A generalised porous medium approach to study thermo-fluid dynamics in human eyes

  • Alessandro Mauro
  • Nicola Massarotti
  • Mohamed Salahudeen
  • Mario R. Romano
  • Vito Romano
  • Perumal Nithiarasu
Original Article
  • 101 Downloads

Abstract

The present work describes the application of the generalised porous medium model to study heat and fluid flow in healthy and glaucomatous eyes of different subject specimens, considering the presence of ocular cavities and porous tissues. The 2D computational model, implemented into the open-source software OpenFOAM, has been verified against benchmark data for mixed convection in domains partially filled with a porous medium. The verified model has been employed to simulate the thermo-fluid dynamic phenomena occurring in the anterior section of four patient-specific human eyes, considering the presence of anterior chamber (AC), trabecular meshwork (TM), Schlemm’s canal (SC), and collector channels (CC). The computational domains of the eye are extracted from tomographic images. The dependence of TM porosity and permeability on intraocular pressure (IOP) has been analysed in detail, and the differences between healthy and glaucomatous eye conditions have been highlighted, proving that the different physiological conditions of patients have a significant influence on the thermo-fluid dynamic phenomena. The influence of different eye positions (supine and standing) on thermo-fluid dynamic variables has been also investigated: results are presented in terms of velocity, pressure, temperature, friction coefficient and local Nusselt number. The results clearly indicate that porosity and permeability of TM are two important parameters that affect eye pressure distribution.

Graphical abstract

Velocity contours and vectors for healthy eyes (top) and glaucomatous eyes (bottom) for standing position.

Keywords

Generalised porous medium model Eye modelling Aqueous humor flow Intraocular pressure (IOP) Patient oriented Heat transfer 

Nomenclature

Cf

Skin friction coefficient

cp

Specific heat (J/g/K)

Da

Darcy number

F

Forchheimer coefficient

g

Gravity (m/s2)

k

Thermal conductivity (W/m K)

L

Characteristic length (m)

n

Normal direction

NuL

Local Nusselt number

p

Pressure (Pa)

Pr

Prandtl number

t

Time (s)

T

Temperature (°C)

u

Velocity (m/s)

Greek symbols

α

Thermal diffusivity (m2/s)

β

Thermal expansion coefficient (1/°C)

ε

Porosity

κ

Permeability (m2)

μ

Viscosity (Pa s)

ρ

Density (kg/m3)

τw

Wall shear stress (Pa)

Acronyms

AC

Anterior chamber

AH

Aqueous humor

CC

Collector channel

HE

Healthy eye

GE

Glaucomatous eye

IOP

Intraocular pressure

SC

Schlemm’s canal

TM

Trabecular meshwork

Subscripts

e

Effective

f

Fluid

m

Mean

o

Ocular

ref

Reference

s

Solid

Notes

Funding information

Alessandro Mauro, Nicola Massarotti and Mario R. Romano gratefully acknowledge the financial support of TeVR SIR project no. RBSI149484, CUP E62I15000760008. Alessandro Mauro also gratefully acknowledges the local program of the University of Napoli “Parthenope” for the support to individual research.

Compliance with ethical standards

The study was conducted according to the tenets of the Declaration of Helsinki.

References

  1. 1.
    Quigley HA, Broman AT (2006) The number of people with glaucoma worldwide in 2010 and 2020. Br J Ophthalmol 90:262–267CrossRefPubMedPubMedCentralGoogle Scholar
  2. 2.
    Johnson M, Mclaren JW, Overby DR (2017) Unconventional aqueous humor outflow: a review. Exp Eye Res 158:94–111CrossRefPubMedGoogle Scholar
  3. 3.
    Bill A, Phillips CI (1971) Uveoscleral drainage of aqueous humor in human eyes. Exp Eye Res 12:275–281CrossRefPubMedGoogle Scholar
  4. 4.
    Fan BJ, Wiggs JL (2010) Glaucoma: genes, phenotypes, and new directions for therapy. J Clin Investig 120:3064–3072CrossRefPubMedPubMedCentralGoogle Scholar
  5. 5.
    Braunger BM, Fuchshofer R, Tamm ER (2015) The aqueous humor outflow pathways in glaucoma: a unifying concept of disease mechanisms and causative treatment. Eur J Pharm Biopharm 95:173–181CrossRefPubMedGoogle Scholar
  6. 6.
    Dautriche CN, Xie Y, Sharfstein ST (2014) Walking through trabecular meshwork biology: towards engineering design of outflow physiology. Biotechnol Adv 32:971–983CrossRefPubMedGoogle Scholar
  7. 7.
    Stamer WD, Braakman ST, Zhou EH, Ethier CR, Fredberg JJ, Overby DR, Johnson M (2015) Biomechanics of Schlemm’s canal endothelium and intraocular pressure reduction. Prog Retin Eye Res 44:86–98CrossRefPubMedGoogle Scholar
  8. 8.
    Acott TS, Kelley MJ (2008) Extracellular matrix in the trabecular meshwork. Exp Eye Res 86:543–561CrossRefPubMedPubMedCentralGoogle Scholar
  9. 9.
    Allingham RR, Dekater AW, Ethier CR, Anderson PJ, Hertzmark E, Epstein DL (1992) The relationship between pore density and outflow facility in human eyes. Investig Ophthalmol Vis Sci 33:1661–1669Google Scholar
  10. 10.
    Grant WM (1958) Further studies on facility of flow through trabecular meshwork. A.M.A Archives of Ophthalmology.  https://doi.org/10.1001/archopt.1958.00940080541001
  11. 11.
    Moses RA (1979) Circumferential flow in Schlemm’s canal. Am J Ophthalmol 88:585–591CrossRefPubMedGoogle Scholar
  12. 12.
    Johnson MC, Kamm RD (1983) The role of Schlemm’s canal in aqueous outflow from the human eye. Investig Ophthalmol Vis Sci 24:320–325Google Scholar
  13. 13.
    Lutjen-Drecoll E (1973) Structural factors influencing outflow facility and its changeability under drugs. Investig Ophthalmol 12:280–294Google Scholar
  14. 14.
    Pedrigi RM, Stamer D, Reed A, Overby DR (2011) A model of giant vacuole dynamics in human Schlemm’s canal endothelial cells. Exp Eye Res 92:57–66CrossRefPubMedGoogle Scholar
  15. 15.
    Scott PA, Overby DR, Freddo TF, Gong H (2007) Comparative studies between species that do and do not exhibit the washout effect. Exp Eye Res 84:435–443CrossRefPubMedGoogle Scholar
  16. 16.
    Braakman ST, Pedrigi R, Read AT, Smith JA, Stamer WD, Ethier CR, Overby DR (2014) Biomechanical strain as a trigger for pore formation in Schlemm’s canal endothelial cells. Exp Eye Res 127:224–235CrossRefPubMedPubMedCentralGoogle Scholar
  17. 17.
    Ayyalasomayajula A, Park RI, Simon BR, Geest JPV (2016) A porohyperelastic finite element model of the eye: the influence of stiffness and permeability on intraocular pressure and optic nerve biomechanics. Comput Meth Biomech Biomed Eng 19:591–602CrossRefGoogle Scholar
  18. 18.
    Eklund A, Linden C, Backlund T, Andersson BM, Lindahl OA (2003) Evaluation of applanation resonator sensors for intra-ocular pressure measurement: results from clinical and in vitro studies. Med Biol Eng Comput 41:190–197CrossRefPubMedGoogle Scholar
  19. 19.
    Avtar R, Srivastava R (2007) Modelling aqueous humor outflow through trabecular meshwork. Appl Math Comput 189:734–745Google Scholar
  20. 20.
    Avtar R, Srivastava R (2006) Aqueous outflow in Schlemm’s canal. Appl Math Comput 174:316–328.  https://doi.org/10.1016/S0096300305004583 Google Scholar
  21. 21.
    Bradley M, Heys JJ (2008) Effect of variable permeability on aqueous humor flow. Appl Math Comput 196:371–380Google Scholar
  22. 22.
    Wessapan T, Rattanadecho P (2014) Influence of ambient temperature on heat transfer in the human eye during exposure to electromagnetic fields at 900 MHz. Int J Heat Mass Transf 70:378–388CrossRefGoogle Scholar
  23. 23.
    Narasimhan A, Kumar JA, Gopal L (2010) Transient simulations of heat transfer in human eye undergoing laser surgery. Int J Heat Mass Transf 53:482–490CrossRefGoogle Scholar
  24. 24.
    Amara EH (1995) Numerical investigations on thermal effects of laser-ocular media interaction. Int J Heat Mass Transf 38:2479–2488CrossRefGoogle Scholar
  25. 25.
    Ferreira JA, Oliveira PD, Pascoal S, Joaquim M (2014) Numerical simulations of aqueous humor flow: from healthy to pathologic situations. Appl Math Comput 226:777–792Google Scholar
  26. 26.
    Chen H, Zhang F, Huang Y, Jiankang W (2015) Numerical investigation of topical drug transport in anterior human eye. Int J Heat Mass Transf 85:356–366CrossRefGoogle Scholar
  27. 27.
    Crowder TR, Ervin VJ (2013) Numerical simulations of fluid pressure in the human eye. Appl Math Comput 219:11119–11133Google Scholar
  28. 28.
    Kapnisis K, Doormaal MV, Ethier CR (2009) Modeling aqueous humor collection from human eye. J Biomech 42:2454–2457CrossRefPubMedGoogle Scholar
  29. 29.
    Villamarin A, Roy S, Hasballa R, Vardoulis O, Reymond P, Stergiopulos N (2012) 3D simulation of the aqueous flow in the human eye. Med Eng Phys 34:1462–1470CrossRefPubMedGoogle Scholar
  30. 30.
    Xiong G, Zhan J, Zuo K, Li J, Rong L, Xu G (2008) Numerical flow simulation in the post-endoscopic sinus surgery nasal cavity. Med Biol Eng Comput 46:1161–1167CrossRefPubMedGoogle Scholar
  31. 31.
    Perez JF, Barea R, Boquete L, Hidalgo MA, Dapena M, Vilar G, Dapena I (2008) Cataract surgery simulator for medical education & finite element/3D human eye model. IFMBE Proc 20:429–432CrossRefGoogle Scholar
  32. 32.
    Deplano V, Bertolotti C, Boiron O (2001) Numerical simulations of unsteady flows in a stenosed coronary bypass graft. Med Biol Eng Comput 39:488–499CrossRefPubMedGoogle Scholar
  33. 33.
    Massarotti N, Nithiarasu P, Zienkiewicz OC (2001) Natural convection in porous medium-fluid interface problems: a finite element analysis by using the CBS procedure. Int J Numer Methods Heat Fluid Flow 11(5):473–490CrossRefGoogle Scholar
  34. 34.
    Arpino F, Massarotti N, Mauro A (2011) Efficient three-dimensional FEM based algorithm for the solution of convection in partly porous domains. Int J Heat Mass Transf 54:4495–4506CrossRefGoogle Scholar
  35. 35.
    Massarotti N, Ciccolella M, Cortellessa G, Mauro A (2016) New benchmark solutions for transient natural convection in partially porous annuli. Int J Numer Methods Heat Fluid Flow 26:1187–1225CrossRefGoogle Scholar
  36. 36.
    Arpino F, Cortellessa G, Mauro A (2015) Transient thermal analysis of natural convection in porous and partially porous cavities. Numerical Heat Transfer Part A: Appl 67:605–631CrossRefGoogle Scholar
  37. 37.
    Arpino F, Massarotti N, Mauro A (2010) A stable explicit fractional step procedure for the solution of heat and fluid flow through interfaces between saturated porous media and free fluids in presence of high source terms. Int J Numer Methods Eng 83:671–692Google Scholar
  38. 38.
    Alazmi B, Vafai K (2001) Analysis of fluid flow and heat transfer interfacial conditions between porous medium and a fluid layer. Int J Heat Mass Transf 44:735–1749CrossRefGoogle Scholar
  39. 39.
    Vafai K, Kim SJ (1990) Fluid mechanics of the interface region between a porous medium and a fluid layer—an exact solution. Int J Heat Fluid Flow 11:1254–1256CrossRefGoogle Scholar
  40. 40.
    Vafai K, Kim SJ (1990) Analysis of surface enhancement by a porous substrate. J Heat Transf 112:700–706CrossRefGoogle Scholar
  41. 41.
    Siddique SS, Suelves AM, Baheti U, Foster CS (2013) Glaucoma and uveitis. Survey Opthalmol 58:1–10CrossRefGoogle Scholar
  42. 42.
    Tandon PN, Autar R (1991) Biphasic model of the trabecular meshwork in the eye. Med Biol Eng Comput 29:281–290CrossRefPubMedGoogle Scholar
  43. 43.
    Irshad FA, Mayfield MS, Zurakowski D, Ayyala RS (2010) Variation in Schlemm’s canal diameter and location by ultrasound biomicroscopy. Ophthalmology 117:916–920CrossRefPubMedGoogle Scholar
  44. 44.
    Ng EY, Ooi EH (2007) Ocular surface temperature: a 3D FEM prediction using bioheat equation. Comput Biol Med 37:829–835.  https://doi.org/10.1016/j.compbiomed.2006.08.023 CrossRefPubMedGoogle Scholar
  45. 45.
    Jooybar E, Abdekhodaie MJ, Farhadi F, Cheng Y (2014) Computational modeling of drug distribution in the posterior segment of the eye: effects of device variables and positions. Math Biosci 255:11–20CrossRefPubMedGoogle Scholar
  46. 46.
    Heys JJ, Barocas VH (2002) A Boussinesq model of natural convection in the human eye and the formation of Krukenberg’s spindle. Ann Biomed Eng 30:392–401CrossRefPubMedGoogle Scholar
  47. 47.
    Siggers JH, Ethier CR (2012) Fluid mechanics of the eye. Annu Rev Fluid Mech 44:347–372CrossRefGoogle Scholar
  48. 48.
    Maus TL, Brubaker RF (1999) Measurement of aqueous flow by fluorophotometry in the presence of a dilated pupil. Investig Ophthalmol Vis Sci 40:542–546Google Scholar
  49. 49.
    McLaren JW (2009) Measurement of aqueous humor flow. Exp Eye Res 88:641–647CrossRefPubMedGoogle Scholar
  50. 50.
    Mäepea O, Bill A (1989) The pressures in the episcleral veins, Schlemm’s canal and the trabecular meshwork in monkeys: effects of changes in intraocular pressure. Exp Eye Res 49:645–653CrossRefPubMedGoogle Scholar
  51. 51.
    Sit AJ, Ekdawi NS, Malihi M, McLaren JW (2011) A novel method for computerized measurement of episcleral venous pressure in human. Exp Eye Res 92:537–544CrossRefPubMedGoogle Scholar
  52. 52.
    Emery AF, Kramar P, Guy AW, Lin JC (1975) Microwave induced temperature rises in rabbit eyes in cataract research. J Heat Transf 97:123–128CrossRefGoogle Scholar
  53. 53.
    Weller HG, Tabor G, Jasak H, Fureby C (1998) A tensorial approach to computational continuum mechanics using object-oriented techniques. Comput Phys 12:620.  https://doi.org/10.1063/1.168744 CrossRefGoogle Scholar
  54. 54.
    Patankar SV, Spalding DB (1972) A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. Int J Heat Mass Transf 15:1787–1806.  https://doi.org/10.1016/0017-9310(72)90054-3 CrossRefGoogle Scholar
  55. 55.
    Passalacqua A, Fox RO (2011) Implementation of an iterative solution procedure for multi-fluid gas-particle flow models on unstructured grids. Powder Technol 213:174–187CrossRefGoogle Scholar
  56. 56.
    Mauro A, Romano MR, Romano V, Nithiarasu P (2018) Suprachoroidal shunts for treatment of glaucoma. Int J Numer Methods Heat Fluid Flow 28(2):297–314Google Scholar
  57. 57.
    Chang WJ, Chang WL (1996) Mixed convection in vertical parallel plate channel partially filled with porous media of high permeability. Int J Heat Mass Transf 39(7):1331–1342CrossRefGoogle Scholar

Copyright information

© International Federation for Medical and Biological Engineering 2018

Authors and Affiliations

  1. 1.Dipartimento di IngegneriaUniversità degli Studi di Napoli “Parthenope”NapoliItaly
  2. 2.Department of Biomedical SciencesHumanitas UniversityRozzano (Milan)Italy
  3. 3.Department of Eye and Vision Science, Institute of Ageing and Chronic DiseaseUniversity of LiverpoolLiverpoolUK
  4. 4.Moorfields Eye HospitalLondonUK
  5. 5.Zienkiewicz Centre for Computational Engineering, College of EngineeringSwansea UniversitySwanseaUK

Personalised recommendations