L Denant-Boèmont
email: laurent.denant-boemont@univ-rennes1.fr

R Petiot

Werner Güth

Gordon Mills

Jean-Christophe Vergnaud

Petiot

Information Value and Sequential Decision-making in a Transport Setting: An Experimental Study

Keywords: C91, D83, R41 Dynamic Choice, Expected Utility, Information Value, Transport Congestion

In economic models of dynamic choice under uncertainty, the "Information Value" is the highest price an individual is willing to pay for information. This amount usually increases as information becomes more specific. Applying experimental economics an experimental game has been constructed in which individuals choose between alternative congested transport modes to try to arrive at a destination at a specified time. To reduce the risk associated with modal choice, each subject can buy information about traffic levels. Two information messages are offered one after the other, the second giving more information. The results

INTRODUCTION

The provision of real-time road traffic information is an important part of present-day transport policy [START_REF] Emmerink | Effects of information in road transport networks with recurrent congestion[END_REF][START_REF] Mahmassani | Dynamic aspects of departure-time choice in a commuting system: Theoretical framework and experimental analysis[END_REF][START_REF] Mahmassani | System performance and user response under real-time information in a congested traffic corridor[END_REF][START_REF] Mahmassani | Availability of information and dynamics of departure time choice: Experimental investigation[END_REF]. Given the growing congestion on urban and interurban roads and the difficulties in financing new facilities, better information technology may improve traffic flow, particularly as it has even been demonstrated that information provision has the potential to reduce excess travel time (Ben-Akiva, de [START_REF] Ben-Akiva | Dynamic network models and driver information systems[END_REF].

However, there have been few analytical studies of individual transport choice with changing levels of traffic information. Some of these studies have aimed to assess the welfare impact of increased knowledge about traffic [START_REF] Arnott | Does providing information to drivers reduce traffic congestion?[END_REF][START_REF] Arnott | Information and usage of free-access congestible facilites with stochastic capacity and demand[END_REF][START_REF] Arnott | Information and time-of-usage decisions in the bottleneck model with stochastic capacity and demand[END_REF]Emmerink et al., 1998).

Such of the models are based on rational expectations and study the impact on choice of three types of information (no information, imperfect and perfect information). Obviously, improved information increases consumer surplus, but the welfare benefit could be reduced in some cases by increased congestion, especially when there are too many informed drivers or when drivers overreact to the information they receive [START_REF] Delvert | Is flexibility in choice option valuable? 9th International Association for Travel Behaviour Research Conference[END_REF][START_REF] Emmerink | Effects of information in road transport networks with recurrent congestion[END_REF]. Note that in most of these models, information is free at all times. Furthermore individual choices are stochastic but remain static (except in [START_REF] Arnott | Information and time-of-usage decisions in the bottleneck model with stochastic capacity and demand[END_REF]) -i.e. once users have made their travel choice (which they may do in several stages) no more information about traffic is available for them to learn and to modify their choice. Such a model is implemented in this paper.

This paper builds an analytical model of Information Value (IV) for transport choice where decision-making is dynamic and information is not free. How do road users make use of new information about traffic levels when congestion is likely to occur on the available routes? How do they change their transport plans when they receive additional information? An experimental economics method [START_REF] Davis | Experimental Economics[END_REF] has been used to test the behavioral responses of players in a transport situation. It is important to distinguish between the usual methods used to assess user preferencei.e. revealed and stated preference techniques -and experimental economics. As [START_REF] Friedman | Experimental Methods: A Primer for Economists[END_REF] have pointed out, the important fact is that economic experimenters try to produce laboratory experimental data, which implies a high level of control of experimental conditions and players' incentives by making payments to players which depend on their choices. There have still only been a few experimental economics studies of transport decision-making [START_REF] Delvert | Behavioural response to traffic variability: an experimental outlook on road pricing as willingness to pay for certainty[END_REF][START_REF] Delvert | Is flexibility in choice option valuable? 9th International Association for Travel Behaviour Research Conference[END_REF][START_REF] Schneider | Against all odds: Nash Equilibria in a Road Pricing Experiment[END_REF]. To the best of our knowledge, this study is the first to implement a method of this type for dynamic choice 1 in the area of transport when individuals are faced with uncertainty about traffic. In the field of transport, experimental economics has only been implemented to test market efficiency (Grether, Issac and Plott, 1989 who dealt with air transport deregulation; [START_REF] Brewer | A Binary Conflict Ascending Price (BICAP) Mechanism for the Decentralized Allocation of the Right to Use Railroad Tracks[END_REF]Plott, 1994, or Nilsson, 1997 who dealt with railway deregulation; Fischer et al., 1992, who dealt with the market for taxi-cab licences).

Models Dynamic choice models with increasing information use a Bayesian learning process where the IV is the additional utility resulting from improved information about the state of nature that is to occur (see the seminal analysis of [START_REF] Marschak | Economic Theory of teams[END_REF]. Experimental studies that deal with individual information valuation are rare [START_REF] Rauchs | Experimental Evidence on the Irreversibility Effect[END_REF] compared to dynamic choice studies, especially ones dealing with temporal inconsistency and myopic behavior [START_REF] Anderhub | On saving, updating and dynamic programming -An experimental analysis[END_REF]. Our purpose is to use an experimental method to observe IV in dynamic choice settings by focusing on sequential transport choice in a context of increasing information. In forming their plans, transport users can be regarded as making sequential choices: to get to their destination point, they first choose a mode and then a departure time. In our experiment the amount of information available is increased as the player goes through these steps. We are thus able to study individual adaptation.

More specifically, two kinds of information are offered at different prices: the second message is more precise than the first, but is provided later in the sequence of decisions -i.e. when the route has already been chosen. In most cases, the individual should be willing to pay more for the second message than for the first.

Two components of our experiment are quite new considering how information learning is implemented and how choices are dynamic, especially by comparison with the papers dealing with information and road transport cited above. First, decision-making is sequential -i.e. (a) the individual is able to reconsider his or her choice as regards information acquisition and (b) the level of available information increases as the decision sequence progresses, which implies Bayesian process for the individual. Second, the information messages are not free of charge.

Section 2 briefly defines the expected utility of increasing information -i.e. the Information Value -and then describes the dynamic transport choice experiment. The third section derives the theoretical results stemming from the model. Lastly, the fourth section gives the experimental results and compares them with theoretical predictions.

AN EXPERIMENT TO INVESTIGATE DYNAMIC TRANSPORT CHOICE WITH INCREASING INFORMATION

How much is a decision-maker willing to pay for additional information before he or she makes his or her decision? The Information Value (IV) 2 is defined as the maximum amount a player is willing to pay in order to obtain a given item of information. In more precise terms, what is involved is an information partition under expected utility maximizing assumption.

Information Value in Expected Utility

A basic theoretical model of Information Value in Expected Utility describes a two-period choice (t = 1, 2) under uncertainty. Let D be the set of possible choices at the beginning of the first period (

I i D di
,..., 1 ;   for t = 1), S the set of states ( s j the state j, j = 1 to J) and Y a set of messages/observations with elements y ( y k the signal k). By definition, an Information Set is the set of different states that the agent is unable to discern 3 . An Information Partition (IP) is therefore a collection of information sets, the information sets being themselves subsets of S which are mutually exclusive and for all incompatible. In Von Neumann-Morgenstern (VNM) Expected Utility, an IP(A) is more informative than an IP(B) if IP(A) is more detailed than IP(B). Moreover, an IP(A) is better than an IP(B) if

    (.) (.), ; ) ( (.) (.), ; ) ( u B IP U u A IP U    2
The following definitions are taken from [START_REF] Willinger | Flexibilité et valeur de l'information[END_REF]. For more details about the axioms which underline the Bayesian information structure, see [START_REF] Laffont | Fondements de l'économie publique. Tome 2[END_REF] or [START_REF] Jones | Flexibility and Uncertainty[END_REF].

3 For instance, if an individual makes an experiment by throwing a dice and announces to an hooded individual that the result is "an even number", this individual is not able to say if the result is "2", "4" or "6" but knows that "1", "3", "5" are impossible states. Then, he or she is not able to discern among states "2", "4", "6" because the experiment do not enable to discriminate between these three states. The information partition associated to this experiment is the union of information sets IP(A)=({1,3,5},{2,4,6}), each set being attached to an experiment ({1,2,3} is attached to "the number is uneven", {2,4,6} is attached to "the number is even"), whereas before throwing the dice, the information partition was IP(B)=({1,2,3,4,5,6}). where (.)

 is the probability function on S before any message is observed, and u(.) is the utility function attached to states s j .

"Increasing information" means that the agent moves from IP1 (a particular collection of subsets of S at t = 1) to IP2 (a collection of subsets of S at t = 2) which is more detailed than IP14 . For example, if we assume that a driver goes from A to B via the same road every morning and that four states of traffic are possible: "very light", "light", "medium" and "heavy". These states are all equally probable. Each level gives a certain level of transport cost for the driver. Before leaving A, he or she could obtain more information about the traffic level, for example by means of a radio-message saying "the traffic between A and B is not heavy". This is one possible message y among other possible messages, and enables the driver to revise his or her probabilities. Here, the state "heavy" becomes impossible, and the probabilities have changed (now 1/3 per possible traffic state). Thus, three kinds of situation have to be considered:

(i) No information (incomplete IP): the decision-maker is not able to to improve his or her information -i.e. is unable to refine his or her information via the set of S and the IP does not change over time. The driver receives no more information before leaving A or receives a message that is of no use.

(ii) Perfect information (Perfect IP): the agent goes from IP(A) to IP(B) the later being singletons. The agent then combines a signal y k and a state s j -if an agent receives the signal y k , he or she knows that state s j will occur in the coming period. In our example, a driver who gets a radio message saying "the traffic is heavy" will be able to change his or her departure time for instance. He or she knows with certainty that the probability is 1 for "heavy" and 0 for the other states;

(iii) Imperfect information (Partial IP): the agent improves his or her information -i.e. refines his or her IP, but remains unable to combine any y k with a single s j . For instance, if the radio message says "The traffic level is changing from very light to light at the moment", the driver will know that the probability of having medium or heavy traffic is 0, but not know with certainty from the message which state will occur.

The IV is the difference between the expected utility with the more detailed IP and the expected utility with the less detailed IP. In VNM expected utility, IV is always positive or equal to zero. Obviously, the imperfect information cases are the most interesting, because it is unusual for the driver to have perfect knowledge about the states which will occur in the coming period.

A dynamic transport choice experiment with increasing information about traffic

The experiment deals with a sequential transport choice and observes how transport choices change when individuals are offered traffic information which is not free of charge. First, if the IV theory is right, subjects should be willing to pay more for the most accurate information. Second, subjects should only buy the information in the case where IV -i.e.

Expected Utility with the information minus Expected Utility without information -exceeds the information price.

(a) Design 5

In the experiment 6 , 3 groups of 10 subjects 7 play 10 rounds. During each round, each of the subjects plans their travel from point A to point B. The arrival time * t at B is constrained (fixed at 5 pm) 8 . In each round, players are first informed about road capacities. In this experiment the capacity is the number of players who can take a given route without incurring congestion (L = 1 means that if 2 or more players choose L the travel time on L will increase).

First, players choose their transport modeeither Train (T) or Road (R). If they choose T, they are sure of getting B at 5 pm. All players have the opportunity to purchase information about traffic level on R, R I giving nR, the actual number of players who has chosen R. All the players know the price of this information, which is different for each round. They can either purchase R I or refuse to do so. The players are unaware of whether the others have purchased R I .

Players who choose R will then select their routeeither a Local road route (L) or a free

Highway (H). The capacity on L route is always lower than on H route. All players have the opportunity to purchase information about the actual traffic on H ( H I gives nH) and the price is different during each round. The players do not know who has purchased the information.

At the end of each round, the players have to choose their departure time d t from among a set of possible departure times. The amount paid to each player depends both on his arrival time 5 The design of the experiment stems from a discussion with Professor Werner Güth, who suggested a design to investigate transport choice.

His initial idea has only been implemented in part, the emphasis being placed on information value.

6 Full instructions are available from the authors on request.

7 Students taking economics and the management courses. There is no difference between the 3 groups.

8 The paper does not directly deal with the morning commute problem. We do not refer explicitly to the morning rush in such a way that in the experiment the subjects only concentrate on the information impact on their own modal and time decision in front of a constraint arrival time whenever during the day.

L. Denant-Boèmont, R. Petiot a t , which may differ from the constrained arrival time * t , and on his or her travel time t . The travel time t is a function of the traffic level on each route -i.e. the number of players who have chosen L or H. In the experiment, the travel time without congestion is 40 minutes on route L and 30 minutes on route H. The travel times with congestion are shown in the table of payments given to the players.

Each player's monetary endowment for the first round is 60 ECUs (1 ECU for 1 FF or 6.55957 €). The endowment for each of the following rounds is the sum of the player's payments for the previous rounds added to the initial endowment. Any communication between players is forbidden during the experiment.

(b) The payoff function

The payoff function is based on a transport cost function [START_REF] Vickrey | Congestion Theory and Transport Investment[END_REF]. It takes into account both the cost of the journey time and the cost of the difference between the actual and target arrival times. Such a function is widely used in transport economics, in the context of both analytical studies and experimental work.

For any departure time d t , in the event of early arrival    

a d t t t  * , the total travel cost is         t t t t t t C d a d d    *   (1)
where  is the cost of a unit of transport time and  is the cost per time unit of being early.

For

any d t , in the event of late arrival     a d t t t  * , the total travel cost is         t t t t t t C d a d d *      ( 2
)
where  is the cost per time unit of being late.

The maximum gain G is fixed. If

  t C Arg d min
is the lowest possible travel cost for all departure time d t , the payoff function is

      2 min t C t C Arg =G+ t P d d d  .
(3)

The payoff for choice T is 1 ECU.

In the experiment 9 , G = 20 ECUs,   1 ECU,   2 ECUs,   3 ECUs. We have also implemented a bonus of 20 ECUs in the case of 9 The values taken for the cost of travel time and schedule delay are not typical of those found in the standard literature [START_REF] Arnott | Departure time and route choice for the morning commute[END_REF].

However, in our experiment, the idea was to give a high value for schedule delay cost in order to encourage players to make an intelligent choice of departure time.

THEORETICAL RESULTS AND MODEL

Basic assumptions

The main assumption is that individuals are Expected Utility maximizers. Thus, when individuals receive perfect information about traffic levels they choose the departure time that maximizes the payment. Moreover, the IVs are determined with respect to the payoff function.

We assume, as do most experimenters in decision-making games, that this payoff function represents the utility function of any player, and that all players are risk-neutrali.e. that expected utility is equivalent to expected payoff. Information Value is in this case equivalent to the gain in expected payoff enabled by increasing information. We have assumed, however, that we could use an uniform probability function. Note that the model implemented here differs from classical discrete choice models [START_REF] Ben-Akiva | Discrete Choice Analysis: Theory and Application to Predict Travel Demand[END_REF] in that we have assumed that neither the decision rules nor the utility function are probabilistic. The utility function is assumed to be known and the choice process is deterministic. Only the traffic levels are uncertain.

In the experimental design, all the subjects know they are playing against the (n-1) other subjects in the group, and this knowledge affects their choice. Then, if the experiment implements a game situation, why not use game theory tools in order to determine the equilibrium instead of decision theory? There seems to be no reason that decision theory does not apply to decision-making in game theory situations, as has been pointed out by [START_REF] Broome | Bolker-Jeffrey Expected Utility Theory and Axiomatic Utilitarianism[END_REF]. This author describes the traditional "twin prisoners' dilemma" 10 (i.e. situations where the players are entirely self-interested, but think very much alike), where "you and your twin can act nice or nasty". Acting nasty is a dominant strategy, so both players will choose it.

What would be the outcome if decision theory were used? Broome says (1990, p. 487):

"Faced with the twin prisoners dilemma, a follower of Savage has two alternatives. She may decline to apply the theory at all, perhaps taking the general view that decision theory does not apply to games. Or she may pick some things to serve as states of nature. She may, for instance, take your twin's acts as states of nature from your point of view. They will then have to be assigned probabilities independent of your own acts. And whatever probabilities they are assigned, acting nasty will come out with a higher expected utility for you".

Broome then goes on to argue that in numerous game situations, using decision theory could give the same result as game theory. For example, in Public Good Games, the best theoretical solution is not to contribute. There is no doubt that in many of the experimental settings associated with such games, the best choice according to expected utility theory is also not to contribute. We have assumed here that the best choice given by decision theory is the same as that given by game theory.

So, for routes H and L, the individual will have to choose the departure time t d . We have assumed that the driver will choose the best t d for route L or route H if he or she have received 10 Two people are arrested for the same crime. The police lack evidence to convict either suspect and consequently need them to give testimony against each other. They are placed in separate rooms and given the opportunity to confess. If only one of them confesses, he or she serves one year and the other receives a ten-year sentence. If both confess, they each serve five-year terms. If neither confesses, each receives a maximum two-year penalty. Thus, both would be better off if neither confesses, but each, aware of each other's incentives to confess in any case, will probably confess. This game, called a "coordination game" by game theorists, demonstrates the failure of spontaneous coordination between non-cooperative players, according to the Pareto-optimality criterion. perfect information about the traffic level on the route. The information alternatives available to the individual are: "purchase no information" or, "purchase R I giving nR" or, "purchase H I giving nH" or, "purchase both R I and H I ". It might be thought that the last case need not be considered, but it may be of value to purchase H I if the individual has already purchased R I (for example, individuals who have purchased R I and choose R would get perfect information with H I ).

Expected gains and information value

We have to calculate how the expected gains are increased if the individuals purchase R I or H I , or R I and H I . We shall first of all compute the expected gains without any information purchase.

(a) The expected gains without any information

The expected gains without any information are as follows : where: i td is the departure time i for choice L (i = 1 to 5); i td ' is the departure time i' for choice H (i' = 1 to 9);

1 ) (  T E (4)   E R Max E L E H ( ) ( ); ( )  (5) with   E L Max e i td ( )  ;   i td e td nL x U i x      (
) ( i td U
is the utility attached to i td at each traffic level for L -i.e. the payoff defined in (3);

) ( ' i td U
is the utility attached to ' tdi at each traffic level for H;

   nL x 
is the probability of the number of players on L (nL) being equal to x (the traffic

level on L is x);    nH y 
is the probability of the number of players on H (nH) being equal to y.

Let us now explain equations ( 5), ( 6) and ( 7). In equation ( 6), a player who chooses L can choose a departure time of 16:00. The expected utility of this departure time given his or her choice L is then the expected payoff of a 16:00 departure time choice. This is the probability of x of players choosing L multiplied by the payoff for every possible level of traffic. Here, the possible states for traffic are nL = 1,…,10 therefore

1 . 0 ) (   x nL 
. Equation ( 7) gives the expected utilityin the experiment, the expected payofffor the choice H. Finally, equation ( 5) means that, if the player is an expected utility maximizer, he or she should choose between L or H by comparing the values he or she expects.

Given the payoff structure, a player who has chosen R has to choose between L and H before choosing a departure time. The payoff of L or H will then depend upon the departure time. For instance, in one of our experiments (where the capacity of L was 2 and the capacity of H was 4), the expected payoff of a 16:00 departure time on L was +13.75, -0.75 for 16:05, etc.

(expected payoff declines with a later departure time). A rational player should choose 16:00 if L has been chosen so, the expected payoff for L is +13.75. For H, the expected payoff is given by the best choice from among the possible departure times on that route. For example, the highest expected payoff on H in this case was -7 ECUs with a departure time of 16:00 on H. So, a rational player having chosen mode R should choose route L and a departure time of 16:00, because, without any information, this is the best choice with regard to the expected payoff.

(b) The expected gains with I H -The Information Value for I H

The expected gains with I H are as follows :

  nH y ; y  0 10 ,...,   nL x ;     y x   10 , 0 ,   ) ( ); ( ) ( y nH I H E y nH I L E Max y nH I R E H H H     (8)
where I H gives the number of players y who have chosen H (nH) and

            10 0 y y H H y nH I R E y nH I R E  (9)
and the IV for  

I H is     0 ) (    R E I R E I IV H H . ( 10
) ( ); ( ) ( z nR I H E z nR I L E Max z nR I R E R R R     (11) so             10 1 z z H R z nR I R E z nR I R E  (12)
and the IV for

I R is     0 ) (    R E I R E I IV R R . ( 13

 

). A player who has information about R can infer which states of nH and nR are not possible. Purchasing H I will then give him or her perfect information about the state of nature for R and H. When buying R I , the player has to choose between L and H. This choice is made by considering the possible states for nH and the probabilities of these states given the state of nR. For any nH, the player makes the best decision (L or H). So, for a given value of nR = z

          td U td U Max z nR y nH z nR I and I R E z nR i i z y y H R ' 0 ;            (14)             10 1 z z H R H R z nR I and I R E z nR I and I R E  . ( 15
)
Finally,

    0 ) (    R E I and I R E I and I IV H R H R . ( 16
)
Because having R I and H I is equivalent to a perfect Information Set, we should have

) ( ) ( ) ( I IV I IV I and I IV R H H R  
(17)

Theoretical results

On the basis of the payoff function, the expected gains and IVs are shown in Table 1. Here, the player receives information about the traffic on mode R (if he or she buys it!) and then chooses between routes L and H. The player then gets further information about the traffic on route H (if he or she buys it) before choosing a departure time. In this case, the player is in a "perfect information" state (if there are 5 R choices there are 5 T choices and if there are 3 H choices there are 2 L choices) -i.e. the player knows at the beginning of the game that he or she will be able to know the state of traffic with certainty, but also that for the time being no information is available to make the first choice between mode T and mode R.

Therefore, to use the terminology of game theory, the player has to use backward induction to assess his or her best choice at the beginning of the game. At this moment, each state of traffic on R is equally probable and the player has to assess each possibility. For example, let us assume that there is 1 R (the probability is 10%). There are then two possible states for the traffic on L and H: either the traffic on L is in state "1", in which case the traffic on H is in state "0", or the traffic on L is in state "0" and the traffic on H is in state "1". More generally, if the traffic level on R is x, there are (x+1) possible states for the distribution of this traffic between L and H, each state being equally probable. For each possible state, the payoff of an R choice is the maximum from among the payoffs for L and the payoffs for H. For instance, if nR=2, there are 3 possible states:

1. H=0 and L=2 (probability equal to 1/3 given a 10% probability that nR=2): this implies that the player on R has chosen L and will receive 35 ECUs (the payoff on L given the best departure time); 2. H=1 and L=1 (probability equal to 1/3): a player who chooses H will get 40 ECUs and a player who chooses L will get 35 ECUs. Given the state, the best choice is H; 3. H=2 and L=0: this implies that the player on R has chosen H and will get 40 ECUs.

Then, in the case where nR=2, the expected payoff is 

     38.3 40 1/3 40 1/3 35 1/3        .
This expected payoff has to be multiplied by 0.1 to obtain the expected payoff for the state R=2 that is to say +3.83. This kind of calculation has to be performed for each possible state of R and at the end the expected payoff of  

I and I R H R is the sum of 34.1 3.83     ECUs.
A rational player (an expected utility maximizing individual) should perform this process for each possible state in order to assess the expected payoff of the R choice with these two levels of information and to assess the R choice in a bayesian manner. Given all the possible messages (i.e. all the possible levels of traffic) and the payoffs attached to each state for each possible choice, the player has to compare the expected value of R without any kind of information (in the example +13.75) with the expected value of R in a "perfect information" state (in our example, the expected payoff of R in a "perfect information" state is +34.1).

Not surprisingly, the first thing we find is that the expected utility of choice R increases as additional information is provided to individuals, i.e. that IVs are strictly positive. Secondly, the IV of H I is higher than the IV of R I . We cannot say that this result is linked to the fact that the information H I is "perfect" compared to R I , which is imperfect. Although H I amounts to perfect information when the player has chosen H, this is not true when the player has chosen L. So, H I is obviously a refinement of R I , but it is not equivalent to perfect information: both R I and H I are "imperfect" information. This explains the small difference between IVs which is confirmed by the large difference between the IV of ( R I and H I ) and IV( H I ), IV( R I ), because both messages are equivalent to a complete IP for any choice R.

It should also be noted that the IVs for any kind of information are greater when the capacities of roads L and H are high than when the capacities are low, which could seem paradoxical.

However, what occurs is that the uncertainty about payments grows as the capacities of both roads increase (the relative capacities of H and L are constant in our experiment), as do the IVs. The level of IV is mainly linked to the dispersion of the results in the different states.

Another explanation relates to the characteristics of the payoff function, which favors an early departure time.

The data in Table 2 enable us to build the different scenarios for the experiment, i.e. the prices for information R I and H I for given infrastructure capacities, which give an information price that is over or under the IV. The aim is to observe different information purchase possibilities for any kind of information in the two differing capacity scenarios.

Descriptive results

Mode R was chosen 295 times and mode T only 5 times (see Fig. 1), which is explained by the low level of payment for choice T. Route L was chosen 206 times (69.8%) and route H 89 times (30.2%). Choice L was dominant within each of the 9 rounds, but the level of this domination varied from one round to another. The choice of departure time should depend on the possible travel time on each route, with reference to the traffic level and capacities. A departure time of before 16:20 for L or before 16:30 for H is considered to be an early departure. Experimental outcomes are given in Figure 2. It shows the share of the chosen departure times for L and H for each round. First, choice L requires an earlier departure time (16:00 for about 90% of the choices L) than choice H because the payoff function means that H is the riskier choice (the expected gain is lower while the variance is higher). The individuals choosing H therefore are adopting a risky strategy (more than 50% of the players choose 16:30 which maximizes the payment if there is no congestion). Some other players are risk averse and play H with an early departure time (16:00 or 16:20). In this case, the dispersion around the departure time is higher for the H choice. The second explanation is that the information about traffic on route H plays a greater part in the choice of H than it does in the choice of L: when the individual receives H I , he or she chooses the best answer, which could be quite different from the choices made by individuals without H I . The behavior of the groups is practically similar and explains 37.7% of the variance for choice L and 23.1% of the variance for choice H. Therefore, a "group effect" is not sufficient to explain the variability of the choices between L and H. There is no bar chart for the choice H for the group B because no player have chosen the route H.

Is this variability explained by the difference in route capacities? The global preference for choice L is again observed in both capacity situations (see Fig. 3). It is obvious that choice H is more frequent when the capacity of the route H is the largest, but choice L seems to be preferred when all the rounds are considered. Perhaps the choices made by players can be explained by some of the characteristics of payments. The dispersion of payments as a result of choice H deters more players from choosing H than the dispersion of payments as a result of choice L. Let us assume that players first try the route which allows them to improve their initial monetary endowment. They are then in a position where they can take the risk of choosing route H (Rounds 5 and 6) as they have enough money to do so. Some of them lose money as a result of choosing route H and return to route L (Rounds 7, 8, 9, 10). Risk aversion analysis might shed some light on this situation, but we shall not attempt this here.

So, a certain "capacity effect" would seem necessary to explain some of the variability that affects route choices during the rounds. Nevertheless, this capacity effect seems to be closely linked to the payment structure for both routes. Clearly, choices should be explained from the point of view of information purchase.

The players more often purchased information about the traffic level on H than on R, 139 times versus 78 times respectively (see Fig. 4). Furthermore, information purchase behavior changed in different rounds. Most players purchased messages in the early rounds ( I R and/or H I ). Towards the end a small number of players bought message. Players who still purchased information preferred to by H I . An initial hypothesis we can make is that the price of information influences the amount bought. However, there is no clear cut relationship between price and information demand.

The elasticity of H I purchase with respect to information price is -0.7 whereas the elasticity of R I purchase is -0.26. If these results are not conclusive enough concerning the price impact on the information purchase, H I purchase seems yet more sensitive than R I purchase.

The players appraisal leading to buy H I with respect to its price seems more accurate than it is for R I . Perhaps this is explained by the fact that R I is offered at the beginning of the round, whereas H I is sold at the end of the round and may thus be more beneficial to the subject. It seems possible that information purchase may depend on the chosen route.

Most of the players who choose L begin to purchase information in the 6 first rounds, but most of them do not purchase any information in the 2 last rounds (94% for the round 9 and 96% for the round 10). According to the R I elasticity, the R I purchase cannot be explained by the R I price, but it seems clear that R I becomes less and less interesting as rounds go along. On the contrary, most of those who choose H purchase H I (more than 57% in the experiment) (see Fig. 5). Nevertheless, the variation in the purchasing behavior of those choosing H seems to be erratic. According to the H I elasticity, it seems obvious that this purchasing behavior cannot be explained without exploring the IV; this is considered below. H i g h w a y w i t h I R a n d I H H i g h w a y w i t h I H H i g h w a y w i t h I R H i g h w a y w i t h o u t i n f o r m a t i o n

A comparison between theoretical results and observations

(a) Information Value in comparison to its price An individual's behavior can be predicted from route capacities. If the IV is higher than its price, the player should buy it. The aim of the experiment is to test whether individuals buy information when and only when its value exceeds its price. The statistical test has to show whether individual demand for information is greater in the rounds where the IV is higher than its price than in the rounds where the IV is lower than its price. We have thus carried out a "within subject" comparison test for non-independent data. We have chosen a nonparametric test (the Wilcoxon test) for which the null hypothesis is: H0: the information demand in the set of rounds where the player should not have bought the information is not significantly different from the demand in the set of rounds where the player should have bought it, versus H1: the demand for information is higher in the set of rounds where the information should be bought than in the set of rounds where the information should not be bought.

Because of the respective levels of price and IV (see Tables 1 and2), H I should not have been bought in rounds 2, 3, 7, 8, 9 and 10, but should have been bought in rounds 1, 4, 5 and 6. The test was made for each group of 10 players (A, B and C). In Table 4, the second column (positive ranks) shows the number of players who bought more information in the rounds where it was profitable to do so than in the rounds where it was not.

The third column (negative ranks) shows the number of players who bought more information in the rounds where it was not worth buying than in the rounds where it was worthwhile. The fourth column shows the number of players who were not sensitive to information price and whose demand for information did not differ between the rounds where information purchase was worthwhile and the rounds where purchase it was not. The fifth column shows the level of statistical significance (the asterisk indicates that H0 should not be accepted for   5% ).

Except for the purchase of H I by group A, it is difficult to accept H0, i.e. the individuals have bought more information when it was worth more than its price. The results favor H1 even more strongly. Another idea is to test the hierarchy between R I and H I , because information about nH would seem, judging from its IV, to be the most interesting. We have therefore performed a "within subject" Wilcoxon test where the nul hypothesis is: H0: the individual demand for R I is not significantly higher than for H I , H1: the individual demand for H I is significantly higher than for R I

The results show that it is difficult to accept H0: the individuals considered that H I was more useful that R I and bought it significantly more frequently. (a) In a context of increasing information where information is not free of charge, players compare the utility of information to its cost and this calculation is an important part of the comparison between risk with regard to travel time and monetary travel cost, (b) The dynamic choice model seems to provide a reasonable approximation of individual valuation. In particular, the willingness to pay (WTP) for more detailed information is higher than for less detailed information. This still applies even when the less detailed information is available before the more detailed information.

(c) Players choose to buy information when the variance in payoffs for the route choice is sufficiently high. This confirms the theoretical result obtained by Emmerink et al. (1998).

(d) The types of choices individuals make change over time. At the beginning of the experiment many players explore the environment by making risky choices and by buying information. Gradually, their strategy changes and most individuals choose a less risky mode without information purchase. This illustrates the way in which individual learning substitutes for exogenous information provided by the experimenter.

Our findings show that, in a transport context, players presented with real-time information tend to be rational in their modal and route choice behavior. Moreover, information matters for players, in the sense that behavioral responses could be significant in an evolving environment. Finally, the experiment also shows that it is important to recognize the diversity of users when faced with information provision, and to discriminate between experienced and inexperienced users. It confirms the findings of [START_REF] Dudek | Human factors design of dynamic displays[END_REF] that inexperienced drivers are more likely to change their route in response to a message than experienced drivers. It should therefore be possible to avoid the decrease in the effectiveness of information that may occur when too many users are informed about traffic levels.

One of the first questions we can ask about this kind of transport-related experimental game is whether the results can be transferred to real-life departure time tradeoffs. The problem is that we are dealing with players and not real users; real life can never be exactly produced in the laboratory and subjects are always liable to behave differently. In real world, we are aware that users do not behave rationally as we assume here. Therefore, further developments should consider bounded rationality theory. This applies yet to any work in the field of experimental economics, because the primary aim of a researcher is to perform a normative study by designing an experiment to investigate a single effect. However, our exploratory experiment Appendix A -42,5 -17,5 -5 -12,5 -17,5 -27,5 -35 -42,5 -50 8 70 -20 -47,5 -15 -22,5 -27,5 -37,5 -45 -52,5 -60 9 70 -20 -47,5 -15 -22,5 -27,5 -37,5 -45 -52,5 -60 10 70 -20 -47,5 -15 -22,5 -27,5 -37,5 -45 -52,5 -60

Tables of payoffs

  possible costs is always 30 (in the case of a choice H with low traffic levels on H (1 or 2 users), and a 16:30 departure time choice). Then, if the player has chosen L and nL being equal to 10, then his or her travel time is 60 minutes and his or her travel cost is 60 ECUs (60 minutes *1 ECU). Moreover, he or she has chosen 16:15 as a departure time, and then he or she arrived at B 60 minutes lateri.e. at 17:15, being late for 15 minutes. His or her schedule delay cost is then equal to 15 minutes * 3 ECUs = 45 ECUs. His or her total travel time cost is then 60 ECUs + 45 ECUs = 105 ECUs. The corresponding payoff is equal to 20 + (30-105)/2 = -17.5 ECUs (see the Tables of payoffs used in the experiment in Appendix A).

  ) (c) The expected gains with I R -The Information Value of I R The expected gains with I R are as follows :

  ) (d) The expected gains with I R and I H -The Information Value of I R and I H It is obvious that the IV of both pieces of information about traffic levels is different from the sum of the IVs (

Fig

  Fig. 1. Aggregated modal and route choices

Fig

  Fig. 2. Departure time frequencies for L and H

Fig

  Fig. 3. Aggregated Route choices and Route capacities

Fig

  Fig. 4. Aggregated information purchase per round

Fig. 5 .

 5 Fig. 5. Aggregated route choices and Information purchase per round

  To illustrate, let us fix the constrained arrival time t * at 17:00. Assume the player's chosen route is L and the player's chosen departure time td is 16:15 and that in view of congestion the actual arrival time  

	t t d a	is 17:15 in the case where	1 L , 	H		2	and	nL		10

* t t a  .

Table 1 Information Values for I R , I H , « I R and I H » (ECUs)

 1

				Capacity	L=2 ; H=4 L=1 ; H=2						Capacity	L=2 ; H=4 L=1 ; H=2
	Expected Gain			Information Value
	  E R							+13.75	+16.75
	  E R R I				+25.9	+19.7	IV R I ( )				+12.1	+2.9
	 E R H I					+28.5	+20.9	IV H I ( )				+14.8	+4.1
	 E R	I	R	and	I	H		+34.1	+31	IV	(	I	R	and	I	H	)	+20.4	+14.2

Table 2 Price levels (ECUs) for I R and I H and theoretical predictions

 2

	Capacities Round	p I R	p I H	Information which should have been purchased
		1	14	4	H I or ( R I and H I )
	L = 2	2	6	16	I	R
		5	14	10	I	H
	H = 4	8	2	16	( R I and H I ) or R I
		9	15	15	none
		3	1	6	( R I and H I ) or R I
	L = 1	4	1	3	( R I and H I ) or R I or H I
		6	13	3	I	H
	H = 2	7	1	15	I	R
		10	6	12	none
				4. EXPERIMENTAL RESULTS

Table 3 Actual information purchasing behaviors

 3

	Rounds 1	2	3	4	5	6	7	8	9 10
	Actual number of buying R I	8	16 20 12	1	0	10	8	1 2
	Actual number of buying H I	19 13 17 21 17 24	4	10 9 5
	Actual number of buying R I and H I	6	7	14 10	1	0	3	4	1 1

Table 5 Wilcoxon Test about individual demand on I R compared to I H

 5

	Group	(+) ranks	(-) ranks	number of discarded results	Critical
	A	6	2	3	probability 0.039*
	B	8	1	1	0.008*
	C	8	1	1	0.006*
			5. CONCLUDING REMARKS
	Our most important findings are as follows:

Table 6 Tables

 6

	of payoffs L=1
	L=1			Payoffs according to time departure
					(ECUs)
	Traffic	Travel time
	(actual number of persons	(minutes)	16:00	16:05	16:10	16:15	16:20
	who have chosen L)
	1	40	-5	0	5	10	35
	2	45	-2,5	2,5	7,5	32,5	5
	3	50	0	5	30	2,5	-5
	4	60	25	-2,5	-10	-17,5	-25
	5	60	25	-2,5	-10	-17,5	-25
	6	60	25	-2,5	-10	-17,5	-25
	7	60	25	-2,5	-10	-17,5	-25
	8	60	25	-2,5	-10	-17,5	-25
	9	60	25	-2,5	-10	-17,5	-25
	10	60	25	-2,5	-10	-17,5	-25

Table 7 Tables of payoffs L=2 L=2 Payoffs according to time departure (ECUs) Traffic (actual number of persons who have chosen L)

 7

		Travel time
		(minutes)	16:00	16:05	16:10	16:15	16:20
	1	40	-5	0	5	10	35
	2	40	-5	0	5	10	35
	3	45	-2,5	2,5	7,5	32,5	5
	4	50	0	5	30	2,5	-5
	5	60	25	-2,5	-10	-17,5	-25
	6	60	25	-2,5	-10	-17,5	-25
	7	60	25	-2,5	-10	-17,5	-25
	8	60	25	-2,5	-10	-17,5	-25
	9	60	25	-2,5	-10	-17,5	-25
	10	60	25	-2,5	-10	-17,5	-25

Table 8 Tables of payoffs H=2 H=2 Payoffs according to time departure (ECUs) Traffic (actual number of persons who have chosen H) Travel time (minutes) 15:50 15:55 16:00 16:05 16:10 16:15 16:20 16:25 16:30

 8

	1	30	-60	-55	-10	-5	0	5	10	15	40
	2	30	-60	-55	-10	-5	0	5	10	15	40
	3	40	-55	-50	-5	0	5	10	35	7,5	0
	4	60	-45	-40	25	-2,5	-10	-17,5	-25	-32,5	-40
	5	65	-42,5 -17,5	-5	-12,5	-20	-27,5	-35	-42,5	-50
	6	70	-20	-47,5	-15	-22,5	-30	-37,5	-45	-52,5	-60
	7	70	-20	-47,5	-15	-22,5	-30	-37,5	-45	-52,5	-60
	8	70	-20	-47,5	-15	-22,5	-30	-37,5	-45	-52,5	-60
	9	70	-20	-47,5	-15	-22,5	-30	-37,5	-45	-52,5	-60
	10	70	-20	-47,5	-15	-22,5	-30	-37,5	-45	-52,5	-60

Table 9 Tables of payoffs H=4 H=4 Payoffs according to time departure (ECUs) Traffic (actual number of persons who have chosen H) Travel time (minutes) 15:50 15:55 16:00 16:05 16:10 16:15 16:20 16:25 16:30

 9

	1	30	-60	-55	-10	-5	0	5	10	15	40
	2	30	-60	-55	-10	-5	0	5	10	15	40
	3	30	-60	-55	-10	-5	0	5	10	15	40
	4	30	-60	-55	-10	-5	0	5	10	15	40
	5	40	-55	-50	-5	0	5	10	35	7,5	0
	6	60	-45	-40	25	-2,5	10	-17,5	-25	-32,5	-40
	7	65

It is possible to say that IP2 is a refinement of IP1 if in IP2, (i) the number of information sets is greater than in IP1 and (ii) if every information set belonging to IP2 is a collection of subsets of any information set belonging to IP1. See[START_REF] Rasmusen | Games and information: An Introduction to Game Theory Blackwell, Scd Edition[END_REF] for further details.

can be considered to be quite adequate in order to explain the potential impact of traffic information for driving with a limited number of modes and a limited number of routes.

Finally, we consider that this paper makes a contribution to the application of experimental economics as a rigorous data production method to test theoretical models in the field of transport behavior analysis.


Anette Gry

Tuset

Preservice teachers' geometrical discourses when leading classroom discussions about defining and classifying quadrilaterals

Keywords: Commognitive framework, Defining and classifying quadrilaterals, Preservice teachers

In this paper I use Sfard's commognitive framework to investigate four geometrical discourses, as they appeared when preservice teachers (PSTs) were leading classroom discussions about defining and classifying quadrilaterals in authentic lower secondary school classrooms. Their tryouts are part of a broader learning teaching project in a practice-based teacher education course in Norway. The focus is on mathematical problems PSTs address in learning-teaching situations, with students and geometrical discourses, in their efforts to promote hierarchical relations between quadrilaterals using pre-constructed dynamic figures in Geogebra. The analysis shows that PSTs create opportunities to talk about hierarchical relations. However, they struggle to move between static and dynamic ways of talking about geometrical objects when needed and it seems challenging to build appropriately on students' prototypical and partitional contributions.

Introduction

The most challenging work preservice teachers (PSTs) face in enacting current reforms is to orchestrate classroom discussions that use students' responses in such ways that they advance the mathematical learning of the whole class [START_REF] Lampert | Using designed instructional activities to enable novices to manage ambitious mathematics teaching[END_REF]. The PST is expected to execute skilled ways to support and challenge students when they are presenting their ideas, and build upon them by contrasting, comparing and connecting their ideas to accepted disciplinary ideas [START_REF] Stein | Orchestrating productive mathematical discussions: Five practices for helping teachers move beyond show and tell[END_REF]. For students, age 13-15, it could mean helping them move from using visual recognition or informal properties check when identifying and comparing geometrical shapes, to using definitions and state logical arguments to justify claims about these shapes [START_REF] Sfard | Thinking as communicating: Human development, the growth of discourses, and mathematizing[END_REF][START_REF] Sinclair | The more it changes, the more it becomes the same: The development of the routine of shape identification in dynamic geometry environment[END_REF].

Defining and classifying quadrilaterals contribute to the development of geometrical thinking and mathematical argumentation and proof [START_REF] Fujita | Learners' understanding of the definitions and hierarchical classification of quadrilaterals: Towards a theoretical framing[END_REF]. Research reports, however, that PSTs and students find it difficult to understand and accept hierarchical relations of geometrical shapes (e.g., [START_REF] Fujita | Learners' understanding of the definitions and hierarchical classification of quadrilaterals: Towards a theoretical framing[END_REF][START_REF] Millsaps | Challenging preservice elementary teachers' Image of Rectangles[END_REF]. A significant reason is their reliance on prototypical images. [START_REF] Hershkowitz | Psychological aspects of learning geometry[END_REF] describes prototype examples as the subset of examples that have all the necessary properties and, in addition, other specific (none-necessary) properties that have strong visual characteristics. Prototype images often lead to what de Villiers (1994) calls partitional definitions, which exclude, for instance, rectangles from being parallelograms. In mathematics, hierarchical definitions, which includes rectangles as parallelograms, are preferred because they are more economical and useful. Consequently, defining and classifying quadrilaterals are highly interrelated activities [START_REF] De Villiers | The role and function of hierarchical classification of quadrilaterals[END_REF].

Leading classrooms discussions is a complex and situated activity [START_REF] Stein | Orchestrating productive mathematical discussions: Five practices for helping teachers move beyond show and tell[END_REF]. One way of looking at this complexity is to see teaching as an activity of problem solving and problem posing [START_REF] Lampert | Using designed instructional activities to enable novices to manage ambitious mathematics teaching[END_REF]. These problems arise in establishing and maintaining relationships between the teacher, students and the mathematics because of the constraints and possibilities the participants generate during instruction [START_REF] Lampert | Using designed instructional activities to enable novices to manage ambitious mathematics teaching[END_REF]. For instance, the way the teacher creates and exploits opportunities for mathematics learning can constrain or promote the kind of mathematics students can engage in [START_REF] Sfard | Ritual for ritual, exploration for exploration: Or, what learners are offered is what you get from them in return[END_REF]. Similarly, students and the mathematics can both constrain and open up what the teacher can do to teach. To solve these problems of practice, the teacher needs to deal with a wide range of issues across social, temporal, and intellectual domains. This paper focus on mathematical problems PSTs address in situations, with students and mathematics, in their efforts to promote hierarchical relations between quadrilaterals using shape makers (i.e. pre-constructed dynamic figures in Geogebra). For this purpose, I use [START_REF] Sfard | Thinking as communicating: Human development, the growth of discourses, and mathematizing[END_REF] commognitive approach to examine the geometrical discourses that developed in four wholeclassroom discussions led by PSTs in authentic classrooms. The framework enables me to make claims about PSTs' geometrical discourses in terms of how they communicate when teaching [START_REF] Sfard | Ritual for ritual, exploration for exploration: Or, what learners are offered is what you get from them in return[END_REF]. In the following, I will first briefly outline key concepts from the commognitive framework related to characteristics of geometrical discourses. Then I will introduce the context of the study and illustrate identified mathematical problems with excerpts from the discussions.

Theoretical background

The movement towards a geometrical discourse that includes hierarchical definitions are, by Sfard (2008, p. 256) called meta-level development, because the rules that governs the participation change and words are used differently than colloquial discourses. Taking a participationist view, these changes must be mediated by an experienced participant in the desired discourse (Sfard, 2008, p. 254). I therefore argue, as Sfard (2017, p. 55), that being able to capture mathematical problems PSTs address in their attempts it is necessary to go into the finest details of the geometrical discourse offered. From a commognitive perspective (Sfard, 2008, p. 93), this means the use of words (e.g., parallelograms) and visual mediators (e.g., shape makers created for the sake of communication) to surface how geometrical objects are perceived and the construction and substantiation of geometrical narratives (stories about geometrical objects) to characterize the routines (e.g., how to identify shapes) employed in the classroom. Sfard (2008, p. 228) describes orders of geometrical discourses, where each subsequent discourse is a meta-discourse of the former and geometrical objects typically arise as a result of saming lower order objects. Drawing on Sfard's work, Sinclair and Moss (2012, p. 31) present three orders of geometrical discourses relevant for school. In first and second discourse, identification routines rely on a continuous transformation of a figure into a prototype and some informal properties check of necessary and specific properties. The words are mainly used as proper names of concrete objects or as family names of exclusive classes of concrete objects. In the third geometrical discourse, transformability is no longer the official identification routine. Here, recognition needs to be followed by a discursively mediated procedure, such as, checking if the conditions described in a definition holds or not. The set of properties listed in a definition are seen as necessary and sufficient conditions for determining membership in a category. This means that the new routine opens up for the possibility to call a figure by several names [START_REF] Sinclair | The more it changes, the more it becomes the same: The development of the routine of shape identification in dynamic geometry environment[END_REF].

Studies (e.g., [START_REF] Millsaps | Challenging preservice elementary teachers' Image of Rectangles[END_REF][START_REF] Sinclair | The more it changes, the more it becomes the same: The development of the routine of shape identification in dynamic geometry environment[END_REF] report that dynamic environments are helpful to move towards more sophisticated geometrical discourses. It is especially the actual witness of continuous transformations of shapes when dragging a figure that is promising because students flex their prototypical range of shapes they are ready to call the same [START_REF] Sinclair | The more it changes, the more it becomes the same: The development of the routine of shape identification in dynamic geometry environment[END_REF]. However, according to [START_REF] Sinclair | To be or to become: How dynamic geometry changes discourse[END_REF], the transitions between dynamic and static media are challenging, because the transitions influence the talk about geometrical objects. For instance, in dynamic geometry, identification routines become time-dependent and human-based operations that invite students to act on the figure. In addition, [START_REF] Millsaps | Challenging preservice elementary teachers' Image of Rectangles[END_REF] found that PSTs struggle to move between the two discourses when they explore shape makers that embody hierarchical definitions.

Methods

The PSTs in this study are undertaking a one-year mathematics course as part of a four-year Norwegian teacher education program for lower secondary school teachers. They were provided opportunities to move back and forth between campus and practicum to collaboratively prepare for, enact and reflect on their teaching. In their preparation, the PSTs investigated seven shape makers. See Figure 1. Each maker generates a category of shapes with all necessary properties of specific quadrilaterals. The PST rehearsed discussions about these makers where they all acted, in turns, as teachers and students. Afterwards, four practicum groups planned a lesson using [START_REF] Stein | Orchestrating productive mathematical discussions: Five practices for helping teachers move beyond show and tell[END_REF] framework to prepare for classroom discussions in authentic classrooms. One PST taught the lesson. Others video recorded the activity. The videos and transcripts of the enactments and the rehearsals are data in this paper, as well as PSTs' portfolios about the tryouts. The focus is the geometrical discourses that surfaced in four classroom discussions with students. In addition, the rehearsals and the portfolios provide a window to PSTs' geometrical discourses before and after the enactments in the classroom.

The transcriptions of the classroom discussions were first organized into mathematical episodes that encompass the whole discussion around a claim about the shape makers. 55 episodes were identified and categorized in three groups; 1)17 episodes about necessary and specific properties; 2)18 episodes about the naming process; 3) 10 episodes addressed hierarchical relations. Since classification is closely related to definitions (de Villiers, 1994), I analyzed all episodes focusing on the use of words and routines related to orders of geometrical discourses [START_REF] Sinclair | The more it changes, the more it becomes the same: The development of the routine of shape identification in dynamic geometry environment[END_REF]. I looked especially for instances in the talk that seemed to constrain or promote the geometrical discourse in order to capture mathematical problems the PSTs address in these complex settings.

Findings and analysis

The analysis shows that the four PSTs, called by pseudonyms Andy, Fay, Joy and Lisa, manage to create opportunities to talk about hierarchical relations in their discussions. However, PSTs struggle to address the problems that arise in the learning-teaching situations.

The problem of moving between dynamic and static ways of talking about geometrical objects

This problem arises in almost all of the hierarchical episodes, which resonates with what [START_REF] Sinclair | To be or to become: How dynamic geometry changes discourse[END_REF] and [START_REF] Millsaps | Challenging preservice elementary teachers' Image of Rectangles[END_REF] report in their studies. For instance, in Lisa's, Joy's and Fay's first hierarchical episode they ask students if the diagonal angles are perpendicular in Figure B (the rectangle maker). The students answer "no." In contrast to squares, perpendicular diagonals is a specific property in rectangles. The PSTs exploit the opportunity offered and similar problems arise. However, they address them differently as illustrated by excerpts from Joy's and Lisa's discussion.

In Joy's discussion, another student disagrees and says; "yes, you can make the diagonals perpendicular. If you make it into a square. Just take it in" [19]. The student uses the terms "make it" and "take it in" to describe the dynamic movement involved in dragging the rectangle maker until it looks like a square. Joy acknowledges the suggestion and demonstrates the transformation-based routine on the smartboard.

Joy:

Then it is not a rectangle anymore, it is a square. But a square is also the same as a rectangle. (3s) It can be a rectangle, but a rectangle cannot be a square.

(2s). [20]

Joy reinforces the dynamic, exclusive way of talking about a rectangle becoming another shape, a square, in the first sentence [START_REF] Millsaps | Challenging preservice elementary teachers' Image of Rectangles[END_REF]. Then she contradicts this statement using an inclusive way of talking. Finally, she concludes with a dynamic and incorrect way of talking about the asymmetric relation between a square and rectangle. In the rehearsals, Joy's group addressed the same problem using the words "can be" and "cannot be" to express these asymmetric relations as well. In contrast, they used "can be" appropriately ("A rectangle can be a square, but a square cannot be a rectangle"). This shows that Joy struggles to articulate hierarchical relations when she moves between dynamic and static ways of talking about figures. Joy does not address these contradictory statements and she avoids hierarchical episodes in the rest of the discussion.

Lisa was in the same rehearsal group as Joy. In her discussion with students, she manages to use the term "can be" appropriately when she explains why diagonals "can be" made perpendicular in Figure B. She states the following inclusive narrative: "This is because a rectangle also can be a square" [43]. However, Lisa does not address the opposite asymmetric relationship, as Joy did. Nor does she verify the claim by using the hierarchical definitions she introduced to the students in the beginning of the lesson. In the other hierarchical episodes, Lisa repeats the same narrative, replacing "rectangle" with "rhombus" and "parallelogram." To conclude, the way the PSTs' address the problem of moving between dynamic and static ways of talking about geometrical objects seems to constrain the kind of mathematics students can engage in during the discussions.

The problem of building on students' prototypical and partitional contributions

In Fay's and Andy's discussions, another problem surfaced. Namely, the problem of building on students' prototypical and partitional contributions. Diagonals "can be" perpendicular: Fay's first hierarchical episode surfaced in the discussion about perpendicular diagonals being a specific property in rectangles. Fay drags the rectangle maker to show that it is possible to make the diagonals perpendicular. When Fay asks if it can be named anything else than rectangles [50], a student suggests that it can "if we take it in" [51]. Fay reinforces the dynamic, exclusive way of talking about shapes when she uses the terms "become" and "could it" in her following questions; "what had it become then?" [52] and "could it be a square?" [54]. A student confirms and Fay concludes by saying "But, then the sides are of equal length, right?"[58].

Diagonals "are not" of equal length: After the discussion about squares and rectangles, Fay asks a student about the diagonals in Figure D (the parallelogram maker). "Are they not of equal length?"[61], a student replies. Fay asks if "they look equal?" [62]. The student revises his answer ("no, no, just kidding"[63]). Fay replies "OK" and concludes with a prototypical claim that diagonals "are not equal" [64]. Equal and perpendicular diagonals are specific properties to parallelograms and necessary properties to rectangles and rhombs, respectively. There are therefore many possible solutions and Fay has to make a decision which one to pursue. In this situation, Fay does not use dragging to explore these "can be" possibilities as she did in the discussion about rectangles "becoming" a square. She chooses to follow the path of "not equal diagonals." In this situation, the student contributes with a prototypical and partitional claim about parallelograms [START_REF] Hershkowitz | Psychological aspects of learning geometry[END_REF]. Fay repeats the student's narrative and draws attention towards the angles in the vertices [78]. The phrasing of the questions has changed from the dynamic "could this be" to a static "will this be" 90 degrees. However, the student's suggestion in [85] provides Fay with an opportunity to search for counter examples where the angles "can be" 90 degrees and therefore flex the student's prototypical claim [START_REF] Sinclair | The more it changes, the more it becomes the same: The development of the routine of shape identification in dynamic geometry environment[END_REF]. She follows the student's suggestion but drags the figure so that the diagonals become perpendicular, not the vertices [86].

Figure
The potential opportunity provided by the student therefore becomes a confirming example, which is not sufficient to substantiate that the vertices "will not be" 90 degrees nor to convince the student that not all parallelograms are slanted. Fay does not address this problem in the discussion, but uses the dragging experience to verify that figure D cannot be a square, at least [93].

In parallelograms, the diagonals are not equal, the vertices are not 90 degrees, and the diagonals are not perpendicular: Fay drags the parallelogram maker back to a prototypical parallelogram.

95 Fay: Okay, we say that the diagonals are not of equal length, and the vertices are not 90 degrees. Are these 90 degrees? The angles inside here? (Points at the diagonal angles). 96 Student:

No 97 Fay:

Not 90 degrees? Yes, maybe this is a recipe for a parallelogram?(2s) Maybe. 98 Student: Maybe 99 Fay:

Yes, because this is the only one we have seen until now that have different lengths on the diagonals. (Points to the diagonals)

During the discussion, Fay has shifted from talking about necessary properties used as conditions for determining membership in a category of specific quadrilaterals, such as squares and rectangles, to talking about conditions that exclude membership in a category of parallelograms. Fays advocates a "recipe" [97] where excluding conditions, such as "diagonals are not equal" and "vertices are not 90 degrees," are sufficient conditions for naming a figure a parallelogram, because parallelograms "is the only one" they have seen that have different length of diagonals [99]. Furthermore, she claims that diagonal angles are not 90 degrees in parallelograms, despite the fact that she recently presented a counter example (the slanted rhombus). Thus, the dynamic perspective is absent when Fay talks about the properties of parallelograms, which I believe happened when she decided to follow the path of "not equal diagonals"[64] instead of the dynamic "can be" talk she promoted earlier. The other critical learning-teaching situation was when Fay decided to build on the prototypical claim about parallelograms being slanted [77]. I have tried to show how these decisions changed the inclusive talk that started in the first hierarchical episode where rectangles could become squares, and ended as a static and exclusive talk where rectangles and squares are excluded as parallelograms. These problems, I believe, affected the conflicting geometrical discourses that surfaced in the final hierarchical episode in Fay's discussion.

Parallelograms cannot be a rectangle, but a rectangle is a parallelogram: In her final hierarchical episode, Fay recalls a peer PST's teaching some weeks ago to sum up and end the discussion.

Concluding remarks

Several studies have used interviews or questionnaires to examine PSTs' hierarchical reasoning and they report that PSTs struggle even though they have taken geometry courses as part of their teacher education program (e.g., Fujiti & Jones, 2007;[START_REF] Millsaps | Challenging preservice elementary teachers' Image of Rectangles[END_REF]. It is therefore no surprise that PSTs in this study meet similar challenges when they are trying to lead classroom discussions about defining and classifying quadrilaterals in authentic classrooms with students. However, what challenges and how they tackle them in this setting has not been sufficiently investigated.

The analysis show that the PSTs are in their early process of individualizing hierarchical relations between quadrilaterals. They manage to create opportunities for students to engage in hierarchical relations. However, none of the PSTs use hierarchical definitions in their reasoning about the shapes. This resonates with [START_REF] Fujita | Learners' understanding of the definitions and hierarchical classification of quadrilaterals: Towards a theoretical framing[END_REF] reporting that PSTs prefer to rely on specific examples when identifying shapes, even though they know definitions. When PSTs need to step in and out of dynamic and static discourses, they struggle to connect and articulate hierarchical relations as illustrated in Lisa's and Joy's discussions. Millsaps (2013, p. 38) reports on similar challenges where experiences with shape makers actually can reinforce some aspects of existing partitional definitions. She therefore highlights discussions about the use of classification narratives to support the transformation from partitional to hierarchical definitions (Millsaps, 2013, p. 40).

Trying to interpret and build on incomplete contributions from students is challenging for PSTs [START_REF] Lampert | Using designed instructional activities to enable novices to manage ambitious mathematics teaching[END_REF]. I believe Fay's discussion illustrates how difficult it is for PSTs to balance between honoring students' participation and promoting mathematical discourse in classrooms.

Especially when the discussion requires a sensitivity to step in and out of geometrical discourses when needed and provide opportunities for students to engage meta-level development [START_REF] Sfard | Thinking as communicating: Human development, the growth of discourses, and mathematizing[END_REF].

Figure 1 :

 1 Figure 1: Four of the seven Shape Makers of specific quadrilaterals

  I use excerpts from Fay's talk about parallelograms (Figure D) to illustrate how students' contributions and the mathematical discourse seem to constrain and open up what PSTs can choose of possible solutions during the discussion.

  Yes, they are parallel and they are parallel. Right, this and this. (Points to the pair sides in the rectangle) They just go upwards and will never intersect each other. Yes. (2s) Figure B, it is in fact a parallelogram as well, but it can also be a rectangle.In this episode, Fay asks students to compare rectangles and parallelograms. First, she asks if parallelograms can be a rectangle[109]. She waits for 7 seconds and there is no response. Fay simplifies the question by drawing attention to the vertices in the slanted parallelogram in FigureDand recalling that in rectangles they must be 90 degrees[109]. Several students reply no, which is correct according to visual recognition and the offered partitional definition that excludes rectangles from being parallelograms. Then Fay prompts for the opposite relation between rectangles and parallelograms asking if a rectangle "can be" a parallelogram[111]. The student claims yes, and she uses the lengths of the opposite sides as a necessary and sufficient property to determine membership in parallelograms[114]. The student uses the definition of parallelograms that most students in lower secondary school know. Fay acknowledges the contribution and she demonstrates what sides are parallel[115]. Fay ends up with two conflicting narratives, which are both endorsed using "accepted" definitions of parallelograms by the class community. The first uses the vertices not being 90 degrees to exclude rectangles from being parallelograms[109]. The second uses the opposite parallel sides to include rectangles as parallelograms[115]. Both suggestions are based on student's contributions and Fay's attempt to build on them. Fay does not identify the problem, which resonates withMillsaps' (2013, p. 38) study that reports on PSTs being capable of holding two conflicting claims about hierarchical relations when they are moving between dynamic and static geometry.

	115 Fay:
	109 Fay:	He talked about a rectangle that could be a parallelogram and the opposite.
		(.) Or, can parallelograms be a rectangle? (7s) Can it? (2s) Can this be a
		rectangle? (3s) (Points at the parallelogram maker) If I say to make this a
		rectangle, it must be 90 degrees at the angles here. (Points at the vertices)
		Can it?
	110 Students:	No
	111 Fay:	No, but can this (Points at the rectangle maker) be a parallelogram?
	112 Student:	Yes
	113 Fay:	Yes. Why?
	114 Students:	Because the opposite sides are parallel.


Camille Krewcun

Emilie Pery

Nicolas Combaret

Pascal Motreff

Laurent Sarry

Émilie Péry

Simulation of Balloon-Expandable Coronary Stent Apposition with Plastic Beam Elements

Keywords: Simulation, Stent deployment, Finite Element Method, Beam element, Plasticity

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Introduction

The coronary artery disease results from a physiological ageing process causing a progressive narrowing (stenosis) of the artery lumen. In this study, we focus on the treatment of the disease by balloon inflation (i.e. angioplasty) and stent deployment. The procedure outcomes highly depend on the stent final geometry and on the accuracy of the deployment. Consequently, numerical simulation, if reliable, can be a valuable asset to minimise complications. We find in literature an important number of studies using the Finite Element Method (FEM) to simulate the deployment of coronary stents. Several studies aim at the stent design optimisation: see for instance [START_REF] Migliavacca | Mechanical behavior of coronary stents investigated through the finite element method[END_REF] (stent alone), [START_REF] Beule | Realistic finite element-based stent design: The impact of balloon folding[END_REF] (stent and balloon) or [START_REF] Zahedmanesh | Simulation of a balloon expandable stent in a realistic coronary arterydetermination of the optimum modelling strategy[END_REF] (stent, balloon, and artery). As design optimisation gives priority to precision over computational time (e.g. 48 hours in [START_REF] Beule | Realistic finite element-based stent design: The impact of balloon folding[END_REF]), the corresponding models can not be used directly in a clinical environment. In this paper, we propose a faster simulation routine to model accurately the deployment of a balloon-expandable stent. Our objective is to reach an execution time compatible with clinical routine, so that our simulation method could be used in practice as a relevant prediction tool (typically in patient-specific applications). We decide to base our model on the discretisation of the stent geometry by 1D serially linked beam elements. Similar work was proposed by Čanić and Tambača in [START_REF] Čanić | Cardiovascular stents as pde nets: 1d vs. 3d[END_REF], where the stent structure is modelled by an assembly of 1D rod elements. Although limited to a linear elastic behaviour (auto-expandable stents), the simulation is much faster than for design optimisation. We decide to use beam elements in a similar way, in order to model the more complex (nonlinear) deformation undergone by a balloon-expandable stent. The use of connected beam elements has already been proved efficient in medical simulation to model slender structures, such as endovascular coils for brain aneurysm in [START_REF] Dequidt | Interactive simulation of embolization coils: Modeling and experimental validation[END_REF] or flexible needles in [START_REF] Adagolodjo | Inverse real-time finite element simulation for robotic control of flexible needle insertion in deformable tissues[END_REF]. The use of beam elements in these studies allows to achieve low computational times (up to interactive simulations), but once again is limited to elastic deformations.

Method

Stent discretisation

In order to benefit from the slender shape of the stent struts, we use serially linked 1D beam elements to discretise the entire stent structure. Each beam element is represented by two nodes, each of them described by 6 Degrees of Freedom (DoF), 3 for position, and 3 for orientation. The approximation of the continuous medium between the nodes can be written in matrix form as:

u(x, y, z) = N(x, y, z)ũ, (1)
where u is the continuous displacement field inside the element, (x, y, z) are the material coordinates, and ũ is the 12 × 1 vector of nodal displacement. N is a 3 × 12 matrix containing Timoshenko interpolation shape functions, which expression is given in [START_REF] Bazoune | Shape functions of three dimensional timoshenko beam element[END_REF]. The Timoshenko beam model notably allows to take into account shear deformations and describe 'thick' beams more accurately. We encounter this type of beams in highly curved parts of the stent. The mesh conception is made with Computer Aided Design. The dimensions of the stent are retrieved from the micro-CT acquisition of a crimped metallic stent. From these measures, we can reproduce a flat version of the stent mesh, composed of 2D edges.The mesh is then wrapped up into a cylindrical shape, each edge corresponding to a beam element.

Finite Element workflow

Our Finite Element (FE) implementation entirely relies on the open simulation framework SOFA 3 . A thorough description of most of the mechanisms involved in the SOFA simulation workflow can be found in [START_REF] Faure | SOFA: A Multi-Model Framework for Interactive Physical Simulation[END_REF].

Very briefly, we base our simulation on the (implicit) Backward Euler numerical scheme. This leads to solving at each time step the following mechanical system:

(M -hB -h 2 K) A ∆v t+h = hf t (x, v) + h 2 Kv t b , (2)
in which M is the mass matrix of the discretised mechanical system (i.e. if the system is described by n DOFs, then M is a n × n matrix), x is the global position vector, and v the velocity. Subscript t indicates that the variable is considered at time t, and h is the time step. f stands for the forces, K = ∂f ∂x for the stiffness matrix and B = ∂f ∂v the damping matrix. The system in ( 2) is solved in ∆v t+h at each time step. From this, the simulation can be progressed forward in time.

Mechanical model

Elasticity In elasticity, the relation between the stress tensor σ and the strain tensor is given by Hooke's law, and can be written in matrix form as:

σ = C . (3)
The explicit expression of C as a matrix can be found in [START_REF] Przemieniecki | Theory of Matrix Structural Analysis[END_REF]. We stress out that and σ are actually second order tensors, which can be written as 6 × 1 vectors without loss of information thanks to symmetry.

From (3) and the virtual work principal, the 12 × 12 element stiffness matrix K e can be expressed as:

K e = Ω B e T CB e dΩ, ( 4
)
where Ω is the 3D domain on which the element is defined, and B e is a 6 × 12 matrix obtained by spatial derivation of N. Expression of B e results from the small strain hypothesis, which is expressed in [START_REF] Przemieniecki | Theory of Matrix Structural Analysis[END_REF]. For details of the computation of K e from the principle of virtual work, we also refer the reader to [START_REF] Przemieniecki | Theory of Matrix Structural Analysis[END_REF].

Plasticity In plasticity theory, the constitutive law (3) varies during the deformation, following two phases:

an elastic phase, during which the internal stress depends linearly on the strain, as in (3), a plastic phase, during which part of the strain can be distinguished as resulting from plastic energy dissipation, and the stress-strain relation becomes nonlinear.

The transition from the elastic to the plastic phase is described by a yield criterion f : R 6 → R, defined on the stress space. In this study, we use the Von Mises yield function, which expression is given in [START_REF] Hughes | Theoretical foundation for large-scale computations of nonlinear material behavior[END_REF].

The computation of an acceptable plastic stress state relies on the decomposition of the strain into a plastic and an elastic component: = el + pl . In the following, we adopt a model of perfect plasticity, as described in [START_REF] Hughes | Theoretical foundation for large-scale computations of nonlinear material behavior[END_REF]. The term 'perfect' indicates that the yield surface, described by f (σ) = 0 in the stress space, also remains unchanged. The evolution of the plastic strain is described by an associative flow rule, expressed as:

d pl = dλ ∂g ∂σ . (5)
We use the radial return algorithm as described in [START_REF] Hughes | Theoretical foundation for large-scale computations of nonlinear material behavior[END_REF], to compute a new plastic stress state σ t+h at each time step. Details of the computation can be found in [START_REF] Hughes | Theoretical foundation for large-scale computations of nonlinear material behavior[END_REF].

Once σ t+h is known, we may finally compute the resulting internal forces, and linearised stiffness matrix, so that they are taken into account in the global mechanical system (2). The new internal forces are simply computed by integration:

f int (σ t+h ) = Ω B e T σ t+h dΩ. ( 6
)
The linearised version of the stiffness matrix (or tangent stiffness matrix) K t can be expressed in a similar analysis way as for (4), leading to:

K t = Ω B e T dσ d B e dΩ. (7)
Starting from Hooke's law, we have:

dσ = C d el = C(d -d pl ) = C(d -dλ ∂g ∂σ ).
Through calculus, and using the consistency condition (∇f T dσ = 0), we obtain an equivalent differential relation between stress and strain in plastic deformation:

dσ =   C - C ∂f ∂σ ∂f ∂σ T C ∂f ∂σ T C ∂f ∂σ   d = C ep d . (8)
This allows to explicitly express the tangent stiffness as:

K t = Ω B e T C ep B e dΩ. (9)
In practice, we use Gaussian reduced integration at each time step to compute the element internal forces with [START_REF] Hughes | Theoretical foundation for large-scale computations of nonlinear material behavior[END_REF], and update the tangent stiffness matrix with [START_REF] Tournier | Stable Constrained Dynamics[END_REF]. Once K e is computed for each element, all the matrices are assembled into the global stiffness matrix K in (2).

Balloon and contact constraints

We simulate the balloon indirectly by attaching a spring to each of the stent nodes. The other end of each spring is attached to a particle on a virtual surface, representing the balloon membrane. At time t = 0, the springs rest lengths are defined to maintain the system in equilibrium.

During the simulation, the surface particles are moved in the radial direction so that the diameter of the virtual membrane corresponds to the one given in the balloon compliance table. During the simulation, we gradually increment a virtual pressure to reproduce the actual increase in pressure, controlled by the physician. The characteristics of the springs (stiffness and damping) are fixed so that the stent diameter in the simulation is coherent with the compliance table . As the compliance table generally starts at pressures close to the nominal pressure, we complete the table with two entries, corresponding to 0 atm, and to the first pressure measured experimentally for which the stent becomes fully cylindrical (p cyl ). We then use a linear-by-parts model for pressure increasing in the simulation: the first part is a linear interpolation between 0 atm and p cyl , while the second part is fitted on the p cyl value and the constructor compliance table.

We handle contacts between the topological primitives composing the stent (edges) and the artery (triangles) with a collision pipeline available in SOFA. At each time step, if a collision between two primitives is detected, a response is computed accordingly in the form of a Lagrangian constraint. Briefly, the addition of constraints involves an additional step in the numerical solution:

-At first, (2) is solved without constraints, giving a new position (free motion).

-Then, a modified constrained system is solved from the free motion:

A∆v t+h = b + hH T λ, ( 10
)
where H is the Jacobian of the constraint expressions, and λ contains the Lagrange multipliers associated to each constraint. In SOFA, this system is typically solved iteratively using a Gauss-Seidel algorithm. A more detailed description of the process is available in [START_REF] Faure | SOFA: A Multi-Model Framework for Interactive Physical Simulation[END_REF].

Experimental validation

To propose a first assessment of the simulation, we conducted an experiment involving the apposition of a stent in a perfectly straight coronary artery phantom.

We chose silicone (Sylgard 184, Dow Corning, Midland, Michigan, U.S.A.) to reproduce the artery, as the material has already been successfully used to create phantom vessels. In order to exactly control the vessel geometry, we used additive manufacturing to create a plastic mould and a water-soluble stick made in PolyVinyl Alcohol (PVA). The mould gives the external shape of the phantom, while the lumen geometry is entirely define by the stick (once dissolved).

Following the phantom creation, we actually expand a stent inside the mock artery, under micro-CT supervision. We proceed by gradually incrementing the inflation pressure of the balloon, and realising a CT acquisition at each incremental step. The resulting 3D reconstructions contain the geometry of the stent in transient states, from a pressure of 0 atm (crimped stent) to the nominal pressure (fully expanded stent). The deformation can be compared to the output of the simulation. As the artery phantom was obtained from 3D-printed elements, the 3D artery geometry to be included in the simulation is known. In this case, we use a surface mesh, on which we add an elastic behaviour [START_REF] Tournier | Stable Constrained Dynamics[END_REF], already implemented in SOFA.

Results

As a first validation, we simulated the expansion of a Synergy II coronary stent (Boston Scientific, Marlborough, Massachusetts, U.S.A) in a perfectly cylindrical artery model, which we reproduced experimentally. The artery diameter is chosen to match the stent nominal diameter (3 mm). Pressure in the experiment was increased from 0 atm to 11 atm, while carrying out micro-CT acquisition at 5, 6, 7, 9 and 11 atm. We increment the pressure in the simulation by following the linear-by-parts model mentioned above. We export the stent geometry at regular pressure increments (every 200 iterations, corresponding to ∆P=0.44 atm) between 0 and 11 atm. Comparison is made between a pointset extracted from the micro-CT 3D reconstruction, and the stent nodes. We register the two pointsets in two steps: first by superimposing the principal axes of inertia and centres of gravity (rigid transform), and then by minimising the Euclidean distance between simulated points and their closest neighbours in the CT pointset (rigid rotation around the common axis). We compare the poinset of each acquisition to all the exported stent geometries. We choose as best correspondence the one minimising the mean radial distance dr and Euclidean distance dE between the pairs of points. Table 1 gives the best matching simulated stent state for each micro-CT acquisition, with the number of iterations (it), the corresponding simulated pressure and diameter, and the minimised metrics values. 3D rendering of the registration output is given in Fig. 1 for a crimped and a deployed configurations.

Discussion and conclusion

The first remark which has to be made regarding our results is the significant difference between the experimental pressure and the simulated pressure in the best corresponding configuration. This discrepancy can be explained by the complete lack of data in the constructor compliance table, for pressure values below 8 atm. Consequently, the first part (P ≤ 5 atm) of the model that simulates the pressure increase is too approximative. This is evidenced by the almost identical diameters measured in the acquisitions for P = 5 atm and P = 6 atm, which can't correspond to a linear model. An issue with developing a more accurate pressure/diameter model for low pressure values is that the stent experience asymmetrical deformation for P ≤ 5 atm. As presented by De Beule et al. in [START_REF] Beule | Realistic finite element-based stent design: The impact of balloon folding[END_REF], this phenomenon, called dog boning can only be reproduced if a realistically folded balloon is simulated.

If we don't consider early transient states, Zahedmanesh et al. showed in [START_REF] Zahedmanesh | Simulation of a balloon expandable stent in a realistic coronary arterydetermination of the optimum modelling strategy[END_REF] that an approximated balloon model is able to accurately retrieve the final configuration of the stent. In this study, it is indeed the case for the nominal pressure of 11 atm, for which the acquisition and the simulation are in accordance. In addition to that, considering only the diameter independently of the pressure, we can see that the simulation is also able to reproduce transient states as long as the stent is fully cylindrical. On the correspondences we obtain, the distance errors are thus in the same order of magnitude as the stent strut thickness (50 µm for the Synergy II).

Regarding the computational time, the plastic mechanical model we imple-mented can be handled at 4 Frames Per Second (FPS), allowing a stent mesh composed of 1622 beam elements to reach the artery diameter in about 15 minutes, without any form of optimisation (single thread CPU execution on an Intel Xeon E3-1270 v5 (8CPUs) 3.6 GHz, 16 GB RAM, Windows10 64 bits). As soon as contact occurs, the number of FPS decreases, but the computational time stays significantly lower than for fully volumetric meshes.

In this paper, we present a complete methodology to realistically simulate the plastic deformation of a balloon-expandable stent with 1D serially linked beam elements, in a reasonable time. Presently, clinical assessment of stent apposition almost exclusively relies on the evaluation by the physicians of the artery diameter, before and after the procedure. In this context, we believe that a stent apposition simulation, fast and reliable enough to be used in clinical routine, could provide additional information to the physician. In order to progress towards this clinical use, we propose to improve the present simulation by optimising our code (addressing the issue of computational time), and by developing simulation and experiments with more realistic artery geometries.

Fig. 1 .

 1 Fig. 1. 3D rendering of the rigid registration between the last micro-CT acquisition (deflated balloon) and the simulation best correspondence. The CT reconstruction is displayed in grey, while the simulation mesh nodes are superimposed as red points.

Table 1 .

 1 Comparison between experimental and simulated stent expansions.

	Experimental Theoretical (compliance)	Simulation best correspondence
	P (atm) ∅ (mm)	∅ (mm)	it P (atm) ∅ (mm) dr (µm) dE (µm)
	5	2.496	-	2000 4.4	2.540	40	150
	6	2.580	-	2000 4.4	2.540	30	150
	7	2.710	-	2200 4.84	2.714	30	80
	9	2.876	2.96	3200 7.04	2.884	30	90
	11	3.004	3.08	4400 9.68	2.992	27	90

www.sofa-framework.org

Acknowledgements

The authors would like to thank Arnaud Briat from the multi-modal imaging facility IVIA (In Vivo Imaging Auvergne), Clermont-Ferrand, France, for their assistance in the acquisition of the micro-CT images. We also thank François Wastable for helping with the phantom manufacturing.