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Multiresponse kinetic modelling of the formation, release, and degradation of allyl isothiocyanate from ground mustard seeds to improve active packaging

This study aims to describe allyl isothiocyanates (AITC) formation from enzymatic sinigrin hydrolysis in ground mustard seeds and its release and degradation in the headspace using multiresponse kinetics modelling. The mechanistic modelling of the steps involved in the packaging system consists of a set of ordinary differential equations established from bio (chemical) reaction models combined with mass transfer models. The estimated parameters consist of the accessible sinigrin fraction, rate constants of sinigrin hydrolysis, AITC degradation in the particles and headspace, and its mass transfer coefficient. The model provides a good fit to experimental results and confirms the proposed mechanism of the AITC formation, degradation, and release inside the packaging system. Fat content has significant effects on AITC formation and release rate constants, while particle sizes significantly affect accessible sinigrin in the particles. These results give an understanding of AITC’s controlled release by manipulating the mustard properties to optimize antimicrobial packaging designs.


Introduction
Antimicrobial packages are currently being developed by the food industries for shelf life extension of food products by preventing the spoilage bacteria to grow at its food surface (Reyes-Jurado et al., 2019). Antimicrobial packages contain compounds that can be slowly released from antimicrobial carriers to food products with direct surface-food contact transfer or indirect headspace transfer (Wong et al., 2020). For indirect contact, the concentration of volatile compounds in the headspace determines the microbial growth rate in the food (Kurek et al., 2017). Therefore, controlling the release rate of the compounds to the headspace to retain sufficient concentrations has a major impact in effectively preventing bacterial growth to reach the desired product shelf life (Lorenzo et al., 2014;Quintavalla and Vicini, 2002).
Allyl isothiocyanate (AITC) is acknowledged as a strong antimicrobial agent and can be formed from the enzymatic hydrolysis of sinigrin, which is the main glucosinolate in mustard seeds (Reyes-Jurado et al., 2019). Once the cells of the mustard seeds are disrupted, e.g. during the grinding process and fat extraction, myrosinase hydrolyzes the sinigrin upon moisture exposure to yield AITC (Hanschen et al., 2018;Okunade et al., 2015). The formed AITC can be subsequently partitioned to phases with different stability; fat phase of the seeds (most stable), headspace (slightly stable), or aqueous phase (unstable). Bahmid et al. (2020) reported that AITC amount released from the ground mustard seeds still containing fat was lower than the AITC release from defatted ground mustard seeds (0% fat). The results indicate that the AITC prefers to move into the fat phase of the mustard particles rather than into the packaging headspace (Dai and Lim, 2014). To understand the AITC partition and stability, the kinetics of the release and degradation of AITC taking place in the seeds and headspace of the package needs to be evaluated by developing such a kinetic model.
Multiresponse modelling could be used as a powerful tool for describing physical and chemical processes using multiresponse data, such as multiple compounds involved in chemical reactions (Boekel, 2009;Verkempinck et al., 2019). This model approach is used to estimate parameters more accurately and is considered as a more comprehensive assessment of a proposed mechanistic model, compared to single response modelling (Quintas et al., 2007;Ziegel and Gorman, 1980). Some examples of studies using multiresponse modelling are on the understanding of the complex reaction mechanisms of lipid digestion (Verkempinck et al., 2019), caramelization, and Maillard reactions (Goncuoglu Tas and Gokmen, 2017;Kocadaglı and Gökmen, 2016). Applying multiresponse kinetic modelling in food packaging, is still limited, whereas food packaging has a high complexity including mass transfer/migration of compounds through the package, between the packaging material and the packaged food and reactions with food components.
A single modelling approach was applied to understand the release of compounds from a film in an active antimicrobial package by Kurek et al. (2017). The single response of mathematical model used to describe the AITC release from the encapsulation of β-cyclodextrin (β-CD) from a polymeric matrix did not include the possible reaction in the film or the headspace, or absorption of the volatile in the packaged (model) food. Ignoring these mechanisms, which will also change the concentration in the film and headspace, will have led to less accurate predictions of the model parameters. A multiresponse kinetic model can, therefore, be a novel way to describe those changes of compound concentration in the whole packaging system, which benefits the understanding of the reactions and mass transfer involved in the packaging system in an integrated way.
The multiresponse kinetic modelling approach in this mechanistic study aims to describe the formation, release, and degradation of the AITC in the packaging system. The experimental data of sinigrin degradation, concentration AITC in mustard particles, and headspace investigated from our previous study (Bahmid et al., 2020) were used. The fits of the developed kinetic model to the measured data of the sinigrin and AITC concentration in each phase of the package were evaluated to determine whether the established model was able to accurately describe degradation and release of AITC. A better understanding of the reactions and mass transfer in the packaging system could help to optimize the mustard seeds properties and packaging design to effectively control the AITC release in antimicrobial packaging.

Theoretical considerations
Sinigrin degradation is initiated upon tissue and cell damage in the mustard seeds (Okunade et al., 2015). Sinigrin is located in protein bodies or vacuoles of S-cells and the hydrolyzing enzyme myrosinase is located in different compartments, called myrosin cells (Kissen et al., 2009;Nakano et al., 2014;Stauber et al., 2012;Torrijos et al., 2019). Once the tissue is disrupted by physical disruption or fat extraction, myrosinase instantaneously hydrolyzes sinigrin to form AITC in the presence of water (Dekker et al., 2009;Tetteh et al., 2019;Xiao et al., 2019). This mechanism of hydrolysis of sinigrin is called the "mustard oil bomb theory" (Stauber et al., 2012). Lüthy and Matile (1984) developed this theory for the subcellular organization of the glucosinolate-myrosinase system in horse-radish roots (Sinapsis alba). This theory explains the stability of the glucosinolate-myrosinase system with a different compartment of the substrate and enzyme. The theory was then critically assessed by Kissen et al. (2009) that clarified that the glucosinolate hydrolysis by myrosinase happened only in the case of tissue damages.
The accessible sinigrin is enzymatically hydrolyzed by myrosinase upon the addition of water to yield AITC in the ground mustard seeds. In the mustard particles, the formed AITC can be degraded due to the water added to the packaging system. The water attacks the sulfhydryl groups of AITC to form other compounds (Encinas-Basurto et al., 2017;Olaimat and Holley, 2016). At the same time, the volatile AITC also partitions into the headspace. From the headspace, the AITC has an antimicrobial effect on the food product in the package. In the headspace, AITC can react with moisture to form other compounds as occurred in the mustard particles as well. These reactions and mass transfer pathways are schematically depicted in Fig. 1a. Fig. 1b shows the release of AITC from mustard particles into headspace, which can be specifically described by the theory of mass transfer from a solid to a gas (Harrison et al., 2006). In the mustard particle phase, AITC diffuses to the interface and is transferred from the ground mustard seeds, across the interface, into the headspace through unsteady state molecular diffusion. The main resistance of this transfer will be in the outer surface layer of the mustard seed particles. This mass transfer occurs until reaching the equilibrium in concentration between the ground mustard seeds and headspace.

Sample preparation
Mustard seeds were freeze-dried (Christ Alpha 1-4 LD plus) and then ground by using a milling machine (Fritsch; Pulverisette 14). The ground particles were sieved (Hosokawa Alpine Air Jet sieve 200 Ls) in different particle sizes (50-100, 200-315, 400-500 and 630-800 μm) and then d efatted using soxhlet extraction till intended fat content = rate constant of sinigrin degradation and AITC formation (h − 1 ) k 2 = rate constant of AITC degradation in the mustard particles (h − 1 ) k 3 = rate constant of AITC degradation in the headspace (m 3 / mol/h) K s/g = equilibrium partition coefficient ground mustard seeds and headspace (dimensionless) MIC = Minimum Inhibitory Concentration m s = mass of mustard particles (g) m t = mass transfer coefficient from mustard particles into headspace (m/h) N = molar flux (mol/m 2 h) p g = breakdown products in the headspace p s = breakdown products in the mustard particles Sin a = concentration of accessible in mustard particles sinigrin (mol/m 3 ) Sin i = concentration of inaccessible sinigrin in mustard particles (mol/m 3 ) Sin tot = concentration of total sinigrin in particles (mol/m 3 ) Bahmid et al. (29.1%, 17.1%, 2.8%, and 0.0%) were reached. Mustard particles (0.16 g) were put inside the Duran bottles (10 mL) and rehydrated with 0.132 mL of Milli-Q water (ratio (w/v); 1:0.825) as shown in Fig. 2. The samples were stored at 20 • C for 48 h and sinigrin and AITC were measured at certain time intervals.

Sinigrin and AITC measurement
Sinigrin (C 10 H 16 KNO 9 S 2 ; 397.46 g/mol) and AITC (C 4 H 5 NS; 99.15 g mol − 1 ) concentration in the mustard particles and AITC in the headspace were determined using the method described by Bahmid et al. (2020). The analysis of sinigrin and AITC in the ground mustard seeds was done by extraction followed by High-Performance Liquid Chromatography (HPLC; Thermo Scientific; UHPLC + focused Dionex Ulti-Mate 3000) according to reported by Oliviero et al. (2012) with minor adjustments. The concentration of sinigrin in mustard particles was calculated with the help of glucotropaeolione as an internal standard and the relative response factor (RRF) of sinigrin (1.053). The AITC concentration in the mustard particles was quantified by the curve of the calibration standards of pure AITC (Allyl isothiocyanate purchased from Sigma Aldrich Chemie GmbH, 97%).
Analysis of AITC in the headspace was performed using Gas Chromatoghraphy -Flame Ionization Detector (GC-FID; Thermo-Scientific, Focus GC) and autosampler (Thermo-Scientific, TriPlus Autosampler) for AITC in the headspace as described by Bahmid et al. (2020). The calibration was done with pure AITC dissolved in Hexane (n-Hexane PEC grade, Actu-All chemicals), with the range of used concentrations 1 ppm-250 ppm.
Data (mol compounds in liter volume of mustard particles or headspace) for modelling the sinigrin degradation and AITC release in the headspace were collected by sampling every 3 h for 48 h in mol/m 3 . Data for modelling AITC remaining in the mustard particles were collected by sampling at 0, 24, and 48 h. Each dataset was obtained using duplicate samples and expressed as the mean values ± standard deviations (SD).

Determination of the partition coefficient of AITC
The partition coefficient (K s/g ), the concentration ratio of AITC in the immiscible phase to the headspace as expressed in Equation (1), was measured using the method modified from Zhang et al. (2010).
where AITC s and AITC g are the concentrations of AITC in the immiscible phase and headspace, respectively.
To investigate the effect of fat content in mustard particles on K s/g , emulsions with different oil (mustard oil extract) content (0, 2.8, 17.1, and 29.1%) were studied as (model) ground seeds. The oil was mixed with Milli-Q and 0.5% of tween 20 as an emulsifier with an Ultra-turrax (IKA Werke GmbH & Co. KG) at 9500 rpm min − 1 within 2 min. The unstable emulsion was subsequently homogenized using a homogenizer (Delta Instruments) for 10 min at 120 bar. Afterward, AITC (100 mg/L) was added to the emulsions. Different volumes (1, 2, 3, 4, and 5 mL) of the emulsion were poured into 20 mL vials and immediately sealed gastightly. The samples were stored at 20 • C for 3 h to reach equilibrium partition between emulsion and headspace. The method used to perform AITC measurement using GC-FID is described in a previous publication (Bahmid et al., 2020). The partition coefficient (K s/g ) of AITC between emulsion and headspace was determined according to phase ratio variation (PRV) method (Zhang et al., 2010), which is the relationship of K s/g , β, and the peak area (A e ) as expressed below; where f 0 is a proportional factor (A e = f 0 ⋅C e ), and C o and C e are the initial concentration of AITC in the seeds and the equilibrium AITC concentration in the emulsion, respectively; A e is gas chromatographic peak area of AITC in the headspace at equilibrium; β is the phase volume ratio of headspace to the emulsion. The value of K s/g can be directly calculated from the ratio of the y-intercept to the slope of the regression line in the plot of 1/Ae versus β (Zhang et al., 2010).

Development of the kinetic model
The mathematical model was restricted to (bio)chemical reactions, solid-gas equilibria, and physical mass transfer processes. It consists of a combination of differential equations, equilibria, and mass balances for Fig. 1. Schematic representation of allyl isothiocyanate (AITC) formation in mustard particles (white area) and AITC release in the headspace (grey area) (a) proposed reaction pathways on multiresponse modelling procedures, and (b) the schematic presentation of mass transfer of released AITC from the mustard seed particles into the headspace. N is molar flux, k 1-3 is the degradation rate constant of each phase, P s and P g are breakdown products of AITC in the solid and gas phase respectively, K s/g is the partition coefficient, and m t is interfacial mass transfer.   Bahmid et al. both the mustard particles and headspace phases. The following assumptions were stated; 1. All hydrolyzed sinigrin is converted in AITC on a molar basis. 2. The mass transfer takes place from the mustard particles phase to the gaseous phase, the mustard particles are loosely packed and the surface area of each particle is in contact with the gaseous phase. 3. The mass transfer to the gaseous phase is described with the stagnant film model. The conditions justify that the mass transfer in the gaseous phase can be assumed not to be a limiting factor. 4. The mustard particles are considered to be one phase, no separate aqueous, solid, and fat sub-phases are considered in the model.
The AITC formation by sinigrin hydrolysis catalyzed by myrosinase can be described by a first-order reaction (Hanschen et al., 2018). Part of the sinigrin can be in inaccessible parts of the mustard seeds that were not damaged by the grinding process. Therefore, this inaccessible sinigrin needs to be included in the first-order reaction to describe well the mechanism of sinigrin hydrolysis. The initial hydrolysis of accessible sinigrin fraction leads to the following differential equation for fractional conversion first-order kinetics; where Sin tot is the concentration of total sinigrin in particles (mol/m 3 ), Sin i and Sin a are the concentration of inaccessible and accessible sinigrin (mol/m 3 ) respectively, t is degradation time (h) and k 1 is the rate constant of sinigrin degradation and AITC formation (h − 1 ). The formation (f) of AITC in the mustard particles is subsequently described by: where AITC s is the concentration of AITC volatiles (mol/m 3 ) within the mustard particles. The formed AITC can either be transferred to the headspace or degrade in the mustard particle. The AITC transfer to the headspace can be calculated from the penetration theory and the equilibrium constant (Harrison et al., 2006). The transfer to the headspace is determined by the mass transfer coefficient (m t ). From the theory, the mass transfer across the interface can be described by the following equation (Backhurst and Harker, 2001); where N is molar flux (mol/m 2 h), m t is the mass transfer coefficient (m/ h). AITC s i and AITC s are the concentration of AITC volatiles at the interface (mol/m 3 ) and within the mustard particles (mol/m 3 ) respectively. As N is an unknown variable, the molar flux in Eq. (5) is integrated into the mass transfer rate (m t ) of AITC in the solid and headspace described by the following equations: where t is time (h); AITC and V are AITC concentrations and volume of each phase (m 3 ), respectively, and the subscripts s and g denote seed particles and gas phases, respectively. The AITC can simultaneously be degraded in mustard particles and packaging headspace. Hydroxyl groups (e.g. from water) attack sulfhydryl groups of AITC to generate another compound, e.g. allyldithiocarbamate and pentasulfide (Tsao et al., 2000;Weerawatanakorn et al., 2015). As the amount of added water is in large excess and hardly changes over time during sinigrin hydrolysis, the mechanism of the AITC degradation in mustard particles can be described with the pseudo-first-order kinetics (Jiang et al., 2006). Yet the amount of water present in mustard particles is known, so the AITC degradation can be described by the following kinetic model equation: In the headspace also AITC degradation takes place due to the presence of water vapor as shown later in the results and discussion, was best described by a second-order reaction: where k 2 and k 3 in m 3 /mol.h are the rate constants of AITC degradation in the seeds and headspace, respectively, and H 2 O is the amount of water per volume of the seeds (mol/m 3 ).

Multiresponse kinetic modelling
The proposed pathway (Fig. 1a) consists of three responses to the experimental data; sinigrin concentration in the mustard particles, AITC concentration in mustard particles, and the headspace. Those three responses were estimated using a set of differential equations of mass transfer and reaction enumerated in Eq. (10)-(12).

d[Sin tot
These three differential equations were simultaneously fitted to the experimental data. The estimated parameter values consist of the inaccessible sinigrin fraction (Sin 1 ), rate constants (k 1 , k 2 , and k 3 ), and mass transfer coefficients (m t ). These parameters were estimated by multiresponse non-linear regression with the determinant criterion (Boekel, 2009) using Athena Visual Studio software (v.14.2) (AthenaVisual Inc.). In the parameter estimation solver control panel, standard options were used; the convergence criterion (= 0.01), the maximum number of iterations (=30), estimation solver options (non-linear least-squares, gradient calculation (forward differences scheme) and relative perturbation step size (10 − 3 ). In total 41 parameters were estimated based on 1836 triplicate measurements from 17 experiments with 36 responses giving 612 data points to fit by the model. The goodness of fit of the kinetic models was then evaluated by examining the generated size and distribution of the residuals, the R-square of predicted vs observed data, and the standard deviations and correlations of the parameters.
The degradation rate in the headspace (k 3 ) was assumed to be independent of the fat content and particle size. The degradation rate in the particles (k 2 ) was assumed only to be independent of the particle size. The dependency of the partition coefficient (K s/g ) on the fat content of the particles was taken from the experimental determination explained in section 3.3. The initial concentrations (shown in Table A1 and A2 in Supplementary data) were set equal to the observed initial concentrations of sinigrin and AITC in the non-hydrated mustard particles, while the initial concentration in the headspace was set equal to zero. Parameters and constants used in the modelling with known values are given in Table A3 in Supplementary data.

Statistical analysis
Significant differences were evaluated to investigate the effects of N.A. Bahmid et al. particle sizes and fat content on parameter estimates. Statistical significance (p < 0.05) was evaluated based on the method developed by Julious (2004) using the confidence intervals around individual means of the parameter estimates to assess statistical significance between two means.

Partition coefficient (K s/g )
The K s/g was thus calculated from the relationship as expressed in Eq.
(2) and summarized in Fig. 3. The phase ratio variation (PRV) method used to calculate K s/g was depicted in Figure A2 in Supplementary Materials. The volume ratio (β) was linearly proportional with the 1/A o in all different samples, which shows a good relationship between β and 1/ A o with R-square > 0.95.
The calculated value of K s/g significantly and linearly increased in the emulsion with higher oil content, as shown in Fig. 3 (p < 0.05). The emulsion containing over 17.1% fat content suddenly reduced the volatility of AITC in the headspace, in which the K s/g value of emulsion containing 29.1% fat was ten times higher than that of the emulsion containing 17.1% fat. It indicates that the higher fat content more effectively suppressed the volatility of the AITC in the headspace. The K s/g value of emulsion with 29.1% mustard oil (1992.7 ± 9.6) is in line with the value observed by Zhang et al. (2010) using 30% canola oil, which was 1962.9 ± 502.4. Furthermore, the linear increase of log 10 k s/g value indicates that AITC has a higher affinity for the bigger volume of fat phase in an emulsion (Keppler et al., 2018) and the low value of K s/g for the low-fat contents indicated the high volatility of AITC in an increase in water volume in the emulsion. These results are a useful parameter for the developed multireponse kinetic modelling.

Mechanistic multiresponse model
In Fig. 4, the multiresponse model in Eq. (10)-(12) describe observed data of sinigrin and AITC concentrations in the headspace well, including AITC concentration in the mustard particles ( Figure A3. Supplementary Materials). Linear regression of the predicted vs observed data plot provided slopes very close to 1.0 (Fig. 5).
There are some considerations on the assessment of the goodness of fit. Fig. 5a shows several horizontal clustering of data, which indicates very similar predicted data while there is considerable fluctuation in the observed data. These predicted data were the result of the fractional conversion first-order kinetics, in which the same predicted data indicated the minimum value of sinigrin fraction accessed by myrosinase, while there is still experimental variation around these values. In   Fig. 5b, the predicted data of AITC in the headspace has two overestimated points at high concentration. These data points are from the observed data of the defatted particles with sizes of 400-500 μm. The results of this particle size show an unexpected and peculiar result which caused a very high standard deviation in the parameter estimates (Table 2) for this sample, compared to the other particle sizes. As discussed in our previous study (Bahmid et al., 2020), the unexpected data shows fast hydrolysis, possibly due to the capillary forces being higher for the microstructure of this particular size, to completely degrade the sinigrin. In addition, the model did not satisfactory fit the peak value (underestimated) and the degradation rate (overestimated) of the measured data of headspace AITC concentration (Fig. 4f), whereas the model did perform well on the sinigrin concentration. These deviations were observed for the bigger mustard particles (400-500 and 630-800 μm) in the 29.1%-fat particles releasing low amounts of AITC. A possible reason is that at low AITC concentration in the gaseous phase and high amounts of fat in the mustard particles, the equilibrium between headspace and mustard particles is competing with the equilibrium between the aqueous and fat phase inside the mustard particles (Keppler et al., 2018), this can result in a homogenous distribution of AITC in the mustard particles resulting in the observed deviations. Overall, the model described the observed data very well. Therefore, the established multiresponse model proposed is well applicable and the reaction and mass transfer pathways (Fig. 1) in the packaging system can describe the antimicrobial active packaging system very well.

Interpretation of the estimated parameters from the established multiresponse kinetic model
The kinetic parameters and their standard deviation for this multiresponse kinetic model are summarized in Tables 1 and 2. A lowstandard deviation of the parameters (Tables 1 and 2) and low parameter correlation coefficients <0.8 ( Figure A4 in Supplementary Materials) were found. The effects of particle size and fat content on the estimated parameters obtained from the established kinetic model are discussed in detail below.

Sinigrin hydrolysis and AITC formation
The estimated kinetic parameters (the rate constants (k 1 ) and inaccessible sinigrin (Sin i )) of the modified first-order model Eq.
(3) are shown in Tables 1 and 2 Table 1 shows that k 1 decreased significantly with defatted particles, compared to full-fat (29.1% fat) particles as presented in Table 1 (p < 0.05) and Table 2 shows no significant effects of particle sizes on k 1 in general (p > 0,05), except for the 400-500 μmsized defatted particles. The lower k 1 in defatted particles was not our expectation, as in our previous study it was observed that there was more cell damage in the defatted particles. A possible reason for slower k 1 is a difference in sensitivity for cell damage for different cell types. Kissen et al. (2009) have shown that sinigrin and myrosinase are located in different compartments, where myrosinase is located inside myrosin cells and sinigrin in S-cells. The fat extraction and particle reduction might not damage the myrosin cells in a similar way as the S-cells, resulting in less myrosinase release in defatted particles.
The cell damages resulted in more sinigrin being accessible to myrosinase and more AITCs are formed (Hanschen et al., 2018;Wang et al., 2011). In Tables 1 and 2 for the smallest defatted particles (p < 0.05) indeed less remaining sinigrin (Sin i ) after 48 h is observed. The presence of the fat in ground seeds causes its intracellular cells to remain intact as the fat content protects the cells from the grinding process, consequently, the higher sinigrin concentration was left in the ground seeds containing fat content. Additionally, a sudden increase in the inaccessible sinigrin for the big particles (400-500 and 630-800 μm) in Table 2 is likely induced by incomplete cell damage. These results confirm the previous study that observed micrographs of the big particles containing less cell damage.

AITC release from mustard particles into packaging headspace
The mass transfer kinetic model Eqs. (6) and (7) described the AITC release from mustard particles into headspace well. Tables 1 and 2 show that the mass transfer coefficient (m t ) was significantly affected by the fat content (p < 0.05), while no effect of particle sizes was observed (p > 0.05). Table 1 shows the m t -values of the fat mustard particles (17.1 and 29.1% fat) were significantly lower than the m t -values of the particles with lower fat content. This result was in line with the result found by Dai and Lim (2014); the release rate constants value of defatted mustard seed meal powder (MSMP) were faster than the normal MSMP. The fat in mustard particles reduced the release of AITC from the fat-particles, as the AITC was solubilized in the lipid phase (Tsao et al., 2000). As particle size did not significantly affect the AITC release, it indicates that the higher AITC concentrations observed in Fig. 4d and f were driven by the higher accessible sinigrin fraction available in the mustard seeds leading to more formation and then release of AITC into the headspace. This indicates that cell damage in the particle is crucial to make sinigrin accessible and get a higher AITC concentration in the headspace. These results may have important consequences for the formulation of mustard particle properties with controlled-release profiles of AITC and its application in the packaging.

AITC degradation in mustard particles and headspace
The reaction kinetics in Eqs. (8) and (9) were applied to describe AITC degradation in mustard particles and headspace, respectively. The degradation rate constant in the particles (k 2 ) was significantly increased in particles without fat (Tables 1 and 2) (p < 0.05). The high solubility of AITC in the fat phase caused the lower degradation rate of AITC in the mustard particles phase with fat. In the continuous phase (aqueous phase of particles) AITC degradation is faster, while the compounds in the dispersed phase (lipid phase) were more stable (Liu and Yang, 2010). This degradation rate of AITC in mustard particles shows that the presence of fat is the best way to stabilize the AITC concentration for a longer time.
The second-order degradation rate constant in the headspace (k 3 ) was estimated at around 3.02 ± 0.30 × 10 5 m 3 /mol.h. As no reaction   Bahmid et al. kinetics for this reaction was described in the literature, zero-order, firstorder, fractional conversion first-order, and second-order kinetics were tried to fit the experimental data of the headspace AITC concentration. When fitting the model to the experimental data, only the second-order model in Eq. (9) can adequately describe the AITC degradation in the headspace without a trend in the residuals and an estimated parameter with a low standard deviation.
The effectiveness of AITC as an antimicrobial agent relies on its concentration in the packaging system (Dufour et al., 2015;Saladino et al., 2017). Once the concentration of AITC in the packaging system is lower than its minimum inhibitory concentration (MIC) against certain bacteria, the bacteria will grow and spoil the foods. AITC degradation is accounted for its stability to maintain the capability of AITC to keep the inhibition of bacteria. The breakdown products of AITC, e.g. allyl-dithiocarbamate and pentasulfide have no antimicrobial activity (Tsao et al., 2000;Weerawatanakorn et al., 2015), so besides the controlled release of AITC in the packaging system, the rate of degradation in the packaging system needs to be hindered to maintain the concentration above MIC. Further study related to other factors reducing the AITC degradation in the headspace, like temperature (Saladino et al., 2017), is required to retain AITC in the headspace to have a longer antimicrobial inhibition in foods.

Conclusion
The multiresponse kinetic model combining the mechanisms of (bio) chemical reactions and mass transfer of AITC inside the packaging system described the experimental observations well. This model produced a good prediction of the enzymatic hydrolysis of sinigrin, AITC formation, its release, and degradation in the mustard particle and headspace. Defatting the mustard particles has positive impacts on some parameters, e.g. increasing the accessible sinigrin and lowering the formation rate constant and increasing the mass transfer coefficient between the particles and the headspace. The particle size has a small effect on the estimated parameters, and no significant effect on the release of AITC into the headspace was found. The results also confirm our previous study that fat content has a significant effect on the partitioning of AITC in the packaging system. The better understanding of the kinetics of the concurrent reactions and mass transfer in the packaging system by this study can help to optimize the mustard seeds properties to effectively control the AITC release for antimicrobial packaging design. A similar multiresponse modelling approach as described in this paper can be applied for future applications, with other components used in the active packaging concept.

Declaration of competing interests
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Values represent means ± standard deviation. Small letters indicate significant differences between particles with different fat content.

Table 2
The estimated parameters from the multiresponse kinetic model for the effect of particle size of the mustard seed particles with 0 and 29.1% fat. Values represent means ± standard deviation. Small letters indicate significant differences between the different sizes of particles with 0% fat, while capital letters indicate significant differences between the different sizes of particles with 29.1% fat.