CHIRAL SEPARATIONS. CHIRAL DYNAMIC CHROMATOGRAPHY
3. MODELS AVAILABLE TO SIMULATE/ANALYZE DYNAMIC CHROMATOGRAMSCHROMATOGRAMS
Several mathematical models have been developed in the last three decades to extract kinetic data from dynamic chromatograms. Thecontinuous flow modelis the oldest model, developed by Horva´th et al.
in 1984 (Melander et al., 1984; Jacobson et al., 1984). Because of its limited flexibility to the development of the required iterative procedures it is currently out-of-date. In fact, rather complex mathematics is involved, i.e., the coupled chromatographic and secondary chemical equilibria are described by suitable differential equations that have to be resolved by nontrivial numerical solutions.
Much more important, from an operative point of view, is the approach known as thetheoretical plates model(TPM) (Bu¨rkle et al., 1984). It portrays the chromatographic column as a discontinuous entity constituted by a defined number,Nth, of elementary chemical reactors (calledtheoretical plates) and approximates the concurrent chromatographic and secondary chemical equilibria as occurring inside each theoretical plate in three successive steps: (1) equilibration between the MP and SP of the species involved in the secondary chemical process; (2) chemical evolution of each species according to the involved kinetic law (frequently, but not necessarily, irreversible first-order kinetics) for a time Dtcorresponding to that of the residence of the MP inside each theoretical plate (Dt¼t0/Nth, witht0
being the dead time); and (3) shifting of the MP from the actual plate to the next one, according to the flow direction imposed on the MP through the column. The whole of these three steps have to be repeated a number of times (n_shift) sufficient to elute all the injected species from the column [in its practical use, this number is calculated by the following equation: n_shif t¼(trmaxþ3W0.5)/Dt, where trmaxis the retention time of the last eluted species and 3W0.5is three times the width at half- height of the last eluted peak]. On the whole, the TPM model proved to be very effective and relatively easy to be implemented into computer programs that automatically iterate the simulation of experi- mental dynamic chromatograms until achieving good agreement. However, because the time of simulation is exponentially related to Nth, only separations with not too many marked efficiencies (smaller than about 10,000 theoretical plates) can be treated in an iterative way yielding reasonable simulation times. Examples of widely employed computer programs implementing the TPM model are SIMUL (Jung, 1992), ChromWin (Trapp and Schurig, 2001b), Auto DHPLC y2k (Gasparrini et al., 2002a; Cabri et al., 2008; Cirilli et al., 2009b), and ChromWin_2D (Trapp et al., 2003; Trapp, 2004).
Theclassical stochastic model(CSM) or simply thestochastic model, which was developed in its current formulation by the joint contributions fromKeller and Giddings (1960)andKramer (1975), has also proved to be very useful. An additional contribution to the approach, called theimproved stochastic model, was more recently reported (Trapp and Schurig, 2001b). The CSM model exclu- sively refers to the instance of monoequilibrium processes featuring first-order kinetics, occurring during a chromatographic separation. In this case, the concurrent chromatographic and secondary chemical equilibria are indirectly taken into account by expressing the profiles of the dynamic chromatograms by means of two typologies of time-dependent distribution functions: the first one, f(t), related to the fraction of molecules of IA and IB never involved in the secondary chemical equilibrium [f(t)¼fIA(t)þfIB(t)], and the other one,4(t), expressing the plateau zone due to the fraction of IA and IB molecules that underwent the transformation at least once during the chro- matographic separation [4(t)¼4IA(t)þ4IB(t)]. Althoughf(t) [and then also its parent components fIA(t) andfIB(t)] is not related to the rate constants of the secondary process and may be successfully
described by a classical Gaussian distribution, the function 4(t) is exponentially related to both kvapp1 andkvapp1 because it may be evinced by its mathematical expression derived by Keller, Giddings, and Kramer as shown inFig. 3.3.
The primary advantage of the CSM model is the very short time of simulation, which results from the fact that it is significantly independent ofNth. This allows a quick treatment of dynamic chro- matograms obtained by employing very efficient techniques, such as high-resolution gas chroma- tography (HRGC) and ultra-high performance liquid chromatography (UHPLC). An example of the relationships between time of simulation (seconds) of dynamic chromatograms and chromatographic efficiency (number of theoretical plates) involved in the TPM and CSM models is given inFig. 3.4.
Computer programs implementing the CSM model, and widely tested on a large variety of first- order processes, involving both constitutional and conformational isomerizations (i.e., enantiomeri- zations, diastereomerizations, tautomerizations, etc.) are ChromWin (Trapp and Schurig, 2001b) and Auto DHPLC y2k (Gasparrini et al., 2002a; Cabri et al., 2008; Cirilli et al., 2009b). To strongly reduce the computational time, which in the aforementioned iterative procedure of comparison between simulated and experimental chromatograms is not negligible, the modification of kinetic and, if desired, chromatographic parameters is automated in the Auto DHPLC y2k program by the use of an algorithm driven by a simplex procedure. In this way, the user is not busy in this tedious step. Moreover, the same program also implements the possibility of taking tailing effects into account, thus extending the FIGURE 3.3
Classical stochastic model.
applicability of the procedure (for both TPM and CSM methods) to the frequent occurrence of nonlinear sample repartition between MP and SP.
Even faster than the just-described model is that based on the derivation of the so-calledunified equation of chromatography(UEC) (Trapp, 2006), which overcomes an earlier approximate version (Trapp and Schurig, 2001a) that only worked in the simplified case of enantiomerization processes. The UEC allows direct calculation of rate constants of secondary chemical first-order reactions by a few iterative steps, without the need of performing a computationally extensive simulation of elution profiles. The only parameters required to perform the calculus are: (1) the retention times of the reacting species, trIAand trIB; (2) the peak widths at half-height, W0.5IAand W0.5IB; (3) the relative height of the plateau, hp; (4) the initial amounts of the reacting species, IA0and IB0; and (5) the equilibrium constant, Keq¼IB/IA. However, although very fast, this approach does not assure estimations with acceptable accuracy when the plateau height approaches that of the residual adjacent peaks (Cirilli et al., 2009b) or when the dynamic chromatograms are affected by marked asymmetry (Uray et al., 2010). Furthermore, because no simulation of the experimental dynamic chromatograms is performed by this model, it is not possible to directly check the reliability of the estimated rate constants by superimposition of simulated and experimental profiles.
A further commonly employed mathematical approach is that often labeled with the generic term of thedeconvolution method, although there are at least three different ways in which this method may practically be addressed. Such an approach, in fact, requires that the dynamic chromatogram is resolved into the components related to the fractions of molecules reacted (area of the plateau zone)
Theoretical Plates Model Classical Stochastic Model
Time of simulation (sec)
Efficiency (n. of plates) 490
390
290
190
90
–10
–1000 1000 3000 5000 7000 9000 11000
FIGURE 3.4
Relationships between time of simulation (seconds) of dynamic chromatograms and chromatographic efficiency (number of theoretical plates) involved in the theoretical plates model and classical stochastic model.
and not reacted (area of the residual peaks on either side of the dynamic profile) during the separation.
Thus, deconvolution can be performed by: (1) the combined use of two or more tools of separation, which give rise to multidimensional hardware systems (Trapp, 2004; Marriott et al., 2001); (2) the Gaussian or exponentially modified Gaussian functions (Krupcik et al., 2000a,b; Oswald et al., 2002a,b); or (3) the combined use of two or more detectors (one of which must be chiro-optical) as monitoring tools of monodimensional chromatographic separations (Mannschreck et al., 1988;
Mannschreck and Kiessl, 1989; Wolf et al., 1995; Allenmark and Oxelbark, 1998;Nishikawa et al., 1997; Kusano et al., 1999).
Finally, a novel stochastic approach to the DC has just been proposed in 2010 (Pasti et al., 2010).
The novelty of the model is a microscopic point of view of the interconnections existing between repartition and secondary chemical equilibria, so that it might be suitably referred to as themicroscopic stochastic model(MSM). In its formalism, the shape of a dynamic chromatogram is calculated in the frequency domain when the reaction follows a simple reversible first-order scheme. Then, the derived solutions are expressed in closed form in the Fourier domain. However, at the moment, the model is not implemented in any dedicated standalone software, and because of the quite complex mathematics involved, it is unlikely that such an approach may routinely be used in DC experiments. Nevertheless, the MSM model could attract specific interest on the basis of its ability to correlate macroscopic classical chromatographic parameters with the behavior properties of individual molecules (Pasti et al., 2005, 2016; Felinger et al., 2005).