Canan G Nebigil

Laurent D Ésaubry

Laurent Désaubry

Author. Canan G Nebigil

The role of GPCR signaling in cardiac epithelial to mesenchymal transformation (EMT)

Congenital heart disease is the most common birth defect, affecting 1.35 million newborns every year. Heart failure is a primary cause of late morbidity and mortality after myocardial infarction. Heart development is involved in several rounds of epithelial-to-mesenchymal transition (EMT) and mesenchymal-to-epithelial transition (MET). Errors in these processes contribute to congenital heart disease, and exert deleterious effects on the heart and circulation after myocardial infarction.

The identification of factors that are involved in heart development and disease, and the development of new approaches for the treatment of these disorders are of great interest. G protein coupled receptors (GPCRs) comprise 40% of clinically used drug targets, and their signaling are vital components of the heart during development, cardiac repair and in cardiac disease pathogenesis. This review focuses on the importance of EMT program in the heart, and outlines the newly identified GPCRs as potential therapeutic targets of reprogramming EMT to support cardiac cell fate during heart development and after myocardial infarction. More specifically we discuss prokineticin, serotonin, sphingosine-1-phosphate and apelin receptors in heart development and diseases. Further understanding of the regulation of EMT/MET by GPCRs during development and in the adult hearts can provide the following clinical exploitation of these pathways.

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2 KeyWords Congenital heart disease; heart development; heart failure; GPCR; epithelial to mesenchymal transformation (EMT); cardiac progenitor cells, cardiac stem cells; prokineticin; apelin, serotonin; sphingosine-1-phosphate

EMT during Heart Development and diseases

Epicardial-to-mesenchymal transition (EMT) plays a critical role in cardiac development [START_REF] Thiery | Epithelial-mesenchymal transitions in tumour progression[END_REF]. The down-regulation of intercellular adhesion junctions and the expression of EMT-inducing transcription factors are the two key events in the EMT process (Fig. 1). All cells in the heart arise from one or more rounds of endocardial-EMT and epicardial-EMT. Endocardial-EMT plays a key role for formation of the cardiac valves and for complete cardiac septation. The early developing heart consists of an endocardial layer surrounded by a myocardial layer. These layers are separated by extracellular matrix, also known as cardiac jelly. The endocardial cushions become visible in the midgestation embryos. These cushions later develop into the atrioventricular (AV) and outflow tract (OFT) valves [2; 3]. Endocardial cells that line the AV cushions undergo EMT in the midgestation embryos to populate them with mesenchymal cells, thus contributing to precursors of the future valve interstitial cells. OFT cushion development also involves endocardial EMT in a similar manner as that of AV but occurs later in development [2; 3] (Fig. 2).

Epicardial-EMT is a physiological process in developing heart, such as myocardial differentiation, development of coronary vascular network, fibrous annulus and valves. Epicardium originates as proepicardium, a cluster of coelomic cells at the caudal end of the developing heart (E9.0 to E9.5 in the mouse). The formation of most nonmyocardial cells starts with the development of the proepicardium [4]. Proepicardial cells are transferred to the myocardial surface, where they attach and spread, forming a continuous monolayered patch of epithelial cells, called epicardium [START_REF] Manner | Munoz-Chapuli The origin, formation and developmental significance of the epicardium: a review[END_REF]. Epicardial cells are important for both myocardium and coronary development (Fig. 2). During formation of the epicardial epithelium, EMT is

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3 initiated, allowing epicardial-epithelial cells to transform into a population of highly invasive, mesothelial epicardium-derived cells (EPDCs) [START_REF] Ruiz-Villalba | Perez-Pomares Characterization of epicardial-derived cardiac interstitial cells: differentiation and mobilization of heart fibroblast progenitors[END_REF]. The EPDCs formed via EMT invade the sub-epicardial matrix and migrate into the myocardium, where they differentiate into several cardiac lineages, including coronary smooth muscle cells [START_REF] Bax | In vitro epithelial-to-mesenchymal transformation in human adult epicardial cells is regulated by TGFbeta-signaling and WT1[END_REF], and perivascular and cardiac interstitial fibroblasts [START_REF] Van Tuyn | Epicardial cells of human adults can undergo an epithelial-to-mesenchymal transition and obtain characteristics of smooth muscle cells in vitro[END_REF], coronary endothelial cells [START_REF] Zangi | Modified mRNA directs the fate of heart progenitor cells and induces vascular regeneration after myocardial infarction[END_REF], possibly cardiomyocytes [10; 11] and adipocytes [START_REF] Yamaguchi | Adipogenesis and epicardial adipose tissue: a novel fate of the epicardium induced by mesenchymal transformation and PPARgamma activation[END_REF]. Importantly, reactivated epicardial cells after heart injury proliferate and undergo EMT or mesenchymal-toepithelial transition (MET), and subsequently differentiate into cardiac cells to contribute to cardiac repair, or protect cardiomyocytes via paracrine interaction [13; 14].

The impact of errors in EMT on the heart

Several studies have suggested that EMT is essential for proper embryogenesis, without which developmental defects or embryonic lethality can occur [START_REF] Thiery | Epithelial-mesenchymal transitions in tumour progression[END_REF]. In the adult heart, disease-mediated activation of EMT may contribute to degenerative valve disease, cardiac fibrosis, and the myocardial injury response [3].

Valve diseases lead to significant morbidity and mortality. Valve diseases are described by expression of mesenchymal markers, such as vimentin (VIM) and smooth muscle actin in the valvular interstitial cells [START_REF] Liu | The emerging role of valve interstitial cell phenotypes in regulating heart valve pathobiology[END_REF]. During development, progenitor cells derived from endocardial-EMT also express these markers indicating that potential reactivation of endocardial-EMT occurs in valve disease. In accord with these findings, enhanced valve endocardial-EMT has been shown to contribute to pathological changes in diseased leaflets in animal models and patients [START_REF] Macgrogan | How to make a heart valve: from embryonic development to bioengineering of living valve substitutes[END_REF].

Endocardial-EMT forms valve progenitor cells that can be differentiated towards osteogenic or chondrogenic phenotypes, therefore contributing to valve calcification a hallmark of adult valve disease [3].

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The disturbance in epicardial-EMT not only causes non-compaction cardiomyopathies, but also early coronary vascular diseases [START_REF] Lie-Venema | Origin, fate, and function of epicardium-derived cells (EPDCs) in normal and abnormal cardiac development[END_REF]. Defective epicardial outgrowth leads to lack of annulus fibrosis formation with electrophysiological evidence of the Wolff-Parkinson-White (WPW) syndrome [START_REF] Kolditz | Schalij Epicardium-Derived-Cells in annulus fibrosis development and persistence of accessory atrioventricular pathways[END_REF].

An Ebstein-like valve phenotype could be related to impaired homing of EPDCs to valves [START_REF] Lie-Venema | Origin, fate, and function of epicardium-derived cells (EPDCs) in normal and abnormal cardiac development[END_REF]. Myocardial infarction reactivates endogenous epicardium which reexpresses embryonic markers, such as WT1, raldh2, Tbx18, tcf21 [10; 19]. Injection of human EPDCs (hEPDCs) into the infarcted myocardium preserves cardiac function and reduces remodeling after the onset of infarction [START_REF] Winter | Preservation of left ventricular function and attenuation of remodeling after transplantation of human epicardium-derived cells into the infarcted mouse heart[END_REF].

Cardiac fibrosis is also defined as an abnormal thickening of the heart valves due to an inappropriate number of fibroblasts. Fibroblast plasticity occurs either by replication of the resident myocardial fibroblasts, migration and transformation of circulating bone marrow cell-derived fibroblast precursor cells (fibrocytes), or transformation of resident non-fibroblast cells such as endothelial and perivascular cells into fibroblasts and myofibroblasts [21; 22]. However, the epicardium is the major source of cardiac fibroblasts, and reactivated by epicardial-EMT during pathologic fibrogenesis after myocardial infarction [START_REF] Fang | Yutzey Epicardium-derived fibroblasts in heart development and disease[END_REF]. Indeed, cardiac adaptive fibrosis is necessary to maintain the cardiac structural integrity and pressuregenerating capacity of the heart. However, transforming into myofibroblasts by pathological stress, cardiac fibrosis leads to the progression to heart failure [3].

Fibrogenesis also contribute to arrhythmogenesis by increasing ventricular stiffness.

After the myocardial infarction, epicardial-EMT increases number of fibroblast that can be differentiated into adipocytes to generate epicardial adipose tissue (EAT) [START_REF] Yamaguchi | Adipogenesis and epicardial adipose tissue: a novel fate of the epicardium induced by mesenchymal transformation and PPARgamma activation[END_REF]. Formation of EAT, myofibroblasts and vasculogenic cells are regulated by EMT and MET in epicardial cells (Fig. 3). EAT plays a key role in the association between obesity and coronary artery disease (CAD). Although anatomically EAT may provide mechanical support for the coronary arteries, adipokines produced by EAT play key

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roles in the development of atherosclerotic plaque [START_REF] Salazar | Epicardial Fat: Physiological, Pathological, and Therapeutic Implications[END_REF]. Since adipogenesis is a mesenchymal developmental process, the suppression of spontaneous EMT and retention of mesothelial cells at the epithelial stage reduces EAT formation in response to adipogenic stimuli [START_REF] Qureshi | Nebigil A prokineticin-driven epigenetic switch regulates human epicardial cell stemness and fate[END_REF].

Although reactivation of epicardial-EMT process contributes to disease pathogenesis, redirection of disease-related epicardial-EMT stimulates cardiac regeneration. G protein-coupled receptor (GPCR) signaling controls EMT during heart development, and may redirect EMT after myocardial infarction to regenerate heart.

GPCR signaling

Another aspect strongly associated with cardiac development and defects is G protein-coupled receptors (GPCRs) that are also called seven transmembrane helical (7TM) receptors [START_REF] Gurevich | GPCRs and Signal Transducers: Interaction Stoichiometry[END_REF]. GPCRs transduce extracellular signals from endogenous hormones and neurotransmitters or from exogenous factors, such as photons of light, odorants, and tastants (chemosensory receptors). GPCRs are classified into 6 subgroups based on the sequence homology [START_REF] Gurevich | GPCRs and Signal Transducers: Interaction Stoichiometry[END_REF]. These subgroups are Class A (rhodopsin-like receptors), Class B (secretin receptor family), Class C (metabotropic, glutamate receptors), Class D (pheromone receptors), Class E (cyclic AMP receptors) and Class F (frizzled receptors). Among the GPCRs, circulating catecholamines (Class A) play an important role in the control of cardiovascular function, via their adrenergic receptors, by mediating the actions of sympathetic nervous system activation. In cardiovascular diseases, the most used GPCRtargeted drugs are the blockers of β-adrenergic (β-AR) and angiotensin (AngII) receptors. AngII receptor blockers have been used for the treatment of hypertension and cardiovascular disorders [START_REF] Devost | Conformational Profiling of the AT1 Angiotensin II Receptor Reflects Biased Agonism, G Protein Coupling, and Cellular Context[END_REF]. During heart failure, β1-ARs are hyper-stimulated, which induces a cardiotoxic effect. However, β2-ARs exert cardioprotective roles,

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counteracting the deleterious effects of chronic sympathetic stimulation in failing hearts. Thus, β1-specific ARs antagonist together with β2-AR agonist have been used for treatment of heart failure [START_REF] Woo | Biased beta2-adrenoceptor signalling in heart failure: pathophysiology and drug discovery[END_REF]. Some of the GPCRs are involved in heart development such as prokineticin receptor-1 (PKR1), sphingosine-1-phosphate (S1P) 1 receptor (S1P 1 R) and serotonin (5-HT)-2B receptors (Fig. 4). Here we discuss mainly these 3 GPCRs that are involved in heart development and diseases.

Prokineticin receptors in heart development and disease

Prokineticins as angiogenic and anorectic hormones, exert their biological activity by binding to their cognate GPCRs; namely, PKR1 and PKR2 [START_REF]Nebigil Prokineticin receptors in cardiovascular function: foe or friend?[END_REF]. A critical role of epicardial-PKR1 signaling during cardiac development has been recently demonstrated. Mice lacking PKR1 in their Wt1 + epicardial cells showed partial embryonic lethality, due to impaired EMT in their hearts [START_REF] Arora | Prokineticin receptor-1 signaling promotes Epicardial to Mesenchymal Transition during heart development[END_REF]. PKR1 signaling promotes the proliferation and EMT of epicardial cells of the Wt1 + lineage and their subsequent differentiation into endothelial and vascular smooth muscle cells during heart development. PKR1 activation or overexpression induces changes in embryonic epicardial cell morphology, actin cytoskeleton remodeling, and EMT-gene expression profile [START_REF] Arora | Prokineticin receptor-1 signaling promotes Epicardial to Mesenchymal Transition during heart development[END_REF].

PKR1 orchestrates epicardial-cardiomyocyte interactions in the developing heart. In mice, inactivation of PKR1 in epicardial-derived cells also causes epicardial detachment and a reduction of the adjacent myocardium compact zone, impairing myocardial growth [START_REF] Arora | Prokineticin receptor-1 signaling promotes Epicardial to Mesenchymal Transition during heart development[END_REF]. Although myocardial expression of PKR1 induces coronary vessel formation by a paracrine process [START_REF] Urayama | Prokineticin receptor-1 induces neovascularization and epicardial-derived progenitor cell differentiation[END_REF], a reciprocal interaction between the epicardium and cardiomyocytes via PKR1 signaling has also been shown to be involved in EPDC function and myocardial development and function [START_REF] Arora | Prokineticin receptor-1 signaling promotes Epicardial to Mesenchymal Transition during heart development[END_REF].

EMT/MET plays important roles in stem cell differentiation and dedifferentiation or reprogramming [START_REF] Chen | Epithelial-mesenchymal transition (EMT): A biological process in the development, stem cell differentiation, and tumorigenesis[END_REF]. Prokineticin-2/PKR1 signaling in hEPDCs

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A N U S C R I P T 7 controls a switch between formation of EAT and vasculogenesis via inhibiting EMT (Fig. 3). Prokineticin/PKR1-mediated inhibition of EMT allows the regaining of epithelial features with self-renewal and rapid generation of vascular precursors [START_REF] Qureshi | Nebigil A prokineticin-driven epigenetic switch regulates human epicardial cell stemness and fate[END_REF].

A newly described epigenetic route involving histone modifications by KDM6A overcomes the intrinsic limitations of differentiating hEPDCs into vasculogenic cells.

Activation of KDM6A by PKR1 signaling plays a key role in the demethylation of H3K27me3, leading to promoter derepression and induction of lineage-specific marker expression for endothelial cells and smooth muscle [START_REF] Qureshi | Nebigil A prokineticin-driven epigenetic switch regulates human epicardial cell stemness and fate[END_REF]. Prokineticin-2/PKR1-mediated KDM6A activity is also essential for the inhibition of adipogenesis in response to adipocyte induction in hEPDCs, by inhibiting EMT, and stabilizing cytosolic β-catenin levels, thereby blocking PPARγ activity. Supporting these in vitro findings, epicardial-specific PKR1 knockout mice have been shown to have an imbalance between adipogenic and vasculogenic transformation following exposure to a high-fat diet (HFD), leading to systolic and diastolic left ventricular dysfunction and severe arrhythmia [START_REF] Arora | Prokineticin receptor-1 signaling promotes Epicardial to Mesenchymal Transition during heart development[END_REF]. Mice lacking PKR1 in their tcf21 + cardiac fibroblast progenitors displayed low capillary formation. However, such mice fed a high fat diet exhibited an increased epicardial adipose tissue (EAT) and a severe regression of blood vessels, leading to cardiac dysfunction [START_REF] Qureshi | Prokineticin receptor-1-dependent paracrine and autocrine pathways control cardiac tcf21+ fibroblast progenitor cell transformation into adipocytes and vascular cells[END_REF].

Overall, these data show that PK2/PKR1 signaling plays an important role in regulating epicardial fate via epigenetic control from adipogenesis towards vasculogenesis in vitro and in vivo. Thus, the control of epicardial-EMT/MET program is a potentially valuable target for future applications in cardiac tissue engineering.

Sphingosine-1-phosphate (S1P) receptor signaling in heart development and disease

S1P is a lysophospholipid that uses GPCRs to promote cellular differentiation, proliferation, migration, cytoskeletal reorganization, and apoptosis [START_REF] Tamminga | Developing novel treatments for mood disorders: accelerating discovery[END_REF]. S1P receptors (S1PRs) are composed of five subtypes, S1P 1 R to S1P 5 R [START_REF] Kihara | Lysophospholipid receptor nomenclature review: IUPHAR Review 8[END_REF]. The deletion of the S1P 1 R in mice (S1P 1 R -/-) leads to embryonic lethality between embryonic days E13.5 and E14.5 due to several defects in the developing heart.

These defects are atrioventricular canal cushion and vascular maturation defects, and malformed myocardium [START_REF] Liu | Role of kinins in chronic heart failure and in the therapeutic effect of ACE inhibitors in kininogen-deficient rats[END_REF]. The heart condition S1P 1 R -/-hearts may also be due to the disruption of fibronectin (FN) in the epicardium due to altered EMT [START_REF] Kono | Sphingosine kinase 1 regulates differentiation of human and mouse lung fibroblasts mediated by TGF-beta1[END_REF].

This disrupted epicardial layer due to impaired EMT may explain the disorganized myocardium observed in the compact layer of S1P 1 R -/-hearts. Further analysis of the role of S1P 1 R signaling during epicardial development is needed to confirm this possibility.

Serotonin receptor signaling in heart development and disease

The serotonin (5-hydroxytryptamine, 5-HT) regulates cardiovascular functions during embryogenesis and adulthood [START_REF] Nebigil | Maroteaux Developmentally regulated serotonin 5-HT2B receptors[END_REF]. There are at least 15 receptor subtypes that belong to four populations namely, 5-HT 1/5 , 5-HT 2 , 5-HT 3 , and 5-HT 4/6/7 subtypes.

The 5-HT 2 receptors are composed of three subunits: 5-HT 2A , 5-HT 2B , and 5-HT 2C [START_REF] Nebigil | A novel role for serotonin in heart[END_REF]. The serotonin 2B receptor and serotonin uptake inhibitors are involved in heart development and diseases. Inactivation of 5-HT 2B receptor leads to embryonic midgestation lethality due to a lack of trabeculae in the heart [40; 41]. All surviving newborn mice display a severe ventricular hypoplasia caused by impaired proliferative capacity of myocytes, leading to neonatal lethality [START_REF] Nebigil | Maroteaux Functional consequence of serotonin/5-HT2B receptor signaling in heart: role of mitochondria in transition between hypertrophy and heart failure[END_REF]. This phenotype is similar to defective EMT-mediated heart malformations. A selective serotonin reuptake inhibitor (SSRIs) fluoxetine used for treatment of gestational depression. It causes dramatic postnatal mortality of the offspring with severe heart failure due to dilated cardiomyopathy in rodents [START_REF] Noorlander | Modulation of serotonin transporter function during fetal development causes dilated heart cardiomyopathy and lifelong behavioral abnormalities[END_REF]. This suggests that the 5-HT system is involved in heart development and that fluoxetine treatment during fetal development

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affects the heart, resulting in dilated cardiomyopathy. Major cardiac malformations have also been associated with prenatal SSRIs in human practice, although most of the cardiac malformations were observed after prenatal SSRI treatment [START_REF]Cuzzell Paroxetine may increase risk for congenital malformations[END_REF].

Moreover, a meta-analysis of clinical trials has shown that prenatal SSRIs significantly increased the risk for spontaneous abortion, which is also observed in mice [START_REF]Cuzzell Paroxetine may increase risk for congenital malformations[END_REF]. Nevertheless, more research is necessary to confirm the effects of SSRI and serotonin receptor ligands on cardiac development-and diseases-related EMTs.

Other GPCRs that regulate heart development and disease

A chemokine, Apelin, and its receptor, Agtrl1b (Angiotensin II receptor-like 1 receptor) has been found to regulate blood pressure [START_REF] Tatemoto | The novel peptide apelin lowers blood pressure via a nitric oxide-dependent mechanism[END_REF] and to induce a positive inotropic effect in humans [START_REF] Berry | Apelin has in vivo inotropic effects on normal and failing hearts[END_REF]. Apelin has also been shown to be required for normal vascular development in frog embryos [START_REF] Inui | Asashima Xenopus apelin and msr are required for cardiovascular development[END_REF]. The loss of Agtrl1b in mice leads to embryonic lethality due to growth retardation and cardiac malformations similar to that observed in mice having defective EMT [START_REF] Zeng | Solnica-Krezel Apelin and its receptor control heart field formation during zebrafish gastrulation[END_REF]. The earliest ligand for Agtrl1b called ELABELA promotes endoderm differentiation and subsequent cardiogenesis.

Loss of ELABELA causes a rudimentary heart or no heart in embryos [START_REF] Chng | ELABELA: A Hormone Essential for Heart Development Signals via the Apelin Receptor[END_REF].

Several orphan GPCRs have emerged as novel therapeutic targets to treat heart failure. Two such candidates, GPR37L1 (downregulated in cardiovascular diseases) and GPR35 (upregulated in myocardial infarction), were identified by correlating gene expression with clinical parameters, such as pulmonary artery pressure, left ventricular ejection fraction, and brain natriuretic peptide mRNA levels [START_REF] Min | Identification of genes related to heart failure using global gene expression profiling of human failing myocardium[END_REF]. However their role in heart development and disease-related EMT have not been studied yet.

Concluding Remarks and Future Perspectives

EMT is a key player for heart development during embryogenesis and after In animals it would be useful for cell tracing studies. The therapeutic manipulation of EMT/MET can also be a therapeutic approach for treatment of heart diseases.

GPCRs are the largest family of signaling receptors and the major targets of new and approved pharmaceuticals. Indeed, GPCR-targeted drug discovery still successfully continues. Newly approved 19% of therapeutics target GPCRs, including the antithrombotic drug (Plavix) that targets platelet P2Y12 adenosine receptors, the antidepressant drugs that are selective serotonin reuptake inhibitors (SSRIs) and act agonists or antagonists of serotonin receptors, and the anti-insomnia drug (Tasimelteon) that is a melatonin receptor agonist.

GPCRs can be a valid target for redirecting and epigenetic reprogramming EMT/MET to stimulate regeneration in the injured heart. Given the central role of the GPCR signaling pathways in both health and disease, their role in tissue engineering and modeling needs to be further explored. The efforts in discovering and developing compounds such as antibodies, allosteric modulators and biased agonists that modulate GPCRs may open a new possibility to more precisely controlling receptor signaling and ultimately therapeutic effects. Recently, the sequencing of the human genome revealed new members of the GPCR family, which brings new opportunities for developing novel therapeutics targeting these new GPCRs.

Disclosure

Authors declared no conflict of interest. We the undersigned declare that this manuscript is original, has not been published before and is not currently being considered for publication elsewhere. We confirm that the manuscript has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship but are not listed. We further confirm that the order of authors listed in the manuscript has been approved by all of us. We confirm that all authors are responsible for the content and have read and approved the

  S C R I P T 10 the cardiac injury, however, without EMT heart defects and impaired cardiac regeneration occur. Targeting cardiac EMT/MET could hold a great potential to redirect the fate of cardiac stem/progenitor cells. Tracking EMT/MET in vivo in humans by imaging techniques could be a new approach to delineate the disease stage.

Fig 2 .

 2 Fig 2. EMT-mediated events during heart development. Endocardial-EMT leads to development of endocardial cushion and valves. Several epicardial-EMT and MET is involved in coronary arteriogenesis and primary cardiomyocyte proliferation (trabeculation).

Fig 3 .

 3 Fig 3.EMT and MET-mediated biological events in adult hearts. Epicardium undergoes EMT to form mesoepithelial EPDCs that can be differentiated into adipocytes or myofibroblasts, thereby inducing arrhythmia and scars. EPDCs can undergo MET leading to the formation of epithelial progenitor cells and vascular and epithelial/endothelial precursors that can further differentiate into vasculogenic cells, thereby contributing to regeneration of heart.

Fig 4 .

 4 Fig 4. GPCRs in heart development and diseases. β1-adrenergic receptor via Gαs activates adenylate cyclase (AC), however β2-adrenergic (β2-AR) and Spingosine-1 phosphate (S1P 1 R) receptors couple to Gαi, and inhibit AC. Angiotensin receptor (ATR1) serotonin (5-HT2B) and prokineticin (PKR1) receptors uses Gαq/11 to activate phosphor lipase C (PLC) and phospho inositol 3 kinase (PI3K). These receptors are involved in heart development, regulation of contractility, survival and hypertrophic pathways.

Fig 1 .

 1 Fig 1. EMT activating signals and gene expression define epithelial and mesenchymal phenotypes.

Fig 2 .

 2 Fig 2. EMT-mediated events during heart development. Endocardial-EMT leads to development of endocardial cushion and valves. Several epicardial-EMT and MET is involved in coronary arteriogenesis and primary cardiomyocyte proliferation (trabeculation).

Fig 3 .

 3 Fig 3. EMT and MET-mediated biological events in adult hearts. Epicardium undergoes EMT to form mesoepithelial EPDCs that can be differentiated into adipocytes or myofibroblasts, thereby inducing arrhythmia and scars. EPDCs can undergo MET leading to the formation of epithelial progenitor cells and vascular and epithelial/endothelial precursors that can further differentiate into vasculogenic cells, thereby contributing to regeneration of heart.

Fig 4 .

 4 Fig 4. GPCRs in heart development and diseases. β1-adrenergic receptor via Gαs activates adenylate cyclase (AC), however β2-adrenergic (β2-AR) and Spingosine-1 phosphate (S1P 1 R) receptors couple to Gαi, and inhibit AC. Angiotensin receptor (ATR1) serotonin (5-HT2B) and prokineticin (PKR1) receptors uses Gαq/11 to activate phosphor lipase C (PLC) and phospho inositol 3 kinase (PI3K). These receptors are involved in heart development, regulation of contractility, survival and hypertrophic pathways.

The publication was made possible by grants from ERA-NET, ERA-CVD, French government managed by Agence Nationale de la Recherche (ANR) Centre National de la Recherche Scientifique.

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A N U S C R I P T 11 manuscript; and that the manuscript conforms to the Uniform Requirements for Manuscripts Submitted to Biomedical Journals published in Annals in Internal Medicine We understand that the Corresponding Author is the sole contact for the Editorial process. He/she is responsible for communicating with the other authors about progress, submissions of revisions and final approval of proofs. Signed by all authors as follows:

Canan G. Nebigil Laurent Désaubry

We would like to thank to all members of our team who contributed to this work.


C Hu

E Lemarchand

L Dormieux

F Skoczylas
email: frederic.skoczylas@centralelille.fr

Quasi isotropic Biot's tensor for anisotropic porous rocks. Experiments and micromechanical modelling

Keywords: Anisotropic rock, Biot's tensor, Poromechanical coupling, Micromechanical model

Several experimental studies, carried out on anisotropic rocks, have evidenced that even though strains, due to isotropic loading and/or internal fluid pressure, are strongly anisotropic, the resulting Biot's tensor is almost isotropic. Those results were found on two different rocks: a clay rock (France-Bure argillite) and a sandstone from the Vosges region (France). Such (a priori) surprising results led us to develop micromechanical modelling in which anisotropy comes either from an anisotropic solid matrix (and isotropic pore space) or from an anisotropic pore space (and isotropic solid matrix). The obtained results have shown that for both cases the Biot's tensor is virtually isotropic or presents a very weak anisotropy. This unambiguously supports the fact that a strongly anisotropic porous material is compatible with experimental measurements of isotropic (or quasi isotropic) Biot's tensor.

Poromechanical behavior of rocks, which accounts for couplings between fluid pressure(s) and material deformation, is crucial to be identified in many practical situations such as petroleum engineering and radioactive waste storage fields. The Biot's poroelastic theory [START_REF] Biot | Theory of elasticity and consolidation for a porous anisotropic solid[END_REF] is usually recognized as the relevant theoretical approach for describing these poromechanical couplings. The latter is a generalization of Terzaghi's theory to the case of a compressible solid matrix. The concept of elastic effective stress, initially introduced by Terzaghi, is then generalized. The Biot's coefficient is introduced to calculate effective stresses for isotropic materials. For non isotropic materials, a Biot's second rank tensor must be used, which makes tricky the experimental measurements of its components. For example a simple transverse isotropic porous material needs eight poromechanical properties to be identified [START_REF] Cheng | Material coefficient of anisotropic poroelasticity[END_REF]. As a consequence there is a lack of laboratory measurements of the latters. One can nevertheless mention the comprehensive work of Wong T.F. [START_REF] Wong | Anisotropic poroelasticity in a rock with cracks[END_REF] who gathered many results on cracked Berea sandstone. The results are diverse as the anisotropy induced by cracks may sometimes lead to significant Biot's components anisotropy or not. Other results on shales can also be found in [START_REF] Suarez-Rivera | Evaluating the poroelastic effect on anisotropic, organic rich, mudstone systems[END_REF] but they are very disperse and strongly dependent on the stress levels and on the methodology used to measure the Biot's components. The present study mainly focuses on this particular point for two isotropic transverse materials, such as the COx argillite and a Vosges sandstone. As it will be detailed further, this case involves two independent components to be identified b 1 = b 2 = b and b 3 according to the material structural axes.

This paper is organized in two parts: The first part sums up experimental measurements, carried out in the LaMCube Laboratory on argillite and sandstone, for which (quasi) identical values of b and b 3 were obtained. This important result was found despite a strong anisotropic effect of the internal pore pressure on the material strains. A theoretical approach has been derived in order to assess this a priori counterintuitive result. The latter constitutes the second part of the present study. It relies on microporomechanics approaches [START_REF] Dormieux | Microporomechanics[END_REF] applied here in the context of poroelasticity. This methodology allows to relate the tensorial Biot's coefficient to the material anisotropy. This modeling unambiguously shows that, despite a strong modeled anisotropy, the Biot's components can virtually be identical.

EXPERIMENTAL PART

MATERIALS USED

Into this part are presented typical results obtained on a clay rock (Bure argillite) and on a high porosity sandstone from the Vosges region in France. The argillite samples were originally drilled from the Callovo-Oxfordian stratum of the Meuse-Haute Marne (MHM) site in France, by URL of the French organization Andra. This argillite is mainly composed of a clay matrix, quartz and carbonate. The average proportions of these components are approximately: 45 ± 7% clay matrix, 23 ± 4% quartz, 27 ± 9% carbonate (calcite) and 5% feldspars, pyrite, and iron oxides [START_REF] Song | Multi-scale pore structure of cox claystone: Towards the prediction of fluid transport[END_REF]. However, the exact mineralogical composition of argillite varies significantly with depth. The clay matrix includes approximately 40% illite, 30% kaolinite, 5% chlorite and 25% swelling minerals (such as interstratified and smectite). Such a material has a swelling capacity that can modify its pore structure during the dehydration and/or hydration processes, which leads to porosity variations depending on its water content. These variations in porosity have a significant effect on the poroelastic properties of argillite during a hydro-mechanical loading process [START_REF] Yang | Dependence of elastic properties of argillaceous rocks on moisture content investigated with optical full-field strain measurement techniques[END_REF].

Argillite has long been identified as a transverse isotropic material [START_REF] Mohajerani | Oedometric compression and swelling behaviour of the callovo-oxfordian argillite[END_REF]. Axes 1 and 2 will characterize the plan of isotropy, which is related to the plan of horizontal layers at great depth in the Bure site. The vertical axis is therefore the revolution axis 3. On a practical point of view the ratio between Young's moduli E 1 (= E 2 )/E 3 is around 2, which is already a pronounced anisotropy. The material porosity is often measured between 15% and 18% whereas its water permeability is generally less that 10 -20 m 2 . This low value makes difficult to perform poromechanical experiments with water, as the porous fluid, since a long time is necessary, between each loading step, for the fluid pressure (or the strain values) to be stable. This also requires the use of small samples to shorten the experimental time. Both gas and water have been used to measure some poromechanical properties of the argillite but, in the results presented throughout this study, only experiments conducted with water are analyzed. The interested reader may find the results obtained with gas in [START_REF] Yuan | Measurement of biot's coefficient for cox argillite using gas pressure technique[END_REF] that give the same conclusions as with water.

[Fig. 1 about here.]

The sandstone used in the present study was obtained from a depleted gas reservoir in Vosges Mountain in the east of France [START_REF] Hu | Poromechanical properties of a sandstone under different stress[END_REF]. The initial porosity measured with distilled water is around 19.4%. The microstructure of this sandstone was observed with X-ray tomography. Several horizontal sedimentary bands were detected from the CT scanning image (see Fig. 2). This structure leads to a slight anisotropy as it will be seen in the following. The material gas permeability is as high as 10 -12 to 10 -13 m 2 . This makes easy to perform poromechanical experiments with water or with gas as the pore fluid.

In the results presented hereafter for the sandstone all the experiments were conducted with gas.

[Fig. 2 about here.]

EXPERIMENTAL METHODS AND PROCEDURE FOR ARGILLITE

As mentioned before, the COx argillite from Bure is a transverse isotropic material. As shown in Fig. 1, the isotropic plane is the bedding plane containing the structural axis 1 and 2, and axis 3 is the in situ vertical axis. The samples used for the poromechanical tests were obtained from a 'T 1 cell' which is 80 mm diameter, 250 mm long cylindrical core. These cylinders are often cored in the horizontal direction, i.e. with a horizontal coring axis (axis 1 or 2). It is thus very difficult to accurately identify axis 3 (see Fig. 2) as the bedding plane is not visible at that scale. Cylindrical samples 20 mm in diameter and 40 mm in height were cored from 'T 1 cell' for the tests. These sample dimensions satisfy the standard requirement of the usual ratio (length-to-diameter ratio=2), thus avoiding end effects and reducing the time needed to complete the poromechanical tests. Each sample was polished and verified, to ensure that it had parallel end surfaces within a tolerance of less than 0.05 mm. Four strain gauges were glued onto the surface of each sample: two longitudinal (1 or 2 axis), and two transverse (n-axis) gauges were used, as shown in Fig. 1. The strain values were measured with a Labview system having an accuracy of ±10 -6 m/m. Within the framework of Biot's theory, the transverse isotropic poroelastic behavior of the tested material can be expressed with respect to its material axis using the following relation:

        ε 11 ε 22 ε 33 γ 12 γ 13 γ 23         =          1 E1 -ν E1 -ν ′ E1 0 0 0 -ν E1 1 E1 -ν ′ E1 0 0 0 -ν ′ E1 -ν ′ E1 1 E3 0 0 0 0 0 0 2(1+ν) E1 0 0 0 0 0 0 1 G13 0 0 0 0 0 0 1 G13                  σ 11 + bp σ 22 + bp σ 33 + b 3 p σ 12 σ 13 σ 23         (1)
where tensile stresses are positive, and b, b 3 are the coefficients accounting for the Biot's tensor in the transverse isotropic case. A convenient approach to the identification of these components relies on the application of a hydrostatic loading Pc and/or a pore pressure p. One interesting case arises when b = b 3 . When this specific case is verified, the following expressions for the three strain terms can be derived:

ε 11 = ε 11 (Pc)(1 -αb) ε 22 = ε 22 (Pc)(1 -αb) ε 33 = ε 33 (Pc)(1 -αb) γ ij = 0 (2)
where α = p/Pc, and

ε 11 (Pc) = ε 22 (Pc) = -Pc E 1 (1 -(ν + ν ′ )) ; ε 33 (Pc) = -Pc 1 E 3 - 2ν ′ E 1 (3)
A gauge Jn, glued in any direction n, will give εnn = n•ε•n. This yields (n = n i e i ):

εnn = (1 -αb) n 2 1 ε 11 (Pc) + n 2 2 ε 22 (Pc) + n 2 3 ε 33 (Pc) (4)
Equation ( 4) shows that, whatever the gauge direction, the Biot's coefficient b can be derived from two measurements, which may be combined: hydrostatic loading with a confining pressure Pc and a change in pore pressure p. This is useful when the direction 3 is not accurately known as it is the case for Cox argillite samples. It can also be underlined that an "isotropic" Biot's coefficient does not mean that a change in pore pressure would lead to isotropic strain state (see relations ( 2) and ( 3)). Another modulus denoted by H i is sometimes used to analyze the different experiments. This modulus is related to a change in pore pressure p according to (no summation on i):

ε ii (p) = p H i (5)
While relation ( 1) is only relevant for poroelastic behavior, initial experiments on argillite have shown that, due to plastic strain effects and/or micro-cracking, its behavior is not reversible. The following method was thus used to identify Biot's coefficient for a given value of Pc:

•

Step 1 is a hydrostatic loading phase during which confining pressure is increased from P 1 c to P 2 c = P 1 c + ∆Pc. Since this step is generally nonlinear and irreversible, unloading steps are needed in order to obtain elastic values of strain.

• Step 2 is a p loading phase, with confining pressure being increased up to P 2 c , following which an increase in pore pressure ∆p is applied, with ∆p = ∆Pc. This loading is elastic, since it is equivalent to hydrostatic pressure unloading. The strain values ε 22 (∆p) and εnn(∆p) are measured after this process. The pore pressure p is then decreased down its initial value.

• Step 3 is a hydrostatic unloading phase, with confining pressure being reduced from P 2 c to P 2 c -∆Pc = P 1 c , allowing ε 22 (∆Pc) and εnn(∆Pc) to be measured. • Step 4 involves comparing ε 22 (∆P c) and εnn(∆Pc) with ε 22 (∆p) and εnn(∆p), therefore allowing Biot's coefficient b to be identified.

When b is not equal to b 3 , further tests, such as axial loading, must be carried out in order to produce new conditions allowing b and b 3 to be determined. This approach was not required in the present study, since the first results confirmed that b = b 3 .

EXPERIMENTAL METHODS AND PROCEDURE FOR SANDSTONE

Contrary to argillite, the structural axis identification is easier for sandstone as the sedimentary layers are clearly visible (see Fig. 2). Two couples of crossed gages are diametrically glued on the cylindrical sample, which dimensions are 37mm diameter and 70mm height. They will allow to measure the axial strain εa (=ε 33 ) and the lateral strain ε l (=ε 11 or ε 22 ). The measurement method used to obtain the strains due to ∆Pc and ∆p are the same than those previously described in section 2.2.

EXPERIMENTAL DEVICE

The device used is composed of a hydrostatic cell that allows both confining pressure and pore pressure to be controlled (see Fig. 3). This system is used either for water or gas injection. A Labview system is connected to this system in order to get the strain measurements and, more specifically, to follow the strain evolution when a long time is needed to assess their stability (see the case for argillite in the following).

[Fig. 3 about here.]

RESULTS FOR ARGILLITE

Numerous tests have been performed on this material but the presentation of the whole series is not the purpose of this study. Hence, only typical and representative results are selected and given in figure 4. Four gages were glued on the sample as drawn in figure 1. These are crossed gages such as J 2 and Jn, and they are therefore located at the same point (or position); two crossed disposals were diametrically opposed: the (axial) average values of the two "J 2 " are plotted in green, while the (transverse) average values of the two "Jn" are in black. A complete cycle of loading-unloading operations, described in section 2.2, is represented in figure 4, which also gives the selected values of Pc and p.

[Fig. 4 about here.]

The total duration of this cycle is around 20 days. This long time comes from the low water permeability of argillite and is necessary to get a complete strain value stabilization. The difference between strains from A to B (or A ′ to B ′ ) and from D to E (or E ′ to D ′ ) is due to the Pc loading or unloading (5-12-5MPa) whereas B -C (B ′ -C ′ ) and C -D (C ′ -D ′ ) comes from the p loading or unloading (2-9-2MPa).

In the following the writing (MN) will indicate the strain difference ε(N ) -ε(M ).

The strong material anisotropy is clearly visible as (AB) is almost twice higher than (A ′ B ′ ), which is also confirmed by the comparison (DE) and (D ′ E ′ ), or (BC) and (B ′ C ′ ) etc. It is also interesting to underline that (AB) or (A ′ B ′ ) and (ED) or (E ′ D ′ ) are virtually the same. This simply means that the first Pc loading step led to elastic strains. (BC) = (DC) and (B ′ C ′ ) = (D ′ C ′ ) were expected results as they are due to an increase in p, which was already assumed to lead to elastic strains (cf. section 2.2). Another crucial information is given by the ratios: ((AB) or (ED))/((CB) or (CD)) and ((

A ′ B ′ ) or (E ′ D ′ ))/((C ′ B ′ ) or (C ′ D ′ )
) that are all virtually identical to 1. This unambiguously means that b = b 3 = 1 (see relation (2) in section 2.2 when α = 1).

RESULTS FOR SANDSTONE

The whole set of results obtained for five different samples are summed up in table 1. Four samples were used at confining pressure of 5, 10, 20 and 30MPa respectively. The Biot's coefficients were measured with the same methodology as for argillite. Then these samples were conducted up to the failure under increasing deviatoric stress (this is why the samples are not the same).

The last column 8, in italic letters, is related to a 5th sample that was submitted to a hydrostatic stress only. The second and third columns give the ratio of mean lateral strain over mean axial strain, respectively for a decreasing confining pressure or an increasing pore gas pressure. Both indicate a slight material anisotropy (10% to 20%), which can also be observed in columns 6 and 7. In columns 4 and 5, the 'apparent' Biot's tensor components are calculated from the ratio ∆ε g l /∆ε c l (resp. ∆ε g a /∆ε c a ) for b 1 (resp. b 3 ). The term apparent is chosen here as these ratios are the real 'b i ' components only in the case where they are equal. As b 1 and b 3 are very close to each other, generally by a difference that is less than 5%, it can be admitted that they are virtually identical. It can be finally observed that these b 1 and b 3 values are also very close to the 'b' values obtained on the same sample at different confining pressures. Once again, despite a (slight) material anisotropy, it is found that the Biot's tensor is virtually composed of a unique component b.

[Table 1

about here.]

As a partial conclusion, two set of experimental results indicate that a (quasi) isotropic Biot's tensor, despite a more or less strong material anisotropy, can be obtained. This result can a priori be surprising, or non logical, but it must be reminded here that this result does not mean that the pore pressure effect leads to isotropic strains. The next section raises the issue of this a priori surprising result from a theoretical point of view through a microporomechanics point of view [START_REF] Dormieux | Microporomechanics[END_REF].

ANISOTROPY OF THE POROMECHANICAL COUPLING: A MICROMECHANICAL MODEL

The purpose of this section is to propose a micromechanical model that supports the fact that a strong material anisotropy is compatible with a quasi-isotropic Biot tensor. The anisotropy of the poromechanical coupling will be examined through three indicators:

• Biot's tensor B;

• The strain tensor ε 1 induced by a change of the confining pressure;

• The strain tensor ε 2 induced by a change of the pore pressure.

From a computational point of view, the starting point is the first state equation of poroelasticity in the form :

σ = C dr : ε -Bp (6)
which is the generalized formulation of (1). C dr is the drained stiffness tensor of the porous material. For the unit change of pore (resp. confining) pressure, this yields:

ε 1 = -C dr -1 : δ ; ε 2 = C dr -1 : B
As regards the drained stiffness tensor, it is convenient to introduce the following anisotropy indicators:

ρ = C dr 1111 C dr 3333 ; χ = C dr 1212 C dr 1313 (7)
For a second rank tensor T such as B, ε (1) and ε (2) , the relevant anisotropy indicator is

r T = T 11 /T 33 (8)
In the sequel, we propose to determine the drained stiffness tensor C dr and Biot's tensor B of a theoretical porous material (acronym TPM) in the framework of an upscaling process. The scale at which the drained stiffness tensor and the Biot tensor are defined and can be measured is referred to as the macroscopic scale: It is the uppermost scale of the upscaling process. This way suggests two options that we shall consider successively:

1. The solid matrix itself is anisotropic. This option is considered in section 3.1.

2. The porous material has an isotropic solid matrix and the anisotropy of C dr and B is the consequence of the anisotropy of the geometry of the pore space. This option is considered in section 3.2

It is assumed in the sequel that the Mori-Tanaka scheme is appropriate for all forthcoming upscaling operations.

THE ANISOTROPIC MATRIX

In this section, we shall consider a porous material having a transversely isotropic solid matrix (symmetry direction along e 3 ). We shall assume that the geometry of the pore space is isotropic, so that the porous material itself will be also transversely isotropic along e 3 . In fact, a transversely anisotropic matrix (symmetry axis along e 3 ) can a priori be defined by any appropriate choice of a set of the five constants

C m 1111 , C m 1122 , C m 1133 , C m
3333 and C m 2323 , chosen in such a way that the condition of definite positiveness of the elastic stiffness tensor C m is satisfied. In this paper, a more physical approach is preferred, in which a micromechanical interpretation of the anisotropy of the matrix is provided. We shall use the standard terminology of homogenization and introduce three levels of geometrical analysis, respectively referred to as microscopic, mesoscopic and macroscopic, associated with increasing length scales: The lowest scale reveals the heterogeneity of the matrix. It will be termed the microscopic scale. At this scale, a representative elementary volume (rev) ω of matrix is regarded as an heterogeneous structure, comprising a solid subdomain and pores. In contrast, at the scale above, referred to as mesoscopic scale, this structure reduces to a material point which mechanical behavior is characterized by the matrix stiffness tensor C m . The latter can be derived by appropriate averaging techniques from the response of the rev (ω) to a mechanical boundary value problem defined at the microscopic scale on ω. In turn, the mesoscopic scale reveals the heterogeneity of the TPM. This means that a rev Ω of the TPM at the mesoscopic scale is a heterogenous structure comprising an elastic solid domain (the matrix with elastic stiffness tensor C m ) and a pore space filled by a pressurized fluid (pressure p). Eventually, the rev of the TPM reduces to a material point at the macroscopic scale. Again, the macroscopic state equation ( 6) can be derived by averaging techniques from the solution to a boundary value problem defined at the mesoscopic scale on Ω by the macroscopic strain tensor ε and the fluid pressure p [START_REF] Dormieux | Microporomechanics[END_REF]. More precisely, the Hashin type boundary conditions consist in prescribing the displacement ε • z at any point z on the boundary ∂Ω, while the pressure p is applied on the whole boundary of the mesoscopic pore space (see Fig. 5).

[Fig. 5

about here.]

The first step consists in defining an anisotropic matrix characterized by a stiffness tensor C m , as the result of a micro → meso scale transition. Depending on assumptions on the pore space at the mesoscopic scale, we shall afterwards compute the macroscopic elastic stiffness (C dr ) and the three indicators of the poromechanical coupling (B, ε 1 and ε 2 ).

DEFINITION OF THE MATRIX

At the microscopic scale, a rev ω of matrix is made up of an isotropic linear elastic solid (elastic stiffness tensor C s ) in which a set of homothetic spheroidal inclusions (same aspect ratio X) is embedded. The volume fraction of the microscopic inclusions in the rev ω is denoted by ϕ. The symmetry axis of the spheroids is along e 3 . As regards the numerical determination of C m , these inclusions will be interpreted as empty pores (C → 0). It should be emphasized that the role of this inclusionary phase is solely to provide an explanation to the anisotropy of the matrix behavior. Many other strategies could have been used alternatively for the same purpose including spheroidal rigid inclusions (C → ∞). Consequently, ϕ may be interpreted as a microscopic occlusive porosity. As such, it will not be taken into account in the determination of the effective porosity that will only consider the mesoscopic pore space. The homogenized elastic stiffness C m of the matrix is determined by means of the Mori-Tanaka homogenization scheme that is dedicated to the particulate composite morphology:

C m = (1 -ϕ)C s : (1 -ϕ)I + ϕ (I -P s (X) : C s ) -1 -1 (9)
where P s (X) is the Hill tensor of a spheroid (aspect ratio X, symmetry axis along e 3 ) embedded in the elastic medium with stiffness C s (see section Appendix). The fourth-rank identity tensor I is defined by

I ijkl = 1 2 δ ik δ jl + δ il δ jk
The Young modulus E s of the isotropic solid phase is taken as reference unit (E s = 1). The Poisson coefficient is arbitrarily fixed: ν s = 0.3 and the matrix porosity is ϕ = 0.5. Two numerical simulations, respectively with an aspect ratio X = 5 (prolate spheroids) and an aspect ratio X = 0.4 (oblate spheroids) are performed. The set of elastic modulii derived from (9) are given in table 2.

[Table 2 about here.]

In the following, the subscripts 5 and 0.4 are used: Hence, C m 5 and C m 0.4 are respectively associated with X = 5 and X = 0.4.

MACROSCOPIC POROELASTIC BEHAVIOR

The material determined at section 3.1.1 from the micro→meso transition is now the (anisotropic) solid phase at the mesoscopic scale (see figure 5). The next step is the meso→macro transition. We again implement the Mori-Tanaka scheme. For the same value f of the mesoscopic porosity (numerical value f = 0.15), two different isotropic geometries of the mesoscopic pore space are tested:

(case a) The pore space is made up of spheres. In this case, the drained macroscopic stiffness tensor is:

C dr a = (1 -f )C m : (1 -f )I + f I -P m sph : C m -1 -1
where P m sph is the Hill tensor of a sphere embedded in the elastic medium with stiffness C m . (case b) The pore space is made up of a set of spheroidal pores (aspect ratio X = 3), the distribution of the orientations of the symmetry axes being isotropic.

The expression of the macroscopic drained stiffness tensor then reads:

C dr b = (1 -f )C m : (1 -f )I + f (I -P m (θ, φ, X) : C m ) -1 -1
where P m (θ, φ, X) is the Hill tensor of a spheroid embedded in the elastic medium with stiffness C m , having a symmetry axis along the radial vector er(θ, φ) in the system of spherical coordinates in which θ is the angle between er and e 3 ) (see section 4). A is the averaging operator over the unit sphere applied on the fourth-rank tensor A:

A = 1 4π 2π 0 dφ π 0 A(θ, φ) sin θ dθ
In both cases, Biot's tensor is given by the same relation:

B = δ : I -C m-1 : C dr 3.1.3 NUMERICAL RESULTS
The set of the elastic modulii defining C dr are given in tables 3 and 6, respectively for a microscopic pore aspect ratio X = 5 and X = 0.4. The anisotropy indicators are gathered in tables 5 and 8.

• matrix elastic stiffness C m 5 :

[Table 3 about here.]

[Table 4 about here.]

[Table 5 about here.] It appears that the numerical results concerning cases a and b (spherical pores or isotropic distribution of spheroidal prolate pores) are very close. The anisotropy of Biot's tensor is very weak (see. table 4) while the anisotropy of the drained stiffness tensor is pronounced (see. table 3). The anisotropy of the characteristic strain tensors ε 1 and ε 2 are significant as well and similar (see. table 4).

• matrix elastic stiffness C m 0.4 .

[Table 6 about here.]

The conclusions in the case X = 0.4 are qualitatively identical to those formulated previously in the case X = 5: strong elastic anisotropy and very weak anisotropy of Biot's tensor. Interestingly, the anisotropy indicators are essentially inverted w.r.t. the previous case. It is therefore reasonable to hope that the similar conclusions drawn in the two considered cases are in fact general.

[Table 7 about here.]

[Table 8 about

THE ANISOTROPIC PORE SPACE

In the second morphological option, it is assumed that the solid matrix of the porous material is a linear elastic isotropic solid (isotropic elastic stiffness tensor C m ). The macroscopic anisotropy is therefore due to the anisotropic shape of the pores. More precisely, we consider a pore space with volume fraction f made up of a set of homothetic spheroids (same symmetry axis along e 3 and same aspect ratio X). This second model is very simple in so far as it considers only two scales: The microscopic scale reveals the heterogeneity of the rev, which comprises a solid domain and a pore space, while the same material is homogenized at the macroscopic scale. The quantitative transition micro→macro is again carried out with the help of the Mori-Tanaka scheme:

C dr = (1 -f )C m : (1 -f )I + f (I -P m (X) : C m ) -1 -1 (10)
where P m (X) is the Hill tensor of a spheroid (aspect ratio X, symmetry axis along e 3 ) embedded in the elastic medium with stiffness C m (see Appendix).

As regards numerical simulations, the value of the porosity f is identical to the mesoscale porosity of the first model (see section 3.1.2), that is: f = 0.15. Figure 8 provides, through the anisotropy indicators ρ and χ (see ( 7)), an estimate of the anisotropy induced for a given pore aspect ratio.

[Fig. 8 about here.]

Figure 9 presents the anisotropy indicators r 1 , r 2 and r B respectively related to ε (1) , ε (2) and B (see ( 8)). With this micromechanical model in which the anisotropy is induced by the geometry of the pore space, it appears that the strain ε (2) induced by a pore pressure change is the most sensitive anisotropy indicator. Again, Biot's tensor exhibits the weakest anisotropy. Nevertheless, it is only slightly below the anisotropy of the strain ε (1) induced by a change in confining pressure. As opposed to the first model, ε (1) and ε (2) have very different sensitivities to the material anisotropy.

[Fig. 9 about here.]

CONCLUSION

This study, involving both experimental and theoretical considerations, was first motivated by poromechanical experiments intended for the measurements of Biot's tensor components of transversely isotropic porous materials. As it is highlighted in the first part of this paper, a priori surprising results were found that showed an isotropic (or quasi isotropic) Biot's tensor. This result does not mean that the pore pressure effect is isotropic as the strains due to a pore pressure variation revealed to be anisotropic. It was therefore very tempting to involve a modelling based on a micromechanical model, able to take in a wide sweep various anisotropic scenarii. Hence two main options were chosen: case 1 -the solid matrix is transversely isotropic and the pore space is isotropic or case 2 -the solid matrix is isotropic and the pore space is anisotropic i.e. designed to obtain a transversely isotropic behavior of the skeleton. The strains calculated with those modellings, either for a confining pressure or for a pore pressure loading, were always indicative of a (sometime) strong anisotropic behavior. Conversely, for both cases the Biot's tensor always exhibited the weakest anisotropy. The anisotropy indicator for this tensor never exceeded 11% in case 1 (and for some extreme case only). As a consequence, the Biot's tensor is quasi isotropic. It is slightly different for case 2 but, in a large range of anisotropic pore space geometries, the Biot's tensor can be seen as virtually isotropic. On an experimental point of view, and taken into account the difficulties to measure the Biot's coefficient, 10% variation (or errors) are very few and can be included into the experimental uncertainties. Hence, the obtained differences in the Biot's tensor components are not significant for case 1, which in our opinion is likely to represent more in situ cases than the case 2.

To conclude, the micro-modelling calculations evidence that an isotropic Biot's tensor is compatible with an anisotropic porous medium behavior.

  -rank stiffness tensor C dr : 4 th -rank drained stiffness tensor δ : 2 nd -rank identity tensor I : symmetrized 4 th -rank identity tensor A : 4 th -rank strain concentration tensor X : spheroidal inclusion aspect ratio P(X) : 4 th -rank Hill tensor of a spheroidal inclusion 1 INTRODUCTION

  here.] [Fig. 6 about here.] For illustrative purposes, the case of an anisotropic matrix of the previous type (see section 3.1.1) with spherical pores (same porosity f = 0.15) is finally considered for the range of values of X ∈ [0.4, 5]. The anisotropy indicators of the drained stiffness tensors thus generated are presented at figure 6. The corresponding anisotropy indicators of ε (1) , ε (1) and B are plotted against X at figure 7. The latter emphasizes that the Biot's tensor is almost isotropic on the whole range of material anisotropy considered herein. [Fig. 7 about here.]

FIGURESFig. 1 FIGURESFig. 3 FigFig. 5 Fig. 6 Fig. 7 XFig. 8 Fig. 9

 1356789 FIGURES

Table 1

 1 , ε (2) and B. X is the aspect ratio of the mesoscopic porosity Note: ∆ε c a and ∆ε c l (resp. ∆ε g a and ∆ε g l ) are the variation of axial and lateral strain caused by confining (resp. pore) pressure. Initial anisotropic properties under hydrostatic stress and before failure test (except for the results in column 8 that is related to the same sample) Matrix modulii for pore aspect ratios X = 5 and X = 0.4

	1 Pc (MPa) 5	2 l /∆ε c ∆ε c a 1.134	3 l /∆ε g ∆ε g a 1.149	4 b 1 0.775	5 b 3 0.765	6 H 1 (GPa) 30.69	7 H 3 (GPa) 35.30	8 'b' 0.767
	10	1.180	1.209	0.754	0.735	38.57	46.65	0.740
	20	1.140	1.231	0.704	0.689	47.70	58.75	0.707
	30	1.100	1.151	0.724	0.692	52.88	60.88	0.706

Table 3

 3 Drained stiffness modulii (microscopic aspect ratio X = 5)

	X = 5 case a	B 11 0.280707	B 33 0.298636	ε -2.756448 (1) 11	ε -1.237511 (1) 33	ε 0.7583025 (2) 11	(2) 33 0.4004709 ε
	case b	0.299152	0.320066	-2.824996	-1.284746	0.8268580	0.4477064

Table 4

 4 Biot's tensor and strain tensors ε (1) and ε (2) (microscopic aspect ratio X = 5)

	X = 0.4 case a	C dr 1111 0.341446	C dr 1122 0.111591	C dr 1133 0.078029	C dr 3333 0.185284	C dr 2323 0.08184
	case b	0.332991	0.107499	0.0755121	0.181577	0.080445

Table 6

 6 Drained stiffness modulii (microscopic aspect ratio X = 0.4)

	TABLES

Table 7

 7 Biot's tensor and strain tensors ε (1) and ε (2) (microscopic aspect ratio X = 0.4)

	X = 0.4	ρ	χ	r B	r	ε (1)	r	ε (2)
	case a	1.84	1.40	1.10	0.361	0.447
	case b	1.83	1.4	1.11	0.366	0.461

Table 8

 8 Anisotropy scalar indicators (microscopic aspect ratio X = 0.4)

Conflict of interest

The authors declare that they have no conflict of interest.

APPENDIX: HILL TENSOR OF AN ELLIPSOID

Consider an ellipsoid defined by the equation:

where S is a positive definite symmetric second rank tensor. This ellipsoid is embedded in an infinite linear elastic medium with elastic stiffness tensor C. Let ξ denote some vector on the unit sphere: |ξ| = 1. The associated acoustic tensor is

The coefficient P ijkl of the Hill tensor reads:

In the above expression, the integral is taken with respect to ξ over the unit sphere. The susbscript (ij), (kℓ) means that the expression is symmetrized w.r.t. the subscripts i and j, and w.r.t. the subscripts k and ℓ:

In section 3.1.2, the Hill tensor P m (θ, φ, X) refers to a spheroid with aspect ratio X, and a symmetry axis along the radial unit vector er(θ, φ):

It is recalled that the matrix is transversely isotropic (symmetry axis along e 3 ).

In the spherical basis (er, e θ , e φ ), the tensor S(θ, φ, X) of this spheroid reads:

The integration variables in (11) are the two angles x and y that define the unit vector ξ: ξ = sin x (cos y e 1 + sin y e 2 ) + cos x e 3

where dS ξ = sin x dxdy.

List of Figures