# Rational Design of Microfluidic Glaucoma Stent

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}using multiple outlets in the hexagonal meshwork of the current stent design, as shown in Figure 2. The cell-sized dimensions of the microchannels are expected to reduce the risk of fibrosis [18]. IOP can be adjusted by varying the dimensions of the outlet tubes of the stent, while the mesh structure and the dimensions of the mesh channels remain fixed. Large cross-sections are adopted for part A and part B to achieve a near-zero pressure drop until the distribution of AH in Part C. For the meshwork of part C in Figure 2, we used a honeycomb structure with thin connections (channels). The hexagonal geometry provides high in-plane stability and high flexibility for out-of-plane deformations [20]. This preserves the channel dimensions and the uniform outflow of AH into the subconjunctiva, while allowing the mesh to conform to the tissue in the out-of-plane direction at low mechanical stress. In the remainder of the paper, we will discuss only Part C of the stent, which we will refer to simply as the stent or meshwork.

## 2. Numerical Methods

^{®}[21] employing the finite element method (FEM). We used the Laminar Flow interface of the Fluid Flow module and the physical model Creeping Flow. The outflow from the stent openings into the subconjunctival tissue was calculated by means of the Darcy’s Law interface of the Porous Media and Subsurface Flow branch in COMSOL. Finite element meshes were generated automatically using the COMSOL option Physics Controlled Mesh. Typical element size was Finer. COMSOL and the Electrical Circuit interface of the AC/DC module were used to verify the resistor network. We always computed stationary solutions.

## 3. Mathematical Models

#### 3.1. Circuit Model of Stent Flow

^{®}[21]. The lowest resistor ${R}_{1}$ contributes little to the total resistance and is a free design parameter. It shall be noted that the channels below the lowest outlet tube in row $1$ (below resistance ${R}_{1}$) are preferably closed to avoid stagnant liquid.

#### 3.2. Numerical Model of Stent Flow

^{®}[21]. Since the flow in the stent meshwork is in the low Reynolds regime $Re\ll 1$, it is a so-called Stokes flow for which the physical model Creeping Flow from COMSOL can be used. This physics omits the inertial term in the Navier Stokes equation: $\varrho \left(u\xb7\nabla \right)u=0$, where $\varrho $ is the fluid density, $u$ the fluid velocity and $\nabla $ the Nabla operator. Tests with the fully laminar formulation did not show any differences. Computation time in this case was, however, 30% longer.

#### 3.3. Model of Drainage to Subconjunctival Tissue

^{®}to compute Darcy’s law inside the tissue. The FEM program solves the following equation for the pressure $p:$

#### 3.3.1. Drainage from Hemispherical Microbleb

#### 3.3.2. Drainage from Bleb Array

#### 3.4. IOP after Surgery

#### 3.5. Study Limitations

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Calculation of Resistance ${R}_{0}$

**Figure A1.**Various possible unit cells of the tubular hexagonal mesh are shown as shaded rectangles. The total resistance R

_{0}in each unit cell is the same.

## Appendix B. Aqueous Humor Production Rate and Uveoscleral Outflow

## Appendix C. Step-by-Step Procedure for a Hexagonal Glaucoma Stent Design

Steps | Input | Input | Output | Equation |
---|---|---|---|---|

1 | $IO{P}_{BS}$ measured before surgery, e.g., $25\mathrm{mmHg}$ | $IO{P}_{AS}$ targeted after surgery, e.g., $14\mathrm{mmHg}$ | Serial outflow resistance ${R}_{sa}$ of stent and bleb array | ${R}_{sa}=\frac{IO{P}_{AS}\left(IO{P}_{BS}-{p}_{EV}\right)}{\left(\dot{Q}-{\dot{Q}}_{UP}\right)\left(IO{P}_{BS}-IO{P}_{AS}\right)}$ Equation (16), e.g., $\left(\dot{Q}-{\dot{Q}}_{UP}\right)=2.1\frac{\mathsf{\mu}\mathrm{L}}{\mathrm{min}}$ |

2 | Hemispherical bleb radius for COMSOL simulation, e.g., ${r}_{b}=30\mathsf{\mu}\mathrm{m}$ | Bleb spacing in array for COMSOL simulation, e.g., ${d}_{a}=300\mathsf{\mu}\mathrm{m}$ | Bleb array drainage resistance ${R}_{a}$ | ${R}_{a}={p}_{b}/{\dot{Q}}_{s}$ where ${p}_{b}$ is the pressure in the blebs from COMSOL simulation and ${\dot{Q}}_{s}$ is the flow rate through the stent, e.g., ${\dot{Q}}_{s}=1.7\frac{\mathsf{\mu}\mathrm{L}}{\mathrm{min}}$ ${R}_{a}>\frac{4\times {10}^{4}\left[\mathrm{mmHg}\frac{\mathrm{min}}{\mathsf{\mu}\mathrm{L}}\xb7\mathsf{\mu}\mathrm{m}\right]}{n\xb7{r}_{b}\left[\mathsf{\mu}\mathrm{m}\right]}$ Equation (13), where $n$ is the number of outlets |

3 | ${R}_{sa}$ (from step 1) | ${R}_{a}$ (from step 2) | Stent flow resistance ${R}_{s}$ | ${R}_{s}={R}_{sa}-{R}_{a}$ |

4 | Number of stent columns, e.g., ${n}_{c}=42$ | Number of stent rows, e.g., ${n}_{r}=20$ | Number of stent outlets and micro blebs | $n={n}_{c}\xb7{n}_{r}$ |

5 | Stent column resistance ${R}_{c}={n}_{c}\xb7{R}_{s}$ | Fix lowest stent outlet resistance, e.g., ${R}_{1}={R}_{0}/10$ | Stent flow resistance ${R}_{0}$ connecting two outlet tubes | ${R}_{0}=\frac{6\left({n}_{r}{R}_{c}-{R}_{1}\right)}{{n}_{r}\left(3{n}_{r}-1\right)}=\frac{6\left(n{R}_{s}-{R}_{1}\right)}{{n}_{r}\left(3{n}_{r}-1\right)}$ Equation (5) |

6 | Hexagonal segment flow resistance $R=\frac{2}{3}{R}_{0}$ | Bleb spacing ${d}_{a}$ | Channel length $L$ and cross section width $D$ of hex. segment | $L={d}_{a}\mathrm{tan}30\xb0$ (geometry of hexagon) $D={\left(28.4\xb7\eta \frac{L}{R}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}={\left(42.7\xb7\eta \frac{L}{{R}_{0}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}$ Equations (1) and (2) |

7 | ${R}_{0}$ (from step 5) | Lowest stent outlet resistance ${R}_{1}$ (from step 5) | Flow resistance ${R}_{k}$ of outlet tube in row number $k$ | ${R}_{k}=\frac{1}{2}k\left(k-1\right){R}_{0}+{R}_{1}$ Equation (4) |

8 | Length ${L}_{K}$ of straight outlet tube, e.g., $100\mathsf{\mu}\mathrm{m}$ | Row number $k$ | Cross section width of outlet tube | ${D}_{k}={\left(28.4\xb7\eta \frac{{L}_{k}}{{R}_{k}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}$ Equation (3) |

## Appendix D. List of Parameters Used for Calculations

Parameter | Value | SI Units | Description |
---|---|---|---|

$\varrho $ | ${10}^{3}$ | $\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}$ | Density of fluid, of AH |

$\eta $ | $7\times {10}^{-4}$ | $\mathrm{Pa}\xb7\mathrm{s}$ | Dynamic viscosity of liquid |

$K$ | $5\times {10}^{-13}$ | $\frac{{\mathrm{m}}^{2}}{\mathrm{Pa}\xb7\mathrm{s}}$ | Hydraulic conductivity in subconjunctival tissue [6] |

$\kappa =K\xb7\eta $ | $3.5\times {10}^{-16}$ | ${\mathrm{m}}^{2}$ | Fluid permeability in subconjunctival tissue, used in Equation (6) |

${S}_{p}=\varrho \xb7{L}_{p}\frac{{S}_{A}}{V}$ | ${10}^{-3}$ | $\frac{\mathrm{s}}{{\mathrm{m}}^{2}}$ | Used in Equation (6) |

${L}_{p}$ | ${10}^{-10}$ | $\frac{\mathrm{m}}{\mathrm{Pa}\xb7\mathrm{s}}$ | Hydraulic permeability of blood vessel wall [6] |

$\frac{{S}_{A}}{V}$ | ${10}^{4}$ | $\frac{1}{\mathrm{m}}$ | Vessel wall area per tissue volume [6] |

$\frac{1}{\sqrt{C}}=\sqrt{\frac{\varrho}{{S}_{p}}\frac{\kappa}{\eta}}$ | $7\times {10}^{-4}$ | $\mathrm{m}$ | Characteristic drainage length, used in Equation (8) |

$\dot{Q}$ | $2.5$ | $\frac{\mathsf{\mu}\mathrm{L}}{\mathrm{min}}$ | Typical AH production rate |

${\dot{Q}}_{UP}\cong 0.15\xb7\dot{Q}$ | $0.4$ | $\frac{\mathsf{\mu}\mathrm{L}}{\mathrm{min}}$ | Constant outflow rate through uveoscleral pathway |

${p}_{EV}$ | $11$ | $\mathrm{mmHg}$ | Episcleral venous pressure, ranging from $9$ to $12\mathrm{mmHg}$ [6] |

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**Figure 1.**(

**a**) Sectional anatomy of the eye with glaucoma stent. Part A is placed in the anterior chamber to collect AH. Part B conducts the AH into the meshwork of part C. Part C drains the fluid into the subconjunctival tissue. (

**b**) Drainage pathways and flow balance in the eye after surgery. AH is produced in the ciliary body and drained by the trabecular meshwork (TM) via Schlemm’s Canal, the uveoscleral pathway (UP) and the stent. At the stent outlets, blebs form in the subconjunctiva on the scleral barrier. The indicated flow rates were used in numerical simulation.

**Figure 2.**The meshwork consists of honeycomb cells and is $12.6\mathrm{mm}$ wide and $5.2\mathrm{mm}$ high. Each hexagonal segment is a microtube (microchannel) with a square internal cross-section $14\mathsf{\mu}\mathrm{m}$ wide. Each honeycomb cell contains an outlet tube with specific dimensions and a corresponding flow resistance (see enlarged detail). The black numbers are the dimensions; the red numbers denote the columns and rows of the meshwork. The liquid flows evenly from part B above into the meshwork.

**Figure 3.**Tubular structure of the stent and fluid resistance model. (

**a**) The honeycomb meshwork geometry is the same everywhere; only the outlet tubes have different dimensions. The star-shaped part, colored in red, can be expressed as a fluidic resistance ${R}_{0}$ between two subsequent outlet tubes. (

**b**) Equivalent circuit diagram of a single column of the stent, from the fluid inlet (top) to the individual outlets. The numbered resistors ${R}_{1}$ to ${R}_{20}$ correspond to the various outlet tubes along the stent column. Due to the different dimensions, the outlet tubes have distinct flow resistances. The dimensions are chosen so that the same rate of liquid flows out of each tube. The diagram in (

**c**) illustrates how the resistance ${R}_{0}$ can be calculated using the flow resistance $R$ of a straight channel segment (marked in blue). Drawing (

**d**) depicts the relevant geometric entities for the calculation of the flow resistance.

**Figure 4.**Pressure and flow velocity field in the microchannel mesh computed with the program COMSOL. The boundary conditions were the inlet flow rate of $1.7\frac{\mathsf{\mu}\mathrm{L}}{\mathrm{min}}$ and the outlet pressure of $0\mathrm{mmHg}$ at each orifice. (

**a**) Pressure distribution in the stent of the leftmost and rightmost part of the mesh. The maximum value is $6.3\mathrm{mmHg}$ at the inlet, above row 20. (

**b**) Flow velocities in the midplane of the meshwork. Shown are flow details at top and bottom of the stent, along the centerline of the honeycomb structure.

**Figure 5.**(

**a**) COMSOL simulation of the outflow from a bleb with radius ${r}_{b}=30\mathsf{\mu}\mathrm{m}$. The inlet boundary condition is the mass flow rate $\dot{M}={\dot{Q}}_{b}\xb7\varrho =3.3\times {10}^{-11}\frac{\mathrm{kg}}{\mathrm{s}}$ at the hemispherical bleb surface. This corresponds to ${\dot{Q}}_{b}={\dot{Q}}_{t}=2.0\frac{\mathrm{nL}}{\mathrm{min}}$, given in Table 2. The outer boundary condition is the pressure ${p}_{d}=0.17\mathrm{mm}\mathrm{Hg}$ at the domain boundary at ${r}_{d}=300\mathsf{\mu}\mathrm{m}$. ${p}_{d}$ is calculated by means of Equations (8) and (11) and used to mimic an infinite domain. The white lines and arrows indicate the flow direction given by the simulation. The axis of rotational symmetry is shown as a vertical, dashed red line. (

**b**) Pressure as a function of distance from the origin. The blue line is the COMSOL result; the red circles are obtained using Equations (8) and (11).

**Figure 6.**(

**a**) Microbleb array of $12.6\times 5.2{\mathrm{mm}}^{2}$ of the current stent meshwork. The microblebs lie on the scleral barrier and are spaced 300 µm apart. All blebs have a radius of ${r}_{b}=30\mathsf{\mu}\mathrm{m}$. AH flows from the bleb surfaces into the $0.6\mathrm{mm}$ thick subconjunctiva. (

**b**,

**c**) Drainage pressure field in the subconjunctival tissue determined by COMSOL simulation. The total flow rate of the bleb array is $1.7\frac{\mathsf{\mu}\mathrm{L}}{\mathrm{min}}$. Figure 6b shows a quarter of the simulation domain. Figure 6c is an enlarged view of 6b. The pressure inside the blebs is $7\mathrm{mmHg}$ and reaches $4.1\mathrm{mmHg}$ at the surface of the subconjunctiva. The pressure decreases rapidly in the plane of the array within the characteristic length $1/\sqrt{C}=0.7\mathrm{mm}$.

**Figure 7.**Drainage resistance ${R}_{a}$ obtained by COMSOL simulations of hexagonal arrays as in Figure 6. (

**a**) ${R}_{a}$ as a function of bleb spacing ${d}_{a}$. The bleb radius is constant ${r}_{b}=30\mathsf{\mu}\mathrm{m}$. The curve reaches the value of $1.5\mathrm{mmHg}\frac{\mathrm{min}}{\mathsf{\mu}\mathrm{L}}$ for large separations, which is consistent with Equation (13) for free blebs. (

**b**) ${R}_{a}$ as a function of bleb radius ${r}_{b}$. The bleb spacing is constant at ${d}_{a}=300\mathsf{\mu}\mathrm{m}$. For bleb radii ${r}_{b}>30\mathsf{\mu}\mathrm{m}$, the curve terminates in a constant value corresponding to the drainage resistance of a shallow macrobleb of $12.6\times 5.2{\mathrm{mm}}^{2}$ surface area.

**Figure 8.**$IO{P}_{AS}$ as a function of microbleb radius ${r}_{b}$. The bleb separation is ${d}_{a}=300\mathsf{\mu}\mathrm{m}$ corresponding to the geometric period in the current honeycomb design. The numbers in the boxes are the IOP values before surgery in units of $\mathrm{mmHg}$. The curves were calculated using Equation (16) and $\dot{Q}.$ $IO{P}_{AS}$ ranges from $12$ and $15\mathrm{mmHg}$ for bleb radii greater than $30\mathsf{\mu}\mathrm{m}$. The green shaded area corresponds to the $IO{P}_{AS}$ for AH production rate varying between $1.5$ and $3.0\mathsf{\mu}\mathrm{L}/\mathrm{min}$ and an IOP before surgery of $25\mathrm{mmHg}$. The green shaded area demonstrates that the IOP is within a healthy range even with daily fluctuating production rates [4].

**Table 1.**Equations to calculate the outlet tube resistances as a function of the connecting resistance ${R}_{0}$ and the bottom outlet resistance ${R}_{1}$. Only the equations for the lowest outlet tubes are shown in the table. The general equation for the outlet resistance of row $k$ is presented in the text (Equation (4)). ${R}_{rest\left(k-1\right)}$ is the flow resistance of the circuit below row $k$.

$\mathbf{Row}\mathit{k}$ | $\mathbf{Resistance}\mathbf{of}\mathbf{Outlet}\mathbf{Tube}\mathbf{in}\mathbf{Row}\mathit{k}$ | $\mathbf{Resistance}\mathbf{of}\mathbf{Circuit}\mathbf{Below}\mathbf{Row}\mathit{k}$ |
---|---|---|

1 | ${R}_{1}$ | |

2 | ${R}_{2}={R}_{rest1}={R}_{0}+{R}_{1}$ | ${R}_{rest1}={R}_{0}+{R}_{1}$ |

3 | ${R}_{3}=2{R}_{rest2}=2\left({R}_{0}+\frac{{R}_{2}{R}_{rest1}}{{R}_{2}+{R}_{rest1}}\right)=2\left({R}_{0}+\frac{1}{2}{R}_{rest1}\right)=3{R}_{0}+{R}_{1}$ | ${R}_{rest2}=\frac{3}{2}{R}_{0}+\frac{1}{2}{R}_{1}$ |

4 | ${R}_{4}=3{R}_{rest3}=3\left({R}_{0}+\frac{{R}_{3}{R}_{rest2}}{{R}_{3}+{R}_{rest2}}\right)=3\left({R}_{0}+\frac{2}{3}{R}_{rest2}\right)=6{R}_{0}+{R}_{1}$ | ${R}_{rest3}=\frac{6}{3}{R}_{0}+\frac{1}{3}{R}_{1}$ |

5 | ${R}_{5}=4{R}_{rest4}=4\left({R}_{0}+\frac{{R}_{4}{R}_{rest3}}{{R}_{4}+{R}_{rest3}}\right)=4\left({R}_{0}+\frac{3}{4}{R}_{rest3}\right)=10{R}_{0}+{R}_{1}$ | ${R}_{rest4}=\frac{10}{4}{R}_{0}+\frac{1}{4}{R}_{1}$ |

**Table 2.**Resistances, flow rates and pressures of the meshwork of Figure 1 and Figure 2. The resistances $R$ and ${R}_{1}$ are obtained from the channel geometry. The stent flow rate ${\dot{Q}}_{s}$ is an example value after surgery. The pressure difference ${p}_{s}$ across the stent is then calculated with the stent fluid model using the mentioned quantities or with COMSOL simulations using the specified geometry.

$R$ | $9.0\times {10}^{13}$ $11.2$ | $\mathrm{Pa}\xb7\mathrm{s}/{\mathrm{m}}^{3}$ $\mathrm{mmHg}\xb7\mathrm{min}/\mathsf{\mu}\mathrm{L}$ | Equation (1) | $D\times D\times L=14\mathsf{\mu}\mathrm{m}\times 14\mathsf{\mu}\mathrm{m}\times 173\mathsf{\mu}\mathrm{m}$ |

${R}_{0}$ | $1.35\times {10}^{14}$ $16.8$ | $\mathrm{Pa}\xb7\mathrm{s}/{\mathrm{m}}^{3}$ $\mathrm{mmHg}\xb7\mathrm{min}/\mathsf{\mu}\mathrm{L}$ | Equation (2) | |

${R}_{1}$ | $1.35\times {10}^{13}$ $1.69$ | $\mathrm{Pa}\xb7\mathrm{s}/{\mathrm{m}}^{3}$ $\mathrm{mmHg}\xb7\mathrm{min}/\mathsf{\mu}\mathrm{L}$ | Equation (3) | ${D}_{1}\times {D}_{1}\times {L}_{1}=14\mathsf{\mu}\mathrm{m}\times 14\mathsf{\mu}\mathrm{m}\times 26\mathsf{\mu}\mathrm{m}$ |

${R}_{c}$ | $1.33\times {10}^{15}$ $166$ | $\mathrm{Pa}\xb7\mathrm{s}/{\mathrm{m}}^{3}$ $\mathrm{mmHg}\xb7\mathrm{min}/\mathsf{\mu}\mathrm{L}$ | Equation (5) | Resistance of a whole column composed of 20 rows |

${R}_{s}$ | $1.16\times {10}^{13}$ $3.95$ | $\mathrm{Pa}\xb7\mathrm{s}/{\mathrm{m}}^{3}$ $\mathrm{mmHg}\xb7\mathrm{min}/\mathsf{\mu}\mathrm{L}$ | $\frac{{R}_{c}}{42}$ | Resistance of a whole stent meshwork composed of 20 rows and 42 columns |

${\dot{Q}}_{s}$ | 1.7 | $\frac{\mathsf{\mu}\mathrm{L}}{\mathrm{min}}$ | specified | typical stent flow rate |

${\dot{Q}}_{c}$ | $40$ | $\frac{\mathrm{nL}}{\mathrm{min}}$ | $={\dot{Q}}_{s}/42$ | Flow rate of a whole column of 20 outlet tubes |

${\dot{Q}}_{t}$ | $2.0$ | $\frac{\mathrm{nL}}{\mathrm{min}}$ | $={\dot{Q}}_{s}/\left(42\times 20\right)$ | Flow rate of a single outlet tube |

${p}_{s}\left\{\mathrm{model}\right\}$ | $6.7$ | $\mathrm{mmHg}$ | $={R}_{c}\xb7{\dot{Q}}_{c}={R}_{s}\xb7{\dot{Q}}_{s}$ | |

${p}_{s}\left\{\mathrm{COMSOL}\right\}$ | $6.3$ | $\mathrm{mmHg}$ | COMSOL simulations | Section 3.2. and Figure 4 |

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**MDPI and ACS Style**

Graf, T.; Kancerevycius, G.; Jonušauskas, L.; Eberle, P.
Rational Design of Microfluidic Glaucoma Stent. *Micromachines* **2022**, *13*, 978.
https://doi.org/10.3390/mi13060978

**AMA Style**

Graf T, Kancerevycius G, Jonušauskas L, Eberle P.
Rational Design of Microfluidic Glaucoma Stent. *Micromachines*. 2022; 13(6):978.
https://doi.org/10.3390/mi13060978

**Chicago/Turabian Style**

Graf, Thomas, Gitanas Kancerevycius, Linas Jonušauskas, and Patric Eberle.
2022. "Rational Design of Microfluidic Glaucoma Stent" *Micromachines* 13, no. 6: 978.
https://doi.org/10.3390/mi13060978