Saurabh Srivastava
email: srivastava.saurabh@nims.go.jp

Hiori Kino

Christian Joachim
email: christian.joachim@cemes.fr

Quantum Half-Adder Boolean Logic Gate with a Nano-Graphene molecule and Graphene Nano-electrodes

Keywords: Graphene, Logic Gates, Quantum Transport, Quantum Computing, Molecular electronics, Half-Adder

A molecule Boolean 1/2-adder is designed and the XOR and AND truth table calculated at +0.1 V using 4 graphene electrodes. It functions with level repulsion and destructive interferences effects using 4 molecule electronic states in a quantum Hamiltonian computing approach (QHC) with the abrupt change of the molecular orbital weight of those 4 calculating states as a function of the logical input configuration. The logical inputs enter rotating the two nitro groups of the central board. With QHC, a complex Boolean digital function can be implemented employing the same graphene material for interconnects and the molecule calculating parts.

Introduction

The continuing miniaturization of electronic circuits triggers the ultimate search for single molecule based electronics where a molecule embedded between many electrodes will perform a digital Boolean function by itself [START_REF] Joachim | Electronics using hybridmolecular and mono-molecular devices[END_REF][START_REF] Zhang | Voltage-driven spintronic logic gates in graphene nanoribbons[END_REF]. One possibility is to interconnect molecular switches [START_REF] Collier | Electronically configurable molecular-based logic gates[END_REF][START_REF] Collier | A [2]catenane-based solid state electronically reconfigurable switch[END_REF], rectifiers [START_REF] Metzger | All about (n-hexadecylquinolin-4-ium-1yl)methylidenetricyanoquinodimethanide, a unimolecular rectifier of electrical current[END_REF][START_REF] Metzger | Electrical rectification by a molecule:the advent of unimolecular electronic devices[END_REF] or amplifiers [START_REF] Joachim | An electromechanical amplifier using a single molecule[END_REF] together using metallic nanowires [START_REF] Ami | Logic gates and memory cellsbased on single c 60 electromechanical transistors[END_REF] to construct the circuit. This approach is now progressively abandoned due to the absence of power gain per molecular device [START_REF] Staddler | A memory/adder model based on single c 60 molecular transistors[END_REF]. Semi-classically designed intramolecular circuits [START_REF] Ellenbogen | Architectures for molecular electronic computers. i. logic structures and an adder designed from molecular electronic diodes[END_REF] have also been explored [START_REF] Jlidat | A semi-classical xor logic gate integrated in a single molecule[END_REF] facing now the problem of the exponential decay of the output current with an increase of the spatial extension of the molecular circuit [START_REF] Lafferentz | Conductance of a single conjugated polymer as a continuous function of its length[END_REF].

To solve those gain and exponential decay problems still with the objective to embark a calculating unit on a single molecule, a new quantum computing approach was proposed [START_REF] Renaud | Quantum design rules for single molecule logic gates[END_REF].When prepared in a non-stationary state, the quantum Hamiltonian computing (QHC) logic gate beneficiates from the spontaneous Heisenberg-Rabi quantum oscillations of the quantum system to run a quantum computation with classical inputs [START_REF] Renaud | Design and stability of nor and nand logic gates constructed with three quantum state[END_REF]. By measuring the effecting Heisenberg-Rabi oscillations frequency using metallic nano-electrodes [START_REF] Faizy | Parallel quantum circuit in a tunnel junction[END_REF], different Boolean logic operations can be performed in parallel and within the same quantum system [START_REF] Dridi | The mathematics of a quantum hamiltonian computing half adder boolean logic gate[END_REF]. The functioning of a Boolean QHC gate is based on the quantum level repulsion effect together with the control of constructive and destructive quantum interferences affecting tunneling transport. Quantum design rules for QHC single molecule logic gates have been reported by Renaud et al. [START_REF] Renaud | Quantum design rules for single molecule logic gates[END_REF]. Following this approach, Soe et al. [START_REF] Soe | Manipulating molecular quantum states with classical metal atom inputs: Demonstration of a single molecule nor logic gate[END_REF] have demonstrated experimentally a NOR logic gate function using a tri-naphthalene molecule whose frontiers molecular orbitals were manipulated by single Au atom contact to each of the naphthyl branch. Formally, A different QHC design was used to structure a single molecule NOR-AND [START_REF] Renaud | A norand quantum running gate molecule[END_REF]gate. For the previous hybrid 1/2-adder Boolean logic functions [START_REF] Duchemin | A quantum digital half adder inside a single molecule[END_REF][START_REF] Duchemin | An intramolecular digital 1/2-adder with tunneling current drive and read-outs[END_REF], rotation of two lateral nitro chemical groups were also providing the classical digital inputs to the QHC gate here with a direct conversion of each rotation angle in quantum information. This conversion can be exactly measured by decomposing each molecular state on its Slater atomic basis set. It shows how a given NO 2 rotation angle is encoded on the normalized atomic orbital coefficients of certain molecular states.

For these QHC molecule Boolean logic gates, metallic nano-electrodes were used to measure the logical outcome of the gates. Such metal-organic hybrids lead to a very low intensity tunneling current per output due to the large differ-ence in electronic structure between the central calculating molecule unit and its metallic measuring nano-electrodes. To design more complex QHC molecule logic gates (for exploring how the QHC approach can do better in term of decoherence as compared to the quantum qubit approach [START_REF] Nielsen | Quantum Computation and Quantum Information[END_REF]) and for beneficiating in application of tunneling current intensities larger than a few nano-ampere, the calculating molecule and the measuring electrodes must be made of the same material. The graphene planar 2D material with its tunable electronic, mechanical and electronic properties [START_REF] Geim | The rise of graphene[END_REF][START_REF] Novoselov | Electric field effect in atomically thin carbon films[END_REF][START_REF] Novoselov | Two-dimensional gas of massless dirac fermions in graphene[END_REF] is an excellent candidate for a monolithic approach of QHC molecular logic gates.

In this letter, we demonstrate theoretically that a double nitro functionalized nano-graphene molecule (NGM) connected using graphene electrodes is performing a 1/2 adder Boolean operation in the QHC way using the rotation of the nitro-groups as the classical digital inputs. The first section is describing the QHC NGM design. Section 2 is recalling the technique used in this letter to calculate all the tunneling current intensity passing through the central NGM. Section 3 is describing the functioning of our 1/2 adder QHC NGM. In conclusion, we compare our design with a classical 1/2 adder electronic circuit.

The Boolean 1/2 adder QHC molecular design

On Fig. 1, a central NGM is chemically bonded to 4 ultra-short 3 phenyls in length and 2 in width graphene nano-ribbon (GNR1 to GNR4). The core of the NGM molecule was designed starting from a coronene molecule extended asymmetrically by phenyl rings to be able to distinguish an AND and an XOR functionality by its MO weight at the measuring point. Four different conjugated boards were tested not to extend too much this central NGM. Each GNR is connected to independent semi-infinite graphene electrode whose electronic band structure is well defined using periodic boundary conditions [START_REF] Srivastava | Contact conductance of a graphene nanoribbons with its graphene nano-electrodes[END_REF]. The central NGM is 2D crystal oriented in the same direction as the graphene electrodes to beneficiate from a complete homo electronic structure between the NGM and its measuring graphene electrodes. Nano-graphene molecules have already been synthesized [START_REF] Dorel | Tetrabenzocircumpyrene: a nanographene fragment with an embedded peripentacene core[END_REF]. The Fig. 1 NGM circuit may be nanoscale fabricated for its interconnection parts using a high precision He + scattering [START_REF] Nakaharai | Conduction tuning of graphene based on defectinduced localization[END_REF] and using scanning tunneling microscope (STM) single atom vertical manipulations for the atomic scale part as already demonstrated at the surface of 2D MoS 2 materials [START_REF] Hosoki | Surface modification of mos 2 using an stm[END_REF]. In Fig. 1, the A and B nitro group inputs were chemically bonded to the NGM for distributing the logical inputs between the XOR and AND part of the QHC circuit without requiring ancillary internal or external molecular wires as commonly performed in standard electronic Boolean logic circuits. In electronics, the output of a given gate along the circuit is generally required to distribute its output signal at many locations of the circuit at the same time. Our present molecule QHC design can be considered as a practical implementation of the formal QHC 1/2-adder [START_REF] Dridi | The mathematics of a quantum hamiltonian computing half adder boolean logic gate[END_REF] where it was shown how to position the logical inputs along the QHC quantum graph to avoid this cascading problem. Two fluorine atoms were also added to the NGM to optimize, as presented in Fig. 4, the distribution of the frontier molecular orbitals over the NGM depending on the logical input configuration as discussed later. With respect to the graphene electrode plane, a planar nitro is encoding for a "0" binary logical input and a perpendicular one for a "1" logical input. This encoding was chosen because of the non-zero switching energy certainly required to input a logical "1" on the NGM by STM rotating a single NO 2 group, as it was already demonstrated for a single ter-butyl group [START_REF] Loppacher | Direct determination of the energy required to operate a single molecule switch[END_REF], the native nitro configuration being planar [START_REF] Domenicano | Molecular structure of nitrobenzene in the planar and orthogonal conformations[END_REF]. A nitro group is a small but very efficient electron acceptor group, very active in interacting with a conjugated molecule while co-planar with it [START_REF] Srivastava | Contact conductance of a graphene nanoribbons with its graphene nano-electrodes[END_REF][START_REF] Patoux | Topological effects on intramolecular electron transfer via quantum interference[END_REF].

The Calculation Technique

Through the Fig. 1 QHC circuit and using Landauer formula [START_REF] Buttiker | Generalized many-channel conductance formula with application to small rings[END_REF][START_REF] Stone | What is measured when you measure a resistance?the landauer formula revisited[END_REF], the current intensity I was calculated between any two graphene electrodes and when applying a bias voltage V using :

I = e πh E f +eV E f T (E)dE, (1)
where E f is the Fermi energy of a given graphene electrode and T (E) = Trace(t(E)t † (E)), the electronic transparency between those two graphene electrodes. t(E) is the top left block of the multi-channel elastic scattering matrix describing the scattering process through the central NGM. The complete multichannel scattering matrix was calculated exactly using the elastic scattering quantum chemistry (ESQC) method [START_REF] Dridi | The mathematics of a quantum hamiltonian computing half adder boolean logic gate[END_REF][START_REF] Doumergue | Conductance of a finite missing hydrogen atomic line on si(001)-(2x1)-h[END_REF][START_REF] Yong | Calculation of the conductance of a finite atomic line of sulfur vacancies created on a molybdenum disulfide surface[END_REF]. A full valence semi-empirical

Hamiltonian was used in ESQC to describe the central molecule along with its 4 short GNR molecular wires and the graphene electrodes. This Hamiltonian was constructed with the full valence extended Hückel molecular orbitals (EHMO) basis set using Hoffmann parameters [START_REF] Hoffmann | An extended hückel theory. i. hydrocarbons[END_REF][START_REF] Wolfsberg | The spectra and electronic structure of the tetrahedral ions mno -4 , cro --4 , and clo -4[END_REF] for all the C, N, O, F, and H atoms. A discussion about the Hoffmann parameters preferred over Cérda parameters [START_REF] Cerdá | Accurate and transferable extended hückel-type tightbinding parameters[END_REF][START_REF] Kienle | Extended hückel theory for band structure, chemistry, and transport. i. carbon nanotubes[END_REF] can be found in [START_REF] Srivastava | Contact conductance of a graphene nanoribbons with its graphene nano-electrodes[END_REF] for graphene. Using those EHMO parameters, the calculated graphene electrode band structure shows a relative energy position of the Dirac cone crossing point at E = E f = -10.53 eV.

The functioning of the nano-graphene QHC 1/2 adder

In Fig. 2, the calculated T(E) spectra for the XOR and AND outputs are presented as a function of the A and B logical input configurations. The cross "leakage" T(E) spectra between one XOR and one AND graphene electrodes, respectively, through the NGM were also calculated and according to equation 1 lead to a less than 0.01 µA tunneling current intensity for a 0.1 V bias voltage as compared with the direct XOR and AND outputs. This is the maximum calculated leakage current from one XOR to one AND graphene electrodes and for only the (1,1) input configuration. A 0.01µA current intensity on the AND for (1, 1) is not very noticeable as compared to the 2µA measured output current for this input configuration. All the others cross leakage current are one order of magnitude lower than 0.01µA at 0.1 V for all the other logical input According to the T(E) spectra in Fig. 2, both the XOR and AND output status must be measured in an energy range about 0.1 eV up the Fermi energy.

As a consequence, the chemical structure of the C1 and C2 contacts have been designed differently to be able to distinguish between the XOR and AND out-

Discussion

Using the Fig. 2 T(E) spectra in equation 1 to calculate the 0. Our nano-graphene QHC 1/2-adder demonstrates how a small molecular nanostructure can performed a rather complex logic function: a binary addition with a carry without using an extended molecular system but more important without requiring any power gain inside the calculating board. The nonnecessity of a power gain comes from the fact that the output status of the gate are measured without passing a current through to the entire molecule. Contrary to previous proposed QHC 1/2-adder design [START_REF] Duchemin | An intramolecular digital 1/2-adder with tunneling current drive and read-outs[END_REF], it results from a local measurement. This new QHC design opens the search for more complex and monolithic QHC intramolecular logic gates based on intramolecular quantum information exchanges where the multiple output status are measured locally. going from (0, 1), (1, 0) to (0, 0) or [START_REF] Joachim | Electronics using hybridmolecular and mono-molecular devices[END_REF][START_REF] Joachim | Electronics using hybridmolecular and mono-molecular devices[END_REF]. This MO manipulation permits to measure the XOR and AND logical output status for exactly the same bias voltage. Blue circles are indicating the C1 and C2 contact position to appreciate the MO manipulation. Accordingly, the AND logical output is better defined than the XOR one for this current design (see also Table 1).

  configurations. The T (E) spectra are showing up some π and π * resonances together with destructive interference patterns i.e. very pronounced s in the T(E) spectrum for example at E = -10.5 eV for the XOR (0, 0) and at E = -10.[START_REF] Yong | Calculation of the conductance of a finite atomic line of sulfur vacancies created on a molybdenum disulfide surface[END_REF] eV for the AND (0, 1)/(1, 0) configurations. The functioning of the Fig.1QHC 1/2 adder is based on the quantum level repulsion effect and on an optimized energy positioning of those tunneling destructive interferences relative to the graphene Fermi energy. As also demonstrated recently with a formal Boolean Hamiltonian, having two outputs for the same QHC system and measured in the same energy range requires that the weight of the molecular orbitals at measurement points (here C1 and C2) be quite exclusive as a function of the input configuration[START_REF] Dridi | The mathematics of a quantum hamiltonian computing half adder boolean logic gate[END_REF].Let us first start with the level repulsion effect. Among all the π and π * molecular orbitals of the central NGM and in an energy range 2 eV around the Fermi energy, only a few are effectively contributing to the 1/2 adder functioning because many have no molecular weight at the C1 and C2 chemical bonding sites of the graphene electrodes. We have identified on Fig.2, the 4 main "calculating" molecular states of the central NGM i.e. the T(E) spectra resonances contributing the most to the 1/2 adder functioning. They have been labeled 1 to 4 for mono-electronic quantum states |1 to |4 (For completion and in the Suppl. Mat. 5, they are also labeled on the corresponding complete spectrum of this central NGM alone). For clarity, Fig. 3 is only showing up the energy position of those 4 states in a quantum level correlation diagram. There is a clear level repulsion effect in action on state |1 shifted up in energy by about 0.3 eV and on state |3 by about 0.2 eV by going from a parallel to a perpendicular nitro conformations for AND and XOR respectively. Based on this level repulsion effect, the functioning of the Fig. 1 1/2 adder QHC gate is also relaying on the control by the nitro angles of the T (E) constructive and destructive interference patterns. Notice that, those patterns also depend on the chemical structure of contact C1 and C2. For a (0, 0) logical input i.e. for two planar nitro's, there is a very pronounced T (E) destructive interference between states |1 and |4 at E = -10.49 eV for the XOR T (E) and at E = -10.51 eV for the AND T (E). Located near the Fermi energy and at low positive bias voltage, this will leads to a very low output current according to equation 1, ensuring the required zero logical output for both the XOR and the AND. Notice that, the two very sharp resonances located just below the Fermi energy are coming from states |1 and |3 which are only weakly coupled to the graphene electrodes for (0, 0). They will not contribute to the XOR and AND logical outputs in this case. The (0, 0) input destructive interference patterns are drastically changed by rotation of the nitro group one after the other from a planar to a perpendicular conformation (See Suppl. Mat. 6 for the detail T(E) interference pattern changes following the progressive rotation of the two nitro groups). For the (0, 1) and (1, 0) input configurations, the (0, 0) XOR central destructive interference observed previously at E = -10.49 eV is now shifted below the Fermi energy and state |3 leads now to a very sharp constructive resonance positioned up the Fermi energy at E = -10.50 eV. This results from the restoration of the electronic coupling between state |3 and the two GNR3 and GNR4 nano-contacts after reaching a perpendicular conformation for only one nitro group having the effect of reintroducing an electronic density expansion of |3 at contact C2 (see also below Fig. 4). The resonance from state |2 is still very sharp and will not contribute to the tunneling current because it is deeply attenuated by the central destructive interference located at E = -10.55 eV. Also for (0, 1) and (1, 0), the AND central destructive interference is shifted up and above the Fermi energy at E = -10.35 eV. Here, state |3 is still leading to a very sharp resonance below the Fermi energy. State |3 will not contribute to the AND tunneling current output. For the (1, 1) input configuration, the XOR central destructive interference is still located below the Fermi energy and state |2 resonance remains below the Fermi energy. More important, the resonance coming from |3 is still very sharp and now shifted up further away from the Fermi energy. This means that at low bias voltage and according to equation 1, it will not contribute to the XOR tunneling current. Also for (1, 1), the AND destructive interference between |1 and |4 which was located for (0, 1) and (1, 0) at E = -13.35 eV is now no more existing. The state |1 resonance has now reached E = -10.49 eV with a large width resonating Lorentzian shape. At low positive voltage and according to equation 1, this will lead to the AND output functioning.

Fig. 4 ,

 4 Fig. 4, this is exactly what was achieved with the Fig. 1 design thanks also to the substitution of 2 hydrogen atoms by 2 fluorine atoms on the central NGM. By representing states |1 and |3 corresponding molecular orbital normalized weight distribution on the NGM, it is clear that only the input configuration (1, 1) is leading to a large orbital weight for |1 at C1 for the AND. Only input configurations (0, 1) and (1, 0) are doing the same for |3 at C2. This exclusive distribution of the molecular orbital spatial weight determines the good functioning of our 1/2 adder QHC gate. It is exactly the chemical implantation of the formal Boolean Hamiltonian studied in Dridi et. al [16].

Figure 1 :Figure 3 :Figure 4 :

 134 Figure 1: The atomic structure of the designed QHC 1/2 adder nano-graphene molecular logic gate with its 4 ultra-short GNR interconnecting molecular wires (GNR1 to GNR4) and its 4 graphene electrodes. The AND is measured at contact C1 and the XOR at contact C2.Contacts C1 and C2 are made of 2 phenyl rings each for the tunneling current to be measured between GNR1 and GNR2 for C1 and between GNR3 and GNR4 for C2. The graphene electrodes are semi-infinite to work with the graphene electronic band structurereproduced here using lateral cyclic boundary conditions. The two fluorine atoms are shown in blue whereas the two rotating nitro groups are shown in red. The presented geometry is for the nitro-groups in a planar configuration encoding for a (0, 0) logical input.

  |2 to moderate the (1, 1) XOR output current to ensure a low output current for this input configuration. Nearby in energy to those 4 states, a few more are moderating the level repulsion acting on |1 and |3 to limit, for example, the shifting up of the resonances |1 and |3 while rotating the nitro groups. All the other NGM quantum states can be considered as structural states ensuring the chemical stability of the central nano-graphene molecule.

	circuit nodes. The presented 232 atoms nano-graphene calculating molecule
	has no transistors nor apparent internal molecular wires like circuitry. Among
	the 168 π and π * quantum states of the Fig. 1 central nano-graphene molecule,
	mainly 4 states are "calculating" as presented in Fig. 3. The NGM quantum
	states |1 and |3 are shifted up because of the rotation of 2 strong electron
	acceptor groups. Those are classical mechanical inputs which cancertainly be
	activated by scanning probe techniques but has not been experimented yet.
	State |4 is very important to create the (0, 0) central destructive interference
	and state Importantly the XOR and AND output current intensities are quite large. First
	because the tunneling electrons have not to be transferred through the complete
	NGM molecular structure to deliver each output status. Second because the
	measuring nano-pads are homo electronic with the central calculating nano-
	graphene, a way to optimize the contact conductance at C 1 and C 2 .
	1 V low
	voltage tunneling current intensity passing through the central NGM at C1 and
	C2, the 1/2 adder Boolean truth table is presented in Table 1. According to the
	standard C. Shannon like design[40], a Boolean 1/2 adder is normally requiring
	the interconnection of a minimum of 4 transistors and a dedicated distribution
	of the voltage logical inputs among those transistors using a few wires and

Acknowledgement SS thanks the MANA-NIMS Theory-Experiment Fusion Research Program and the NIMS computation facilities. CJ thanks the WPI MANA program for financial support and the constant encouragements of Pr. M. Aono during this work.

The nitro rotation angle encoding is also recalled. The XOR and AND logical output current intensities were calculated using 1 for the same +0.1 V bias voltage. The Fig. 1 molecule is functioning like an 1/2 adder Boolean logic gate with a very large ON/OFF ratio for both the XOR and the AND. As anticipated according to Fig. 4, the XOR logical "1" output current is a bit small in intensity as compared to the AND one with almost one quantum of conductance at C1.


Jorge Echeverria

Serge Monturet

Christian Joachim

One-way rotation of a molecule-rotor driven by a shot noise

Keywords: Single molecule motor, rotation mechanism, shot noise, excited states

The shot noise of the tunneling current passing through a molecule-motor can sustain its rotor one-way rotation when virtually populating its excited states by tunneling inelastic excitations. No need for a ratchet like ground state rotation potential energy of the rotor.

Just a relative shift in energy between the maxima of this ground state and the minima of the excited states used is required. The rotor speed of rotation and its rotation direction are both controlled by this shift pointing out the need of a careful design of both the ground and excited states of the next generation of molecule-motor to be able to generate a motive power.

INTRODUCTION

Since the first observation of the rotation of a single molecule adsorbed on a surface, [START_REF] Gimzewski | Rotation of a single molecule within a supramolecular bearing[END_REF] studies on single molecule mechanical machineries on a surface have expanded in many directions such as molecule-gears, [START_REF] Manzano | Step-by-step Rotation of a Molecule-gear Mounted on an Atomic-scale Axis[END_REF][START_REF] Frantz | Molecular Spur Gears Comprising Triptycene Rotators and Bibenzimidazole-Based Stators[END_REF] molecule-motors [START_REF] Delden | Unidirectional Molecular Motor on a Gold Surface[END_REF][START_REF] Zheng | Dipolar and Nonpolar Altitudinal Molecular Rotors Mounted on an Au(111) Surface[END_REF] and molecule nano-vehicles. [START_REF] Shirai | Recent progress on nanovehicles[END_REF][START_REF] Joachim | Molecule Concept Nanocars: Chassis, Wheels, and Motors[END_REF] Nowadays, there are several possibilities to drive the rotation of a single molecule rotor when adsorbed on a metallic surface, like pushing on the rotor by an STM or AFM tip, [START_REF] Manzano | Step-by-step Rotation of a Molecule-gear Mounted on an Atomic-scale Axis[END_REF][START_REF] Pawlak | Directed Rotations of Single Porphyrin Molecules Controlled by Localized Force Spectroscopy[END_REF] feeding the rotor using inelastic electron tunnelling effects via an STM tip [START_REF] Nickel | Moving Nanostructures: Pulse-Induced Positioning of Supramolecular Assemblies[END_REF][START_REF] Kudernac | Electrically Driven Directional Motion of a Four-wheeled Molecule on a Metal Surface[END_REF] , applying an oscillating electric field to the rotor to synchronise its rotation with the large oscillation period of this field, [START_REF] Neumann | Driving and Controlling Molecular Surface Rotors with a Terahertz Electric Field[END_REF][START_REF] Seldenthuis | van der An All-Electric Single-Molecule Motor[END_REF][START_REF] Horinek | Surface-mounted Altitudinal Molecular Rotors in Alternating Electric Field: Single-molecule Parametric Oscillator Molecular Dynamics[END_REF] or powering the rotor with light. [START_REF] Delden | Unidirectional Molecular Motor on a Gold Surface[END_REF] Aside from a tip pushing, i.e. when the operator decides the rotation direction of rotor, 2 in the others cases mentioned above the conditions for a one way rotation of a single molecule-rotor in a molecule-motor have not yet been clarified. Currently, the literature is fairly following two directions: (a) the introduction of an intramolecular ratchet effect which is usually characterized by plotting the potential energy variation of the rotor as a function of its rotation angle [START_REF] Perera | Controlled Clockwise and Anticlockwise Rotational Switching of a Molecular Motor[END_REF] and (b) the control of the interactions between the rotor and a specific external driving force to break micro-reversibility. [START_REF] Neumann | Driving and Controlling Molecular Surface Rotors with a Terahertz Electric Field[END_REF] In this article, following the recent demonstration of a control rotation of a single molecule motor driven by a tunnelling current, [START_REF] Perera | Controlled Clockwise and Anticlockwise Rotational Switching of a Molecular Motor[END_REF] we demonstrate by constructing a general model from this example that the conditions for a one-way rotation of a single molecule motor are between (a) and (b). From (a), there is a need to break the symmetry of the rotational potential energy surface of the rotor as a function of its rotation angle and from (b) there is a need for the rectification of some random excitation process of the rotor to drive the rotation. At the origin of this study, our molecule motor is a piano-stool organometallic complex composed of a five-arm rotor mounted on a tripod stator as presented in Figure 1. [START_REF] Perera | Controlled Clockwise and Anticlockwise Rotational Switching of a Molecular Motor[END_REF][START_REF] Vives | Directed Synthesis of Symmetric and Dissymmetric Molecular Motors Built Around a Ruthenium Cyclopentadienyl Tris(indazolyl)borate Complex[END_REF][START_REF] Vives | The Chemistry of 1,2,3,4,5pentaphenylcyclopentadienyl Hydrotris(indazolyl)borate ruthenium(II) Complexes, Building Blocks for the Construction of Potential Organometallic Molecular Motors[END_REF] This molecular motor was recently operated by STM on a Au(111) surface yielding a step-by-step one-way rotation, both clockwise and counter-clockwise, under a tunnelling current. The direction of rotation was selected by positioning the STM tip either on a truncated or a non-truncated arm of the rotor.

In this paper, we first build up a simple model for the Figure 1 motor to analyse the general requirements for the rotor of a single molecule-motor to rotate one-way in a controllable manner. Then, a complete numerical analysis of the rotation dynamics of the corresponding rotor is performed, paying special attention to the average driving force and to the role played by the electronic excited states of the molecule. Finally, we demonstrate how to drive a one-way rotation using the shot noise of the tunnelling current passing through the molecule-motor explaining also how to control the way of rotation. Despite many others degrees of freedom contribute to the mechanics of this motor, their role is minor as compared to those 9 principal ones. Even with only those 9 coordinates, the corresponding E o (,  1 , … 5 ,  1 , .., 3 ) ground state potential energy surface is considerably difficult to master. Starting from one of the possible E o (,  1 , … 5 ,  1 , .., 3 ) energy minima, a simple way to model the rotation mechanisms is to step by step increase  while minimizing the molecule motor potential energy for each . This delivers a  parametric rotation trajectory on the E o (,  1 , … 5 ,  1 , .., 3 ) manifold. For the Fig. 1 motor, this trajectory was calculated using the ASED+ semi-empirical technique. Along this rotation trajectory, the resulting E o () potential energy curve is highly asymmetric, with a potential barriers height of V = 0.25 eV as presented in Fig. 2. On E o , the combination of the rotor upper C5 and the stator lower C3 concurrent symmetries [START_REF] Echeverría | Concurrent Symmetries: The Interplay Between Local and Global Molecular Symmetries[END_REF] leads to 15 potential wells per turn. This periodic series of barriers is due to the interaction between the 5 phenyl rings of the rotor arms and the 3 motor legs bound to the surface. During the rotation and every time one arm interacts with one leg, the corresponding phenyl ring flips in order to avoid the leg and this leg is consequently deformed. On the Fig. 2 curve, this results in a ratchet-like potential energy variation.

DISCUSSION

Calculations with a 4 arms rotor are qualitative identical (see Fig. S1 in Supporting Information) MAY BE NOT NECESSARY and Also THE CORRESPONDING Supporting Information. However, this asymmetry, constructed along the rotation trajectory, is artificial because it is created by forcing the rotation in a given direction.

There is no asymmetry when passing from one minimum to the next on E o (,  1 , … 5 ,  1 , .., 3 ) and, thus, no breaking of the micro-reversibility principle. Therefore, by forcing a  rotation, we are anticipating the good functioning of the molecule motor but not demonstrating it. In fact, with the initial objective of determining the conditions for a one-way rotation, an ideal rotation trajectory was created on E o (,  1 , … 5 ,  1 , .., 3 ) as if the rotor was already rotating one way instead of finding the optimized excitation process to approach this ideal rotation trajectory.



To illustrate how a saw-tooth like rotation trajectory can come out from an a priori symmetric potential energy surface, let us consider the rotation of a single phenyl of the 2,2'-dibromobiphenyl molecule around its central C-C bond (Figure 3a). The  rotation angle is here defined by the dihedral C1, C2, C3, C4 angle and the torsion angle by the C2, C3, C4, Br5 dihedral angle (Figure 3a). To build up the E o (, ) potential energy surface presented in Fig. 3b, the molecule potential energy was calculated as a function of (, ) without optimization of the molecular geometry by means of the AM1 semi-empirical technique. This surface is periodic in (, ) and provides a map of one of those periods including also two energy minima A and B.

There is a central energy maximum corresponding to the Br-Br repulsion. This is also obtained when the partial optimization involves all the molecular degrees of freedom but the rotation angle. However, if this torsion angle is the only degree of freedom allowed during the optimization, the rotation energy profile is then symmetric.

This example illustrates the extreme care required in selecting the good degree of freedom to describe the mechanics of a single molecule motor. Since the number of degrees of freedom is generally quite large, the ones to consider, the ones to minimize and the ones to freeze must be very well chosen to model in a full generality the intramolecular physics involved. For the Fig. 1 like molecule-motor (and for many others of the same kind), a minimum of two degrees of freedom, here the  rotation angle and a generic torsion angle , must be considered and not only  as used in Fig. 2.

For the Fig. 1 motor, this torsion angle  can be defined by the dihedral angle between a given phenyl ring of the rotor and the central cyclopentadienyl ring. In this case, a model E o (, ) ground state can be written:

E 0 (,)  V 0 sin 2 ( )cos 2 ( )e  2  k 6 (1)
Instead of the 15 minima of the Fig. 1 molecule motor, we have here limited the number of minima to 4 per turn to simplify the analysis. It can be easily extended to 15 or more for molecule motor with a larger number of arms. During the rotor rotation, the steric repulsion between a rotor phenyl and one of the three stator legs is taken into account in

(1) by a  6 term (with k = 5•10 -3 ). V0 models the energy barrier height and k the strength of the deformation. This E o (, ) periodic potential energy surface is presented in Figure 4. When a  rotation is forced on this surface, the energy variation along the rotation trajectory minimized by using the Nudged Elastic Band (NEB) method [START_REF] Sheppard | Optimization Methods for Finding Minimum Energy Paths[END_REF] is similar to the one presented in Figure 1b (see Figure S2 in Supporting Information).

The En electronic excited states involved in the rotation are certainly composed of several unoccupied molecular orbitals and the corresponding En(,) potential energy surfaces associated to the rotation are rather complex. [START_REF] Perera | Controlled Clockwise and Anticlockwise Rotational Switching of a Molecular Motor[END_REF] Since our goal here is to understand how the excited states can be involved to reach an unidirectional rotation, we can select (1) for the analytic expression of the first excited state E1(,) and include a shifting term in this E1(,) to change the angular position of its minima relative to

E0(,).

The molecule-motor equation of motion. After the construction of a general model for the potential energy surfaces of a molecule motor, the next step is to construct its general system of motion equations. A preparation of the molecule motor in a coherent non-stationary quantum superposition of states by mixing its ground and some molecule electronic excited states must be enough to provide the energy required to drive the motion. [START_REF] Joachim | The Driving Power of the Quantum Superposition Principle for Molecule-machines[END_REF] This preparation has to be cyclic in time since the rotor will lose rapidly the coherence of this initial preparation due to the large number of internal degrees of freedom of the molecule-motor and to the finite lifetime of the excited states. For the two angles used in our molecular motor model, the system of semi-classical equations of motion can be written:

I d 2  dt 2    c n t   2  E n ,      a 2 d  dt  F t  a (2)
I d 2  dt 2    c n t   2  E n ,      a 2 d  dt  F t  a (3)
where the molecular rotor is described by a disk of mass m = 600 amu and of radius a = 1 nm, with an extremely small I = 10 -42 kg m 2 momentum of inertia with respect to  and  being the friction coefficient of the rotor motion. Notice that in (3), the momentum of inertia of the rotation along  is supposed to be smaller (around 10 -44 kg m 2 ) than in (2). However, we have observed that this difference has very little effect on the dynamic behavior of the system. Accordingly, we have used the same value of I = 10 -42 kg m 2 in both (2) and (3).

The coherent non-stationary preparation is described in Considering the E0(,) ground state, the E1(,) excited state and the decay rate , a new system of equations can be derived from ( 2) and ( 3) to describe molecule motor dynamics taking into account the total decoherence of the superposition included in ( 2) and (3): [START_REF] Zheng | Dipolar and Nonpolar Altitudinal Molecular Rotors Mounted on an Au(111) Surface[END_REF] where (t-ti) is a time shifted step function that permits the force F to act at a given time series ti and to reset the excited state preparation with a ground state total depopulation.

I d 2  dt 2  (1e t )(t t i ) i    E 0 (,) e t i  (t t i )  E 1 (,)a 2 d dt  Fa i  (t t i ) (4) I d 2  dt 2  (1e t )(t t i ) i    E 0 (,) e t i  (t t i )  E 1 (,)a 2 d dt  Fa i  (t t i )
Therefore, there is in (4) and ( 5) no more coherence between the ground and the excited states as compared to (2) and (3).

Before going to the full optimisation of F(t) to drive the rotor one way, let us comment on what happens when this rotor is driven only in its ground state by an external random force F(t). In this case, the system (4) and ( 5) simply reduces to the system:

I d 2  dt 2    E 0 (,) a 2 d dt F(t)a (6) I d 2  dt 2    E 0 (,) a 2 d dt F(t)a (7)
With this system, we have tested whether a one-way rotation trajectory can be stabilized on E0(,) using a simple Gaussian distributed F(t) driving force. This force was applied in average, half time on ( 6) and half time on (7). The system ( 6) and ( 7) was solved using a Verlet-like algorithm based on the finite differences method. Due to the extremely small I value, much care have been taken to ensure the stability of the numerical solutions. 4) and ( 5) can be constant for all random distributed t i or be also random in strength following for example a Gaussian distribution. The first case is equivalent to consider that each electron transfer event through the molecule is inelastically active. However, this is certainly not the case in a molecular tunnel junction. A Gaussian distribution for F along the random t i represents better the low efficiency of an inelastic tunneling excitation. By selecting a Poisson time-dependent excitation distribution at t i , the energy is provided to the molecule-motor by the shot noise. Furthermore, each time t i a zero mean Gaussian distributed F is applied to the molecule, it must act on  or  with an equal probability over time.

Unlike ( 6) and ( 7) using only the ground state of a molecule-motor, a one-way rotation trajectory is now obtained with ( 4) and (5) using the above described Poisson noise and with both E0(,) and E1(,) fully symmetric in  and . As presented in Fig. 5, the necessary condition for such a rotation is a shift of the E1(,) minima relative to the E0(,) ones. The rotation stops when there is a coincidence between the

E1(,) minimum and E0(,) maximum or between the E1(,) and E0(,) minima.

Notice that such behavior was previously reported in another context for the directed transport of Brownian particles in a double symmetric potential [START_REF] Sanchez-Palencia | Directed Transport of Brownian Particles in a Double Symmetric Potential[END_REF] .

According to Fig. 5, a change in the rotation direction is obtained by changing the min-max relative shift between E0(,) and E1(,). This was observed experimentally with the Fig. 1a molecule-motor by locating the tip apex of the STM on different rotor arms to be coupled to different excited states of the molecule at the same positive bias voltage range. [START_REF] Perera | Controlled Clockwise and Anticlockwise Rotational Switching of a Molecular Motor[END_REF] This gives a design rule for new molecule-motors with a possible reversible choice of the rotation direction since excited states with different min-max relative shifts with respect to the ground state can be reached depending on the location of the excitation. Knowing the detailed topology of the excited electronic states potential energy surface of a molecule-motor becomes now an important part of its design. For instance, an elegant example of a molecule able to shift its min-max using hydrogen bonds was recently reported by L. M. Frutos, D. Sampedro et al. [START_REF] García-Iriepa | Chiral Hydrogen Bond Environment Providing Unidirectional Rotation in Photoactive Molecular Motors[END_REF] With the same barrier height (0.25 eV) in E0(,) and E1(,), a moleculemotor of the type presented Fig. 1 is able to rotate one way for an average Gaussian F between 0.02 and 0.2 pN. The necessary condition is that the E0(,) and E1(,) minmax must be shifted as discussed above and presented in Fig. 5. The average time interval between two ti that forges the Poisson distribution in ( 4) and ( 5) must also be tuned in order to get a stable unidirectional rotation. For example, according to the solution of ( 4) and ( 5) and for an excited state decay time  -1 of 1 ns, the rotor of the molecule-motor rotates one-way for an average time separation between two ti larger than 0.35 ns, reaching a maximum rotation speed at a 1.25 ns ti interval. After this threshold, the average speed decays exponentially due to the decrease in the energy supplied to the molecule per unit of time as presented in Fig. 6a. Therefore, when  decreases, that is for very long E1(,) life times, it is necessary to increase the average excitation time further in order to keep control of the rotation. The control of the rotation is better when the average time interval between two ti is long enough as compared to the lifetime of the excited state (here longer than 2.5 ns). For example, an average interval of 0.35 ns and of 7 ns between 2 consecutive ti leads to similar rotation velocities (~7.5•10 6 turns/s). This corresponds to a tunneling current intensity below 1 nA for this life time. Notice that the energy provided to the molecule-motor in the first case is twenty times larger than in the second (VERY GOOD, HOW TO EVALUATE THIS ENERGY? SEE THE NEXT RED REMARK BELOW). A representation of the two cases is presented in Fig. 6b.

Our model assumes that the relaxation process occurs between E1(,) and E0(,) whatever their energy difference. Therefore, it is not the absolute energy difference between E0(,) and E1(,) in ( 4) and ( 5) which provide the energy for a one way rotation (AND SO What it is ..?). Furthermore, the barrier height along their respective potential energy surface of the states controls the rotor average velocity. We have employed for the simulations the value previously calculated with the ASED+ semi-empirical method, which is 0.25 eV for the ground state, for a calculated average We have demonstrated that a molecule without an intrinsic asymmetry can reconstruct a one-way rotation by using its internal characteristics. These findings will help the optimization of the chemical structure of new molecule-motors, not only to rotate one-way but certainly also to develop a true motive power to be able to move atoms, small molecules or to drive cooperatively a train of solid state nano-gears.

METHODS

The rotation potential energy profile of the molecule motor ground state was obtained by means of the ASED+ semi-empirical technique. [START_REF] Ample | A semi-empirical study of polyacene molecules adsorbed on a Cu(110) surface[END_REF] Rotation energy profile of 2,2'-dibromobiphenyl, partial geometry optimizations and scan calculations on this molecule to build the potential energy surfaces were done with the semi-empirical AM1 method [START_REF] Dewar | Ground States of Molecules. 38. The MNDO Method. Approximations and Parameters[END_REF] as implemented in Gaussian 09. 26 Numerical simulations were carried out with a home-built program that solves the dynamics equations by means of a Verlet-like algorithm based on the finite differences method.

A

  simple mechanical model for a molecule-motor. The electronic ground and first excited states potential energy surfaces of the Fig. 1 motor mainly depend on 9 angles: the top rotor rotation angle  around the central Ru axis, the 5 flipping angles  1 to  5 of the 5 rotor phenyl arms and the 3 stator legs deformation angles  1 to  3 .

  maximum is symmetric, as expected from the original symmetry of the molecule. With no external forces applied, this molecule remains in one of the E o (, ) energy minima. When a  rotation imposes to rotate from A to B and the molecule energy is minimized at each  steps of 1°, the Fig. 3b top surface trajectory is obtained. If now, starting from B, the molecule is forced to return back to A, the Fig. 3b down trajectory is obtained. Following both trajectories by plotting the corresponding potential energy variations as a function of , the Fig. 3c saw-tooth like curves are obtained. Of course, a unique minimum energy path is expected after a complete search with a full optimization of the reaction coordinate of a non-forced rotation. Notice that an asymmetric energy rotation trajectory with two different pathways between A and B

( 2 )

 2 and (3) by the cn(t)summation over the En(θ,) ground and electronic excited states potential energy surfaces. If the superposition coherence is step by step preserved along the rotation time, a semi-classical motion will result based on the multiple potential energy surfaces En(θ,) employed for the dynamics and according to the cn(t) amplitudes of the superposition. But such a coherence is very difficult to preserve on the long run even by a cyclic re-preparation of the superposition. All the cn(t) for n > 0 will progressively decay in time towards the ground state with an average decay rate β. In (2) and (3), the external driving force F(t) is there to compensate for this decoherence and relaxation to sustain a rotation.

  For a given Vo value and exploring different friction parameters relative to the average strength of F(t), it was not possible to construct a  trajectory on E0(,) corresponding to a one-way rotation of the rotor even if F(t) is time correlated. This fact is in good agreement with the general behavior of Brownian motors, which cannot undergo directed motion in the presence of only one source of anisotropy. 20 Driving a one-way rotation with a shot noise. In a molecular tunneling junction, the tunneling current intensity is the result of billions of electron transfers per second happening at random time. The temperature independent statistical time distribution of this random process is a Poisson distribution.21 Its first momentum gives the tunneling current intensity through the molecule depending on the bias voltage used whereas its second momentum measures the time fluctuations of this intensity and is characteristic of a shot noise. The force F during each excitation in (

velocity of 3 .

 3 43•10 7 turns/s. Increasing the energy barrier of both states up to 0.44 eV leads to a slightly slower rotation (3.28•10 7 turns/s). However, when decreasing the barrier in E0(,) to 0.25 eV while keeping E1(,) at 0.44 eV, a smaller average velocity is calculated (2.25•10 7 turns/s). In contrast, a barrier of 0.44 eV in E0(,) and of 0.25 eV in E1(,) leads to a large average velocity of 4.11•10 7 turns/s. Further modifications of the energy barriers must be accompanied by a new optimization of the average time interval between 2 ti and of the average Gaussian force in order to keep the one-way rotation.CONCLUSIONSFormally equivalent to a particle in solution, which is subject to a thermal noise due to random collisions with solvent molecules, 20 a surface-mounted molecule in ultra high vacuum can be exposed to a shot noise coming from the STM tip. This noise has been recently used to drive the motion of a single molecule motor.[START_REF] Perera | Controlled Clockwise and Anticlockwise Rotational Switching of a Molecular Motor[END_REF] Inspired by this recent achievement, we have carried out a systematic study of the dynamics of a single molecule motor model driven by a shot noise to model a tunneling current inelastic excitation. The use of an excited state in driving the rotation of the molecule motor allows the conversion of a Poisson time-distributed applied force to a controlled unidirectional motion. For a barrier height of 0.25 eV, the motor reaches a maximum speed of 4.11•10 7 turns/s. Furthermore, the direction of rotation of a molecule-motor can be reversed depending on the characteristics of the excited state involved in the process.

Figure 1 :

 1 Figure 1: The detailed chemical structure (a) and the surface adsorption conformation

Figure 2 .

 2 Figure 2. Asymmetric potential energy profile for a forced rotation of the molecule

Figure 3 .

 3 Figure 3. a) 2,2'-dibromobiphenyl with the significant atoms labelled. Rotation angle is

Figure 4 .

 4 Figure 4. Potential energy surface, given by Equation 1, used in numerical simulations

Figure 5 .

 5 Figure 5. Effect of the electronic states shifting on the rotation efficiency and direction

Figure 6 .

 6 Figure 6. a) Average velocity as a function of the average time between two hits, for a


E Chereul

D Grenier

A.-L Perrier

F Taborik

L Mahieu-Williame

K Tse

Ve Koon

T Chuzel

S Martin

L Magnier

X Pesesse

L Mahieu

; T Chuzel

Sandra Pietri

H Contamin

O Beuf

Morphological assessment of non-human primate models of osteoarthritis using HR-MRI and µCT arthrography

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

Introduction

Small animal models of osteoarthritis (OA) do not mimic perfectly the complex conditions occurring in human OA. OA that closely resembles the human condition occurs naturally in primate making non-human primates (NHP) useful to model the human disease. Non-invasive techniques such as 3D HR-MRI have been validated to directly assess the cartilage thickness on guinea pigs (1) and instrumental developments allowed volume quantification in the different compartments of the cartilage can be achieve on rat models of OA (2)(3). Nonetheless, spatial resolution is limited compared to CT scanner that however needs contrast agent injected in the joint to depict cartilage limits. The aim of this work was, based on morphological parameters assessed on MRI and µCT arthrography (CTA) acquisitions, to characterize an induced model of OA by transection of the anterior cruciate ligament (ACL).

Material and Method

The ethical guidelines for animal experimental investigations were followed and the experimental protocol was approved by the Animal Ethics Committee of our institution. Three groups of young four year old female primates were constituted. (i) Group 1 (n=3) and Group 3 (n=3) with control animals where only the right knee was injected with contrast agent (Hexabrix 320mg/ml); (ii) Group 2 (n=6) with ACL transection of the right knee; Group 1 and 3 both allow to assess the potential µCTA protocol impact on the model (comparing right and left knee thicknesses by MRI) and the model characterization (comparing right knee thicknesses of all groups). Primates were examined at baseline, 30, 60, 90 and 180 days after surgery. For µCTA, an additional measurement point was performed at D15. MRI was performed on a 1.5T Siemens Sonata system using a dedicated home-made pair of two-channel array coil. Each dual-channels array coil was interfaced with a flex interface from Siemens and a coil configuration file was created to drive the interface in array mode. HR-MRI was performed in the sagittal plane using a 3D water excitation FLASH sequence. A total of 120 partitions (220 µm thick) were acquired with an in-plane pixel of 112 x 131 µm2. The scan time was 20 min. µCTAs were performed on a GE Locus µ-CT at standard voltage and amperage parameters with an isotropic resolution of 90µm. The scan time was 15 min. For each animal, both knees were scanned in the same FOV. For both modalities, 3D thicknesses of the lateral and medial cartilage tibial plateaus were assessed using the same image processing protocol. First a double segmentation procedure was achieved with a rough and manually handled contour segmentation to isolate the cartilage regions of interest (ROI) followed by a regional automatic global segmentation procedure extracting medial and lateral cartilage ROIs. Inside the segmented ROIs, the quantification of cartilage thicknesses was performed using the method described by Hildebrand et al. (4).

Results

The µCTA protocol (repeated injection of contrast agent in the knee) did not interfere with the model development since comparisons of cartilage thickness MRI measurements between injected and non-injected knees in control animals showed a mean residual of R1 = 0.015±0.134 mm, a value well below MRI spatial resolution. Additionally; the cartilage thickness µCTA measurements (acquisition and processing protocols) did not show any bias with respect to MRI-based cartilage thickness in control animals: the mean residual between both measurements was R2 = 0.036±0.068 mm, a value well below MRI or µCTA resolution too. Both imaging modalities showed superimpose 3D thickness distributions measurements. Mean cartilage thickness of medial tibia plateaus of the right joint were found constant for G1 but decreased significantly for G2 from D15 (-2.1±5.5 %) to D90 (-25.8±6.1 %) and with intermediate values on D30 (-12.6±8.1 %) and D60 (-20.2±4.8 %).

Conclusion

The cartilage thickness µ-CTA measurement on a NHP model of OA by transection of the ACL was validated with respect to the MRI approach since no residual measurement between both methods were found above the resolution of these techniques. Moreover the NHP model of OA was characterized by both imaging methods showing a monotone progression of the cartilage thinning up to -25.8±6.1 % on D90.


Cristina Nica
email: nica.cristina87@gmail.com

Agnès Braud
email: agnes.braud@unistra.fr

Florence Le Ber
email: florence.leber@engees.unistra.fr

Exploring Heterogeneous Sequential Data on River Networks with Relational Concept Analysis

Nowadays, many heterogeneous relational data are stored in databases to be further explored for discovering meaningful patterns. Such databases exist in various domains and we focus here on river monitoring. In this paper, a limited number of river sites that make up a river network (seen as a directed graph) is given. Periodically, for each river site three types of data are collected. Our aim is to reveal userfriendly results for visualising the intrinsic structure of these data. To that end, we present an approach for exploring heterogeneous sequential data using Relational Concept Analysis. The main objective is to enhance the evaluation step by extracting heterogeneous closed partially-ordered patterns organised into a hierarchy. The experiments and qualitative interpretations show that our method outputs instructive results for the hydro-ecological domain.

Introduction

In Europe, according to the recomandations of Water Framework Directive [START_REF]European Union: Directive 2000/60/ec of the European parliament and of the council of 23 october 2000 establishing a framework[END_REF], a special attention should be given to preserving or restoring the good state of waterbodies. Monitoring and assessing the effect of pollution sources and the one of restoration processes must be done in order to improve domain knowledge and to define guidelines for stakeholders.

During an interdisciplinary research project, namely REX1 , many and various hydro-ecological data have been collected periodically between 2002 and 2014 from a river network (seen as a directed graph). These data are about past restoration projects, temporal evolution of aquatic ecosystems and land use pressures. The REX data have been studied with statistical methods, but relational information could not be taken into account (e.g. effect of upper restoration).

Therefore, in this paper we deal with heterogeneous sequential data and we try to make sense of them by means of hierarchies of heterogeneous closed partially-ordered patterns (cpo-patterns, [START_REF] Casas-Garriga | Summarizing sequential data with closed partial orders[END_REF]) that exhibit the natural structure of these data. Indeed, a cpo-pattern is compact, contains the same information as the set of sequential patterns it synthesises and is user-friendly thanks to its representation as a directed acyclic graph. Moreover, a hierarchy provides a convenient way for navigating to interesting heterogeneous cpo-patterns.

To that end, we extend our self-contained approach RCA-Seq -introduced in [START_REF] Nica | Extracting hierarchies of closed partially-ordered patterns using relational concept analysis[END_REF] and based on Relational Concept Analysis (RCA, [START_REF] Rouane-Hacene | Relational concept analysis: Mining concept lattices from multi-relational data[END_REF]) -for exploring classical sequential data to exploring heterogeneous sequential data. We propose to manipulate the data as a directed graph that has heterogeneous itemsets (i.e. a set of itemsets of different domains) as vertices and binary spatial relations as edges. Accordingly, we show that RCA-Seq is robust and can be appropriate for exploring graphs and networks, as well.

The paper is structured as follows. Section 2 gives the theoretical background of our work. Section 3 describes the analysed heterogeneous hydro-ecological data. Section 4 introduces a data model used to encode the data into the RCA input. Section 5 presents the RCA-based exploration step. Section 6 explains our proposal for directly extracting hierarchies of heterogeneous cpo-patterns from the RCA output. In Sect. 7 experimental results are discussed. Section 8 presents related work. Finally, we conclude the paper in Sect. 9.

Preliminaries

Heterogeneous CPO-Patterns

Let I = {I 1 , I 2 , ..., I m } be a set of items. An itemset IS = (I j 1 ...I j k ), where I j i ∈ I, is an unordered subset of I. An itemset IS with k items is referred to as k-itemset. Let IS be the set of all itemsets built from I. A sequence S is a non-empty ordered list of itemsets, S = IS 1 IS 2 ...IS p where IS j ∈ IS. The sequence S is a subsequence of another sequence S = IS 1 IS 2 ...IS q , denoted as S s S , if p ≤ q and if there are integers j 1 < j 2 < ... < j k < ... < j p such that IS 1 ⊆ IS j 1 , IS 2 ⊆ IS j 2 , ..., IS p ⊆ IS j p .

Sequential patterns have been defined by [START_REF] Agrawal | Mining sequential patterns[END_REF] as frequent subsequences discovered in a sequence database. A sequential pattern is associated with a support θ, i.e. the number of sequences containing the pattern.

Suppose now that there is a partial order (i.e. a reflexive, antisymmetric and transitive binary relation) on the items, denoted by (I, ≤). We say that (I, ≤) is a poset. A multilevel itemset IS ml = (I j 1 ...I j k ), where I j i ∈ I and I ji , I j i ∈ IS ml such that I ji ≤ I j i , is a non-empty and unordered set of items that can be at different levels of granularity (i.e. items from different levels of poset (I, ≤)). We denote by IS ml the set of all multilevel itemsets built from (I, ≤). The partial order on the set of all multilevel itemsets (IS ml , ⊆ ml ) is defined as follows:

IS ml ⊆ IS ml if ∀I j ∈ IS ml , ∃I j ∈ IS ml , I j ≤ I j and ∀I l = I j , ∃I l = I j such that I l ≤ I l .
To illustrate this, let us consider I 1 = {a, b, c, d, e, Consonants, Vowels, Letters} a set of items and (I 1 , ≤) a partial order depicted in Fig. 1, where an edge represents the binary relation is-a, denoted by ≤. Let H = {I 1 , I 2 , ..., I n } be a set of distinct sets of items, where I j with j ∈ {1, ..., n} represents a domain. We note that I j can be a poset or an unordered set. Let IS j be the set of all itemsets built from I j ∈ H. A heterogeneous itemset IS H = {IS 1 , IS 2 , ..., IS n }, where IS j ∈ IS j , is a non-empty and unordered set of itemsets built from distinct sets of H. In addition, a multilevel heterogeneous itemset (hereinafter referred to as heterogeneous itemsets) is a set of itemsets that has at least one multilevel itemset.

Let IS H be the set of all heterogeneous itemsets built from H. The partial order (IS H , ⊆ H ) is defined as follows:

IS H ⊆ H IS H if ∀IS k ∈ IS H , ∃IS k ∈ IS H such that IS k ⊆ IS k , where IS k , IS k ∈ IS k , k ∈ {1, .
.., n}. The order on heterogeneous itemsets is defined accordingly relying on ⊆ ml .

To illustrate this, let us consider H = {I 1 , I 2 }, where I 1 is partially ordered as shown in Fig. 1 and I 2 = { , ♦, } is an unordered set of shapes. Furthermore, let be two multilevel heterogeneous itemsets IS H1 = {(Vowels c), (♦)} and IS H2 = {(a c), ( ♦)}, then IS H1 ⊆ H IS H2 since (Vowels c) ⊆ ml (a c) (that is a ≤ Vowels and c ≤ c) and (♦) ⊆ ( ♦).

A heterogeneous sequence S H = IS H1 IS H2 ...IS Hr , where IS Hi ∈ IS H with i ∈ {1, ..., r}, is a non-empty ordered list of heterogeneous itemsets. In addition, a heterogeneous sequence that has at least one multilevel heterogeneous itemset represents a multilevel heterogeneous sequence (hereinafter referred to as heterogeneous sequence). A heterogeneous sequence S H is a subsequence of another heterogeneous sequence S H = IS H1 IS H2 ...IS Hq , denoted by S H s H S H , if r ≤ q and if there are integers j 1 < j 2 < ... < j k < ... < j r such that IS H1 ⊆ H IS Hj 1

, IS H2 ⊆ H IS Hj 2 , ..., IS Hr ⊆ H IS Hj r . A frequent heterogeneous subsequence is called a heterogeneous sequential pattern.

To illustrate this, let be two heterogeneous sequences on the aforementioned

H = {I 1 , I 2 }: S1 H = {(a Consonants), ( ♦)} {(Letters), ∅} and S2 H = {(a d), ( ♦)} {(a c), ( )} . Then S1 H s H S2 H since -{(a Consonants), ( ♦)} ⊆ H {(a d), ( ♦)}, i.e. a ≤ a, d ≤ Consonants, ( ♦) ⊆ ( ♦), -{(Letters), ∅} ⊆ H {(a c), ( )}, i.e. a ≤ Letters (or c ≤ Letters), ∅ ⊆ ( ).
Partially ordered patterns, po-patterns, have been introduced by [START_REF] Casas-Garriga | Summarizing sequential data with closed partial orders[END_REF], to synthesise sets of sequential patterns. Formally, a po-pattern is a directed acyclic graph G = (V, E, l). V is the set of vertices, E is the set of directed edges such that E ⊆ V × V, and l is the labelling function mapping each vertex to an itemset. With such a structure, we can determine a strict partial order on vertices u and v such that u = v : u < v if there is a directed path from tail vertex u to head vertex v. However, if there is no directed path from u to v, these elements are not comparable. Each path of the graph represents a sequential pattern, and the set of paths in G is denoted by P G . A po-pattern is associated to the set of sequences S G that contain all paths of P G . The support of a po-pattern is defined as

Support(G) = |S G | = |{S ∈ D S |∀M ∈ P G , M s S}|. Furthermore, let G and G be two po-patterns with P G and P G their sets of paths. G is a sub po-pattern of G, denoted by G g G, if ∀M ∈ P G , ∃M ∈ P G such that M s M . A po-pattern G is closed, referred to as cpo-pattern, if there exists no po-pattern G such that G ≺ g G with S G = S G .
A cpo-pattern whose paths are heterogeneous sequential patterns is called heterogeneous cpo-pattern.

RCA

RCA extends the purpose of Formal Concept Analysis (FCA, [START_REF] Ganter | Formal Concept Analysis: Mathematical Foundations[END_REF]) to relational data. RCA applies iteratively FCA on a Relational Context Family (RCF). An RCF is a pair (K, R), where K is a set of object-attribute contexts and R is a set of object-object contexts. K contains n object-attribute contexts

K i = (G i , M i , I i ) , i ∈ {1, ..., n}. R contains m object-object contexts R j = (G k , G l , r j ) , j ∈ {1, ..., m}, where r j ⊆ G k × G l is a binary relation with k, l ∈ {1, ..., n}, G k = dom(r j
) the domain of the relation, and G l = ran(r j ) the range of the relation. G k and G l are the sets of objects of the object-attribute contexts K k and K l , respectively. RCA relies on a relational scaling mechanism that is used to transform a relation r j into a set of relational attributes that extends the object-attribute context describing the set of objects dom(r j ). A relational attribute ∃r j (C), where ∃ is the existential quantifier, and C = (X, Y ) is a concept whose extent contains objects from ran(r j ), is owned by an object g ∈ dom(r j ) if r j (g) ∩ X = ∅. Other quantifiers can be found in [START_REF] Rouane-Hacene | Relational concept analysis: Mining concept lattices from multi-relational data[END_REF]. RCA process consists in applying FCA first on each object-attribute context of an RCF, and then iteratively on each object-attribute context extended by the relational attributes created using the learnt concepts from the previous step. The RCA result is obtained when the families of lattices of two consecutive steps are isomorphic and the object-attribute contexts are unchanged.

Heterogeneous Hydro-Ecological Data

We focus on hydro-ecological data concerning Rhine river. These data have been collected during REX project. A number of 15 river sites (i.e. fixed points) in the Alsace plain were monitored between 2002 -2014. These sites make up the In addition, a river site is included in a river segment that can be restored at one or more locations during the whole monitored period of time. There are two types of restoration: global and wetland. According to the number of restorations i undertaken during 2002 -2014, there are three levels of the type of restoration as follows:

L1 if i ∈ (0, 2], L2 if i ∈ (2, 5] and L3 if i ∈ (5, ∞).
For instance, by analysing Fig. 4, the heterogeneous itemset {(NITRITE red ), (IBGN green ), (FORESTS low 500m INDUSTRIAL AREAS high 500m )} is associated with river site S7742 in period 2010 -2014; the itemset (Wetland L1 Global L2 ) is associated with river segment 20165 in the whole monitored period.

Data Modelling

Our purpose is to highlight how the ecological state of aquatic ecosystem and the land use in upstream river sites impact the aquatic ecosystem in downstream river sites and, thus determine the necessity of the restorations of river segments. The set of river sites contains all ordered pairs given by the Cartesian product of the river sites shown in Fig. 2 and the three monitored periods of time. BIO indicators, PHC parameters and land use correspond to the taxonomies depicted in Fig. 3. In addition, let us mention that we consider, firstly, river segments as target objects since these are restored; and secondly, river sites as non-target objects since these are assessed to understand the necessity of restorations.

Table 1: Analysed sets of objects; the set of river sites is given by the Cartesian product of the river sites given in Fig. 2 and the three monitored periods of time The links between objects are highlighted by using binary relations as follows:

Exploration of Heterogeneous Data by Using RCA

In this section we briefly recall and slightly adapt the RCA-exploration step of sequential data presented in our previous paper [START_REF] Nica | Extracting hierarchies of closed partially-ordered patterns using relational concept analysis[END_REF]. Firstly, the RCA input (RCF) -an excerpt is depicted in Tab. 2 -is built by relying on the data model shown in Fig. 5 and on the sets of objects given in Tab. 1. Basically, this RCF encodes all hydro-ecological data collected during the whole monitored period 2002 -2014. There is an object-attribute context for each rectangle out of the data model, precisely KSegments (river segments), KSites (river sites), KRT (restoration types), KBIO (BIO indicators), KPHC (PHC parameters) and KLU (land use). KSites has no column since river sites are described only by using the has indicator, has parameter and is surrounded by relations. Similarly, KSegments has no column since river segments are described by using the has restoration relations. As shown in Tab. 2, a nominal scaling is Fig. 5: Modelling heterogeneous hydro-ecological data used to build KRT in order to obtain a partial order over the unordered set of restoration types. In contrast, an ordinal scaling is used to build KBIO, KPHC and KLU in order to encode the taxonomies given in Fig. 3. In addition, there are 21 object-object contexts, one for each relation out of the data model, e.g. in Tab. 2 RSite-ds-Site (river site is downstream of river site) and RSite-red-BIO (river site has red BIO indicator ).

Secondly, RCA is applied3 to the RCF shown in Tab. 2 and a family of concept lattices is obtained after four iterations. The RCA output comprises six concept lattices, one for each object-attribute context, as follows: target lattice L KSegments (river segments), non-target lattice L KSites (river sites), lattice of restoration types L KRT and the taxonomy lattices L KBIO , L KPHC , L KLU that correspond to the taxonomies illustrated in Fig. 3. The concepts of the latter three lattices are used to describe river sites by means of the revealed qualitative relational attributes. Similarly, the concepts of L KRT are used to describe river segments.

It is worthwhile to mention that the RCA-based exploration step employs a relational scaling mechanism that relies on quantifier ∃ because the objective is to capture all the relations between the analysed objects. The target lattice and non-target one contain respectively 860 and 4554 concepts.

Table 2: Excerpt of the RCA input composed of object-attribute contexts (KSites, KRT and KPHC) and object-object contexts (RSite-ds-Site and RSite-red-BIO).

KSites (S7743,I) (S7743,II) (S7743,III) (S8735,I)) (S8735,II) KRT Wetland Global Wetland × Global × KPHC PHC NITROGEN NITRITE PHOS PHC × NITROGEN × × NITRITE × × × PHOS × × RSite-ds-Site (S7872,I) (S7872,II) (S9515,I) (S7743,I) × (S7743,II) × (S7743,III) (S8735,I)) × × (S8735,II) × RSite-red-PHC PHC NITROGEN NITRITE (S7743,I) (S7743,II) × (S7743,III) (S8735,I)) (S8735,II) ×

Extraction of Heterogeneous CPO-Patterns Organised into a Hierarchy

To extract a hierarchy of heterogeneous cpo-patterns from the RCA output (obtained as explained in Sect. 5), we apply and slightly modify the RCA-Seq approach presented in [START_REF] Nica | Extracting hierarchies of closed partially-ordered patterns using relational concept analysis[END_REF]. Let us note that the hierarchy is directly obtained since each concept of the target lattice is associated with a heterogeneous cpopattern.

Briefly, starting with a concept from the target lattice, a heterogeneous cpopattern is extracted by navigating interrelated concept intents. For each navigated concept intent, a vertex (labelled with an itemset) is derived from all (spatial-)qualitative relational attributes (hereinafter referred to as qualitative relational attributes) whereas an edge is derived from a spatial relational attribute. A qualitative/spatial relational attribute highlights a qualitative/spatial relation.

In fact, in this paper a vertex derived from a concept intent of the non-target lattice (river sites) is actually a heterogeneous vertex labelled with a heterogeneous itemset. Basically, an itemset of the heterogeneous itemset is built for each set of qualitative relational attributes, which define the same qualitative relation, out of the concept intent. Therefore, for a concept intent we analyse the qualitative relational attributes, which are built using a qualitative relation r q and concepts from a taxonomy lattice L Ktax = (C Ktax , Ktax ), to derive items as follows:

-from a qualitative relational attribute ∃r q (C tax ), where C tax ∈ C Ktax , is derived an item, denoted by "item q ", where extent(C tax ) = {item} and q is the item quality according to r q ; -if there is no qualitative relational attribute that highlights relation r q and the information introduced by this relation is mandatory, then is derived an item, denoted by "item ? " where extent( (L Ktax )) = {item}, that constitutes the 1-itemset obtained for this type of information; conversely, if the information introduced by this relation is not mandatory, then no item is derived and, thus ∅ is obtained for this type of information.

Let us mention that a vertex derived from a concept intent of the target lattice (river segments) is labelled with a multilevel itemset. As described in [START_REF] Nica | Extracting hierarchies of closed partially-ordered patterns using relational concept analysis[END_REF], this itemset can contain: the abstract item ("? ? " -different types of restoration at distinct number of locations), qualitative abstract items (e.g. "? L1 " -different types of restoration at most 2 locations) and/or concrete items (e.g. "Global L3 " -global restorations at more than 5 locations).

Results and Evaluation

In this section, we present some interesting results obtained with the RCA-Seq approach applied to the heterogeneous data collected between 2002 -2014 from the river network shown in Fig. 2. The evaluation relies on the positive feedback given by a hydro-ecologist who is well acquainted with cpo-patterns.

By navigating the lattices starting from the target concepts of L KSegments we obtain a hierarchy of 859 heterogeneous cpo-patterns (the bottom concept of L KSegments is not considered since generally it is too specific and associated with no river segment). It is worthwhile to mention that a smaller hierarchy of cpo-patterns can be extracted by varying the quantifiers employed by the relational scaling mechanism. In addition, various measures [START_REF] Nica | Exploring temporal data using relational concept analysis: An application to hydroecological data[END_REF] can be used to select relevant heterogeneous cpo-patterns.

Figure 6 depicts an excerpt from this hierarchy, precisely the organised 1 , 2 , 3 , 4 , 5 , 6 and 7 heterogeneous cpo-patterns.

A cpo-pattern is associated with a set of river segments whose number (support) is shown in . The restoration types of these river segments are illustrated in , e.g. Global L1 meaning that the river segments have at most 2 locations with global restoration. A vertex ( ) is associated with a set of river sites and it is labelled with PHC parameters and their qualitative values. A vertex can have additional information: land use ( ) and BIO indicators (♦). In the following, we focus on the cpo-patterns 1 , 4 and 6 .

CPO-Pattern 1 is associated with 11 ( in Fig. 6) river segments that contain at most 2 locations with global restoration. In addition, itemset (PHC blue ) reveals locally (i.e. in the associated river segments) a very good PHC state of water.

CPO-Pattern 4 is associated with 5 river segments that contain at most 2 locations with global restoration. Itemset (IBD green ) (♦ in Fig. 6) reveals locally a good ecological state of water based on the analysis of diatom species. In addition, the PHC state of water is very good for temperature, biological oxygen demand and nitrogen that represent a part of the abiotic characteristics suitable for diatom species [START_REF] Raibole | Impact of physico-chemical parameters on microbial diversity: Seasonal study[END_REF].

CPO-Pattern 6 , which is a more concrete specialisation of 5 , is associated with 3 river segments that contain at most 2 locations with global and wetland restorations. Itemset (BIO green ) (♦ in Fig. 6) reveals locally a good ecological state of the aquatic ecosystem. Since BIO is an abstract item, we cannot In contrast, in the upstream rivers at 500 m buffer on the one hand the land use pressures of industrial areas and arable lands are low ; on the other hand, a high percentage of the area is covered with forests that lead to a good ecological state of the aquatic ecosystem in the surroundings. Indeed, by analysing the E vertex, itemset (IBGN green ) (♦, Fig. 7) reveals a good ecological state of the aquatic ecosystem in the period 2010 -2014 based on the analysis of macro-invertebrates. Moreover, water temperature is very good; organic matter (dissolved oxygen, biological oxygen demand and oxygen saturation) is good and very good; nitrogenous parameters (nitrite and ammonium), which are related to organic matter, are as well good and very good; and nutrients (total phosphorous and nitrate) are very good.

By comparing the E vertex with the A , B , C and D vertices, it is noted a degradation up to one level regarding the qualitative values of PHC parameters probably caused by the medium building pressures at 500 m buffer, e.g.:

-AMMONIUM blue and DISSOLVED OXYGEN blue (very good) from E are measured when the surroundings are covered with a low percentage of industrial areas and arable lands (i.e. the land use pressures are low), while AMONIUM green and DISSOLVED OXYGEN green (good) from A , B , C and D are measured when the surroundings are covered with a medium percentage of buildings (i.e. the land use pressures are medium); -TOTAL PHOSPHORUS blue (very good) from E is measured when in the surroundings the land use pressures are low; TOTAL PHOSPHORUS green (good) from B , C and D is measured when in the surroundings the land use pressures are medium.

Furthermore, the cpo-pattern shown in Fig. 7 reflects that BIO indicators seem to be more sensitive (up to two levels of their qualitative values) to land use pressures [START_REF] Villeneuve | Can we predict biological condition of stream ecosystems? a multi-stressors approach linking three biological indices to physico-chemistry, hydromorphology and land use[END_REF][START_REF] Wasson | Quelle limite de " bon état écologique " pour les invertébrés benthiques en rivières ? Apport des modèles d'extrapolation spatiale reliant l'indice biologique global normalisé à l'occupation du sol[END_REF]. For instance, IBGN green in upstream rivers ( E ) in contrast to BIO yellow and IBGN orange locally ( C and D , respectively).

To sum up, by means of cpo-patterns, we can help hydro-ecologists to check well-known correspondences among the analysed ecological factors as well as to consider lesser-known facts.

Related Work

Classical sequential pattern mining approaches deal with sequences whose items are homogeneous and, therefore cannot be applied to heterogeneous sequences (i.e. sequences whose items are different in nature). To our knowledge, [START_REF] Pinto | Multi-dimensional sequential pattern mining[END_REF] proposed the first work for exploring multidimensional sequential data. A multidimensional sequence takes the form (d 1 , d 2 , ..., d m , S), where S is a sequence of itemsets and d i represents the i th type of information associated with S. The authors proposed three methods for extracting multidimensional sequential patterns that rely on classical sequential pattern algorithms (e.g. PrefixSpan [START_REF] Pei | Prefixspan: Mining sequential patterns efficiently by prefix-projected pattern growth[END_REF]).

A key drawback of such multidimensional sequences is the additional information that is constant for all itemsets of sequence S.

In [START_REF] Plantevit | Mining multidimensional and multilevel sequential patterns[END_REF], a multidimensional sequence is defined as an ordered list of multidimensional items. A multidimensional item takes the form (d 1 , d 2 , ..., d n ), where d k is an item of the k th dimension. Furthermore, each considered dimension is represented at different levels of granularity by means of partial orders. Therefore, multilevel sequential patterns can be discovered, as explained in [START_REF] Srikant | Mining sequential patterns: Generalizations and performance improvements[END_REF]. The authors proposed the M3SP algorithm that searches for multidimensional and multilevel sequential patterns in two steps. First, the most specific frequent multidimensional items, referred to as maf-sequences, are found. Second, the mafsequences are used to remodel the original multidimensional sequences and then these sequences are mined by using algorithm SPADE [START_REF] Zaki | Spade: An efficient algorithm for mining frequent sequences[END_REF].

Nevertheless, [START_REF] Egho | A contribution to the discovery of multidimensional patterns in healthcare trajectories[END_REF] highlighted a limitation of M3SP, i.e. the multidimensional items do not allow itemsets whose items are of k th dimension. The authors proposed the MMISP algorithm that considers complex and heterogeneous sequences, where a sequence contains elementary sequences (ES), i.e. itemsets whose items can be of two types: atomic and different in nature taken from user-defined taxonomies or subsets of unordered sets of items. MMISP does not discover directly sequential patterns in heterogeneous data since a preprocessing step is involved, i.e. the original sequences are encoded into classical sequences. In contrast, RCA-Seq directly searches for cpo-patterns (rather than sequential patterns) in complex and heterogeneous data and, besides, reveals how these patterns relate to each other. Moreover, our approach generalises the ES proposed in [START_REF] Egho | A contribution to the discovery of multidimensional patterns in healthcare trajectories[END_REF] by considering its atomic items as 1-itemsets.

Conclusion

RCA-Seq is an approach for exploring classical sequential data. In this paper, we have presented an extension of RCA-Seq that highlights its generality, i.e. the capability to explore sequential data regardless of their complexity. Given heterogeneous sequential data on river networks, we have shown that hydroecologists can draw valuable insights by exploiting the "richness" (e.g. the additional information captured by concept extents and the revealed abstract items) of the RCA-Seq output. In the future, we plan to (i) improve our extension in order to be applicable to large volumes of heterogeneous sequential data and (ii) explore with RCA-Seq complex relational data such as social networks and knowledge graphs.

Fig 1 =

 1 Fig. 1: An example of a partial order on I 1 = {a, b, c, d, e, Consonants, Vowels, Letters}

Fig. 2 :

 2 Fig. 2: River network

Fig. 3 :

 3 Fig. 3: Excerpts from the taxonomies over the three analysed domains

Fig. 4 :

 4 Fig. 4: River segment 20165 ; period III: river site S7742

Fig. 6 :

 6 Fig. 6: Excerpt from the hierarchy of heterogeneous cpo-patterns discovered in the analysed hydro-ecological data. 1 , 2 , 3 , 4 , 5 , 6 and 7 identify the cpo-patterns. is the support (number of river segments) of a cpo-pattern; represents the types of river segment restoration; represents land use; ♦ represents BIO indicators; represents PHC parameters

Figure 7

 7 Figure7depicts a more complex heterogeneous cpo-pattern. This is associated with the river segments 8674 and 19949 that contain at most 2 locations with global restoration.

Fig. 7 :

 7 Fig. 7: A complex heterogeneous cpo-pattern extracted from the analysed hydroecological data. A , B , C , D , E and F identify the vertices; is the support (number of river segments) of the cpo-pattern; represents the types of river segment restoration; represents land use; ♦ represents BIO indicators; represents PHC parameters

http://obs-rhin.engees.eu

http://rhin-meuse.eaufrance.fr/IMG/pdf/grilles-seq-eau-v2.pdf

using http://dolques.free.fr/rcaexplore

Acknowledgement

The REX data were provided by Prof. Jean-Nicolas Beisel and the interpretation of cpo-patterns was done with the help of Corinne Grac (ENGEES Strasbourg).


Isabelle Saniour

Anne-Laure Perrier

Gwenaël Gaborit

Jean Dahdah

Lionel Duvillaret

Olivier Beuf

Direct optical measurement of the RF electrical field for MRI

Purpose

 and may disturb the electromagnetic field with bias measurements. Others optical probes do not measure directly the Efield 5 . In the following, the E-field measurement experiment using a sub-cm electro-optical (EO) probe in a preclinical 4.7-T MR system is presented.

Methods

The EO probe is basically formed from a monocentrosymetric crystal (ZnTe, BSO…) that changes its refractive indexes proportionally to the applied electrical field (Pockels' effect). In fact, this index variation leads to a modification of the polarization state of the distributed feedback (DFB) laser (wavelength λ=1.55 µm), according to the electrical field. This DFB laser is emitted toward the crystal through a 15 m optical fiber. Then, this optical polarization is transmitted, using the same optical fiber, to be analyzed in real time and to be converted to an analogue electrical signal due to optoelectronic components. This small size probe (sub-cm) is non-invasive and insensitive to the variation of the temperature or the magnetic field (static or dynamic) [START_REF] Gaborit | Single Shot and Vectorial Characterization of Intense Electric Field in Various Environments with Pigtailed Electrooptic Probe[END_REF] which is suitable for MRI environment. It can operate in a frequency range from 30 Hz to 18 GHz and the electrical field is directly proportional to the measured output signal. The experiment was performed using a 4.7-T preclinical MRI using a 60 cm inner diameter transceiver birdcage linear coil (figure 1). A moving EO probe was inserted into the transceiver coil to localize E-field concentrations. The optical detected signal was transmitted via an optical fiber to the signal processing unit in order to be visualized by an oscilloscope. The trigger signal was provided by MR system. For this experiment, a RARE sequence with hermite shape RF pulse was used. The imaging parameters are: TR/TE =6020.6ms/14ms, flip angle=90° and turbo factor=8.

Results

First of all, in order to demonstrate that the probe is completely dielectric and non-perturbative; the EO probe was placed on the NiSO 4 phantom .The presence of the probe had no impact on the image quality (absence of artefacts and similar signal to noise ratio). Second, the probe was moving (axial and radial movement) inside the RF coil to detect the presence of E-field. Three hot spots of the E-field were detected clearly at proximal and distal sides of the transceiver coil corresponding to three different probe's locations (figure 2). Their magnitudes vary between 10000 V/m and 20000 V/m. Figure 3 shows the shape of the detected EO signal and related electric field during the emission of RF signals. The E-field is linearly proportional to the EO signal with a factor of linearity around 38000 and with a dynamic range exceeding 120 dB. Using the Fast Fourier Transform (FFT), the frequency of the detected signal is 200 MHz that corresponds exactly to the proton resonant frequency at 4.7-T (figure 4).

Discussion

The EO probe can detect directly the E-field in real time at Larmor frequency with a transverse spatial resolution lower than 5 mm. The analysis of hot spots' magnitudes shows that the E r field is not uniform and it depends on both the position of the EO probe inside the transceiver and the orientation of the probe itself. Further experiments will be performed to study this correlation. The results exhibit the effectiveness of measuring directly the E-field in a preclinical 4.7-T MRI using an optical probe without alteration of environment. Due to availability, the experiments were performed at 4.7-T but will be further addressed in a clinical 3-T MRI system.

Conclusion

The objective of this study was to validate the use of a specific EO probe to determine the local magnitude of the E-field associated to the RF magnetic field. During the RF emission, a nonperturbative EO probe can be used to give a real time measurement of the electrical field in order to protect the patient from any safety risk. Such probe could be used to provide localized E 1 -field map.

Figure 1 .

 1 Figure 1. Schematic set-up of electrical field measurement experiment.

Figure 2 .

 2 Figure 2. Locations of the detected hot spots at proximal and distal sides of the transceiver coil.

Figure 4 .Figure 3 .

 43 Figure 4. The spectrum of the EO signal was obtained using the fast Fourier transform. The frequency corresponds to the proton resonant frequency at 4.7 T MRI

Acknowledgment

This work was supported by the Région Rhône-Alpes and performed within the framework of the LABEX PRIMES (ANR-11-LABX-0063) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).


I Saniour

R Aydé

A L Perrier

G Gaborit

H Dorez

L Duvillaret

J Dahdah

R Sablong

O Beuf

Optical for both active decoupling and conversion/transmission for a MR endoluminal coil Type: Scientific Session communications Topic: Preclinical Studies and Basic Science / MR hardware (RF, magnets, gradients)

Purpose / Introduction An accurate and early detection of the stages of colorectal cancer increases patient survival rates. Spatial resolution of MR images can be improved by using endoluminal coils close to the region of interest [1]. Unfortunately, the clinical use of such coils, deeply introduced in the patient is compromised due to local heating and safety issues in presence of galvanic wires [2]. In the following work, adequate optical solutions both for active decoupling and conversion/transmission of MR signals is proposed and demonstrated.

Subjects and Methods

To be free of any galvanic and wired connection, an optical active decoupling circuit was added to the coil: two photodiodes generate a DC current to change the state of the PIN diode and ensure coil decoupling [START_REF] Aydé | ISMRMESMRMB[END_REF]. The experiments were performed on a 3T MR750 GEHC system to assess the decoupling efficiency based on images. The transmission of the NMR signal was done herein by a coaxial cable. On the other hand and on an optical bench, the optical conversion of the RF signal is realized by associating the endoluminal coil to an electrooptical (EO) Ti:LiNbO3 waveguide [START_REF] Aydé | [END_REF]. A RF magnetic field was generated and applied to the coil resonating at 128 MHz (figure 1). The optical properties of the waveguide vary according to the induced electrical field (Pockels effect) and thereby the polarization of the emitted laser toward the waveguide is modified. This optical polarization state is maintained during transmission and finally treated to have an analogue electrical output proportional to the field to be measured.

Results

The presence of a galvanic or an optical decoupling doesn't affect significantly the image in term of signal intensity uniformity (figure 2). Besides, the graph in the figure 3 shows an excellent linearity of the results and a dynamic range of the input power exceeding 100 dB. The magnetic field is ranging between 0.3 pT (electronic noise) to 2.10 5 pT.

  Optical for both active decoupling and conversion/transmission for a MR endoluminal coil http://abstracts.webges.com/submission/preview/print_publication.php?publication_id=281 2/2 Discussion / Conclusion Based on these satisfactory results, a fully optical endoluminal receiver coil, combining the two described systems, is in development.

	10/6/2015	#281:

Acknowledgments : The authors thank the RhôneAlpes region and DGA for financial support. This work was performed within the framework of the LABEX PRIMES ANR11LABX0063/ANR11IDEX0007.