OPTIMAL DESIGN OF A MICROFABRICATED DIFFUSION-BASED EXTRACTION DEVICE Mark R. Holl1, Paul Galambos† , Fred K. Forster† , James P. Brody, and Paul Yager Center for Bioengineering University of Washington Seattle, Washington
ABSTRACT The miniaturization of micro-fluidic chemical assays of fluid mixtures containing particles such as biological molecules and cells is a technically and commercially significant objective. Successful automation of chemical analysis of small samples requires seamless integration of several subsystems that perform tasks routinely carried out by a skilled technician. In this paper is presented a methodology for the optimal design of a novel micro-flow diffusion-based constituent extraction device based on parallel fluid flow through a microchannel. The streamwise distance required for the constituent being extracted to achieve an average concentration across the micro-channel that is a fixed percentage of the equilibrium concentration is defined as the equilibration length. The constituent concentration within the micro-channel is calculated using a 1-D analytical diffusion model. The equilibration length is used to construct a family of process space design curves specific to the extracted constituent. An optimization objective function is specified to identify the design that maximizes the volume flow rate of product stream. The methodology is applied to the design of a device for the extraction of albumin (a protein present in blood) from a carrier sample stream with viscosity approximately that of water. The device is specified for a length sufficient for the mean albumin concentration in the product stream to reach a value that is 99% of the equilibrium concentration of albumin for an infinite length device. This process sensitivity information provides design requirements for upstream and downstream fluidic components, and is essential for integration of the device into a “lab on a chip” chemical analysis system.
1
†
Direct correspondence to M. R. Holl, Center for Bioengineering, Box 352141, University of Washington, Seattle Washington 98195,
[email protected] Department of Mechanical Engineering, University of Washington
NOMENCLATURE
V˙ss V˙es V˙
Sample stream flow rate (m3/s)
V˙bps V˙ind V˙ds vy ci ,ss ci ,es ci ,bps ci , ps cdye,ind ci ,ds d
By-product stream flow rate (m3/s)
ps
Product stream flow rate (m3/s)
Indicator dye stream flow rate (m3/s)
%
Detection stream flow rate (m3/s) Mean extraction channel flow velocity (m/s) Sample stream constituent i concentration (kg/kg) Extraction stream constituent i concentration (kg/kg) By-product stream constituent i concentration (kg/kg) Product stream constituent i concentration (kg/kg) Indicator stream dye concentration (kg/kg) Detector stream constituent i concentration (kg/kg) Diffusion direction extraction channel depth (m) Extraction channel width (m) Extraction channel length (m) Device length required to achieve α % (m)
%
Normalized equilibration length (dimensionless)
w
L Lα α% L˜α xs xp
Extraction stream flow rate (m3/s)
Percentage of equilibrium concentration
Interface streamline location between sample and extraction streams at the extraction channel entrance (m) Interface streamline location between the by-product and product streams (m)
P ∆p Di µ ρ ξ c˜
x
y
z x˜ , y˜ Pe Re
Absolute pressure within the fluid stream (Pa) Differential pressure between the entrance and exit of the extraction channel (Pa) Binary diffusion coefficient of constituent i (m2/s) Fluid viscosity (Pa s) Fluid density (kg/ m3) Equilibrium normalized constituent concentration for an infinite length extraction channel (dimensionless) Normalized constituent concentration (dimensionless) Extraction channel depth coordinate (primary diffusion direction) Extraction channel length coordinate (flow direction) Extraction channel width coordinate Non-dimensional normalized variables (dimensionless) Peclet number Reynolds number
⋅
INTRODUCTION Chemical Analysis in Micro-fluidic Total Analytical Systems (TAS)
Several steps commonly performed in the chemical assay of a fluid mixture are: 1) precise mixture dilution, 2) extraction of a specific constituent, 3) precise mixing of indicator reagents or test probes (e.g. fluorescently tagged polymer beads), and 4) non-invasive detection of the indicator or probe (e.g. absorbance or fluorescence spectroscopy). The development of micro-fluidic system components required to realize a “lab-on-a-chip” capable of providing an accurate chemical assay of microliter sized samples is an area of aggressive research (Brody and Yager 1996). The prototypical lab-on-a-chip system should incorporate both sample preparation and analyte detection subsystems exploiting the advantages of micro-scale systems integration (Ramsey et al. 1995). Optimal design of system components, micro-scale integration of micro-fluidic components, and surface interaction effects are the most significant problem areas to be addressed (Gravesen et al. 1993; Manz et al. 1993; Cefai et al. 1994; Elwenspoek et al. 1994; Zengerle and Richter 1994; Ramsey et al. 1995). This paper presents a methodology for the optimal design of a constituent extraction device intended for use in an integrated micro-fluidic chemical assay system. Such a device constitutes a critical component for a broad class of microfabricated chemical analysis instruments. The detection strategy presented in Fig. 1 requires constituent extraction from the particulate laden sample, fluorescent indicator mixing with the diluted analyte, and fluorescent optical detection. Critical to the precise operation of the inference technique is the precise regulation of all stream flow rates in the system. In the system shown in Fig. 1 the sample, extraction, and indicator dye streams enter the system. By-product and detection streams exit the system. Optical detection control inputs and fluorescent signal outputs are also shown. Using a calibration between fluorescence intensity and constituent concentration and information precisely defining the constituent extraction and indicator mixing dilution ratios, the concentration of constituent in the original sample stream is estimated. A similar approach to a different micro-fludic problem was presented by Afromowitz (Afromowitz and Samaras
1989). Precise flow control in integrated TAS systems might in part be achieved using on-chip micro-pumps (Gravesen et al. 1993; Elwenspoek et al. 1994; Forster et al. 1995). 3-input, 2-output "Lab-on-a-chip" By-product stream, (bps)
Sample stream, (ss)
V˙ss , c i,ss Extraction stream, (es)
V˙es
Diffusion-Based Extraction Device
V˙bps , ci, bps Product stream, (ps)
V˙ps , c i,ps
Indicator dye stream, (ind)
V˙ind , cdye, ind
Indicator Mixing
Detector stream, (ds)
Optical detector excitation actuation
Optical Detection
V˙ds , ci,ds , cdye,ds Fluorescence signal
Fig. 1 Fluorescence detection “lab-on-a-chip” for the assay of constituents present in a particulate or cell laden sample stream. The “Diffusion-based Extraction Device,” is the topic of this paper. The extraction of constituents from a particle laden sample stream using diffusion to transport the constituents to a separate stream was presented by Williams in 1992 (Williams et al. 1992). Williams used a “SPLITT” device 15 cm long, 3 cm in width, and 381 µm thick in the diffusion extraction direction. The device had a total volume of 1.71 ml. In the Williams experiments excellent agreement with theory was demonstrated for fractional retrieval of γ-globulin, BSA, cytochrome c, and sodium benzoate. The retrieval objectives were accomplished using a volume flow rate resolution of 2.59 ml/min ± 10µl/min (±166µl/s). In 1994 Yue et al. (Yue et al. 1994) published a study of a slightly smaller field-flow fractionation device intended for use in separating the individual blood cell populations present in whole blood. Yue et al. concluded that elution rates of red blood cells (RBC) were dependent on sample age as well as device design and operational parameters. They observed that further research on fractionating devices was essential to identifying potential biomedical applications for such devices. Brody, in a novel set of experiments, demonstrated the potential for using diffusion based extraction to separate diffusing constituents from a particle laden sample stream using micron sized devices microfabricated in silicon (Brody and Yager 1996). See Fig. 2 (a). Fluorescein dye was extracted from a sample stream containing 0.5 µm fluorescent polystyrene spheres and fluorescein dye. Operation was demonstrated with zero contamination of the extraction stream by fluorescent spheres. The Brody device had a total extraction channel fluid volume of approximately 1 femtoliter; this was far too small to use in the extraction of diffusing constituents from whole blood (40-50% RBC by volume) having RBC with ellipsoidal shape and 8µm major axis dimension and WBC with nominal diameter of approximately 15-25µm. Even so, the femtoliter scale device demonstrated that separation was possible at the femtoliter scale given appropriate attention to precise flow stream regulation. Further, Brody demonstrated that efficient separation was possible in extraction channels with aspect ratios ( a R = w d ) much less than 50 and in channels with
diffusion direction dimension much less than 100µm. Both the aspect ratio requirement and diffusion direction requirement had previously been claimed to be essential to successful extraction device operation (Williams et al. 1992). Extraction stream inlet
Product stream outlet
Width, w
Extraction channel length, L
Channel depth, d
Non-diffusing particles
x
Diffusing particles
(0, 0)
y Coordinate system origin By-product stream outlet
Sample stream inlet
(a) Sample stream inlet
Product stream outlet
By-product stream outlet
W
id
th
di
m
en
sio
n,
w
Extraction stream inlet
silicon channel, 850µm Pyrex cover slip, 350µm Extraction channel length, L
x
(0, 0)
Channel depth, d
y Coordinate system origin
(b) Extraction channel length, L
Extraction ˙ stream Ves Sample ˙ stream Vss
xp xs
V˙ps Product stream Channel depth, d V˙bps By-product stream
x
(0, 0)
y Coordinate system origin
(c) Fig. 2 Diffusion-based extraction devices and device nomenclature: (a) configuration for smaller sample processing volumes w d ≤ 1, d < 100µm , (b) planar extraction device configuration for larger processing volumes 1 < w d V˙ps,min .
(22)
RESULTS AND DISCUSSION The methodology is applied to the design of an optimal device for the extraction of albumin (a protein constituent present in human blood) from a carrier sample stream with viscosity approximately that of water. Figure 4 presents the equilibration length and required flow pressure differential process space for a family of diffusion extraction devices designed for a α % = 99% equilibration length threshold. The precision of the series solution (Eq. 12) increases as the number of series summation terms is increased. In general, more terms are required to achieve a desired level of precision the for positions in the channel length closer to the sample and extraction stream inlet, y˜ = 0 . We evaluated this position dependent precision and determined that 50 terms were adequate to insure 0.01% precision in the use of Eq. 12 for channel length coordinate positions y˜ = 0.1 . The product stream flow rate, Reynolds number, and Peclet number are plotted on the abscissa of Fig. 4 (a) and (b). The physical constants used in generating the process space are Di =
7 ⋅ 10 −11 m 2 / s (albumin), µ = 10 −3 Pa / s (water), and ρ = 10 3 kg / m 3 (water). These physical constants are
approximately correct for a dilute aqueous solution of albumin. The parameters are; α % = 99% , x˜ s = 0.5 , x˜ p = 0.5 , and w = 4 mm . In Fig. 4 (a) the equilibration length, Lα % =99 , is shown to be a
linear function of V˙ps at a given channel depth, d . Equation 12 demonstrates the exponential dependence of concentration as a function of position y˜ within the extraction channel and the Peclet number. This exponential dependence may be actively controlled by changing the mean extraction channel flow velocity, v y , or through device design by changing the channel depth, d . If the channel depth, d , is reduced as v y is increased in proportion (resulting in the same value of 1 / Pe ) then Lα % =99 will remain
unchanged. If only the mean velocity, v y is increased then longer channel lengths are required and convection effects increase. The design Reynolds number and Peclet number is determined by the design objective function and the design constraints. If the Reynolds and/or Peclet number were added to the design constraints the optimal design geometry will change if the new constraint functions further restrict the process space. In this study maximum throughput is desired and a unique boundary optimum defines the parameters for such a device. This optimum design point falls on the border of the acceptable design space. Uncertainties in model parameters such as constituent diffusivity, flow rate control, and deviations of model behavior from true behavior are expected. Therefore, the “optimal design” may not be a good balance of these uncertain factors. The monotonic nature of the process space curves will however not change unless a discontinuous unmodeled phenomenon occurs (such as cell aggregation). In our example we use a single viscosity fluid for both the sample and extraction streams and such phenomenon is not expected. Therefore, the underlying conclusion that for optimal performance (as specified by the objective function) the device should have maximum length and minimum depth will remain unchanged in the presence of these uncertain factors. Operation of this device has been presented for a single viscosity fluid with a single diffusing constituent. Ultimately, the device is intended for use with particulate laden sample streams exhibiting non-Newtonian characteristics. Theoretical and experimental
examination of the effects of high particulate volume fraction fluids with arbitrary carrier fluid viscosity and arbitrary carrier fluid density are the focus of continuing work by this group. w = width = 4 mm
50
L>Lmax, Design violates silicon real estate constraint
45
99% Equilibration Length [mm]
40
h=100 µm
h>h max, Micro-channel
80 µm
90 µm
35 greater than maximum
60 µm
70 µm
allowable for efficient diffusion.
30
50 µm
The "Optimal" Design
40 µm
25
30 µm
20
The constrained process space
15
h
