A model of the mechanics of airway narrowing

To examine the interaction between airway smooth muscle shortening and airway wall thickening on changes in pulmonary resistance, we have developed a model of the tracheobronchial tree that allows simulation of the mechanisms involved in airway narrowing. The model is based on the symmetrical dichotomous branching tracheobronchial tree as described by Weibel and uses fluid dynamic equations proposed by Pedley et al. to calculate inspiratory resistance during quiet tidal breathing. To allow for changes in lung volume, we used the airway pressure-area curves developed by Lambert et al. The model is easily implemented with a spreadsheet and personal computer that allows calculation of total and regional pulmonary resistance. At each airway generation in the model, provision is made for airway wall thickness, the maximal airway smooth muscle shortening achievable, and an S-shaped dose-response relationship to describe smooth muscle shortening. To test the validity of the model, we compared pressure-flow curves generated with the model with measurements of pulmonary resistance while normal subjects breathed air and 20% O2-80% He at a variety of lung volumes. By simulating progressive airway smooth muscle shortening, realistic pulmonary resistance vs. dose-response curves were produced. We conclude that this model provides realistic estimates of pulmonary resistance and shows potential for examining the various mechanisms that could produce excessive airway narrowing in disease.     

A theoretical analysis of interrupter technique for measuring respiratory mechanics

The application of the flow interrupter technique to series and parallel models of the respiratory system is examined theoretically, assuming instantaneous transmission of pressures and incompressible gases in the lung air spaces. The initial pressure change observed immediately after occlusion divided by the preocclusion flow gives an initial resistance (Rinit) equal to that of the airway tree when the model consists of compartments connected in parallel. When the compartments are connected in series, Rinit is the resistance of the most proximal airway only. In general, the initial pressure change is followed by a second slower change, reflecting equilibration of pressures between the compartments. The total postocclusion pressure change divided by the flow gives a steady-state resistance (Rss) whose value depends on the ventilation history before occlusion. When this history consists of a relaxed expiration Rss asymptotes from Rinit to a value higher than the zero-frequency resistance of the model as the expiratory time increases. However, the relative contributions of serial and parallel pendelluft and viscoelasticity to Rss cannot be determined from pressure and flow measurements made at the airway opening. Therefore in disease, the interrupter method does not permit one to say whether ventilation inhomogeneity or alteration in lung tissue properties is the predominant abnormality.     

Density dependence of respiratory system impedances between 5 and 320 Hz in humans

For respiratory system impedance (Zrs), the six-element model of DuBois et al. (J. Appl. Physiol. 8: 587-594, 1956) suggests three resonant frequencies (f1,f2,f3), where f1 is the result of the sum of tissue and airway inertances and tissue compliance and f2 is the result of alveolar gas compression compliance (Cg) and tissue inertance (Iti). Three such resonant frequencies have been reported in humans. However, the parameter estimates resulting from fitting this model to the data suggested that f2 and f3 were not associated with Cg and Iti but with airway acoustic properties. In the present study, we measured Zrs between 5 and 320 Hz in 10 healthy adult humans breathing room air or 80% He-20% O2 (HeO2) to gain insight as to whether airway or tissue properties are responsible for the f2 and f3. When the subjects breathed room air, f2 occurred at 170 +/- 16 (SD) Hz, and when they breathed HeO2 it occurred at 240 +/- 24 Hz. If this resonance were due to Cg and Iti it should not have been affected to this extent by the breathing of HeO2. We thus conclude that f2 is not due to tissue elements but that it is an airway acoustic resonance. Furthermore, application of the six-element model to analyze Zrs data at these frequencies is inappropriate, and models incorporating the airway acoustic properties should be used. One such model is based on the concept of equivalent length, which is defined as the length of an open-ended, cylindrical tube that has the same fundamental acoustic resonant frequency.(ABSTRACT TRUNCATED AT 250 WORDS)     

New perspectives on the mechanical basis for airway hyperreactivity and airway hypersensitivity in asthma.

We revisit the airway wall model of Lambert et. al. (Lambert RK, Wiggs BR, Kuwano K, Hogg JC, and Pare PD. J Appl Physiol 74: 2771-2781, 1993). We examine in detail the notion of a general airway bistability such that the airway lumen can suddenly decrease from a relatively open to a relatively closed condition without needing additional increase in active airway smooth muscle (ASM) tension during the stimulation. The onset of this bistability is an emergent consequence of the balance of forces associated with airway wall properties, parenchymal tissue properties, maximum lung elastic recoil, and the maximum stress that the ASM can generate. In healthy lungs, we find that all these properties reside in conditions that largely prevent the emergence of the bistability even during maximum ASM stimulation. In asthmatic airways, however, the airway wall and ASM remodeling conditions can tip the balance so as to promote the onset of the bistability at a lower dose of ASM stimulation (enhanced sensitivity) and then work to amplify the maximum constriction reached by each airway (enhanced reactivity). Hence, a larger fraction of asthmatic airways can display overall airway hyperreactivity. Simulations studies examine the role of increasing ASM maximum tension, airway wall stiffening, reduced lung volume, and decreased parenchymal tethering. Results predict that the single most important factor causing this airway hyperreactivity is amplified maximum ASM tension and not a thickening of the airway wall per se.     

Quasi-likelihood estimation for relative risk regression models

For a prospective randomized clinical trial with two groups, the relative risk can be used as a measure of treatment effect and is directly interpretable as the ratio of success probabilities in the new treatment group versus the placebo group. For a prospective study with many covariates and a binary outcome (success or failure), relative risk regression may be of interest. If we model the log of the success probability as a linear function of covariates, the regression coefficients are log-relative risks. However, using such a log-linear model with a Bernoulli likelihood can lead to convergence problems in the Newton-Raphson algorithm. This is likely to occur when the success probabilities are close to one. A constrained likelihood method proposed by Wacholder (1986, American Journal of Epidemiology 123, 174-184), also has convergence problems. We propose a quasi-likelihood method of moments technique in which we naively assume the Bernoulli outcome is Poisson, with the mean (success probability) following a log-linear model. We use the Poisson maximum likelihood equations to estimate the regression coefficients without constraints. Using method of moment ideas, one can show that the estimates using the Poisson likelihood will be consistent and asymptotically normal. We apply these methods to a double-blinded randomized trial in primary biliary cirrhosis of the liver (Markus et al., 1989, New England Journal of Medicine 320, 1709-1713).     

Examples of Mathematical Modeling: Tales from the Crypt

Mathematical modeling is being increasingly recognized within the biomedical sciences as an important tool that can aid the understanding of biological systems. The heavily regulated cell renewal cycle in the colonic crypt provides a good example of how modeling can be used to find out key features of the system kinetics, and help to explain both the breakdown of homeostasis and the initiation of tumorigenesis.We use the cell population model by Johnston et al. (2007) Proc. Natl. Acad. Sci. USA 104, 4008-4013, to illustrate the power of mathematical modeling by considering two key questions about the cell population dynamics in the colonic crypt. We ask: how can a model describe both homeostasis and unregulated growth in tumorigenesis; and to which parameters in the system is the model most sensitive? In order to address these questions, we discuss what type of modeling approach is most appropriate in the crypt.We use the model to argue why tumorigenesis is observed to occur in stages with long lag phases between periods of rapid growth, and we identify the key parameters.     

Prevention of abnormal pulmonary mechanics in the postmortem guinea pig lung

Severe postmortem bronchoconstriction has been shown previously in guinea pig lungs and linked to pulmonary blood loss during exsanguination (Lai et al., J. Appl. Physiol. 56: 308-314, 1984). To reexamine this phenomenon we measured postmortem airway function in anesthetized open-chest guinea pigs after sudden circulatory arrest. Animals were divided into 4 groups of 10 and ventilated for 15 min postmortem with different gases: 1) room air, 2) conditioned air, 3) dry 5% CO2-21% O2-74% N2, and 4) conditioned 5% CO2-21% O2-74% N2. In room air-ventilated lungs there was a 50% decrease in dynamic compliance (Cdyn) by 15 min and marked gas trapping compared with control lungs. Conditioning the room air did not attenuate these changes, but when 5% CO2 was added to the conditioned postmortem inspirate, gas trapping was eliminated and the fall in Cdyn was almost abolished. Ventilation with a dry 5% CO2 gas mixture at room temperature resulted in a 31% fall in Cdyn at 15 min but no gas trapping. We conclude that marked abnormalities of airway function occur postmortem in room air-ventilated guinea pig lungs in the absence of pulmonary blood loss. The changes are mainly due to airway hypocarbia, a known cause of bronchoconstriction, but a reduction in Cdyn can also occur if there is marked airway cooling and drying. Acute postmortem airway dysfunction can be prevented in the guinea pig by maintaining normal airway gas composition.     

The First Knockout Mouse Model of Retinoblastoma

The retinoblastoma susceptibility gene (RB1) was the first tumor suppressor gene identified in humans (Friend, et al., 1986) and the first tumor suppressor gene knocked out by targeted deletion in mice (Jacks, et al., Clarke, et al., Lee, et al., 1992). Children with a germline mutation in one of their RB1 alleles are likely to experience bilateral multifocal retinoblastoma; however, mice with a similar disruption of Rb1 do not develop retinoblastoma. The absence of a knock-out mouse model of retinoblastoma has slowed the progress toward developing new therapies and identifying secondary genetic lesions that occur after disruption of the Rb signaling pathway. Several advances have been made, over the past several years, in our understanding of the regulation of proliferation during retinal development (Zhang, et al., 2004; Dyer J, 2004; Dyer, Cepko, 2001) and we have built upon these earlier studies to generate the first nonchimeric knock-out mouse model of retinoblastoma. These mice are being used as a preclinical model to test new therapies for retinoblastoma and to elucidate the downstream genetic events that occur after inactivation of Rb1 or its related family members.     

Comparison of theoretical and experimental particle diffusion data within human airway casts

The dose delivered to airway cells is a critical factor whether one is addressing the therapeutic (i.e., positive) effects of inhaled pharmacologic agents or the toxic (i.e., negative) effects of pollutants. In this study, theoretical models describing particle deposition have been compared with experimental data from the literature. In the simulations, airways can be either roughor smooth-walled to be consistent with human lungs which can be either lined by cartilaginous rings (i.e., rough) or muscle (i.e., smooth). Particle motion for rough-walled airways within generations I=1–6 is calculated using the formula proposed by Martonen et al. (1). For smooth-walled airways within generations I=7–10, particle motion is calculated using the formula proposed by Martonen et al. (2). Theoretical predictions of particle deposition efficiencies are not only in agreement with the overall best fit empirical correlation presented by Cohen and Asgharian (3) over a wide range of dimensionless diffusion parameters, but also match individual experimental measurements (only available in I=1–6) with regard to effects of the parameters of particle size, flow rate, and airway dimensions. The mean difference in the ratio of experimental-to-theoretical particle diffusion values is 0.9 for a flow rate of 18 L/min and 1.1 for a flow rate of 34 L/min (i.e., the difference is only about 10%) within the upper airways of the casts (airway generations I=1–6), the mean difference for the whole casts was much greater. This may be attributed to the uncertainty of flow conditions in the peripheral airways as a result of the trimmmed nature of the casts. Overall, the findings suggest that the model can be a valuable component of aerosol therapy and risk assessment protocols, especially to address effects of enhanced deposition of pharmacologic drugs and radionuclides at sites within the human tracheobronchial tree.     

Mean airway pressure and mean alveolar pressure during high-frequency jet ventilation in rabbits

Mean airway pressure underestimates mean alveolar pressure during high-frequency oscillatory ventilation. We hypothesized that high inspiratory flows characteristic of high-frequency jet ventilation may generate greater inspiratory than expiratory pressure losses in the airways, thereby causing mean airway pressure to overestimate, rather than underestimate, mean alveolar pressure. To test this hypothesis, we ventilated anesthetized paralyzed rabbits with a jet ventilator at frequencies of 5, 10, and 15 Hz, constant inspiratory-to-expiratory time ratio of 0.5 and mean airway pressures of 5 and 10 cmH2O. We measured mean total airway pressure in the trachea with a modified Pitot probe, and we estimated mean alveolar pressure as the mean pressure corresponding in the static pressure-volume relationship to the mean volume of the respiratory system measured with a jacket plethysmograph. We found that mean airway pressure was similar to mean alveolar pressure at frequencies of 5 and 10 Hz but overestimated it by 1.1 and 1.4 cmH2O at mean airway pressures of 5 and 10 cmH2O, respectively, when frequency was increased to 15 Hz. We attribute this finding primarily to the combined effect of nonlinear pressure frictional losses in the airways and higher inspiratory than expiratory flows. Despite the nonlinearity of the pressure-flow relationship, inspiratory and expiratory net pressure losses decreased with respect to mean inspiratory and expiratory flows at the higher rates, suggesting rate dependence of flow distribution. Redistribution of tidal volume to a shunt airway compliance is thought to occur at high frequencies.(ABSTRACT TRUNCATED AT 250 WORDS)     

Multiple unloadings by nectar foragers in honey bees: a matter of information improvement or crop fullness?

Summary When a honey bee forager returns to her hive and unloads nectar, she sometimes transfers her entire load to one bee, but other times she makes a series of unloadings to several bees. One intriguing hypothesis for why foragers make multiple unloadings is the Information Improvement hypothesis: multiple unloadings improve a foragers estimate of the difficulty in finding a receiver bee, and thus of the allocation of labor between nectar collecting and nectar processing. In this paper, we discuss a possible weakness in the empirical evidence in support of the Information Improvement hypothesis. We also present a competing hypothesis, the Crop Fullness hypothesis: multiple unloadings arise from a mismatch between the amount of nectar a forager has to unload and the amount of nectar a receiver can imbibe. Finally, we test the two hypotheses by checking their predictions regarding the conditions under which multiple unloadings occur and which bee (forager or receiver) breaks off the first unloading when a forager makes multiple unloadings. We find that multiple unloadings are common only at times of high nectar influx and that most often it is the receiver, not the forager, who breaks off the first unloading. We argue that both of these findings are contradictory to the Information Improvement hypothesis but are consistent with the Crop Fullness hypothesis. Furthermore, we relate our findings to a recent theoretical study (Gregson et al., 2003) which shows, by means of a simulation model, that the level of multiple unloadings observed can be accounted for by a mismatch between the crop loads of foragers and the crop capacities of receiver bees. We combine our measurements with the Gregson et al. model to identify the rule used by receiver bees in deciding when to stop receiving more nectar. We conclude that receivers make this decision during the course of an unloading, not after completing an unloading. Finally, with this conclusion in hand, we test the Gregson et al. model by comparing predictions and observations on how full receivers are when they decide to break off an unloading. We find a remarkable agreement (prediction: 60%, observation: 52–59%), in strong support of the model.     

Calculation of the reflection coefficient from measurements of endogenous vascular indicators

The solvent drag reflection coefficient (sigma) for total proteins can be estimated by comparing the relative degrees of concentration of erythrocytes and plasma proteins that occur during fluid filtration in an isolated perfused organ. In this analysis, we evaluated the accuracy of equations proposed by Pilati and Maron [Am. J. Physiol. 247 (Heart Circ. Physiol. 16): H1-H7, 1984] and Wolf et al. [Am. J. Physiol. 253 (Heart Circ. Physiol. 22): H194-H204, 1987] to calculate sigma from these concentration changes. We calculated sigma with each equation using data generated from a mathematical model of fluid and solute flux in membranes with known sigma's. We found that the equation of Wolf et al. provided the closest approximation to the true sigma over the entire range of filtration fractions tested (0.1-0.6), with the differences between the two equations increasing with filtration fraction. At low filtration fractions, the difference in sigma obtained using either approach was found to be inconsequential. At larger filtration fractions, a closer approximation of the true sigma can be obtained using the equation of Wolf et al.     

Growth of a root system described as diffusion. II. Numerical model and application

In simulation models for water movement and nutrient transport, uptake of water and nutrients by roots forms an essential part. As roots are spatially distributed, prediction of root growth and root distribution is crucial for modelling water and nutrient uptake. In a preceding paper, De Willigen et al. (2002; Plant and Soil 240, 225–234) presented an analytical solution for describing root length density distribution as a diffusion-type process. In the current paper, we present a numerical model that does the same, but which is more flexible with respect to where root input can occur. We show that the diffusion-type root growth model can describe well observed rooting patterns. We used rooting patterns for different types of crops: maize, gladiolus, eastern white cedar, and tomato. For maize, we used data for two different types of fertiliser application: broadcast and row application. In case of row application, roots extend more vertically than horizontally with respect to the broadcast application situation. This is reflected in a larger ratio of diffusion coefficients in vertical versus horizontal direction. For tomato, we considered tomatoes grown on an artificial rooting medium, i.e. rockwool. We have shown that, in principle, the model can be extended by including reduction functions on the diffusion coefficient in order to account for environmental conditions.     

How Trp repressor binds to its operator.

We propose that the generally accepted model of a single Trp repressor dimer binding to a center of symmetry in the natural trp operator (Otwinowski et al., 1988) is wrong. We show here that the Trp repressor binds to a sequence whose center is located four base pairs either to the right or to the left of the central axis of symmetry that was previously identified. We show that: (i) the oligonucleotide used by Otwinowski et al. is not retarded by the Trp repressor in a mobility shift assay under conditions wherein a shorter oligonucleotide carrying our consensus sequence is retarded, (ii) that methylation protection experiments on the full natural operator sequence and the short oligonucleotide protect similar patterns and (iii) that by varying every base in the shorter oligonucleotide, we can demonstrate an optimal sequence for Trp repressor binding.     

Pressure, flow, and density relationships in airway models during constant-flow ventilation

Adequate CO2 elimination and normal arterial PCO2 levels can be maintained in dogs during apnea by delivering a continuous flow of inspired gas at high flow rate (1-3 l.min-1.kg-1) through tubes placed in the main-stem bronchi. However, during constant-flow ventilation (CFV) the mean alveolar pressure is increased, causing increased lung volume despite low pressures in the trachea. We hypothesized that the increased dynamic alveolar pressures during CFV were due to momentum transfer from the high-velocity jet stream to resident gas in the lung. To test this, we simulated CFV in straight tubes and in a branched airway model to determine whether changes in gas flow rate (V), gas density (rho), and tube diameter (D) altered the pressure difference (delta P) between alveoli and airway opening in a manner consistent with that predicted by conservation of momentum. Momentum analysis predicts that delta P should vary with V2, whereas measurements yielded a dependence of V1.69 in branched tubes and V1.9 in straight tubes. Substitution of heliox (80% He-20% O2) for air significantly reduced lung hyperinflation during CFV. As predicted by momentum transfer, delta P varied with rho 1.0. Momentum analysis also predicts that delta P should vary with D-2.0, whereas measurements indicated a dependence on D-2.02. The influence of V and rho on depth of penetration of the jet down the airway was explored in a straight tube model by varying the flow rate and gas used. The influence of geometry on penetration was measured by changing the ratio of jet-to-airway tube diameters.(ABSTRACT TRUNCATED AT 250 WORDS)     

Assessment of survival prediction models based on microarray data

MOTIVATION: In the process of developing risk prediction models, various steps of model building and model selection are involved. If this process is not adequately controlled, overfitting may result in serious overoptimism leading to potentially erroneous conclusions. METHODS: For right censored time-to-event data, we estimate the prediction error for assessing the performance of a risk prediction model (Gerds and Schumacher, 2006; Graf et al., 1999). Furthermore, resampling methods are used to detect overfitting and resulting overoptimism and to adjust the estimates of prediction error (Gerds and Schumacher, 2007). RESULTS: We show how and to what extent the methodology can be used in situations characterized by a large number of potential predictor variables where overfitting may be expected to be overwhelming. This is illustrated by estimating the prediction error of some recently proposed techniques for fitting a multivariate Cox regression model applied to the data of a prognostic study in patients with diffuse large-B-cell lymphoma (DLBCL). AVAILABILITY: Resampling-based estimation of prediction error curves is implemented in an R package called pec available from the authors.     

Traffic through the Golgi apparatus.

The role of vesicles in cargo transport through the Golgi apparatus has been controversial. Large forms of cargo such as protein aggregates are thought to progress through the Golgi stack by a process of cisternal maturation, balanced by a return flow of Golgi resident proteins in COPI-coated vesicles. However, whether this is the primary role of vesicles, or whether they also serve to transport small cargo molecules in a forward direction has been debated. Two papers (Martínez-Menárguez et al., 2001; Mironov et al., 2001, this issue) use sophisticated light and electron microscopy to provide evidence that the vesicular stomatitis virus membrane glycoprotein (VSV G)* is largely excluded from vesicles in vivo, and does not move between cisternae, whereas resident Golgi enzymes freely enter vesicles as predicted by the cisternal maturation model. Both papers conclude that vesicles are likely to play only a minor role in the anterograde transport of cargo through the Golgi apparatus in mammalian tissue culture cells.     

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.     

Flow oscillation-A measure to predict the surgery outcome for obstructed sleep apnea (OSA) subject

Obstructed sleep apnea (OSA) is a common disorder which may need surgery to widen the airway; however the success rate of surgery is limited. Here we report a finding that could be used to predict the outcome of the OSA surgery. We found that inspiratory flow oscillates due to flow separation near the larynx, and the resulting periodic signal (3-5Hz) is an intrinsic property of breathing. This flow oscillating signal may be the afferent stimulus to trigger respiratory events. It is found that the flow oscillation is attenuated for the OSA subjects. The computational fluid dynamics (CFD) simulation reveals that there exists flow separation near larynx and this separation is severely weakened in the OSA upper airway model. It is believed that the flow oscillating signal can serve as the measure to quantify the breathing quality of an OSA subject. This makes it possible to predict the surgery outcome of the OSA subject by applying CFD simulation.