Mathematical Modeling of Respiratory System: A Review
Content
Indian Journal of Biomechanics: Special Issue (NCBM 7-8 March 2009)
Mathematical Modeling of Respiratory System: A Review
Devdatta, V.K.Katiyar, Pratibha Department of Mathematics, IIT Roorkee, India, 247667. Abstract Respiration is the transport of Oxygen from the outside air to the cells within tissues and the transport of carbon dioxide in the opposite direction. Respiratory mechanics represent a complex system, yet their understanding is essential for the diagnosis, prognosis, and treatment of various respiratory disorders. This study is focused on the review of various mathematical models for respiratory mechanics, which provides the basis for the most clinically applied methods of respiratory mechanics analysis. 1. Introduction Breathing consists of two phases, inspiration and expiration. During inspiration, the diaphragm and the intercostals muscles contract. During expiration the diaphragm and muscles relax. When a breath is taken, air passes in through the nostrils, through the nasal passages, into the pharynx, through the larynx, down the trachea, into one of the main bronchi, then into smaller bronchial tubules, through even smaller bronchioles, and into a microscopic air sac called an alveolus. It is here that external respiration occurs. Humans need a continuous supply of oxygen for cellular respiration, and they must get rid of excess carbon dioxide, the poisonous waste product of this process. Gas exchange supports the cellular respiration by constantly supplying oxygen and removing carbon dioxide. The oxygen we need is derived from the Earth's atmosphere, which is 21% oxygen. This oxygen in the air is exchanged in the body by the respiratory surface. In humans the alveoli in the lungs serve as the surface for gas exchange. 2. Review One of an important topic to human health is the control of the cardiovascular and respiratory system. The knowledge of this control mechanism will be very helpful for improving diagnostics and treatment of cardio-respiratory diseases. A number of dynamic models of the human cardio-respiratory system have been proposed since the 1950s. Most of them arise from the compartmental theory [F. Kappel and R.O. Peer (1993), F. Kappel et. al. (1997), S. Timischl-Teschl (2004)]. These models consist in solving control optimal problems of nonlinear differential equations with cumbersome terms, leading to unstable solutions. The equation of motion (EOM) is the commonly accepted mathematical model of the respiratory system which provides the basis for the most clinically applied methods of respiratory mechanics analysis. S. Ganzert et. al. (2007) derives the equations for model identification in respiratory mechanics under conditions of mechanical ventilation. This was the first application of an equation discovery technique to measured respiratory data from intensive care medicine. S.Sepehari (2007) developed physical based model of human respiration. He modeled the slow deep breathing by tunnel diode oscillator. 56
Nonlinearity of respiratory mechanics is a well known phenomenon, e.g. observable in the sigmoid shape of the static pressure-volume curve. In the dynamic situation the tidal volume is small compared to inflated volume in the static pressure-volume curve and intra tidal nonlinearity of respiratory mechanics was in the past a matter more of scientific than of clinical interest. This has changed with the upcoming of an increasing number of methods to analyze respiratory mechanics under the dynamic conditions of uninterrupted i.e. maneuver free mechanical ventilation. The mechanical properties of the respiratory system can be schematized in terms of resistance, compliance, and iterance. The latter is generally considered less important for the overall mechanical behavior at normal breathing frequencies. As air passes through the different airway generations, the overall cross-sectional area of all airways increases. Air flow can be considered essentially laminar in the small airways, while it is both laminar and turbulent in the upper airways. The total flow resistance is determined by the friction of gas flowing into the airways and the friction due to movement of tissues of the lung and of the chest wall. Since pressure applied to the airways is first transmitted to the lung and then to the chest wall, these elements are considered in series and thus additive, even if the chest wall contribution to total respiratory system resistance is considered relatively modest. Elastance (the reciprocal of compliance) is a measure of respiratory system distensibility. Being the lung and the chest in series, overall elastance is the sum of lung and chest wall elastance. H. Zhu et. al. (2007) demonstrates that predictive neural network may prove to be valuable as a tool in automatic mechanical ventilation support. In his study a method suitable for an adaptive continuous control of nonlinear pulmonary mechanics has been proposed. The method is based on the application of a neural network predictive control approach that incorporates the nonlinear mechanics of an individual patient. Assessment of the mechanical properties of the lung is important for the management of mechanically ventilated patients with acute respiratory distress syndrome (ARDS), and can be achieved through continuous measurement of pressure and flow at the entrance to the end tracheal tube and pressure in the esophagus. Mathematical models are fit to the measured signals, and the estimated values of the model parameters taken as indices of corresponding physiological quantities. The mechanical status of the respiratory system in incubated mechanically ventilated patients is conveniently assessed from the dynamic relationships between the pressure and flow of gas measured at the entrance to the endotracheal tube by J.H.T. Bates and P. Goldberg (1999). A model of respiratory mechanics was fit to the data, and the parameters of the model are then taken to be measures of important physiological quantities. They showed that the respiratory system of the ARDS patient during normal mechanical ventilation exhibited significant nonlinear mechanical behavior that indicated a nonlinear dynamic pressure-volume relationship and non-linear behavior of the respiratory system in such patients may be important for protecting against possible barotrauma caused by increases in tidal volume. Due to the mechanical properties and mutual coupling of the lung and thorax, respiratory mechanics represent such a complex system, yet their understanding is essential for the diagnosis, prognosis, and treatment of various respiratory disorders. M. Kuebler (2007) developed a two-component simulation model for respiratory mechanics. A comprehensive understanding of respiratory mechanics is pivotal for the accurate 57
diagnosis and treatment of lung disease, for adequate artificial or assisted ventilation, and for the analysis of environmental effects, e.g., in diving or under hyperbaric conditions on lung mechanics and function. Yet, understanding of lung mechanics is difficult due to the complex and dynamic nature of pressure-volume relationships of the respiratory system. Basically, two types of control systems exist in the human body. These are (a) servo systems and (b) regulators. A disturbance, such as a step input of CO2 inhalation, will cause this system to respond in a much different way. Brain tissue Pco2 will rise, but respiration will also increase to "blow off' CO2 so that the new steady-state value reached is much lower than it would have been if respiration had not increased. The respiratory control system can also, under proper conditions, exhibit damped and sustained oscillation. H.T.Milhorn et. al. (1965) investigated the human respiratory control system by mathematical modeling. They derive the basic equations for respiratory control in the human being and to obtain transient and steady-state solutions for both positive and negative step input disturbances of inspired CO2 and O2 concentrations. An insight into the importance of this type of analysis is given by the study of the effectiveness of the respiratory system as a regulator. Its purpose has been, first, to derive the basic equations of the system and, secondly, to investigate the system as a biological regulator. They assumed that the system consists of three reservoirs (the lungs, the brain tissues, and the body tissues). Blood flow to the brain is dependent upon cerebral-arterial Pco2 and Po2 which controls alveolar ventilation. F.T Tehrani (1997) developed a mathematical model for the respiratory system of infants at different stages of maturity; the effects of respiratory parameters on the infant respiratory response have been determined by a simulation model. The analysis of infant respiratory system is helpful in early diagnosis of sudden infant death syndrome. The contributions of different mechanisms to particle deposition in local airway segments vary with effective particle size, density, and local airflow rate and gravity angle. In 56regions, where the Reynolds numbers are low. Computational fluid–particle dynamics (CFPD) simulations for gravitational deposition started in the 1990s, focusing on small bronchial airways and alveolar ducts or sacs. For example, W. Hofmann et. al. (1995) simulated the gravitational settling of 10 micrometer particles in an asymmetric, single bifurcating airway model representing generations 15 and 16. It was found that the gravity angle is significant for determining particle distributions and localized doses. Using a 2-D symmetric six-generation model, C. Darquenne and G.K.Prisk (2003) stressed that gravity is important in the deposition of 0.5 and 1 micrometer particles in the human acinus. S. Haber et. al. (2003) incorporated gravitational sedimentation with wall movements, employing a 3-D hemispherical alveolus model. Recently, L. Harrington et. al. (2006) simulated trajectories of 1–5 micrometer particles in 3-D alveolated ducts representing generations 18–22 with different gravity angles. They concluded that the total deposition can be a function of the gravity angle and the ratio of the terminal settling velocity to mean lumen flow velocity. In summary, most of the computational analyses for gravitational deposition focused on the alveolar region where sedimentation may play a dominant role. The effects of gravity on the aerosol deposition in the bronchial airways, especially in the medium-size airways where sedimentation and impaction may occur simultaneously, have been thoroughly investigated by C. Kleinstreuera et. al.(2007). 58
The expiratory flow pattern during tidal breathing is the result of the continuing inspiratory muscle activity during the first part of expiration, expiratory muscle activity and the mechanical properties of the respiratory system. D. Walraven et. al. (2003) developed a mathematical model based on physiological properties to describe the entire expiratory flow pattern in spontaneously breathing, anesthetized cats. Such a model requires besides modeling of the mechanical properties of the respiratory system at least a description of the continuing inspiratory muscle activity in early expiration. As a result, the model fit to the data provides estimates for the relevant respiratory parameters. Expiratory activity was not since it was usually absent or only present at the end of expiration, which part was then excluded from the model fit. It is good modeling practice not to use a more complicated model than needed to give a good fit of the measured data, and it turned out in that the simplest mechanical model of the respiratory system could be used for that purpose. This is the one compartment model characterized by the time constant of the respiratory system (TRS), which is the product of resistance (RRS) and compliance (CRS) of the respiratory system. The continuing inspiratory muscle activity was described by a sigmoid function characterized by another time constant. The derivation of all equations used is explained in the methods section. In this study the model provides an accurate description of the expiratory flow profile in anesthetized tracheostomized cats during tidal breathing and that the parameter estimates obtained for TRS and Tmus are comparable with values obtained using different methods. Jean Marie Ntaganda et al (2007) design a mathematical model for determining blood pressures response to cardiac and respiratory parameters. Reference 1. C. Kleinstreuer, Z. Zhang, C. Kim, “Combined inertial and gravitational deposition of micropartical in small model airways of a human respiratory system” Aerosol Science 38 ,1047-1061, 2007. 2. C. Darquenne and G. K. Prisk, “Effect of gravitational sedimentation on simulated aerosol dispersion in the human acinus”, Journal of Aerosol Science, 34, 405–418, 2003. 3. D. Walraven, C.P.M Vander Grinten, J.N.Bogard, C.K Vander Ent, S.C.N Luijendijk, “Modeling of expiratory flow pattern of spontaneously breathing”, Respiratory Physiology and Neurobiology 134,23-32, 2003. 4. F. Kappel, R.O. Peer, A mathematical model for fundamental regulation processes in the cardiovascular model, J. Math. Biol. 31 (6), 611–631, 1993. 5. F. Kappel, S. Lafer, R.O. Peer, “A model for the cardiovascular system under an ergometric workload”, Surv. Math. Ind. 7, 239–250, 1997. 6. F. T. Tehrani, “A Model Study of periodic breathing stability of neonatal respiratory system and causes of sudden infant death syndrome”, Med. Eng. Phys, 19,547-555, 1997. 7. H. T. Milhorn,Jr., R. Benton , R. Ross and A. Guytom., “A mathematical model of the human respiratory control system”, , Journal Biophysical 5, 27-46, 1965. 8. H. Zhu, J. Guttmann, K. Moller, “Control of respiratory mechanics with artificial neural networks”, IEEE, 1202-1205, 2007. 9. J. H.T Bates and P. Goldberg, “Fitting non linear time domain models of respiratory 59
mechanics to pressure flow data from an intubated patient”, Serving Humanity Advancing Technology Medical Research of Canada, 1999. 10. J. M. Ntaganda , B. Mamapassi, D. Seck, “Modeling blood partial pressure of the human respiratory system”, Applied Mathematics and Computation, 187, 1100-1108, 2007, 11. L. Harrington, G. K.Prisk and C.Darquenne, “Importance of the bifurcation zone and branch orientation in simulated aerosol deposition in the alveolar zone of the human lung”, Journal of Aerosol Science, 37(1), 37–62, 2006. 12. M. Kuebler,M. Mertens and R. Axel, A two component simulation model to teach respiratory mechanics”, Advance Physiology Education 31,218-222, 2007. 13. S. Ganzert, K. Moller, Kristian, L. D. Readt and J. Guttmann, “Equation discovery for model identification in respiratory mechanics under condition of mechanical ventilation”, ICML07 USA, 24June 2007. 14. S. Sepehris, “Physical model of human respiration”, Young Researchers club, Islamic Azad university of Shiraz. 12-17 ,2007 15. S. Haber, D.Yitzhak and A. Tsuda, “Gravitational deposition in a rhythmically expanding and contracting alveolus”. Journal of Applied Physiology, 95, 657–671, 2003. 16. S. Timischl-Teschl, “Modeling the human\ cardiovascular-respiratory control system: An optimal control application to the transition to non-REM sleep”, Math. Biosci. Eng., 7 (7), 2004. 17. W. Hofmann, I. Balásházy and L.Koblinger, “The effect of gravity on particle deposition patterns in bronchial airway bifurcations”. Journal of Aerosol Science, 26, 1161–1168, 1995.
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Mathematical Modeling of Respiratory System: A Review
Devdatta, V.K.Katiyar, Pratibha Department of Mathematics, IIT Roorkee, India, 247667. Abstract Respiration is the transport of Oxygen from the outside air to the cells within tissues and the transport of carbon dioxide in the opposite direction. Respiratory mechanics represent a complex system, yet their understanding is essential for the diagnosis, prognosis, and treatment of various respiratory disorders. This study is focused on the review of various mathematical models for respiratory mechanics, which provides the basis for the most clinically applied methods of respiratory mechanics analysis. 1. Introduction Breathing consists of two phases, inspiration and expiration. During inspiration, the diaphragm and the intercostals muscles contract. During expiration the diaphragm and muscles relax. When a breath is taken, air passes in through the nostrils, through the nasal passages, into the pharynx, through the larynx, down the trachea, into one of the main bronchi, then into smaller bronchial tubules, through even smaller bronchioles, and into a microscopic air sac called an alveolus. It is here that external respiration occurs. Humans need a continuous supply of oxygen for cellular respiration, and they must get rid of excess carbon dioxide, the poisonous waste product of this process. Gas exchange supports the cellular respiration by constantly supplying oxygen and removing carbon dioxide. The oxygen we need is derived from the Earth's atmosphere, which is 21% oxygen. This oxygen in the air is exchanged in the body by the respiratory surface. In humans the alveoli in the lungs serve as the surface for gas exchange. 2. Review One of an important topic to human health is the control of the cardiovascular and respiratory system. The knowledge of this control mechanism will be very helpful for improving diagnostics and treatment of cardio-respiratory diseases. A number of dynamic models of the human cardio-respiratory system have been proposed since the 1950s. Most of them arise from the compartmental theory [F. Kappel and R.O. Peer (1993), F. Kappel et. al. (1997), S. Timischl-Teschl (2004)]. These models consist in solving control optimal problems of nonlinear differential equations with cumbersome terms, leading to unstable solutions. The equation of motion (EOM) is the commonly accepted mathematical model of the respiratory system which provides the basis for the most clinically applied methods of respiratory mechanics analysis. S. Ganzert et. al. (2007) derives the equations for model identification in respiratory mechanics under conditions of mechanical ventilation. This was the first application of an equation discovery technique to measured respiratory data from intensive care medicine. S.Sepehari (2007) developed physical based model of human respiration. He modeled the slow deep breathing by tunnel diode oscillator. 56
Nonlinearity of respiratory mechanics is a well known phenomenon, e.g. observable in the sigmoid shape of the static pressure-volume curve. In the dynamic situation the tidal volume is small compared to inflated volume in the static pressure-volume curve and intra tidal nonlinearity of respiratory mechanics was in the past a matter more of scientific than of clinical interest. This has changed with the upcoming of an increasing number of methods to analyze respiratory mechanics under the dynamic conditions of uninterrupted i.e. maneuver free mechanical ventilation. The mechanical properties of the respiratory system can be schematized in terms of resistance, compliance, and iterance. The latter is generally considered less important for the overall mechanical behavior at normal breathing frequencies. As air passes through the different airway generations, the overall cross-sectional area of all airways increases. Air flow can be considered essentially laminar in the small airways, while it is both laminar and turbulent in the upper airways. The total flow resistance is determined by the friction of gas flowing into the airways and the friction due to movement of tissues of the lung and of the chest wall. Since pressure applied to the airways is first transmitted to the lung and then to the chest wall, these elements are considered in series and thus additive, even if the chest wall contribution to total respiratory system resistance is considered relatively modest. Elastance (the reciprocal of compliance) is a measure of respiratory system distensibility. Being the lung and the chest in series, overall elastance is the sum of lung and chest wall elastance. H. Zhu et. al. (2007) demonstrates that predictive neural network may prove to be valuable as a tool in automatic mechanical ventilation support. In his study a method suitable for an adaptive continuous control of nonlinear pulmonary mechanics has been proposed. The method is based on the application of a neural network predictive control approach that incorporates the nonlinear mechanics of an individual patient. Assessment of the mechanical properties of the lung is important for the management of mechanically ventilated patients with acute respiratory distress syndrome (ARDS), and can be achieved through continuous measurement of pressure and flow at the entrance to the end tracheal tube and pressure in the esophagus. Mathematical models are fit to the measured signals, and the estimated values of the model parameters taken as indices of corresponding physiological quantities. The mechanical status of the respiratory system in incubated mechanically ventilated patients is conveniently assessed from the dynamic relationships between the pressure and flow of gas measured at the entrance to the endotracheal tube by J.H.T. Bates and P. Goldberg (1999). A model of respiratory mechanics was fit to the data, and the parameters of the model are then taken to be measures of important physiological quantities. They showed that the respiratory system of the ARDS patient during normal mechanical ventilation exhibited significant nonlinear mechanical behavior that indicated a nonlinear dynamic pressure-volume relationship and non-linear behavior of the respiratory system in such patients may be important for protecting against possible barotrauma caused by increases in tidal volume. Due to the mechanical properties and mutual coupling of the lung and thorax, respiratory mechanics represent such a complex system, yet their understanding is essential for the diagnosis, prognosis, and treatment of various respiratory disorders. M. Kuebler (2007) developed a two-component simulation model for respiratory mechanics. A comprehensive understanding of respiratory mechanics is pivotal for the accurate 57
diagnosis and treatment of lung disease, for adequate artificial or assisted ventilation, and for the analysis of environmental effects, e.g., in diving or under hyperbaric conditions on lung mechanics and function. Yet, understanding of lung mechanics is difficult due to the complex and dynamic nature of pressure-volume relationships of the respiratory system. Basically, two types of control systems exist in the human body. These are (a) servo systems and (b) regulators. A disturbance, such as a step input of CO2 inhalation, will cause this system to respond in a much different way. Brain tissue Pco2 will rise, but respiration will also increase to "blow off' CO2 so that the new steady-state value reached is much lower than it would have been if respiration had not increased. The respiratory control system can also, under proper conditions, exhibit damped and sustained oscillation. H.T.Milhorn et. al. (1965) investigated the human respiratory control system by mathematical modeling. They derive the basic equations for respiratory control in the human being and to obtain transient and steady-state solutions for both positive and negative step input disturbances of inspired CO2 and O2 concentrations. An insight into the importance of this type of analysis is given by the study of the effectiveness of the respiratory system as a regulator. Its purpose has been, first, to derive the basic equations of the system and, secondly, to investigate the system as a biological regulator. They assumed that the system consists of three reservoirs (the lungs, the brain tissues, and the body tissues). Blood flow to the brain is dependent upon cerebral-arterial Pco2 and Po2 which controls alveolar ventilation. F.T Tehrani (1997) developed a mathematical model for the respiratory system of infants at different stages of maturity; the effects of respiratory parameters on the infant respiratory response have been determined by a simulation model. The analysis of infant respiratory system is helpful in early diagnosis of sudden infant death syndrome. The contributions of different mechanisms to particle deposition in local airway segments vary with effective particle size, density, and local airflow rate and gravity angle. In 56regions, where the Reynolds numbers are low. Computational fluid–particle dynamics (CFPD) simulations for gravitational deposition started in the 1990s, focusing on small bronchial airways and alveolar ducts or sacs. For example, W. Hofmann et. al. (1995) simulated the gravitational settling of 10 micrometer particles in an asymmetric, single bifurcating airway model representing generations 15 and 16. It was found that the gravity angle is significant for determining particle distributions and localized doses. Using a 2-D symmetric six-generation model, C. Darquenne and G.K.Prisk (2003) stressed that gravity is important in the deposition of 0.5 and 1 micrometer particles in the human acinus. S. Haber et. al. (2003) incorporated gravitational sedimentation with wall movements, employing a 3-D hemispherical alveolus model. Recently, L. Harrington et. al. (2006) simulated trajectories of 1–5 micrometer particles in 3-D alveolated ducts representing generations 18–22 with different gravity angles. They concluded that the total deposition can be a function of the gravity angle and the ratio of the terminal settling velocity to mean lumen flow velocity. In summary, most of the computational analyses for gravitational deposition focused on the alveolar region where sedimentation may play a dominant role. The effects of gravity on the aerosol deposition in the bronchial airways, especially in the medium-size airways where sedimentation and impaction may occur simultaneously, have been thoroughly investigated by C. Kleinstreuera et. al.(2007). 58
The expiratory flow pattern during tidal breathing is the result of the continuing inspiratory muscle activity during the first part of expiration, expiratory muscle activity and the mechanical properties of the respiratory system. D. Walraven et. al. (2003) developed a mathematical model based on physiological properties to describe the entire expiratory flow pattern in spontaneously breathing, anesthetized cats. Such a model requires besides modeling of the mechanical properties of the respiratory system at least a description of the continuing inspiratory muscle activity in early expiration. As a result, the model fit to the data provides estimates for the relevant respiratory parameters. Expiratory activity was not since it was usually absent or only present at the end of expiration, which part was then excluded from the model fit. It is good modeling practice not to use a more complicated model than needed to give a good fit of the measured data, and it turned out in that the simplest mechanical model of the respiratory system could be used for that purpose. This is the one compartment model characterized by the time constant of the respiratory system (TRS), which is the product of resistance (RRS) and compliance (CRS) of the respiratory system. The continuing inspiratory muscle activity was described by a sigmoid function characterized by another time constant. The derivation of all equations used is explained in the methods section. In this study the model provides an accurate description of the expiratory flow profile in anesthetized tracheostomized cats during tidal breathing and that the parameter estimates obtained for TRS and Tmus are comparable with values obtained using different methods. Jean Marie Ntaganda et al (2007) design a mathematical model for determining blood pressures response to cardiac and respiratory parameters. Reference 1. C. Kleinstreuer, Z. Zhang, C. Kim, “Combined inertial and gravitational deposition of micropartical in small model airways of a human respiratory system” Aerosol Science 38 ,1047-1061, 2007. 2. C. Darquenne and G. K. Prisk, “Effect of gravitational sedimentation on simulated aerosol dispersion in the human acinus”, Journal of Aerosol Science, 34, 405–418, 2003. 3. D. Walraven, C.P.M Vander Grinten, J.N.Bogard, C.K Vander Ent, S.C.N Luijendijk, “Modeling of expiratory flow pattern of spontaneously breathing”, Respiratory Physiology and Neurobiology 134,23-32, 2003. 4. F. Kappel, R.O. Peer, A mathematical model for fundamental regulation processes in the cardiovascular model, J. Math. Biol. 31 (6), 611–631, 1993. 5. F. Kappel, S. Lafer, R.O. Peer, “A model for the cardiovascular system under an ergometric workload”, Surv. Math. Ind. 7, 239–250, 1997. 6. F. T. Tehrani, “A Model Study of periodic breathing stability of neonatal respiratory system and causes of sudden infant death syndrome”, Med. Eng. Phys, 19,547-555, 1997. 7. H. T. Milhorn,Jr., R. Benton , R. Ross and A. Guytom., “A mathematical model of the human respiratory control system”, , Journal Biophysical 5, 27-46, 1965. 8. H. Zhu, J. Guttmann, K. Moller, “Control of respiratory mechanics with artificial neural networks”, IEEE, 1202-1205, 2007. 9. J. H.T Bates and P. Goldberg, “Fitting non linear time domain models of respiratory 59
mechanics to pressure flow data from an intubated patient”, Serving Humanity Advancing Technology Medical Research of Canada, 1999. 10. J. M. Ntaganda , B. Mamapassi, D. Seck, “Modeling blood partial pressure of the human respiratory system”, Applied Mathematics and Computation, 187, 1100-1108, 2007, 11. L. Harrington, G. K.Prisk and C.Darquenne, “Importance of the bifurcation zone and branch orientation in simulated aerosol deposition in the alveolar zone of the human lung”, Journal of Aerosol Science, 37(1), 37–62, 2006. 12. M. Kuebler,M. Mertens and R. Axel, A two component simulation model to teach respiratory mechanics”, Advance Physiology Education 31,218-222, 2007. 13. S. Ganzert, K. Moller, Kristian, L. D. Readt and J. Guttmann, “Equation discovery for model identification in respiratory mechanics under condition of mechanical ventilation”, ICML07 USA, 24June 2007. 14. S. Sepehris, “Physical model of human respiration”, Young Researchers club, Islamic Azad university of Shiraz. 12-17 ,2007 15. S. Haber, D.Yitzhak and A. Tsuda, “Gravitational deposition in a rhythmically expanding and contracting alveolus”. Journal of Applied Physiology, 95, 657–671, 2003. 16. S. Timischl-Teschl, “Modeling the human\ cardiovascular-respiratory control system: An optimal control application to the transition to non-REM sleep”, Math. Biosci. Eng., 7 (7), 2004. 17. W. Hofmann, I. Balásházy and L.Koblinger, “The effect of gravity on particle deposition patterns in bronchial airway bifurcations”. Journal of Aerosol Science, 26, 1161–1168, 1995.
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