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• W2109191874 abstract "IlluminationsThe role of equal pressure points in understanding pulmonary diseasesP. J. G. M. Voets, and H. A. C. van HelvoortP. J. G. M. VoetsRadboud University Nijmegen Medical Centre, Nijmegen, The Netherlands; and , and H. A. C. van HelvoortDepartment of Pulmonary Diseases, Radboud University Nijmegen Medical Centre, Nijmegen, The NetherlandsPublished Online:01 Sep 2013https://doi.org/10.1152/advan.00014.2013MoreSectionsPDF (91 KB)Download PDF ToolsExport citationAdd to favoritesGet permissionsTrack citations ShareShare onFacebookTwitterLinkedInWeChat the respiratory system is composed of a conducting part and a respiratory part. The conducting airways, i.e., the trachea, bronchi, and bronchioles, differ in architecture. Both trachea and bronchi are surrounded by rings of hyaline cartilage for support, whereas the bronchiolar wall does not contain any cartilage and therefore tends to collapse in response to changes in intrapleural or airway pressure. As a result of airflow resistance, pressure that is generated in the alveoli drops along the airways during expiration (1, 3). This is also known as friction loss. When airway pressure has dropped to a level where it equals intrapleural pressure during forced expiration, an equal pressure point (EPP) is reached (4). At the EPP, airways that are not supported by cartilage will collapse (Fig. 1). In healthy lungs, the EPP will be reached in cartilaginous airways as a result of sufficient alveolar driving pressure and only a gradual drop in pressure due to minimal airway resistance (4). In the case of (severe) airway obstruction, resistance to airflow will be much greater, and the pressure drop will be much steeper. The EPP will move upstream toward the alveoli and will be reached in the thin-walled bronchioles, causing airway collapse and the typical depression in the flow-volume curve (2, 3).Fig. 1.Schematic representation of the respiratory system, enclosed by the intrapleural cavity. Alveolar pressure (Palv) is generated as a result of elastic recoil due to wall tension (T) and intrapleural pressure (Ppl). Where the pressure drop (ΔP) over a certain distance (L) from the alveoli equals recoil pressure, an equal pressure point will arise. Cartilaginous airways (on the right side) are barred.Download figureDownload PowerPointAlthough the clinical importance of EPPs, especially in obstructive airway diseases, is unquestionable, relatively little effort has been put into a comprehensible description of the dynamics involved in this phenomenon. This is unfortunate, because correctly applying complex physics in a clinical context can be a difficult task. Therefore, this report aimed to provide a straightforward mathematical approach to the concept of EPPs in the airways by combining important principles of airflow dynamics in a conceivable way. This results in an equation (Eq. 1) that can be used to get a better understanding of how airflow and airway properties alter the tendency of airways to collapse: L=Tr24ηRv(1) where L is length, T is wall tension, r is the cross-sectional radius of a segment, η is viscosity, R is the alveolar radius, and v is velocity.The derivation of Eq. 1 will be discussed stepwise in this article. Table 1 shows a glossary of the parameters.Table 1. Glossary of parametersParameterDefinitionηViscosityΦAirflowACross-sectional area of the airwaysEPPEqual pressure pointLLengthPalvAlveolar pressurePrecRecoil pressurerCross-sectional radius of the segmentRAlveolar radiusRAWResistance to airflowTWall tensionvVelocityMathematical approach.Alveolar pressure (Palv) is the result of recoil pressure (Prec), created by expansion of the alveoli, and intrapleural pressure (Ppl) and produces the driving force for airflow through the respiratory system (2–4). According to Pascal's law, pressure will be dispersed equally in all directions throughout the enclosed intrapleural cavity. Therefore, Ppl exerted on the alveoli will equal Ppl exerted on the airways. Assuming that the alveoli are perfect spheres, Laplace's law (Eq. 2) can be used to describe Prec in terms of R and T, which is the result of lung tissue elasticity and surface tension (6): Prec=2TR(2) The equation for Palv (Eq. 3) can now be written as follows: Pclv=Ppl+Prec=Ppl+2TR(3) As air flows during expiration, pressure inside the airways will drop as a result of friction loss. The relationship between the pressure drop (ΔP), resistance to airflow (RAW), and airflow (Φ) is given by Poiseuille's law (Eq. 4), which can be applied to the laminar flow of Newtonian fluids (4, 6). Airflow becomes more turbulent as flow velocity and airway diameter increase. Therefore, larger airways are much more prone to turbulent flow than bronchiloles (5). To apply Poiseuille's law, laminar flow is assumed in the smallest airways in this model, although even in bronchioles airflow will not be entirely laminar during forced expiration. RAW depends on η, the length of the airway segment through which air flows (L), and the cross-sectional radius of this segment (r): ΔP=RAWΦ=8ηLπr4Φ(4) The continuity equation for Φ (Eq. 5) essentially states that what goes in at one end of the airway segment must come out at the other end. The volume of air (ΔV) that passes a certain point per unit of time (Δt), i.e., the flow, will not change. Therefore, an increase in cross-sectional area of the airways (A) is accompanied by a decrease in v (6): Φ=ΔVΔt=AΔxΔt=Av=πr2v=constant(5) This result (Eq. 6) can be combined with Poiseuille's law (Eq. 4) as follows: ΔP=RAWΦ=8ηLπr4πr2v=8ηLvr2(6) As mentioned above, an EPP will arise where airway pressure has dropped to a level where it equals Ppl (4). Airway pressure at distance L from the alveoli is defined as Palv minus ΔP over L, as follows: Palv−ΔP=Ppl+2TR−ΔP=Ppl(7) Because Ppl on the left side of the equation is the same as on the right side of the equation (Pascal's law), the EPP will occur where Prec equals friction loss: 2TR=ΔP=8ηLvr2(8) Rearranging Eq. 8 produces the relationship between the distance from the alveolus to the EPP (L) and the factors that influence it: L=Tr24ηRvDiscussion and conclusions.This report provides a comprehensible mathematical approach to the dynamics of a pulmonary EPP. The equation that has been derived in this article (Eq. 1) shows that as the value of L decreases, the EPP moves away from the cartilaginous airways toward the alveoli, causing an obstruction to airflow as soon as the EPP moves to a collapsible region of the airway. This can be illustrated by the altered airflow dynamics in a number of pulmonary diseases (3, 4). In asthma and chronic obstructive pulmonary disease, for instance, airway inflammation reduces r (increasing RAW) and therefore reduces the distance from the alveolus to the EPP. The same effect on L can be seen in emphysema, as a result of the decrease in T and an increase in r, due to the disintegration of alveolar septa. Conversely, pursed lip breathing (which is commonly used instinctively by persons suffering from chronic obstructive pulmonary disease) decreases v, increasing the airway pressure, which moves the EPP downstream toward the cartilaginous airways, preventing collapse.Taken together, it is important to note that it is not the aim of this equation to calculate EPP positions exactly, and it should not be used for this purpose. Rather, the derived equation should be considered a means for students or clinicians to get a better understanding of how (changes in) relevant parameters influence the tendency of airways to collapse in pulmonary disease.DISCLOSURESNo conflicts of interest, financial or otherwise, are declared by the author(s).AUTHOR CONTRIBUTIONSAuthor contributions: P.V. conception and design of research; P.V. analyzed data; P.V. and H.v.H. interpreted results of experiments; P.V. prepared figures; P.V. and H.v.H. drafted manuscript; P.V. and H.v.H. approved final version of manuscript; H.v.H. edited and revised manuscript.REFERENCES1. Boron WF, Boulpaep EL. Mechanics of ventilation. In: Medical Physiology, edited by , Boron WF. Philadelphia, PA, Elsevier Saunders, 2012, p. 637–641.Google Scholar2. Frew AJ, Holgate ST. Respiratory diseases. In: Kumar & Clark Clinical Medicine, edited by , Kumar P, Clark M. Edinburgh: Elsevier Saunders, 2009, p. 815–817.Google Scholar3. Jordanoglou J, Pride NB. Factors determining maximum inspiratory flow and maximum expiratory flow of the lung. Thorax 23: 33–37, 1968.Crossref | ISI | Google Scholar4. Mead J, Turner JM, Macklem PT, Little JB. Significance of the relationship between lung recoil and maximum expiratory flow. J Appl Physiol 22: 95–108, 1967.Link | ISI | Google Scholar5. Rhoades RA. Ventilation and the mechanics of breathing. In: Medical Physiology, edited by , Rhoades RA, Bell DR. Baltimore, MD: Lippincott, Williams & Wilkins, 2009, p. 342–345.Google Scholar6. Van Oosterom A, Oostendorp TF. Wet van Poiseuille. In: Medische Fysica, edited by , Van Oosterom A, Oostendorp TF. Amsterdam: Reed Business, 2008, p. 8–11.Google ScholarAUTHOR NOTESAddress for reprint requests and other correspondence: H. A. C. van Helvoort, Dept. of Pulmonary Diseases (454), Radboud Univ. Nijmegen Medical Centre, PO Box 9101, Nijmegen 6500 HB, The Netherlands (e-mail: H.[email protected]umcn.nl). Download PDF Previous Back to Top Next FiguresReferencesRelatedInformation Cited ByImpact of chronic systolic heart failure on lung structure-function relationships in large airways14 July 2016 | Physiological Reports, Vol. 4, No. 13The “Inverted Square Root Sign”Journal of Bronchology & Interventional Pulmonology, Vol. 23, No. 1Airway clearance techniques, pulmonary rehabilitation and physical activity More from this issue > Volume 37Issue 3September 2013Pages 266-267 Copyright & PermissionsCopyright © 2013 the American Physiological Societyhttps://doi.org/10.1152/advan.00014.2013PubMed24022774History Received 4 February 2013 Accepted 14 May 2013 Published online 1 September 2013 Published in print 1 September 2013 Metrics" @default.
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