Dr Aurélien Perrin
email: aurelien.perrin@ext.inserm.fr

Corinne Métay

Marco Savarese

Rabah Ben Yaou

German Demidov

Isabelle Nelson

Guilhem Solé

Yann Péréon

Enrico Silvio Bertini

Fabiana Fattori

Adele D ' Amico

Federica Ricci

Mira Ginsberg

Andreea Seferian

Odile Boespflug-Tanguy

Laurent Servais

Françoise Chapon

Emmeline Lagrange

Karen Gaudon

Adrien Bloch

Robin Ghanem

Lucie Guyant-Maréchal

Mridul Johari

Charles Van Goethem

Michel Fardeau

Raul Juntas Morales

Casie A Genetti

Minttu Marttila

Michel Koenig

Alan H Beggs

Bjarne Udd

Gisèle Bonne

Dr Mireille Cossée
email: mireille.cossee@inserm.fr

Titin copy number variations associated with dominant inherited phenotypes

Background

Titinopathies are caused by mutations in the titin gene (TTN). Titin is the largest known human protein; its gene has the longest coding phase with 364 exons. Titinopathies are very complex neuromuscular pathologies due to the variable age of onset of symptoms, the great diversity of pathological and muscular impairment patterns (cardiac, skeletal muscle or mixed) and both autosomal dominant and recessive modes of transmission. Until now, only few CNVs in TTN have been reported without clear genotype-phenotype associations.

Methods

Our study includes eight families with dominant titinopathies. We performed next-generation sequencing or comparative genomic hybridization array analyses and found CNVs in the TTN gene. We characterised these CNVs by RNA sequencing (RNAseq) analyses in six patients' muscles and performed genotype-phenotype inheritance association study by combining the clinical and biological data of these eight families.

Results

Seven deletion-type CNVs in the TTN gene were identified among these families. Genotype and RNAseq results showed that five deletions do not alter the reading frame and one is outof-reading frame. The main phenotype identified was distal myopathy associated with contractures. The analysis of morphological, clinical and genetic data and imaging let us draw new genotype-phenotype associations of titinopathies.

Conclusion

Identifying TTN CNVs will further increase diagnostic sensitivity in these complex neuromuscular pathologies. Our cohort of patients enabled us to identify new deletion-type CNVs in the TTN gene, with unexpected autosomal dominant transmission. This is valuable in establishing new genotype-phenotype associations of titinopathies, mainly distal myopathy in most of the patients.

--

INTRODUCTION

Titinopathies are caused by mutations in the titin (TTN) gene. Titin is the largest human protein (3800 kDa); its gene has the largest known coding phase with 364 exons.1 Titinopathies are complex neuromuscular pathologies for several reasons: age at onset is variable, there is great diversity of phenotypes with various muscular impairments (cardiac, skeletal muscle or mixed impairments) and both autosomal dominant and recessive transmission modes are possible.2 Other reasons, notably limited understanding of the pathophysiological mechanisms, limit the interpretation of TTN variants and confirmation of the cause of muscle pathology. The size of the gene, titin interactions with other proteins as well as the development of more sensitive sequencing and TTN variant analytical technologies are improving our understanding of the diversity of described phenotypes and leading to delineation of new phenotypes associated with TTN variants.3 So far, few TTN CNVs have been reported. One published TTN CNV is a recessive large deletion of exons 34-41, associated in trans with a frameshift mutation p.(Lys3596Asnfs*), reported in a family with a severe phenotype (loss of ambulation before the age of 40 years and marked hyperCKemia). Roggenbuck et al45 described a deletion of exons 346-362 (A-band) predicted to cause a frameshift with early-onset skeletal muscle involvement and dilated cardiomyopathy (DCM) developing in later life. [START_REF] Roggenbuck | A novel TTN deletion in a family with skeletal myopathy, facial weakness, and dilated cardiomyopathy[END_REF] We previously reported a dominant deletion of TTN exons 11-18 resulting in a 16 nucleotide long intron retention between exons 10 and 19 in skeletal muscle transcripts, in two families with a distal titinopathy. [START_REF] Perrin | Novel dominant distal titinopathy phenotype associated with copy number variation[END_REF] The standard method for CNV detection is comparative genomic hybridisation (CGH) array,7 although the short size of the oligoprobes and the repetitive structure of TTN are possible limitations to be considered. Recently, several studies have shown that next-generation sequencing (NGS) combined with the use of powerful bioinformatics methods improves the CNV detection. [START_REF] Valipakka | Copy number variation analysis increases the diagnostic yield of NGS studies in muscle disease patients[END_REF] 8 Small CNVs, in particular single-exon deletions and duplication, are still difficult to detect and TTN triplicate region is particularly challenging. [START_REF] Valipakka | Copy number variation analysis increases the diagnostic yield of NGS studies in muscle disease patients[END_REF]

MATERIALS AND METHODS

Genetic analysis, bioinformatics analysis and variant interpretation

For probands from families A, C, D and E, gDNAs (extracted from blood samples) were used for custom-targeted gene enrichment and DNA library preparation was performed using Nimblegen EZ choice probes and Kappa HTP Library preparation kits according to the manufacturer's instructions (Nimblegen, Roche Diagnostics, Madison, Wisconsin, USA). Paired-end sequencing was performed using the Illumina MiSeq platform (Illumina, Santa Cruz, California, USA). NGS data were analysed using in-house pipelines for detection of single nucleotide variants and CNVs. Detection of CNVs was performed after coverage normalisation by computing the ratio of a target's coverage of a given individual over the mean coverage of this target across all patients of the same sequencing run.9

For family B, as part of the Myocapture Project, whole-exome sequencing (WES) was performed by the Centre National de Recherche en Génomique Humaine, Evry, France on DNA samples from patients II-1, II-2, III-1 and IV-3. Variants were annotated and filtered using an in-house software system (PolyWeb). The CNV analysis was performed within the Solve-RD (https://solve-rd.eu/) Project after uploading fastq files in RD-Connect (https://rd-connect.eu/). Trio analysis of family F used CGH array and Myocap V.6 (targeted panel covering TTN exons and introns).10 In family G, a CGH array, using the platform SurePrint G3 60K platform (Agilent, California, USA), was performed on the proband and his parents. Proband's DNA was also analysed by Myocap V.6.

In family H (local ID BOS1210), the TTN deletion was initially identified by read depth analysis on a next-generation 43-gene sequencing panel as described.11 Trio WES was also performed at the Broad Institute of MIT and Harvard and analysed via the Boston Children's Hospital Rare Disease Cohorts (CRDC) Project, as described previously.12 Long-range whole-genome sequencing was performed on a PacBio, Sequel II platform at GeneDx (Gaithersburg, Maryland, USA).

CNV confirmation on proband's DNA and variant familial segregation analyses were performed by targeted Sanger sequencing, qPCR or digital droplet PCR (ddPCR).

Transcript analyses

mRNAs were extracted from muscle biopsies and prepared as described in Zenagui et al9 or using Qiagen RNeasy Plus Universal Mini Kit (Qiagen, Hilden, Germany).

RNA sequencing (RNAseq) analyses were performed as described in Perrin et al. [START_REF] Perrin | The importance of an integrated genotypephenotype strategy to unravel the molecular bases of titinopathies[END_REF] For families F and H, RNAseq was performed by Novogene (Durham, North Carolina, USA) using an Illumina Ribo-Zero Gold kit for ribodepletion followed by NEBNext Ultra II Stranded Library Prep Kit and sequenced to 100 M 2×150 bp paired-end reads. Data processing was performed by the CRDC12 for family H and Illumina NovaSeq6000 for family F. RNA sequence alignments for all datasets were performed using splice-aware RNAseq aligner (STAR) software14 and analysed using Integrative Genomics Viewer15 targeted on titin transcripts, NM_001267550.2 and represent as sashimi plot with ggsashimi algorithm. [START_REF] Garrido-Martín | Ggsashimi: Sashimi plot revised for browser-and annotation-independent splicing visualization[END_REF] For the sequence variant nomenclature, we followed the Human Genome Variation Society recommendations and used the TTN transcript reference NM_001267550.2, and sequence alignment against the human reference genome (Hg19) was performed.

RESULTS

Clinical data

Pedigrees of all families are represented in figure 1. Main clinical, histological and imaging data are summarised in online supplemental table 1.

Family A is characterised by four affected members, all males, in two generations. The proband (II-3) was referred for distal myopathy of inferior limbs since teenage years. The evolution of muscular damages was gradual and it was associated with cardiomyopathy (arrhythmias on ECG atrial hyperexcitability). Presenting symptoms included pes cavus, running difficulties and steppage gait. Muscle CT scan was normal in the upper limbs, but showed bilateral and symmetrical atrophy of the rectus abdominis, semitendinosus, triceps sural and extensor digitorum of the toes. Electromyography (EMG) revealed a myogenic pattern in the tibialis anterior (60 years old). Muscle biopsy performed at early 60s in the tibialis anterior was normal and electron microscopy showed abnormalities of Z-discs and presence of elongated non-quadrilateral structures extending along the sarcomeres evoking pseudo-rods. His brother (II-2) presented with distal myopathy, pes cavus, head drooping but no limitation in walking. Symptoms started in the teenage years with pes cavus, claw toes and steppage gait. Muscle CT scan (early 60s) showed no abnormality in the upper limbs, but fatty atrophy of the lower limbs and paravertebral muscles were observed. Muscle biopsy was not performed. No cardiac abnormality was noticed (echography and effort test in early 60s). Their brother (II-1) had myopathy and sudden death in late 50s. Their deceased father exhibited camptocormia and cardiac features. No genetic analysis was performed for these two family members.

Family B includes two affected subjects, a father (II-1) and his son (III-3). Patient II-1 acquired walking late (after 18 months) and immediately with tiptoeing. Overt climbing stairs difficulties appeared at 3-4 years of age. The disease evolved with predominant distal weakness and progression to loss of ambulation at the end of adolescence. Muscle CT scan assessment during the late third decade of life (no image available) revealed pelvifemoral involvement in addition to selective involvement of tibialis anterior muscles. In late 40s, a clinically prominent distal involvement in the four limbs was evident. More recent assessments during the fifth and sixth decades of life revealed a more diffuse muscle pattern of weakness and wasting in the four limbs as well as axial muscles (figure 2). Additional features observed at these late stages included joint contractures (hips, knees, Achilles tendons), wrist and finger hyperlaxity and facial weakness. Creatine kinase (CK) was moderately elevated in the young ages. Cardiac assessment was normal throughout disease progression. Conversely, respiratory function was altered in late 20s, requiring nocturnal non-invasive ventilation from late 30s and became permanent after 50 years old. Muscle biopsy performed after 30 years (deltoid muscle) showed a mild fibre atrophy, type 1 fibre semiuniformity with core-like intermyofibrillar network abnormalities.

In patient III-3, the onset of the disease occurred during childhood when parents observed clumsy walking and running and fatigability. At the first assessment at the end of adolescence, there was a predominant distal muscle weakness in the four limbs associated with hamstring contractures, wrist/finger hyperlaxity and facial weakness, although muscle CT scan revealed only mild gluteus maximus muscle fatty infiltration. Muscle biopsy, done at the end of adolescence, showed non-specific findings including fibre size variation, increased number of internalised nuclei and type 1 fibre predominance. However, immunofluorescence studies showed reduced and irregular dystrophin (Dys1 and Dys3 antibodies) and dysferlin stainings. Muscle western blot analysis revealed normal dystrophin amounts with very slightly reduced molecular weight. At that time, a hypothesis was raised that III-1 and his father (II-1) had two different diseases:III-1 possibly having dystrophinopathy and II-1 a myopathy with core-like abnormalities. Patient III-1' s exome analysis through Myocapture and Solve-RD Projects did not show any DMD or other sarcoglycans and dysferlin gene pathogenic variants. Furthermore, DMD gene mRNA did not show any mutation thus definitely infirming the dystrophinopathy hypothesis in III-1 patient and suggesting that dystrophin abnormalities were probably secondary to an unidentified alteration. Motor difficulties evolved progressively, the patient being unable to climb stairs and having difficulties to get up from the chair at the end of 20s. In early 30s, muscle weakness was more diffuse but still predominating in the distal parts of the four limbs. Conversely, to his father who lost walking at the end of adolescence, the disease course seemed less incapacitating as he is still walking at 30 years old, although with frequent falls. Cardiac assessment at 17 years revealed DCM. Last assessment in early 30s confirmed the DCM which responded favourably to perindopril and bisoprolol treatments. Respiratory function at age 30 years was moderately decreased (vital capacity 2.75 L, 58%) not requiring any respiratory support.

Family C includes three affected subjects with variable disease course and severity.

In the mother (I-1), extensive but mild joint contractures were observed since young adult age, but no muscle weakness was observed. The older affected son (II-1) developed ankle joint contractures during childhood. Last assessment in early 10s revealed mild lower limb proximal weakness after prolonged walking or exercise. Joint contractures affected mainly ankles and mildly shoulders and hips. A moderately elevated CK (five times the normal upper value) was noticed during childhood. Muscle imaging was normal during this period. Cardiac assessments in the mother and the eldest son were normal. The younger affected son (II-3) showed delayed motor milestones. When assessed at before 3 years old, the phenotype was considered consistent with type 2 spinal muscular atrophy (SMA-II) with preserved distal motor functions and the presence of knee contractures. A biallelic SMN1 gene exon 7 homozygous deletion was found and the patient was treated with nusinersen. The disease course was characterised by unusual diffuse joint contractures (elbows, wrists, fingers, ankles) that appeared after 6 years of age.

No SMN1 gene deletion was found in the oldest affected brother thus suggesting that SMN1 gene deletion does not explain the disease in the oldest brother or the mother.

Family D reported previously6 includes four affected female patients in two generations, with an adult age of onset (40-52 years old) and asymmetric distal myopathy. Patient II-1 and her mother (I-2) are more severely affected than II-2 and II-3 who had a mild right anterior lower leg muscle atrophy.

In family E already reported,6 patient III-1 developed difficulty walking and, in early 50s, developed difficulty climbing stairs. Symptoms progressed after 65 years old with the use of a cane, and in early 70s, he had weakness in the right upper limb, frequent falling and used a wheelchair soon after. The last neurological examination, in early 80s, showed considerable proximal (in the posterior thighs) and distal muscle atrophy in both lower and upper limbs and comparable proximodistal weakness. The daughter (patient IV-1) evaluated at mid-50s, initially reported as asymptomatic, presented with discrete clinical abnormalities: anterior distal leg weakness predominantly on the right and childhood Achilles tendon contracture (has never been able to squat with feet on the ground) with hammer toe retraction, predominantly on the right.

Family F is a sporadic case. Currently, the proband is a child, born after uneventful pregnancy and labour to non-consanguineous parents. In the first months of life, asymptomatic noncompaction cardiomyopathy was diagnosed. Since the age of 3 years, she exhibited impaired motor skills, frequent falls and progressive muscle weakness of proximal and distal muscles, mostly of the lower limbs. In addition, there were some facial dysmorphic features, mild microcephaly and learning difficulties. Her gait was asymmetric with left tendon Achilles contraction that required surgical correction. Serum CK levels were normal. EMG demonstrated mild myopathic features. She complains of fatigability, exercise intolerance and myalgia. Muscle MRI showed only minor fat replacement changes of the left calf muscles, with smaller volume than in the right calf, and with severe contracture. Degenerative muscle tissue changes are present in the tibialis anterior muscles, more on the right side and in the left semimembranosus.

Family G includes a sporadic severely affected patient, with congenital arthrogryposis and weakness and delayed motor development during childhood. Family history is negative for neurological disorders. He was a full-term first born, with transient hyperbilirubinaemia after an uneventful pregnancy with scheduled CT for breech presentation, with birth weight 2.890 kg at 39 weeks' gestation. At birth, upper limbs were flexed and leaning against the trunk. Supplemental O2 was provided for the first 2 days of life because of respiratory distress and some swallowing disturbances. On the third day, transitory hyperCKemia was found up to 1187 IU/L. The child had arthrogryposis with flexed elbows, bilateral camptodactyly of the fifth finger and contractures of his left foot. At age 1 month, worsening respiratory function with hypercapnia led to temporary O2 therapy and a nasogastric (NG) tube for swallowing difficulties. At that time, CGH array analysis documented a deletion at 2q encompassing part of the TTN gene. A muscle biopsy was carried out showing variability in fibre size diameter, with the presence of some small and regular-shaped fibres. Hypospadias was detected with recurrent urinary tract infections during the follow-up. The child was discharged at 3 months of life with the NG tube and O2 support. In the following months, the child progressively improved until he no longer required respiratory support. He had some upper respiratory tract infections, but never had pneumonia.

Family H is characterised as a sporadic case with no family history of neuromuscular disease. The male proband presented at birth with bilateral clubfeet and generalised weakness. The pregnancy was otherwise normal with the exception of reduced fetal movement. Gross motor milestones were generally delayed, with walking attained at 19 months. The child had a clinical diagnosis of arthrogryposis multiplex congenita and underwent left quadricepsplasty during childhood to improve his gait as part of treatment for decreased joint mobility. Weakness and reduced muscle bulk in the lower limbs was noted, particularly of the left quadriceps as well as mild bilateral ptosis which together indicated an underlying neuromuscular disorder. Muscle biopsy of the left quadriceps before 10 years old was consistent with congenital fibre-type disproportion. Mobility in his knee, ankle, wrist, hip, jaw and shoulder joints has decreased with age. In late adolescence, he is fully ambulant and independent; however, he experiences muscle fatigue after long periods of standing. Weakness is predominantly proximal and more pronounced in the lower than the upper limbs. He has difficulty with raising his hands above his head as well as sitting on the floor due to shoulder and hip tightness. Pulmonary function is subnormal, but respiratory support is not required. Cardiac status was normal at last evaluation in early 10s.

Imaging data

In family A, muscle CT scan of II-3 was normal in the upper limbs, but showed bilateral and symmetrical atrophy of the rectus abdominis, semitendinosus, triceps sural and extensor digitorum of the toes. CT scans of patient B-III-3 showed advanced fatty replacement of all lower limb, pelvic and paravertebral muscles with less severe degenerative changes in upper body parts.

In family D, imaging showed for patient D-I-2 severe fatty replacement of hamstrings and vastus lateralis in distal thigh, and of tibialis anterior and gastrocnemius in the lower legs; for patient D-II-1, early hypotrophy and fatty degeneration in adductor magnus and semimembranosus muscles of the thigh and asymmetric medial gastrocnemius, tibialis anterior and patchy soleus on the left lower leg; and for patient D-II-2, patchy fatty degeneration in the right soleus.

In patient E-III-1, MRIs identified severe fatty replacement in adductor longus and magnus, all hamstrings, rectus femoris and parts of vastus lateralis and intermedius of the thigh, total fatty replacement of tibialis anterior and right soleus on the lower legs with less changes in left soleus, right tibialis posterior and left extensor digitorum longus.

For patient G-II-1, MRIs showed severely hypotrophic gluteus maximus compared with medius and severe fatty involvement of pelvic obturators and less in adductor longus.

Molecular analysis

All CNVs are novel and not previously reported in the database of genomic variants (http://dgv.tcag.ca/dgv/app/home) except for families D and E. NGS analyses did not detect other TTN variants of interest, nor variants in other neuromuscular disease genes that might explain the patient's presentation.

Family A

Targeted NGS performed in patients II-2 and II-3 identified a heterozygous deletion encompassing exons 334-340. The boundaries of this deletion, defined by Sanger sequencing, are located within exons 334 and 340: chr2:g.179412605_179418808del; c.89030_93748del (figure 3A). This CNV results in truncation of part of exon 334 to part of exon 340 that predict an in-frame truncated transcript: c.89030_93748del without amino acid change at the junction and loss of 1573 amino acids (173 kDa), located in the A-band. RNAseq coverage analysis did not demonstrate nonsense-mediated mRNA decay (NMD) of transcripts carrying the CNV (approximately 50% of the reads) (figure 3A).

Family B

In this family, WES identified in affected patients II-1 and III-3 a heterozygous deletion with boundaries located by DNA genomic sequencing in introns 297 and 312: g.179448846_179459004del; c.58 151-35_65275+157 del. RNAseq analysis showed transcripts with deletion of exons 298-312, keeping an open reading frame. This CNV leads to the truncation of a part of the A-band, loss of 2375 amino acids (263 kDa). Read depth analysis of the RNAseq data did not suggest NMD of transcripts carrying the mutated allele (figure 3B).

Family C

The mother (I-1) and the two children (II-1 and II-3) have a heterozygous deletion from exon 94 to intron 128. Interestingly, the sequence breakpoints obtained by Sanger sequencing revealed the presence of identical sequences TTCA at the 3' and 5' junctions. The breakpoint sequences result in two possibilities: chr2:c.26786_c.32311+321 del or chr2:c.26783_c.32311+317 del, leading in both cases to deletion of the donor splice site of exon 94 that predicts exon 94 skipping from the transcripts. Unfortunately, due to the lack of patient muscle biopsy, RNA analysis was not possible to evaluate whether activation of a cryptic donor splice site occurred. If we consider the hypothesis of a deletion of exons 94-128, the reading frame would be preserved and the CNV would result in a deletion localised in the titin I-band which partially truncates the PEVK domain. The predicted protein translated would have a loss of 1850 amino acids (207 kDa). In this family, patient II-3 is also affected by SMA due to biallelic exon 7 deletion of the SMN1 gene, with three copies of this exon in the SMN2 gene.

Families D-E

NGS analyses identified in families D and E a heterozygous deletion of TTN exons 11-18, with breakpoints in intron 10 and 18 determined by Sanger sequencing to be very close to exon-intron junctions, chr2:c.1662+15_3101-3del, resulting in 16 nucleotides long recombinant intron between exons 10 and 19.6 RNAseq analysis confirmed the retention of the 16-nucleotide recombinant intron in the mutant allele (figure 3C). The sequencing depth of intron retention showed a proportion of deleted transcripts close to 50%, suggesting that NMD of the deleted allele is probably absent or minimal. [START_REF] Perrin | Novel dominant distal titinopathy phenotype associated with copy number variation[END_REF] The predicted protein translated would be 60 kDa (figure 3C).

Family F

CGH array analysis identified a de novo heterozygous intragenic TTN deletion. A trio NGS analysis confirmed the deletion on the maternal allele and refined the breakpoints in intron 295 and in exon 327 (chr2:179434372_179462179del). RNAseq analysis indicated the presence of two populations of aberrant transcripts in phase: a major one with deletion of exons 296-327, resulting in a loss of 9759 amino acids (1077 kDa) (NM_00126 7550.1:r.57545_86821del p.(Asp19182_Thr28940del)), and a minor one with a deletion of exons 295-327 (figure 3D). NMD of the deleted allele is probably absent or minimal, as indicated by balanced allelic ratios for 10 phased SNPs (data not shown).

Family G

CGH array analysis identified a large de novo heterozygous intragenic TTN deletion. Sequencing analysis confirmed the heterozygous deletion and mapped the breakpoints in intron 47 and intron 220 (chr2:g.179506480_179617668del). RNAseq analysis indicated the presence of a transcript with the deletion of exons 48-220, resulting in a loss of 8426 amino acids (934 kDa) (NM_001267550.1: r.11312_40558del p.(Glu3771_ Ser13519del) (figure 3E). Based on allelic ratio for five SNPs, NMD of the deleted allele is probably absent or minimal (data not shown).

Family H

Read depth analysis of an NGS 43-gene sequencing panel identified a large heterozygous intragenic deletion of the TTN I-band. Trio ddPCR of sequences within exon 55 confirmed the heterozygous deletion as de novo. Long-read PacBio sequencing confirmed a 119 383 bp deletion (chr2:g.179517492-179636871) encompassing exons . RNAseq analysis of gastrocnemius mRNA revealed a prominent abnormal splice variant predicting production of an internally deleted in-frame protein lacking exons 34-200 from roughly half the sequenced transcripts and loss of 8918 amino acids (990 kDa) (figure 3F).

DISCUSSION

Here, we demonstrate phenotypical diversity among dominant titinopathies resulting from intragenic CNV deletions of the TTN gene. Location in the gene and size of the deletions are variable and there is evident clinical heterogeneity between families but with a predominance of distal involvement in five families of our cohort (figure 4). It is interesting to note that affected members of three families, C, G and H, in whom arthrogryposis is a prominent feature of the disease, all had deletions involving the Ig-like domains of the I-band including the metatranscript-only exons (exons not included in any of the cardiac and skeletal muscle isoforms). More cases will be needed to be identified in order to determine whether these preliminary observations hold as reliable genotype-phenotype associations.

In two of our families, some patients had additional abnormalities. In family B, although both patients (II-1 and his son III-3) share a prominent distal muscle involvement, patient III-3, in whom moderate dystrophin abnormalities were identified, also has proximal muscle weakness and DCM. However, no mutation was detected in the DMD gene nor transcripts, nor in genetic coding for proteins that interact directly with dystrophin. A secondary dystrophin defect due to CNV TTN is unlikely since the father (II-1) does not have a dystrophin defect. This defect could be due to a DMD variant not identified with the techniques used, or be secondary to another genetic defect.

In family C, patient II-3 was first diagnosed as SMA-II with homozygous SMN1 deletion. This patient then developed diffuse joint contractures (elbows, wrists, fingers, ankles) that were considered as unusual for typical SMA-II and lead to genetic reanalysis that revealed the TTN CNV deletion. Similarly, his older brother (II-1) as well as mother (I-2) developed also joint contractures. The latter two carried the TTN CNV deletion but not the homozygous SMN1 deletion. Although we cannot rule out a gene interaction, the most likely mechanism is the addition of the phenotypes of the two pathologies in the patient, with the delayed motor milestones due to SMN1 mutation and the diffuse joint contractures caused by the TTN CNV deletion.

The diagnosis yield of NGS for primary myopathies varies between 30% and 50%.17 Efficient detection of CNVs is a major challenge for improving the diagnostic rate in rare neuromuscular diseases, even more in the TTN gene which is characterised by the presence of three repeated regions (for which the relative quantification lacks reliability). Due to the complex nature of TTN gene, several orthogonal approaches are recommended be used.18-21 Improved detection combined with an increase in the number of CNVs in genetic databases could promote better understanding of new genotype-phenotype associations. Long-read DNA, whole-genome and RNA sequencing approaches, as well as optical genomic mapping performed in a more routine manner should help to identify CNV rearrangements.22 Concerning the mechanisms of these large rearrangements, identification in family C of identical sequences TTCA at 3' and 5' junctions suggests a non-homologous joining event as they frequently occur at microhomology sites. [START_REF] Luis | Mechanobiology of muscle and myofibril Morphogenesis[END_REF] The RNAseq approach is very useful to define the transcripts resulting from the identified CNV. We used polyA or ribodepletion to analyse muscle transcripts. PolyA capture of transcripts is limited by less efficient capture and therefore less sequencing coverage of the 5' parts of very large transcripts such as for titin. Ribodepletion significantly enriches for mRNA, and thus has very high sensitivity, with the disadvantage of a high background of aberrant transcripts. In this cohort, the reading frame was maintained in transcripts for families A, B, F, G and H but not for families D and E. The reading frame of resulting transcripts in family C could not be determined in the absence of available muscle for RNAseq analyses.

Since haploinsufficiency is not a reasonable pathomechanism considering the high frequency of titin truncating variants in the general population, we believe that the dominant effect of the in-phase deleted CNVs could be due to a recombinant protein with aberrant domain alignment that would alter its conformation. It would lead to aberrant protein interactions (toxic gain-offunction effect) and/or interact with the wild-type titin to inhibit its normal function (dominant-negative effect).6 This mechanism would require that a stable truncated protein is produced. NMD mechanism of mutated transcripts appeared to be absent or minimal in these families, whether the phase is maintained or not. For families A, B, F, G and H, in the absence of a predicted premature stop codon and of NMD, we can assume the translation of internally deleted proteins (but due to the priority for RNA analysis, western blot analyses were not performed). Similar mechanisms can be suggested with another giant protein, nebulin, where a truncated protein is detected as a result of large dominant deletions in the NEB gene. 24 25 In families D and E, the deletion of exons 11-18 results in the loss of the reading frame due to partial intronic retention of introns 10 and 18 and thereby a premature stop codon. In the absence of detectable NMD, the deletion is predicted to lead to a titin protein of 60 kDa lacking a large part of the I-band, all A-band and M-band. The predicted truncated protein would preserve the insertion domains in the Z-disk, suggesting a possible dominant-negative effect. However, western blot analyses with current techniques did not detect any truncated protein. [START_REF] Perrin | Novel dominant distal titinopathy phenotype associated with copy number variation[END_REF] Peculiarly, the phenotype has an adult onset and a clear distal component (figure 4). The best known dominant titinopathies with distal preference are missense variants in exon 364 causing late-onset tibial muscular dystrophy26 and some of the known missenses in exon 344 causing hereditary myopathy with early respiratory failure.27 In our series of dominant deletions, the phenotype surprisingly often had a distal presentation irrespective of deletion position which may be compatible with the alternative of the mutant harming the wild-type protein.

Titin interacts with a large number of proteins28 and these interactions are essential for proper muscle structure and function. Some of these interactions are essential for cell signalling,29 30 others for an elasticity component31 or in the transduction of the mechanical signal, in particular by its interactions with myosin and actin. 23 31 The diversity of CNVs in our cohort suggests that different interaction domains may be affected when truncated titin proteins are integrated into the sarcomere. The integration of these proteins could destabilise the titindependent protein network. For families A, B and F, the CNVs affect different myosin interaction domains in the A-band (exons 334-340 and exons 297-313, 296-327). For families G and H, the deletion is very large, removing domains going from I-band to A-band (the actin interaction domains, the PEVK and part of the myosin interaction domains). Recent genotype-phenotype association studies have shown that variants affecting the PEVK domain were associated with arthrogryposis-like phenotypes as in families G and H,32-38 which can be understood as a consequence of amyoplasia caused by mutated metatranscript exons with preferential fetal expression.

The presence of cardiomyopathy in some but not all patients in this study is of interest. When the deletion co-segregates with cardiomyopathy, the most likely explanation is that it affects isoforms important for cardiac muscle. In families without complete co-segregation, we postulate the involvement of a modifying gene or a second mutated gene (digenism).

Animal and cellular models would be useful to evaluate the presence, location and behaviour of the deleted protein and to advance in the understanding of the pathogenic dominant mechanisms of TTN CNV and genotype-phenotype associations. Also, the finding of smaller deletions being able to cause a distal phenotype changes the diagnostic approach in patients with a distal myopathy as CNVs need to be considered in addition to pathogenic variants in the C-terminal TTN exons and in the already known other genes.

Correction notice

This article has been corrected since it was published online first. The name of one of the authors has been amended.

Twitter

Mridul Johari @mriduljohari and Gisèle Bonne @BonneGisele

  8 Despite the aforementioned technical challenges, we present in this study 17 patients from 8 families with dominant CNV in the TTN gene, identified by NGS or CGH array.

Acknowledgements

The authors thank AFM-Telethon for their financial support, French titin consortium for their discussion about patients, and Heng Li, Jason Chin, Joseph Devaney and the team at GeneDx for long-range sequencing and interpretation of data for family H. In particular, heartfelt thanks to all the patients and their families for their participation without which these research studies would not have been possible.

Funding

This work was funded by AFM 21381 and 24259 grants (the French Muscular Dystrophy Association (AFM-Téléthon)); the Délégation à la Recherche Clinique et à l'Innovation du Groupement de Coopération Sanitaire de la Mission d'Enseignement, de Recherche, de Référence et d'Innovation (DRCI-GCS-MERRI) de Montpellier-Nîmes and the Solve-RD Project have received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement no 779257. Analysis of family H was supported by Muscular Dystrophy Association (USA) grant MDA602235, Boston Children's Hospital IDDRC Molecular Genetics Core Facility funded by P50HD105351 from the National Institutes of Health of USA, the Boston Children's Hospital CRDC Initiative and by a sponsored research agreement with GeneDx. Some coauthors are members of the ERN NMD Network (AD'A, FF, OB-T, MS, RBY, IN, GS, YP, ESB, AS, MJ, BU, GB).

Data availability statement

All data relevant to the study are included in the article or uploaded as supplemental information.

Contributors

AP, CM, MS, AB, BU, GB and MC contributed to the conception and design of the study. All authors contributed to the acquisition and analysis of data. AP, CM, MS, RBY, CVG, BU and MC contributed to drafting a significant portion of the manuscript or figures. MC and AP are the guarantors of this study.

Competing interests

None declared.

Patient consent for publication

Obtained.

Ethics approval

The study was approved by the ethical guidelines issued by our institutions for clinical studies in compliance with the Helsinki Declaration. Patients gave informed consent for the genetic analysis according to the French legislation (Comité de Protection des Personnes OUEST 6-CPP1128HPS3 IDRCB-2018-AO2287-48).

Provenance and peer review

Not commissioned; externally peer reviewed.


Maksim Berezov
email: maksim.berezov@mines-paris.fr

Corinne Ancourt
email: corinne.ancourt@mines-paris.fr

Mikhail Kashchenko

Tiling parameters prediction using Machine Learning techniques

Keywords: code optimization, compilation, source-to-source transformations, machine learning, loop tiling, data-dependencies, feature space design

The tiling transformation is one of the most crucial code optimization techniques to expose data locality and parallelism. The main idea is to split the initial iteration space into blocks and traverse them in a special order. This transformation is parametric and very sensitive to parameter tuning. Poor parameter tuning can lead to much lower performance than the initial code. Existing state-of-the-art solutions consider a restricted list of parameters to handle this issue and guarantee safe solutions. Our work proposes solutions that go beyond current state-of-the-art techniques and gain additional speedup considering a larger set of options for tiling. Our approach is based on Machine Learning methods and automatically derives heuristics to tune tiling parameters. We can predict: 1) the optimal partitioning matrix of the iteration space 2) the tile sizes, 3) the optimal directions for scanning inter-tiles, 4) the optimal directions for scanning intra-tile elements. The optimal selection of these parameters is crucial especially for programs that have data dependencies. We introduce sets of features that feed our models in their predictions. The first set encodes data dependencies, the second one captures the level of parallelism and data locality in a code. The third one aggregates information about the iteration space. Our approach surpasses existing feature spaces for tiling parameters prediction. Moreover, it could be used in conjunction with auto-tuners to iterate through the iterative search.

Introduction

The loop tiling transformation is considered to be one of the more complex code transformations. Its application can provide significant execution time gains because it allows targeting different memory levels including cache levels. Loop tiling depends on data dependencies and impacts many code aspects such as the temporal and spatial locality of data, the parallelism.

Many factors explain the complexity of predicting the tiling parameters including the large space of possible solutions. Typically, researchers only predict the sizes of rectangular tiled shapes. In addition to tile sizes, our work investigates the profitability of choosing different tile shapes and tile scanning directions for code generation. These options are poorly reflected in existing state-of-the-art solutions, but can potentially provide significant performance gains. We classify the methods of parameter selection for code transformations as:

-Analytical models derived by an expert [START_REF] Chame | A tile selection algorithm for data locality and cache interference[END_REF], [START_REF] Coleman | Tile size selection using cache organization and data layout[END_REF], [START_REF] Rivera | A comparison of compiler tiling algorithms[END_REF], [START_REF] Shirako | Analytical bounds for optimal tile size selection[END_REF] [START_REF] Lam | The cache performance and optimizations of blocked algorithms[END_REF], [START_REF] Mehta | Tile size selection revisited[END_REF], [START_REF] Hsu | A quantitative analysis of tile size selection algorithms[END_REF], [START_REF] Yotov | Is search really necessary to generate high-performance blas[END_REF],

[8] -Iterative auto-tuners [START_REF] Teixeira | Locus: a system and a language for program optimization[END_REF], [START_REF] Tiwari | A scalable autotuning framework for compiler optimization[END_REF], [START_REF] Pouchet | Combined iterative and model-driven optimization in an automatic parallelization framework[END_REF], [START_REF] Hoste | Cole: compiler optimization level exploration[END_REF], [START_REF] Ansel | Opentuner: An extensible framework for program autotuning[END_REF], [START_REF] Qasem | Profitable loop fusion and tiling using model-driven empirical search[END_REF], [START_REF] Knijnenburg | The effect of cache models on iterative compilation for combined tiling and unrolling[END_REF] -Static analytical models derived by Machine Learning algorithms [START_REF] Yuki | Automatic creation of tile size selection models[END_REF], [START_REF] Liu | An efficient tile size selection model based on machine learning[END_REF], [START_REF] Rahman | Neural network assisted tile size selection[END_REF], [START_REF] Malik | Optimal tile size selection problem using machine learning[END_REF] Analytical models derived by experts provide correct solutions but are more difficult to implement than the others because they must be adapted very precisely for each parameters to the target machine. Increasing the set of parameters degrades the quality and stability of these solutions. This is also the case with auto-tuners, moreover, the search time increases in proportion to the size of search parameters. Our study focuses on the third approach.

This paper is organized as follows. Section 2 introduces the context of tiling transformation and its parameters. Section 3 presents the proposed features that would be used for ML modelling. Section 4 highlights our experiments in Machine Learning modelling. It contains steps like synthetic data generation, model training and model evaluation.

Loop Tiling Parameters

The potential performance that you can get by applying loop tiling depends on the choice of the loop tiling parameters but also on the quality of the code generated after tiling. Indeed, once the tile parameters have been chosen, there are several possible execution paths to generate the tiled code which respect the semantics of the initial program. The questions of the choice of the directions for scanning the tiles and the ordering of the computational iterations in the tiles arise.

This section introduces important criteria: parameters and code generation options that have an impact on the loop tiling transformation performance.

Kernels of interest

Our methodology allows optimizing kernels with any kind of loops.

But the highest performance gain could be achieved for kernels with constant data dependencies. It is intuitive since some options to optimize tiling parameters rely on the fact that dependencies exist. Hence, for this case, we have a more restricted set of optimizations and less potential benefit. Thus, this paper takes place in the context of kernels with constant data dependencies. The methodology could be applied to a kernel of any kind. But we conclude that more effective code is generated under this context.

Scanning directions for tiles and its elements

The loop tiling transformation implies the partitioning of the iteration space into blocks according to the chosen partitioning matrix [START_REF] Irigoin | Supernode partitioning[END_REF]. However, the order in which the blocks are traversed is not unique. Data dependencies define the correctness of execution (block traversal). Likewise, the order of traversal of the points inside the block is not unique. Since several possibilities are possible, we can consider this choice as a parameter to predict. PIPS compiler [START_REF] Paristech | Pips: Automatic parallelizer and code transformation framework[END_REF] offers nine possible choices for generating tiled code. It is a combination of the three possible choices for scanning the tiles and the three for scanning the elements inside the tiles.

Figure 1 illustrates one possible way of scanning the tiles of the computational domain according to its data dependency vectors. Let us define d1 and d2 as the extreme rays of the dependence cone which summarizes the set of dependencies of the computational kernel [START_REF] Yang | Minimal data dependence abstractions for loop transformations[END_REF]. The red vector represents their sum (the sequential hyperplane direction), the blue vector is perpendicular to the red vector and represents a potential parallel direction [START_REF] Lamport | The parallel execution of do loops[END_REF]. The red and blue vectors describe a possible basis of scanning the tiles that we refer to as T P arallel tile direction. The sequential hyperplane direction carries all dependencies, while blocks along the parallel direction can be executed concurrently. So, the sequence of blocks 1-4-2-7-5-3-8-6-9 represents a legal execution if we apply T P arallel scanning; among others blocks 3-5-7 can be run in parallel.

The second possible choice for the directions scanning tiles is named T Shape . Its scanning directions are parallel to the partitioning vectors of the tiling shape (rectangular shape in this case). Thus the sequence of blocks 1-2-3-4-5-6-7-8-9 is an example of legal execution in this case. The third possible scanning direction T Initial corresponds to the initial iteration space and matches the second case for a rectangular shape.

We use the same strategy to define possible scanning directions for the elements inside each tile:

-L Initial : relative to the initial basis, -L P arallel : corresponding to the hyperplane sequential direction and its relative orthogonal vector, -L Shape : relative to the partitioning vector directions.

Figure 2 illustrates scanning directions for inter tile. The same idea of possible scanning vectors is applied here.

In the general case, we have nine combinations of scanning directions. However, some of them may not be legal due to data dependencies.

Data Dependence Abstraction for Tiling

The second concept that we would like to introduce is the abstraction used to encode dependencies. Yang et al. [START_REF] Yang | Minimal data dependence abstractions for loop transformations[END_REF] have defined the minimal abstraction for a loop transformation. The dependence cone turned out to be the minimal abstraction to legally apply the tiling transformation, which means that among the studied abstractions, it is the smallest abstraction which is enough precise to check whether the transformation is legal. In our study, we encode this abstraction to a feature vector for Machine Learning issues. The dependence cone is the convex hull of the set of points that are positive linear combination of dependence distance vectors. More formally, the dependence cone is defined as

DC(L) = {v = k i=1 λ i d i ∈ Z n |d i ∈ D(L), λ i ≥ 0, k i=1 λ i ≥ 1}, where D(L) is the set of all distance vectors d i in loop nest L.
Figure 3 shows an example of dependence cone for a code with d1, d2, d3 dependence vectors. Note, that vectors d3 and d1 form the extreme rays of this cone.

Tiling sizes

The right choice of the tiling sizes is a crucial component of the tiling transformation. Large tile sizes can result in a large number of slow high-level caches uses. In contrast small tiles may not fully benefit from improved data locality.

The selection of tile size is a complex problem that depends on many criteria such as the application, the architecture, the cache hierarchy. In our study, we only vary the application characteristics such as the size of the iteration domain, the access functions for array elements, and the data dependencies. The architecture is fixed. After applying tiling and exploiting parallelism in the nest of loops, several loops can be parallelized. In OpenMP, it is not beneficial to generate several levels of parallel loops because the creation of new parallel threads in threads has an additional cost. It is more efficient to keep a single parallel loop externally and a vector loop internally.

The choice of which parallel loop to keep when there are several possible is tricky. We must take into account the number of iterations of these loops; it must be greater than the number np of processors and be a multiple of it, if possible, to take full advantage of the potential number of architectural parallelism. The outermost loop should be preferred for GPU or coarse grain architectures.

In the framework of this study, we exploit the parallelism of loops by generating OpenMP parallel directives for the outermost parallel loop and OpenMP vectorial directives for the innermost loop if it is parallel.

The number of threads has been chosen to use the maximum number of parallel cores available and multi-threading when possible.

Feature space design

Machine Learning techniques draw conclusions based on some characteristics of a phenomenon under consideration. Usually, it is a vector of fixed size, and each value represents relevant properties of the phenomenon [START_REF] Blum | Selection of relevant features and examples in machine learning[END_REF]. A choice of irrelevant features implies that the characteristics of the phenomenon are not sufficiently represented. Machine Learning algorithms cannot generalize based on them. It would lead to unsatisfactory results and poor performance. Therefore, the design of a suitable function space comes to the fore.

We distinguish two main concepts which are crucial for the selection of tiling parameters. These are 1) the data dependencies, 2) the array access functions, and information about the iteration domain. Data dependencies define the legality of the tiling and motivate the use of different scanning directions [START_REF] Ancourt | Scanning polyhedra with do loops[END_REF].

Array access functions associated with the iteration domain characterize the spatial and temporal data locality and the parallelization/vectorization opportunities [START_REF] Yuki | Automatic creation of tile size selection models[END_REF], [START_REF] Liu | An efficient tile size selection model based on machine learning[END_REF]. Array access functions are important criteria to choose the correct tile sizes.

Encoding of dependencies

Encoding dependencies is not an easy task. The main problem is that although the concept of data dependencies is well-defined, they can be represented or approximated in many ways. Moreover, the information about data dependencies needs to be transferred to a vector of a fixed size for ML purposes. To find the proper level of abstraction, we investigate research about the suitable abstractions for data dependencies to apply tiling.

Abstractions to encode data dependencies in our ML model We use three abstractions to encode the data dependency information. We illustrate them in the example of Listing 1.1.

Listing 1.1: Original code 1 for ( int i = 2 ; i < 1 0 2 2 ; i ++) 2

for ( int j = 2 ; j < 5 1 0 ; j ++)

3 A [ i ] [ j ] = A [ i -2 ] [ j -1 ] + A [ i -1 ] [ j -1 ] + A [ i -1 ] [ j -2 ] ;
1 -Dependence cone The dependence cone is the minimal abstraction to verify the legality of the loop tiling application. The dependence cone can be represented as a set of extreme rays. The first abstraction encodes the areas of the iteration space where we observe these extreme rays. For each extreme ray, we compute its angle with the x-axis in degrees. Here, x represents the first dimension of the dependency vector.

Our abstraction to encode the information on the extreme rays is a vector of fixed size (8 for 2-dimensional iteration domain). Let's describe the elements of the vector from 0 to 7. The i-th element of this vector gives how many extreme rays of the dependence cone form the angle θ with the x-axis, such as θ ∈ [45× i; 45 × (i+1)).

The mapping function for 3-dimensional iteration domains is pretty similar, except that it goes through the coordinates of two polar angles for the dependency vectors instead of one. The vector summing these types of vectors is of size 32 (c.f. section 3.3).

-Summation vector

The second abstraction related to the data dependencies is a vector which is the sum of all data dependency vectors. We extract a unique value -the angle of this vector with the x-axis. This information is required because this vector is used as a basis vector for some scanning directions (T P arallel and L P arallel directions). In the example 1.1, the sum of three dependence vectors results in vector [START_REF] Blum | Selection of relevant features and examples in machine learning[END_REF][START_REF] Blum | Selection of relevant features and examples in machine learning[END_REF], which makes an angle of 45 degrees with the x-axis. Fig. 4: Encoding of the extreme rays 3 -Encoding of uniform data-dependencies In the same way, we encode the extreme rays 3.1, we encode the uniform data dependence vectors. Our abstraction to encode the information on the uniform dependencies is a vector of fixed size (8 for 2 dimension iteration domain). The i-th element of this vector gives how many uniform data dependencies form the angle θ with the x-axis, such as θ ∈ [45× i; 45 × (i+1)).

This vector for listing 1.1 equals [1,2,0,0,0,0,0,0].

Encoding of the iteration domain

The iteration domain can have an impact on the prediction of all the parameters considered. We propose to encode it with two simple features: 1) the number of iterations for each loop and their total 2) the presence of constant loop bounds for all loops.

The first feature vector is a vector of the size that equals the number of nested loops + 1. The first value represents the number of iterations for the first nested loop (NaN, if it is unknown, e.g. variables of functions), the second value represents the number of iterations for the second nested loop, etc. The last value represents the total number of iterations for the loop nest. This vector for listing 1.1 equals [1020, 508, 518160].

The second feature vector is a vector of the size that equals the number of nested loops. The first value represents the presence of constant loop bound [START_REF] Ancourt | Scanning polyhedra with do loops[END_REF] or their absence (0) for the first nested loop, etc. This vector for listing 1.1 equals [START_REF] Ancourt | Scanning polyhedra with do loops[END_REF][START_REF] Ancourt | Scanning polyhedra with do loops[END_REF] since the loop bounds are constant.

Generalization for 3-D case

In the proposed encoding of the dependence cone, data dependencies can be easily generalized for 3-D tiling and for higher dimensions. Let θ be the angle between a data-dependence vector/extreme ray of a dependence cone and the Z-axis. And ϕ is the angle between the projection of the same data-dependence vector/extreme ray to the XY plane and X-axis. Let V denotes the vector, which contains all the subareas of 3-D space to be encoded.

V = v 1 v 2 ... v K
where v i is a particular subarea of 3-D space, i ≤ K, K is the number of chosen subareas of 3-D space (e.g. we have experimented that 32 is a good choice). Then we can define the mapping function that assigns the subarea of the 3-D space to each value of θ and ϕ angles. Actually, this mapping function could be considered as the mapping from polar coordinates to the subareas of 3-D space.

f (x) :< ϕ, θ >→ v i ∈ V In the 3-D case, we need also two angles θ and ϕ to encode the information about the summation vector.

Encoding of array accesses

There is a bunch of research investigating the proper encoding of array accesses.

Yuki et al. [START_REF] Yuki | Automatic creation of tile size selection models[END_REF] investigate the problem of automatic tiling selection using machine learning approaches. The authors consider cubic tiling on three nested loops with 2D data. They describe each loop according to the references inside the arrays. Liu et al. [START_REF] Liu | An efficient tile size selection model based on machine learning[END_REF] propose a slightly similar approach. In the context of this paper, we use the concatenation of two vectors to properly encode the array accesses.

Machine Learning modeling

Our methodology is presented in Figure 5. It consists of four main steps: feature extraction, synthetic data generation, data labeling, machine learning pipeline and code generation process.

Data collection

We used the code generator [START_REF] Berezov | Cola-gen: Automatic code generator for program optimization using active learning techniques[END_REF] to generate around 1100 synthetic kernels. The main concept that we targeted during the generation was the creation of uniform data dependencies. The generated programs had from 1 to 5 true uniform data dependencies (constant write-before-read dependencies). We tried to generate kernels with the most different possible dependence cones. The methodology was to generate kernels whose second outermost loop could be parallel after the tiling transformation.

Autotuning process We used LOCUS Autotuner [START_REF] Teixeira | Locus: a system and a language for program optimization[END_REF] to label the data for two different settings. First, we asked the Autotuner to find the optimal combination of parameters for the square tiling considering 4 different scanning directions: TI-LI, TI-LP, TP-LI, and TP-LP. TS and LS scanning directions are respectively identical to TI and LI in the case of square tiling. Second, we asked the Autotuner to tune the same kernel with diamond tiling considering 9 possible scanning directions. Scanning directions related to the partitioning shape (TS, LS) have been added for -Intra and -inter tiles. That gives 9 combinations in total. All the collected executions and corresponding execution times were collected for the learning process. Note that not all of these 9 combinations could be legal, so we check the legality and skip not legal executions.

ML modelling

We have an imbalance in our classes. The original tiling shape and scanning are the best options in almost 80% of cases. Hence, we think it reasonable to create a two-stage ML model that would consist of two parts. The first model will predict whether we consider only the original tile settings or the extended settings, the best option for the given code. The second model will predict the tile sizes if the answer is positive (not profitable for extended settings), or all the parameters for a given kernel (tiling shape+scanning directions+ tile size) otherwise. The pipeline is illustrated in Figure 6. Our first-step model is a binary classification model. It takes the input features presented in section 3 and predicts whether this kernel is potentially a good candidate for restricted parameters of tiling or not (extended parameters). We do modeling using the stacked classifier. We use Logistic regression [START_REF] Wright | Logistic regression[END_REF] as a meta-classifier to aggregate the predictions.

Train/test split We evaluated the model on the test set. These kernels were not involved in the training phase. Kernels from the train set were involved in the training of all models in this subsection. The train/Test split ratio is 0.85/0.15 with the same class balance of the minority class in both splits.

Model evaluation

We face a problem of imbalanced classification. We consider it reasonable to use the ROC-AUC score to evaluate the quality of the predictions. The ROC-AUC curve is a graph showing the performance of a classification model at all classification thresholds. We achieved a ROC-AUC score of 0.9, we conclude that our model has a good generalization ability to distinguish whether a kernel benefits from extended tiling parameters or not.

Second-step models We distinguish two models for the second step prediction. If the data sample was marked as beneficial for the extended tiling parameters then we predict the tile shape, scanning directions, and tile size. Otherwise, we only predict tile sizes. We consider it as a regression problem that predicts the speedup and we sample the best parameters that maximize the speedup for the prediction. We used single CatBoost [START_REF] Dorogush | Catboost: gradient boosting with categorical features support[END_REF] to handle this task.

Our approach to predict (tile sizes or shapes/scanning directions/tile size) is presented in Figure 7. The idea is to predict the speedup for a given code based on shapes/scanning directions/tile size and a feature vector. We sample all possible code parameters for a given kernel and predict the speedup for each parameter. We select the parameters with the highest predicted speedup. It should be noted that the speedup we predict cannot be used as an accurate estimate of the real speedup. There are many different factors that impact the speedup and we do not take them into account, but the predicted "speedup" helps differentiate the impact of the tiling parameters on real speedup. This approximation helps to choose the more appropriate combinations of parameters.

Evaluation of the results

We propose the following method for the evaluation of our model. Firstly, we get predictions for the kernels from the test set that benefit from advanced tiling. Second, we generate optimized code according to the predicted tiling parameters. Finally, we measure the average absolute speedup of the mentioned above kernels.

We perform the same procedure for our two-stage pipeline and "simpler" ML model that predicts only cubic tile sizes. Both models were trained on the same dataset. The comparison of these two models helps us to understand how much performance gain we achieve if we increase the complexity of our model (predicted parameters). It is not always true that the increase in complexity leads to an increase in performance. Fig. 7: Tile size prediction based on speedup prediction pipeline Figure 8 shows the comparison of our approach and the "simpler" model. The average absolute speedup was normalized by the performance of the "simpler" model. This experiment shows that our methodology could bring up to 5% speedup for the kernels that benefit from it. The increase in model complexity leads to a gain in performance.

The relative speedup for these kernels is about 84% (out of the theoretical maximum). It is the ratio of the speedup that we archived with our method (one-shot prediction) to the speedup that was found by the LOCUS autotuner during the exhaustive search. This paper presents a methodology to predict the optimal parameters for loop tiling transformation. We do not limit our predictions just with tile size but consider extended set of parameters. It helps to archive up to 5% of additional absolute two dimensional loops.

Our second contribution concerns the design of the feature space for the tiling prediction problem using Machine Learning. We propose different sets of features. They encode data dependencies, iteration domain and information about data locality and parallelism in the initial code. To our knowledge, our study is the first to use explicit information about data dependencies for tiling parameter prediction.

Our future work concerns the improvement of the training/validation sets. Our methodology supports the usage of tiling of arbitrary perfectly-nested loopnests, but training data contains just 2-dimensional loops. Moreover, we used synthetically generated kernels for model evaluation. Collection of a large benchmark of real-worlds kernels would make our results more reliable. Also, we would like to perform our experiments on different architectures (GPU, other CPUs) and build the architecture-dependent model. It would use architecture-dependent features (e.g. memory hierarchy features) to be able to generalize across different architectures. The model presented does not benefit from these features because the experiments were performed on one architecture.

Fig. 1 :

 1 Fig. 1: Inter-tile scanning

Fig. 3 :

 3 Fig. 3: Dependence cone

Fig. 5 :

 5 Fig. 5: Methodology

Fig. 6 :

 6 Fig. 6: Prediction pipeline

Fig. 8 :

 8 Fig. 8: Acceleration of classical pipeline


Abdo-Rahmane Anas

Gurvan Jodin
email: gurvan.jodin@ens-rennes.fr

Mario Costanza
email: mario.costanza@femto-st.fr

Damien Hoareau
email: damien.hoareau@ens-rennes.fr

Samuel Margueron
email: samuel.margueron@femto-st.fr

Nicolas Bideau
email: nicolas.bideau@univ-rennes2.fr

Florence Razan
email: florence.razan@ens-rennes.fr

A low cost, self-powered, plantar pressure distribution sensing insole +

Keywords: low power, smart insole, Bluetooth low energy, piezoresistive sensors, plantar arch

Introduction

Nowadays, using smart insoles to monitor plantar motion is becoming increasingly important. These insoles are renowned for their importance in the sports and medical fields [START_REF] Drăgulinescu | Smart Socks and In-Shoe Systems: State-of-the-Art for Two Popular Technologies for Foot Motion Analysis, Sports, and Medical Applications[END_REF]. Some devices are already commercially available (Moticon, Pedar, F-scan etc…), however, these solutions are lacking in terms of energy consumption, comfort, cost, and mechanical robustness. Thus, researchers are making significant effort in the development of this kind of device [2]. In this context, a study of a low-cost, low-power, and self-powered wearable intelligent pressure monitoring system based on flexible piezoresistive sensors is presented in this paper. A synoptic diagram of the system is presented in Figure 1.

Materials and Methods

To measure the pressure distribution on the foot, we have designed an insole with an 8 × 2 sensors matrix [START_REF] Laaraibi | Flexible dynamic pressure sensor for insole based on inverse viscoelastic model[END_REF]. The structure of this insole is multi-layered with a single layer of carbon black (Velostat). The electrode structure is based on lines on one side, and on columns on the other side of the Velostat layer, as shown in Figure 1a. It is then sufficient to measure the resistance between a line and a column, mainly at the intersection to measure the change in resistance due to applied pressure.

The insole is encapsulated in a shoe size 27 cm mold with a flexible, and biocompatible polydimethylsiloxane (PDMS), RTV 615, to protect the electrode and insole from external influences such as moisture, mechanical stress, fluctuating voltage, temperature, or vibration.

The insole is connected to an electronic board. The data acquisition is done by an Arduino nano 33 BLE and the data is transmitted through the BLE radio integrated within the microcontroller. The electrical energy provided by the flexible solar panel (MPT3.6-150 PowerFilm ©, 100mA@3.6V), is stored in a capacitor called Cstorage, which powers the entire electronic board (Vstor).

The collected data is processed by a computer using an inverse viscoelastic model [START_REF] Laaraibi | Flexible dynamic pressure sensor for insole based on inverse viscoelastic model[END_REF] to estimate the plantar force. The error of the measurements is below 2% of the full scale.

Discussion

Figure 2 shows the voltage Vstor across the capacitor Cstorage using indoor lighting as the PV panel is designed to. The value chosen for this capacitor was 100 𝜇F, to allow some margin and extra energy capacity to supply the whole system, and 𝐼 𝑠𝑦𝑠𝑡 the current flowing through the system. The developed smart system features a power consumption Pq ≈ 62.8 𝑚𝑊 given by Equation (1):

𝑃 𝑞 = 1 τ ∫ 𝑉 𝑠𝑡𝑜𝑟 (𝑡) τ 0 ⋅ 𝐼 𝑠𝑦𝑠𝑡 (𝑡) ⋅ d𝑡, (1)
where, 𝜏 is the time of the measurements and the data transmission over BLE.

The average power consumption of the insole is 113 𝜇W in active mode; in fact, the system takes full advantage of the low-power characteristics of the components used.

Figure 1 .

 1 Figure 1. System diagram: (a) Self-powered encapsulated insole using a flexible solar panel; (b) Inverse viscoelastic model; (c) Plantar pressure distribution sensing.

Figure 2 .

 2 Figure 2. Functional evolution of the supply voltage Vstor versus time.

Funding: This study is funded by the ANR within the framework of the PIA EURDIGISPORT project (ANR-18-EURE-0022).