A. Fluid flow inside MNAs

Fluid flow in MNAs is generally viscous-dominated and highly laminar, mainly due to small length scales. In addition, the Reynolds number is small (often on the order of unity or smaller) so that the convective terms of the flow equations are negligible.7373. F. M. White, Fluid Mechanics, 8th ed. (McGraw Hill, 2016), p. 773. Thus, flow can be well described by the incompressible Stokes equations with no-slip boundary conditions at the wall, and classical solutions are often relevant and useful.7474. J. D. Zahn, D. Trebotich, and D. Liepmann, “Microdialysis microneedles for continuous medical monitoring,” Biomed. Microdevices 7(1), 59–69 (2005). https://doi.org/10.1007/s10544-005-6173-9 As a result, the typical velocity profile in MNAs is parabolic in simple geometries when the flow is driven by an applied pressure difference, as is the case in all MNA applications. The geometrical parameters of the microneedle, the number of microneedles on the array, and fluid characteristics such as fluid viscosity affect the required pressure difference for fluid flow through the microneedle. In a research on 3D printing of microneedles,7575. M. Rezapour Sarabi et al., “Machine learning-enabled prediction of 3D-printed microneedle features,” Biosensors 12(7), 491 (2022). https://doi.org/10.3390/bios12070491 three primary geometric parameters for a microneedle were studied: needle base diameter, needle height, and angle of the draft. While 3D-printed cylindrical microneedles were fabricated more efficiently, manufacturing conical microneedles with a design draft angle of 5° or 10° showed considerable resolution limits. For MNA biosensors that intend to continuously monitor analytes, achieving effective and reliable skin penetration and microneedle-tissue interlocking is crucial, especially regarding the variation in skin thickness originated from differences in age, gender, and body mass index, and the elasticity of the skin, which results in a counteracting force to the penetration.3131. H. Teymourian et al., “Lab under the skin: Microneedle based wearable devices,” Adv. Healthcare Mater. 10(17), 2002255 (2021). https://doi.org/10.1002/adhm.202002255 To overcome this challenge, a precise optimization of the microneedle geometrical parameters, such as the tip radius, height, needle density, base diameter, and spacing, which determine the insertion and fracture forces, leads to reliable skin penetration.7676. R. F. Donnelly et al., Microneedle-mediated Transdermal and Intradermal Drug Delivery (Wiley, 2012). The ability to penetrate via hollow silicon MNAs for glucose monitoring with tapered and straight profiles was studied both theoretically and experimentally.7777. B. Chua et al., “Effect of microneedles shape on skin penetration and minimally invasive continuous glucose monitoring in vivo,” Sens. Actuators, A 203, 373–381 (2013). https://doi.org/10.1016/j.sna.2013.09.026 Two important parameters for penetration calculation and pain estimation were reported for the mentioned microneedle profiles: (i) the kinetic energy of the prototype, which has a direct effect on both penetration and pain sensation for the human subject; (ii) the stopping distance of MNAs inside skin tissue that is related to the mechanical properties of the skin. The experimental results revealed that the existence of a cutting edge in the straight profile MNAs dramatically reduced the minimum required penetration pressure, despite the fact that the tapered profile MNAs were more likely to penetrate into most skin types given the same amount of kinetic energy.7777. B. Chua et al., “Effect of microneedles shape on skin penetration and minimally invasive continuous glucose monitoring in vivo,” Sens. Actuators, A 203, 373–381 (2013). https://doi.org/10.1016/j.sna.2013.09.026 Microneedles can be made of a variety of materials, including polymers and metals, depending on the patch’s design and component parts.7878. J. H. Jung and S. G. Jin, “Microneedle for transdermal drug delivery: Current trends and fabrication,” J. Pharm. Investig. 51(5), 503–517 (2021). https://doi.org/10.1007/s40005-021-00512-4 Although silicon has sufficient mechanical strength for skin insertion and can be precisely manufactured with short, sharp tips using deep reactive ion etching and photolithography methods, silicon microneedles can cause a risk to patient safety if they break away from the skin and pieces remain in the tissue.7979. M. G. McGrath et al., “Determination of parameters for successful spray coating of silicon microneedle arrays,” Int. J. Pharm. 415(1-2), 140–149 (2011). https://doi.org/10.1016/j.ijpharm.2011.05.064 The most popular metal used to make hollow, coated, and solid microneedles that can readily pass through the skin is stainless steel, but it corrodes more quickly than Ti alloys.8080. D. Amalraju and A. Dawood, “Mechanical strength evaluation analysis of stainless steel and titanium locking plate for femur bone fracture,” Eng. Sci. Technol., An Int. J. 2(3), 381–388 (2012). The polymers used to manufacture microneedles should be soluble in water, biocompatible, and strong enough to penetrate the skin.7878. J. H. Jung and S. G. Jin, “Microneedle for transdermal drug delivery: Current trends and fabrication,” J. Pharm. Investig. 51(5), 503–517 (2021). https://doi.org/10.1007/s40005-021-00512-4 Various types of polymers, such as hydroxypropyl methylcellulose (hypromellose), hyaluronic acid (HA), carboxymethyl cellulose (CMC), polyvinyl pyrrolidone (PVP), and poly-lactic-co-glycolic acid (PLGA), can be used in the solvent casting procedure to fabricate dissolving or hydrogel microneedles. On the other hand, due to the micro-scale diameters of microneedles, flow resistance is high. Measurements and predictions of fluid dynamics are critical when developing microneedles for transdermal drug delivery so that the microneedle is small enough to avoid pain while being sharp enough to penetrate into the skin easily and large enough to achieve the appropriate flow rate.8181. D. W. Bodhale, A. Nisar, and N. Afzulpurkar, “Structural and microfluidic analysis of hollow side-open polymeric microneedles for transdermal drug delivery applications,” Microfluid. Nanofluid. 8(3), 373–392 (2010). https://doi.org/10.1007/s10404-009-0467-9

1. Modeling flow in microneedle

The drug is usually delivered through a straight circular or rectangular channel in the microneedle and the flow can be well approximated by a fully developed uni-directional laminar flow except for the inlet and exit sections. Figure 5(a) illustrates a schematic representation of such a microchannel with a circular cross section. Several analytical solutions in the fully developed region for a wide range of cross sections exist.8282. F. M. White, Viscous Fluid Flow, 3rd ed. (McGraw Hill, 2006), p. 629. For an incompressible steady flow in a circular channel, applying the energy conservation equation between two locations in Fig. 5(a) yields79–8179. M. G. McGrath et al., “Determination of parameters for successful spray coating of silicon microneedle arrays,” Int. J. Pharm. 415(1-2), 140–149 (2011). https://doi.org/10.1016/j.ijpharm.2011.05.06480. D. Amalraju and A. Dawood, “Mechanical strength evaluation analysis of stainless steel and titanium locking plate for femur bone fracture,” Eng. Sci. Technol., An Int. J. 2(3), 381–388 (2012).81. D. W. Bodhale, A. Nisar, and N. Afzulpurkar, “Structural and microfluidic analysis of hollow side-open polymeric microneedles for transdermal drug delivery applications,” Microfluid. Nanofluid. 8(3), 373–392 (2010). https://doi.org/10.1007/s10404-009-0467-9P1−P2ρ+v12−v222+g(z1−z2)=fLdv22+ΣKv22,(1)where subscripts 1 and 2 describe the up- and downstreams of the control volume, respectively. The terms on the left-hand side denote the work done by pressure difference, changes in the kinetic energy and the potential energy, respectively, with P,ρ,v,g, and z being pressure, fluid density, average velocity in the channel, the gravitational acceleration, and elevation, respectively. The terms on the right-hand side represent the major and minor losses, respectively, where L and d are length and diameter of the microchannel, respectively, while f is the Darcy–Weisbach friction factor and K is the minor loss coefficient. For a fully developed laminar flow in a circular pipe, the Darcy–Weisbach friction factor is given by f=64Red, where the Reynolds number is defined as Red=ρvdμ with μ being the fluid viscosity.4848. Dabbagh, S. R., M. R. Sarabi, M. T. Birtek, N. Mustafaoglu, Y. S. Zhang, and S. Tasoglu, “3D bioprinted organ-on-chips,” Aggregate 4(1), e197 (2022). https://doi.org/10.1002/agt2.197 The minor loss coefficient, K, is defined to account for the additional losses due to entrance/exit flows, expansion/contraction, bends, tees, and the like. In a typical microneedle, the change in the potential energy is negligible compared to the other terms. In addition, for a fully developed flow in a constant cross-sectional channel, the mass conservation requires v1=v2. Thus, Eq. (1) can be further simplified by neglecting the gravity effects in comparison with the rest of the equation, and that the channel has a constant circular cross-sectional area,ΔP=μ128πqLd4+ρ8π2Q2d4(K1+K2),(2)where ΔP=P1−P2 is the applied pressure difference, Q=πd24v is the volume flow rate, and K1 and K2 are the minor loss coefficients associated with the entrance and exit flows, respectively.5151. A. K. Waljee and P. D. Higgins, “Machine learning in medicine: A primer for physicians,” Am. J. Gastroenterol. 105(6), 1224–1226 (2010). https://doi.org/10.1038/ajg.2010.173 Assuming that losses at the needle’s entrance and exit are negligible, Eq. (2) reduces to Poiseuille's law given bySimilar results can be obtained for channels with different cross sections. Here, the rectangular channel will be discussed as it is more relevant for the microneedle applications, and we refer to Ref. 8282. F. M. White, Viscous Fluid Flow, 3rd ed. (McGraw Hill, 2006), p. 629. for other cross sections.

2. Fluid flow in rectangular ducts

When the microneedle's cross section is rectangular rather than circular,84,8584. D. Trebotich, J. Zahn, and D. Liepmann, “Complex fluid dynamics in BioMEMS devices: Modeling of microfabricated microneedles,” in Proceedings of the 2002 International Conference Modeling Simulation Microsystems, San Juan, Puerto Rico (Citeseer, 2002).85. J. D. Zahn et al., “Microfabricated polysilicon microneedles for minimally invasive biomedical devices,” Biomed. Microdevices 2(4), 295–303 (2000). https://doi.org/10.1023/A:1009907306184 analytical solutions for the velocity profile and associated flow rate can be obtained for a fully developed laminar flow. Adopting a Cartesian coordinates of x,y, and z and considering a rectangular cross section with −a≤y≤a and −b≤z≤b as shown in Fig. 5(b), the axial velocity profile and flow rate can be given by8282. F. M. White, Viscous Fluid Flow, 3rd ed. (McGraw Hill, 2006), p. 629.u(y,z)=16a2μπ3(−dPdx)[∑i=1,3,5,…∞⁡(−1)i−12[1−cosh(iπz/2a)cosh(iπb/2a)]]cos(iπy/2a)i3,(4)Q=4ba33μ(−dPdx)[1−192aπ5b∑i=1,3,5,…∞tanh(iπb/2a)i5],(5)where 2a is the width of the microchannel, 2b is the height of the microchannel, dPdx is an applied pressure gradient in the x direction, and μ is the viscosity of the fluid.8282. F. M. White, Viscous Fluid Flow, 3rd ed. (McGraw Hill, 2006), p. 629. For a rectangular channel with height h=2a and width w=2b, the volume flow rate is correlated to the pressure drop ΔP over a length of L approximately as8686. H. A. Stone, A. D. Stroock, and A. Ajdari, “Engineering flows in small devices: Microfluidics toward a lab-on-a-chip,” Annu. Rev. Fluid Mech. 36(1), 381–411 (2004). https://doi.org/10.1146/annurev.fluid.36.050802.122124Q=wh312μ(ΔPL)[1−O(h/w)],(6)where O(h/w) denotes the error term of order of h/w. This term can be approximated as O(h/w)=192π5hw yields less than 10% error when h/w≤0.7. For h/w≪1, Eq. (6) becomes Q≈wh312μ(ΔPL).