Device design and test
The design concept and a photograph of the whole microfluidic device are shown in Fig. 1A and 1B. The microfluidic device was manufactured by bonding the CO2 layer (upper) on the cell cultivation layer (bottom) using air plasma. Throughout the experiment, 5% CO2 was pumped into the upper layer and diffused into the bottom layer. Meanwhile, the whole microfluidic system was heated up to 37℃. These methods could maintain the vitalities of single cells trapped in microvessels.
Specifically, cell cultivation layer consists of four parallel identical microchannels with inlet, filter, trapping area and outlet as detailed in Fig.1C. The filter structures (20 μm in width) are designed to block cell clusters and large debris. And the trapping area is composed of a main square-wave shaped loop channel (40 μm in width, 20 μm in height, 40 loop and 7.2 mm long for each loop) and repeated triplet capillary-like straight channels positioned along both sides of the axis. For microvessels, there are two sizes available in our experiments. The width, height and the length of capillary-like channels are 7.5μm×6μm×40μm and 6μm×5μm×40μm respectively, which are comparable to conventional micropipette studies.
The experimental design is shown in Fig.1D. The flow through the narrow channels will carry single cells into the trap based on the principle of flow resistance [24]. In our design, cells with the diameters of less than 16μm could be captured when they flow on the side of trap units. On the other hand, the geometric parameters of microfluidic chip are fine-tuned to ensure that the inequation, , is satisfied, where is the flow resistance of narrow straight channels, is the flow resistance of loop channels. Therefore, the pressure differences applied to the cells in the narrow straight channels could be kept the same whether other narrow straight channels blocked by cells or not. After the period of cell loading, around 15 minutes in our experiments, sufficient numbers of single cells could be captured. Then predefined pressures would be applied to cells for 4 hours and their dynamic behaviors would be recorded using time-lapse microscopy. In our experiments, we applied pressure differences , 50 mbar, 100 mbar, 200 mbar or 400 mbar, to the four channels in one device, corresponding to pressure drops , around 63 Pa, 125 Pa, 250 Pa or 500 Pa on captured cells. As a comparison, the blood pressure in the human body ranges from 3 mmHg to 120 mmHg, and the lymph pressure could be as low as 1 mmHg, that is, 133 Pa.
To ensure the effectiveness of microfluidic device, preliminary experiments and simulations were performed to test the occupied rate of trap units and estimate the pressure drops on isolated cells. As shown in Fig.S1A, the occupied rate of trap units increases with the cell density during the period of cell loading. As illustrated in Fig. S1B, C, simulations of fluid flow using COMSOL (COMSOL Multiphysics) software indicates that as the pressure difference applied on the whole chip increases geometrically, the pressure drops exerted on single cells increase in a similar manner with an acceptable deviation from defined expected pressure drops. In our experiments, the occupied rate can be controlled at about 20-50% using suitable cell density (Fig S1A). Under these conditions, the design of our microfluidic device could meet the experiment requirements to compare the behaviors of cells under predefined pressure differences in this work.
Under the fixed pressure, cells enter the microvessels, clog the flow, and bear the force. The cell width could fill the microvessel width in most cases to guarantee the force exerted on cells as expected. Using the microfluidic chips demonstrated previously, a large amount of data about the dynamic behaviors of single cells at fixed pressures were recorded. Customized MATLAB codes were used to identify and analyze these dynamic behaviors of the single cells across the microvessels. In our experiments, the protrusion length L of single cell is defined as the length of the trailing edge of the single cell into the microvessels. Focusing on the period that cells enter the microvessels and reach the ends, two kinds of patterns of dynamic behavior were recognized through data analysis as demonstrated in Fig. 1E, F through linear fitting (see Movie S1 and S2 for details). One pattern displays excellent linearity (R-squared about 0.96) suggesting the Newtonian droplet state, and the other elucidates some cells might behave in a complex and nonlinear manner after adhesion with microvessels.
Dynamic patterns under fixed pressure drops
All the dynamic behaviors of cells under all different pressure drops could be observed in our experiments as depicted in Fig. 2 and Fig. S2. As described above, most of the data seems in a mess and has no obvious pattern, yet detailed analysis indicates that a portion of single cells show excellent linearity and the rest behave nonlinearly. In previous researches about the responses of single cells under high stresses in seconds, most results have emphasized that single cells should be considered as elastic or viscoelastic material [5, 25]. That is, the linear elastic solid model was applied widely and worked well in this field. Nevertheless, our research demonstrates viscosity plays a dominant role instead of elasticity under low mechanic stresses for minutes to hours. A classification was performed based on R-squared of linear fitting, as demonstrated in Fig. 2. The classification threshold 0.85 was chosen to discriminate the “linear” cells from the “nonlinear” cells. Obviously, the ratio of the “linear” cell number to the number of total cells decreased from 99% to 48% when the pressure difference decreased from 400 mbar to 50 mbar. More cells exhibit complicated and nonlinear behaviors under lower pressure drop.
For most adherent cell types such as cancer cells, adhesion time scale ranges from tens of minutes to several hours [26]. And the dynamic behaviors of adherent cells could be tuned by related proteins, such as integrin, talin [2-3]. It seems to indicate that in our experiments, cells are likely to switch from the suspended state to the adherence state. The biochemical reactions during adherence and random migration might make a significant contribution to cell traverse-vessel behaviors for the “nonlinear” cases.
Modified Newtonian droplet model
Basic Newtonian droplet model regards the single cell as the Newtonian droplet [25], and the behavior of these cells could be elucidated using a linear differential equation. It means that the viscosities of these “linear” cells remain constant during the whole dynamic process. This model could only explain the linear behaviors of single cells and have no explanatory ability for the other “nonlinear” cells. As the concept shown in Fig. 3A, these “nonlinear” cells elongate in the microvessels and may adhere to the microvessels with time, then they behave more randomly.
To describe these complicated and nonlinear dynamic patterns, we illustrated the dynamic behavior mechanism as shown in Fig. 3B and modified the original Newtonian droplet model (details given in the Supplementary Information):
![](data:image/png;base64,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)
to:
![](data:image/png;base64,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)
where is the protrusion length of the cell into the microvessel, is the applied pressure to the cell, is the area of the cross section of the microvessel, Rf is the resistance corresponding to the critical excess suction pressure because of the capillary effect, ηint is the intrinsic viscosity of the cell, is the equivalent radius of the cross section of the microvessel.
In the modified model, α·A(t) is introduced into the formula according to the cell adhere effect may greatly contribute to overcome the pressure difference, where A(t) is defined as the degree of cell adhesion over time (Supplementary Fig. S3A), and α is the constant coefficient between cell adhesion and additional viscosity. f(t)·g(L) is used to describe the active force generation during the whole traverse-vessel behavior, where f(t) is the value of the active force, and g(L) is tunable factor. Here, we assumed that the active forces f(t) generated by cells are distributed randomly in the allowable range from –A(t) to A(t). And g(L) depicts the effect of the protrusion length on active force generation. The simulation parameters are determined based on the previous studies (details given in the Supplementary Table S1).
As shown in Fig.3C, these simulation results are in agreement with our experiments mostly. Under low pressure drop, cells behave like Newtonian droplet before adhesion and the nonlinearity of cells become more pronounced during adhesion. Most cells behave in an excellent linear manner under high pressure drop. That is, the active force generation sourced from biochemical reactions has prominent influence on the cell behaviors during the traverse-vessel process. In order to confirm our conjectures, this parameterized model was used to simulate cell behaviors in narrower microvessels. As expected, Fig. 3D illustrates that more cells behave “nonlinear” in narrower microvessels under the same pressure drop. Detailed data analyses demonstrated that our model could explain most experimental phenomena for two sizes of microvessels (Supplementary Fig. S4, Fig. S5).
Distribution analysis of cells’ behaviors illustrated mode switch
As discussed in the model and equations (1) and (2), the cells may act like Newton droplet phase and transform to adhesion/migration phase with the increase of contact time. Besides R-squared, we also used apparent viscosity(ηapp) to indicate the cell behavior changes. The apparent viscosity(ηapp) is defined as:
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As shown in Fig. 4A, under the constant pressure drop, the major peak (blue) shifts towards higher apparent viscosity indicating the apparent viscosity of cells gradually rises. Meanwhile, a weak peak (pink) at greater apparent viscosity occurs and increases in the proportion of total over time. This behavior is more remarkable under the lower constant pressure drop, while under the higher pressure drop cells often traverse through the capillary before they could adhere to their surrounding environment. The R-squared distribution in Fig. 4B could also offer support for this conjecture. Under the lower consistent pressure drop, cells mostly show a relatively great linear behavior within dozens of minutes, then cells progressively make the transitions from Newtonian droplet state to adhesion/migration state. Finally, cells completed the transitions and exhibited non-linear behaviors. All the distributions of apparent viscosity and R-squared within four hours can be found in the Supplementary Fig. S6 (experiment) and Fig. S7 (simulation).
To further confirm the transition and compare experiments with theories, the numbers of cells with specific R-squared and apparent viscosities in experiments and simulations are drawn in Fig. 5A, B. It is worth noting that Fig. 5A counts cells between 20 and 40 minutes and Fig. 5B counts cells between 40 and 80 minutes both under the pressure difference 50 mbar. These results indicate a negative correlation between the apparent viscosities of single cells and R-squared regardless of in experiments or simulations. Besides, the heatmaps also show two evident clusters in the upper left and the lower right suggesting the existence of two potential distinct states for cells. One state with lower apparent viscosities and higher R-squared close to 1, and the other with higher apparent viscosities and lower R-squared close to 0. To eliminate the doubt about whether the number of single cells is enough for proving this conjecture, a heatmap depicting the accumulation of the numbers of cells during different periods is shown in Fig. S8. And the comparison of Fig. S8A and Fig. S8F could elucidate the transition between these two states over time. Similar results are also observed in simulations as Fig. S9 shown. Taken together, it is assured that cancer cells could switch their state from Newtonian droplet state to adhesion/migration state.
Throughout these transitions, apparent viscosities of cells increased initially and then leveled off with time indicating cells attaches to the surrounding environments completely as shown in Fig. 5C. Meanwhile, Fig. 5D suggests that the dynamic behaviors of cells are influenced by cell adhesion and become more nonlinear than before. Finally, considering the residual cells before traversing the microvessels, the ratio of these cells to total implies two potential distinct stages, a rapid decline at the initial stage and a steady decline at the latter stage, as plotted in Fig. 5E using dotted lines, which also suggests the mode switch. From the beginning to the end, the biochemical reactions have little effect initially while dominate these transitions finally. Experiments and simulations both verified these phenomena qualitatively, although they might not match perfectly from a view of quantitative because elapsed time of cell loading cannot be guaranteed to be the same in all experiments and parameters chosen in simulations might not be best suited for all experiments.
Further discussions for pressures and vessel sizes effect for the cell behaviors
The dynamic behavior of cells in the microvessels with different sizes under different pressure applied is summarized and compared in Fig. 6 to verify our parameterized model further.
In order to compare the R-squared and the apparent cell viscosity of the cell traverse-vessel behaviors at different conditions in experiments with simulations, simulations were performed densely under the pressure drop which ranged from 63 Pa to 500 Pa. Fig. 6A shows that the median value of the fitted R-squared of all cell behaviors with different pressure drops. With the pressure drops decreased, the R-squared decreased from about 1.0 to a pretty low value, ~0.8 for microvessels with 7.5μm×6μm×40μm and ~0.4 for microvessels with 6μm×5μm×40μm. The smaller the microvessel size was, the larger decrease of the median value of R-squared would be. In contrast, Fig. 6B demonstrates the median of the apparent viscosities ηapp of cells increase greatly from ~3000Pa·s as the applied pressure decreased. The value of cell apparent viscosities matches with previous studies approximately [27-29]. Experiments are in good agreement with simulation results in Fig. 6A, B.
To illustrate the core concept, the schematics of two kinds of dynamic patterns are depicted as shown in Fig. 6C. Before adhesion, cells could linearly traverse the capillary-like vessels and their apparent viscosities ηapp are approximate to real viscosities ηint of these cells. On the other hand, if pressure is too low or the microvessel is much smaller, the cells would adhere to the microvessels and generate active force to speed up or slow down their migration. After adhesion, these cells behave randomly and nonlinearly, then the apparent viscosities ηapp of these cells are much larger than their real viscosities ηint. In this state, these cells could maintain their physiological conditions and seed in current sites, which means that cells might become more aggressive and form a metastatic tumor. In other words, if we could reduce the viscosities of cancer cells or the adhering force between cancer cells and the microvessels, the cancer metastasis might be restrained to some degree.