Keywords: Source Detection, Source Term Estimation, variational methods, 3D-Var, air pollution, neural network, advection-diffusion

Source detection is a field of study gaining interest due to environmental concerns about air quality in populated areas. We developed a machine learning framework inspired by previous works on road traffic estimation, and compared it to a classical variational method under a unidimensional and stationary problem. We tested source reconstruction with datasets coming from 12 and 50 sensors with and without noise. Noise was set to follow a gaussian law with a dependent variance from the maximum measured value of a concentration profile. Both methods are reasonably robust to noise. The results reveal that the Neural Network used here, a multilayer perceptron, performs very well compared to the classical 3D-Var method.

INTRODUCTION

To the knowledge of the authors, machine learning is not much used for Source detection. [START_REF] Hutchinson | A review of source term estimation methods for atmospheric dispersion events using static or mobile sensors[END_REF] cite several works where Machine Learning methods were applied, but none of them aims to identify a source of pollution from a set of geolocalised data set.. Another work dealt with building of 2D maps of pollutant concentrations using mobile sensors measurements [START_REF] Ma | A deep autoencoder model for pollution map recovery with mobile sensing networks[END_REF]), but no method on source term estimation was proposed.

Recently machine learning has been applied on inverse problem dealing with road traffic estimation [START_REF] Georges | Machine Learning for Receding Horizon Observer Design: Application to Traffic Density Estimation[END_REF]) and has inspired the present work, where a feed forward neural network (NN) is used with a few hidden layers. This study proposes a method to reconstruct a source of pollution in an urban area. In the case of a unidimensional domain and stationary conditions, one could imagine the reconstruction of a unique source of pollution located on a boulevard of a town surrounded by an urban area. The machine learning approach is compared with a model-based variational framework (3D-Var) which acts as a reference.

Physical model used in machine learning and variational frameworks

In both frameworks, a stationary physical model was used to create source and concentration data of pollutants.

In a simple one dimensional model where a stationary source has emitted long enough to reach stationarity, the ⋆ This work is supported by The French Alternative Energies and Atomic Energy Commission displacement of pollutants is governed by the following advection-diffusion law, c∂ x u(x) -b∂ 2 xx u(x) = q(x) ,

(1) with c [m/s], b [m 2 /s], parameters corresponding to the wind field and turbulent diffusion respectively, both considered as scalars. u denotes pollutant concentrations in g/m 3 or in mol/m 3 depending on the considered source rate. q denotes the source with units of concentration per time and is defined on the physical domain (x in one dimension). The source profile q used in the experiments of this work is defined as follows,

q (x) = A s ∆ s (x -x s ) , (2) with
∆ s (x -x s ) = exp -σ 2 disp (x -x s ) 2 ,
where A S , x s , σ 2 disp [∅, km, km -2 ] represent the amplitude, the position and the inverse of the variance of a unique source of pollution. ∆ S denotes the function of dispersion chosen for the source around its centre. The partial differential equation can be solved in three dimensions by a function called gaussian puff if a pointwise source is considered. In this study, the equation is solved by finite differences on a domain Ω = x ∈ [0, 1]. Boundaries are chosen as Dirichlet conditions imposing a null concentration at the edges of x which model the absence of pollution in the background. The chosen unit of length is kilometre [km].

The model M is solved with spatial finite differences and leads to,

u = M(θ) = A -1 (B 1 q(θ) + B 2 C), (3)
where, A and C are the representative matrices of discretisation and boundary conditions respectively. B 1 , B 2 are matrices that map the discretised source profile q and the boundary conditions C towards the target system to inverse.

Simulations Parameters

As stated at the beginning of section 1, this study aims to reconstruct a source of pollution in an urban area where the wind field velocity generally does not exceed ∼ 10 m/s as seen in [START_REF] Wang | Wind field reconstruction for the dispersion modeling of accidental chemical spills on complex geometry[END_REF] and [START_REF] Wang | Wind speed estimation in urban areas based on the relationships between background wind speeds and morphological parameters[END_REF].

The latter studied a technique to assess wind field without using CFD models in Seoul. The turbulent coefficient is set to b = 25000 m 2 /s as found in [START_REF] Winiarek | Estimation of errors in the inverse modeling of accidental release of atmospheric pollutant: Application to the reconstruction of the cesium-137 and iodine-131 source terms from the Fukushima Daiichi power plant: ESTIMATION OF ERRORS IN INVERSE MODEL-ING[END_REF] for the horizontal components of b in a problem of source identification at a continental scale. In urban scale, this coefficient is often imposed by the chosen wind turbulence model with,

b = D m + ν t Sc t , (4)
where D m is the molecular diffusion of the pollutant, v t the turbulent viscosity and Sc t the Schmidt Number.

According to [START_REF] Lin | Eulerian RANS simulations of near-field pollutant dispersion around buildings using concentration diffusivity limiter with travel time[END_REF] this number is chosen as a constant in atmospheric study with a value usually set to Sc t = 0.3 whereas v t is determined by the turbulence model of Navier-Stockes which is written in Kikumoto ( 2020) is for a k -ϵ RANS model,

v t = C µ kT e where, v t = C µ k 2 ϵ
and T e = k ϵ .

(5)

Noise considered on sensors

Many studies consider measurements with a dedicated measurement sampling function to take into account that a sensor i, 1 ≤ i ≤ N C (N C denoting the number of sensors) measures average concentration in its surrounding, leading to the sensor function for the i th sensor (See Georges (2019), [START_REF] Hammond | PBDW: A non-intrusive Reduced Basis Data Assimilation method and its application to an urban dispersion modeling framework[END_REF], [START_REF] Krysta | Source reconstruction of an accidental radionuclide release at European scale[END_REF]),

H i : u, x i → Ω ϕ i (x -x i ) u (x) dx, 1 ≤ i ≤ N C , (6)
where ϕ i : x → ϕ (x -x i ) is a sampling function associated to a sensor i at a position x i . In this work ϕ is chosen as a Gaussian kernel but studies can also consider a rectangular function as in [START_REF] Kovalets | Inverse identification of unknown finite-duration air pollutant release from a point source in urban environment[END_REF]. The overall observation operator H which maps the outputs of the physical model to measurements is thus an operator which returns,

H : u → [H 1 (u), . . . , H N C (u)] ( 7
) Gaussian noise is also considered for several experiments in this paper. This noise measurements is applied to each of the N C sensors following the expression,

u m i = H i (u) + ϵ obs i with, ϵ obs i ∼ N (0, σ obs 2 ) , (8)
where ϵ obs i is a gaussian noise corresponding to a i th sensor. Sensor noise is often defined in literature by two contributions: background noise and relative noise. The last depends on the magnitude of the measurement. Thus noise can be written as follows,

ϵ obs i = n R + n b , (9)
with n b ∼ N (0, σ 2 base ) a basal noise which would command the limit of blank of the sensor (according to the definitions of [START_REF] Armbruster | Limit of Blank, Limit of Detection and Limit of Quantitation[END_REF]) and n R ∼ N (0, (σ relative × H i (u))

2 ) the measure-dependant noise. However this formulation would modify the expression of the covariance matrix of measurement errors written in (13) and hence making the problem non quadratic. For this reason, we apply a constant variance of measurements as written in (8). Consequently, for each measurement of a concentration profile u s , a constant standard deviation with σ obs s = 5% × max (H(u s )) is chosen. It constitutes also a conservative hypothesis on the amount of noise for lower measured values. The basal noise is neglected. The choice of gaussian errors of measurements is motivated by [START_REF] Defforge | Improving Numerical Dispersion Modelling in Built Environments with Data Assimilation Using the Iterative Ensemble Kalman Smoother[END_REF] and [START_REF] Asch | Data assimilation: methods, algorithms, and applications[END_REF] where gaussian errors are advised to be considered in order to build proven optimal filters such as the Kalman filter.

METHODS

The goal of this source identification study is to reconstruct the source term written in (2) given measurements.

To carry this study two methods are applied and compared on the stationary problem described in 1.1: a method based on a Neural Network model (NN) trained with a synthetic database and a classical model-based method (3D-Var).

Machine Learning

Let us name, θ = A S , x s , σ 2 disp a vector representing source parameters. Alternatively, we could also consider that the source contains as much parameters as grid points of simulation, i.e. θ i = q(i∆x) with 0 ≤ i ≤ N x -1, N x being the total number of discretised points in the x axis. A multilayer perceptron with one or several hidden layers is built to learn source parameters given concentration profiles and wind-field inputs. This type of architecture was shown to be reliable to approximate an inverse function as done in Georges (2020) and as proven by mathematicians (see [START_REF] Hornik | Multilayer feedforward networks are universal approximators[END_REF]).

We consider three tensors C, c and Θ. C contains n examples of stationary concentration profiles induced by: 1-n wind coefficients contained in the tensor c (here a 1D vector) and 2-by a gaussian stationary source of parameters θ k ;

1 ≤ k ≤ n. [C, c] and Θ play the role of X and Y traditionally written in machine-learning framework, where X is the input tensor which is estimated by the Neural Network given a target tensor Y :

(X, Y ) ≡ ([C, c], Θ).
(10) To train the model, a given number n train ∈ N of simulations of pollutant dispersion are performed using the model written in (1) with various wind speeds c and source parameters θ. To cover all the space of possible training examples, wind speeds and training source parameters are generated with a Sobol sequence. Sobol sequence has the property to be a pseudo-random generator with low discrepancy so that each training example is not far in Euclidean distance from another set of parameters (see [START_REF] Niederreiter | Low-discrepancy and lowdispersion sequences[END_REF]). Its python implementations used in this work are borrowed from an open-source module written by [START_REF] Ng | crankycoder/sobol seq[END_REF].

Training data are generated from n train wind and source parameters c k and 50, 10 4 ]. The sources of pollution are considered to be defined by a bell curve of amplitude A S such as a source is defined by,

θ k , 1 ≤ k ≤ n train , with c k ∈ [-10, 10] m/s and θ k ∈ [1e -2 , 2] × [0.1, 0.9] × [
q k (x) = θ k (1) exp -θ k (3) 2 (x -θ k (2)) 2 , 1 ≤ k ≤ n train (11)
The target tensor corresponds to

Θ k = [θ k (1), θ k (2), θ k (3)],
1 ≤ k ≤ n samples , with n samples the number of training or validation examples. Source parameters need to be scaled to the same order of magnitude. To carry this we build the following, adimensional training parameters,

θ(j) = θ(j) -min(Θ(j)) max(Θ(j) 1 ≤ j ≤ 3, ( 12
)
for each set of the n train parameters used for the training.

Thus the model returns an estimation of the adimensional parameters as an output. A rescaling is applied to the output estimate using min(Θ(j)) and max(Θ(j)) the scaling parameters used in training, such that a scaled estimation of source parameters is performed on the validation set. To avoid learning biases, the rescaling still uses the extrema of parameters from training to rescale validation examples.

Variational methods

Variational methods rely on the minimisation of a cost function using a known model of transport of pollutants. These kinds of methods are widely used and considered as a standard way to estimate parameters given some measurement especially with the use of adjoint estimation [START_REF] Georges | A Variational Calculus Approach to Wildfire Monitoring Using a Low-Discrepancy Sequence-Based Deployment of Sensors[END_REF], [START_REF] Nguyen | Adjoint-based approach for estimation & sensor location on 1D hyperbolic systems with applications in hydrology & traffic[END_REF]). Here we propose to apply a variational method to a stationary problem where u can be expressed as an explicit function of θ, thus avoiding the use of the adjoint. The 3D-Var cost function is chosen. A regularisation term [START_REF] Tikhonov | Solution of Illposed Problems[END_REF]) is provided by an initial guess such that the entire cost function is written,

J (θ) = 1 2 (θ -θ b ) T B -1 (θ -θ b ) + 1 2 (F(θ) -u m ) T R -1 (F(θ) -u m ) (13)
with,

u = M(θ) = A -1 (B 1 q(θ) + B 2 C), (14)
the concentration profile written by the physical model M written in (1). And with,

F(θ) = H • M(θ), (15)
a forward operator which is a composition of the observation operator H with M where H maps the concentrations returned by the model on the whole domain to the space of the measurements. Matrices B and R respectively contain the covariance errors of initialisation and the covariance errors of measurements. We assume here that these errors are uncorrelated so that B and R are diagonal-and with an equal variance. σ b and σ obs are the standard deviations associated with initial guess and observations. To proceed gradient descent, the following gradient is given to the cost function,

∇J (θ) = B -1 θ -θ b + Ω N C i=1 ϕ (x -x i ) A -1 B 1 ∇q(θ)dx T R -1 (F(θ) -u m ) .
(16) The optimisation of the cost function to find the optimal set of source parameters θ * is carried out by a quasi-Newton descent using the L-BFGS-B algorithm.

CASE STUDY : SOURCE IDENTIFICATION IN A STREET

Source detection gains an increasing interest for the estimation sources of pollution in urban areas (e.g. [START_REF] Defforge | Improving Numerical Dispersion Modelling in Built Environments with Data Assimilation Using the Iterative Ensemble Kalman Smoother[END_REF], [START_REF] Kovalets | Inverse identification of unknown finite-duration air pollutant release from a point source in urban environment[END_REF]). The approach proposed here could consider an emissive vehicle stopped on a boulevard with its motor turned on and where stationarity has been reached.

Parameters

Database is built by building the set of parameters,

p k = A S k , x S k , σ 2 disp k , c k ; 1 ≤ k ≤ n train , (17) with A S , x S , σ 2 disp , c k ∈ [1e -2, 4]×[0.1, 0.9]×[50, 2000]× [-10e -3, 10e -3]
generated by Sobol sequence, with the units ∅, km, km -2 , km/s respectively. The validation set is created in the same manner with n test samples, choosing

A S , x S , σ 2 disp , c ∈ [5e -2, 3] × [0.1, 0.9] × [70, 1500] × [-10e -3, 10e -3].
Source position is purposely generated so that x S ∈ [0.1, 0.9] to avoid to generate sources which would be located too close from domain boundaries leading to a non-physical solution of our dispersion model written in (1). We remind that every scaling parameter involved in ( 12) is based on training only. Our database contains n train = 15000 training samples, whilst validation is applied on n test = 200 samples. As a metric of performance, the relative root-square error (RRSE) is chosen, defined by,

RRSE = ∥q -q T ∥ 2 ∥q T ∥ 2 , ( 18
)
where (q T , q) ∈ R Nx 2 are respectively the real source profile and its estimate. To be fair with variational method, a fair initialisation is given with a vector,

θ b k = [0, argmax(u m k ), 0] , 1 ≤ k ≤ n test ;
(19) so that the initial guessed source position is located at the highest measured concentration.

The best neural network architecture has been chosen after having performed some experiments. Hence the chosen Neural Network contains 2 dense hidden layers of 50 neurons each, whose tanh is the activation function. The optimisation is carried out using the L-BFGS-B algorithm. L2 regularisation is set at a low value : 10 -12 . The machine learning framework is built using keras and tensorflow libraries. Some estimates returned by the neural network and by the variational method are not physical because on one hand, the learning of source parameters in the Neural Network is not strongly constrained by the source model, on the other hand, the variational method can stop gradient descent on local minima. We will exclude these few examples from our results.

Comparison without noise

Experiment with 12 sensors

Simulations are performed without noise to compare how the machine learning framework performs next to the variational method. Several non-physical estimates returned by the neural network and bad estimates from the variational method are excluded: 16 out of 200, leading to the study of 184 estimates.

On these examples, the neural network is able to estimate a source with an RRSE in the same order of magnitude than with the standard 3D-Var, with a mean RRSE and a standard deviation of µ N N RRSE = 0.22 and σ N N RRSE = 0.12 and µ var RRSE = 0.31 and σ var RRSE = 0.21 using neural network and variational method respectively. Overall the Neural Network performs slightly better.

The root squared errors obtained from machine learning and variational framework shew to follow a log-normal law confirmed by the use of a Kolmogorov-Smirnov test (p -value = 0.39 > 0.05 and 0.44 > 0.05 respectively). The distributions of RRSE are shown in Fig. 2.

The machine learning framework method is data-driven and thus does not contain any physical model whereas variational framework relies on the knowledge of the physics and on a good guess. The guess written in ( 19) specifies the position of the sensor which measures the maximum concentration. We notice that the RRSE metric is sensitive to a slightly error of estimated position. For example in the case of the 65 th sample of validation illustrated in Fig. 1, the neural network estimates well the amplitude and the dispersion, but a slight error of position leads to an RRSE N N = 0.16 against RRSE var = 0.04 for the neural network and the variational framework respectively. We will get back to the metric choice at the end of the paper.

Overall, without noise and applied to 12 sensors in total (10 regularly spaced in the domain and 2 which measure the boundary conditions), the machine learning framework performs slightly better than the variational framework regarding the Relative Root Square Error. The neural network gave sometimes non-physical estimates such as negative sources or non-Gaussian sources. Fig. 3 show that the performance of the two methods are close.

Experiment with 50 sensors: Simulations with 50 sensors are carried out to observe a sensitivity of the estimations towards measurement density. Several non-physical estimates returned by the neural network or by the variational framework are excluded of the network: 9 out of 200 (but not the same as with 12 sensors), leading to the study of 191 estimates. Both methods return similar results as previously, with mean and standard deviation of µ N N RRSE = 0.13 and σ N N RRSE = 0.08 compared to µ var RRSE = 0.24 and σ var RRSE = 0.15 for neural network and variational frameworks respectively. This means that the error decreased by about a half for the NN and by one third for 3D-Var comparing to the use of 12 sensors. Again the Neural Network performs better than 3D-Var. As with 12 sensors RRSE distributions shew to follow log-normal law with a Overall, source reconstruction using 50 noiseless sensors (48 regularly spaced in the domain and 2 which measure the boundary conditions) achieves 30 to 50% improvements compared to the case with 12 noiseless sensors. Fig. 3 shows that the performance of the neural network is clearly better than the variational framework, its distribution being narrower and closer to 0 on RRSE axis. On the other hand, both methods improve with the use of 50 sensors instead of 12.

Comparison with noise

In this experience, a gaussian noise n s is applied to each sample s of data-set of concentration profiles after having applied a characteristic sensor function as described in 1.3, with n s following the law :

n s ∼ N (0, σ obs = 5% × max(H(u s )), 1 ≤ s ≤ n test or n train . (20)
Thus the gaussian noise of a set of measurement has a constant variance, which depends of the maximum of the noiseless-measured concentration profile. This hypothesis is reasonably conservative because the lower measured values of a concentration profile will be relatively noisier than its maximum value.

To be fair with variational method, the initial guess error was modified from σ b = 1 × 10 6 to σ b = 5 × 10 4 to to help to make the problem more convex.

Experiment with 12 sensors: Simulations are performed with noise in the same manner as previously. Several nonphysical estimates returned by the NN or 3D-Var are excluded of the results: 9 out of 200. Leading to the study of 191 estimates. Both methods are close in the presence of noise with mean and standard deviation of µ N N RRSE = 0.48 and σ N N RRSE = 0.26 using the NN and µ var RRSE = 0.52 and σ var RRSE = 0.23 using 3D-Var. Fig. 4 shows that the distribution of Relative Root Squared Errors of both methods are close to each other. timates. However, the machine learning framework showed greater improvement than the variational framework when increasing sensor density. We have shown that the machine learning approach is able to reconstruct sources even in the presence of noisy measurements. Sensor noise decreases severely the performance of Source Term Estimation for both methods. Overall, the performance of the Neural Network is as good or superior to the variational approach as demonstrated by table 1. These results revealed that machine learning approaches are promising for source detection in the context of urban air pollution. Future work will consider another metric such as Wasserstein distance to evaluate the performance of source reconstruction. Indeed, Wasserstein distance is more adapted to compare distributions than RRSE. The former is far less punishing with slight errors of source position than the latter. Nevertheless, the behavior of error distribution for NN and 3D-Var remain similar across all the cases adressed in this study using RRSE or either Wasserstein metric.

It would be interesting to develop this approach for real world cases in urban area. Indeed, once trained, a neural network model could be useful to monitor online urban pollution given an urban network of sensors. We could also imagine a framework where a neural network could provide a proper initial guess to a variational method such as 3D-Var or 4D-Var.

  -Ferber * S. Leirens * * D. Georges * * *

Fig. 1 .

 1 Fig. 1. Example of source reconstruction : left, with a neural network; right, with a variational method Kolmogorov-Smirnov test (p -value = 0.19 > 0.05 for the NN and p -value = 0.90 > 0.05 for 3D-Var).Overall, source reconstruction using 50 noiseless sensors (48 regularly spaced in the domain and 2 which measure the boundary conditions) achieves 30 to 50% improvements compared to the case with 12 noiseless sensors. Fig.3shows that the performance of the neural network is clearly better than the variational framework, its distribution being narrower and closer to 0 on RRSE axis. On the other hand, both methods improve with the use of 50 sensors instead of 12.

Fig. 2 .

 2 Fig. 2. Compared distribution of Relative Root Squared Error for obtained by Variational and Neural Network Estimates. No noise, 12 sensors

Table 1 .

 1 Measured mean and standard deviation of Relative Root Square Error of validation samples for each case of study

	Case			NN		3D-Var
	N	N°sensors noise mean	std	mean	std
	184	12	no	0.22	0.12	0.31	0.21
	191	50	no	0.13	0.08	0.24	0.15
	191	12	yes	0.48	0.26	0.52	0.23
	191	50	yes	0.40	0.21	0.45	0.22

ACKNOWLEDGEMENTS

The authors acknowledge the french nuclear and alternative energies commission for the funding of this work and John D. Garrett (2021) for providing IEEE plot styles thanks to his open-source module.


Laetitia Balaresque

Gabriela De

Carvalho Bezerra

Gabriela De Carvalho

How does the local and regional socio-political context fit into the energy transition

• The local energy transition seems to be on the road towards depolitization.

• Solar is expected to cover the energy needs of Tucson, at the price of minimizing technical and environmental challenges, with an increasing impact on landscapes.

• Additional incentives for distributed generation, which is more expensive than utility-scale generation, should be implemented to help assist the transition.

How does the local and regional sociopolitical context fit into the energy transition process in Pima County?

The many and varied roadblocks explain why the solar potential is still under-exploited

CITY OF TUCSON

In

Actors of the energy transition in Pima County

INTRODUCTION

Interviews: what are the blocking points and challenges to the transition?

In Pima County, Arizona, where the consequences of global warming are particularly sensible, and where the solar potential is substantial, the deployment of solar energy has accelerated over the last years. This process of energy transition is inscribed in a complex and multi-scale network of actors, comprising of public and private sectors. Source: Tucson Electric Power

• An energy transition focused on solar power

• High GHI (global horizontal irradiance)

Transition Roadmap: toward 70% solar energy by 2035

A complex and multi-scalar interlocking of public and private actors Energy transition amalgamates a number of challenges and issues

  v e st m e n t Ta x C re d it R e n e w a b l e e n e r g y s t a n d a r d ( R E S ) S U M E R -P R O D U C E R C O N S U M E R A d m i n i s t r a t i v e a n d f i n a n c i a l h e l p C a n c e l a t i o n o f n e tm e t e r i n g ( 2 0 1 6 )

  Who are the protagonists of this process and what are their perceptions on the transition?

MATERIALS AND METHODS DISCUSSION RESULTS

	• An under-exploited solar
	potential, despite a burgeoning
	market
	• Significant demographic growth
	and increasing energy needs
		• Documentation from the utility
		company
		• Ethnography and participant
		observation with local solar
		companies and associations
		• Exploratory and semi-structured
		interviews with numerous local
		stakeholders
				FEDERAL GOVERNMENT
		ARIZONA CORPORATION COMMISSION (ACC)		STATE OF ARIZONA
		SOLAR
		COMPANIES		TUCSON ELECTRIC
				POWER
		A c t i v e r o l e		(TEP) manages
		i
		n
		s o
			l
			a
			r
			a
			d
			o
			p t
			i
			o n p r o	INDIVIDUALS
			c e s s
		DISTRIBUTED GENERATION			UTILITY-SCALE
	"Arizona has been very, very stingy, even with tax credits (for solar)."	Cost	Blocking points	"People want this transition to occur without noticing any change or without having to sacrifice anything."
		Political will	Technology	Social acceptance
				"It takes time to
				rebuild energy
				systems."

"Most of the challenges (..), they're political right? When you have a conflict in the political sensibilities, that's where everything stagnates" "The big solar farms I think, are where we will find conflict, especially if you have people living nearby." "It's almost always true that it (solar) costs more upfront and then has radically lower operating costs…but if you don't have good credit or access to credit, that's a big roadblock." CONTEXT Laetitia Balaresque, Master's candidate at ENS de Lyon Gabriela de Carvalho Bezerra, Master's candidate at EHESS Paris PIMA COUNTY


Clément Alix

Grégoire Ramé

Grégoire Ramé Connection

Connection/ Disconnection: Mexican Wastewaters for the American riparian ecosystems


Cultural reconnection

• Tucson birthplace The foot of the A-Mountain (Sentinel Peak) is what the locals consider to be the birthplace of the town, especially with the Hohoham and the agriculture developed around the Santa Cruz River for centuries.

• San Juan Day, June 24 is for Mexicans the feast of St John The Baptist (according to the catholic tradition), which heralds the beginning of the monsoon. The choice of this date for the inauguration of the Heritage project is particularly appropriate and symbolic.

• The Living River of Words is an art and science program promoting contact and connection with the Santa Cruz River for hundreds of young people around Pima County's schools.

The research presented on this poster focuses on the Santa Cruz River socio-hydrosystem, a tributary of the Colorado River, located in the Southwestern United States in Arizona. Subject to anthropogenic pressures, the Santa Cruz River is an essential resource for the supply of urban areas, such as the city of Tucson in the Sonoran Desert. In the 1940s, the river dried up and became intermittent, particularly because of excessive groundwater pumping. Only during the rainy season does a surface flow appear. Today, various initiatives are carried out in the Tucson area to "bring back to life" the Santa Cruz River and thus make visible a river that had become invisible. This social reconnection is all the more important as Arizona is currently facing drought and water shortage that is affecting the water supply of its residents. Through the projects studied -social network mobilization, recreational reconnection, socio-cultural reconnection -local actors aim to reconnect residents to the river and to the ecological functioning of the Santa Cruz watershed.

What are the vectors of social reconnection to a dried up urban watershed ?

Louise Biville, Coline Clergeau Laurans, Manon Delègue. Master 1 Gestion de l'eau et développement local, Université Paris-Nanterre.

Limits and perspectives

-Very artificial restoration and human dependent.

-One of the challenges of water management is the risk of pollution from numerous landfills.

-Green gentrification.

-The illusion of an oasis in the desert ?

Mobilizing social medias

Word cloud from @TucsonWater and @Sonoraninst on Twitter : • "Environmental influencer" "I didn't even wanna do social medias, and then I thought, oh yeah, the stories are actually very useful for communicating !" (M. Bogan, 2022).

Introduction

In a drought and urban growth context, treated effluents are a sustainable way to recharge diminishing groundwater table levels. On this effluent dominated river, we focus on two groundwater recharge projects led by Tucson Water that recharge groundwater levels and, by reconnecting groundwater and surface water, restore riparian areas, providing wildlife habitat and amenities for the community.

The Santa Cruz River : an effluent-dependent river

Methods

• Semi-structured interviews with managers, non-profits, scientists.

• Participatory observations : seminars, bird-watching field trips, exhibitions, workshops, collaborative worksites.

• Counting and analysis of the frequentation of the studied sites.

• Online participatory observations and social medias analysis.

Recreational reconnection

Recreation activities are deeply linked to the environment and the outdoors due to the unique ecosystem of the region. The Sonoran Desert hosts a rich biodiversity, enhanced by the Sky Islands mountains ranges, connecting people to even more diverse biomes.

• The Loop "By September 2020, about 22,300 pedestrians and 45,500 cyclists will use this stretch of the path" Living River-2020 Water Year, Sonoran Institute The 130-mile length of this bike path is a great asset to the city, as it connects users to the various rivers that run through Tucson, making it a must for residents.

• Bird-watching

The Tucson area hosts a large biodiversity of bird species and offers an ecological corridor for migration thanks to the Santa Cruz River's riparian areas. The Sweetwater Wetlands Park is a well-known hot-spot for birding.

• Dragonflies

It's only few days after the beginning of the Heritage Project that dragonflies have taken up residence where the flow began. It became such an important species that a festival is organized in their honor.

• Native plants are a big part of the Tucsonian culture. Here, heat mitigation is a huge deal. Native plants, such as mesquites are praised as they create shade and a cooling effect without needing irrigation.

Heritage Project

Confronted with excess effluent and in order to reconnect residents to their river, Tucson Water and Pima County chose what can be considered the birthplace of Tucson to discharge effluent and thus revive a portion of the river. Launched in June 2019, the project has attracted a crowd of professionals, enthusiasts and residents. In a short time, this formerly dry area has become a ecosystem again and the site has become a favourite place for various species. Three years after its implementation, the Heritage project remains a popular place, thanks to its loop cycle path, making it a must for cyclists.

©Coline Clergeau Laurans, picture from the riverbed and down the A Mountain at the Heritage Project, March 2022

This constructed wetlands depends on highly treated effluents to provide wildlife habitat and recreational amenities to visitors. Opened in 1996, the facility is managed by Tucson Water, while non-profits, such as Tucson Audubon Society, organize field trips in this "birding hot-spot".

After moving through the wetlands, treated effluents are send to infiltration basins surrounding the park to recharge groundwater.

Sweetwater Wetlands Park

©Manon Delègue, Sweetwater Wetlands Park, March 2022.

• Waste removal workshops and initiatives along the Santa Cruz River are promoting a way to restore and preserve the river ecosystem. Volunteers can reconnect to the river's natural resources and its cultural heritage.

• Invasive plants removal

Arundo is an invasive specie from Middle East. The non-profit Watershed Management Group has initiated participative restoration around "Arundo removal" to allow native plants to grow. The participants come in the morning at Tanque Verde Creek and they pull out Arundo right from the riverbed.

• Citizen science is a way to reconnect to natural resources. WMG organizes Wet/Dry surveys with volunteers to track surface flow in the Santa Cruz River watershed. While Tucson Audubon Society promotes the use of e-Bird, an online data base of bird sightings.

Citizen reconnection

• Castoriando is another participative restoration program hosted by RRN. It takes place in Ciénega Creek and participants "think like beavers" in order to build wood dams to slow stormwater flow.

  generated with Voyant tools by Louise Biville • Environmental communication

  DunbarSpringNeighborhoodForester.org ©Louise Biville, March 2022 Map of the Loop, website of the city tucson.com ©Manon Delègue, April 2022, Sweetwater Wetlands, birding with Audubon Society

©Louise Biville, April 2022, Tanque Verde Creek, arundo removal


G Dardenne

C Hamitouche

E Stindel

C Roux

ULTRASOUND IMAGING-BASED PROCEDURE TO INTEGRATE THE DYNAMIC BEHAVIOR OF THE PELVIS IN TOTAL HIP ARTHROPLASTY PLANNING

ABSTRACT

One of the main constraint in Total Hip Arthroplasty (THA) is to optimize the position of the acetabular implant in order to avoid postoperative instability. The best orientation of the cup has been studied by several authors in order to avoid it, but for most of them, they don't take into account the dynamic behavior of the pelvis during daily activities. This dynamic acts on the orientation of the acetabulum and therefore, not taking into account this behavior can lead to postoperative instability of the implant in some positions (sitting, standing or supine position). To assess the dynamic of the pelvis and the effects on the acetabular orientation, a rapid, accurate and easy new method has been developed. A functional parameter representing the pelvic flexion is first measured in different position (sitting, standing and supine position). Then, a software allowing us to assess the behavior of an acetabular implant in these positions is used. A study on twenty healthy patients has been carried out and shows that taking into account this method during THA planning might be very crucial to improve postoperative results.

Index Terms-Biomedical imaging, Surgery, Digital control, Prosthetics.

INTRODUCTION

The dynamic behavior of the pelvis has an important role for the orientation of the acetabulum during daily activities like the sitting, standing or supine positions. These orientations can make easier, for example, the flexion of the hip in the sitting position or the extension in the standing position. Taking into account this dynamic behavior of the pelvis for Total Hip Replacement (THR) could improve postoperative stability. Several authors have studied this dynamic behavior [START_REF] Nishihara | Measurements of pelvic flexion angle using three-dimensional computed tomography[END_REF][START_REF] Di Gioia | Functional pelvic orientation measured from lateral standing and sitting radiographs[END_REF][START_REF] Lazennec | Hip-spine relationship : a radio-anatomical study for optimization in acetabular implant cup positioning[END_REF][START_REF] Ala Eddine | Variations of pelvic anteversion in the lying and standing positions of 24 control subjects and implications for ct measurement of position of a prosthetic cup[END_REF][START_REF] Mayr | The frontal pelvic plane provides a valid reference system for implantation of the acetabular cup[END_REF]. However, for all of them, the procedure is difficult, takes a long time and is sometimes invasive due to patient exposure to radiation (radiography or CT scan). The proposed method is based on a calibrated ultrasound (US) probe, on a navigation system (Surgetics R station) and can measure in 3D with accuracy, reproducibility rapidity and without any radiation exposure the pelvic flexion in several positions. For our study, only three positions are used: the supine, the standing and the sitting positions. Then, the functional orientation (functional anteversion and inclination) of an acetabular implant thanks to the pelvic flexion is estimated. A study on twenty patients was realized.

MATERIALS AND METHODS

Experimental setup

The material used to assess the pelvic flexion and the actetabular implant orientation is composed of a navigation system (Surgetics R station) which uses stereoscopic infrared cameras: two cameras emit infrared signals and receive those sent back by retro reflective markers located on the references that are fixed on the objects we want to track. These trackers are called rigid bodies. Two kinds of trackers are used (fig. 1): reference trackers and a pointer tracker which allows us to digitize 3D targeted structures. A linear ultrasound (US) Fig. 1. Rigid bodies (left) and Navigation station (right) probe HL7.5/40/128D and an US machine EchoBlaster R from Telemed R is integrated into the navigation system. A rigid body is attached to this US probe in order to obtain its position and orientation in real time. A calibration step has been previously performed in order to obtain the 3D localization of any point located on the US image [START_REF] Mercier | A review of calibration techniques for freehand 3-d ultrasound systems[END_REF]. This step was carried out thanks to a plane for the simplicity of the procedure [START_REF] Dardenne | A new optimization approach for the calibration of an ultrasound probe using a 3d optical localizer[END_REF][START_REF] Prager | Rapid calibration for 3-d freehand ultrasound[END_REF][START_REF] Rousseau | A fully automatic calibration procedure for freehand 3d ultrasound[END_REF][START_REF] Rousseau | Robust and automatic calibration method for 3d freehand ultrasound[END_REF].

Obtaining the pelvic flexion

In order to optimize the position of an acetabular implant in THA, a reference plane called Anterior Pelvic Plane (APP) defined by the two anterior superior iliac spines and the pubic symphysis (fig. 2) is used in computer aided THA [START_REF] Lewinnek | Dislocations after total hipreplacement arthroplasties[END_REF][START_REF] Jaramaz | Computer assisted measurement of cup placement in total hip replacement[END_REF]. To assess the pelvic behavior, the APP is obtained in different Fig. 2. Definition of the Anterior Pelvic Plane (APP) positions and the pelvic flexion is measured between the Anterior Pelvic Plane and a reference [START_REF] Nishihara | Measurements of pelvic flexion angle using three-dimensional computed tomography[END_REF]. This APP is obtained with the calibrated localized US probe. The two anterior superior iliac spines and the pubic symphysis are scanned in a given position by the localized US probe (fig. 3). The three reasons (fig. 4). The APP is then built in 3D thanks to the calibration and the pelvic flexion is computed between this plane and a reference. The reference is, in our case, an horizontal plane set in front of the navigation station when the APP is scanned (fig. 5). In the sitting and standing positions, Fig. 5. The reference (surrounded with a circle) and the localized US probe the pelvic flexion is measured between the APP and a vertical obtained with the plane reference. Concerning the supine position, the angle is directly measured between the APP and the horizontal plane reference.

Obtaining the functional acetabular orientation

In order to assess the behavior of the acetabular implant for different positions, we have developed a software allowing us to estimate the functional orientation [START_REF] Lazennec | Hip-spine relationship : a radio-anatomical study for optimization in acetabular implant cup positioning[END_REF] (functional anteversions and inclinations) of a cup in different daily positions of the patient (in our case: standing, sitting and supine positions)(fig. 6). To achieve this task, the pelvic flexion in these positions and the anatomical orientation (anteversion and inclination) of the acetabulum with respect to the vertical frontal plane are needed. The functional anteversion is computed in a transverse plane, parallel to the horizontal reference, between Fig. 6. Orientations of the acetabulum in the standing and sitting positions the projected vector and the frontal vector. The functional inclination is computed between the vertical reference and the normal vector of the acetabular implant (fig. 7). Fig. 7. Functional anteversion (left) and inclination (right).

RESULTS

Reliability of the pelvic flexion measurements

The accuracy of the US probe calibration was assessed and was less than 1mm [START_REF] Dardenne | A new optimization approach for the calibration of an ultrasound probe using a 3d optical localizer[END_REF]. The reliability of the procedure which determines the pelvic flexion was also assessed on one patient. This patient was a 24 years old man which weighted 85Kg for 1.8m. Eight measurements of the pelvic flexion were carried out on each position (Table 1 were needed to obtain the pelvic flexion in the three positions for each patient.

Effects on the acetabular orientation

The procedure used to assess the behavior of an acetabular implant has been described. For our study, the implant is oriented with an anatomical inclination of 45 o and an anteversion of 15 o with respect to the vertical frontal plane for the 20 patients. Knowing their pelvic flexion in the three positions (sitting, supine and standing), the functional acetabular orientation is computed in these positions. of the literature [START_REF] Chen | Implant position calculation for acetabular cup placement considering pelvic lateral tilt and inclination[END_REF] proving their correctness.

DISCUSSION

Up to now the pelvic flexion was studied using mainly radiographics [START_REF] Nishihara | Measurements of pelvic flexion angle using three-dimensional computed tomography[END_REF][START_REF] Di Gioia | Functional pelvic orientation measured from lateral standing and sitting radiographs[END_REF][START_REF] Ala Eddine | Variations of pelvic anteversion in the lying and standing positions of 24 control subjects and implications for ct measurement of position of a prosthetic cup[END_REF]. However, the determination of the APP is difficult and most often inaccurate. The symphysis can be actually difficult to localize [START_REF] Lazennec | Hip-spine relationship : a radio-anatomical study for optimization in acetabular implant cup positioning[END_REF] and 2D projection can lead to inaccurate results. One other author used a digitized arm [START_REF] Mayr | The frontal pelvic plane provides a valid reference system for implantation of the acetabular cup[END_REF] but this does not take into account the depth of the intervening tissue i.e. the skin and subcutanous tissue and can lead also to inaccurate results. Nevertheless, concerning our application, the APP is determined in 3D with accuracy, simplicity, rapidity and with no radiation exposure thanks to a procedure based on an 2.5D ultrasound probe and on a navigation station. The results observed concerning the pelvis are similar to those of the literature [START_REF] Nishihara | Measurements of pelvic flexion angle using three-dimensional computed tomography[END_REF][START_REF] Di Gioia | Functional pelvic orientation measured from lateral standing and sitting radiographs[END_REF]. There are significant interpersonal variations. However, for a given patient, in a given position, the pelvic flexion is unchanging and reproducible. This different pelvic flexion following the positions modifies the functional orientation of the actetabulum. This modification can introduce postoperative instabilities. Hence, using this knowledge, a new procedure can be performed to optimize the placement of a hip prosthesis: First, the pelvic flexion is measured preoperatively with the described method. Then, the surgeon can visualize peroperatively the functional orientation of the implant following the positions and its orientation with respect to the APP. The surgeon will therefore obtain postoperative information during the surgery about the implant's behavior if we assume that the postoperative flexion is equal to the preoperative flexion as shown in [START_REF] Nishihara | Measurements of pelvic flexion angle using three-dimensional computed tomography[END_REF][START_REF] Di Gioia | Functional pelvic orientation measured from lateral standing and sitting radiographs[END_REF].

Fig. 3 .

 3 Fig. 3. Procedure allowing us to compute the pelvic flexion for the standing position

Fig. 4 .

 4 Fig. 4. US images of an iliac spine (left) and of the symphysis (right) and manual detection with a cursor.

Table 1 .

 1 ). Reliability of the procedure3.2. Pelvic flexion measurementsA study was carried out on 20 healthy men with the described procedure in the supine, standing and sitting positions. The morphological description of the patients are given in the table 2. The results are shown in the table 3. Fifteen minutes

		Supine ( o ) Standing ( o ) Sitting ( o )
	Average flexion	4.9	-4.8	-29.4
	Std. Dev.	0.6	1.1	1.2
	Min.	3.9	-6.2	-30.1
	Max.	5.7	-3.4	-27.5
	Max. Variation	1.8	2.8	3.4

Table 2 .

 2 Morphological description of the patients Supine ( o ) Standing ( o ) Sitting ( o )

	Average	6.8	2.4	-25.2
	Std. Dev.	3.5	5.1	5.8
	Min.	0.9	-4.8	-39.6
	Max.	15.2	15.5	-11.2

Table 3 .

 3 Pelvic flexion measured on 20 patients

Table 4 .

 4 Results are shown in the table 4 and 5. These results are in coherence with those Functional anteversion measured on 20 patients.

		Supine ( o ) Standing ( o ) Sitting ( o )
	Average	8.2	12.6	34.1
	Std. Dev.	3.7	5.2	3.3
	Min.	-0.7	-1.1	24.9
	Max.	14.1	19.5	40.9
		Supine ( o ) Standing ( o ) Sitting ( o )
	Average	43.7	44.7	55.9
	Std. Dev.	0.4	1	3.2
	Min.	43.1	43.1	48.8
	Max.	44.8	46.4	64.6

Table 5 .

 5 Functional inclination measured on 20 patients.

ACKNOWLEDGMENTS

This work was funded by the ANR (Agence Nationale de la Recherche) with the RNTS (Réseau National des Technologies pour la Santé), and is related to the project "Surgetique Minimalement Invasive".


Deepak Gupta
email: deepak.gupta@telecom-bretagne.eu

Shaban Almouahed

Cyril Lahuec

Matthieu Arzel

Michel Jezequel

Chafiaa Hamitouche

In-Vivo Polyethylene Wear and Knee Prosthesis Longevity Estimation

Keywords:

This paper describes an in-vivo polyethylene wear and knee prosthesis longevity estimation approach using an im plantable ligament imbalance metric system. Four piezoelectric elements are used to measure the axial force components, on the tibial tray surface and their center of pressure is calculated.

This can be used, along with a knee simulator, to estimate polyethylene wear, and the state of the implanted knee pros thesis. Surgeons can exploit this to improve the post operative rehabilitation measures, whereas prosthesis manufacturers can use it to improve prosthesis's strength, geometry and life. The proposed self-powered measurement and telemetry system does not require any external power.

I. INTRODUCTION

One of the most common problem with the knee joint is the deterioration of cartilage (meniscus) between femoral and tibial bones. These tissues have limited capacity of self restoration and their wear and tear is one of the principal reason of osteoarthritis. The severe cases are usually treated through a Total Knee Arthroplasty (TKA) during which an artificial knee prosthesis made up of a femoral component, a polyethylene (PE) insert emulating the meniscus and a lower tibial tray is fitted. However, among the millions of TKA conducted worldwide, over a period of time the PE component starts wearing due to various reasons such as malalignment, erosion etc. This results in an asymmetrical PE surface which ultimately induces the collateral ligament laxity. As a consequence, the point on the tibial tray where the net axial force is exerted, offsets further. This may result in change of contact area between the condyle contact point and the tibial plateau surface. This has an impact on the contact stress and consequently the polyethylene wear rate [START_REF] Liau | The influence of contact alignment of the tibiofemoral joint of the prostheses in vitro biomechanical testing[END_REF] - [START_REF] Kretzer | Effect of joint laxity on polyethylene wear in total knee replacement[END_REF]. Hitherto, most of the methods used, to estimate the PE wear rate and status quo of the already implanted prosthesis, are indirect and approximative.

In this work, we propose an approach to estimate and predict the PE wear and the longevity of the already im planted knee prosthesis. For this, an implantable ligament imbalance measurement system can be used along with a knee simulator. The results will help prosthesis manufacturer to improve overall prosthesis design and the surgeons to identify a potential prosthesis failure and take corrective or precautionary measures. In section II the existing measure ment techniques are briefly described. Section III explains the ligament imbalance measurement system and its usage Retrieval analysis is a straightforward technique in which a used or failed prosthesis is evaluated to measure the polyethylene wear using statistical methods. Since this tech nique analyzes already "failed" specimens, are of little use for an early assessment of a newly implanted prosthesis. Furthermore, it is generally accepted these analyses results are only approximative and may overestimate wear [START_REF] Sanzen | Radiographic wear assessment in a total knee prosthesis: 5-to 9-year follow-up study of 158 knees[END_REF].

B. Radiography

In this, relatively simple method, plain X-rays of the implanted prosthesis are taken for different angles and po sitions. These are then studied for any aberration from the normal silhouette of the implanted prosthesis. However, the accuracy is limited due to parallax error [START_REF] Hoshino | Accurate in vivo measure ment of polyethylene wear in total knee arthroplasty[END_REF]. Improved X ray techniques extracts three-dimensional (3-D) images of an implant from a single perspective and are then matched against highly accurate geometrical model of the implant, which is known, a priori. However, the disadvantage is inaccuracy and the subject has to be X-rayed many times which could be hazardous. Another in-vivo measurement techniques for PE wear, based on 3-D image and Roentgen Sterephotograrnmetric Analysis (RSA) is reported in [START_REF] Gill | In vivo measurement of volumetric wear of a total knee replacement[END_REF].

C. Knee motion analysis

Knee motion analysis techniques are used to measure the bone, cartilage positions and their relative motions. The results are matched against their pre-defined mathemati cal models. These techniques include the electrical linkage method in which one or more potentiometers are used to measure voltage changes caused due to rotation or linear translation and thus predicting the bone movements. For this, both in-vivo and in-vitro methods have been developed using implanted markers or anatomical landmarks. Video techniques are also used to measure joint motion, in which markers are attached to the limb segments and their relative motion is captured with different cameras. In fluoroscopy, images are matched against a pre-registered library of three dimensional images of the tibial and femoral components of the prosthesis. These techniques largely rely on the accuracy of biomechanical analysis and a rough estimation of PE wear is achievable, though it is difficult to recover the volumetric losses.

D. Biomechanical Measurement

Other set of techniques include, methods focusing on biomechanical parameters such as axial force, ligament laxity measurement and then infer their impact on PE wear. An in vivo system for distribution of axial force measurement was developed by Kaufman et al [START_REF] Kaufman | Instrumented implant for measuring tibiofemoral forces[END_REF]. Another technique for the in-vivo knee load measurement with telemetry were proposed in [START_REF] D'lima | In vivo knee moments and shear after total knee arthroplasty[END_REF] [START_REF] Graichen | Implantable 9-channel telemetry system for in vivo load measurements with orthopedic implants[END_REF]. In this the stem of the artificial prosthesis was instrumented with strain gauges to measure the axial forces and moments. A micro-transmitter was also housed in it, which processes the sensed data and transmit it to an external receiver. Though interesting, the biggest inconvenience of this method is an external induction coil wrapped around the knee to power the implanted electronic system. Hence, making it usable only under strict supervision. In addition, knee simulators are used to emulate the knee motion and observe their impact on the artificial prosthesis longevity by measuring the wear and tear rate of PE. A recent in-situ study by Kretzer et al [START_REF] Kretzer | Effect of joint laxity on polyethylene wear in total knee replacement[END_REF] attempted to quantify the effect of joint laxity on polyethylene wear.

In order to facilitate an early assessment of a newly im planted prosthesis, estimate and predict the PE wear and the longevity of the implanted prosthesis, the in-vivo autonomous ligament imbalance measurement system in combination with in-situ experiments is envisaged.

III. AUTONOMOUS LIGAMENT IMBALANCE MEASUREMENT SYSTEM

A. Metrics

In [START_REF] Lahuec | A ligament laxity telemetry system architecture for a knee replacement prosthesis[END_REF], the authors have calculated the center of pressure (COP), as the barycenter of the axial force applied at the tibial tray, using force sensing elements (Fig. 1). COPx and COPy are calculated in the medio-Iateral (ML) and in anterior-posterior(AP) directions respectively. Generic ex- pressions for respective measurements are recalled here in (1) and ( 2):

COPx COPy L�=l FiXi L�=l Fi L�=l FiYi L�=l Fi (1) (2)
where Fi can represent either the average force or the peak value component obtained over one gait cycle at the ith force sensing device located at coordinates (Xi,Yi ).

B. Measurements

A MTS 858 MINI BIONIX I knee simulator was used to emulate the knee motion. Four piezoelectric ceramic elements were placed in the four compartments of the tibial tray. Forces Fi were respectively measured in the form of proportionate voltages Vi, During in-vitro or in-situ experiments it is possible to emulate the ligaments behavior and calculate the collateral ligament tension [START_REF] Crottet | Development of a force amplitude-and location-sensing device designed to improve the ligament balancing procedure in tka[END_REF] but that is not the case for in-vivo experiments or implanted prosthesis. Hence, the medial offset (MO) which is the displacement of the line of application of the axial force (Fig. 2), was used as an input to the knee simulator (ISO 14243) and COP values were calculated using (1) and ( 2). The results obtained for average and peak based fixed point COP calculations for different values of MO are shown in Fig. 3.

A strong correlation is obtained between the COPx and the medial offset in the medio-Iateral (ML) direction. This establishes the credibility of COPx as a metric that can correctly represent the displacement of the line of application of the axial force on the tibial. As the observed dynamic range for COPy was not sufficiently large to effectively distinguish two different medial offset points, it is not considered here. a::

4 '0 ID 2 C Q) 0 0 -2 0 -COPxAvgFixedPoint _ COPxPeakFixedPoint 2 3 4
Medial Offset (mm) 5 Fig. 3. Center of pressure displacement for average values of force components force components, with respect to the applied medial offset along the X axis.

C. Autonomous Measurement and Telemetry Architecture

A self-powered architecture conceived for COP measure ment and transmission via an integrated telemetry system is shown in Fig. 4. The system is autonomous because the piezo ceramics, used to measure the axial force components, also generate power through piezoelectric effect. Power Har vesting Unit harvests this power and supplies to the rest of the system. The signal processing implements the logic to calculate the metric. It can be a digital circuit based data acquisition and processing system or the signals recovered from the piezo elements can be directly used to calculate the metric with the analog computation cells. The Clock Generation Unit (CGU) generates one or two different clocks depending on the digital or analog based systems. The Data Transmission Unit (DTU) transmits the data outside the body, to an external receiver and conforms to the Medical Implant Communication Service (MICS) standard working in 402-405 MHz frequency band. System and Power Management block takes care of system management. The feasibility study for such a system was done in [START_REF] Lahuec | A ligament laxity telemetry system architecture for a knee replacement prosthesis[END_REF]. In addition, COP can be calculated based on average or peak values, though they have considerable differences in terms of architectural efficiency. Both digital and analog based signal processing cores can be imagined, though area and power consumption may differ considerably. The expected power consumption of the system using a 0.35 /-lm CMOS fabrication process is 1.7 mW [START_REF] Gupta | Ligament imbalance metrics and an autonomous measurement system for post tim[END_REF].

D. Discussion

It is clearly observed that the COP is a good read out for the collateral ligament imbalance metric as represented by the medial offset. Thus, COP values received via telemetry can be subsequently used to find the the medial offset using a linear fit analysis. The calculated medial offset versus the actual medial offset for fixed point implementation is shown in Fig. 5. A good correspondence is observed between the actual and the calculated values. This medial offset repre sents the collateral ligament imbalance or ligament laxity as described earlier in [START_REF] Gupta | Ligament imbalance metrics and an autonomous measurement system for post tim[END_REF]. Moreover, the COP value can be used to estimate and predict the PE wear and the life of the already implanted prosthesis as described in next section.

The other advantage of the system vis-a-vis the state-of-the art technique is in being self-powered and less cumbersome due to the removal of external induction coil used to power the implanted system [START_REF] Graichen | Implantable 9-channel telemetry system for in vivo load measurements with orthopedic implants[END_REF].

IV. PERSPECTIVES

So far mostly in-vitro and in-situ techniques have been de veloped to monitor and estimate PE wear in knee prosthesis [START_REF] Liau | The influence of contact alignment of the tibiofemoral joint of the prostheses in vitro biomechanical testing[END_REF] [START_REF] Mazzucco | Effects of contact area and stress on the volumetric wear of ultrahigh molecular weight polyethylene[END_REF]. Similarly few studies have been conducted to under stand how parameters such as ligament laxity influences the joint kinematics post TKA [START_REF] Walker | Characterizing the motion of total knee replacements in laboratory tests[END_REF] and its effect on polyethylene wear [START_REF] Kretzer | Effect of joint laxity on polyethylene wear in total knee replacement[END_REF]. However, the principal drawback of these methods is, that they are applicable under in-situ scenario only and approximative, since it is not possible to exactly replicate the knee joint motion. But in this work, the proposed approach goes a step ahead. After implanting the prosthesis with the system it would be possible to obtain real-time data under in vivo conditions. Once the prosthesis is fitted with the system, via the medial offset metric, the ligament imbalance can be observed during different ambulatory activities such as walking, squatting etc. Furthermore, fatigue test data correlating the PE wear and the number of cycles for different medial offsets can be observed using knee simulators [START_REF] Mazzucco | Effects of contact area and stress on the volumetric wear of ultrahigh molecular weight polyethylene[END_REF]. In this way, a set of charts can be generated for PE component types with different shapes, materials and loading conditions. Consider a scenario in which a patient has been fitted with a knee prosthesis implanted with the proposed system. After a period of usage, during follow-up, a change in the metric value is observed i.e. medial offset is different from its previous position. This would be a clear indication that the position where the net axial force is applied on the tibial plateau surface has been displaced from its initial position. This implies a non-uniform tibial plateau surface and there is a PE wear at the site. Now, for the observed value of medial offset from the system, the fatigue test data can be referred for similar duration of usage (number of cycles) and for a similar type of prosthesis, an estimated volumetric loss of PE wear can be obtained. And by further extrapolating the data, the life of the implanted knee prosthesis can be predicted. In the long run, it might be possible to establish a relationship between the observed medial offset, the PE wear and ultimately the longevity of the prosthesis. It should be pointed out that COP calculations here do not take into account of dynamic parameters such as shear forces and torque. Though they don't make any significant part of total in-vivo forces exerted on tibial surface [START_REF] D'lima | In vivo knee moments and shear after total knee arthroplasty[END_REF]. Using COP measurement, along with finite element analysis, it should be possible to calculate contact stresses which also influences the PE wear [START_REF] Mazzucco | Effects of contact area and stress on the volumetric wear of ultrahigh molecular weight polyethylene[END_REF].

This technique can pave the way for an integrated mea surement and monitoring system to evaluate ligament im balance metric, PE wear and longevity for TKA operated knees. In such a system, an individual case can be monitored by calculating the imbalance metric under various physical activities. According to the observed data, the surgeon can recommend physical exercises which might aid in better adaptability of the prosthesis and a faster rehabilitation of the patient. This information could help prosthesis manufac turing houses to study and predict the implant longevity and subsequently improve their geometry and strength.

V. CONCLUSION

An approach for in-vivo estimation for polyethylene wear and the knee prosthesis longevity has been presented. The conceived method consists of utilization of an autonomous post TKA ligament imbalance measurement system and knee simulators, to provide an early assessment of the prosthesis state. Knee axial force components are measured to calculate their center of pressure, which also indicates ligament imbalance. This metric, along with in-situ exper iments, can be exploited to estimate the PE wear of the already implanted prosthesis and predict its potential life. Moreover, the information can be further used to ameliorate rehabilitation programs, establish relation between in-vivo forces and contact stresses and improve prosthesis design with better longevity.

Fig. 1 .

 1 Fig. 1.Tibial tray with X and Y axes taking the center of the stem as origin. Sensing devices are located to obtain the axial force distribution in the medio-lateral (M-L) and the anterior-posterior (A-P) directions.

Fig. 2 .

 2 Fig. 2. The Medial Offset (MO) represents the displacement of the line of application of the axial force.

Fig. 4 .

 4 Fig. 4.An autonomous self-powered system architecture for ligament imbalance metrics calculation and their RF transmission[START_REF] Lahuec | A ligament laxity telemetry system architecture for a knee replacement prosthesis[END_REF].

4 Acutal medial offset (mm) 5 Fig. S .

 45S Fig. S. Computed medial offset versus the applied (actual) medial offset. A good correspondence between them provides accurate measurement up to I mm precision.