratory and expiratory valving are required to produce the high efficiencies of flows observed in avian lungs. We further show that the efficacy of the inspiratory and expiratory valving depends on airsac compliances and airflow resistances that may not be located in the immediate area of the valving. Our model provides additional novel insights; for example, we show that physiologically realistic resistance values lead to efficiencies that are close to maximum, and that when the relative lumped compliances of the caudal and cranial airsacs vary, it affects the timing of the airflow across the gas exchange area. These and other insights obtained by our study significantly enhance our understanding of the operation of the avian respiratory system.

Birds and mammals have similar metabolic demands and cardiovascular systems, but they have evolved drastically different respiratory systems. A key difference in birds is that gas exchange occurs in rigid tubes, through which air flows unidirectionally during both inspiration and expiration. How this unidirectional flow is generated, and the factors affecting it, are not well understood. It has been hypothesized that the unidirectional flow is due to aerodynamic valving resulting from the complex anatomical structure. To test this hypothesis we have constructed a novel mathematical model that, unlike previous models, produces unidirectional flow through the lungs consistently even when the amplitude and frequency of breathing change. We have investigated the model both analytically and computationally and shown the importance of aerodynamic valving for generating strong airflow through the lungs. Our model also predicts that the timing of airflow through the lungs depends on the relative compliances of the different airsacs that exist in birds. The lumped parameters approach we use means that this model is generally applicable across all birds.

Anatomical and biomechanical traits of broiler chickens across ontogeny. Part I. Anatomy of the musculoskeletal respiratory apparatus and changes in organ size

Genetic selection for improved meat yields, digestive efficiency and growth rates have transformed the biology of broiler chickens. Modern birds undergo a 50-fold multiplication in body mass in just six weeks, from hatching to slaughter weight. However, this selection for rapid growth and improvements in broiler productivity is also widely thought to be associated with increased welfare problems as many birds suffer from leg, circulatory and respiratory diseases. To understand growth-related changes in musculoskeletal and organ morphology and respiratory skeletal development over the standard six-week rearing period, we present data from post-hatch cadaveric commercial broiler chickens aged 0, 2, 4 and 6 weeks. The heart, lungs and intestines decreased in size for hatch to slaughter weight when considered as a proportion of body mass. Proportional liver size increased in the two weeks after hatch but decreased between 2 and 6 weeks. Breast muscle mass on the other hand displayed strong positive allometry, increasing in mass faster than the increase in body mass. Contrastingly, less rapid isometric growth was found in the external oblique muscle, a major respiratory muscle that moves the sternum dorsally during expiration. Considered together with the relatively slow ossification of elements of the respiratory skeleton, it seems that rapid growth of the breast muscles might compromise the efficacy of the respiratory apparatus. Furthermore, the relative reduction in size of the major organs indicates that selective breeding in meat-producing birds has unintended consequences that may bias these birds toward compromised welfare and could limit further improvements in meat-production and feed efficiency.

Inspiratory valving in avian bronchi: aerodynamic considerations

The presence of unidirectional flow in the avian lung is thought to be effected by aerodynamic 'valves'. First we review the history of this hypothesis and summarize existing evidence. Second, we present a semiquantitative treatment of the various fluid dynamic factors that may be involved in directing fluid flow. The resulting calculations show in some detail how the inspiratory valve may work, and upon what mechanisms it may depend. Our calculations suggest that gas convective inertial forces are sufficient to effect inspiratory valving. Finally, we give some heuristic arguments regarding the mechanisms of expiratory valving.

Single breath CO2 measurements of deadspace in ducks

We measured deadspace (VD) in ducks using CO2 expirograms (plots of expired PCO2 vs expired volume) obtained during artificial ventilation at different tidal volumes (VT) and respiratory system volumes (VRS). Conventional analysis of the expirograms for Bohr and Fowler VD indicated both were larger than anatomic VD. Most expirograms at VT less than or equal to 100 ml had terminal slopes greater than predicted for lung gas and violated the usual assumptions of the Fowler calculation. Bohr VD was not affected by VRS but increased with VT. This can be explained by expired PCO2 not reaching lung values at low VT and an expiratory mesobronchial ventilatory shunt. We propose a measure of mesobronchial shunt corresponding to a volume of gas exhaled in one breath from caudal air sacs through the mesobronchus (VM). VM/VT changes with pump vs constant flow ventilation indicating sensitivity of VM to flow pattern. We estimate mesobronchial shunting is greatest at the beginning of expiration and approaches zero only near the end of a 200 ml expiration with constant flow ventilation.

Lung volume changes during respiration in ducks

The avian lung has been considered to be rigid and to remain isovolumetric during the respiratory cycle. We tested this hypothesis by implanting radiopaque markers of tantalum on the dorsal pulmonary surfaces and ventral pulmonary aponeuroses of Pekin ducks (Anas platyrhynchos) and measuring changes in lung thickness during the respiratory cycle using high speed cineradiography. We found small but regular changes in lung thickness that were synchronous with respiratory phase. Lung thickness was greatest at mid-inspiration (0.6% greater than mean) and least at mid-expiration (0.8% less than mean). Measurements made on ostrich (Struthio camelus) respiratory structures suggest that the maximal force that could be generated by the muscles (Mm. costopulmonales) at the margins of the ventral pulmonary aponeurosis is more than two orders of magnitude greater than would be required to resist pressure-induced changes in lung volume during respiration at rest. The action of these muscles could account for the very small magnitude of the volume changes measured during the respiratory cycle.

Ventilation-perfusion inequally in avian lungs

We assessed ventilation-perfusion inequality in 8 anesthetized, tidally ventilated geese in terms of continuous V/Q distributions using the multiple inert gas elimination technique modified for cross-current avian lungs (Powell and Wagner, 1982). Thirty-four data sets were collected. Allowing for differences in solubility, high molecular weight gases (Enflurane, SF6) were not retained in the blood to any greater extent than the other gases, suggesting that diffusion in the gas phase is functionally complete. Shunt averaged only 0.4 +/- 0.1% (SEM) of cardiac output and areas of low V/Q were seldom seen. Twenty-nine of the 34 data sets had bimodal V/Q distributions with 10.6 +/- 1.4% of expired ventilation and 0.3 +/- 0.1% of cardiac output in a high V/Q mode; the physiological basis of the high mode is unknown. The log-standard deviation of the main Q mode averaged 0.56 or slightly greater than that for healthy men, dogs, or earlier estimates from unidirectionally ventilated birds. It is predicted that CO2 will be more impaired by such V/Q inequality than O2, but that increased ventilation will overcome the CO2 impairment more easily than that of O2 transport.

Control of air flow in bird lungs: radiographic studies

The complex pattern of air flow in the respiratory system of birds suggests that certain sites function as valves. To examine the possibility of mechanical valving, rather than aerodynamic valving, we recorded radiographic images of the orifices where the medioventral secondary bronchi branch from the primary bronchus in resting Pekin ducks. Analysis of the images indicated that the orifices do not change size or shape during the respiratory cycle, suggesting that they function as aerodynamic rather than mechanical valves in directing air flow through the lung.

Airflow in the avian respiratory tract: variations of O2 and CO2 concentrations in the bronchi of the duck

Variations of CO2 and O2 concentrations within a respiratory cycle were recorded at various sites in the bronchial system of anesthetized, spontaneously ventilating ducks, using small metal cannulae introduced into the main bronchus (MB), a medioventral (MV) or mediodorsal (MD) secondary bronchus and connected to a mass spectrometer for continuous gas analysis. The following results were obtained and conclusions drawn. (1) Since during inspiration, CO2 concentration (FCO2) was close to zero all along MB and since FCO2 was nearly constant throughout the respiratory cycle in MV, it must be inferred that on inspiration, no significant amount of air passes directly either from MV to MB or in the opposite direction, there being thus a complete functional valving of the MV orifices. In particular the Hazelhoff loop mechanism (inspiratory reflux of lung gas into the MB) is not operative. (2) During expiration, FCO2 in MV was only slightly higher than that in the trachea, but substantially above FCO2 deep in MB. This suggests that most of the expiratory flow from caudal air sacs is diverted through the paleopulmo and only little exits directly through MB. It is shown that the functional valving of bronchial air flow is advantageous for gas exchange as it reduces air shunts and provides a nearly steady lung ventilation.

Incomplete gas mixing in air sacs of the duck

During normal breathing, the CO2 concentration in caudal air sacs of birds is higher, and the O2 concentration lower, than expected on the basis of the known air flow pattern. We have experimentally tested two hypotheses which could explain this finding: (1) Preferential shunting of re-inspired dead space gas into caudal air sacs; (2) Incomplete mixing of inspired and residential air sac gas. - Different portions of the inspired air in anesthetized ducks were labeled by injecting a small bolus of argon (Ar) into the trachea. The resulting Ar concentration was recorded continuously in the caudal thoracic air sac at the ostium and in deeper regions.-The amount of Ar entering the sac was found to be independent of the volume inspired prior to injection of the label, and hypothesis (1) thus dismissed. However, during inspiration and subsequent expiration the Ar bolus was found to be neither perfectly mixed within the inspired gas nor with the air sac residential gas. More than 10 sec of breath-hold were necessary for air sac gas to approach an equilibrium value. Gas layering (stratification) in caudal air sacs gas is proposed to cause the high CO2 and low O2 levels during steady state breathing, as air sac residential gas equilibrates with a layer of dead space gas that enters the air sac on each breath and contains a higher CO2 and lower O2 concentration than the mixed inspirate.

Ventilatory response to the PCO2 profile in chicken lungs

We investigated the influence of intrapulmonary chemoreceptors (IPC) on ventilatory movements in anesthetized chickens when PCO2 profiles along the parabronchi were changed. In all experiments the right lung was denervated, both lungs unidirectionally ventilated, and PaCO2 kept constant. In series 1 (7 birds), gas flow and the PCO2 profile in the left lung were reversed. PaCO2, PECO2 and ventilatory movements did not change. In Series 2 (4 birds), PCO2 in caudal regions of the innervated lung was elevated by increasing gas flow and P1CO2 from 0 to 21 Torr. Ventilatory movements did not change. In Series 3 (4 birds), either lung was over-ventilated with 7 or 49 Torr P1CO2, alternating the gases between lungs every 100 sec. Ventilatory movements changes with P1CO2 but much less than predicted from P1CO2 effects in the non-perfused, innervated lung. From the longitudinal distribution of IPC and PCO2 profiles in the lung we predicted moderate to large changes in ventilatory movements in all series. The discrepancy between predicted and observed results in Series 1 and 2 indicates that IPC in caudal regions of the lung have little effect on ventilation under the conditions examined. In Series 3, observed ventilatory movements were less sensitive to P1CO2 than predicted, indicating that IPC sense a different PCO2 than the PCO2 profile in the parabronchial lumen and that IPC have a significant sensitivity to pulmonary blood PCO2.

Corrections to the Hazelhoff model of airflow in the avian lung

The ventilatory activity of the anterior and posterior groups of air sacs was simulated in unidirectionally-ventilated geese and the resultant flow of air in the mediodorsal secondary bronchi was used as an indicator of the route which air followed through the lung. The results were used to isolate the roles of the respective groups of air sacs in the shaping of the unidirectional pattern of airflow known to exist during normal respiration. Findings indicated that, in contrast to Hazelhoff's model, the anterior and not the posterior sacs are responsible for producing the caudo-cranial flow of air through the parabronchi during inspiration. The posterior sacs, as predicted by Hazelhoffs model, and primarily responsible for driving the caudo-cranial current through the parabronchi during expiration.

Theory of gas exchange in the avian parabronchus

To understand the distribution of oxygen and carbon dioxide in the avian lung, a theoretical treatment of gas exchange in the parabronchus of the avian lung is described. The model is modified after Zeuthen (1942). In addition to bulk flow through the parabronchial lumen, diffusion through the air spaces of both the parabronchial lumen and air capillaries is treated. The relationship of PO2 and PCO2 within the blood capillaries, air capillaries, and parabronchial lumen to parabronchial blood flow and ventilation is graphically shown. The results indicate that the variations of PO2 and PCO2 along an air capillary are less than one torr under resting conditions. Removal of diffusion resistance within the air space of the air capillaries increases calculated parabronchial gas exchange by less than 0.1% at rest. At high or resting ventilation rates the partial pressure profile along the parabronchial lumen calculated considering bulk flow only agrees well with the profile calculated considering bulk flow and axial diffusion, but as the ventilation rate decreases there is increasingly large disagreement. Forward diffusion of O2 toward the parabronchus reduces pre-parabronchial PO2 and backward diffusion of CO2 from the parabronchus increases PCO2. Neglecting diffusion within the air spaces of both the lumen and the air capillaries increases calculated parabronchial gas exchange by less than 2% (CO2) or 6% (O2) at rest.

Estimation of effective parabronchial gas volume during intermittent ventilatory flow: theory and application in the duck

The difference in gas exchange performance between continuous and intermittent ventilatory flow is theoretically studied in the alveolar lung model. When measurements obtained with intermittent flow are analyzed assuming continuous flow, an apparent diffusing capacity. Dapp, results which is an underestimate of the true value, D. This true value, D, may be assessed from measurements at continuous flow. With decreasing effective lung gas volume, Veff, Dapp increasingly deviates D. The dependence of Dapp/D on Veff in the parabronchial lung seems to be similar to that in the alveolar lung. Experimental data of Dapp/D (Scheid et al., 1977) are used to assess Veff for the duck lung. The average value of Veff thus obtained, 93 ml, exceeds the anatomical estimate of parabronchial gas volume. Gas transfer across the open parabronchial ends may contribute in enlarging the parabronchial gas volume to the volume, Veff, that is effective as gas capacity in conditions of non-steady ventilatory flow.

Respiratory responses of ducks to simulated altitude

Domestic ducks were exposed to simulated altitudes of 0, 3000, 6000, and 9000 m in order to study the respiratory changes that take place. We found that the respiratory minute volume (VE,BTPS) increased with altitude, the increase being due to increased respiratory frequency while tidal volume (VT, BTPS) showed only minor changes. The quantity of air moved (VE, STPD), however, remained nearly unchanged with increasing altitude. The oxygen extraction, calculated as 1--(FIN2FEO2)/(FEN2FIO2), remained constant at about 0.28 up to 6000 m and declined to 0.17 at 9000 m. The fractional gas concentrations (FO2 and FCO2) in exhaled air and in the interclavicular and posterior thoracic air sacs changed only little up to 6000 m, but at 9000 m FO2 increased and FCO2 decreased. The relative constancy of expired and air sac gas up to 6000 m seems remarkable. However, when applied to current models of air flow in the avian respiratory system the results seem fully explainable and permit a detailed analysis of the functioning of the avian respiratory system.

Response of avian intrapulmonary chemoreceptors to venous CO2 and ventilatory gas flow

Avian intrapulmonary chemoreceptor activity is reduced by increasing airway PCO2 from 0 to 60 torr. Using extracellular electrodes, we recorded discharge of individual intrapulmonary chemoreceptor cell bodies in the left nodose ganglion of the rooster (Gallus domesticus) during unidirectional ventilation of the lungs. All receptors recorded were in the left lung. To vary pulmonary arterial PCO2 independently of ventilation, we ventilated the two lungs separately and supplied the left pulmonary circulation with systemic arterial blood. When the PCO2 in the pulmonary arterial blood was increased, discharge frequency decreased in all 21 receptors studied. Sensitivity to pulmonary arterial PCO2 was similar to sensitivity to airway PCO2. When PCO2 of ventilatory gas was lower than that of pulmonary arterial blood, discharge frequency of the receptor increased when pulmonary blood flow was stopped. Discharge frequency also increased when PCO2 at the receptor site was lowered by increased ventilatory gas flow. We conclude that intrapulmonary chemoreceptors respond to the delivery and removal of CO2 by blood and ventilatory gas. This suggests that the receptors are located within the respiratory gas exchange region of the lung. Because these receptors have a strong inhibitory effect on ventilation, they may serve to (1) adjust minute ventilation to the rate of metabolic CO2 production and (2) to regulate individual breath size.

Ventilatory and circulatory O2 convection at 4000 m in pigeon at neutral or cold temperature

Awake domestic pigeons, either maintained at 22 degrees C (series I) or acutely exposed at 2 degrees C (series II), were studied in a hypobaric chamber at 140 m and at various stages during a 4-week exposure to 4000 m. Steady-state pulmonary ventilation (Vg) and breathing pattern (VT, fr), oxygen consumption (MO2), O2 concentrations and pressures in the arterial (a) and mixed venous blood (v), hematocrit (Ht) and acid-base status in arterial blood, systolic blood pressure and heart frequency (fH) were measured. From these data cardiac output (Vb) and stroke volume (Vs), ventilatory and circulatory requirements (Vg/MO2, Vb/MO2), extraction of O2 from inspired air (EgO2) and blood EbO2), and capacitance coefficient of blood for oxygen (betabo2) were calculated. At 140 m, by comparison with predicted values for mammals of same body weight, pigeons at 22 degrees C extracted more O2 from the inspired gas, with lower fR, larger VT, similar Vg; they extracted O2 from the blood like mammals, with lower fH, larger VS, greater Vb, similar betabO2=70 mumol-L-1-torr-1. Acute exposure to 2 degrees C provoked a two-fold increase in MO2 which was achieved by doubling Vg and increasing O2 extraction from the blood. At 4000 m, in both series, pigeons hyperventilated within the first 30 min, with a resultant hypocapnic alkalosis comparable to that in mammals. Further hyperventilation with consequent greater hypocapnia and increase of arterial PO2 was complete beyond 3 hr. After a few weeks, the pH remained 0.07 above control normoxic value, Ht increased from 45 to 52%, betabO2 reached about 172 mumol-L-1-torr-1. At 2 degrees C, Vb also increased, mainly due to tachycardia.

Effects of CO2 on pulmonary air flow resistance in the duck

Effects of CO2 on pulmonary smooth muscle were assessed by measuring the air flow resistance of secondary bronchi and parabronchi in ducks unidirectionally ventilated with a constant gas flow through the parabronchial lung, the bypass of the primary bronchus being occluded by a blocking catheter. Pressure differences across the blocking balloon (deltaP), corresponding to the pressure drop in the gas flowing through the mediodorsal and medioventral secondary bronchi (MD and MV) and parabronchi, were measured at flow rates (V) varied from 0.5 to 3 L-min-1 and at CO2 concentrations of ventilating gas (FICO2) varied from 0 to 10%. 1) deltaP increased more than linearly with V. The resulting flow resistance R(= deltaP/V) averaged 43 and 95 cm H2O-L-1-sec at V = 0.5 and 3 L-min-1, respectively. 2) Step changes in FICO2 at constant V were followed within 0.5 to 5 sec by changes in R. 3) Lowering FICO2 from 5% resulted in marked increases in R, the value at FICO2 = 0% being more than twice the average value at 5%. Raising FICO2 from 5% up to 10% was followed by only slight changes in R. 4) Vagotomy did not consistently change R at any level of CO2; it did, however, slightly increase the delay time for changes in R on step changes of FICO2. 5)The medioventral secondary bronchi and their orifices into the primary bronchus appeared to be mainly responsible for the resistance measured and its changes with CO2. The resistance offered by the parabronchi appeared to be much smaller and much less dependent on CO2. The results suggest importance of lung gas CO2 in aerodynamic valving of respiratory flow in avian lungs during normal breathing and particularly during thermal panting to prevent alkalosis.

Observations on the sites of respiratory evaporation in the fowl during thermal panting

The rate of respiratory water loss (RWL) was investigated in domestic fowls by the open-flow method, and the relative importance of the surfaces of the upper and lower respiratory tract was assessed by cannulating the trachea and by recording the temperatures at the potential evaporating sites. Birds were exposed to Ta from 20 to 40 degrees C and RWL examined at rectal temperatures (Tre) from 41 to 44 degrees C. Overall, the increase in RWL from the whole tract, and from the upper and lower divisions, was by about 1.1, 1.0 and 0.3 mg (g-hr. degrees C)-1, respectively. There was a rapid increase in V and in RWL from the whole and from the upper tract at Tre 41.5-42.5 degrees C, but no comparable change from the lower tract. Temperatures significantly below Ta and Tre (both 43 degrees C) were detected in the trachea and in the nasal and buccal cavities, but not in the air sacs. It was concluded that respiratory evaporation occurs mainly from the upper tract during panting and that the air sacs are unlikely to be involved.

Oxygen and carbon dioxide dissociation of duck blood

Oxygen and CO2 dissociation of duck blood was studied in blood samples equilibrated with known gas mixtures at the bird's body temperature (41 degrees C) and analyzed in the Van Slyke manometric apparatus and in pH electrodes. At various pH values between 7.38 and 7.55 the Hill plots yielded straight and parallel lines over a wide range of O2 saturation, the Hill coefficient being 2.9. Half saturation pressure P50 at pH = 7.50 was 36 torr. The Bohr effect factor was -0.53. Buffering properties were analyzed by equilibrating blood samples with gas mixtures of different PCO2 at 41 degrees C. The buffer value for whole blood in the range of 3-7% CO2 was 19.3 mMol-L-1-pH-1, the buffer value for true plasma 22.9 mMol-L-1-pH-1. The CO2 dissociation curve constructed using the buffer values had a slope of 0.17 mMol-L-1-torr-1 in the PCO2 range from 40 to 50 torr. The CO2 content of oxygenated blood at PCO2 = 40 torr was 21.7 mMol-L-1. The Haldane effect factor at PCO2 = 35 torr equalled 0.30 mMol of combined CO2 per mMol HbO2. With the values of PO2, PCO2 and pH measured in arterial blood of undisturbed and unrestrained, resting ducks effective dissociation curves for both O2 and CO2 were constructed assuming a metabolic R.Q. of 0.8. These curves are expected to resemble closely the actual in vitro dissociation curves of resting ducks.