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

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.

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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 s1)

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 min1)

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

AH outflow rate through TM (uL min1)

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

Fluid volume loss rate via surgical pathway into subconjunctival tissue (uL min1)

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

Fluid outflow rate through uveoscleral pathway (uL min1)

EVP :

Episcleral venous pressure (Pa)

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

Outflow facility via trabecular pathway (uL min1 Pa1)

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

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 s2)

\(C\) :

Drug concentration (mol m3)

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

Initial drug concentration (mol m3)

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

Drug concentration in the blood (mol m3)

\(D\) :

Diffusion coefficient of the drug (m2 s1)

Dp, DHb :

Diffusion coefficient of free oxygen and oxyhemoglobin in plasma and red blood vessels, respectively (m2 s1)

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

Mean particle diameter of packed bed (µm)

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

Inertial resistance factor

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

Drug clearance rate from the tissues (mol m3 s1)

\(m\) :

Reaction rate coefficient (s1)

\(\overrightarrow{n}\) :

Normal vector pointing outwards from a surface

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

Added convectional velocity (m s1)

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

Mass transfer coefficient at the scleral surface (m s1)

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 ml1)

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 (s1)

\(\sigma\)  :

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 m2)

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

Relative viscosity for a fixed discharge hematocrit (0.45) (Ns m2)

\(\upbeta\) :

Thermal expansion coefficient of AH (K1)

\(\varnothing\) :

Porosity

\(\gamma\) :

Drug transport rate constant across choroidal blood vessels (s1)

\(\varphi\) :

Kozeny and solubility constant, respectively

\(\in\) :

Parameter to vary effective outflow facility

\(\alpha\) :

Permeability (m2)

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

Oxygen solubility coefficient (ml O2 ml1 mmHg1)

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

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)

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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

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KEY WORDS

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