Laurent Seppecher

Elie Bretin

Pierre Millien

Lorena Petrusca

Elisabeth Brusseau
email: elisabeth.brusseau@creatis.insa-lyon.fr

Reconstructing the spatial distribution of the relative shear modulus in quasi-static ultrasound elastography: plane stress analysis

Keywords: Ultrasound imaging, Quasi-static ultrasound elastography, Inverse problem, Shear modulus contrast

Quasi-static ultrasound elastography (QSUE) is an imaging technique that mainly provides axial strain maps of tissues, when the latter are subjected to compression. In this article, a method for reconstructing the relative shear modulus distribution within a linear elastic and isotropic medium, in QSUE, is introduced. More specifically, the plane stress inverse problem is considered. The proposed method is based on the variational formulation of the equilibrium equations and on the choice of adapted discretization spaces, and only requires displacement fields in the analyzed media to be determined.

Results from plane stress and 3D numerical simulations, as well as from phantom experiments, showed that the method is able to reconstruct the different regions within a medium, with shear modulus contrasts that unambiguously reveal whether inclusions are stiffer or softer than the surrounding material.

More specifically, for the plane stress simulations, inclusion-to-background modulus ratios were found to be very accurately estimated, with an error lower than 3%. For the 3D simulations, for which the plane stress conditions are no longer satisfied, these ratios were, as expected, less accurate with an error that remained lower than 10% for two of the three cases analyzed, but which was around 34% for the last case. Concerning the phantom experiments, a comparison with a shear wave elastography technique from a clinical ultrasound scanner was also made. Overall, the inclusion-to-background shear modulus ratios obtained with our approach were found to be closer to those given by the phantom manufacturer, than the ratios provided by the clinical system.

Introduction

Many diseases are known to be associated with changes in the mechanical properties of tissues [START_REF] Krouskop | Elastic moduli of breast and prostate tissues under compression[END_REF]; [START_REF] Mazza | The mechanical response of human liver and its relation to histology: an in vivo study[END_REF]). Accessing local values of elasticity-or viscoelasticity-related parameters can therefore provide useful information for diagnosis, and has led to the development of elastography techniques [START_REF] Ophir | Elastography: a quantitative method for imaging the elasticity of biological tissues[END_REF]). Various elastographic approaches have been described in the literature, all sharing the same underlying principle: imaging and analyzing the tissue response to a mechanical perturbation, to extract a parameter of interest. Depending on the imaging modality employed (e.g., ultrasound imaging, magnetic resonance imaging), the type of mechanical stress used (e.g., compression, harmonic excitation), and the parameter to be determined (e.g., axial strain, shear modulus), different techniques have been introduced. For a more detailed overview of the variety of developments achieved in elastography, we refer the reader to the following references [START_REF] Bamber | EFSUMB guidelines and recommendations on the clinical use of ultrasound elastography. Part 1: basic principles and technology[END_REF]; [START_REF] Mariappan | Magnetic resonance elastography: a review[END_REF]; [START_REF] Parker | Imaging the elastic properties of tissue: the 20 year perspective[END_REF]; [START_REF] Wang | Optical coherence elastography for tissue characterization: a review[END_REF]).

This article focuses more particularly on quasi-static ultrasound elastography (QSUE), where the deformation of tissues subjected to some compression is analyzed [START_REF] Varghese | Quasi-static ultrasound elastography[END_REF]). Although the information produced is generally limited to strain images, QSUE has proven to be a valuable technique for distinguishing regions that differ in stiffness within a medium, and its evaluation as a diagnostic tool has been the subject of many studies [START_REF] Cosgrove | EFSUMB guidelines and recommendations on the clinical use of ultrasound elastography. Part 2: clinical applications[END_REF]; [START_REF] Gong | Real-time elastography for the differentiation of benign and malignant breast lesions: a meta-analysis[END_REF]; [START_REF] Itoh | Breast disease: clinical application of US elastography for diagnosis[END_REF]; [START_REF] Lyshchik | Thyroid gland tumor diagnosis at US elastography[END_REF]).

In particular, different work comparing strain and shear wave elastography reported similar diagnostic performances for the two techniques [START_REF] Chang | Comparison of Shear-Wave and Strain Ultrasound Elastography in the Differentiation of Benign and Malignant Breast Lesions[END_REF]; [START_REF] Kim | Comparison of strain and shear wave elastography for qualitative and quantitative assessment of breast masses in the same population[END_REF]; [START_REF] Seo | Comparison and Combination of Strain and Shear Wave Elastography of Breast Masses for Differentiation of Benign and Malignant Lesions by Quantitative Assessment: Preliminary Study[END_REF]; [START_REF] Youk | Comparison of Strain and Shear Wave Elastography for the Differentiation of Benign From Malignant Breast Lesions, Combined With B-mode Ultrasonography: Qualitative and Quantitative Assessments[END_REF]). A major application of QSUE is the differentiation between benign and cancer tissues, which has led to the investigation of specific criteria. As an example, the strain ratio between a suspicious area and a reference region represents a straightforward indicator to compare these two tissues [START_REF] Lyshchik | Thyroid gland tumor diagnosis at US elastography[END_REF]; [START_REF] Cho | Sonoelastographic strain index for differentiation of benign and malignant nonpalpable breast masses[END_REF]; [START_REF] Waage | Endorectal elastography in the evaluation of rectal tumours[END_REF]). Such a ratio, but computed this time from the elastic modulus estimates, has also been reported to be helpful in distinguishing benign from malignant lesions in shear wave elastography [START_REF] Berg | Shear-wave elastography improves the specificity of breast US: the BE1 multinational study of 939 masses[END_REF]; [START_REF] Au | Diagnostic performance of quantitative shear wave elastography in the evaluation of solid breast masses: determination of the most discriminatory parameter[END_REF]). In QSUE, however, the strain ratio only partially reflects the contrast in the elastic modulus between the selected regions, as the stress distribution is not uniform within the scanned area.

Generally speaking, the inverse problem of determining the elastic modulus or the stiffness contrast within a medium has been widely studied [START_REF] Doyley | Model-based elastography: a survey of approaches to the inverse elasticity problem[END_REF]; [START_REF] Ammari | Stability analysis for magnetic resonance elastography[END_REF]; [START_REF] Widlak | Stability in the linearized problem of quantitative elastography[END_REF]; [START_REF] Barbone | Elastic modulus imaging: some exact solutions of the compressible elastography inverse problem[END_REF]). The main difference between solving this problem in dynamic and quasi-static elastography is that the acceleration term is null in the second case. Among the methods used in QSUE is the direct inversion, as proposed by Nitta and Shiina to recover the spatial distribution of the Young's modulus E in a 3D body [START_REF] Nitta | A method of tissue elasticity estimation based on threedimensional displacement vector[END_REF]). The method consists of substituting, in the equilibrium equations, the stress terms using the material constitutive law. Assuming that the Poisson's ratio is known and constant throughout the medium, this results in equations in which the unknowns are the spatial derivatives of E, divided by E. With this method, modulus values (in kPa) can be accessed, provided that additional information (e.g., the surface Young's modulus) is known. In a previous study, [START_REF] Sumi | Estimation of shear modulus distribution in soft tissue from strain distribution[END_REF] similarly investigated the reconstruction of the shear modulus µ for a linear elastic and isotropic material through 1D and 2D approaches.

As no initial information on modulus values is available a priori for biological tissues, the authors suggested positioning some material with known mechanical properties on the surface of the region to be examined, so as to provide a reference for the method. Because of its simplicity, direct inversion is an attractive method. Nevertheless, it requires differentiating the displacement but also the strain fields, which will negatively impact the results if the displacements are initially affected by noise.

Iterative methods have also been developed for the estimation of the material constitutive law parameters. They are based on the formulation of the inverse problem as an optimization problem in which the modulus values are iteratively varied until minimizing an error between the displacements or strains determined by solving the forward problem and those measured experimentally [START_REF] Doyley | Evaluation of an iterative reconstruction method for quantitative elastography[END_REF]; [START_REF] Baldewsing | Assessment of vulnerable plaque composition by matching the deforma-tion of a parametric plaque model to measured plaque deformation[END_REF][START_REF] Baldewsing | Young's modulus reconstruction of vulnerable atherosclerotic plaque components using deformable curves[END_REF]). Solving the forward problem requires knowledge of the boundary conditions, which makes this method difficult to use for clinical applications. Moreover, the computational cost associated with this approach can be heavy, as a solution of the forward problem needs to be computed at each iteration.

Recently, preliminary results have been reported using machine learning [START_REF] Hoerig | An information-based machine learning approach to elasticity imaging[END_REF][START_REF] Hoerig | Data-driven elasticity imaging using cartesian neural network constitutive models and the autoprogressive method[END_REF]). This field, which can be applied in countless areas, is also employed in numerous applications in medical imaging. In Ho-erig et al. (2017Ho-erig et al. ( , 2019)), a neural network-based approach is used to learn the stress-strain relationship of the material, and the mechanical parameters of choice are then deduced from the estimated stresses and strains. However, this technique also involves finite element modeling of the experiment and requires a set of force-displacement measurements as input data.

The present study focuses on the shear modulus contrast reconstruction of linear elastic and isotropic media in quasi-static ultrasound elastography. Recently, mathematical developments for analyzing the inverse elasticity problem were performed, and some results presented [START_REF] Ammari | A direct linear inversion for discontinuous elastic parameters recovery from internal displacement information only[END_REF]; [START_REF] Brusseau | Reconstructing the shear modulus contrast of linear elastic and isotropic media in quasi-static ultrasound elastography[END_REF]). In this paper, the plane stress inverse problem is more particularly investigated. The proposed method is based on the variational formulation of the equilibrium equations and on the use of a suitable finite element discretization technique, and requires only the displacement fields within the examined media to be determined beforehand. This article is organized as follows. In the next section, the relative shear modulus reconstruction method is described in detail, as well as the media used for the method assessment and the ultrasound data acquisition.

Results obtained from numerical simulations and phantom experiments are then presented and discussed. In particular, a comparison is made with a shear wave elastography technique from a clinical ultrasound scanner for the experimental data. Concluding remarks are provided in the last section.

Methods

This study focuses on the reconstruction of shear modulus maps up to a multiplicative constant, in linear elastic and isotropic media, from displacement fields obtained in quasi-static ultrasound elastography. A main difficulty associated with this inverse problem is that, under compression, biological tissues undergo 3D deformation, whereas the data generally available in ultrasound imaging are 2D images, giving access to in-plane motion only.

To overcome this problem, we assume that the plane stress conditions are satisfied, conditions that have already been chosen in various other elastography studies, such as in [START_REF] Seidl | The coupled adjoint-state equation in forward and inverse linear elasticity: Incompressible plane stress[END_REF] and in [START_REF] Karimi | A novel fast full inversion based breast ultrasound elastography technique[END_REF]. It should be emphasized that neither plane stress nor plane strain makes it possible to exactly describe the medium deformation when it is compressed with the ultrasound transducer. However, the choice of plane stress versus plane strain was made here, as the displacement fields measured experimentally were found to be closer to those obtained in the plane stress case, when modeling the corresponding experiments using a finite element method-based simulation software (COMSOL Multiphysics, COMSOL AB, Stockholm, Sweden, www.comsol.com). The method developed is fully described below.

Relative shear modulus map reconstruction

Let us consider a linear elastic and isotropic medium. The constitutive equation of such a medium is:

σ ij = 2µε ij + λδ ij ε kk (1)
with λ and µ the Lamé parameters, σ ij and ε ij the components of the stress and strain tensors, σ and E, respectively, δ ij the Kronecker delta, and ε kk , the trace of E. We assume that a displacement field u or a sequence of displacement fields u 1 , . . . , u n , is measured inside this medium. We also assume that these fields result from a quasi-static elastography experiment and that they all satisfy the same system of equilibrium equations. Neglecting the body forces, these equations write

∇ • σ = 0, (2)
and combining (1) and (2) gives

∇ • 2µE(u) + ∇(λ∇ • u) = 0, (3)
with E(u) := ∇u + (∇u) ⊤ )/2, the strain tensor, and ∇• and ∇, the divergence and gradient operators, respectively. Note that if u is known, then

(3) is a linear and homogeneous system of three equations with respect to the unknown pair (λ, µ). As a homogeneous system, (3) can be solved only up to a global multiplicative constant, additional scalar information being necessary for the Lamé parameters to be determined.

Under the plane stress conditions, the three stress tensor components σ xz , σ yz , and σ zz are equal to 0. Consequently, (3) is reduced to two equations and the two strain tensor components, ε xz and ε yz , are null as well. Moreover, σ zz = 0 gives:

2µε zz + λ(ε xx + ε yy + ε zz ) = 0, (4)
i.e.,

ε zz = - λ 2µ + λ (ε xx + ε yy ), (5)
and

∇ • u = 2µ 2µ + λ (ε xx + ε yy ), (6)
which leads to

λ∇ • u = 2µλ 2µ + λ (ε xx + ε yy ). ( 7
)
At this step, we can note that estimating displacements in the imaging plane (u 2D ) will provide all the information needed for the inverse problem to be solved. Furthermore, this method is intended for the examination of biological tissues, which are characterized by values of λ much higher than those of the shear modulus. With λ ≫ µ, (3) and ( 7) lead to

∇ • µE(u 2D ) + ∇(µ∇ • u 2D ) ≈ 0, (8)
and reconstructing the spatial distribution of the relative shear modulus will finally be performed by considering the following homogeneous system of two equations

∇ • µE(u 2D ) + ∇(µ∇ • u 2D ) = 0, (9)
with µ(x, y), the only unknown and u 2D (x, y), the 2D displacement estimated from ultrasound images. For simplification purposes, u 2D will be denoted u in the remainder of the article.

General principle

Let us consider a 2D domain Ω, which can be any region of interest in the 2D image, within which the displacement field u has been estimated.

As the unknown elastic parameter µ can be discontinuous, we look for it as an element of the parameter space M := L 2 (Ω). Moreover, a variational approach is used here, which consists of choosing a Hilbert space of test functions, to give a weak formulation of the equilibrium equations ( 9). As no boundary information is available, we propose to use a class of functions that vanish at this boundary. We therefore introduce V := H 1 0 (Ω, R 2 ) that is a classic Sobolev space defined as

V := v ∈ L 2 Ω, R 2 | ∇v ∈ L 2 Ω, R 2×2 , v| ∂Ω = 0 . ( 10
)
Multiplying ( 9) by v ∈ V and integrating by parts lead to

Ω µ(E(u) : E(v) + (∇ • u)(∇ • v)) = 0 ∀v ∈ V, (11)
with

E(u) : E(v) := ij E(u) ij E(v) ij .
To build a finite dimensional system of equations, we approach the two function spaces M and V by finite dimensional discretization subspaces M h ⊂ M and V h ⊂ V , where h > 0 is a parameter of discretization. Denoting

e M h 1 , . . . e M h dim M h a basis of M h , and e V h 1 , . . . e V h dim V h a basis of V h , we de- compose any function m ∈ M h as m = j m j e M h j , (12)
and from (11), we deduce the following finite dimensional system of equations

j µ j Ω e M h j (E(u) : E(e V h i ) + (∇ • u)(∇ • e V h i )) = 0 ∀i ∈ {1, . . . , dim V h }. (13)
This is a homogeneous linear system of dim V h equations involving dim M h unknowns. Denoting the real vector µ := (µ 1 , . . . , µ dim M h ) ⊤ , this system is written in a matrix form as

Aµ = 0, ( 14
)
where A is a large sparse matrix, given by

A ij := Ω e M h j (E(u) : E(e V h i ) + (∇ • u)(∇ • e V h i )). ( 15
)
The problem is now reduced to finding non-zero solutions of ( 14), which is equivalent to finding the null space of the matrix A. As this problem may not have non-zero solutions, the determination of the relative shear modulus spatial distribution can be reformulated as a minimization problem, such as

minimize µ ∥Aµ∥ 2 2 s.t. µ k = 1, ∀k ∈ K (16)
with, here, the constraint that the value of µ k is equal to 1, ∀k ∈ K, with K the set of indices corresponding to any region running along the image borders, and assumed to be part of the background. In practice, with our data, a region of interest will be selected within which the reconstruction will be performed. All indices outside this region will belong to K.

Determining the null space of the matrix A is equivalent to determining the null space of A ⊤ A, which is a symmetric square matrix of smaller size than A as soon as the system is overdetermined. Thus, in this work, the problem solved is finally

minimize µ A ⊤ Aµ 2 2 s.t. µ k = 1, ∀k ∈ K. ( 17
)

Use of multiple data

For multiple data, that is, when a sequence of displacement fields u 1 , . . . , u n is available, it is very easy to take these different fields into account. For each displacement field u ℓ , we build the corresponding sparse system defining the matrix A ℓ , as described above ( 15). The multiple data problem is then formulated as

minimize µ A ⊤ Aµ 2 2 s.t. µ k = 1, ∀k ∈ K with A :=      A 1 . . . A n      . ( 18
)

Honeycomb space discretization

The choice of a pair of discretization spaces M h and V h is crucial, and is certainly a key question for this inversion. We found that a very efficient discretization technique with excellent numerical stability is the honeycomb space discretization.

The domain Ω ⊂ R 2 is covered by a hexagonal honeycomb tiling of edge size h > 0, as illustrated in Figure 1. Thus, Ω is decomposed as

Ω = N h k=1 Ω h k , (19)
where Ω h k are the open hexagons. We choose the parameter approximation space M h as the class P 0 {Ω h k } of functions that are constant in each hexagon. More precisely,

M h := µ ∈ L 2 (Ω)| µ| Ω k is constant for each k . (20) A canonical basis of M h is simply given by e M h k := 1 Ω k for k = 1, . . . , N h .
We now choose a corresponding test function subspace V h ⊂ V that connects the hexagons together. To do so, sub-discretization {τ h k } of Ω is performed using an adapted equilateral triangulation (Fig. 1). We define V h as a subset of the classic P 1 {τ h k }, R 2 finite element class. More precisely, for each intersection node p i of three adjacent hexagons, φ i is defined as the unique function of H 1 (Ω) that is linear in each triangle, and satisfies φ i (p i ) = 1 and cancels at any other nodes of the triangular sub-mesh (Fig. 2). We then define the vector-valued test functions

e V h ik (x, y) := φ i (x, y)   δ 1k δ 2k   k = 1, 2, (21)
for i such that p i is an intersection node of three hexagons. The space V h is the subspace of V generated by these functions.

V h := span e V h ik | i ∈ I, k ∈ {1, 2} , (22)
where I is the set of indices of all intersection nodes of three adjacent hexagons.

Finally, the method described constructs an approximation of the relative shear modulus in M h , as a piecewise constant function on the honeycomb discretization. In order to obtain a smoother result, we project the reconstruction on P 1 {τ h k } , the class of continuous piecewise linear functions on the triangulation. It should be noted that for all of the results presented later in this article, the edge size h of the honeycomb tiling was set at 0.7 mm.

This value was found to be adapted to the sampling of the data used.

Solution computation

Different efficient methods can be used to solve ( 17) or ( 18), and in this work, we used CVX (Grant andBoyd, 2014, 2008). For the results shown below, the computation time required for this step was only a few seconds on a standard laptop.

Displacement field estimation and regularization

The reconstruction method uses displacement fields as input data. The initial displacement fields are estimated using our method previously developed for strain imaging [START_REF] Brusseau | In vivo response to compression of 35 breast lesions observed with a two-dimensional locally regularized strain estimation method[END_REF]. The basic principle of this method is briefly recalled here. Additional information can be found in the cited paper.

Let us consider two radiofrequency ultrasound images, I 1 and I 2 , that are acquired before and after, or during, medium compression. The first image is partitioned into many regions of interest (2D ROIs), regularly spaced and of equal size, and for each one of these ROIs, its deformed replica is searched for in I 2 . To describe the ROI transformation between images, 2D translation and axial scaling are considered. For each ROI, these parameters are determined via the maximization of the correlation coefficient between the preand post-deformation regions. It should be noted that, with this approach, the axial strain ε can be directly deduced from the axial scaling factor α (ε = α -1), without requiring any derivative computation.

During experiments, rather than only two images, a sequence of radiofrequency frames is generally acquired, which can be used to estimate the required fields. In that case, medium-compression related parameters are computed for pairs of successive images, and the resulting information is combined to provide the final displacement maps.

An example of displacement field u obtained with this method is shown in Figure 3, resulting from a phantom experiment (CIRS model 049, Type III inclusion, corresponding to case #8 later in the article). The axial and lateral components are denoted u y and u x , respectively. We can clearly see a difference in quality between these two components, which is typical of ultrasound elastography measurements. The axial displacement appears to be very smooth and usable, whereas the lateral displacement needs to be regularized before being used. For a more in-depth analysis of this difference, it is also relevant to provide the images of the derivatives of u (Fig. 4), which are, furthermore, necessary for the construction of the matrix A.

We note that the inclusion is easy to locate in the fields corresponding to the derivatives of u y , contrary to the derivatives of the lateral displacement field u x , which are dominated by noise.

To improve the lateral displacement, both the axial and lateral components are used, while keeping in mind that the field that really needs to be modified is the lateral component. More precisely, the approach consists of minimizing the following energy J := J reg + J data,u , where

J reg (w) := Ω 2µ * (∥E(w)∥ 2 + (∇ • w) 2 ), (23)
is an elastic regularization term, and

J data,u (w) := 1 r x Ω (u x -w x ) 2 + 1 r y Ω (u y -w y ) 2 , ( 24
)
represents the discrepancy functional.

In this work, µ * is set at 1 and the parameters r x and r y associated with each component of u are adjusted accordingly. In particular, to preserve the axial field while allowing more important variations in the lateral field, the value of r y should be much lower than r x . From a numerical point of view, the minimum w of J is computed via finite element discretization, and only requires the resolution of a linear system. An example of results obtained with the fields shown in Figures 3 and4 is given in Figure 5, using 1/r x = 15 • 10 -2 , and 1/r y = 15. We observe that w x is a smooth version of u x , and now smooth enough to make the inclusion easily detectable on its derivatives.

Simulation and phantom description, ultrasound data acquisition

The method for mapping the relative stiffness within a medium was applied to simulated and experimental data.

Numerical simulations were performed using Comsol Multiphysics. Two kinds of situations were analyzed, considering the plane-stress (cases #1-3) and 3D (cases #4-6) problems. For the 3D problem, three parallelepipedshaped media were built, measuring 60 mm x 40 mm x 40 mm (length x width x height) and which all consist of a 10-mm-diameter spherical inclusion embedded in the middle of a homogeneous background, the different regions being made of a linear elastic and isotropic material. In one case (case #4), the inclusion is softer than the background, with a shear modulus of 4 kPa vs. 9 kPa for the surrounding material, whereas in the two other cases, the inclusions are stiffer, with shear moduli of 16 kPa (case #5, modulus ratio = 16/9) and 27 kPa (case #6, modulus ratio = 27/9). For all regions, the Lamé parameter λ was adjusted to obtain a Poisson's ratio of 0.495, a value typically used for the simulation of media in elastography studies [START_REF] Nayak | Principal strain vascular elastography: simulation and preliminary clinical evaluation[END_REF]; [START_REF] Poul | Fat and fibrosis as confounding cofactors in viscoelastic measurements of the liver[END_REF]; [START_REF] Thittai | On the advantages of imaging the axial-shear strain component of the total shear strain in breast tumors[END_REF]).

To get closer to the experimental conditions, the configuration of compression of the medium with the ultrasound probe was reproduced, resulting in only part of the medium top surface being subjected to displacement (Fig. 6). The probe was positioned above the inclusion and displaced vertically, downward. To determine whether the results could be influenced by the level of compression applied, three displacement values were considered, 0.4 mm, 0.8 mm and 1.2 mm, corresponding to 1%, 2% and 3% axial strain, respectively. The bottom surface was fixed, whereas the vertical ones were free to move. In this work, relative shear modulus maps are reconstructed from the 2D displacements estimated in the ultrasound imaging plane. With the simulations performed, the axial and lateral displacements retained were those of the vertical plane crossing the center of the spherical inclusion, as illustrated in Figure 6. Finally, the same media, but this time considering the plane stress problem, were also simulated (cases #1-3). In all cases, no noise was added to the data. When reconstructing the relative shear modulus maps, the estimated inclusion-to-background shear modulus ratios are therefore expected to be close to 0.44 (cases #1 and #4), 1.78 (cases #2 and #5), and to 3 (cases #3 and #6), whatever the applied compression.

Experimental tests were carried out with two CIRS phantoms (Computerized Imaging Reference Systems, Norfolk, VA, USA), the models 049 and 059. The CIRS model 049 (Elasticity QA) consists of a parallelepiped-shaped medium, within which several spherical inclusions are embedded. The inclusion position, size, and Young's modulus as well as the modulus of the background material are specifications provided by the manufacturer. For this phantom, two different regions were scanned, both containing a 10-mm-diameter spherical inclusion. In the first region, the inclusion of Young's modulus of 13 kPa (Type II inclusion) is softer than the surrounding medium with a modulus of 26.5 kPa (case #7), whereas in the other case, the 47-kPa inclusion (Type III inclusion) is stiffer than the background (case #8). The second CIRS phantom (model 059 or breast elastography phantom) is characterized by an overall shape that mimics the breast of a patient in the supine position. This phantom contains several spherical inclusions that are stiffer than the surrounding material. The Young's moduli of the inclusions and the background are 43.3 kPa and 13 kPa, respectively. Two different areas were also scanned, one showing the presence of a single inclusion (case #9), the other, two inclusions one below the other (case #10). For each region examined, a typical quasi-static elastography experiment was performed, i.e., the operator cautiously compressed the medium with the hand-held probe while radiofrequency images were acquired. Data were collected using an Ultrasonix ultrasound scanner (Ultrasonix Medical Corporation, Richmond, BC, Canada), equipped with an L14-5W/60 linear array transducer. The sampling frequency was 40 MHz. For these phantom data, the inclusion-tobackground shear modulus ratios estimated from the reconstructed maps, are expected to be close to 0.49 (cases #7), 1.77 (cases #8), and to 3.33 (cases #9 and #10).

Analysis of the results and comparison with a shear wave elastography technique

For all the media described above, the relative shear modulus maps were reconstructed and the inclusion-to-background stiffness ratios computed. These ratios were obtained by selecting circular regions (ROIs) of the same size inside the inclusion and the background (excluding the transition zone), and by computing the ratio of their mean values. For each case, two ratios were determined, R 1 and R 2 , from two different regions in the background. As is done in clinical practice, the ROI selection was performed manually, which inevitably has an impact on the results. For that reason, a third ratio was reported, R tb , considering this time the total background. As the media analyzed in this study consist of inclusions embedded in homogeneous background materials, R tb appears particularly adapted for the method assessment.

To complete the analysis of the experimental results, a comparison with a shear wave elastography technique was also made, using an Aixplorer ul-trasound scanner (SuperSonic Imagine, Aix-en-Provence, France) equipped with an SL15-4 linear array transducer [START_REF] Bercoff | Supersonic shear imaging: a new technique for soft tissue elasticity mapping[END_REF]). This clinical system provides images of the Young's modulus of tissues computed as three times the shear modulus (3µ), considering that the media investigated are linear elastic, isotropic, and (quasi-) incompressible [START_REF] Tanter | Quantitative assessment of breast lesion viscoelasticity: Initial clinical results using supersonic shear imaging[END_REF]).

Any Young's modulus ratio is therefore equal to the shear modulus ratio, which allows a direct comparison with the results from our technique. The selection of circular ROIs was here also performed manually using the tools available on this ultrasound system. These tools do not allow to extract information concerning the whole background, which prevents the ratio R tb from being determined.

Results and Discussion

Results from the numerical simulations and experiments with phantoms are detailed below. Relative shear modulus images are shown for all the cases described in the previous section. It should be specified that, as the reconstructed maps are obtained up to a multiplicative constant, we chose to divide them by their minimum value before display. These maps are, therefore, without unit.

Simulation results

For each case, three relative shear modulus maps were reconstructed from the three different data sets corresponding to distinct levels of compression.

To quantify the variations between the reconstructed maps, the absolute difference was computed. In all cases, this difference remains locally lower than 10 -8 , which explains that no visual differences between the three maps can be observed and that the inclusion-to-background modulus ratios do not vary. For those reasons, in the following, results will be presented without making any distinction regarding the applied compression.

The reconstructed maps obtained with the plane stress and 3D simulations are displayed in Figures 7 and8, respectively. In all cases analyzed, the inclusions are easily detectable in the images, and they appear stiffer or softer than the background in agreement with the mechanical properties chosen for the different media. However, some differences can be observed between

Figures 7 and8. The shape of the inclusions, first, is well preserved with the plane stress simulations (Fig. 7), whereas in Figure 8, a weak deformation of these inclusions (with a clearer illustration in Fig. 8a) can be noted.

Then, the different regions appear relatively homogeneous in the plane stress results whereas more variations are visible for the 3D simulations. To provide a more quantitative assessment of the results, inclusion-to-background shear modulus ratios are reported here (Table 1). For the plane stress simulations, these ratios are R 1 = 0.43, R 2 = 0.43 and R tb = 0.44 for case #1, R 1 = 1.81, R 2 = 1.81 and R tb = 1.81 for case #2, and R 1 = 3.11, R 2 = 3.11 and R tb = 3.08 for case #3, and are in perfect agreement with the mechanical description of the chosen numerical models. The relative error expressed as a percentage, also called the percentage error and computed as 100 × |1-R tb /R theo | with R theo the theoretical ratio, was determined and found to be lower than 3% in the three cases examined. For the 3D simulations, the ratios are R 1 = 0.55, R 2 = 0.58 and R tb = 0.60 for case #4, R 1 = 1.60, R 2 = 1.69 and R tb = 1.68 for case #5, and R 1 = 2.62, R 2 = 2.79 and R tb = 2.74 for case #6. The different values of ratios for each case illustrate the stronger variations in the background observed in the reconstructed maps. However, the difference in the values remains limited.

Moreover, compared with the plane stress simulation results, these ones show a higher deviation from the theoretical values, which remains lower than 10%

for cases #5 and #6, but reaches around 34% for case #4. Such a higher deviation could be expected, as in this work, a plane-stress-based approach was developed for reconstructing maps of the relative shear modulus within media. When plane stress conditions are no longer satisfied, as with the 3D simulations, errors in the determined ratios occur. However, the proposed method allows us to clearly identify the different regions within a medium, and despite a reduction in the elastic contrast observed in cases #4-6, this method should remain useful for elastography purposes.

Experimental results

Relative shear modulus maps obtained with the experimental data are presented in Figure 9. As with the simulation results, the inclusions can be easily identified in the elastograms and two inclusions, even spatially close, can remain clearly distinguishable, as can be seen in Figure 9d. For each case, inclusion-to-background modulus ratios are also provided here (Table 2), with an illustration of ROI selection in Figure 11. For the model 049, the following ratios were obtained, R 1 = 0.50, R 2 = 0.52 and R tb = 0.52 for case #7, and R 1 = 1.73, R 2 = 1.77 and R tb = 1.76 for case #8. These values are very close to those expected, 0.49 and 1.77, respectively. For the breast phantom (model 059), however, the ratios were found to be lower than the target value, 3.33. Indeed, for the area containing a single inclusion (case #9), R 1 = 2.51, R 2 = 2.69 and R tb = 2.60, and for the two inclusion case (#10), these ratios were: R 1 = 2.48, R 2 = 2.70 and R tb = 2.35 for the top inclusion, and R 1 = 2.46, R 2 = 2.68 and R tb = 2.49 for the bottom one.

For a more complete analysis of the results, a comparison with a shear wave elastography technique, the one available in the Aixplorer ultrasound scanner, was conducted. Elastograms obtained with this technique are presented in Figure 10, along with an illustration of ROI selection for modulus ratio computation in Figure 11. For the CIRS model 049, the modulus ratios here were found to be 0.49 and 0.50 for case #7 and 1.38 and 1.59 for case #8.

For the breast phantom, these values were 2.11 and 2.19 for the region containing a single inclusion (case #9), and for the other region 2.01 and 2.33

for the top inclusion and 2.30 and 2.34 for the bottom one (case #10). It is interesting to note that except for case #7, the ratios provided by this technique are further from the expected values than those obtained with our method.

The relative shear modulus maps presented in this article are the very first ones reconstructed with the proposed method. The latter is able to clearly reveal regions differing in stiffness within a medium, and the results were found not to be affected by a variation in the level of compression applied.

For all media analyzed, inclusions appeared stiffer or softer than the background in agreement with the mechanical properties chosen for the numerical models or the Young's modulus values provided by the manufacturer for the phantoms. Moreover, the inclusion-to-background modulus ratios obtained with the plane stress simulations were very close to the actual values. For the model 049 also, these ratios were in perfect agreement with the target values, whereas for the 3D simulations and experimental results with the breast phantom, some differences were observed between estimated and expected ratios. In this work, to overcome the lack of 3D data, the plane stress case was used to solve the inverse problem, and as previously discussed, this

does not make it possible to describe exactly the biological medium deformation when compressed with the ultrasound probe, which is inherently a 3D

problem. When compression-induced displacement fields deviate from those that would be obtained under plane stress conditions, errors inevitably occur, as clearly illustrated with the 3D simulations. The choice between plane stress and plane strain was made here, as the displacement fields measured experimentally were found to be closer to those obtained in the plane stress case, when performing a modeling of the experiments using COMSOL Multiphysics. These observations, however, were made from a few experiments using only two different phantoms. Therefore, it will be of interest to also investigate the plane strain-based reconstruction method and to conduct a thorough comparison between the plane stress-and plane strain-based approaches, as well as to go deeper into the analysis of the 3D problem. Such a study is beyond the scope of this paper, but will be part of future work.

For tests with phantoms, a comparison of the results with those from a commercial ultrasound scanner (Aixplorer) was made. This comparison was performed to better appreciate the results obtained with the proposed method and to determine whether this method deserves to be further developed. The Aixplorer ultrasound scanner, which comprises a shear wave-based elastography approach (supersonic shear imaging) was used, as the only clin- Contrary to shear wave techniques, which give access to modulus values (in kPa), our reconstruction method only provides maps of the relative shear modulus of the examined media. However, such maps enable computation of the stiffness ratio between regions, which is also a criterion of interest for diagnosis. Indeed, when modifications occur locally within tissues, an increase in stiffness at that location will directly result in a higher ratio, when compared with a reference region. Many studies have shown that using qualitative and quantitative parameters from shear wave elastography, among which the modulus ratio, can be helpful for distinguishing between tissues [START_REF] Berg | Shear-wave elastography improves the specificity of breast US: the BE1 multinational study of 939 masses[END_REF]; [START_REF] Au | Diagnostic performance of quantitative shear wave elastography in the evaluation of solid breast masses: determination of the most discriminatory parameter[END_REF]; [START_REF] Brunel | Focal nodular hyperplasia and hepatocellular adenoma: The value of shear wave elastography for differential diagnosis[END_REF]). Among the other parameters studied are the mean and maximum elastic moduli, both directly providing information on the mechanical properties of the selected tissue area. These three parameters have been widely investigated, notably for the assessment of breast lesions. Results revealed that the mean modulus, maximum modulus and modulus ratio are all significantly higher for malignancies than for benign lesions, the best performing parameter varying with the studies [START_REF] Berg | Shear-wave elastography improves the specificity of breast US: the BE1 multinational study of 939 masses[END_REF] The developments presented are very recent, and additional work will be necessary to fully assess this approach. Although illustrating simplified situations, this first evaluation of the method with simulated and phantom data remains an essential step, keeping in mind that this method is developed for diagnostic purposes. Therefore, many other tests including in vivo biological tissues will need to be conducted, which can potentially lead to method modifications.

The proposed method is based on the variational formulation of the equilibrium equations. This approach avoids computing the spatial derivatives of the stress components. Unlike some methods discussed in the introduction, it therefore allows us to get rid of displacement second-order derivatives, which is particularly interesting in ultrasound elastography, as the lateral displacement fields estimated are typically more affected by noise than the axial fields. Moreover, since no spatial derivative of the stress components, and consequently of the Lamé parameters, needs to be computed, this approach also allows us to get rid of the medium local homogeneity assumption, which can help improve the results as this assumption is not adapted to the examination of heterogeneous media like biological tissues [START_REF] Scott | Artificial neural networks for magnetic resonance elastography stiffness estimation in inhomogeneous materials[END_REF]).

The developments described in this article could therefore benefit other elastography techniques that use the local homogeneity assumption.

Conclusions

In this paper, an inversion method to reconstruct the stiffness contrast in QSUE has been presented, along with preliminary results from simulations and phantom experiments. This method is able to recover the different regions within a medium, with shear modulus ratios that clearly reveal the stiff and soft inclusions. Moreover, a comparison with a shear wave elastography technique available in a clinical ultrasound scanner showed that the inclusion-to-background shear modulus ratios obtained with the proposed approach were closer to the expected values. Future work will focus on a more thorough assessment of this approach, including in vivo biological tissues.

shear wave elastography technique were obtained using an Aixplorer ultrasound scanner from the Centre Léon Bérard (Lyon, France). Finally, part of Axes are in mm. Tables Table 1: Inclusion-to-background shear modulus ratios: results from plane stress and 3D simulations.

Case

  ical system we have access to. Nevertheless, the wide use of this scanner in clinical practice and studies makes the given comparison of interest. The Aixplorer provides values of the local shear wave speed or of the Young's modulus, the latter being, as explained earlier, computed as three times the shear modulus considering that the media examined are linear elastic, isotropic, and (quasi-) incompressible. Consequently, any Young's modulus ratio provided by this system is equal to the shear modulus ratio. Moreover, when examining biological tissues with shear wave elastography techniques, it is well known that there is a frequency dependence of the modulus measurements, linked to the viscoelastic properties of the medium. For purely elastic materials, the modulus value remains the same, whatever the frequency of the shear wave. The phantoms used in this study and manufactured by CIRS are made of Zerdine hydrogel, described as being elastic, as shown also with some specific measurements and analysis performed in different studies[START_REF] Andoh | Multifrequency magnetic resonance elastography for elasticity quantitation and optimal tissue discrimination: A two-platform liver fibrosis mimicking phantom study[END_REF];[START_REF] Oudry | Comparison of four different techniques to evaluate the elastic properties of phantom in elastography: is there a gold standard?[END_REF]). These different elements allow us to conduct a direct comparison of the ratios from the two techniques.

  this work was supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).

Figure Captions Figure 1 :

 Captions1 Figure Captions

Figure 2 :

 2 Figure 2: Graph of the test function φ i .

Figure 3 :

 3 Figure 3: Example of a displacement field before regularization, obtained from a phantom experiment (CIRS model 049, Type III inclusion). (a) axial displacement, u y (in mm), and (b) lateral displacement, u x (in mm). In (a), axes are in millimeters.

Figure 4 :

 4 Figure 4: Derivatives of the displacement components before regularization, obtained from the fields displayed in Figure 3. (a) axial strain (∂ y u y ), (b) lateral-shear strain (∂ y u x ), (c) axial-shear strain (∂ x u y ), and (d) lateral strain (∂ x u x ).

Figure 5 :

 5 Figure 5: Displacement components after regularization and their derivatives for the CIRS model 049, Type III inclusion. (a) axial displacement, w y (in mm), (b) lateral displacement, w x (in mm), (c) axial strain (∂ y w y ), (d) lateral-shear strain (∂ y w x ), (e) axial-shear strain (∂ x w y ), and (f) lateral strain (∂ x w x ). In (a), axes are in mm.

Figure 6 :

 6 Figure 6: 3D simulations -Illustration of the configuration of a medium compressed with the ultrasound transducer. The dashed lines indicate the positions of the axial and lateral displacement images selected for relative shear modulus reconstruction.

Figure 7 :

 7 Figure 7: Relative shear modulus maps obtained with the plane stress simulations, for media characterized by inclusion-to-background shear mod-

Figure 8 :

 8 Figure 8: Relative shear modulus maps obtained with the 3D simulations, for media characterized by inclusion-to-background shear modulus ratios of (a) 4/9 (case #4), (b) 16/9 (case #5), and (c) 27/9 (case #6).

Figure 9 :

 9 Figure 9: Experimental results obtained with the CIRS phantoms. (a) Model 049, Type II inclusion (case #7), (b) model 049, Type III inclusion (case #8), (c) model 059, single inclusion (case #9), and (d) model 059, two inclusions (case #10). Axes are in mm.

Figure 10 :

 10 Figure 10: Young's modulus images (in kPa) obtained with the Aixplorer ultrasound scanner. (a) Model 049, Type II inclusion (case #7), (b) model 049, Type III inclusion (case #8), (c) model 059, single inclusion (case #9), and (d) model 059, two inclusions (case #10).

Figure 11 :

 11 Figure 11: Illustration of region selection for modulus ratio computation with the scanned area containing a single inclusion in the breast elastography phantom (case #9). Regions selected in the map reconstructed with (a) our method, and (b) and (c) the Aixplorer.

Figure 2 :

 2 Figure 2: Graph of the test function φ i .

Figure 3 :Figure 4 :Figure 5 :

 345 Figure 3: Example of a displacement field before regularization, obtained from a phantom experiment (CIRS model 049, Type III inclusion). (a) axial displacement, u y (in mm), and (b) lateral displacement, u x (in mm). In (a), axes are in millimeters.

Figure 6 :

 6 Figure 6: 3D simulations -Illustration of the configuration of a medium compressed with the ultrasound transducer. The dashed lines indicate the positions of the axial and lateral displacement images selected for relative shear modulus reconstruction.

Figure 7 :Figure 8 :

 78 Figure 7: Relative shear modulus maps obtained with the plane stress simulations, for media characterized by inclusion-to-background shear modulus ratios of (a) 4/9 (case #1), (b) 16/9 (case #2), and (c) 27/9 (case #3). Axes are in mm.

Figure 9 :

 9 Figure 9: Experimental results obtained with the CIRS phantoms. (a) Model 049, Type II inclusion (case #7), (b) model 049, Type III inclusion (case #8), (c) model 059, single inclusion (case #9), and (d) model 059, two inclusions (case #10). Axes are in mm.

Table 2 :

 2 Inclusion-to-background shear modulus ratios: experimental results and comparison with the Aixplorer ultrasound scanner.Figure 1: Honeycomb space discretization, with h the edge size (black), and adapted triangular sub-mesh (dashed blue).

			Target
		#	ratio	R 1	R 2	R tb
		1	0.44	0.43	0.43	0.44
	Plane stress
		2	1.78	1.81	1.81	1.81
	simulations
		3	3.00	3.11	3.11	3.08
		4	0.44	0.55	0.58	0.60
	3D simulations	5	1.78	1.60	1.69	1.68
		6	3.00	2.62	2.79	2.74
			Target	Our method	Aixplorer
		Case #
			ratio	R 1	R 2	R tb	R 1	R 2
	CIRS 049	7	0.49	0.50	0.52	0.52	0.49	0.50
	CIRS 049	8	1.77	1.73	1.77	1.76	1.38	1.59
	CIRS 059	9	3.33	2.51	2.69	2.60	2.11	2.19
	CIRS 059 (top)	10	3.33	2.48	2.70	2.35	2.01	2.33
	CIRS 059 (bottom)	10	3.33	2.46	2.68	2.49	2.30	2.34

Acknowledgments

The authors thank H. Ammari for fruitful discussions. The quasi-static ultrasound elastography data were acquired with equipment provided by PI-LOT (INSA-Lyon, France), and the images and modulus ratios from the

Data availability statement

Because of valorization strategies under study, the source code cannot currently be made available. The raw phantom data can be accessed at: https://www.creatis.insa-lyon.fr/ ~brusseau/Elastography_UMB2022

Conflict of interest disclosure

The authors declare no conflicts of interest.


Orlando Santoro

Lorenzo Piola

Karl Mc Cabe

Olivier Lhost

K Den Dauw

Alvaro Fernandez

Alexandre Welle

Laurent Maron
email: laurent.maron@irsamc.ups-tlse.fr

J.-F Carpentier

Evgueni Kirillov
email: evgueni.kirillov@univ-rennes1.fr

Katty Den Dauw

Jean-François Carpentier
email: jean-francois.carpentier@univ-rennes1.fr

François Carpentier

Comonomer-controlled synthesis of long-chain branched (LCB)-polyethylene

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

Introduction

During the last three decades, metallocene-catalyzed ethylene/diene copolymerization has emerged as an efficient method towards post-functionalized- [START_REF] Marathe | Regioselective Copolymerization of 5-Vinyl-2-norbornene with Ethylene Using Zirconocene-Methylaluminoxane Catalysts: A Facile Route to Functional Polyolefins[END_REF][START_REF] Chung | Synthesis and Functionalization of Unsaturated Polyethylene: Poly(ethylene-co-1,4-hexadiene)[END_REF][START_REF] Lee | Copolymerizations of olefins and dienes with homogeneous and heterogeneous catalysts[END_REF][START_REF] Hou | Double-Linear Insertion Mode of α,ω-Dienes Enabled by Thio-iminoquinoline Iron Catalyst[END_REF][START_REF] Li | Binuclear titanium complexes coordinated by rigid p-phenylene linked bis-β-carbonylenamine: Synthesis, structure, ethylene polymerization and copolymerization with 1,5-hexadiene[END_REF][START_REF] Kateb | Synthesis of 1hexene/1,7-octadiene copolymers using coordination polymerization and postfunctionalization with triethoxysilane[END_REF] and long-chain branched (LCB) polyethylene (PE) [START_REF] Mehta | Long-Chain Branched polymers and their production[END_REF][START_REF] Kokko | Long-chain-branched polyethene by the copolymerization of ethene and non-conjugated α, ω-dienes[END_REF][START_REF] Pietikäinem | Copolymerization of ethylene and nonconjugated dienes with Cp 2 ZrCl 2 /MAO catalyst system[END_REF][START_REF] Pietikäinem | Copolymerization of ethylene and nonconjugated dienes with Cp 2 ZrCl 2 /MAO catalyst system: effect of polymerization temperature on the copolymer structure[END_REF][START_REF] Paavola | Propylene copolymerization with nonconjugated dienes and α-olefins using supported metallocene catalyst[END_REF][START_REF] Naga | Copolymerization of ethylene and 1,7-octadiene, 1,9-decadiene with zirconocene catalysts[END_REF][START_REF] Williamson | Alternating copolymers of ethylene and diolefins containing pendent functional groups[END_REF][START_REF] Sarzotti | Microstructural Characterization of Molecular Weight Fractions of Ethylene/1,7-Octadiene Copolymers Made with a Constrained Geometry Catalyst[END_REF][START_REF] Mehdiabadi | Production of Ethylene/α-Olefin/1,9-Decadiene Copolymers with Complex Microstructures Using a Two-Stage Polymerization Process[END_REF][START_REF] Karimkhani | Revisiting the long-chain branch formation mechanism in metallocene catalyzed polyethylenes[END_REF][START_REF] Liu | Assessing 1,9-Decadiene/Ethylene Copolymerization with Ziegler-Natta Catalyst to Access Long Chain-Branched Polyethylene[END_REF]. The latter aspect proved particularly interesting, since LCB-PEs most often exhibit significantly better processability than their linear counterparts, albeit retaining the excellent mechanical properties (i.e. impact resistance) typically found in metallocene-PE [START_REF] Liu | A Comprehensive Review on Controlled Synthesis of Long-Chain Branched Polyolefins: Part 1, Single Catalyst Systems[END_REF][START_REF] Lai | Elastic substantially linear olefin polymers[END_REF][START_REF] Brant | Long chain branched polymers and a process to make long chain branched polymers[END_REF][START_REF] Kim | Melt rheological and thermodynamic properties of polyethylene homopolymers and poly(ethylene/α-olefin) copolymers with respect to molecular composition and structure[END_REF]. In this scenario, we have recently identified Al-, Zn-and B-alkenyl species as suitable LCB promoters in the ethylene polymerization catalyzed by rac-{EBTHI}ZrCl 2 -and (nBuCp) 2 ZrCl 2 -based systems (Zr-1 and Zr-2, respectively, Chart 1), proposing the incorporation of its alkenyl moiety into the growing PE chain as a crucial step for LCB formation [START_REF] Santoro | Long-Chain Branched Polyethylene via Coordinative Tandem Insertion and Chain-Transfer Polymerization Using rac-{EBTHI}ZrCl 2 /MAO/Al-alkenyl Combinations: An Experimental and Theoretical Study[END_REF][START_REF] Santoro | Al-alkenyl-induced formation of long-chain branched polyethylene via coordinative tandem insertion and chain-transfer polymerization using (nBuCp) 2 ZrCl 2 /MAO systems: An experimental and theoretical study[END_REF][START_REF] Santoro | Group 12 and 13 metal-alkenyl promoted generation of long-chain branching in metallocene-based polyethylene[END_REF]. Herein, we disclose the ethylene copolymerization with two α,ω-dienes, namely 1,7-octadiene and 1-10-undecadiene, catalyzed by the previously investigated Zr-1 and Zr-2-based catalyst systems, both under homogeneous and heterogeneous (slurry) conditions. The rheological properties of the produced PEs as well as the difference between Al-alkenyl co-reactants and α,ω-dienes comonomer in the formation of LCB-PE are discussed. As for our prior studies, these zirconocene catalysts were chosen because of their opposite tendency to naturally form LCB as well as for their different ability to incorporate comonomers. Indeed, Zr-1 was shown to afford LCB-PEs by macromonomer incorporation along with displaying high comonomer incorporation ability. On the other hand, Zr-2 proved less prone to comonomer insertion and capable of producing, in the absence of a suitable LCB-promoted, essentially linear PEs [START_REF] Kokko | Influence of the catalyst and polymerization conditions on the long-chain branching of metallocene-catalyzed polyethenes[END_REF].

With respect to the nature of the α,ω-diene, the in house-prepared 1,10-undecadiene was employed for the tests under homogeneous conditions to favor the formation of branches over that of cycles embedded into the growing chain [26], thus avoiding complications in the characterization of the resulting materials and in the subsequent analysis of the results. Given that supported catalysts are largely employed in industrial PE production, it was decided to use the commercially available 1,7-octadiene to implement a more easily scalable (and economically viable) process.

Chart 1. Structure of the metallocenes used in this study: rac-{EBTHI}ZrCl 2 (Zr-1) and (nBuCp) 2 ZrCl 2 (Zr-2).

Results and Discussion

Ethylene polymerization in the presence of 1,10-undecadiene: homogeneous conditions

Complexes Zr-1 and Zr-2, activated by methylalumoxane (MAO), were tested in the ethylene polymerization under homogeneous conditions in the presence of 1,10-undecadiene (Table 1). For both catalyst systems, the presence of the diene had a positive effect on the productivity of the process [START_REF] Barranco-García | New Insight into the Comonomer Effect through NMR Analysis in Metallocene Catalysed Propene-co-1-Nonene Copolymers[END_REF]. However, while for Zr-2 the productivity progressively increased with [diene] 0 , a detrimental effect was observed with Zr-1 at comonomer concentrations higher than 0.5 mM. This was attributed to the formation of insoluble (allegedly cross-linked) PE hampering the diffusion of the monomer in the reaction medium (fouling). For Zr-1 a progressive drop of T m of the resulting polymer was observed upon increasing the amount of diene, while such values for the samples synthesized with Zr-2 (PE5PE8) were found in a very narrow range (133134 °C), regardless of the diene concentration.

None of the PEs isolated with Zr-1/MAO/diene systems, as well as PE7 (Table 1) could be analyzed by GPC due to their insolubility in dichlorobenzene at 135 °C. Such poor solubility was thought to be due to the formation of "H-type" cross-linked polymers (Chart 2) at this initial diene concentration, [Diene] 0 (vide infra) [START_REF]Gelation of the reaction medium and related mass-transfer issues occurred for[END_REF].

Chart 2. "H-type" cross-linked polyethylene.

In order to gain more information on the kinetics aspects of the LCB/cross-links formation, a polymerization test at shorter reaction time (3 instead of 15 min) was performed with the Zr-2/MAO system in the presence of 5 mM of diene (Table 1, entry 8). No vinyl moieties were detected in the 1 H NMR spectrum of the corresponding polymer (PE8, see the Supporting information, Figure S2), suggesting that all of the diene incorporated was involved in the formation of "H-type" branched structures/cross-links (Chart 2) [START_REF] Mehta | Long-Chain Branched polymers and their production[END_REF][START_REF]formation of cycles was excluded due to the size of the diene employed[END_REF]. The amount of unreacted diene was determined by 1 H NMR in the presence of an internal standard (2-methoxynaphtalene) in the concentrated quenching solution of the polymerization (see the Supporting information, Figure S3). Since <2% of unreacted diene was detected, it was assumed that nearly quantitative comonomer consumption occurred and that, given the short reaction time, the formation of LCB/cross-linking takes place in the early stages of the polymerization. The effect of [Diene] 0 on the rheological behavior of the samples was investigated. Samples PE1PE3 exhibited similar values of viscosity at low frequency and the enhancement of shear thinning upon increasing the comonomer concentration (Figure 1, top).

On the other hand, PE4 (prepared at [Diene] 0 = 5 mM) showed values of viscosity at low frequency ca. one order of magnitude higher than those of PE1PE3 and more pronounced shear thinning. These observations indicate that the extent of LCB/cross-linking is proportional to the amount of diene introduced; this is consistent with previous reports [START_REF] Stadler | Detecting very low levels of long-chain branching in metallocene-catalyzed polyethylenes[END_REF][START_REF] Takeh | Analytical rheology of metallocene-catalyzed polyethylenes[END_REF][START_REF] Stadler | Correlations between the Characteristic Rheological Quantities and Molecular Structure of Long-Chain Branched Metallocene Catalyzed Polyethylenes[END_REF]. Further evidence supporting this statement could be achieved by analyzing the van Gurp-Palmen plots [START_REF] Trinkle | Van Gurp-Palmen-plot: a way to characterize polydispersity of linear polymers[END_REF][START_REF] Trinkle | Van Gurp-Palmen Plot IIclassification of long chain branched polymers by their topology[END_REF] for PE1PE3 (Figure 1, bottom). In fact, all curves tend to phase angle values lower than 90 ° (90 ° is indicative of purely linear polymers, vide infra) [START_REF] Trinkle | Van Gurp-Palmen Plot IIclassification of long chain branched polymers by their topology[END_REF]; moreover, the inflection points observed for PE2 and PE3 are diagnostic for highly branched structures [START_REF] Stadler | Correlations between the Characteristic Rheological Quantities and Molecular Structure of Long-Chain Branched Metallocene Catalyzed Polyethylenes[END_REF][START_REF] Stadler | Correlation between the shape of viscosity functions and the molecular structure of long-chain branched polyethylenes[END_REF][START_REF] Stadler | On "modulus shift" and thermorheological complexity in polyolefins[END_REF][START_REF] Yan | Classification of thermorheolegical complexity for linear and branched polyolefins[END_REF]. It has to be noted that with Zr-1, a remarkable variation of the corresponding PE rheological properties was observed even for concentrations of diene as low as 0.5 mM, while a similar response was attained with the Zr-2-based system only at higher diene feed (5 mM).

This could be accounted to the different co-monomer incorporation ability of the two metallocenes, which is in the order Zr-1 >> Zr-2 [START_REF] Lehtinen | A comparison of (n-butCp) 2 ZrCl 2 and other simple metallocenes with bridged Et(Ind) 2 ZrCl 2 and Me 2 Si(Ind) 2 ZrCl 2 catalysts in ethene/propene copolymerization[END_REF].

Ethylene polymerization in the presence of 1,7-octadiene: heterogeneous conditions

In order to assess the industrial viability of the process, polymerization tests catalyzed by heterogeneous supp-Zr-1 and supp-Zr-2, synthesized from Zr-1 and Zr-2 according to reported procedures [START_REF] Cirriez | Dual Catalyst Composition[END_REF][START_REF] Giffin | Meso-and rac-[bis(3-phenyl-6-tert-butylinden-1-yl)dimethylsilyl]zirconium dichloride: Precatalysts for the production of differentiated polyethylene products with enhanced properties[END_REF], were conducted in the presence of 1,7-octadiene [44] as the LCB/cross-linking-promoter and by using triisobutylaluminum (TIBAL) as the scavenger (Table 2) [45]. The productivity of the supp-Zr-1-based system proved only slightly dependent on the diene concentration. On the other hand, while the polymers' T m , M n and dispersity values were also found in a very narrow range for [1,7-octadiene] 0 up to 300 ppm (Table 2, entries 15), a PE insoluble in dichlorobenzene at 135 °C was obtained upon increasing the diene concentration to 500 ppm (PE14, Table 2, entry 6) [START_REF]Fouling and mass-transfer issues occurred for[END_REF]. A significant and progressive drop of the melt flow index (MI2) [START_REF]The Melt Flow Index (MI2) is defined as the number of grams of polymer extruded in ten minutes as measured by ASTM Method D[END_REF], indicative of the increase of polymer viscosity, was also observed.

The van Gurp-Palmen plots for such samples (PE9PE14, Figure 3) showed a drop of the phase angle at low |G*| consistent with the amount of 1,7-octadiene employed in the polymerization; they also featured clear inflection points. This clearly indicates the presence of LCB/cross-linking and, remarkably, that its extent can be tuned by selecting an appropriate concentration of diene. As regards the tests performed with supp-Zr-2 (Table 2, entries 713), the presence of the diene exerted a detrimental effect on the catalyst productivity; as expected, the latter drop was more pronounced in the runs performed without 1-hexene (Table 2, entries 9, 11 and 13) [START_REF] Barranco-García | New Insight into the Comonomer Effect through NMR Analysis in Metallocene Catalysed Propene-co-1-Nonene Copolymers[END_REF]. Both thermal-and molecular features of the PEs proved independent of the diene concentration and only affected by the presence/absence of 1-hexene. In fact, the drop of T m obtained in the presence of 1-hexene is due to the formation of short-chain branching (SCB) derived from insertion of the comonomer into the growing chain. Moreover, a slight broadening of the molecular weight distributions was observed, tentatively accounted to chain-transfer events occurring more often after the comonomer insertions [START_REF] Galland | Linear low-density polyethylene synthesis promoted by homogeneous and supported catalysts[END_REF]. This is in agreement with the larger distributions obtained with supp-Zr-1, which exhibits higher comonomer incorporation ability. The increase of MI2 values could be accounted for to the presence of larger content of SCB arising from the incorporation of 1,7-octadiene not evolving into LCB/cross-link formation (vide infra).

In contrast to the homogeneous Zr-2, the van Gurp-Palmen data for the samples synthesized with supp-Zr-2 (Figure 4) were compatible with that of purely linear PEs, regardless of diene concentration. This indicated that the insertion of 1,7-octadiene did not evolve into the formation of branched/cross-linked structures (as suggested by the increase of the Melt Flow Index). Interestingly, the same outcome was observed both in the presence and in the absence of 1-hexene; this indicates that the insertion of such comonomer, detrimental to LCB formation, is not in competition with that of 1,7-octadiene. The opposite tendency towards branching formation displayed by homogeneous-and supported Zr-2 (i.e., LCB vs. SCB, respectively) was also observed with the Al-alkenyl LCB-promoter [START_REF] Santoro | Al-alkenyl-induced formation of long-chain branched polyethylene via coordinative tandem insertion and chain-transfer polymerization using (nBuCp) 2 ZrCl 2 /MAO systems: An experimental and theoretical study[END_REF]. Again, this could be possibly rationalized in terms of steric effects exerted by the silica support [START_REF] Galland | Linear low-density polyethylene synthesis promoted by homogeneous and supported catalysts[END_REF],

which reduces accessibility to the metal center, eventually hampering the insertion of the larger comonomer 1,7-octadiene.

On the difference between Al-alkenyl species and α,ω-diene as LCB promoters

The comparison between the results reported herein and those reported in our previous contributions [START_REF] Santoro | Long-Chain Branched Polyethylene via Coordinative Tandem Insertion and Chain-Transfer Polymerization Using rac-{EBTHI}ZrCl 2 /MAO/Al-alkenyl Combinations: An Experimental and Theoretical Study[END_REF][START_REF] Santoro | Al-alkenyl-induced formation of long-chain branched polyethylene via coordinative tandem insertion and chain-transfer polymerization using (nBuCp) 2 ZrCl 2 /MAO systems: An experimental and theoretical study[END_REF][START_REF] Santoro | Group 12 and 13 metal-alkenyl promoted generation of long-chain branching in metallocene-based polyethylene[END_REF] indicates that -dienes are more efficient LCB/cross-linking promoters than iBuAl(oct-7-en-1-yl) 2 (Al-1). In fact, PEs exhibiting rheological properties compatible with LCB were obtained in the presence of rather low amounts of diene (ca. 50 equiv. with respect to the catalyst), while the amount of Al-1 required to achieve similar outcomes was found to be at least one order of magnitude higher. It is assumed that such different efficiency depends on the pathways towards LCB formation and the nature of LCB/cross-linking generated with the two types of co-reactants (Scheme 1).

Scheme 1.

Putative pathways for the formation of LCB/cross-linking in the presence of Alalkenyl species or α,ω-diene co-monomers.

In both cases, the first step of the mechanism is assumed to imply the insertion of a vinyl group into the Zr-C(alkyl) bond of a growing polymer chain. The second part of the mechanism involves a transmetallation to zirconium in the case of Al-1 and the insertion of the second vinyl group in another Zr-C(alkyl) bond in the case of α,ω-dienes. In both cases, further insertion of monomers/propagation leads to the formation of LCB./cross-linking. We assume that the rate of insertion of the pendant double bond of the α,ω-diene co-monomer is much higher than the rate of transmetallation form Al to Zr. Hence, the higher efficiency of dienes would be justified mostly by this kinetic aspect. Similarly, the rapid formation of crosslinks leading to more intricate macromolecular structures also likely occurs.

In order to evaluate this hypothesis, the effect of the reaction time on the LCB density was investigated. The results of the polymerization of ethylene catalyzed by supp-Zr-1 in the presence of both Al-1 and 1,7-octadiene are summarized in Table 3. In the benchmark reactions with 1000 ppm of TIBAL only, the productivity of the catalyst in the 20-min reaction was found nearly twice of that in the test performed over 60 min (Table 3, entries 1 and 2, respectively). The same outcome was obtained when a 1:1 TIBAL/1,7-octadiene mixture was employed (Table 3, entries 3 and 4). These results suggest that, both in the presence and in the absence of the diene, the catalyst undergoes rather fast deactivation (inversely to a "comonomer effect" [START_REF] Santoro | Long-Chain Branched Polyethylene via Coordinative Tandem Insertion and Chain-Transfer Polymerization Using rac-{EBTHI}ZrCl 2 /MAO/Al-alkenyl Combinations: An Experimental and Theoretical Study[END_REF][START_REF] Santoro | Al-alkenyl-induced formation of long-chain branched polyethylene via coordinative tandem insertion and chain-transfer polymerization using (nBuCp) 2 ZrCl 2 /MAO systems: An experimental and theoretical study[END_REF][START_REF] Santoro | Group 12 and 13 metal-alkenyl promoted generation of long-chain branching in metallocene-based polyethylene[END_REF]) and that most of the LCB/cross-linked polymer is produced during the early stages of the polymerization. Remarkably, when the reaction was performed in the presence of Al-1 as the only co-reactant, the productivity of the catalyst was found to be constant, regardless of the reaction time (60, 20 or 10 min, Table 3, entries 5, 6 and 7, respectively). These results can be rationalized considering the formation of stable Zr/Al heterobimetallic species which stabilize the catalyst, preventing its early deactivation. The melting and crystallization temperatures of the polymers synthesized in the presence of Al-based co-reactants were found in a narrow range (133134 °C and 117118 °C, respectively) and proved to be independent from the reaction time. Slightly lower T m and T c values were detected for the polymers produced in the presence of 1,7-octadiene.

No significant effect of the reaction time on the rheological behavior of the polymers prepared in the presence of TIBAL (PE22 and PE23) was observed. In fact, the van Gurp-Palmen curve of both curves tends to a phase angle at low |G*| of ca. 80° (Figure 5).

On the other hand, the rheological properties of PE24 and PE25, obtained by using 1,7-octadiene, showed that higher LCB/cross-linking densities could be achieved at shorter reaction times. Indeed, the van Gurp-Palmen plot curve for PE25 (20-min reaction) showed a lower phase angle at low |G*| than that of PE24 (60-min test). These results, further suggest that consumption of the diene occurs during the early stages of the polymerization, affording

PEs with high LCB/cross-linking density. At longer reaction times, since no more α,ω-diene is present in the reaction medium, formation of essentially linear polymer chains occurs, contributing to a dilution of the LCB/cross-linking density within the whole polymer. Remarkably, all curves were found in a rather narrow range, regardless of the reaction time.

These results suggest that, contrary to the case of α,ω-dienes, the LCB density remains constant throughout the polymerization process. As aforementioned, we assume that this could be accounted for by the formation of stabilizing Zr/Al bimetallic species. Following the computational protocol that has been outlined in our previous studies [START_REF] Santoro | Long-Chain Branched Polyethylene via Coordinative Tandem Insertion and Chain-Transfer Polymerization Using rac-{EBTHI}ZrCl 2 /MAO/Al-alkenyl Combinations: An Experimental and Theoretical Study[END_REF][START_REF] Santoro | Al-alkenyl-induced formation of long-chain branched polyethylene via coordinative tandem insertion and chain-transfer polymerization using (nBuCp) 2 ZrCl 2 /MAO systems: An experimental and theoretical study[END_REF][START_REF] Santoro | Group 12 and 13 metal-alkenyl promoted generation of long-chain branching in metallocene-based polyethylene[END_REF], the coordination energies of 1,7-octadiene was investigated with either Zr- (Scheme 3), the products that are produced favor a linear polymer. This is similar to the observed Zr-Al-1 environment studied previously [START_REF] Santoro | Long-Chain Branched Polyethylene via Coordinative Tandem Insertion and Chain-Transfer Polymerization Using rac-{EBTHI}ZrCl 2 /MAO/Al-alkenyl Combinations: An Experimental and Theoretical Study[END_REF][START_REF] Santoro | Al-alkenyl-induced formation of long-chain branched polyethylene via coordinative tandem insertion and chain-transfer polymerization using (nBuCp) 2 ZrCl 2 /MAO systems: An experimental and theoretical study[END_REF]. The 1,7-octadiene insertion events are plausible but rare in comparison. This is beneficial for increased propagation and consequently may influence branching at shorter chain lengths; having an impact on the formation of LCB and/or cross-links during growth. The fact that the branched products could arguably be in some state of competition thermodynamically in the first and second insertions, suggests a plausible environment for more products with pendant olefinic functions. The latter could warrant sufficient conditions to rapidly incorporate into other growing chains, enabling cross-linked structures. Unfortunately, difficulties in unequivocal characterizing LCB vs cross-links concentrations by melt rheology enforces the general conclusion that the presence of a diene (1,7-octadiene) is agreeably beneficial for formations of cross-links recalling samples PE2PE4.

The same computational analysis was conducted using the [Zr-2R] + system.

Interestingly, the experimental observations indicate a diene moiety had no influence on LCB PE formation favoring a more linear PE product regardless of 1,7-octadiene concentrations.

Computationally, one would expect the data to result in a kinetic preference for monomer insertions relative to diene coordination mechanisms. In fact, the first and second insertions (Schemes S4 and S5) indicate that the coordination of the diene molecule provides a slightly more stabilizing interaction to the [Zr-2R] + catalyst (<1.4 kcal•mol 1 for R = Me).

Observing the two transition states located, the insertion of the diene is preferred. The reactivity presented would indicate the 1,7-octadiene insertion event is theoretically possible and kinetically driven. As the chain propagates to longer lengths, e.g. [Zr-2pentyl] + (Scheme 3), a growing propensity for linear product formations occurs, which is in agreement with a more linear rheology fingerprint presented (Figure 2). The energy profile demonstrates the growing preference for ethylene monomer coordination and insertion as the chain propagates. The coordination of the diene molecule remains slightly stronger relative to monomer coordination but highlights the competition that is present about the [Zr-2pentyl] + system. In contrast to initial insertion steps, the ethylene insertion seems to be becoming increasingly favorable at longer polymeryl chain lengths. Furthermore, the products formed indicate a preference and thermodynamic control for a linear PE material.

Though the initial competition to the [Zr-2R] + system may give rise to preferred insertions of 1,7-octadiene, at longer polymeryl chains on the catalyst, an ethylene coordination/insertion is both more kinetically and thermodynamically favored giving rise to a linear product.

Conclusions

In conclusion, we have investigated the ability of rac-ethylenebis(4,5,6,7-tetrahydro-1indenyl)]zirconium dichloride (Zr-1)-and bis(butylcyclopentadienyl)zirconium dichloride (Zr-2)-based catalyst systems to produce "LCB-PE" upon using an α,ω-diene comonomer as LCB/cross-linking promoter. Under homogeneous conditions, 1,10-undecadiene proved to be such an effective promoter in combination with the Zr-1,2/MAO catalyst systems. In fact, the rheological properties of the corresponding PEs proved to be sensibly affected, even by the use of quite small amounts of α,ω-diene. On the other hand, a similar response was attained with the Cp-based catalyst Zr-2 only at higher [Diene] 0 concentration (5 mM). This was accounted to the different co-monomer incorporation ability of the two metallocenes (Zr-1 >> Zr-2). [START_REF] Lehtinen | A comparison of (n-butCp) 2 ZrCl 2 and other simple metallocenes with bridged Et(Ind) 2 ZrCl 2 and Me 2 Si(Ind) 2 ZrCl 2 catalysts in ethene/propene copolymerization[END_REF] Under heterogeneous (slurry) conditions, supp-Zr-1 exhibited a behavior similar to that of its homogeneous version, while in the presence of supp-Zr-2, only linear PEs were isolated regardless of the amount of diene introduced.

Rheological and NMR spectroscopy studies proved that the consumption of the α,ωdiene and the formation of LCB/cross-linked structures take place during the early stages of the polymerization, inducing concentration of branches and cross-links in a limited fraction of macromolecules. Hence, in order to achieve even distribution of LCB, the α,ω-diene should be progressively introduced in the reaction medium. Differently, the formation of branches promoted by Al-alkenyl species constantly occurs during the whole polymerization process allowing for the regular distribution of LCB, even if all of the co-reactant is introduced in the initial feed. These experimental observations were in line with DFT computational results.

Figure 1 .

 1 Figure 1. Rheological properties of PE1PE4: |η*| = f(ω) (top), van Gurp-Palmen plot

Figure 2 .

 2 Figure 2. Rheological properties of P5PE8: |η*| = f(ω) (top), van Gurp-Palmen plot

Figure 3 .

 3 Figure 3. Rheological properties of P9-PE14: van Gurp-Palmen plot. Shear strain (γ) 10%, angular frequencies (ω) from 0.1 to 250 rad⋅s 1 , T = 190 °C, N 2 atmosphere.

Figure 4 .

 4 Figure 4. Rheological properties of P15-PE21:van Gurp-Palmen plot. Shear strain (γ) 10%, angular frequencies (ω) from 0.1 to 250 rad⋅s 1 , T = 190 °C, N 2 atmosphere.

Figure 5 .

 5 Figure 5. van Gurp-Palmen plot for the PEs obtained with supp-Zr-1/TIBAL/1,7-octadiene

Figure 6 .

 6 Figure 6. van Gurp-Palmen plot for the PEs obtained with supp-Zr-1/TIBAL/Al-1 systems -

Scheme 3 .

 3 Scheme 3. A comparison between the third insertions of ethylene monomer vs. 1,7-octadiene vinyl group coordination/insertion in [Zr-2pentyl] + .

Table 1 .

 1 Ethylene polymerization in the presence of 1,10-undecadiene catalyzed by

Zr-1- and Zr-2/MAO systems

  (homogeneous conditions).

	Entry	Sample ID	Catalyst	[Diene] 0 (mM)	Productivity (kg PE /g Zr . h)	T m a (°C)	M n b (kDa)	M w b (kDa)	b M w /M n	Crystallinity c (%)
	1	PE1		0	35	136	43	140	3.3	48
	2 3	PE2 PE3	Zr-1	0.5 1.5	60 46	134 133	insoluble sample insoluble sample	51 51
	4	PE4		5.0	31	130	insoluble sample	47
	5	PE5		0	42	134	30	109	3.6	47
	6 7	PE6 PE7	Zr-2	0.5 5.0	50 55	133 133	36 insoluble sample 145 4.0	57 53
	8 d, e	PE8		5.0	100	134		n.d.		44
	Reaction conditions: [Zr] = 10 mM, Al MAO /Zr= 5000, T = 60 °C, 15 min, toluene = 150 mL, P Et = 4 barg. a Determined by DSC from second run. b Determined by HT-GPC. c Calculated from
	[(ΔH m /ΔH 0 ΔH 0 m is the theoretical heat of fusion for 100% crystalline PE (293 J/g) [30]. d The reaction was m ) × 100], where ΔH m is the heat of fusion of the sample (in J/g) determined by DSC and stopped after 3 min. Diene left in the quenching solution < 2%. e No vinyl moieties detected in the 1 H
	NMR spectrum of the polymer.

Table 2 .

 2 Ethylene polymerization in the presence of 1,7-octadiene and 1-hexene[45] catalyzed by

supp-Zr-1-and supp-Zr-2/TIBAL systems

  (heterogeneous conditions).

	Entry PE ID	Catalyst	[Diene] 0 (ppm)[mM]	Productivity (kg PE /g cat . h)	T m a (°C)	M n b (kDa)	M w b (kDa)	b M w /M n	MI2 (g/10 min)
		PE9		0	5.1	120	28	79	2.8	1.06
		PE10		50 [0.5]	5.3	119	28	82	2.9	0.77
		PE11 PE12	supp-Zr-1	100 [1.0] 200 [2.0]	4.9 5.3	119 120	27 28	84 88	3.1 3.1	0.53 0.38
		PE13		300 [3.0]	5.3	120	28	94	3.4	0.18
		PE14		500 [5.0]	5.4	120	insoluble sample	n.d.
		PE15		0	7.8	122	24	53	2.2	10.8
		PE16		500 [5.0]	6.2	123	19	46	2.3	15.1
	9 c	PE17		500 [5.0]	4.0	132	23	60	2.6	6.60
	10	PE18	supp-Zr-2	1500 [15]	5.9	124	19	44	1.7	16.2
	11 c	PE19		1500 [15]	3.3	131	21	54	2.5	10.4
	12	PE20		3000 [30]	5.3	123	18	43	1.9	18.8
	13 c	PE21		3000 [30]	3.2	131	20	54	2.5	14.5
		Reaction conditions: supported catalysts = 2 mg, T = 85 °C, 60 min, iso-butane = 75 mL, P Et = 25
		barg, H

2 = 800 ppm, 1-hexene = 2.44 wt%. a Determined by DSC from second run. b Determined by HT-GPC. c No 1-hexene.

Table 3 .

 3 Polymerization

	Entry	PE ID	TIBAL (ppm)	Co-reactant (ppm)	Time pol (min)	Productivity (kg pol /kg cat . min)	T m (°C)	T cryst (°C)
	1	PE22	1000	none	60	2.25	134	119
	2	PE23	1000	none	20	5.25	134	118
	3	PE24	1000	1,7-octadiene (1000)	60	3.15	131	117

of ethylene with supp-Zr-1/TIBAL/Al-1 or 1,7-octadiene -effect of the reaction time.

  1 orZr-2 systems. The first coordination and insertion of the diene moiety to [Zr-1methyl] + catalyst indicates an initial stabilizing interaction of 6.3 kcal•mol 1 (See the Supporting information, Scheme S2). Kinetically, either first insertion is inconclusive with a ΔH Transition_state of 0.2 kcal•mol 1 , highlighting the competitive environment of [Zr-1methyl] + catalyst is subjected to when the diene moiety is present in the reaction mixture.Of great interest regarding the product formations observed thus far, is the fact that the 1,7octadiene insertion product is initially thermodynamically competitive with respect to the insertion of ethylene monomer, although at the cusp of the methodology (2 kcal•mol 1 ), ΔH Product = 1.8 kcal•mol 1 . Further studies were conducted in order to explore if the same trends are reproduced at longer polymeryl chain lengths: C3 and C5, with the respective [Zr-1propyl] + (Scheme S3) and [Zr-1pentyl] + catalysts (Scheme 2).The second insertion (Scheme S3) demonstrates a similar reactivity to the first coordination. A ΔH Transition_State of 2.4 kcal•mol 1 could insight some doubt regarding the kinetic propensity of linear product formations whilst postulating the ability of a more stable 1,7-octadiene insertion product to form. As the chain propagates affording [Zr-1pentyl] +

	Scheme 2. A comparison between the third insertion of ethylene monomer vs. 1,7-octadiene
	vinyl group coordination/insertion to [Zr-1pentyl] + following a dissociation of the diene.

Acknowledgements: LM is a senior member of the Institut Universitaire de France. CalMip is acknowledged for a generous grant of computing time.

Funding sources: This work was gratefully supported by TotalEnergies One Tech Belgium (postdoctoral and PhD fellowships to OS, LP and KMC).

Appendix A: Supplementary data

Supporting information to this article (NMR spectroscopy and rheological analyses of polymers, computational details, Cartesian coordinates) can be found online at DOI: