Droplet-based microdialysis—Concept, theory, and design considerations

Abstract

The capability of continuously sampling the extracellular fluid opens up a wide range of applications of microdialysis in biological, pharmaceutical, and clinical studies. Existing microdialysis, however, faces challenges in sampling analytes with fast clearance and limited diffusivity because sampling resolution is limited by device size. Size reduction in probes and interconnected cannulae is a promising solution to improve temporal and spatial resolution. But the back pressure produced by resistance to laminar flows will be magnified in smaller channels, raising a concern as to whether it is feasible to operate continuous perfusion for miniaturized microdialysis. We demonstrate that a 10-fold smaller channel will exhibit 100-fold larger back pressure in response to the increase in the flow rate to maintain the relative recovery. In order to overcome the foreseen back pressure issue, this paper discusses a new concept using discrete droplets instead of continuous flows to operate dialysis in a miniaturized probe. This conceptual design is referred to as droplet-based digital microdialysis, in which droplets are produced, controlled and advanced within microchannels at a rate that in theory should allow for analytes to equilibrate with the extracellular fluid under no flow conditions. Expecting that a digital droplet design will entirely eliminate back pressure by introducing air between droplets, we numerically compare the equilibration kinematics of droplets to that of continuous flow. Results suggest equilibration of low molecular weight analytes between intermittently stationary droplets and the extracellular fluid in a few seconds. Considerations in design, prototyping, calibration and quantification, and the integration with other devices are suggested.

1. Introduction

The ability to continuously sample the extracellular compartment has made microdialysis widely applied in biological sample cleanup [1]; observation of metabolic activity in tumors, brain, and other tissues in humans [2]; and for monitoring neurotransmitters in the brain [3] since this catheter-type technique was first presented in 1966 [4]. The existing microdialysis technique uses controlled perfusate flow to continuously flush analytes, which are sampled by diffusion from the extracellular fluid through a semipermeable membrane. When coupled with analytical separation techniques that simplify identification of sample analytes, microdialysis enables online monitoring of targeted bioactive analytes. Microdialysis also allows for delivery of compounds into targeted extracellular sites, facilitating administration of pharmacological agents for focused applications to neural recording [5,6]. The microdialysis technique is in high demand in part because it is more cost-effective than other non-invasive methods, such as the positron emission tomography [7].

The sampling performance of microdialysis, however, is limited by the perfusion flow rate and device size. Current microdialysis typically provides temporal and spatial resolution of about 600 s and 0.1 mm³, respectively [7]. Although a higher sampling rate is achievable by raising the perfusion flow speed, the recovery will be compromised and the raised back pressure may induce ultrafiltration [8] through the membrane. Although back pressure can be reduced by lowering and/or shortening the outlet flow tubing [9] or by the push–pull mode [10], we shall show shortly that the back pressure, an inevitable nature in continuous flow, will be substantially increased in a smaller channel. On the other hand, problems associated with the rough spatial resolution include (relatively) large dead volumes and traumatic tissue damage associated with probe implantation which can hamper interpretation of results [11,12]. The poor spatial resolution and prolonged temporal resolution are a concern for glutamate detection because of the presumed rapid clearance and short diffusion distances associated with glutamatergic synapses [13–15]. Given these limitations, existing microdialysis devices often fail to access synaptic pools of neurotransmitters [16]. In addition, the probe technology has been barely revised since its inception in 1966 (except for the use of smaller cannulae for mice). There is a strong need for miniaturized microdialysis devices to improve temporal and spatial resolution, and meet the stringent requirements of accessing extracellular space in brain and other tissues.

A possible solution to improve sampling resolution is to miniaturize microdialysis devices. Recent work with small carbon fibers [17] suggests that smaller probes may minimize traumatic tissue damage resulting from probe implantation. Micro-fabrication technologies now can easily shrink integrated microchannels and semipermeable surfaces to 30–40 μm [18–20], suggesting that microdialysis probes could be downscaled as well to a similar range of dimensions. Designed to mimic capillary function, existing microdialysis probes are perfused with continuous flow. The route to miniaturizing the existing microdialysis technique, however, is not straightforward by merely down-scaling the probe dimensions while still operating a continuous flow for perfusion, for the following reasons. First, Jiang et al. [21] demonstrated the influence of the inertia of fluid on the dynamic response of flow in microchannels by showing a time lag between the pressure (input) and the flow volume rate (output). Such a time-lag phenomenon will induce a pulse-like flow whenever the pressure source is disturbed, although this inertia effect is less in smaller microchannels. Second, as the probe size is reduced, the laminar flow will be subject to a relatively higher frictional force, making precise fluid control more rigorous [22]. Third, fluctuations in the flow rate (as well as pharmacological treatments and temperature) influence the relative recovery [23,24]. Although the influence of flow rate fluctuations can be reduced by operating microdialysis at very low flow rates and extrapolating to zero flow and the zero-flow and zero-net-flux methods are also available to estimate actual extracellular concentrations of analyte [7,25], calibration often loses its precision over time in the case of the zero-net-flux method; re-calibration is thus needed but interrupts continuous sampling. Furthermore, in the case of very low, “quantitative” flow rates (e.g., when flow rate is stopped [26]), the ability to detect rapid changes in extracellular concentrations of analyte is compromised because of time required for analyte to equilibrate with a large internal volume of conventional microdialysis probes.

In this paper we introduce a new concept toward miniaturization of the existing microdialysis technique by operating droplets for sampling in a miniaturized probe; we also numerically estimate its sampling performance and compare it with conventional microdialysis. In the remaining sections, we discuss the back pressure issue arisen in operating a continuous flow in microchannels, followed by introducing the concept of the proposed droplet-based microdialysis. Emphasis is placed on numerical estimation of the sampling performance of the droplet-based microdialysis and comparison to that of the continuous flow-based microdialysis. Finally, we summarize our work and address some concerns for future study regarding the needs for prototyping, calibration and quantification of the probe performance of the digital microdialysis, and its integration with analytical techniques.

2. Scaling effect on the back pressure

Fluid viscosity and flow boundary conditions such as the wall ruggedness cause the so-called back pressure that gradually reduces flow momentum along the stream. The back pressure effect is more significant for a fluid flowing in a smaller channel, as its surface-to-volume ratio becomes larger and thus exaggerates the frictional effect [27,28].

The back pressure gradient of a laminar flow in an enclosed channel is often described by the Hagen–Poiseuille equation [29]:

$$\frac{\mathrm{\Delta }p}{\mathrm{\Delta }L}=f\frac{1}{W3}\frac{\rho V_{\text{avg}}^2}{2}$$ (1)

where W3 is the hydraulic diameter (or the equivalent dimension), $\rho V_{\text{avg}}^2/2$ the dynamic pressure (ρ the density of the fluid and V_avg the average speed of the flow), and f is the Darcy’s friction factor. For a fully developed laminar flow in a circular pipe, f is usually expressed by [29]:

$$f=\frac{64}{Re}$$ (2)

where Re is short for the Reynolds number. By expanding Re according to Eq. (7) below, the Hagen–Poiseuille equation can also be expressed in terms of the flow volume rate Q:

$$\frac{\mathrm{\Delta }p}{\mathrm{\Delta }L}=128\frac{\mu Q}{\pi {(W3)}^4}$$ (3)

where μ is the dynamic viscosity of the fluid. One can easily estimate the gradient of back pressure, Δp/ΔL, for Q being constant. For quantitative microdialysis, however, it will be more useful to relate Δp/ΔL for a perfusate flow subject to constant relative recovery. For this reason we built a two-dimensional (2D) model (Fig. 1) with an attempt to quantify the scaling effect of probe dimensions on Δp/ΔL subject to a constant relative recovery. It is noteworthy that the schematic drawing shown in Fig. 1 outlines a portion of the microdialysis probe near the semipermeable membrane, in which the membrane is deliberately placed on one side of the microchannel in the 2D model (namely, the membrane is on one face of the channel in a 3D model), a consideration adapting to the micro-fabrication process. This configuration is different from the circumferential one as seen in conventional microdialysis probes. Nevertheless, the membrane configuration should not affect the theoretical analysis as follows.

Fig. 1.

Schematic of conventional microdialysis in vitro. Microdialysis performance is determined by the variables shown: V_avg is the averaged velocity of the perfusate flow; μ, ρ, and D are the viscosity, density, and diffusion coefficient of the perfusate, respectively; W3 designates one dimension of the cross-section of the chamber (height or width); and A is the membrane area; c and c_∝ are the concentrations of analyte at equilibrium in the probe chamber and in the tissue far away from the membrane, respectively.

Among many parameters influencing the relative recovery of microdialysis [30,31], only six parameters pertain to the microdialysis probe for they have a direct impact on the probe’s instrumental response. We use a brace {} following each parameter to designate the generic unit of each parameter via three basic dimensions: mass (M), length (L), and time (t). (For generic we mean that, for example, M represents any mass unit.) These six parameters are: V_avg {Lt⁻¹} the averaged velocity of the perfusate flow; μ {ML⁻¹}, ρ {ML⁻³}, and D {L²t⁻¹} the viscosity, density, and diffusion coefficient of the perfusate, respectively; W3 {L} describes one dimension of the cross-section of the chamber (height or width); and A {L²} is the membrane area. In addition, we use c {ML⁻³} and c_∝ {ML⁻³} to indicate the concentration of analyte at equilibrium in the probe chamber and in the tissue (far away from the membrane), respectively.

Interpretation of microdialysis results is typically based on proportional changes in analyte following the definition of relative recovery, which is the ratio of dialysate over actual extracellular concentrations. For quantitative microdialysis using analytefree perfusate for sampling, the relative recovery is an implicit function of the six parameters previously mentioned:

$$\frac{C}{C_{\infty }}=f(V_{\text{avg}},\mu ,\rho ,D,W3,A)$$ (4)

Although this function excludes the diffusion coefficients of particles in the semipermeable membrane and in the extracellular fluid, Eq. (4) sufficiently describes the instrumental response of micro-dialysis devices.

The implicit relation on the right hand side of Eq. (4) can be further grouped into three dimensionless parameters by applying the Buckingham Π theorem [29], which outlines a procedure for dimensional analysis of a similarity problem. With W3, V_avg, and ρ as the “repeating parameters”, three dimensionless parameters can be found as:

$${\mathrm{\Pi }}_1=\frac{D}{(W3)V_{\text{avg}}}$$ (5)

$${\mathrm{\Pi }}_2=\frac{A}{{(W3)}^2}$$ (6)

$${\mathrm{\Pi }}_3=\frac{\mu }{\rho (W3)V_{\text{avg}}}\triangleq \frac{1}{Re}$$ (7)

The relative recovery can be accordingly expressed in terms of the three Πs:

$$\frac{C}{C_{\infty }}=f\left(\frac{D}{(W3)V_{\text{avg}}},\frac{A}{{(W3)}^2},\frac{\mu }{\rho (W3)V_{\text{avg}}}\right)$$ (8)

Eq. (8) implicitly quantifies the dependence of the relative recovery on the probe-related parameters. It shows that the relative recovery can be sustained at a constant level once all the three Π parameters remain constant regardless of the individual changes in the six parameters. For example, the first Π parameter (Eq. (5)) will remain constant if W3 is 10-fold smaller and V_avg is 10-fold faster.

To understand the effect of shrinking the dimensions A and W3 on the back pressure gradient while keeping the relative recovery constant, we deliberately assume that the perfusion properties (i.e., μ, ρ, and D) are constant on the right-hand side of Eq. (8). Under this condition, keeping the relative recovery constant in a downscaled probe requires that:

$${\mathrm{\Pi }}_2=\frac{A}{{(W3)}^2}=\text{constant}$$ (9)

$$(W3)V_{\text{avg}}=\text{constant}$$ (10)

The above equations address that, for a constant relative recovery, the average flow of perfusate must be l-fold faster when a probe is miniaturized by downscaling W3 by l times. In other words,

$$(W3)~l^{-1}\text{and}{V}_{\text{avg}}~l$$ (11)

where “~” means “proportional to” and l (>1) is a scaling factor (e.g., l = 10 corresponds to the miniaturization of a probe 10-fold smaller in one dimension of its cross-section). By introducing Eq. (11) into Eq. (3) we found that:

$$\frac{\mathrm{\Delta }p}{\mathrm{\Delta }L}~l^3$$ (12)

The increase in the back pressure gradient in a smaller probe is in response to the increase in the flow rate for maintaining the relative recovery (in the steady state). Eq. (12), for example, predicts that the back pressure gradient will be 1000-fold larger in a miniaturized probe which has a diameter 10-fold smaller and pertains both the membrane-to-probe area ratio (Eq. (9)) and the relative recovery the same as those of its unscaled counterpart. In other words, the back pressure will be 100-fold larger in a 10-fold miniaturized probe. Apparently back pressure will be tremendously exaggerated in a miniaturized probe, thus imposing challenges to maintain continuous flow.

3. Concept of droplet-based digital microdialysis

To circumvent high back pressure in miniaturized probes we propose the digital microdialysis concept. For “digital” we mean the use of discrete droplets envisioned to move within a microchannel integrated with a semipermeable membrane at the probe top for sampling. According to the operation principle schematically shown in Fig. 2, droplets are sequentially produced and conveyed to the membrane; reside there for a period of time to allow for analytes to diffuse from the extracellular fluid to equilibrate with the droplet; then droplets are transported to the outlet for further analysis. (The figure illustrates, but is not necessarily limited to, the use of a tapered capillary to manually transport analyte-laden droplets for analysis.) With droplets sequentially produced, conveyed, and placed on the membrane to wait to equilibrate with the extracellular fluid, the proposed digital microdialysis creates conditions for sampling under no-flow, stationary conditions. The key feature of the proposed method is on its potential of eliminating back pressure. Droplets will be produced under ambient pressure and sequentially transported to the probe through a microchannel. In the channel any two consecutive droplets are thus separated by air whose pressure is very close to the ambient pressure. It thus envisions that the air pressure increase in the channel between two neighboring droplets will be insignificant when two consecutive droplets are transported at the same pace. The surface tension and capillary force between the droplet and membrane are expected to be sufficiently strong to prevent perfusate from leaking (into tissues through membrane, the so-called membrane ultrafiltration effect [8]).

Fig. 2.

Operation principle of droplet-based dialysis. Droplets are sequentially formed under ambient pressure and transported to the semipermeable membrane for sampling. Droplets are separated by air in the microchannel. A droplet will reside on the semipermeable membrane as it reaches there, waiting for analytes to diffuse through the membrane to fully or partially equilibrate between the droplet and the extracellular fluid. (A tapered capillary is illustrated to manually move the droplet from the outlet for analysis.)

The temporal resolution of the digital microdialysis in sampling will be determined by two factors, the duration of droplet residence (on the membrane) and the time of transporting a droplet to the outlet. The former will be estimated numerically in the following section, while the latter is determined by the probe design and the droplet control scheme, the discussion of which is not our emphasis here. On the other hand, by its very nature, a miniaturized microdialysis probe will also improve spatial resolution and reduce tissue damage. Thus, digital microdialysis can potentially achieve a fast sampling rate (i.e., a high marching rate of droplets, as opposed to the flow rate of a continuous fluid), more consistent relative recovery and enhanced feasibility of achieving 100% recovery in each droplet (by adjusting the droplet residence times on the membrane).

4. Theory of droplet-based digital microdialysis—equilibration kinematics

A number of mathematical models have been reported to quantify conventional microdialysis under continuous perfusion [30,32–39]. These endeavors include modeling of the effect of pore structures (e.g., the anatomical interstices and the semipermeable membrane) and perfusion rates on the microdialysis performance, and diffusion that reflects binding, uptake, release and other processes to account for metabolism and transport across the microenvironment of cells and the intercellular space. To contrast the equilibration kinematics between the conventional dialysis process (with continuous flow) and the proposed digital dialysis (with stationary droplets), we modeled the mechanistic aspect of analyte transportation and deliberately excluded production and depletion of these molecules.

Our models start with formulating the flux q⃗, the analyte transport rate per unit area with a unit {ML⁻²t⁻¹}, along a path across the extracellular space (ECS), through the semipermeable membrane, and then into the chamber of the probe:

$$\overrightarrow{q}=-D\nabla c+c\overrightarrow{\upsilon }$$ (13)

where D {L⁻²t⁻¹} is the diffusion coefficient of the model analyte, c {ML⁻³} the analyte concentration, and υ⃗ {Lt⁻¹} is the flow speed of the medium carrying the analytes. Given no ultrafiltration, the flow speed must be zero in both the ECS and semipermeable membrane. In the probe chamber, analytes move under combined diffusion and drift in a “piggy-back” fashion relative to the perfusate flow [40], a phenomenon referred to as dispersion [41]; υ⃗ should thus be determined by the velocity field of the perfusate flow in Eq. (13).

The flux q⃗ in Eq. (13) can be further expressed in terms of the concentration c by applying the Reynolds transport theorem [42], leading to the following equation for governing analyte transportation:

$$\frac{\partial c}{\partial t}+\nabla ·(c\overrightarrow{\upsilon })=\nabla ·(D\nabla c)$$ (14)

where “∇” is the divergence operator. Below we justify the use of this equation for quantifying equilibration kinematics for the proposed digital microdialysis and for conventional microdialysis, respectively:

Droplet equilibrium kinematics

The model setup for simulating the equilibration kinematics of a droplet is shown in Fig. 3a which depicts a scenario that a droplet fully covers the membrane and resides there, motionless, for equilibrium. The equilibrium kinematics is governed by dropping the velocity term υ⃗ from Eq. (14).

Fig. 3.

2D models for dialysis based on (a) droplet and (b) continuous flow. The dimensions of the simulated domain are: chamber length H = 50 μm, chamber width W3 = 10 μm, membrane width W2 = 6 μm, and ECS width W1 = 10 μm. Both models are subject to a segmental line source of glutamate distributing in ECS along a segment of line 50-μm long on the leftmost side of the problem domain. Along the unspecified boundaries no constraint is imposed on the diffusion and dispersion of analytes.

Equilibration kinematics of a continuous perfusate flow

Fig. 3b illustrates that the perfusate flow is approximated as a Poiseuille flow entering from the bottom of the probe chamber. Eq. (14) governs the equilibrium kinematics by introducing the parabolic velocity profile of the flow to the velocity term υ⃗ in the chamber.

The models in Fig. 3 are two-dimensional and describe the sampling of a constant line source of glutamate diffusing through the passages (highlighted in gray). The 2D simulation can provide a relevant result for the probe response to a well-stirred solution in vitro because a constant line source to a 2D diffusion model is mathematically equivalent to a constant planar source to a 3D model [43]. The problem domain in Fig. 3 is identical to each other and comprises three parts: the ECS, a sandwich-like semipermeable membrane, and the probe chamber. Below we describe each part of the model and the needed boundary/initial conditions for simulations.

ECS

A porous ECS was constructed to simulate true extracellular space in brain [39]. The numerically constructed ECS has a volume fraction of 20% as the diffusion passage and a tortuosity 1.6 and 1.7 in the x and y directions, respectively [44]. We arbitrarily chose 10 μm for parameter W1 (see Fig. 3). Although the value of W1 has no influence on the comparison of the equilibrium kinematics; the larger the value for W1, the longer the analytes will diffuse to the probe chamber, which in turn will require more computational effort.

Membrane

Our model considers a sandwich-like membrane, which is commonly seen in photolithographic micro-fabrication [18]. This membrane model resembles an electron beam-fabricated semipermeable membrane made of layers of materials with a 30–50 nm gap between each layer. The gaps (as indicated by the bars/lines within the membrane model in Fig. 3) are interconnected by fabricating a lattice pattern of holes and serve as diffusion passages. The membrane is 6-μm thick (=W2) and 50-μm long (=H). We modeled that a droplet fully covers the membrane, which minimizes the equilibration time for sampling.

Probe chamber

The probe chamber simulates a volume of 50 μm (H) × 10 μm (W3) × 50 μm (=0.025 nL) in which we have implicitly assigned the third dimension 50 μm (in the z-direction) for simulations. The simulations also considered that the chamber has a rectangular cross-sectional area to accommodate the micro-fabrication features.

Boundary conditions

A constant line source of analytes, which has a constant concentration of 100% over time, is placed on the leftmost side of the problem domain. A reflective boundary condition was imposed at the chamber walls (on the rightmost side of the problem domain) to consider analytes reflecting from the walls and thus being confined in the chamber. We also imposed a zero-concentration condition at the two opposite edges of the membrane boundaries (see Fig. 3) to consider the non-adsorbing feature of the membrane structure.

Initial conditions

The initial distribution of concentration is zero everywhere in the problem domain except the constant line source at the leftmost side.

It is worth mentioning that the ECS and membrane are full of non-penetrable substances or pores that analytes cannot diffuse in. The influence of the pore structure on diffusion can be quantified by the tortuosity factor [45]. (Readers may refer to Nicholson’s work [39,46] on the tortuosity of ion diffusion in the brain extra-cellular space.) Homogenizing (averaging) D over a heterogeneous domain can help simplify the complexity of detailed but nontrivial descriptions of the porous-medium diffusion problem. We have considered that the diffusion coefficient D varies place from place in porous media (e.g., ECS and membrane) and applied the random-walk principle [44] to calculate the effective diffusion coefficients of analytes in the ECS and semipermeable membrane to reduce the computational rigor. By using glutamate as a sample molecule which has a diffusion coefficient of 760 μm²/s in a bulky aqueous fluid [47], the effective diffusion coefficient for glutamate in the extracellular space and the semipermeable membrane is 367 and 108 μm²/s, respectively. Thus, the coefficient D in Eq. (14) can be termed outside the parentheses.

The simulation was conducted in Matlab by the finite difference method to quantify the distribution of analyte concentration over the period of 0–1.25 s. We adopted the explicit space-forward, time-central scheme with a spatial increment of 0.5 μm and a step size of 20 μs to discretize the space-time problem in Eq. (14) with test-good numerical stability and convergence. Fig. 4 compares the histograms of analyte concentration in the probe chamber, not achieving their equilibrium conditions yet. Each rectangular slot in Fig. 4 outlines the probe chamber (refer to Fig. 3). The concentration level is represented by the scales shown, where the darker the scale is, the higher the concentration.

Fig. 4.

Histograms of analyte concentration in a probe chamber of 10 μm (W3) × 50 μm (H) (refer to Fig. 3) subject to 2D modeling. (a) Droplet-based dialysis. A perfusate droplet fills the entire chamber to simulate a full coverage of the membrane (located on the left side of the chamber, not shown) to minimize the equilibration time. Sitting still in the chamber during equilibration, the droplet is approximately equal to the chamber in size. (b) and (c) Continuous-flow based dialysis. A perfusate flow flushes through the chamber from the inlet (bottom) to the outlet (top) at a constant volume rate of 15 nL/s for (b) and 1.5 nL/s for (c). The scale shown on the rightmost side numerates 1 for saturation and 0 for no analyte.

The droplet-based stationary equilibrating process (Fig. 4a) shows a more uniform and symmetrical distribution in the analyte concentration, in which analytes distribute more sparsely near the top and the bottom sides of the probe chamber due to the zero-concentration boundary condition imposed in the membrane. For the continuous flow-based microdialysis, Figs. 4b and c shows the equilibrating process for a perfusate flow entering the probe chamber from its bottom side at a constant volume rate of 15 and 1.5 nL/s, respectively. A higher concentration can be seen near the walls toward the outlet in Fig. 4b because analytes are perfused toward the outlet and the perfusate flows slower along the streamlines closer to the wall. The perfusate flow rate in Fig. 4c is 10 times slower than that in Fig. 4b, rendering a more uniform distribution of analyte in the chamber which is very similar to Fig. 4a. Given the volume rate and the chamber size modeled, the average flow speed is 4.5 and 0.45 cm/s for Fig. 4b and c, respectively. At room temperature, given the viscosity and density of an aqueous perfusate 8.9 × 10⁻⁴ Pa s and 1 g/cm³, respectively, the results in Fig. 4b is associated with a back pressure gradient as high as 1.3 × 10⁷ Pa/m (or 46.4 psi/in.), according to Eq. (3). If a miniaturized microdialysis probe were to use a capillary that is 2-m long, the end-to-end back pressure would be as high as 2.6 × 10⁷ Pa (or nearly 1900 psi). It means that the syringe pump, as commonly used in the conventional microdialysis, must have a minimum output pressure of 1900 psi to pump the perfusate. Similarly, Fig. 4c is associated with a much lower back pressure gradient (190 psi end-to-end difference along a 2-m capillary tube) and a 10-fold slower perfusion. These results suggest that the droplet-based microdialysis can achieve an equilibrating process similar to the ultraslow (conventional) micro-dialysis; despite the benefit gained in reducing back pressure, slow perfusion means prolonged sampling times in conventional micro-dialysis.

The droplet size critically determines the time for achieving equilibrium in sampling. By varying the dimension W3 of the chamber, Fig. 5 compares the evolution of the volume-averaged concentration in each droplet which covers a membrane H = 50-μm long. By fixing the probe dimension in the z (out-of-plane) direction as 50 μm, the volumes for the three droplets are 0.025, 0.100, and 0.200 nL, respectively. For the case of W3 = 10 μm, the 0.025-nL droplet has nearly achieved equilibrium at 2.75 s.

Fig. 5.

Equilibration time versus droplet size. The volume-averaged concentration is calculated by averaging the distribution of concentration over the droplet volume. By fixing the third dimension (in the out-of-plane direction) as 50 μm, the droplet volume is 0.025, 0.100, and 0.200 nL for W3 = 10, 40, and 80 μm, respectively. The distribution of concentration in the 0.025-nL droplet has nearly achieved equilibrium at 2.75 s.

We have also conducted other parametric studies such as changing the dimensions W1, W2, W3, H, and the flow volume rate Q of the Poiseuille flow (results not shown) and summarize our observations of the results below:

1. At any time, the concentration of analyte in a stationary droplet is always higher than that of the perfusate flow in the same chamber. The difference in the concentration levels becomes less as the perfusion speed of the conventional microdialysis is slower.

2. Analyte equilibration is always faster in a droplet than in a flowing perfusate regardless how far the probe is placed away from the analyte source in ECS (i.e., the parameter W1) and how thick the membrane thickness (W2). However, the larger the dimension in W1 and/or W2, the longer the equilibration time (since analyte takes a longer journey, by diffusion, in the extracellular space and the membrane).

5. Other considerations

Besides the numerical emphasis on predicting the temporal performance of the proposed method, we suggest some considerations for future enhancement to this work.

5.1. Design

To be more efficient in sampling, a fully functional digital micro-dialysis probe should be capable of advancing an array of droplets to the membrane site, equilibrating each droplet there, and then transport the analyte-laden droplets, one by one, for further chemical analysis. The droplet control scheme should enable droplets to move at a constant pace in the go-stop-go manner to avoid air pressure increase in the gaseous space between any two consecutive droplets.

5.2. Prototyping

The interior of the microchannel should be coated with a hydrophobic material to reduce the contact angle (and thus the contact area) between the droplet and the channel wall to facilitate droplet transportation. A thin hydrophilic coating may be applied to the exterior of the digital microdialysis probe to reduce protein adsorption. It is noteworthy that, among many techniques available to produce and move droplets in microchannels [48], the electrowetting principle has been successfully demonstrated to operate droplets on a dielectric layer with a hydrophobic coating [49,50]. One may also employ the capillary effect to manipulate droplets in a microchannel with a non-uniform cross-section [51,52]. The two techniques can be coupled in the digital microdialysis design [48] (see Fig. 6) to reduce the number of electropads and thus lower the complexity in circuit design and droplet controls. In fact, the technique for droplets formation and control has been widely demonstrated [53].

Fig. 6.

Capillary-assisted electrowetting for droplet transportation. (a) Stationary droplet in a uniform, hydrophobic channel. (b) Electrowetting-driven movement. Biased voltage which is applied sequentially to charge the electropads (in gray) across the droplet breaks the surface tension balance, thus causing a net force (toward the + side) to move the droplet. (c) A droplet automatically moves toward the widening side within a hydrophobic microchannel due to the capillary effect. (d) The droplet stops moving at a narrowing gate of the microchannel, again, due to the capillary effect. (e) Sufficient energy provided by electrowetting enables the droplet to pass the narrowing gate and move toward the next gate.

5.3. Calibration and quantification

The period of retrieving an analyte droplet in digital microdialysis will be determined by two factors, how fast a probe can produce and transport a droplet to the membrane and how long the droplet will reside on the membrane for equilibration. Although the period limits overall analysis time, it does not impact the temporal resolution of events in the tissue because one may adjust the “lag” time by subtracting it from that of real-time observations. On the other hand, since droplets are stationary on the semipermeable membrane for equilibration, our proposed method is conceptually equivalent to the zero-flow-rate microdialysis. The no-net-flow method may also be implemented in the digital microdialysis by using perfusate droplets which are concentrated, to certain various levels, with the analyte of interest and analyzing the concentration difference between the perfusate and dialysate droplets.

5.4. Automatic droplet handling and integrated chemical analysis

We estimate that each droplet is smaller than 1 nL in volume (1 nL = [100 μm]³). The analysis frequency can be greatly improved by integrating the proposed digital microdialysis with other analytical methods (e.g., capillary electrophoresis) on a single chip. The on-chip integration of sampling, separation, and detection will allow for analysis automation.

5.5. Influence of aqueous-dissolved gases

Sample molecules with vapor pressure higher than aqueous-dissolved gases will not contribute to vapor pressure of the air in the microchannel because these small molecules (e.g., O₂, N₂) diffuse readily through the semipermeable membrane. No other molecules with vapor pressure higher than aqueous liquid are known to exist in biological systems.

6. Conclusions

Miniaturization of microdialysis is promising to greatly improve sampling resolution in vivo. Toward successful realization of this idea it will first encounter the back pressure issue which exists in a continuous flow, regardless of its flow speed, and will be magnified in downscaled channels. In this paper we have suggested a new concept of droplet-based microdialysis to resolve this issue. The droplet-based dialysis is advantageous for its capability of alleviating the challenges of maintaining constant flow rates as well as eliminating back pressure in microchannels. By using stationary and small droplets for dialysis, analytes are expected to rapidly equilibrate (under a stationary condition) between the droplet and the extracellular fluid making this technique quantitative. We have compared the equilibration kinematics of droplets to that of continuous flow, showing that a stationary droplet can acquire a higher concentration at a faster pace and that the equilibration kinematics of droplets is similar to that of a slow perfusion. Thus, the droplet-based dialysis essentially approximates a periodic stop-and-go scenario in the conventional microdialysis but can advantageously eliminate the need for hydraulic pressure, thus circumventing the back pressure concern. Enhancement to this work includes prototyping and experimental proof-of-concept.

Nomenclature

Generic expressions of three basic dimensions length (L), mass (M), and time (t) are used to define the unit of the parameter if not otherwise expressed in the SI unit.

A

membrane area (L²)

C, C_∝

concentration of particles in the probe chamber and in the ECS (ML⁻³)

D

diffusion coefficient (L²t⁻¹)

f

Darcy’s friction factor (dimensionless)

H

length of the probe chamber (μm)

l

scaling factor (dimensionless)

q⃗

flux of analyte (ML⁻²t⁻¹)

Δp

pressure difference (ML⁻²t⁻²)

Re

Reynolds number (dimensionless)

υ⃗

flow speed (Lt⁻¹)

V_avg

mean flow speed in microchannels (Lt⁻¹)

W1, W2, W3

dimensions of the problem domain for simulations (μm)

Greek letters

μ

dynamic viscosity (ML⁻¹)

Π₁, Π₂, Π₃

dimensionless parameters

ρ

density (ML⁻³)