Simple algorithms for fitting and optimisation for multilinear gradient elution in reversed-phase liquid chromatography

doi:10.1016/j.chroma.2007.04.059

Journal of Chromatography A

Volume 1157, Issues 1–2, 20 July 2007, Pages 178-186

https://doi.org/10.1016/j.chroma.2007.04.059 Get rights and content

Abstract

The theory of the multilinear gradient elution in reversed-phase liquid chromatography (RP-HPLC) presented in [P. Nikitas, A. Pappa-Louisi, P. Agrafiotou, J. Chromatogr. A 1120 (2006) 299] is modified to increase its flexibility. In addition, it is embodied to simple algorithms for fitting gradient data and especially for resolution optimisation under multilinear gradient conditions. In particular, two new algorithms for fitting and one for optimisation are tested and compared with conventional algorithms. Their performance was examined using 13 o-phthalaldehyde (OPA) derivatives of amino acids with mobile phases modified by acetonitrile. It was found that the new proposed algorithms, a repeated application of the Levenberg–Marquardt (LM) method for fitting (R_LM) and a modified descent algorithm for optimisation (RND_D), in combination with the modified theory of the multilinear gradient elution can lead to high quality predictions of the retention times and optimisation results.

Introduction

Gradient elution is a powerful method for separation and peak identification of many branches of chromatography provided that the optimum gradient profile, that is the profile that yields the best separation of the chromatographic peaks of a mixture of analytes, can be easily determined [1], [2], [3]. This task can be achieved by the use of proper optimisation algorithms, which usually presume that (a) there are explicit expressions describing the retention time of each solute in terms of the gradient mode characteristics, and (b) the dependence of the retention factor, k, upon the mobile phase composition is known via a fitting procedure.

In reversed-phase liquid chromatography (RP-HPLC) explicit expressions of the retention time are available when ln k varies linearly with the volume fraction φ of an organic modifier in the water–organic mobile phase, and φ is programmed to vary linearly with time. The combination of linear gradient with a linear dependence of ln k upon φ is called linear solvent strength gradient [4], [5], [6], [7], [8], [9], [10], [11]. This approach constitutes the base of DryLab, the most widely published HPLC simulation package to date [12], [13]. Due to the importance of gradient optimisation, several other approaches have already been developed [14], [15], [16], [17], [18], [19], [20], [21], [22], [23].

In recent papers [24], [25], we have presented a new mathematical treatment that yields analytical expressions for the retention time when the programmed gradient profile is linear or multilinear and ln k is not a linear function of φ. According to this treatment, the experimental ln k versus φ curve is subdivided into a finite number of linear portions, that is, the domain of the ln k versus φ function is divided into m equidistant portions, (φ₀, φ₁, …, φ_m), and the solute gradient retention time is calculated by means of an analytical expression arising from the fundamental equation of gradient elution [2], [26], [27], [28], [29], [30], [31]. Here, φ₀ and φ_m are the minimum and the maximum values of φ used in the experiments. However, this approach has the following drawback, which limits considerably the flexibility of treatment in practical applications: the φ coordinates of the multilinear gradient should necessarily belong to the vector (φ₀, φ₁, …, φ_m).

In what concerns the dependence of the retention factor, k, upon the mobile phase composition and more precisely the dependence of ln k upon φ, this is easily determined from isocratic data. However, if we use gradient data to determine the function ln k = f(φ), the appearance of convergence problems cannot be ruled out [25].

In the present paper, we (a) modify the theory of the multilinear gradient elution in RP-HPLC presented in [24], [25] to increase its flexibility and (b) present and examine simple algorithms for fitting gradient data and especially for reliable separation optimisation under multilinear gradient conditions.

Section snippets

Analytical expressions of the gradient retention time

The retention time t_R of a solute under gradient conditions may be calculated from the fundamental equation [1], [31] $\int_{0}^{t_{R} - t_{0} - t_{in} - t_{D}} \frac{d t}{t_{0} k_{φ}} = 1 - \frac{t_{D} + t_{in}}{t_{0} k_{φ_{in}}}$ where k_φ = (t_φ − t₀)/t₀ is the solute retention factor which corresponds to a constant mobile phase composition equal to φ, t_φ ≡ t_R(φ) is the corresponding isocratic retention time, t₀ is the column hold-up time, t_D is the dwell time, i.e. the time needed for a certain change in the mixer to reach the beginning of the chromatographic column, t_in is an

Cost functions

The problems of fitting and optimisation are interrelated since in both cases a cost function is defined and its global minimum (maximum) is determined using a proper algorithm. The cost function we adopted for fitting was $CF = \sum_{j = 1}^{N} {(t_{R_{j}, exp} - t_{R_{j}, calc})}^{2}$ where $t_{R_{j}, exp}$ is the experimental retention time of a certain solute under the j-th gradient profile and $t_{R_{j}, calc}$ is the corresponding calculated value from the theory presented above. The fitting algorithm looks for the vector with coordinates c₁, c₂

Experimental

The liquid chromatography system consisted of a Shimadzu LC-20AD pump, a model 7125 syringe loading sample injector fitted with a 20 μL loop, a MZ-Analysentechnik column, PerfectSil Target ODS-3 (5 μm, 250 mm × 4.6 mm) thermostatted by a CTO-10AS Shimadzu column oven at 25 °C, and a Shimadzu UV-visible spectrophotometric detector (Model SPD-10A) working at 338 nm. The mobile phases were aqueous phosphate buffers (pH 2.5) modified with different volume fractions, φ, of acetonitrile (MeCN). Their total

Results and discussion

For the optimisation of the separation of the 13 o-phthalaldehyde derivatives of amino acids adopted in the present study, we first determined the adjustable parameters c₁, c₂, c₃ of Eq. (15) by using the six gradient profiles shown in Table 1. The experimental retention times that correspond to these gradients are listed in Table 2. Based on these data, we have first proceeded to extensive preliminary tests using all the fitting algorithms, RND_LM, R_LM, RND_D_LM and GA_LM, with a variety of

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Simple algorithms for fitting and optimisation for multilinear gradient elution in reversed-phase liquid chromatography

Abstract

Introduction

Section snippets

Analytical expressions of the gradient retention time

Cost functions

Experimental

Results and discussion

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