Inertial Focusing for the Preferential Sorting of Tumor Spheroids - Owen O'Connor, Michael Beauregard, Uday Prakhya
Inertial focusing is a passive technique used in microfluidics for particle ordering or separation. As opposed to passive techniques, active techniques require an externally applied force which causes a time delay at higher flow rates. One major advantage of inertial focusing is its ability for high throughput (high flow rates). Inertial focusing has many applications including the ordering or separation of cells needed for biological research. Particularly, circulating tumor cells (CTCs) found in the blood are predictive signs of metastasis which will help the doctor utilize the appropriate treatment. Due to the low concentration of CTCs in the blood, the process of filtering the CTCs out of the blood needs to be precise. Inertial focusing has served as a possible solution to detect CTCs in the blood . Another use of this technique is to sort groups of cancer cells to be used for high throughput cancer drug testing. Inertial focusing could be used to produce groups of cells of the same size in order to provide for more consistent results.
Explanation of Forces
Wall Interaction Forces
A particle flowing through a channel will exhibit interactions with the wall. As the particle gets close to wall, the flow near the wall is restricted, building up pressure. This forces the fluid to go around the particle at a faster velocity away from the wall. This increase in velocity decreases the pressure creating a pressure gradient force the particle away from the wall. This force scales with particle diameter raised to the sixth power. In addition, this force increases inversely with the normalized distance of the particle from the wall. The walls also cause the particle to lag behind the flow and possibly rotate .
Shear Lift Gradient Force
When a fluid is flown through a channel, a parabolic velocity profile will form due to no-slip conditions on the wall. The flow will be greatest in the center of the channel and weakest at the walls. As a result, a particle will experience different forces laterally. The magnitude of the particle velocity relative to the fluid velocity increases as the particle gets closer to the wall. This increase in relative velocity creates a pressure gradient away from the wall. As a result, a particle will get pushes away from the center of the channel until it reaches an equilibrium force with the wall .
Inertial Focusing in straight channels In straight channels, only the lift forces are present. As a result, equilibrium positions will form where the shear lift gradient force and wall interaction force are equally opposing. Particles will focus in equilibrium positions on the channel faces. Since the wall interaction force scales more with particle diameter, larger particles will focus closer to the channel center .
Secondary Flow Drag Force - Deans Drag
A fluid flowing through a channel will impart a secondary flow drag force that pushes the particle. This force acts parallel with the fluid velocity whereas the wall interaction force and shear lift gradient force act perpendicular to the flow. This force scales with the particle diameter and particle migration velocity. When a curved channel is introduced, two symmetrical counter rotating vortices will form due to this Dean drag force. Due to the centrifugal effects, the fluid on the outside of the channel travels faster than the fluid on the inside. This causes a decrease in pressure on the outside of the channel creating a pressure gradient. Due to the fluid being encapsulated, it recirculates around the outside of the channel. The Dean drag force will act towards the inside and outside of channel depending where the particle is in the y-direction. At the top and bottom of the channel, the Dean drag will push the particle towards the inside of the channel. At the middle of the channel, the Dean drag will push the particle towards the outside of the channel .
Inertial Focusing in straight circular channels
In 1961, the effects of inertial focusing was first tested by Sigre and Silberberg . Particles with diameters varying between 0.8 mm and 1.6 mm were flown through a round pipe with a diameter of approximately 1 cm. It was demonstrated that the particles would focus to an equilibrium position 0.6 times the radius of the pipe. In addition, it was shown that this focusing effect was proportional to the length of the pipe, to the velocity of the fluid and to the fourth power of the ratio of particle diameter to pipe diameter.
Inertial Focusing in straight rectangular channels
Kagalwala et al. designed a microfluidic device with straight rectangular channel to separate particles by size . Since the channels are straight, no Dean vortices exist at any flow rate. As a result, the particles will equilibrate at positions where the wall interaction force and the shear gradient lift force equally oppose each other. Since the wall interaction force scales more with particle diameter, the larger particles with focus closer to the middle of the channel. Over 1500 µm, Kagalwala et al. was able to successfully separate 2 µm and 4.16 µm beads. Since the 4.16 µm beads had a larger diameter, they focused closer to the channel center.
Zhou et al. investigated the positive and negative lift coefficients in straight rectangular channels to better understand why particles focus to two equilibrium positions within low aspect ratio channels . A two-stage mechanism was proposed to explain this phenomenon. In the first stage, the particles find equilibrium positions at the top and bottom of the channel due to negative lift. In the second stage, a rotation induced lift force, also known as the Saffman lift force, slowly pushes the particle to the center of the channel. This rotation induced lift force becomes significant near the channel wall where there is increased shear. Near the channel center, this force is negligible. When the shear lift gradient force and wall interaction force cancel out, the rotation induced lift force becomes dominant pushing the particle towards the center of the channel. This theory provides an explanation for particles focusing in four equilibrium positions on the channel faces in straight channels and particles focusing in two equilibrium positions in the center of the channel in low aspect ratio straight channels.
Multiple studies have been conducted in the use of asymmetric channels for inertial focusing. Asymmetry in the channel geometry, coupled with a low aspect ratio, leads to an imbalance between the shear lift gradient force and the wall interaction force. This imbalance causes particles to focus to one point along the channel, rather than the two points observed in low aspect ratio, symmetric channels. Figure 8 shows the relationship between the dean number of the flow regime and the ratio of the particle diameter to the hydraulic diameter. The yellow shaded region indicates the properties that yield focusing to one or two points .
Figure 8. Size dependence on particle ordering. Results for focusing to a single streamline as a function of the Dean number of the flow and the ratio of particle diameter to channel hydraulic diameter (a/D h) are plotted. No focusing or focusing to four streams corresponds to filled squares, focusing to two streams corresponds to open triangles, focusing to a single stream is represented by open circles, and more complex behavior is shown as filled triangles. Data for this graph were collected by using various-sized particles and channel geometries with a fixed length of 3 cm. 
Channel Expansion and Contraction for Particle Sorting
Park et al. describe a method of particle sorting that works by inducing vortices caused by changes in channel width. A spherical particle that is neutrally buoyant encounters hydrodynamic lift forces caused by inertia, thus the particle experiences a velocity perpendicular to the primary direction of flow, pushing the particle toward the wall. As the particle moves down the channel it achieves an equilibrium position approximately 0.62R away from the center of the channel. This phenomenon is called the tubular pinch. Manipulating the width of the channel creates a low pressure zone on the wall side of the particle, creating a vortex which sweeps particles toward the walls of the channel (Figure 9). Results are shown for experiments run at different flow rates with particles of constant size in (Figure 10). At low particle reynolds numbers, particles focus to two streams after multiple expansions and contractions .
Asymmetric curved channels for separation of CTCs from Blood
Circulating tumor cells CTCs are shed from primary and metastatic tumors and sent through the bloodstream to attach to other parts of the body. CTCs are the mechanism by which cancer metastasizes to other organs. Capture and analysis of CTCs could improve the capacity for early detection of aggressive cancers and improve treatments.Once detected, characterization of CTCs has been applied to prognosis of breast, prostate, and colon cancers, and to predictive markers for targeted drug therapy in lung cancer. However, the limited sensitivity of current approaches combined with the complexity of the disease has prevented the broad use of CTC-based diagnostics. Ozkumur et al. created a microfluidic device for the isolation of CTCs from blood. Their device, the CTC-iChip, (Figure 11) combines hydrodynamic focusing, inertial focusing and magnetophoresis to separate CTCs treated to display magnetic beads through specific antigen binding, from whole blood. The first stage separates CTCs and white blood cells from red blood cells, platelets and other components. The buffer containing white blood cells and CTCs are then inertially focused using a curved channel to align then to a single focal point. Upon exiting the asymmetric channel, the CTCs and while blood cells are already inclined to separate to do the imbalance of forces and the differences in size. A magnetic field is then applied to supplement the separation by interacting with the magnetic beads bound to the CTCs. They were able to achieve sorting comparable to that of standard CTC identification methods. In fact, in cases of low CTC concentration in blood samples, the CTC-ichip outperformed the control method in 22 of 36 trials, with the remaining 14 having CTC concentrations too low for either method to detect .
In spiral channels, two counter rotating dean vortices form in the channel that are directed towards the inside of the channel at the top and bottom of the channel and towards the outside of the channel in the middle of the channel (Figure 12). On the inner half of the channel, the shear lift gradient force opposes the Dean drag force creating an equilibrium position. Since these equilibrium positions are far away from the wall, it is assumed that the wall interaction force is negligible. No focusing occurs in the outer half of the channel since the shear lift gradient force and Dean drag force are both directed towards the outside of the channel .
Spiral Rectangular Channels
Martel et al. investigated how channel geometry affected the dynamics of inertial focusing in spiral channels . The width of the channel was varied from 50 µm to 400 µm while the height stayed constant at 50 µm. The aspect ratio had a range of 1:1 to 1:8. As the DH increases, the shear lift gradient force decreases due to the velocity profile flattening out whereas the Dean drag force increases. It was observed that the minimum flow rate for particle focusing occurred in the device with the smallest DH. As the flow rate was increased, particle focusing was seen in all the devices. In addition, the increase in flow rate cause the focusing position to shift up away from the wall of the channel. This occurred since Dean drag force scales faster with flow rate than the shear gradient lift force does. In the wider devices, at high velocities (~2 m/s), the streak width increased inferring a decrease in particle focusing. This result was not fully understood. However, a correlation was found between streak width and streak position. Streak width was minimized towards the channel wall due to the shear lift gradient force dominating the Dean drag force.
Most papers that used inertial focusing in spiral channels used the inside of the spiral as the inlet and the outside as the outlets. Due to the geometry of a spiral, the radius is smallest in the beginning and increases with each consecutive turn. As a result, the Dean’s number will decrease as fluid is flown through the device. Alternatively, if the middle of the spiral is the outlet and the outside of spiral was the inlet, then the Dean’s number would increase with each turn. Currently, scientists use the middle of the spiral as the inlet due to the ease of fabrication of the outlets where extra space is needed for the channels to be splayed. Martel et al. also investigated the effect of increasing and decreasing Dean’s number . The increasing Dean’s number device had smaller streak widths which can be accounted for the strong Dean drag force at near the outlets. In addition, the decreasing Dean’s number device achieved particle focusing at lower flow rates in the wider devices. Overall, there were little differences between the increasing and decreasing Dean’s number devices.
In another paper, Martel et al. examined the effects of Reynolds number and Deans number . Particles of three sizes were tested in devices while varying the radius of the curvature and increasing flow rate or Reynolds number. The independent variables included the confinement ratio (between 0.066 and 0.225), the curvature ratio (between 0 and 0.0166) and Reynolds number (between 0 and 400). The particles were observed to migrate equilibrium positions either closer to channel wall or centerline. As the curvature ratio increased at a constant Reynolds number, the particles migrated closer to the channel centerline in the y-direction where the Dean drag force is directed towards the outside. This occurred due to the increase in Deans number altering the velocity profile decreasing the shear lift gradient force. The particle moved outward until Dean drag force was equilibrated with the shear lift gradient force. Since the shear lift gradient force scales with particle diameter to the third power, a large curvature ratio is necessary to decrease the shear lift gradient force in order to push the larger particles towards center of the channel in the y-direction. As Reynolds number was increased at a constant curvature ratio, the particles migrated closer to the channel wall. Since the particles were in the Dean drag regime that was directed towards the inner wall, the increase in flow rate pushed the particles closer to the wall which was equilibrated with the wall interaction force. These results seemed to contradict previous results where all the focusing occurred in the middle of the channel in the y-direction where the Dean drag regime was directed towards the outer wall. Martel et al. explained the anomaly by proposing that the particles remained in the dean drag regime at the top and bottom of the channel due to the strong shear gradient. However, this does not agree will all the data but does provide a possible explanation to the results.
Hasni et al. was able to separate large-sized particles with a diameter of 40 µm and 60 µm . The microfluidic device consisted of a 5 loop spiral design with a height of 220 µm and a width of 500 µm. The minimum flow rate for particle focusing was observed at 1 mL/min. Particle focusing was observed on the inner half of the channel. The 60 µm beads focused closer to the channel wall. As the flow rate was increased, the beads focused further away from the channel wall.
Russom et al. compared the particle focusing experiments on asymmetric curved channels from DiCarlo et al. with spiral channels . Similar results were obtained. Guidelines for spiral channels include a/ DH>0.1 and 20<De<1. At low Dean’s number, no focus was observed most likely due to insufficient lift forces and Dean drag forces. At high Dean’s number, multiple streams were observed. At intermediate Dean’s number, focusing occurred at one equilibrium position due to the shear gradient lift force dominating the Dean drag force. When flow rate was increased, particles eventually unfocused due to the Dean drag force increasing faster with flow rate than the shear gradient lift force. Also, the density of the particle and fluid was found to be negligible confirming the results from DiCarlo et al.
Kuntaegowdanahalli et al. tested the effects of channel height and Deans number on particle focusing . 10 µm beads were flown through a device with channel dimensions of 500 µm wide, the height varying between 90 µm and 140 µm. As the flow rate increased, the equilibrium positions were shifted towards the center of the channel. This occurred because the shear lift gradient force and the Dean drag force scales with fluid velocity but the lift coefficient decreases. As a result, the Dean drag forces dominate the shear lift gradient force pushing the particle closer to the channel center. As the channel height was increased, the equilibrium positions was also shifted away from the channel wall due to the Dean drag force scaling with hydraulic diameter. Kuntaegowdanahalli et al. was able to use the device to separate 10 µm, 15 µm and 20 µm beads with a 90% efficiency. Consistent with the theory, the 20 µm beads focused closest to the channel wall due to its large particle diameter.
Spiral Trapezoidal Channels
Most spiral channels used for inertial focusing have a rectangular channel geometry. Guan et al. discovered that a trapezoidal channel geometry creates a higher resolution of separation of particles at a critical flow rate . At low flow rates, the particles focused on the inner side of the channel similar to the rectangular channels. However, when the flow rate reached a critical flow rate, the particles focused on the outside of the channel. At the high flow rates, the particles get trapped in the center of the dean vortices on the outside of the channel. The transition from equilibrium positions on the inner half of the channel to the outer half is size dependent. For example, the critical flow rate for the 15 µm beads was 2-2.6 mL/min while 20 µm beads had a critical flow rate of 2.6-3.4 mL/min. Since equilibrium positions are found on the inner and outer half of the channel, separation resolution is increased dramatically. 15.5 µm and 18.68 µm beads were separated with 92% efficiency. Although the trapezoidal geometry has shown the best separation resolution, it is new and little studies have been done on it.
Current limitations and Future Work
Current limitations include the need for low particle concentration (~1-2% by volume). All the force balances considered above assume no particle-particle interactions.
Future work would include more experiments on particle focusing in spiral channels. Although a lot has been done on spiral channels, the force balances are still not fully understand. Also, no one has made an empirical equation to predict where along the channel the particles will focus.
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