Maria Marques

Rui Neves-Silva

Decision Support for Life-cycle Optimization Using Risk Assessment

Keywords: Risk Assessment, Decision Support, Life-cycle optimization

The key idea of this work, is to use risk assessment to support the user in deciding which service should be used, from a set of services developed to support life-cycle optimization, in a specific situation. The risk of a specific situation affecting an industrial plant, characterized by the symptoms, is estimated from the information stored on the system concerning the probability of occurrence of the consequence and its impact. It is expected that this knowledge grows along the life-cycle of a industrial plant. Then, depending of the knowledge available and on the risk of the situation, the adequate service is suggested for promptly reaction in eliminating the problem or avoiding critical situations.

Introduction

Life Cycle Management (LCM) has been developed as a business approach for managing the total life cycle of products and services. By learning how to more effectively manage this cycle, a company or organisation can uncover a wealth of business, environmental and social value -and make the choice to engage in more sustainable activities and production patterns.

Life Cycle Management is all about making more informed business decisionsand chances are that life cycle considerations are already influencing the decisions currently made. Life Cycle Management is simply about helping making these decisions in a more deliberate and systematic way -so a more sustainable production and consumption is possible, and clearly define and measure the business value gained by doing so [START_REF]Draft Final Report of the LCM Definition Study Version 3.6. Life Cycle Initiative[END_REF]. Life-cycle management of industrial plants is about taking decisions on a daily basis. What configuration is best, what production pattern, when to replace a part, who should be involved in a maintenance process, etc. On the other side, an industrial plant can produce, through the available instrumentation, a huge amount of information from where decisions can be supported if well interpreted.

Thus the central problem is on how to combine the existing information so that we can foresee what impact a specific decision will have. Most of the times the information is not structured and it is scattered along the plant making extremely difficult to correlate it and reach a consistent conclusion, i.e. a decision.

In terms of decision theory reaching a conclusion is choosing among a set of alternatives. The parameters involved in this process are treated as subjective judgments, expressing the decision-maker's knowledge, experience, and intuition [START_REF] Raiffa | Decision Analysis: Introductory Lectures on Choices under Uncertainty[END_REF]. Nevertheless, some research using naturalistic methods shows, that in situations with higher time pressure, higher stakes, or increased ambiguities, experts give a more important role to intuitive decision making rather than structured approaches, following recognition primed decision approach to fit a set of indicators into the expert's experience and immediately arrive at a satisfactory course of action without weighing alternatives [START_REF] Baker | Guidebook to Decision-Making Methods[END_REF]. Due to the large number of considerations involved in many decisions, computer-based decision support systems have been developed to assist decision makers in considering the implications of various courses of thinking. These systems can help reduce the risk of human errors due to cognitive and temporal limitations [START_REF] Holsapple | Decision Support Systems[END_REF]. All these aspects become clear when the decision must be made in a noisy environment or in a distressing situation, which are very common conditions in industrial plants.

The work here presented was developed under the scope of the InLife project. One of the objectives of InLife was to develop a system that uses the data coming from different sources along the industrial plant to support decisions for lifecycle management [START_REF] Marques | Life-Cycle Management of Complex Manufacturing and Assembly Lines[END_REF]. From the analysis of the received data, and in case there are changes that may cause an increase of the risk level, the system is able to fire adequate services that suggest actions helping the user deal with the situation. Additionally the incoming data is used to compute life-cycle parameters, defined by the user. If any of the defined parameters exceed a predefined threshold, than the same strategy is applied, which will provide support to the life-cycle management.

Contribution to Technological Innovation

Approach for system development. The system comprises a service platform, which constitutes the user interface. Thus, the existing services will interact with the user presenting the information available in a structured and suitable way. The system supports the user by providing suggestions in terms of the most suitable service to be used in a specific situation, as well as in making a decision considering the relevant information for each case. Accordingly, the system is responsible for analysing the data received, process it and transform it into knowledge that is used to deliver services to the user. The generation/collection of knowledge is based on the use of Ambient Intelligence technology (AmI) [START_REF]Scenarios for Ambient Intelligence in 2010[END_REF], [START_REF]Ambient Intelligence: from Vision to reality[END_REF], Knowledge Management (KM) techniques [START_REF] Bray | Literature Review -Knowledge Management Research at the Organizational Level, Social Science Research Network[END_REF] and the developed service platform [START_REF]InLife Public Concept-Report. InLife Project: Integrated Ambient Intelligence and Knowledge-Based Services for Optimal Life-Cycle Impact of Complex Manufacturing and Assembly Lines, the contract[END_REF].

Among the set of services offered three are in the core of the decision support approach: Condition-based Maintenance, Online Remote Maintenance and Diagnostics, and Prevention of Hazardous Situations. They were selected since they are seen as services to be used on a daily basis operation, especially the first two. Thus the level of decision involved and the impact those decisions will have is more interesting in terms of life-cycle management then the services used occasionally.

The services main function is to provide support for actions to achieve a specific service goal through recommendations to the human user of the system. These recommendations result from the understanding of the plant operation by the system which is the result of modelling cause effect relations (cause events/actions to consequence effects) achieved by an effective life-cycle parameter monitoring.

These relations are used for the establishment of cases that describe the effects of a certain cause. The collection of these cases is then used to select the best option, from the set of already available, for a specific situation. This is the strategy used to implement the decision support approach based on the utilization of the three above mentioned services. System Entities. In order to have the system installed and running in several industrial plants there is the need for developing a model, general enough that comprises all the relevant information on plant operation. Nevertheless, and despite the generic character of the model it is also required that it includes several aspects of the operation. Thus the result requires a good balance between simplification of some aspects and the complexity that still is associated to the production process. Since the developed model to support the entire system includes some aspects that are not relevant for the work here described, we will focus on the entities that are vital for the application of the proposed approach.

Fig. 1 shows part of the developed model simply showing the entities that are directly involved on the risk assessment and in support to decision process. The entity Problem is the central part of the entire system, since any deviation on the defined normal behaviour of the plant will be stored has an instance of this entity. This entity is connected to Symptom, representing the firing event for the risk assessment process, and to Probable Cause, which is a set of possible causes for the occurred situation (among a list of predefined causes). The Symptom entity makes the connection between the lower level part of the model and the upper one. In the lower level we can find entities to represent the operating parts of the plant like Generic (representing machines involved in production, production processes or product parts) and Life-Cycle Parameters (representing the defined parameters that are seen as relevant for the plant). Additionally, MetaData is used to represent the a priori knowledge related with the plant (e.g. in terms of which Generics are related to which Life-Cycle Parameters so that an abnormal value of these latest generate a Symptom). In the upper part of the model the aim is to store the information associated to causes and actions that are related to the problems. Additionally the information on which are the probable causes for a specific problem and the probable consequences for each cause are also stored.

All these entities have several fields that are used to store the associated information, once again, and for the sake of simplicity, here we will only refer the ones that are used in the context of this work. Thus, and in what refers to Problem, the field frequency is used to represent the certainty of knowledge about a problem. The problems with frequency higher than 1 did not occurred, but they were artificially introduced at setup stage and are reference expected problems for each symptom.. Thus, when a similar problem occur the probability of the cause to be the same defined for the problem with higher frequency is higher than for other different problems that only occurred once.

In what concerns Probable Cause the fields probability and risk are used. The first one will store the probability of that cause to be the correct one, whereas the second will store the value of the calculated risk during risk assessment. Note that a Probable Cause is unique but two probable causes can be associated to the same cause. The Probable Consequence has a field to store the probability of a specific cause to induce a specific consequence. Finally, Consequence has a field to store the impact that that consequence will have (if occurs). On the other part of the model we find that Symptom has a field to identify the Generic where the abnormal life-cycle parameter value occurred (given by the connection between Generic, Life-Cycle Parameter and MetaData).

In terms of innovation three aspects are considered as major contributions: (i) the use of AmI and KM to collect and manage the information generated by the plant and its users; (ii) the developed model; and (iii) the use of the information to develop the risk assessment and provide support to decision to the user based on the results.

Risk Assessment

Risk is a concept that denotes a potential negative impact to an asset or some characteristic of value that may arise from some present process or future event. In everyday usage, "risk" is often used synonymously with "probability" of a loss or threat. Nevertheless, risk should combine the probability of an event occurring with the impact that the event would have and with its different circumstances [START_REF] Holton | Defining Risk[END_REF], [START_REF] Stamatelatos | Probabilistic Risk Assessment: What Is It And Why Is It Worth Performing It?[END_REF]. Risk assessment is considered as the initial and periodical step in a risk management process. Risk assessment is the determination of quantitative or qualitative value of risk related to a concrete situation and a recognised threat [START_REF] Anderson | Intelligence-Based Threat Assessments for Information Networks and Infrastructures: A White Paper[END_REF]. Thus, risk R is defined as the result of the product between the probability of occurrence of a specific incident E and the impact I of that incident (in money or injuries), i.e.

) ( ) ( E I E P R o × =
If any of the relevant parameters defined for the plant crosses its threshold, the information provided by the system allows the probability of an incident to be estimated. Also, the estimated consequences of the incident allow the estimation of the level of risk that we are dealing with. The functions that describe both incident probability and losses are supposed to be known (even if with some level of uncertainty). This information is collected during set-up phase and during operation using operator knowledge in order to compute the risk. Assessment Method. In what concerns InLife the calculated risk is presented in cost units. The values of threshold between risk levels are to be defined by the industrial users since they may not be adequate for all cases [START_REF] Marques | Risk Assessment to Support Decision on Complex Manufacturing and Assembly Lines[END_REF].

The use of Event Tree Analysis, using Bayesian networks [START_REF] Ben-Gal | Bayesian Networks In F[END_REF], for assessing the probability of occurrence is seen as the most suitable method to achieve the final consequences. An Event Tree starts from an undesired initiator (loss of critical supply, component failure etc) and follows possible further system events through to a series of final consequences. As each new event is considered, a new node on the tree is added with a split of probabilities of taking either branch. The probabilities of a range of 'top events' arising from the initial event can then be seen [START_REF] Russel | Artificial Intelligence -A modern Approach[END_REF]. The method described corresponds to the so called "Probabilistic Risk Assessment" and as an analytical tool includes the consideration of the following:

Identification of the combinations of events that, if occur, could lead an undesired event;

Estimation of the chance of occurrence for each combination; and Estimation of the consequences associated with each combination. For instance, consider that an alarm is fired indicating that there is a deviation on the expected value of "mean-time-between-failure", expressing that this value exceeded some threshold that is considered normal. Assuming that the occurrence of this type of situation is known as generating a specific loss, that must be quantified, and that the probability of incident associated to the fired alarm is also known, than it is possible to calculate the risk associated to this situation.

Fig. 2. Risk assessment based on probabilities.

Fig. 2 shows the process of calculating the probabilities of consequences associated to the detection of a specific symptom (S). The symptom is the entity that enables the detection of some abnormal situation. Note that it is not the cause of the situation nor the situation it self. Thus the symptom may be caused by one or more possible causes (C). Each cause has one or more consequences (Q). For the sake of simplicity we establish that the measurement of impact I is associated with the consequences Q. The identification of the situation with the higher level of risk is done by navigation on the tree following the higher probabilities. The calculation of risk is done by using the formula:

ij ij i i i i I Q C P C S P R * ) , ( ) , ( × =
For each combination the method computes a value of risk. The combination with the higher risk value is the one that must be considered, i.e. treated, in first place. Thus if we only considerer the higher risk value than:

} max{ i R R =
Depending on the options made at the set-up phase there is also the possibility of presenting the several values of risks computed for the other consequences in order to guide the user on solving each of the associated causes [START_REF] Marques | Decision support system using risk assessment for life-cycle management of industrial plants[END_REF]. This is an option for each company to decide since it is highly dependent on company policies. Example. Let us consider two problems P 1 , P 2 that are stored in the system, and classified as solved problems (i.e. the cause for each of them is already found and the problem was efficiently eliminated). For each one we also have information on "frequency" (f) and "Probable Cause" (PC i ). Now, Let us consider a new third problem P 3 which has just occurred and about which we only know that has the same symptoms of P 1 , P 2 . Assuming that: f 1 >f 2 .

P 1 P 2 P 3 f f 1 f f 2 f 1 PC i PC 1 PC i PC 3 PC i PC 2 PC i PC 4
Lets say that for problem P 1 we are pretty sure about its cause but in what concerns P 2 the possible causes are equally probable. Thus when defining the probabilities we must reflect these considerations, admitting that p 1 >p 2 and p 1 +p 2 =1 and p 3 = 0.5.

P 1 P 2 p(PC 1 ) p 1 p(PC 3 ) p 3 p(PC 2 ) p 2 p(PC 4 ) p 3 Note that p(PC i )=P(S i ,C i ).

Lets now associate the "Cause" (C i ) to each Probable Cause, assuming that two probable causes are associated to the same cause. This must be considered when computing the probable causes for the new problem.

PC i C i PC 1 C 1 PC 2 C 2 PC 3 C 3 PC 4 C 4
Finally we have to define the probability of "Cause" to degenerate in a specific "Consequence" (Q ij ) with an associated impact (I ij ). Assuming that: p 4 +p [START_REF] Marques | Life-Cycle Management of Complex Manufacturing and Assembly Lines[END_REF] The selected risk value, following the defined methodology is R 1 since, considering the conditions imposed, it presents the higher risk value.

Reaching this point we can present the result to the user in terms of what is the most risky situation and which procedure should be applied to avoid it. Remember that we kept tracking of the cause associated to R 1 , thus it is possible to suggest actions that will correct the situation.

In terms of system we need to keep track of what were the causes involved in the problem evaluation, since even if they were not elected, it is useful to have this information. But, how should this information be stored? Remember that P 1 and P 2 have four probable causes but two of them are associated to the same cause, C 1 . Thus to set the probable causes of P 3 we have to consider that the probability associated to C 1 is higher that the other two. If we consider the four different probable causes the probability for each of them would be 0.25. Then, we will have 0.25 for the associated to C 2 and C 3 and 0.5 to the associated to C 1 .

Support to Decision

Through the services developed the users have access to specific functionalities of the system prepared to provide a prompt acquisition and provision of knowledge using an adequate presentation for each type of user. Thus, let us consider again the case shown on Fig. 2 where it would be possible to compute the risk associated to a specific situation. Nevertheless we are able to improve this knowledge with the one that has already been gathered along the life-cycle of the plant. In fact, if we have specific knowledge of the plant it is possible to develop a more accurate and more useful support to decision. For instances if the system stores not only the data that constitutes the case but also the action that was performed to solve it, then it is possible to provide the user with this information which will help him to make a decision on what should be done. In this point of view the support to decision system correlates the past cases that match the current one, and assess the involved risk accordingly. On the top of this strategy each user has to establish what is considered as critical in the considered plant. This information is crucial since it is used to assess the risk for hazardous situations. Fig. 3. Decision support using risk assessment Fig. 3 shows the flowchart that describes how the risk is assessed and how this information combined with the information available is used to select the appropriate service. The situation begins while plant is operating normally and a symptom is detected by monitoring systems. After providing the system with a description of the detected symptom, the system will start searching for stored situations that involve symptoms with the same description provided. The stored situations have associated to them one or more symptoms. Each symptom has one or more probable causes (probability) associated and each cause one or more probable consequences (probability). This cascade of information relating the existing knowledge on the situation will be used to compute risk by means of Probabilistic Risk Assessment. Furthermore the existence of knowledge on how to solve the situation, i.e. actions, is also important for the launching of services since they are launched based on this knowledge.

Finally the system will support the user in solving the situation by providing information on which is the adequate service to be used. If the situation is well known, and was not assessed as critical, the system will select "Condition-based maintenance" service. Further the system will suggest the appropriate maintenance action that must be developed based on the information that was already available to cope with that situation. The service of Condition-based maintenance will also support the user to decide when the suggested action should be performed. This scheduling module will also use information of production in order to suggest non producing slots for scheduling the necessary action.

If the situation involves any of the parameters that were previously established as being critical for the plant and their operators, the system will select Prevention of hazardous situations service. Note that the main idea is to prevent the accidents from occurring thus, the actions should be preventive (e.g. operators training). If the situation does not fit in any of the previous categories the system will select Online remote maintenance and diagnostics service. This service aims not only at developing maintenance but also to provide diagnostics to the user. Thus the service uses InLife problem solving module to collect and correlate all the information available that may be related with the situation described. This information is presented to the user in a structured way in order to support him on finding an appropriate solution. If the user was able of finding a solution for the situation, which includes an action to be developed, than the system stores this information and use it if another symptom with the same description is detected or introduced by the user. The calculation of probabilities is highly dependent on this information update since the objective is to use the knowledge acquired during system operation to optimize system performance.

Conclusions and Future Work

This paper proposes a approach on how to collect information that is scattered along the manufacturing system. This information is then processed and the knowledge derived from it can be used to improve life-cycle management of the industrial plant. With the knowledge acquired it is possible to improve the decision making process by developing a set of services specifically focused on its support. The work here presented shows that by assessing the risk associated to the current status of the plant, it is possible to identify the cause for a certain problem and, based on that, propose, actions that support the user on the decision process. The impact of this kind of system is not assessable on a short term basis. Being a life cycle management system, the results need to be measured along time, on a long term approach. Nevertheless the

Fig. 1 .

 1 Fig. 1. Simplified developed model for saving data from plant.

  +p 6 =1, p 7 +p 8 =1, p 9 +p 10 =1, and: p 4 >p 5 >p 6 >p 7 >p 8 and p 10 >p 9 . The calculated values must be normalised so that the resulting sum is 1. Note that p 11 >p 12 . Remember that PC 4 is related with C 1 , what results in having three different causes, thus we can sum the values p 11 and p 13 associated to p(PC 4 ) and create a new value p 14 . Let us now build the table of the probability of specific consequence associated to a cause. Considering the assumptions that we made previously, we have: p 15 >p 16 >p 17 >p 21 >p 18 >p 20 >p 19 . At this point we just have to multiply the defined impacts for each consequence, and we will have the different risk values. Note that each consequence Q ij has an associated impact I ij . Assuming that: I 11 >I 12 >I 13 >I 21 >I 22 >I 31 >I 32 we have:

	Note that p(Q ij )=P(C i ,Q i ).
	We are now able of computing the risk associated to this case. Let's start by
	building the table of the probability of the probable causes that depends o the values
	established for the frequency:
		P 3
		p(PC 1 )	f 1 p 1 =p 11
		p(PC 2 )	f 1 p 2 =p 12
		p(PC 3 )	f 2 p 3 =p 13
		p(PC 4 )	f 2 p 3 =p 13
			p(Q 11 ) p 14 p 4 =p 15
		C 1	p(Q 12 ) p 14 p 5 =p 16
			p(Q 13 ) p 14 p 6 =p 17
		C 2	p(Q 21 ) p 12 p 7 =p 18 p(Q 22 ) p 12 p 8 =p 19
		C 3	p(Q 31 ) p 13 p 9 =p 10 p(Q 32 ) p 13 p 10 =p 21
		R1 p 15 I 11	R5 p 19 I 22
		R2 p 16 I 12	R6 p 20 I 31
		R3 p 17 I 13	R7 p 21 I 32
		R4 p 18 I 21
	C 1		C 2		C 3
	p(Q 11 )	p 4	p(Q 21 )	p 7	p(Q 31 )	p 9
	p(Q 12 )	p 5	p(Q 22 )	p 8	p(Q 32 )	p 10
	p(Q 13 )	p 6

1 This work was partially supported by Commission of European Union within the project InLife (Integrated Ambient Intelligence and Knowledge-Based Services for Optimal Life-Cycle Impact of Complex Manufacturing and Assembly Lines), under the Sixth Research Framework Program of the European Union, with the contract NMP2-CT-2005-517018.

level of structured information that is being gathered and stored and the support to decision offered is seen as very promising for a enhanced life cycle management of the industrial plant.

Future work, will include refinement of the risk assessment algorithm especially in what concerns the interrelation of symptoms and the inclusion of the time effect. Additionally, authors are also interested in exploring the relation of the method with the adaptive systems theory.


Luis Ribeiro

José Barata

João Ferreira

The Meaningfulness of Consensus and Context in Diagnosing Evolvable Production Systems

Keywords: Diagnosis, Evolvable Production Systems, Multi-agent Systems, Interaction Diagnosis, Hidden Markov Models

An Evolvable Production System (EPS) is a complex and lively entity composed of intelligent modules that interact, through bio-inspired mechanisms, to ensure high system availability and seamless reconfiguration. The diagnosis of such dynamic systems, characterized by constant change, presents new diagnostic challenges and opportunities that can hardly be tackled by traditional approaches. On the one hand, given the decoupled nature of the system, fault interaction and propagation are harder to detect and contain, as is the development of a global diagnostic model, on the other hand local intelligence and careful characterization of the interactions, between the modules, can be explored to emerge the diagnostic functionalities. The impact of simple consensus mechanisms (majority voting) and fault context analysis (module and its current interactions states) is assessed in a multiagent-oriented application in the assembly domain to understand the validity and contribution of this approach in emerging useful self-diagnostic properties in EPS.

Introduction

Mass Customization has been perceived as the excellence paradigm in industry and services however, as pointed out by [START_REF] Christopher | The Agile Supply Chain: Competing in Volatile Markets[END_REF], "we are now entering the era of network competition". This has led enterprises to seek innovative organizational paradigms [START_REF] Christopher | The Agile Supply Chain: Competing in Volatile Markets[END_REF][START_REF]Establishing the Foundation of Collaborative Networks[END_REF]: Supply chains, Extended Enterprises, Virtual Enterprises, Collaborative Networks, etc. In this context, industrial installations require improved agility to face the socioeconomic challenges ahead and effectively contribute to network competitiveness. Researchers in production paradigms, acknowledging this dynamic business environment, increasingly suggest and envision lively shop floors which are aggregations of intelligent and distributed interacting entities. Evolvable Production Systems (EPS) [START_REF] Alsterman | Evolvable Assembly Systems Platforms: Opportunities and Requirements[END_REF][START_REF] Barata | Evolvable Production Systems: Enabling Research Domains[END_REF] relying is such intelligent environments will allow quicker ramp-up times once both product design and shop-floor reconfiguration adjust to meet a balanced compromise These systems, however, are inherently complex and present significant diagnostic challenges and opportunities. Fault interaction and propagation are harder to detect and contain as it becomes more difficult to devise the diagnostic model of a system that, in the limit, can be in constant change and is highly heterogeneous. Nonetheless, local intelligence can be explored to circumvent these problems and enhance the diagnosis capabilities of the system by indirectly inferring the local state from the behaviour fault behaviour of other devices.

Being agile is also a matter of preventing breakdowns, incidents and developing sustainably. Intelligent automation such as required for future production systems must involve diagnostic capabilities that match the performance of the control solution.

Traditional diagnostic approaches often require a full model of the system and are not directly applicable to modern production paradigms like EPS. In fact, to avoid corrupt the control functionalities, the diagnostic system should be embedded in the paradigm and designed accordingly.

This has not been the case. Most diagnostic methods have been developed targeting a specific shop-floor component or a subsystem (manipulators, electrical motors, rotary equipment), or even the entire plant, rather than an evolving entity. The approach proposed in this paper is being developed under the framework of EPS whose control and design principles been investigated in the EUPASS1 project. Intelligent interaction is explored as a fundamental diagnostic feature in EPS. The main concept is the design of local (module level) diagnosers that evaluate their internal status is respect to a given interaction type taking into consideration their operational context and their sensorial perceptions. The objective is to, through this local interaction, emerge polarized consensus regarding disturbances in the shop floor. The presented experiments report some preliminary results of the ongoing research.

Contribution to Technological Innovation

Networked information systems are the cornerstone of today's society [START_REF] Amaral | Complex networks: Augmenting the framework for the study of complex systems[END_REF][START_REF] Newman | The structure and function of complex networks[END_REF]. The pervasive nature of such system ensures seamless, and often unnoticed, everyday-life interaction. The natural evolution of these human-machine networks is towards complexity. This is manifested in the size of the network and in the interconnectivity degree. Industrial networks are not immune to this effect as the complexity of the installation grows more and more devices require communications and more processes need to be synchronized. There is a considerable set of industrial network standards that ease the integration of traditional logic based control devices. However, in the era of network competition, the flux of information between shop floor components and higher level management tools is increasingly important [START_REF] Colombo | Industrial Agents: Towards Collaborative Production Automation, Management and Organization[END_REF]. To meet these requirements there is an ongoing generalized research effort to change the nature and interfacing of the industrial devices in order to make them more pluggable and seamlessly available to other systems.

Research in this area will enable industry to effectively instantiate powerful production paradigms such as EPS at the cost of increased interaction complexity between the architectural building blocks (automation modules). As earlier detailed conventional shop floor diagnostic methods, alone, will not encompass this dramatic shift. This is the context addressed by the present work.

The main contributions of the ongoing research can be summarized as follows: propose a diagnostic architecture that is suitable for intelligent automation system instantiating the EPS concept and in that context ensure the co-evolution of the diagnostic system while improving the overall sustainability of the installation; Capture the dynamics of fault propagation and interaction in distributed industrial systems composed of a high number of interaction modules and Enhance the traditional device-customized diagnostic approaches by introducing the network dimension and incorporating it in the local diagnosis.

Related Literature

A complete review on diagnostic methods derived from the automatic control community can be found in [START_REF] Isermann | Fault Diagnosis Systems: An Introduction from Fault Detection to Fault Tolerance[END_REF] where the application of: parameter estimation, evaluation of parity relations, state estimation and principal component analysis methodologies is properly covered.

A review of quantitative and qualitative history based methods where diagnosis is performed based on the previous system's faulty behaviour can be found in [START_REF] Venkatasubramanian | A review of process fault detection and diagnosis Part III: Process history based methods[END_REF] where the application of artificial neural networks, probabilistic inference methods and expert system is discussed. Qualitative logic based diagnostic methods are covered in [START_REF] Venkatasubramanian | A review of process fault detection and diagnosis Part II: Qualitative models and search strategies[END_REF].

The research being reported in this paper, however, has a close connection with the authors' previous work developed under the Inlife Project2 where web services were applied to a pilot assembly cell and functionalities including selfmonitoring/diagnosis/reporting were implemented at device and process levels [START_REF] Barata | Diagnosis using Service Oriented Architectures (SOA)[END_REF][START_REF] Ribeiro | A Diagnostic Infrastructure for Manufacturing Systems[END_REF]. A preliminary version of ACORDA [START_REF] Lopes | Prospective Programming with ACORDA. Empirically Successful Computerized Reasoning[END_REF], a prospective logic engine, that enables the revision of results, trough the encoding of preference rules was used. Although the engine provided an efficient platform to formalize the diagnostic models, the used version did not support modeling uncertainty. Probabilistic methods applied in the diagnosing industrial systems are reported in [START_REF] Zhou | Fault diagnosis in industrial processes using principal component analysis and hidden markov models[END_REF], [START_REF] Rodrigues | Learning and diagnosis in manufacturing processes through an executable Bayesian network[END_REF] and in [START_REF] Son | An integrated knowledge representation scheme and query processing mechanism for fault diagnosis in heterogeneous manufacturing environments[END_REF]. The first work proposes the application of Hidden Markov Models and principal component analysis to diagnose chemical processes. The second work uses Bayesian Networks to diagnose and study processes in a caravan manufacturing line. The third solution relies on a structured representation of the domain to tackle fault scenarios with multiple causes and fault propagation. The present work rather than considering the entire shop floor or subsystems focuses on the intelligent modules and their interactions with purpose of emerging at network level consistent fault propagation patterns.

A Diagnostic Method Proposal

The main goal of the proposed method is to, through stigmergy, emerge a coherent diagnostic perspective at network level. For that purpose each agent will evaluate its own internal state based on a sequence of observations that it performs as the fault event develops in the system. Each individual agent should be only able to observe its direct neighbours and its sensorial equipment. This constraint is to ensure that the diagnostic system scales in highly dynamic environments with a considerable number of nodes.

A common interaction semantic is required to enable consistent diagnosis. In this context, the interactions between the agents have to be characterized. For the problem considered a suitable representation is the following ) , ( N D I = where I denotes an interaction, D an interaction's direction and N its nature. The system can therefore be envisioned as a directed graph where each agent is a node and each link is a specified interaction. While the nature of the interaction can be any symbol with an associated semantic, the direction is either: inbound, outbound or both (

{ } B O I d b b , , ∈ ).
Distinct networks are considered for each interaction nature (n).

Internally each agent can be in one of the following states: OK -the module abstracted by the agent is working normally; NOK -the module has a fault, PFOthe module is propagating a fault, that he has generated, through its outbound connections; PFOther -the module is affected by a propagating fault on its inbound connections and PFOPFOther -the module is being affected by a fault that is propagating through its inbound connections and that it is propagating over it is outbound connections. The last state may raise some confusion of whether the propagated fault is from an inbound neighbour and is transmitting to outbound neighbours or is the summed effect of this interaction with a self-generated fault however, the relevant aspect to consider is the propagation itself. Once the origin of the fault is eliminated the agent will exhibit a state that is either OK or PFO accordingly to the fault's context and disambiguate the previous occurrence as depicted in Fig 1. state. The agent's observations enable the emergence of a polarized consensus on the network's state. A note on how the agents perform their observations is worth at this point. Rather than polling information from its direct neighbours they push it asynchronously when their internal state changes.

The quality and type of the observations play a major role on the convergence of the system to a useful diagnosis (the network emerging a consensus for any given fault context). In the next section the present diagnostic method will be formalized using Hidden Markov Models (HMM) and the impact of the observations in ensuring the converge of the network will be assessed in two scenario where the influence of other agents state weighted differently.

Preliminary Experiments and Results

The diagnostic model proposed can be formalized as HMM [START_REF] Lawrence | A tutorial on hidden Markov models and selected applications in speech recognition[END_REF]

= = = + ) | ( 1 , B is a M N × matrix (M is the number of observation symbols) that encloses the observation probabilities ) ( ) | ( k b S q k O P i i t t = = =
(i.e. the probability that the observation k happens given the state S i ), π represents the starting state probabilities (in the model considered in this paper the initial state of the model is always OK P=1). In the present case N=5 where each state has the semantic described in the previous section. Concerning M, two scenarios are considered M=8 (Table 1) and M=18 (Table 2). For brevity not all the observation symbols are represented in the second case.

0 OOF 0 0 1 OFO 0 1 0 OFF 0 1 1 FOO 1 0 0 FOF 1 0 1 FFO 1 1 0 FFF 1 1 1
To clarify the reading of the tables in Table 1 the symbol OOF implies that the agent observes that its sensor readings are normal as well as its neighbours with inbound connections. On the contrary, the neighbours with outbound connections are affected with faults. Both models share the A matrix. For brevity both A, B 1 and B 2 matrices have been omitted. Table 2 considers more observation symbols that enrich the model and allow agents a finer granularity control over the inbound and outbound observations. Further they allow the proper balance of B matrices supporting changes in opinions as larger number of neighbours are affected by a fault.

Table 2.

Observation alphabet for M = 18 note that some of the symbols have been omitted and some combinations denoting simultaneously inbound minority and majority are impossible. To test both models a simple experiment comprising four agents was set up. Two pneumatic valves (V 1 and V 2 ) controlling the compressed air flow from a robotic manipulator (R 1 ) which provides air to a gripper (G 2 ). One of the valves allows air in the gripper and the other expels the air when it is not used as shown in Fig. 2. Tables 3 and4 summarize the sequence of observations each agent performs for the cases where M=8 and M=18 respectively The robot detects a fault using its sensor information. That information is passed on to its direct neighbours V 1 and V 2 . Soon V 1 and V 2 will also fail causing a change in the agents' observations. In this example the differences in the observations are felt by G 1 which in the first model is limited basic inbound fault detection. As shall be seen in Table 5 ignoring minorities and majorities, ultimately the fault context, impacts the final consensus on the network. 5 clearly illustrates that the M = 18 model better captures the fault propagation dynamic in the network. The final result is always biased by the B matrix set up. When learning the model parameters from sample data the second diagnostic model has also the advantage of being able to capture system's details such as the relation between the reliability of the agent's sensor and indirect fault inference (inferring the agent's own fault behaviour considering its neighbours response to faults.). In the present test the first model failed in diagnosing G 1 as it is highly unlikely that if both valves fail the gripper will remain operational (state OK) whether in the second model G 1 detected that it was under the effect of fault propagation. If G 1 had been connected to another device the error would propagate and affect the emerged network consensus.

Symbol

Conclusions and Future Work

The presented preliminary experiments testing the proposed diagnostic method suggest that there significant differences in the emergence of coherent diagnostic consensus at network level when ignoring the fault context. Further testing is required to study the behaviour of the proposed method in networks with a high number of interacting agents and distinct levels of connectivity. Also, the fault update dynamics of the agents (propagation of error messages) needs to be addressed to understand of synchronous versus asynchronous update can influence the overall outcome. Although is still early to conclude the results obtained are encouraging and suggest that the proposed method may be suitable to support diagnosis under the framework of the EPS paradigm.

Fig. 1 .

 1 Fig. 1. The agents' states during a fault (a) and after that fault has been resumed in its origin (b) where the system can emerge two distinct configurations according the presence of fault summing effects.

Fig. 1

 1 Fig.1also details a fundamental aspect which is the polarization of the agents' state. The agent's observations enable the emergence of a polarized consensus on the network's state. A note on how the agents perform their observations is worth at this point. Rather than polling information from its direct neighbours they push it asynchronously when their internal state changes.

Fig. 2 .

 2 Fig. 2. Network of agents under study.

Table 1 .

 1 Observation alphabet for M = 8.

	Symbol Agent Internal State Inbound Interactions Outbound Interactions
	OOO	0	0

Table 3 .

 3 Sequences of observations per agent with M = 8 for eight consecutive instants.

	Agent	t 0	t 1	t 2	t 3	t 4	t 5	t 6	t 7	t 8
	R 1	OOO FOO FOO FOO FOF FOF FOF FOF FOF
	V 1 OOO OOO OFO FFO FFO FFO FFO FFO FFO
	V 2 OOO OOO OFO OFO FFO FFO FFO FFO FFO
	G 1 OOO OOO OOO OFO OFO OFO OFO OFO OFO

Table 4 .

 4 Sequences of observations per agent with M = 18 for eight consecutive instants.

	Agent t 0 t 1	t 2	t 3	t 4	t 5	t 6	…	t 8
	R 1	Ok OF OF	OF	OF_OMin OF_OMaj	OF_OMaj	…	OF_OMaj
	V 1 Ok Ok IMin OF_IMin OF_IMin OF_IMin OF_IMin_OMin … OF_IMin_OMin
	V 2 Ok Ok IMin	IMin	OF_IMin OF_IMin OF_IMin_OMin … OF_IMin_OMin
	G 1 Ok Ok Ok	Ok	IMin	IMaj	IMaj

… IMaj

Table 5 .

 5 Evolution of agents' internal as new observations are processed.

	Observations t 0	t 1	t 2	t 3	t 4	t 5	t 6	…	t 8
	M = 8
	R 1	Ok NOk NOk	NOk	PFO	PFO	PFO	…	PFO
	V 1	Ok Ok Ok PFOther PFOther PFOther	PFOther	…	PFOther
	V 2	Ok Ok Ok	Ok	PFOther PFOther	PFOther	…	PFOther
	G 1	Ok Ok Ok	Ok	Ok	Ok	Ok	…	Ok
	M = 18
	R 1	Ok NOk NOk	NOk	PFO	PFO	PFO	…	PFO
	V 1	Ok Ok Ok PFOther PFOther PFOther PFOPFOther … PFOPFOther
	V 2	Ok Ok Ok	Ok	PFOther PFOther PFOPFOther … PFOPFOther
	G 1	Ok Ok Ok	Ok	Ok	PFOther	PFOther

… PFOther

Table

EUPASS project http://www.eupass.org/

InLife project http://www.uninova.pt/inlife/


Jean-Marie Favreau

Simone Hemm

Christophe Nuti

Jérôme Coste

Vincent Barra

Jean-Jacques Lemaire

A Tool for Topographic Analysis of Electrode Contacts in Human Cortical Stimulation

Keywords: Electrodes, Humans, Magnetic resonance imaging, Pain, Surfaces, Magnetic analysis, Computed tomography, Biomembranes, MONOS devices, Image analysis 2

Electric chronic stimulation of the human motor cortex (ECSM) has been reported to alleviate chronic severe pain. However the mechanism of action of ECSM is still hypothetical. This is due mainly to the poor knowledge of, 1) the electric diffusion through the multiple structures beneath the epidural contacts (i.e. dura matter, cerebrospinal fluid space, arachnoid membrane, grey and white matter layers, pie mere and vascular tree), 2) the absence of consensus concerning the stimulation parameters (mono versus bipolar stimulation, cathodic or anodic current) and 3) the detailed cortical topography of the contacts. In this study we focused on the precise Favreau JM et al., IEEE 11th International Conference on Computer Vision 2007 indirectly of the pathophysiologic process of chronic pain, this new tool might be used in the future for image guided electrode positioning.

identification of the cortical areas covered by the electric contacts in a series of twelve patients operated on for ECSM. We propose a new automatic tool for topographic analysis able to compute 2D maps from the 3D anatomic MRI with bijective transformation (point-to-point correspondance). Anatomical regions of interest (AROIs) were visually identified, manually outlined and extracted (Iplan, BrainLab, Germany) for further analysis: 1) for the anatomic structures, on pre operative T1-weigthed magnetic resonance imaging (MRI), the frontal (superior or F1, intermediate or F2 and inferior or F3), the pre central and the post central gyrus;

2) for the electrode contacts (Resume, Medtronic, USA), on post operative computerized tomography (CT). After getting white and gray matter membership maps by automatic segmentation, we produced a cortical mask to build a triangular mesh. We defined a homeomorphism between the 3D mesh and a subset of ℝ 2 and could apply in consequence the circle packing algorithm. We built depth maps (distance to the skull), distance-to-contact maps (distance to a given electrode contact) and anatomic structure maps. Results showed that it was easier to accurately define the location of the contact projection on the cortex allowing physicians to correlate the benefit with the topography. In particular, because of the unfolding, it was easier to integrate the cytoarchitectonics (i.e. the manually identified AROIs) knowledge in the analysis. Beyond the better understanding of ECSM and

Introduction

Electric chronic stimulation of motor cortex (ECSM) through epidural electrodes has been reported to control chronic pain [START_REF] Carroll | Motor cortex stimulation for chronic neuropathic pain: a preliminary study of 10 cases[END_REF][START_REF] Nuti | Motor cortex stimulation for refractory neuropathic pain: four year outcome and predictors of efficacy[END_REF]. However the mechanism of action of ECSM is still hypothetical because of our poor knowledge of both biophysical and pathophysiologic mechanisms. The electric diffusion through the multiple structures beneath the epidural contacts is extremely complex and simplified in order to attempt to analyse the clinical effects [START_REF] Manola | Anodal vs cathodal stimulation of motor cortex: a modeling study[END_REF]. On the other hand the pathophysiologic mechanisms of central neuropathic chronic pain rely on several anatomic, physiologic and pathophysiologic knowledge, in both clinical and experimental domains, not enough to conclude. Recent advances in neuroimaging show the complexity of the phenomenon [START_REF] May | Neuroimaging: visualising the brain in pain[END_REF]. In practice the ECSM is supposed to act through a neuromodulation of the primary motor area located on the pre central gyrus and the frontal slope of the central sulcus as repetitive transcranial stimulation [START_REF] Leo | Repetitive Transcranial Magnetic Stimulation (rTMS) in Experimentally Induced and Chronic Neuropathic Pain: A Review[END_REF].

Beyond the biophysical considerations, the absence of reproducible effect [START_REF] Coffey | Neurostimulation for chronic noncancer pain: an evaluation of the clinical evidence and recommendations for future trial designs[END_REF] raises questions on the influence of the reorganization of functions on the surface of the brain [START_REF] Butefisch | Neurobiological bases of rehabilitation[END_REF] and on the exact location of electric contacts. This latter is still imprecise because of the complexity of gyration and sulcation which makes difficult the interpretation of topography in 3D even with the most recent surgical navigation tools.

Most people consider that the final electrode position determined at the end of the surgery using the neuronavigation system is precise enough for further analysis.

Afterwards the position of each electric contact is projected on a schematic representation of the central region, the mean position of the sulci and fissures seen on superficial computer tomography (CT) reconstructions [START_REF] Nguyen | Motor cortex stimulation in the treatment of central and neuropathic pain[END_REF]. Another possibility is the projection to a classical brain atlas such as the Talairach atlas [START_REF] Talairach | Atlas d'anatomie stereotaxique. Reperage radiologique indirect des noyaux gris centraux des regions mesencephalo-sous-optiques et hypothalamiques de l'homme[END_REF] or the normalization of the patient information to Talairach space with a subsequent data projection onto a representative Magnetic Resonance Imaging (MRI) slice [START_REF] Schlereth | Lefthemisphere dominance early nociceptive processing in the human parasylvian cortex[END_REF]. Both approaches are hindered by the low anatomic precision due to the schematic representation for the former and to the relative probabilistic location without reference to the individual patient's brain for the latter. Furthermore, the complex structure of the cortex makes it difficult to define cortical areas implicated in the stimulation. Postoperative CT or MRI scans are still rarely performed. A tool for the topographic analysis of the cortex is thus essential for neurosurgeons not only to understand brain's topological structure but also to localize the final anatomic position of each stimulating contact. It should have a precise location system, specific for each brain and taking into account morphological specificities.

Furthermore, the presentation mode should be adapted in order to compensate for the irregular cortex convolutions. Several authors in different domains have proposed 2D reconstructions of the cortical surface instead of 3D representations.

Most of them worked on retinotopic maps with fMRI studies, using cortex unfolding.

Van Essen et al. [START_REF] Essen | Functional and structural mapping of human cerebral cortex: solutions are in the surfaces[END_REF] built a smoothed surface and mapped cortical surface onto an ellipsoid. This representation is a good preview of cortical surface, but has some limitations. An ellipsoid cannot be easily drawn in a plane without distortion and this kind of map cannot be printed with good precision. Warnking et al. [START_REF] Warnking | fMRI retinotopic mappingstep by step[END_REF] proposed a flattening algorithm to unfold the visual cortex. After projecting nodes on the plane using representative nodes as anchors, their iterative method minimized geodesic length distortion. The main limitation of these methods is the non-bijectivity between the original surface and the resulting map: two points on the original cortical surface can be computed at the same location on the unfolded surface. A connection tool between both 3D data and resulting map is then infeasible. Haker et al. [START_REF] Haker | Conformal surface parameterization for texture mapping[END_REF] used differential geometry to produce a quasi-conformal bijective mapping of surfaces homeomorphic to a ball. However in our application, cortical surface is not homeomorphic to a ball, first because we are only working on a small part of the brain, second because data we are using are MRI: in this context, gaps between two parts of the cortex may be less than the voxel size, producing a connection on the resulting surface (PVE). Hurdal et al. [START_REF] Hurdal | Quasiconformally flat mapping the human cerebellum[END_REF] computed a bijective flattening of a cerebellum using 27 MRI acquisitions from a single subject. In clinical routine it is not realistic to obtain so many images and the method should be automatic. As in Haker's method, Hurdal assumed that the segmented surface was homeomorphic to a ball, without local topology defaults. Both methods are not appropriate for clinical data.

Herein we propose an automatic tool for topographic analysis able to produce 2D maps from a single 3D anatomic MRI with bijective transformation. In the context of the chronic stimulation of motor cortex for analgesic intention, this tool offers to our knowledge a new and unique approach for anatomical analysis and automatic localization. In the following, we first present our three steps method .Then we describe different maps that can be constructed in ECSM and that can be visualized in connection with 3D anatomical data. Finally, we apply our algorithm to twelve patients treated by ECSM and we discuss the interest of this tool in neurosurgery.

Method

Fig. 1 presents an overview of our method. Given an MRI data set and a region of interest (ROI) we detected the cortical surface, then unfolded it into the plane. This resulting map was used to display information such as topological areas or electrodes' position. Each step is detailed in this section. In our application, maps are among others based on a manual segmentation of contacts and brain cortical areas.

1.Segmentation

For bone and brain structure we used a fuzzy segmentation algorithm. We first performed a skull-stripping using the Brain Surface Extraction (BSE) [START_REF] Shattuck | Magnetic resonance image tissue classification using a partial volume model[END_REF], which recently proved to be the most powerful algorithm compared to the classical ones [START_REF] Fennema-Notestine | Quantitative evaluation of automated skull-stripping methods applied to contemporary and legacy images: Effects of diagnosis, bias correction, and slice location[END_REF].

Given this brain mask, brain segmentation was performed using a Possibilistic Clustering Algorithm [START_REF] Krishnapuram | A possibilistic approach to clustering[END_REF] which has proved to compute real memberships of voxels to tissue classes [START_REF] Barra | Tissue segmentation on MR images of the brain by possibilistic clustering on a 3D wavelet representation[END_REF], and not degrees of sharing across all classes as the classical fuzzy C-means [START_REF] Bezdek | Pattern Recognition with Fuzzy Objective Function Algorithms (Advanced Applications in Pattern Recognition)[END_REF].

Both white and gray matter fuzzy membership maps were then used to produce a cortical mask, used to build a triangular mesh and to select a ROI for unfolding.

Mesh computation

Given a coefficient δ ∈ [0,1], the criterion uGM,x + uW M,x ≥ δ. resulted in a a binary volume describing parenchyma membership. Some resulting defaults can be removed using a morphological operation like opening. This last step produced a binary volume that defined the brain mask. For practical purposes we used a fixed δ value (δ = 0.70). We then applied the marching cubes algorithm [START_REF] Lorensen | Marching cubes: A high resolution 3d surface construction algorithm[END_REF] to build the cortical surface mesh, using isosurface defined by the mask edges. The resulting surface was a triangular oriented mesh, defined by a set of vertices V ⊂ ℝ 3 and a set of triangles T ⊂V ×V ×V.

Since our application (see part 4) only needed a partial cortical mapping (the central sulcus and its neighborhood), we selected this ROI by manually setting the center and the radius of this region. This solely manual step allows to only keep triangles contained in the ROI to perform unfolding.

The unfolding method we used was based on topological properties of Riemann surfaces. Thus the initial surface had to be homeomorphic to a disc. A final selection step was thus needed to define a surface able to be correctly unfolded. Triangles included in the final selected region were computed using the given center. We first searched for the nearest triangle to the center and added it to the final mesh. We then successively added neighbor triangles t, and make sure they did not modify the topology of the resulting mesh. If so, the incriminated t were removed from the resulting surface, and added in a fifo structure processed later if feasible (Fig. 2).

The resulting surface was included into the ball of interest defined by the center and the radius. Furthermore this surface was homeomorphic to a disc and could be fully unfolded.

Figure 2. Construction of the region of interest (iterative method).

Unfolding using Circle Packing

The unfolding process aimed at building a planar map representing the cortical surface, i.e. we wanted to define an homeomorphism between the surface mesh in 3D from the ROI and a subset of ℝ 2 .

Many available methods do not produce bijective maps: one point on the resulting mesh may not have a single corresponding point in the original surface mesh in 3D. This limitation would be a major disadvantage for our application.

Because of the homeomorphic property, we applied the Circle Packing algorithm [START_REF] Collins | A circle packing algorithm[END_REF][START_REF] Hurdal | Quasiconformally flat mapping the human cerebellum[END_REF] to produce a quasiconformal and bijective mapping. A conformal mapping f:A→B is a continuous locally bijective function preserving all angle measures. When A is a manifold defined by a triangular surface, there is not necessarily a conformal mapping between A and a subset of ℝ 2 .

The circle packing algorithm is an iterative method that minimizes angular distortions and produces a bijective map. A circle is first associated to each vertex of the original mesh, with a radius value arbitrarily fixed. Radii on the edges of the mesh are defined using euclidean distance between boundary vertices. Radii of the vertices vi ∈ V contained in the mesh but not in the boundary are proceeded iteratively. Then the following algorithm is computed until convergence of the radius values: for each circle c of V, the new radius value is readjusted using radii of its neighbors ci ,i = 0...n . First, we computed the sum of all angles ciccj defined by neighbor triangles (ci,c,cj), afterwards radius of c is adjusted (increased Fig. 3(a A planar location was finally computed for each vertex using circles radii. For each vertex a projection πV: V → ℝ 2 can be defined using ℝ 2 computed location and ℝ 3 original location. Given πV and triangular surface, a projection π can be defined by π: C→𝐶 ̅ where C ⊂ ℝ 3 was the cortical surface and 𝐶 ̅ ⊂ ℝ 2 its planar projection. This function was defined by π(V)πV(V) and by linear interpolation inside triangles. Circle packing guarantees that C and 𝐶 ̅ are locally homeomorph, without foldings.

Building maps using projections

Quantitative maps were then easily computable from π. 3D data were projected on a resulting planar map summarizing much information and producing a useful tool for surgeons. Three maps were more particularly studied A distance map mp0, computed using the Euclidean distance |•,•| between a chosen point p0 and each point of C. This information was projected to the plane using π:∀x ∈ 𝐶 ̅ , mp0(x) = | p0,π -1 (x) |. This first example produced a map with an hot spot corresponding to the part of the surface nearest to p0. A distance map mS to a subset S of space was also produced using an approximation of Euclidean distance ( e.g. chamfer)dS. This distance information was projected to the plane using π:∀x ∈ 𝐶 ̅ , mS (x) = dS(p0,π -1 (x)). Based on a subset S corresponding to a global surface overhead C, this second example produced a grayscale map that can be understood as a depth map.

Manual surrounding or automatic classifications can also be displayed on the unfolded surface using a simple projection. Given a set of classes C, and a classification c: ℝ 3 → C of MRI voxels, a classification map mC can be defined by:

∀x∈̄ 𝐶 ̅ ,mC (x)c(π -1 (x)).
Application

1.Data

We applied our algorithm to twelve patients presenting neuropathic pain and treated by CSM. Intra operative anatomical localization of the target was performed using a 3D navigation system (ISIS, St Martin d'Hères, France).The final positioning of electrode relied on empiric intra operative tryings, driven by various data from litterature (e.g. [START_REF] Tsubokawa | Chronic motor cortex stimulation in patients with thalamic pain[END_REF]). In order to cover the largest surface above the motor cortex two electrodes (Resume, Medtronic,USA), each with four contacts (diameter: 4 mm, center to center distance: 10 mm), were secured on the dura mater.

Retrospectively, in the frame of a research protocol, pertinent AROIs were visually identified, manually outlined and extracted (Iplan, BrainLab, Germany)

for further analysis: 1) for the anatomic structures, on pre operative T1-weigthed MRI (voxel size 0.49×0.49×1mm 3 ), the frontal (superior or F1, intermediate or F2

and inferior or F3), the pre central and the post central gyrus; 2) for the electrode contacts (Resume, Medtronic, USA), on post operative computerized tomography (CT, voxel size 0.53×0.53×1.25mm 3 ). MRI and CT were matched using the mutual information algorithm [START_REF] Wells | Multi-modal volume registration by maximization of mutual information[END_REF] (IPlan, BrainLab, Germany) for further analysis. Because of the surgical technique used (epidural implantation, CSF-proof) there was no brain shift which could have influenced the analysis of geometric relationships between contacts and anatomic structures.

2.Results

We then determined the map center selecting the vertex of the cortical surface closest to the barycenter of the eight surrounded contacts. The only manual part of our method in this context was thus the choice of the map radius. The complete and fully automatic unfolding method, from segmentation to circle packing, was applied to the twelve patients.

Three different maps were produced, representing available data on the unfolded surface. First we computed for each contact i the point pi in the MRI reference system by determining the barycenter of each contact. Distance-to-contact maps m pi were computed for each contact I and summed into an image (Fig.

4(c)).

We then built a depth map mS by using the distance to the skull S (Fig. 4(d)).

We inversed the BSE mask and applied our segmentation step to extract the skull Our tool provides a graphical user interface displaying together the preoperative MRI, the corresponding surrounded structures and the three maps described above.

At the end of the unfolding step, an unique cursor is represented in all views, updated in real time following the user actions. The preliminary results showed that it was easier to accurately define the location of the contact projection on the cortex, allowing physicians to correlate the benefit with the topography. In particular, 2D maps simplify the interpretation of effects taking into account the functional organization of the motor cortex according to the gold standard pioneering works on the cyto-architectonics classifications [START_REF] Nieuwenhuys | The human nervous system. A synopsis and atlas[END_REF].

Discussion

The tool we proposed here is an automatic method for topographic analysis to produce 2D maps from a 3D anatomic MRI acquired from a clinical protocol for cortical stimulation. It fills the clinical needs minimizing manual steps; the bijective unfolding offers a useful and ergonomic tool for the clinical procedure to navigate in the different views.

Until recently, neurosurgeons in our research group analyzed the final electrode contact position on a 3D view merging MRI and CT data including the surrounded structures and contacts (Fig. 4(b)). This was already a more sophisticated approach compared to the usual contact visualization on a schematic representation of the central region [START_REF] Nguyen | Motor cortex stimulation in the treatment of central and neuropathic pain[END_REF].Nevertheless, the 3D analysis was complex and time consuming (about one hour for the definition of AROIS). In spite of the anatomic knowledge, it is always difficult even impossible to determine the exact location of contacts mostly for sulcus folders which support in special areas most of the primary motor cortex.

Our method does not need the intervention of a clinician to produce maps and offers some synthetic views of the cortical surface. After the automatic computation, only a few minutes are required to locate electrode contacts according to the structures, with a better precision than the 3D method notably inside the sulci: a contact overhead a suclus (Fig. 5(a)) cannot be easily localized without a map view (Fig. 5(b)).

The planar representation increasing precision because of disambiguation of the electrode location onto the surface. Further information were offered by assorted maps, like a skull depth map or the probabilistic location of the electrodes on the cortical surface. The understanding of this new representation is provided by an interactive interface using the bijection properties for navigation. Depth maps are also a major aid to understand topological properties displayed by this tool.Beyond these preliminary results, the main benefit of our tool is the precision of the location system. It offers many prospects in the clinical application of ECSM: localization of the contacts, identification of anatomical structures, correlation between anatomical data and clinical results, identification of the optimal target and surgical correction of the location of the electrodes. Meanwhile the complex topology of the cortex in some patients is not optimally processed during the construction of the ROI. It involves shattered maps that for the moment can not easily be explored.

Conclusion

We developed a promising and original tool with an ergonomic representation format. This tool is a quasiconformal optimal approach minimizing angles distortion.The main contribution of our bijective method is delivrance from topology constraint: original surface does not need to be homeomorphic to a disc or a ball, and unfolding is proceeded preserving global shape. In the context of ECSM, our approach offers an original and complete tool for location, with many possibilities like localisation or data synthesis.

The next step will be to increase the quality of the maps by improving the construction of the ROI. For this, we are introducing an algebraic topological method.

Our tool might be applied to compare the final anatomic position of each contact with the corresponding clinical results. Furthermore we intend to improve the preprocessing by building new maps like the electric field computed from the electrode position and from the stimulation parameters. Neurosurgeons intend to benefit from the higher precision introduced by the planar maps to define subparts within each cortical area to detail the clinical interpretation. In the long term, the different maps should be a helpful tool as from the surgical planning in order to optimize the contact positioning according to each individual patient's brain.

Figure 1 .

 1 Figure 1. Overview of the method.

  ) or decreased Fig. 3(b)) to obtain a 2 π sum of angle then having a local planar piece of plane.

Figure 3 .

 3 Figure 3. Increasing (a) and decreasing (b) angles according to neighbors' radii.

  subset. Furthermore an anatomic structure map m C was constructed from the surrounded areas (Fig.4(e)).

Figure 4 .

 4 Figure 4. (a) axial T1-weigthed MRI slice used for intra operative navigation. (b) 3D view with the AROIs (greyscale labelling, from white to black, post central gyrus, F2, pre central gyrus and F1) plus the 4 contacts (black full circle) of two parallel electrodes placed above the motor cortex and resulting unfolded maps of a patient suffering of neuropathic pain secondary to thalamic ischemic stroke. Maps: (c) distance-to-contact map, (d) depth map (e) and anatomic structure map. The common cursor (cross) is visible in all views.

Figure 5 .

 5 Figure 5. Simplistic outline of the localization problem: (a) cross section of a suclus and (b) the corresponding unfolded map. In this figure, the black box is a modelling of the electrode, the black line is the segmented cortical surface, and the grey regions are segmented anatomical areas.

Acknowledgments

The authors wish to acknowledge Pascale Picard (MD) for patient selection and follow-up.