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How much can we learn from a single chromatographic experiment? A Bayesian perspective Pawel Wiczling, and Roman Kaliszan Anal. Chem., Just Accepted Manuscript • DOI: 10.1021/acs.analchem.5b03859 • Publication Date (Web): 26 Nov 2015 Downloaded from http://pubs.acs.org on December 7, 2015
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How much can we learn from a single chromatographic experiment? A Bayesian perspective Paweł Wiczling*, Roman Kaliszan Department of Biopharmaceutics and Pharmacodynamics, Medical University of Gdańsk, Gen. J. Hallera 107, 80-416 Gdańsk, Poland *Corresponding author: tel. ++48 58 349 3260; fax ++48 58 349 3262; e-mail:
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Abstract In this work we proposed and investigated a Bayesian inference procedure to find the desired chromatographic conditions based on known analyte properties (lipophilicity, pKa, and polar surface area) using one preliminary experiment. A previously developed nonlinear mixed effect model was used to specify the prior information about a new analyte with known physicochemical properties. Further the prior (no preliminary data) and posterior predictive distribution (prior + one experiment) were determined sequentially to search towards the desired separation. The following isocratic high-performance reversed-phase liquid chromatographic conditions were sought: 1) retention time of a single analyte within the range of 4-6 min, 2) baseline separation of two analytes with retention times within the range of 4-10 min. The empirical posterior Bayesian distribution of parameters was estimated using the "slice sampling" Markov Chain Monte Carlo (MCMC) algorithm implemented in Matlab. The simulations with artificial analytes and experimental data of ketoprofen and papaverine were used to test the proposed methodology. The simulation experiment showed that for a single and two randomly selected analytes there is 97 % and 74 % probability of obtaining a successful chromatogram using none or one preliminary experiment. The desired separation for ketoprofen and papaverine was established based on a single experiment. It was confirmed that the search for a desired separation rarely requires a large number of chromatographic analyses at least for a simple optimization problem. The proposed Bayesian-based optimization scheme is a powerful method of finding a desired chromatographic separation based on a small number of preliminary experiments.
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Introduction Reversed-phase high-performance liquid chromatography (RP HPLC) is one of the most widely used chromatographic techniques. It is a consequence of its universality, relatively low costs and general simplicity of analytical procedures.1 Still, the development of any method with the desired separation might be long and resource-demanding2 – partially due to large number of chromatographic settings that might be adjusted (temperature, mobile phase composition, pH etc.). The most popular trial-and-error approach involves performing sequential analysis (one at a time) with conditions depending on the previously gathered data. However, this approach has several disadvantages: it is often time-consuming, usually requires a large number of preliminary experiments and might not be fully efficient, especially when conducted in a naïve way without appropriate weighting of all the available information. Moreover model-based techniques (or automated screening) can be used in the process of searching for the desired separation.3 These methods usually provide a very accurate theoretical chromatogram based on a series of preliminary experiments. Therefore a relatively large number of carefully designed experiments is needed (at least equal to the number of model parameters).3,4 We believe that the number of experimental data may be reduced by utilizing the knowledge obtained from other analytes or literature information using Bayesian inference methods. In fact, we rarely analyze completely unknown sample. Very often we know the chemical structure or physicochemical properties of compounds present in the sample. We also know the system-specific parameters and the properties of stationary phases that we plan to use, i.e. from literature, manufacturer or previous experiments. The analyst often utilizes that information intuitively during the process of method development using a common-sense thinking, i.e. by using a high methanol content in the eluent for a compound with high lipophilicity, or conducting the analysis at pH ensuring the presence of non-dissociated form of analyte. The Bayesian inference methods are ideally suited for putting this intuitive thinking into the quantitative framework. Basically, they allow to combine prior knowledge about the problem with new experiments providing the most likely predictions (with uncertainty around those) for a new separation condition that is yet to be tested. In this work we will provide several examples of how this technique can be used to predict analytes’ retention. The Bayesian predictions are based on (i) the knowledge gathered from previously analyzed compounds (a priori information) and (ii) a series of preliminary experiments so
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that information on the analyzed compound can be obtained from other structurally similar compounds analyzed previously. To specify the prior knowledge, a very general statistical model describing retention of analytes can be considered. In this work we will use a nonlinear mixed effect model (NLME) developed by our group.5 This model allows to predict analyte retention time for a broad spectrum of organic modifier content (methanol) and mobile phase pH changes. It was developed based on dataset of 66 structurally-diverse analytes. This model consists of 1) a structural model quantifying the effect of pH and methanol content on retention, 2) Quantitative Structure Retention Relationships based equation (QSRR) relating four basic physicochemical properties of the analyte (lipophilicity of nonionized form, aqueous pKa, polar surface area (PSA) and acid/base type) and model parameters, 3) intra- and inter-compound variability. In principle such model can be developed once for a given stationary phase and may be used to predict analyte retention for new compounds. In our recent work5 we used this model to estimate all analyte-specific parameters from limited experimental design by a Maximum Bayesian a posteriori approach. We demonstrated that four experiments are sufficient to obtain accurate parameter estimates and retention time predictions. Nevertheless, we do not need to estimate all model parameters to make accurate predictions, especially for simple problems.6 We hypothesized that one experiment is sufficient to obtain a desired retention of a single known analyte and to provide a desired resolution between two known analytes. This Bayesian-based optimization strategy will be illustrated using the experimental data on ketoprofen and papaverine. In this example we will search for conditions ensuring the desired retention time of a single analyte and the baseline separation within a given retention window for these two analytes. In the next step artificial analytes with randomly selected values of log P, PSA, pKa, and Acid/Base type will be simulated. The probability of achieving a desired chromatogram with one or two randomly selected analytes will be calculated to assess the general applicability of the Bayesian-based optimization strategy. Theoretical The model used in this work is described in the Supporting Information and previous article.5 Briefly, the statistical model for the observed tR,ij of compound i under the experimental design Dij, is given by t R ,ij = f ( Dij , Ri ) + ε ij , where Ri is the p-vector of the individual parameters (it includes log kw, pKa, S and others), Dij is a vector of all adjustable system parameters influencing analyte retention, i.e. organic modifier content, pH, flow rate
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etc., and εij is the residual error. The standard structural model (f) was used to describe the relationship between analyte retention time and mobile phase composition under isocratic and gradient conditions.7,8,9 The individual (analyte-specific) parameter (Ri) follows a multivariate normal distribution around a set of QSRR relationships that depend on dissociation constant pKa, lipophilicity (log P) and polar surface area (PSA). Posterior predictions For a new analyte x for which a set of nx experiments was performed the posterior distribution of its parameter (up to the proportionality constant) can be calculated according to the equation:
p( Rx|{t R, x1 ,...t R, xnx },{Dx ,1 ,...Dx ,nx }) ∝ p({t R x }|Rx ,{Dx }) p( Rx )
(1)
where {tR,x} denotes a set of new retention time measurements under a set of experimental designs {Dx}. The conditional distribution of Rx given data (i.e. p(Rx|{tR,x},{Dx})) denotes the posterior distribution. The p({tR,x}|Rx,{D,x}) is the distribution of {tR,x}, which is a likelihood function when viewed as a function of model parameters, assuming that Rx is known, The p(Rx) is the distribution of Rx without any knowledge on data and is referred to as prior distribution. Eq (1) demonstrates that the posterior distribution of Rx is proportional to the product of the likelihood of Rx given {tR,x} and the prior distribution of Rx. The likelihood was further calculated as: nx
p({t R x }|Rx ,{Dx }) = ∏ MVN (t R , xj − f ( Dxj , Ri ), varxj ) (2) j =1
Since the priors p(Rx) are replaced with their maximum likelihood estimates from our model (The details about the model are presented in the Supporting Information), p(Rx|{tR,x},{Dx}) is called the empirical posterior Bayesian distribution. The empirical posterior Bayesian distribution was estimated using the "slice sampling" algorithm implemented in Matlab® Software version 2015b slicesample function based on the Markov Chain Monte Carlo (MCMC) algorithm. MCMC is an iterative simulation based approach used to make inferences about posterior distributions of the parameters of interest. Convergence of the MCMC chains was assessed using the trace history of MCMC samples for all parameters, in which a ‘fuzzy caterpillar’ suggested that MCMC chains reached stationary distribution. The MCMC chains were run for 1000 samples (excluding the 100 samples that were discarded during the burn-in phase).
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Knowing the empirical posterior Bayesian distribution, p(Rx|{tR,x},{Dx}), allows to determine all the statistical properties of a given system, i.e. to calculate the probability of achieving desired retention time, desired resolution or any other measure of interest. In this work, the simulating draws of parameters from p(Rx|{tR,x},{Dx}) were obtained (and noted as ()
()
) along with retention time predictions (, ) where S denotes the number of draws equal to 1000. From this simulated values, we estimated the posterior distribution of several measures of interest, calculating the proportion of S simulations which are true. In this work we considered two simple cases: Condition 1: The probability of obtaining an isocratic retention time of a compound x in a ()
given retention time window Pr(4 min
