# A Parametric Analysis of Capillary Height in Single-Layer, Small-Scale Microfluidic Artificial Lungs

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{c}) between 10 and 100 μm are compared for a fixed set of performance specifications (rated blood flow and blood-side pressure drop). All designs contained blood distribution channels and artificial capillaries and were constrained to fit on a 6″ silicon wafer. Rated blood flow and total pressure drop were fixed at 0.8 mL/min and 50 mmHg for all designs. Surface area (gas exchange and blood contacting), blood priming volume in capillaries and distribution channels (V

_{prime}) capillary wall shear rate and number of bifurcations were determined for each capillary height and compared. Blood flow was modeled in three designs (H

_{c}= 30, 60, 100 μm) using CFD, implemented via soft lithography and tested in vitro with bovine blood to verify performance. A preliminary version of this work was accepted as a conference abstract and presented at the ASAIO 66th Annual Conference in June 2021 [33].

## 2. Materials and Methods

#### 2.1. Materials

#### 2.2. Mathematical Models of μALs with 10–100 μm Tall Capillaries

^{−2}(43–260 s

^{−1}), and arteriole wall shear stress spans 10–70 dynes∙cm

^{−2}(434–3038 s

^{−1}) [7]. Ultimately, the blood-side network of each μAL was designed to contain two distribution channels (input and output) connected by capillaries, achieve equivalent rated flows and pressure drops and fit within 6” diameter circles. Surface area, priming volume and average capillary wall shear were calculated and compared for each capillary height.

_{c,g}for the target rated blood flow, where Q

_{R}is defined as the highest blood flow rate at which an inlet blood oxygen saturation of 70% can be increased to 95% upon exiting from the outlet [9]. A

_{c,g}includes the portion of the capillaries that actively contributes to gas exchange and excludes distribution channels. Once A

_{c,g}was determined, the target pressure drop and Equation (3) were used to solve for the number and length of artificial capillaries via Solver, a Microsoft Excel Add-in.

_{c,g}is the available capillary gas exchange area, S

_{B,O2}is the average effective solubility of oxygen in blood and R

_{D,O2}is the membrane’s effective resistance to diffusion. PO2

_{B,i}, PO2

_{B,o}and PO2

_{G}are the partial pressures of oxygen in the blood entering and exiting capillaries and in the sweep gas, respectively. In normal human blood, S

_{B,O2}is a constant, at approximately 7.9 × 10

^{−4}mL-O

_{2}∙mL-blood

^{−1∙mmHg−1}[34,40]. A sweep gas of pure oxygen meant PO2

_{G}was defined as 760 mmHg. PO2

_{B,i}and PO2

_{B,o}were defined as 36.4 mmHg and 79.2 mmHg, corresponding to SO

_{2}values of 70% and 95%, respectively. Thus, R

_{D,O2}was calculated to solve for A

_{c,g}.

_{M}), channel height (H), membrane permeability to oxygen (P

_{M,O2}), effective diffusivity of oxygen in blood (D

_{B,O2}) and S

_{B,O2}. Devices were designed to contain 30 μm-thick membranes constructed from PDMS, which corresponds to a P

_{M,O2}of 3.6 × 10

^{−7}mL-O

_{2}∙cm

^{−1}∙min

^{−1}∙mmHg

^{−1}. The FDA Guidance for Cardiopulmonary Bypass Oxygenators 510(k) Submissions defines normal human blood as having a D

_{B,O2}of 1.4 × 10

^{−6}cm

^{2}∙s

^{−1}[40].

_{1}bifurcates from its respective parent vessel with radii r

_{p}at an angle α. r

_{2}is the radius of the parent vessel that continues on after the daughter channel branches off (Figure 2). Since the channels used in blood side designs were all rectangular, the various hydraulic radii (r

_{H}) were converted to channel width (W) and height (H) using ${\mathrm{r}}_{\mathrm{H}}=\mathrm{H}\xb7\mathrm{W}/\left(\mathrm{H}+\mathrm{W}\right)$.

_{c}= 30, 60, 100 μm) spanning the mathematically modeled range were selected for blood flow simulations in CFD. After which, the three designs were implemented via soft lithography and testing in vitro to verify performance and CFD simulations.

#### 2.3. Computer Aided Drawing and 3D Modeling

_{c}= 30, the first and last 30 capillaries in the design were also curved to allow each capillary to extend their full, assigned length. Corresponding gas side designs were also made in DraftSight. In each design, pillars were 300 μm wide, and the number and spacing of pillars were limited such that pillars occupied 11% of the gas side surface area.

#### 2.4. Blood and Gas Mold Construction

#### 2.5. Device Construction

#### 2.6. In Vitro Flow Experiments

_{c}= 30 devices, so H

_{c}= 30 devices were re-made and coated with polyethylene glycol (PEG) to ensure gas exchange and pressure drop data would be reflective of complete capillary filling. PEG coating followed our previously established methods [20]. Given that the primary variables of interest (pressure drop and gas exchange) would depend on complete filling of all capillaries, H

_{c}= 60 and H

_{c}= 100 devices were left uncoated, since their capillary beds perfused easily at all experimental flow rates.

#### 2.7. Statistics

## 3. Results

#### 3.1. Mathematical Modeling for H_{c} = 10 to 100 μm

_{b,c}) and V

_{prime}exhibited a maximum and capillary wall shear rate (τ

_{w}) exhibited a minimum around H

_{c}= 40 μm.

#### 3.2. Implementation of 3 Specific Designs (H_{c} = 30, 60, 100 μm)

_{c}of 30, 60 and 100 μm were selected. Resulting channel dimensions for H

_{c}= 30, 60 and 100, respectively, are: capillary widths of 120, 240 and 400 µm; capillary lengths of 0.195, 2.001 and 5.610 cm; distribution channel heights of 239, 210 and 241 µm; distribution channel widths between 27–957, 62–841 and 120–964 µm; distribution channel lengths of 9.144, 1.4623 and 0.700 cm. We achieved channel heights within 3% of intended dimensions. For the sake of brevity, individual branching angles are not included; however, branching angles for H

_{c}= 30, 60 and 100 ranged from 85.2–54.0°, 79.0–70.9° and 73.7–54.0°, respectively. Implemented branching angles were within 1–12% of what was theorized using Murray’s Law (Equation (5)).

_{c}= 30, 60 and 100 required gas exchange surface areas of 1.2, 2.1 and 3.1 cm

^{2}, respectively, to meet rated flow, pressure drop and wall shear rate goals (Table 2). Despite A

_{c,g}increasing with increasing capillary height, H

_{c}= 60 required the smallest V

_{prime}and caused the smallest A

_{b,c}in mathematical modeling (Table 2). Additionally, all three designs produced τ

_{w}within the physiological range, but τ

_{w}in H

_{c}= 30 and H

_{c}= 100 were 43 and 29% less than in H

_{c}= 60, respectively. Computer-aided drawings (CAD) of each design are shown below (Figure 5).

#### 3.3. CFD Simulations

_{c}= 30 μm design achieved 51.5 mmHg, so the pressured drop goal was reduced to 50 mmHg and the H

_{c}= 60 and H

_{c}= 100 μm designs were adjusted so that all designs would have similar pressure drops (Table 2). By revising the capillary lengths in H

_{c}= 60 from 2.4428 to 2.0012 cm and H

_{c}= 100 μm design from 7.3879 to 5.6099 cm, the range in pressure drop across the three designs narrowed substantially (51.5–53.8 mmHg). Revised designs were used for all subsequent steps.

_{c}= 100, but a Level 4 global mesh was necessary in H

_{c}= 60 and H

_{c}= 30 to capture shear rate details in capillary height-width cross-sections. A Level 4 local mesh provided no more insight than Level 5 in all designs. Ultimately, this meant computational domains for H

_{c}= 30, 60 and 100 had 845,767, 1,226,957 and 1,723,605 fluid cells, respectively.

^{−1}in H

_{c}= 30, 60 and 100, respectively (Figure 8). In planar Poiseuille flow, shear rate is highest at capillary walls and approaches zero at the center of capillary lumens [55], so maximum shear rates in cross-sections should be equivalent to capillary wall shear rate (as validated in Figure 8, below). Despite achieving physiologic wall shear rates that matched mathematical modeling results (Figure 4), the cross-sections of individual capillaries revealed better symmetry in H

_{c}= 60 and H

_{c}= 100 than in H

_{c}= 30 (Figure 8). Reasons for these differences are discussed in 4.3.

#### 3.4. In Vitro Experimental Results

_{c}= 30, 60 and 100 devices were 47%, 27% and 32% lower than theory.

_{c}= 30 vs. H

_{c}= 60 at all flow rates, and between H

_{c}= 30 vs. H

_{c}= 100 at flow rates except 1.2 mL/min (Figure 10). This may be due to the fact that the H

_{c}= 30 design was coated with PEG (see Discussion below).

_{c}= 100 at 0.6 mL/min. The one-way ANOVA test for comparison at 0.6 mL/min resulted in p = 0.032; a post-hoc Tukey-Kramer test was performed, revealing significant differences between only H

_{c}= 60 and H

_{c}= 100 (Figure 11). Observing similar gas exchange for all three designs validated theoretical modeling and design goals.

## 4. Discussion

#### 4.1. Mathematical Modeling for H_{c} = 10 to 100 µm

_{c}= 30 requires hundreds of short capillaries, the overall design is long and narrow; H

_{c}= 60 requires a few dozen capillaries, so the design resembles a square; H

_{c}= 100 contains approximately one dozen long capillaries and looks like a short rectangle (Figure 5). All three overarching design goals were mathematically achieved, as demonstrated in Table 2.

#### 4.2. CFD Simulations

_{c}= 30’s capillaries appear blue (<1041 s

^{−1}), as is similarly notable in H

_{c}= 100’s capillaries, while H

_{c}= 60’s capillaries appear green (1041–2039 s

^{−1}) (Figure 7). This agrees with the mathematical modeling which shows that wall shear rate approaches a maximum when capillaries are 40–60 µm tall (Figure 4, Table 2).

_{c}= 30’s shear rate cross-section appears less symmetric than the two other designs (Figure 8); this is potentially due to artifacts from low resolution. Although efforts were made to increase the local mesh refinement in H

_{c}= 30 from Level 4 to Level 5, computation time became excessive, such that higher-resolution results could not be collected. Furthermore, capillary wall shear simulated via CFD was only 3% higher than predicted by mathematical modeling, so we did not pursue higher refinement.

#### 4.3. In Vitro Results

_{c}= 30 may be due to the increased wettability caused by PEG coating. We have found in previous work that a hydrophilic surface reduces pressure drop in small diameter flow channels [20]. Future experiments should control for surface hydrophilicity by applying the same surface modification, such as PEG, to all devices regardless of channel height. Furthermore, CFD simulations assumed devices were constructed from a non-deformable solid, whereas PDMS is inherently flexible, so this may have contributed to in vitro pressure drops being lower than predicted [56]. In vitro pressure drops in H

_{c}= 60 and H

_{c}= 100 were not significantly different (Figure 10), despite both experimental pressure drops being less than theory. This suggests that the mathematical modeling and CFD of pressure (Table 2) were accurate in predicting similar pressure drops across designs containing differing capillary heights.

_{c}= 60 and H

_{c}= 100 at 0.6 mL/min (Figure 11). Change in SO

_{2}generally decreased with increasing flow rates, a trend that agrees with other previously reported microfluidic artificial lungs [22,38]. Overall, O2 exchange in test devices met or exceeded theoretical expectations.

#### 4.4. Limitations

_{c}= 30 µm design molds. Before committing a mold for use in device fabrication, preliminary perfusion tests were performed with dyed water. Perfusion tests revealed that although a vast majority of capillaries perfused immediately, flow through H

_{c}= 30’s first and last few dozen capillaries was delayed. This delay in perfusion may be due to partially obstructed access between capillaries and distribution channels, a potential result of slight misalignment. Or, as discussed previously, it may have been due to the first/last few dozen capillaries having branching angles that varied from Murray’s law.

#### 4.5. Continuing Efforts

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Table A1.**Photolithography settings used to construct blood side and gas side silicon molds. Actual channel height reported as mean ± standard deviation. PEB = post exposure bake.

Intended Channel Height (μm) | Actual Channel Height (μm) | Spin Speed (rpm) | Soft Bake (65 °C) (min) | Soft Bake (90 °C) (min) | UV Exposure (s) | PEB (65 °C) (min) | PEB (90 °C) (min) | |
---|---|---|---|---|---|---|---|---|

Capillary | 30 | 30.8 ± 1.8 | 2800 | 2 | 5 | 25 | 2 | 3 |

60 | 60.2 ± 3.7 | 1480 | 3.5 | 7 | 70 | 2 | 3 | |

100 | 101.7 ± 5.9 | 730 | 5 | 13 | 80 | 2.5 | 3.5 | |

Distribution | 239 | 238.8 ± 10.4 | 1450, 1450 | 4, 12 | 5, 30 | 450 | 6 | 15 |

210 | 215.4 ± 5.6 | 1600, 2000 | 4, 5 | 5, 15 | 300 | 5 | 12 | |

241 | 237.1 ± 15.8 | 900, 2100 | 4, 5 | 5, 20 | 300 | 5 | 12 |

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**Figure 1.**A cross-section drawing of a microfluidic artificial lung. Blood flows in the Z direction and gas flows in the X direction. Image reused with permission from Thomson et al., 2019 [38].

**Figure 4.**(

**a**) Gas exchange surface area by capillaries (A

_{c,g}), total blood contacting surface area (A

_{b,c}); (

**b**) priming volume in capillaries and distribution channels (V

_{prime}); and (

**c**) wall shear rate in capillaries (τ

_{w}) with respect to capillary height (H

_{c}).

**Figure 6.**(

**a**) Pressure drop cut plots half-way through capillary heights of H

_{c}= 30; (

**b**) H

_{c}= 60; and (

**c**) H

_{c}= 100. Dark blue corresponds to lower pressures (760 mmHg) and red corresponds to higher pressures (814 mmHg). Scale bars = 1 cm.

**Figure 7.**(

**a**) Shear rate cut plots halfway through capillary heights of H

_{c}= 30; (

**b**) H

_{c}= 60; and (

**c**) H

_{c}= 100. Dark blue and red corresponds to the lower and upper bounds of physiological shear rate, 43 and 3038 s

^{−1}(1 and 70 dyn·cm

^{2}), respectively. Scale bars = 1 cm.

**Figure 8.**(

**a**) Shear rate cut plots at the midpoint of the centermost capillaries in of H

_{c}= 30; (

**b**) H

_{c}= 60; and (

**c**) H

_{c}= 100. Local mesh Level 4 was applied to maximize resolution. Dark blue and red corresponds to the lower and upper bounds of physiological shear rate, 43 and 3038 s

^{−1}(1 and 70 dyn·cm

^{2}), respectively.

**Figure 9.**(

**a**) H

_{c}= 30; (

**b**) H

_{c}= 60; and (

**c**) H

_{c}= 100 perfused with blood. All images are oriented such that the inlet is positioned at the bottom left (dark circle) and the outlet and the top right (bright triangle). Scale bars = 1 cm.

**Figure 10.**Experimental and computational pressure drops for each design at various flow rates (n = 3) when tested with bovine whole blood. At each flow rate, bars represent the following from left to right: H

_{c}= 30, H

_{c}= 60, H

_{c}= 100 and mathematical theory. Red diamonds represent theoretical CFD results. Error bars = mean ± standard deviation. * = statistically significant at alpha of 0.05.

**Figure 11.**Experimental and theoretical change in oxygen saturation achieved by each design at various flow rates (n = 3) when tested with bovine whole blood. At each flow rate, bars represent the following from left to right: H

_{c}= 30, H

_{c}= 60, H

_{c}= 100 and mathematical theory. Error bars = mean ± standard deviation. + n = 1, # n = 2. * = statistically significant at alpha of 0.05.

Blood Density (kg∙m ^{−3}) | Maximum Dynamic Viscosity (Pa∙s) | Minimum Dynamic Viscosity (Pa∙s) | Power-Law Index | Time Constant (s) |
---|---|---|---|---|

1060 | 0.056 | 0.00345 | 0.3568 | 3.313 |

**Table 2.**Mathematical modeling results (gas exchange surface area of capillaries (A

_{c,g}), blood contacting surface area (A

_{b,c}), priming volume in capillaries and distribution channels (V

_{prime}) and capillary wall shear rate (τ

_{w}) of H

_{c}= 30, 60 and 100 at rated flow (Q

_{R}) compared to pressure drop (ΔP) and τ

_{w}from CFD simulations.

Mathematical Theory | CFD | |||||||
---|---|---|---|---|---|---|---|---|

Designs | Q_{R}(mL/min) | A_{c,g}(cm ^{2}) | A_{b,g}(cm ^{2}) | V_{prime}(µL) | ΔP (mmHg) | τ_{w}(s ^{−1}) | ΔP (mmHg) | τ_{w}(s ^{−1}) |

H_{c} = 30 | 0.8 | 1.2 | 5.4 | 25 | 60 | 1458 | 53.8 | 1500 |

H_{c} = 60 | 0.8 | 2.1 | 5.0 | 15 | 60 | 2546 | 51.2 | 2588 |

H_{c} = 100 | 0.8 | 3.1 | 7.3 | 33 | 60 | 1818 | 51.5 | 1729 |

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**MDPI and ACS Style**

Ma, L.J.; Akor, E.A.; Thompson, A.J.; Potkay, J.A.
A Parametric Analysis of Capillary Height in Single-Layer, Small-Scale Microfluidic Artificial Lungs. *Micromachines* **2022**, *13*, 822.
https://doi.org/10.3390/mi13060822

**AMA Style**

Ma LJ, Akor EA, Thompson AJ, Potkay JA.
A Parametric Analysis of Capillary Height in Single-Layer, Small-Scale Microfluidic Artificial Lungs. *Micromachines*. 2022; 13(6):822.
https://doi.org/10.3390/mi13060822

**Chicago/Turabian Style**

Ma, Lindsay J., Emmanuel A. Akor, Alex J. Thompson, and Joseph A. Potkay.
2022. "A Parametric Analysis of Capillary Height in Single-Layer, Small-Scale Microfluidic Artificial Lungs" *Micromachines* 13, no. 6: 822.
https://doi.org/10.3390/mi13060822