Sample Preparation Meets Farey Sequence: A New Design ...

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Sample Preparation Meets Farey Sequence: A New Design Technique for Free-Flowing Microfluidic Networks Tapalina Banerjee * , Sudip Poddar , Tsung-Yi Ho , and Bhargab B. Bhattacharya § * Department of Computer Science and Engineering, JIS University, Kolkata, India Institute for Integrated Circuits, Johannes Kepler University Linz, Austria Department of Computer Science, National Tsing Hua University, Hsinchu, Taiwan § Department of Computer Science and Engineering, Indian Institute of Technology Kharagpur, India {tapalinabanerjee1, sudippoddar2006, bhargab.bhatta}@gmail.com, and [email protected] Abstract—Design of microfluidic biochips has led to newer challenges to the EDA community due to the availability of various flow-based architectures and the need for catering to diverse applications such as sample preparation, personalized medicine, point-of-care diagnostics, and drug design. The ongoing Covid-19 pandemic has increased the de- mand for low-cost diagnostic lab-on-chips manifold. Sample preparation (dilution or mixing of biochemical fluids) is an indispensable step of any biochemical experiment including sensitive detection and successful assay execution downstream. Although for valve-based microfluidic biochips various design automation tools are currently available, they are expensive, and prone to various manufacturing and operational defects. Additionally, many problems are left open in the domain of free-flowing biochips, where only a single layer of flow-channels is used for fluid-flow devoid of any kind of control layer/valves. In this work, we present a methodology for designing a free-flowing biochip that is capable of performing fluid dilution according to user’s requirement. The proposed algorithm for sample preparation utilizes the Farey-sequence arithmetic of fractions that are used to represent the concentration factor of the target fluid. We also present the detailed layout design of a free-flowing microfluidic architecture that emulates the dilution algorithm. The network is simulated using COMSOL multi-physics software accounting for relevant hydrodynamic parameters. Experiments on various test-cases support the efficacy of the proposed design in terms of accuracy, convergence time, reactant cost, and simplicity of the fluidic network compared to prior art. Index Terms—Free-flowing, Fluidic network, Farey sequence, Lab-on- Chip, Microfluidics, Sample preparation. I. I NTRODUCTION Microfluidic lab-on-chips (LoCs) have fueled the automation of biochemical protocols on a tiny device by enabling the move- ment of fluids on a minuscule scale through microchannels as in Continuous-Flow Microfluidic Biochips (CFMBs), or on patterned- array of electrodes as discrete droplets for Digital Microfluidic Biochips (DMFBs). An LoC (a.k.a. biochip) encapsulates several fluidic modules that are capable of performing the desired sequence of fluidic functions needed to execute a biochemical protocol (assay), and sensors that report the outcome. LoCs provide fast and low- cost solutions to various applications such as sample-preparation, pathological diagnostics, DNA-sequencing, polymerase chain reac- tion (PCR), cell sorting, drug design, therapeutics, and synthetic biology, to name a few. The demand for biochips as a diagnostic kit has skyrocketed with the ongoing Covid-19 pandemic. Continuous-flow microfluidic biochips (CFMBs) manipulate fluids in a network of microchannels using pressure-driven microvalves. CFMBs can be of two types: (i) valve-based and (ii) valve-free (free- flowing). The former type uses multi-layer technology to fabricate flow-layer and control-layer. Micro-valves are actuated at the inter- section of the control and the flow layer to enable or stop fluid- flow through the latter. Many complex fluidic modules such as mixer, multiplexer, separator, pump, and storage units can be built on a single chip utilizing thousands of micro-channels and valves [1]. Although the use of control-valves provides the benefit of programmability, these chips suffers from various fluidic errors due to various defects in elastic micro-channels and micro-valves [2], [3]. A simpler technology for designing CFMBs is to use free-flowing fluid through rigid channels, where no control layer or valves are needed [4]–[7]. Free-flowing biochips only require a switching mech- anism to inject or stop fluids at the inlets (or input ports). The fluid flow through the micro-channels is fully controlled by their global hydraulic resistance and injection velocities at input ports. Tradi- tionally, these chips were more popular among biochemists, as they are inexpensive, easy to fabricate, and suitable for high-throughput applications. However, free-flowing chips are mostly fully customized and cannot be re-programmed easily for general usage. Since there is almost no control on fluid navigation, it is not straightforward to implement protocols such as dilution or mixture preparation, which require high degree of programmability, flexibility, and precision on concentration factors. Additionally, the layout design of a free- flowing chip should satisfy certain hydrodynamical requirements in order to avoid turbulence through the channels. The ubiquity of sample preparation (SP), which refers to the task of preparing dilution or mixture of fluids in a certain ratio, is felt in almost all biochemical protocols. Many algorithms for automated sample preparation are known both for DMFBs [8]–[12] as well as for CMFBs [12]–[16]. Most of the on-chip microfluidic modules support the traditional (1:1) mix-split model where two equal-volume fluids are mixed and then split into two equal parts. Dilution (mixing) of fluids to achieve a desired target concentration factor (CF) (or a ratio of CFs) is implemented using a sequence of such operations. Note that CF is expressed as a fraction 0 CF 1. Clearly, the CF of the resulting mixture following one (1:1) mix operation is the arithmetic mean of the two parent-CFs. As a consequence, in most of the SP-algorithms, a target-CF is first approximated as an n-bit binary fraction, where n is a positive integer that determines the desired accuracy of CFs. The reliability of sample preparation depends on the correctness of (1:1) mixing steps and balancing of split operations during implementation [17]–[21]. SP-algorithms usually aim to minimize the number of mix-split operations (assay- time) [9], reactant-usage (cost) [11], or waste production [22], and their complexities largely depend on the value of n, the required output volume, and the availability of physical resources on-chip. Free-flowing biochips are easy to fabricate and more reliable as only a few control ports are used. However, for these biochips, SP- algorithms are mostly unexplored because in the absence control valves, volume-metering during mixing or splitting of bulk fluid is extremely difficult. No electronic design automation (EDA) tool is currently available that is capable of synthesizing such chips. One has to model several fluidic parameters (e.g., density, viscosity, inertial arXiv:2107.12463v1 [cs.ET] 26 Jul 2021

Transcript of Sample Preparation Meets Farey Sequence: A New Design ...

Page 1: Sample Preparation Meets Farey Sequence: A New Design ...

Sample Preparation Meets Farey Sequence: A New DesignTechnique for Free-Flowing Microfluidic Networks

Tapalina Banerjee∗, Sudip Poddar†, Tsung-Yi Ho‡, and Bhargab B. Bhattacharya§∗Department of Computer Science and Engineering, JIS University, Kolkata, India†Institute for Integrated Circuits, Johannes Kepler University Linz, Austria

‡Department of Computer Science, National Tsing Hua University, Hsinchu, Taiwan§Department of Computer Science and Engineering, Indian Institute of Technology Kharagpur, India{tapalinabanerjee1, sudippoddar2006, bhargab.bhatta}@gmail.com, and [email protected]

Abstract—Design of microfluidic biochips has led to newer challengesto the EDA community due to the availability of various flow-basedarchitectures and the need for catering to diverse applications suchas sample preparation, personalized medicine, point-of-care diagnostics,and drug design. The ongoing Covid-19 pandemic has increased the de-mand for low-cost diagnostic lab-on-chips manifold. Sample preparation(dilution or mixing of biochemical fluids) is an indispensable step ofany biochemical experiment including sensitive detection and successfulassay execution downstream. Although for valve-based microfluidicbiochips various design automation tools are currently available, theyare expensive, and prone to various manufacturing and operationaldefects. Additionally, many problems are left open in the domain offree-flowing biochips, where only a single layer of flow-channels is usedfor fluid-flow devoid of any kind of control layer/valves. In this work,we present a methodology for designing a free-flowing biochip that iscapable of performing fluid dilution according to user’s requirement. Theproposed algorithm for sample preparation utilizes the Farey-sequencearithmetic of fractions that are used to represent the concentrationfactor of the target fluid. We also present the detailed layout designof a free-flowing microfluidic architecture that emulates the dilutionalgorithm. The network is simulated using COMSOL multi-physicssoftware accounting for relevant hydrodynamic parameters. Experimentson various test-cases support the efficacy of the proposed design in termsof accuracy, convergence time, reactant cost, and simplicity of the fluidicnetwork compared to prior art.

Index Terms—Free-flowing, Fluidic network, Farey sequence, Lab-on-Chip, Microfluidics, Sample preparation.

I. INTRODUCTION

Microfluidic lab-on-chips (LoCs) have fueled the automation ofbiochemical protocols on a tiny device by enabling the move-ment of fluids on a minuscule scale through microchannels as inContinuous-Flow Microfluidic Biochips (CFMBs), or on patterned-array of electrodes as discrete droplets for Digital MicrofluidicBiochips (DMFBs). An LoC (a.k.a. biochip) encapsulates severalfluidic modules that are capable of performing the desired sequence offluidic functions needed to execute a biochemical protocol (assay),and sensors that report the outcome. LoCs provide fast and low-cost solutions to various applications such as sample-preparation,pathological diagnostics, DNA-sequencing, polymerase chain reac-tion (PCR), cell sorting, drug design, therapeutics, and syntheticbiology, to name a few. The demand for biochips as a diagnostickit has skyrocketed with the ongoing Covid-19 pandemic.

Continuous-flow microfluidic biochips (CFMBs) manipulate fluidsin a network of microchannels using pressure-driven microvalves.CFMBs can be of two types: (i) valve-based and (ii) valve-free (free-flowing). The former type uses multi-layer technology to fabricateflow-layer and control-layer. Micro-valves are actuated at the inter-section of the control and the flow layer to enable or stop fluid-flow through the latter. Many complex fluidic modules such as mixer,multiplexer, separator, pump, and storage units can be built on a singlechip utilizing thousands of micro-channels and valves [1]. Although

the use of control-valves provides the benefit of programmability,these chips suffers from various fluidic errors due to various defectsin elastic micro-channels and micro-valves [2], [3].

A simpler technology for designing CFMBs is to use free-flowingfluid through rigid channels, where no control layer or valves areneeded [4]–[7]. Free-flowing biochips only require a switching mech-anism to inject or stop fluids at the inlets (or input ports). The fluidflow through the micro-channels is fully controlled by their globalhydraulic resistance and injection velocities at input ports. Tradi-tionally, these chips were more popular among biochemists, as theyare inexpensive, easy to fabricate, and suitable for high-throughputapplications. However, free-flowing chips are mostly fully customizedand cannot be re-programmed easily for general usage. Since thereis almost no control on fluid navigation, it is not straightforward toimplement protocols such as dilution or mixture preparation, whichrequire high degree of programmability, flexibility, and precisionon concentration factors. Additionally, the layout design of a free-flowing chip should satisfy certain hydrodynamical requirements inorder to avoid turbulence through the channels.

The ubiquity of sample preparation (SP), which refers to the taskof preparing dilution or mixture of fluids in a certain ratio, is feltin almost all biochemical protocols. Many algorithms for automatedsample preparation are known both for DMFBs [8]–[12] as well as forCMFBs [12]–[16]. Most of the on-chip microfluidic modules supportthe traditional (1:1) mix-split model where two equal-volume fluidsare mixed and then split into two equal parts. Dilution (mixing)of fluids to achieve a desired target concentration factor (CF) (ora ratio of CFs) is implemented using a sequence of such operations.Note that CF is expressed as a fraction 0 ≤ CF ≤ 1. Clearly, theCF of the resulting mixture following one (1:1) mix operation isthe arithmetic mean of the two parent-CFs. As a consequence, inmost of the SP-algorithms, a target-CF is first approximated as ann-bit binary fraction, where n is a positive integer that determinesthe desired accuracy of CFs. The reliability of sample preparationdepends on the correctness of (1:1) mixing steps and balancingof split operations during implementation [17]–[21]. SP-algorithmsusually aim to minimize the number of mix-split operations (assay-time) [9], reactant-usage (cost) [11], or waste production [22], andtheir complexities largely depend on the value of n, the requiredoutput volume, and the availability of physical resources on-chip.

Free-flowing biochips are easy to fabricate and more reliable asonly a few control ports are used. However, for these biochips, SP-algorithms are mostly unexplored because in the absence controlvalves, volume-metering during mixing or splitting of bulk fluid isextremely difficult. No electronic design automation (EDA) tool iscurrently available that is capable of synthesizing such chips. One hasto model several fluidic parameters (e.g., density, viscosity, inertial

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force, etc) in order to understand the underlying fluid dynamics.Moreover, the functionality of such chips is highly sensitive tothe dimension of microfluidic channels, input pressure, and theoverall fabric of the network, which determines the global hydraulicresistance, and in turn, the output flow. All such limitations slowdown the cycle of “Design-Prototyping-Testing”, thereby increasingthe cost of such chips and turn-around-time.

In this paper, our contribution is two-fold: (i) first, we develop anew SP-algorithm for producing dilution that can be easily emulatedwith a free-flowing chip, and (ii) present a methodology for designingthe chip with details of its physical design. This chip is programmablein the sense that by choosing appropriate fluid inputs, we can producean output flow with a desired concentration factor.

The proposed algorithm introduces a non-traditional mixing modelfor the first time, where target-CFs are represented by rationalnumbers based on Farey sequence [23]–[25] instead of using classicalbinary fractions (i.e., where the denominator is a power of two only).We show that such a model supports different fluid-mixing strategies,which are suitable for implementation with a free-flowing biochip.We next construct a 3D-model of the proposed channel architectureand study its behavior via COMSOL simulation framework [26]. Thechip does not need any fluidic split operation and thus errors dueto unbalanced splitting are completely avoided. Unused fluids arerecycled efficiently and the output flow achieves quick convergenceto the desired target-CF. The accuracy of CF obtained by this methodcompares favorably with prior art.

As stated above, we introduce a new method to discretize theCF-range using a Farey sequence Fm of order m, which is anincreasing sequence of all irreducible rational numbers a

b, where

a, b, and m are positive integers, and 0 < a < b ≤ m [23]–[25]. The two boundary values 0 and 1 are also included in thesequence. Thus, given a value of a positive integer m, this allowsus to approximate a target-CF using a fraction a

b. The value of m

can be chosen depending on the accuracy level in CF as close tothat used in traditional methods such as twoWayMix [27] or REMIA[11]. It is known that the number of elements in Fm asymptoticallyapproaches to 3m2

π2 [23], and thus the closed interval [0, 1] for theCF-range can be discretized by choosing a suitable subset of nearlyequi-spaced fractions among the elements of the Farey sequence ofa given order. Based on the above discretization scheme, we proposea generalized mixing model (p : q) and a new dilution algorithmnamely FSD (Farey-Sequence based Dilution). In order to performhomogeneous mixing operations, we used flow-equalizers followedby inter-digitized orthogonal obstacles inside the microchannels. Suchmicrofluidic channels can be easily manufactured using availabletechniques [28]. We validate the functionality of the network usingCOMSOL Multiphysics Software [26] and report performance resultsfor several random and real-life test-cases.

The rest of the paper is organised as follows: Section II presentsthe basics of sample preparation; Section III describes prior-workrelated to this research. We explain the motivation behind this workin Section IV. The proposed methodology is discussed in SectionV followed by architectural design detailing the fluidic network inSection VI. Simulation framework, experimental results on randomand real-life target-CFs, and comparisons with earlier approaches arereported in Section VII. Finally, Section VIII draws the conclusion.

II. PRELIMINARIES OF SAMPLE PREPARATION

In dilution, a raw sample fluid is mixed with a buffer fluid in acertain volumetric ratio to achieve desired target concentration factors(CF, 0 ≤ CF ≤ 1) of a sample, where the CF of a raw sample

(buffer) is considered as 1 (0) [9], [27]. In the (p : q) model, a mixingoperation is performed between p-unit and q-unit volume fluids withCF = C1 and C2, respectively. Thus, the CF of the resultant mixturebecomes p·C1+q·C2

p+q. It reduces to the classical (1 : 1) mix-split

model when the value of p and q become 1. Given a target-CF,a specific sequence of mix-split operations need to be performed(utilizing sample and buffer fluids) to achieve the desired target-CFof a sample. The complete sequence of such mix-split operations canbe graphically visualized with a directed acyclic-graph called mixingtree/dilution graph [9], [27]. The depth d (d > 0) of the mixingtree represents the accuracy n (n > 0) of a target-CF, which isdetermined by a user-specified error-tolerance limit ε (0 ≤ ε < 1).In order to limit the CF-error in the target-droplet by 1

2n+1 , each CFis represented as x

2n(n is a positive integer, and x is a non-negative

integer such that 0 ≤ x ≤ 2n), for the special case of the (p : q)mixing model where p = q.

III. RELATED WORK

SP-algorithms for implementation on valve-based CFMB chips wereproposed in [14], [16] where valve-states (open/close) need to becontrolled online following pre-scheduled dynamics. Albeit they offerthe advantage of programmability to some extent, malfunctioning ofcontrol-valves may lead to errors in fluidic operations or performancedegradation. For example, valve expansion might affect the pressuregradient in the microchannels of PDMS based CFMBs. Asynchronousvalve actuations may cause errors in fluidic operations and slow downthe response time [29], [30]. Any defect in elastic micro-valves ormicro-pumps may also introduce various microfluidic errors in valve-based CFMBs [2], [3]. Free-flowing programmable architectures thatobviate the need for valves as well as split operations were proposedin [4], [20], [21] to overcome these limitations. The work in [4]studies a 2D microfluidic network for implementing dilution on afree-flowing biochip. In order to capture subtleties of mixing anddiffusion phenomena in a more realistic fashion [31], a 3D-model forthe fluidic network was presented later [20]. These designs, however,need variable injection-velocities at input ports and precise timingsequence for fluid injection, which again are hard to implement inpractice. Free-flowing biochips resembling a binary tree was proposedin [21] to mitigate these problems. However, on the negative side,they require a special set of concentration values to be fed as inputs.

IV. MOTIVATION

The impact of several parameters that govern fluid flow through anetwork of channels needs to be considered while designing thechannel structures and physical layout of a free-flowing biochip.Examples of such parameters include (i) density, (ii) dynamic andkinematic viscosities, (iii) fluid-injection velocities at input ports,and (iv) the volume of fluid injected into the system per unittime. Moreover, in sample preparation (e.g., dilution), some of theseparameters may also depend on the CF of the solution. Hence, thehydraulic resistance, and consequently, the rate of fluid flow as wellas miscibility also depend on the CF-value of the fluid.

Note that the volume and injection velocity of fluids in a channelnetwork can be controlled by choosing appropriate physical dimen-sions of input ports and input pressure. In previous approaches topreparing dilution with free-flowing biochips [4], [20], a large volumeof fluids is wasted initially before the stablity of flow is reached at thesink-outlet(s). Also, fluid-injection velocities at different input portsalong with their corresponding injection times, play a vital role inachieving the stability and accuracy of target-CF. Adjustment of suchvelocities and timing is a challenging task [4], [20].

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In digital microfluidics as well as in valve-based CFMBs, volu-metric errors that often occur during split operations cause errorsin target-CFs [17]–[19]. Additionally, split operations may lead tothe wastage of costly fluids. In free-flowing chips, this problem isavoided at the cost of consuming additional input fluids and producinga larger amount of target fluids. The combination of density, dynamicviscosity, fluid-velocity, and channel length determines the Reynoldsnumber [32] of the flow, which in turn, dictates the nature of fluid-motion (such as turbulance or laminar).

Two issues thus need to be addressed while designing free-flowingCFMBs:

� off-line issues such as the selection of underlying dilutionalgorithms, physical layout of the chip, optimization parameters,design complexities;

� online issues such as handling of microvalves, the choice ofinjection velocities and timing management so as to achievefaster stability of flow and desired accuracy of target-CF.

In the proposed approach, we need to set the states of each inletport a-priori (offline), i.e., before executing the dilution assay on thefree-flowing biochip. Thus, it does not require any real-time control ofports. We also present a new architecture based on Farey sequencethat supports programmability, and at the same time, obviates theneed for different injection velocities and precise online injectiontiming. This improves upon other free-flowing fluidic architecturesproposed earlier [4], [20], [21]. We construct a 3D-model of theproposed channel architecture considering shear stress between thewall and fluid layer as well as those between adjacent fluid layers inboth parallel and perpendicular directions, and study its behavior viaCOMSOL simulation framework.

V. PROPOSED METHODOLOGY

In this section, we will describe a new methodology for designingthe free-flowing 3D microfludic biochips for sample preparation.A. Representation of CF-values using Farey sequence

As stated earlier, in almost all prior state-of-the-art dilution algo-rithms both in DMFB and CFMB domains, each target-CF (tCF) isrepresented as x

2n, where 2 appears in the denominator because of the

underlying (p : p) mixing-model; n is a positive integer. Thus, theCF-values ranging from 0% to 100% are discretized as 0

2n, 1

2n, · · ·

2n−1

2n, 2n

2n, n denoting the accuracy level [4], [20], [21]. We call this

as binarized sequence BS of order n (i.e., BSn). For example, thepossible target-CF values of a binarized sequence of order 3 (BS3)are shown in the bottom row in Fig. 1(a) for accuracy level = 3. It canbe seen that the target-CFs appear following a gap of 1

2nimplying

that the CF-error in the target-CF lies within ± 0.52n

. Therefore, thetarget-CF = 44.375

64(see Fig. 2(b)) will be approximated as 44

64. Hence,

the error in the target-CF becomes −0.37564

.In this work, we relax the constraint on the denominator part of

each tCF, and propose a dilution algorithm for flow-based biochipsbased on Farey-sequence arithmetic [23], [33]. More precisely, werepresent each target-CF using an element of a Farey-sequence. TheFarey sequence [23] of order m = 1 to 8 are listed below:F1 = { 0

1, 11},

F2 = { 01, 12, 11},

F3 = { 01, 13, 12, 23, 1

1},

F4 = { 01, 14, 13, 12, 23, 34, 11},

F5 = { 01, 15, 14, 13, 25, 12, 35, 23, 34, 45, 11},

F6 = { 01, 16, 15, 14, 13, 25, 12, 35, 23, 3

4, 4

5, 56, 11},

F7 = { 01, 17, 16, 15, 14, 27, 13, 25, 37, 12, 47, 35, 23, 57, 34, 45, 56, 67, 11},

F8 = { 01, 18, 17, 16, 15, 14, 27, 13, 3

8, 25, 37, 12, 47, 35, 58, 23, 57, 34, 45, 56, 67, 78, 11}.

For example, the set of CFs derived from F13 is shown in Fig. 1(a).It has been observed from Fig. 1(c) that the number of representableCF-values by the proposed method is much higher than binarizeddiscretization for accuracy level = 3. Thus, for target-CF = 44.375

64,

the approximated CF becomes 913

(Fig. 2 (b)), and hence, the error inCF becomes smaller (i.e., 0.067

64) compared to binarized discretization.

The rationale behind choosing a value for m lies in the physicaldesign of the proposed diluter architecture (shown in Fig. 4) whichwill have n inlets for injecting the sample fluid and n inlets forinjecting the buffer fluid. Furthermore, the cross-section of theseinlets will be progressively doubled. Thus, the width of the rect-angular inlets would be 1x, 2x, 22x, 23x, ..., 2n−1x (x representsthe unit). Hence, both for sample and buffer fluids, the minimum(maximum) amount that can be injected into the channel is 0 (2n -1) units. Now, starting from the Farey sequence F2n+1−3, we canconstruct a shorter sequence yet having a larger number of termsthan those in BSn. We name this as FSD2n . Note that FSD2n is aproper subset of Fm, where m = 2n+1 − 3. Algorithm 1 describedlater how to construct FSD2n from F2n+1−3. In the beginning (Line

ALGORITHM 1: Generation of FSD2n .Input: Order (accuracy) n (≥ 1) of BSnOutput: FSD sequence of order 2n.

1 Generate the general sequence of CFs (BSn) of order n ; i.e.; BSn= { 0

2n, 12n

, 22n

, .... , 2n−1

2n, 2n

2n}

2 Generate the numbers following Farey sequence based arithmetic oforder m (Fm), where m = 2n+1 − 3 ; i.e.; Fm = { 0

m, 1m−1

,1

m−2, .... , m−1

m, mm}

3 Replace boundary values of Fm, i.e., { 01} and { 1

1} with the values

of buffer ( 02n

) and sample ( 2n

2n), respectively;

4 Select a set of numbers RF (of order 2n) from Fm (i.e., RF2n ⊂Fm) as follows:

(a) RF2n = {x | x ∈ { 02n

, φψ

, 2n

2n} }, where φ

ψ∈ Fm, (0 < (φ,

ψ) < 2n), and ((ψ - φ) < 2n)5 Select a set of numbers FSD2n from RF2n (i.e., FSD2n ⊂

RF2n ) based on:(a) FSD2n = {x | x ∈ 0

2n, φψ

, 2n

2n}, where φ

ψ∈ {BSn}

⋃{ y2n

< φψ< 2y+1

2n+1 | where φ is minimum among all numerators of

( y2n

, 2y+12n+1 )}

⋃{ 2y+12n+1 < φ

ψ< y+1

2n| where φ is minimum among

all numerators of ( 2y+12n+1 , y+1

2n)}

6 return FSD2n

1), Algorithm 1 creates the binarized sequence of CF-values BSn(Fig. 1 (d)). In Line 2, all elements of Farey sequence of order m(Fm) are created. In Line 3, the boundary values of Fm, i.e., { 0

1}

and { 11} are replaced with 0

2n(buffer CF) and 2n

2n(sample CF),

respectively. For n = 3, we have shown a Farey sequence of order13 (F13 = 2(3+1) − 3) in Fig. 1 (a) which comprises 59 elements.First, the fractions are eliminated for which the numerators are greaterthan 7 (=23 - 1) or the difference between the numerator and thedenominator exceeds 13 (=2(3+1) - 3). This set is called reducedFarey sequence (RF8) as shown in Fig. 1 (b) (Step 3). This reducedFarey sequence is equivalent to { a

a+b| a, b ∈ {0, 1, · · · , 2n−1}},

where a and b are positive integers such that 0 ≤ a, b ≤ 2n − 1,(a + b) 6= 0. Note that the underlying architecture exactly supportsthe same sequence: that is mixing of a units of sample with b unitsof buffer producing target-CF = a

a+b. After execution of Step 4, we

form the sequence FSD8. Note that BS3 has fewer elements thanFSD8 between the open interval (0, 1). In general, first includeall numbers of BSn into FSD2n . Then, select a set of numbersfrom RF2n within the range of ( 1

2n, 2n−1

2n) and insert them into

FSD2n as follows. Suppose two consecutive elements of FSD2n

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(b) RF8

(c) FSD8

(d) BS3

(a) F13

0/8

8/8

0/8

1/8

1/4

3/8

1/2

5/8

3/4

7/8

8/8

0/8

1/8

1/4

3/8

1/2

5/8

3/4

7/8

8/8

1/6

1/5

2/7

1/3

2/5

4/9

5/9

3/5

2/3

5/7

4/5

5/6

Fig. 1. Derivation of FSD8 sequence from Farey sequence (F13).

064

6464

464

3264

4864

4464

error= ±0.5

64

allowable

164

264

364

4364

4564

064

6464

464

3264

4864

target-CF = 913

164

143

133

1116

710

132

(a) BS 6 sequence

(b)FSD64 sequence

target-CF =

Fig. 2. (a) Representable CF -values for n = 6 used in traditional SP-Algorithms; (b) by the proposed method, which offers more dense discretiza-tion for the same value of n.

are ( y2n

, y+12n

) (y = 1, 2, 3, · · · 2n−2). Then, the algorithm selects twonumbers (among a set of numbers between every two consecutivenumbers ( y

2n, y+1

2n)) from RF2n and insert them into FSD2n . Note

that one number, with minimum numerator value is selected withinthe range ( y

2n, 2y+1

2n+1 ) and another with minimum numerator valueis selected within the range ( 2y+1

2n+1 , y+12n+1 ). Here, minimum values

are selected for reducing the sample consumption. All CF-values ofFSD8 generated by the proposed method is shown in Fig. 1 (c).Thus, for the same CF-range, the proposed method generates moreCF-values (21) compared to binary methods (8 CF-values for n = 3).Hence, the proposed method approximates a given CF with higheraccuracy for a given value of n. Here, another interesting thing isthat RF2n = { a

a+b| a, b ∈ {0, 1, · · · , 2n−1} is a sub-sequence of

Farey sequence F2n+1−3 and FSD2n is a sub-sequence of RF2n .Theorem 1: The numerator of each irreducible fraction element in

FSD2n is less than 2n. Also, the difference between denominatorand numerator is also less than 2n.

Proof: Numerator of each φψ

(i.e., φ) must be within the openinterval (0, 2n). In the proposed diluter network, the activation of allsample inlets at a time can inject

∑n−1i=0 2i = (2n−1) units of sample

fluid. Thus φ < 2n. Also, φ > 0, since some units of sample fluidshould be present in the dilution. The difference between denominatorψ and numerator φ represents the number of buffer fluid units usedin the dilution to achieve tCF = φ

ψ. Note that the activation of all

buffer inlets at a time can inject∑n−1i=0 2i = (2n − 1) buffer units.

Thus, (ψ - φ) < 2n.Theorem 2: |FSD2n | > |BSn|, i.e., the number of elements in

FSD2n is always greater than the number of elements in BSn.Proof: This follows from step 5 of Algorithm 1. Let us consider

two consecutive elements from BSn sequence t2n

and t+12n

. The

middle element of these two is 2t+12n+1 (= { t

2n+ t+1

2n}/2). Next, we

choose φψ

in such a way that the numerator φ is minimum amongall those lying between the open interval of the lower half, i.e.,( t2n

, 2t+12n+1 ). Find another φ

ψwhere the numerator φ is minimum

among all those lying between the open interval of the upper half,i.e., ( 2t+1

2n+1 , t+12n

). Thus |BSn| = 2n and |FSD2n | = 3(2n − 1).Theorem 3: The set FSD2n includes the set BSn, i.e., FSD2n

⊃ BSn.Proof: Follows easily from the construction of FSD2n as de-

scribed in Algorithm 1.

B. Mixing tree based on Farey-approximation of tCF

The proposed method first maps the target-CF (in the form φψ

)to its nearest element of farey sequence FSD2n for creating themixing tree (which represents the sequence of mixing steps forachieving the desired target-CF). In order to accomplish this task, itadopts the FindClosestFast technique [33]. For example, if we invokeFindClosestFast, the nearest element of tCF = 44.375

64(= 69.3%)

becomes 913

(i.e., with CF-error = 0.06764

) for n = 6, as shown inFig. 2 (b). Here, we skip the algorithm details due to the limitationof allowable pages.

Buffer

Sample

Intermediate

V = Unit volume

Required Sample Units = (9)10 = (001001)2Required Buffer Units = (13 − 9)10 = (4)10 = (000100)2

Fluid

Fluid

Fluid

Fluid

1 V

1 V

8 V

5 V

4 V

13V

CF = 1 CF = 15

CF = 913

For tCF= 69.3%, the nearest value in FSD64 sequence is 69.23% = 913%

Fig. 3. FSD tree for tCF= 913

.

We will now demonstrate how given a target-CF, Farey arithmeticcan be used to determine the required sequence of mixing operations.Fig. 3 shows the mixing tree to achieve the target-CF= 9

13. In

order to produce this target concentration, we need to mix 9 unitsof sample/reactant with 4 (= 13-9) units of buffer. We design afluidic network that comprises six inlets each for injecting sampleand buffer as shown in Fig. 4. Also, the cross-sections of the inletpipes are progressively doubled. The inlets can be made open orclosed under external control as desired. For this target-CF, 6-bitbinary representations of φ (9)10 and ϕ (4)10 become (001001)2and (000100)2, respectively. The states (fluid-in/closed) of the inletports from left-to-right in Fig. 4 needed to feed (stop) sample (buffer)into the mixing channel correspond to the respecting bits of φ (ϕ) as

Page 5: Sample Preparation Meets Farey Sequence: A New Design ...

Si = Inlets for sample fluid

Bi = Inlets for buffer fluid

B5Flow at Outlet

Flowequalizer

Inter-digitizedobstacles

B5

chamber ofbuffer fluid

Reusechamber ofsample fluid

B1 B2B3

B4B5

B6

S1 S2 S3 S4 S5 S6

S5direction towards

OPEN CLOSE Reuse Chamber

State of State of Resultant flow path

CLOSE OPEN Mixing Chamber

Reuse

red port green port

Fig. 4. 3D-view of the proposed microfluidic serpentine network.

seen from the right-to-left. The progressive doubling of cross-sectionsof inlet ports from left-to-right takes care of the positional weightsof the binary bits. In other words, by controlling the inlet ports, onecan enable φ (ϕ) units of sample (buffer) to enter into the mixingchannel so that the desired target concentration is produced at theoutput port.

VI. LAYOUT DESIGN

A. Proposed Layout

We will now describe the architecture and the layout design of theproposed free-flowing fluidic network, and required input assignmentsto produce a given target-CF . A three-dimensional model of theproposed fluidic network is presented that is tailored to implementthe intrinsic subtleties of FSD-algorithm. The design parameters ofthe network include height, width, and length of channels, the numberof input and output ports and their cross-sections, layout geometry(how the channels are interconnected and resistive fins are shaped),and pressure-specs (fluid-injection velocities and timing). Most of thedesign parameters are pre-determined and invariant regardless of thetarget-CF. However, the number of input ports will depend on thedesired accuracy-level (n) of target concentration. For demonstrationpurpose, we choose n = 6, thus the error in target-CF will beless than 1

128. The overall geometric layout of the corresponding

architecture is shown in Fig. 4, which has six inlet ports for injectingsample and buffer, each, and one output port, Thus, altogether it is a12-input, 1-output free-flowing microfluidic network with no controlvalves in the mixing channel. The serpentine channel is built usinginter-digitized obstacles that facilitate homogeneous mixing of fluids.The physical dimensions of the channel and obstacles are shown inFig. 5 (b) - Fig. 5 (g).

The cross-sections of input ports are made progressively doubledso that the volume of fluid injected per unit time through them underconstant pressure, increases two-fold at every consecutive inlets. Also,a controlling port is placed at every inlet which may be kept openor closed depending on which input needs activation. Unlike othervalve-based chips, these ports require only off-line control, i.e., theyare set at the beginning of the assay depending on the target-CF, anddo not need any further control during the execution of the assay.We keep the height of the entire channel network fixed at 25µm.On application of constant pressure to all inlets, the rate of fluidflow (speed) will remain constant throughout the entire length ofthe mixing channel. The fluid-flux (volume flowing per unit time)increases in geometric progression along its path until the flow finallyreaches the output port. We select input pressure in such a waythat laminar flow involving miscible fluids is guaranteed through thechannel network. Proposed method maintains laminar flow in the

A, B 27.5µmC, D 52.5µm

ZoneTurningRadius

E, F 102.5µm

AB

CD

E

F

(a)

16 µm

31.5 µm

50 µm

25 µm

67 µm

5 µm

Zone A16 µm

31.5 µm

67 µm50 µm

25 µm5 µm

Zone B

(b) (c)

Zone C

43 µm

57 µm

100 µm

25 µm

67 µm5 µm

Zone D

67 µm25 µm

5 µm

43 µm57 µm

100 µm

(d) (e)Zone E

5 µm

25 µm

200 µm116 µm

67 µm

87 µm

Zone F

25 µm

200 µm116 µm

87 µm

67 µm5 µm

(f) (g)Fig. 5. (a) Varying turning radii in the fluidic network; (b-g) the geometryof inter-digitized obstacles in different zones of the channel network.

fluidic network by controlling value of the Reynolds number (R) [32]using Equation 1:

Reynolds number R =Inertial force

V iscous force=ρul

µ(1)

A high (low) value of R indicates the existence of turbulent (laminar)flow in the network. In the proposed model, R is kept lower than 2000by proper choice of l (length of the channel), u (fluid-injection rate)which assure laminar flow throughout the fluidic network for givenvalues of ρ (density of fluid) and µ (dynamic viscosity).

The proposed network architecture comprises two sets of flowpaths. One of them corresponds to the regular mixing channel asdescribed above. The other one (re-use channel) enables the sampleand buffer fluids that are not needed at some inlets to return totheir respective source reservoirs. As shown in Fig. 4 and Fig. 6, anadditional open/close port controller is placed on a by-pass channelattached to each inlet. Fluid injected at an inlet arm can flow througheither the mixing channel or the re-use channel depending on the stateof the corresponding port controller, and their states are determinedby the given target-CF.

Note that two different re-use channels are incorporated in thefluidic network – one for buffer-fluid outflow and the other one forsample-fluid outflow with a separate port attached to each inlet. Portsattached towards the mixing channel and those towards the re-use

Page 6: Sample Preparation Meets Farey Sequence: A New Design ...

Flow at

X

Y

Port is open or “ON” state

Port is close or “OFF” state

Flowequalizer

Inter-digitizedobstacles

Reusechamber ofbuffer fluid

Reusechamber ofsample fluid

B1 B2 B3 B4 B5 B6

S1

S2 S3 S4 S5 S6

B5

S5

Outlet

Si = Inlets for sample fluid

Bi = Inlets for buffer fluid

Inter-digitizedobstacles

Fig. 6. States of inlet ports for the target CF = 913

(≡ 69.3%).

channel are operated in complementary fashion for each inlet (seeFig. 6). The rationale behind it is two-fold: first, they help equalize thehydraulic resistance confronted per unit area for every inlet; secondly,fluid-injection time through different inlets can be made concurrentirrespective of the target-CF. The components of input fluids that arerequired in the dilution process are allowed to enter into the mixingchamber, whereas those not needed are fully diverted towards there-use channel so that they can return to their respective sources.

In order to enable balanced entry of fluids towards the mixingzone and to facilitate homogeneous diffusion-based mixing, flow-equalizers (septums) are used followed by a series of inter-digitizedorthogonal obstacles (fins) in the mixing channel as shown in Fig.5. The first pair of inlets used for injecting sample and buffer areoriented akin to a Y -junction with its two arms converging with aseparation angle of 120◦ ; it is then cascaded with a bisecting straightmixing channel. After that, each inlet (either carrying sample orbuffer) arm is joined with each of its incoming and outgoing mixingchannels, with 120◦ orientation (Fig. 4).

B. Procedure for generating a target CF using the proposed layout

The network topology shown in Fig. 6 demonstrates the runningexample target CF = 9

13(≡ 69.3%). For each inlet port, the flow-

controlling state should be set properly in order to produce a giventarget-CF. As discussed, to achieve a target CF φ

ψ, φ unit of raw

sample/reactant needs to be mixed with ϕ(= ψ − φ) unit of bufferfluid. For example, for the target CF 9

13, the 6-bit binary forms of

φ and ϕ becomes 001001, and 000100, respectively. The weightedvalues of these binary bits correspond to the amount of bufferand sample to be injected, respectively. Therefore, a bit value 1(0) indicates that the corresponding inlet port towards the mixingchamber is set as open/ON (closed/OFF) to let the fluid enter into themixing chamber (re-use branch). The controller attached to the re-usebranch is set in opposite fashion. In Fig. 4, an open (closed) port iscolored as green (red). For the example target CF = 9

13, the states of

the inlets towards the mixing chamber would be the following whenviewed from left-to-right in Fig. 6: sample: (on, off, off, on, off, off);buffer: (off, off, on, off, off, off). As stated earlier, the controllerstowards the corresponding re-use branch are set in opposite states.Depending on the target CF, each inlet is set at the beginning of thedilution assay, and no further real-time control is required.

C. Throughput-sensitive choice of input flows

The proposed device also has another useful feature: For a giventarget-CF, the values of a and b can be chosen in different waysdepending on the required output flow-rate. The minimum throughputcorresponds to the elements of the FSD-sequence, where each of themis an irreducible fraction. However, for other choices of a and b thatyield the same value of a

a+b, we can obtain larger throughput. Some

examples are shown in Table I.

TABLE ITHROUGHPUT-SENSITIVE FLOW-RATE

Target-CF Sample Buffer Rate of output-flow

1 2 313

2 4 63 6 91 3 4

14

2 6 83 9 124 1 58 2 10

45

12 3 1516 4 2020 5 25

VII. EXPERIMENTAL RESULTS

We have simulated the proposed fluidic network using COMSOLMultiphysics software [26] and studied the performance of the FSD-algorithm for various synthetic and real-life test-cases. We observeless than 1% error in the target-CF with respect to the theoretical CFvalue in each instance.

A. Simulation framework

During simulation, we consider different fluidic properties such asdensity, viscosity, surface tension, diffusion coefficient, electricalconductivity, and thermal conductivity of each fluid. Also, channelsof the proposed 3D fluidic network model are initially assumed tobe filled with air similar to wet-lab experiments.

TABLE IICOMPUTATION PLATFORM

CPU Intel Core i7−3770CPU @ 3.40 GHz, 4 coresOperating System Windows 10

Software COMSOL Multiphysics 5.2Model Dimension 3D

Materials(incompresibleNewtonianFluid)

Water used as bufferSample Diffusion co-efficient of

sample in water [m2/sec]

Ethanol 1.24× 10−9

Glycerol 0.94× 10−9

Used Physics Laminar flow (SPF: Single Phase Flow)Transport of diluted species (TDS)

Meshing Predefined element size at extra fine whilecalibrating fluid dynamics

Study Time dependent(Time range: 0 sec - 200 sec; time step 1 sec)

Solver Direct PARDISO solver

Throughout all experiments, we consider the computational plat-form as summarized in Table II. We have used the following twophysics of COMSOL Multiphysics software to study the physicalprofile of fluids flowing through the channels.

• Single-phase laminar flow (SPF): It is utilized for studying fluidvelocity, fluid pressure, etc.

• Transport of diluted species (TDS): It is used for studying theconcentration profile.

For analyzing the dynamics of the fluids, SPF uses the Navier-Stokes Equation that captures the conservation of momentum of

Page 7: Sample Preparation Meets Farey Sequence: A New Design ...

fluids [32]. Equation 2 represents the Navier-Stokes equation forincompressible Newtonian fluids:

ρ(δu

δt+u ·5u) = −5p+5·(µ(5u+(5u)T )− 2

3µ(5·u)I)+F

(2)where velocity, pressure, density, and dynamic viscosity of fluid arerepresented by u, p, ρ, and µ, respectively. In Equation 2, inertialforces are represented by the left-hand-side term; the right-hand-sideterm represents the total effect of pressure force (− 5 p), viscousforce (5 · (µ(5u + (5u)T ) − 2

3µ(5 · u)I)), and external forces

(F ). Equation 3 represents the continuity which follows from theconservation of mass of fluid.

δρ

δt+5 · (ρu) = 0 (3)

This continuity equation [32] helps to solve these forces. Fick’s law[34] is used to compute the relative concentration of the sample. Also,the net transport is computable in the proposed fluidic network model.We have successfully used TDS physics along with the transportequation (Equation 4):

5 · (−Di · 5ci) + u · 5ci = Ri (4)

where Di represents the diffusion coefficient, ci is the concentrationfactor for fluid i, and Ri stands for the rate of reaction of the fluids.

B. Results and Discussions

We performed experiments for comparing the accuracy of theFSD based discretization and binarized discretization (BS) sequence,and observed that for an accuracy value n, FSD provides denserdiscretization of the concentration range [0, 1]. For example, foraccuracy level 6, binarization based discretization splits the interval[0, 1] into 64 segments as shown in Table III; whereas in FSD, it issplit into 188 intervals. Therefore, it is evident that in the proposedFSD framework, a given target-CF is approximately more accuratelycompared to the conventional binary discretization method.

We further performed experiments on 200 test-cases coveringseveral random target-CFs, some real-life test-cases such as BradfordProtein Assay, Sucrose Gradient Analysis, and Harmonic GradientAnalysis, and on some special dilution gradients e.g., Gaussian, sin x,√x, x2, Linear, 2-fold Log [35] (shown in Table IV). We notice that

CF-errors generated by the proposed method become much smallerthan those in other methods [4], [27]. This is also reflected fromFig. 7 (a) where CF-errors generated by the proposed method liescloser to the Y -axis. We also define differential cost of a reactantas follows: the absolute difference between reactant requirement asmandated by user’s specified target-CF and that actually neededwhen its nearest approximation on the discretized concentration rangeis used in computation. For both FSD- and BS-based methods, we

TABLE IIICOMPARISON AMONG THE NUMBER OF ELEMENTS IN THREE SERIES

Accuracy(n)

# elements inFarey sequenceof orderm (= 2n+1 − 3)

# elements inFSD sequenceof order 2n

(3× (2n − 1))

# elements in bi-narized sequence oforder n (2n + 1)

2 F5 = 11 FSD4 = 9 BS2 = 53 F13 = 59 FSD8 = 21 BS3 = 94 F29 = 271 FSD16 = 45 BS4 = 175 F61 = 1163 FSD32 = 93 BS5 = 336 F125 = 4797 FSD64 = 189 BS6 = 65

TABLE IVTARGET CFS FOR DIFFERENT DILUTION GRADIENTS [35]

Dilution System Target-CFs Accuracylevel (n)

Specialtestcases

Gaussian { 7128

, 24128

, 61128

, 107128}

7

sin x { 40128

, 76128

, 104128

, 122128}

x12 { 58

128, 81128

, 99128

, 114128}

x2 { 5128

, 20128

, 46128

, 82128}

Linear { 26128

, 51128

, 77128

, 102128}

2-fold Log { 8128

, 16128

, 32128

, 64128}

Real-life testcases

Bradford proteinassay

{ 10256

, 20256

, 30256

,40256

, 50256} 8

Sucrose gradientanalysis

{ 10256

, 15256

, 20256

,25256

, 30256

, 35256

, 40256}

Harmonic gradi-ent analysis

{ 128256

, 85256

, 64256

, 51256

,42256

, 36256

, 32256

, 28256}

[4, 27] FSD

(a) Scatter-plot (error vs target-CF) of 200 experimental data.

FSD

[4, 27]

Sample Buffer0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

(b) Average differential cost of sample and buffer is shown alongthe Y -axis with respect to theoretical requirement.

Fig. 7. Comparison between FSD and BS-based methods [4], [27] for accuracy9.

record the differential cost of sample and buffer averaged over alldiscretized values of target-CF s (Fig. 7(b)). The results clearly showthat FSD outperforms BS-based methods [4], [27].

We also compare the theoretical CF-errors of randomly gener-ated synthetic test-cases {3%, 3.25%, 12.5%, 31%, 41.75%, 54.5%,69.3%, 70%, 87.5%, 95%} by FSD method and BS-based method.As before, we observe FSD provides better accuracy then BS-basedmethod. We run COMSOL simulation on these synthetic test-casesand found that experimentally observed CF -errors favorably matchwith theoretically estimated CF-errors of FSD (see Fig. 8).

Moreover, we compare the performance (w.r.t. accuracy) of FSD

Page 8: Sample Preparation Meets Farey Sequence: A New Design ...

Predicted CF (in %)

Concentrationerror(in%)

FSD error in %(Theoretical)

FSD error in %

(Simulated)

Error in % of [4, 27]

Fig. 8. Plot showing theoretical and experimental absolute error values inconcentration factors for several test-cases.

sin x

x12

Linear

x 2

2-fold

Log

Gussian

[4]

[20]

FSD

[21]

Fig. 9. Radar chart representation of errors in tCFs of some gradient testcases for the proposed method and three other methods with accuracy 7.

with prior approaches [4], [20], [21]. To accomplish this task, wefurther run COMSOL simulation on several concentration-gradienttest-cases for accuracy 7 (as listed in Table IV). A radar-chart isshown in Fig. 9, where the radial axis reflects the errors in observedCF-values. This chart demonstrates that the proposed FSD networkefficiently generates the target CFs with much less error comparedto existing methods [4], [20], [21].

twoWayMix [27]

Proposed method

Fig. 10. Comparison with respect to number of mixing steps required togenerate different target-CFs by [27] and the proposed method.

We perform additional experiments on 200 randomly generatedtarget-CFs to compare the number of mixing steps required by theproposed method and twoWayMix [27].We show the results in Fig.10 from which it can be noticed that the FSD method requires fewermixing steps in most of the target-CFs compared to twoWayMix [27].

Also, to investigate the concentration stability in different zones(zones A-E in Fig. 11) of the proposed layout, we captured the flow-dynamics of the mixing channels while producing the target-CF =913

using COMSOL Multiphysics Software. We show the convergenceof CFs with time at different zones (A-E) in Fig. 12. We observedthat the concentration stability in Zone A occurs approximately after80 seconds. Whereas, in Zone B and Zone C, stability is achievedapproximately after 125 seconds and 140 seconds, respectively. ZoneD and Zone E reach the steady-state after 200 seconds, followingwhich flow with the desired target concentration factor is steadilyobserved at the outlet.

A B

CD E

Concentration (in mol/m3) profile

0 20 40 60 80 100

Outlet for reuse

Outlet for reuse

chamber of sample fluid

chamber of buffer fluid Outlet ofdiluted sample

Fig. 11. Flow dynamics in the mixing channel.Concentration

inmol/m

3

Time in second

Fluid injection start time

Time to reach steady state

Zone A

Concentrationin

mol/m

3

Time in second

Fluid injection start time

Time to reach steady state

Zone B

(a) CF-stability at zone A. (b) CF-stability at zone B.

Con

centration

inmol/m

3

Time in second

Time to reach

Zone C

steady state Con

centrationin

mol/m

3

Time in second

Time to reach

Zone D

steady state

(c) CF-stability at zone C. (d) CF-stability at zone D.

Con

centrationin

mol/m

3

Time in second

Time to reach

Zone E

steady state

(e) CF-stability at zone E.Fig. 12. (a)-(e) Convergence of concentration factor with time at differentzones (Fig. 11) of the fluidic network for tCF = 69.3%.

Page 9: Sample Preparation Meets Farey Sequence: A New Design ...

VIII. CONCLUSIONS

In this paper, we have presented a novel architecture of free-flowing biochip that can be used to perform dilution of sample fluidsaccurately. This work mainly has two contributions: (i) in contrastto traditional binary discretization of the concentration interval [0, 1],we first propose a new technique for approximating a target-CF basedon Farey-sequence (FS) arithmetic, and (ii) secondly, we present thedetailed design of a befitting microfluidic architecture that mimicsthe proposed FS-based CF-approximation rule and subsequentlyleads to an efficient implementation of the diluter. The networkis free-flowing in nature, i.e., it does not need any control valvefor effecting conditional navigation of fluids through the interiorlabyrinth of channels. Flow-controllers are required only at the inletports to enable the fluid to enter either into the mixing channel ortowards re-use branches. The state of each controller can be set apriori (off-line) depending on the given target-CF. Additionally, nodifferential pressure/velocity control is needed for injecting input-fluids into the network, and thus sustenance of constant-pressurewould suffice. Being free-flowing in nature, no controlled splittingor metering of sample/buffer units is required during the executionof the dilution assay. The network supports programmability in thesense that dilution of fluid targeting any concentration factor withinthe range [0, 1] can be achieved with no more than 1% error usingthis chip. We simulated the network architecture using COMSOLMultiphysics Software and studied its performance for various test-cases. We observe that error in target-CF, convergence time, andsample/buffer cost for the FSD-based method compare favorablyagainst those achievable with other competing flow-based systems.The proposed fluidic network can thus provide a potential platformfor automating sample preparation.

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