Modelling and

analysis of complex

food systems: State

of the art and new

trends

N. Perrota,*, Le. Treleaa,l ,

e. Baudrita,l , G. Trystramb,2

and P. BourgineC,3

aINRA, UMR782 GMPA, AgroParisTech, INRA, 78850 Thiverval-Grignon, France (Te!.: +331 3081 5379;

fax: + 331 30 81 55 97; e-mails: cedric.baudrit@ grignon.inra.fr; cristian.trelea@agroparistech.fr;

nathalie.perrot@grignon.inra.fr)

bAgroParisTech, UMR GENIAL, AgroParisTech, INRA, 1 avenue des Olympiades, 91744 MASSY Cedex,

France (e-mail: gilles.trystram@agroparistech.fr)

'ISC PIF (Institut des Systemes complexes Paris lie de France), UMR 7656, CREA, Centre de Recherche en

Epistemologie Appliquee, Ecole Polytechnique/CNRS, 32, boulevard Victor, 75015 Paris, France (e-mail:

paul.bourgine@polytechnique.edu)

The aim of this review is tvvofold. Firstly, we present the state of the art in dynamic modelling and model-based design, op­timisation and control of food systems. The need for nonlinear, dynamic, multi-physics and multi-scale representations of food systems is established. Current difficulties in building such models are reviewed: incomplete, piecewise available knowl­edge, spread out among different disciplines (physics, chemis­try, biology and consumer sciencel and contributors (scientists, experts, process operators, process managersl, scarcity,

* Corresponding author. 1 Tel.: +331 30 81 53 79; fax: +331 3081 5597. 2 Tel.: +331 69 93 50 69; fax: +331 6993 51 85. 3 Tel.: +33145 5264 11; fax: +331 45 52 64 55.

uncertainty and high cost of measured data, complexity of phenomena and intricacy of time and space scales. Secondly, we concentrate on the opportunities offered by the complex systems science to cope with the difficulties faced by food sci­ence and engineering. Newly developed techniques such as model-based viability analysis, optimisation, dynamic Bayes­ian netvvorks etc. are shown to be relevant and promising for design and optimisation of foods and food processes based on consumer needs and expectations.

Introduction Food engineering covers a large spectrum of applica­

tions that include, but are not limited to: product engineer­ing, process engineering, control, optimisation and decision support systems. Some 25 years ago, modelling and simu­lation of food processing was mostly dedicated to product preservation with safety considerations, most of the studies focused on time-temperature diagrams for predicting and limiting residual spores or microorganisms in foods. Due to increased process understanding and computing power, applications emerged where other quality attributes were considered: moisture content, colour, viscosity, sometimes food composition. More recently, food structure was also considered (e.g. viscosity, porosity) and models became available to represent the evolution of such structure (Theys, Geeraerd, & Van Impe, 2009). In parallel, progress in observation and analytical methods (imaging techniques, magnetic and electronic beams) allowed investigating dif­ferent structural scales and interactions between chemical species, mainly between macromolecules and small mole­cules. Food starts to be viewed as a complex system, with various possible interactions between key variables at dif­ferent scales (from nano scale to macroscopic one) (Baudrit, Sicard, Wuillemin, & Perrot, 2010).

It is now recognised in most scientific domains that dynamic modelling and computer sirnulations are valuable tools for product and process understanding, design, optimisation and control. Dynamic modelling vs. static modelling is modelling of processes over time. In dynamic modelling new attributes are computed as a function of attribute changes over time.

The purpose of a dynamic mathematical model is to capture relevant features (in a given context) of a complex object or process, based on existing theoretical lll1derstand­ing of the phenomena and available measurements and

there evolution over time. Current industrial applicationsusually rely on extremely simplified, stationary modelsthat cannot produce a realistic evaluation of transient ef-fects on plant performance, quality and safety conditionsand environmental impact. The modelling and simulationresearch efforts should be directed towards main phenome-nological aspects, such as heat, mass, momentum, popula-tion balance coupled with chemical reactions andcoupling different scales, such as nanostructure, macro-structure and texture of products.

Design of new foods as ‘intelligent’ vectors for targetmolecules responsible for nutritional or sensory propertiesbecame a major goal for food industry. These target mole-cules can be sapid (for example NaCl concentration) oraroma compounds, micro-nutrients or microorganisms ofinterest (technological flora used in the fermented products)whose controlled release or digestion satisfies physiologicalobjectives of bioavailability. E. Windhab suggested in 2004an integrating concept (Preference, Acceptance Need)taken over by the platform ‘Food for life’, expressing theneed to establish a compromise between all these proper-ties. Up to now, few studies were able to work in sucha complex design space. Existing reverse engineering pub-lications focus either on safety or sensory questions. Sus-tainability and environmental impact are additionalfactors to be taken into account.

The emerging field of complex systems science, situ-ated at the crossroads of mathematics and artificial intel-ligence (cf. the living roadmap for complex systemhttp://cssociety.org/tiki-download_wiki_attachment.php?attId¼123), develops methods and tools to comprehendand describe instable and changing environments, sys-tems that evolve and adapt through internal and externaldynamic interactions and are not predictable withina conventional scientific framework. Our hypothesis isthat techniques developed in complex systems scienceare applicable and useful to tackle difficulties encoun-tered in food systems.

Understanding and modelling of complex foodsystems: state of the art

Model-based approaches in food science, technologyand engineering have received great attention during thepast three decades (Banga, Balsa-Canto, & Alonso, 2008;Datta, 2008; Sablani, Datta, Rahman, & Mujumdar, 2007)and numerous academic works have been dedicated tomodelling and its applications (Bimbenet, Schubert, &Trystram, 2007). The demand for models is now clearly es-tablished; as an example, the European Food for Life plat-form (www.ciaa.be) presents modelling as a key tool for thedevelopment of European food Industries. Compared tochemical engineering, where modelling is now part of vir-tually any scientific and technical development, food engi-neering follows a similar trend, with considerable (w20years) delay. In the authors’ view, one of the main reasonsfor this delay is the increased complexity of food systems,

including physical, chemical and biological phenomena ona wide range of time and space scales (Banga et al., 2008;Christakos, 2002; Georgakis, 1995; Perrot, Bonazzi, &Trystram, 1998).

Dynamic models for food systemsThis review makes a particular emphasis on dynamic

models, able to describe transient process operation. Typi-cal examples are batch processes, which always operatein transient state. For continuous processes, optimisingstart-up, shutdown or recipe change regimes can be impor-tant for reducing costs and environmental impact. On-linecontrol of continuous processes also require dynamicmodels for unavoidable disturbance compensation, suchas variations in raw materials (Trystram & Courtois, 1996).

First principles vs. data-driven modelsWhen modelling approach is primarily guided by the

knowledge of the underlying mechanisms, the resultingmodel is usually termed as ‘first principles’ or ‘whitebox’. Classical examples include heat, mass and momen-tum transfer, chemical and biochemical conversions, etc.The scales covered by first principles range from atomicto macroscopic ones. A lot of innovative work today is ded-icated to micro and meso scale. As an example, SAFES(Fito, LeMaguer, Betoret, & Fito, 2007) illustrates the useof thermodynamics to understand the evolution of food dur-ing processing. Multiphase approaches viewed as a generalbackground by Datta (2008) cover similar scales. Availablemolecular tools become increasingly relevant for food ma-trices but the connection with macroscopic scales remainsdifficult.

In contrast with first principles, empirical ‘data-driven’or ‘black-box’ models describe observed tendencies in ex-perimental data by arbitrary mathematical functions such aspolynomials or artificial neural networks (ANN). Quick andeasy-to-use when sufficient experimental data is available,such models also encounter important limitations when ap-plied to food systems: risk of over-parameterisation, inter-pretation difficulty, lack of generalisation ability whenfood composition or process parameters are changed out-side the range of the initial experimental design (Bangaet al., 2008; Karim et al., 1997). Last but not least, the num-ber of required measurements increase exponentially withthe number of studied factors.

A quite efficient intermediate approach consists in de-signing a model structure based on first principles and com-plete missing information by empirical relationshipsderived directly from experimental data. Such models aresometimes called ‘grey box’. A dynamic research field isthe development of artificial intelligence-based approaches(Allais, Perrot, Curt, & Trystram, 2007; Perrot et al., 2004;Davidson, 1994; Linko, 1998; Ndiaye, Della Valle, &Roussel, 2009) taking into account the human expertknowledge. Many applications especially for food qualitycontrol (for a review see Perrot, Ioannou, Allais, Curt &

305N. Perrot et al. / Trends in Food Science & Technology 22 (2011) 304e314

Hossenlopp, 2006) were reported, mostly based on the the-ory of fuzzy sets. Nevertheless, the bottleneck of these ap-proaches is the difficulty to capture the dynamic of thesystem using the expert knowledge. This difficulty wasalso pointed out by the community of cognitive science(Farrington-Darby & Wilson, 2006; Hoffman, Shadbolt,Burton, & Klein, 1995).

Building of the food modelsA typical approach for model development is sche-

matically shown in Fig. 1. On the basis of literature re-view, previous scientific or expert background andexperimental evidence, a first set of hypothesis, mecha-nisms, state variables and parameters is defined. Gener-ally, one space and/or time scale is explicitly taken intoaccount. Other scales are usually lumped into some ap-parent or average material properties. Uncertainty israrely considered. When it is, it can be taken into ac-count explicitly, e.g. via fuzzy numbers (Ioannou,Mauris, Trystram, & Perrot, 2006) or implicitly by con-sidering statistical distributions of model parameters. Se-lected model structure primarily depends on the planneduse of the model: hypothesis testing, simulation, state es-timation and software sensors, control design optimisa-tion, etc.

Model parameters are determined from classical experi-mental designs or from specifically designed optimal ones(Banga et al., 2008). Once the model is build and its param-eters determined, a range of tools is available for indenti-fiablity, sensitivity and uncertainty analysis, both structuraland parametric (Walter & Pronzato, 1997, 413pp.). The

outcome of these procedures may be the reconsideration ofmodel hypothesis and structure, and/or the design of addi-tional experiments to allow reliable parameter identification.

Limitations of current modelling approaches for foodsystems

Model-based approaches in food engineering are usuallysubject to one or more of the limitations synthesised in thefirst column of Table 1 (Baudrit, H�elias, & Perrot, 2009;Bimbenet et al., 2007; Fito et al., 2007; Ioannou et al.,2006; Perrot et al., 2006; Van Impe, 1996). Moreover, sev-eral of these difficulties often arise simultaneously in foodtechnology and biotechnology (Van Impe, 1996).

In many domains, existing knowledge of food scientistshas led to specific models, valid in a tiny domain, either ofcomposition or of physicochemical environment. More-over, their conceptual framework does not allow easy inte-gration of results coming from other existing models(Rodriguez-Fernandez, Balsa-Canto, Egea, & Banga,2007). For instance, very detailed heat and mass transfer(H&M) studies, possibly including spatial distribution,and described by (partial) differential equations, are oftencoupled with oversimplified chemical and biological kinet-ics. On the other hand, detailed modelling of reaction net-works, possibly involving a molecular level, often ignoreheat and mass transfer aspects. Coupling process andmolecular scales is expected to significantly increase thecomplexity of the modelling task; coupling those is some-times possible but not easy or general. Moreover nonhomogeneous scales can increase the complexity of themodelling task.

1

Physical, biochemical,

… laws

Human expertise

Classical experimental trials

Analysis and development of the model:

Data basis

Food Process

Past knowledge (data, expertise)

Hypothesis Structure and identifiability

Known and unknown parameters

Uncertainty on the measurements and

knowledge

Experiments

Optimal experimental design

Food model

mecanistic

And tools to study it (sensibility, viability, uncertainty)

Uses from state estimation

Sensor

Control

Decision help

Optimization

Systemic analysis

From « pure » empiric to « pure »

Fig. 1. A typical approach for model development in food engineering.

306 N. Perrot et al. / Trends in Food Science & Technology 22 (2011) 304e314

Furthermore, experimental data in food science andtechnology is often limited in amount and quality. On-line sensors are currently available for technologicalmeasurements only, such as temperature, pressure, veloc-ity, etc. Measurements related directly to food quality(microbial count, desired or undesired compound concen-tration, texture.) are still performed by off-line labora-tory analysis and are slow, costly, and labour-intensive.In large projects, a rule of thumb is that one laboratoryanalysis is ultimately obtained per full-time equivalentof the personnel involved in the project and per day.Compared to measurements performed in other fields(mechanics, electronics and even chemistry), laboratoryanalysis in food science are subject to significant uncer-tainty. Differences of �0.5 logarithmic units on replicatemicrobial counts, for example, are considered normal,while this represents a factor of 3. In sensory analysis,30 or 50% variations between replicates are usual. Test-ing mechanism hypothesis and building reliable modelsbased on scarce and uncertain data is obviously a difficulttask.

To cope with the bottlenecks bring by the study of foodcomplex systems, some ways of research appear to bepromising (second column in Table 1).

Co-operation between disciplinesMany scientific fields share the challenge of unifying

complex and dissimilar data (Desiere, German, Watzke,Pfeifer, & Saguy, 2001) and deal with multiple physicsmodels. As shown by Datta (2008), food structure develop-ment is not just a function of current parameters like tem-perature and moisture, but of their entire history, when thecomplex physical structure develops, changes porosity andtransport properties.

One of the research streams is related to the develop-ment of reliable models integrating different sources andformat of knowledge is so-called knowledge integration.The principle is to deal with the different pieces of the puz-zle of knowledge represented under different formalisms:data, models, expertise. One of the problems that must beaddressed (Stuurstraat & Tolman, 1999) is how to cope

with the conflicting requirements of each particular subsys-tem, optimised for its own knowledge domain (Varela,1979). No easy solutions are available by now. The keypoint is the ability to cope with knowledge of different na-ture, at different scales, expressed in different formalisms(conservation laws and human rules of expertise for exam-ple) and to be able to take them into account in a unifiedmanner. Nevertheless, this issue is a key for the future, en-abling us to exploit the different sources of knowledge thatwe are developing in our laboratories today. Interactionsbetween various fields of science was pointed out in con-nection with environmental and natural resource issues(Christakos, 2002) biological issues (Olivier et al., 2010),nutrition (McLachlan & Garrett, 2008), molecules andgene interaction (Kasabov, 2005) etc.

UncertaintyAnother key issue in food processes is the management of

the uncertainty. Explicit integration of uncertainty has be-come crucial in industrial applications and consequently indecision making processes (Baudrit, Dubois, & Guyonnet,2006). In food processes, few contributions are available in-cluding uncertainty on model parameters or on model struc-ture itself (Baudrit et al., 2009; Perrot et al., 2006;Petermeier et al., 2002). However, taking into account thecomplexity of microbiological and/or physicochemical trans-formations in food processes, available knowledge is oftentainted with vagueness, imprecision and incompleteness.Furthermore, for use in industrial applications, models andespecially mechanistic models should be studied upon theirsensitivity to this uncertainty (Banga et al., 2008;Bimbenet et al., 2007).

Computing powerComputationally demanding tasks are increasingly used

in food processes. These include for example simulation ofspatially distributed models, stochastic migration of mole-cules to determine diffusion and partition properties incomplex media (Vitrac & Hayert, 2007), mathematical vi-ability calculations (Sicard, Perrot, Baudrit, Reuillon &Bourgine 2009), dynamic optimisation (Banga, Balsa-

Table 1. Difficulties for the development and analysis of the models in food engineering (column 1) and possible solutions (column 2).

Difficulties Possible solutions

Diversity of the mechanisms (physicochemical reactions, microbialreactions)

Multidisciplinary research teamKnowledge integration through appropriate formalisms

Different and non homogeneous scales for variables and differenttype of knowledge

Unifying mathematical formalisms

Nonlinear connections between the variablesTime scale coupled with space scale

Adapted formalismsIncreased computing power

Uncertainty on the measurements and inconsistency in data Formalisms able to cope with epistemic and stochastic uncertaintiesEmpiricism and fragmented knowledgeCost and duration of experiments

Co-operation between scientists and experts from differentdisciplinesModular modelling approach, able to integrate building blocks ofdifferent nature

307N. Perrot et al. / Trends in Food Science & Technology 22 (2011) 304e314

Canto, Moles, & Alonso, 2003), global sensitivity analysisetc. These tasks require new calculation methods on com-puter grids to be tested and implemented (Reuillon,Chuffart, Leclaire, Faure, & Hill, 2010).

A representative example: modelling of a cheesemaking process

To illustrate previous considerations, consider the caseof the modelling and simulation of a cheese making pro-cess. The quality of soft mould cheese depends on environ-mental factors during ripening (relative humidity,temperature, gas composition) and on interactions betweeninoculated microorganisms and curd substrates. The con-centrations of these substrates are subject to variations inmilk quality and cheese making conditions (Helias,Mirade, & Corrieu, 2007). Over the last 10 years, morethan 112 studies (FSTA and ISI web of sciences sources)have been carried out to understand this process in a micro-bial, physicochemical, biochemical and sensory points ofview. About 52% of those models were empirical. For ex-ample Bonaiti, Leclerc-Perlat, Latrille and Corrieu (2004)developed an RSM approach to predict the pH and substrateevolution versus time for a soft cheese. Sihufe et al. (2010)used the principal component analysis to predict the opti-mal ripening time, while Jimenez-Marquez, Thibault, andLacroix (2005) have proposed a neural network to predictthe ripening state of a cheese.

Nearly 46% of the studies fell into the first principlescategory. 44% were mechanistic approaches based onmass transfer laws, e.g. for syneresis prediction (Heliaset al., 2007; Tijskens & De Baerdemaeker, 2004), some-times combined with microbial growth laws (Guillier,Stahl, Hezard, Notz, & Briandet, 2008; Riahi et al.,2007). In the remaining 2% of the publications, expert sys-tems were developed.

Most of the analysed publications were focused on onespecific phenomenon, were limited to the experimentallyexplored domain without any generalisation ability andwithout taking into account the inherent uncertainty. Forexample the mass loss model presented in (Helias et al.,2007) is developed under the hypothesis of average waterand convective heat transfer coefficients fixed for air veloc-ity upper than 0.2 m s�1 while for some ripening chamberin the industry this velocity is lower than 0.2 m s�1. Wateractivity is also supposed to be constant while it is true insome specific configurations of the process. Integratingother type of information, such as expert knowledge ordealing explicitly with the uncertainty of the process couldhave enhanced the results. Each of those studies, constitutea part of the puzzle of knowledge that were built to under-stand the cheese making process but are not sufficient,taken alone, (1) to understand it in its global behaviour in-cluding all the scales and (2) to use it in decision makingsystems.

Some recent studies have nevertheless proposedapproaches for modelling the links between different scales

and different type of knowledge, including uncertainty(Arguelles, Castello, Sanz, & Fito, 2007; Baudrit et al.,2010; Thomopoulos, Charnomordic, Cuq, & Abecassis,2009). Quite a few such integrating approaches are avail-able up to now. Knowledge is still missing to model com-plex processes such as cheese making. Considerableexperimental effort, large databases and progress in micro-bial physiology are needed to understand numerous vari-ables relevant for cheese making and their interactions.

New opportunities: complex system science for foodengineering

It follows from previous considerations that remarkableopportunities are now open for theories and techniquesdeveloped in the field of complex systems science, to be ap-plied and adapted to food science and technology. The restof this review will concentrate on knowledge integration,management of the uncertainty and model analysis for re-verse engineering purposes.

Knowledge integrationKnowledge integration has been reported in several ap-

plication fields, including food science. Quintas,Guimaraes, Baylina, Brandao, and Silva (2007) studiedcomplex caramelisation reactions. Alternative reactionpathways have been suggested, each described by a differ-ent set of differential equations. Automatic model selectionwas performed based on parameter identification results.Allais et al. (2007) illustrate how mechanical laws can becoupled with an expert knowledge database to better com-prehend a sponge finger batter process. Hadiyanto et al.(2007) applied similar ideas to quality prediction of bakeryproducts.

A Systematic Approach for Food Engineering Systems(SAFES) based on the theoretical framework of irreversiblethermodynamics has been proposed by Fito et al. (2007).The principle is to define a simplified and unifying spaceof structural features, called ‘structured phases and compo-nents’. These features are grouped in a composition matrixand are time dependant. The approach has been applied todifferent processes, e.g. prediction of the change in proteinconformation during ripening (Arguelles et al., 2007). Acentral hypothesis is the identifiability of the resultingmodel. This hypothesis is not always satisfied, however,when establishing relationships between food compositionand structure, in realistic foods.

The contribution presented by Thomopoulos et al. (2009)concentrates on durum wheat chain analysis. The developedinformation system allows the integration of experimentaldata, expert knowledge representation and compilation aswell as reasoning mechanisms, including the decision treelearning method. The principle is to encode the existingknowledge about a given food chain in a unified language.The uncertainty pertaining to the expert knowledge is takeninto account in the form of fuzzy sets. The information

308 N. Perrot et al. / Trends in Food Science & Technology 22 (2011) 304e314

system can be used in assisting decision makers but cannothandle numeric approaches, like model-based optimalcontrol.

As a last example, Baudrit et al. (2010) have shown thatby introducing expert knowledge, a good prediction on themicrobial and physicochemical kinetics during the ripeningof a camembert type cheese was possible, based on limitedexperimental data set. The theoretical framework used hereis that of Dynamic Bayesian Networks (DBNs) proposed byMurphy (2002). DBNs are classical Bayesian networks(Pearl, 1988) in which nodes representing random variablesare indexed by time (Eq. (1)). In the considered example,the average adequacy rate in predicting microscopic andmacroscopic scales was of 85%, on a test data basis of 80measurements.

PðXð1Þ;.XðtÞÞ ¼Yt

t¼1

YN

i¼1

PðXiðtÞjPaðXiðtÞÞÞ ð1Þ

where XðtÞ ¼ fX1ðtÞ;.XNðtÞg and PaðXiðtÞÞ denotes theparents of XiðtÞ in the graphical structure of the DBN.This probability represents the beliefs about possible trajec-tories of the dynamic process X(t). Fig. 2 illustrates a DBNrepresenting a network applied on the example of cheeseripening.

Management of the uncertaintyUncertainty, as explained in detail by Datta (2008), is

usually of significant concern in food processing, perhapsmore than in other domains. Uncertainties are often cap-tured within a probabilistic framework. It is particularlytrue in food engineering for risk assessment (Aziza,Mettler, Daudin, & Sanaa, 2006). Generally, uncertainty

pertaining to the parameters of mathematical models repre-senting physical or biological processes can be describedby a single probability distribution. However, this methodrequires substantial knowledge to determine the probabilitylaw associated with each parameter. It is more and more ac-knowledged that uncertainty concerning model parametershas two origins (Ferson & Ginzburg, 1996).

It may arise from randomness (often referred to as ‘sto-chastic uncertainty’) due to natural variability of observa-tions resulting from heterogeneity or the fluctuations ofa quantity over time.

Alternatively, uncertainty may be caused by imprecision(often referred to as ‘epistemic uncertainty’) due to a lack ofinformation. This lack of knowledge may arise from a partiallack of data or because experts provide imprecise information.For example, it is quite common for experts to estimate the nu-merical values of parameters in the form of confidence inter-vals according to their experience and intuition.

The uncertainty affecting model parameters is thus dueboth to randomness and incomplete knowledge. This is typ-ically the case in presence of several, heterogeneous sourcesof knowledge, such as statistical data and expert opinions.The most commonly used theory for distinguishing incom-pleteness from randomness is the imprecise probabilities cal-culus developed at length by Walley (1991, 720pp.). In thistheory, sets of probability distributions capture the notionof partial lack of probabilistic information. While informa-tion regarding variability is best conveyed using probabilitydistributions, information regarding imprecision is more ac-curately represented by families of probability distributions.Examples of tools to encode probability families includeprobability boxes (Ferson&Ginzburg, 1996), possibility dis-tributions (also called fuzzy intervals) (Dubois, Nguyen, &Prade, 2000) or belief functions introduced by Dempster(1967) and elaborated further by Shafer (1976) and Smetsand Kennes (1994) make it possible to encode such families.

As an illustration, consider mass loss model during a rip-ening process, developed by Baudrit et al. (2006). The ideaof this contribution is to take into account the imprecise na-ture of available information about the heat and water trans-fer coefficients and to jointly propagate variability andimprecision to the estimation of cheese mass loss throughthe ripening process. In order to do this, the most faithfullyavailable knowledge and the associated form of uncertaintywere implemented (Table 2). For the measurements, spatialvariations of humidity and temperature due to climate con-trol were taken into account. Due to low airflow velocity in-side ripening chambers, imprecision about the heat andmass transfer coefficients reported in the literature was in-corporated and represented by means of a possibility distri-bution. The joint propagation of these uncertainties,coupling random sampling with interval calculus, has ledthe authors to provide key information for improving thecontrol of the mass loss of cheeses under industrial condi-tions. A further step forward would be the integration of theuncertainty as part of the model equations.

Fig. 2. An example of DBN applied to cheese ripening presented inBaudrit et al. (2010), with la, Gc and Ba microorganisms concentra-tions, la and lo substrate and product concentration, T temperatureof the ripening cell, colour, coat, humidity, odour and under-rind mac-

roscopic sensory evolutions.

309N. Perrot et al. / Trends in Food Science & Technology 22 (2011) 304e314

Analysis of the models for reverse engineeringpurposes applied to complex food systemsModel-based optimisation for identification andcontrol

Model-based optimisation is usually implemented for threemajor areas in food technology (Banga et al., 2008): optimalidentification of model parameters, building reduced-orderedmodels for faster simulation and selection of optimal operat-ing policies (model predictive control). A worked-out exam-ple in the first category is given by Balsa-Canto, Rodriguez-Fernandez, and Banga (2007), where the identification of ki-netic parameters for thermal degradation of microorganismsis considered. Authors show how well-designed time-varyingexperiments can achieve an accurate and robust identificationof model parameters, with a reduced experimental effort. Inmodelling of fermentation kinetics, optimal experimental de-signwas applied by Bernaerts, Versyck, and Van Impe (2000),Smets, Versyck, and Van Impe (2002), with similarconclusions.

A comprehensive review of optimal control for food pro-cesses was provided by Garc�ıa, Balsa-Canto, Alonso, andBanga (2006), Lutton, Pilz and Levy (2005), Kennedyet al. (2001) and Trelea, 2003. Global optimisation methodslike evolutionary algorithms, scatter search and particleswarm optimisation ensure robust convergence towards op-timal control profile despite the presence of constraints andlocal optima. An interesting contribution can be found ap-plied to the alcoholic fermentation of a beer productionprocess (Trelea, Titica, & Corrieu, 2004). The results dem-onstrate the possibility of obtaining various desired finalaroma profiles and reducing the total process time using dy-namic optimisation of three control variables: temperature,top pressure and initial yeast concentration in the tank. Ap-plied to the alcoholic fermentation, it has led to the reduc-tion of the production cost (reducing the process residencetime from 121 h to 95 h) for an existing sort of beer withoutaltering its aroma profile (Fig. 3). Compared to classical se-quential quadratic programming optimisation (SQP), PSOoptimisation, as well as other stochastic search algorithms,require much less conditions on the dynamic model, objec-tive function and constraints (continuity, derivability) andcan thus be applied to almost any existing process modelwithout further reformulation.

Viability theory for decision help or control purposesGiven the dynamics of a complex process, a ‘viable’

control is sequences of actions driving the process along

admissible evolutions. Admissible evolutions are suchthat the industrial production constraints are satisfied andthe consumer expectations, expressed as targets, arereached. The main purpose of the viability theory is to ex-plain the evolution of a system (model exploration), deter-mined by given non deterministic dynamics and viabilityconstraints, to reveal the concealed feedbacks which allowthe system to be regulated and provide selection mecha-nisms for implementing them. Cost function can also be as-sociated to trajectories in the state space. The aim is toreach a target with an optimal trajectory (minimal cost).If we denote SF(x), the set of evolutions governed by thecontrolled dynamical system x0(t) ¼ f(x(t), u(t)), the viabil-ity kernel is defined by (Eq. (2)):

ViabFðKÞ :¼ fx˛Kjdxð:Þ˛SFðxÞ; ct > 0; xðtÞ˛Kg ð2ÞThis is a variant of the viability problem called capture

basin. Numerical schemes to solve ‘viability’ or ‘capture’problems were first proposed by Saint Pierre (1994) andAubin (1991).

As in model-based optimisations methods, an optimalcontrol can be calculated on the basis of the dynamicmodel. The advantage of the viability approach com-pared to the previous one is that the exact calculus ofthe frontier of the admissible evolutions is included in

Table 2. Type of uncertainties propagated in a mechanistic model of cheese mass loss during a ripening process.

Sources of information Character of knowledge Mode of representation

Input variables Respiration rates ro2, rCo2 Measurements Precise Fixed valuesClimate control Rh(t), TN(t) Measurements Spatial variability Probability distribution

Model parameters Transfer coefficients h,k Expert opinion þ literature Imprecise Fuzzy setsLiterature physical constantss, l, a, wco2, wo2, 3, s, C, aw

Literature Precise Fixed value

Ethyl acetate [30 mg/L]

Ethyl hexanoate [0.25 mg/L]

Isoamyl acetate [4.5 mg/L]

Isoamyl alcohol[110 mg/L]

Phenyl ethanol[50 mg/L]

010

13

16

Temperature [°C]

0 100

1013

1413

1813

Pressure [mbar]

Time [h] Time [h]

tf = 95 h X0 = 20 • 106 mL-1

100

Fig. 3. Fermentation time reduction of an existing beer without chang-ing the final aroma profile. Top: aroma concentrations at the end of thealcoholic fermentation. Bottom: operating conditions for the alcoholic

fermentation process.

310 N. Perrot et al. / Trends in Food Science & Technology 22 (2011) 304e314

the viability scheme (Martin, 2004). It is also possible,by evaluation of the distance of each evolution to thecalculated frontier at each time step, to quantify the ro-bustness of each control trajectory in the state space(Alvarez, Martin, & Mesmoudi, 2010). Indeed, neareris the evolution to the frontier of the tube, less robustis the selected viable trajectory. Nevertheless viabilitysuffers from the curse of dimensionality, with a needfor an exhaustive search in the state space, in contrastto stochastic calculus. Such a bottleneck is in pass tobe solved with research led in computer science and in-creased availability of powerful computer systems(Reuillon et al., 2010).

A pioneering application of viability theory to food pro-cesses was the optimisation of Camembert cheese massloss during ripening, while preserving an equilibrategrowth of ripening microorganisms (expressed using the

expert knowledge). The control variables taken into ac-count in the algorithms were the relative humidity andthe temperature of the ambient air of the ripening chamber(Sicard et al., 2009). In this study, the computation wasachieved by the distribution of the algorithm on a clustercomposed of 200 CPU (Central Processing Units). An ex-ample of viability kernel calculated for 12 days of ripeningis presented Fig. 4. The distance of the determined viabletrajectory to the boundary (frontier) of the viability tube isshown. An optimal ripening control trajectory calculatedusing the viability algorithm was implemented and vali-dated experimentally. The gain in ripening time with a tra-jectory selected in the viability kernel for a given qualityof the cheese, was of 5 days, to be compared with the res-idence time in the ripening chamber of around 12 days fora standard control policy (92% relative humidity and12 �C).

Fig. 4. An example of viability tube for 12 days of a cheese ripening process. Distance square map for each point is presented in colour: from whitenear the boundary of the viable tube, to black at the heart of the tube. 3 dimensions are taken into account for the calculus of the viable state: mass,

respiration rate of the microorganisms and temperature of the surface of the cheese.

311N. Perrot et al. / Trends in Food Science & Technology 22 (2011) 304e314

Finally, both optimal control and viability theory are rel-evant approaches for reverse engineering purposes and canintegrate global requirements encountered in food industry(nutritional, organoleptic, economical, technical, environ-mental, etc.). Nevertheless, their main limitation is theavailability of dynamic models sufficiently representativeof the complex phenomena involved in food processes.

ConclusionThe paper reviews current trends in modelling, design

and control of foods and manufacturing processes, bypointing out modern promising approaches to tackle com-plexity, uncertainty, lack of complete first principles under-standing and of reliable data and its high cost. Considerableopportunities are now open to capture and manage the com-plex dynamics of food systems, coupling different scalesand reduce the associated uncertainty. Tight collaborationwith various disciplines is needed to unify complex and dis-similar data and knowledge. Fundamental tools developedin complex systems science appear to be able to dealwith the identified bottlenecks:

� Develop high-dimension models, integrating all relevanttime and space scales, without reduction.

� Develop approaches for decision making and reverse en-gineering, integrating various sources of information andassociated uncertainty.

Key issues towards these goals are knowledge integra-tion, unifying mathematical formalisms, uncertainty rep-resentation and management, optimal control, viabilityand increased computing power. Complex system scienceprovides appealing research directions for these issues andhas proven some efficiency to tackle such complex prob-lems as multi-scale reconstruction in embryogenese(Olivier et al., 2010). Nevertheless, it is obvious that fur-ther interdisciplinary work is required at the frontier ofcomplex system science, which is on its own at the bound-ary of mathematics, physics and computer science, andfood science. A generic structure for this modelling ap-proach could lead in the future to intelligent systemsable to guide the user in defining a model, coupling differ-ent mathematical tools and solving the problem by bring-ing together available knowledge, irrespective of itsformat and scale.

AcknowledgementsThe financial support of government (French ANR pro-

ject INCALIN) and the funding received from the EuropeanCommunity’s Seventh Framework Programme (FP7/2009-2013) under grant agreement DREAM n� 222654-2 and(FP7/2007-2013) under grant agreement CAF�E n� KBBE-212754 are kindly acknowledged as well as the reviewersfor their relevant remarks that help us to significantly im-prove the paper.

References

Allais, I., Perrot, N., Curt, C., & Trystram, G. (2007). Modelling theoperator know-how to control sensory quality in traditional pro-cesses. Journal of Food Engineering, 83(2), 156e166.

Alvarez, I., Martin, S., & Mesmoudi, S. (2010). Describing the result ofa classifier to the end-user: Geometric-based sensitivity. In: 19thEuropean conference on artificial intelligence, Lisbon, August2010.

Arguelles, A., Castello, M., Sanz, J., & Fito, P. (2007). Application ofthe SAFES methodology in manchego-type cheese manufacture.Journal of Food Engineering, 83(2), 229e237.

Aubin, J. P. (1991). Viability theory. Boston, Basel: Birkhauser.Aziza, F., Mettler, E., Daudin, J. J., & Sanaa, M. (2006). Stochastic,

compartmental, and dynamic modeling of cross-contaminationduring mechanical smearing of cheeses. Risk Analysis, 26(3),731e745.

Balsa-Canto, E., Rodriguez-Fernandez, M., & Banga, J. R. (2007).Optimal design of dynamic experiments for improved estimation ofkinetic parameters of thermal degradation. Journal of Food Engi-neering, 82(2), 178e188.

Banga, J. R., Balsa-Canto, E., & Alonso, A. A. (2008). Quality andsafety models and optimization as part of computer-integratedmanufacturing. Comprehensive Reviews in Food Science and FoodSafety, 7, 168e174.

Banga, J. R., Balsa-Canto, E., Moles, C. G., & Alonso, A. A. (2003).Improving food processing using modern optimization methods.Trends in Food Science & Technology, 14, 131e144.

Baudrit, C., Dubois, D., & Guyonnet, D. (2006). Joint propagation andExploitation of probabilistic and Possibilistic information in riskassessment models. IEEE Transactions on Fuzzy Systems, 14(5),593e608.

Baudrit, C., H�elias, A., & Perrot, N. (2009). A Joint treatment ofimprecision and variability in food engineering: application tocheese mass loss during ripening. Journal of Food Engineering,93, 284e292.

Baudrit, C., Sicard, M., Wuillemin, P. H., & Perrot, N. (2010). Towardsa global modelling of the Camembert-type cheese ripening processby coupling heterogeneous knowledge with dynamic Bayesiannetworks. Journal of Food Engineering, 98(3), 283e293.

Bernaerts, K., Versyck, K. J., & Van Impe, J. (2000). On the design ofoptimal dynamic experiments for parameter estimation of a Rat-kowsky-type growth kinetics at suboptimal temperatures. Interna-tional Journal of Food Microbiology, 54(1e2), 27e38.

Bimbenet, J. J., Schubert, H., & Trystram, G. (2007). Advances in re-search in food process engineering as presented at ICEF9. Journalof Food Engineering, 78, 390e404.

Bonaiti, C., Leclercq-Perlat, M. N., Latrille, E., & Corrieu, G. (2004).Deacidification by debaryomyces hansenii of smear soft cheesesripened under controlled conditions: relative humidity and tem-perature influences. Journal of Dairy Science, 87(11), 3976e3988.

Christakos, G. (2002). On the assimilation of uncertain physicalknowledge bases: Bayesian and non Bayesian techniques.Advances in Water Resources, 25, 1257e1274.

Datta, A. K. (2008). Status of physics-based models in the design offood products, processes, and equipment. Comprehensive Reviewsin Food Science and Food Safety, 7(1), 121e129.

Davidson, V. J. (1994). Expert systems in process control. FoodResearch International, 27, 121e128.

Dempster, A. P. (1967). Upper and lower probabilities induced bya multivalued mapping. Annals of Mathematical Statistics, 38,325e339.

Desiere, F., German, B., Watzke, H., Pfeifer, A., & Saguy, S. (2001).Bioinformatics and data knowledge: the new frontiers for nutritionand foods. Trends in Food Science & Technology, 12(7), 215e229.

Dubois, D., Nguyen, H. T., & Prade, H. (2000). Possibility theory,probability and fuzzy sets: misunderstandings, bridges and gaps. In

312 N. Perrot et al. / Trends in Food Science & Technology 22 (2011) 304e314

D. Dubois, & H. Prade (Eds.), Fundamentals of fuzzy sets(pp. 343e438). Boston, Mass: Kluwer.

Farrington-Darby, T., & Wilson, J. R. (2006). The nature of expertise:a review. Applied Ergonomics, 37, 17e32.

Ferson, S., & Ginzburg, L. R. (1996). Different methods are needed topropagate ignorance and variability. Reliability Engineering andSystems Safety, 54, 133e144.

Fito, P., LeMaguer, M., Betoret, N., & Fito, P. J. (2007). Advancedfood process engineering to model real foods and processes: the‘‘SAFES” methodology. Journal of Food Engineering, 83,390e404.

Garc�ıa, M. S. G., Balsa-Canto, E., Alonso, A. A., & Banga, J. R. (2006).Computing optimal operating policies for the food industry. Journalof Food Engineering, 74(1), 13e23.

Georgakis, C. (1995). Modern tools of process control: the case ofblack, gray and white models. Entropie, 194, 34e48.

Guillier, L., Stahl, V., Hezard, B., Notz, E., & Briandet, R. (2008).Modelling the competitive growth between Listeria monocyto-genes and biofilm microflora of smear cheese wooden shelves.International Journal of Food Microbiology, 128(1), 51e57.

Hadiyanto, A., Asselman, A., Van Straten, G., Boom, R. M.,Esveld, D. C., & Van Boxtel, A. J. B. (2007). Quality prediction ofbakery products in the initial phase of process design. InnovativeFood Science Emerging Technologies. Innovative Food Science &Emerging Technologies, 8(2), 285e298.

Helias, A., Mirade, P. S., & Corrieu, G. (2007). Modeling ofcamembert-type cheese mass loss in a ripening chamber: mainbiological and physical phenomena. Journal of Dairy Science, 90,5324e5333.

Hoffman, R. R., Shadbolt, N. R., Burton, A. M., & Klein, G. (1995).Eliciting knowledge from experts - a methodological analysis.Organizational Behavior and Human Decision Processes, 62(2),129e158.

Ioannou, I., Mauris, G., Trystram, G., & Perrot, N. (2006). Back-propagation of imprecision in a cheese ripening fuzzy model basedon human sensory evaluations. Fuzzy Sets and Systems, 157,1179e1187.

Jimenez-Marquez, S. A., Thibault, J., & Lacroix, C. (2005). Predictionof moisture in cheese of commercial production using neural net-works. International Dairy Journal, 15(11), 1156e1174.

Karim, M. N., Yoshida, T., Rivera, S. L., Saucedo, V. M., Eikens, B., &Gyu-Seop, O. (1997). Global and local neural network modelsin biotechnology: application to different cultivation processes.Journal of Fermentation and Bioengineering, 83(1), 1e11.

Kasabov, N. (2005). Discovering rules of adaptation and interaction:from molecules and gene interaction to brain functions. In: Con-ference HIS’04, 4th international conference on hybrid intelligentsystems. Japan (pp. 3).

Kennedy, J., Eberhart, R. C., & Shi, Y. (2001). Swarm intelligence. SanFrancisco: Morgan Kaufmann Publishers.

Linko, S. (1998). Expert Systems: what can they do for the food in-dustry. Trends in Food Science & Technology, 9, 3e12.

Lutton, E., Pilz, M., & Levy, J. (2005). The Fitness Map Scheme.Application to interactive multifractal image denoising. In:Conference CEC2005, IEEE congress on evolutionary computationin Edinburgh, UK, September 2nde5th.

McLachlan, M., & Garrett, J. (2008). Nutrition change strategies: thenew frontier. Public Health Nutrition, 11(10), 1063e1075.

Martin, S. (2004). The cost of restoration as a way of definingresilience: a viability approach applied to a model of lake eutro-phication. Ecology and Society, 9(2), 8.

Murphy, K. P., (2002). Dynamic Bayesian networks: Representationinference and learning. Ph.D. thesis, University of California,Berkeley.

Ndiaye, A., Della Valle, G., & Roussel, P. (2009). Qualitative model-ling of a multi-step process: the case of French breadmaking. ExpertSystems with Applications, 39(2), 1020e1038.

Olivier, N., Luengo-Oroz, M., Duloquin, L., Faure, E., Savy, T.,Veilleux, I., et al. (2010). Cell lineage reconstruction of earlyZebrafish embryos using label-free nonlinear microscopy. Science,329, 967e971.

Pearl, J. (1988). Probabilistic reasoning in intelligent systems: Net-works of plausible inference. San Diego: Morgan Kaufmann.

Perrot, N., Bonazzi, C., & Trystram, G. (1998). Application of fuzzyrules-based models to prediction of quality degradation of rice andmaize during hot air drying.Drying Technology, 16(8), 1533e1535.

Perrot, N., Agioux, L., Ioannou, I., Mauris, G., Corrieu, G., &Trystram, G. (2004). Decision support system design using theoperator skill to control cheese ripening. Journal of Food Engi-neering, 64, 321e333.

Perrot, N., Ioannou, I., Allais, I., Curt, C., Hossenlopp, J., & Trystram, G.(2006). Fuzzy concepts applied to food product quality control:a review. Fuzzy Sets and Systems, 157, 1145e1154.

Petermeier, H., Benning, R., Delgado, A., Kulozik, U., Hinrichs, J., &Becker, T. (2002). Hybrid model of the fouling process in tubularheat exchangers for the dairy industry. Journal of Food Engineering,55, 9e17.

Quintas,M.,Guimaraes, C., Baylina, J., Brandao, T. R. S., & Silva, C. L.M.(2007). Multiresponse modelling of the caramelisation reaction. In-novative Food Science & Technology, 8(2), 306e315.

Reuillon, R., Chuffart, F., Leclaire, M., Faure, T., & Hill, D. (2010).Declarative task delegation in OpenMOLE. In: Proceedings of theIEEE international conference on high performance computing&Simulation (HPCS 2010), Caen, France, June 2010. 7pp.

Riahi, M. H., Trelea, I. C., Picque, D., Leclerq-Perlat, M. N., Helias, A., &Corrieu,G. (2007).Model describingDebaryomyces hansenii growthand substrate consumption during a smear soft cheese deacidificationand ripening. Journal of Dairy Science, 90(5), 2525e2537.

Rodriguez-Fernandez, M., Balsa-Canto, E., Egea, J. A., & Banga, J. R.(2007). Identifiability and robust parameter estimation in foodprocess modelling: application to a drying model. Journal of FoodEngineering, 83, 374e383.

Sablani, S., Datta, A. K., Rahman, M. S., & Mujumdar, A. S. (2007).Handbook of food and bioprocess modelling techniques. InA. Mujumdar, A. Datta, S. Sablany, & S. Rahman (Eds.). BoccaRaton Fla: CRC Press.

Saint Pierre, P. (1994). Approximation of the viability kernel. AppliedMathematics and Optimization, 29, 187e209.

Sicard, M., Perrot, N., Baudrit, C., Reuillon, R., Bourgine, P.,Alvarez, I., et al. (2009). The viability theory to control complexfood processes. European Conference on Complex Systems(ECCS’09). UK: University of Warwick.

Sihufe, G. A., Zorrilla, S. E., Perotti, M. C., Wolf, I. V., Zalazar, C. A.,Sabbag,N.G., et al. (2010).Accelerationof cheese ripeningat elevatedtemperature. An estimation of the optimal ripening time of a traditionalArgentinean hard cheese. Food Chemistry, 119(1), 101e107.

Shafer, G. (1976). A mathematical theory of evidence. PrincetonUniversity Press.

Smets, P., & Kennes, R. (1994). The transferable belief model. ArtificialIntelligence, 66, 191e234.

Smets, I. Y. S., Versyck, K. J. E., & Van Impe, J. (2002). Optimalcontrol theory: a generic tool for identification and control of (bio-)chemical reactors. Annual Reviews in Control, 26(1), 57e73.

Stuurstraat, N., & Tolman, F. (1999). Product modelling to buildingknowledge integration.Automation inConstruction, 8(3), 269e275.

Theys, T. E., Geeraerd, A. H., & Van Impe, J. F. (2009). Evaluation ofa mathematical model structure describing the effect of (gel)structure on the growth of Listeria innocua, Lactococcus lactis andSalmonella Typhimurium. Journal of Applied Microbiology,107(3), 775e784.

Thomopoulos, R., Charnomordic, B., Cuq, B., & Abecassis, J. (2009).Artificial intelligence-based decision support system to managequality of durum wheat products. Quality Assurance and Safety ofCrops & Foods, 1(3), 179e190.

313N. Perrot et al. / Trends in Food Science & Technology 22 (2011) 304e314

Tijskens, E., & De Baerdemaeker, J. (2004). Mathematical modell ing of syneresis of cheese curd. Mathematics and Computers in Simula­tion, 65(1-2), 165-175.

Trelea, I. C. (2003). The particle swarm optimization algorithm: con­vergence analysis and parameter selection. Information Processing Letters, 85, 317-325.

Trelea, I. c., Titica, M., & Corrieu, G. (2004). Dynamic optimisation of the aroma production in brewing fermentation. Journal of Process Control, 14,1-16.

Trystram, G., & Courtois, F. (1996). Food process modelling and sim­ulation. In G. S. Mittal (Ed.), Computerized control systems in the food industry (pp. 55-85). New York: M. Dekker.

Van Impe, 1. F. (1996). Power and limitations of model based bioprocess optimization. Mathematics & Computers in Simulation, 42, 1 59-169.

Varela, F. (1979). Principles of biological autonomy. New York: Elsevier North Holland.

Vitrac, 0., & Hayert, M. (2007). Effect of the distribution of sorption sites on transport diffusivities: a contribution to the transport of medium-weight-molecules in polymeric materials. Chemical Engineering Science, 62(9),2503-2521.

Walley, P. (1991). Statistical reasoning with imprecise probabilities. New York: Chapman and Hall.

Waiter, E., & Pronzato, L. (1997). Identification of parametric models from experimental data. Heidelberg: Springer-Verlag.