1642 Volume 53, Number 12, 1999 APPLIED SPECTROSCOPY0003-7028 / 99 / 5312-1642$2.00 / 0

q 1999 Society for Applied Spectroscopy

Peak Fitting and Deconvolution for Determination of theImpulse Response Function of the Sample IntroductionSystem in Flow Injection Inductively Coupled PlasmaSpectrometry

ROBIN RATTRAY, DAVID H. BURNS,* and ERIC D. SALIN *Department of Chemistry, McGill University, 801 Sherbrooke St. West, Montreal, Quebec, Canada H3A 2K6

A technique for determining the impulse response function for an

inductively coupled plasma atomic emission spectrometry sample

introduction system is demonstrated. The input response functionmay then be useful for the determination of outputs, and conse-

quently the performances, of ¯ ow injection or other sample treat-

ment systems, which precede the sample introduction system. In aprocedure to generate a known input response, a bolus of analyte

was proceeded and followed by an air bubble, which reduced dis-

persion to a minimum. The impulse response function was deter-mined and used to deconvolute other inputs of known concentration

and dispersion. The deconvolution of the outputs using the impulse

response function returned the expected result, indicating that thetechnique and impulse response function were correct. When used

with an unsegmented traditional ¯ ow injection sample introduction

valve, the system demonstrated clearly that signi® cant dispersiontakes place even in very compact introduction arrangements.

Index Headings: Inductively coupled plasma; Sample introduction;

Deconvolution.

INTRODUCTION

Since it was ® rst described in the mid-1970s,1,2 ¯ owinjection analysis (FIA) has emerged as a popular ap-proach for sample manipulation and chemical analysis bytechniques such as spectrophotometry and electrochem-istry.3 More recently, ¯ ow injection (FI) procedures havebeen implemented in conjunction with several atomicspectrometric techniques4 including inductively coupledplasma atomic emission spectrometry (ICP-AES) andmass spectrometry (ICP-MS). With the use of FI sampleintroduction with these techniques, several disadvantagesassociated with conventional sample introduction systemshave been overcome. Some of the bene® cial aspects ofFI sample handling can be an increased sample through-put, a higher tolerance of dissolved solids, and the pos-sibility of on-line sample treatment prior to analysis.5 Inthe simplest application, the sample is loaded into a loopand then injected into a continuously ¯ owing carrierstream, which transports it to the sample introduction sys-tem of the instrument. FI preconcentration applicationscan be more sophisticated in that they can perform an-alyte preconcentration and matrix elimination by the in-clusion in the manifold of a microcolumn ® lled with asuitable material to preferentially retain analyte species.The analyte is then eluted in a small volume with anappropriate eluant.6

Pneumatic nebulizers are commonly used for aerosol

Received 6 April 1998; accepted 21 July 1999.* Authors to whom correspondence should be sent.

generation in combination with spray chambers prior tointroduction of the aerosol into the plasma. These com-ponents, however, affect the shape of the response of thesystem to the sample introduced from an FIA manifold.Normally it is impossible to determine the output froman FI-ICP system because the observed ICP signal is aconvolution of the FI input with the sample introductionsystem impulse response function. This feature makes un-derstanding and optimization of the FI component dif® -cult; however, by deconvolution of the output with aknown input signal, the impulse response function shouldbe determinable. The impulse response function, once ob-tained, may then be useful for determining the outputfrom FI systems.

The theoretical description of peak shapes in FIA hasreceived considerable attention.7±9 In general, the FIAoutput takes the form of a skewed Gaussian peak. Theeffect of manifold components on the dispersion char-acteristics and therefore peak shape in FIA has also beenreported. Van Nugtereren-Osignga et al. applied a decon-volution process to the response of the system with andwithout the manifold component of interest to determinethe impulse response function of that component.9 A fastFourier transform algorithm was used to perform the nec-essary convolution/deconvolution. These authors notedthe detrimental effect that signal noise had on such anapproach and, in a follow-up publication, described theuse of the Hunt ® lter, which operates in the Fourier do-main, to decrease the in¯ uence of noise.10 This approachwould not be applicable to our situation, since the neb-ulizer/spray chambers are essential to operation of thesystem.

In this report we describe the use of nonlinear least-squares modeling of the output signal for the determi-nation of the impulse response function of a typical ICP-AES sample introduction system. The impulse responsedetermined for the ICP-AES system is then used to de-convolute the output signal from a series of segmentedand unsegmented bolus injections using a damped least-squares deconvolution procedure. The results demon-strate that a reasonably accurate reconstruction of the in-put boli can be achieved.

THEORY

System Convolutions. In this section we review thestandard method for systems evaluation, introduce the no-tation, and establish a basis for the deconvolution pro-cedures.

APPLIED SPECTROSCOPY 1643

FIG. 1. Output intensity from boli of 3 (a), 6 (b), 12 (c), and 24 (d )second durations where bubbles have been introduced at the start andthe output intensity from an unsegmented input with a Rhedoyne valve(e).

TABLE I. Results from computations with three samples for eachbolus length.

Im-pulse

(s)

Sigma (s)

Mean Std. dev.

Tau (s)

Mean Std. dev.

Width (s)

Mean Std. dev.

36

1224

3.093.313.302.50

0.07. 0.01. 0.01

0.04

3.323.403.454.37

0.290.17

. 0.010.16

3.106.04

12.6025.11

0.090.42

. 0.010.20

FIG. 2. Estimated impulse response function as determined by the non-linear regression of curves a through d of Fig. 1 using an exponentiallymodi® ed skewed Gaussian model and the mean values for Sigma andTau from Table I.

For linear systems such as the output from the sampleintroduction system, the concentration measured can bedesignated i(t 9 ) as a superposition of the input to the sys-tem, o(t 9 ), with the systems impulse function h(t,t 9 ) ofthe time span measured from a to b. This can be ex-pressed as

b

i(t 9 ) 5 2 o (t 9 )h(t, t 9 ) dt. (1)Ea

For the sample introduction system, the impulse responsefunction should remain constant over time; thus, the formof H will be time invariant so that h(t, t 9 ) 5 h(t 2 t 9 ).In this form, Eq. 1 represents the convolution of o(t 9 )with h(t 2 t 9 ). The object of the analysis is then to es-timate o(t 9 ) given the measured i(t 9 ) and h(t 2 t 9 ). Fordiscrete concentration measurements, Eq. 1 can be re-stated as

I 5 HO (2)

where the vector I 5 {ij} is the set of measured inten-sities at a sequence of times indexed by j, the vector O5 {o k} is the set of known concentrations of the input tothe system at a set of sequenced times indexed by k. His the circulant matrix (square and diagonally symmetric)generated by the system impulse response function,which must be known.

Impulse Response Determination. There have beenmany methods developed to estimate the impulse re-sponse function; however, it has been shown that themost stable solution is obtained when the response func-tion is constrained to a known physical model appropriatefor the system. For modeling the response from ICP-AESmeasurements, the model should take into account effectscaused by mixed retention mechanisms and nonequilib-rium mass transfer, which cause tailing in the measuredsignal. One expression that has been shown to satisfac-torily approximate a wide variety of peak shapes is theexponentially modi® ed Gaussian,11 which is expressed as

A2h (t) 5 exp(( 2 (t 2 R 2 t 9 ) /2s) 2 (t 9 /t)) (3)O t2p ts

where Eq. 3 models the peak of area A, as a Gaussianfunction convoluted with an exponential function. Threeparameters govern the peak shape, with two pertinent toa Gaussian form: s, the standard deviation of the Gauss-ian, and Rt, the retention time. In addition there is onefurther parameter, t 9 , the time constant, which models theexponential decay function in the model. The output fromthe system should then be a convolution for the functionin Eq. 3 with given input according to Eqs. 1 and 2.

To estimate the parameter values for the exponentiallymodi® ed Gaussian function, it is useful to measure theoutput response for the system given a known input func-tion. The output response can then be estimated by usingEq. 1 with the known input and estimates for the impulseresponse function. The optimal parameters can then befound by using a nonlinear least-squares optimization ofthe measured data. In this procedure, the variables areinitially estimated and used to determine the changeneeded with regard to each variable to better approximatethe observed data. The process is iterated until the chang-es in the parameters are at a minimum. A method thathas been used extensively for searching for the optimalparameters is the simplex method.12 A detailed discussionand computer program for this method have been pro-vided as functions in the Matlab program library as theroutine FMINS.

Deconvolution. With the impulse response functionknown, it is well appreciated that the recovery of O bydirect inversion of Eq. 2 is not generally possible becauseof the noise in the results, and the frequency-bandwidth

1644 Volume 53, Number 12, 1999

FIG. 3. Damped least-squares deconvolution results for representativesegmented bolus lengths of 6 (A), 12 (B), and 24 (C ) seconds in du-ration. The experimental measurements are presented as circles. The® nal estimated input is labeled Input Function, and the estimated outputis presented as the Data Fit. The Data Fit is generated as a convolutionof the Input Function with the impulse response function of Fig. 2.Ideally, the Data Fit should closely match the Actual Data.

limitation of the impulse response function leads to anill-conditioned problem. Details of this issue are dis-cussed by Meinel.13 However, if the solution is known toobey certain physical constraints, a stable estimate of Ocan often be found. For the concentration measurements,an obvious constraint is that the concentration is alwayspositive or zero. In addition, it is frequently known thatthe input is a smooth function of limited frequency ex-tent. Under these conditions, it is reasonable to seek anestimate of O that obeys the constraints by using themethod of damped least-squares.14 In essence, an inverse® lter can be determined from the impulse response func-tion then used to reconstruct any given input from themeasured output response. In the frequency domain, theinverse ® lter is given by

W k 5 H*/([H ]2 1 ek) (4)

where H is the discrete Fourier transform of the impulseresponse function mentioned in Eq. 2, and H* is its com-

plex conjugate. If the noise-to-power spectral density ra-tio is approximated by a suitable constant ek, then thereconstruction can be given in the form of a Wiener ® lteras

O 5 I W k. (5)

However, because of the ambiguity due to noise the re-construction is ill-conditioned. This problem can lead tohigh-frequency noise in the reconstruction. To reduce theeffects of noise, one can use an iterative approach. Therestored estimate for each iteration is given by

O k 1 1 5 O k 1 W k[I 2 HO] (6)

where e k for each W k is given by e k 5 ck e0 and with theinitial estimate O0 5 0. Additionally, constraints of non-negativity and median ® ltering are useful in the time do-main at each cycle of the iteration. Therefore, at eachiteration, the inverse Fourier transform is taken of O k 1 1,constraints applied and then the Fourier transform takenof the result. The iteration is stopped when the changebetween iteration for O k drops below a predeterminedthreshold.

EXPERIMENTAL

All experiments were carried out on a Thermo JarrellAsh Model 61 ICP spectrometer. This instrument was® tted with a LeÂgeÁ re V-groove nebulizer and spray cham-ber (270 mL). The spray chamber was conical in shapeand did not contain a baf¯ e.15 The 324.5 nm line forcopper was monitored without background correction. A10 ppm solution was used for all experiments.

A six-port two-position rotary injection valve (Rheo-dyne, Model 5020) was used in the experiments involv-ing injection into a liquid carrier, using a conventionalsingle line manifold. A Gilson Minipuls 3 peristalticpump (Mandel Scienti® c, Guelph, Ontario) operated at 1mL min 2 1 was used throughout.

RESULTS AND DISCUSSION

A series of experiments were ® rst performed by usingloops of varying volume to inject the sample into a dis-tilled water carrier. The volumes of the loops were chosento yield sample boli of 3, 6 and 12 duration at the 1 mLmin 2 1 pumping rate used corresponding to volumes of50, 100, and 200 m L. These times were not long enoughto attain a steady-state signal level in the plasma, butsuf® ciently long to minimize the effects of valve switch-ing and imprecision in valve actuation. Narrow-bore ethylvinyl acetate tubing (0.5 mm i.d., Microline, Cole Pal-mer) was used to minimize dispersion of the sample plug.The length of tubing between the injection valve and thenebulizer was kept to a minimum (2 cm) for the samereason. The initial output from the ICP (100 points, 1integration per point) is shown in Fig. 1, curve a (see® gure caption for identi® cation of curves). In order tominimize dispersion, the sample was therefore introducedinto the carrier stream while being separated from thecarrier by two air bubbles (Fig. 1, curves b, c, and d ).This approach was previously used to reduce tailing ef-fects and increase sample throughput with ¯ ame atomicabsorption spectrometry.16 The data obtained are in sub-stantial agreement with the behavior observed by Ruzicka

APPLIED SPECTROSCOPY 1645

FIG. 4. Damped least-squares deconvolution results for a representa-tive unsegmented bolus of 24 duration.

and Hansen.3 The volume of sample introduced was var-ied by manually inserting the tube alternately into thecarrier and the sample solution for the required period oftime.

The impulse response for the sample introduction sys-tem (spray chamber and nebulizer) was determined bythe method described above, a nonlinear least-squares ® t.The skewed Gaussian parameters were determined withthe use of the results from the bubble-segmented samplesassuming a square input concentration pro® le. Table Iprovides the results from the computation with three sam-ples for each bolus length. The calculated width for thebolus lengths calculated in Table I is in good agreementwith actual bolus length used. The mean Sigma and Tauagree well between bolus lengths. The average Sigma andTau are 3.048 and 3.634, respectively. With the use ofthese values, the input response function can be calcu-lated with Eq. 3. The input response function is illustratedin Fig. 2. This response approximates a 5 s time constantfor this speci® c combination of nebulizer and spraychamber.

With the use of the iterative approach with the air seg-mented data, the results were signi® cantly improved, asindicated by Fig. 3, curves A , B, and C . The experimen-tal measurements are presented as circles. The experi-mental measurements can be deconvoluted by using theimpulse response function to provide the input function.In turn, the input function convoluted with the impulseresponse function produces the data ® t, which should ap-proximate the actual data. Figure 3 demonstrates that thisis indeed true for all cases.

The deconvoluted data are marked as the input func-tion. Note that the top (plateau regions) of the 6, 12, and24 s input functions are very close in duration to theexpected times, as indicated in Table I. The average erroron the width estimate is less than 4%. As expected, thereis a certain amount of tailing; however, the rising edge isquite sharp. It is also interesting to note that even bubblesegmentation fails to provide a perfect leading or trailingedge. The calculated input response function for the spraychamber is provided in Fig. 2. The Sigma and Tau valueswere determined from averaging the results presented inTable I for the various bolus lengths. Application of thisfunction to the original data set (Fig. 4) indicates thatconsiderable dispersion took place with the use of thestandard Rheodyne valve, probably as the sample tra-versed the injection loop, since there was only 2 cm oftubing connecting the injection valve and the nebulizer.

CONCLUSION

Nonlinear least-squares deconvolution appears to becapable of determining convolution functions for ICP-AES, if a skewed Gaussian function is assumed. Usageof the impulse response function indicates that consid-erable dispersion takes place even in relatively simple FIsystems. This dispersion may be so severe that the noiseof the conversion process may make the deconvolutedresponse function only minimally useful for predicting orobserving processes that are taking place in the FI sys-tem.

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