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0Sensors 2010, 10, 6730-6750; doi:10 3390/s100706730
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Article
ComputationalM ethodoLogy for Absolute Calibration Curves ibr M icrofluidic OpticalAnaL^sss
Chia-Pin Chang *,David J. Nageland M ona E . ZaghLoul
Departnentof Electrical and ComputerEngiesring, The George W ashington University, W ashington DC 20052, USA; E-M ail!: nagel@ gwuedu D jJN .); zaghJoul® gwuedu M EZ.)
* Authortowhom cbrtespondence should be adcfre^d;E-M ail: chiapinc® gwuedu; 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
Abstract: Optical fluorescence and absorption are two of the prim any techniques used for analytical m icrofluidics. W e provide a thorough yet tractable m ethodfor com puting the perform ance of divert optical m icro-analytical ^stem s. Sam ple sizes range from nano- to m any m ido-Uiters and concentrations from nano- to m illi-m olar. Equations are provided to trace quantitatively the flow of the fundam ental entities, nam ely photons and electrons, and the conversion of energy from the source, through optical com ponents, scmplfes and gpectral-setective components, to the detectors and beyond. The equations perm it facie com putations of caQioration curves that relate the concentrations or num bers of m oLecules m easuued to the absolute signals from the ^stem . This m ethodology provides the basis for both detailed understanding andim proved design of m iclbfluiia optical analytical ^stem s. It saves prototype turn-around tm e, and ismuch sm pU^and fasterto use than ray tracing program s. Over two thousand gprteadsheet com putations were perform ed during this study. W e found that som e design variations produce higher signal levels and, for constant noi^ levels, low erm inim um detection limits. Im provem ents of m ore than a facfeorof 1,000 were realized.
K eyw ords: m icrofluidic; chem ical analysis; bio-chem ical analysis; optical fluorescence; optical absorption
1. Introduction
The qualitative and quantitative Chem ical and bic-chem ical analyses of m icrc-liter and sm aller volum es of diverse fluids constitute one of the m ain applications of m icrcfluidic system s [1]. There are several approaches to obtaining signals from m icrcm eter-scale volUm es inthe process of perform ing analyses [2-6]. Electrical m easurem ents eoe com m onfcr sam pies that have an ionic com ponent DC conductivity) or polarizable m olecules AC im pedance). Optical techniques, notably fluorescence and also absorption, are also widely used for sam p]es that are optically active [78].
As part of an experimental study on the lim its of detection foranalyte molecules in m icro-channels or thin film s, we are concemedw ithrIlating the absolute numberof molecules accessible to optical em isson and absorption equipm ent to the absolute signal strengths (usually in volts) that are available from analytical instrum ents. This paper provides the set of linked equations for such relationships for both optical em isson and absorption m easurem ems. There is considerable literature on chem ical and biological analytical calibration curves form icrofluidic system s, butm ost calb>ratbn curves are not on an absolute basis. Further, no papers provide a com plete description of the com ponents and geom etries employed. Ihthis paper, we present anduse a new andstcaii^tf:^]wactdcomp^taticnalapE^roach for quantitative optical analysis of m icroscalle fluids. Absolute calibration curves were calculated for 216 varying designs and concentrations.
There are several advantages to the technique we have developed for optical m icro-analyses of fluids. M ost fundam entally, it deals w ithindividual pntitipR. These are the m olecules, w hich are the objectof using m icrcfluidic analytical system si, and quanta, specifically photons and electrons, that are employed for the analyses. Our approach focuses on the individual components in an optical m icro-analytical system , each with associated specifications, efficiency and geom etry, which 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 transport of photons through the system . Overall, use of the equations relates the num ber of m olecules in the analytical volum e to the m easured signal. This approach m akes it relatively easy to determ ine the com ponents orgeom etries that are m ost am enable to significant im provem ents during design of an analyzer system . m fact, the varaticnof the m easured siignal w ith change in any of the com ponent param eters is straightforward to com pute, if the geom etricalandotherparam eters are knownor estimated. Calculations basedohthe m ethodcan be m ade using sim ple com puter program s or even spreadsheets.
This paper provides three benefits. First, we developed and utilized a com prehensive, yet efficient, m eans of com puting absolute caliDra^tion curves for m icrcfluidic optical analysis system si. Second, the num erous results reported anddiscussed clearly dem onstrate the advantages of this m ethbdolbgy for exam ining the efficacy of alternative optical com ponents and designs. Finally, we have a com putational basis for com parison with experim ents.
M ore specifically, the m ain features of ournew m ethbdolbgy can be sum m arized as follow s:
• It is absolute, and relates m olecular concentrations or num bers to realistic detector signals.
• It is com plete, including all com ponents and geom etrical factors that affect the m easured signal for a given anaHyte concentration.
• The m ethbdolbgy is aim ost entirely algebraic, except for the case of fiber optic coupling to m icrcchannels, which is notvery im portant practically.
• The effects of the various param eters needed for com putatons are quite clear.
• Being mathematicallysimple, the method makes possible fast calculations and thorough param etric studies.
• The technique perm its exam ination of realistic designs without the tm e and expense of m aking and using prototypes.
• The calculation of calibration curves is much more efficient than to measuring them in the laboratory.
• The m ethodology is testable by com parsons of its predictions with the results of experm ents using the sam e com ponentsand geom etries.
• The methodology is scale-independent. It can be used for macroscopic, mesosoopic and m iroscopic optical system s.
Our interest in em ission and absorption m ethods of opticalm icro-analysis has another basis, nam ely their sm ilarity.This is indicated sdhem atically in Figure 1. In both cases, a source of light is needed. In the em issbn case, the night is absorbed, and that stimulates fluorescence from m olecules in the sam pie or from tags attached to them . In the absorption case, the source provides the photons that probe the sample and are fractionally absorbed within it. Both em issbn and absorption methods usually involve a variety of optics betw een the source and sam pie in orderto cbl]ecslight from the source and focus it on the sample. Similarly, optics are commonly used between the sample and detector to collect emission or unabsorbedphotons from the sam ple and focus them ona detector. O ptics inbothpositions m ay give spectral diacrim ination to provide m olecular specificity or give btherbenefis, notably background reduction. The quantitative transport of photons from the source to the sample to the measuring equipm ent depends on the optical efficiency of the individual com ponents, and m any geom etrial and Spectral factors.
Figure 1. The sequence of m ajor com ponents inan optical m icro-analytical ^stem . For em ission measure ents, the source light goes as far as the sample, where the new fluorescent liht originates. For absorptionm easurem ents, the two sets of optics and the sam pie can be thoughtof as the entire optical system coupling the source to the detector-.
The next sectionpresents our com putational m ethodology for quantitative analysis of sam pies in micro-channels or thin films by absorption or fluorescence. Section 3 provides many illustrative computed calibration curves, which were obtained using the m ethodology. These results are diacusssd in the follow ing section. The last sectionsketches what is needed for future experim ental workon quantitative m icrofluidic optical analyses.
Optics
Optics
2. C om putatonal M ethodology
W e seek to com pute the output of the detector in a m icrofluidic optical analytical system as a functon of the concentration or the num ber of m oHecutes accessible to the syst^ . Such a rela^tibnship constates the useable part of the calibraton cuive for the instrum ena That is the part of a calibraton cuive above the noiœ level of the signal and below the saturation of the system outpua The com putaton requîtes linking the source of photons for stmulating fluorescence or probing absorption to the analytical sam ple and detector throughall intermediate optics and spact:ral]ysэnsjjve com ponents. G eom etry plays a dom inant role in the efEciency w ith w hich all the com ponents are coupled. In tthis œction, we provide equations and diagram s for the nœded calculatons. C oncatenation of all the equations for a particular œt of com ponents and their arrange ent yields the desir^l calibration curve. W e em phasize that w e are sacrificing som e detail for com pHeteiess. W e protide relaLvelysimple, but useful equations fora complète linkage. Uncertainties inour results aie small com parM to the large variations in optical design, which can change calibration curves by m ore than three orders of m agnitudes for the ssm e concentration of the analyte.
The quantitative présentation of our methodology is for both fluorescence and absorption m easur^ ents of sam pies in both m icrochanneJs and tthin film s w ith lens, no optics or fiber optic coupling of the source to the sam pie and the sam pie to the detector'. The light from the source will be asEni ed to strike the sam pies in the channeJs or film s norm ally, withone exception. That is coaxial fiber coupling into and out of the ends of m icrochanneJs. It is relatively difficult and improductive to use lensss to couple light from a source into the axis of a m irochannel. After ransidering the prim arry aspectsof lens coupling, we will turn to the po^ibility of digpensing with geometrical optics entirely. Then, w e consider the use of fiber optics. Fiers also m eke it po^nble to do either fluorescence or absorption m easurïm ents along the length or perpendicular to the axis of a m icrochannel. The use of optical Sbers with tthin film sam pies is usually not reasonable bacause either very little of the sam pie film is vi^ ed or the insLum entbaobm es relatively large. However, ourm etthodology can be appli^ to that case ato.
The follow ing paragraphes trace the source or fluorescent photons from com ponent to com ponent, for sam pies in m icrochannels or tthin film s. It is asaim ed throughout that the com ponents of the syst^ are properrlyalign^. Achieving alignment is chall^ging but must be done expermentally, if perform ance is to be optim ized, or if com parisbns of com puted and m easurl signals are to be m ade.
Again, we em phasize that, for-absorption m easurm ents, the So to Sa and Sa to D axes on both sides of the sam plees m usa be co-linœr. H ow ever, that is neither neœssarry nor desirable for fluorescence m easurm ents because light from the source that transits the sam ple m ight strike andstim ulate the detector as a very undesirable backgiound. W e will not expliciaiy triât the very diverse geom etres for fluorescence measurïm ents in which the So to Sa and Sa to D axes on the opposite or sam e side of the sam pies are not co-linœr. D oing so for a specifi ^stem design (com ponents and geom etries) is straighabrward.
21. Source Strength
The specifications forthe intensity of LED s are com m only given in the photom etric units of lum ens. Equation (1) can be used to convert lum ens to watts.
Lumens
Power (W ) = ■
683 lumens perwattx Luminous Effcacy)
where Hum inous efficacy is wavelength-dependent [9].
Laser specifications aie generally given in watts. Equation 2) isused forcomputing the photons per second from the watts.
Photons 15 „, % . ,
Ps =-= 503x 1015x Power(W )xX (nm )
Second
where X is the wavelength of the laser light.
Some light source specifications give the full conical emission beam angle (20). The corresponding solid angle QS) in unitsof steradians isgiven by Equation (3):
Qs - 2^1- cos6>) 0)
2 2. Source to Lens to Sample
A diagram useful for com puting the fraction of the photons em ittted by the initial source that gets to the plane of the m irofluidic sample containing the analyte isgiven in Figure 2. There aie two prim any cases. in the first, sorneof the source photons miss the lens and aie wasted. Then, Equation (4) perm its com putation of the fraction of the photons that strikes the lens and gets focused onto the sam pie plane. Otherwite all of the photons hit the lens. The am all loss of photons due to the lens itself is ignored.
P (So ^ L1)= PS x
PS is num ber of photons per second the light source generates, R L1 is the radius of Lens L1, X1 is the distance betw een light source and Lens L1, and Qs is the source em ission solid angle in steradians.
Figure 2. Schem atic of a largersource like an LED (dotted box) ora m all source such as a laser black line), and a lens that collects the light and focuses it on the sam pie.
2 3. Focal C onditons on the M icro-channel
There are three possibilities for the relative size of the focal spot on the plane of the channel and the size of the channel. Sim ilarly, there are three cases for the view of the detector backw ards to thatplane. The nine com binations are indicated inFigure 3. The essential factors are (a) the size of the source focal spot at the sam ple and (b) the sam ple area from which photons can get to the detector, both relative to (c) the w idthof the channel and each other. The focal spot tor the source andthe region viewed by the detector or spectrom eter are commonly circles, although they m ay have rectangular or other shapes.
The area of the focused source spot on the plane of the m icro-channelA c can be com puted from the source area AS, the lens focal length F1 andthe distances between components shownin Figure 3. Equations (5) and (6) apply for a thin lens.
Ac = AS X
1 - 1 + 1 F X X,,
X1 and X 2 are both >F1. If they are equal and equal to 2F1, the area of the spot on the channel is the sam e as the source size. Then, ignoring the sm all losses in the lens, the arm photon density is the sam e at the source and channel, when the lens intercepts all of the em itled photons.
Figure 3. Schematic showing the relative sizes of the source focal spot, the detector acceptance regionandthe m icro-channel, which is indicated by the two parallel vertical lines. The num bers to the left of each of the nine sketches indicate w hat fractionof the source photons can m ake it into the analyte fluid in the channel. The num bers to the right of the sketches indicate the fraction of the illum inated area at the channel, which be seen by the detector-.
As already noted for both em issron and absorption., the coHinear directions of So to Sa and Sa to D can be either (a) norm al to a channel ora thin film sam pie;, or (b) parallel to and within a channel. The frEti. is best w ith lens coupling with either one-dim ensional (channel) or two-dim ensional (film ) sam pies, and it will be treated next. Then, the second, which is best for fiber optic coupling, will be considered near the end of this action. Other geom etries are possible, but those two lim iting cases are generallym ost advantageous. The prim ary exceptionis to have the So to Sa andthe Sa to D axes at e angle to each other in order to prevent source photons from directly entering the detector during fluorescence m eastern ents.
2.4. Transm itted Light Perpendicular to a Channel or Film
For absorption, the incident and transm itted radiation can be normal to the channel or film . In that case, the num berof transm itted photons PT is given be E quaton (7), again for the optically thin case.
PT = Pc e~EC<" 7)
where Pc is number of photons striking the fluid in the channel or film, s is the molar absorption coefficient [10] w ith units of L m oie1 cm _1 when 1 is the sam pie thickness, in centim eters, in the direction on a line to the source. C is the volum etric concentration (m olarity) of the solution.
2 5. Fluorescence Perpendicular to a Channelor Film .
The num ber of em itted fluorescent photons is equal to the num ber of absorbed photons tm es the quantum yield. Equations (8), (9) and (10) apply.
Pabs = Pc 1- e"^ ) (8)
For the com m on case that the sam pie fluid is optically thin, that is, eCl is sm all com pared to unity,
Pabs- PcC X£ (9)
PSa - Pabs X QY 10)
where PSa is num berof photons sample emitted and QY isthe quantum yield.
2 6. Sam ple to Lens to Filter and Detector
The sam pie is Effectively a source of radiation with an em ission solid angle of 4n steradians for the restof the system , when the light from the sample is fluorescence. As was the case with the source, it is 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 4.
Figure 4. Schematic of the path for radiation from the sample, either fluorescence or transm issbn, through a lens and spectral filter to the detector..
Sam pl
Lens L2 ■•••»-A........
...........
X3-------*\t-------X4
Detector
A relation sim ilar to Equation (4) is em ployed to com pute the fraction of the fluorescent radiation from the sample that strikes the second lens L2. It isgiven in Equation (11).
P Sa ^ L2) = PSa x
where RL2 is the radls of Lens L2 and X3 is the distance between sample and Lens L2. W e note that, if a large area detector can be placed close to the sample, the lens L2 is not needed. However, for florescence m easursm ents, this w ill Dead to the detector intercepting andresponding to unabsorbed source photons. M ost of that unwanted radiation can be intercepted and absorbed by a narrow bandw idth filter in frontof the detector.
For absorption computations, the angle at which transm itted radiation em erges from the analyte fluid can be determined by either (a) its entrance angle, when absorption is measured acro^ a m icrochannel, or b) the confines of the m icro-channel, when absorption ismeasured along the length of a channel. That is, the ratio of the channel w idth to the length over which incident radiation propagates within the channel can determ ine the em ergence angle.
2.7. Spectral Discrimination
Although a spectrometer is the best spectral diacrim ination tool, it will not be quantitatively considered in this m ethodology. In order to com pute the output of a spectrom eteron an absolute basis, both the wavelength-dependent input and overall speotroom eter efficiency m ust be know n. The latter: is rarely available.
A useful filer i us ially a narrow band interference device with peak wavelength very close to the peak wavelength of the florescence spectrum . The transm ission characteristics of weH-designed and manufactured filters perm it 50% to nearly 100% transm i^aon within a passband that includes some or all of the entire w idtthof the florescence lines, or the transm itted radiationfor the absorptioncase. Transm itted florescence photons after filter can be com puted as:
FW HM of filerx Transmision Efficiency
= P Sa ^ L2)x-
Bandwidth of Emisin Spectrum
A quantitative determ ination of the filter pass band and the fluorescent line width can be done by auxiliary m easureem ents with a speotroom eter, if they are not available from the m anufactureer. Doing ao
will determ ine if any correction has to be applied in the com putation of the num berof photons reaching the detector [8].
2 8. Detector Signals
In som. e cases, the active area of a detector is ot aller than the exposed area in the detector plane, which is irradiated by fluorescent photons. Equation (13) gives the number of photons striking the detector, nam ely PD :
Active area of Detector , %
PD = P aftf? ^^^-■ ^ ^ ^ ^ (13)
Exposed Area in Detection Plane
The electronic signals from the detector depend on the num ber of photons incident on it, the waelength-dependent efficiency and the electronic gain, if any. Equation (14) applies.
Ed = PD xQExG (14)
where ED is the number of electrons per second from the detector, Pd is the number 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 conversbnof photons to electrons is less thanunity. Quantum efficiencies are usually available from the detector manufacturer. M any detectors do not cause m uliplicatbn of the num berof electrons that result. from photon absorption in the detector. That is, they have no gain. However, avalanche photo diodes, and either solid-state or vacuum photomulipliers, do provide gain. The value of the gain can be high, w ithas m any as one m illion electrons emerging from the detector for eveiyelecttron initially generated by photo absorption. However, detectors that provide very high gains involve high voltages, to which the gain is veiy ^nsitive. Also, they are relatively expensive and, in the case of vacuum tubes, are significantly larger than solid-state detectors without gain. The latter are com m only PN or PIN diodes, which are relatively ot all and cheap, and require only low voltages. H ow ever, they do not have gainw ithinthe detector elem ent, Solid state phottom ulipliers em ploy interm ediate voltages and still offer substaintialgains.
Photo sensitivity (also known as responsivity) is commonly expressed as amps (coulombs per second) per watt (joules per second) of the incident light. Hence the definition of a Coulomb and Equation (2) m ust.be em ployed for conversion of units. 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 output signal of the detectoris:
Output Signal--D-x (cesponsivity atGain -1) x G 15)
5.03 x10 x X ^^
2 9. Post-Detection Electronics
The signals directly from individual detectors or arrays of detectors are com m only quite sm all and they m ay contain noise that is often am enable to electronic filtering. m general, signals from photon detectors are handled in eitherof two m odes, pulse counting or current m easurem ents. In the firstcase, pulses due to absorption of individual photons in a detector, usually with gain, can be counted. Then,
there are som e bœeficialposibilites to reject noiœ. Electronic filters can be used to diacrim inate against noise with frequencies that are either too low , orelse too high relative to the photon arrivai and electrón production rates. Electronics, which determ ine the height or integral of the pulses, are com m only use to rejectpulses that are ttoo sm all. Such electrolices can perfora analyses of the shape of the pulses to insure that, even if the pulses pass the size screening, they have the proper characteristics to be caused by photons. However, the very fast electrolices 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, thencurrent m easurem ents are made. In thiscase, it is possible to employ electrolices afterthe detector to amplify the analog current. Then, the final signal is given by Equation (16) :
Ea = ED x Amplification) (16)
where EA is the num berof electrons after am plificat-on. The electron arrival rate is a current of courra. Transim pedance am pliffiers turn currentvaljes intto voltage. For the case of pulse counting of photons, digitalm etthods are used forcom puterrecording of the photon arrival rates. For the analog currentcase, without or w itham pliEica^tio^, analog-to-(^i!^ital converters are usuallyused to obtaindata in digital form for recording and m aripulation by com puters.
W hatevertthe m eans of gcectral disorim ination or photon detection and am pl^fi•at-br^f in or after the detector, both for digital photon counting and for analog current m easurem ents, there usually results a digital signal related to the photon arrival rate atthe detector:.
210. No Optics
The preceding m etthodology dealt withlens 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:. Analytical microsystems w ithout lens coupling are also possible. Their performance (calbration curves) can be com puted using the equations already presented. System s w ithout interm ediate optics can handle sam pies sizes over a wide range. Also, they are sim pler than the case with lenses because there are ffew ercom ponents to procure, align and hold in place. W ithout the constraint of the lens focal lengths, system s with no lenses can also be m ore com pact. However, as will be seen in Sectibh 3, the no-optics case has loweroutput signalsfiorparticularcbncehtratbhscompared to system s with lenses.
211. Fiber Optics
The equations above provide the m eans to com pute the calbration curves for m icrofuiidic optical anal/tical system s using lens or no optics. As noted earlierv fiber optics can be employed to transport photons from the source to the sam pie and, thence, to a gcectraly-sensitive com ponent and detector:. There are some notable advantages to using fiber optics with microflu^id^ic systems. Because the external and core diam eters of fibers can be com parable to the wid^th^s and depths of m icro-chahnels, it is possible and relatively easy to integrate fiber optics into such flu^id^ic platíonm s. This can be done by using ordinary fibers and putting tthem into the m icroflu^id^ic platíiorm , or by building optical channels, as wellas fluidic chamela, into a substrate. Either way, it is possible to closely couple an off-chip source to a fiber optic, which ends close to the fliid charnel. Sim ilarly, the gpaœ betw een the sam pie
and a second fiberoptic to take the fUorescentorunabsorbed photons to the filter before a detector, or to a spectrom eter, can be sm all and geom etrically effic^ien^t. It m ust be noted that fiber coupling is not attractive for single-use m icrofluidic platform s unless the fiber can also be disposable and easily (cheaply) connected to the unit containing the source, detector and electronics.
The coupling of a source to a fiberoptic is shownschem aticalyinm ore detail inFigure 5. Two steps are needed to calculate the fractionof the em ited photons that enter the fiber:. The first is to com pute the fraction of the source area that is within the acceptance angle of the fiber:. The next step is to calculate the fraction of the photons em ited from that area that fall on the core of the fiber optic:. The result is Equation (17).
PF = PS
O,- D'
Qs • D-
= p x - F F
As "fis
where PF is number of photons entering the fiber optics, PS is number of photons the source em ited, is the acceptance solid angle of the fiber optics and the RF is the core radius of fiber optics. D is the distance between light source and fiber:. W hen ^FD2 is larger than source area AS, then ^F-D2/AS is equal to 1.
Figure 5. D iagram show ing the partof the source (heavy vertical line) that is w ithintthe acceptance angle of a nearby fiber optic indicated by the bracket, and the solid. angle of light from one partof thatregion, which is intercepted by the core of a fiber (stippled).
If the optical fiber acceptance specificationis givenas a num erical aperture NA), Equation (18) perm is calculation of half acceptance angle of fiberoptic, 0f:
NA = n-sin(0F) (18)
The refractive index n is 1 for air, 133 for water and 136 for ethanol. Equation (3) enables calculationof from 0F.A s intthe case of a lens accepting radiationfrom a source, the em issbn pattern (solid angle) of the source enters the calculation. However, the very sm all fiber cores (on the order of 10 to 100 m icrom eters in diam eter), rather than the lens diam eter (on the order of 10 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 viewed by the fiber increases as D2 while the area intercepted by the fiber core decreases as 1/D2. H ence, as long as the area of the source within the fiber acceptance angle is less than the overall source area, increasing D does not decrease the num ber of photons that get into the fiber:. This presum es a source that em its uniform ly over its area and over its solid. angle.
There are two primary geom etries fortthe coupling of light into and out of m icrochannels using fiber optics. They are orthogonal to the channel or co-axial with the channel. The transm issdon of incident photons for absorption m easurm ents in the cross-the-channel case is relatively easy to com pute using Equation (7). The beam com ing from the fiber optic coupled to the source does not spread m uch when crossing a am all channel..
The calculation of the number of unabsorbed photons is more complex in the co-axial case. Sim ilarly, com putation of the num ber of fluorescent photons generated, and the fractbncapturBd by the fiber optic going to the detector, is not as sm ple in the coaxial case as in the lens coupling case. Calculation of both transm isson and fluorescent signals for channels of varying lengths requires a single integration over the channel length.. That is straightforward, but stillmore complex tthantthe algebraic equations presented above. Coaxial couplings of m icrochannels and fiber optics are little used. B ecause of that fact, and because of their greaterm atthem aticalcom plexity, we do not present the integral equations for the coaxial case. However, results based on the use of these equations are given in the next section. It can be seen that coaxial coupling of m icrochannels and fiber optics leads to non-linear calibration curves athigh concentrations and to very poor system efficiency.
The transm ission efficiency of fiber optics is wavelength dependent. That efficiency m ay be gotten either from m easursm ent or from the m anufactursrfe specifications in order to com pute the fraction of the flux of photons from the source or sam pie that gets to the nextpartof the system .
212. M ixed Optics Systems
In the first part of this section, we dealtw ith system s having two lenses, one on each side of the sam pie in the m icro-channel. Next, we dealtwitth the no-optics case. Then, we outlined the behaviorof fiber optics that can be used in lieu of either of the lenses. It is possible to have optical m io-analytical system s that have m ixed optics, with lenses, fiber optics or no optics either before or after the sam pie. For exam ple, a lens m ightbe used for an LED wittha relatively broadem issonsaidangle to focus m ost of the source photons ontthe analytical fliidin a channel. Then, if a spectrom eter w ittha fiber optic input is being used, it w ould couple the fluorescence from the sam ple into the spectrom eter.
213. Overall Signal Calculation
The final expression, which relates the m easuiBd signal to the concentration or num ber of analfe m olecules, can be gotten by successively linking the individual equations given above for the particular com bination of com ponents in any m irofliidic analytical instrum ent. This is true for lens, no optics or fiberoptics cases. Forboth fluorescence and absorption experim ents, the signal depends linearly on the source strength. If the analy^ fluid is opticallythin to both incident and either fluorescent or transm ited radiation, then the signal also depends linearly on the number of molecules that are both irradiated by the source and view ed by the detector, whether it is an individual device behind a filteror built into a spectrom eter.
The sensitivity of the signal to any of the geom etrical and other param eters in the overall equation can be com puted by taking the partial derivative of the signal strength with respect to the param eter of interest. In particular:, the derivative of the signal with respect to the num ber of m olecules is the slope of the calibration curve, that is, the instrum ental responsE/ity, which is particularly im portant. A large
derivative, that is, a high responsi/ity of the signal to the num ber of m olecules, generally m eans that the precision of the analysis can be high, but the dynam ic range will be relatively sm all. C onversely, a sm all slope and rsponsivty may m ake itpossible for the instnm ent to give useful values over a broad range of m olecularnum bers (concentrations), butwiith less precision.
3. C om puted Calibration Curves
The com putational m etodology just presented has been used to calculate the calibration curves for a w ide variety of com binations of source, optics, sam pies, detectors and geom etries. W hile the m ethodology can be used for absorption analysis as wel as for fluorescence situations, we concentrate on the fluorescence approach. M ost of the published papers on m irofliidic optical analysis use fluorescence rather than absorption. And, the measurements we are planning to test the new com putational m etthodology are based on fluorescence and not absorption. Besides, the com putation of the source absorption in the proces of estim ating the fluorescence intensity is esentially the sam e as the calculation of signals for absorption experm ent.
The results of our calculations of fluorescence calibration curves presented in this action are based on particular optical com ponents and their specifications. The specific com ponents for which we have done calculations and are doing experm ents will be cited in detail in experim entalpapers.
Since the optical coupling and geom etry are m ajor variables in both the design and perform ance of m irofliidic optical cncHytical system s,we employed three different cases, which are presented in Figure 6. The com putational results are basedbnthese three geom etrcal cases, and on using three different light sources, three different optics, two different sam pies and two detectors. The detector outputs for six concentrations w ere com puted for each sam ple and com binationof com ponents and geom etry. H ence, the inform ationpresented here is the result of over 600 individual calculations of detector output for specific com binations of com ponents, geom etries and concentrations all done using an EXCEL spreadsheet. Over two thousand com putations were done with the spreadsheet in order to exam ine alternative geom etries. This testifies to the facility wihwhich calibration curves for optical m icro-analysis can be com puted using ourm ethodology.
Figure 6. The three cases forwhich calculated calibration curves are presented. The first is lens coupling to and from etherm irochannels or thin film s. The second case has the sam e types of sam pie holders, but without optics. The last case deals with fiber optics coupling to a micro-channel either within (co-axial)) the channel or efe orthogonal (cross) the channel.
The volum es of the samples, which are analyze for these three types of optics, are plotted in Figure 7. It is noteew orthy tthatourm etthodology has handle sam p]es that range in volum e from 1 nL to about 1 m L. Com putation of calibration curves for sm aler or larger sam p]es is al^oo posrble with this m ethodology. W e used fluorescein for the illustrative calculations because ithas been widely em ployed in experiments with microfli^it^ic analytical system s [11-17].
Figure 7. The volum es of samples forwhich the results in this section were obtained. 1 nL isa cube 100 m icaom eteerson a side. 1 mL is 1 centim eteercubed. Fiberoptics are sm all and interrogate only m all volum es. Systam s with no optics can probe a wide range of volum es, including relatively large sam pies.
Fiber Optics-■ Lens Optics — No Optics ■■■■
1nL 10nL 100nL 10^L 100^L 1m L
Range of Sam ple Volum es
Calibraton curves were com puteed as a function of both concentration (m olariy) and the num bers of molecules accessible in the analysis. Concentrations are commonly desired, but the numbers of m olecules are useful for com paring the efficiencies of optical ana]yt±ae instrrum ents. The calibration curves for the three geom etrcal ca^ of Figure 6, andm any component variations, are presented in Figures 8 and 9. The calculated curves have the sam e slopes because allpartsof the ^stem sare linear. The use of log-log scales is necessary because of the veryw ide ranges of conc^trations andoutput signals. These graphs clearly show the absolute andrelatve perform ance of the various com ponents and geom etries. Vert±ae lines at specific conc^trations w ould show that the signals from the detectors can vary by over three orders of m agniude fora particular concentraton. Horizontal lines can be used to bracket the detector outputs ranging from the noite level to the sauiaticnsignaL The m inim um detectable lim it and the dynam ic range varygreatly depending on the optical com ponents and their arrangem ents.
The com puted signals for specific cbnc:entratbns ornum berof m o]ecu]es vary m ore than 103 for the different com ponents and geom etries. It is clear that the case for the analyte in m icrochannels and coaxal light transm ission gives relatively poor perform ance. C onver^ly, having the sam pie in a thin film with both the incident excitation light and fluorescence at 90 degrees to the film provides m uch greater signals than the other ca^s.
Figure 8. Computed calibration curves as a function of the molar concentration of fluorescein for several geom etries, sources, optics and detectors: (a) lens (15 cm focal lengtthanddiam eter) cbupling to a 100 ym square m iaiochannel, (b) lens 15 cm focal length and diam eter) cbupling olthbgonal to a 100 ym thin film , (c) light from sources to 100 ym square m icrcchannels and fluorescence to detectors without intervening optics and (d) light from sources to 100 ym thin film s and fluorescence to detectors w itthout intervening optics, (e) fiber optic (100 ym diam eter ) coupled orthogonal (cross) to a 100 ym square m icrochannel, (f) fiberoptic (100 ym diam eter cbuplBd within (co-axial) a 1,000 ym length of a 100 ym square channel. The sources are blue LEDs with either 10 or 150 degr^ full em ission angles ora UV LED with a 120 degree full em isson angle. A filter-was em ployed and the traansm isson loss through the filter was calculated, as described in Section 2.7. The detectors are either an am plified photo diode Am PD) ora Silicon phottom i iltiplir SiPM ). The straight lines are draw n through the com puted points in these graphs.
Lens to M icrochannel Signalvs C bncвntraticn
— 10 Deg Blu Li
— 10 Deg Blu Li
- 150 Deg Blu L
- 150 Deg Blu L
^ 120 Deg UV L — 120 Deg UV L
Lens to Thin Film Sample Sigalvs Concentration
I- 120 Deg UV Lens ThiFilAPD — 120 Deg UV Lens ThiFil SiPM
Concentration Log M )
Concente^on (Log M )
No Optics to M icrochannel Signalvs Concentration
— 10 Deg Blu NoOp uCha i
— 10 Deg Blu NoOp uCha !
— 150 Deg Blu NoOp uCha A
— 150 Deg Blu NoOp uCha i
120 Deg UV NoOp uCha 120 Deg UV NoOp uCha
No O ptics to Thin Film Sample Signal vs Concentration
— 10 Deg Blu NoOp ThiFilAPD _ 10 Deg Blu NoOp ThiFilSiPM
— 150 Deg Blu NoOp ThiFilAPD ' _ 150 Deg Blu NoOp ThiFilSiPM •
120 Deg UV NoOp ThiFil APD -120 Deg UV NoOp ThiFilSiPM
Concente^on Log M )
Concentration Log M)
FO to 100 um wide fcross ) M icrochannel Signalvs Concentration
— 10 Deg Blu FO uCha ;
— 10 Deg Blu FO uCha !
— 150 Deg Blu FO uCha ,
— 150 Deg Blu FO uCha :
n120 Deg UV FO uCha A n120 Deg UV FO uCha i
FO to 1000 um long Co-Axial M icrochannel Signalvs Concentration
Concentration Log M )
Concentration Log M)
10 Deg Blu FO 10 Deo Blu FO
150 Deg Blu FO uCh 150 Deg Blu FO uCh
120 Deg UV FO uCh 120 Deg UV FO uCh
Figure 9. Computed calibration curves as a function of the number of fluorescein molecules for several geometries, source, optics and detectors. This figure is made by converting the concentration into number of molecules in different volumes shown in Figure 7. Because of the sample volumes are different in various geometrical arrangem ents, the num berof m oIbcuIbs is
Lens to M icrochannel Signal vs Num ber of Molecules
— 10 Deg Blu Li
— 10 Deg Blu Li
- 150 Deg Blu L
- 150 Deg Blu L
^120 Deg UV L "120 Deg UV L
Lens to Thin Film Sample Sigalvs Num ber of Molecules
-10 Deg Blu Lens ThiFilAPD -10 Deg Blu Lens ThiFil SiPM
— 150 Deg Blu L
— 150 Deg Blu L
120 Deg UV Lens ThiFil APD 120 Deg UV Lens ThiFil SiPM
Number ofMolecules Log)
Number ofM^^^!^ Log)
No O ptics to M iclbchannвl Signalvs Num ber of Molecules
— 10 Deg Blu NoOp uCha j
— 10 Deg Blu NoOp uCha ;
— 150 Deg Blu NoOp u1
— 150 Deg Blu NoOp u1
^120 Deg UV NoOp uCha ; n120 Deg UV NoOp uCha ;
No Optics to Thin Film Sample Signalvs Num ber of M olecules
10 Deg Blu NoOp ThiFil APD — 10 Deg Blu NoOp ThiFil SiPM
— 150 Deg Blu NoOp ThiFilAPD
— 150 Deg Blu NoOp ThiFilSiPM
120 Deg UV NoOp ThiFil APD n120 Deg UV NoOp ThiFil SiPM
Number ofMolecules Log)
Number ofM^^^!^ Log)
FO to 100 um fcross) M icrochannel Signalvs Number of Molecules
— 10 Deg Blu FO uCl
— 10 Deg Blu FO uCh
— 150 Deg Blu FO uCh
— 150 Deg Blu FO uCh
^120 Deg UV FO uCha , — 120 Deg UV FO uCha ;
FO to 1000 um Co-AxialM icrochannel Signalvs Number of Molecules
— 10 Deg Blu FO uCh
— 10 Deg Blu FO uCh
^120 Deg UV FO uCha ; n120 Deg UV FO uCha ;
Number of Molecules (Log)
Number of Molecules (Log)
150 Deg Blu FO 150 Deg Blu FO
We em phasize the fact that our m etthodology does not provide absolute com puted noise lim its for calibration curves nor absolute computed maximum available signals. However, obtaining these characteristics is not a practical problem . The intersections of the calibration curves, which are com puted, with the m inim um detector signal gives the m inim um detectable concentration (M D L). The M D L w ill be improved if the detector has a lower noise floor. Sim ilarly, the intersections of the calibrations curves w itth the m axim um detector outputs give the highest concentration that can be assayed, andhence, the dynam ic range of the system . The M D L anddynam ic range for the various
calibration curves were determ ined using the published characteristics for the two detectors;. The silicon photomultiplier SPM M icro1000X01A1from SensL) has a noise fOorof 1 mV and a maximum signal of 500 m V . The am plified photodiode M odel ODA-6W B -500M from OptoD iode) has the sam e noise floor and a m axim um output of 5 V when supplied w ithvolages equal to+5 V . These values were employed in determ ining the M DL and dynam ic ranges for the 36 combinations in Figures 8 and 9. The results are presented in Table 1.
Table 1. The m inim um detection limits MDL) in nM and dynam ic ranges factors above the MDL in parentheses) for the calibration curves for the three sources, three optics
options, three sam pie geom etries, and two detectors.
Optics Sample Detectors Light Sources
10 Deg Blue LED 150 Deg Blue LED 120 Deg UV LED
Lenses 100 iam W ide M icrochannel SiPM 111 (473) 27.58 (11,700) 148.53 (63 200)
Lenses 100 iam W ide M icrochannel Amplified Photodiode AmPD ) 0.03 (173) 0.86 (4,300) 3.98 (19,900)
Lenses 100 iam Thin Film SiPM 017 71) 4.14 (1,760) 28 37 (12,100)
Lenses 100 iam Thin Film Amplified Photodiode AmPD ) 0003 (13) 0.06 (320) 2.65 526)
None 100 iam W ide M icrochannel SiPM 32 67 (139,020) 958 52 (408,000) 46622 98 20,000,000)
None 100 iam W ide M icrochannel Amplified Photodiode AmPD ) 0 51 (2,550) 14.93 (75,000) 726.11 (144,000)
None 100 iam Thin Film SiPM 1.78 (760) 2 64 (1,100) 128.06 55,000)
None 100 iam Thin Film Amplified Photodiode AmPD ) 003 (140) 0.04 200) 1.99 (400)
Fiber Optics 100 im M icrochannel (cross) SiPM 5 25 (2,630) 1000 523,800) 8128.3 (4,065670)
Fiber Optics 100 im M icrochannel (cross) Amplified Photodiode AmPD ) 0.48 2510) 95.5 (490,000) 758.58 (3,900,000)
Fiber Optics 1,000 iam M icrochannel (co-axial) SiPM 17 78 (10,450) 3801.89 2235,000) 28183.83 (14,100,000)
Fiber Optics 1,000 iam M icrochannel (co-axial) Amplified Photodiode AmPD ) 155 (9,770) 346.73 2,190,000) 2630.27 (13,180,000)
4. Discussion of the Results
The tabulation of M D L values and dynam ic ranges m akes easier the evaluation of the results of the com putations com pared to use of the log-log graphs already presented. C onsidering the M D L for the
■ ■ ... 7
various cases, the values range from 3 pico-m olar to 46 m icro-m olar, a variation of over 10 . The facility with which these calculations were done and the wide varatbn in results instates the value of our methodology for m iicrc-analy1±ae ^stem design and comparison. The narrower em ission angle LED light source is m ore efficient for delivering the photons to excite the fluorescence em i^ion compared to the same LED with little collinatbn. Lens coupling shows better incident photon transm issbn from an LED light source to the sam p]e along wihbeter fluorescence photondelvery from the sam p]e to the detector. However, itm ust be re-em phasized that we put the source on one side of the a^um ed-transparent substrate containing the channel or thin fim and the detector on the other side. This is not a practical geom etry because light from the source w ould enter the detector. Placing the source and detector on the sam e side of the substrate w ould ^^entially rem ove this problem , but decrease the geom etric coupling sightly.
The M D L values in Table 1 show that the thin fim sam p]e geom etry is substantially beter for all com binatbns of sources, sam p]es and detectors. This is because the useful part of the thin fim sam pHe contains m ore m olecules due to having a bigger volum e. It is a good trade-off to use larger volum e of sample (that is m icro-l iters, rather than nano-l iters) in order to reach lowerMDL. One ordinary drop contains about50 pL.
5. Conclusions
Optical m irofluidic system s are widelyemployed inm :C:rc-ana]yt±ae research andindustry [18]. Exam inatbn of the alternatives we considered leads to an appreciation of the large num ber of posible optical m i::ro-ana]yticae^'stem s. We disarmed m ultiplle photon sources; llen^s, fioerand no optics for photontransPort• fluorescence and absorptintechniques for probing sample inm icro-channels and thin film s; filters and gpectrom eters tor epochal discrm inatbn; and various detectors w ith anallary electro^ic^^. There are m any specific choices in each of these categories. H ence, there are hundreds of Specific system s. AH of these can be analyzed and com pared quantiatvely using our m ethodology.
The largest photon loss in an optical ana]yt±ae ^stem occurs when there is no efficient way to collect florescence photons from the sam p]e to the detector. W hether the ^stem has len^s or fiber optics, there is a num eri:ae aperture (NA) or acceptance angle for each optic:. It determ ines the efficiency for gathering the fluorescence photons from the sample that are emitted into 4n steradians. Light gathering efficiency is a key factor indesigning m irofluidic analysis system s that can provide low er MDLs. Use of ellipsoidal or other m irrors to gather fluorescent photons was not com puted for this paper:. However, this m ethodology can be confidently em ployed for those cases.
Com parsons of com puted and m easured calibratbn curves, both with the sam e units, should prove especially useful. W e note the central im portance of the absorption coefficients in both fluorescent and absorptbnm etthods andof the fluorescent yields influorescent m easursm ents. ]tm ay be possible to obta^rela^tive or absolute experim ental estim atees of these param eters ibr particular com binatbns of analyte molecules and wavelengths using our m ethodology:. This requires that all the geom etrcal and
otherparam eters are know n, or can be independently m easuri^, with sufficient accuracy. In particular, the absc]utв source strength, and the quantitative performance of the detector and subsequent electronics, must all be known. Det^imination of absorption and fluorescence parameters is challenging. However, if such values are not available, comparison of the computed and measured signal strengths could give estim atts for the absorption coefficients and fluorescent yields. It. rem ains to be seen if this approach has usefully sm all errors. Det^rm ining that w ould be one of the m otivations fcrperonm ing an experim enttal assessm entof the m etthodolog^.
A cknow ledgem ents
Comm ents by .belGollden on an early version of this paper are appreciated.
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Appendix: Paiam ettric Relationships
Theie aie five paiam eteis that aie ielevantto the analyte in a miaofluidic plla^ífoim . They aie 1) the num beiof m oHles and (2) the analyte volm e that aie within the acceptance geom etiy of the optical ^stOT . Togethei, these deteim ine 0) the concEn^tia^íion of the m oledles of inteiest in the sam p]e. If (4) the moleculai weight is known, then it and the numbei of molecules give (5) the mass of the m olecHes within the vi^ ^ pait of the sam pHe. B ecause of the ю^йс^!;« betw een these quantities, they can be show n togetheigiap^ically^, as indicated in Figuie A-1.
Figuie A -1. Tw o sets of gisphs ielating the num bei of analyte molecules to the sample volum e and concentntion (bottom ) and to the m o]ecu]arand totalweights (top).
The num ber of m olecules is m ost im portant and., henoe, is com m on to the two sets of graphs. That num ber and the volum e give the concentration (m olarty) in the bottom of the figure. The num ber and m olecularweight give the absolute weight of the analyte m olecules in the top of the figure. The overall weight can also be related to the volum e and density of a dry, solid partile of the m olecules of interest thatm ightbe captured and dissolved form irofLiiic analysis.
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