David Margery

Lucas Nussbaum

Estimation of Costs and Pay per Use on a Large-scale Shared Computer Science Testbed: the Grid'5000 Case

Grid'5000 is a large-scale science testbed: multiple academic organizations contribute to some of the costs associated to operating and maintaining it, to the point many approximations must be made when attempting to build a view of the overall costs associated to it. However, such an approximated cost has been estimated, first in 2015, and more recently in 2017 and 2018 as part as a standardized infrastructure cost evaluation for all large scientific instruments on France's national research infrastructure roadmap. From this approximated cost, we can derive pricing.

However, this does not imply that we can make users pay for their usage of Grid'5000, as: (1) many users are affiliated in a way or another to the organizations that are contributing to supporting the costs of the infrastructure;

(2) the way costs were approximated does not qualify Grid'5000 usage as an eligible cost in the context of funded projects as it is not auditable.

In practice, this only leaves as potential paying users: (1) foreign academics with no ongoing collaboration with their French colleagues, and (2) private sector users. Being a minority of the user base, we cannot expect a significant revenue stream from such categories of users. In theory, we could also hope to invoice organizations who benefit more than they contribute to costs, but in practice, such a scheme would be difficult to operate.

In conclusion, we do not believe that we can transition to a pay-for-use model for a majority of Grid'5000 users without a major increase in bureaucracy to associate usage to affiliation and a reduction to the number of organizations supporting the costs of the scientific instrument. This would certainly have a negative impact on usage, increasing the number of custom-built mini-testbeds and therefore the overall cost to the research ecosystem.

5.1.3 Annual cost estimation . . . . . .

Introduction

When discussing the costs of research infrastructures, it is often suggested to work on estimating costs in order to develop a pay-per-use model, that supposedly would be able to solve funding issues. This document discusses this issue in detail, detailing possible motivations, outlining some limitations and challenges, and illustrating this with the specific case of Grid'5000's attempts to estimate costs and establish a pay-per-use model.

This document is organized as follows. Section 2 provides a short description of the Grid'5000, in order to provide some context and some background for the assumptions made throughout the document. Section 3 discusses motivations for estimating costs and establishing various kinds of pay-per-use models. Section 4 then articulates limitations and challenges raised by those models. Finally, 5 describes Grid'5000's experience in that context, before the document is concluded in 6.

2 Context: The Grid'5000 Testbed Grid'5000 is a distributed computer science testbed. It is composed of 8 sites, hosted by several institutions (nationwide research institutions, and several universities, sometimes in joint labs between research centers and universities). Seven sites are located in France, one is in Luxembourg. Each site is composed of several clusters (homogeneous sets of machines that are usually purchased at the same time). Different funding sources are used for the clusters: regional projects (sometimes combined with European funding), own resources of one of the institutions, own resources of one research team at one of those institutions, etc. The engineering workforce is composed of a mix of permanent staff, and staff on short-term temporary contracts. Engineers are employed by the institutions that also host sites.

The major sources of cost for the infrastructure are:

• The purchase of hardware resources;

• The hosting of hardware resources (such as the operation of the server room itself, and electricity costs);

• The engineering workforce required to develop and maintain the infrastructure's software stack;

• The engineering workforce required to operate the testbed and its services.

Motivations

The motivations for estimating costs are described below.

Objective 1: Advertising costs to users to increase self-awareness The use of the infrastructure would remain free from the point of view of users, but, in order to increase the self-awareness of users and their teams about the important costs generated by their use of the infrastructure, some kind of usage report or virtual billing would be done, with an estimation of the generated costs, without resulting in actual billing.

Objective 2: Billing some users of the infrastructure The use of the infrastructure would be billed to some categories of users, such as:

(Objective 2.1) Users from the private sector (Objective 2.2) Academic users affiliated to institutions that do not contribute to the infrastructure (Objective 2.3) All users (including those affiliated to institutions that contribute to the infrastructure)

Objective 3: Using the estimated costs for reimbursement in the context of contracts The costs generated by the use of the infrastructure could be considered in the context of contracts (industrial collaborations, collaborative projects funded by ANR or UE), in other to be compensated.

Challenges

Accurate Data Collection

(Affects all objectives, but especially Objective 3) Collecting correct data about costs can be extremely difficult. Some costs are trivial to identify (for example, the initial purchase costs) but raise questions on how to amortize them over the lifetime of a resource. Some costs are general to the infrastructure (for costs related to the purchase and operation of the network infrastructure, or of the service machines) and must be shared by all resources. Other costs are hard to evaluate, such as electricity costs in a data center that hosts both machines for the testbed, and machines for other purposes. More examples of difficulties are provided in Section 5.

This might not be a severe issue for Objective 1, as long as the costs provide a good estimate. It might also not be a problem for Objective 2, as long as the costs are fairly accurate. However, it might be a severe problem when costs must be justified to a third party such as a funding agency.

Academic Culture of Invisible Costs

(Affects Objectives 2.2 and 2.3) Most costs related to the life of research teams are generally hidden from the research teams themselves, and amortized over the whole research institution. For example, costs related to accessing publications, or to hosting people, are generally not billed to the teams. Now let's suppose that the use of the testbed is billed to a researcher, who is a member of a French public research organization that does not contribute to funding the infrastructure. That researcher could be tempted to build her own testbed, hosting it in her lab, with local human resources. It is likely to be a less expensive solution for her, because many of the costs will be covered transparently by her lab. However, the overall cost for the French public research ecosystem would be higher due to the duplicate work and the lack of consolidation between testbeds.

Other Incentives for not Using Pay-per-Use Infrastructures (Affects Objective 2.2 and 2.3)

There are other incentives for not using pay-per-use infrastructures. First, researchers have an incentive to build their own infrastructure, because the building of testbeds (even private testbeds that duplicate the functionality of other pre-existing testbeds) is usually seen favorably, for example when the researcher's lab is evaluated.

It might even be possible for the researcher to get paid to use other infrastructures. For example, many research projects on testbeds design have Open Calls where prospective users of those testbeds can be funded to use the testbed (and provide early feedback to the testbed operators).

Loss of Visibility for the Infrastructure

(Affects Objectives 2.1 and 2.2) Providing free access to the testbed helps lower the barrier to entry, and increasing the visibility (especially the international visibility) of the testbed. Free access to the infrastructure can also act as a honeypot to attract collaborations with industry. One could even question the interest of asking industrial users for pay-per-use if it is possible, instead, to build mutually fruitful collaborations.

This could, of course, be mitigated by providing a free preview mode (enabling prospective users to explore the infrastructure capabilities by providing free access to the infrastructure for limited amounts of time and resources), but then care must be taken to ensure that users do not abuse the system by creating multiple accounts.

Quantifying Contributors' Contributions (Affects Objectives 2.2 and 2.3)

The split between users from institutions contributing to the infrastructure, and users from institutions that do not contribute to the infrastructure, is not that simple.

First, there is a wide spectrum of contribution levels. If all contributions are considered the same, then this acts as an incentive for institutions to minimize their costs and avoid renewing hardware (if an old cluster is sufficient to be considered a contributing institution, what is the point of replacing it?). Therefore it is important to evaluate and differentiate contributions.

Second, not all affiliations are the same. Many users have multiple affiliations due to, for example, joint teams between several institutions. Also, there are nationwide research institutions with several sites (locations): if some sites contribute to an infrastructure, should users from other sites be identified as users from contributing institutions?

Third, not all institutions have the same user base. How should contributions be matched with the usage volume?

This could be solved by measuring the value of contributions and providing some kind of virtual billing even for users of contributing institutions, to ensure that the costs resulting from the usage of the infrastructure do not exceed the contributions (or else, the usage would be billed as for users of non-contributing organizations). But this is a complex scheme, which raises additional questions, such as which users should be billed when the institution's quota is exceeded.

Splitting of the Revenue Among Contributors

(Affects Objectives 2 and 3) Another problem is the splitting of revenue among institutions contributing to the infrastructure. Should it be split according to the value of contributions, or to the actual resources used? The latter is harder to implement (especially since some elements are shared over all resources, such as development costs), but can act as an incentive to provide more interesting resources.

Legal Challenges Related to the Funding Sources

(Affects Objectives 2 and 3) Some sources of funding that could have been used to purchase parts of the infrastructure have additional requirements, such as: forbidding to generate revenue from the resources, or forbidding commercial use. Special care would have to be taken to respect those additional rules, for example when allocating usage to resources.

Complex Billing Scheme

(Affects Objective 3) When attempting to justify costs in order to claim their reimbursement from a funding agency, there are two different institutions at play: (A) the institution that provides the infrastructure (assuming, for simplification, that a single institution is in charge of the billing on behalf of all institutions providing the infrastructure); (B) the institution of the user of the infrastructure.

If (A) and (B) are different, and if the user also has free access to the infrastructure, there is no real incentive for the user and (B) to declare their use of the infrastructure to the funding agency (and thus pay for the use of the infrastructure), because it would not benefit (B), but (A). (B) also has an incentive not to claim costs for the use of the infrastructure, because projects generally operate on a budget fixed in advance, so reimbursement going to (A) is funding lost for (B).

If (A) and (B) are the same institution, it requires the funding agency to accept the evaluation of the costs made by (A/B) as correct, because (A) cannot invoice itself to present an invoice to the funding agency. Such a scheme might be an issue for some funding agencies, even if the French ANR seems to accept such scheme 1 .

Finally, the only case where this works well is if (A) and (B) are different institutions, and (B) is always paying for its use of the infrastructure. But in that case, the fact that (B) gets reimbursed by the funding agency is invisible from the point of view of (A).

The Grid'5000 experience

This section describes the Grid'5000 experience of evaluating its costs (Section 5.1), and then for implementing a pay-per-use model for a company (Section 5.2).

Evaluation of Costs

Data available

The Grid'5000 testbed has a database with data that are crucial to estimate the costs of running Grid'5000.

Annual cost estimation

For the last 2 years, the French Ministry of research has made an attempt at standardizing the way the annual costs of all research infrastructures on the national roadmap. For this reason, we have an total annual cost for Grid'5000 since 2016 (published in 2017) and 2017 (published in 2018). This cost takes inflation into account but not networking costs as the National Research and Education Network (RENATER) also presents its costs, and that networking costs for Grid'5000 are included in RENATER's costs.

For 2016, the reported annual cost for Grid'5000 was 1.904 Me (in 2015 e). In the same year, 47 993 263 core.hours were used on the infrastructure, putting the cost of each core.hour used that year at 0.0396 e. Even if networking was factored in, we would not reach the 0.1 e for each core.hour computed using the previous method.

For 2017, the reported annual cost for Grid'5000 was 1.845 Me (in 2017 e). In the same year, 41 903 708 core.hours were used on the infrastructure, putting the cost of each core.hour used that year at 0,0440 e, when network is not factored in. 2

Definition of a Usage Cost for Pay-per-Use

Using the data from the initial cost estimation, Grid'5000 governing body, the Conseil de Groupement du GIS Grid'5000, has formally approved selling access to Grid'5000 at the price of 0.1e for each core-hour used.

Because of that decision, Inria, hosting the governing body, is able to invoice users of Grid'5000 provided they have agreed to use Grid'5000 on a Pay-per-Use basis.

Formalizing that agreement is the difficult part. A contract has been written to do so, and signed once with a private company. The contract allowed the company to buy the right to use a pre-defined number of corehours. Grid'5000 would be allowed to block usage should they exceed that amount, but only real usage would be invoiced.

For history, it should be noted that because of the expertise required to run proper experiments, they started financing a PhD in a collaboration with a university involved in Grid'5000. They therefore acquired the right to use Grid'5000 freely before the first invoice.

We therefore have a contract to sell access to Grid'5000 on a Pay-per-Use basis, but no identified clients.

Discussion and conclusions

This document outlines the challenges of estimating costs and building a pay-per-use model. It is worth stressing that moving forward in that direction would require a lot of expertise that is not currently found in the current Grid'5000 workforce: collaboration with services responsible for building management, and with legal and financial services to understand all the legal consequences. Also, if significant revenue is expected from paying customers, it would also require the help of commercial staff to advertise the infrastructure and extend the customers base. Finally, administrative staff would be required to operate the scheme on a daily basis. If continued, the current scheme (free use) could nevertheless be improved. Specifically, we could develop schemes that encourage institutions to contribute to Grid'5000, by providing higher priorities to users from contributing institutions. Such a scheme was outlined in [START_REF] Nussbaum | Usages et utilisateurs de Grid'5000: stratégie pour l'accès aux ressources[END_REF], but has not been implemented yet, mainly due to lack of development resources in order to implement the required changes in the Grid'5000 services.

See ANR -Règlement financier, section 3.1.3

Total of core.hours used for each day, with complete data going back to 2007 and partial data for years 2005 and 2006.

Price and date of purchase for all equipment since the start of the project. For the initial clusters, the price of purchase is estimated using the amount of budget available for the purchase. Errors should be minimal.

For every engineer having worked on the project, dates and proportion of time devoted to the project, and whether he or she worked as an admin, a developer, a manager or for a project connected to Grid'5000. This dataset is, however, missing some crucial information:

The cost of the network interconnection between Grid'5000 sites, and between Grid'5000 and the rest of the internet. It has been guesstimated at 100ke a year for each 10G lambda in 2015.

A trace of the involvement of the different scientists in the management of the project. For all that follows, this cost has been neglected.

The salaries and overhead costs for all the staff having worked on the project. There are too many organizations involved to be able to track that information, rather confidential by nature, in the project's stats database.

The annual costs of hosting a Grid'5000 site: cooling infrastructure, maintenance of the cooling infrastructure, electricity, network link to the Renater's Point of Presence). No research institution or university hosting Grid'5000 is organized to estimate hosting costs at the granularity of a rack or to monitor electricity consumed. In the best case, we have a one-time measure of the proportion of electricity consumed by Grid'5000 over a one-month period on the overall consumption of the buildings.

Even if all that information was available, some difficult choices need to be made to compute a cost.

Choosing the units of costs. For this document, the unit chosen in the core.hour. This makes it free to store data on Grid'5000, and does not differentiate between cores based on their performance or when they give access to a large amount of memory, accelerator cards, specific local storage performance, or high performance network interconnect. Should the cost be passed on to users, optimization based on this simple unit of cost would surely lead to a lot of undesirable side effects.

Choosing how to depreciate the value of the hardware and of the engineering invested to build the infrastructure. When staff is only seen as an operating cost, the figures fail to capture the value of all the processes, software infrastructure and know-how required to build a research testbed. Value of depreciated but still used hardware must also be captured.

Initial cost estimation

First in 2015, and then at the start of 2016, a cost for Grid'5000 was computed based on the Grid'5000 history since 2005, without taking into account inflation. This as lead to an initial estimation of a cost of 0,1 e per core.hour, using the following data extracted on February 11, 2016 from our statistics database: 264 333 207 core.hours had been used and registered on Grid'5000 since 2005. Note that these are not normalized to account for different processor performance, and data for older jobs is not complete. 6 093 870 e was spent on personnel, outside European projects. This is an estimated value, with an average cost of 5000 e per person.month. 7 738 911 e was spent on hardware. For older purchases, the funded amount was used instead of the final purchase price.

12 000 000 e spent by RENATER on network. This is the less precise estimate. Each site has a 10 Gb/s uplink to the core network, and Grid'5000 has its own uplinks (two) to the core network.

457 420 e had been spent on travel and small equipment. Exact figures are used between 2009 and 2014, and the average of these figures is used for the other years taken into consideration. When all those costs are added, and divided by the number of core.hours, you get a cost of 0.0995 e per core.hour.


Bates Clark

Irving Fisher

Alvin S Johnson

Keywords: natural capital, history of economic thought, monetary valuation, theory of capital, John

Introduction

In a classic article published in 2003, then republished in 2005, Maria Åkerman describes the origins of the natural capital concept, starting her story in the 1980s. As she explains, David W. Pearce was the first economist to use the expression "natural capital" to refer to natural resources and services, or, to use his own words, "the set of all environmental assets" (Pearce, 1988, p. 599). 1 A few years later, the nascent ecological economics movement (see Costanza, 1991, p. 8, 16-17, 329-330) adopted the concept to support the incorporation of environmental constraints into the economic lexicon, and therefore into economic analysis. Ecological Economics was the journal in which the natural capital concept developed, in many articles dealing with sustainability. For example, in 1992, Fikret Berkes and Carl Folke discussed the relationship between human-produced capital, cultural capital and natural capital, defining the last (p. 2) as the sum of (1) exhaustible resources, (2) renewable resources, and (3) what are called today regulating ecosystem services (climate, hydrological cycles, etc.). This triad became a classic definition of natural capital for many scholars, both inside and outside ecological economics. And, so far, it has achieved wide success -i.e. the natural capital concept has spread throughout the economic literature.

This spread could have been seen as a victory for ecological economists. But, from the middle of the 1990s, some scholars involved in a radical criticism of mainstream economics (see [START_REF] Harribey | Développement soutenable et réduction du temps de travail[END_REF][START_REF] O'connor | Ecological Distribution and Distributed Sustainability[END_REF] warned their colleagues about the distortion of natural capital as an operational concept: instead of widening economic analysis to include environmental constraints, the concept was accused of encouraging a narrow vision of the environment, reduced to mere assets with an economic value. Maria Åkerman herself notes this development (Åkerman 2003 p. 437; 2005, p. 42).

This controversy actually opposes two different uses of the natural capital concept, which correspond to the strong and weak positions in the sustainability debate (see [START_REF] Neumayer | Weak versus strong sustainability[END_REF]. On the one hand, natural capital can be seen as a set of immutable natural items satisfying basic needs. It is then defined as critical -i.e. non-substitutable. As a result, since it is immeasurable using common methods, it cannot be compared in monetary terms with other types of capital, and thus cannot be used in cost-benefit analysis. This strong interpretation of the natural capital concept aims to incorporate natural constraints into economic analysis, through the use of the term "capital", but endowing this capital with particular characteristics (non-substitutability, nonreproducibility, etc.). On the other hand, natural capital can be considered as a set of natural resources or services that need to be consciously managed, but that can be used in standard costbenefit calculations. In this framework, limited substitutability and reproducibility still exist. But this does not prevent monetary valuation and cost-benefit analysis, with theoretical tools coming from capital theory (amortisation, discount rates, etc.). This second approach has led to many monetary valuation studies of resources and ecosystem services, starting with [START_REF] Costanza | The value of the world's ecosystem services and natural capital[END_REF] famous paper published in Nature in 1997. On the other hand, research into critical natural capital also developed, bringing together ecological economists attached to a strong interpretation of the natural capital concept (see for instance [START_REF] Ekins | A framework for the practical application of the concepts of critical natural capital and strong sustainability[END_REF].

Such a controversy about the correct use of the natural capital concept, or rather about the consequences of its use by economists engaged in monetary valuation, needs to be enlightened by a new inquiry into the roots of the concept. In fact, contrary to Åkerman's observations, the notion of natural capital, in its modern meaning, appeared in economic literature long before the 1980s, being used for the first time in a textbook that is almost unknown today, written by Alvin S. Johnson in 1909. This historical moment, which is very little documented, is worth examining because it sheds light on the relationship between the concept of natural capital and economic theory, and it puts current controversies into perspective. The objective of this paper is thus twofold: (i) to complete This article is organised as follows: Section 2 reports the historical inquiry into the natural capital concept. This inquiry produces interesting results with respect to Alvin S. Johnson's writings (Section 3). Section 4 gives some theoretical explanation for these results in relation to the Austrian and American discussions about capital theory. Section 5 shows how this narrative helps interpret current controversies, and makes some concluding remarks about this investigation.

Searching for Natural Capital

Searching for a concept in the history of ideas may involve two processes: searching for the idea behind the words; or searching for the words themselves. Linguistically speaking, this equates to searching either for the signified, or the signifier. As far as natural capital is concerned, this inquiry sits at the meeting-point of these two perspectives. In other words, it seeks, in the history of economic ideas, the moment when the expression "natural capital" appeared simultaneously with its modern meaning.

As far as the signified is concerned, DesRoches (2015) produced a remarkable piece of work, describing the philosophical characteristics of the conception of nature which is conveyed by the natural capital concept, defined as a set of resources and autonomous productive services. DesRoches carried out his own historical inquiry, and he emphasises the role of classical economists, beginning with Adam Smith and David Ricardo, in changing economists' view of nature. His inquiry is worth reading, but it ignores the role of the word itself. DesRoches reports different authors who do not refer to natural capital as a signifier, but only as a signified. In order to construct a parallel between past and present conceptions of natural capital, it is necessary to go beyond DesRoches' analysis, and to search for the expression itself.

As regards the signifier, searching for the natural capital concept in 18th-, 19th-and 20thcentury economic corpuses in various languages -English, German, French and Spanish 2 -gives contrasting results. As far as historical encyclopaedias and digitised books and articles may give accurate results for such an inquiry, 3 the natural capital concept did not appear in a significant manner before the middle of the 19th century.

In the English-language literature, one of the first occurrences of the term may be ascribed to Ebenezer Jones, a British publicist and lampoonist, who, in 1849, wrote an essay on the concentration of landed property and its consequences for social justice and economic efficiency. Jones was close to the socialist movement, and he developed his criticism to denounce the excessive concentration of wealth in the hands of a few people. The expression "natural capital" is used a few times, as a synonym for land (p. 6, 19, 21). In his rhetoric, this expression has the advantage of echoing other kinds of productive capital (machines, tools) owned by the same people. It reinforces the idea that capitalism leads to concentration of wealth in all its forms. Jones's intention and definition of natural capital -as land -remains different from modern meanings. Some years later, in 1864, reporting on the 5th International Statistical Congress held in Berlin the year before, Samuel Brown uses the term "natural capital" to refer to "land, etc." (p. 205), but places "water and steam power, [and] the principal metals" (p. 205) outside this category. He gives an approximate definition of the concept, simply listing it among many statistical categories requiring discussion. The same issue arises in a note from the Royal Statistical Society published in 1904, in which the author alerts 2 These languages are the most relevant, since most Scandinavian and Russian authors wrote in German until English took the lead in economic literature after World War II. The Italian "capitale naturale" gives no result.

3 Searches were made on dictionaries and document databases including JSTOR, BSC, Google Books, the New Palgrave and the Dictionnaire de l'économie politique by Charles Coquelin and Gilbert Guillaumin. This investigation was not designed to be totally comprehensive, but to obtain relevant results about old usage of the concept. -published in Ecological Economics (2018), 143, 90-96 -4 readers to the degradation of "national resources" (p. 688), including men and "natural capital" (land and mines). Mining comes into consideration, but once again without extensive theorisation legitimising the analogy between natural resources and capital.

In the German-language literature, the expression "Naturkapital" is just as scarce. Albert Schäffle, one of the main pioneering thinkers in the Austrian tradition, before Carl Menger and Eugen Böhm-Bawerk, employs it on several occasions in his Die Nationalökonomie (1861), and he does give it some theoretical basis. In his view, "Naturkapital" refers to all natural resources and forces, such as water, light and air, which furnish productive services (p. 43). Here the modern meaning of natural capital comes closer. But Schäffle lacks precision about which resources he places in his "Naturkapital" category. Collective natural goods are included (p. 47), but nothing is stated about a potential link between resource management and asset management. Although Schäffle is cited by Menger (1871, p. 54n, 277n, 288, 290, 293-294, 300-301, 307, 312) and Böhm-Bawerk (1889a, p. 54n, 93n, 292n, 347), his use of the natural capital concept has not been followed up in the Austrian tradition.

During the 19th century, it was probably the French-language literature that made the most use of the natural capital concept. Bastiat (1854, p. 256-257), a leading liberal figure, indicates that the socialist Victor P. Considérant regularly makes use of the expression "capital naturel". In his Théorie du droit de propriété et du droit au travail (1848), Considérant indeed defines "capital naturel" as a synonym of primitive capital ("capital primitif") to refer to the value of land (p. 20-21). This looks like Ebenezer Jones' narrow definition of natural capital -not enough to draw parallels between past and present. For their part, Jean-Baptiste Say and Antoine Augustin Cournot, both eminent 19th century French economists, do not use the natural capital concept. 4 In the French-language literature, the most significant uses of the natural capital concept are to be found in the writings of Auguste and Léon Walras. In the 8th lesson of his Evreux course, given in 1832-1833, Auguste Walras, besides his criticism of classical political economy, introduces the notion of "capital naturel" for the first time, referring to original productive forces: land and labour (A. Walras, 1833, p. 170). The "natural" epithet here is a synonym of "raw" or "crude", when human beings have not yet transformed their innate capacities and available materials into useful goods. Auguste Walras uses this terminology to differentiate between natural and artificial items, a distinction that he would maintain throughout his writing. 5 In this respect, as in many others (see [START_REF] Dockès | Introduction générale[END_REF], Léon Walras, the future proponent of general equilibrium, followed his father. In 1860, in his L'économie politique et la justice and in an essay entitled "De la cherté des loyers à Paris", he makes a distinction between artificial capital and natural capital, placing land and working men's capacities into the category of natural capital (Walras 1860a, p. 93-94;1860b, p. 241-242, 252, 265). This view fits into a more complex constellation of categories of capital (see Lallement, 2000, p. 460). Refining his nomenclature, Walras (1860b, p. 242) separates land and personal skills, noticing that land may be handed down, unlike personal faculties. But he makes no assumption about a potential link between natural resources and assets, even in his Éléments d'économie politique pure [START_REF] Walras | Éléments d'économie politique pure[END_REF] when he continues to talk about "capital naturel" (p. [269][270]352) to make a contrast with artificial items, 6 in the same fashion as Auguste Walras did before him (labour being part of natural capital).

Finally, in the Spanish-language literature, the word "capital natural" was employed on very few occasions. Influenced by the French liberal economists, Carreras y Gonzalez (1865, p. 84) reports some uses of the natural capital concept, defined as a synonym of primitive between capital and natural items (Cournot, 1863, p. 83). 5 See A. Walras 1837, p. 350-361; 1844, p. 436; 1849, p. 57; 1864, p. 386, 398. 6 See also Walras 1861, p. 366;1865, p. 331.

-Author's post-print --published in Ecological Economics (2018), 143, 90-96 -5 capital -i.e. exactly the same as Considérant's use, reported by Bastiat. A few years later, in an essay entitled "Economia politica", Ignacio Ramirez, the famous Mexican writer better known as El Nigromante, employs the expression "capital natural" to refer to the single capital that poor people can make use of, namely their labour (1889, p. 6). Ramirez's definition of natural capital therefore does not include natural resources, but is limited to labour.

These various contributions are summarised in the following table (Table 1). It appears that the natural capital concept was used in various language traditions, by authors coming from many different intellectual and social positions. The definitions of natural capital were however not very close to the modern sense -i.e. productive assets.

To summarise, whatever the language tradition and school of thought, searching for the natural capital concept in the 19th century seems to give no satisfactory result to enable us to compare past and present. The beginning of the 20th century offers better results, in particular in the English-language literature, with the writing of Alvin S. Johnson.

Johnson's Introduction to Economics (1909)

Alvin S. Johnson is essentially known for his participation in the first developments of the New School for Social Research in New York during the 1920s and 1930s. At the beginning of his career as an economist, after graduating from Columbia University under the supervision of John Bates Clark, one of the major American economists of the time, he published three books. The first, Rent in Modern Economic Theory: An Essay in Distribution (1902), was taken from his PhD dissertation. The second, Introductory Economics (1907), was a textbook like many others published at the time (for a review, see McCrea, 1909, p. 328-330). The third, Introduction to Economics (1909), was an update of the 1907 textbook, with more detailed arguments and examples. Johnson's footprint in the history of economic thought is modest, but there is one criterion whereby he deserves much more interest than he has received: giving the natural capital concept a meaning which turns out to be close to that used today. In 1902, Johnson is not yet talking about natural capital, simply referring to "artificial instruments of production" (p. 41), a term that calls for its complement: natural instruments. He also mentions an analogy between ore stocks and capital (p. 23-24), but he does not use the natural capital concept itself. In 1907, natural capital is still not mentioned, but land is considered as a kind of capital good (p. 194-195), and productivity gains are related to "increase in artificial capital or in available natural resources" (p. 133), a clause that suggests common points between artificial and natural items. It is in 1909 that Johnson finally takes the plunge:

"A generation ago practically all economists restricted the term "capital" to productive wealth that has been produced by industry, such as machines, stocks of materials, etc. Productive wealth, the origin of which cannot be traced to man's industry, was usually classified under the heading "natural agents", or simply under "land", since land is by far the most important good in this class. This terminology is still widely used by economists. In everyday language men speak of investing capital in land, as of investing capital in buildings or machinery. This usage will be followed in this book; wherever it is necessary to distinguish between the two classes of productive wealth, we shall call the one artificial capital, the other natural capital." (Johnson, 1909, p. 197) At first sight, this is no revolution. In this quotation, however, Johnson not only makes a distinction between artificial capital and natural capital, but he also associates the natural capital concept with the productive power of natural agents, without including labour in the same category. By considering natural capital as the set of natural items that furnish productive services to human beings, Johnson is much closer to recent authors, such as [START_REF] Berkes | A systems perspective on the interrelations between natural, human-made and cultural capital[END_REF], than his predecessors.

He further remarks that industrial development implies a rise in the number of natural agents available for economic production. On this point, he explains that industrial development comes with an increase, or an accumulation, of natural capital: "Until the frontiersmen crossed the Appalachian Mountains, the land of the Mississippi Valley, with the timber upon it and the coal and other minerals beneath its surface, was scarcely to be classed as wealth at all. At best, it was potential wealth, not actual wealth. The settlement of the country and the development of means of communication transformed this potential wealth into an immense fund of productive wealth, or capital. Every increase in population, every improvement in methods of agricultural production, increases the importance, and with it the value, of the natural resources of a country. Measuring the capital represented by these natural resources in terms of value, we see that it is constantly growing with the progress of society." (Johnson, 1909, p. 199) This rhetoric, completed with other quotations along the same lines (Johnson, 1909, p. 213-214), seems much more familiar to capital theory than to natural resource issues. And it echoes modern debates about the increase or decrease in natural capital -and total capital -in the context of sustainable development (see for instance [START_REF] Pearce | Capital theory and the measurement of sustainably development: an indicator of 'weak' sustainability[END_REF]. In that sense, Johnson's contribution seems to provide an accurate first use of the natural capital concept in its current meaning.

At the beginning of the 20th century, Johnson's distinction between artificial capital and natural capital was not perceived with much interest at first. In his review of Johnson's book, published in The Economic Bulletin in 1910, Thomas Nixon Carver, one of the main 1910s Conservationists (see [START_REF] Carver | Conservation of Human Resources[END_REF], indicates that the natural capital concept is indeed new, but does not provide any great theoretical gains, apart from creating new words: "An interesting development of doctrine is found in the author's conception of capital. [...] [he] squares himself with the more widely accepted view by dividing capital into two classes, natural and artificial, natural capital being nothing more nor less than that which is generally called "land and natural agents". There can be no possible objection to this procedure except that it increases the number of terms and adds nothing to our knowledge." (Carver, 1910, p. 22) This severe judgment might be interpreted in several ways. Perhaps Carver did not perceive the importance of such a shift, namely from natural resources to natural capital. This shift is not only a form of rhetoric, but also opens up the possibility of using analytical tools that would be placed out of scope by focusing on natural resources or agents (on that point, see Missemer, 2017, chapter 4). Perhaps Carver found Johnson's innovation unsurprising, and therefore not worth praising. This explanation is plausible, since Johnson's proposal did not come out of the blue (see below).

In the 1910s, Frank W. Taussig was one of the first influential economists who appropriated the natural capital concept. In his Principles of Economics (1911), he cites Johnson's Introduction to Economics, and shows no hesitation in using the concept, since it enables a better understanding of natural resource valuation. Taussig's work provides an interesting echo to modern controversy about the monetary valuation of natural agents:

"Pursuing this train of thought further, we might say that capital is of two kinds, artificial and natural. Natural capital is that which has been classed under the general head of "land" or "natural agents"; artificial capital includes all instruments made by man. Natural capital may be highly useful and highly valuable, as in the case of a rich mine or a deep-harbor site. [...] Their valuation is high; their capitalization indicates the existence of a large volume of capital." (Taussig, 1911, vol. 2, p. 118) In the 1920s, reading Taussig for his own research, Raymond T. Bye, professor of economics at the University of Pennsylvania, considers the expression "natural capital" almost clumsy (1925, p. 22), but he adopts it (p. 393). In subsequent years, the use of the natural capital concept continues to develop. In his review of Bye's book, Jen P. [START_REF] Jensen | Review of principles of economics[END_REF] argues that it is part of the "treatment and the conclusions [which] conform substantially to recent orthodox masters" (p. 158). Three years later, in a paper published in the American Economic Review, Clark Warburton places it in the category of "common economic terminology" (1928, p. 70-71). The natural capital concept therefore became progressively integrated into economic 92 -Author's post-print --published in Ecological Economics (2018), 143, 90-96 -7 theory in the first third of the 20th century. However, despite this relative success, the concept apparently stopped developing, and remained nearly unused, from the 1930s to the 1980s, when Pearce finally gave it a successful re-launch. Although this fall from grace might be explained by a shift in economic literature from conservation issues to macroeconomics and econometrics in the 1930s, it would need further investigation to provide a complete explanation.

Returning to Johnson's initial innovation, in addition to the fact that the natural capital concept was indeed coined in the 1900s, the question is: why it did appear at that time? And why in Johnson's work? To answer these questions, it is necessary to place Johnson's contribution in its historical and theoretical context, in order to explain why that first use of the natural capital concept took place at that moment, under those conditions.

Some Explanations for Johnson's Innovation

The end of the 19th century set the stage for profound changes in economic theory. A few years after the appearance of marginalism in the 1870s, a whole range of economic mechanisms were updated through a new theory of capital. On this point, the Austrian tradition played a significant role, starting with Menger's Grundsätze der Volkswirtschaftslehre (1871) in which commodities are distinguished according to their situation along the productive chain: consumer goods are considered as 1st-order goods, intermediate goods to produce consumer goods as 2nd-order goods, etc. and original factors of production (labour and land) as the highest-order goods 7 (1871, p. 56-58). This nomenclature is employed by [START_REF] Böhm-Bawerk | Capital and interest. A Critical History of Economical Theory[END_REF], 1889a, 1889b) when considering the 7 Menger uses this terminology to propose a new theory of value, called imputation theory, whereby the value of consumer goods, determined by supply and demand (and therefore by marginal utility), gives the value of 2nd-order goods, that determines the value of 3rd-order goods, and so on. The imputation theory runs in the opposite direction to the classical theory of value that starts with labour costs, and then continues down to consumer goods. definition of capital in economic theory. Since labour and land are part of the highest order of goods, the third standard factor of production, namely capital, must be a secondary factor in economic processes, different from highest-order goods (1889b, p. 105). As a consequence, capital is defined as goods that are produced to produce other goods: "[...] capital signifies a complex of produced means of acquisition -that is, a complex of goods that originate in a previous process of production, and are destined, not for immediate consumption, but to serve as means of acquiring further goods." (Böhm-Bawerk, 1884, p. 6) In Menger's terminology, this means that capital refers to all the goods of intermediate orders, between 1st-order goods and highestorder goods. This is an extensive definition of capital, because it includes heterogeneous goods that do not enter economic processes at the same stage. However, Böhm-Bawerk stresses a common characteristic of all capital goods: they all constitute what he names "roundabout ways" of producing (1889a, p. 18). This famous Austrian definition of capital makes a link between the use of capital goods and productivity: the longer the detour, the more numerous are the capital goods, and the higher is the productivity of the final production process (on this point, see [START_REF] Buechner | Roundaboutness and Productivity in Böhm-Bawerk[END_REF].

To respond to these theoretical innovations, American economists proposed their own definitions of capital. Merging all these different concepts in a single phrase has no sense, but two authors deserve attention since they actively participated in the constitution of an American economic tradition, debating with Austrian authors (see Tobin, 1985, p. 28), and because they explicitly influenced Johnson's own vision of capital: John Bates Clark and Irving Fisher.

John Bates [START_REF] Clark | Capital and its earnings[END_REF][START_REF] Clark | Distribution as determined by a law of rent[END_REF][START_REF] Clark | The Distribution of Wealth[END_REF] is well known for having introduced marginal productivity into economic theory. Taking inspiration from the Ricardian rent differential principle, he explains that marginal differences in productive efficiency can determine the payment gaps between capital goods. In other words, the Ricardian principle is generalisable to all factors of production:

-Author's post-print --published in Ecological Economics (2018), 143, 90-96 -8 "Labor and capital, in current theories, are the antithesis of the typical rent-producer, land. Yet wages in the aggregate constitute the income derived by society from its entire fund of pure labor energy; and interest is, in like manner, the product of a fund of pure capital. Both are differential gains, and completely amenable to the Ricardian law." (Clark, 1891, p. 300) Unlike Böhm-Bawerk, Clark considers that the productivity of capital does not depend mainly on the duration of production periods, but on the quantity of capital already in existence (1899, p. 139). His allusion to the law of diminishing returns, through the Ricardian rent differential principle, is stressed here. And it results in a very different theory of interest in Clark's scheme compared to that of the Austrians'. 8 With regard to the classical definition of capital -as an instrument of production, Clark asserts that it causes confusion, placing "tools", "raw materials" and "food for labourers" in the same group (1899, p. 123), whereas they are intrinsically different. To overcome this confusion, he states that capital takes two forms: "capital goods" and the "fund of capital" (1888, p. 9; 1899, p. 117). The former provides the best definition: "capital consists of instruments of production [that] are always concrete and material" (1899, p. 117). This differs from the classical economists' concept inasmuch as "acquired abilities of workmen" (p. 116) are not part of capital. The fund of capital consists of the permanent, abstract and monetary form of capital, and capital goods are its concrete, temporary and physical forms. Ever since this period, this duality has been a permanent feature of controversies about capital in the history of economic thought (see [START_REF] Cohen | Introduction. Capital theory controversy: scarcity, production, equilibrium and time[END_REF]. Beyond Clark's various innovations, it is noteworthy that he uses a wide yet plural definition of capital, not including food or raw materials, but covering productive tools and machinery, and also productive wealth in general (through the "fund of capital"). Johnson's view of natural capital is not far removed from such a conception: since natural agents constitute productive wealth, they can be included in capital. There is no explicit textual evidence of the shift from Clark's wide view of capital to Johnson's natural capital concept, but what is sure is that Clark exerted a great influence on Johnson, since the latter was a student, then a friend, of the former. In his autobiography, [START_REF] Johnson | Pioneer's progress. An Autobiography[END_REF] writes that he felt "unreserved admiration and loyalty" for his teacher when he was young (p. 123), and that Clark was "the greatest of American theorists" of his time (p. 153). In the prefaces of his early works, he already noted Clark's influence on his own writings (Johnson 1902, p. v;1907, p. iv-v;1909, p. v), and this influence was clearly perceptible through all his arguments, as reported by McCrea (1909, p. 328). Johnson's own choice of an extensive definition of capital, finally arriving at the natural capital concept, is therefore not surprising.

In the controversies about capital at the turn of the 20th century, Irving [START_REF] Fisher | What is capital?[END_REF]Fisher ( , 1897aFisher ( , 1897b[START_REF] Fisher | The Nature of Capital and Income[END_REF] also played an important part, challenging Clark's conception.9 According to Fisher, defining capital with concrete or abstract forms does not work, since capital has many features and cannot be correctly understood in this way. He argues that it is better to define capital as a mere stock of wealth -i.e. as a stock of "material objects owned by human beings" (1906, p. 3). Following Cannan's (1903, p. 14) proposal, Fisher gives an even more extensive definition of capital than Clark. All material objects that are owned at a moment in time are considered to be capital, whatever their intrinsic nature. There is no more connection here with production processes, and that is partly why Fisher and Clark do not share the same definition of capital. Despite his filiation to Clark, Johnson did not completely ignore Fisher's work when developing his own view. In the preface to his 1909 Introduction to Economics, he praises Fisher's writings on "rent and capitalisation" (p. v).

Paving the way for the natural capital concept, Fisher's definition of capital leads to a new conception of natural resources in economic theory. Since natural resources are material objects that can be possessed, they are part of wealth. At a given moment in time, a stock of such resources can then be considered as capital (Fisher 1896, p. 534;1897a, p. 199). On the same grounds, the usual boundaries between economic terms become blurred. In his 1906 book, Fisher suggests that the words "resources" and "assets" are synonymous. He does not specifically talk about natural resources, but, retrospectively, such a semantic convergence might be relevant, in particular as regards Pearce's 1988 definition of natural capital as the "set of all environmental assets" (p. 599). Johnson used the legacy of both Clark's and Fisher's views of capital. He had in mind an extensive definition, with a potential connection to the notion of wealth. In some ways, he synthesised Fisher's matching of resources and assets and Clark's attachment to production processes. As a consequence, capital could be seen as a productive stock of wealth, and natural agents could form part of this definition. The natural capital concept had only to be coined.

In his 1909 contribution, Johnson does not explicitly indicate why natural capital is a meaningful notion, but the present article helps to explain that all the possible conditions, including the Austrian controversy about capital and its aftermath in America, had been met at the time he wrote his textbook. He was the right man at the right moment.

Conclusion: Past and Present

As already mentioned, in the middle of the 1990s, the natural capital concept was the subject of recurrent debates concerning its correct use in economic theory. Of course, this concept does not belong to any school of thought. And it can be interpreted in various ways. Yet this controversy raises the following questions: Does the natural capital concept stand at the core, or at the periphery, of economics? In other words, is it a challenging concept as Pearce, Berkes, Folke and others consider? Or is it the artefact of a monetary perspective applied to environmental issues? These questions can be answered in different ways. The historical inquiry conducted in the preceding sections sheds new light on these issues.

Obviously, Johnson's natural capital and today's natural capital do not cover exactly the same range of environmental items. Johnson focused on natural "agents" -i.e. resources and forces. Today, ecosystem services are included; but Johnson did not consider them. However, convergences still exist, since, in both cases, the natural capital concept is a way of encapsulating natural items in a stock of wealth managed by human beings. And both the old and current terms "natural capital" include the idea of something having a productive nature.

Exploring what happened to the natural capital concept in the 1910s and 1920s, after Johnson's initial impetus, gives some insight into present controversies. The natural capital concept did not experience the same long-term success as today, probably because of differences in political and scientific agendas; but when it was employed, it was not at the periphery of economic theory, but at the core. [START_REF] Taussig | Principles of Economics[END_REF] was one of the main American economists of the first third of the 20th century. [START_REF] Bye | Principles of Economics[END_REF] book was a canonical neoclassical textbook for many American students, and it was republished several times between the 1920s and the 1950s. As [START_REF] Jensen | Review of principles of economics[END_REF] and [START_REF] Warburton | Economic terminology: factors of production and distributive shares[END_REF] underlined, the natural capital concept had become almost a mainstream expression, rather than a revolutionary term.

Going back to the term's origins, Johnson himself was not a revolutionary author. He had been the student of one of the most influential neoclassical economists of his time, John Bates Clark. And Johnson's, 1909 textbook, in which the natural capital concept appeared, was considered as one of the most useful textbooks of the 1900s for all university students, merging Clark's, Marshall's, and Austrian teachings (see Carver, 1910, p. 22-23). Once again, the early history of the natural capital concept did not take place at the periphery of economics, but at its core. The emergence of the concept was the consequence of an extensive definition of capital, initiated by Clark and Fisher, who were -Author's post-print --published in Ecological Economics (2018), 143, 90-96 -10 themselves far from the margins of economic theory. Capital was seen as a stock of wealth, with a monetary value. Natural capital was probably perceived in the same way, insofar as Johnson talks about "measuring the capital represented by these natural resources in terms of value " (1909, p. 199), and about potential "revaluation of natural capital goods" (p. 214).

Bearing this narrative in mind, it should not be surprising that a monetary-valuation shift has been observed in the use of the natural capital concept since the middle of the 1990s. To a certain extent, it has been part of the concept itself since the beginning, though its rebirth in the 1980s occurred with no reference to those ancient origins. As a result, historically speaking, perhaps the ecological economists have been bolder, in wanting to give a strong, challenging interpretation to the natural capital concept, whereas mainstream economists have opted for a monetary view that is, in the end, more consistent with the deep roots of the concept. But, hopefully, the history of economic thought does not forbid new uses and conceptions that are as legitimate as old ones; it simply helps to understand all that surrounds current issues.

-

  s post-print --published in Ecological Economics (2018), 143, 90-96 -6

  Åkerman's work by paying tribute to Johnson for his neglected role in the history of environmental economics, and (ii) to build on that ancient historical moment to give a new interpretation of contemporary debates.

	-Author's post-print -
	-published in Ecological Economics (2018), 143, 90-96 -
	3

As Nadal (2016, p.65) states, the concept of natural capital already existed in the 1970s, for instance in[START_REF] Schumacher | Small is Beautiful. Economics as if People Mattered[END_REF], but with no clear definition.

It may be noted that Say's definition of the "productive service" provided by "natural agents"(Say, 1814, p. 28) can be related to modern conceptions of nature (see[START_REF] Boisvert | Les services écosystémiques: un nouveau concept?[END_REF], and Cournot sometimes alludes to potential connections

Clark emphasises the fact that the rate of interest depends on the marginal product of capital, while the Austrian economists focus on discounting the future (see[START_REF] Rebeyrol | Répartition, intérêt et théorie du capital[END_REF]. On other points of difference between Clark and Böhm-Bawerk, see[START_REF] Cohen | The mythology of capital or of static equilibrium? The Böhm-Bawerk/Clark controversy[END_REF]

For more on the place of Fisher's conception of capital in the history of economic thought, see[START_REF] Meacci | Irving fisher and the classics on the notion of capital: upheaval and continuity in economic thought[END_REF]. 94

Acknowledgements

A previous version of this paper was presented at CIRED in November 2015. I thank the participants of this seminar, in particular Philippe Quirion, for their comments. I also thank Sylvère Mateos for assistance in some bibliographical matters. Finally, I am grateful to the editors and reviewers of the journal who gave me sound directions for finishing the article.

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Tania S Vergara Gomez

Marc Dubois

Stanislav Glybovski

Benoit Larrat

Julien De Rosny

Carsten Rockstuhl

Monique Bernard

Redha Abdeddaim
email: redha.abdeddaim@fresnel.fr

Stefan Enoch

Frank Kober

Vergara Gomez

Benoît Larrat

Wireless coils based on resonant and nonresonant coupled-wire structure for small animal multinuclear imaging

Keywords: RF receive coils < Metasurface, RF coils, 19 F MRI, computational electromagnetics, whole-body imaging, metamaterials ABBREVIATIONS SNR: signal to noise ratio VNA: vector network analyzer FOV: field of view fast low-angle shot rapid imaging with refocused

Volume coils such as birdcage coils [START_REF] Hayes | An efficient, highly homogeneous radiofrequency coil for whole-body NMR imaging at 1.5 T[END_REF][START_REF] Tropp | The theory of the bird-cage resonator[END_REF] provide uniform B 1 magnetic field in large volumes.

However, when a birdcage is used for both transmission and reception of the RF field, it provides relatively low sensitivity and therefore poor SNR [START_REF] Doty | Radio frequency coil technology for small-animal MRI[END_REF] due to generally suboptimal volume filling.

On the other hand, small surface coils usually implemented as flat copper loops, due to their superior sensitivity, can provide high SNR [START_REF] Keltner | Electromagnetic fields of surface coil in vivo NMR at high frequencies[END_REF], but with the main drawback that the B 1 magnetic field decays in depth as 1 /

, where R is the radius of the loop and d is the distance from the loop 𝑅 2 + 𝑑 2 plane [START_REF] Hoult | The NMR receiver: a description and analysis of design[END_REF]. Consequently, the high efficiency of small surface coils is only available in a small area close to the coil. The currently most effective and established solution is the use of surface coil arrays to extend the in-plane coverage while keeping high SNR [START_REF] Roemer | The NMR phased array[END_REF]. However, surface arrays are costlier, require multi-channel drivers and they are in most cases only used for signal reception. They also require specific element-wise signal recombination procedures [START_REF] Fujita | New horizons in MR technology: RF coil designs and trends[END_REF], which are not trivial when phase-sensitive information is to be kept in the final image or spectrum. Finally, phased arrays are also more complicated to use for RF transmission, especially for localized spectroscopy [START_REF] Ohliger | An introduction to coil array design for parallel MRI[END_REF].

Recently, metamaterial-inspired RF coils have been developed to bring more flexibility into the design of MRI coils for preclinical imaging. First, a volume coil based on hybridized wires has been proposed [START_REF] Jouvaud | Volume coil based on hybridized resonators for magnetic resonance imaging[END_REF]. Later on, metasurfaces based on a parallel array of inductively coupled wires were explored. These wire arrays also rely on the hybridization mechanism [START_REF] Dubois | Kerker Effect in Ultrahigh-Field Magnetic Resonance Imaging[END_REF] in order to obtain a homogeneous B 1 magnetic field. To do so, the wires used need to be either of a resonant length [START_REF] Dubois | Kerker Effect in Ultrahigh-Field Magnetic Resonance Imaging[END_REF] or miniaturized by high-permittivity loading [START_REF] Shchelokova | Locally enhanced image quality with tunable hybrid metasurfaces[END_REF], or by capacitive interconnections [START_REF] Glybovski | Capacitively-loaded metasurfaces and their application in magnetic resonance imaging[END_REF]. These studies demonstrated the possibility of using such metasurface structures to achieve a dual-frequency coil for proton and fluorine imaging [START_REF] Hurshkainen | A novel metamaterial-inspired RF-coil for preclinical dual-nuclei MRI[END_REF] and a coil for whole-body imaging with higher SNR than a volume coil [START_REF] Zubkov | Smallanimal, whole-body imaging with metamaterial-inspired RF coil[END_REF].

It has been shown that the optimal impedance matching of such metasurface-based coils is obtained with a non-resonant feed loop when one of the metasurface resonances coincides with the targeted Larmor frequency. This configuration offers the best compromise between SNR and large sensitive volume [START_REF] Zubkov | Smallanimal, whole-body imaging with metamaterial-inspired RF coil[END_REF]. However, the highest local SNR was still obtained with a small matched loop coil [START_REF] Hoult | The NMR receiver: a description and analysis of design[END_REF].

The objective of this work was to explore whether advantages can be obtained by combining both strategies. Due to mutual coupling, however, the simple combination of a resonant metasurface with a small matched loop would lead to shift of both resonances away from the original frequency [START_REF] Mispelter | NMR probeheads for biophysical and biomedical experiments: theoretical principles & practical guidelines[END_REF], and a new approach had to be developed. To assess the mutual coupling issue, we simulated the current amplitude and phase in both the loop and the coupled-wire structure.

For this purpose, we designed, simulated and built an elementary structure with two parallel wires of the same length [START_REF] Glybovski | Metasurfaces: From microwaves to visible[END_REF]. This is indeed the minimum number of wires allowing to excite a mode with currents flowing in opposite directions in the wires. This mode behaves similarly to a long loop and allows a large coverage in the plane of the structure. The structure was studied numerically and experimentally on a 7 T preclinical scanner to verify its potential advantages over existing solutions.

Namely, we hypothesized that beyond their use in a resonant regime, coupled-wire structures can be used off resonance and can be combined with a commercial surface coil instead of a non-resonant feed loop to achieve a significant improvement of sensitive volume while maintaining a high efficiency over the surface coil area. We further explored feasibility of 19 F MRI with the same structure on a phantom and in mice in vivo.

MATERIALS AND METHODS

Numerical Study

Three configurations were numerically studied (Figure 1) using CST Microwave Studio 2017 (Computer Simulation Technology GmbH, Darmstadt, Germany): (case A) a surface coil tuned and matched with lumped capacitors, (case B) a resonant coupled-wire coil with two wires of length L 0 coupled to a non-matched feed loop and (case C) a detuned coupled-wire coil with variable wire length L combined with a matched loop.

All results were obtained at a frequency of 300 MHz (Larmor frequency of 1 H at 7 T). In all cases, the loop had a diameter of 3 cm, and the input power used was 0.5 W. In the coupled-wire based coils, the wires were 3 cm apart, parallel to the MRI bore axis, and the loop used was placed above the wires at a distance of 1 mm and centered at the median position of the wire length. The material used for wires and loop was copper. Simulations were carried out without phantom and without RF shield.

In the first case (Figure 1A), tuning and matching of the surface coil was achieved with a matching network. For the second case (Figure 1B), impedance matching was obtained by modifying the distance between the loop and the wire plane. The resonance of the coupled-wire structure was tuned to 300 MHz by adjusting the wire length to L 0 , close to a half wavelength (≈50 cm). The last case (Figure 1C) required both surface coil and coupled-wires to be detuned from the Larmor frequency to correctly tune the whole coil structure. Different wire lengths were considered, while keeping L < L 0 to preserve the coupled-wires in a non-resonant regime. For each length, the capacitor values of the surface coil's matching circuit were adapted to reach impedance matching at the Larmor frequency. The current was simulated in all cases as a function of wire length (Figure 1D). Finally, the spatial B 1 amplitude profile was obtained for each case along the vertical axis in the center of the structure (Figure 1E) and along the wire length at 1 cm above them (Figure 1F).

To optimize the wire length and separation as well as analyzing the reflection coefficient (|S11|) for both frequencies (Figure 3), further simulations were carried out in the presence of a homogeneous phantom of 35x35x70 mm 3 with a relative dielectric permittivity (ε) of 76 and a conductivity (tan δ) of 0.98 S/m. The RF shield was simulated using a copper tube with 1000 mm length and 100 mm internal diameter.

Coils setups

Three different coils setups were compared as single channel transmit-receive antenna for proton and fluorine imaging: (case A) a 30 mm diameter surface coil (Bruker 1 H-19 F, model 1P T957 8V), (case B) the resonant coupled-wire coil and (case C) the non-resonant coupled-wire coil. A 70 mm diameter commercial 1 H birdcage coil (Bruker Biospin) was included in the tests as an additional reference, but only allowed proton imaging.

A prototype of the simulated coupled-wire structure was built using two telescopic brass tubes [START_REF] Shchelokova | Volumetric wireless coil based on periodically coupled split-loop resonators for clinical wrist imaging[END_REF].

The wires were separated by 3 cm and placed on a plastic board of 2 mm thickness as shown in Figure 2D. Their length (L) was adjustable to tune the resonance to the desired frequency. Both wires had the same length at the end of each tuning procedure.

To build the resonant coupled-wire coil (case B), an unmatched copper loop of 3 cm diameter was printed on a 0.5 mm thick FR-4 circuit board which had an ε = 4.4 and a tan δ = 0.02 S/m (Figure 2B).

The loop was fed by a coaxial cable. The matching of this coil relied on the symmetry and the distance between loop and wires. Tuning for both frequencies (300 MHz for 1 H or 282.6 MHz for 19 F) was achieved by adjusting the length of the wires to L 0 at which the coupled-wires resonate at the Larmor frequency.

To reproduce the non-resonant coupled-wire coil (case C), a commercial surface coil of 3 cm diameter (Bruker 1 H-19 F, model 1P T957 8V) (Figure 2A) was used in combination with the described coupled-wire structure. Since the commercial surface coil allowed frequency-switching from 1 H to 19 F, this coil's matching circuit could be used for final tuning and matching at both frequencies. This only required an adjustment of the wire length L such that L < L 0 to allow the redistribution of currents in the coil structure. On-bench measurements of the reflection coefficient were done only for the resonant coupled-wire coil (case B) using a vector network analyzer (VNA MS2036C, Anritsu, Kanagawa, Japan) to ensure proper tuning and impedance matching for both frequencies. On-bench measurement could not be performed for the other configurations as they could not be interfaced with the VNA.

Phantom experiments

Validation of the experimental coils was done using a PharmaScan 7T MR system (Bruker Biospin, Ettlingen Germany) running ParaVision 6.0.1 software. The phantom used contained a small volume fraction of 2-2-trifluoroethanol leading to a concentration of 19 F nuclei on the order of 100 mmol/L (Figure 2C). The phantom was placed horizontally, parallel to the MRI bore axis, on the center of the corresponding coil.

For every tested coil, a FLASH localizer sequence was used after wobble, iterative shimming and frequency adjustment. Using the coronal slice from the localizer series, the reference power was adjusted automatically (Spin Echo/stimulated echo comparison/iteration) in a slab positioned directly adjacent to the coil surface, except for the birdcage coil, for which the reference power was adjusted without volume selection.

Single slice FLASH sequences were run in the three orientations to obtain 1 H images of the phantom. The parameters of the sequence were: TE/TR = 2.54/500 ms, number of averages (NA) = 1, flip angle α = 60°, slice thickness = 0.9 mm, in-plane spatial resolution = 0.7 x 0.7 mm 2 and field of view (FOV) = 9 x 9 cm 2 with a total acquisition time of 1 min 4 sec. SNR maps were obtained from the proton images using MATLAB (MathWorks, Natick, MA, USA). Noise was calculated as the standard deviation of pixel values in a region of interest (ROI) in the lower left corner of the image where residual signal is absent. The SNR maps were calculated as the ratio of each voxel signal over the noise value of the image.

After tuning to the 19 F frequency, a single pulse FID sequence (TR = 3000 ms, spectral bandwidth = 7 kHz, number of points = 2048, NA = 4) was used to obtain the exact resonance frequency of 2-2trifluoroethanol (offset -21636Hz). 19 F images were acquired with a T 2 -TurboRARE-3D sequence with effective TE = 68.22 ms, TR = 3000 ms, RARE factor = 32, NA = 1, slice thickness = 40 mm, matrix = 64 x 64 x 32 (in-plane spatial resolution = 1 x 1 mm 2 ), FOV = 6 x 6 cm 2 and a total acquisition time of 20 minutes. Fluorine images were processed in MATLAB (MathWorks, Natick, MA, USA) to overlay them on 1 H FLASH images using a transparency of 0.6, where 0 is completely transparent and 1 is opaque.

In vivo MRI experiments

In vivo validation was performed using the same coils setups. All experiments were conducted according to a protocol approved by the University's animal experimentation committee (APAFIS#10547-2017071009112930 v4). Three C57Bl6 mice (age 9 weeks) were used for this in vivo demonstration. Mice were anesthetized with 3% isoflurane in an induction chamber. The animals were held under anesthesia using 1.5-1.7% isoflurane in a stream of 1 L/min room air administered through a face mask. They were placed in prone position with the abdominal region in the center of the corresponding coil (Figure 2E). Great care was taken to reproduce the same position across the different coil setups tested. Post-processing of the 1 H and 19 F images was similar to the phantom study.

The 2D FLASH sequence used for proton imaging had the following parameters: TE/TR = 2.23/50 ms, NA = 8, α = 30°, slice thickness: 0.9 mm, in-plane spatial resolution = 0.23 x 0.23 mm 2 , FOV = 6 x 6 cm 2 and a total acquisition time of 2 minutes.

Despite negligible natural endogenous fluorine concentrations in the body [START_REF] Ruiz-Cabello | Fluorine ( 19 F) MRS and MRI in biomedicine[END_REF], 19 F imaging of the mouse abdomen in vivo was possible by using the signal of isoflurane, used here as anesthetic.

Isoflurane is indeed known to accumulate in certain lipid compartments in anesthetized animals as shown in a recent study [19]. 19 F imaging was done using a FLASH sequence with TE/TR = 2.23/800 ms, NA = 32, α = 90°, slice thickness = 3.9 mm, in-plane spatial resolution = 1.8 x 1.8 x 3.9 mm 3 , frequency offset = -23537 Hz, FOV = 6 x 6 cm 2 and a total acquisition time of 13 minutes.

RESULTS

Numerical study

In figure 1D, the maximum current in the loop (red curve) and in the coupled-wire structure (blue curve) are plotted. The current in the loop was taken at the feed point as its amplitude remains almost constant due to the loop's small diameter. We monitored the wire currents in the center of the wire length where the current amplitude was maximum. It can be seen that case (B), were the resonance of the coil structure was due to the coupled-wire only, featured an extremely low current in the loop, which continuously increased when moving to a configuration where there are no wires, similar to case (A). With the wires getting shorter (L < L 0 ), the resonance shifted. Note that this change was compensated for each value of L by adjusting the capacity in the matching circuit at the loop's input.

Nonetheless, the maximum current in the wires decayed continuously as the wire length was reduced. maximal for case (A) and minimal for case (B). On the other hand, the horizontal extension of the FOV was linked to the current amplitude in the wires. The non-resonant behavior, case (C), shown here for a wire length of 45 cm allowed us to obtain simultaneously a large sensitivity near the loop and to still benefit from the increased B 1 extent from the wires.

Coils setups

According to the simulations with phantom and RF shield, the reflection coefficient |S11| was lower than -20 dB in all cases for both frequencies (300 MHz for 1 H and 282.6 MHz for 19 F) as shown in Figure 3.

With the VNA, it was found that the optimal length L 0 of the wires in case (B) was 48.0 cm for 1 H (300 MHz) and 50.5 cm for 19 F (282.6 MHz), which in both cases corresponds approximately to a half wavelength in air.

In cases (A) and (C), the commercial surface coil could not be directly connected to the VNA and therefore tuning could only be done in the scanner. For case (C) a wire length of 39.0 cm was used for both nuclei, 19 F and 1 H, as the limit of tuning/matching circuit capabilities of the Bruker surface coil was reached. Nevertheless, tuning and matching of the assembled coil was good enough to perform MRI experiments.

Phantom MRI experiments

Table 1 shows the reference power values (power required for reaching a flip angle of 90° with a 1 ms pulse) obtained on the scanner for each coil with the phantom located in the center of the coil (center of wires and loop aligned). As expected, the birdcage coil required higher power than the other coils (3.89 W). The surface coil required the lowest power (0.14 W). Note that in case (C) the reference power (0.22 W) was comparable to that of the surface coil alone. SNR maps and profiles for every coil tested with the phantom are presented in Figure 4. Figures 4I-K show that there was almost a five-fold ratio in SNR between birdcage coil/case (B) and surface coil/ case (C). The color scale has been adapted for each group, and the images have been grouped in boxes of different colors, blue for low SNR and orange for high SNR. As expected, the birdcage coil produced the lowest SNR, although it was the most homogeneous in all orientations. Case (B) provided a large imaging FOV, comparable to that of the birdcage coil but around 1.5 times higher SNR in shallow coronal slice (Figure 4B). However, Figure 4J shows that the SNR gain, as expected, was only present for a shallow depth (1.5 cm). For case (C) (Figure 4 4.5 times was encountered at the loop location compared to case (B), as predicted by the numerical analysis above. When compared to the surface coil alone, case (C) presents a reduction of SNR of 20%. However, we observed that the FOV was enlarged in the coronal-sagittal profile (Figure 4I) giving access to areas invisible with the surface coil. At the same time, the presence of the wires in case (C) yielded a much more homogeneous SNR distribution in the axial-coronal profile (Figure 4K). 19 F images are presented in Figure 5. We show that the coils present the same SNR distributions as for proton imaging with the only difference that the signal was much lower due to the weak concentration of 19 F nuclei in the phantom. Nonetheless, the figure confirms that the SNR provided by the two coupled-wire coils (cases B and C) was enough to produce a 3D 19 F image from a diluted phantom within 20 minutes. For fluorine, it was found that the Larmor frequency of 19 F nuclei in isoflurane was 282.577 MHz.

In vivo MRI experiments

The images are shown in Figure 7. We can see that case (C) and the surface coil provided enough sensitivity to enable a clear localization of isoflurane with the given scan parameters (Figure 7B,C).

For case (B), the measured signal was comparable to the noise level making the analysis more difficult with the given scan parameters (Figure 7A).

DISCUSSION

Numerical studies of the sensitivity of coupled-wire coils show that when a resonant length (case previously considered in the literature) is used (case B), the coil performance is dominated by the wires. In this case, the current in the loop is extremely low, and, consequently a low amplitude B 1 field is generated. On the contrary, the current along the wires is almost constant leading to a wide FOV [20]. This explains why B 1 remains almost constant along the wires as shown in Figure 1F. A simple approach to benefit from the loop's and the wire's maximum contributions would be to place the resonant coupled-wire structure on top of a loop matched with lumped capacitors. However, this option would not actually work since the interaction of the two resonant parts would inevitably affect both resonances due to strong inductive coupling. This usually leads to two hybrid resonances which frequencies are shifted away from the initially targeted frequency [START_REF] Mett | MRI surface-coil pair with strong inductive coupling[END_REF]. Our results show that we were able to combine the coupled-wire and the matched loop by tuning the wires resonance away from the Larmor frequency together with a small adjustment in the surface coil's matching circuit.

Using a non-resonant coupled-wire structure with adjustable wire length coupled to a matched loop (case C), a strong increase in the loop current could be clearly observed. The main benefit is that the length of the wires can be adapted to rearrange the distribution of currents within the different parts of the coil (loop and wires). The fact that the current in the wires dropped slowly as the wire length became shorter (blue curve in Figure 1D) allowed us to benefit from both contributions (wire and loop) when using such a configuration. This benefit can be observed more clearly in the B 1 profile of case (C) with a wire length of 45.0 cm (simulation size), shown in Figure 1E andF, where a strong increase in sensitivity at the loop position can be observed, while maintaining 75 % of the sensitivity along the wires on average. This redistribution of currents has a significant impact on the sensitivity and the field of view that can be achieved by the coil. It gives the ability to adapt the coil in terms of sensitivity and accessible volume, according to the targeted MRI application. Indeed, this new configuration was able to combine the advantages of both coupled-wire based coil with large coverage and surface coils. It may become an interesting alternative to conventional MR coil designs.

In order to explore such a concept experimentally, two parallel coupled-wire structure was designed, and a prototype was built to compare the properties in resonant and non-resonant mode with those of a standard birdcage coil and a matched surface coil. For the sake of simplicity, we chose a two-wire configuration rather than multiple wires (i.e. more than 2). Indeed, two wires is the minimum number allowing to excite the surface mode with a large coverage in the plane of the structure which behaves similarly to a large loop. More than two wires produce higher-order mode which are out of the scope of the present work. The results of the experimental study demonstrate the flexibility of the coupled-wire based coils in terms of FOV and SNR. Namely, Figures 4A-B show that case (B) yielded higher SNR than the birdcage coil, although only in the first 1.5 cm depth in the normal direction from the wires plane (Figure 4J). This depth may, however, be sufficient to obtain a satisfactory whole-body image of a mouse as shown in the in vivo experiments.

With case (C) it was possible to achieve high SNR at the location of the loop, with only 20% loss compared with the surface coil alone. Simultaneously, we can see that the wires contributed to enlarge the accessible volume of the surface coil by 25% in sagittal section (Figure 4I) and its SNR homogeneity in the coronal section (Figure 4K). Equivalent performances were obtained when imaging 19 F nuclei contained within the phantom.

The experimental results perfectly corroborate the numerical predictions from Figure 1. We demonstrated here that the coupled-wire structure can be used in both resonant and non-resonant regimes depending on the requirements of the preclinical MRI study. Moreover, the comparison of reference power shown in Table 1 directly illustrates the strong gain obtained in terms of efficacy of the coil. This is interesting for applications that have stronger RF power requirements such as sequences involving inversion [START_REF] Weigel | Inversion recovery prepared turbo spin echo sequences with reduced SAR using smooth transitions between pseudo steady states[END_REF] or labeling pulses [START_REF] Alsop | Recommended implementation of arterial spin-labeled perfusion MRI for clinical applications: A consensus of the ISMRM perfusion study group and the European consortium for ASL in dementia[END_REF]. Finally, it is also important to stress out that the noise measured in presence of the wires was almost identical for both configurations, as presented in Table 1.

To verify the consistency of results in in vivo situations, proton and fluorine images were acquired from the abdomen of three mice in identical configurations as those used for the phantom scans and simulations. The in vivo results demonstrate that the suggested coils can indeed be used for preclinical research and comply with standard equipment requirements. The enlargement of the FOV resulting from the case (C) can be better seen in the in vivo images than in the phantom, especially when comparing Figure 6G that shows mainly abdominal organs and Figure 6H where heart, lungs and lower abdominal regions can be clearly seen. The in vivo fluorine images of the two cases (Figures 7A,C) show that both were able to detect 19 F within the entire mouse body, whereas the surface coil is limited to 19 F present in the abdomen. However, due to very low SNR, localization of the signal was difficult with the case (B) (Figure 7A).

In conclusion, we showed that coupled-wire structures can be used as parts of RF coils in both resonant and non-resonant regimes. Each of these regimes led to different current amplitude distributions within the different parts of the coil. The experiments obtained on phantom and living mice confirmed the numerical predictions on the sensitivity of the coupled-wire structure. We further showed that the coupled-wire based coils provided enough SNR to obtain 19 F images from isoflurane accumulated in the mouse body in vivo with the chosen sequence parameters. The performance of the coupled-wire based coils was not affected by the Larmor frequency difference. We believe that such structures bring a new alternative to the conventional choice between volume and surface coils. The concept of combining a carefully detuned coupled-wire structure and surface coils may help in the design of future versatile RF coils for preclinical MRI. It is worth noting that using the proposed approach, a commercial surface coil can be modified by expanding the FOV and keeping almost the same SNR in the original area. Here the expanded region followed directly the shape of the wires. Moreover, our approach could be generalized to more complex metasurface designs including larger number of wires and different geometries.

AC K N O W L E D G E M E N T S

INTRODUCTION

  In clinical and preclinical magnetic resonance imaging (MRI), the commonly used radiofrequency (RF) coils belong to one out of two categories: volume and surface coils.

Figure

  Figure1Eshows that the maximum B 1 magnitude depends mainly on the loop current, it was

  D,H), a large enhancement of the SNR of http://mc.manuscriptcentral.com/nbm NMR in Biomedicine -For Peer Review Only

Figure 6

 6 Figure 6 presents the results obtained for the mouse proton MRI experiments. The behavior of the different coil setups remains consistent with the previous phantom study. Like in the phantom study, we can see that the birdcage coil has low and homogeneous SNR. Case (B) yielded higher SNR near the wires than the birdcage coil. The surface coil provided the highest SNR but the smallest FOV compared with case (C). Indeed, case (C) allowed imaging of organs away from the loop thanks to the wire contribution. At the same time, the SNR values were preserved close to the loop in the center of the FOV.

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Figure 1 :

 1 Figure 1: Numerical analysis of the three configurations: Sketch of (A) surface coil with a matching circuit, (B) resonant coupled-wire coil conformed by two wires with length L 0 (49 cm) coupled with an unmatched feed loop and (C) non-resonant coupled-wire coil that combines a matched surface coil with two wires of length L smaller than the resonant length L 0 . The diameter of the loop in all cases is 3 cm and the wires are 3 cm apart. (D) shows the plot of the current amplitudes in the loop (red) and in the wires (blue) with the different configurations. In each of the configurations the coil was tuned and matched to 300 MHz, and 0.5 W of input power was used. (E) shows the B 1 profiles obtained along a vertical line through the center of the coil and (F) is the B 1 evaluated on a horizontal line along the wires at 1 cm above the wire plane.

Figure 2 :

 2 Figure 2: Coupled-wire inspired coil for 7 T small animal imaging: (A) picture of the commercial surface coil (Bruker, model 1P T957 8V); (B) picture of the printed loop of 3 cm diameter on circuit board (FR-4, 0.5 mm thickness) and fed through a coaxial cable; (C) Bruker 19 F/ 1 H phantom of 3.5x3.5x9 cm 3 used during on-bench measurements and MRI scans, concentration of 19 F of 100 mmol/L; (D) picture of the coupled-wire structure with 2 telescopic wires separated by 3 cm and placed on a plastic board; (E) picture of the in vivo experimental setup.

Figure 3 :

 3 Figure 3: |S11| simulation and on-bench measurements for the case (B) and |S11| simulations for the surface coil and the case (C) ( 1 H and 19 F Larmor frequencies).

Figure 4 :

 4 Figure 4: Coronal and sagittal SNR maps in gray scale (dash lines locate the profile cuts). (A),(E) birdcage coil; (B),(F) case B; (C),(G) surface coil; (D),(H) case C. The cyan line gives the coronalsagittal profiles (I); the red line gives the sagittal-axial profiles (J) and the yellow line gives the axialcoronal profiles (K). Noise standard deviation is given on top of each map.

Figure 5 :

 5 Figure 5: 19 F T 2 -turboRARE-3D image in jet color map acquired with: (A) case B ; (B) surface coil and (C) case C in sagittal orientation. The images were overlaid on top of 1 H FLASH images in gray scale.

Figure 6 : 17 (

 617 Figure 6: In vivo 1 H SNR maps. Sagittal and coronal-plane images with: (A), (E) birdcage coil; (B),

Figure 7 :Figure 1 :Figure 2 :Figure 3 :

 7123 Figure 7: In vivo 19 F FLASH coronal-plane images in jet scale obtained with: (A) case B; (B) surface coil and (C) case C. The images were overlaid on top of 1 H FLASH images in gray scale.

Figure 4 :Figure 5 :

 45 Figure 4: Coronal and sagittal SNR maps in gray scale (dash lines locate the profile cuts). (A),(E) birdcage coil; (B),(F) case B; (C),(G) surface coil; (D),(H) case C. The cyan line gives the coronal-sagittal profiles (I); the red line gives the sagittal-axial profiles (J) and the yellow line gives the axial-coronal profiles (K).Noise standard deviation is given on top of each map.160x115mm (300 x 300 DPI)

Figure 6 :Figure 7 :

 67 Figure 6: In vivo 1 H SNR maps. Sagittal and coronal-plane images with: (A), (E) birdcage coil; (B), (F) case B; (C), (G) surface coil and (D), (H) case C. Low SNR images are in a blue box with a scale of 0 to 60. High SNR images are in an orange box with a scale of 0 to 130. 160x78mm (300 x 300 DPI)

Table 1 :

 1 This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 736937 and from the Ministry of Education and Science of the Russian Federation (project No. 14.587.21.0041 with the unique identifier RFMEFI58717X0041). The preclinical MRI scanner was acquired with help of the France Life Imaging National Programmegrant ANR-11-INBS-0006. 14 accumulation and localization of isoflurane in the C57BL/6 mouse and assessment of its potential contamination in 19 F MRI with perfluoro-crown-etherlabeled cardiac progenitor cells at 9.4 Tesla. Journal of Magnetic Resonance Imaging 2017;45(6):1659-1667. Analysis of 1 H results. The table shows the reference powers obtained for the different coils after radiofrequency autocalibration of the magnet, the average standard deviation of noise calculated from both coronal and sagittal images, and the comparison of the FOV and SNR obtained from each coil.

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European Union's Horizon 2020 Research and Innovation programme under Grant Agreement No 736937, France Life Imaging National Programme grant ANR-11-INBS-0006 and the Ministry of Education and Science of the Russian Federation (project No. 14.587.21.0041 with the unique identifier RFMEFI58717X0041).