# Finger-Actuated Microneedle Array for Sampling Body Fluids
## 2 Methods
### Boundary Conditions (BCs) and Simulation  
The design in this work consists of two inlets and one outlet, which are (i) the input for the mechanical pressure via finger, (ii) the inlet for the target fluid, and (iii) the outlet to the reservoir. For simulating the process of human finger pressing, we used experimental data of a previously designed finger-powered microfluidics device [55]. The average pressure created by human finger pressing is \(\approx\)4500 Pa. For pressing with a time interval of every 5 s, the following pressure-time function can be developed:  
\[P(t)=2250\times\cos\left(\frac{2\pi}{5}\times t\right)+2250 \tag{1}\]  
where \"\(t\)\" is time in seconds and \"\(P\)\" is pressure in Pascals. Equation (1) was the base function for the pressure boundary condition (BC) at the finger pressure input. To study the effect of the force input by pressing with fingers, we modified the function of Equation (1) into a version where it defines pressing 5 times faster:  
\[P(t)=2250\times\cos(2\pi\times t)+2250 \tag{2}\]  
Equation (2) means that the user is pressing the dome-shaped chamber every second. In this equation, \"\(t\)\" is time in seconds and \"\(P\)\" is pressure in Pascals. Human blood was chosen as the first target fluid, aimed to be collected with the designed device. Blood is a non-Newtonian fluid in which the viscosity has been reported to be modeled with several mathematical equations [56]. The power-law model (Equation (3)) was used for the viscosity equation in the simulation:  
\[\mu=k\dot{\gamma}^{n-1} \tag{3}\]  
where \"\(\mu\)\" is the viscosity of the blood and \"\(\dot{\gamma}\)\" is the shear rate. The coefficients of \"\(k\)\" and \"\(n\)\" for blood are 0.42 and 0.61, respectively [56,57]. In addition, the blood density was defined as an amount equal to 1060 kg.m\({}^{-1}\). High-resolution blood flow velocity in the human finger was reported [58] with measurement of arterial blood flow velocities ranging from 0.049 to 0.19 m.s\({}^{-1}\), and venous blood flow velocity in the range of 0.015 to 0.071 m.s\({}^{-1}\). Moreover, the blood flow rate in the arm at rest was reported to be 0.21 \(\pm\) 0.04 L.min\({}^{-1}\)[59]. In addition, the amounts of blood pressure in the human arms in sitting and supine position were reported [60], with systolic blood pressure (SBP) 135.5 \(\pm\) 24.0 mmHg and diastolic blood pressure (DBP) 79.2 \(\pm\) 9.7 mmHg for sitting position and amounts of SBP = 145.0 \(\pm\) 23.2 mmHg and DBP = 84.1 \(\pm\) 9.0 mmHg for supine position. On the other hand, the normal human heart rate at rest is in the range of 60 to 100 beats per minute [61,62]. Taking 80 beats per minute with SBP of 135.5 mmHg (\(\approx\)18,000 Pa) and DBP of 79.2 mmHg (\(\approx\)10,000 Pa) as the average amounts, we were able to derive an equation of blood pressure in the vessels:  
\[P(t)=4000\times\cos\left(\frac{13\pi}{5}\times t\right)+14000 \tag{4}\]  
where \"\(t\)\" is time in seconds and \"\(P\)\" is pressure in Pascals. Equation (4) and alternatively a simplified constant pressure value were used as the BC for the blood input. A second series of simulations were performed for ISF collection. The viscosity of the ISF in the literature was reported to be in the range between 1.2-1.5 \(\times 10^{-3}\) Pa.s [63] and 3.5 \(\times 10^{-3}\) Pa.s [64]. In this work, an average amount of 2.5 \(\times 10^{-3}\) Pa.s was used. In addition, the density of ISF was reported to be 1000 kg.m\({}^{-3}\)[64]. The input BC for ISF was defined with ISF pressure, which is in the range of \(-\)0.5 to \(-\)8.0 mmHg [63], with the negative sign attributing to the action of lymphatics [65]. An average value of 4 mmHg was utilized in the simulation. Additionally, the BC for the reservoir for all the cases was set to be pressure of zero. The simulation of the fluid transport across the designed device was performed via finite element analysis, generating the flow curves at the reservoir outlet along with pressure and velocity gradients across the device.