Dr Charlotte Proudhon
email: charlotte.proudhon@curie.fr

Dr Marc-Henri Stern
email: marc-henri.stern@curie.fr

Guillaume Bataillon

Luc Cabel

Bruno Buecher

Jean-Yves Pierga

Samia Kasperek

Marie-Laure Melaabi

Manuel Tanguy

Amanda Bortolini Rodrigues

Francois-Clement Silveira

Amélie Bidard

Amanda Bortolini Silveira

François-Clément Bidard

Amélie Kasperek

Samia Melaabi

Marie-Laure Tanguy

Manuel Rodrigues

Jean-Yves Pierga1

High-accuracy determination of microsatellite instability compatible with liquid biopsies Running title: MSI detection in tissue and plasma

Keywords: repair deficiency, mismatch, microsatellite instability, liquid biopsy droplet digital PCR microsatellite instability, mismatch repair deficiency, droplet digital PCR, liquid biopsy Clinical Chemistry

Microsatellite instability (MSI) Circulating tumor DNA (ctDNA) Droplet digital PCR (ddPCR) DNA mismatch repair (MMR) Microsatellite stable (MSS) Next-generation sequencing (NGS) Immunohistochemistry staining (IHC) Cell-free DNA (cfDNA) Formalin-fixed, Paraffin-embedded (FFPE) Limit of detection (LOD) Limit of blank (LOB) Mutant Allele Frequency (MAF) Colorectal cancer (CRC) Wild-type (WT) Human genes: KRAS (KRAS proto-oncogene, GTPase) BRAF (B-Raf proto-oncogene, serine/threonine kinase) EGFR (epidermal growth factor receptor) ACVR2A (activin A receptor type 2A) DEFB105A/B (defensin beta

Introduction

Tumors with DNA mismatch repair (MMR) deficiency accumulate high numbers of DNA replication errors, particularly deletions at repetitive DNA sequences termed microsatellites. This phenotype, referred to as microsatellite instability (MSI), can occur in several human cancers, but is more frequently observed in colorectal, endometrial and gastric tumors [START_REF] Hause | Classification and characterization of microsatellite instability across 18 cancer types[END_REF][START_REF] Cortes-Ciriano | A molecular portrait of microsatellite instability across multiple cancers[END_REF].

Until recently, MMR status testing was performed in the context of newly diagnosed colorectal cancers and as a first screening tool in cancers suspected to be related to Lynch syndrome, an inherited cancer predisposition condition associated with MMR deficiency and MSI cancer formation [START_REF] Umar | Testing guidelines for hereditary non-polyposis colorectal cancer[END_REF][START_REF] Buza | Mismatch repair deficiency testing in clinical practice[END_REF][START_REF] Latham | Microsatellite instability is associated with the presence of lynch syndrome pancancer[END_REF]. Following the approval of pembrolizumab (anti-programmed cell death-1) in 2017 for the treatment of unresectable or metastatic MSI solid tumors, MSI testing became critical for all advanced cancers regardless of their tissue of origin [START_REF] Le | Mismatch repair deficiency predicts response of solid tumors to pd-1 blockade[END_REF][START_REF] Overman | Nivolumab in patients with metastatic DNA mismatch repair-deficient or microsatellite instabilityhigh colorectal cancer (checkmate 142): An open-label, multicentre, phase 2 study[END_REF][START_REF] Overman | Durable clinical benefit with nivolumab plus ipilimumab in DNA mismatch repairdeficient/microsatellite instability-high metastatic colorectal cancer[END_REF][START_REF] Kim | Comprehensive molecular characterization of clinical responses to pd-1 inhibition in metastatic gastric cancer[END_REF].

Routine clinical testing relies on immunohistochemistry of MMR machinery components (mutL homolog 1 [MLH1], mutS homolog 2 [MSH2], mutS homolog 6 [MSH6], and PMS1

homolog 2 [PMS2]) and PCR amplification of reference microsatellite markers, followed by capillary electrophoresis [START_REF] Buza | Mismatch repair deficiency testing in clinical practice[END_REF]. The presence of alleles in the tumor that are not found in matched normal tissue are indicative of MSI. The most widely used commercial kit, referred to as the "pentaplex assay", interrogates 5 microsatellite markers (BAT-25, BAT-26, NR-21, NR-24, and MONO-27), comprising stretches of 21 to 27 poly-A repeats. Tumors with instability in two or more markers are classified as MSI, whereas those with one or no detectable alterations are classified as microsatellite stable (MSS). This panel has good clinical sensitivity for MSI status characterization in Lynch-spectrum tumors [START_REF] Bacher | Development of a fluorescent multiplex assay for detection of msi-high tumors[END_REF][START_REF] Maruvka | Analysis of somatic microsatellite indels identifies driver events in human tumors[END_REF].

However, its relevance for MSI status characterization in other tumor types is largely unknown. Recently, algorithms that can detect MSI using next-generation sequencing (NGS) data were developed [START_REF] Hause | Classification and characterization of microsatellite instability across 18 cancer types[END_REF][START_REF] Cortes-Ciriano | A molecular portrait of microsatellite instability across multiple cancers[END_REF][START_REF] Maruvka | Analysis of somatic microsatellite indels identifies driver events in human tumors[END_REF][START_REF] Bonneville | Landscape of microsatellite instability across 39 cancer types[END_REF][START_REF] Middha | Reliable pancancer microsatellite instability assessment by using targeted next-generation sequencing data[END_REF]. Such approaches showed that MSI is much more prevalent than previously thought. For example, using exome sequencing data to investigate more than

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200,000 microsatellites, Hause et al. identified MSI in 14 different cancer types, including those with low MSI incidence [START_REF] Hause | Classification and characterization of microsatellite instability across 18 cancer types[END_REF]. Moreover, these studies have also identified new highly informative MSI markers, opening avenues for the development of panels more relevant for cancer types not classically associated with MSI.

The current trend towards systematic detection of MSI in metastatic cancer patients has also stimulated the development of non-invasive diagnostic methods capable of detecting MSI in circulating tumor DNA (ctDNA). Such strategies require methodologies with high sensitivity, capable of detecting abnormal MSI alleles highly diluted in fragmented DNA from normal cells. Here, we describe for the first time a quantitative droplet-digital PCR (ddPCR) approach for highly sensitive detection of MSI (<0.1% mutant allelic frequency or MAF). The method is compatible with both tissue and body fluid samples and provides a cost-effective, simple and fast diagnostic tool for the high-throughput screening of patients and the monitoring of responses to therapy.

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Materials and methods

Cell lines and clinical samples

Human cancer cell lines HCT-116, HCT-15 and LoVo were used as MSI positive controls. SW480, SW1116, PSN1, and buffy coat from healthy donors were used as MSS controls.

Genomic and cell-free DNA (cfDNA) were extracted using the QIAamp DNA mini and Circulating Nucleic Acid Kits, respectively. Archived clinical samples were retrieved from Institut Curie Pharmacogenomics. A waiver of informed consent was obtained given the retrospective nature of the study. Plasma samples from healthy donors were obtained from Etablissement Français du Sang, under French and European ethical practices.

ddPCR conditions and data analysis

Primers and probes are described in online Supplemental Table 1. The KRAS proto-oncogene (KRAS) drop-off assay is described elsewhere [START_REF] Decraene | Multiple hotspot mutations scanning by single droplet digital pcr[END_REF]. The KRAS Q61L assay was ordered from BioRad. ddPCR reactions were performed using the Bio-Rad QX100, with 900 nM of each primer, 250 nM of each probe, up to 5000 genome equivalents of DNA, and under the following conditions : 95°C for 10 min followed by 40 cycles of 94°C for 30 s, 61°C for 3 min (BAT-26) or 59°C for 3 min (defensin beta 105A/B [DEFB105A/B]) or 55°C for 3 min (activin A receptor type 2A [ACVR2A]) or 60°C for 1 min (B-RAF proto-oncogene [BRAF V600E ] and phosphatidylinositol-4,5-bisphosphate 3-kinase catalytic subunit alpha [PIK3CA H1047R ]) or 63°C for 1 min (KRAS G12/13 ) or 55°C for 1 min (KRAS Q61L ); final hold at 98°C for 10 min. Cluster thresholding and quantification were performed with QuantaSoft v1.7.4. For the drop-off assays, droplets were manually assigned as wild type (WT) or mutant based on their fluorescence amplitude: WT=VIC + /FAM + ; MSI mutant=VIC + /FAM -/low and mutant KRAS=VIC -/low /FAM + . Samples were run in one or multiple replicates depending on sample availability, cfDNA concentration and ctDNA fraction. Samples were considered to be

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positive if at least one replicate presented ≥2 positive MSI droplets and if the mean MAF was higher than the defined limit of detection (LOD) of each assay.

Limit of blank (LOB) and LOD

The false-positive rate of each assay was estimated using ≥53 replicates of WT DNA. The LOB was defined as the upper 95% confidence limit of the mean false-positive measurements. The analytical sensitivity was estimated using serial dilutions of HCT-116 in WT DNA, with MAFs ranging from 10% to 0.01%. The number of replicates per dilution point ranged from 3 to 8. LOD was estimated as the lowest mutant concentration that could be reliably distinguished from the LOB [START_REF] Decraene | Multiple hotspot mutations scanning by single droplet digital pcr[END_REF][START_REF] Bidshahri | Quantitative detection and resolution of braf v600 status in colorectal cancer using droplet digital pcr and a novel wild-type negative assay[END_REF].

Statistical analysis

The discriminatory power of each marker was investigated using receiver operating characteristic analyses. Areas under the curve were: 0.95 (95% confidence interval, 0.9-0.99) for BAT-26 and 0.99 (0.99-1) for ACVR2A and DEFB105A/B. Optimized cut-offs were calculated to minimize classification errors. Analytical sensitivity and specificity for the chosen cut-offs were: 99% [confidence interval, 88%-100%] and 95% [96%-99%] for BAT-26, 99% [96%-100%] and 100% for ACVR2A, and 99% [96%-100%] and 96% [91%-100%] for DEFB105A/B, respectively. Global error rates were 3.2% for BAT-26, 0.8% for ACVR2A and 2.4% for DEFB105A/B.

Results

Proof of principle

We recently described drop-off ddPCR strategies for the accurate detection and quantification of all nucleotide variations present in a given region, for example in KRAS exon 2 and EGFR exon 19 hotspots [START_REF] Decraene | Multiple hotspot mutations scanning by single droplet digital pcr[END_REF]. The drop-off ddPCR method relies on the design of two hydrolysis

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probes within the same PCR fragment. The reference probe (REF) is complementary to a nonvariable region, and the target probe (DROP-OFF) is complementary to the WT sequence of the hotspot region where mutations are frequently observed. The presence of variants in the hotspot region results in a mismatch with the DROP-OFF probe and the absence of hydrolysis during amplification. Therefore, in a drop-off ddPCR assay three clusters of droplets can be observed: droplets with no template (REF -/DROP-OFF -), droplets containing WT alleles (REF + /DROP-OFF + ) and droplets containing mutant alleles (REF + /DROP-OFF -/low ).

We predicted that such drop-off ddPCR strategy could be instrumental to detect the microsatellite length variations that characterize MSI tumors. To determine the feasibility of this approach, we first developed an assay targeting the BAT-26 reference microsatellite, one of the five markers of the pentaplex assay [START_REF] Bacher | Development of a fluorescent multiplex assay for detection of msi-high tumors[END_REF][START_REF] Suraweera | Evaluation of tumor microsatellite instability using five quasimonomorphic mononucleotide repeats and pentaplex pcr[END_REF]. The BAT-26 assay included a VIC-labeled REF probe and a FAM-labeled drop-off probe (MSI probe), which covered the entire A27 homopolymer plus 4 and 2 bases upstream and downstream of the repeat, respectively (Figure 1A). After adjustments of the annealing temperature and extension time, we reached a clear separation between WT and MSI droplet clouds (Figure 1). The assay could specifically detect MSI in DNA from the MSI colorectal cancer (CRC) cell lines HCT-116 and HCT-15, which carry shortened BAT-26 alleles (Figure 1B and online Supplemental Figure 1A).

Although the separation of the WT and MSI-positive clouds was clear in both cell lines, it was less pronounced for HCT-15 (3 nucleotide [nt] deletion) than for HCT-116 (12 nt deletion), suggesting a low level of non-specific binding of the MSI probe to BAT-26 alleles containing short deletions (Figure 1B). It is noteworthy that no instability was observed in DNA from MSS cell lines, such as the CRC cell lines SW480 and SW1116 or the pancreatic cancer cell line PSN1.

To demonstrate the flexibility of the MSI-ddPCR approach, we next developed assays targeting the two most discriminatory microsatellite markers associated with MSI-H cancers

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identified by Hause et al. [START_REF] Hause | Classification and characterization of microsatellite instability across 18 cancer types[END_REF]. The two loci comprised the A8 and A9 repeats located within the ACVR2A gene and DEFB105 A and B paralogous genes, respectively. These shorter markers usually show deletions of only 1 or 2 nt and are difficult to interrogate with the PCRcapillary electrophoresis approach. Similar to what was observed for BAT-26, the ACVR2A and DEFB105A/B assays specifically detected MSI in HCT-116 and HCT-15, while no instability was observed in MSS cell lines (Figure 1B and online Supplemental Figure 1). Moreover, both assays detected MSI in the LoVo CRC cell line, which carries a homozygous deletion encompassing the BAT-26 marker (online Supplemental Figure 1B andC). The DEFB105A/B assay discriminates MSI alleles with different deletion sizes, as observed for HCT-116 and LoVo, which harbor MSI alleles shortened by 1 nt and 2 nt (online Supplemental Figure 1C). For ACVR2A, MSI alleles shortened by a single nt were observed in the three cell lines (Supplemental Figure 1C). Remarkably, the three assays accurately quantified MSI in 1/2 and 1/10 dilutions of HCT-116, HCT-15 and LoVo DNA in WT DNA (online Supplemental Figure 1D). Indeed, MAFs quantified by MSI-ddPCR assays correlated with r ≥ 0.99 (p<0.001) with the ones reported by a commercial ddPCR assay that targets KRAS G13D , a heterozygous mutation shared by the three cell lines. Taken together, these results demonstrate the feasibility of developing ddPCR assays targeting long and short repetitive regions of the genome and its applicability to MSI status determination and accurate quantification.

Analytical sensitivity for MSI detection and quantification accuracy

We subsequently used serial dilutions ranging from 10% to 0.01% of HCT-116 DNA in WT DNA and pure WT DNA to estimate the analytical sensitivity of each MSI-ddPCR assay. The LOB was estimated at 0.01% for the three assays (online Supplemental Figure 2A, online Supplemental Table 2). The LOD was found to be 0.04% for BAT-26 and 0.08% for both

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ACVR2A and DEFB105A/B markers (online Supplemental Figure 2B-D). For the three assays, the Pearson's correlations were r=0.99 p<0.0001 between the expected and observed MAFs, suggesting the accurate quantification of MSI alleles at a wide range of allelic frequencies.

This in vitro performance of the MSI ddPCR assays prompted us to assess their efficiency in clinical samples.

Comparison MSI-ddPCR versus pentaplex-PCR

BAT-26, ACVR2A and DEFB105A/B markers are recurrently unstable among MSI CRC samples [START_REF] Hause | Classification and characterization of microsatellite instability across 18 cancer types[END_REF][START_REF] Maruvka | Analysis of somatic microsatellite indels identifies driver events in human tumors[END_REF][START_REF] Suraweera | Evaluation of tumor microsatellite instability using five quasimonomorphic mononucleotide repeats and pentaplex pcr[END_REF]. To test the clinical sensitivity of this 3-marker panel for detection of MSI in this cancer type, we blindly tested a series of 126 formalin-fixed paraffin-embedded (FFPE) samples previously classified as MSI (n=70) or MSS (n=56) using the pentaplex test and immunohistochemistry (IHC) for the 4 major MMR machinery components (online Supplemental Table 2, FFPE-1 to 126). As expected, MSI MAFs estimated by MSI-ddPCR assays showed a bimodal distribution, concordant with the MSI status reported for these samples (Figure 2A and online Supplemental Tables 3 and4, FFPE-1 to 126). Median MAFs obtained for MSI and MSS samples were 41% (interquartile range, 30%-54%) and 0% (0%-0%) for BAT-26; 36% (23%-48%) and 0.08% (0%-0.2%) for ACVR2A; and 21% (17%-29%) and 0.13% (0%-0.4%) for DEFB105A/B, respectively. Conversely, MAFs estimated by ddPCR assays targeting known BRAF (n=34), KRAS (n=23) or PIK3CA (n=1) tumor mutations displayed a unimodal distribution among MSI and MSS samples, with median values of 22% (16%-29%). Using receiver operating characteristic analyses, we next determined optimized MAF cut-offs of ≥ 6% for BAT-26 and ≥ 1.2% for both ACVR2A and DEFB105A/B to classify these markers as unstable. Considering a sample as MSI when at least 2 markers were reported as unstable, the 3-marker MSI-ddPCR test achieved 100% clinical sensitivity and 100% clinical specificity in comparison with the pentaplex

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classification (online Supplemental Table 3). This criterion was used in order to avoid misclassification of MSS samples as MSI due to germline polymorphisms, as observed for BAT-26 in samples FFPE-55, 65 and 91 (online Supplemental Table 3). Finally, we tested an additional sample (FFPE-127) corresponding to a metastatic tumor highly infiltrated by lymphocytes (cellularity < 10%). This sample was classified as dMMR by IHC (loss of expression of MSH2 and MSH6), but as MSS by pentaplex PCR. MSI-ddPCR results supported the MSI phenotype with MAFs close to the defined thresholds (BAT-26, ACVR2A and DEFB105A/B markers at 2.9%, 2.5% and 0.8% MAF, respectively; online Supplemental Tables 3 and4), which could be explained by the higher analytical sensitivity of this approach.

Based on these results, we concluded that the 3-marker MSI-ddPCR performed as well as the pentaplex-PCR for MSI status characterization in CRC, with the additional advantage of allowing MSI testing in tumor samples highly contaminated with WT DNA.

We next evaluated the assay ability to detect MSI in other tumor types. We screened a collection of 59 previously characterized FFPE samples (25 MSI and 34 MSS), including endometrial (20 MSI and 28 MSS), ovarian carcinomas (1 MSI and 3 MSS), sebaceoma (1 MSI), cholangiocarcinoma (1 MSI), gastric carcinoma (1 MSI), small bowel adenocarcinoma (1 MSI), pancreatic ductal adenocarcinomas (2 MSS) and appendix carcinoma (1 MSS) (Figure 2B and online Supplemental Tables 3 and4, FFPE-128 to 186). Using the criteria described above, we found concordant results between the pentaplex-PCR and the 3-marker MSI-ddPCR. 92% of the samples defined as MSI by the pentaplex test were classified as MSI by MSI-ddPCR (23/25), whereas all the MSS samples were correctly classified as MSS. We further evaluated 8 additional samples (endometrial carcinoma, n=4; ovarian carcinoma, n=1; chordoma, n=1; glioma, n=1; glioblastoma, n=1) that were classified as dMMR by IHC but MSS by pentaplex PCR. As expected, BAT-26 was reported stable by MSI-ddPCR in these 8

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samples. However, MSI was detected at ACVR2A and DEFB105A/B markers in 4 and 5 samples, respectively (online Supplemental Tables 3 and4). Taken together, these results supported the development of assays targeting additional markers identified by cancer exome sequencing data to improve the clinical sensitivity of MSI testing for different cancer types.

Quantification of tumor content by MSI-ddPCR

One of the advantages of ddPCR over the standard pentaplex-PCR is its quantitative capability. Using the data obtained for the 40 MSI CRC samples mutated for BRAF (n=31), KRAS (n=8) or PIK3CA (n=1) oncogenes (Figure 2A; online Supplemental Tables 3 and4), we evaluated the correlation between MAFs estimated by MSI-ddPCR assays and the ones reported by ddPCR assays targeting these mutations. Pearson's correlations of r=0.78 and r=0.7, respectively; p<0.0001 (Figure 2C) were observed for the BAT-26 and ACVR2A markers, which accurately estimated the tumor fraction. No correlation was found for DEFB105A/B across the different samples (Pearson's r=0.25 p=0.12; Figure 2C), which may be explained by frequent variations in the number of copies of the DEFB105 genes among individuals (2 to 12 per haploid genome) [START_REF] Hollox | Extensive normal copy number variation of a betadefensin antimicrobial-gene cluster[END_REF], or by clonal heterogeneity of DEFB105 instability. Taken together, these results supported the utility of the MSI-ddPCR assays for accurate quantification of tumor fractions in patient samples.

MSI status characterization and tumor fraction in blood samples by MSI-ddPCR

MSI-ddPCR is a highly clinically sensitive and quantitative assay. We thus hypothesized that it would be relevant as a liquid biopsy tool to simultaneously detect MSI and quantify ctDNA.

We therefore characterized 30 archival blood samples (plasma or serum) collected at baseline and/or during treatment from 14 patients with locally advanced or metastatic CRC or endometrial tumors, defined as MSI by the pentaplex assay. The three ddPCR assays detected

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MSI in blood samples from all the patients tested (online Supplemental Figure 3, online Supplemental Tables 5 and6). Moreover, for patients with BRAF or PIK3CA mutated tumors (P1, P2, P4, P5, and P8), ctDNA fractions quantified by the MSI-ddPCR assays were validated by ddPCRs specifically targeting these mutations (BRAF V600E and PIK3CA H1047R , Pearson's r=0.99 p<0.0001, Figure 3B, online Supplemental Tables 5 and6). These results supported the reliability of the MSI-ddPCR assays for accurate ctDNA quantification. Indeed, for patients with sequential samples collected at baseline and at different time points during treatment, ctDNA kinetics estimated by MSI-ddPCR assays mirrored the tumor response to therapy (Figure 3A). Of note, patients P1, P4, P5, P6, and P7 were treated with pembrolizumab. Strikingly, patients P1, P4, and P7 responded to pembrolizumab and showed undetectable or dramatically decreased ctDNA concentrations (>90% MAF decrease) over the first weeks of treatment. On the other hand, patients P5 and P6 did not respond to therapy and showed increased or stable ctDNA concentrations. Similarly, decreased ctDNA concentrations for the patient P9 followed well the patient's partial response to chemoradiotherapy (Figure 3A). Finally, when the MSI-ddPCR assays were applied to 38 serum and 4 plasma samples collected from 28 patients with advanced MSS tumors, none of the samples were positive for MSI, even though 74% (29/39) of the samples from patients with KRAS (n=21) or BRAF (n=5) mutated tumors were positive for ctDNA (KRAS or BRAF median MAF 4.2%, min=0.2%, max=71.4%). Similarly, plasma samples collected from healthy donors (n=48) were all negative for MSI, yielding a clinical specificity of 100% (online Supplemental Tables 5 and6). Based on these data, we concluded that the MSI-ddPCR assays were both highly clinically sensitive and specific, as well as suitable for the quantitative detection of ctDNA in longitudinal studies.

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C o n f i d e n t i a l Discussion

Here we present a new ddPCR test for fast, clinically sensitive and cost-effective detection of MSI. Most importantly, the high analytical and clinical sensitivity reached by this method allows for its use as a liquid biopsy tool to simultaneously detect MSI and quantify ctDNA.

The method is also well suited to screen FFPE tumors and does not require matched normal samples for reliable MSI identification. As a proof-of-concept, we set up ddPCR assays targeting the microsatellites located inside MSH2 (BAT-26), ACVR2A and DEFB105A/B genes. While BAT-26 is one of the five NCI reference markers widely used in clinical practice, ACVR2A and DEFB105A/B are novel discriminatory markers identified as frequently unstable among MSI cancers [START_REF] Hause | Classification and characterization of microsatellite instability across 18 cancer types[END_REF][START_REF] Cortes-Ciriano | A molecular portrait of microsatellite instability across multiple cancers[END_REF]. As expected based on the frequent instability of these markers in CRC [START_REF] Hause | Classification and characterization of microsatellite instability across 18 cancer types[END_REF][START_REF] Maruvka | Analysis of somatic microsatellite indels identifies driver events in human tumors[END_REF][START_REF] Suraweera | Evaluation of tumor microsatellite instability using five quasimonomorphic mononucleotide repeats and pentaplex pcr[END_REF], the MSI-ddPCR test showed 100% concordance with the reference pentapex assay for MSI status characterization in this cancer type. The concordance decreased to 90% for endometrial cancers, with both panels failing to classify all dMMR samples as MSI (20/24 correctly classified as MSI by pentaplex and 21/24 by MSI-ddPCR). Indeed, pan-cancer identification of MSI remains challenging when using reduced panels of markers. This limitation is currently being addressed by more comprehensive NGS-based approaches [START_REF] Hause | Classification and characterization of microsatellite instability across 18 cancer types[END_REF][START_REF] Cortes-Ciriano | A molecular portrait of microsatellite instability across multiple cancers[END_REF][START_REF] Maruvka | Analysis of somatic microsatellite indels identifies driver events in human tumors[END_REF][START_REF] Bonneville | Landscape of microsatellite instability across 39 cancer types[END_REF][START_REF] Middha | Reliable pancancer microsatellite instability assessment by using targeted next-generation sequencing data[END_REF]. A recent study reported the use of a 98-kb pan-cancer panel, which included microsatellites, to detect MSI in blood [START_REF] Georgiadis | Noninvasive detection of microsatellite instability and high tumor mutation burden in cancer patients treated with pd-1 blockade[END_REF]. With a panel of 99 microsatellite sequences, Willis et al. described MSI detection across 16 cancer types [START_REF] Willis | Validation of microsatellite instability detection using a comprehensive plasma-based genotyping panel[END_REF]. However, NGS assays are costly and time-consuming when compared to targeted approaches such as MSI-ddPCR. Moreover, MSI-ddPCR achieved 100% clinical sensitivity in blood samples compared to 87% [START_REF] Willis | Validation of microsatellite instability detection using a comprehensive plasma-based genotyping panel[END_REF] and 78% [START_REF] Georgiadis | Noninvasive detection of microsatellite instability and high tumor mutation burden in cancer patients treated with pd-1 blockade[END_REF] for the above studies. MSI-ddPCR is therefore more suitable for routine clinical practice screening and monitoring of response to treatment. Systematic investigation of MSI in cancer exomes has uncovered shared and cancer-type specific patterns of instability and therefore is setting the ground for future development of

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more sensitive panels of markers [START_REF] Hause | Classification and characterization of microsatellite instability across 18 cancer types[END_REF][START_REF] Cortes-Ciriano | A molecular portrait of microsatellite instability across multiple cancers[END_REF][START_REF] Maruvka | Analysis of somatic microsatellite indels identifies driver events in human tumors[END_REF][START_REF] Bonneville | Landscape of microsatellite instability across 39 cancer types[END_REF][START_REF] Middha | Reliable pancancer microsatellite instability assessment by using targeted next-generation sequencing data[END_REF]. In this context, MSI-ddPCR stands as a promising approach for the next generation of panel-based MSI tests. Indeed, the method provides robust direct measurements of MSI for both long non-coding markers (such as BAT-26), as well as for coding microsatellites (such as ACVR2A), which usually show indels of only a single nucleotide. Therefore, MSI-ddPCR should be easily adapted to different panels of markers, depending on the clinical need, and could help to improve MSI testing in extracolonic tumors for which the reference pentaplex panel has more limited clinical sensitivity. Moreover, high order multiplexing of ddPCR assays should become feasible with the release of multicolor digital PCR platforms.

The analytical sensitivity of MSI-ddPCR is at least 2 orders of magnitude higher than the detection threshold of the reference pentaplex method (≥10% MAF) [START_REF] Lee | Molecular testing for gastrointestinal cancer[END_REF][START_REF] Ladas | Enhanced detection of microsatellite instability using pre-pcr elimination of wild-type DNA homo-polymers in tissue and liquid biopsies[END_REF], opening up the possibility of non-invasive MSI detection in ctDNA. CtDNA has demonstrated its clinical value as a liquid biopsy tool [START_REF] Pantel | Clinical applications of circulating tumor cells and circulating tumor DNA as liquid biopsy[END_REF][START_REF] Siravegna | Integrating liquid biopsies into the management of cancer[END_REF], and has been approved in metastatic non-small cell lung cancer as a non-invasive surrogate to tumor biopsies for the detection of activating EGFR mutations [START_REF] Douillard | Gefitinib treatment in egfr mutated caucasian nsclc: Circulating-free tumor DNA as a surrogate for determination of egfr status[END_REF]. In face of the unprecedented approval of anti-checkpoint inhibitors for the treatment of metastatic solid tumors regardless of tumor site, a similar situation could be envisaged for MSI [START_REF] Le | Mismatch repair deficiency predicts response of solid tumors to pd-1 blockade[END_REF][START_REF] Overman | Nivolumab in patients with metastatic DNA mismatch repair-deficient or microsatellite instabilityhigh colorectal cancer (checkmate 142): An open-label, multicentre, phase 2 study[END_REF][START_REF] Overman | Durable clinical benefit with nivolumab plus ipilimumab in DNA mismatch repairdeficient/microsatellite instability-high metastatic colorectal cancer[END_REF][START_REF] Kim | Comprehensive molecular characterization of clinical responses to pd-1 inhibition in metastatic gastric cancer[END_REF]. Recently, an alternative strategy was proposed for highly sensitive MSI detection [START_REF] Ladas | Enhanced detection of microsatellite instability using pre-pcr elimination of wild-type DNA homo-polymers in tissue and liquid biopsies[END_REF]. The method uses targeted nuclease digestion to enrich for altered microsatellite sequences prior to PCR amplification. This procedure reduces confounding stutter bands generated by amplification of unaltered microsatellites, therefore increasing the resolution of fragment analysis. However, pre-analytic enrichment renders this approach nonquantitative. MSI-ddPCR in contrast provides absolute quantification of MSI alleles and therefore is also suitable for other clinical applications of ctDNA, including patient follow-up [START_REF] Cabel | Clinical potential of circulating tumour DNA in patients receiving anticancer immunotherapy[END_REF]. We anticipate that MSI-ddPCR will prove its usefulness for the monitoring of immune

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checkpoint inhibitor efficacy. Early detection of non-responders should improve patient management and significantly reduce costs for the healthcare system. In summary, MSI-ddPCR allows robust, non-invasive detection of MSI with high clinical sensitivity. The method is straightforward and has the potential to be routinely applied clinically in laboratories equipped with a digital PCR platform. MSI-ddPCR is compatible with high throughput screening of patients, including those with cancers with low MSI prevalence, promising increased access to personalized treatments.

Figure 1 .

 1 Figure 1. Microsatellite instability (MSI)-ddPCR drop-off assays. A. Schematic

Figure 2 .

 2 Figure 2. Microsatellite instability (MSI) status characterization in tumor specimens by

Figure 3 .

 3 Figure 3. Microsatellite instability (MSI) detection in blood samples of advanced cancer

Acknowledgements:

We thank Aurore Rampanou, Caroline Hego, Chloe Derouet, Pascal Portois and Frederique Maraone for invaluable technical support. We thank Charles Decraene and Olivier Lantz for discussions and Christopher R Muller for critical reading of the manuscript. We also thank the group of Valérie Taly for providing us with MSI cell lines (LoVo, HCT-15).

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Xavier Tellier
email: xavier.tellier@enpc.fr

Sonia Zerhouni

Guillaume Jami

Alexandre Le Pavec

Thibault Lenart

Mathieu Lerouge

Nicolas Leduc

Cyril Douthe

Laurent Hauswirth

Olivier Baverel

The Caravel heX-Mesh pavilion, illustration of a new strategy for gridshell rationalization

Keywords: Architectural geometry, gridshell, hexagonal mesh, free-form surface, rationalization, pavilion, node repetition

The fabrication of a freeform structural envelope is usually a highly complex task. The costliest aspect is often the connections between the constitutive parts. The Caravel heX-Mesh Pavilion is a prototype that demonstrates a new rationalization strategy. Its structure, composed of a hexagonal grid of beams and cladding panels, is based on a geometry that rationalizes connections at two levels: firstly, nodes are free of geometrical torsion and are repetitive: only two types of nodes are used. Secondly, panels can easily be connected to the support beams as they are orthogonal to each other. The mechanical behavior is validated by finite-element analysis. We generate these meshes by numerical optimization from a smooth target surface, with an initialization derived from the asymptotic case and surface theory. The pavilion shows an alternative way of rationalizing a gridshell beyond the popular planar-quad meshes and circular/conical meshes. It also demonstrates a way to generate hexagonal gridshells which are not necessarily synclastic, this limitation being typically imposed to achieve planarity of cladding panels.

Introduction

Free-form architectural structural envelopes have become increasingly popular in the past decades. The cost of these projects is usually strongly impacted by the fabrication of the nodes, which often need to be all unique. The connection between beams and panels is also problematic, as the kink angles between adjacent panels often vary significantly.

Two main geometrical strategies have been studied and used to simplify node fabrications in order to reduce their cost. The first one is to use a geometry that allows for torsion-free nodes. In such nodes, all the median planes of the incoming beams meet on a common axis. This property can be achieved by having all beams in a vertical plane or by using circular or conical meshes [START_REF] Liu | Geometric modeling with conical meshes and developable surfaces[END_REF][START_REF] Pottmann | The focal geometry of circular and conical meshes[END_REF]. Planar hexagonal meshes also have torsion-free nodes, but the hexagons are necessarily non-convex (shaped like a bow-tie) in anticlastic surfaces [START_REF] Wang | Hexagonal Meshes with Planar Faces[END_REF].

The second one is node repetition. As detailed by Eike Schling in [START_REF] Schling | Repetitive Structures[END_REF], node repetition can be achieved by two means. The first one is to have the exact same geometry for each node, or for some groups of nodes.

For example, this can be achieved with meshes of revolution, or with isogonal molding surfaces [START_REF] Mesnil | Isogonal moulding surfaces: A family of shapes for high node congruence in free-form structures[END_REF]. The second one is to use construction tolerances or adjustable nodes to allow one or more degrees of angle variation with a same physical connector. This last option was for example used in the Neckarsulm swimming pool dome [START_REF] Bergermann | Freizeitbad aquatoll neckarsulm[END_REF].

One could then name a third aspect that is very important for the detailing of façade: the beam-panel connection. Indeed the variation of the kink angle between the panel and the top surface of the beams renders impossible to make the connection structural, and thus to use the cladding system as a structural element. Furthermore, complex joining systems are needed if the king angle is too high and sealing becomes hard to achieve. The optimization for a structural layout regarding the kink angle between beam and panel was hence at the starting point of the thought on Caravel meshes which were first presented in [START_REF] Tellier | Gridshells without kink angle between beams and cladding panels[END_REF], extended in [START_REF] Tellier | Caravel meshes: a new mesh family for rationalized design of gridshells[END_REF] and give birth to the Caravel Pavilion shown in Figure 1. In this article, we propose a new geometric configuration, based on a hexagonal mesh, which solves the three issues. In section 2, we detail this geometric structure, discuss its potential applications, and prove its existence. A generation method is then presented in section 3. Finally, in section 4, we present how this process has been applied to design and rationalize the Caravel pavilion.

A new torsion free geometrical configuration

Description of Caravel mesh characteristics

The proposed geometrical configuration is a continuation of the work presented in [START_REF] Tellier | Gridshells without kink angle between beams and cladding panels[END_REF], and is shown in This mesh can be used to design a hexagonal gridshell covered by tri-folded hexagonal panels. In that case, the properties that we just described can significantly simplify the fabrication, in particular with respect to the connections between the structural elements, as shown in Figure 3. Firstly, the contact between panels and beams top surface is perfect and the angle between beam webs and panels is 90°.

Thanks to these properties, standard low-cost connections can be used to structurally connect beams to panels. Secondly each node is torsion-free: medial axes of beams meet on a common axis. As a result, structural depth can easily be given to the grid. Thirdly, one half of the nodes are planar, and for the other half, beam planes intersect at 120°, thus allowing a standardization of all beam connections.

The thus formed gridshell is a cladded honeycomb structure. We note that Jiang et al. [START_REF] Jiang | Freeform honeycomb structures[END_REF] proposed a method to design this type of structure without torsion on an arbitrary target surface. However, they do not constrain the node axes to be normal to the surface. This is major difference with our configuration, which is therefore much more constrained geometrically. The proposed properties (a), (b) and (c) form a new type of geometrical structure, based on a mesh in which face normals and vertex normals are coplanar. The proposed application is based on a hexagonal pattern, but many other types of patterns and structural applications are possible [START_REF] Tellier | Gridshells without kink angle between beams and cladding panels[END_REF][START_REF] Tellier | Caravel meshes: a new mesh family for rationalized design of gridshells[END_REF]. For clarity, we propose to name this type of mesh a Caravel mesh, a name which invites to explore meshes with CoplanAR fAce and VErtex normaLs.

Proof of existence on any smooth surface approximation

In [START_REF] Tellier | Caravel meshes: a new mesh family for rationalized design of gridshells[END_REF], we prove by construction the existence of the geometric structure described in the previous sections in the asymptotic case, i.e. in the case where a mesh approximates a surface with smaller and smaller face size. In this case, properties can then be described in the setting of smooth differential geometry. It can be shown that it is asymptotically possible to construct our geometry from a planar hexagonal mesh approximating a smooth surface S without umbilics and such that one family of hexagon edges is aligned with curvature directions.

Let us start by introducing a classical local approximation model of a surface. A surface S can be locally approximated at any point P at the second order by a paraboloid. This paraboloid is elliptic if the Gaussian curvature K is positive (i.e. the surface is synclastic), cylindrical if K=0, and hyperbolic if K<0 (i.e. the surface is anticlastic). The equation of this paraboloid is, in the tangent plane at P:

= 1 2 ( + )
Where k1 and k2 are the principal curvatures. The unit normal of the paraboloid at a point (x,y,z) neighboring P is given by :

( , ) = ( , )(-; -; 1 )

The application ( , ) ↦ ( , ) is referred to as the surface Gauss map. The value of the real factor ( , )will not be important here. Considering a neighborhood such that | | ≪ 1/ and | | ≪ 1/ , ( , ) belongs at the first order to a horizontal plane, and is then an orthotropic dilatation in the directions

x and y. The ratios of the dilatation in the x and y directions are k1 and k2, up to an homothety (the scaling of the Gauss map is not important in our construction).

After this preliminary considerations, we are now going to construct a mesh and its vertex normals satisfying the properties described in the previous sections. We will use a capital N to describe the mesh normals, as they often differ from the surface normals n.

Let us first consider a series of planar hexagonal meshes with decreasing faces sizes approximating S, and with one family of edges aligned with a principal curvature direction. As mesh size tends towards zero, each hexagon tends to have central symmetry and to be inscribed in a homothetic copy of the Dupin indicatrix, as explained in [START_REF] Wang | A Note on Planar Hexagonal Meshes[END_REF] -the Dupin indicatrix being the conic resulting from the intersection between the paraboloid and the plane z = 1. As shown in Figure 4 (left), the surface Gauss map of the Dupin indicatrix is also a conic, with equation + = 1 (up to a homothety).

Let us choose an hexagon ABCDEF in Dupin ellipse with AB and DE aligned with the principal direction (ie with the axes of Dupin ellipse) and pick three vertices to form a triangle ACE. We build the Fermat center P of ACE, i.e. the point such that , ! =

, " ! = " , ! = 120°. P will be a flat node of the mesh, and A, C and E will be 3D nodes. AP, CP, and EP will correspond to beams. Since adjacent hexagons are congruent in the limit case, the angles between beams at E is also 120°. Therefore this construction yields properties (b) and (d).

In order to fulfill property (b), the mesh normal at P, NP, must be the normal of triangle ABC. This normal is, in the space of normals, the center of the Gauss map of the Dupin indicatrix. As we want no torsion along PA, PE and PC (property (a)), the mesh normal at A, NA, must lie on a line lA parallel to (AP) passing through NP. In the general case, this line does not include the surface Gauss map at A, therefore % & ≠ ( (this is a second order difference, as the normals are equal at the limit). Now, considering adjacent hexagons Figure 4 (middle), we observe that the lines previously built intersect at one point with the same lines from the neighboring hexagons if and only if edges AB and DE are aligned with a principal direction (i.e. parallel to an axis of the conic). These intersections points NQ1, NQ2… are the normals of the 3D nodes.

The hexagonal network (in red) along with the attached normals described by the hexagonal mesh on the Gauss map (also in red) verify all the geometrical properties described in the previous sections.

Generation method

We now give a method to generate the geometrical configuration described in section 2. In a first step, we compute an initial mesh close to this configuration. In a second step, we optimize node position and normals orientation such that properties (a) to (d) are satisfied within given tolerances.

Mesh initalization

The designer starts hence by defining a target geometry, in the present case a channel like smooth surface (see Figure 5 i ). The surface is then meshed by its principal curvature lines (see Figure 5 ii ). One then chooses one of the two principal directions to build a two-colorable triangular mesh on the surface picking out one out of two nodes of the curvature network (see Figure 5 iii ). The nodes of this triangular mesh are the future 3D-nodes of the caravel mesh. After what, the Fermat points of one half of the triangular faces (the colored ones in Figure 5 iii ) are constructed. Those nodes are planar and will be the future 2D-nodes of the caravel mesh. Connecting finally 2D-nodes and 3D-nodes, one gets a hexagonal mesh, aligned with principal curvature directions where one out of two nodes is planar (see Figure 5 iv ). One needs then to assign a normal to each node. By default one chooses the normal to the triangular faces for the 2D-nodes and the normal to the surface for the 3D-nodes.

i.

Defining the target smooth surface ii.

Plotting the principal curvature lines

iii.

Paneling the surface with 3D-nodes iv.

Building the hexagonal mesh v.

Optimizing the mesh for torsion free nodes, coplanarity, orthogonality and 120° property Figure 5: Shape generation method for the Caravel Pavilion, from smooth Surface to optimized mesh.

Geometrical optimization

Based on the asymptotic analysis, this mesh and its oriented normals is close to a Caravel mesh, but it does not fulfill the properties (a) to (d) exactly yet. To meet those properties, an optimization is necessary.

A non-linear optimization is conducted within the framework of Rhinoceros © 's plugin Kangaroo2. The underlying algorithm is made of multiple geometrical criteria which are expressed as a projection [START_REF] Bouaziz | Shape-up: Shaping discrete geometry with projections[END_REF].

The first criterion forces every node of valence three to be torsion-free (property (a)), by requiring the normals of the three neighboring nodes to be such that the median planes of the beams coming to the node meet on a common line corresponding to its normal. The second criterion makes the 2D nodes and their respective neighboring nodes coplanar (property (b)). The third one constraints the normals of the 2D nodes to be respectively aligned with the normals of the planes defined by their three neighboring nodes (property (c)). Finally, the fourth one forces 3D nodes to be, when projected on the triangles defined by their three neighboring nodes, the Fermat points of these triangles (property (d)). Complementary criteria are also set for smoothing purposes, such as proximity to the target surface, and to treat the boundaries.

This optimization problem is highly non-linear and non-convex. Hence, there is no theoretical guarantee that the mesh obtained after iterations of the optimization algorithm perfectly meets all the criteria or reach a global minimum of the problem. However, empirically, the resulting mesh tends to satisfy all the geometrical properties within very low tolerances as shown in Figure 5 v :

• less than 0.5° for torsion angle at nodes (a),

• less than 1 mm for planarity default at 2D-nodes (b), for member sizes around 500 mm in the Caravel Pavilion case,

• less than 1° distortion between panel normal and beam normals (c) and

• less than 0.5° tolerance on the 120° intersections between beam planes on 3D-nodes (d).

Thus, this two-step method (initialization-optimization) allows in practice to generate a geometrical configuration as described in section 2 within acceptable tolerances.

Comparison with other hexagonal meshing methods

There exist many other methods to generate hexagonal meshes with different geometrical properties. In architecture, the generation of hexagonal meshes with planar faces has raised a lot of attention [START_REF] Wang | Hexagonal Meshes with Planar Faces[END_REF][START_REF] Wang | A Note on Planar Hexagonal Meshes[END_REF][START_REF] Troche | Planar hexagonal meshes by tangent plane intersection[END_REF][START_REF] Vaxman | Dupin Meshing : A Parameterization Approach to Planar Hex-Dominant Meshing[END_REF]. A major limitation of planar hexagon meshes is that they have non-convex faces when curvature is negative [14, p705], which is not the case with Caravel meshes. Planar hexagonal meshes are preferably aligned with curvature directions to avoid skewed faces [START_REF] Li | Planar hexagonal meshing for architecture[END_REF]. The proposed method then yields the same distribution and size of faces, except that faces are always convex.

Other popular methods to generate meshes with mostly hexagonal faces include Voronoi meshes [START_REF] Kunze | Geodesic voronoi diagrams on parametric surfaces[END_REF][START_REF] Tonelli | Statics aware voronoi grid-shells[END_REF],

methods based on duals of triangular meshes [START_REF] Troche | Planar hexagonal meshes by tangent plane intersection[END_REF], and subdivision schemes [START_REF] Claes | A corner-cutting scheme for hexagonal subdivision surfaces[END_REF]. However, they often yield unstructured meshes, and they offer a poor control of edge orientation, which is critical in the present approach. They also do not offer the Caravel properties.

The X-Mesh pavilion

In order to demonstrate the innovation and the adaptability of the workflow hereby presented, it was applied to the design and realization of a pavilion for the IASS 2019 Pavilion contest.

Design

The shape of the pavilion is a doubly curved envelope, of which transversal sections are horseshoe arches.

The shape has both synclastic and anticlastic portions, as shown in Figure 6, and demonstrates the ability of our method to generate structures on a broad class of forms. The pavilion is high enough to walk under (see Figure 6).

The starting point of the design is a horseshoe arch surface, chosen for its aesthetic and its functionality relatively to the implantation site at the IASS exhibition. Such shape cannot be funicular. It therefore highlights the potential of the proposed constructive system to resist bending. Bending stiffness and resistance is allowed by the structural depth of the structure. Usually, structural depth in gridshells amplifies fabrication issues due to geometrical torsion at nodes. This is not the case with Caravel meshes thanks to their torsion-free nodes.

This surface was discretized by a hexagonal mesh aligned with curvature lines. There are two ways to align a hexagonal mesh with curvature directions. The alignment with the horizontal curvature lines is chosen in order to create the graphic impression of an ascending movement, as the triangular panels are pointing upwards diagonally.

The concept of this pavilion focuses on highlighting the geometrical properties through the architectural treatment of the elements, especially the connections. The conception of each type of node is entirely based on the geometrical rationalization that simplifies its fabrication, and takes advantage of each geometrical strategy to define its technology.

Surface optimization for mesh alignment with ground level

For aesthetic and fabrication reasons, it was desirable to align the mesh with the ground. Since the mesh follows curvature directions, we modified the target surface so that its bottom boundaries (verifying z=0) are curvature lines. By the Joachimsthal theorem, this property is achieved if the surface has a constant slope along the lines z=0. As our target surface is a B-spline, the slope at the ground boundary is given by the bottom two rows of control points. We therefore optimized the position of these control points so that the slope of the surface is constant at the ground level (see figure 5 ii ).

Detailing

Beams

The torsion free nodes allowed us to conceive the main hexagonal grid as a beam structure. Indeed, since adjacent node axes are coplanar, their common plane defines the median planes of beams. This allowed to build all elements out of sheet materials. After structural analysis (see section 4.5), it was hence decided to build the hexagonal grid 4 mm thick laser-cut aluminum plates and to paint them pink to differentiate them from the cladding (see figure 1).

Two standard connectors

We took advantage of the node repetition to design two standards types of connections, one for each type of geometrical node (2D and 3D). These nodes are showed in Figure 7 and 8. 2D nodes (figure 7) offered us the possibility to create a structural connection through the common plane of the three incoming beams.

This connection was realized by two 1 mm thick galvanized steel panels on the superior and inferior edge of the three beams. Structurally, the height of the beams can be adjusted to have sufficient vertical inertia.

The horizontal inertia is guaranteed by the cladding panels. Panels are fixed to the edges of the three beams thanks to locking tabs in which the carved panels are pushed into. As beams are orthogonal to panels, all the cuts are orthogonal to the sheet planes, and can thus be realized by 2D laser cutting. The technological treatment of the 3D nodes (figure 8) is based on the existence of an axis assigned to each node and on the fact that beam planes always intersect at 120°. For 3D nodes, a hinged connection is built from two pieces, cut out from 4 mm aluminum sheets and swaged in a stamping press. These formed parts are all identical and guarantee the 120° angle between each beam plane of the 3D node. They are maintained in place thanks to a single screw, materializing the direction of the normal axis of the node.

A lug has been added on every member end as well as corresponding adjusted square holes in the stamped aluminum placket in order to lock the rotation of the beams in the nodes.

Edge details

The two types of junctions are adapted on the free edges of the pavilion; thus it is unnecessary to develop an entirely different solution for the nodes on the boundary (see Figure 9). Only two beams are joined on 3D nodes on naked edges, so in order to keep the same technological solution, we added a short stub that closes the node. 2D nodes with only two beams can work the same way on naked edges than on interior edges, as long as the panel is designed accordingly. This degree of freedom offers the opportunity to adapt the general design of the edge.

Figure 9 : Free edge treatment 3D node on the left, 2D node on the right

Supports

The pavilion was designed to be autostable so that it does not require ground anchors. Practically, the ground supports are connected between them to form a rigid frame through a floor beam on each side with four ties triangulating the whole, all made of 4 mm thick aluminum plates (see Figure 6). The supports themselves which are modeled as hinges from a mechanical point of view, are made of two parallel plates slotted into the floor beams that maintain two plates fitting the beam plane and allowing rotation along the desired inclined axes (see Figure 10). A set of lugs and stops are then designed in order to prevent lifting. This detail can accommodate wide angle variations in two directions, such that it could be used at each of the ten supports. Furthermore, it requires only one bolt to tighten, thus accelerating the assembly and disassembly of the pavilion.

Figure 10: Detail of the hinged supports and principle of assembly

Structural check

To ensure the safety and structural behavior of the pavilion, we implemented a finite element model of the structure and verified its behavior under dead load and accidental load (0.2kN applied horizontally at 1 m from the ground). The structural check follows then the approach of Eurocode: no damage at ELU and reasonable displacements at ELS (here <4 cm under the lateral push). The whole pavilion (i.e. beams, panels, basement) is modeled considering hinged supports between structural members and ridge beams of the support (see figure 11ª). Releases at 2D-nodes and 3D-nodes have then be model carefully:

• At 2D node, the forces and moments are transferred from one beam to the other essentially by the panels. Therefore, they have to go through the connectors. In the Caravel Pavilion, these connectors work by contact, through the four L-shaped stops on both top and bottom edges of each beam that slide into corresponding rectangular holes in the panels until they hit one of their edges (see figure 7). Holes are accurately cut such that the stops are also laterally in contact with the panels within very low tolerances. For simplicity reasons, the panels are thus represented by sets of galvanized steel strips on the top and bottom sides of the beams (1 mm thick and 20 mm wide) as can be seen in Figure 11 a . To connect the strips with the beams neutral axis, rigid links span between the ends of the hinged strips and the beams' axis. Finally considering that the screw/washer system prevents the beams back sliding, relative translations at beams end have been blocked. To account for buckling of the strips, only tension forces are considered in the analyses of the strips.

• At 3D nodes, all forces and moments are transferred through the stamped aluminum plackets which are not perfectly stiff compared to the members. Therefore, a plate finite element model of the placket was used to estimate the stiffness of the connector in the six directions (see Table 1). Linear static analyses have then been conducted under gravity load and horizontal push. With the chosen cross sections for the aluminum beams (100 mm height and 4 mm width), the pavilion was found very safe: the maximum stress equals 80 % of limit stress, the maximum displacement equals 3.9 cm (2.1 cm vertically) and the eigenfrequency equals 1.7 Hz.

The analysis of displacements (shown in Figure 11 b ) reveals that the structure works as an arch defined by a cut along the longitudinal plane YZ. Indeed the displacements are almost equals for all the points on a transversal cut along any plane XZ (except close to the support on the ground where the chosen narrowing is a handicap). The double curvature of the structure ensures thus a certain stiffness in the transversal direction and the transfer of forces from the free edges toward the supports This arch behavior is confirmed by the analysis of member forces. When seen from aside the prototype forms a horseshoe arch, which resists forces through a mix of strong axis bending moment Mz (Figure 11 d ) and normal force Nx (Figure 11 c ). The structure is clearly not a funicular shell dominated by membrane forces but still it can described as a gridshell because all the nodes transfer bending moments along the strong axis giving out-of plane bending stiffness to the structure while the bracing by the panels at the planar nodes gives membrane shear stiffness.

It is worth also remarking that the six directions were solicited. Indeed, due to the hexagonal arrangement of the grid and to the curvature of the global shape, normal forces are transformed into normal forces and shear forces, and bending moments in the strong direction are transformed as well into bending moment in both directions and torsion (see Figure 11 e and 11 f ). Despite the low values of these moments, their contribution to the overall compliance of the structure is high. Indeed, the bending stiffness of the strong axis is 500 times higher than the weak axis stiffness and 160 times higher than the torsion stiffness.

It seems hence that for further developments of the Caravel meshes, it will be necessary to investigate the connection between beams and panels so that it can transfer shear and that the system works as an I-beam and change the order of magnitude of the torsional stiffness and weak axis bending stiffness. This could not be done for the presented pavilion which gave priority to assembly methods which were driven here by the fact that the pavilion had to be assembled and disassembled in one day. Besides, it is noticed here that the pavilion was then rebuilt for a three days' exhibition on February 2020 in the main building of

Conclusions

In this article, a new family of meshes combining several rationalization properties was introduced:

Caravel meshes. Their potential for structural applications was demonstrated by designing a full scale pavilion which was exhibited at the 60th IASS Symposium in October 2019, rebuilt for a temporary exhibition in Paris-Malaquais and will be reassembled a third times for the pavilion exhibition of the "Advances in Architectural Geometry 2020" conference. It was a unique experience of collaboration between mathematics, structural engineering and architecture students and researchers working together on the design of a prototype from differential geometry theory to fabrication. The main issues of the design process of caravel meshes have been tackled: geometric characteristics of caravel meshes, proof of existence on any smooth surface, generation method, structural analysis and detailing. The proposed approach for surface rationalization which only uses two types of nodes (here only two types: 2D flatnode and a 3D 120°-node) seems very promising and opens up a lot of perspectives for innovative design of gridshell structures.

The geometrical and structural principles developed here were meant to be scalable at other structural scale. Indeed from a geometrical point of view, nodal axes of the prototype are clearly identifiable through the bolt axes: the height of the members is a parameter that can be freely adjusted by the designer who can therefore adjust it to the scale and required structural depth. From a technological point of view, the details shown here are adapted to the scale of the prototype and to the fact that it should be dismountable.

Should it be built in another context or at another scale, the details should be changed but thanks to the geometrical properties of the structure, those changes are relatively simple or remain in the framework of standard detailing. Two variants of this prototype have indeed been built by the authors: one 8m span wooden structure and its 1/10 MDF mockup shown in Figure 12.

Figure 1 :

 1 Figure 1: Three quarter view of the pavilion built for the IASS symposium in Barcelona, 2019

Figure 2 .

 2 Figure 2. The configuration is based on a hexagonal mesh with non-planar faces. Each of its nodes is

Figure 2 :

 2 Figure 2: Geometrical configuration

Figure 3 :

 3 Figure 3: Geometry of connections structural elements

Figure 4 :

 4 Figure 4: Asymptotic construction

Figure 6 :

 6 Figure 6: Shape and dimensions of the Caravel Pavilion.

Figure 7 :

 7 Figure 7: Technological solution for the 2D nodes

Figure 8 :

 8 Figure 8: Technological solution for the 3D nodes

theFigure 11 :

 11 Figure 11: Structural analysis of the pavilion

Figure 12 :

 12 Figure 12: Another application of caravel meshes for a 8m wooden structure and its 1/10th mockup

Table 1 :

 1 Identified characteristics of the 3D-connectors in the member local axes.

	These stiffness's are then introduced in the global structural model following the local axes of the
	various members. It appeared that half of the structure displacement comes from the connectors
	stiffness.
	Force	Normal X	Shear Y	Shear Z	Torsion X	Moment Y	Moment Z
	Stiffness	21 MN/m	10 MN/m	21 MN/m	25 kNm/rad 25 kNm/rad	∞

Acknowledgement

The authors would like to thank all funding partners as well as Eike Schling and Daniel Piker for fruitful discussions.

Funding

The design and fabrication of the Pavilion were financially supported by the I-Site Impulsion grant "Games", by the Steel contractor VIRY who gently provided the laser-cut panels as well as technical support and by ENSA Paris-Malaquais. This work is also part of the PhD thesis of the first author who is funded by Labex MMCD (http://mmcd.univ-paris-est.fr/).

Conflict of Interest

The authors declare that they have no conflict of interest.