Alexia Missoffe
email: alexia.missoffe@univ-lorraine.fr

Julie Kabil

Pierre-André Pav Vuissoz

Pierre-André Vuissoz

Jacques Felblinger

New transmission line model of an insulated cable embedded in gel for MRI radiofrequency interaction hazard evaluation as an alternative

Keywords: radiofrequency, transmission line, transfer function, MRI safety, heating, pacemaker radiofrequency, transmission line, transfer function, MRI safety, heating, pacemaker

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INTRODUCTION

Magnetic Resonance Imaging is an essential imaging modality for soft tissue imaging. Nevertheless, it presents a risk for patients implanted with a medical device. In particular, there is a risk concerning the radiofrequency field for active implants with long leads such as neurostimulators, pacemakers or defibrillators. Indeed, the induced currents along the lead can lead to heating of the tissues around the bare electrode at the end of the lead and induced voltages at the input impedance of the active device. Simulation plays an important role in the study of lead interactions with the radiofrequency field as it facilitates the collection of important information without the cost of MRI time. Currently the leads of the implanted devices mentioned are generally too complex to be simulated by the numerical resolution of full-wave Maxwell equations [START_REF] Talcoth | Electromagnetic Modeling of Pacemaker Lead Heating during MRI[END_REF]. The problematic of simulating coaxial helicoidal pacemaker leads has been studied by Talcoth et al. [START_REF] Talcoth | Electromagnetic Modeling of Pacemaker Lead Heating during MRI[END_REF] leading to the conclusion that the approximations necessary to allow pacemaker lead simulation are too coarse. Indeed because of the consequence of the proximity effect on current for two adjacent windings, the full 3D geometry of the winded wires has to be taken into account which would lead to unacceptable simulation times. An alternative to the full-wave simulation is to use the transfer function model proposed by Park et al. [START_REF] Park | Calculation of MRI-Induced Heating of an Implanted Medical Lead Wire with an Electric Field Transfer Function[END_REF] that can be evaluated experimentally. This model, once determined, allows to evaluate in a very straight forward manner the heating at the electrode for any distribution of the incident field along the path of the lead. Especially, one can run a full-wave simulation of anatomical models without the lead to extract the possible incident field along the path of a device lead. The transfer function model then allows the evaluation of the heating risk in an anatomical environment. The transfer function model is all the more important so as the phase distribution of the incident field in a human body model has a crucial importance on the heating effect at the electrode. These so called "phase effects" were brought to light by Yeung et al. (3) in 2002 and this work was at the origin of the transfer function concept proposed later [START_REF] Park | Calculation of MRI-Induced Heating of an Implanted Medical Lead Wire with an Electric Field Transfer Function[END_REF]. The realistic phase distribution of the incident field in a human body is usually not experimentally reproducible in phantoms with standard radiofrequency MRI coils. Therefore it is essential to have a theoretical model to link the heating at the electrode to the real incident field distribution.

In the article by Yeung et al.(3), the fact that the heating at the electrode depends linearly on only the incident electrical field along the path of the cable comes directly from the formulation of the numerical resolution of the problem by the method of moments. Park et al. [START_REF] Park | Calculation of MRI-Induced Heating of an Implanted Medical Lead Wire with an Electric Field Transfer Function[END_REF] suggested that the work by King (4) on transmission line models of antennas embedded in matter could be useful to determine the transfer function of a simple cable for which analytical expression of the parameters exist. Also, according to Feng et al. [START_REF] Feng | A Technique to Evaluate MRI-Induced Electric Fields at the Ends of Practical Implanted Lead[END_REF] the underlying assumption under the transfer function model is that a transmission line model is valid to describe the behavior of the cable embedded in gel at the frequency of interest.

The work of King (6) inspired the modified transmission line model proposed by Acikel et al. [START_REF] Acikel | Modeling of Radio-Frequency Induced Currents on Lead Wires during MR Imaging Using a Modified Transmission Line Method[END_REF]. This model considers the ambient conducting medium to be the return path for the current. The main adjustment made from the model by King et al. was to integrate the fact that the excitation was no more a point excitation at the end of the insulated cable but a field distribution all along the cable generated by another antenna. They evaluate the transmission line parameters for bare and insulated wires using a theoretical analytical analysis and show that their model correctly predicts the full-wave simulation current distribution along different wires. In this work [START_REF] Acikel | Modeling of Radio-Frequency Induced Currents on Lead Wires during MR Imaging Using a Modified Transmission Line Method[END_REF], for the insulated wire, the current distributions fall progressively to zero at the ends suggesting that they are also isolated at the end, the end loads being considered as infinite.

An issue of this transmission line model is actually to correctly model the bare electrode at the end of the insulated cable which corresponds neither to a short-circuit nor to an open circuit. In (4), King considers that the impedance of a bare portion of the cable at the end of an insulated cable can be calculated from the expression of the impedance of a bare monopole in a homogeneous media determined in previous work [START_REF] King | Cylindrical Antennas Immersed in Arbitrary Homogeneous Isotropic Media[END_REF]. Alternatively Acikel et al. [START_REF] Acikel | Modeling of Electrodes and Implantable Pulse Generator Cases for the Analysis of Implant Tip Heating under MR Imaging[END_REF] propose an electrode model which is a voltage source associated to an impedance. The voltage source models the incident field influence and is normalized to it. Nevertheless, it seems non physical to have to model a completely passive element such as an electrode embedded in gel by a voltage source as if it acted as an incident field amplifier. In the work presented here, we present a transmission line model with an electrode model which is a simple complex load impedance meaning a strictly passive element as suggested by King (4). This model has a physical meaning and might even be a measurable parameter in the future.

An issue that also appears here is the definition of the voltage along the transmission line. Here the second conductor, which is the return path for the current, is made of the slightly conducting ambient medium. At the surface of the insulation, the electrical field decays exponentially in the gel with a characteristic distance which is not negligible therefore necessitating a new definition of the voltage drop between the two conductors. Acikel et al. [START_REF] Volkan | inProc Intl Soc Mag Reson Med[END_REF] explain that the voltage is a scaled version of the charge distribution at the surface of the insulated cable. It seems nevertheless necessary to give a physical definition of the transmission line voltage that can be quantitatively calculated from the field distributions. This work proposes a definition. The proposed definition of the voltage along the line allows the simple extraction of the load impedance from a full-wave simulation.

To sum up, this work presents a model based on the modified transmission line method (MoTLim) presented by Acikel et al. [START_REF] Acikel | Modeling of Radio-Frequency Induced Currents on Lead Wires during MR Imaging Using a Modified Transmission Line Method[END_REF] inspired from earlier work by King (6) but with a strictly passive model for the end electrodes. It specifically focuses on showing that this model is indeed an alternative to the transfer function model having the advantage of being described by a set of more physical parameters than the mathematical transfer function model. We concentrated on a wire similar to the standard S-AIMD 1 wire from the ISO-TS 10974 norm [START_REF]Assessment of the Safety of Magnetic Resonance Imaging for Patients with an Active Implantable Medical Device[END_REF] for active implants and we varied the length and termination conditions. We first studied the data extracted from full-wave simulations at 64 MHz (1.5T MRI), to evaluate if indeed an insulated wire embedded in gel can be described by the modified transmission line theory meaning that the incident field, voltage and current distribution along the cable respect the modified transmission line equations. A second part focused on the equivalence with the transfer function model and how the transmission line model can predict experimental and simulated transfer functions. A third part shows the transmission line model can predict the experimental and full-wave simulated resonant behavior of the heating at the electrode as a function of length of two different cables with two different distal electrode termination conditions.

METHODS

Determination of a transmission line model from a full-wave numerical simulation

The wire studied was a steel wire with a 1.5 mm diameter isolated by a 0.5 mm thick heat-shrinkable tube made of polyolefin. Steel was considered to be a perfect electrical conductor at the working frequency of 64 MHz. The measured insulation permittivity at 64MHz was 2.35+/-0.01. The wire studied to extract the transmission line parameters was a 20 cm wire of the type described above with 1 cm of insulation removed at both ends.

The full-wave simulations were performed using the software SEMCAD-X v14.8 (SPEAG, Zurich, Switzerland). The cable was placed in an American Society for Testing and Materials (ASTM) [START_REF] Astm F | Standard Test Method for Measurement of Radio Frequency Induced Heating On or Near Passive Implants During Magnetic Resonance Imaging[END_REF] phantom filled with gel with a relative permittivity ε r of 80 and a conductivity of 0.47 S/m. The excitation used was a plane wave excitation in a volume of 208 x 110 x 640 mm encompassing the cable. The boundary conditions were set up such as to imitate an infinite volume of the ASTM gel. The cable was placed along the z-axis of the phantom. The plane-wave excitation had a propagation direction along the direction x, the electrical field direction being along z meaning it was parallel to the cable direction which is necessary to induce currents therefore heating at the bare electrodes. The simulation was a 20 period harmonic simulation at 64 MHz. A simulation without the cable was performed to extract the incident field along the cable which is the entry to the transmission line model. We worked in the frequency domain, all variables were phasors which are the complex representative of the sinusoidally time varying fields characterized by an amplitude and a phase.

The power deposited at the electrode was considered to be proportional to the active power at the load impedance modeling it. The actual output wanted from the model is the voltage or the current at the load impedance. Nevertheless, a transmission line model taking for input the incident field along the cable will allow the prediction of the voltage and current distribution all along the cable. And the voltage and current distribution along the cable extracted from full-wave simulations were necessary to extract the transmission line parameters which are the impedance per unit length of the cable , the admittance per unit length of the cable , and the load impedances at both ends and . The current distribution was extracted directly from the current density map. The voltage distribution was considered to be the integration of the electrical field component perpendicular to the cable in the x-direction until it reaches less than 1% of its maximum value which is at the interface with the perfect electrical conductor. If the cable embedded in gel follows a transmission line model, the distribution of and should follow the transmission line model equation 1 and 2 associated to the two boundary conditions 3 and 4, z 1 and z 2 being the z coordinate of the ends of the cable. These boundary conditions introduce the simple passive electrode model proposed in this work. Therefore, knowing these distributions allows to extract and which should be constant if the hypothesis of a transmission line model is right.

and can be extracted as well.

[1]

[2]

[3]

[4]

Parameters extracted from the full-wave simulations were then used in a finite difference model of 1 to 4 to predict voltage and current distributions of the original simulation.

Hertel et al. [START_REF] Hertel | The Insulated Linear Antenna-Revisited[END_REF] give analytical expression of the parameters and for simple geometries of insulated antennas. These parameters were compared to those extracted from the full-wave simulation. Associated to the values of the loads extracted from these same simulations, they will be used as an input for the finite difference model to compare the results to the original full-wave distributions.

Different lengths and different termination conditions of the same type of cable were considered to check that the transmission line model has indeed a predictive value for other cables than the one used to extract the model. We considered a 30 cm cable uncapped at both ends as well as a 20 cm cable capped at one end. For the load of the capped cable at the capped end, we used two different values of . First 1 MΩ to simulate an open circuit as it is isolated from the gel, second a load extracted from the full-wave simulation as for the first cable studied. The extracted parameters were used at inputs for the transmission line finite difference model for both cables and the results were compared to the full-wave simulations.

From the modified transmission line model to the transfer function model.

Finite difference model

Solving equation ( 1) to (4) for an incident field Einc with a finite difference method can be written

Wave propagation model

Another way of considering the problem is to consider that the incident field at a certain position propagates in both ways with a propagation constant k related to the transmission line parameters by [START_REF] King | The Many Faces of the Insulated Antenna[END_REF] and is reflected at each end with a reflection coefficient ρ that can be calculated by the following classical transmission line equations 5 and 6.

[5] [START_REF] King | The Many Faces of the Insulated Antenna[END_REF] Considering that the origin of the z axis is at the electrode we want the scattered electrical field from, that the reflection coefficient at this end is ρ1 and ρ2 at the distal end and that the length of the isolated part of cable is l, the transfer function at position z can be written: [START_REF] Acikel | Modeling of Radio-Frequency Induced Currents on Lead Wires during MR Imaging Using a Modified Transmission Line Method[END_REF] For a 16 cm capped wire of the type SAIMD-1 of the ISO-TS 10974 norm [START_REF]Assessment of the Safety of Magnetic Resonance Imaging for Patients with an Active Implantable Medical Device[END_REF] for active implants the value of the transfer function converges with a error of less than 1% if one considers 9 terms and is in accordance with the transfer function derived from the finite difference model described above with an difference of about 7%. In this work, this wave propagation model was considered to be the more reliable because analytic and was used to calculate the transfer functions from the transmission line model.

Validation Comparison with experimental and simulated transfer functions

The cable tested was the same as the uncapped cable one used in numerical study but with a total length of 53 cm.

Direct comparison

The transmission line model was determined partly from analytical formulas for the impedance and admittance per unit length and Z and Y (13) and from the previous SEMCAD-X simulation for the load impedances. Using the wave propagation model, the transmission line model was used to predict a transfer function.

We have measured the transfer function with a set up based on the reciprocity approach described by Feng et al. [START_REF] Feng | A Technique to Evaluate MRI-Induced Electric Fields at the Ends of Practical Implanted Lead[END_REF]. We excited the electrode we wanted to know the power deposited with a simple monopole at and measured the current along the cable. The cable was placed in an ASTM phantom as described on Figure 1. The transfer function was given by the transmission parameter S21 between the excitation probe and the current probe measured with a vector network analyzer. It was measured every cm.

The transfer function predicted by the transmission line model was also compared to a simulated transfer function based on the reciprocity approach. The excitation scheme of the experimental setup was reproduced and the current distribution along the cable directly extracted from the simulation results. The simulation software was CST MICROWAVE STUDIO® (CST® MWS®, Darmstadt, Germany). Nyenhuis et al. [START_REF] Nyenhuis | 2015 Comput. Cardiol. Conf. CinC[END_REF] demonstrate that the measured transfer function with the direct approach depends on the boundaries of the phantom it is measured in. It is also the case for the simulated transfer function based on the reciprocity approach. The transmission line model extracted from the full-wave SEMCAD simulations is a transmission line of the cable in an infinite volume of gel. Therefore the results on the transfer functions are likely to be different.

Solving the inverse problem

As mentioned earlier the transfer function is fully determined by the transmission line propagation constant k and two reflection coefficients through equation 7. The inverse problem was solved for both the experimental and simulated transfer function first letting the two parameters k, and ρ (the reflection coefficient being in our case the same at both ends) free and second by fixing the constant propagation to the value given by the analytical formulas [START_REF] Hertel | The Insulated Linear Antenna-Revisited[END_REF] and letting the reflection coefficient free. In the second case the analytical formulas give also the characteristic impedance of the cable therefore from the reflection coefficient one can deduce the load impedance Z LOAD . The parameters k, ρ and Z LOAD extracted to best fit the experimental and simulated transfer functions were compared to the analytical parameters and the full-wave extracted Z LOAD . The inverse problem was also solved using the finite difference model and not the wave propagation model for comparison.

Resonant behaviour of a capped and uncapped cable in an ASTM phantom as a function of cable length.

Cable configurations

The cable type tested was the same as the one used for the transfer function study. Different lengths of the capped cable and uncapped cable were tested. The cables were placed at a position x=130 mm at the right of an ASTM phantom [START_REF] Astm F | Standard Test Method for Measurement of Radio Frequency Induced Heating On or Near Passive Implants During Magnetic Resonance Imaging[END_REF] when looking from the bottom of the torso. The phantom was positioned head first in a birdcage and the static 1.5 T field pointed to the head of the phantom. The center of the torso was placed at the isocenter of the MRI bore on the x and z axis and at y=-55 mm, the y axis pointing up. The cable was positioned 45 mm under the surface of gel which was 90 mm deep. The uncapped end of the cable where the temperature was measured was always positioned at the bottom of the torso at z=215mm (the z axis pointing to the bottom of the torso). The heating at the electrode was measured for 14 different lengths ranging from 16 cm to 53 cm for each of the capped and uncapped cables. The set of experiments were conducted twice for the capped cable giving an idea of the uncertainty of the measurement results.

Sequence

The sequence run was the Fast Spin Echo sequence described in the ASTM F2182 norm (12) on a 1.5T GE scanner with the number of slices fixed to 42, the transmit gain to 130 and a nex of 2 which makes a total duration of 6 min 34 s. The mean SAR in the ASTM phantom corresponding to this transmit gain is 1.38 +/-0.11 W/kg according to the calorimetric measurements made following the ASTM heating norm [START_REF] Astm F | Standard Test Method for Measurement of Radio Frequency Induced Heating On or Near Passive Implants During Magnetic Resonance Imaging[END_REF].

Full-wave simulation

The exact experiments were reproduced using the full-wave simulator CST MWS. The 1.5T birdcage model used was validated by comparison of the B1 map in the ASTM phantom. The result extracted was the mean loss on a 2 mm cube of gel at the uncapped end.

Transmission Line Model

The same 1.5T birdcage model as for the full-wave simulation was used to simulate the incident field at the position of the cables which was the input to the model. The transmission line model was the one determined from solving the inverse problem on the simulated transfer function fixing k to the analytical values and calculating the best fitting value for the reflection coefficient. The reflection coefficient for the capped end was considered to be one. The wave propagation model with the transmission line parameters will be used to predict the transfer function for each length of cable. The transfer function associated to the simulated incident electrical field allows to predict the scattered electrical field at the electrode. The temperature increase is proportional to the square of the scattered electrical field.

Scaling

A scaling has to be applied corresponding to the calibration procedure for the transfer function [START_REF] Park | Calculation of MRI-Induced Heating of an Implanted Medical Lead Wire with an Electric Field Transfer Function[END_REF]. The scaling factor for the full-wave simulation results and the transmission line model results were calculated such that the error with the experimental results was minimized.

RESULTS

Determination of a transmission line model from a full-wave simulation

Figure 2 shows the current and voltage distribution along the uncapped and capped 20 cm cables extracted from the full-wave simulation with a plane-wave excitation. One can notice the asymmetry induced by the different termination conditions at both ends for the capped cable compared to the uncapped one.

Figure 3 shows the norm and phase of and along the uncapped and capped 20 cm cables extracted using equation 1 and 2. The load impedances model the bare part of the cable for uncapped ends, so the uncapped cable is considered to be situated between -0.09 m and 0.09 m and the capped cable between -0.09m and 0.1m. The results of Figure 3 show that as well as seem to be approximately constant along the cables therefore demonstrating that the hypothesis of a transmission line behavior of a cable embedded in gel holds. If one looks more precisely, one can notice that varies quite notably along the length of the cables. The relative difference between the value of along the uncapped cable compared to the value at the center reaches 30%. For the admittance per unit length , ignoring the singularity at the center due to the small voltage values , the relative difference along the cable can reach 20%.

The load impedance of the bare electrode is calculated by using equation 3 and 4 at the positions z=+/-0.09m. The load impedance for this electrode is found to be 51.3-j 32.7 ohm.

Although the characteristic parameters of the transmission line seem to vary along the uncapped cable, we fixed them to the value at the center for and to the mean value of the values at +/-0.05 m for . Associated to the value of the load impedance, we used them as an input to the finite difference transmission line model. Table 1 sums up the results on the error on the voltage and current distribution for the full-wave simulation extracted model (Model 1) and the one using analytical formulas for Z and Y (13) associated to from the simulation (Model 2).

One can notice that the fact of fixing the characteristic parameters to a certain constant value allows to predict the current and voltage distribution of the full-wave simulation with an error of less than 5%. The parameters extracted from the simulation are close to the parameters given by the analytical formulas [START_REF] Hertel | The Insulated Linear Antenna-Revisited[END_REF], and these allow to retrieve the full-wave distributions with a relative quadratic error of less than 10%.

Figure 3 also shows the amplitude and phase of and for a 20 cm capped cable to be compared to the uncapped case. The results are in accordance. The differences can be attributed to numerical errors.

The load impedance at the capped end extracted from the full-wave simulation was 2203 -j 1969 ohm. Considering that the capped end corresponds to an infinite impedance means that the reflection coefficient is 1. With the full-wave simulation load impedance and the analytical transmission line parameters the reflection coefficient is found to be 0.965-j0.026.

The results concerning the ability of the transmission line model determined from the simulation on the 20 cm uncapped cable to retrieve the current and voltage distribution of the full-wave simulation for a 30 cm uncapped cable and a 20 cm capped cable are summarised in Table 2.

For both the 20 cm capped and 30 cm uncapped cable, the full-wave extracted model and the mixed analytical/ full-wave model predict the current and voltage distribution of the full-wave simulation with an error on the distribution of less than 10%. Fixing the reflection coefficient of the capped end to 1 makes the error on the current distribution increase to slightly above 10%, the voltage error remaining under 5%. The mixed analytical/full-wave model gives results as good as the full-wave extracted model.

Validation Comparison with experimental and simulated transfer functions

Direct comparison

Figure 4 shows the experimental and full-wave simulated transfer function. The green line shows for comparison, the transfer function predicted by the transmission line model with the analytical propagation constant and the full-wave load impedance. The phase distribution fits nearly perfectly. The amplitude distribution has the same global trend but one can observe a shift of the local maximum and minimum towards the distal end.

Solving the inverse problem

The red line of Figure 4 corresponds to the transmission line model extracted by best fitting the simulated transfer function according to equation 7 using the analytical propagation constant. Table 3 shows that the transfer function perfectly fits a transmission line model but with a load impedance slightly different from the one determined from the full-wave simulation.

Table 3 also sums up the results on the propagation constant k, the reflection coefficient ρ and the load impedance Z LOAD of solving the inverse problem from the experimental and simulated transfer functions. The fit leaving the propagation constant k free gives a result very close to the analytical propagation constant for the simulated transfer function and slightly different for the experimental transfer function. The finite difference model also allows to solve the inverse problem and gives results comparable to the wave propagation model.

Resonant behaviour of a capped and uncapped cable in an ASTM phantom as a function of cable length.

Figure 5 shows the resonant behavior of the capped and uncapped cable placed in an ASTM phantom in a 1.5T birdcage as a function of length. The full-wave simulation results (green line) coincide well with the results predicted by the transmission line model extracted from the simulated transfer function (black line). There is a shift of a few cm of the resonant length. The trend of the experimental data is the same as the trend of the full-wave model and the two different transmission line extracted model. There is a high uncertainty of the temperature increase measurement for higher temperature. This is due to the fact that for higher temperature increases the temperature gradients around the electrode are higher inducing a greater sensitivity of the measurement on the temperature probe position.

The results on the resonant behavior of the uncapped are also consistent. Especially the shift of the resonant length between the capped and uncapped cable is correctly predicted by the different models. The sharper increase of the temperature elevation with the length for the capped cable compared to the uncapped cable is also correctly predicted.

The full-wave simulation results slightly differ from the results predicted by the transmission line model extracted by solving the inverse problem on the simulated transfer function. One has to recall that the simulate transfer function depends on the position of the cable inside the phantom. The fullwave simulations were made with the cable placed in the position it was in the experiments which is different from the position used for the transfer function simulation. This probably explains the difference in the predicted temperature elevation as a function of the length of the cable.

DISCUSSION

Full-wave simulations

It has been shown using full-wave simulations of an insulated cable embedded in gel that the current and voltage distribution along the cable do follow approximately a transmission line behavior. Supposing it strictly follows a transmission line behavior still gives results consistent with the fullwave simulation.

From a transfer function model to a transmission line model

It was shown that simulated and experimental transfer function can fit nearly perfectly a transmission line model showing the underlying assumption under the transfer function model is indeed a transmission line model [START_REF] Park | Calculation of MRI-Induced Heating of an Implanted Medical Lead Wire with an Electric Field Transfer Function[END_REF][START_REF] Feng | A Technique to Evaluate MRI-Induced Electric Fields at the Ends of Practical Implanted Lead[END_REF]. Therefore the behavior of a simple cable submitted to the radiofrequency field of a 1.5T MRI can be entirely described by a physical set of parameters instead of a mathematical transfer function model. Especially as the termination conditions play a crucial role [START_REF] Langman | Pacemaker Lead Tip Heating in Abandoned and Pacemaker-Attached Leads at 1.5 Tesla MRI[END_REF], it can link the heating at the bare electrode to the reflection coefficient at the distal end. More generally, this can help in the design of safe cables as this model enables to link directly the geometrical and physical properties of the cable to the heating risk in MRI for any incident field. For example, it was possible to predict the experimental resonant behavior as a function of length of a capped and uncapped cable from the simple set of very few parameters without having to evaluate experimentally or with full-wave simulations the transfer function for each length.

Load impedance model

This work combined the work by Acikel et al. [START_REF] Acikel | Modeling of Radio-Frequency Induced Currents on Lead Wires during MR Imaging Using a Modified Transmission Line Method[END_REF] which extended the work by King (4,[START_REF] King | The Many Faces of the Insulated Antenna[END_REF] for a cable excited by a field distribution with the simple model of the bare electrode proposed by King (4,[START_REF] King | Cylindrical Antennas Immersed in Arbitrary Homogeneous Isotropic Media[END_REF]. Instead of being calculated analytically considering it as a monopole as in ( 8), the load model was first extracted from full-wave simulation data then determined from solving the inverse problem from transfer function data. The results showed that the load was not exactly the same. The most relevant method in the future seems to be to get the load or reflection coefficient from the experimental or simulated transfer function. Indeed the high field distribution around the bare electrode will be very sensitive to numerical errors. Further work would be to compare for simple cables, the load model determined from the transfer function and the analytical value from King [START_REF] King | Cylindrical Antennas Immersed in Arbitrary Homogeneous Isotropic Media[END_REF].

Transmission line model for pacemaker leads

One has to remind oneself that the transfer function concept was thought mainly for cables one cannot simulate. Therefore it is important to be able to get the information from experimental data for example, an experimentally measured transfer function. There are nevertheless two problems arising for leads of typical medical devices such as pacemakers. If someone wants to reduce the heating at the electrode by varying the reflection coefficient at the distal end meaning at the case, it is necessary in the future to develop tools to directly evaluate this coefficient either theoretically or experimentally without measuring the new transfer function each time. Nevertheless, the transmission line will still give a better physical insight of the behavior of the pacemaker lead in the radiofrequency field. A second problem arising comes from the fact that pacemaker leads are usually made of two coaxial helicoidal wires. The transmission line model developed here was for cables made of one wire only. Further work needs to be carried on to see if it is possible to consider a coupled transmission line model for cables made of multiple wires which will be a model with supplementary coupling parameters such as a distributed mutual inductance and a distributed mutual capacitance.

CONCLUSION

A new transmission line model was developed for simple cables made of one wire submitted to a radiofrequency excitation field at 64 MHz and embedded in tissue properties imitating gel. The termination conditions were modeled by a simple passive load impedance that has a physical meaning. It was shown that it has a strict equivalence with the transfer function model and it brings physical insight into it. This physical insight can help in the design of safe cables in MRI. 3 Results on the propagation constant k, the reflection coefficient of the uncapped end ρ, and the load impedance of the uncapped end solving the inverse problem of getting these parameters from the simulated and experimental transfer function of an uncapped 53 cm cable.

  under the following matrix form inc . The submatrices Dz1, Z, Y and Dz2 have a dimension NxN with N the number of discretization points along the cable. The boundary conditions are implemented at the Nth and last line of the whole matrice. The value that is directly related to the scattered electric field at the electrode is the voltage at the load modeling the electrode. To solve the finite difference model we inversed the square matrix. The transfer function seen from the left electrode corresponds to the N-1 values of the first line of the inverse matrix, and the transfer function seen from the right electrode corresponds to the N-1 values of the Nth line of the inverse matrix.

Figure 1

 1 Figure 1 Setup for the measurement and simulation of the transfer function.

Figure 2

 2 Figure 2 Distribution of amplitude and phase of current and voltage for an uncapped 20 cm cable (solid blue line) and a

Figure 3

 3 Figure 3 Amplitude and phase of the impedance per unit length Z and admittance per unit length Y as a function of position along the cable if one makes the transmission line model assumption. The blue solid line represents the results for an 20 cm uncapped cable, the green dashed line represent the results for a 20 cm capped cable isolated at the position +0.1m.a) amplitude of impedance per unit length, b) phase of impedance per unit length, c) amplitude of admittance per unit length, d) phase of admittance per unit length.

Figure 4

 4 Figure 4 Experimental and different models amplitude and phase distribution of the transfer function of a 53 cm uncapped cable. a) Transfer function normalized amplitude, b) Transfer function Phase. The electrode the heating will be evaluated at is at position d=0 cm.

Figure 5

 5 Figure 5 Temperature increase at the uncapped end of capped and uncapped cable placed in an ASTM phantom in a 1.5T MRI as a function of length of the cable. The prediction of the transmission line models best fitting the simulated and experimental transfer function are compared to experimental data. a) capped cable, b) uncapped cable.

Table 1

 1 Error of the transmission line models with constant parameters on the full-wave current and voltage distribution for a 20 cm uncapped cable.

			Z	Y		ZLOAD	Error I	Error V
	Model 1 (full_wave)	91.5+j 324.6 0.0071+j 0.0972	51.3-j 32.7	3.17%	2.97%
	Model 2	85.1+j 361.8 0.0073+j 0.0939	51.3-j 32.7	6.02%	7.56%
	(analytical,	full-wave)
			20 cm capped cable	30 cm uncapped cable
			Error I	Error V	Error I	Error V
	Model 1(full-wave)	8.34%	3.64%	3.04%	5.2%
	Model 2 (analytical, full-wave)	6.82%	3.37%	7.99%	5.3%
	Model 1 (full-wave/ infinite load 1 MΩ)	12.65%	3.29%	x	x
	Model 2 (analytical/ infinite load 1 MΩ)	11.4%	4.69%	x	x

Table 2

 2 Error on the voltage and current distribution of a 20 cm capped and 30 cm uncapped cables of the transmission line model with constant parameters extracted from full-wave simulation on a 20 cm uncapped cable. The reference distributions are from full-wave simulations.

Table

ACKNOWLEDMENTS

The author thanks Julie Kabil, my aunt Colette Thomson for correcting the English. And finally Cédric Pasquier for the funding through Region Lorraine and FEDER.


Naturalistic decision-making in expert badminton players Naturalistic decision-making

Keywords: Naturalistic decision-making, Expert performance, Badminton

This paper reports on a study of naturalistic decision-making in expert badminton players. These decisions are frequently taken under time-pressured conditions, yet normally lead to successful performance. Two male badminton teams participated in this study. Selfconfrontation interviews were used to collect data. Inductive data analysis revealed three types of intentions during a rally: to maintain the rally; to take the advantage; and to finish the point. It also revealed eight types of decision taken in this situation: to ensure an action; to observe the opponent's response to an action; to realize a limited choice; to influence the opponent's decision; to put pressure on an opponent; to surprise the opponent; to reproduce an efficient action; and to play wide. A frequent decision was to put pressure on the opponent. Different information and knowledge was linked to specific decisions. The results are discussed in relation to research that has considered naturalistic decision-making.

Introduction

In the training domain, it is considered that athletes apply decision rules that are implemented during the training (e.g. [START_REF] French | Adaptations in response selection processes used during sport competition with increasing age and expertise[END_REF]McPherson 1999, McPherson and[START_REF] Mac Pherson | Tactics, the neglected attribute of expertise[END_REF]. This rules contain one or more conditions that involve a particular action ('if… then'), (e.g. [START_REF] Williams | Knowledge in sport: a by-product of experienced or a caracteristic of expertise[END_REF]. These ideas are supported by classical models of cognition. These models insist on information processes and knowledge bases (e.g. [START_REF] Chase | Perception in chess[END_REF]). This focus on information processes and knowledge bases has had two consequences: the former has focused on human functioning and the latter has assigned task concepts. In these models, the human system is compared to a computer-metaphor that processes information, handles knowledge and symbols, and relies on diagrams and algorithms. These models focus on quantitative parameters, such as reaction time, movement time or error (in comparison to a target). Therefore, the classical cognitive approach has used a mechanistic approach that determines that action is a direct result of cognition [START_REF] Sanders | Energical states underlying task performance[END_REF]).

The human system contains a knowledge base gained from learning and experience, which is available for use to direct action. Therefore, an individual can be considered as a rational and omniscient actor who possesses all the useful knowledge required to act.

According to rational decision-making models (e.g., [START_REF] Pitz | Risk taking, design and training[END_REF]) individuals assess the usefulness of each possible response in order to maximize behaviour. It is proposed that one approach is optimal and corresponds to each situation; it belongs to an individual repertoire and contains rules and strategies that have to be learned and used advisedly. These strategies allow individuals to anticipate actions. They rely on knowledge, competencies and specific game analysis. According to this model an athlete's strategies are considered to be preregulated; they function like algorithms and result in a minimum number of errors (e.g., understanding error in a current situation, choice error or realization error). An athlete's expertise, which results from a hyper-adaptation to the task, limits realization error (Ericcson 1996), but they cannot avoid decision-making error or understanding error. Decision making, according to this model, seems not to be able to adapt to a current situation.

The second consequence focuses on task concepts. This has typically discredited the observable components of human behaviour and consequently has not taken into account its dynamic and evolutionary nature [START_REF] De Keyser | Les systèmes de report d'incidents [Systems of incidents carried forward[END_REF]. Observation of sporting situations has, therefore, lead to doubts concerning rational decision-making. Expert sports decisions do not allow for control over situations where the athlete is fully committed. Frequently, their control is limited; in the sport of badminton, it leads to a continuation of the rally but not to the ultimate success in winning it. Decisions can vary markedly between situations and can change considerably with experience, space, and biological constraints. One athlete may not make the same decision every time he/she encounters a similar situation. Similarly, decisions can vary between individuals. In these conditions, the validity of rationalistic decision making can be questioned to aid in the study of the heuristic impact of naturalistic decision-making models.

Naturalistic decision-making

Natural situation

In high-level sport performance analysis, it is sometimes difficult to define task components.

For example, De [START_REF] De Keyser | Les systèmes de report d'incidents [Systems of incidents carried forward[END_REF], states that the concept of 'task' does not appear sufficient to fully understand what is happening in any given situation. It does not take into consideration the context, or the individual who performs the task (with limits and work constraints). The 'situation' concept seems more appropriate to an understanding of individual activity during the course of action. Situations can be termed static (they evolve with individual activity) or dynamic (they evolve with non-individual factors). [START_REF] Amalberti | Analyse des activités cognitives en situations dynamiques: pour quels buts ? Comment ? [Cognitive activities analysis in dynamic situations : for which goals ? How ?[END_REF] have considered the situational context through the parameters of temporality, uncertainty and control. Temporality refers both to the available time to act and to the evolution of a part of the situation g.i.e., its process. Control can be partial or total. Uncertainty can be spatial, temporal or factual. In sport, these 'natural' situations consist of situations arising during coaching, training or competing. Research must, therefore, consider task contingency, as well as consideration of the continuous processes which are used in naturalistic situations. Naturalistic research uses a situation-action perspective [START_REF] Suchman | Plans and Situated Actions[END_REF], and focuses on how individuals match their activity to a particular context. Action is not regulated by an a priori plan.

Studies concerned with sport have focused on: the temporal components of coaches' actions (e.g., Sève and Durand 1999); the temporal and conceptual organization of interaction between coaches and athletes during competition (e.g., d'Arripe-Longueville et al. 2001); knowledge building during the course of an action (e.g., Sève et al. 2003); action organization according to athlete performance level (e.g., [START_REF] Hauw | Elite athletes' differentiated action in trampolining: a qualitative and situated analysis of different levels of performance using retrospective interviews[END_REF]; goal differentiation in consideration of competition type (e.g., [START_REF] Saury | L'action d'un entraîneur expert en voile en situation de compétition: étude de cas [action of expert competition sailing coach: case study[END_REF]; and naturalistic decision-making [START_REF] Macquet | Le contrôle des situations chez les volleyeurs experts: informations perçues et connaissances mobilisées ou construites[END_REF]. These studies considered opportunistic and situational cognition and focused on local decisions, considered in a diachronic and non-synchronic way. The selection of verbalized decisions depended upon their meaning for the athlete; decisions do not naturally follow one another over time. The present study aimed to highlight a synchronic and a diachronic issue.

Decision in natural situation

Previous work in the area has considered emergency decision making in the domains of nuclear power, transportation, industrial management, military operations, and emergency services. These studies are relative to two approaches which are developed independently, but are nevertheless very similar. The model of Naturalistic Decision Making (e.g., Klein et al. 1993, Zsambok andKlein 1997) focuses on: ill-structured problems; dynamic and uncertain conditions; ill-defined and evolving goals; action and feed-back loops; temporal pressure; high risk situations; high numbers of individuals committed to a situation; and organizational standards. The model of sufficiency has been developed by [START_REF] Amalberti | La conduite des systèmes à risques [The control of risky systems[END_REF][START_REF] Amalberti | La maîtrise des situations dynamiques [Control of dynamic situations[END_REF], who examined ill-defined problems and dynamic problems. These models refers to decisionmaking. All the research above considers decisions which were adapted to the situation and the person. Adaptive decision-making allows control over part or all of a situation. In contrast to cognitive conceptions, adaptive decision-making is more satisfying for the individual and corresponds to actual individual competencies. Adaptive decision-making refers to the athletes' own goals for a situation in a specific context. Under these conditions, decisions do not seem rational, but do seem appropriate and sufficient for the decision-maker. This sufficiency is linked to limited capacities or to contingent rationality [START_REF] Simon | Models of Man: Social and Rational[END_REF].

According to [START_REF] Amalberti | De l'erreur humaine au risque : évolution des concepts en psycho/ergonomie [From human error to risk : concepts evolution in pyschology/ergonomics[END_REF], an individual is not a passive agent of this rationality and organizes behaviour to cope cognitively. The athlete takes risks and proceeds by anticipation to reduce situational complexity. The athlete commits to a more autonomous level of behaving instead of a reflexive and logical process to save resources.

Naturalistic decision-making therefore focuses on a continual process in a natural situation. It takes into account task contingency. It considers both the situation arena (objective spatial and social environment), and the product of an activity setting, that part of environment marked by action, see [START_REF] Lave | Cognition in Practice[END_REF]. It is argued here that this framework will affect the continual process that is used in naturalistic situations.

Decision-making process

Unconditional, rational analysis concerning an action is not always possible or desirable. An individual may function in such a way so as to achieve an optimal cognitive working mode. In this case, optimal implies a situation of compromise, allowing goal(s) attainment in a dynamic way with appropriate and effective performance [START_REF] Amalberti | De l'erreur humaine au risque : évolution des concepts en psycho/ergonomie [From human error to risk : concepts evolution in pyschology/ergonomics[END_REF]. This concept suggests three ideas. The first conforms to a notion of sufficiency; it corresponds to a response that is adapted to the environment according to an individual's goals, knowledge and competence. An individual athlete cannot cognitively and consciously understand all actions and behaviours. Similarly, the athlete cannot physically perform all actions; an optimal response that is the product of these abilities must be produced. The second idea relates to the concept of dynamic adaptation. It comprises two aspects: attained performance level over time; and acceptable global performance level at any given moment (final result or local/intermediate result). The third idea refers to meta-knowledge. This concept considers the management of acceptable and accepted risks.

The notion of sufficiency [START_REF] Amalberti | La maîtrise des situations dynamiques [Control of dynamic situations[END_REF]) considers the capability for understanding, decision-making, control and execution. These factors contribute to a global plan and activity emerges through environmental coupling and the use of natural affordances [START_REF] Gibson | The Ecological Approach to Visual Perception[END_REF][START_REF] Suchman | Plans and Situated Actions[END_REF]). This sufficiency framework therefore suggests a dynamic and contextual building of cognitive activities. It aims at a more effective response to situation evolution and according to limited individual resources (Amalberti and[START_REF] Amalberti | Analyse des activités cognitives en situations dynamiques: pour quels buts ? Comment ? [Cognitive activities analysis in dynamic situations : for which goals ? How ?[END_REF][START_REF] Hoc | Analyse des activités cognitives en situations dynamiques: d'un cadre théorique à une méthode [Analysis of cognitive activities in dynamic situation : from a theoretical frame to a method[END_REF]. To manage complex and dynamic situations, it would not be ideal to have a solution that was completely predetermined [START_REF] Amalberti | La sécurité écologique et la maîtrise des situations : concepts et stratégies mis en jeu par les professionnels pour assurer leur propre sécurité et celle du système [Ecological security and situation control : concepts and strategies used by professionals to ensure their own security and those of the context[END_REF]. Individuals require compromises to deal with situations presenting conflicting dimensions. Action must be opportunistic rather than rational. Decision-making should appear as a continual process that is coupled to the environment. It could be compared to a process of natural drift: decision-making would continue until it would no longer be satisfying for the individual to be committed to a situation. In these conditions, the individual would 'wait' for situation evolution, this would allow further understanding in a way sufficient to act. Behaviour would then proceed to risks limits that would represent an afforded zone (viz. the concept of meaningful as used by [START_REF] Gibson | The Ecological Approach to Visual Perception[END_REF].

The research that has considered the decision making process shows two key stages.

The first relates to situation assessment: the definition of the problem; awareness of risks; and of the time available. The second is the choice of a course of action that depends upon available situation possibilities. In consideration of these stages, Hollnagel (1998) has proposed concern for the cyclic character of human cognition and on the difference between observation and inference. In his context control model, Hollnagel hypothesizes that human performance is the result of a controlled use of competencies according to a situation not the result of response sequences that are predetermined by events. The control can be described in reference to a continuum, going from no or limited control over events, to complete control. [START_REF] Hollnagel | Cognitive Reliability and Error Analysis Method[END_REF] proposed four modes of control: scrambled control; opportunistic control; tactical control; and strategic control. In scrambled control, the choice of action is unpredictable or haphazard. The individual does not consciously think, or thinks only a little about what to do. This mode appears when task demands are high, when the situation changes in an unexpected way, when it is unfamiliar, or when there is a loss of situational awareness.

The extreme case of scrambled control is a panic situation. Opportunistic control involves action that is determined by the salient features of the environment, instead of more stable intentions or goals. There is little planning or anticipating, because context is not clearly understood or because time pressure is high. Individual behaviour is lead by perception of the interface of dominant features, experience, and habits. In tactical control, behaviour is based on planning and on the use of rules or known procedures. The latter is limited to a particular context. If planning is often used, then performance refers to rule-based behaviour. In strategic control, the individual takes into consideration the global situation, according to long term goals. This mode is predicted to lead to more efficient performance and is influenced by knowledge and skills. The person, such as a high-level athlete, moves from one control mode to another depending on the perceived characteristics of the situation. In order to predict the expected level of performance, it seems important to know how actions are chosen and subsequently performed.

Decision and coordination

This review has focused on research that has used motor function limited to movement of handles and joysticks or the press of button. There is a limited amount of research that has considered some of the over-learned, highly coordinated skills that are present in high-level sport. In these studies, coordination will have little influence on decision-making. According to Hollnagel's model (1998), elite sport behaviour corresponds only to the tactical control mode, opportunistic mode, or strategic mode. Coordination can, therefore, affect decisionmaking if the athlete is involved in an unfamiliar situation and if coordination is not adapted to the situation. In this case, the athlete has to create a new coordination pattern and use scrambled control.

This study aimed to determine how high-level performers make decisions during the course of an action. Decision-making was studying in a synchronic and in a diachronic way.

The research also considered decision-making within the temporal constraints of the match to reveal any similarities and differences between situations and to understand features structuring decision-making.

Method

Participants

Four male expert badminton players participated to this study. All were ranked in the top 23 players in the world at the moment of the competitions. They completed a written informed consent form prior to participating in the experiment and the study was approved by the local ethics committee.

Data collection

Two types of qualitative data were collected: data observed and recorded during competitive matches; and data elicited during self-confrontation interviews.

3.2.1. Data observed and recorded during the competitive matches. Two matches were recorded from international qualifying tournaments for the Athens 2004 Olympic Games. A digital video camera was placed within the stands of the court, in the middle of the court's width. The researcher noted other forms of data that were considered important for the research and were not recorded directly by the camera.

Data collected during self-confrontation interview. Verbal feedback was collected

after each competition during the self-confrontation interview. These were conducted with both team players. The self-confrontation interview is a method developed by Von Cranach and Harré (1982). It aims to elicit ongoing cognitions. The athlete is presented with their activity as soon as possible after the recording. Players were asked to describe and explain their own behaviour and activity. The duration of each interview was limited to 150 minutes.

Experienced coaches proposed that the first four actions were judged the most significant in the course of the rally, since they typically determined the outcome of the rally. Therefore, players had to explain the decisions behind the first four actions of each rally. These choices were constrained by the available time for the interview. Only the first two sets were retained because of time constraints (badminton players have to win two sets to win the match). The interviews were conducted during two periods to avoid tiredness and loss of concentration.

Each period covered one set. Interviews took place during the third and the fifth day after the match for Team 1 and during the match day and third day after the match for Team 2. Players could stop the video at any time to ask questions. Interviewer questions were concerned with specific actions and sought to avoid general answers. For example, a participant might be asked "what do you want to do here?" or "what did you see here?" "what were you thinking about there?". Questions were limited to decision-making; they did not attempt to cover the participants' perceptions regarding the ultimate success of their decisions.

Data processing.

The interviews were transcribed verbatim and provided for the participants to confirm the validity of transcription. The transcripts were analyzed through a process of inductive analysis containing three phases: identification of extrinsic data (score, description of the situation) and intrinsic data (short accounts of decision, perceptions and knowledge used in decision making); labelling decisions (the situation referred to) and intentions (attributable to the rally); labelling perceptions and knowledge. The coding reliability procedure was assessed by two independent researchers in sport ergonomics. The intentions and decisions labels were judged by two coaches.

Results

Qualitative analysis produced 312 short accounts (145 for Team 1 and 167 for Team 2).

Decisions and subsequent intentions, features that structured decision-making and decisionmaking efficiency related to behaviour and action were analyzed.

Players' decisions and subsequent intentions during the rally

Local decisions (defined as the course of action) and intentions (attributed to the progression of the rally) were labelled. Three types of intentions were identified: 'to maintain the rally'; 'to take an advantage to build the rally'; and 'to finish the point'. Eight types of decisions were elicited. The sub-category 'to maintain the rally' incorporated decisions such: to ensure his action; to observe the opponent response to his own action; to realize a limited choice. The intention 'to take an advantage to build the rally' referred to influence opponent decision; to put pressure on opponents; to surprise opponents; to reproduce an efficient action; and to play wide.

Intentions and decisions labelling can be explained through the players' verbal responses. Intention to maintain the rally meant that the player wanted to continue or engage in the rally, he aimed to delay any attack but he did not want to lose the point. The decision to ensure his action signified a neutral action, the player did not want to take any risk with the shot. For example: "I aimed to produce a classical serve" (match 1, set 1, decision 75). The decision to observe an opponent response to his action meant the player aimed to test the Naturalistic decision-making 14 opponent's abilities. In another example the decision was, "to produce a short serve and observe the return" (match 1, set 1, decision 9). The decision to realize a limited a choice meant decision-making was made under pressure to keep the rally going, "I hadn't any other choice than to do a high lift" (match 2, set 2, decision, 27).

The intention to take advantage to build the rally meant that the player tried to go on the attack, to be in a good position to finish the point later in the rally. The decision to influence the opponent's decision was intended to guide the opponent's action in a particular way or to dissuade the opponent from making a specific action later in the rally. In a further example a player referred to an action, "to take the attack by playing under the tape to make them to lift the shuttle" (match 2, set 2, decision 32). The decision to put pressure on the opponents intended to increase the shuttle's speed and accuracy and players used terms such as to, "block at net to take the attack" (match 1, set 2, decision 15). For the decision to surprise opponents the behaviour was designed to create an unexpected trajectory for the opponent and led to phrases such as to "cross my serve to surprise him" (match 1, set 2, decision 21). Reproducing an efficient action referred to a previous effective action and players commented "to serve by doing my best" (match 2, set 2, decision 48). By playing wide, the players intended to shoot along alleys or the back of the court. In describing these conditions, players used phrases such as, "play wide by shooting in the alley" (match 1, set 1, decision 2).

The intention to finish the point referred to winning the rally by playing with speed and accuracy. For example, players commented that they should, "finish the point by shooting to half-court with the most powerful and steepest trajectory possible" (match 1, set 1, decision 55). This intention refers to both the current and a future situation because the intended result was to change the situation and finish the rally.

Naturalistic decision-making 15

Analysis of the players' intentions revealed that the players often intended to take advantage to build the rally (Figure 2). Analysis of decisions revealed similarities between players. For example, the three most frequent decisions were: to put pressure on opponents; to surprise opponents; and to influence opponent's decision. However, decisions concerning observing or ensuring an action were seldom made.

For Team 1, decisions were more varied in the first set than in the second set. For Team 2, there were fewer differences between the sets. Results suggested that, at first, Team 1 altered their game to test their opponents, and to gain an understanding of their play. They then moved to decision-making that was more focused on the outcome of the rally. Team 2 tried to impose their style of game on their opponents and as long as their decisions appeared effective they continued with this style. When they became less effective they found other solutions to finish the rally (from decision 30 after 5 min play for set 1; and from decision 20 after 3'19 play for set 2). Team 2 did not aim to observe opponent's reactions to their actions.

A particular local decision, in situation, corresponded to several categories of decision. For example, a straight clear was used with the aim of realizing a limited choice; to produce an efficient action to play wide and to influence the opponent's action.

'[insert figure 1 about here]'

Decision-making in prototypical situations

A number of typical situations were observed. These were serve, return of serve and third or fourth shot. Analysis of decision-making in prototypical situations revealed similarities and differences. These are discussed below.

Serve.

Results, shown in Figure 3, indicated a predominance of intention to build an advantage in the rally. The most frequent decision was concerned with attempting to surprise opponents. This was more frequent for Team 1 than Team 2, and more frequent in the second set than in the first set for both of teams. A further trend revealed two decisions: to reproduce an efficient action and to influence the opponent's decision. Decisions were more varied for the first set for Team 1, and for the second set for Team 2.

'[insert figure 2 about here]' 4.2.2. Return of serve. The results, shown in Figure 4, indicated the players' intention to build an advantage. The most common decision was to put pressure on the opponents. This was more frequent in the first set than in the second for both teams. Decisions were more varied for Team 2 than for Team 1. They also achieved more returns. Team 2 tried to finish the rally and to influence and to surprise their opponents. They were more varied in the first set than in the second set. Team 1, essentially, aimed to put pressure on their opponents. '[insert figure 3 about here]' 4.2.3. Third or fourth shot. Figure 5 highlights the player's attempts to build an advantage.

The second intention varied between teams. Team 1 aimed to maintain the rally, whereas Team 2 intended to finish the point. The most common decision was to put pressure on the opponents. This trend was more pronounced in the second set for Team 1. For Team 2 it was more marked in the first set. In addition, differences were observed between other trends. For example, Team 1 attempted to realize a limited choice for their opponents, especially during the second part of the first set. They also aimed to keep play wide. Team 2 aimed to surprise their opponents, and also to realize a limited choice. The decisions of Team 2, therefore, showed greater difference.

'[insert figure 4 about here]'

The structured components of decision making

Inductive analysis of the transcripts highlighted two categories; information and knowledge.

These can be subdivided into eleven sub-categories. The sub-category 'information' incorporated: player's actions and position; partner's actions and position; opponent's action and position; and the shuttle trajectory. Knowledge referred to the player's abilities, behaviours, efficiency in action and trust; to the partner's abilities; and to the opponent's abilities (e.g., "he always serve to the same place" match 1, set 1, decision 36). Knowledge also pertained to the individual and collective or situational rules (e.g., "the only way to stay in the rally is to lift" match 1, set 1, decision 5). In addition, a preceding event was also included in the sub-section 'knowledge'(e.g., "it's the same situation as before", match 2, set 2, decision 87), as was expectation about the opponent's action (e.g., "I expected a serve to the T position", match 2, set 1, decision 1), and consequences of the player's actions (e.g., "it leads to move the player away from the net and causes the back player to have to lift and give the attack back" match 1, set 1, decision 38).

The most frequent associations were found to be between decisions and some structured components of decision-making. No evidence was found for rare or infrequently made decisions. Sometimes, there were differences between teams. The decision to ensure his action was rarely taken by Team 1. However, Team 2 required information about the player and his opponents. In the case of the decision to observe the opponent's response to his own action, Team 1 often reported rules (especially individual-based rules). Whilst in relation to realizing a limited choice, the large majority of information referred to the player himself, with knowledge concerning rules about a situation (e.g., "I am in late, my only solution is…").

Naturalistic decision-making 18

To influence the opponent's decisions, the majority of responses were directed at the opponents; they concerned knowledge for Team 1 and information for Team 2. Fewer responses referred to the consequences of actions for Team 1, and (collective) rules for Team 2. The decision to put pressure on the opponent was indicated by information associated with the player and his opponents and the majority of knowledge was assigned to expectations.

Knowledge concerning consequences and collective rules was often reported by Team 1, but not by Team 2. In order to surprise opponents, the decisions revealed a large amount of information relating to opponents and knowledge concerning expectations. Team 1 also verbalized knowledge about a preceding event, whereas this was seldom the case for Team 2.

In justifying their decision to reproduce an efficient action, Team 1 often reported knowledge about individual rules, whereas Team 2 referred to previous events or to a player. With reference to playing wide, much information was attributed to the player and his opponents.

Team 2 frequently reported decisions relating to shuttlecock trajectory, unlike Team 1.

Decisions regarding finishing the point were linked to information concerning the player and his opponents.

Discussion

The results of this study showed that players did not try to immediately win a rally. In most cases they created the optimal conditions to end it successfully. The findings will be discussed from three perspectives: the situational understanding for building the rally; the contextual building of the rally; and the implications for coaching.

Situational understanding to build the rally

The results showed that situational understanding was progressive. It depended on the available information at different times in the rally (e.g., position, behaviour of the players, shuttle trajectory). This ability to quickly take account of evolving features of the situation corresponds well with the concept of situation awareness [START_REF] Wickens | Workload and situation awareness: an analogy of history and implications[END_REF][START_REF] Endsley | Toward a theory of situation awareness in dynamic systems[END_REF][START_REF] Endsley | Designing for situation awareness[END_REF]. The environment is continuously assessed with the aim of including this information in the player's actions. Understanding appeared to be dynamic and contextual. It was linked to uncertainty about the environmental evolution. The player tried to understand the situation. When this understanding seemed sufficient for the player to act, he made a decision whilst still continuing to assess the environment. The situation was considered through the coupling between himself and all other players. Players showed that they had a shared knowledge and understanding of each other's competencies and they took account of these features for decision-making. Decisions related to realizing a limited choice, putting pressure on an opponent, or observing an opponent's response to his action showed a shared knowledge and understanding between partners. Decisions related to surprising opponents were linked to a previous event or to expectations about the opponent's action. Shared knowledge and competencies correspond to a common frame of reference [START_REF] Hoc | A cognitive approach to cooperation in dynamic situations[END_REF] which takes account of prescriptive data available from the players' behaviours and actions. It anticipates changes according to the player's own features (e.g., competencies, knowledge) and situations (e.g., specific previous situations either practiced or observed). The frame of reference can also evolve according to significant events (e.g., earlier situations, player competencies). A change of decision during on-going play suggests that the reference values evolve from one momentary situation assessment to another. The reference value for a future situation is not a random fluctuation around a constant, predetermined strategy. Instead it depends on an antecedent reference value; the successive situational assessments created from the player's experience. The decision does not seem to depend on a constant component of the studied system, such as a personality or predetermined strategies. These findings are consistent with those obtained for decision-making situations in volleyball [START_REF] Macquet | Le contrôle des situations chez les volleyeurs experts: informations perçues et connaissances mobilisées ou construites[END_REF]). In the volleyball study players employed knowledge of current and previous matches, concerning teammate and opponents competencies, with the aim of increasing the competition.

Analysis of the intentions and decisions in the present study revealed the players took account of various time-based factors: situation; rally; and set. Intentions were primarily concerned with issue relating to the rally and set. Whereas, decisions were strongly linked to situation control. Knowledge categories showed an emphasis on these time-based issues too.

For example, players took account of past events, current player competencies and rules, and anticipated future expectations and action consequences. The use of various temporal-based decisions is consistent with results found in a volleyball context [START_REF] Macquet | Le contrôle des situations chez les volleyeurs experts: informations perçues et connaissances mobilisées ou construites[END_REF], where decision and intention categories were found to be similar to the present study. Decisionmaking clearly depends on previous, current and anticipated future events. The reference to past events is also consistent with data from the sport of table tennis (Sève et al. 2003). In the process of decision-making, players use knowledge that has been built up during the match and/or earlier matches. They make links between current and past situations to produce an efficient decision for any given moment.

Player's assessments of the situations exhibited differences in understanding. Some situations were poorly understood, or were understood too late (because of the unexpected evolution of the current situation). In these cases, the player tried to adapt by also performing unexpectedly. Where the situation had a higher time length (e.g., the serve), situational assessment was easier. In all cases, the player did not seem to reach a complete understanding of any given situation. He focused on the salient features for him, but may have missed other opportunities. Information linked to players was often verbalized, unlike information Naturalistic decision-making 21 associated with shuttle trajectory that was understood through opponent behaviour information.

There were also differences in explained knowledge which were more numerous and more varied when the time available to act was greater. In emergency situations the player made decisions from a limited repertoire, these were based on the salient features about the player himself and rules about the situation. In less time-constrained situations, previous or constructed knowledge was more common and more varied. For the decisions made to influence the opponent's decision, knowledge about the opponents, expectation, action consequence, and rules were used. These results are also consistent with those seen in [START_REF] Macquet | Le contrôle des situations chez les volleyeurs experts: informations perçues et connaissances mobilisées ou construites[END_REF] volleyball study. When time pressure was perceived to be high (for example in a defensive situation), the decision-making of volleyball players proceeded essentially from information. When the available time to act was longer, the players verbalized more knowledge-based decision-making. The difficulties of situation assessment under high temporal pressure are consistent with the idea of sufficient understanding to act in a dynamic situation [START_REF] Amalberti | La maîtrise des situations dynamiques [Control of dynamic situations[END_REF]. Players do not try to understand everything to do with a situation. Instead, they attempt to control their lack of understanding by focusing on salient features that will help them to make the most appropriate decision.

Contextual building of the rally

The results indicated various decisions made during the course of the match. They can be categorized as belonging to different levels. The first can be attributed to player verbalizations of decision-making. The player indicated both his decision and in the technical and tactical gesture how he would build the rally (e.g., to block at the net). The player also noted the expected effects of the situation evolution. This was seen at a very local level. The second level related to a desired objective to manipulate the current situation in order, for example, to surprise the opponent. The third level related to intention; the desired decision on the rally evolution, for example, to take advantage during the build up of the rally. Specifically, these levels concerned action, situation or rally. They referred to various time-based events and players did not try to win the rally immediately. They attempted to create progressively good conditions to win by building advantage in the rally. Only when situational conditions were perceived to be right did they try to finish the point. Sometimes they accept a delay in creating winning conditions and maintain the rally. All intentions and decisions showed a contextual building of the rally according to the evolution of the situation. The most common intention was to take the advantage and thus build up the rally. The most common decisions varied between game situations. At service and return of serve, decisions aimed at putting pressure on the opponent. At the third or fourth shot, decisions were more varied. This can be explained by the nature of the situations. At the beginning of the rally, uncertainty is deemed to be low so the player can often impose his play. In the third or fourth shot, the rally is more advanced with each team influencing its evolution. At this stage it appears more difficult for players to impose their game because their decision is a result of the previous and anticipated situations. This study was focused on the temporal-nature of decision-making. Decisions were viewed in a synchronic and a diachronic way. Synchrony concerns the first four shots of the rally; whereas diachrony concerns the period from the third or the fourth shot of one rally to the first shot of the next rally. Therefore, this study proved innovative since prior research has focused only on diachronic decisions during the course of a match. In the present study, the four first shots were verbalized. This allowed greater understanding of the build up of each rally from its beginning, where important decisions were made regarding the development of the rally. situations can be anticipated and/or easy to understand by the players than some others. The players appear to be rational actors: they use rules that have been overtrained. This use of rules is consistent with McPherson and Kernodle's (2003) results. Nevertheless these situations were not the most frequently reported in the matches studied.

Consequences for coaching

The building of the badminton rally appears to be dynamic and contextual. It is founded on a sufficient understanding of the current situation information from a player's past as well as prediction about the future events. This type of situation is rarely considered during badminton coaching. Instead, players work on skill development and regimented routines. This type of training would seem at odds with the findings of this study. Coaches typically ask players to link action sequences. Consideration of errors within the rally development are rare and the opponents are considered as 'sparing partners' doing little more than helping to develop routines. The findings of this study suggest that routines are not always used as players try to build up the rally by exploiting the opponent's actions, position, and behaviour frequently by employing a surprise effect. If training is focused on routines, the conditions for analysis and to exploit the opponent's game may not be in place. Routine training relies on an ideal player frame; one that is explanatory and prescriptive. During a competitive situation, players frequently use a compromise model. This model suggests that to win the match, there will be occasions when the player must accept a deferral of the win when difficulties of play appear. In easily understandable situations, players will attempt to take the advantage in the rally or to finish the point quickly. In difficult situations, a full understanding of the situation does not exist; information is not available and the mental load is high. In this case the findings of this study would suggest the use of a lower level of understanding, to play by routines, for example, in an emergency situation players should lift the shuttle to create more time for decision making and understanding. Understanding will not be optimal, but it allows the player to act with some efficiency in pursuit of the maintenance of the rally. This compromise requires essential knowledge about ones partner and the opponents. This knowledge seems to be built and updated during the match itself. In addition, it is possible that it could be developed during training; a specific framework where the player changes role to become an accomplice for the opponent. The player would use specific techniques to research an opponent's decisions for example, using video analysis of the opponent's strategies. Through theses techniques, the opponent's strategies would become better understood to the player and then increasingly exploited progressively by the player.

Conclusion

This study aimed to reveal the contextual decision-making of high-level badminton players during competition and to help coaches to understand the preferential mode of player cognitive functioning. Decision-making was shown to depend highly on understanding of the immediate situation. The aim, for coaches, is to increase understanding of the conditions of the situation in order to facilitate performance. The study revealed that taking account of context and previous, current and anticipated events during a rally was essential for understanding and naturalistic decision-making. This study could be continued through a longitudinal approach by consideration of play at different stages of the match and in extended rallies or indeed at different stages of a competitive season. These studies may show further differences in decision-making that depend on the advantage gained during the match and performance-based fluctuations throughout a season.

Figure captions Figure 1 .Figure 2 .Figure 3 .

 captions123 Figure captions

Figure 4 .

 4 Figure 4. Third or fourth shot decision evolution during the match

Acknowledgments

This study was supported by a grant from the French Ministry of Youth and Sports. The authors are grateful to the coaches and the athletes of the French national badminton teams for their participation.

The results of this study indicated that the same local decision could be used for different aims. For example, the local decision of a cross-court drop referred to the decision to play wide, surprise opponents, and influence an opponent's decision. Therefore, local decisions were perceived as an effective way to reach a goal. They were considered as specific skills from an individual's repertoire.

The decisions elicited from the analysis were similar with the control modes proposed by [START_REF] Hollnagel | Cognitive Reliability and Error Analysis Method[END_REF]. According to the situations, the decision to realize a limited choice referred to scrambled control or opportunistic control. A scrambled control occurred when a situation evolved in an unexpected way and when there was very little time to act. Whereas, opportunistic control occurred when the choice of actions was limited due to the constraints in movement, time or space. In contrast to Hollnagel, situations did not appear to be unfamiliar to players; they were simply unexpected. This was different to scrambled control that was typified by a loss of situational awareness with emergency decision-making.

The decision to play wide, to surprise the opponents, to realize a limited choice, and the intention of finish the point, belongs to opportunistic control. In the situation, the decision is made from salient features of the situation instead of stable intentions or goals. In these situations, players were waiting for relevant information to make their decision, such as information linked to the behaviour or the position of their opponents. The decision to ensure his action, to observe the opponent's response to his action, and influence the opponent's decision was based on a tactical control. The decisions were planned. Players used known rules but which were limited to a restricted situational context. These rules could be individual (e.g., to reproduce an action, or to observe an action), collective (e.g., to influence opponent decision) or situational (e.g., to ensure his action). These types of decisions show that all the situations are not ambiguous, ill-defined or unsteady, as has been suggested by the Natural Decision Making Model (e.g. Klein et al. 1993, Zsambok andKlein 1997). Some Decision categories: 1 to ensure his action; 2. to realize a limited choice; 3 to observe opponent response to his own action; 4 to influence opponent decision; 5 to put pressure on opponents; 6 to surprize opponents; 7 to reproduce an efficient action; 8 to play wide; 9 to finish the point. Decisions from 1 to 3 correspond to the intention of maintaining the rally; decisions from 4 to 8 belong to the intention of taking the advantage.

Decision categories: 1 to ensure his action; 2. to realize a limited choice; 3 to observe opponent response to his own action; 4 to influence opponent decision; 5 to put pressure on opponents; 6 to surprise opponents; 7 to reproduce an efficient action; 8 to play wide; 9 to finish the point. Decisions from 1 to 3 correspond to the intention of maintaining the rally; decisions from 4 to 8 belong to the intention of taking the advantage.

Naturalistic decision-making 34

Decision categories: 1 to ensure his action; 2. to realize a limited choice; 3 to observe opponent response to his own action; 4 to influence opponent decision; 5 to put pressure on opponents; 6 to surprise opponents; 7 to reproduce an efficient action; 8 to play wide; 9 to finish the point. Decisions from 1 to 3 correspond to the intention of maintaining the rally; decisions from 4 to 8 belong to the intention of taking the advantage.

Decision categories: 1 to ensure his action; 2. to realize a limited choice; 3 to observe opponent response to his own action; 4 to influence opponent decision; 5 to put pressure on opponents; 6 to surprise opponents; 7 to reproduce an efficient action; 8 to play wide; 9 to finish the point. Decisions from 1 to 3 correspond to the intention of maintaining the rally; decisions from 4 to 8 belong to the intention of taking the advantage.