Nomenclature
W,	micro-channel width, m;	u,	height-wise averaging velocity of the fluid, m/s;
W1,	width of Solution A in the mixing micro-channel, m;	Pe,	Petlet number, Pe = uW/D;
W2,	width of Solution B in the mixing micro-channel, m;	Deff,	effective solute diffusivity coefficient, m2/s;
ε,	ratio of W2 to W1;	p,	pressure, Pa;
H,	height of the mixing microchannel, m;	Q,	total flow rate in the microchannel, m3/s;
η,	fluid viscosity, Pa·s;	QA,	flow rate at Inlet A, m3/s;
D,	solute diffusivity, m2/s;	QB,	flow rate at Inlet B, m3/s;
u,	fluid velocity in the z-direction, m/s;	fv,	frequency of the pulsatile flow, Hz;
ϕ(x, z, t),	height-wise averaging concentration of the solution, mol/m3;	τw,	wall shear stress, Pa;
ϕ(x, y, z, t),	solution concentration, mol/m3;	ϕ0,	a reference value of the height-wise averaging concentration of the solution at Inlet A, mol/m3;
ϕ0(t),	height-wise averaging concentration of the solution at Inlet A, mol/m3;	fϕ,	dynamic biochemical signal frequency,

1 Introduction

The biological cells in vivo live in complex dynamic microenvironments, which include time-varying interstitial fluid flows and time-dependent concentrations of various biochemical substances^[1]. Therefore, they are exposed to different time-varying wall shear stresses induced by interstitial fluid flows, and are used to dynamically stimulate biochemical signals. Biological cells can recognize dynamic biomechanical and biochemical signals, and transmit them to the interior of the cells, which leads to different cellular responses involving the changes in intra-and inter-cellular signaling, cell morphology, gene expressions, and protein synthesis. These responses are regarded to be closely correlated with the cellular functions and behaviors such as cell spreading, division, proliferation, differentiation, apoptosis, and migration^[2-12].

Since in vivo microenvironments are very complicated, in vitro investigations can exclude many interfering factors. To date, microfluidics has become an effective experimental platform for simulating dynamic biomechanical and/or biochemical microenvironments and investigating the interactions between cultured cells and dynamic signals in their microenvironments in vitro^{[5-10, 13-20]}. Many types of microfluidic devices have been developed to generate the biomechanical and/or biochemical signals for specific cellular biological applications^[13-20], among which the shallow Y-shaped microfluidic channel (see Fig. 1) with high aspect ratio (width to height) has usually been adopted to generate the fluid shear stress and switch the biochemical stimuli between two types of biochemical substances^[19-20].

The understanding of the transfer characteristics of the dynamic signals being transported in channels is important for precisely loading the biochemical signals on the desired cells and cultured on the bottom of the mixing micro-channel. Li et al.^[21] analyzed the transportation of dynamic biochemical signals input from Inlet A in the mixing flow in a shallow Y-shaped microfluidic channel (see Fig. 1) by analytically solving the governing equations for the time-dependent Taylor-Aris dispersion and molecular diffusion in steady flows. It was concluded that, while dynamic biochemical signals were being transported in the steady flow, the mixing Y-shaped micro-channel was acting as a low-pass filter. It reduced the amplitudes of the dynamic biochemical signals being transported into the micro-channel and cut off their high frequency components due to the longitudinal dispersion. The low frequency components of the input signals could pass through the steady flow in the micro-channel, and no new frequency components could be found in the output signals. With transverse molecular diffusion, the magnitudes of the output dynamic signal were reduced compared with those without transverse molecular diffusion. However, if the dynamic biochemical signals are being transported into time-dependent flows, the interactions between the dynamic biochemical signals and the time-dependent flows will modulate the dynamic biochemical signals and affect their output. The transport and modulation of the dynamic biochemical signals in time-dependent flows still remain elusive. Understanding this point is helpful for the long-range hormone transmission, which is being released periodically to the bloodstream in pulsatile blood flows^[22]. Moreover, it will benefit our understanding of the interaction pulsatile blood flow and dynamic drug delivery^[23].

In this paper, in order to understand the dynamic biochemical signal transport in time-dependent flows and their modulation by these flows in a shallow Y-shaped microfluidic channel, the dynamic biochemical signals with different frequencies are input from Inlet A, and the non-reversing pulsatile flows in the mixing micro-channel are modulated by the input pulsatile flows at Inlet B. The output dynamic biochemical signals at the desired positions of the mixing micro-channel are derived by numerically solving the governing equations for the time-dependent Taylor-Aris dispersion and molecular diffusion in the time-varying flows in the mixing microchannel of the Y-shaped microfluidic channel. The modulation of the transporting dynamic biochemical signals by the flows in the mixing micro-channel is also numerically investigated by changing their frequencies and the pulsatile flows, which are also validated by experimental investigations.

2 Methods 2.1 Mathematical model and equations

The geometry and coordinate system of the shallow micro-channel used in this study are illustrated in Fig. 1. The cross section of the mixing micro-channel C is rectangular with high aspect ratio, and its length is L. The dynamic biochemical signals are input through at Inlet A by the pressure-driven steady flow with the flow rate Q_A, and the input flows without biochemical signals through Inlet B are non-reversing pulsatile flows with the flow rate Q_B (t)

2.1.1 Equation governing the pulsatile flows in the mixing micro-channel C

It is assumed that both of these two fluids, i.e., A and B, are Newtonian fluids with identical viscosity. By neglecting the end effects, the pulsatile flows of the Newtonian fluid within the long mixing micro-channel C are fully developed laminar flows. With the assumption of high aspect ratio, the fluid velocity is only along the longitudinal z-direction by neglecting the side-wall effect. Since the cell dimension is much smaller than the channel height, the cell effect is also neglected. The equation governing the pulsatile flows is^[24]

(1)

where u=u(y, t) is the flow velocity along the z-direction, p=p(z, t) is the pressure, t is the time, η is the fluid viscosity, and ρ is the fluid density.

With the quasi-steady flow assumption and the consideration of the pulsatile flow characteristics in the shallow micro-channel, the velocity profile u=u(y, t), the height-wise averaging velocity u(t), and the shear stress τ_w (t) can be given as follows^[24]:

(2)

(3)

(4)

where Q(t)=Q_A +Q_B(t) is the total flow rate through the mixing micro-channel C.

2.1.2 Control of two stream flow widths in the mixing micro-channel C

The flow velocity u_A of Solution A is the same as the flow velocity u_B of Solution B, which satisfies^[24]

(5)

where Q(t)=Q_A+Q_B(t). The volumetric flow rates Q_A and Q_B(t) satisfy^[24]

(6)

where W₁ and W₂ are the widths of Solution A and Solution B, respectively, and

Equation (6) shows that the width ratio W₁/W₂ of the two streams in the mixing channel is uniquely determined by the externally controlled flow rate ratio Q_A/Q_B(t). In this research, the flow rate ratio is

Therefore, the width of Solution B in the Y-shaped channel is εW, which primarily determines the inlet boundary (z=0) of the biochemical flow in the mixing micro-channel C.

2.1.3 Taylor-Aris dispersion in the mixing micro-channel C

For an input spatiotemporal biochemical signal, the concentration ϕ of the biochemical substance in the mixing micro-channel C (see Fig. 1) is governed by^{[21, 24-25]}

(7)

where D is the diffusivity. By taking the average of Eq. (7) over the height, we can obtain the governing equation for the time-dependent Taylor-Aris in the z-direction dispersion with the transverse molecular diffusion in the x-direction as follows^{[21, 24-25]}:

(8)

where the first term on the right-hand side is induced by the transverse molecular diffusion, and the second term stands for the effect of the longitudinal Taylor-Aris dispersion. In Eq. (8), ϕ and u (see Eq. (3)) are, respectively, the height-wise average of ϕ and u (see Eq. (2)). The effective diffusivity coefficient is expressed as follows^{[21, 24-25]}:

(9)

where Pe is the Peclet number defined by

which describes the ratio of advection to diffusion of a solute in the channel.

The average concentration of the biochemical solution from Inlet A, ϕ₀(t), provides a spatially uniform boundary condition, while no biochemical solution comes from Inlet B. Thus, the boundary conditions of Eq. (8) are

(10)

where an infinity boundary condition is used at the end of the channel to neglect the reflection. The initial condition is

(11)

2.2 Numerical simulation

For numerical simulation studies, Eq. (8) is solved by the finite difference method. An Euler explicit discretization is used for the temporal derivation. The first-order and second-order central differences are adopted to approximate the first-order and second-order spatial derivations, respectively. Therefore, a typical time step of the concentration ϕ(x, z, t) reads

(12)

where the subscripts i and j represent the spatial mesh points for x and z, respectively. The superscript n represents the step-in time. △x, △z, and △t are the mesh sizes in the x, z and time domains, respectively. The selection of the mesh sizes △x, △z, and △t satisfies the Courant-Friedrichs-Lewy condition, and reduces the computational cost^[26].

The given boundary conditions (10), i.e., the flow rates Q_A and Q_B(t), satisfy

(13)

and the input signal ϕ_A(t), and the spatiotemporal dynamic biochemical signal in the mixing micro-channel C is numerically simulated with the MATLAB software.

A real dynamic flow signal may contain many frequency components, which can be expressed by the Fourier series. For the sake of simplification, the flow rate at the inlet is assumed to have only one term with the basic frequency in the simulation studies. Once we have established that how a flow signal with one harmonic component corresponding to a certain frequency affects the biochemical transport, it is not difficult to understand the effect of a real pulsatile blood flow with many harmonic components in terms of the Fourier series on the biochemical signal transport.

All the simulation results are normalized by a constant reference value. In the numerical simulations, the default values for the model parameters are listed in Table 1.

2.3 Experimental validation 2.3.1 Microfluidic device fabrication and experimental setup

A polydimethylsiloxane (PDMS)-glass Y-shaped microfluidic device is designed, which is shown in Fig. 2(a). The height H of all the micro-channels is 80 μm while the width W and the length L of the mixing micro-channel C are 1 mm and 4 cm, respectively. The detailed protocol for fabricating PDMS-glass microfluidic is similar to that in Ref. [24].

As shown in Fig. 2(b), the fabricated Y-shaped microfluidic chip is connected with three syringe pumps (NE-1000, New Era Pump Systems) for controlling the dynamic biochemical signal and the magnitude of the wall shear stress in the mixing micro-channel C by regulating the flow rates from these pumps. More specifically, Inlet A is connected to two syringe pumps with T-bend and silicone tubes. The dynamic biochemical signal is generated by controlling the flow rates of the input solution and solvent from two syringe pumps, simultaneously. Inlet B is connected to the third syringe pump to generate the time-varying laminar flows without the biochemical factor. An inverted microscope (CKX41, Olympus Corporation) equipped with a CCD camera (DS126431, Canon Inc.) is adopted to observe the fluorescent signal intensity in the mixing channel C in real time (see Fig. 2(b)).

2.3.2 Experimental protocol

For actual experimental validation, the fluorescent solution (Rhodamine-6, Sigma-Aldrich) with time-dependent concentrations is used to simulate the biochemical signal. The fluorescent signal ϕ_A(t) at 1/30 Hz or 1/6 Hz is input through Inlet A of the Y-shaped microfluidic chip at a constant volumetric flow rate Q_A = 1× 10^-9 m³/s. The volume flow rate Q_B(t) of Solution B changes as a square wave at 1/30 Hz or 1/6 Hz between Q_A (2× 10^-9 m³/s) and 2Q_A (2× 10^-9 m³/s). The time-varying images for the dynamic fluorescent signals at any position in the mixing channel can be observed and detected by the fluorescence microscope with the CCD camera (see Fig. 2(c)). The dynamic fluorescent intensities are then extracted from the images by use of the MATLAB software. When the fluorescent intensities are calculated at each time point, the grey-values from the image background are subtracted. All the experimental results are normalized by a constant reference value.

3 Results 3.1 Simulation results and experimental validation for tempo-spatial profiles of biochemical signals

To understand the discrepancy between the temporal-spatial profiles of biochemical signals being transported in the steady flow and pulsatile flows, it is assumed that a biochemical signal ϕ₀(t)=ϕ₀ (1.0+0.5sin (2πf_ϕt)) is transported into a steady flow with the flow rate Q=2× 10^-9 m³/s or a square-wave-like pulsatile flow with the flow rate between 2× 10^-9 m³/s and 3× 10^-9 m³/s and the frequency f_v=1/30 Hz in the mixing micro-channel C. Figure 3 exhibits the numerically simulated spatial concentration profiles of the signal transported in the steady flow (see Fig. 3(a)) and the square-wave-like pulsatile flow (see Fig. 3(b)), respectively, at t=110 s in the mixing micro-channel C. It is evident from Fig. 3 that, for steady or pulsatile flows, the biochemical signals have amplitude attenuation at any position along the channel length (the z-direction), the dynamic signal keeps wave-like at the region near the side wall at x=1 mm, while the signal amplitude at the region around the interface between two streams from Inlets A and B decreases due to the transverse molecular diffusion. Moreover, the interface between the two streams remains unchanged under the steady flow (see Fig. 3(a)), while its position changes dynamically under the pulsatile flows (see Fig. 3(b)).

To clearly observe the temporal profiles of the dynamic biochemical signals being transported into the pulsatile flows, numerical simulation results are shown in Figs. 4(a) and 4(b) for the biochemical signal

with different frequencies (f_ϕ=1/30 Hz, 1/6 Hz) in the square-wave-like pulsatile flows with different frequencies (f_v=1/6 Hz, 1/30 Hz) at x =W/8, W5/8, and W7/8, respectively, and the central regime (z =2 cm) in the mixing micro-channel. It is clear from Figs. 4(a) and 4(b) that, the biochemical signals at different transverse locations (x = W/8, 5W/8, and 7W/8) exhibit complicated dynamic behaviors, while they are being transported into the pulsatile flows in the mixing micro-channel C. When no biochemical signals exist around x=W/8, the biochemical signals around x=5W/8 and 7W/8 are significantly affected by the pulsatile flows from Inlet B.

It is worth noting that, the nonlinear amplitude-frequency modulations are clearly observed in the biochemical signals (see Fig. 4(b)). Besides, the biochemical signals around the interface between the two streams from Inlets A and B, i.e., x = 5W/8, are directly affected by the pulsatile flow from Inlet B. The simulation results (see Figs. 4(a) and 4(b)) are qualitatively validated by the experimental data (see Figs. 4(c) and 4(d)).

3.2 Simulation studies for the interaction between pulsatile flows and dynamic biochemical signals

To better understand the nonlinear interaction between the pulsatile flows and the dynamic biochemical signals, the effects of different variables associated with the pulsatile flows and dynamic biochemical signals on this nonlinear interaction are numerically investigated. To exclude the direct effects of the pulsatile flows from Inlet B on the biochemical signals around the interface of the two streams induced by Solution A and Solution B, only the biochemical signals at the location x=7W/8 are analyzed (see the next subsections).

3.2.1 Effects of biochemical signal frequency

Figure 5 depicts the temporal profiles and the corresponding frequency spectrogram of the input dynamic biochemical signal

with f_ϕ =1 Hz, 0.5 Hz, 0.1 Hz, 0.05 Hz, and 0.033 Hz, respectively, which are transported into the pulsatile flow

with f_v =0.1 Hz. It is evident from Fig. 5(a) that, the temporal profiles of the biochemical signals at the observing site depend on the frequency of the input biochemical signals. The biochemical signals with higher frequencies (f_ϕ =1 and 0.5 Hz) have larger amplitude attenuation, while those with lower frequencies (f_ϕ =0.05 and 0.033 Hz) have smaller amplitude attenuation. When the biochemical signal frequency is bigger than that of the pulsatile flow, nonlinear amplitude-frequency modulations can be obviously observed. The higher the biochemical signal frequency is, the more significant modulations are found. The phenomena of the signal amplitude attenuation and the nonlinear interactions are clearly demonstrated in the frequency spectrogram (see Fig. 5(b)). It is shown in Fig. 5(b) that, much more new frequency components of biochemical signals are observed while the nonlinear interactions are significant.

3.2.2 Effects of flow frequency

To assess the effects of the pulsatile flow frequency on the modulation of the biochemical signal

the frequency of the biochemical signal from Inlet A is kept at 1 Hz. Meanwhile, the frequencies of the pulsatile flow

are alternating at various frequencies of 5 Hz, 2 Hz, 1 Hz, 0.1 Hz, 0.05 Hz, and 0 Hz, respectively via Inlet B. The temporal profiles of the biochemical signals are observed at x=7W/8 (see Fig. 6(a)). It is seen that, no significant nonlinear modulation is found when the flow frequency is larger than the biochemical signal frequency, while significant nonlinear modulations are observed when the flow frequency is smaller than the biochemical signal frequency. These phenomena also exhibit in the frequency spectrogram (see Fig. 6(b)). As the flow frequency decreases from 5 Hz to 0.05 Hz, much more frequency components of the biochemical signals are found. However, when the flow frequency continuously decreases to 0.05 Hz, the nonlinear modulation becomes weaker. For a special case, when the flow frequency is 0 Hz, i.e., the steady flow case, no flow modulation is observed at all.

3.2.3 Effects of transporting distance

It is noticed from Figs. 3(a) and 3(b) that the dynamic biochemical signals have longitudinal amplitude attenuation while they are being transported along the mixing micro-channel C. To verify the effects of different transporting distances on the interplay between the pulsatile flow

and the biochemical signal

the temporal profiles of the biochemical signals and their frequency spectrogram at different transporting distances are shown in Figs. 7 and 8.

In Fig. 7(a), the frequencies of the input biochemical and pulsatile flow signals are 0.5 Hz and 0.1 Hz, respectively. When the biochemical signal is transported at z=1 cm, the temporal profile is almost identical to the initial input biochemical signal. When the distance is z=2 cm, the biochemical signal becomes a little bit different. When the distances are z=5 cm, 10 cm, and 20 cm, respectively, the obvious modulation effects induced by the pulsatile flows are found. It is also clearly seen in the frequency spectrogram (see Fig. 7(a)) that, when z≥ 2 cm, there are more frequency components. It is worth noting that, when the distance increases, the biochemical signals have longer delay.

In Figs. 7(b) and 7(c), the frequencies of the input biochemical and pulsatile flow signals are identical at 0.1 Hz or the frequencies of the input biochemical and pulsatile flow signals are 0.1 Hz and 0.5 Hz, respectively. It is obvious in Figs. 7(b) and 7(c) that, when the distance increases, the biochemical signal with the frequency at 0.1 Hz also has longer delay but smaller amplitude attenuation. In this case, no significant nonlinear modulation is found (see Figs. 7(b) and 7(c)).

4 Discussion 4.1 Longitudinal dispersion and transverse diffusion

It has been suggested by Li et al.^[21] that the longitudinal dispersion and transverse molecular diffusion play important but different roles in the dynamic biochemical signal transportation of the steady flows in the mixing microchannel of a Y-shaped microfluidic device. The current study also shows that, when the dynamic signals are transmitted into pulsatile flows through the mixing channel, they will be affected by both the longitudinal dispersion and the transverse diffusion. The initial widths of Inlets A and B provide a primary tempo-spatial distribution of signal concentration but the precise signal profiles are determined by both the longitudinal dispersion and the transverse diffusion (see Fig. 3). When the longitudinal dispersion smoothens the signal in the longitudinal z-direction, the transverse diffusion tends to average the signal in each cross section. As shown in Fig. 3(b), at Inlet A, the locations away from the interface have a higher signal concentration than those closer to the interface, while at Inlet B, the locations away from the interface have a lower signal concentration than those closer to the interface. Therefore, the concentrations of biochemical signals are different for the cells at different transverse locations (see Figs. 3 and 4).

4.2 Signal filtering and nonlinear amplitude-frequency modulation

As discussed above, the mixing microchannel acts as a low-pass filter due to the longitudinal dispersion. A low-pass filter causes not only the amplitude attenuation of a signal with high frequency but also the phase delay of the signal (see Fig. 5). It suggests that, if a dynamic biochemical signal in a microfluidic channel contains a high frequency component, the signal cannot be perfectly preserved. As a result, in practice, it is hard to assure the cells cultured on the bottom of the channel to be stimulated by the complicated signal patterns with high frequencies (see Fig. 8).

It is worth noting that the frequency of a pure tone signal, which is transported into the non-reversing pulsatile flows in the mixing microchannel, is not well preserved in addition to the signal filtering (see Figs. 5 and 6). Some new frequency components of the signals are generated by the interaction between the dynamic biochemical signals and pulsatile flows. In contrast, the signal frequencies, which are being transported into the steady flows, can be preserved and passed to the cells with reduced amplitude^[21]. The pulsatile-flow-induced nonlinear amplitude-frequency modulation of signals is affected by the biochemical signal frequency (see Fig. 5), the flow signal frequency (see Fig. 6), and the signal transporting distance (see Fig. 7), may distort the signals between the input and the output (see Figs. 5-8), and poses difficulties on the input-signal control. As shown in Fig. 8, when a specific signal at 1/60 Hz is transported into the pulsatile flows at 0.01 Hz, 0.2 Hz, and 1 Hz, respectively, it can be well preserved at z =2 cm and 5 cm; however, when a similar signal at 1/6 Hz is in the pulsatile flows at 0.01 Hz, 0.2 Hz, and 1 Hz, respectively, it is evidently filtered and modulated at z =2 cm and 5 cm.

4.3 How to better load desired dynamic biochemical signals on cells

The Y-shaped channel is designed to transport the desired dynamic wall shear stress and dynamic biochemical signals to cultured cells^[24-25]. These desired extracellular dynamic biochemical signals may be transferred into the intracellular dynamic biological signals, and thus regulate the diverse cellular behaviors including messenger ribonucleic acid (mRNA) transcription^[27], gene expression^[28], and inflammatory cytokine expression^[29]. However, the signal filtering and nonlinear amplitude-frequency modulation make it difficult to preserve the signals released at the inlets. Although the signal filtering and nonlinear amplitude-frequency modulation cannot be prevented, their effects can be reduced by adjusting the influencing factors, e.g., the biochemical signal frequency (see Fig. 5), the flow signal frequency (see Fig. 6), and the signal transporting distance.

We have seen that, when the frequencies of the biochemical signals are relatively low, which are far smaller than those of the pulsatile flows, the signal filtering and nonlinear amplitude-frequency modulation can be decreased (see Fig. 8). In practice, we can use the biochemical signal with low frequency (see Figs. 8(a), 8(b), and 8(c)) instead of the biochemical signal with high frequency (see Figs. 8(d), 8(e), and 8(f)). Besides, the pulsatile flows with lower or higher frequencies away from those of the biochemical signals can be preferred (see Figs. 8(a) and 8(d)). Therefore, the balance between the biochemical signal frequency and the flow frequency should be considered for optimal signal loading. Since the biochemical signals cannot be affected by the signals filtering at the positions closer to the inlet (see Figs. 7 and 8), the cells close to the inlet are better positioned than those far from the inlet do, and the corresponding dynamic signals are relatively better preserved. However, around the inlet, the fully-developed laminar pulsatile flow assumption does not hold. As a result, the signals applied on the cells are not well controlled. Therefore, for better signal loading, multiple factors should be taken into consideration.

5 Conclusions

In this study, the transfer characteristics of the dynamic biochemical signals in non-reversing pulsatile flows in the mixing micro-channel of a Y-shaped microfluidic device are investigated by numerical simulations and experimental validation. The results show that, the mixing micro-channel acts as a low-pass filter, and the nonlinear amplitude-frequency modulation of the biochemical signals depends on multiple factors, e.g., the biochemical signal frequency, the flow signal frequency, and the signal transporting distance. It is concluded that, for better signal loading on the cells in the microfluidic channel, signal filtering and nonlinear amplitude-frequency modulation should be carefully considered.

Acknowledgements The authors would like to appreciate Dr. Hong TANG for the insightful advice.