Device Characterisation by Simulation

To start the characterisation of the our microfluidic device, both the measuring chamber as well as the mixing channel were simulated using COMSOL Multi-physics^iv. For the geometry of the combined device, the reader is referred back to Fig. 3.3. The measuring chamber was investigated towards its velocity profile and the expected shear rates above the gel. This shear rate was then taken as the shear rate experienced by the platelet during the experiments and the goal here was to be in a shear rate regime as found in larger human veins. Later on, the experiments were conducted using three different flow rates, namely 300 µL/h, 500 µL/h and 700 µL/h. Another flow rate, 1000 µL/h, was tested concerning the attachment rate but not employed during experiments. In Table 7.2, the an-alytically calculated mean velocities within the measuring chamber are given for all four tested flow rates. Here, we assumed that the gel was 3 mm wide and 30µm high while the chamber itself was 4 mm wide and 110µm high, which cor-responds to the average dimensions of both the substrate as well as the chamber.

For the same dimensions, the Reynolds number is included in the table.

ivCOMSOL AB, Stockholm, Sweden

Blood Platelets Under Flow Conditions 7.3 Simulation of the Velocities and Shear Rates Inside the Measuring Chamber

To be able to calculate the velocity profile, the Navier-Stokes equation (Eq. (3.17)) was solved using the finite element method. Given the analytically calculated Reynolds numbers, we expected the flow profile to be laminar,i.e. resembling that of a parabola. As the substrate did not span the entire chamber, two different profiles were anticipated, one between the substrate’s surface and the chamber ceiling and another between the glass slide and the ceiling. For simplicity, we assumed that the substrate was completely centred inside the chamber’s chan-nel. This resulted in the geometry to become symmetric with respect to a vertical plane through the chamber’s middle. Hence, only half of the chamber had to be simulated, compare Fig 7.8A. For all simulations, we assumed that the platelet-thrombin-buffer solution, from a fluid dynamics point of view, behaved as water with its corresponding viscosity and density. This was justified by the fact that water was the major component of the liquid. Note that water is a Newtonian fluid. For the simulations, the physical properties of the chamber and substrate are found in Table 7.3. We assumed a rectangular chamber, disregarding the rounded endings of the actual chamber. As these endings were avoided during assembly of the chamber to reduce the chance of the substrate being positioned underneath the PDMS cast, this simplification was warrantable. Overall, the geometry of the ending had no influence on the flow profiles at the measuring site in the middle of the chamber.

Table 7.3.: The values used for the simulation of the measuring chamber. Note that the actual geometry size is given while the simulation was conducted on half of the geometry, using symmetrical boundary condition.

Physical Property Value Chamber height 110µm Chamber width 4 mm Chamber length 29 mm Substrate height 30µm Substrate width 3 mm Substrate length 24µm Inlet radius 0.19 mm

Solving the Navier-Stokes equation for our geometry, we did indeed observe a laminar flow regime. For the flow rate of 700µL/h, the flow profile around the border of the substrate is depicted in Fig. 7.8 B. The laminar flow is seen both before the substrate as well as above it. As we already noted in Table 7.2, the

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Figure 7.8.: A The geometry used for the theoretical characterisation of the ow chamber. The upper sketch shows the cross section along the width of the channel (gel - yellow with green beads, channel walls - gray and glass slide - blue). Note that the problem is symmetric towards a plane in the middle of the channel, denoted here by the black line. In the lower sketch, the cross-section along the length is sketched, denoting the geometry used for the simulations, containing the inlet and outlet (blue with black arrow). B The laminar ow velocity prole at the border of the substrate for a ow rate of 700µL/h. The substrate edge is marked by white lines. The shear rates directly above the gel (height 30µm) along the centre of the chamber, from inlet to outlet, is shown for 300 µL/h and 700 µL/h in C and D, respectively. The high peaks in the graph denote the positions of, from left to right, the inlet, the rst gel edge, the second gel edge and the outlet. The linear scaling of the shear rate with the ow rate is illustrated when comparing the values above the substrate for C and D. Note that panel C was adapted from Hanke et al. [36].

velocity scales linearly with the flow rate. Hence, all velocities scale accordingly in the flow profiles and are thus not shown here. From the velocity profiles, the shear rate was directly derived according to Eq. (3.19). For the lowest and highest flow rate utilised during the measurements, the shear rates directly above the gel are shown in Fig. 7.8 C and D, respectively. The shear rate was taken along the length of the entire chamber, from one narrow side to the other. The dimensions from the substrates are noted by the inner high peaks in the graphs, from 2.5 mm to 26.5 mm. From the simulations, the shear rates are calculated to be approximately between 14 s⁻¹ and 33 s⁻¹ for the flow rates of 300 µL/h and 700 µL/h, respectively. Again, the shear rate scales linearly with the flow rate.

Blood Platelets Under Flow Conditions 7.3 Previously, the shear rates in larger veins have been reported to start at values of

5 s⁻¹ to 10 s⁻¹ [91, 99]. Hence, the chamber dimensions and the chosen flow rates allowed us to mimic the flow situation found in human veins. At the same time, the size of the substrate was large enough to guarantee that spots for recording of attached platelets were found.

Simulation of the Diusion and Transport Processes Inside the Mixing Channel

We have now shown that the measuring chamber we designed fulfilled our phys-ical criteria such as the desired flow rate. Its practphys-ical application is presented in the following section. Before the actual experiments, the mixing device was also characterised. The mixing channel was introduced into the set-up to satisfy two purposes. On one hand, within the device, the thrombin was mixed with the platelet solution such that the end concentrations were the same as used for the static experiments. On the other hand, by mixing the substances in an external device in lieu of the syringe ensured that the reaction time between them was of the order of minutes instead of hours. Due to the controlled mixing process, all platelets recorded at a given place on the substrate had about the same amount of time to interact with the thrombin, independent at which time point they were recorded. Here, we defined the time from the complete mixing of the substances to the entrance of the measuring chamber as the reaction time. Due to the chosen flow rates and the needed length of tubing to connect both devices, the reaction time was estimated to be under 5 min for all flow rates. In particular, given the inner radius of the connecting tubing as stated in Table 7.3 and length of about 12 cm needed to link the devices on the microscope stage, the dwell time was estimated to be about 2.7 min for the slowest flow velocity at 300µL/h. The dwell time in the mixing channel (the dimensions are noted in Table 7.4) after complete mixing was in the order of a few seconds. The reaction time created with this set-up correlated with the time difference we observed in the static experiments between the mixing of thrombin and platelets and the first attached cells on the substrate.

To simulate the mixing process within the mixing channel, we combined the Navier-Stokes equation with the convection-diffusion equation (3.21). All com-bined flow rates from the side inlets and the central inlet summed up to the flow rates used during the experiments. The thrombin solution injected into the side inlets was set to a concentration of 40 u/mL while the platelets were added at a concentration of 2·10⁷ cells/mL, similar to the static case. Hence, to regain the

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Table 7.4.: The values used for the simulation of the mixing channel. Note that the rst three entries are given for the actual device while the simulation was conducted on half of the geometry, using symmetrical boundary condition along the length.

Physical Property Value

Channel height 180µm

Channel width 200µm

Channel length from mixing region 22.3 mm Channel length in simulation 15 mm

Platelet radius 1.5µm

Diffusion coefficient platelets 1.63·10⁻¹³m²/s Platelet concentration 3.32·10⁻¹¹ mol/m³ Diffusion coefficient thrombin [40] 8.7·10⁻¹¹m²/s Thrombin concentration 3·10⁻⁴mol/m³

original mixing concentrations of 4 u/mL thrombin in the platelets solution, the ratio between the flow rates of the single inlet was set to 1:18,i.e.a ration of 35-630-35µL/h for the highest flow rate used in the experiments. All physical quantities needed for the diffusion process, such as the diffusion coefficients, inlet concen-trations and the assumed radius of an un-spread platelet, are found in Table 7.4.

The diffusion coefficient for the platelets was estimated according to Eq. (3.24). To calculate the concentration of the thrombin, note that 1 NIH unit corresponds to 0.324µg of active enzyme and the molecular weight is 36 kDa.

The results of such a simulation can be found in Fig. 7.9. A sketch of the ge-ometry is found in A. Again, the problem was symmetric along the the central plane in length. As the channel is comparatively long, only a part was simulated as marked in the figure. First, we intended to ensure that the platelets were not exposed to such a high shear stress within the channel as to get activated without the addition of thrombin. The most critical flow rate here was the highest one, namely 700 µL/h. For the first part of the channel, around the position of the side inlet, the shear rate was studied as depicted in panel B. We observed that the maximal shear rate did not exceed 300 s⁻¹. This is a shear rate expected in smaller veins [77] and well below the shear rates found in healthy arteries by an order of magnitude. We thus concluded that an activation by pure shear stress was unlikely. Thus, we investigated the mixing process. We remarked earlier that we expect the platelets to diffuse from the central line towards the outer wall much slower than the thrombin for the reverse direction due to their difference in size. This was well seen in the simulations, compare panelC. While the thrombin

Blood Platelets Under Flow Conditions 7.3

Figure 7.9.: A The general structure of the mixing channel. The thrombin solutions enter from the side inlets, the platelets from the central inlet. We have again a symmetric problem, the symmetry line being along the length of the channel. The geometry is reduced for the simulations to the central part denoted with the dashed rectangle, suciently long to see complete mixture from the thrombin. B The shear rate distribution for the highest ow rate of 700µL/h. The shear rate does not exceed 300 s⁻¹. We thus assume no activation due to shear stresses in the mixing channel. C The concentration distribution of thrombin (upper part) and the platelets (lower part) for the rst few millimetres. While the thrombin solution diuses fast, the platelets do not show any noticeable diusion. D The concentration of thrombin along the central line of the channel. The stabilising concentration marks the even distribution of thrombin within the entire channel. All gures are taken for a ow rate of 700µL/h. The image is adapted from [36].

rapidly changes its concentration locally, the platelet solution does not. We thus define the point of complete mixture as the point where the thrombin concentra-tion stabilises along the central line of the channel. As shown in Fig 7.9Dfor the highest flow rate, this point is always reached within the first centimetre after the position of the side inlet. Hence, we can conclude that the solutions are always mixed completely inside the channel before entering the connecting tubing.

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