For the second set of experiments, we include a fluid flow within a channel to the force measurements of the platelets. The analysis of the forces exerted by the cells is conducted with the theory described above. However, to be able to design our flow chamber, we first need to understand the physical concepts that are of relevance in our set-up. This means we have to understand the fluid motion within a channel as well as the mass transport and diffusion under flow. In particular, we here use the concepts of microfluidics, the study of flow on small length scales.
Briefly, for our specific problem, our microfluidic unit is a combination of two different channels connected in series, sketched in Fig. 3.3A. In one channel, a sub-strate is embedded on which the platelets are able to attach and spread, enabling us to measure the exerted forces, compare Fig. 3.3BandC. This chamber mimics the "blood vessel". In an additional channel preceding the measuring chamber, the platelets are mixed with the trigger substance thrombin in a controlled fash-ion. This helps to avoid pre-mature triggering within a shared syringe or delayed activation within the measuring chamber.
3.3.1 Fluid Flow in Microuidic Channels
First, we study the fluid flow or fluid motion within an arbitrary channel. The fundamental equation describing the flow of incompressible Newtonian fluids is the so calledNavier-Stokes equationgiven as
ρ ∂v
∂t + (v· ∇)v
=−∇p+η∇2v+fext, (3.17) subject to the incompressibility condition ∇ ·v [44]. Here, ρ denotes the density, p the pressure, η the viscosity, fext the external forces and v andt as previously the velocity and time, respectively. On the left hand side of the equation, the single terms correspond to the fluid acceleration and the convection, representing the inertial forces. On the right hand side, the terms correspond to the pressure gradient and the viscous forces.
Depending on whether the viscous or inertial forces dominate the flow process, different phenomena within the fluid are observed such as laminar flow or the presence of turbulences. The relationship between these forces is characterised by
Fluid Dynamics 3.3
Figure 3.3.: A A sketch of the entire device designed. The platelets enter through the central inlet of the left device, thrombin is added through the side inlets. The mixing channel is connected through a connecting tube (dark grey) to the measuring chamber (light grey), containing a PAA substrate (brown with green beads). The measuring chamber is sketched in prole in B. On a glass slide (light blue), a PAA substrate stripe (brown) containing uorescent beads (green) is situated. Above it, a microuidic chamber is build from PDMS (gray, channel denoted with dashed lines). On the top, tubing is inserted into the PDMS, acting as inlet and outlet. On the gel, coated with brinogen, platelets can adhere (red). C A cross-section through the channel.
This gure is partially taken from Ref. [36].
theReynolds number
Re:= ρvDH
η , (3.18)
where DH denotes the hydraulic diameter. Note that the Reynolds number is dimensionless. ForRe>2300, turbulent flow may be observed while lower values indicate laminar flow [6].
From the velocity field of a Newtonian fluid, several other physical quantities can be derived. One of them is the shear rate, γ, which describes how one layer of fluid passes over a neighbouring fluid layer. Hence, it is proportional to the gradient of the velocityv
γ= ∇v+ (∇v)T (3.19)
and is measured in reciprocal seconds, s−1 [44]. We define the scalar norm of the shear rate as γ =
q1
2γ:γ such that a : b = ∑n∑manmbnm [15]. Further, from the shear rate, we can calculate the shear stress, τ. The shear stress represents,
Chapter 3 THEORY
analogous to the shear rate, the frictional forces between fluid layers [44]. Simul-taneously, if a body is inserted within the fluid flow, the shear stress results in a force acting on the body to deform it as depicted in Fig. 3.1 C. For Newtonian fluids, this corresponds to the linear relationship
τ=ηγ. (3.20)
For our specific case, we study a mixed solution of buffer, platelets and thrombin.
From a fluid dynamics point of view, due to the relatively low content of all other substances, this mixture may be approximated by its major component, namely water. Water is considered a Newtonian fluid, meaning we can apply Eq. (3.17) to study its behaviour. At the same time, the concept of microfluidics is used,i.e. the study of flow on small length scales. Generally, the usage of microfluidics has two major advantages. First, only limited amounts of substance is needed to measure the desired properties of the species. Second, due to the small length scales, the Reynolds numbers are often so low that a laminar flow regime is observed [6, 30].
These facts are employed in the construction of our measuring devices. In partic-ular, we later show that the chamber we design exhibits a Reynolds number of no more than 0.09 for the highest used flow rate during the experiments. While this is on the larger side of the Reynolds number usually associated with microfluidics (orders of 10−2or 10−3), it is still fairly low. Thus, we are well within the laminar flow regime.
3.3.2 Mass Transport in Microuidic Channels
Next, let us consider the mass transport within our device. In particular, as men-tioned above, we want to mix a cell solution with a trigger substance within an-other microfluidic device. Here, we use a three inlet, one outlet device. On one hand, this device mixes the solutions as controlled as possible. At the same time, the shear rate remains reasonable low as to not trigger platelet activation which is known to happen at high shear rates.
To study the mixing of the both substances, in addition to the fluid flow as described in the previous section, we also have to consider the transport of our chemicals within the channel. Here, two processes play a role; first the active mass transport orconvectiondue to the fluid flow and secondly the diffusion within the solution. The convection-diffusion equation for mass transfer is generally given
Fluid Dynamics 3.4 by
∂c
∂t =∇ ·(D∇c)− ∇ ·(vc) +R, (3.21) wherecdenotes the concentration of the studied substance,Dits diffusion coeffi-cient andR additional sources or sinks of c, meaning if a substance is created or used up during the process ([113], Chapter 3). The velocity fieldv is in our case given by Eq. (3.17). Note that the lower the diffusion coefficient is, the more the convection part dominates the behaviour of the substance. The diffusion coeffi-cient describes the movement of the particles in flow not driven by an active mass transport. It was previously stated by Einstein and Smoluchowski that it is given by
D=µkBT, (3.22)
where T is the temperature, kB the Boltzmann constant and µ is the mobility of the studied particle or molecule. At low Reynolds numbers, so in flow regimes as studied here, the mobility is proportional to the drag coefficient cd such that µ= c−d1. If we further approximate the studied particles by small spheres, it was derived by Stokes that
cd =6πηrp, (3.23)
withrp being the radius of our particle. Hence, it follows that D= kBT
6πηrp. (3.24)
For our experiments, we can directly see that the diffusion coefficient for the platelets is considerably larger than for thrombin due to their difference in size. In particular, the diffusion process of the platelets is so slow that an even concentra-tion within the channel is not reached. For the thrombin, this is not the case and an its even distribution over the entire channel width is taken as the point of equal activation for the cells. The thorough characterisation of this process is found in the results, Section 7.3.1.
Chapter 3 THEORY