Oussama Derbel

Tamas Peter
email: peter.tamas@mail.bme.hu

Benjamin Mourllion

Michel Basset

GENERALIZED VELOCITY-DENSITY MODEL BASED ON MICRO-SCOPIC TRAFFIC SIMULATION

Keywords: Velocity-Density Model, microscopic traffic simulation, Adaptive Cruise Control sented in the environmental parameter vector e: V=v(ρ, e)

]. In our calculations also the applied model parameter values remained in the range of the model parameters used in the literature.

This paper presents a new approach for Velocity-Density Model (VDM) synthesis. It consists in modelling separately each of the density and the velocity (macroscopic parameter). From this study, safety time headway (microscopic parameter) can be identified from macroscopic data by mean of interpolation method. By combining the density and the velocity models, a generalized new VDM is developed. It is shown that from this one, some literature velocity-density models, as well as their properties, can be derived by fixing some of its parameters.

Introduction

Traffic modelling can be classified into the three following classes.

The first class includes the microscopic models where the traffic is viewed as a system of interacting particles (vehicles). This interaction is modelled by differential equations as done by [START_REF] Bando | Analysis of optimal velocity model with explicit delay[END_REF]], [Helbing & Tilch (1998)], [START_REF] Treiber | Microscopic simulation of congested traffic[END_REF]], [Molina (2005)], [START_REF] Ge | Two velocity difference model for a car following theory[END_REF]], [Kesting (2008)], [START_REF] Rakha | [END_REF]].For interested readers, a general overview of microscopic traffic models is presented by [Derbel, Mourllion & Basset (2012)].

The second class includes the macroscopic models where the traffic is viewed as one group of particles. Macroscopic parameters are traffic density, traffic flow and velocity which are used for graphic representation of the fundamental diagram. Several works are based on this approach such as those ones developed by [Holden & Risebro (1995)], [Herty & Klar (2003)], [START_REF] Treiber | Microscopic simulation of congested traffic[END_REF]], [Peter (2012)], [Peter & Szabo (2012)], [Bede & Peter (2013)], [START_REF] Peter | Model for the analysis of traffic networks and traffic modelling of gyor[END_REF]], [START_REF] Peter | Model for the analysis of traffic networks and traffic modelling of gyor[END_REF]] [START_REF] Bede | [END_REF]].

The third class includes the mesoscopic models, which appears to be an intermediate between the two last classes. Here, the traffic is viewed as clusters of vehicles. Several studies have been realized in this field such as those done by [START_REF] Prigogine & Herman ; Prigogine | Kinetic theory of vehicular traffic[END_REF]], [START_REF] Paveri-Fontana | On boltzmann-like treatments of traffic flow: A critical review of the basic model and an alternative proposal for dilute traffic analysis[END_REF]] and [START_REF] Mahnke | [END_REF]].

The presented research work deals with microscopic and macroscopic approaches, where the traffic is represented by one cluster. The goal is to predict traffic density and velocity with a minimum of errors. Most of the literature velocity-density models are developed and identified by mean of experimental macroscopic traffic data. The lack of these ones leads to work under simulation. Here, the idea is to generate macroscopic traffic data from microscopic traffic model simulation. Nonetheless, this microscopic model must faithfully reproduce the traffic behaviours and be validated by the fundamental diagram study.

In this paper, the developed Velocity-Density Model (VDM) is based on the Intelligent Driver Model (IDM) ([Kesting (2008)], [treiber et al. (2000)]). Our new method proceeds by the identification of the simulated velocity and density data separately. Among the advantages of this method is the ability to identify each microscopic parameter from macroscopic ones by interpolation method. Then, a new VDM is computed with a mathematical development of the velocity and density models. And it will be shown later that some of existing models in the literature, as well as their properties can be derived from this new VDM by fixing some of its parameters.

In this paper, Section 1 presents a brief state-of-the art of the existing velocity-density models. Section 2 identifies the velocity and the density functions. Section 3 presents the synthesis method of the new VDM. Section 4 presents the generalized VDM and section 5 concludes and gives outlooks.

State-of-the-art of the Velocity-Density Models

Two classes of Velocity-Density Models can be distinguished: the stochastic els and the deterministic models. This section presents a brief state-of-the-art of deterministic VDM considering logical order.

GREENSHIELDS MODEL

Up to our knowledge, the first ministic VDM was proposed by [greenshields (1934)]. This affine model is given by the following expression

) 1 ( ) ( max D D V D V Max   (1)
where D max is the maximum density which is the jam density. This model is identified by linear regression method using seven experimental observations.

Recently, [START_REF] Wang | Stochastic modeling of the equilibrium speed-density relationship[END_REF]] have showed that, with more than seven data, the Greenshields model is not enabling the prediction of velocity and density.

GREENBERG MODEL

By the analogy with fluid flow, [Greenberg (1959)] has developed a logarithmic velocity-density relationship given by

) log( ) ( max D D V D V Max  (2)
The main drawback of this model is its inability to predict velocity for low densities. Indeed, the velocity tends to infinity when the density tends to zero, which is unrealistic.

UNDERWOOD MODEL

[ [START_REF] Underwood | Speed, volume and density relationship: quality and theory of traffic flow[END_REF]] has derived an exponential model in order to overcome the drawbacks of the Greenberg and Greenshields models for the free traffic flow condition. This model is given by

) exp( ) ( max D D V D V Max   (3)
The main drawback of the Underwood model is that velocity becomes zero only when density D reaches infinity and not D=D max . Hence, this model cannot be used for predicting velocities at high densities.

NEWELL MODEL

The VDM of [Newell (1961)] is expressed by

                    ) 1 1 ( exp 1 ) ( max max D D V V D V Max  (4)
where λ>0 is the slope of inter-distance-velocity curve at V=0 km/h. Here, when D=D max then V=0 km/h. When D=0 then V=V max km/h. Therefore, the limit conditions are verified by the Newell model.

1.5 DRAKE MODEL [START_REF] Drake | A statistical analysis of speed-density hypothesis[END_REF]] have enhanced the Greenberg model by studying various macroscopic traffic models. By estimating the density from velocity and flow data, they propose the new VDM expressed by

                  2 max 2 1 exp ) ( D D V D V Max (5)
According to [START_REF] Ardekani | Macroscopic speed-flow models for characterization of freeway and managed lanes[END_REF]], this model presents a better fitting than the above Greenshields, Greenberg and Underwood models for non-congested conditions. In case of congested conditions, the Drake model presents a poor data fitting.

1.6 PIPES MODEL [Pipes (1967)] has generalized the Greenshields model leading to a new velocity-density relationship given by

m r Max D D V D V                   max 1 ) ( (6)
By varying the values of r and m, a faily of models can be developed. For example, Greenshields model is obtained for r=1 and m=1.

1.7 DREW MODEL [Drew (1968)] has proposed another model expressed as follows:

                     2 1 max 1 ) ( p Max D D V D V (7)
According to [START_REF] Ardekani | Macroscopic speed-flow models for characterization of freeway and managed lanes[END_REF]], at free-flow phase, the Drew model presents an underestimated velocity, but in the congested phase, the velocity is overestimated.

1.8 DEL CASTILLO MODEL [Del Castillo & Benitez (1995)] have developed a velocity-density model which is given by the following expression:

                   ) 1 ( exp 1 ) ( max max D D V C V D V j Max (8)
where C j is the kinematic wave speed given by max max

D D j dD dV D C   
According to [START_REF] Macnicholas | A simple and pragmatic representation of traffic flow[END_REF]], the drawback of this model is the large kinematic wave speed range which makes difficult its estimation.

If D max =1 and C j = λ, then we have the Newell model.

1.9 VAN AERDE MODEL [Aerde (1995)] developed a new velocity-density model which is based on a simple car-following model. This last one is depends on the free and current velocity and a calibrated constants. The minimum desired distance headway is the output of this model. The velocity-density model is given by

V C V V C C V D 3 max 2 1 1 ) (     (9)
where C 1 , C 2 , and C 3 are constants which can be calibrated by nonlinear regression.

1.10 MAC NICHOLAS MODEL [START_REF] Macnicholas | A simple and pragmatic representation of traffic flow[END_REF]] have proposed the following velocity-density model

q q q q Max D m D D D V D V     max max ) ( (10)
where m and q are real constants. By varying these constants, a family of models can be developed. For example, if D max =1, m=0 and 2 1   p q the Drew model expression is found.

POWER FUNCTION MODEL

The power function proposed by [Del Castillo (2012)] is given by

                   1 ) 1 ( ) ( ) ( 1 ) (     b a b a b D V (11)
where θ is a shape parameter,

EXPONENTIAL MODEL

The exponential model is given by Del Castillo (2012)as follows

                  1 log 1 ) ( 1 ) ( ) 1 (       

NEGATIVE POWER MODEL

The negative power model is presented by [Del Castillo (2012)] as

                1 1 1 ) (    f u D V (13)
Where w is a shape parameter.

ENVIRONMENTAL

PARAMETERIZATION MODEL [START_REF] Peter | [END_REF]] have enhanced the environmental parameterization model by studying various macroscopic traffic models. The classical literature does not deal with the definition of the environmental vector, but the velocity is determined not only by vehicle density, but by other environmental parameterization, as well: this refinement can be implemented with the modification of V Max , or via the modification of the function itself considering the weather, visibility, road quality, width of the road. These environmental, seasonal factors can be repre- The following table demonstrates the favourable and unfavourable parameter domains. The internal domain is located between the two distinct domains, in most of the cases the practical parameter comes from this internal interval. The borders of the intervals are empirical values, in a given case the coordinates of the e = [e 1 , e 2 , e 3 , e 4 , e 5 ] parameter-vector are determined via regression analysis after the velocitydensity measurement.

Table 1 Demonstration of the e parameter vector e i

Meaning of the parameter

Unfavorable cases

Favorable cases This result shows that Greenshields' linear function has parameter values from the mid-range, so that provides really an average v(ρ) velocity-density function relationship in practice.

SYNTHESIS

Table 2 summarizes the references with the macroscopic traffic models cited in this section. V max is the maximum velocity, D max the jam density, λ the slope of interdistance-velocity curve at V = 0, C 1 , C 2 , C 3 , m, p, q and r are constants and C j is the kinematic wave speed at jam density.

Table 2 Velocity Density model summary

Reference Model [Greenshields (1934)]

) 1 ( ) ( max D D V D V Max   [Greenberg (1959)] ) log( ) ( max D D V D V Max  [Underwood (1961)] ) exp( ) ( max D D V D V Max   [Newell (1961)]                     ) 1 1 ( exp 1 ) ( max max D D V V D V Max  [Drake et al. (1967)]                   2 max 2 1 exp ) ( D D V D V Max [Pipes (1967)] m r Max D D V D V                   max 1 ) ( [Drew (1968)]                      2 1 max 1 ) ( p Max D D V D V [Del Castillo & Be- nitez (1995)]                    ) 1 ( exp 1 ) ( max max D D V C V D V j Max [Aerde (1995)] V C V V C C V D 3 max 2 1 1 ) (     [MacNicholas & Michael (2008)] q q q q Max D m D D D V D V     max max ) ( [Del Castillo (2012)] (power model)                    1 ) 1 ( ) ( ) ( 1 ) (     b a b a b D V [Del Castillo (2012)] (exponential model)                   1 log 1 ) ( 1 ) ( ) 1 (        b a e e b a b D V [Del Castillo (2012)] (negative power model)                 1 1 1 ) (    f u D V [Peter & Fazekas (2014)] environmen- tal parameterization model) 1 5 1 ) , ( 2 3 4 e e Max e e V e e V                
It was necessary to review the evolution of the Velocitydensity laws in the literature, since these functions have a decisive role in the fundamental equation in case of traffic models. They shall satisfy two conflicting needs: to be as simple as possible, so that the large complex simulation models provide fast numerical calculations, while ing the real practical situations as widely as possible. A common feature of the velocity -density laws is that they result from non-linear regression analysis based on urements and hypotheses (preconceptions). The common flaw of the classical results is that because they do not refer to other environmental impacts, they only examine the effect of speed on the vehicle density, so essentially any of them can be suitable for a given modelling. [START_REF] Peter | [END_REF]] goes much further and examines also the environmental parameters. In case of the IDM-models the comprehensive examination of the speed-density tions is an interesting problem, because without ception an expanded set of functions can be examined from the point of view of which known function types, or other new models are considered as valid by this model. Of course, neither the classic IDM-models contain the environmental parameters mentioned above, so the studies in this area are far from closed. The next section is dedicated to generate microscopic and macroscopic simulated traffic data which are the density and the velocity. In addition, the impact of microscopic parameters to macroscopic ones is investigated.

Macroscopic traffic data generation

Based on microscopic traffic simulation, density-time and velocity-time functions will be computed in this section.

Paragraph 2.1 presents the simulation assumptions and the Intelligent Driver Model intended to represent the microscopic longitudinal vehicle motion. Paragraph 2.2 introduces the used mathematical formula to compute macroscopic parameters (velocity and density). Paragraph 2.3 presents the simulation results and discusses it and paragraph 2.4 studies the impact of microscopic parameter on macroscopic ones.

MICROSCOPIC TRAFFIC MODEL: INTELLIGENT DRIVER MODEL (IDM)

The Intelligent Driver Model (IDM) is an Adaptive Cruise Control system intended for adjusting the driver's longitudinal desired velocity and safety time gap. The IDM model, developed by [START_REF] Kesting | Adaptive cruise control design for active congestion avoidance[END_REF]

]), is ex- pressed by . , 4 2 * ( , ) 1 0 n n v s v v n n n a s v n n v                              (15)
where a n is the maximum acceleration of the vehicle n (m/s 2 ), v 0 n the desired velocity of the vehicle n (m/s), and the distance gap (m).

1     n n n l x s
The desired minimum gap of the vehicle n, s* n is given by * 0 ( , ) 2

v n v n s v v s v T n n n n n av nn      (16)
where b n is the desired deceleration of the vehicle n (m/s 2 ), s 0 n the jam distance of the vehicle n (m) and T n the safety time gap of the vehicle n (s).

Compared to the other ACC models in the literature, the IDM shows more advantages in terms of easy implementation, calibration and the intuitive and the availability of its parameters. These ones are the desired velocity v 0 n and the safety time gap T n which are fixed as system inputs.

The next subsection details the density and velocity formula for computing macroscopic data under microscopic traffic simulation using the IDM model.

DENSITY AND VELOCITY FORMULA

Density formula

As shown in subsection 2.14 in table 1, the quotient between the traffic density and the jam density are used in all models in the literature for density normalization. In this paper, the density is already normalized. Here, the traffic road section is limited by the first and the last vehicle. Then, the density formula can be written as where D max =1 in this case. For this purpose, in the rest of this paper, the density d will be noted as D.

Velocity formula

The platoon velocity v is the average velocity of the vehicles in the platoon which is given by 1

1 ( ) ( ) N n n v t t N v    (18)
where v n is the velocity of the vehicle n.

SIMULATIONS AND RESULTS

The simulation model

The following system of non-linear matrix differential equations shows the structure of the IDM-model under investigation. For a multi vehicle system, the Intelligent Driver Model is given by: 1

)) ( ( )) ( ( ) ( 2 1 1 1      t x f S t x f V t x A    (19) with N a a a A 1 ,..., 1 , 1 2 1 1   ; 4 4 2 4 1 1 ,..., 1 , 1 1 N v v v V   ; N s s s S ,..., , 2 1  fixed s s i i   0 or: ) , ( 1 i i i i x x s s     (i=1,2,…,N).                4 4 2 4 1 1 ... )) ( ( N x x x t x f     ,                          2 1 2 2 1 2 1 0 2 ) ( 1 ... ) ( 1 ) ( 1 )) ( ( N n x x x x x x t x f ,              1 ... 1 1 1
This model examines the longitudinal dynamics in the direction of travelling in case of vehicles travelling in a lane and is specifically capable of analyzing the evolution of the speed-density law. The movement is performed on a line, which is a given lane. We have dealt with the ther generalization of the model in [START_REF] Derbel | Modified intelligent driver model[END_REF]] and [START_REF] Derbel | Modified intelligent driver model for driver safety and traffic stability improvement[END_REF]].

Simulation assumptions

In our simulations, the following assumptions are taken into account:

 The AAC model, which is our microscopic traffic model, is already validated and its parameters are identified based on experimental data ( [START_REF] Kesting | Adaptive cruise control design for active congestion avoidance[END_REF]]).  The traffic is homogeneous. It means that the longitudinal motion of all vehicles is controlled by the same ACC (IDM) with the same parameter values. A heterogeneous traffic includes vehicles with manual and automatic driving modes.  The road has only one lane.

 Only the longitudinal vehicle motion is studied.

The lane change and the lateral motion are neglected.

The simulator

The development of traffic simulators has received much attention in recent years. Many companies have invested in such projects and have made simulators integrating various dynamic and kinematic models of vehicles, driver and road architectures available. These simulators are known by their complexities in terms of software development. For example, the simulator ARCHISIM adopts multi-agent approach and techniques from artificial intelligence to simulate complex phenomena. Our motivations for the development of a mixed traffic simulator (coexistence of automated and manual driving style in the same traffic section) are justified according to the following two aspects:

-The high cost of traffic simulators in the market.

-The collision management problem. For example, the traffic simulators CarMaker and Microsimulation of Road Traffic Flow (MRTF) do not perfectly manage collisions between vehicles.

Our mixed traffic simulator is developed using the C language and the OpenGL library (Open Graphics brary) for graphical interface. Figure 2 shows the structure of mixed traffic simulator. This simulator has inputs that are used by its different modules to save the output of the simulation in different formats such as video and/or data required for post-processing.

The inputs of the simulator are the following three:

-"XML File" in which some parameters can be set: the simulation number, the initial vehicle number, the road length and the minimum and maximum percentage of automated vehicles.

-"Global Settings": these parameters are fixed throughout the simulation (e.g. road width).

-"OpenGL library": This library is responsible for the GUI management such as scenarios and architectures used in the simulation.

In the simulator body, the following modules are developed:

-"Scenario Development Module": This module cludes the different developed scenarios. They take their input parameters through the "main" function. It selects the script to run from the choice made in the XML file. In this module, there are useful models: the IDM model for longitudinal automated driving style, the Two Velocity Difference Model (TVDM) [START_REF] Ge | Two velocity difference model for a car following theory[END_REF]] for longitudinal manual driving style and the Minimizing Overall Braking change Induced by Lane change (MOBIL) model [START_REF] Kesting | Adaptive cruise control design for active congestion avoidance[END_REF]] for the lane change management. These models need to be updated in each simulation step through the numerical integration module (Runge Kutta).

-"GUI Management Module": This module is based on the OpenGL library and the measurements are recorded to generate and update the simulator graphical interface.

The outputs of this simulator are the following three:

-"Screen": Displays the architecture and the scenario.

-"Video": to save the simulation -"File .txt or .bin": saves the data needed for postprocessing Figure 3 shows the position of the vehicles in a platoon. The IDM simulation model with the parameters given by table 3 is applied to simulate the fully automated traffic using our developed microscopic traffic simulator. The presented parameter values in this table are originated from an identification step using experimental data and used to validate the IDM with the fundamental diagram by [START_REF] Kesting | Adaptive cruise control design for active congestion avoidance[END_REF]]. At each simulation step, the simulated microscopic traffic data such as the position, the velocity and the acceleration of each vehicle are computed. Then, the density and the velocity are computed according to equations ( 17) and ( 18) respectively. Simulation parameters are given in table 4. In this table, the jam distance s 0 n is set to zero in order to have the maximum density equal to 1.

Figure 2 Structure of the developed simulator

To study the sensitivity of macroscopic parameters (velocity and density) according to the microscopic parameters (a n , b n , T n) , traffic simulation is performed made for each of the maximum acceleration a n values (the other microscopic parameters are constants), the desired deceleration b n (the other microscopic parameters are constants) and the safety time headway T n (the other microscopic parameters are constants). Figure 4 shows the density versus the time (first curve) and the velocity versus the time (second curve) during 1000 s of simulation with the microscopic parameters given by table 3 and the simulation parameters given by Table 4. Here as the density increases, the velocity decreases until the stable state. The ranges are empirical intervals given by [START_REF] Kesting | Adaptive cruise control design for active congestion avoidance[END_REF]]. In this paragraph, it is noted that all vehicles have the same parameters (homogeneous traffic).

Simulation results

Figure 3 Positions of the vehicles in the platoon

Impact of a n , b n and T n on density

As shown in figure 3, the maximum acceleration and the desired deceleration have no impact on traffic density. Indeed, when the traffic becomes stable, 0   n v

, ∀n ∈ {1,...,N} and then

0 2     n n n n b v a v v .
The safety time headway T n has an impact on traffic density i.e when T n increases, traffic density decreases. In fact, vehicles tend to increase inter-distance when T n is high, then the traffic density decreases.

Impact of a n , b n and T n on velocity

As shown in figure 6, the maximum acceleration and the desired deceleration have no impact on the velocity. The same reason cited in 3.4 is the cause of this result: when the traffic becomes stable,

0   n v ∀ n ∈ {1...N} and then 0 2     n n n n b v a v v
. The safety time headway T n has an impact on the velocity i.e. when T n decreases vehicles velocity increases. Vehicles accelerate to reach the desired velocity until the safety time headway is guaranteed.

Figure 7 shows the impact of the time headway T n on the macroscopic parameters. Now the curves of this figure will be considered as reference to us. In case of unknown time headway, from the density and the velocity values, we can identify the mean value of time headway between vehicles way by mean of interpolation.

New velocity-density model synthesis method

The new method consists in studying the variation of each of the two macroscopic parameters (velocity and density given by the figure 2) separately along the simulation time. To carry out this study, a family of functions is computed to approximate the density-time and velocitytime functions. For a given γ, this family of candidate functions is expressed by

  2 1 3 2 1 exp ) ( ; ) , , ( / F t F t F F F F H f f                   
For reasons which will be later explained, γ is fixed for all candidate functions of the two macroscopic parameters. The function D ∈ H γ , which is a candidate to approximate the density-time function, is given by

t vt V V v          (21)
where V 1 <0, and τ v >0, V 2 >0 are constants (V 2 ≥ïV 1 ½). γ is fixed for the two candidate functions to have an analytical time independent velocity-density relationship from equations ( 20) and ( 21). D and v functions are used to fit separately the density and the velocity data in the least square sense. Figure 6 shows the fitted curves of the density and the velocity together with the macroscopic data given by IDM simulation. Parameters of these two functions, obtained with setting γ = 0.8, are summarized in table 5. The mean, the maximum and the minimum absolute errors are given by table 6. The absolute error between data given by microscopic simulation and the computed functions D and v is small compared to the maximum values of each one. The IDM is written as

. , 4 2 * ( , ) 1 0 n n v s v v n n n a s v n n v                               and * 0 ( , ) 2 v n v n s v v s v T n n n n n av nn      .
The solution for the velocity is given by ( ) exp 1 2

t vt V V v          When traffic become stable, we obtain 0  n v  and 0   v t e   . Then,     2 4 4 2 4 2 2 0 0 . . n n n n T v V V v s         (22)
And, as shown in equation ( 22), V 2 depends on the interdistance and the safety time headway T n and not from a n and b n .

Velocity-density model synthesis

From the density model defined by equation ( 20) and the velocity model defined by equation ( 21), the VDM is developed analytically in this section.

MODEL SYNTHESIS

Equations ( 20) is given by

( ) exp 1 2 t Dt D D d          (23)
The function v ∈ H γ , which is a candidate to approximate velocity-time function, is given by

( ) exp 1 2 t vt V V v          (24)
From these two equations, we get

                             1 2 1 2 ) ( ln ) ( ln V V t v t D D t D t v d     (25) 2 1 2 1 ) ( )) ( ( V D D t D V t D v v d              (26) Let V, the function defined as 2 1 2 1 ) ( : V D D t D V D V v d              (27)
With slight abuse of notation, we have v (D (t)) = V (D), Then, the analytic velocity-density relationship can be written as

2 1 2 1 ) ( V D D D V D v v d              (28)
To calibrate this model, the limit conditions are applied in the next sub-section.

GENERALIZED VELOCITY-DENSITY MODEL

Applying the limit conditions given by

            max 2 1 2 ) ( ; 0 1 V V D D V V V D V V   (29) to equation (28), we have                                max 2 1 2 2 1 2 1 2 1 1 V V D D D V V D D V v d v d V      (30)
Equations ( 30) is given by ( 31) and ( 32)

max 2 V V  (31) v d v d D D D D V V V                           1 2 1 2 max 1 1 1 (32)
using ( 28), ( 31) and ( 32)

max 1 2 1 2 1 2 max 1 1 ) ( V D D D D D D D V D V v d v d v d V                                                        (33)
Then, the generalized VDM can finally be expressed as

v d v d D D D D D D V D V V                                        2 2 2 2 max 1 1 1 ) ( ] , 1 [ 2 D D  (34)
Where τ d >0, τ v >0, D 1 >0, D 2 >0 are constants, and D 1 +D 2 =1. From this calibrated model, some of models defined in section 2 can be derived. Next subsections show which of these models can be obtained.

SUB MODELS

From the model defined by equation ( 34), some of existing models in the literature can be derived by fixing some of its parameters.

If D 2 =0 and e V =0

In this case, 20) and ( 21) can be written as

v d D V D V   ) 1 ( ) ( max   (

If

D 2 =0, e V =0 and τ d = τ v In this case, ) 1 ( ) ( max D V D V   (36) Equations (
( ) exp 1 2 t Dt D D d          (37) ( ) exp 1 2 t Vt V V v         (38) Then ( ) 1 1 2 1 D VD V V D       (39)
Here, the relationship between velocity and density remains linear. In case of the Greenshields model, V max =V 2 and D max =1. → 0, then only lower densities can be predicted. Therefore, velocity can be estimated only in free flow condition. Thus, the model cannot predict lower velocities. This result has been proven by [START_REF] Ardekani | Macroscopic speed-flow models for characterization of freeway and managed lanes[END_REF]], empirically and proved analytically now by our method of velocity-density synthesis and through our generic model.

Conclusions

The Velocity-density functions known from the literature are results of non-linear regressions, which are obtained by the determination of functions that fit the measured coherent speed-density values with the minimum error. The first step of the analysis we have performed in the field of IDM-models is also a non-linear regression, which was developed based on the speed and density changes over time. Next we have described a direct function relation for the speed-density function using a time parameter. There is no defined speed-density law in the IDM-model, so it was an important achievement to show what speeddensity function is followed by the IDM-model. A separate new interesting result is that the law presented by us has defined a more general class of function compared to what was known earlier. At the same time this has also integrated three known speed-density relationships as special cases (with the proper selection of parameters) [Greenshields (1934)], [Pipes (1967)] and [Drew (1968)].

We emphasize that the analysis -in terms of the development of simulation models -led to another important question. The known speed-density functions are univariate functions of the vehicle densities. The significant formal differences between these types of functions raise the question whether the differences can significantly be predetermined by the environmental parameters, as well. An interesting question is whether more general classes of functions that integrate more known function types can be determined by broader measurement and analysis, and further IDM analysis. These known function types are multivariate functions that depend on the density and environmental parameters, as well, for example the one presented in [START_REF] Peter | [END_REF]]. The simulator discussed in this article is capable of carrying out these further investigations.

ACKNOWLEDGEMENT

Authors thank the Hungarian Government and the European Social Fund for their financial support to the "Smarter Transport" -IT for co-operative transport system project. TÁMOP-4.2.2.C-11/1/KONV-2012-0012.

  Figure 1 V(ρ) velocitydensity function with e 1 =2; e 2 =1;e 3 =1;e 4 =1;e 5 =1 parameters

  the introduced V=v(ρ,e) function is that it gives the same results as the linear function of Greenshields, if every parameters' value equals to 1 (Fig.2.).

Figure 2 V

 2 Figure 2 V(ρ) velocitydensity function with e 1 =1; e 2 =1;e 3 =1;e 4 =1;e 5 =1 parameters

  length of the vehicle (k=1,2,…,N), N the number of vehicle in the platoon and L platoon the length of the platoon. And the density d can be normalized in the interval [

Figure 4

 4 Figure 4 Traffic density and velocity time functions 2.4 FROM MACROSCOPIC PARAMETERS TO MICROSCOPIC PARAMETERS In this section, two goals are fixed: the first one consists in studying the dependency of the shape of the velocitytime and the density-time functions to microscopic parameters. The second goal is to study the ability to identify microscopic parameters from macroscopic parameters. Microscopic parameters are the maximum acceleration a n , the desired deceleration b n and the safety time headway T n . In this step the following assumptions are made: the maximum acceleration a n ∈ [2, 7] m/s 2 the desired deceleration b n ∈ [2, 5] m/s 2 and the safety time headway T n ∈ [1.5, 4] s. The ranges are empirical intervals given by [Kesting et al. (2008)]. In this paragraph, it is noted that

Figure 5

 5 Figure 5 Density for different a n , b n and T n sets

Figure 6 Figure 7

 67 Figure 6 Velocity for different a n , b n and T n sets

  where D 1 , τ d and D 2 are constants. The function v ∈ H γ , which is a candidate to approximate the velocity-time function,

Figure 8

 8 Figure 8 Density-time function with D function and velocitytime function with v function

  D max =1. One of the advantages of the new model is the finding of the physical meaning of the parameter m.

  which is the Drew model given by equation (7), where D max =1 . Since τ d and τ v are two time constants, p+1 ≥ 0 and then p ≥ -1. In this study, the density is

Table 3 IDM simulation parameters

 3

	Parameter	Mean value	Unit
	Maximum acceleration a n	3	m/s 2
	Desired deceleration b n	3	m/s 2
	Safety time headway T n	1.5	s
	Table 4 Simulation parameters
	Parameter	Value	Unit
	Vehicle number	20	vehicle
	Simulation time	1000	s
	Simulation time step	0.01	s
	Initial inter distance	0	m
	Initial acceleration	0	m/s 2
	Desired velocity	50	km/h
	Initial velocity	0	km/h
	The jam distance	0	m
	Vehicle length	5	m

Table 5 Parameters of the D and v functions with γ = 0.8

 5

	Parameter	Mean value	Unit
	D 1	0.8971	-
	τ d	0.1205	s
	D 2	0.1029	-
	V 1	-48.4583	km/h
	τ v	0.0813	s
	V 2	48.48205	km/h
	R 2 ( Velocity)	0.9788	-
	R 2 ( Density)	0.9399	-

Table 6 Density and velocity error

 6

	Parameter	Mean	Max	Min
	Density error	0.0157	0.0786	0
	Mean velocity error	0.6402	6.3786	0


Cécile Batailler

Sébastien Parratte
email: sebastien.parratte@gmail.com

Assistive technologies in knee arthroplasty: fashion or evolution? Rate of publications and national registries prove the Scott Parabola wrong

Keywords: Knee arthroplasty, Scott parabola, Robotic surgery, Computer-assisted system, Assistive technologies

Purpose Most opponents of assistive technologies in orthopedic surgery consider them as a marketing ruse or fashion. Our hypothesis was that many innovations in modern knee arthroplasty are not following the Scott Parabola. This parabola rep-resents the visual curve of a procedure or therapy showing great promise at the beginning, becoming the standard treatment after reports of encouraging results, only to fall into disuse due to adverse outcome reports. This study aimed to assess the interest in these assistive technologies by (1) their number of publications/year and (2) their actual surgical use reported in the National Joint Registries. Methods The search was performed through PubMed, EMBASE, and MEDLINE databases from 1997 to 2021 inclusive to identify all available literature that described the use and results of assistive technologies or new surgical techniques in knee arthroplasty. In the Australian and Norwegian registries, the number of cases performed with these techniques in knee arthroplasty has been quantified year by year. Results Following the initial online search, a total of 4085 articles was found. After the assessment mentioned above, 2106 articles were included in the study. The orthopedic techniques assessed in this study are not following the "Scott's parabola" in the literature. Computer-assisted knee arthroplasty and patient-specific instrumentation have increased quickly to have reached a plateau, with a stable number of publications over the last 6 years. The number of publications concerning robotic surgery, accelerometers and sensors continue to rise. In the Australian registry, the proportion of primary TKA performed by computer-assisted systems increased from 2.4% in 2003 to 32% in 2019. In the Norwegian registry, the proportion of computer-assisted TKA remained between 8 and 12% of primary TKA since 2007. Conclusion Most of the innovations in modern knee arthroplasty are not following the Scott Parabola. After a fast rise, these techniques do not disappear but continue to evolve. Their evolution is synergistic, and techniques appeared to be linked to each other's. Despite persisting concerns about the cost-efficiency of assisting technologies in knee arthroplasties, the inter-est and use do not decrease and seems to be directly linked to an exponential increase in interest for a better understanding of alignment targets and improved functional recovery.

Introduction

Orthopedic surgery is one of the most dynamic surgical specialties, and knee arthroplasty surgery is one of its fastest-growing sub-segments. In fact, the number of potential candidates for knee arthroplasty is estimated to grow exponentially during the upcoming 20 years [START_REF] Kurtz | Projections of primary and revision hip and knee arthroplasty in the United States from 2005 to 2030[END_REF]. In parallel of this growth, the assistive technologies have been progressively developed to increase the accuracy and the reproducibility of knee arthroplasties [START_REF] Gao | New technologybased assistive techniques in total knee arthroplasty: a Bayesian network meta-analysis and systematic review[END_REF]. In early 2000, computer-assisted surgery (CAS) started to gain interest. CAS was a logical solution to improve implants position through smaller incisions. After a decade of CAS use, new tools arrived in the surgical armamentarium, such as patient-specific instrumentation and accelerometers-based navigation [START_REF] Kawaguchi | Comparison of an accelerometer-based portable navigation system, patient-specific instrumentation, and conventional instrumentation for femoral alignment in total knee arthroplasty[END_REF][START_REF] Liow | Computer-assisted stereotaxic navigation improves the accuracy of mechanical alignment and component positioning in total knee arthroplasty[END_REF]. At the same time, new concepts concerning patient's alignment and more anatomic solutions came-up too. The latest assistive technology that is gaining interest is the robotically assisted surgery in knee arthroplasty [START_REF] Van Der List | Current state of computer navigation and robotics in unicompartmental and total knee arthroplasty: a systematic review with meta-analysis[END_REF][START_REF] Robinson | A systematic review of robotic-assisted unicompartmental knee arthroplasty: prosthesis design and type should be reported[END_REF][START_REF] Onggo | Roboticassisted total knee arthroplasty is comparable to conventional total knee arthroplasty: a meta-analysis and systematic review[END_REF]. Clear advantages and superiority of these innovations or assistive technologies have never been proven [START_REF] Klasan | Conventional instruments are more accurate for measuring the depth of the tibial cut than computer-assisted surgery in total knee arthroplasty: a prospective study[END_REF][START_REF] Yamamura | Evaluation of the accuracy of resected bone thickness based on patient-specific instrumentation during total knee arthroplasty[END_REF]. Most of the opponents of these techniques are claiming that these solutions are just trends coming one after another that will fade. Several surgeons state, for example, that robotic surgery as a fashion and a marketing ruse [START_REF] Defrance | The impact of author financial conflicts on robotic-assisted joint arthroplasty research[END_REF][START_REF] Booth | Robotics in hip and knee arthroplasty: real innovation or marketing ruse[END_REF]. They often use the Scott Parabola as a reference to describe the phenomenon.

The Scott Parabola describing the rise and the fall of a new surgical technique. This scheme of great craze for innovation then its progressive abandonment has been described by Scott. In 2001, J.W. Scott, a British gynecologist, published in the British Medical Journal a brief paper entitled "Scott's parabola: the rise and fall of a surgical technique" [START_REF] Scott | Scott's parabola: the rise and fall of a surgical technique[END_REF].This parabola represents a procedure or therapy that shows great promise at the outset, becoming the standard treatment after reports of encouraging results, only to fall into disuse due to adverse outcome reports (Fig. 1). The MIS low-profile tibial component developed for minimally invasive knee arthroplasties is a good example for the Scott Parabola. These implants with a low-profile keel have known a high interest in early 2000, until the publication of disappointing results and early aseptic loosening [START_REF] Foran | Early aseptic loosening with a precoated low-profile tibial component: a case series[END_REF][START_REF] Kajetanek | Mid-term survivorship of mini-keel versus standard keel in total knee replacements: differences in the rate of revision for aseptic loosening[END_REF]. The active robotically assisted system ROBODOC presents also a similar evolution in some countries. Recent study did not demonstrate a real benefit of this device [START_REF] Jeon | Robot-assisted total knee arthroplasty does not improve long-term clinical and radiologic outcomes[END_REF]. Nevertheless, this device has contributed to the development of current advanced technologies in knee arthroplasties; and the CUREXO system remains commonly used in Korea. If it is true that some trends in the arthroplasty world are just fashion (such as metal-on-metal bearing surfaces or modular femoral stems for hip arthroplasty) with a real decline, the definition of trend could also be "a general direction in which something is developing or changing"; i-e an evolution.

It was our hypothesis that most of the assistive technologies in modern knee arthroplasties are not following the Scott Parabola. To demonstrate our hypothesis, we aimed in this study to evaluate (1) the interest for the technique as measured by the number of publication/year on each of these assistive techniques and (2) the actual surgical use as measured by the number of cases performed with these techniques reported in the National Joint registries.

Materials and methods

Article identification and selection process

A search in February 2021 was performed to identify all available literature that described the use and results of assistive technologies or new surgical techniques in knee arthroplasty. The search was performed through PubMed, EMBASE, and MEDLINE databases from 1997 to 2021 inclusive.

Inclusion criteria for the search strategy were all English studies reporting information regarding the use and results of assistive technologies or new surgical techniques in knee arthroplasty. The included articles were randomized controlled studies, cohort studies, case-controlled studies, cadaveric studies, systematic reviews, and meta-analyses. The following terms were used: "knee arthroplasty" or "knee replacement" associated with each following groups "PSI" or "patient-specific instruments" or "patient-specific Fig. 1 Scott's parabola instrumentation" or "patient-specific guide"; "navigation" or "navigated" or "computer-assisted"; "robotic" or "robotically assisted"; "accelerometer"; "sensor" or "sensing technology". Exclusion criteria consisted of (1) comments on previous articles, (2) articles evaluating joints other than the knee. The abstracts from all identified articles were independently reviewed by two senior orthopedic surgeons. Articles were excluded based on the title and abstract. The number of included articles describing each technique was assessed year by year.

For the articles on CAS, articles have been allocated into two groups. The first one included all articles assessing CAS in knee arthroplasty (comparative studies, cadaveric studies, and reviews or meta-analysis on the outcomes of CAS). The second group included all comparatives in vivo studies in which CAS was used during the surgery, but was not the main topic assessed in the study. The aim of this second group was to assess if this device was always used in current practice.

National registries

In each National Joint Registry available via Internet, the number of cases performed with these techniques in knee arthroplasty has been quantified year by year. The searched techniques included: CAS, PSI, robotically assisted systems, sensors, accelerometers-based navigation in total knee arthroplasty (TKA) or unicompartmental knee arthroplasty (UKA). The assessed national joint registries were: American Joint Replacement Registry, Australian Orthopaedic Association National Joint Replacement Registry, National Joint Registry for England Wales Northern Ireland, and the Isle of Man, New Zealand Joint Registry, Canadian Joint Replacement Registry, Swedish Knee Arthroplasty Register, Norwegian Arthroplasty Register.

Results

Evolution of surgical techniques in knee arthroplasty in the literature

Following the initial online search, a total of 4085 articles were found. After the assessment mentioned above, 2106 articles were included in our study. In the literature, the publications progression evolves in two different ways: growth and plateau phase or continuous growth. The assistive technologies where a plateau phase was observed were: CAS and PSI; and the ones with a continuous growth: accelerometers, sensor technologies, and robotically assisted knee arthroplasties.

The first group comprises: computer-assisted knee arthroplasty and PSI have increased quickly during 5-8 years to then reach a plateau, with a stable number of publications/ year over the last six years (Fig. 2a andb). Articles on "computer-assisted knee arthroplasty" represent the highest proportion of articles on assistive technologies in knee arthroplasty. Concerning the articles on CAS, 2 phases can be observed: from early 2000 until 2013-2014, the articles mainly reported studies on the assessment of the efficiency of CAS with publications reporting randomized or non-randomized comparative studies (with or without CAS). The second phase after 2013 until today shows a persistent interest with more than 60 new publications by year (Fig. 3) quoting CAS. The content of these publications, however, has changed as CAS is not the main topic evaluated in the study, but just the tool used to perform the operation.

The second group with a continuous increase of the number of publications/year comprises: accelerometers, sensor technologies, and robotically assisted knee arthroplasties. For the first two technologies, the number of publications continuously increased over the last 10 years (Fig. 2b) with more than 20 articles reporting their results by year currently. The number of publications concerning robotically assisted knee arthroplasty is also rising quickly since 2000, with a rapid increase since 2012 and more than 80 publications in 2020. Nevertheless, currently we did not find the same transition between the papers assessing these devices and the papers quoting them, as for CAS.

Evolution of surgical techniques in knee arthroplasty in national registries

Only two registries have detailed the surgical techniques in knee arthroplasty: the Australian and Norwegian registries. They reported similar results than in the literature, with the lack of decrease in new techniques.

In the Australian registry, the proportion of primary TKA performed by computer-assisted systems increased from 2% in 2003 to 32% in 2019 [START_REF]Hip, knee and shoulder arthroplasty: 2020 annual report[END_REF] (Fig. 4). In 2019, a computerassisted system was used in 32% of all primary TKA. The robotically assisted system follows the same increase for the UKA. The proportion of UKA using robotic assistance had risen from 7% in 2015 to 30% in 2019 [START_REF]Hip, knee and shoulder arthroplasty: 2020 annual report[END_REF]. In 2019, 30% of UKA used robotic assistance.

In the Norwegian registry, the proportion of computerassisted TKA remained between 8 and 12% of primary TKA since 2007 [START_REF] Bergen | Norwegian National Advisory Unit on arthroplasty and hip fractures 2020 annual report[END_REF]. In 2019, computer navigation was used in 8% of all primary TKA [START_REF] Bergen | Norwegian National Advisory Unit on arthroplasty and hip fractures 2020 annual report[END_REF]. The number of cases operated with a computer-assisted system remains steady, even if there are some fluctuations from year to year.

Discussion

The main findings of this study were first that publications concerning these assistive technologies experienced a high rise during their development phase and maintained an important interest year by year. Second, the usage as described in the joint registries remains stable without observing any real decline.

Based on the literature screening and the analysis of publications per year, the assistive techniques assessed in this study are not following the "Scott's parabola". For earlier innovations, such as CAS, the publications number remained high, but the type of publications has evolved. The articles assessing the efficiency of these techniques have decreased drastically. Only few articles have been published to evaluate these techniques with a strong study design or a long follow-up [START_REF] Petursson | Computer-assisted compared with conventional total knee replacement: a multicenter parallel-group randomized controlled trial[END_REF][START_REF] Cip | Twelve-year follow-up of navigated computer-assisted versus conventional total knee arthroplasty: a prospective randomized comparative trial[END_REF]. Most current publications, quoting CAS use it as a tool now within the study, where CAS is no longer the objective of that study. This "consolidation phase" can be interpreted as a form of maturity of the technique. It's also interesting to consider Fig. 3 Evolution of the publications number by year on computer-assisted knee arthroplasties over the last 20 years, with the articles assessing the computer-assisted system (assessed) and the articles where this system was used usually (quoted) Fig. 4 Proportion of primary TKA performed by computer-assisted system or PSI, and proportion of primary UKA performed by robotically assisted system in the Australian registry between 2013 and 2019 the synergy between different techniques and concepts in the literature linked to each other like the pieces of a puzzle. Indeed, the real need for CAS has exponentially grown as a tool to improve the position of the implants when surgeons started to reduce the length of their incisions [START_REF] Pagnano | Minimally invasive total knee arthroplasty meets computer navigation[END_REF]. Then, CAS was considered accurate, but time consuming and cumbersome. Furthermore, it asked for a capital investment from surgeons and hospitals. Therefore, PSI and accelerometers have been developed to answer these needs. Thanks to the use of PSI, surgeons discovered the interest of limiting the number of surgical tools to improve surgical efficiency. The concept of optimizing OR efficiency increased. In parallel, interest for new alignment concepts started to gain interest and this interest never declined. Bone cuts were improved but surgeons realized that the analysis of the ligament tension was an important factor to integrate, aiming for less releases. That is when the interest for the sensors started to grow. Modern robotic platforms are integrating all these parameters and will probably continue to evolve integrating augmented reality to be able to visualize 3D and to use intra-operatively a 3D planning without having to look at a screen while operating. AI tools will also probably become a part of the platforms as most of these systems are giving direct feedbacks on a significant number of intricated surgical parameters such as component position, bone resections, femoral rotation [START_REF] Murgier | Variable rotation of the femur does not affect outcome with patient specific alignment navigated balanced TKA[END_REF], ligament balancing [START_REF] Moore | How to quantitatively balance a total knee? A surgical algorithm to assure balance and control alignment[END_REF], and global alignment [START_REF] Mooney | The effect of the native kinematics of the knee on the outcome following total knee arthroplasty[END_REF].

The registries' data are limited, but they confirmed the persistent use of some surgical technique or technology, such as computer-assisted. Contrary to the publications, the registries report the practice of all surgeons in a country and not only in highly specialized surgical centers. Boylan et al. confirmed this evolution for the robotic and computer navigation technology in the New York State [START_REF] Boylan | Technology-assisted hip and knee arthroplasties: an analysis of utilization trends[END_REF]. They reported that the proportion of cases using technological assistance grew each year in the New York State, increasing from 4.3% in 2008 to 11.6% in 2015. The proportion of hospitals and surgeons using robotic assistance also increased from 16.2% of hospitals and 6.2% of surgeons in 2008 to 29.2% of hospitals and 17.1% of surgeons in 2015. Similar results have been reported in the Nationwide Inpatient Sample database from 2005 through 2014, corresponding to the largest nationwide all-payer hospital inpatient care database in the United States (approximately 20% sampling of all hospital discharges) [START_REF] Antonios | Trends in computer navigation and robotic assistance for total knee arthroplasty in the United States: an analysis of patient and hospital factors[END_REF]. The proportion of technology-assisted TKAs steadily increased in this database from 1.2% in 2005 to 7.0% in 2014. Even if these studies were on a smaller scale, they confirmed the persistence, and even the growth, of the use of these surgical techniques.

The opponents of assistive technology in orthopedic surgery present these innovations as a fashion, sometimes dangerous due to unknown complications. Indeed, some innovations in recent years have known this decline, such as metal-on-metal bearing surfaces [START_REF] Klasan | Are serum metal ion levels a concern at midterm followup of revision knee arthroplasty with a metal-on-metal Hinge design?[END_REF], modular femoral stems for hip arthroplasty [START_REF] Hamilton | Scott's parabola and the rise and fall of metal-on-metal hip replacements[END_REF], or MIS low-profile tibial component [START_REF] Foran | Early aseptic loosening with a precoated low-profile tibial component: a case series[END_REF]. Nevertheless, these devices or innovations did not represent the majority of new surgical technologies in orthopedic surgery. Scott's parabola, described for the metal-on-metal bearing surfaces [START_REF] Hamilton | Scott's parabola and the rise and fall of metal-on-metal hip replacements[END_REF], does not represent the evolution of all surgical techniques. The "consolidation phase" described for CAS is not yet reported for other assistive technologies. Some could also decrease, as for example the PSI described by several authors as disappointing [START_REF] Thienpont | Efficacy of patientspecific instruments in total knee arthroplasty: a systematic review and meta-analysis[END_REF][START_REF] Thienpont | A systematic review and meta-analysis of patient-specific instrumentation for improving alignment of the components in total knee replacement[END_REF]. However, the PSI conserves a high interest for other reasons than the surgical accuracy. The PSI is recently used with customized implants [START_REF] Bonnin | Custom total knee arthroplasty facilitates restoration of constitutional coronal alignment[END_REF], or to improve the cost-efficiency ratio in operating room with the concept of singleuse instrumentation [START_REF] Giannotti | Single-use, patient-specific instrumentation technology in knee arthroplasty: a comparative study between standard instrumentation and PSI efficiency system[END_REF][START_REF] Moerenhout | Advantages of patient-specific cutting guides with disposable instrumentation in total knee arthroplasty: a case control study[END_REF]. Even if clear advantages and superiority of these assistive technologies are not yet proven, these innovations allow to progress slowly. Each technique contributes partially to a better understanding of what we are doing and how to achieve it to improve functional results and patient satisfaction after knee arthroplasty. And the robotic surgery would also be a step through achieving a more efficient technique, such as artificial intelligence and machine learning. The goal is thus to improve the surgical tools in knee arthroplasty continuously and not to replace them.

Several limitations can be outlined in our study. First, evaluating the interest for a technique trough the number of publications might not be the ideal method; however, this has been used in the past for similar purposes [START_REF] Dalton | Quantitative analysis of technological innovation in knee arthroplasty: using patent and publication metrics to identify developments and trends[END_REF]. Second, only some surgical techniques have been assessed and they are relatively recent, such as sensors or accelerometers. Potentially, their follow-up was not enough to assess their decline if any would happen. Nevertheless, CAS, or robotic surgery have sufficient follow-up to evaluate their evolution after the initial craze. Third, marketing has probably an essential part in the publication system. The pressure of some orthopedic device companies could increase the motivation to publish the assistive technologies advantages. That is also the reason why the second goal of the study was to evaluate the actual use of these technologies in the national registries which are independent of these marketing issues. To know the evolution and the use of a surgical technique, it would be preferable to assess the exact number of patients operated with each surgical technique all over the world. There is to date no data bank integrating all these data. On seven available national registries, only two reported some data on the surgical techniques in knee arthroplasty. The registries' follow-up is short, and their completeness is not absolute. For each registry, there are some missing data, whose proportion is not always known. The exactitude of these data is based only on the surgeons' conscientiousness. Despite these limitations, our study based on the analysis of a large amount of articles of the literature and the available data from the registries is only the first of its kind including the latest technologies [START_REF] Dalton | Quantitative analysis of technological innovation in knee arthroplasty: using patent and publication metrics to identify developments and trends[END_REF].

Conclusion

Most of the innovations in modern knee arthroplasties are not following the Scott Parabola. After a fast rise, these assistive technologies and new techniques do not disappear but continue to evolve. Their evolution is synergistic and combined to enhance the knee knowledge on how to restore the anatomy closer to a native knee after knee arthroplasty to improve the patients' clinical outcomes.

Fig. 2 a

 2 Fig. 2 a Evolution of the publications number by year on the orthopedic over the last 20 years, including MIS, computerassisted system and robotically assisted system. b Evolution of the

Funding No funding was received.

Declarations

Conflict of interest CB: Grant from SoFCOT. SP: Royalties from Zimmer Biomet and Newclip; Consultant for Zimmer Biomet; Treasurer for European Knee Society.

Ethical approval Not applicable (publication metrics).

Informed consent Not applicable (publication metrics).