Gabin Rolland

Romain Vuillemot

Wouter Bos

Nathan Rivière

Characterization of Space and Time-Dependence of 3-Point Shots in Basketball

Understanding characteristics of 3-point shots is paramount for modern basketball success, as in recent decades, 3-point shots have become more prevalent in the NBA. They accounted for 33,6% of the number of total shots in 2017-2018, compared to only 3% in 1979-1980 [1]. In this paper, we aim at better understanding the connections between the type of 3-point shooting (catch-and-shoots and pull-ups) and the timing for shooting, using two distinct space-time models of player motion. Those models allow us to identify individual behavior as a function of specific defensive settings, e.g. shot-behavior when a player is guarded closely. We assess our models using SportVU data for specific NBA players and our code is open-source to enable more players and teams explorations, as well as to support further research and application of those models, beyond basketball and sport.

Introduction

When the 3-point-line was introduced in 1979, on average 2.8 shots were taken per game. Since then, this number increased roughly by a factor of ten, up to ~27 shots per game [START_REF]NBA & ABA League Index[END_REF]. The influence of 3point shots has thereby radically changed the NBA and has become a major asset for the most successful teams. An illustrative example are the Golden State Warriors for whom 3-point shots following a counter-attack were decisive during the 2017-2018 season. They won that season taking 654 3-point shots in the first seven seconds after a change of possession for a success rate of 43% [START_REF] Arnovitz | How the NBA got its groove back[END_REF]. Understanding the factors which influence the 3-point performance is therefore extremely valuable as illustrated by either predictive or descriptive carried out in sports analysis.

Our work relates to players' influence zone models, and their dependence on players' velocity. In [START_REF] Fujimura | Geometric Analysis and Quantitative Evaluation of Sport Teamwork[END_REF], [START_REF] Taki | Visualization of dominant region in team games and its application to teamwork analysis[END_REF] and [START_REF] Chawla | Classification of Passes in Football Matches Using Spatiotemporal Data[END_REF] for each player a dominant region was calculated using Voronoi tessellation, adapting the metric using different motion models adapted to the players' dynamics. Fernandez and Bornn used statistical techniques to characterize players' space occupation [START_REF] Fernández | Wide Open Spaces: A statistical technique for measuring space creation in professional soccer[END_REF]. Other investigations focused on events occurring in a match, such as passes or shots. For instance, different methods were developed to predict shooting and passing possibilities in basketball [START_REF] Yue | Learning Fine-Grained Spatial Models for Dynamic Sports Play Prediction[END_REF], or describing important plays in football matches [START_REF] Perin | SoccerStories: A Kick-off for Visual Soccer Analysis[END_REF]. A different analysis tool [START_REF] Stein | Revealing the Invisible: Visual Analytics and Explanatory Storytelling for Advanced Team Sport Analysis[END_REF] based its assessment of football passes on space occupation models. Several studies are related to shot performance in basketball: [START_REF] Goldsberry | CourtVision : New Visual and Spatial Analytics for the NBA[END_REF] proposed an interface allowing to compare shooting performance from different players; [START_REF] Cervone | POINTWISE: Predicting Points and Valuing Decisions in Real Time with NBA Optical Tracking Data[END_REF] introduces a metric associated with the possible number of points scored in an action; [START_REF] Bailey | Understanding Features of Successful 3 Point Shots in the NBA[END_REF] describes a statistical study on the influences of different factors which influence 3-point performance and tries to classify shooters behavior; Finally, [START_REF] Chang | Quantifying shot quality in the NBA[END_REF] introduces two metrics to respectively characterize shot quality and shot ability.

In this paper we aim at better understanding 3-point shots in basketball, as those are frequent and decisive situations during NBA games. In particular we focus on catch-and-shoot and pull-up shots and their dependence on defense settings. We first introduce our methodology to extract 3-point attempts (discrete events) from SportVU spatio-temporal data (players and ball position timeseries). We then present our space-time models designed to quantify available space for a single player relative to the space occupied by other players. One of our models measures only the distances between players (static model, Figure 2 (a)), while the other one takes into account the speed and direction of the players (dynamic model, Figure 2 (b)). Those two models allow us to characterize defensive dynamics and to analyze its impact on 3-point efficiency. All our code and analysis are available as an open-source project https://github.com/amigocap/MecaSportStats/ to support further research and application of those models.

Extracting Pull-up and Catch-and-Shoot Data from SportVU

Our study relies on motion data of player and ball positions during 632 matches from the 2015-2016 NBA season, supplied by STATS SportVU. We were able to detect 3-point attempts by tracking back the ball paths that crossed a circular horizontal surface of radius 5 feet above the basket (at 10 feet height) and looking at the position of the latest ball owner at shot time. Figure 1 illustrates the distribution of such shots (circles) attempted by Stephen Curry. Shot time was defined as the time just before the ball's vertical angle was higher than 70 degrees tracking back from the moment ball's height reaches 10 feet. Figure 1 distinguishes hits and miss shots (respectively blue and red) which were differentiated by evaluating the trajectory of the ball and its intersection (or not) with the plane of the basket. After identifying 3-point attempts, we differentiate two subclasses of shots, catch-and-shoot and pullup shots. A catch-and-shoot shot consists of the play where a player receives a pass and shoots less than two seconds after the pass without dribbling. Conversely, a pull-up shot is a shot taken after a dribble. To differentiate these shots, we had to set the release time which is the delay between the time the player catches the ball (𝑡 "#$"% ) and the ball releases the shooter's hands (𝑡 &%'$ ). To do so, we determine the time the shooter catches the ball by looking at when the ball was within 1.6 feet of the player for the first time. Then a shot was considered as catch-and-shoot if release time was less than 2 seconds and if the shooter did not dribble before shooting, e.g. if the ball did not pass under 2 feet.

As a result, we found 26 332 valid shots; we discarded shots with noisy or data gaps. 72% of these shots were catch-and-shoot ones. We found 9136 hits and 17196 misses, corresponding to an overall success rate of 34,7%. We obtained a catch-and-shoot success rate of 36% against 31,5% for the pullup ones which is consistent with previous studies [START_REF] Cheema | The Dying Art of the Catch-And-Shoot Shot[END_REF]. For all these shots, we recovered the position of the shooter and his closest defender three seconds before the shot in order to evaluate several features such as the distance of the closest defender at the shot time illustrated by black segments in Figure 1. This quantity is useful to characterize the opening of a shot and allows to compare to the definitions of the NBA stats website [START_REF]NBA Stats[END_REF] defining 4 levels of opening for a shot according to the distance of the nearest defender: from 0 to 2 feet it is a very tight shot, from 2 to 4 feet it is a tight shot, from 4 to 6 feet it is an open shot and more than 6 feet it is a wide-open shot.

Players Occupation Models

To identify the occupation zones of a player, one needs to take into account the positions of all players from both teams. The simplest approach is to determine for every point in space the closest player. This allows to subdivide the court into 10 Voronoi cells, defined as follows: a point belongs to a player's cell if and only if it is closer to the player than to any other player [START_REF] Voronoi | Nouvelles applications des paramètres continus à théorie des formes quadratiques. Deuxième Mémoire. Recherches sur les paralléloèdres primitifs[END_REF]. This approach is however limited by its discrete, binary nature and does not account for other players' orientation and inertia (i.e. direction and motion). 2). An animated version of the dynamic model is available at https://amigocap.github.io/MecaSportStats/video.mp4.

Occupation Map

We introduce a continuous measure, by taking into account the relative distance of a point to a player.

This quantity, 𝛿 &)#"* (𝑥, 𝑦) is calculated as the difference between the distance 𝑑 from the point (𝑥, 𝑦)

to the closest player and the distance from the same point to the closest opponent,

𝛿 &)#"* (𝑥, 𝑦) = 𝑑 "4'&*&$ '))'6*6$ (𝑥, 𝑦) -𝑑 "4'&*&$ )4#8*9 (𝑥, 𝑦) (1)
This quantity does therefore not only take into account the distance to a player, but also the fact that only players of the opposite team will dispute the control of a certain area. On the boundaries of a

Dynamic Players Occupation Model

The previous approach gives a straightforward way to quantify how players occupy a basketball court. However, it does not take into account that players have inertia (i.e. players in motion). Clearly, for a fast-moving player, it is more difficult to control the area behind him, and some delay is induced by the fact that players have a finite force and can therefore not instantaneously change their velocity to a different value. It is thus necessary to develop a model characterizing the acceleration of the players. However, the distance to a point does not fully define the control of a point, but it is rather the time it takes for a player to reach the point which defines the controlled area. Therefore, the boundaries between a defense-controlled zone and an attack-controlled zone are more precisely defined by a quantity 𝛿 $01* ,

𝛿 $01* (𝑥, 𝑦) = 𝑡 "4'&*&$ '))'6*6$ (𝑥, 𝑦) -𝑡 "4'&*&$ )4#8*9 (𝑥, 𝑦) (2)
where 𝑡 "4'&*&$ )4#8*9 is the time it takes for the closest attacker (in seconds) to join the point (𝑥, 𝑦), and 𝑡 "4'&*&$ '))'6*6$ the same time for a defender. In the unphysical case of players with zero mass, or infinite force, this 𝛿 $01* should behave as 𝛿 &)#"* . However, inertia is expected to change this. The model we use to determine the time is similar in spirit, but not in detail and implementation, to previously investigated models [START_REF] Fujimura | Geometric Analysis and Quantitative Evaluation of Sport Teamwork[END_REF], [START_REF] Taki | Visualization of dominant region in team games and its application to teamwork analysis[END_REF] and [START_REF] Chawla | Classification of Passes in Football Matches Using Spatiotemporal Data[END_REF]. In particular, the fact that we have an analytical solution for the time-interval makes its implementation fast compared to iterative procedures such as the one proposed in [START_REF] Taki | Visualization of dominant region in team games and its application to teamwork analysis[END_REF], which makes the current procedure more tractable for real-time applications. Details on time calculation are available in the Appendix.

With 𝛿 $01* we can again determine an occupation-map, one of which is illustrated in Figure 2(b).

When comparing both maps Figure 2(a) and 2(b) clear visual differences are observed. Consider for instance the position on the court indicated by a green circle. In a simple (non-inertial) model considering only the distance of the players to a point, this part of the court is controlled by the defensive team. However, in reality, it is the offensive player (1) which will arrive at this point before the player (2) since the velocity of player ( 1) is directed towards this point while the defensive player (2) is moving away from this point. This example clearly shows how the inertia of the different players can be important in the determination of the influence zones of the teams.

Offset Induced by Inertia Between Space and Time

We now apply our models to characterize space and time free-space dependencies. To achieve this, . This difference clearly shows how inertia introduces a delay in defensive play. When a defender observes that the offensive player he is guarding changes his position, he needs to adapt his velocity to adjust to the best possible defensive position. The time lag 𝜏 ? is of the order of 0.3 seconds. The precise value of the delay is obtained by measuring for which value 𝜏 ? the maximum of the correlation coefficient 𝜌 ? is observed. This correlation coefficient is defined as,

we
𝜌 ? (𝜏) = ⟨𝛿 &)#"* B (𝑡 + 𝜏)𝛿 $01* B (𝑡)⟩ ⟨𝛿 &)#"* BE ⟩ F/E ⟨𝛿 $01* BE ⟩ F/E (5)
with 𝛿′ = 𝛿 -⟨𝛿⟩ and where ⟨. ⟩ denotes a time-average. The maximum of the correlation is shown in Figure 4(a). The maximum correlation is found to be 𝜏 ? = 0.3𝑠. It thus takes on average 0.3𝑠 for a defensive player to adapt his velocity to the velocity of the offensive player just before a 3-point shot.

We can further compare this specific measure of the adaptation time of a defender to approach the shooting attacker to a global, team-based delay. Indeed, the necessity to rapidly approach an offensive player starting his shot leads probably to a much faster adaptation to the defensive positions than the global delay of the full team with respect to its inertia-corrected position. In order to check this we computed the temporal correlation of 𝛿 &)#"* and 𝛿 $01* as in Equation ( 5), but now not only for the distance to the closest defender to the holder of the ball but computing the maximum correlation between the full occupation map such as the one in Figure 1(a) and the inertia-corrected occupation map (cf. The result is obtained by averaging over all the events during one quarter of a game.

Characterizing Free Space and 3-point Shots

This section uses the previously introduced occupation models to analyze and compare shooters' behavior before 3-point attempts. three seconds before and one second after 50 randomly chosen 3-point shots (grouped either by pull-up shots or catch-andshoot ones). Two different trends can be noticed between these shots. Before shooting, a catch-andshoot shot requires substantial free space for the shooter who needs time to both receive the ball and shoot while pull-up shots only require time to shoot. This is why catch-and-shoot curves generally lay above pull-up ones in this representation. Moreover, pull-up shots reveal an oscillation in the player's free space as he is dribbling, while it appears that before a catch-and-shoot shot the shooter was already free so his free space only decreases before the shot. While we mainly performed a visual analysis to separate shots, we now divide catch-and-shoot shots into two subclasses thanks to the k-means clustering algorithm. We then plot the average evolution of shooter's free space for the two catch-and-shoot classes and for pull-up shots. The results are available in Figure 6. As expected, pull-up shots reveal a global decrease as the shooter dribbles over time while catch-and-shoot shooters need some free space to have time to receive the ball. However, two behaviors stand out between players who are already free three seconds before the shot and those who have to move to free themselves. Considering the two ways of evaluating free space we can see that the global trend is the same, but the time offset calculated in section 4 appears again.

Free Space Evolution Before a 3-point Shot

Free Space and 3-point Shot Efficiency

As explained in section 2, the opening of a shot can be quantified thanks to the distance to the closest defender. Does this remark hold if we take into account the free space of the player at the time of the shoot defined as 𝛿 $01* * (𝑡 &%'$ )? We have computed the evolution of success rate with shooter's free space at a shot time using a moving average procedure and the results are shown in Figure 7. Those results are similar to the results by [START_REF] Lucey | How to Get an Open Shot": Analyzing Team Movement In[END_REF] (for Figure 7(a)) and confirm that 6 feet is a key distance for successful shots. The trend in Figure 7(b) is the same: the more time a defender needs to join the shooter, the more efficient the shooter is. The equivalent of 6 feet in free space calculated with 𝛿 $01* * (𝑡 &%'$ ) is 0.4s: if the closest defender is within 0.5s, the shooter's accuracy decreases drastically.

Comparing Catch-and-shoot Shooters

We have seen that catch-and-shoot shooters need a certain amount of free space to receive the ball and shoot. One key question is: how does this free space influence the player's behavior once they receive the ball? The correlation between release time and free space at the catch time is a good indicator. But in that case, it is more valuable to use 𝛿 $01* * to calculate free space to take into account whether a defender is moving in the shooter's direction at catch time. Figure 8 shows for 16 different players, ordered by success rate, the link between 𝛿 $01* * at catch time and release time. We can remark that players can have a similar success rate but have different behavior. For instance, Korver does not seem to adapt his release time to the free space he has at catch time, while Matthews, Nowitzki or Thompson clearly take more time if they are more free when they catch the ball.

Moreover, this figure reveals that players like Porter or Russell only shoot if 𝛿 $01* * (𝑡 "#$"% ) is superior to 0.5s while Curry or Korver take shots even if they are closely guarded. Finally, let's note that Kobe Bryant's release time can be really high even if his free space at catch time is low. This represents the fact that he shoots even when a defender is very close to him.

As mentioned in section 2, shot difficulty can be measured by the opening of a shot e.g, the distance of the closest defender 𝛿 &)#"* * at the shot time. However, we can refine the shot characteristic looking at release time to compare players' behavior and check if players shoot quickly or not and if they take open or closed attempts. Figure 9 reveals different behaviors. Players like Ilyasova, Gordon Johnson, and Leonard only take open shots (closest player at least at 4 feet). Players with a success rate higher than 40% globally shoot quicker than the median. This suggests that good shooters need to be able to shoot rapidly after receiving the ball. these players, we can focus on Leonard and Thompson noting that Thompson's attempts can be very tight or wide-open for the same release time while Leonard mainly takes shots when the nearest defender is 4 to 8 feet away from him. Two players can be mentioned as they have a particular behavior. Looking at Russell's distribution we can note that he misses almost all the shots he tries if the nearest defender is within 6 feet of him. However, he still has a global percentage of 33%, therefore, he would benefit from taking only wideopen shots. Then Bryant has also a particular behavior as most of his shots are taken with the player closer than 6 feet. https://amigocap.github.io/MecaSportStats/video.mp4.

Analysis of a Shot

Conclusion and Perspectives

In this section, we summarize our contribution and main takeaways based on the occupation models we introduced in this paper. Visual analysis reveals that the dynamic model is a better way to characterize court occupation. A more detailed quantitative comparison enabled to quantify the inertia with a measurable time-lag in the defensive dynamics. We have computed this time-lag, which is of the order of 0.3s focusing on 3-point attempts and 1.26s for the global team behavior, which is a fairly important amount of time in a high-frequency sport such as basketball.

The study of 3-point shooters free space evolution 3 seconds before shot time revealed three different behaviors. Firstly, we noticed that catch-and-shoot attempts require more free space than pull-up ones as the player needs time to catch the ball before taking his shot. Secondly, we were able to differentiate two types of catch-and-shoot shots as some of them ask the shooter to free himself

Appendix

We want to compute, for a moving player on a domain how much time it takes to reach a point (𝑥, 𝑦). Without loss of generality, we will determine the time it takes for a player at a given position and with a given speed to reach the point (𝑥, 𝑦) = (0,0). Initially the velocity of the player at position (𝑥(0), 𝑦(0)) is (𝑢(0), 𝑣(0)). We will consider that the player will use a constant force (per unit mass) in a given direction, of strength |𝐹| E = 𝐹 U E + 𝐹 8 E . This is an assumption which allows to find a simple analytical solution. In particular, it allows to consider the two directions separately.

Newton's law writes:

𝑑 $ E 𝑥 = 𝐹 U (6)
so that we have:

𝑥(𝑡) = 𝑥(0) + 𝑢(0)𝑡 + 1 2 𝐹 U 𝑡 E (7)
We evaluate this expression at 𝑥 = 0 and want to determine at which time this point is reached. Let us first determine the force per mass 𝐹 U ,

𝐹 U = -2 𝑥(0) + 𝑢(0)𝑡 𝑡 E (8)
This equation has formally 4 solutions. However, only one of these is the shortest physical time for a player to reach the origin. The constraints to choose the correct solution are that the time needs to be the smallest positive and real root of equation [START_REF] Stein | Revealing the Invisible: Visual Analytics and Explanatory Storytelling for Advanced Team Sport Analysis[END_REF].

The present model contains one adjustable control parameter, the value of 𝐹. A previous investigation [START_REF] Fujimura | Geometric Analysis and Quantitative Evaluation of Sport Teamwork[END_REF] suggests that the value of 𝐹 should be of order 10m/s.

In principle, since we have not bounded the velocity of a defender and fixed its acceleration, nonphysically large velocities can be developed. Thereto in a previous model [START_REF] Fujimura | Geometric Analysis and Quantitative Evaluation of Sport Teamwork[END_REF] the dynamics were refined introducing a drag, which limits the increase in velocity. However, for the present application, the time it takes for a defender to reach a shooter rarely exceeds 1 second, so that the velocities do not reach non-physical values. Therefore, and for the sake of simplicity, we have chosen not to refine the model any further.

Figure 1 :

 1 Figure 1: 380 Curry's shots from Season 2015-2016. Blue dots represent successful shots location and red ones miss shots location. Black segments represent the distance of the closest defender.

Figure 2 :

 2 Figure 2: Occupation-maps calculated with (a) static model 𝛿 &)#"* (𝑥, 𝑦) and (b) dynamic model 𝛿 $01* (𝑥, 𝑦) at each point of the court. Black lines represent Voronoi diagrams. Taking into account inertia best fits with the intuition that offensive player (1) will reach the area inside the green circle before the defensive player (2). An animated version of the dynamic model is available at https://amigocap.github.io/MecaSportStats/video.mp4.

  Voronoi cell between players of different teams the value of 𝛿 &)#"* is per definition 𝛿 &)#"* = 0. By calculating 𝛿 &)#"* for each point of the court we were able to determine an occupation-map describing the occupation of the court by the two teams. We show an illustration of such an occupation-map in Figure 2 (a).

  first calculate 𝛿 &)#"* and 𝛿 $01* at the player position ;𝑥 )4#8*9 , 𝑦 )4#8*9 < as 𝛿 &)#"* * = 𝛿 &)#"* ;𝑥 )4#8*9 , 𝑦 )4#8*9 < = 𝑑 "4'&*&$ '))'6*6$ ;𝑥 )4#8*9 , 𝑦 )4#8*9 < (3) 𝛿 $01* * = 𝛿 $01* ;𝑥 )4#8*9 , 𝑦 )4#8*9 < = 𝑡 "4'&*&$ '))'6*6$ ;𝑥 )4#8*9 , 𝑦 )4#8*9 < (4)Those calculations allow us to represent the evolution of players' free space (Figure3). A clear trend is that free space 𝛿 $01* * lags free space 𝛿 &)#"* *

Figure 3 :

 3 Figure 3: Example of Kawhi Leonard's free space evolution during an 8s time sequence of the game opposing San Antonio Spurs to Washington Wizards that played on November 4, 2015. Comparison of free space measured using 𝛿 &)#"* *

Figure 1 (Figure 4 :

 14 Figure 4: (a) Time-correlation between ⟨𝛿 &)#"* *

Figure 5 (

 5 a) and 5(b) respectively show the evolution of 𝛿 &)#"* * and 𝛿 $01* *

Figure 5 :

 5 Figure 5: Evolution of 50 shot-free-space indicators calculated with the static model 𝛿 &)#"* *

Figure 6 :

 6 Figure 6: Average evolution of a shooter's free-space measured by (a) 𝛿 &)#"* *

Figure 7 :

 7 Figure 7: (a) Shot efficiency as a function of maximum available space 𝛿 &)#"* * at the release-time of the ball. If the closest defender is within 4 feet, the shooting percentage is 27,9%, while it goes to 33,6% if he is within 8 feet. (b) Shot efficiency as a function of maximum available time 𝛿 $01* * (free space) at the release of the ball. Around t=0.4s a significant change in behavior is observed.

Figure 8 :

 8 Figure 8: Shot release time function of free space at catch time calculated with 𝛿 $01* *

11

 11

Figure 9 :

 9 Figure 9: Distance to the closest defender at shot time 𝛿 &)#"* * function of release time. Vertical dashed line shows median release time over all catch-and-shoot shots while horizontal dashed lines delimit degrees of a shot opening defined by NBA stats website [16]: very tight, tight, open and wide open above the thickest dashed line. Hits and miss are respectively associated with blue and red dots.

Figure 10 Figure 10 :

 1010 Figure 10 illustrates the step-by-step analysis of a single, catch-and-shoot 3-point shot by Thompson. This is an example where the shooter has to free himself from his defender to be able to receive the ball. At the beginning of the action, Thompson is guarded by two players (Figure 10 (a)) which is confirmed by respective values of 𝛿 &)#"* *

  Analogous expressions to (6)-[START_REF] Yue | Learning Fine-Grained Spatial Models for Dynamic Sports Play Prediction[END_REF] are written for 𝑑 $ E 𝑦, 𝑦(𝑡) and 𝐹 8 . Since |𝐹| E = 𝐹 U E + 𝐹 8 E , we have,

	𝐹 E =	4 𝑡 E WX 𝑥(0) + 𝑢(0)𝑡 𝑡 E	Y	E	+ X	𝑡 E 𝑦(0) + 𝑣(0)𝑡	Y E	Z	(9)
	Yielding a 4 $% order polynomial for t,
	𝑡 E -	4 𝐹 E WX 𝑥(0) + 𝑢(0)𝑡 𝑡 E	Y	E	+ X	𝑦(0) + 𝑣(0)𝑡 𝑡 E	E Y	Z = 0

Acknowledgments

The authors are grateful for the pioneering work by Marc Mattis, which constituted the foundation of this study. We are also grateful to STATS LLC for releasing its SportVU player tracking data. This research was partially funded by the M2I project http://www.mob2i.fr/, Projet Investissement d'Avenir on urban mobility by the French Environment Agency (ADEME) and by a BQR (Bonus Qualité Recherche) project funding of École Centrale de Lyon.

from defenders while he can only stay in his position when he is free enough both to receive and shoot the ball three seconds before the attempt.

Then, considering shot accuracy, we were able to determine key values: when the closest defender is within 6 feet or 0.4s the percentage of success decreases drastically.

Eventually, the study of 16 individual behaviors reveals that some players, like Matthews, adapt their release time to the free space they have when they catch the ball while others prefer to keep a constant release time. We were also able to note that some players only take their chance if they have enough free space at time catch while players like Curry can shoot even if they are closely guarded. Finally, a focus on release time indicates that players with a success rate higher than 40% are always able to shoot quicker than average suggesting it is a necessary condition to be a good shooter.

The main perspective for our work is to take into account in our models the players' characteristics such as their height, speed or endurance. This will require additional meta-dataset to be combined with SportVU data. We sought to look at the correlation between types of shots and de-aggregated data over space and time, e.g. shot clock, current score or decisive game or not. Another improvement of our model is to take into account the direction of the nearest defender in addition to his distance since a defender not on the firing line has less influence on the shooter's performance.

We release code and analysis as an open-source project with a permissive license https://github.com/AmigoCap/MecaSportStats to support further research and application of those models. In particular we are interested in applying those models to transportations systems in cities, to let urban planners optimize bus and metro lines locations and schedules based on the share of population they cover.


Dr Vincent Planche
email: vincent.planche@u-bordeaux.fr

Pierrick Coupé

Catherine Helmer

Mélanie Le Goff

Helene Amieva

François Tison

Jean-François Dartigues

Gwénaëlle Catheline

Evolution of brain atrophy subtypes during aging predicts long-term cognitive decline and future Alzheimer's clinical syndrome

Keywords: Hippocampus, Cortex, Imaging, Alzheimer, Aging, Cohort study

It is currently unknown whether brain atrophy subtypes defined in Alzheimer's disease are clinically relevant during aging. We investigated participants (n=368) from a population-based cohort of non-demented older adults who received longitudinal neuropsychological assessments during 12 years. MRI scans at baseline and 4 years later were used to define participants with "hippocampal predominant atrophy", "cortical predominant atrophy", "homogenous atrophy" and "no evidence of brain subtype atrophy" based on the dynamics of hippocampal-to-cortical volume ratio evolution. After adjustment on age, gender, educational level and ApoE4 genotype, participants with "hippocampal predominant atrophy" declined faster regarding global cognition, verbal fluency and verbal episodic memory. In Cox proportional-hazards models, "hippocampal predominant atrophy" was associated with an increased risk of developing Alzheimer's clinical syndrome over time (HR=5.73; 95%CI 2.71-12.15), independently of age and ApoE4 genotype, the two other significant predictive factors.

As a possible surrogate of confined tauopathy and early Alzheimer's disease pathology, future studies should consider the definition of "hippocampal predominant atrophy" based on hippocampal-to-cortical volume ratio evolution rather than hippocampal volume alone.

Introduction

The neurodegenerative process causing brain atrophy and cognitive impairment in Alzheimer's disease involves both the limbic system and neocortical areas. Hence, MRI plays a key role in the clinical assessment of patients with suspected Alzheimer's disease because regional atrophy can provide positive diagnostic information [START_REF] Scheltens | Alzheimer's disease[END_REF]. However, the pattern and dynamics of brain atrophy in Alzheimer's disease are somewhat different according age at onset, clinical presentation, neuropsychiatric comorbidities, vascular risk factors and rate of decline [START_REF] Dickerson | The cortical signature of Alzheimer's disease: regionally specific cortical thinning relates to symptom severity in very mild to mild AD dementia and is detectable in asymptomatic amyloid-positive individuals[END_REF][START_REF] Dickerson | Alzheimer's disease: The influence of age on clinical heterogeneity through the human brain connectome[END_REF].

Studies combining pre-mortem neuroimaging and post-mortem neuropathology have suggested that patterns of gray matter atrophy can be related to the topographic distribution and progression of tau neurofibrillary tangles [START_REF] Gosche | Hippocampal volume as an index of Alzheimer neuropathology: findings from the Nun Study[END_REF][START_REF] Jack | Antemortem MRI findings correlate with hippocampal neuropathology in typical aging and dementia[END_REF]. Interestingly, recent cross-sectional MRI studies have been able to capture distinct neuropathologically defined subtypes of Alzheimer's disease regarding to the Murray-Dickson definition [START_REF] Whitwell | Neuroimaging correlates of pathologically defined subtypes of Alzheimer's disease: a casecontrol study[END_REF][START_REF] Byun | Heterogeneity of Regional Brain Atrophy Patterns Associated with Distinct Progression Rates in Alzheimer's Disease[END_REF][START_REF] Risacher | Alzheimer disease brain atrophy subtypes are associated with cognition and rate of decline[END_REF]. This categorization of Alzheimer's disease is based on an algorithm which classifies Alzheimer's disease cases into "typical", "hippocampal sparing" and "limbic predominant" patterns of neurofibrillary tangles distribution, using the ratio of hippocampal-to-cortical neurofibrillary tangles density [START_REF] Murray | Neuropathologically defined subtypes of Alzheimer's disease with distinct clinical characteristics: a retrospective study[END_REF]. In these studies, an hippocampal sparing pattern of atrophy on MRI (correlating to an hippocampal sparing distribution of neurofibrillary tangles) was associated with a faster cognitive and functional decline [START_REF] Murray | Neuropathologically defined subtypes of Alzheimer's disease with distinct clinical characteristics: a retrospective study[END_REF][START_REF] Whitwell | Neuroimaging correlates of pathologically defined subtypes of Alzheimer's disease: a casecontrol study[END_REF][START_REF] Risacher | Alzheimer disease brain atrophy subtypes are associated with cognition and rate of decline[END_REF]. However, it is currently unknown whether this rather simple categorization of brain atrophy subtypes is also clinically relevant in asymptomatic preclinical Alzheimer's disease [START_REF] Dubois | Advancing research diagnostic criteria for Alzheimer's disease: the IWG-2 criteria[END_REF]. Furthermore, none of these previous studies have really captured the dynamics of brain atrophy subtypes because they relied on cross-sectional analyses.

Although the precise timing remains elusive, due to the lack of longitudinal studies long enough at the presymptomatic stage of the disease, the current models of Alzheimer's disease pathophysiology postulate that neurofibrillary tangles appear years before the symptomatic phase of the disease [START_REF] Braak | Neuropathological stageing of Alzheimer-related changes[END_REF][START_REF] Jack | Hypothetical model of dynamic biomarkers of the Alzheimer's pathological cascade[END_REF][START_REF] Sperling | Toward defining the preclinical stages of Alzheimer's disease: recommendations from the National Institute on Aging-Alzheimer's Association workgroups on diagnostic guidelines for Alzheimer's disease[END_REF]. Thus, based on the dynamics of hippocampal-to-cortical volume ratio evolution, this study aims at determining whether different subtypes of brain atrophy in older adults could predict differential cognitive decline and an increased risk of Alzheimer's clinical syndrome over time.

For that purpose, we studied a well-defined population-based cohort of older adults who underwent two MRI examinations at 4-year intervals and a neuropsychological and clinical follow-up during 12 years.

Methods

Study sample

The data used in the following analyses were obtained from a subset of the Bordeaux sample of the three-city (3C) study, a longitudinal population-based cohort designed to evaluate risk factors of dementia (3C Study [START_REF] Group | Vascular factors and risk of dementia: design of the Three-City Study and baseline characteristics of the study population[END_REF]. During the 1999-2000 inclusion period, noninstitutionalized individuals aged 65 and over were randomly recruited from electoral lists and followed prospectively for up to 12 years. From the initial cohort of participants with baseline MRI (n=663), only non-demented participants who agreed to have a second MRI 4 years later were included in the present analyses (n=368). Information regarding demographical characteristics and APoE4 genotype (carriers/non carrier: at least one allele) were also collected at baseline. All participants gave written informed consent to participate and the study protocol was approved by the ethics committee of Kremlin-Bicêtre University Hospital (Paris, France).

Neuropsychological assessment and diagnosis of incident Alzheimer's clinical syndrom

During the 12-year follow-up period, neuropsychological assessments were administered by The diagnosis of dementia was pre-specified at home by a neuropsychologist, at each visit (2, 4, 8, 10 and 12 years). After this first assessment, a definitive diagnosis of dementia and of possible or probable Alzheimer's disease was made by a panel of independent neurologists according to the DSM-IV criteria and the NINCDS-ADRDA criteria [START_REF] Mckhann | Clinical diagnosis of Alzheimer's disease: report of the NINCDS-ADRDA Work Group under the auspices of Department of Health and Human Services Task Force on Alzheimer's Disease[END_REF].

They were finally labeled "Alzheimer's clinical syndrome" according to the recent NIA-AA research framework recommendations (Jack et al., 2018a).

MRI acquisition and processing

MRI examinations were performed on a 1.5T Gyroscan Intera system (Philips Medical Systems) with a quadrature head coil. The morphological protocol consisted of threedimensional (3D) high-resolution T1-weighted images acquired using magnetisation prepared rapid gradient echo (MPRAGE, TR=8.5 ms, TE=3.9 ms, α=10°, FOV=240 mm, voxel size=0.94 x 0.94 x 1 mm 3 ). The same scanner and the same sequence were used for the baseline and the 4-year follow-up MRI.

For cortical and hippocampal volumetric analyses, T1-weighted images were processed using the volBrain system (http://volbrain.upv.es) [START_REF] Manjón | volBrain: An Online MRI Brain Volumetry System[END_REF]. Recently, volBrain pipeline was compared with well-known tools used on MR brain analysis (SPM, FSL and Freesurfer) showing significant improvements in terms of both accuracy and reproducibility for intrascanner and interscanner scan-rescan acquisition (Manjón et al., 2010a[START_REF] Naess-Schmidt | Automatic thalamus and hippocampus segmentation from MP2RAGE: comparison of publicly available methods and implications for DTI quantification[END_REF], even if it was not specifically designed to reduce intra-individual variability and noise with longitudinal registration approach. After denoising (Manjón et al., 2010b), images were corrected for inhomogeneity [START_REF] Tustison | N4ITK: improved N3 bias correction[END_REF], intensity-normalized [START_REF] Nyúl | On standardizing the MR image intensity scale[END_REF] and affine-registered into the Montreal Neurological Institute (MNI) space [START_REF] Avants | A reproducible evaluation of ANTs similarity metric performance in brain image registration[END_REF]. Then, intracranial cavity was segmented using NICE method [START_REF] Manjón | Nonlocal intracranial cavity extraction[END_REF] and tissue classification (including cortical segmentation) was performed using TMS method (Manjón et al., 2010b). The cortical gray matter volume was calculated as the global gray matter volume minus the deep gray matter volumes (i.e., caudate, thalamus, accumbens, globus pallidus, putamen, hippocampus, and amygdala) [START_REF] Coupé | Towards a unified analysis of brain maturation and aging across the entire lifespan: A MRI analysis[END_REF]. Hippocampus was automatically segmented with a patch-based multi-template method that uses expert manual segmentations in MNI space as priors [START_REF] Coupé | Patch-based segmentation using expert priors: application to hippocampus and ventricle segmentation[END_REF]. Anatomical boundaries of the hippocampus were defined according to the European Alzheimer's Disease Consortium and Alzheimer's Disease Neuroimaging Initiative (EADC-ADNI) Harmonized Protocol [START_REF] Frisoni | EADC-ADNI Working Group on The Harmonized Protocol for Manual Hippocampal Volumetry and for the Alzheimer's Disease Neuroimaging Initiative[END_REF]. To control for variation in head size, hippocampal and cortical volumes were normalized using the intracranial cavity volume (ICV) of each subject, in order to express volumes as a proportion of ICV.

The definition of longitudinal evolution of hippocampal-to-cortical volume ratio (HV/CTV ratio) was adapted from previous cross-sectional studies having investigated brain atrophy subtypes in Alzheimer's disease (also known as Murray-Dickson subtypes) because this MRIbased algorithm is able to reliably track the distribution of neurofibrillary tangles in Alzheimer's disease [START_REF] Whitwell | Neuroimaging correlates of pathologically defined subtypes of Alzheimer's disease: a casecontrol study[END_REF][START_REF] Risacher | Alzheimer disease brain atrophy subtypes are associated with cognition and rate of decline[END_REF]. These previous studies defined "limbic predominant'', "hippocampal sparing'', and "typical'' patterns of atrophy according to cross-sectional measures of brain volumes (a 2-steps procedure based on splitting participants according to: (1) the 25 th and 75 th percentiles of HV/CTV ratio and (2) the normalized median hippocampal and cortical volumes). Because in the present study, we performed our analyses on a population-based sample including healthy people, we categorized brain atrophy as follows: "hippocampal predominant atrophy'', "cortical predominant atrophy'', "homogenous atrophy'' and "no evidence of specific brain subtype atrophy'' based on the longitudinal evolution of this ratio. We calculated HV/CTV ratio on baseline MRI and 4-year follow-up MRI. The difference between these 2 ratios (∆(HV/CTV ratio)) was considered as a measure of the dynamics of preferential brain atrophy. Participants with ∆(HV/CTV ratios) below the 25 th percentile were considered as having "cortical predominant atrophy" if their normalized cortical volume after 4-year follow-up was less than the median value of the whole group and if their normalized hippocampal volume was greater than the median volume of the whole group of participants. Participants with ∆(HV:CTV ratios) above the 75 th percentile were considered as having "hippocampal predominant atrophy" if their normalized hippocampal volume after 4-year follow-up was less than the median value of the whole group and if their normalized cortical volume was greater than the median volume of the whole group of participants. Participants with ∆(HV:CTV ratios) between the 25th and the 75th percentile were considered as having "homogenous atrophy" if their normalized cortical volume after 4-year follow-up was less than the median value of the whole group and if their normalized hippocampal volume was less than the median volume of the whole group of participants. Finally, participants were considered as having "no evidence of specific brain subtype atrophy" if they were not classified in the previous categories (Fig. 1).

Statistical analyses

Statistical analyses were performed with Prism software 6 (Graphpad) and XLstats 19.4 (Addinsoft). The distribution of all continuous data was tested with the Shapiro-Wilk test. We first compared clinical and imaging characteristics at baseline between the 4 groups of participants by using the Chi-squared test for categorical variables, and ANOVA or Kruskal-Wallis test (when assumptions of ANOVA were not met) for continuous variables, followed by appropriate post-hoc multiple comparisons tests (Tukey-Kramer or Dunn test, respectively).

Second, annual cognitive decline for each test has been calculated for each participant using linear mixed model and compared between groups using analyses of covariance (ANCOVA, using age, gender, educational level and ApoE4 genotype as covariates). Third, the association between brain atrophy subtype and incident Alzheimer's clinical syndrome at follow-up was tested using a log-rank test for trend comparing estimates of the Kaplan-Meier survival analysis.

Finally, brain atrophy subtypes and usual risk factors of Alzheimer's disease (age, gender, educational level and APoE4 genotype) were tested to predict time to occurrence of possible or probable Alzheimer's disease using Cox proportional hazard models. All tests were two-sided, with a type I error set at α=0.05.

Results

Demographical, neuropsychological and MRI characteristics at baseline

Out of the 368 participants included in the analyses, 34 (9.2%) met the criteria defining the "hippocampal predominant atrophy'' subtype, 43 (11.7%) met the criteria for "cortical predominant atrophy'' and 46 (12.5%) for "homogeneous atrophy''. Then, 245 participants (66.6%) were classified as having "no evidence of specific brain subtype atrophy''. The characteristics of the sample by atrophy subtypes at baseline are summarized in table 1.

Participants with "hippocampal predominant atrophy" were significantly older than all the other groups (p=0.0026) and had poorer performances on verbal episodic memory tests (both the free and total recall of the FCSRT, η 2 =0.065, p<0.001 and η 2 =0.089, p<0.001 respectively) and verbal fluency tests (both IST-15 and 60, η 2 =0.024, p=0.033 and η 2 =0.019, p=0.043 respectively). Participants with "hippocampal predominant atrophy" and "homogeneous atrophy" already had significant lower normalized hippocampal volumes at baseline (especially the homogeneous atrophy group) (η 2 =0.11, p<0.001). In contrast, the "cortical predominant atrophy" group did not differ from the other groups at baseline for cortical volumes (Table 1).

MRI volumes after 4 years

As expected on the MRI performed at 4-year follow-up, the "hippocampal predominant atrophy" and the "homogeneous atrophy" groups had significantly lower normalized hippocampal volumes (both 0.48% of intracranial cavity volume, compared to 0.56% and 0.54% in the two other groups, η 2 =0.24, p<0.001). The "cortical predominant atrophy" and the "homogeneous atrophy" groups had significantly lower normalized cortical volumes on this follow-up MRI (37.1% and 38.3% respectively, compared to 42.7% and 40.0% in the "hippocampal predominant atrophy" group and the group with all other participants, η 2 =0.13, p<0.001). Individual trajectories of normalized hippocampal and cortical volumes are depicted in figure 2.

Cognitive decline over 12 years

Out of the 368 participants included in the analyses, 343 (93.2%) were seen at least one time after the second MRI for a new neuropsychological assessment. Regarding participants loss to follow-up, there was no difference between brain atrophy subtypes, with 2 participants lost in the "hippocampal predominant atrophy'' group, 3 in the "hippocampal predominant atrophy'' group, 4 in the "homogeneous atrophy'' group and 16 in the "no evidence of specific brain subtype atrophy'' group (Chi-square test, p=0.95) Among the 25 participants who were not seen after the second MRI, 18 had died.

For each neuropsychological test, the slope of cognitive decline over 12 years (assessment at baseline and after 2, 4, 8, 10 and 12 years) was modeled for each participant using mixed effect models and compared between groups after adjustment for age, gender, educational level and ApoE4 genotype (Fig. 3). We found that participants with "hippocampal predominant atrophy" decline faster than all other groups regarding global cognition (MMSE, ß=0.13, p=0.013), verbal episodic memory (FCSRT free recall, ß=0.12, p=0.023 and FCSRT total recall, ß=0.15, p=0.009) and verbal fluency ß=0.11,ß=0.13,p=0.011).

Participants with "homogeneous atrophy" decline faster than the other groups on a visuospatial perception and memory test (BVRT, ß=0.016, p=0.015). No atrophy subtype was associated with faster decline on attention or executive functions, regarding the TMT-A and TMT-B.

Incident Alzheimer's disease

After 12 years, we identified 37 cases of Alzheimer's clinical syndrome, 3 cases of Parkinson's disease, 3 cases of possible or probable Lewy-body dementia, and 2 cases of probable frontotemporal lobar degeneration. The proportion of participants who develops Alzheimer's clinical syndrome was 44% in the "hippocampal predominant atrophy" group, 9% in the "cortical predominant atrophy" group, 17% in the "homogeneous atrophy" group and 7% in the "no evidence of specific brain subtype atrophy" group. Kaplan-Meier survival curves of time to incident Alzheimer's clinical syndrome comparing the four groups of brain atrophy subtypes are shown in Figure 4. Comparing to the 3 other groups, the log-rank test was significant for participants with "hippocampal predominant atrophy" (Chi 2 =38.0, p<0.001).

The results of Cox proportional hazards models investigating the risk of developing Alzheimer's clinical syndrome in our cohort regarding age, gender, educational level, ApoE4 genotype and brain atrophy subtype are shown in table 2. "Hippocampal predominant atrophy" was associated with increased risk of developing Alzheimer's clinical syndrome over time (HR=5.73; 95%CI 2.71 -12.15), independently of age and ApoE genotype, the two other significant predictive factors (respectively HR=1.19; CI95% 1.08 -1.30 and HR=2.56; 95%CI 1.27 -5.16).

As a sensitivity analysis, we also studied the proportion of participants who develop Alzheimer's clinical syndrome only after the 4-year follow-up visit (i.e after the second MRI).

The log-rank test was still significant for participants with "hippocampal predominant atrophy", compared to the 3 other groups (Chi 2 =20.3, p<0.001). Regarding the Cox proportional hazards models investigating the risk of developing Alzheimer's clinical syndrome over time, "hippocampal predominant atrophy" was associated with increased risk after the 4-year followup visit (HR=6.00; 95%CI 2.67 -13.49) independently of the age and ApoE4 genotype.

When hippocampal and cortical atrophy rates were included as a continuous variable into the Cox proportional hazards models (instead of brain atrophy subtypes), these MRI measures were not significant to predict the incidence of Alzheimer's clinical syndromes, alone or in association with confounders.

Discussion

In this study, we have adapted the cross-sectional Murray-Dickson algorithm defining "typical", "hippocampal sparing" and "limbic predominant" pathology in patients with Alzheimer's disease [START_REF] Murray | Neuropathologically defined subtypes of Alzheimer's disease with distinct clinical characteristics: a retrospective study[END_REF][START_REF] Whitwell | Neuroimaging correlates of pathologically defined subtypes of Alzheimer's disease: a casecontrol study[END_REF][START_REF] Byun | Heterogeneity of Regional Brain Atrophy Patterns Associated with Distinct Progression Rates in Alzheimer's Disease[END_REF][START_REF] Risacher | Alzheimer disease brain atrophy subtypes are associated with cognition and rate of decline[END_REF] to longitudinal MRI data gathered from a population-based cohort of non-demented elderly people with a long-term prospective neuropsychological follow-up. We found that participants with "hippocampal predominant atrophy" decline faster than all other groups regarding global cognition, verbal fluency and verbal episodic memory (even when compared to participants with "homogenous atrophy" who had smaller hippocampal volumes at baseline).

Furthermore, we found that the "hippocampal predominant atrophy" group had a much higher risk to develop Alzheimer's clinical syndrome over time, independently of age, gender, educational level and ApoE4 genotype. MRI measures of differential hippocampal-to-cortical atrophy have been demonstrated to be a surrogate of neurofibrillary tangles deposits in neuropathologically-defined patients with Alzheimer's disease [START_REF] Whitwell | Neuroimaging correlates of pathologically defined subtypes of Alzheimer's disease: a casecontrol study[END_REF]. Thus, we can assume that the differential hippocampal vulnerability measured in the "hippocampal predominant atrophy" group is linked to the early tauopathy defining Braak stage I and II. In this study, which allows a real unbiased prospective assessment of the preclinical phase of Alzheimer's disease, we demonstrated that this differential and specific hippocampal atrophy (related to cortical atrophy) can precede the diagnosis of dementia for up to 8-12 years. In this context, our results are a rare longitudinal and in vivo support of current models of Alzheimer's disease pathophysiology [START_REF] Jack | Hypothetical model of dynamic biomarkers of the Alzheimer's pathological cascade[END_REF][START_REF] Sperling | Toward defining the preclinical stages of Alzheimer's disease: recommendations from the National Institute on Aging-Alzheimer's Association workgroups on diagnostic guidelines for Alzheimer's disease[END_REF]. Indeed, these models postulate that the spreading of neurofibrillary tangles occurs years before dementia, with a stereotypical pattern of early medial temporal lobe involvement (entorhinal cortex and hippocampus), followed by progressive neocortical damage, according to Braak staging [START_REF] Braak | Neuropathological stageing of Alzheimer-related changes[END_REF]. However, these conclusions were to date mostly based on cross-sectional studies, in which longitudinal changes were inferred by studying individuals at different stages of the disease (i.e., controls, mild cognitive impairment and Alzheimer's disease groups) [START_REF] Fotenos | Normative estimates of cross-sectional and longitudinal brain volume decline in aging and AD[END_REF][START_REF] Tabatabaei-Jafari | Cerebral atrophy in mild cognitive impairment: A systematic review with meta-analysis[END_REF].

Previous works studying the differential hippocampal-to-cortical distribution of neurofibrillary tangles or the hippocampal-to-cortical volume ratio in patients with Alzheimer's disease or Alzheimer's clinical syndrome concluded that a hippocampal sparing disease process was linked to poorer cognitive and functional prognoses, especially if cortical atrophy predominate in parietal lobes [START_REF] Na | Malignant progression in parietal-dominant atrophy subtype of Alzheimer's disease occurs independent of onset age[END_REF][START_REF] Risacher | Alzheimer disease brain atrophy subtypes are associated with cognition and rate of decline[END_REF][START_REF] Ten Kate | Atrophy subtypes in prodromal Alzheimer's disease are associated with cognitive decline[END_REF]. Indeed, "hippocampal sparing" damage define atypical variants of Alzheimer's disease, known to have faster evolutions due to impairment in executive functions, language or visuospatial abilities, which strongly impact autonomy [START_REF] Scheltens | Alzheimer's disease[END_REF]. In contrast, in our cohort of elderly subjects, "hippocampal predominant atrophy" (rather than "cortical predominant" or "homogeneous atrophy") is associated with a poorer prognosis because it announces a typical form of Alzheimer's disease, by far the most common. Finally, we can conclude that the evolution of hippocampal-to-cortical volume ratio on MRI can be seen as both an early and a late marker of the disease process [START_REF] Frisoni | The clinical use of structural MRI in Alzheimer disease[END_REF], with a two stages evolution. First, hippocampal predominant damage is deleterious during aging or the prodromal stage of the disease because it announces future Alzheimer's disease and then, predominant cortical involvement is deleterious in already diagnosed patients because it signs an atypical evolution or a cortical spreading of the pathophysiological process. Another explanation could be that our study "missed" future Alzheimer's disease with "cortical predominent atrophy" because of its inclusion criteria (individuals aged >65). Indeed, previous studies showed that "hippocampal sparing" Alzheimer's disease is associated with early-onset dementia (63±10 years [START_REF] Murray | Neuropathologically defined subtypes of Alzheimer's disease with distinct clinical characteristics: a retrospective study[END_REF]).

Regarding neuropsychological functioning, "hippocampal predominant atrophy" group declined faster than all other groups on the IST, a test known to be one of the first to decline in the prodromal phase of Alzheimer's disease [START_REF] Amieva | Prodromal Alzheimer's disease: successive emergence of the clinical symptoms[END_REF] (even before tests such as the FCSRT that measures episodic memory and define the classical "amnestic syndrome of hippocampal type" in Alzheimer's disease [START_REF] Sarazin | Amnestic syndrome of the medial temporal type identifies prodromal AD: a longitudinal study[END_REF][START_REF] Auriacombe | Validity of the free and cued selective reminding test in predicting dementia: the 3C study[END_REF].

Because IST is a multidetermined test, it has been postulated that its early decline in elderly people could be linked to executive functions and information processing speed impairment due to accelerated age-related damage or to semantic memory impairment due the insidious accumulation of neurofibrillary pathology in the temporal area [START_REF] Amieva | Prodromal Alzheimer's disease: successive emergence of the clinical symptoms[END_REF]. Our results argue for the second hypothesis.

In our study, it is interesting to note that the "homogeneous atrophy" group had a lower hippocampal volume at baseline than the "hippocampal predominant atrophy" group. However, the "homogeneous atrophy" group is not at higher risk to develop Alzheimer's clinical syndrome than the "cortical predominant atrophy" group or the group of all other participants, because hippocampal atrophy may not be related to Alzheimer's disease tauopathy in these cases. Furthermore, the rate of hippocampal atrophy alone was not a significant predictor of the incidence of Alzheimer's clinical syndrom over time in Cox proportional hazards models (instead of brain atrophy subtype). Thus, it highlights that small hippocampal volumes per se are not real predictors of future Alzheimer's clinical syndromes, but that the measure of a dynamic process leading to the differential atrophy of the hippocampus comparing to the neocortex could be a very good marker instead. These findings are in accordance with clinical practice where hippocampal atrophy in the elderly is known to be poorly specific of Alzheimer's disease or prodromal Alzheimer's disease because it also occurs in neurovascular diseases, neuro-inflammatory diseases, bipolar disorder and schizophrenia or other neurodegenerative processes, such as frontotemporal lobar dementia and Lewy body dementia [START_REF] Harper | An algorithmic approach to structural imaging in dementia[END_REF]. Then, we can envision that the measure of the evolution of hippocampalto-cortical volume ratio in therapeutic clinical trials for preclinical Alzheimer's disease could become a better biomarker than hippocampal volumetry alone, by assessing the dynamics of neurofibrillary tangles spreading. Furthermore, such anatomical classifications of brain atrophy subtypes could also help physicians to identify at-risk people with subjective memory impairment or mild cognitive impairment [START_REF] Jung | Classifying anatomical subtypes of subjective memory impairment[END_REF][START_REF] Kim | Cortical atrophy pattern-based subtyping predicts prognosis of amnestic MCI: an individual-level analysis[END_REF].

The strengths of this study are the large sample size, the 12-year follow-up period along with a large neuropsychological battery, the few lost to follow-up participants, the longitudinal measure of brain atrophy progression with the same scanner across both time points and the population-based, natural history design. Because the inclusion period of this study was 1999-2000, its first limitation is the lack of assessment of amyloid pathology using PET-imaging or CSF biomarkers, to explore the temporal link between hippocampal-to-cortical volume ratio evolution and amyloid-β deposits. In addition, these biomarkers would have specified the diagnosis of Alzheimer's disease, which was in this study only based on clinical criteria. However, the diagnosis of Alzheimer's clinical syndrome (possible or probable Alzheimer's disease according to 1984 and 2011 criteria) was done in our cohort by a panel of independent and expert neurologists, and are congruent with observations from the literature regarding for instance the larger ratio of subjects with incident Alzheimer's disease presenting at least one ApoE ε4 allele. We also acknowledge that we have not clearly assessed what are the distinct characteristics of the "homogeneous atrophy" and "cortical predominant atrophy" groups, as well as the cause of their distinct pattern of brain atrophy. Indeed, it could be explained by other ongoing neurodegenerative processes or by neurovascular or psychiatric diseases, known to drive slower cognitive decline than Alzheimer's disease. Future studies will need to address these points.

Our approach based on an indirect measure of the neurofibrillary tangles spreading in aging will be probably overcome by the use of PET-Tau imaging in the coming decade, if studies with a similar inclusion criteria and follow-up are set up. Regarding recent PET-Tau studies, it is indeed interesting to note the strong correlation between localized 18 F-AV-1451 uptake and the longitudinal measures of medial temporal lobe atrophy while the correlation was rather weak regarding 18 F-AV-1451 uptake and transversal volumetric analyses [START_REF] Das | Longitudinal and crosssectional structural magnetic resonance imaging correlates of AV-1451 uptake[END_REF].

These data support the notion that in vivo measures of tau pathology are tightly linked to the local rate of neurodegenerative change measured with longitudinal MRI measures, and strongly support our findings. Furthermore, future longitudinal PET-Tau studies will be able to address fundamental questions regarding early Alzheimer's disease pathophysiology (Jack et al., 2018b). For instance, recent transversal multimodal studies on Alzheimer's disease patients or elderly people had reported exciting findings linking distant Aß and neurofibrillary tangles interactions with brain hypometabolism and atrophy [START_REF] Sepulcre | In Vivo Tau, Amyloid, and Gray Matter Profiles in the Aging Brain[END_REF][START_REF] Whitwell | Imaging correlations of tau, amyloid, metabolism, and atrophy in typical and atypical Alzheimer's disease[END_REF].

Conclusion

The monitoring of "hippocampal predominant atrophy" using hippocampal-to-cortical volume ratio on MRI appears to be a strong predictor of cognitive decline and incident Alzheimer's clinical syndrome, independently of age, gender, educational level and ApoE4 genotype. This rather simple morphometric analysis probably captures early lesions of tauopathy defining Braak stage I and II, a decade before Alzheimer's disease onset in non-demented people.

Longitudinal hippocampal-to-cortical volume ratio evolution could become a strong marker to disentangle brain changes in normal aging from the earliest signs of Alzheimer's disease.

  trained psychologists at baseline and after 2, 4, 8, 10 and 12 years. The battery consisted of the Mini Mental State Evaluation (MMSE: global cognitive functions), the Free and Cued Selective Reminding Test (FCSRT: verbal episodic memory (sum of the number of words retrieved at the three free or cued trials)), the Benton Visual Retention Test (BVRT: visuospatial working memory), the Isaacs Set Test (IST: semantic fluency), and the Trail-Making Test part A and B (TMT-A and TMT-B: attention, information processing speed and executive functions ((number of correct moves/total time)x10)).

FiguresFigure 1 .

 1 Figures

Figure 2 :

 2 Figure 2: Spaghetti plots of hippocampal and cortical volumes at baseline and 4 years later. Each line represents one participant. ICV: Intracranial Volume

Figure 4 .

 4 Figure 4. Kaplan-Meier survival curves representing the incidence of Alzheimer's disease during the 12-year follow-up period, regarding the dynamics of initial brain subtype atrophy. The dashed lines represent standard errors.

Table 1 : Clinical, neuropsychological, and MRI features of the studied populations at baseline.

 1 Benson Visual Retention Test ; FCSRT : Free and Cued Selective Reminding Test ;

	Tables
		Hippocampal predominant atrophy (n=34)	Cortical predominant atrophy (n=43)	Homogeneous (n=46) atrophy	(n=245) Others	p value b
	Demographical variables at baseline
	1.6Age, mean (SD)	74.3 (3.9)* † ‡	71.1 (3.1)	72.0 (3.6)	72.2 (3.9)	0.0026
	Gender, women, %	50.0%	48.8%	42.6%*	63.4%	0.029
	ApoE (ε4 +/-or +/+), %	29.4%	18.6%	11.1%	22.4%	0.21
	Education level, high a , %	44.1%	39.5%	50.0%	48.0%	0.43
	Neuropsychological tests at baseline
	MMSE, median [range]	28 [24 -30]	29 [25 -30]	29 [24 -30]	28 [24 -30]	0.091
	FCSRT, free recall, mean (SD) 20.5 (7.2) ** † † † ‡ ‡	27.0 (5.1)	24.8 (6.3)	25.1 (5.6)	<0.001
	FCSRT, total recall, median [range]	43 [19 -48]* †	46 [37 -48]	46 [33 -48]	46 [30 -48]	<0.001
	BVRT, median [range]	11 [6 -15]	12 [8 -15]	12 [7 -15]	12 [6 -15]	0.067
	Isaac set test 15s, mean (SD)	28.9 (5.1)*	30.5 (4.4)	30.8 (5.6)	31.8 (6.1)	0.033
	Isaac set test 30s, mean (SD)	43.4 (8.0)	47.5 (6.9)	47.9 (8.4)	47.8 (9.7)	0.074
	Isaac set test 60s, mean (SD)	63.2 (14.5)*	70.4 (10.6)	71.0 (13.8)	70.9 (15.4)	0.043
	TMT-A, mean (SD)	4.7 (1.4)	5.1 (1.5)	5.0 (1.4)	4.8 (1.5)	0.49
	TMT-B, mean (SD)	2.4 (1.1)	2.0 (1.0)	2.4 (1.1)	2.3 (1.2)	0.31
	MRI volumes at baseline
	Hippocampal volume, mean % ICV (SD)	0.54 (0.07) †	0.58 (0.03)	0.51 (0.05) ** † †	0.56 (0.01)	<0.001
	Cortical volume, mean % ICV (SD)	40.1 (7.2)	42.1 (2.2)	40.1 (2.6)	39.8 (6.3)	0.12
	ICV : IntraCranial Volume; MMSE : Mini Mental State Examination ; SD : Standard
	deviation ; TMT : Trail Making Test. a : Education level was considered as high or low
	according to French baccalaureate (equivalent to A-level). b: p-values refer to Chi-squared test
	for categorical variables, and ANOVA or Kruskal-Wallis test for ordinal variables. *p<0.05
	and **p<0.01 vs ''others''; † p<0.05, † † p<0.01 and † † † p<0.001 vs ''cortical predominant
	atrophy''; ‡ p<0.05 and ‡ ‡ p<0.01 vs ''homogeneous atrophy'' (Chi square test or post-hoc
	Tukey-Kramer or Dunn tests, as appropriate).

Table 2 : Predictive values of age, gender, educational level, ApoE4 genotype alone (model 1) and in combination with brain atrophy subtype (model 2) on time to incident Alzheimer

 2

	Hazard ratio	95% CI

's disease (Cox models, n = 368).

Acknowledgements and Funding

The 3C Study is conducted under a partnership agreement among the Institut National de la Santé et de la Recherche Médicale (INSERM), Bordeaux University, and Sanofi. The Fondation pour la Recherche Médicale funded the preparation and initiation of the study. The 3C Study is also supported by Caisse Nationale Maladie des Travailleurs Salariés, Direction Générale de la Santé, Mutuelle Générale de l'Education Nationale, Institut de la Longévité, Conseils Régionaux d'Aquitaine et Bourgogne, Fondation de France, and the Ministry of Research-INSERM Programme "Cohortes et collections de données biologiques". The 10-year and 12-year follow-up have been funded by ANR 2007LVIE 003 and "Fondation Plan Alzheimer". This study was also achieved within the context of Human Brain Project, the Laboratory of Excellence TRAIL ANR-10-LABX-57 for the BigDataBrain project and the Investments for the future Program IdEx Bordeaux (ANR-10-IDEX-03-02, HL-MRI Project),

Cluster of excellence CPU and the CNRS. VP also received grants from Fondation Bettencourt Schueller (CCA-Inserm-Bettencourt). The sponsors did not participate in any aspect of the design or performance of the study, including data collection, management, analysis, and the interpretation or preparation, review, and approval of the manuscript.

Author Contributions

Conception and design of the study: VP, FT, JFD and GC. Acquisition and analysis of data: VP, PC, CH, MLG, HA, JFD and GC. Drafting the manuscript: VP and GC. Revision of the manuscript for important intellectual contents: all authors.

Potential Conflicts of Interest

The authors declare no competing financial interests relative to the present study.