Alessandro De Sire
email: alessandro.desire@unicz.it

Roberto De Sire
email: roberto.desire@libero.it

Claudio Curci
email: claudio.curci@asst-mantova.it

Fabiana Castiglione

Walter Wahli
email: walter.wahli@unil.ch

Role of Dietary Supplements and Probiotics in Modulating Microbiota and Bone Health: The Gut-Bone Axis

Keywords: bone health, osteoporosis, microbiota, gut microbiota, dietary supplements, probiotics

Osteoporosis is characterized by an alteration of bone microstructure with a decreased bone mineral density, leading to the incidence of fragility fractures. Around 200 million people are affected by osteoporosis, representing a major health burden worldwide. Several factors are involved in the pathogenesis of osteoporosis. Today, altered intestinal homeostasis is being investigated as a potential additional risk factor for reduced bone health and, therefore, as a novel potential therapeutic target. The intestinal microflora influences osteoclasts' activity by regulating the serum levels of IGF-1, while also acting on the intestinal absorption of calcium. It is therefore not surprising that gut dysbiosis impacts bone health. Microbiota alterations affect the OPG/RANKL pathway in osteoclasts, and are correlated with reduced bone strength and quality. In this context, it has been hypothesized that dietary supplements, prebiotics, and probiotics contribute to the intestinal microecological balance that is important for bone health. The aim of the present comprehensive review is to describe the state of the art on the role of dietary supplements and probiotics as therapeutic agents for bone health regulation and osteoporosis, through gut microbiota modulation.

Introduction

Poor bone health is a relatively silent condition that can develop in both adults and the elderly, inducing bone microstructural alterations and osteoporosis, with a decreased bone mineral density (BMD), leading to fragility fractures [START_REF] Marini | Epigenetic Mechanisms in Bone Biology and Osteoporosis: Can They Drive Therapeutic Choices?[END_REF]. Around 200 million people worldwide are affected by osteoporosis [START_REF] Pisani | Major osteoporotic fragility fractures: Risk factor updates and societal impact[END_REF]. Fragility fractures, due to low-energy trauma, might necessitate hospitalization and surgical interventions, with detrimental consequences, such as permanent disabilities and an increased risk of death [START_REF] Kanis | A new scorecard for osteoporosis in Europe[END_REF]. Osteoporosis is associated with several risk factors-primarily endocrine disorders, medication, immobility, lifestyle, and hereditary genetic factors [4].

Today, altered intestinal homeostasis is being investigated as a potential additional risk factor for reduced bone health, and also as a novel therapeutic target [START_REF] Zhang | The impact of the intestinal microbiome on bone health[END_REF]. Indeed, the intestinal microbial populations that live in symbiosis with their human host are collectively defined as the human gut microbiota [START_REF] Peterson | The NIH Human Microbiome Project[END_REF][START_REF] Qin | A human gut microbial gene catalogue established by metagenomic sequencing[END_REF], composed of more than 1000 different microorganisms, with the majority particular to each healthy subject, along with a common, stable, and abundant component of Bacteroidetes, Prevotellaceae, and Faecalibacterium species [START_REF] Nagpal | Gut microbiota: The next-gen frontier in preventive and therapeutic medicine?[END_REF][START_REF] Frost | Long-term instability of the intestinal microbiome is associated with metabolic liver disease, low microbiota diversity, diabetes mellitus and impaired exocrine pancreatic function[END_REF].

Alterations of the gut microbiota-called dysbiosis-which result in an imbalance between the types of microorganisms present in a person's natural microflora, can dysregulate metabolic processes, leading to inflammatory bowel diseases [START_REF] Wright | Recent advances in characterizing the gastrointestinal microbiome in Crohn's disease: A systematic review[END_REF], diabetes [START_REF] Ussar | Interactions between host genetics and gut microbiome in diabetes and metabolic syndrome[END_REF], malignancies [START_REF] Minemura | Gut microbiota and liver diseases[END_REF], cardiovascular diseases [START_REF] Pevsner-Fischer | The gut microbiome and hypertension[END_REF], rheumatoid arthritis, and systemic lupus erythematosus [START_REF] Rosser | A clinical update on the significance of the gut microbiota in systemic autoimmunity[END_REF].

Microbiota dysbiosis is a multifactorial condition that can be triggered by conditions such as stress, inflammation, and aging. In general, after healing of a diseased state, dysbiosis tends to be corrected naturally [START_REF] Ozaki | Association between gut microbiota, bone metabolism, and fracture risk in postmenopausal Japanese women[END_REF].

Environmental conditions associated with genetic factors of the host can modify their immunological profile and microbiota composition, which can unbalance the microbial community. For example, in animal models, quantitative trait loci encoding for interferons, interleukin-1 receptor-associated kinase 4 (IRAK-4),and transforming growth factor-beta 3 (TGF-β3) modulate the ratio of Bacteroidetes, Rikenellaceae, and Prevotellaceae, respectively [START_REF] Zhang | The impact of the intestinal microbiome on bone health[END_REF][START_REF] Mcknite | Murine gut microbiota is defined by host genetics and modulates variation of metabolic traits[END_REF].

Furthermore, the gut microbiota participates in the bone metabolic signaling pathways through the generation and translocation of metabolites. Bone is impacted by gut microbial dysbiosis that both impairs the intestinal absorption of calcium and dysregulates osteoclasts' activity via the serum levels of insulin-like growth factor 1 (IGF-1) [START_REF] Yan | Gut microbiota induce IGF-1 and promote bone formation and growth[END_REF]. Moreover, microbiota alterations also reduce bone strength and quality [START_REF] Hathaway-Schrader | Specific Commensal Bacterium Critically Regulates Gut Microbiota Osteoimmunomodulatory Actions During Normal Postpubertal Skeletal Growth and Maturation[END_REF][START_REF] Castaneda | Alterations to the gut microbiome impair bone tissue strength in aged mice[END_REF], and affect the OPG/RANKL pathway in osteoclasts [START_REF] Novince | Commensal Gut Microbiota Immunomodulatory Actions in Bone Marrow and Liver have Catabolic Effects on Skeletal Homeostasis in Health[END_REF]. Furthermore, dysbiosis in mice also interferes with skeletal muscle mass and physical performance, altering the production of rapsyn and Lrp4-two crucial proteins in the functioning of neuromuscular junctions [START_REF] Lahiri | The gut microbiota influences skeletal muscle mass and function in mice[END_REF]. Similarly, the antibiotic and antiparasitic drug metronidazole causes gut dysbiosis, inducing skeletal muscle atrophy and impacting muscle chronometabolism [START_REF] Manickam | Metronidazole Causes Skeletal Muscle Atrophy and Modulates Muscle Chronometabolism[END_REF].

Although gaps exist in the understanding of the pathophysiological mechanisms underpinning the interaction between gut microbiota and bone, it has been hypothesized that dietary supplements, prebiotics (non-digestible dietary fibers stimulating selective bacterial growth), and probiotics (microorganisms that might confer a health benefit to the host) contribute to the intestinal microecological balance, which is important for bone health [START_REF] Ratajczak | Do Only Calcium and Vitamin D Matter? Micronutrients in the Diet of Inflammatory Bowel Diseases Patients and the Risk of Osteoporosis[END_REF][START_REF] Behera | The role of gut microbiota in bone homeostasis[END_REF][START_REF] Floch | Probiotics and Prebiotics[END_REF]. Indeed, dietary supplements and probiotics might play a role in the management of osteoporosis, in conjunction with the already-known medical and lifestyle interventions.

For this purpose, we wrote the present comprehensive review after analyzing both animal and human-based model studies published in the past 20 years. Our survey focuses on the roles that dietary supplements and probiotics play as therapeutic agents for the regulation of bone health and osteoporosis, through gut microbiota modulation.

The Gut-Bone Axis

On the one hand, the gut microbiota shows variability depending on several conditions, such age, sex, diet, environmental conditions, and diseases [START_REF] Rizzoli | Microbiota and Bone Health: The Gut-Musculoskeletal Axis[END_REF][START_REF] Biagi | Through ageing, and beyond: Gut microbiota and inflammatory status in seniors and centenarians[END_REF]. On the other hand, bone is also a dynamic organ undergoing constant remodeling, maintaining bone mass and calcium serum level homeostasis [START_REF] Rizzoli | Microbiota and Bone Health: The Gut-Musculoskeletal Axis[END_REF].

Gut physiology and the gut microbial population are both involved in the regulation of bone metabolism at different levels. Gastrectomy causes a loss of BMD in both animal models and humans [START_REF] Heiskanen | Bone mineral metabolism after total gastrectomy[END_REF][START_REF] Tovey | A review of postgastrectomy bone disease[END_REF], and impaired gastric acid influences calcium metabolism and uptake [START_REF] Schinke | Impaired gastric acidification negatively affects calcium homeostasis and bone mass[END_REF]. In addition, several neuropeptides secreted by the gastrointestinal system (i.e., glucagon-like peptide-1, glucagon-like peptide-2, and ghrelin) enhance bone formation or inhibit bone resorption, whereas others (i.e., gastric inhibitory polypeptide, gastrin, and serotonin), in contrast, act as pro-osteoporotic factors [START_REF] Hansen | Alliances of the gut and bone axis[END_REF][START_REF] Karsenty | Regulation of bone mass by serotonin: Molecular biology and therapeutic implications[END_REF]. Bacterial variance and the prevalence of individual microbial species in the gut of the host can alter micronutrients, calcium, and vitamin D absorption, and cause systemic inflammation [START_REF] Chen | Association Between Gut Microbiota and Bone Health: Potential Mechanisms and Prospective[END_REF].

The relationship between the gut-bone axis and immunity is complex; under physiological conditions, the gut microbiota stimulates and develops the host's innate and adaptive immune systems [START_REF] Peterson | Immune homeostasis, dysbiosis and therapeutic modulation of the gut microbiota[END_REF]. By contrast, gut microbiota dysbiosis can provoke inflammatory cytokine overproduction, as shown in animal models. Under dysbiosis, bacterial metabolites are delivered to the liver by the blood circulation and upregulate the hepatic immune response. In parallel, they also alter the bone marrow CD4+ T cells, with a resulting pro-osteoclastic effect [START_REF] Novince | Commensal Gut Microbiota Immunomodulatory Actions in Bone Marrow and Liver have Catabolic Effects on Skeletal Homeostasis in Health[END_REF] (see Figure 1 for further details). In a recent study involving 181 postmenopausal women, divided into 3 groupsnormal BMD, osteopenia, and osteoporosis-fecal microbiota sample analysis revealed that the most prevalent microbial species in these conditions statistically correlated with altered bone metabolism [START_REF] Das | Gut microbiota alterations associated with reduced bone mineral density in older adults[END_REF]. In particular, Actinomyces and Clostridium species were more prevalent in subjects with osteoporosis, whereas Bacteroidetes and Firmicutes were more prevalent in healthy individuals. Interestingly, Actinomyces might be involved in the development of osteonecrosis of the jaw, whereas Firmicutes contribute to an estrogen analogue conversion, with anabolic effects on bone [START_REF] Rafii | The role of colonic bacteria in the metabolism of the natural isoflavone daidzin to equol[END_REF]. Interestingly, bacterial populations vary in osteopenia, primary osteoporosis, or secondary osteoporosis, but the relationship and prevalence between microbial species under these conditions are still debated [START_REF] Tu | The microbiota-gut-bone axis and bone health[END_REF]. To date, several substances-including trimethylamine N-oxide, N-acetyl-mannosamine, L-threonate, L-lysine, and short-chain fatty acids (SCFAs)-have been shown to modulate the gut microbiota, with a consequent impact on bone [START_REF] Tu | The microbiota-gut-bone axis and bone health[END_REF][START_REF] Liu | Gut microbiota-dependent trimethylamine N-Oxide are related with hip fracture in postmenopausal women: A matched case-control study[END_REF]. Therefore, the potential therapeutic role of dietary supplements and probiotics is intriguing and worthy of assessment in people affected by osteoporosis; such interventions might alter the gut microbiota composition, promoting bone health [START_REF] Behera | The role of gut microbiota in bone homeostasis[END_REF].

Effects of Dietary Supplements on the Gut Microbiota and Bone Health

Diet is the main determinant of the types and proportions of microorganisms in the gut microbiota, influencing bone health and wellness [START_REF] Claesson | Gut microbiota composition correlates with diet and health in the elderly[END_REF]. For example, a high-fat diet reduces microbiota biodiversity in mice, while a highly sweetened diet causes glucose intolerance [START_REF] Suez | Artificial sweeteners induce glucose intolerance by altering the gut microbiota[END_REF]. From a nutritional point of view, the Mediterranean diet provides fibers, fermented dairy products, and polyphenols, which have beneficial effects on the gut microbiota in humans, reducing the Firmicutes and Bacteroidetes ratios and enhancing the levels of SCFAs [START_REF] Gentile | The gut microbiota at the intersection of diet and human health[END_REF], Together, these effects promote bone health and reduce fracture risk [START_REF] Byberg | Mediterranean Diet and Hip Fracture in Swedish Men and Women[END_REF][START_REF] De Filippis | High-level adherence to a Mediterranean diet beneficially impacts the gut microbiota and associated metabolome[END_REF][START_REF] Palomeras-Vilches | Adherence to the Mediterranean Diet and Bone Fracture Risk in Middle-Aged Women: A Case Control Study[END_REF][START_REF] Noel | Dietary Approaches to Stop Hypertension, Mediterranean, and Alternative Healthy Eating indices are associated with bone health among Puerto Rican adults from the Boston Puerto Rican Osteoporosis Study[END_REF].

Compared to the known beneficial effects of the Mediterranean diet, it is still uncertain whether supplementation with prebiotics, proteins, peptides, amino acids, and micronutri-ents might modulate the gut microbiota composition to induce beneficial effects on bone health [START_REF] Zaiss | The gut-bone axis: How bacterial metabolites bridge the distance[END_REF]. Current knowledge on this subject is summarized and discussed below.

Roles of Prebiotics

Prebiotics are non-digestible food ingredients that stimulate the growth and/or activity of bacteria colonizing the large bowel by acting as substrate for them [START_REF] Whisner | Bone and Mineral Metabolism[END_REF]. In most cases, they are products of the enzymatic conversion of sugars, such as galactooligosaccharides (GOSs), fructooligosaccharides (FOSs), inulin, digestion-resistant starch, xylooligosaccharides (XOSs), and lactulose.

GOSs are composed of a chain of galactose (typically 2-8 units) with a terminal glucose, which fosters the growth of Bifidobacteria and Lactobacillus [START_REF] Whisner | Bone and Mineral Metabolism[END_REF], which are known to increase the quantity and activity of osteoblasts [START_REF] Schepper | Probiotic Lactobacillus reuteri Prevents Postantibiotic Bone Loss by Reducing Intestinal Dysbiosis and Preventing Barrier Disruption[END_REF][START_REF] Zhang | Loss of Bone and Wnt10b Expression in Male Type 1 Diabetic Mice Is Blocked by the Probiotic Lactobacillus reuteri[END_REF]. Moreover, GOSs increase the intestinal absorption of calcium and magnesium, along with bone mineralization in animal models [START_REF] Weaver | Galactooligosaccharides improve mineral absorption and bone properties in growing rats through gut fermentation[END_REF]. In postmenopausal women, their consumption leads to increased calcium absorption and stable urinary calcium excretion [START_REF] Van Den Heuvel | Transgalactooligosaccharides stimulate calcium absorption in postmenopausal women[END_REF]. Lastly, in a double-blind trial, adolescent girls receiving 5 g of GOSs twice a day for 3 weeks increased both calcium absorption in the lower gut and fecal Bifidobacteria [START_REF] Whisner | Galacto-oligosaccharides increase calcium absorption and gut bifidobacteria in young girls: A double-blind cross-over trial[END_REF].

FOSs are composed of 3-10 fructose units, with the terminal fructose linked to a glucose residue. FOSs show similar effects to those of GOSs concerning the growth stimulation of Bifidobacteria [START_REF] Bornet | Nutritional aspects of short-chain fructooligosaccharides: Natural occurrence, chemistry, physiology and health implications[END_REF], associated with an increase in colon concentrations of butyrate-an SCFA that, along with propionate, regulates the gut microbiota, in particular with respect to its action on bone metabolism. FOS ingestion has been correlated with increased bone strength, mineralization, and reduced bone resorption in animal models [START_REF] Yan | A fructooligosaccharide from Achyranthes bidentata inhibits osteoporosis by stimulating bone formation[END_REF][START_REF] Mathey | Fructooligosaccharides maximize bone-sparing effects of soy isoflavone-enriched diet in the ovariectomized rat[END_REF][START_REF] Tanabe | Dietary Fructooligosaccharide and Glucomannan Alter Gut Microbiota and Improve Bone Metabolism in Senescence-Accelerated Mouse[END_REF]. In another, more recent animal study, FOSs increased the maximum amount of bone mass-the so-called "peak bone mass"-and rised the serum level of butyrate in ovariectomized rats affected by bone loss caused by estrogen deficiency [START_REF] Porwal | A prebiotic, short-chain fructo-oligosaccharides promotes peak bone mass and maintains bone mass in ovariectomized rats by an osteogenic mechanism[END_REF]. In humans, FOS supplementation (3.6 g/day for 12 months) coupled with calcium supplementation reduced the serum levels of bone turnover markers in postmenopausal women, with no effect on BMD [START_REF] Slevin | Supplementation with calcium and short-chain fructo-oligosaccharides affects markers of bone turnover but not bone mineral density in postmenopausal women[END_REF].

Resistant starches are a subgroup of dietary fibers that reach the colon undigested, as the small intestine is not able to alter their composition. These fibers are utilized as substrates by the microbiota in the colon, and their fermentation produces SCFAs [START_REF] Topping | Short-chain fatty acids and human colonic function: Roles of resistant starch and nonstarch polysaccharides[END_REF]; they also promote soya isoflavone production, and increase the ratio of Bifidobacteria, Lactobacillus species, and Bacteroides [START_REF] Tousen | Resistant starch promotes equol production and inhibits tibial bone loss in ovariectomized mice treated with daidzein[END_REF][START_REF] Tousen | The combined effects of soya isoflavones and resistant starch on equol production and trabecular bone loss in ovariectomised mice[END_REF]. These starches reduce bone loss in ovariectomized mice [START_REF] Tousen | The combined effects of soya isoflavones and resistant starch on equol production and trabecular bone loss in ovariectomised mice[END_REF], decrease inflammation, and interfere with the RANKL/OPG pathway [START_REF] Tousen | Resistant Starch Attenuates Bone Loss in Ovariectomised Mice by Regulating the Intestinal Microbiota and Bone-Marrow Inflammation[END_REF].

XOSs are prebiotics composed of 2-7 xylose molecules fostering Bifidobacteria over Lactobacillus [START_REF] Li | In vitro study of the prebiotic xylooligosaccharide (XOS) on the growth of Bifidobacterium spp and Lactobacillus spp[END_REF]; in mice, they increase calcium uptake by upregulating the expression of transient receptor potential vanillin receptor 6 (TRPV6) and Na + /Ca 2+ exchanger in the duodenum, and increase BMD [START_REF] Gao | Effect of Xylo-Oligosaccharides Supplementation by Drinking Water on the Bone Properties and Related Calcium Transporters in Growing Mice[END_REF]; they also reduce bone loss in high-fat-diet insulinresistant mice [START_REF] Eaimworawuthikul | Altered gut microbiota ameliorates bone pathology in the mandible of obese-insulin-resistant rats[END_REF], and decrease inflammation markers.

Finally, oral supplementation of lactulose (20 gr/kg per 6 weeks) inhibits osteoclastogenesis, reduces bone resorption, and prevents ovariectomy-induced bone loss in mice [START_REF] Chen | Lactulose Suppresses Osteoclastogenesis and Ameliorates Estrogen Deficiency-Induced Bone Loss in Mice[END_REF], while increasing calcium and magnesium absorption in both men and postmenopausal women [START_REF] Karakan | Low-Dose Lactulose as a Prebiotic for Improved Gut Health and Enhanced Mineral Absorption[END_REF][START_REF] Seki | Effect of lactulose on calcium and magnesium absorption: A study using stable isotopes in adult men[END_REF].

The prebiotics have in common the ability to be converted to SCFAs-including acetate, propionate, and butyrate-by the gut microbiota, increasing both their intestinal and serum levels, and lowering the intestinal pH. In an acidic environment, most minerals-such as magnesium and calcium-become more soluble; hence, their absorption is increased [START_REF] Scholz-Ahrens | Effects of prebiotics on mineral metabolism[END_REF][START_REF] Mccabe | Prebiotic and Probiotic Regulation of Bone Health: Role of the Intestine and its Microbiome[END_REF][START_REF] Wolever | Short chain fatty acid absorption from the human distal colon: Interactions between acetate, propionate and calcium[END_REF]. Butyrate also acts as a growth factor for enterocytes and colonocytes [START_REF] Den Besten | The role of short-chain fatty acids in the interplay between diet, gut microbiota, and host energy metabolism[END_REF].

Moreover, SCFAs generated from prebiotics regulate the number and function of regulatory T cells in the colon, thereby controlling inflammation [START_REF] Furusawa | Commensal microbe-derived butyrate induces the differentiation of colonic regulatory T cells[END_REF], and modulate the synthesis of IGF-1 involved in bone remodeling [START_REF] Nath | Biological Activities of Lactose-Based Prebiotics and Symbiosis with Probiotics on Controlling Osteoporosis, Blood-Lipid and Glucose Levels[END_REF]. Therefore, supplementation with saccharides might have beneficial effects on bone metabolism, although further studies are warranted to generalize these findings (see Table 1 for further details).

Xylooligosaccharides reduce bone resorption in systemic inflammation

Abbreviations-SCFA: short-chain fatty acid; CRP = C-reactive protein; IL-7R: interleukin-7 receptor; IL-10: interleukin-10; RANKL: receptor activator of nuclear factor kappa-B ligand; TRPV6: transient receptor potential vanilloid subfamily member 6; TNFα: tumor necrosis factor-alpha; IL-17: interleukin-17; ↓ = lower; ↑ = higher.

Proteins, Peptides, and Amino Acids

Intake of dietary proteins-in particular those from dairy products-has been linked with bone size, bone mass, and strength [START_REF] Liu | Bioactive peptide isolated from casein phosphopeptides promotes calcium uptake in vitro and in vivo[END_REF]. Some peptides contained in these products solubilize calcium and promote its absorption by the intestine [START_REF] Liu | Bioactive peptide isolated from casein phosphopeptides promotes calcium uptake in vitro and in vivo[END_REF]. Furthermore, the production of hepatic IGF-1 is stimulated by aromatic amino acids that are abundant in fermented dairy products, and this growth factor regulates bone growth [START_REF] Dawson-Hughes | Comparative effects of oral aromatic and branched-chain amino acids on urine calcium excretion in humans[END_REF]. Indeed, a recent study suggested that consumption of fermented dairy products might slow the age-related cortical bone loss in non-weight-bearing bone sites, independent of total energy, calcium, or protein intake, in a cohort of healthy postmenopausal women [START_REF] Biver | Fermented dairy products consumption is associated with attenuated cortical bone loss independently of total calcium, protein, and energy intakes in healthy postmenopausal women[END_REF]. Furthermore, ingestion of fermented dairy products was associated with increased bone mass and reduced risk of fracture in the elderly [START_REF] Laird | Greater yogurt consumption is associated with increased bone mineral density and physical function in older adults[END_REF][START_REF] Ong | Fermented Milk Products and Bone Health in Postmenopausal Women: A Systematic Review of Randomized Controlled Trials, Prospective Cohorts, and Case-Control Studies[END_REF], partly due to its content in lactulose-derived products which, together with probiotics, reduce the serum levels of parathyroid hormone (PTH), promoting bone resorption and, consequently, bone resorption markers [START_REF] Rizzoli | Dairy products and bone health[END_REF].

Among dairy products, kefir is a complex fermented product generated through the symbiotic fermentation of milk by lactic acid bacteria and yeasts encased in a protein-andpolysaccharide matrix. Kefir supplementation for 8 weeks in ovariectomized mice caused increased BMD, increased the number and thickness of bone trabeculae, increased bone volume, and promoted better mechanical properties and fracture toughness [START_REF] Chen | Kefir improves bone mass and microarchitecture in an ovariectomized rat model of postmenopausal osteoporosis[END_REF][START_REF] Tu | Kefir Peptides Prevent Estrogen Deficiency-Induced Bone Loss and Modulate the Structure of the Gut Microbiota in Ovariectomized Mice[END_REF]. In the same study, effects on the variability of gut microbiota were also shown, with a normalization of fecal levels of Alloprevotella, Anaerostipes, Parasutterella, Romboutsia, Ruminococcus, and Streptococcus species that were all more abundant compared to the sham group [START_REF] Tu | Kefir Peptides Prevent Estrogen Deficiency-Induced Bone Loss and Modulate the Structure of the Gut Microbiota in Ovariectomized Mice[END_REF]. Bacterial species might hydrolyze proteins in the intestine, although this process can be modified by diet [START_REF] Diether | Microbial Fermentation of Dietary Protein: An Important Factor in Diet-Microbe-Host Interaction[END_REF]. A higher gut bacterial production of tyrosine and tryptophan, resulting in elevated levels of these amino acids, might increase osteoporosis [START_REF] Guo | Succinate and its G-proteincoupled receptor stimulates osteoclastogenesis[END_REF], while reduced gut degradation of the branched-chain amino acids valine, leucine, and isoleucine-and, consequently, higher serum levels of themwas associated with the prevalence of specific gut microbiota species [START_REF] Wu | Bioregional Alterations in Gut Microbiome Contribute to the Plasma Metabolomic Changes in Pigs Fed with Inulin[END_REF], and was inversely correlated with the occurrence of osteoporosis [START_REF] Jennings | Amino Acid Intakes Are Associated with Bone Mineral Density and Prevalence of Low Bone Mass in Women: Evidence from Discordant Monozygotic Twins[END_REF][START_REF] Ling | The Association of Gut Microbiota with Osteoporosis Is Mediated by Amino Acid Metabolism: Multiomics in a Large Cohort[END_REF] (see Table 2 for further details).

Micronutrients

Several micronutrients other than calcium might play a role in bone mineralization and growth [START_REF] Karaaslan | Comparison of bone tissue trace-element concentrations and mineral density in osteoporotic femoral neck fractures and osteoarthritis[END_REF], and in some cases their levels are strictly linked to the gut microbiota composition [START_REF] Rayman | Selenium intake, status, and health: A complex relationship[END_REF].

Selenium deficiency can affect bone growth during development [START_REF] Beukhof | Selenium Status Is Positively Associated with Bone Mineral Density in Healthy Aging European Men[END_REF], and low serum values of this micronutrient have been correlated with low BMD in the elderly [START_REF] Wang | Association between dietary selenium intake and the prevalence of osteoporosis: A cross-sectional study[END_REF]. Supplementation with selenium modifies the gut microbiota composition, promoting a dose-dependent increase in bacterial variety [START_REF] Kasaikina | Dietary selenium affects host selenoproteome expression by influencing the gut microbiota[END_REF]. Moreover, zinc is a micronutrient that participates in bone metabolism, as it affects collagen synthesis and alkaline phosphatase activity [START_REF] Palacios | The role of nutrients in bone health, from A to Z[END_REF]; its deficiency is correlated with lower BMD in postmenopausal women [START_REF] Mutlu | zinc and copper status in osteoporotic, osteopenic and normal post-menopausal women[END_REF]. In animal models, zinc supplementation increased the number of Lactobacillus, while its deficiency caused a reduced ratio of Firmicutes, with lower SCFA production [START_REF] Starke | The impact of high dietary zinc oxide on the development of the intestinal microbiota in weaned piglets[END_REF][START_REF] Reed | Chronic Zinc Deficiency Alters Chick Gut Microbiota Composition and Function[END_REF]. Lastly, both iron deficit and excess were linked to osteoporosis [START_REF] Balogh | Influence of Iron on Bone Homeostasis[END_REF]. Low iron levels increased the expression of the fibroblast growth factor 23 (FGF23) gene [START_REF] Imel | Serum fibroblast growth factor 23, serum iron and bone mineral density in premenopausal women[END_REF]. Transgenic mice overexpressing FGF23 in osteoblasts presented a rachitic bone phenotype with a bone mineralization deficit, and showed increased levels of osteoblast markers, but the numbers of osteoclasts were slightly reduced or unchanged [START_REF] Hu | Adamts1 is highly induced in rachitic bones of FGF23 transgenic mice and participates in degradation of non-mineralized bone matrix collagen[END_REF]. Adamts1 was highly induced in rachitic bones of FGF23 transgenic mice, and participated in the degradation of non-mineralized bone matrix collagen [START_REF] Hu | Adamts1 is highly induced in rachitic bones of FGF23 transgenic mice and participates in degradation of non-mineralized bone matrix collagen[END_REF]. Conversely, a blockade of FGF23 signaling improved the rachitic bone phenotypes [100]. Unsurprisingly, therefore, serum iron levels are linked to BMD, and an adequate intake of this micronutrient is recommended [101]. Iron intake modifies the gut microbiota, increasing Bacteroidetes and Bifidobacterium [102,103], when introduced with the diet. When supplemented at high doses, iron decreases Bifidobacterium, resulting in increased bacterial pathogenicity [104,105]. Therefore, among the several micronutrients with beneficial bone effects, selenium, zinc, and iron can impact the gut microbiota, strengthening the interaction between bone and gut microbial populations (see Table 3 for further details).

Impact of Probiotics on the Gut Microbiota

Probiotics are defined as "live micro-organisms which, when given in adequate amounts, confer a health benefit to the host" [106], and are made available via yogurts, milkbased foods, powders, capsules, or oral solutions. Their main characteristics include acid and bile tolerance that confers survival in the gastrointestinal tract (GIT), phenotypic and genotypic stability, adhesion to the mucosal surface, antibiotic resistance, and production of antimicrobial factors able to inhibit known pathogens [107]. The more frequently administered probiotics include Lactobacillus, Bifidobacterium, Escherichia, Enterococcus, and Bacillus subtilis-but also yeasts such as Saccharomyces-at a concentration of approximately of 10 7 to 10 8 cells per gram of probiotic supplement, with a serving size of 100 to 200 mg [108]. Given the mechanism of action of probiotics, most studies have failed to report any impact on gut microbiota composition. In fact, the effects of probiotics do not lie in their ability to directly restore an altered gut microbiota composition, but rather in contributing genes and metabolites that directly influence epithelial and immune cells, promoting gut health through regulation of the intestinal pH and permeability, along with resistance to colonization by unwanted bacteria [109][110][111][112].

The beneficial effects of the administration of probiotics on bone tissue are widely recognized, such that they have been named by some authors "the new calcium and vitamin D" for bone health [108]. Several studies have demonstrated that probiotics (i.e., L. reuteri, L. paracasei, and L. helveticus) prevent bone loss in the ovariectomy-induced postmenopausal mouse model [113][114][115][116]. Lactobacillus reuteri administration prevents type-1diabetes-induced osteoporosis by inhibiting TNFα-mediated suppression of Wnt10b; it also enhances bone density via increased osteoblast activity and decreased bone marrow adiposity [117]. Similarly, it was documented that Lactobacillus reuteri administration ameliorates trabecular bone loss, restoring Wnt10b suppression in the bones of mice with glucocorticoidinduced osteoporosis [118]. Wnt10b is thought to be an endogenous regulator of bone formation; one way whereby it increases osteoblastogenesis is by downregulating the expression of PPAR-γ [119].

A randomized double-blind controlled trial showed that the administration of a multispecies probiotic supplement for six months significantly decreased bone-specific alkaline phosphatase and collagen type 1 crosslinked C-telopeptide levels in postmenopausal women with osteopenia, slowing down the rate of bone turnover [120]. Similarly, another study reported beneficial effects of a novel red clover extract rich in isoflavone aglycones, and probiotics against postmenopausal osteopenia, characterized by an attenuation of estrogen-deficiency-related BMD loss and an improvement of bone turnover in women [121]. Furthermore, in a double-blind placebo-controlled study, 12-month daily administration of Lactobacillus reuteri decreased the loss of tibia total volumetric BMD in elderly women [122].

Therefore, probiotics might play a key role in bone tissue, through modulation of the gut microbiota, providing a health benefit in frail subjects, although their role still needs to be characterized further (see Table 4 for currently known roles).

Multispecies probiotic reduces bone turnover

Abbreviations-RANKL: receptor activator of nuclear factor kappa-B ligand; TNFα: tumor necrosis factor-alpha; Trap5: serum band 5 tartrate-resistant acid phosphatase; CD4 + T -: cluster differentiation 4 positive cT helper cells; IL-1β: interleukin-1 beta; OPG: osteoprotegerin; Wnt10b: Wnt family member 10B; PTH: parathyroid hormone; ↓ = lower; ↑ = higher.

Interventional Microbiota Modulation to Reduce Bone Loss

GIT and bone tissue interact with one another through a complex network modulated by the gut microbiota, in which osteoblasts, osteoclasts, and receptor activator of nuclear factor kappa-B ligand (RANKL) participate [123]. Recently, several studies have focused on establishing the role of the gut microbiota in the onset of osteoporosis, considering its modulation as a potential therapeutic strategy to reduce bone loss [124][125][126]. For instance, restoring gut microbiota eubiosis has positive effects on the treatment of dysbiosisrelated extraintestinal disorders-for instance, in bones and muscles-such as osteoporosis, osteoarthritis, and sarcopenia [123,124,127-132]. In particular, gut microbiota modulation can be achieved primarily through diet, but also with probiotics, prebiotics, synbiotics (mixtures of probiotics and prebiotics that beneficially affect the host), postbiotics (bioactive compounds produced by food-grade microorganisms during fermentation), and obviously also by antibiotics and fecal microbiota transplantation (FMT) [132][133][134][135].

Diet represents the main environmental factor contributing to the interindividual differences in gut microbiota composition in healthy people [134][135][136][137]. A balanced diet is fundamental for human health, to which a well-balanced complex microorganism ecosystem composed of more than 10 14 bacteria-as well as fungi, viruses, phages, parasites, and archaea, which colonize the GIT-is a major contributor [START_REF] Gentile | The gut microbiota at the intersection of diet and human health[END_REF]. On the other hand, high-fat diets lead to a decrease in the numbers of "good bacteria", such as Bacteroidetes, and an overgrowth of opportunistic pathogens, able to promote a disruption of the intestinal epithelial tight junctions. This leads to an increase in gut permeability, allowing the translocation of bacterial lipopolysaccharides into the systemic blood circulation (so called "leaky gut syndrome"), promoting systemic inflammation and insulin resistance [138,139]. In contrast, high-fiber diets and fructooligosaccharides are associated with an increased number of Bifidobacteria species producing SCFAs, such as acetate, propionate, and butyrate, which enhance glucose uptake and insulin sensitivity [140]. SCFAs also play a role in regulating bone homeostasis by downregulating both osteoclast activity that causes bone resorption and inflammation sustained by T-reg lymphocytes [141]. In this context, the adherence to a Mediterranean diet-rich in fibers, fermented dairy products, and polyphenols-is associated with lower hip fracture risk [142]. In association with diet, physical exercise has an impact on several cortical bone parameters and bone mechanical properties-possibly by modulating the intestinal microbial balance toward a decrease in the Firmicutes/Bacteroidetes ratio and an increase in the number of Actinobacteria phylum members, such as Bifidobacteriaceae. Modulating the gut microbiota with probiotics can promote the overgrowth of not only Lactobacillus and Bifidobacterium species, but also Faecalibacterium, Akkermansia, Ruminococcus, and Roseburia species, leading to a stimulation of SCFA production [143][144][145][146]. Furthermore, prebiotics exert a direct antiadhesive effect against pathogens by interacting with bacterial receptors, thereby inhibiting pathogen colonization [147,148]. Several experimental studies [149-152], both in animals and in humans, support the hypothesis that prebiotics play a key role in the intestinal absorption of calcium and magnesium by causing a decrease in the cecal pH, which results in an increase in BMD.

In the fast moving "-biotics" field, postbiotics have been defined recently as "bioactive compounds produced by food-grade micro-organisms during a fermentation process" [153]. Postbiotics is an umbrella term that includes metabolites, SCFAs, microbial cell fractions, functional proteins, extracellular polymeric substances (EPSs), cell lysates, teichoic acid, peptidoglycan-derived muropeptides, and pili-type structures, which may be used to promote health by modulating the gut microbiota [153]. In this context, only a few very recent studies [154][155][156] have shown how the administration of postbiotics ameliorates osteopenia and/or osteoporosis. A study of the effects of cell lysates and supernatants of five native probiotic strains (L. acidophilus, L. reuteri, L. casei, B. longum, and B. coagulans) in ovariectomized rats showed that the Bacillus-coagulans-derived-postbiotics had the best impact on bone homeostasis, ameliorating several parameters, such as reduced bone area, bone mineral content, and BMD caused by ovariectomy [153]. Furthermore, a corroborating study [155] using a murine model of postmenopausal-related osteoporosis reported an inhibitory effect of the postbiotic Lactobacillus curvatus 38-CS on RANKL-induced osteoclast differentiation and bone loss, by downregulating the TRAF6/NF-κB/MAPKs axis (see Figure 2 for further details).

At this point, it is of interest to discuss the effects and consequences of antibiotics on the gut microbiota. Antibiotics are considered to be deep modulators of the gut microbiota, and the characteristics that can influence their negative or positive impact relate to antibiotic class, dosage, duration, and administration route [156]. On the one hand, their abuse causes several intestinal and extraintestinal disorders associated with gut microbiota impairmentespecially Clostridium difficile infection. On the other hand, antibiotic therapy can provide a so-called "eubiotic" effect, increasing the abundance of beneficial bacteria, and therefore appears to be a reliable therapeutic strategy for several non-communicable disorders, such as hepatic encephalopathy or irritable bowel syndrome [156]. Antibiotic treatment, by modulating the gut microbiota, can influence bone mass. Animal studies have shown that short-term administration of low doses of antibiotics at weaning results in increased BMD and adiposity, with substantial taxonomic changes in terms of the gut microbiota [157]. Subsequent supporting results showed that a low dose of penicillin delivered from birth to weaning in mice was sufficient to transiently perturb the gut microbiota, inducing sustained effects on body composition with an increase in BMD [158]. In addition, treatment with tetracyclines was found to prevent ovariectomy-induced bone loss, reducing bone resorption modestly and stimulating bone formation substantially [159]. A more recent therapeutic strategy for the modulation of the gut microbiota is the transfer of donor stools to a recipient's GIT-the so-called fecal microbiota transplants (FMTs), which have gained much recognition recently [160]. Considered to be an established treatment for recurrent Clostridium difficile infection, FMT is a potential therapy for inducing remission of mild-moderate ulcerative colitis [161]. Furthermore, its use has been debated in the literature for improving several extraintestinal conditions, including metabolic syndrome, modulation of responses to chemotherapy, eradication of multidrug-resistant organisms, and osteomuscular disorders [161]. In the context of bone disorders, FMT from young to aged rats improved senile osteoporosis in terms of bone volume, trabecular bone volume fraction, trabecular number, and trabecular thickness by restoring gut dysbiosis at the phylum and family levels [162]. Furthermore, FMT improved leaky gut syndrome in aged rats, upregulating the expression of the tight-junction proteins occludin, claudin, and ZO-1 [162]. Interestingly, transplantation of fecal material from mice treated with the corticosteroid prednisolone into naïve, untreated recipients caused bone loss, supporting a role of the gut microbiota in the development of glucocorticoid-induced osteoporosis [118]. However, further studies are required in order to clarify the emerging role of gut microbiota modulation in reducing bone loss to prevent osteoporosis.

Future Research Avenues and Perspectives for Osteoporosis Treatment

The current available osteoporosis treatments consist of antiresorptive agents that inhibit osteoclast activity, bone-forming compounds that stimulate osteoblasts, and dual-acting drugs that simultaneously inhibit bone resorption and enhance bone accrual [163][164][165]. In bone homeostasis, the Wnt/β-catenin and OPG/RANKL/RANK signaling pathways play crucial roles in bone anabolism and catabolism, respectively [166]. Antiresorptive treatments comprise bisphosphonates and denosumab-an antibody against RANKL [167,168].

Animal studies recently aimed to develop molecules capable of interfering even with the osteoclast-bone matrix proteins (αvβ3 integrin antagonists), acting as bone resorption inhibitors [169,170]. Furthermore, the anabolic bone accrual process can be stimulated by parathyroid hormone analogues (amino acids [171]. Moreover, romosozumab is a novel antibody that inhibits sclerostin and, hence, interferes with the sclerostin/Wnt signaling pathway, and might have both anabolic and antiresorptive effects on bone [172]. In the same pathway, frizzled-related proteins (FRPs) are considered to be novel therapeutic targets, and miRNA-based therapies targeting FRPs are under development to prevent and treat osteoporosis [173].

The gut microbiota interferes to some extent with the Wnt signaling pathway, in particular promoting the renewal of hematopoietic stem cells [174]; moreover, it also participates in the OPG pathway and inhibits RANKL [175], possibly similarly to denosumab, but its mechanism is not fully understood. As discussed above, the gut microbiota enhances the intestinal absorption of calcium; interestingly, it might regulate bone metabolism through a calcium-dependent interaction with the Wnt/β-catenin signaling pathway, as demonstrated in animal models [176].

To date, knowledge on gut microbial interaction with the host-and even more so its potential regulatory role in bone health-remains partial [177]. Furthermore, we are aware that the interpretation of animal-model-based studies requires caution with respect to the extrapolation of results to humans, especially considering the clinical management of osteoporosis.

However, several dietary supplements and probiotics highlighted in this review might modulate bone health by acting on the same biological pathways used by some currently utilized drugs. Hence, it might be suggested that the exploration of combined supplementdrug approaches might help to maximize the anti-osteoporotic effects of these interventions.

Finally, it is worth noting that systemic inflammation and immune system effects are often neglected as therapeutic targets in osteoporosis, even though many natural and synthetic substances exert bone effects by acting on T cells or by decreasing the levels of proinflammatory cytokines.

In this review, we highlighted how gut microbiota modulation is appealing as a therapeutic intervention to counteract bone loss. In this context, dietary supplements, prebiotics, probiotics, postbiotics, and synbiotics should be explored further as useful adjuvant treatments for osteoporosis, especially with regards to targeting the most relevant biochemical and signaling pathways. Future human studies are warranted to confirm the hypotheses discussed, in order to build and provide the right support for the clinical management of the gut-bone axis and sustain bone health.

Figure 1 .

 1 Figure1. The gut-bone axis: pathways and factors starting from gut dysbiosis that determine bone metabolism alterations favoring osteoclasts. Abbreviations-SCFAs: short-chain fatty acids; PAMPs: pathogen-associated molecular patterns; D Cell: somatostatin-producing cells; T Cell: type of leukocyte that is an essential part of the immune system; OB: osteoblast; OC: osteoclast; RNK-L: receptor activator of nuclear factor-κB ligand; OPG: osteoprotegerin; VDR: vitamin D receptor; IGF-1: insulin-like growth factor one.

Figure 2 .

 2 Figure 2. Lifestyle intervention and supplementation that might reduce bone loss; these modulators might exert their therapeutic effects through different known biological pathways, such as OPG, RANK, and TRAF. Abbreviations-OPG: osteoprotegerin; RANK-L: receptor activator of nuclear factor-κB ligand; RANK: receptor activator of nuclear factor κB; TRAF: tumor necrosis factor receptorassociated factor.

Table 1 .

 1 Characteristics of studies investigating bone effects and gut microbiota modulation of supplementation with prebiotics.

	Authors	Journal	Year	Dietary Supplement	Subjects	Pathways Investigated	Gut Microbiota Modulation	Main Findings
								Fructooligosaccharides
	Porwal et al. [55]	Biomedicine & Pharmacotherapy	2020	Fructooligosaccharides	Animals	SCFA production	Not investigated	prevent ovariectomy-induced
								bone loss
	Yan et al. [52]	Carbohydrate Polymers	2019	Fructooligosaccharides	Animals	Osteoblast stimulation	Not investigated	Fructooligosaccharides increase bone mineral density and strength
	Tanabe et al. [54]	Journal of Agriculture and Food Chemistry	2019	Fructooligosaccharides	Animals	↓ Serum CRP; SCFA production	↑ Lactobacillus ↑ Bacteroides ↓ Clostridium	Fructooligosaccharides reduce bone resorption and systemic inflammation
	Slevin et al. [56]	The Journal of Nutrition	2014	Fructooligosaccharides	Humans	Reduction in the number of bone turnover markers	Not investigated	Fructooligosaccharides reduce postmenopausal bone loss
	Mathey et al. [53]	Calcified Tissue International	2004	Fructooligosaccharides	Animals	SCFA production	Not investigated	Fructooligosaccharides increase bone mineral density and strength
	Whisner et al. [50]	British Journal of Nutrition	2013	Galactooligosaccharides	Humans	SCFA production	↑ Bifidobacteria	Galactooligosaccharides increase calcium absorption
	Weaver et al. [48]	Journal of Agricultural and Food Chemistry	2011	Galactooligosaccharides	Animals	SCFA production	↑ Bifidobacteria	Galactooligosaccharides increase calcium absorption
	Van den Heuvel et al. [49]	The Journal of Nutrition	2000	Galactooligosaccharides	Humans	SCFA production	Not investigated	Galactooligosaccharides increase calcium absorption
	Karakan et al. [65]	Frontiers in Nutrition	2021	Lactulose	Humans	SCFA production	↑ Bifidobacteria ↑ Lactobacillus ↓ Clostridium	Lactulose increases calcium absorption
	Chen et al. [64]	Aging and Disease	2020	Lactulose	Animals	↓ TNFα ↓ RANKL ↓ IL-17 ↑ IL-10	↓ Firmicutes ↑ Bacteroides	Lactulose inhibits osteoclastogenesis and bone resorption

Table 1 .

 1 Cont.

	Authors	Journal	Year	Dietary Supplement	Subjects	Pathways Investigated	Gut Microbiota Modulation	Main Findings
	Seki et al. [66]	Journal of Nutritional Science and Vitaminology	2007	Lactulose	Humans	Not investigated	Not investigated	Lactulose increases calcium and magnesium absorption
	Tousen et al. [60]	Nutrients	2019	Resistant starch	Animals	↓ IL-7R mRNA↑ IL-10 mRNA↓ RANKL	↑ Bifidobacteria	Resistant starch prevents post-ovariectomy bone loss
	Tousen et al. [59]	British Journal of Nutrition	2016	Resistant starch	Animals	↓ IL-7R mRNA	↑ Bifidobacteria	Resistant starch prevents post ovariectomy bone loss
						Increased isoflavone		Resistant starch prevents
	Tousen et al. [58]	Metabolism	2011	Resistant starch	Animals	availability and	↑ Bifidobacteria	post-ovariectomy bone
						estrogenic activities		loss
						↑ Na + /Ca 2+		Xylooligosaccharides
	Gao et al. [62]	Nutrients	2020	Xylooligosaccharides	Animals	exchanger 1	Not investigated	increase calcium
						↑ TRPV6		absorption
	Eaimworawuthikul et al. [63]	European Journal of Nutrition	2020	Xylooligosaccharides	Animals	Osteoclast inhibition	Not investigated

Table 2 .

 2 Characteristics of studies investigating bone effects and gut microbiota modulation of supplementation with proteins, peptides, and amino acids.

	Authors	Journal	Year	Dietary Supplement	Subjects	Pathways Investigated	Gut Microbiota Modulation	Main Findings
	Ling et al. [86]	The Journal of Clinical Endocrinology and Metabolism	2021	Amino acids	Humans	Not determined	↓ Actinobacillus, ↓ Blautia, ↓ Oscillospira	Valine, leucine, and isoleucine serum levels are inversely related to the occurrence of osteoporosis
		Journal of Bone and						Alanine, arginine, glutamic acid,
	Jennings et al. [85]	Mineral	2016	Amino acids	Humans	↑ IGF1	Not investigated	leucine, lysine, and proline
		Research						increase BMD
	Dawson-Hugler et al. [75]	Osteoporosis International	2007	Amino acids	Humans	↑ IGF1	Not investigated	Phenylalanine and histidine increase calcium absorption
								Casein phosphopeptides
	Liu et al. [74]	Food & Function	2018	Casein Phosphopeptides	Animals	↑ TRPV6	Not investigated	increase calcium absorption and prevent bone
								resorption
	Ong et al. [78]	Advances in Nutrition	2020	Fermented dairy products	Humans	↓ TNF-α ↓ IL-6 ↓ RANKL	Not investigated	Fermented dairy products might reduce hip fracture risk
	Biver et al. [76]	Osteoporosis International	2018	Fermented dairyproducts	Humans	Action on calcium balance and decrease in secondary hyper parathyroidism.	Not investigated	Fermented dairy products attenuate postmenopausal bone loss
	Laird et al. [77]	Osteoporosis International	2017	Fermented dairy products	Humans	Modulation of osteoclast numbers and activity	Not investigated	Fermented dairy products increase bone mineral density
							↑ Alloprevotella,
							↑ Anaerostipes,
	Tu et al. [81]	Nutrients	2020	Kefir peptides	Animals	↓ TNF-α ↓ RANKL	↑ Parasutterella, ↑ Romboutsia,	Kefir peptides prevent ovariectomy-induced bone loss
							↑ Ruminococcus,
							↑ Streptococcus
	Chen et al. [80]	Osteoporosis International	2014	Kefir peptides	Animals	↑TRPV6	Not investigated	Kefir peptides prevent ovariectomy-induced bone loss

Abbreviations-BMD: bone mineral density; RANKL: receptor activator of nuclear factor kappa-B ligand; TRPV6: transient receptor potential vanilloid subfamily member 6; TNFα: tumor necrosis factor-alpha; IL-6: interleukin-6; IGF1: insulin growth factor-1; ↓ = lower; ↑ = higher.

Table 3 .

 3 Characteristics of studies investigating bone effects and gut microbiota modulation of supplementation with micronutrients.

	Authors	Journal	Year	Dietary Supplement	Subjects	Pathways Investigated	Gut Microbiota Modulation	Main Findings
								Favorable effects of iron
	Qasem et al. [100]	BMC Pediatrics	2017	Iron	Humans	Lowering inflammation	↑ Bifidobacteria ↑ Bacteroides	on bone might be mediated by the gut
								microbiome
								Selenium deficiency is
	Wang et al. [89]	BMC Musculoskeletal Disorders	2019	Selenium	Humans	Not determined	↓ Parabacteroides ↓ Firmicutes	correlated with a higher prevalence of
								osteoporosis
						SCFA		Favorable effects of zinc
	Reed et al. [94]	Nutrients	2015	Zinc	Animals	production Lowering of	↑ Lactobacillus	on bone might be mediated by
						inflammation		the gut microbiome

Abbreviations-SCFA: short chain fatty acid; ↓ = lower; ↑ = higher.

Table 4 .

 4 Characteristics of the studies investigating bone effects and gut microbiota modulation of supplementation with probiotics.

	Authors	Journal	Year	Dietary Supplement	Subjects	Pathways Investigated	Gut Modulation Microbiota	Main Findings
								Lactobacillus
	Narva et al. [113]	Annals of Nutrition and Metabolism	2007	Lactobacillus helveticus	Animals	Increasing bone formation	Not investigated	helveticus prevents ovariectomy-induced bone
								loss

Table 4 .

 4 Cont.

	Authors	Journal	Year	Dietary Supplement	Subjects	Pathways Investigated	Gut Modulation Microbiota	Main Findings
								Lactobacillus
	Ohlsson et al. [114]	PLoS ONE	2014	Lactobacillus paracasei	Animals	↓ TNF-α ↓ IL-1β ↑ OPG	Not investigated	paracasei prevents ovariectomy-induced bone
								loss
	Schepper et al. [46]	The Journal of Bone and Mineral Research	2020	Lactobacillus reuteri	Animals	↑ Wnt10b	↓ Clostridium	Lactobacillus reuteri prevents glucocorticoid-induced bone loss
	Nilsson et al. [120]	Journal of Internal Medicine	2018	Lactobacillus reuteri	Humans	Not determined	Not investigated	Lactobacillus reuteri prevents bone loss
								Lactobacillus reuteri
	Zhang et al. [115]	Endocrinology	2015	Lactobacillus reuteri	Animals	↓ TNF-α	Not investigated	prevents type-1-diabetes-induced
								bone loss
	Britton et al. [111]	Journal of Cellular Physiology	2014	Lactobacillus reuteri	Animals	↓ Trap5 ↓ RANKL ↑ CD4 + T-lymphocytes	Promoting gut microbiota diversity	Lactobacillus reuteri prevents Ovariectomy-induced bone loss
		Journal of the
	Jafarnejad et al. [118]	American Nutrition	2017	Multispecies probiotic	Humans	↓ PTH ↓ TNF-α	Not investigated
		Association

Funding: This research received no external funding.

Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.

Data Availability Statement: Not applicable.

Conflicts of Interest:

The authors declare no conflict of interest.


Ahmed Kaci Boukellal

Morgane Rouby

Jean-Marc Debierre

Tip dynamics for equiaxed Al-Cu dendrites in thin samples : phase-field study of thermodynamic effects

published or not. The documents may come

Tip dynamics for equiaxed Al-Cu dendrites in thin samples : phase-field study of thermodynamic effects

Introduction

In order to confer isotropic physical properties to industrial metallic parts like cast engine blocks, equiaxed solidification is preferred in practice. Many materials that are used in such applications belong to the large family of aluminum alloys [1,[START_REF] Kaufman | Aluminum Alloy Castings: Properties, Processes and Applications[END_REF]. We focus here on Al-Cu diluted alloys that have been studied quite intensively over the last decades [START_REF] Mondolfo | Aluminum Alloys: Structure and Properties[END_REF]. One key point concerning the dynamics of growing equiaxed grains is to predict and control the interactions between two grains that grow toward each-other. Scaling laws that govern the grain growth and the grain-grain interactions in thin samples were recently proposed in reference [START_REF] Boukellal | [END_REF] (hereafter denoted as BEA). Some preliminary results in [START_REF] Boukellal | Isothermal solidification of Si and Al-Cu : 3D phasefield simulations[END_REF] have shown that a sample of Al-1wt.%Cu, 1000µm in width and length, presents confinement effects for a thickness below 500µm. This thickness is approximately five times the solutal diffusion length l s = D/V m where D is the solute diffusion coefficient in the liquid phase and V m is the maximum tip growth velocity. The samples considered in the present study are 100µm in thickness, thus definitely confined in this direction. The scaling laws are of great relevance because they can be extrapolated to predict the dynamics of equiaxed dendrites in materials containing higher copper concentrations that are not easy to simulate directly. In the present study, phase-field simulations in three dimensions are performed to explore the implications of experimentally relevant thermodynamic effects that were not included originally in BEA, specifically how the growth and interaction scaling laws are affected.

Most of the phase-field simulations of Al-Cu solidification reported in the literature were performed for directional solidification [START_REF] Badillo | [END_REF][7][8] or in the case of a constant applied undercooling [9]. Conversely, simulations of solidification in samples cooled homogeneously at a constant rate, as we consider here, are less common [START_REF] Boukellal | [END_REF]10]. In section 2, we introduce the phase-field model used to simulate isothermal solidification at a constant cooling rate. The constitutive physical equations are discussed together with the corresponding phase-field equations, and their implementation in the numerical code is described. The phase diagram of a diluted alloy is very often assumed linear in numerical studies, while it is usually curved, even for rather low concentrations [START_REF] Okamoto | Phase Diagrams for Binary Alloys[END_REF]. It is the first task of the present study to consider the curved Al-Cu phase diagram provided by thermodynamic calculations. The corresponding results are presented and analyzed in section 3. During dendritic growth, the development of secondary branches may have a more or less important influence on the final polycrystalline structure of the material [START_REF] Bőttger | [END_REF]13], thus on the final properties of the material [START_REF] Kaufman | Aluminum Alloy Castings: Properties, Processes and Applications[END_REF]. In the phase-field simulations, it is necessary to introduce thermodynamic temperature fluctuations by means of a controlled spatio-temporal noise in order to produce realistic dendritic branching. Section 4 is devoted to this second task. The central question of the present study is to quantify the influence of the two thermodynamic effects that are introduced. To do so, the growth and interaction scaling growth laws obtained with and without these two effects are compared in section 5. Finally, section 6 summarizes our main conclusions.

Phase-field model

Physical equations

We consider aluminum alloys that contain a low copper concentration c 0 . For these alloys the partition coefficient k < 1, the liquidus slope m < 0, so m(k -1) > 0.

At a given temperature T 0 , the equilibrium concentration at the planar interface taken on the liquid side is given by

c 0 l = (T 0 -T M )/m, (1)
where T M is the Al melting temperature. In the reference frame where the material is at rest, the copper diffusion equation reads

∂c ∂t = D∇ 2 c, (2)
where D is the solute diffusion coefficient that will be assumed zero in the solid phase (one-sided model). In addition, two local equations must be verified at any interface point. The first one expresses solute conservation and it imposes the interface normal velocity,

V n = -D ∂c ∂r , (3)
the derivative being taken on the liquid ( ) side, in the direction perpendicular to the solid-liquid interface. The second one is the Gibbs-Thomson equation that expresses local thermodynamic equilibrium and that reads in terms of temperatures,

T i = T M + mc i - Γ ρ - V n µ . (4)
In this equation, ρ is the average radius of curvature, µ the linear kinetic coefficient, and the Gibbs-Thomson constant Γ is related to the chemical capillary length d 0 through

d 0 = Γ m(k -1)c 0 l . (5)
In the present case, solidification results from a homogeneous cooling at a constant rate R = 0.5 K/min. Then the interface temperature T i = T 0 -Rt, where T 0 is set to be the liquidus temperature T L . Thus, c 0 l = c 0 and the Gibbs-Thomson equation in terms of concentrations reads

U * i = c i -c 0 (1 -k)c 0 = - d 0 ρ -β kin V n + Rt m(k -1)c 0 , (6)
where

β kin = 1/[m(k -1)c 0 µ] is the corresponding kinetic coefficient.
In the following, we will use the notation

Θ(t) = - Rt m(k -1)c 0 (7)
for the negative undercooling that decreases in time as a result of the sample cooling.

Phase-field equations

We adopt the quantitative phase-field model introduced by Karma et al to simulate the solidification of diluted alloys [14,15]. Instead of the usual phase-field -1 ≤ ϕ ≤ 1, we use the preconditioned phase-field

ψ = √ 2 tanh -1 (ϕ) (8)
that was shown to increase numerical precision for a given mesh size [16]. A number of recent phase-field studies confirmed that quantitative results are obtained when using the preconditioned phase-field [START_REF] Boukellal | [END_REF][17][18][19][20].

We follow the usual convention where lengths are scaled by the interface thickness W 0 and times by the relaxation time τ 0 . The resulting nondimensional evolution equation for ψ is

[1 -(1 -k)Θ]a 2 s ∂ψ ∂t = a 2 s ∇ 2 ψ - √ 2ϕ( ∇ψ) 2 + √ 2 ϕ -λ(1 -ϕ 2 )(U + Θ) + 2a s ∇a s ∇ψ + √ 2 (1 -ϕ 2 ) ∇ A. (9)
As in the case of directional solidification, the imposed undercooling Θ(t) is also introduced on the left hand side of this equation to prevent the kinetic coefficient β kin from deviating in time from its imposed value (zero here) [15]. The nondimensional concentration field U is related to the physical concentration field c through

U = f U (ϕ)c -c 0 (1 -k)c 0 , (10)
with f U (ϕ) = 2 (1 + k) -(1 -k)ϕ . ( 11
)
This is the diffuse interface version of the physical definition given in [START_REF] Badillo | [END_REF]: in the liquid phase, ϕ = -1.0, so f U (ϕ) = 1.0. Al-Cu is a material with an atomically rough solid-liquid interface. Thus, the kinetic coefficient β kin should be set to zero for the low solidification velocities considered here. In practice, this is ensured by imposing λ = (75/47)Dτ 0 /W 2 0 for the constant that couples the non-dimensional concentration field U to the phase-field [21]. Since the interface width W 0 = ξd 0 is adjusted by varying the numerical parameter ξ, this fixes the relaxation time to τ 0 = (47 √ 2/120)(d 2 0 /D)ξ 3 . In the present work, as in [START_REF] Boukellal | [END_REF]19,22,23], the equation for the time evolution of the phase-field is written in terms of the anisotropy vector A. This vector appears when, in the original phase-field equation for solidification of a pure melt [21], one rewrites the following sum of three terms as a vector divergence, η=x,y,z

∂ η | ∇ϕ| 2 a s ∂a s ∂ϕ η = ∇. A. (12)
It is worth mentioning that this formulation can be easily implemented for most anisotropy functions of the surface energy, a s (see Appendix A). For instance, the case of a surface energy anisotropy that leads to facetted growth shapes was recently considered by using this anisotropy vector formulation [23].

Crystalline anisotropy governs the expression of the anisotropy function a s . Here, we choose to align the Al-Cu cubic crystal axes [1 0 0], [0 1 0] and [0 0 1] with the x, y, and z axes of the numerical mesh. The corresponding anisotropy function reads then [24]

a s = (1 -3 4 ) + 4 4 (n 4 x + n 4 y + n 4 z ), (13)
where n x , n y , n z are the components of the unit vector n along the normal to the solid-liquid interface and 4 is the anisotropy strength of the interface energy. As shown in Appendix A, the three components of the anisotropy vector A are then

A x,y,z = 16 4 (1 -ϕ 2 ) √ 2 | ∇ψ|a s (n 4 x + n 4 y + n 4 z ) -n 2 x,y,z n x,y,z . (14)
As we neglect solute diffusion in the solid (one-sided model), we must introduce a corrective solute current

j at = W 0 2 √ 2 c 0 (1 -k)G n, (15)
where

G = 1 -ϕ 2 √ 2 [1 + (1 -k)U ] ∂ψ ∂t , (16)
in order to counterbalance the excess of solute trapping and other spurious corrections due to the finite interface thickness W 0 [14,15]. The evolution equation for the nondimensional concentration field U is then [14,15] (

1 + k) -(1 -k)ϕ ∂U ∂t = (1 -ϕ)D * ∇ 2 U - (1 -ϕ 2 ) √ 2 D * ∇ψ ∇U - 1 √ 2 n ∇G + G 1 - ∇ n √ 2 , (17)
where

D * = Dτ 0 W 2 0 . (18)

Implementation

We very closely follow the implementation described in BEA. The main physical parameters of the Al-Cu alloys are gathered in Table 1. It was shown in BEA that the choice W 0 = 24.0d 0 for the interface width gives well converged results, so we also use this value here.

The first step of the work consists in bringing an initial germ to equilibrium for the imposed parameters. To do so, we start from a sphere of radius R 0 . We set the phase-field ψ( r, t = 0) = -| r -R 0 |/W 0 , the supersaturation

U ( r, t = 0) = -d 0 /R 0 and Θ(t = 0) = d 0 /R 0 .
Then ψ and U are calculated while varying up and down Θ until the V x along the x axis cancels out. The final state of this step is used as the initial condition in what follows. Mirror (no-flux) conditions are constantly imposed at all the domain boundaries. Thus, it is sufficient to simulate a quarter of the whole sample plane xz, let us say x ≥ 0 and z ≥ 0. For the same reason, only one half of the sample thickness is necessary, let us say y > 0. Altogether, we only simulate an eighth of the sample and complete it by applying symmetries: the origin of the numerical domain is thus the center of the solid. Due to the no-flux boundary conditions, mirror dendrites develop and progress toward the reference dendrite from the right and from above, causing dendrite-dendrite collisions after some time. As the interface width W 0 is inversely proportional to the copper concentration, the physical mesh size δs varies the same way. We take here δs = hW 0 with the numerical grid spacing h = 1.0. As shown in [16], this choice gives very accurate results for a pre-conditioned phase-field. The dimensions L × L × H of the simulated physical domains are the same as in BEA, i. e., L = 800, 1000, 1200 µm and H = 100 µm. These dimensions are chosen to reproduce the geometry of recent experiments in thin samples, that we want to compare with our simulations [START_REF] Boukellal | [END_REF]25]. We use the simple Euler time integration scheme and a simple finite difference scheme for the spatial derivates. Then the time step must satisfy the stability condition δt ≤ 75 282

h 2 λ τ 0 (19)
in three dimensions. The finite difference scheme is accelerated more than 30 times by coding it on a Graphics Processing Unit (GPU). A general purpose GPU is used that is optimized for 32-bit numbers, so all the real numbers we use are implemented as simple precision floats. As discussed in Appendix B, this is not without consequences when a small thermodynamic noise is to be introduced in the code.

Thermodynamic phase diagram

Concentration dependence

In BEA, average estimates of the partition coefficient, k = 0.14, and of the liquidus slope, m = -2.6K/wt.%Cu, were used. Both estimates were taken from an early representation of the whole Al-Cu phase diagram where the liquidus and the solidus are assumed globally linear for c 0 < 16wt.%Cu [START_REF] Hultgren | Selected Values of the Thermodynamic Properties of Binary Alloys[END_REF]. However, the phase diagram of Al-Cu alloys is actually not linear, even for concentrations of a few wt.%Cu. The reason is that the assumption of a diluted alloy rapidly loses its statistical justification: for a concentration of c 0 = 2.0 wt.%Cu, the probability that two Cu atoms in a randomly diluted alloy are in contact is already about 10 percent. According to recent thermodynamic data [START_REF][END_REF], the liquidus and solidus are definitely curved, as can be seen in Fig. 1. In order to assess the robustness of the BEA results, the magnitude of the corrections induced by these deviations from linearity has to be carefully estimated.

The first step is to estimate k and m as functions of the copper concentration c. To do so, the available thermodynamic data [START_REF][END_REF] for the liquidus temperature T L and for the solidus temperature T S are plotted in Fig. 1 as functions of c. Both curves are fitted to third-order polynomials,

T L (c) = a L c 3 + b L c 2 + c L c + T M , (20)
and

T S (c) = a S c 3 + b S c 2 + c S c + T M , (21)
The values of (a L , b L , c L ) and (a S , b S , c S ) are listed in Table 2.

For the alloy nominal concentration c = c 0 , (20) is used to obtain the corresponding liquidus temperature T 0 = T L (c 0 ). Then ( 21) is used to find c 0 such that T S (c 0 ) = T 0 (see Fig. 1), and the partition coefficient is given by k = c 0 /c 0 . For the same concentration c 0 , the liquidus slope is given by ∂T

L /∂c c=c 0 , that is m = 3a L c 2 0 + 2b L c 0 + c L . (23)
The resulting values of k and m are listed in Table 3 for a few copper concentrations c 0 . In the range of concentrations considered in the present study (c 0 ≤ 2 wt.%Cu), the partition coefficient is roughly constant, k = 0.098, and the liquidus slope decreases slightly but remains close to m = -2.7 K/wt.%Cu. Both values definitely differ from the estimates used in BEA.

Regarding the imposed undercooling defined in (7), one sees that, at given time t and concentration c 0 , Θ(t) is roughly 8 percent smaller in the present case than in BEA, so dendritic growth is expected to be slower.

Results

Figure 2 shows six snapshots illustrating the time evolution of a dendrite obtained in a 1000 × 1000 × 100 µm 3 sample for a concentration c 0 = 1.0 wt.%Cu. As in BEA, the production of secondary arms is very limited because thermal fluctuations are not included. One recovers the three main growth regimes already observed in BEA:

1. slow growth (0-60 s): the undercooling is proportional to time and cooling rate R, and so it is very low at the beginning of the growth. 2. fast growth (60-90 s): the nearly spherical seed turns into a crossshaped dendrite with two primary arms along x and z axes. The two primary arms grow fast and almost freely because the undercooling is high and the available space is large enough. 3. poisoned growth (90-150 s): the two primary arms reach the domain boundaries. They stop growing and they broaden. Secondary arms develop and fill the space.

Figure 3 represents the dendrite tip velocity V x measured along the horizontal direction x as a function of time. In comparison with the curve that was obtained in BEA, the maximum velocity is lower and it arises at later time in the present study because the instantaneous undercooling is smaller. Besides these quantitative differences, the two curves are very similar.

(∆T ) 2 = k B T 2 0 c p ∆V , (24)
with k B the Boltzmann constant, c p the Al specific heat (table 1), and ∆V, the fluctuation volume [START_REF] Landau | Statistical Physics Course of Theoretical Physics[END_REF]. At the scale of one mesh element, ∆V = (δs) 3 16.2 µm 3 , one expects thermodynamical temperature fluctuations to be of the order of 5 × 10 -4 K for the alloy considered here. So far, we neglected the thermal fluctuations, so secondary branches did not appear near the dendrite tip during the fast growth regime. Including physically reasonable thermal fluctuations should provoke branching. Moreover, since branching is not deterministic, an asymmetry between the two growth directions x and z may arise. The question we want to decide is wether this asymmetry would ultimately lead to different growth laws in the two directions and/or in different simulations.

In our code, the fluctuations of the sample temperature T (t) = T 0 -Rt are imposed by the means of a numerical spatiotemporal noise,

T ( r, t) = (T 0 -Rt){1 + F 0 [2η( r, t) -1)]}, ( 25
)
where F 0 is the noise amplitude and η( r, t) is a pseudo-random number that is picked for each mesh point r and each time t from a uniform distribution over the range [0, 1). As a consequence, the dimensionless undercooling introduced in equation ( 7) now becomes

Θ( r, t) = -Rt + A 0 (T 0 -Rt) (k -1)mc 0 , ( 26
)
where

A 0 = F 0 [2η( r, t) -1)
] is a random number between -F 0 and +F 0 .

In the following, we impose a noise amplitude F 0 = 1.0 × 10 -4 . Noting that Rt remains negligible as compared to T 0 here, this leads to temperature fluctuations of the order of F 0 T 0 0.1 K per time step. Taking the time correlations into account [START_REF] Karma | [END_REF]30], we obtain independent fluctuations of typical amplitude F 0 T 0 δt/τ 0 0.0035 K which is about 7 times higher than the thermodynamic value calculated from equation (24). This enhancement of the thermal fluctuations is deliberately made here because we want to decide wether significant effects will result or not. As shown in Appendix B, studying the case of physical thermal fluctuations would in addition require to use the double precision representation of real numbers. The code would thus become much slower, unless 64-bit GPUs are used. Let us stress that the present noise differs in nature from the one that is usually imposed in phase-field simulations. While the latter is imposed on the phase and/or the concentration fields that are computed from the time evolution equations [START_REF] Karma | [END_REF]30], the former concerns the fluctuations of a control parameter (the imposed temperature).

Figure 4 shows six snapshots illustrating the evolution of a dendrite obtained with the thermal noise just discussed and for the k and m values used in BEA (linear phase diagram). Here again, the three main growth regimes are obtained. It is worth noting that during regime 2, in which primary arms grow freely, secondary arms do appear now because of the thermal fluctuations (see panel d). During this regime, secondary arms did not appear in our previous simulations without noise.

Growth velocity

Figure 5 shows the velocities of two dendrite tips when thermal noise is included and when it is not. This figure also shows the velocity differences ∆V x,z between the two curves. Both velocity curves (with and without noise) show the same global behaviour: a steady increase of the velocity until it reaches a maximum, then a sharp decrease followed by a slow convergence to zero. Obviously, the spatiotemporal noise affects more the tip velocity near its maximum. In the specific case shown in figure 5, the tip that progresses along the x axis is accelerated while the tip along the z axis is decelerated as compared to the case without noise. We obtained nine comparable figures by performing a single simulation for each choice of (L, c 0 ). It would have been interesting to average ∆V x,z for each (L, c 0 ) over a number of simulations with different realizations of the spatiotemporal noise but this represented a very time-consuming task that was beyond the scope of the present study. For the different choices of (L, c 0 ), one sometimes observes the same effect as in figure 5 and sometimes the opposite one, so the average of the velocity differences ∆V x,z on the different (L, c 0 ) simulations is close to zero (see figure 6). It is interesting to note that, even for the large noise amplitude considered here, there is no dramatic change of the tip dynamics but rather fluctuations around the mean behavior obtained without noise.

Growth scaling laws

In BEA, we characterized the time evolution of the dendritic tip velocity V through the maximum velocity reached, V = V m , and through the time ∆t m necessary for V to increase from V m /2 to V m (see figure 3). We obtained scaling laws giving both quantities as functions of the scaling parameters L a /c α 0 and L b /c β 0 , respectively. The scaling exponents (a, α) were obtained by fitting the numerical values of V m . On the other hand, (b, β) were deduced from the result Λ = L m /L =cst, where L m = V m ∆t m and L is the box length. In the present work, we plot the values of V m and ∆t m as functions of the same scaling parameters as in BEA, both for a linear phase diagram in the

ΔV

x (µm/s)

Figure 5: Time evolution of the tip velocities (red disks) along x and z for the dendritic growth with thermal noise illustrated in Fig. 4. The represented data are obtained for the parameter choice (L, c 0 ) = (1000 µm, 1.0 wt.%Cu). For comparison, the results obtained without noise are represented by continuous curves (black lines). The irregular sawtooth curves represent the differences between the velocities obtained with and without noise (dashed blue lines).

case of imposed thermal fluctuations and for a realistic curved phase diagram. Figure 7 shows the variations of V m and ∆t m with the same scaling parameters as in BEA. As could be expected, the scaling laws derived previously fit the present data as well, with slight deviations due to numerical reasons. On one hand, in the case of a linear phase-diagram, one can notice that the results in presence of a thermal noise are in good agreement with those in

ΔV m,x /V m ΔV m,z /V m ΔV m,xz /V m Figure 6: Velocity difference ∆V x /V m = (V m,x -V m )/V m
, where V m represents the maximum growth velocity without noise and V m,x represents the maximum growth velocity along the x axis, for an imposed thermal noise of amplitude F 0 = 1.0 × 10 -4 . The different cases correspond to the nine couples (L, c 0 ) considered in this work. ∆V z is the corresponding velocity difference along the z axis and ∆V xz the average over the x and z axes.

BEA. Indeed, taking into account the thermal noise in the present work does not affect the growth regimes described in section (4.2). One can see that although the amplitude of the fluctuations imposed in the code is 7 times higher than the thermodynamic one predicted by Equation ( 24), the noise induces only slight deviations of V m and ∆t m . On the other hand, we also notice a good agreement with the results in BEA when a thermodynamic phase diagram is considered. Let us remark that for the dilute Al-Cu alloys considered here, the product m(k -1) is only increased by a factor of 1.088 for a given c 0 in the present case as compared to BEA. This small variation is thus expected to weakly affect the dynamics of the system as W 0 and Θ(t) keep almost the same values in both cases. Considering the three series of the data altogether, one finds that Λ ≈ 0.537 instead of Λ ≈ 0.546 quoted in BEA (that was readjusted after a more systematic data analysis to Λ ≈ 0.550). This result strengthens the conclusions drawn previously stating that V m ∆t m is only a function of the distance 2L between the grains whose primary arms grow toward each other. Altogether, the results obtained in this section confirm that extrapolation of our scaling laws to the concentrations of industrial or experimental interest remains possible even in presence of thermal fluctuations and in the case of a thermodynamic phase diagram. As a general remark from a numerical point of view, the results obtained in the case of a thermodynamic phase diagram can be reproduced by the code developed in BEA provided that the product m(k -1)c 0 and W 0 are adjusted so as to be identical in both cases. However, the results will correspond to different physical systems as c 0 will be shifted on the phase diagram.

Summary and conclusion

We have presented the numerical results of 3D phase-field simulations of Al-Cu equiaxed growth. Two thermodynamic effects have been considered and their implications on the scaling behaviour of the dendrite tip growth dynamics has been explored. First, a curved thermodynamic phase diagram has been considered, instead of the linear diagram approximation that is usually taken. The partition coefficient k and the product m(k -1) are lower in the former case (m is the liquidus slope). As a result, growth is slower than in the linear approximation. This shows that taking into account a thermodynamic phase diagram in the phase-field model is required to make a very quantitative comparison with experiment.

Second, keeping the linear phase diagram, spatiotemporal noise has been imposed to describe the thermal fluctuations that renders growth less deterministic. Developed secondary branches are indeed obtained and the curves giving the growth velocity as a function of time differ from one dendrite tip to the other. This partly explains the unique character of the dendrite evolutions observed in experiments [START_REF] Boukellal | [END_REF]25].

Finally, the scaling laws derived in BEA, giving the maximum dendritic growth rate V m and the characteristic growth time ∆t m as functions of the copper concentration c 0 and the available space L have been independently tested for both thermodynamic effects. Although consideration of a thermodynamic phase diagram is important at higher concentrations and the presence of thermal fluctuations is essential at lower concentrations, the scaling laws remain valid even in the presence of these thermodynamic effects. A natural extension of this work would be to combine both thermodynamic effects and check the accuracy of the scaling laws in these conditions that are closer to the experimental ones.

where F 0 is the fluctuation amplitude and a is a pseudo-random number picked from a uniform distribution over the range [-1, 1). For the sake of the argument, let us assume that one exactly has This means that the corresponding range of small fluctuations is always out of reach in numerical simulations. Our numerical simulations have been performed on a GPU that is very fast for simple precision (s = 6) but much slower for double precision (s = 15). For this reason, we used simple precision floats. Thus, for the noise amplitude we used (m = 4), the fluctuation range -0.01 < a < 0.01 is not described correctly. Physical fluctuations are roughly 10 times lower (m = 5). In this case, 10 percent of the fluctuation interval would not be described correctly, which is not acceptable. The solution would be to switch to double precision but, as just said, the computing time would be increased significantly. It is thus advisable, when one needs to simulate physical thermal fluctuations, to use GPUs that are optimized for double precision.

Figure 1 :

 1 Figure 1: A part of the Al-Cu phase diagram. Markers represent available data from a thermodynamic data base[START_REF][END_REF]. Continuous lines are fits to third order polynomials giving approximate equations for the liquidus (L) and the solidus (S) lines.

Figure 2 :

 2 Figure 2: Time evolution of a dendrite obtained in a 1000 × 1000 × 100 µm 3 sample for a concentration c 0 = 1.0 wt.%Cu (see the second line in Table 3 for the phase diagram parameters k and m).

4. Thermal fluctuations 4 . 1 .

 41 Spatiotemporal noiseOur second point of interest is the role of thermal fluctuations the mean square of which is given by

Figure 3 :

 3 Figure 3: Velocity V x of the dendrite tip as a function of time (curve with markers). A unique concentration, c 0 = 1.0 wt.%Cu, and a unique sample size L = 1000 µm is considered. The curve corresponding to BEA (k = 0.14) is shown for comparison (without markers).

Figure 4 :

 4 Figure 4: Time evolution of a dendrite obtained in a 1000 × 1000 × 100 µm 3 sample for c 0 = 1.0 wt.%Cu, k = 0.14 and m = -2.6 K/wt.%Cu. Noise amplitude F 0 = 1.0 × 10 -4 .

Figure 7 :

 7 Figure 7: Scaling of the data obtained for different phase diagrams and different noise amplitudes F 0 . The solid curves are fits of the data for the linear phase diagram without noise (from BEA). These fits give the following rounded values for the scaling exponents: a = 2, α = 2/3, b = -1, and β = -2/3.

F 0

 0 = 10 -m , (B.2) with m a positive integer. If s represents the number of significative digits of the floating-point representation used for real numbers, S = 10 -s (B.3)is the smallest number that can be added to or subtracted from 1 without totally falling in rounding errors. Thus, the imposed fluctuations will be masked by rounding errors if|aF 0 | < S,

Table 1 :

 1 Physical parameters of the Al-Cu alloys considered in the simulations.

	Symbol	Physical parameter	Value	Unit
	D	copper diffusion coefficient	3000.0	µm 2 /s
		in the liquid
	T M	aluminum melting	933.0	K
		temperature
	c p	aluminium specific heat	2.8 × 10 -12 J/K/µm 3
		at melting temperature
	c 0	copper concentration	1.0, 1.5, 2.0 wt.%Cu
	4	crystal cubic anisotropy	0.01	-
	Γ	Gibbs-Thomson coefficient	0.236	Kµm

Table 3 :

 3 Partition coefficient k and liquidus slope m for Al-Cu alloys at different copper concentrations c 0 up to the eutectic concentration (last entry).

	c 0 (wt.%Cu)	k	m (K/wt.%Cu)
	0.5	0.098	-2.697
	1.0	0.098	-2.692
	1.5	0.098	-2.688
	2.0	0.098	-2.685
	4.0	0.099	-2.687
	10.0	0.103	-2.811
	20.0	0.118	-3.415
	33.1	0.172	-4.956

Acknowledgments We wish to express our gratitude to Guillaume Reinhart for directing us to the Thermo-Calc data base, and to Damien Touret for his 20 critical reading of the manuscript. The project leading to this publication has received funding from Excellence Initiative of Aix-Marseille University A*MIDEX, a French Investissements d Avenir programme. We thank the French Centre for Spatial Research (CNES) for constant financial support.

Appendix A. Anisotropy vector for the surface free energy

In this appendix, we give the general expression of the anisotropy vector A as a function of the preconditioned phase-field ψ. We obtain explicit expressions of three components A x , A y , and A z for the case of a cubic anisotropy. Using the preconditioned phase-field

and its first derivatives

where η = x, y, z, one obtains

Introducing the components

of the unit vector n normal to the solid-liquid interface, the last term in eq. A.3 becomes

with δ η,ζ = 1 if η = ζ, and 0 otherwise. Thus,

The anisotropy vector can thus be derived for any continuous function a s (n x , n y , n z ) representing the anisotropy function. In the present work, we study the growth of Al-Cu alloys, that adopt a cubic symmetry in the solid phase. In this case, the anisotropy function simply reads

and its partial derivatives are

Then the three components of the anisotropy vector are

with η = x, y, z.

Appendix B. Significative range of fluctuations for a uniform spatiotemporal noise of amplitude F 0

In this appendix, we obtain bounds for the range of random fluctuations that it is possible to represent on a computer and we give advices to implement fluctuations on GPUs. In equation (25), the thermal fluctuations of the imposed temperature T are introduced by multiplying T by a factor 1 + aF 0 , (B.1)