Aplicació de metodologies quimiomètriques a l'estudi de l'efecte del solvent sobre els aspectes termodinàmics i estructurals dels equilibris àcid-base dels polinucleòtids

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(1)Aplicació de metodologies quimiomètriques a l'estudi de l'efecte del solvent sobre els aspectes termodinàmics i estructurals dels equilibris àcid-base dels polinucleòtids Anna de Juan Capdevila. Aquesta tesi doctoral està subjecta a la llicència ReconeixementSenseObraDerivada 4.0. Espanya de Creative Commons.. NoComercial. –. Esta tesis doctoral está sujeta a la licencia Reconocimiento - NoComercial – SinObraDerivada 4.0. España de Creative Commons. This doctoral thesis is licensed under the Creative Commons Attribution-NonCommercialNoDerivs 4.0. Spain License..

(2) em o. 1. DEPARTAMENT DE. PROGRAMA. o. m. 0. o. QuiMICA ANALInCA DE LA UNIVERSITAT DE BARCELONA. QuiMICA ANALIncA DEL MEDI AMBIENT I DE LA POL·LUCIO. (BIENNI 1988-1990). DE DOCTORAT:. APLICACIO DE L'EFECTE. o. m. METODOLOGIES. QUIMIOMETRIQUES A L'ESTUDI DE. DEL SOLVENT SOBRE ELS ASPECTES. ESTRUCTURALS DELS EQUILIBRIS. Memoria. ACID-BASE. TERMODINAMICS. DELS. POLINUCLEOTIDS. presentada per Anna de Juan i Capdevila per optar al Ciencies Quimiques.. Directors: Enric Casassas i Sima i Gemma Fonrodona. I. grau de Doctor. en. Baldajos.. Barcelona, julio1 de 1997.. "alliilillii�TliIMr 0700452390.

(3) CHAPTER 6.. THE ACID-BASE. EQUILIBRIA OF POLYNUCLEOTIDES IN WATER-DIOXANE MIXTURES.

(4) 6.1. The. use. monitoring. of curve resolution. techniques. of biochemical processes:. of the chemometric. procedures.. to. interpret. improvement. the multivariate. and. understanding.

(5) ANALYTICA. CHIMICA ACTA ELSEVIER. Analytica. Assessment of. Chirnica Acta 18087. constraints. new. (1997). 1-12. applied. the. to. alternating. least. squares method A. de. Heyden",. Y. Vander. R.. Tauler",. D.L.. Massart". Qufmica Analitica; Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Spain Farmaceutisch Instituut, Vrije Universiteit Brussel, Laarbeeklaan 103, B·1090 Brussel, Belgium. "Departametu. bChemoAC,. Juana,b,*, de. Received 6 December 1996; received in revised form 6 December. 1996; accepted. 16 December 1996. Abstract The introduction of constraints in multivariate. curve. resolution methods, such. as. the. used to limit the span of possible solutions, guiding the iterative process to true situation. In the present work, two modifications of the unimodality constraint and. commonly. concentration. profiles. related to the. prevention. been used to evaluate the effect of these. goodness. of the constraints. Keywords: XXX; XXX;. are. of. have been checked. Simulated data sets. fronting. Least. as. well. as. real data have. constraints in the resolution results. The parameters measured to assess the related to the recovery of the concentration profiles and to the quality of the data fit. new. XXX. 1. Introduction. Within the. application of constraints is a common opera tion employed in iterative curve resolution methods, such as the Alternating Least Squares (ALS) [1-3], to The. narrow. Squares (ALS), is final result as close as possible to the a new constraint for chromatographic Alternating. the span of feasible solutions to those chemi. cally meaningful [4 6].. family of constraints, selectivity is the most important to resolve any kind of data sets [1]. If this feature exists for all the compounds, the ambiguity associated with the factor analysis decomposition of bilinear matrices disappears and unique solutions can be attained [1,8]. However, having partial or null selectivity is the most frequent situation in real data. wide sense, the concept of constraint would include any general feature of the data sets translated. kind of constraints related to the chemical features of. into mathematical. concentration. In. a. language. However, a careful atti tude has to be adopted in order to avoid false general izations (e.g. not all the chromatographic peaks have a Gaussian shape) and the researcher has to be aware that the effectiveness of a constraint can be strongly affected by the way it has been implemented.. B.Y. All. rights. reserved.. 11-63. role of. some. other. profiles and to the instrumental responses acquires relevance and helps to decrease significantly the domain of possible solutions. The development of new constraints belonging to this last group becomes then specially interesting for the reso lution of the most complex data sets. Numerous works have reported the usefulness of constraints for. 'Corresponding author. E-mail:annaj@zeus.qui.ub.es. 0003-2670/97/$17.00 © 1997 Elsevier Science PI/ S0003-2670(97)00145-1. sets. When the latter occurs, the. curve. resolution methods like. non-.

(6) A. de Juan. 2. et. al.IAnalytica Chimica. negativity (applied to both concentration profiles and instrumental responses) [1-7,9-14], unimodality (i.e. presence of an only maximum in each concentration profile) [1-3,5-7,10--14] and closure (i.e. the total amount of the reactive constituents is forced to be. in all the stages of a process) [ 1 ,2, 1 0-13,15,16]. In the present work, two modifications of the unimodality constraint are examined, as well as a. the. same. chromatographic concentration profiles (called hereafter symmetry) related to the prevention of front-tailed peaks. A two-level full factorial design has been used to generate simulated data sets which could yield a reliable picture of the effect of these new constraints in the resolution procedure. The responses measured to assess the goodness of the constraints are related to the recovery of the qualitative information and the quality of the data fit. All conclusions inferred from this basic study have been afterwards confirmed work ing with a real example. new. constraint for. Acta 18087. 2.2. Presentation. of the. 2.2.1. Horizontal. unimodality. Tbeory. 2.1.. this it. variety differs from the original version in the way is implemented. The common steps in both implementations can be. summarized. belongs. to. the. .... gence criterion is reached. The matrices of the con centration profiles and spectra obtained in the last. cycle. are. the definitive solutions of the. reso. lution method when convergence is achieved. A more detailed explanation of the ALS method is out of the scope of this paper and can be found in of the authors [1-3,10--14].. follows:. largest. maximum in the. concen. tration 2. 3.. profile (rn). Suppression of the left local maxima. Suppression of the right local maxima.. The difference lies in the elimination of the. secondary. unimodality (a) sets the non equal to zero and the new imple. maxima: the classical unimodal elements. mentation element. (b) equals these elements. keeping. c(rn. -. i) i). the nearest. the unimodal condition. In. mic notation, steps 2 and 3 -. to. >. r x. c(rn. ::::::!. 0. (for. c(rn. -. can. i +. be. algorith expressed as:. 1). classical. unimodality). i + 1) (b) c(rn i) (for horizontal unimodality). ALS rnethod. .... iterative. as. 1. Location of the. -. family of iterative curve resolution methods. All these techniques share a gen eral working procedure consisting of the refining of some initial estimates (either chromatographic or spectral) by using constraints related to the intrinsic features of the data (i.e. selectivity, zero-concentration regions, ) or to the chemical characteristics of the experimental system (i.e. non-negative concentration profiles or spectra, unimodality, ). In the present work, the initial estimates to be input in the ALS method are always built in the chromatographic direc tion applying the results obtained in the needle algo rithm [17]; then, a constrained alternating least squares optimization procedure runs till the conver The ALS. constraints. in concept to the classical unimodal con straint (i.e. no more than one maximum is allowed),. (a) c(rn. Brief description of the. new. Equal. 2. if 2.. (1997) 1-12. previous works. 11-64. 3. if. c(rn +. (a) c(rn +. =. i + i +. 1) 1). > ::::::!. -. r x. 0. c(rn + i) (for. classical. unimodality). (b) c(rn + i + 1) c(rn + i) (for horizontal unimodality) =. where. c(rn) is the maximum value of the concentration. profile. and the. [c(rn +. i +. pairs [c(rn. 1), c(rn + i)]. are. -. i), c(rn. -. i +. the consecutive. 1)]. and. concen. tration values to be. compared when looking for left and right local maxima, respectively. The parameter r can be optionally larger than 1; if this is the case, small departures of the unimodality are accepted. From a graphical point of view, the classical con straint cuts the non-unimodal part of the concentration profile vertically, whereas the new modification does it. horizontally. Fig. I illustrates the effect of both vari eties of unimodality on a concentration profile. The plot stresses the positive behaviour of the new imple mentation for noisy peaks, normally related to minor compounds. Such peaks contain noisy spikes that are.

(7) A. de Juan et. al.lAnalytica. Chimica Acta 18087. a). (1997) 1-12. 3. maxima for the different. previously determined by using methods like the needle algo rithm [17], OPA [18] or SIMPLISMA [9]. The application sequence of this constraint is. •. .. •. •. ·. '. • ". ·. ,. ,. ,. •. ,. ,. ,. ,. ·. ,. •. ,. .. tional step, absent in the normal unimodality, for which the algorithmic notation is also included.. ". •. ,. •. ". ·. " •. ,. ,,,. ,. '. •. ,. .. •. ". ,. ,. •. ... ,. r. • ,. •• -. r ,. 1. Location of the. ,. •. •. are. detailed below. The text in italics indicates the addi. •. •. compounds. ,. ,. ·. ... •. •. .. ,. . .. ·. ... .. ... ". ·. • •. , ,. ,. ,. ,. ,. •. ••. • ". ,. '. ._.;._;Qtri'.. maximum. '. .. .' .'. '.. ,. •. ,. '. ". ,. profile (m). Comparison between. •. '. ". ,. '.. ,. ". ,. ". ,. •. ,. •. 2.. ,. .. ,. maximum in the. concen. tration. ... ,. largest. resolution. position of the reference (mt) and the position obtained from the profiles (m). Relocation of the peak the. maximum if necessary if. b). abs(m. m. 3,. • ". •. •. ·. I. ". ,. ,. •. 4,. ,. ·. ,. =. -. mt). >. rpeak. mi.. Suppression Suppression. of the left local maxima,. of the. right. local maxima,. ,. ,. •. •. •. •. The maximum shift allowed in the. -. .. ... ,. '. .. ,. ,. ,. peak position is. represented by the parameter rpeak, whose value is selected by the user, Analogously to the other varieties. •. •. , ,. , .. .. of this. ,. • ... constraint, small departures of the unimodal. ... condition. ... • •. • •. ,. • .. .. ,. ,. ·. ";'". · .. , • •. . .. �,. •. •. ".. ,. ,. ,. ... .,. ". •. •. .. ,. '.. ,. '. '. ". Fig,. 1. Effect. •. -:--. I'. .'. .. .. • ••. I. ••. •. _.. ':'. .. �... 2.2.3.. •. .. ,. ,. detected. its name, the present constraint does not transform all the concentration profiles into symme. Despite. ,. of the two. signals. Tailed peaks are accepted, provided that this asymmetry in the profile is placed after the peak maxima. Thus, real chromatographic situations where column ageing or other causes produce the distortion of the theoretical Gaussian peaks can be correctly reproduced. trical. peak.. secondary maxima, this fact leading to the wrong suppression of a big part of the concentration profile when the classical implementation is applied. as. In contrast, the horizontal elimination of these. ima allows with. a. to. shape. keep. a. ever. much closer to the. implementation applied.. profile. are. max. arises from the resolution results, the concentration profile of the affected compound is forced to be. original peak.. version of the normal. only maximum. The symmetry constraint is focused on the suppres sion of front-tailed peaks. When such a peak shape. constrained concentration profile. 2.2.2. Localized unimodality This constraint has to be considered. demanding. Symmetry. •. implementations of the unimodal constraint on a concentration profile: (a) classical unimodality; (b) horizontal unimodality, Dotted line: original peak. Solid line: constrained. be. f. ,. I.,.......J. optionally accepted. Steps 3 and 4 applied as in the horizontal unimodaiity.. can. '... ..... ,. .. J. .'. .'. ' .. .,' •. ...._.. ._. , ,. .. .... .'. ,. ,. ,. have been. ,.. ·. •. .,_-. ... ... .' ,. ,. ·. 1. ". .. ,. ". r. �,. •••. ,. •. •. ... •. •. '. , •. symmetrical. Although phenomena of band fronting can occasionally be found in real data, their occur a. more. unimodality,. what. as. Both the existence of. rence. common. tailing ion-pair chromatography [19]. Visual evidence of band fronting in a chromatogram would indicate the need of a modification in the separation parameters. an. and its. controlled.. than the appearance of band and it is practically reduced to the domain of. is far less. position in the concentration The positions taken as reference. 11-65.

(8) A. de Juan et aU Analytica Chimica Acta 18087. 4. (1997). 1-12. 3. Data sets 3.1. Simulated data. sets. Two-compounds simulated data sets have been used assess the constraints presented above. The choice of this kind of systems is related to the role they playas reference models in peak purity problems and to the fact that many real multicompound samples can be resolved analyzing submatrices of peak clusters which do not contain usually more than two or three over lapped substances. Furthermore, the simple structure of these systems allows a clear interpretation of the to. Fig. 2. profile.. Effect of the symmetry constraint on a concentration Dotted line: original peak; solid line: constrained peak.. effects of the constraints tested, in contrast with the vague conclusions inferred when very complex simu lations are employed. In all the. and there is. examples the two peaks are slightly tailed a major and a minor compound. Severely have been chosen. purpose for 3). Since the. (i.e. increase of temperature, decrease of the amount injected sample, ). The non-adequacy of the. overlapped spectra. symmetry constraint would be limited. constraints to be assessed affect the concentration. of. .... to those rare. fronting appears and cannot be chromatographically suppressed. Therefore, the pro posal of such a constraint is chemically reasonable in by far the most cases. The symmetry constraint reshapes the front-tailed peaks by making the back half of the peak symme trical to the front (see Fig. 2). Gaussian and tailed peaks are not modified. The mathematical formulation of this constraint is very simple and can be explained situations where band. in two steps:. all the simulated data sets. profiles,. a. on. (see Fig.. better evaluation of their real effect. be carried out when. can. influ. spectral selectivity procedure. The general validity of the conclusions related to the quality of the new constraints has been ensured through the non-arbitrary selection of the simulations employed in the testing procedure. A two-level fac torial design has been used to determine the features of the generated data sets [20]. Table 1 shows the rele vant information concerning the factors and their ence. largest maximum in the concen profile (m). Detection and suppression of band fronting:. 1. Location of the. can. no. the resolution. O.07��-_-. -_-·-'----·I·--. ,"-. I. tration 2.. if. c(m. -. c(m + i). i) =. where the. >. r x. c(m. -. c(m + i) i). pair [c(m. concentration values. -. i), c(m + i)] represents. equidistant. to the. peak. two. maxima. 0.02. Band. fronting is present when the values in the left peak are bigger than those in the right half of the peak. The parameter r can be optionally bigger than 1; departures of the symmetry constraint are then accepted.. half of the. 0.01 i I L-.. _l. 10. 15. --'-----'. 20. 25. .. _�,.-�:::::;=""="""''''''''' 30. 311. WavoIeng1I1s. Fig.. 1I-66. 3.. Spectra. used in the simulated data sets.. .co. I �6.

(9) A. de Juan. et. Table I. sets.. wide span of real situations. Thus, the con be checked is introduced as a factor, whose. cover a. Two-level fulJ factorial List of the coded. design used to generate the simulated properties related to each simulation. data. straint. 2. B. D. C. E. +. 3. +. 4. +. +. 5. +. 6. +. +. 7 8. +. +. +. +. +. 9. +. 10. +. +. 11. +. +. 12. +. +. +. 13 14. 16. +. +. +. +. +. +. +. +. +. +. +. 15. +. to. qualitative levels are absence (-) and presence (+). The negative level of this factor has been differently defined according to the constraint to be checked (detailed explanations in Section 4.1.2 and next). The remaining factors are features of the chromato graphic system, namely the resolution between peaks, the concentration ratio between major and minor compound, the noise pattern and the signal-to-noise ratio for the minor compound. The heteroscedastic noise has been simulated by adding to a homoscedas tic background a scaled contribution of this base noise proportional to the square root of the intensity of the signal, i.e. for each ijth point of the data matrix, the. Simulation Factors A. 5. aU Analytica Chimica Acta 18087 (1997) 1-12. noise added total. can. be described. noise.,. Jfree. +. 18. +. +. 19 20. +. +. +. 21 22. +. 23. +. 24. +. +. +. +. +. +. +. 26. +. 28. +. 30 31 32. +. %. measurement. +. SIN ratio for the small. +. +. +. +. +. +. +. +. +. +. +. +. +. Factors. levels. (-). (+). Constraint (A) Resolution (B) Ratio minor/major. Absennce. Presence. 0.2. 0.8. species has been considered general parameter to evaluate the distortion of the minor signals, strongly tied to the possibility of resolving these compounds. For both homoscedastic and heteroscedastic systems, the sig nal-to-noise ratio for the minor compound follows the expression below: more. informative than. max. SN= /. (free. -. a. noise. signal minor compound) ,. L;/homosc. noise)ij. I: 10. 100. common. +. +. :. the. using. +. +. I. instead of. pound. +. +. +. (1). +. +. +. 29. signalij. The concept of noise level has been represented through the signal-to-noise ratio for the minor com. +. +. 27. noise. of noise with respect to the largest absorbance. In systems containing major and minor compounds, the. +. +. 25. -. (homosc. noise)ij.. x. +. +. follows:. (homosc. noise)ij. =. +. 17. as. ixj. compound (C) Noise pattern (D) SIN ratio minor. Homosecdastic. Heteroscedastic. 20. 50. (2) where the free-noise. compound. comes. signal. the simulated concentration. levels,. the selected. design. and the coded. properties of. all the simulated data sets.. The factors included in the. design. are. parameters. Whose influence on the results of a resolution method is either proven or at least potential and the two levels set. for each of the factors have been chosen. trying. to. II-67. for the minor. from the outer product Cm. x Sm,. profile. compound being Cm and Sm. and spectrum. asso. ciated with this constituent, respectively. Please note that the apparently large SIN values used in the present work. are. actually. the maxima allowed for this para the position of the. meter in the data sets, since far from. signal. maximum the ratio between. becomes. considerably. signal. and noise. lower. Variations of the minorl.

(10) A. de Juan et aU Analytica Chimica Acta 18087. 6. major concentration ratios have been simulated keep ing fixed the signal related to the minor compound and modifying the signal of the major compound appro priately. Thus, all the simulations having the same SIN ratio for the minor compound have the same amount of added noise as well, since the free-noise signal for the minor compound remains invariant. This strategy simplifies the simulation process and the further inter pretation of the results. 3.2. Real data. results. (1997). 1-12. related to the recovery of the qualitative (i.e. dissimilarities between actual and. are. information. recovered concentration. and to the. profiles). error. associated with the definitive solution (i.e. standard deviation of the residuals, a, and the lack of fit). The mathematical are. responses. dissimilarity. sets a=. associated. expressions. with. the. shown below:. J. =. 1. (correlation coefficientj'",. -. Lij(dij-d:/ ixj. Two-compound data sets, reported systems for peak purity studies [21],. as. reference. are. used to. confirm the conclusions inferred from the simulated data about the. goodness. =. Lij(dij -dijf. Lijdb. of the checked constraints.. These real systems contain hydrocortisone as major compound and prednisone as minor constituent. The. spectra of these species. lack of fit. are. simulated data sets, shown in. the. same. Fig.. 3.. used in the. where. dij. the. experimental data and dij the repro by using the ALS optimization. Subscripts. are. duced data. i and j are referred to matrix, respectively.. rows. and columns of the data. Tables 2-4 list the numerical responses obtained in study of each of the constraints tested. These data. the. 4. Results and discussion. have been used to calculate the main effects of the factors and the 2, 3 and 4-factor interactions [20]. The information associated with the latter calculations is. 4.1. General remarks To test each of the. constraints, the ALS method has. been. applied to the series of 32 experiments designed according to the information in Table 1. ALS has always been run forcing non-negative concentration profiles and spectra. Even though the use of selectivity is essential in this resolution. method, the information. this. point has not been taken into account in any of the systems analyzed. In the spectral direction, no selectivity can be found as pointed out above, whereas in the chromatographic direction the detec tion of this feature is only evident in some systems where the resolution between peaks is large (Rs=0.8) and the signal-to-noise ratio for the minor compound quite favourable. The application of selectivity in these last cases would be unquestionable in a normal analysis; no explicit use of this constraint has been done in the present study because in the cases where it. regarding. is present its strong effect would mask the influence in the ALS solution coming from the constraints to be checked. The responses collected to analyze the effects of the constraints in the quality of the resolution. proposed. 11-68. graphically presented in the normal probability plots shown in Figs. 4--6 [20]. These graphs represent the value of each of the calculated effects. vs.. the. expected. probability it should have if all the effects were normally distributed. The vertical scale in the plots has been transformed, so that the plotted effects could be fitted with a straight line when changes in the levels cause any notice able variation in the responses or, if they do, this variation is random and does not follow any definite. of all the factors considered do not. tendency.. Positive. or. negative points falling. this line indicate either factors. or. far from. interactions having. important influence on the responses. The graphical concerning the significance of the effects have been statistically confirmed through the applica tion of a t-test in which it is investigated whether or not an effect is significantly different from zero [20]. To do so, the numerical value of each potentially sig nificant effect is compared with a critical effect value, calculated as the product between the averaged value corresponding to the 3-factor interaction effects (such higher-order interactions are assumed to measure an. conclusions.

(11) A. de Juan et al.ZAnalytica Chimica Acta 18087. Table 2. 1-12. 7. Table 3. Responses obtained from. the ALS. runs. used. to test. the horizontal. Responses obtained unimodality. unimodaIi ty. Simulations. (1997). Responses. Dis(cl)". Simulations. Dis(c2)b. from the ALS. runs. used to test the localized. Responses. Lack of. Dis(cl)". Dis(c2)b. 2.09E-Ol. L49E-Ol. 0.0023. Lack of. fit (%). fit. (%). L60E-01. 5.26E-Ol. 0.005. 2. L49E-Ol. 2.09E-Ol. 0.0023. 3.06. 2. 2.09E-Ol. L49E -01. 0.0023. 3.06. 3. 2.51E-03. 4.35E-Ol. 0.0004. 0.6. 3. U8E-Ol. 2.18E-03. 0.00027. 0.37. 4. 2.18E-03. U8E-Ol. 0.00027. 0.37. 4. U8E -01. 2.18E-03. 0.00028. 0.37. 5. 2.31E-Ol. 4.07E-01. 0.0007. 8.6. 5. L72E-Ol. 1.10E-Ol. 0.00034. 4.2. 4.22. 6.31. 3.06. 6. LlOE-Ol. L72E-Ol. 0.00034. 4.2. 6. L72E-Ol. 1.10E-Ol. 0.00034. 7. 3.23E-02. 4.15E-Ol. 0.00037. 4.9. 7. L55E-Ol. 3.52E-02. 0.00026. 3.45. 8. 3.52E-02. 1.55E-Ol. 0.00026. 3.45. 8. 1.55E-Ol. 3.52E-02. 0.00026. 3.45. 13. 9. 2.21E-Ol. 7.78E-0 1. 0.0098. 9. 3.09E-Ol. 2.2lE-01. 0.0055. 7.31. 10. 2.21E-Ol. 3.09E-OI. 0.0055. 7.31. 10. 4.28E-Ol. 2.13E-Ol. 0.0056. 7.51. 11. 3.76E-03. 3.25E-Ol. 0.00047. 0.62. 11. 3.31E-03. 0.00035. 0.48. 12. DIE-03. L52E-Ol. 0.00035. 0.48. 12. 1.52E-Ol. 3.31E-03. 0.00035. 13. 6.91E-Ol. 6.55E-Ol. 0.0042. 52.1. 13. 4.6JE-Ol. 3.44E-Ol. 0.0016. 19.86. 14. 3.44E-Ol. 4.61E-Ol. O. 0016. 19.86. 14. 3.22E-Ol. 3. 44E-Ol. 0.0017. 21.18. 15. 6.77E-02. 7.78E-Ol. 0.00082. 10.8. 15. 3.14E-Ol. 7. 16E-02. 0.00057. 7.57. 16. 7. 16E-02. 3.14E-01. 0.00057. 7.57. 16. 3.14E-Ol. 7. 16E-02. 0.00057. 7.57. 17. 1.12E-Ol. 2.47E-Ol. 0.0012. L63. 17. 7.96E-02. I.06E-01. O. 0006. 0.79. 18. L06E-Ol. 7.96E-02. 0.0006. 0.79. 18. 7.96E-02. L06E-Ol. 0.00059. 0.79. 19. L17E-03. 8.02E-02. 0.00012. 0.16. 19. 6.72E-02. 0.0001. 0.16. 20. 1.19E-03. 6.72E-02. 0.0001. 0.16. 20. 6.72E-02. 1.19E-03. 0.00012. 0.16. 21. L55E-01. 7.05E-02. 0.00028. 3.5. 21. L65E-01. L29E-01. 0.00014. 1.71. 22. L29E-Ol. L65E-01. 0.00014. 1.71. 22. L65E-Ol. L29E-01. 0.00014. 1.71. 23. L04E-02. 8.91E-02. 0.00013. 1.7. 23. 6.68E-02. LOlE-02. 0.00011. 1.54. 24. LOIE-02. 6.68E-02. 0.00011. 1.54. 24. 6.68E-02. L01E-02. 0.00011. 1.54. 25. UIE-Ol. L97E-Ol. 0.0018. 2.4. 25. UOE-OI. L49E-01. 0.0012. 1.56. J.52E-Ol. 1.19E-03. 0.48. 26. L49E-Ol. 1�OE-01. 0.0012. 1.56. 26. UOE-Ol. L49E-01. 0.0012. 1.56. 27. 2.36E-03. 8.04E-02. 0.00019. 0.26. 27. 7.82E-02. L82E-03. 0.00016. 0.21. 28. L82E-03. 7.82E-02. 0.00016. L82E-03. 29. 7.35E-Ol. 6.47E-01. 0.0027. 30. 2.78E-Ol. L82E-01. 0.00044. 31. 2.28E-02. U4E -01. 32. 2.34E-02. 9. 11E-02. •. Dissimilarity. minor b. 28. 7.82E-02. 29. L82E-OI. 5.41. 30. L85E-Ol. 0.00025. 3.37. 31. 0.00022. 2.9. 32. between the actual and the recovered. 0.21 33. profile. 0.21 5.41. 2.61E-01. 0.00044. 5.5. 9. 11E-02. 2.34E-02. 0.00022. 2.9. U4E-Ol. 2.28E-02. 0.00022. 2.9. a. Dissimilarity between the actual and the recovered profile compound. b Dissimilarity between the actual and the recovered profile major compound.. for the. for the. minor. compound.. Dissimilarity. 0.00016 0.00044. 2.78E-01. between the actual and the recovered. profile. for the. major compound.. differences arising from experimental. for the. and statistical. diagnostic are also labelled corresponding capital letters (see Table 1). improvement of the ALS results is always. [20]) and. inspection. the t-value corresponding to a 95% significance level and a number of degrees of freedom equal to the number of 3-factor interactions in the design. When. with their. the value of. the responses. Indeed, better recoveries of the quali tative information are reached when the dissimilarities. an. effect is. larger. error. The. connected with. than the critical refer. ence. value, the tested effect is found to be significant. In all normal probability plots, the main effect caused by the constraint is identified with the letter A. Effects found to be both visual. significant through. decrease in the numerical value of. between the actual and the calculated concentration. profiles. become lower and achievements of. accurate. fits. ish.. II-69. a. occur. when the. error. more. parameters dimin.

(12) A. de Juan et aU Analytica Chimica Acta 18087. 8. (1997) 1-12 3�----. 3---. Table 4. Responses. obtained from the ALS. runs. used to test the symmetry. constraint. i�:d�Cl)/. i}. Simulations. Dis(cl)a. 0-. i. Responses. Dis(c2)b. Lack of. a. fit. (%). 1-'. A. •. ';. "'_2·. 1.49E-01. 0.0023. 3.06. 2. 1.41E-01. 3.39E -01. 0.0069. 9.2. 3. 1.38E-Ol. 2. 18E-03. 0.00027. 0.37. 1.38E-01. 4. -0.1. 1.l0E-01. 0.00034. 4.2. �. 1.72E-01. 1.10E-01. 0.00034. 4.2. io. 7. 1.55E-01. 3.52E-02. 0.00026. 3.45. l'. 8. 1.46E-01. 4.00E-02. 0.0003. 4.02. 9. 3.09E-Ol. 2.21 E-01. 0.0055. 7.31. 10. 2.94E-01. 2.53E-01. 0.0066. 8.72. 11. 1.5 2E-01. 3.31E-03. 0.00035. 0.48. 12. 2.59E-01. 3.34E-03. 0.0004. 0.53. 3.44E-01. 0.0016. 19.86. 4.72E-Ol. 3.24E-01. 0.0029. 13.9. 15. 3.14E-01. 7. 16E-02. 0.00057. 16. 2.50 E-Ol. 8. 15E-02. 0.00086. 8.69. 17. 7.96E-02. 1.06E-01. 0.0006. o. 79. 18. 7.41E-02. 2.02E-01. 0.0023. 3.04. 19. 6.72E-02. 1.19E-03. 0.0001. 0.16. 20. 1.07E-01. 1.27E-03. 0.00014. 0.19. 21. 1.65E-01. 1.29E-01. 0.00014. 1.71. 22. 1.65E-01. 1.29E-01. 0.00014. 1.7. 23. 6.68E-02. 1.01E-02. 0.00011. 1.54. 24. 6.88E-02. 1.13E-02. 25. l.30E-01. 1.49E-Ol. 0.0012. 1.14E-0 1. 3.58E-Ol. 0.0065. 8.59. 7.82E-02. 1.82E-03. 0.00016. 0.21. 28. 7.22E-02. 1. 82E-03. 0.00016. 29. 6.47E-Ol. 7.35E-Ol. 0.0027. 30. 2.38E-Ol. 3.47E-Ol. 0.00055. 6.75. 31. 1.34E-01. 2.28E-02. 0.00025. 3.37. 32. 1.34E-01. 3.94E-02. 0.00032. 4.22. Dissimilarity between minor compound. b Dissimilarity between major compound.. the actual and the recovered. profile. 3. .: !. di9{c2). d,S(c')/·�-�l. 1. A:. 0. i. .. l'. �.'. E. "'_2. ·0.'. -0.05 0 0.05 Observed etlecl. effects indicate when. �. for the. an. improvement. level in the simulations. On the. 0-. 0. �. for the. I I. E. 11-". B. of the ALS results. Fig.. level to the. positive contrary, significant. meaning. B B. -3 -------------' -C.2 -0.1 0. Ob_effect. � Ii 6il. 1. �_..J. -o.s. 0. 0.5. D. lack/oIflt(%) ••. A. 0. w.2 ·1. :. 2. B". �.. .3':...._---� -5. 0. Observed effect. probability plots related to the assessment of the unimodality constraint. Significant effects are labelled capital letters (see Table 1 for identification) and the. 5. Normal. localized with. always identified with the letter A. The the dissimilarity between the actual and the recovered concentration profile for the minor compound, dis(c1l, and the major compound, dis(c2l, and the lack of fit. constraint effect is responses. of the results have to be the chemical. �-,. }-,. 9. •. Observed _ (x 1000). positive effects are associated with factors or interac tions causing improvements in the ALS results when going from the positive to the negative level. There fore, the interpretation performed according to. I�� CD. A. 0. 1iJ·2. 0.1. ,/. 1. showing significant negative. going from the negative. i. 3. sigma. 2. w_2'. interactions. f. ---j. -3. -1.5. or. 10. Oboervod oll8<:t. 4. Normal. .3Li Factors. ·3�-------10 -5 0 5. ,. horizontal. 0.21. profile. 0. -,. •. probability plots related to the assessment of the unimodality constraint. Significant effects are labelled with capital letters (see Table 1 for identification) and the constraint effect is always identified with the letter A. The responses analyzed are the dissimilarity between the actual and the recovered concentration profile for the minor compound, dis(c1), and the major compound, dis(c2), and the lack of fit.. Fig.. 33. the actual and the recovered. -2'. -2. ,. �. !-'. •. ••. 01>_ gffgef (x 1000). 51 a. B. ·3. 1.56. 26. O·. i. -3�--�-------'. 1.53. 27. Sklma/A f.-:/lackoffit(%) ... 1. iJJ.2. 7.57. 0.00011. � =. 1.72E-01. 4.61E-Ol. OboelVed effect 3�---------. 6. 14. 0.2. 2. .!Ii. 5. 13. 0.'. 0. OI>&9MJdell9ct. 0.37. 0.00027. 2. 18E-03. -3·---------0 0.1 0.2 -0.2 -0_'. ·3'--�--------' -0.2. 2.09E-01. of. II-70. analyzed. are.

(13) A. de Juan. 3 ... �. 0. i-I. .ll_2. the. •. ... -�-.-. -0.1. ... more. Observed effect. Be. dissimilarities. only. i-I. ·3�---�--� -3 -2 -, 0 ,. Obstjrved. Fig.. 6. Normal. be. probability plots related to interpreted taking into account. the main effects of the factors.. However,. a. related to the interactions resolution-noise pattern (BD) and minor/major ratio-noise pattern (CD) is included, since the effects of these combinations. �. 8. can. comment. 0. w.2. difficult.. Most of the normal. t�!�a/. i. in the simulated data sets.. major signals lead globally to a more merged noisy binary system, where the distinction and correct modelling of the compounds becomes. -3·-------� -0.' �.2 0 0.'. --.. 0 0.' Db_effect. .. major compound signal. and. 3-------� ,. 9. both minor and. B. -3. (1997) 1-12. The consequent diminution of the signal-to-noise ratio for this constituent and the comparable contribution of. /0. ,. a". aUAnalytica Chimica Acta 18087. 3·. ---.-�--.. d. 2. et. have been found. significant in the dissimilarity of the major compound in Fig. 5. Both interactions show that variations in the chromatographic resolution (B) and in the minor/major concentration ratio (C) affect differently the quality of the ALS results when occur ring in systems with homoscedastic or with hetero. affect (x '000). probability plots. related to the assessment of the. symmetry constraint. Significant effects. are. labelled with. capital. letters (see Table 1 for identification) and the constraint effect is always identified with the letter A. The responses analyzed are the. dissimilarity between the actual and the recovered concentration profile for the minor compound, dis(cl), and the major compound, dis(c2), and the lack of fit. scedastic noise pattern. Thus, the BD interaction implies that a decrease in the resolution between peaks is. more. profile. critical in the recovery of the concentration compound when the noise is. of the main. the factors and the definition of their levels in the. heteroscedastic, whereas the CD interaction tells that. experimental design.. decreases in the. 4.1.1. Comments about the data. the system affect more strongly the modelling of the concentration profile of this compound in the presence. features effects. on. the ALS results. Figs.. 4-6 show. some. general. quality. of the ALS results. As. factors. are. concerning on. out above,. pointed peaks,. the resolution between. heteroscedasticity. In both cases the explanation is directly related to the stronger diminution of the SIN ratio for the major compound induced by the presence of heteroscedasticity when the resolution decreases (BD) and when the proportion of major compound decreases (CD). An examination of Eq. (1) allows to. the the. these. the minor/. major concentration ratio, the noise pattern and the S/ N ratio for the minor compound and they are identified capital respectively.. letters B, C,. D and E. understand the intense effect of the heteroscedastic. (see Table 1),. pattern in both kind of interactions. This expression consists of two terms, the first. As it is shown in the. recovery of the. one including a homo background (added to all systems, whatever noise pattern they have) and the second involving properly the heteroscedastic contribution. In data sets with equal peaks and different resolutions, the term. scedastic. the. mentioned. figures, lately qualitative information is positively. influenced by increases in the resolution between. peaks and in the signal-to-noise ratio for the minor compound (the first factor being clearly the most. important. in the. in. of. trends. effects of the factors related to the data features. with the. proportion of the major compound. shape modelling. of the. major. referred to the homoscedastic contribution remains invariant, whereas the second term, scaled according. com. the square root of the signal, increases locally in systems with low resolution because the global signal. pound and the second in the modelling of the minor).. to. Bigger dissimilarities between actual and recovered concentration profiles appear when the noise pattern is. larger due to the big overlap between com pounds. comparison between systems with dif ferent minor/major concentration ratios shows that the becomes. The. heteroscedastic and when the minor/major concentra tion ratio in the binary system increases. The negative action of the latter factor is linked with the decrease of. variation of the total noise added to the data sets when. II-71.

(14) A. de Juan et aU Analytica Chimica Acta 18087. 10. going from the low level to the high level of this factor is also larger in systems with heteroscedastic noise pattern because of the contribution of the second in. term. Eq. (1). A fast examination of the normal. related to the. error. parameters (and. probability plots lack of. fit). (1997) 1-12. magnitude of the elements of the original data matrix they are associated with. The rest of statistically significant effects are less important than those men tioned above and do not deserve an exhaustive expla nation.. shows. significant effects are less pronounced in these responses. The smaller variability of the error para meters in the different ALS runs analyzed is simply explained because of the rotational ambiguity asso ciated with the decomposition of the bilinear matrices. 4.1.2. Comments. when selective information is not available. concentration. that the. products. 4.1.2.1. Horizontal been. (i.e. many profiles,. are. run. unimodality The. of the tested. has been. linear combinations of the actual. ALS method has. unimodality for the experiments with Horizontal unimodality. classical. forcing profiles. negative constraint. between matrices of concentration. whose columns. the assessment. on. constraints. applied. in. level.. the. instead when the constraint level. positive. Small departures of the unimodal. and spectra matrices, whose rows are linear combinations of the actual spectra, can reproduce the. is. similar fit) [1]. The latter original statement reveals the dissimilarities as a more sensi. implementations (tolerance parameter, r=1.1).. tive indicator of changes in the ALS results and therefore, simulated studies for which actual and. the assessment of the horizontal. profiles,. data matrix with. recovered profiles. starting point tion in In. a. spite. that. are. available constitute. error. parameters. are. the. reason. studying why the knowledge. only. results); in the. error. same sense as. ones. real systems of the effects. straint effect is found. compound. affects a. in the. and. to. be. averaged.. This. causes. as. such,. the constraint is also detected in the normal probability plot associated with the lack of fit. The value of the constraint effect for this response is not statistically significant, but it is clearly negative (i.e. the use of the horizontal unimodality produces a decrease in the lack. are. that small residuals. associated with small values of the. original. be. original signal. The constraint effect is not so essential shape recovery of major compounds, since their signals detach clearly from the noise and can be more easily modelled. A remarkable positive influence of. differently the a and the lack of fit. The noisy binary system arisen from the merged decrease in the major signal causes an expected worsening in the lack of fit. The apparently incon sistent improvement of the residual standard deviation (a) is a consequence of the absolute character of this taken. to. was. applied, whereas the horizontal unimodality preserves much better the shape of the. ratio influence. parameter. The values of the residuals,. probability plots related to unimodality con. unimodal constraint is. heteroscedastic pattern yields to a worsening of the results. Variations in the minor/major concentration more. both. statistically significant. already commented in the theory section about the effect of both implementa tions of the unimodal constraint in minor peaks. The noisy signals associated with these minor constituents are often reduced to narrow peaks when the classical. This confirms what. parameters in the final. well, whereas the existence of. in. point associated with analyzed responses. Negative values of the constraint effect in all the plots indicate the improvement of the resolution results with the inclusion of the horizontal unimodality constraint compared with the classical implementation. Such a positive influence is specially noticeable in the shape modelling of the minor compound, where the con. by the different factors on them is necessary. In agreement with the dissimilarity studies, the resolu tion between peaks is the factor having the clearest negative effect (i.e. an increase in the chromatographic the SIN ratio for the minor. 4 shows the normal. allowed. the effect of this constraint in the. caused. resolution reduces the. been. have. straint. The letter A marks the. advisable. influence of any modifica technique onto the final solutions.. be determined when. and this is the. an. Fig.. to assess the. resolution. of this, the. can. constraint. a. data. matrix appear to be better than slightly bigger resi duals associated with much larger numerical values of. of. fit). This. last conclusion is. specially important. to. confirm the goodness of the analyzed constraint in real data sets, where the lack of fit can be determined in. the. original data matrix. This fact proofs the danger of analyzing the residuals without taking into account the. II-72.

(15) A. de Juan et aU Analytica Chimica Acta 18087. Table 5. Comparison of the lack of fit for several application using different constraints. real data sets after ALS. % Minor cornp.. Resolution. 1. (%). a. Fig.. 5. 0.93. 2.39. 0.1. 10. 0.58. 0.86. 0.2. 0.51. 1.72. 0.3. 0.21. 0.55. 0.4. 0.79. 0.80. 0.5. 0.5. 0.23. 0.37. 0.8. 0.5. 0.20. 0.33. 1.0. 0.5. 0.30. 1.91. applied in the ALS profiles and spectra + b Constraints applied in the ALS concentration profiles and spectra + Constraints. concentration. 11. departures. of the unimodal constraint. allowed in both varieties. 5 shows the normal. probability plots related to unimodality constraint. a significant effect of this. None of the. plots. shows. constraint in the resolution results, not even a notice able positive or negative effect. Hence, in its current. method:. in. non-negativity unimodality. method: non-negativity classical unimodality. horizontal. implementation and for the conditions spanned by the designed experiments, the localized unimodaIity does not seem to affect the quality of the resolution results. Additional studies revealed that no improvements were obtained when the constraint was only applied to the minor compound.. in. 4.1.2.3.. Symmetry. The. experiments. with. constraint level have been resolved horizontal contrast. with the dissimilarities between actual and. recovered concentration. (tolerance. the assessment of the localized. 2b. 0.1. a. been. 1-12. parameter, r=1.1).. Lack of fit. Data sets. Small. positive. have. (1997). profiles.. negative. applying. the. tolerance parameter, unimodaIity r= 1.1. When the constraint level is positive, ALS is. run. Table 5 includes the lack of fit related to the resolu. with. a. by using. a. the horizontal. unimodality. and the. symmetry constraint. Small departures from both. tion of several real data sets, already described in a previous section. The lower values obtained for this. constraints. parameter when the unimodal condition is applied according to the horizontal modality confirm the. with the symmetry constraint. None of the responses is affected significantly by the introduction of the sym metry constraint; so that, no clear modifications can be. usefulness of this. inconsistencies table. new. can. implementation.. Some apparent. be observed when the values in the. examined from top to bottom (i.e. comparing systems with the same amount of minor compound, lower lack of fit can be occasionally observed in are. systems with less chromatographic resolution). These. unexpected. reversals often. occur. when. data with low concentration of minor. experimental. compound. of the. comparison. pairs of values compared (i.e. lack of fit applying ALS runs. both. implementations). of the ALS method. allowed (tolerance parameter, r= 1.1). plots connected. noticed in the resolution results. The effect of this constraint. changes its sign according to the response (positive for sigma and negative for the rest of responses); however, this opposed behaviour is not relevant because of the small magnitude of the con straint effect in all the plots examined. observed. 5. Conclusions. between the ALS results. obtained using the classical unimodality and the hor izontal unimodality is not questionable because the. using. are. 6 shows the normal probability. are. used, since the noise patterns and levels in the different samples are not reproducible. Despite this fact, the. validity. Fig.. on. are. the. referred to. same. pairs. of. The proposal of new constraints related to the experimental features of the data sets has been used as a strategy to improve the solutions coming from the. resolution methods. All the constraints related to the. data matrix.. modelling. presented are profiles.. of the concentration. Among the constraints, the horizontal unimodality has 4.1.2.2. Localized unimodality The ALS method has been run forcing horizontal unimodality for the concentration. profiles. in. the. experiments. with. negative constraint level. Localized unimodality has been applied instead when the constraint level is. a more. general application. tems.. An exhaustive. study. data sets showed the. II-73. and the localized unim. and symmetry constraint are more focused on the resolution of hyphenated chromatographic sys. odality. on a. wide span of simulated of the horizontal unim-. goodness.

(16) A. de Juan et. 12. al./Analytica. Chimica Acta 18087 (1997) 1-12. Despite the theoretical validity of the constraints proposed for most of the real chromatographic data sets, the introduction of these constraints in the. lution. procedure. supported. on. is. the. reso. and must be. always optional chemical knowledge. researcher about his data. Evidence of. of. the. weird beha. viours (i.e. fronting phenomena) justify completely the non-application of any of the constraints proposed.. 0.1. 0.05. �o. 25. 30. 35. 55. 4S. 40. References. GO. RelBf11lon time. [I] R. Tauler, A.K. Smilde and B.R. Kowalski, J. Chemometrics,. Fig. 7. Comparison of the recovered normalized concentration profiles related to a binary chromatographic system (Rs=0.2, ratio minor: major compound 1 : 100 and SIN for the minor compound equal to 20) by using different implementations of the unimodal constraint. True concentration profiles (single lines), profiles recovered using horizontal unimodality (thick lines) and profiles recovered using vertical unimodality (dashed lines). The minor compound is the first eluting.. odality. 9 (1995) 31. [2] R. Tauler, Chemom. Intell. Lab. Sys., 30 (1995) 133. [3] R. Tauler and D. Barcelo, TrAC, 12 (1993) 319. [4] W.H. Lawton and E.A. Sylvestre, Technometrics, 13 (1971) 617.. [5] B.G.M. Vandeginste, W. Derks and G. Kateman, Anal. Chim. Acta, 173 (1985) 253. [6] PJ. Gemperline, Anal. Chem., 58 (1986) 2656. [7] J. Craig Hamilton and P.J. Gemperline, J. Chemometrics, 4 (1990) 1. [8] O.M. Kvalheim and Y.Z. Liang, Anal. Chem., 64 (1992) 936. [9] W. Windig and J. Guilment, Anal. Chem., 63 (1991) 1425. [10) R. Tauler, A. Izquierdo-Ridorsa, R. Gargallo and E. Casassas, Chemom. Intell. Lab. Sys., 27 (1995) 163. [II] A. de.Juan, G. Fonrodona, R. Gargallo, A. Izquierdo-Ridorsa, R. Tauler and E. Casassas, J. Inorg. Biochem., 63 (1996) 155. [12] J. Saurina, S. Hernandez-Casson and R. Tauler, Anal. Chern., 67 (1995) 3722. [13] E. Casassas, R. Tauler and I. Marques, Macromolecules, 27 (1994) 1729. [14] S. Lacorte, D. Barcelo and R. Tauler, J. Chrom A, 697 (1995). in both the recovery of the shape of the profiles and the error associated with. concentration. the final solutions. The better. quality of the profiles using this kind of unimodality is shown in Fig. 7, where profiles obtained using this new imple mentation are compared with those obtained using the classical vertical unimodality. The clear decrease of the lack of fit detected in the study performed with. recovered. simulated data. sets. allowed the confirmation of the. usefulness of this constraint with real data.. 345.. Since the horizontal. unimodality is exactly equal in concept to the classical unimodality, this better imple mentation has been applied in the test of the more demanding localized unimodality and symmetry con. Gampp, M. Maeder, C. Meyer and A.D. Zubergiibler, Talanta, 32 (1985) 1133. [16] H. Gampp, M. Maeder, C. Meyer and A.D. Zubergiibler, Anal. Chim. Acta, 193 (1987) 287. [17] A. de Juan, B. van den Bogaert, F. Cuesta Sanchez and D.L. Massart, Chemom. Intell. Lab. Sys., 33 (1996) 133. [15]. straints. Neither the localized. unimodality nor the symmetry improvement on study performed. [18] F. Cuesta Sanchez, J. Toft, B. van den Bogaert and D.L. Massart, Anal. Chem., 68 (1996) 79. [19] J.W. Dolan and L.R. Snyder, Troubleshooting LC Systems: A. constraint offered any kind of visible the resolution results according to the in the in the. comprehensive approach to troubleshooting and separation, Humana Press, US, 1989.. present work. Since the real information. included in these constraints. modelling. can. be. potentially helpful profiles, future more. Model. for these latter constraints. The pos constraints based on different proper. new. ties of the. experimental. data could also be. equipment. Experimenters: An introduction to Design, Data Analysis and Building, John Wiley and Sons, US, 1978. [21] H.R. Keller and D.L. Massart, Anal. Chem., 58 (1993) 471.. effective. implementations tulation of. LC. [20] G.E.P. Box, w.G. Hunter and J.S. Hunter, Statistics for. of concentration. research could be oriented to find. H.. explored.. 11-74.

(17) Comparison Least. between the Trilinear. Squares (ALS). Anna de. Juan':",. Decomposition (TLD). and the. Alternating. methods for the resolution of three-way data sets.. Sarah C.. Rutan", Roma Tauler'. and D. Luc. Massart'.. 1.. Departament de Quimica Analitica. Universitat de Barcelona, Barcelona, Spain. Department of Chemistry. Virginia Commonwealth University, Richmond, US. 3. ChemoAC. Farmaceutisch Instituut. Vrije Universiteit Brussel, Brussel, Belgium. 2.. Abstract. Trilinear of the most on. Decomposition (TLD). Alternating Least Squares (ALS). representative three-way resolution procedures.. the resolution of the. focused. and. on. the. The. methods. are two. former, non-iterative,. is based. generalized eigenvector/eigenvalue problem. optimization. of initial estimates. by using data. and the latter, structure. iterative,. is. and chemical. constraints.. TLD and ALS have been tested common sources. on a. of variation in real response. variety. profiles,. of. three-way simulated. such. distortions caused by noise. The effect of these factors evaluated and. through the analysis of several parameters. on. signal shift, broadening. having. or. shape. the resolution results has been. related to the recovery of both. qualitative. quantitative information and to the quality of the overall data description.. Conclusions inferred from the simulated both methods. on a. real. example. and to. provide. the potential of each method.. *. as. data sets. Corresponding author. II-75. examples help some. to. clarify. general guidelines. the to. performance. of. understand better.

(18) Introduction. Curve resolution methods Extended work has been necessary conditions to. in two theorems matrix is. [1].. usually applied. are. the real response. When these. requirements. subject to ambiguities [2,3]. together [4]. Three-way resolution recovery of the true response. not. methods. single. matrices and the. recently been. stated. single. matrix. are. (the so-called three-way data sets) introduce. Manne. partially are. or. treated. significant improvement. a. and have the additional benefit of providing. profiles. by. not obtained.. are. the resolution of a. always. Analysis.. fulfilled, the decomposition of a bilinear. and the true solutions. when several matrices. overcome. have. profiles. are. Many of the limitations associated with. completely. the resolution of. reported concerning. recover. in the framework of Mixture. in the. quantitative. information.. Two tendencies. non-iterative. within the. prevail. procedures, the solutions of which and the. eigenvalue/eigenvector problem [5,6] optimization of initial. Trilinear are. estimates. advantages furnishes. and drawbacks. unique. concentration it is not. matrices. possible. are. process and. recognised. and spectra for the. to. input. optionally. sharing only. are. and. on. the resolution of. Alternating. Least. one. not. more. species. trilinearity. order in. common. procedure is. always guaranteed,. the iterative. rows. II-76. can. or. [3,7].. common. appended matrices) and meaningless. work with non-trilinear. columns). Input. Despite. optimization. demanding user intervention is required.. the. fast, user-friendly and. always assumed (i.e.. in the different. also allowed.. on. tendencies, respectively. General. and therefore. (either. of. Squares [3] (ALS) methods. data structure and chemical information to prevent assumes. use. generalized. data structure and chemical constraints. for both methods. TLD is. same. a. of iterative methods, focused. of the former and the latter. information in the resolution. solutions. based. solutions. However, trilinear data structure is. profiles. solutions. ALS. are. application. by using suitable. Decomposition [6] (TLD). good representatives. resolution methods: the. family of three-way. these. of external. advantages, unique. slows down the resolution.

(19) Successful chemical but there is. a. applications. of TLD. lack of comparative literature about the. work presents the results of both methods. including or. [8,9] and. common sources. distortions caused. of noise.. performance. Knowledge. of simulated. real. study. are. example [8]. three-way. data sets. signal shifts, broadening. synthetic. data sets allows. a. Conclusions inferred from this. the TLD and ALS results when. interpret. general guidelines. some. reported,. of both methods in terms of recovery of the response. used afterwards to. and. as. of these. profiles, quantitation and quality of the overall data description. theoretical. have been. of these two methods. This. of variation in real chemical data, such. by the presence. sound assessment of the. [10-13]. performance. variety. on a. ALS. are. suggested. to. take. applied. advantage. to a. of the. capabilities of both procedures.. Theory. Both TLD and ALS work with the. three smaller matrices. (X,. Y and Z for TLD and. associated with the evolution of the pure the initial. several. original. three-way data. tensor or stack of. spectra and. a. a. first matrix. third. one. to obtain. in each of the three directions of. compounds profiles. high-performance liquid chromatography with UV. samples would be. (D). C, S and Z for ALS) containing information. Thus, the results coming from. set.. matrices. a. typical. data set formed. by. diode array detection. (HPLC-DAD). second. including pure. having pure chromatographic profiles,. a. with the information about the relative concentration of each. compound in the different samples.. The. shown in Z. three-way. Figure. provide only. data set. 1. The one. picture. profile. per. independent modelling for the. decomposition is. algorithms. are. differently by. shows how. trilinearity is inherently assumed. compound. for all the stacked. concentration. profiles. is introduced in the resolution process. Detailed. their related. carried out. reported. matrices),. in Dr, Dz,. explanations. .... study.. II-77. a. better. in TLD. (X,. as. Y and. whereas ALS allows. unless the trilinear constraint. about TLD. in the literature and will not be. the main steps of each method will be mentioned for. TLD and ALS,. [6,14],. given. ALS. [3,4] and. in this section;. only. understanding of the comparative.

(20) a) a>. �qj. .�. -s:. "'-/_.. 03. I. 1�------------�0�2�. _7. z __. 01 y =. x. -. -. column space. b) 5 C1. 01. 02. C2. =. .. 03. C3. column-wise unfolded matrix. Figure 1. Three-way data decomposition according Alternating Least Squares (ALS) resolution methods. The. (i). a) Trilinear Decomposition (TLD) and b). working procedure of Trilinear Decomposition can be. Application. of. Singular. wise unfolded matrices. Value. (V). to the. sets. formed. by. the. row. tensor. of two on. the. (iv) Determination of. and. Y. scores. scores. representative pseudosamples G1. (U, V, W) basis. X. as. follows:. row-wise, column-wise. space. and the first two vectors of the tube space. (iii) Determination original. Decomposition. summarized. and tube. coming from the original tensor.. (ii) Construction of the basis scores. to. (U),. the column space. (W).. and G2. by projection. of the. sets.. matrices. from. the. eigenvalue/eigenvector problem for matrices G1 and G2• (v) Least-squares estimation of Z, given X and Y.. II-77. resolution. of. the. generalized.

(21) Alternating Least Squares operates following the (i). sequence below:. Determination of the number of compounds of the column-wise unfolded matrix.. (ii) Building of the initial estimates matrix (either concentration profiles (C) Selection of the constraints to be. (iii). input in the iterative resolution. or. spectra (S)).. (e.g.. process. non-. negativity, unimodality, selectivity, trilinearity, ). .... of the initial estimates. (iv) Optimization. by using. calculation and constraint of the C and S matrices. (v) Determination data matrices a. the. matrix C1 and the. same. simultaneous. Both TLD and ALS. as. the ratio between the. area. of the. same. cycle. includes the. (see Figure 1).. of the relative concentration of each. (Z matrix). least squares. alternating. till the convergence criterion is fulfilled. Each iterative. procedure. in. constrained. a. particular compound. area. compound. in the different. of its resolved concentration. in another data matrix. Cj. profile. included in. analysis and taken as reference.. algorithms. have been. implemented. in. a. set of MATLAB routines. [15].. Data sets. Simulated data sets. All the 110. x. 44. three-way. representing. of the concentration. matrix. The 5 channel. a. data sets. are. formed. by. two. appended. typical binary chromatographic system.. profiles. for. compounds. 1 and 2. are. matrices. In all the. (DA. and. examples,. DB) sized the. areas. 25:75 in DA matrix and 70:40 in DB. chromatographic profiles have been simulated as slightly tailed gaussian peaks (o. units). matrices, unless. and. a. =. 0.01% Amax homoscedastic noise level has been added to all data. stated otherwise.. F our basic trilinear systems. are. proposed. to. represent the variety of real situations,. namely: System. 1: resolution between peaks, Rs. System. 2: Rs. System 3:. Rs. =. 0.2, dissimilar spectra.. =. 0.8, similar spectra.. =. II-78. 0.2, similar spectra..

(22) 4: Rs. System. Figure 2 shift effects,. =. 0.8, dissimilar spectra.. shows the spectra used in the different simulations. Variations in real data like. signal broadening or noise. addition. DA in the chromatographic direction according No. than. more. simplicity in the. one cause. in the. spectral. are. generated by modifying DB. to the effect to be studied as. of variation is simulated in each. interpretation. three-way data. or. both DB and. explained. below.. set for the sake of. of the results. None of the simulated data sets includes. changes. direction.. a) 0.08 en. �. 0.06. .s; .. .;::;. e-0.04 0 en. �. 0.02. 10. 30. 20. 40. Wavelengths. b) en. Q) +:::. 0.1. .s; +:::. eo. �. 0.05. <. 0. 0. 10. 20. 30. 40. Wavelengths Figure. 2.. Spectra used. in the simulations of a) systems 1 and 3 and. Simulations related to. generated.. compounds. In all of. on. three-way examples representing different signal shift levels. them, D A matrix is. 1 and 2 in DB matrix. channels, depending. and 4.. signal shift. For each basic system, four are. b) systems 2. the shift. are. not. modified, whereas the peaks related. shifted apart from their initial. intensity.. The. peak shapes. appended matrices.. II-79. are not. position by 1, 2,. 4. or. to. 6. modified in any of the two.

(23) Simulations related. to. changes. in width. For each basic system, four. changes and 2. generated.. are. Thus, .1cr. =. -1 and -2. are. by changing. note that .10' is. introduced in any of the. Simulations related to. two. signal distortions. width. signal. invariant, whereas width signals of compounds the. cr. value used in the. reproduce situations of signal narrowing and .1cr. signal broadening (please. positions. three-way examples representing different. D A matrix remains. modified in matrix DB. are. signal. expressed. in channel. units).. generation =. +1. No. of the. 1. peaks.. and +2 account for. changes in. the. peak. appended data matrices.. caused by noise.. The effect of noise pattern and noise level is taken into account in the different. simulations. Peak. shapes. and. positions. levels. (0.01 % Amax and 0.5% Anax). signal. and heteroscedastic. basic system,. as. oc. in matrices DA and DB. and three noise patterns. "signal). are. are. not. modified. Two noise. (homoscedastic,. combined in the six simulations. heteroscedastic. oc. generated. for each. compounds. in all the. shown below:. hoi. �. noise level: 0.01 % Amax. Noise pattern: homoscedastic.. hsl. �. noise level: 0.01 % Amax. Noise pattern: heteroscedastic. hql. �. noise level: 0.01 % Amax. Noise pattern: heteroscedastic. hoh. �. noise level: 0.5% Amax. Noise pattern: homoscedastic.. hsh. �. noise level: 0.5% Amax. Noise pattern: heteroscedastic. hqh. �. noise level: 0.5% Amax. Noise pattern: heteroscedastic. signal.. oc. oc. "signal.. signal.. oc. oc. "signal.. Results and discussion. Comments. on. the simulated data. Both TLD and ALS have been. three-way. data sets. Trilinear. run. assuming the existence. Decomposition. has been. directions and the results presented in this work best data fit. The. negativity. profiles. as. applied. in the different. those related to the direction. are. unfolding giving. Alternating Least Squares method has always been used introducing. constraint in both. constraint in the. of two. chromatographic. chromatographic profiles.. and. All the. initial estimates. The initial concentration. spectral. examples. profiles. the. the. non. directions and the unimodal have. adopted chromatographic. for systems 1,3 and 4 have been. built applying the needle search methodology [16]. The spectral selectivity detected by using 11-80.

(24) local rank detection. methods, such. as. FSMW EFA. [17], has. -. been taken into account to build. the initial estimates of system 2. In this case, column vectors. spectral. selective. zones. of the. chromatographic profiles.. The. data matrices have been selected to be. original. difference between the results. coming from. and ALS will be used hereafter to refer. constraint in the iterative. The. performance. methods have been. quality and. these two different. of Trilinear. as. is assessed. Decomposition. or. initial. twice for each. run. doing. so.. To make. the acronyms ALSf. absence of the trilinear. and the. Alternating. Least. Squares. the observation of several parameters related to the. qualitative information,. i.e.. chromatographic. at the dissimilarities between the true response. by looking. the response. are. profile, respectively,. calculated. the presence. as. profiles obtained by using the resolution methods.. where Xcalc and Xtrue. The. options,. input. in. optimization procedure.. of the final results. The recovery of the. and the. has been. data set and second, without. respectively to. compared through. spectral profiles,. profiles. Squares method. Least. Alternating. example: first, forcing trilinear structure in the a. (chromatograms) placed. quality. profile. calculated with ALS. or. TLD and the true response. and r is the correlation coefficient between these two factors.. of the overall data. description is. related to the percent of lack of fit,. follows:. LrJk. % lack of fit. =. 100. i,j,k. x. Ld�k. i.j.k. where rijk is the residual of the. between the element in the. ijkth. original. results of the resolution method and The. quantitation ability. element of the. a. data set, found. tensor and the same element. dijk is. the. ijkth. has been evaluated with. the true concentration ratios of. three-way. compound II-81. a. element of the. as. the difference. reproduced by using. original three-way. relative parameter of. comparison. the. data set. between. in the D A and DB matrices and the same.

(25) concentration ratios obtained from the resolution results. The mathematical last parameter is shown below. (the ith subscript indicates the. different. expression. compounds in. of this. the data. set).. % rsdz. 100. =. x. The conclusions inferred from the. Alternating. Least. Squares. methods. way data sets is associated with. signal position). As. trilinearity The best. if it is not. general rule,. a. is broken,. are no. owing. as. ambiguities. (please. correct. to its. in the. sets. coming. of the. binary systems,. 2). 3.. or. sets. coming from system 3),. in both of them. still. performs better in most of the. this. tendency. departures. moderate. nor. profiles. the conditions. matrix. are. by ALS [3]. be found when the. are. associated with. shown in. Figures. 4 and 5.. increase in the. are. overlapping. In the latter case, there is. a. on. ALS and. by. the loss of. trilinearity. on. as. Only. profiles.. TLD and ALSf. This distortion in the II-82. some. between. critical. profiles shape. reversals in. small. or. when. compounds (i.e.. zone. are. for both. in system 1, ALS. extremely. (+2. >. .10'. > +. 1). chromatographic. comparable.. behaviour is specially noticed in the recovered response profiles, which combinations of the pure. theorems for the. selectivity. where the negative effects caused by the increase of correlation in the. direction. coming. and the minimal. required by Manne's. departures of trilinearity. an. positive signal broadening).. as. sets. due to the noise added to all data. chromatographic selectivity exist,. cases,. (data. in. Indeed, when selectivity is present, there. augmented. true and the calculated. compounds). When neither spectral. these. from system. (data. resolution of data matrices coincide with the presence of. can. in width and. specially clear when the analyzed three-way. direction. example of Figure. decomposition. note that for. (i.e. changes. greater flexibility in the modelling of the response profiles.. in these situations is. shown in the. dissimilarities between the sets. loss of the trilinear data structure. selectivity in the chromatographic. 4),. if the variation introduced in the three. ALS works much better than TLD and ALSf in all aspects when. spectral direction (data. from system. change completely. and. (i.e. signal distortions produced by noise).. performance of ALS. data sets have the. or. a. comparison of the Trilinear Decomposition. are. Such. actually. a. linear. also influences the.

(26) in the Trilinear. quantitation ability On the. procedures. modify. the. variations. one. also associated with. parameter being used to the. Least. Alternating. Squares. hand, variations in the form of the profiles (i.e. in matrices X and Y). least-squares calculation. are. and the. Decomposition. of the Z matrix in TLD and. changes. in the. peak. area. of the different. compounds, this. in the ALS and ALS£ In contrast. quantitative information. to obtain the. the other hand, these. on. parameters above, the lack of fit is only severely affected by losses of trilinearity. As. TLD and ALSf. reproduce. only. the data. use. the trilinear information present in the first two the. matrix,. compounds. to. non-trilinear contribution excluded from the model. is. responsible for the increase in the lack of fit. Owing to the greater flexibility in the modelling of the. profiles,. ALS. always inputs. more. information in the data. combinations of the real solutions when there is therefore the data fit is. behaviour and the. always. possibility. no. selective information in the system, and. better than for methods with trilinear structure to. when the. same. Figures. always. have. to. signal. a. significantly greater. 4 and 5 show. shift. clearly. the factors that affect. or. in the. mainly. of width. changes. influenced. provides. an. data sets since. lack of fit for TLD than for ALS. signal,. in the ALS results. compounds (i.e. big signal. shifts. are. or. whatever their. associated with. signal narrowing). increases in the correlation between profiles. negatively. by trilinearity. variations in the correlation between the response. improvements. three-way. This. number of compounds is used in the resolution process.. methods. Whereas TLD and ALSf are in the. imposed.. calculate this parameter for real data sets. additional method to check the existence of trilinear structure in real non-trilinear systems will. linear. reproduction, though using. a. losses. sign),. profiles. and the. (i.e. by increases. ALS is. more. sensible. of the data set.. decrease in the. opposite. (i.e. signal broadening).. 1I-83. the different resolution. overlapping effect is. Thus,. between. promoted by.

(27) Chromatographic 0.90. dissimilarities. .-----------------------------------------------------�. 0.80 III. 0.70. �. 0.60. 19. 0.50. ·e. 0.40. .�. 0.30. _. _-. 2i 0.20 0.10 0.00 o. 2. 3. -TLD(1A) -TLD(2A) �TLD(1B) --=--TLD(2B) 0 c ALS(1 B) ALS(2B) - ALS(1 A)I ALS(2A)1 -. -. -. -. -. -. Spectral. 0.025. -. 6. 5. 4 •. --c. -. -. ALS(1A) ALS(1 B)I. -. •. -. .....0. -. ALS(2A) ALS(2B)1. dissimilarities. -r-----------------------------------------------�. 0.020. .; =E 0.015 .!!!. :i. 0.010. 2i 0.005. o. 2. 3. 4. 5. 6. Shift. I-TLD. (1). -TLD. (2). -. •. -. ALS(1). -. •. -. ALS. (2). --. ALS(1)f. Error in. % lack of fit. _-. ALS. (2)fl. quantitation. 60. 30. 50 25. ; -. ,,-. ,,-. 20. 0 .II:. � rft.. �. 15. N. "C. /". I!! � 0. ,,",. 10. 40. 30 20 10. ", �. 5. ",. 0. -i. 0. 2. 0. 0. I-TLD. -. -. ALS. -. -. 6. Shift. Shift -. 4. 6. 4. 2. I-TLD. ALSfl. -. -. -. ALS. -. -. ALSf. I. 3. Effect of the signal shift on the resolution of the three-way data sets generated by modifying the basic system 3. Solid lines: TLD. Pointed lines: ALS. Dashed lines: ALSf. Compound one: squares. Compound two: circles. Compounds in DA matrix: filled symbols. Compounds in DB matrix: empty symbols. Legends: e.g., TLD(lA), the number and the letter between parentheses when existing indicate. Figure. the. compound (1. or. 2). and the data matrix. (DA. or. determined.. II-84. DB) for. which. a. certain parameter has been.

(28) Chromatographic dissimilarities 0.90. �------------------------=... 0.80. :n. 0.70. :e. 0.60. ..!!! 0.50. ·e. 0.40. .�. 0.30. is 0.20 0.10 0.00 2. o. 6. 5. 4. 3 hif. -TLD(1A) -TLD(2A) --O-TLD(18) 0 c ALS(18) ALS(28) _. ALS(1 A)f •. •. •. •. ---TLD(28). •••. _.. ......[].. ALS(2A)f. ALS(1A) ALS(18)f. ••• ......0.. ALS(2A) ALS(28)f. Spectral dissimilarities 0.045. -.-----------------------------,. 0.040 UI Q). 0.035 -_-. 0.030 E ... ..!!! 0.025. .--_. ------. :§ 0.020 :g. 0.015. is 0.010 -. 0.005. •••. _._ •••••• •••• _._._._ ••• O.OOO�----+---��--�-·�·�·�·�·�·�� _. _. o. 2. 3. 4. 6. 5. Shift. I-TLD. (1) -TLD (2). •••. ALS(1). •••. ALS (2). _.. ALS(1)f. Error in. % lack of fit. _.. ALS. (2)fl. quantitation. 60. 25. 50 ,. 20. ",. -. ;: -. 0. "C UI. ",. ..IC: u. ..!!!. N. ",. 15. .... cfl.. ",. 10. ,. 40. 30 20. ",. � 0. 5. ,. ,... 0. 2. 4. 0. 6. 2. I-TLD. •••. ALS. 6. Shift -. •. I-TLD. ALSfl. 4. Effect of the. the basic system 1.. ,.. .. 4. Shift. Figure. -. 0. �. 0. .. 10. ,... -. -. •. ALS. -. •. ALSfl. signal shift on the resolution of the three-way data sets generated by modifying Legends, line and symbol patterns are the same as in Figure 3.. 11-85.

(29) Chromato. raphic. dissimilarities. 0.35 0.30. 0.25. -2. -1.5. -1. -0.5. o. 2. 1.5. 0.5. Width. -TLD(1A) -TLD(2A) �TLD(1B) -o-TLD(2B) c ALS(1B) -0- ALS(2B) _. ALS(1A)f _. ALS(2A)f •. -. Spectral. ••• -0.. ALS(1A) ••• ALS(2A) ALS(1B)f......()· ALS(2B)f. dissimilarities. 0.040. .................. 0.035. 0.030 0.025. .. -2. .. -. ....... ....... -. o. -1. 2. Width. I-TLD. (1). -TLD. (2). -. •. -. ALS(1). •••. ALS. 6. \.. -. 0 ..x: (). \.. .!!!. -2. -2. 2. 0. -1. -1. Width. I-TLD. •. -. •. quantitation. /. \.. "#.. (2)fl. ALS. /. 3. .... _-. /. 4. .... ALS(1)f. /. 5. :;:. _.. Error in. % lack of fit. -. (2). ALS. -. •. o. 1. 2. Width. ALSfl. I-TLD. •••. ALS. -. •. ALSf. 1. Figure 5. Effect of width signal changes on the resolution of the three-way data sets generated by modifying the basic system 1. Legends, line and symbol patterns are the same as in Figure 3.. II-86.

(30) In all the comments referred to non-trilinear systems, TLD and ALSf have been put in. the. group when. same. compared. with ALS. However, noticeable differences between them. derive from the distinct way to include the trilinear condition in their resolution processes. TLD works. best fit the. on. the whole data set to obtain the trilinear combination of response. original. the response. data. This strategy. concentration matrix to get the best In contrast to TLD, ALSf is. than. the. on. global. observed in. quantitation,. fit. The. more. common. one. on. each column of the. and constrained. profile. for each. the. quality. above. are. in agreement with the. whereas TLD shows lower lack of fit. The. compound [18].. bigger lack of fit in the. profiles. tendencies. general. smaller dissimilarities and. provides. for. augmented. of the individual response. on. focused. explanations. by. one. 4 and 5, where ALSf. figures. that. gives priority to the data fit over the recovered shape. ALSf operates. profiles.. profiles. in. errors. ALSf models is. related to the removal of both the non-trilinear contributions and the trilinear information not. in agreement with the constraints and. negative parts. provided that it keeps. applied. are. optimization procedure (e.g. non-unimodal. in the. larger distortions in. the response. due to the acceptance of any kind of. profile shape,. of the concentration. found when TLD is. profiles. imposed. profile, ). .... The. the trilinear condition.. An unavoidable element in the real data sets is the presence of noise. This factor. itself does not induce. variation. sources. trilinearity. losses unless it appears. mentioned above. The simulated. the resolution methods. simultaneously. examples. are. used to. against the only introduction of different noise. with. of the data. some. analyze. by. the. stability of. levels and patterns in the. data sets.. All the tested level than more. by. methods. variations in the noise pattern.. satisfactory results. The best the. three-way resolution. than ALS for the. performance. 3-way data set,. as. Figure. affected. Generally speaking,. by increases. in the noise. both TLD and ALSf provide. original noisy data sets.. of TLD and ALSf is. shown in. are more. more. 6. In this case,. evident when there is. owing. no. selectivity. to the intrinsic trilinear structure. of the data sets and the forced trilinear character of the resolution results, both methods able to reach the real with the. unique. decomposition. solutions. In contrast, ALS suffers from the. of the bilinear matrices. lacking. II-87. in. are. ambiguity associated. selective information and the. profiles.