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2015, 12(5): 1107-1126. doi: 10.3934/mbe.2015.12.1107

Cilium height difference between strokes is more effective in driving fluid transport in mucociliary clearance: A numerical study

1. 

Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN 46556, United States

2. 

Department of Mathematics and Statistics, Georgia State University, Atlanta, GA 30303, United States

Received  January 2015 Revised  March 2015 Published  June 2015

Mucociliary clearance is the first line of defense in our airway. The purpose of this study is to identify and study key factors in the cilia motion that influence the transport ability of the mucociliary system. Using a rod-propel-fluid model, we examine the effects of cilia density, beating frequency, metachronal wavelength, and the extending height of the beating cilia. We first verify that asymmetry in the cilia motion is key to developing transport in the mucus flow. Next, two types of asymmetries between the effective and recovery strokes of the cilia motion are considered, the cilium beating velocity difference and the cilium height difference. We show that the cilium height difference is more efficient in driving the transport, and the more bend the cilium during the recovery stroke is, the more effective the transport would be. It is found that the transport capacity of the mucociliary system increases with cilia density and cilia beating frequency, but saturates above by a threshold value in both density and frequency. The metachronal wave that results from the phase lag among cilia does not contribute much to the mucus transport, which is consistent with the experimental observation of Sleigh (1989). We also test the effect of mucus viscosity, whose value is found to be inversely proportional to the transport ability. While multiple parts have to interplay and coordinate to allow for most effective mucociliary clearance, our findings from a simple model move us closer to understanding the effects of the cilia motion on the efficiency of this clearance system.
Citation: Ling Xu, Yi Jiang. Cilium height difference between strokes is more effective in driving fluid transport in mucociliary clearance: A numerical study. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1107-1126. doi: 10.3934/mbe.2015.12.1107
References:
[1]

B. A. Afzelius, Ultrastructural basis for ciliary motility,, Eur J Respir Dis Suppl., 128 (1983), 280. Google Scholar

[2]

S. A. Baba, Regular steps in bending cilia during the effective stroke,, Nature, 282 (1979), 717. doi: 10.1038/282717a0. Google Scholar

[3]

C. Barton and S. Raynor, Analytical investigation of cilia induced mucous flow,, Bull Math Biophys., 29 (1967), 419. doi: 10.1007/BF02476581. Google Scholar

[4]

J. R. Blake, A note on the image system for a stokeslet in a no slip boundary,, Proc. Gamb. Phil. Soc., 70 (1971), 303. doi: 10.1017/S0305004100049902. Google Scholar

[5]

J. R. Blake, A model for the micro-structure in ciliated organisms,, J. Fluid Mech., 55 (1972), 1. Google Scholar

[6]

B. Button, L.-H. Cai, C. Ehre, M. Kesimer, D. B. Hill, J. K. Sheehan, R. C. Boucher and M. Rubinstein, A periciliary brush promotes the lung health by separating the mucus layer from airway epithelia,, Science, 337 (2012), 937. doi: 10.1126/science.1223012. Google Scholar

[7]

A. Braiman and Z. Priel, Efficient mucociliary transport relies on efficient regulation of ciliary beating,, Respir Physiol Neurobiol., 163 (2008), 202. Google Scholar

[8]

D. L. Brown, R. Cortez and M. L. Minion, Accurate projection methods for the incompressible Navier Stokes equations,, J. Comput Phys., 168 (2001), 464. doi: 10.1006/jcph.2001.6715. Google Scholar

[9]

M. A. Chilvers and C. O'Callaghan, Analysis of ciliary beat pattern and beat frequency using digital high speed imaging: comparison with the photomultiplier and photodiode methods,, Thorax, 55 (2000), 314. doi: 10.1136/thorax.55.4.314. Google Scholar

[10]

M. A. Chilvers, A. Rutman and C. O'Callaghan, Ciliary beat pattern is associated with specific ultrastructural defects in primary ciliary dyskinesia,, J Allergy Clin Immunol., 112 (2003), 518. doi: 10.1016/S0091-6749(03)01799-8. Google Scholar

[11]

J. R. Clamp, Chemical aspects of mucus. General considerations,, Br Med Bull., 34 (1978), 25. Google Scholar

[12]

R. Cortez, The method of regularized Stokeslets,, SIAM J. Sci. Comput., 23 (2001), 1204. doi: 10.1137/S106482750038146X. Google Scholar

[13]

R. Cortez, L. Fauci and A. Medovikov, The method of regularized Stokeslets in three dimensions: Analysis, validation, and application to helical swimming,, Phys. Fluids, 17 (2005). doi: 10.1063/1.1830486. Google Scholar

[14]

R. H. Dillon, L Fauci, C. Omoto and X. Yang, Fluid dynamic models of flagellar and ciliary beating,, Ann N Y Acad Sci., 1101 (2007), 494. Google Scholar

[15]

S. H. Donaldson, W. D. Bennett, K. L. Zeman, M. R. Knowles, R. Tarran and R. C. Boucher, Mucus clearance and lung function in cystic fibrosis with hypertonic saline,, N. ENGL. J. MED., 354 (2006), 241. Google Scholar

[16]

J. Elgeti and G. Gompper, Emergence of metachronal waves in cilia arrays,, PNAS, 110 (2013), 4470. doi: 10.1073/pnas.1218869110. Google Scholar

[17]

D. Eshel and Z. Priel, Characterization of metachronal wave of beating cilia on forg's palate epithelium in tissue culture,, J. Physiol., 388 (1987), 1. Google Scholar

[18]

L. Fauci, C. Peskin, A Computational Model of Aquatic Animal Locomotion,, J. Comput Phys., 77 (1988), 85. Google Scholar

[19]

H. Flores, E. Lobaton, S. Mendez-Diez, S. Tlupova and R. Cortez, A study of bacterial flagellar bundling,, Bull Math Biol., 67 (2005), 137. doi: 10.1016/j.bulm.2004.06.006. Google Scholar

[20]

W. M. Foster, E. Langenback and E. H. Bergofsky, Measurement to tracheal and bronchial mucus velocities in man: relation to lung clearance,, J Appl Physiol Respir Environ Exerc Physiol., 48 (1980), 965. Google Scholar

[21]

M. Friedman, R. Dougherty, S. R. Nelson, R. P. White, M. A. Sackner and A. Wanner, Acute effects of an aerosol hair spray on tracheal mucociliary transport,, Am Rev Respir Dis., 116 (1977), 281. Google Scholar

[22]

L. Gheber, A. Korngreen and Z. Priel, Effect of viscosity on metachrony in mucus propelling cilia,, Cell Motil Cytoskeleton, 39 (1998), 9. Google Scholar

[23]

J. Kim and P. Moin, Application of a fractional step method to incompressible Navier-Stokes equations,, J. Comput Phys., 59 (1985), 308. doi: 10.1016/0021-9991(85)90148-2. Google Scholar

[24]

M. R. Knowles and R. C. Boucher, Mucus clearance as a primary innate defense mechanism for mammalian airways,, J Clin Invest., 109 (2002), 571. Google Scholar

[25]

M. C. Lai and C. S. Peskin, An immersed boundary method with formal second-order accuracy and reduced numerical viscosity,, J. Comput Phys., 160 (2000), 705. doi: 10.1006/jcph.2000.6483. Google Scholar

[26]

R. L. Leopold, M. J. O'Mahony, X. J. Lian, A. E. Tilley, B. G. Harvey and R. G. Crystal, Smoking is associated with shortened airway cilia,, PloS One, 4 (2009). doi: 10.1371/journal.pone.0008157. Google Scholar

[27]

R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, Steady and Time Dependent Problems,, {SIAM}, (2007). doi: 10.1137/1.9780898717839. Google Scholar

[28]

A. Livraghi and S. H. Randell, Cystic fibrosis and other respiratory diseases of impaired mucus clearance,, Toxicol Pathol, 35 (2007), 116. doi: 10.1080/01926230601060025. Google Scholar

[29]

P. M. Low, C. L. Luk, M. J. Dulfano and R. J. Finch, Ciliary beat frequency of human respiratory tract by different sampling techniques,, Am Rev Respir Dis., 130 (1984), 497. Google Scholar

[30]

M. R. Marino and E. Aiello, Cinemicrographic analysis of beat dynamics of human respiratory cilia,, Cell Motility, 2 (1982), 35. doi: 10.1002/cm.970020709. Google Scholar

[31]

H. Matsui, S. H. Randell, S. W. Peretti, C. W. Davis and R. C. Boucher, Coordinated clearance of periciliary liquid and mucus from airway surfaces,, J Clin Invest., 102 (1998), 1125. doi: 10.1172/JCI2687. Google Scholar

[32]

M. Salathe, Cilia and Mucus, from Development to Respiratory Defense,, CRC Press, (2001). Google Scholar

[33]

S. M. Mitran, Metachronal wave formation in a model of pulmonary cilia,, Comput Struct., 85 (2007), 763. doi: 10.1016/j.compstruc.2007.01.015. Google Scholar

[34]

P. G. Noone, M. W. Leigh, A. Sannuti, S. L. Minnix, J. L. Carson, M. Hazucha, M. A. Zariwala and M. R. Knowles, Primary ciliary dyskinesia: Diagnostic and phenotypic features,, Am J Respir Crit Care Med, 169 (2004), 459. doi: 10.1164/rccm.200303-365OC. Google Scholar

[35]

C. S. Peskin, Numerical analysis of blood flow in the heart,, J. Comput Phys., 25 (1977), 220. doi: 10.1016/0021-9991(77)90100-0. Google Scholar

[36]

E. M. Purcell, Life at low Reynolds number,, AIP Conf. Proc., 28 (1976). doi: 10.1063/1.30370. Google Scholar

[37]

W. S. Sale and P. Satir, Direction of active sliding of microtubules in Tetrahymena cilia,, PNAS, 74 (1977), 2045. doi: 10.1073/pnas.74.5.2045. Google Scholar

[38]

M. J. Sanderson and M. A. Sleigh, Ciliary activity of cultured rabbit tracheal epithelium: Beat pattern and metachrony,, J Cell Sci., 47 (1981), 331. Google Scholar

[39]

P. Satir, Studies on cilia: II. examination of the distal region of the ciliary shaft and the role of the filaments in motility,, J Cell Biol., 26 (1965), 805. doi: 10.1083/jcb.26.3.805. Google Scholar

[40]

A. Schmid and M. Salathe, Ciliary beat co-ordination by calcium,, Biol Cell, 103 (2011), 159. doi: 10.1042/BC20100120. Google Scholar

[41]

P. R. Sears, K. Thompson, M. R. Knowles and C. W. Davis, Human airway ciliary dynamics,, Am J Physiol Lung Cell Mol Phyiol., 304 (2013). doi: 10.1152/ajplung.00105.2012. Google Scholar

[42]

M. A. Sleigh, Ciliary function in transport of mucus,, Eur J Respir Dis. Suppl., 128 (1983), 287. Google Scholar

[43]

M. A. Sleigh, Adaptations of ciliary systems for the propulsion of water and mucus,, Comp Biochem Physiol A Comp Physiol., 94 (1989), 359. doi: 10.1016/0300-9629(89)90559-8. Google Scholar

[44]

G. Taylor, Analysis of the Swimming of Microscopic Organisms,, Proc. R. Soc. Lond. A, 209 (1951), 447. Google Scholar

[45]

E. O. Tuck, A note on a swimming problem,, J. Fluid Mech., 31 (1968), 305. Google Scholar

[46]

E. Tuomanen, The surface of mammalian respiratory cilia: interactions between cilia and respiratory pathogens,, Ciliary and Flagellar Membranes, (1990), 363. doi: 10.1007/978-1-4613-0515-6_14. Google Scholar

[47]

X. Yang, R. H. Dillon and L. J. Fauci, An integrative computational model of multiciliary beating,, Bull Math Biol., 70 (2008), 1192. doi: 10.1007/s11538-008-9296-3. Google Scholar

show all references

References:
[1]

B. A. Afzelius, Ultrastructural basis for ciliary motility,, Eur J Respir Dis Suppl., 128 (1983), 280. Google Scholar

[2]

S. A. Baba, Regular steps in bending cilia during the effective stroke,, Nature, 282 (1979), 717. doi: 10.1038/282717a0. Google Scholar

[3]

C. Barton and S. Raynor, Analytical investigation of cilia induced mucous flow,, Bull Math Biophys., 29 (1967), 419. doi: 10.1007/BF02476581. Google Scholar

[4]

J. R. Blake, A note on the image system for a stokeslet in a no slip boundary,, Proc. Gamb. Phil. Soc., 70 (1971), 303. doi: 10.1017/S0305004100049902. Google Scholar

[5]

J. R. Blake, A model for the micro-structure in ciliated organisms,, J. Fluid Mech., 55 (1972), 1. Google Scholar

[6]

B. Button, L.-H. Cai, C. Ehre, M. Kesimer, D. B. Hill, J. K. Sheehan, R. C. Boucher and M. Rubinstein, A periciliary brush promotes the lung health by separating the mucus layer from airway epithelia,, Science, 337 (2012), 937. doi: 10.1126/science.1223012. Google Scholar

[7]

A. Braiman and Z. Priel, Efficient mucociliary transport relies on efficient regulation of ciliary beating,, Respir Physiol Neurobiol., 163 (2008), 202. Google Scholar

[8]

D. L. Brown, R. Cortez and M. L. Minion, Accurate projection methods for the incompressible Navier Stokes equations,, J. Comput Phys., 168 (2001), 464. doi: 10.1006/jcph.2001.6715. Google Scholar

[9]

M. A. Chilvers and C. O'Callaghan, Analysis of ciliary beat pattern and beat frequency using digital high speed imaging: comparison with the photomultiplier and photodiode methods,, Thorax, 55 (2000), 314. doi: 10.1136/thorax.55.4.314. Google Scholar

[10]

M. A. Chilvers, A. Rutman and C. O'Callaghan, Ciliary beat pattern is associated with specific ultrastructural defects in primary ciliary dyskinesia,, J Allergy Clin Immunol., 112 (2003), 518. doi: 10.1016/S0091-6749(03)01799-8. Google Scholar

[11]

J. R. Clamp, Chemical aspects of mucus. General considerations,, Br Med Bull., 34 (1978), 25. Google Scholar

[12]

R. Cortez, The method of regularized Stokeslets,, SIAM J. Sci. Comput., 23 (2001), 1204. doi: 10.1137/S106482750038146X. Google Scholar

[13]

R. Cortez, L. Fauci and A. Medovikov, The method of regularized Stokeslets in three dimensions: Analysis, validation, and application to helical swimming,, Phys. Fluids, 17 (2005). doi: 10.1063/1.1830486. Google Scholar

[14]

R. H. Dillon, L Fauci, C. Omoto and X. Yang, Fluid dynamic models of flagellar and ciliary beating,, Ann N Y Acad Sci., 1101 (2007), 494. Google Scholar

[15]

S. H. Donaldson, W. D. Bennett, K. L. Zeman, M. R. Knowles, R. Tarran and R. C. Boucher, Mucus clearance and lung function in cystic fibrosis with hypertonic saline,, N. ENGL. J. MED., 354 (2006), 241. Google Scholar

[16]

J. Elgeti and G. Gompper, Emergence of metachronal waves in cilia arrays,, PNAS, 110 (2013), 4470. doi: 10.1073/pnas.1218869110. Google Scholar

[17]

D. Eshel and Z. Priel, Characterization of metachronal wave of beating cilia on forg's palate epithelium in tissue culture,, J. Physiol., 388 (1987), 1. Google Scholar

[18]

L. Fauci, C. Peskin, A Computational Model of Aquatic Animal Locomotion,, J. Comput Phys., 77 (1988), 85. Google Scholar

[19]

H. Flores, E. Lobaton, S. Mendez-Diez, S. Tlupova and R. Cortez, A study of bacterial flagellar bundling,, Bull Math Biol., 67 (2005), 137. doi: 10.1016/j.bulm.2004.06.006. Google Scholar

[20]

W. M. Foster, E. Langenback and E. H. Bergofsky, Measurement to tracheal and bronchial mucus velocities in man: relation to lung clearance,, J Appl Physiol Respir Environ Exerc Physiol., 48 (1980), 965. Google Scholar

[21]

M. Friedman, R. Dougherty, S. R. Nelson, R. P. White, M. A. Sackner and A. Wanner, Acute effects of an aerosol hair spray on tracheal mucociliary transport,, Am Rev Respir Dis., 116 (1977), 281. Google Scholar

[22]

L. Gheber, A. Korngreen and Z. Priel, Effect of viscosity on metachrony in mucus propelling cilia,, Cell Motil Cytoskeleton, 39 (1998), 9. Google Scholar

[23]

J. Kim and P. Moin, Application of a fractional step method to incompressible Navier-Stokes equations,, J. Comput Phys., 59 (1985), 308. doi: 10.1016/0021-9991(85)90148-2. Google Scholar

[24]

M. R. Knowles and R. C. Boucher, Mucus clearance as a primary innate defense mechanism for mammalian airways,, J Clin Invest., 109 (2002), 571. Google Scholar

[25]

M. C. Lai and C. S. Peskin, An immersed boundary method with formal second-order accuracy and reduced numerical viscosity,, J. Comput Phys., 160 (2000), 705. doi: 10.1006/jcph.2000.6483. Google Scholar

[26]

R. L. Leopold, M. J. O'Mahony, X. J. Lian, A. E. Tilley, B. G. Harvey and R. G. Crystal, Smoking is associated with shortened airway cilia,, PloS One, 4 (2009). doi: 10.1371/journal.pone.0008157. Google Scholar

[27]

R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, Steady and Time Dependent Problems,, {SIAM}, (2007). doi: 10.1137/1.9780898717839. Google Scholar

[28]

A. Livraghi and S. H. Randell, Cystic fibrosis and other respiratory diseases of impaired mucus clearance,, Toxicol Pathol, 35 (2007), 116. doi: 10.1080/01926230601060025. Google Scholar

[29]

P. M. Low, C. L. Luk, M. J. Dulfano and R. J. Finch, Ciliary beat frequency of human respiratory tract by different sampling techniques,, Am Rev Respir Dis., 130 (1984), 497. Google Scholar

[30]

M. R. Marino and E. Aiello, Cinemicrographic analysis of beat dynamics of human respiratory cilia,, Cell Motility, 2 (1982), 35. doi: 10.1002/cm.970020709. Google Scholar

[31]

H. Matsui, S. H. Randell, S. W. Peretti, C. W. Davis and R. C. Boucher, Coordinated clearance of periciliary liquid and mucus from airway surfaces,, J Clin Invest., 102 (1998), 1125. doi: 10.1172/JCI2687. Google Scholar

[32]

M. Salathe, Cilia and Mucus, from Development to Respiratory Defense,, CRC Press, (2001). Google Scholar

[33]

S. M. Mitran, Metachronal wave formation in a model of pulmonary cilia,, Comput Struct., 85 (2007), 763. doi: 10.1016/j.compstruc.2007.01.015. Google Scholar

[34]

P. G. Noone, M. W. Leigh, A. Sannuti, S. L. Minnix, J. L. Carson, M. Hazucha, M. A. Zariwala and M. R. Knowles, Primary ciliary dyskinesia: Diagnostic and phenotypic features,, Am J Respir Crit Care Med, 169 (2004), 459. doi: 10.1164/rccm.200303-365OC. Google Scholar

[35]

C. S. Peskin, Numerical analysis of blood flow in the heart,, J. Comput Phys., 25 (1977), 220. doi: 10.1016/0021-9991(77)90100-0. Google Scholar

[36]

E. M. Purcell, Life at low Reynolds number,, AIP Conf. Proc., 28 (1976). doi: 10.1063/1.30370. Google Scholar

[37]

W. S. Sale and P. Satir, Direction of active sliding of microtubules in Tetrahymena cilia,, PNAS, 74 (1977), 2045. doi: 10.1073/pnas.74.5.2045. Google Scholar

[38]

M. J. Sanderson and M. A. Sleigh, Ciliary activity of cultured rabbit tracheal epithelium: Beat pattern and metachrony,, J Cell Sci., 47 (1981), 331. Google Scholar

[39]

P. Satir, Studies on cilia: II. examination of the distal region of the ciliary shaft and the role of the filaments in motility,, J Cell Biol., 26 (1965), 805. doi: 10.1083/jcb.26.3.805. Google Scholar

[40]

A. Schmid and M. Salathe, Ciliary beat co-ordination by calcium,, Biol Cell, 103 (2011), 159. doi: 10.1042/BC20100120. Google Scholar

[41]

P. R. Sears, K. Thompson, M. R. Knowles and C. W. Davis, Human airway ciliary dynamics,, Am J Physiol Lung Cell Mol Phyiol., 304 (2013). doi: 10.1152/ajplung.00105.2012. Google Scholar

[42]

M. A. Sleigh, Ciliary function in transport of mucus,, Eur J Respir Dis. Suppl., 128 (1983), 287. Google Scholar

[43]

M. A. Sleigh, Adaptations of ciliary systems for the propulsion of water and mucus,, Comp Biochem Physiol A Comp Physiol., 94 (1989), 359. doi: 10.1016/0300-9629(89)90559-8. Google Scholar

[44]

G. Taylor, Analysis of the Swimming of Microscopic Organisms,, Proc. R. Soc. Lond. A, 209 (1951), 447. Google Scholar

[45]

E. O. Tuck, A note on a swimming problem,, J. Fluid Mech., 31 (1968), 305. Google Scholar

[46]

E. Tuomanen, The surface of mammalian respiratory cilia: interactions between cilia and respiratory pathogens,, Ciliary and Flagellar Membranes, (1990), 363. doi: 10.1007/978-1-4613-0515-6_14. Google Scholar

[47]

X. Yang, R. H. Dillon and L. J. Fauci, An integrative computational model of multiciliary beating,, Bull Math Biol., 70 (2008), 1192. doi: 10.1007/s11538-008-9296-3. Google Scholar

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