77

© 1992 Elsevier SciencePublishers B.V. All rights reserved U022-1759/92/$05.0t)

JIM 06420

Microplate washing: process description and improvements T o m B e u m e r , E d d y Stoffelen ~, Joost Smits ~ and W i m Carpay Methodology Researeh Unit, Organon Teknika BV, Boseind 15. tloxtel. Netherlands

(Rcceivcd31 July 1991, revised received 19 February 1992.accepted 23 April 1992)

Heterogeneous immunoassays require wash steps in order to separate bound from free constituents. In this paper we demonstrate that in microplate assays the washing process includes two separate physical processes: (l) a rapid and wash volume-dependent direct dilution of the droplet-shaped residual volume, and (2) a diffusion-limited and strongly time-dependent dilution of a residual layer of liquid, which necessitates the use of time-consuming soak times in the immunoassay. We have shown that optimizing the motion of the wash fluid effectively reduces the residual layer thickness that results in extended soak times. This results not only in improved washing efficiency and reduced background variance in the immunoassay, it also yields a significantly improved immunoassay sensitivity. Key words: Heterogeneousimmunoassay; Microplate;Washing; Diffusion;Convection

Introduction Heterogeneous solid phase immunoassays of the reagent-excess type (Gosling, 1990) involve washing steps, to remove unbound analyte. The fluid present in the well is generally removed using vacuum and wash fluid is then added to each well. This sequence is repeated several times and finally, before adding the next reagent, all microplate wells are emptied. In manually performed washing, the user generally has control over the number of wash cycles

Correspondence to: T. Beumer, Organon Teknika BV, BOF4216, P.O. Box 84, 5280AB Boxtel, Netherlands. Tel.: 4116.54558; Fax: 4116.54427. I E. Stoffelenand J. Smits cooperated in this work during their stay in Organon Teknika as students in the Eindhoven Universityof Technology,Netherlands. Abbreciations: HBsAg, hepatitis-B surface antigen; HRP, horseradish peroxidase.

and the wash fluid volume. Using automated washing equipment, the user may also vary the period of time the wash fluid remains in the wells ('soak time'), and sometimes the liquid dispensing speed. The proper combination of washing conditions is either prescribed by the immunoassay kit supplier, or is obtained by careful optimizing experiments. It is thus generally determined empirically. Washing away an excess of labelled immunoreagent requires even more care than washing away sample fluids because of the relatively high concentrations of label. Modern, highly sensitive assays generally include such high concentrations of conjugate, that empirically proven long soak times are used in washing sequences in combination with at least three or four washing cycles. In the case of microplates this may give a total wash time of 3 - 6 min per plate. This washing period is the major limiting factor in an automated ELISA environment (Spermon, 1990).

Although there have been many detailed studies of incubation and detection in immunoassay development (e.g. Tijsen, 1985; Stenberg and Nygren, 1988), the importance of washing in heterogeneous assays has only been addressed with respect to the development of dedicated equipment (Ruitenberg, 1977; Carlier et al., 1979; Stobbs, 1990) or the chemical requirements of immunomolccules during washing (Tijsen, 1985). We have shown that washing, including soak time requirements, is largely a quantifiable physical process that is an important cause of well-to-well assay variation. We have attempted to develop an adequate description of all the relevant physical processes that are involved and we have demonstrated that substantial assay acceleration and quality improvement can be achieved using a considerably faster and more reproducible method of washing microplates rather than standard empirical methods.

Materials and methods

Washing model experiments Experiments were performed studying the washing process only, without regard to complete immunoassay performance. All these model experiments were performed using uncoated flat bottom Organon Teknika MicroElisa 12-well stripplates (Organon Teknika, Netherlands). Various washers were used in parallel: (1) A programmable Organon Teknika Washer-500 with a 12-channel washing head and with modified software to permit controlled wash fluid volumes and soak times. (2) A manually operated Organon Teknika Microwasher for emptying wells; in these cases washing buffer was added manually using a 12channel pipette. (3) A 96-channel Skatron Microwash-ll washer (Skatron, Norway) was used to study the effect of high-speed and high-volume wash fluid addition. In order to investigate the washing process in microplates, a series of experimental parameters were measured. The total residual volume value of the washer was quantified colorimetrically (dye: Bayer's Levafix Brilliant Red, measured at 540 nm) using a

manually filled reference plate containing 0 and 5 #l/well absorbance equivalents. This method was validated gravimetrically, using a 0.I mg resolution Sartorius balance. Both methods matched within 5%. A variation on the above method was used to estimate the thickness of the residual boundary layer: after emptying the well under vacuum, a small ring-shaped droplet was left in the lower corner of each well. This droplet was removed with help of a small vacuum connected tip. The remaining fluid was assumed to be uniformly distributed over the wetted part of the well. An antibody against the hepatitis B surface antigen (a-HBs-lgG), that was produced in-house, was conjugated with horseradish peroxidase (HRP), and was stored at a concentration of approximately 100 mg HRP/ml. Working concentrations were obtained by diluting this stock IgG-HRP in 0.9% (w/w) physiological saline to which 30 g/I polyethylene glycol (PEG-20M) had been added. The PEG increases fluid viscosity by a factor of approximately 2, which is comparable to a normal value for human serum. The PEG also prevents the IgG-HRP from spontaneously coating the polystyrene of the well and thus permits studies of the behaviour of free IgG-HRP alone. Initial dilution factors for the IgG-HRP varied from 1/25 to 1/100. In all experiments, 100 #1 of this diluted conjugate were introduced into the microplate wells. A special measurement and data analysis method was developed for quantifying the fraction of the input material left after washing, based on a kinetic reading of the plate. Analysis was performed, based on the assumption that the initial absorption increase rate dA/dt[,~o was directly proportional to the enzyme concentration [enzyme] left behind in the well (Muller, 1976; Murthy et al., 1986):

d A / d t l,=o ~ [enzyme]

(1)

Due to equation (1), application of this procedure to a serial dilution series of the starting material yielded a direct relation between d A / d t [t=q~ values and known IgG-HRP fractions. This curve was then used to indicate the IgG-HRP fractions left after washing. 100 #1 of chromogenic sub-

strate (3,3',5,5'-tetramethylbenzidine or TMB, Or#anon Teknika) were added to each well as rapidly as possible using a 12-channel pipette. While filling the last strip the reader was started. For validation purposes, we checked the residual fraction of conjugate measured after emptying the well. A colour reading at 650 nm was obtained every 15 s over 5 rain, and thereafter every minute up to a total incubation time of 20 rain. Under proper working conditions, these enzymatically measured residual fractions matched gravimetrically determined fractions to within 5%. In each plate a separate strip containing 10 #l of a dilution series of the applied conjugate was incubated with substrate, and measured under the same conditions as the other wells. This provided data for estimating the residual fractions of conjugate. To investigate the actual motion of fluid in microplate wells during orbital or longitudinal shaking, video recordings were made on SuperVHS (Panasonic, Japan) using an illumination of 1/1000 s. The tape was studied frame by frame. For more detailed information on the streamline patterns macroscopic PMMA beads (diameter-0.3 mm) were added to the well and their motion was studied from tape. Optimizing the shaking amplitude and frequency, and the corresponding shaking period required for fastest reagent removal, was based on the time taken for wash fluid to penetrate the residual volume boundary layer. This penetration time was estimated as follows. Plates were coated with a water-insoluble dye and dried. Then such a plate was filled with an aqeous fluid with a relative viscosity comparable to that of serum or conjugate (¢' --- 2). After removing this liquid the wet plate was filled with an organic fluid. Due to the intermediate residual water layer, the organic fluid cannot readily dissolve the dye. The plate was then shaken at constant frequency and amplitude for approx. 5 s, followed by a reading of the absorbance in the wells a~ the appropriate wavelength. For each well a sudden increase of absorbance was monitored, indicating the moment the organic material had dissolved the dye and thus broken through the residual water layer. The optimal shaking period was given by the shortest

period required to maximisc this colour development. All viscosity measurements were performed u~ ing a Haake Mikroviskosimeter (Haakc, Germany). Two shakers were u ~ d : (1) an orbitally moving Sarstedt-TPM-2 microplate shaker with an additional display showing the exact rotations-perminute; (2) a longitudinally moving dispensing module of Organon Teknika's Microplate Processor-3000 (Spermon, 1990). This was able to address the steppermotor directly through Microsoft's BitBus controller, in all experiments the actual shaking frequency was monitored using a VTl-type piezo aecelerometer (Giravia, Brus~ls, Belgium). All mieroplate readings were performed using a ,software controlled Or#anon Teknika Reader510. Absorbance data were collected in Lotus" comma separated value-format (.PRN files), and analysed with Lotus-123 spreadsheet .software using an IBM-compatible XT computer. l m m u r l o a s,~u y.~

Data wer~ ,~b~ned using Org.':mon Teknika's Hcpanos~ika HBsAg-Monoclonal assay as a model. The validity of our findings were demonstrated when performing immunoassay washing. Negative and positive control samples were used containing 0 and I U / m l HBsAg respectively. The assay was performed in two series. (a) According to the test insert: 100 /xl of control sample per well was incubated for l h at 37°C, washed four times with 300 # l / w e l l of Or#anon Teknika's washing buffer (phosphatebuffered saline, 0.05% Tween) using a 60 s s o ~ time per cycle. The conjugate (100 #! per well) was incubated for 1 h at 37°C, and the plate again washed as after sample incubation, but now varying the number of washing cycles per strip from one to five. Finally the plate was filled with 100 #l of TMB suhstratg per well, incubated at ambient temperature for 30 min in darkness ( < 1 lux) and, after stopping the enzyme reaction with 100 #1 of 1 mole/! sulphurie acid per well, samples were read spectrophotometricaUy at 450 nm. (b) The same reaction scheme as under (a) was followed. However, the 60 s soak period was replaced by a 15 s shaking period, operated at

optimized frequency and amplitude, and using only 150 p.I of washing buffer per well. Data were analysed as follows. Of each series of identically washed negative (N) and positive (P) control samples the average A450 value and standard deviation were calculated. From the assumption that for any number of washings the conjugate contribution in negative control samples is the best possible estimate of the amount of free material left ('noise') we define the statistical difference between negative and positive control samples as given by Student's t value: t = ( e - N)IiI(o'gI,,N + ,~1,,,,)

-I log(residual IgG fraction)

(2) .s

With improved removal of residual conjugate, N and trr~ will reduce, and t will increase.

Results

Quantifying residual volume and residual layer thickness generally gives values that (a) depend on the initial volume of the well and (b) show large variances. With the wash devices used in this study we typically found for sample volumes of 100 /~1 having e' = 2 that V r = 1.5 p,l _+0.5 pi/well and V~--~ 1.0/zl/weli. The latter leads to an estimated average layer thickness of approximately 6 = 10 + 3 p.m. The 8 value strongly increases with e', and decreases rapidly with the aspirate time of the washer (data not shown). Using a fully software controlled washer, Organon Teknika's Washer-500, the effect of soak time variation on the residual amount of free labelled compound was studied using a 300 p.I washing volume. As Fig. 1 clearly demonstrates, increasing the soak time may help reduce the residual levels of free molecules by approximately a factor of 10. For soak periods over 1 min, the practical removal of free molecules gets close to the best achievable result of a pure dilution. Fig. 2 clearly shows that the soak time is largely a matter of diffusion. When fitting the data shown in Fig. 1 to equation (A4), using an approximate layer thickness of 10/~m, the estimated D value

I

I

o

1

washing cycles

L

10"

-o-

o"

÷" - -

ideal till.

theory

i

Fig. I. Residual fractions of lgG-HRP in a single washing step for various soak time periods using 300 ~l/well washing volumes. The lower straight line represents the ideal case of perfect mixing as described in equation (Ai). Based on a residual volume of I ~1 from a 100 ~1 sample we postulated the residual fractions f ( N w = 0 ) after emptying a well to satisfy f ~ 0.01. x axis: number of wash cycles; y axis: J°log of the residual fraction of input material.

for IgG-HRP in a PEG solution (e' = 2, T--- 20°C) satisfies: D = 4.10-ttm2/s and agrees well with the data in the literature (Berg, 1986). Having demonstrated that the efficiency of classical microplate washing procedures is basically diffusion-limited due to the (sample-viscosity-dependent) residual fluid layer at the solid phase, methods for approximating to a pure dilution must be sought. The fastest way to mix boundary layers with the hulk fluid, is to rapidly reduce the thickness ~ of the layer (eq. (A3)) using for example convective forces. A rapid theoretical estimate (Appendices 2 and 3) suggests that, in order to reduce the flow-induced boundary layer efficiently enough, the bulk of the fluid

relative residual I g G - H R P fraction

0.8 0.6

0.4

O,

10

20

30

40

.~0

60

70

washed using normal p r e s s u r e / n o r m a l flow settings (data not shown). Not being able to reduce 8 during the filling of the well, we investigated shaking of the filled plate as a means of introducing convection in the well. Two methods of moving the plate were tested: (1) orbitally, with the plate being moved in small circles, and (2) longitudinally, with the plate moving along a single axis. As it is hard to theoretically predict the effect of shaking on the fluid motion inside the well, we first investigated the optimal relationship between shaking properties (radius of shaking motion and shaking frequency) and well properties (radius and filling volume). Closely studying video recordings of fluid motion during shaking demonstrated that the effect of shaking was two-fold (as schematically visualized in Fig. 3). A t optimal shaking frequency and

soak time (s) J [3

m~sureddata

" - - fittedcurve

~1

I

Fig. 2. Diffusion-based fractional dilution of the residual 10 #m layer as a function of th~ total soak time. Data are scaled relative to the residual fraction at zero soak time. The individual points were measured colorimetrically, the curve drawn illustrates a fitted curve as described in Appendix I for a diffusion coefficient of D= 4.10 -ll m2/s in a fluid with relative viscosityof 2. x axis: soak time (s); y axis: nqog of the relative residual fraction of IgG-HRP.

then requires velocities which are in the order of 3 m / s (Patterson, 1983). During any wash cycle convection can be applied twice: (1) while filling the well with wash buffer, making use of the intrinsic flow, and (2) after filling the well with wash buffer, when moving the wash fluid using an external force. Our analysis showed that none of the wash devices used exhibited a high enough dispensing velocity of the wash fluid to obtain efficient bulk flow in the well (v < 1 m / s ; other data not shown) and on theoretical grounds it also became clear that the time taken to fill the well is too short to equilibrate flow and thereby reduce ~$ values efficiently. This was fully confirmed when washing immunoassays with a Skatron washer: no significant differences were found between high p r e s s u r e / h i g h speed dispensing and assays

real

liquid surface velocity

apparent tangential surface velocity

Fig. 3. Impression of the relevant fuid motion properties of the oscillating liquid surface in an individual well during rotational shaking motion of the microplate. Open arrows indicate the apparent velocity of small fluid elements (a suggestion of tangential flow), the black arrows indicate the real, practically vertical velocity vector of small liquid elements, as described in detail in the text.

amplitude, a standing surface wave of the fluid was observed in the well. Clearly, the shaking frequency that matches this resonance best must be a function of well geometry and fluid properties (viscosity, density). Although one would expect fluid motion to move along the wall of the well tangentially,our video analysis clearly showed that during this oscillation the vertical fluid velocity vector was by far larger than the tangential velocity vector. These observations clearly showed that optimal efficiency of wash fluid convection induced by shaking could only be obtained by carefully matching the shaking properties to well and fluid properties. When optimizing shaking frequency and amplitude in both shaking modes, using dye-coated mieroplates, a single optimum combination was observed (Fig. 4). This optimum combination of amplitude and frequency was selected for all further convection-based experiments on model systems and immunoassays. Using Sarstedt's TPM-2 shaker a similarly optimized result was found (data not shown) at a shaking frequency of 850 rpm. The shaking amplitude of this shaker cannot be varied.

removed percentage of residual layer (%)

6OO

;~ 800 900 1000 II00 linear shaking frequency (steps/see)

[ ~ l an,q~itune. . . . .

1~10

IIFa.

l,m

i

Fig. 4. Experimental optimization of shaking amplitude and frequency for a longitudinal shaking motion and using 150 p.l of wash liquid in each well. The plate was moved using a stepper motor at amplitudes of 10, 20, 30 or 40 steps ( ---0.11 ram/step) and at frequencies of 600 up to 1100 steps/s, x axis: relative amplitude (*0.11 mm); y axis: frequency (steps/s); z axis: fraction of coated dye removed after 5 s shaking period (%).

residual conjugate fraction II

!

0.1

0.01

i

i

i

i

i

i

i

I

i

0

is

30

4s

60

7s

9o

10s

120

13s

soak- or shake time (s) single wash, soak --3(-- singlewash, shake

"-4"- double wash, soak

i

-~-

I

double wash, shake

Fig. 5. Residual amount of free conjugate after washing, as expressed relative to the amount left at zero soak time, and plotted as a function of soak or shaking time. x axis: soak or shaking period (s); y axis: residual fraction of free conjugate compared to the amount left at zero soak time.

When applying IgG-HRP conjugates to study the efficiency of various washing methods, the advantages associated with convection-based washing compared to diffusion-based washing were clearly demonstrated (Fig. 5): shaking for approximately 10 s or more was as effective as a 60-120 s soak period for the removal of residualfree IgG-HRP. For both wash methods we found that the first washing cycle was more efficient than the second. This is easily explained bearing in mind that after one wash cycle we could no longer guarantee a homogeneous concentration profile in the residual i~yer. Applying these findings to a commercially available HBsAg immunoassay the same trend was observed {cf. Fig. 6). When shaking the wash fluid for 15 s, the net responses of negative control sera did not significantly decrease after two washings. Under conditions of diffusion-limited washing, however, the absorbances of the negative control sera ~'ould still be significantly re-

duced after four cycles of 61) s soak time each. This largely explains the drastically increased Student's t value between N and P due to the decreased variance in the test results (cf. Fig. 7). In these experiments the coefficient of variation was decreased by 50% for the negative control sera and by 2(1% for the positive control sera when comparing diffusion-based and convectionbased washing procedures.

log A450 O0 min substrate incubation) ~0

)o

Discussion and conclusions

1

2

3

Number of washing cycles

I

.......

Fig. 6. Absorbance of control samples in HBsAg ELISA (details: see text) for two washing methods: 60 s soak time vs. 15 s shaking. In all cases PBS/Tween washing buffer was used. x axis: number of washings; y axis: average end point response of 6 controls at 45(1nm (N = negative, P = positive, approx. 1 U/ml HBsAg).

~udenes t-value for (P-N)

i

2

3

4

washing cycles

Fig. 7. Student's t value for comparing P and N values in convection- and diffusion-based washing procedures (15 s shake vs 50 s soak time), x axis: number of washings; y axis: Student's t value.

We have developed colorimctric and enzymatic methods that allow rapid and detailed analysis of residual volumes and residual conjugate fractions. The value of t h e ~ methods is partly determined by the variance of the properties that are to be measured. The colorimetric measurement of residual volumes deviates from the gravimetric measurement by up to approx. 5%. This measurement error is considerably smaller than the variance found in residual volumes or when estimating the re.~idual layer thickness. The latter measurement is a l ~ affected by the tinting of analysis: during the 30-60 s time interval between aspiration of the bulk fluid and the manual removal of the droplet, values considerably smaller than 10 ttm were found. Correcting for such relaxation effects, the observed values have an estimated accuracy of approximately 25%. Enzymatic analysis has some built-in procedure-based causes of error. Because of the time taken to fill the wells with chromogenic T M B substrate the colour development at very high residual IgG-HRP concentrations may already be out of the proportional range (Bally, 1990), giving underestimated values of actual residual fractions. Nevertheless, by filling the plate from low to high concentrations of lgG it was possible to minimize the risk of poor [IgG] estimates. Because of practical limitations in the possible input conjugate concentrations, the lower detection limit was restricted to a fraction of approximately l0 -~' at a given lower level of detectable HRP molecules in the order of 106/well. Increasing the sensitivity of this method can only be achieved by using higher IgG-HRP input concert-

trations or by using other assay labels such as, e.g., radio isotopes. For the purpose of defining the physical origin of the washing process, however, the enzymatic method proved quite satisfactory. By using the model experiments as de~ribed, we demonstrated that a soak time-based washing of microplates can properly be described in terms of a two-compartment residual volume. (1) A residual layer. Using a given sample volume and aspirate cycle time, the lowest possible residual volume to be achieved is determined by the fluid properties. The removal of material from this layer largely depends on soak time and layer thickness, and thereby implicitly on liquid properties. (2) A residual droplet. This volume is a matter of imperfectly emptying the walls, e.g., by choosing too short an aspirate time or by choosing bad aspiration properties. The volume in this compartment is thus generally washer dependent, and its time-independent dilution is merely a matter of the wash volume. Both compartments involve two separate, and indc~endent physical processes involved in mie~aplate washing: slow, diffusion-based emptying of the layer, and rapid, dilution-based emptying of the dro~|e~,. The fastest way to overcome diffusion limitations of molecules leaving the residual layer, is to apply convection° It is important, however, that convecti~'e forces never reach the solid phase completely, in the region where protein-interaction-based immune reactions take place (over several tens of nanometres (Tyn and Gusak, 1990)) shear forces due to too high fluid velocity gradients potentially disrupt the binding reactions between coating, analyte and conjugate. We decided to maintain a minimum flow boundary thickness of the order of several hundreds of nanometres, a value that was based on measured fluid flow properties and assuming a laminar flow pattern. The validity of this experimental decision can also be supported from a diffusion point of view: considering that convection not only physically r¢~noves IgG-HRP molecules from the residual layer, but at the same time homogenizes the bulk wash fluid, there will always be a maximum concentration gradient in this flow bound-

ary layer area. From Fick's law it is readily seen that the conjugate efflux from this layer will also be optimal. We have clearly demonstrated that during shaking of the wash fluid convective fluid forces not only rapidly and strongly reduce residual lgG fractions compared to diffusion-based washing, but also that the removal of lgG is more reproducible, approximating to the theoretical optimal ideal dilution and becoming more equipment and liquid type independent. The method permits significantly reduced assay variation, despite the reduced number of washing cycles and soak time periods. Being driven by external forces, this procedure more readily corrects for variations in residual volume due to liquid viscosity differences. Finally, the method permits the use of smaller amounts of wash fluid. Though neither investigated nor optimized, the procedure described here for shaking may indeed reduce concentration gradients during incubation, and thereby shorten incubating periods for immunoreagents. This suggestion has previously been made by others (e.g. Stenberg et al., 1988). In conclusion three statements can be made regarding immunoassay washing. On the basis of a better understanding of the physical processes involved, we have demonstrated that improvements in microplate immunoassay washing are feasible: there was also a higher throughput of equipment due to the use of fewer washing cycles and, even more interesting, a reduced assay variation yielding higher assay sensitivity. Washing equipment and washing procedures for immunoassays should be judged on their ability to reduce the residual conjugate fraction to a minimum. Being primarily based on known physical phenomena, our method can rapidly be adapted to washing other well or even tube systems for a wide range of assays.

Acknowledgements We gratefully ~rcknowledge the stimulating discussions with Dr. R. Bally on diffusion modelling, Ir. B. Konings and Mr. R. van Beuningen for

85 their generous supply of IgG-HRP and Dr. P. Kamps for reviewing the text of this manuscript.

List of symbols D

diffusion coefficient (m2/s)

dA/dtlt=oinitial rate of absorbance change per f(N~) HRP IgG kB N nN (na) Nw P R t TMB c Vd Vr V~ V~ a ¢'

time unit (absorbance units/s) fraction of free molecules after Nw washing cycles horseradish peroxidase immunoglobulinof class G Boltzman constant average absorbance of negative control samples (absorbance units) number of samples involved in calculating N (P) number of washing cycles average absorbance of positive control samples (absorbance units) radius of microplate well (m) times (s) 3,Y,5,5'-tetramethylbenzidine fluid velocity vector (m/s) volume of residual droplet in well (m 3) residual volume (m3) sample volume (m 3) washing volume (m 3) volume of residual fluid layer (m 3) fraction of molecules in residual layer left despite diffusion the viscosity relative to water: E'=

droplet with volume Vj and a thin layer along the wall, having thickness ~ and volume V~. Let the total washing fluid volume be V,If the washing process were a matter of ideal mixing, the fraction f ( N w) of free molecules left after Nw washing steps equals:

f(N,,) = ( v j v , ) . ( v j v , , ) N"

(Al)

If the liquid layer were inaccessible to the washing fluid, a worst case estimate for f(N~)would be:

f(Nw) = (vd/V,)

(A2)

However, in practice a fraction ( I - o r ) of the molecules in the thin residual layer will be mixed with the bulk wash fluid, giving the experimental fraction f(Nw): f(Nw) = ( v j v , ) • (1 - a ) ~'~

+ ( Vd/ Vw) " ( Vd/ Vw) Nw

(A3)

The fraction of the layer content that empties into the washing fluid is deterndned by diffusion. Considering this layer as analogous to a capacitor (C = mass content of layer) that empties, a special solution of Fick's law yields a value of (1 - a) according to: ( I - a) = [I - exp( - Ot/~ 2)]

(A4)

~/6water ~rN

thickness of residual fluid layer in well (m) standard deviation of negative control absorbance standard deviation of positive control absorbance liquid density (kg/m 3) fluid viscosity (Pa- s)

Appendix 2

Concectice cleaning of the boundary layer

Layer and droplet colume

Assuming laminar flow along the vertical walls and neglecting bottom effects, the flow boundary layer when filling the well with wash fluid will penetrate the residual layer in the well. A simple flow model leads to a fraction /3 being removed from the boundary layer volume Vz. The thickness of the boundary, layer is generally defined as the distance from the solid wall at which goes (Patterson, 1983)

Consider that the residual volume Vr in a microplate well consists of two compartments: a

c,~(x) = 0.99c~(x)

~rp F /~

Appendix h Removal of residual volume

(B1)

86 which compares to the initial layer volume 27rRh8 as a fraction f that we estimate (with t = h / l , ) as satisfying:

While filling the well with liquid it is unrealistic to assume a fully developed laminar flow profile and therefore we consider the well (a) bottomless and (b) at the beginning of an infinitely long cylinder. For such geometry the thickness of the flow boundary layer starts at zero in the well bottom and grows with increasing height h. In these circumstances 6(h) satisfies approximately:

f = 0.33(Ft,s62/p,h)=/3

~$(h)

Estimating c s max

--

4.64((#h/Ft'~)

(B2)

Writing ~ = ~(h) in a series expansion of (B2) yields:

t'/c~ = 1.5x/6

(B3)

from which we combine towards

h ( x , c ) = O . l ( k / c 2 ) x 2 (k = rt,;'/~)

(B4)

(a9)

Appendix 3

Assuming the removal of at least 90% of the initial boundary layer content, and with an estimated initial thickness 6, we estimate the laminar sublayer of the turbulent flow to approximate to 0.1 & Assuming that this problem may be approximated by a flat plate analogue we then require (Patterson, 1983): t$ = 72yRe~-°''~ --- t,~"~L~

(CI)

a parabolic line of constant fluid velocity. Between two such parabolas with respective constant fluid velocities c and c + dc (for the total height of the well) the total volume dV equals:

For 8 = 10 #.m this yields:

l / c ~ n'~= l

(C2)

d V = - 2 . 0 9 R 8 dh

(B5)

cy = 1 m / s

(C3)

Applying a mass balance to a thin ring-shaped layer in the boundary layer, the net efflux out of the boundary layer can be calctdated:

References

dV=

- 1.02R(t k / h ) 32 at

(B6)

Assuming (B5) and the integral of (B6) to be equal we find:

hg = 0.81( kt52t2) I/3

(B7)

in which hg is the distance from the bottom of the well to the point where the fluid between parabolas c and c + dc has left the boundary layer with thickness & This relation can then be used to estimate the total amount V of material that leaves the boundary layer in a single washing: V = 2.05R(k85t2) t/3

(B8)

Bally, R. and Gribnau, T. (1989) Some aspects of the chromogen 3,3',5,5'-tetramethylbenzidineas hydrogen donor in a horseradish peroxidase assay. J. Clin. Chem. Clin. Biochem. 27, 791. Berg, O. and yon Hippel, P. (1985) Diffusion controlled macromolecular interactions. Annu. Rev. Biophys. Biophys. Chem. 14, 131. Carlier, Y., Bout, D. and Capron, A. (1979) Automation of ELISA. J. lmmunol. Methods31,237. Conradie, J., Vorster, P. and Kirk, R. q981) A simple and rapid method of washing and drying micro titre plates used in ELISA. J. lmmunoassay2, 1(19. Gosling, LP. (1990) A decade of development in immune assay methodology.Clin. Chem. 36, 1408. Muller, B. (1976) Der EnzymatisehenTest im klinischen Labor. BehringwerkeAG, Frankfurt. Murthy, V., Freundlich,L. and Kamen,A. (1986) Anal. Chem. 58, 1898. Patterson, A.R. (1983) A First Course in Fluid Dynamics. Cambridge UniversityPress, Cambridge.

Ruitenberg, J. (1977) Mechanisation of the enzyme linked immunosorbent assay (ELISA) for large scale screening of sera. J. Immunol. Me~:hods 16, 351. Spermon, J, (1990) An Automated Microplate Processor. Oak Ridge Conference on Advanced Analytical Concepts for the Clinical Laboratory, Tampa, FL. Stenberg, M. and Nygren, H. (1988) Kinetics of antigen-antibody interactions at solid-liquid interfaces. J. immunol. Methods 113, 3. Stenberg, M., Werthcn, M., Thenader, S. and Nygren, H.

(1988) Diffusion limited theory for a micro titer plate assay. J. lmmunol. Methods 112, 23. Stobbs. L.W. (1990) Construction of a simple, inexpensive, multiple enzyme-linked immunosorbent assay microdilution plate washer. Appl. Environ. Microbiol. 56, 1963. Tyn, M.T. and Gusak. T.W. (1990) Predicticn of diffusion coefficients of proteins. Biotechnol. Bioeng, 35. :327. Tijssen. P. (1985) Practice and Theory of Enzyme Immunoassay. Elsevier, Amsterdam.