A G Chavez Castillo

B Gaume

Y Rouizi

O Quéméner

P Glouannec

Identification of insulating materials thermal properties by inverse method using reduced order model

Keywords: Reduced model, Branch eigenmodes reduction method, inverse problem, thermal characterization, thermal properties, insulating materials, thermal contact resistance

The analytical models used by hot-wire type probes are no longer suitable for the characterization of insulating materials. This paper proposes a solution, which is based on a numerical reduced order technique named AROMM (Amalgamated Reduced Order Modal Method) coupled to an inverse procedure. We demonstrate that a single reduced order model provides precise results for insulated materials characterized by different thermal properties, with the advantage of computing 250 times faster than a classical finite element modelization. Such model is then tested through multiple scenarios in order to evaluate the accuracy of the proposed methodology. Results show the importance of the sensitivity of the measurement regarding the sought parameters, which later intervene during the identification process. A statistical study allows us to access a satisfying confidence interval for a common measurement noise. At last, a study on the influence of an eventual thermal contact resistance is conducted.

INTRODUCTION

Nowadays, environmental constraints are increasing at a fast rate, leading to the appearance of new materials in the building world, in particular thermal insulators.

However, their use stays limited due to non-availability of databases in reference to their intrinsic properties. Moreover, we lack hindsight regarding their sustainability (humidity sensitivity, fluctuation of micro-structure, fungal growth).

The purpose of this paper is to present a method which enables to identify on site thermal properties of insulating materials, in order to maintain an active monitoring of their properties overtime.

In literature, some characterization methods exist :

The most simple techniques which enable to determine an isotropic conductivity k [W.m -1 K -1 ] in equilibrium is the hot plate method [START_REF] Astm | Standard Test Method for Steady-State Heat Flux Measurements and Thermal Transmission Properties by Means of the Guarded-Hot-Plate Apparatus[END_REF]. Jannot et al. [START_REF] Jannot | Thermal conductivity measurement of insulating materials with a three layers device[END_REF] created a three-layer device combined with an inverse method aimed at improving the precision on the estimated values specifically for very low conductivity materials (k < 0.15 W.m -2 .K -1 ).

It is often coupled with a DSC calorimetric technique to determine the heat capacity Cp [J. kg -1 .K -1 ] or c [J. m -3 .K -1 ].

Another technique widely used in the laboratory is the flash method [START_REF] Nagpure | Thermal diffusivity study of aged li-ion batteries using flash method[END_REF][START_REF] Boutinguiza | Thermal properties measurement of slate using laser flash method[END_REF][START_REF] Akoshima | Experimental verification to obtain intrinsic thermal diffusivity by laser-flash method[END_REF]. Initially developed to measure diffusivity a = k c [m 2 /s], the analytical relations of integral transforms type allow by inverse method to go back quickly to the two parameters k and c [START_REF] Jeon | Thermal conductivity measurement of anisotropic material using photothermal deflection method[END_REF][START_REF] Pawlowski | The least square method in the determination of thermal diffusivity using a flash method[END_REF][START_REF] Milosević | Simultaneous estimation of the thermal diffusivity and thermal contact resistance of thin solid films and coatings using the two-dimensional flash method[END_REF]. This experimental principle has also made it possible to use numerical models to determine non-homogeneous properties: Brouns et al. [START_REF] Brouns | Diagnostic de structures de génie civil : Identification des propriétés spatiales et de la surface d'un défaut[END_REF] relied on surface temperature measurements by infrared thermography, paired with a Finite Element Method (FEM) to evaluate the thermal conductivity of a specimen and find the presence of a faulty area. Reulet et al. [START_REF] Reulet | Méthode inverse pour la détermination expérimentale des conductivités thermiques et de la capacité calorifique de matériaux orthotropes en fonction de la température[END_REF] applied a numerical conduction model linked to the iterative method of Levenberg-Marquardt to estimate simultaneously the set of thermal properties {Cp, k xx , k yy ,k zz } of an orthotropic sample. Adamczyk et al. performed several studies ( [START_REF] Adamczyk | Retrieving thermal conductivities of isotropic and orthotropic materials[END_REF][START_REF] Adamczyk | Retrieving thermal conductivity of the solid sample using reduced order model inverse approach[END_REF]) based on the flash method with the purpose of measuring thermal conductivities on isotropic and orthotropic materials.

Flat shock probes, of which the best known is the HotDisc® [START_REF] Zhang | Theoretical accuracy of anisotropic thermal conductivity determined by transient plane source method[END_REF][START_REF] Jeon | Thermal conductivity measurement of anisotropic material using photothermal deflection method[END_REF], allow here again by inverse method the determination of the 2 parameters k and c. Raji et al. [START_REF] Raji | Thermophysical characterization of a laminated solid-wood pine wall[END_REF] characterized a wall made of stacked laminated solid-wood planks by implementing a thermal effusivity relation in the analytical equations of the hot strip method for both directions of the fiber (perpendicular and parallel). The results pointed out that thermal properties of wood depend greatly on the water content inside the wood fiber, as well as the important role that permeability and the diffusion coefficient could have. Lagüela et al. [START_REF] Lagüela | Thermal conductivity measurements on wood materials with transient place source technique[END_REF] provided a theoretical derivation of the hot-disc method to obtain the thermal diffusivity and conductivity in anisotropic materials, specifically in their case for wood.

Searching for a easier way to characterize insulating materials, Jannot et al. [START_REF] Jannot | Simplified estimation method for the determination of the thermal effusivity and thermal conductivity using a low cost hot strip[END_REF] developed a simplified estimation method based on the hot-plate method by using a 1.2 cm wide hot-strip and a thermocouple. They then deducted simultaneously the effusivity and conductivity from a temperature recording by using a linear regression of the heat equation.

A major inconvenience with most of these techniques is that they are essentially used in laboratory : Even if the flash method has been tested in situ (Kruczek et al. [START_REF] Kruczek | In situ measurement of thermal diffusivity in anisotropic media[END_REF]), the difficulties are numerous: the profile of the laser source that creates the heating must be precisely known. In addition, given the relatively large and fragile equipment, this technique is not well suited to frequent travel. Finally, even if recent work proposes to go back to multilayer characteristics [START_REF] Rassy | Unconventional flash technique for the identification of multilayer thermal diffusivity tensors[END_REF], the characterization of a sandwiched insulating material seems to be difficult. As far as the HotDisc is concerned, its principle is not adapted to an in situ measurement and moreover, even if its major interest is its experimental simplicity, a major defect remains: its fragility.

Another technologie it's the "hot wire or needle probe method", which consists of an infinite line source with constant power per unit length, placed inside an infinite medium, having a thermocouple to record the temperature transient evolution at an x distance. From the point of view of the experimental protocol, this solution meets the specifications, since the equipment is light, easily transportable, requiring a moderate measurement time, and inexpensive.

Initially intended to determine the conductivity from more or less simple analytical laws (Jeager [START_REF] Jaeger | Conduction of heat in an infinite region bounded internally by a circular cylinder of a perfect conductor[END_REF], Blackwell [START_REF] Blackwell | A transient flow method for determination of thermal constants of insulating materials in bulk part I -Theory[END_REF]), it has been used in many fields [START_REF] Pereira | Thermal conductivity measurements and predictive models for frozen guava and passion fruit pulps[END_REF][START_REF] Nahor | Optimization of the temperature sensor position in a hot wire probe set up for estimation of the thermal properties of foods using optimal experimental design[END_REF][START_REF] Kömle | In situ methods for measuring thermal properties and heat flux on planetary bodies[END_REF]. For all these models, the resulting plot of the temperature evolution against the natural logarithm of time is characterized by a linear zone, for which the slope gives direct access to the thermal conductivity value. In addition, the use of inverse techniques also allows, as for the other techniques, to go back to the 2 parameters simultaneously k and c, always based on analytical laws [START_REF] Nagihara | Improved data reduction algorithm for the needle probe method applied to in-situ thermal conductivity measurements of lunar and planetary regoliths[END_REF][START_REF] Goodhew | Analysis of thermal-probe measurements using an iterative method to give sample conductivity and diffusivity data[END_REF].

But at the same time, studies are questioning the validity of these models for materials with low conductivity [START_REF] Batty | Use of the thermal-probe technique for the measurement of the apparent thermal conductivities of moist materials[END_REF][START_REF] Pilkington | Thermal conductivity probe length to radius ratio problem when measuring building insulation materials[END_REF][START_REF] Humaish | Assessment of long time approximation equation to determine thermal conductivity of high porous materials with NSS probe[END_REF][START_REF] Marmoret | Limit of validity of the log-linear model for determining thermal properties of light insulation materials with cylindrical hot probe[END_REF]. For these authors, the analytical model presents a limitation for the characterization of insulating materials even if it is used in an inverse procedure. This is due to a strong simplification because the model does not allow to take into account the inertia of the probe, the non homogeneity of temperature, and the heat flows that exist along the sensor. This phenomena, often negligible with a conductive sample, must be taken into account in the case of insulation. This is why numerical models have been developed. Given the limits of the number of degrees of freedom of the numerical models that can be applied in an inverse method, all models have so far been simulated only in 2D axisymmetric, considering the probe perfectly homogeneous and dissipating a uniform power density [START_REF] Humaish | Assessment of long time approximation equation to determine thermal conductivity of high porous materials with NSS probe[END_REF][START_REF] Bording | Determination of thermal properties of materials by monte carlo inversion of pulsed needle probe data[END_REF].

In this paper, our goal is to use a numerical model associated to an inverse procedure, in order to broaden the application of a classical heat-wire type probe which could be used in situ for an highly insulating materials, in order to determine their thermal conductivity and thermal volumetric capacity.To overcome the numerical difficulty caused by the large number of degrees of freedom of the model, incompatible with the iterative inverse procedure, a reduced modal model is used instead. The challenge lies then in establishing one single reduced model suitable to a variety of values of thermal properties of the insulating materials. Therefore, we seek to show the effectiveness of such procedure.

The first part of this paper introduces the geometry of an existing hotwire probe and the conditions affecting the material to be studied. The construction of a numerical model is conducted as well. We then move on to the use of a modal reduction method, where the previously numerical model is reduced; thus showing the capacity of the method. Section 4 addresses the inverse procedure. It thoroughly explains each of the stages taken in order to conduct an identification, and presents the results obtained for the study. And finally, conclusions and perspectives are discussed.

PHYSICAL PROBLEM

Studied configuration

Here we consider Ω I as one portion of a slab of insulating material (Figure 1) whose thermal conductivity k I [W.m -1 .K -1 ] and capacity c I [J.m -3 .K -1 ] are to be determined. This insulator is shaped like a cube, where each side measures 10 cm. The system uses an existing hot-wire probe Ω P , developped in a laboratory, composed by a heating wire and two thermocouples initially conceived to identify solely thermal conductivities. They are embedded in a steel needle-like body filled with glue (Figure 2). This probe is particularly thin, since its diameter is D = 2 mm for a length L = 9.3 cm . This probe Ω P is composed of different materials, whose thermal conductivities (k i ) and volumetric thermal capacities (c i ) are presented on Table 1. The insulating materials (Ω I ) will undergo a convective condition on the side (Γ h ) where the probe (Ω P ) is inserted and which is characterized by a convective coefficient h = 10 [W.m -2 .K -1 ] and an environmental temperature T env = 0 [ • C] (temperature elevation). Starting from an uniform initial temperature T 0 = 0 [ • C], the heating wire Ω HW will dissipate a known power π = 0.06 [W ]. All other surfaces (noted Γ 0 ) are insulated. This thermal problem is described by the following equations (with Ω = Ω I ∪ Ω P )

No. of domain in Ω

P k P i [W.m -1 .K -1 ] c P i [J.m -3 .K -1 ] #1 -
             ∀M ∈ Ω , t > 0 c ∂T ∂t = - → ∇(k - → ∇T ) + Π ∀M ∈ Γ h , t > 0 k - → ∇T. - → n = -hT ∀M ∈ Γ 0 , t > 0 - → ∇T. - → n = 0 ∀M ∈ Ω , t = 0 T = T 0 = 0 (1)
where:

k and c gather all the values of thermal conductivities and heat capacities, including those regarding the probe and the insulating material, and Π is equal to π in the heating wire Ω HW and zero elsewhere.

The main goal will be to seek the values of the thermal properties of the insulating material, being the thermal conductivity k I and the volumetric heat capacity c I . This is achieved by previously knowing the materials conforming the probe, as well as the thermal loads that the insulator will undergo.

Numerical model

The resolution of the problem defined by Eq.( 1) is done from its variational formulation, in which the test function f ∈ H 1 (Ω) is defined:

Ω c dT dt f dΩ = - Ω k ∇T • ∇f dΩ - Γ h hT f dΓ + Ω Πf dΩ (2)
By expressing each of the preceding volume integrals as a sum of two terms, on the one hand as the volume of the probe Ω P and on the other hand, as that of the insulating material Ω I , it is possible to isolate the unknown sought terms, that is to say, the thermal conductivity k I and the heat capacity c I of the insulator:

c I Ω I dT dt f dΩ + Ω P c P dT dt f dΩ = -k I Ω I ∇T • ∇f dΩ - Ω P ∇T • k P ∇f dΩ - Γ h h T f dΓ + Ω HW π f dΩ (3)
Once a spatial discretisation is performed by the Finite Element Method (FEM), considering a linear interpolation law (type P1 elements), we obtain the following matrix expression (respecting the same order as in Eq.( 3)):

(c I C I + C P ) Ṫ = (k I K I + A)T + U (4)
The size of these matrices are [N , N ], where N = 61, 591 taking into consideration the degree of precision for the representation of the geometry.

A numerical simulation was executed, using a second-order solver working at a variable time-step, to evaluate the temperature field of the formerly discussed insulator. With the same set of thermal characteristics (k I = 0.041 [W.m -1 .K -1 ] and c I = 1.15 × 10 5 [J.m -3 .K -1 ]). Our simulation has been validated with the modeling software COMSOL Multiphysics®.

Figure 3 shows the evolution of the temperature at the 2 measuring points A and B of the probe over time, as well as the results obtained by the Blackwell analytical model. One can thus note the important difference between these two models. In addition, the inexistence of a linear law as a function of the logarithm of time in the case of the numerical model. It is thus this one which must be used in the case of the identification of an insulating material. Figure 4 shows the numerical temperature profiles at both measuring points A and B of the probe. Depending on the axis orientation, the temperature profile changes. There is a presence of a notable non-homogeneity regarding the temperature inside the probe, both radially (concerning the heating wire), as well as longitudinally (thermal flow along the probe). The difference in temperature ∆T equals to 0.56 [ • C]. This proves how essential it is to know the probe's geometry and thermal properties with precision, and not just simply consider it as an uniform object englobing all components. We note that ∆T could vary according to the insulating material being tested, and in case of a lower conductivity, the gap in temperature between both measuring points will become more noticeable. For the rest of the study, this numerical model called complete model will be used for 2 things:

It will make it possible to obtain virtual measurements at the results obtained at measurement points A and B, by adding Gaussian noise: of zero mean, constant standard deviation and uncorrelated along time.

As will be shown in the following, it will serve as a starting point for the construction of an accurate model for the whole range of insulators that we wish to characterize with this probe.

Modal reduction

General principle of modal reduction

The modal reduction method is based on the decomposition of the temperature field into a small number ñ of known spatial functions Ṽi (M ) called eigenmodes. We are then able to approximate the sought temperature T (M, t), defined as:

T (M, t) T (M, t) = ñ<<N i=1 xi (t) Ṽi (M ) (5)
where xi are the unknown excitation states of the eigenmodes Ṽi (M ).

Based on this principle, different techniques exist, such as the Modal Identification Method (MIM) [START_REF] Rouizi | Numerical model reduction of 2D steady incompressible laminar flows : Application on the flow over a backward-facing step[END_REF][START_REF] Rouizi | Model reduction by the modal identification method in forced convection : Application to a heated flow over a backward-facing step[END_REF], the Proper Generalized Decomposition (PGD) [START_REF] González | Proper generalized decomposition based dynamic data driven inverse identification[END_REF], and the most well-known the Proper Orthogonal Decomposition (POD) [START_REF] Fic | Solving transient non linear heat conduction problems by proper orthogonal decomposition and the finiteelement method[END_REF][START_REF] Park | On the solution of inverse heat transfer problem using the karhunen loève galerkin method[END_REF],

For this study, we chose to apply another method called the Amalgam Reduction Order Modal Model (AROMM) method. Initiated by Neveu et al. [START_REF] Neveu | Simulation de la conduction non linéaire en régime variable: décomposition sur les modes de branche[END_REF], this technique is adapted to problems with variable parameters, and has so far been used for problems of identification of sources or heat flows [START_REF] Videcoq | Heat source identification and on-line temperature control by a branch eigenmodes reduced model[END_REF][START_REF] Videcoq | Online temperature prediction using a branch eigenmode reduced model applied to cutting process[END_REF][START_REF] Carmona | Estimation of heat flux by using reduced model and the adjoint method. application to a brake disc rotating[END_REF][START_REF] Carmona | Spatio-temporal identification of heat flux density using reduced models. application to a brake pad[END_REF]. The main difference between POD and AROMM is the way in which eigenvectors are obtained Ṽi : Both techniques use several reference temperature fields, the POD makes a statistical treatment of them by using a spatial autocorrelation function, while the AROMM method uses the heat equation to calculate a large generic initial base, which is then reduced from the reference cases according to a minimization process.

Very few studies have used these different modal reduction techniques for the identification of the conductivity of materials: Adamczyk [START_REF] Adamczyk | Retrieving thermal conductivity of the solid sample using reduced order model inverse approach[END_REF] uses a POD type model, et Girault [START_REF] Girault | Parametric low-order models in transient heat diffusion by MIM. estimation of thermal conductivity in a 2d slab[END_REF] develops the MIM technique.

The following paragraphs specify the different steps in the construction of the modal reduced base using the AROMM method.

The complete base of the AROMM method

The complete basis is obtained by the resolution of a particular eigenvalues problem which takes up the operators associated with the heat equation (eq 1) :

∀M ∈ Ω , k 0 - → ∇.( - → ∇ V i ) = z i c 0 V i ∀M ∈ Γ h , k 0 - → ∇ V i . - → n = -z i ζ V i (6)
The feature of this problem is that the eigenvalues z i are present in the boundaries condition. The variational form of ( 6) is written, separating the domain of the probe Ω P from that of the insulator Ω I :

-

Ω I ∇V i • k I 0 ∇f dΩ - Ω P ∇V i • k P 0 ∇f dΩ = z i Ω I V i c I 0 f dΩ + Ω P V i c P 0 f dΩ + Γ h ζ f V i dΓ (7)
The choice of unique values for the different parameters involved in this eigenvalue problem is as follows:

The probe domain corresponding to Ω P had every single one of its components with a certain well-known thermal property. The properties k P 0 and c P 0 of the different elements of the probe correspond to the known real values (table 1).

With respect to the insulator for which there is no fixed value in the heat equation (3) the value of the parameters k I and c I , we opted here for an arbitrary value, near to the range of the materials being studied:

∀M ∈ Ω I , k I 0 = 0.01 [W.m -1 .K -1 ] , c I 0 = 10000 [J.m -3 .K -1 ] (8)
The quantity ζ [J.m -2 K -1 ] is called the Steklov parameter and it is a simple coefficient that allows to maintain the physical dimensions of the quantity in the boundaries condition equations. To balance the two terms linked to the eigenvalue in (eq. 7), an appropriate choice of the Steklov coefficient ζ is given by:

ζ Ω I c I 0 dΩ + Ω P c P dΩ Γ h dΓ = 3528 [J.m -2 .K -1 ] (9)
The contribution of these Branch vectors is that they form a basis for all types of thermal problems, whatever the set of values of the thermophysical characteristic of the isolant k I and c I :

T (M, t) (c I , k I ) = N i=1 x i (t) V i (M ) (c I 0 , k I 0 , ζ) (10)
These Branch basis are characterized by the orthogonality property:

∀i, j ∈ N, Ω V i c 0 V j dΩ + Γ V i ζ V j dΓ = δ ij (11)
where δ ij corresponds to the Kronecker function.

The numerical calculations for this Branch basis are achieved by using the Arnoldi Method [START_REF] Lehoucq | Arpack User's Guide Solution of large scale eigenvalue problem with implicit restarted Arnoldi method[END_REF], which allows to calculate the first modes characterized by their smallest value of z i (absolute value). The whole (100%) base is computed, consisting of 61,591 modes, needing approximately 3 hours and 20 minutes of CPU time on a Dell Precision 7530 with Intel Xeon.

Basis reduction

The reduction of the complete base calculated in the previous stage can be easily done by removing some modes, according to a criteria that can be time-based [START_REF] Marshall | An approximation method for reducing the order of linear system[END_REF] or energetic [START_REF] Joly | Modal reduction of an advectiondiffusion model using a branch basis[END_REF]. The method chosen for this study is a more developed technique, known by the name of Amalgam Reduction [START_REF] Quéméner | The generalized amalgam method for modal reduction[END_REF], that is based on a distribution of the eigenmodes space into ñ subspaces that are orthogonal to each other. Every subspace is conformed by a main mode V j,0 , and ñj minor modes. The latter will provide additional thermal information to each main mode, by creating a mix of modes, hence the name of the method.

The core of the technique relies in rebuilding new modes Ṽj by linear combinations of initial modes V i , with the particularity that each eigenvector is used only once, either as a main mode or as a minor mode:

∀j ∈ 1, ñ Ṽj = V j,0 + ñj p=1 α j,p V j,p (12)
The distribution of initial modes in the sub-spaces of the amalgamated basis, as well as determining the amalgam coefficients α, are achieved by minimizing a criteria J R based on the difference between the temperature fields obtained respectively by the complete model T and the reduced model T. These calculations are then made from known temperature fields, characteristic of the simulated process.

In the case of the insulating material, since we cannot predict the exact set of thermal properties, we are forced to take into account a wider spectrum instead of limiting it to just one set of values. Therefore, we will build a compound scenario where the reference temperature fields T (R i ) , i = [1, ..., 4], are derived from simulations presenting different sets of thermal properties. Table 2 presents the selected reference cases for the basis reduction.

No. Reference case k I Ref [ 10 -2 W.m -1 .K -1 ] c I Ref [10 4 J.m -3 .K -1 ] R 1 1 1.5 R 2 1 150 R 3 6 1.5 R 4 6 150
Table 2: Amalgam reduction reference points

The criteria J R is built in order to take advantage of the orthogonality properties of the complete base (Eq.( 11)). It leads to a fast and sequential optimization process as a function of the number of modes of the reduced base.

J R = 4 i=1 τ Ω c 0 T (R i ) -T(R i ) 2 dΩ + Γ ζ T (R i ) -T(R i ) 2 dΓ dt (13)
In order to determine the order of the reduced model used afterwards, the projection of the known temperature fields T on the reduced base Ṽ makes it possible, thanks to the orthogonality properties, to obtain the states of excitation of every mode:

xi = Ω Ṽi c 0 T dΩ + Γ Ṽ ξ T dΓ (14)
Through Eq.( 5) we are able to obtain the states that will allow to determine the integrality of the sought temperature field and then error the between the complete model and the reconstructed reduced model. For this study, only the measurement points A and B will be considered:

P ∈ [A, B],
(P, t) = T (P, t) -

ñ i=1 xi (t) • Ṽi (P ) (15)
The mean and maximum values of these errors at the two measuring points over the entire duration of the test are then deduced: 16)) as a function of the order of the model. We note that according to the order of reduction chosen, the different reference cases do not have the same behaviour. Indeed the optimization of the reduced model (for a given ñ order) is based on a global criterion (13) at the same time on the whole spatial and temporal domain and for the 4 reference cases. It is therefore not possible to rigorously link this criterion to the punctual errors used thereafter.

     P ∈ [A, B], max (
We will then choose a model of order 100, which leads to a maximum error of less than 0.07[ • C]. The temporal evolution of the temperature obtained by the complete and reduced models for the worst case scenario is shown in Figure 6. We can see that the maximum error is obtained during the first instants and it is evident that it will continue to decrease. Error related to the first instants is typical of model reduction, where the errors will be more related a priori to fast dynamics (high frequencies). As shown in Table 3, the mean error is largely lower regarding the maximum error at each measuring point.

Insulator Point max [10 -2 • C] ¯ [10 -2 • C] T max [ • C] R1 A 5

State equation

The modal model is obtained by introducing the modal approximation of temperature (Eq.( 5)) in the heat equation (Eq.( 1)). Projecting the discrete heat equation (Eq.( 3)) over this reduced basis by using the test function g as the basis eigenvectors:

∀j ∈ N, ñ i=1 Ω Ṽj c Ṽi dΩ d xi dt = - ñ i=1 Ω k - → ∇ Ṽj . - → ∇ Ṽi dΩ + Γ Ṽj h Ṽi dΓ xi + Ω HW Ṽj Π dΩ (17)
By making Ṽ the matrix grouping the entire ñ of the reduced modes Ṽi (expressed under a discrete form), and X being the vector of ñ associated excitation states xi , the reduced model equation corresponding to Eq.( 4) is expressed as:

(c I V t C I V + V t C P V) ˙ X = (k I V t K I V + V t A V) X + V t U (18)
that becomes under compact form:

(c I L I + L P ) ˙ X = (k I M I + M) X + N (19)
By solving this problem, it is possible to determine the transient evolution of the ñ excitation states and also rebuild the temperature field T (M, t) (Eq.( 5)).

This simulation by reduced model is then used for 5 samples of insulating materials frequently used, which have different thermal properties (see Table 4) from those used to build the reduced model (see Table 2). We note that the range of thermal properties of these samples is significantly smaller regarding the wider range used for the reference cases. This choice was made with a view of conducting experimental validations.

The maximum errors at the two measurement points of these reduced simulations compared to the results obtained by the complete model are presented in Table 5. It is worth pointing out that the results present the same order of error as that obtained by projection for the reference materials (Table 2). These results show the capacity of these reduced models obtained by the AROMM method to be used for a wide range of the insulator's physical properties. In terms of computation time, the use of this reduced model (ñ = 100) compared to a classic finite element type model (N ≈ 62,000), leads to a gain of 250 times faster. It is this result that justifies the use of this type of reduced model for an inverse process to identify parameters. The goal here is to replace the numerical model by a reduced order model previously built:

No. of sample k

I [W.m -1 .K -1 ] c I [J.m -3 .K -
Sample Point max [10 -2 • C] ¯ [10 -2 • C] T max [ • C] S1 A
Y = ET ≈ E Ṽ X (20)
The iterative identification process is based on the minimization of a quadratic criterion built on the difference between the measurement Y ij at the measurement points and the result of the simulation Y ij for the estimated parameters β at each iteration. Here it is important to note that the measurements Y ij are calculated with a complete model with added noise and the simulation Y ij is performed by the reduced model.

J ( β) = 1 2 Nt i=1 Nmes j=1 Y ij -Y ij ( β) 2 (21)
where N t is the number of time steps of the simulation.

As for the optimization algorithm, a series of approaches can be implemented at different degrees of complexity in accordance with the sought parameters. This will be addressed in Section 4.3.

Sensitivity study

In order to assess the feasibility of the procedure to identify thermal properties, we conducted a sensitivity study to ensure the relevance of the measurements points, since the sensitivity to the variation of parameters is scaled. They were evaluated by using:

S * k I = k I ∂Y ∂k I S * c I = c I ∂Y ∂c I ( 22
)
This study focused on the wide range of thermal properties within the reference cases and on the more restrained range of thermal properties in the samples listed for future experimental studies.

First of all, in order to quantify the expected accuracy of this identification, the evolution of these sensitivities as a function of time is plotted. Regarding the reference cases, we noted on Figure 8 a remarkable difference in magnitude of the sensibility for each parameter according to the reference cases 1 and 4. Furthermore, both sensitivities for the heat capacity are considerably smaller than those for the conductivity. Even if the reduced model allows the reproduction of a wide range of materials by simulation, the identification could eventually pose a problem.

Figure 9 shows the results regarding the samples. The sensitivities are enclosed in inside the domain previously obtained by the reference cases, and their variation for the same parameter are closer. We noticed that the sensitivity in relation to the heat capacity is once again lower than that of the conductivity. Thus, we could expect less accurate results during the identification procedure.

Next, in order to study an eventual correlation between these two parameters, one traces them one in relation to the other. Figure 10 represents the evolution of the sensitivity of the capacity as a function of that of the conductivity. It is chosen here to adimentionnalize these values as a function of the maximum values obtained, in order to obtain the same scale for the two reference cases studied R1 and R4. It can thus be noted that none of these evolutions follows a linear law: there is no correlation between these sensitivities and the simultaneous identification of thermal conductivity and heat capacity is therefore theoretically possible by this hot-wire method.

Minimization method

There are a variety of methods to carry out a function minimization. In this study we explored a stochastic method by the name of Particle Swarm Optimization, that is a population-based stochastic technique inspired by the social and cognitive behavior of a bird flock [START_REF] Clerc | The particle swarm-explosion, stability, and convergence in a multidimensional complex space[END_REF][START_REF] Clerc | L'optimisation par essaim particulaire[END_REF].

Its principle is based on a random particle population that updates its positions and velocity at every iteration, looking to converge at the minimum of the specified function. In our case, being the thermal set of parameters the targeted value, the algorithm will update the thermal conductivity and capacity until a minimum is found (See annexe A for details).

The iterative process comes to an end when:

-the functional Eq.( 21) stops evolving, -the maximum number of iterations is achieved, -the Morozov discrepancy principle is reached [START_REF] Beck | Inverse Heat Conduction: Ill-Posed Problems[END_REF].

One can appreciate a series of captures of the PSO algorithm for Sample 1 in Figure 11. The first iteration allows to visualize the different initial values of the particle population, which are randomly chosen in the possible field of research of the parameters (k I , c I ).

. Identification without measurement noise

From virtual measurements obtained by numerical simulations carried out using a complete model (Eq.( 4)), we are seeking the set of thermal characteristics of different materials. On the one hand, those who have been used for the construction of the reduced model (Table 2) and, on the other hand, those that are commonly used (Table 4).

For a given parameter β i , we define σ β i as the relative error of the average value, and σ total as the quadratic error between the measurement and the temperature built with the parameters identified:

σ β i = | βi -β i | β i (23)
σ total = 1 2 N t N mes Nt i=1 Nmes j=1 Y ij -Y ij ( β) 2 (24)
The set of results obtained with an absence of measurement noise is presented on Table 6. Regarding the reference materials (Ref. , the accuracy in the results is highly satisfactory since the overall of the identified parameters show an error under 1%, except for the heat capacity of Reference 4 sample, for which the error has an order of magnitude of 6%. This result is completely justified by the low sensitivity observed previously in Figure 8.

k i k I σ k I c i c I σ c I σ total t cpu [10 -2 W.m -1 .K -1 ] [%] [10 4 J.m -3 .K -1 ] [%] [ • C] [s] S1 3
With regard to the conventional insulators (samples 1 to 5), it is noted that the identification errors are globally satisfactory. In terms of the thermal conductivity, identification error remains under 1%, whereas the error for the heat capacity reaches 6% at its maximum.

We note that the quadratic error σ total corresponds to the error found between the reduced and complete model.

With respect to the calculations time, they are always under 5 minutes. The benefit of using a reduced model within this iterative procedure, which needs a large number of simulations, then becomes obvious. As an example, in the case of Sample 1, it took 200 iterations, and taking into consideration that the number of particles used in each iteration equals 20, this leads to 4,000 simulations. By using a simple extrapolation, an identification using a complete model will take approximately 20 hours of computation time.

Influence of a measurement noise

In this section we studied Sample 1 for which a stochastic noise with zero mean, and constant standard deviation σ B 1 = 0.05[ • C], and uncorrelated along time is added (Gaussian noise). In order to characterize the dispersion of the results linked to this noise, a large number of identifications are carried out (n I = 300) which allows to evaluate for every identified magnitude β i :

-the average value defined by: βi

= 1 n I n I j=1 βj i (25)
-the standard deviation characterized by the dispersion of the estimation βj i of parameter β i around the expectation of the estimator:

std β i = 1 n I n I j=1 βj i -βi 2 (26)
-the average error:

σβ i = | βi -β i | (27)
-the confidence interval corresponds to the maximum error made on one measurement with a probability of 99.7%:

σmax = σβ i + 3 std β i (28)
The results dispersion is illustrated in Figures 12 and13, as well as in more detail in Table 7. At first we saw that each of the 300 identifications gave highly satisfying results. Secondly, for each parameter the average value βi of these identifications corresponds exactly to the values obtained without measurement noise (see Table 6). Thus, the average error σβ i gives insight to the error between the complete and the reduced model. Moreover, the standard deviation std β i remains entirely reasonable compared to the real value β i . The confidence interval σ max , chosen voluntarily wide, gives 0.8% for the thermal conductivity and 7% for the heat capacity, respectively. The relative errors σβ i /β i are 0.4% for the conductivity and 4.6% for the heat capacity.

β i k I [10 -2 W.m -1 .K -1 ] c I [10 4 J.m -3 .K -1 ] Exact β i 3
It is possible to obtain theoretically standard deviation values without performing the 300 estimations. It depends on the sensitivities of the measurements at each point A and B, in relation to the parameter to be identified k I and c I , grouped together in the form of a sensitivity matrix:

S = ∂Y ∂β i (29)
To quantify the quality of the estimation procedure, one can calculate the variance-covariance matrix defined by:

cov β = σ 2 B S T S -1 (30)
This covariance matrix has diagonal elements which enables us to approximate the theoretical standard deviation:

std β i ≈ cov(i, i) (31)
In the case of Sample 1 we obtained:

Sample std k [10 -2 W.m -1 .K -1 ] std c [10 4 J.m -3 .K -1 ] S1 0.004 0.050
These results are compared to those obtained from the 300 estimations shown on Table 7. In terms of the thermal conductivity, the approximation is accurate. As for the heat capacity, there is a difference of 20%, and one can appreciate that the standard deviation std c remains smaller regarding the real value of the heat capacity c.

This technique is applied to approximate the confidence interval σ max (Eq.( 28)), since the average value σβ i corresponds to the value without noise (see Table 6). Table 8 regroups these estimations for all the samples for two different noises.

Conductivity: σ max k [%]
Heat capacity: Overall results for these estimations are satisfactory. As noted before, the estimation of the heat capacity is less precise when compared to that of the conductivity. Moreover, a bigger measurement noise σ B leads to a slight increase of the confidence interval σ max .

σ maxc [%] Sample σ B = 0.05[ • C] σ B = 0.1[ • C] σ B = 0.05[ • C] σ B = 0.1[ • C] S1 0.
One single estimation is conducted for each sample for both of the measurement noises. Table 9 presents the corresponding error, which remains within the confidence interval.

Conductivity: σ k [%] Heat capacity: σ c [%] Sample σ B = 0.05[ • C] σ B = 0.1[ • C] σ B = 0.05[ • C] σ B = 0.1[ • C] S1 0.

Thermal contact resistance influence

In this section, we address the question of the presence of a thermal contact resistance (TCR) between the probe Ω P and the insulating material Ω I . In the reduced model used within the inverse procedure, this TCR was not taken into account, insofar as we consider that the probe is inserted in the insulator by breaking through the material, thus creating a perfect contact. However, in the literature, some authors refer to the existence of an imperfect contact between the probe and the material: Bording et al [START_REF] Bording | Determination of thermal properties of materials by monte carlo inversion of pulsed needle probe data[END_REF] analyzes the influence of TCR over a very broad spectrum (between 10 -4 and 3 [K.m 2 .W -1 ] ) without experimental justification, and shows that TCR has little influence on the determination of the conductivity k but leads to errors on the capacity c. Two authors give TCR values depending on the material in which the probe is implanted: Wenlong et al [START_REF] Cheng | Simultaneous measurement of thermal properties by thermal probe using stochastic approximation method[END_REF] displays a TCR ≈ 10 -3 [K.m 2 .W -1 ] for ceramics. Goodhew et al [START_REF] Goodhew | Analysis of thermal-probe measurements using an iterative method to give sample conductivity and diffusivity data[END_REF] proposes values of TCR = 5.10 -3 [K.m 2 .W -1 ] in clay straw and TCR = 2.10 -3 [K.m 2 .W -1 ] in Agar. They do not specify the procedure for implanting the probe in the material (with or without pre-drilling before introduction of the probe).

In this study, we will evaluate the identification error when an TCR that appears in reality is not taken into account in the identification procedure. The use of a numerical model which takes into account a contact resistance between the probe and the insulating material allows to obtain new virtual measurements from which the identification process is done with the previously reduced model (see Eq.( 19)). This process is conducted without a measurement noise for the most sensitive sample S2 and for different values of TCR.

Table 10 shows the identification errors of the two parameters k I and c I for different TCR values. The table specifies the correspondence in terms of equivalent air gap thickness around the probe, taking into account both the circular geometry of the probe and its very small size (The appendix presents the model used).

T CR [10 We can thus see that we find the trends presented in the previous studies [START_REF] Bording | Determination of thermal properties of materials by monte carlo inversion of pulsed needle probe data[END_REF]:

On the one hand, the value of the identified thermal conductivity remains satisfactory whatever the quality of the contact between the probe and the air gap. Only a strong resistance (T CR = 5 10 -3 [K.m 2 .W -1 ]) which corresponds to a pre-drilling a diameter significantly larger than the probe, leads to a significant error (2 [%]), of the same order of magnitude as the error related to the measurement noise (according to the table 8, an average measurement error of 0.1 o C leads to an identification error of 1.44 %).

On the other hand, the results obtained show a very high sensitivity of the thermal capacity to the quality of the contact between the probe and the material. The difficulty to manage this parameter greatly degrades the quality of the identification of this capacity, which already depends significantly on measurement noise (the table 8 shows that an average measurement error of 0.1 o C leads to an identification error of about 11 %).

This study brings to light the need of conducting a characterization for an imperfect contact between the probe and the material. In this case, the reduced model must then be adapted to take into account the estimated TCR.

Conclusion

This study evaluates the possibility of taking advantage of recent progress in reducing thermal models to extend the use of thermal characterization probes on site such as the hot-wire type. We wanted to be able to use them, on the one hand on strong insulating materials, which until now were difficult to characterize by this type of probes, and moreover to carry out a complete determination of the thermal properties, including the heat capacity.

This characterization requires a large numerical model to be able to perform a precise simulation taking into account the integrality of the thermal phenomena with the complex geometry of the probe. Given the incompatibility faced by large models in the face of an iterative process that constitutes the inverse problem posed, a solution is a reduced model, provided that the latter can be independent of the intrinsic properties of the model intended to be identified.

This study examined modal reduced models of the AROMM type. The advantage of these models is that they are built from complete families of eigenvectors, that form a base in H 1 . These modal models are then independent of intrinsic parameters, such as the thermal conductivity and heat capacity. The construction of the reduced model was optimized from reference cases defining a wide range of insulating materials having a thermal conductivity varying from 1 to 6 [10 -2 W.m -1 .K -1 ] and, a thermal capacity value ranging from 1.5 to 150 [10 4 J.m -3 .K -1 ]. Direct simulation tests for different materials with a single reduced model showed significant reduction in computation time (250 times) with a maximum error at the measurement points of less than 0.08 [ • C].

Therefore, the reduced model avoids a prohibitive computation time for an identification process. In order to estimate the influence of the reduced model within the inverse procedure, the first tests were carried out with virtual measurements obtained from a finite element complete model in the absence of noise. The results were highly satisfactory, even if it appeared that the low sensitivity of the temperatures measured with respect to the heat capacity generated an error reaching 6%.

A measurement noise study made it possible to show the feasibility of the method. The standard deviation, which is a function of the measurement noise, makes it possible to predict the confidence interval likely to be reached for a given probability. The tests conducted for all the materials examined had satisfactory results, with the worst case scenario resulting in 1.5% error on thermal conductivity and 8.7% error on heat capacity, thus producing a total measurement noise of σ B = 0.05[ • C].

Furthermore, given the speed of the numerical calculations (< 5 min.) during the identification process, that employs a reduced model, one can imagine a continuous process of repeated identifications that overrides the influence of measurement noise, only to obtain an error solely related to the use of the reduced model.

Finally, the effect of an eventual imperfect contact between the probe and the insulating material was addressed by using virtual temperature measurements that took into account a given air gap. It was found that if there was thermal resistance, it could generate important additional errors for the heat capacity but acceptable for the thermal conductivity. Additional research will be required to assess this thermal resistance either by analyzing the state of the insulating material around the probe, or by identifying the resistance itself. The question then arises how to handle imperfect contact in the reduced model.

This study has shown the strength of reduced models to simulate complex thermal phenomena, always getting closer to reality.

This work was conducted for a real hot-wire type probe originally dedicated to the determination of the conductivity in an analytical way. This probe configuration is probably not optimal in terms of measurements sensitivity to the parameters to be identified, particularly to the heat capacity. Hence, the elaboration of a new probe configuration, which would allow greater sensitivity to heat capacity could generate better results. Regardless of the design of the probe, the use of the AROMM method in the inverse procedure would be efficient.

Finally, this technique could be extended to identify the same parameters for an orthotropic geometry, typical of thermal insulators, by implementing a new technological tool.

Annexe B : Correspondence between TCR and equivalent air gap

We consider here the case a homogeneous probe that dissipates a power P uniformly distributed in the volume. Placed in a cylindrical material and neglecting the ends of the probe, the axial symmetry allows the use of two analytical models : the case of the air gap uniformly surrounding the probe is represented on Figure 14 [START_REF] Videcoq | Heat source identification and on-line temperature control by a branch eigenmodes reduced model[END_REF]

Figure 1 :

 1 Figure 1: Studied system

Figure 2 :

 2 Figure 2: Geometry of the studied system

Figure 3 :Figure 4 :

 34 Figure 3: Temperature evolution : analytical and numerical models

Figure 5 :

 5 Figure 5: Maximum error at measuring points max max (A), max (B) for each reference cases

Figure 5

 5 Figure 5 presents the maximum error between max (A) and max (B) (Eq.(16)) as a function of the order of the model. We note that according to the order of reduction chosen, the different reference cases do not have the same behaviour. Indeed the optimization of the reduced model (for a given ñ order) is based on a global criterion (13) at the same time on the whole spatial and

Figure 6 :

 6 Figure 6: Reduced model of ñ = 100 modes : results for Reference n • 4 at point A

  From the temperature field obtained, by using an inverse technique, we are able to identify the thermal properties (k I and c I ) of the targeted insulating material.

Figure 7 :

 7 Figure 7: Inverse procedure scheme

Figure 7

 7 Figure 7 demonstrates the iterative process aimed at identifying the best solution. In practice, the inverse approach focuses on N mes specific points, through a selection matrix E, that creates an observable vector Y to compute an approximation of β = [ k I , c I ].The goal here is to replace the numerical model by a reduced order model previously built:

Figure 8 :Figure 9 :

 89 Figure 8: Sensitivity study for Reference R1 and R4

Figure 10 :

 10 Figure 10: Correlation between the adimensionnalised sensivity for Reference R1 and R4

Figure 11 :

 11 Figure 11: Particles evolution for PSO algorithm for sample No.1

Figure 12 :Figure 13 :

 1213 Figure 12: Identification of k for a σ B = 0.05 • C

  (a), and a simple power balance allows to write : an equivalent thermal resistance T CR which imposes a temperature jump at the interface between the probe and the insulation leads to (Figure14(b)) :The equality of the different resistances of the equations (34) and[START_REF] Park | On the solution of inverse heat transfer problem using the karhunen loève galerkin method[END_REF] gives:

Table 1 :

 1 Physical parameters of the hot-wire probe

	Ω HW Heating wire	19.5	3.47×10 6
	#2 -Ω HS Heating wire sheath	0.25	2.25×10 6
	#3 -Ω Ch Chromel	19	3.81×10 6
	#4 -Ω A Alumel	29.7	4.50×10 6
	#5 -Ω T h Thermocouple	24.35	4.15×10 6
	#6 -Ω T hS Thermocouple sheath	0.2	2.25×10 6
	#7 -Ω S Steel	16.3	3.97×10 6
	#8 -Ω G Glue/filling	0.1	3.15×10 5
	#9 -Ω C Copper	390	3.38×10 6

Table 3 :

 3 Efficacy of the reduced order model of ñ = 100 modes of the reference cases

			.31	1.61	31.50
		B	5.28	1.96	33.82
	R2	A B	6.37 6.36	0.76 1.80	15.82 16.10
	R3	A B	5.50 5.54	2.55 1.22	7.90 8.47
	R4	A B	6.97 6.92	2.04 2.20	4.54 4.56

Table 4 :

 4 Thermal properties of the tested insulators

	1 ]

Table 5 :

 5 Reduced model of ñ = 100 modes : errors of the samples

Table 6 :

 6 Identification without measurement noise

Table 7 :

 7 Statistical study with measurement noise

	7.2

Table 8 :

 8 Estimations of the confidence interval σ max for different measurement noises σ B

		70	1.06	6.66	8.73
	S2	0.96	1.44	8.70	11.31
	S3	1.43	1.71	6.41	8.58
	S4	1.32	1.57	5.98	7.87
	S5	0.72	0.99	3.12	4.81

Table 9 :

 9 One single identification for different measurement noises σ B

		33	0.10	4.30	1.87
	S2	0.40	0.78	5.38	7.28
	S3	1.04	1.25	4.23	6.25
	S4	0.91	0.64	4.47	2.17
	S5	0.06	0.06	1.36	2.03

  -3 K.m 2 .W -1 ] e [mm] σ k I [%] σ c I [%]

	0	0	0.51	5.83
	0.25	0.015	0.76	9.13
	0.50	0.03	0.62	11.82
	1.00	0.03	0.39	16.71
	1.64	0.10	0.73	22.54
	5.00	0.34	2.14	44.98

Table 10 :

 10 Influence de la TCR sur la precision de l'identification : materiau S2

Initialize velocity v = 0 for each particle; while termination condition is not fullfilled do foreach particle do compute velocity v (Eq.32); compute position β (Eq.33); if position in feasible space then evaluate position; end end end

Eq.( 32) allows to update the particle velocities. The first term concerns the inertia, it makes the particle move in the same direction and with the same velocity. The second term is linked to the personal influence, it improves the individual by making the particle return to a previous position better than the current one. And the third term regards the social influence, meaning that the particle will follow the best neighbors direction.

Eq.( 33) moves each particle to their new position where:

β represents the particle position, -v the path direction, -χ the weight of local information, also considered the inertia factor, it defines the exploration capacity of each particle, -λ 1 and λ 2 are the weight of global information, also known as construction factors, -pBest is the best position of the particle, -gBest is the best position of the swarm, -rand is a random number between [0,1] For this study, the PSO parameters are as follows: number of particles p = 20, χ = 0.7298 and λ 1 = λ 2 = 1.46618. These parameters were chosen in order to assure the convergence of the algorithm [START_REF] Eberhart | Comparing inertia weights and constriction factors in particle swarm optimization[END_REF]. We can see here that the expression of the equivalent contact resistance is no longer as simple as the T CR = e/k air model used frequently. This contact resistance depends not only on the cylindrical geometry, but also on the conductivity of the material in which the probe is inserted.


Dr Louis-Marie Desroche
email: lmdesroche@hotmail.fr

MD Olivier Milleron

MD Benjamin Safar

MD Phalla Ou

MD. PhD Eric Garbarz

MD Yoan Lavie-Badie

MD Jérémie Abtan

MD Damien Millischer

MD Atul Pathak

MD PhD Isabelle Durand-Zaleski

PhD Simon Cattan

MD Thibault Ronchard

PharmD Guillaume Jondeau

Cardiovascular Magnetic Resonance May Avoid Unnecessary Coronary Angiography in Patients With Unexplained Left Ventricular Systolic Dysfunction: A Retrospective Diagnostic Pilot Study

Keywords: coronary artery disease, cardiovascular magnetic resonance imaging, left ventricular systolic dysfunction

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Reduced left ventricular ejection fraction (LVEF) is present in 3-7% of the general population, which is expected to increase with the aging of the population [START_REF] Mosterd | Prevalence of heart failure and left ventricular dysfunction in the general population; The Rotterdam Study[END_REF][START_REF] Van Riet | Epidemiology of heart failure: the prevalence of heart failure and ventricular dysfunction in older adults over time. A systematic review[END_REF]. Its most common cause is ischemic cardiomyopathy [START_REF] Ponikowski | ESC Guidelines for the diagnosis and treatment of acute and chronic heart failure: The Task Force for the diagnosis and treatment of acute and chronic heart failure of the European Society of Cardiology (ESC)Developed with the special contribution of the Heart Failure Association (HFA) of the ESC[END_REF], but the most favorable process of identifying patients benefiting from invasive coronary angiography (CA) remains unclear [START_REF] Ponikowski | ESC Guidelines for the diagnosis and treatment of acute and chronic heart failure: The Task Force for the diagnosis and treatment of acute and chronic heart failure of the European Society of Cardiology (ESC)Developed with the special contribution of the Heart Failure Association (HFA) of the ESC[END_REF][START_REF] Yancy | ACCF/AHA guideline for the management of heart failure: a report of the American College of Cardiology Foundation/American Heart Association Task Force on Practice Guidelines[END_REF]. CA is almost systematically performed [START_REF] Yancy | ACCF/AHA guideline for the management of heart failure: a report of the American College of Cardiology Foundation/American Heart Association Task Force on Practice Guidelines[END_REF][START_REF] Bradley | Patient Selection for Diagnostic Coronary Angiography and Hospital-Level PCI Appropriateness: Insights from the NCDR®[END_REF] to identify coronary artery disease (CAD): indeed, in the STICHES study [START_REF] Velazquez | Coronary-Artery Bypass Surgery in Patients with Ischemic Cardiomyopathy[END_REF], the patients with ischemic cardiomyopathy had their survival improved by revascularization. However in the absence of chest pain, three-fourths of patients with reduced LVEF do not have CAD [START_REF] Mccrohon | Differentiation of heart failure related to dilated cardiomyopathy and coronary artery disease using gadolinium-enhanced cardiovascular magnetic resonance[END_REF][START_REF] Assomull | Role of cardiovascular magnetic resonance as a gatekeeper to invasive coronary angiography in patients presenting with heart failure of unknown etiology[END_REF] but nevertheless undergo CA with its cost and complications [START_REF] Laskey | Multivariable model for prediction of risk of significant complication during diagnostic cardiac catheterization. The Registry Committee of the Society for Cardiac Angiography & Interventions[END_REF]. A method for identifying individuals with reasonable probability of having CAD prior to CA would be beneficial. Furthermore CA may not correctly identify all ischemic cardiomyopathies: post-mortem [START_REF] Roberts | Idiopathic dilated cardiomyopathy: analysis of 152 necropsy patients[END_REF] and cardiovascular magnetic resonance imaging (CMR) [START_REF] Mccrohon | Differentiation of heart failure related to dilated cardiomyopathy and coronary artery disease using gadolinium-enhanced cardiovascular magnetic resonance[END_REF][START_REF] Assomull | Role of cardiovascular magnetic resonance as a gatekeeper to invasive coronary angiography in patients presenting with heart failure of unknown etiology[END_REF] studies have shown that 12-17% of "idiopathic dilated cardiomyopathies" may actually be ischemic following myocardial infarction with non-obstructive coronary arteries (MINOCA) [START_REF] Reynolds | Mechanisms of myocardial infarction without obstructive coronary artery disease[END_REF].

The diagnostic performance of subendocardial late gadolinium enhancement (LGE) on CMR for the diagnosis of myocardial infarction is very high [START_REF] Kim | Performance of delayed-enhancement magnetic resonance imaging with gadoversetamide contrast for the detection and assessment of myocardial infarction: an international, multicenter, doubleblinded, randomized trial[END_REF][START_REF] Wagner | Contrast-enhanced MRI and routine single photon emission computed tomography (SPECT) perfusion imaging for detection of subendocardial myocardial infarcts: an imaging study[END_REF]. Some series [START_REF] Mccrohon | Differentiation of heart failure related to dilated cardiomyopathy and coronary artery disease using gadolinium-enhanced cardiovascular magnetic resonance[END_REF][START_REF] Assomull | Role of cardiovascular magnetic resonance as a gatekeeper to invasive coronary angiography in patients presenting with heart failure of unknown etiology[END_REF][START_REF] Casolo | Identification of the ischemic etiology of heart failure by cardiovascular magnetic resonance imaging: diagnostic accuracy of late gadolinium enhancement[END_REF][START_REF] Valle-Muñoz | Late gadolinium enhancement-cardiovascular magnetic resonance identifies coronary artery disease as the aetiology of left ventricular dysfunction in acute new-onset congestive heart failure[END_REF] which evaluate CMR for the detection of ischemic cardiomyopathy in patients with LVEF reduction have been encouraging (sensitivity of 85-100%), but based on a limited number of patients (60-120 patients), and they did not formally assess the benefits and risks of a strategy based on CMR in first-line.

This study was based on the hypothesis that at least one myocardial segment is expected to have an infarction pattern on CMR in patients with significant reduction in LVEF due to CAD. We aimed to evaluate the potential medical benefits of using CMR as a first-line investigation to identify ischemic cardiomyopathy in patients with unexplained reduced LVEF. Our main hypothesis was that the NPV of CMR for predicting ischemic cardiomyopathy approaches 100% which would allow to avoid CA when no ischemic pattern is found on CMR. The main objective of this study was to examine the agreement between CA and cardiovascular magnetic resonance(CMR) among a cohort of patients with unexplained reduced LVEF, and estimate what would have been the consequences of using CMR as the first-line exam. Our secondary objectives were to compare the diagnostic value of CMR with those of ECG and transthoracic echocardiography(TTE) in predicting the presence of significant CAD, and to evaluate the economic benefits of the "alternative strategy" consisting in a systematic first-line CMR, with CA performed only when an ischemic scar is found on CMR.

Material and Methods

Patients with unexplained reduced LVEF from 2 institutions, hospitalized between March 2009 and October 2015 were eligible to be included in this retrospective study, if CMR and CA were performed less than 3 months apart. LVEF ≤ 45% on 2D TTE was chosen as indicative of incipient systolic dysfunction; this threshold was selected taking into consideration the European Society of Cardiology definition (LVEF ≤50%) [START_REF] Ponikowski | ESC Guidelines for the diagnosis and treatment of acute and chronic heart failure: The Task Force for the diagnosis and treatment of acute and chronic heart failure of the European Society of Cardiology (ESC)Developed with the special contribution of the Heart Failure Association (HFA) of the ESC[END_REF] and the 5% inter-and intra-observer variability of the LVEF on TTE [START_REF] Wood | Left Ventricular Ejection Fraction and Volumes: It Depends on the Imaging Method[END_REF], and has also been used in large clinical studies on heart failure [START_REF] Taylor | Combination of isosorbide dinitrate and hydralazine in blacks with heart failure[END_REF][START_REF] Prabhu | Catheter Ablation Versus Medical Rate Control in Atrial Fibrillation and Systolic Dysfunction: The CAMERA-MRI Study[END_REF]. Patients with an obvious ischemic (myocardial infarction or typical angina), valvular ("moderate-to-severe" or "severe" left valvular regurgitation, or "severe" aortic stenosis, according to the European Association of Cardiovascular Imaging [START_REF] Lang | Recommendations for cardiac chamber quantification by echocardiography in adults: an update from the American Society of Echocardiography and the European Association of Cardiovascular Imaging[END_REF]), or rhythmic (arrhythmia greater than 120 bpm on initial assessment) cause of the LVEF impairment were excluded.

This observational retrospective study did not interfere with the usual care of patients.

CMR and CA were performed in the context of care. Since the study met the criteria of the reference methodology MR-005 of French regulations, no approval from a local commission or consent was required.

ECG and TTE data were collected; left ventricular dilatation was defined by a left ventricular end diastolic volume index ≥ 87ml/m 2 for men and 74ml/m 2 for women [START_REF] Felker | A standardized definition of ischemic cardiomyopathy for use in clinical research[END_REF].

CMR was performed using 1.5-T scanners: MAGNETOM Avanto (Siemens) in the first institution and Signa TwinSpeed (General Electric) in the second. Contiguous short-axis locations encompassing the left ventricle were acquired in the cine steady-state free precession sequence. Each slice was acquired during one short breath-hold. Delayed enhanced images were obtained 10 minutes after a gadolinium-based contrast agent injection (DOTAREM ® , Guerbet; 0.4 mL/kg) using a 3D segmented inversion-recovery gradient-echo pulse sequence, with the slice position identical to that of the cine images (field of view 300-360 mm, slice thickness 6.0 mm, inversion time 200-280 ms). LGE assessment was qualitative, describing the number of segments with LGE according to the 17-segment model of the American Heart Association and the myocardial layer involved. Subendocardial LGE was defined with areas of bright signal, and an increasing degree of transmural extension, depending on the extent of the infarct. The four CMR readers had the national diploma required for CMR reading, and each experienced over 1000 CMR analyses in 2009; two of whom had a Ph.D.

Classification of patients

Patients were classified as CMR + if subendocardial (or transmural depending on the extension) LGE was present in at least one segment; they were otherwise categorised as being "CMR -" . The gold standard exam was CA. They were classified as CA + if significant CAD was present according to Felker's criteria concerning prognostically significant CAD [START_REF] Pencina | Evaluating the added predictive ability of a new marker: from area under the ROC curve to reclassification and beyond[END_REF]: i.e., ≥75% stenosis of left main or proximal left anterior descending artery (LAD), or ≥75% stenosis of two or more epicardial vessels. Otherwise they were classified as CA -. Eight patients had only one coronary stenosis > 75% not including left main or proximal LAD and they have thus been classified as CA-. Their CMR result and medical care are described in a specific paragraph of the results section.

Felker's criteria was used because the benefit of revascularization of a stenosis not meeting these criteria is doubtful. The benefit of revascularization in patients with LVEF reduction was even difficult to demonstrate in patients with stenosis meeting Felker's criteria: STICH trial [START_REF] Viera | Understanding interobserver agreement: the kappa statistic[END_REF] did not show any significant benefit after 5 years. After an extended-follow up, STICHES [START_REF] Mccrohon | Differentiation of heart failure related to dilated cardiomyopathy and coronary artery disease using gadolinium-enhanced cardiovascular magnetic resonance[END_REF] finally showed a benefit after 10 years.

Statistical analysis

This was a proof of concept study(pilot study); no calculation of the number of subjects required has been performed in this retrospective study. Indeed, this pilot study was designed for the CAMAREC prospective and multicenter study (Clinical Trials Registration No.NCT03231189). A large number of patients was therefore required to have an accurate estimate of the diagnostic performance of CMR in this situation.

Statistical analyses were performed using XLSTAT. In this diagnostic study, the patients were separated into four groups depending on CA and CMR results: CA -CMR -, CA -CMR + , CA + CMR + , and CA + CMR -. Sensitivity, specificity, positive and negative predictive value, positive and negative likelihood ratio, and diagnostic accuracy, with confidence intervals of 95%, were calculated to evaluate the diagnostic value of CMR for detecting CA + patients. This calculation was also performed for the electrocardiogram, regional wall motion abnormalities on TTE, and the combination of both. The net reclassification improvement [START_REF] Tavakol | Risks and complications of coronary angiography: a comprehensive review[END_REF] was calculated to evaluate the additional benefit of using CMR for the detection of CA+ patients compared to the combination of electrocardiogram and regional wall motion abnormalities on TTE.

To test the reproducibility of the first real-life interpretation of the CA and CMR, all exams were reinterpreted by both an interventional cardiologist and a cardio-radiologist who had not participated in the original interpretations; they were blinded to clinical, echocardiographic, CMR and CA data. Agreement with the original interpretation was calculated for both CA and CMR using kappa statistics, with κ > 0.6 indicating substantial agreement [START_REF] Hong | Remodeling of small intramyocardial coronary arteries distal to a severe epicardial coronary artery stenosis[END_REF].

Cost and complication analysis between the current and the alternative strategy

The cost per patient of the "alternative strategy" consisting of performing CA only in CMR + patients was compared to the cost per patient of the "current practice". The "current practice" was evaluated according to number of CMR and CA performed in the initial real-life samples of patients with newly diagnosed and unexplained reduced LVEF.

We first took the standpoint of the French healthcare system with €4,854 for CA (including the procedure and physicians' fees), and the cost of CMR based on physician fees and materials costs, for an out-patient (€303: €69 for physician fees, €173 for equipment cost, and €61 per dose of gadolinium). Then we used the Centers for Medicare Medicaid Services cost schedule for 2018, and used $512 for CMR and $12,276 for CA.

The incidence of complications was also compared between the current and the alternative strategy, using a major complication rate of 1.5% for CA [START_REF] Horstick | Critical single proximal left arterial descending coronary artery stenosis to mimic chronic myocardial ischemia: a new model induced by minimal invasive technology[END_REF] comprising vascular complication, stroke, renal injury, cholesterol emboli syndrome and death.

Results

Population

On 604 patients admitted to investigate a newly diagnosed reduced LVEF ≤ 45%, by TTE(figure 1), 202 were excluded because they did not undergo CMR, because CMR was not easily accessible mainly in the early years of the study; thereafter, CMR became part of the systematic investigation of any unexplained alteration of LVEF, regardless of CA results. CA was not proposed to 2/604 patients due to a low cardiovascular risk and the absence of LGE on CMR. 22 patients were excluded because CA and CMR were not performed within 3 months apart, 63 because etiology of the LVEF reduction was known, and 10 because CMR interpretation was not possible (7 because of a technical problem and 3 because of claustrophobia)(figure 1).

Ultimately, 305 patients were included in the study. The mean delay between CA and CMR was 15.4 days, the CMR having been done first in 41% of the cases. Twenty patients (6.5%) underwent revascularization: 16 (5%) received percutaneous coronary intervention and 4 (1.5%) coronary artery bypass grafting. No patients were revascularized prior to CMR.

Two patients had a pacemaker, and none had an Implantable-Cardioverter-Defibrillator. Table 1 summarizes the characteristics of these patients. Supplemental Table S1 compares our population to the patients excluded because no CA or CMR was performed.

Classification of patients

In the reproducibility testing of the interpretations of CA and CMR, the agreement rate between the reinterpretation and the original interpretation was high (κ = 0.85 and κ = 0.76, respectively). One patient initially classified as CA -CMR -was reclassified as CA -CMR + and another initially classified as CA + CMR + was reclassified as CA -CMR + .

Figure 2 shows the distribution of patients according to the results of CA and CMR.

Figure 3 represents the alternative strategy using first-line CMR. All CMR-patients were also CA-, except 2 patients (0.7% of 216 CMR-patients) who had a CAD too distal to explain such LV dysfunction (15% and 25% LVEF) and to have revascularization. Therefore, no revascularization was proposed in any CMR-patient, and LV systolic dysfunction was not explained by ischemic cardiomyopathy in any of them.

Among the 89 CMR+ patients, CA identified significant CAD according to Felker's criteria in 52 CA+ patients; and revascularization was performed in 20 patients. 37 CMR+ were CA-. They were younger (59 years vs. 62 years), and more likely to be female (21% vs. 16%), with less dyslipidaemia (27% vs. 50%).

CMR sensitivity for detecting significant CAD(CA+) was 96%, specificity 85%, negative predictive value 99%, and positive predictive value 58% (Table 2). Negative predictive value was 100% for identification of coronary artery stenosis requiring revascularization and explaining the reduced LVEF. Positive and negative likelihood ratio of CMR for predicting CA+ patients were respectively 6.53 (95% confidence interval 4.83-8.84) and 0.04 (95% confidence interval 0.01-0.17). Conversely, if CMR was considered the gold standard, the sensitivity of CA for detecting CMR+ patients would only be 64% (95% CI: 53%-74%).

Cost savings and complications avoided through the alternative strategy

If the "alternative strategy" had been applied (CMR performed on all patients, and CA only for CMR + patients), CA would have been performed on only 89 patients (52 CA+CMR+ and 37 CA-CMR+). 216/305(71%) CAs would have been avoided. Using a 1.5% complication rate for CA, the alternative strategy based on CMR would avoid 1 complication every 94 patients investigated for LVEF reduction. We also calculated the cost difference of the two approaches : the "current practice", with the number of CMRs and CAs observed in the initial population of 604 patients evaluated for reduced LVEF (400 with CMR and CA, 202 with CA alone and 2 with CMR alone), and on the other side the "alternative strategy" (CA performed only in the 29% CMR + patients). This alternative strategy would have saved €3,320 /patient in France and $8,482 /patient in the USA.

Thus, for every 100 patients investigated for LVEF reduction, this alternative diagnostic strategy would have avoided 1,065 major complication, associated with savings of €332,000 in France and $848,200 in the USA.

Additional value of CMR over ECG and TTE

The diagnostic value of the ECG and wall motion abnormalities on TTE for predicting CA+ patients, are shown in Table 2. The net reclassification improvement of CMR, compared to the combination of electrocardiogram and regional wall motion abnormalities on TTE, was 0,62(respectively 0,15 and 0,47 for the reclassification of the CA + and CA -patients).

Additional findings on CMR

CMR revealed diverse non-ischemic etiologies for LVEF reduction in 72 patients:

myocarditis

Discussion

This multicenter study suggests that the use of CMR as a guide in the decision-making process of the need for CA in patients with unexplained systolic dysfunction (alternative strategy) may be superior to current practice with near-systematic CA.

It reliably identifies patients with significant CAD according to Felker's criteria(96% sensitivity), especially those requiring revascularization(100% sensitivity), while reducing the number of CAs performed(-71%), resulting in substantial savings. Furthermore, this strategy allows to identify some etiologies responsible for the LV systolic dysfunction including amongst others, MINOCA, and occasional access to additional information guiding care (e.g., apical thrombi not identified by TTE).

The most important initial fear of using the "alternative strategy" (CA performed only for CMR + patients) was to be unable to diagnose all patients with significant CAD requiring revascularization. In our study all the patients requiring revascularization were identified.

This seems to suggest that hibernation alone can rarely be responsible for a severe LVEF reduction without at least 1 segment showing necrosis, even subendocardial. Besides myocardial hibernation animal models failed to significantly reduce LVEF without causing some necrosis [START_REF] Giorgetti | Baseline/post-nitrate Tc-99m tetrofosmin mismatch for the assessment of myocardial viability in patients with severe left ventricular dysfunction: comparison with baseline Tc-99m tetrofosmin scintigraphy/FDG PET imaging[END_REF][START_REF] De Jong | Abnormal contractile responses during dobutamine stress echocardiography in patients with idiopathic dilated cardiomyopathy[END_REF].

Imaging and diagnostic strategy

Current guidelines state that in the absence of angina, CA should be considered on patients suitable for revascularization, with LVEF dysfunction, intermediate to high pre-test probability of CAD, and after a positive non-invasive stress test [START_REF] Ponikowski | ESC Guidelines for the diagnosis and treatment of acute and chronic heart failure: The Task Force for the diagnosis and treatment of acute and chronic heart failure of the European Society of Cardiology (ESC)Developed with the special contribution of the Heart Failure Association (HFA) of the ESC[END_REF][START_REF] Yancy | ACCF/AHA guideline for the management of heart failure: a report of the American College of Cardiology Foundation/American Heart Association Task Force on Practice Guidelines[END_REF]. Our study suggests that CAD is difficult to predict with basic tests, which is underlined by the high net reclassification improvement of CMR compared to the combination of electrocardiogram and TTE. Furthermore the diagnostic performance of stress imaging is questionable when LVEF is reduced, whether with single-photon emission computed tomography [START_REF] Darlington | Costeffectiveness of computed tomography coronary angiography versus conventional invasive coronary angiography[END_REF] or stress TTE [START_REF] Heywood | High prevalence of renal dysfunction and its impact on outcome in 118,465 patients hospitalized with acute decompensated heart failure: a report from the ADHERE database[END_REF]. About coronary computed tomography angiography, its low specificity (50%) for identification of hemodynamically significant CAD [START_REF] Srichai | Clinical, imaging, and pathological characteristics of left ventricular thrombus: a comparison of contrast-enhanced magnetic resonance imaging, transthoracic echocardiography, and transesophageal echocardiography with surgical or pathological validation[END_REF] would lead to perform many unnecessary CAs if used as a first-line screening test in these patients, over the average age of 60, and 23% diabetic. Above the economic impact, this would lead to multiply iodine contrast injections, while 20% of these patients with heart failure have a Glomerular Filtration Rate <30 mL/min/1,73m2 [START_REF] Allman | Myocardial viability testing and impact of revascularization on prognosis in patients with coronary artery disease and left ventricular dysfunction: a meta-analysis[END_REF]. Coronary computed tomography angiography may be relevant for low cardiovascular risk patients; however, it will be less efficient than CMR in diagnosing MINOCA.

CMR and ischemic cardiomyopathy

CMR reclassified 11% of the cardiomyopathies as MINOCA, instead of "idiopathic dilated cardiomyopathies", which is close to the ratio observed in post-mortem or CMR studies [START_REF] Mccrohon | Differentiation of heart failure related to dilated cardiomyopathy and coronary artery disease using gadolinium-enhanced cardiovascular magnetic resonance[END_REF][START_REF] Assomull | Role of cardiovascular magnetic resonance as a gatekeeper to invasive coronary angiography in patients presenting with heart failure of unknown etiology[END_REF][START_REF] Roberts | Idiopathic dilated cardiomyopathy: analysis of 152 necropsy patients[END_REF]. This can lead to modify the treatment, introducing aspirin and statins, or in some cases, to raise the hypothesis of a coronary artery spasm.

CMR allowed the detection of apical thrombi, 25% of which had been missed by TTE, a similar proportion reported by Srichai [START_REF] Velazquez | Coronaryartery bypass surgery in patients with left ventricular dysfunction[END_REF]. This helped prevent cardioembolic events introducing an anticoagulant therapy.

Lastly, CMR prevented futile revascularization in 60% of CA + patients, by reporting the absence of myocardial viability in the territory of the stenosis, since revascularization presents no benefit in this situation [33].

Limitations

Even if this study was sizable, its main limitation was its retrospective design and the not blind initial reading of CA and CMR. However, we assessed this risk with a new blind reading of CMR and CA, and the suitable agreement between the first real-life reading and the new blind reading was reassuring.

A selection bias may have arisen due to the exclusion of 202 patients without CMR and 2 without CA. However, these patients were similar to those included in the study (Table S1). The non-completion of the second examination (majority CMR) was mainly due to heterogeneous practices between cardiologists, especially in the early years of the study when CMR was not easily accessible.

The rate of significant CAD(18%) was lower than in previous studies [START_REF] Mccrohon | Differentiation of heart failure related to dilated cardiomyopathy and coronary artery disease using gadolinium-enhanced cardiovascular magnetic resonance[END_REF][START_REF] Assomull | Role of cardiovascular magnetic resonance as a gatekeeper to invasive coronary angiography in patients presenting with heart failure of unknown etiology[END_REF](21%-44%) but in our study we excluded obvious etiologies for systolic dysfunction, including typical angina, which was not the case in others.

The mean age of our population with reduced LVEF was 60.2 years, lower than in some large epidemiological studies (66 years in the MAGGIC meta-analysis in 41,972 patients [34]). This may have under-represented the number of CAD.

During this study, FFR was not systematic, and therefore the accuracy of the decision to perform revascularization may be questioned.

Conclusion

An "alternative strategy" using CMR as a screening tool for performing CA seems reliable, more informative, and less expensive than a strategy using CA as a first-line investigation for CAD screening in patients with an unexplained reduction in LVEF. Further

Declarations

  studies are needed, and a prospective multicenter study has been launched to confirm these results ("Prospective Multicentric Study for the Diagnostic Performance of CArdiac MAgnetic REsonance imaging used first for the diagnosis of Coronary artery disease as the etiology of left ventricular dysfunction"; CAMAREC trial; Clinical Trial Registration No.NCT03231189).

FIGURE 1 .FIGURE 2 .FIGURE 3 .

 123 FIGURE 1. Flowchart of patient selection

TABLE 1 .

 1 Population characteristics

	Clinical features	n
	Male gender	214 (70%)

SD: Standard deviation; TTE: transthoracic echocardiography.

TABLE 2 .

 2 Diagnostic value of basic tests and CMR for predicting ischemic cardiomyopathy

		Se	Sp	NPV	PPV	D.A.
	T abnormality	17 (8-29)	64 (57-70) 78 (75-80)	9 (5-15)	55 (50-61)
	Q-wave	33 (21-47)	96 (93-98) 87 (85-89)	67 (49-81) 85 (81-89)
	WMA	62 (51-72)	18 (13-24) 54 (44-64)	23 (20-26) 30 (25-36)
	Q-wave or WMA 74 (60-85)	65 (58-70) 92 (88-95)	31 (26-36) 66 (61-72)
	CMR	96 (87-100) 85 (80-89) 99 (96-100) 58 (51-66) 87 (83-91)
	All values are expressed as the number and percentage, n (%), (95% confidence interval).
	A single vessel disease not including left main or proximal LAD, was present in 8
	patients (classified as CA-above). 4 were CMR+ and 4 CMR-. No revascularization was
	performed on these 4 CMR-patients with single vessel disease. Of the 4 CMR+ patients with
	single vessel disease (and not including left main or proximal LAD), 1 percutaneous
	revascularization was performed on a 80% right coronary artery stenosis. Therefore, if CAD

CA: coronary angiography; CMR: cardiovascular magnetic resonance imaging; D.A.: diagnostic accuracy; ECG: electrocardiogram; ECG+WMA: combination of ECG and WMA; NPV: negative predictive value; PPV: positive predictive value; Se: sensitivity; Sp: specificity; WMA: regional wall motion abnormalities on transthoracic echocardiography.

  in 23, LV noncompaction (not reported by TTE) in 20, LV hypertrophy (not reported by TTE) in 11, left intraventricular dyssynchrony (not reported by TTE) in 9, Takotsubo cardiomyopathy in 8, and Chagas disease in 1. CMR also influenced care in 14/305(5%) patients: 12 LV apical thrombi not detected by TTE received anticoagulation, 1 atrial septal defect was referred for percutaneous closure, and 1 patient with constrictive pericarditis underwent surgery. Finally, CMR reported the absence of myocardial viability in the stenosis territory in 31/52(60%) of CA + patients, thus avoiding revascularization.

Funding

The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript

Competing interests

The authors declare that they have no competing interests.