Laurent Yon

Lyne Breault

GRE ´ GORY BELLANCOURT ‡ Vincent Contesse

Grégory Bellancourt

Catherine Delarue

Alain Fournier

Jean-Guy Lehoux

Hubert Vaudry

AND Nicole Gallo-Payet

Receptors in the Fetal Human Adrenal Gland during the Second Trimester of Gestation*

Keywords: Localization, Characterization, and Second Messenger Coupling of Pituitary Adenylate Cyclase-Activating Polypeptide

P ITUITARY adenylate cyclase-activating polypeptide (PACAP) is a 38-amino acid ␣-amidated neuropeptide initially isolated from the ovine hypothalamus because of its ability to stimulate adenylyl cyclase activity in rat anterior pituitary cells [START_REF] Miyata | Isolation of a novel 38-residues hypothalamic polypeptide which stimulates adenylate cyclase in pituitary cells[END_REF]. PACAP38 possesses an internal cleavageamidation consensus sequence (Gly 28 -Lys 29 -Arg 30 ) and can thus generate a 27-residue ␣-amidated peptide (PACAP27) that exhibits the same biological activity as PACAP38 [START_REF] Miyata | Isolation of a neuropeptide corresponding to the N-terminal 27 residues of the pituitary adenylate cyclase-activating polypeptide with 38-residues (PACAP 38)[END_REF][START_REF] Arimura | Pituitary adenylate cyclase-activating polypeptide (PACAP): discovery and current status of research[END_REF]. The NH 2 -terminal region of PACAP shows 68% sequence similarity with vasoactive intestinal polypeptide (VIP), identifying PACAP as a member of the VIP/secretin/glucagon superfamily. The primary structure of PACAP has been remarkably conserved during evolution [START_REF] Chartrel | Primary structure of frog pituitary adenylate cyclase-activating polypeptide (PACAP) and effects of ovine PACAP on frog pituitary[END_REF], indicating that the peptide plays important biological functions.

Two types of PACAP-binding sites have been characterized so far (see Ref. 5 for review). Type I binding sites have a high affinity (dissociation constant (K d ) ϳ 0.2 nmol/L) for PACAP38 and PACAP27 and a much lower affinity (K d ϳ 1 mol/L) for VIP [START_REF] Rawlings | Pituitary adenylate cyclase-activating polypeptide (PACAP) and PACAP/vasoactive intestinal polypeptide receptors: action on the anterior pituitary gland[END_REF][START_REF]Type I receptors for PACAP (a neuropeptide even more important than VIP?)[END_REF], whereas type II binding sites have similar affinity for PACAP38, PACAP27, and VIP [START_REF] Arimura | Pituitary adenylate cyclase-activating polypeptide (PACAP) and its receptors: neuroendocrine and endocrine interaction[END_REF]. Both receptor subtypes belong to the seven-transmembrane domain G protein-coupled receptor superfamily (8 -10) and are differentially coupled to adenylyl cyclase and phospholipase C [START_REF] Spengler | Differential signal transduction by five splice variants of the PACAP receptor[END_REF]. PACAP and PACAP-binding sites are widely distributed in the central nervous system and in peripheral organs [START_REF] Arimura | Pituitary adenylate cyclase-activating polypeptide (PACAP) and its receptors: neuroendocrine and endocrine interaction[END_REF][START_REF] Ghatei | Distribution, molecular characterization of pituitary adenylate cyclase-activating polypeptide and its precursor encoding messenger RNA in human and rat tissues[END_REF]. In particular, the occurrence of PACAP and its recognition sites has been demonstrated in various endocrine glands, including the pituitary, pancreas, testis, ovary, and adrenal [START_REF] Arimura | Pituitary adenylate cyclase-activating polypeptide (PACAP): discovery and current status of research[END_REF][START_REF] Rawlings | Pituitary adenylate cyclase-activating polypeptide (PACAP) and PACAP/vasoactive intestinal polypeptide receptors: action on the anterior pituitary gland[END_REF][START_REF] Arimura | Pituitary adenylate cyclase-activating polypeptide (PACAP) and its receptors: neuroendocrine and endocrine interaction[END_REF][START_REF] Yon | Localization, characterization and activity of pituitary adenylate cyclase-activating polypeptide in the frog adrenal gland[END_REF][START_REF] Tabarin | Pituitary adenylate cyclaseactivating peptide in the adrenal gland of mammals: distribution, characterization and responses to drugs[END_REF]. ‡ Recipient of a studentship from the Conseil Re ´gional de Haute-Normandie.

§ Affiliated Professor at the Institut National de la Recherche Scientifique-Sante ´, Montre ´al.

Developmental studies have shown that, in the brain, the expression of PACAP and PACAP-binding sites undergoes important variations during ontogenesis. For instance, a high concentration of PACAP has been detected in the rat cerebellum between postnatal days 4 and 20 [START_REF] Masuo | Ontogeny of pituitary adenylate cyclase-activating polypeptide (PACAP) and its binding sites in the rat brain[END_REF][START_REF] Tatsuno | Developmental changes of pituitary adenylate cyclase-activating polypeptide (PACAP) ands its receptors in the rat brain[END_REF]. Similarly, intense expression of PACAP receptors has been observed in the proliferative zone of the cerebellar cortex during the postnatal period [START_REF] Basille | Localization and characterization of PACAP receptors in the rat cerebellum during development: evidence for a stimulatory effect of PACAP on immature cerebellar granule cells[END_REF][START_REF] Basille | Ontogeny of pituitary adenylate cyclase-activating polypeptide (PACAP) receptors in the rat cerebellum: a quantitative autoradiographic study[END_REF], suggesting that PACAP exerts neurotrophic activities during development. In support of this hypothesis, it has been shown that PACAP promotes cell survival and neurite outgrowth in cultured cerebellar granule cells [START_REF] Gonzalez | Pituitary adenylate cyclase-activating polypeptide (PACAP) promotes cell survival and neurite outgrowth in rat cerebellar neuroblasts[END_REF] and stimulates proliferation of folliculo-stellate cells [START_REF] Matsumoto | Pituitary folliculostellate-like cell line (TtT/GF) responds to novel hypophysiotropic peptide (pituitary adenylate cyclase-activating polypeptide), showing increased adenosine 3Ј, 5Јmonophosphate and interleukin-6 secretion and cell proliferation[END_REF].

The occurrence of PACAP and PACAP receptors in the adult adrenal gland (13, 20 -22) and the observation that PACAP modulates proliferation and differentiation of adrenal chromaffin cells [START_REF] Fro ¨din | Neuronal localization of pituitary adenylate cyclase-activating polypeptide-38 in the adrenal medulla and growth inhibitory effect on chromaffin cells[END_REF][START_REF] Tischler | Mitogenic and antimitogenic effects of pituitary adenylate cyclase-activating polypeptide (PACAP) in adult rat chromaffin cell cultures[END_REF][START_REF] Wolf | Phenotypic development of neonatal rat chromaffin cells in response to adrenal growth factors and glucocorticoids: focus on pituitary adenylate cyclase-activating polypeptide[END_REF] suggest that PACAP may play a role in the ontogeny of the adrenal gland. The fetal human adrenal gland is composed of morphologically and functionally distinct steroid-secreting tissues: a thin neocortex that produces cortisol and a large fetal zone that produces massive amounts of dehydroepiandrosterone sulfate [START_REF] Winter | Fetal and neonatal adrenocortical development[END_REF][START_REF] Bocian-Sobkowska | Cytological aspects of the human adrenal cortex development in the course of intra-uterine life[END_REF]. During fetal life, cells derived from the ectoderm migrate inside the gland to form the medulla [START_REF] Cooper | Histogenesis of the human adrenal medulla[END_REF][START_REF] Ehrhart-Bornstein | 17␣-hydroxylase and chromogranin A in 6th week human fetal adrenals[END_REF]. After birth, the fetal zone involutes, and the cortex develops and differentiates into the three characteristic zones present in the adult adrenal gland. These profound changes in the organization of the gland suggest that trophic factors, such as PACAP, may control the modeling of the tissue during ontogenesis. However, the distribution and function of PACAP receptors have not yet been studied in the adrenal during development.

In the present study, we have localized PACAP-binding sites in the fetal human adrenal gland by autoradiography, and we have investigated the effect of PACAP on secondmessenger systems in suspended and cultured fetal adrenal cells.

Materials and Methods

Reagents and peptides

Myo-[ 3 H]inositol (10 -20 Ci/mmol) and [ 3 H]adenine (24 Ci/mmol) were purchased from Amersham (Oakville, Ontario, Canada). Na 125 I (2000 Ci/mmol) was obtained from Amersham (Les Ulis, France). ATP, cAMP, and deoxyribonuclease were supplied by Sigma (St. Louis, MO). Eagle's MEM, collagenase, and Opti-MEM medium were obtained from Gibco (Burlington, Ontario, Canada). PACAP27 was from Bachem (Marina Delphen, CA). BSA was from Boehringer (Mannheim, Germany). PACAP38 and VIP were synthesized by the solid-phase methodology previously described [START_REF] Chartrel | Characterization of melanotropin release-inhibiting factor (melanostatin) from frog brain: homology with human neuropeptide Y[END_REF].

Monoiodinated PACAP27 was used as a tracer. Iodination of synthetic PACAP27 was performed by means of the lactoperoxidase technique, and the radioligand was purified by reverse-phase high-performance liquid chromatography, as previously described [START_REF] Laquerrie `re | Distribution of somatostatin receptors in the brain of the frog Rana ridibunda: correlation with the localization of somatostatin-containing neurons[END_REF][START_REF] Yon | Pituitary adenylate cyclaseactivating polypeptide stimulates both adrenocortical cells and chromaffin cells in the frog adrenal gland[END_REF]. The specific radioactivity of the radioligand was approximately 800 Ci/mmol.

Tissue preparation

Fetal human adrenal glands were obtained from fetuses between 14 and 20 weeks old (post conception) at the time of therapeutic abortion. Fetal ages were estimated by foot length and time after the last menstruation, according to Streeter [START_REF] Streeter | Weight, sitting height, head size, foot length and menstrual age of the human embryo[END_REF]. After removal, glands were kept on ice in McCoy's medium and transported within 4 h to the laboratory. The project was approved by the human subject review committees at our institutions. Immediately after collection, glands were cleansed of fat and were either frozen at Ϫ80 C or processed for cell dispersion.

Autoradiographic studies

Frozen adrenal glands were sliced into 10-m sections on a cryostat (Frigocut, Reicher-Jung, Germany). Sections were thaw-mounted on gelatin-coated slides, placed in a desiccator overnight at 4 C, and processed for autoradiography, as previously described [START_REF] Basille | Localization and characterization of PACAP receptors in the rat cerebellum during development: evidence for a stimulatory effect of PACAP on immature cerebellar granule cells[END_REF]. Tissue slices were preincubated for 30 min at 24 C in 50 mmol/L Tris buffer (pH 7.4) containing 1% BSA, 5 mmol/L MgCl 2 , 32 mmol/L sucrose, and 0.5 g/mL bacitracin. The sections were incubated with [ 125 I]PACAP27 (50 pmol/L) for 2 h at 24 C in the same buffer, supplemented with 2% BSA and 100 KIU/mL aprotinin (Hoechst Laboratories, Puteaux, France). Nonspecific binding was determined by adding 1 mol/L PACAP27. For competition studies, the sections were incubated with 50 pmol/L tracer in the presence of various concentrations of unlabeled PACAP27, PACAP38 (10 Ϫ10 to 10 Ϫ6 mol/L) or VIP (10 Ϫ9 to 10 Ϫ6 mol/L). The slices were washed three times for 5-min periods at 4 C in 50 mmol/L Tris buffer containing 0.1% BSA, 5 mmol/L MgCl 2 , and 0.5 g/mL bacitracin. Finally, the sections were dried under a cold-air stream and apposed onto Amersham Hyperfilm-3 H for 10 days.

The gray levels of the autoradiograms were measured by means of a computer-assisted image analysis station BIO 500 (Biocom, Les Ulis, France), as previously described [START_REF] Leroux | Autoradiographic study of somatostatin receptors in the rat cerebellum[END_REF]. The concentration of ligand which causes 50% inhibition of the binding values (IC 50 ) and Hill coefficients were determined using the Sigma Plot program (Jandel Scientific, Corte Madera, CA).

Histochemical and immunological procedures

After exposure for autoradiography, adrenal sections were fixed with vapors of paraformaldehyde (24 h, 60 C). Tissue slices were either stained with hematoxylin-eosin or immunostained with antibodies against human chromogranin A (CgA; Dako, Glostrup, Denmark), a marker of chromaffin cells [START_REF] Ehrhart-Bornstein | 17␣-hydroxylase and chromogranin A in 6th week human fetal adrenals[END_REF]. Briefly, tissue sections were incubated overnight at 4 C with the CgA antiserum, diluted 1:100 in phosphate buffer (PB; 0.1 mol/L, pH 7.4) containing 0.3% Triton X-100 and 1% BSA. After several rinses in PB, the sections were incubated for 90 min at room temperature with fluorescein isothyocyanate-conjugated goat antirabbit ␥-globulins (GAR/FITC; Caltag Laboratories, San Francisco, CA) diluted 1:100. Finally, the sections were rinsed in PB and mounted in PB/glycerol (1:1). The preparations were examined on a Leitz Orthoplan (Heidelberg, Germany) microscope equipped with a photographic system.

Dissociation of adrenal cells

The adrenal glands from 18-to 20-week-old fetuses were finely chopped into 1-to 2-mm pieces. The tissue fragments were subjected to enzymatic dissociation in Eagle's MEM containing collagenase (2 mg/ mL) and deoxyribonuclease (25 g/mL), as previously described [START_REF] Breault | The angiotensin AT2 receptor is present in the fetal human adrenal gland throughout the second trimester of gestation[END_REF]. The tissue was mechanically dissociated by gentle aspiration, and dispersed cells were filtered and centrifuged for 10 min at 100 ϫ g. The cell pellet was resuspended in Opti-MEM containing 2% FBS, 100 U/mL penicillin, and 100 mg/mL streptomycin. Cells were plated in 35-mm Petri dishes at a density of about 2.5 ϫ 10 5 cells per dish and were grown in a humidified atmosphere of 95% air-5% CO 2 at 37 C. The culture medium was changed 24 h after seeding, and cells were used after 5 days of culture. For studies using suspended cells, cells were incubated in culture medium for a resting period of 12 h. After centrifugation (10 min; 100 ϫ g), cells were suspended in Hank's buffered saline (HBS)-glucose (HBS: NaCl, 130 mmol/L; KCl, 3.5 mmol/L; CaCl 2 , 1.8 mmol/L; MgCl 2 , 0.5 mmol/L; NaHCO 3 , 2.5 mmol/L; HEPES, 5 mmol/L) at a density of 5 ϫ 10 5 cells/tube.

Cyclic AMP determination

Cyclic AMP production was determined by measuring the conversion of [ 3 H]ATP into [ 3 H]cAMP, as previously described [START_REF] Gallo-Payet | Excitation-secretion coupling: involvement of potassium channels in ACTH-stimulated rat adrenocortical cells[END_REF]. Briefly, cultured cells were incubated at 37 C for 1 h in Opti-MEM culture medium containing 2 Ci/mL [ 3 H]adenine. The cultures were washed with HBS buffer and incubated in the same buffer, containing 1 mmol/L isobutyl methylxanthine, for 15 min at 37 C. Cultured cells were incubated with PACAP38 for an additional 15 min at 37 C. The reaction was stopped by aspiration of the medium and addition of 1 mL 5% trichloroacetic acid. Cells were scraped with a rubber policeman, and 100 L of cold solution of ATP and cAMP (5 mmol/L each) was added to the mixture. Cell membranes were pelleted at 5,000 ϫ g for 15 min, and the supernatants were sequentially chromatographed on Dowex and alumina columns, as previously described [START_REF] Salomon | A highly sensitive adenylate cyclase assay[END_REF], allowing the separation of

Measurement of inositol phosphate (InsP) formation

The effect of PACAP on polyphosphoinositide metabolism was investigated as described previously [START_REF] Gallo-Payet | Vasopressin induces breakdown of membrane phosphoinositides in adrenal glomerulosa and fasciculata cells[END_REF]. Briefly, cells were grown for 2 days in Opti-MEM culture medium containing 2 Ci/mL myo-[ 3 H]inositol. The radioactive medium was then discarded, and the cells were incubated in isotope-free and serum-free culture medium. After 30 min, cells were washed, incubated for 15 min in HBS-glucose/LiCl (10 mmol/L) medium, and incubated for another 15 min at 37 C with PACAP38 in the same medium. The incubation was stopped by aspiration of the medium and addition of 1 mL of 5% (vol/vol) perchloric acid and 200 L BSA (20 mg/mL). Total InsPs were separated by ion exchange chromatography on (1 ϫ 8 cm) Dowex columns. The radioactivity found in the InsP fractions was determined by scintillation counting in gel phase in a Beckman ␤ counter.

Data analysis

The data are presented as means Ϯ se. Statistical analysis was performed using the one-way ANOVA test. Homogeneity of variance was assessed by Bartlett's test, and P values were obtained from Dunnett's tables.

Results

Localization of PACAP-binding sites

The autoradiographic localization of [ 125 I]PACAP27-binding sites was performed in the adrenal of fetuses at 14 -20 weeks of gestation. The distribution of the binding sites in a 16-week-old fetus is illustrated in Fig. 1A. Labeling of consecutive sections with antibodies against human CgA demonstrated that the autoradiographic labeling was restricted to the chromaffin tissue (Fig. 1, A andB). Histochemical staining of tissue sections with hematoxylin-eosin showed that chromaffin cells formed radial clusters, penetrating the eosinophilic cells of the fetal zone (Fig. 1C). At all developmental stages studied, PACAP-binding sites were found only in the fetal medullary zone.

Characterization of PACAP-binding sites

Figure 2 illustrates the displacement of [ 125 I]PACAP27 binding (Fig. 2A) by increasing concentrations of synthetic PACAP27 (Fig. 2, B-D), PACAP38 (Fig. 2, E-G), or VIP (Fig. 2, H-J) on consecutive sections of a fetal adrenal gland at 15 weeks of gestation. Incubation of tissue slices with 10 Ϫ8 mol/L PACAP27 (Fig. 2C) or PACAP38 (Fig. 2F) completely abolished the autoradiographic labeling. In contrast, at the same dose, VIP induced only a slight displacement of [ 125 I]PACAP27 binding (Fig. 2I).

Similar experiments were performed on adrenal slices from 14-to 20-week-old fetuses, and the autoradiographic labeling was quantified using a computer-assisted image analyzer. At all stages studied, PACAP27 and PACAP38 were more potent than VIP in displacing [ 125 I]PACAP27 binding (Fig. 3, A-F). Scatchard plot analysis of PACAP27 inhibition curves (Fig. 3B inset) revealed that the affinity of PACAP-binding sites and the density of sites were relatively constant throughout the gestation period studied (Table 1). PACAP27 and PACAP38 were equally potent in displacing [ 125 I]PACAP27, whereas VIP was 20 to 500 times less potent than PACAP27 in competing with the tracer during the whole period studied (Table 2). The values of the Hill coefficient were close to 1 for PACAP27 and slightly lower for PACAP38, but systematically inferior to 0.6 for VIP (Table 2).

Effect of PACAP on second-messenger coupling

The capacity of PACAP38 to stimulate cAMP production was measured both in cell suspensions (used after a 12-h resting period) and in cells cultured for 5 days. In suspended cells, PACAP38 induced a dose-dependent increase in the conversion of ATP into cAMP (Fig. 4A). The ED 50 value was 0.07 Ϯ 0.02 nmol/L, and the maximum effect was reached at a dose of 10 nmol/L, with a stimulation ratio of 2.55 Ϯ 0.25 (n ϭ 7, P Ͻ 0.01) over basal value. In cultured cells, the basal conversion of ATP into cAMP was significantly lower than in suspended cells. The ED 50 value was 5.4 Ϯ 1.8 nmol/L, and the maximal effect was reached at a dose of 100 nmol/L, with a stimulation ratio of 3.4 Ϯ 0.40 (n ϭ 9, P Ͻ 0.001) over basal value. In contrast, PACAP38 did not affect InsP accumulation in suspended adrenal cells from 20-week-old fetuses but only induced a modest increase of InsP content (1.30-fold over control) in cells from 15-to 16-week-old fetuses (data not shown).

Discussion

The presence of PACAP in the adrenal gland and the effect of the neuropeptide on catecholamine and steroid secretion have been documented in various mammalian species. In contrast, little is known concerning the possible role of PACAP in the human adrenal gland. The present study provides the first evidence for the occurrence of functional PACAP receptors in fetal chromaffin cells.

Localization and characterization of PACAP-binding sites

Autoradiographic labeling, using [ 125 I]PACAP27 as a radioligand, revealed the presence of a high density of binding sites in the central zone of the gland. Immunohistochemical staining of adrenal slices with antibodies against human CgA, a selective marker of neuroendocrine cells [START_REF] Ehrhart-Bornstein | 17␣-hydroxylase and chromogranin A in 6th week human fetal adrenals[END_REF], demonstrated that PACAP-binding sites are exclusively borne by chromaffin cells. The strong autoradiographic signal observed at the earlier stage studied (14-week-old fetuses) is consistent with previous reports indicating that, in the developing human adrenal gland, chromaffin cells establish their phenotype as early as 6 weeks of gestation, as evidenced by the expression of CgA and tyrosine hydroxylase [START_REF] Ehrhart-Bornstein | 17␣-hydroxylase and chromogranin A in 6th week human fetal adrenals[END_REF][START_REF] Molenaar | Early fetal acquisition of the chromaffin and neuronal immunophenotype by human adrenal medullary cells. An immunohistological study using monoclonal antibodies to chromogranin A, synaptophysin, tyrosine hydroxylase, and neuronal cytoskeletal proteins[END_REF]. The fetal human adrenal cortex is composed of a thin outer zone (neocortex) and a large fetal zone that represents over 80% of the gland [START_REF] Winter | Fetal and neonatal adrenocortical development[END_REF][START_REF] Bocian-Sobkowska | Cytological aspects of the human adrenal cortex development in the course of intra-uterine life[END_REF]. The present study showed that both of these cortical zones are virtually devoid of PACAP-binding sites. In agreement with this observation, autoradiographic labeling, membrane binding assay, and in situ hybridization immunohistochemistry studies indicate that, in the adult rat adrenal gland, PACAP-binding sites are located only on chromaffin cells (20 -22).

Scatchard plot analysis of PACAP-binding in the fetal human chromaffin tissue showed the existence of highaffinity binding sites with K d values ranging from 0.32-0.74 nmol/L. Displacement experiments demonstrated that PACAP27 and PACAP38 were equally potent in competing with the radioligand at all developmental stages studied, whereas VIP was a much weaker competitor. These data revealed that the binding sites evidenced with [ 125 I]PACAP27 correspond predominantly to type I PACAP receptors. The fact that the Hill coefficients measured with PACAP27 were close to 1 suggested that the peptide interacts with a single class of receptors. However, the lower Hill coefficient value calculated with VIP as a competitor would indicate the existence of a heterogeneous population of PACAP/VIP receptors, as previously reported in rat [START_REF] Shivers | Two high-affinity binding sites for pituitary adenylate cyclase-activating polypeptide (PACAP) have different tissue distribution[END_REF][START_REF] Mazzochi | Vasoactive intestinal polypeptide (VIP) stimulates hormonal secretion of the rat adrenal cortex in vitro: evidence that adrenal chromaffin cells are involved in the mediation of the mineralocorticoid, but not glucocorticoid secretagogue action of VIP[END_REF][START_REF] Hinson | Vasoactive intestinal peptide is a local regulator of adrenocortical function[END_REF] and human adult adrenal medulla [START_REF] Bornstein | Cellular communication in the neuro-adrenocortical axis: role of vasoactive intestinal polypeptide (VIP)[END_REF].

Second-messenger coupling of PACAP receptors

There is now clear evidence that type I PACAP receptors can be coupled to adenylyl cyclase and phospholipase C (6 -7, 10). We have thus investigated the transduction mechanisms associated with the PACAP binding sites in fetal human adrenomedullary cells. We found that PACAP38 stimulated, in a dose-dependent manner, cAMP production from both isolated and cultured adrenal cells. The concen- tration of ligand which causes 50% inhibition of the binding (IC 50 ) for PACAP27 binding and the ED 50 for cAMP formation were in the same range (0.3 and 0.07 nmol/L, respectively), showing the existence of a good correlation between binding and coupling potency. These data indicate that the binding sites visualized by autoradiography actually correspond to functional receptors positively coupled to adenylyl cyclase. However, although PACAP38 stimulates both adenylyl cyclase and phospholipase C activity in porcine adrenochromaffin cells [START_REF] Isobe | Ca 2ϩ -dependent stimulatory effect of pituitary adenylate cyclase-activating polypeptide on catecholamine secretion from cultured porcine adrenal medullary chromaffin cells[END_REF], we found that PACAP causes only a slight increase in InsP formation. The modest effect of PACAP on phospholipase C activity suggests that fetal human adrenal cells express either the SV1 or the SV3 splice variants of the PACAP receptor, which exhibit a low efficacy for phospholipase C activation [START_REF] Pisegna | Cloning and characterization of the signal transduction of four splice variants of the human pituitary adenylate cyclaseactivating polypeptide receptor[END_REF]. Alternatively, the Gq protein, which is required for receptor coupling to phospholipase C, may be expressed at a very low level at this stage of fetal develop-ment in human adrenochromaffin cells. In support of this latter hypothesis, we found that fluoroaluminate, a nonspecific activator of all heterotrimeric G proteins, induces a 2-fold increase in InsP production in fetal adrenal cells, compared with a 15-fold increase in adult cells (data not shown).

Functional implications

The role of PACAP in the human adrenal during development is currently unknown. In vivo and in vitro studies on animal models have shown that PACAP stimulates tyrosine hydroxylase activity and catecholamine secretion from chromaffin cells [START_REF] Watanabe | Pituitary adenylate cyclase-activating polypeptide provokes cultured rat chromaffin cells to secrete adrenaline[END_REF][START_REF] Isobe | Ca 2ϩ -dependent stimulatory effect of pituitary adenylate cyclase-activating polypeptide on catecholamine secretion from cultured porcine adrenal medullary chromaffin cells[END_REF][START_REF] Marley | Activation of tyrosine hydroxylase by pituitary adenylate cyclase-activating polypeptide (PACAP-27) in bovine adrenal chromaffin cells[END_REF]. It has recently been reported that PACAP also enhances catecholamine release from adult human adrenal explants [START_REF] Neri | Pituitary adenylate cyclase-activating polypeptide enhances aldosterone secretion of human adrenal gland: evidence for an indirect mechanism, probably involving the local release of catecholamines[END_REF]. These data suggest that PACAP may also exert a stimulatory effect on catecholamine secretion by fetal adrenochromaffin cells. In support of this hypothesis, immunohistochemical studies have shown the presence of both tyrosine hydroxylase and dopamine ␤-hydroxylase in human chromaffin cells at early stages of development [START_REF] Cooper | Histogenesis of the human adrenal medulla[END_REF][START_REF] Molenaar | Early fetal acquisition of the chromaffin and neuronal immunophenotype by human adrenal medullary cells. An immunohistological study using monoclonal antibodies to chromogranin A, synaptophysin, tyrosine hydroxylase, and neuronal cytoskeletal proteins[END_REF].

In the adult human adrenal gland, Neri et al. [START_REF] Neri | Pituitary adenylate cyclase-activating polypeptide enhances aldosterone secretion of human adrenal gland: evidence for an indirect mechanism, probably involving the local release of catecholamines[END_REF] have shown that PACAP stimulates aldosterone and 18-OH-corticosterone secretion, probably via an indirect mechanism involving the release of catecholamines by chromaffin cells. This paracrine effect of PACAP in the adult adrenal tissue is facilitated by the existence of islets and medullary rays within the adrenal cortex [START_REF] Gallo-Payet | On the presence of chromaffin cells in the adrenal cortex: their possible role in adrenocortical function[END_REF][START_REF] Bornstein | Intimate contact of chromaffin and cortical cells within the human adrenal gland forms the cellular basis for important intraadrenal interactions[END_REF]. In the fetal human adrenal, a paracrine mode of action is even more plausible, inasmuch as, during ontogenesis, chromaffin cells migrate centripetally through the cortex towards their central destination [START_REF] Ehrhart-Bornstein | 17␣-hydroxylase and chromogranin A in 6th week human fetal adrenals[END_REF][START_REF] Molenaar | Early fetal acquisition of the chromaffin and neuronal immunophenotype by human adrenal medullary cells. An immunohistological study using monoclonal antibodies to chromogranin A, synaptophysin, tyrosine hydroxylase, and neuronal cytoskeletal proteins[END_REF].

PACAP is known to act as a growth factor, modulating proliferation, survival, and differentiation of various types of neuronal cells [START_REF] Gonzalez | Pituitary adenylate cyclase-activating polypeptide (PACAP) promotes cell survival and neurite outgrowth in rat cerebellar neuroblasts[END_REF][START_REF] Deutsch | 38-amino acid form of pituitary adenylate cyclase-activating polypeptide induces process outgrowth in human neuroblastoma cells[END_REF][START_REF] Vaudry | PACAP stimulates both c-fos gene expression and cell survival in rat cerebellar granule neurons through activation of the protein kinase A pathway[END_REF]. In particular, PACAP has been shown to induce neurite outgrowth and tyrosine hydroxylase gene expression in cultured rat chromaffin cells [START_REF] Tischler | Mitogenic and antimitogenic effects of pituitary adenylate cyclase-activating polypeptide (PACAP) in adult rat chromaffin cell cultures[END_REF][START_REF] Wolf | Phenotypic development of neonatal rat chromaffin cells in response to adrenal growth factors and glucocorticoids: focus on pituitary adenylate cyclase-activating polypeptide[END_REF] and PC12 cells [START_REF] Deutsch | The 38-amino acid form of pituitary adenylate cyclaseactivating polypeptide stimulates dual signaling cascades in PC12 cells and promotes neurite outgrowth[END_REF]. It is thus conceivable that PACAP also may affect the migration and/or differentiation of immature human chromaffin cells.

In conclusion, the present study has shown that type I PACAP receptors are expressed in human chromaffin cells in 14-to 20-week-old fetuses and that these receptors are functionally coupled to adenylyl cyclase. These data suggest that PACAP may play a role in the differentiation and/or secre- tory activity of chromaffin cells in the fetal human adrenal gland. Sections of fetal adrenal glands were processed for autoradiography as described in the Materials and Methods section. IC 50 and Hill coefficients obtained with PACAP27 are the mean Ϯ SE of at least three independent experiments. For PACAP38 and VIP, results are the mean of two independent experiments.

  Address all correspondence and requests for reprints to: Dr. Hubert Vaudry, European Institute for Peptide Research (IFRMP 23), Laboratory of Cellular and Molecular Neuroendocrinology, Institut National de la Sante ´ et de la Recherche Me ´dicale U413, UA CNRS, University of Rouen, 76821 Mont-Saint-Aignan, France. E-mail: hubert.vaudry@ univ-rouen.fr. * This work was supported by the Medical Research Council of Canada (MRC MT13679, to N.G.-P. and J.-G.L.), the Institut National de la Sante ´ et de la Recherche Me ´dicale (INSERM U413), a France-Que ´bec exchange programme (Coope ´ration Scientifique et Technologique Franco-Que ´bécoise (PVP-73-9, to N.G.-P. and H.V.), and the Conseil Re ´gional de Haute-Normandie. † Recipient of studentship from the Fonds de la Recherche en Sante ´ du Que ´bec et Fonds pour la Formation de Chercheurs et l'Aide a `la Recherche.

  [ 3 H]ATP nucleotide from [ 3 H]cAMP. cAMP formation was expressed as: % conversion ϭ ([ 3 H]cAMP/([ 3 H]cAMP ϩ [ 3 H]ATP) ϫ 100 per 15 min.

FIG. 1 .

 1 FIG. 1. Microphotographs of sections of an adrenal gland from a 16week-old human fetus. A, Autoradiographic localization of [ 125 I]PACAP27-binding sites; B, immunohistofluorescent localization of chromogranin A-containing cells; C, hematoxylin-eosin staining; CC, chromaffin cells; FZ, fetal zone; NC, neocortex. Bars: 1.4 mm (A); 0.6 mm (B); 0.2 mm (C).

FIG. 2 .

 2 FIG. 2. Typical autoradiograms showing the displacement of [ 125 I]PACAP27 binding (A) by graded doses of unlabeled PACAP27 (B-D), PACAP38 (E-G), or VIP (H-J). All autoradiograms were obtained from 10-m consecutive sections of an adrenal gland from a 15week-old human fetus. Bar: 2 mm (A-J).

FIG. 3 .

 3 FIG. 3. Representative competition curves showing the displacement of [ 125 I]PACAP27 binding on adrenal slices from human fetuses by graded concentrations of unlabeled PACAP27 (E-E), PACAP38 (F-F), or VIP (-). The curves were obtained by quantification of autoradiograms similar to those shown in Fig. 2. The six graphs correspond to six different ages of gestation: A, 14 weeks; B, 15 weeks; C, 16 weeks; D, 18 weeks; E, 19 weeks; and F, 20 weeks. Inset, Scatchard representation of the displacement curve by PACAP27, shown in B.

FIG. 4 .

 4 FIG. 4. Effect of PACAP38 on cAMP production by adrenal cells from 18-to 20-week-old human fetuses. Suspended cells, used after a 12-h resting period (A) or cells cultured for 5 days (B), were labeled with [ 3 H]adenine and then incubated for 15 min at 37 C in the absence (control) or in the presence of graded concentrations of PACAP38. Conversion of ATP into cAMP was calculated as described in Materials and Methods. Results are the mean (Ϯ SE) of triplicate determinations from one representative experiment out of three. When no error bars are shown, they are contained within the symbols.

TABLE 1 .

 1 Affinity and binding capacity of [ 125 I]PACAP27 binding sites in the fetal human adrenal Materials and Methods. K d and Bmax values were determined from Scatchard plot analysis of competition studies using PACAP27 as a competitor. Results are the mean Ϯ SE. Bmax, binding capacity.

	Weeks of gestation	K d (nmol/L)	Bmax (pmol/mg wet tissue)
	14	0.39 Ϯ 0.10	0.43 Ϯ 0.14
	15	0.37 Ϯ 0.07	0.30 Ϯ 0.06
	16	0.54 Ϯ 0.01	0.35 Ϯ 0.02
	17	0.74 Ϯ 0.04	0.46 Ϯ 0.05
	18	0.54 Ϯ 0.06	0.81 Ϯ 0.10
	19	0.32 Ϯ 0.03	0.35 Ϯ 0.07
	20	0.59 Ϯ 0.04	0.69 Ϯ 0.09
	Sections of fetal adrenal glands were processed for autoradiogra-
	phy as described in

TABLE 2 .

 2 Half-maximum displacement and Hill coefficients of [ 125 I]PACAP27 binding sites in the fetal human adrenal gland

	Weeks of gestation	PACAP27	IC 50 (nmol/L) PACAP38	VIP	PACAP27	Hill coefficients PACAP38	VIP
	14	0.35 Ϯ 0.15	0.22	311	0.99 Ϯ 0.04	0.82	0.23
	15	0.28 Ϯ 0.04	0.23	4	0.93 Ϯ 0.10	0.80	0.26
	16	0.46 Ϯ 0.02	0.52	141	1.04 Ϯ 0.11	0.90	0.19
	18	0.64 Ϯ 0.08	0.15	362	0.92 Ϯ 0.05	0.74	0.58
	19	0.29 Ϯ 0.03	0.81	11	0.91 Ϯ 0.02	0.69	0.28
	20	0.51 Ϯ 0.04	0.29	41	0.93 Ϯ 0.03	0.82	0.27

Acknowledgments

The authors would like to thank Ms. Huguette Lemonnier (INSERM U413) and Lucie Chouinard (Service of Endocrinology, Sherbrooke) for skillful technical assistance. We appreciate the advice from Drs. Magali Basille and Bruno Gonzalez (INSERM U413) for autoradiography.


T Chaumont-Frelet

D Pardo

Á Rodr Íguez-Rozas

FINITE ELEMENT SIMULATIONS OF LOGGING-WHILE-DRILLING AND EXTRA-DEEP AZIMUTHAL RESISTIVITY MEASUREMENTS USING NON-FITTING GRIDS

We propose a discretization technique using non-fitting grids to simulate magnetic field based resistivity logging measurements. Non-fitting grids are convenient because they are simpler to generate and handle than fitting grids when the geometry is complex. On the other side, fitting grids have been historically preferred because they offer additional accuracy for a fixed problem size in the general case. In this work, we analyse the use of non-fitting grids to simulate the response of logging instruments that are based on magnetic field resistivity measurements using 2.5D Maxwell's equations. We provide various examples demonstrating that, for these applications, if the finite element matrix coefficients are properly integrated, the accuracy loss due to the use of non-fitting grids is negligible compared to the case where fitting grids are employed.

Introduction

Electromagnetic (EM) logging methods are widely used in oil engineering for their ability to distinguish between oil and water-saturated rocks [START_REF] Liu | Principles and applications of well logging[END_REF]. Applications include logging while drilling (LWD) [START_REF] Pardo | Two-dimensional high-accuracy simulation of resistivity logging-while-drilling (lwd) measurements using a self-adaptive goal-oriented hp finite element method[END_REF], deep and extra-deep azimuthal logging [START_REF] Bittar | A new azimuthal deep-reading resistivity tool for geosteering and advanced formation evaluation[END_REF][START_REF] Tilsley-Baker | The Importance of Extra-Deep Azimuthal Resistivity in Reservoir Development: An Update on Recent Field Experiences[END_REF] and crosswell tomography [START_REF] Wilt | Crosswell electromagnetic tomography: System design considerations and field results[END_REF]. In these applications, an EM field is generated by a logging instrument placed inside a well. The tool is equipped with EM receivers in order to measure how the generated field is impacted by the resistivity of surrounding rocks. The raw receiver measurements are rarely provided, but instead, they are often post-processed to compute a quantity called "apparent resistivity", which is defined as the resistivity of the infinite homogeneous medium that reproduces the measurements [START_REF] Liu | Principles and applications of well logging[END_REF]. The apparent resistivity is recorded as the logging instrument is moved along the well, and the resulting curve in which the apparent resistivity is plotted against the tool position is called a "resistivity log".

In simple geometries where the logging trajectory is almost perpendicular to the discontinuities in resistivity, practitioners can often identify the EM rock properties nearby the well directly from the resistivity log. When the formation geometry is more complex, however, the relation between apparent resistivity and the actual resistivity distribution is not obvious. In addition, borehole effects [START_REF] Nam | Simulation of eccentricity effects on shot-and long-normal logging measurements using a fourier-hp-finite-element method[END_REF][START_REF] Nimmer | Effects of borehole fill on resistivity investigations[END_REF], and invasion by drilling mud [START_REF] Howard | Electromagnetic borehole fields in a layered, dipping-bed environment with invasion[END_REF][START_REF] Salazar | Quantitative comparison of processes of oil-and water-based mud-filtrate invasion and corresponding effects on borehole resistivity measurements[END_REF] can further affect the measurements.

To properly interpret borehole resistivity measurements in all possible situations, numerical inversion and simulation algorithms are required. Hence, there is a need for accurate, robust and efficient discretization schemes that are able to reproduce resistivity logs. This is especially true in geosteering applications, in which inverted resistivity measurements are used to guide the drilling well trajectory in real time.

Nowadays, real-time fully 3D simulations are out of reach. But they are feasible if additional assumptions are made on the resistivity distribution (see e.g. [START_REF] Pardo | Fast 1d inversion of logging-while-drilling resistivity measurements for improved estimation of formation resistivity in high-angle and horizontal wells[END_REF] or [START_REF] Muga | Semi-analytical response of acoustic logging measurements in frequency domain[END_REF]). For simplicity of computations, we focus on the case where the resistivity distribution is invariant along one space direction, say y. Notice, however, that most results presented here can be trivially applied to fully 3D scenarios.

Given a 2D conductivity distribution, we apply a Fourier transform along the y direction, and we obtain a set of independent 2D problems, one per Fourier mode ξ. The strategy then consists in sampling a finite number of Fourier modes {ξ j } N j=1 for which the 2D problems are solved. We then recover the electromagnetic fields by performing an approximate inverse Fourier transform using the sampled Fourier modes. As this technique amounts to approximate a 3D problem by a sequence of 2D problems, it is usually called 2.5D [START_REF] Abubakar | 2.5d forward and inverse modeling for interpreting low-frequency electromagnetic measurements[END_REF][START_REF] Abubakar | An integral equation approeach for 2.5-dimensional forward and inverse electromagnetic scattering[END_REF].

For each sampled Fourier mode, one Maxwell-type 2D problem has to be numerically approximated. Different families of numerical methods are available, including finite differences [START_REF] Abubakar | 2.5d forward and inverse modeling for interpreting low-frequency electromagnetic measurements[END_REF][START_REF] Davydycheva | An efficient finite-difference scheme for electromagnetic logging in 3d anisotropic inhomogeneous media[END_REF], finite elements [START_REF] Pardo | Simulation of marine controlled source electromagnetic measurements using a parallel fourier hp-finite element method[END_REF] and integral equations [START_REF] Abubakar | An integral equation approeach for 2.5-dimensional forward and inverse electromagnetic scattering[END_REF]. In this work, we focus on finite elements discretizations. We partition the computational domain into a finite element mesh over which we define basis functions. We employ tensor product meshes, which greatly simplify the implementation. In addition, efficient linear solvers and domain decomposition techniques can be easily designed for this type of meshes.

Unfortunately, if used directly, tensor product grids are not well adapted to handle complex geometries. Indeed, if the conductivity distribution does not exhibit a Cartesian structure, the conductivity parameter will take different values inside some mesh elements. In particular, inclined layers produce the so-called stair-case approximations [START_REF] Cangellaris | Analyses of the numerial error caused by the stair-stepped approximation of a conducting boundary in FDTD simulations of electromagnetic phenomena[END_REF]. More generally, we shall refer to "nonfitting meshes" to describe those cases in which the physical interfaces of the conductivity model are not aligned with the element edges. Several approaches have been proposed to overcome difficulties due to the use of non-fitting grids, both for finite differences and finite elements. These approaches include homogenization [START_REF] Habashy | A generalized material averaging formulation for modelling the electromagnetic fields[END_REF] and immersed interface [START_REF] Leveque | The immersed interface method for elliptic equations with discontinuous coefficients and singular sources[END_REF] methods.

In this work, we consider finite elements set on non-fitting meshes. The main justification for using non-fitting grids is the analysis performed in [START_REF] Chaumont-Frelet | Finite element approximation of electromagnetic fields using non-fitting meshes for geophysics[END_REF], where the authors consider 3D Maxwell's equations in a propagation medium that exhibits a constant magnetic permeability, and investigate the approximation properties provided by non-fitting grids for first-order edge finite elements. They show that the use of non-fitting grids (as compared to fitting grids) severely deteriorates the electric field approximation. However, the optimal convergence rate in terms of element size h is preserved for the magnetic field as long as accurate quadrature schemes are employed to integrate the finite element linear system. As a result, the use of non-fitting grids seems appealing to simulate the responses of logging instruments that are based on magnetic field measurements.

The main contribution of the present paper is to assess the performance and accuracy of nonfitting grids to simulate resistivity logs in practical applications. To this end, we consider realistic examples that are representative of LWD and extra-deep azimuthal applications. We propose three different quadrature techniques and discuss their efficiency and accuracy. In addition, the results obtained with non-fitting grids are compared with those resulting from using standard finite element discretizations with fitting meshes.

The paper is organized as follow. Section 2 describes the model problem we solve. The discretization strategy is presented in Section 3. Section 4 is devoted to numerical experiments, and is followed by the conclusions.

Model Problem

2.1. Logging-While-Drilling instrument. For Logging-While-Drilling applications, we consider a logging device equipped with two transmitters and two receivers. As depicted on Figure 1, the tool has a symmetric configuration. The receivers are located at a distance d RX = 0.1 m from the center of symmetry. For the transmitters, the distance is d T X = 0.6 m. The transmitters generate alternating currents (AC) with frequency f = 2 MHz. The transmitter coils are represented by punctual sources. Similarly, we use the value of the magnetic field at the receivers location to modelize the receivers measurements.

We denote by x, ŷ, ẑ a coordinate system aligned with the logging instrument, ẑ being the direction of the tool axis. We denote by H ij ab (where a, b = x, ŷ or ẑ and i, j = 1 or 2) the magnetic field value recorded by the j-th receiver in the b direction when the field is generated by the i-th transmitter in the a direction. The attenuation A i ab and the phase difference P i ab measured when the i -th transmitter is active are then defined via post processing as

A i ab = Re log H i1 ab H i2 ab , P i ab = Im log H i1 ab H i2 ab .
The final attenuation and phase difference are defined as the average of the values produced by the two transmitters:

A ab = 1 2 A 1 ab + A 2 ab , P ab = 1 2 P 1 ab + P 2 ab .
We compute the apparent resistivity by post processing either the attenuation or the phase difference. Since analytical solutions are available in homogeneous media, for a value of ρ, one easily estimates the attenuation and phase corresponding to the tool configuration. Hence, the curves ρ → Āab (ρ) and ρ → Pab (ρ) in a homogeneous medium can be computed once the tool configuration is known. This is an "off-line" operation that needs to be performed only once per tool configuration.

The apparent resistivities are then obtained as the values ρ A ab and ρ P ab such that Āab (ρ A ab ) = A ab or Pab (ρ P ab ) = P ab , assuming both functions Āab and Pab are bijective. If any of them is not injective, it leads to a multiple definition of apparent resistivity, while the lack of surjectivity produces the so-called "horns" on the resistivity log.

In applications, all transmitter and receiver directions (a, b = x, ŷ, ẑ) are of interest to help identify the resistivity distribution of the formation [START_REF] Davydycheva | Triaxial induction tool with electrode sleeve: FD modeling in 3D geometries[END_REF]. In this work, however, for the sake of simplicity we will focus on the coaxial component a = b = ẑ. Nonetheless, all triaxial components can be naturally handled.

Deep-azimuthal logging instrument.

In deep-azitmuthal logging applications, we consider a logging device equipped with one transmitter and one receiver that are spaced by 25 m, as depicted on Figure 2. The transmitter operates at the frequency f = 2 KHz. We use the same modelization techniques than for the LWD tool to represent the transmitter and the receiver. We denote by x, ŷ, ẑ a coordinate system aligned with the logging instrument, ẑ being the direction of the tool axis. The transmitter is oriented along the tool axis. We consider two configurations for the receiver: one where the ẑ component of the magnetic field is recorded (H zz ), and one where the x component is measured (H zx ).

2.3. The 2.5D Maxwell's system. As the logging instrument emits alternating currents (AC) at frequency f , time-harmonic Maxwell's equations are used to modelize the electromagnetic fields. The magnetic field H : R 3 → C 3 is then characterized as the solution to the reduced wave equation given by ( 1)

iωµ 0 H + ∇ × (ρ∇ × H) = M in R 3 ,
where M is a magnetic dipole representing the transmitter. We consider a transverly isotropic (TI) medium, so that

ρ =   ρh 0 0 0 ρh 0 0 0 ρv   =   iωε 0 + ρ h 0 0 0 iωε 0 + ρ h 0 0 0 iωε 0 + ρ v   ,
and ρ h , ρ v : R 3 → R * + are the horizontal and vertical resistivies, respectively. ε 0 = 8.854 × 10 -12 Fm -1 and µ 0 = 4π × 10 -7 NA -2 are the vacuum permittivity and permeability, ω = 2πf is the angular frequency and i is the imaginary unit (i 2 = -1). In addition to (1), the magnetic field H should decay (exponentially fast) as approaching infinity. To simplify the mathematical analysis, we represent the magnetic dipole by a volumetric source. Here, we select a Gaussian load to represent such magnetic dipole. Thus, M = gd, where g is a 3D Gaussian function and d a unit vector that represents the direction of the transmitter. The actual amplitude of the source is irrelevant, since phase differences and attenuations are insensitive to source amplitude variations.

We assume a two dimensional (2D) resistivity distribution. Hence, we write ρ(x, y, z) = ρ(x, z) where the y direction is arbitrary. We perform a Fourier transform with respect to y. Because ρ does not depend on y, the resulting equations decouple for each Fourier mode. Thus, for ξ ∈ R, we introduce

(2) Ĥξ (x, z) = +∞ -∞
H(x, y, z)e iξy dy.

For each ξ ∈ R, Ĥξ is a vector with three components depending on variables x and z. We introduce H ξ y = Ĥξ y and H ξ = ( Ĥξ x , Ĥξ z ), so that H ξ y is a 2D scalar function and H ξ is a 2D vector field.

Starting from 3D problem (1), we have that for each fixed ξ ∈ R, the pair (H ξ , H ξ y ) is solution to the following 2D problem:

(3)

iωµ 0 I 2 + ξ 2 ρ H ξ + curl ρh curl H ξ -iξ ρ∇H ξ y = M ξ iωµ 0 H ξ y -∇ • ρ∇H ξ y + iξ∇ • ρH ξ = M ξ y , where M ξ = (d x , d z )ĝ ξ and M y = d y
ĝξ and ĝξ stands for the Fourier transform of the Gaussian load g. Also, we have kept the notation ρ, and introduced the notation I 2 to denote the 2 × 2 matrices ρ = ρh 0 0 ρv , I 2 = 1 0 0 1 .

In addition to (3), a decaying condition is satisfied at infinity. To mimic that condition, we artificially bound the domain of computation to numerically approximate the problem with finite elements. The artificial boundary only introduces a small modelization error when placed sufficiently far from the magnetic dipole. We thus introduce a domain Ω which includes the magnetic dipole. We select the boundary ∂Ω such that it is sufficiently far from the source. Then, we prescribe the following boundary conditions on ∂Ω (4)

H ξ • n = 0, ∇H ξ y • n = 0,
where n is the unit vector normal to ∂Ω pointing outward Ω.

Taking into account boundary conditions (4), we reformulate (3) into the following variational problem:

Find (H ξ , H ξ y ) ∈ H(curl, Ω) × H 1 (Ω) such that (5) iωµ 0 I 2 + ξ 2 ρ H ξ , ψ + (ρ h curl H ξ , curl ψ) -iξ(ρ∇H ξ y , ψ) = (M ξ , ψ) ∀ψ ∈ H(curl, Ω), iωµ 0 H ξ y , φ + (ρ∇H ξ y , ∇φ) -iξ(ρH ξ , ∇φ) = (M ξ y , φ) ∀φ ∈ H 1 (Ω).
For general geometries, one can not solve (5) analytically. It is thus required to numerically approximate its solution. In this work, we use a finite element method described in Section 3.

In addition, we do not approximate the solution to problem (5) for all Fourier modes ξ ∈ R. Instead, we restrict to a finite number of selected Fourier modes and recover the magnetic field using an approximate inverse Fourier transform: [START_REF] Cangellaris | Analyses of the numerial error caused by the stair-stepped approximation of a conducting boundary in FDTD simulations of electromagnetic phenomena[END_REF] H(x, y, z)

N f j=1 w j Ĥξj (x, z)e -ξj y ,
where {ξ j } N f j=1 are N f selected Fourier modes, and {w j } N f j=1 are associated weights.

3. The finite element method 3.1. Finite element spaces. We use a finite element method to approximate variational formulation (5) of 2.5D Maxwell's equations. The tangential and normal components of the approximated field respectively belong to H(curl, Ω) and H 1 (Ω). As a result, we employ Nédélec edge elements [START_REF] Nédélec | Mixed finite elements in R 3[END_REF] to approximate the tangential component, and Lagrange nodal elements [START_REF] Ciarlet | The finite element method for elliptic problems[END_REF] to approximate the normal component (in the y direction).

For the sake of simplicity, we consider tensor product grids composed of squared elements. The presented method can be naturally extended to curved and/or triangular elements, as discussed in Section 3.2.

For a selected mesh T h , the Lagrange nodal element space for H ξ y is defined as

S h,p = v h,p ∈ H 1 (Ω) | v h,p ∈ Q pp (K) ∀K ∈ T h .
where p ≥ 1 is the polynomial degree. For the tangential component H ξ , we introduce the Nedelec edge element space

V h,p = v h,p ∈ H(curl, Ω) | v h,p ∈ Q p(p-1) (K) × Q (p-1)p (K) ∀K ∈ T h .
In the above definitions, Q rq stands for the space of scalar polynomials of degree at most r in the x variable and at most q in the z variable.

The discrete version of problem ( 5) is obtained by replacing the continuous functional spaces H(curl, Ω) and H 1 (Ω) by their discrete counterparts V h,p and S h,p .

3.2.

Assembly of the linear system and integration. We employ possibly non-fitting meshes. Authors of [START_REF] Chaumont-Frelet | Finite element approximation of electromagnetic fields using non-fitting meshes for geophysics[END_REF] demonstrate that non-fitting meshes are able to produce accurate approximations of the magnetic field. Specifically, they show that for first-order Nedelec's elements, the magnetic field approximation converges linearly in L 2 -norm, both for fitting and non-fitting meshes. This legitimate further investigations around the use of non-fitting grids to simulate the response of logging instruments that are based on magnetic measurements.

The analysis presented in [START_REF] Chaumont-Frelet | Finite element approximation of electromagnetic fields using non-fitting meshes for geophysics[END_REF] assumes that the coefficients of the finite element matrix are integrated exactly. For fitting meshes, performing the integration of the linear system coefficients is simple. Indeed, since the resistivity parameter is constant (or at least smooth) inside each cell, it is sufficient to select a quadrature scheme that integrates polynomials of sufficiently high degree over a square. For non-fitting meshes, however, the resistivity parameter can jump inside mesh cells, and the behaviour of traditional quadrature schemes in terms of accuracy is unclear.

In order to simplify the discussion, we focus on one particular term. In each element K, we need to integrate the contributions of basis functions φ i , φ j ∈ S h,p that involve the quantities [START_REF] Chaumont-Frelet | Finite element approximation of electromagnetic fields using non-fitting meshes for geophysics[END_REF] K ρh curl φ j curl φ i .

For a fitting mesh, ρh | K ∈ R is constant inside K. As a result, we can extract ρh from integral [START_REF] Chaumont-Frelet | Finite element approximation of electromagnetic fields using non-fitting meshes for geophysics[END_REF], and then, it simply remains to integrate a polynomial quantity over a square. This can be done analytically. Furthermore, optimized strategies involving a "master element" and affine mappings make it possible to compute [START_REF] Chaumont-Frelet | Finite element approximation of electromagnetic fields using non-fitting meshes for geophysics[END_REF] fast (see [START_REF] Ciarlet | The finite element method for elliptic problems[END_REF][START_REF] Monk | Finite element methods for Maxwell's equations[END_REF] for instance).

On the other hand, when the mesh is non-fitting, computing ( 7) is more challenging, since ρ can take several values inside K. A popular approach to compute the integral is then to introduce a homogenized (or, in some sense, averaged) parameter ρ K that is constant over K. However, it is challenging to select a particular averaging strategy, as several of them are available [START_REF] Habashy | A generalized material averaging formulation for modelling the electromagnetic fields[END_REF][START_REF] Kristensson | Homogenization of the Maxwell equations in an anisotropic material[END_REF][START_REF] Smith | Homogenization of metamaterials by field averaging (invited paper)[END_REF]. In addition, most averaging strategies are rigorously established under some restrictive assumption of the parameter ρ. Examples include one dimensional variations [START_REF] Babuška | Special finite element methods for a class of second order elliptic problems with rough coefficients[END_REF][START_REF] Moskow | A finite difference scheme for elliptic equations with rough coefficients using a cartesian grid nonconforming to interfaces[END_REF] or small-scale periodicity [START_REF] Kristensson | Homogenization of the Maxwell equations in an anisotropic material[END_REF]. While the resulting averaging formula can be used in more general media, the accuracy of the resulting homogenized approximation is not guaranteed.

In this work, we employ a different strategy to approximate integral ( 7) that is purely based on quadrature techniques. We do not modify the parameter ρ, but instead, we use a sufficiently accurate quadrature scheme to integrate (7) directly. In the following, we investigate three different strategies:

(a) "Constant parameter integration": we approximate ρh by a constant value over each cell. Specifically, we employ the value of ρh at the center x K of the element K. Thus, constant parameter integration reduces to (8)

K ρh curl φ j curl φ i ρh (x K ) K curl φ j curl φ i ,
This strategy is simple and can be rapidly evaluated using affine mappings from a "master element" (see [START_REF] Monk | Finite element methods for Maxwell's equations[END_REF]). As depicted in Figure 3, this method amounts to introduce a so-called "staircase" approximation. (b) "Gaussian integration": we use the formula

(9) K ρh curl φ j curl φ i N k=1 w k ρh (x k ) curl φ j (x k ) curl φ i (x k ),
where {x k } N k=1 and {w k } N k=1 are Gaussian quadrature points and weights that exactly integrate polynomials of Q 2p+1,2p+1 (K). Actually, ( 9) is just the widely used Gaussian quadrature technique. When the parameter ρh is smooth, it is known that the quadrature error is small and below the order of approximation. Thus, the quadrature error does not deteriorate the accuracy of the finite element solution. However, for non-fitting grids, the error caused by the quadrature rule might be more important, as the parameter ρh we consider features discontinuities. Gaussian integration (9) has the advantage to work for non-affine mappings, however, it is more costly than (8) in the case of affine mappings. In addition, there is no clear geometrical interpretation of the approximation committed on the resistivity as in the case of constant parameter integration. (c) "Subelements integration technique": we introduce a set of squared subelements {C s } S s=1 for each element K. Then, the integration is performed as

(10) K ρh curl φ j curl φ i S s=1 ρh (x s C ) Cs curl φ j curl φ i ,
where x s C is the center of C s . The last integral in the right-hand-side is easily computed, since the subelements C s are squares. In addition, if affine mappings are used, an efficient strategy involving "master subelements" has been designed in [START_REF] Barucq | Stability analysis of heterogeneous Helmholtz problems and finite element solution based on propagation media approximation[END_REF] to compute [START_REF] Davydycheva | Triaxial induction tool with electrode sleeve: FD modeling in 3D geometries[END_REF]. We 10) is thus a generalization of [START_REF] Ciarlet | The finite element method for elliptic problems[END_REF], where the parameter S can be tuned to achieve the desired integration accuracy. Figure 4 shows that this strategy provides a refined staircase approximation of the resistivity distribution.

The presented quadrature techniques can actually be employed on more general meshes. As presented in [START_REF] Barucq | Stability analysis of heterogeneous Helmholtz problems and finite element solution based on propagation media approximation[END_REF], the Constant parameter, and Subelement integration techniques naturally extend to triangular (or tetrahedral) meshes, as long as the elements are straight (e.g. defined with affine mappings). On the other hand, general curved quadrangular elements can be used together with the Gaussian integration technique.

3.3.

Comparison with 3D approaches. In this section, we discuss the computational cost of the proposed 2.5D approach compared to a standard 3D FEM discretization. We assume here that a direct solver is used to factorize the finite element linear system(s), and that the factorization(s) represent(s) most of the computational cost.

Each factorization requires O (N (d-1)/d dof

) 3 + N dof p 2d floating point operations (FLOPs), where d is the number of spatial dimensions, p is the polynomial degree, and N dof is the number of degrees of freedom in the linear system [START_REF] Garcia | The value of continuity: Refined isogeometric analysis and fast direct solvers[END_REF][START_REF] Pardo | The value of continuity: Refined isogeometric analysis and fast direct solvers[END_REF].

If we assume that each spatial dimension is subdivided into n elements, then N dof = O((np) d ). As a result, for a 3D FEM problem, the factorization requires O(n 6 p 6 ) FLOPs, while the cost for a 2D FEM problem is O(n 3 p 3 + n 2 p 6 ) FLOPs.

Since we must solve one 2D problem for each Fourier mode, we obtain that the cost of all factorizations for the 2.5D approach is O(N f n 3 p 3 + N f n 2 p 6 ), where N f is the number of Fourier modes.

Since the solution is expected to converge exponentially fast as a function of the number of Fourier modes, we have N f n in applications (for instance, in our LWD experiments n 100 and N f 30), the 2.5D approach employs less than O(n 4 p 3 + p 6 n 3 ) FLOPs. As a result, the 2.5D approach requires at least n 2 less FLOPs than the 3D approach. In addition, the computations associated with each Fourier mode are independent, so that one can easily implement fully parallel algorithms for the 2.5D approach.

4.

Numerical experiments for logging-while-drilling 4.1. Grid design. Our aim is to compare the accuracy of fitting grids versus non-fitting grids, and, for the case of non-fitting grids, to evaluate the errors committed by the use of non-exact integration techniques. To this end, we will use the coarsest possible grids that provide physically meaningful solutions. Indeed, for coarser grids, the non-fitting effect is more important. Thus, coarse grids represent the "worst case scenario".

In order to use coarse grids, we employ high order finite elements. In the following, we select the value p = 4. In addition, we use a different grid for each tool position. Such grids are more refined nearby the tool center than close to the domain boundary.

For a particular experiment, we consider the highest resistivity value ρ max present in the propagation medium. We adapt the grid so that it is able to correctly reproduce the resistivity log in a homogeneous medium of resistivity ρ max with an accuracy of 1%.

The grid is also adapted to each Fourier mode. We have empirically observed that using grids twice larger (both the domain and elements sizes are doubled) for the lowest Fourier mode than for the largest one lead to satisfactory results. Therefore, assuming that the Fourier modes are sampled in the interval [0, ξ max ], we first fix a grid for the Fourier mode zero. Then, for a higher Fourier mode ξ, we apply the scaling

x 0 → x ξ = 1 - 1 2 ξ ξ max x 0
to the grid. Finally, the original grid for the Fourier mode zero is obtained following a manual "trial and error" approach. In our validation experiments, the maximum resistivity is 10Ω•m. In this case, we end up with the grid size:

h(x) =        0.08m if |x| < 0.5m 0.1m if 0.5m < |x| < 0.8m 0.11m if 0.8m < |x| < 0.9m 0.8m if 0.9m < |x| < 2m.
In addition, we truncate the y direction with the length 2π, and we select the Fourier modes ξ j = 0, . . . , 25.

In the more realistic experiment we consider, the maximum resistivity is 100Ω•m, and the corresponding grid size is:

h(x) =        0.05m if |x| < 0.5m 0.075m if 0.5m < |x| < 0.75m 0.1m if 0.75m < |x| < 1m 2.0m if 1m < |x| < 4m.
We also select a larger length for y-direction in this experiment: 4π. As a result, we employ the Fourier modes

ξ j = j 2
, j = 0, . . . , 30.

Validation experiments.

We first consider isotropic layers with parallel interfaces. The logging tool follows a straight trajectory and crosses the layers interfaces with an angle θ.

In order to evaluate the accuracy loss due to the use of non-fitting grids, we propose two different scenarios, which are physically equivalent, but that are handled differently by numerical approximations. Figure 5 depicts these cases.

In the first scenario, the layers interfaces are horizontal, and the tool follows a straight path in the direction of the unit vector (cos(θ), sin(θ)). In this case, since the interfaces are horizontal, they are easily fitted by a Cartesian grid.

In the second case, the tool position is horizontal, but the interfaces follow the direction (cos(-θ), sin(-θ)). Since the relative angle between the tool trajectory and the physical interfaces is the same as before, the physical situation is identical. However, the numerical approximation will be different. 4.2.1. Two half-spaces. We consider a propagation medium composed of two half-spaces. In the upper half-space, the resistivity value is 1 Ω•m, while the value of 10 Ω•m is prescribed in the lower layer. The angle between the tool trajectory and the layer interface is θ = 10 o . Figure 6 shows the resistivity logs corresponding to fitting meshes and non-fitting meshes using different quadrature techniques. We employ a finer "overkilling" fitting grid to estimate the errors committed on the A-log and P-log. Figures 7 and8 describe those errors as a function of the tool position.

The "constant parameter" quadrature technique leads to inaccurate logs when the tool is close to the interface. We further see that for the cases of Gaussian or Subelements integration techniques, non-fitting meshes provide a level of accuracy comparable to that achieved by fitting meshes. the simulated A-and P-logs, respectively. We observe that all quadrature methods produce similar logs. Nevertheless, a zoom on a particular area shows that those obtained with the constant parameter integration technique are more "noisy". In this experiment, we select p = 2. We use grids that are refined around both the transmitter and receiver. Specifically, if we denote by

D(x) = min(|x -x RX |, |x -x T X |),
the distance between the point x and the receiver and transmitter, we design the grid so that the local element size satisfies

h(x) =    1m if D(x) < 5m 2m if 5m < D(x) < 10m 10m if 10m < D(x) < 50m.
We select the length 32π 100 m in the y-direction and the Fourier modes Figures 14 and16 represent the simulated values of H zz and H zx , respectively. The absolute errors (compared to an "overkill" solution computed with a very fine fitting grid) are displayed on Figures 15 and17. We observe that non-fitting grids provide sufficiently accurate results (with errors below the expected noise level), regardless of the integration method.

Due to the large depth-of-investigation of the considered logging instrument, in this example we realize that even the "constant integration" technique leads to highly accurate results in terms of the required engineering precision. We emphasize that in other deep-azimuthal examples, the use of a more precise technique (either Gaussian or sub-integration) may be necessary. Indeed, it is possible to consider a deep-azimuthal example whose results are identical to those of the LWD problems shown in Section 4 (this can be done by employing a simple re-scaling technique, see [START_REF] Rodríguez-Rozaz | Advances toward the fast simulation of 2.5d lwd and deep azimuthal resistivity tools[END_REF]).

Conclusions

Non-fitting grids are easier to generate than fitting grids, and thus lead to simpler and possibly more efficient implementations. For the particular case of 3D Maxwell's equations with a constant magnetic permeability, we know from [START_REF] Chaumont-Frelet | Finite element approximation of electromagnetic fields using non-fitting meshes for geophysics[END_REF] that the optimal convergence of the magnetic field is preserved for first-order edge finite elements.

In this work, we evaluate the impact of non-fitting grids on the accuracy of the simulated resistivity logs in the context of borehole logging applications. To this end, we consider 2.5D Maxwell's equations that we discretize using high-order edge and nodal elements (for the tangential and normal components of the magnetic field with respect to the Fourier plane, respectively). We present four numerical experiments, which are representative of LWD and deep-azimuthal applications. In all considered examples, the use of fitting and non-fitting grids provide approximations of similar quality, provided that an appropriate quadrature scheme is employed for non-fitting grids. We compare three different strategies that we call "Constant parameter", "Gaussian" and "Subelement".

Both Gaussian and Subelement integration techniques provide accurate solutions, while the fastest Constant integration technique often (but not always) lead to inaccurate results. The Gaussian integration technique seems more convenient for quadrangular elements, as non-affine mappings are easily handled. On the other hand, the Subelement integration technique may be more suitable for straight triangular elements.

Figure 1 .

 1 Figure 1. Sketch of the tool configuration for LWD

Figure 2 .

 2 Figure 2. Sketch of the tool configuration for deep-azimuthal

Figure 3 .

 3 Figure 3. Transformation of a physical interface by the constant parameter integration technique

Figure 4 .

 4 Figure 4. Transformation of a physical interface by the subelements integration technique

Figure 5 .

 5 Figure 5. Fitting (left) and non-fitting (right) grids for the validation experiments

Figure 6 .

 6 Figure 6. Resistivity log simulations for two half spaces

Figure 9 .Figure 10 .

 910 Figure 9. Resistivity log simulations for a thin layer

  Figures 12 and 13 describe the simulated A-and P-logs, respectively. We observe that all quadrature methods produce similar logs. Nevertheless, a zoom on a particular area shows that those obtained with the constant parameter integration technique are more "noisy".

Figure 11 .Figure 12 .Figure 13 . 5 . 5 . 1 .

 111213551 Figure 11. Synthetic example comprised of a formation and a geosteering well trajectory

2 .

 2 Validation example. We consider a single experiment that consists of two half-spaces. The top layer is anisotropic, with ρ h = 40Ω•m and ρ v = 200Ω•m. The bottom layer consists of an isotropic material, and we have ρ h = ρ v = 2Ω•m. The tool follows a straight path, and crosses the interface with an angle θ = 45 o .

Figure 14 .Figure 15 .Figure 16 .Figure 17 .

 14151617 Figure 14. H zz component for the deep-azimuthal logging experiment

Acknowledgements

The authors have received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 644602, the Projects of the Spanish Ministry of Economy and Competitiveness with reference MTM2016-76329-R (AEI/FEDER, EU), and MTM2016-81697-ERC/AEI, the BCAM "Severo Ochoa" accreditation of excellence SEV-2013-0323, and the Basque Government through the BERC 2014-2017 program, and the Consolidated Research Group Grant IT649-13 on "Mathematical Modeling, Simulation, and Industrial Applications (M2SI)".

4.2.2.

A thin layer. We now consider a 5 cm thick layer of resistivity 10Ω•m. The layer is embedded into a constant 1Ω•m resistivity background. The angle between the tool trajectory and the layer interfaces is θ = 10 o . Figure 9 presents the resistivity logs obtained using fitting and non-fitting meshes.

Figure 10 presents the error committed on the resistivity logs (compared to a reference solution computed on a finer mesh). As in the case of the two half-spaces, we see that constant parameter integration lead to an important accuracy loss. On the other hand, fitting and non-fitting meshes provide comparable accuracy when using Gaussian or Subelement quadrature techniques.

4.3.

A realistic synthetic example. We now consider the geophysical medium depicted on Figure 11. The medium contains a 5 m deep reservoir that is partially saturated with oil and water. In addition, the medium features two geophysical faults that are separated by 30 m. All materials are modelized as isotropic. The tool follows a trajectory of 90 m that is representative of applications in which one tries to keep the geosteering tool inside the oil saturated part of the reservoir.