Soleiman Galeshi
email: soleiman.galeshi-mooziraji@g2elab.grenoble-inp.fr

David Frey

Yves Lembeye

Efficient and scalable power control in multi-port active-bridge converters

Keywords: control», «Modelling», «Estimation technique», «

Power converters for EV», «Scalable control».

Introduction

Application of multi-port active-bridge converter (MAB) in energy systems is increasing due to several advantages of this type of converters. The multi-port transformer in MAB provides galvanic isolation and voltage level compatibility. Soft switching of active-bridges allows for increasing the switching frequency and reducing the size of magnetic components. Another advantage is that MAB converters exchange energy between several sources and loads at the same time through the same medium, which is the magnetic core of the transformer. Many researches and experiments have been done on application of triple port active-bridge converter (TAB) [START_REF] Delmonte | Development of a GaN Based Triple-Active-Bridge for DC Nanogrid[END_REF][START_REF] Katagiri | Decoupling Power Flow Control of Triple-Active Bridge Converter with Voltage Difference between Each Port for Distributed Power Supply System[END_REF]. Quad-active-bridge converters (QAB) have also been studied and designed for several applications such as interfacing three-phase and dc networks [START_REF] Costa | Quad-Active-Bridge DC-DC Converter as Cross-Link for Medium-Voltage Modular Inverters[END_REF][START_REF] Vermulst | Quad-Active-Bridge Single-Stage Bidirectional Three-Phase AC-DC Converter With Isolation: Introduction and Optimized Modulation[END_REF], electric vehicles [START_REF] Nair | A Quad Active Bridge Based on-Board Power Electronic Interface for an Electric Vehicle[END_REF][START_REF] Vishal | Quad-Active-Bridge DC-DC Converter based on-board Power Electronic Interface for Electric Vehicle[END_REF], and micro-grids [START_REF] Falcones | LQR control of a quad-active-bridge converter for renewable integration[END_REF][START_REF] Galeshi | Application of Clustered Multi-port Active-bridge Converters in Microgrids[END_REF]. Two main challenges of MAB converters are the complexity of their design and the associated control. As the number of ports increases, control and design parameters increase as fast and power transfer equations become more and more complex. Considering the complex calculations and high number of parameters in control and switching of a MAB, closed-form analytical solutions become almost impossible to find. Therefore, numerical optimization methods are required to find the optimum possible solution for each power flow. The problem is that not only numerical optimizations take a long time to find a solution, but also, it is not possible to implement them on industrial control chips with limited processing power and memory. Some researches consider special constraints or scenarios that lead to direct solution of the power flow problem [START_REF] Costa | Quad-Active-Bridge DC-DC Converter as Cross-Link for Medium-Voltage Modular Inverters[END_REF]. Although they manage to efficiently solve the problem, they still miss the ability to put forward a general control scheme for MAB. Other works employ voltage or current feedback loops to adjust the phase shifts of each active bridge [START_REF] Nair | A Quad Active Bridge Based on-Board Power Electronic Interface for an Electric Vehicle[END_REF][START_REF] Vishal | Quad-Active-Bridge DC-DC Converter based on-board Power Electronic Interface for Electric Vehicle[END_REF]. Employing feed-back loops alone, brings about challenges such as stability issues and the need for correct adjustment of speed and accuracy. Another problem is that large dynamics may occur in those systems, because the flow of power between every two ports are intertwined and the feed-back control loops could affect each other.

Considering the above-mentioned challenges in control of a MAB, this work seeks to propose a method that has acceptable accuracy and dynamic behavior in finding an optimum operating point, while being compatible with limited resources of an industrial control target and assuring stability.

Power flow problem in MAB converters

A multi-port converter, whatever its application is, connects several generators and consumers of energy in one place. The role of the converter is to flow power between its different ports, i.e., the generators and the consumers, with a certain direction and amount. Energy system of a smart building, as a common application of multi-port converters, includes several resources such as photovoltaic panels, wind turbines, storage devices, and several consumers on ac and dc networks [START_REF] Galeshi | Application of Clustered Multi-port Active-bridge Converters in Microgrids[END_REF]. There is a higher-level controller in these systems, which takes into account several inputs (such as solar irradiance, wind speed, state of charge of the batteries, instantaneous demand, and electricity tariffs), and based on predefined strategies, determines how the power must flow in the energy system. The power flow indicates how much power should be fed to or drawn from each port and the converter is responsible for its realization.

The power flow between ports of a MAB can be controlled through applying phase shifts between switching of different legs. Two types of phase shifts can be defined in an active-bridge converter: internal phase shift, which is the phase shift between switching of two legs of a single active-bridge; and external phase shift, which is the phase shift between first harmonics of output voltages of two active-bridges. Fig. 1 shows internal and external phase shifts in a dual-active-bridge converter (DAB). It can be concluded that the control system has two domains of freedom for realization of the desired power flow: internal phase shifts, and external phase shifts. In this study, square-wave voltages (Fig. 2-a) are approximated by their first harmonics, and represented by phasors (Fig. 2-b). Magnitudes of first harmonics of output voltages are controlled through adjusting the internal phase shifts, so they can be considered equivalent to internal phase shifts. In this regard, solving the power flow problem means determining a set of magnitudes (internal phase shifts) of voltage phasors and angles between them (external phase shifts) for of every port, which corresponds to the desired power flow.

Using modular matrixes to solve power flow problem

As explained in the previous section, power flow controller must determine magnitude of voltage phasors of every port and the phase difference between them. In an active bridge converter with n active bridges, there are n magnitudes and n-1 phase shifts (one phasor is considered as phase reference) to be determined. Complexity of the power flow problem increases quickly with the number of ports, and there is no direct analytical solution for more than two ports. This work proposes a modular method of solving power flow problem in MAB with any number of ports. Application of modular matrixes for solving power flow problem of a MAB was previously presented in [START_REF] Vermulst | Scalable multi-port active-bridge converters: modelling and optimised control[END_REF] for a special structure of MAB. The current paper presents the solution for a general structure.

A QAB based on general MAB structure is displayed in Fig. 3-a. The first step to determine power transfer is to distribute the inductors between ports. The star-delta transform illustrated in Fig. 3-b is employed to do this. This transform can be generalized for n ports, as

, = ∑ (1)
It is calculated based on superposition theorem. For example, in Fig. 3-b would be

, = = (2)
Through generalizing power transfer of DAB for a MAB with n ports, the power output of port k would be = .

, .

. sin( -) + .

, .

. sin( -) + ⋯ + . , .

. sin ( -)

where, and are magnitude and phase of first harmonic voltage phasor of port , respectively; and is the angular frequency, 2πf, f being the switching frequency. The equation for power transfer can be simplified using approximate assumption of ( ) = as = .

, .

( -) + .

, .

( -) + ⋯ + .

, .

( -) Maximum power each port can transfer, (the maximum value for power transfer when ( ) = 1), is an important factor in design of MAB and depends inversely on the size of the inductors [START_REF] Galeshi | Design Procedure of DC-DC Multi-Port Active-Bridge Converters[END_REF]. If inductors are large, i.e., is relatively small, the instantaneous transferred power might often get close to and current magnitudes will increase nonlinearly, leading to large losses. Choosing a very small inductor, on the other hand, has its own challenges such as sharp changes in current waveform, higher voltages across the transformer windings, and limited range of soft switching. Considering this trade-off, active-bridge converters are usually designed in a way that their nominal power transfer is around 60% of [START_REF] Galeshi | Design Procedure of DC-DC Multi-Port Active-Bridge Converters[END_REF]. Under this design condition, the phase shift between the ports will be in the range of ±37 degrees, and assumption of ( ) = in (4) will have maximum 7.3% error, which is acceptable. Rearranging (4) gives

where, , does not have a real representation and is only added for keeping symmetry in the equation. It will be omitted from calculations in each line of the final matrix. All the power equations can be put into one matrix equation as [START_REF] Vishal | Quad-Active-Bridge DC-DC Converter based on-board Power Electronic Interface for Electric Vehicle[END_REF].

… = . , -∑ , . , … . , . , . , -∑ , … . , … … … … . , . , … . , -∑ ,
× … (6)
The matrix of coefficients in [START_REF] Vishal | Quad-Active-Bridge DC-DC Converter based on-board Power Electronic Interface for Electric Vehicle[END_REF] may not be reversible under some conditions; for instance, when all inductances and voltage magnitudes are equal. In order to modify this matrix and make it reversible, can be considered as the reference for phase; therefore, will be zero. The first column of the matrix of coefficients can be omitted without changing the outcome, because all of its elements will be multiplied by , which is zero. Additionally, the last row can be omitted because can be determined as = -∑ , based on conservation of power. The final equation would be

It is now possible to determine phase shifts as

… = … (8)
where, is the matrix of coefficients in [START_REF] Falcones | LQR control of a quad-active-bridge converter for renewable integration[END_REF]. Equation ( 8) gives an approximate solution for the power flow problem, i.e., given a desired value of power transfer on each port, (8) can determine a set of phase shifts that if applied to switching of the active bridges, leads approximately to the desired power flow. It should be noted that voltage magnitudes (internal phase shifts) have to be determined before this step. This is usually done by another part of the converter control scheme, whose task is to maintain soft switching conditions. Depending on the variations in dc link voltages and the amount of power on each port, the soft switching control determines the internal phase shifts, and the method proposed in this work uses them for its calculations.

Simulation and experimental results

Fig. 4 shows simulation results of open-loop control of power flow in a QAB. The thick lines are the reference values of powers, changing at 0.2 ms and 0.4 ms moments, and thin lines represent the actual measured powers. The actual powers follow the changes in reference values but with some steady state error. The coupling of powers can be seen on the second port at 0.2 ms and 0.4 ms, where its reference value (P2*) does not change but the actual power (P2) varies. The steady state errors, that are large in some cases, are due to the approximations in the control scheme, the fact that the control is open-loop, and the losses that were not considered in the calculations. Fig. 4: performance of the proposed power controller when in a QAB. Thick lines show the reference values, and thin lines show actual powers. Fig. 5 shows the lab-scale prototype that was used to validate the proposed control system experimentally. The details of the converter are provided in Table I. Several experimental tests were performed to validate correct operation of the proposed method. Closed-loop control scheme with PI controllers was employed to validate effective elimination of steady state errors. It should be noted that the reference values of the power sum up to zero due to conservation of power, ignoring the losses in the converter. In other words, it not possible to have all the actual powers equal to their reference value. Therefore, control of one port is always open-loop to allow it to provide the losses. Fig. 6 shows experimental results. The open-loop results show that the proposed power controller is able to realize the desired power but with some steady state error. The closed-loop results show that the steady state error can be effectively eliminated using PI feedback loops. Due to the limited number of input ports of the oscilloscope and analog outputs of the controller, it was not possible to capture and show all the powers. Measurement of powers on the other ports also showed effective elimination of steady state error, similar to that of the second port in Fig. 6. Fig. 5: The quad active-bridge converter that was built in G2ELab for experimental validation of the proposed power control method. The black boxes, numbered 1 and 2, indicate the controller and signal conditioning circuit, respectively. The green and the blue boxes indicate four active bridges and inductors, respectively. The red box is a four-winding transformer. Colors of the boxes correspond to the ones in Fig. 3-a.

Table I: Details of the prototype converter

Performance of the proposed method

An iterative method of solving the problem is presented in [START_REF] Galeshi | Lembeye: Modular Modeling and Control of Power Flow in A Multi-Port Active-Bridge Converter[END_REF]. It begins with finding a first approximate solution by assuming the currents to be zero. The solution found this way could be far from the correct solution, but through performing some iterative calculations and updating magnitudes of the current, this method can get very close to the desired power flow. The method presented in [START_REF] Galeshi | Lembeye: Modular Modeling and Control of Power Flow in A Multi-Port Active-Bridge Converter[END_REF] and the method presented in this work do the same task: given the desired power flow and voltage magnitudes as inputs, these methods determine a set of phase shifts that, if applied as external phase shifts between switching of the active bridges, will correspond to the desired power flow. The major difference between these methods is the approximations they use for solving the problem. These two methods will be compared in terms of accuracy of their final solution and amount of calculations they require on a controller chip.

Since the reference control signal is the power transfer of each port, a good parameter for comparison of accuracy is the average difference between the desired power transfer and the one measured after applying the phase shifts. Fig. 7 shows the defined error of consecutive iterations of the method proposed in [START_REF] Galeshi | Lembeye: Modular Modeling and Control of Power Flow in A Multi-Port Active-Bridge Converter[END_REF] (labeled as "zero currents"), and the error from the method proposed in the current work (Labeled as "matrix"). The "zero currents" method starts with a large error in its first guess, but quickly reduces the error after an additional iteration. The "matrix" method leads to a smaller error, but does not include a mechanism to minimize it. It is noticeable that the error of the proposed method is larger when powers are higher (Fig. 7-a) compared to lower powers (Fig. 7-b). This is due to the ( ) = approximation that is used in this method. The phase shifts increase as the powers increase and the difference between ( ) and becomes larger, reaching up to around 7% at the nominal power, where is around 37 degrees. Fig. 7 also shows that two iterations of the "zero currents" method is enough for reaching negligible errors. It should be noted that a feedback PI controller is usually present in control systems that depend on models, such as the one presented in this work. The main goal of using a PI feedback is to eliminate the steady state errors that occur due to the mismatch between the values in the model and the real system. This feedback loop can be counted upon for eliminating the steady state error caused by the approximation of the proposed method, too. Performance of the proposed method with a closed-loop control was illustrated in Fig. 6-b. In this regard, the steady state errors of the proposed method, although larger than those of the "zero currents" method, are within acceptable range and can be eliminated using a feedback loop.

The second comparison is the amount of calculations involved with each method. The final goal of designing a MAB is to build and use it in a real energy system. The control system for this converter should be able to work with limited resources of low-end controller chips used in industrial products. In this regard, it is important to assess performance of the presented methods on a controller and validate that they run correctly. NI myRIO 1900 package, with a 40MHz processor and a Xilinx Z-7010 chip, was used as the controller. Both methods were implemented on the target chip and run on its processor. Fig. 8 illustrates the results. The time required to do the calculations of the proposed method is 4 µs, compared to 127 µs for two iterations of the "zero currents" method of [START_REF] Galeshi | Lembeye: Modular Modeling and Control of Power Flow in A Multi-Port Active-Bridge Converter[END_REF]. Considering the overhead processing time of 18 µs, the total runtime would be 22 µs for the proposed method, compared to 145 µs for two iterations of "zero currents". The comparison shows that the proposed method is computational cost efficient and a good candidate for real-time control and optimization purposes. Fig. 8: comparing the time needed to run the proposed method and the method presented in [START_REF] Galeshi | Lembeye: Modular Modeling and Control of Power Flow in A Multi-Port Active-Bridge Converter[END_REF]. The first section on the left corresponds to runtime of an empty program and indicates the overhead processing time, which is 18 µs. The second section corresponds to the proposed method, which requires 4 µs to run plus the additional 18 µs overhead processing time. The third and the fourth sections indicate the runtime of the method presented in [START_REF] Galeshi | Lembeye: Modular Modeling and Control of Power Flow in A Multi-Port Active-Bridge Converter[END_REF] for one and two iterations, respectively. The total time required for running two iterations is 145 µs.

Conclusions

A modular method is proposed in this paper for control of power flow in multi-port active-bridge converters. Application of these types of converters is increasing, especially in energy systems based on renewable resources. These converters facilitate simultaneous exchange of energy between multiple resources and loads, with lower costs, smaller size, good efficiency, and isolation. The challenge of these converters is their control, which becomes more and more complex when number of their ports increases. The proposed method solves the power flow control problem using an approximate mathematical model of the converter. Compared to a method previously presented, this method has a larger, yet acceptable error due to its approximation. The proposed method acts as a feed-forward loop and it was shown with experimental results that in combination with a feed-back loop, it can effectively control the power flow with no steady state errors. This method requires less mathematical calculations and runs about 7 times faster, compared to a previously proposed method in the literature. This fact makes it applicable for industrial control systems with limited processing resources. Several simulation and experimental results are presented to validate correct performance of the method.

Fig. 1 :

 1 Fig. 1: internal and external phase shifts in switching of a DAB. Switches on the same arm are switched complementarily.

Fig. 2 :

 2 Fig. 2: voltages and phase shifts in a quad-active-bridge converter: (a) output voltages of active bridges; (b) phasor-space representation.

Fig. 3 :

 3 Fig. 3: a quad-active-bridge converter: (a) the structure, with a four-port transformer (the red box), four active bridges (green box), and four inductors (blue box); (b) ac representation and star-delta transform.

Fig. 6 :

 6 Fig. 6: experimental results of power control in a quad active-bridge converter: (a) open-loop results; (b) closed-loop results. The first port (P1) is always open-loop, to be able to provide the losses. The rest are closed-loop, including P2 that is shown here. Due to the limited number of available analog outputs of the controller and inputs on the oscilloscope, the reference and measured powers of the third and fourth port are not shown.

Fig. 7 :

 7 Fig. 7: Comparing convergence of different power control methods in a simulated 2 kW quad activebridge converter: (a) ideal converter with all powers and voltage at their nominal values; (b) non-ideal converter with ±10% voltage variation.

This work has been partially supported by the CDP Eco-SESA receiving fund from the French National Research Agency in the framework of the "Investissements d'avenir" program (ANR-15-IDEX-02). The authors also thank the Carnot Institute Energies du Futur for supporting this work.


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À Monsieur le Professeur Denis AGOSTINI, de me faire l'honneur de présider ce jury et de juger mon travail. Veuillez trouver ici toute ma reconnaissance et l'expression de mon profond respect. À Monsieur le Professeur Nicolas AIDE, pour avoir accepté́ de diriger mon travail de thèse, pour m'avoir transmis son savoir et encadré durant ces années d'internat. Enfin, merci pour m'avoir fait sourire si souvent.

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Table des matières

Tableaux et figures

Introduction

Over the last decade, Positron emission tomography (PET) has been marked by technological involvements, including improved reconstruction algorithms and hardware evolutions [START_REF] Van Der Vos | Quantification, improvement, and harmonization of small lesion detection with state-of-the-art PET[END_REF], the most recent one being digital PET that is now available from the major PET vendors [START_REF] Fuentes-Ocampo | Digital vs. analog PET/CT: intra-subject comparison of the SUVmax in target lesions and reference regions[END_REF][START_REF] Lopez-Mora | Comparison of image quality and lesion detection between digital and analog PET/CT[END_REF][START_REF] Pan | Performance evaluation of the 5-Ring GE Discovery MI PET/CT system using the national electrical manufacturers association NU 2-2012 Standard[END_REF][START_REF] Rausch | Performance Evaluation of the Vereos PET/CT System According to the NEMA NU2-2012 Standard[END_REF][START_REF] Van Sluis | Image quality and semi-quantitative measurements of the Siemens Biograph Vision PET/CT: Initial experiences and comparison with Siemens Biograph mCT PET/CT[END_REF]. Fully digital PET cameras, such as the Vereos system (Philips Heathcare®, Ohio, US), feature small digital silicon photomultipliers instead of much larger photomultiplier tubes, providing true digital photon counting with 1-to-1 crystal coupling. The properties of digital systems, in particular the enhancement in the time-of-flight capabilities, have been shown to improve signal-to-noise ratio and ultimately image contrast and lesion detectability compared to analog PET [START_REF] Lopez-Mora | Comparison of image quality and lesion detection between digital and analog PET/CT[END_REF][START_REF] Aljared | BSREM Reconstruction for Improved Detection of In-Transit Metastases With Digital FDG-PET/CT in Patients With Malignant Melanoma[END_REF][START_REF] Salvadori | Head-to-head comparison of image quality between brain (18)F-FDG images recorded with a fully digital versus a last-generation analog PET camera[END_REF]. From a daily routine perspective, these PET advancements are usually used to decrease injected activity and reduce acquisition time, the decrease in the injected activity being made with the aim of reducing radiation exposure but also for obvious economic reasons. However, evaluation of the physics performance of modern PET system with regards to noise equivalent count rate curves (NECR), which account for useful information (true coincidences) and events creating noise (random and scattered coincidences), show that recent PET scanners could handle high activity in the FOV without compromising image quality (i.e noise level).

An issue faced by PET unit teams is the management of bedridden patients experiencing pain or dyspnea. Also challenging is the scan of uncompliant pediatric patients when premedication is not efficient or feasible. Positioning of the patient on the table and PET acquisition being the bottleneck of the total scan time, the fastest PET acquisition would be very useful in all of these situations, provided the diagnostic performances are maintained.

The aim of this study was to seek the minimal duration per bed position being feasible with a recent PET system without compromising image quality and tumour or other target detection in patients for whom fast or even ultra-fast PET imaging may be useful. We focused on cancer patients with dyspnea or in pain, as well as paediatric patients in whom recommendation for injected activity (PEDdose: https://www.eanm.org/publications/dosage-calculator/) lead to activity per unit of weight higher than those used in adult.

The minimal time per bed position was investigated by reconstructing list mode data with decreasing time down to 10 s per bed position. We also evaluated whether modifications had to be made regarding reconstruction parameters when using ultra-fast imaging (i.e in images acquired with less than 60 seconds per bed position), in order to maintain an acceptable noise level. A comprehensive evaluation of the impact of reducing acquisition time was performed on simulated data, including image quality, lesion detectability as well as inter-and intra-observer agreement.

Materials and methods

Patients' selection

From January 2018 to September 2018, patients experiencing pain or dyspnea, in whom a standard 18 F-FDG PET scan duration (around 20 min) was deemed to be infeasible, received a higher injected activity (arbitrary set to 5 MBq/Kg) and underwent a faster scan, as described below in the materials and methods section. Also included in this series were pediatric patients in whom the injected activity, according to the EANM guidelines (using the PETdose recommendations), usually leads to an injected activity around 5-6 Mbq/kg, which is almost twice the injected activity per unit weight in adults at our PET centre. This study was part of the NUMANATEP project, meant to evaluate digital PET systems at our institution, which was approved by the regional Ethics Committee (registration number: CERES 27669) and authorization to use additional reconstructions for research purpose was sought from every patient. Anonymised data were collected and used as per the European mutually agreed General Data Protection Regulation (GDPR) with the French committee on data privacy registration number CNIL 2204611 v 0.

PET/CT acquisition and reconstruction parameters

The 18 F-FDG usual injected activity was increased from 3 MBq/kg to 5 MBq/kg in adult patients and the acquisition time was decreased from 2 min to 1.5 min. Pediatric patients' activities were computed using the EANM PEDdose application) and these patients were also scanned at 1.5 min per bed position.

Data were acquired in list mode on a Vereos digital PET/CT system. At our centre, the standard reconstruction is as follows: 2 iterations, 20 subsets (2i20s), point spread function modelling and time-of-flight enabled, no post filtering and voxel size of 2mm 3 . In addition, frames of 60s, 30s, 20s and 10s per bed position were reconstructed with the same parameters. Given that an increase in the noise level was expected for reconstructions shorter than 60s per bed position, we also reconstructed 30, 20 and 10s per bed simulated acquisitions with (i) the same number of iterations but with PSF modeling disabled and (ii) a two-fold decrease in the number of iterations (1i10sec) and PSF disabled. Therefore, a total of 15 reconstructions per patient were studied (Figure 1). Within this complete manuscript, the reconstructions are defined and labeled as follows: xixsxSec where xi, xs and xsec stand for the number of iterations, the number of subsets and the time per bed position in seconds. PSF: point spread function modeling.

Evaluation of noise and tumour/local background when decreasing the acquisition time

Images were analyzed using the MIM software. For all analysis, region of interest (ROI) and volume of interest (VOI) were drawn by one PET reader, using a gradient-based contouring method (PET edge) on images obtained with the standard reconstruction and then copied and pasted to the ten other set of images to avoid any intra-observer variability.

Noise was evaluated in large reference organs and in the vascular background as follows:

-Liver: a 3 cm diameter VOI centred on the right liver lobe -Aorta: a 1.5 cm VOI centre on the descending aorta -Muscles: a 1.5 cm diameter VOI placed on the right ilio-psoas muscle at the level of the 4th lumbar vertebra

In order to assess the potential lesion detectability as a function of lesion/surrounding background visual ratio, a doughnut shaped VOI surrounding the tumours' VOI was used and tumour/background ratios were computed using the SUVmean values within the tumours and doughnut VOIs (supplemental figure 1). As most of included patients had multiple lesions, we focused on the most obvious and less obvious lesions on the standard reconstruction.

Comparison of short and standard acquisitions

Based on the analysis of noise level described below, we selected three reconstructions for which the noise was not statistically different, compared to the standard reconstruction (2i10sPSF90sec):

1i10s20sec, 1i10s30sec, and 2i10sPSF60sec, which were randomly assigned to 4 PET readers with at least a 12-month experience in the reading of PET images using the Vereos system. A fifth reader was asked to read all these reconstructions as well as the standard reconstruction, starting with the supposedly worst reconstruction (in terms of noise level) and increasing the time per bed position up to the standard acquisition time. The order of review was (1) 1i10s10sec, (2) 1i10s20sec, (3) 1i10s30sec, (4) 2i10sPSF60sec and finally the standard of reference 2i10sPSF90sec.

PET readers rated the image quality according to a 3-point scale (1=poor; 2=fair; 3= good), recorded the number of malignant foci (0, 1, 2-5, >5) and the number of anatomical sites (0, 1, 2, 3, 4, ≥5). The 5 cutoff value was chosen as this figure is commonly used by the oncology community to define oligometastic disease. We also focused on the number of anatomical sites, which is even more important than the number of lesions by organ in the staging system of many cancer types. Readers were aware of the reasons why the patients had been referred to an 18 F-FDG PET examination, in order to avoid bias in interpretations.

Statistical analysis

Quantitative data are presented as mean ± standard deviation (SD).

Noise (in the liver, aorta and psoas), lesion/background ratios and mean image quality were compared using the non-parametric rank-sum Friedmann test for multiple paired sample. For that test, the mean rank of each reconstruction was compared to the standard acquisition and reconstruction set (i.e 90s per bed, 2i20s, PSF modeling enabled), which was used as a control reconstruction. The aim was to seek reconstruction settings where the noise level was not statistically significant, compared to the standard reconstruction, which would therefore be useable in clinical practice with an acceptable loss in image quality. The reported P values were corrected for multiple comparison testing using the Dunns' method. Inter-observer agreement was evaluated using the Cohen's Kappa. Kappa values were reported using the benchmarks of Landis and Koch (0.81-1 almost perfect agreement, 0.61-0.80 substantial agreement, 0.41-0.60 moderate agreement and 0.21-0.40 fair agreement). Graphs and statistical analysis were performed using Prism 8 (Graphpad, graphpad Software, San Diego, CA, USA) and XLSTAT (addinsoft, Paris France).

Results

Comparison of short and standard acquisitions: noise level, contrast and visual image quality

Over a nine-month period, twenty-eight patients were included. During this period, 873 patients were referred for 18 F-FDG PET. Therefore, the prevalence of patients potentially requiring fast PET imaging was 3.4%. Their characteristics are displayed on Table 1.

Noise level, expressed as the coefficient of variation (CoV) in the blood pool (aorta), liver and muscle was 0.12 (0.04), 0.12 (0.04) and 0.17 (0.08), respectively for the standard reconstruction (2i10sPSF90sec) (Figure 2). If the same reconstruction parameters were maintained (2 iterations, 10 subsets and PSF modeling), only the 60s per bed reconstruction displayed a noise level comparable to the standard reconstruction, with CoV-AORTA, CoV-LIVER and CoV-MUSCLE of 0.14 (0.04), 0.15 (0.04) and 0.20 (0.09), respectively. For reconstruction with a shorter time per bed, maintaining a noise level not significantly higher than that of the standard reconstruction required disabling PSF and decreasing the number of iterations by a two-fold factor in images with less than 60s per bed position. Using these parameters, the 30s (1i10s30sec) and 20s (1i10s20sec) per bed reconstructions displayed CoV-AORTA, CoV-LIVER and CoV-MUSCLE of 0.11 (0.03), 0.12 (0.03) and 0.14 (0.05) and 0.13 (0.04), 0.14 (0.03) and 0.17 (0.07), respectively. Despite adaptation of the reconstruction parameters, the shortest reconstruction used (10s per bed) showed a significantly higher noise, compared to the standard clinical reconstruction (Figure 2).

Based on this analysis of noise level, we selected three reconstructions for comparison of lesion to background ratios and visual image quality: 1i10s20sec, 1i10s30sec, and 2i10sPSF60sec (Figure 4). Only the 60s per bed reconstruction displayed a mean lesion/background ratio comparable to the standard reconstruction: 3.19 (1.51) versus 3.22 (1.57). For the 30s and 20s time per bed reconstructions, mean lesion/background ratios for the most obvious lesion were still both higher than 2.0 {2.43 (0.93) and 2.43 (0.91)} but represented a decrease of 22% and 23%, respectively compared to the standard reconstruction (Figure 3a). For the less obvious lesion, mean lesion/background ratios were almost similar to those obtained with the most obvious lesion {1.99 (0.54) and 2.01 (0.56)} and represented a decrease of 21% and 22%, respectively compared to the standard reconstruction (Figure 3b). Concerning mean quality scores, the only reconstruction displaying no significant difference as compared to the standard reconstruction is the 1i10s30sec reconstruction. It showed a mean quality score equal to 2.6 (0.5) versus 2.7 (0.5) for the standard reconstruction. All other reconstructions showed significantly slightly lower mean quality scores (Figure 4a).

Impact of acquisition time on Lesion detectability

Intra observer agreement

As described in the Material and Methods section, intra rater agreement was evaluated by a PET reader who interpreted all reconstructions, starting with the supposedly worst reconstruction (in terms of noise level) and increasing the time per bed position up to the standard acquisition time.

The best intra rater agreements were observed with the 2i10sPSF60s for both the number of detected lesions and the number of detected anatomical sites. Regarding the number of detected lesions, the Cohen kappa coefficients were equal to 0.78, 0.72 and 0.89 for 1i10s20sec, 1i10s30sec, and 2i10sPSF60sec, respectively. For the number of detected anatomical sites, the Cohen kappa coefficients were equal to 0.72, 0.72 and 0.95 for 1i10s20sec, 1i10s30sec, and 2i10sPSF60sec, respectively. However, all Cohen kappa coefficients were ranged at least good, whatever the reconstruction used.

Regarding the number of detected lesions, 3 (11%) patients had fewer lesions detected using the 1i10s20sec, 1i10s30sec reconstructions as compared to the 2i10sPSF60sec and/or the standard reconstruction (Figure 4b). In line with those findings, 6 (22%) had fewer anatomical sites detected with the 1i10s20sec, 1i10s30sec reconstructions (Figure 4c). Representative images of one of these patients are displayed on Figure 5. Of note, in line with noise level evaluation (Figure 1), no false positive findings related to noise level were observed on the 1i10s20sec and 1i10s30sec reconstructions.

Inter observer agreement

As described in the Material and Methods section, the three selected reconstructions (1i10s20sec, 1i10s30sec, and 2i10sPSF60sec) were randomly assigned to 3 other PET readers. Inter-rater agreement for the number of involved anatomical sites and detected lesion was good or almost perfect (Kappa: 0.64-0.91) for all the selected acquisitions (Table 2). In particular, kappa coefficients for the 30s per bed acquisition were 0.79 and 0.91 for lesion and anatomical sites number, respectively. Regarding the number of detected lesions, discordances occurred in 6, 4 and 4 patients for 1i10s20sec, 1i10s30sec, and 2i10sPSF60sec reconstructions, respectively. Regarding the number of detected anatomical sites, discordances occurred in 7, 2 and 4 patients for 1i10s20sec, 1i10s30sec, and 2i10sPSF60sec reconstructions, respectively.

Discussion

For a PET centre, being able to perform fast imaging may be a challenge, as shown in Figure . 6 where a lymphoma patient with a bulky mediastinal mass and a superior vena cava syndrome had to be scanned with several pillows to cope with dyspnea. This study was conducted as a non-inferiority study to seek the fastest acquisition time possible with our system, focusing on a specific population likely to benefit from fast or ultra-fast imaging: patients in pain, experiencing dyspnea and pediatric patients. Claustrophobic patients were not included but could also benefit from fast imaging.

This type of study could be applied to modern PET scanners capable of high-count rate. Hence, a recent evaluation of the Vereos system showed that its NECR is 513.4 Kcps @ 54.9 KBq/cc [START_REF] Rausch | Performance Evaluation of the Vereos PET/CT System According to the NEMA NU2-2012 Standard[END_REF]. This can be compared to a NECR of 185 Kcps @ 29 KBq/cc and 266.3 Kcps @ 20.8 KBq/cc for the Siemens mCT [START_REF] Rausch | Performance evaluation of the Biograph mCT Flow PET/CT system according to the NEMA NU2-2012 standard[END_REF] and the GE Discovery MI (5 ring version) [START_REF] Pan | Performance evaluation of the 5-Ring GE Discovery MI PET/CT system using the national electrical manufacturers association NU 2-2012 Standard[END_REF], respectively.

The liver, which is the organ of reference in many circumstances (Deauville score, and marker of quality for PERCIST, where absolute and relative variation in liver uptake have to remain below 0.3 units and 20%, respectively [START_REF] Wahl | From RECIST to PERCIST: Evolving Considerations for PET response criteria in solid tumors[END_REF]) was used, together with the vascular background in the aorta and muscle. Concordant results were obtained when comparing the level of noise in these three reference organs amongst simulated short acquisitions, provided that the shortest acquisitions were improved by decreasing the number of iterations and disabling PSF modeling. TOF was enabled for all reconstructions, as it has been shown that the improved TOF capabilities of modern digital PET systems leads to a dramatically enhanced image quality, as compared with analog PET, which may prove particularly useful for PET exams performed with reduced injected activities and/or recording times [START_REF] Salvadori | Head-to-head comparison between digital and analog PET of human and phantom images when optimized for maximizing the signal-to-noise ratio from small lesions[END_REF].

Based on noise in reference organs, we studied intra-and inter-observer analysis amongst the reconstructions deemed to be best suitable for fast imaging, i.e 1i10s20sec, 1i10s30sec, and 2i10sPSF60sec. The 30s per bed position reconstruction appeared to be suitable for clinical practice, with intra observer agreement of 0.79 and 0.90 for lesion and anatomical sites number and a mean image quality comparable to that of the standard reconstruction (2.6 +/-0.5 versus 2.7 +/-0.5, P ns).

Of note, while we focused on adapted reconstructions displaying a noise level comparable to a standard reconstruction, one could decide to perform ultra-fast PET acquisitions despite an increase in noise level. As shown in Figure 1, an acquisition with 10 s per bed position would visually lead to a significant level of noise in the liver but still allows visualization of the tumour. This increase in noise level in an organ frequently used as a reference (for example for PERCIST [START_REF] Pinker | Evaluating tumor response with FDG PET: updates on PERCIST, comparison with EORTC criteria and clues to future developments[END_REF], Deauville scoring [START_REF] Nanni | Report of the 6th International Workshop on PET in lymphoma[END_REF])

and being part of the metastatic spread of many cancer types [START_REF] Viadana | The metastatic spread of cancers of the digestive system in man[END_REF] was 73% on average when comparing the 1i10s10sec and 1i10s30sec reconstructions (Figure 2).

The relatively small number of patients included in this study could be regarded as a limitation of it, but in fact reflects the prevalence of patients potentially requiring fast PET imaging (3.4%) at our institution. This figure is related to the population scanned, in our case 90% of the PET scans performed in our department are oncology-related [START_REF] Lasnon | Patient's weight: a neglected cause of variability in SUV measurements? A survey from an EARL accredited PET centre in 513 patients[END_REF]. Secondly, we arbitrarily decided to increase the injected activity. One might assume that a clinical situation requiring fast imaging may occur after the 18 F-FDG has been injected. Data regarding this situation are provided as supplementary file and show similar results for noise level but tumour to background ratio smaller than 2 for the less obvious lesions, requiring further studies before being applied.

Our data could be useful to other centers facing the situation of patients experiencing pain, dyspnea or any other disease requiring a decreased acquisition time. In our experience, from a practical point of view, fast scanning requires a sufficient number of technologists and physicians to position the patient on the table of the PET scanner and remove him/her promptly and efficiently (with the appropriate equipment in the case of bedridden patients) so that the time spared during acquisition (and the gain in patient' comfort) is not lost during these phases. Median estimated total PET acquisition time for the 1i10s30sec was 4 min, compared to 12 min for the standard reconstruction.

It is noteworthy that PET systems with very long axial field-of-view such as total body PET [START_REF] Cherry | Total-Body PET: Maximizing Sensitivity to Create New Opportunities for Clinical Research and Patient Care[END_REF][START_REF] Badawi | First Human Imaging Studies with the EXPLORER Total-Body PET Scanner[END_REF] would obviously allow ultra-fast imaging but this new technology is unlikely to be widely available given its cost. Perspectives of fast PET imaging using modern PET systems will require adaptation of the reconstruction parameters for a given system, and are likely to be impacted by evolution of reconstruction capabilities, especially implementation of deep learning reconstruction [START_REF] Cui | PET image denoising using unsupervised deep learning[END_REF][START_REF] Lu | An investigation of quantitative accuracy for deep learning based denoising in oncological PET[END_REF][START_REF] Wang | Image Reconstruction is a New Frontier of Machine Learning[END_REF],

enabling handling of noise level at low-count statistics.

Conclusion

Acquisition time per bed on the Vereos system can be reduced down to 30s without significant impact on quantitative and visual image quality and with preservation of a good detectability as compared to the standard reconstruction. These fast acquisitions require optimization of reconstruction parameters.

Résultats:

Les reconstructions suivantes, avec un bruit non significativement supérieur à la reconstruction standard, ont été sélectionnées: 1itération/10 sous-ensembles/20sec (1i10s20sec), 1i10s30sec et 2i10sPSF60sec. Seuls les 60sec par pas avaient des ratios cible/arrière-plan similaires à ceux de la reconstruction standard, mais les ratios moyens étaient toujours supérieurs à 2,0 pour la reconstruction des 30. L'accord inter-évaluateur et intra-évaluateur pour le nombre de sites anatomiques et de lésions détectées était bon voir presque parfait, en particulier pour les acquisitions de 30sec. Le temps d'acquisition total estimé de TEP pour le 1i10s30sec et la reconstruction standard étaient respectivement de 4 et 12 min.

Conclusion:

L'imagerie rapide est possible avec des acquisitions de 30sec par position de lit sur le système Vereos, nécessitant une optimisation des paramètres de reconstruction. Purpose:

To seek the minimal duration per bed position with a digital PET system without compromising image quality and lesion detection in patients requiring fast 18F-FDG PET imaging.

Materials and methods:

28 patients were scanned on a Vereos system. List mode data were reconstructed with decreasing time frame down to 10s per bed position. Noise was evaluated in the liver, blood pool and muscle, and using target-to-background ratios. 5 PET readers recorded image quality, number of clinically relevant foci and of involved anatomical sites in reconstructions ranging from 60 to 10s per bed position, compared to the standard 90s reconstruction.

Results:

The following reconstructions, which harboured a noise not significantly higher than the standard reconstruction, were selected: 1iteration/10subsets/20sec (1i10s20sec), 1i10s30sec, and 2i10sPSF60sec.

Only the 60s per bed displayed similar target-to-background ratios compared to the standard reconstruction, but mean ratios were still higher than 2.0 for the 30s reconstruction. Interrater agreement for the number of involved anatomical sites and detected lesion was good or almost perfect (Kappa: 0.64-0.91). In particular, kappa for the 30s per bed acquisition was 0.78 and 0.91 for lesion and anatomical sites number, respectively. Intra-rater agreement was also excellent for the 30s reconstruction. Estimated total PET acquisition time for the 1i10s30sec, and the standard reconstruction were 4 and 12 min, respectively.

Conclusion:

Fast imaging is feasible with state-of-the-art PET systems. Acquisitions of 30s per bed position are feasible with the Vereos system, requiring optimization of reconstruction parameters.

KEY WORDS: PET; Digital; 18 F-FDG; pain; dyspnea

  CoV: coefficient of variation EANM: European association of nuclear medicine FOV: field of view KBq/cc: Kilo Becquerel/centimètre cube Kcps: Kilo coups MBq: Mega Becquerel Min: minute MIP: maximum intensity projection NECR: noise equivalent count rate PERCIST: PET evaluation response criteria in solid tumors PET/CT: positron emission tomography / computed tomography PSF: point spread function SD: standard deviation SUV: standard uptake value VOI: volume of interest ROI: region of interest

Table 1 :

 1 Patient characteristics (n = 28) )*: 63 (6,67) Glycaemia (g/l)*: 1,03 (0,27) * Mean (SD)

Figure 1 :

 1 Figure 1: Illustration of the multiple reconstructions performed to simulate fast acquisition and to

Figure 2 :

 2 Figure 2: Noise in reference organs (a: aorta, b: liver, c: muscle) expressed as the coefficient of

Figure 3 :

 3 Figure 3: Lesion to background ratio using a doughnut VOI for background definition. Fast

Figure 4

 4 Figure 4 Comparison of the number of image quality (a), anatomical sites (b) and number of lesions

Figure 5 :

 5 Figure 5: 63-year-old female patient with severe pain due to a myeloma recurrence. A suspicious

Figure 6 :

 6 Figure 6: 62-year-old female lymphoma patient with bulky mediastinal mass and superior vena cava

  MOTS CLES : TEP; Numérique; 18 F-FDG; douleur; dyspnée ANNEE DE SOUTENANCE : 2020 NOM ET PRENOM DE L'AUTEUR : COUDRAIS NICOLAS TITRE DE LA THESE EN ANGLAIS: How fast can we scan patients with modern (digital) PET/CT systems? RESUME DE LA THESE EN ANGLAIS :

Table 2 :

 2 Intra-observer agreement evaluated for a PET reader who interpreted all reconstructions, starting with the supposedly worst reconstruction (in terms of noise level) and increasing the time per bed position up to the standard acquisition time.

	< 0.2: poor agreement, 0.2 -0.4: fair agreement, 0.41 -0.6: moderate agreement, 0.61 -0.8: good
	agreement, 0.8 -1: very good
	Reconstructions	Number of lesions Number of sites
	1i10s20sec	0.64	0.68
	1i10s30sec	0.79	0.91
	2i10sPSF60sec	0.78	0.81
	2i10sPSF90sec	0.83	0.84

Table 3 :

 3

	Inter-observer agreement: interpretation was compared between one reader who read the four
	reconstructions, and four other observers who interpreted one reconstruction each.
	< 0.2: poor agreement, 0.2 -0.4: fair agreement, 0.41 -0.6: moderate agreement, 0.61 -0.8: good
	agreement, 0.8 -1: very good

  Matériels et méthodes:28 patients ont été scannés sur un système TEP numérique Vereos. Les données ont été reconstruites avec un laps de temps décroissant jusqu'à 10 secondes par position de lit. Le bruit a été évalué en utilisant des ratios cible/bruit de fond, dans le foie, les vaisseaux et les muscles. 5 médecins ont enregistré la qualité d'image, le nombre de foyers et de sites anatomiques impliqués dans des reconstructions allant de 60 à 10sec par pas, par rapport à la reconstruction standard de 90sec.

	ANNEE DE SOUTENANCE : 2020
	NOM ET PRENOM DE L'AUTEUR : COUDRAIS NICOLAS
	TITRE DE LA THESE: À quelle vitesse pouvons-nous scanner des patients avec des
	systèmes TEP/TDM modernes (numériques) ?
	RESUME DE LA THESE EN FRANÇAIS :
	Objectif:	VU, le Doyen de la Faculté
	Rechercher la durée minimale d'acquisition avec un système TEP numérique sans
	compromettre la qualité d'image et la détection des lésions chez les patients nécessitant une
	imagerie rapide TEP 18F-FDG.
		VU et permis d'imprimer
		en référence à la délibération
		du Conseil d'Université
	en date du 14 Décembre 1973
		Pour le Président
	de l'Université de CAEN et P.O
		Le Doyen

Remerciements

Supplemental Figure 1: Methodology for delineation of a doughnut shaped VOI surrounding the tumours' VOI used for computation of the tumour/background: a: gradient-based tumour VOI, b: 3D VOI including a 1 cm, c: doughnut VOI (red colour) obtained by substracting the tumour VOI to the tumour+ surrounding noise VOI