Springer Nature is making Coronavirus research free. View research | View latest news | Sign up for updates

Numerical study on the dynamics of primary cilium in pulsatile flows by the immersed boundary-lattice Boltzmann method


An explicit immersed boundary-lattice Boltzmann method is applied to numerically investigate the dynamics of primary cilium in pulsatile blood flows with two-way fluid–structure interaction considered. To well characterize the effect of cilium basal body on cilium dynamics, the cilium base is modeled as a nonlinear rotational spring attached to the cilium’s basal end as proposed by Resnick (Biophys J 109:18–25, 2015. After several careful validations, the fluid–cilium interaction system is investigated in detail at various pulsatile flow conditions that are characterized by peak Reynolds numbers (\(Re_{{\rm peak}}\)) and Womersley numbers (\(Wo\)). The periodic flapping of primary cilium observed in our simulations is very similar to the in vivo ciliary oscillation captured by O’Connor et al. (Cilia 2:8, 2013. The cilium’s dynamics is found to be closely related to the \(Re_{{\rm peak}}\) and \(Wo\). Increase the \(Re_{{\rm peak}}\) or decrease the \(Wo\) bring to an increase in the cilium’s flapping amplitude, tip angular speed, basal rotation, and maximum tensile stress. It is also demonstrated that by reducing the \(Re_{{\rm peak}}\) or enhancing the \(Wo\) to a certain level, one can shift the flapping pattern of cilium from its original two-side one to a one-side one, making the stretch only happen on one particular side. During the flapping process, the location of the maximum tensile stress is not always found at the basal region; instead, it is able to propagate from time to time within a certain distance to the base. Due to the obstruction of the primary cilium, the distribution of wall shear stress no longer remains uniform as in the absence of cilia. It oscillates in space with the minimum magnitude which is always found near where the cilium is located. The presence of cilium also reduces the overall level of wall shear stress, especially at the region near the cilium’s anchor point.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17


  1. Boo YC, Jo H (2003) Flow-dependent regulation of endothelial nitric oxide synthase: role of protein kinases. Am J Physiol Cell Physiol 285:C499–C508.

  2. Connell BSH, Yue DKP (2007) Flapping dynamics of a flag in a uniform stream. J Fluid Mech 581:33–68.

  3. Davies PF (2009) Hemodynamic shear stress and the endothelium in cardiovascular pathophysiology. Nat Clin Pract Cardiovasc Med 6:16–26.

  4. Downs ME, Nguyen AM, Herzog FA, Hoey DA, Jacobs CR (2014) An experimental and computational analysis of primary cilia deflection under fluid flow. Comput Methods Biomech Biomed Eng 17:2–10.

  5. Goetz SC, Anderson KV (2010) The primary cilium: a signalling centre during vertebrate development. Nat Rev Genet 11:331–344.

  6. Guo Z, Zheng C, Shi B (2002) Discrete lattice effects on the forcing term in the lattice Boltzmann method. Phys Rev E 65:046308.

  7. Hagiwara H, Ohwada N, Aoki T, Suzuki T, Takata K (2008) The primary cilia of secretory cells in the human oviduct mucosa. Med Mol Morphol 41:193–198.

  8. Han YF, Ganatos P, Weinbaum S (2005) Transmission of steady and oscillatory fluid shear stress across epithelial and endothelial surface structures. Phys Fluids 17:13.

  9. Hanasoge S, Hesketh PJ, Alexeev A (2018) Metachronal motion of artificial magnetic cilia. Soft Matter 14:3689–3693.

  10. Hassounah NB, Bunch TA, McDermott KM (2012) Molecular pathways: the role of primary cilia in cancer progression and therapeutics with a focus on hedgehog signaling. Clin Cancer Res 18:2429–2435.

  11. Huang W-X, Shin SJ, Sung HJ (2007) Simulation of flexible filaments in a uniform flow by the immersed boundary method. J Comput Phys 226:2206–2228.

  12. Khayyeri H, Barreto S, Lacroix D (2015) Primary cilia mechanics affects cell mechanosensation: a computational study. J Theor Biol 379:38–46.

  13. Kruger T, Holmes D, Coveney PV (2014) Deformability-based red blood cell separation in deterministic lateral displacement devices—a simulation study Biomicrofluidics 8:15.

  14. Ladd AJC (1994a) Numerical simulations of particulate suspensions via a discretized Boltzmann-equation. 1. Theoretical foundation. J Fluid Mech 271:285–309.

  15. Ladd AJC (1994b) Numerical simulations of particulate suspensions via a discretized Boltzmann-equation. 2. Numerical results. J Fluid Mech 271:311–339.

  16. Lim YC, McGlashan SR, Cooling MT, Long DS (2015) Culture and detection of primary cilia in endothelial cell models. Cilia 4:11.

  17. Liu W, Murcia NS, Duan Y, Weinbaum S, Yoder BK, Schwiebert E, Satlin LM (2005) Mechanoregulation of intracellular Ca2+ concentration is attenuated in collecting duct of monocilium-impaired orpk mice. Am J Physiol Ren Physiol 289:F978.

  18. Masyuk AI, Masyuk TV, LaRusso NF (2008) Cholangiocyte primary cilia in liver health and disease. Dev Dyn 237:2007–2012.

  19. McDonald DA (1955) The relation of pulsatile pressure to flow in arteries. J Physiol 127:533.

  20. McGlashan SR, Cluett EC, Jensen CG, Poole CA (2008) Primary cilia in osteoarthritic chondrocytes: from chondrons to clusters. Dev Dyn 237:2013–2020.

  21. Menzl I et al (2014) Loss of primary cilia occurs early in breast cancer development Cilia 3:7.

  22. Nauli SM et al (2003) Polycystins 1 and 2 mediate mechanosensation in the primary cilium of kidney cells. Nature Genet 33:129–137.

  23. Nguyen AM, Jacobs CR (2013) Emerging role of primary cilia as mechanosensors in osteocytes. Bone 54:196–204.

  24. Nguyen AM, Young YN, Jacobs CR (2015) The primary cilium is a self-adaptable, integrating nexus for mechanical stimuli and cellular signaling. Biol Open 4:1733–1738.

  25. Niu XD, Shu C, Chew YT, Peng Y (2006) A momentum exchange-based immersed boundary-lattice Boltzmann method for simulating incompressible viscous flows. Phys Lett A 354:173–182.

  26. O’Connor AK et al (2013) An inducible CiliaGFP mouse model for in vivo visualization and analysis of cilia in live tissue. Cilia 2:8.

  27. O’Connor J, Revell A, Mandal P, Day P (2016a) Application of a lattice Boltzmann-immersed boundary method for fluid-filament dynamics and flow sensing. J Biomech 49:2143–2151.

  28. O’Connor J, Revell A, Mandal P, Day P (2016b) Application of a lattice Boltzmann-immersed boundary method for fluid-filament dynamics and flow sensing. J Biomech 49:2143–2151.

  29. Peskin CS (2003) The immersed boundary method. Acta Numer 11:479–517.

  30. Praetorius HA, Spring KR (2001) Bending the MDCK cell primary cilium increases intracellular calcium. J Membr Biol 184:71–79.

  31. Praetorius HA, Spring KR (2003) Removal of the MDCK cell primary cilium abolishes flow sensing. J Membr Biol 191:69–76.

  32. Praetorius HA, Frokiaer J, Nielsen S, Spring KR (2003) Bending the primary cilium opens Ca2+-sensitive intermediate-conductance K+ channels in MDCK cells. J Membr Biol 191:193–200.

  33. Resnick A (2015) Mechanical properties of a primary cilium as measured by resonant oscillation. Biophys J 109:18–25.

  34. Resnick A (2016) HIF stabilization weakens primary cilia. PLoS ONE 11:15.

  35. Russell D, Jane Wang Z (2003) A cartesian grid method for modeling multiple moving objects in 2D incompressible viscous flow. J Comput Phys 191:177–205.

  36. Rydholm S, Zwartz G, Kowalewski J, Kamali-Zare P, Frisk T, Brismart H (2010) Mechanical properties of primary cilia regulate the response to fluid flow. Am J Physiol 298:F1096.

  37. Schwartz EA, Leonard ML, Bizios R, Bowser SS (1997) Analysis and modeling of the primary cilium bending response to fluid shear. Am J Physiol Renal Physiol 272:F132–F138

  38. Sen Gupta P, Prodromou NV, Chapple JP (2009) Can faulty antennae increase adiposity? The link between cilia proteins and obesity. J Endocrinol 203:327–336.

  39. Tian F-B, Luo H, Zhu L, Liao JC, Lu X-Y (2011) An efficient immersed boundary-lattice Boltzmann method for the hydrodynamic interaction of elastic filaments. J Comput Phys 230:7266–7283.

  40. Xu S, Wang ZJ (2006) An immersed interface method for simulating the interaction of a fluid with moving boundaries. J Comput Phys 216:454–493.

  41. Yoder BK (2007) Role of primary cilia in the pathogenesis of polycystic kidney disease. J Am Soc Nephrol 18:1381–1388.

  42. Yoder BK, Hou XY, Guay-Woodford LM (2002) The polycystic kidney disease proteins, polycystin-1, polycystin-2, polaris, and cystin, are co-localized in renal cilia. J Am Soc Nephrol 13:2508–2516.

  43. Young YN, Downs M, Jacobs CR (2012) Dynamics of the primary cilium in shear flow. Biophys J 103:629–639.

  44. Yuan HZ, Niu XD, Shu S, Li MJ, Yamaguchi H (2014) A momentum exchange-based immersed boundary-lattice Boltzmann method for simulating a flexible filament in an incompressible flow. Comput Math Appl 67:1039–1056.

  45. Zhu LD, Peskin CS (2003) Interaction of two flapping filaments in a flowing soap film. Phys Fluids 15:1954–1960.

Download references


Support to J.Y. Cui by PolyU RKC1 and supports given by PolyU G-UACM and G-YBG9 are gratefully acknowledged.

Author information

Correspondence to Yang Liu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Cui, J., Liu, Y. & Fu, B.M. Numerical study on the dynamics of primary cilium in pulsatile flows by the immersed boundary-lattice Boltzmann method. Biomech Model Mechanobiol 19, 21–35 (2020).

Download citation


  • Immersed boundary
  • Lattice Boltzmann method
  • Primary cilium
  • Fluid–structure interaction
  • Pulsatile flow