MD Ruben Coronel
email: rubencoronel@gmail.com

PhD Bastiaan J Boukens

MD Vichai Benjacholamas

MSc Shirley Van Amersfoort

PHD Veronique M Meijborg

Cees Schumacher Msc

PhD Bjarke Jensen

MD PhD Michel Haissaguerre

MD Arthur Wilde

MD, MSC Somchai Prechawat

MD Anurut Huntrakul

MD Koonlawee Nademanee

Structurally abnormal myocardium underlies ventricular fibrillation storms in a patient diagnosed with the early repolarization pattern

Keywords: Conduction, Early Repolarization, Ablation, Ventricular Fibrillation

Introduction: The early repolarization pattern refers to abnormal ending of the QRS complex in subjects with structurally normal hearts and has been associated with ventricular fibrillation (VF). The aim of this study was to investigate the mechanism underlying QRS slurring in a patient with the early repolarization pattern in the electrocardiogram (ECG) and VF storms. Methods and Results: We studied a patient with slurring of the QRS complex in leads II, III and aVF of the ECG and recurrent episodes of VF. Echocardiographic and imaging studies did not reveal any abnormalities. Endocardial mapping was normal but subxyphoidal epicardial access was not possible. Open chest epicardial mapping showed that the inferior right ventricular free wall activated the latest with local J-waves in unipolar electrograms. The last moment of epicardial activation concurred with QRS-slurring in the ECG whereas the Jwaves in the local unipolar electrograms occurred in the ST segment of the ECG. Myocardial biopsies obtained from the late activated tissue showed severe fibrofatty alterations in the inferior right ventricular wall where fractionation and local J-waves were present. After ablation, the early repolarization pattern in the ECG disappeared and arrhythmias have been absent since (follow-up 18 months). Conclusion: In this patient, the electrocardiographic early repolarization pattern was caused by late activation due to structurally abnormal myocardium. The late activated areas were marked by J-waves in local electrograms. Ablation of these regions prevented arrhythmia recurrence and normalized the ECG.

Condensed Abstract

The early repolarization pattern refers to abnormal ending of the QRS complex and has been associated with ventricular fibrillation in subjects with structurally normal hearts. We studied a patient with the early repolarization pattern and ventricular fibrillation storms, who underwent epicardial mapping during open chest surgery. Biopsies at the sites of electrical abnormality demonstrated extensive fibrosis in the right ventricular inferior wall. Epicardial ablation of these sites resulted in long-term normalization of the cardiac rhythm and of the QRS-complex. Abbreviations ECG = Electrocardiogram VF = ventricular fibrillation ICD = implantable cardioverter defibrillator RVFW = right ventricular free wall RVOT = ventricular outflow tract ECGI = electrocardiographic imaging

Introduction

The early repolarization pattern refers to slurring or notching at the end of the QRS complex and is associated with sudden cardiac arrest due to ventricular fibrillation (VF) in patients with structurally normal hearts. [START_REF] Haïssaguerre | Sudden cardiac arrest associated with early repolarization[END_REF][START_REF] Tikkanen | Long-term outcome associated with early repolarization on electrocardiography[END_REF] In patients with documented cardiac arrest and the early repolarization pattern, implantation of an automated implantable cardioverter defibrillator (ICD) is the recommended therapy. Identification of patients with the early repolarization pattern at risk for arrhythmias is challenging because it is common in the healthy general population. Previous syncope or a family history of sudden cardiac death have been identified as risk factors but their predictive power is low. [START_REF] Haïssaguerre | Sudden cardiac arrest associated with early repolarization[END_REF] Recent reports indicate that ablation of myocardium with fractionated potentials can prevent VF in a subset of patients with the early repolarization pattern. [START_REF] Haïssaguerre | Depolarization versus repolarization abnormality underlying inferolateral J-wave syndromes: New concepts in sudden cardiac death with apparently normal hearts[END_REF][START_REF] Nademanee | Mapping and Ablation of Ventricular Fibrillation Associated with Early Repolarization Syndrome[END_REF] This implies that a localized and potentially arrhythmogenic substrate underlies life-threatening arrhythmias in patients with the early repolarization pattern.

We present the first open-chest epicardial electrophysiological mapping of a patient with the early repolarization pattern and VF storms in whom myocardial biopsies were obtained from tissue with fractionated potentials. Our data provide electrophysiological and histological evidence that localized myocardial structural abnormalities can be the underlying arrhythmogenic pathological substrate for life-threatening ventricular arrhythmias and early repolarization in the electrocardiogram (ECG). Epicardial ablation of the structurally abnormal myocardium prevented recurrence of VF and normalized the ECG. We propose that in similar patients with the early repolarization pattern in whom a structural substrate is present, ablation may be the therapy of choice.

Methods

This study was performed after written informed consent of the patient, and was executed in accordance with the declaration of Helsinki. The patient underwent standard medical diagnostics and care, and standard resuscitation procedures were followed.

Electrophysiological study

First the patient underwent an electrophysiologic study involving endocardial mapping using CARTO-mapping system (Biosense Webster Inc). Subsequently, electrocardiographic imaging (ECGI) mapping was executed with the CardioInsight system (Medtronic, St. Paul, MN). The ECGI methodology has been described before. [START_REF] Nademanee | Mapping and Ablation of Ventricular Fibrillation Associated with Early Repolarization Syndrome[END_REF][START_REF] Ramanathan | Noninvasive electrocardiographic imaging for cardiac electrophysiology and arrhythmia[END_REF] Body surface ECG recordings were acquired with a 252 electrode-vest wrapped around the torso. Low-dose computed tomography was used to localize the vest electrode positions relative to the heart and the torso. The vest remained in the same position during the electrophysiologic study and programmed stimulation. Body surface ECG recordings from the vest electrodes were acquired before and during the invasive mapping procedure. A 3D model of the heart was created using dedicated software (CardioInsight, Medtronic, St. Paul, MN). As a result of pericardial adhesions it was not possible to gain access to the pericardial space for detailed invasive mapping of the epicardium.

The CardioInsight system automatically displays epicardial wavefront patterns on the 3D reconstruction of the patient's heart based on the solution of the inverse problem. [START_REF] Ramanathan | Noninvasive electrocardiographic imaging for cardiac electrophysiology and arrhythmia[END_REF] In this manner, activation mapping was performed during spontaneous or induced VF. After adequate filtering and phase mapping, dynamic wavefront propagation maps were generated.

Cardioversion was performed if VF lasted > 10 seconds.

The wavefront maps display the electrical wavefront at the pi/2 phase value of each ECGIcalculated unipolar electrogram morphology, serving as a surrogate for local activation. We analyzed the VF maps during an initial organized period of VF (the initial 5 seconds), as previously described.(6) VF drivers were defined as either focal breakthrough or full reentrant activity with a high activation frequency. Focal breakthroughs are detected when centrifugal activation originated from a given site. Rotations are detected when the rotational core, or singularity point, of a rotating wavefront is within a 2.5-cm area for ≥1.5 rotations.

We then created spatiotemporal density maps displaying the number, location, and spatial extent (of reentry trajectories) of VF drivers. We marked (colored hexagons) the number and location of epicardial focal breakthrough.

Open chest multi electrode mapping A midsternal sternotomy was performed. After opening of the pericardium and careful dissection of the pericardial adhesions, sequential unipolar epicardial mapping was executed using a rectangular 8x8 multi-electrode grid (64 electrodes, 5 mm interelectrode distance).

The electrode grid was sequentially placed on 9 locations of the epicardium to cover the entire surface of the heart. The reference electrode was placed in the thoracic wound. A three lead ECG was recorded simultaneously. During the entire mapping procedure, a train of 8 S1 stimuli were delivered from the right atrium (cycle length 800 msec), followed by a single S2 from the left ventricular apex (coupling interval 580). Primary data analysis was performed in the operating theatre to guide ablation. Local moments of repolarizations were defined as maximum positive dV/dt of the local T-wave [START_REF] Coronel | Monophasic action potentials and activation recovery intervals as measures of ventricular action potential duration: experimental evidence to resolve some controversies[END_REF].

Radiofrequency ablation

Ventricular epicardial ablation was performed by radiofrequency energy (20-45 Watt, duration ranging 10-60 seconds) delivered by a ThermoCool catheter (Biosense Webster, California USA).

Histology and 3D reconstruction

Biopsies were obtained from selected sites in the right ventricular inferior wall and basal free wall and directly snap frozen in liquid nitrogen. The biopsies were cryo-sectioned (8 µm) and stained with picro-sirius red. The 3D model was generated in Amira (version 6.5, Thermo Fischer Scientific, USA). We imported the images (2560x1920 pixels at 600 dpi, resolution of 1.95µm/pixel) of 49 histological sections (collagen is red, myocardium yellow) and a 3D reconstruction was generated as published previously.( 8) The images were imported and converted to grayscale images which rendered collagen dark grey and myocardium light grey, with a high contrast between the two. We then automatically aligned the images. Alignment errors were corrected manually. Next, we surveyed the image stack for clusters of myocardium, the core of which could be reliably traced on at least 10 consecutive sections (on every section this core of myocardium also had peripheral parts which could not always be traced on the neighbouring sections). Myocardial clusters were labelled green.

Results

Diagnosis of the early repolarization pattern

The patient, a 36-year-old man, suffered aborted sudden cardiac arrest at age 30 and was transferred to the nearest hospital after cardiopulmonary resuscitation in 2012. During the hospital admission, the patient had documented VF and was defibrillated by a DC shock. An ICD was then implanted. VF episodes recurred one year later, increasing in frequency from 1 to 4 episodes per month. The patient then received oral propranolol (10 mg q8h). In May 2018, the patient experienced palpitations and dizziness while sitting at home, 10 seconds before the ICD delivered a shock. Four more appropriate shocks were delivered at the same day and the patient was referred to our hospital.

We observed an apparently healthy young man with a midline sternotomy scar resulting from prior chest surgery performed because of a chest trauma when he was 6 years old. The ECG showed slurring of the QRS complex (≥1 mV) in leads II, III and aVF, typical of the early repolarization pattern (Figure 1A). An ECG recorded during childhood was not available. Intravenous bolus administration of Ajmaline (1 mg/kg; total 60 mg) did not provoke right precordial ST-segment elevation, a characteristic of the Brugada ECG pattern.

Echocardiographic and imaging studies did not reveal any abnormalities. Family screening revealed that an uncle had (unspecified) heart disease. The patient's parents, his 3 brothers, his 2 children did not have a history of arrhythmias. The patient's symptoms and unexplained cardiac arrest meet the standards for diagnosis of a malignant early repolarization pattern according to the Shanghai Score System [START_REF] Antzelevitch | J-Wave syndromes expert consensus conference report: Emerging concepts and gaps in knowledge[END_REF].

An electrophysiological percutaneous catheter study did not reveal endocardial electrical abnormalities (low voltage or fractionated potentials, Supplemental Figure 1).

During the electrophysiological study, spontaneous short-coupled premature ventricular pacing (Figure 4A, only anterior view shown). The anterior wall of the right ventricle activated the earliest whereas the right inferior wall activated last, slightly later than the RVOT (85 vs 83 msec, respectively). Repolarization was the earliest in the anterior wall of the right ventricle. The RVOT and apex of the left ventricle repolarized last. Repolarization in the right ventricular inferior wall was heterogeneous; parts repolarized relatively early (location a) and other parts relatively late (location b). However, the earliest repolarization times occurred much later than the slurring of the ECG (figure 4B).

The last activated myocardium in the inferior RVFW showed local J-waves in the unipolar electrograms which occurred during the ST-segment of lead aVF, but after the moment of slurring or of the notch in the QRS complex (Figure 4B). Figure 4C (left panel)

shows that the largest J-waves in the local electrograms occurred in the last activated tissue. Indeed, the amplitude and timing of the peak of the J-wave in the local electrograms correlated with the moment of local activation (Figure 4C mid and right panel).

Epicardial ablation of structural abnormal right ventricular inferior wall

Two transmural right ventricular biopsies were obtained before epicardial ablation was performed. Fractionation at these sites was confirmed by unipolar and bipolar electrogram recordings obtained with the ablation electrode (Figure 5A). One biopsy was taken from the last activated tissue of the inferior RVFW, showing local J-waves or fractionation (Figure 5A), and one from the basal right ventricular wall. The biopsy of the right ventricular inferior wall showed extensive fibrosis (Picrosirius Red staining, collagen is red) with fragments of surviving myocardium. In contrast, the basal right ventricle only showed increased diffuse fibrosis without loss of myocardium (Figure 5B). Three-dimensional reconstruction of the fibrotic tissue showed a fine network of interconnecting individual bundles of myocardium (Figure 5C, green).

Immediately after radiofrequency ablation of the epicardial myocardium that showed fractionated potentials and/or J-waves, the QRS notching in the ECG leads was absent (not shown). The early repolarization pattern has remained absent during the follow-up period (publication date) and the patient has been free of VF since the procedure (Figure 1B).

Discussion

In this case study, we combined high resolution epicardial electrophysiological mapping and histological analysis in a patient with the documented early repolarization pattern. Although endocardial abnormalities were not detected, we show that localized structural abnormalities in the subepicardium were the cause for the electrocardiographic pattern and for VF-storms.

Ablation of these regions with structural abnormalities normalized the electrocardiogram and prevented arrhythmia recurrence.

We describe severe but localized interstitial fibrosis in RV myocardium and recorded fractionated unipolar electrograms from the same locations. This resembles post-infarcted myocardium, where surviving myocardial strands within the fibrotic infarcted tissue constitute a substrate for life threatening arrhythmias. [START_REF] Kawara | Activation delay after premature stimulation in chronically diseased human myocardium relates to the architecture of interstitial fibrosis[END_REF] The loose myocardial network within a fibrotic area may well form the basis for delayed conduction or failure of conduction by current-to-load mismatch [START_REF] Hoogendijk | Mechanism of right precordial STsegment elevation in structural heart disease: excitation failure by current-to-load mismatch[END_REF][START_REF] Rohr | Paradoxical improvement of impulse conduction in cardiac tissue by partial cellular uncoupling[END_REF]. If local fibrosis is present in already late activated myocardium-as is the case for the subepicardium of the right and left ventricular wall of the described patient-local activation may be delayed beyond the end of the QRS complex and become visible as slurring or notching. In this patient, the structural abnormalities provided the substrate for arrhythmias as confirmed by ablation of these sites that resulted in an arrhythmia-free follow up.

Our data also indicate that a loose myocardial network within a fibrotic area is associated with local J-waves. J-waves in unipolar electrograms are caused phase-1 repolarization at the action potential level, particularly at the subepicardium. The common mechanistic explanation of the early repolarization pattern is that enhanced phase-1 repolarization causes J-waves in local electrograms which are visible in the ECG as notching or slurring of the QRS complex [START_REF] Antzelevitch | J-Wave syndromes expert consensus conference report: Emerging concepts and gaps in knowledge[END_REF]. However, in our patient local J-waves recorded in late activated myocardium did not coincide with the QRS slurring and were not visible in the ECG. This observation challenges the common view that all cases of early repolarization pattern are caused by enhanced phase-1 repolarization.

Recent reports have shown that idiopathic VF is associated with fractionated local unipolar electrograms or with electrograms with a J-wave. [START_REF] Haïssaguerre | Depolarization versus repolarization abnormality underlying inferolateral J-wave syndromes: New concepts in sudden cardiac death with apparently normal hearts[END_REF][START_REF] Nagase | Local left ventricular epicardial J waves and late potentials in Brugada syndrome patients with inferolateral early repolarization pattern[END_REF] Our patient had both local fractionated potentials and local J-waves, indicating that these conditions can co-exist. We have previously argued that local J-waves can be recorded at epicardial sites that are activated late. [START_REF] Boukens | J-Waves in Epicardial Electrograms Can Guide Ablation of Arrhythmogenic Substrates[END_REF] The plots in Figure 3C confirm that both the time and the amplitude of local J-waves correlated with activation time. Our data show that the choice of potential ablation sites can be guided by local fractionation activity and local-J waves, as expression of late activation. [START_REF] Boukens | J-Waves in Epicardial Electrograms Can Guide Ablation of Arrhythmogenic Substrates[END_REF] We show that the inferior wall activated late and repolarized relatively early compared to the surrounding myocardium (Figure 3A). Consequently, the activation-recovery-interval (ARI) -surrogate for action potential duration -was short in this region. Short ARIs in the right inferior wall have been described before in patients with a similar early repolarization pattern in the ECG as the case we describe here.( 15) However, it is unlikely that the brief action potentials alone explain the early repolarization pattern in the ECG as they are only present in regions that are activated late. Also, in our patient the earliest moment of repolarization occurs much later than the moment of QRS-slurring in the ECG. The mechanism of the relatively early repolarization in this late activation myocardium is unclear.

We speculate that adjacent unexcited (or not yet) myocardium electrotonically shortens local repolarization. Our observations do not allow speculation about the transmural repolarization gradient.

Considerations

We cannot exclude the possibility that the chest surgery that the patient has undergone in his youth has contributed to the localized structural changes. Neither can be ruled out the possibility that the antiarrhythmic effects were mediated by a suppression of the triggering mechanism associated with the scar tissue. A common view is that a J-wave syndrome is only truly a J-wave syndrome by exclusion of a scar or other structural abnormalities. Our observations, however, show that small structural abnormalities can be present in patients diagnosed with the early repolarization syndrome. In our patient the diagnosis of early repolarization syndrome (or early repolarization pattern) was made based on aborted sudden cardiac death, documented VF or polymorphic tachycardia in combination with the early repolarization pattern in the ECG (Shanghai score system) [START_REF] Antzelevitch | J-Wave syndromes expert consensus conference report: Emerging concepts and gaps in knowledge[END_REF]. The patient does not meet the criteria for any pathology listed as other causes for ERP. This means that current (clinical) detection methods may not be sufficient to exclude small regional structural abnormalities that can underlie VF in patients with the early repolarization syndrome. This makes the mechanistic insight into the early repolarization syndrome provided by this study relevant for patient management.

Conclusion

We show that the early repolarization pattern can result from localized structural abnormalities. These abnormalities were diffuse and not detected with standard clinical tools.

We do not imply that structural abnormalities are present in all patients with the early repolarization pattern. However, our observation does challenge the definition of a structurally normal heart and the notion that a single functional pathophysiological mechanism can be established for all patients with the early repolarization pattern.

Perspectives

Competency in Medical Knowledge

The current view is that patients diagnosed with the early repolarization syndrome have structurally normal hearts. We demonstrate that clinically concealed right ventricular subepicardial structural abnormalities cause ventricular fibrillation in a patient with the early repolarization syndrome.

Translational Outlook

Localized structural abnormalities (fibrosis) can cause the early repolarization pattern and can form an arrhythmogenic substrate. Detailed electrophysiological and anatomic imaging should be performed to identify a substrate that can be ablated in patients with the early repolarization syndrome. 2B) and showed local J-waves with the highest amplitude (Voltage map of J-wave amplitude at the right). Activation times correlated with both the moment (R=0.5) and the amplitude (R=0.7) of the J-wave peak. ERP, early repolarization pattern; AO, aorta; RA, right atrium; RV, right ventricle; PT, pulmonary trunk; LA, left atrium. Figure 5. Activation delay of the right ventricular inferior wall is due to structural abnormalities. A) The traces (from the ablation catheter) show unipolar and bipolar recordings from the right inferior wall before ablation. B) Schematic drawing shows where biopsies were taken and the epicardial region that was ablated (pink). The sections of the biopsies were stained using picro-sirius Red. C) The 3D model of the biopsy is based on 17 serial sections. The clustered myocardium is labelled green and connected resulting in a 3D network of myocardial fibers. Supplemental Figure 1.

CARTO-merge map of biventricular endocardium. The cardiac computed tomography of the patient's heart is merged with the electroanatomic maps of the RV and LV endocardium recorded during electrophysiologic studies. Areas in the CARTOmerge are color-coded according to the local EGM voltage, ranging from red (lowest local signals with ≤ 0.5 mV amplitude) to magenta (highest voltage, i.e., ≥ 1.5 mV). The insets display samples of the eletrograms (EGMs) recorded from the Decapolar Navistar (DecaN) catheter at various sites of RV and LV epicardium. The voltage map and the representative bipolar and unipolar tracings are shown. Note that the endocardial EGMs of the RV are normal without signs of late activation. The right panel-inset displays recording from the LV septum and shows the His -bundle recording on DecaN9-10. DecaN5-6 and DecaN7-8, exhibited low voltage signals but they were recorded just below the aortic valve with poor contact and EGMs duration < 70 ms. Because also the unipolars 5 and 7 were normal we interpreted these normal EGMs with poor contact artefacts on electrodes 6 and 8. Another explanation could be that the activation front was perpendicular to the normal between the electrodes. We cannot exclude that a scar was present in this region. However, this region was not ablated and yet the ERP was resolved and the arrhythmias prevented, showing that this region is not causal. RVAp = Right ventricular apex; Central figure. The early repolarization pattern can result from localized structural abnormalities. At the right, a 3D model of a biopsy from the right inferior wall of a patient diagnosed with the early repolarization syndrome. The biopsy shows separated myocardial bundles because of interstitial fibrosis. Activation of the right inferior wall was delayed and coincided with the early repolarization pattern in the electrocardiogram. J-waves present in local electrograms recorded from the late activated right inferior wall occurred after the early repolarization pattern in the electrocardiogram.

Figure 1 .

 1 Figure 1. A) Twelve-lead electrocardiogram of the patient at rest. B) Twelve-led electrocardiogram 10 days after ablation. ECG, electrocardiogram.

Figure 2 .

 2 Figure 2. Electrocardiographic imaging predicts arrhythmogenic focus at the inferior right ventricular free wall. A) Lead II showing a spontaneous polymorphic ventricular tachycardia. B) Reconstructed activation patterns during sinus rhythm and apical stimulation based on ECGi. Sinus rhythm: Note late activation in the RV inferior wall and local crowding of isochrones. Pacing: crowding of isochrones in RV inferior wall. C) Three dimensional ECGI (anterior-posterior view on the left, caudal view of the inferior aspect in the middle, and left posterior oblique view on the right, see mannequins in corner) during the first 5 second of VF. The numbers on the ECGI map represent the number of rotations at each point on the heart. The colored hexagons represent the number and location of epicardial focal breakthroughs. Note that the majority of rotational and focal activities are located in the inferolateral and the bottom of the right ventricle. ECGI, electrocardiographic imaging.

Figure 3 .

 3 Figure 3. QRS slurring coincides with late activation of the right ventricular inferior wall. A) Schematic representation of the anterior side of the heart showing the 7 locations of the multielectrode (8x8) (2 locations at the posterior side are not shown). B) Activation maps of the RVOT (upper) and right ventricular inferior wall (lower) during right atrial stimulation (left and middle) and premature ventricular stimulation (right). C) Local electrograms (upper 7) from selected locations indicated in panel B. The lower trace shows lead II. The early repolarization pattern in the ECG coincided with the last moment of activation in the right ventricular inferior wall. Note that during left ventricular premature stimulation the right ventricular inferior wall remained late activated despite inversion of the QRS complex. RVOT, right ventricular outflow tract.

Figure 4 .

 4 Figure 4. Early local repolarization does not coincide with the early repolarization pattern in the ECG. A) The activation (upper) and repolarization (lower) patterns based on local electrograms recorded with the multielectrode (8x8) from 9 different locations. The signals are aligned to the onset the QRS complex in lead aVF. B) Lead aVF and 4 local electrograms recorded from locations depicted in panel A. The moment of repolarization in the local electrograms are marked with a red star. QRS-slurring in the ECG coincided with the last activation in the right inferior wall but not with the time of peak of the local J-wave or local repolarization. C) The right ventricular inferior wall was activated last (Figure2B) and showed local J-waves with the highest amplitude (Voltage map of J-wave amplitude at the right). Activation times correlated with both the moment (R=0.5) and the amplitude (R=0.7) of the J-wave peak. ERP, early repolarization pattern; AO, aorta; RA, right atrium; RV, right ventricle; PT, pulmonary trunk; LA, left atrium.

This study was supported by the Leducq foundation (16CVD02 RHYTHM, RC). BJB received funding from the Dutch Heart Foundation (2016T047). We thank Qing Lou for assisting with the analysis of the ECGI data.

contractions and polymorphic ventricular tachycardias occurred frequently (Figure 2A).

Subxyphoïdal access was impossible due to pericardial adhesions. Non-invasive ECGI during sinus rhythm and of ventricular fibrillation suggested an arrhythmogenic substrate at the right ventricular free wall epicardium (Figure 2B/C).

The patient then underwent open chest surgery to identify a possible epicardial arrhythmogenic substrate and to perform epicardial ablation, if necessary.

Late activation of the right ventricular inferior wall

The electrode grid was placed over 7 anterior locations (Figure 3A) as well as 2 posterior locations of the heart. Figure 3B shows the activation patterns of three subsequent beats recorded from the right ventricular outflow tract (RVOT) and RV inferior wall (red in panel A). Figure 3C shows unipolar electrograms (a-g indicate electrode positions in 3B) and lead II of the ECG. The earliest onset of the body surface ECG was taken as the reference time.

During S1 pacing, the inferior right ventricular free wall (RVFW) activated last, also relative to all other grid positions (Figure 4A) and later than the right ventricular outflow tract (Figure 3A/B). Activation of the inferior RVFW concurred with the QRS-slurring in lead II of the ECG, both during the basic (S1, dotted line) and the premature (S2) activation (Figure 3C). Following S2 (coupling interval 580 ms) activation delay occurred and a drastic reduction in the voltage of sites b, d, f and g, and fractionated local potentials in the same tissue became evident. Note that following S2, a notch has appeared in lead II, coinciding with last activation in the right ventricular inferior wall, and that the RV inferior wall remained activated late despite the difference in activation patterns between the atrial and ventricular beats (see QRS-morphology in lead II).

Early repolarization did not coincide with the early repolarization pattern in the ECG

Based on local multi-electrode recordings from 9 different locations we reconstructed the activation and repolarization patterns of the entire epicardial surface of the heart during atrial


Linyuan Liu

Jens Carsten

ON THE COHOMOLOGY OF LINE BUNDLES OVER CERTAIN FLAG SCHEMES II

Keywords: Mathematics Subject Classification. 05E05, 05E10, 14L15, 20G05 . cohomology, line bundles, flag schemes, Weyl modules, symmetric functions

Over a field K of characteristic p, let Z be the incidence variety in P d × (P d ) * and let L be the restriction to Z of the line bundle

. When f = 0, using the first author's description of H d (Z, L ) and Jantzen's sum formula, we obtain as a by-product that the sum of the monomial symmetric functions m λ , for all partitions λ of 2p -1 less than (p -1, p -1, 1) in the dominance order, is the alternating sum of the Schur functions S p-1,p-1-i,1 i+1 for i = 0, . . . , p -2.

Introduction. This paper is an addition to the first author's paper [START_REF] Liu | On the cohomology of line bundles over certain flag schemes[END_REF]. We now consider the group scheme G = SL d+1 over an arbitrary field K of characteristic p > 0. For m, n ∈ N, set µ m,n = mω 1 -(n + d)ω d and write simply µ n instead of µ n,n . We describe the H i (µ m,n ) when p > n (thus recovering and extending [START_REF] Liu | On the cohomology of line bundles over certain flag schemes[END_REF], Cor. 2) and for n = p + f with 0 ≤ f ≤ p -2 we prove that H d (µ n ) is the simple G-module L(λ f ) = L(f ω 1 + (p -2 -f )ω 2 + (f + 1)ω 3 ). Further, when f = 0 we express the character of L(λ f ), using Jantzen sum formula, as an alternating sum of Weyl characters. Comparing this with the character of H d (µ p ) given in [START_REF] Liu | On the cohomology of line bundles over certain flag schemes[END_REF], Cor. 3, we obtain as a by-product that the sum of the monomial symmetric functions m λ , for all partitions λ of 2p-1 less than λ f in the dominance order, is the alternating sum of the Schur functions S p-1,p-1-i,1 i+1 for i = 0, . . . , p -2.

1.1. Notation. We keep the notation of [START_REF] Liu | On the cohomology of line bundles over certain flag schemes[END_REF], except that we now consider the group scheme G = SL d+1 over an arbitrary field K of characteristic p > 0 and denote simply by P the maximal parabolic subgroup P 1 . Let W P be the Weyl group of (P, T ) and define W Q similarly. Let ch V denote the character of a T -module V . For each simple root α i , let s i ∈ W be the corresponding simple reflection. Let w 0 (resp. w P , resp. w Q ) be the longest element of W (resp. W P , resp. W Q ) and set N = (w Q ). Then (w P ) = N + d -1.

Let ρ Q (resp. ρ P ) denote the half-sum of the positive roots of Q (resp. P ). Then one has:

(1.1.1) 2ρ Q = (2-d)(ω 1 +ω d )+2 d-1 i=2 ω i , 2ρ P = (1-d)ω 1 +2 d i=2 ω i .
Recall that, since 2ρ

-2ρ Q = d(ω 1 + ω d ), the dualizing sheaf on Z = G/Q is L (-dω 1 -dω d ). Hence, by Serre duality on G/Q, one has (1.1.2) H d (µ m,n ) H d-1 (-(m + d)ω 1 + nω d ) * .
Further, let τ be the (involutive) automorphism of (G, T ) induced by the automorphism of the Dynkin diagram which swaps α i and α d+1-i for i = 1, . . . , d. Note that τ also acts on X(T ) and preserves X(T ) + . For any G-module V , let τ V denote the corresponding module twisted by τ . For example, for a Weyl module V (λ) (resp. a simple module L(λ)), one has τ V (λ) V (τ λ) (resp. τ L(λ) L(τ λ)). Then, (1.1.2) can be rewrited as:

(1.1.3)

H d (µ m,n ) H d-1 (τ µ n,m ) * .
Recall (see [START_REF] Carsten | Representations of algebraic groups[END_REF], II.4.6 and its proof) that if P ⊂ P are parabolic subgroups containing B and if V is a P -module, one has for all i ≥ 0:

(1.1.4)

H i (P /B, V ) H i (P /P , V ).
This will be used several times, without always mentioning it.

A description of H

i (µ m,n ). For each i ∈ N, we denote by H i Q (-) the functor H i (Q/B, -) and define H i P (-) similarly. Since -µ m,n is trivial and µ m,n -2ρ Q is anti-dominant with respect to the Levi subgroup of Q one has: (1.2.1) H i Q (-µ m,n ) -µ m,n if i = 0, 0 if i > 0
and, using Serre duality on Q/B:

(1.2.2)

H i Q (µ m,n -2ρ Q ) 0 if i < N, H 0 Q (-µ m,n ) * µ m,n if i = N.
Consider induction from B to P . Thanks to (1.2.2), the spectral sequence of composite functors (see [START_REF] Carsten | Representations of algebraic groups[END_REF], I.4.5) degenerates and gives (together with (1.1.4) applied to P = Q and P = P ) isomorphisms for each i ≥ 0:

(1.2.3)

H i P (µ m,n ) H i+N P (µ m,n -2ρ Q ) and since µ m,n -2ρ Q = (m+d-2)ω 1 -2 d-1 j=2 ω j -(n+2
)ω d is anti-dominant with respect to P , the latter group is zero unless i + N = dim(P/B), i.e. i = d -1. Moreover, by Serre duality on P/B, one has:

H N +d-1 P (µ m,n -2ρ Q ) H 0 P (-µ m,n + 2ρ Q -2ρ P ) * = H 0 P ((1 -m)ω 1 + nω d ) * . Set ν m,n = (1 -m)ω 1 + nω d and π m,n = -w P ν m,n . Since w P = s 2 • • • s d w Q , one has (1.2.4) π m,n = (m -n -1)ω 1 + nω 2
and since H 0 P (ν m,n ) * is isomorphic with V P (π m,n ), the Weyl module for P with highest weight π m,n , one obtains: Lemma 1.2.1. For each m, n ∈ N and i ∈ N, one has

H i P (µ m,n ) V P (π m,n ) if i = d -1, 0 if i = d -1.
Now, consider induction from B to G. Thanks to the lemma, the spectral sequence of composite functors degenerates and gives (together with (1.1.4) applied to P = P and P = G) isomorphisms H i (µ m,n ) H i-d+1 (V P (π m,n )) for each i ≥ 0. Since the former is zero for i ∈ {d-1, d} (see [START_REF] Liu | On the cohomology of line bundles over certain flag schemes[END_REF], Section 2), this gives: Proposition 1.2.2. For each m, n ∈ N and i ∈ N, one has

H i (µ m,n )        H 0 (V P (π m,n )) if i = d -1, H 1 (V P (π m,n )) if i = d, 0 if i = d -1, d.
This allows us to recover and extend the result of [START_REF] Liu | On the cohomology of line bundles over certain flag schemes[END_REF], Corollary 2:

Corollary 1.2.3. Suppose that p > n. (i) If m ≥ n, then H d (µ m,n ) = 0 and H d-1 (µ m,n ) H 0 (π m,n ). In particular, H i (µ n ) = 0 for all i ≥ 0. (ii) If n ≥ m, then H d-1 (µ m,n ) = 0 and H d (µ m,n ) H 1 (π m,n ) H 0 (π n,m ). (iii) Furthermore, if p > m ≥ n then H d-1 (µ m,n ) L(π m,n ) H d (µ n,m ).
Proof. Suppose p > n. Then V P (π m,n ) is irreducible as a P -module, being isomorphic as a module over the Levi subgroup of P to the GL d -module S n K d which is irreducible since n < p. Therefore, V P (π m,n ) H 0 P (π m,n ). Thus, the proposition gives for all i ≥ 0 that [START_REF] Carsten | Representations of algebraic groups[END_REF], II.5.4), which proves (ii).

H i (µ m,n ) H i-d+1 (π m,n ). If m ≥ n then π m,n belongs to C := X(T ) + -ρ and hence H j (π m,n ) = 0 for j > 0 (and also for j = 0 if m = n). This proves (i). If m < n, then H 0 (π m,n ) = 0. Further, π n,m = (n-m-1)ω 1 +mω 2 belongs to C and one has π m,n = s α 1 • π n,m . Since 0 ≤ (π n,m + ρ, α ∨ 1 ) = n -m < p, one has H 1 (π m,n ) H 0 (π n,m ) (see
Suppose now that p > m ≥ n. By (i) and Serre duality on G/Q, one has

H 0 (π m,n ) H d-1 (µ m,n ) H d (-(m + d)ω 1 + nω d ) * = H d (τ µ n,m ) * .
Now, using the automorphism τ of (G, T ), one deduces from (ii) that

H d (τ µ n,m )) * H 0 (τ π m,n ) * .
Since the latter is the Weyl module V (π m,n ), one obtains that H 0 (π m,n ) L(π m,n ). This proves (iii).

Our goal in the next subsection is to determine H d (µ p ). Since for d = 2 the SL 3 -modules H 2 (m, -n -2) have been described in [START_REF] Liu | Cohomologie des fibrés en droites sur SL3/B en caractéristique positive: deux filtrations et conséquences[END_REF], where it is proved in particular (see [START_REF] Liu | Cohomologie des fibrés en droites sur SL3/B en caractéristique positive: deux filtrations et conséquences[END_REF], Th. 3) that H 2 (p, -p -2) is the Weyl module V (0, p -2), which is simple, we will henceforth assume that d ≥ 3.

Computation of H

d (µ n ) for p ≤ n ≤ 2p -1. Suppose now that d ≥ 3 and n = p + f with 0 ≤ f ≤ p -1.
For each weight ν which is dominant (resp. dominant with respect to P ), denote by L(ν) (resp. L P (ν)) the irreducible G-module (resp. P -module) with highest weight ν. Let (1.3.1)

λ f = f ω 1 + (p -2 -f )ω 2 + (f + 1)ω 3 if f ≤ p -2, (p -1)ω 1 + (p -2)ω 3 + ω 4 if f = p -1 (with the convention ω 4 = 0 if d = 3), and set L f = L(λ f ) and N f = L P (λ f ). Proposition 1.3.1. One has H d-1 (µ n ) L f H d (µ n ).
Remark 1.3.2. This generalizes Corollary 3 in [START_REF] Liu | On the cohomology of line bundles over certain flag schemes[END_REF], which was the case f = 0. Note that if f > 0 and d > p-f +1 then µ p+f does not belong to the closure of the facet containing µ p . Indeed, denoting by α 1 , . . . , α d the simple roots and setting

β = α 1 + • • • + α p-f +1 , one has µ p + ρ, β ∨ = 2p -f + 1 whereas µ p+f +ρ, β ∨ = 2p+1.
Hence the result for µ p+f cannot be deduced from the one for µ p by applying a functor of translation.

Proof. Set π n = -ω 1 + nω 2 and M = V P (π n ). According to Doty [Dot85], § §2.3-2.4, one has exact sequences of P -modules:

(1.3.2) 0 / / N f / / M / / C / / 0 and (1.3.3) 0 / / C / / H 0 P (π n ) / / N f / / 0
where C = L P (π n ). Applying the functor H 0 to (1.3.3) and using that:

(1.3.4)

H i (H 0 P (π n )) = H i (π n ) =
0 for all i ≥ 0, one obtains H 0 (C) = 0 and isomorphisms H i (C) H i-1 (N f ) for all i ≥ 1. Taking this and Proposition 1.2.2 into account and applying the functor H 0 to (1.3.2), one obtains isomorphisms:

(1.3.5) H d-1 (µ n ) H 0 (M ) H 0 (N f ),
an exact sequence:

(1.3.6)

0 / / H 1 (N f ) / / H d (µ n ) / / H 0 (N f ) / / H 2 (N f ) / / 0 and isomorphisms (1.3.7) H i (N f ) H i+1 (C) H i+2 (N f ) for i ≥ 1. Since H i (N f ) = 0 for i > |R + |, one obtains H i (N f ) = 0 for all i ≥ 1.
Together with (1.3.6) and (1.3.5), this gives isomorphisms:

(1.3.8)

H d (µ n ) H 0 (N f ) H d-1 (µ n ).
On the other hand, by [START_REF] Liu | On the cohomology of line bundles over certain flag schemes[END_REF], Cor. 4,

λ f has multiplicity 1 in H d (µ n ), which is therefore non-zero. Thus H d (µ n ) H d-1 (µ n ) H 0 (N f ) is a non-zero submodule of H 0 (H 0 P (λ f )) = H 0 (λ f ). Now, using the automorphism τ of (G, T ), one obtains that H d (µ n ) H d-1 (τ µ n ) * is a quotient of H 0 (τ λ f ) * V (λ f ). Since any non-zero mor- phism V (λ f ) → H 0 (λ f ) has image L f (see [Jan03] II.6.16 Remark), then (1.3.8) gives that H d (µ n ) L f H d-1 (µ n ).
Remark 1.3.3. 1 Seitz has shown ([Sei87], Prop. 6.1) that if a simple SL d+1module L(µ) with µ p-restricted has one-dimensional weight spaces then either µ is a fundamental weight ω i or a multiple of ω 1 or ω d , or µ = aω i + (p -1 -a)ω i+1 for some i ∈ {1, . . . , d -1} and a ∈ {0, . . . , p -1}. Since λ f is not in that list when f > 0, it follows that H d (µ p+f ) L(λ f ) never has all its weight spaces of dimension 1 when f > 0. This improves on Remark 3 (2) of [START_REF] Liu | On the cohomology of line bundles over certain flag schemes[END_REF].

1.4. Jantzen sum formula and consequences. Now, consider the case f = 0, i.e. n = p. Then λ 0 = (p -2)ω 2 + ω 3 . We shall use Jantzen sum formula ([Jan03], II.8.19) to express ch L 0 in terms of Weyl characters. Set r = min(d, p). In addition to λ 0 , define for i = 1, . . . , r -2 the dominant weights:2

(1.4.1)

λ i = iω 1 + (p -2 -i)ω 2 + ω 3+i
(with the convention ω d+1 = 0) and set L i = L(λ i ) and

N i = L P (λ i ). In other words, if d > p the sequence ends with λ p-2 = (p -2)ω 1 + ω p+1 whilst if d ≤ p it ends with λ d-2 = (d -2)ω 1 + (p -d)ω 2 + ω d+1 = (d -2)ω 1 + (p -d)ω 2 .
Lemma 1.4.1. Consider SL n+1 for some n ≥ 2 and for k = 1, . . . , n consider the weight For α ∈ R + let α ∨ be the corresponding coroot. Then, for m ∈ Z, let s α,mp be the affine reflection defined for all λ ∈ X(T ) by

θ k = ω 1 -kω k + (k -1)ω k+1 . Note that θ 1 = 0. (i) For k ≥ 2, one has s k • θ k = θ k-1 . (ii) For k ≥ 2, one has ω 1 -kω k + kω k+1 = s k • • • s 2 • ω k+1 . Proof. One has s k • θ k = θ k + (k -1)α k = θ k-1 . Thus s 2 • • • s k • θ k = θ 1 = 0, whence θ k = s k • • • s 2 • 0. Next,
s m,α (λ) = λ -((λ, α ∨ ) -m)α. Further, one sets s α,m • λ = s m,α (λ + ρ) -ρ.
For each Weyl module V (λ), Jantzen has defined a decreasing filtration

V (λ) ⊃ V (λ) 1 ⊃ V (λ) 2 ⊃ • • •
i≥1 ch V (λ) i = α∈R + m 0<mp<(λ+ρ,α ∨ ) v p (m)χ(s α,mp • λ)
where v p denotes the p-adic valuation and χ is the Weyl character. Recall that χ(µ) = 0 if µ is singular for the dot action of W , i.e. if there exists α ∈ R + such that (µ + ρ, α ∨ ) = 0, and otherwise there exists a unique couple (w, µ + ) ∈ W × X(T ) + such that µ = w • µ + and then χ(µ) = (-1) (w) ch V (µ + ). In particular, if the right-hand side, to which we shall refer as "Jantzen's sum" (relative to λ), equals ch L for some simple module L (resp. equals 0), then one has ch V (λ) = ch L(λ) + ch L (resp. V (λ) = L(λ)).

Proposition 1.4.2. For i = 0, . . . , r -2 one has the equality:

(1.4.2) ≥1 ch V (λ i ) = r-2 j=i+1 (-1) j-i-1 ch V (λ j )
and the exact sequences:

(1.4.3) 0 / / L i / / H 0 (λ i ) / / L i+1 / / 0 and (1.4.4) 0 / / N i / / H 0 P (λ i ) / / N i+1 / / 0
with the convention L r-1 = 0 = N r-1 . Therefore, one has exact sequences:

(1.4.5) 0 → L(λ 0 ) → H 0 (λ 0 ) → H 0 (λ 1 ) → • • • → H 0 (λ r-2 ) → 0 and (1.4.6) 0 → L P (λ 0 ) → H 0 P (λ 0 ) → H 0 P (λ 1 ) → • • • → H 0 P (λ r-2 ) → 0. Proof. Fix i ∈ {0, . . . , r -2} and for 1 ≤ j ≤ k ≤ d, set α j,k = α j + • • • + α k and c i,j,k = (λ i + ρ, α ∨ j,k
). We are going to prove that all terms in Jantzen's sum (for both G and P ) are zero, except the ones given in (1.4.2). Fix j ≤ k such that c i,j,k > p, let m ∈ N * such that mp < c i,j,k and set t = -mp and ν m = s α j,k ,mp • λ i = λ i -tα j,k . There are four cases to consider.

Case 1: c i,j,k = k -j + 1 (this occurs only j > i + 3 and k ≥ j + p). Then the expression of ν m in terms of the fundamental weights contains the "sequence" -tω j -tω k , the coefficients of the ω for j < < k being 0, and hence ν m + ρ is orthogonal to both α ∨ j,j+t-1 and α ∨ k-t+1,k . Therefore ν m gives no contribution to Jantzen's sum, neither for G nor for P .

Case 2: c i,j,k = k -j + 2 (this occurs only if j ≤ i + 3 ≤ k, including the case k = i + 3, j = 2 and i = p -2). Assume first that j < i + 3 < k. Then the expression of ν m in terms of the fundamental weights contains the "sequence" -tω j + ω i+3 -tω k , the coefficients of the ω for j < < k and = i + 3 being 0. For s = 0, . . . , i + 2 -j, i + 3 -j, . . . , k -j -1, (ν m + ρ, α ∨ j,j+s ) takes all the values from 1 -

t to k -j + 1 -t, except i -j + 4 -t. Since 1 -t ≤ 0 and k -j + 1 -t > 0 (since t ≤ k -j + 2 -p ≤ k -j), the value 0 is obtained unless t = i -j + 4. Similarly, for s = 0, . . . , k -i -4, k -i -3, . . . , k -j -1, (ν m + ρ, α ∨ k-s,k ) takes all the values from 1 -t to k -j + 1 -t, except k -i -2 -t. Hence the value 0 is obtained, unless t = k -i -2. This shows that ν m + ρ is singular, except possibly if t = i -j + 4 = k -i -2.
But in this case one has 2t = k -j + 2 and hence:

(ν m + ρ, α ∨ j,k ) = k -j + 2(1 -t) = 0. Consider now the "boundary" cases j = i + 3 or k = i + 3. If j = i + 3 then, since k -t + 1 > j, one has k -t + 1 > i + 3 and hence (ν m + ρ, α ∨ k-t+1,k ) = 0. If k = i + 3, then j + t -1 < k = i + 3
and hence (ν m + ρ, α ∨ j,j+t-1 ) = 0. Thus, in any case ν m gives no contribution to Jantzen's sum, neither for G nor for P .

To close this case, note that since i

+ 3 ≤ p + 1 ≤ k -j + 2, the case k = i + 3 can occur only if j ≤ 2, in which case c i,j,k = k -j + 2 implies i = p -2. Case 3: j = 2 and c i,j,k = k + p -2 -i and i < p -2. (Note that c i,2,i+2 = p -1 hence the hypothesis c i,2,k > p implies k ≥ i + 3.) Then one has ν m = (i + t)ω 1 + (p -2 -i -t)ω 2 + ω i+3 -tω k + tω k+1 .
For s = 0, . . . , i, i + 1, . . . , k -3, (ν m + ρ, α ∨ 2,2+s ) takes all the values from

p -i -1 -t to p + k -i -3 -t, except p -t. Since the last value taken is ≥ p -1 > 0, the value 0 is obtained except if t = p or if the initial value p -i -1 -t is > 0, i.e. t ≤ p -i -2. Similarly, for s = 0, . . . , k -i -4, k -i -3, . . . , k -3, (ν m + ρ, α ∨ k-s,k ) takes all the values from 1 -t to k -1 -t, except k -i -2 -t. Moreover one has 1 -t ≤ 0 < k -1 -t, hence the value 0 occurs unless t = k -i -2. Thus, ν m + ρ is singular except possibly if t = k -i -2 belongs to {1, . . . , p -i -2} or if t = k -i -2 = p. In the latter case, one has 2t = p + k -i -2 and hence (ν m + ρ, α ∨ 2,k ) = k -2 + p -i -2t = 0. In the former case, one has k = i + 2 + t with t = 1, . . . , p -i -2, whence m = 1. In fact, since k = i + 2 + t is ≤ d, we have t ∈ {1, . . . , r -i -2}, recalling that r = min(p, d). For t = 1, . . . , r -i -2, set θ t = s α 2,i+2+t ,p • λ i = (i + t)ω 1 + (p -2 -i -t)ω 2 + ω i+3 -tω i+2+t + tω i+3+t .
Using Lemma 1.4.1 with a shift of i + 2 in the indices, one obtains that θ 1 = λ i+1 and that

θ t = s i+2+t • • • s i+4 • λ i+t for t ≥ 2.
Denote by G P the Levi subgroup of P containing T and recall ([Jan03], II.5.21) that V P (λ i ) is just the corresponding Weyl module for G P on which the unipotent radical of P acts trivially. Therefore, applying Jantzen's sum formula for G P , one already obtains the equality:

(1.4.7)

≥1 ch V P (λ i ) = r-i-2 t=1 (-1) t-1 ch V P (λ i+t ).
To prove the analogous equality for G we must consider the last case, where j = 1.

Case 4: j = 1. Note that the assumption c i,1,k > p implies k ≥ 3. If k ≤ i + 3 then c i,1,k = p + k -2 + δ k,i+3 is ≤ 2p (since i ≤ p -2), hence m = 1 and t = k -2 + δ k,i+3
and the expression of ν m in terms of the fundamental weights contains the "sequence" (p -i -2)ω 2 -(k -2)ω k , the coefficients of the ω for 2 < < k being 0. Then (ν m + ρ, α ∨ 3,k ) = 0 hence ν m gives no contribution to Jantzen's sum. Suppose now that k > i + 3. Then c i,1,k = k -1 + p and

ν m = (i -t)ω 1 + (p -2 -i)ω 2 + ω i+3 -tω k + tω k+1 . For s = 0, . . . , k -i -4, k -i -3, . . . , k -3, (ν m + ρ, α ∨ k-s,k ) takes all the values from 1 -t to k -1 -t, except k -i -2 -t. Moreover one has 1 -t ≤ 0 ≤ k -1 -t, hence the value 0 occurs unless t = k -i -2.
Let us assume henceforth that t = k -i -2. Then t > p, for otherwise one would have

k -i -2 ≤ p hence k ≤ 2p (since i ≤ p -2) whence m ≤ 2; but m = 1 gives k -i -2 = t = c i,1,k -p = k -1, a contradiction, whereas m = 2 gives k -i -2 = t = c i,1,k -2p = k -p -1, a contradiction too, since i ≤ p -2.
Now, for s = 0, . . . , i, i + 1, . . . , k -3, (ν m + ρ, α ∨ 1,2+s ) takes all the values from p -t to p + k -2 -t, except p + i + 1 -t. Moreover since p < t ≤ k -1, the initial term is < 0 and the final term > 0, hence the value 0 occurs unless t

= p + i + 1. Now, if t = k -i -2 = p + i + 1 then 2t = p + k -1 and hence (ν m + ρ, α ∨ 1,k ) = p + k -1 -2t = 0.
Thus, in any case ν m gives no contribution to Jantzen's sum. This proves (1.4.2).

It follows from (1.4.2) that H 0 (λ r-2 ) = L r-2 ; then for λ r-3 the Jantzen sum equals ch L r-2 hence ch H 0 (λ r-3 ) = ch L r-3 + ch L r-2 . By decreasing induction one obtains that ch H 0 (λ i ) = ch L i + ch L i+1 for i = r -3, . . . , 0, whence the exact sequences (1.4.3). Similarly, using (1.4.7) one obtains that H 0 P (λ r-2 ) = N r-2 and ch H 0 P (λ i ) = ch N i + ch N i+1 for i = r -3, . . . , 0, whence the exact sequences (1.4.4). This completes the proof of Proposition Let us derive the following corollary (which is not used in the sequel).

Corollary 1.4.3. For i = 0, . . . , r -2, one has H 0 (N i ) = L i and H j (N i ) = 0 for j > 0.

Proof. Applying the functor H 0 to each exact sequence (1.4.4) gives an exact sequence:

0 / / H 0 (N i ) / / H 0 (λ i ) / / H 0 (N i+1 ) / / H 1 (N i ) / / 0
and isomorphisms H j (N i+1 ) H j+1 (N i ) for j ≥ 1. Since H j (N 0 ) = 0 for j ≥ 1, one obtains H j (N i ) = 0 for all i ≥ 0 and j ≥ 1, hence the previous exact sequence becomes:

(

1.4.8) 0 / / H 0 (N i ) / / H 0 (λ i ) / / H 0 (N i+1 ) / / 0 .
Since H 0 (N 0 ) L 0 , the exact sequences (1.4.3) then imply, by induction on i, that H 0 (N i ) L i for i = 0, . . . , r -2.

On the other hand, in [START_REF] Liu | On the cohomology of line bundles over certain flag schemes[END_REF], Cor. 3 and 4, the first author proved, using a result of Suprunenko pointed out by one of the referees, that the dominant weights of L(λ 0 ) (resp. of L(λ 0 ) = L((p -2)ω 1 + ω 2 )) are exactly the dominant weights ≤ λ 0 (resp. λ 0 ), each occuring with multiplicity one. Let us now switch to representations of GL d+1 and identify each λ i (resp. λ i = (p -2 -i)ω 1 + ω 2+i ) with the partition (p -1, p -1 -i, 1 i+1 ) (resp. (p -1 -i, 1 i+1 )). Recall that (see for example [START_REF] Macdonald | Symmetric Functions and Hall Polynomials[END_REF], Chap. I) for a dominant weight λ of GL d+1 , identified with a partition with at most d + 1 parts, the Weyl character ch V (λ) (resp. the orbit sum ν∈W λ e ν ) corresponds to the Schur function S λ (resp. the monomial symmetric function m λ ). Let us assume that d ≥ 2p -2, then the dominant weights smaller than λ 0 correspond to all partitions of 2p -1 which are smaller than (p -1, p -1, 1) in the dominance order. Thus, one deduces from Propositions 1.4.2 and 1.3.1 (or Corollary 3 in [START_REF] Liu | On the cohomology of line bundles over certain flag schemes[END_REF]) the following: where the sum on the left hand-side of the first equality (resp. second equality) is taken over all partitions λ of 2p -1 (resp. p) such that λ ≤ (p -1, p -1, 1) (resp. λ ≤ (p -1, 1)) in the dominance order.

Remark 1.4.5. One may conjecture that the previous equality holds for any integer n ≥ 2 (not only for prime numbers). Using a Sage program, we have checked this up to n = 23. 3

  for any weight λ and w ∈ W , one has w • λ = wλ + w • 0, which equals λ + w • 0 if wλ = λ. Applying this to w = s k • • • s 2 and λ = ω k+1 gives assertion (ii).

Corollary 1.4. 4 .

 4 For each prime number p, one has the equality:λ≤(p-1,p-1,1) i S (p-1,p-1-i,1 i+1 ) i S (p-1-i,1 i+1 )

We are grateful to one of the referees for this remark.

These are different from the weights λ f considered in section 1.3.

On August 23, 2019 (the day after the first version of this article was posted on arXiv), a proof of the first equality of this conjecture was kindly sent to us by Darij Grinberg, see[START_REF] Grinberg | Petrie symmetric functions[END_REF].

Acknowledgements. We thank one of the referees for several very useful comments.


Cindy Linn Rountree

Damien Vandembroucq

Mehdi Talamali

Elisabeth Bouchaud

Stéphane Roux

Plasticity-induced structural anisotropy of silica glass

Keywords: numbers: 62, 20, F, 81, 05, Kf

Plasticity of amorphous media, which can be easily evidenced via indentation or scratch tests [1], has a very different nature from its counterpart for crystalline media, since no elementary entities such as dislocations whose evolution controls plastic flow can be easily defined [2,3]. The current view is that spatially distributed local restructuring rather than extended defect motion (such as dislocation) are responsible for irreversible strains in amorphous materials [4,5]. At a very local scale, under load, a small group of atoms (called a Transformation Zone or TZ) may undergo rearrangements, a change of conformation eventually affecting the topology of the atomic bonds which will contribute to an elementary increment in irreversible strain. Although a complete description of these TZ is extremely complex, and cannot be cast into simple categories, a statistical analysis capturing the key properties of these zones is an attractive route for relating the macroscopic mechanical behavior to the underlying microstructural counterpart [4].

In contrast with dislocations which naturally lead to isochoric plastic deformation, transformation zones may densify or dilate as they rearrange. Indeed at a macroscopic scale, plasticity of silicate glasses is known to exhibit permanent densification [6,7] from a few percents for soda-lime glasses [8] to values as large as 20% in the extreme case of amorphous silica [9,10]. This densification naturally affects shear plasticity, and hence pressure and shear stress are to be coupled in the yield criterion of amorphous silica [11].

Plasticity of structural glasses is furthermore characterized by a significant hardening behavior [9,10]. The yield surface evolves with the mechanical loading. This means in particular that, when applying stress in a given direction (pure shear, pure hydrostatic pressure, etc) the value of the elastic limit depends on the history of the loading.

To account for this dependence on mechanical history a proper description of plasticity thus requires the use of additional internal variables. The first one is obviously density and indeed a recent study of densification with pressure allows one to characterize the density hardening of silica [10]. The necessity to include more internal variables than the mere density is a difficult question to address. Experimentally, plasticity of amorphous media calls for a high level of stress and confinement which can either be met in indentation-type test (with the complication of the very strong spatial heterogeneity of the stress state) or in anvil diamond cell where shear cannot be imposed. Thus it appears very difficult to follow a particular stress path which would allow one to answer this question. Hence, in the present study, we will resort to Molecular Dynamics simulations where homogeneous loadings with arbitrary stress path can be imposed.

As for granular media plasticity, shear reversal experiments show that a simple scalar internal variable is not sufficient to account for the mechanical behavior, and a tensor-valued internal variable, characterizing the geometry packing is needed [START_REF] Radjai | The Physics of Granular Media[END_REF]. For this purpose, a natural candidate is the so-called fabric tensor, F = n ⊗ n , which captures the mean orientation of the contact normals, n, through the spatial average of their dyadic product. This tensor characterizes an anisotropic texture of the medium. Experimental and numerical studies of granular media [START_REF] Oda | [END_REF]14], have shown that such was the case after shearing. Indeed, more contacts are oriented along the direction(s) of compression. Similarly, the same tools have recently been used to study the rheology of foams [15].

The fabric tensor can be characterized by its three eigenvalues, λ i , for 1 < i < 3. The fact that n is a unit vector implies that the trace of F is equal to unity, or i λ i = 1 Thus, for an isotropic medium, the three eigenvalues are equal to 1/3. In case of anisotropy, degeneracy is lifted and different eigenvalues λ i are measured. In order to give a quantitative scalar estimate of this anisotropy, the following scalar α parameter, propor- 6) from +17.5 % (red) and ( 7) from +25 % (blue). (Note: the deformation rate is 10 -2 per ps.)

tional to the norm of the deviatoric part of F , is defined:

α = 3 2 3 i=1 λ i - 1 3 2 . (1)
For an isotropic medium, α = 0, and the prefactor has been chosen such that in case of full anisotropy, {λ i } = {1; 0; 0} we have α = 1. This quantitative parameter, which is built in the same spirit as an effective shear stress allows one to compare results obtained in different stress geometries. Pursuing our analogy, it is natural to consider a similar fabric tensor for amorphous silica. We will show that indeed, besides the usual (reversible) anisotropy induced by an elastic strain, plasticity induces a series of structural rearrangements which after unloading endow silica with a remnant anisotropic structure. The latter state can be regarded as a novel phase of amorphous silica.

In the case of silica, chemistry imposes that each silicon atom is encaged in a tetrahedron of four oxygen atoms forming a SiO 4 elementary unit. These tetrahedra are bounded to each other through a common oxygen atom. They are extremely robust and remain virtually unaltered under mechanical loadings. However, pairs of tetrahedra have much more freedom in their relative orientations. Henceforth, it is natural to build the equivalent of a fabric tensor from the "contacts" between neighboring tetrahedra. Thus for all Si-O-Si triplets, we consider the unit vector n which connects all next to nearest neighbors, NNN, (i.e. Si-Si atoms see Fig. 2a). The fabric tensor F as defined above is built up by averaging over all NNN Si atoms. The fact that second neighbors are to be considered is a challenge to study this anisotropy experimentally using classical tools for structural analysis.

Amorphous silica was studied via Molecular Dynamics (MD) computer simulations. This technique allows us to impose an extended (simple) shear along one direction under constant volume, and reverse it. This particular stress path is able to reveal whether plasticity is accompanied by any structural change and in particular anisotropy. MD simulations are performed on amorphous silica (a-SiO 2 ), using the empirical interatomic potential developed by Vashishta et al [16,[START_REF] Vashishta | Amorphous Insulators and Semiconductors[END_REF]18]. This potential incorporates steric repulsion, charge transfer, and electronic polarizability of atoms through pair-wise interaction terms. Covalent effects in silica are included through bond-bending and bond-stretching three-body terms. The parameters of the potential have been adjusted by measurements of structural correlations, elastic moduli, and fracture toughness [16,19]. The a-SiO 2 samples were first prepared by melting (i.e. heating to 4000 K) 117,912 (cubic box of length 12.088 nm) atoms of ideal β-cristobalite crystal. The sample was cooled to 2500 K at a rate of 1 K/∆t (where ∆t is the time step) and allowed to relax for 60000 ∆t (large specimen). The cooling process was repeated at 1500 K, 600 K, 300 K, and 5 K. Afterwards a conjugate gradient method is used to cool the sample to 0 K. The temperature is once again elevated to 300 K in order to conduct simulations at room temperature. Periodic boundary conditions have been invoked in all simulations.

Shear plasticity of this model silica glass has been studied on 12 independent samples. Each sample was sheared by applying an external shear in small increments such that the volume was conserved (NVT ensemble). Between each subsequent shear the sample was allowed to relax. Also during the simulations the temperature was held fixed at 300 K using a thermostat. Two different shear strain rates have been used, 10 -4 ps -1 and 10 -2 ps -1 . It is to be stressed that these shear rates are considerably higher than the highest ones that may be considered experimentally. However lower shear rates cannot conceivably be imposed within the present framework of Molecular Dynamics. Therefore the effect of long time relaxation processes such as atom diffusion is out of reach of the present study.

Figure 1 depicts a typical stress strain curve. The initial behavior corresponds to linear elasticity; for the faster strain rate here represented, a stress overshoot is visible before a plateau corresponding to the stationary plastic regime. As discussed in [20], the amplitude of this stress overshoot is dependent on the strain rate and almost disappears for the slower rate. In addition to this monotonic test, unloading to zero shear stress has been performed starting from different values of the maximum total strain: 2.5%, 5%, 7.5%, 12.5%, 17.5% and 25%. As can be seen on Fig. 1, beyond 5% strain (which corresponds to a shear stress of about 2.25 GPa), permanent Evolution of the eigenvalues of the fabric tensor when sheared from 0 % to 30 % (b) and unloaded from: 2.5% (c); 5% (d); 7.5% (e); 12.5% (f); 17.5% (g) and 25% (h). Before loading the material is isotropic, the 3 eigenvalues should be equal within the errors bars. Upon loading (b), degeneracy is lifted and anisotropy first increases reversibly (c-d) due to the elastic strain. Then plasticity sets in (e-f-g) and after unloading, the material is left with a remnant structural anisotropy. plastic deformation sets in. During the simulations the pressure was monitored, and it was found to increase from ∼ 0GP a during the elastic loading stage up to ∼ 2.5GP a during plastic flow. Further details can be found in [21].

In Fig. 2 the evolution of the eigenvalues of the fabric tensor is shown along shear loading and unloading from various values of total strain. At rest, isotropy of the glass is almost perfectly obtained; the three eigenvalues are equal within a precision of 10 -3 . Shear loading then induces first an elastic strain which renders the medium anisotropic. The elastic shear strain naturally leads to the lifting of the degeneracy: the eigenvalues take different values with a growing shift. In the elastic regime, a change in fabric tensor can be shown to be proportional to the strain. Note that we recover here more or less the behavior of the shear stress: a linear regime related to elasticity followed by a bump and a stationary elasto-plastic regime. In this regime, the shearinduced anisotropy is marked and eigenvalues are clearly separated. In this plastic flow regime, where the elastic strain remains invariant, so does the fabric tensor. When unloading, the fabric tensor eigenvalues linearly return to their original value showing that the initial rise of anisotropy is perfectly reversible, and hence of elastic nature.

For ε max = 2.5% and 5%, isotropy is approximately obtained at zero deformation; the glass has recovered its original state. However, unloading from larger deformations ε max = 7.5%, 12.5%, 17.5% and 25% we observe that isotropy is never fully recovered. The beginning of the unloading stage shows a linear evolution somewhat comparable to the first elastic loading, consistent with the fact that the incremental strain during unloading is expected to be purely elastic.It is however noteworthy that a significant plastic strain occurs during this unloading phase showing a strong kinematic hardening, and a very severe shrinkage of the elastic domain. The permanent plastic shear deformation obtained at zero shear stress can be associated to the minimal gap between the eigenvalues of the fabric tensor but this gap is clearly growing well beyond the numerical uncertainty.

The same analysis can be performed in a more quantitative manner by following the evolution of the scalar anisotropy index α under these shear loading and unloading tests and is presented in Fig. 3. The anisotropy index decreases under unloading, reaches a minimum value, α min , approximately when the shear stress is null. If the load is reversed to negative values, α rises again to saturate at a similar level as for direct shear. For a maximum deformation ε max = 2.5%, 5%, we recover after unloading α min ≈ 10 -3 : no change is noticeable when compared ) FIG. 4: Anisotropy parameter after unloading αmin vs. maximum strain εmaxfor two different imposed strain rates 10 -2 ps -1 (black circles) and 10 -4 ps -1 (red squares). After shear unloading, normal stresses persist due to the plane deformation geometry and amplify the structural anisotropy (see insert). After relaxation of these stresses with an additional NPT step of 20 ps, a stress free state is obtained while the silica structure still exhibits a significant anisotropy index.

with the original state. For larger deformations, however, a clear increase of α min is observed up to values close to 10 -2 , well above the numerical uncertainty.

These results are summarized in Fig. 4 which shows the evolution of the minimum gap, α min , versus the maximum deformation, ε max . The results obtained with a much slower strain rate 10 -4 ps -1 , (one hundred times slower than the previous case) are also shown in the same figure. It is remarkable that both curves almost superimpose each other. Once silica has experienced shear plasticity, a significant anisotropy persists even after unloading. Note that an additional relaxation towards zero stress state (NPT ensemble) during 20 ps after unloading was performed. This ensures to eliminate normal stresses due to the shearing in plane deformation. As shown in the insert, the latter naturally induce an additional anisotropy of trivial elastic origin which has to be eliminated before attesting for the presence of the plasticity induced anisotropy discussed in the present study.

On the basis of molecular dynamics study of plasticity of amorphous silica under simple shear, a non-reversible anisotropy sets in and appears stable (at the time scale of MD). Detailed structure investigation could not reveal any significant deviation from the amorphous structure based on e.g. interatomic distances The eigenvectors of these Si-O-Si directions are aligned with the principal strain/stress directions. Because of the relatively short time scale naturally associated with MD simulations, an experimental validation of this observation is needed to validate our conclusions, in particular concerning the stability of this phase. Experimentally, anisotropy is ex-pected to give rise to birefringence, a property which might be more easily accessible than structural analyses. Using standard photoelasticity parameters for silica and the maximum anisotropy obtained in the present work α = 510 -3 , a crude estimate of the birefringence gives an index contrast ∆n = 10 -3 . Such a phenomenon may for instance be tested in the framework of rotational anvil cell experiments [22,23].

FIG. 1 :

 1 FIG.1:The evolution of the average stress of 12 independent samples, composed of 117,912 atoms each, when sheared: (1) from 0 % to 30 % (black); unloaded (2) from +2.5 % (sky blue) ; (3) from +5 % (lavender); (4) from +7.5 % (pink); (5) from +12.5 % (green); (6) from +17.5 % (red) and (7) from +25 % (blue). (Note: the deformation rate is 10 -2 per ps.)

FIG. 2 :

 2 FIG.2: Sketch of the "contacts" Si-Si between neighbouring SiO4 tetrahedra used to build the fabric tensor F = n ⊗ n . Evolution of the eigenvalues of the fabric tensor when sheared from 0 % to 30 % (b) and unloaded from: 2.5% (c); 5% (d); 7.5% (e); 12.5% (f); 17.5% (g) and 25% (h). Before loading the material is isotropic, the 3 eigenvalues should be equal within the errors bars. Upon loading (b), degeneracy is lifted and anisotropy first increases reversibly (c-d) due to the elastic strain. Then plasticity sets in (e-f-g) and after unloading, the material is left with a remnant structural anisotropy.

FIG. 3 :

 3 FIG.3: Evolution of the anisotropy index α under shear loading and unloading (same as above). The minimum value of α, black circles, is associated with the state of the structure under zero shear stress. Permanent deformation is characterized by a non-zero value of the anisotropy index: the glass structure has become anisotropic.

Acknowledgments We acknowledge D. Bonamy. R. Kalia, and L. Van Brutzel. CLR wishes to acknowledge the National Science Foundation Graduate Research Fellowship under grant no. 0401467 for financial support. DV, SR and MT acknowledge the financial support of ANR grant "Plastiglass" no. ANR-05-BLAN-0367-01.


Marjorie Aubanel
email: marjorie.aubanel@chu-lyon.fr

Aurélie Swalduz
email: aurelie.swalduz@lyon.unicancer.fr

Virginie Avrillon
email: virginie.avrillon@lyon.unicancer.frl.doublet@ramsaygds.fr

Louis Doublet

Bénédicte Mastroianni
email: benedicte.mastroianni@lyon.unicancer.fr

Eve-Marie Neidhardt-Bérard

Maurice Pérol
email: maurice.perol@lyon.unicancer.fr

Marie Neidhardt-Bérard

Combining EGFR and MET Inhibition With Crizotinib in EGFR-mutated Lung Adenocarcinoma Harboring MET Amplification: A Brief Report

Keywords: Lung cancer, Combination targeted therapies, crizotinib, EGFR TKI, Resistance

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

INTRODUCTION

Epidermal growth factor receptor (EGFR) activating mutations are found in 10 to 15% of advanced non-squamous non-small cell lung carcinoma (NSCLC) for Caucasian patients and are associated with dramatic response to EGFR tyrosine kinase inhibitors (EGFR TKIs). All patients treated with 1 st or 2 nd generation EGFR TKIs experience disease progression due to secondary mutations (T790M mutation in exon 20 that confers sensitivity to third-generation EGFR TKI), phenotypic changes and activation of bypass signaling pathways [START_REF] Pao | Acquired resistance of lung adenocarcinomas to gefitinib or erlotinib is associated with a second mutation in the EGFR kinase domain[END_REF][START_REF] Mok | Osimertinib or Platinum-Pemetrexed in EGFR T790M-Positive Lung Cancer[END_REF][START_REF] Sequist | Genotypic and histological evolution of lung cancers acquiring resistance to EGFR inhibitors[END_REF][START_REF] Yang | Osimertinib in Pretreated T790M-Positive Advanced Non-Small-Cell Lung Cancer: AURA Study Phase II Extension Component[END_REF] . MET amplification is one of the bypass signaling pathways that promotes resistance to first-, second-and third-generation EGFR TKI, in approximately 5-20% of acquired resistances [START_REF] Westover | Mechanisms of acquired resistance to first-and second-generation EGFR tyrosine kinase inhibitors[END_REF][START_REF] Papadimitrakopoulou | Analysis of resistance mechanisms to osimertinib in patients with EGFR T790M advanced NSCLC from the AURA3 study[END_REF][START_REF] Ramalingam | Mechanisms of aquired resistance to first line osimertinib: Preliminary data from the phase III FLAURA study[END_REF] . Preclinical data suggest that inhibition of both EGFR and MET is required to obtain tumor regression. Crizotinib is a potent MET inhibitor showing activity in NSCLC with MET amplification or MET exon 14 skipping mutations [START_REF] Camidge | MET Copy Number as a Secondary Driver of Epidermal Growth Factor Receptor Tyrosine Kinase Inhibitor Resistance in EGFR -Mutant Non-Small-Cell Lung Cancer[END_REF][START_REF] Moro-Sibilot | Activity of crizotinib in MET or ROS1 positive (+) NSCLC: results of the AcSé trial[END_REF] . Responses to dual MET and EGFR inhibition with a combination of crizotinib and EGFR TKI have been occasionally reported in patients with EGFR activating mutations and MET amplification [START_REF] Baldacci | Outcome of EGFR-mutated NSCLC patients with METdriven resistance to EGFR tyrosine kinase inhibitors[END_REF][START_REF] Ou | Phase I Results from a Study of Crizotinib in Combination with Erlotinib in Patients with Advanced Nonsquamous Non-Small Cell Lung Cancer[END_REF][START_REF] Gainor | Dramatic Response to Combination Erlotinib and Crizotinib in a Patient with Advanced, EGFR-Mutant Lung Cancer Harboring De Novo MET Amplification[END_REF][START_REF] Li | Combination therapy of erlotinib/crizotinib in a lung adenocarcinoma patient with primary EGFR mutation plus secondary MET amplification and a novel acquired crizotinib-resistant mutation MET G1108C[END_REF][START_REF] York | Tolerable and Effective Combination of Full-Dose Crizotinib and Osimertinib Targeting MET Amplification Sequentially Emerging after T790M Positivity in EGFR-Mutant Non-Small Cell Lung Cancer[END_REF][START_REF] Dietrich | Response to Crizotinib/Erlotinib Combination in a Patient with a Primary EGFR-Mutant Adenocarcinoma and a Primary c-met-Amplified Adenocarcinoma of the Lung[END_REF][START_REF] Wang | Clinical analysis by next-generation sequencing for NSCLC patients with MET amplification resistant to osimertinib[END_REF][START_REF] Scheffler | Spatial Tumor Heterogeneity in Lung Cancer with Acquired Epidermal Growth Factor Receptor-Tyrosine Kinase Inhibitor Resistance: Targeting High-Level MET-Amplification and EGFR T790M Mutation Occurring at Different Sites in the Same Patient[END_REF][START_REF] Yoshimura | Successful crizotinib monotherapy in EGFR -mutant lung adenocarcinoma with acquired MET amplification after erlotinib therapy[END_REF] .

Herein, we report 5 patients' cases presenting with advanced lung adenocarcinoma harboring both EGFR mutation and de novo or acquired MET amplification treated in our institution with combination of a first-or third-generation EGFR TKI and crizotinib.

RESULTS

We report five cases of EGFR-mutated patients harboring MET-amplification which was defined as a MET copy number gain (CNG) ≥ 6 with MET/centromere ratio (MET/CEP7) ≥2 and was detected by immunohistochemistry (IHC) or Next-Generation Sequencing (NGS) and was confirmed by Fluorescent In Situ Hybridization in all patients. PD-L1 expression was assessed with Ventana SP263 antibody before initiation of combination TKIs.

Median age at the time of diagnosis was 51 years (range 36-66). All patients except one (#5) who was a former light smoker with 2.5 packs-year smoking history were non-smokers. All patients had stage IV adenocarcinoma, two depending on a L858R mutation in exon 21 and three on an exon 19 deletion. A. Crizotinib was administrated at the dose of 250 mg twice daily, osimertinib was administrated at the dose of 80 mg once daily and erlotinib at the doses of 75, 100 or 150 mg once daily, based on the previous erlotinib dosage and according to the MET/centromere ratio: 2.1) and was treated with erlotinib. Restaging CT after 1 month demonstrated thoracic and mediastinal progressive disease (PD). No T790M was detected on ctDNA and lung biopsy. After 9 months of chemotherapy, erlotinib 150 mg once daily and crizotinib were initiated with response in some lung metastases but emergence of new pulmonary lesions. After disease progression under two subsequent lines of chemotherapy and nivolumab, patient #1 was rechallenged with erlotinib 100 mg once daily and crizotinib with partial response. After 5 months of treatment, the systematic brain MRI revealed 3 asymptomatic infracentimetric lesions without evidence of extra cerebral relapse. Brain metastases were treated with stereotactic radiotherapy and combination therapy is ongoing (Figure A).

Clinical and biological characteristics are reported in Table

Patient #2 was diagnosed with a metastatic (brain and lung) lung adenocarcinoma harboring an EGFR L858R mutation and was treated with erlotinib 150 mg once daily for 4 months. Lung rebiopsy at disease progression (lung, mediastinal lymph nodes and new bone lesions with stable brain metastases) did not detect T790M mutation but MET-amplification (CN: 12.6; MET/centromere ratio: 5.6). She then received a second-line chemotherapy with carboplatin and pemetrexed with thoracic and brain disease progression after 2 cycles of pemetrexed maintenance. She was treated with brain stereotactic radiotherapy for the two brain metastases and erlotinib 75 mg once daily and crizotinib were subsequently initiated with a dissociated response consisting in an improvement of carcinomatous lymphangitis and irradiated brain metastases but appearance of new lung nodules. ctDNA revealed T790M mutation and erlotinib was then replaced by osimertinib whereas crizotinib was continued. First disease evaluation at 3 months of treatment revealed partial response but the patient experienced rapid subsequent cerebral disease progression and died one month later.

Patient #3: lung adenocarcinoma was revealed by a single brain metastasis treated with stereotactic radiotherapy. An EGFR exon 19 deletion was found on the primary lung tumor biopsy. She experienced PD (supra-clavicular, jugular-carotid node) without detection of T790M mutation neither on supra-clavicular rebiopsy, nor on ctDNA after 6 months of gefitinib treatment. MET-amplification (CN: 6.4; MET/centromere ratio: 2.3) was detected after failure of a second line chemotherapy Tolerance of the addition of crizotinib to EGFR TKI was quite good, without emergence of unexpected toxicities. No significant prolongation of QTc was observed on systematic ECG. Adverse events observed were rash, transaminase elevation, nausea, edema, visual disorders and diarrhea. Most of them were grade 1 or 2 and crizotinib dose was only reduced to 200 mg twice daily for patient #4 due to nausea.

DISCUSSION

In vitro, MET amplification results in the activation of PI3K/AKT downstream pathway through EGFR-independent phosphorylation of ERBB3, occurring also in the presence of EGFR-TKI, reinforcing the rational for a concomitant inhibition in the EGFR mutation landscape [START_REF] Engelman | MET amplification leads to gefitinib resistance in lung cancer by activating ERBB3 signaling[END_REF] . Its occurrence varies from 3% at diagnosis to at least 15 to 20% at progression under osimertinib first-line, probably underestimated due to evaluation of resistance mechanisms based on circulating tumor DNA in FLAURA and AURA3 trials [START_REF] Papadimitrakopoulou | Analysis of resistance mechanisms to osimertinib in patients with EGFR T790M advanced NSCLC from the AURA3 study[END_REF][START_REF] Ramalingam | Mechanisms of aquired resistance to first line osimertinib: Preliminary data from the phase III FLAURA study[END_REF] .

Identification of biomarkers remains the cornerstone to predict the efficacy of targeted therapies; however the definition of MET amplification is heterogeneous across the different studies. In some trials, MET-amplification is defined only as MET copy number gain (CNG), evaluating the copy number of gene per nucleus and potentially identifying polysomy. Others take into consideration the ratio of MET gene to centromeric portion of chromosome 7 (CEP7), which detect "true amplification" but the cut-off used for this ratio is not standardized. The interpretation is all the more difficult as polysomy and amplification can coexist on the same tumor. MET-amplification is probably more a subclonal event, and a special attention is required with next-generation sequencing results, which reflects circulating tumor DNA as a part of a tumor at a non-cellular level, which makes interpretation difficult.

In our cohort, MET-amplification was defined by MET CNG ≥ 6 and ratio MET/CEP7 ≥ 2, as often defined in previous studies as high-level of MET-amplification. We only associated crizotinib with EGFR TKIs as it is the only routinely available MET-inhibitor in France. Herein, combining crizotinib with first-or third-generation EGFR TKI, according to T790M status, demonstrated encouraging results with three almost complete responses, one objective complete response and importantly durable responses. Interestingly, two patients were rechallenged with osimertinib + crizotinib directly after failure under combination of 1 st generation EGFR-TKI and crizotinib due to emergence of T790M resistance mutation, producing a new objective response suggesting that tumor was still depending on this oncogenic pathway. However, central nervous system (CNS) was a frequent site of relapse under combination therapy probably due to the lower intracranial penetration of crizotinib; association of EGFR TKIs with selective MET-inhibitors with high CNS activity should be promising.

Studies evaluating the association of EGFR and different MET-inhibitors in treatment-naïve patients

or in the context of resistance are ongoing, showing promising results. A phase Ib/II study evaluated gefitinib associated with capmatinib in EGFR-mutated patients with acquired resistance to EGFR-TKI and MET-dysregulation defined as MET amplification (ratio≥2 and/or GCN≥5) and/or MET overexpression (IHC 2+ or 3+). Objective response rate (ORR) in the overall population was 27%.

Responses better correlated with GNC (ORR of 32% for IHC 3+ and 47% in patients with GCN≥6) [START_REF] Wu | Phase Ib/II Study of Capmatinib (INC280) Plus Gefitinib After Failure of Epidermal Growth Factor Receptor (EGFR) Inhibitor Therapy in Patients With EGFR-Mutated, MET Factor-Dysregulated Non-Small-Cell Lung Cancer[END_REF] .

Tepotinib, a potent selective MET-TKI demonstrated significant median progression free survival (PFS) and overall survival (OS) improvement (16.6 vs 4.2 months and 37.3 vs 13.1 months respectively) compared with chemotherapy in MET amplified (ratio≥2 GCN≥5) EGFR-mutated patients at acquired resistance to gefitinib. Three patients with MET amplification are still receiving treatment with tepotinib for more than 27 months [START_REF] Yang | NSCLC): Predefined subgroup analysis of a Phase Ib/II study[END_REF] . Savolitinib, another highly potent and selective MET-inhibitor, was also assessed in association with osimertinib in the TATTON Phase Ib study in patients with evidence of acquired resistance to EGFR TKI and MET-alteration defined as following: IHC 3+ on ≥50% of cells or FISH+ (ratio≥2 GCN≥5) or NGS (≥20% tumor cells, ≥200x sequencing depth of coverage and ≥5 copies over tumor ploidy). Patients enrolled in cohort B received osimertinib 80 mg and savolitinib 600 mg daily and were divided in 3 groups: patients previously treated with third generation TKI (B1); not previously treated with third generation TKI and T790M negative (B2) or T790M positive (B3). Patients in cohort D had the same characteristics than in cohort B2 but received savolitinib 300 mg daily. Median PFS were 7.6 and 9.1 months with ORR of 48 and 64% in cohorts B and D, respectively [START_REF] Sequist | Osimertinib plus savolitinib in patients with EGFR mutationpositive, MET-amplified, non-small-cell lung cancer after progression on EGFR tyrosine kinase inhibitors: interim results from a multicentre, open-label, phase 1b study[END_REF] . In these studies, the combination of a MET-inhibitor and an EGFR TKI was well tolerated with a low rate of additional toxicities. SAVANNAH (NCT03778229), a single arm phase II study of osimertinib combined with savolitinib for patients who experienced progression after prior osimertinib is ongoing.

Associating MET-inhibitors to the standard EGFR-inhibition in patients progressing with a METamplification seems to be a valid therapeutic option based on a strong preclinical rationale, showing minimal and acceptable overlapping toxicity and leading to deep and prolonged responses [START_REF] Wu | Phase Ib/II Study of Capmatinib (INC280) Plus Gefitinib After Failure of Epidermal Growth Factor Receptor (EGFR) Inhibitor Therapy in Patients With EGFR-Mutated, MET Factor-Dysregulated Non-Small-Cell Lung Cancer[END_REF][START_REF] Lara | Preclinical Evaluation of MET Inhibitor INC-280 With or Without the Epidermal Growth Factor Receptor Inhibitor Erlotinib in Non-Small-Cell Lung Cancer[END_REF][START_REF] Scagliotti | Phase III Multinational, Randomized, Double-Blind, Placebo-Controlled Study of Tivantinib (ARQ 197) Plus Erlotinib Versus Erlotinib Alone in Previously Treated Patients With Locally Advanced or Metastatic Nonsquamous Non-Small-Cell Lung Cancer[END_REF] . Low CNS crizotinib penetration is a clear limitation and new potent MET-inhibitors with a better CNS penetration are needed [START_REF] Solomon | Intracranial Efficacy of Crizotinib Versus Chemotherapy in Patients With Advanced ALK-Positive Non-Small-Cell Lung Cancer: Results From PROFILE 1014[END_REF][START_REF] Costa | CSF concentration of the anaplastic lymphoma kinase inhibitor crizotinib[END_REF] . This underscores the need for rebiopsy even after acquired resistance to osimertinib, notably when given in frontline, as MET inhibition in addition to EGFR inhibition can achieve durable disease control.

DISCLOSURE

The authors Aubanel M., Doublet L., Neidhardt EM., declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

The authors Swalduz A., Perol M., Avrillon V., Mastroianni B., declare the following financial interests/personal relationships which may be considered as potential competing interests:

• Swalduz A.: advisory boards for Roche, Bristol-Myers Squibb, Takeda, Lilly, Pfizer and lectures for Roche, AstraZeneca, Boehringer-Ingelheim, Bristol-Myers Squibb, Pfizer, Takeda • Perol M.: advisory boards for Roche, Genentech, Eli Lilly, Pfizer, Boehringer-Ingelheim, Clovis Oncology, MSD, Bristol-Myers Squibb, Novartis, Pierre Fabre, AstraZeneca, Takeda.

Institutional grants for Roche, AstraZeneca, Chugai, Takeda and lectures for Eli Lilly, Roche, AstraZeneca, Pfizer, Amgen, Boehringer-Ingelheim, Bristol-Myers Squibb, Takeda, Chugai

  Abbreviations : CEP7= centromeric portion of chromosome 7 ; CN = copie number ; CNS = central nervous system ; CNG = copie number gain ; IHC = immunohistochemistry; MET = Mesenchymal-Epithelial Transition factor ; NGS = nextgeneration sequencing ; RNAseq = RNA sequencing ; TKI = tyrosine kinase inhibitor ; ORR = objective reponse rate

  (carboplatin-paclitaxel-bevacizumab, maintenance with pemetrexed, then docetaxel 4 cycles) on a retrospective analysis of the previous supra-clavicular biopsy. Combination of erlotinib 75mg once daily and crizotinib was initiated leading to an almost complete response (Figure B.1). After almost one year of combination treatment, PET-scanner showed evidence of oligo-progression on the primary tumor without recurrence of the disease on other sites (bone metastases, mediastinal and supraclavicular nodes) leading to propose local stereotactic radiotherapy with continuation of combination therapy. Five months later, she experienced progression disease with brain and liver metastases.ctDNA revealed T790M mutation and erlotinib was replaced by osimertinib with continuation of crizotinib. PET-scanner and brain MRI performed 1 month later showed complete response. Treatment is ongoing. Patient #4 was diagnosed with stage IVa lung adenocarcinoma depending on EGFR exon 19 deletion and was treated by gefitinib 250 mg once daily. After 17 months of treatment, she experienced cerebral and lung disease progression leading to give chemotherapy with cisplatin-pemetrexedbevacizumab followed with pemetrexed-bevacizumab as maintenance treatment. An almost complete response was observed on brain metastases. After 18 months, she experienced a subsequent thoracic disease progression leading to gefitinib reintroduction then replaced by osimertinib based on ctDNA revealing a T790M mutation. Lung and mediastinal PD without brain disease progression occurred after 21 months of osimertinib with evidence of MET-amplification (CN: 12; ratio MET/centromere: 4.1) on lung rebiopsy. After failure of subsequent chemotherapy and immunotherapy with nivolumab, she received a combination of crizotinib and osimertinib leading to almost complete response after 1 month. After 7 months of treatment, disease progression (new brain metastases, lung carcinomatous lymphangitis and meningitis) was observed and patient died one month later. Patient #5 was treated with gefitinib for a lung adenocarcinoma revealed by a single bone metastasis, with evidence of EGFR exon 19 deletion. She experienced bone and lung disease progression after 30 months of treatment with detection of a T790M mutation on bone biopsy. As osimertinib was not available yet, she received cisplatin-pemetrexed-bevacizumab and continuation chemotherapy with pemetrexed during 3 years. After rechallenge with gefitinib for a lung and mediastinal lymph nodes symptomatic progression, the patient subsequently received osimertinib with disease stabilization as best response and mediastinal lymph node progression at 4 months. Mediastinal lymph node cytology revealed exon 19 deletion, disappearance of T790M mutation but MET-amplification (CN: 8.1; MET/centromere ratio: 2.1) leading to addition of crizotinib to osimertinib with almost complete response lasting one year (Figure B.2). Systematic brain MRI then revealed two asymptomatic brain metastases without systemic disease progression. She was treated with stereotactic brain radiotherapy and combination therapy is ongoing.

•

  Avrillon V.: congress and boards financing MSD, Bristol-Myers Squibb, Roche, AstraZeneca• Mastroianni B.: Roche, AstrazenecaFUNDINGSThis research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Figure A :

 A Figure A: Treatment sequence and patient's outcomes Arrows indicate patients who had an ongoing response at the time of the last evaluation. The symbol † indicates that patient is dead. The symbol indicates PR, * indicates brain PD and brain radiotherapy and ** thoracic PD and thoracic radiotherapy