Model simulation of heat and water transport dynamics in an airway

Heat and water transport processes in the respiratory tract depend on environmental conditions, breathing patterns, and the physiological state of the respiratory system. To study these processes, we have developed a mathematical model of the dynamics of temperature and water vapor in the radial and axial directions of an idealized trachea. The model is expressed as two implicit finite-difference equations and solved using an alternating-direction algorithm. Using these equations, we simulated the effects of inspired gas temperature and humidity, velocity profile, and flow rate on heat and water transport between the gas and airway wall. Under inspired gas conditions of low temperature or high relative humidity, supersaturation occurs. Increasing either the velocity gradient at the wall or the flow rate increases the heat and water transport rates. However, these rates change by only 10 percent when the velocity gradient is doubled, and by about 35 percent when flow rate undergoes a two-fold change. The model can be used with in-vivo data from the trachea to test hypotheses concerning normal and abnormal heat and water transport.     

A theoretical model of localized heat and water vapor transport in the human respiratory tract

A steady-state, one-dimensional theoretical model of human respiratory heat and water vapor transport is developed. Local mass transfer coefficients measured in a cast replica of the upper respiratory tract are incorporated into the model along with heat transfer coefficients determined from the Chilton-Colburn analogy and from data in the literature. The model agrees well with reported experimental measurements and predicts that the two most important parameters of the human air-conditioning process are: the blood temperature distribution along the airway walls, and the total cross-sectional area and perimeter of the nasal cavity. The model also shows that the larynx and pharynx can actually gain water over a respiratory cycle and are the regions of the respiratory tract most subject to drying. With slight modification, the model can be used to investigate respiratory heat and water vapor transport in high stress environments, pollutant gas uptake in the respiratory tract, and the connection between respiratory air-conditioning and the function of the mucociliary escalator.     

Indirect assessment of mucosal surface temperatures in the airways: theory and tests

We developed and tested a method, based on conduction heat transfer analysis, to infer airway mucosal temperatures from airstream temperature-time profiles during breath-hold maneuvers. The method assumes that radial conduction of heat from the mucosal wall to inspired air dominates heat exchange during a breath-hold maneuver and uses a simplified conservation of energy analysis to extrapolate wall temperatures from air temperature vs. time profiles. Validation studies were performed by simultaneously measuring air and wall temperatures by use of a retractable basket probe in the upper airways of human volunteers and intrathoracic airways of paralyzed intubated dogs during breath holding. In both protocols, a good correlation was demonstrated between directly measured wall temperatures and those calculated from adjacent airstream temperature vs. time profiles during a breath hold. We then calculated intrathoracic bronchial wall temperatures from breath-hold airstream temperature-time profiles recorded in normal human subjects after cold air hyperpnea at 30 and 80 l/min. The calculations show airway wall temperatures in the upper intrathoracic airways that are below core body temperature during hyperpnea of frigid air and upper thoracic airways that are cooler than more peripheral airways. The data suggest that the magnitude of local intrathoracic heat/water flux is not represented by heat/water loss measurements at the airway opening. Both the magnitude and locus of heat transport during cold gas hyperventilation vary with changes in inspired gas temperature and minute ventilation; both may be important determinants of airway responses.     

Longitudinal distribution of canine respiratory heat and water exchanges

We assessed the longitudinal distribution of intra-airway heat and water exchanges and their effects on airway wall temperature by directly measuring respiratory fluctuations in airstream temperature and humidity, as well as airway wall temperature, at multiple sites along the airways of endotracheally intubated dogs. By comparing these axial thermal and water profiles, we have demonstrated that increasing minute ventilation of cold or warm dry air leads to 1) further penetration of unconditioned air into the lung, 2) a shift of the principal site of total respiratory heat loss from the trachea to the bronchi, and 3) alteration of the relative contributions of conductive and evaporative heat losses to local total (conductive plus evaporative) heat loss. These changes were not accurately reflected in global measurements of respiratory heat and water exchange made at the free end of the endotracheal tube. Raising the temperature of inspired dry air from frigid to near body temperature principally altered the mechanism of airway cooling but did not influence airway mucosal temperature substantially. When local heat loss was increased from both trachea and bronchi (by increasing minute ventilation), only the tracheal mucosal temperature fell appreciably (up to 4.0 degrees C), even though the rise in heat loss from the bronchi about doubled that in the trachea. Thus it appears that the bronchi are better able to resist changes in airway wall temperature than is the trachea. These data indicate that the sites, magnitudes, and mechanisms of respiratory heat loss vary appreciably with breathing pattern and inspired gas temperature and that these changes cannot be predicted from measurements made at the mouth. In addition, they demonstrate that local heat (and presumably, water) sources that replenish mucosal heat and water lost to the airstream are important in determining the degree of local airway cooling (and presumably, drying).     

Heat and water rate transfer processes in the human respiratory tract at various altitudes

The process of the respiratory air conditioning as a process of heat and mass exchange at the interface inspired air-airways surface was studied. Using a model of airways (Olson et al., 1970) where the segments of the respiratory tract are like cylinders with a fixed length and diameter, the corresponding heat transfer equations, in the paper are founded basic rate exchange parameters-convective heat transfer coefficient h(c)(W m(-2) degrees C(-1)) and evaporative heat transfer coefficient h(e)(W m(-2)hPa(-1)). The rate transfer parameters assumed as sources with known heat power are connected to airflow rate in different airways segments. Relationships expressing warming rate of inspired air due to convection, warming rate of inspired air due to evaporation, water diffused in the inspired air from the airways wall, i.e. a system of air conditioning parameters, was composed. The altitude dynamics of the relations is studied. Every rate conditioning parameter is an increasing function of altitude. The process of diffusion in the peripheral bronchial generations as a basic transfer process is analysed. The following phenomenon is in effect: the diffusion coefficient increases with altitude and causes a compensation of simultaneous decreasing of O(2)and CO(2)densities in atmospheric air. Due to this compensation, the diffusion in the peripheral generations with altitude is approximately constant. The elements of the human anatomy optimality as well as the established dynamics are discussed and assumed. The square form of the airways after the trachea expressed in terms of transfer supposes (in view of maximum contact surface), that a maximum heat and water exchange is achieved, i.e. high degree of air condition at fixed environmental parameters and respiration regime.     

Finite difference analysis of respiratory heat transfer

A numerical computer model of heat and water transfer within the tracheobronchial tree of humans was developed based on an integral formulation of the first law of thermodynamics. Simulation results were compared with directly measured intraluminal airway temperature profiles previously obtained in normal human subjects, and a good correlation was demonstrated. The model was used to study aspects of regional pulmonary heat transfer and to predict the outcomes of experiments not yet performed. The results of these simulations show that a decrease in inspired air temperature and water content at fixed minute ventilation produces a proportionately larger increase in heat loss from extrathoracic airways relative to intrathoracic, whereas an increase in minute ventilation at fixed inspired air conditions produces the opposite pattern, with cold dry air penetrating further into the lung, and that changes in breathing pattern (tidal volume and frequency) at fixed minute ventilation and fixed inspiratory-to-expiratory (I/E) ratio do not affect local air temperature profiles and heat loss, whereas changes in I/E ratio at fixed minute ventilation do cause a significant change.     

A method for the approximation of the relative humidity in the upper human airways

In order to determine the growth of inhaled aerosol particles in the human respiratory tract the relative humidity in a lung model has been calculated using a numerical method. The computations take into account different types of airflows, enhanced transport mechanisms and an optimized wall temperature profile in the upper airways. These parameters are varied to fit experimental temperature data. Under certain conditions the corresponding relative humidity shows a maximum near the first bifurcation, which exceeds the final humidity in the alveoli. This high humidity forces dry NaC1 particles with diameters less than 0.5 μm to grow to their maximum size in the first bronchi. Thereafter the droplets loose water and reach their final size in the terminal bronchioles.     

A Model of Respiratory Heat Transfer in a Small Mammal

A steady-state model of the heat and water transfer occurring in the upper respiratory tract of the kangaroo rat, Dipodomys spectabilis, is developed and tested. The model is described by a steady-state energy balance equation in which the rate of energy transfer from a liquid stream (representing the flow of heat and blood from the body core to the nasal region) is equated with the rate of energy transfer by thermal conduction from the nose tip to the environment. All of the variables in the equation except the flow rate of the liquid stream can be either measured directly or estimated from physiological measurements, permitting the solution of the equation for the liquid stream flow rate. After solving for the liquid stream flow rate by using data from three animals, the energy balance equation is used to compute values of energy transfer, expired air temperature, rates of water loss, and efficiency of vapor recovery for a variety of ambient conditions. These computed values are compared with values measured or estimated from physiological measurements on the same three animals, and the equation is thus shown to be internally consistent. To evaluate the model's predictive value, calculated expired air temperatures are compared with measured expired air temperatures of eight additional animals. Finally, the model is used to examine the general dependence of expired air temperature, of rates of water loss, and of efficiency of vapor recovery on ambient conditions.     

Desaturation of exhaled air in camels

We have found that camels can reduce the water loss due to evaporation from the respiratory tract in two ways: (1) by decreasing the temperature of the exhaled air and (2) by removal of water vapour from this air, resulting in the exhalation of air at less than 100% relative humidity (r.h.). Camels were kept under desert conditions and deprived of drinking water. In the daytime the exhaled air was at or near body core temperature, while in the cooler night exhaled air wat at or near ambient air temperature. In the daytime the exhaled air was fully saturated, but at night its humidity might fall to approximately 75% r.h. The combination of cooling and desaturation can provide a saving of water of 60% relative to exhalation of saturated air at body temperature. The mechanism responsible for cooling of the exhaled air is a simple heat exchange between the respiratory air and the surfaces of the nasal passageways. On inhalation these surfaces are cooled by the air passing over them, and on exhalation heat from the exhaled air is given off to these cooler surfaces. The mechanism responsible for desaturation of the air appears to depend on the hygroscopic properties of the nasal surfaces when the camel is dehydrated. The surfaces give off water vapour during inhalation and take up water from the respiratory air during exhalation. We have used a simple mechanical model to demonstrate the effectiveness of this mechanism.     

Measurement of local mass transfer coefficients in a cast model of the human upper respiratory tract

Local mass transfer coefficients measured using the naphthalene sublimation technique in an acrylic cast model of the human upper respiratory tract are reported as the Sherwood numbers for the corresponding regions. A steady air flow rate of 12 L per min was used for all measurements. Values of the Sherwood number are seen to be highest in the nasal cavity and proximal nasopharynx while a minimum value occurs just downstream from the larynx. Local values of the Nusselt number obtained in the trachea and proximal nasal cavity assuming a complete heat and mass transfer analogy agree well with in-vivo physiological measurements. The mass transfer coefficients found can be incorporated into an analytical model of respiratory heat and water vapor transfer or into a model of pollutant gas uptake in the respiratory tract.     

Radial heat and water transport across the airway wall

The thermal profiles in the airways of healthy human volunteers and patients with asthma differ after cessation of hyperpnea. The asthmatic patients rewarm their airways more rapidly. To identify thermal properties and processes that could account for the difference between these populations, we developed a model describing the radial transport of heat and water across the trachea. A distinctive feature of the model is a variable parameter describing blood supply to the mucosal and submucosal layers. Simulations performed with the model are initiated by a breath-hold maneuver and are propagative in time. Blood perfusion rates in the airway wall, the thickness of the layer of airway surface liquid, and the mucosa-submucosa thickness, all thought to be more pronounced in asthmatic patients, were varied by changing model parameters and initial conditions. Increasing the thickness of the liquid layer by more than an order of magnitude had little effect on the temperature or water content in the airway lumen. Doubling the blood flow to the mucosa-submucosa resulted in a slight increase in airway temperature. When this effect was coupled, however, with an increase in the thickness of the mucosa-submucosa layer, the increase in temperature was more pronounced. Because the bronchial circulation is the major source of heat to the airway, these results indicate that differences in airway wall thickness coupled with differences in the magnitude or responsiveness of the bronchial microcirculation could account for the differences in intra-airway temperature between the two populations.     

Respiratory heat loss of Holstein cows in a tropical environment

In order to develop statistical models to predict respiratory heat loss in dairy cattle using simple physiological and environmental measurements, 15 Holstein cows were observed under field conditions in a tropical environment, in which the air temperature reached up to 40°C. The measurements of latent and sensible heat loss from the respiratory tract of the animals were made by using a respiratory mask. The results showed that under air temperatures between 10 and 35°C sensible heat loss by convection decreased from 8.24 to 1.09 W m^–2, while the latent heat loss by evaporation increased from 1.03 to 56.51 W m^–2. The evaporation increased together with the air temperature in almost a linear fashion until 20°C, but it became increasingly high as the air temperature rose above 25°C. Convection was a mechanism of minor importance for respiratory heat transfer. In contrast, respiratory evaporation was an effective means of thermoregulation for Holsteins in a hot environment. Mathematical models were developed to predict both the sensible and latent heat loss from the respiratory tract in Holstein cows under field conditions, based on measurements of the ambient temperature, and other models were developed to predict respiration rate, tidal volume, mass flow rate and expired air temperature as functions of the ambient temperature and other variables.This paper forms part of A. S. Campos Maias doctoral thesis.     

A mathematical model of the dynamic heat transfer from the respiratory tract of a chicken

A mathematical model of the dynamic (periodic) heat exchange from the respiratory tract of a chicken is postulated and solved analytically. The model expresses the periodic respiratory heat loss as a function of respiration rate, respiratory air velocity, ambient temperature and humidity ratio, and body (trachea) temperature. It is unique in that previous models have been formulated for steady state heat transfer. The processes of sensible and latent heat exchange are considered as uncoupled processes.     

Modeling of unstable heat-mass air exchange in the lungs

A mathematical model of the unsteady-state heat and mass exchange of expired air in the bronchial tree is suggested. The model includes heat and mass exchange between air and bronchial walls, and heat exchange between blood circulation and bronchial tree. A problem has been numerically solved as a unidimensional one in the quasi-steady-state formulation. It is shown that air conditioning occurs through the whole length of a respiratory tract. During inspiration bronchial walls are cooled, that in its turn induces a decrease of air temperature and water vapour content in time. That process depends on the intensity of lung blood circulation and character of air velocity changes during inspiration.     

Simulation of a steady-state mass-heat exchange in the respiratory tract

On the basis of Weibel respiratory tract model the mathematical model of mass and heat transfer in the lungs was solved for steady-state one-dimensional case. Coefficients of mass and heat transfer were taken from empirical expressions for canals. The model shows that independent water vapour or air heat saturation in the lungs occurs in 12-14 generations of the bronchial tree. The saturation site depends upon volume velocity of the air and functioning of the upper respiratory tract.     

Respiratory heat loss in the sheep: a comprehensive model

A model is presented for the respiratory heat loss in sheep, considering both the sensible heat lost by convection ( C(R)) and the latent heat eliminated by evaporation ( E(R)). A practical method is described for the estimation of the tidal volume as a function of the respiratory rate. Equations for C(R) and E(R) are developed and the relative importance of both heat transfer mechanisms is discussed. At air temperatures up to 30 degrees C sheep have the least respiratory heat loss at air vapour pressures above 1.6 kPa. At an ambient temperature of 40 degrees C respiratory loss of sensible heat can be nil; for higher temperatures the transfer by convection is negative and thus heat is gained. Convection is a mechanism of minor importance for the respiratory heat transfer in sheep at environmental temperatures above 30 degrees C. These observations show the importance of respiratory latent heat loss for thermoregulation of sheep in hot climates.     

Computation of respiratory heat exchanges

Two sets of equations have been proposed to estimate the convective or sensible (WCV) and the evaporative or insensible (WEV) respiratory heat exchanges. They are applicable both at sea-level barometric pressure with air breathing (SLA) and in hypo- or hyperbaria in both air (HA) and artificial (HAA) atmospheres. The only environmental parameters required are in SLA: the temperature (TI) and the partial pressure of water vapor of the inspired air (PIH2O); in HA: TI, PIH2O, and the actual barometric pressure (PB); and in HAA: IT, PIH2O, PB, the volumetric mass (rho mix), and the specific heat (cp mix) of the breathed gas mixture. When no physiological data are available the results are expressed in energy units per liter of pulmonary ventilation (WCV/V and WEV/V) in J X dm-3 BTPS. If the ventilation value (V) is known the results are obtained in units of power (W).     

Dynamic BTPS correction factors for spirometric data

Because it is often difficult to completely control ambient temperature, a study was conducted to investigate dynamic body temperature pressure saturated (BTPS) correction factors for spirometric data. A forced expiratory simulator system was heated to 37 degrees C and loaded with air saturated with water vapor. This air was then forced from the simulator into a dry rolling-seal spirometer maintained at various ambient temperatures from 3 to 32 degrees C. Errors in forced expiratory volume in 1 s (FEV1) and peak flow from assuming a constant BTPS correction ranged from 7.7 and 14.1% at 3 degrees C to 2.1 and 4.6% at 23 degrees C. Differences between errors observed when saturated and dry air were forced into the spirometer indicate that water vapor condensation introduces an added heat load to the spirometer, adding approximately one percent to the error in FEV1 at lower temperatures. By use of a model to estimate the dynamic BTPS correction factor, errors in FEV1 at all temperatures between 3 and 32 degrees C were reduced to less than 1.5%.     

A Simple Dynamic Model of Respiratory Pump

To study the interaction of forces that produce chest wall motion, we propose a model based on the lever system of Hillman and Finucane (J Appl Physiol 63(3):951–961, 1987) and introduce some dynamic properties of the respiratory system. The passive elements (rib cage and abdomen) are considered as elastic compartments linked to the open air via a resistive tube, an image of airways. The respiratory muscles (active) force is applied to both compartments. Parameters of the model are identified in using experimental data of airflow signal measured by pneumotachography and rib cage and abdomen signals measured by respiratory inductive plethysmography on eleven healthy volunteers in five conditions: at rest and with four level of added loads. A breath by breath analysis showed, whatever the individual and the condition are, that there are several breaths on which the airflow simulated by our model is well fitted to the airflow measured by pneumotachography as estimated by a determination coefficient R ² ≥ 0.70. This very simple model may well represent the behaviour of the chest wall and thus may be useful to interpret the relative motion of rib cage and abdomen during quiet breathing.