16.3 A Progressive-ILP Based Rouing Algorihm for Cross-Referencing Biochips Ping-Hung Yuh 1, Sachin Sapanekar 2, Chia-Lin Yang 1, Yao-Wen Chang 3 1 Deparmen of Compuer Science and Informaion Engineering, Naional Taiwan Universiy 2 Deparmen of Elecrical and Compuer Engineering, Universiy of Minnesoa 3 Graduae Insiue of Elecronics Engineering and Deparmen of Elecrical Engineering, Naional Taiwan Universiy {r91089, yangc}@csie.nu.edu.w; sachin@umn.edu; ywchang@cc.ee.nu.edu.w ABSTRACT Due o recen advances in microfluidics echnology, digial microfluidic biochips and heir associaed CAD problems have gained much aenion, mos of which has been devoed o direc-addressing biochips. In his paper, we solve he drople rouing problem under he more scalable crossreferencing biochip paradigm, which uses row/column addressing scheme o acivae elecrodes. We propose he firs drople rouing algorihm ha direcly solves he problem of rouing in cross-referencing biochips. The main challenge of his ype of biochips is he elecrode inerference which prevens simulaneous movemen of muliple droples. We firs presen a basic ineger linear programming (ILP) formulaion o opimally solve he drople rouing problem. Due o is complexiy, we also propose a progressive ILP scheme o deermine he locaions of droples a each ime sep. Experimenal resuls demonsrae he efficiency and effeciveness of our progressive ILP scheme on a se of pracical bioassays. Caegories and Subjec Descripors B.7.2 [Inegraed Circuis]: Design Aids General Terms Algorihm, Performance, Design Keywords Microfluidics, biochip, rouing, progressive-ilp 1. INTRODUCTION Recenly, here have been many significan advances in microfluidic echnologies [7]. Microfluidic biochips show numerous advanages over convenional assay mehods, including porabiliy and sample/reagen volume reducion, and offer a plaform for developing clinical and diagnosic applicaions, such as healhcare of infans and poin-of-care diagnosics of disease. This breadh of applicabiliy implies ha microfluidic biochips are increasingly used in laboraory procedures in molecular biology. Laely, a new ype of biochips, which are based on digializing coninuous liquid flow ino discree liquid paricles, called This work was parially suppored by he Naional Science Council of Taiwan under Gran No s. NSC 96-2221-E-002-250, NSC 96-2752-E-002-008-PAE, NSC 96-2628-E-002-248-MY3, NSC 96-2628-E- 002-249-MY3, NSC 96-2221-E-002-245, and by he Excellen Research Projecs of Naional Taiwan Universiy, 96R0062-AE00-07. Permission o make digial or hard copies of all or par of his work for personal or classroom use is graned wihou fee provided ha copies are no made or disribued for profi or commercial advanage and ha copies bear his noice and he full ciaion on he firs page. To copy oherwise, o republish, o pos on servers or o redisribue o liss, requires prior specific permission and/or a fee. DAC 2008, June 8 13, 2008, Anaheim, California, USA. Copyrigh 2008 ACM 978-1-60558-115-6/08/0006...$5.00. droples, have been proposed [5]. These ype of biochips are more suiable for large-scale and scalable sysems due o heir reconfigurabiliy. Each drople can be independenly conrolled by he elecrohydrodynamic forces generaed by elecrodes. In his way, droples movemen can be conrolled by a sysem clock. Due o his parallel wih digial elecronic sysems, his ype of biochips is referred o as digial microfluidic biochips. Typically, a digial microfluidic biochip has a 2D microfluidic array, which consiss of a se of elecrodes for fundamenal operaions, such as drople mixing and ransporaion. In he simples and mos common drople conrol scheme, each elecrode is direcly addressed and conrolled by a dedicaed conrol pin, which allows each elecrode o be individually acivaed. In his paper, we refer o hese ypes of digial microfluidic biochips as direc-addressing biochips. While his archiecure provides very flexibiliy for drople movemen, i suffers from he major drawback ha he number of conrol pins rapidly increases as he sysem complexiy (i.e., he size of he array) increases. The large number of conrol pins affecs produc cos and he conrol wire rouing problem complicaes he design process, and herefore, his archiecure is only applicable o small-scale biochips [9]. To overcome hese limiaions, recenly a new digial microfluidic biochip archiecure has been proposed [3]. This archiecure uses a row/column addressing scheme, where a se of elecrodes in one row/column is conneced o a conrol pin. Therefore, he number of conrol pins is grealy reduced: his number is now proporional o he perimeer of he chip raher han he area of he chip. We refer o his ype of biochip archiecure as crossreferencing biochips. However, his archiecure also inroduces a new se of limiaions. Since an elecrode can poenially conrol he movemen of all droples in a row/column a he same ime, his archiecure incurs higher drople movemen complexiy han ha of direc-addressing biochips. Moreover, he manipulaion of more han wo droples causes elecrode inerference among droples, which prevens muliple droples o move a he same ime. This performance limiaion is a major drawback o highperformance applicaions, such as large-scale proein analysis. In his paper, we ackle he problem of drople rouing in crossreferencing biochips. The main challenge of his rouing problem is o ensure he correcness of drople movemen; he fluidic propery which avoids unexpeced mixing among droples needs o be saisfied, and elecrode inerference paerns ha preven muliple droples from moving a he same ime mus be avoided. The goal of drople rouing is o minimize he maximum drople ransporaion ime, and has several moivaions. Firs, his minimizaion is criical o real-ime applicaions, such as monioring environmenal oxins. Second, he minimized drople ransporaion ime leads o shorer ime a sample spen on a biochip, which is desirable o mainain he bioassay execuion inegriy. 1.1 Previous Work Drople rouing is a criical sep in biochip design auomaion. Previous rouing approaches mainly focus on direc-addressing biochips [4, 6, 10]. Recenly, he problem of manipulaing droples on a cross-referencing biochip has araced some aenion, bu o our knowledge, all exising mehods begin wih a direc-addressing rouing resul, and ransform i o a cross-referencing rouing re- 284
sul. Griffih e al. [4] proposed a graph coloring based mehod ha is applied o each successive cycle of he direc-addressing soluion. Each node in an undireced graph represens a drople, and wo nodes are conneced by an edge if hese wo droples canno move a he same ime. Therefore, he minimum ime o move all droples is equivalen o he minimum number of colors o color his undireced graph. Xu e al. [8] proposed a clique pariioning based mehod ha is also sequenially applied o he direc-addressing soluion. Each clique is a se of droples whose desinaion cells, i.e., he cell o which he droples are supposed o move, are in he same row or column. The number of cliques is he maximum ime o move all droples. However, droples wih differen desinaion cells can move a he same ime. Therefore, heir algorihm may poenially increase drople ransporaion ime. Any algorihm ha uses he direc-addressing soluion as a saring poin is limied by ha soluion, and so far, we believe ha here is no rouing algorihm ha is direcly argeed o cross-referencing biochips. 1.2 Our Conribuion In his paper, we propose an ineger linear programming (ILP) based drople rouing algorihm for cross-referencing biochips. We derive he basic ILP formulaion o simulaneously perform drople rouing and assign volages o he cross-referencing elecrodes, while minimizing he maximum drople ransporaion ime. Moreover, we also model muli-pin nes in our ILP formulaion for pracical bioassays, where muliple droples are merged during heir ransporaion. To overcome he compuaional cos of he ILP,we also propose a progressive ILP rouing scheme, which is used o find he min-cos drople locaions a each ime sep using an ILP. Unlike he progressive ILP scheme proposed in [2], which divides he original problem spaially, our algorihm divides he original rouing problem emporally. In his way, he original problem is reduced o a manageable size, and we can pracically apply an ILP-based mehod o find a good soluion wihin reasonable CPU ime. The major conribuions of his paper include he following: We propose he firs rouing algorihm ha direcly solves he rouing problem in cross-referencing biochips. In conras wih previous works ha sar wih an iniial direcaddressing rouing soluion, our algorihm has higher flexibiliy and can obain beer soluions for drople rouing on cross-referencing biochips. To ackle he complexiy of he basic ILP formulaion, we propose he progressive ILP rouing scheme. We ieraively deermine he locaions of droples a each ime sep by ILP formulaion. Therefore, our algorihm can obain a highqualiy soluion wihin reasonable CPU ime. Unlike previous works ha only move a subse of droples a each ime (for example, he algorihm proposed in [8] only moves droples whose desinaion cells are in he same row/column), our algorihm maximizes he number of droples ha can simulaneously move a he same ime, even he desinaion cells are no in he same row/column. Therefore, our algorihm can obain a rouing soluion wih lower drople ransporaion ime. This minimizaion is especially imporan for cross-referencing biochips due o he elecrode inerference problem. Experimenal resuls demonsrae he efficiency of our progressive rouing scheme compared wih he basic ILP formulaion. The ILP formulaion needs more han five days while he progressive ILP rouing scheme needs a mos 15.36 seconds for one bioassay. Experimenal resuls also demonsrae he effeciveness of our algorihm compared wih previous work. For example, for he proein assay, our algorihm obains 45.83% smaller maximum drople ransporaion ime han he nework-based mehod proposed in [10], plus he clique pariioning based algorihm proposed in [8]. The reminder of his paper is organized as follows. Secion 2 deails rouing on cross-referencing biochips and formulaes he drople rouing problem. Secion 3 presens he basic ILP formulaion for drople rouing problem. Secion 4 deails he progressive ILP rouing scheme, while Secion 5 shows he experimenal resuls. Finally, concluding remarks are provided in Secion 6. Y X V Drople Top elecrode Boom elecrode Figure 1: Top view of a cross-referencing biochip. 1 2 3 High volage Low volage Acivaed cell Exra acivaed cell Figure 2: Illusraion of elecrode inerference. Unseleced rows/columns are lef floaing. 2. ROUTING ON CROSS-REFERENCING BIOCHIPS In his secion, we firs show he archiecure of cross-referencing biochips. Then we deail he unique elecrode inerference ha resuls in incorrec drople movemen. Finally, we presen he problem formulaion of he drople rouing problem. 2.1 Cross-Referencing Biochips Figure 1 shows he op view of a cross-referencing biochip. A drople is sandwiched beween wo plaes. A se of elecrodes spans a full row (column) in he X-dimension (Y -dimension), and is assigned eiher a driving or reference volage. Two ses of elecrodes are orhogonally placed, one se each on he op and boom plaes as shown in Figure 1, and each elecrode can be se o eiher a high or a low volage. A grid poin is addressed if here is a poenial difference beween he upper and lower elecrode, i.e., one is high while he oher is low. This causes a drople a a neighboring grid poin o move ino his locaion. The advanage of cross-referencing biochips is ha we need only Ŵ +Ĥ insead of Ŵ Ĥ conrol wires for drople movemen, where Ŵ (Ĥ) is he widh (heigh) of a biochips, measured in erms of he number of elecrodes in each dimension. Therefore, he package and fabricaion coss are reduced. 2.2 Elecrode Consrain However, his improvemen comes a he cos of reduced flexibiliy for drople movemen, as compared o direc-addressed biochips. When moving muliple droples simulaneously, we mus se poenial levels on muliple rows/columns as driving elecrodes. However, since each se of elecrodes in he X (Y ) direcion spans he enire row (column), seing elecrode volage values o move a se of droples could imply ha some droples may be inadverenly and incorrecly moved. To see his, we presen an illusraion in Figure 2, where our goal is o move hree droples a he same ime. The picure shows a se of volage assignmens o he elecrodes ha are required o move he hree droples o heir desired locaion by acivaing he corresponding rows and columns and by seing one of he lines o high and he oher o low volage. However, due o he grid srucure, several oher locaions are also accidenally acivaed. If eiher of hese is adjacen o a drople, i will cause an unwaned effec. For example, in his scheme, drople 2 is now araced o he locaions jus above and jus below i, and is likely o spli ino wo. This scenario is referred o as elecrode inerference, andimus be avoided during drople ransporaion. The resricion ha avoids elecrode inerference is referred o as an elecrode consrain. Noe ha no every exra acivaed cells causes incorrec drople movemen: for example, he op-righ exra cell has no effec on drople movemen since i has no neighboring droples. Only he exra cells ha are around a drople and is desinaion cell may cause incorrec drople movemen. The elecrode consrain imposes serious resricion on he movemen of muliple droples. When more han wo droples are moved, we may need o sall some droples o saisfy he elec- 285
(a) Deacivaed ce l Drople Cel(x+1,y) Cel(x,y) Acivaed ce l Figure 3: Modeling of elecrode consrain when a drople moves (a) or says a is original locaion (b). rode consrain, which will resul in longer drople ransporaion imes. For example, in Figure 2, we can sall drople 3 and move he oher wo, and his will avoid elecrode inerference. The manner in which elecrode consrains are handled is a criical rouing consideraion in cross-referencing biochips, and is imporan in minimizing he drople ransporaion ime. 2.3 Problem Formulaion In his paper, we focus on he drople ransporaion problem. As saed in [6, 10], he drople ransporaion problem can be represened in a 3D space, where he hird dimension corresponds o ime. A each ime sep, he problem reduces o working wih he drople movemen problem in a 2D plane. We focus he problem formulaion on one 2D plane, noing ha similar consrains may be added for every oher 2D planes. When rouing droples on one 2D plane, we consider he modules (and he surrounding segregaion cells) ha are acive as obsacles. Besides he elecrode consrain, we also need o saisfy he saic and dynamic fluidic consrains for correc drople movemen [6]. The saic fluidic consrain saes ha he minimum spacing beween wo droples is one cell. Noe ha once elecrode consrain is saisfied, he dynamic fluidic consrain is auomaically saisfied, since he neighboring cells of he desinaion cell and he cell where a drople is originally a will no be acivaed. Therefore, we do no need o explicily handle he dynamic fluidic consrain on cross-referencing biochips. Besides he above consrain, for pracical bioassays, we mus be able o handle 3- pin nes o represen he fac ha wo droples may have o be merged during heir ransporaion for efficien mix operaions [6]. Therefore, he drople rouing problem for each 2D plane can be formulaed as follows: Inpu: Anelisofm nes N = {n 1,n 2,...,n m}, whereeach ne n i is a 2-pin ne (one drople) or a 3-pin ne (wo droples), and he locaions of pins and obsacles. Objecive: Roue all droples from heir source pins o heir arge pins while minimizing he maximum ime o roue all droples. Consrain: Boh fluidic and elecrode consrains mus be saisfied. 3. ILP FORMULATION FOR DROPLET ROUTING In his secion, we presen he basic ILP formulaion for one 2D plane. We show how he ILP opimizes drople rouing and scheduling and volage assignmen on he elecrodes, wih he consideraion of 3-pin nes. 3.1 Basic ILP Formulaion One of he major challenges in he formulaion of his problem is in modeling he elecrode consrain wihin an ILP. Figure 3 illusraes he elecrode consrain. When a drople moves from cell (x, y) o cell (x +1,y)aime + 1, as shown in Figure 3 (a), all neighboring cells of (x, y) and(x + 1,y) mus be deacivaed, excep for cell (x+1,y), which mus be acivaed. Noe ha in he case ha a drople says a is original locaion, cell (x, y) does no necessarily need o be acivaed and all he neighboring cells of (x, y) mus be deacivaed, as demonsraed in [3]. Figure 3 (b) shows his case. Therefore, he elecrode consrain can be modeled by he following hree rules: 1. If a drople is a cell (x, y)aime, hen he four diagonally adjacen cells canno be acivaed a ime +1. 2. If a drople moves o cell (x,y )aime+1, hen he eigh neighboring cells of (x,y ) canno be acivaed a +1. (b) Table 1: Noaions used in our ILP formulaion. Ŵ /Ĥ biochip widh/heigh T maximum drople ransporaion ime N se of 3-pin nes D se of droples d i j j-h drople of ne n i; j = {1, 2} (s i,j x,si,j y ) locaion of he source of di j (ˆ i x, ˆ i y ) locaion of he sink of ne ni (x i,j ) locaion of drople d i j a ime C se of available cells E(x, y) se of cell (x, y) and is four adjacen cells E (x, y) se of cell (x, y) s four adjacen cells Ê(x, y) se of available neighboring cells of (x, y) E x (x, y)/e y (x, y) se of available cells adjacen o (x, y) wih he same x- (y-) coordinae ˆD(x, y) se of available diagonal cells of (x, y) T l maximum drople ransporaion ime p i j (x, y, ) a 0-1 variable o represen ha drople di j locaes a (x, y) aime L c x () a 0-1 variable represens ha column x is se o volage low a ime H c x () a 0-1 variable represens ha column x is se o volage high a ime L r y () a 0-1 variable represens ha row y is se o volage low a ime H r y () a 0-1 variable represens ha row y is se o volage high a ime a 0-1 variable represens ha (x, y) is acivaed a ime m i a 0-1 variable represens ha d i 0 and di 1 are merged a ime 3. If a drople is a cell (x, y) aime, amosonecellcan be acivaed among cells (x, y) and is four adjacen cells. Noe ha boh rule #1 and #2 sae ha he cells diagonally adjacen o cell (x, y) canno be acivaed a ime +1. Thisredundancy reduces he size of he basic ILP formulaion; oherwise, we would need exra variables o represen he moving direcion of each variable p i j (x, y, ) and exra consrains o deermine he values of hese exra variables. In he following secions, we inroduce he objecive funcion and consrains of our basic ILP formulaion. The noaions used in our ILP formulaion are shown in Table 1. 3.1.1 Objecive Funcion The goal is o minimize he laes ime a drople reaches is sink. Therefore, he objecive funcion is defined by he following equaion: Minimize : T l. (1) 3.1.2 Consrains There are oal eigh consrains in our basic ILP formulaion. 1. Objecive funcion compuaion: If a drople reaches is sink a ime + 1, hen he ime i reaches is sink can be compued as + 1 imes he difference of he wo variables p i j (ˆ i x, ˆ i y, +1) and pi j (ˆ i x, ˆ i y,). Therefore, he objecive funcion can be compued by he following consrain: ( +1)(p i j (ˆ i x, ˆ i y,+1) pi j (ˆ i x, ˆ i y,)) Tl, di j D, 0 <T. (2) 2. Source and sink requiremens: We assume ha a ime zero, all droples are a heir source locaions. All droples mus reach heir sinks. Once a drople reaches is sink, i remains here. Therefore, he above requiremens can be represened by he following consrains: p i j (si,j x,si,j y, 0) = 1, di j D (3) T 1 p i j (ˆ i x, ˆ i y,) 1, di j D (4) =0 p i j (ˆ i x, ˆ i y,) pi j (ˆ i x, ˆ i y,+1) 0, di j D, 0 <T. (5) 286
3. Exclusiviy consrain: The exclusiviy consrain saes ha a each ime sep, a drople only has one locaion and can be represened by he following consrain: p i j (x, y, ) =1, di j D, 0 <T. (6) x,y 4. Saic fluidic consrain: To saisfy he saic fluidic consrain, he minimum spacing beween wo droples mus be one cell. In oher words, here are no oher droples in he 3 3 region cenered by a drople. The saic fluidic consrain can be modeled by he following consrain: p i j (x, y, ) +pi j (x,y,) 1, d i j,di j D, (x, y) C, (x,y ) Ê(x, y), 0 <T. (7) 5. Volage assignmen and cell acivaion: Each row/column can be assigned one volage (high or low), or lef floaing, a any ime. A cell is acivaed if and only if i is in he inersecion of a row wih high (low) volage and a column wih low (high) volage. Therefore, his consrain can be modeled by he following consrains: L c x () +Hc x () 1, 0 x<ŵ,0 <T (8) L r y () +Hr y () 1, 0 y<ĥ,0 <T (9) (L c x () and Hr y ()) or (Hc x () and Lr y ()) ax,y, (x, y) C 0 <T. (10) 6. Drople movemen consrain: A drople can only move o an adjacen cell, which needs o be acivaed before he movemen occurs. On he oher hand, if a drople is o say a is original locaion, he corresponding cell may or may no be acivaed. Therefore, he drople movemen consrain can be represened by he following consrains: p i j (x,y,+1) p i j (x, y, ) 0, di j D, (x,y ) E(x,y) (x, y) C, 0 <T 1 (11) p i j (x, y, +1)+ p i j (x,y,) +1 1, (x,y ) E (x,y) d i j D, (x, y) C, 0 <T 1. (12) Noe ha an acivaed cell does no imply ha a drople will move o his cell, as can be seen in he case of he oprigh exra acivaed cell in Figure 2. Therefore, consrain (12) does no sae an if-and-only-if relaion beween cell acivaion and drople movemen. 7. Elecrode consrain: The hree rules explained earlier can be represened as he following consrains: p i j (x, y, ) +ax,y +1 1, d i j D, (x, y) C, (x,y ) ˆD(x, y), 0 <T 1 (13) 9p i j (x, y, ) + a x,y 9, (x,y ) Ê(x,y) d i j D, (x, y) C, 0 <T (14) 6p i j (x, y, ) + a x,y +1 7, d i j D, (x, y) C, (x,y E(x,y) 0 <T 1, (15) where consrain (13) represens rule #1, consrain (14) represens rule #2, and consrain (15) represens rule #3. 8. 3-pin nes: We use he following hree consrains o handle he 3-pin nes: (p i 0 (x, y, ) pi 1 (x, y, )) = 0 mi =1, T 2 =1 (x,y) n k N, (x, y) C, 1 <T (16) T 1 m i 1, nk N (17) =1 ( +1) (m i +1 mi pi 0 (ˆ i x, ˆ i y,+1)+pi 0 (ˆ i x, ˆ i y,)) 1. (18) Algorihm: Progressive ILP rouing Le T l be he maximum Manhaan disance of all droples from heir sources o sinks; begin 1 while no all droples reach heir sinks 2 Compue drople movemen cos(); 3 Consruc ILP(); 4 Solve ILP(); 5 Updae drople posiion(); 6 if i is no possible o roue all droples wihin T l 7 T l += α maximum Manhaan disance; end Figure 4: Overview of progressive ILP mehod. Drople Figure 5: Illusraion of he localiy of drople movemen and elecrode consrain when locaions of droples are known. Consrain (16) is used o deermine wheher wo droples are merged; i.e., in he same physical locaion. Consrain (17) is used o guaranee ha wo droples mus be merged during heir ransporaion by resricing ha heir physical locaion mus be he same for a lea one ime sep. Consrain (18) saes ha hese wo droples mus be merged before reaching heir sink. Noe ha once hese wo droples are merged, hey will always move ogeher by he drople movemen consrain. Therefore, hese wo droples are no splied once hey are merged. 4. PROGRESSIVE ILP ROUTING SCHEME Alhough he basic ILP formulaion presened in he previous secion can solve he drople ransporaion problem, i may incur high run imes. Hence, i may be hard o direcly apply he basic ILP formulaion o pracical bioassays. In his secion, we presen a progressive ILP rouing scheme o solve he drople ransporaion problem. The main idea is o divide he original problem ino a se of subproblems corresponding o each ime sep. The goal of each subproblem is o find min-cos locaions of he droples a nex ime sep by ILP. In he following subsecions, we firs overview our progressive ILP rouing scheme, and hen presen he ILP formulaion. Then, we presen how o handle 3-pin nes and how o deermine drople movemen cos. 4.1 Progressive Rouing Algorihm Overview Figure 4 shows he overview of our progressive rouing scheme. The essenial inuiion is ha his scheme finds he opimal movemen of droples progressively, one ime sep a a ime. We firs se T l as he maximum Manhaan disance of all droples from heir sources and sinks. Unil all droples reach heir sinks, we firs calculae he drople movemen cos, and hen we consruc he progressive ILP formulaion and solve i by an ILP solver. Finally, if i is found ha some droples canno be roued wihin T l, T l is increased by α imes he maximum Manhaan disance, where α is a user-specified consan and is se o be 0.2. 4.2 Progressive ILP Formulaion The key idea of he progressive ILP mehod is o reduce he problem o a manageable size by solving a series of subproblems. We can furher reduce he problem size by observing ha he elecrode and saic fluidic consrains and drople movemen have only local effecs when he locaions of he droples are known. Figure 5 shows a biochip wih wo droples, where he arrows represen possible direcions in which he droples can move. Since he movemen of a drople d i j can only be affeced by is neighbor- 287
ing cells and heir adjacen cells, he progressive ILP does need no o consider he effecs of acivaed cells ouside he 5 5region around a drople. To saisfy he saic fluidic consrain, we observe ha afer a drople moves, wo droples are adjacen o each oher only if a cell wih more han one droples nearby is acivaed. Therefore, we do no acivae his cell in our progressive ILP formulaion. Finally, each drople d i j can eiher move o one of is four adjacen cells, or say a is original locaion. Tha is, he possible posiion of a drople can be represened as a cross shown in Figure 5. Therefore, we only need variables +1 o represen he cell (x i,j ) and is four adjacen cells. Oher cells eiher mus be deacivaed (wihin he 5 5 region) or have no effec on drople movemen (ouside he 5 5 region). In his way, unnecessary variables can be eliminaed. We now presen he progressive ILP formulaion in he following secions. The noaions used in he progressive ILP formulaion are also lised in Table 1. 4.2.1 Objecive Funcion The goal of he progressive ILP formulaion is o find he mincos drople movemen. Therefore, he objecive funcion can be represened as he following equaion: Minimize d ij D (x,y) E(x i,ji ) wi j (x, y)pi j (x, y, +1),(19) where wj i (x, y) is he cos when a drople di j moves o cell (x, y). 4.2.2 Consrains There are five consrains in he progressive ILP formulaion. 1. Drople movemen consrain: A drople can move o one of is four adjacen cells if and only if his cell is acivaed. Therefore, his consrain can be expressed by he following consrain: p i j (x, y, +1) ax,y +1, di j D, (x, y) E (x i,j ). (20) Noe ha since only he four adjacen cells are associaed wih cell acivaion variables, he cell acivaion and he drople movemen mus be modeled as a if-and-only-if relaion, unlike he if relaion in he basic ILP formulaion. Moreover, a cell says a is original locaion if and only if all is four adjacen cells are deacivaed. Therefore, we can use he following consrains o represen his siuaion: p i j (xi,j 4p i j (xi,j,+1)+,+1)+ (x,y) E (x i,j ) (x,y) E (x i,j ) +1 1, di j D (21) +1 4, di j D. (22) 2. Elecrode consrain: Rule #1, which saes ha he diagonal cells canno be acivaed, can be represened by he following consrains: L c x ( +1)+Hr i,j y ( +1) 1, (x, y) ˆD(x ) (23) H c x ( +1)+Lr i,j y ( +1) 1, (x, y) ˆD(x ). (24) The following wo consrains, similar o consrain (14), are used for rule #2: if p i j (x, y, +1)=1 L c x ( +1)+H r y ( +1) 1orH c x ( +1)+L r y ( +1) 1 d i j D, (x, y) E x (xi,j ), (x,y ) E y (x, y) (25) if p i j (x, y, +1)=1 L c x ( +1)+H r y ( +1) 1orH c x ( +1)+L r y ( +1) 1 d i j D, (x, y) E y (xi,j ), (x,y ) E x (x, y). (26) As shown in Figure 3, if d i j moves o cell (x +1,y), hen cells (x +2,y), (x +2,y 1), and (x +2,y + 1) canno be acivaed. A similar condiion holds for cell (x 1,y). Consrain (25) is used o represen hese wo cases, and consrain (26) is used o represen he case when d i j moves o cell (x i,j +1) or (x i,j 1). Finally, for rule #3, we impose he following consrain: a x,y +1 1, d i j D. (27) (x,y ) E(x i,j ) 3. Saic fluidic consrain: Since he locaions of he droples are known, we can model he saic fluidic consrain in a simpler way. The fluidic consrain is violaed only if a cell wih more han one drople nearby is acivaed. Therefore, insead of using consrain (7), we represen he fluidic consrain as follows: if more han one drople are around cell (x, y) +1 =0, (x, y) E (x i,j ), d i j D. (28) 4. Sink requiremen: Once a drople reaches is sink, i mus say here. The sink requiremen can be modeled by he following consrain: p i j (xi,j,) p i j (xi,j +1,yi,j +1,+1) 0, di j D, (29) which means ha if d i j reaches is sink a ime, heni mus say here a ime +1. 5. Volage assignmen and cell acivaion: This consrain is he same as ha in he basic ILP formulaion. 4.3 Drople Movemen Cos A key issue in he progressive ILP rouing scheme is he deerminaion of he drople movemen cos. Since our goal is o minimize he maximum drople ransporaion ime, our objecive is o move a drople oward is sink a each ime sep. Furhermore, we need o avoid congesion among droples; oherwise, we may eiher have o deour or sall some droples for cerain ime, o saisfy boh he fluidic and elecrode consrains. Therefore, he drople ransporaion ime is poenially increased. In his subsecion, we deail how o deermine he drople movemen cos. The drople movemen cos w i j (x, y) when a drople di j moves o cell (x, y) consiss of rouing and congesion coss and is defined as follows: w i j (x, y) =βri j (x, y, ˆ i x, ˆ i y )+γgi j (x, y, ˆ i x, ˆ i y ), (30) where rj i (x, y, ˆ i x, ˆ i y ) is he rouing cos and gi j (x, y, ˆ i x, ˆ i y )ishe congesion cos. β and γ are user-specified consans. In his paper, we empirically se β = 15 and γ = 0.1. The rouing cos is he real disance compued by he maze rouing algorihm from cell (x, y) o he sink of d i j over he area of a biochips (for normalizaion). We consider all oher droples excep d i j as 3 3 obsacles when performing he maze rouing algorihm due o he saic fluidic consrain. The goal of he congesion cos componen is o reduce he probabiliy of he violaions of he elecrode and fluidic consrains. In his paper, we use he concep of he idle inerval presened in [10] for congesion cos compuaion. The idle inerval Ij i(x, y) =[Tm (x, y, d i j ),TM (x, y, d i j )] represens all possible imes a drople d i j will be a cell (x, y) wih a ime limiaion on he laes ime when d i j mus reach is sink. Here, T m (d i j ) represens he earlies ime when d i j reaches (x, y) fromissource and T M (d i j ) represens he laes ime when di j can say a (x, y) wihou violaing he ime limiaion. If we se T l as he ime limiaion, he congesion cos of a cell can be measured as he lengh of he idle inerval of d i j on cell (x, y) over he summaion of he lengh of he idle inervals of oher droples on cell (x, y). Insead of using he source locaion of a drople, as in [10], we use he possible desinaion cell (x, y) o compue he idle inerval. The values of T m (d i j )andtm (d i j ) for a subproblem a ime can be compued by he following wo equaions: T m (d i j )= + dm (x, y, ˆ x, ˆ y ) (31) T M (d i j )=Tl dm (x, y, ˆ x, ˆ y ), (32) where d m (x, y, x,y ) is he Manhaan disance beween cells (x, y)and(x,y ). The congesion informaion is updaed a every ime sep for he laes congesion informaion. To obain more accurae congesion informaion, i is no sufficien o only consider he congesion informaion of he possible desinaion cell (x, y). In his paper, we consider all cells congesion informaion wihin he bounding box defined by cell (x, y) 288
Table 2: Comparison beween basic ILP and progressive ILP formulaions. Circui max. #vari. max. #cons. CPU ime basic prog. basic prog. basic prog. ILP ILP ILP ILP ILP ILP in-viro 1 24889 138 176580 388 > 7200 min. 2.55 sec. in-viro 2 18061 134 126983 398 > 7200 min. 2.53 sec. Proein 1 49561 134 350645 357 > 7200 min. 15.36 sec. Proein 2 22238 132 252989 385 > 7200 min. 6.70 sec. obsacle High volage Low volage Table 3: Rouing resul of he wo bioassays. Circui [10] + [8] [10] + [4] Progressive ILP Max/avg CPU Max/avg CPU Max/avg CPU T l ime T l ime T l ime (cycle) (sec.) (cycle) (sec.) (cycle) (sec.) in-viro 1 40 / 16.72 0.05 47 / 20.18 0.06 24 / 13.09 2.55 in-viro 2 35 / 13.46 0.05 52 / 16.80 0.06 22 / 11.00 2.53 proein 1 48 / 19.32 0.26 55 / 24.40 0.30 26 / 16.15 15.36 proein 2 36 / 11.00 0.20 53 / 14.33 0.21 26 / 10.23 6.70 and is sink. Noe ha we use he real bounding box compued by he maze rouing algorihm. Therefore, he congesion cos gj i(x, y, ˆ i x, ˆ i y) when drople d i j moves o cell (x, y) is defined by he following equaion: g i j (x, y, ˆ i x, ˆ i y )= I i j (x,y ) (x,y ) b i j (x,y) d i Ii j D j (x, y), (33) where b i j (x, y) is he bounding box obained by he maze rouing algorihm and Ij i (x, y) is he lengh of he idle inerval defined as he difference of is wo end poins plus one. Noe ha if T m (d i j ) >TM (d i j )hen Ii j (x, y) is zero. If he congesion cos is large, i is likely ha he elecrode and fluidic consrains will be violaed if d i j moves o cell (x, y). Therefore, our rouer will no end o move a drople o cell (x, y). 4.4 Handling 3-pin Nes For a 3-pin ne n i, we need o merge hese wo droples during heir ransporaion. To guaranee ha hese wo droples are merged, we firs roue hese wo droples oward each oher. Afer merged, we consider hem as one drople and roue hem oward heir sink. Therefore, for drople movemen cos compuaion, he sink of d i 0 (di 1 ) is he locaion of di 1 (di 0 ) before merged. 4.5 Implemenaion We now presen several implemenaion deails associaed wih obaining a fas and reasonable soluion o his problem. To furher reduce he problem size, we add a resricion ha a drople can only move oward is sink excep he following wo cases. Firs, if a drople canno move oward is sink due o obsacles, we allow he drople o move in all possible direcions. Second, if he cell wih minimum drople movemen cos is no oward he sink, we allow d i j o move o his cell. 5. EXPERIMENTAL RESULTS Our progressive ILP algorihm was implemened in he C++ language, and GLPK [1] was used as our ILP solver. For he purpose of comparison, we also implemened he clique pariioning based [8] and he graph coloring based [4] algorihms. All of he above algorihms and ILP formulaions were execued on a 1.2 GHz SUN Blade-2000 machine wih 8GB memory. We evaluaed our rouing algorihm on he wo pracical bioassays used in he previous work [10]: he in-viro diagnosics and he colorimeric proein assay. We performed wo experimens o verify he efficiency and effeciveness of our progressive ILP algorihm. In he firs experimen, we compared he basic ILP and progressive ILP algorihms. We generaed boh he basic and progressive ILP formulaions for each 2D plane. The experimenal resuls are lised in Table 2. Columns 2 and 3 (Columns 4 and 5) lis he maximum number of variables (consrains) of all problem insances. For he basic ILP, he problem insance is one 2D plane, and for he progressive ILP, he problem insance is one ime sep. The resuls show hahebasicilpneedsaleasfivedaysosolveall2dplanes of one benchmark, which is no feasible for his problem; in conras, our progressive ILP algorihm needs a mos 15.36 sec due Figure 6: The resul of he in-viro 1 benchmark. o he significanly smaller problem size. For example, as shown in Table 2, for he proein 1 benchmark, we can reduce he number of variables (consrains) by 99.72% (99.89%) for he larges problem insance. This resul demonsraes he efficiency of our progressive ILP formulaion. Our progressive ILP formulaion can obain a near-opimal soluion wih much less CPU ime compared wih he basic ILP formulaion. For example, for one 2D plane of he proein 1 benchmark wih four droples, he progressive ILP formulaion obains a soluion of 21 cycles in 0.40 sec while he basic ILP formulaion obains a 20-cycle soluion in 19632 sec. Since i is no feasible o solve he basic ILP for pracical bioassays, we are no able o compare he soluion qualiy of hese wo algorihms direcly. Neverheless, we observe ha he difference in he soluion qualiy for he wo algorihms is a mos 3 cycles for 41 2D planes compleed by he basic ILP formulaion, and he average soluion difference is only 1.68%. The resul shows ha he progressive ILP algorihm is close o he opimal soluion. In he second experimen, we verified he qualiy of our progressive ILP rouing scheme. Since previous works require a direcaddressing rouing soluion, for fair comparison, we firs generaed he direc-addressing rouing soluion using he neworkflow based algorihm [10] and applied he previous approaches o obain he final rouing soluion. Moreover, we modified he nework-flow based algorihm o minimize he drople ransporaion ime. Table 3 shows he experimenal resul. We repor he maximum and average T l of all 2D planes and he CPU ime o roue all 2D planes. The CPU imes for [10]+[8] and [10]+[4] are he oal CPU imes o obain he cross-referencing rouing soluion. As shown in his able, our progressive ILP rouing algorihm can obain much smaller drople ransporaion imes (in cycles) han he previous approaches, under reasonable CPU imes. This resul demonsraes ha our algorihm is very effecive for drople rouing on cross-referencing biochips. Figure 6 shows he rouing resul of one 2D plane of he in-viro 1benchmark a cycle 8. 6. CONCLUSION In his paper, we have proposed he firs drople rouing algorihm ha operaes direcly on cross-referencing biochips wihou a direc-addressing rouing soluion. We have also presened he basic ILP formulaion o minimize he drople ransporaion ime and he progressive ILP rouing scheme o ieraively deermine he minimum-cos posiions of he droples a each ime sep. Experimenal resuls have shown he efficiency and effeciveness of our algorihm. 7. REFERENCES [1] hp://www.gnu.org/sofware/glpk/. [2] M. Cho and D. Z. Pan. Boxrouer: a new global rouer based on box expansion and progressive ilp. In DAC, pages 373 378, July 2006. [3] J. Gong, S.-K. Fan, and C.-J. Kim. Porable digial microfluidics plaform wih acive bu disposable lab-on-chip. In MEMS, pages 355 358, 2004. [4] E. J. Griffih, S. Akella, and M. K. Goldberg. Performance characerizaion of a reconfigurable planar-array digial microfluidic sysem. 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