Practical Identifiability Analysis for the Characterization of Mass

Article pubs.acs.org/IECR

Practical Identifiability Analysis for the Characterization of Mass Transport Properties in Migration Tests Brais Martinez-Lopez,† Stéphane Peyron,*,† Nathalie Gontard,† and Miguel Mauricio-Iglesias‡ †

UMR IATE 1208 cc023, Université de Montpellier II Place Eugène Bataillon 34095 Montpellier CEDEX 05, France Department of Chemical Engineering, University of Santiago de Compostela, 15782 Santiago de Compostela, Spain

‡

ABSTRACT: The common assumption of considering external mass transfer resistance as negligible can lead to a systematic bias that underestimates the actual diffusivity value. In this context, the suitability of two methods and the possible experimental setups (global or local concentration measurements) for determination of both diffusivity and mass transfer coefficient are discussed. The assessment was based on the experimental results of the desorption of Uvitex OB from low density polyethylene into the food simulant Miglyol 829. A practical identifiability analysis performed for each of the methods and experimental setups demonstrated that the increased amount of information provided by local measurement methods allows a better identifiability of the parameters, thereby reducing the experimental work and providing an unbiased method for determination of diffusivity.

1. INTRODUCTION The use of numerical simulation to replace or complement migration tests in polymers was first introduced in the European regulation on plastic materials and articles intended to come into contact with food more than ten years ago (regulation 2002/72/EC), thus replacing costly experiments in cases where the predicted migration is well below the regulatory threshold. Yet, the following physical parameters are generally needed to simulate mass transfer in the food/ packaging system: diffusivity of the migrant in the polymer (D), food/polymer partition coefficient (K), and for liquid or semisolid food products, the mass transfer coefficient (k). The lack of a diffusivity database fostered the development of methods to determine the diffusivity of migrants in polymers, especially solid−solid contacts,1−3 and making use of a variety of analytical techniques, like FRAP4 or vibrational spectroscopy.5 In most experiments aimed at determining diffusivity, it is assumed that the external mass transfer resistance is negligible (i.e., k is high), which is a good approximation in case the food or food simulating liquid is a stirred liquid with relatively low viscosity. Otherwise, the determined parameter (often called apparent diffusivity, Dapp) is systematically lower than the actual diffusivity value, since it represents the inverse of the sum of the resistances, both to external and internal transfer. Hence, Dapp is not a worst case estimation of D. If Dapp is used to simulate migration in conditions where k is higher than those when Dapp was determined, the level of migration will be underestimated. As a consequence, it is of great importance to estimate both k and D in experiments aiming at the determination of the diffusivity. The determination of k in desorption experiments is, however, not straightforward, which may partly explain why this parameter has been frequently overlooked in migration modeling. One of the reasons of this difficulty lies in the fact that variations of k and D give similar macroscopic results (i.e., more desorption when they increase and less when they decrease). In other words, k and D have a poor practical © 2015 American Chemical Society

identif iability when estimated from desorption kinetics measuring the amount of migrant that has been transferred (either as uptake by the liquid medium or loss by the polymer). To improve the simultaneous estimation of D and k, Vitrac et al.6 proposed a reparametrization of the mass transfer equation, which aims at isolating the effect of k at the beginning of the desorption experiment. More recently, Vitrac et al.7 analyzed the determination of both parameters as well as the partition coefficient for low-density polyethylene (LDPE)/ethanol systems with gas chromatography with a flame ionization detector (GC-FID) and concluded that the determination of the mass transfer was fairly feasible for Biot number below 50. Nevertheless, the influence of the analytical setup on the determination of D and k and their identifiability is rare in literature: Azevedo et al.8 studied the mass transfer coefficients for the acidification of raw cut vegetables in a sequential way, and more recently, they also studied9 the feasibility of the simultaneous determination of both parameters from simulated data. They concluded that the simultaneous estimation of D and k was only possible at some intermediate values of the Biot number. The analysis was merely theoretical and restricted to two measurements of the global concentration, which is not enough to draw general conclusions, as more information can be gathered with other experimental setups or increasing the number of measurements. In effect, Petersen et al.10 reported cases where even if a set of parameters is highly correlated, they can be identified provided that a informative enough experiment is designed. Nevertheless, data available in literature for mass transfer coefficients in food packaging systems are even scarcer: Gandek et al.11 and others12 studied migration of antioxidants from polyolefins into water, theoretically and in the practice, while Vergnaud13 dealt with the transfer of diethylhexyl phthalate Received: Revised: Accepted: Published: 4725

July 18, 2014 April 9, 2015 April 16, 2015 April 23, 2015 DOI: 10.1021/ie505057a Ind. Eng. Chem. Res. 2015, 54, 4725−4736

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Industrial & Engineering Chemistry Research

substance, so the system is said to follow Fickean kinetics. Eq 1 can be solved with the initial and boundary conditions that apply to the case to obtain an expression for the concentration distribution:

from PVC into hexane. The aforementioned contributions have focused on experimental setups that measure the evolution of the global concentration along the experiment, either in the sample or in the surrounding liquid. However, other setups that provide the evolution of the concentration profile in the solid sample have been reported, for example, using Raman microspectroscopy14 or by depth-profiling.15 The aim of this contribution is to explore the use of two different analytical methods for the simultaneous determination of D and k in a polymer/food simulating liquid system. The desorption kinetics is monitored by measuring the remaining concentration of migrant in the polymer by Fourier transform infrared (FTIR) spectroscopy (global measuring) or the profile of concentration along the polymer thickness direction by Raman microspectroscopy16 (local measuring). This approach allows us to (i) compare our results with previous literature since the reported experimental setups measure global concentration, (ii) demonstrate for the first time the use of local measuring to the determination of the mass transfer coefficient, and (iii) compare the increase of accuracy and precision of the determination gained by using a highly informative experiment such as local measuring compared to global measuring. A model system composed of linear low density polyethylene (LLDPE) including Uvitex OB, an optical brightener and UV stabilizer commonly used in polyolefin, was set in contact to Miglyol used as fatty food simulating liquid. The paper is organized as follows: first, there is a brief explanation of the mass transfer model taking into account both internal and external transfer. A short information to the principles of a practical identifiability analysis is also given in section 3.6. In section 4, the results of the parameter determination for the two experimental setups are discussed; the two methods are compared in section 5 along with a discussion of their practical limitations.

⎧ t>0 ∂C ∂ 2C = D 2 ; with⎨ ⎩− L ≤ x ≤ ∂t ∂t

⎫ ⎬ ⎭ L

(1)

The rate at which the diffusing substance is transferred into the liquid is equal to the rate at which this substance reaches the surface of the polymer sheet by internal diffusion. This rate can be expressed by a simple mass balance at the polymer/ liquid interface (eq 2): −D

⎧ t>0 ⎫ ∂C ⎬ = k(Cx = L , t − Cx = L , ∞); with⎨ ⎩− L ≤ x ≤ L ⎭ ∂x

(2)

−1

where k is the mass transfer coefficient (m s ), Cx=L,t is the concentration at the interface at time t, and Cx=L,∞ is the concentration at the interface at equilibrium (kg m3). The relation between Cx=L,∞ and the concentration in the liquid CL is given by eq 3: Cx = L , ∞

K=

CL , ∞

(3)

where K is the partition coefficient between the polymer and the liquid. For the system studied, the migrant is more soluble in the liquid than in the polymer, and more importantly, the liquid volume is around 350 times larger than the polymer volume. The liquid can be considered to have infinite capacity to absorb the migrant, hence, CL = 0 and K can be safely supposed to have a value of 1000. For convenience, the following dimensionless numbers were introduced, namely the Fourier number (Fo), or dimensionless time, and the Biot number (Bi), which represents the ratio of the external over the internal transfer:

2. MASS TRANSFER MODELING The mass transfer in a polymer−liquid system is a combination of transport phenomena, governed by the molecular diffusion (Figure 1). Assuming no interactions between the liquid and

Fo =

Dt L2

c=

C C0

X=

x L

Bi =

kL DK

Dimensionless time Dimensionless concentration

Dimensionless space External/internal transfer ratio

(4)

(5) (6)

(7)

Since it is assumed that CL = 0, due to the large liquid mass surrounding the sample, a variation in the partition coefficient will not change the equilibrium concentration. However, the desorption kinetics will be influenced as indicated by a different value of the Biot number. These dimensionless magnitudes can be substituted in eqs 1and 2, obtaining:

Figure 1. Schematic representation of mass transport phenomena in a polymer−liquid system.

the polymer, the one-dimensional transport of a molecule can be represented by the mass balance of the diffusing migrant, expressed by eq 1,17 where t is the time (s), x is the distance (m), L is the thickness (m), C is the concentration in diffusing substance (mass diffusing substance/mass of polymer), and D is the diffusivity of the molecule in the sheet material (m2 s−1). D is assumed independent of the concentration of the diffusing

∂c ∂ 2c =D 2 ∂Fo ∂X

(8)

−∂c = Bi(c − c∞) ∂X

(9)

Eq 1 was solved numerically for every boundary condition by the method of lines, that is, discretization of the space (x, the polymer thickness), and numerical approximation of the space derivatives, to obtain a system of ordinary differential equations 4726

DOI: 10.1021/ie505057a Ind. Eng. Chem. Res. 2015, 54, 4725−4736

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thickness of the Uvitex OB/LLDPE confirmed this hypothesis: 2.31 ± 0.92 wt % and 231.83 ± 39.5 μm for the unstirred contact; 2.37 ± 0.77 wt % and 251.83 ± 36.9 μm for the stirred contact on the basis of a triplicate for each data point at 3, 6, 15, and 24 h. 3.5. Raman Spectroscopy. Uvitex OB concentration profiles were determined as follows. Thin slices of LLDPE were prepared using a razor blade and stuck on a microscope slide. Raman spectra were recorded between 95 and 3500 cm−1 Raman shift wavenumber using a confocal Raman microspectrometer Almega (Thermo-Electron) with the following configuration: excitation laser He−Ne 633 nm, grating 500 grooves/mm, pinhole 25 μm, objective ×50. The collection time was about 1 min 40 s (five scans of 20 s each). Measurements were carried out in the cross-section of the sample with a spacing of 5 or 15 μm. All spectra pretreatments were performed with Omnic v 7.1 (Thermo-Electron). Processing included a multipoint linear baseline correction, normalization according to the area of the LLDPE specific band at 1129 cm−1 representing the symmetric C−C stretching of alltrans PE chains. The relative content of Uvitex OB was assessed using the area of the specific doublet (1569−1614 cm−1) assigned to the aromatic CC and CN bands. Two concentration profiles, with a spacing of 5 or 15 μm, were taken from each stirred and unstirred duplicate, making a total of four concentration profiles for each desorption experiment. For the experiment at high stirring, the sample thicknesses were 200 and 165 μm, while for the unstirred experiment, they were of 200 and 210 μm. 3.6. Practical Identifiability Analysis. To be able to identify the parameter set of a model, it must fulfill two conditions. First, the output of the model must be sensitive enough to changes on each of the parameters of the set. Second, changes in the model output due to changes in single parameters may not be canceled by changing the values of other parameters. If the sensitivity of a model with relation to a parameter is defined as the variation of the model output (y) with relation to the variation of a predictor variable (θ), a sensitivity matrix Sa can be defined (eq 10) as matrix conformed by the variation of the model output with relation to the variations of each of the parameters:18

(ODEs). A number of 100 was found to be sufficient. The model was implemented and solved in Matlab R2013a (The MathWorks, Natick, MA).

3. MATERIALS AND METHODS 3.1. Materials. Linear low-density polyethylene (LLDPE) pellets (density 920 kg m−3) were purchased from SigmaAldrich. 2,5-Bis(5-tert-butyl-benzoxazol-2-yl)-thiophen (Uvitex Optical Brightener, 430.6 g mol−1) was purchased from Fluka. Miglyol 829, a triglyceride of the fractionated plant fatty acids C8 and C10 combined with succinic acid, with a density between 1 and 1.02 g cm−3 and a viscosity about 230 mPa s was purchased from Sasol. 3.2. Films Fabrication. LLDPE pellets were mixed with Uvitex OB at 140 °C (50 rpm) for 5 min. The dough material obtained after mixing was then thermoformed using a hot press at 150 bar for 5 min at 140 °C. The nominal concentration of the films was 2% wt, measured by FTIR. After fabrication, films were placed at 60 °C for 48 h to ensure a homogeneous distribution through the film. The actual film thickness was 240 ± 40 μm and was measured by using a micrometer (Braive Instruments, Chécy, Fr) in quintuplicate. 3.3. Experimental Setup. LLDPE disks of 3.5 cm2 were fully immersed and held in suspension in 15 mL of Miglyol 829 at 40 °C on a ∼25 mL test cell as shown in Figure 2. For each

Figure 2. Experimental design used on desorption tests.

analysis, one duplicate was just immersed, while the other was placed under a stirring of 350 rpm. Each sample was wiped with a soft tissue containing ethanol to eliminate adhered remains of Mygliol. FTIR measures were done four times after 0, 3, 6, 15, and 24 h, in triplicate, while Raman measurements were performed in duplicate at two different times: once after the thermal treatment at 60 °C, to check the homogeneity and obtain the signal value of CC0, and once after 1 h of contact with the simulant to obtain the experimental profiles. 3.4. FTIR Measurements. LLDPE film samples were analyzed by transmission FTIR. Spectra were recorded using a Nexus 5700 spectrometer (ThermoElectron Corp.) equipped with He−Ne beam splitter and cooled MCT detector. Spectral data were accumulated from 128 scans with a resolution of 4 cm−1 in the range of 800−4000 cm−1. Each stirring level was analyzed in triplicate, recording three spectra for each of them. All spectra treatements were performed using Omnic 7.1 and TQ Analyst v7.2 software (ThermoElectron). Processing included a multipoint linear baseline correction and a normalization according to the area of the LLDPE doublet (1369−1378 cm−1) due to the CH3 symmetric deformation vibration. To avoid imprecisions derived from the inhomogeneity of the initial concentration and the lack of sensitivity of the FTIR, a different sample was used for each data point, its concentration being measured before and after the contact, and then normalized in relation to the concentration value before the contact (C0). The dispersion of the concentrations and

Sa = {Sa , ij} where Sa , ij =

∂yi ∂θj

(10)

This sensitivity matrix can be nondimensionalized and normalized according to eqs 11 and 12, respectively: Snd = {Snd, ij} where Snd, ij =

Δθ δyi SCi δθj

Snorm = {Snorm, ij} where Snorm, ij =

(11)

Snd, ij || Snd, ij ||

(12)

where Δθj is the reasonable span of parameter θj, SCi is a scale factor with the same physical dimension as the corresponding observation, and ∥Snd,ij∥ is the Euclidean norm of S. Since the parameters D and k can span over several decades, their logarithm was used as a parameter instead of their actual values. Hence, the initial guesses for their values were log (D) = −12.2 and log (k) = −3.5 and the scaling factors (Δθj) equal to 6 for both. Since the only output was the measured migrant 4727


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Figure 3. Desorption curves in function of the Fourier number or dimensionless time obtained with FTIR. (a) Stirred contact used to obtain diffusivity (5.3 × 10−13 m2 s−1). (b) Unstirred contact used to evaluate the mass transfer coefficient (1.3 × 10−4 m2 s−1) by fixing the value of diffusivity obtained from the stirred contact.

experimental and predicted profiles and by using a nonlinear least-squares minimization method (lsqnonlin command predefined from Matlab software) combined with the multistart routine, which uses different starting guesses to ensure that global solutions were obtained:

concentration, the scaling factor used was equal to the initial concentration. The derivatives ((∂yi)/(∂θj)) were obtained by finite differences. If a column of the dimensionless sensitivity matrix Sj is linearly or nearly linearly dependent, the parameters are said to be collinear or nearly collinear. A common metric for characterizing the collinearity of parameters is the collinearity index γk ; which can be defined according to eq 13:18 γk =

1 where λm = eigen(SnormT Snorm) min λm

SSQR = (ypred − yexp )2

To ensure the robustness of the parameters estimated and determine their confidence intervals, each determination was iterated 200 times following a bootstrap method with Monte Carlo resampling algorithm. The precision of each estimated parameter was calculated by a standardized half-width (SHW), defined as half of the 95% confidence interval, divided by the value of the estimated parameter. The accuracy was expressed by the ratio of the absolute value of the fractional error of the estimated parameter relative to the true value of the parameter, as described by Azevedo et al.9 3.8. Generation of Data for Virtual Experiments. Virtual data were generated mimicking the experimental protocol. In the case of global kinetics, five measures were taken at 0, 3, 6, 15, and 24 h of contact. For local concentration profiles, 12 equally spaced nodes separated by 15 μm were measured after 1 h of contact. Diffusivity was considered constant and equal to D = 5 × 10−13 m2 s−1, as this value represents the typical diffusivity of a migrant in polyethylene, the most frequently used packaging material. The thickness of the film was taken as L = 240 um and the partition coefficient as K = 1000. Several mass transfer coefficient values were taken so that the Biot number would equal 1, 4, 50, 300, and 1000. 3.9. Simulation of Migration Tests. A migration test was simulated to show the difference on the time required to reach the specific migration limit (SML), derived from using Dapp instead of D and k to predict the concentration of the migrant in food. To set realistic conditions, a packaging of 6 dm2 into contact with 1 kg of food was considered. Instead of a commercial concentration of 5000 ppm of Uvitex OB, the packaging was considered spiked enough so the maximum concentration attainable in the food is SML of Uvitex OB, of 0.6 ppm. The diffusion coefficients used for the simulations were the upper and lower confidence bounds from the

(13)

The effect of the variation of parameter θj on the measurable output yi can be compensated up to a value of 1/γ by a change in another parameter θj. For example, a value of γ = 20 would indicate that a change in a parameter can be compensated up to 1/20 or 5% by changing another parameter. Exceeding a value of γ of 10−15 is often considered as an indication of a poorly identifiable parameter set.19 Minimizing the collinearity index is equivalent to the criterion of E-optimality in optimal experimental design (maximize the minimum eigen value of the information matrix). Another useful metric is the so-called determinant measure ρm,20 the determinant measure ρm is defined as in eq 14: ρm = det(SaT Sa)1/2

(15)

(14)

where m is the number of parameters considered. This metric combines the information about the sensitivity of the outputs to the parameters and their collinearity. For ρ to be large, the sensitivities must be large, and γ must be low (since γ is the inverse of the smallest eigenvalue λ, a large γ would imply a low λ and consequently a low ρ). Unfortunately, since ρ depends on the scaling factors used, a threshold cannot be given. It should be used instead to compare different parameter subsets or experimental setups. Maximizing the determinant measurement is equivalent to the criterion of D-optimality in optimal experimental design. 3.7. Parameter Estimation. Uvitex diffusivity and mass transfer coefficient were identified from experimental data by minimizing the sum of the squared residuals (eq 15) between 4728


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Figure 4. Concentration profiles in the thickness of the LLDPE films. Data from (○) stirred contacts and (□) unstirred contacts. The solid line represents the best fit as follows: (a) single regression of D under stirring, (b) single regression of k using the value of D from panel a; (c) simultaneous regression of D and k using the same experimental data as in panel a; (d) simultaneous regression of D and k using the same experimental data as in panel b.

Table 1. True Values of Diffusivity and Mass Transfer Coefficient, as Well as Their Associated 95% Confidence Intervals and Significance, Measured from the Difference between a Stirred and an Unstirred Contact Dapp stirred contact (m2 s−1) global measuring (FTIR) local measuring (Raman)

−13

5.3 × 10

−13

, (3.7, 14.6) × 10

(a)

9.1 × 10−13, (8.4, 9.8) × 10−13 (a)

accuracy Dapp

precision Dapp

k unstirred conctact (m s−1)

accuracy k

precision k

Bi

0.12

0.91

1.3 × 10−4, (0.85, 1.9) × 10−4 (b)

0.39

0.29

36

0.04

0.08

3.0 × 10−4, (2.8, 3.3) × 10−4 (c)

0.06

0.09

31

profile of migrant concentration along the thickness of the polymer (so-called local concentration method) by Raman microspectroscopy. 4.1. Sequential Determination of D and k. 4.1.1. Global Measuring. The estimation of apparent diffusivity was done at high stirring rate (Figure 3a) by following the desorption by FTIR. By fixing the value at D = 5.3 × 10−13 m2 s−1, the mass transfer coefficient without stirring was determined (Figure 3b); this results on a Bi = 36. Despite the lack of sensitivity of FTIR, the effect of stirring on the desorption rate is appreciated on the desorption curves. However, the scattered experimental data lead to a wide confidence interval of the estimated apparent diffusivity. Taking into account that a more accurate or precise determination is reflected by lower values of accuracy and precision, the determination of the mass transfer coefficient is less accurate but more precise than the determination of the diffusion coefficient. This can be regarded as an inherent limitation of this method: while precise, the error in the estimation of the apparent diffusivity is a burden for an accurate estimation of the mass transfer coefficient, since k is determined fixing a previously estimated value of D. 4.1.2. Local Measuring. The local concentration profiles were determined after 1 h of contact time with Raman microspectroscopy, 24 times shorter than required by FTIR. As for the global measurements, the local concentration profiles were fitted to estimate the apparent diffusivity (Figure 4a), and by fixing the estimated value, the mass transfer coefficient was determined for the unstirred case (Figure 4b). All the estimated values are gathered in Table 1. The values found by local measuring are not significantly different from those determined by global measuring, resulting on a similar Bi = 31. It is nevertheless noticeable that confidence intervals are significantly narrower when determined by local measuring, reflected

bootstrapping of the virtual data for both global and local measuring.

4. RESULTS The implemented approach consists on characterizing the mass transfer coefficient following two distinct methods: the sequential estimation method and the simultaneous estimation method. The sequential estimation method relies on using conditions when one of the resistances to mass transfer (internal or external) is negligible. The simultaneous estimation method requires the measured outputs to be sensitive to both parameters in the experimental conditions. Sequential Estimation of D and k. The rationale behind this approach is that k depends strongly on the stirring rate of the fluid, and, as consequence, D can be estimated at a high stirring rate so that the resistance to external transfer is negligible. Then, the stirring rate is increased until the estimation of D converges to a given value assumed to be the true diffusivity. The experiment is then repeated at the actual stirring rate (or with stagnant liquid) to determine k with known diffusivity. Simultaneous Estimation of D and k. To apply this method, eqs 1 and 2 are integrated, and the variation of concentration with time and space is determined and dependent on both D and k. The sequential and the simultaneous estimation of the parameters were carried out in this work by using two different experimental setups. In one of the setups, the average concentration (so-called global concentration method) of migrant remaining in polymer was followed by FTIR. Global concentration methods, where the migrant is measured in the liquid or in the polymer, are probably the most common way to determine diffusion coefficients. The other setup consisted on measuring the 4729


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Figure 5. SSQR contour plot for global measuring with FTIR (left), stirred contact (middle), and unstirred contact (right) local measuring with Raman.

Figure 6. Contour plots made according to the sampling rate and spacing of the data used in the simultaneous determination: five measures in 24 h, taken at 0, 3, 6, 15, and 24 h for global measuring with FTIR and 12 measures equally spaced (15 μm) taken at the end of the experiment (1 h) for local measuring with Raman. Panels a and c collinearity plot for local and global measuring, respectively. (Panels b and d show determinant measure for local and global, respectively. The solid dots represent the experimental data: ■ and • indicate the unstirred and the stirred contact with Raman, whereas ⧫ points to all of the data points in the sequential global determination.

on the accuracy and precision obtained for each of the parameters, which happen to be surprisingly low for either of the coefficients compared to their global measuring counterparts. This may be caused by several reasons, partly due to the sensitivity difference between FTIR and Raman and partly due to the higher number of measurements obtained. In general,

since more measurements are collected by the local experiments, the variance of the estimates should decrease. However, the information collected by the FTIR and the Raman microspectroscopy are inherently different and cannot be compared straightforwardly. Rigorously, the variance of estimates obtained by different methods can be compared by 4730


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Table 2. Values of Diffusivity and Mass Transfer Coefficient from Simultaneous Regression of Locally Measured Data from Stirred and Unstirred Contacts. Significant Differences Are Found between Both Types of Contacts D (m2 s−1) stirred contact unstirred contact

−13

accuracy D −13

17.1 × 10 , (13.8, 21.4) × 10 (a) 6.2 × 10−13, (5.3, 7.6) × 10−13 (b)

0.015 1.7 × 10−3

k (m s−1)

accuracy k

precision k

Bi

6.5 × 10 , (5.3, 8.1) × 10−4 (c) 3.4 × 10−4, (3, 3.8) × 10−4 (d)

8.3 × 10−4 1.9 × 10−3

0.22 0.12

34 54

precision D 0.22 0.18

−4

4.2.3. Validity of the Estimated Parameters and Methods. The main hypothesis supporting the sequential method is that it is possible to reach a stirring rate that makes external mass transfer resistance negligible. While this is a common statement, to confirm or reject this hypothesis, the evolution of Dapp along with the stirring level should have been observed. Regarding the simultaneous determination, stirring should influence only k and not D. Since there is not data at more than one stirring level, k and D data may be validated by comparing the sequential and the simultaneous determination. This way, D values obtained by simultaneous determination should not be significantly different from the Dapp determined sequentially. As well, k values obtained from unstirred contacts (simultaneous or sequential) should be different from the one obtained from the stirred contact (only simultaneous determination). Results show that all the unstirred mass transfer coefficients (1.3, 3, and 3.4 × 10−4 m s−1) are significantly different by very small margins; while very different from the stirred one, which almost doubles the value of the greatest of them (6.5 × 10−4 m s−1). It might be then concluded that stirring increases as much as twice the value of the mass transfer coefficient. These differences are noticeable even if with the lack of accuracy and precision of the sequential determination, derived from the scatter of the data taken by global measuring. Contrary to longaccepted ideas, the resistance to transfer appears to be not negligible in the case of a viscous liquid. Regarding the internal transport, apparent diffusivity determined by sequential fitting under stirring from both global and local measurements can be considered the same (5.3 and 8.7 × 10−13 m2 s−1) and similar to the value of true diffusivity determined from local measures by double regression from an unstirred contact (6.2 × 10−13 m2 s−1). On the other hand, values of true diffusivity obtained by simultaneous regression are different by almost a factor of three (17 and 6.2 × 10−13 m2 s−1 stirred and unstirred conditions, respectively), which suggests that stirring might influence the internal mass transport as well. This result is also confirmed by the contour plot of SSQR w.r.t. D and k. The contours close to the minimum SSQR also show that an ellipse is formed with a long semiaxis almost parallel to the horizontal axis (where D is represented). In principle, the error would be comparatively larger when estimating D than when estimating k, but no evidence allows extrapolating this observation to other conditions. This difference may be explained by the effect of the solvent uptake, which has been reported to influence diffusivity of migrants in polymers. The evidence of such an effect is nonconclusive for polyolefins and olive oil, while literature using miglyol as a simulant is still scarce.21−23 As well as in the case of the determination of the mass transfer coefficient, the differences in terms of accuracy and precision between global and local measuring are remarkable for comparable Bi. Regarding the global mass transfer coefficient or apparent diffusivity, values are in the range of those determined by the same global and local methodologies in ref 16 using olive oil as simulant (8 ± 0.25 × 10−14 m2 s−1). More difficult is the

comparing the elements of the Fisher Information Matrix (FIM), which are related to the sensitivity of the observables to the model parameters. The elements of the FIM will increase because more measurements are available or because the measurements are more sensitive to a change in the parameters; hence, it seems likely that the Raman microspectroscopy can outperform the FTIR. In any case, the properties of the FIM for each of the methods have not been analyzed here but will be the object of upcoming work. 4.2. Determination of Transport Properties by Simultaneous Regression. 4.2.1. Global Measuring. Using the same experimental data issued from global measuring and presented previously, D and k were determined by simultaneous regression. It was observed that the value of the estimates depended strongly on the initial guess, which indicates that identifiability problems could be present. The problem of the unique estimation of D and k was further investigated mapping the residuals (Figure 5 left). The plot of SSQR (eq 15) for different estimates of D and k revealed a region of infinite pairs of parameters that minimize the SSQR. As a consequence, in the conditions of this experiment, the parameters are nonidentifiable. This analysis can be generalized if the collinearity index (γ) is represented for numbers of Fourier and Biot that virtually represent the whole space of application of plastics in food packaging (Figure 6a). It can be seen that for an experimental setup equivalent to this one (five global measurements distributed in time), no region presents a γ < 15, which can be considered as the threshold for practical identifiability. As discussed previously, the identification of the parameters becomes difficult and largely dependent on the initial guess, and hence, the obtained values of D and k only calibrate the model but cannot be extrapolated in any other system. 4.2.2. Local Measuring. The same estimation steps were done for local measurements using the data previously presented. For the local measurements, it was possible to determine a pair of D and k that did not depend of the initial guesses. In effect, representing the residuals (Figure 5), it can be seen that the SSQR contours converge to a point with a unique minimum. The value of the coefficients lead to Bi = 34 and Bi = 54 for the stirred and the unstirred contacts, respectively. In more general terms, the collinearity index of the local measuring setup is lower than for the global measuring one. As shown in Figure 6, panel c, there is a region of space corresponding to high Bi and low Fo, where γ < 15, which indicates that, a priori, the parameters may be identifiable. The values estimated are displayed in Table 2. Indeed, the identifiability of the parameters can be improved if more information is provided about the system. As an example, decreasing the spacing between the local measures then leading to a higher number of measurements within the polymer results in a larger region where γ < 15. However, the information provided about the system does not grow indefinitely increasing the number of measurements, whether global or local, due to sensitivity limitations of every analysis method. 4731


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Figure 7. Evolution of the accuracy and precision of the fit of generated data with only Dapp (continuous line for global measuring, dashed line for local measuring). Accuracy and precision of the experimental determinations (values obtained from Tables 1 and 2).

low diffusion coefficients (e.g., polystyrene, polyethylene terephthalate, polycarbonate), or when a low-viscosity liquid is well stirred. Hence, and based on the previously presented results, this section tries to answer to the two following questions: (1) Under which conditions must the two coefficients (D and k) be determined since the approximation Dapp ≈ D would lead to an unacceptable large error? (2) And, for those conditions, what experimental setup would allow the identification of the two parameters to accurately determine D? Of course, the answers to these two questions are linked. If approximating Dapp to D leads to a large error, that means that external resistance to mass transfer influences the desorption of the migrant. The measurements are thus sensitive to the value of k, and it is likelier that both parameters can be identified. However, it does not imply that if the impact of a change in k is relevant for the results of a migration test, the change in k is enough to identify both parameters. Fortunately, it is possible to design highly informative experiments (e.g., local measuring profiles) that enhance the identifiability of both parameters before the impact on migration test results (which are indeed global measuring profiles) is unacceptably large. Without loss of generality, we consider a representative case of food/packaging migration: a migrant with a diffusivity of D = 5 × 10−13 m2 s−1, polymer thickness L = 240 μm, and a test lasting for 24 h (the expected remaining concentration of the migrant in the polymer at the end of the experiment is 10% (or similar)). Measurements are taken at t = 0, 3, 6, 15, and 24 h. Figure 7 represents the plot of the evolution of the accuracy and the precision of the fit of the synthetic data using only Dapp along with an increasing Bi number. The figure includes the values of accuracy and precision for the sequential and joint determination depicted in Tables 1 and 2 so the experimental setups can be compared between them. As supposed, the greater the Bi number, the better the fit with one parameter describes the synthetic data, which represented by decreasing values of accuracy and precision. By comparing Figure 7 to Figure 6, it is noticeable that the ranges of Bi and Fo with better (while not optimal) identifiability correspond to the coefficients presenting the best values of accuracy and precision. Consequently, the joint

comparison of the mass transfer coefficient, since literature values are really scarce for food-packaging systems.7 Report values spread between approximately 6−20 × 10−8 m s−1 and diffusion coefficients between 10−14 and 10−12 m2 s−1 for low molecular weight families of alkanes, alcohols, and other commercial molecules in a LDPE/ethanol system followed by global concentration measuring with GC-FID until equilibrium. While diffusivities stay in range, mass transfer coefficients are significantly lower for a less viscous liquid, which appears theoretically contradictory. This might be explained because of all the three coefficients being determined by global measuring until equilibrium was reached. If, as it is shown by the practical identifiability analysis (Figure 6), global measures taken at the beginning of the kinetics increase the sensitivity to the mass transfer coefficient, measures taken at very high numbers of Fo (near equilibrium) may increase the sensitivity to the partition coefficient but may also decrease the sensitivity to the mass transfer coefficient dramatically, leading to a less robust estimation, which might be reflected by this difference of approximately three orders of magnitude. More in agreement are values from ref 12 for desorption of BHT (220.4 g mol−1) from LLDPE into water, going from 10−5 to 10−7 m s−1 with a model that also takes into account the BHT degradation reaction. 4.3. Influence of the External Transfer. As it has been stated, it is common practice to neglect the mass transfer coefficient, especially for high Bi numbers, indicative of a small resistance to the external transfer compared to the internal. To evaluate the influence of this external mass transfers, two different analyses have been made: first, the accuracy and precision of a fit of just one parameter were evaluated by calculating confidence bounds at several Bi numbers. Then, at a single Bi number, the influence of the error of the determination in migration was estimated. 4.3.1. Accuracy and Precision. A sound first-principles description of migration would consist at least of two phenomena: diffusion in the polymer (characterized by D) and mass transfer in the liquid phase (characterized by k). However, in a number of situations, diffusion is the limiting step, and migration can be characterized solely by Dapp without introducing a significant error. This is the case with migrants of high molecular weight, certain polymers that have inherently 4732


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Figure 8. Time required to reach the SML of Uvitex OB according to the widest confidence bound found for Bi = 30. The vertical dot-dashed lines represents the shortest and the longest time required to reach equilibrium.

and 5. Because of this, it has been chosen not to include the evolution of the precision in the figure, since it is considered unrealistic. Again, a comparison between Figures 6 and 7 reveals that the range of Bi numbers that allow the most accurate and precise identifiability while still needing both coefficients lies between Bi = 30 and Bi = 100, always for low Fo numbers. 4.3.2. Influence on Migration. Figure 8 represents the required time to reach the SML according to the widest confidence different bound of the intervals obtained for Bi = 30. As stated in the Materials and Methods section, although the simulation represents a typical migration test with a packaging of 6 dm2 in contact with 1 kg of food, the packaging is considered spiked enough, and the partition coefficient low enough so that the maximum concentration attainable in the food is the SML of the Uvitex OB (0.6 ppm). As said before, the confidence intervals computed for Bi = 1 and Bi = 5 are absolutely unrealistic, for the case of local measuring, due to the neglect of the partition effect. At Bi = 30, they span an order of magnitude and can be representative of a maximum error of an experimental determination. The plot of the migration kinetics with these diffusion coefficients shows that the shortest time (corresponding to the biggest diffusion coefficient) is of more than 5 days, while the longest takes more than three months. While these kinetics are not representative of reality, the time difference between the lowest and the highest migration time illustrates the importance that the mass transfer coefficient may attain for moderate values of Bi.

determination under unstirring conditions, presenting a lower Bi number than joint determination under stirring conditions, exhibits slightly worse accuracy for k, while the accuracies for the diffusivity remain in the same range of values. The sequential determination by global measuring results in bad accuracy for either of the coefficients, specially for k. This is probably a consequence of the data points laying in regions with poor identifiability (Figure 6c) and containing little information (Figure 6d). According to Figure 7, panel b, the joint determination of D and k under stirring conditions is the most precise (0.1). Values determined under unstirring conditions are less precise (between 0.15 and 0.22) than the fit with one coefficient (0.12). Local measuring presents one particularity: while the coefficients determined either sequentially or jointly present outstanding values for accuracy and precision, the description of the system with just one parameter leads to very bad fitting of the data, for Bi numbers from 1−300. This is the result of the model being unable to represent the points at the interface at low Bi numbers. In effect, the approximation Dapp ≈ D is based on the hypothesis of a large Bi number. However, it can hold at moderate Bi for global measuring results to the large collinearity of both k and D, it poorly represents the evolution of the concentration profile in the polymer. At large Bi, the concentration in the outer layer of the polymer equals the concentration of the liquid for any t > 0. However, as the Bi decreases, the difference between the concentration in the polymer interface and the liquid increases, which cannot be captured by the structure of the approximated model. Had the concentration in the liquid been taken into account, the misfit of the data would have been represented by relatively wide confidence intervals for both Dapp and K instead of confidence intervals of up to three orders of magnitude for values of Bi of 1

5. RESULT DISCUSSION AND COMPARISON OF METHODS Global and local measurements give comparable results if the coefficients are determined by the sequential method. Each of the methods has its own pros and cons that make them more 4733


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one coefficient; the second, what are the conditions that provide a good identifiability; and the third, the hypothetical existence of a match between the regions with better identifiability and the most accurate and precise determinations. Azevedo et al.9 concluded that the joint determination of both parameters should, in general, be avoided, especially for high Bi above 100. While this analysis is only performed on global kinetics, their synthetic data show a clear loss of accuracy and precision for Bi above 50. Results of the present work can enrich this discussion. For representative experimental setups like those presented, the regions with better identifiability (Figure 6) (represented by a low collinearity index and a high determinant measure) range between Bi = 30 and Bi = 1000. Below 30, the diffusion coefficient might be too low, and above 1000, the mass transfer coefficient might be to high to be properly identificated. As well, the regions on which the system can be properly described by just using one parameter (Figure 7) start on Bi = 50 according to the synthetic data. As well, it has been shown that at Bi = 30, the error introduced by using just one coefficient may lead to significant underestimation of the migration levels. Nevertheless, it is important to remark that the Bi numbers with better identifiability found during this work can only be compared with those from Azevedo et al. for the global measuring case because of the additional information the local measuring provides. As said, while identifiability depends strongly on Bi for both global and local measuring setups, for the global measuring setup, every data point is taken at a different Fo number; differently from local measuring, for which all data points are obtained at the same value of Fo. Because of this, the local measuring setup allows to improve identifiability by taking several data sets at different Fo numbers. It may be concluded that, for the system and stirring levels studied in this work, which lead to Bi between 30 and 50, both coefficients can be determined with the local measuring setup, since the identifiability, while not optimal, is good, laying in the region of Fo between 0.22 and 0.72. This is not the case for the global measuring, for which three out of four data points lay in regions with either too low Bi or too high Fo. If the global measuring technique had been more sentitive, it would allow us to take measures at more informative Bi/Fo regions and improve identifiability. In other words, the error on a migration test derived for the use of just Dapp instead of D and k depends on the precision on the determination of Dapp, which is determined by Bi. This is not necessarily linked to the identifiability of D and k, which depends on D, Bi, and the experimental setup.

suitable in a case-by-case basis. For instance, global methods require more samples and follow the kinetics at different desorption times for a comparative accuracy in the estimation of D. On the other hand, measurements can be taken online or in an automatized way and are adapted not only to FTIR, but also to a number of well-known analytical techniques such as UV, fluorescence spectroscopy, chromatography, etc., making it very versatile. Local measurements, such as those obtained by Raman,15 FRAP,24 or gravimetry after cutting the sample with a microtome,25 give valuable insight from the inside of the polymer, such as crystallinity or solvent uptake, and can also be used in a nondestructive way. However, the sample preparation and experimental setup are usually more complex. The main disadvantage of the sequential method is that, to be reliable, it requires a number of repetitions of the experiment until the estimated diffusivity value converges to a given value. Even then, it cannot be ensured that the experimental setup will lead to a completely negligible external mass transfer resistance if the fluid is very viscous or if diffusivity is very high. In principle, the simultaneous method is more suitable since the determination of both parameters does not require any repetition of experiments. On the contrary, an obvious limitation is a potential poor identifiability, which in this case has been shown for the global measuring method. In theory, the sequential method would always be able to determine both parameters, also with global concentration measurements, since it is assumed that the true diffusivity can be known. In practice, the relatively large errors that involve the determination of diffusivity may be a serious obstacle to determine the mass transfer coefficient, in particular at large Bi. A low collinearity index γ is a necessary but not a sufficient condition for practical parameter determination. It is also important that the measurements are sensitive to the outputs what makes the determinant measure (ρ) a more interesting metric to elucidate the differences between the global and local measuring methods. Even though it is not possible to give absolute thresholds to ρ, it can be regarded as a comparison among methods provided that the measured outputs and parameters have the same dimensions. The region where ρ is high for global measurements (Figure 6d) indicates conditions where the internal and the external resistances limit mass transfer similarly (centered in Bi = 102 and Fo = 0.8) at a time scale where the measurements provide diverse information (i.e., different concentrations). As for the local measuring method (Figure 6b), it outperforms the global measuring method for experiments with a similar time demand given that ρ has a higher value for the whole space of Bi and Fo. It is important to note that the previous discussion assumes the validity of Fickean diffusion, which is accepted as the main framework for modeling migration in food packaging. Currently, the effect of solvent uptake and plasticization in polymers remains unclear. While there is evidence of plasticization effects in PVC,26 reports are not conclusive for polyolefins.16 If plasticization caused by solvent uptake is possible, the sequential method becomes less reliable. In effect, the solvent uptake would depend on the diffusivity of the solvent and the mass transfer coefficient, which would, in turn, affect the diffusivity of the migrant. Changing the mass transfer coefficient would influence the migrant diffusivity. To what extent or which experimental setup would allow to determine the parameters of such a complex system is not tackled here. This way, there seem to be three discussion points: the first, whether the system may be similarly described by using just

6. GENERAL CONCLUSIONS Methods for parameter estimation and the corresponding experimental setup were compared with the aim of elucidate the best conditions to determine the mass transfer coefficient and the diffusivity of a compound commonly used by packaging industry. The system subject of study was optical brightener Uvitex OB from LLDPE into food simulant Miglyol 829 at 40 °C. The methods were based on utilization conditions where the external where only the internal mass transfer is limiting (sequential method) or treating both simultaneously (simultaneous estimation method). Each of the methods can be based on measuring the average concentration in the polymer (global measuring) or the concentration profile along the thickness (local measuring). 4734


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(3) Helmroth, I. E.; Dekker, M.; Hankemeier, T. Additive Diffusion from LDPE Slabs into Contacting Solvents as a Function of Solvent Absorption. J. Appl. Polym. Sci. 2003, 90 (6), 1609. (4) Tseng, K. C.; Turro, N. J.; Durning, C. J. Tracer Diffusion in Thin Polystyrene Films. Polymer 2000, 41 (12), 4751. (5) Lagaron, J.; Cava, D.; Gimenez, E.; Hernandez-Munoz, P.; Catala, R.; Gavara, R. On the Use of Vibrational Spectroscopy To Characterize the Structure and Aroma Barrier of Food Packaging Polymers. Macromol. Symp. 2004, 205, 225. (6) Vitrac, O.; Hayert, M. Identification of Diffusion Transport Properties from Desorption/Sorption Kinetics: An Analysis Based on a New Approximation of Fick Equation during Solid−Liquid Contact. Ind. Eng. Chem. Res. 2006, 45 (23), 7941. (7) Vitrac, O.; Mougharbel, A.; Feigenbaum, A. Interfacial Mass Transport Properties Which Control the Migration of Packaging Constituents into Foodstuffs. J. Food Eng. 2007, 79 (3), 1048. (8) Azevedo, I. C. A.; Oliveira, F. A. R. A Model Food System for Mass Transfer in the Acidificaacid of Cut Root Vegetables. Food Sci. Technol. 1995, 30, 473. (9) Azevedo, I. C. A.; Oliveira, F. A. R.; Drummond, M. C. A Study on the Accuracy and Precision of External Mass Transfer and Diffusion Coefficients Jointly Estimated from Pseudo-experimental Simulated Data. Math. Comput. Simul. 1998, 48, 11. (10) Petersen, B.; Gernaey, K.; Vanrolleghem, P. A. Practical Identifiability of Model Parameters by Combined RespirometricTitrimetric Measurements. Water Sci. Technol. 2001, 43 (7), 347. (11) Gandek, T. P.; Hatton, T. A.; Reid, R. C. Batch Extraction with Reaction: Phenolic Antioxidant Migration from Polyolefins to Water. 1. Theory. Ind. Eng. Chem. Res. 1989, 28 (7), 1030. (12) Gandek, T. P.; Hatton, T. A.; Reid, R. C. Batch Extraction with Reaction: Phenolic Antioxidant Migration from Polyolefins to Water. 2. Experimental Results and Discussion. Ind. Eng. Chem. Res. 1989, 28 (7), 1036. (13) Vergnaud, J. M. General Survey on the Mass Transfers Taking Place between a Polymer and a Liquid. J. Polym. Eng. 1995, 15, 57. (14) Martinez-Lopez, B.; Chalier, P.; Guillard, V.; Gontard, N.; Peyron, S. Determination of Mass Transport Properties in Food/ Packaging Systems by Local Measurement with Raman Microspectroscopy. J. Appl. Polym. Sci. 2014, 131, 40958. (15) Mauricio-Iglesias, M.; Guillard, V.; Gontard, N.; Peyron, S. Raman Depth-Profiling Characterization of a Migrant Diffusion in a Polymer. J. Membr. Sci. 2011, 375 (1−2), 165. (16) Mauricio-Iglesias, M.; Guillard, V.; Gontard, N.; Peyron, S. Application of FTIR and Raman Microspectroscopy to the Study of Food/Packaging Interactions. Food Addit. Contam. Part Chem. Anal. Control 2009, 26 (11), 1515. (17) Fick, A. Ueber Diffusion. Ann. Phys. 1855, 170 (1), 59. (18) Brun, R.; Reichert, P.; Künsch, H. R. Practical Identifiability Analysis of Large Environmental Simulation Models. Water Resour. Res. 2001, 37 (4), 1015. (19) Sin, G.; Meyer, A. S.; Gernaey, K. Assessing Reliability of Cellulose Hydrolysis Models To Support Biofuel Process Design Identifiability and Uncertainty Analysis. Comput. Chem. Eng. 2010, 34 (9), 1385. (20) Brun, R.; Kühni, M.; Siegrist, H.; Gujer, W.; Reichert, P. Practical Identifiability of ASM2d ParametersSystematic Selection and Tuning of Parameter Subsets. Water Res. 2002, 36 (16), 4113. (21) Begley, T.; Biles, J.; Cunningham, C.; Piringer, O. Migration of a UV Stabilizer from Polyethylene Terephthalate (PET) into Food Simulants. Food Addit. Contam. 2004, 21 (10), 1007. (22) Begley, T. H.; Hsu, W.; Noonan, G.; Diachenko, G. Migration of Fluorochemical Paper Additives from Food-Contact Paper into Foods and Food Simulants. Food Addit. Contam. 2008, 25 (3), 384. (23) Franz, R.; Brandsch, R. Migration of Acrylic Monomers from Methacrylate PolymersEstablishing Parameters for Migration Modelling. Packag. Technol. Sci. 2013, 26 (8), 435. (24) Pinte, J.; Joly, C.; Ple, K.; Dole, P.; Feigenbaum, A. Proposal of a Set of Model Polymer Additives Designed for Confocal FRAP Diffusion Experiments. J. Agric. Food Chem. 2008, 56, 10003.

The sequential method can be used with both local and global measuring analyses. It requires a number of experiment repetitions to ensure that only the internal resistance to diffusion is limiting, which can be cumbersome or impossible for very viscous liquids. The simultaneous method cannot be used with global measuring analyses because of poor parameter identifiability, except for narrow experimental conditions and a large number of samples. The analytical method should also be relatively sensitive. Local measuring analyses, such as Raman microspectroscopy, could be used in both sequential and simultaneous methods, as the information provided by the measurements gave place to good identifiability (γ) and sensitivity (ρ). Regarding the experimental setup, it can be concluded that local measuring methods should be selected whenever possible. As discussed previously, it is possible to achieve narrower confidence intervals, a better identifiability thanks to both higher sensitivity and lower collinearity and hence carry out a simultaneous identification of D and k, hence ensuring that the external mass transfer resistance is taken into account. Simulations have also shown that the conditions under which it may be desirable to take into account both coefficients to avoid the underestimation of the migration levels lay in regions with good, while not optimal identifiability, leading to accurate and precise determinations. Since mathematical modeling supposed to replace experimentation in the determination of migration levels, an accurate description of the system is of vital importance. The design of the experimental conditions (number of measurements, when they must be taken) has not been tackled in this paper, but it was shown to be of great importance. In an upcoming contribution, these results will be formalized and systematized to provide the conditions needed for an optimal experimental design for the determination of the mass transfer and diffusion coefficient.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work has been possible thanks to the funding provided by the French funding agency Association Nationale de la Recherche within the framework of the Research Project SafeFoodPack Design. M.M.-I. gratefully acknowledges the financial support provided by the People Program (Marie Curie Actions) of the European Union’s Seventh Framework Programme FP7/2007-2013 under REA Agreement No. 627475 (GREENCOST). M.M.-I. belongs to the Galician Competitive Research Group GRC 2013-032, programme cofunded by FEDER.

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REFERENCES

(1) Moisan, J. Y. Diffusion Des Additifs Du Polyethylene: Influence de La Nature Du Diffusant. Eur. Polym. J. 1980, 16 (10), 979. (2) Ferrara, G.; Bertoldo, M.; Scoponi, M.; Ciardelli, F. Diffusion Coefficient and Activation Energy of Irganox 1010 in Poly(propyleneco-ethylene) Copolymers. Polym. Degrad. Stab. 2001, 73 (3), 411. 4735


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Industrial & Engineering Chemistry Research (25) Reynier, A.; Dole, P.; Feigenbaum, A. Integrated Approach of Migration Prediction Using Numerical Modelling Associated to Experimental Determination of Key Parameters. Food Addit. Contam. 2002, 19 (Suppl. 1), 42. (26) Fankhauser-Noti, A.; Grob, K. Migration of Plasticizers from PVC Gaskets of Lids for Glass Jars into Oily Foods: Amount of Gasket Material in Food Contact, Proportion of Plasticizer Migrating into Food, and Compliance Testing by Simulation. Trends Food Sci. Technol. 2006, 17 (3), 105.

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Practical Identifiability Analysis for the Characterization of Mass

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