Robust Unidirectional Airflow through Avian Lungs: New Insights from a Piecewise Linear Mathematical Model
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{"title"=>"Robust Unidirectional Airflow through Avian Lungs: New Insights from a Piecewise Linear Mathematical Model", "type"=>"journal", "authors"=>[{"first_name"=>"Emily P.", "last_name"=>"Harvey", "scopus_author_id"=>"36704525400"}, {"first_name"=>"Alona", "last_name"=>"Ben-Tal", "scopus_author_id"=>"10138847200"}], "year"=>2016, "source"=>"PLoS Computational Biology", "identifiers"=>{"pmid"=>"26862752", "sgr"=>"84959530907", "doi"=>"10.1371/journal.pcbi.1004637", "scopus"=>"2-s2.0-84959530907", "pui"=>"608854521", "issn"=>"15537358"}, "id"=>"72321836-ea3f-3e6d-beb0-ed434715cad3", "abstract"=>"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.", "link"=>"http://www.mendeley.com/research/robust-unidirectional-airflow-through-avian-lungs-new-insights-piecewise-linear-mathematical-model", "reader_count"=>3, "reader_count_by_academic_status"=>{"Researcher"=>1, "Student > Bachelor"=>1, "Other"=>1}, "reader_count_by_user_role"=>{"Researcher"=>1, "Student > Bachelor"=>1, "Other"=>1}, "reader_count_by_subject_area"=>{"Biochemistry, Genetics and Molecular Biology"=>1, "Agricultural and Biological Sciences"=>1, "Mathematics"=>1}, "reader_count_by_subdiscipline"=>{"Agricultural and Biological Sciences"=>{"Agricultural and Biological Sciences"=>1}, "Biochemistry, Genetics and Molecular Biology"=>{"Biochemistry, Genetics and Molecular Biology"=>1}, "Mathematics"=>{"Mathematics"=>1}}, "group_count"=>0}

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Figshare

  • {"files"=>["https://ndownloader.figshare.com/files/2636521"], "description"=>"<p>The caudal and cranial airsacs have pressures <i>P</i><sub>1</sub> and <i>P</i><sub>2</sub> respectively, and compliances <i>C</i><sub>1</sub> and <i>C</i><sub>2</sub> respectively. The pressure outside both sets of airsacs, <i>P</i><sub><i>ext</i></sub>(<i>t</i>), varies periodically due to the respiratory muscles, which causes the airsacs to inflate and deflate. The pressure <i>P</i><sub><i>atm</i></sub> is atmospheric pressure. Between each node in the system there is resistance to flow, <i>R</i><sub><i>i</i></sub>, and airflow, <i>q</i><sub><i>i</i></sub>. Blue arrows represent the flow during inspiration, and green arrows represent the flow during expiration. The grey shaded area indicates the parabronchi, where gas exchange occurs.</p>", "links"=>[], "tags"=>["expiratory valving", "resistance", "Model Avian lungs", "gas exchange area", "air flows unidirectionally", "airflow", "inspiratory", "Robust Unidirectional Airflow", "model"], "article_id"=>1643881, "categories"=>["Biological Sciences"], "users"=>["Emily P. Harvey", "Alona Ben-Tal"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1004637.g001", "stats"=>{"downloads"=>0, "page_views"=>0, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Schematic_model_of_the_avian_respiratory_system_/1643881", "title"=>"Schematic model of the avian respiratory system.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2016-02-10 10:48:40"}
  • {"files"=>["https://ndownloader.figshare.com/files/2636522"], "description"=>"<p><b>A:</b> The resistance <i>R</i><sub>1</sub> is discontinuous at <i>P</i><sub>1</sub> = <i>P</i><sub><i>J</i></sub>, which is when <i>q</i><sub>1</sub> = 0. For <i>q</i><sub>1</sub> > 0, <i>R</i><sub>1</sub> = <i>R</i><sub>1,<i>insp</i></sub>, while for <i>q</i><sub>1</sub> < 0, <i>R</i><sub>1</sub> = <i>R</i><sub>1,<i>exp</i></sub> ≥ <i>R</i><sub>1,<i>insp</i></sub>. <b>B:</b> The resistance <i>R</i><sub>2</sub> is discontinuous at <i>P</i><sub>2</sub> = <i>P</i><sub><i>J</i></sub>, which is when <i>q</i><sub>2</sub> = 0. For <i>q</i><sub>2</sub> > 0, <i>R</i><sub>2</sub> = <i>R</i><sub>2,<i>exp</i></sub>, while for <i>q</i><sub>2</sub> < 0, <i>R</i><sub>2</sub> = <i>R</i><sub>2,<i>insp</i></sub> ≥ <i>R</i><sub>2,<i>exp</i></sub>.</p>", "links"=>[], "tags"=>["expiratory valving", "resistance", "Model Avian lungs", "gas exchange area", "air flows unidirectionally", "airflow", "inspiratory", "Robust Unidirectional Airflow", "model"], "article_id"=>1643882, "categories"=>["Biological Sciences"], "users"=>["Emily P. Harvey", "Alona Ben-Tal"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1004637.g002", "stats"=>{"downloads"=>2, "page_views"=>0, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_The_resistances_R_1_and_R_2_are_discontinuous_and_depend_on_the_flow_direction_/1643882", "title"=>"The resistances <i>R</i><sub>1</sub> and <i>R</i><sub>2</sub> are discontinuous and depend on the flow direction.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2016-02-10 10:48:40"}
  • {"files"=>["https://ndownloader.figshare.com/files/2636523"], "description"=>"<p><b>A:</b> the pressure differences from atmospheric pressure, <i>x</i><sub>1</sub> = <i>P</i><sub>1</sub> − <i>P</i><sub><i>atm</i></sub> and <i>x</i><sub>2</sub> = <i>P</i><sub>2</sub> − <i>P</i><sub><i>atm</i></sub>, which are the outputs of the model. <b>B:</b> the flow rates <i>q</i><sub><i>T</i></sub>, <i>q</i><sub><i>P</i></sub>, <i>q</i><sub>1</sub>, and <i>q</i><sub>2</sub> in one side of the respiratory system. <b>C:</b> the volumes of the caudal set of airsacs, <i>V</i><sub>1</sub>, and the cranial set of airsacs, <i>V</i><sub>2</sub>, on one side of the respiratory system. Inspiration and expiration phases are labelled (INSP and EXP).</p>", "links"=>[], "tags"=>["expiratory valving", "resistance", "Model Avian lungs", "gas exchange area", "air flows unidirectionally", "airflow", "inspiratory", "Robust Unidirectional Airflow", "model"], "article_id"=>1643883, "categories"=>["Biological Sciences"], "users"=>["Emily P. Harvey", "Alona Ben-Tal"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1004637.g003", "stats"=>{"downloads"=>0, "page_views"=>0, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Model_outputs_for_the_chosen_default_parameter_values_Table_1_/1643883", "title"=>"Model outputs for the chosen default parameter values (Table 1).", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2016-02-10 10:48:40"}
  • {"files"=>["https://ndownloader.figshare.com/files/2636524"], "description"=>"<p>The variables are <i>x</i><sub>1</sub> = <i>P</i><sub>1</sub> − <i>P</i><sub><i>atm</i></sub> and <i>x</i><sub>2</sub> = <i>P</i><sub>2</sub> − <i>P</i><sub><i>atm</i></sub>. The line <i>x</i><sub>2</sub> = <i>x</i><sub>1</sub> marks all the possible pressures for which the flow <i>q</i><sub><i>P</i></sub> = 0. Above this line <i>q</i><sub><i>P</i></sub> < 0 (marked in shaded grey), and below this line <i>q</i><sub><i>P</i></sub> > 0. Inspiration is marked by the light blue region and expiration is marked by the green region. Dark blue curves show the solutions to the system from different initial conditions, when the external pressure is constant. Superimposed in red is an example of a solution to the system when the external pressure changes (along the vector [1, 1]<sup><i>T</i></sup>) due to the respiratory muscles during breathing. This solution is ‘trapped’ below the line <i>x</i><sub>1</sub> = <i>x</i><sub>2</sub> in the region <i>q</i><sub><i>P</i></sub> > 0 and hence the flow is unidirectional. See <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004637#sec014\" target=\"_blank\">Methods</a> and <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004637#pcbi.1004637.g013\" target=\"_blank\">Fig 13</a> for a detailed analysis.</p>", "links"=>[], "tags"=>["expiratory valving", "resistance", "Model Avian lungs", "gas exchange area", "air flows unidirectionally", "airflow", "inspiratory", "Robust Unidirectional Airflow", "model"], "article_id"=>1643884, "categories"=>["Biological Sciences"], "users"=>["Emily P. Harvey", "Alona Ben-Tal"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1004637.g004", "stats"=>{"downloads"=>3, "page_views"=>0, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Sketch_of_the_system_dynamics_not_to_scale_showing_conditions_for_unidirectional_flow_when_there_is_effective_inspiratory_valving_/1643884", "title"=>"Sketch of the system dynamics (not to scale), showing conditions for unidirectional flow when there is effective inspiratory valving.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2016-02-10 10:48:40"}
  • {"files"=>["https://ndownloader.figshare.com/files/2636525"], "description"=>"<p>In this figure we plot the flow through the parabronchi, <i>q</i><sub><i>P</i></sub>, against time for a range of <i>P</i><sub><i>amp</i></sub> values. The flow rates increase linearly as the amplitude of breathing, <i>P</i><sub><i>amp</i></sub> increases. Inspiration and expiration are labelled (INSP and EXP). All parameters, except <i>P</i><sub><i>amp</i></sub>, are as in <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004637#pcbi.1004637.t001\" target=\"_blank\">Table 1</a>.</p>", "links"=>[], "tags"=>["expiratory valving", "resistance", "Model Avian lungs", "gas exchange area", "air flows unidirectionally", "airflow", "inspiratory", "Robust Unidirectional Airflow", "model"], "article_id"=>1643885, "categories"=>["Biological Sciences"], "users"=>["Emily P. Harvey", "Alona Ben-Tal"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1004637.g005", "stats"=>{"downloads"=>3, "page_views"=>0, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Unidirectional_flow_is_robust_to_changes_in_amplitude_/1643885", "title"=>"Unidirectional flow is robust to changes in amplitude.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2016-02-10 10:48:40"}
  • {"files"=>["https://ndownloader.figshare.com/files/2636526"], "description"=>"<p>The flow through the parabronchi, <i>q</i><sub><i>P</i></sub>, during a single breath, is plotted for breathing period, <i>T</i>, ranging from 1–6 seconds (frequency ranging from 1–1/6 Hz respectively). The traces are aligned such that phase = 0 is at the beginning of inspiration. Inspiration and expiration are labelled (INSP and EXP). All the parameters, except the breathing period, are as in <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004637#pcbi.1004637.t001\" target=\"_blank\">Table 1</a>.</p>", "links"=>[], "tags"=>["expiratory valving", "resistance", "Model Avian lungs", "gas exchange area", "air flows unidirectionally", "airflow", "inspiratory", "Robust Unidirectional Airflow", "model"], "article_id"=>1643886, "categories"=>["Biological Sciences"], "users"=>["Emily P. Harvey", "Alona Ben-Tal"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1004637.g006", "stats"=>{"downloads"=>1, "page_views"=>0, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Unidirectional_flow_is_robust_to_changes_in_frequency_/1643886", "title"=>"Unidirectional flow is robust to changes in frequency.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2016-02-10 10:48:40"}
  • {"files"=>["https://ndownloader.figshare.com/files/2636527"], "description"=>"<p>Flow rates <i>q</i><sub><i>T</i></sub>, <i>q</i><sub><i>P</i></sub>, <i>q</i><sub>1</sub>, and <i>q</i><sub>2</sub> against time for the parameters <i>R</i><sub>1,<i>insp</i></sub> = <i>R</i><sub>2,<i>exp</i></sub> = 3 cmH<sub>2</sub>O/L⋅s, <i>C</i><sub>1</sub> = <i>C</i><sub>2</sub> (<i>γ</i> = 1), and <i>C</i><sub><i>tot</i></sub> = 450 mL/cmH<sub>2</sub>O. Panel <b>A</b> shows the case where there is effective inspiratory valving: <i>R</i><sub>2,<i>exp</i></sub> = <i>γR</i><sub>1,<i>exp</i></sub> and <i>R</i><sub>2,<i>insp</i></sub> = 100 × <i>R</i><sub>1,<i>insp</i></sub>, with <i>R</i><sub>1,<i>exp</i></sub> = <i>R</i><sub>1,<i>insp</i></sub>. Panel <b>B</b> shows the case with effective expiratory valving: <i>R</i><sub>1,<i>insp</i></sub> = <i>γR</i><sub>2,<i>insp</i></sub> and <i>R</i><sub>1,<i>exp</i></sub> = 20 × <i>R</i><sub>2,<i>exp</i></sub>, with <i>R</i><sub>2,<i>insp</i></sub> = <i>R</i><sub>2,<i>exp</i></sub>. All other parameters are given in <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004637#pcbi.1004637.t001\" target=\"_blank\">Table 1</a>.</p>", "links"=>[], "tags"=>["expiratory valving", "resistance", "Model Avian lungs", "gas exchange area", "air flows unidirectionally", "airflow", "inspiratory", "Robust Unidirectional Airflow", "model"], "article_id"=>1643887, "categories"=>["Biological Sciences"], "users"=>["Emily P. Harvey", "Alona Ben-Tal"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1004637.g007", "stats"=>{"downloads"=>1, "page_views"=>0, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Inspiratory_and_expiratory_valving_both_produce_unidirectional_flow_q_P_0_/1643887", "title"=>"Inspiratory and expiratory valving both produce unidirectional flow, <i>q</i><sub><i>P</i></sub> > 0.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2016-02-10 10:48:40"}
  • {"files"=>["https://ndownloader.figshare.com/files/2636528"], "description"=>"<p>Plot of the overall efficiency when <i>R</i><sub>1,<i>insp</i></sub> / <i>R</i><sub>2,<i>exp</i></sub> is varied whilst keeping the total resistance of the system constant (<i>R</i><sub>1,<i>insp</i></sub> + <i>R</i><sub>2,<i>exp</i></sub> = 6 cmH<sub>2</sub>O/L⋅s). The effect is similar for a range of <i>γ</i> values. All other parameters are given in <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004637#pcbi.1004637.t001\" target=\"_blank\">Table 1</a>.</p>", "links"=>[], "tags"=>["expiratory valving", "resistance", "Model Avian lungs", "gas exchange area", "air flows unidirectionally", "airflow", "inspiratory", "Robust Unidirectional Airflow", "model"], "article_id"=>1643888, "categories"=>["Biological Sciences"], "users"=>["Emily P. Harvey", "Alona Ben-Tal"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1004637.g008", "stats"=>{"downloads"=>0, "page_views"=>0, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Changing_the_relative_resistance_of_R_1_insp_R_2_exp_affects_the_efficiency_of_the_system_/1643888", "title"=>"Changing the relative resistance of <i>R</i><sub>1,<i>insp</i></sub> / <i>R</i><sub>2,<i>exp</i></sub> affects the efficiency of the system.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2016-02-10 10:48:40"}
  • {"files"=>["https://ndownloader.figshare.com/files/2636529"], "description"=>"<p>As <i>γ</i> increases (<i>C</i><sub>1</sub> increases relative to <i>C</i><sub>2</sub>) more air flows through the parabronchi during expiration. The dashed line at 1 indicates when the total flow during expiration and inspiration are equal. All the parameters are as given in <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004637#pcbi.1004637.t001\" target=\"_blank\">Table 1</a>. The vertical dotted line shows the selected default <i>γ</i> value.</p>", "links"=>[], "tags"=>["expiratory valving", "resistance", "Model Avian lungs", "gas exchange area", "air flows unidirectionally", "airflow", "inspiratory", "Robust Unidirectional Airflow", "model"], "article_id"=>1643889, "categories"=>["Biological Sciences"], "users"=>["Emily P. Harvey", "Alona Ben-Tal"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1004637.g009", "stats"=>{"downloads"=>1, "page_views"=>0, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Varying_the_ratio_of_compliances_947_C_1_C_2_changes_the_timing_of_the_flow_through_the_parabronchi_/1643889", "title"=>"Varying the ratio of compliances <i>γ</i> = <i>C</i><sub>1</sub>/<i>C</i><sub>2</sub> changes the timing of the flow through the parabronchi.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2016-02-10 10:48:40"}
  • {"files"=>["https://ndownloader.figshare.com/files/2636530"], "description"=>"<p>This figure plots the flow rate <i>q</i><sub><i>P</i></sub> versus time, for <i>γ</i> = 1/4 (red), <i>γ</i> = 1 (green), and <i>γ</i> = 4 (blue). The inspiratory period (INSP) is shaded blue, and the expiratory period (EXP) is shaded green. All other parameters are as given in <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004637#pcbi.1004637.t001\" target=\"_blank\">Table 1</a>.</p>", "links"=>[], "tags"=>["expiratory valving", "resistance", "Model Avian lungs", "gas exchange area", "air flows unidirectionally", "airflow", "inspiratory", "Robust Unidirectional Airflow", "model"], "article_id"=>1643890, "categories"=>["Biological Sciences"], "users"=>["Emily P. Harvey", "Alona Ben-Tal"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1004637.g010", "stats"=>{"downloads"=>0, "page_views"=>0, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_The_ratio_of_compliances_947_C_1_C_2_changes_the_shape_of_the_oscillatory_flow_q_P_/1643890", "title"=>"The ratio of compliances <i>γ</i> = <i>C</i><sub>1</sub>/<i>C</i><sub>2</sub> changes the shape of the oscillatory flow <i>q</i><sub><i>P</i></sub>.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2016-02-10 10:48:40"}
  • {"files"=>["https://ndownloader.figshare.com/files/2636531"], "description"=>"<p>Here we plot the ratio of the inspiration and expiration phase durations (I:E time ratio) against the relative resistance of <i>R</i><sub>1,<i>insp</i></sub>/<i>R</i><sub>2,<i>exp</i></sub> whilst keeping the total resistance constant (<i>R</i><sub>1,<i>insp</i></sub> + <i>R</i><sub>2,<i>exp</i></sub> = 6 cmH<sub>2</sub>O/L⋅s). The same effect is seen for a range of <i>γ</i> values. The dashed line indicates where the period of expiration and inspiration are equal, <i>T</i><sub><i>e</i></sub> = <i>T</i><sub><i>i</i></sub>.</p>", "links"=>[], "tags"=>["expiratory valving", "resistance", "Model Avian lungs", "gas exchange area", "air flows unidirectionally", "airflow", "inspiratory", "Robust Unidirectional Airflow", "model"], "article_id"=>1643891, "categories"=>["Biological Sciences"], "users"=>["Emily P. Harvey", "Alona Ben-Tal"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1004637.g011", "stats"=>{"downloads"=>0, "page_views"=>0, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_The_relative_resistance_of_R_1_insp_R_2_exp_affects_the_duration_of_the_expiration_and_inspiration_phases_/1643891", "title"=>"The relative resistance of <i>R</i><sub>1,<i>insp</i></sub>/<i>R</i><sub>2,<i>exp</i></sub> affects the duration of the expiration and inspiration phases.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2016-02-10 10:48:40"}
  • {"files"=>["https://ndownloader.figshare.com/files/2636532"], "description"=>"<p>The caudal and cranial airsacs have pressures <i>P</i><sub>1</sub> and <i>P</i><sub>2</sub>, respectively. The direction of positive flow rate is indicated by the red arrows. If we assume symmetry, we can reduce the model to consider only one side, which gives the model in <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004637#pcbi.1004637.g001\" target=\"_blank\">Fig 1</a>, with <i>R</i><sub><i>T</i></sub> = 2<i>R</i><sub><i>trachea</i></sub> + <i>R</i><sub><i>EPPB</i></sub>. The flows found in the reduced model will be for a single side of the animal, and will need to be doubled to find the total flow in the whole animal.</p>", "links"=>[], "tags"=>["expiratory valving", "resistance", "Model Avian lungs", "gas exchange area", "air flows unidirectionally", "airflow", "inspiratory", "Robust Unidirectional Airflow", "model"], "article_id"=>1643892, "categories"=>["Biological Sciences"], "users"=>["Emily P. Harvey", "Alona Ben-Tal"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1004637.g012", "stats"=>{"downloads"=>0, "page_views"=>0, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Schematic_of_the_full_avian_model_including_left_and_right_sides_of_the_respiratory_system_/1643892", "title"=>"Schematic of the full avian model including left and right sides of the respiratory system.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2016-02-10 10:48:40"}
  • {"files"=>["https://ndownloader.figshare.com/files/2636533"], "description"=>"<p>The grey shaded region above the long dashed line <i>x</i><sub>2</sub> = <i>x</i><sub>1</sub> is where <i>q</i><sub><i>P</i></sub> < 0, and the unshaded region below the line <i>x</i><sub>2</sub> = <i>x</i><sub>1</sub> is where <i>q</i><sub><i>P</i></sub> > 0. The blue shaded region indicates where <i>q</i><sub><i>T</i></sub> > 0, and the green shaded region where <i>q</i><sub><i>T</i></sub> < 0. The value of <i>R</i><sub>1</sub> changes on the line <i>q</i><sub>1</sub> = 0 such that: <i>R</i><sub>1</sub> = <i>R</i><sub>1,<i>insp</i></sub> in region (1), and <i>R</i><sub>1</sub> = <i>R</i><sub>1,<i>exp</i></sub> in regions (2), (3), and (4). The value of <i>R</i><sub>2</sub> changes on the line <i>q</i><sub>2</sub> = 0 such that: <i>R</i><sub>2</sub> = <i>R</i><sub>2,<i>exp</i></sub> in region (4), and <i>R</i><sub>2</sub> = <i>R</i><sub>2,<i>insp</i></sub> in regions (1), (2), and (3).</p>", "links"=>[], "tags"=>["expiratory valving", "resistance", "Model Avian lungs", "gas exchange area", "air flows unidirectionally", "airflow", "inspiratory", "Robust Unidirectional Airflow", "model"], "article_id"=>1643893, "categories"=>["Biological Sciences"], "users"=>["Emily P. Harvey", "Alona Ben-Tal"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1004637.g013", "stats"=>{"downloads"=>1, "page_views"=>0, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_The_position_of_zero_flow_points_q_T_0_q_1_0_and_q_2_0_shown_in_the_phase_plane_/1643893", "title"=>"The position of zero flow points <i>q</i><sub><i>T</i></sub> = 0, <i>q</i><sub>1</sub> = 0, and <i>q</i><sub>2</sub> = 0 shown in the phase plane.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2016-02-10 10:48:40"}
  • {"files"=>["https://ndownloader.figshare.com/files/2636534"], "description"=>"<p>This figure plots the flow through the parabronchi <i>q</i><sub><i>P</i></sub> against time for a range of <i>C</i><sub><i>tot</i></sub>, with all traces aligned such that the beginning of inspiration is at <i>t</i> = 0. The parameter <i>γ</i> = 1, other parameters are given in <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004637#pcbi.1004637.t001\" target=\"_blank\">Table 1</a>. Note: the transition from inspiration to expiration happens at close to the same time for <i>C</i><sub><i>tot</i></sub> ≥ 90 mL/cmH<sub>2</sub>O and is shown as a black dashed line. The time of the transition for <i>C</i><sub><i>tot</i></sub> = 9 mL/cmH<sub>2</sub>O is shown with a dark blue dot-dashed line.</p>", "links"=>[], "tags"=>["expiratory valving", "resistance", "Model Avian lungs", "gas exchange area", "air flows unidirectionally", "airflow", "inspiratory", "Robust Unidirectional Airflow", "model"], "article_id"=>1643894, "categories"=>["Biological Sciences"], "users"=>["Emily P. Harvey", "Alona Ben-Tal"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1004637.g014", "stats"=>{"downloads"=>0, "page_views"=>0, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_The_shape_of_the_flow_q_P_smoothes_out_as_C_tot_increases_/1643894", "title"=>"The shape of the flow <i>q</i><sub><i>P</i></sub> smoothes out as <i>C</i><sub><i>tot</i></sub> increases.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2016-02-10 10:48:40"}
  • {"files"=>["https://ndownloader.figshare.com/files/2636535"], "description"=>"<p>The experimentally measured ratio [<a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004637#pcbi.1004637.ref038\" target=\"_blank\">38</a>] in spontaneous breathing (dashed line) and artificial ventilation (dot-dashed line) ducks are shown. For different <i>C</i><sub><i>tot</i></sub> values, we would require slightly different <i>γ</i> values to match the experimental findings. The required <i>γ</i> values in each case for <i>C</i><sub><i>tot</i></sub> = 450 mL/cmH<sub>2</sub>O are shown with vertical dotted lines. All other parameters can be found in <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004637#pcbi.1004637.t001\" target=\"_blank\">Table 1</a>.</p>", "links"=>[], "tags"=>["expiratory valving", "resistance", "Model Avian lungs", "gas exchange area", "air flows unidirectionally", "airflow", "inspiratory", "Robust Unidirectional Airflow", "model"], "article_id"=>1643895, "categories"=>["Biological Sciences"], "users"=>["Emily P. Harvey", "Alona Ben-Tal"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1004637.g015", "stats"=>{"downloads"=>1, "page_views"=>0, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_The_ratio_of_caudal_to_cranial_airsac_ventilation_increases_as_the_ratio_of_airsac_compliances_947_C_1_C_2_increases_i_e_the_volume_of_the_caudal_airsacs_changes_more_than_that_of_the_cranial_airsacs_/1643895", "title"=>"The ratio of caudal to cranial airsac ventilation increases as the ratio of airsac compliances <i>γ</i> = <i>C</i><sub>1</sub>/<i>C</i><sub>2</sub> increases, i.e. the volume of the caudal airsacs changes more than that of the cranial airsacs.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2016-02-10 10:48:40"}
  • {"files"=>["https://ndownloader.figshare.com/files/2636536", "https://ndownloader.figshare.com/files/2636537", "https://ndownloader.figshare.com/files/2636538"], "description"=>"<div><p>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.</p></div>", "links"=>[], "tags"=>["expiratory valving", "resistance", "Model Avian lungs", "gas exchange area", "air flows unidirectionally", "airflow", "inspiratory", "Robust Unidirectional Airflow", "model"], "article_id"=>1643896, "categories"=>["Biological Sciences"], "users"=>["Emily P. Harvey", "Alona Ben-Tal"], "doi"=>["https://dx.doi.org/10.1371/journal.pcbi.1004637.s001", "https://dx.doi.org/10.1371/journal.pcbi.1004637.s002", "https://dx.doi.org/10.1371/journal.pcbi.1004637.s003"], "stats"=>{"downloads"=>1, "page_views"=>0, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Robust_Unidirectional_Airflow_through_Avian_Lungs_New_Insights_from_a_Piecewise_Linear_Mathematical_Model_/1643896", "title"=>"Robust Unidirectional Airflow through Avian Lungs: New Insights from a Piecewise Linear Mathematical Model", "pos_in_sequence"=>0, "defined_type"=>4, "published_date"=>"2016-02-10 10:48:40"}

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  • {"unique-ip"=>"16", "full-text"=>"18", "pdf"=>"1", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2019", "month"=>"3"}
  • {"unique-ip"=>"17", "full-text"=>"24", "pdf"=>"4", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2019", "month"=>"4"}
  • {"unique-ip"=>"6", "full-text"=>"5", "pdf"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"2", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2019", "month"=>"5"}
  • {"unique-ip"=>"8", "full-text"=>"13", "pdf"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2019", "month"=>"8"}

Relative Metric

{"start_date"=>"2016-01-01T00:00:00Z", "end_date"=>"2016-12-31T00:00:00Z", "subject_areas"=>[]}
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