Véronique Fonteille
email: veronique.fonteille@inserm.fr

Jérôme Redouté
email: redoute@cermep.fr

Pierre Lamothe
email: pierre.lamothe@ch-le-vinatier.fr

Dominique Straub
email: dominique.straub@ch-roanne.fr

Frank Lavenne
email: lavenne@cermep.fr

Didier Le Bars
email: lebars@cermep.fr

Véronique Raverot
email: veronique.raverot@chu-lyon.fr

Virginie Moulier
email: vmoulier@gmail.com

Jean-Jacques Marchand
email: jeanjacquesmarchand@yahoo.fr

Aurélie Vittoz
email: aurelie.vittoz@ch-le-vinatier.fr

Charlotte Leriche
email: chleriche@yahoo.fr

Michel Pugeat
email: michel.pugeat@chu-lyon.fr

Serge Stoléru
email: serge.stoleru@inserm.fr

Brain processing of pictures of children in men with pedophilic disorder: A positron emission tomography study

Keywords: pedophilic disorder, Positron Emission Tomography, visual sexual stimuli, brain, temporal cortex. Abbreviations. Ch: children, Ad: adults, DSA: Desire for Sexual Activity, PE: Perceived Erection, SA: sexual arousal, VSSc: visual sexual stimuli depicting children, VSSa: visual sexual stimuli depicting adults

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

Introduction

In their review of the rates of child sexual abuse (CSA), Barth and colleagues (2012) concluded that the prevalence of CSA in children and adolescents under 18 years of age ranged from 8 to 31% for females and from 3 to 17% for males. Several studies of the consequences of CSA on the victims' mental health show that a history of CSA is associated with a substantially higher prevalence of various psychiatric disorders in adulthood [START_REF] Kendler | Childhood sexual abuse and adult psychiatric and substance use disorders in women. An epidemiological and cotwin control analysis[END_REF][START_REF] Nelson | Association between self-reported childhood sexual abuse and adverse psychosocial outcomes: results from a twin study[END_REF][START_REF] Pérez-Fuentes | Prevalence and correlates of child sexual abuse: a national study[END_REF]. It is, therefore, justified to consider CSA as a public health problem.

Importantly, not all men with histories of sexual offenses against children meet the diagnostic criteria for pedophilic sexual disorder. As pointed out by [START_REF] Mohnke | Brain alterations in paedophilia: A critical review[END_REF], failure to distinguish child sexual offending from pedophilic sexual interest may have biased earlier studies of the neural correlates of pedophilia. In addition, recent studies have shown that clinical characteristics as well as both structural and functional neuroimaging findings differed across offending and non-offending patients with pedophilia [START_REF] Gerwinn | Clinical characteristics associated with paedophilia and child sex offending-Differentiating sexual preference from offence status[END_REF][START_REF] Kärgel | Evidence for superior neurobiological and behavioral inhibitory control abilities in non-offending as compared to offending pedophiles[END_REF][START_REF] Schiffer | Gray matter anomalies in pedophiles with and without a history of child sexual offending[END_REF].

At least half of the perpetrators of CSA do meet the diagnostic criteria of pedophilia [START_REF] Seto | Pedophilia and sexual offending against children: Theory, assessment, and intervention[END_REF].

Although its causes are poorly known, pedophilic orientation has repeatedly been found associated with a history of CSA [START_REF] Seto | Pedophilia and sexual offending against children: Theory, assessment, and intervention[END_REF]. In addition, evidence supports a neurodevelopmental etiology, at least for pedophilic child sexual offending. Compared with teleiophilic controls, men with pedophilia with a history of child sexual offenses are more likely to report head injuries before the age of 13 [START_REF] Blanchard | Selfreported head injuries before and after age 13 in pedophilic and nonpedophilic men referred for clinical assessment[END_REF], obtain lower IQ scores on average [START_REF] Cantor | Intelligence, memory, and handedness in pedophilia[END_REF][START_REF] Cantor | Quantitative reanalysis of aggregate data on IQ in sexual offenders[END_REF], show increased rate of non-right-handedness [START_REF] Cantor | Intelligence, memory, and handedness in pedophilia[END_REF] and alterations in fronto-executive and/or temporal verbal skills [START_REF] Fabian | Neuropsychology, neuroscience, volitional impairment and sexually violent predators: A review of the literature and the law and their application to civil commitment proceedings[END_REF][START_REF] Joyal | The neuropsychology and neurology of sexual deviance: A review and pilot study[END_REF].

Given the major role of the brain in sexual arousal (SA) [START_REF] Stoléru | Functional neuroimaging studies of sexual arousal and orgasm in healthy men and women: a review and meta-analysis[END_REF], several recent studies have investigated the possible implication of structural and/or functional brain abnormalities in the pathophysiology of pedophilic disorder. The identification of specific dysfunctions of brain areas or brain networks might lead to novel approaches to the prevention of first sexual offences and of recidivism.

Several studies using voxel-based morphometry have reported widely diverging results. As reviewed by [START_REF] Mohnke | Brain alterations in paedophilia: A critical review[END_REF], the only replicated finding is a reduction in amygdalar volume [START_REF] Poeppl | Association between brain structure and phenotypic characteristics in pedophilia[END_REF][START_REF] Schiltz | Brain pathology in pedophilic offenders: evidence of volume reduction in the right amygdala and related diencephalic structures[END_REF]. Although there may not be any consistent structural brain abnormality in pedophilia, sexual attraction to children may have functional neural correlates. In seven functional magnetic resonance imaging (fMRI) studies, brain responses to erotic pictures were compared across patients and teleiophilic controls (for a review, [START_REF] Mohnke | Brain alterations in paedophilia: A critical review[END_REF]. Within-group analyses have shown that regions activated in men with pedophilic disorder in response to VSSc were similar to those activated in teleiophiles in response to VSSa [START_REF] Poeppl | Functional cortical and subcortical abnormalities in pedophilia: a combined study using a choice reaction time task and fMRI[END_REF]Schiffer et al., 2008aSchiffer et al., , 2008b)). A preliminary meta-analysis [START_REF] Polisois-Keating | Functional neuroimaging of sexual arousal: a preliminary metaanalysis comparing pedophilic to non-pedophilic men[END_REF] confirmed this result by demonstrating robust activations in both groups within the fusiform gyrus, occipital cortex, cerebellum, anterior cingulate cortex, and substantia nigra in response to the subjects' preferred sexual stimuli. Activated regions were distributed over the four components of the neurophenomenological model of SA, i.e., the cognitive, motivational, emotional and bodily components [START_REF] Stoléru | Functional neuroimaging studies of sexual arousal and orgasm in healthy men and women: a review and meta-analysis[END_REF].

The second category of analyses has directly compared patients and controls (between-groups analyses). In response to pictures representing children, the right medial frontal gyrus [START_REF] Poeppl | Functional cortical and subcortical abnormalities in pedophilia: a combined study using a choice reaction time task and fMRI[END_REF]Schiffer et al., 2008a), the hippocampus, and the thalamus [START_REF] Poeppl | Functional cortical and subcortical abnormalities in pedophilia: a combined study using a choice reaction time task and fMRI[END_REF]Schiffer et al., 2008b) were repeatedly found more strongly activated in men with pedophilic disorder than in teleiophiles. However, locations of maximal activation in the activated clusters were divergent across studies, except for the hippocampus. For some other regions, as reviewed by [START_REF] Mohnke | Brain alterations in paedophilia: A critical review[END_REF], diverging results were reported; for instance, Schiffer et al. (2008b) found a greater anterior cingulate cortex activation in response to images of children in patients than in teleiophiles, a result not replicated by [START_REF] Poeppl | Functional cortical and subcortical abnormalities in pedophilia: a combined study using a choice reaction time task and fMRI[END_REF].

In a study comparing patients attracted to boys with heterosexual healthy controls, patients exhibited more activation in the right amygdala in response to pictures of both boys and girls wearing swimsuits or underwear, with the highest activation observed for boys [START_REF] Sartorius | Abnormal amygdala activation profile in pedophilia[END_REF].

In response to VSSa, the findings in teleiophiles of a higher activation in the insula [START_REF] Poeppl | Functional cortical and subcortical abnormalities in pedophilia: a combined study using a choice reaction time task and fMRI[END_REF][START_REF] Walter | Pedophilia is linked to reduced activation in hypothalamus and lateral prefrontal cortex during visual erotic stimulation[END_REF] and in the right cuneus (Schiffer, 2008b;[START_REF] Walter | Pedophilia is linked to reduced activation in hypothalamus and lateral prefrontal cortex during visual erotic stimulation[END_REF] were replicated.

Finally, [START_REF] Ponseti | Assessment of pedophilia using hemodynamic brain response to sexual stimuli[END_REF] showed that it was possible to diagnose pedophilic attraction on the sole basis of fMRI responses to child versus adult stimuli. They found preference-specific activity in areas involved in processing sexually arousing stimuli [START_REF] Stoléru | Functional neuroimaging studies of sexual arousal and orgasm in healthy men and women: a review and meta-analysis[END_REF], such as the caudate nucleus, cingulate cortex, insula, and fusiform gyrus.

Reviews of the literature have suggested some methodological improvements, including larger sample sizes, control for the recruitment setting (in-patient vs. out-patient status), stricter inclusion/exclusion criteria to better limit potential confounds, control of sexual orientation (gender orientation and/or exclusivity of pedophilic preference), and validation of stimulus sets [START_REF] Mohnke | Brain alterations in paedophilia: A critical review[END_REF][START_REF] Tenbergen | The neurobiology and psychology of pedophilia: Recent advances and challenges[END_REF]. In the present study, we incorporated several of these recommendations. In addition, we used penile plethysmography as an objective index of SA in response to visual sexual stimuli (VSS).

Our objective was to study male outpatients with pedophilic disorder in order (i) to identify through Positron Emission Tomography (PET) their brain responses to validated VSS, and (ii) to compare them with healthy controls matched on sexual orientation (to female or male adults), age, and handedness.

In the neurophenomenological model developed by [START_REF] Stoléru | Functional neuroimaging studies of sexual arousal and orgasm in healthy men and women: a review and meta-analysis[END_REF], the cognitive component of SA comprises a process of appraisal through which specific stimuli, e.g., adults and/or children, males and/or females, are categorized as sexual incentives and quantitatively evaluated as such. Our main hypothesis was that brain activation in men with pedophilic disorder in response to VSSc would comprise regions mediating the cognitive component of SA, including the lateral part of the right orbitofrontal cortex and the caudal part of the fusiform gyri (for a review, [START_REF] Stoléru | Functional neuroimaging studies of sexual arousal and orgasm in healthy men and women: a review and meta-analysis[END_REF]. Conversely, we hypothesized that VSSc would elicit lower, or no, activation in these regions in controls. In addition, as our proposed model of SA posits that the activation of the right lateral orbitofrontal cortex is the starting point of "downstream" components, we hypothesized that a higher activation would be found in patients than in controls in the insula bilaterally (emotional component), the left anterior cingulate gyrus (motivational component), and the hypothalamus (bodily component).

Methods

Participants

Fifteen patients were recruited through centers specialized in the treatment of persons with sexual offense histories and of patients with paraphilias. The latter included patients who had not committed CSA, but used child pornography or remained totally abstinent. Fifteen controls were recruited through a web platform dedicated to informing potential participants about current investigations in psychology and neuroscience.

Inclusion criteria common to the two groups were male gender and age between 18 and 65. Specific inclusion criteria for patients were: (i) to meet the criteria for pedophilia according to the DSM-IV-TR (American Psychiatric Association, 2000) and/or according to the International Classification of Diseases (World Health Organization,1992); the latter classification allows for the inclusion of patients attracted to children at the early stage of puberty; patients who had not committed CSA could be included as they met the second clause of the second criterion of the DSM IV-R, "The person has acted on these urges, or the sexual urges or fantasies cause marked distress or interpersonal difficulty" and/or the corresponding clause of the ICD-10 definition, "Acts on the urges or is markedly distressed by them"; (ii) to acknowledge sexual attraction to prepubescent children; (iii) willingness to participate in treatment to prevent first, or further, sexual offenses. As a specific inclusion criterion, each control had to match an already recruited patient on age (± 13 years), on the Edinburgh Handedness Inventory score (± 50 points), and on sexual orientation (to adult women, men, or to women and men for controls matching patients attracted to girls, to boys, or to children of both genders, respectively).

Exclusion criteria common to both groups were: (i) IQ<70; (ii) brain disorders; and (iii) current detention. Specific exclusion criteria for patients were: (i) current treatment with antihormonal therapy or with antidepressant drug of the selective serotonin reuptake inhibitor family; (ii) endocrine disorder affecting the hypophysis, the hypothalamo-pituitary-gonadal axis or the adrenal glands. Specific exclusion criteria for controls were: (i) psychiatric disorders, including sexual and substance-related disorders; (ii) psychotropic medication;

(iii) drug treatment with potential sexual side-effects.

Protection of subjects' intimacy was ensured following guidelines [START_REF] Rosen | Concerns involving human subjects in sexual psychophysiology[END_REF]. Subjects received financial compensation for participating. The study was approved by the local ethics committee, and written informed consent was obtained after complete description of the study. All procedures performed in the study were in accordance with the ethical standards of the 1964 Helsinki Declaration and its later amendments.

Assessment of participants

A psychiatrist interviewed each patient to confirm the diagnosis of pedophilia and each control to assess potential psychiatric disorders or medication. A psychologist conducted a standardized psychological assessment with all participants. Handedness was assessed with the Edinburgh Handedness Inventory [START_REF] Oldfield | The assessment and analysis of handedness: The Edinburgh Inventory[END_REF], and IQ was evaluated with a seven-subtest short form of the Wechsler Adult Intelligence Scale-III [START_REF] Axelrod | Evaluation of seven-subtest short forms of the Wechsler Adult Intelligence Scale-III in a referred sample[END_REF]. Sexual function was assessed as follows in all participants, unless otherwise specified: (1) to further characterize the sexual orientation of all participants and identify potential interest in children in controls, all subjects were presented with standardized questions, redacted by our team, on the age and sex of persons to whom they felt attracted; [START_REF] Reynolds | Assessment of sexual function in depressed, impotent, and healthy men: Factor analysis of a Brief Sexual Function Questionnaire for men[END_REF] to assess the severity of pedophilia and identify potential comorbidity of several paraphilias [START_REF] Raymond | Psychiatric comorbidity in pedophilic sex offenders[END_REF], patients were presented with the Sex History Questionnaire Revised (SHQ-R; [START_REF] Langevin | Sex History Questionnaire for Males -Revised[END_REF]; [START_REF] Cantor | Quantitative reanalysis of aggregate data on IQ in sexual offenders[END_REF] to obtain a quantitative assessment of the current level of sexual interest, two instruments were used: (i) the questionnaire on Sexual Fantasies, Desires and Activity (SFDAQ; [START_REF] Stoléru | Développement du Questionnaire sur l'intensité des Fantasmes, des Désirs et des Comportements Sexuels chez les auteurs de violences sexuelles sur enfants[END_REF] which focuses on the last 30 days; while its first part assesses the levels of fantasies, of desires and of sexual activity whatever their targets (adults, adolescents and/or children), the second focuses on the levels of fantasies, of desires and of sexual activity exclusively directed to prepubescent children; Fantasies and Desires scores range theoretically from 1 to 18, Activities scores from 1 to 24, and the means from 1 to 20; (ii) the Sexual Interest Score (SIS, [START_REF] Bancroft | The control of deviant sexual behaviour by drugs: I. Behavioural changes following oestrogens and anti-androgens[END_REF], which does not specify the targets of sexual interest; (4) in order to assess sexual function directed to adults during the last month, all participants were presented with the Brief Sexual Function Questionnaire (BSFQ; [START_REF] Reynolds | Assessment of sexual function in depressed, impotent, and healthy men: Factor analysis of a Brief Sexual Function Questionnaire for men[END_REF]; (5) finally, to compare the controls with those of previously published studies, control subjects were presented with the Sexual Arousal Inventory (SAI; [START_REF] Hoon | An inventory for the measurement of female sexual arousability[END_REF], adapted to men.

Other aspects of psychopathology were assessed using the Symptom Checklist-90-Revised (SCL-90R; [START_REF] Derogatis | The SCL-90-R Manual. I: Scoring, administration and procedures for the SCL-90[END_REF], the 21-item Beck Depression Inventory version [START_REF] Beck | Depression Inventory. Center for Cognitive Therapy[END_REF], and the Barratt Impulsiveness Scale (BIS-10; [START_REF] Patton | Factor structure of the Barratt impulsiveness scale[END_REF].

A physician performed a medical examination and screened participants for exclusion criteria. A blood sample was obtained for hormonal investigation. Testosterone, sex-hormone binding globulin (SHBG), bioavailable testosterone (not bound to SHBG), and estradiol were measured by using previously reported methods [START_REF] Déchaud | Radioimmunoassay of testosterone not bound to sex-steroid-binding protein in plasma[END_REF] and commercially available kits were used for routine measurements of LH, FSH, prolactin, and cortisol.

The above-mentioned assessments were distributed over two to three sessions in patients and over one or two sessions in controls. On the following session, functional images were acquired. At the end of each PET scan, subjects were presented with a series of rating scales, ranging from 1 (null or extremely low) to 9 (extremely high), to assess seven kinds of subjective responses to each of the picture categories: desire for sexual activity (DSA), perceived erection (PE), beauty of presented characters, pleasure, displeasure, attention, tenderness. Participants indicated their answers through an audio system connected to the acquisition room. The functional imaging acquisitions were coupled with penile response measurements performed by using a volumetric penile plethysmograph (described in [START_REF] Mouras | Activation of mirror-neuron system by erotic video clips predicts degree of induced erection: An fMRI study[END_REF].

Positron Emission Tomography

Stimuli and design

Six categories of pictures were presented: (i) undressed, or lightly dressed, female children; (ii) undressed, or lightly dressed, male children; (iii) ordinarily dressed female children; (iv) ordinarily dressed male children; (v) undressed, or lightly dressed, adults of the participant's preferred gender; and (vi) ordinarily dressed adults of the preferred gender. Pictures of adults of the non-preferred gender were not presented for radiation exposure limitation considerations.

All pictures showed only one person. Most were obtained free of rights, or, for some pictures representing children, from a photographic agency. Most photographs of undressed or lightly dressed children showed children in swimming suits, with two back-view photographs showing a nude boy, and two back-view photographs a nude girl. Pictures of adult women were the same as those used in our previous studies (e.g., [START_REF] Mouras | Activation of mirror-neuron system by erotic video clips predicts degree of induced erection: An fMRI study[END_REF]. Pictures of children and those of adult males were selected out of a large series of photographs by 15 control subjects and 11 men with histories of sexual offenses against children, and not otherwise involved in the experiment. They rated the level of SA and of negative emotion induced by each picture on scales ranging from 1 to 9. Then, the sets of selected pictures of undressed or lightly dressed adults and children were matched on the average SA ratings, level of nudity and view of the portrayed person (front, back, profile or three-quarter front views). Pictures of dressed adults and children were also matched in the same manner, except for nudity degree. Pictures retained as stimuli did not induce negative emotions.

Subjects were scanned in six experimental conditions, each of which was defined by the category of presented pictures. For each condition, two scans were performed to ameliorate the signal/noise ratio (12 acquisitions per subject). Conditions were presented in a pseudorandom order across participants, as each patient and matched control had the same assigned order. Each condition comprised 9 photographs, each presented for 10 s. Pictures were displayed through a mirror and a screen disposed behind the scanner. For each scan, the following procedure was followed: presentation of three pictures (30 s) in order to induce SA; start of PET acquisition simultaneously with the presentation of six photographs (60 s). To ensure participants paid attention to the stimuli, they were requested to press a button of a keypad when a brief orange screen (0.5 s) appeared, once for each scan and at a random time. A ten-minute interval separated successive PET scans to allow for sufficient decay of radioactivity, for the collection of subjective ratings, and for the presentation of a two-minute emotionally neutral documentary to help participants regain their baseline emotional state.

Neuroimaging data acquisitions

Prior to functional acquisitions, a structural MRI examination was performed in order both to identify potential brain abnormalities and to record the anatomical volume used for PET data processing. Structural MRIs were acquired on a Siemens Magnetom Sonata 1.5 T scanner (Siemens, Erlangen, Germany), using a 3D anatomic T1-weighted sequence covering the whole brain, with the following parameters: 176 slices; slice thickness = 1mm; TE = 3.93 ms; TR = 1970 ms; 1mm x 1mm pixel size, 256 x 256 matrix).

PET data were acquired on a PET-CT tomograph (Siemens Biograph mCT/S 64, Erlangen, Germany).

Subjects were surrounded by a curtain isolating them from staff's view. Head position was maintained by a vacuum cushion. To check that anxiety levels were low before beginning data acquisition, a short version of the State-Trait Anxiety Inventory [START_REF] Van Knippenberg | Shortening the State-Trait Anxiety Inventory[END_REF] was presented. At the beginning of the session, a low-dose CT scan (<0.3 mSv) was carried out in order to estimate tissue attenuation. For each scan, the regional cerebral blood flow (rCBF) was measured by recording the radioactivity distribution following a bolus injection of 333 MBq (9 mCi) of [15O] H2O. Emission data were collected in 3D list-mode The first 60 s of each 90 s PET scan corresponding to visual stimulation were then reconstructed using the 3D OPSEM iterative algorithm, incorporating point spread function and time of flight, with a Gaussian filter of 3 mm and after correction for scatter and attenuation. Reconstructed volumes consisted in 109 contiguous slices (2.03-mm thickness) of 200 × 200 voxels in-plane (1.36 x 1.36 mm2). The actual resolutions for reconstructed images were approximately 2.6 mm in full width at half maximum in the axial direction and 3.1 mm in full width at half maximum in the transaxial direction measured for a source located 1 cm from the field of view [START_REF] Jakoby | Physical and clinical performance of the mCT time-of-flight PET/CT scanner[END_REF][START_REF] Martí-Climent | Contribution of time of flight and point spread function modeling to the performance characteristics of the PET/CT Biograph mCT scanner[END_REF].

PET data preprocessing

Statistical Parametric Mapping software (SPM12, Wellcome Department of Cognitive Neurology, London, UK) was used for image preprocessing and subsequent analysis. The PET rCBF volumes of each subject were realigned with the first acquired volume, in order to correct for subject's motion during acquisition. The T1 structural MRI image was then coregistered with PET volumes, using the mean functional image as a reference, and spatially normalized into the standard MNI space (Montreal Neurological Institute/International Consortium for Brain Mapping stereotactic space) using the segment function of SPM12. Spatial normalization parameters were applied to the realigned PET volumes, and finally, images were smoothed using an isotropic 8 mm full width at half maximum Gaussian kernel to reduce the variance due to inter-individual anatomical variability and to improve the signal/noise ratio in individual data.

Contrast analyses

Preprocessed PET volumes were analyzed with SPM using flexible factorial designs, defined by a Subject factor, a six-level Condition factor and, for between-groups comparisons, a Group (Patients vs. Controls) factor.

The models were adjusted using the Global Mean (ANCOVA).

In order to increase statistical power, we pooled into the same group heterosexual and homosexual patients on the one hand, and heterosexual and homosexual controls on the other hand. Pooling was conducted as heterosexual and homosexual patients did not show any differential responses to the pictures of their respective preferred gender [(Girlsundressed -Girlsdressed)heterosexual patients -(Boysundressed -Boysdressed)homosexual patients, and opposite contrast].

Contrasts are listed in table 1. Regional responses to the Dressing style factor (Dressed versus Undressed) were analyzed on a voxel-wise basis for pictures showing prepubescent girls (i.e., Girlsresponse = Girlsundressed -Girlsdressed), prepubescent boys, as well as adult females or adult males. All contrasts were based on stimuli with the preferred gender.

As planned before acquiring the data, we compared patients and controls on both their behavioral and brain responses (1) to pictures of children (Ch) of the patients' preferred gender; [START_REF] Reynolds | Assessment of sexual function in depressed, impotent, and healthy men: Factor analysis of a Brief Sexual Function Questionnaire for men[END_REF] VSSa and then compared patients with controls on the obtained differences. ( 6) Finally, in an exploratory analysis, we compared patients who had committed child sexual offenses to patients with no histories of child sexual offenses on their responses to the nudity of preferred children.

To further compare brain responses across groups, contrasts obtained in one group were used as masks for contrasts obtained in the other group. Exclusive masks reveal clusters in one group that do not overlap with those from the other group.

Correlational analyses

In order to study the correlation between psychological (DSA, PE, beauty of presented characters, pleasure, displeasure, attention, tenderness) or physiological (induced erection) responses of subjects and the PET signal, we conducted regression analyses by introducing each of these parameters as covariates in the SPM statistical model (Flexible factorial design, subject factor, 1 or 2-level group factor, 1 covariate, adjusted on global mean).

Regarding plethysmographic data, for each participant presenting an erectile response and each PET acquisition, the Area Under the Curve (AUC) was computed from the corresponding dataset, using as a baseline the mean plethysmographic value recorded during the first 15 seconds of visual stimuli presentation.

For all these regression analyses, within and between-groups analyses were performed, assessing both positive and negative correlations.

While correlational analyses between hormonal levels, in particular those of hormones of the pituitarygonadal axis, and brain activity would have been of great interest, we could not perform such exploration for the following reasons: (i) PET analyses are performed following a fixed-effects methodology (Group factor). As we acquired 12 scans per subject (2 repetitions per condition * 6 conditions), a random-effects approach, i.e., a second-level analysis as performed with fMRI data, is impossible because of lack of statistical power at the Brain responses in pedophilic disorder 11 individual level; (ii) exploring the correlation between any hormonal level and brain activity in such a fixedeffects design would require one hormone measurement for each PET scan entered in the model, i.e., 12 hormonal assays, but in our experiment, we performed one hormonal assay per subject.

PET data exploratory and hypothesis-driven analyses

Regarding statistical significance, inferences were made based on cluster-wise significance (p<0.05 FWEcorrected). Accordingly, although peak MNI coordinates and T-value are reported in the tables, it can only be inferred that there is signal somewhere within the cluster and no inferences can be made about the statistical significance of specific locations within the cluster [START_REF] Woo | Cluster-extent based thresholding in fMRI analyses: pitfalls and recommendations[END_REF]. To minimize the risk of false-positive (type I) errors [START_REF] Eklund | Cluster failure: why fMRI inferences for spatial extent have inflated false-positive rates[END_REF], the cluster-forming height threshold was set at P = 0.001, uncorrected, as recommended in [START_REF] Roiser | Minimum statistical standards for submissions to Neuroimage: Clinical[END_REF] and [START_REF] Woo | Cluster-extent based thresholding in fMRI analyses: pitfalls and recommendations[END_REF]. This also applied to results of masking procedures, with masks thresholded at p< 0.001, uncorrected. All analyses were conducted in the following two spaces.

Exploratory analyses. We performed traditional exploratory analyses in the whole brain.

Hypothesis-driven analyses. We also used a set of regions previously found to show activation or deactivation in response to VSS in healthy adult males in two meta-analyses [START_REF] Poeppl | The functional neuroanatomy of male psychosexual and physiosexual arousal: A quantitative meta-analysis[END_REF][START_REF] Stoléru | Functional neuroimaging studies of sexual arousal and orgasm in healthy men and women: a review and meta-analysis[END_REF]. Those regions were hypothesized to respond to images of children in pedophiles. For each region, we specified its shape, center, radius, and dimensions based on information available in the two meta-analyses [START_REF] Poeppl | The functional neuroanatomy of male psychosexual and physiosexual arousal: A quantitative meta-analysis[END_REF][START_REF] Stoléru | Functional neuroimaging studies of sexual arousal and orgasm in healthy men and women: a review and meta-analysis[END_REF] and thus defined a sphere or a box centered on a peak voxel. In order to apply SPM12 small-volume correction procedure, we used WFU PickAtlas software [START_REF] Maldjian | An automated method for neuroanatomic and cytoarchitectonic atlas-based interrogation of fMRI data sets[END_REF] to define two masks of activated and deactivated regions of interest, respectively. These masks were then used to apply small-volume corrections (SVC) on activation and deactivation results obtained in the whole brain analyses. The same procedure was used to define two masks and perform SVC analyses for brain regions positively or negatively correlated with indices of SA.

T-tests or chi-square tests were performed to compare patients and controls on sociodemographic, clinical and behavioral variables. Repeated measures analyses of variance of subjective ratings and of erectile responses were performed using SPSS® 19 software (Armonk, NY). Regarding erectile responses, the dependent measure was the mean area under the two curves obtained for each category of pictures (mean of the two acquisitions performed for each category of pictures).

Results

Sociodemographic and clinical characteristics

Sociodemographic and sexological characteristics of participants are presented in table 2. Ten patients had sexually offended children, while five had used child pornography or remained totally abstinent. Seven patients were attracted to prepubescent girls and five to prepubescent boys; these patients are called heterosexual and homosexual patients, respectively. Three were attracted to children of both genders; these patients were assigned to the subgroup corresponding to the children of their preferred gender, which was the homosexual subgroup for all three. Twelve patients were non-exclusive, i.e., were also attracted to adults. Six heterosexual patients and two bisexual patients were attracted to adult women; four homosexual patients were attracted to adult men; and three patients -one heterosexual, one homosexual and one bisexual -declared no sexual attraction to adults. Ten patients were attracted to children outside of the family (six with a history of child sexual offenses), one was incestuous (with a history of child sexual offenses), and four belonged to both categories (three with a history of child sexual offenses). Among controls, eight were heterosexual (here "heterosexual" refers to adult women), four were homosexual, and three bisexual. One bisexual control was predominantly homosexual and was presented with male VSSa, the two other bisexual controls predominantly heterosexual and presented with female VSSa. For the three bisexual controls the gender of their "preferred" children was male, in accordance with the orientation of their matched patients. It should be noted that one heterosexual control was matched with a bisexual patient (predominantly attracted to boys) who was more attracted to adult women than to adult men and was presented with female VSSa; in PET contrast analyses (see 3.3.1. below) where patients' brain responses to children of their preferred gender were compared to those of controls, the brain responses of this heterosexual control were those to male children, in accordance with the orientation of his matched patient.

On the BSFQ, patients had lower scores than controls on Sexual Activity, Sexual Satisfaction, and Physiological Component, while Sexual Interest did not differ significantly across groups. On the BDI and on the SCL-90R, patients had significantly higher scores than controls (see Inline Supplementary Table 1). Twelve patients were receiving some form of psychological treatment at the time of the study, mostly individual supportive psychotherapy, for their pedophilic disorder.

Patients and controls did not differ significantly on intelligence quotient (table 2 and see Inline Supplementary Table 1). Regarding hormonal measurements (see Inline Supplementary Table 2), the plasma levels of total testosterone and of bioavailable testosterone, i.e., testosterone unbound to the Sex-Hormone Binding Globulin, were lower in patients.

Measures of sexual arousal during PET scans

We first provide the results of the analyses of the behavioral indexes of SA, based on the same contrasts (table 1) as those subsequently reported for the neural responses.

3.2.1.

Ratings of Stimuli Here we detail the results of analyses focused on two rating scales, DSA and PE (table 3). Analyses related to the other scales are presented in Inline Supplementary Table 3.

For both DSA and PE, compared with controls patients showed a higher difference between ratings for undressed children and for normally dressed children (Contrast 1). This Group by Condition interaction is important as it represents the behavioral correlate of the C1 contrast tested on PET data.

The comparison of patients and controls on their ratings of the pictures of characters with both the age and the gender preferred by each group (Contrast 3) showed a Group by Condition interaction only for DSA, with a lower difference between the conditions in patients.

Finally, the comparison of patients who had committed child sexual offenses to patients with no history of child sexual offenses on their ratings of pictures of preferred children did not reveal any significant differences (Contrast 5).

Penile plethysmography

One homosexual patient could not be included in analyses due to technical problems affecting plethysmographic data collection. Results are presented in Inline Supplementary Table 4. The comparison of the two groups on their responses to pictures showing children of the patients' preferred gender showed a Group by Condition interaction, with a higher difference in patients between responses to images of undressed and dressed children. Finally, the comparison of patients who had committed child sexual offenses to those with no history of child sexual offenses on their responses to pictures of preferred children did not reveal any significant difference (Contrast 5).

PET Imaging

Contrast analyses

Contrast 1. In response to preferred VSSc, patients showed an activation in the right inferior temporal gyrus and in the left middle occipital gyrus (table 4; figure 1). After masking (exclusive mask) the previous contrast with activations observed in patients in response VSSa, the two clusters were no longer present, as they were also activated in response to images of nude adults. Thus, these activations were not specifically related to the age of characters displayed. By contrast, when the same contrast was masked (exclusively) by the controls' response to children's nudity, the two clusters persisted. In other words, these responses to images of children's nudity were characteristic of the patients' group. Finally, when Contrast 1 was masked (exclusively) by the controls' response to adults' nudity, the clusters were no longer apparent.

Regarding controls, analysis of contrast 1 showed an activation in the right middle occipital gyrus. In addition, an SVC analysis, limited to the space of regions that get deactivated in healthy subjects in response to VSSa, showed a cluster in the left fusiform gyrus (table 4). Thus, it is only in controls that a region considered as inhibitory in the neurophenomenological model was found activated in response to images of undressed children.

After masking (exclusive mask) the previous contrast with activations observed in controls in response to nude adults of the preferred gender, both clusters survived. Thus, these clusters of activation characterized the controls' responses to images of children's nudity, and not to adults' nudity. Similarly, when the contrast was masked exclusively by the patients' response to adults' nudity, both clusters survived. Finally, when the same contrast was masked exclusively by the patients' response to children's nudity, both clusters persisted. In other words, these controls' responses were both stimulus-specific and group-specific. Inline Supplementary Table 5 shows which regions showed both group-and stimulus-specific activations. Conversely, regarding "deactivations" (Dressed -Undressed contrast), in patients there was a deactivated cluster in the right lingual gyrus in response to pictures of undressed children (table 4).

The between-group analysis of the contrast [(Chresponse)Controls -(Chresponse)Patients] showed a cluster in the right inferior temporal gyrus (p<0.05, corrected) (figure 2 and table 4). This cluster in the right inferior temporal lobe lay in a distinctly more rostral position (BA 20) than the above-mentioned cluster found activated in patients (BA 37).

Contrast 2. Regarding the within-group analyses, in response to VSSa, patients showed a significant activation in the right and left middle occipital gyri (table 5). When this contrast was masked with activations of patients in response to VSSc (Chresponse), activation in the left middle occipital gyrus persisted. Thus, this cluster characterized the patients' responses to images of adults' nudity, and not to children's nudity. In addition, when the patients' response to adults' nudity was masked (exclusively) by the controls' response to adults' nudity, the bilateral activation in the middle occipital gyri persisted. Finally, when the patients' response to adults' nudity was masked (exclusively) by the controls' response to children's nudity, these two clusters survived. In other words, the response in the left middle occipital gyrus characterized the response of patients only (groupspecificity) to images of undressed adults only (stimulus-specificity).

Brain responses in pedophilic disorder 15

Conversely, in patients the analysis of the opposite contrast [Addressed -Adundressed] showed a deactivation in both inferior temporal gyri and in the left inferior frontal gyrus (table 5).

The between-groups analysis of the contrast (Adresponse)controls-(Adresponse)patients showed a cluster in the medial orbitofrontal cortex (table 5), with a peak in the olfactory sulcus.

Contrast 3. In response to their preferred stimuli, no cluster was more activated in patients than in controls.

The same was true for the opposite contrast.

Contrast 4. The analysis of the patients' responses to children's versus adults' nudity showed a cluster in the posterior cingulate cortex [x, y, z = 2, -56, 20; PFWE-corr. <0.05; k=145 voxels; t=4.63); it was related to a higher deactivation in response to VSSa than to VSSc.

Contrast 5. Analysis did not show any brain regions responding differentially to VSSc in patients who had committed child sexual offenses (n=10) compared to those with no history of child sexual offenses (n=5). The same was true for the comparisons between each subgroup of patients and the controls.

Correlational analyses

Ratings of pictures. In patients the ratings of DSA (whatever the age and gender of characters in pictures)

was positively correlated with the level of activation in widespread areas including (i) the right inferior temporal gyrus and the left inferior occipital gyrus, (ii) two clusters located bilaterally in the insula and claustrum, (iii) a cluster at the junction of the anterior insula and the orbital operculum, (iv) the pregenual anterior cingulate gyrus, (v) and a cluster in the cerebellar uvula (table 6; figure 3). The ratings of PE were positively correlated with the level of activation in similar areas, including the right inferior temporal gyrus (BA 37), as well as in the supplementary motor area of the right medial prefrontal gyrus. Conversely, ratings were negatively correlated with rCBF in the right inferior temporal gyrus (BA 20) for Desire for Sexual Activity and with rCBF in the middle temporal gyrus (BA 20) for Perception of Erection.

Correlations of DSA ratings with rCBF found in controls are presented in Inline Supplementary Table 6. The results of the correlational analysis of ratings of PE were similar to those for DSA.

The between-groups analysis showed that the ratings of DSA were differentially correlated across groups in the anterior cingulate gyrus (BA 24; x, y, z = -8, 10, 34; PFWE-corr.<0.05, k=195), with a higher correlation in patients; conversely, in the cerebellar pyramis (x, y, z = 14, -76, -34; PFWE-corr.<0.05, k=163) and in the right gyrus rectus (BA 11;x,y,z = 4,42,; PFWE-corr.<0.01, k=319), the correlation was lower in patients than in controls. Similarly, the between-groups analysis showed that the ratings of PE were differentially correlated across groups in the anterior cingulate gyrus (BA 32;x,y,6,44;k=889), and in the central sulcus (BA 3/BA 4; x, y, z =-52, -12, 46; PFWE-corr.<0.05, k=207), with a higher correlation in patients; conversely, in the cerebellar declive (x, y, z =16, -76, -32; PFWE-corr.<0.01, k=255), the correlation was higher in controls.

Penile plethysmography. Twelve patients and 10 controls presented penile responses and were included in the analyses. The regions where the rCBF showed a correlation with the variations of patients' penile volume are presented in Inline Supplementary Table 7. Positive correlation was found in the right inferior frontal gyrus, precentral and postcentral gyri, claustrum and putamen, midbrain as well as in the left supplementary motor area, and right anterior cingulate gyrus. By contrast, in the 10 controls with an erectile response, no region showed a correlation between rCBF and penile volume. The between-groups comparison of correlations showed that in patients there was a higher correlation between rCBF and penile volume in the left anterior cingulate cortex (x,y,14,40;k=312;corrected) and in the supplementary motor area bilaterally (x, y, z=0, -8, 58; k=150; PFWE-corr.<0.05, corrected).

Discussion

Visual stimuli induced the behavioral effects that we intended and whose correlates were investigated through PET analyses. For instance, compared with controls, patients showed a higher difference between DSA ratings for undressed and for normally dressed children. The same Group by Condition interaction was found for erectile responses to VSSc.

The lower plasma levels of testosterone in patients was unexpected. To our knowledge, decreased basal plasma testosterone has rarely been reported in pedophilia [START_REF] Shostakovich | A comparative evaluation of the biochemical and psychopathological characteristics in subjects with signs of pedophilia[END_REF]. Most studies focus on pharmacologically induced decreased testosterone as a means to control sexual desire, but do not compare its basal levels in patients and healthy controls (e.g., [START_REF] Rösler | Treatment of men with paraphilia with a long-acting analogue of gonadotropin-releasing hormone[END_REF]. Although lower than in controls, testosterone levels were not pathological. They could not be caused by antihormonal therapy as such treatment was a criterion for non-inclusion. In normal circumstances, aging males show a progressive decrease in testosterone levels with some increase in SHBG that contributes to lower the bioavailable testosterone, as reported by several groups, including ours [START_REF] Déchaud | Radioimmunoassay of testosterone not bound to sex-steroid-binding protein in plasma[END_REF]. However, age could not explain this finding as it was similar in both groups and the difference between groups remained significant both for total and for bioavailable testosterone when age was included as a covariate in the comparison.

Subtractive analyses

Activations found in patients in response to VSSc were located in lateral temporal and occipital areas that respond in healthy males to VSSa [START_REF] Stoléru | Functional neuroimaging studies of sexual arousal and orgasm in healthy men and women: a review and meta-analysis[END_REF], which was consistent with one of our hypotheses and with the patients' higher behavioral responses to pictures of undressed children than to pictures of dressed children (DSA and PE ratings, erectile responses; table 3 and Inline Supplementary Table 4). Observed activations are also consistent with some previous reports [START_REF] Poeppl | Functional cortical and subcortical abnormalities in pedophilia: a combined study using a choice reaction time task and fMRI[END_REF]Schiffer et al., 2008a and2008b).

However, by contrast with Poeppl et al.'s study (2011), we did not observe any significant activation in the right middle frontal gyrus, the posterior cingulate gyrus, the right hippocampus-amygdala complex, the right postcentral gyrus, the right precentral gyrus and the right calcarine sulcus. However, the comparability of analyses is limited as [START_REF] Poeppl | Functional cortical and subcortical abnormalities in pedophilia: a combined study using a choice reaction time task and fMRI[END_REF] used neutral scrambled images as reference conditions whereas we used images of dressed characters.

Unlike the patients, controls did not show any significant activation of lateral occipital and temporal areas in response to pictures of undressed children. This is consistent with the non-significant difference between their Brain responses in pedophilic disorder 18 ratings of pictures of undressed and normally dressed children (table 3). Activation was located at the occipital pole, right above the calcarine fissure. We speculate that the location of this activation in controls, i.e., near their primary visual cortex, could correspond to the processing of the low-level visual features distinguishing dressed from undressed children, whereas the more lateral location of activation in patients could correspond to their attribution of a sexually arousing character to images of undressed children. However, in contrast with our results, bilateral occipital activations have previously been reported in controls (Schiffer et al., 2008a).

The activation in the caudal part of the fusiform gyrus (BA 18) of controls in response to images of nude children was located in a region where healthy subjects show a deactivation in response to VSSa [START_REF] Redouté | Brain processing of visual sexual stimuli in human males[END_REF]; such deactivation in response to VSSa occurring in the context of unfolding SA has been understood as the release of an inhibition of SA. Hence, the activation of the fusiform gyrus in response to nude children could be a correlate of the ongoing inhibition of SA in controls.

As regards the between-groups analysis, the analysis of the [(Chresponse)Controls -(Chresponse)Patients] can theoretically be related to higher activations in controls than in patients and/or to higher deactivations in patients than in controls. In order to investigate these alternative mechanisms regarding the response in the right inferior temporal gyrus (BA 20), we examined the responses of this area in within-group analyses performed separately in controls and in patients. While no significant responses were observed in controls, a deactivation was found in patients (Undressed < Dressed), albeit at an uncorrected threshold (p<0.001, unc.). This maintained activation in the right inferior temporal gyrus (BA 20) of controls in response to images of nude children also occurred in a region that plays an inhibitory role on SA according to the neurophenomenological model [START_REF] Stoléru | Functional neuroimaging studies of sexual arousal and orgasm in healthy men and women: a review and meta-analysis[END_REF].

This part of the right inferior temporal gyrus (BA 20; y=-34) is distinct from the activated cluster found in patients in a more caudal location (BA 37;. According to the neurophenomenological model, while the more caudal part is positively related to SA, the more rostral part is inhibitory. The between-group difference is related to the fact that, whereas the level of activation in right BA 20 decreased in patients in response to undressed children (at an uncorrected threshold), it did not change significantly in controls. Thus, the betweengroup difference appeared clearly for an inhibitory region (no need for masking operations), while it was less obvious and needed masking procedures for regions that mediate SA: in other words, according to the neurophenomenological model, patients may be said to have "released the brake", but not the controls. This finding can be related to the fact that in supposedly nonpedophilic samples of male participants, the prevalence of sexual attraction to children has been reported to be non-negligible [START_REF] Briere | University males' sexual interest in children: Predicting potential indices of "pedophilia" in a non-forensic sample[END_REF]. We speculate that in the men of these samples who report attraction to children, the level of activation in right BA 20 in response to the sight of children may be weaker than in men who do not feel such attraction.

Surprisingly, we did not find any higher activations in patients than in controls in response to VSSc despite the significant Condition by Group interaction on ratings of DSA and of PE and on erectile responses. This is in line with results reported by [START_REF] Poeppl | Functional cortical and subcortical abnormalities in pedophilia: a combined study using a choice reaction time task and fMRI[END_REF] when, like in the present study, they used a corrected probability threshold. In heterosexual pedophiles, Schiffer et al. (2008b) reported a higher activation of the left hippocampus (PFDR-corrected <0.05, > 10 voxels) in response to nude female children, which we did not observe at the same probability threshold.

Regarding the patients' responses to VSSa, the bilateral activation in the right and left lateral occipital and temporal cortices is consistent with previous findings in heterosexual and/or homosexual patients (Schiffer et al., 2008a;Schiffer et al., 2008b). These regions are also activated in healthy males in response to VSSa [START_REF] Stoléru | Functional neuroimaging studies of sexual arousal and orgasm in healthy men and women: a review and meta-analysis[END_REF]. In the control group, these regions were activated in response to VSSa. Conversely, the "deactivation" observed in patients in the right and left inferior temporal gyri (BA 20 and BA 39, respectively), i.e., the higher activation in response to images of dressed than of undressed adults, is consistent with findings in healthy males who showed a negative correlation between markers of SA and activation in these areas [START_REF] Redouté | Brain processing of visual sexual stimuli in human males[END_REF].

These findings are likely to be related with the fact that most patients (80%) were non-exclusive, i.e., were also attracted to adults.

As regards the between-groups analysis of responses to adults' nudity (table 5), the analysis of the [(Adresponse)Controls -(Adresponse)Patients] can theoretically be related to higher activations in controls than in patients and/or to higher deactivations in patients than in controls. In order to investigate these alternative mechanisms regarding the response in the medial orbital gyrus (BA 11), we examined the responses of this area in withingroup analyses performed separately in controls and in patients. While no significant responses were observed in controls, a deactivation was found in patients (Undressed < Dressed), albeit at an uncorrected threshold (p<0.001, unc.). The lack of deactivation in controls stands in contrast to the neurophenomenological model [START_REF] Stoléru | Functional neuroimaging studies of sexual arousal and orgasm in healthy men and women: a review and meta-analysis[END_REF]. In contrast with [START_REF] Walter | Pedophilia is linked to reduced activation in hypothalamus and lateral prefrontal cortex during visual erotic stimulation[END_REF] and using the same threshold as in their study (p<0.005, uncorrected, k>10 voxels), we did not find a higher activation in healthy males in the dorsal midbrain, the left dorsolateral prefrontal cortex, the right parietal cortex, the right ventrolateral frontal cortex, the right occipital cortex and the left insula. However, the comparability of analyses is limited as [START_REF] Walter | Pedophilia is linked to reduced activation in hypothalamus and lateral prefrontal cortex during visual erotic stimulation[END_REF] compared groups on the contrast "sexual arousal > emotional arousal".

Brain responses in pedophilic disorder 20

Whereas [START_REF] Poeppl | Functional cortical and subcortical abnormalities in pedophilia: a combined study using a choice reaction time task and fMRI[END_REF] reported that several regions showed higher activation (using an uncorrected threshold) in men with pedophilia than in controls in response to their respective age-preferred VSS (Contrast 3),

we found no such region (also using uncorrected threshold). However, in [START_REF] Poeppl | Functional cortical and subcortical abnormalities in pedophilia: a combined study using a choice reaction time task and fMRI[END_REF] control subjects were men with nonsexual offense histories, which limits the comparability with our study. A meta-analysis comparing men with pedophilia to non-pedophilic men (mostly non clinical subjects) on their brain responses to their age-preferred VSS failed to reveal any significant difference in either direction (men with pedophilia > nonpedophilic men, or non-pedophilic men > men with pedophilia ) (Polisois-Keating and Joyal, 2013).

Correlational analyses

Several regions where rCBF was found positively correlated with the patients' erectile response have previously been identified as neural correlates of erection in healthy men [START_REF] Kühn | A quantitative meta-analysis on cue-induced male sexual arousal[END_REF][START_REF] Poeppl | The functional neuroanatomy of male psychosexual and physiosexual arousal: A quantitative meta-analysis[END_REF]. This is the case for the right and left anterior cingulate cortex, right putamen/claustrum, and the left insula. In addition, the cluster observed in the right inferior frontal gyrus is consistent with a similar finding reported by [START_REF] Mouras | Activation of mirror-neuron system by erotic video clips predicts degree of induced erection: An fMRI study[END_REF], also in a study of healthy men.

Such replication of results previously reported in healthy subjects was also observed for regions where activation was negatively correlated with penile tumescence, i.e., the bilateral middle temporal gyri, left inferior and superior temporal gyri, and left medial orbitofrontal cortex [START_REF] Redouté | Brain processing of visual sexual stimuli in human males[END_REF]. By contrast, and surprisingly, a negative correlation was also found in the middle part of the right orbitofrontal cortex, a region where rCBF was reported as positively correlated with penile volume in healthy subjects [START_REF] Moulier | Neuroanatomical correlates of penile erection evoked by photographic stimuli in human males[END_REF][START_REF] Mouras | Activation of mirror-neuron system by erotic video clips predicts degree of induced erection: An fMRI study[END_REF].

As regards ratings of stimuli, in patients the clusters showing a positive correlation between rCBF and ratings of DSA were located in the lateral occipital and temporal cortices as well as in the anterior cingulate gyrus. This is consistent with previous meta-analyses which found these regions positively related to SA in healthy men [START_REF] Poeppl | The functional neuroanatomy of male psychosexual and physiosexual arousal: A quantitative meta-analysis[END_REF][START_REF] Stoléru | Functional neuroimaging studies of sexual arousal and orgasm in healthy men and women: a review and meta-analysis[END_REF]. Similarly, the negative correlation of the rCBF in the right inferior temporal gyrus (BA 20) and in the left supramarginal gyrus (BA 40) is consistent with similar negative correlation found previously in healthy men in response to VSSa [START_REF] Poeppl | The functional neuroanatomy of male psychosexual and physiosexual arousal: A quantitative meta-analysis[END_REF][START_REF] Redouté | Brain processing of visual sexual stimuli in human males[END_REF][START_REF] Stoléru | Functional neuroimaging studies of sexual arousal and orgasm in healthy men and women: a review and meta-analysis[END_REF].

The rCBF in the left anterior cingulate gyrus was positively correlated with the three markers of the patients' SA, i.e., the erectile response, and ratings of both DSA and PE.

The scarcity of statistically significant between-groups differential responses to VSSc found in this study may be related to a true lack of differential responses or to methodological limitations. Regarding limitations, it Brain responses in pedophilic disorder 21 is possible that some patients felt embarrassment and even some guilt while watching pictures of undressed children. Twelve (80%) were engaged in psychotherapy and may have had ambivalent feelings during these presentations, including the fear of potential relapse because of sexual stimulation. In addition, it is also possible that the moderate level of SA induced by pictures of children in swimming suits (only four photographs presented naked children) limited the level of activation. While the level of DSA induced by their preferred stimuli in the undressed condition was similar in patients and controls, the difference between ratings induced by undressed and dressed characters was higher in controls, with a significant Condition by Group interaction. Here, the experimental need to induce strong SA in patients was conflicting with the ethical and clinical concerns to limit such arousal, especially because the outpatients recruited in the present study were left unaccompanied after the experimental sessions. In addition, although the two groups did not differ significantly on age, it should be noted that the age range used for matching was wide (± 13 years), due to the difficulty to match simultaneously on three variables (age, handedness, and sexual orientation). Finally, from a statistical point of view, the sizes of the two groups may be insufficient to demonstrate limited effect sizes.

Conclusion

Several lines of evidence implicate the right inferior temporal gyrus (BA 37) as a possible candidate area mediating SA of patients with pedophilic disorder in response to VSSc. Firstly, this region was more activated in response to pictures of undressed than of dressed children of the preferred gender. Secondly, in patients the rCBF in this area was positively correlated with the ratings of Desire for Sexual Activity and of Perceived Erection.

Thirdly, this area of the right inferior temporal cortex was activated by pictures of undressed children in patients but not in controls. This temporal area is distinct from another area, BA 20, also located in the right temporal lobe but more rostrally, which is thought to exert an inhibitory function on SA in healthy males and whose rCBF was (i) lower in patients than in controls and (ii) negatively correlated with the ratings of Desire for Sexual Activity and of Perceived Erection in the patients of the present study. Altogether, these results implicate the right inferior temporal gyrus as a possible candidate region mediating sexual arousal in patients with pedophilic disorder and suggest that two of its areas play opposite, i.e., activating and inhibitory, roles.

  to pictures of adults (Ad) of the patients' preferred gender; (3) to pictures of characters with both the preferred age and gender. Other analyses were performed post hoc. (4) We compared patients' responses to images of prepubescent children with their responses to images of adults: [(Chresponse)patients -(Adresponse)patients]. (5) In line with Ponseti et al. (2012), we subtracted in each group the responses to female (resp., male) VSSc from the responses to female (resp., male)

Figure 1 .Figure 2 .Figure 3 .

 123 Figure 1. Patients' brain activation in response to pictures of undressed children of their preferred gender in (a) the right inferior temporal (Brodmann area 37) and (b) the left middle occipital (Brodmann area 19) gyri. Note. Activated clusters (PFWE<0.05, corrected) are rendered on the normalized whole sample mean structural MRI in an axial section, 6 mm below the bicommissural plane; the anterior part of the brain is in the upper part of the figure and the right hemisphere is on the right. For display, height threshold: T=3.14; extent threshold: k=0. Rostral clusters failed to reach statistical significance threshold. [Color should be used for figure 1 in print] [Single column fitting image] Figure 2. Compared with controls, patients' Brodmann area 20 in the right inferior temporal gyrus shows a lower response to undressed children of their preferred gender. Note. The lesser-activated cluster (PFWE<0.05, corrected) is rendered on the normalized whole sample mean structural MRI in an axial section, 16 mm above the bicommissural plane; the anterior part of the brain is in the upper part of the figure and the right hemisphere is on the right. For display, height threshold: T=3.86, extent threshold: k=0. The rostral cluster failed to reach statistical significance threshold. The blue color, instead of the yellow, is used to show that activation was lower in patients than in controls. [Color should be used for figure 2 in print] [Single column fitting image]

Table 1 .

 1 Statistical contrasts tested both for behavioral variables and PET data Note. Ch: children; Ad: adults; "Response" refers to the effect of dressing style, i.e., undressed or lightly dressed minus ordinarily dressed. For every "(X-Y)" contrast, its opposite, "(Y-X)", was also tested.

	Variable	Patients n=15	Controls n=15	P
	Age	42.0 ± 12.6	41.2 ± 16.2	NS
	Years of education	16.6 ± 5.5	14.9 ± 4.1	NS
	Sexual partnership with adult (M/PS/PNS/NoP)	4/2/2/7	3/6/3/3	NS
	SHQ-R
	Female Child Frequency (heterosexual patients)	57.5 ± 27.5	NP	NA
	Male Child Frequency (homosexual patients)	65.5 ± 26.6	NP	NA
	SFDAQ
	Fantasies, any target	7.9 ± 6.6	4.3 ± 2.5	NS
	Desires, any target	15.5 ± 5.0	16.7 ± 3.1	NS
	Sexual activity, any target	9.1 ± 4.7	8.6 ± 1.5	NS
	Mean score, any target	10.8 ± 3.3	9.9 ± 1.7	NS
	Fantasies, children	5.5 ± 4.7	1.0 ± 0.0	<0.01
	Desires, children	10.9 ± 7.3	1.0 ± 0.0	<0.001
	Sexual activity, children	5.4 ± 3.3	1.0 ± 0.0	<0.001
	Mean score, children	7.3 ± 4.6	1.0 ± 0.0	<0.001
	Sexual Interest Score	3.0 ± 1.1	2.8 ± 0.6	NS
	BSFQ Factors
	Sexual Activity	40.4 ± 31.3	85.3 ± 28.5	<0.001
	Sexual Interest	37.8 ± 29.7	45.8 ± 27.2	NS
	Sexual Satisfaction	3.9 ± 2.7	7.5 ± 2.8	<0.01
	Physiological Component	12.7 ± 6.2	20.5 ± 6.7	<0.01

Table 2 :

 2 Sociodemographic and sexological characteristics of subjects Note. Figures are means ± SDs, except for sexual partnership where figures are frequencies; BSFQ: Brief Sexual Function Questionnaire; M: Married or living together; NA: not applicable; NP: not presented; NoP: no current adult partner; NS: Not significant; PNS: Partner, Not Stable; PS: lasting relationship, Same Partner, not living together; SFDAQ: Sexual Fantasies, Desires and Activities Questionnaire; SHQ-R: Sex History Questionnaire-Revised; figures are percentiles referring to a sample of sexual offenders gathered by the SHQ-R developers. "Female/Male Child Frequency" refer to the overall frequency of sexual activities of various types with children over the lifetime of the patients.

	Variable	Patients n=15	Controls n=15	Group effect	Condition effect	Group by Condition
	Desire for Sexual Activity
	Children, undressed Children, normally dressed	4.8 ± 1.9 2.2 ± 1.3	1.0 ± 0.1 1.0 ± 0.0	<0.01E-4 <0.01E-5	<0.01E-5
	Adults, undressed Adults, normally dressed	5.0 ± 2.5 1.2 ± 0.5	5.9 ± 1.7 1.8 ± 0.8	NS	<0.01E-7	NS
	Characters with preferred age, undressed Characters with preferred age, dressed	4.8 ± 1.9 2.2 ± 1.3	5.9 ± 1.7 1.8 ± 0.8	NS	<0.01E-9	<0.05
	Perception of Erection
	Children, undressed Children, normally dressed	3.1 ± 2.1 1.4 ± 1.0	1.0 ± 0.0 1.0 ± 0.0	<0.01	<0.001	<0.001
	Adults, undressed Adults, normally dressed	2.9 ± 2.1 1.0 ± 0.0	3.5 ± 2.0 1.2 ± 0.5	NS	<0.01E-3	NS
	Characters with preferred age, undressed Characters with preferred age, dressed	3.1 ± 2.1 1.4 ± 1.0	3.5 ± 2.0 1.2 ± 0.5	NS	<0.01E-3	NS

Table 3 .

 3 Comparison of patients and controls on Desire for Sexual Activity and Perceived Erection ratings of stimuli of their preferred gender Note. Figures are means ± SDs. Condition effect: Pictures of undressed vs. dressed characters.

Acknowledgments

We thank V. Berthier, F. Cazala, H. Dechaud, H. Leca, A. Maurin, E. Nischelwitzer, C. Orset, C. Pelletier, F. Vey, and C. Vighi for their help in data collection.

Brain responses in pedophilic disorder 23

Funding

This research project has been made possible by a grant from the Programme Hospitalier de Recherche Clinique (PHRC 2008 n° 27.42; Promoter: Hospices Civils de Lyon; Principal Investigator: Michel Pugeat).

Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any personal, professional or financial relationships that could be construed as a potential conflict of interest. Unless P is associated with "SVC" , Pcorr. refers to cluster-wise significance (cluster-forming threshold: 0.001 unc.;

cluster-extent threshold: FWEc=150 and FWEc=157 for positive and negative correlations, respectively). SVC+ , SVC-: Small Volume Correction, based on areas showing a positive or a negative correlation with markers of sexual arousal in healthy males. (SVC+) Region identified both within whole brain and with Small Volume Correction; for these regions, Pcorr refer to analysis in whole mask. k: Cluster extent; when no k-value is provided, the region belongs to the cluster indicated on the line above. Abbreviations: see Note of table 4.


Zaccharie Ramzi

Philippe Ciuciu

HYBRID LEARNING OF NON-CARTESIAN K-SPACE TRAJECTORY AND MR IMAGE RECONSTRUCTION NETWORKS

Compressed sensing (CS) in Magnetic resonance Imaging (MRI) essentially involves the optimization of 1) the sampling pattern in kspace under MR hardware constraints and 2) image reconstruction from the undersampled k-space data. Recently, deep learning methods have allowed the community to address both problems simultaneously, especially in the non-Cartesian acquisition setting. This paper aims to contribute to this field by tackling some major concerns in existing approaches. Regarding the learning of the sampling pattern, we perform ablation studies using parameter-free reconstructions like the density compensated (DCp) adjoint operator of the nonuniform fast Fourier transform (NUFFT) to ensure that the learned k-space trajectories actually sample the center of k-space densely. Additionally we optimize these trajectories by embedding a projected gradient descent algorithm over the hardware MR constraints. Later, we introduce a novel hybrid learning approach that operates across multiple resolutions to jointly optimize the reconstruction network and the k-space trajectory and present improved image reconstruction quality at 20-fold acceleration factor on T1 and T2-weighted images on the fastMRI dataset with SSIM scores of nearly 0.92-0.95 in our retrospective studies.

INTRODUCTION

Speeding up magnetic resonance imaging (MRI) acquisitions involves jointly optimizing how the k-space is undersampled and how image quality is preserved (or aliasing artifacts are discarded) during image reconstruction. Based on the CS theory, the k-space has to be undersampled according to a variable density sampling (VDS) [START_REF] Puy | On variable density compressive sampling[END_REF][START_REF] Chauffert | Variable density compressed sensing in mri. theoretical vs heuristic sampling strategies[END_REF][START_REF]Variable density sampling with continuous trajectories[END_REF][START_REF] Adcock | Breaking the coherence barrier: asymptotic incoherence and asymptotic sparsity in compressed sensing[END_REF][START_REF] Boyer | Compressed sensing with structured sparsity and structured acquisition[END_REF], whose shape depends on the underlying anatomy. In 2D imaging, 2D VDS patterns can only be efficiently achieved by using non-Cartesian sampling. Recently, SPARKLING [START_REF] Lazarus | SPARKLING: variable-density k-space filling curves for accelerated T2*-weighted MRI[END_REF][START_REF]3D variable-density SPARKLING trajectories for highresolution T2*-weighted magnetic resonance imaging[END_REF][START_REF] Chaithya | Optimizing full 3D SPARKLING trajectories for high-resolution T2*-weighted Magnetic Resonance imaging[END_REF] was introduced as an iterative procedure to optimize a non-Cartesian k-space sampling pattern according to a prescribed target sampling density (TSD) with each k-space trajectory being compliant with MR hardware constraints (particularly maximum gradient Gmax and slew rate Smax). Further, the algorithm results in locally uniform sampling patterns and thus avoids holes and clusters in k-space. However, a major limitation of SPARKLING is the need for a TSD as an input in the optimization process. To address this issue, the TSD was learned in a deep learning setting in [START_REF]Learning the sampling density in 2D SPARKLING MRI acquisition for optimized image reconstruction[END_REF] using LOUPE [START_REF] Bahadir | Deep-learning-based optimization of the under-sampling pattern in MRI[END_REF] as an acquisition model. While this method provided adequate reconstruction performances, there was still a mismatch in the learning process. Indeed, the gridded TSD was learned in the Cartesian domain, while the actual trajectory being optimized was non-Cartesian. As this could lead to suboptimal results, there is a need to jointly learn both the TSD and the reconstruction network in a non-Cartesian setting.

More recent methods [START_REF] Weiss | PILOT: Physics-informed learned optimal trajectories for accelerated MRI[END_REF][START_REF] Wang | B-spline parameterized joint optimization of reconstruction and k-space trajectories (BJORK) for accelerated 2d MRI[END_REF][START_REF] Vedula | 3D FLAT: Feasible Learned Acquisition Trajectories for Accelerated MRI[END_REF] bypass the need for estimating a TSD, by jointly learning the trajectories and the reconstruction network in a data-driven manner. In [START_REF] Weiss | PILOT: Physics-informed learned optimal trajectories for accelerated MRI[END_REF][START_REF] Vedula | 3D FLAT: Feasible Learned Acquisition Trajectories for Accelerated MRI[END_REF], the authors use a gridding step followed by a U-net to reconstruct the MR images and jointly learn the network parameters and k-space sampling trajectory. This method, however, relies on auto-differentiation of the NUFFT operator, which may not be very accurate (see [START_REF] Guanhua | Efficient NUFFT Backpropagation for Stochastic Sampling Optimization in MRI[END_REF]), thus resulting in suboptimal local minima. This was actually observed as the final learned trajectories only slightly deviated from their initialization. In [START_REF] Wang | B-spline parameterized joint optimization of reconstruction and k-space trajectories (BJORK) for accelerated 2d MRI[END_REF], the authors use [START_REF] Guanhua | Efficient NUFFT Backpropagation for Stochastic Sampling Optimization in MRI[END_REF] to obtain a more accurate Jacobian approximation of the NUFFT operator. Both above referenced approaches [START_REF] Weiss | PILOT: Physics-informed learned optimal trajectories for accelerated MRI[END_REF][START_REF] Wang | B-spline parameterized joint optimization of reconstruction and k-space trajectories (BJORK) for accelerated 2d MRI[END_REF] enforce the hardware constraints with the help of a penalty on the cost function. Although a viable option, this does not guarantee that the optimized trajectories will meet these constraints. Additionally this requires tuning the hyperparameter associated with this additional penalty in the cost function. Further, this penalty could affect the overall gradients, thereby resulting in suboptimality. To overcome these issues, the authors in [START_REF] Wang | B-spline parameterized joint optimization of reconstruction and k-space trajectories (BJORK) for accelerated 2d MRI[END_REF] parameterized the trajectory with B-spline curves, which reduce the number of inequality constraints and hence the influence of this penalty. However, such parameterization could severely limit the trajectories and prevent them from better exploring the k-space. Finally, both methods do not make use of density compensation (DCp) which plays a crucial role in obtaining cleaner images in the non-Cartesian deep learning setting [START_REF] Ramzi | NC-PDNet: a Density-Compensated Unrolled Network for 2D and 3D non-Cartesian MRI Reconstruction[END_REF].

Given these results, in this work, we first present a generic model for jointly optimizing the non-Cartesian trajectories and the corresponding reconstruction network. More precisely, we introduce a method to learn the trajectories in a data-driven manner while embedding a projected gradient descent algorithm [START_REF] Chauffert | A projection algorithm for gradient waveforms design in MRI[END_REF] to fulfill the hardware constraints. Second, we show experimentally that a novel generalized hybrid learning scheme performs better compared to a joint or alternated learning of the trajectory and the reconstruction network. Further we compare the trajectories obtained under different reconstruction settings (DCp Adjoint, DCp Adjoint + U-Net and NC-PDNet) and evaluate their performances in terms of image quality metrics (SSIM [START_REF] Wang | Image quality assessment: from error visibility to structural similarity[END_REF], PSNR) in a retrospective evaluation framework.

GENERIC MODEL

In this section we provide details (see Fig. 1) about the different components of the general model that is used for optimizing the non-Cartesian sampling trajectories as well as the image reconstruction network.

Fig. 1: A generic learning-based framework for joint optimization of the MRI acquisition and reconstruction models.

Acquisition

Data: In this work, we relied on the brain k-space data from the fastMRI dataset [START_REF] Zbontar | fastMRI: An open dataset and benchmarks for accelerated MRI[END_REF] for learning the k-space trajectories. We applied the virtual coil method [START_REF] Parker | Phase reconstruction from multiple coil data using a virtual reference coil[END_REF] to combine per-channel images to obtain a complex-valued single channel image, which was used in our training stage. This was done to account for the phase accrual during k-space data acquisition and make the learned trajectories more realistic. Following the formulation developed in [START_REF] Chaithya | Optimizing full 3D SPARKLING trajectories for high-resolution T2*-weighted Magnetic Resonance imaging[END_REF], we denote an MR image x ∈ C N ×N , over a field of view F × F. Then the 2D k-space of the image is defined in [-Kmax, Kmax] 2 , with Kmax = N 2F . In all our trajectories, we kept N = 320 and F = 0.23 m. For the sake of simplicity, we normalize the k-space to

Ω = [-1, 1] 2 .
Trajectory Specification: In our work, we directly learn the kspace trajectory locations unlike BJORK [START_REF] Wang | B-spline parameterized joint optimization of reconstruction and k-space trajectories (BJORK) for accelerated 2d MRI[END_REF] where the coefficients of B-spline curves were learned. We believe that such parameterizations could potentially limit the exploration in the k-space. Instead we optimize the 2D k-space sampling pattern K which is composed of Nc shots, K = (ki) Nc i=1 . Each shot can be played by the scanner hardware at the pace of gradient raster time ∆t, throughout the readout time T obs , resulting in Ns = T obs ∆t samples per shot. Thus the overall sampling pattern is K ∈ Ω Nc×Ns . Throughout this work, we used Ns = 512 (T obs ≈ 5ms, for T1-w and T2-w imaging) and Nc = 16, leading to an acceleration factor AF = N Nc = 20. Forward Model: For the sake of simplicity, here we model the MR acquisition process using the NUFFT modeled by F K for the corresponding k-space sampling pattern K. In practice, the k-space data is sampled at a δt-pace, with δt the dwell time used by the analog to digital converter (ADC). Hence a more realistic modeling of k-space data y ∈ C Nc×Ns× ∆t δt is as follows:

y = F S(K) x (1)
where S(K) ∈ Ω Nc×Ns× ∆t δt is the sampling pattern obtained after linearly interpolating each shot in K by a factor of ∆t δt . In this work we used δt = 2 µs. This interpolation is done to model the ADC hardware in place at the scanner. Note that at a cost of increased complexity more realistic forward models can be considered to take off-resonance effects as well as T * 2 -decay into account. However, they were not included in this work.

Reconstruction

Throughout this work, we denote the reconstruction network as R K θ (y), which is a parameterized by θ and depends on the k-space trajectory K being optimized (through the interpolated trajectory S(K)) and its corresponding k-space measurements y.

DCp Adjoint: As a first step, we rely on a simple DCp adjoint NUFFT to perform image reconstruction. This ensures that the given network actually results in trajectories which are data-aware and samples the center of k-space more densely as expected by traditional CS theory [START_REF] Puy | On variable density compressive sampling[END_REF][START_REF] Chauffert | Variable density compressed sensing in mri. theoretical vs heuristic sampling strategies[END_REF][START_REF]Variable density sampling with continuous trajectories[END_REF][START_REF] Adcock | Breaking the coherence barrier: asymptotic incoherence and asymptotic sparsity in compressed sensing[END_REF][START_REF] Boyer | Compressed sensing with structured sparsity and structured acquisition[END_REF]. Note that in this case the reconstruction network is parameter-free and satisfies the following reconstruction formula:

x dc = R K (y) = F H S(K) D S(K) y (2)
where F H is the adjoint of NUFFT operator F and D S(K) is the density compensators for the interpolated sampling pattern S(K) obtained iteratively using the method described in [START_REF] Pipe | Sampling density compensation in MRI: Rationale and an iterative numerical solution[END_REF]. U-Net: We then used a U-net architecture [START_REF] Ronneberger | U-net: Convolutional networks for biomedical image segmentation[END_REF] directly on the density-compensated adjoint image x dc . This network can be trained to reduce aliasing and other artifacts in the final reconstructed image.

NC-PDNet: Finally, we made use of the state-of-the-art DCp unrolled network called NC-PDNet [START_REF] Ramzi | NC-PDNet: a Density-Compensated Unrolled Network for 2D and 3D non-Cartesian MRI Reconstruction[END_REF]. This network is known to provide improved image quality compared to U-Net and remain robust to changes in trajectories, which is crucial in our approach as the sampling pattern is learned and updated in every gradient descent step. A key difference of this network as compared to that used in BJORK [START_REF] Wang | B-spline parameterized joint optimization of reconstruction and k-space trajectories (BJORK) for accelerated 2d MRI[END_REF] is that we use DCp to better condition the reconstruction problem. Moreover, in the underlying CNN-based denoiser, we use different parameters across iterations, giving improved reconstruction performance at the cost of larger memory footprint.

OPTIMIZATION SCHEME

We trained our networks for nearly 1.5k steps with batch size of 64 on the fastMRI training data, which was split into training and validation in a 90%-10% ratio. This enabled early stopping to prevent overfitting. The original fastMRI validation dataset was used only for later evaluation.

Costs, gradients and optimizers

We optimize for the k-space trajectory K and reconstruction network R K θ as follows:

( K, θ) = arg min

(K∈Q Nc ,θ) L x, R K θ F S(K) x (3)
where the hardware constraints on the trajectories are specified by the feasible trajectory space Q Nc as described in Sec. 3.4. The loss function L used in this study was inspired by [START_REF] Pezzotti | An adaptive intelligence algorithm for undersampled knee mri reconstruction[END_REF] which is a weighted sum of 1, 2 and multi-scale structural similarity index (S) [START_REF] Wang | Multiscale structural similarity for image quality assessment[END_REF]:

L(x, x) = α(1 -S(x, x)) + ᾱ||x -x||1 + ᾱ2 2 ||x -x||2
with ᾱ = 1 -α and the value of α was tuned to 0.998 to give nearly equally balanced terms.

Optimizing the trajectory in (3) requires computing the gradient of L with respect to K:

∂L(x, x) ∂K = ∇L(x, x) ∂ x ∂K = ∇L(x, x) ∂R K θ (y) ∂K (4)
In order to simplify this gradient calculation and reducing its computational complexity, we neglect the contribution of gradients from density compensators D S(K) . For the case of DCp Adjoint reconstruction, Eq. ( 4) reads:

∂L ∂K = ∇L ∂ x ∂D S(K) y D S(K) ∂ F S(K) x ∂K + ∂F H S(K)

∂K

In order to obtain the gradient of NUFFT operators F S(K) and F H S(K) with respect to the k-space trajectory K, we implemented [START_REF] Guanhua | Efficient NUFFT Backpropagation for Stochastic Sampling Optimization in MRI[END_REF] in tfkbNUFFT. As these underlying gradients vary extremely in norm depending on the k-space region (as noted in [START_REF] Gournay | Spurious minimizers in non uniform Fourier sampling optimization[END_REF]), we used the ADAM optimizer for learning the trajectories, while we relied on rectified-ADAM for optimizing the image reconstruction network R K θ . We used a learning rate of 10 -3 .

Multi-resolution

Following [START_REF] Lebrat | Optimal Transport Approximation of 2-Dimensional Measures[END_REF], we used a multi-resolution approach to optimize the k-space trajectory and allow for the algorithm to reach faster convergence. Consequently, the sampling pattern was first optimized on down-sampled trajectories. After optimizing over N ds ≈ 250 steps, the solution K n1 at the resolution level n1 was then interpolated and further used as a warm restart for the up-sampled problem n2 = 2 n1. We used dyadic scaling and trained our trajectory over 6 decimation levels (i.e. n1 = Nc × Ns/2 5 up to n6 = 2 5 n1). At any decimation level n k , the solution K n k used in the forward model ( 1) was still linearly interpolated as S( K n k ) to match the final trajectories shape and ensure that the reconstructor sees the same amount of data at every n k .

Learning schemes

Here there are two sub-models that are being optimized: The k-space trajectory K and the image reconstruction network R K θ . These submodels can be trained in the following ways:

Joint Learning (JL): In an ideal scenario, we would like to jointly optimize both sub-models. This idea was pushed forward pretty successfully in both PILOT and BJORK [START_REF] Weiss | PILOT: Physics-informed learned optimal trajectories for accelerated MRI[END_REF][START_REF] Wang | B-spline parameterized joint optimization of reconstruction and k-space trajectories (BJORK) for accelerated 2d MRI[END_REF] at the cost of producing trajectories close to their initialization. Here, as we rely on an unconstrained parameterization of K this scheme may lead to poor local minimizers. The reason is that the reconstruction network R K θ is initially naive, and thus not trained on various types of trajectories, hence it may give poor reconstructed images associated with high losses L for certain configurations of S(K). This initial bad conditioning of R K θ cannot be addressed by updating K. Consequently, the gradient of L with respect to K (see ( 4)) is corrupted leading to trajectory updates in wrong directions.

Alternative Descent (AD): In this scenario, we first optimize the trajectory using the DCp Adjoint reconstruction method. As the latter is parameter-free, the optimized sampling pattern K is purely data-driven and it densely samples the center of k-space, hence providing reasonable image reconstruction. Later on, we learn R K θ . Although this method can provide improved results, it still does not truly learn the sub-models in a joint fashion.

Hybrid Learning (HL): In this third scenario, we try to retain the best of both previous schemes, by slowly moving from AD to JL. As a first step, we start off by only learning the trajectory K using DCp adjoint as reconstruction. Following this, we carry alternative descent between K and R K θ while slowly increasing the number of steps to move from noisy to more accurate gradients over time. In practice, this was done at different resolution levels as prescribed in Tab. 1.

Constraints and Projection

In order to obtain realistic learned trajectories which can be run by a scanner gradient hardware, we need to impose Gmax and Smax constraints. Under the normalized sampling domain Ω = [-1, 1] 2 , we obtain the feasible constraint set Q Nc from [8, Eq. ( 2)]. To enforce these constraints, we project all shots in K using the projector Π Q Nc developed in [8, Sec. S1] after each gradient descent step (for more details, see [START_REF] Chauffert | A projection algorithm for gradient waveforms design in MRI[END_REF]). With this in mind, given a step size η t , each update of the trajectory along the learning process reads:

Kt+1 ← Π Q Nc Kt -η t ∂L(x, x) ∂K (5)
Given this projection step our trajectories strictly meet the constraints on Gmax and Srmax in contrast to earlier works [START_REF] Weiss | PILOT: Physics-informed learned optimal trajectories for accelerated MRI[END_REF][START_REF] Wang | B-spline parameterized joint optimization of reconstruction and k-space trajectories (BJORK) for accelerated 2d MRI[END_REF][START_REF] Vedula | 3D FLAT: Feasible Learned Acquisition Trajectories for Accelerated MRI[END_REF] which handle these constraints on the trajectories through a penalty term in the loss L. Importantly, this projection step is parameterfree. Additionally, the projector has no influence over the gradient of L with respect to K, allowing it to improve image quality at the reconstruction stage.

NUMERICAL STUDIES

We compare our results for AF = 20, and 2 imaging contrasts (T1-w, T2-w) using AD and HL as learning strategies. We did not compare with JL startegy as our experiments led to low SSIM scores (between 0.7 and 0.8 after training). Further, for comparison with an earlier baseline, we use SPARKLING trajectories generated with the learned TSD using LOUPE as obtained in [START_REF]Learning the sampling density in 2D SPARKLING MRI acquisition for optimized image reconstruction[END_REF]. 1 . We denote this approach by SP. The learned models were tested on 512 slices from the fastMRI validation dataset. We used the SSIM and PSNR scores on each slice as image quality metrics for evaluation.

Quantitative

We first present the trajectory and quantitative comparisons for each of the learning strategies in the form of box plots in Fig. 2. One key aspect of all our learned trajectories is that they are extremely curvy (in contrast to [START_REF] Weiss | PILOT: Physics-informed learned optimal trajectories for accelerated MRI[END_REF][START_REF] Wang | B-spline parameterized joint optimization of reconstruction and k-space trajectories (BJORK) for accelerated 2d MRI[END_REF]) and maximally explore the k-space, which ideally will lead to better performance. We notice that for the AD strategy, where the trajectory was learned with parameter free DCp adjoint as reconstructor, we do sample the center of k-space densely as expected by CS theory.

We observe that NC-PDNet consistently outperforms U-Net as it is an unrolled network. Further, we see that HL learning strategy clearly outperforms all the other methods with SSIM scores in the range of 0.9-0.95. The only exception to this is PSNR values for T1-w contrasts, where SP presents equivalent results to NC-PDNet. However, note that the SP trajectories were obtained by learning the density, followed by obtaining trajectories with SPARKLING and finally learning the NC-PDNet network for it. HL strategy on the other hand learns jointly in a single training.

Qualitative

In order to assess image quality and the residuals across the different learning schemes (SP, AD, HL), we present a single reconstructed slice for T1-w images in Fig. 3 and T2-w images in Fig. 4. We clearly observe lesser residual structures in the HL/NC-PDNet approach, which can also be verified quantitatively with highest SSIM and PSNR scores for 0.926 and 34.241, respectively for T1-w images. However, it is notable to observe that SP approach gives at par results as observed previously. For T2-w, we see HL/NC-PDNet approach clearly outperforms with SSIM and PSNR scores of 0.95 and 34.865 respectively.

CONCLUSIONS

In this work, we present a hybrid framework for jointly learning the trajectory and image reconstructor through projected gradient descent, which provides trajectories comparable to those generated by SPARKLING with learned density. Through restrospective studies on the fastMRI validation dataset, we showed that this novel learning scheme works across multiple resolutions and leads to superior performance of the trajectories and improved image quality overall.

Future prospects of this work include prospective implementations through modifications of T1 and T2-w imaging sequences. This could lead to additional constraints on the trajectory arising from the MR sequencing or contrasts.

Fig. 2 :

 2 Fig. 2: (a) The optimized k-space trajectories with SPARKLING on learned density (top, blue) and by joint optimization (bottom). For joint optimization, results with AD (orange) and HL (green) methods are presented for (AX)T1/(AX)T2 contrasts. Finally, trajectories optimized with U-Net and NC-PDNet as reconstructors shown for HL method. (b) Box plots summarizing the corresponding colormatched image reconstruction results on a retrospective study using 512 slices (fastMRI validation dataset). SSIMs/PSNRs appear on top/at the bottom.

Fig. 3 :

 3 Fig. 3: Top: Reference and reconstruction results for a single slice from file brain AXT1 202 2020283.h5 with SP, AD and HL strategies and NC-PDNet reconstructor. Bottom: The residuals, scaled to match and compare across methods.

Fig. 4 :

 4 Fig. 4: Top: Reference and reconstruction results for a single slice from file brain AXT2 209 6001069.h5 with SP, AD and HL strategies and NC-PDNet reconstructor. Bottom: The residuals, scaled to match and compare across methods.

Table 1 :

 1 Hybrid learning strategy

	Decimation levels	Learning Method
	32 → 4	Alternative descent for 20 steps each
	4 → 1	Alternative descent for N ds /2 steps each
	1	True joint optimization throughout

Comparisons with PILOT[START_REF] Weiss | PILOT: Physics-informed learned optimal trajectories for accelerated MRI[END_REF] and BJORK[START_REF] Wang | B-spline parameterized joint optimization of reconstruction and k-space trajectories (BJORK) for accelerated 2d MRI[END_REF] could not be done due to lack of time, although we thank the authors for sharing their trajectories with us. We will compare them in future work.

Acknowledgments

This work was granted access to the HPC resources of IDRIS under the allocation 2021-AD011011153 made by GENCI. Chaithya G R was supported by the CEA NUMERICS program, which has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No 800945.