Handling Uncertainty in Model-Based Optimal Experimental Design

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Ind. Eng. Chem. Res. 2010, 49, 5702–5713

Handling Uncertainty in Model-Based Optimal Experimental Design Tilman Barz,* Harvey Arellano-Garcia, and Gu¨nter Wozny Chair of Process Dynamics and Operation, Berlin Institute of Technology, Sekr. KWT-9, Strasse des 17. Juni 135, D-10623 Berlin, Germany

In contrast to the majority of published works in the field of model-based optimal experimental design which focused on numerical studies so as to demonstrate the validity of the OED approach or the development of new criteria or numerical approaches, this work is mainly concerned with the experimental application and practical insights gained from the adaption of an optimal design framework. The presented work is discussed based on the determination of protein ion-exchange equilibrium parameters. For this purpose, special attention is paid to the explicit modeling of all laboratory steps so as to prepare, implement, and analyze experiments in order to have a realistic definition of the numeric design problem and to formally include experimental restrictions and sources of uncertainties in the problem formulation. Moreover, whereas the effect of erroneous assumptions in the initially assumed parameter values have been covered by various authors, in this work, uncertainties are considered in a more general way including those which arise during an imprecise implementation of optimal planned experiments. To compensate for uncertainty influences, a feed-back based approach to optimal design is adopted based on the combination of the parallel and sequential design approaches. Uncertainty identification is done by solution of an augmented parameter estimation problem, where deviations in the experimental design are detected and estimated together with the parameter values. It has been shown that uncertainty influences vanish along with the iterative refinement of the experiment design variables and estimated parameter values. 1. Introduction Optimal experimental design (OED) techniques have become widely adopted for the development of mechanistic process models in systems engineering. The decision on appropriate experiments to be conducted is a critical issue in order to obtain an accurate parameter estimation and to reduce time and experimental effort as well. A recent overview over model-based design of experiments and a critical state of the art analysis is given by Franceschini and Macchietto.1 The generation of an OED comprises primarily the identifiability of the parameters to be estimated. Furthermore, when several possible process models are available, additional aspects such as model distinguishability should also be considered. Most of the previous works in OED are based on numerical examples in order to demonstrate their validity or the development of new numerical approaches. There are rather few publications available with experimental results which provide relevant insights on the appropriate formulation and solution of OED problems for the practical engineer in order to cope principally with practical limitations, for example, limited instrument accuracy or restrictions in the operating range and/ or equipment. For instance, a practical case study is presented by Franceschini and Macchietto2 for a biodiesel production process, where parameters of a complex kinetic network are identified. For this purpose, strategies are proposed to cope with special problems, which can occur in the parameter estimation of complex and highly nonlinear systems such as those which make use of Arrhenius’ equations. In this work, a reliable determination of adsorption isotherm parameters for a bioprocess is presented for which data generation is a very time-consuming, labor-intensive, and costly job. Since there is no theoretical tool available for the prediction of adsorption isotherms based on physic-chemical data, equi* To whom correspondence should be addressed. E-mail: tilman.barz@ tu-berlin.de.

librium models have to be determined experimentally. Moreover, despite the fact that there are several experimental methods available, the experimental determination of isotherms is still far away from being a routine job.3 Therefore, especial attention has to be paid to the compensation of uncertainties as they exist while putting the experimental design framework into practice and dealing with experimental results. The remainder of the paper is organized as follows, first the general systematic procedure and basic numerical principles from planning to implementation and experiment analysis are briefly reviewed. Second, special attention is paid to the influence of uncertainties during this procedure. In the subsequent section, the impact of various uncertainties is first demonstrated using a numerical example. Afterward, the formulation of an OED-problem for the determination of adsorption isotherm parameters is described in detail. Moreover, the interaction of uncertainties is studied and a need for their compensation can be demonstrated based on a feed-back strategy (sequential approach) together with the identification of the most relevant uncertainties. 1.1. Optimal Experimental Design Problem Formulation. OED is aimed at selecting conditions of experiments that generate a maximum information content for the determination of specific parameters, θ, of an underlying general nonlinear process model, g. g(x, u, θ, p, t) ) 0

(1)

In eq 1, x and u represent dependent state and free design variables, respectively, θ denotes the parameter set to be determined and p are all other independent model and/or experimental design parameters, which are constant (and have a known and assumed value). The estimation of parameters θ is realized by fitting the corresponding simulated model output y to the measurement data ymeas. Assuming an absence of systematic errors, the nonlinear regression reads: where measurement errors ξy are assumed to be zero-mean white noise

10.1021/ie901611b  2010 American Chemical Society Published on Web 05/11/2010

Ind. Eng. Chem. Res., Vol. 49, No. 12, 2010

max φ{F(u, θ, p)}

(7b)

u

characterized by the measurement-covariance matrix MV. When considering more than one experiment (NExp > 1), the objective function of the parameter estimation problem subject to the model equations in eq 1 is given as follows: NExp

min θ

∑ (y

meas k

- yk(uk, θ, pk))T · MVk-1 · (ymeas - yk(uk, θ, pk)) k

k)1

(3) The accuracy of the estimated parameters θ is characterized by the parameter covariance matrix θV. NExp

θV g F-1 ) (

∑F )

-1

k

(4)

k)1

Equation 4 gives a linear approximation of θV based on the so-called Fischer information Matrix F.4 Its value results from summing over all individual contributions Fk. For one single experiment, Fk is calculated on the basis of the sensitivities of measured variables with respect to the parameters weighted with the inverse measurement-covariance matrix MVk as defined in eq 5.4

( ) ∂yk ∂θ

Fk )

T

· MVk-1 ·

( ) ∂yk ∂θ

(5)

It is though sometimes convenient to use the normalized covariance matrix θV, which is obtained using the normalized sensitivities according to eq 6.

( )

∂yk ∂yk ) ·θ ; ∂θi ∂θi i

∀i ∈ Nθ

(6)

Equation 4 gives exact values for θV in the case of a process model, which is linear in states and parameters. However, for a general nonlinear process model (see eq 1), the approximation represents only a lower bound on the true parameter covariance matrix θV.5 Together with the assumption that all uncertainties in measurement and model parameters can be represented by a Gaussian probability distribution, eq 4 is a generally accepted approach in parameter estimation and OED problem formulations.1 It can be seen from eq 4 and eq 5 that θV depends not only on the sensitivities of the model output with respect to the parameters, but also on the measurement accuracy defined by MV. In the multivariate case, correlations in sensitivities and measurements are also considered. Generally, high and uncorrelated sensitivities as well as exact and uncorrelated measurements (indicated by small values in MV) will enable an accurate parameter estimation characterized by small values in θV. The formulation of an OED-problem is aimed at maximizing the parameter accuracy by minimizing the values in θV. Considering the dependencies of y in eq 2, as well as eq 4 and eq 5, this can be accomplished by selecting optimal design variables u for the actual parameter estimate θ and all other constant parameters p. The objective function in eqs 7 can be formulated using different standard metrics φ which provide a scalar indicator for parameter accuracy. min φ{θV(u, θ, p)} u

(7a)

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The metrics φ can be applied either to the approximated parameter covariance matrix θV or to the Fischer information matrix F. Common metrics are the so-called A-, D-, and E-optimal criteria, whose definition and discussion on their suitability can be found elsewhere.1 It is anyway important to note that the review on the application of different criteria presented by Franceschini and Macchietto1 is only restricted to the case defined in eq 7b. In this work, the definition in eq 7a is used instead, where φ applies directly to θV.6,7 The use of this definition can be interpreted graphically by considering the confidence region of the estimated parameters. The limits of this region are defined by θV and have an ellipsoid shape. The size of the ellipsoid represents uncertainties in θ. The A-optimal criterion corresponds to the minimization of the sum of the ellipsoids principle axes. Its value is given by the matrix trace. In contrast, the often referenced D-optimal criterion means to minimize the determinant of θV. Its value is equivalent to the surface area of the confidence ellipsoid. Applied to θV, the D-optimal criterion can lead to thin confidence regions, but still possessing a significant length in one single direction of the major axis of the confidence ellipsoid because of the correlations between parameters. Comparing the A- and D-optimal criteria applied to θV, the A-optimal criterion is deemed to be more suitable for the minimization of parameter cross-correlation, which is especially important when parameters with a physical meaning have to be determined. Finally, the E-optimal criterion aims at minimizing the major principal axis of the confidence ellipsoid, which is obtained from the largest eigenvalue, λimax. For strong correlated parameters (with a very large major axis compared to the other axes) the optimal result is similar when applying the A- or the E-optimal criterion.8 The advantage of the A-optimal criterion is that it does not include discontinuities, which can result if the orientation of the major axis switches during optimization. On the basis of these facts and using eq 7a instead of eq 7b, the stated drawbacks related to the application of the A-optimal criterion by Franceschini and Macchietto,1 which could cause an appreciable loss of information in the case of high crosscorrelation between parameters, are not valid here. When applied to the Fischer information matrix F, the matrix off-elements are not considered, and thus, correlations are neglected. In contrast, the parameter covariance matrix θV is based on the inverse of F, and thus, correlations are considered in the A-optimal criterion. Equation 8 shows the A-optimal criterion applied throughout this work. Nθ

φ{θV(u, θ, p)} ) trace[θV(u, θ, p)] )

∑ λ (θV(u, θ, p)) i

i)1

(8) 1.2. The Role of Uncertainties in the Optimal Experimental Design Approach. The determination of model parameters can generally be seen as a sequence of three consecutive steps: (1) computation of the OED variables, u0, based on the current parameter estimate, θ0, and based on some assumptions about constant design variables, p0; (2) implementation of the planned experiments, analysis of the measured variables, ymeas, and (if possible) verification of the assumed experimental conditions, u0 f u* and p0 f p*; (3) estimation of new model parameters, θ0 f θ* and statistical assessments, and (if possible) estimation of the values of the implemented variables u0 f u* and p0 f p* (see Figure 1). For the solution of the design problem in eq

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Figure 1. General solution approach to parameter determination using an OED framework and possible sources of uncertainties. The use of the feed-back loop corresponds to the sequential strategy.

7a, an initial guess for the parameters, θ0, is adopted, that is, the best currently available parameter values. On the basis of the general nonlinear process model in eq 1, the quality of the computed experimental design depends on the accuracy of θ0. Therefore, the OED-solution is usually termed as local optimum1,9 and is a function of the error in the initial parameter guess, ξθ. After planning and implementation of optimal experiments, the parameter estimation problem in eq 3 is solved, and the parameters are then updated, θ* f θ0. As shown in Figure 1, besides uncertainties in the parameter values, ξθ, additional uncertainties related to the practical realization of the experiments can have an influence on the design procedure. The problem of an imprecise implementation of desired decisions is a common phenomenon in control applications (e.g., the realization of a set-point in a control loop, which is subject to disturbances). According to Skogestad, et al.,10 the term implementation error is adopted to indicate that certain design values deviate from originally planned or assumed process conditions. These errors usually affect certain experimental conditions with an assumed and fixed value, p0 + ξp, as well as the computed OED variables, u0 + ξu, which can not be implemented exactly as desired.

regions) with a drastic less experimental effort in the presence of uncertainties.6,16 However, when multiple equipment pieces are available, parallel planned experiments can be advantageous in terms of time and use of resources. The concept of modelbased design of parallel experiments is presented by Galvanin, et al.,17 together with a novel design criterion, which aims at maximizing complementary information by the consideration of different eigenvalues in the information matrix. For the robustness with respect to uncertainties, a combination of both strategies, the so-called parallel/sequential approach, can be used, where parallel planned experiments are designed in a subsequential manner. Additionally, it has to be noted that in the case of dynamic experiments, the consequent application of the sequential approach means to exploit available information as soon as possible, and consequently, to start the redesign of experiments online.18,19 Generally, the application of the sequential design approach means that the design of a certain number of n optimal experiments is considered together with m already planned experiments. Accordingly, eq 10 represents the sum over constant and variable parts of the FischerInformation Matrix: m

θ* ) θ + ξ u* ) u0 + ξu p* ) p0 + ξp 0

F)

θ

(9)

Depending on their sensitivities with respect to the adopted design criterion, all considered uncertainties may generally cause large additional losses in the optimality of a specific design. Thus, the need for a robust design, which is insensitive to parametric uncertainties, has commonly been discussed.1,6,11-15 However, the source of uncertainties is usually assigned to unreliable initial guesses of θ. The most widely used approach to cope with uncertainties is the indirect method,13 which relies on an iterative refinement of the experimental design. The indirect method, also referred to as sequential design strategy,16 is described in Figure 1, where a feed-back loop is used. Here, experiments are alternately designed, experimental data are collected, and parameters are estimated. The robustness is based on the identification of uncertainties from previous implemented experiments and the stepwise improvement of the values and the confidence regions of the parameters. The sequential approach can provide reliable estimates (high parameter accuracy with small confidence

∑F

k

k)1

m+n

+

∑

Fk(uk, θ, pk)

(10)

k)m+1

In contrast to the indirect method, which is based on a feedback-based approach, in the direct method, uncertainties are defined and considered a priori (feed-forward approach). The consideration of all uncertainties leads then to the maximization of the probability density function of the design objective. As a result, the solution of a single design problem shows already robustness against uncertainties, for example, deviations from initially assumed parameters or optimally planned design variables. Consequently, dependent and independent model variables cannot be separated, and thus, a more general implicit or so-called error-in-Variables model formulation is needed.15 However, recent approaches for a robust design focus on unreliable initial parameter guesses. Two different criteria are used: first, the expected value criterion, which optimizes the design criterion on average, and second, the minimax criterion, which accounts for the worst possible performance.13 The drawback of the direct method is that both criteria rely on the description of the size of the parameter uncertainty (here the size of the error in ξθ) using either a probability distribution in the first case, or the knowledge of an admissible parameter


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domain in the second one. This information can either result from basic assumptions or when experiments are designed sequentially. The distribution may be obtained from prior experiments. For the expected value criterion, the objective function reads in the case of eq 7a as follows: min E[φ{θV(u, θ, p)}]; u

with θ ∈ Θ

(11)

where E[ · ] is the expected value and Θ is the probability space of θ indicating the realizable set of parameter values, which can be defined, for example, by a uniform or Gaussian distribution function. Applications of this criterion have been presented by Walter,13 for a hypothetical first order example, and by Asprey and Macchietto,12 for a small semicontinuous bioreactor model. For the minimax criterion the following objective function is used in the case of eq 7a: min max φ{θV(u, θ, p)}; u

θ

with θ ∈ Θ

(12)

Here, the OED problem is solved for the worst case parameter set obtained from the solution of the inner maximization problem. In contrast to eq 11, where the design tries to ensure optimality on average (but can be even very poor for certain parameters), here the algorithm focuses on parameter sets which represent always the worst case in the domain Θ. Applications of the minimax design have been presented by Dette et al.,11 for a classical biological Monod growth model, by Asprey and Macchietto,12 for the example of a small semicontinuous bioreactor model, by Bock et al.,6 for a biochemical problem from enzyme kinetics, and finally by Ko¨rkel et al.,7 for the reaction of urethane. In the last two examples, the authors used a modification of eq 12 based on a Taylor expansion of the parameters in the min-max objective function, which is then solved more efficiently. The main advantage of the minimax approach is certainly the relatively simple definition of parameter uncertainty by simple bounds of the admissible region. The computational time for solving eq 12 is greater than the time needed to compute a nonrobust experiment (eq 7a). However, the application of the robust design allows the reduction of the number of real experiments, and thus, the time necessary to identify the parameters. Finally, it should be noted that for the parameter determination (eq 3), the term robustness is linked to a solution which is nonsensitive to outliers in data. For numeric experiments, Kostina,20 presents results which perform better using a l1-based parameter estimation rather than the traditional l2-based approach. 2. Impact of UncertaintiessAn Illustrative Example The influence of various uncertainties on the information content of an experimental design is now discussed first using a simple dynamic process model. The system of ordinary differential equations is taken from Espie and Macchietto21 and describes a semicontinuous (fed-batch) fermentation of baker’s yeast. Using Contois kinetics together with a constant specific death rate, the following equations describe the consumption of biomass and substrate in the reactor: dx1 ) (r - p2 - u1) · x1 dt r · x1 dx2 )+ u1 · (u2 - x2) dt p1 θ1 · x2 r) θ2 · x1 + x2

(13)

Figure 2. OED results based on the “true” parameter set, θ*, and yielding the maximum information content for this design problem (theoretically best design).

where x1 and x2 denote biomass and substrate concentrations, respectively. Both are given in g/L. The parameters for the Contois kinetics, θ ) [θ1, θ2]T, are estimated using OED. Here, the “true” parameter set is given with θ* ) [0.30, 0.03]T. The parameters, p0 ) [0.55, 0.03]T, are assumed to be known constants. Whereas the initial substrate concentration is fixed with x2(t ) 0) ) 0.01 g/L, the initial biomass concentration, x1(t ) 0), can be chosen between 1.0 and 10.0 g/L. Together with the dilution factor of the feed, u1 (range 0.05-0.20 h-1) and the substrate fed to the reactor, u2 (range 5-35 g/L), the experimental design vector is u ) [u1, u2, x1(t ) 0)]T. Furthermore, it is assumed that the state variables can be measured at equidistant points in time, Mi; with i ∈ {1, · · · , 10} and the measurement y ) [x1, x2]T. An OED problem is formulated which minimizes the A-optimal criterion as defined in eq 8. However, it has to be noted that the resulting problem shows plenty of local minima such that a combination of a global and local search algorithm may be suitable in order to solve this kind of problems.8 Figure 2 shows the solution when no uncertainties are considered and the true parameter set is known a priori. As discussed above, the assumption that θ0 is equal to θ* is not realistic. Instead, the initially assumed θ0 will deviate from the “true” estimated parameter set θ*. As a result, the computed optimal experimental design (using θ0) is not optimal for θ*. In other words, with larger deviations of θ0 from θ*, the OED differs from the design shown in Figure 2 and the accuracy of θ* deteriorates. To account for these deviations, the minimax problem: minu maxθ0 trace[θV(u, θ0, p)] (see also eq 12) has been solved for four different scenarios: The first scenario corresponds to the ideal case θ0 ) θ*, where the minimax problem is reduced to a deterministic minimization problem. The other three scenarios consider increasing uncertainties in θ0, being θ0 ) θ* + ξiθ with the corresponding uncertainty intervals ξθ ∈ {(10%, (20%, (30%} · θ*, respectively. The problems are solved by a uniform sampling of the uncertain parameter values in their respective intervals together with a repeated solution of the OED-problem and incorporating the corresponding worst case result. Figure 3 shows 95% confidence regions of the normalized “true” parameters θ* for all four scenarios. The confidence regions indicate the accuracy of θ*. It can be seen that the parameter accuracy, and thus, the quality of the experimental design depends directly on the uncertainties attached to the initially assumed parameter values θ0. Besides the uncertainties in θ0, implementation errors are also considered, which result from an imprecise realization of

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Figure 3. The 95% confidence region of the normalized “true” parameter j *, when the “true” parameter set, θ*, is known a priori (θ0 ) θ*) and set, θ the worst cases (w.c.) when θ0 deviates from the “true” parameter value θ*. Table 1. Numeric Results for OED with an Increasing Uncertainty Level A-optimal criterion worst case considered uncertainties nominal (0% θ* + ξ u0 + ξu p0 + ξp

θ

0.001759

(10%

(20%

(30%

0.006226 0.012190 0.101032 0.004318 0.015401 0.024500 0.002140 0.003020 0.003704

the optimally planned decisions u0. Thus, the worst case (w.c.) is again calculated for four different scenarios, namely: u* ) u0 (with no implementation errors) and u* ) u0 + ξiu with ξu ∈ {(10%, (20%, (30%} · u0. The respective parameter accuracy is obtained by a simulation based on a uniform sampling using perturbed values of u0. In the same way, the impact of uncertainties in the assumed parameters p has been studied. Table 1 shows the A-optimal criterion as a scalar indicator for the parameter accuracy for all three considered sources of uncertainties: ξθ, ξu, ξp. The example demonstrates that for small uncertainties (deviations from the true parameter set, θ*, from the planned decisions, u0, and from the assumed design conditions, p0), the information content of an OED can degrade significantly. Summing up, it can be concluded that the OED framework will loose its significant effect whenever uncertainty is not explicitly incorporated and/or compensated. The main sources of uncertainties as they occur during the different steps of an OED have been depicted in Figure 1. 3. Case StudysDetermination of Protein Adsorption Isotherms Errors in the initially assumed parameter values will always exist at the beginning of an optimal design for parameter precision. For a strong nonlinear process model, initial guesses can vary over several orders of magnitude from the finally estimated values, for example, in equilibrium relations or reaction kinetics.2,8 Accordingly, it is not always possible to define reasonable intervals for the uncertainty size in initial parameter guesses. Moreover, implementation errors (deviations from initially assumed or optimally planned process conditions) can generally be identified using measurement data, which is available after the implementation of optimal planned experiments. Thus, from a practical point of view, the a priori consideration of uncertainties, as preconditioned in the direct approach and which implies the solution of a worst-case or minimax OED problem, shall be deemed to be not appropriate. In this work we use the indirect method (sequential approach),

where erroneous assumptions and uncertainties are compensated by an iterative refinement and repeated solution of OED and parameter estimation problems. For this purpose, the parameter estimation problem is augmented for the identification and partial compensation of implementation errors. The following case study describes the experimental determination of adsorption isotherms which are generally used in modeling, scale-up, and control of high performance liquid chromatography. The proteins considered in this work (βlactoglobulin A and B) are part of the potential important commercial proteins, which can be extracted from milk whey.22,23 In bioprocess engineering, ion-exchange chromatography is used for their separation and purification.24 The adsorbent material used here is a strong anion exchanger Source 30Q from GEHealthcare (Munich, Germany). It is composed of rigid, monodispersed, spherical serine particles with a diameter of 30 µm. However, the wide range of retentivities of the macromolecules makes it difficult to separate them under isocratic conditions. Thus, using an additional component (here salt ions) with a variable concentration increases the eluent strength, i.e. the retentivities of the eluites, and thus, the chromatographic separation can be improved. This is the so-called nonisocratic operation, which is a relevant operation mode in liquid chromatography.24 For more detailed information on the biological system, the experimental setup, the resulting parameter estimation problem, and the evaluation of the results, we refer to Barz et al.25 3.1. Equilibrium Model and Parameters. Adsorption isotherms describe the equilibrium for different components, NC, between pore surface and fluid phase in the macropores of an adsorbent. Thus, equilibrium models link the adsorbed stationary phase concentration, qjeq, to the free liquid concentration, cjeq. Accordingly, the equilibrium model consists of NC equations and some constant parameters θ: eq qeq j ) f(cj , θ);

with j ∈ {1, · · · , NC}

(14)

For the explicit consideration of nonisocratic operation in chromatographic protein separation, Brooks and Cramer26 developed a steric mass (SMA) ion-exchange equilibrium formalism, which explicitly accounts for the steric hindrance of salt counterions upon protein binding in multicomponent equilibria. In analogy to the variable coefficient multicomponent Langmuir isotherms, the SMA formalism can be represented as follows: qeq j )

Λ · Rj,1 · ceq j

∑R

i,1

i

· (σi + νi) · ceq i

;

with j ∈ {1, ..., NC} ) ˆ {Cl-, β-LgA,β-LgB} (15)

Equation 15 gives the equilibrium concentration in the liquid and stationary phase for NC components. Besides the proteins, salt ions are here used as modulator in the nonisocratic system and designated as the first component. The SMA parameters are the stationary phase capacity for the salt counterions, Λ [mM], the dimensionless constant for the characteristic charge, νj, and the dimensionless steric hindrance factor, σj. The separation factor, Rj, 1, is the variable coefficient and defined as follows: Rj,1 ) k1,j ·

() qeq 1 ceq 1

νj-1

;

with j ∈ {1, · · · , NC}

(16)

Ind. Eng. Chem. Res., Vol. 49, No. 12, 2010 26

According to Brooks and Cramer salt ions (Cl-) hold the values k1,1 ) 1;

σ1 ) 0;

the SMA parameters for ν1 ) 1

(17)

Thus, the unknown parameter space is given by θ ) [Λ, k1,j, νj, σj]T ;

with j ∈ {2, 3} and σ2 ) σ3 (18)

Since the proteins β-LgA and β-LgB have a similar structure, the steric factor of the proteins, σj, is assumed to be equal for both proteins,27-29 and thus, the parameter space is then reduced by 1, being then Nθ ) 6. 3.2. Experimental Design of “Static” Batch Experiments. In the well-tried batch method, thermodynamic equilibrium points are obtained, which characterize fluid phase concentrations, ceq, and the loading of the solid, qeq. For one batch, preset amounts of solutes are equilibrated in a closed vessel, which is filled by parts with adsorbent. It has to be noted that despite the intrinsic dynamic behavior of batch processes, here, initial and final equilibrium states are only analyzed, and therefore, the method is considered a static method. A general overview over recent (alternative) methods for the determination of adsorption isotherms is given by Seidel-Morgenstern.3 The static batch method is well suited for the measurement of multicomponent equilibrium data. The major advantage is the exclusion of kinetic effects during the adsorption process, such as the intraparticular mass transfer, and thus no additional parameters have to be determined. On the other hand, there is a relatively high effort of laboratory work due to the timeconsuming preparation of each batch and the concentration analysis, as well as the required time until equilibrium is reached. Because of the limited accuracy, a systematic consideration of uncertainties is compulsory for a precise determination of the adsorbent amount. The liquid volume in the batch consists of both the free liquid (external) around the adsorbent particle, and the liquid in the particle pores (internal). The adsorbed phase concentration, q, refers to the molecules attached to the adsorbent. The volumetric liquid part of the batch volume, εtot, is calculated as follows: εtot ) εext + (1 - εext) · εint

(19)

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so as to implement a certain equilibrium point. The batch volume, Vbt, is usually determined by the minimum amount of liquid needed for the analysis of the dissolved concentrations. The equilibrium conditions can be selected by choosing the initial amount of solutes and adsorbent in the batch. Thus, the resulting degree of freedom is then: NC + 1. 3.3. Preparation, Analysis, And Uncertainties of a Batch Experiment. The inclusion of practical restrictions is decisive for the formulation of the experimental design problem. Furthermore, the necessary decisions in order to obtain a desired equilibrium point are not obvious, because of the reduction from the dissolved initial protein concentration to the equilibrium concentration. Therefore, the proposed model is extended by additional equations, which principally describe the basic steps done by the laboratory assistant for the batch preparation, which then lead to a desired equilibrium point. To get a maximal accuracy using precise pipettes, components and adsorbent are added as volumetric liquid solution or suspension to the batch (Figure 4), which defines the total batch volume, Vbt. NC

V )V + bt

sl

∑V

std i

(21)

i)1

In eq 21, the superscripts “sl” and “std” denote the slurry with the adsorbent in the solvent and the dissolved standard protein concentrations as well as the solvent enriched with a desired salt concentration, correspondingly. The total porosity sl of the slurry, εtot , is defined in the same way as in eq 19. Its value is influenced by changing the amount of adsorbent in the sl slurry. For a given εtot , the volumetric liquid fraction of the batch is calculated as follows: NC

εsltot · V sl +

∑V

std i

i)1

εtot )

(22)

Vbt

With the corresponding standard concentrations, cjstd; with j ∈ {1, 2, 3}, and a constant solvent salt concentration, csol 1 , which results from the specific pretreatment during the batch preparation, the initial liquid salt concentration in the batch is NC

where εext and εint are the external or interstitial porosity and the internal or intraparticle porosity, respectively. The intraparticular adsorbent porosity is εint ) 0.57, taken from Wekenborg et al.30 The external porosity, and thus the value of εtot, is varied by changing the amount of adsorbent in the batch. During the adsorption process each component has to fulfill the mass balance: ini eq eq εtot · cini j + (1 - εtot) · qj ) εtot · cj + (1 - εtot) · qj ; with j ∈ {1, · · · , NC} (20)

In eq 20, initial concentrations and final equilibrium states are denoted by the superscripts “ini” and “eq”, respectively. The adsorption process is initialized by filling a certain amount of dissolved proteins in the batch, which is partially filled with pure adsorbent. Thus, the initial adsorbed concentration of the proteins is qjini ) 0; with j ∈ {2, 3}. In contrast, the initial salt ion concentration is set to the total adsorbent capacity, qini 1 ) Λ, because of the special pretreatment of the adsorbent.25 Each specific experiment aims to adjust the initial solute and adsorbent amount in the batch

std sol cstd 1 · V 1 + c1 · (

∑V

std i

+ εsltot · Vsl)

i)2

cini 1 )

εtot · Vbt

(23)

and the initial protein concentrations are cini j )

std cstd j · Vj

εtot · Vbt

;

with j ∈ {2, 3}

(24)

For all experiments, the standard protein concentration is set to cjstd ) 0.37 mM; with j ∈ {2, 3}, which together with the corresponding standard volume determines the maximal initial protein concentration, cjini. According to the above-mentioned steps (see also Barz et al.25), a batch experiment is now std determined by the three volumes Vsl, Vstd 2 , V3 , and the salt std concentration, c1 , (Figure 4). Owing to the small batch size of 4 mL, a measurement of salt ion concentration has not been carried out. The analysis of the protein concentration is done using an analytical HPLC by peak area integration of the photometric outlet signal. The concentrations are obtained from

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Figure 4. Preparation of initial and equilibrium state for one single batch experiment.

the specific peak areas, Ajeq, using a linear calibration with the constant parameters mj and nj as follows: eq ceq j ) mj · A j + nj ;

with j ∈ {2, 3}

NC

)

∑A

eq j

(26)

j)2

eq eq T The measured variables, yk ) [Aeq 2 , A3 , Asum] , represent the result of one single batch experiment, k. In addition, some uncertain parameters can be considered, which result from an imprecise preparation of the batch experiments. Uncertainties can here mainly be ascribed to the prepared standard concentrastd std sl tions cstd 1 , c2 , c3 , and the void fraction, εtot. As shown in Figure 4, values of the protein standard concentrations can be verified by measurements. This is done through analysis of the corresponding protein peak areas by using the same linear calibration as in eq 25:

std cstd j ) mj · A j + n j ;

with j ∈ {2, 3}

variable

(25)

A total separation of the specific areas of each individual protein has not been achieved.25 For their independent determination, the overlapping peaks have to be divided manually, which introduces a certain error. However, the sum of the protein concentrations eq Asum is not affected by this division and possesses a much higher accuracy than each single Aeq j . Therefore, it is convenient to consider also Aeq sum as a measurement variable (eq 26), which provides some extra information with a relatively high accuracy. eq Asum

Table 2. Uncertainties Defined as Uncorrelated Zero Mean White Noise for Measured Variables, Uncertain Experimental Conditions, and Design Parameters

(27)

Uncertainties in the protein concentration measurement are assigned to the corresponding protein peak areas. The statistical analysis of several repeated calibration curves revealed that a combination of relative and absolute measurement error reproduces the resulting deviations in the protein peak areas quite sl well. In contrast, the values for εtot and cstd 1 represent empirical approximations of their respective uncertainties. They are based on analysis of the laboratory steps for the preparation of the slurry and the standard salt solution. The corresponding standard deviations used to define the diagonal elements of the measurement covariance matrix, MVk, are shown in Table 2.

standard deviation, σ

Ajeq, Ajstd; j ∈ {2,3}

σ ) σabs + σrel σabs ) 10; σrel ) 1/80 · Aj

eq Asum

σ ) σabs + σrel σabs ) 5; σrel ) 3/400 · Asum

sl εtot cstd 1

σ ) 1/800 σ ) 3/4

3.4. Preliminary Studies with Different Experimental Design Strategies. OED aims at minimizing the number of batch experiments, in other words, the number of certain equilibrium points which are necessary in order to obtain an adequate accuracy of the estimate of the SMA parameters, θ. At a first glance, a simultaneous design of experiments seems to be the first choice for the described problem as it corresponds to the standard procedure in the regular lab work, where several batches are produced and analyzed in parallel. Here, first of all, different design strategies including also the sequential and parallel/sequential approach (see also Figure 1) are compared for a given and fixed initial parameter set, θ0. Using the A-optimal criterion (see eq 8), the general OED problem for k experiments with a restricted maximal available amount of proteins, mjmax, is shown in eq 28. While the number of parameters remains constant, with Nθ ) 6, the number of variables, equations, and inequality constraints depends on the number of experiments to be planned. Accordingly, the number of model equations and states is Nx ) 14 · NExp, the number of measurement equations and measured variables is Ny ) 3 · NExp, the number of design variables is Nu ) 4 · NExp, and the number of constant parameters for each experiment is Np ) 9 · NExp. The diagonal elements of the measurement covariance matrix, MVk, (see eq 5) are defined by their respective values given in Table 2. As seen in Figure 4, neither the values, nor the uncertainties of concentrations and void fractions of the standard volumes as well as the calibration


min φ{θV(u , · · · , u Exp, θ, p , · · · , p Exp) 1 N 1 N u

{

eqs 15, 16, 20-26 std 0 e Vsl + Vstd 1 + V2 e 4 mL

s.t.

0e

0e

with

{

cstd 1

∑

sl

e 400 mM; 0 e V ; 0 e NExp k

{

Vstd j

+

}

cstd j k T

e

mmax ; j

Vstd 2 ;

j∈

0e

{2,

Vstd 3

3}

θ ) [Λ, k1, j, νj, σj] ; σ2 ) σ3 ; j ∈ {2, 3}

std T eq ini { } xk ) [Rj, 1, ceq j , qj , cj , εtot, V1 ] ; j ∈ 1, 2, 3 eq eq T yk ) [Aeq 2 , A3 , Asum] std std T uk ) [Vsl, Vstd 2 , V3 , c1 ] T std sl std pk ) [Vbt, cstd 1 , εtot, c2 , c3 , m2, m3, n2, n3]

}

}

5709

∀k ∈ {1, · · · , NExp}

(28)

∀k ∈ {1, · · · , NExp}

parameters, mj, nj, are necessarily independent. These correlations were neglected here. This is because in some cases neither the implementation nor the analysis of simultaneously planned experiments could be accomplished parallel-wise. Figure 5 shows the results of theoretical studies concerning the evolution of the A-optimal criterion for different design strategies and fixed parameters θ0. It can be seen that a minimum of four parallel planned batch experiments is necessary in order to reach identifiability of all six SMA parameters, θ. Thus, up to a total of 20 experiments were additionally planned. However, highdimensional OED problems exhibit commonly several local minima (e.g., for 10 parallel planned experiments the number of design variables reaches 40). Thus, each problem has been solved repeatedly by a gradient-based SQP algorithm using random start values of the decision variables, u, and choosing the best result.8 It should be noted that in Figure 5 the planning of 4 + 1 + · · · + 1 experiments follows in essence the sequential design strategy, while the planning of 10 + 10 experiments can be seen as nearly a parallel strategy. The remaining ones in Figure 5 comply with a combined parallel/sequential strategy. Surprisingly, the evolution of the A-optimal criterion is not influenced by the strategies applied. Even if experiments are repeated using exactly four times the same “optimal” design variables, which were obtained for five parallel planned experiments (4 · 5 strategy), the evolution of the A-optimal criterion does not differ significantly. Moreover, arising deviations can be assigned to local optima rather than to factual differences in the A-optimal criterion because of a specific design strategy. In contrast to this, examples can be found, for example, in Scho¨neberger et al.,8 where the planning of different experimental designs offers already an advantage for a two-dimensional problem. In this case study, the A-optimal criterion seems to be independent from the applied strategy. This is mainly because of the low accuracy of the measured data. Simply spoken, the information content of the first five measured equilibrium points is already relatively high in terms of identifiability, but the measured data accuracy is still low. For all applied strategies, it can be concluded that for an increasing experiment number, the decrease of the A-optimal criterion is mainly because of an increased measured data accuracy based on repeated measurements rather than because of new information (e.g., analysis of the adsorption behavior at a different equilibrium point). 3.5. Experimental Results for a Combined Parallel/Sequential Strategy. In practice, after each set of parallel planned and implemented optimal experiments, the parameters are updated, θ0 f θ*, using available measurements and solving a parameter estimation problem such as in eq 29. The updated parameters, θ*, are then used for the solution of subsequent OED-problems. For an exact determination of the SMA-parameters, implementation errors have to be considered.15 For this purpose, uncertain design and experimental conditions, p′k, of each batch experiment are added as free decisions and are now adjustable parameters in the problem defined in eq 29. It is by this means that the size estimation of so-called implementation errors is now part of the parameter estimation problem: NExp

min θ′

∑ (y

meas k

∼

meas - yk(θ′, pk))T · MV-1 - yk(θ′, pk)) k · (yk

k)1

s.t. eqs 15, 16, 20-27 with θ′ ) [θ, p1′, p2′, · · · , pN′ Exp]T θ ) [Λ, k1,j, νj, σj]T ; σ2 ) σ3 ; std sl std T pk′ ) [cstd 2 , c3 , εtot, c1 ]

{

eq ini std T xk ) [Rj,1, ceq j , qj , cj , εtot, V1 ] ;

yk ) pk )

∀k ∈ {1, · · · , NExp} j ∈ {2, 3} j ∈ {1, 2, 3}

eq eq std std std sl T [Aeq 2 , A3 , Asum, A2 , A3 , c1 , εtot] std std sl bt std [V , V2 , V3 , V , c1 , m2, m3, n2, n3]T

}

(29) ∀k ∈ {1, · · · , NExp}

sl where the measurement vector, yk, is now augmented first by assumed values for cstd 1 and εtot, and second by the measured peak std std std areas, Astd and A , which represent the protein standard concentrations, c and c , using the linear calibration defined in eq 27. 2 3 2 3 The number of parameters and estimated uncertain experimental conditions is then Nθ′ ) 6 + 4 · NExp, the number of states and model equations is Nx ) 14 · NExp, the number of the measurement equations and variables is Ny ) 7 · NExp, and the number of constant parameters is Np ) 9 · NExp, respectively. It should be noted that the constant design parameter in the OED-problem in eq std std sl p u 28, cstd 1 , as well as the design variables, c2 , c3 , εtot, are affected by implementation errors, here ξ and ξ , respectively. The values

5710


Figure 5. Theoretical studies using different experimental design strategies for a fixed parameter guess, θ0, (e.g., 4 + 1 + · · · + 1 means: 4 parallel planned experiments and subsequently several single planned experiments).

Figure 6. Experimental results of the parallel/sequential design approach over the implemented number of experiments. Each value of the A-optimal criterion is shown before (superscript 0) and after (superscript /) the update of the SMA-parameters.

Figure 7. Influence of implementation errors, ξu, ξp, on the experimental results for the parallel/sequential design approach.

for the definition of MVk are again taken from Table 2. It has however to be noticed that a careful selection of the uncertainty size attached to yk (Table 2) is crucial in order to obtain a meaningful solution of the problem defined in eq 29. In

Figure 9. Reduction of dissolved concentrations from initial to equilibrium state for different adsorbent void fractions, εtot, with a constant initial salt concentration, cini 1 ) 100 mM, and constant initial protein concentrations, ini cini 2 ) c3 ) 0.1 (computation based on the SMA-parameters from Table 3).

particular, the considered uncertain experimental conditions, p′k, depend strongly on the uncertainty attached to their assumed values. So, for instance σ(Astd 2 ) has an influence on the result of sl sl cstd 2 and so has σ(εtot) on εtot, respectively. In other words, if the respective uncertainty values of a variable in p′k is selected too high, a significant deviation from the assumed experimental conditions is possible. This can no longer be explained by correction of a possible implementation fault, but it means an artificial change in the experimental conditions, which is directly used for the parameter fitting problem in eq 29. It is however apparent that this is not desirable for an exact determination of the unknown SMA-parameter set, θ. Figure 6 shows the experimental results from the parallel/ sequential (5 + 5 + 5) strategy. For the first and second five experiments, the initial parameter guess θ0 used for the experiment planning differs significantly from those parameters θ* obtained after implementation and parameter estimation (see also Figure 1). The optimally planned experiment design gives only a maximal parameter accuracy (represented by a small value of the A-optimal criterion) for the initially assumed parameters θ0. For the different values θ* after implementation the accuracy degrades and the A-optimal criterion increases as depicted in Figure 6. However, the quality of the estimate of θ0 improves, and thus, the error of an imprecise initial parameter estimate diminishes with an increasing number of experiments. Moreover, as discussed above, the source of uncertainty cannot only be assigned to uncertainties in the initial guess of the SMAparameters, ξθ. The theoretical influence of the considered possible implementation errors ξp and ξu is obtained by sampling. Corresponding standard deviations are given in Table 2. In Figure 7, the resulting influence after parameter update is shown with respect to the obtained A-optimal criterion. Here again, the possible error on the design criterion is reduced for an increasing number of experiments. It should anyway be noted

Figure 8. Liquid concentrations for the equilibrium points of conventional planned experiments (left) and those which result from the solution of the OED problem (right).


5711

Table 3. Experiment Results with Initially Assumed (0) and Updated (/) Parameters σ j i,i [%]

protein amount [mg] no. of expt. pure A/pure B/mix ceq 1 close to 100 mM

A

B

θ, u, p A-criterion φ jemax [%]

k1,2

k1,3

V2

V3

σ2 ) σ3

40.6 60.7 35.0 49.1 36.3 36.8

29.4 54.4 26.2 26.8 22.0 25.3

32.5 58.7 29.3 46.2 31.4 28.6

1.3 2.2 1.2 1.3 0.8 0.9

1.8 3.5 1.7 2.1 0.7 0.8

3.5 4.0 2.4 2.7 2.3 2.9

3.1 5.3 3.5 4.2 2.6 3.5

2328

1943 979

5.2 5.5 377

381

42.2

27.9

33.4 0.8 1.0 3.4

3.5

Λ

Optimal Experiments 1-5

2/0/3

4

25.0

8.7

1-10

5/1/4

7

50.0

23.2

1-15

6/4/5

9

55.8

43.9

40

10/10/25

0

123.1

55

16/14/30

9

0 * 0 * 0 *

0.174 0.647 0.157 0.289 0.148 0.148

Conventional Experiments 106.2

*

516

Optimal and Conventional Experiments 178.9

150.1

that the size of the implementation error can partially be detected by the solution of the augmented parameter estimation problem in eq 29.15 This approach improves the quality of the SMAparameter estimate, but it is not a prevention of the degradation of the experimental design criterion. 3.6. Comparison of Conventional and Optimal Planned Experiments. Figure 8 shows both conventional planned experiments at two different salt ion concentrations (a total of 40 batches), and the optimal planned experiments (15 experiments). It can be seen that the optimally placed equilibrium points cannot be selected by an intuitive reasoning, while the conventional planned experiments aim at keeping certain liquid concentrations or their relation constant for several batch experiments. The optimal planned experiments cover a wider range regarding the salt concentration ceq 1 , whereas the conventional experiments are performed only at 130 and 150 mM. It eq can also be noted that the protein concentrations ceq 2 and c3 in the optimal planned experiments are generally smaller in comparison to conventional experiments. This is because of the and mmax restrictions related to the total amount of proteins mmax 2 3 (see also eq 28). For a further characterization of all conducted experiments, the reduction of protein concentrations is analyzed ini starting from initial values cini 2 and c3 to the equilibrium protein eq eq concentrations c2 and c3 for different adsorbent void fractions εtot, and a constant initial salt concentration, cini 1 , in each batch. In Figure 9, it can thus be seen that liquid and adsorbed eq eq eq equilibrium protein concentrations ceq 2 , c3 , q2 , q3 , get different ini ini values for constant values of c2 , c3 depending on εtot. This is

*

0.193

basically because of the higher adsorption affinity of β-lgA. The variation of the equilibrium points with increasing values of εtot (i.e., decreasing amounts of adsorbent in the batch) is calculated by using the final parameter estimation from Table 3. The experimental results for the different batches are depicted in Figure 10. It should be noted that while for the conventional planned experiments (Figure 10, left) the reduction of liquid concentrations is relatively small and nearly constant, it is rather much higher for the optimally planned experiments because of higher amounts of adsorbent in each batch. However, because ini of the limited accuracy of the measured values of cini 2 , c3 and eq eq c2 , c3 , both their absolute values and their difference have to be sufficiently high in order to obtain significant results. The clear advantages of the OED are the reduced experimental effort due to the smaller number of experiments and the higher parameter accuracy because of the higher information content of the measured concentrations. It can be seen from Table 3 that a value of φopt ) 0.148 is obtained for the 15 optimal designed experiments with a much smaller protein consumption, whereas φconv ) 516.0 is achieved based on the conventional planned experiments. The corresponding relative standard deviations σ j i, i, which are adopted from the diagonal elements, and the maximal correlation of the parameter jemax taken from the off-elements of the approximated parameter covariance matrix in eq 4 confirm these results. It has to be noted that the OED procedure combines equilibrium points with single (pure A and B) and also mixed protein concentrations. Table 4 shows the initially assumed and finally estimated parameter values θ.

Figure 10. Experimental results for the reduction of the dissolved protein concentrations with different salt concentrations and adsorbent amounts for conventional planned experiments (left) and those which result from the solution of the OED problem (right). Table 4. Initially Assumed and Finally Estimated Parameter Values model parameter values, initially assumed (0) and final estimate (/) k1,2 0

θ θ*

k1,3 -3

1.45 × 10 1.73 × 10-5

-3

3.5 × 10 5.00 × 10-6

V2

V3

σ2 ) σ3

Λ

6.38 8.75

5.14 8.57

40.0 36.8

803.0 818.8

5712


In Table 5 are listed the initially assumed and finally verified experimental conditions p as well as the corresponding design variables u, which result from eq 29.

it has been demonstrated that both variances and correlations have significantly been minimized in comparison to conventional planned experiments. However, independent from the discussed approaches here for uncertainty compensation, model-based OED suffers from one drawback: its unsystematic nature. The optimal experimental conditions are rarely evident in complex models. They are often located at the limits of the operating region omitting those regions with a low information content for the underlying process model. Consequently, in the case of uncertainties related to the model structure, the experiment design is seldom easily interpreted (e.g., selection of the appropriate adsorption mechanism), and it does not substitute a systematic study over the entire operating range.

Conclusions

Acknowledgment

When designing experiments, the practical engineer will generally encounter different sources of uncertainties such as implementation errors, deviations from general process conditions (which are assumed to be constant), and erroneous assumptions in initial parameter guesses. The identification of main sources of uncertainties and their size are not known a priori. They will rather be identified after the implementation of experiments. Moreover, when dealing with strongly nonlinear process models, the initially assumed parameter values can vary over several orders of magnitude from the finally estimated value. On the basis of these reasons, an inclusion of uncertainties in the OED problem formulation so as to directly account for uncertainties is often not possible. The proposed feed-back-based approach (indirect method), which follows the sequential design of experiments, appears to be the most appropriate procedure to partially compensate for possible degradations during the design procedure. While the update of a parameter set is an inherent step in the sequential strategy, special attention should be paid to the identification of deviations in the originally planned design variables and conditions. This can be done either directly by a measurement, which verifies the implemented value, or indirectly by solving an augmented estimation problem, which accounts simultaneously for model parameters and uncertain experimental conditions. In the latter case, the size estimation of the so-called implementation errors becomes part of the parameter estimation problem. In this work, the obtained results of the case study show a significant improvement in terms of lab work load, number of experiments required, as well as reduced use of costly chemicals. By an explicit consideration of all experimental steps beginning from error-prone procedures in the experiment preparation (e.g., feed preparation) up to the result analysis and verification of the realized optimal experimental conditions, the experimental effort can additionally be reduced by a standardization of the lab work. This is in particular interesting when performing parameter identification for different components, but using the same equipment. Moreover, it has been shown that the setting of specific and especially informative process conditions (e.g., a desired equilibrium point or a specific reaction temperature) can often only be influenced indirectly by the experiment design variables. Accordingly, without an inclusion of these relations in the OED problem formulation and without an iteratively process model refinement, deviations from the desired experiment conditions are to be expected. Lastly, the A-optimal criterion becomes a valuable choice when applied to the approximation of the parameter covariance matrix. On the basis of the values of the parameter correlations,

The authors gratefully acknowledge the support of Knauer GmbH (Berlin, Germany) and the financial support of BMBF (Federal Ministry of Education and Research of Germany), support code 03WOPAL4.

Table 5. Estimated Deviations in Initially Assumed (0) and Finally Estimated (/) Constant Experimental Conditions pk As Well As Planned Design Variables uk Including Minimum/Maximum/Mean Values for All Conducted Experiments estimated deviations/implementation errors in constant experimental conditions (|pk0 - p* k |) c2std min max mean

c3std -4

5.8 × 10 2.3 × 10-2 6.5 × 10-3

design variables (|uk0 - u* k |) εtotsl

-4

3.7 × 10 1.4 × 10-2 4.0 × 10-3

c1std -5

7.3 × 10 1.9 × 10-2 4.6 × 10-3

0.0 7.24 1.36

Nomenclature Latin j ) component index j ) {1,2,3} ) ˆ {Cl-, β-LgA, β-LgB} Aj ) integrated area of HPLC signal (eq, equilibrium, ini, initial state, std, standard, sum, sum signal), maU min cj ) mobile phase concentration (eq, equilibrium; ini, initial state; sol, solvent; std, standard), mM jemax ) maximal relative parameter correlation F ) Fischer information matrix k1, j ) equilibrium constant, dimensionless mj, nj ) linear calibration parameter mj ) available protein amount (max, maximum), mg MV ) measurement covariance matrix N ) number of: C, components; P, proteins; Exp, experiments; θ, parameters p ) constant model parameters/experimental conditions (0, initially assumed; /, true value) qj ) stationary phase concentration (eq, equilibrium; ini, initial state), mM t ) time u ) design variables (0, initially planned; /, true value) V ) volume (bt, batch; sl, slurry; std, standard volume), mL x ) state variables y ) measured state variables (meas, measurement values) Greek Letters and Symbols Rj ) variable Langmuir coefficient, dimensionless σ ) standard deviation (abs, absolute; rel, relative; {i, i}, diagonal element) σj ) steric factor, dimensionless ε ) porosity (tot, total; ext, external; int, intraparticular; sl, slurry), dimensionless λ ) eigenvalues Λ ) stationary phase capacity (for monovalent salt counterions), mM θ ) model parameters (0, initially assumed; /, true value) Θ ) variable space/domain of θ θV ) parameter covariance matrix νj ) characteristic charge, dimensionless ξ ) error in: θ, initial parameter guess; p, experimental conditions; u, design variables φ ) OED functional


Literature Cited (1) Franceschini, G.; Macchietto, S. Model-based design of experiments for parameter precision: State of the art. Chem. Eng. Sci. 2007, 63, 4846– 4872. (2) Franceschini, G.; Macchietto, S. Validation of a model for biodiesel production through model based experiment design. Ind. Eng. Chem. Res. 2007, 46, 220–232. (3) Seidel-Morgenstern, A. Experimental determination of single solute and competitive adsorption isotherms. J. Chromatogr. A 2004, 1037, 255– 272. (4) Bard, Y. Nonlinear Parameter Estimation; Academic Press: New York, 1974. (5) Benabbas, L.; Asprey, S. P.; Macchietto, S. Curvature-based methods for designing optimally informative experiments in multiresponse nonlinear dynamic situations. Ind. Eng. Chem. Res. 2005, 44, 7120–7131. (6) Bock, H. G.; Ko¨rkel, S.; Kostina, E.; Schlo¨der, J. P. Robustness aspects in parameter estimation, optimal design of experiments and optimal control. In ReactiVe Flows, Diffusion and Transport; Ja¨ger, W., Rannacher, R., Warnatz, J., Eds.; Springer: New York, 2007; pp 117-146. (7) Ko¨rkel, S.; Kostina, E.; Bock, H.; Schlo¨der, J. Numerical methods for optimal control problems in design of robust optimal experiments for nonlinear dynamic processes. Optimization Methods and Software 2004, 19, 327–338. (8) Scho¨neberger, J. C.; Arellano-Garcia, H.; Wozny, G.; Ko¨rkel, S.; Thielert, H. Model-based experimental analysis of a fixed-bed reactor for catalytic SO2 oxidation. Ind. Eng. Chem. Res. 2009, 48, 5165–5176. (9) Atkinson, A. C.; Bogacka, B. Compound and other optimum designs for systems of nonlinear differential equations arising in chemical kinetics. Chemom. Intell. Lab. Syst. 2002, 61, 17–33. (10) Skogestad, S. Plantwide control: The search for the self-optimizing control structure. J. Process Control 2000, 10, 487–507. (11) Dette, H.; Melas, V. B.; Pepelyshev, A.; Strigul, N. Robust and efficient design of experiments for the Monod model. J. Theoret. Biol. 2005, 234, 537–550. (12) Asprey, S. P.; Macchietto, S. Designing robust optimal dynamic experiments. J. Process Control 2002, 12, 545–556. (13) Walter, E. Identifiability of Parametric Models; Pergamon Press, Inc.: Elmsford, NY, 1987. (14) Chu, Y.; Hahn, J. Integrating parameter selection with experimental design under uncertainty for nonlinear dynamic systems. AIChE J. 2008, 54, 2310–2320. (15) Dovı´, V. G.; Reverberi, A. P.; Maga, L. Optimal design of sequential experiments for error-invariables models. Comput. Chem. Eng. 1993, 17, 111–115. (16) Ko¨rkel, S.; Bauer, I.; Bock, H. G.; Schlo¨der, J. P. A sequential approach for nonlinear optimum experimental design in DAE systems. Sci. Comput. Chem. Eng. II 1999, 338. (17) Galvanin, F.; Macchietto, S.; Bezzo, F. Model-based design of parallel experiments. Ind. Eng. Chem. Res. 2007, 46, 871–882.

5713

(18) Galvanin, F.; Barolo, M.; Bezzo, F. Online model-based redesign of experiments for parameter estimation in dynamic systems. Ind. Eng. Chem. Res. 2009, 48, 4415–4427. (19) Ko¨rkel, S.; Arellano-Garcia, H. Online experimental design for model validation. In Computer-Aided Chemical Engineering; De Brito Alves, R. M., Oller Do Nascimento, C. A., Biscaia, E. C., Eds.; Elsevier, 2009; pp 528-533. (20) Kostina, E. Robust parameter estimation in dynamic systems. Optim. Eng. 2004, 5, 461–484. (21) Espie, D.; Macchietto, S. The optimal design of dynamic experiments. AIChE J. 1989, 35, 223–229. (22) Gadam, S. D.; Cramer, S. M. Salt effects in anion exchange displacement chromatography: Comparison of pentosan polysulfate and dextran sulfate displacers. Chromatographia 1994, 39, 409–418. (23) Pedersen, L.; Mollerup, J.; Hansen, E.; Jungbauer, A. Whey proteins as a model system for chromatographic separation of proteins. J. Chromatogr. B 2003, 790, 161–173. (24) Natarajan, V.; Ghose, S.; Cramer, S. M. Comparison of linear gradient and displacement separations in ion-exchange systems. Biotechnol. Bioeng. 2002, 78, 365–375. (25) Barz, T.; Lo¨ffler, V.; Arellano-Garcia, H.; Wozny, G. Optimal experimental design for the determination of protein ion-exchange equilibrium parameters. In Computer-Aided Chemical Engineering.; De Brito Alves, R. M., Oller Do Nascimento, C. A., Biscaia, E. C., Eds.; Elsevier, 2009; pp 309-314. (26) Brooks, C. A.; Cramer, S. M. Steric mass-action ion exchange: Displacement profiles and induced salt gradients. AIChE J. 1992, 38, 1969– 1978. (27) Susanto, A. Untersuchung und Modellierung intrapartikularer Stofftransportmechanismen bei der Proteinaufreinigung durch Lonenaustauschchromatographie (InVestigation and Modeling of Intraparticular Mass Transport Mechanisms in Protein Purification by Ion Exchange Chromatography); VDI Verlag: Du¨sseldorf, 2006; Vol. 865. (28) Wekenborg, K.; Susanto, A.; Schmidt-Traub, H. Modelling and Validated Simulation of Solvent-Gradient Simulated Moving Bed (SG-SMB) Processes for Protein Separation. Comput.-Aided Chem. Eng. 2005, 20, 313– 318. (29) Frederiksen, S. S. Ph.D. Thesis. IVC-SEP, Department of Chemical Engineering, Center for Phase Equilibria and Separation Processes, Technical University of Denmark, 2004. (30) Wekenborg, K.; Susanto, A.; Frederiksen, S.; Schmidt-Traub, H. Nicht-isokratische SMBTrennung von proteinen mittels ionenaustauschchromatographie. Chem. Ing. Technol. 2004, 76, 815–819.

ReceiVed for reView October 15, 2009 ReVised manuscript receiVed March 24, 2010 Accepted April 21, 2010 IE901611B

Handling Uncertainty in Model-Based Optimal Experimental Design

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