Study of a tunable MEMS capacitor: influence of fluids

N. Habbachi, H. Boussetta, A. Boukabache, M. A. Kallala, P. Pons, K. Besbes

In this letter we evaluate the effect of a dielectric liquid on the tunability of capacitor operating in RF domains. The RF measurement shows a high variations of the resonant frequency accompanied with a low insertion loss. Moreover, the fluid positions between electrodes modifies the capacitance value up to Tr = 6660% at 600MHz. The quality factor decreases in response of water filling from Qmax = 51.9 when it is empty to Qmin = 1.49 when it is fully filled. According to the FEM analysis, the change of the dielectric permittivity influences the capacitor performances. Essentially, the tuning range of the capacitance and the quality factor could reach respectively: Tr = 7660% and Qmin = 35.

-/ Introduction: The RF MEMS capacitors present many advantages comparatively to the conventional varactors. To enhance their performances or to introduce some agility in their operating mode, several actuation techniques have been proposed [START_REF] Pu | RF MEMS Zipping Varactor With High Quality Factor and Very Large Tuning Range[END_REF][START_REF] Pulskamp | High-Q and Capacitance Ratio Multilayer Metal-on-Piezoelectric RF MEMS Varactors[END_REF][START_REF] Elshurafa | Low voltage RF MEMS variable capacitor with linear C-V response[END_REF]. Three setting parameters are highlighted: capacitance tuning range, quality factor, and resonant frequency. Many microwave circuits or components have used microfluidic as a powerful tool to reconfigure or to upgrade some RF circuits [START_REF] Zhou | Novel microfluidically tunable bandpass filter with precisely-controlled passband frequency[END_REF][START_REF] Morishita | Two-octave tunable liquid-metal monopole antenna[END_REF][START_REF] Fayad | Broadband liquid antenna[END_REF].

The presence of deionized water (DI) in microwave structure could increase the effective permittivity of the capacitor, giving hence an adaptable enlargement of the corresponding electrical length. The practical result can be expressed as a semi-continuous capacitance variations or a smooth shift of the resonant frequency with a stable insertion loss.

In this work we present some results about the performances of a microfluidically tunable microwave capacitor actionable by DI water. This fluid is inserted in microchannels and it can occupy different positions between the metallic electrodes of the MEMS structure. The resulting permittivity modifications impact the device performances, particularly its electrical and frequency characteristics.

We study the impact of the fluid presence by investigating the capacitance variations, insertion loss, quality factor and the resonant frequency.

Structure design: The spiral electrodes are completely doubled by fluidic channels which would increase their capacitance variations as shown in Fig. 1. They are made with gold layer due to its good resistance to water corrosion and to its high electrical conductivity. The fluidic parts are structured by using the SU-8 photoresist: on the one hand, this material allows having high aspect ratio for fluidic structures and on the other hand it presents some interesting microwave properties used to tune specific components and circuits [START_REF] Gmati | RF MEMS continuous reversible variable inductor based on a microfluidic network[END_REF][START_REF] Pinon | Development of a microsystem based on a microfluidic network to tune and reconfigure RF circuits[END_REF]. The substrate BR33 glass is lossless dielectric and presents a good candidate to be used as a mechanical/electrical support. The physical characteristics of these different materials are summarized in Table 1. Fabrication process: The realized capacitor is presented in Fig. 2. The fabrication process can be divided into two important parts: fabrication of electrodes and structuration of the SU8 photoresist to obtain fluidic channels. At the first part, we patterned a layer of positive photoresist (AZ4562) by using photolithography techniques in order to realize a mold for the capacitor structure. Thereafter, by using an electrolysis process we obtain metallic electrodes (Au) having a typical thickness of 7 µm, and 15 µm of width. Finally, the photoresist layer is removed to let the dual circular electrodes with typical dimensions: separating distance 20 µm, inner diameter close to 200 µm.

For the second part, the walls of the channel are obtained by spin coating 4 ml of SU-8 3050 on a glass substrate (BR33), and structured by photolithography. The realized channels are separated within 100 µm with 50 µm of height, and 180 µm of width. The cover layer is constituted by an SU-8 dry film having a thickness of 20 µm: it is laminated on the top of the channels network. The last step to finalize the device is to open the access holes allowing thereafter to introduce fluids in the device. To evaluate the influence of the DI water on the RF responses we have used the S-parameters to extract the resonant frequency and the insertion loss. Their variations according to the six water positions are represented in Fig. 4(a): the resonant frequency decreases continuously with the water penetration on the microchannels while the insertion loss presents a good stability. When the capacitor is empty the resonant frequency present a maximal value Fres = 3 GHz and reaches a minimal value Fres = 690 MHz when it is fully filled. Globally, these results show an important variation of the resonance frequency up to Tr = 330% accompanied with an insertion loss lower than -1 dB. 4 Regrouped results of the measurements and simulations realized for the six DI water positions in microchannels a: resonant frequency and insertion loss b: capacitance and quality factor at 600 MHz On Fig. 4(b) we have reported the capacitance measurements and the quality factor for six DI water positions at 600 MHz. As it can be observed, the capacitance value increases according to the water positions in microchannels. When the last are empty Cmin = 0.52 pF and it can reach the maximal value Cmax = 35.2 pF when they are fully filled. Hence, we obtain a high tuning range of the order of Tr = 6660%. This result could be explained by the increase of the capacitance value of the device by the presence of a high permittivity liquid between its electrodes. Furthermore, the quality factor decreases from Qmax = 51.9 when the capacitor is empty to Qmin = 1.49 when the channel is full.

To complete our experimental approach, we have used a finite element method tool (HFSS) to simulate this tunable device for the six DI water positions in channels. The simulation results reported on Fig. 4(a-b) show that the global behavior matches correctly with that obtained from experiments; the difference between measurements and simulations is essentially related to the permittivity value of the global structure instead that of DI water.

Permittivity effect:

In this part we have used the same FEM tool (HFSS) to investigate the influence of the global permittivity variations including those of the fluid and the substrate on capacitor performances. We have modified the substrate permittivity from 3 to 5 and that of the liquid from 10 to 80. The obtained results are reported in Fig. 5 for the case where the channels are fully filled with a fluid. The first color map (Fig. 5(a)) presents the effect of the permittivity value on the capacitance tuning range. The red area indicates that the variation is maximal for higher permittivity value of fluid and substrate, giving a tuning range from Tr = 120% to Tr = 7660%, i.e. sixty times. However, the second color map (Fig. 5(b)) shows that the quality factor is inversely proportional to the permittivity of dielectrics materials when the microchannels are full. The quality factor decreases from Q = 35.9 when the fluid permittivity is minimal to Q = 1.1 for higher permittivity, i.e thirty times.

The third color map (Fig. 5(c)) shows that the variation of the resonant frequency has a strong permittivity dependence from 85% to 338%. The maximal variation of the resonant frequency is reached for a higher fluidic permittivity upper than ε = 60. The insertion loss variation is showed in Fig. 5(d): it value decreases when permittivity rises, varying from -0.52 dB to -0.31 dB. It is obvious to remark that the RF performances are less influenced than the electrical one by medium permittivity change.

Conclusion:

A widely tunable microwave capacitor using microfluidically actuation has been fabricated and analyzed. The measurement and simulation results have a quasi-concordance, showing very high capacitance and resonant frequency variations when the DI water occupies different positions in the channels. Nevertheless, these advantages are accompanied by quality factor degradation.

Fig. 1

 1 Fig. 1 3D view of the projected capacitor structure.

Fig. 2 Fig. 3

 23 Fig. 2 Photography of the finalized MEMS device.Experimental and theoretical analysis: We have characterized the microwave capacitor by using Agilent 8510 and GSG probes. We have evaluated its performances for the six DI water positions illustrated in Fig.3by varying the frequency between 10 MHz and 4 GHz.

Fig. 5

 5 Fig. 5 Maps showing of the influence of the global variations of permittivity on the capacitor characteristics a: tuning range, b: quality factor, c: frequency resonant d: insertion loss

Table 1 :

 1 physical characteristics of the different materials used for the MEMS capacitor.

	Material Relative Permittivity	Dielectric loss	Bulk Conductivity S.m -1	Relative permeability
	Br33	4.6	0.0037	0	1
	Au	1	0	41E6	0.99996
	Di water	80	0.01	0.002	1
	SU-8	2.9	0.04	0	1

Acknowledgments: This work was initiated in the framework of the cooperation between DGRS (Tunisia) and CNRS (France). N. Habbachi, H. Boussetta, M. A. Kallala, K. Besbes (Microelectronics and Instrumentation Laboratory, Faculty of Sciences of Monastir, 5000 Monastir, University of Monastir, Tunisia.) A. Boukabache, P. Pons (Laboratory for Analysis and Architecture of Systems, CNRS, LAAS, 7 avenue du Colonel Roche, 31077 Toulouse, France.) E-mail: habbachinizar@yahoo.fr. K. Besbes: also with (Center for Research on Microelectronics & Nanotechnology, CRMN Sousse TechnoPark, Tunisia.)


Isabelle Guérin

Solène Morvant-Roux

Jean-Michel Servet

Understanding the Diversity and Complexity of Demand for Microfinance Services: Lessons from Informal Finance

There is growing consensus that microfinance supply is too standardised, inflexible and inadequate given the diversity of financial needs. As a result, microfinance is a very partial substitute for informal financial services and their comparative advantages.

This paper aims to deepen understanding of financial service demand by learning from informal finance. Based on economic anthropology, our analysis shows that microfinance does not substitute informal finance for many reasons: because money and informal finance are multidimensional and context specific; because the boundary between saving and borrowing is blurred; because money circulates in small quantities and quickly in village economies; because informal finance is more flexible; and, last but not least, because informal finance is a vector of social inclusion.

The first part proposes a general analytical framework for understanding the social dimension of money, debt and saving. The second part describes some fundamental mechanisms of informal finance, whilst the third part uses this analytical framework to understand how people appropriate microfinance.

The Social Meaning of Money, Debt and Saving

Money is the economic object of study par excellence; the great equalizer, money is considered a means of individualization, destroyer of hierarchies, obliterator of statutory privileges. The fundamental role of money, defined as a unit of calculation and a standardised means of payment, is ostensibly to create contractual relationships between equals. But ethnological and historical analysis of monetary practices reveals that money's impersonality and anonymity is illusory [START_REF] Baumann | L'argent des anthropologues, la monnaie des économistes[END_REF][START_REF] Bazin | Monnaie: Pluralités -contradictions[END_REF]Guyer, 1995;[START_REF] Villarreal | Striving to make capital do "economic things" for the impoverished: On the issue of capitalization in rural microenterprises[END_REF]Bloch and Parry, 1989;[START_REF] Zelizer | The Social Meaning of Money[END_REF][START_REF] Zelizer | The Purchase of Intimacy[END_REF][START_REF] Servet | Nomismata. Etat et origines de la monnaie[END_REF][START_REF] Servet | Banquiers aux pieds nus: La microfinance[END_REF]. Money, and the practices stemming from it, are above all a social construct. Money is embedded in preexisting relationships governed by rights and obligations; relationships it can influence, but never destroy.

Money: A source of tension between the individual and the group

Money and finance are social institutions in that their access and use depends on conventions, norms and formal rules [START_REF] Commons | Institutional Economics. Its place in Political Economy[END_REF][START_REF] Polanyi | Primitive, Archaich and Modern Economies[END_REF][START_REF] Servet | Nomismata. Etat et origines de la monnaie[END_REF][START_REF] Servet | Banquiers aux pieds nus: La microfinance[END_REF]. Consequently, money is characterised by a permanent tension between the individual and the group, between personal aspirations and collective responsibilities. This ongoing tension takes several forms:

-Although economists generally define saving and indebtedness in terms of time, with the purpose of securing material gain, finance is oftena means of relating to the group or creating interpersonal bonds of dependence and domination. When individual behaviours are not reduced solely to their economic dimension, going into debt or lending money is a sign of social inclusion. Indebtedness and saving reinforces a sense of social belonging, whether characterized by domination, dependence or equality. As a result, in some societies, it is important for money to change hands quickly. The poor often accumulate debt and credit and repay loans according to their own informal hierarchies [START_REF] Shipton | The nature of entrustment. Intimacy, exchange and the sacred in Africa[END_REF] and frameworks of calculation [START_REF] Villarreal | Striving to make capital do "economic things" for the impoverished: On the issue of capitalization in rural microenterprises[END_REF]. Such phenomena transcend material or self-centred motivations and reflect issues of status, honour, power, and individual and group identity. -Hoarding goes directly against the rationale that money must change hands, which is why non-monetary saving practices are so common, whether in kind or in the form of exchanges or loans to others. This same rationale explains why people tend to rank savings possibilities not only in terms of security but social considerations. -The social dimension of finance does not preclude financial reasoning, in the economic sense of the term. Quite the opposite. The poor more than anyone need to keep accounts, calculate and anticipate. But they are not necessarily sensitive to the same criteria, constraints or rationale that apply to the wealthy. Interest rates, for example, are often addressed in a Manichean manner ("Are the poor sensitive or not to high interest rates?"). In fact, there are many ways to interpret interest rates. Some vernacular languages have no specific terms to designate the surplus a debtor pays a lender. In such cases, debt is seen not in relationship to time but in terms of commercial margin [START_REF] Baumann | Chez nous, c'est 15% par mois. Coût du crédit et représentation du temps en milieu populaire à Dakar[END_REF] or as a fee [START_REF] Collins | Portfolios of the Poor: How the World's Poor Live on $2 a Day[END_REF]. -Financial transactions imply inclusion in one or more social groups, but the nature of this affiliation is far from simple. Not only can an individual belong to several social groups, ranging from the customary (family, ethnicity, caste, gender, religion) to the constructed (professional, neighbourhood, associative groupings), but membership is constantly evolving. The variety and vibrancy of financial practices reflect this diversity. -Understanding financial behaviours requires a temporal perspective.

There are immediate needs associated with daily survival, needs associated with life-cycle events, needs associated with social and religious rituals, and needs that are investments over a lifetime or even several generations.

The claim that microfinance clients are "financially excluded" and in need of financial inclusion and money management skills only holds true from a formal sector standpoint. Take into account informal finance, and it becomes altogether dubious.

Informal finance: Diverse practices and landscapes

Informal finance has kept pace with the monetarisation and financialisation of contemporary societies [START_REF] Servet | Banquiers aux pieds nus: La microfinance[END_REF], remaining vibrant and extraordinarily diverse. Indeed, the term "informal finance" -with its monolithic connotation -is almost meaningless: it makes more sense to refer to informal financial practices instead. Because informal financial practices evolve with society,1 they are as diverse as the social settings where they are found. Their social embeddedness makes them consistently distinctive, their diversity a tribute to humankind's imaginative and adaptive capacities.

Informal finance takes many forms, both collective and individual. Examples include rotating savings and credit associations (ROSCAs) and community organisations designed to cover funeral costs, collective celebrations or large community projects.

A multitude of private intermediaries also exist. We now know that the cliché of exploitative and greedy usurers, a caricature much-favoured by the media, decision makers and many MFIs, does not stand up to factual analysis. Professional lenders are only one category of a mosaic of lenders. First, there is the inner circle of neighbours, friends and family, probably used everywhere to varying degrees. The principles of solidarity and reciprocity prevail in such exchanges, organised such that everyone takes their turn as debtor and creditor.

Next, there are individuals who lend by virtue of their status or privileged access to cash. These include private specialised lenders, pawnbrokers, shopkeepers who accord credit, traders who pay for harvests in advance, manual labourer employers or recruiters who advance payment, and salespeople who sell goods on credit and then immediately buy them back for less. Also in this category are wholesalers who sell on credit to small shopkeepers, artisans or travelling vendors, and employers who lend to former apprentices wanting to start their own activity. Local elites -landowners, employers, teachers, civil servants, migrants or migrants' wives, local elected officials, religious leaders, doctors -may also wish to invest liquidity surpluses and/or extend their social network.

There are also those who safeguard money. Some are mobile and make house calls. Others have shops or simply belong to the saver's social network. Often, savings collectors are a more secure alternative to hoarding.

The prevalence and complementarily of these informal practices varies across regions and cultures according to legal, technical, cultural and social constraints. Hence the expression 'financial landscape' [START_REF] Bouman | Roscas and Ascras: Beyond the Financial Landscape[END_REF], which seeks to account for the diversity and interconnectedness of different practices even between the so-called formal financial sector and informal financial practices. Indeed, most societies do not function in terms of financial dualism, with the formal sector on one side and the informal on the other. People often move fluidly between the two, not only the poor, but middle and high-income populations as well.

The diversity of so-called "informal financial practices" stems from all these elements combined: the social embeddedness of money, multiple group affiliation, the permanent tension between the individual and the group, and evolution of financial needs throughout the life-cycle. These elements help better understand how people perceive finance in their daily life, whether in the context of credit, saving or risk management. They also shed light on how people appropriate financial services offered by outsiders, and in particular microfinance institutions.

In What Ways Do People Perceive and Experience

Finance?

Analysing so-called informal finance practices brings to light la pensée sauvage (the savage mind) -to employ Lévi-Strauss's expression -with respect to money, debt and saving. It reveals conventions, habits and local categories of thought, i.e., the ways people perceive and use finance.

Saving and credit: A false dichotomy

For a long time, the "poor" were considered incapable of saving. While it is true that monetary hoarding is often limited, there is no doubt the poor do save to protect themselves from future risk and anticipate certain expenses [START_REF] Collins | Portfolios of the Poor: How the World's Poor Live on $2 a Day[END_REF][START_REF] Lelart | Les circuits parallèles de financement: État de la question[END_REF][START_REF] Rutherford | The Poor and their Money[END_REF][START_REF] Servet | Community relations, individual, social and economic constraints in the savings and loans associations[END_REF]. These include recurring expenditures (school fees, religious festivals) and life-cycle costs (housing, birth, coming of age ceremonies, marriages, funerals, pilgrimages, etc.). On the other hand, their savings options -and the criteria they use to assess them -vary. Reasons for saving are also diverse and sometimes contradictory, given the permanent tension between social obligations and individual desires. The result is a plethora of complementary practices, sometimes impossible to substitute. What accounts for the decision to use a particular form of saving? The following list is neither exhaustive nor definitive, reflecting only tendencies observed in several regions of the world. The importance of each factor varies according to context, social groups and individual behaviours.

-Security/thrift: saving's primary function. Goods acting as a form of saving must be able to be stored safely without risk of degradation or depreciation.

-Access to liquidity: in urgent need, cash should be quickly and easily accessible. -Social identity: some acts of saving are more an expression of one's social identity than an individual act. -Anonymity and discretion: when saving is for a personal project, anonymity and discretion come into play. Even in families where individual members pool part of their income for common expenses, individuals often have savings practices and networks of their own. The desire for discretion is particularly great amongst women, who often seek to preserve a space, however tiny, for themselves, free from male intervention. -Illiquidity and incentives: saving is difficult not only because cash is short and income irregular; the poor often face pressure to spend. Torn by the demands from family members and a constantly needy entourage, people often say that cash "burns their fingers". Given this pressure, families often look for mechanisms which incite, or even force them to save.2 This need can be so pressing that people are prepared to pay to save: hence the success of itinerant savings collectors in many countries.

The same logic holds true for ROSCA members who prefer to receive their sum at the end of the cycle. To account for the tensions created by the ongoing balancing act between liquidity and illiquidity, individualism and group responsibility, [START_REF] Shipton | How Gambians save: Culture and economic strategy at an ethnic crossroad[END_REF] conceptualizes this mechanism as the "squawk factor" which he describes in the Gambian context: "[. . . ] saving strategies are mainly concerned with removing wealth from the form of readily accessible cash without appearing antisocial" (Shipton, 1995: 257). -Speculation: "Money needs to multiply", declare Indian women, explaining their membership in seetu (ROSCAs) or gold purchases. Similar ideas can be found in the practices of Cameroonian tontines or Chinese hui. Speculation often explains the preference for saving forms likely to appreciate over time, such as precious metals, cereals or cattle.

While saving forms vary from one context to another, monetary hoarding is a rarity. This is as much a question of safety as it is an effort to resist the temptation to spend and to ward off requests from one's entourage; furthermore, immobilized money serves no purpose. In the English language and particularly in the work of David Riccardo, the term "circulation" is used in reference to money. Indeed in Senegal, money is often said to "burn", it circulates so quickly (Guérin, 2003). Women frequently joke about the way money circulates without stopping: "Money, yes, we see a lot of it, but it never stays still for long". "As soon as it arrives, it leaves." No sooner is it received, it is spent, released into the community as an "investment",the women's choice of term -liable to be recovered at any moment in the case of "pressing need" or "problem". To the question "do you save?", it is common for people to sincerely reply that they lend; it is considered a form of savings. In the indigenous communities of Mexico, all forms of wealth (not only coins and notes but also bricks, food products or cattle) may be loaned if the owner does not have an immediate need for them [START_REF] Morvant-Roux | Accès au microcrédit et continuité des dynamiques d'endettement au Mexique: Combiner anthropologie et économétrie[END_REF]. [START_REF] Shipton | How Gambians save: Culture and economic strategy at an ethnic crossroad[END_REF] has made the same observation in rural Gambia, where the slightest riches, whether in cash or in kind, are loaned to conceal ownership and cement social bonds. The logic of constant circulation of debt and credit is neither exotic nor archaic and the blurring of savings and loans is found throughout the world (Guyer, 1995;Lont and Hospes, 2004). In fact, borrowing is simply a means to force oneself to save in the future [START_REF] Rutherford | The Poor and their Money[END_REF], just as lending is a form of saving that presupposes the right to borrow later. Microfinance institutions are but one of many strategies clients use when juggling various formal and informal finance opportunities accessible.

ROSCAs: Individual projects and collective constraints

ROSCAs are a common financial form that exemplify the melding of saving and credit simultaneously. ROSCAs exist around the world, but their modalities, function and nomenclature are specifically local. A vast body of literature confirms their extraordinary diversity and capacity to adapt to very different and rapidly evolving environments [START_REF] Ardener | The Comparative Study of Rotating Credit Associations[END_REF]Ardener and Burman, 1996;[START_REF] Bouman | Indigenous Savings and Credit Societies in the Third World: A Message[END_REF]Servet, 1995;[START_REF] Lelart | Les circuits parallèles de financement: État de la question[END_REF][START_REF] Lelart | De la finance informelle à la microfinance. Paris: AUF Editions, collection "Savoirs francophones[END_REF]. However, the success of ROSCAs probably stems largely from their ability to address individual financial constraints collectively [START_REF] Servet | Community relations, individual, social and economic constraints in the savings and loans associations[END_REF]. Hence the low rates of default as members force themselves to make regular payments.

Members who receive their sum early in the cycle essentially receive uncollateralized loans from the others. If a debtor stops paying his or her share, s/he may be banished from the community, thus the importance of carefully selecting members, or at least the organiser. The organiser is responsible for defaulting members, even if this means tracking down the defaulter, or calling on the police and courts. This is where group and individual rationales intersect. The organiser must ensure the ROSCA's proper functioning while exercising his/her power to ensure the defaulter eventually repays. The obligation to pay, to use Marcel Mauss's expression from his famous Essay on the Gift (1950), is stronger than most legal constraints. Not honouring one's word would amount to social suicide. ROSCA members know they must respect their obligations if they want to participate in other tontines or simply extend to other contexts the benefits of solidarity and protection that come with being a member in good standing. In ROSCAs, the act of saving is not about an individual relationship to time but rather a social relationship where reciprocal obligations are bound and unbound [START_REF] Lelart | Les circuits parallèles de financement: État de la question[END_REF][START_REF] Baumann | Vulnerability and micro-insurance. Reflections on post-adjustment Africa[END_REF].

The subtle balance between individual and group needs takes other forms. ROSCAs permit members to ward off loan requests from family members, whilst still allowing them, at least in some cases, to respect social obligations (pilgrimages, ceremonies, support family members, etc.). In broad terms, ROSCAs express and contribute to the restructuring of social relationships. Indeed, various case studies on ROSCAs show how family ties are sometimes used and reinforced, but also avoided, substituted or even weakened, depending on the region and community (Ardener, 1995;[START_REF] Guérin | Women and Money: Multiple, Complex and Evolving Practices[END_REF].

The circulation of money and juggling practices

Money is constantly "lacking", but it also circulates with astonishing intensity. The tendency to take on debt clearly arises from a mismatch between income and expenses, but it is also a matter of maintaining credibility, one's reputation and social networks. Lending presupposes the two parties already share a relationship of trust, but it also serves to maintain, reinforce and renew this relationship. When describing their financial practices, Senegalese women state: "Sab bukki, sulli bushidô" (take a hyena, bury a hyena) or "sab-sul" (dig and bury), meaning they take new debt to pay off old debt. They also speak of "drawers", whereby all the people or groups they lend to or do a favour for represent a "drawer" they can pull at any moment. In Mexico, Magdalena Villarreal describes women's credit chains, in which any income is largely used to repay old debts, to maintain credibility and thus borrow again later (Villarreal, 2002). [START_REF] Morvant-Roux | Processus d'appropriation des dispositifs de microfinance: Un exemple en milieu rural mexicain[END_REF] demonstrates how in rural Mexico, more than a third of 239 interviewees were concurrently indebted to at least two distinct financial sources.

Permanent juggling practices should not always be understood as a sign of over-indebtedness or poor management. In many cases, they reflect deliberate choices and strategies geared to multiply and reinforce social relationships and maintain a certain balance, considering the inherent ambiguity of all debt relations. Ambiguous, because while debt is a source of protection and solidarity, as well as a means of expressing reciprocal trust and respect, when it is not honoured or is too imbalanced, it can be a source of humiliation, shame, exploitation and servitude. It is both a lifeline and a death knot, to employ the expression of Charles Malamoud (1980). Marcel Mauss speaks of gift as the "poison", which he assimilates to a form of loan. Hence the subtle game to regularly reduce one's debt whilst taking on debt elsewhere.

"Saving-loans" are always reciprocated (the lender eventually becomes the borrower and vice versa) and address both short-term survival needs and long-term social ones, like religious rituals, marriage, puberty or godparenting ceremonies, annual festivals or funerals. The exchange may take the form of cash or goods with social and symbolic value: jewels, clothing (pagnes in Senegal, sarees in India) or animals (cows among the Fulbe in Sub-Saharan Africa, pigs in Papua New Guinea and also in certain tribes in India [START_REF] Thanuja | Relevance of microfinance and empowerment in tribal areas: A case study of Konda Reddis[END_REF], turkeys in Mexico [START_REF] Morvant-Roux | Processus d'appropriation des dispositifs de microfinance: Un exemple en milieu rural mexicain[END_REF]).

Flexibility and negotiability

Many studies on how populations compare financial services have highlighted the importance of "negotiability", defined as the possibility to negotiate transaction modalities and, particularly, defer repayment deadlines [START_REF] Rutherford | The Poor and their Money[END_REF][START_REF] Johnson | Gender norms and financial markets: Evidence from Keyna[END_REF][START_REF] Servet | Banquiers aux pieds nus: La microfinance[END_REF].

Negotiability is important for two reasons. The possibility to adapt repayments to irregular incomes and expenses is a key advantage, particularly if income fluxes are seasonal and uncertain (as in agricultural production or migration). But negotiability is also a way to personalise a social relationship.

The principle of standardised prices and terms allows for a contractual relationship between equals. The principle of negotiability, in contrast, expresses a personal relationship: here, the nature of the relationship and the relative statuses of the two parties influence the terms of the exchange. Anthropological literature has shown the extent to which not fixing prices a priori is standard practice (Bloch and Parry, 1989). What at first glance appears to be a lack of transparency is ultimately a form of protection against the anonymity of a commercial relationship in which the same price and terms apply to everyone [START_REF] Toren | Drinking cash: The purification of money through ceremonial exchange in Fiji[END_REF]. In Africa, during the second half of the 20th, century, Sarah [START_REF] Berry | Stable prices, unstable values: Some thoughts on monetization and the meaning of transactions in West African economies[END_REF] demonstrates that negotiability responds to two sources of major uncertainties: uncertainty linked to economic crisis and hyperinflation, but also uncertainty linked to the redefinition of the notion of "value", status, hierarchies and identities as a result of monetarisation and "modernisation" (education, migration). There is no shortage of information regarding the allocation of resources and wealth creation; what is lacking is an understanding of the meaning behind this information; after all, the permanent negotiation of prices is first and foremost a negotiation of value.

In Southern India, a study on quality of financial services involving 170 families showed that "negotiability" is the most valued criteria (highlighted by over a third of those interviewed). Next was cost (26 percent) followed by "discretion" (17 percent). Negotiability is a major component of informal finance, and "contracts" between creditors and debtors are often flexible: cost and duration are not always specified at the outset and are likely to evolve over time. Repayment methods can be adapted to the borrowers' or lenders' constraints and the latter may reclaim their due in an emergency. In vertical or hierarchical financial relationships (for instance loans from an employer, a landowner or member of the local elite), the debt relationship is but one component of a larger relationship, which often resembles a form of patronage. The lender's generosity and flexibility demonstrates his or her role as protector, but this protection is "repaid" by material and symbolic compensations (favours or free assistance at a moment's notice, patronage of a shop controlled by the lender, recognition and gratitude). In the case of itinerant lenders, transaction terms vary greatly on the basis of loyalty and trust (not unlike commercial relationships in the West which employ client loyalty strategies). Regular clientele benefit from the best terms (such as low prices or greater flexibility), as do those who act as guarantor for other clients -a very common practice, as trust is the main collateral. Benefiting from the best terms is implicit compensation for favours accorded. Negotiability also applies to the loan amount. Microcredit providers usually demand regular interest payments along with the principal. Some informal private lenders demand only regular interest payments, whilst retaining their rights over the principal. Such arrangements can last several years. The term "rent" is more fitting than "interest" in this kind of financial relationship. As a strategy, it compares to owners renting out a house that they have no interest in selling, even if they sometimes change tenants. What matters is that the balance is periodically paid off. 3 The way the debt is settled depends on the context, era and nature of the relationship between the creditor and the debtor. Sometimes the debt is never settled and is even passed on to the following generation. Sometimes, it is ultimately cancelled, if the debtor manages to show that the total interest paid is enough (in India, for example, some creditors cancel debt when interest payments represents two or three times the principal). In other cases, the principal is paid back after a few months or years.

Our purpose is not to idealise this personalisation of debt relationships, which can affirm solidarity but also reinforce subordination. Sometimes extreme flexibility or debt cancelation conceals relationships of subjugation and exploitation. Rather, we seek to demonstrate that people perceive finance and manage their finances in this way.

The Role of Informal Financial Practices in the Appropriation of Microfinance

Analysing informal finance gives us a better understanding of how microfinance is used. Interestingly, the fluid boundaries between saving and lending show up in loan use. For example, clients commonly put aside a portion of microcredit for saving (in the form of liquidity, gold purchases, or loans to others, etc.). 4 It is not surprising that some MFIs struggle to collect savings: in many cases, informal practices better correspond to people's constraints and motivations. Some noteworthy innovations do exist. For short-term saving, SEWA's home savings programme in Gujarat, operating successfully since the 1970s, comes to mind, as does SafeSave in Bangladesh, betting on flexibility and proximity since the 1990s. For long-term saving, SEWA offers saving accounts remunerated in gold. There are probably other innovations worthy of documentation. But much remains to be done to design services adapted to how people perceive saving. Moreover, many informal saving practices are so much more advantageous, that the attempt to substitute is often in vain.

Analysing informal finance also reveals how microcredit is used. Clients do not passively consume microcredit services. They translate and interpret them according to their own frame of reference, adjusting and adapting them, often bypassing the rules to do so. A process of appropriation takes place, both at the individual and collective level [START_REF] Morvant-Roux | Processus d'appropriation des dispositifs de microfinance: Un exemple en milieu rural mexicain[END_REF].

Borrowing to onlend: Abuse or a normal part of the system?

Some studies have found clients use microcredit for money lending. Instead of investing in a so-called productive activity or covering a family expense, clients onlend to their entourage, often with interest. MFIs antagonistic to usurers naturally condemn such practices. In our opinion, however, onlending should not be automatically judged as a digression or anomaly of the system, but rather a natural (even rational) outcome. For example, [START_REF] Perry | Microcredit and women moneylenders. The shifting terrain of credit in rural Senegal[END_REF] has shown that in some rural zones in Senegal, the major impact of microfinance has been to create a new category of informal private lenders: middle class and poor women. The zone studied is characterised by reduced availability of credit due to a slowdown of cooperative banks and landowners moving away from agriculture to go into business and who no longer have liquidity to lend. In contrast, for several years now, microfinance has targeted women. A large number of loans are in fact onlent locally, mostly to the borrowers' extended male kin. Several reasons explain this phenomenon. Productive opportunities are limited and risky (entry barriers, weak local demand, activities highly segmented along gender lines), whilst men's demand for credit is high. At the same time, the women explain that they are fulfilling a social obligation by helping their kin and maintaining bonds of reciprocity which they will be able to rely on in times of need. This type of activity is lucrative, but not considered "immoral" by the community. b980-ch04

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Understanding the Diversity and Complexity of Demand for MFS 99

In India, the principle of the self-help group also leads to parallel lending. Borrowing to onlend is relatively common, but difficult to quantify because women know that MFIs condemn this practice. Disbursement dates do not necessarily correspond to a precise need, however, and women clearly state that they do not know what to do with their money and that onlending is a way to put it to good use. Group leaders and loan officers are also active lenders, but their role is in fact much broader and includes a number of additional services. Clients use the term "adjustment" to refer to these additional services.

Adjustment is when the group leader or loan officer introduces a degree of negotiability. It is difficult to obtain reliable data on repayment rates but it is likely that on-time repayment rates are relatively low (around 50 percent), and readjustments the rule rather than the exception. Official rates of 95 percent refer to nine or 12 months. Loan officers and group leaders negotiate and jointly decide readjustments. Adjusting can involve helping clients find additional credit sources when microcredit is insufficient (in the case of certain investments or health/ceremonial expenses) or disbursement delays are too long. Adjustment may be necessary when clients have trouble repaying their loan. In the absence of an official grace period, the loan officer or group leader's occasional assistance helps the borrower "save face" and maintain credibility. This may involve facilitating access to other credit sources by acting as guarantor for dealings with private lenders or physically accompanying women to certain transactions. This assistance may also consist of direct loans. The source of funds is not clear, but it appears common that money comes from the group itself. What might appear an abuse of the system is deemed legitimate by group members as long as they have regular access to liquidity. Members even expect this practice: efficient group leaders and loan officers should be able to make loans; it is their duty!

Group lending: Internal arrangements and the reproduction of pre-existing practices

Once considered a major innovation, group lending is increasingly criticized as overly rigid, incapable of adapting to diverse needs, a source of hidden costs, and catering to the better-off.5 Our purpose is not to compare individual and group loans. Either one is likely to prevail, depending on context, target population, and the objective. Instead, we aim to show how group members adjust to the group ethos and adapt it to their own habits.

In Mexico, a study of Grameen-type groups highlighted various arrangements between members [START_REF] Morvant-Roux | Microfinance institution's clients borrowing strategies and lending groups financial heterogeneity under progressive lending: Evidence from a Mexican microfinance program[END_REF]. Whilst officially individual loan amounts are determined by the MFI, the actual distribution is often very different. For example, clients tend to ask for the maximum amount even if they do not need it; the difference is given to another group member (who is in charge of repayment). "This amount must not be wasted", justify clients. In India, analyses of SHGs gave similar results: whether for internal or external loans, financial circulation within groups is more intense than the account books indicate. Such lack of transparency is deliberate: the women are all implicated in multiple and mutual debt relationships they prefer not to reveal, to avoid suspicion or jealousy.

These appropriation practices occur all the more readily, given some groups tend to mirror pre-existing social and therefore financial networks. In Mexico as in India, our observations show that borrower groups are grafted over existing debt and credit relationships. To form groups, leaders select members based on need and solvency -information only available to people already in similar financial circuits. Such mutual knowledge is exploited when the groups are created. The goal is not to contort financial supply, but rather to address the diverse needs within the same borrower group.

This appropriation process is not entirely surprising. After all, the principle of group lending is to exploit borrowers' mutual knowledge to compensate for information asymmetry. However, this process has two major implications. On the one hand, it renders financial service supply more flexible and adaptable through informal arrangements among group members. On the other, it has the potential to increase preexisting inequalities in terms of financial access. It is not unusual for group leaders themselves to be or to become moneylenders. Similarly, several quantitative studies indicate microfinance clients already have the best access to informal finance (see the example of Thailand [START_REF] Coleman | Microfinance in Northeast Thailand: Who Benefits and How Much?[END_REF], Mexico [START_REF] Morvant-Roux | Processus d'appropriation des dispositifs de microfinance: Un exemple en milieu rural mexicain[END_REF], Bangladesh [START_REF] Sinha | Informal Credit Transactions of Microcredit Borrowers in Rural Bangladesh[END_REF]). The ambivalence of group lending -flexibility and negotiability but also the reproduction of power relationships -is also highlighted by Susan [START_REF] Johnson | Institutionalized suspicion: The management and governance challenge in user-owned microfinance groups[END_REF]

Substitution or leverage effect?

Ample literature demonstrates that microfinance does not substitute informal finance. Not only are microfinance clients already well integrated into informal circuits, they are agile jugglers of various forms of financing and cross-financing: microfinance is used to pay back the informal, and vice versa [START_REF] Sinha | Informal Credit Transactions of Microcredit Borrowers in Rural Bangladesh[END_REF]Roesch and Héliès, 2007). In rural Mexico, juggling is the rule rather than the exception. Informal lenders regularly make "bridging loans" which clients reimburse with part of their microcredit [START_REF] Morvant-Roux | Processus d'appropriation des dispositifs de microfinance: Un exemple en milieu rural mexicain[END_REF][START_REF] Morvant-Roux | Accès au microcrédit et continuité des dynamiques d'endettement au Mexique: Combiner anthropologie et économétrie[END_REF]. These various studies analyse cross-indebtedness and the coexistence of microfinance and informal finance from a predominantly economic and financial perspective (transaction costs, flexibility, risk). We would add two further arguments.

Firstly, substitution is limited for economic and financial but also social reasons: cutting oneself off from certain sources of indebtedness would amount to social suicide. Exclusive dependence on a single provider is unfeasible -especially when the provider is a foreign institution. Such dependence would require unlimited confidence in the institution's ability to satisfy all financial needs in the long-term.

Indebtedness can occur in response to a need, but can also serve to maintain, reinforce or create a social tie. The choice of creditor can partly be explained by the desire to maintain, reinforce or create a bond, sometimes regardless of the cost: for instance, being indebted to an employer or a labour recruiter as a guarantee of getting work; being indebted to a member of the local elite to facilitate access to various forms of assistance (like governmental programmes); becoming indebted to a itinerant lender because s/he favours loyal clients; or quite simply becoming indebted to maintain a desired link for what it is (and not for what it can bring). Conversely, refusing a loan with a priori advantageous terms to avoid dependence is also common practice, particularly amongst certain employers, suppliers or private lenders but also with respect to the entourage or family. Some people prefer to become indebted to a costly private lender rather than "beg" from family members.

Secondly, it is our hypothesis that in certain contexts and for certain clients, microfinance not only does not substitute the informal, but has a leverage effect and thus increases informal access. Our inquiries in Southern India indicate that clients do not think in terms of substitution, but rather multiplication of finance sources (Guérin et al., 2009a[START_REF] Guérin | Microfinance and informal finance: Substitution or leverage effects? RUME Working Paper[END_REF]. Before microfinance, households juggled with two or three different options; now they have five, six or even more. There is a true leverage effect via various routes:

• The repayment of past debts (depending on the zone and population, this represents between a tenth and a quarter of microcredit use6 ) can lead to contradictory effects. Using microcredit to repay an informal lender can have a substitution effect, but it is often temporary. It also serves to preserve the borrower's reputation and recover jewels or land pledged, allowing him or her to borrow again later from the same creditor. • Improved creditworthiness among potential creditors. The indirect role of microfinance is confirmed both by SHG members and informal financial providers. Some women say that they remain microfinance clients to maintain creditworthiness vis à vis other creditors. Even if amounts are limited compared to what they can obtain elsewhere, some women use their status as a SHG members to convince private lenders.7 Some door-to-door moneylenders explicitly state that members of SHGs are privileged clients for two reasons. Firstly, they know that the clients can request a microcredit in the event of repayment difficulties; conversely, they know that some SHG members will require their services in order to pay into their obligatory savings or for loan repayment. Studies conducted in Kerala confirm these findings [START_REF] Sunil | Microfinance, informal finance and empowerment of the poor. In: Microfinance Challenges: Empowerment or Disempowerment of the Poor?[END_REF]. Some door-to-door lenders even choose to visit the village on the day of the SHG meeting. • Improved understanding of local financial markets. The financial market is diversified and dynamic. But it is also segmented and opaque. Information circulates through the SHG. (For instance, who are the "good" and the "bad" lenders? What arguments should one use during negotiations?) Women who are mobile and accustomed to borrowing stand as guarantors for others (in the case of door-to-door lenders or pawnbrokers, for example) -thus increasing their access. While such mutual assistance is common in SHGs, repayment obligations and joint responsibility reinforce this type of practice.

Conclusion

The empirical results discussed in this article highlight the limits of the dichotomized approach to informal and formal finance based on economic criteria (transaction costs, risk, interest rates, etc.). The enmeshed relationship between informal and formal financial practices explains, in certain contexts, the overindebtedness of populations who juggle more or less deftly available finance sources. Understanding this relationship calls for studies that address all facets of finance: not just economic, but cultural, political and psychological as well. Microfinance practitioners are not totally unaware of these practices. It is not uncommon for MFIs to use ROSCA-style expressions in advertising new products, to name their institution after the village granary, or use logos that evoke the idea of a "good" community finance institution that does its clients favours and builds relationships between individuals. At the same time, "bad finance", epitomized by "usurers" who are, incidentally, also community-based, is vilified. Many MFIs have the mission of eradicating this "bad finance", even when the latter are themselves very good MFI clients. This discourse has always resonated amongst decision makers [START_REF] Bouman | Small, Short and Unsecured. Informal Rural Finance in India[END_REF] and continues to do so today. For example, the World Bank report Finance For All considers financial inclusion a process that allows for the gradual suppression of informal finance, deemed both inefficient and unfair (World Bank, 2007: 66). It is true that some forms of informal finance are a source of poverty entrapment and servitude, and that microfinance can help reduce unhealthy dependency by offering contract-based working capital and risk coverage. But microfinance can also reproduce preexisting local hierarchies, for instance, when monopolized by the better-off.

More broadly, to truncate informal finance to the exploitative usurer or greedy landowner is extremely reductive. Moreover, it ignores the questions of social construction, social embeddedness and social meaning of informal finance.

Microfinance and informal finance should be considered as complementary and not as substitute. One of the main challenges of the microfinance industry is to better identify market niches where informal finance is not competitive and produces social and financial exclusion. As we have seen, people are permanently borrowing and lending tiny amounts, both for economical and social reasons. If these transactions were to be intermediated by microfinance institutions, transaction costs would be astronomical, and interest rates would be even higher. In theory, microfinance is supposed to make money available for medium and long-term financial needs (especially "productive" investment). Some innovations are worth noting, such as contractual savings (Churchill, 2007), commitment saving plans [START_REF] Collins | Portfolios of the Poor: How the World's Poor Live on $2 a Day[END_REF], rural leasing or warrantage (Morvant-Roux, 2009b;[START_REF] Bouquet | Rice inventory credit in Madagascar: Diversity of rural household strategies around an hybrid financial and marketing service[END_REF]. However these services are more the exception than the rule. Microfinance's ability to deal with risk in the long run still remains to be proved (especially in rural areas). In many cases, microcredit is still mainly used for short-term financial needs. A lot remains to be done to design financial services that could better complement the weaknesses of informal finance. This volume will certainly help to move this forward.

  and Complexity of Demand for MFS 95

  regarding SHGs in Kenya.

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  Understanding the Diversity and Complexity of Demand for MFS 103

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Before almost universal salarisation took place, informal financial ties were extremely common in European societies(Fontaine,

2008).

This type of behaviour was studied by anthropologists in the 1980s and 1990s (see, for example, Guyer (1995),[START_REF] Shipton | How Gambians save: Culture and economic strategy at an ethnic crossroad[END_REF]. Over the last few years, an increasing number of economists have been looking at this "preference for illiquidity". See for instance[START_REF] Bauer | Behavioral Foundations of Microcredit: Experimental and Survey Evidence From Rural India[END_REF]. See also[START_REF] Collins | Portfolios of the Poor: How the World's Poor Live on $2 a Day[END_REF],[START_REF] Vonderlack | Women, Microfinance, and Savings: Lessons and Proposals[END_REF],[START_REF] Guérin | Women and Money: Multiple, Complex and Evolving Practices[END_REF].

This analogy has basis in fact. In India, for example, it is possible for a tenant to pay an owner a large sum of money in exchange for the occupation of a house. The owner must return this sum when the tenant leaves the property. If he does not, the tenant can retain possession of the house and can sublet it until the owner repays the debt.

This is common practice in southern India.[START_REF] Sebstad | Microfinance, risk management and poverty[END_REF] point to the same phenomenon in several regions of the world.

See, for example,[START_REF] Coleman | Microfinance in Northeast Thailand: Who Benefits and How Much?[END_REF];[START_REF] Harper | What's wrong with groups? In What's Wrong with Microfinance[END_REF]; Mayoux(2001);[START_REF] Molyneux | Gender and the silences of social capital[END_REF];[START_REF] Morvant-Roux | Microfinance institution's clients borrowing strategies and lending groups financial heterogeneity under progressive lending: Evidence from a Mexican microfinance program[END_REF];[START_REF] Rankin | Social Capital, Microfinance and the Politics of Development[END_REF];[START_REF] Wright | The darker side to microfinance: Evidence from Cajamarca, Peru[END_REF].

These figures come from various studies carried out by the Institut Français de Pondichéry team in 2006 and 2007, which covered 1395 families, clients of various microfinance organisations, and recipients of 3457 loans.

 7 In Orissa, David[START_REF] Mosse | Cultivating Development. An Ethnography of Aid Policy and Practice[END_REF] has observed exactly the same phenomenon.


Thierry Bastogne
email: thierry.bastogne@cran.uhp-nancy.fr

Sophie Mézières-Wantz

Nacim Ramdani
email: nacim.ramdani@lirmm.fr

Pierre Vallois
email: pierre.vallois@iecn.u-nancy.fr

Muriel Barberi-Heyob

Parameter Estimation of Pharmacokinetics Models in the presence of Timing Noise

Keywords: parameter estimation, pharmacokinetics models, timing errors, bounded errors, biological systems

The problem addressed in this paper deals with the parameter estimation of in vitro uptake kinetics of drugs into living cells in presence of timing noise. Effects of the timing noise on the bias and variance of the output error are explicitly determined. A bounded-error parameter estimation approach is proposed as a suited solution to handle this problem. Application results are presented and emphasize its effectiveness in such an experimental framework.

I. INTRODUCTION

Pharmacokinetics is the study of the bodily absorption, distribution, metabolism and excretion of drugs by bodies. In chemical kinetics, reactions are generally described by differential equations which link the reaction rate with concentrations or pressures of reactants. In molecular cell biology, because of the complexity of systems, the nature of some reactions is still unclear. This paper is focused on the intracellular uptake kinetics of a photosensitizing drug (PS), i.e. the rate of photosensitizing molecules incorporated and accumulated by living cancer cells according to incubation terms [START_REF] Barberi-Heyob | Wild-type p53 gene transfer into mutated p53 HT29 cells improves sensitivity to photodynamic therapy via induction of apoptosis[END_REF]. The delivery control of the photosensitizing agent into the cancer cells is one the major factor on the therapeutic efficiency of the photodynamic therapy (PDT) [START_REF] Moser | Photodynamic Tumor Therapy: 2nd and 3rd Generation[END_REF]. Most of the PS uptake kinetics models are non-parametric, the temporal evolution of the PS intracellular concentration is described by step responses. The purpose of this study is the estimation of kinetics model parameters from data collected during in vitro kinetics experiments. These parameters are crucial information to both improve the modalities of the drug delivery process in photodynamic therapy and discriminate the uptake characteristics of different photosensitizers. Few papers have been reported for the application of system identification techniques to pharmacokinetics modeling problems [START_REF] Evans | Compartmental modelling of the uptake kinetics of the anti-cancer agent topotecan in human breast cancer cells[END_REF]- [START_REF] Ali | A PK-PD model of cell cycle response to topotecan[END_REF].

Unfortunately, these approaches cannot be applied to the PS uptake kinetics. The intracellular PS concentration ([P i ]) is measured by a spectrofluorimeter but the latter induces a photobleaching process of the PS. The term photobleaching refers to the process by which the chromophoric structure of the PS is degraded by absorbed light energy [START_REF] Niedre | In vivo tests of the validity of singlet oxygen luminescence measurements as a dose metric in photodynamic therapy[END_REF]. As PS can be photobleached after light exposure, repeated experimentations for the same biological sample are not conceivable. In other terms, one biological sample with PS cannot be used for consecutive measurements of [P i ]. Collecting N t data points of the kinetics then requires to repeat N t times the same experiment (N t biological samples) with identical initial conditions. To avoid the time consuming and the too high cost of such an experiment set up, N t is generally kept small, i.e. N t ≤ 10. This limitation on N t is the first problem to overcome for estimating kinetics parameters. The second difficulty is the low signal-to-noise ratio. The latter is due to a great measurement variability when working on living cells which are very sensitive to external disturbances. Thirdly, the choice of the stimulus signal is restricted to step signals which correspond to the amount of PS injected into the culture medium wells at time t = 0. A last issue is a timing offset error in the measurement samples. This timing error is bounded and can reach until ±15mn for a measurement time sequence {t j } = {1, 2, 4, 6, 8, 14, 18, 24h}. This error is not negligible and has rarely been taken into account in parameter estimation problems [START_REF] Kusuma | Signal parameter estimation in the presence of timing noise[END_REF], particularly in this restricted experimental framework.

The problem addressed in this paper deals with the parameter estimation of pharmacokinetic models in presence of timing errors. Parameter estimation algorithms in system identification methods are often based on the minimization of a quadratic function of the output error, i.e. the difference between the system and the model outputs. The sensitivity of the output error to timing noise comparatively to input and output noise is unknown. The contributions of this paper are twofold:

• the stochastic effects of the timing noise on the output error are explicitly determined. They are compared with the ones of input and output noise. These results are obtained by assuming that there is no modeling error between the model and the biological system; • a solution suited to this system identification problem is proposed and applied to in vitro data sets in a second part of this paper in order to assess its effectiveness in practice. Fig. 1 depicts the basic material used in in vitro experiments for studying the uptake kinetics of a photosensitizing drug into living cells. Cells are seeded in 250µl culture wells and are exposed at time t 0 = 0 to a photosensitizing drug D. Let us consider the uptake phenomenon as a dynamic system. Its input variable u(t) is a step signal which corresponds to the amount of drug injected into the well at time t = 0. Its output variable y(t) is the amount of drug absorbed by the cells. ŷ(t) is the measurement of y(t) given by

x(t) y(t) k r k u u(t) Fig. 2. in vitro compartmental model a spectrofluorimeter at times {t j } with j ∈ {1, • • • , N t }.
However, the spectrofluorimeter affects the biological state of the photosensitizing drug through a photobleaching process. Each culture well then becomes unusable after measurement. Consequently, to measure the intracellular PS concentration at N t different time instants, it is necessary to repeat the same experiment in N t different culture wells. Moreover, N r identical wells are handled by the experimenter at each time instant t j to a posteriori estimate the repeatability of the measurements. Globally, N r × N t wells are handled during the whole experiment. In practice, such an experiment is also repeated for other PS and different concentrations of protein in the medium. Consequently, the total number of wells to handle can be much larger than N r × N t . All the wells are prepared in the same initial conditions.

III. in vitro PS UPTAKE MODELING

The in vitro uptake of the PS agent into cancer cells can be described by a compartmental modeling approach. In this paper, a linear two compartments model presented in Fig. 2, is used. The two compartments are associated with the extracellular and intracellular volumes respectively. x(t) denotes the amount of extracellular PS. Parameters k u and k r are the uptake and release rates respectively. Differential equations of this compartmental model are defined as follows

dx dt = k r y(t) -k u x(t) + du dt (1)
dy dt = k u x(t) -k r y(t), (2)
with x(0) = y(0) = 0. After substitution of x(t) from ( 1) in ( 2), it comes that

1 k u + k r dy dt + y(t) = k u k u + k r u(t), or (3)
T dy dt + y(t) = ku(t), (4)
where T = 1/(k u + k r ) and k = k u /(k u + k r ) are the time constant and the static gain of the PS uptake model described by a linear first-order differential equation. In [START_REF] Bastogne | Modélisation système de la thérapie photodynamique[END_REF], it is shown that a first-order transfer function is indeed a parsimonious model structure for describing the uptake kinetics of the chlorin e6 photosensitizing drug into HT29-A4 cancer cells (human colon cancer cell line).

IV. MODEL AND ERRORS DESCRIPTIONS

The determination of a parametric model describing the uptake kinetics of a photosensitizing drug into living cells by extracting information from observations of u and y is a system identification problem [START_REF] Walter | Identification of Parametric Models from experimental data[END_REF]. At this point and thereafter, it is assumed that the system and the model are identical M(p) ≡ S(p). However, as depicted in Fig. 3, three kinds of uncertainties are examined and are represented by output, input and timing noise (errors). Output and intput noise (n y and n u ) are described by stationary stochastic processes added to the output and input signals. The timing noise (n t ) is a sequence of random variables added to the timing sequence {t j } controlling the sampling process of the output signal. tj is the real time instant at which the output variable y is measured while t j represents the theoretical measurement time instant noted by the experimenter. e y (t j ) denotes the output error between the system and the model outputs (ŷ( tj ) and y M (t j ) respectively). Table I sums up the main notations used in the sequel.

A. Model structure

For the sake of simplicity, it is assumed in the sequel that M(p) and S(p) both rely on a first-order transfer function, inspired from (4),

S(p) : T • dy dt + y(t) = k • û(t) (5)
M(p) : T • dy M dt + y M (t) = k • u(t), (6)
with y(0) = y M (0) = 0. p = (T , k) is the parameter vector where T and k denote the time constant and the static gain respectively. From a biological point of view, T and k inform the biologist about the uptake rate and yield respectvely. u(t) is a step signal of magnitude u 0 defined in [START_REF] Walter | Identification of Parametric Models from experimental data[END_REF]. As a result, the intracellular concentration of the photosensitizing drug y follows a mono-exponential kinetics defined by

y(t) = k • û0 • (1 -e -t/T ). (7)

B. Output noise

Conjugated effects of measurement noise and disturbances are usually described by a stochastic variable n y added to y such that

ŷ(t j ) = y(t j ) + n y (t j ), (8)
where y and ŷ denote the real biological response and its measurement respectively.

C. Input noise u is a step signal defined by

u(t) = 0 t < 0 u 0 t ≥ 0 (9)
The step magnitude (u 0 ) represents the amount of the injected drug. The duration of injection is not significant compared to the duration of the experiment. The drug administration is usually carried out by multichannel micropipettes.

For technical reasons, the real filling levels of drug in the cones are not identical and do not match with the dose stipulated by the experimenter. This error is represented by an input noise n u added to u such that

û(t) = 0 t < 0 û0 = u 0 + n u t ≥ 0 (10)
where u 0 and û0 denote the prescribed dose and the effectively administrated dose respectively.

D. Timing noise

Each experimentation is carried out in a sterile framework in which each well is carefully handled by the experimenter. For each time instant t j of the experimental set up, the living cells have to be washed, removed, lysed and diluted in ethanol by the experimenter. Such operations can take about 10mn during which each well, among the N r ones, is individually handled. But the longest stage is the first one, i.e. the stage where the PS is administrated into all the wells; such an operation can take 20mn. In practice, only the time instants t - j and t + j corresponding to the beginning and the end of the experiment are noted by the experimenter. The nominal measurement time instant t j noted by the experimenter in his table is an average time instant defined by t j = (t + j + t - j )/2. The real time instant tj at which the uptake kinetics is stopped and measured, is unknown. This lack of precision in the timing of experiments is described by a timing noise n t .

V. STOCHASTIC MODELING

In this section it is assumed that {n y (t j )} is an independent and identically distributed sequence of Gaussian variables

n y (t j ) = σ y • G j y , (11)
where σ y denotes the standard deviation of n y and G j y ≈ N (0, 1). n u is supposed to be a Gaussian variable defined by

n u = σ u • G u , (12)
where σ u denotes the standard deviation of n u and G u ≈ N (0, 1). The timing noise sequence {n tj } is supposed to be independent and identically distributed sequence of uniform variables such that

n tj ≈ U(t - j , t + j ) (13) ≈ - τ j 2 + τ j • U j t , (14)
with τ j = t + jt - j and U j t ≈ U(0, 1). τ j denotes the width of the timing uncertainty interval for the time instant t j . {n y (t j )}, n u and {n tj } are supposed to be independent. Given the previous assumptions about the input, output and timing noise, the expression of e y (t j ) becomes e y (t j ) = ŷ( tj )y M (t j ) (15)

= k • (u 0 + σ u G u ) • (1 -e -1 T (tj - τ j 2 +τj U j t ) ) + σ y G j y -k • u 0 • (1 -e -t j T ), (16)
where k, T are given.

The mathematical expectation of e y (t j ) is defined in Proposition 5.1, its demonstration is developed in appendix II.

Proposition 5.1:

E{e y (t j )} = k • u 0 • e -t j T 1 -sinhc( τ j 2T ) , ( 17
)
with sinhc(x) = sinh(x)/x denotes the hyperbolic sinus cardinal function of x. Since x → sinhc(x) is increasing on R + , equation ( 17) shows that E{e y (t j )} < 0. This systematic bias is only due to the timing noise. The absolute value of the mean output error increases with respect to τ and is null only for τ = 0.

The variance of e y (t j ) is given in Proposition 5.2, its demonstration is developed in appendix III.

Proposition 5.2: u . To take into account both the bias and the variance of e y (t j ), its mean square error defined by ε(t j ) = E 2 {e y (t j )} + V ar{e y (t j )} is examined thereafter. Three specific values of ε(t j ), noted ε ny (t j ), ε nu (t j ) and ε nt (t j ), are determined to emphasize the contribution of each kind of noise.

V ar{e y (t j )} = σ 2 y + k 2 σ 2 u + k 2 e -t j T • sinhc( τ j 2T )(A + B), (18
• σ u = 0, τ j = 0:

ε ny (t j ) = σ 2 y , (19)
• σ y = 0, τ j = 0:

ε nu (t j ) = k 2 σ 2 u e -t j T -1 2 , (20)
• σ y = 0, σ u = 0:

ε nt (t j ) = k 2 u 2 0 e -2t j T 1 -2sinhc( τ j 2T ) (21) +sinhc( τ j 2T ) cosh( τ j 2T ) . (22)
The effect of the timing noise on the output error is estimated as significant if there exists a time instant t j such that ε nt (t j ) > (ε nu (t j ) + ε ny (t j ))/10. For instance, if u 0 = 1, k = 0.3, T = 5, t j = 1, τ j = 0.5, σ y = 0.01, σ u = 0.1 1 1 These values have been chosen from empirical knowledge of biologists and experimental results reported in [START_REF] Bastogne | System identification of photosensitiser uptake kinetics in photodynamic therapy[END_REF].

then ε nt (t j ) ≈ 5 • 10 -5 , ε nu (t j ) ≈ 3 • 10 -5 and ε ny (t j ) ≈ 1 • 10 -4 . Consequently, the effect of the timing noise on the output error cannot be neglected for the time instant t j = 1 (h). The impact of n t decreases as t j and becomes negligible from t j 3 (h). This example emphasizes that n t could significantly influence the estimation of the time constant which mainly depends on the first measurement samples. Since the consequences of n t cannot be reasonably ignored, usual parameter estimation methods (those assuming only the presence of output noise) and error-in-variables approaches are not appropriate to solve this estimation problem. In the next section, a bounded-error parameter estimation approach is proposed as a suited solution to handle timing errors.

VI. BOUNDED-ERROR ESTIMATION WITH in vitro DATA

Bounded-error or set-membership approaches allow to estimate parameters and their uncertainty in inverse problem contexts in which all uncertain quantities are assumed as unknown but bounded with known bounds. No further hypothesis about probability distributions is stated. Several algorithms have been developped to solve estimation problems stated in the bounded-error context. When models are non linear, most approaches use interval analysis and constraint propagation techniques. Allied with global algorithms and a reliable numerical implementation, they derive guaranteed computations, i.e. they provide a numerical proof of property or non-property. They are rather mature techniques and have already been successfully applied for solvling problems in biology, chemical or thermal engineering, economics, computer vision or robotics, when guaranteed computations were essential [START_REF] Jaulin | Applied Interval Analysis[END_REF], [START_REF] Braems | Guaranteed characterization of thermal conductivity and diffusivity in presence of model uncertainty[END_REF].

In this part, we assume that all the uncertain quantities satisfy this bounded-error property and bounded-error estimation techniques are applied to experimental data collected during in vitro uptake kinetics experiments of a PS into human malignant glioma cells. The experimental protocol is defined by N t = 5, N r = 6 and u 0 = 25mol. Fig. 4 presents the experimental data of six PS uptake kinetics carried out in the same experimental framework. Each cross corresponds to one measurement. The output variable measured by the spectrofluorimeter is given in arbitrary unit. Prior intervals [ŷ(t j )] and [t j ] on the output measurements and the time instants are given in table II. Bounds of [t j ] have been measured during the kinetics experiment. [ŷ(t j )] has been determined from the minimum and maximum values of measurements. The uncertainty associated with each pair of output and time data is materialized by a box. The set P

(p, t) = k • u 0 • (1 -e -t/T ) (23)
goes through all five boxes of Fig. 4. P is defined as

P = {p ∈ R +2 |∃t ∈ R +5 , (g(p, t), p, t) ∈ X} (24) with g(p, t) = g(p, t 1 ) • • • g(p, t Nt ) , (25)
and

X = [ŷ(t 1 )] × • • • × [ŷ(t 5 )] × [ ǩ] × [ Ť ] × [t 1 ] × • • • × [t 5 ], (26)
and the prior box for the parameters given by

P = [ ǩ] × [ Ť ] = [1, 4] × [1, 40]. (27)
P can be estimated in a guaranteed way using a set inversion algorithms based on parameter space partionning, interval analysis and constraint propagation techniques (see [START_REF] Jaulin | Applied Interval Analysis[END_REF] and the references therein). Fig. 5 presents the estimate of P when the partionning algorithm is set not to partition boxes with a size smaller than 0.01 In Fig. 5, the paving form associated with P is composed of grey and black boxes. The grey boxes have been proved to be included in P but no conclusion has been reached for the black ones. The external envelope of P is defined by k ∈ [1.37; 3.49] and T ∈ [1.7; 33]. This results shows that the estimation uncertainty on the time-constant is larger than the one on the static gain. Fig. 6 depicts the a posteriori estimate of the output set Y, a set of time trajectories defined by

Y = (t, y) ∈ R + × R| y(t) = ku 0 (1 -e -t/T ), with (k, T ) T ∈ P . (28)
This figure points out a wide variation of the initial slope of the step response which explains the large uncertainty on the time-constant estimate. In this study case, the wide variation of the initial slope is mainly due to the height of the boxes rather than their width. In other terms, in this application, the uncertainty on the time-constant estimate is mainly caused by the output noise rather than the timing noise.

VII. CONCLUSION

This paper focuses on consequences of timing errors in collected data on the parameter estimation of kinetics models and more precisely their effects on the output error. The contribution of the timing noise on the output error is compared with the ones induced by input and output noise in terms of bias and variance. Mathematical expressions of the bias and variance of the output error with respect to the parameters of input, output and timing noises are established. It is shown that the influence of the timing noise on the output error can be significant, particularly for the first measurement time instants (t j 3 h). An application to in vitro data is developed in the second part of this paper. It is shown how the timing noise can be taken into account by boundederror estimation algorithms based on interval analysis. The timing noise is described as a bounded error and no further hypothesis about probability distributions is stated. The results presented herein emphasize the effectiveness of such an bounded-error estimation approach in such an experimental framework.

Fig. 3 .

 3 Fig. 3. Description of uncertainties

Fig. 4 .

 4 Fig. 4. Experimental data, intervals and boxes

Fig. 5 .Fig. 6 .

 56 Fig. 5. a posteriori estimate of the parameter set b P

APPENDIX I

Lemma 1.1:

where a is a non-null constant and U is a random variable distributed according to a uniform law on [0, 1].

APPENDIX II PROOF OF THE PROPOSITION 5.1

Proof: Since G u and G j y are centered and since U j t and G u are independent, then

It can be deduced from Lemma 1.1 that

APPENDIX III PROOF OF THE PROPOSITION 5.2 Proof: From ( 16), e y (t j ) is rewritten such that

with

Since X 1 and X 2 are independent, it can be deduced that

Let us compute the expectation of X 1 .

according to Lemma 1.1. The expectation of X 2 1 is given by

where

We have

We finally obtain (45) Equation (18) in proposition 5.2 is then a direct consequence of (38).