Thin-Film Coupled Fluid-Solid Analysis of Flow Through the Ahmed Glaucoma Drainage Device

Transcription

1 Matthew S. Stay Department of Biomedical Engineering, University of Minnesota, Tingrui Pan Department of Electrical and Computer Engineering, University of Minnesota, J. David Brown Department of Opthalmology, University of Minnesota, Babak Ziaie Department of Electrical and Computer Engineering, University of Minnesota, Victor H. Barocas 1 Department of Biomedical Engineering, University of Minnesota, Thin-Film Coupled Fluid-Solid Analysis of Flow Through the Ahmed Glaucoma Drainage Device The Ahmed glaucoma valve (AGV) is a popular glaucoma drainage device, allowing maintenance of normal intraocular pressure in patients with reduced trabecular outflow facility. The uniquely attractive feature of the AGV, in contrast to other available drainage devices, is its variable resistance in response to changes in flow rate. As a result of this variable resistance, the AGV maintains a pressure drop between 7 and 12 mm Hg for a wide range of aqueous humor flow rates. In this paper, we demonstrate that the nonlinear behavior of the AGV is a direct result of the flexibility of the valve material. Due to the thin geometry of the system, the leaflets of the AGV were modeled using the von Kármán plate theory coupled to a Reynolds lubrication theory model of the aqueous humor flow through the valve. The resulting two-dimensional coupled steady-state partial differential equation system was solved by the finite element method. The Poisson s ratio of the valve was set to 0.45, and the modulus was regressed to experimental data, giving a best-fit value 4.2 MPa. Simulation results compared favorably with previous experimental studies and our own pressure-drop/flow-rate data. For an in vitro flow of 1.6 L/min, we calculated a pressure drop of 5.8 mm Hg and measured a pressure drop of 5.2±0.4 mm Hg. As flow rate was increased, pressure drop rose in a strongly sublinear fashion, with a flow rate of 20 L/min giving a predicted pressure drop of only 10.9 mm Hg and a measured pressure drop of 10.5±1.1 mm Hg. The AGV model was then applied to simulate in vivo conditions. For an aqueous humor flow rate of L/min, the calculated pressure drops were 5.3 and 6.3 mm Hg. DOI: / Keywords: Lubrication Theory, von Kármán Plate Theory, Finite Element Method, Intraocular Pressure 1 To whom correspondence should be addressed. Contributed by the Bioengineering Division for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received by the Bioengineering Division April 7, 2004; revision received May 9, Associate Editor: David J. Beebe. 1 Introduction Aqueous humor AH is the clear liquid secreted into the eye to nourish the tissues of the anterior segment, including the lens and cornea. AH originates in the ciliary processes and, due to pressure-driven flow, is transported between the lens and iris, through the pupil, and eventually through the trabecular meshwork, and into Schlemm s canal. The density and viscosity of AH are similar to those of saline, and normal physiological flow rates range between 2 and 3 L/min 1. If the permeability of the trabecular meshwork decreases, the intraocular pressure IOP increases to maintain relatively constant flow rate, and glaucoma optic nerve damage usually associated with elevated IOP can arise. The most common form is primary open-angle glaucoma, in which the outflow pathway is accessible but of high pressure 2. Drug therapies or laser trabeculoplasty are used to treat primary open-angle glaucoma, but if elevated IOP persists, surgical treatment is usually advised. The most common surgical options are trabeculectomy or implantation of a glaucoma drainage device, which directs the AH around the increased-resistance trabecular meshwork 2. Several glaucoma drainage devices have been studied including the following Table 1 : 1 the Molteno implant, 2 the Baerveldt implant, 3 the OptiMed glaucoma pressure regulator, 4 the Krupin eye valve, and 5 the Ahmed glaucoma valve AGV implant. Each device includes a tube and a plate. The tube is inserted into the anterior chamber and drains the AH from the eye to a plate placed in the posterior subconjunctiva space. A fibrous capsule forms around the plate and provides the resistance to AH flow, ideally maintaining an IOP between 8 and 20 mm Hg 3. Devices 3, 4, and 5 have added internal resistances to prevent hypotony overfiltration of the AH and dangerously low IOP until the fibrous capsule forms. Prata et al. 4 were the first to characterize the performance of glaucoma drainage devices in vitro. They constructed an experimental apparatus to supply steady-state flow rates of both saline and plasma through drainage devices 1 5, and measured the resulting pressure drop for flow rates ranging from 2 to 100 L/min. All devices, with the exception of the Ahmed valve, exhibited a linear relation between flow-rate and pressure drop, i.e., all had constant resistances R= P/F, where P is the pressure drop and F is the volumetric flow rate. Eisenberg et al. 5 and Francis et al. 6 confirmed Prata s trends for selected devices Table 1. Constant resistance was expected for the Molteno and Baerveldt drainage devices since their designs do not include valves. Resistance in these devices, due solely to flow through the tube, can be estimated with Poiseuille s law for steady laminar flow in a rigid pipe 7. The OptiMed pressure regulator adds an array of microtubes between the tube and plate. The microtubes are smaller than the drainage tube of either the Molteno and Baerveldt designs, and so the resistance for the OptiMed drainage device is greater. The Krupin device uses a slit valve at the end of the tube to regulate AH flow. It is unclear why the Krupin valve has a constant resistance, given its design, unless the 776 / Vol. 127, OCTOBER 2005 Copyright 2005 by ASME Transactions of the ASME

2 Device Table 1 Summary of glaucoma experiments Internal resistance Prata 4 Experimental studies Eisenberg 5 Francis 6 Molteno plate None X Baerveldt implant None X X OptiMed regulator Microtube array X X X Krupin valve Slit valve X X X Ahmed valve Deformable ceiling X X Fig. 1 The AGV. Cartoon a shows where the AGV is attached to the eye, and how the valve inlet tube is used to redirect aqueous humor around the trabecular meshwork. Cartoon b shows the basic AGV components construction: AH is fed through an inlet tube into a channel formed by two opposing sheets of silastic material. Drawing c defines the dimensions of the two-dimensional channel formed by the silastic sheets or leaflets. flaps of tubing that form the slit-valve fold completely open even under low pressure. In this scenario, flow resistance would be due only to the tube, making the device similar to the Baerveldt and Molteno designs. Both Francis et al. 6 and Prata et al. 4 measured in vitro pressure-drop/flow-rate curves. In vitro studies of the AGV demonstrated a sublinear pressuredrop response to increased flow rate, an effect hypothesized to be due to the flexibility of the valve material and the geometry of the valve. The AGV is constructed from two opposed deformable silastic sheets pinned together along the side walls, which run parallel to the direction of AH flow. Figure 1 a shows a cartoon of the AGV packaging sutured to the eye, Fig. 1 b shows just the valve, and Fig. 1 c shows the dimensions of the valve channel and leaflets. When AH passes through the channel formed by the two membranes, the fluid pressure pushes the two opposing sheets apart. If the AH flow rate increases, the valve leaflets move farther apart. The hypothesis tested by this study was that the pressuredrop/flow-rate curve for the AGV is nonlinear because the valve leaflets continue to deform for increased flow rate, i.e., increased upstream pressure. The relatively narrow range of pressure drops that result from a relatively large range of AH flow is, in part, the reason for the AGVs increased popularity 8. Previous in vitro experiments 4,6 have shown that small AH flow rates, approximately 2 5 L/min, produce pressure drops between 7 and 8 mm Hg while large flow rates, those between 25 and 50 L/min, produce pressure drops of 12 to 14 mm Hg. In this paper we use our coupled Reynolds-von Kármán RVK model to predict the sublinear behavior of the AGV. An experimental setup was constructed to measure flow-curve data of water through the AGV, and a comparison between the theoretical predictions and experimental results is presented. 2 Methods Mathematical Model The AGV can be viewed as a channel formed from the enclosure of two opposite-facing thin rubber sheets, or leaflets. As AH moves through the channel, the leaflets part, and for a given steady-state flow, an equilibrium pressure drop forms across the length of the channel. If the AH flow rate increases, the equilibrium pressure increases, and the elastic leaflets displace farther. The pressure does not rise as high as it would if the leaflets were fixed, since increased parting of the leaflets increases the channel height and decreases channel resistance. Thus, coupled fluid-solid interactions between the valve leaflet and the AH flow generate a resistance that is a function of the flow rate through the valve. Since AH is Newtonian and incompressible, the fluid mechanics of the AGV are governed by the well-known Navier Stokes equations 7, while the mechanics of leaflet deformation are described by the solid equilibrium relations 9,10. To perform a full three-dimensional computer simulation of the AGV dynamics would require coupling the Navier Stokes equations and the solid equilibrium relations, a difficult and expensive computation. Fortunately, scaling of the Navier Stokes equations and solid equilibrium relations produces a simplified, but accurate, twodimensional coupled RVK model that can predict valve performance. The RVK model has two advantages: reduced dimensionality which results in reduced computational cost and a static domain which results in reduced complexity. The characteristic length L of the channel and leaflet domains is between 2 and 3 mm. The leaflet thickness t is approximately 125 m, and given the 5 20 mm Hg range of reported in vitro equilibrium pressure drops across the AGV 4, we expect leaflet displacements w to be less than 100 m. This implies that the characteristic channel height H is on the order of 100 m, making the channel height-to-length aspect ratio H/ L=0.1 mm/ 2.5 mm =0.04. The valve leaflet thickness-to-length aspect ratio t/ L =0.125 mm/ 2.5 mm=0.05, suggesting the AGV fluid and solid domains are thin and planar. It is well established that if the channel curvature is small dh/dx 2 1 and viscous drag dominates Re H/L 2 1, then the lubrication assumptions 11 apply: 1 the fluid velocity is parabolic with respect to the fluid film thickness z, and 2 the fluid pressure is uniform across the channel thickness. The channelflow Reynolds number is Re= UH/ where is the fluid density, is the fluid viscosity, and U is the characteristic velocity. For AH at room temperature, the viscosity and density are approximately =1.1 cp and =1.0 g/cm 3, respectively. For an AH flow rate of 25 L/min, the characteristic velocity is U=F/ HW 0.1 cm/s, and assuming this flow rate produces the maximum expected leaflet deformation w=100 m, the ratio of fluid inertia to viscous drag becomes Re H/L We know from our characteristic length and transverse displacement that the channel curvature is small w/l 2 = 10 2 cm/10 1 cm These results suggest that the lubrication, or Reynolds, equation accurately describes the AH flow in the AGV. Written for a two-dimensional domain, the steady-state Reynolds equation is H 0 + w 3 p =0 1 3 where p is the lubrication pressure. The channel height is given by H 0 +w, where H 0 is the initial height and w is the transverse displacement z of the valve at position x,y. The initial height is taken as zero, or less than a few microns, since the valve is initially closed. Equation 1 states that the local volumetric flow rate per unit width f Journal of Biomechanical Engineering OCTOBER 2005, Vol. 127 / 777

3 f = H 0 + w 3 p 3 remains constant for steady flow throughout the length of the channel. The outlet pressure P x=l,y is taken to be zero while the inlet pressure is set to P x=0,y = P IN. Scaling of the solid momentum balance and the subsequent derivation of the von Kármán equations is tedious, and so only the basic details underlying the equations are discussed here. More detailed descriptions can be found in a good textbook, e.g., Ref. 9, and in our other paper 12 on coupled lubrication-plate systems. Von Kármán theory is a large deformation theory for an elastic plate, which requires that 1 the solid plate is thin t/l 1, 2 the deflection w of the plate is the same order of magnitude as the plate thickness, and 3 the curvature of the plate after deflection is small w/ x 1 and w/ y 1. Given these conditions, von Kármán reasoned that all strain components are small, the solid is Neo Hookean, and in the strain-displacement relationship, only nonlinear terms that depend on w/ x and w/ y are retained. The final assumption is Kirchhoff s hypothesis that normals to the midplane surface before deformation remain normal after deformation, and that the displacements vary linearly with z through the plate thickness. From the Kirchhoff hypothesis, we can rewrite the threedimensional displacement field u x, u y, and u z in terms of the leaflet mid-plane displacements u, v, and w the rotations of the midplane normal about the x and y axis, denoted by x and y,as 2 ux = u x,y z x x,y u y = v x,y z y x,y 3 u z = w x,y Substituting Eq. 3 into Green s strain tensor and imposing assumption 3, the in-plane portion of the Lagrangian strain tensor E= simplifies to E= = e= e= N zk= 4 where the linear strain e=, the nonlinear strain e= N, and the curvature k= tensors are e= = 1 2 u + u T 5 e= N = w w T 6 k= = T 7 where u = uvw T and = x y T. The von Kármán equations are a simplification of the solid equilibrium relations, where the stress balance is averaged over the plate thickness. Averaging the solid equilibrium stress balance gives a differential force balance on the plate midplane N= = 0 where N= is the result of stress-averaging the second Piola Kirchhoff stress tensor through the plate thickness. The twodimensional in-plane stress resultant tensor is N= = Et 1 2 tr e= e= N I= + Et + N 1+ e= 1 2 e= Stress averaging of the momentum equations inherently results in a loss of information. For example, if the stress varied linearly through the thickness of the plate, the stress-averaged resultant would not account for the moment acting on the plate. Thus, the von Kármán equations include a moment balance, using the z axis as the moment arm. The differential moment balance on the midplane surface is given by 8 9 M= = Q 10 where M= is moment tensor and Q is transverse shear vector. The two-dimensional moment tensor and two-dimensional shear vector are M= = 12 t3 E E 1 2 tr k= I= + k= Et Q = + w The final equation in the von Kármán analysis is the averaged moment balance in the plate-thickness z direction. It relates the change in the plate s transverse shear plus in-plane stretching to the pressure difference across the plate. Since the pressure loading on the valve leaflet is the fluid pressure, fluid-solid coupling in the AGV is captured in the final governing balance Q + w N= = p 13 If the pressure loading from the AH is large enough to stretch the midplane surface and create an appreciable mid-plane area change, the nonlinear terms w N= in the Eq. 13 will dominate, and the solid deformation problem becomes a membrane stretches, as opposed to a bending, problem. This occurs if the ratio of midplane displacement to thickness w/t exceeds unity. Equations 1, 8, 10, and 13 were solved, with appropriate substitutions, for the unknown variables p, u, v, x, y, and w with the Galerkin finite element method. The six unknowns were interpolated with biquadratic basis functions. Integration of the solid residual and Jacobian equations was performed with a two-point reduced Gauss quadrature; a three-by-three point quadrature was used to integrate the lubrication residual and Jacobian. The algebraic problem was solved with an ILUT preconditioned GMRES solver 13. Since silicone rubber is nearly incompressible, a Poisson s ratio of 0.45 was used. The elastomer s Young modulus was not known a priori, and was determined through a curve fit to the experimental data. A modulus of 4.2 MPa minimized the squared difference between the theoretical and experimental flow curves. The saline viscosity was 1.1 cp. The valve dimensions were as given in Fig. 1, and the leaflet thickness was 125 m. For the physiological simulations, the viscosity was scaled by the same factor as pure water 25 / 37 = to give an AH viscosity of 0.77 cp. For an ideal elastomer, the modulus is proportional to the absolute temperature. We therefore estimated that the modulus of the valve leaflet in vivo would be 4.2 * 310 K/298 K =4.4 MPa. The AH flow rate was varied between 0.0 and 5.0 L/min. 3 Methods Experiment Figure 2 shows a cartoon of the microfluidic system used to test the performance of the AGV. The experimental setup consisted of a syringe pump, a thin graduated glass column, an AGV mounted in a sealed glass test-tube, a plastic outlet tube, and a three-way stopcock. The syringe pump fed a steady flow of filtered saline through one of the three ports attached to the three-way connector. The viscosity and density of saline were =1.1 cp and =1.00 g/cm 3. The second port of the stopcock was open to the graduated column while the third directed saline to the AGV via a 27 gauge needle. Given a measured base steady flow rate, the height of saline in the graduated cylinder was used to measure the inlet pressure to the AGV. After working its way through the AGV, the AH exited through a 21 gauge needle into the outlet tube. Taking the outlet pressure as equal to ambient pressure, the measured inlet pressure was equal to the pressure drop across the valve. Six separate flow rates were used. For each flow rate, three runs were performed, and the equilibrium pressure drop was measured to account for the resistance of the apparatus. Before each run, the 778 / Vol. 127, OCTOBER 2005 Transactions of the ASME

4 Fig. 2 The Ahmed valve in vitro experimental setup. The experimental setup consists of a syringe pump, a pipe stand and manometer, and an AGV sealed in a saline bath. The saline viscosity is 1.1 cp, and tubing size was chosen so that its resistance was negligible compared to the AGV. AGV was removed from the line, and the baseline height h init was measured. The AGV was then reconnected and a new height measured. The difference in column heights is related to a pressure drop through the hydrostatic relationship p= g h final h init, where g is the gravitational constant and h final is the equilibrium column height. 4 Results Solutions to the RVK model were calculated on a finite element mesh with 256 elements 16 by 16, which resulted in 1089 nodes and 6534 degrees of freedom. To determine the optimal balance between computational time and solution accuracy, mesh refinements were performed until the AH flow rate plateaued for a given pressure drop. The difference in AH flow rate using a finer mesh with 1681 nodes and 10,086 degrees of freedom was approximately 1%. Figure 3 a shows the calculated midplane displacement of an AGV leaflet and the AH pressure contours for an AH flow rate of 1.6 L/min very low physiological conditions. Each valve leaflet deformed, at its maximum, 45.4 m. The displacement-tothickness aspect ratio w/ t=0.363 suggests that the leaflet was beginning to stiffen from nonlinearities in the transverse strain Eq. 13. Channel heights were greatest and therefore channel resistances were lowest along the midline, leading to greater flow rates there. The RVK model predicted an equilibrium pressure drop of 5.8 mm Hg 0.77 kpa for a flow rate of 1.6 L/min; the measured pressure drop for the same flow rate was within the range mm Hg kpa. Figure 4 shows the results for an AH flow of 20.0 L/min. This flow rate was unrealistically large, but served to demonstrate the AGVs predicted performance in an extreme upper flow-rate limit. The predicted equilibrium pressure drop was 10.9 mm Hg 1.47 kpa and the measured pressure drop was mm Hg kpa. The maximum leaflet displacement was 77.9 m. In Figure 5, a comparison between the predicted and measured pressure drop/flow-rate curves was made. The qualitative features of the experimental curve were predicted well by the RVK model. The experiments showed slightly sharper curvature, with the model underpredicting experimental pressure drops for F=3 10 L/min and overpredicting for F=20 25 L/min. Figure 6 plots the results of the in vivo simulations. Normal physiological AH flow rates rarely exceed 3 L/min, so we only report pressure drops for flow rates ranging from 0 to 5 L/min. Fig. 3 Predicted in vitro results: Physiologically low AH flow. For a 1.6 L/min flow rate, subfigure a shows the displacement of the midplane surface and the contours of the transverse displacement w. b plots the pressure contours of AH as it moves the through the valve. Figure 6 demonstrates the effects of the decreased viscosity and stiffened membrane on the pressure drop/flow rate curve. 5 Discussion and Conclusions We have shown that the RVK model, given good measurements of the initial channel dimensions, can predict accurately the coupled fluid-solid physics of the AGV. It reduces the difficult task of solving the three-dimensional Navier Stokes and solid equilibrium relations on a deforming domain to a more tractable two-dimensional problem with a negligible loss in accuracy. The close match between the theoretical predictions and in vitro experimental results confirmed our assumptions that the AGV was thin and planar, and that the deformed leaflet had small curvature. The agreement also provides strong evidence in support of our hypothesis that the nonlinear behavior of the AGV arises from the fluid-solid interactions. The model was also used to predict the performance of the AGV in vivo. As expected based on clinical studies 14,15, the Journal of Biomechanical Engineering OCTOBER 2005, Vol. 127 / 779

5 Fig. 4 Predicted in vitro results: Severe overfiltration. For a 20.0 L/min flow rate, subfigure a shows the displacement of the midplane surface and the contours of the transverse displacement w. b plots the pressure contours of AH as it moves the through the valve. AGV maintains a low pressure drop of 6.2 mm Hg even at a fairly high AH flow rate of 3.0 L/min. The AGV also has the advantage that it reduces the risk of hypotony pathologic al reduction in IOP by means of its large resistance at low flow rates. For an AH flow rate of 1.0 L/min, the pressure drop across the AGV is 4.7 mm Hg. The nonlinear response of the AGV is seen to be quite advantageous. Consider that the goal of the device is to maintain pressure above 5 mm Hg but below 10 mmhg. The AGV is able to achieve that goal for AH flow rates between 1.3 and 19.8 L/min, a factor of 15.2 in range. A constant-resistance device could only accommodate a factor of 2 range in flow rates, barely encompassing the reasonable physiological limits of L/min. Thus, the added flexibility of the AGV allows use on a wider range of patients with greater confidence that glaucoma and hypotony will be avoided. The design of the AGV is limited by numerous extraneous factors e.g., material restrictions and the desire for a small overall Fig. 5 In vitro AGV performance. a compares the predicted RVK model results against the in vitro experimental measurements. Error bars show the range of measured results for the three experimental runs. b shows a power curve fit to the experimental data. The power fit exponent is 0.23, a favorable match to the RVK model value of The simulations were performed for an AH viscosity of 1.1 cp, a Poisson s ratio of 0.45, and a Young s modulus of 4.2 MPa. size, but some insight into the characteristic performance of the valve can be drawn from our model. Given a fixed pressure drop and viscosity, lubrication theory implies that the AH flow rate is proportional to the pressure drop times the channel height cubed, and for the AGV the channel height is equal to the leaflet deformation. Thus, AGV performance is characterized by the leaflet deformation, which is captured by the following two relationships w W 4 t 3 P Ave E w 1 2 P Ave W 4 1/3 Et F P Avew 3 4 P Ave, w/t F P Ave w 3 2 P Ave, w/t where W is the leaflet width and P Ave is the average pressure across the valve. The first relationship occurs during leaflet bending w/t 0.2 and the second happens when the leaflet is stretched w/t 1. In the bending regime, the deflection is linear 780 / Vol. 127, OCTOBER 2005 Transactions of the ASME

6 Fig. 6 Predicted in vivo AGV valve performance. AGV performance from Fig. 5 is affected by increased temperature, due to decreased AH viscosity and increased leaflet stiffness. The predicted in vivo pressure-drop/flow-rate results are plotted along with valve resistance R= P/F for a range of physiological flow rates. in the pressure, which implies a cubic dependence of flow conductivity on pressure, which in turn implies a quartic dependence of flow rate on applied pressure. A power fit to the experimental in vitro and theoretical in vivo data is shown in Fig. 5 b ; the 0.23 exponent is consistent with the scaling analysis and suggests that under physiological conditions, bending is the dominant mode. This is true for even the largest measured flow rate 25 L/min, where w/t=81.5 m/125 m=0.65. In the stretching regime, the deflection depends only on the 1/3 power of the pressure, implying only a quadratic dependence of flow rate on applied pressure. From relationships 14 and 15, we can see that increasing the modulus of the leaflets, for example, would tend to increase the resistance to AH flow and shift the flow curve Fig. 6 up. Increasing leaflet thickness would also tend to increase resistance, and would tend to keep the leaflets in bending rather than stretching mode longer, leading to a wider range of flow rates over which the flow curve bends sharply. Narrowing the valve would increase resistance dramatically since it would involve both a reduction in cross section requiring a higher velocity for the same total flow rate and a reduction in deflection of the leaflets. Lengthening the valve would increase the resistance linearly by increasing the distance the AH must travel, but it would have relatively little effect on the leaflet mechanics. Acknowledgments This work was supported in part by the U.S. Department of Energy through Sandia National Laboratories Albuquerque, NM. Computations were made possible by a supercomputing resources grant from the University of Minnesota Superconducting Institute for Digital Simulation and Advanced Computation. This work was also supported by the University of Minnesota s Biomedical Engineering Institute through the Ophthalmic Engineering Interest Group. AGV samples were provided by New World Medical. Nomenclature AGV Ahmed glaucoma valve AH Aqueous humor IOP Intraocular pressure RVK Reynolds-von Kármán dh/dx 2 Leaflet curvature, unitless E= Green s strain tensor e= AGV leaflet linear strain tensor e= N AGV leaflet non-linear strain tensor E Young s modulus of the AGV leaflet f Local thickness-averaged lubrication flow vector F Volumetric flow rate of aqueous humor, L/min H Characteristic AGV height, cm H 0 Initial AGV channel height, cm I Identity tensor k= AGV leaflet linear strain tensor L Characteristic length of the AGV, cm M= von Kármán moments N= von Kármán in-plane stress resultants p Lubrication/Reynolds pressure, mm Hg P Pressure drop across the AGV, mm Hg Q von Kármán transverse shear R Resistance of AGV, mm Hg/ L/min Re Dimensionless Reynolds number, UH/ t Characteristic AGV leaflet thickness, m U Characteristic aqueous humor velocity in the AGV u von Kármán midplane displacements u, v, and w x,y In-plane coordinates of the AGV z Plate-thickness coordianate von Kármán midplane rotations x and y Density of aqueous humor Viscosity of aqueous humor References 1 Hart, W. M., 1992, Adler s Physiology of the Eye, Mosby Year Book, St. Louis. 2 Epstein, D. L., Allingham, R. R., and Schuman, J. S., 1997, Chandler s and Grant s Glaucoma, 4th ed. Williams and Wilkins, Baltimore. 3 Tong, L., Frazao, K., LaBree, L., and Varma, R., 2003, Intraocular Pressure Control and Complications with Two-Stage Insertion of the Baerveldt Implant, Ophthalmology, 110 2, pp Prata, J. A., Mermoud, A., LaBree, L., and Minckler, D. S., 1995, In Vitro and In Vivo Flow Characteristics of Glaucoma Drainage Implants, Ophthalmology, 102, pp Eisenberg, D. L., Koo, E. Y., Hafner, G., and Schuman, J. S., 1999, In Vitro Flow Properties of Glaucoma Implant Devices, Ophthalmic Surg. Lasers, 30 8, pp Francis, B. A., Cortes, A., Chen, J., and Alvarado, J. A., 1998, Characteristics of Glaucoma Drainage Implants during Dynamics and Steady-State Flow Conditions, Ophthalmology, 105, pp Bird, R. B., Stewart, W. E., and Lightfoot, E. N., 1960, Transport Phenomena, Wiley, New York. 8 Wilson, M. R., Mendis, U., Smith, S. D., and Paliwal, A., 2000, Ahmed Glaucoma Valve Implant Vs Trabeculectomy in the Surgical Treatment of Glaucoma, Am. J. Ophthalmol., 130, pp Fung, Y. C., and Tong, P., 2001, Classical and Computational Solid Mechanics, World Scientific, Singapore. 10 Malvern, L. E., 1969, Introduction to the Mechanics of a Continuous Medium, Prentice-Hall Inc., Englewood Cliffs, NJ. 11 Panton, R. L., 1992, Incompressible Flow, Wiley, New York. 12 Stay, M. S., and Barocas, V. H., 2004, Coupled Fluid-Solid Thin-Film Analysis of Microscale Pumping. 13 Saad, Y., and Schultz, M. H., 1986, Gmres: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems, SIAM Soc. Ind. Appl. Math. J. Sci. Stat. Comput., 7, pp Beck, A. D., Sharon, F., Kammer, J., and Jin, J., 2003, Aqueous Shunt Devices Compared with Trabeculectomy with Mitomycin-C for Children in the First Two Years of Life, Am. J. Ophthalmol., 136 6, pp Nouri-Mahdavi, K., and Caprioli, J., 2003, Evaluation of the Hypertensive Phase After Insertion of the Ahmed Glaucoma Valve, Am. J. Ophthalmol., 136 6, pp Journal of Biomechanical Engineering OCTOBER 2005, Vol. 127 / 781