### Overview

Modelling was conducted to analyze the flow of ISF through microneedles, determining that a 15-20-minute collection period would be required to collect 287nL of fluid. This is reasonable given the minimally invasive nature of our collection method. This flowrate calculation accounts for ISF flow through the dermis and then uptake and flow in the microneedle itself, as can be seen in the microneedle diagram below:

Flow was characterized by determining important non-dimensional numbers, such as the Reynolds and Bond numbers, using fluid mechanics equations. The Hagen Poiseuille equation was used to estimate the rate of flow of ISF through capillary action, confirming that it would be a sufficient driving force for the device, such that additional vacuum suction methods would not be required.

### Obtaining a Sufficient Amount of ISF

**Figure 1:** Representative hollow stainless steel microneedle (750 μm tall) with a single hollow capillary to draw fluid out of skin [1]

Only extremely low volumes of interstitial fluid 20 nL/mm2 in the epidermis, 800 nL/mm2 in the dermis)[2] can be found on the skin, complicating the process of ISF extraction. Demonstrated by previous work, ISF may be extracted by penetrating superficial skin layers to access ISF in small amounts of up to 200 nL per microneedle by capillary action over periods of 15 to 20 minutes[2]. Larger volumes, 1–10 μL, may be obtained using a combination of a glass microneedle and vacuum suction over 2–10 minutes[2], however, such additional equipment would not be ideal for implementation in this device with a goal of minimal invasiveness and user optimization.

Based on a hollow 25G microneedle, literature predicts an ISF volume of 0.041 μL over 20 minutes [1]. Thus, it can be assumed that a user would be able to produce this as a mean value. Using this and the device design concept of five microneedle arrays with 7 microneedles each, the total anticipated volume of ISF collected was calculated to be 1.435µL, or 287nL per microneedle array for each analyte, as shown:

A 15-20-minute collection period would be reasonable and achievable, given the minimally invasive nature of the microneedle approach. Ideally, the user could apply the device 30 minutes after a meal and continue with daily activities unencumbered, a vast improvement from in-office blood collection. As the microneedles being used could vary, further research could be conducted to confirm the flow rate of ISF using the final microneedle mechanism.

### Characterizing Flow

#### Reynolds Number

Reynolds number is the ratio of inertial forces to viscous forces, used to predict fluid flow patterns. To analyze the flow of the ISF using Reynolds number, the dynamic viscosity and density of ISF are required, in addition to the diameter of the microneedle. The viscosity of ISF is assumed to be similar to plasma viscosity, 0.012 Pa·s [1] and has a density of 1000 kg∙m-3 [3]. The diameter of a typical hollow 25G microneedle is 260µm.

Reynolds number is defined as Re = ρDV/μ, where ρ is the density of the fluid (kg∙m-3), D is the diameter of the inlet (microneedle in this case) (m), V is the velocity of the fluid (m/s), and μ is the dynamic viscosity of the fluid (Pa∙s). Thus, to determine the exact Reynolds number, a fluid velocity is required or a desired Reynolds number may be assigned to calculate the necessary velocity. Based on these known values, it can be readily assumed that all ISF flow within the microneedles corresponds to laminar flow, wherein Re<2100. This value can be assigned to calculate the velocity of the ISF as follows:

This calculation holds at speeds of up to 96.92 m/s, while actual fluid speeds within the microneedles will be in the order of magnitude of picometers per second. Given that the Reynolds number would be extremely small as a result, Stokes flow, or creeping flow, is applicable to this situation (Re << 1), where advective inertial forces are small in comparison to viscous forces. This indicates that flow within the microneedles will take on a parabolic flow profile, as shown below.

### Pressure Needed to Translate ISF

#### Collection by Capillary Action

As described by Samant and Prausnitz [1], collection of ISF by capillary action involves first flow through the dermis to the microneedle and then uptake and flow in the microneedle itself.

The Hagen Poiseuille equation may be employed to estimate the rate of flow of ISF through capillary action, in order to confirm that no external forces, such as vacuum suction, would be required for collection. The equation is defined as Q = (∆P )/L (πR^4)/8μ, where Q is fluid (ISF) flow rate, ∆P is pressure drop across the needle, L is the length of the needle (500 µm), R is the needle radius (130 µm, corresponding to a 25G needle), and µ is viscosity of ISF (assumed to be similar to plasma viscosity, as above), 0.0012 Pa·s [1].

The Lucas Washburn equation can be used to estimate the pressure drop, defined by ∆P = 2γ/R cosθ, where 𝛾 is the surface tension of ISF (assumed to be similar to water 0.072 N/m at 37°C) and θ is the contact angle between ISF and the microneedle (assumed to be 0°) [1].

Thus, flow after 20 minutes can be estimated via the following calculations

With the anticipated experimental ISF collection values of course much lower than these theoretical values, it is assumed that ISF absorption at the dermis-microneedle interface is instantaneous. As a result, concentration of ISF at this interface is 0. Furthermore, the flow of ISF in the dermis can be modelled using Fick’s second law of diffusion, ∂C/∂t=D_derm (∂^2 C)/(∂x^2 ) . This law follows the boundary conditions C(x, 0) = C0, C(0,t) = 0, C(∞,t) = C0 where C is ISF concentration in the dermis, 𝐷derm is ISF diffusivity in dermis (8.72 x 10-10 m2/s, estimated using values for permeability of water through dermis) and C0 is ISF concentration in unperturbed dermis (0.7 g/mL) [1].

These conditions can be substituted in the equation, giving flowrate into the microneedle of Q ̇_ISF= -D ∂C/∂x |_(x=0) A=√(D/πt) C_0 A , where, based on an inner diameter of 260 µm, the area of the open tip of a 25G microneedle is 5.3 x 10-8 m2¬ [1]. Substituting these values in the flux equations, the ISF concentration profiles and collection rate for a hollow 25G microneedle over 20 minutes can be predicted as is 0.041 µL, as shown by Samant and Prausnitz below [1].

**Figure 2: **Models predicting ISF collection by capillary flow into hollow 25G microneedles. (a) Concentration profile of ISF in the dermis and microneedle at time 0 min (grey line) and 20 minutes (dashed line). (b) ISF volume entering hollow microneedles over time [1]

Of note, literature provides experimentally observed flow rates for a hollow 25G microneedle at 0.025±0.019 µL [1]. The predicted value being 1.6 times larger than the experimental value is reasonable, given the simplifying assumptions made in this analysis. With the rate of transport through microneedle capillaries nearly instantaneous compared to through dermis, this concludes that the rate of ISF uptake is limited by the diffusion of ISF through the dermis.

#### Bond Number

The Bond (Bo) number can be used to further characterize the driving force of fluid, measuring the importance of gravitational forces compared to surface tension forces to characterize the shape of drops. To calculate the Bond number, the density of the microneedle material is required, along with the other variables identified previously. The density of 7480-8000 kg∙m-3 was used for a range of stainless steel, the material typically used for 25G microneedles [4].
The Bond number (Bo) equation is defined by Bo=(∆ρgL^2)/γ, where Δρ is the density difference between the microneedle material and the fluid (kg∙m-3), g is the gravitational constant (m/s2), D is the diameter of the microneedle (m), and γ is the surface tension of the ISF (N/m).

Calculated for the high and low ranges of stainless steel density, the Bond number for the microneedles was calculated as shown below.

The low value (less than one) given by both densities indicates that surface tension dominates gravitational forces [5]. This means that stainless steel is an appropriate material choice for the needles, especially with the lack of external driving forces other than capillary action.

Relating to the work above, for a sufficiently narrow tube (ie. low bond number), the capillary pressure induced within is balanced by the height change, which will be positive for wetting angles less than 90˚. By rearranging the Young-Laplace equation and setting it equal to the gravitational forces, the height achieved at this hydrostatic equilibrium can be calculated, as shown below.

Using a microneedle surface that is just hydrophilic with a wetting angle of 89˚, the height achievable by capillary action is shown below.

Thus, it can be confirmed that capillary action is a sufficient driving force of the moving ISF for the height of the microneedles. The contact angle necessary to propel the ISF the length of 1mm (as in our design) can be calculated by rearranging the above equation, as shown.

Since the required height can be achieved even at such a high wetting angle, additional efforts to increase the wettability of the surface will not be required.

Of note, for cost-effectiveness of the disposable microneedle array component, a polymer would likely be used in place of stainless steel. Additional analysis could be conducted in future iterations of the project to confirm that the above observation remains.

### References

- P. P. Samant and M. R. Prausnitz, “Mechanisms of sampling interstitial fluid from skin using a microneedle patch,” Proc. Natl. Acad. Sci., vol. 115, no. 18, pp. 4583–4588, May 2018, doi: 10.1073/pnas.1716772115.
- S. A. Ranamukhaarachchi, C. Padeste, M. Dübner, U. O. Häfeli, B. Stoeber, and V. J. Cadarso, “Integrated hollow microneedle-optofluidic biosensor for therapeutic drug monitoring in sub-nanoliter volumes,” Sci. Rep., vol. 6, Jul. 2016, doi: 10.1038/srep29075.
- W. Yao, Z. Shen, and G. Ding, “Simulation of Interstitial Fluid Flow in Ligaments: Comparison among Stokes, Brinkman and Darcy Models,” Int. J. Biol. Sci., vol. 9, no. 10, pp. 1050–1056, 2013, doi: 10.7150/ijbs.7242.
- “Metals and Alloys - Densities.” https://www.engineeringtoolbox.com/metal-alloys-densities-d_50.html (accessed Jun. 24, 2020).
- S. Li, M. Liu, D. Hanaor, and Y. Gan, “Dynamics of Viscous Entrapped Saturated Zones in Partially Wetted Porous Media,” Transp. Porous Media, Feb. 2018, doi: 10.1007/s11242-018-1113-3.