Skip to main content

Ocular Fluid Mechanics and Drug Delivery: A Review of Mathematical and Computational Models

Pharmaceutical Research volume 38, pages 2003–2033 (2021)Cite this article

Abstract

The human eye is a complex biomechanical structure with a range of biomechanical processes involved in various physiological as well as pathological conditions. Fluid flow inside different domains of the eye is one of the most significant biomechanical processes that tend to perform a wide variety of functions and when combined with other biophysical processes play a crucial role in ocular drug delivery. However, it is quite difficult to comprehend the effect of these processes on drug transport and associated treatment experimentally because of ethical constraints and economic feasibility. Computational modeling on the other hand is an excellent means to understand the associated complexity between these aforementioned processes and drug delivery. A wide range of computational models specific to different types of fluids present in different domains of the eye as well as varying drug delivery modes has been established to understand the fluid flow behavior and drug transport phenomenon in an insilico manner. These computational models have been used as a non-invasive tool to aid ophthalmologists in identifying the challenges associated with a particular drug delivery mode while treating particular eye diseases and to advance the understanding of the biomechanical behavior of the eye. In this regard, the author attempts to summarize the existing computational and mathematical approaches proposed in the last two decades for understanding the fluid mechanics and drug transport associated with different domains of the eye, together with their application to modify the existing treatment processes.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

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

Abbreviations

\({L}_{p}\) :

Permeability of the equivalent membrane (m Pa−1 s−1)

\(p\) :

Intraocular pressure (Pa)

\({p}_{a}\) :

Ciliary body capillary pressure (Pa)

\({p}_{0}, {p}_{m}\) :

Pressure at inner and outer wall of TM, respectively (Pa)

\(\overrightarrow{{\varvec{v}}}\) :

AH velocity (m s−1)

G :

Shear Modulus (kPa)

\({K}_{TM}\) :

Permeability tensor of TM (m2)

\({S}_{TM}\) :

Specific surface area of TM

\({F}_{in}\) :

AH production rate by the CB (uL min−1)

\({F}_{out}\) :

AH outflow rate through TM (uL min−1)

\({F}_{drain}\) :

Fluid volume loss rate via surgical pathway into subconjunctival tissue (uL min−1)

\({F}_{u}\) :

Fluid outflow rate through uveoscleral pathway (uL min−1)

EVP :

Episcleral venous pressure (Pa)

\({C}_{trab}\) :

Outflow facility via trabecular pathway (uL min−1 Pa−1)

\({p}_{tissue}\) :

Fluid pressure in the bleb (Pa)

\(k\) :

Thermal conductivity (W \({m}^{-1}{K}^{-1}\))

c:

Specific heat capacity (\({\mathrm{J }Kg}^{-1}{K}^{-1}\))

\({c}_{b}\) :

Specific heat capacity of the blood (J \({Kg}^{-1}{K}^{-1}\))

\({h}_{amb}\) :

Convective heat transfer coefficient between cornea and ambience (\(\mathrm{W }{m}^{-2}{K}^{-1}\))

\(h_{bl}\)  :

Convective heat transfer coefficient between eye and surrounding blood vessels (\({\mathrm{W }m}^{-2}{K}^{-1}\))

\(T\) :

Temperature distribution inside the eye (K)

\({T}_{amb}\) :

Ambient temperature (K)

\({T}_{bl}\) :

Blood temperature (K)

\({T}_{ref}\) :

Reference temperature (K)

\(E\)  :

Tear evaporation rate (\({\mathrm{W }m}^{-3}\))

\({Q}_{blood}\) :

Blood perfusion term (\(\mathrm{W }{m}^{-3}{K}^{-1}\))

\({Q}_{met}\) :

Metabolic heat generation rate (\({\mathrm{W }m}^{-3}\))

\({Q}_{ext}\) :

External heating source (\({\mathrm{W }m}^{-3})\))

\(\overrightarrow{g}\) :

Acceleration due to gravity (m s−2)

\(C\) :

Drug concentration (mol m−3)

\({C}_{0}\) :

Initial drug concentration (mol m−3)

\({C}_{bl}\) :

Drug concentration in the blood (mol m−3)

\(D\) :

Diffusion coefficient of the drug (m2 s−1)

Dp, DHb :

Diffusion coefficient of free oxygen and oxyhemoglobin in plasma and red blood vessels, respectively (m2 s−1)

\({D}_{p}\) :

Mean particle diameter of packed bed (µm)

\({C}_{2}\) :

Inertial resistance factor

\({R}_{d}\) :

Drug clearance rate from the tissues (mol m−3 s−1)

\(m\) :

Reaction rate coefficient (s−1)

\(\overrightarrow{n}\) :

Normal vector pointing outwards from a surface

\({k}_{act}\) :

Added convectional velocity (m s−1)

\({k}_{sc}\) :

Mass transfer coefficient at the scleral surface (m s−1)

Pb :

Partial pressure of oxygen (Pa)

P50 :

Half partial pressure of oxygen saturation in hemoglobin (Pa)

n:

Hill exponent

CHb :

Hemoglobin carrying capability of oxygen in the blood (ml O2 ml−1)

S:

Oxyhemoglobin saturation function

W:

Shape of viscosity dependence on hematocrit

\(\rho\)  :

Aqueous humor (AH) density (kg/m3)

\({\rho }_{b}\) :

Blood density (kg/m3)

\({\omega }_{b}\) :

Blood perfusion rate (s−1)

\(\sigma\)  :

Stefan-Boltzmann constant (\({\mathrm{W }m}^{-2}{K}^{-1}\))

\(\epsilon\)  :

Emissivity of corneal surface

\(\delta\) :

Displacement of the iris (µm)

\(\upmu\) :

Dynamic viscosity of AH (Ns m−2)

\({\mu }_{0.45}\) :

Relative viscosity for a fixed discharge hematocrit (0.45) (Ns m−2)

\(\upbeta\) :

Thermal expansion coefficient of AH (K−1)

\(\varnothing\) :

Porosity

\(\gamma\) :

Drug transport rate constant across choroidal blood vessels (s−1)

\(\varphi\) :

Kozeny and solubility constant, respectively

\(\in\) :

Parameter to vary effective outflow facility

\(\alpha\) :

Permeability (m2)

\({\alpha }_{b}\) :

Oxygen solubility coefficient (ml O2 ml−1 mmHg−1)

\({\sigma }_{p}, {\sigma }_{s}\) :

Protein and low molecular component osmotic reflection coefficient, respectively

\({\Delta \pi }_{p}, \Delta {\pi }_{s}\)  :

Protein and low molecular component osmotic pressure difference, respectively (Pa)

References

  1. Bourne RRA, Adelson J, Flaxman S, Briant P, Bottone M, Vos T, et al. Global prevalence of blindness and distance and near vision impairment in 2020: progress towards the vision 2020 targets and what the future holds. Invest Ophthalmol Vis Sci. 2020;61:2317.

    Google Scholar 

  2. Flaxman SR, Bourne RRA, Resnikoff S, Ackland P, Braithwaite T, Cicinelli MV, et al. Global causes of blindness and distance vision impairment 1990–2020: a systematic review and meta-analysis. Lancet Glob Heal. 2017;5:e1221–34.

  3. Shell JW. Pharmacokinetics of topically applied ophthalmic drugs. Surv Ophthalmol. 1982;26:207–18.

    CAS  PubMed  Google Scholar 

  4. Lang JC. Ocular drug delivery conventional ocular formulations. Adv Drug Deliv Rev. 1995;16:39–43.

    CAS  Google Scholar 

  5. Gaudana R, Jwala J, Boddu SHS, Mitra AK. Recent perspectives in ocular drug delivery. Pharm Res. 2009;26:1197–216.

    CAS  PubMed  Google Scholar 

  6. Gaudana R, Ananthula HK, Parenky A, Mitra AK. Ocular drug delivery. AAPS J. 2010;12(3):348–60. https://doi.org/10.1208/s12248-010-9183-3.

    CAS  Article  PubMed  PubMed Central  Google Scholar 

  7. Urtti A. Ocular drug delivery. Adv Drug Deliv Rev. 2006;58:1129–30.

    CAS  Google Scholar 

  8. Mishima S, Gasset A, Klyce SD Jr, Baum JL. Determination of Tear Volume and Tear Flow. Invest Ophthalmol Vis Sci. 1966;5:264–76.

    CAS  Google Scholar 

  9. Chrai SS, Makoid MC, Eriksen SP, Robinson JR. Drop size and initial dosing frequency problems of topically applied ophthalmic drugs. J Pharm Sci. 1974;63:333–8.

    CAS  PubMed  Google Scholar 

  10. Chrai SS, Patton TF, Mehta A, Robinson JR. Lacrimal and instilled fluid dynamics in rabbit eyes. J Pharm Sci. 1973;62:1112–21.

    CAS  PubMed  Google Scholar 

  11. Prausnitz MR. Permeability of cornea, sciera, and conjunctiva: A literature analysis for drug delivery to the eye. J Pharm Sci. 1998;87:1479–88.

    CAS  PubMed  Google Scholar 

  12. Yokoi N, Komuro A. Non-invasive methods of assessing the tear film. Exp Eye Res. 2004;78:399–407.

    CAS  PubMed  Google Scholar 

  13. Zhu H, Chauhan A. A mathematical model for ocular tear and solute balance. Curr Eye Res. 2005;30:841–54.

    PubMed  Google Scholar 

  14. Ranta VP, Mannermaa E, Lummepuro K, Subrizi A, Laukkanen A, Antopolsky M, et al. Barrier analysis of periocular drug delivery to the posterior segment. J Control Release. 2010;148:42–8 (Elsevier B.V.).

    CAS  PubMed  Google Scholar 

  15. Thrimawithana TR, Young S, Bunt CR, Green C, Alany RG. Drug delivery to the posterior segment of the eye. Drug Discov Today [Internet]. 2011;16:270–7. https://doi.org/10.1016/j.drudis.2010.12.004 (Elsevier Ltd).

    CAS  Article  Google Scholar 

  16. Yadav D, Varma LT, Yadav K. Drug Delivery to posterior segment of the eye: conventional delivery strategies, their barriers, and restrictions. In: Patel J, Sutariya V, Kanwar J, Pathak Y, editors. Drug delivery for the retina and posterior segment disease. Cham: Springer; 2018. https://doi.org/10.1007/978-3-319-95807-1_3.

  17. Varela-Fernández R, Díaz-Tomé V, Luaces-Rodríguez A, Conde-Penedo A, García-Otero X, Luzardo-álvarez A, et al. Drug delivery to the posterior segment of the eye: Biopharmaceutic and pharmacokinetic considerations. Pharmaceutics. 2020;12:1–39.

    Google Scholar 

  18. Smith SJ, Smith BD, Mohney BG. Ocular side effects following intravitreal injection therapy for retinoblastoma: A systematic review. Br J Ophthalmol. 2014;98:292–7.

    PubMed  Google Scholar 

  19. Siggers JH, Ethier CR. Fluid mechanics of the eye. Annu Rev Fluid Mech. 2011;44:347–72.

    Google Scholar 

  20. Mircioiu C, Voicu V, Anuta V, Tudose A, Celia C, Paolino D, et al. Mathematical modeling of release kinetics from supramolecular drug delivery systems. Pharmaceutics. 2019;11:1–45.

    Google Scholar 

  21. Tojo K. A pharmacokinetic model for ocular drug delivery. Chem Pharm Bull (Tokyo) Japan. 2004;52:1290–4.

    CAS  Google Scholar 

  22. Kim SH, Lutz RJ, Wang NS, Robinson MR. Transport barriers in transscleral drug delivery for retinal diseases. Ophthalmic Res. 2007;39(5):244–54. https://doi.org/10.1159/000108117.

    CAS  Article  PubMed  Google Scholar 

  23. Agrahari V, Mandal A, Agrahari V, Trinh HM, Joseph M, Ray A, et al. A comprehensive insight on ocular pharmacokinetics. Drug Deliv Transl Res. 2016;6:735–54.

    CAS  PubMed  PubMed Central  Google Scholar 

  24. Shell JW. Ocular drug delivery systems - a review. Cutan Ocul Toxicol. 1982;1:49–63.

    CAS  Google Scholar 

  25. Le Bourlais C, Acar L, Zia H, Sado PA, Needham T, Leverge R. Ophthalmic drug delivery systems - Recent advances. Prog Retin Eye Res. 1998;17:33–58.

    CAS  PubMed  Google Scholar 

  26. Sultana Y, Jain R, Aqil M, Ali A. Review of Ocular Drug Delivery. Curr Drug Deliv. 2006;3:207–17.

    CAS  PubMed  Google Scholar 

  27. Thassu D, Chader GJ, editors. Ocular drug delivery systems: barriers and application of nanoparticulate systems. 1st edn. CRC Press; 2012. https://doi.org/10.1201/b12950.

  28. Jaswal P, Sharma RB, Agarwal S. Recent trends in ocular drug delivery system. Int J Pharm Sci Rev Res. 2016;38:119–24.

    CAS  Google Scholar 

  29. Lee VHL, Robinson JR. Topical Ocular Drug Delivery: Recent Developments and Future Challenges. J Ocul Pharmacol. 1986;2:67–108.

    CAS  PubMed  Google Scholar 

  30. Barar J, Javadzadeh AR, Omidi Y. Ocular novel drug delivery: Impacts of membranes and barriers. Expert Opin Drug Deliv. 2008;5:567–81.

    CAS  PubMed  Google Scholar 

  31. Kompella UB, Kadam RS, Lee VHL. Recent advances in ophthalmic drug delivery. Ther Deliv. 2010;1:435–56.

    CAS  PubMed  Google Scholar 

  32. Molokhia SA, Thomas SC, Garff KJ, Mandell KJ, Wirostko BM. Anterior eye segment drug delivery systems: Current treatments and future challenges. J Ocul Pharmacol Ther. 2013;29:92–105.

    CAS  PubMed  Google Scholar 

  33. Maulvi FA, Soni TG, Shah DO. A review on therapeutic contact lenses for ocular drug delivery. Drug Deliv. 2016;23:3017–26.

    CAS  PubMed  Google Scholar 

  34. Rodríguez Villanueva J, Navarro MG, Rodríguez Villanueva L. Dendrimers as a promising tool in ocular therapeutics: Latest advances and perspectives. Int J Pharm [Internet]. 2016;511:359–66. https://doi.org/10.1016/j.ijpharm.2016.07.031 (Elsevier B.V.).

    CAS  Article  Google Scholar 

  35. Bertens CJF, Gijs M, van den Biggelaar FJHM, Nuijts RMMA. Topical drug delivery devices: A review. Exp Eye Res. 2018;168:149–60.

    CAS  PubMed  Google Scholar 

  36. Peng CC, Burke MT, Carbia BE, Plummer C, Chauhan A. Extended drug delivery by contact lenses for glaucoma therapy. J Control Release [Internet]. 2012;162:152–8. https://doi.org/10.1016/j.jconrel.2012.06.017 (Elsevier B.V.).

    CAS  Article  Google Scholar 

  37. Amrite AC, Edelhauser HF, Kompella UB. Modeling of corneal and retinal pharmacokinetics after periocular drug administration. Investig Ophthalmol Vis Sci. 2008;49:320–32.

    Google Scholar 

  38. Geroski DH, Edelhauser HF. Transscleral drug delivery for posterior segment disease. Adv Drug Deliv Rev. 2001;52:37–48.

    CAS  PubMed  Google Scholar 

  39. Tojo K, Isowaki A. Pharmacokinetic model for in vivo/in vitro correlation of intravitreal drug delivery. Adv Drug Deliv Rev. 2001;52:17–24.

    CAS  PubMed  Google Scholar 

  40. Hughes PM, Olejnik O, Chang-Lin JE, Wilson CG. Topical and systemic drug delivery to the posterior segments. Adv Drug Deliv Rev. 2005;57:2010–32.

    CAS  PubMed  Google Scholar 

  41. Ranta VP, Urtti A. Transscleral drug delivery to the posterior eye: Prospects of pharmacokinetic modeling. Adv Drug Deliv Rev. 2006;58:1164–81.

    CAS  PubMed  Google Scholar 

  42. Barocas VH, Balachandran RK. Sustained transscleral drug delivery. Expert Opin Drug Deliv. 2008;5:1–10.

    CAS  PubMed  Google Scholar 

  43. Missel PJ. Computer modeling for ocular drug delivery. In: Ocular Drug Delivery Systems: Barriers and Application of Nanoparticulate Systems. CRC Press; 2012. p. 59–92.

  44. Paola C, Francesca M. Mathematical and Numerical Methods for Modeling Drug Delivery to the Posterior Segment of the Eye. J Ophthalmic Res Ocul Care. 2017;1:4–11.

    Google Scholar 

  45. Ethier CR, Johnson M, Ruberti J. Ocular Biomechanics and Biotransport. Annu Rev Biomed Eng. 2004;6:249–73.

    CAS  PubMed  Google Scholar 

  46. Alm A, Nilsson SFE. Uveoscleral outflow - A review. Exp Eye Res [Internet]. 2009;88:760–8. https://doi.org/10.1016/j.exer.2008.12.012 (Elsevier Ltd).

    CAS  Article  Google Scholar 

  47. Siggers JH, Ethier CR. Fluid Mechanics of the Eye. Annu Rev Fluid Mech. 2012;44:347–72.

    Google Scholar 

  48. Goel M, Picciani RG, Lee RK, Bhattacharya SK. Aqueous humor dynamics: a review. Open Ophthalmol J. 2010;4:52–9. https://doi.org/10.2174/1874364101004010052.

    CAS  Article  PubMed  PubMed Central  Google Scholar 

  49. Goldmann VH. Minute volume of the aqueous in the anterior chamber of the human eye in normal state and in primary glaucoma. Ophthalmologica. 1950;120:19–21.

    CAS  PubMed  Google Scholar 

  50. van Zyl T, Yan W, McAdams A, Peng YR, Shekhar K, Regev A, et al. Cell atlas of aqueous humor outflow pathways in eyes of humans and four model species provides insight into glaucoma pathogenesis. Proc Natl Acad Sci U S A. 2020;117:10339–49.

    PubMed  PubMed Central  Google Scholar 

  51. Lyubimov GA, Moiseeva IN, Stein AA. Dynamics of the intraocular fluid: Mathematical model and its main consequences. Fluid Dyn. 2007;42:684–94.

    Google Scholar 

  52. Kiel JW, Hollingsworth M, Rao R, Chen M, Reitsamer HA. Ciliary blood flow and aqueous humor production. Prog Retin Eye Res [Internet]. 2011;30:1–17. https://doi.org/10.1016/j.preteyeres.2010.08.001 (Elsevier Ltd).

    CAS  Article  Google Scholar 

  53. Dobler B, Bendl R. Precise modelling of the eye for proton therapy of intra-ocular tumours. Phys Med Biol. 2002;47:593–613.

    PubMed  Google Scholar 

  54. Heys JJ, Barocas VH, Taravella MJ. Modeling passive mechanical interaction between aqueous humor and iris. J Biomech Eng. 2001;123(6):540–7. https://doi.org/10.1115/1.1411972.

    CAS  Article  PubMed  Google Scholar 

  55. Heys JJ, Barocas VH. Computational evaluation of the role of accommodation in pigmentary glaucoma. Investig Ophthalmol Vis Sci. 2002;43:700–8.

    Google Scholar 

  56. Huang EC, Barocas VH. Active iris mechanics and pupillary block: Steady-state analysis and comparison with anatomical risk factors. Ann Biomed Eng. 2004;32:1276–85.

    PubMed  Google Scholar 

  57. Huang EC, Barocas VH. Accommodative microfluctuations and iris contour. J Vis. 2006;6:653–60.

    PubMed  Google Scholar 

  58. Amini R, Barocas VH. Reverse pupillary block slows iris contour recovery from corneoscleral indentation. J Biomech Eng. 2010;132:1–6.

    Google Scholar 

  59. Dvoriashyna M, Repetto R, Romano MR, Tweedy JH. Aqueous humour flow in the posterior chamber of the eye and its modifications due to pupillary block and iridotomy. Math Med Biol. 2018;35:447–67.

    CAS  PubMed  Google Scholar 

  60. Wang W, Qian X, Song H, Zhang M, Liu Z. Fluid and structure coupling analysis of the interaction between aqueous humor and iris. Biomed Eng Online BioMed Central. 2016;15:569–86.

    Google Scholar 

  61. Johnson MC, Kamm RD. The role of Schlemm’s canal in aqueous outflow from the human eye. Invest Ophthalmol Vis Sci. 1983;24:320–5.

    CAS  PubMed  Google Scholar 

  62. Johnson M, Shapiro A, Ethier CR, Kamm RD. Modulation of outflow resistance by the pores of the inner wall endothelium. Investig Ophthalmol Vis Sci. 1992;33:1670–5.

    CAS  Google Scholar 

  63. Avtar R, Srivastava R. Aqueous outflow in schlemm’s canal. Appl Math Comput. 2006;174:316–28.

    Google Scholar 

  64. Avtar R, Srivastava R. Modelling aqueous humor outflow through trabecular meshwork. Appl Math Comput. 2007;189:734–45.

    Google Scholar 

  65. Tandon PN, Autar R. Biphasic model of the trabecular meshwork in the eye. Med Biol Eng Comput. 1991;29:281–90.

    CAS  PubMed  Google Scholar 

  66. Crowder TR, Ervin VJ. Numerical simulations of fluid pressure in the human eye. Appl Math Comput [Internet]. 2013;219:11119–33. https://doi.org/10.1016/j.amc.2013.04.060 (Elsevier Inc.).

    Article  Google Scholar 

  67. Merchant BM, Heys JJ. Effects of variable permeability on aqueous humor outflow. Appl Math Comput. 2008;196:371–80.

    Google Scholar 

  68. Gong H, Ruberti J, Overby D, Johnson M, Freddo TF. A new view of the human trabecular meshwork using quick-freeze, deep-etch electron microscopy. Exp Eye Res. 2002;75:347–58.

    CAS  PubMed  Google Scholar 

  69. Kumar S, Acharya S, Beuerman R, Palkama A. Numerical solution of ocular fluid dynamics in a rabbit eye: Parametric effects. Ann Biomed Eng. 2006;34:530–44.

    PubMed  Google Scholar 

  70. Brubaker RF. Flow of aqueous humor in humans [The Friedenwald Lecture]. Invest Ophthalmol Vis Sci. 1991;32(13):3145–66.

    CAS  PubMed  Google Scholar 

  71. Bill A. Uveoscleral drainage of aqueous humor: physiology and pharmacology. Prog Clin Biol Res. 1989;312:417–27.

    CAS  PubMed  Google Scholar 

  72. Nilsson SFE. The uveoscleral outflow routes. Eye. 1997;11:149–54.

    PubMed  Google Scholar 

  73. Sit AJ, Nau CB, McLaren JW, Johnson DH, Hodge D. Circadian variation of aqueous dynamics in young healthy adults. Investig Ophthalmol Vis Sci. 2008;49:1473–9.

    Google Scholar 

  74. Gardiner BS, Smith DW, Coote M, Crowston JG. Computational modeling of fluid flow and intra-ocular pressure following glaucoma surgery. PLoS One. 2010;5(10):e13178. https://doi.org/10.1371/journal.pone.0013178.

  75. Phelps CD, Armaly MF. Measurement of episcleral venous pressure. Am J Ophthalmol. 1978;85(1):35–42. https://doi.org/10.1016/s0002-9394(14)76662-0.

    CAS  Article  PubMed  Google Scholar 

  76. Liu D, Wood NB, Witt N, Hughes AD, Thom SA, Xu XY. Computational analysis of oxygen transport in the retinal arterial network. Curr Eye Res. 2009;34:945–56.

    PubMed  Google Scholar 

  77. Bryant MR, Szerenyi K, Schmotzer H, McDonnell PJ. Corneal tensile strength in fully healed radial keratotomy wounds. Invest Ophthalmol Vis Sci. 1994;35:3022–31.

    CAS  PubMed  Google Scholar 

  78. Hjortdal J. Regional elastic performance of the human cornea. J Biomech. 1996;29:931–42.

    CAS  PubMed  Google Scholar 

  79. Stitzel JD, Duma SM, Cormier JM, Herring IP. A nonlinear finite element model of the eye with experimental validation for the prediction of globe rupture. SAE Tech Pap. 2002;46:81–102.

    Google Scholar 

  80. Heys J, Barocas VH. Mechanical characterization of the bovine iris. J Biomech. 1999;32:999–1003.

    CAS  PubMed  Google Scholar 

  81. Villamarin A, Roy S, Hasballa R, Vardoulis O, Reymond P, Stergiopulos N. 3D simulation of the aqueous flow in the human eye. Med Eng Phys [Internet]. 2012;34:1462–70. https://doi.org/10.1016/j.medengphy.2012.02.007 (Institute of Physics and Engineering in Medicine).

    Article  Google Scholar 

  82. Stay MS, Pan T, Brown JD, Ziaie B, Barocas VH. Thin-film coupled fluid-solid analysis of flow through the AhmedTM glaucoma drainage device. J Biomech Eng. 2005;127:776–81.

    PubMed  Google Scholar 

  83. Siewert S, Sämann M, Schmidt W, Stiehm M, Falke K, Grabow N, Guthoff R, Schmitz KP. Gekoppelte Analyse der Fluid-Struktur-Interaktion eines mikromechanischen Ventils für Glaukomdrainageimplantate [Coupled Analysis of Fluid-Structure Interaction of a Micro-Mechanical Valve for Glaucoma Drainage Devices]. Klin Monbl Augenheilkd. 2015;232(12):1374–80. https://doi.org/10.1055/s-0041-107940.

    CAS  Article  PubMed  Google Scholar 

  84. Kudsieh B, Fernández-Vigo JI, Agujetas R, Montanero JM, Ruiz-Moreno JM, Fernández-Vigo JÁ, et al. Numerical model to predict and compare the hypotensive efficacy and safety of minimally invasive glaucoma surgery devices. PLoS One. 2020;15:1–17.

    Google Scholar 

  85. Martínez Sánchez GJ, Escobar del Pozo C, Rocha Medina JA, Naude J, Brambila Solorzano A. Numerical simulation of the aqueous humor flow in the eye drainage system; a healthy and pathological condition comparison. Med Eng Phys. 2020;83:82–92.

    PubMed  Google Scholar 

  86. Lagendijk JJ. A mathematical model to calculate temperature distributions in human and rabbit eyes during hyperthermic treatment. Phys Med Biol. 1982;27(11):1301–11. https://doi.org/10.1088/0031-9155/27/11/001.

    CAS  Article  PubMed  Google Scholar 

  87. Scott JA. A finite element model of heat transport in the human eye. Phys Med Biol. 1988;33(2):227–41. https://doi.org/10.1088/0031-9155/33/2/003.

    CAS  Article  PubMed  Google Scholar 

  88. Scott JA. The computation of temperature rises in the human eye induced by infrared radiation. Phys Med Biol. 1988;33:243–57.

    CAS  PubMed  Google Scholar 

  89. Emery AF, Kramar P, Guy AW, Lin JC. Microwave induced temperature rises in rabbit eyes in cataract research. ASME J Heat Transfer. 1975;97(1):123–8. https://doi.org/10.1115/1.3450259.

    Article  Google Scholar 

  90. Ciceki U. Computational Model for heat Transfer in the Human Eye Using the Finite Element Method. Louisiani University; 2003.

  91. Neelakantaswamy PS, Ramakrishnan KP. Microwave-induced hazardous nonlinear thermoelastic vibrations of the ocular lens in the human eye. J Biomech. 1979;12:205–10.

    CAS  PubMed  Google Scholar 

  92. Karampatzakis A, Samaras T. Numerical model of heat transfer in the human eye with consideration of fluid dynamics of the aqueous humour. Phys Med Biol. 2010;55:5653–65.

    PubMed  Google Scholar 

  93. Chen H, Zhang F, Huang Y, Wu J. Numerical investigation of topical drug transport in the anterior human eye. Int J Heat Mass Transf [Internet]. 2015;7:237–43. https://doi.org/10.1016/j.ijheatmasstransfer.2015.01.142 (Elsevier Ltd).

    Article  Google Scholar 

  94. Heys JJ, Barocas VH. A Boussinesq model of natural convection in the human eye and the formation of Krukenberg’s spindle. Ann Biomed Eng. 2002;30:392–401.

    PubMed  Google Scholar 

  95. Canning CR, Greaney MJ, Dewynne JN, Fitt AD. Fluid flow in the anterior chamber of a human eye. IMA J Math Appl Med Biol. 2002;19:31–60.

    CAS  PubMed  Google Scholar 

  96. Fitt AD, Gonzalez G. Fluid mechanics of the human eye: Aqueous humour flow in the anterior chamber. Bull Math Biol. 2006;68:53–71.

    CAS  PubMed  Google Scholar 

  97. Ooi EH, Ng EYK. Simulation of aqueous humor hydrodynamics in human eye heat transfer. Comput Biol Med. 2008;38:252–62.

    PubMed  Google Scholar 

  98. Ooi EH, Ng EYK. Effects of natural convection within the anterior chamber on the ocular heat transfer. Int J Numer Meth Biomed Engng. 2011;27:408–23. https://doi.org/10.1002/cnm.1411.

    Article  Google Scholar 

  99. Ng EYK, Ooi EH. FEM simulation of the eye structure with bioheat analysis. Comput Methods Programs Biomed. 2006;82:268–76.

    CAS  PubMed  Google Scholar 

  100. Ooi EH, Ng EYK, Purslow C, Acharya R. Variations in the corneal surface temperature with contact lens wear. Proc Inst Mech Eng Part H J Eng Med. 2007;221:337–49.

    CAS  Google Scholar 

  101. Ng EYK, Ooi EH. Ocular surface temperature: A 3D FEM prediction using bioheat equation. Comput Biol Med. 2007;37:829–35.

    CAS  PubMed  Google Scholar 

  102. Tiang KL, Ooi EH. Effects of aqueous humor hydrodynamics on human eye heat transfer under external heat sources. Med Eng Phys Phys. 2016;38:776–84 (Elsevier Ltd).

    Google Scholar 

  103. Amara EH. Numerical investigations on thermal effects of laser-ocular media interaction. Int J Heat Mass Transf. 1995;38:2479–88.

    Google Scholar 

  104. Chua KJ, Ho JC, Chou SK, Islam MR. On the study of the temperature distribution within a human eye subjected to a laser source. Int Commun Heat Mass Transf. 2005;32:1057–65.

    Google Scholar 

  105. Narasimhan A, Jha KK, Gopal L. Transient simulations of heat transfer in human eye undergoing laser surgery. Int J Heat Mass Transf [Internet]. 2010;53:482–90. https://doi.org/10.1016/j.ijheatmasstransfer.2009.09.007 (Elsevier Ltd).

    Article  Google Scholar 

  106. Jha KK, Narasimhan A. Three-dimensional bio-heat transfer simulation of sequential and simultaneous retinal laser irradiation. Int J Therm Sci [Internet]. 2011;50:1191–8. https://doi.org/10.1016/j.ijthermalsci.2011.02.005 (Elsevier Masson SAS).

    Article  Google Scholar 

  107. Narasimhan A, Sadasivam S. Non-Fourier bio heat transfer modelling of thermal damage during retinal laser irradiation. Int J Heat Mass Transf [Internet]. 2013;60:591–7. https://doi.org/10.1016/j.ijheatmasstransfer.2013.01.010.

    Article  Google Scholar 

  108. Chen B, Zhao Y, Li D. Numerical simulation of ophthalmic laser surgeries by a local thermal non-equilibrium two-temperature model. Int J Numer Methods Heat Fluid Flow. 2019;29:4706–23.

    Google Scholar 

  109. Shafahi M, Vafai K. Human eye response to thermal disturbances. ASME J Heat Transfer. 2011;133(1):011009. https://doi.org/10.1115/1.4002360.

  110. Wessapan T, Rattanadecho P, Wongchadakul P. Effect of the body position on natural convection within the anterior chamber of the human eye during exposure to electromagnetic fields. Numer Heat Transf Part A Appl. 2016;69:1014–28.

    Google Scholar 

  111. Wessapan T, Rattanadecho P. Influence of ambient temperature on heat transfer in the human eye during exposure to electromagnetic fields at 900 MHz. Int J Heat Mass Transf [Internet]. 2014;70:378–88. https://doi.org/10.1016/j.ijheatmasstransfer.2013.11.009 (Elsevier Ltd).

    Article  Google Scholar 

  112. Wessapan T, Rattanadecho P. Aqueous humor natural convection of the human eye induced by electromagnetic fields: in the supine position. J Med Bioeng. 2014;3:241–58.

    Google Scholar 

  113. Wessapan T, Rattanadecho P. Specific absorption rate and temperature increase in human eye subjected to electromagnetic fields at 900 MHz. J Heat Transfer [Internet]. 2012;134:091101. https://doi.org/10.1115/1.4006243.

    Article  Google Scholar 

  114. Mauro A, Massarotti N, Salahudeen M, Romano MR, Romano V, Nithiarasu P. A generalised porous medium approach to study thermo-fluid dynamics in human eyes. Med Biol Eng Comput. 2018;56:1823–39.

    PubMed  Google Scholar 

  115. Huang D, Chen Y-S, Rupenthal ID. Overcoming ocular drug delivery barriers through the use of physical forces. Adv Drug Deliv Rev [Internet]. 2018;126:96–112. https://doi.org/10.1016/j.addr.2017.09.008 Elsevier B.V.

    CAS  Article  Google Scholar 

  116. Hämäläinen KM, Kananen K, Auriola S, Kontturi K, Urtti A. Characterization of paracellular and aqueous penetration routes in cornea, conjunctiva, and sclera. Investig Ophthalmol Vis Sci. 1997;38:627–34.

    Google Scholar 

  117. Freddo TF. Shifting the paradigm of the blood-aqueous barrier. Exp Eye Res. 2001;73:581–92.

    CAS  PubMed  Google Scholar 

  118. Keister JC, Heidmann PS, Missel PJ. Transient analysis of ocular drug delivery: Zero-volume effect. J Pharm Sci. 1997;86:1040–5.

    CAS  PubMed  Google Scholar 

  119. Zhang W, Prausnitz MR, Edwards A. Model of transient drug diffusion across cornea. J Control Release. 2004;99:241–58.

    CAS  PubMed  Google Scholar 

  120. Deng F, Ranta VP, Kidron H, Urtti A. General Pharmacokinetic Model for Topically Administered Ocular Drug Dosage Forms. Pharm Res [Internet]. 2016;33:2680–90. https://doi.org/10.1007/s11095-016-1993-2.

    CAS  Article  Google Scholar 

  121. Wyatt HJ. Modelling transport in the anterior segment of the eye [erratum appears in Optom Vis Sci. 2004, 81(6):478]. Optom Vis Sci. 2004;81:272–82.

    PubMed  Google Scholar 

  122. Wyatt HJ. Ocular pharmacokinetics and convectional flow: Evidence from spatio- temporal analysis of mydriasis. J Ocul Pharmacol Ther. 1996;12:441–59.

    CAS  PubMed  Google Scholar 

  123. Wyatt HJ, Lustgarten M. Probing anterior segment kinetics with focally applied mydriatics. J Ocul Pharmacol Ther. 2001;17(5):461–73. https://doi.org/10.1089/108076801753266848.

    CAS  Article  PubMed  Google Scholar 

  124. Lin CW, Yuan F. Numerical simulations of ethacrynic acid transport from precorneal region to trabecular meshwork. Ann Biomed Eng. 2010;38:935–44.

    PubMed  Google Scholar 

  125. Naghipoor J, Jafary N, Rabczuk T. Mathematical and computational modeling of drug release from an ocular iontophoretic drug delivery device. Int J Heat Mass Transf [Internet]. 2018;123:1035–49. https://doi.org/10.1016/j.ijheatmasstransfer.2018.03.021 (Elsevier Ltd).

    Article  Google Scholar 

  126. Loke CY, Ooi EH, Salahudeen MS, Ramli N. Segmental aqueous humour outflow and eye orientation have strong influence on ocular drug delivery. Appl Math Model [Internet]. 2018;57:474–91. https://doi.org/10.1016/j.apm.2018.01.007 (Elsevier Inc.).

    Article  Google Scholar 

  127. Bhandari A, Bansal A, Sinha N. Effect of aging on heat transfer, fluid flow and drug transport in anterior human eye: A computational study. J Control Release [Internet]. 2020;328:286–303. https://doi.org/10.1016/j.jconrel.2020.08.044 (Elsevier).

    CAS  Article  Google Scholar 

  128. Lamminsalo M, Taskinen E, Karvinen T, Subrizi A, Murtomäki L, Urtti A, Ranta VP. Extended pharmacokinetic model of the rabbit eye for intravitreal and intracameral injections of macromolecules: quantitative analysis of anterior and posterior elimination pathways. Pharm Res. 2018;35(8):153. https://doi.org/10.1007/s11095-018-2435-0.

    CAS  Article  PubMed  Google Scholar 

  129. Pimenta AFR, Valente A, Pereira JMC, Pereira JCF, Filipe HP, Mata JLG, et al. Simulation of the hydrodynamic conditions of the eye to better reproduce the drug release from hydrogel contact lenses: experiments and modeling. Drug Deliv Transl Res [Internet]. 2016;6:755–62. https://doi.org/10.1007/s13346-016-0303-1.

    CAS  Article  Google Scholar 

  130. Ferreira JA, De Oliveira P, Da Silva PM, Murta JN, Oliveira PDE, Murta JN. Numerical simulation of aqueous humor flow: from healthy to pathological situations. Appl Math Comput [Internet]. 2014;226:777–92. https://doi.org/10.1016/j.amc.2013.10.070.

    Article  Google Scholar 

  131. Bhandari A, Bansal A, Sinha N. Numerical modeling of therapeutic lens drug delivery in the anterior human eye for the treatment of primary open-angle glaucoma. Proc Inst Mech Eng H. 2020;234(9):942–54. https://doi.org/10.1177/0954411920934960.

    Article  PubMed  Google Scholar 

  132. Li CC, Chauhan A. Modeling ophthalmic drug delivery by soaked contact lenses. Ind Eng Chem Res. 2006;45:3718–34.

    CAS  Google Scholar 

  133. Le Goff MM, Bishop PN. Adult vitreous structure and postnatal changes. Eye. 2008;22:1214–22.

    PubMed  Google Scholar 

  134. Repetto R, Ghigo I, Seminara G, Ciurlo C. A simple hydro-elastic model of the dynamics of a vitreous membrane. J Fluid Mech. 2004;503:1–14. https://doi.org/10.1017/S0022112003007389.

    Article  Google Scholar 

  135. Modarreszadeh A, Abouali O. Numerical simulation for unsteady motions of the human vitreous humor as a viscoelastic substance in linear and non-linear regimes. J Nonnewton Fluid Mech [Internet]. 2014;204:22–31. https://doi.org/10.1016/j.jnnfm.2013.12.001 (Elsevier B.V.).

    CAS  Article  Google Scholar 

  136. Tojo KJ, Ohtori A. Pharmacokinetic model of intravitreal drug injection. Math Biosci. 1994;123:59–75.

    CAS  PubMed  Google Scholar 

  137. Friedrich S, Cheng YL, Saville B. Finite element modeling of drug distribution in the vitreous humor of the rabbit eye. Ann Biomed Eng. 1997;25:303–14.

    CAS  PubMed  Google Scholar 

  138. Friedrich S, Saville B, Cheng YL. Drug distribution in the vitreous humor of the human eye: The effects of aphakia and changes in retinal permeability and vitreous diffusivity. J Ocul Pharmacol Ther. 1997;13:445–59.

    CAS  PubMed  Google Scholar 

  139. Tojo K, Nakagawa K, Morita Y, Ohtori A. A pharmacokinetic model of intravitreal delivery of ganciclovir. Eur J Pharm Biopharm. 1999;47:99–104.

    CAS  PubMed  Google Scholar 

  140. Stay MS, Xu J, Randolph TW, Barocas VH. Computer simulation of convective and diffusive transport of controlled-release drugs in the vitreous humor. Pharm Res. 2003;20:96–102.

    CAS  PubMed  Google Scholar 

  141. Xu J, Heys JJ, Barocas VH, Randolph TW. Permeability and diffusion in vitreous humor: Implications for drug delivery. Pharm Res. 2000;17:664–9.

    CAS  PubMed  Google Scholar 

  142. Missel PJ. Hydraulic flow and vascular clearance influences on intravitreal drug delivery. Pharm Res United States. 2002;19:1636–47.

    CAS  Google Scholar 

  143. Missel PJ. Hydraulic flow and vascular clearance influences on intravitreal drug delivery. Pharm Res. 2002;19:1636–47.

    CAS  PubMed  Google Scholar 

  144. Missel PJ. Simulating intravitreal injections in anatomically accurate models for rabbit, monkey, and human eyes. Pharm Res. 2012;29:3251–72.

    CAS  PubMed  PubMed Central  Google Scholar 

  145. Park J, Bungay PM, Lutz RJ, Augsburger JJ, Millard RW, Roy AS, et al. Evaluation of coupled convective-diffusive transport of drugs administered by intravitreal injection and controlled release implant. J Control Release. 2005;105:279–95.

    CAS  PubMed  Google Scholar 

  146. Kim H, Lizak MJ, Tansey G, Csaky KG, Robinson MR, Yuan P, et al. Study of ocular transport of drugs released from an intravitreal implant using magnetic resonance imaging. Ann Biomed Eng. 2005;33:150–64.

    PubMed  Google Scholar 

  147. Kathawate J, Acharya S. Computational modeling of intravitreal drug delivery in the vitreous chamber with different vitreous substitutes. Int J Heat Mass Transf [Internet]. 2008;51:5598–609. https://doi.org/10.1016/j.ijheatmasstransfer.2008.04.053 (Elsevier Ltd).

    Article  Google Scholar 

  148. Lee TWY, Robinson JR. Drug delivery to the posterior segment of the eye III: The effect of parallel elimination pathway on the vitreous drug level after subconjunctival injection. J Ocul Pharmacol Ther. 2004;20:55–64.

    CAS  PubMed  Google Scholar 

  149. Wai-Yip Lee T, Robinson JR. Drug delivery to the posterior segment of the eye IV: Theoretical formulation of a drug delivery system for subconjunctival injection. J Ocul Pharmacol Ther. 2009;25:29–37.

    Google Scholar 

  150. Robinson MR, Lee SS, Kim H, Kim S, Lutz RJ, Galban C, et al. A rabbit model for assessing the ocular barriers to the transscleral delivery of triamcinolone acetonide. Exp Eye Res. 2006;82:479–87.

    CAS  PubMed  Google Scholar 

  151. Zhang F, Chen H, Huang Y. Computer modeling of drug delivery in the anterior human eye after subconjunctival and episcleral implantation. Comput Biol Med [Internet]. 2017;89:162–9. https://doi.org/10.1016/j.compbiomed.2017.07.016 (Elsevier Ltd).

    CAS  Article  Google Scholar 

  152. Balachandran RK, Barocas VH. Computer modeling of drug delivery to the posterior eye: Effect of active transport and loss to choroidal blood flow. Pharm Res. 2008;25:2685–96.

    CAS  PubMed  Google Scholar 

  153. Kotha S, Murtomäki L. Virtual pharmacokinetic model of human eye. Math Biosci [Internet]. 2014;253:11–8. https://doi.org/10.1016/j.mbs.2014.03.014 (Elsevier Inc).

    CAS  Article  Google Scholar 

  154. Mac Gabhann F, Demetriades AM, Deering T, Packer JD, Shah SM, Duh E, et al. Protein transport to choroid and retina following periocular injection: Theoretical and experimental study. Ann Biomed Eng. 2007;35:615–30.

    PubMed  Google Scholar 

  155. del Amo EM, Rimpelä AK, Heikkinen E, Kari OK, Ramsay E, Lajunen T, et al. Pharmacokinetic aspects of retinal drug delivery. Prog Retin Eye Res [Internet]. 2017;57:134–85. https://doi.org/10.1016/j.preteyeres.2016.12.001 (Elsevier Ltd).

    CAS  Article  Google Scholar 

  156. Causin P, Malgaroli F. Mathematical assessment of drug build-up in the posterior eye following transscleral delivery. J Math Ind [Internet]. Causin and Malgaroli; 2016;6. https://doi.org/10.1186/s13362-016-0031-7.

  157. Narasimhan A, Vishnampet R. Effect of choroidal blood flow on transscleral retinal drug delivery using a porous medium model. Int J Heat Mass Transf [Internet]. 2012;55:5665–72. https://doi.org/10.1016/j.ijheatmasstransfer.2012.05.060 (Elsevier Ltd).

    Article  Google Scholar 

  158. Jooybar E, Abdekhodaie MJ, Farhadi F, Cheng YL. Computational modeling of drug distribution in the posterior segment of the eye: Effects of device variables and positions. Math Biosci [Internet]. 2014;255:11–20. https://doi.org/10.1016/j.mbs.2014.06.008 (Elsevier Inc).

    CAS  Article  Google Scholar 

  159. Kavousanakis ME, Kalogeropoulos NG, Hatziavramidis DT. Computational modeling of drug delivery to the posterior eye. Chem Eng Sci [Internet]. 2014;108:203–12. https://doi.org/10.1016/j.ces.2014.01.005 (Elsevier).

    CAS  Article  Google Scholar 

  160. Zhang Y, Bazzazi H, E Silva RL, Pandey NB, Green JJ, Campochiaro PA, et al. Three-dimensional transport model for intravitreal and suprachoroidal drug injection. Investig Ophthalmol Vis Sci. 2018;59:5266–76.

    CAS  Google Scholar 

  161. Abootorabi S, Tripathi A, Yu HW, Dávila LP. Computational modeling of intraocular drug delivery supplied by porous implants. Drug Deliv Transl Res. 2021;11:2134–43.

    CAS  PubMed  Google Scholar 

  162. Penkova A, Zhang S, Humayun M, Fraser S, Moats R, Sadhal S. Measurement of the hydraulic conductivity of the vitreous humor. J Porous Media. 2020;23:195–206.

    PubMed  PubMed Central  Google Scholar 

  163. Tsuboi S. Measurement of the volume flow and hydraulic conductivity across the isolated dog retinal pigment epithelium. Investig Ophthalmol Vis Sci. 1987;28:1776–82.

    CAS  Google Scholar 

  164. Fatt I, Hedbys BO. Flow of water in the sclera. Exp Eye Res. 1970;10:243–9.

    CAS  PubMed  Google Scholar 

  165. Engler CB, Sander B, Larsen M, Dalgaard P, Lund-Andersen H. Fluorescein transport across the human blood-retina barrier in the direction vitreous to blood: Quantitative assessment in vivo. Acta Ophthalmol. 1994;72:655–62.

    CAS  Google Scholar 

  166. Palestine AG, Brubaker RF. Pharmacokinetics of fluorescein in the vitreous. Investig Ophthalmol Vis Sci. 1981;21:542–9.

    CAS  Google Scholar 

  167. Roos MW. Theoretical estimation of retinal oxygenation during retinal artery occlusion. Physiol Meas. 2004;25:1523–32.

    PubMed  Google Scholar 

  168. Avtar R, Tandon D. Modelling the transmural transport of oxygen to the retina. Appl Math Comput. 2007;186:540–7.

    Google Scholar 

  169. Avtar R, Tandon D. Mathematical modelling of intraretinal oxygen partial pressure. Trop J Pharm Res. 2008;7:1107.

    Google Scholar 

  170. Cassani S. Blood circulation and aqueous humor flow in the eye: multi-scale modeling and clinical application. Purdue; 2016.

  171. Causin P, Guidoboni G, Malgaroli F, Sacco R, Harris A. Blood flow mechanics and oxygen transport and delivery in the retinal microcirculation: multiscale mathematical modeling and numerical simulation. Biomech Model Mechanobiol. 2016;15:525–42 (Springer Berlin Heidelberg).

    PubMed  Google Scholar 

  172. Dissertation A. Computational Modeling of Drug Transport in the Posterior Eye. 2010.

  173. He Z, Nguyen CT, Armitage JA, Vingrys AJ, Bui BV. Blood pressure modifies retinal susceptibility to intraocular pressure elevation. PLoS One. 2012;7(2):e31104. https://doi.org/10.1371/journal.pone.0031104.

  174. Guidoboni G, Harris A, Cassani S, Arciero J, Siesky B, Amireskandari A, et al. Intraocular pressure, blood pressure, and retinal blood flow autoregulation: A mathematical model to clarify their relationship and clinical relevance. Investig Ophthalmol Vis Sci. 2014;55:4105–18.

    Google Scholar 

  175. Carlson BE, Arciero JC, Secomb TW. Theoretical model of blood flow autoregulation: Roles of myogenic, shear-dependent, and metabolic responses. Am J Physiol Hear Circ Phyiol. 2008;295:1572–9.

    Google Scholar 

  176. Arciero J, Harris A, Siesky B, Amireskandari A, Gershuny V, Pickrell A, et al. Theoretical analysis of vascular regulatory mechanisms contributing to retinal blood flow autoregulation. Investig Ophthalmol Vis Sci. 2013;54:5584–93.

    Google Scholar 

  177. Takahashi T, Nagaoka T, Yanagida H, Saitoh T, Kamiya A, Hein T, et al. A mathematical model for the distribution of hemodynamic parameters in the human retinal microvascular network. J Biorheol. 2009;23:77–86.

    Google Scholar 

  178. Liu D, Wood NB, Witt N, Hughes AD, Thom SA, Xu XY. Assessment of energy requirement for the retinal arterial network in normal and hypertensive subjects. J Biomech Eng. 2012;134:1–7.

    Google Scholar 

  179. Ganesan P, He S, Xu H. Development of an image-based network model of retinal vasculature. Ann Biomed Eng. 2010;38:1566–85.

    CAS  PubMed  Google Scholar 

  180. Ganesan P, He S, Xu H. Analysis of retinal circulation using an image-based network model of retinal vasculature. Microvasc Res [Internet]. 2010;80:99–109. https://doi.org/10.1016/j.mvr.2010.02.005 (Elsevier Inc).

    CAS  Article  Google Scholar 

  181. Ganesan P, He S, Xu H. Development of an image-based model for capillary vasculature of retina. Comput Methods Programs Biomed. 2011;102:35–46.

    CAS  PubMed  Google Scholar 

  182. Malek J, Azar AT, Nasralli B, Tekari M, Kamoun H, Tourki R. Computational analysis of blood flow in the retinal arteries and veins using fundus image. Comput Math with Appl [Internet]. 2015;69:101–16. https://doi.org/10.1016/j.camwa.2014.11.017 (Elsevier Ltd).

    Article  Google Scholar 

  183. Malek J, Azar AT, Tourki R. Impact of retinal vascular tortuosity on retinal circulation. Neural Comput Appl. 2014;26:25–40.

    Google Scholar 

  184. Dziubek A, Guidoboni G, Harris A, Hirani AN, Rusjan E, Thistleton W. Effect of ocular shape and vascular geometry on retinal hemodynamics: a computational model. Biomech Model Mechanobiol. 2016;15:893–907 (Springer Berlin Heidelberg).

    PubMed  Google Scholar 

  185. Rebhan J, Parker LP, Kelsey LJ, Chen FK, Doyle BJ. A computational framework to investigate retinal haemodynamics and tissue stress. Biomech Model Mechanobiol [Internet]. 2019;18:1745–57. https://doi.org/10.1007/s10237-019-01172-y (Springer Berlin Heidelberg).

    Article  Google Scholar 

  186. Abouali O, Modareszadeh A, Ghaffarieh A, Tu J. Investigation of saccadic eye movement effects on the fluid dynamic in the anterior chamber. J Biomech Eng. 2012;134:1–9.

    Google Scholar 

  187. Modarreszadeh S, Abouali O, Ghaffarieh A, Ahmadi G. Physiology of aqueous humor dynamic in the anterior chamber due to rapid eye movement. Physiol Behav [Internet]. 2014;135:112–8. https://doi.org/10.1016/j.physbeh.2014.05.017 (Elsevier Inc).

    CAS  Article  Google Scholar 

  188. Repetto R, Pralits JO, Siggers JH, Soleri P. Phakic iris-fixated intraocular lens placement in the anterior chamber: effects on aqueous flow. Invest Ophthalmol Vis Sci [Internet]. 2015;56:3061–8. https://doi.org/10.1167/iovs.14-16118.

    CAS  Article  Google Scholar 

  189. Dvoriashyna M, Repetto R, Tweedy JH. Oscillatory and steady streaming flow in the anterior chamber of the moving eye. J Fluid Mech. 2019;863:904–26.

    Google Scholar 

  190. David T, Smye S, Dabbs T, James T. A model for the fluid motion of vitreous humour of the human eye during saccadic movement. Phys Med Biol. 1998;43:1385–99.

    CAS  PubMed  Google Scholar 

  191. Repetto R, Stocchino A, Cafferata C. Experimental investigation of vitreous humour motion within a human eye model. Phys Med Biol. 2005;50:4729–43.

    PubMed  Google Scholar 

  192. Ferroni M, Cereda MG, Boschetti F. A combined approach for the analysis of ocular fluid dynamics in the presence of saccadic movements. Ann Biomed Eng. 2018;46:2091–101.

    PubMed  Google Scholar 

  193. Ferroni M, De Gaetano F, Cereda MG, Boschetti F. A drug delivery analysis of large molecules in ocular vitreous chamber: Dependency on saccadic movements after intravitreal injection. Med Eng Phys [Internet]. 2020;82:49–57. https://doi.org/10.1016/j.medengphy.2020.06.005 (Elsevier Ltd).

    Article  Google Scholar 

  194. Silva AF, Pimenta F, Alves MA, Oliveira MSN. Flow dynamics of vitreous humour during saccadic eye movements. J Mech Behav Biomed Mater. 2020;110:103860. https://doi.org/10.1016/j.jmbbm.2020.103860.

  195. Repetto R. An analytical model of the dynamics of the liquefied vitreous induced by saccadic eye movements. Meccanica. 2006;41:101–17.

    Google Scholar 

  196. Repetto R, Siggers JH, Stocchino A. Mathematical model of flow in the vitreous humor induced by saccadic eye rotations: Effect of geometry. Biomech Model Mechanobiol. 2010;9:65–76.

    CAS  PubMed  Google Scholar 

  197. Stocchino A, Repetto R, Cafferata C. Eye rotation induced dynamics of a Newtonian fluid within the vitreous cavity: The effect of the chamber shape. Phys Med Biol. 2007;52:2021–34.

    PubMed  Google Scholar 

  198. Stocchino A, Repetto R, Siggers JH. Mixing processes in the vitreous chamber induced by eye rotations. Phys Med Biol. 2010;55:453–67.

    PubMed  Google Scholar 

  199. Repetto R, Tatone A, Testa A, Colangeli E. Traction on the retina induced by saccadic eye movements in the presence of posterior vitreous detachment. Biomech Model Mechanobiol. 2011;10:191–202.

    CAS  PubMed  Google Scholar 

  200. Balachandran RK, Barocas VH. Contribution of saccadic motion to intravitreal drug transport: Theoretical analysis. Pharm Res. 2011;28:1049–64.

    CAS  PubMed  PubMed Central  Google Scholar 

  201. Abouali O, Modareszadeh A, Ghaffariyeh A, Tu J. Numerical simulation of the fluid dynamics in vitreous cavity due to saccadic eye movement. Med Eng Phys [Internet]. 2012;34:681–92. https://doi.org/10.1016/j.medengphy.2011.09.011 (Institute of Physics and Engineering in Medicine).

    Article  Google Scholar 

  202. Modareszadeh A, Abouali O, Ghaffarieh A, Ahmadi G. Saccade movements effect on the intravitreal drug delivery in vitreous substitutes: A numerical study. Biomech Model Mechanobiol. 2013;12:281–90.

    PubMed  Google Scholar 

  203. Cabrera Fernández D, Niazy AM, Kurtz RM, Djotyan GP, Juhasz T. Finite element analysis applied to cornea reshaping. J Biomed Opt. 2005;10:064018.

    PubMed  Google Scholar 

  204. Deenadayalu C, Mobasher B, Rajan SD, Hall GW. Refractive change induced by the LASIK flap in a biomechanical finite element model. J Refract Surg. 2006;22:286–92.

    PubMed  Google Scholar 

  205. Pandolfi A, Fotia G, Manganiello F. Finite element simulations of laser refractive corneal surgery. Eng Comput. 2009;25:15–24.

    Google Scholar 

  206. Sinha Roy A, Dupps WJ Jr. Patient-specific modeling of corneal refractive surgery outcomes and inverse estimation of elastic property changes. J Biomech Eng. 2011;133(1):011002. https://doi.org/10.1115/1.4002934.

  207. Sinha Roy A, Dupps WJ, Roberts CJ. Comparison of biomechanical effects of small-incision lenticule extraction and laser in situ keratomileusis: Finite-element analysis. J Cataract Refract Surg [Internet]. 2014;40:971–80. https://doi.org/10.1016/j.jcrs.2013.08.065 (ASCRS and ESCRS).

    Article  Google Scholar 

  208. Seven I, Vahdati A, De Stefano VS, Krueger RR, Dupps WJ. Comparison of patient-specific computational modeling predictions and clinical outcomes of lasik for myopia. Investig Ophthalmol Vis Sci. 2016;57:6287–97.

    Google Scholar 

  209. Ariza-Gracia M, Zurita J, Piñero DP, Calvo B, Rodríguez-Matas JF. Automatized patient-specific methodology for numerical determination of biomechanical corneal response. Ann Biomed Eng. 2016;44:1753–72.

    CAS  PubMed  Google Scholar 

  210. Bao FJ, Wang JJ, Cao S, Liao N, Shu B, Zhao YP, et al. Development and clinical verification of numerical simulation for laser in situ keratomileusis. J Mech Behav Biomed Mater [Internet]. 2018;83:126–34. https://doi.org/10.1016/j.jmbbm.2018.04.016 (Elsevier Ltd).

    Article  Google Scholar 

  211. Fang L, Ma W, Wang Y, Dai Y, Fang Z. Theoretical analysis of wave-front aberrations induced from conventional laser refractive surgery in a biomechanical finite element model. Invest Ophthalmol Vis Sci. 2020;61(5):34. https://doi.org/10.1167/iovs.61.5.34.

    Article  PubMed  PubMed Central  Google Scholar 

  212. Suen WLL, Wong HS, Yu Y, Lau LCM, Lo ACY, Chau Y. Ultrasound-mediated transscleral delivery of macromolecules to the posterior segment of rabbit eye in vivo. Investig Ophthalmol Vis Sci. 2013;54:4358–65.

    CAS  Google Scholar 

  213. Wang C, Seo SJ, Kim JS, Lee SH, Jeon JK, Kim JW, et al. Intravitreal implantable magnetic micropump for on-demand VEGFR-targeted drug delivery. J Control Release [Internet]. 2018;283:105–12. https://doi.org/10.1016/j.jconrel.2018.05.030 (Elsevier).

    CAS  Article  Google Scholar 

  214. Otto KJ, Schmidt CE. Neuron-targeted electrical modulation. Science (80- ) [Internet]. 2020;367:1303–4. https://doi.org/10.1126/science.abb0216 (American Association for the Advancement of Science).

    CAS  Article  Google Scholar 

  215. Lajunen T, Nurmi R, Kontturi L, Viitala L, Yliperttula M, Murtomäki L, et al. Light activated liposomes: Functionality and prospects in ocular drug delivery. J Control Release [Internet]. 2016;244:157–66. https://doi.org/10.1016/j.jconrel.2016.08.024 (Elsevier B.V.).

    CAS  Article  Google Scholar 

  216. Grimaudo MA, Pescina S, Padula C, Santi P, Concheiro A, Alvarez-Lorenzo C, et al. Poloxamer 407/TPGS Mixed Micelles as Promising Carriers for Cyclosporine Ocular Delivery. Mol Pharm. 2018;15:571–84.

    CAS  PubMed  Google Scholar 

  217. Kabiri M, Kamal SH, Pawar SV, Roy PR, Derakhshandeh M, Kumar U, et al. A stimulus-responsive, in situ-forming, nanoparticle-laden hydrogel for ocular drug delivery. Drug Deliv Transl Res. 2018;8:484–95.

    CAS  PubMed  PubMed Central  Google Scholar 

  218. Lin X, Wu X, Chen X, Wang B, Xu W. Intellective and stimuli-responsive drug delivery systems in eyes. Int J Pharm [Internet]. 2021;602:120591. https://doi.org/10.1016/j.ijpharm.2021.120591 (Elsevier B.V.).

    CAS  Article  Google Scholar 

  219. Zahn D, Klein K, Radon P, Berkov D, Erokhin S, Nagel E, et al. Investigation of magnetically driven passage of magnetic nanoparticles through eye tissues for magnetic drug targeting. Nanotechnology [Internet]. 2020;31:495101. https://doi.org/10.1088/1361-6528/abb0b4 (IOP Publishing).

    CAS  Article  Google Scholar 

  220. Erokhin S, Berkov D. Magnetic Targeted Drug Delivery to the Human Eye Retina: An Optimization Methodology. IEEE J Electromagn RF Microwaves Med Biol. 2019;3:3–8 (IEEE).

    Google Scholar 

  221. Eljarrat-Binstock E, Raiskup F, Frucht-Pery J, Domb AJ. Transcorneal and transscleral iontophoresis of dexamethasone phosphate using drug loaded hydrogel. J Control Release. 2005;106:386–90.

    CAS  PubMed  Google Scholar 

  222. He X, Yuan Z, Gaeke S, Kao WWY, Li SK, Miller D, et al. Laser-Activated Drug Implant for Controlled Release to the Posterior Segment of the Eye. ACS Appl Bio Mater. 2021;4:1461–9.

    CAS  PubMed  Google Scholar 

  223. Rimpelä AK, Reinisalo M, Hellinen L, Grazhdankin E, Kidron H, Urtti A, et al. Implications of melanin binding in ocular drug delivery. Adv Drug Deliv Rev [Internet]. 2018;126:23–43. https://doi.org/10.1016/j.addr.2017.12.008 (Elsevier B.V.).

    CAS  Article  Google Scholar 

  224. Urtti A, Salminen L, Kujari H, Jäntti V. Effect of ocular pigmentation on pilocarpine pharmacology in the rabbit eye. II. Drug response. Int J Pharm. 1984;19(1):53–61. https://doi.org/10.1016/0378-5173(84)90132-7.

    CAS  Article  Google Scholar 

  225. Shibata T, Mishima H, Kurokawa T. Ocular pigmentation and intraocular pressure response to forskolin. Curr Eye Res [Internet]. 1988;7:667–74. https://doi.org/10.3109/02713688809033195 (Taylor & Francis).

    CAS  Article  Google Scholar 

  226. Leblanc B, Jezequel S, Davies T, Hanton G, Taradach C. Binding of drugs to eye melanin is not predictive of ocular toxicity. Regul Toxicol Pharmacol. 1998;28(2):124–32. https://doi.org/10.1006/rtph.1998.1243.

    CAS  Article  PubMed  Google Scholar 

  227. Raghavan PR, Zane PA, Tripp SL. Calculation of drug-melanin binding energy using molecular modeling. Experientia [Internet]. 1990;46:77–80. https://doi.org/10.1007/BF01955422.

    CAS  Article  Google Scholar 

  228. Agrahari V, Mandal A, Agrahari V, Trinh HM, Joseph M, Ray A, et al. A comprehensive insight on ocular pharmacokinetics. Drug Deliv Transl Res [Internet]. 2016;6:735–54. https://doi.org/10.1007/s13346-016-0339-2.

    CAS  Article  Google Scholar 

  229. Hollemans M, Elferink RO, De Groot PG, Strijland A, Tager JM. Accumulation of weak bases in relation to intralysosomal pH in cultured human skin fibroblasts. Biochim Biophys Acta. 1981;643(1):140–51. https://doi.org/10.1016/0005-2736(81)90226-1.

    CAS  Article  PubMed  Google Scholar 

  230. Lowrey AH, Fameini GR, Loumbev V, Wilson LY, Tosk JM. Modeling Drug-Melanin Interaction With Theoretical Linear Solvation Energy Relationships. Pigment Cell Res [Internet]. 1997;10:251–6. https://doi.org/10.1111/j.1600-0749.1997.tb00684.x (John Wiley & Sons, Ltd).

    CAS  Article  Google Scholar 

  231. Radwa A, Fra̧ckowiak T, Ibrahim H, Aubry A-F, Kaliszan R. Chromatographic modelling of interactions between melanin and phenothiazine and dibenzazepine drugs. Biomed Chromatogr [Internet]. 1995;9:233–7. https://doi.org/10.1002/bmc.1130090509.

    Article  Google Scholar 

  232. Reilly J, Williams SL, Forster CJ, Kansara V, End P, Serrano-Wu MH. High-throughput melanin-binding affinity and in silico methods to aid in the prediction of drug exposure in ocular tissue. J Pharm Sci. 2015;104(12):3997–4001. https://doi.org/10.1002/jps.24680.

    CAS  Article  PubMed  Google Scholar 

  233. Manzanares JA, Rimpelä A-K, Urtti A. Interpretation of ocular melanin drug binding assays. Alternatives to the model of multiple classes of independent sites. Mol Pharm [Internet]. 2016;13:1251–7. https://doi.org/10.1021/acs.molpharmaceut.5b00783.

    CAS  Article  Google Scholar 

  234. Rimpelä A-K, Schmitt M, Latonen S, Hagström M, Antopolsky M, Manzanares JA, et al. Drug Distribution to Retinal Pigment Epithelium: Studies on Melanin Binding, Cellular Kinetics, and Single Photon Emission Computed Tomography/Computed Tomography Imaging. Mol Pharm [Internet]. 2016;13:2977–86. https://doi.org/10.1021/acs.molpharmaceut.5b00787 (American Chemical Society).

    CAS  Article  Google Scholar 

  235. Pitkänen L, Ranta VP, Moilanen H, Urtti A. Binding of betaxolol, metoprolol and oligonucleotides to synthetic and bovine ocular melanin, and prediction of drug binding to melanin in human choroid-retinal pigment epithelium. Pharm Res. 2007;24:2063–70.

    PubMed  Google Scholar 

  236. Menon IA, Trope GE, Basu PK, Wakeham DC, Persad SD. Binding of timolol to iris-ciliary body and melanin: An in vitro model for assessing the kinetics and efficacy of long-acting antiglaucoma drugs. J Ocul Pharmacol Ther [Internet]. 1989;5:313–24. https://doi.org/10.1089/jop.1989.5.313 (Mary Ann Liebert, Inc., publishers).

    CAS  Article  Google Scholar 

  237. Borrelli E, Sacconi R, Brambati M, Bandello F, Querques G. In vivo rotational three-dimensional OCTA analysis of microaneurysms in the human diabetic retina. Sci Rep. 2019;9:1–8.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Indian Institute of Technology (Indian School of Mines) Dhanbad, Jharkhand, 826004, India

    Ajay Bhandari

Authors
  1. Ajay Bhandari
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ajay Bhandari.

Ethics declarations

Competing Interests

None.

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

Bhandari, A. Ocular Fluid Mechanics and Drug Delivery: A Review of Mathematical and Computational Models. Pharm Res 38, 2003–2033 (2021). https://doi.org/10.1007/s11095-021-03141-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11095-021-03141-6

KEY WORDS

  • computational and mathematical models
  • fluid mechanics
  • non-invasive methods
  • ocular drug delivery