Mucus Transport in the Human Lung Airways: Effect of Porosity Parameter and Air Velocity

Transcription

1 Mucu Tranport in the Human Lung Airway: Effect of Poroity Parameter and Air Velocity V.S. Verma 1, Vikah Rana Department of Mathematic and Statitic, DDU Gorakhpur Univerity, Gorakhpur-73009, U.P. (India) Abtract: In thi paper, a planar two layer teady tate fluid model i propoed to tudy the mucu tranport in the human lung airway by taking the effect of air-velocity due to air-motion and poroity parameter due to forming of porou cilia bed by certain immotile cilia by conidering mucu a a vico-elatic fluid. It i hown that mucu tranport increae a the preure drop, air velocity due to air-motion increae. It i alo oberved that mucu tranport decreae a the vicoity of erou layer fluid or that of mucu increae, but any increae in mucu vicoity at it larger value doe not eem to affect the mucu tranport. It i alo found that for given total depth of erou layer and mucu layer, there exit a erou fluid layer thickne for which mucu tranport i maximum. It i alo hown that mucu tranport decreae a it elatic modulu increae. Keyword: Mucu tranport, poroity parameter, vico-elatic mucu, air velocity. 1. Introduction The muco-ciliary ytem i one of the mot important primary defene mechanim of the human lung airway for cleaning the inpired air of contaminant and for removing entrapped particle uch a bacteria, virue, cellular debri, carcinogen in tobacco moke, etc. from the lung through mucu tranport. It conit of three layer namely: a mucu layer, a erou layer and the cilia which are mall hair-like projection lining with the epithelium of the bronchial repiratory tract. The erou layer fluid i conidered a a Newtonian fluid while mucu a a vico-elatic fluid. It ha been pointed out that, in general, mucu tranport depend upon the tructure of cilia, the function imparted by cilia tip in the erou ub-layer fluid, the thicknee and the vicoitie of the erou fluid and mucu, the interaction of mucu with the erou layer fluid alongwith the vico-elatic parameter of modulu i.e. elatic modulu of mucu. In recent decade, the mucu tranport in the human lung ha been tudied by everal reearcher. In particular, Barton and Raynor [3] preented an analytical model for mucu tranport by conidering cilium a an ocillating cylinder with a greater height during the effective troke and maller height during the recovery troke. Blake [5] conidered a two-layer Newtonian fluid model, one erou layer fluid and the other mucu and pointed out the importance of gravity and effect of air flow on mucu tranport. Another mathematical analyi of two-layer fluid model i given by Blake and Winet [6]. They uggeted that if cilia jut penetrate the upper, much more vicou layer, then the mucu tranport rate would be ubtantially enhanced. tudy tate and ocillatory airflow condition and pointed out the importance of vicoity of mucu gel on tranport. Agarwal et al.[] have tudied the mucu tranport by airflow interaction in a miniaturized imulated cough machine and found that mucu tranport increae a the vicoity of the erou layer imulant decreae or a the mucu filance (pinnability) decreae. King et al. [1] have preented a planar two-layer fluid model for muco-ciliary tranport in the repiratory tract due to cilia beating and air motion by conidering mucu a vico-elatic fluid and have hown that mucu tranport increae a hear tre due to air motion, preure drop and mean velocity of cilia tip increae. They have alo hown that mucu tranport rate i maximum at ome value of erou fluid thickne for fixed total depth of erou layer fluid and mucu. Agarwal and Verma [1] preented a two layer teady tate mathematical model to tudy the mucu tranport in the repiratory tract due to air-flow by conidering cilia bed a porou matrix. In their paper, the effect of air motion i incorporated by precribing hear tre at the mucu-air interface. They howed that mucu tranport increae a the preure drop, hear tre due to air motion and poroity parameter increae. It wa alo oberved that mucu tranport decreae a the vicoity of erou layer or that of mucu increae, but at higher value of mucu vicoity doe not affect mucu tranport. It wa alo found that for a fixed total thickne of mucu and erou layer, there exit a erou layer thickne for which the mucu tranport i maximum. Though the air flow reitance in bronchial airway ha been tudied by Schroter and Sudlow [17] and Pedley et al.[13], the role of mucu interaction with mucu in bronchial clearance ha been emphaized by Clarke [7], Clarke et al. [8] and many other, including Puchelle et al. [14], Zahm et al. [], King et al. [9,10,11] in their experimental tudie. Scherer and Burtz [16] conducted experiment relevant to coughing and howed the importance of vicoity of the fluid. King et al. [1] alo tudied the interaction of airflow with the mucu gel in a imulated cough machine under In thi paper, a two layer teady tate mathematical model i preented to tudy the mucu tranport in the human lung airway by taking the following apect into account: (i) The erou layer fluid i conidered a incompreible Newtonian fluid while mucu layer i conidered a a vicoelatic fluid. (ii) The erou layer fluid i divided into two ub-layer, one in contact with the epithelium and the other in contact with the mucu. It i aumed that cilia are immotile and form a Paper ID: SUB

2 porou matrix bed in erou ub-layer, where flow may occur due to preure gradient a conidered by Beaver and Joeph [4] and due to poroity parameter a conidered by Verma [1]. No net flow i aumed in the erou ub layer in contact with the epithelium. (iii) The effect of air motion i incorporated by precribing air-velocity at the mucu-air interface a a boundary condition. (iv) The effect of preure gradient and gravity are alo incorporated in the model.. Mathematical Model The phyical ituation of the tranport of erou fluid and mucu in the human lung airway may be repreented by a planar two-layer fluid model a hown in Fig.1: In the erou ub-layer 0 y h e, no net flow of the fluid i aumed. However, in the erou ub-layer h e y h and in the mucu layer h y h m, the flow of repective fluid i governed by the interaction of cilia, air- motion in contact with the mucu, preure gradient preent in the fluid layer and acceleration due to gravity. The equation governing the motion of the erou layer fluid and the mucu under teady tate and low Reynold number flow approximation by taking the effect of acceleration due to gravity in the direction of flow, can be written a follow (Blake [5], King et al. [1]): Figure 1: Mucu tranport in the human lung airway Region I: Serou layer (h e y h ): μ u y = p x ρ g co α Region-II: Mucu layer (h y h m ): τ m y μ m u m y = p (1) x ρ m g co α () = τ m 1 + λ u m y where p i the preure that i contant acro the layer; u and u m are the velocity component of erou ub-layer fluid and mucu in x- direction repectively; ρ, µ, ρ m and µ m are their repective denitie and vicoitie; g i the acceleration due to gravity and α i the angle by which the airway in the human lung i inclined with the vertical. Here, h e i the mean thickne meaured from the urface of the epithelium to the tip of cilia during beating i.e. the interface between the two erou ub-layer; h i the thickne meaured from the urface of the epithelium to the (3) interface between erou ub-layer and mucu and h m i the thickne meaured from the urface of the epithelium to the mucu air-interface, λ (= μ m ) i the relaxation time, G i the G elatic modulu of mucu and τ m i the hear tre in the mucu layer. Equation (3) give the relationhip between the hear tre and velocity gradient for vico-elatic fluid in the cae of one dimenional flow (Tanner [18], King et al. [1]). The following boundary and matching condition are taken for the ytem of equation (1) - (3): Boundary Condition u = β u y, y = h e (4) u m = U a, y = h m (5) where β i the poroity parameter due to formation of porou matrix bed by immotile cilia in the erou ub- layer in contact with epithelium. The condition (5) implie that the air-velocity i continuou at the mucu- air interface and incorporate the effect of air- motion imilar to the analyi of Blake [5]. Matching Condition u = u m = U 1, y = h (6) μ u y = τ m, y = h (7) where U 1 i the mucu-erou ub layer interface velocity to be determined by uing equation(7). The condition (6) and (7) imply that the velocitie and the hear tree are continuou at mucu-erou layer interface. 3. Analytical Solution Solving (1)-(3) and uing boundary and matching condition (4)-(7), we get u = φ μ y h h e +βh e β +h h e y β h + h h e h e β +h h e β +y h e Paper ID: SUB and +U 1 β +h h e u m = φ m y h μ m y h m + φ m 3 {(y h 64μ m G m )+ y h } 4 h m h 4 + U a (9) where U 1 = φ β + h μ h e h h e φ m β + h μ h e h m h (10) φ = p ρ x g co α and φ m = p ρ x m g co α (11) The volumetric flow rate i.e. fluxe in the two layer are repectively defined a follow: h h m Q = u dy and Q h m = u m dy e h which after uing (8) and (9) are found a: Q = φ h h 4 e 3μ β +h h e and φ 6μ h β h h e β + h h e 8 h h e + 6β h h e + 3h e h (8) φ m 4μ β + h h e h m h h h e (1)

3 Q m = φ m 3 80μ m G φ m 4μ β + h h e h m h φ 4μ h m h 5 φ m 1μ m h m h 3 β + h h e h h e h m h + U a h m h (13) It can be een by uing equation of fluid continuity that Q and Q m are contant, therefore, from equation (1) and (13), we note that ( p ) i alo contant. Hence, replacing it x by the preure drop over the mean length L of the cilia forming porou matrix bed, the expreion for the fluxe may be written a: Q = φ 0 h h 4 e 3μ and β + h h e β h h e β + h h 8 h h e + 6β h h e e + 3h e h + φ m 0 β + h 4μ h e h m h h h e (14) + φ 0 6μ Q m = φ m0 3 80μ m G h m h 5 1μ m h m h 3 4μ β + h h e h m h + φ 0 4μ β + h h e h h e h m h + U a h m h (15) φ 0 = ( Δp L + ρ g co α) and φ m 0 = ( Δp L + ρ mg co α) (16) where Δp = p 0 p L, p = p 0 at x = 0, Δp = p L at x = L. It i noted that the effect of acceleration due to gravity i imilar to that of the preure drop. Now, when φ m0 = 0 and φ 0 = 0, then expreion for volumetric flow rate become: Q = 0 (17) Q m = U a h m h (18) Again, when U a = 0, the expreion for volumetric flow rate become: + φ 0 6μ Q m = φ m0 3 80μ m G Q = φ 0 3μ h h e 4 β + h h e β h h e β +h h e 8 h h e + 6β h h e + 3h e h + φ m 0 4μ β + h h e h m h h h e (19) h m h 5 1μ m h m h 3 4μ β + h h e h m h + φ 0 4μ β + h h e h h e h m h (0) Remark: The following remark can be made by cloe obervation of equation (14)-(15) and (17)-(0) regarding tranport rate Q and Q m : 1) From equation (15) and (0), we note that the effect of G on mucu tranport i dependent on U a, φ m0 and φ 0. When thee quantitie are zero, Q m doe not depend on either G or μ m. In general, when U a 0, φ 0 0 and φ m0 0, then mucu tranport decreae a it elatic modulu G increae. ) When φ 0 = 0 and φ m0 = 0, then from (17), we oberve that Q = 0 and from (18), we oberve that Q m increae a the air velocity U a at the mucu-air interface increae. It i alo een that Q m increae a the mucu thickne increae. The mucu tranport remain relatively independent of mucu vicoity, implying that mucu move a an elatic lab which i in the line with the finding of Ro and Corrin [16]. 3) When U a = 0, i.e. in the abence of air motion, from equation (19)-(0), we clearly note that Q and Q m both increae a the preure drop, acceleration due to gravity and poroity parameter increae. It i alo een in thi cae, that mucu tranport decreae a it elatic modulu G increae. Further, in thi cae, Q and Q m decreae a the vicoitie of mucu and erou layer increae. 4) Alo, when h h e i.e. for negligible thickne of erou layer, then from (19), we get Q = 0 and from (0), we note that Q m increae a the mucu thickne increae. Alo, for φ m0 = 0, Q m doe not depend on mucu vicoity. However, for φ m0 > 0, Q m decreae a mucu vicoity increae. Thi particular cae correpond with experimental tudie of King et al. [09, 10]. The prediction of the mathematical model are in general agreement with thoe obtained experimentally, i.e. poitive dependence on vicoity in the abence of erou layer. 5) To ee the effect of mucu thickne on mucu tranport in the general cae, we find from (15) the rate of change of Q m for a fixed total thickne of mucu and erou layer a follow: Paper ID: SUB Q m h = φ 0 4μ β + h h e h m h β + h h e h h e 4μ β + h h e + h m h h m h φ m φ m 0 4μ m G h m h h m h U a (1) From equation (1), we note that Q m can be negative, zero h or poitive depending on the value of h and other parameter. Thi implie that there may exit a critical value h 0 of h for which Q m may be maximum. Thu, for fixed total thickne of mucu and erou layer and for ome value of h > h 0, mucu tranport may increae with decreaing erou layer thickne (i.e., with increaing mucu thickne), while for the other value of h < h 0, mucu tranport may decreae with decreaing thickne of the erou layer. The former reult i in line with the experimental obervation of King et al. [10] a pointed out earlier, while the latter reult i imilar to that obtained by Ro and Corrin [15]. 6) From equation (15), we notice that the coefficient of 1 i G alway poitive; hence, the mucu tranport Q m increae a G decreae for given value of variou parameter. Thi implie that mucu tranport increae a it elatic modulu decreae in the general cae alo. The former reult i in line with the experimental obervation of

4 King et al. [09,10] for mucu gel imulant. Similar reult have been obtained by Verma and Tripathee [0]. 4. Reult and Dicuion To tudy the effect of variou parameter on mucu tranport rate quantitatively, the expreion for Q m given by (15) can be written in non-dimenional form a: φ 3 m0 Q m = 80μ mg + U a 1 h 5 1μ m 4μ 1 h 3 β + h h e 1 h + φ 0 β + h 4μ h e h h e 1 h 1 h () by uing the following non-dimenional parameter: β = β, h h e = h e, h m h = h, μ m h = μ, μ m = μ m, m μ 0 μ 0 φ 0 = φ 0 h m 1 μ 0 U 0, φ m0 = φ m 0 h m Q m h m U 0. (3) μ 0 U 0, U a = U a U 0, G = Gh m μ 0 U 0, λ 0 =, Q G m = where μ 0 i the vicoity of the erou ub layer fluid in contact with epithelium. Expreion for Q m given by () i plotted in Fig. to 7 uing the following et of parameter which have been calculated by uing typical value of variou characteritic related to airway (King et al.[1], Agarwal and Verma [1]): β = , h e = 0.1, h = , φ 0 = 1, φ m0 = 5 0, μ = 1 10, μ m = , U a = , λ 0 = (4) Figure 3: Variation of Q m with φ m0 for different value of μ m Fig.3 illutrate that for the fixed value of β = 0.0, h e = 0.10, h = 0.0, φ 0 = 1, φ m0 = 5, U a = 0.00 and λ 0 = 0.0, mucu tranport increae a the preure drop in mucu layer or acceleration due to gravity increae, but it decreae with increae in it vicoity, the relative decreae being larger at larger value of the preure drop or acceleration due to gravity. Thi reult i in line with the analytical reult of Agarwal and Verma [1], Verma [19,1] King et al. [1] and the experimental finding of King et al. [09]. Fig. illutrate that for the fixed value of β = 0.0, h e = 0.10, h = 0.0, φ 0 = 1, φ m0 = 5, U a = 0.00, and λ 0 = 0.0, mucu tranport decreae a the vicoity of the erou layer fluid or that of the mucu increae. However, increae in mucu vicoity at larger value do not have any ignificant effect on it tranport. Thi correpond to the reult that mucu move a an elatic lab (Ro and Corrin [15] and King et al. [10, 11]). Figure 4: Variation of Q m with U a for different value of μ m Figure : Variation of Q m with μ m for different value of μ Fig. 4 illutrate that for the fixed value of β = 0.0, h e = 0.10, h = 0.0, μ = 1, φ 0 = 1, φ m0 = 5 and λ 0 = 0.0, mucu tranport increae a the air-velocity (due to airmotion) at the mucu air-interface increae, but it decreae a it vicoity increae. Thi i in line with the analytical reult of Verma [1]. Paper ID: SUB

5 Fig.7 illutrate that for the fixed value of β = 0.0, h e = 0.10, μ = 1, φ 0 = 1, φ m0 = 5 and U a = 0.00, mucu tranport deceae a h increae or a the mucu vicoity increae. Thi figure alo illutrate that the mucu tranport become independent with λ 0 for a fixed value of mucu vicoity, from which we conclude that the mucu tranport decreae a it elatic modulu increae. Thi i in line with the analytical reult of King et al. [1]. Figure 5: Variation of Q m with h for different value of μ m Fig 5 illutrate that for the fixed value of β = 0.0, h e = 0.10, μ = 1, φ 0 = 1, φ m0 = 5, U a = 0.00 and λ 0 = 0.0, mucu tranport increae a h increae upto a critical value of h (approximately equal to 0.36) after which it tart decreaing with increaing h. Since Q m approache to unity, thi implie that for a fixed total thickne of mucu and erou layer, there exit an optimum value of Q m for ome value of erou layer thickne. The concluion correponding to decreae in mucu tranport with decreae in erou layer thickne i in line with the analyi of Ro and Corrin [15], Agarwal and Verma [1] and Verma [19]. Fig. 6 how the variation of Q m with β for different value of μ m and h and fixed value of h e = 0.1, μ = 1, U a = 0.0, φ 0 = 1, φ m0 = 0 and λ 0 = 0.0. Thi figure illutrate that mucu tranport rate Q m increae a h increae upto a maximum value of h after which it tart decreaing with increaing value of h. Thi figure alo illutrate that the mucu tranport decreae a it vicoity increae. It i alo clear from the figure that the mucu tranport increae a the poroity parameter increae. Figure 6: Variation of Q m with β for different value of μ m and h Figure 7: Variation of Q m with λ 0 for different value of μ m and h. 5. Concluion In thi paper, we have preented a planar two-layer mathematical model to tudy mucu tranport in the human lung airway under teady tate condition by incorporating the effect of poroity parameter due to formation of porou matrix bed by certain immotile cilia and air-velocity due to air-motion by conidering mucu a a vico-elatic fluid. The effect of air-motion i conidered by precribing air-velocity at the mucu air interface. The governing equation of motion are written and have been olved analytically and the effect of variou parameter on the mucu tranport rate have been dicued. Furthermore, the effect of value of variou parameter on mucu tranport rate have been computed numerically and have been explained graphically. It i hown that mucu tranport increae a the preure drop, air velocity due to air-motion and poroity parameter due to formation of porou matrix bed by certain immotile cilia increae. It i alo noted that the effect of acceleration due to gravity i imilar to that of the preure drop. It i alo oberved that mucu tranport decreae a the vicoity of erou layer fluid or that of mucu increae, but any increae in mucu vicoity at it larger value doe not eem to affect the mucu tranport. It i alo found that for given total depth of erou layer and mucu, there exit a erou fluid layer thickne for which mucu tranport i maximum. It i alo een that mucu tranport decreae a it elatic modulu increae. Reference Paper ID: SUB

6 [1] Agarwal, M. and Verma, V.S. : A planar model for mucociliary tranport: Effect of air motion and poroity. Proc. Acad. Sci. India, 67(A) II, pp , [] Agarwal, M., King, M. and Shukla, J.B.: Mucu tranport in a miniaturized imulated cough machine: Effect of contruction and erou layer imulant. Biorheology 6, , [3] Barton, C. and Raynor, S.: Analytic invetigation of cilia induced mucu flow. Bull. Math. Biophy. 9, pp , [4] Beaver, G. S. and Joeph, D. D. : Boundary condition at a naturally permeable wall. J. Fluid. Mech. 30(1), pp , [5] Blake, J. R.: On movement of mucu in lung. Journal of Biomechanic. 8(3-4), pp , [6] Blake, J. R. and Winet, H. (1980): On the Mechanic of muco-ciliary tranport. Biorheology. 17(1-), pp [7] Clarke, S.W. : The role of two phae flow in bronchial clearance. Bull. Eur. Phyiopath. Repir. 9, pp ,1973. [8] Clarke, S.W.; Jone, J.G. and Oliver, D.R.: Reitance two-phae ga-liquid flow in airway. J. Appl. Phyiol. 9, pp , [9] King, M.: Mucu, mucociliary clearance and coughing: Section 3.3. In: Repiratory function in dieae. Bate, D.V. (Ed.) Philadelphia, aunder, [10] King, M.; Brock, G. and Lundell, C. : Clearance of mucu by imulated cough. J. Appl. Phyiol. 58, pp , [11] King, M.; Chang, H.K. and Weber, M. E.: Reitance of mucu-lined tube to tudy ocillatory airflow. J. Appl. Phyiol. 5, pp , 198. [1] King, M.; Agarwal, M. and Shukla, J. B.: A planer model for muco-ciliary tranport: Effect of mucu vico-elaticity. Biorheology 30, 49-61,1993. [13] Pedley, T. J., Schroter, R. C. and Sudlow, M. F.: The prediction of preure drop and variation of reitance within the human bronchialairway. Re. Phyiol. 9, pp , [14] Puchelle, E.; Zahm, J.M. and Duvivier, C.: Spinnability of bronchial mucu: Relationhip with vicoelaticity and mucu tranport propertie. Biorheology 0, pp. 65-7, [15] Ro, S.M. and Corrin, S.: Reult of an analytical model of mucociliary pumping. J. Appl. Phyiol. 37, pp , [16] Scherer, P. W. and Burtz, L.: Fluid mechanical experiment relevant to coughing. J. Biomech.11, ,1978. [17] Schroter, R. C., and Sudlow, M. F.: Flow pattern in model of human bronchial airway. J. Biomech. 11, pp , [18] Tanner, R.I.: Full-film lubrication theory for a Maxwell liquid, Int. J. Mech. Sci. 1, 06-15, [19] Verma, V. S. : A planar model for mucu tranport in human repiratory tract: Effect of air-flow, poroity and and mucu vicoelaticity. J. Nat. Acad. Math. 4, pp.53-60, 010. [0] Verma, V. S. and Tripathee, S. M.: A planar model for muco-ciliary tranport in the human lung: Effect of mucu vicoelaticity, cilia beating and poroity. IJMRS International Journal of Mathematical Modeling and Phyical Science. Vol.01,Iue 01, pp.19-5, 013. [1] Verma,V.S.: A mathematical tudy on mucu tranport in the lung. J. Nat. Acad. Math.Vol.1, pp , 007. [] Zahm, J.M.;King,M.;Duvivier, C.; Pierot, D. and Puchelle, E.: influence of airway urface liquid (ol. phae) on clearance by cough. Biorheology 6, pp ,1989. Author Profile Dr. Vijai Shanker Verma, Aociate Profeor, Department of Mathematic and Statitic, DDU Gorakhpur Univerity, Gorakhpur ha received hi M.Sc. and Ph.D. degree in Mathematic from Lucknow Univerity, Lucknow. He i recipient of National Merit Scholarhip from U.P. Board, Allahabad, Ram Ballabh Memorial Scholarhip from Lucknow Univerity, Lucknow and ha been honoured for hi ocial and academic achievement by Hon'ble Prime Miniter Sri Chandra Shekhar in 004. He i life-member of variou academic bodie namely; Bharat Ganita Parihad, National Academy of Mathematic, Indian Society of Mathematic and Mathematical Science and The Tenor Society. He ha publihed everal reearch paper in variou Journal of National and International repute. He ha alo publihed everal text book. Paper ID: SUB