Particles’ motion simulation

The key idea of the dynamic sensing method is to exploit the different diffusion rate among bare magnetic particles and after bonding with an analyte (for example tau protein for AD or beta-amyloid for PD), because they have different hydrodynamic radius. For a solute in a fluid medium, the diffusion coefficient is:

𝐷 = 𝑘𝑇

6𝜋𝜂𝑎, (3.1)

where 𝑘𝑇 is the thermal energy, 𝜂 the fluid viscosity and 𝑎 the hydrodynamic radius of the dispersed particles’ population. The diffusion profile of a mix of populations can be studied by letting them move freely in a microfluid channel above a magnetoresistive sensors array (passivated with few nanometer of insulating material). In Figure 3.1, a sketch is drawn for the case of linear diffusion: it can be noticed that, since different particles’ populations have different diffusion rates, the bare (smaller) ones will travel longer than the bigger (antigen-bounded) ones on average after the same time, obtaining different profiles depending on the ratio between their quantities.

Figure 3.1: Sketch of the dynamic detection principle developed through MADIA project.

52 This theoretical expectation has been validated before moving to the fabrication phase by the quantitative calculation through the development of a computational model. In fact, to study this phenomenon in a systematic way, various Finite Elements Method (FEM) simulations have been performed with the software Comsol MultiPhysics, in particular employing the modules Creeping flow and Transport of diluted species, which led to an idea of the needed sensitivity for the magnetoresistive structures for the concentration under study and speed of response.

Calculation with the Finite Elements Method employs the reduction of a differential equation in its weak form, meaning an integral one in which the mathematical requirements for the functions under study are less pressing. Furthermore, considering the additive property, the system can be divided in small domains where the parameters are considered constant and only small changes are admitted between adjacent ones, to have a reasonably continuous solution.

With the hypothesis of water at 25 °C as solvent, for particles with 10 nm of radius, it results 𝐷 ≃ 2.12 ⋅ 10 𝑚 ⋅ 𝑠. Regarding the initial target concentration to be detected, it was supposed to be 10 𝑓𝑔 ⋅ 𝑚𝐿 for Tau protein (molecular weight of about 60 𝑘𝐷𝑎), which leads to 𝐶 ≃ 2 ⋅ 10 𝑘𝑔 ⋅ 𝑚 . Since the MADIA project partner CNR-ISMN worked on an aggregation and concentration chamber able to confine 1 𝑚𝐿 of solution into 0.1 𝑛𝐿, so with an enhancement factor of 10 , as simulation parameter has been considered 𝐶 ≃ 2 ⋅ 10 𝑘𝑔 ⋅ 𝑚 into the reservoir. There was also planned to realize the microfluidic channels in PDMS or SU8 so, for their mechanical stability, it’s not advisable to have less than 10 𝜇𝑚 as height and this constraint has been taken into account during calculations. To optimize the computation, it was not employed the classical free tetrahedral mesh, which is very useful for complex designs on three axes, since with standard lithography procedure the patterning regards only 𝑥𝑦 plane with slightly vertical projection. For this reason, a free quad mesh has been employed at the 𝑧 = 0 figures and then swept until the height proposed (10 𝜇𝑚 for any layout).

Three geometries were considered for the channels, to evaluate different possible situations:

linear diffusion, lateral diffusion and hydrodynamic focusing, sorted by increasing complexity.

In any of the approach, the goal of the simulation was to calculate how many particles are directly above or very near the surface of the elements of an equal-spaced magnetoresistive array at different times: this quantity (concentration times Avogadro’s number) should be proportional to the signal measured by the magnetoresistive structures with good approximation. This assertion is not exactly true considering the magnetic field generated by a

53 dipole, since it is not limited in space, but it can be a good approximation, because its magnitude falls down with the third power of the distance, so it is possible to assume that a consistent part of the signal is generated by the particles situated near the sensing surface and the contribution from the others is negligible.

The first two configurations implement only the module Transport of diluted species, which solves Fick’s diffusion equation

𝜕𝑐

𝜕𝑡 + ∇ ⋅ (−𝐷∇𝑐) = 𝑅

(3.2)

for any of the species in solution, to obtain the time dependent concentration in any mesh element of the system, which size has been set to obtain a good spatial resolution, since their dimension goes from 0.356 𝜇𝑚 to 1.89 𝜇𝑚 (preset “fine mesh” for fluid dynamics in COMSOL).

The case of linear diffusion is the simplest among the simulated ones: the idea is to put a droplet of the solution to be sensed at the beginning of a linear channel through the use of a peristaltic pump and after stopping the flow, the particles can diffuse freely without external drag. The optimized final layout is shown in Figure 3.2 (a): it consists of a single channel of 50 𝜇𝑚 of width, 10 𝜇𝑚 of height and 800 𝜇𝑚 of length, including a 200 𝜇𝑚 long (volume of 0.1 𝑛𝐿) reservoir that simulates the initial confinement of the droplet before the diffusion starts, so with an homogenous concentration of 0.2 𝜇𝑚𝑜𝑙/𝑚 . The ten parallelepipeds 50 𝜇𝑚 x 20 𝜇𝑚 x 10 𝜇𝑚 (W x L x h) after the reservoir are supposed to be the sensing area with the same number of magnetoresistive structures. The color map in Figure 3.2 (a) represents particles’

concentration respect to the position along the channel in 𝑚𝑜𝑙/𝑚 , resulting from the initial conditions just exposed The diffusion profile for a single 10 𝑛𝑚 particles’ population, after 30 minutes from the release is shown in Figure 3.2 (b), where it’s possible to notice that, as expected, the particles’ number, calculated by integrating the concentration in a volume domain and then multiplying this value for Avogadro’s number, decreases linearly moving forth the array with an excursion of about 50 any 20 𝜇𝑚, so any sensor should be able to measure few tens of magnetic moment nearby its surface.

After the dimensional optimization to obtain a diffusion profile with reasonable variation along the channel for a single diffusion coefficient, it was also calculated the response expected with

54 the same total number of particles with two mixed populations in the reservoir, so an approximation of the real detection method. For the aim of simplicity, it has been considered as a good indicator the difference between the number of particles located above the first sensor and another sensor in the array instead of studying the whole diffusion profile. The discriminant is the hydrodynamic radius of 10 𝑛𝑚 and 20 𝑛𝑚, furthermore the mix has been varied from 0% to 100%, composition in terms of bigger particles with ten possible pair (0-100%, 10-90%

etc.). Figure 3.2 (c) reports the obtained plot. It’s possible to notice that the excursion increases with the distance, without arriving at a saturation point with a “sensitivity” of 1.46 N/% after 1800 𝑠 of free diffusion.

Figure 3.2: Linear diffusion geometry (a) and calculated diffusion (b and c).

This geometry is very easy to fabricate and align on a sensing array, the only tricky part could be the exact time of release for the particles, since, according to calculations, in the first seconds the number of particles at the beginning of the channel, so just after the reservoir, increases very sharply. Nevertheless, this effect can be considered negligible after 30 minutes of diffusion, since the particles found in the different domains from 1787 𝑠 to 1800 𝑠 after the release is almost constant.

(a) (b)

(c)

55 Regarding the lateral diffusion, the final layout is shown in Figure 3.3 (a). There are two inlets of 50 𝜇𝑚 of width: the buffer solution comes from top and the sample from bottom, they join in a central 100 𝜇𝑚 x 400 𝜇𝑚 channel, that is provided with ten reference and the same number of sensing domains (50 𝜇𝑚 x 20 𝜇𝑚) just after the junction. The difference between the number of particles in two adjacent volumes can be interpreted as an indicator of the lateral flow of the solution into the medium. In other words, with reference to Figure 3.3 (b), the variable of the x axis refers to the position index of a pair of domains having in common the same x extension, as sketched in Figure 3.3 (a), but located after sample inlet and buffer inlet respectively. The farer particles travel along the channel, the more uniformly they will distribute in transversal direction, obtaining an equilibrium after about 200 𝜇𝑚, where the difference is zero, as reported in Figure 3.3 (b) for 10 𝑛𝑚 of diameter.

The geometrical parameters have been optimized to maximize this effect and the speed of the response but taking into account the constraint imposed by the fluidic fabrication procedure and also the desire to use UV lithography patterning for magnetoresistive structures and their connections.

Figure 3.3: Lateral diffusion geometry (a) and calculated diffusion (b and c). (a) (b)

(c)

56 After this calculation, also in this case, the diffusion of two species with different hydrodynamic radius has been simulated, in terms of difference between the quantities obtained for different pairs of references/sensing transducers, holding as reference the first one (close to the end) evaluating any time the values on another pair respect to the fraction of bounded with analytes particles and the total introduced. It’s interesting to note that after 100 𝜇𝑚 from the junction of the inlets, the results obtained from the pairs six to ten are slightly similar, suggesting that there is no need to build very long channels to increase the resolution of the system, since at this distance there will be an equilibrium in both sides, so with reference to Figure 3.3 (c), the signal Delta16 shows a sensitivity of 1.7 𝑁/%.

In this case it has also been calculated that is not necessary to be extremely precise for the particles’ release, but the “safety delay” which doesn’t affect the measurement is about a half of the one in the previous method.

Figure 3.4: Velocity field calculated for the hypnotized hydrodynamic focusing scenery (a) and its magnitude in transversal direction respect to the central channel (b).

(a)

(b)

57 The hydrodynamic focusing is a technique based on a three-inlet geometry: in the central channel it flows the solution to be analyzed, while the external ones provide the buffer able to spatially concentrate the sample in transversal direction (perpendicular to the walls). To simulate this approach, it was necessary to implement first the module Creeping flow, which solves the following equation:

𝜚𝜕𝒖

𝜕𝑡 = ∇ ⋅ [−𝑝𝐈 + 𝜇(∇𝒖 + (∇𝒖) ] + 𝐅, (3.3)

that represents the Navier-Stokes relationship without the convection term, which is reasonable in this case considering that the Reynold’s number of the system is less than 1 with dimension and speed introduced for this situation. At the end, the results are the velocity field 𝒖 (which magnitude is plotted in Figure 3.4) and the pressure 𝑝 of the fluid, which drive the diffusion equation. During this procedure, the species experience a strong gradient of concentration from the central zone to the walls, which results in a flow assisted lateral migration of the particles that should amplify the effect shown in the previous case.

The best layout obtained is shown in Figure 3.5 (a). During the development of the model related to this situation, it was necessary to study in a systematic way two parameters: the angle between the focusing inlets and the channel, for which 60° has resulted the best option and also the ratio between the flow rates of buffer solution and sample. About the geometry, the dimension has been kept in the range of effectiveness of fast and simple UV lithography for fabrication, also because by reducing the flowing size, the speed of the fluid increases and longer channels would be necessary to see the defocusing effect, since the drag pushes the solution strongly. As in the previous case, the difference between the number of particles between two adjacent domains (sensing and reference) has been considered as the indicator of the lateral migration of the MNPs, but in this case better results have been obtained with more spaced array elements, since there was no saturation of the signal until hundreds of microns.

The calculation depending upon the fraction of bounded particles was also the same, as it is shown in Figure 3.5 (c). It can be noticed that the excursion reach its maximum for the pair 1-5, so for a distance of 300 𝜇𝑚 between the array elements, but in any case the sensitivity obtained is so low compared to the other scheme simulated, since it is 0.32 N/% in this case,

58 furthermore it has been evaluated that this model is very sensitive to the instant of particles’

release, as resulted from many calculations performed, in which the time of particles’

introduction has been changed, obtaining different results. On the other hand, this layout is probably more stable in the “real world”, since in this case the system is always at the same pressure in a dynamic equilibrium and the introduction of low concentrated particles does not affect it.

Figure 3.5: Hydrodynamic focusing geometry (a) and calculated diffusion (b and c).

It is important to notice that not only the performances itself can help in the choose of which is the best fluidic layout for this project, since other factors should be considered. First, the magnetic field due to the particles should not saturate the sensors, in order to be in a response region with enough sensitivity to discriminate between different mix of populations. This point is also related to other part developed through MADIA project, as the before mentioned aggregation and concentration chamber. The stability of the measurements is also important, for example regarding possible imperfection during the time sampling, since it can lead to avoidable errors. Another point is related to the potential mass production of the device, since, for example, it is better to have a single readout module that switches along the array elements than many instruments that read at the same time any magnetoresistive structure if the

(a) (b)

(c)

59 difference in terms of performance is negligible. For this reason, the hydrodynamic focusing layout has been discarded, since it implies more complexity than the others.