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Kelsey Green, William Norkett, Algis Grybauskas, Chetan Velagapudi, John R Samples, Louis R Pasquale, Paul A Knepper; Mathematical modeling of shear stress and shear rate: implications for NTG and POAG. Invest. Ophthalmol. Vis. Sci. 2015;56(7 ):2765.
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© ARVO (1962-2015); The Authors (2016-present)
Blood flow abnormalities occur in both primary open angle glaucoma (POAG) and normal tension glaucoma (NTG). Color Doppler imaging studies demonstrate decreased blood velocities and increased flow resistance in the short posterior ciliary arteries (SPCAs) for both POAG and NTG. In this study, we mathematically modeled the shear rate and shear stress in SPCAs in controls, POAG, and NTG and the subsequent effects on platelet aggregation.
The shear rate (γ) modeling was based on a derivation of Poiseuille’s equation. If blood’s viscosity is constant and the fluid is incompressible, then: γ=∂v/∂r. Assuming that there is no slip, implying that velocity at the capillary wall is zero: γ=v/r where v is the velocity r is the radius and γ is the shear rate. Shear stress was then modeled: τ=μ*γ where γ is the shear rate, µ is the viscosity, and is τ the shear stress. Velocities, radius of the SPCAs and the relative viscosity of blood were obtained from literature sources. Vascular stenosis, either by compression of SPCAs in POAG or vasospasm in NTG, was assumed to be present in the disease state and values were calculated accordingly.
Shear rates varied between groups, especially when varying levels of stenosis were assumed in the diseased state. When 40% stenosis in both POAG and NTG were assumed, shear rates exceeded 10,000 s-1. See Table 1. At 10,000 s-1, platelet aggregation is induced without the presence of agonists and could potentially lead to hemorrhage as back pressure increases. Minimal changes were found when no stenosis was assumed. Both shear rate and shear stress decreased by 20% in POAG and 11% in NTG compared to control.
Calculated shear rates for the SPCAs are pathologically high when stenosis and/or vasoconstriction are assumed in either NTG or POAG. High shear rates cause platelets to aggregate, leading to thrombosis and subsequent hemorrhages. Given the anastomotic features, the superior and inferior SPCA are the most distal end and vulnerable watershed arterioles and predicative of arcuate field defects in NTG and POAG.
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