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Anal. Chem. 1987, 5 9 , 2420-2423
Off-Line Optimization of Gas Chromatographic Temperature Programs Eric V. Dose Environmental Science a n d Engineering, Inc., Gainesville, Florida 32602
The optlmlzation of gas chromatographic temperature programs uslng simulated retention tlmes and peak widths is descrlbed and studled In practlce. A time/resolutlon compromise term is included in the chromatographlc response function and the effects of changing the weight ot that term are Investigated. The ablHty to set limits on the legal temperature program settings is provided for by a new simplex boundary-handling algorithm. The complexity of behavior when the method Is applied to analytes of slgnlflcantly differlng molecular shape Is discussed. Convergence to the experimental opttmum temperature program Is demonstrated in one case by searchlng the settlng space about the determined optimum.
A current trend in capillary gas chromatography methods used in modern environmental trace analysis is the inclusion of more target analytes in a single chromatogram. Given the greatly increasing volume of sample screening to be done, multianalyte analysis time is a t a premium: 45-min temperature programs are now often considered unacceptably long, and determination of 16 analytes from a 20-min chromatogram is not unknown in practice. The conflicting goals of short analysis time and achievement of minimum acceptable resolution (1) make for difficult and often time-consuming optmization of the temperature program. With control afforded by modern gas chromatographs, general stability of bonded-phase capillary columns, and where the target analytes are stable, the temperature program is by far the easiest set of conditions to change and that most amenable to automated management. In previous work, the author demonstrated the ability of a numerical-integration model to simulate gas chromatographic behavior under temperature programming when given the target analytes' retention times and peak widths under isothermal elution at a few temperatures (2). This computer simulation was performed off-line, that is without connection to the chromatograph except through entry of initial isothermal data. While a user, guided by the results of the model, could manually enter successively better trial temperature programs as a means of optimizing the temperature program, computer generation and scoring of trial temperature programs seem a sensible alternative. To demonstrate the ability of a carefully constrained optimization algorithm to optimize automatically the temperature program for a given time/resolution compromise, two classes of analytes are studied under isothermal conditions and the results are subjected to simplex optimization of the temperature program. The behavior of this optimization will be described, and the calculated optimum will be shown to be the experimental optimum. EXPERIMENTAL SECTION Chromatograms were obtained on HewlettPackard 5890A gas chromatographs with electron-capture detection. Temperatures from the chromatographs' controllers were Apparatus.
used directly since fits of -2' In It to T typically had linear correlation coefficients greater than 0.9999, No corrections for heating lag time ( 3 ) were needed. Peak retention times were measured as the crest time indicated by the Hewlett-Packard 3392A integrator. No dependence of retention time on peak area was seen (4). Peak widths were calculated from the integrator's area/height ratio. Assuming Gaussian or moderately skewed Gaussian peak shape, the full width at half height values given in the tables are calculated from the area/height ratios by multiplying by [(2 In 2 ) / ~ ] ' / (5). ~ Sixteen chlorinated hydrocarbons (2)were eluted from a 30-m, 0.32 mm i.d. DB-17 capillary column (J&W Scientific) of film thickness 0.25 Fm. Six nitroaromatic analytes (2)were eluted from a 30-m 0.53 mm i.d. DB-210+ capillary column (J&W) of film thickness 1.0 pm. Helium was used as carrier gas for all experiments. Computation. AT-class microcomputers with math coprocessors were used for all computations. All software was written in C language and is highly structured (40 functions or subprograms). Fifteen decimal digits of accuracy were retained (double precision) (6). Typical simulation time for a 16-analyte, 30-min chromatogram is 20-30 s, and 40-60 simulations were required for convergence in most cases. Optimization. The modified simplex algorithm (7,8) as altered by Aberg and Gustavsson (9) was used. Expansions were twice and contractions were half the normal projection length from the best-face centroid. The initial n-dimensional simplex in each optimization was constructed from a set of n + 1vertices, one representingthe initial user-selected temperature program and one displaced in each dimension by a user-specifiedincrement from the original vertex. Each item in the temperature program could be fixed to a selected value (for example, the final isothermal-hold temperature) or included as a simplex dimension for adjustment. The simplex was considered to have converged when the standard deviation in each adjusted item was below a user-specified value for that item. This convergence criterion was normally taken to be half of the smallest increment allowed by the instrument for that item (1.0 "C for oven temperatures, 0.01 min for isothermal hold times, and 0.1 deg/min for heating ramp rates).
THEORY The simulation/prediction method used in this work is described elsewhere (2). The method accounts for temperature dependence of carrier-gas flow rate (10) and assumes additivity of squared peak widths rather than of peak widths directly ( 3 , I I ) . Ninbthe theoretical plate count at infinite retention, is assumed to be temperature-independent (2,12).The average pressure down the column is used (13) since previous retention times obtained from the simulation model (2) were sufficiently accurate. In the cases presented, retention times were predicted to about 1%accuracy, and peak widths were predicted to about 10-15% accuracy with no general bias towards higher or lower values. The present need calls for a chromatographic response function (CRF) which includes all target peaks, since they can be simulated regardless of overlap, rather than one based on statistical methods (14). The chromatographic time/resolution response function (TRRF, a subclass of the general definition of CRF) used in this work to score each temperature program is given in eq 1,where t , is the retention time of the last peak,
0003-2700/87/0359-2420$01.50/00 1987 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 59, NO. 19, OCTOBER 1, 1987
TRRF =
tn + Ce-Rv/Rcrit
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(1)
If1
tcrltis a user-selected time-cost weighting or compromise factor, and R,, is the resolution between peaks i and j , defined as usual in terms of retention times t, and t, and base-line widths (4g) w,and wlof peaks i and j . Rcrltis a user-selected resolution target value. The first term in eq 1penalizes to a degree specified by tcnt for long analysis times. The second term in eq 1 penalizes for peak-to-peak overlap. The interaction between each peak is taken to be the negative exponential of the resolution R divided by Rcrit,a term which discriminates strongly against overlapping peaks while it is continuous and asymptotic to zero a t large R. Summation of the resolution-negative-exponential terms tends in a conceptually simple and easily implemented manner to emphasize the least resolved peak pairs without totally ignoring the other peak pairs. There need be no arbitrary limit on a peak pair’s resolution above which the pair ceases to contribute to the sum. The inclusion of all possible peak-to-peak interactions maintains continuity when peaks exchange positions under two temperature programs, and it strongly discourages programs which cause more than two peaks to elute in rapid succession. Continuity is necessary to well-behaved vertex projection under super modified simplex (8) or controlled weight centroid simplex (8, 15) algorithms. However these accelerated algorithms were not adopted because the TRRF surfaces were expected to be strongly nonquadratic. The small extra computation cost of including all interactions rather than nearest-neighbor interactions only (16) is partially offset by eliminating the need to sort peaks by retention time in each cycle. Equation 1also avoids excessive importance being placed on one difficult separation among several others (17). Also, optimization of sensitivity is effected indirectly by the peak sharpening caused by this term. Real gas chromatographs have limits on temperatures, heating/cooling ramp rates, etc., and these limits should be represented by boundaries on the temperature-program space searched. Since simplex optimization on complex response surfaces tends to be unstable, restricting searches to the optimal region tends both to assure convergence to the optimum and to avoid searching regions of no interest to the user. The approach of a projected vertex to flat boundaries in setting space must be handled very carefully since ”flattening” of the resulting simplex against a boundary causes loss of degrees of freedom which cannot be recovered except by expansion of the simplex. This expansion should be avoided since it would likely be arbitrary in scale and direction and would surely be very inefficient if the simplex were already close to the optimum. Thus a means of avoiding projection of a vertex onto a boundary is required. If a trial vertex is projected through a boundary into illegal space, the present method calculates 20 randomly placed points along the line segment from the centroid to the illegal trial vertex. The new trial vertex is the random point farthest from the centroid but still within the boundary. If even the closest random point is outside the boundary, that point is used as the terminus of the line segment along which 20 new random points are calculated, and the process is repeated recursively. This algorithm will fail (1)if the centroid is itself outside the boundary, which happens if a vertex of the initial simplex is outside the legal space, or (2) if the boundary surface is anywhere convex toward the legal space. Both situations are unnecessary in optimizations considered here, and they should be avoided. Major advantages of such a boundary-handling algorithm are ease of implementation and the extreme unlikelihood that two or more vertices can be projected so close to the boundary
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Cl
Figure 1. Simplex behavior for the 16 organochlorine pesticides on a DB-17 capillary column. Starting temperature program is marked by the circle at lower left.
that a degree of freedom is lost and the optimization fails. No extra projections, surface searching, or boundary-following algorithms are necessary, and thus all that is required of the boundary definition is a subprogram which answers whether a given point is within the legal space. The boundaries can then be as complex as desired (so long as they are convex outward). Other boundary-handling algorithms flatten simplices more often than one might expect since users tend to choose round numbers for all the values representing starting vertices, starting simplex edge lengths, and boundaries, and in many cases simplices project exactly onto the boundary surface or are clipped back to the boundary. The present algorithm’s use of double-precision random numbers virtually eliminates this problem while allowing the projected vertex to approach the boundary rapidly if the real optimum is nearby.
RESULTS AND DISCUSSION To employ the optimization method described above, one specifies the starting temperature program. initial simplex size, parts of the program to optimize, and boundaries of the legal search space. The progress of initial temperature and heating ramp rate optimization for 16 organochlorine pesticides (OCPs) eluted from a capillary column under a simple oneramp temperature program is shown in Figure 1. As in each such figure in this article, the choice of two adjustable settings facilitates simplex representation, although the method has been successfully applied to the optimization of up to four settings. In Figure 1, all four operations under the modified simplex method are visible: expansion (simplices 1 and 2), reflection (2 and 3), negative contraction (4 and 5), and positive contraction (11and 12). Simplices generally avoid the lower-left corner of these plots because the very long analysis times cannot be compensated for sufficiently in the TRRF by decreased overlap between peaks. Also, in most such cases, cycles after the first few reside in a diagonal “valley” where decreasing heating rates are nearly counterbalanced by increasing initial temperatures. The TRRF for an isothermal separation of the six nitroaromatics is well-behaved and shows a single minimum at 235 “C (Figure 2) as a result of the compromise of the decreasing time and increasing overlap terms (relative to increasing temperature). However, isothermal separation of the OCPs shows two local minima in addition to the global minimum at 230 “C (Figure 3). The local maxima are caused by retention-time crossing of DDE and dieldrin at 197 “C and of endosulfan I and DDE at 210 “C. Optimization of nonisothermal temperature programs becomes very complicated
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ANALYTICAL CHEMISTRY, VOL. 59, NO. 19, OCTOBER 1, 1987
--
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3
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9
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21
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TEMPERATURE
Flgure 4. Dependence of the optimal analysis time on t,,,, including localsptima crossing discontinuities, for the six nitroaromatic analytes (R,,, = 1.5).
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Figure 2. TRRF dependence (upper curve) on isothermal temperature for the six nitroaromatic analytes on a DE210+ capillary column. The lower curve decreasing with increasing temperature is the contribution from the first (analysis time) term in eq 1 and the other lower curve is the contribution from the second (peak overlap) term.
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Figure 5. Convergence of simplices from disparate starting points (circles) for the 16 OCP analytes on DB-17.
165
160
195
210
TEMPPERATURE
225
240
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Figure 3. TRRF dependence (upper curve) on isothermal temperature for the 16 organochlorine pesticide (OCP) analytes on a DE17 capilllary column. Lower curves as for Figure 2.
where minima are separated by TRRF “ridges” representing classes of temperature programs each of which cause two or more chromatogrpahic peaks to cross. This complication of the TRRF surface has been observed when analytes of different molecular shapes (18) are operated on together by this method, and it has not been observed when all the analytes are similar in structure. An optimization can converge a t a local minimum if care is not taken to use either a sufficiently large initial simplex (9) or to start from a simplex with one vertex placed where the program’s initial temperature is the optimal isothermal temperature and the ramp rate is zero (so that the simplex is encouraged to search down the valley). The latter approach, borrowed from nonlinear least-squares optimization experience (19), has worked well in the author’s hands, but proof that it necessarily leads to convergence a t the global minimum would be difficult. Convergence a t a strictly local minimum can also occur if the user alters his relative valuation of time and resolution (reflected in a change in tcrit)and then expects to find the new optimum by starting from the old one. The danger of this seemingly reasonable and time-saving strategy is indicated by the abrupt changes in the analysis time at optimum TRRF (Figure 4). The discontinuities for the 16 OCPs plotted vs.
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Figure 6. Convergence of simplices from disparate starting points (circles) for the six nitroaromatic analytes on DB-210+.
in the curve are caused not by any sudden changes in the valley in which the optimum lies but rather in the identity of that valley, since as tcritchanges, the TRRF values of two local minima cross. Each plateau in Figure 4 represents a tcrit region in which a given local minimum is also the global optimum. Evidence that convergence to the global optimum within a legal space has occurred is obtained when several simplices initially dispersed through the legal space translate and contract to the same point. In Figure 5 , four disparate simplices converge to the same optimum temperature program for the 16 OCPs. Similarly, eight disparate simplices converge
I4NALYTICAL CHEMISTRY, VOL. 59, NO. 19, OCTOBER 1, 1987 12
Table I. Evaluation of Experimental TRRF about Optimuma
0
9
W
B
ramp rate, OC/min
TRRF
150
5.6
0.718
130 130 150 150 170 170
5.6 7.6 3.6 7.6 3.6 5.6
0.786 0.761 0.760 0.968 0.896 0.820
initial temp,
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2423
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INITIAL TEMPERATURE
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140
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pendent values, each row describing one analyte's retention and theoretical-plate-number (reversibility) behavior on that stationary phase. By selecting target analyte's values from such data compiled in advance and applying this simulation model and optimization method, one could compare the ability of several phases ut the optimum temperature program to effect a given separation (and, by adding dimensions to the simplex method, optimize the column geometry and carrier flow rate (20)). These methods could be readily extended to simulate or optimize temperature gradients down the column (21)or to optimize the relative lengths of stationary components in serially coupled columns (22, 23). It may be possible by analogy to the last case to optimize stationary phases composed of mixtures of previously studied phases for separation of a given set of analytes.
(dag C)
Flgure 7. (A) Convergence to a boundary (upper Mal temperature limit 140 "C) for the 16 OCP analytes on DB-17. (B) Enlargement of A about the optimum.
to the same program for the nitroaromatic analytes (Figure 6). Such consistency of convergence was found in cases except those where two local optima have very nearly equal TRRF values, as when applying tcritvalues near the sort of discontinuity seen in Figure 4. Boundary handling by the random vector truncation algorithm was not observed to fail in several hundred optimizations. A typical optimization (Figure 7) expands and traslates rapidly to the upper temperature limit at 140 "C and converges rapidly. The reversal of direction and reconvergence (see Figure 7B) to a better position on the boundary is quite typical of this algorithm's behavior and demonstrates its resistance to simplex flattening and false convergence at a boundary. Optimization methods based upon simulated results rather than upon experimentally measured responses should be tested for convergence to a real minimum. The TRRF for the experimental chromatogram resulting from the temperature program yielded by the method is compared in Table I to TRRF values for temperature programs in the space around the predicted optimum. One could in principle describe the behavior of analytes on a given stationary phase as a matrix of temperature-inde-
LITERATURE CITED (1) Stan, H.J.; Steinbach, B. J. Chromatogr. 1984. 290, 311. (2) Dose, E. V. Anal. Chem., preceding paper in this Issue. (3) Bartu, V. J. Chromatogr. 1983, 260, 255. (4) Conder, J. R.; Rees, G. J.; McHale, S. J. Chromatogr. 1983, 258, 1. (5) Seferovic, W.; Hinshaw, J. V., Jr.; Ettre. L. S. J. Chromatogr. Sci. 1986, 2 4 , 374. (6) LeSage, J. P.; Simon, S. D. Comput. Stat. Data Anal. 1985, 3 . 47. (7) Morgan, S. L.; Deming, S. N. Anal. Chem. 1974, 46, 1171. (8) Routh, M. W.; Swartz, P. A.; Denton, M. B. Anal. Chem. 1977, 4 9 , 1422. (9) Aberg, E. R.; Gustavsson, G. T. Anal. Chim. Acta 1982, 144, 39. IO) Davies, N. W. Anal. Chem. 1984. 6 7 , 2600. 11) Bartu, V.; Wlcar, S. Anal. Chim. Acta 1983, 150, 245. 12) Ceulemans, J. J. Chromatogr. Sci. 1988, 2 4 , 147. 13) Keliey, J. D.; Walker, J. Q. Anal. Chem. 1969, 41, 1340. 14) Martin, M.; Guiochon, G. Anal. Chem. 1985, 5 7 , 289. 15) Ryan, P. B.; Barr, R. L.; Todd, D. L. Anal. Chem. 1980, 5 2 , 1460. 16) Debets, H. J. G.; Bajema, B. J.; Dornbos, D. A. Anal. Chim. Acta 1983, 151, 131. (17) Wegscheider, W.; Lankmayr, E. P.; Otto, M. Anal. Chim. Acta 1983,
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(18) Saxton, W. L. J. Chromatogr. 1986, 357, 1. (19) Kragten, J.; deJagher, P. C.; Decnop-Weever, L. G. Anal. Chim. Acta 1986, 180, 457. (20) Ogan, K.; Scott, R. P. W. HRC CC, J, H@h Resolut. Chromatogr. Chromatogr. Commun. 1986, 7 , 382. (21) Berezkin, V. G.; Chernysheva, T. Y.; Buzayev, V. V.; Koshevnik, M. A. J. Chromatogr. 1986, 373, 21. (22) Purneli, J. H.; Rodriguez, M.; Williams, P. S. J. Chromatogr. 1986. 358, 39. (23) Hinshaw, J. V., Jr.; Ettre, L. S. Chromatographia 1986, 2 1 , 561.
RECEIVED for review February 9, 1987. Accepted June 22, 1987.
