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Georges Guiochon - One of the best experts on this subject based on the ideXlab platform.
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Theoretical comparison of the performance of gradient Elution Chromatography at constant pressure and constant flow rate.
Journal of Chromatography A, 2012Co-Authors: Fabrice Gritti, Georges GuiochonAbstract:Abstract The theory of gradient Elution usually assumes a constant flow rate. This works extends it to gradients performed under constant pressure drop and variable flow rates. The peak capacity is derived under both constant flow rate and constant pressure gradient Chromatography by integrating the rate of increase of the peak capacity from the hold-up time to the end of the gradient time. Assuming that the eluent mixture is incompressible, the chromatographic system isothermal, the pressure has no effect on the retention pattern, and neglecting the contributions of the instrument to the total pressure drop and the total peak width, it is found that both modes of gradient Elution Chromatography are strictly equivalent, provided that the Elution time of the last eluted compound and the volume gradient are kept the same in both cases.
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performance of columns packed with the new shell kinetex c18 particles in gradient Elution Chromatography
Journal of Chromatography A, 2010Co-Authors: Fabrice Gritti, Georges GuiochonAbstract:The performance of columns packed with the new 2.6 microm Kinetex-C(18) shell particles was investigated in gradient Elution Chromatography and compared with those of the 2.7 microm Halo-C(18) shell particles and the 1.7 microm BEH-C(18) totally porous particles. The peak capacities P(c) of these columns were derived from the resolution of the components of a peptide mixture (beta-Lactoglobulin digest) and of a mixture of two biomolecules (insulin and lyzozyme).The three columns exhibit the same peak capacities for the peptides at low linear velocity (u(0) < 0.05 cm/s) and at any gradient steepness (0.8 < G < 10). When the linear velocity is increased 10-fold, the peak capacity of the Kinetex column remains nearly unchanged while those of the Halo-C(18) and the BEH-C(18) columns decrease by 20%, approximately. This result confirms the very flat HETP curve, the very low C term of the Kinetex column and its ability to successfully operate at high flow rates while experiencing less efficiency loss than other columns. Despite its smaller average mesopore size (96 A versus 130 A), the column packed with 2.6 microm shell Kinetex-C(18) particles gives an equivalent or even slightly better separation of biomolecules having a size and a mass around 40 A and 15 kDa, respectively, than the column packed with 1.7 microm BEH-C(18) totally porous particles. This result demonstrates the advantages of the shell versus the conventional particle technology when it comes to resolve mixtures of large and slow diffusive biomolecules.
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Exact peak compression factor in linear gradient Elution. I. Theory.
Journal of chromatography. A, 2008Co-Authors: Fabrice Gritti, Georges GuiochonAbstract:The only existing expression for the peak compression factor in linear gradient Elution Chromatography assumes that the linear-solvent-strength model (LSSM) applies to the retention of the compound studied, that the column efficiency is independent of the mobile phase composition, and that, during gradient Elution, the relative retention factor of a compound inside its band varies linearly with the distance from the band center. Because the retention factors of many analytes in reversed-phase liquid Chromatography do not rigorously follow the LSSM, we extend the theoretical approach of Poppe et al. to the prediction of peak compression factors in linear gradient Elution Chromatography for any retention model, when column efficiency varies with the mobile phase composition. Only the contribution of the chromatographic column to the peak compression was taken into account, the contribution of the dwell volume being neglected. A second restriction is the linearity of the relative retention factor as a function of the position along the band width inside the column. These constraints could be the sources for the difference observed between experimental and theoretical values of peak compression factors. When the retention factor varies steeply with the mobile phase composition, such as with proteins or large peptides in RP-HPLC, it is found that the thermodynamic compression term, which tends to sharpen the peak, is coupled with the column dispersion term, which tends to broaden the peak. This coupling term acts as an apparent dispersion term, contributing to broaden the peak. This result is consistent with the measurements of peak compression factors found in the literature.
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the ultimate band compression factor in gradient Elution Chromatography
Journal of Chromatography A, 2008Co-Authors: Fabrice Gritti, Georges GuiochonAbstract:The equations predicting the ultimate time band compression factor, G=(t(R)-t(F))/t(p) in linear gradient Elution Chromatography, for an infinitely narrow injection (injection time t(p)-->0) were derived for an ideal-model column (dispersionless Chromatography, H=0) assuming the Linear Solvent Strength Model for the retention behavior of the analyte. Numerical solutions can readily be obtained when the LSSM model does not apply. The results can be generalized to any retained organic modifier (k'(A)) in the mobile phase. The stronger the retention of the organic modifier, the more effective the band compression. Dispersion in real chromatographic column (H not equal 0) affects the limits that can be reached in linear gradients but poorly in step gradients. Examples based on a conventional HETP of about 12 microm using a 5 microm particle packed column reveal that the best time compression factor that could be expected is twice the one predicted with an ideal column.
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the bandwidth in gradient Elution Chromatography with a retained organic modifier
Journal of Chromatography A, 2007Co-Authors: Fabrice Gritti, Georges GuiochonAbstract:The variance of a chromatographic band is derived in the case of RPLC gradient Elution when the organic modifier is significantly retained onto the stationary phase. This derivation is based on the extension of a model due to Poppe et al. [H. Poppe, J. Paanakker, M. Bronckhorst, J. Chromatogr., 204 (1981) 77] which assumes that the gradient front remains unchanged and propagates along the column at the same speed as the mobile phase, following piston flow. Theoretical and experimental results are compared in the case of caffeine on a C1-silica stationary phase eluted with an acetonitrile gradient. The actual retention behaviors of caffeine and acetonitrile were implemented in the theoretical calculations. The model predicts compression factors between 0.71 and 0.34 for relatively smooth gradient steepness, βt0βt0, between 0.009 and 0.054 while the corresponding experimental band compression factors vary between 1.01 and 0.43 for the very same gradient steepness. The model underestimation of these factors arises likely from the strong deviation of the actual retention behavior from the prediction of the Linear Solvent Strength Model (LSSM).
Peter W. Carr - One of the best experts on this subject based on the ideXlab platform.
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instrument parameters controlling retention precision in gradient Elution reversed phase liquid Chromatography
Journal of Chromatography A, 2014Co-Authors: Ayse Beyaz, Peter W. Carr, Adam P. SchellingerAbstract:Abstract The precision of retention time in RPLC is important for compound identification, for setting peak integration time windows and in fundamental studies of retention. In this work, we studied the effect of temperature ( T ), initial ( ϕ o ) and final mobile phase ( ϕ f ) composition, gradient time ( t G ), and flow rate ( F ) on the retention time precision under gradient Elution conditions for various types of low MW solutes. We determined the retention factor in pure water ( k ′ w ) and the solute-dependent solvent strength ( S ) parameters of Snyder's linear solvent strength theory ( LSST ) as a function of temperature for three different groups of solutes. The effect of small changes in the chromatographic variables ( T , ϕ o , ϕ f , t G and F ) by use of the LSST gradient retention equation were estimated. Peaks at different positions in the chromatogram have different sensitivities to changes in these instrument parameters. In general, absolute fluctuations in retention time are larger at longer gradient times. Drugs showed less sensitivity to changes in temperature compared to relatively less polar solutes, non-ionogenic solutes. Surprisingly we observed that fluctuations in temperature, mobile phase composition and flow rate had less effect on retention time under gradient conditions as compared to isocratic conditions. Overall temperature and the initial mobile phase composition are the most important variables affecting retention reproducibility in gradient Elution Chromatography.
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peak capacity optimization of peptide separations in reversed phase gradient Elution Chromatography fixed column format
Analytical Chemistry, 2006Co-Authors: Xiaoli Wang, Dwight R Stoll, And Adam P Schellinger, Peter W. CarrAbstract:The optimization of peak capacity in gradient Elution RPLC is essential for the separation of multicomponent samples such as those encountered in proteomic research. In this work, we study the effect of gradient time (tG), flow rate (F), temperature (T), and final eluent strength (φfinal) on the peak capacity of separations of peptides that are representative of the range in peptides found in a tryptic digest. We find that there are very strong interactions between the individual variables (e.g., flow rate and gradient time) which make the optimization quite complicated. On a given column, one should first set the gradient time to the longest tolerable and then set the temperature to the highest achievable with the instrument. Next, the flow rate should be optimized using a reasonable but arbitrary value of φfinal. Last, the final eluent strength should be adjusted so that the last solute elutes as close as possible to the gradient time. We also develop an easily implemented, highly efficient, and effecti...
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peak capacity optimization of peptide separations in reversed phase gradient Elution Chromatography fixed column format
Analytical Chemistry, 2006Co-Authors: Xiaoli Wang, Dwight R Stoll, And Adam P Schellinger, Peter W. CarrAbstract:The optimization of peak capacity in gradient Elution RPLC is essential for the separation of multicomponent samples such as those encountered in proteomic research. In this work, we study the effect of gradient time (tG), flow rate (F), temperature (T), and final eluent strength (phi(final)) on the peak capacity of separations of peptides that are representative of the range in peptides found in a tryptic digest. We find that there are very strong interactions between the individual variables (e.g., flow rate and gradient time) which make the optimization quite complicated. On a given column, one should first set the gradient time to the longest tolerable and then set the temperature to the highest achievable with the instrument. Next, the flow rate should be optimized using a reasonable but arbitrary value of phi(final). Last, the final eluent strength should be adjusted so that the last solute elutes as close as possible to the gradient time. We also develop an easily implemented, highly efficient, and effective Monte Carlo search strategy to simultaneously optimize all the variables. We find that gradient steepness is an important parameter that influences peak capacity and an optimum range of gradient steepness exists in which the peak capacity is maximized.
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Isocratic and gradient Elution Chromatography: A comparison in terms of speed, retention reproducibility and quantitation
Journal of Chromatography A, 2006Co-Authors: Adam P. Schellinger, Peter W. CarrAbstract:Chromatographers are cautioned to avoid gradient Elution when isocratic Elution will do. In this work, we compared the analytical properties of gradient and isocratic separations of a sample which can be done quite readily under isocratic conditions. We found that gradient Elution gave a shorter overall analysis with similar resolution of the critical pair compared to isocratic Elution without sacrificing repeatability in retention time, peak area and peak height or linearity of the calibration curve. We also obtained acceptable repeatability in peak area/height and linearity of calibrations curves for a sample that required gradient Elution using a practical baseline subtraction technique. Based on these results and related work which show that columns can be reequilibrated by flushing with less than two column volumes of the initial eluent, we conclude that many of the reasons given to avoid gradient Elution deserve serious reconsideration, especially for those samples which are easily separated isocratically. However, we believe isocratic Elution will remain preferable when: (1) the sample contains less than 10 weakly retained components (i.e. the last peak elutes with k' < 5) or (2) the gradient baseline impedes trace analysis.
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analysis of the separability of plate height into overload and intrinsic contributions using the kinetic model of non linear Chromatography
Journal of Chromatography A, 1991Co-Authors: Charles A Lucy, Peter W. CarrAbstract:Abstract In developing optimization strategies for preparative-scale Chromatography it is very convenient, if not entirely valid, to the represent overall peak broadening in terms of the sum of two distinct, independent contributions to the plate height: that portion due to band broadening under linear chromatographic conditions and that due to the effect of mass overload. The kinetic model of non-linear Elution Chromatography is used to demonstrate that this separation of terms is a reasonable approximation under a wide range of chromatographic conditions and to define the limits of this approximation.
Fabrice Gritti - One of the best experts on this subject based on the ideXlab platform.
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Theoretical comparison of the performance of gradient Elution Chromatography at constant pressure and constant flow rate.
Journal of Chromatography A, 2012Co-Authors: Fabrice Gritti, Georges GuiochonAbstract:Abstract The theory of gradient Elution usually assumes a constant flow rate. This works extends it to gradients performed under constant pressure drop and variable flow rates. The peak capacity is derived under both constant flow rate and constant pressure gradient Chromatography by integrating the rate of increase of the peak capacity from the hold-up time to the end of the gradient time. Assuming that the eluent mixture is incompressible, the chromatographic system isothermal, the pressure has no effect on the retention pattern, and neglecting the contributions of the instrument to the total pressure drop and the total peak width, it is found that both modes of gradient Elution Chromatography are strictly equivalent, provided that the Elution time of the last eluted compound and the volume gradient are kept the same in both cases.
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performance of columns packed with the new shell kinetex c18 particles in gradient Elution Chromatography
Journal of Chromatography A, 2010Co-Authors: Fabrice Gritti, Georges GuiochonAbstract:The performance of columns packed with the new 2.6 microm Kinetex-C(18) shell particles was investigated in gradient Elution Chromatography and compared with those of the 2.7 microm Halo-C(18) shell particles and the 1.7 microm BEH-C(18) totally porous particles. The peak capacities P(c) of these columns were derived from the resolution of the components of a peptide mixture (beta-Lactoglobulin digest) and of a mixture of two biomolecules (insulin and lyzozyme).The three columns exhibit the same peak capacities for the peptides at low linear velocity (u(0) < 0.05 cm/s) and at any gradient steepness (0.8 < G < 10). When the linear velocity is increased 10-fold, the peak capacity of the Kinetex column remains nearly unchanged while those of the Halo-C(18) and the BEH-C(18) columns decrease by 20%, approximately. This result confirms the very flat HETP curve, the very low C term of the Kinetex column and its ability to successfully operate at high flow rates while experiencing less efficiency loss than other columns. Despite its smaller average mesopore size (96 A versus 130 A), the column packed with 2.6 microm shell Kinetex-C(18) particles gives an equivalent or even slightly better separation of biomolecules having a size and a mass around 40 A and 15 kDa, respectively, than the column packed with 1.7 microm BEH-C(18) totally porous particles. This result demonstrates the advantages of the shell versus the conventional particle technology when it comes to resolve mixtures of large and slow diffusive biomolecules.
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Exact peak compression factor in linear gradient Elution. I. Theory.
Journal of chromatography. A, 2008Co-Authors: Fabrice Gritti, Georges GuiochonAbstract:The only existing expression for the peak compression factor in linear gradient Elution Chromatography assumes that the linear-solvent-strength model (LSSM) applies to the retention of the compound studied, that the column efficiency is independent of the mobile phase composition, and that, during gradient Elution, the relative retention factor of a compound inside its band varies linearly with the distance from the band center. Because the retention factors of many analytes in reversed-phase liquid Chromatography do not rigorously follow the LSSM, we extend the theoretical approach of Poppe et al. to the prediction of peak compression factors in linear gradient Elution Chromatography for any retention model, when column efficiency varies with the mobile phase composition. Only the contribution of the chromatographic column to the peak compression was taken into account, the contribution of the dwell volume being neglected. A second restriction is the linearity of the relative retention factor as a function of the position along the band width inside the column. These constraints could be the sources for the difference observed between experimental and theoretical values of peak compression factors. When the retention factor varies steeply with the mobile phase composition, such as with proteins or large peptides in RP-HPLC, it is found that the thermodynamic compression term, which tends to sharpen the peak, is coupled with the column dispersion term, which tends to broaden the peak. This coupling term acts as an apparent dispersion term, contributing to broaden the peak. This result is consistent with the measurements of peak compression factors found in the literature.
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the ultimate band compression factor in gradient Elution Chromatography
Journal of Chromatography A, 2008Co-Authors: Fabrice Gritti, Georges GuiochonAbstract:The equations predicting the ultimate time band compression factor, G=(t(R)-t(F))/t(p) in linear gradient Elution Chromatography, for an infinitely narrow injection (injection time t(p)-->0) were derived for an ideal-model column (dispersionless Chromatography, H=0) assuming the Linear Solvent Strength Model for the retention behavior of the analyte. Numerical solutions can readily be obtained when the LSSM model does not apply. The results can be generalized to any retained organic modifier (k'(A)) in the mobile phase. The stronger the retention of the organic modifier, the more effective the band compression. Dispersion in real chromatographic column (H not equal 0) affects the limits that can be reached in linear gradients but poorly in step gradients. Examples based on a conventional HETP of about 12 microm using a 5 microm particle packed column reveal that the best time compression factor that could be expected is twice the one predicted with an ideal column.
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the bandwidth in gradient Elution Chromatography with a retained organic modifier
Journal of Chromatography A, 2007Co-Authors: Fabrice Gritti, Georges GuiochonAbstract:The variance of a chromatographic band is derived in the case of RPLC gradient Elution when the organic modifier is significantly retained onto the stationary phase. This derivation is based on the extension of a model due to Poppe et al. [H. Poppe, J. Paanakker, M. Bronckhorst, J. Chromatogr., 204 (1981) 77] which assumes that the gradient front remains unchanged and propagates along the column at the same speed as the mobile phase, following piston flow. Theoretical and experimental results are compared in the case of caffeine on a C1-silica stationary phase eluted with an acetonitrile gradient. The actual retention behaviors of caffeine and acetonitrile were implemented in the theoretical calculations. The model predicts compression factors between 0.71 and 0.34 for relatively smooth gradient steepness, βt0βt0, between 0.009 and 0.054 while the corresponding experimental band compression factors vary between 1.01 and 0.43 for the very same gradient steepness. The model underestimation of these factors arises likely from the strong deviation of the actual retention behavior from the prediction of the Linear Solvent Strength Model (LSSM).
Giorgio Carta - One of the best experts on this subject based on the ideXlab platform.
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analysis of gradient Elution Chromatography using the transport model
Chemical Engineering Science, 2020Co-Authors: Giorgio Carta, Shamsul Qamar, Nazia Rehman, Andreas SeidelmorgensternAbstract:Abstract A transport model is considered to describe gradient Elution in liquid Chromatography in packed beds with the linear isotherm dependent on the mobile phase modulator. By applying a coordinate transformation, the model is solved analytically using the Laplace transform approach. The moment generating property of the Laplace domain solution is used to derive analytical expressions for the first three moments of the response to rectangular injections. These moments are instructive for analyzing the retention time, band broadening and asymmetry of Elution profiles. Compared to isocratic Elution, the derivation of analytical solutions and moments for gradient Elution is more complicated, because the retention behavior of the solutes depends on the varying mobile phase modulator. Several case studies are evaluated theoretically. To gain confidence on the derived analytical results, a high-resolution finite volume scheme is also applied to solve the same model equations numerically. The analytical solutions and moments provided are utilized to predict the effects of starting and ending times of gradient, magnitude of modulator concentration variation, gradient slopes, and mass transfer coefficient on retention and peak shape. The analytical moment expressions derived can be used to determine retention and mass transfer parameters from experimental peaks and to predict Elution behaviors if these parameters are known.
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separation of monoclonal antibody monomer dimer mixtures by gradient Elution with ceramic hydroxyapatite
Journal of Chromatography A, 2020Co-Authors: Yiran Wang, Giorgio CartaAbstract:Modeling the chromatographic separations of proteins at manufacturing scale is important since downstream processing costs are often dominant. At such scales, the columns are highly overloaded heightening the challenge of predicting performance. In this work, the separation of a monoclonal antibody monomer-dimer mixture is conducted by gradient Elution Chromatography with ceramic hydroxyapatite (CHT) columns Type I and Type II under overloaded conditions. Phosphate gradients are shown to be preferable over sodium chloride gradients since the latter result in undesirable pH transitions generated within the column itself. Using sodium phosphate gradients separation is obtained with both CHT types, achieving approximately 90% recovery at 99% monomer purity starting with a mixture containing 30% dimer at total protein loads up to 30 mg/mL. Because of its higher binding capacity, even higher loadings can be obtained with CHT Type I without monomer breakthrough. A hybrid model is developed to describe the separation. The model, based on an empirical description of two-component, competitive isotherms at low sodium phosphate concentration coupled with the stoichiometric displacement model at higher sodium phosphate concentrations, is in good agreement with the experiments using the linear driving force (LDF) approximation to describe adsorption/desorption kinetics. The same LDF rate coefficient predicts the separation at loadings between 0.8 and 30 mg/mL. The model developed in this work can be used as a general tool to optimize operating conditions, understand what factors limit performance, and compare different operating modes.
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systematic interpolation method predicts antibody monomer dimer separation by gradient Elution Chromatography at high protein loads
Biotechnology Journal, 2019Co-Authors: Arch Creasy, Jason M Reck, Timothy M Pabst, Alan K Hunter, Gregory Barker, Giorgio CartaAbstract:A previously developed empirical interpolation (EI) method is extended to predict highly overloaded multicomponent Elution behavior on a cation exchange (CEX) column based on batch isotherm data. Instead of a fully mechanistic model, the EI method employs an empirically modified multicomponent Langmuir equation to correlate two-component adsorption isotherm data at different salt concentrations. Piecewise cubic interpolating polynomials are then used to predict competitive binding at intermediate salt concentrations. The approach is tested for the separation of monoclonal antibody monomer and dimer mixtures by gradient Elution on the cation exchange resin Nuvia HR-S. Adsorption isotherms are obtained over a range of salt concentrations with varying monomer and dimer concentrations. Coupled with a lumped kinetic model, the interpolated isotherms predict the column behavior for highly overloaded conditions. Predictions based on the EI method shows good agreement with experimental Elution curves for protein loads up to 40 mg mL-1 column or about 50% of the column binding capacity. The approach can be extended to other chromatographic modalities and to more than two components.
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adsorption equilibrium and kinetics of monomer dimer monoclonal antibody mixtures on a cation exchange resin
Journal of Chromatography A, 2015Co-Authors: Jason M Reck, Timothy M Pabst, Alan K Hunter, Xiangyang Wang, Giorgio CartaAbstract:Abstract Adsorption equilibrium and kinetics are determined for a monoclonal antibody (mAb) monomer and dimer species, individually and in mixtures, on a macroporous cation exchange resin both under the dilute limit of salt gradient Elution Chromatography and at high protein loads and low salt based on batch adsorption equilibrium and confocal laser scanning microscopy (CLSM) experiments. In the dilute limit and weak binding conditions, the dimer/monomer selectivity in 10 mM phosphate at pH 7 varies between 8.7 and 2.3 decreasing with salt concentration in the range of 170–230 mM NaCl. At high protein loads and strong binding conditions (0–60 mM NaCl), the selectivity in the same buffer is near unity with no NaCl added, but increases gradually with salt concentration reaching high values between 2 and 15 with 60 mM added NaCl. For these conditions, the two-component adsorption kinetics is controlled by pore diffusion and is predicted approximately by a dual shrinking core model using parameters based on single component equilibrium and kinetics measurements.
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protein Chromatography process development and scale up
2010Co-Authors: Giorgio Carta, Alois JungbauerAbstract:Preface DOWNSTREAM PROCESSING OF BIOTECHNOLOGY PRODUCTS Introduction Bioproducts and their Contaminants Bioprocesses Role of Chromatography in Downstream Processing References INTRODUCTION TO PROTEIN Chromatography Introduction Basic Principles and Definitions Modes of Operation Performance Factors Separation Performance Metrics Chromatography MEDIA Introduction Interaction Types and Chemistry Buffers and Mobile Phases Physical Structure and Properties LABORATORY AND PROCESS COLUMNS AND EQUIPMENT Introduction Laboratory-scale Systems Process Columns and Equipment ADSORPTION EQUILIBRIA Introduction Single Component Systems Multi-component Systems References ADSORPTION KINETICS Introduction Rate Mechanisms Batch Adsorption Kinetics DYNAMICS OF Chromatography COLUMNS Introduction Conservation Equations Local Equilibrium Dynamics Multi-component Systems Displacement Development EFFECTS OF DISPERSION AND ADSORPTION KINETICS ON COLUMN PERFORMANCE Introduction Empirical Characterization of Column Efficiency Modeling and Prediction of Column Efficiency GRADIENT Elution Chromatography Introduction General Theory for Gradient Elution with Linear Isotherms LGE Relationships for Ion Exchange Chromatography LGE Relationships for RPC and HIC Separations with pH Gradients Modeling Gradient Elution with Non-linear Isotherms References DESIGN OF CHROMATOGRAPHIC PROCESSES Introduction Chromatographic Process Steps and Constraints Design for Capture Design for Chromatographic Resolution SMB Design References INDEX
Chunhua Yang - One of the best experts on this subject based on the ideXlab platform.
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a max min control problem arising in gradient Elution Chromatography
Industrial & Engineering Chemistry Research, 2012Co-Authors: Qinqin Chai, Ryan Loxton, Kok Lay Teo, Chunhua YangAbstract:Gradient Elution Chromatography is an industrial process used to separate and purify multi-component chemical mixtures. In this article, we consider an optimal control problem in which manipulative...
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a max min control problem arising in gradient Elution Chromatography
Industrial & Engineering Chemistry Research, 2012Co-Authors: Qinqin Chai, Ryan Loxton, Kok Lay Teo, Chunhua YangAbstract:Gradient Elution Chromatography is an industrial process used to separate and purify multi-component chemical mixtures. In this article, we consider an optimal control problem in which manipulative variables in the chromatographic process need to be determined to maximize separation efficiency. This problem has two nonstandard characteristics: (i) the objective function is nonsmooth, and (ii) each state variable is defined over a different time horizon. The final time for each state variable, the so-called retention time, is not fixed and actually depends on the control variables. To solve this optimal control problem, we first introduce a set of auxiliary decision variables to govern the ordering of the retention times. Then, we approximate the control by a piecewise-constant function and apply a novel time-scaling transformation to map the retention times and control switching times to fixed points in a new time horizon. The retention times and control switching times become decision variables in the ne...