M Chammas
email: marc.chammas@universite-paris-saclay.fr

Yu Ch

Lin Hh

Comparison of synthetic exposure and sequential exposure for multiple exposure speckle imaging of blood flow

Comparison of synthetic exposure and sequential exposure for multiple exposure speckle imaging of blood flow

Introduction

Multiple exposure speckle imaging (MESI) has emerged as an efficient and reliable tool for in vivo relative blood flow changes monitoring and tissue perfusion assessment. It has been found as a powerful diagnostic tool in several medical fields such as rheumatology, dermatology, ophthalmology, and neurology among others [START_REF] Heeman | Clinical applications of laser speckle contrast imaging: a review[END_REF]. MESI has been proven to be more accurate than the previous laser-speckle imaging tools when it comes to flow quantification and the capability of distinguishing the contribution of static scatters (i.e. bones, cartilages, etc.) from that of the moving scatterers (i.e. blood cells) in the speckle pattern. The MESI system developed previously in our lab relied on a posttreatment of speckle images followed by a nonlinear fit to the MESI model in the aim of extracting the functional blood flow maps [START_REF] Soleimanzad | Multiple speckle exposure imaging for the study of blood flow changes induced by functional activation of barrel cortex and olfactory bulb in mice[END_REF]. However, the acquisition of multiple exposure relied on a fixed camera exposure associated to laser pulses varying in duration and intensity. This implies the use of an acousto-optic modulator (AOM) associated to an arbitrary waveform generator (AWG) and the synchronization of the camera to the laser pulses. We investigate here a much simpler instrumental approach by creating synthetic exposure by the summation of images acquired at a short exposure time (a 10ms synthetic exposure is obtained by the summation of 10 images obtained at 1ms). Such approach has been proposed and validated using high sensitivity, low noise SPAD detectors but at the expense of a limited spatial resolution [START_REF] Dragojević | Compact, multiexposure speckle contrast optical spectroscopy (SCOS) device for measuring deep tissue blood flow[END_REF]. Further studies have used CMOS cameras but no comparison to the "classical" AOM was performed as a validation [START_REF] Sun | Multi-exposure laser speckle contrast imaging using a high frame rate CMOS sensor with a field programmable gate array[END_REF][START_REF] Hultman | A 15.6 frames per second 1-megapixel multiple exposure laser speckle contrast imaging setup[END_REF]. On one hand, the camera noises (dark, read-out and shot noise) add linearly with the exposure time in the synthetic exposure scheme, thus decreasing the speckle contrast. On the other hand, the synthetic exposure scheme is likely to be much less sensitive to the intensity calibration bias observed in the AOM multiple exposure approach. Indeed, in this approach, not only the duration of pulses is modulated, but also their amplitude to keep a similar light level at the detector for all exposure times. The present study was designed to compare the synthetic exposure approach to the laser modulation approach in typical conditions associated with speckle blood flow imaging in the brain of rodents [START_REF] Soleimanzad | Obesity in Midlife Hampers Resting and Sensory-Evoked Cerebral Blood Flow in Mice[END_REF].

Materiel and Methods

The imaging set-up used in this study is showed in figure 1. The laser used is a 634 nm laser diode (Shangai laser, China) with a maximum output power of 419 mW. A genI sCMOS camera (Orca Flash 2.8, Hamamatsu, Japan) was used. It has a 42% spectral sensitivity at 634 nm, a typical read noise of 3e-and a dark noise of 1e-. The microfluidic channels and set-up were described in details previously [START_REF] Soleimanzad | Multiple speckle exposure imaging for the study of blood flow changes induced by functional activation of barrel cortex and olfactory bulb in mice[END_REF]. Speckle images were recorded for microfluidic channels with a rectangular section of 100, 300 and 500 µm of widths and 75 µm of length. We used 2% intralipid as the flowing media since its optical scattering properties are close to that of blood. Flows from 1 µl.min-1 up to 7 µl.min-1 were controlled using a pressure controller with inline flow feedback (Fluigent, France). For all conditions, contrast images were computed using the following equation:

𝐾 = 𝛔 $%& (1)
Where σ is the standard deviation of the intensities in the acquired image, and <I> is the average intensity in the same image. A spatial window of 5x5 is used for the calculation of contrast. The assumption behind the speckle contrast imaging of biological flows is that the flow is inversely proportional to the decorrelation time tc of the scatterers. Using multiple exposure contrast images, it is possible to derive tc while considering the contribution of static scatterers. Multiple exposure speckle contrast data as a function of exposure are fitted to the following equation [START_REF] Parthasarathy | Robust flow measurement with multi-exposure speckle imaging[END_REF]: 𝐾(𝑇, 𝜏𝑐) = -𝛽 𝜌 1 2 + 𝜐 ?@AB2 C 7/1

(2)

In the present study, we have acquired and compared multiple exposure speckle contrast data at Texposure of 1, 2, 5, 10, 25 and 50ms using two different approaches. In a first approach, called laser modulation method, the acquisition of multiple exposure relies on a camera exposure fixed at 60ms associated to laser pulses varying in duration and intensity using an acousto-optic modulator. The width of each pulse is set to the desired exposure duration (same values as the previously chosen 6 exposure times), while its amplitude is set so that the areas for all the pulses (in other words the pulses energies) are equalized. The shortest exposure time requires the maximum amplitude whereas the longest exposure is associated with the lowest amplitude. Contrast images are then calculated for the 6 exposure times using equation ( 1) and tc is derived from a fit of the contrast data to equation [START_REF] Soleimanzad | Multiple speckle exposure imaging for the study of blood flow changes induced by functional activation of barrel cortex and olfactory bulb in mice[END_REF]. In a second approach, called synthetic exposure, contrast images are generated using synthetic exposures as follows: the camera exposure time is set at 1ms and the laser diode power is adjusted to ensure that the maximum dynamic of the sensor is used. 50 independent frames at 1ms are acquired and contrast images for Texposure : 1, 2, 5, 10, 25, and 50ms are computed as the sum of the corresponding number of 1ms-frames. In other words, short exposure frames (1ms) are summed to create longer exposure times. In order to limit the bias due to summation of noise (dark, readout and shot-noise), noise corrections are applied and contrast images are computed using equation (1). Both acquisition set-ups are shown in figure 1.

Preliminary results

Fig. 1 .

 1 Fig. 1. Set-ups used for laser modulation method (A) and synthetic exposure (B) acquisition of multiple exposure speckle images.

Fig. 2 .

 2 Fig. 2. Speckle contrast images calculated for a channel of 300 µm width and a flow of 3 µl.min-1 using the synthetic multi-exposure approach. (A) 1 image acquired at 1ms; (B) 2 images acquired at 1ms thus a total exposure of 2ms; (C) 5 images acquired at 1ms thus a total exposure of 5ms; (D) 10 images acquired at 1ms thus a total exposure of 10ms; (E) 25 images acquired at 1ms thus a total exposure of 25ms; (F) 50 images acquired at 1ms thus a total exposure of 50ms.


S Moindjie

D Munteanu

M Dentan

P Moreau

F P Pellissier

J Bucalossi

G Borgese

V Malherbe

T Thery

G Gasiot

P Roche

J L Autran

Real-Time Characterization of Neutron-induced SEUs in Fusion Experiment at WEST Tokamak during D-D Plasma Operation

We conducted a real-time SER characterization of CMOS bulk 65nm SRAMs subjected to fusion neutrons during deuterium-deuterium plasma operation at WEST tokamak. Neutron spectrometry and numerical simulation have been also performed to analyze experimental results.

I. INTRODUCTION

Magnetic and inertial confinement fusion devices studied to develop future carbon-free source of energy are based on deuterium-tritium (D-T) nuclear fusion reactions [START_REF] Morse | Nuclear Fusion[END_REF]. Deuterium-deuterium (D-D) nuclear fusion reactions are also used to study their operation before introducing tritium fuel. Both D-D and D-T operations produce a high flux of energetic neutrons, creating a residual neutron field outside the reaction chamber. All future fusion devices will contain a large amount of electronics. Most of it will be installed outside the radiation field. The limited amount of electronics which will stay close to the reaction chamber will be exposed to nuclear radiation and negatively affected by this environment condition. This work, which is part of a study conducted by ITER on the effects on electronics of the radiation environment in certain areas of the ITER tokamak building, is a first tentative to characterize neutron induced SEUs (Single Event Upsets) in SRAM memories exposed to the radiation field in the vicinity of a nuclear fusion reaction. The experiment was conducted at WEST tokamak during a test campaign under D-D plasma operation. WEST (acronym derived from W -tungsten -Environment in Steady-state Tokamak) is an experimental nuclear fusion device operated by the IRFM (Institute for Magnetic Fusion Research) institute from CEA (French Alternative Energies and Atomic Energy Commission) [2]. Its 50 m 3 toroidal vacuum chamber is equipped with superconducting toroidal magnets (3.7 T) which allow steady-state plasma operation. The previous configuration of the machine, named Tore Supra, was operated between 1988 and 2011. The upgrade to WEST in 2013-2016 transformed the tokamak from a limiter to a divertor configuration [START_REF] Bucalossi | [END_REF], a key-transformation to test and validate the divertor architecture and operation for the international experimental fusion device ITER currently in construction in south of France [4]. Our experiment is based on the use of a well-known and proven real-time soft-error rate (RTSER) testbench with CMOS bulk 65nm SRAMs, previously operating in altitude on the ASTEP platform, underground at the Laboratoire Souterrain de Modane (LSM) and at sea level in Marseille [5][START_REF] Autran | Soft Errors: from particles to circuits[END_REF][START_REF] Moindjie | [END_REF]. These RTSER measurements have combined with a real-time neutron spectrometry to simultaneously characterize the neutron flux and energy distribution during machine shots. In the following, the experiment is detailed, measurements are reported and also discussed with respect to additional numerical simulations, physics of plasma and machine operation.

II. EXPERIMENTAL SETUP AND CONFIGURATION

The experimental test equipment embeds 380 memory circuits [START_REF] Bucalossi | [END_REF]226 Gbit) manufactured by STMicroelectronics in CMOS 65 with a fabrication processes based on a BPSG-free BEOL that eliminates the major source of 10 B in the circuits and drastically reduces the possible interaction between 10 B and very low (thermal) energy neutrons. The test chips contain 8.5 Mbit of single-port SRAM (without deep-Nwell) with a bit cell area of 0.525 µm 2 . The technology core voltage is 1.2 V. The circuits have been assembled on printed boards and tested several years in altitude, underground and at sea level using a dedicated equipment, fully compliant with all the specifications of the JEDEC standard JESD89. Details about the setup and previous RTSER experiments can be found in Ref. [START_REF] Moindjie | [END_REF]. Just before this experiment at WEST facility, the equipment has been continuously working since March 2015 in Marseille, showing a consolidated and stable soft-error rate equal to 226 FIT/Mbit due to atmospheric neutrons (NYC conditions) and 1148 FIT/Mbit due to internal chip radioactivity.

The neutron flux and energy characterization during reactor shots was performed using a commercial DIAMON neutron spectrometer. DIAMON (Direction-aware Isotropic and Active neutron MONitor) is a novel detection system based on a polyhedral moderator body made of high-density polyethylene (HDPE) which hosts a matrix of semiconductor-based thermal neutron sensors [8]. The instrument was carefully calibrated at the neutron calibration facility of Politecnico di Milano using a 241 Am-Be source.

During the experiment, the test equipment has been placed in the facility hall of the WEST tokamak, successively at three positions (P1 = 7 m, P2 = 6 m and P3 = 5.2 m from the center of the machine). In this publication, after discussing the SEEs of natural origin measured in these 3 positions, we focus our analysis on the SEEs induced by the plasma neutrons at the position P3, in which the test equipment has received the highest neutron fluence. The SRAM circuits with their upper face oriented towards the center of the machine, were placed in the equatorial plane of the tokamak tanks to a mechanical support. The computer controlling the experiment was placed on the machine basement, shielded from neutrons under a borated concrete slab (thickness 80 cm). Figure 1 shows different views and schematics of the implementation of the experiment in the WEST hall.

III. EXPERIMENTAL RESULTS DURING WEST D-D OPERATION

A. WEST reactor operation and fusion neutrons

The WEST C5 SEE measurement campaign started on December 8, 2020 and finished on January 27, 2021 (50 days). It consisted in a series of several hundreds of machine shots (ranging from 1 to 18s in duration, see Figure 2 Total neutron production during the shots was measured using a network of fission chamber distributed around the tokamak in the WEST hall (Figure 2 right). Neutrons transported outside the machine produce, by absorption, reflection and attenuation mechanisms, a complex radiation field outside the machine dominated by thermal and low energy neutrons. Figure 3 shows a typical neutron energy distribution measured during a machine shot by the DIAMON spectrometer. The "high" energy peak is around 1.6 MeV: it corresponds to 2.45 MeV plasma neutrons transported outside the tokamak. Note that a significant neutron flux is measured up to 10 MeV, corresponding to initial 14 MeV neutrons, also transported and moderated through the tokamak structure.

B. SEU measurements

For the whole duration of the C5 campaign, 54 SEU events for a total of 66 bit flips has been measured: 5 events (4 single bit upsets -SBU -and 1 multiple cell upset -MCU -of multiplicity 3) have been recorded during the machine shutdown (nights, week-ends) or machine stops (> 30 minutes) between consecutive machine shots and 49 events (42 SBU, 4 MCU(2) and 3 MCU(3)) have been detected in perfect coincidence with a shot (taking into account the duration of the reading/verification cycle of the test equipment). For these 49 events induced by the machine operation, 41 on 49 have been observed after January 17, 2020, i.e. in the last ten days of the C5 campaign during which steady-state plasma conditions and a maximal efficiency have been obtained in terms of D-D fusion. Figure 4 shows the occurrence of all bit flips as a function of time for the whole duration of the C5 campaign.

IV.RESULTS ANALYSIS AND DISCUSSION

The events observed during the whole campaign (50 days = 1200 h) are the result of two radiation sources working in parallel: the natural radiation background and the artificial machine environment. For the first one, the observations perfectly agree with the neutron (226 FIT/Mbit) and alpha SER (1148 FIT/Mbit) previously measured. Indeed, the number of expected events for 1200 h is 226 ´ 3226 ´ (1200/10 9 ) = 0.87 » 1 for atmospheric neutrons and 1148 ´ 3226 ´ (1200/10 9 ) = 4.44 » 4 for chip internal radioactivity, that corresponds to a total of 5 events. We effectively observed 4 SBU and 1 MCU not correlated with a machine shot. For the artificial machine environment, the analysis of our results has been performed for steady-state conditions of the machine, i.e. for the 41 SEU events correlated to machine shots and observed in the ten last days of the C5 campaign, at position P3 = 5.2 m from the centre of the machine. They correspond to 36 SBU, 3 MCU(2) and 2 MCU(3) representing a total of 48 bit flips. For these SEU events, an accurate neutron fluence has been determined combining DIAMON and fission chamber measurements. Figure 5 shows the cumulated bit flip distributions for these events as a function of the cumulated neutron fluence (the total cumulated duration for the corresponding shots is 1252 s). This distribution exhibits a very regular slope after 7´10 6 n/cm 2 when reproducible steadystate conditions have been reached. From this slope, it results a bit flip SER = 4.28´10 7 FIT/Mbit and an acceleration factor of the WEST radiation field with respect to the natural background of about AF » 190,000. To understand the quite surprising occurrence of MCU events in such a low energy neutron irradiation, we performed Geant4 and TIARA numerical simulations to compute: i) a complete set of interaction databases of monoenergetic neutrons with a natural silicon target (1 cm 2 ´ 20 µm) for energies ranging from 1 MeV to 14 MeV by step of 1 MeV and ii) a corresponding set of SEE event databases for the 65nm SRAM subjected to 1 to 14 MeV neutrons. All details and extensive results of these simulations will be given in the final paper for both nuclear and circuit-response databases. Figures 6 and 7 respectively show the distribution of elastic, inelastic and nonelastic events and the number of events per reaction type resulting from the interaction of 5´10 8 millions of monoenergetic neutrons with a natural silicon target of 1 cm 2 ´ 20 µm. Below 4-5 MeV, quasi only elastic and nonelastic interactions occur, producing Si recoils with short ranges in the target that are not able to induce MCUs in the simulated SRAM (this point will be detailed and discussed in the final paper). Consequently, our results suggest that the observed MCUs may be due to residual number of neutrons effectively detected above 5 MeV (Figure 3). This hypothesis is supported by TIARA Monte Carlo simulations performed on a matrix of 15´15 SRAM circuit (Figure 8) showing that MCU events effectively occur in significant proportion only for neutrons above »5 MeV. Considering now the upper part (E >1 MeV) of the WEST spectrum of Figure 3 as an input for Geant4 and TIARA simulations, additional simulation results (Figure 9) seem to confirm this hypothesis, suggesting the non-negligible impact on electronics of D-T neutrons in D-D fusion experiment despite of their relative low flux. Additional results including the analysis of SER values will be included in the extended paper, as well as projections and discussion for other fusion machine environments (JET, ITER).

  left) during which plasma conditions (with different characteristics: temperature, density) are met for D-D fusion following the main reactions: D + D ® 3 He(0.82 MeV) + n(2.45 MeV) (50%) and D + D ® T(1.01 MeV) + p(3.02 MeV) (50%). Once D-D fusion is correctly initiated at the beginning of a machine shot, the concentration of tritium (produced in the second D-D reaction) gradually increases in the plasma and renders possible D-T fusion: D + T ® 4 He(3.52 MeV)+ n(14.07 MeV). However, this last reaction should remain marginal compared to the first two, given the dilution of tritium in the plasma essentially composed of deuterium.

Figure 1 .

 1 Figure 1. Global and detailed views of the RT-SER test equipment installed in the vicinity of the tokamak in the WEST experimental hall. The schematics represents a top view of the installation showing the location of the test bench, at 5.2 m from the center of the tokamak.

Figure 2 .

 2 Figure 2. Distribution of shot duration (left) and of total number of neutrons produced per shot (right) as a function of time. Shots corresponding to P3 position start after 1,150 h; shots in coincidence with SEU detection are in red.

Figure 3 .

 3 Figure 3. Differential neutron spectrum measured by the DIAMON spectrometer during shot #56359 at the level of the rack of the SRAM circuits.

Figure 4 .

 4 Figure 4. Distribution of bit flips as a function of time for the whole WEST C5 campaign.

Figure 5 .

 5 Figure 5. Distribution of bit flips as a function of the cumulated neutron fluence after January 18, 2021 and until the end of the WEST C5 campaign (test equipment at position P1 = 5,20 m).

Figure 6 .

 6 Figure 6. Distribution of elastic, inelastic and nonelastic events in monoenergetic neutron-silicon interactions resulting from Geant4 (version 4.10.07) simulations.

Figure 7 .

 7 Figure 7. Number of n-Si interaction event per type of reactions in a natural silicon target (1 cm 2´ 20 µm) subjected to 5´10 8 millions of monoenergetic neutrons with energy ranging from 1 to 14 MeV.

Figure 8 .

 8 Figure 8. Distribution of SBU and MCU events obtained by TIARA simulation for the 65nm SRAM subjected to neutrons from 1 to 14 MeV.

Figure 9 .

 9 Figure 9. TIARA simulation considering the upper part (E > 1 MeV) of the WEST spectrum of Figure 3 as the input neutron source and comparison with the real-time experiment presented in this paper.

  in a natural silicon target(1 cm2 x 20 µm, 500 millions of incident neutrons, Geant4 simulations)Total Si(n,n)Si elastic Total Si(n,n')Si inelastic Total Si(n,γ)Si capture Total Si(n,αTIARA Simulation vs Real-Time ExperimentReal-Time experiment TIARA simulation


Gerardo Kenny Rumindo

Jacques Ohayon

Pierre Croisille

Patrick Clarysse

In vivo estimation of normal left ventricular stiffness and contractility based on routine cine MR acquisition

Keywords: left ventricle model, finite element analysis, cine MRI, myocardial stiffness, myocardial contractility, ischemic heart disease Beginning diastole, ESP: end-systolic pressure, EDP: end-diastolic pressure, EDV: end-diastolic volume, EDS: end-systolic volume, FT: feature tracking, CMR: Cardiac Magnetic Resonance, NA: not applicable. FG: Fung-Guccione law [9], HO: Holzapfel-Ogden Bestel-Clement-Sorine model [11], Time-varying elastance model [12], h, t: spatial helix/transverse angle distribution of the myocardial fibers

Post-myocardial infarction remodeling process is known to alter the mechanical properties of the heart. Biomechanical parameters, such as tissue stiffness and contractility, would be useful for clinicians to better assess the severity of the diseased heart. However, these parameters are difficult to obtain in the current clinical practice. In this paper, we estimated subject-specific in vivo myocardial stiffness and contractility from 21 healthy volunteers, based on left ventricle models constructed from data acquired from routine cardiac MR acquisition only. The subject-specific biomechanical parameters were quantified using an inverse finite-element modelling approach. The personalized models were evaluated against relevant clinical metrics extracted from the MR data, such as circumferential strain, wall thickness and fractional thickening. We obtained the ranges of healthy biomechanical indices of 1.60 ± 0.22 kPa for left ventricular stiffness and 95.13±14.56 kPa for left ventricular contractility. These reference normal values can be used for future model-based investigation on the stiffness and contractility of ischemic myocardium.

Introduction

Ischemic heart disease occurs due to the constriction of the coronary arteries, causing reduced supply of blood and oxygen to the heart muscles. This eventually leads to myocardial functional impairment and scarring. Post-infarction remodeling modifies the mechanical behavior of the myocardium [START_REF] Richardson | Physiological implications of myocardial scar structure[END_REF], making stiffness and contractility good potential prognostic parameters. Unfortunately, clinicians normally lack information on these biomechanical parameters, since direct measurement is very challenging, if not impossible. Mechanical models capable of predicting subject-specific stiffness and contractility would be a useful tool to assess the condition of an ischemic patient. However, before performing any measurement on ischemic hearts, it is necessary to first establish the range of normal LV stiffness and contractility.

The determination of subject-specific constitutive material parameters is a challenging inverse problem involving several key components: hearts shape, tissue structure and motion representation, myocardial material modeling, definition of appropriate boundary and loading conditions and optimization strategy [START_REF] Wang | Image-based predictive modeling of heart mechanics[END_REF]. Cardiac imaging modalities such as cardiac magnetic resonance (CMR) now provide access to several of the complementary information needed to build cardiac models e.g. geometry, tissue characteristics, and motion. Several finite element (FE)-based studies have addressed the identification of diastolic mechanical parameters of the myocardium within an optimization process in reference to myocardial displacement [START_REF] Xi | The estimation of patient-specific cardiac diastolic functions from clinical measurements[END_REF][START_REF] Wang | Modelling passive diastolic mechanics with quantitative MRI of cardiac structure and function[END_REF], strains [START_REF] Walker | MRI-based finite-element analysis of left ventricular aneurysm[END_REF], strains and volumes either simultaneously [START_REF] Gao | Changes and classification in myocardial contractile function in the left ventricle following acute myocardial infarction[END_REF] or separately [START_REF] Genet | Distribution of normal human left ventricular myofiber stress at end diastole and end systole: a target for in silico design of heart failure treatments[END_REF][START_REF] Pena | Unraveling changes in myocardial contractility during human fetal growth: a finite element analysis based on in vivo ultrasound measurements[END_REF]. Estimation of myocardial contractility requires the incorporation of subject-specific active contraction, which often results in the estimation of one active parameter of various active tension formulations. Table 1 presents a non-exhaustive list of models (with initial geometry, active and passive constitutive laws, and boundary conditions used) developed to estimate the passive and active mechanical properties of the myocardium. The variety of estimated passive parameters in various conditions is visible in Table 4 for human studies. Genet et al. proposed a personalization method to construct a reference left ventricular stress map [START_REF] Genet | Distribution of normal human left ventricular myofiber stress at end diastole and end systole: a target for in silico design of heart failure treatments[END_REF]. Their study managed to find a normal range of myocardial contractility, but it was limited to five healthy subjects who underwent extensive CMR acquisitions, which are time-consuming, costly, and not suitable for every individual.

In this study, we aim to estimate left ventricular stiffness and contractility of 21 healthy subjects based on routine cardiac cine CMR acquisition.

Methods

Subjects datasets

Two cardiac MR datasets of healthy volunteers were used in this study. The first dataset (n=11, 9 males) is open-access, available within the context of a MICCAI STACOM Challenge [START_REF] Tobon-Gomez | Benchmarking framework for myocardial tracking and deformation algorithms: an open access database[END_REF]. The second dataset was part of the MAR-VEL cohort run by CHU Saint-Etienne, France (n=10, 4 males). The MARVEL study (ID RCB 2016-A00913-48) was approved by the local ethical committee (URB #6/052) and is registered at ClinicalTrials.gov (NCT03064503). All patients provided written informed consent. Unfortunately, no pressure measurements were recorded during the examinations for these two datasets.

Processing of MR data

The MICCAI dataset has a pixel size of 1.25 mm and a slice thickness of 8 mm, whereas the MARVEL dataset has a pixel size of 0.55 mm and a slice thickness of 7 mm. LV segmentation was performed on the short-axis, the 2-chamber and 4-chamber long-axis images within CVI42 software (©Circle Cardiovascular Imaging Inc., Figure 1-top). The end-systolic (ES) and end-diastolic (ED) volumes of each subject were computed from the segmented endocardial surfaces. The LV wall was meshed with hexahedral elements in the FE software ABAQUS (©Dassault Systemes) (Figure 2). The mesh for each model consisted of 3745 ± 450 C3D8R elements and 4521 ± 539 vertices. We verified our chosen mesh density via a mesh convergence analysis. The ES strains for each subject were computed using the Feature Tracking (FT) technique available in CVI42 (Figure 1-bottom). The ES strain was calculated with respect to ED configuration.

LV myocardial modelling

In the present work, the ES geometry has been chosen as the reference configuration ; it was directly accessible from clinical MRI data (with the enddiastolic one). We therefore assumed that the ES geometry was closer to the one at the end of isovolumic relaxation configuration [START_REF] Asner | Estimation of passive and active properties in the human heart using 3D tagged MRI[END_REF][START_REF] Asner | Patient-specific modeling for left ventricular mechanics using datadriven boundary energies[END_REF], the state where both volume and pressure are minimal and the closest to unloaded configuration. Figure 3 illustrates the fiber arrangement attributed to the LV myocardium according to the description in [START_REF] Bovendeerd | Determinants of left ventricular shear strain[END_REF]. Appendix A provides the mathematical formulation of the myocardial fiber arrangement. Myocardial passive behavior was modeled by transversely-isotropic hyperelastic material [START_REF] Guccione | Passive material properties of intact ventricular myocardium determined from a cylindrical model[END_REF], while active behavior was defined by a stretch-dependent active law [START_REF] Guccione | Mechanics of active contraction in cardiac muscle: Part IIcylindrical models of the systolic left ventricle[END_REF]. They were implemented in the user material (UMAT) subroutine for the FE solver ABAQUS/Standard.

The passive material behavior is parameterized by 4 material parameters

(C 0 , b f , b t , b f t ).
C 0 is the linear constant that determines the overall isotropic stiffness of the myocardium, b f , b t determines the stiffness in the local myocardial fiber directions, and the transverse directions, respectively, whereas b f t defines the shear stiffness. The ratio of b f to b t to b f t was kept fixed at 1.0 to 0.4 to 0.7, in accordance to the finding of Wenk et al. [START_REF] Wenk | First evidence of depressed contractility in the border zone of a human myocardial infarction[END_REF]: b t = 0.4b f ; b f t = 0.7b f . This approach limits the material parameters identification to 2 parameters, C 0 and b f . The active stress law is linearly-scaled by its maximal value T max , which needs to be identified for each subject. Further details can be found in Appendix B.

The base ring was kept fixed (Figure 2), while the epicardium was boundaryfree. In the normal heart, the base plane exhibits a large longitudinal displacement toward the apex (up to 1cm) while the apex moves much less. However, the imposed constraint (adopted in most of the 3D finite element simulations) helps to stabilize the numerical solution and should not drastically alter the intraparietal distributions of stresses and strains, which are the mechanical responses of interest in this study. FE Simulations were performed by imposing a prescribed volume change to the LV cavity using an integrated optimization process available in the ABAQUS software, which iteratively calculates the amplitude of the cavity pressure that must be imposed in order to satisfy the required LV volume.

Quasi-incompressibility is enforced by setting the bulk modulus to be close to that of water [START_REF] Holzapfel | Nonlinear Solid Mechanics: A Continuum Approach for Engineering[END_REF].

Model personalization

Three parameters were to be identified for each subject; C 0 and b f for the passive law, and T max for the active law. We identified the passive and active parameters separately using a 2-point simulation approach [START_REF] Genet | Distribution of normal human left ventricular myofiber stress at end diastole and end systole: a target for in silico design of heart failure treatments[END_REF][START_REF] Sun | A computationally efficient formal optimization of regional myocardial contractility in a sheep with left ventricular aneurysm[END_REF].

The passive parameters C 0 and b f were identified so that the distance between the simulated end-diastolic pressure-volume relationship (EDPVR) and that of Klotz et al. [START_REF] Klotz | Single-beat estimation of end-diastolic pressure-volume relationship: a novel method with potential for noninvasive application[END_REF] was minimized. Note that the value of the ED pressure results from this optimization process (close to 9mmHg = 1.2kP a). Once the passive parameters C 0 and b f were identified for each subject, the active parameter T max was identified by matching a normal ES pressure of 120mmHg(16kP a).

The parameters identification was performed using the Nelder-Mead optimization available in MATLAB (©MathWorks) [START_REF] Lagarias | Convergence properties of the Nelder-Mead simplex method in low dimensions[END_REF]. The cost function for the diastolic personalization was the sum of squared error between the "Klotz" curve pressure P klotz and the predicted pressure P sim for the corresponding normalized volume values i being compared. For the systolic personalization, the active material parameter was optimized to result in the targeted ES pressure.

Model evaluation

In vivo ES circumferential strain values measured with FT technique on cine

MR images were interpolated on the finite-element mesh. These interpolated strains were compared with the personalized simulation strains. Only circumferential strains were compared, as radial strain extracted from FT technique was found to not correlate well with other LV strain measurement techniques [START_REF] Augustine | Global and regional left ventricular myocardial deformation measures by magnetic resonance feature tracking in healthy volunteers: comparison with tagging 29 and relevance of gender[END_REF][START_REF] Padiyath | Echocardiography and cardiac magnetic resonance-based feature tracking in the assessment of myocardial mechanics in Tetralogy of Fallot: An intermodality comparison[END_REF], and to be less robust and reproducible [START_REF] Lu | Strain measurement by cardiovascular magnetic resonance in pediatric cancer survivors: validation of feature tracking against harmonic phase imaging[END_REF][START_REF] Schuster | The intraobserver reproducibility of cardiovascular magnetic resonance myocardial feature tracking strain assessment is independent of field strength[END_REF]. Longitudinal strains were not compared as only 2-long-axis images were acquired, leading to insufficient number of calculated strain points for our validation purpose.

A set of global LV function metrics were calculated from the subject-specific models: ED wall thickness (EDW T ), systolic fractional thickening (SF T ), global longitudinal strain (GLS) and segmental rotations (ROT ). These LV metrics were obtained referring to Figure 4 as follows:

• EDWT: wall thickness at ED (mm)

• SFT = 100 * (ESW T -EDW T )/EDW T : systolic fractional thickening (%), i.e. the relative difference between the wall thickness at ES and ED with respect to ED • GLS = 100 * (EDL -ESL)/EDL: global longitudinal strain (%), the relative difference between the length along the myocardial longitudinal direction at ES and ED with respect to ED.

• ROT base and ROT apex : segmental rotation (degrees), which are calculated by observing the angular displacement of a point at basal and apical levels with respect to the LV axis, denoted as θ base and θ apex in Figure 4, respectively. We assumed counter-clockwise rotation to be positive as seen from apex.

As an additional method evaluation, the metrics obtained from the midventricle region of the personalized models were compared to the ones measured from cine MR images and to the normal values found in literatures. The modelpredicted EDW T and SF T values were compared to the values computed from the MR images. Moreover, all four metrics from our personalized models were also compared to the normal values found in literature.

LV stiffness and contractility indices

We use the initial Young's modulus of the passive myocardium in the fiber direction to represent the subject-specific LV stiffness with one scalar index.

The mathematical expression of this parameter is detailed in Appendix B. This stiffness index (SI ) reflects the behavior of the transversely isotropic incompressible passive myocardium at its initial configuration (i.e. zero-stress state).

We considered the optimized T max parameter as the subject-specific index of myocardial contractility (CI ). Both indices are expressed in kPa.

Statistical analysis

The correlations between the ejection fraction (EF), myocardial volume, enddiastolic or end-systolic volume, contraction index (CI ) and stiffness index (SI )

were analyzed by multiple linear regression using a commercially available software package (SigmaStat 3.5, Systat Software, Inc., Point Richmond, CA, USA).

Regressions with probability values p < 0.001 were considered statistically significant.

Results

Subject-specific MR data

Table 2 displays the LV metrics for all subjects. 'Measured' lists the metrics obtained from the MR data. ED wall thickness is established to be different between male and female: in our study we found ED wall thickness of 8.1 ± 1.3 mm and 6.1 ± 0.8 mm for male and female subjects, respectively.

Model personalization evaluation

The one-parameter (CI ) personalization process conducted to the targeted ES pressure of 16kP a for all the cases. The average optimized ED pressure in the SI identification process was 1.28kP a. 2 also reports on the goodness-offit between imaging data and model i.e. ED wall thickness, systolic fractional thickening and circumferential strain for the short axis direction and the global longitudinal strain for the long-axis direction. Figure 6 shows the Bland-Altman respectively. E f and E cf are the initial Young's modulus in the fiber and cross-fiber direction, respectively. (see Appendix B for details). The parameter α is defined as plots comparing the measured and simulated ES circumferential strains, midventricular ED wall thickness and systolic fractional thickening. Figure 7 shows the segmental circumferential strain error of our personalized models from the midventricular slices of four cases.

α = b f + bt + b f t . Study C 0 b f b t b f t α SI = E f E cf E f /E cf CI (kPa) (kPa) (kPa) (kPa
On average, our personalized models had an ED wall thickness of 8.7 ± 0.9 mm (basal), 8.9 ± 0.7 mm (medial), 8.8 ± 0.9 mm (apical) for male subjects, and 6.7 ± 0.5 mm (basal), 6.3 ± 1.0 mm (medial), 5.7 ± 0.8 mm (apical) for female subjects (Figure 8). Similarly, we divided the results from LV systolic fractional thickening into basal, medial and apical positions on the left ventricles (Figure 9(a)). On average, our personalized models resulted in SFT of 72.0 ± 17.7 % (basal), 69.2 ± 8.5 % (medial) and 56.9 ± 7.6 % (apical). In addition, our personalized models lead to an average GLS values of -15.1 ± 1.9 % (Figure 9(b)). With respect to the ED condition, average segmental rotations of +11.8°± 0.7°and -3.0°± 2.6°for basal and apical levels, respectively, were observed in our models. shown in [START_REF] Wang | Left ventricular diastolic myocardial stiffness and end-diastolic myofibre 30 stress in human heart failure using personalised biomechanical analysis[END_REF].

Subject-specific biomechanical indices

Discussion

Despite the clear advantages for clinicians and biomedical engineers in estimating the in vivo left ventricular mechanical properties, no studies on a non-invasive estimation method have been developed and validated solely based on data acquired from routine clinical data. The present work is one of the first that use inverse FE modeling and measurements from routine cine CMR acquisitions to estimate subject-specific left ventricular stiffness and contractility on twenty-one healthy subjects. Our personalization approach was extensively evaluated against measured regional LV metrics in the form of circumferential strain values and against measured global LV metrics such as ED wall thickness and systolic fractional thickening. The range of biomechanical indices found in this study (Table 3) could serve as a reference for in silico diagnosis of potential cardiac pathologies. Table 4 presents the results of the estimation of mechanical parameters performed on the 21 healthy volunteers, with a comparison with some values reported in the literature. Because of the large variability observed in the values of the constitutive parameters, it remains difficult to compare them among the published works. This is notably due to the wide variety of approaches used to solve the inverse biomechanical problem. In addition, the obtained parameter values strongly depend on the optimization scheme used, the solution of which is generally not unique [START_REF] Xi | An automatic data assimilation framework for patient-specific myocardial mechanical parameter estimation[END_REF]. The lack of a standardized characterization of the passive properties of the myocardium (through e.g. 'normal' parameter value range) represents a major problem that limits the use of such patient-specific biomechanical modeling approaches for clinical diagnosis.

One group has recently suggested that by simultaneously considering two cost functions, one based on energy and the other on geometry, it was possible to uniquely determine the set of solutions (C 0 , α = b f + b t + b f t ) in the case of the constitutive law of the transversely isotropic material [START_REF] Nasopoulou | Improved identifiability of myocardial material parameters by an energy-based cost function[END_REF]. This is a promising way to unify and compare stiffness measurements obtained by different authors and to extract clinically meaningful reference values. In a sensitivity study (not reported in this paper), the impacts of C 0 , b f , CI , myofiber helix angle α h , and transverse angle α t onto myocardial mechanical strains (E cc , E rr , E ll , and E rc )

has been assessed at basal, mid and apical levels. Their values were varied between 0.7 and 1.3 times the reference value. It was observed that the evolution of the strains was globally consistent and smooth. In particular, both stiffness indices (C 0 and b f ), and therefore SI , had a negative correlation with strains.

CI had a positive correlation with the radial strain E rr .

The circumferential strains obtained with the FT technique were within the normal range as found in literature [START_REF] Kawel | Normal left ventricular myocardial thickness for middle-aged and older subjects with steady-state free precession cardiac magnetic resonance[END_REF][START_REF] Lang | Recommendations for chamber quantification: A report from the american society of echocardiographys guidelines and standards committee and the chamber quantification writing group, developed in conjunction with the european association of echocardiography, a branch of the european society of cardiology[END_REF][START_REF] Rodrigues | The relationship between left ventricular wall thickness, myocardial shortening, and ejection fraction in hypertensive heart disease: Insights from cardiac magnetic resonance imaging[END_REF][START_REF] Schuster | The intraobserver reproducibility of cardiovascular magnetic resonance myocardial feature tracking strain assessment is independent of field strength[END_REF][START_REF] Ubachs | Normal values for wall thickening by magnetic resonance imaging[END_REF]. In total, 771 ± 132 strain points were computed via the FT technique on cine MR images per subject for circumferential strain. The shift between measured and simulated values are in average low for ED wall thickness (-0.56 ± 0.98mm), systolic fractional thickening (-1.8 ± 24.4%) and circumferential strain (-0.89 ± 3.8%), indicating a good match in the radial-circumferential directions (Table 2). The average shift for GLS is however larger (5.3 ± 4.3%). The average measured value is significantly high -9.8 ± 3.7 %, and far from the range for normal human values [START_REF] Kalam | Prognostic implications of global lv dysfunction: a systematic review and meta-analysis of global longitudinal strain and ejection fraction[END_REF][START_REF] Moore | Threedimensional systolic strain patterns in the normal human left ventricle: characterization with tagged MR imaging[END_REF]. This probably results from the estimation that was based only on 2 long-axis images, which led to 29 ± 7 longitudinal strain points computed on average per subject. On the contrary, the average simulated GLS value is closer to the normal range (-15.1 ± 2%).

We also compared the global LV pump functions from the personalized models with the normal values found in literature. Figure 8 and Figure 9 show these comparisons for ED wall thickness, systolic fractional thickening and global longitudinal strain. The box-plots represent the values from our personalized model and the arrows represent the reference values found in literature. These LV metrics obtained from the MR data were also in agreement with literature.

The segmental rotations predicted by our personalized models were in agreement with [START_REF] Lorenz | Delineation of normal human left ventricular twist throughout systole by tagged cine magnetic resonance imaging[END_REF][START_REF] Young | Two-dimensional left ventricular deformation during systole using magnetic resonance imaging with spatial modulation of magnetization[END_REF]. Since LV torsion is highly dependent on endocardial and epicardial contraction, LV geometry and myocardial fiber arrangement, this showed that the rule-based myocardial fiber architecture in our model was able to produce realistic physiological torsional deformation of the LV.

No significant bias is observed for both ES circumferential strain and systolic fractional thickening. The bias for ED wall thickness is about -0.5mm expressing a slight tendency to higher MRI-based measurements (Figure 6(b)).

There seems to have a magnitude dependence of the systolic fractional thickening (Figure 6(c)); this is not the case for the other measurements. While limits of agreement remain acceptable for circumferential strain and wall thickness (Figures 6(a) and 6(b)), they are significantly high for systolic fractional thickening (Figure 6(c)). Our personalized approach was able to produce similar circumferential strains to the measured ones (Figure 6(a)). However, two of the male subjects -#11 and #19 -were found to be outliers (Figure 6(a)). Table 5 lists in details the image-based measured ED wall thickness of these subjects, which were relatively thicker in comparison with the remaining male subjects and from the literature [START_REF] Kawel | Normal left ventricular myocardial thickness for middle-aged and older subjects with steady-state free precession cardiac magnetic resonance[END_REF]. Furthermore, subject #11 had a reduced global circumferential strain of -11.2 ± 6.7 % as measured with the FT technique, which might be an indication of hypertensive cardiomyopathy [START_REF] Rosen | The prognostic value of global circumferential strain in patients withsuspected myocardial disease[END_REF]. The limitations of the present study are mainly related to modelling assumptions. The exclusion of the right ventricle in the model definitely affects the distribution of the circumferential strains. We also neglected the multiphysics aspects of a functional heart, i.e. hemodynamics and electrophysiology.

Another limitation is that, due to the absence of pressure measurements, our model personalization was based on normal ED and ES pressures found in literature. A very interesting study has considered the impact of the uncertainty in the pressure values based on their modelling approach [START_REF] Xi | Understanding the need of ventricular pressure for the estimation of diastolic biomarkers[END_REF]. They found that the PV-based approach (Klotz like) lead to reasonable stiffness results on 18 healthy and pathological cases. We therefore assumed that this approximation can be considered valid for healthy subjects, although probably not for patients. Moreover, our validated models were limited to reflect the mechanical properties of

W V = 1 D (J - 1 J )
where D determines the material incompressibility and J is the Jacobian of the deformation gradient F: J = det(F). To enforce quasi-incompressibility, D is set to 0.001 in our study, so that the bulk modulus of the tissue is close to that of water.

The deviatoric part of the strain energy density function is as follows:

W I = C 0 2 (e Q -1) Q = b f Ē2 f f + b t ( Ē2 ss + Ē2 nn + Ē2 sn + Ē2 ns ) + b f t ( Ē2 f s + Ē2 sf + Ē2 f n + Ē2 nf )
where C 0 is the linear passive parameter constant, b f , b t and b f t are the exponential parameter constants. In this study, we kept the exponential ratio fixed, i.e. b f : b t : b f t is equal to 1.0 : 0.4 : 0.7, in accordance to the findings by Wenk et al [START_REF] Wenk | First evidence of depressed contractility in the border zone of a human myocardial infarction[END_REF]. E is the Green-Lagrange strain tensor, where the prefixes f , s and n denote the myofiber, sheet and sheet-normal directions, respectively. Ē is the deviatoric part of the Green strain, which is defined by:

Ē = 1 2 ( FT . F -I) with F = J -1 3 F
The stiffness index proposed in our study is defined in accordance to the linearization of the stress-strain relationship described in [START_REF] Payan | Biomechanics of Living Organs: Hyperelastic Constitutive Laws for Finite Element Modeling[END_REF] (Chapter 1, equation 23 and 24):

E (1) = C0 2 (2b f + b t ) = SI (1)
E (2) = E (3) = C0bt(2b f +bt) b f +bt (2)
where E (i) denotes the linearized stiffness in the i direction. The stiffness index SI (Eq. 1) was therefore defined as the amplitude of the initial Young's modulus of the passive myocardium in the fiber direction E f (i.e., SI = E f = E (1) ).

Using a transversely-isotropic constitutive law and assuming b t = 0. During active contraction, the active tension is added to the passive second Piola-Kirchhoff stress tensor T as follows:

T = T passive + T active
where T passive is the first derivation of the strain energy density function with respect to the Green strain tensor, i.e. ∂W ∂E . The active tension is dependent on the stretch in the local myofiber direction as follows:

T active = T max Ca 2 0 Ca 2 0 + ECa 2 50 C t
where T max is the active parameter need to be identified, representing the maximum isometric tension the tissue is able to generate. Ca 0 is the peak intracellular calcium concentration. The length-dependent calcium sensitivity (ECa 2 50 ), the time-related scaling (C t ), and the current sarcomere length (l) are given by: ECa 50 = (Ca 0 ) max √ e B(l-l0) -1 and where (Ca 0 ) max is the maximum peak intracellular calcium concentration, B is a constant that governs the shape of peak isometric tension-sarcomere length relation, l 0 is the sarcomere length when no active tension develops, t 0 is the peak tension time, t r is the duration of the relaxation, m and b are constants that govern the relation of the relaxation duration and sarcomere length, and l r is the stress-free sarcomere length. The values of the parameters are listed in Table 7 in accordance to [START_REF] Guccione | Mechanics of active contraction in cardiac muscle: Part IIcylindrical models of the systolic left ventricle[END_REF]. We defined the patient specific contractility index as : CI = T max .

C t = 1 2 ((1 -cos(ω)) ω =            0 ≤ t <

  These biomechanical parameters are quantified by personalizing a FE mechanical model of the LV. The models were validated individually through comparison against measured strains and other clinically relevant measurements, i.e. enddiastolic wall thickness, systolic fractional thickening, global circumferential and longitudinal strains. The main goal of this study is to propose an initial step toward estimating and verifying subject-specific left ventricular stiffness and contractility based on routine CMR acquisitions.

Figure 1 :

 1 Figure 1: Top: short-axis, 2-chamber and 4-chamber long-axis cine images (from left to right). Bottom: Feature Tracking results; boundary points, tracking mesh, circumferential strain map (from left to right)

Figure 2 :

 2 Figure 2: Meshed geometries of the left ventricles at end systole from four subjects

Figure 3 :

 3 Figure 3: Left: definition of helix (α h ) and transverse (αt) angles within the normalized coordinate system (Appendix A). Right: spatial distribution of helix angle (top) and transverse angle (bottom)

Figure 4 :

 4 Figure 4: Illustration for the LV metrics. See text for the complete formula (EDWT and ESWT: end-diastolic and end-systolic wall thickness, EDL and ESL: end-diastolic and end-systolic length, θ base and θapex: basal and apical rotations)

Figure 5

 5 shows an example of the personalized LV dynamics of a healthy volunteer, where the top figures show the distribution of ES myofiber strain and stress, and the bottom figures shows a midventricular slice of deformed simulated LV geometry at ED and ES superimposed with the respective MR images. The comparison of anatomical and functional parameters given in Table

Figure 5 :

 5 Figure 5: Personalized model from one healthy volunteer data (subject #5). Top: myofiber strain (left) and stress (right) at ES. Bottom: deformed midventricular slice of one LV model in yellow superimposed with the MR images at ED (left) and ES (right)

Figure 6 :

 6 Figure 6: Bland-Altman plots: simulated vs MRI-based measurements for (a) circumferential strain (arrows indicate the outliers), (b) end-diastolic wall thickness, and (c) systolic fractional thickening.

Figure 7 :

 7 Figure 7: Segmental circumferential strain error (FT -personalized simulation) from a midventricular slice of four cases. The error is represented as a mean of circumferential strain point-to-point difference (simple subtraction) in each segment

Figure 8 :Figure 9 :

 89 Figure 8: ED wall thickness calculated from the personalized models at basal (a), medial (b) and apical (c) levels for 21 subjects (13 males). The arrows are reference values from Lang et al. [29] (red) and Kawel et al. [30] (black)

left ventricle in 2 -

 2 points of the cardiac cycle, end-diastole and end-systole. The unicity of the estimated stiffness and contractility parameters was not mathematically proven. However, by imposing the proportionality between b f , b t and b f t as in[START_REF] Wenk | First evidence of depressed contractility in the border zone of a human myocardial infarction[END_REF], the estimation for the passive part is constrained to two parameters only (namely C 0 and b f ), enforcing a local convergence to convenient solutions. Furthermore, the co-dependence of the stiffness and contractility parameters is limited by solving sequentially for the two passive parameters (C 0 and b f ) and the contractility index (CI = T max ). Despite remaining issues, the study demonstrates promising value for clinical research as the proposed approach allows the quantification of LV myocardial rigidity and contractility based on a personalized model constructed only from routine cardiac MR data. Conclusion Left ventricular stiffness and contractility from 21 normal subjects were estimated using routine cardiac cine MR data and personalized cardiac modelling. The modelling approach was extensively validated by comparison of measured/estimated local circumferential strain and other global LV function metrics. The range of biomechanical indices obtained in this study could serve as reference normal values to investigate the degree of severity of ischemic myocardium. The prognosis values of these indices should be explored further within a clinical longitudinal study.

  4b f and b f t = 0.7b f , the expressions of the Youngs moduli in the fiber and cross-fiber(E cf ) directions become: E f = E (1) = 1.2 C 0 b f and E cf = E (2) = E (3) = 0.69 C 0 b f .

Table 2 :

 2 Mean and standard deviation (SD) of LV metrics across all subjects, as well as the differences between measured and personalized simulated values (Ecc: circumferential strain, GLS: global longitudinal strain).

	Subject metrics	Measured	Simulated
				meas-sim
	n = 21, 13 male	(meas)	(sim)
	End-diastole	137.3 (25.8)
	Volume (ml)			-
	End-systole	58.8 (11.6)
	End-diastole	1.2 (prescribed)	1.28 (0,0014)	0.08 (0,0014)
	Pressure (kPa)
	End-systole	16		-
	Ejection fraction (%)	57.1 (3.7)	-
	ED wall thickness (mm)	7.3 (1.5)	7.9 (1.6)	-0.56 (0.98)
	Systolic fractional thickening (%)	67.4 (23.7)	69.2 (8.5)	-1.8 (24.4)
	Ecc (%)	-17.4 (2.6)	-17.0 (2.6)	-0.89 (3.8)
	GLS (%)	-9.8 (3.7)	-15.1 (1.9)	5.3 (4.3)

Table 3 :

 3 Subject-specific stiffness (SI ) and contractility (CI ) indices for the 21 subjects. The unit of the indices are in kP a

		#1	#2	#3	#4	#5	#6	#7	#8	#9	#10	#11
	SI	1.61	1.46	1.69	1.63	1.73	1.97	1.24	1.56	1.46	1.67	1.38
	CI 92.92 82.58 82.66 79.79	89.40 97.16 78.29 81.19 86.86 84.20	124.83
		#12	#13	#14	#15	#16	#17	#18	#19	#20	#21 Mean(SD)
	SI	1.50	1.56	1.56	1.81	1.84	1.56	1.92	1.01	1.62	1.75	1.60(0.22)
	CI 98.41 100.42 88.22 115.61 115.76 99.71 109.27 87.95 121.47 80.95 95.13(14.56)

Table 4 :

 4 Estimated constitutive law parameters for healthy cases from similar studies in the literature. SI and CI are our proposed stiffness and contractility indices,

Table 3

 3 lists the personalized material parameters and indices. The average LV stiffness index SI was 1.60 ± 0.22 kPa across all subjects, while the average contractility index CI was 95.13 ± 14.56 kPa. Those values are compared to values obtained by similar studies in Table4. Multiple linear regression analyses were carried out on the 21 cases to look for potential correlations between the five

following parameters: stiffness index (SI , unit: kPa), ejection fraction (EF ), end-diastolic volume (EDV, unit: ml) or end-systolic volume (ESV, unit: ml), myocardial volume (MV, unit: ml), and the contractility index (CI , unit: kPa).

Only two independent variables (namely, SI and CI ) appear to be statistically correlated to EF estimation. We found that the ejection fraction increases with CI and decreases with SI : EF = 0.133CI -19.782SI + 70.765, with R 2 = 0.96 and p < 0.001. This shows a relationship between SI , CI and EF although the very small range of EF values (57.1 ± 3.7%) for this population of normal subjects. It is expected that SI and CI would be able to differentiate subjects with reduced EF from controls and even subjects with preserved EF as nicely

Table 5 :

 5 MR-based ED wall thickness and circumferential strains of subjects #11

	and #19 compared with the remaining 11 male subjects and literature values [30]
		(Ecc: circumferential strain)
	Measurements Mean (SD)	basal	ED wall thickness (mm) medial apical	mean	Ecc (%)
	#11	10.8	8.0	8.5	9.1	-11.2 (6.7)
	#19	8.8	9.2	6.7	8.2	-20.0 (7.2)
	Others (n=11)	8.5 (1.1)	8.1 (1.3)	6.8 (1.2)	7.8 (1.0)	-17.6 (2.4)
	Kawel et al. [30] (n=300)	8.3 (1.0)	7.2 (1.0)	5.6 (1.0)	7.03 (1.0)	-

Table 6 :

 6 Parameter values for the rule-based myocardial fiber arrangement[START_REF] Rijcken | Optimization of cardiac fiber orientation for homogeneous fiber strain during ejection[END_REF]

	Parameters Values
	h 10	0.362 rad
	h 11	-1.16 rad
	h 12	-0.124 rad
	h 13	0.129 rad
	h 14	-0.0614 rad
	h 22	0.0984
	h 24	-0.0501
	t 11	-0.626
	t 12	0.502
	t 21	0.626 rad
	t 23	0.211 rad
	t 25	0.038 rad
	with:

  t 0 π t t0 t 0 ≤ t < (t 0 + t r ) π t-t0+tr

tr (t 0 + t r ) ≤ t 0 t r = ml + b and l = l r 2E f f + 1

Table 7 :

 7 Parameter values of the active tension constitutive law[START_REF] Guccione | Mechanics of active contraction in cardiac muscle: Part IIcylindrical models of the systolic left ventricle[END_REF]

	Parameters Values
	Ca 0	4.35 µM
	(Ca 0 ) max	4.35 µM
	B	4.75 /µm
	l 0	1.58 µm
	t 0	0.1 s
	m	1.0489 s/µm
	b	-1.429 s
	l r	1.85 µm

Statements

• Conflicts of Interest: None • Funding: Funding: European Commission H2020 MSCA Training Network VPH-CaSE (www.vph-case.eu), grant agreement No 642612. • Ethical Approval: Not required

Acknowledgements

GK Rumindo was supported by the European Commission H2020 MSCA Training Network VPH-CaSE (www.vph-case.eu), grant agreement No 642612. This work was performed within 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.

Appendix A: rule-based myocardial fiber orientation

The rule-based method [START_REF] Bovendeerd | Determinants of left ventricular shear strain[END_REF] formulates the spatial distribution of both the helix and the transverse angles based on the local left ventricular coordinates.

Within a pseudoprolate LV coordinate system, the local longitudinal coordinate u is normalized to be u = +0.5 at the basal level and u = -1 at the apex, whereas the radial coordinate is normalized to be v = -1 at the endocardium and v = +1 at the epicardium. The helix angles (α h ) and the transverse angles (α t ) are described as follows:

where h ii and t ii are scaling parameters for helix and transverse angles, and L i are normalized Legendre polynomials of order i. The scaling parameters are optimized based on [START_REF] Rijcken | Optimization of cardiac fiber orientation for homogeneous fiber strain during ejection[END_REF], and are listed in Table 6.

Appendix B: myocardial constitutive model

The myocardial mechanical behavior is divided into passive and active parts.

The passive part is formulated as a quasi-incompressible transversely-isotropic Fung-type law, originally introduced in [START_REF] Guccione | Passive material properties of intact ventricular myocardium determined from a cylindrical model[END_REF], whereas the active tension is defined by fiber stretch-dependent active force law [START_REF] Guccione | Mechanics of active contraction in cardiac muscle: Part IIcylindrical models of the systolic left ventricle[END_REF]. The constitutive laws have been widely used in previous studies [START_REF] Genet | Distribution of normal human left ventricular myofiber stress at end diastole and end systole: a target for in silico design of heart failure treatments[END_REF][START_REF] Wang | Modelling passive diastolic mechanics with quantitative MRI of cardiac structure and function[END_REF].

The strain energy density W is decomposed into two parts: the volumetric part that relates to the change in tissue volume (W V ), and the deviatoric part that relates to the change in tissue shape (W I ):