Comparison of Two Objective Functions for Optimization of Simultaneous Multi-element Determinations in Inductively Coupled Plasma Spectrometry CHRISTINE SARTOROS AND ERIC D. SALIN* Department of Chemistry, McGill University,Montreal, Quebec, Canada H3A 2K6 Two objective functions for multi-element optimization in ICP- Leary et al.6 developed and tested several objective functions AES were compared using signal-to-background ratios as a based on SBRs, the best one being represented by the sum of figure of merit.Complete three-dimensional response surfaces the reciprocals of the SBRs for the elements studied: were generated for a number of elements (Ca, Cu, Al, Na, Ni, Mn and Ba) and two artificial ‘elements’ to evaluate the F= n .n i=1 (S/B)i-1 performance of both objective functions in locating the optimum compromise instrumental operating conditions in multi-element determinations. In the determination of the best where n is the number of elements studied for the optimization compromise instrument operating conditions for most and (S/B)i are the SBRs of each of the ith elements. This combinations of the elements used, both objective functions equation generates a value for each set of instrumental performed equally well; however, one occasionally performed operating conditions using the SBRs of all the elements.Ebdon significantly better than the other. and Carpenter8 used a modified version of Leary’s objective function in their study.Kalivas9 also used Leary’s objective Keywords: Multi-element optimization; inductively coupled function in the optimization of operating conditions for mini- plasma atomic emission spectrometry ; objective functions mal interferences. Instead of SBRs, Kalivas used selectivity, sensitivity and accuracy, as derived by Lorber,13 as response Optimization of instrumental operating conditions may improve functions. Moore et al.10 also used Leary’s objective function analytical accuracy and precision.The instrumental operating with SBRs in addition to ionization interference as the figure conditions of an ICP that may be optimized are rf power, flow of merits. Belchamber et al.11 developed an objective function rates of gases (the outer or coolant, intermediate or plasma and based on a measure of the magnitude of the matrix effects injector gas flow rates), observation region in the plasma and such as to minimize or remove these matrix effects. solution pump rate to the nebulizer.1 The primary optimization We tested another approach towards satisfying the two techniques that have been used with ICPs are simplex2–11 and the conditions required for obtaining optimum compromise Davidon–Fletcher–Powell12 algorithm.Traditionally the response operating conditions: (1) obtaining the maximum compromise functions used have been signal-to-background ratios (SBRs), SBRs for all elements and (2) emphasizing the maximization signal-to-noise ratios (SNRs), precision and accuracy.Thomas of the SBR of the elements close to their detection limits. This and Collins12 also used detection limits as the response function. objective function is called the combined ratio method (CRM) Any of these response functions maybe directlyusedfor determinand is given by: ing optimum instrument operating conditions for single-element analysis.2–5 Signal-to-background ratios are easily obtained and require the fewest measurements of the response functions listed above.The SBR is also a good figure of merit since it can be CRM= .n i=1 (S/B)i .k j=1 Rj correlated to detection limits. Therefore, SBRs are used throughout this work as calculated using the following equation: where n is the number of elements, k is (n-1)+(n-2)+...+1, S/B=total signal-background background (S/B)i are the SBRs of each of the ith elements and Rj is the ratio of the SBRs of two given elements ( jth combination) where where S/B is the SBR.the maximum SBR of the two is in the numerator such that The difficulty arises in simultaneous multi-element analysis Rj1. For example, given the following SBRs of three elements because optimization techniques generally require a single Zn, Na and K, the values for R1, R2 and R3, would be: value representing each set of operating conditions but, using (S/B)Zn=1; (S/B)Na=2; (S/B)K=10 any of the response functions mentioned above, multiple values (one for each element) are obtained for each set of conditions.R1=(S/B)K/(S/B)Zn=10/1=10 Galley et al.,5 in their automated simplex optimization of R2=(S/B)K/(S/B)Na=10/2=5 multi-element solutions, used the following choices for optimization criteria: (1) the maximization of the net signal or signal- R3=(S/B)Na/(S/B)Zn=2/1=2 to-background noise ratio, (2) the minimization of the relative and the CRM would be calculated as: standard deviation of the background and (3) the maximization of the ratio of atomic or ionic lines.An objective function, CRM=(S/B)Zn+(S/B)Na+(S/B)K R1+R2+R3 which by definition would result in a single value for a set of operating conditions, is needed. It should comprise the SBRs of all the elements studied with emphasis on the elements =1+2+10 10+5+2=13 17=0.76 closer to the detection limits. Journal of Analytical Atomic Spectrometry, January 1997, Vol. 12 (13–19) 13The CRM performs a weighted average on the sum of the SBRs and maximizes the individual SBRs while minimizing the difference among these ratios (i.e., minimizing .Rj ). In this work, we performed a comparison of Leary’s objective function and the CRM by examining the response surfaces generated by each function and evaluating the performance of each function in determining the optimum instrument operating conditions. EXPERIMENTAL The instrument used was a Thermo Jarrell Ash (Franklin, MA, USA) Model 25 sequential scanning spectrometer.The instrument functionsare all automated and controlled by an independent computer via an RS-232 port. The solution pump rate to the nebulizer was 0.9 ml min-1. A signal integration time of Fig. 1 Surface of SBRs of Al. 1.0 s was used. The background for each line was selected at 0.05 nm on both sides of each spectral line peak. An average of the background was used for all readings. All operating conditions were held constant except for observation height and rf power.The observation height was varied from 3 to 24 mm above the top of the load coil (ATOLC) in steps of 3.0 mm and the rf power was varied from 750 to 1550 W in steps of 200 W. All these combinations generated a response surface which characterized the parameter space. While any combination of rf power, observation region in the plasma, flow rates of gases and solution pump rate to the nebulizer could be optimized, we chose to vary only two of these parameters for simplicity of the graphical representation of the response surfaces.A stock standard solution was prepared from Fisher (Pittsburgh, PA) certified sodium, calcium, copper, aluminum, nickel, manganese and barium 1000 ppm standard solutions Fig. 2 Surface of SBRs of Ca. and the concentrations and spectral lines of these seven elements studied are listed in Table 1 along with their ionization potentials. These elements have both hard and soft lines. The SBRs were determined for all seven elements at each set of instrumental operating conditions.Based on these SBRs, a response surface was generated for each element. Using these seven response surfaces for the seven elements, two other surfaces were generated, one using Leary’s objective function and the other using the CRM. The optimum operating conditions were determined from these latter surfaces and used in the comparison of the two objective functions. In addition, theoretical models were used to compare the two objective functions further.RESULTS AND DISCUSSION Three-dimensional response surfaces were obtained for each Fig. 3 Surface of SBRs of Cu. element by plotting the SBRs of each element against the observation heights and rf powers (Figs. 2–8). The SBRs of each element at each set of operating conditions are listed in Tables 2–8. The optimum instrumental operating conditions for each element are listed in Table 9. Low power seems to be the best choice for these elements at their present concentrations.This is expected since an increase in power increases Table 1 Elements used Concentration Wavelength/ Ionization potential/ Element (ppm) nm eV14 Al I 10 309.28 5.99 Ca II 10 317.93 11.87 Cu I 10 324.75 7.73 Ba II 2 455.40 10.00 Ni I 10 232.00 7.64 Mn II 2 257.61 15.64 Na I 10 589.59 5.14 Fig. 4 Surface of SBRs of Ba. 14 Journal of Analytical Atomic Spectrometry, January 1997, Vol. 12Table 2 SBRs of aluminum Rf power/W Observation height/mm 750 950 1150 1350 1550 3 0.66 0.75 0.57 0.38 0.34 6 1.68 1.03 0.81 0.54 0.40 9 4.10 2.01 1.48 0.78 0.56 12 8.18 4.14 2.77 1.57 1.07 15 13.02 7.09 4.67 2.69 1.97 18 11.96 7.99 7.42 3.86 2.84 21 9.83 10.57 9.89 6.00 3.26 24 10.09 10.50 10.30 8.47 7.05 Table 3 SBRs of calcium Fig. 5 Surface of SBRs of Ni. Rf power/W Observation height/mm 750 950 1150 1350 1550 3 0.59 0.75 0.75 0.75 0.61 6 1.55 1.76 1.68 1.36 1.12 9 4.01 3.59 2.99 2.15 1.70 12 7.66 5.89 4.92 3.40 2.66 15 7.81 7.44 6.78 4.80 3.93 18 5.30 6.92 6.55 6.31 5.21 21 2.54 4.61 5.73 5.62 5.65 24 1.19 1.81 3.08 3.85 4.78 Table 4 SBRs of copper Rf power/W Observation height/mm 750 950 1150 1350 1550 3 4.05 2.46 1.78 1.18 0.92 6 6.92 3.87 2.99 1.79 1.32 Fig. 6 Surface of SBRs of Mn. 9 14.02 7.65 4.93 2.73 1.76 12 25.38 13.73 9.41 5.13 3.32 15 33.97 21.41 16.19 8.72 6.06 18 35.04 26.98 22.53 14.28 9.80 21 33.95 32.95 30.77 23.27 16.28 24 32.83 35.57 38.44 31.99 28.40 Table 5 SBRs of barium Rf power/W Observation height/mm 750 950 1150 1350 1550 3 0.39 0.29 0.27 0.16 0.13 6 0.85 0.55 0.42 0.26 0.19 9 1.50 0.73 0.52 0.24 0.15 12 3.94 1.61 1.09 0.49 0.32 15 7.65 3.75 2.30 1.05 0.77 18 10.32 5.87 4.20 2.18 1.38 Fig. 7 Surface of SBRs of Na. 21 9.16 8.64 6.64 3.72 2.57 24 6.35 8.23 7.70 5.68 4.73 Table 6 SBRs of nickel Rf power/W Observation height/mm 750 950 1150 1350 1550 3 1.28 1.15 0.73 0.67 0.47 6 2.38 2.23 1.55 1.20 0.93 9 6.64 5.51 4.08 2.11 1.54 12 12.61 9.55 6.77 3.84 2.22 15 14.48 12.05 9.03 6.56 4.12 18 8.45 10.55 10.05 6.14 3.83 21 3.23 5.65 5.83 4.99 2.21 24 2.50 2.25 2.35 2.99 3.78 Fig. 8 Surface obtained using the CRM method. Journal of Analytical Atomic Spectrometry, January 1997, Vol. 12 15Table 7 SBRs of manganese Table 10 Best compromise conditions for all the combinations of the elements Rf power/W Observation CRM Leary height/mm 750 950 1150 1350 1550 Power/ Height/ Power/ Height/ 3 5.47 6.17 5.60 4.60 3.26 Line Combination W mm W mm 6 13.38 12.19 9.95 8.34 6.56 9 26.31 24.57 19.61 12.66 8.70 1 Al–Ca 750 12 750 15 2 Al–Cu 750 15 750 15 12 41.05 32.97 28.80 19.58 14.98 15 36.87 39.03 36.83 26.53 20.32 3 Al–Ba 750 18 750 18 4 Al–Ni 750 15 750 15 18 27.46 35.66 36.65 32.85 26.49 21 14.08 22.28 22.34 22.73 18.74 5 Al–Mn 1150 24 750 15 6 Al–Na 750 15 750 18 24 4.44 6.63 10.57 9.51 11.21 7 Ca–Cu 950 15 750 15 8 Ca–Ba 750 15 750 15 9 Ca–Ni 750 12 750 15 Table 8 SBRs of sodium 10 Ca–Mn 750 15 750 12 11 Ca–Na 950 15 750 15 Rf power/W 12 Cu–Ba 750 18 750 18 Observation 13 Cu–Ni 750 15 750 15 height/mm 750 950 1150 1350 1550 14 Cu–Mn 750 15 750 15 3 3.20 1.69 1.27 0.75 0.58 15 Cu–Na 950 24 950 24 6 3.90 2.05 1.37 0.81 0.64 16 Ba–Ni 750 18 750 15 9 5.10 2.45 1.60 0.94 0.63 17 Ba–Mn 750 21 750 18 12 9.50 4.54 2.94 1.60 1.21 18 Ba–Na 750 18 750 18 15 14.24 7.59 5.31 3.08 2.59 19 Ni–Mn 750 15 750 15 18 15.98 12.37 8.65 5.32 4.09 20 Ni–Na 750 15 750 15 21 14.87 14.26 11.97 8.82 7.25 21 Mn–Na 750 21 750 15 24 16.17 17.73 15.81 11.75 10.84 22 Al–Ca–Cu 750 15 750 15 23 Al–Ca–Ba 750 15 750 15 24 Al–Ca–Ni 750 15 750 15 25 Al–Ca–Mn 750 15 750 15 Table 9 Optimum operating conditions for each element 26 Al–Ca–Na 750 15 750 15 27 Al–Cu–Ba 750 18 750 15 Observation 28 Al–Cu–Ni 750 15 750 15 Element Power/W height/mm 29 Al–Cu–Mn 750 15 750 15 Al 750 15 30 Al–Cu–Na 750 15 750 18 Ca 750 15 31 Al–Ba–Ni 750 18 750 15 Cu 1150 24 32 Al–Ba–Mn 750 21 750 18 Ba 750 18 33 Al–Ba–Na 750 18 750 18 Ni 750 15 34 Al–Ni–Mn 750 15 750 15 Mn 750 12 35 Al–Ni–Na 750 15 750 15 Na 950 24 36 Al–Mn–Na 750 18 750 15 37 Ca–Cu–Ba 750 15 750 15 38 Ca–Cu–Ni 750 15 750 15 39 Ca–Cu–Mn 750 15 750 15 the background more than the signal with a subsequent 40 Ca–Cu–Na 750 15 750 15 decrease in SBR.1 For example, looking at the combination of 41 Ca–Ba–Ni 750 15 750 15 manganese and sodium, the optimum operating conditions for 42 Ca–Ba–Mn 750 15 750 15 manganese are 750W rf power and 12 mm ATOLC, whereas 43 Ca–Ba–Na 750 15 750 15 those for sodium are 950 W rf power and 21 mm ATOLC. 44 Ca–Ni–Mn 750 15 750 15 45 Ca–Ni–Na 750 15 750 15 However, using either of these operating conditions in the 46 Ca–Mn–Na 750 15 750 15 simultaneous determination of these two elements would pro- 47 Cu–Ba–Ni 750 15 750 15 duce poor results for one of them. 48 Cu–Ba–Mn 750 18 750 18 Using the data in Tables 2–8, the optimum compromise 49 Cu–Ba–Na 750 18 750 18 instrumental operating conditions were determined for all 50 Cu–Ni–Mn 750 15 750 15 combinations (Table 10) of the seven elements studied by 51 Cu–Ni–Na 750 15 750 15 52 Cu–Mn–Na 750 18 750 15 applying Leary’s objective function and the CRM.The appli- 53 Ba–Ni–Mn 750 18 750 15 cation of these two objective functions to any combination of 54 Ba–Ni–Na 750 15 750 15 the elements for all sets of operating conditions produces two 55 Ba–Mn–Na 750 18 750 18 response surfaces (one for Leary’s objective function and the 56 Ni–Mn–Na 750 15 750 15 other for the CRM).For example, in the optimization of 57 Al–Ca–Cu–Ba 750 15 750 15 operating conditions over all seven elements, the resulting 58 Al–Ca–Cu–Ni 750 15 750 15 59 Al–Ca–Cu–Mn 750 15 750 15 surfaces are depicted in Figs. 8 and 9. The maximum point on 60 Al–Ca–Cu–Na 750 15 750 15 a surface indicates the best compromise operating conditions 61 Al–Ca–Ba–Ni 750 15 750 15 given by each method. Considering the combination of these 62 Al–Ca–Ba–Mn 750 15 750 15 seven elements, the resulting surfaces are similar to each other 63 Al–Ca–Ba–Na 750 15 750 15 and give the same set of operating conditions as the optimum 64 Al–Ca–Ni–Mn 750 15 750 15 compromise. 65 Al–Ca–Ni–Na 750 15 750 15 66 Al–Ca–Mn–Na 750 15 750 15 The optimum compromise settings were determined for each 67 Al–Cu–Ba–Ni 750 15 750 15 combination of the elements using both objective functions 68 Al–Cu–Ba–Mn 750 18 750 18 and are listed in Table 10.In many cases, both objective 69 Al–Cu–Ba–Na 750 18 750 18 functions give the same set of instrumental operating conditions 70 Al–Cu–Ni–Mn 750 15 750 15 as the best compromise.In the cases where they give different 71 Al–Cu–Ni–Na 750 15 750 15 operating conditions (Table 11), the CRM puts a greater 72 Al–Cu–Mn–Na 750 15 750 15 73 Al–Ba–Ni–Mn 750 18 750 15 emphasis on decreasing the difference between the SBRs (i.e., 16 Journal of Analytical Atomic Spectrometry, January 1997, Vol. 12Table 10 (continued) Table 11 SBR of elements for the best compromise conditions obtained using the two methods CRM Leary Line Method SBR of elements Power/ Height/ Power/ Height/ 1 Al Ca Line Combination W mm W mm CRM 8.18 7.66 Leary 13.02 7.81 74 Al–Ba–Ni–Na 750 15 750 15 75 Al–Ba–Mn–Na 750 18 750 18 5 Al Mn CRM 10.30 10.57 76 Al–Ni–Mn–Na 750 15 750 15 77 Ca–Cu–Ba–Ni 750 15 750 15 Leary 13.02 36.87 6 Al Na 78 Ca–Cu–Ba–Mn 750 15 750 15 79 Ca–Cu–Ba–Na 750 15 750 15 CRM 13.02 14.24 Leary 11.96 15.98 80 Ca–Cu–Ni–Mn 750 15 750 15 81 Ca–Cu–Ni–Na 750 15 750 15 7 Ca Cu CRM 7.44 21.41 82 Ca–Cu–Mn–Na 750 15 750 15 83 Ca–Ba–Ni–Mn 750 15 750 15 Leary 7.81 33.97 9 Ca Ni 84 Ca–Ba–Ni–Na 750 15 750 15 85 Ca–Ba–Mn–Na 750 15 750 15 CRM 7.66 12.61 Leary 7.81 14.48 86 Ca–Ni–Mn–Na 750 15 750 15 87 Cu–Ba–Ni–Mn 750 15 750 15 10 Ca Mn CRM 7.81 36.87 88 Cu–Ba–Ni–Na 750 15 750 15 89 Cu–Ba–Mn–Na 750 18 750 18 Leary 7.66 41.05 11 Ca Na 90 Cu–Ni–Mn–Na 750 15 750 15 91 Ba–Ni–Mn–Na 750 18 750 15 CRM 7.44 7.59 Leary 7.81 14.24 92 Al–Ca–Cu–Ba–Ni 750 15 750 15 93 Al–Ca–Cu–Ba–Mn 750 15 750 15 16 Ba Ni CRM 10.32 8.45 94 Al–Ca–Cu–Ba–Na 750 15 750 15 95 Al–Ca–Cu–Ni–Mn 750 15 750 15 Leary 7.65 14.48 17 Ba Mn 96 Al–Ca–Cu–Ni–Na 750 15 750 15 97 Al–Ca–Cu–Mn–Na 750 15 750 15 CRM 9.16 14.08 Leary 10.32 27.46 98 Al–Ca–Ba–Ni–Mn 750 15 750 15 99 Al–Ca–Ba–Ni–Na 750 15 750 15 21 Mn Na CRM 14.08 14.87 100 Al–Ca–Ba–Mn–Na 750 15 750 15 101 Al–Ca–Ni–Mn–Na 750 15 750 15 Leary 36.87 14.24 27 Al Cu Ba 102 Al–Cu–Ba–Ni–Mn 750 15 750 15 103 Al–Cu–Ba–Ni–Na 750 15 750 15 CRM 11.96 35.04 10.32 Leary 13.02 33.97 7.65 104 Al–Cu–Ba–Mn–Na 750 18 750 18 105 Al–Cu–Ni–Mn–Na 750 15 750 15 30 Al Cu Na CRM 13.02 33.97 14.24 106 Al–Ba–Ni–Mn–Na 750 15 750 15 107 Ca–Cu–Ba–Ni–Mn 750 15 750 15 Leary 11.96 35.04 15.98 31 Al Ba Ni 108 Ca–Cu–Ba–Ni–Na 750 15 750 15 109 Ca–Cu–Ba–Mn–Na 750 15 750 15 CRM 11.96 10.32 8.45 Leary 13.02 7.65 14.48 110 Ca–Cu–Ni–Mn–Na 750 15 750 15 111 Ca–Ba–Ni–Mn–Na 750 15 750 15 32 Al Ba Mn CRM 9.83 9.16 14.08 112 Cu–Ba–Ni–Mn–Na 750 15 750 15 113 Al–Ca–Cu–Ba–Ni–Mn 750 15 750 15 Leary 11.96 10.32 27.46 36 Al Mn Na 114 Al–Ca–Cu–Ba–Ni–Na 750 15 750 15 115 Al–Ca–Cu–Ba–Mn–Na 750 15 750 15 CRM 11.96 27.46 15.98 Leary 13.02 36.87 14.24 116 Al–Ca–Cu–Ni–Mn–Na 750 15 750 15 117 Al–Ca–Ba–Ni–Mn–Na 750 15 750 15 52 Cu Mn Na CRM 35.04 27.46 15.98 118 Al–Cu–Ba–Ni–Mn–Na 750 15 750 15 119 Ca–Cu–Ba–Ni–Mn–Na 750 15 750 15 Leary 33.97 36.87 14.24 53 Ba Ni Mn 120 All 750 15 750 15 CRM 10.32 8.45 27.46 Leary 7.65 14.48 36.87 73 Al Ba Ni Mn CRM 11.96 10.32 8.45 27.46 Leary 13.02 7.65 14.48 36.87 better compromise operating conditions over the other for all seven elements.When fewer element responses are combined, such as for the combination of Al, Ba and Mn (line 32 in Tables 10 and 11), Leary’s objective function provides better compromise operating conditions than the CRM since the CRM puts more emphasis on minimizing the difference between the SBRs of these three elements.However, for other combinations, such as that of Al, Ba and Ni (line 31 in Tables 10 and 11), it is a question of which is of greater importance, Fig. 9 Surface obtained using Leary’s objective function. maximizing the smallest SBR obtained (i.e., minimizing the difference between the SBRs of the elements) (CRM) or maximizing the total of the SBRs (Leary’s objective function). When decreasing Rj ). The similarities between the surface produced with Leary’s objective function (Fig. 8) and that produced with considering SBRs which can vary widely, these data suggest that the Leary approach is better, as significant improvements the CRM (Fig. 9) indicate that neither function would provide a better surface over the other for use with simplex optimization are sometimes found with relatively small losses compared with the CRM approach. techniques or the Davidon–Fletcher–Powell algorithm. It is also difficult to tell whether one of the two functions provides Given that the response surfaces were relatively similar for Journal of Analytical Atomic Spectrometry, January 1997, Vol. 12 17the various elements, several artificial ‘elemental’ surfaces were model (Fig. 11) is similar to the first except that the SBRs of the two elements at the maximum peak height are completely generated to compare the two approaches. Two of these models depicting extreme situations are presented (Figs. 10 and 11). different. Again, the surfaces obtained using the CRM and Leary’s objective function are very similar. All models gener- The first model (Fig. 10) illustrates SBR surfaces of two elements that peak under completely different instrumental ated gave the same operating conditions or produced the same situation described earlier where Leary’s objective function operating conditions but with approximately the same SBR at the top of the peak. The surfaces obtained using both the performed better since the CRM decreased the difference between the SBRs.CRM approach and Leary’s objective function are very similar, peaking under the same operating conditions. The second Signal-to-background ratios are convenient to use for Fig. 10 Theoretical model of two elements with similar SBRs. Fig. 11 Theoretical model of two elements with dissimilar SBRs. 18 Journal of Analytical Atomic Spectrometry, January 1997, Vol. 12optimization since fewer measurements are required compared acknowledges financial support from the Fonds pour la Formation de Chercheurs et l’Aide a` la Recherche.with SNRs and they often are easily related to detection limits. While one expects SBRs to vary widely, SNRs should be relative similar given concentrations well above the detection REFERENCES limit. In this case the CRM may be advantageous. With many 1 Inductively Coupled Plasma Emission Spectroscopy. Part I: ICP-MS instruments one tends to adjust a variety of operating Methodology, Instrumentation, and Performance, ed.Boumans, parameters (e.g., lens settings) to obtain a roughly uniform P. W. J. M., Wiley-Interscience, New York, 1987, pt. 1, ch. 4. pp. 100–257. sensitivity for all elements. Because of its tendency to promote 2 Ebdon, L., Cave, M. R., and Mowthorpe, D. J., Anal. Chim. 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The authors gratefully acknowledge financial support from the Paper 6/06319E Received September 13, 1996 National Sciences and Engineering Council of Canada. C.S. Journal of Analytical Atomic Spectrometry, January 1997, Vol. 12 19