Sensors 2010,10,6730-6750;doi:10.3390/s100706730 O PEN A C C ESS sensors ISSN 1424-8220 w w w m dpi.com /journal/sensors Article C om putationalM ethodology for A bsolute C alibration C urves for M icrofluidic O pticalA nalyses C hia-Pin C hang *,D avid J.N ageland M ona E.Zaghloul D epartm entofElectricaland Com puterEngineering,The G eorge W ashington U niversity, W ashington D C 20052,U SA ;E-M ails:nagel@ gw u.edu (D J.N );zaghloul@ gw u.edu (M E.Z.) * A uthorto w hom correspondence should be addressed;E-M ail:chiapinc@ gw u.edu; Tel.:+1-202-994-5293;Fax:+1-202-994-5505 Received:10 M ay 2010;in revised form :12 July 2010 /Accepted:12 July 2010 / Published:12 July 2010 A bstract: O pticalfluorescence and absorption are tw o of the prim ary techniques used for analytical m icrofluidics W e provide a thorough yet tractable m ethod for com puting the perform ance ofdiverse opticalm icro-analyticalsystem s.Sam ple sizes range from nano-to m any m icro-liters and concentrations from nano-to m illi-m olar.Equations are provided to trace quantitatively the flow ofthe fundam entalentities,nam ely photons and electrons,and the conversion of energy from the source, through optical com ponents, sam ples and spectral-selective com ponents, to the detectors and beyond The equations perm it facile com putations of calibration curves thatrelate the concentrations or num bers of m olecules m easured to the absolute signals from the system This m ethodology provides the basis for both detailed understanding and im proved design of m icrofluidic optical analytical system s.Itsaves prototype turn-around tim e,and is m uch sim plerand fasterto use than ray tracing program s.O vertw o thousand spreadsheetcom putations w ere perform ed during this study.W e found thatsom e design variations produce highersignallevels and,forconstant noise levels,low erm inim um detection lim its.Im provem ents ofm ore than a factorof1,000 w ere realized K eyw ords: m icrofluidic; chem ical analysis; bio-chem ical analysis; optical fluorescence; opticalabsorption Sensors 2010,10 6731 1.Introduction The qualitative and quantitative chem ical and bio-chem ical analyses of m icro-liter and sm aller volum es ofdiverse fluids constitute one ofthe m ain applications ofm icrofluidic system s [1].There are severalapproaches to obtaining signals from m icrom eter-scale volum es in the process of perform ing analyses [2-6].Electrical m easurem ents are com m on for sam ples that have an ionic com ponent (D C conductivity) or polarizable m olecules (A C im pedance).O pticaltechniques,notably fluorescence and also absorption,are also w idely used forsam ples thatare optically active [7,8] A s partofan experim entalstudy on the lim its ofdetection foranalyte m olecules in m icro-channels or thin film s,w e are concerned w ith relating the absolute num ber of m olecules accessible to optical em ission and absorption equipm entto the absolute signalstrengths (usually in volts)thatare available from analyticalinstrum ents.This paper provides the setof linked equations for such relationships for both opticalem ission and absorption m easurem ents.There is considerable literature on chem icaland biologicalanalyticalcalibration curves form icrofluidic system s,butm ostcalibration curves are noton an absolute basis.Further,no papers provide a com plete description ofthe com ponents and geom etries em ployed In this paper, w e present and use a new and straightforw ard com putational approach for quantitative opticalanalysis of m icroscale fluids.A bsolute calibration curves w ere calculated for 216 varying designs and concentrations There are several advantages to the technique w e have developed for optical m icro-analyses of fluids.M ost fundam entally,it deals w ith individual entities.These are the m olecules,w hich are the objectofusing m icrofluidic analyticalsystem s,and quanta,specifically photons and electrons,thatare em ployed for the analyses O ur approach focuses on the individual com ponents in an optical m icro-analyticalsystem ,each w ith associated specifications,efficiency and geom etry,w hich determ ine the overall perform ance of the system W e present sim ple and useful equations that link the com ponents optically.They determ ine the transportofphotons through the system O verall,use ofthe equations relates the num ber of m olecules in the analytical volum e to the m easured signal This approach m akes itrelatively easy to determ ine the com ponents orgeom etries thatare m ostam enable to significantim provem ents during design of an analyzer system In fact,the variation of the m easured signal w ith changes in any of the com ponent param eters is straightforw ard to com pute, if the geom etrical and other param eters are know n or estim ated.Calculations based on the m ethod can be m ade using sim ple com puterprogram s oreven spreadsheets This paper provides three benefits.First,w e developed and utilized a com prehensive,yetefficient, m eans of com puting absolute calibration curves for m icrofluidic opticalanalysis system s.Second,the num erous results reported and discussed clearly dem onstrate the advantages of this m ethodology for exam ining the efficacy of alternative optical com ponents and designs Finally, w e have a com putationalbasis forcom parison w ith experim ents M ore specifically,the m ain features ofournew m ethodology can be sum m arized as follow s: x Itis absolute,and relates m olecularconcentrations ornum bers to realistic detectorsignals x Itis com plete,including allcom ponents and geom etricalfactors thataffectthe m easured signal fora given analyte concentration x The m ethodology is alm ost entirely algebraic, except for the case of fiber optic coupling to m icrochannels,w hich is notvery im portantpractically Sensors 2010,10 6732 x The effects ofthe various param eters needed forcom putations are quite clear x Being m athem atically sim ple, the m ethod m akes possible fast calculations and thorough param etric studies x The technique perm its exam ination ofrealistic designs w ithoutthe tim e and expense ofm aking and using prototypes x The calculation of calibration curves is m uch m ore efficient than to m easuring them in the laboratory x The m ethodology is testable by com parisons of its predictions w ith the results of experim ents using the sam e com ponents and geom etries x The m ethodology is scale-independent It can be used for m acroscopic, m esoscopic and m icroscopic opticalsystem s O urinterestin em ission and absorption m ethods ofopticalm icro-analysis has anotherbasis,nam ely theirsim ilarity.This is indicated schem atically in Figure 1.In both cases,a source oflightis needed.In the em ission case,the lightis absorbed,and thatstim ulates fluorescence from m olecules in the sam ple or from tags attached to them In the absorption case,the source provides the photons thatprobe the sam ple and are fractionally absorbed w ithin it.Both em ission and absorption m ethods usually involve a variety ofoptics betw een the source and sam ple in orderto collectlightfrom the source and focus iton the sam ple.Sim ilarly,optics are com m only used betw een the sam ple and detector to collectem ission or unabsorbed photons from the sam ple and focus them on a detector.O ptics in both positions m ay give spectraldiscrim ination to provide m olecularspecificity orgive otherbenefits,notably background reduction The quantitative transport of photons from the source to the sam ple to the m easuring equipm entdepends on the opticalefficiency of the individualcom ponents,and m any geom etricaland spectralfactors Figure The sequence of m ajor com ponents in an optical m icro-analytical system For em ission m easurem ents, the source light goes as far as the sam ple, w here the new fluorescent light originates.For absorption m easurem ents,the tw o sets of optics and the sam ple can be thoughtofas the entire opticalsystem coupling the source to the detector Sam ple (Sa) Source (So) O ptics D etector (D ) O ptics The next section presents our com putational m ethodology for quantitative analysis of sam ples in m icro-channels or thin film s by absorption or fluorescence Section provides m any illustrative com puted calibration curves,w hich w ere obtained using the m ethodology.These results are discussed in the follow ing section The last section sketches w hat is needed for future experim ental w ork on quantitative m icrofluidic opticalanalyses Sensors 2010,10 6733 2.C om putationalM ethodology W e seek to com pute the output of the detector in a m icrofluidic optical analytical system as a function ofthe concentration orthe num berofm olecules accessible to the system Such a relationship constitutes the useable partofthe calibration curve forthe instrum ent.Thatis the partofa calibration curve above the noise level of the signal and below the saturation of the system output The com putation requires linking the source of photons for stim ulating fluorescence or probing absorption to the analytical sam ple and detector through all interm ediate optics and spectrally sensitive com ponents G eom etry plays a dom inant role in the efficiency w ith w hich all the com ponents are coupled.In this section,w e provide equations and diagram s forthe needed calculations.Concatenation of all the equations for a particular set of com ponents and their arrangem ent yields the desired calibration curve W e em phasize that w e are sacrificing som e detail for com pleteness W e provide relatively sim ple,but useful equations for a com plete linkage U ncertainties in our results are sm all com pared to the large variations in opticaldesign,w hich can change calibration curves by m ore than three orders ofm agnitudes forthe sam e concentration ofthe analyte The quantitative presentation of our m ethodology is for both fluorescence and absorption m easurem ents of sam ples in both m icrochannels and thin film s w ith lens, no optics or fiber optic coupling ofthe source to the sam ple and the sam ple to the detector.The lightfrom the source w illbe assum ed to strike the sam ples in the channels or film s norm ally,w ith one exception.Thatis coaxial fiber coupling into and outof the ends of m icrochannels.Itis relatively difficultand unproductive to use lenses to couple lightfrom a source into the axis ofa m icrochannel.A fterconsidering the prim ary aspects oflens coupling,w e w illturn to the possibility of dispensing w ith geom etricaloptics entirely Then,w e consider the use of fiber optics Fibers also m ake it possible to either fluorescence or absorption m easurem ents along the length or perpendicular to the axis of a m icrochannel.The use of opticalfibers w ith thin film sam ples is usually notreasonable because either very little of the sam ple film is view ed orthe instrum entbecom es relatively large.H ow ever,ourm ethodology can be applied to thatcase also The follow ing paragraphs trace the source or fluorescent photons from com ponent to com ponent, forsam ples in m icrochannels orthin film s.Itis assum ed throughoutthatthe com ponents ofthe system are properly aligned A chieving alignm ent is challenging but m ust be done experim entally, if perform ance is to be optim ized,orifcom parisons ofcom puted and m easured signals are to be m ade A gain,w e em phasize that,forabsorption m easurem ents,the So to Sa and Sa to D axes on both sides of the sam ples m ust be co-linear H ow ever, that is neither necessary nor desirable for fluorescence m easurem ents because light from the source that transits the sam ple m ight strike and stim ulate the detectoras a very undesirable background.W e w illnotexplicitly treatthe very diverse geom etries for fluorescence m easurem ents in w hich the So to Sa and Sa to D axes on the opposite or sam e side of the sam ples are not co-linear D oing so for a specific system design (com ponents and geom etries) is straightforw ard Sensors 2010,10 6734 2.1.Source Strength The specifications forthe intensity ofLED s are com m only given in the photom etric units oflum ens Equation (1)can be used to convertlum ens to w atts Pow er(W ) Lum ens 683 lum ens perw attu (Lum inous Efficacy) (1) w here lum inous efficacy is w avelength-dependent[9] Laserspecifications are generally given in w atts.Equation (2)is used forcom puting the photons per second from the w atts Ps Photons Second 5.03 u 1015 u Pow er(W )u O (nm ) (2) ZKHUHȜLVWKHZDYHOHQJWKRIWKHODVHUOLJKW 6RPHOLJKWVRXUFHVSHFLILFDWLRQVJLYHWKHIXOOFRQLFDOHPLVVLRQEHDPDQJOH ș The corresponding solid angle (:S)in units ofsteradians is given by Equation (3): :s 2S (1 cosT ) (3) 2.2.Source to Lens to Sam ple A diagram usefulforcom puting the fraction ofthe photons em itted by the initialsource thatgets to the plane ofthe m icrofluidic sam ple containing the analyte is given in Figure 2.There are tw o prim ary cases.In the first,som e ofthe source photons m iss the lens and are w asted.Then,Equation (4)perm its com putation ofthe fraction ofthe photons thatstrikes the lens and gets focused onto the sam ple plane O therw ise allofthe photons hitthe lens.The sm allloss ofphotons due to the lens itselfis ignored P (So o L1) PS u SRL12 :S X (4) PS is num ber of photons per second the lightsource generates,R L1 is the radius of Lens L1,X is the distance betw een lightsource and Lens L1,and :s is the source em ission solid angle in steradians Figure 2.Schem atic ofa largersource like an LED (dotted box)ora sm allsource such as a laser(black line),and a lens thatcollects the lightand focuses iton the sam ple Sensors 2010,10 6735 2.3.FocalConditions on the M icro-channel There are three possibilities forthe relative size ofthe focalspoton the plane ofthe channeland the size ofthe channel.Sim ilarly,there are three cases forthe view ofthe detectorbackw ards to thatplane The nine com binations are indicated in Figure 3.The essential factors are (a) the size of the source focal spot at the sam ple and (b) the sam ple area from w hich photons can get to the detector, both relative to (c) the w idth of the channel and each other.The focal spot for the source and the region view ed by the detector or spectrom eter are com m only circles,although they m ay have rectangular or othershapes The area ofthe focused source spoton the plane ofthe m icro-channelA C can be com puted from the source area A S, the lens focal length F1 and the distances betw een com ponents show n in Figure Equations (5)and (6)apply fora thin lens AC ASu X2 X1 (5) F1 1 X1 X (6) X and X are both >F1.Ifthey are equaland equalto 2F1,the area ofthe spoton the channelis the sam e as the source size.Then,ignoring the sm alllosses in the lens,the area photon density is the sam e atthe source and channel,w hen the lens intercepts allofthe em itted photons Figure Schem atic show ing the relative sizes of the source focal spot, the detector acceptance region and the m icro-channel, w hich is indicated by the tw o parallel vertical lines.The num bers to the left of each of the nine sketches indicate w hat fraction of the source photons can m ake itinto the analyte fluid in the channel.The num bers to the right of the sketches indicate the fraction of the illum inated area atthe channel,w hich be seen by the detector Sensors 2010,10 6736 A s already noted for both em ission and absorption,the collinear directions of So to Sa and Sa to D can be either(a)norm alto a channelora thin film sam ple,or(b)parallelto and w ithin a channel.The first is best w ith lens coupling w ith either one-dim ensional (channel) or tw o-dim ensional (film ) sam ples,and itw illbe treated next.Then,the second,w hich is bestfor fiber optic coupling,w ill be considered nearthe end ofthis section.O thergeom etries are possible,butthose tw o lim iting cases are generally m ost advantageous.The prim ary exception is to have the So to Sa and the Sa to D axes at som e angle to each otherin orderto preventsource photons from directly entering the detector during fluorescence m easurem ents 2.4.Transm itted LightPerpendicular to a Channelor Film For absorption,the incidentand transm itted radiation can be norm alto the channelor film In that case,the num beroftransm itted photons PT is given be Equation (7),again forthe optically thin case PT Pc eHC A (7) w here Pc LV QXPEHU RI SKRWRQV VWULNLQJ WKH IOXLG LQ WKH FKDQQHO RU ILOP İ LV WKH PRODU DEVRUSWLRQ coefficient [10] w ith units of L·m oleí1·cm í1 w hen l is the sam ple thickness, in centim eters, in the direction on a line to the source.C is the volum etric concentration (m olarity)ofthe solution 2.5.Fluorescence Perpendicular to a Channelor Film The num ber of em itted fluorescentphotons is equalto the num ber of absorbed photons tim es the quantum yield.Equations (8),(9)and (10)apply Pabs Pc (1 e HC A ) (8) Forthe com m on case thatthe sam ple fluid is optically thin,thatis,HClis sm allcom pared to unity, Pabs Pc u H u C u A (9) and PSa Pabs u QY (10) w here PSa is num berofphotons sam ple em itted and Q Y is the quantum yield 2.6.Sam ple to Lens to Filter and D etector The sam ple is effectively a source ofradiation w ith an em ission solid angle ofʌVWHUDGLDQVforthe restofthe system ,w hen the lightfrom the sam ple is fluorescence.A s w as the case w ith the source,itis necessary to com pute the fraction of the photons from the sam ple that are intercepted by the second lens.This is illustrated in Figure Sensors 2010,10 6737 Figure Schem atic of the path for radiation from the sam ple, either fluorescence or transm ission,through a lens and spectralfilterto the detector Lens L2 Filter Sam ple D etector X3 X4 A relation sim ilar to Equation (4) is em ployed to com pute the fraction of the fluorescentradiation from the sam ple thatstrikes the second lens L2.Itis given in Equation (11) R P (Sa o L2) PSa u L 2 4X (11) w here R L2 is the radius ofLens L2 and X is the distance betw een sam ple and Lens L2.W e note that,if a large area detector can be placed close to the sam ple, the lens L2 is not needed H ow ever, for fluorescence m easurem ents,this w ill lead to the detector intercepting and responding to unabsorbed source photons M ost of that unw anted radiation can be intercepted and absorbed by a narrow bandw idth filterin frontofthe detector For absorption com putations, the angle at w hich transm itted radiation em erges from the analyte fluid can be determ ined by either (a) its entrance angle, w hen absorption is m easured across a m icrochannel,or(b)the confines ofthe m icro-channel,w hen absorption is m easured along the length of a channel That is, the ratio of the channel w idth to the length over w hich incident radiation propagates w ithin the channelcan determ ine the em ergence angle 2.7.SpectralD iscrim ination A lthough a spectrom eter is the best spectral discrim ination tool, it w ill not be quantitatively considered in this m ethodology.In orderto com pute the outputofa spectrom eteron an absolute basis, both the w avelength-dependentinputand overallspectrom eterefficiency m ustbe know n.The latteris rarely available A usefulfilter is usually a narrow band interference device w ith peak w avelength very close to the peak w avelength of the fluorescence spectrum The transm ission characteristics of w ell-designed and m anufactured filters perm it50% to nearly 100% transm ission w ithin a pass band thatincludes som e or all of the entire w idth of the fluorescence lines,or the transm itted radiation for the absorption case Transm itted fluorescence photons afterfiltercan be com puted as: Pafter filer t P (Sa o L2)u FW H M of filter u Transm ission Efficiency Bandw idth of Em ission Spectrum (12) A quantitative determ ination of the filter pass band and the fluorescentline w idth can be done by auxiliary m easurem ents w ith a spectrom eter,ifthey are notavailable from the m anufacturer.D oing so Sensors 2010,10 6738 w illdeterm ine ifany correction has to be applied in the com putation ofthe num berofphotons reaching the detector[8] 2.8.D etector Signals In som e cases,the active area of a detector is sm aller than the exposed area in the detector plane, w hich is irradiated by fluorescent photons Equation (13) gives the num ber of photons striking the detector,nam ely PD : PD P after filteru A ctive areaof D etector Exposed A rea in D etection Plane (13) The electronic signals from the detector depend on the num ber of photons incident on it, the w avelength-dependentefficiency and the electronic gain,ifany.Equation (14)applies ED PD u Q E u G (14) w here ED is the num ber of electrons per second from the detector, PD is the num ber of photons received by detector per second, Q E is the quantum efficiency of detector, and G is the gain of the detector For alm ost all detectors, the efficiency for conversion of photons to electrons is less than unity Q uantum efficiencies are usually available from the detector m anufacturer M any detectors not cause m ultiplication ofthe num berofelectrons thatresultfrom photon absorption in the detector.That is, they have no gain H ow ever, avalanche photo diodes, and either solid-state or vacuum photom ultipliers, provide gain The value of the gain can be high, w ith as m any as one m illion electrons em erging from the detector for every electron initially generated by photo absorption H ow ever, detectors that provide very high gains involve high voltages, to w hich the gain is very sensitive.A lso,they are relatively expensive and,in the case of vacuum tubes,are significantly larger than solid-state detectors w ithoutgain.The latterare com m only PN orPIN diodes,w hich are relatively sm alland cheap,and require only low voltages.H ow ever,they not have gain w ithin the detector elem ent.Solid state photom ultipliers em ploy interm ediate voltages and stilloffersubstaintialgains Photo sensitivity (also know n as responsivity) is com m only expressed as am ps (coulom bs per second) per w att (joules per second) of the incident light H ence the definition of a Coulom b and Equation (2)m ustbe em ployed forconversion ofunits.The responsivity converts the photons received by detector per second into the signal output of the detector w ithout the use of Equation (14) If responsivity inform ation is provided,then outputsignalofthe detectoris: Output Signal PD u (responsiv ity atGain 1)u G 5.03 u 1015 u Ȝ (15) 2.9.Post-D etection Electronics The signals directly from individualdetectors or arrays of detectors are com m only quite sm alland they m ay contain noise thatis often am enable to electronic filtering.In general,signals from photon detectors are handled in eitheroftw o m odes,pulse counting orcurrentm easurem ents.In the firstcase, pulses due to absorption of individualphotons in a detector,usually w ith gain,can be counted.Then, Sensors 2010,10 6739 there are som e beneficial possibilities to reject noise Electronic filters can be used to discrim inate againstnoise w ith frequencies thatare eithertoo low ,orelse too high relative to the photon arrivaland electron production rates Electronics, w hich determ ine the height or integral of the pulses, are com m only use to rejectpulses thatare too sm all.Such electronics can perform analyses ofthe shape of the pulses to insure that,even ifthe pulses pass the size screening,they have the propercharacteristics to be caused by photons.H ow ever,the very fastelectronics for capture and exam ination of individual pulses in realtim e are relatively large and expensive If the pulses arrive at rates that preclude their individual analysis,then current m easurem ents are m ade.In this case,itis possible to em ploy electronics afterthe detector to am plify the analog current Then,the finalsignalis given by Equation (16): (16) EA ED × (A m plification) w here EA is the num berofelectrons afteram plification.The electron arrivalrate is a current,ofcourse Transim pedance am plifiers turn currentvalues into voltages.Forthe case ofpulse counting ofphotons, digitalm ethods are used forcom puterrecording ofthe photon arrivalrates.Forthe analog currentcase, w ithout or w ith am plification, analog-to-digital converters are usually used to obtain data in digital form forrecording and m anipulation by com puters W hateverthe m eans ofspectraldiscrim ination orphoton detection and am plification,in orafterthe detector,both fordigitalphoton counting and foranalog currentm easurem ents,there usually results a digitalsignalrelated to the photon arrivalrate atthe detector 2.10.No O ptics The preceding m ethodology dealt w ith lens coupling of the source photons to the sam ple and the coupling of either the transm itted source photons or generated fluorescent photons to the detector A nalytical m icrosystem s w ithout lens coupling are also possible Their perform ance (calibration curves) can be com puted using the equations already presented.System s w ithout interm ediate optics can handle sam ples sizes over a w ide range.A lso,they are sim pler than the case w ith lenses because there are few ercom ponents to procure,align and hold in place.W ithoutthe constraintofthe lens focal lengths,system s w ith no lenses can also be m ore com pact.H ow ever,as w illbe seen in Section 3,the no-optics case has low eroutputsignals forparticularconcentrations com pared to system s w ith lenses 2.11.Fiber O ptics The equations above provide the m eans to com pute the calibration curves for m icrofluidic optical analyticalsystem s using lens or no optics.A s noted earlier,fiber optics can be em ployed to transport photons from the source to the sam ple and,thence,to a spectrally-sensitive com ponentand detector There are som e notable advantages to using fiber optics w ith m icrofluidic system s Because the externaland core diam eters offibers can be com parable to the w idths and depths ofm icro-channels,it is possible and relatively easy to integrate fiberoptics into such fluidic platform s.This can be done by using ordinary fibers and putting them into the m icrofluidic platform ,or by building opticalchannels, as w ell as fluidic channels, into a substrate Either w ay, it is possible to closely couple an off-chip source to a fiberoptic,w hich ends close to the fluid channel.Sim ilarly,the space betw een the sam ple Sensors 2010,10 6740 and a second fiberoptic to take the fluorescentorunabsorbed photons to the filterbefore a detector,or to a spectrom eter,can be sm alland geom etrically efficient.Itm ustbe noted thatfiber coupling is not attractive for single-use m icrofluidic platform s unless the fiber can also be disposable and easily (cheaply)connected to the unitcontaining the source,detectorand electronics The coupling of a source to a fiber optic is show n schem atically in m ore detail in Figure 5.Tw o steps are needed to calculate the fraction of the em itted photons that enter the fiber The first is to com pute the fraction ofthe source area thatis w ithin the acceptance angle ofthe fiber.The nextstep is to calculate the fraction of the photons em itted from thatarea thatfallon the core of the fiber optic The resultis Equation (17) 2 :F SR F :F D SR F (17) PS u PF PS u u A S :S AS :S D w here PF is num ber of photons entering the fiber optics,PS is num ber of photons the source em itted, ȍF is the acceptance solid angle ofthe fiberoptics and the R F is the core radius offiberoptics.D is the distance betw een lightsource and fiber.W hen ȍFÂ'ðLV ODUJHUWKDQVRXUFHDUHD A S,then ȍFÂD ²/A S is equalto Figure 5.D iagram show ing the part of the source (heavy vertical line) that is w ithin the acceptance angle of a nearby fiber optic (indicated by the bracket),and the solid angle of lightfrom one partofthatregion,w hich is intercepted by the core ofa fiber(stippled) If the optical fiber acceptance specification is given as a num erical aperture (N A ),Equation (18) perm its calculation ofhalfacceptance angle offiberoptic,șF: 1$ QÂVLQ șF) (18) The refractive index n is for air, 1.33 for w ater and 1.36 for ethanol Equation (3) enables calculation of ȍF from șF.A s in the case of a lens accepting radiation from a source, the em ission pattern (solid angle) of the source enters the calculation.H ow ever,the very sm allfiber cores (on the order of 10 to 100 m icrom eters in diam eter), rather than the lens diam eter (on the order of10 m illim eters),are the acceptance areas It is interesting to note that,as the source-to-fiber distance D is increased,the area of the source view ed by the fiber increases as D w hile the area intercepted by the fiber core decreases as 1/D H ence,as long as the area ofthe source w ithin the fiberacceptance angle is less than the overallsource area,increasing D does not decrease the num ber of photons that get into the fiber.This presum es a source thatem its uniform ly overits area and overits solid angle Sensors 2010,10 6741 There are tw o prim ary geom etries forthe coupling oflightinto and outofm icrochannels using fiber optics.They are orthogonalto the channelor co-axialw ith the channel.The transm ission of incident photons forabsorption m easurem ents in the cross-the-channelcase is relatively easy to com pute using Equation (7).The beam com ing from the fiberoptic coupled to the source does notspread m uch w hen crossing a sm allchannel The calculation of the num ber of unabsorbed photons is m ore com plex in the co-axial case Sim ilarly,com putation of the num ber of fluorescent photons generated, and the fraction captured by the fiber optic going to the detector,is notas sim ple in the coaxialcase as in the lens coupling case Calculation of both transm ission and fluorescent signals for channels of varying lengths requires a single integration over the channel length That is straightforw ard, but still m ore com plex than the algebraic equations presented above Coaxial couplings of m icrochannels and fiber optics are little used.Because ofthatfact,and because oftheirgreaterm athem aticalcom plexity,w e notpresentthe integralequations for the coaxialcase.H ow ever,results based on the use ofthese equations are given in the next section It can be seen that coaxial coupling of m icrochannels and fiber optics leads to non-linearcalibration curves athigh concentrations and to very poorsystem efficiency The transm ission efficiency of fiber optics is w avelength dependent.Thatefficiency m ay be gotten eitherfrom m easurem entorfrom the m anufacturer's specifications in orderto com pute the fraction of the flux ofphotons from the source orsam ple thatgets to the nextpartofthe system 2.12.M ixed O ptics System s In the first partof this section,w e dealt w ith system s having tw o lenses,one on each side of the sam ple in the m icro-channel.N ext,w e dealtw ith the no-optics case.Then,w e outlined the behaviorof fiberoptics thatcan be used in lieu ofeitherofthe lenses.Itis possible to have opticalm icro-analytical system s thathave m ixed optics,w ith lenses,fiberoptics orno optics eitherbefore orafterthe sam ple For exam ple,a lens m ightbe used for an LED w ith a relatively broad em ission solid angle to focus m ost of the source photons on the analytical fluid in a channel.Then,if a spectrom eter w ith a fiber optic inputis being used,itw ould couple the fluorescence from the sam ple into the spectrom eter 2.13.O verallSignalCalculation The finalexpression,w hich relates the m easured signalto the concentration or num ber of analyte m olecules,can be gotten by successively linking the individualequations given above forthe particular com bination ofcom ponents in any m icrofluidic analyticalinstrum ent.This is true forlens,no optics or fiberoptics cases.Forboth fluorescence and absorption experim ents,the signaldepends linearly on the source strength If the analyte fluid is optically thin to both incident and either fluorescent or transm itted radiation,then the signalalso depends linearly on the num ber of m olecules thatare both irradiated by the source and view ed by the detector,w hetheritis an individualdevice behind a filteror builtinto a spectrom eter The sensitivity of the signalto any of the geom etricaland other param eters in the overallequation can be com puted by taking the partialderivative ofthe signalstrength w ith respectto the param eterof interest.In particular,the derivative ofthe signalw ith respectto the num berofm olecules is the slope ofthe calibration curve,thatis,the instrum entalresponsivity,w hich is particularly im portant.A large Sensors 2010,10 6742 derivative,thatis,a high responsivity of the signalto the num ber of m olecules,generally m eans that the precision ofthe analysis can be high,butthe dynam ic range w illbe relatively sm all.Conversely,a sm allslope and responsivity m ay m ake itpossible forthe instrum entto give usefulvalues overa broad range ofm olecularnum bers (concentrations),butw ith less precision 3.C om puted C alibration C urves The com putationalm ethodology justpresented has been used to calculate the calibration curves for a w ide variety of com binations of sources, optics, sam ples, detectors and geom etries W hile the m ethodology can be used forabsorption analysis as w ellas forfluorescence situations,w e concentrate on the fluorescence approach M ost of the published papers on m icrofluidic optical analysis use fluorescence rather than absorption A nd, the m easurem ents w e are planning to test the new com putationalm ethodology are based on fluorescence and notabsorption.Besides,the com putation of the source absorption in the process of estim ating the fluorescence intensity is essentially the sam e as the calculation ofsignals forabsorption experim ents The results ofourcalculations offluorescence calibration curves presented in this section are based on particularopticalcom ponents and theirspecifications.The specific com ponents forw hich w e have done calculations and are doing experim ents w illbe cited in detailin experim entalpapers Since the opticalcoupling and geom etry are m ajorvariables in both the design and perform ance of m icrofluidic optical analytical system s, w e em ployed three different cases, w hich are presented in Figure The com putational results are based on these three geom etrical cases, and on using three different light sources, three different optics, tw o different sam ples and tw o detectors The detector outputs for six concentrations w ere com puted for each sam ple and com bination of com ponents and geom etry.H ence,the inform ation presented here is the result of over 600 individual calculations of detectoroutputforspecific com binations ofcom ponents,geom etries and concentrations alldone using an EX CEL spreadsheet.O ver tw o thousand com putations w ere done w ith the spreadsheetin order to exam ine alternative geom etries.This testifies to the facility w ith w hich calibration curves for optical m icro-analysis can be com puted using ourm ethodology Figure 6.The three cases forw hich calculated calibration curves are presented.The firstis lens coupling to and from eitherm icrochannels orthin film s.The second case has the sam e types of sam ple holders,butw ithoutoptics.The lastcase deals w ith fiber optics coupling to a m icro-channel, either w ithin (co-axial) the channel or else orthogonal (cross) the channel Lenses D M icroChannelorEdge V iew ofThin Film D N o O ptics In Fiber O ptics M icroChannel O ut Sensors 2010,10 6743 The volum es of the sam ples, w hich are analyzed for these three types of optics, are plotted in Figure 7.Itis notew orthy thatourm ethodology has handled sam ples thatrange in volum e from nL to about1 m L.Com putation ofcalibration curves forsm allerorlargersam ples is also possible w ith this m ethodology.W e used fluorescein forthe illustrative calculations because ithas been w idely em ployed in experim ents w ith m icrofluidic analyticalsystem s [11-17] Figure 7.The volum es ofsam ples forw hich the results in this section w ere obtained.1 nL is a cube 100 m icrom eters on a side.1 m L is centim etercubed.Fiberoptics are sm alland interrogate only sm allvolum es.System s w ith no optics can probe a w ide range ofvolum es, including relatively large sam ples Fiber O ptics Lens O ptics N o O ptics 1nL 10nL 100nL 1µL 10µL 100µL 1m L R ange ofSam ple Volum es Calibration curves w ere com puted as a function ofboth concentration (m olarity)and the num bers of m olecules accessible in the analysis Concentrations are com m only desired, but the num bers of m olecules are useful for com paring the efficiencies of optical analytical instrum ents.The calibration curves for the three geom etrical cases of Figure 6,and m any com ponent variations,are presented in Figures and 9.The calculated curves have the sam e slopes because allparts ofthe system s are linear The use of log-log scales is necessary because of the very w ide ranges of concentrations and output signals.These graphs clearly show the absolute and relative perform ance of the various com ponents and geom etries.V erticallines atspecific concentrations w ould show thatthe signals from the detectors can vary by overthree orders ofm agnitude fora particularconcentration.H orizontallines can be used to bracket the detector outputs ranging from the noise level to the saturation signal The m inim um detectable lim it and the dynam ic range vary greatly depending on the optical com ponents and theirarrangem ents The com puted signals forspecific concentrations ornum berofm olecules vary m ore than 10³forthe different com ponents and geom etries It is clear that the case for the analyte in m icrochannels and coaxiallighttransm ission gives relatively poor perform ance.Conversely,having the sam ple in a thin film w ith both the incidentexcitation lightand fluorescence at90 degrees to the film provides m uch greatersignals than the othercases Sensors 2010,10 6744 Figure Com puted calibration curves as a function of the m olar concentration of fluorescein for several geom etries, sources, optics and detectors: (a) lens (1.5 cm focal length and diam eter) coupling to a 100 µm square m icrochannel, (b) lens (1.5 cm focal length and diam eter) coupling orthogonal to a 100 µm thin film , (c) light from sources to 100 µm square m icrochannels and fluorescence to detectors w ithout intervening optics and (d) light from sources to 100 µm thin film s and fluorescence to detectors w ithout intervening optics, (e) fiber optic (100 µm diam eter ) coupled orthogonal (cross) to a 100 µm square m icrochannel,(f)fiberoptic (100 µm diam eter)coupled w ithin (co-axial) a 1,000 µm length of a 100 µm square channel The sources are blue LED s w ith either10 or150 degree fullem ission angles ora U V LED w ith a 120 degree fullem ission angle.A filterw as em ployed and the transm ission loss through the filter w as calculated,as described in Section 2.7.The detectors are either an am plified photo diode (A m PD ) or a Silicon photom ultiplier (SiPM ).The straightlines are draw n through the com puted points in these graphs Lens to M icrochannelSignalvs C oncentration 10 D eg B lu Lens uC APD 150 D eg B lu Lens uC APD 120 D eg U V Lens uC APD 10 D eg B lu Lens uC SiPM 150 D eg B lu Lens uC SiPM 120 D eg U V Lens uC SiPM (b) 0 -1 -1 Signal(Log V) Signal(Log V) (a) -2 Lens to Thin Film Sam ple Sigalvs C oncentration 10 D eg B lu Lens ThiFilAPD 150 D eg B lu Lens ThiFilAPD 120 D eg U V Lens ThiFilAPD 10 D eg B lu Lens ThiFilSiPM 150 D eg B lu Lens ThiFilSiPM 120 D eg U V Lens ThiFilSiPM -2 -3 -3 -4 -4 -10 -9 -8 -7 -6 -5 -4 -10 -3 -9 -8 C oncentration (Log M ) N o O ptics to M icrochannelSignalvs C oncentration 10 D eg B lu N oO p uC APD 150 D eg B lu N oO p uC APD 120 D eg U V N oO p uC APD 10 D eg B lu N oO p uC SiPM 150 D eg B lu N oO p uC SiPM 120 D eg U V N oO p uC SiPM (d) 0 -1 -1 Signal(Log V) Signal(Log V) (c) -2 -6 -5 -4 -3 N o O ptics to Thin Film Sam ple Signalvs C oncentration 10 D eg B lu N oO p ThiFilAPD 150 D eg B lu N oO p ThiFilAPD 120 D eg U V N oO p ThiFilAPD 10 D eg B lu N oO p ThiFilSiPM 150 D eg B lu N oO p ThiFilSiPM 120 D eg U V N oO p ThiFilSiPM -2 -3 -3 -4 -4 -10 -9 -8 -7 -6 -5 -4 -10 -3 -9 -8 (e) -7 -6 -5 -4 -3 C oncentration (Log M ) C oncentation (Log M ) FO to 100 um w ide (cross )M icrochannelSignalvs C oncentration 10 D eg B lu FO uC APD 150 D eg B lu FO uC APD 120 D eg U V FO uC APD 10 D eg B lu FO uC SiPM 150 D eg B lu FO uC SiPM 120 D eg U V FO uC SiPM (f) 0 -1 -1 Signal(Log V) Signal(Log V) -7 C oncentation (Log M ) -2 FO to 1000 um long C o-AxialM icrochannelSignalvs C oncentration 10 D eg B lu FO uC APD 150 D eg B lu FO uC APD 120 D eg U V FO uC APD 10 D eg B lu FO uC SiPM 150 D eg B lu FO uC SiPM 120 D eg U V FO uC SiPM -2 -3 -3 -4 -4 -10 -9 -8 -7 -6 C oncentration (Log M ) -5 -4 -3 -10 -9 -8 -7 -6 C oncentration (Log M ) -5 -4 -3 Sensors 2010,10 6745 Figure Com puted calibration curves as a function of the num ber of fluorescein m olecules for several geom etries, sources, optics and detectors This figure is m ade by converting the concentration into num ber of m olecules in different volum es show n in Figure Because of the sam ple volum es are different in various geom etrical arrangem ents,the num berofm olecules is differentatany concentration Lens to M icrochannelSignalvs N um ber ofM olecules 10 D eg B lu Lens uC APD 150 D eg B lu Lens uC APD 120 D eg U V Lens uC APD 10 D eg B lu Lens uC SiPM 150 D eg B lu Lens uC SiPM 120 D eg U V Lens uC SiPM (b) 0 -1 -1 Signal(Log V) Signal(Log V) (a) -2 Lens to Thin Film Sam ple Sigalvs N um ber ofM olecules 10 D eg B lu Lens ThiFilAPD 150 D eg B lu Lens ThiFilAPD 120 D eg U V Lens ThiFilAPD 10 D eg B lu Lens ThiFilSiPM 150 D eg B lu Lens ThiFilSiPM 120 D eg U V Lens ThiFilSiPM -2 -3 -3 -4 -4 10 11 12 13 14 10 N um ber ofM olecules (Log) N o O ptics to M icrochannelSignalvs N um ber ofM olecules 10 D eg B lu N oO p uC APD 150 D eg B lu N oO p uC APD 120 D eg U V N oO p uC APD 10 D eg B lu N oO p uC SiPM 150 D eg B lu N oO p uC SiPM 120 D eg U V N oO p uC SiPM (d) 0 -1 -1 Signal(Log V) Signal(Log V) (c) -2 12 13 14 N o O ptics to Thin Film Sam ple Signalvs N um ber ofM olecules 10 D eg B lu N oO p ThiFilAPD 150 D eg B lu N oO p ThiFilAPD 120 D eg U V N oO p ThiFilAPD 10 D eg B lu N oO p ThiFilSiPM 150 D eg B lu N oO p ThiFilSiPM 120 D eg U V N oO p ThiFilSiPM -2 -3 -3 -4 -4 10 11 12 13 14 10 N um ber ofM olecules (Log) 11 12 13 14 N um ber ofM olecules (Log) (f) (e) FO to 100 um (cross)M icrochannelSignalvs N um berofM olecules FO to 1000 um C o-AxialM icrochannelSignalvs N um ber ofM olecules 10 D eg B lu FO uC APD 150 D eg B lu FO uC APD 120 D eg U V FO uC APD 10 D eg B lu FO uC APD 150 D eg B lu FO uC APD 120 D eg U V FO uC APD 10 D eg B lu FO uC SiPM 150 D eg B lu FO uC SiPM 120 D eg U V FO uC SiPM 10 D eg B lu FO uC SiPM 150 D eg B lu FO uC SiPM 120 D eg U V FO uC SiPM 0 -1 -1 Signal(Log V) Signal(Log V) 11 N um ber ofM olecules (Log) -2 -3 -2 -3 -4 -4 10 11 N um ber ofM olecules (Log) 12 13 14 10 11 12 13 14 N um ber ofM olecules (Log) W e em phasize the factthatour m ethodology does notprovide absolute com puted noise lim its for calibration curves nor absolute com puted m axim um available signals H ow ever, obtaining these characteristics is not a practical problem The intersections of the calibration curves, w hich are com puted,w ith the m inim um detectorsignalgives the m inim um detectable concentration (M D L).The M D L w ill be im proved if the detector has a low er noise floor Sim ilarly, the intersections of the calibrations curves w ith the m axim um detector outputs give the highest concentration that can be assayed,and hence,the dynam ic range of the system The M D L and dynam ic range for the various Sensors 2010,10 6746 calibration curves w ere determ ined using the published characteristics for the tw o detectors The silicon photom ultiplier(SPM M icro1000X 01A 1from SensL)has a noise floorof1 m V and a m axim um signalof500 m V The am plified photodiode (M odelO D A -6W B-500M from O ptoD iode)has the sam e noise floor and a m axim um outputof V w hen supplied w ith voltages equalto ±5 V These values w ere em ployed in determ ining the M D L and dynam ic ranges forthe 36 com binations in Figures and The results are presented in Table Table 1.The m inim um detection lim its (M D L) in nM and dynam ic ranges (factors above the M D L in parentheses) for the calibration curves for the three sources, three optics options,three sam ple geom etries,and tw o detectors LightSources 150 D eg Blue LED 27.58 (11,700) O ptics Sam ple D etectors Lenses 100 µm W ide M icrochannel SiPM Lenses 100 µm W ide M icrochannel A m plified Photodiode (A m PD ) 0.03 (173) 0.86 (4,300) 3.98 (19,900) Lenses 100 µm Thin Film SiPM 0.17 (71) 4.14 (1,760) 28.37 (12,100) Lenses 100 µm Thin Film A m plified Photodiode (A m PD ) 0.003 (13) 0.06 (320) 2.65 (526) N one 100 µm W ide M icrochannel SiPM 32.67 (139,020) 958.52 (408,000) 46622.98 (20,000,000) N one 100 µm W ide M icrochannel A m plified Photodiode (A m PD ) 0.51 (2,550) 14.93 (75,000) 726.11 (144,000) N one 100 µm Thin Film SiPM 1.78 (760) 2.64 (1,100) 128.06 (55,000) N one 100 µm Thin Film A m plified Photodiode (A m PD ) 0.03 (140) 0.04 (200) 1.99 (400) SiPM 5.25 (2,630) 1000 (523,800) 8128.3 (4,065,670) A m plified Photodiode (A m PD ) 0.48 (2,510) 95.5 (490,000) 758.58 (3,900,000) SiPM 17.78 (10,450) 3801.89 (2,235,000) 28183.83 (14,100,000) A m plified Photodiode (A m PD ) 1.55 (9,770) 346.73 (2,190,000) 2630.27 (13,180,000) FiberO ptics FiberO ptics FiberO ptics FiberO ptics 100 µm M icrochannel (cross) 100 µm M icrochannel (cross) 1,000 µm M icrochannel (co-axial) 1,000 µm M icrochannel (co-axial) 10 D eg Blue LED 1.11 (473) 120 D eg U V LED 148.53 (63,200) Sensors 2010,10 6747 4.D iscussion ofthe R esults The tabulation ofM D L values and dynam ic ranges m akes easierthe evaluation ofthe results ofthe com putations com pared to use of the log-log graphs already presented.Considering the M D L for the various cases, the values range from pico-m olar to 46 m icro-m olar, a variation of over 107 The facility w ith w hich these calculations w ere done and the w ide variation in results illustrates the value of our m ethodology for m icro-analytical system design and com parison The narrow er em ission angle LED light source is m ore efficient for delivering the photons to excite the fluorescence em ission com pared to the sam e LED w ith little collim ation Lens coupling show s better incident photon transm ission from an LED lightsource to the sam ple along w ith better fluorescence photon delivery from the sam ple to the detector.H ow ever,itm ustbe re-em phasized thatw e putthe source on one side of the assum ed-transparentsubstrate containing the channelor thin film and the detector on the other side.This is nota practicalgeom etry because lightfrom the source w ould enter the detector.Placing the source and detector on the sam e side of the substrate w ould essentially rem ove this problem ,but decrease the geom etric coupling slightly The M D L values in Table show thatthe thin film sam ple geom etry is substantially better for all com binations ofsources,sam ples and detectors.This is because the usefulpartofthe thin film sam ple contains m ore m olecules due to having a biggervolum e.Itis a good trade-offto use largervolum e of sam ple (thatis m icro-liters,rather than nano-liters) in order to reach low er M D L.O ne ordinary drop contains about50 µL 5.C onclusions O ptical m icrofluidic system s are w idely em ployed in m icro-analytical research and industry [18] Exam ination ofthe alternatives w e considered leads to an appreciation ofthe large num berofpossible opticalm icro-analyticalsystem s.W e discussed m ultiple photon sources;lenses,fiberand no optics for photon transport; fluorescence and absorption techniques for probing sam ples in m icro-channels and thin film s; filters and spectrom eters for spectral discrim ination; and various detectors w ith anallary electronics.There are m any specific choices in each of these categories.H ence,there are hundreds of specific system s.A llofthese can be analyzed and com pared quantitatively using ourm ethodology The largest photon loss in an optical analytical system occurs w hen there is no efficient w ay to collectfluorescence photons from the sam ple to the detector.W hether the system has lenses or fiber optics, there is a num erical aperture (N A ) or acceptance angle for each optic It determ ines the efficiency for gathering the fluorescence photons from the sam ple thatare HPLWWHGLQWRʌVWHUDGLDQV Lightgathering efficiency is a key factor in designing m icrofluidic analysis system s thatcan provide low er M D Ls.U se of ellipsoidalor other m irrors to gather fluorescentphotons w as notcom puted for this paper.H ow ever,this m ethodology can be confidently em ployed forthose cases Com parisons ofcom puted and m easured calibration curves,both w ith the sam e units,should prove especially useful.W e note the centralim portance ofthe absorption coefficients in both fluorescentand absorption m ethods and of the fluorescent yields in fluorescent m easurem ents.It m ay be possible to obtain relative or absolute experim ental estim ates of these param eters for particular com binations of analyte m olecules and w avelengths using our m ethodology.This requires thatallthe geom etricaland Sensors 2010,10 6748 otherparam eters are know n,orcan be independently m easured,w ith sufficientaccuracy.In particular, the absolute source strength, and the quantitative perform ance of the detector and subsequent electronics, m ust all be know n D eterm ination of absorption and fluorescence param eters is challenging H ow ever, if such values are not available, com parison of the com puted and m easured signalstrengths could give estim ates for the absorption coefficients and fluorescentyields.Itrem ains to be seen ifthis approach has usefully sm allerrors.D eterm ining thatw ould be one ofthe m otivations forperform ing an experim entalassessm entofthe m ethodology A cknow ledgem ents Com m ents by JoelG olden on an early version ofthis paperare appreciated R eferences N guyen, N T.; W ereley, S.T Fundam entals and Applications of M icrofluidics; A rtech H ouse: Boston,M A ,U SA ,2006 Petersen,K E.;M cM illan,W A ;K ovacs,G T.A ;N orthrup,M A ;Christel,L.A ;Pourahm adi,F Tow ard next generation clinical diagnostic instrum ents: scaling and new processing paradigm s Biom ed.M icrodev.1998,1,71-79 D ickert, F.L.; Lieberzeit, P.;H ayden, O Sensor strategies for m icroorganism detection²From physicalprinciples to im printing procedures.Anal.Bioanal.Chem 2003,377,71-79 M ogensen, K B.; K lank, H ; K utter, J.P Recent developm ents in detection for m icrofluidic system s.Electrophoresis 2004,25,3498-3512 V iskari, P.J.; Landers, J.P U nconventional detection m ethods for m icrofluidic devices Electrophoresis 2006,27,1797-1810 Sheehan, P.E.; W hitm an, L.J D etection lim its for nanoscale biosensors Nano Lett 2005, 5, 803-807 Ligler, F.S.; Taitt, C.R O ptical Biosensors: Today and Tom orrow; Elsevier Science Ltd: N ew Y ork,N Y ,U SA ,2008 D andin, M ; A bshire, P.; Sm ela, E O ptical filtering technologies for integrated fluorescence sensors.Lab.Chip.2007,7,955-977 Lum inous Efficacy A vailable online: http://hyperphysics.phy-astr.gsu.edu/hbase/vision/ efficacy.htm l(accessed on 27 A pril2010) 10 M olar absorption coefficient A vailable online: http://en.w ikipedia.org/w iki/M olar_absorptivity (accessed on 27 A pril2010) 11 N ovak,L.;N euzil,P.;Pipper,J.;Zhang,Y ;Lee,S.A n integrated fluorescence detection system forlab-on-a-chip applications.Lab.chip.2007,7,27-29 12 de Jong, E.P.; Lucy, C.A Low -picom olar lim its of detection using high-pow er light-em itting diodes forfluorescence.Analyst2006,131,664-669 13 K uo,J.S.;K uyper,C.L.;A llen,P.B.;Fiorini,G S.;Chiu,D T.H igh-pow erblue/U V lightem itting diodes as excitation sources forsensitive detection.Electrophoresis 2004,25,3796-3804 Sensors 2010,10 6749 14 D asqupta,P.K ;Eom ,I.Y ;M orris,K J.;Li,J.Lightem itting diode-based detectors:A bsorbance, fluorescence and spectroelectrochem icalm easurem ents in a planarflow -through cell.Anal.Chim Acta.2003,500,337-364 15 Chabinyc, M L.; Chiu, D T.; M cD onald, J.C.; Stroock, A D ; Christian, J.F.; K arger, A M ; W hitesides, G M A n integrated fluorescence detection system in poly(dim ethylsiloxane) for m icrofluidic applications.Anal.Chem 2001,73,4491-4498 16 Chediak,J.A ;Luo,Z.;Seo,J.;Cheung,N ;Lee,L.P.;Sands,T.D H eterogeneous integration of CdS filters w ith G aN LED s for fluorescence detection m icrosystem s.Sensor.Actuator-A.2004, 111,1-7 17 K am ei, T.; Paegel, B.M ; Schrer, J.R.; Skelley, A M ; Street, R.A ; M athies, R.A Integrated hydrogenated am orphous Si photodiode detector for m icrofluidic bioanalytical devices Anal Chem 2003,75,5300-5305 18 Chang, C.P.; N agel, D J; Zaghloul, M E G o w ith the (m icro) flow IEEE Potentials 2008, 27, 17-25 A ppendix:Param etric R elationships There are five param eters thatare relevantto the analyte in a m icrofluidic platform They are (1)the num berofm olecules and (2)the analyte volum e thatare w ithin the acceptance geom etry ofthe optical system Together,these determ ine (3) the concentration of the m olecules of interestin the sam ple.If (4) the m olecular w eight is know n, then it and the num ber of m olecules give (5) the m ass of the m olecules w ithin the view ed partofthe sam ple.Because ofthe relationships betw een these quantities, they can be show n togethergraphically,as indicated in Figure A -1 Figure A -1.Tw o sets of graphs relating the num ber of analyte m olecules to the sam ple volum e and concentration (bottom )and to the m olecularand totalw eights (top) Sensors 2010,10 6750 The num berofm olecules is m ostim portantand,hence,is com m on to the tw o sets of graphs.That num berand the volum e give the concentration (m olarity)in the bottom ofthe figure.The num berand m olecularw eightgive the absolute w eightofthe analyte m olecules in the top ofthe figure.The overall w eightcan also be related to the volum e and density ofa dry,solid particle ofthe m olecules ofinterest thatm ightbe captured and dissolved form icrofluidic analysis © 2010 by the authors; licensee M D PI, Basel, Sw itzerland This article is an O pen A ccess article distributed under the term s and conditions of the Creative Com m ons A ttribution license (http://creativecom m ons.org/licenses/by/3.0/) Copyright of Sensors (14248220) is the property of MDPI Publishing and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission However, users may print, download, or email articles for individual use